modeling of viscoelastic dampers by e r

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Computers and Structures 88 (2010) 1–17

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Identiﬁcation of the parameters of the Kelvin–Voigt and the Maxwell fractional models, used to modeling of viscoelastic dampers _ R. Lewandowski *, B. Chora˛zyczewski Poznan University of Technology, ul. Piotrowo 5, 60-965 Poznan, Poland

a r t i c l e

i n f o

Article history: Received 15 June 2008 Accepted 3 September 2009 Available online 14 October 2009 Keywords: Parameters identiﬁcation Fractional rheological models Viscoelastic dampers

a b s t r a c t Fractional models are becoming more and more popular because their ability of describing the behaviour of viscoelastic dampers using a small number of parameters. An important difficulty, connected with these models, is the estimation of model parameters. A family of methods for identification of the parameters of both the Kelvin–Voigt fractional model and the Maxwell fractional model are presented in this paper. Moreover, the equations of hysteresis curves are derived for fractional models. One of the methods presented used the properties of hysteresis curves. The validity and effectiveness of procedures have been tested using artificial and real experimental data. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Viscoelastic (VE) dampers have often been used in controlling the vibrations of aircrafts, aerospace and machine structures. Moreover, VE dampers are used in civil engineering to reduce excessive oscillations of building structures due to earthquakes and strong winds. A number of applications of VE dampers in civil engineering are listed in [1]. The VE dampers could be divided broadly into fluid and solid VE dampers. Silicone oil is used to build the fluid dampers while the solid dampers are made of copolymers or glassy substances. Good understanding of the dynamical behaviour of dampers is required for the analysis of structures supplemented with VE dampers. The dampers behaviour depends mainly on the rheological properties of the viscoelastic material the dampers are made of and some of their geometric parameters. These characteristics depend on the temperature and the frequency of vibration. Temperature changes in dampers can occur due to environmental temperature fluctuations and also on the internal temperature rising due to energy dissipation. A classic shift factor approach is commonly used to capture the effect of temperature changes. Therefore, only the frequency dependence of damper characteristics is considered in this paper. In a classic approach, the mechanical models consisting of springs and dashpots are used to describe the rheological properties of VE dampers [2–7]. A good description of the VE dampers requires mechanical models consisting of a set of appropriately connected springs and dashpots. In this approach, the dynamic * Corresponding author. Tel.: +48 61 6652 472; fax: +48 61 6652 059. E-mail address: roman.lewandowski@put.poznan.pl (R. Lewandowski). 0045-7949/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2009.09.001

behaviour of a single damper is described by a set of differential equations (see [5,6]), which considerably complicates the dynamic analysis of structures with dampers because the large set of motion equation must be solved. Moreover, the cumbersome, nonlinear regression procedure, described for example in [8,9], is propose to determine the parameters of the above mentioned models. The rheological properties of VE dampers are also described using the fractional calculus and the fractional mechanical models. Currently, this approach has received considerable attention and has been used in modeling the rheological behaviour of linear viscoelastic materials [10–15]. The fractional models have an ability to correctly describe the behaviour of viscoelastic material using a small number of model parameters. A single equation is enough to describe the VE damper dynamics, which is an important advantage of the discussed models. In this case, the VE damper equation of motion is the fractional differential equation. The fractional models of VE ﬂuid dampers are proposed in [16,17]. The parameters of the model proposed in [17] are complex numbers which has added to the complexity of the dynamic analysis of structures with VE dampers, especially when the in time domain analysis must be performed. However, fractional models of which the parameters are real numbers are also proposed, for example, in [10,21,22]. The dependence of the above mentioned model parameters on excitation frequency could lead to major complications of analysis of structures with VE dampers in a time domain. Fortunately, efﬁcient time-domain approaches have been proposed recently. The methods, based on Prony series and Biot model, are proposed in papers [4,29,30]. Additionally, the method which used the generalized Maxwell model and the Laguerre polynomial approximation is suggested in [5]. Very recently, seismic analysis of structures with VE dampers modeled by the Kelvin chain and

R. Lewandowski, B. Chora˛z_ yczewski / Computers and Structures 88 (2010) 1–17

the Maxwell ladder is presented in [6]. Moreover, the non-viscous state-space model, described in [31], could be used in this context. The analysis in the frequency domain is possible and presented, for example, in [3]. An important problem, connected with the fractional rheological models, is an estimation of the model parameters from experimental data. In the past, different methods have been tested for the estimation of model parameters from both static and dynamic experiments [4,8,9,16–19,21]. The process of parameter identification is an inverse problem which is overdetermined and can be ill conditioned (see, for example [8,19]) because of noises existing in the experimental data. The mathematical difficulties may be overcome by the regularization method which is described, for example, in [8,18]. It is the aim of this paper to describe a few methods of identification of the parameters of VE dampers. The parameters are estimated using the results obtained from dynamical tests. The fractional rheological models are used to describe the dynamic behaviour of dampers. The Kelvin–Voigt model and the Maxwell model are discussed. Three kinds of identification methods are suggested. Equations of hysteresis curves are derived for both fractional models and some of the properties of these curves are used to develop the first-kind identification procedure. The second- and third-kind identification procedures are based on time series data. Each identification procedure consists of two main steps. In the first step, for the given frequencies of excitation, experimentally obtained data are approximated by simple harmonic functions while model parameters are determined in the second stage of the identification procedure. The results of a typical calculation are presented and discussed. Three symbols will be used in the paper to describe the damper force. The symbol ue ðtÞ denotes the results of experimental mea~ ðtÞ is the function which is an analytical approximasurements, u tion of experimental results while uðtÞ is the solution to the ~ðtÞ and qðtÞ are damper equation of motion. The symbols qe ðtÞ; q used in the same way to describe the damper displacement.

2. General explanation of fractional rheological models Due to their simplicity, the simple rheological models, such as the Kelvin–Voigt model or the Maxwell model, are used very often to describe the dynamic behavior of VE dampers installed on various types of civil structures. For example, the Kelvin–Voigt model is used in papers [2,3,22–24], while the Maxwell model is used in papers [2,3,25–28]. The Kelvin–Voigt model consists of the spring and the dashpot connected in parallel, while the Maxwell model is built from the serially connected spring and dashpot. The discussed simple rheological models have a small number of parameters but do not have enough parameters to accurately capture the frequency dependence of damper parameters. More sophisticated classical rheological models which, however, contain many parameters, can be used to correctly describe the dynamic behavior of VE dampers. Models of this kind are developed in [6,7]. The next group of models used to describe the behaviour of VE dampers are the fractional derivative models. Using the fractional calculus, a number of rheological models, e.g., the fractional Kelvin–Voigt model [32], the fractional Zener model [10,11], the fractional Jeffreys model [12], or the fractional Maxwell model [17] can be developed. It has been proved in [4,15] that the fractional derivative models can better capture the frequency dependent properties of VE dampers. The simple fractional models discussed in [4,13,15] are sufﬁcient to correctly capture the VE dampers properties. These models contain only a few parameters but the dynamic analysis requires knowledge of the fractional calculus.

The simple Kelvin–Voigt model seems to be more appropriate to describe the solid dampers behavior, while the simple Maxwell model is mainly used to describe the liquid dampers behavior. Recently, it has been shown in [6] that both the generalized Kelvin model and the generalized Maxwell model are also useful models of the solid VE damper. One important problem, considered here and connected with these models, is how to determine model parameters from experimental data in an efﬁcient way. This problem is solved in this Section for the fractional Kelvin–Voigt model and the fractional Maxwell model. 2.1. Fractional models equation of motion and the steady state solution to the motion equation In order to construct the fractional models equation of motion, we introduce the fractional element called the springpot which satisﬁes the constitutive equation:

uðtÞ ¼ ~ca Dat qðtÞ ¼ cDat qðtÞ;

ð1Þ

where c ¼ ~ca and a; 0 < a 6 1, are the springpot parameters and Dat qðtÞ is the fractional derivative of the order a with respect to time t. There are a few deﬁnitions of fractional derivatives which coincide under certain conditions. Here, symbols such as Dat qðtÞ denote the Riemann–Liouville fractional derivatives with the lower limit at – 1 (see [34]). Some valuable information about fractional calculus can be found in [34]. The springpot element, also known as the Scott–Blair’s element (see [33]), is schematically shown in Fig. 1a. The considered element can be understood as an interpolation between the spring element ða ¼ 0Þ and the dashpot element ða ¼ 1Þ. The fractional Kelvin–Voigt model consists of the spring and the springpot connected in parallel, while the fractional Maxwell model is built of the serially connected spring and springpot. These models are shown schematically in Fig. 1b and c, respectively. The motion equation of the above mentioned fractional Kelvin– Voigt model, obtained in a usual way, is in the following form:

uðtÞ ¼ kqðtÞ þ ksa Dat qðtÞ;

ð2Þ

~ca =k

where ¼ ¼ c=k. The following relationships can be written for the fractional Maxwell model (Fig. 1c)

uðtÞ ¼ kq1 ðtÞ;

uðtÞ ¼ ~ca Dat ðqðtÞ q1 ðtÞÞ:

ð3Þ

Eliminating q1 ðtÞ from the above relationships, we get the equation in the form:

uðtÞ þ sa Dat uðtÞ ¼ ksa Dat qðtÞ:

ð4Þ

It is easy to recognize that both of the considered fractional models have three real and positive value parameters: k, c, and a. In the case of a harmonically excited damper, i.e. when

qðtÞ ¼ q0 expðiktÞ;

ð5Þ

the steady state solution of the motion equation of both of the fractional models is assumed in the form:

uðtÞ ¼ u0 expðiktÞ:

ð6Þ

Taking into account that in our case (see [34, p. 311])

Dat expðiktÞ ¼ ðikÞa expðiktÞ;

ð7Þ

for k > 0, we obtain the following equations

u0 ¼ k½1 þ ðiskÞa q0 ;

u0 ¼ k

ðiskÞa q0 ; 1 þ ðiskÞa

ð8Þ

R. Lewandowski, B. Chora˛z_ yczewski / Computers and Structures 88 (2010) 1–17

Fig. 1. Diagram of fractional rheological models.

from (2) and (4), respectively. Moreover, after introducing the fola lowing formula i ¼ cosðap=2Þ þ i sinðap=2Þ, we can rewrite relationships (8) in the form 00

u0 ¼ ðK 0 þ iK Þq0 ¼ K 0 ð1 þ igÞq0 ; 00

ð9Þ

where gðkÞ ¼ K ðkÞ=K ðkÞ is the loss factor, K ðkÞ is the storage modulus and K 00 ðkÞ is the loss modulus. These quantities are deﬁned as:

K 0 ¼ k½1 þ ðskÞa cosðap=2Þ ;

K 00 ¼ kðskÞa sinðap=2Þ;

ð10Þ

g¼

ðskÞ sinðap=2Þ ; 1 þ ðskÞa cosðap=2Þ

ð11Þ

for the fractional Kelvin–Voigt model and

K 0 ¼ kðskÞa K 00 ¼ kðskÞ

ðskÞa þ cosðap=2Þ

1 þ ðskÞ2a þ 2ðskÞa cosðap=2Þ sinðap=2Þ a

;

1 þ ðskÞ2a þ 2ðskÞa cosðap=2Þ sinðap=2Þ g¼ ; ðskÞa þ cosðap=2Þ

;

ð12Þ ð13Þ

qðtÞ ¼ qc cos kt þ qs sin kt;

for the fractional Maxwell model. If the classical Kelvin–Voigt model (i.e. a ¼ 1Þ is used as the VE dampers model the above mentioned quantities are given by

K 0 ðkÞ ¼ k;

K 00 ðkÞ ¼ ksk;

gðkÞ ¼ K 00 ðkÞ=K 0 ðkÞ ¼ sk

ð14Þ

while for the classical Maxwell model we have

s2 k2 sk 1 K ðkÞ ¼ k ; K 00 ðkÞ ¼ k ; gðkÞ ¼ : sk 1 þ s2 k 2 1 þ s2 k 2 0

The storage modulus grows with non-dimensional frequency for all values of the parameter a. However, for 0 < sk < 1 the function K 0 ðaÞ increases for decreasing values of a. An opposite tendency is evident for sk > 1. The function of the loss modulus could be flat (for instant when a ¼ 0:4Þ. Moreover, gð0Þ ¼ tgðap=2Þ which means that the loss factor has a finite value for k ¼ 0. This is an important difference in comparison with the loss factor function of the classic Maxwell model of which the values approach infinity if k approaches zero and in comparison with the both versions of the Kelvin–Voigt model for which the loss factor is equal to zero for sk ¼ 0. According to results presented by Lion in [35] the fractional rheological models fulfill the second law of thermodynamics when values of the storage modulus K 0 ðkÞ and the loss modulus K 00 ðkÞ, given by formulae (10)–(13), are positive for all possible values of frequency of excitation k. It can easily be demonstrated that both models fulfill the second law of thermodynamics when 0 6 a 6 1; s > 0 and k > 0. If the damper displacement varies harmonically in time and is described using the trigonometric functions, i.e.:

the steady state solution to the fractional Kelvin–Voigt model equation of motion (2) is given by

uðtÞ ¼ uc cos kt þ us sin kt;

The dependence of the above mentioned model parameters on excitation frequency could significantly complicates in time domain analysis of structures with VE dampers. According to the classical Kelvin–Voigt model the storage modulus is the constant function of k, while the loss modulus and the loss factor linearly increases with k. Behaviour of the fractional Kelvin–Voigt model is substantially different in comparison with the classic Kelvin–Voigt model. The storage modulus is increased significantly when the non-dimensional frequency increases and when the parameter a decreases. Moreover, the loss modulus and the loss factor decrease for an increasing non-dimensional frequency and the decreasing values of the parameter a. The properties of the fractional Maxwell model, for different values of the parameter a are shown in Figs. 2 and 3. In the figures the non-dimensional frequency is defined as sk. The calculation is made using the value of the k parameter equal to 1,00,000.0 N/m.

ð17Þ

where

uc ¼ u1 qc þ u2 qs ; ð15Þ

ð16Þ

us ¼ u2 qc þ u1 qs ;

u1 ¼ k þ ck cosðap=2Þ; u2 ¼ ck sinðap=2Þ:

ð18Þ ð19Þ

The steady state solution to the equation of motion (4) of the fractional Maxwell model is also given by (16) and (17) and the coefﬁcients qc ; qs ; uc and us are interrelated in the following way:

qc ¼ /1 uc /2 us ;

qs ¼ /2 uc þ /1 us ;

ð20Þ

where

/1 ¼

1 h api ðskÞa þ cos ; 2 kðskÞa

/2 ¼

1 ap sin : 2 kðskÞa

ð21Þ

2.2. Hysteresis loops of the damper models The equation of the hysteresis loops of the fractional Kelvin– Voigt model could be derived if the damper’s kinematical excitation is given by

qðtÞ ¼ q0 sin kt:

ð22Þ

R. Lewandowski, B. Chora˛z_ yczewski / Computers and Structures 88 (2010) 1–17

Fig. 2. The storage modulus of the fractional Maxwell model for different values of a parameter.

Fig. 3. The loss modulus of the fractional Maxwell model for different values of a parameter.

Taking into account that (see [34, p. 311]) a

Dt qðtÞ ¼ k q0 sin½kt þ ðap=2Þ ;

which means that

ð23Þ

and introducing (22) and (23) into Eq. (2) we can write

ð24Þ 2

Next, using (22) and the identity sin ðap=2Þ þ cos2 ðap=2Þ ¼ 1, we can rewrite (24) in the form of the following equation

ð25Þ

which describes the ﬁrst version of the hysteresis loop of the fractional Kelvin–Voigt model. Derivation of the second version of the hysteresis loop started with an assumption that the damper’s displacement is described by

qðtÞ ¼ q0 sin½kt ðap=2Þ ;

ð27Þ

Introducing relationship (26) into the motion Eq. (2) we can write

uðtÞ ¼ k½1 þ ðskÞa cosðap=2Þ q0 sin kt þ kðskÞa q0 sinðap=2Þ cos kt:

2 2 uðtÞ k½1 þ ðskÞa cosðap=2Þ qðtÞ qðtÞ þ ¼ 1; q0 kðskÞa q0 sinðap=2Þ

Dat qðtÞ ¼ ka q0 sin kt:

ð26Þ

uðtÞ ksa Dat qðtÞ ¼ kq0 ½sin kt cosðap=2Þ cos kt sinðap=2Þ :

ð28Þ

Using (27) it is possible to write the second version of the hysteresis loop equation in the following form:

2 a 2 uðtÞ kk a ½ðskÞa þ cosðap=2Þ Dat qðtÞ D qðtÞ þ ta ¼ 1: kq0 sinðap=2Þ k q0

ð29Þ

In a case of the classical Kelvin–Voigt model we have:

2 2 uðtÞ kqðtÞ qðtÞ þ ¼ 1; kcq0 q0 ¼ 1; instead of (25) and (29).

_ uðtÞ cqðtÞ kq0

2 þ

2 _ qðtÞ kq0 ð30Þ

R. Lewandowski, B. Chora˛z_ yczewski / Computers and Structures 88 (2010) 1–17

Let us now proceed to developing the equation describing the hysteresis loop of the fractional Maxwell model. The excitation is described by

uðtÞ ¼ u0 sin kt;

which could be rewritten as

ð38Þ

which is the second version of the hysteresis loop equation searched for. It is obvious that, for the classical Maxwell model ða ¼ 1Þ we obtain:

2 _ uðtÞ 2 cqðtÞ uðtÞ þ ¼ 1; u0 sku0

ðskÞ2

kqðtÞ uðtÞ u0

uðtÞ u0

2 ¼ 1; ð39Þ

ð34Þ

This is the ﬁrst version of the hysteresis loop equation of the fractional Maxwell model. To obtain the second version of the hysteresis loop equation the motion Eq. (4) must be fractionally integrated. This operation is formally deﬁned in [34] and here it will be denoted using the syma bol D t uðtÞ if the function uðtÞ is integrated. Moreover, it can be a a demonstrated that D t ðDt uðtÞÞ ¼ uðtÞ. After the fractional integration of Eq. (4) we obtain

ð35Þ

Now, if uðtÞ is given by (31) then

Introducing (31) into (4) we can write Eq. (35) in the form:

ð32Þ

ð33Þ

D t a ðu0 sin ktÞ ¼ u0 k a sin½kt ðap=2Þ :

ð37Þ

2 2 kðskÞa qðtÞ ½ðskÞa þ cosðap=2Þ uðtÞ uðtÞ þ ¼ 1; u0 sinðap=2Þ u0

uðtÞ ksa Dat qðtÞ ¼ ðskÞa u0 ½sin kt cosðap=2Þ þ cos kt sinðap=2Þ ;

D t a uðtÞ þ sa uðtÞ ¼ ksa qðtÞ:

cos kt sinðap=2Þ :

Next, with the help of relationships (31) and sin ðap=2Þþ cos2 ðap=2Þ ¼ 1 we can transform the above equation to

Introducing (32) into the motion Eq. (4) we obtain the following relationship

2 a a 2 ks Dt qðtÞ ½1 þ ðskÞa cosðap=2Þ uðtÞ uðtÞ þ ¼ 1: a u0 ðskÞ u0 sinðap=2Þ

kðskÞa qðtÞ ðskÞa uðtÞ ¼ u0 ½sin kt cosðap=2Þ

ð31Þ

what means that

Dat uðtÞ ¼ ka u0 sin½kt þ ðap=2Þ :

ð36Þ

instead of (34) and (38), respectively. The hysteresis loops of the fractional Kelvin–Voigt model and the fractional Maxwell model are shown in Figs. 4 and 5, respectively, for different values of a and for k = 1,00,000.0 N/m, c = 1,00,000.0 Ns/m, k = 10.0 rad/s. It is evident that for both models the damper’s damping abilities decrease as the values of a decrease. 3. General remarks concerning identification methods The problem of determination of the parameters of the fractional derivative rheological models is, more or less extensively, discussed in papers [10,13,17,20,21,37–40]. In two papers [10,13], Pritz describes the method of parameters identification of two fractional derivative rheological models with four and five parameters, respectively. The method utilizes some asymptotic properties of the storage and loss modulus functions, experimentally

Fig. 4. Hysteresis loops of the fractional Kelvin–Voigt model for different values of a.

R. Lewandowski, B. Chora˛z_ yczewski / Computers and Structures 88 (2010) 1–17

Fig. 5. Hysteresis loops of the fractional Maxwell model for different values of a.

obtained over a certain range of excitation frequencies. However, no systematic procedure of identification is presented. Similar methods are developed in [20,38] where more detailed description of identification procedure is also presented. The above mentioned methods require experimental data from a large range of excitation frequencies. Very recently, in paper [39], the identification procedure of parameters of fractional derivative model of viscoelastic materials is also presented. The method uses an optimization procedure together with the experimentally and numerically obtained frequency response functions to determine the parameters of the considered model VE materials. The differences between the measured and calculated frequency response functions are minimized in order to estimate the true values of the searched parameters. Moreover, the least squares method, in which the error between the model and the experimental complex modulus 00 K ðkÞ; ðK ðkÞ ¼ K 0 ðkÞ þ iK ðkÞÞ is minimized in order to obtain the parameters of the fractional model of viscoelastic materials is suggested in [21,37,40]. Unfortunately, details of the used least squares method are presented in [40] only. There are many papers containing a description of the method of identification of parameters of the classic rheological models of VE materials and VE dampers. In paper [4] Park used the Prony series together with the least squares method to determine parameters of the generalized Maxwell model of the VE solid and the VE liquid damper. The normalized error between the experimentally obtained and the calculated complex modulus is minimized. The optimization problem is linear because the number of Prony series and the relaxation times are fixed. The last square method is also used in [6] to determine the parameters of two mechanical models of VE dampers. The models are the Kelvin chain and the Maxwell ladder. Some information concerning identification of the parameters of the so-called GHM model and the ADF model used for modeling viscoelastic materials are presented in [41]. The advanced numerical methods for parameter identification of VE materials are presented in [8,9,42] where the least squares method together

with the regularization technique is used to solve the considered identification problem. In this paper, the parameters of the fractional Kelvin model and the fractional Maxwell model are determined using results of dynamic tests. The proposed identification methods differ substantially in comparison with the previously mentioned ones. First of all, instead of using the experimentally obtained complex modulus the suggested methods utilize the measured steady state responses of damper. The previous methods, presented in [10,13,20,38], require experimental data from a large range of excitation frequency while the suggested methods could be also used when data are available only from a narrow range of excitation frequency. Moreover, the error functional minimized in the least squares method and adopted in our identification methods is different from one used in previous papers. The results of calculation show that the proposed methods are not sensitive to measurements noises and, therefore, it is not necessary to use regularization techniques. Additionally, a detailed description of the proposed identification method and the identification procedure are presented. Three kinds of identification methods are suggested in this paper to determine three parameters of the fractional Kelvin–Voigt model and the fractional Maxwell model. Each identification procedure consists of two main steps. In the first step, for the given frequencies of excitation, experimentally obtained data are approximated by simple harmonic functions while model parameters are determined in the second stage of the identification procedure. 4. Identification of model parameters for fractional Kelvin– Voigt model 4.1. Identification procedures based on hysteresis loop (first method) First of all, the first kind method based on the hysteresis loop and applied to the fractional Kelvin–Voigt model will be described.

R. Lewandowski, B. Chora˛z_ yczewski / Computers and Structures 88 (2010) 1–17

Eq. (25) seems to be more useful in comparison with (29). If, for the given frequency of excitation and t ¼ t 1 , we have qðt 1 Þ ¼ q0 > 0 and uðt1 Þ ¼ u1 > 0 then from Eq. (25) it follows

4.2. Identiﬁcation procedures based on time series data (second and third method)

k½1 þ ðskÞa cosðap=2Þ ¼ k þ cka cosðap=2Þ ¼ u1 =q0 :

The identification procedure based on time series data is also developed. The second-kind method will be described first. The experimentally measured damper displacement is approximated ~s are ob~c and q using the harmonic function (A1). The parameters q tained from the set of Eq. (A3). Moreover, it is assumed that the experimentally obtained steady state solution represented by the harmonic function (A1) approximately fulfills the steady state Eq. (18) of the fractional ~s are introduced in a place of qc and ~c and q Kelvin model where q qs , respectively. Now, for a given excitation frequency k, the quantities u1 and u2 , appearing in (18), are determined as described below. The right values of u1 and u2 are the ones which minimize the functional

ð40Þ

Moreover, for t ¼ t 2 when qðt 2 Þ ¼ 0 and uðt 2 Þ ¼ u2 > 0 from (25) we obtain

kðskÞa sinðap=2Þ ¼ cka sinðap=2Þ ¼ u2 =q0 :

ð41Þ

For a given frequency k, relationships (40) and (41) constitute a set of two nonlinear equations with three unknowns: k; c; a or k; s; a. The quantities u1 ; u2 and q0 have clear physical meanings and their values can easily be obtained from the experimental data. Alternatively, these quantities could be calculated from trigonometric functions (A1) and (A6) used for the approximation of experimental data. Details are given in Appendix A. For a given a, the above set of equations are linear with respect to k and c. During experiments the damper is several times harmonically excited and in each case the excitation frequency, denoted here as ki , ði ¼ 1; 2; ::; nÞ, is different. The steady state response of the damper is measured, which means that the experimental damper displacements qei ðtÞ and the experimental damper forces uei ðtÞ ~i ðtÞ u ~ i ðtÞ which approximate the experimental data and functions q are known for each excitation frequency ki . Moreover, the above mentioned quantities, such as u1 ; u2 and q0 can easily be determined for each excitation frequency. These quantities obtained from experimental data are denoted as u1i ; u2i and q0i . Now it is assumed that resulting quantities u1i ; u2i and q0i approximately fulﬁll relationships (40) and (41). For each excitation frequency we can write

J KV ðu1 ; u2 Þ ¼

a22 ¼

where i ¼ 1; 2; ::; n. Symbols r i and si denote residuals obtained after introducing u1i ; u2i and q0i into (42) and (43). The above equations constitute a set of overdetermined nonlinear equations with respect to k; c and a. A pseudo-solution to the above system of equation is chosen in such a way that it minimizes the following functional

Icq ¼

@eJ KV ðk; c; aÞ ¼ 0; @k

@eJ KV ðk; c; aÞ ¼ 0: @c

ð45Þ

n X

kai cosðap=2Þ ¼

i¼1

n X i¼1

ki cosðap=2Þ þ c

n X i¼1

n X u1i ; q0i i¼1

k2i a ¼

n X i¼1

kai

~c q ~2s q ~2c ÞIcs ; ~s ðIcc Iss Þ þ ðq a12 ¼ a21 ¼ q

~2c Iss q

~2s Icc ; q

~c q ~s Ics þ 2q

~c Icu þ q ~s Isu ; b1 ¼ q

ue ðtÞ cos ktdt;

t1 Z t2

Isu ¼

qe ðtÞ cos ktdt;

Isq ¼

~c Isu þ q ~s Icu ; b2 ¼ q ð50Þ

ue ðtÞ sin ktdt;

ð51Þ

qe ðtÞ sin ktdt:

ð52Þ

b1 a21 þ b2 a11 : a11 a22 a12 a21

ð53Þ

t1 Z t2

The solution to Eq. (48) is

u1 ¼

b1 a22 b2 a12 ; a11 a22 a12 a21

u2 ¼

During experiments, the data are measured for different excitation frequencies ki ; ði ¼ 1; 2; ::; nÞ and subsequently the values of u1 and u2 , denoted now as u1i and u2i , respectively, are calculated from (48) or (53). For each particular excitation frequency ki the following equations, very similar to (42) and (43), could be written:

ri ¼ k þ ckai cosðap=2Þ u1i ¼ 0;

si ¼ ckai sinðap=2Þ u2i ¼ 0: ð54Þ

give us the following set of equations, which are linear with respect to k and c

kn þ c

ð48Þ

ð49Þ

Icu ¼

If we assume that the parameter a is known, then stationary conditions:

ð47Þ

a21 u1 þ a22 u2 ¼ b2 ;

~2c Icc þ 2q ~c q ~2s Iss ; ~s Ics þ q a11 ¼ q

ð43Þ

ð44Þ

½ue ðtÞ uðtÞ 2 dt:

where coefﬁcients a11 ; a12 ; a21 ; a22 ; b1 and b2 are given by

ð42Þ

i¼1

a11 u1 þ a12 u2 ¼ b1 ;

si ¼ ckai sinðap=2Þ u2i =q0i 0;

n X ðr 2i þ s2i Þ:

considered here as the functional of u1 and u2 . Stationary conditions of the above-mentioned functional give us the following linear equations with respect to u1 and u2

r i ¼ k þ ckai cosðap=2Þ u1i =q0i 0;

eJ KV ðk; c; aÞ ¼

1 t2 t1

u1i u2i cosðap=2Þ þ sinðap=2Þ : q0i q0i ð46Þ

The right value of a is obtained using the systematic searching method. The set of values of a, denoted as aj ðj ¼ 1; 2; ::; mÞ, where aj ¼ aj 1 þ Da is chosen from a given range of a. For each aj the corresponding values of k and c (denoted as kj and cj Þ are determined from (46) and the value of functional (44) is calculated. These values of aj ; kj and cj for which the functional (44) has a minimum value are the searched parameters of the fractional Kelvin–Voigt damper model.

The solutions to a set of overdetermined equations (i.e. Eq. (54) written for i ¼ 1; 2; ::; nÞ are found in a similar way as in the first identification procedure described above. The searched pseudosolution minimizes the functional (44) with r i and si defined by formulae (54). For a given value of a, the parameters k and c fulfill the equations

kn þ c

n X

kai cosðap=2Þ ¼

i¼1

n X i¼1

kai cosðap=2Þ þ c

n X

u1i ;

i¼1 n X i¼1

k2i a ¼

n X

kai ½u1i cosðap=2Þ þ u2i sinðap=2Þ ;

i¼1

ð55Þ and the parameter a is obtained with the help of the systematic searching method described earlier in this paper. One can see that Eq. (55) are very similar to Eq. (46).

R. Lewandowski, B. Chora˛z_ yczewski / Computers and Structures 88 (2010) 1–17

Please note, in this procedure only the data concerning the damper displacement qe ðtÞ are approximated using the trigonometric function. The second identiﬁcation procedure based on in-time data series (called here the third-kind method) will also be formulated for the fractional Kelvin–Voigt model. In this procedure, experimentally measured variations of both the damper force and the damper displacement are approximated using the harmonic functions (A1) and (A6), respectively. Moreover, it is assumed that Eq. (18) ~s ; u ~c ; q ~ c and u ~ s are introduced are approximately fulﬁlled when q in a place of qc ; qs ; uc and us , respectively. Now, from Eq. (18) we obtain

~u ~u ~ þq ~ q u1 ¼ c~2c ~2s s ; qc þ qs

~u ~u ~ q ~ q u2 ¼ s ~2c ~c2 s : qc þ qs

ð56Þ

The quantities u1 and u2 obtained for a particular frequency of vibration ki , will be denoted as u1i and u2i , respectively. For a given set of excitation frequencies ki ; ði ¼ 1; 2; ::; nÞ, the overdetermined set of Eq. (54) could be written. The values of model parameters k, c and a are chosen in such a way that the functional (44) will be minimized (ri and si are defined by relationships (54)). The procedure described in this subsection could be used without being modified at all. The right value of a is obtained using the systematic searching method and the values of the associated parameters k and c are determined from the set of Eq. (55). It is easy to notice that both of the methods described above differ only in the way u1i and u2i are calculated. The first identification procedure based on in-time series approximate the experimental data concerning the damper displacement with the help of the trigonometric function while the third-kind method used this approximation for both experimental data. 5. Identification of model parameters for fractional Maxwell model

eJ M ðc; s ~; aÞ ¼

kðskÞ q1 ðskÞ u0 u0 cosðap=2Þ ¼ 0;

ð57Þ

n X

~ k2i a ðq21i þ q22i Þ s

i¼1

~ka u0 u0 cosðap=2Þ ¼ 0: cka q1 s

ð58Þ

Moreover, at t ¼ t 2 when the state of the damper is described by uðt 2 Þ ¼ 0 and qðt 2 Þ ¼ q2 < 0, from the hysteresis loop Eq. (38) it follows:

kðskÞa q2 þ u0 sinðap=2Þ ¼ 0;

ð59Þ

which subsequently could be rewritten in the form:

cka q2 þ u0 sinðap=2Þ ¼ 0:

ð60Þ

If we have experimental data for a given set of excitation frequencies ki ; ði ¼ 1; 2; ::; nÞ then the following set of equations (with re~ and aÞ spect to c; s

~kai u0i u0i cosðap=2Þ ¼ 0; r i ¼ ckai q1i s a

si ¼ cki q2i þ u0i sinðap=2Þ ¼ 0;

ð61Þ

could be written. As in the previous model, a pseudo-solution to the above set of equations is chosen as one which minimized the functional:

n X

k2i a u0i q1i

i¼1

n X

ki u0i ½q1i cosðap=2Þ q2i sinðap=2Þ ;

i¼1

n X

~ k2i a u0i q1i þ s

i¼1

n X

k2i a u20i ¼

i¼1

n X

kai u20i cosðap=2Þ:

ð63Þ

i¼1

For particular values of a, the values of damper parameters resulting from (61) could be negative. These solutions do not have physical meaning and must be rejected. As described in Section 4.1, the right value of a is obtained using ~ and c for the method of systematic searching. The values of a, s which the functional (62) has a minimum value are the searched parameters of the fractional Maxwell model. 5.2. Identiﬁcation procedures based on time series data (second and third method) In the second-kind method, for the given frequency of excitation the experimentally measured damper force ue ðtÞ is approxi~ s are determined from ~ c and u mated using function (A6) where u Eq. (A8). The steady state solution to the equation of motion (4) is given by (16) and (17) and the coefﬁcients qc ; qs ; uc and us are interrelated as it is given by relation (20). ~ s , obtained from the experimental data and for the ~ c and u Next, u given frequency of excitation k, are introduced in relationships (20) in the place of uc and us and the functional

1 t2 t1

½qe ðtÞ qðtÞ 2 dt;

ð64Þ

is minimized with respect to /1 and /2 . The stationary conditions of this functional give us the following set of equations

a11 /1 þ a12 /2 ¼ b1 ;

a21 /1 þ a22 /2 ¼ b2 ;

ð65Þ

where

~c u ~ 2s Iss ; ~ 2c þ 2u ~ s Isc þ u a11 ¼ Icc u ~ 2c ðu

which could also be written in the form:

ð62Þ

If we assume that the parameter a is known, the stationary conditions of (62) give us the following system of equations:

5.1. Identiﬁcation procedures based on hysteresis loop (ﬁrst method)

ðr 2i þ s2i Þ:

i¼1

J M ð/1 ; /2 Þ ¼

A similar method to the one described in Section 4.1 is developed for the fractional Maxwell model. For a given excitation frequency k and t ¼ t1 , when uðt 1 Þ ¼ u0 > 0 and qðt1 Þ ¼ q1 > 0, from Eq. (38) we obtain:

n X

a12 ¼ a21 ¼ ~ c Icq þ u ~ s Isq ; b1 ¼ u

~2c Iss 2u ~c u ~2s Icc ; ~ s Isc þ u a22 ¼ u

~ 2s ÞIsc u

~ s ðIss Icc Þ; ~c u þu ~c Isq u ~ s Icq : b2 ¼ u

ð66Þ ð67Þ ð68Þ

The values Iss ; Isc ; Icc are deﬁned by the relationships (A4) and Isq ; Icq by (A5). The values of /1 and /2 , determined from the set of Eq. (65), is given by

/1 ¼

b1 a11 b2 a12 ; a11 a22 a12 a21

/2 ¼

b1 a21 þ b2 a11 : a11 a22 a12 a21

ð69Þ

The model parameters c, k and a will be determined using the leastsquare method. As previously, the error functional (62), where

ri ¼

1 1 a ap þ k cos /1i ; k c i 2

si ¼

1 a ap k sin /2i ; c i 2

ð70Þ

is minimized with respect to model parameters. The residuals r i and si are obtained from relationships (21) after introducing ki instead of k and taking into account that (21) could only approximately be satisﬁed. As previously, the index i of such quantities as /1i emphasizes their dependence on the i-th frequency of excitation used in the experiments.

R. Lewandowski, B. Chora˛z_ yczewski / Computers and Structures 88 (2010) 1–17

The parameter a is determined with the help of the searching method described earlier, while the values of c and k are obtained from the following set of equations:

þ c nk

n X

k i a cosðap=2Þ ¼

i¼1

n X

a k 2 i

k i a ½/1i cosðap=2Þ þ /2i sinðap=2Þ ;

ð71Þ

¼ 1=k and c ¼ 1=c. where k The third-kind method, applied to the fractional Maxwell mod~ðtÞ and u ~ ðtÞ, described, for k ¼ ki , by relael, utilizes the functions q ~ci and qsi ~ si ; q ~ ci ; u tionships (A1) and (A6). The parameters u ~c and qs approximately fulﬁll ~c ; u ~s ; q introduced now in a place of u the following equations

~si ¼ /2i u ~ ci þ /1i u ~ si ; q

~ci u ~si u ~ ci þ q ~ si q ; ~ 2si ~ 2ci þ u u

/2i ¼

~si u ~ci u ~ ci q ~ si q : ~ 2si ~ 2ci þ u u

for i=1,...,n Determine q , u 1i , u

for i=1,...,n Determine uoi , q1i and q 2i

Determine k, c and α using Eqns (46) and the searching method

Determine c, τ and α using Eqns (63) and the searching method

Stop

ð72Þ

Fig. 6. Flowchart of the procedure of the ﬁrst-kind identiﬁcation method.

resulting from Eq. (20). From the equations above we have

/1i ¼

Maxwell model

Kelvin-Voigt model

i¼1

~ci ¼ /1 u ~ ci /2 u ~ si ; q

for i=1,...,n Approximate q e (t) and u e (t) using Eqns (A1) and A(6)

/1i ;

i¼1 n X

Start

i¼1

k i a cosðap=2Þ þ c

i¼1

ð73Þ

Model parameters are determined with the help of the procedure described above, where the error functional (62) and the identical searching procedure together with relationships (70) and (71) are used. For the fractional Maxwell model both of the methods described in this subsection differ only in the way /1i and /2i are calculated. The reciprocal comparison of all proposed identification methods makes it possible to formulate the following remarks: (a) as proved in Section 6, none of the proposed methods is sensitive to measurements noises, (b) for both rheological models the second-kind method and the third-kind method differ only in the way in which /1i and /2i or u1i and u2i are calculated, (c) the second-kind method analytically approximates experimental data obtained for the damper displacement (the Kelvin–Voigt model) or only for the damper force (in the case of the Maxwell model) while the third-kind method analytically approximates the experimental data for both the damper displacement and the damper force, (d) similar calculation efforts are needed when each of the presented methods is used to identify the dampers parameters, (e) any existing differences between the discussed methods have no significant influence on the effectiveness of the methods and on the results of identification.

6. Algorithms of identification methods and results of typical calculation 6.1. Algorithms of identification methods All of the identification procedures have some similarities which were mentioned, to some extent, in previous Subsections. Generally speaking, all algorithms consist of three steps. Approximation of the experimental data using the trigonometric functions is made in the first step. In the second step, some quantities, for example, u1i u2i ; /1i and /2i are calculated for all frequencies of excitation. Finally, the model parameters are determined in the third step. For the readers’ convenience, the flowcharts of all of the algorithms are presented in Figs. 6–8.

Start Kelvin-Voigt model

Maxwell model

for i=1,...,n Approximate qe(t) using Eqn (A1)

for i=1,...,n Approximate u e (t) data using Eqn (A6)

for i=1,...,n Calculate ϕ1i and ϕ 2i using Eqn (53)

for i=1,...,n Calculate φ 1i and φ 2i using Eqn (69)

Determine k, c and α using Eqns (55) and the searching method

Determine k, c and α using Eqns (71) and the searching method

Stop Fig. 7. Flowchart of the procedure of the second-kind identiﬁcation method.

Start for i=1,...,n Approximate qe(t) and ue(t) using Eqns (A1) and A(6)

Kelvin-Voigt model

Maxwell model

for i=1,...,n Determine ϕ 1i and ϕ 2i

for i=1,...,n Determine φ 1i and φ 2i

Determine k, c and α using Eqns (55) and the searching method

Determine k, c and α using Eqns (71) and the searching method

Stop Fig. 8. Flowchart of the procedure of the third-kind identiﬁcation method.

R. Lewandowski, B. Chora˛z_ yczewski / Computers and Structures 88 (2010) 1–17

6.2. Results of typical calculation – the fractional Kelvin–Voigt model The identification procedure described above is applied to artificial data. The artificial experimental steady state solutions are generated on the basis of the solution to the motion Eq. (2). These solutions are given by functions (16) and (17). If, for example, qc ¼ 0, then

qðtÞ ¼ qs sin kt;

u1 ¼ qs k½1 þ ðskÞa cosðap=2Þ ;

u2 ¼ qs kðskÞ sinðap=2Þ:

ð74Þ

The artificial data obtained from relationships (74) are modified by applying small, randomly inserted perturbations. The noises of 1% intensity of the damper force or the damper displacement are applied. The following data are used to generate artificial experimental data: qs = 0.001 m, qc = 0, k = 2,90,000.0 N/m, c = 68,000.0 Ns/m, a = 0.6, n = 9. The values of excitation frequencies ki chosen in this example and the values of u1i ; u2i and q0i determined from the artificial data are given in Table 1. After application of the identification procedure based on the hysteresis loop the following results are obtained: a = 0.601, k = 2,89,936.0 N/m, c = 67,646.8 Ns/m, which is in good agreement with exact values. Results of calculation performed for different noise levels are presented in Fig. 9. It is easy to notice that the rel-

ative errors of values of a, k and c parameters are of the order of noises. The third-kind method is also applied to artificial experimental data described above. The following results are obtained: a = 0.609, k = 2,92,115.0 N/m, c = 64,984.0 Ns/m when 3% noises are randomly introduced to artificial data. Relative errors of damper parameter values performed for different noise levels are shown in Fig. 10. Very good results are obtained using the second-kind method. Following the method presented in [36], random noises is added to the artificial data uðtÞ and qðtÞ using the formulae:

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ue ðtÞ ¼ uðtÞ þ exðtÞ v arðuðtÞÞ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qe ðtÞ ¼ qðtÞ þ ezðtÞ v arðuðtÞÞ;

ð75Þ

where e is the noise level, xðtÞ and zðtÞ are the standard distribution vectors with zero mean ant unit deviation, v arðuðtÞÞ and v arðqðtÞÞ are the variation of uðtÞ and qðtÞ, respectively. Results of calculation performed for different noise levels are presented in Fig. 11. In the considered case, the inﬂuence of noises on damper parameters is very small. The discussed fractional Kelvin–Voigt model was also used to determine the parameters of the small size VE damper. The damper consists of three steel plates and two layers made from viscoelastic material VHB 4959 manufactured by 3M. The thickness of each viscoelastic layer is 3 mm. A harmonically varying excitation with dif-

Table 1 Artiﬁcially generated experimental data. Frequency (Hz)

0.5 1.0 2.0 4.0 6.0 8.0 10.0 12.5 15.0

Fractional Kelvin–Voigt model

Fractional Maxwell model

q0 (m)

u1 (N)

u2 (N)

u0 (N)

q1 (m)

q2 (m)

0.991010e 3 0.991010e 3 1.004310e 3 1.005330e 3 1.000410e 3 0.990801e 3 0.997754e 3 1.007520e 3 0.991348e 3

366.564 414.464 471.856 564.810 637.696 702.286 761.647 834.779 902.471

108.520 165.182 250.977 382.750 489.893 580.127 653.921 753.747 836.230

297.306 302.356 300.441 299.111 298.085 297.083 300.699 301.900 301.531

0.232195e 2 0.191155e 2 0.160286e 2 0.140054e 2 0.131743e 2 0.127103e 2 0.125184e 2 0.121994e 2 0.121300e 2

0.177817e 2 0.119442e 2 0.787455e 3 0.517613e 3 0.401366e 3 0.338493e 3 0.295670e 3 0.261395e 3 0.234501e 3

Fig. 9. Relative errors of the a parameter (small rhombs), the k parameter (small cross marks) and the c parameter (small triangles) versus the level of noises (the ﬁrst-kind identiﬁcation method applied to the Kelvin–Voigt fractional model).

R. Lewandowski, B. Chora˛z_ yczewski / Computers and Structures 88 (2010) 1–17

Fig. 10. Relative errors of the a parameter (small rhombs), the k parameter (small cross marks) and the c parameter (small triangles) versus the level of noises (the third-kind identiﬁcation method applied to the Kelvin–Voigt fractional model).

Fig. 11. Relative errors of the a parameter (small rhombs), the k parameter (small cross marks) and the c parameter (small triangles) versus the level of noises (the secondkind identiﬁcation method applied to the Kelvin–Voigt fractional model).

ferent excitation frequencies taken from the range 0.5–15.0 Hz is used during the experiments. The Kelvin–Voigt model is adopted to describe the VE damper’s behaviour. At the beginning, the classical Kelvin–Voigt model, i.e. the fractional model with a ¼ 1, was used to determine the dampers parameters. In this case, the values of k and c parameters can be determined independently for each frequency of excitation from Eqs. (54). The results of calculation based on a times series are presented in Table 2 where values of the parameters k and c in dependence on the values of excitation frequency are given. As expected, the model parameters strongly depend on excitation frequency. A comparison of the experimentally obtained hysteresis curve with ones resulting from the identiﬁcation procedure is made in Fig. 12 where the comparison is given for k ¼ 1:0 Hz. The small cross marks show the experimental data while the solid line presents identiﬁcation results. The very good agreement between the both curves is obvious.

Table 2 Stiffness and damping factors of small damper. Frequency (rad/s)

k (N/m)

c (Ns/m)

3.14 6.28 12.56 25.12 37.68 50.24 62.80 78.50 94.20

73303.1 86615.5 98416.0 106055.0 106111.0 105254.0 101254.0 98644.0 86823.0

14676.5 9659.6 5927.0 3362.0 2295.0 1702.0 1373.0 1085.0 851.0

The results obtained using the fractional Kelvin–Voigt model and the third-kind method will be brieﬂy described. The following values of model parameters: a = 0.3755, k = 2603.1 N/m i c =

R. Lewandowski, B. Chora˛z_ yczewski / Computers and Structures 88 (2010) 1–17

Fig. 12. A comparison of the experimentally obtained hysteresis curve (small cross marks) and the hysteresis curve resulting from the identiﬁcation procedure (solid line).

Fig. 13. Storage modulus – comparison of experimental data (small cross marks) with results obtained from the identiﬁcation procedure (solid line) – the fractional Kelvin– Voigt model.

30,333.9 Ns/m are obtained after application of the identiﬁcation procedure. A comparison of the storage modulus resulting from the identiﬁcation procedure (the solid line) with the storage modulus obtained experimentally (the small cross marks) is presented

in Fig. 13. A similar comparison for the loss modulus is shown in Fig. 14. The presented results indicate that the three parameter fractional Kelvin–Voigt model could reasonably well describe the behaviour of solid VE dampers.

R. Lewandowski, B. Chora˛z_ yczewski / Computers and Structures 88 (2010) 1–17

Fig. 14. Loss modulus – comparison of experimental data (small cross marks) with results obtained from the identiﬁcation procedure (solid line) – the fractional Kelvin–Voigt model.

6.3. Results of typical calculation – the fractional Maxwell model As previously, the identification procedure described above is applied to artificial data. The artificial experimental steady state solutions are generated on the basis of the solution to Eq. (4). If, for example, uc ¼ 0 and u0 ¼ us then

u0 ap sin ; 2 kðskÞa h u0 api a : q1 ¼ qs ¼ a ðskÞ þ cos 2 kðskÞ

q2 ¼ qc ¼

ð76Þ

The artificial data are modified by applying small, randomly inserted perturbations. The following data are used to generate artificial data: uc = 0, us = u0 = 300.0 N, k = 2,90,000.0 N/m, c = 68,000.0 Ns/m, a = 0.6, n = 9. The values of excitation frequencies chosen in this example and values of u0i ; q1i and q2i are given in Table 1. After application of the identification procedure of first-kind the following results are obtained: a = 0.601, k = 2,90,469.0 N/m, c = 67,786.2 Ns/m which is in good agreement with exact values. The results of calculation performed for different noise levels (taken in the range of 0–5 %) are presented in Fig. 15. It is easy

Fig. 15. Relative errors of the a parameter (small rhombs), the k parameter (small cross marks) and the c parameter (small triangles) versus the level of noises (the ﬁrst-kind identiﬁcation method applied to the Maxwell fractional model).

R. Lewandowski, B. Chora˛z_ yczewski / Computers and Structures 88 (2010) 1–17

Fig. 16. Relative errors of the a parameter (small rhombs), the k parameter (small cross marks) and the c parameter (small triangles) versus the level of noises (the third-kind identiﬁcation method applied to the Maxwell fractional model).

Fig. 17. Error functional eJ M of the fractional Maxwell model versus the a parameter for artiﬁcial data with noise intensities of 3%, 5% and 10%.

to notice that the relative errors of values of a, k and c parameters are of the order of noises. The third-kind method is also applied to the artificial experimental data. The following results are obtained: a = 0.610, k = 2,84,543.0 N/m, c = 68,096.0 Ns/m when 3% noises are randomly introduced to artificial data. The results of calculations performed for different noise levels are shown in Fig. 16. Additionally, in Fig. 17 the plot functional eJ M (given by relationship (62)) of the fractional Maxwell model versus the parameter a is presented for three levels of noises. In the range of values of interest of the parameter a we have one minimum of the functional. Very similar results to those presented above are obtained using the second-kind method. Moreover, a similar plot to the one presented in Fig. 17 is obtained for the fractional Kelvin–Voigt model. The results of calculations performed for artificial data showed that if the noise levels are not too high then all of the suggested

methods of parameters identification are not sensitive to the noises. Errors in the values of the parameters obtained in the identification procedures are of the order of the noise levels. The next step is to apply the identification procedure to real experimental data. The experimental data presented by Makris and Constantinou [17] are chosen and used in this example. In their investigations, Makris and Constantinou were using a damper manufactured by GERB Schwingungsisolierungen GmbH & Co. KG. The following values of parameters of the fractional Maxwell model are determined: a = 0.77, k = 503.350 kN/m, c = 13.823 kNs/m. In Figs. 18 and 19 a comparison of the experimental and approximated storage modulus K 0 and the loss modulus K 00 is presented. The values of K 0 and K 00 resulting from the identification procedure are calculated from relationships (12) and (13). The presented model works satisfactorily. The identification procedure of the parameters of the fractional Maxwell model is simple, well applicable and efficient.

R. Lewandowski, B. Chora˛z_ yczewski / Computers and Structures 88 (2010) 1–17

Fig. 18. Storage modulus – ﬂuid damper.

Fig. 19. Loss modulus – ﬂuid damper.

7. Concluding remarks A family of parameters identification methods for the Kelvin– Voigt and the Maxwell fractional models are proposed in the paper. Three kinds of identification methods are suggested to determine three parameters of the fractional Kelvin–Voigt model and the fractional Maxwell model. Each identification procedure consists of two main steps. In the first step, for the given frequencies of excitation, experimentally obtained data are approximated by simple harmonic functions while model parameters are determined in the second stage of the identification procedure. The parameters

of fractional models are determined using results from dynamical tests. The suggested identiﬁcation methods differ substantially in comparison with previously proposed method. Rather than using the experimentally obtained complex modulus, the discussed methods directly utilize the measured steady state responses of damper. The methods could be also used when data are available only from a narrow range of excitation frequency. The results of calculation show that the proposed methods are not sensitive to the measurement noises and, therefore, it is not necessary to use regularization techniques. Similar calculation efforts are needed when each of the presented methods are used to identify the dam-

R. Lewandowski, B. Chora˛z_ yczewski / Computers and Structures 88 (2010) 1–17

per parameters and the existing differences between the discussed methods have no significant influence of the effectiveness of the methods and on results of identification. All of the proposed identification procedures are simple, well applicable and efficient. Moreover, hysteresis loop equations are derived for both models. Some properties of the hysteresis curves are used to develop one of the identification procedures. The validity and effectiveness of the identification procedures have been successfully tested using both artificially generated and real experimental data. It was found that they are real VE dampers to which this model can be fitted with a satisfactory accuracy. Acknowledgments The authors wish to acknowledge the financial support received from the Poznan University of Technology (Grant No. DS 11-018/ 08) in connection with this work. The authors would like to express their appreciation to the reviewers for their valuable suggestions.

Appendix A. Approximation of experimental data by trigonometric functions In this Appendix A we assume that, during the experimental tests, two output functions ue ðtÞ (function of force in a time domain) and qe ðtÞ (function of displacement in a time domain) which represent the steady state behavior of damper were obtained. Both functions could be approximated using simple trigonometric functions, as described below. Two procedures of the parameters identiﬁcation based on time series data are presented in this Subsection. Experimentally measured displacements of the damper are approximated using the function:

~ðtÞ ¼ q ~c cos kt þ q ~s sin kt: q

ðA1Þ

~c and q ~s of function Using the least-square method, parameters q (A1) are determined. This method requires minimization of the following functional:

~c ; q ~s Þ ¼ J 1 ðq

1 t2 t1

t2 t1

~ðtÞ 2 dt; ½qe ðtÞ q

ðA2Þ

where the symbols t 1 and t 2 denote the beginning and the end of the time range in which the damper’s displacements were measured. Part of the measuring results related to steady state vibration are used as data in this identiﬁcation procedure. From the stationary conditions of the functional (A2), the following system of equa~s are received: ~c and q tions with respect to parameters q

~c þ Isc q ~s ¼ Icq ; Icc q

~c þ Iss q ~s ¼ Isq ; Isc q

ðA3Þ

where

Icc ¼ ¼ Icq ¼

t1 Z t2 t1 Z t2

cos2 ktdt;

Iss ¼

sin ktdt;

Ics ¼ Isc

sin kt cos ktdt; qe ðtÞ cos ktdt;

ðA4Þ Isq ¼

qe ðtÞ sin ktdt:

ðA5Þ

In a similar way the experimentally measured damper force is approximated using the following harmonic function:

~ s sin kt: ~ ðtÞ ¼ u ~ c cos kt þ u u

ðA6Þ

~ c and u ~ s are determined with the help of the lastThe parameters u square method. The functional and the system of equations with re~ s are: ~ c and u spect to u

~c ; u ~s Þ ¼ J 2 ðu

1 t2 t1

~ c þ Isc u ~ s ¼ Icu ; Icc u

~ ðtÞ 2 dt; ½ue ðtÞ u

ðA7Þ

~ c þ Iss u ~ s ¼ Isu ; Isc u

ðA8Þ

where Icc ; Iss and Ics ¼ Isc are given by relationships (A4). Moreover,

Icu ¼

ue ðtÞ cos ktdt; t1

Isu ¼

ue ðtÞ sin ktdt:

ðA9Þ

References [1] Christopoulos C, Filiatrault A. Principles of passive supplemental damping and seismic isolation. Pavia, Italy: IUSS Press; 2006. [2] Singh MP, Moreschi LM. Optimal placement of dampers for passive response control. Earthquake Eng Struct Dyn 2002;31:955–76. [3] Shukla AK, Datta TK. Optimal use of viscoelastic dampers in building frames for seismic force. J Struct Eng 1999;125:401–9. [4] Park SW. Analytical modeling of viscoelastic dampers for structural and vibration control. Int J Solids Struct 2001;38:8065–92. [5] Palmeri A, Ricciardelli F, De Luca A, Muscolino G. State space formulation for linear viscoelastic dynamic systems with memory. J Eng Mech 2003;129:715–24. [6] Singh MP, Chang TS. Seismic analysis of structures with viscoelastic dampers. J Eng Mech 2009;135:571–80. [7] Chang TS, Singh MP. Mechanical model parameters for viscoelastic dampers. J Eng Mech 2009;135:581–4. [8] Gerlach S, Matzenmiller A. Comparison of numerical methods for identification of viscoelastic line spectra from static test data. Int J Numer Meth Eng 2005;63:428–54. [9] Syed Mustapha SMFD, Philips TN. A dynamic nonlinear regression method for the determination of the discrete relaxation spectrum. J Phys D 2000;33:1219–29. [10] Pritz T. Analysis of four-parameter fractional derivative model of real solid materials. J Sound Vib 1996;195(1):103–15. [11] Atanackovic TM. A modified Zener model of a viscoelastic body. Continuum Mech Thermodyn 2002;14:137–48. [12] Song DY, Jiang TQ. Study on the constitutive equation with fractional derivative for the viscoelastic fluids – modified Jeffreys model and its application. Rheol Acta 1998;37(5):512–7. [13] Pritz T. Five-parameter fractional derivative model for polymeric damping materials. J Sound Vib 2003;265:935–52. [14] Bagley RL, Torvik PJ. Fractional calculus – a different approach to the analysis of viscoelastically damped structures. AIAA J 1989;27:1412–7. [15] Schmidt A, Gaul L. Finite element formulation of viscoelastic constitutive equations using fractional time derivatives. J Nonlinear Dyn 2002;29:37–55. [16] Aprile A, Inaudi JA, Kelly JM. Evolutionary model of viscoelastic dampers for structural applications. J Eng Mech 1997;123:551–60. [17] Makris N, Constantinou MC. Fractional-derivative Maxwell model for viscous dampers. J Struct Eng 1991;117:2708–24. [18] Honerkamp J. Ill-posed problem in rheology. Rheol Acta 1989;28:363–71. [19] Hansen S. Estimation of the relaxation spectrum from dynamic experiments using Bayesian analysis and a new regularization constraint. Rheol Acta 2007;47:169–78. [20] Beda T, Chevalier Y. New method for identifying rheological parameter for fractional derivative modeling of viscoelastic behavior. Mech Time-Depend Mater 2004;8:105–18. [21] Gaul L, Schmidt A. Parameter identification and FE implementation of a viscoelastic constitutive equation using fractional derivatives. PAMM (Proc Appl Math Mech) 2002;1(1):153–4. [22] Galucio AC, Deu JF, Ohayon R. Finite element formulation of viscoelastic sandwich beams using fractional derivative operators. Comput Mech 2004;33:282–91. [23] Castello DA, Rochinha FA, Roitman N, Magulta C. Modelling and identification of viscoelastic materials by means of a time domain technique. In: Proceedings of sixth world congress of structural and multidisciplinary optimization, Rio de Janeiro, Brazil; 30 May–03 June. p. 1–10. [24] Matsagar VA, Jangid RS. Viscoelastic damper connected to adjacent structures involving seismic isolation. J Civil Eng Manage 2005;11:309–22. [25] Lee SH, Son DI, Kim J, Min KW. Optimal design of viscoelastic dampers using eigenvalue assignment. Earthquake Eng Struct Dyn 2004;33:521–42. [26] Park JH, Kim J, Min KW. Optimal design of added viscoelastic dampers and supporting braces. Earthquake Eng Struct Dyn 2004;33:465–84. [27] Singh MP, Verma NP, Moreschi LM. Seismic analysis and design with Maxwell dampers. J Eng Mech 2003;129:273–82. [28] Hatada T, Kobori T, Ishida M, Niwa N. Dynamic analysis of structures with Maxwell model. Earthquake Eng Struct Dyn 2000;29:159–76. [29] Makris N, Zhang J. Time-domain viscoelastic analysis of earth structures. Earthquake Eng Struct Dyn 2000;29:745–68. [30] Spanos PD, Tsavachidis S. Deterministic and stochastic analyses of a nonlinear system with a Biot visco-elastic element. Earthquake Eng Struct Dyn 2001;30:595–612.

R. Lewandowski, B. Chora˛z_ yczewski / Computers and Structures 88 (2010) 1–17 [31] Adhikari S. Dynamics of non viscously damped linear systems. J Eng Mech 2002;128:328–39. [32] Papoulia KD, Panoskaltsis VP, Korovajchuk I, Kurup NV. Rheological representation of fractional derivative models in linear viscoelasticity. Rheol Acta; in press. [33] Yin Y, Zhu KQ. Oscillating flow of a viscoelastic fluid in a pipe with the fractional Maxwell model. Appl Math Comput 2006;173:231–42. [34] Podlubny I. Fractional differential equations. Academic Press; 1999. [35] Lion A. Thermomechanically consistent formulations of the standard linear solid using fractional derivatives. Arch Mech 2001;53:253–73. [36] Zhu XQ, Law SS. Orthogonal function in moving loads identification on a multispan bridge. J Sound Vib 2001;245:329–45. [37] Eldred LB, Baker WP, Palazotto AN. Kelvin–Voigt vs fractional derivative model as constitutive relations for viscoelastic materials. AIAA J 1995;33:547–50.

[38] Soula M, Vinh T, Chevalier Y. Transient responses of polymers and elastomers deduced from harmonic responses. J Sound Vibr 1997;205:185–203. [39] Kim SY, Lee DH. Identification of fractional-derivative-model parameters of viscoelastic materials from measured FRF’s. J Sound Vib 2009;324:570–86. [40] Makris N. Complex-parameter Kelvin model for elastic foundations. Earthquake Eng Struct Dyn 1994;23:251–64. [41] Vasques CMA, Moreira RAS, Rodrigues JD. Experimental identification of GHM and ADF parameters for viscoelastic damping modeling. In: Mota Soares CA et al., editors. Proceedings of the III European conference on computational mechanics. Solids, structures and coupled problems in engineering, Lisbon, Portugal; 5–8 June 2008. p. 1–25. [42] Gerlach S, Matzenmiller A. On parameter identification for material and microstructural properties. GAMM-Mitt 2007;30:481–505.

Computers and Structures 88 (2010) 18–24

Contents lists available at ScienceDirect

Computers and Structures journal homepage: www.elsevier.com/locate/compstruc

Comparison on numerical solutions for mid-frequency response analysis of ﬁnite element linear systems Jin Hwan Ko a, Doyoung Byun b,* a b

School of Mechanical and Aerospace Engineering, Seoul National University, Daehak-dong, Gwanak-gu, Seoul 151-742, South Korea Department of Aerospace Information Engineering, Konkuk University, 1 Hwayang-dong, Kwangjin-Gu, Seoul 143-701, South Korea

a r t i c l e

i n f o

Article history: Received 28 October 2008 Accepted 26 September 2009 Available online 28 October 2009 Keywords: Mid-frequency response analysis Algebraic substructuring Frequency sweep algorithm Modal acceleration method

a b s t r a c t Mid-frequency response analysis often faces computational difficulties when a conventional modal approach is used for a finite element linear system. In this paper, the computational burden is relieved by frequency sweep algorithm or mode acceleration method for a reduced-order system constructed by algebraic substructuring, a variant of model order reduction. The two methods are compared with the help of numerical experiments and their computational complexity. As demonstrated by the finite element simulations, in which proportional damping is assumed, of a turbo-prop aircraft and a ring resonator, the frequency sweep algorithm for reduced-order systems shows the best performance among all considered numerical solutions, including the conventional approach. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction

symmetric positive-deﬁnite. The input–output behavior of the model (1.1) is characterized by the frequency response function:

Frequency response analysis is a crucial tool in characterizing the dynamic behavior of a mechanical system; it provides not only the resonance frequencies of the system, but also the amplitudes of the responses under unit force excitations. Responses for a low frequency range have conventionally been computed using a deterministic approach such as the ﬁnite element method, and those in the high-frequency range have been solved by an energy approach such as the Statistical Energy Method. However, the development of numerical methods for solving the responses in the mid-frequency range has remained a frontier area yet to be conquered. In this paper, numerical solutions of ﬁnite element linear systems are considered for the mid-frequency response analysis. The discretized model of a structure for a continuous single-input single-output second-order system can be written as follows:

1 T T HðxÞ ¼ l x2 M þ ixD þ K b ¼ l GðxÞ 1 b;

(

_ þ KxðtÞ ¼ buðtÞ; M€xðtÞ þ DxðtÞ T

yðtÞ ¼ l xðtÞ;

ð1:1Þ

_ where xð0Þ ¼ x0 and xð0Þ ¼ v 0 . Here, t is the time variable, xðtÞ 2 RN is a state vector, and N is the order of the system. uðtÞ is the input excitation force and yðtÞ is the output measurement function. b 2 RN and l 2 RN are the input and output distribution vectors, respectively, and M; K, and D 2 RN N are the system mass, stiffness and damping matrix, respectively. It is assumed that M and K are * Corresponding author. Tel.: +82 2 450 4195; fax: +82 2 444 6670. E-mail address: dybyun@konkuk.ac.kr (D. Byun). 0045-7949/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2009.09.009

ð1:2Þ

where x is the frequency, i is the imaginary unit, and GðxÞ is the system dynamic matrix. Mathematically, the mid-frequency response analysis pertains to the computation of HðxÞ for x in the range ½xmin ; xmax . A conventional numerical solution for the frequency response analysis has been solved by the mode superposition method on a finite element linear system. However, the mode superposition method faces two types of computational difficulties in regard to mid-frequency response analysis. First, a finely discretized model is required to represent the modes for which the natural frequencies are close to the mid-frequency range, leading to a dramatic increase in computational cost. The computational burden of a large order system can be relieved by model order reduction. So far, substructuringbased model order reductions have been developed for improving efficiency in large systems. Automatic multilevel substructuring [1] and algebraic substructuring [2] are included in this type. Moreover, a variant of algebraic substructuring has recently been developed for solving only the modes of the interior eigenvalues [3], which contribute more to the mid-frequency responses than the extreme eigenvalues do. The second difficulty is due to an increase in the number of retained modes. Modal superposition requires two or three times maximum frequency for a given range to achieve a desired accuracy; therefore, when the frequency range under consideration shifts to mid-range, the number of modes increases significantly. Residual

J. Hwan Ko, D. Byun / Computers and Structures 88 (2010) 18–24

flexibility modes play a role in decreasing the number of retained high-frequency modes, considering the static response to a given load [4], but all of the low frequency modes are still required. Therefore, numerical techniques that consider low- and high-frequency mode truncations are also required. There have been two recentlyintroduced methods to compensate for low- and high-truncations; the first is the frequency sweep algorithm and the second is the modal acceleration method. The frequency sweep algorithm was initially introduced by Bennighof and Kaplan [1], and its convergence was verified by Ko and Bai when it was applied to compensate for the both truncation errors [5]. The modal acceleration method, as introduced in Ref. [6], was also used to compensate for both errors for a system on FE subspace. In this paper, modal superposition is modified for solving a reduced system on the algebraic substructuring subspace, referred to as AS subspace. In the present study, the performances of frequency sweep and modal acceleration in AS subspace are compared in detail in terms of their computational complexity and numerical experiments done on the finite element model of a turbo-prop aircraft and a ring resonator. Comparison of both methods with a conventional method on the FE subspace also included. 2. Frequency response analysis Assuming Rayleigh damping D ¼ aM þ bK, and the introduction of the shift r, which is used for obtaining the modes of the interior eigenvalues, the frequency response function HðxÞ on the FE subspace, of column dimension N, can be written thus: 1

HðxÞ ¼ l ½c1 K r þ c2 M b ¼ l GðxÞ 1 b; T

ð2:1Þ

where K r ¼ K rM, c1 ¼ c1 ðxÞ ¼ 1 þ ixb and c2 ¼ c2 ðx; rÞ ¼ x2 þ r þ ixða þ rbÞ. The AS subspace is composed of column vectors in Am , constructed by retained substructure modes in the algebraic substructuring (AS), and typically has a dimension m that is much smaller than N [5]. Projecting Eq. (1.1) onto the AS subspace by xðtÞ ¼ Am zðtÞ yields the following equation:

(

M m€zðtÞ þ Dm z_ ðtÞ þ K m zðtÞ ¼ bm uðtÞ; yðtÞ ¼

T lm zðtÞ;

ð2:2Þ

where K m ¼ ATm KAm ; Mm ¼ ATm MAm ; lm ¼ ATm l and bm ¼ ATm b; The corresponding frequency response function is

1 T T Hm ðxÞ ¼ lm c1 K rm þ c2 M m bm ¼ lm pm ðxÞ;

ð2:3Þ

where K m ¼ K m rMm . pm ðxÞ is solved by the parameterized linear system of order m:

Gm ðxÞpm ðxÞ ¼ bm ;

ð2:4Þ

where Gm ðxÞ ¼ c1 K rm þ c2 Mm . In this paper, Hm ðxÞ is used as an approximation for HðxÞ. 2.1. Modal superposition method In the AS method, the order m typically remains too high for a direct method of computing frequency responses to be applied. Therefore, the modal superposition method is used. Here, the pn ðxÞ values occur in the subspace spanned by the retained global modes Un from a reduced eigensystem K rm / ¼ hr M m / and the pt ðxÞ values occur in the subspace spanned by the low- and high-frequency truncated modes. Given pn ðxÞ ¼ Un gn ðxÞ for some coefﬁcient vector gn ðxÞ, the Eq. (2.4) then follows:

Gm ðxÞðUn gn ðxÞ þ pt ðxÞÞ ¼ bm :

ð2:5Þ

The modal mass and stiffness matrices becomes: T n Mm

Un ¼ I and UTn K rm Un ¼ Hrn ;

ð2:6Þ

where I is a diagonal unit matrix and Hrn is a diagonal matrix because the global modes satisfy the mass and stiffness orthogonality conditions. In this paper, the global modes are normalized so that the modal mass /Ti Mm /i ¼ 1, where i ¼ 1; . . . ; n. When Eq. (2.5) is pre-multiplied by UTn ; UTn Gm ðxÞpt ðxÞ, where pt ðxÞ is expressed by a linear combination of the truncated modes, is eliminated due to the orthogonality and the vector pn ðxÞ is then given by the n uncoupled equations:

1 1 pn ðxÞ ¼ Un UTn Gm ðxÞUn UTn bm ¼ Un c1 Hrn þ c2 I UTn bm ¼ Un gn ðxÞ:

ð2:7Þ

From (2.3), the frequency response function is solved by T

Hm ðxÞ ¼ Hn ðxÞ ¼ lm pn ðxÞ:

ð2:8Þ

Hn ðxÞ contains errors due to the truncated modes in the lowand high-frequency ranges, which are compensated for by the next two methods. 2.2. Frequency sweep algorithm An algorithm known as the frequency sweep (FS) algorithm was introduced [1] to compensate for the error of the truncated modes. The FS algorithm is an iterative scheme that includes the modal superposition method. Eq. (2.5) can be written as a parameterized linear system for pt ðxÞ:

Gm ðxÞpt ðxÞ ¼ bm Gm ðxÞpn ðxÞ:

ð2:9Þ

Using the Galerkin subspace projection based on the orthogoaround nality of Un and Ut , and the Taylor expansion of G 1 m ðK rm Þ 1 , the following iteration is derived [5] for computing the vector pt ðxÞ:

p‘t ðxÞ ¼ pt‘ 1 ðxÞ þ

1 T i ‘ 1 1 h r 1 Km Un Hrn Un r m ðxÞ;

ð2:10Þ

‘ 1 where r m ðxÞ ¼ bm Gm ðxÞðpn ðxÞ þ p‘ 1 ðxÞÞ, and is the ð‘ 1Þ-th t residual vector for ‘ ¼ 1; 2; . . ., with an initial guess of p0t ðxÞ. p0t ðxk Þ is determined by a linear extrapolation of the computed vector at a previous frequency if k > 2; otherwise, p0t ðx0 Þ ¼ 0 and p0t ðx1 Þ ¼ pt ðx0 Þ. A practical stopping criterion is set to test the relative residual error:

kr m ðxÞk2 =kbm k2 6 ;

ð2:11Þ

for a given tolerance . The iteration (2.10) guarantees its convergence based on the derivation in [5] by satisfying the condition that the contraction ratio n is smaller than one when the global cutoff values are determined by

krmin ¼ dmax =n and krmax ¼ dmax =n;

ð2:12Þ

where dðx; rÞ ¼ j c2 =c1 j, and dmax ¼ maxfdðxk ; rÞ; 1 6 k 6 nf g, in which nf is the number of sampling frequencies required to obtain the frequency response curve. In order to compare the performance of the frequency sweep with that of the modal acceleration method, the computational complexities of each are analyzed in detail. First, the computational complexity of the FS is shown in Table 1, in which niter denotes the total number of frequency sweep iterations. Here, K rm ; M m ; Un ; Hrn , and bm are real, but the other vectors or matrices are complex. The value of niter depends on the tolerance of the stopping criterion and the values of the variable n, which stands for its convergence rate. Complex–real multiplications are counted twice and complex–complex multiplications are counted four times when compared to real–real multiplications. A real–complex division is considered as six times as expensive as a real–real

J. Hwan Ko, D. Byun / Computers and Structures 88 (2010) 18–24

Table 1 Cost of operations for the frequency sweep algorithm. Step

Task

Cost

(i)

UTn bm

m n

for k ¼ 1; nf

gn ðxk Þ ¼ ðc1 Hrn þ c2 IÞ 1 ðiÞ pn ¼ Un gn ðxk Þ ifðk > 2Þ; p0t ðxk Þ by linear extrapolation

(ii-1) (ii-2) (ii-3) (iii-1)

10n 2m n 2m 2m c þ 10m

r 0m ðxk Þ ¼ bm ðc1 K rm þ c2 Mm Þp0m ðxk Þ; where p0m ðxk Þ ¼ pn ðxk Þ þ p0t ðxk Þ

(iii-2)

ifðkr0m ðxk Þk2 =kbm k2 6 Þ break for ‘ ¼ 1; 2; . . . 1 T ‘ 1 Un Hrn Un r m ðxk Þ

(iv-1)

4m n þ 2n

(iv-2)

ðK rm Þ 1 r ‘ 1 m ðxk Þ

(iv-3)

p‘t ðxk Þ ¼ p‘ 1 ðxk Þ þ c1 ½ðiv-2Þ—ðiv-1Þ t

2m 4m

r ‘m ðxk Þ ¼ bm ðc1 K rm þ c2 M m Þp‘m ðxk Þ; where p‘m ðxk Þ ¼ pn ðxk Þ þ p‘t ðxk Þ if kr ‘m ðxk Þk2 =kbm k2 6 break end

(v-1) (v-2)

2m c þ 10m 5m

(vi)

Hm ðxk Þ ¼ lm pm ðxk Þ end

Main cost

nf m ð2n þ 2cÞ þ niter m ð4n þ 2cÞ

multiplication. Scalar–scalar operations, such as 1=c1 , are not considered in the table. Note that the (iv-2) value takes only 2m operations because K rm is a diagonal matrix. This is another big advantage of using the AS subspace. The multiplication of M m to a complex vector in (iii-1) and (v-1) can be estimated by 2m c, where c is an integer determined by the sparsity of Mm . The main cost in Table 1 is split into two parts: an nf -dependent part from (ii), (iii), and (vi), and an niter-dependent part from (iv) and (v). Generally, c; n, and nf are much smaller than m. Moreover, the convergence rate of the FS is known to be very fast [5], implying that niter is expected to be much smaller. 2.3. Modal acceleration method The modal acceleration (MA) method, another iterative scheme which is used in addition to the modal superposition method, was also developed to compensate for the errors caused by the truncated modes. Here, the method introduced in [6] is adopted. When it is applied to solve Hm ðxÞ, the following equation is used:

pm ðxÞ p1 ðxÞ þ p2 ðxÞ ‘ L r 1 X c2 r 1 ¼ c 1 K M ðK Þ b m ð xÞ m 1 m m ‘¼0

n X j¼1

c2 c1 hrj

!Lþ1 ðgn ðxÞÞj /j ;

ð2:13Þ

where p1 ðxÞ is the ﬁrst term, p2 ðxÞ is the second term, and ðgn ðxÞÞj is the j-th component of gn ðxÞ. If L ¼ 1; pm is equal to pn in Eq. (2.7) of the modal superposition method. As L increases, accuracy is expected to improve. The value of L is typically smaller than 5; it is given an identical value at every frequency [6]. Note from the vectors that p1 is computed n o r 1in Krylov subspace c2 , where A b ¼ M K m . Because the vectors bm ; Ar bm ; . . . ; AL m r m r c1 tend very quickly to become almost linearly dependent during the computation, methods relying on Krylov subspace frequently involve some orthogonalization scheme, but the conventional MA does not involve it; thus, this may cause bad behavior in convergence of the MA, which will be shown in Section 3.3.1. In this paper, Eq. (2.13) is reformulated into an alternative recursive form allowing us to use the same stopping criterion (2.11) and to compare it with the FS. The equation where L ¼ 0

is considered in the initial step, which is consistent with the linear extrapolation of the FS. The ﬁrst and second terms in this case become: r 1 p01 ðxÞ ¼ c 1 1 ðK m Þ bm ðxÞ; 0

p02 ðxÞ ¼ Un g0n ðxÞ;

ð2:14Þ

where xÞ ¼ bm ðxÞ and g xÞ ¼ c1 Hrn 1 gn ðxÞ. Next, the following reﬁnement iterations for computing the two terms are reformulated from (2.13) 0 bm ð

0 nð

r 1 ‘ p‘1 ðxÞ ¼ p1‘ 1 ðxÞ þ c 1 K m bm ðxÞ; 1 where

‘ bm ð

xÞ ¼

p‘2 ðxÞ ¼ Un g‘n ðxÞ;

ð2:15Þ

1 ‘ 1 1 ‘ 1 M m K rm bm ðxÞ and g‘n ðxÞ ¼ cc2 Hrn gn ðxÞ 1

for ‘ ¼ 1; 2; . . .,. At each frequency xk , iterations are run until p‘m ðxk Þ ¼ p‘1 ðxk Þ þ p‘2 ðxk Þ satisfies (2.11), in which r m ðxk Þ ¼ bm ðxk Þ Gm ðxk Þp‘m ðxk Þ. The convergence of the modal acceleration method is guaranteed so long as the condition of (2.12) is satisfied according to the convergence condition in Ref. [6]. The computational complexity of MA is described in Table 2, in which only M m ; K rm ; krr ; /r , and bm are real, but the other vectors or matrices are complex. Scalar–scalar operations such as 1=c1 and c2 =c1 are not considered. The main cost is also split into two parts, the nf -dependent part from (ii), (iii), and (vi), and the niter-dependent part from (iv) and (v). nf is given by the user, and c and m are determined after constructing the projected system. Consequently, the number of global modes n and the convergence rate are the most important factors in the performance comparison between the MA and the FS. The convergence rate of the MA is also known to be very fast [6]. According to the main costs in Tables 1 and 2, the first terms are equal and the second terms are expected to be close to each other, provides the convergence rates of both methods are similar, as are n and c. Detailed comparisons are represented in the next section. 3. Numerical experiment 3.1. Numerical methods The previously presented methods on the AS subspace are implemented based on ASEIG [2]. The multilevel partitioning is done by METIS [7]. The eigenpairs of the sparse matrices are computed by the shift-invert Lanczos method of ARPACK [8] with SuperLU [9], and those of the dense matrices are solved by LAPACK. The methods for the experiments are listed below:

J. Hwan Ko, D. Byun / Computers and Structures 88 (2010) 18–24 Table 2 Cost of operations for the modal acceleration method. Step

Task

Cost

(i)

UTn bm

m n

for k ¼ 1; nf (ii-1) (ii-2) (ii-3) (ii-4) (iii-1) (iii-2)

gn ðxk Þ ¼ ðc1 Hrn þ c2 IÞ 1 ðiÞ g0n ðxk Þ ¼ cc12 Hrn 1 gn ðxk Þ

10n

r 1 0 1 bm ð k Þ; k Þ ¼ 1 ðK m Þ 0 where bm ð k Þ ¼ bm ð k Þ p02 ð k Þ ¼ Un 0n ð k Þ r 0m ð k Þ ¼ bm 1 K rm þ 2 M m p0m ð k Þ, 0 0 where pm ð k Þ ¼ p1 ð k Þ þ p02 ð k Þ if kr0m ð k Þk2 =kbm k2 6 break

p01 ð

x g x c x

x x

(iv-2)

‘ bm ð

(v-2) (vi)

2m n 2m c þ 10m

1 ‘ 1 gn ðxk Þ r 1 ‘ 1 xk Þ ¼ c1 Mm K m bm ðxk Þ r 1 ‘ 1 p‘1 ðxk Þ ¼ p‘ 1 Km bm ðxk Þ 1 ðxk Þ þ c1

g‘n ðxk Þ ¼ cc12 Hrn

(v-1)

c x

(iv-1)

(iv-4)

x x

for ‘ ¼ 1; 2; . . .

(iv-3)

p‘2 ðxk Þ ¼ Un g‘n ðxk Þ r‘m ðxk Þ ¼ bm c1 K rm þ c2 Mm p‘m ðxk Þ, where p‘m ðxk Þ ¼ p‘1 ðxk Þ þ p‘2 ðxk Þ if kr ‘m ðxk Þk2 =kbm k2 6 break end T

5m 6n 2m c þ 6m 6m 2m n 2m c þ 10m 5m 2m

Hm ðxk Þ ¼ lm pm ðxk Þ end

Main cost

nf m ð2n þ 2cÞ þ niter m ð2n þ 4cÞ

Direct: This computes the frequency responses from Hðxk Þ using SuperLU as a direct sparse solver. BL + RFM: This computes the frequency responses from Hðxk Þ using mode superposition with residual ﬂexible modes [4] and global modes from a Block Lanczos method of BLZPACK [10], which is the package of choice for computing a relatively large number of eigenvectors. BLZPACK uses the subroutines of MA47 [11], which is a direct sparse solver in multi-frontal scheme. The maximum cutoff value is given by kmax ¼ ð1:2xmax Þ2 and the minimum limit is zero. AS + FS: This computes the frequency responses from Hm ðxk Þ using the frequency sweep algorithm of Table 1. AS + MA: This computes the frequency responses from Hm ðxk Þ using the modal acceleration method of Table 2.

The analysis employs the FE model with shell elements of the simplified aircraft shown in Fig. 1, of which the order is 75,918. Matrices, K and M, from the finite element method are symmetric and positive-definite. Assume that region A in Fig. 1 is under an excited load and region B is for sensing the dynamic responses. The diameter of the fuselage is 2.5 m, which is close to the geometry of the Bombardier, and it has a thickness of 3 mm. Material properties of aluminum are used. The eigenvalue of the mode near 60 Hz in Fig. 1b is 580-th, which is located in the mid-frequency range. Frequency responses are computed at 201 frequencies which are discretized in [60 2, 60 + 2] Hz: nf ¼ 201. For proportional damping, Craig defines the damping factor to be f ¼ 0:5ðax þ b=xÞ [13]. Damping coefficients, which make f ¼ 0:2% at f = 60 Hz, are used.

3.2. Numerical examples

3.2.2. Micro-scale resonator The FE simulation of a micro-scale ring resonator is used for designing a high-frequency band-pass ﬁlter, e.g. surface acoustic wave devices in a cell phone. Recently-introduced ring resonators employ the so-called extensional wine-glass mode [14], which is depicted in Fig. 2b. The corresponding eigenvalue of the mode is the 306-th of the ﬁnite element model in Fig. 2a, which is located in the mid-frequency range. The corresponding natural frequency

3.2.1. Turbo-prop aircraft A turbo-prop aircraft, such as the Bombardier Dash 8, has a main excited frequency of 60Hz; the noise due to the excitation has been analyzed and controlled by experimental approaches [12]. Computation of the interior noise requires a mid-frequency response analysis for an unconstrained structure.

Fig. 1. A simpliﬁed turbo propeller aircraft.

J. Hwan Ko, D. Byun / Computers and Structures 88 (2010) 18–24

is 630 MHz and the order of the FE model with solid elements is 92,220. The frequency responses are computed at 201 frequencies which are discretized in [630 3, 630 + 3] MHz: nf ¼ 201. Assume that region A in Fig. 2 is under an excited load and region B is for sensing the dynamic responses. The ends of four beams are set to the clamped boundary condition. The geometry and materials are from the 634.6 MHz resonator of Ref. [14]. The damping coefﬁcients are set for a 0.01% damping factor at 630 MHz due to the high sensitivity of the resonator. All experiments were conducted on a platform, that has 4 GB of physical memory, utilized 2.66 GHz Intel Xeon processor, and uses the Red Hat Linux operating system. 3.3. Numerical results and discussion 3.3.1. Convergence and computational cost (1) Convergence vs. Tolerance First, the convergence rate is explored by varying the tolerance of the stopping criterion with the contraction ratio n ﬁxed at 0.5. The results of these variations are depicted in Fig. 3 with 10 1 ; 10 2 ; 10 3 ; 10 4 ; and 10 5 tolerances. As seen in Fig. 3, the convergence rates of AS + FS are much faster than those of AS + MA. Moreover, AS + FS can obtain more accurate results because AS + MA diverges at 10 3 for the turbo-prop aircraft and at 10 4 for the ring resonator, but AS + FS converges at all given tolerances.

(2) Convergence vs. Contraction ratio Next, the effect of n-variation is explored in a situation where the tolerance is ﬁxed at ¼ 10 3 . To explore the effect of n, the dimension of the AS subspace, m, is kept constant at 2,362 for the turbo-prop aircraft and 1,345 for the ring resonator. The effects on the number of the iterations, niter, and the number of global eigenmodes, n, are depicted in Fig. 4. As shown in Fig. 4, while n increases in both methods, n becomes smaller and niter becomes larger. Fig. 4 also indicates that the rate of increase of niter with the FS is smaller than with the MA, as n increases. Moreover, the divergence that occurs at n less than 1 for the MA, but not for the FS. It is speculated that this divergence is attributed mainly to the linear dependency of the vectors in p1 computation of Eq. (2.13). (3) Computational cost vs. Contraction ratio It is clear that the computational cost should increase as the tolerance decreases. However, the effect of n-variation on computational cost is complex. The time periods for computing the global modes and the iterations are depicted in Fig. 5. The sparsity of Mm ; c, is 352 for the turbo-prop aircraft and 146 for the ring resonator. Thus, n in Fig. 4 is smaller than c except 0.1 with regard to the turbo-prop aircraft and then the computational cost per iteration for the FS is a little smaller than that for the MA, according to the niter-dependent parts in Tables 1 and 2, in case that n is smaller than c. Moreover, the FS requires a much smaller

Fig. 2. A micro-scale ring resonator.

Fig. 3. Inﬂuence on the number of iterations when varying the tolerance.

J. Hwan Ko, D. Byun / Computers and Structures 88 (2010) 18–24

number of iterations than the MA. Subsequently, the computing time periods of the MA iteration exceed those of the FS as seen in Fig. 5. Fig. 5 also indicates that as n increases, the rate at which time for the global modes decreases is faster than the rate at which the time for the iterations increases, especially in the case of the turbo-prop aircraft. Subsequently, larger n, which makes n smaller, is advantageous. Thus, the following values are adopted in the next comparison: 0.9 for the FS and 0.2 for the MA of the turbo-prop aircraft, and 0.9 for FS and 0.6 for MA of the ring resonator.

3.3.2. Performance comparison In cases such as the previous section where the parameters of the AS-based methods are given, the frequency response curves of the two methods are depicted in Fig. 6, in which those of BL + RFM and the direct method are included. As shown, the frequency responses of AS + FS and AS + MA are in good agreement with those of the direct method. The frequency responses of BL + RFM for the turbo-prop aircraft have some discrepancies, but are very close to those of the direct method near the regions of

Fig. 4. Inﬂuence on n and niter when varying the contraction ratio.

Fig. 5. Inﬂuence on computing time when varying the contraction ratio.

Fig. 6. Frequency response curves.

J. Hwan Ko, D. Byun / Computers and Structures 88 (2010) 18–24

Table 3 The numbers of iterations and global modes, the elapsed time in seconds and speedup over BL + RFM. Methods

niter

TAS

Turbo-prop

AS + FS

226

77.32

8.94

3.94

90.20

8.4

Aircraft

AS + MA BL + RFM

338 –

232 759

77.32 –

49.14 752.69

6.76 4.99

133.22 757.68

5.7 –

Ring resonator

AS + FS AS + MA BL + RFM

165 643 –

7 10 397

85.36 85.36 –

1.04 1.40 704.03

0.84 2.44 1.46

87.24 89.20 741.49

8.5 8.3 –

resonance. For the ring resonator, those of BL + RFM also are in good agreement. Next, the number of iterations and global modes, the computing time, and speedup over BL + RFM for performance comparison are listed in Table 3 where TAS is the construction of the AS subspace, TGM is computation of the global modes, TSOL is the time needed to solve the frequency responses, and TTOT is the total time. Table 3 shows that the time for the FS or the MA iteration is short when compared to the total time. AS-based methods retain a smaller number of global modes than BL + RFM; the order is reduced from 75,918 to 2,362 for the turbo-prop aircraft and from 92,220 to 1,345 for the ring resonator. Subsequently, the speedup over BL + RFM is higher than 5.5 for the turbo-prop aircraft and 8 for the ring resonator. The timings in the comparison with BL + RFM can be slightly changed if difference direct linear system solvers are used. When comparing two AS-methods, AS + FS needs a smaller number of global modes and iterations than AS + MA, due to its better convergence behavior, detailed in the previous section. Consequently, AS + FS shows a better performance among the considered numerical methods in this paper. Meanwhile, the MA has advantages in terms of extending applications, as it has been applied to sensitivity analysis etc. [6], whereas the FS has not. Hence, it is critical to improve the convergence of the MA and it is also essential to extend the FS to these applications. 4. Conclusion Solving mid-frequency responses has remained a frontier research area in structural dynamics. A conventional numerical solution has been obtained by a modal approach to a finite element linear system, which often faces computational difficulties due to a large increase in the order of the system and a high number of retained modes. The computational burden of such a large order can be relieved by using the model order reduction, and a variant of this, based on algebraic substructuring, was employed in this paper. Meanwhile, the number of modes can be reduced by numerical methods that use the modes of the interior eigenvalues, such as the frequency sweep algorithm and the modal acceleration method. Comparisons between these two methods were made in this paper through numerical experiments that provided a detailed analysis of their computational complexity. It might be expected that the two methods would show similar performance if their convergence rates were similar. However, for the finite element simulations, in which proportional damping is assumed, of a turbo-prop aircraft and a ring resonator, the frequency sweep algorithm showed a faster convergence rate and was more robust

TGM

TSOL

TTOT

Speedup

than the modal acceleration method. Consequently, the frequency sweep algorithm for a reduced-order systems showed the best performance among the considered numerical solutions including conventional modal approach. Future work may involve the use of the frequency sweep algorithm, along with the algebraic substructuring, for sensitivity, transient, vibro-acoustic analysis and also the structural problems considering nonproportional damping. Acknowledgment This research was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (Grant No. 20090083068 and 2009-0074875). References [1] Bennighof JK, Kaplan MF. Frequency sweep analysis using multi-level substructuring, global modes and iteration. In: Proceedings of the 39th AIAA/ ASME/ASCE/AHS structures, structural dynamics and materials conference; 1998. [2] Gao W, Li S, Yang C, Bai Z. An implementation and evaluation of the AMLS method for sparse eigenvalue problems. ACM Trans Math Software 2008;34(4). [3] Ko JH, Jung SN, Byun DY, Bai Z. An algebraic substructuring using multiple shifts for eigenvalue computations. J Mech Sci Technol 2008;22:440–9. [4] Thomas B, d Gu RJ. Structural-acoustic mode synthesis for vehicle interior using finite-boundary elements with residual flexibility. Int J Vehicle Des 2000;23:191–202. [5] Ko JH, Bai Z. High-frequency response analysis via algebraic substructuring. Int J Numer Meth Eng 2008;76(3):295–313. [6] Qu ZQ. Accurate methods for frequency responses and their sensitivities of proportionally damped system. Comput Struct 2001;79:87–96. [7] Karypis G, Kumar V. METIS: unstructured graph partitioning and sparse matrix ordering system. Technical Report 1995. Department of Computer Science, University of Minnesota, <http://www-users.cs.umn.edu/karypis/metis/metis/ index.html>; 1995. [8] Lehoucq R, Sorensen DC, Yang C. ARPACK user’s guide: solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods. Philadelphia: SIAM; 1998. [9] Demmel JW, Eisenstat SC, Gilbert JR, Li XS, Liu JWH. A supernodal approach to sparse partial pivoting. SIAM J Matrix Anal Appl 1999;20(3):720–55. [10] Marques OA, BLZPACK users guide, <http://crd.lbl.gov/~osni/>; 2001. [11] Duff IS, Reid JK. MA47, a Fortran code for direct solution of indenite symmetric systems of linear equations. Report RAL 95-001. Oxfordshire, England: Rutherford Appleton Laboratory; 1995. [12] Yonsefi-Koma A, Zimcik DG. Applications of smart structures to aircraft for performance enhancement. Can Aeronaut Space J 2003;49(4):163–72. [13] Craig RR. Structural dynamics: an introduction to computer methods. John Wiley and Sons; 1981. [14] Xie Y, Li SS, Lin YW, Ren Z, Nguyen CTC. UHF micromechniacal extensional wine-glass mode ring resonators. Technical digest. In: IEEE international electron devices meeting, Washington DC; 2003.

Computers and Structures 88 (2010) 25–35

Contents lists available at ScienceDirect

Computers and Structures journal homepage: www.elsevier.com/locate/compstruc

Aluminium foams structural modelling M. De Giorgi *, A. Carofalo, V. Dattoma, R. Nobile, F. Palano Dipartimento di Ingegneria dell’Innovazione, Università del Salento, Via per Arnesano, 73100 Lecce, Italy

a r t i c l e

i n f o

Article history: Received 24 February 2009 Accepted 3 June 2009 Available online 30 June 2009 Keywords: Aluminium foams Mechanical characterisation FEM Microstructural model

a b s t r a c t The aim of this work is the development of microstructural numerical models of metallic foams. In particular, attention is focused on closed cell foam made of aluminium alloy. By means of a finite elements code, the material cellular structure was shaped in different ways: firstly the Kelvin cell, with both plane and curved walls; finally an ellipsoidal cell defined by random dimensions, position and orientation has been adopted as base unit. In order to validate the foam numerical models, static tests were performed to obtain the typical stress–strain curves and then compared with the numerical analysis results. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Metallic foams are a new class of materials that turn out to be promising in different engineering fields [1–7]. They show a cellular structure consisting in a solid and a gaseous phase and their classification is based on the cells morphology. A certain number of manufacturing processes are currently used and other innovative methods are being developed in order to obtain a more reliable and uniform production. Technological process determines also if the morphology of the component is made of open or closed cells. Metallic foams show an interesting combination of physical and mechanical properties that make them particularly versatile: the low apparent density, for example, allows obtaining a high stiffness/specific weight ratio, the presence of cavities and the essential in-homogeneity provide them acoustic and thermal insulation properties, besides the possibility to absorb impact loads and to damp vibrations. Finally, the metallic structure potentially gives those good electromagnetic shielding properties and inalterability in time. The combination of these physical properties makes metallic foams able to be competitive in terms of performances and costs in several unusual advanced applications. For example, the realisation of structures able to shield efficaciously the electromagnetic fields is very interesting. Within this context, one of the most frequently used solutions consists in holding the volume to be shielded in a metallic surface. In some situations, it is required that the capability to support relevant load levels without excessively

* Corresponding author. E-mail addresses: marta.degiorgi@unile.it (M. De Giorgi), alessio.carofalo@unile.it (A. Carofalo), vito.dattoma@unile.it (V. Dattoma), riccardo.nobile@unile.it (R. Nobile), fania.palano@unile.it (F. Palano). 0045-7949/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2009.06.005

resting on the whole structure weight. Furthermore, electromagnetic shielding device transparent to light is sometimes required. In these applications, metallic foams seem to have interesting utilisation perspective, because their mechanical properties guarantee the structural auto-supporting. The absence of well-established procedure for the structural verification of innovative materials represents a relevant obstacle to their effective use, even if they could assure the best performances. The use of components made of metallic foams or their use as a filler for hollow structures, in fact, requires a deeper knowledge of structural behaviour. Designer needs not only experimental data, but also reliable and relatively simple analytical or numerical methods for calculation of applied stress and for prediction of failure of metallic foam components. In perspective, these design tools must guarantee a reliability level that would be comparable to the ones used for traditional materials. Intrinsic inhomogeneities of metallic foam represent a serious problem for calculation needed in design phases. In particular, it would be desirable to establish a link between microstructural and macroscopic behaviour of this kind of materials. It is beyond doubt that microscopic geometry and properties affect macroscopic structural behaviour, which is essential for the design of components, but it is very difficult to derive macroscopic properties from microstructure. The problem of deriving macroscopic from microscopic properties is very complex and probably its resolution will lead to partial solutions that will be applicable in well-defined and restricted context. Nevertheless, this step will represent an important advancing in this research field. This paper is inserted in this general perspective, since the main aim is to try to evaluate macroscopic mechanical behaviour starting from a micro-structural approach for particular metallic foam. The purpose is using different numerical methodologies to model a

M. De Giorgi et al. / Computers and Structures 88 (2010) 25–35

specific class of metallic foams in order to evaluate advantages, limitations and simplicity of use. The metallic foam class that was selected for this study is the closed cell aluminium foam, since its high relevance in industrial context. In the literature, there are some studies regarding the way to foresee the foams macroscopic behaviour by cell geometric data. The references [8–14] represent some significant examples of structures formed by closed cells. Hanssen et al. [8] present results from an experimental programme for the general validation of constitutive models for aluminium foam. The obtained database is applied in order to investigate how different existing constitutive models for aluminium foam cope with quasi-static and low-velocity dynamic loading. Compared with the experiments in the validation programme, none of the models managed to represent all load configurations with convincing accuracy. One reason for this is that fracture of the foam is a very likely local failure mechanism, which was not taken into account in any of the models considered. Belingardi et al. in [9] show the results of some finite elements simulations of a set of experimental compression tests, both static and dynamic, in order to verify the quality of the material models. For the load phase, they found that the numerical results agreed with the experimental ones, both in the static and dynamic cases for which the strain rate effects are taken into account. Moreover, the authors develop a new material model for the unload phase. Roberts and Garboczi [10] have computed the density and microstructure dependence of the Young’s modulus and Poisson’s ratio for several different isotropic random models based on Voronoi tessellations and level-cut Gaussian random fields. The results, which are best described by a power law E / qn (1 < n < 2), show the influence of randomness and isotropy on the properties of closed cell cellular materials. In [11], Lu et al. analyse the deformation behaviour of two different types of aluminium alloy foam (with closed and semi-open cells) under tension, compression, shear and hydrostatic pressure. The influence of relative foam density, cell structure and cell orientation on the stiffness and strength of foams is studied; the measured dependence of stiffness and strength upon relative foam density is compared with analytical predictions. Loading paths are compared with predictions from a phenomenological constitutive model. It is found that the deformations of both types of foams are dominated by cell wall bending, attributed to various process induced imperfections in the cellular structure. The closed cell foam is found to be isotropic, whereas the semi-open cell foam shows strong anisotropy. Another possibility is to use X-ray tomography to produce a data file containing all the geometrical information on a relatively large foam specimen. In this case the model is realised through brick elements associated to each voxel identified by X-ray scan [12,13]. Numerical results show a large influence of scan spatial resolution, which influence directly computational time. Finally, an interesting approach is presented in [14], where a spectral analysis carried out on X-ray tomography of open cell foams allows to derive statistical parameters used to build solid finite element models. Although the variety of approaches that haves been briefly presented, all the model could be classified into two different categories. The first one tries to identify a simple geometrical feature that can constitute the repetitive cell used to built a foam model [8–11]. The unit cell, which is not necessarily close to the real foam geometry, is chosen on the basis of energy or morphology consideration and leads to models generally based on shell elements. On the contrary, the efforts of the other works are concentrated to obtain an exact reproduction of foam geometry by means of experimental techniques [12–14]. In this case, solid models are generally obtained.

In this work the first approach has been followed. The use of elements like beam or shell, in fact, presupposes an idea of foam structural behaviour at an intermediate scale, being in authors opinion the better compromise nowadays possible to study foam structural properties both at microscopic and macroscopic scale. The problems to solve in the micro-structural model definition of closed cell metallic foams can be resumed as follows: – choice of a cell morphology to be employed as a repetitive unit for the model construction; – definition of geometric dimensions, which characterise the cell form; – selection of cells assembly criteria; – metallic foam block numerical model realisation through the repetition of several cells in the three directions of the space. In this paper, two different cell morphologies were considered and numerical results were compared to experimental data. In particular, the model adequacy was evaluated by comparison of the predicted and actual compressive elastic modulus. 2. Survey of proposed models The first model considered was referred to the ideal Kelvin cells (tetrakaidecahedron) with planar walls. This cell morphology has been already used by other authors, since it corresponds to the minimum surface energy for a fixed volume [1,15]. Models based on Kelvin cells were various: the most simple considered regular Kelvin cell having planar faces, while more complex models were obtained introducing the cells walls curvature and thickness variation. These modifications could improve the mechanical behaviour approximation. Further refinements of this approach could give a better approximation, but limitation due to the regularity of the resulting cellular structure was not removable and constituted an important starting error. According to other authors [10,16–18] the structure regularity could be considered as the origin of the higher model stiffness with respect to the real foam behaviour. For this reason a different approach was used to develop a second model that considered the real cellular structure irregularity. In order to reproduce the pores feature and to consider its random variability, an elementary ellipsoidal feature was chosen as unitary cell. The assembly of a fixed number of ellipsoidal cells, having dimensions, position and orientation varying parametrically in a random way, could constitute a better approximation of the foam real structure.

3. Foam mechanical characterisation Experimental data used for comparison between numerical models were obtained through compressive tests. These data represented the necessary experimental term of comparison used until now for the numerical model improvement. The uniaxial compression tests represented the easiest and most efficacious way to determine the metallic foams mechanical behaviour under static conditions. The resulting data, particularly the applied load versus the overall strain trend, defined the macroscopic behaviour of the foam cellular structure. Then, r e compressive curve was the term of comparison used to evaluate the studied numerical model efficiency and reliability. The analysed foam was made of AlSi10Mg aluminium alloy and produced by Alulight Company. Table 1 resumes the general properties of this aluminium foam. The material was available in forms of 10-mm-thick sheets that were cut to obtain the specimens. It was observed that sheets showed a central region with a foam

M. De Giorgi et al. / Computers and Structures 88 (2010) 25–35 Table 1 Mechanical properties of AlSi10Mg base alloy. Aluminium alloy

Solidus density qS (kg/m3)

Elastic modulus ES (GPa)

Yield strength ry (MPa)

Poisson modulus

AlSi10Mg

2860

250

0.33

density quite uniform, while external zones were characterised by higher values of density. It was probably due to the different initial distribution of the gas particles in the matrix during manufacturing process. Therefore, specimens having different density were easily obtained from the same sheet. Mechanical experiments were planned in the following way: initial tests were carried out to establish the influence of the specimen dimension on mechanical properties: 40 40 mm (D40), 70 70 mm (D70), 100 100 mm (D100) specimens having a similar density were considered; extensive experimental tests were then carried out on D40 specimens having different density, with the aim to highlight the influence of apparent density on the stress–strain curve; these compressive tests were executed into two steps, as described in the following, in order to evaluate constitutive behaviour of aluminium foam; Tests were carried out on a servo-hydraulic MTS 810 machine having 100 kN load capacity equipped with two planar plates, according to the indication reported in [1]. Since the crosshead speed was fixed at 0.005 mm/s and foam thickness was 10 mm, the initial compressive deformation rate of the test was 0.5 10 3 mm/mm s. Strain was calculated dividing the crosshead displacement by the specimen thickness. As a first result, the specimen dimensions seemed to be not relevant with regard to mechanical foam properties, as it was evident by comparing the plots reported in Fig. 1a and obtained on specimens D40, D70 and D100 having different dimensions and similar densities. On the basis of this observation, further tests were carried out on specimen D40. Stress–strain curves deduced from the experimental tests were similar to those reported in the literature [1–5,17,19] and showed the existence of three different and characteristic zones: – a first part, corresponding to little deformations, in which the material presented an elastic behaviour and the cells walls changed their curvature; – a second very wide zone, characterised by a plateau with a rather constant stress, phase in which the cells underwent phenomena of local instability, yielding and rupture;

– finally a third region in which the material showed a stiffness increment due to the remarkable material densification and to the strain hardening phenomena. It is also important to notice that aluminium foam showed a different behaviour between the first load application and the subsequent ones: in fact, the material seemed to undergo an initial arrangement, which gave rise to a different elastic modulus when comparing the first and the following loading cycles. This fact is generally explained invoking an initial rearrangement and localised plastic strain of cell walls that is realised in the first compression phases and responsible of the subsequent drastic Young’s modulus increment. The Young modulus measured in the load cycles subsequent to the first one is assumed to be the most important constitutive parameter of the metallic foam. As a consequence, compressive test were executed into two steps: – a first loading step up to about 75% of plateau stress: this step leaded to the determination of the first cycle Young modulus (Eload), followed by an unloading phase that allowed to determine a characteristic Young modulus (Eunload); – a second loading step up to foam densification. Fig. 1b represents the trends obtained by testing three samples having different densities and allows drawing attention to the density influence on the foam mechanical behaviour. The curves showed that, when the specimen density decreased, also the elastic modulus, the compressive strength and the curve slope reduced, while the strain corresponding to the densification beginning raised. Fig. 2 reports the linear interpolation of experimental data of the loading and unloading phases that was used for the evaluation of the compressive Young’s modulus Eload and Eunload, respectively. Data reported in Table 2 allowed to evaluate the difference existing in the load and unload phases. The stress–strain curve reported in Fig. 2 could be taken as reference in order to describe the general behaviour of the examined foam and it was achieved imposing firstly two subsequent loading cycles and then loading the sample until its compacting. Testing some specimens with only one loading cycle, we noted that the compacting phase trend was not influenced by the presence of a possible unloading step. The reference loading curve employed for the numerical results verification was then obtained with a first loading phase until 75% of plateau stress and a following unloading step. In fact, such a curve was repeatable and independent from initial arrangement phenomenon, which is characteristic of cellular materials.

Fig. 1. Uniaxial compression tests: (a) effect of specimen size; (b) effect of specimen density.

M. De Giorgi et al. / Computers and Structures 88 (2010) 25–35

Fig. 2. Loading and unloading behaviour for the determination of elastic modulus.

Table 2 Experimental elastic modulus in loading and unloading phases. Specimen number

Specimen type

Maximum stress smax (MPa)

Density (kg/m3)

Eload (MPa)

Eunload (MPa)

1 2 3 4 5 6 7 8 9 10 11 12 13

D70 D100 D100 D40 D40 D40 D40 D40 D40 D40 D70 D40 D70

3.9 4.7 4.9 4.1 12.5 5.0 7.8 4.9 4.4 9.6 4.8 4.2 5.1

423.2 473.7 445.3 433.1 771.2 432.5 546.8 432.9 435.6 568.2 412.3 555.6 418.1

109 78 111 103 243 146 229 155 152 256 97 222 181

Loading only

577 436 433 481 453 215 493 522

4. Numerical analysis 4.1. Kelvin cell model A simple microstructural model reproducing a cellular material can be based on the regular space repetition of a unit cell with proper geometry and dimensions. The tetrakaidecahedron or Kelvin cell is the most utilised in literature as unit cell to model the foam material [1–2,15], since it is the lowest surface energy unit cell known consisting of a single polyhedron. It is deﬁned by six planar square faces and eight hexagonal faces that could be planar, with a mean curvature equal to zero (Fig. 3), or convex, with a positive mean curvature. In order to simulate the actual foam structure, FEM 3D model based on the Kelvin unit cell was reﬁned in the course of the work introducing the following improvements: 1. planar faces model 2. non-zero faces curvature 3. thickness faces variability FEM analysis was performed using the Ansys software adopting a discretisation in shell element named Shell93 having eight nodes and a quadratic formulation. The study was performed in the large displacements range. The material model was isotropic with elastic perfectly plastic stress–strain behaviour. The material properties, reported in Table 1, were the actual mechanical properties measured directly on Alulight foam by micro-hardness characterisation [17]. The unit cell apparent density was assumed equal to the actual density of the foam, of about 500 kg/m3. On the contrary, the actual cell size was assumed based on a statistical analysis reported in

Fig. 3. Tetrakaidecahedral geometry used in the Kelvin cell model.

[18]. In this work, the authors affirm also that the cell size is independent from direction. The used foam properties are summarised in Table 3. A sample load of 1 MPa was used to perform a mesh sensitivity analysis, considering that the model was used to determine macroscopic properties of the foam. In particular the evaluation of the upper surface cell displacement was directly used to calculate foam Young modulus. For this reason, the Von Mises stress in the central node of the upper face and the mean displacement of the central upper face were considered in the sensitivity analysis.On the basis of the results resumed in Fig. 4, authors decided to divide each cell edge in four parts. In this manner, the unit cell consisted in 352 elements. The global model was obtained replicating the unit cell in all the space directions. In order to optimise the solution efficiency, the model presented nine cells on the basis and four along the thickness. The uninfluence of the cells number on the macroscopic mechanical properties of the cellular structure was preliminarily verified through a model having a number of 16 cells on the basis. Boundary conditions were applied on the lower face, imposing a zero value to z-direction displacement and rigid body boundaries; pressure load was applied to nodes of the square faces on the upper surface and it was set to 4 MPa. Fig. 5 shows the node displacements in the load direction and a limited edge effect due to absence of adjacent cells can be observed. However, the displacement of the middle node of the central face is considered as representative of the global behaviour of the foam. Foam macroscopic strain, calculated as displacement/ model thickness ratio, was recorded at each load increment to display the stress–strain curve. Young’s modulus value of this model resulted unacceptably higher than experimental one. This discrepancy was due to the excessive stiffness associated with the idealised structure, which resulted unrealistically ordered. In fact, the real structure presents imperfections originated by the production process and that are typical of the foam. The case study based on Kelvin cell was refined trying to reduce its inadequacy: the face curvature of the unit cell was then introduced in the model. The face curvature can be calculated from measurement of the chord length L and the triangular area A indicated in Fig. 6 [17]. Considering the angle h between the normals at the end of the cell wall, it can obtain the relationship:

M. De Giorgi et al. / Computers and Structures 88 (2010) 25–35 Table 3 Geometrical properties and results of Kelvin cell models. Average cell dimension (mm) Alulight (experimental) Planar faces model Curved faces model

2.6

Average curvature L/2R

0.37

Edge length L (mm)

Density (kg/m3)

Wall thickness tf (mm)

Edge thickness te (mm)

Edge volume fraction U

Young’s modulus (MPa)

0.9192

418.1–555.6 500 500 432.5 432.5

– 0.1360 0.1178 0.113 0.0973 0.0876 0.0778 0.0681

– 0.1360 0.1178 0.113 0.2350 0.2465 0.2575 0.2680

– – – – 50 55 60 65

433–577 3287 1944 1265 2340 2198 1947 1714

Curved faces with variable thickness model

could be explained considering that the curved wall was less stiff than the planar face; moreover, cell faces were characterised by a lower thickness that involved a further decrease of the stiffness. In the more realistic model with curved walls, the effect of the density on the Young’s modulus has been analysed. The comparison of models with planar faces and curved faces having the same density showed that the stiffness of the two models were comparable if the density was higher than 560 kg/m3. On the contrary, if the density approached 500 kg/m3 or less the difference of the two models became very signiﬁcant. For example, in the interest density range of 432.5–500 kg/m3, because of the introduction of the walls curvature curved face model stiffness decreased at about 60% of the planar cell model. The last reﬁnement of the model consisted in the introduction of the wall thickness variability. In the real structure, in fact, during the foaming process the liquid foam drains toward the cell edges causing a thickening of these. According to [2], the solid fraction at the edge results:

/¼

t 2e

ð4Þ

te þ Zn f t f l

is the average number of the edges of each face, Zf the numwhere n ber of faces converging in each edge, te and tf the edge and the face thickness, respectively, and l the edge length. The relative foam density is:

q f n t2e 1 tf ¼ þ qS C 4 2Z f l2 2 l

ð1Þ

and the normalised cell curvature, L/2R, results [17]:

L h ¼ sin 2R 2

ð2Þ

In order to keep constant the density value, the face thickness was recalculated referring to the following equation:

AC t C ¼ A t

ð5Þ

where C4 is the constant related to the cell volume and f the number of faces per cell. For the most foam, Zf = 3, n 5, f 14 and C4 10, the face and edge thickness related to the chosen solid fraction could be obtained with good accuracy [2]:

Fig. 4. Mesh sensitivity analysis of Kelvin cell model.

4A h ¼ 4 tan 1 L2

ð3Þ

where AC and tC were, respectively, the area and the thickness of the curved wall, A and t were, respectively, the area and the thickness of the planar wall. Table 3 reports the geometric parameters of the unit cell with curved walls. The curved walls model presented a less evident edge effect respect to the planar walls one. Analysing the data reported in Table 3, it could be observed that the new model showed stiffness 40% lower than the initial model with the same density. This behaviour

q t2 tf ¼ 1:2 2e þ 0:7 qS l l tf q ¼ 1:4ð1 /Þ l qS 12 1 te q ¼ 0:93ð/Þ2 l qS

ð6Þ ð7Þ ð8Þ

In order to assign a different thickness value to edge and face, the model was divided in different areas. Fig. 7 shows the mesh performed on the model with thickening edges. Cell dimensions were the same of the previous model but the mesh resulted denser and consisting in 792 elements per cell. The U effect was evaluated simulating the compressive test using different U values and an applied load of 4 MPa. Results are reported in Fig. 8, where Young’s modulus was diagrammed versus solid fraction parameter U. Elastic modulus was generally higher respect to the model with the curved face only. Probably, the greater tendency to instability of the cell walls, due to thickness face decrease, was compensate by the stiffening

M. De Giorgi et al. / Computers and Structures 88 (2010) 25–35

Fig. 5. Map of the z-displacement for the Kelvin cell planar faces model.

Fig. 6. Parameters of a curved cell wall.

Fig. 8. Dependence of elastic modulus against solid fraction U.

originated by solid concentration at the edge, at least for the U values considered in this analysis. Therefore, considering the solid concentration, it resulted a useless complication. 4.2. Ellipsoidal cell model

Fig. 7. Mesh of Kelvin cell model with variable thickness cell face.

The Kelvin cell model, discussed in the previous section, resulted inadequate to describe the foam structure. In particular, Young’s modulus of the foam was overestimated and the typical plateau region of the foam compression curve was not reproduced by Kelvin cell model. This difference was originated by the regular disposition of Kelvin cell into the model, which was far from the real aspect of the metallic foam. For this reason, the efforts were focalised to obtain a model that would be able to conjugate the simplicity of a geometrical elementary cell and the possibility to build a random assembly. In order to deﬁne a schematisation with highly random cell morphology, authors’ attention focused on a simple ellipsoidal unit cell, having the same length for two of the three axes. It was possible and very easy to build a parametric model of this elementary cell (Fig. 9) by the deﬁnition of the following parameters:

M. De Giorgi et al. / Computers and Structures 88 (2010) 25–35

Fig. 9. Geometrical parameters of ellipsoidal cell unit.

xc, yc, zc: cartesian coordinates of the ellipsoid centre; r: length of the major semi-axis; t: ratio of major/minor semi-axis; h, u: orientation angles of major semi-axis respect to a global coordinates system;

By this way, the dimension and the position of the elementary cell could be easily deﬁned using an algorithm for the generation of set of random real number. The choose of a probability density function for the random parameter generation is the object of a large number of papers in the literature. Readers can refer to fundamental work of Kaminski [20] for an overview about probabilistic approach in computational mechanics. For the purpose of this study, an algorithm based on a random uniform distribution into the parameter range variability was considered adequate. Uniform random distribution is characterised by a mean value and a variance equal respectively to:

aþb 2 ðb aÞ2 variance ¼ 12 mean value ¼

ð9Þ

Length of major axis r was related, but not equal, to the half of the average cell dimension. Variability range of r was assumed to be 2–3 mm. In this manner the single ellipsoidal cell has dimensions that were twice with the respect to the mean pore dimension of the foam. When ellipsoidal cell were assembled to build the model, interceptions that occurred between cells produced pores. It means that dimension would be near the half of ellipsoidal cell dimension. Ratio of major/minor semi-axis t was chosen in the range 1–2. Lower limit of t = 1 corresponded to a spherical shape. For the choice of upper limit, some trials suggested that values of ratio higher than 2 originated some troubles in the discretisation of ellipsoid cells in elements. Finally, orientation angles h, u of major semi-axis varied in the range 0–180° in order to obtain all the possible orientation in the space. The complete definition of the model required also the choice of two other parameters that were not completely independent. These parameters were the number of cells used to build the model and the thickness associated to cell walls that was considered constant for all the cells. The number of cells had to be sufficient to regularly fill the model and not excessive in order to obtain pores having dimension similar to foam morphology. On the other hand, the number of cells determined also the amount of areas that constitutes the model. Since the volume of material involved was proportional to the foam density, an increase of cell number requires a decrease of the cell wall thickness. Also in this case, several trails were considered and the best solution for the reference cubic volume 10 mm side was obtained considering a number of 35 cells and a corresponding thickness of 0.11 mm, which was not far from reality. Finally, since the foam was delimited by two solid surfaces (Fig. 10), two planes corresponding to the upper and lower surfaces of the cubic volume of the foam were added. On the basis of the previous observations, the ellipsoidal cell model has been built following this operative steps: – definition of a parameter set assigning to each factor a random value in their range variability: a random uniform distribution, provided by a Matlab routine, was used for this purpose; the parameter set generated are reported in Table 4, with the indications of the material volume and the foam density corresponding to the amount of all the ellipsoidal cells; – modelling of a single ellipsoid corresponding to each parameter set: a CAD parametric software, in particular PRO-EÒ WildFire 2.0 modeller, used the input data to generate the ellipsoid surface feature in a neutral format that is easily recognised by several FEM software;

ð10Þ

where a and b are the interval minimum and maximum values. It is important to observe that the random generation of dimension, position and orientation of the different cells produces in most cases the co-penetration of the cells. In this manner, pores of various shape, which will exalts the irregularity of the model, are obtained. Range variability parameter could be deﬁned on the basis of simple considerations. Cartesian coordinates of the ellipsoid centre were a direct consequence of the dimension chosen for the numerical model: in order to reduce number of nodes and elements of the model to an acceptable dimension, the choice was to model a metallic foam region constituted by a 10 mm side cube. Therefore, variability range of xc, yc, zc was ﬁxed to 1 to 11 mm in order to consider also ellipsoid having centre beyond the cubic region but a consistent surface interior to model.

Fig. 10. Aluminium foam specimen.

M. De Giorgi et al. / Computers and Structures 88 (2010) 25–35

Table 4 Geometrical parameters used for the generation of ellipsoidal cell model. Thickness (mm)

Parameters

0.11 Cell number

xc [ 1;11]

yc [ 1;11]

zc [ 1;11]

r [2;3]

t [1;2]

h [0;180]

/ [0;180]

Volume (mm3)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

0.11 0.27 0.55 0.56 0.74 1.62 1.79 1.89 2.3 2.36 2.93 3.19 3.69 3.78 4.18 4.19 4.24 4.34 4.73 4.76 4.98 5.03 5.14 5.16 5.65 5.66 5.68 5.78 5.83 6.16 6.4 6.44 6.5 6.64 6.8

5.09 6.76 9.15 5.78 1.84 9.41 7.28 2.8 1.49 4.04 3.84 9.62 2.88 9.99 0.78 9.62 9.99 1.42 6.56 0.27 4.55 1.86 8.8 4.15 4.91 3.77 2.83 1.49 8.17 7.75 1.19 1.81 4.9 5.65 2.91

1.78 5.48 6.43 0.98 9.29 9.87 4.03 4.72 4.81 3.2 4.11 7.38 1.44 2.37 4.76 3.91 6.73 8.6 2.06 4.07 5.61 8.31 9.03 2.54 6.61 3.28 7.65 7.83 5.04 8.66 6.19 1.11 1.25 5.07 3.13

2.86 2.38 2.59 2.6 2.39 2.67 2 2.7 2.58 2.75 2.35 2.24 2.02 2.73 2.33 2.4 2.89 2.68 2.4 2.7 2.59 2 2.99 2.49 2.47 2.66 2.64 2.96 2.54 2.37 2.83 2.24 2.73 2.3 2.84

1.59 1.21 1.3 1.47 1.23 1.84 1.19 1.23 1.17 1.23 1.44 1.31 1.92 1.43 1.18 1.9 1.98 1.44 1.11 1.26 1.41 1.59 1.26 1.6 1.71 1.22 1.12 1.3 1.32 1.42 1.51 1.09 1.26 1.8 1.03 Total volume Foam density

14.56 139.9 162.92 96.08 19.65 148.65 60.86 52.92 134.34 1.86 120.22 108.62 124.66 127.31 140.43 100.2 131.35 133.43 140.65 8.72 11.09 132.18 94.7 109.42 123.07 87.34 51.84 143.81 98.92 71.37 132.29 15.93 18.87 21.15 60.77 (mm3) (kg/m3)

81.31 109.77 10.69 56.85 139.09 125.36 22.56 23.43 16.62 1.41 76.16 118 130.13 95.62 19.59 113.72 22.77 24.17 17.75 25.56 30.29 35.32 57.15 56.96 39.16 45.19 160.73 126.58 100.03 33.2 38.17 13.92 164.48 127.21 100.4 168.2 481.0

4.303 5.105 5.257 4.144 4.982 2.792 3.694 6.393 6.439 6.637 3.512 3.846 1.448 4.838 5.139 2.106 2.834 4.594 6.171 6.092 4.469 2.069 7.501 3.202 2.758 6.303 7.365 6.903 4.900 3.675 4.669 5.556 6.231 2.151 10.107

– import of all the ellipsoidal cells in the chosen FEM code by neutral .iges format and subsequent assembly in FEM assembly module (Fig. 11a): since ellipsoidal cell model was more complex than Kelvin cell model, the possibility to use parallel computing resources advised authors to prefer Abaqus software in this case; – generation of planes delimiting the cubic volume of the model: only the top and the bottom face of the cube are considered material rigid planes (Fig. 11b), while the remaining were used as trimmer planes (Fig. 11c); – trim and elimination of the surface regions that reverted outside the cube (Fig. 11d): this operation reduced the amount of surfaces that constitutes the model, therefore volume material and foam density reported in Table 4 decreased; however the use of a wall thickness of 0.11 mm allowed to obtain a model characterised by a ﬁnal density of 438 kg/m3 that is very close to experimental mean value; – application of boundary condition and pressure load on the lower and upper plane, respectively. Fig. 12a reports the assembly of the ellipsoidal cells and the rigid surfaces. Fig. 12b, which reports the z-displacement map, allows to evaluate the discretisation utilised for the mesh. In particular, the elements were dominant quadratic shell S8R, having eight nodes and reduced integration. The mesh consisted in 133,428 nodes and 93,228 elements. The geometrical complexity of the structure in correspondence of intersection line between

ellipsoidal cells did not allow meshing the model without admitting the presence of degenerate elements. Material model was isotropic with elastic perfectly plastic stress–strain behaviour. The material properties were the same of the previous model (Table 1). Boundary conditions were applied on the lower face, imposing a zero value to z-direction displacement and rigid body boundaries; load was applied to the upper surface as a 10 MPa pressure load distributed in three load steps to overcome convergence troubles. Displacement map in Fig. 12b was quite uniform in the z-direction and the upper nodes presented the highest values of displacement since they were in proximity of the loaded surface. Displacement distribution that has been obtained in the model reflected the expected behaviour, confirming the approach validity. The ratio of upper surface displacement to model thickness allowed to calculate the global strain of the foam and, then, the stress–strain curve was obtained dividing the displacement by model thickness (Fig. 13). In order to carry out a reliable comparison of numerical and experimental behaviour, not only the variability of foam density but also the scatter of experimental data must be considered. For this reason, mean values of elastic modulus and plateau stress of the specimens, having the same apparent density of the numerical model, have been calculated and used for a direct comparison (Table 5). It is important to observe that this model was capable to predict the first stage of the stress strain curve plateau. Besides, experimental and numerical maximum stress were practically coincident, while a difference still existed

M. De Giorgi et al. / Computers and Structures 88 (2010) 25–35

Fig. 11. Model assembly steps: (a) import of ellipsoidal cells, (b) generation of upper and lower planes, (c) generation of trimmer planes and (d) removing of the exceeding surfaces.

Fig. 12. (a) Random model based on ellipsoidal cells. (b) Mesh and z-displacements map.

in the elastic modulus. This difference, that was however the best approximation that has been obtained, seemed to be quite large, but the direct comparison of numerical and experimental stress– strain curve revealed a good agreement. In particular, the graph in Fig. 14, where both the numerical curve and the experimental one of Specimen 6 are plotted, suggests that a simple comparison of elastic modulus values could be not adequate to evaluate the real accuracy of the model that is instead evident through the direct comparison of the two curves. However, this model presented some limitation during the simulation of compaction phase at strain higher than 3.5%. In these physical conditions the cells begin to collapse and a large number

of internal contact region is created. Due to its complexity, it was not possible to accomplish this phenomenon into the model. As a consequence, model accuracy rapidly decreased and ﬁnally convergence problems appeared Ellipsoidal cell model could be also used to evaluate the inﬂuence on the numerical results when limited changes of apparent density were considered. It was possible, in fact, to consider the same cell geometry but different values of wall cell thickness. This change produced a variability of the foam apparent density, without the need to change the whole model. Four different values of apparent density have been considered in the range 438–550 kg/m3, obtaining the stress–strain curves reported in

M. De Giorgi et al. / Computers and Structures 88 (2010) 25–35

Fig. 15. As expected, density increase produced an increase of elastic modulus and plateau stress, even if there is not a direct relationship between them. Further and minor changes appeared in the shape of the curves. 5. Conclusions

Fig. 13. Numerical stress–strain curve for the model with ellipsoidal cells.

Table 5 Comparison of experimental and numerical results of ellipsoidal cell model. Specimen number

Specimen type

6 D40 8 D40 9 D40 Experimental mean values Ellipsoidal cell model

Maximum stress smax (MPa)

Density (kg/m3)

Eunload (MPa)

5.0 4.9 4.6 4.8 5.3

432.5 432.9 435.6 433.7 438

577 433 481 497 906

A comparative study of different metallic foams micro-structural modelling was carried out. Two different approaches were considered, which differed for the cell morphology. The easiest model was based on the regular repetition of a Kelvin cell. This cell has been largely used [1,15] because it corresponds to the minimum surface energy for a constant volume. The simple planar walls model was refined introducing the walls curvature and material thickening along the faces edges. The curvature introduction remarkably improved the initial elastic region approximation, but the model was still too rigid with respect to the experimental data. Moreover, it was impossible to recognise the plateau region characteristic of the foam compression curve. Then, when the solid volume fraction concentrated in the edges exceeded the 60%, the wall thickness variation further improved the approximation, but not in a significant way. The comparison with the experimental data obtained for commercial AlSi10Mg foam produced by Alulight showed the excessive stiffness of the Kelvin cell models. The elastic modulus calculated through the last model resulted more than the double respect to the experimentally one. The best results were obtained with the model realised by assembling the ellipsoidal cells having random dimensions, orientation and position. The model was able to reproduce also the plateau region, where cells collapsed and densification phenomena became evident. Even if the elastic modulus was overestimated, agreement between experimental and numerical stress–strain curves was excellent.

References

Fig. 14. Comparison between numerical and experimental stress–strain curves.

Fig. 15. Initial behaviour of stress–strain curves for different density values.

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M. De Giorgi et al. / Computers and Structures 88 (2010) 25–35 [16] Bart-Smith H, Hutchinson JW, Fleck NA, Evans AG. Inﬂuence of imperfections on the performance of metal foam core sandwich panels. Int J Solid Struct 2002;39:4999–5012. [17] Andrews E, Sanders W, Gibson LJ. Compressive and tensile behaviour of aluminium foam. Mater Sci Eng 1999;A270:113–24. [18] Simone AE, Gibson LJ. The effect of cell face curvature and corrugations on the stiffness and strength of metallic foam. Acta Mater 1998;46:3929–35.

[19] Dattoma V, Nobile R, Panella FW, Tafuro R. Comportamento meccanico di pannelli sandwich in schiuma di alluminio. Milan, Italy: XXXIV Convegno Nazionale dell’Associazione Italiana per l’Analisi delle Sollecitazioni (AIAS); 2005. [20] Kaminski MM. Computational mechanics of composite materials. London: Springer-Verlag; 2005.

Computers and Structures 88 (2010) 36–44

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Metamodel-based lightweight design of B-pillar with TWB structure via support vector regression Feng Pan, Ping Zhu *, Yu Zhang The State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China

a r t i c l e

i n f o

Article history: Received 28 February 2009 Accepted 24 July 2009 Available online 20 August 2009 Keywords: B-pillar Tailor-welded blank Crashworthiness Metamodeling Support vector regression Lightweight

a b s t r a c t Vehicle lightweight design becomes an increasingly critical issue for energy saving and environment protection nowadays. Optimum design of B-pillar is proposed by using tailor-welded blank (TWB) structure to minimize the weight under the constraints of vehicle roof crush and side impact, in which support vector regression (SVR) is used for metamodeling. It shows that prediction results ﬁt well with simulation results at the optimal solution without compromising the crashworthiness performance, and the weight reduction of B-pillar reaches 27.64%. It also demonstrates that SVR is available for function approximation of highly nonlinear crash problems. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Continuous strict standards for fuel efﬁciency and gas emission have been a great boost to vehicle lightweight design. As the autobody structure weighs about 30% of whole vehicle, the weight reduction in auto-body structure plays a rather important role in decreasing the mass of full vehicle. Generally, mass savings can be achieved by using high-strength steels and other lightweight materials, or utilizing optimization techniques to develop a more robust and lightweight structure. However, what is noteworthy is that the lightweight vehicle should maintain, or even improve the required performance compared to the original one, basically including stiffness performance, crashworthiness performance, noise, vibration, harshness (NVH) performance as well as durability performance etc. In other words, the main target of lightweight design is to minimize the weight of auto-body without compromising the performance of vehicle. Besides alternative materials and optimization techniques, tailor-welded blank (TWB) is also one of the promising approaches to achieve weight reduction and to meet vehicle performance targets since TWB can combine different thicknesses or different types of steels into a single one prior to the forming process, which has contributed to a decrease in body-in-white weight and the number of components [1]. The use of TWB in automobile industry is generally known. An inner door panel using TWB was developed through topology, size and shape optimization [2], while Song and * Corresponding author. Tel./fax: +86 21 34206787. E-mail address: pzhu@sjtu.edu.cn (P. Zhu). 0045-7949/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2009.07.008

Park [3] conducted multidisciplinary design optimization of an automotive front door considering stiffness, natural frequency, and side impact performance with a TWB structure, in which response surface method (RSM) was utilized to approximate the complicated nonlinear problems. Recently, optimum design of an inner door panel using TWB was proposed integrating finite element analysis, artificial neural network (ANN) and genetic algorithm, aiming at reducing the weight and enhancing occupant safety in the case of side impact collision [4]. TWB structure can improve the local performance of component such that no additional reinforcements are needed. Besides front door, B-pillar also plays a critical role in protecting occupant during side impact accident. Additionally, the proposed new rollover regulations are forcing manufacturers to significantly strengthen the B-pillar. While Bpillar can be strengthened either by heavier materials (increasing the gauges) or by stronger but lighter materials (high-strength, extra high-strength or ultra high-strength steel), other advanced manufacturing technique like TWB can also be implemented. Usually, the B-pillar contains some reinforcement parts to enhance stiffness and rigidity in order to resist deflection in side impact or roof crush. So it is available to utilize TWB to improve the performance of B-pillar, while weight reduction can be achieved through optimal distribution of thicknesses in different areas considering its contributions to roof crush and side impact after these reinforcements removed. Meanwhile, there is a growing interest in using metamodel to approximate the complicated highly nonlinear problems to deal with analysis and optimization process, and the computational cost can also be reduced. This method is referred to as metamodel-based

F. Pan et al. / Computers and Structures 88 (2010) 36–44

optimization or surrogate-based optimization [5]. In this work, surrogate model is synonymous with metamodel. There are various metamodeling techniques like response surface method (RSM), artificial neural network (ANN), radial basis functions (RBF), kriging (KR) and support vector regression (SVR) available for engineering design. A number of the literatures can be found dealing with engineering optimization on vehicle crashworthiness problems by RSM, KR and RBF etc. [6–10], while application has been found rarely for approximation on these problems by using SVR. Metamodeling is an approximation method so that there is always the issue of model accuracy. It is found RBF and KR has their own characters for approximating crashworthiness problems or slightly nonlinear responses [9,11]. However, SVR achieves more accurate and robust function approximations than RSM, KR and RBF through many cases validation in their study [12]. And it is also concluded that SVR performs very well in highly nonlinear vehicle crash problems according to our researches, both on accuracy and on efficiency [13]. Hereby, it is utilized as an alternative technique to approximate functions of responses of interest in vehicle crashworthiness problems. In this study, optimum design of B-pillar is performed with a TWB structure subjected to roof crush and side impact. Finite element (FE) models for vehicle roof crush and side impact were modeled firstly and validated with physical test. Then design of experiment is performed to obtain the response values of interest, where Latin hypercube sampling (LHS) is selected for sampling. And metamodels are constructed through SVR. The optimal solution is solved by the sequential quadratic programming (SQP) algorithm. Finally, the analysis of optimization results is compared to the original one, confirming the availability and feasibility of optimal solution.

2. Generate a sample of the design variables space as a training data set, also called Design of Experiment (DOE). These include Latin hypercube design, Uniform design, Orthogonal arrays, Central composite designs, face-centered cubic design, factorial designs. 3. Conduct numerical simulations using the sampled points from step 2 and extract the values of responses of interest. 4. Construct metamodel based on SVR or other metamodeling techniques, using the data obtained previously. The basic theory of SVR will be discussed later in this section. 5. Assess the predictive capabilities of the surrogate model in unsampled design space. Usually it is measured by some error metrics, including R square, relative average absolute error, root mean square error, etc. 6. Solve the constructed metamodes, to make sure the result is convergence in the design space. If achieved, stop; or else, reﬁne design space by increasing additional sample points to update the metamodel. 7. Iterate until convergence. 2.2. Theory of support vector regression SVR is derived from support vector machine (SVM) technique, which is a method from the statistical learning disciplinary. The SVM has been introduced for dealing with classiﬁcation (pattern recognition) and regression (function approximation) problems. The algorithmic principle of SVM is to create a hyperplane, which separates the data into two classes by using the maximum margin principle. The linear separator is a hyperplane which can be written as

f ðxÞ ¼ hw xi þ b 2. Methodology and theory 2.1. Methodology of metamodel-based optimization Metamodel-based optimization is an effective approach for engineering design [14–16]. This approach involves the following steps typically, also shown in Fig. 1. 1. Deﬁne the optimization problem and targets, including objective(s), constraints and design variables, etc.

s:t:

Sampling of the design space

Numerical simulation at sampled points

ð1Þ

where w is the parameter vector that defines the normal to the hyperplane and b is the threshold, and hw xi is the dot product of w and x. In order to produce a prediction which generalizes well, the basic two aims of SVR are to find a function f(x) that has at most e deviations from each of the targets of the training inputs, and to have this function to be as flat as possible. So it is available to minimize the model complexity by minimizing the vector norm |w|2, that is, the flatter the function the simpler it is, the more likely it is to generalize well. Therefore, the optimization problem can be formed as

Min Definition of optimization problem

1 2 jwj 2 yi hw xi i b 6 e

ð2Þ

hw xi i þ b yi 6 e

An assumption above is that the function f(x) can approximate all the yi training points within e precision. However, such a solution may not actually exist, and two slack variables could be introduced to yield a modiﬁed formulation to obtain better predictions. Thus, the optimization problem can be described as

Refine design space No

l X 1 2 ðni þ n i Þ jwj þ C 2 i¼1 8 > < yi hw xi i b 6 e þ ni hw xi i þ b yi 6 e þ n i s:t: > : ni ; n i P 0

Min

Construction of metamodel

Convergence

Model validation

Perform optimization

Yes

Fig. 1. Metamodel-based optimization procedure.

stop

ð3Þ

where C determines the tradeoff between the ﬂatness and tolerance. This is referred as e-insensitive loss function, which enables a sparse set of support vectors to be obtained for regression. After applying Lagrangian principle and substituting Karush–Kuhn–Tucker

F. Pan et al. / Computers and Structures 88 (2010) 36–44

conditions into Lagrangian function, we can write the optimization problem in dual form as

8 l P > 1 > > < 2 ðai ai Þðaj aj Þhxi xj i i;j

Max

l l > P P > > : e ðai þ a i Þ þ yi ðai a i Þ i¼1 i¼1 8 l > < Pða a Þ ¼ 0 i

s:t:

ð4Þ

i¼1 > :

ai a 2 ½0; C i

Fig. 2. Components of B-pillar assembly.

Finally, the weights w and the linear regression f(x) can be calculated through

l X

ai a i xi

i¼1

f ðxÞ ¼

l X

ð5Þ

ai a hxi xi þ b

i¼1

For the nonlinear regression, approximation can also be achieved by replacing the dot product of input vectors with kernel function. Typical choices for the kernel function include linear nonlinear polynomial, Gaussian and Sigmoid, etc. After applying the kernel function to the dot product of input vectors, we can obtain

8 l P > 1 > > < 2 ðai ai Þ aj aj kðxi xÞ i;j

Max

l l > P P > > : e ai þ a i þ yi ðai a i Þ i¼1 8 i¼1 l > < P a a ¼ 0 i

s:t:

i¼1 > :

ð6Þ

ai a 2 ½0; C i

And the ﬁnal result of SVR model for function estimation is obtained

f ðxÞ ¼

l X

ai a i kðxi xÞ þ b

ð7Þ

i¼1

In this study, a Gaussian function is chosen for kernel function since it is used for dealing with engineering problem mostly.

kx x0 k2 Kðx; x Þ ¼ exp 2r 2

ð8Þ

3. Finite element models The B-pillar of vehicle is the structure located between the front and rear doors of compartment. It houses electrical wiring and connections spots for the passenger seatbelts, also does provide structural support for the compartment in the case of lateral or roof impact. This work focuses on optimum design of the B-pillar, which has reinforcement in it. It consists of four components, namely, B-pillar inner, B-pillar reinforcement, seatbelt ﬁxing support, and B-pillar outer, which is shown in Fig. 2. For simplicity, B-pillar outer is called B-pillar later in this study. It is available to apply TWB technique into B-pillar and remove the reinforcements in order to reach mass saving by optimal distribution of thickness of B-pillar. Compared with B-pillar’s contributions to bending or torsion stiffness and frequency performance of bodyin-white (BIW), its effect on crashworthiness is more critical and important. The B-pillar can hold the rigidity of the compartment greatly so that structural intrusion into the compartment can be reduced in roof crush or side impact [17,18]. Therefore, the optimization for B-pillar should consider these crashworthiness perfor-

mances primarily. Hence, FE model of vehicle roof crush and side impact is presented firstly, and validation experiments are carried out to ensure the availability of the finite element (FE) model. In order to improve the simulation accuracy, a detailed fullscale FE model of vehicle has been established, including body assembly, engine, drive system and tires. The FE model, containing 520,205 shell elements, 4302 solid elements, 5479 beam elements and 3723 mass elements, consists of 564 parts, which is shown in Fig. 3. The basic material model is type 24 (piecewise linear plasticity), and the major contact algorithms are automatic single surface and automatic surface to surface. Usually, reduced integration is employed to save the computational cost in an explicit crash analysis code like LS-DYNA, whereas it may cause the elements with spurious zero energy models, which should not be used in practical [19]. Hourglass control is used to avoid these phenomena, consequently. Among shell elements, nearly 97.6% is quadrangle elements with average mesh size of 10 mm. Some other mesh quality indexes are also checked, including the degree of warping, aspect ratio, skew, and the maximum and minimum interior angles of quadrangle and triangle elements. And stricter indexes are employed to some critical parts, such as front side rail. The crash mode or deformation mode of front side rail can be influenced by its meshing quality largely. These indexes include mesh refinement with smaller size, meshing lines perpendicular or parallel with force passing direction as much as possible, and different triangle elements not sharing with the same nodes etc. In addition, an effective meshing method, transition region, is applied to some large-size components such as floor panel, roof panel, and windscreen, which deform largely on one side and slightly on the other side simultaneously in the case of roof crush or side impact collisions so that computational time can be cut down obviously due to the number of grids decreases largely. And the front and rear end of floor and roof panels should be with the average mesh size according to their characters in frontal and rear crash. For example, Fig. 4 shows the FE model of windscreen, in which two layers of transition regions are used to connect meshes of different sizes.

Fig. 3. Finite element model of vehicle.

F. Pan et al. / Computers and Structures 88 (2010) 36–44

Fig. 4. Finite element model of windscreen using two transition regions (3137 nodes).

The number of shell elements is 2936 for this windscreen model, whereas the number is 10,255 if average mesh size of 10 mm is used only. More detailed information about this meshing method can be found in Shi et al. [20]. Vehicle roof crush intends to enhance passenger protection during rollover accident. The test procedure is based on the updated Federal Motor Vehicle-Safety Standards 216 (FMVSS 216), which is a quasi-static test. The auto-body is located on a horizontal ground and a rectangular plate with 1829 mm by 762 mm is added on the driver’s side top of the body as speciﬁed by the FMVSS 216 with roll angle (a = 25°) and pitch angle (b = 5°). The lower surface is tangent to the surface of the vehicle and initial contact point is on the longitudinal centerline of the lower surface of the plate and 254 mm from the forward most point of the centerline. The force–displacement relationship of the plate is deemed as roof strength [21]. Fig. 5 is the FE model of roof crush, where the plate

Fig. 5. Finite element model of roof crush.

is assumed to be rigid. The physical experiment is performed by crushing the roof slowly, at about 8.9 mm/s and for about 10– 30 s [22], so that numerical simulation is time-consuming if LSDYNA is adopted for this case. Some potential approaches are provided in LS-DYNA to reduce the computational time by using mass scaling or increasing the velocity of crushing plate. While the analytical result through mass scaling is inﬂuenced by the choice of scaling parameter mostly and it is still time-consuming anyway, another approach is employed by increasing the velocity of plate in this work, consequently. Bathe et al. [22] proposed that a speed of 0.5 mph can obtain the reasonable results compared with the one of 10 mph for vehicle roof crush, and also pointed out that the ratio of kinetic energy to strain energy of the model should not be too large in their study. However, the velocity of 0.5 mph still has computational burden for optimization in which the model is calculated many times for design of experiment. Recently, it was reported that the velocity of 5 mph (2235.2 mm/s) is also available through comparing the simulation results with physical test [23,24]. In addition, we concludes that the speed is reasonable for roof crush if the ratio of kinetic energy to strain energy is less than 15% based on our benchmark study on Ford Taurus provided by National Crash Analysis Center. Hence, from the viewpoint of computational efﬁciency and accuracy, the speed of 5 mph (2235.2 mm/s) is used for roof crush simulation. The normal vector of plate is related with roll angle (a) and pitch angle (b), which can be written as

~ n ¼ fcos a sin b; sin a; cos a cos bg

ð9Þ

For side impact protection, the National Crash Legislation side impact test configuration (GB20071-2006) is considered, which is very close to European Enhanced Vehicle-Safety Committee side impact procedure. According to GB20071-2006, a side impact simulation was performed with a movable barrier impacting the vehicle perpendicularly at a speed of 50 km/h. Fig. 6 shows the FE model of side impact. LS-DYNA version 971 is used to solve the model. The physical crash test was also carried out, and comparison of structure deformation on left side between test and simulation is illustrated in Fig. 7. Occupant safety is the main concern in side impact, which includes head injury criterion (HIC), chest viscous criterion (VC), rib deflection criterion (RDC) of upper, middle and lower ribs, abdomen load and pubic symphysis force. The last two indexes are not considered in this study, however. Fig. 8 compares the simulation results of head acceleration, lower rib deflection and VC of dummy with test results, which indicate that they fit well both on peak values and on changing trends. The difference of peak values on head accelerations is due to lack of interior trim parts in

Fig. 6. Finite element model of side impact.

F. Pan et al. / Computers and Structures 88 (2010) 36–44

Fig. 7. Comparison of side structure deformation. (a) simulation; (b) test.

the FE model. Anyway, the FE models of roof crush and side impact are available for further studies. 4. Lightweight design of B-pillar via metamodels 4.1. Partitions of B-pillar and design targets The B-pillar with tailor-welded blank (TWB) technique can meet requirements in all partial regions as described before, such that no additional individual parts are required. In this case, B-pillar reinforcement which is described in Fig. 2 can be removed, and TWB technique can be applied into B-pillar not only to ensure the crashworthiness performance, but also to reduce structural weight and manufacturing cost. Probably, there may be small reinforcement plates welding with B-pillar for local rigidity at the places of door hinges and locks, and these small reinforcements are not included and discussed here, however. The partitions of B-pillar for TWB design is based on its functions in roof crush and side impact. The B-pillar must have a high level of resistance and rigidity in the region of door hinges and door locks where the intrusion would reduce the survival space and hurt the occupant directly in the event of a lateral crash, and also have enough strength to resist deformation under roof crush. Another principal for partition is that the welding line should not be located at the areas with complex structure, such as beads, because tearing near the welding seam often occurs in forming process induced by the material properties in the heat-affected zone in welding areas. Hence, TWB structure of B-pillar can be simply divided into three segments according to the aforementioned requirements and distribution of the original reinforcement, which is illustrated in Fig. 9. The solid lines represent the positions of the weld lines. The vertical lengths l1, l2 and l3 of the three segments are 290 mm, 560 mm, and 300 mm, respectively. The rest work is to ﬁnd the optimal thickness of three segments, provided that the type of steel is the same with the original one. Therefore, the thickness of three segments are deﬁned as design variables, denoted by t1, t2 and t3, respectively (1.0 mm 6 ti 6 2.5 mm, i = 1, 2, 3). The objective of optimization is to minimize the weight of B-pillar while structural crashworthiness and occupant safety should be guaranteed primarily subjected to roof crush and side impact. For

Fig. 8. Performances comparison between test and simulation. (a) head acceleration; (b) lower rib deﬂection; (c) lower rib viscous criterion.

roof crush, the increase of gauge design variables tends to get a higher resistance force, whereas the increase of B-pillar’s weight is undesirable, which is also the same for the case of side impact. As deﬁned on the updated FMVSS216, the force generated by vehicle resistance must be greater than 2.5 times the unloaded vehicle’s weight with a maximum allowable displacement of 112 mm in this study such that it won’t touch or hurt the male dummy (50%) with seatbelt placed in the compartment. The resistant force (PRoof) of roof crush, which is the peak force in the force–displacement relationship of the crushing plate, was set to be 27 kN, where the unloaded vehicle’s weight M = 1100 kg. In side impact, responses of HIC, RDC and VC of upper, middle and lower rib are considered. Besides, the intrusion velocity of B-pillar at middle

F. Pan et al. / Computers and Structures 88 (2010) 36–44

l1=290mm

l2=560mm

l3=300mm

Fig. 9. Partition of B-pillar with tailor-welded blank.

point (VB-pillar) is also included. The design targets are listed in Table 1. 4.2. Metamodeling by support vector regression Metamodel is widely used to approximate the highly nonlinear problems, such as vehicle crashworthiness problem. In this study, support vector regression (SVR) is employed to construct metamodels for approximating responses of interest. Latin hypercube sampling (LHS) is adopted for selecting the initial design points in the exploratory design space. It can provide an efﬁcient estimate

of the overall mean of the response than the estimated based on random sampling [25]. LHS is one of ‘‘space-filling” methods, which can treat all regions of design space equally and the sampled points should be chosen to fill the entire design space for computer experiments [26]. Meanwhile, the accuracy of metamodel is also related with the number of design points in the training data set. Too less of design points may not represent the relationship between inputs variables and outputs functions accurately, while too many of design points would increase the computational burden. Based on the research in [27], 5N sample points generated by LHS are used in this study, where N is the total number of design variables. That is, a training data set with 15 sample points is generated as there are three design variables. It is noted that the number of sampling points can vary according to the availability of computing capability. Since the lower rib was always found to experience the largest deflection and highest VC in side impact compared to other two ribs and the values of middle and upper ribs (RDCMiddle, RDCUpper, VCMiddle, VCUpper) are much lower than the design targets, the responses of RDCLower and VCLower are considered in the optimization process only. In addition, because displacement of the plate at peak force is always much less 112 mm, it is also unconsidered in the optimization process. Therefore, metamodels for five responses should be constructed only, namely PRoof, HIC, RDCLower, VCLower, and VB-pillar. The design matrix and values of the responses obtained from FE simulations are given in Table 2. Then Matlab toolbox LSSVM was used to approximate the functions of responses obtained from the training data set. And it is well known that SVR generalization performance depends on a good setting of hyper-parameters C, e, and the kernel parameters. Gaussian kernel function was selected due to its ‘‘good features and strong learning capability” [28]. And e is the default value of 0.001, while C is 1 .The radius of Gaussian kernel is automatically

Table 1 Regulation requirements and design targets. Index

Symbol

Requirement

Original

Target

Roof resistance force (kN) Roof crush displacement (mm) Head injury criterion Viscous criterion (m/s)

PRoof DRoof HIC VCUpper VCMiddle VCLower RDCUpper RDCMiddle RDCLower VB-pillar

P26.95 6112 61000

25.52 65.2 327 0.278 0.333 0.531 21.38 23.02 37.07 8.16

P27 6112 6330

Rib deﬂection Criterion (mm)

Intrusion velocity of B-pillar (m/s)

61.0

642 –

60.53

636 67.9

Table 2 Design matrix and responses. No.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Design variables

Responses

t1 (mm)

t2 (mm)

t3 (mm)

PRoof (kN)

HIC

RDCLower (mm)

VCLower (m/s)

VB-pillar (m/s)

2.1485 2.0031 1.2647 1.3968 1.4177 2.2851 1.0674 2.3026 1.6700 1.7972 1.5769 1.8738 1.1636 1.9304 2.4771

2.4700 2.0647 1.4851 2.1771 1.1770 1.5485 1.9636 1.2177 1.6026 1.8031 2.3738 1.0968 1.7972 1.3304 2.2674

1.8968 1.0026 1.9700 1.1771 1.6636 2.2674 1.3972 1.7647 2.3031 1.4769 2.1738 2.0485 1.5851 1.2177 2.4304

31.37 30.31 26.00 27.19 26.47 30.34 24.98 29.58 28.35 29.16 28.48 28.29 25.51 28.87 32.15

326 351 326 315 315 368 291 340 360 331 299 318 293 387 433

34.86 36.93 33.97 35.98 35.98 35.37 36.32 35.33 34.61 34.80 37.85 33.67 38.93 32.94 37.19

0.503 0.527 0.528 0.534 0.535 0.538 0.535 0.541 0.563 0.520 0.518 0.607 0.504 0.548 0.527

7.858 7.638 8.011 7.936 7.986 8.059 8.083 8.128 8.029 7.940 7.991 8.375 8.063 7.944 7.886

F. Pan et al. / Computers and Structures 88 (2010) 36–44

optimized based on the given training data, aiming at minimizing the generalized mean square error using leave-one-out cross-validation. The parameters tuning process can be achieved in LSSVM toolbox. All the design variables and responses are normalized quantitatively according to

xn ¼ 0:2 þ 0:6

xi xmin xmax xmin

ð10Þ

where xn is the normalized value, xi is the actual value, and xmax and xmin are the maximum and minimum values of that data column, respectively. The vector of Lagrange multipliers ðai a i Þ, offset b, and parameter r for five responses are listed in Table 3. With these results, the explicit analytical formulas can be expressed according to Eqs. (7) and (8) for each response. For accuracy, the goodness of fit obtained from training data set is not sufficient to assess the accuracy of newly predicted points [26]. For this reason, another set of 8 sample points generated randomly by Matlab are used to verify the accuracy of the approximation functions. Three different prediction metrics are used to assess the accuracy: R Square (R2), Relative Average Absolute Error (RAAE), and Root Mean Square Error (RMSE) [29]. The equations for these three measures are given below, respectively.

^ i Þ2 ðyi y R2 ¼ 1 Pi¼1 n 2 i¼1 ðyi yÞ Pn ^ i¼1 jyi yi j RAAE ¼ P n i¼1 jyi j sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Pn 2 ^ ðy y Þ i i¼1 i RMSE ¼ n

ð11Þ ð12Þ

ð13Þ

^i is the corresponding predicted value for the tested value where y is the mean of the tested values. The larger the value of R2, yi; y the more accurate the metamodel is; the larger the value of RAAE or RMSE, the less accurate the metamodel is. Validation results of selected error metrics for different metamodels are shown in Table 4. The model ﬁt of PRoof, HIC, RDCLower

Table 4 Accuracy assessment. Criterion

RAAE

RMSE

RMSE (%)

PRoof HIC RDCLower VCLower VB-pillar

0.9603 0.9681 0.9113 0.8470 0.9072

0.1649 0.1456 0.2746 0.3387 0.2499

0.0197 0.0070 0.0100 0.0041 0.0258

4.0811 1.9670 2.2268 1.1400 5.7233

and VB-pillar are very good with high values of R2 (P0.9). Although the metamodel of VCLower is constructed with the R2 value of 0.8470, it is still acceptable with the RMSE% value of 1.14%. Therefore, metamodels for PRoof, HIC, RDCLower, VCLower, and VB-pillar are reasonable to be used for design optimization. 4.3. Optimization formulation and results In this case, the objective is to minimize the weight of B-pillar under constraints of roof crush and side impact. According to the design targets in Table 1, the problem can be formulated as follows

Min MðtÞ s:t:

F Roof ðtÞ P 27 kN HICðtÞ 6 330 RDC Lower ðtÞ 6 36 mm

ð14Þ

VC Lower ðtÞ 6 0:53 m=s V B-pillar ðtÞ 6 7:9 m=s 1:0 mm 6 ti 6 2:5 mm; i ¼ 1; 2; 3 where M(t) is the weight of B-pillar, which is a linear function with the thickness of three segments. And M(t) can be formulated as follows:

MðtÞ ¼ 0:518 t 1 þ 0:8733 t 2 þ 1:0413 t3

ð15Þ

It is to be noted that M(t) should also be normalized according to Eq. (10) for optimization further.

Table 3 Results of parameters obtained by support vector regression. Index

FRoof

HIC

RDCLower

VCLower

VB-pillar

ai a i

0.9176 0.6492 0.6648 0.7144 0.6497 0.4898 0.0081 0.2611 0.0551 0.3463 0.3445 0.2340 1.1063 0.0926 1.2024 0.2329 1.2576

0.0638 0.3343 0.0027 0.1562 0.3058 0.6238 0.4964 0.0541 0.5221 0.1428 0.6083 0.3663 0.4296 1.0269 0.0055 0.1252 1.0434

0.3040 0.7989 0.7793 0.2744 0.3357 0.0254 0.4595 0.0284 0.4101 0.3170 1.2850 0.9264 0.0102 1.3076 0.9346 0.1449 0.3645

0.9900 0.1383 0.0906 0.1242 0.1842 0.2712 0.5566 0.3643 1.1815 0.3798 0.4587 0.0228 1.1565 0.6336 0.1246 0.1939 0.3721

2.9076 11.555 2.8199 3.2276 5.6302 1.6382 4.1215 6.1206 2.1558 2.0121 1.6418 0.1119 1.6173 2.9576 1.9475 0.8398 12.1195

Fig. 10. Force–displacement optimization.

relationship

roof

crush

before

and

after

Table 5 Comparisons of predicted solutions to ﬁnite element analysis (FEA). Index Predicted optimum FEA validation % error

PRoof (kN) 28.15 27.86 1.04%

HIC 329.96 326 1.21%

RDCLower (mm) 35.43 35.83 1.12%

VCLower (m/s) 0.526 0.518 1.54%

VB-pillar (m/s) 7.89 7.81 1.02%

F. Pan et al. / Computers and Structures 88 (2010) 36–44

For this simple optimization problem with three design variables, sequential quadratic programming (SQP) (function, fmincon in MATLAB) is used to search for optimal point. Because the SQP is dependent on the initial guess of the optimal point, a series of trials with 10 points have been performed before getting the ﬁnal optimal point. Finally, the optimal point of design variables and objective with de-normalized form are as follows: t1 = 1.615 mm, t2 = 1.822 mm, t3 = 1.276 mm, Mmin = 3.77 kg.

The original mass of B-pillar (including B-pillar reinforcement) is 5.21 kg, while the mass of B-pillar with TWB structure is 3.77 kg, with mass saving of 1.44 kg, due to the use of TWB technique and the removal of reinforcement. It achieved 27.64% weight saving less than the original structure. Not only raw material cost can be saved, but also manufacturing cost (i.e., welding, assembly) would be reduced. In order to validate the feasibility of the optimal results, ﬁnite element (FE) simulations for roof crush and side impact were also

Fig. 11. Comparison of time histories of rib deﬂection and viscous criterion for (a) upper rib; (b) middle rib; (c) lower rib.

F. Pan et al. / Computers and Structures 88 (2010) 36–44

carried out based on the optimal results obtained via metamodels constructed through SVR. The results are given in Table 5. The differences between the predicted responses and FE analytical results are all less than 2%; this indicates that the metamodels fit well too. The maximum error (VCLower) is due to the less accuracy of surrogate model than others partially, where R2 value is 0.8470. The comparison of force–displacement relationship before and after optimization shows in Fig. 10. The optimum solution by using TWB gives an increase of 9.2% on resistance force in the case of roof crush. For vehicle side impact, the solution gives reductions of 4.3% on velocity of B-pillar (VB-pillar), while keeping the same level on response of HIC. The rib deflection and VC of upper, middle, and lower ribs were also compared to those of the original design, which are shown in Fig. 11. It can be observed that only few increases on rib deflection and VC of upper and middle ribs, whereas decreases on lower ribs is also tiny. The results satisfy the design targets, also indicate the optimization can meet the requirements of targets and standards, including the updated FMVSS216 and GB20071-2006. Therefore, it can be confirmed that the optimization results are reasonable and practicable, which can reduce the weight of B-pillar, while guaranteeing the structural crashworthiness and occupant safety in the cases of side impact and roof crush. All the results above demonstrate the successful implementation of TWB structure on B-pillar in this study. And it also indicates that SVR shows great potential for metamodeling applications for vehicle structural crashworthiness design.

5. Conclusions Aiming at reducing the structural weight, lightweight design of B-pillar by using TWB is performed in this paper. A full-scale finite element model was used in performing vehicle crash simulations on roof crush and side impact. The B-pillar is divided into three segments according to its contributions and the position of the original reinforcement. Response functions of the dummy’s hurt injury criterion, rib deflection and viscous criterion of lower rib, intrusion velocity of B-pillar, and vehicle’s roof crush force were approximated by using DOE and SVR, where Latin hypercube sampling was employed for sampling. And SQP is used to search for optimum. The optimal solution shows that the weight of B-pillar can be reduced by 27.64%, while other constraints, including structural crashworthiness and occupant safety are either improved or kept in comparison with the original design. Metamodels used in the optimization process fit well according to the low error between the predicted responses and FE analytical results at the optimal point. It can also be concluded SVR is a promising metamodeling technique for function approximation of vehicle crash problems with high accuracy, and can be further applied to structural design optimization. Acknowledgments The work presented in this paper was supported by National Natural Science Foundation of China (Grant # 50875164). The authors also acknowledge Altair for providing the educational license of Hyperworks for this work. We acknowledge the anonymous reviewers for their erudite comments and constructive criticism that immensely helped us improve the paper.

References [1] Bayley CJ, Pilkey AK. Influence of welding defects on the localization behavior of an aluminum alloy tailor-welded blank. Mater Sci Engng A J 2005;403(1– 2):1–10. [2] Shin JK, Lee KH, Song SI, Park GJ. Automotive door design with the ULSAB concept using structural optimization. Struct Multidisc Optim 2002;23(4):320–7. [3] Song SI, Park GJ. Multidisciplinary optimization of an automotive door with a tailored blank. Proc IMechE Part D: J Automobile Engng 2006;220(2):151–63. [4] Zhu P, Shi YL, Zhang KZ, Lin ZQ. Optimum design of an automotive inner door panel with a tailor-welded blank structure. Proc IMechE Part D: J Automobile Engng 2008;222(8):1337–48. [5] Wang GG, Shan S. Use of metamodeling techniques in support of engineering design optimization. ASME J Mech Des 2007;129(4):370–80. [6] Gu L, Yang RJ, Tho CH, Makowskit M, Faruquet O, Li Y. Optimisation and robustness for crashworthiness of side impact. Int J Vehicle Des 2001;26(4):348–60. [7] Youn BD, Choi KK, Yang RJ, Gu L. Reliability-based design optimization for crashworthiness of vehicle side impact. Struct Multisidc Optim 2004;26(3– 4):272–83. [8] Yang RJ, Wang N, Tho CH, Bobineau JP, Wang BP. Metamodeling development for vehicle frontal impact simulation. ASME J Mech Des 2004;127(9):1014–20. [9] Fang H, Rais-Rohani M, Liu Z, Horstemeyer MF. A comparative study of metamodeling methods for multiobjective crashworthiness optimization. Comput Struct 2005;83(25–26):2121–36. [10] Zhang Y, Zhu P, Chen GL, Lin ZQ. Study on structural lightweight design of automotive front side rail based on response surface method. ASME J Mech Des 2007;129(5):553–7. [11] Simpson TW, Mauery TM, Korte JJ, Mistree F. Kriging models for global approximation in simulation-based multidisciplinary design optimization. AIAA J 2001;39(12):2233–41. [12] Clarke SM, Griebsch JH, Simpson TW. Analysis of support vector regression for approximation of complex engineering analysis. ASME J Mech Des 2005;127(11):1077–87. [13] Zhu P, Zhang Y, Chen GL. Metamodel-based lightweight design of automotive front-body structure using robust optimization. Proc Inst Mech Engrs Part D: J Automobile Engng 2009. doi:10.1243/09544070JAUTO1045. [14] Zerpa LE, Queipo NV, Pintos S, Salager JL. An optimization methodology of alkaline-surfactant-polymer flooding processes using field scale numerical simulation and multiple surrogates. J Petrol Sci Engng 2005;47(2–3):197–208. [15] Goel T, Dorney DJ, Haftka RT, Shyy W. Improving the hydrodynamic performance of diffuser vanes via shape optimization. Comput Fluids 2008;37(6):705–23. [16] Forrester AIJ, Keane AJ. Recent advances in surrogate-based optimization. Prog Aerospace Sci 2009;45(1–3):50–79. [17] Hamza K, Saitou K, Nassef A. Design optimization of vehicle B-pillar subjected to roof crush using mixed reactive taboo search 2003; ASME 2003 Design Engineering Technical Conferences and Computers and Information in Engineering Conference DETC2003/DAC-48750. [18] Wang DZ, Dong G, Zhang JH, Huang SL. Car side structure crashworthiness in pole and moving deformable barrier side impacts. Tsinghua Sci Technol 2006;11(6):725–30. [19] Bathe KJ. Finite element procedures. Prentice-Hall; 1996. [20] Shi YL, Zhu P, Zhang Y, Shen LB, Lin ZQ. Methods of the mesh dimension constraint for enhancing the simulation efficiency of vehicle crash. J Shanghai Jiao Tong Univ 2008;42(6):905–9. [21] Bathe KJ, Guillermin O, Walczak J, Chen H. Advances in nonlinear finite element analysis of automobiles. Comput Struct 1997;64(5–6):881–91. [22] Bathe KJ, Walczak J, Guillermin O, Bouzinov PA, Chen H. Advances in crush analysis. Comput Struct 1999;72(1–3):31–47. [23] Mao M, Chirwa EC, Chen T, Latchford J. Static and dynamic roof crush simulation using LS-DYNA3D. Int J Crashworthiness 2004;9(5):495–504. [24] Chen T, Chirwa EC, Mao M, Latchford J. Rollover far side roof strength test and simulation. Int J Crashworthiness 2007;12(1):29–39. [25] Chuang CH, Yang RJ, Li G, Mallela K, Pothuraju P. Multidisciplinary design optimization on vehicle tailor rolled blank design. Struct Multidisc Optim 2008;35(6):551–60. [26] Jin R, Chen W, Simpson TW. Comparative studies of metamodeling techniques under multiple modeling criteria. Struct Multidisc Optim 2001;23(1):1–13. [27] Yang RJ, Gu L. Experience with approximate reliability-based optimization methods. Struct Multidisc Optim 2003;25:1–9. [28] Wang WJ, Xu ZB, Lu WZ, Zhang XY. Determination of the spread parameter in the Gaussian kernel for classification and regression. Neurocomputing 2003;55(1):643–63. [29] Hamas H, Al-Smadi A. Space partitioning in engineering design via metamodel acceptance score distribution. Engng Comput 2007;23(3):175–85.

Computers and Structures 88 (2010) 45–53

Contents lists available at ScienceDirect

Computers and Structures journal homepage: www.elsevier.com/locate/compstruc

A practical method for proper modeling of structural damping in inelastic plane structural systems Farzin Zareian a,*, Ricardo A. Medina b a b

Department of Civil and Environmental Engineering, University of California, Irvine, CA 92697, United States Department of Civil Engineering, University of New Hampshire, Durham, NH 03824, United States

a r t i c l e

i n f o

Article history: Received 5 March 2009 Accepted 3 August 2009 Available online 28 August 2009 Keywords: Rayleigh damping Inelastic responses Structural damping Dynamic analysis Seismic response Seismic evaluation

a b s t r a c t This study addresses some of the pitfalls of conventional numerical modeling of Rayleigh-type damping in inelastic structures. A practical modeling approach to solve these problems is proposed. Conventional modeling of Rayleigh-type damping for inelastic structures generates responses in which unrealistic damping forces are present that results in underestimation of peak displacement demands, overestimation of peak strength demands, and underestimation of buildings’ collapse potential. The approach proposed in this paper avoids these problems by modeling each structural element with an equivalent combination of one elastic element with stiffness-proportional damping, and two springs at its two ends with no stiffness proportional damping. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction One component that can arguably be considered overlooked in current methods for estimation of response of structural systems is the appropriate modeling of structural damping. This shortcoming becomes even more apparent when compared with the advances made in modeling of the inelastic response of structural and nonstructural components during the past decades. These advancements include computational and modeling techniques that can reasonably capture material and geometric nonlinearities, as well as the evolution of damage in a structure. In this context, the term structural damping refers to energy dissipation mechanisms present in a structure due to structural and nonstructural component responses to dynamic excitation other than the energy dissipated in inelastic excursions. Traditionally, linear viscous damping has been utilized to model the energy dissipation characteristics of a structural system exposed to dynamic excitation. In such model, the effect of damping is accounted for at a global scale – the energy dissipated through friction and slippage in joints for structural and nonstructural components is represented by means of equivalent linear viscous damping. However, the use of a linear viscous damping model in many cases produces inaccurate estimates of displacements and internal forces in members. These inaccurate estimates of internal forces are related to nodes in the structural model in which unrealistic damping forces are generated. * Corresponding author. Tel.: +1 949 824 9866; fax: +1 949 824 2117. E-mail address: zareian@uci.edu (F. Zareian). 0045-7949/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2009.08.001

The failure in proper modeling of structural damping is compounded by (1) the lack of reliable experimental data to validate the structural damping models used to represent the energy dissipation characteristics of structural systems in inelastic regimes; and (2) the decreased identiﬁcation accuracy for damping ratios when compared to that of elastic natural frequencies and mode shapes. Results from full-scale tests show that changes in damping are much greater than those for frequency over a similar amplitude range [1]. In the absence of more accurate structural damping models, linear viscous damping is usually utilized for convenience. The most common model of viscous damping used in modeling of multi-degree-of-freedom (MDOF) systems is the Rayleigh-type damping where the damping matrix, C, is composed of the superposition of a mass-proportional damping term (i.e., aM) and a stiffness-proportional damping term (i.e., bK) [2].

C ¼ aM þ bK

ð1Þ

Physically, mass-proportional damping (MPD) is equivalent to having externally supported dampers attached to the dynamic (inertial) degrees of freedom while stiffness-proportional damping (KPD) implies the presence of viscous dampers (i.e., dashpots) that join two adjacent dynamic degrees of freedom. MPD has the effect of having modal damping ratios that are inversely proportional to the frequencies of vibration of the system, while KPD produces modal damping ratios directly proportional to the modal frequencies of the structure. In both cases, very little experimental veriﬁcation exists for such model, particularly for structural systems that undergo signiﬁcant levels of inelastic deformations.

F. Zareian, R.A. Medina / Computers and Structures 88 (2010) 45–53

In the last few years, several researches have studied and identified important limitations of Rayleigh-type damping as it applies to inelastic systems. As shown by Bernal [3], Medina and Krawinkler [4], and Hall [5], when Rayleigh-type damping based on the initial stiffness matrix is used, unrealistic damping forces are generated at joints in which structural elements undergo abrupt changes in stiffness. This is due to the tendency of degrees of freedom with small inertias to undergo abrupt changes in velocity once the stiffness of the element changes during the inelastic response [3]. Thus, unrealistic damping forces develop at these degrees of freedom resulting in an underestimation of the peak displacement demands in the structure and an overestimation of internal forces for elements in the system that do not undergo changes in stiffness [4]. Moreover, as shown in this paper, inappropriate model of structural damping may also cause underestimation of the collapse potential of buildings. The objective of this paper is to present a more appropriate and easy-to-apply numerical modeling approach for implementing Rayleigh-type damping in structures. This approach eliminates the presence of unrealistic damping forces in inelastic time history responses. The implication is that the proposed approach will provide improved inelastic dynamic response predictions. An illustration of the benefits of using this new approach in the context of seismic performance assessment is presented. 2. Viscous damping, Rayleigh damping, and inelastic responses Although damping is considered the primary energy dissipation mechanism for elastic structures, the focus on inelastic structures is warranted by the presence of unrealistic damping forces when Rayleigh-type damping based on initial stiffness is assigned to structural elements that experience inelastic responses. In addition, for relatively large levels of inelastic behavior, the energy dissipated via structural damping, as predicted by numerical models with Rayleigh-type damping, still constitutes a significant percentage (e.g., 25%) of the total dissipated energy. Fig. 1 shows the energy dissipated due to linear viscous damping relative to the total input energy for a 4-story MDOF model, which represents a non-ductile reinforced-concrete building, at various levels of ground motion intensity measure (gray line with diamond markers). The MDOF model has fundamental period T = 0.4 s, and the global strength of the structure corresponds to a response modification factor of 6 assuming an over-strength factor of 2 for this

Relative Dissipated Energy to Input Energy Mean IDA curves N=4, T1=0.6, R μ = 3.0 , ξ=0.05, Peak-Oriented model, Northridge EQ

structure (i.e., R = 6, X = 2, Rl = 3). Rayleigh-type damping based on initial stiffness is used and damping ratios at the first and third mode are set to 5%. On the same plot, the energy dissipated due to hysteretic action of structural components that enter the inelastic regime relative to the total input energy is shown with a black line with square markers. The vertical axis shows the level of ground motion intensity in terms of Sa/g (spectral acceleration at the first mode period of the structure). The ground motion recording used belongs to the 1994 Northridge earthquake. This building has been modeled with a concentrated plasticity approach in which the hysteretic response at the end of beams and columns exhibits moderate levels of monotonic and cyclic deterioration based on the model developed by Ibarra et al. [6]. As illustrated in Fig. 1, when the structural system begins to experience inelastic deformations (i.e., Sa/g > 0.75), the fraction of the total input energy dissipated through viscous damping is reduced and the energy dissipated in structural components hysteresis loops is increased. However, the energy dissipated through viscous damping has a lower-bound of approximately 27% of the total input energy. In this case, even when cyclic deterioration and relatively large levels of inelastic behavior are present (around Sa/g = 4.0), the contribution of viscous damping in dissipating the input energy to the structural system is increased as the hysteretic energy dissipation capacity of structural components is exhausted due to damage and deterioration. Therefore, inappropriate modeling of structural damping has the potential to provide erroneous demand prediction for structures exposed to strong dynamic excitations. This example utilized the most commonly used linear viscous damping model, which is the Rayleigh-type damping based on initial stiffness, i.e., a time invariant stiffness matrix. Muto and Beck [7] have also highlighted the importance of appropriate modeling of viscous damping when applying system identification techniques to estimate hysteretic structural parameters of systems subjected to earthquake loading. They showed that excluding viscous damping from identification models will significantly modify the value of the identified hysteretic parameters if viscous damping is present in the structure. Studies conducted by Medina and Krawinkler [4] on regular moment-resisting frame structures exposed to far-field ground motions have shown that improper modeling of 5% critical structural damping using the Rayleigh model based on initial stiffness results in the underestimation of peak-drift demands, on average, in the order of 10%. However, the overestimation in peak strength demands can be in the order of 30% depending upon the structural properties and ground motion characteristics. This phenomenon is illustrated in Fig. 2, which presents representative results corresponding to a numerical model of a moment-resisting frame

60%

NR94cnp-5% damping ratio NR94cnp-10% damping ratio LP89cap-10% damping ratio

50%

(Mc -Mp,b)/Mp,b

Sa/g

40% 30% 20% 10%

0 0

0.25 0.5 0.75 Relative Dissipated Energy to Input Energy Damping Energy

Hysteretic Energy

Fig. 1. Total damping and hysteretic energy dissipated relative to the input energy for the 4-story case study building.

0% 0

[Sa/g] / γ Fig. 2. Relative difference between maximum column moment (Mc) and beam plastic moment (Mp,b) at the top ﬂoor of a single-bay, moment-resisting frame.

F. Zareian, R.A. Medina / Computers and Structures 88 (2010) 45–53

Model A

Model B

(a)

(b)

Fig. 4. Idealized SDOF structures (a) ﬂexible beam with elastoplastic moment– rotation relationship at both ends and (b) ﬂexible beam + semi-rigid rotation springs to represent elastoplastic behavior.

Displacement Ductility Ratio, SDOF, Model A 6 Mass Proportional Damping

Displacement Ductility Ratio

Initial-Stiffness Proportional Damping

-3 0

Time (sec.)

Column-end Moment Demand, SDOF, Model A 1.5

Mcolumn / M p,beam

structure exposed to two different ground motion records. One of the records is from the 1994 Northridge Earthquake – Canoga Park Station (NR94cnp) and the other one from the 1989 Loma Prieta Earthquake – Capitola Station (LP89cap). The frame was designed such that plastic hinges have the potential to form at the end of each beam and at the bottom of the first-story columns. All columns are infinitely strong. Modal damping ratios were varied and inelastic dynamic analyses were conducted to quantify the relative difference between the maximum column moment and the beam plastic moment at the top floor. In concept, this relative difference must be negligible. This is clearly not the case when initial stiffness proportional damping is assigned to elements whose stiffness changes during the time history (i.e., inelastic beam elements). As expected, the relative difference between column moment and beam moment is negligible in the elastic range (see Fig. 3). In addition, as both damping ratio and level of inelastic behavior increase, the relative difference between the maximum column moment and the beam plastic moment increases. The level of inelastic behavior is quantified by the ratio of the pseudo-spectral acceleration at the fundamental period of the structure to the base shear strength coefficient, i.e., [Sa/g]/c. Such errors in overestimation of demand forces can lead to over-designed structural members that are sized using force-based methods. Consequently, these elements have the potential to attract larger forces into the joint, which may cause the structure to be more vulnerable to damage and collapse once deterioration of structural components is modeled. Similarly, underestimation of deformation demands may lead to unsafe structural designs that use displacement-based methods such as performance-based design. A better understanding of the consequences of modeling viscous damping based on initial stiffness can be obtained with the model in Fig. 4a, which depicts a Single-Degree-Of-Freedom (SDOF) structure in which two rigid columns are joined by a flexible beam (Model A). The beam is modeled based on a concentrated-plasticity approach in which elastoplastic moment– rotation relationships are assigned to the beam ends. The period of the system is 0.4 s and the plastic moment capacity of the beam is 3000 kip.-in. (3518.7 N m). To better show the inaccuracy of Rayleigh damping modeling based on initial stiffness, a damping ratio of 10% is assumed, and damping in the SDOF system is modeled based on mass-proportional damping (Case 1) and initial stiffness proportional damping (Case 2). Both models are subjected to a recorded ground motion that is scaled such that a displacement ductility ratio of 4 (l = 4) is attained. Fig. 5 shows the results of this analysis in terms of displacement ductility ratios and normalized column-end moments, i.e., column-end moment divided by the beam plastic moment. Fig. 5a demonstrates that by using Model

0.5

-0.5

-1 Mass Proportional Damping Initial-Stiffness Proportional Damping

-1.5 0

2500 Column Moment

2000

Moment (kip -in.)

Time (sec.)

Beam Moment

1500

Fig. 5. Comparison between response histories of the idealized SDOF system (Model A) whose 10% critical damping is modeled using ‘‘mass-proportional damping” and ‘‘initial stiffness proportional damping” subjected to a ground motion recording from Northridge Earthquake: (a) displacement ductility ratio, and (b) normalized column-end moment.

1000 500 0 -500 -1000 -1500 -2000 -2500 0

Time (s) Fig. 3. Moment time history at the top ﬂoor of the moment-resisting frame of Fig. 2 exposed to ground motion LP89cap; damping ratio = 10%; [Sa/g]/c = 8.

A the displacement ductility ratio response is identical no matter which method was used to model structural damping, i.e., Cases 1 and 2. However, Fig. 5b shows that the column-end moments for Case 2 (i.e., initial stiffness proportional damping) exceeds the beam plastic moment of 3000 kip.-in. (3518.7 N m). Once again, this is due to the presence of unrealistic damping moments in the response. In this particular case, these unrealistic damping moments do not affect the displacement response signiﬁcantly

F. Zareian, R.A. Medina / Computers and Structures 88 (2010) 45–53

because they act for relatively short periods of time (spikes in Fig. 5a) as compared to the total strong motion duration. 3. Current approaches to overcome the limitations of Rayleightype damping based on initial stiffness Solutions to this problem have been proposed by Bernal [3], Hall [5], and Charney [8]. Bernal [3] proposed a solution in which Caughey-type damping should be used and the damping matrix should be assembled by restricting the exponents of the Caughey series (l terms in Eq. (2)) to zero or negative. Rayleigh-type damping is a special case of Caughey-type damping.

C¼M

al ½M 1 K l

ð2Þ

This proposed solution has the effect of not assigning stiffnessproportional damping to degrees of freedom without mass. For example, if values of l = 0 and 1 are used in Eq. (2), the corresponding Caughey-type matrix becomes:

C ¼ a0 M þ a 1 MK 1 M

ð3Þ

It is evident from Eq. (3) that for typical structural models used in earthquake engineering practice for which masses are lumped at floor levels, the mass matrix will be diagonal, and hence, the damping matrix will only have non-zero terms for degrees of freedom associated with the inertial masses. Thus, the potential for unrealistic damping forces at rotational degrees of freedom is eliminated. However, this approach does not avoid the presence of unrealistic damping forces at rotational degrees of freedom when masses are assigned to them. The implementation of such Caughey-type damping compromises the numerical efficiency of the solution of the equations of motion because for l < 0 the calculation of the inverse of the stiffness matrix K will be required. In addition, as demonstrated by Oliveto and Greco [9], Caughey-type damping with l < 0 does not keep the same Caughey coefficients once a change in the support conditions of the structure takes place. An alternative could be to use only mass-proportional damping (l = 0) in Eq. (3), i.e., Eq. (4). Although this approach will eliminate spurious damping forces, the displacement response of the multidegree-of-freedom structure will exhibit significant higher-frequency content that is not present in the response of real structures. This is due to the presence of small damping ratios at the higher modes once this solution is devised. Studies such as those conducted by Otani [10] demonstrate that damping models that incorporating stiffness-proportional terms provide a better correlation with experimental results.

C ¼ a0 M

ð4Þ

Hall [5] suggested the elimination of mass-proportional damping contribution and the incorporation of an artiﬁcial cap (or bound) to the stiffness proportional damping component. In the authors’ opinion, this approach would require modiﬁcations to the numerical solution of the equations of motion. Moreover, Charney [8] proposed an extension to Bernal’s approach in which the stiffness-proportional component of the Rayleigh-type damping matrix is based only on the diagonal terms of the initial stiffness matrix, i.e., terms that correspond to the dynamic degrees of freedom, in order to avoid assigning stiffness-proportional damping to degrees of freedom without mass. Alternatively, one can assemble a Rayleigh-type damping matrix based on the tangent stiffness of the system, i.e., the damping matrix is updated at each time step, Eq. (5), in which Kt(t) is the tangent stiffness matrix. Petrini et al. [11], based on test results of reinforced concrete bridge piers, showed that using tangent stiffness proportional damping is more appropriate and results in

an increase in estimates of displacement demands compared with predictions based on initial stiffness or mass-proportional damping. However, the application of this approach may cause numerical solution instabilities once signiﬁcant changes in stiffness values occur, e.g., changes due to material strength and stiffness deterioration. This approach is also computationally more expensive than that in which the initial stiffness matrix is used.

CðtÞ ¼ aM þ bKt ðtÞ

ð5Þ

Leger and Dassault [12] proposed a solution in which Rayleightype damping with variable coefficients and the tangent stiffness matrix are used, Eq. (6). Leger and Dassault argue that this particular model provides a more rational control of the amount of energy dissipated by viscous damping in nonlinear seismic analyses. However, the calculation of the scalar coefficients at each time step, while preserving modal orthogonality, is computationally expensive. In addition, the application of this approach is questionable for structural systems that experience significant degradation in stiffness because of material strength and stiffness deterioration.

CðtÞ ¼ aðtÞM þ bðtÞKt ðtÞ

ð6Þ

4. Proposed approach for proper modeling of Rayleigh-type damping in inelastic structures The approach proposed in this study deals with a formulation of a Rayleigh-type matrix with a time invariant stiffness matrix that is assembled by assigning zero stiffness-proportional damping to structural elements that have the potential to experience inelastic deformations. This approach requires an increase of the stiffness proportional damping term to those elements that remain in the elastic range throughout the response to enforce damping energy conservation. The implication is that the structural model will be composed of a combination of elastic and inelastic elements, which is a common approach in current earthquake engineering simulation studies, but Rayleigh damping is solely applied to the elastic elements. As it will be shown in this section, this modeling approach provides results that are consistent with those obtained when Rayleigh-type damping based on the tangent stiffness of the system is used. The examples presented in this paper will incorporate models in which concentrated (localized) plasticity is used. The application of these concepts to other types of models, e.g., fiber models, is the subject of current research by the authors. In the first step, we propose an approach for proper modeling of Rayleigh damping in the form explained here for beam elements whose moment gradient is time invariant. Next, a general approach for elements whose moment gradient is time variant will be presented. 4.1. Structural elements with time invariant moment gradient For structural elements with time invariant moment gradient, a two-dimensional, prismatic beam element with six degrees of freedom (see Fig. 6) is to be replaced with a two-dimensional, prismatic beam element composed of semi-rigid rotational springs at the ends and an elastic beam element in the middle (see Fig. 6 and Model B in Fig. 4b). The 6-degree-of-freedom beam element is referred to as the original beam element and the 8-degree-offreedom beam element as the modified beam element. If the rotational stiffness at the end of the original beam element without transverse loads is denoted as K0 = 6EI/L (where E is the modulus of elasticity, I the moment of inertia, and L the length of the beam), and the ratio of the rotational spring stiffness, KS, to the elastic

F. Zareian, R.A. Medina / Computers and Structures 88 (2010) 45–53

Ordinary beam element i

5 1

Degrees of freedom

Modeling

Equivalent elastic beam element with end springs j

5 1

is Modeling

Degrees of freedom

Fig. 6. Beam element and equivalent model that consists of an elastic beam element with springs at both ends.

KSKe ¼ KS þ Ke

n Ke nþ1

ð7Þ

In this approach, when stiffness-proportional damping is used, zero stiffness-proportional damping is assigned to the semi-rigid springs and the stiffness-proportional damping multiplier (see Eq. (1)) of the modiﬁed elastic beam element is varied from b to 0 0 b . The value of b is calculated by equating the damping work done by the elastic beam of the modiﬁed element plus the damping work done by the rotational springs with the damping work done by the original elastic beam, i.e., Eq. (8). In this equation, WD is the total damping work done by the element; W De is the damping work done by the elastic beam; W Ds is the damping work done by the rotational springs; M is the bending moment at the end of the element; he is the rotation at the end of the elastic beam; hs is the spring rotation; h_ e is the rotational velocity at the end of the elastic beam; and h_ s is the spring rotational velocity. Given that hs = 1/nhe 0 and h_ s ¼ 1=nh_ e one can utilize Eq. (9) for calculating b .

1 1 1 W D ¼ 2W De þ 2W Ds ¼ 2 Mhe þ 2 Mhs ¼ 2 bK e he h_ e 2 2 2 1 _ þ 2 bK s hs hs 2 b0 ¼ ½ð1 þ nÞ=n b

Displacement Ductility Ratio, SDOF, Model B 6

Mass Proportional Damping Initial-Stiffness Proportional Damping (Proposed Approach) Tangent-Stiffness Proportional Damping

-3 0

Time (sec.)

ð8Þ

Column-end Moment Demand, SDOF, Model B 1.5

ð9Þ

It is important to note that in this approach the semi-rigid spring of the modified beam element has a post-yield stiffness tuned to provide the target hysteretic response at the end of the beam. Evidently, one may advocate the use of a fully rigid spring 0 at the end of the element, which will make the estimation of b a 0 moot issue, i.e., b = b . However, this solution is not recommended in order to avoid numerical instabilities in the response, especially when piece-wise linear hysteretic models are used. The requirement that the moment gradient on the beam element be time invariant guarantees that the behavior of the modified beam element is identical to the behavior of the original beam element. Another implication of this approach is that the computer program used to conduct the numerical studies should have the capability of assigning a stiffness-proportional damping multiplier to individual structural elements. This is a capability common to many computer programs currently available. By using the approach proposed hereby for modeling Rayleigh damping of elements, the displacement response is significantly different from that of initial stiffness proportional damping or

Mcolumn / M p,beam

K0 ¼

mass-proportional damping. Model B in Fig. 4b is an SDOF system modeled using the proposed approach and Fig. 7 shows the response of this system exposed to the same ground motion used in Fig. 5. It can be seen that when the normalized column-end moment is plotted, unrealistic damping moments are no longer pres-

Displacement Ductility Ratio

beam stiffness, Ke, of the modiﬁed beam element is deﬁned as n = KS/Ke then:

0.5

-0.5

-1

Mass Proportional Damping Initial-Stiffness Proportional Damping (Proposed Approach) Tangent-Stiffness Proportional Damping

-1.5 0

Time (sec.) Fig. 7. Comparison between response histories of the idealized SDOF system (Model B) whose 10% critical damping is modeled using ‘‘mass-proportional damping”; ‘‘initial stiffness proportional damping” based on the proposed approach in this paper; and ‘‘tangent stiffness proportional damping” subjected to a ground motion recording from Northridge Earthquake: (a) displacement ductility ratio and (b) normalized column-end moment.

F. Zareian, R.A. Medina / Computers and Structures 88 (2010) 45–53

ent in responses corresponding to the proposed approach and those corresponding to damping based on the tangent stiffness. The conclusion is that a conventional formulation with initial stiffness proportional damping in elements that undergo changes of stiffness throughout the response overestimates the amount of damping when the response is inelastic. This constitutes a major drawback of the current implementation of Rayleigh-type damping based on initial stiffness. This conclusion is consistent with the observations made by Petrini et al. [11], who suggest that numerical models based on initial stiffness proportional damping tend to underestimate inelastic displacement demands obtained from test conducted with reinforced-concrete piers.

beam element with the two end springs. The upper portion of Fig. 6 shows the prismatic beam element with 6 degrees of freedom whose stiffness matrix, K0, can be expressed as shown in Eq. (10). In this equation, A is the cross sectional area, and L is the length of the beam element. The gray numbers around the stiffness matrix show the associated degrees of freedom in Fig. 6.

ð10Þ 4.2. Structural elements with time variant moment gradient An extension to the approach for proper modeling of viscous damping using the Rayleigh model that can be applied to beam/column elements whose moment gradient is time variant is proposed. The proposed approach is useful in modeling elements that experience transverse loads, and elements for which the location of the point of inflection is expected to change considerably during their inelastic dynamic responses. The varying moment gradient results in varying elastic stiffness of the structural element making the stiffness ratio of the end springs to elastic element, n, a variable. The solution presented for structural components with time invariant moment gradient can be interpreted as a special case of this more general one that is applicable to structural elements for which changes in the location of the point of inflection are not expected. The approach for proper modeling of Rayleigh-type damping in structural elements with time variant moment gradient involves modifying the stiffness matrix of the elastic internal beam element explained previously such that the effect of fixed stiffness of the springs at its two ends is compensated. The modified stiffness matrix of the internal elastic beam element is obtained by equating the stiffness matrix of a general prismatic beam element with the condensed form of the stiffness matrix of an assembly that consists of an elastic

The lower portion of Fig. 6 shows the assembly consisting of an elastic beam element and two end springs (i.e., 8 degrees of freedom). The stiffness matrix of the assembly can be expressed by Ka:

Ka ¼

Kbb

Kbc

Kcb

Kcc

ð11Þ

In Eq. (11), Kbb is a 2 2 stiffness matrix that corresponds to the degrees of freedom #3 and #6 (i.e., to be eliminated through static condensation), Kcc is a 6 6 stiffness matrix that corresponds to degrees of freedom to be kept after condensation, and Kbc and Kcb are 2 6 and 6 2 stiffness matrices generated through partitioning of Ka. Eqs. (12)–(14) express the components of Ka as a function of stiffness coefﬁcients Sii, Sjj, and Sij; and the elastic element’s moment of inertia Ie. The parameters A and E are those deﬁned for the prismatic beam element in Eq. (10). The gray numbers around the stiffness matrix show the associated degrees of freedom in Fig. 6.

ð12Þ

ð13Þ

ð14Þ

F. Zareian, R.A. Medina / Computers and Structures 88 (2010) 45–53

elastic beam element with moment of inertia obtained from Eq. (17), stiffness coefficients obtained from Eqs. (18) and (19), and two end springs with initial stiffness obtained from Eq. (16). The variation of the stiffness coefficients Sii and Sij with respect to n is plotted in Fig. 8. Values of Sii and Sij asymptotically reach 4.0 and 2.0 for large values of n. The equivalent stiffness proportional damping coefficient, 0 b , for the elastic beam element is found by using Eq. (9). The aforementioned modification of stiffness coefficients Sii, Sjj, and Sij for the elastic beam element in the assembly guarantees the response of the assembly is identical to the elastic response of its equivalent prismatic beam. The inelastic parameters of the end springs in the assembly are tuned such that the responses of the two equivalent systems are identical once the springs in the assembly go inelastic. The backbone characteristics of the end springs are shown in Fig. 9. The constitutive models considered for these nonlinear springs not only include strength and stiffness degradation (represented by hp, hpc/hp, and Mc/My in Fig. 9) but also gradual deterioration of strength and stiffness under cyclic loading (represented by the parameter k), considering a peak-oriented hysteretic model, based on the energy dissipated in each cycle [13]. By tuning the values of hp, hpc/hp, and k of end spring elements in order to obtain the target inelastic behavior of the assembly, one can obtain proper modeling of inelastic behavior with proper modeling of damping in the component. The general approach proposed in this paper for modeling Rayleigh damping of elements has the advantage that it can be applied to beam/column elements whose moment gradient varies with time. In addition, this approach includes the use of a constant damping matrix, which avoids the additional computational effort required to calculate damping forces based on the tangential stiffness of inelastic members.

The objective is to ﬁnd the values of Sii, Sjj, and Sij such that the ^ cc , shown condensed form of the stiffness matrix of the assembly, K in Eq. (15), is equal to K0.

^ cc ¼ Kcc Kcb K 1 Kbc K bb

ð15Þ

By assuming predefined values for Ks, and Ie shown in Eqs. (16) and (17), respectively, as a function of n, one can find the values of Sii, Sjj, and Sij identified in Eqs. (18) and (19).

6EIe L nþ1 Ie ¼ I n 6ð1 þ nÞ Sij ¼ Sji ¼ 2 þ 3n 1 þ 2n Sij Sii ¼ Sjj ¼ 1þn

Ks ¼ n

ð16Þ ð17Þ ð18Þ ð19Þ

This implies that, in the elastic range of response, the original beam element could be modeled with an assembly that consists of an

Stiffness coefficients for equivalent elastic beam springs at both ends of elastic element

stiffness coefficient

3.5

S ii = S jj S ij = S ji

5. Implications for seismic performance evaluation

2.5

Improper modeling of structural damping has signiﬁcant implications in terms of the reliability of seismic design and assessment procedures. Inadequate estimates of force demands in current design methods as a result of improper modeling of structural viscous damping can lead to inefﬁcient designs. Similarly, proper estimation of deformation and acceleration demands in a building due to seismic excitation is fundamental to improve the reliability of

n Fig. 8. Variation of stiffness coefﬁcients Sii and Sij with n for equivalent elastic beam element.

Post-Yielding Elastic Pre-Capping

Basic Parameters

Initial Stiffness

Yield Moment

Mc My

Capping moment ratio

θp

Plastic Hinge Rotation Capacity

Post-Capping

My Yielding Point

θ pc Post-Capping Rotation Capacity Ratio θp

Capping Point

Mc My

θp

θy

θ pc θc

θu

Derived Parameters

θy =

Yield Rotation

θc = θ y + θp

Capping Rotation

Mc =

Mc My My

Capping Moment

θu = θc + θpc

Fig. 9. Component backbone curve and its parameters.

F. Zareian, R.A. Medina / Computers and Structures 88 (2010) 45–53

building loss estimates that are part of performance-based assessment methods [14,15]. Appropriate estimation of the collapse potential of buildings is another important component of current performance-based assessment methods. In this section, inconsistencies in estimation of seismic demands and collapse potential of structures with conventional modeling of structural damping are evaluated and compared to values obtained from the viscous damping modeling methodology proposed in this paper. An evaluation of the displacement response of a MDOF system with Rayleigh-type damping based on the initial stiffness modeled using the approach proposed in this paper, as well as conventional Rayleigh-type damping, provides a more comprehensive illustration of the differences between both approaches. Fig. 10 depicts mean values of 1st story drift ratio response for the 4-story reinforced-concrete frame used in Fig. 1 exposed to a set of 40 ground motion recordings. Modal damping ratios of 5% are applied to the first and third modes. Each record was scaled and the peak 1st story drift ratio responses were plotted as a function of the intensity of the records, Sa/g. Scaling was conducted until the limit state of collapse was imminent. It is evident from this figure that the structural damping model significantly influences the estimates of peak drift ratio demands. The influence of the proposed modeling approach for Rayleightype damping based on initial stiffness can also be evaluated in the

1st Story Drift Mean IDA curves N=4, T1=0.6, Rμ = 3.0 , ξ=0.05, Peak-Oriented model

6 5

Sa/g

4 3 2 1 0 0

1st story drift (inch) Conventional damping model

Improved damping model

Fig. 10. Mean 1st story drift IDA curves for a 4-story moment-resisting frame modeled with conventional and improved Rayleigh stiffness proportional damping.

Effect of Damping Model on Collapse Fragility Curves N = 4, T1 = 0.6, Rμ = 3.0, ξ = 0.05, Peak Oriented Model

0.75

Probability of Collapse

Collapse fragility curve from Conventional damping model: datapoints analytical fragility curve Collapse fragility curve from Improved damping model: datapoints analytical fragility curve

0.5

0.25

context of seismic collapse assessment by estimating collapse fragility curves [16]. A collapse fragility curve expresses the probability of collapse as a function of Sa. In Fig. 11, the data points show the smallest value of Sa at which the nonlinear response history solution of the building subjected to a given ground motion has converged, i.e., collapse capacity. The solid squares show this value for a building whose damping is modeled using the modeling approach proposed in this paper whereas the diamonds show the collapse capacities obtained using a conventional Rayleigh-type damping model based on initial stiffness. The cumulative distribution function, assuming a lognormal distribution, of these spectral acceleration values that correspond to structural collapse is defined as the ‘‘collapse fragility curve” and is shown with heavy lines in Fig. 11. It can be seen that in this case conventional modeling of Rayleigh damping based on the initial stiffness of the system can lead to underestimation of the median of collapse capacity by 30%. 6. Summary and conclusions This study demonstrates that conventional modeling of linear viscous damping via the implementation of a Rayleigh-damping matrix with initial stiffness proportional damping results in inelastic dynamic responses that exhibit unrealistic damping forces. The presence of these unrealistic damping forces is more prevalent when both the damping ratio and the level of inelastic behavior of the structural system increase. For relatively small levels of inelastic behavior, deformation demands are not significantly affected by conventional modeling of Rayleigh-type damping based on initial stiffness. This is not the case for strength demands, as peak column moment demands may increase considerably for the case of moment-resisting frames. For levels of inelastic behavior consistent with structural systems approaching the limit state of collapse, the collapse capacity of a system with Rayleigh-type damping based on initial sitffness is overestimated, i.e., the probability of collapse for a given ground motion intensity is underestimated. This is due to an overestimation of the energy dissipation capacity of the system once conventional modeling of Rayleigh-type damping based on initial stiffness is implemented. This latter issue is significant because contrary to the common notion that damping energy dissipation in structural models may be de-emphasized in the inelastic range, the numerical results from this study showed that even when cyclic deterioration and relatively large levels of inelastic behavior are present, the contribution of viscous damping in dissipating the input energy to the structural system is significant (on the order of 25%) as the hysteretic energy dissipation capacity of structural components is exhausted due to damage and deterioration. Therefore, inappropriate modeling of structural damping has the potential to provide erroneous demand prediction for structures exposed to strong dynamic excitations. This study proposed a modeling approach developed by the authors to mitigate these effects and obtain more reasonable estimates of seismic demands and seismic collapse capacities for plane structural systems modeled with localized (concentrated) plasticity approaches, as well as Rayleigh-type damping based on initial stiffness. The proposed approach proved to be viable even for cases in which cyclic strength and stiffness deterioration was present in the response. Acknowledgements

0 0

Sa(T1 )/g Fig. 11. Effects of damping model on collapse fragility curves.

This study originated from discussion among the authors while the ﬁrst two authors of the paper were doctoral students at Stanford University working under the supervision of Prof. Helmut Krawinkler. His guidance and helpful suggestions are appreciated. The ﬁnancial support provided by the National Science Foundation

F. Zareian, R.A. Medina / Computers and Structures 88 (2010) 45–53

through the Paciﬁc Earthquake Engineering Research (PEER) Center is gratefully acknowledged. The authors would also like to give special thanks to Dr. Luis Ibarra for sharing his ideas and providing much valuable input at the beginning stages of this study. References [1] Jeary AP, Wong J. The treatment of damping for design purposes. In: Proceedings, international conference on tall buildings and Urban Habitat, Kuala Lumpur; 1999. [2] Chopra AK. Dynamics of structures. 2nd ed. Prentice Hall: New Jersey; 2001. [3] Bernal D. Viscous damping in inelastic structural response. ASCE J Struct Eng 1994;120(4):1240–54. [4] Medina RA, Krawinkler H. Seismic demands for nondeteriorating frame structures and their dependence on ground motions. PEER Report 2003/15. Berkeley (CA): Paciﬁc Earthquake Engineering Research Center; 2004. [5] Hall JF. Problems encountered from the use (or misuse) of Rayleigh damping. Earthquake Eng Struct Dynam 2005;35(5):525–45. [6] Ibarra L, Medina RA, Krawinkler H. Hysteretic models that incorporate strength and stiffness deterioration. Earthquake Eng Struct Dynam 2006;34(12):1489–511. [7] Muto M, Beck JL. Bayesian updating and model class selection for hysteretic structural models using stochastic simulation. J Vibr Control 2008;14(1–2):7–34.

[8] Charney FA. Unintended consequences of modeling damping in structures. ASCE J Struct Eng 2008;134(4):581–92. [9] Oliveto G, Greco A. Some observations on the characterization of structural damping. J Sound Vibr 2002;256(3):391–415. [10] Otani S. Nonlinear dynamic analysis of reinforced concrete building structures. Canadian J Civil Eng 1980;7(2):333–44. [11] Petrini L, Maggi C, Priestley N, Calvi M. Experimental verification of viscous damping modeling for inelastic time history analyses. J Earthquake Eng 2008;12(1):125–45. [12] Leger P, Dussault S. Seismic-energy dissipation in MDOF structures. ASCE J Struct Eng 1992;118(5):1251–69. [13] Zareian F, Krawinkler H. Simplified performance-based earthquake engineering. Report No. 169, John A. Blume Earthquake Engineering Center, Department of Civil and Environmental Engineering, Stanford University, Stanford, CA. [14] Krawinkler H, editor. Van Nuys hotel building tested report: exercising seismic performance assessment. Report No. PEER 2005/11, Pacific Earthquake Engineering Research Center. Berkeley, California: University of California at Berkeley; 2005. [15] ATC-58 35% draft guidelines for seismic performance assessment of buildings. Redwood City, CA: Applied Technology Council; 2005. [16] Zareian F, Krawinkler F. Assessment of probability of collapse and design for collapse safety. Earthquake Eng. Struct. Dynam. 2007;36(13):1901–14.

Computers and Structures 88 (2010) 54–64

Contents lists available at ScienceDirect

Computers and Structures journal homepage: www.elsevier.com/locate/compstruc

Structural ﬁnite element model updating using transfer function data A. Esfandiari a,b, F. Bakhtiari-Nejad a,*, M. Sanayei b, A. Rahai a a b

Amirkabir University of Technology, 424, Hafez Ave., Tehran, Iran Tufts University, Department of Civil and Environmental Engineering, 200 College Ave, Medford, MA, 02155 USA

a r t i c l e

i n f o

Article history: Received 16 March 2009 Accepted 11 September 2009 Available online 21 October 2009 Keywords: Structure Structural health monitoring Finite element model updating Parameter estimation Frequency Response Function

a b s t r a c t A new method is presented for the ﬁnite element model updating of structures at the element level utilizing Frequency Response Function data. Response sensitivities with respect to the change of mass and stiffness parameters are indirectly evaluated using the decomposed form of the FRF. Solution of these sensitivity equations through the Least Square algorithm and weighting of these equations has been addressed to achieve parameter estimation with a high accuracy. Numerical examples using noise polluted data conﬁrm that the proposed method can be an alternative to conventional model updating methods even in the presence of mass modeling errors. Ó 2009 Published by Elsevier Ltd.

1. Introduction Identifying structural damage by using nondestructive test data has been investigated by many researchers during the past two decades. This identification plays an important role in better understanding structural behavior and management. Monitoring of structural responses and characteristic parameters such as natural frequencies, mode shapes, Frequency Response Function (FRF), and static displacements provides a great index to assess and identify structural integrity and is addressed by many researchers [1–4]. In this paper, ‘‘damage” is defined as an actual damage, deterioration, or errors in initial section stiffness and mass properties. Many researchers are interested in using dynamic response for model updating, which can be classified into modal-based and response-based methods. The modal-based model updating technique relies on the modal characteristics data obtained from an experimental modal analysis that is extracted from the measured FRF data indirectly. This numerical extraction process inherently introduces errors and inaccuracies over and above those already present in the measured data. Sestieri and D’Ambrogio [5] emphasize that the numerical procedures used for modal identification using experimental vibration data can introduce errors exceeding the level of required accuracy to update FE models. In particular, if the tested structure exhibits close modes or regions of high modal density, traditional updating tools will fail to give reliable results as the extracted modal properties are associated with high levels of measurement errors as well as processing errors. * Corresponding author. Tel.: +9821 64543417; fax: +9821 66419736. E-mail address: baktiari@aut.ac.ir (F. Bakhtiari-Nejad). 0045-7949/$ - see front matter Ó 2009 Published by Elsevier Ltd. doi:10.1016/j.compstruc.2009.09.004

In response-based finite element model (FEM) updating methods, the measured FRF data are directly utilized to identify the unknown structural parameters. In this class of model updating methods the FE models are updated in view of the fully damped response along a frequency axis and not an estimated set of modal properties. Also, the amount of available test data is not limited to a few identified eigenvalue and eigenvectors, and FEM updating can be performed using many more data points in the FRF. Natke [6] and Cottin et al. [7] state the benefits of FRF-based over modal-based updating algorithms. Most of the FRF-based model updating techniques are used to minimize a residual error between analytical and experimental input force or output response [8–11]. The input error (nodal error force) formulations are characteristically different compared to many other FRF model updating formulations since the linear design parameters (such as material properties and section geometric properties) remain linear in the updating. Cottin et al. [7] show that these formulations have a tendency to result in more biased parameter estimations than those done by output residual methods. Larsson and Sas [12] developed a model updating technique utilizing an exact dynamic condensation in which the objective function did not require the computation of the impedance matrix. They emphasized that the desired frequency range, which can be updated, is inherently limited by the condensation procedure. Incomplete measurements and their implication on the FRF model updating formulation seem to severely restrict the method’s ability to update larger FEMs. Modak et al. [13] tuned structural models by a direct model updating method and an iterative model updating method based

A. Esfandiari et al. / Computers and Structures 88 (2010) 54–64

on the FRF data through computer and laboratory experiments. The predictions by the directly updated model are reasonably accurate in the lower frequency range but the predictions contain a significant error in the higher frequency range. Arau´jo dos Santos et al. [14] proposed a damage identification method based on the FRF sensitivities. They indicate that better identification results can be obtained in the lower frequency range using excitation points where there are no modal nodes. It is demonstrated in this research that for small damage, the measurement errors are the main influence in the identification quality, whereas for large damage the incompleteness becomes the most important factor. In contrast to using objective functions based on input residual methods, minimization of the output-residuals can be used for finite element model updating [15–18]. Using output-based methods, the difference between the measured and predicted structural response is minimized. The algebraic nonlinearity of model updating algorithm makes this approach more challenging. Lin and Ewins [19] developed a model updating method by directly using measured FRF data. They state that providing much more information in a desired frequency range is a major advantage of using FRF data over using modal data in model updating. Conti and Donley [20] updated a full finite element model using response data by minimizing the residual error between the analytical and experimental response. In order to overcome coordinate incompatibility, the analytical system is reduced to the size of the experimental data set using a reduction technique. Wang et al. [21] derived a damage detection algorithm that can be used to determine a damage vector by indicating both location and magnitude of damage from perturbation equations of FRF data. An iterative scheme and an effective weighting technique has been introduced to overcome incomplete measurements and reduce the adverse effects of measurement errors. Park and Park [22] proposed a damage detection technique that utilizes an accurate analytical finite element model based on the incomplete FRF data in numerical and experimental environments. Authors also discuss frequency regions where the suggested method works satisfactorily. De Sortis et al. [23] investigated the dynamic behavior at low vibration levels of an existing masonry building subjected to forced vibration sine dwell or sine sweep using an output error equation. Possibly, on account of weak nonlinearities of the tested building, the measurements obtained with sine dwell tests seemed more suited than those obtained by sine sweep tests for the application of identification techniques. Hwang and Kim [24] used FRF data to find the location and severity of damage in a structure by considering only a vector subset of the full set of FRFs. Zimmerman et al. [25] extended their previous developments in minimum rank perturbation theory (MRPT) for damage detection using experimental and numerical FRF data. The FRF-based results are shown to be less sensitive to noise if proper frequency points are used, and provide a damage assessment similar to that obtained using identified modal parameters, but at a substantially reduced level of effort. Model updating techniques ordinarily assume that damage is uniformly distributed along element body and consider a stiffness reduction factor for the whole element. This is an effective crosssectional property leading to the same effective stiffness. It is an idealized assumption while damage might be partially distributed along an element body or appears as a localized stiffness reduction by a crack or a notch. At such cases, model updating techniques result in an equivalent stiffness reduction factor for the partially distributed damage. For a more accurate prediction of the damage locations and severity, an appropriately fine finite element mesh should be adopted. This will increase the number of unknown parameters for the model updating. As an alternative, parameter grouping can be used to decrease the number of unknown param-

eters. Also by a two stage method, a coarse mesh can be used to find the region of damage and recognizing undamaged parts, and then an accurate prediction can be done by using a fine mesh for the damaged components [18]. A relatively fine mesh is necessary for reliable predictions of the analytical response for comparison with the modal responses in the measured frequency range. After determining the mesh size, the number of analytical degrees of freedom (DOF) always exceeds the number of measured coordinates and consequently FRF formulations based on both the input-residuals and the output-residuals cannot be solved directly. Expansion of the measurement vector or reduction of the full finite element is necessary due to incompatibility between the number of measured DOFs and the number of analytical DOFs. No matter how sophisticated is it, the expansion or reduction schemes may introduce numerical errors FRF model updating as an inevitable consequence. While using a full set of measurements at all degrees of freedom, the model updating problem is algebraically linear and can be solved in one iteration. Reducing the model or expanding the data converts model updating to an algebraically nonlinear problem due to matrix inversion containing unknown parameters. Although the Taylor series expansion of the objective function can be used without model reduction or data expansion, model updating using the Taylor series linearization will still be nonlinear. In this study, a structural model updating technique is presented using FRF data and measured natural frequencies of the damaged structure without any expansion of the measured data or reduction of the finite element model. To decrease nonlinearity of the model updating process, the change of FRF of a structure is correlated to the change of stiffness, mass and damping through sensitivity equations, which have been derived using the change of eigenvectors and measured natural frequencies of the damaged structure. The change of eigenvector is also expressed as the linear combination of the original eigenvectors, while eigenvector participation factor is a function of the perturbation of the stiffness and mass. The sensitivity equation set is solved by the Least Square method through a proper weighting procedure. The effect of excitation frequency and weighting factor on finite element model updating has been successfully addressed through a truss model example. 2. Theory Using a finite element modeling, equation of motion of a linear elastic time-invariant structure with n degree of freedoms is given by:

_ þ KxðtÞ ¼ f ðtÞ M€xðtÞ þ C xðtÞ

ð1Þ

where M, C and K are the mass, damping and stiffness matrices of the structure, respectively. f(t) is the vector of applied forces and x(t) is the vector of structural response to the applied force. For a harmonic input, as an external force, displacement response can be expressed as:

f ðxÞ ¼ FðxÞejxt

and xðxÞ ¼ XðxÞejxt

ð2Þ

where x is the frequency of the excitation load. Substituting (2) into (1) yields the input–output relationship using transfer function H(x):

XðxÞ ¼ HðxÞFðxÞ

ð3Þ

where transfer function H(x) (or the impedance matrix) in terms of system properties, can be deﬁned as:

HðxÞ ¼ ð x2 M þ jxC þ KÞ 1

ð4Þ

by a spectral decomposition [3], the response of the structure to a unit harmonic load can be expressed as:

A. Esfandiari et al. / Computers and Structures 88 (2010) 54–64

Hil ðxÞ ¼

n X

/ir /lr

r¼1

x2r x2 þ 2jfr xr x

ð5Þ

where Hil(x) is the displacement of the ith degree of freedom when subjected to the applied unit force at the lth degree of freedom. Hij(x) represents the entire impedance matrix where i represents the measurement points and j is the excitation point. The rth mode shape (eigenvector), natural frequency and damping loss factor are represented as ur, xr and fr, respectively. The higher mode shapes whose natural frequencies are far from the excitation frequency of the applied load x can be excluded from (5) with some impact due to using incomplete measurements. Due to damage, the rth mode shape of the structure changes by the amount of dur, therefore:

/rd ¼ /r þ d/r

For cases such as a large structure where not all eigenvalues and eigenvectors are available, truncated forms of eigenvector derivation can be used [27]. In comparison to using a Taylor series expansion of the transfer function, the ﬁrst order expansion in (9) is weakly nonlinear. This is due to the fact that (9) does not require derivation of the denominator in (5). Using the formulations given in (9) and (10), the measured response is correlated to the perturbation of the stiffness and mass matrices as:

Hild ðxÞ ¼

Hild ﬃ

r¼1

ð7Þ

nm X

/ir /lr

r¼1

þ þ

/ir d/lr d/ir /lr

x2rd x2 þ 2jfrd xrd x

r¼1 n X

/ir /lr

ð8Þ

x2r x2 þ 2jfr xr x r¼nmþ1

The ﬁrst term of this equation can be evaluated using the eigenvector of the intact structure, the measured natural frequencies, and damping loss factors of the damaged structure. Eigenvectors of a structure introduce an orthogonal space vector that can be used to represent any vector of the same order by their linear combination. Therefore, the modal method [26] makes use of the assumption that, the rate of change of modal vector for any mode shape can be approximated as a linear combination of eigenvectors for all modes. Therefore, change of mode shapes of the structure due to the damage, can be evaluated using a ﬁrst order series as [26]:

d/i ﬃ

n X

aiq /q

ð9Þ

q¼1

for q – i;

ð11Þ

The stiffness and mass matrices of an individual structural element such as a bar or beam can be described as follows [28]:

and M e ¼ AMe PMe ATMe

ð12Þ

where PSe and PMe contain nonzero eigenvalues of stiffness and mass matrices as their diagonal entries and ASe and AMe contain corresponding eigenvectors of nonzero eigenvalues of the stiffness and mass matrices. The stiffness and mass matrices of the structure are constructed by assembling the elemental stiffness and mass matrices as:

K¼

ne X

T Tei ASei PSei ATSei T ei ¼ AS PS ATS

i ne X

and

T Tei AMei P Mei ATMei T ei ¼ AM PM ATM

ð13Þ

where Tei is the transformation matrix of the ith element from the local coordinate to the global coordinate. Matrices AS(n nPS) and AM(n nPM) are deﬁned as the stiffness and mass connectivity matrices, and the diagonal matrices PS and PM have the same elemental stiffness and mass parameters as their diagonal entries. Numbers of parameters are dependent on element type: while for truss elements there is only one axial rigidity parameter, for beam-column elements there are two parameters for axial and ﬂexural rigidity. Since the global stiffness matrix is a linear function of the elemental stiffness parameters, (13) can be perturbed to get:

K þ dK ¼ AS ðPS þ dPS ÞATS

and M þ dM ¼ AM ðPM þ dPM ÞATM

aii ¼

/Ti dM/i 2

for q ¼ i ð10Þ

ð14Þ

where dPS and dPM are the changes of elemental stiffness and mass parameters caused by the damage. Expanding (14) and subtracting (13) from it yields a parameterized form of the perturbed global stiffness and mass matrices as:

dK ¼ AS dPS ATS

and dM ¼ AM dPM ATM

ð15Þ

Using these deﬁnitions for dK and dM in (15) and substituting into (11) and by a mathematical manipulation, the change in the FRF of the structure is correlated to the damage as:

Hil ðxÞ ¼ SSil ðxÞdPS þ SM il ðxÞdP M

where

/T dK x2i dM /i aiq ¼ q x2i x2q

x2r x2 þ 2jfr xr x

x2rd x2 þ 2jfrd xrd x

nm X

/ir /lr

r¼nmþ1

/ir /Tq dK/r /lr þ2 2 2 2 xi x2q r¼1 q¼1 xrd x þ 2jfrd xrd x nm X n /ir /Tq dM/r /lr X 2 2xi 2 2 2 xi x2q r¼1 q¼1 xrd x þ 2jfrd xrd x

M¼

x2rd x2 þ 2jfrd xrd x

r¼1 nm X

n X

K e ¼ ASe PSe ATSe

The approximation given in (7) is realistic because we can measure natural frequencies with high accuracy. The second term is related to the unmeasured part of natural frequencies and damping loss factors and is used to alleviate the effects of incomplete measurements. Numerical simulation shows this part increases the accuracy of (7) and its convergence rate. The second part of (7) corrects itself as the optimization process updates the parameters of the structure. Substituting (6) into (7) and neglecting the second order term yields:

Hild ðxÞ ﬃ

r¼1

nm X n X

ð6Þ

n X /ird /lrd /ir /lr þ x2rd x2 þ 2jfrd xrd x r¼nmþ1 x2r x2 þ 2jfr xr x

/ir /lr

x2rd x2 þ 2jfrd xrd x

where the index d indicates a damage case. Substituting (6) into (5) and using the ﬁrst nm measured natural frequencies of the damaged structure; the response (displacement) of the damaged structure can be evaluated as: nm X

nm X

ð16Þ

where

e ild ðxÞ Hil ðxÞ ¼ Hild ðxÞ H and

ð17Þ

A. Esfandiari et al. / Computers and Structures 88 (2010) 54–64

e il ðxÞ ¼ H

nm X r¼1

n X /ir /lr /ir /lr þ x x þ 2jfrd xrd x r¼nmþ1 x2r x2 þ 2jfr xr x 2 rd

ð18Þ SSil ð

SM il ð

xÞ and xÞ are (1 np) row vectors of the stiffness and mass parameters sensitivity for ith degree of freedom subjected to the applied unit load at the lth degree of freedom as: SSil ð

xÞ ¼ 2

nm X n X r¼1 q¼1

/ir /Tq AS diag ATS /r /lr 2 xrd x2 þ 2jfrd xrd x x2r x2q

and

SM il ðxÞ ¼ 2

nm X n X r¼1 q¼1

/ir /Tq AM diag ATM /r /lr 2 xrd x2 þ 2jfrd xrd x x2r x2q

ð19Þ

Operator diag represents the entries of a diagonal matrix as a vector and vice versa. dP S and dPM are the vectors of stiffness and mass parameters. Sensitivity equations of all measurements at the interested frequency ranges can be represented by a set of equations as:

" HðxÞ ¼ ½ S

dP S dPM

# ¼ SðxÞdP

ð20Þ

where S(x) is the total sensitivity matrix for all measurements and dP is the vector of all stiffness and mass parameters changes. One source of problems in the sensitivity-based model updating method is the noise-induced measurement error in the sensitivity matrix that will cause a convergence to a local minimum. In the proposed method, measured natural frequencies and damping loss factors are necessary to construct sensitivity equations. Using modern electronic transducers, measured natural frequencies are very accurate and some researchers have assumed noise-free measurement of natural frequencies. Also, any inevitable error does not effect results if the excitation frequency of the applied load is not in the vicinity of the nearest measured natural frequencies. For undamped or lightly damped structures, the denominators of (7), and consequently S(x), are dominated by x2id x2 . If the excitation frequency is selected close to the resonance frequency a small error in measured frequency introduces a significant change in x2id x2 , causing large changes in the response. By moving away from the resonance frequency, this error sensitivity is significantly reduced, resulting in less vulnerability to measurement errors. Although damping loss factors measurement is not as accurate as natural frequency measurement, appropriate selection of excitation frequency can alleviate measurement error. Moving the excitation frequency away from resonance frequency will result in responses that are less sensitive to damping and errors in damping measurement. Eq. (20) can be solved by several methods such as Least Square method (LS), Non-Negative Least Square method (NNLS) and Singular Value Decomposition method (SVD). The quality of predicted damage by expression (20) depends on the quality of the measured FRF data and on a weighting technique which needs to be applied to the equations. This weighting technique uses a weighting factor without which the Least-square solution would be dominated by equations with the largest coefficients. This means that some equations would overshadow the information from other equations. Several methods have been suggested in the literature for weighting the equation. One may normalize each equation by its second norm so all equations have the same weight in parameter estimation. If an equation associated with the frequency x has values of Hd(x) and Hd ðxÞ of similar magnitudes, the measurement errors may be significantly magnified after the weighting. To overcome this problem, such an equation should be removed [29].

Kwon and Lin [30] state that weighting the sensitivity equation by x 1 decreases inaccuracy of the finite element modeling at higher frequencies [30]. In this study, sensitivity of the FRF (20) is derived using first order derivation of the mode shapes using (9). Inaccuracies in mode-shape changes using first order approximations increase in higher frequency ranges, causing less accurate sensitivity equations. However, the sensitivity of FRFs increases in higher frequency ranges. From a sensitivity point of view, therefore, it is necessary to amplify the sensitivity equation in higher frequency ranges. Due to more approximations at higher frequencies, the weight of these sensitivity equations should be decreased. Numerical simulation shows that sensitivity equation weighting by x 1 creates a balance between the use of higher sensitivity of FRF to change of parameters and the lower accuracy of first order derivation. The following example shows a successful application of the proposed method using a bowstring truss. 3. Bowstring truss example The presented damage detection algorithm was applied to a truss structure as shown in Fig. 1. The structure is modeled numerically using the finite element method with basic structural elements, i.e. bar elements. The finite element model of the structure consists of 25 elements and 21 DOFs. Truss elements are made from steel martial with Young’s modulus of 20 MPa, and cross-sectional areas are given in Table 1. Kinematics degrees of freedom of this truss model are shown in Fig. 2. The unknown parameters are axial rigidity of elements, EA where A is the cross-section area of truss element and E is the Young’s modulus. Several damage cases are considered to investigate the influence of location, severity and number of the damaged elements on the results. Table 2 shows specifications of these

Fig. 1. Geometry of truss model.

Fig. 2. Degrees of freedom of truss model.

Table 1 Cross-sectional area of truss members. Member

Area (cm2)

1–6 7–12 13–17 18–25

18 15 10 12

A. Esfandiari et al. / Computers and Structures 88 (2010) 54–64

Table 2 Percent of stiffness and mass reduction of elements. Case number

Element number and percent of damage

Element number Damage

4 30%(K)

10 50%(K)

– –

Element number Damage

3 40%(K)

9 50%(K)

22 60%(K)

– –

Element number Damage

3 20%(K)

9 30%(K)

20 30%(K)

25 20%(K)

– –

Element number Damage

5 40%(K)

10 40%(K)

13 50%(K)

20 40%(K)

24 30%(K)

Element number Damage

5 20%(M)

11 30%(M)

17 30%(M)

– –

Element number Damage

3 20%(M)

14 30%(M)

19 30%(M)

23 30%(M)

– –

Element number Damage Element number Damage

4 30%(K) 8 30%(M)

8 40%(K) 16 30%(M)

12 30%(K) 19 30%(M)

24 40%(K) – –

– – – –

Element number Damage Element number Damage

4 30%(K) 8 30%(M)

12 30%(K) 16 30%(M)

19 40%(K) 23 30%(M)

– – – –

Element number Damage Element number Damage

4 30%(K) 9 30%(M)

12 30%(K) 20 30%(M)

16 30%(K) 23 30%(M)

25 40%(K) – –

– – – –

-10

Exact nm =15 nm =10

-15

log(FRF)

damage cases. In reality, for the structure shown in Fig. 1, the FRF data should be extracted from an experiment setup. For this research, representative FRF data were simulated numerically using the ﬁnite element method with superimposed measurement errors. Some sources of errors exist in the proposed damage detection algorithm. One is caused by adopted linear approximation to evaluate the change of mode shape in (9). The adverse effects of this approximation can be reduced by an iterative model updating to move from the intact model to damaged model of the structure. Another source of error in the presented method is the incomplete measurements of the natural frequencies of the damaged structure. The proposed method does not work well over all frequency regions; therefore, the selection of excitation frequencies is very important to successful identiﬁcation. Adverse effects can be decreased by a proper selection of the excitation frequencies. As an example, damage case #4 is considered to describe the procedure for selection of the excitation frequency points in detail. Due to practical limitations, it is assumed that only a few of the natural frequencies are measurable, since amplitude of oscillation at higher mode shapes decreases and accurate measurements become impractical. This incompleteness is a source of error in the mathematic of sensitivity equation (20). To investigate this impact, by a numerical simulation the FRF of the truss structure at DOF #10 subjected to the applied load at DOF #15 is plotted in Fig. 3. These FRFs are evaluated using responses at DOFs 10 and 15, and also using all 21 structural mode shapes data. As Fig. 3 shows, the impact of incomplete measurement of natural frequencies on the calculated FRF of the structure depends on excitation frequency of the applied harmonic load. As mentioned by Dos Santos et al. [14], to achieve an accurate evaluation of the FRF of the structure it is necessary that the natural frequency of the included mode shapes be large enough compared to the frequency of excitation. For excitation frequency in low mode shape frequency ranges, the evaluated FRF is very accurate even by a few measured natural frequencies. Acceptable evaluation of the FRF using a truncated set of the measured natural frequencies is possible for excitation frequencies close to the natural frequencies

-20

-25

-30

-35 0

100

150

200

250

300

350

f (Hz) Fig. 3. FRF of the intact structure at 10th degree of freedom.

of the structure even at higher modes frequency range. This is due to the fact that for the excitation frequencies close to the natural frequencies, H(x) is dominated by the nearest resonances. In this paper a set of sensitivity equations is derived to relate the change of structural parameters to the subtraction of the measured response of the structure Hd(x) and the calculated value of e ild ðxÞ, using 10, e d ðxÞ. Fig. 4, illustrates the calculated value of H H 15 and 21 mode shapes along with exact Hd(x) for the 10th DOF e ild ðxÞ is approxsubjected to the harmonic loads at the 15th DOF. H imated in (18) using natural frequencies of the damaged structure and mode shapes of the intact structure. It is very important that the differences between the response e xÞ in (19) be large of the damaged structure Hd(x) in (11) and Hð enough to increase the chance of successful prediction of the damage location and severity. Fig. 4 shows, regardless of the excitation frequency of the applied load, change of the structural response due to damage at lower frequency range is small, and this change may be overshadowed by measurement errors. Note that the main

A. Esfandiari et al. / Computers and Structures 88 (2010) 54–64 -10

, nm=21 , nm=15

-15

, nm=10

log (FRF)

the structural response is more sensitive to the mass changes. An increase of natural frequencies at these ranges indicates that the structural response is dominated by mass reduction. It should be noted that the model updating process diverges for cases 5–9 at a frequency range same as cases 1–4. At these frequency ranges, changes of the FRF are controlled by stiffness and are less sensitive to mass changes. In addition to the error present in mathematical modeling, measurement and data processing errors are also existed in the experiment. To simulate these experimental inaccuracies, measurement error is considered by adding random error to the simulated finite element data. A 5% uniform random error has been added to the exact simulated FRF by the finite element method and natural frequency measurement is considered to be noise-free. Figs. 5–13 show results of the proposed finite element model updating algorithm. Figs. 5–13 show that the proposed method is capable of detecting the damage location, and also quantify the severity using the incomplete noisy measured FRF data. In order to quantify the accuracy of the predicted results, some indices are used to evaluate the confidence level of the results. The Mean Sizing Error defines an average value of the absolute discrepancies between the parameters true damage values dP t and the predicted damage values dPp [31]:

FRF of Damaged Case

~ H(ω) ~ H(ω) ~ H(ω)

-20

-25

-30

-35 0

100

150

200

250

300

350

f (Hz) e ild ðxÞ at 10th degree of freedom. Fig. 4. FRF of the damaged structure and H

e xÞ and the exact FRF of difference between the formulation of Hð the damaged structure Hd(x) is the used mode shapes. This is e d ðxÞ, mode shapes of the intact structures due to the fact that in H are used instead of the mode shapes of the damaged structures as e d ðxÞ indidone in Hd(x). Low discrepancy of value of Hd(x) and H cates that the change of mode shapes of the structure due to damage is small. As mentioned before, when using close value of Hd(x) e d ðxÞ, noise-induced errors in the model updating process beand H come more signiﬁcant and parameter estimation process show less stability and robustness. Therefore, the excitation frequency of applied loads must be moved to a higher frequency range with a sige d ðxÞ. niﬁcant difference between Hd(x) and H e xÞ also occurs Although large relative difference of Hd(x) and Hð at anti-resonance frequencies, these regions should not be selected for model updating. At these regions amplitude of vibration is very low and therefore the response is easily contaminated. Anti-resonance frequencies are also a local characteristic of the structures and may not be sensitive to damage in a complex and large structure. Excitation frequencies were selected at a region close to the natural frequencies of the damaged structure and not exactly at its natural frequencies. Natural frequencies of the damaged cases and frequency ranges of model updating are given in Tables 3 and 4. In these frequency ranges, model updating is applied with frequency increments of 1 Hz. A single harmonic excitation is applied at DOFs 3, 9, 15 and 19 for each load case, and DOFs 1, 8, 14, 15, 20 and 21 are selected as six response measurement locations. Table 4 shows that for damage cases 5–9, where there are some changes in mass matrices, selected frequency ranges for model updating should be adopted at higher ranges, because at these ranges contribution of x2M become large in compare to K and

MSE ¼

ne 1 X jdPt dPp j for 0 MSE 1 ne e¼1

ð22Þ

the relative error,

Pne e¼1 jdP t j e¼1 jdP p j Pne jdP j t e¼1

Pne RE ¼

for 0 RE 1

ð23Þ

and closeness index,

CI ¼ 1

kdPt dPp k kdPt k

for 0 CI 1

ð24Þ

Table 4 Selected frequency ranges for model updating. Damage case

1, 2, 3, 4

5, 6

7, 8, 9

Frequency range

80–90 100–120 185–200 230–245 280–295 – – –

230–245 290–315 335–345 385–400 – – – –

80–90 100–120 185–200 230–245 280–295 325–335 385–390 420–435

Table 3 Natural frequencies of the damaged truss. Mode number

Intact

1 2 3 4 5 6 7 8 9 10 11 12 13

30.3 69.0 96.3 181.8 223.2 275.6 321.6 352.0 357.7 373.0 414.7 460.5 465.7

Damage case 1

29.2 67.4 93.1 178.7 210.2 270.9 320.0 349.5 357.5 369.7 409.2 443.8 463.5

28.1 65.3 93.2 168.9 215.7 252.2 313.9 322.1 342.1 362.1 399.9 438.4 452.5

29.3 67.2 95.7 176.2 215.7 265.4 318.2 340.9 355.1 361.5 410.7 446.0 457.6

28.8 66.9 93.0 168.0 213.6 263.9 310.5 324.9 336.8 361.6 410.7 422.6 429.1

30.7 70.7 98.2 186.0 225.9 281.0 324.5 354.3 362.4 377.9 421.3 465.6 471.1

31.2 70.1 98.6 185.2 228.2 282.1 333.7 358.9 366.9 383.6 418.3 471.8 481.7

29.0 65.5 97.0 178.6 214.3 275.1 321.5 336.7 365.3 375.5 412.3 436.8 470.0

30.2 69.0 96.5 178.2 213.9 272.6 330.5 340.8 360.2 381.2 410.2 452.3 471.1

30.4 69.0 95.9 180.0 212.3 273.3 314.8 347.1 365.2 376.6 403.1 453.0 473.3

A. Esfandiari et al. / Computers and Structures 88 (2010) 54–64 0.6

0.6

Actual Damage

Predicted Damage

Damage Percent

Predicted Damage

0.5

0.4

0.3

0.2

0.4

0.3

0.2

0.1

0 1

-0.1

Element No.

Fig. 5. Actual and predicted damage of case 1 using noisy data.

Fig. 8. Actual and predicted damage of case 4 using noisy data.

0.7

0.4 Actual Damage Predicted Damage

0.6

Actual Damage Predicted Damage

0.3

Damage Percent

0.5 0.4 0.3 0.2

0.2

0.1

0.1 0 0

1 1

-0.1

Element No.

-0.1

Fig. 6. Actual and predicted damage of case 2 using noisy data.

Fig. 9. Actual and predicted damage of case 5 using noisy data.

0.4

0.4 Actual Damage

Actual Damage

Predicted Damage

0.3

Damage Percent

0.2

0.1

0.2

0.1

0 1

-0.1

Element No.

-0.1

Element No.

Fig. 7. Actual and predicted damage of case 3 using noisy data.

Fig. 10. Actual and predicted damage of case 6 using noisy data.

gives an index of the distance between the true and estimated damage parameter vectors. An element is identiﬁed as damaged if jdP p j 2 MSE [31]. The smaller values of the MSE and RE and larger values of the CI indicate better results. The damage indices of predicted results are given in Table 5.

As Figs. 5–13 and Table 5 show, the proposed ﬁnite element model updating method is capable of localization and quantiﬁcation of damage for all elements. Additionally, a few elements appeared as false positive slight severity of damage. This is due to the presence of noise in the FRF data, and sometimes to selected

A. Esfandiari et al. / Computers and Structures 88 (2010) 54–64

excitation frequency. Comparison of predicted damages shows the stability and success of detection despite increasing severity of damage. Increasing the severity of damage and difference of FRF responses of the intact and damaged elements smeared undesirable affects of noisy measurements.

(a) 0.5

Averages of the estimated parameters do not reflect the robustness and confidence of the parameters estimation process. To investigate robustness of a method, it is necessary to evaluate the standard deviation and/or coefficient of variation (COV) of the predicted unknown parameters in the Monte Carlo simulations. Low

(b) 0.4

Actual Damage

Predicted Damage

0.4

Damage Percent

0.3 Damage Percent

0.3

0.2

0.1

0.2

0.1

0 1

-0.1 -0.1

Element No.

Fig. 11. Actual and predicted damage of case 7 using noisy data: (a) mass; (b) stiffness.

(a) 0.5

(b) 0.4

Actual Damage

Predicted Damage

0.4

Damage Percent

0.3 0.3

0.2

0.1

0.2

0.1

0 1

-0.1

Element No.

-0.1

Fig. 12. Actual and predicted damage of case 8 using noisy data: (a) mass; (b) stiffness.

(a) 0.5

(b)

Actual Damage

0.4 Actual Damage

Predicted Damage

0.4

Damage Percent

0.3 0.3

0.2

0.1

0.2

0.1

0 1

-0.1

Element No.

-0.1

Element No.

Fig. 13. Actual and predicted damage of case 9 using noisy data: (a) mass; (b) stiffness.

A. Esfandiari et al. / Computers and Structures 88 (2010) 54–64

standard deviation and COV indicate less scatter in the predicted parameters. As a template, COVs of the estimated unknown parameters for the seventh damage scenario are plotted in Fig. 14. Except for element number 13, COVs of all predicted stiffness parameters of elements are low and indicate a robust solution. High COV of element 13 can be interpreted as a low observability of this element compared to other elements. This issue can be resolved by rearranging excitation and response measurement stations. Accuracy and confidence in the predicted parameters heavily depends on the ratio of measurement error to the observed change of the monitored structural response. Monitored structural response can be natural frequencies, mode shapes, or FRFs. For a low change of response which can be related to the adopted structural response for monitoring, or level of damage, model updating even using low error contaminated measurement can yield a low confidence and high scattered prediction. As stated by Ren and Beards [29], weighting of equations related to the measurement points with a low level of change in response can adversely affect predicted results. At such cases using higher frequency ranges can improve the confidence level in parameter estimation, since; generally higher frequency ranges and mode shapes are more localized and might be able to observe small change. To investigate low level damage scenarios a 5% random stiffness variation is added to all truss members as a deterioration case and also a 10% random stiffness variation of the bottom chord (elements 7–12) is considered as another deterioration case. Model updating is done at frequency set 1 given by Table 6. The predicted parameters and their COV are plotted in Figs. 15 and 16, respectively. Figs. 15 and 16 show that the proposed method is generally capable of detection of light distributed damage cases, although some elements are detected by low accuracy. Elements 13 and 7 have high coefficients of variations and show false stiffness changes. This might be due to low observability of these elements using the frequency set 1 in Table 6. Parameters estimation results can be improved by adopting model updating frequency points at higher frequency ranges. For verification, the above deterioration cases are redone using higher frequency ranges as given by set 2 of Table 6. The estimated parameters and COVs are plotted in Figs. 17 and 18. Fig. 17 shows that overall 5% random stiffness variations are well captured with COVs well below 5%. Fig. 18 shows that 10% random stiffness variations are well captured with COVs well below 15%. Both cases show that measurement set 2 can better observe the overall changes in the stiffness. Better parameters estimation and lower COV of the predicted parameters prove that

Table 5 Comparison of damage indexes. Case number

Index MSE

1 2 3 4 5 6 7 8 9

0.009 0.011 0.016 0.040 0.003 0.005 0.012 0.016 0.018

0.220 0.169 0.310 0.381 0.066 0.070 0.456 0.302 0.397

0.85 0.89 0.81 0.77 0.96 0.94 0.86 0.79 0.80

Table 6 Frequency ranges for stiffness deterioration case.

Frequency ranges

Set 1

Set 2

80–93 100–120 165–177 185–200 210–220 228–235 – –

80–93 100–120 165–177 185–200 210–220 228–235 305–317 325–335

COV

0.2

0.1

0 1

Element No. Fig. 14. Coefﬁcient of variations of the estimated parameters at case 7.

(a) 0.075

(b) 0.25

Actual Damage

Damage Percent

Predicted Damage

0.05

0.025

0 1

0.2 0.15 0.1 0.05

25 0 1

-0.025

Element No.

Fig. 15. Actual and predicted stiffness for 5% random deterioration of truss using frequency set 1: (a) stiffness estimation; (b) coefﬁcient of variations of estimated parameters.

A. Esfandiari et al. / Computers and Structures 88 (2010) 54–64

(a)

0.1

(b) 0.35

Actual Damage Predicted Damage

Damage Percent

0 1

-0.05

Damage Percent

0.3

0.05

0.25 0.2 0.15 0.1 0.05 0

-0.1

15 17 19 21 23 25

Element No.

Fig. 16. Actual and predicted stiffness for 10% random deterioration of bottom chord using frequency set 1: (a) stiffness estimation; (b) coefﬁcient of variations of estimated parameters.

Actual Damage Predicted Damage

0.075

Damage Percent

(b) 0.05

0.1

0.05 0.025 0

-0.025

Damage Percent

(a)

-0.05 -0.075

0 1

-0.1

Element No.

Fig. 17. Actual and predicted stiffness for 5% random deterioration of truss using frequency set 2: (a) stiffness estimation; (b) coefﬁcient of variations of estimated parameters.

(a)

0.1

(b) 0.15

Actual Damage

0.05

0 1

-0.05

Damage Percent

Predicted Damage

0.1

0.05

0 1

-0.1

Element No.

Fig. 18. Actual and predicted stiffness for 10% random deterioration of bottom chord using frequency set 2: (a) stiffness estimation; (b) coefﬁcient of variations of estimated parameters.

using higher frequency ranges can improve the quality of predictions. These results and the low number of iteration illustrates that this is a highly robust method for parameter estimation. Although in most real cases, the mass matrices of the structures are not usually changed by damage, some deviation in stiffness identiﬁcation is possible due to an inaccurate assumption regarding the mass of the intact and damaged elements. This inaccuracy introduces some errors in eigenvector of the undamaged structure

that will be used to construct sensitivity equations. In this study, numerical simulation proves that, considering up to a 10% random error in mass matrices, the parameter estimation process is still robust. Results of such a case of stiffness parameters estimation are given in Fig. 19 and the coefﬁcient of variation of the predicted parameters is given in Fig. 20. Low COVs of the estimated parameters indicates stability and robustness of the method against mass modeling errors.

A. Esfandiari et al. / Computers and Structures 88 (2010) 54–64

References

0.6 Actual Damage Predicted Damage

0.5

Damage Percent

0.4

0.3

0.2

0.1

0 1

-0.1

Element No. Fig. 19. Actual and predicted damage considering mass modeling errors.

C.O.V

0.1

0 1

Element No. Fig. 20. Coefﬁcient of variations of the estimated parameters considering mass modeling errors.

4. Conclusions A structural damage detection method is presented using Frequency Response Function (FRF) and measured natural frequencies. In a decomposed form, change of the FRF of the structure due to damage is evaluated by measured natural frequencies and derivation of mode shapes with respect to mass and stiffness matrices. The change of mode shapes is expressed as a linear combination of original eigenvectors of the intact structures. Elemental level sensitivity of the FRF of the structure to occurrence of damage is characterized as a function of stiffness, mass, and damping parameters change. The sensitivity is weighted by x 1 to improve the stability of the method and obtain more conﬁdence results. Sensitivity equations are solved by the Least Square method to achieve change of structural parameters. Results of truss model show the ability of this method to identify location and severity of parameters change at the elemental level in a structure. It was also shown that, the ﬁnite element parameter estimation results are improved using higher excitation frequencies.

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Upper and lower bound limit analysis of plates using FEM and second-order cone programming Canh V. Le a,*, H. Nguyen-Xuan b, H. Nguyen-Dang c a

Department of Civil and Structural Engineering, University of Shefﬁeld, Shefﬁeld S1 3JD, United Kingdom Department of Mechanics, Faculty of Mathematics and Computer Science, University of Science, VNU-HCM, 227 Nguyen Van Cu, District 5, Ho Chi Minh City, Viet Nam c LTAS, Division of Fracture Mechanics, University of Liège, Bâtiment B52/3 Chemin des Chevreuils 1, B-4000 Liège 1, Belgium b

a r t i c l e

i n f o

Article history: Received 17 March 2009 Accepted 7 August 2009 Available online 12 September 2009 Keywords: Limit analysis Upper and lower bounds Displacement and equilibrium models Criterion of mean Second order cone programming

a b s t r a c t This paper presents two novel numerical procedures to determine upper and lower bounds on the actual collapse load multiplier for plates in bending. The conforming Hsieh–Clough–Tocher (HCT) and enhanced Morley (EM) elements are used to discrete the problem fields. A Morley element with enhanced moment fields is used. The constant moment fields is added a quadratic mode in which the pressure is equilibrated by corner loads only, ensuring that exact equilibrium relations associated with a uniform pressure can be obtained. Once the displacement or moment fields are approximated and the bound theorems applied, limit analysis becomes a problem of optimization. In this paper, the optimization problems are formulated in the form of a standard second-order cone programming which can be solved using highly efficient interior point solvers. The procedures are tested by applying it to several benchmark plate problems and are found good agreement between the present upper and lower bound solutions and results in the literature. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction The yield line theory has been proved to be an effective method to perform plastic analysis of slabs and plates [1,2]. This wellknown method can predict very good upper-bound of the actual collapse multiplier for many practical engineering problems. However, this hand-based analysis method encounters difficulties in problems of arbitrary geometry, especially in the problems involving columns or holes. Consequently, over last few decades various numerical approaches based on bound theorems and mathematical programming have been developed [3–10]. Numerical limit analysis generally involves two steps: (i) numerical discretization; and (ii) mathematical programming to enable a solution to be obtained. The finite element method, which is one of the most popular numerical methods, is often employed to discrete velocity or stress fields. Of several displacement and equilibrium elements that have been developed for Krichhoff plates in bending, the conforming Hsieh–Clough–Tocher (HCT) [11] and equilibrium Morley elements [12] are commonly utilized in practical engineering. The original HCT element will be used in the paper without any modification while the Morley element will be modified by adding a complementary field. Once the stress or displacement fields are * Corresponding author. E-mail addresses: Canh.Le@sheffield.ac.uk (C.V. Le), nxhung@hcmuns.edu.vn (H. Nguyen-Xuan), H.NguyenDang@ulg.ac.be (H. Nguyen-Dang). 0045-7949/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2009.08.011

approximated and the bound theorems applied, limit analysis becomes a problem of optimization involving either linear or nonlinear programming. Problems involving piecewise linear yield functions or nonlinear yield functions can respectively be solved using linear or non-linear programming techniques [13,14,5,15,16]. However, difficulty exists in the upper-bound optimization problem is that the objective function is convex, but not everywhere differentiable. One of the most efficient algorithms to overcome this singularity is the primal-dual interior-point method presented in [17,18] and implemented in commercial codes such as the Mosek software package [19], such as second-order cone programming. The algorithm is also suitable for solving lowerbound limit analysis since most of yield conditions can be cast as a conic constraint [20]. These limit analysis problems can then be solved by this efficient algorithm [21–23]. In this paper two numerical procedures for upper and lower bound limit analysis of rigid-perfectly plastic plates governed by the von Mises criterion are proposed. A second degree moment field proposed by Debongnie and Nguyen-Xuan [24–26] is added to Morley moment fields to achieve exact equilibrium relations when applying a uniform pressure to plates. The enhanced Morley (EM) element will be adopted in the lower-bound limit analysis of plate problems. Attention is also focused on treating the performance of yield condition in numerical limit analysis. The criterion of mean proposed in [27] will be used instead of the exact criterion which is required to strictly satisfy. Due to this weakness of the yield

C.V. Le et al. / Computers and Structures 88 (2010) 65â&#x20AC;&#x201C;73

condition we expect to obtain only an approximation of lowerbound in the statically admissible limit analysis. Attempts are also made by formulating both upper and lower bound limit analysis problems in terms of a standard second-order cone programming (SOCP). To illustrate the method it is then applied to a series of plate bending problems, including those for which solutions already exist in the literature. 2. Limit analysis formulations 2.1. Limit analysis duality theorems Consider a rigid-perfectly plastic body of volume X 2 R3 with boundary C. Let Cu and Cg denote, respectively, an essential boundary (Dirichlet condition) where displacement boundary conditions are prescribed and a natural boundary (Neumann condition) where stress boundary conditions are assumed, Cu [ Cg Âź C. The external loads which are denoted by g and f, respectively subject to surface and volume of the body. Let u_ be a plastic velocity or ďŹ&#x201A;ow ďŹ eld that belongs to a space Y of kinematically admissible velocity ďŹ elds and r be a stress ďŹ eld belonging to an appropriate space of symmetric stress tensor X. The mathematical formulations for limit analysis will be brieďŹ&#x201A;y described in this section. More details can be found in [28,22,23]. The external work rate of forces Ă°g; f Ă&#x17E; associated with a virtual plastic ďŹ&#x201A;ow u_ is expressed in the linear form as

_ Âź FĂ°uĂ&#x17E;

f u_ dX Ăž

g u_ dC:

Ă°1Ă&#x17E;

The internal work rate for sufďŹ ciently smooth stresses r and velocity ďŹ elds u_ is given by the bilinear form

_ Âź aĂ°r; uĂ&#x17E;

_ dX; _ uĂ&#x17E; rT Ă°

Ă°2Ă&#x17E;

_ are strain rates. _ uĂ&#x17E; where Ă° The equilibrium equation is then described in the form of virtual work rate as follows:

_ Âź FĂ°uĂ&#x17E;; _ aĂ°r; uĂ&#x17E;

8u_ 2 Y and u_ Âź 0 on Cu :

Ă°3Ă&#x17E;

both the maximum of all lower bounds k and the minimum of all upper bounds kĂž coincide and are equal to the exact value kexact . 2.2. Formulations for plates Considers a plate bounded by a curve enclosing a plane area A with kinematical boundary Cw [ Cwn and static boundary Cm [ Cmn , where the subscript n stands for outward normal. The general relations for limit analysis of thin plates associated with Kirchhoffâ&#x20AC;&#x2122;s hypothesis are given as follows. Equilibrium: Collecting the bending moments in the vector mT Âź Â˝mxx myy mxy , the equilibrium equations can be written as

Ă°r2 Ă&#x17E;T m Ăž kp Âź 0;

where p is the transverse load and the differential operator r is deh 2 iT @ @2 @2 2 @x@y . ďŹ ned by r2 Âź @x 2 @y2 Compatibility: If w denotes the transverse displacement, the curvature rates can be expressed by relations

j_ Âź Â˝j_ xx

_ j_ yy 2j_ xy T Âź r2 w:

wĂ°mĂ&#x17E; Âź

pďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192; mT Pm mp 6 0;

2 PÂź

1 0

16 4 1 2 0

2 0

7 0 5: 6

_ Âź DĂ°jĂ&#x17E;

Z Z

t=2

pďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;

r0 _ T Q _ dz dA Âź mp

t=2

Q ÂźP

_ Âź kFĂ°uĂ&#x17E;; _ 8u_ 2 Yg kexact Âź maxfkj9r 2 BaĂ°r; uĂ&#x17E; _ Âź max min aĂ°r; uĂ&#x17E;

3. Finite element discretization

Ă°7Ă&#x17E;

_ Âź min DĂ°uĂ&#x17E;;

Ă°8Ă&#x17E;

_ u2C

r2B

2 3 4 2 0 16 7 Âź 4 2 4 0 5: 3 0 0 1

Ă°14Ă&#x17E;

Ă°15Ă&#x17E;

Details on the derivation of the dissipation for plate problems can be found in [6,29].

Ă°6Ă&#x17E;

_ Âź min max aĂ°r; uĂ&#x17E; _ u2C

Ă°13Ă&#x17E;

_ xx 6 7 _ _ Âź 4 _ yy 5 Âź zj;

where sij denotes stress deviator tensor and k is a parameter depending on material properties. _ Âź 1g, the exact collapse multiplier If deďŹ ning C Âź fu_ 2 YjFĂ°uĂ&#x17E; kexact can be determined by solving any of the following optimization problems

_ u2C

Z pďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192; j_ T Q j_ dA;

where

r2B

Ă°12Ă&#x17E;

The dissipation rate: The internal dissipation power of the twodimensional plate domain A can be written as a function of curvature rates as

c_ xy

Ă°5Ă&#x17E;

Ă°11Ă&#x17E;

where mp Âź r0 t 2 =4 is the plastic moment of resistance per unit width of a plate of uniform thickness t; r0 is the yield stress and

Ă°4Ă&#x17E;

B Âź fr 2 Xjsij sij 6 2k g;

Ă°10Ă&#x17E;

Flow rule and yield condition: In framework of a limit analysis problem, only plastic strains (curvatures) are considered and are as@w , where the plastic sumed to obey the normality rule j_ Âź l_ @m multiplier l_ is non-negative and the yield function wĂ°mĂ&#x17E; is convex. In this study, the von Mises failure criterion in the space of moment components is used

Furthermore, the stresses r must satisfy the yield condition for assumed material. This stress ďŹ eld belongs to a convex set, B, obtaining from the used ďŹ eld condition. For the von Mises criterion,

Ă°9Ă&#x17E; 2

_ Âź maxr2B aĂ°r; uĂ&#x17E; _ is the plastic dissipation rate. Problems where DĂ°uĂ&#x17E; (5) and (8) are knows as static and kinematic principles of limit analysis, respectively. The limit load of both approaches converges to the exact solution. Herein, a saddle point Ă°r ; u_ Ă&#x17E; exists such that

3.1. Lower-bound formulation In numerical lower-bound limit analysis problem, a statically admissible stress or moment ďŹ eld for an individual element is chosen so that equilibrium equations and stress continuity requirements within the element and along its boundaries are met. The well-known equilibrium Morley element with constant varying moment is the simplest model for practical engineering. It is, therefore, advantage to extent the use of the element to lower-

C.V. Le et al. / Computers and Structures 88 (2010) 65–73

bound limit analysis problem in this paper. The moment ﬁeld m is assumed to vary constantly within an element and expressed as

m ¼ Ib;

ð16Þ T

where I is an identity matrix and b ¼ ½b1 b2 b3 is an unknown vector. The generalized loads comprise three corner loads Z 1 ; Z 2 ; Z 3 and three normal moments bending along edges m12 ; m23 ; m31 as shown in Fig. 1. All generalized loads can be expressed in terms of moment parameters, if G denotes the generalized vector, the relations are written as

G ¼ Cb;

ð17Þ

where

G ¼ ½ Z1 Z2 Z3 2 c 3 s3 c 1 s1 6c s c s 2 2 6 1 1 6 6 c 2 s2 c 3 s3 C¼6 6 c2 L 6 1 12 6 4 c22 L23 c23 L32

m12

m23

c1 s1 c3 s3 c2 s2 c1 s1 c3 s3 c2 s2 s21 L12 s22 L23 s23 L32

m31 T ; c21 s21 c23 þ s23

c22 s22 c21 þ s21 7 7 7 c23 s23 c22 þ s22 7 7 7 c1 s1 L12 7 7 5 c2 s2 L23

ð18Þ

ð19Þ

c3 s3 L32

in which the direction cosines of the outward normal to the element boundary ðci ; si Þ are determined as

ci ¼

yj yi ; Lij

si ¼

x i xj ; Lij

ij ¼ 12; 23; 31

ð20Þ

and Lij is the length of edge ij. It is important to note that, in the case when a uniform pressure is applied, the Morley element does not result in an exact equilibrium relation. This is because Eq. (9) does not hold with the use of the constant moment fields. It is, therefore, necessary to add to the constant moment fields by a particular higher degree solution which has to be such chosen such that side loads are compatible with the original element. A second degree moment field which can be added to equilibrium elements of either degree one or degree zero has been proposed by [24–26] and can be expressed as

mc ¼ kpae T;

ð21Þ

where ae is the area of an element and T ¼ ½T xx ven as

T yy

T xy and is gi-

3 3 X 2 Þ 2 k2 X 3 ðX k3 þ 2a1e X 2 X 22 k2 X 23 k3 XY 33 k1 þ X 3Y X X Y 3 3 2 7 6 7 16 7: T¼ 6 YX 32 k3 þ 2a1e Y 2 Y 23 k3 7 6 34 5 2X 3 X 2 1 1 1 2 k1 þ 2 k2 2X 2 k3 þ 2ae ðXY X 3 Y 3 k3 Þ 2

ð22Þ This complementary mode is constructed based on a particular system of axes as shown in Fig. 2, in which the side 1–2 is chosen to be the X axis and Y must go through node 1 and is orientated so that Y 3 is positive. Three area coordinates are denoted by k1 ðX; YÞ; k2 ðX; YÞ and k3 ðX; YÞ. The modiﬁed Morley element was called as enhanced Morley (EM) element by [26]. Similarly, the three generalized loads at corners of the triangular element are added by a3e p. The equilibrium equation Eq. (17) is then rewritten as

G ¼ Cb;

ð23Þ

where

¼ ½ b1 b2 b3 k ; b 2 c 3 s3 c 1 s1 c 1 s1 c 3 s3 6c s c s c s c s 2 2 2 2 1 1 6 1 1 6 6 c 2 s2 c 3 s3 c 3 s3 c 2 s2 C¼6 6 c2 L s21 L12 6 1 12 6 2 4 c2 L23 s22 L23 c23 L32

s23 L32

c21 s21 c23 þ s23

pa3e

c22 s22 c21 þ s21 c23 s23 c22 þ s22

pa3e pa3e

c2 s2 L23

7 7 7 7 7: 0 7 7 7 0 5

c3 s3 L32

c1 s1 L12

ð24Þ

The overall equilibrium for the structure can be obtained by assembling all local equilibrium equations of elements and expressed as Fig. 1. Morley equilibrium element.

Cs bs ¼ 0

Fig. 2. Relations between global system (Oxy) and local system (OXY).

ð25Þ

C.V. Le et al. / Computers and Structures 88 (2010) 65–73

with bs ¼ ½b1 b2 b3 nele k ; nele is the number of elements. Notes that boundary conditions are also imposed here in the assemble scheme. is not allowed to Furthermore, the modiﬁed moment ﬁeld m violate the yield condition

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mp 6 0; T Pm m

¼ wðmÞ

ð26Þ

where

ð27Þ

However, in numerical analysis it is not always possible to satisfy this requirement since the yield condition is commonly fulﬁlled at Gauss points or nodes. Instead of strictly satisfying the exact criterion, Nguyen–Dang proposed the criterion of mean [27,30] which is satisﬁed locally within element domains. For plate problem the criterion of mean can be expressed as

Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 da mp 6 0: T Pm m ae ae

ð28Þ

the Eq. (28) can be rewritten Introducing the smoothed value of m as

pffiffiffiffiffiffiffiffiffiffiffi wðqÞ ¼ qT Pq mp 6 0;

ð29Þ

and given by where q is the smoothed version of m

1 ae

ðkÞ wðkÞ ðfÞ ¼ NðkÞ e ðfÞ þ N0 ðfÞF qe ;

k ¼ 1; 2; 3;

ð32Þ

ðkÞ

¼ b þ kpae T: m

q¼

the transverse displacements and 2 the rotation components at each corner node ðwi ; hxi ¼ @wi =@xji ; hyi ¼ @wi =@yji ; i ¼ 1; 2; 3Þ and normal rotations at three mid-side nodes ðhi ¼ @wi =@nji ; i ¼ 4; 5; 6Þ. The displacement expansion wðkÞ can be expressed in terms of area coordinates f ¼ ðf1 ; f2 ; f3 Þ over each sub-triangle as

Z ae

da ¼ b þ kp m

T da ¼ b þ kpS

ð30Þ

in which S is the exact integration of ae T da in the local coordinate OXY. If deﬁning Bi ¼ fqi jwðqi Þ 6 0g is the set of admissible discrete moments for each element, the lower-bound limit analysis (5) can be now written in terms of discrete moment space as

k ¼ max s:t

k 8 > < Cs bs ¼ 0; qi ¼ bi þ kpSi ; > : qi 2 Bi ; i ¼ 1; 2; . . . ; nele

where the partitions NðkÞ e ðfÞ and N0 ðfÞ, respectively, represent the interpolation functions associated with element displacements qe and internal nodal displacements and F is the matrix of elimination obtained by applying compatible requirements at internal nodes 7, 8, 9. The plastic dissipation for a sub-element is now formulated as

DðkÞ ðjðkÞ Þ ¼ mp

ng qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X pffiffiffiffiffiffiffiffiffiffiffiffi _ j Þ; nj j_ T ðfj ÞQ jðf j_ T Q j_ dA ¼ mp

Ase

ð33Þ

j¼1

where ng ¼ 3 is the number of Gauss integration points in each subelement AðkÞ ; nj is the weighting factor of the Gauss point fj and jðkÞ ðfj Þ are curvatures at the Gauss point fj

3 3 2 ðkÞ ðkÞ Ne;xx ðfj Þ þ N0;xx ðfj ÞF j_ ðkÞ xx ðfj Þ 6 7 6 7 ðkÞ ðkÞ 7 7 6 ðkÞ j_ ðkÞ ðfj Þ ¼ 6 4 j_ yy ðfj Þ 5 ¼ 4 Ne;yy ðfj Þ þ N0;yy ðfj ÞF 5 q_ e : ðkÞ ðkÞ ðkÞ j_ xy ðfj Þ Ne;xy ðfj Þ þ N0;xy ðfj ÞF 2

ð34Þ

By summing all dissipations of all sub-elements and elements, the plastic dissipation of the whole plate is

D ¼ mp

ng qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nele X 3 X X _ j Þ: nj j_ T ðfj ÞQ jðf

ð35Þ

j¼1

Similarly, the work rate of applied loads can be expressed as

F¼

ng nele X 3 X X

_ ðkÞ ðfj Þ: nj pw

ð36Þ

j¼1

ð31Þ

The upper-bound limit analysis of plate bending is now written as

kþ ¼ min

and accompanied by appropriate boundary conditions.

ng qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nele X 3 X X _ jÞ nj j_ T ðfj ÞQ jðf j¼1

3.2. Upper-bound formulation

s:t In numerical upper-bound limit analysis of plate problem, the velocity field with an element is represented by a continuous function expressed in terms of spatial coordinates and nodal values. For Krichhoff plates, an element of class C 1 should be employed to approximate the velocity field. The conforming Hsieh–Clough–Tocher (HCT) triangular element will be utilized and briefly summarized in this section. A triangular element is subdivided into three sub-elements using individual cubic expansions over each sub-element as shown in Fig. 3. The element has 12 degrees of freedom:

8 nele 3 ng > < P P P n pw ðkÞ ðfj Þ ¼ 1 j _ j¼1 > : q_ ¼ 0 on Cw :

ð37Þ

4. Second-order cone programming 4.1. Conic programming The general form of a Second-Order Cone Programming (SOCP) problem with N sets of constraints is written as follows:

Fig. 3. HCT element.

C.V. Le et al. / Computers and Structures 88 (2010) 65–73 T

min

f x

s: t:

kHi x þ v i k 6

4.2. Lower-bound programming

yTi x

þ zi

for i ¼ 1; . . . ; N;

ð38Þ

where x 2 Rn are the optimization variables, and the problem coefﬁcients are f 2 Rn ; Hi 2 Rm n ; v i 2 Rm ; yi 2 Rn , and zi 2 R. For optimization problems in 2D or 3D Euclidean space, m ¼ 2 or m ¼ 3. When m ¼ 1 the SOCP problem reduces to a linear programming problem. In framework of limit analysis problems, the two most common second-order cones are the quadratic cone

vffiffiffiffiffiffiffiffiffiffiffiffiffi 8 9 u kþ1 < = X u Cq ¼ x 2 Rkþ1 jx1 P t x2j ¼ kx2!kþ1 k : ; j¼2

ð39Þ

( x 2 Rkþ2 jx1 x2 P

kþ2 X

n o Cq ¼ q 2 R4 jq4 P kJT1 q1!3 k; q4 ¼ mp ;

q 2 Cq ;

ð41Þ

where J1 is the so-called Cholesky factor of P

3 0 0 p ffiffiffi 16 7 J1 ¼ 4 1 3 0 5: pffiffiffi 2 0 0 2 3 2

ð42Þ

The lower-bound limit analysis of plates is then cast in the form of a second-order cone programming as

and the rotated quadratic cone

Cr ¼

Since the matrix P is a positive deﬁnite matrix, the constraint (29) can be cast in terms of a conic quadratic constraint as

)

k ¼ max

x2j ¼ kx3!kþ2 k2 ; x1 ; x2 P 0 :

ð40Þ

j¼3

s:t

k 8 > < Cs bs ¼ 0; qi ¼ bi þ kpSi ; > : q 2 Ci ; i ¼ 1; 2; . . . ; nele i

ð43Þ

and accompanied by appropriate boundary conditions. 4.3. Upper-bound programming In order to cast the optimization problem (37) in the form of a standard second-order cone programming, its objective function is ﬁrstly formulated in a form involving a sum of norms as

ng ng qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nele X 3 X nele X 3 X X X _ j Þ ¼ mp _ j Þk; nj j_ T ðfj ÞQ jðf nj kJT2 jðf j¼1

ð44Þ

j¼1

where J2 is the Cholesky factor of Q

2 3 2 0 0 pffiffiffi 1 6 7 J2 ¼ pffiffiffi 4 1 3 0 5: 3 0 0 1

Fig. 4. Square plate clamped along edges and loaded by a uniformly pressure.

ð45Þ

By introducing auxiliary variables t 1 ; t 2 ; . . . ; tnele 3 ng the present upper-bound optimization problem can be rewritten in the form of a standard SOCP problem as

60 SQP Matlab SOCP Mosek

46 0

28 0

100 150 200 number of elements

250

300

SQP Matlab SOCP Mosek 50

100 150 200 number of elements

Fig. 5. Comparison the performance of SQP and SOCP.

250

300

C.V. Le et al. / Computers and Structures 88 (2010) 65–73

26.5

λ+ (HCT element)

λ+ (HCT element) −

λ (EM element) The average value

−

λ (EM element) The average value

26 25.5

25 * **

45 * ** ***

24.5

(*) Upper−bound in [6] (**) Mixed approach [5] (***) Lower−bound in [3]

40 35

(*) Upper−bound in [6]

(**) Lower−bound in [3] 23.5

30 25

23 0

500

1000

1500 2000 2500 3000 number of elements

3500

500

1000

4000

1500 2000 2500 3000 number of elements

3500

4000

Fig. 7. Bounds on the collapse multiplier vs. number of elements using SOCP. Fig. 6. Bounds on the collapse multiplier vs. number of elements using SOCP.

k ¼ min

nele 3 ng X

ables of this optimization problem is sdof þ 4 3 ng nele; sdof is the degrees of freedom of system.

nk t k

s:t

8 ng nele 3 P PP > > _ ðkÞ ðfj Þ ¼ 1; > nj p w > > > j¼1 < q_ ¼ 0 on Cw ; > > > > _ ri ¼ JT2 j; > > : kri k 6 ti ; i ¼ 1; 2; . . . ; nele 3 ng

5. Numerical examples

ð46Þ

in which kri k 6 t i expresses quadratic cones and ri are additional variables, where every ri is a 3 1 vector. The total number of vari-

The numerical performance of the procedures are illustrated by applying it to uniformly loaded plate problems for which, in most cases, solutions already exist in the literature (the method is applicable to problems of arbitrary geometry). For all the examples considered the following was assumed: length L ¼ 10 m; plate thickness t ¼ 0:1 m; yield stress r0 ¼ 250 MPa. Quarter symmetry was assumed when appropriate. Note that, solutions obtained in

Fig. 8. Mesh reﬁnements for a quarter of the circular plate.

C.V. Le et al. / Computers and Structures 88 (2010) 65–73

free edge

λ (HCT element) λ− (EM element) The exact solution

L/2

simply supported 12

L/2 10

9 0

500

1000

1500 2000 2500 3000 number of elements

3500

4000

L/2

Fig. 9. Bounds on the collapse multiplier vs. number of elements using SOCP (circular plate).

Fig. 10. L-shaped geometry.

the static problems are approximations of lower-bound due to criterion of the mean was used. However, as the discretization is sufficiently fine, increasingly close approximations of the true plastic collapse load multiplier can be expected to be obtained. The first examples is a square plate with clamped supports and subjected to uniform out-of-plane pressure loading. This problem was solved by the top-right quarter of the plate and uniform mesh generation was used, see Fig. 4. Matlab optimization toolbox 3.0 and Mosek version 5.0 optimization solvers were used to obtain solutions (using a 2.8 GHz Pentium 4 PC running Microsoft XP).

The efﬁcacy of various optimization algorithms was ﬁrstly considered. The limit analysis problems (31) and (37) are typically non-linear optimization problems and it can be solved using a general non-linear optimization solver, such as a sequential quadratic programming (SQP) algorithm (which is generalization of Newton’s method for unconstrained optimization) [31]. Fig. 5 shows that solutions obtained using SQP and SOCP algorithms are in very good agreement. However, the SOCP algorithm produced solutions very much more quickly and somewhat more accurate, despite the fact that the number of variables involved was much greater

Fig. 11. Mesh reﬁnement for L-shape plate.

C.V. Le et al. / Computers and Structures 88 (2010) 65–73

(sdof þ 4 3 ng nele cf. sdof when using SQP). To compute solutions for a mesh of 288 elements, the SOCP algorithm typically took only 5–30 s, compared with 1280–7000 s when using SQP. Moreover, the SOCP algorithm is able to solve problems up to 152,148 of variables with less than 400 s CPU time (for the mesh of 4050 elements). It is also important to note that the SOCP algorithm can be guaranteed to identify globally optimal solutions, whereas SQP cannot. The performance of the presented numerical limit analysis procedures is further investigated in convergence analysis as shown in Fig. 6. It can be observed that both upper and lower bounds converge to the actual collapse multiplier when the size of elements tends to zero. A upper-bound of 45.12 was achieved by present method, which is slightly smaller than the solution previously obtained in [6]. In comparison with previously obtained lower-bound solution, the present method provides higher solutions than in [3] where quadratic moment fields were used, by 0.6%. The next example comprises a square plate with simply supported on all edges. Convergence analysis of collapse load multipliers is shown in Fig. 7. It can be seen from the figure that the upperbound converges to the actual collapse multiplier when relatively small number of elements was used; and the gap between upper and lower bound is considerably smaller than the clamped case. This may be explained by the fact that the displacement filed in this problem does not exhibit a singularity in the form of a so-called hinge along boundary. The solutions obtained by the proposed method are in good agreement with previously achieved bounds. Considering previously obtained upper-bound solutions, the present method provides lower solutions than in [3,6], by 6.16% and 0.01%, respectively. Furthermore, a computed lowerbound of 24.93 was found, which is 0.3% higher than the best lower-bound found in [3] where quadratic moment fields were used. In the two examples examined above, the computed upperbounds are slightly higher than solution in [29] where the Element-Free Galerkin method was used to approximate the displacement filed. However, the presented method can provide very tight lower-bound solutions and based on the computed bounds the actual collapse multiplier can be estimated, e.g. taking the mean value of the obtained upper and lower bounds. For these examples, the computed mean values are in excellent agreement with solutions in [5]. Further illustration of the method can be made by examining a clamped circular plate, for which the exact solution exists [32], m k ¼ 12:5 R2p where R is the radius. Mesh refinements for a quarter of the plate are shown in Fig. 8.

7.2 λ+ (HCT element)

−

λ (EM element) The average value

6.8 6.6 6.4 6.2 6 5.8 5.6 5.4 5.2 0

1000

2000 3000 number of elements

4000

5000

Fig. 12. Bounds on the collapse multiplier vs. number of elements using SOCP (L-shape plate).

Fig. 9 shows the improvement in the computed collapse load as the problem is refined uniformly. Due to the singularity of the displacement field along the boundary of the plate, the displacement model (HCT) results in a slower convergence than when using the equilibrium model (EM). When 4050 elements were used, the lower-bound was found to be 12.42, just 0.64% different to the exact solution. Finally, an L-shape plate subject to a uniform load was considered. The plate geometry and uniform mesh refinements are shown in Figs. 10 and 11, respectively. Collapse load multipliers for various numbers of elements are plotted in Fig. 12. The L-shape plate problem exhibits both stress and displacement singularities at the re-entrant corner. This evidently results in a slow convergence and the gap between upper and lower bounds are large despite that fact that a large number of elements was used. For this example, the computed upper-bound was found to be 6.289 which is lower than the best solution obtained previously in [29]. 6. Conclusions The performance of the two novel numerical limit analysis procedures using finite element method in conjunction with secondorder cone programming has been investigated. It has been shown that when limit analysis problems are cast in the form of a SOCP, the resulting optimization problems can be solved rapidly by such a efficient interior point algorithm, even though for cases when a very large number of variables involves. The proposed procedures are enable to provide relatively good bounds on the actual collapse load multiplier since most solutions in existing references were improved. Moreover, the proposed procedures can handle efficiently problems of arbitrary geometry. The only drawback is that the solutions are highly sensitive to the geometry of the finite element mesh, particularly in the region of stress or displacement singularities. An automatically adaptive mesh refinement scheme can be performed to increase the accuracy of solutions. A well-known benefit from dual structure of limit analysis is that both the stress and velocity fields of the upper and lower bound problem can be determined. It is, therefore, relevant to investigate the performance of an adaptive scheme based on a posteriori error estimate using elemental and edge contributions to the bound gap [22,23]. References [1] Johansen KW. Yield-line theory. London: Cement and Concrete Association; 1962. [2] Wood RH. Plastic and elastic design of slabs and plates. London: Thames and Hudson; 1961. [3] Hodge PGJ, Belytschko T. Numerical methods for the limit analysis of plates. Trans ASME, J Appl Mech 1968;35:796–802. [4] Christiansen E, Larsen S. Computations in limit analysis for plastic plates. Int J Numer Methods Eng 1983;19:169–84. [5] Andersen KD, Christiansen E, Overton ML. Computing limit loads by minimizing a sum of norms. SIAM J Sci Comput 1998;19:1046–62. [6] Capsoni A, Corradi L. Limit analysis of plates – a finite element formulation. Struct Eng Mech 1999;8:325–41. [7] Krabbenhoft K, Damkilde L. Lower bound limit analysis of slabs with nonlinear yield criteria. Comput Struct 2002;80:2043–57. [8] Yan AM, Nguyen-Dang H. Limit analysis of cracked structures by mathematical programming and finite element technique. Comput Mech 1999;24:319–33. [9] Yan AM, Jospin RJ, Nguyen-Dang H. An enhance pipe elbow element – application in plastic limit analysis of pipe structures. Int J Numer Methods Eng 1999;46:409–31. [10] Phan-Hong Q, Nguyen-Dang H. Limit analysis of 2D structures using gliding line mechanism generated by rigid finite elements, Collection of papers from Prof. Nguyen-Dang Hungs former students. Vietnam National University, Ho Chi Minh City Publishing House; 2006. p. 447–60. [11] Clough R, Tocher J. Finite element stiffness matrices for analysis of plates in bending. In: Proceedings of the conference on matrix methods in structural mechanics, Ohio,Wright Patterson A.F.B.; 1965. [12] Morley LSD. The triangular equilibrium problem in the solution of plate bending problems. Aero Quart 1968;19:149–69. [13] Gaudrat VF. A Newton type algorithm for plastic limit analysis. Comput Methods Appl Mech Eng 1991;88:207–24.

C.V. Le et al. / Computers and Structures 88 (2010) 65–73 [14] Zouain N, Herskovits J, Borges LA, Feijoo RA. An iterative algorithm for limit analysis with nonlinear yield functions. Int J Solids Struct 1993;30:1397–417. [15] Yan AM, Nguyen-Dang H. Kinematical shakedown analysis with temperaturedependent yield stress. Int J Numer Methods Eng 2001;50:1145–68. [16] Nguyen-Dang H, Yan AM, Vu DK. Duality in kinematical approaches of limit and shakedown analysis of structures, complementary, duality and symmetry in nonlinear mechanics. In: Gao David, editor. Shanghai IUTAM symposium; 2004. p. 128–48. [17] Andersen KD, Christiansen E, Overton ML. An efﬁcient primal-dual interiorpoint method for minimizing a sum of Euclidean norms. SIAM J Sci Comput 2001;22:243–62. [18] Andersen ED, Roos C, Terlaky T. On implementing a primal-dual interior-point method for conic quadratic programming. Math Program 2003;95:249–77. [19] Mosek, The MOSEK optimization toolbox for MATLAB manual, Mosek ApS; 2008. <http://www.mosek.com>. [20] Krabbenhoft K, Lyamin AV, Sloan SW. Formulation and solution of some plasticity problems as conic programs. Int J Solids Struct 2007;44:1533–49. [21] Makrodimopoulos A, Martin CM. Upper bound limit analysis using simplex strain elements and second-order cone programming. Int J Numer Anal Methods Geomech 2006;31:835–65. [22] Ciria H, Peraire J, Bonet J. Mesh adaptive computation of upper and lower bounds in limit analysis. Int J Numer Methods Eng 2008;75:899–944. [23] Munoz J, Bonet J, Huerta A, Peraire J. Upper and lower bounds in limit analysis: adaptive meshing strategies and discontinuous loading. Int J Numer Methods Eng 2009;77:471–501.

[24] Debongnie JF, Applying pressures on plate equilibrium elements. Technical report, University of Liege, Belgium. [25] Nguyen-Xuan H, Debongnie JF, The equilibrium finite element model and error estimation for plate bending. International Congress Engineering Mechanics Today 2004, Ho Chi Minh City, Vietnam; August 16–20, 2004. [26] Debongnie JF, Nguyen-Xuan H, Nguyen-Huy C, Dual analysis for finite element solutions of plate bending. In: Montero G, Montenegro R, editors. Proceedings of the eighth international conference on computational structures technology, B.H.V. Topping, Stirlingshire, Scotland: Civil-Comp Press; 2006. [27] Nguyen-Dang H. Direct limit analysis via rigid-plastic finite elements. Comput Methods Appl Mech Eng 1976;8:81–116. [28] Christiansen E, Limit analysis of collapse states, Handbook of numerical analysis, vol. IV, Amsterdam: North-Holland; 1996. p. 193312 [chapter II]. [29] Le CV, Gilbert M, Askes H. Limit analysis of plates using the EFG method and second-order cone programming. Int J Numer Methods Eng 2009;78:1532–52. [30] Nguyen-Dang H, Konig JA. A finite element formulation for shakedown problems using a yield criterion of the mean. Comput Methods Appl Mech Eng 1976;8:179–92. [31] Le CV, Nguyen-Xuan H, Nguyen-Dang H. Dual limit analysis of bending plates. In: Proceeding of third international conference on advanced computational methods in engineering, Ghent – Belgium; 2005. [32] Hopkins H, Wang A. Load-carrying capacities for circular plates of perfectlyplastic material with arbitrary yield condition. J Mech Phys Solids 1954;3:117–29.

Computers and Structures 88 (2010) 74–86

Contents lists available at ScienceDirect

Computers and Structures journal homepage: www.elsevier.com/locate/compstruc

Uncertain linear structural systems in dynamics: Efﬁcient stochastic reliability assessment H.J. Pradlwarter, G.I. Schuëller * Institute of Engineering Mechanics, University of Innsbruck, A-6020 Innsbruck, Austria

a r t i c l e

i n f o

Article history: Received 2 April 2009 Accepted 14 June 2009 Available online 25 July 2009 Keywords: Finite element models Linear structures Dynamics Gaussian excitation Uncertain structural parameters Reliability

a b s t r a c t A numerical procedure for the reliability assessment of uncertain linear structures subjected to general Gaussian loading is presented. In this work, restricted to linear FE systems and Gaussian excitation, the loading is described quite generally by the Karhunen–Loève expansion, which allows to model general types of non-stationarities with respect to intensity and frequency content. The structural uncertainties are represented by a stochastic approach where all uncertain quantities are described by probability distributions. First, the critical domain within the parameter space of the uncertain structural quantities is identified, which is defined as the region which contributes most to the excursion probability. Each point in the space of uncertain structural parameters is associated with a certain excursion probability caused by the Gaussian excitation. In order to determine the first excursion probability of uncertain linear structures, an integration over the space of uncertain structural parameters is required. An extended procedure of standard Line sampling [P.S. Koutsourelakis, H.J. Pradlwarter, G.I. Schuëller, Reliability of structures in high dimensions, part I: algorithms and applications, Probabilistic Engineering Mechanics 19(4) (2004) 409–417; G.I. Schuëller, H.J. Pradlwarter, P.S. Koutsourelakis, A critical appraisal of reliability estimation procedures for high dimensions, Probabilistic Engineering Mechanics 19(4) (2004) 463–474] is used to perform the conditional integration over the space of uncertain parameters. The suggested approach is applicable to general uncertain linear systems modeled by finite elements of arbitrary size by using modal analysis as exemplified in the numerical example. Special attention is devoted to the efficiency of the proposed approach when dealing with realistic FE models, characterized by a large number of degrees of freedom and also a large number of uncertain parameters. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction The potentials and abilities of modeling the behavior of fluids, solids and complex materials subjected to various forces and boundary conditions by finite element analyses (FEA) provides a key technology for further developments in the industrialized world. Supported by the available relatively inexpensive and continuously growing computer power, the expectations on FEA are shifting towards reliable and robust computational simulations and predictions of physical events. For achieving this goal, answers to many open issues in computational mechanics need to be provided as discussed e.g. in [10]. The need to design, given uncertain material properties, production processes, operating conditions and fidelity of mathematical computational (FE) models to represent reality, leads to the concern of reliability and robustness of computer-generated predictions. Without some confidence in the validity of simulations, their value is obviously diminished. Since * Corresponding author. E-mail address: Mechanik@uibk.ac.at (G.I. Schuëller). 0045-7949/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2009.06.010

uncertainty is always present in non-trivial realistic applications, uncertainty propagation through the FEA is one of the important issues which must be addressed for further developments. In this paper, a computational efficient reliability estimation procedure for uncertain linear systems subjected to dynamic stochastic loading is presented. The approach is designed to cope with large FE-models in terms of degrees of freedom, a large number of uncertain input quantities and high variabilities in terms of coefficient of variation (e.g. P10%). Uncertainties with respect to dynamic loading and of the parameters describing the mechanical properties of the FE-model are propagated by a stochastic approach: Inherent irreducible (aleatory) uncertainties as well as reducible (epistemic) uncertainties due to insufficient knowledge or modeling capabilities are translated into a probability distribution defining the input of the stochastic analysis. The reliability will be assessed in terms of the first excursion probability of critical responses. For this purpose, Line sampling [7,15] is further developed and extended. Efficient solutions of the first excursion probability for deterministic systems are exploited to compute failure probabilities conditioned on discrete realizations of the uncertain

H.J. Pradlwarter, G.I. Schuëller / Computers and Structures 88 (2010) 74–86

structural properties. Computationally efficient procedures to determine the stochastic response in terms of the Karhunen–Loève representation of the conditional critical response are suggested by using modal analysis, impulse response functions in combination with Fast Fourier Transforms (FFT). The associated conditional first excursion probabilities are evaluated by Line sampling. The shown procedure is applicable for any Gaussian distributed excitation represented by a Karhunen–Loève expansion and for an arbitrarily large FE-model. In a further step, the domain of uncertain structural parameters which contributes most to the failure probability is identified by a recently developed estimation procedure which allows to identify the most influential uncertain parameters among a large number. Finally, the unconditional total failure probability is estimated by a novel Line sampling procedure, covering the important failure domain and efficiently integrating over the whole parameter uncertainty space. To demonstrate the applicability of the proposed approach for general FE-models, a realistic FE-model for a twelve story building with more than 24,000 DOF’s and 200 uncertain quantities is analyzed.

2. Methods of analysis 2.1. General outline This section, containing the theoretical developments, is subdivided into six subsections. The first subsection addresses the Karhunen–Loève representation of the stochastic excitation. The second describes the treatment of uncertain structural properties by a stochastic approach. The third considers impulse response functions and its use to evaluate the critical response for linear systems conditioned on specific realizations of the uncertain structural properties. The fourth subsection shows how Line sampling can be applied to estimate the conditional first excursion probability. In the fifth subsection, the domain within the parameter space of uncertainties is identified which contributes most to the first excursion probability. In the last and sixth subsection, the integration over the entire high dimensional parameter space is carried out by a novel Line sampling procedure. 2.2. Representation of uncertain excitation Dynamic excitations acting on the structure are in many cases uncertain. However, although it might be impossible to describe these excitations in a deterministic sense, information on the expected range and its variability is usually available. Such information can be described by the mean value, the standard deviation of the fluctuation, and also the correlation in space and time. The dynamic excitation f ðx; a; AðtÞÞ is a function of the spatial coordinates x, the direction a and the amplitude AðtÞ as a function of time t. This complex dependency on eight scalar quantities fx1 ; x2 ; x3 ; a1 ; a2 ; a3 ; A; tg is considerably simplified by the use of finite element analysis (FEA). In FEA, the spatial coordinates x and the direction of actions a are specified by the degrees of freedom and all forces are reduced to nodal forces. The excitation f can therefore be interpreted as vector with m components, where the structure is assumed to be specified by m degrees of freedom. Hence it suffices in FEA, and as implied here, to specify the dynamic excitation by the m-dimensional vector f ðtÞ as a function of time t. The uncertainties of the excitation f ðtÞ are described mathematically by a stochastic process [8,1,16] characterized as a function of independent random variables N ¼ ðN1 ; N2 ; . . . ; Nn Þ and by deterministic (vector) functions f ðiÞ ðtÞ of time t. Most conveniently, each

of the independent random variables is assumed to follow a standard normal distribution

! 1 n2i qNi ðni Þ ¼ pffiffiffiffiffiffiffi exp 2 2p Z ni P½Ni 6 ni ¼ qNi ðxÞdx ¼ Uðni Þ;

ð1Þ ð2Þ

in which Uð Þ denotes the cumulative standard normal distribution. Any Gaussian distributed process, is most conveniently described by the so called Karhunen–Loève presentation (see e.g. [9,6,5,14]).

f ðt; nÞ ¼ f

ð0Þ

ðtÞ þ

n X

ni f

ðiÞ

ðtÞ:

ð3Þ

i¼1

In the above, all vectors on the right hand side do have deterministic properties and the independent random variables assume the following relations,

E½n2i ¼ 1;

E½ni ¼ 0;

E½ni nj ¼ 0 for i–j;

ð4Þ

where E½ denotes the mean or expectation. The representation (3) speciﬁes uniquely the mean lf ðtÞ and the variance r2fk ðtÞ or standard deviation rfk ðtÞ for each degree of freedom k, and the correlation of the uncertain excitation.

lf ðtÞ ¼ f ð0Þ ðtÞ n h i2 X ðiÞ r2fk ðtÞ ¼ fk ðtÞ

ð5Þ ð6Þ

i¼1

E½fj ðt1 Þfk ðt 2 Þ ¼

n X

ðiÞ

fj ðt 1 Þfk ðt2 Þ:

ð7Þ

i¼1

However, the deterministic vector valued functions f ðiÞ ðtÞ are usually not quantiﬁed a priori. They need to be determined from the symmetric covariance matrix C f ðtj ; t k Þ. Assuming that the excitation is discretized at equidistant instants 0 P tk ¼ kDt P T, the covariance matrix is deﬁned as given below,

C f ðt j ; tk Þ ¼ E½ðf ðtj Þ lf ðtj ÞÞðf ðt k Þ lf ðtk ÞÞ0 ;

0 6 t 1 ; t 2 6 T;

ð8Þ

where a prime ‘‘ ‘‘ denotes the transposed vector and T is the considered duration. After solving the eigenvalue problem,

C f U ¼ UK;

ð9Þ

where U contains the eigenvector and K is a diagonal matrix of eigenvalues, the eigen-solution for the n highest eigenvalues k1 P k2 P P kn are used. The deterministic vector values functions f ðiÞ ðtÞ are then determined by

ðjÞ

ðtk Þ ¼

pffiffiffiffi kj /½k j ;

ð10Þ

where ½k denotes the rows associated with t k and j the column of the eigenvector matrix. In practice it would be extremely difﬁcult, if not unfeasible, to establish the covariance matrix C f for all m degrees of freedom simultaneously and all K time steps tk ; k ¼ 1; . . . ; K. This would lead to a quadratic matrix of the size m K, which might be beyond a manageable size. For a tractable solution, independent excitation processes are considered separately. A typical example for such a separation could be an earthquake excitation in a speciﬁed direction,

f ðtÞ ¼ MI aðtÞ

ð11Þ

in which M is the mass matrix and I is vector with 1’s for all degrees of freedom in the considered direction and zeros elsewhere. Hence the nodal forces f ðtÞ are for any time t fully correlated, and only the correlations at different times t j – t k need to be described. Hence, after a discretization of the time t by t k ¼ kDt; k ¼ 1; . . . ; K,

H.J. Pradlwarter, G.I. Schuëller / Computers and Structures 88 (2010) 74–86

the covariance matrix of the scalar acceleration aðt k Þ has only the size K, for which the eigen-solution poses no difﬁculty, leading to the representation

aðtk Þ ¼ að0Þ ðt k Þ þ pffiffiffiffi a ðt k Þ ¼ ki Uki "

n X

ni aðiÞ ðtk Þ

ð12Þ

i¼1

ðiÞ

f ðt k Þ ¼ MI að0Þ ðt k Þ þ

n X

ð13Þ

# ni aðiÞ ðt k Þ :

ð14Þ

i¼1

In practical applications, each of the deterministic Karhunen–Loève excitation terms f ðiÞ ðtÞ can be described by a constant vector F ðiÞ deﬁned over the range of degrees of freedom and a scalar function ðiÞ h ðtÞ as function of time t,

ðiÞ

ðtÞ ¼ F ðiÞ h ðtÞ:

ð15Þ

As shown above, such a representation applies for earthquake excitation, with F ¼ MI. 2.3. Uncertain structural systems

H ¼ fH1 ; H2 ; . . . ; HS g

ð16Þ

such that for any fixed set H ¼ h, these structural matrices are uniquely specified. Hence, for any fixed set (vector) h, the structural matrices MðhÞ; DðhÞ and KðhÞ can be treated deterministically. It is common practice, to assume the random variables Hi to be standard normally distributed, i.e. with zero mean E½Hi ¼ 0 and unit variance E½H2i ¼ 1. This assumption, however, does not imply that the structural components follow a Gaussian distribution or that correlations among structural parts do not exist. Non-Gaussian distributions are realized by non-linear relations of these basic standard normally distributed random variables. Correlations among different structural parts are established by representing different parts as a functions of one or several random variables. 2.4. Impulse response functions Since the structural system is assumed to be linear, the law of superposition is valid. This law implies, that the response uðt; hÞ for fixed structural properties h and dynamic excitation f ðtÞ has, analogous to (3), also a Karhunen–Loève representation, n X

ni uðiÞ ðt; hÞ;

ð17Þ

i¼1

where each deterministic term uðiÞ ðt; hÞ; i ¼ 0; 1; . . . ; n, is the solution of a deterministic dynamic analysis, involving the constant symmetric mass matrix MðhÞ, damping matrix DðhÞ and stiffness matrix KðhÞ in the equation of motion.

€ ðiÞ ðt; hÞ þ DðhÞu_ ðiÞ ðt; hÞ þ KðhÞuðiÞ ðt; hÞ ¼ f MðhÞu

80 6 i 6 n:

v_ ðiÞ ð0Þ ¼ M 1 F ðiÞ :

ð19Þ

For illustration, Fig. 1 shows an example of the impulse response function for the velocity and displacement response of a 2-DOF system. Since the system is linear, also efﬁcient modal analysis applies. After solving the eigenvalue problem,

KðhÞUðhÞ ¼ MðhÞUðhÞKðhÞ the impulse response zðt; hÞ

v ðt; hÞ

ð20Þ is represented in modal coordinates

v ðiÞ ðt; hÞ ¼ UðhÞzðiÞ ðt; hÞ:

ð21Þ

Then, the modal coordinates have initial values ðiÞ z_ j ð0; hÞ is speciﬁed by the vector

ðiÞ zj ð0; hÞ

¼ 0, and

z_ ðiÞ ð0; hÞ ¼ UT ðhÞF ðiÞ :

ð22Þ

The free motion in modal coordinates has the explicit solution [8]

Linear structural systems are described by a constant symmetric mass matrix M, damping matrix D and stiffness matrix K. These matrices are realistically considered to be associated with uncertainties. In the proposed stochastic parametric approach [1], the uncertainties of these matrices are also conveniently modeled as functions of independent random variables

uðt; h; nÞ ¼ uðhÞð0Þ ðtÞ þ

function v ðiÞ ðtÞ and the Fast Fourier transform for computing the convolution. The impulse response function is computed as free motion with initial zero displacement and the initial velocity:

ðiÞ

zj ðt; hÞ ¼

ðiÞ z_ j ð0; hÞ

x0j

efj xj t sinðx0j tÞ

ðiÞ ðiÞ z_ j ðt; hÞ ¼ z_ j ð0; hÞefj xj t cosðx0j tÞ

fj xj t €zðiÞ _ ðiÞ j ðt; hÞ ¼ zj ð0; hÞe

Hence, to specify the variance of the displacement response, n þ 1 deterministic analyses are required. However, the dynamic response might depend considerably on the structural parameters speciﬁed by the vector h. One of the efﬁcient ways to compute the associated dynamic displacement response uðiÞ ðtÞ is the use of the impulse response

ð23Þ f j xj

x0j

sin x0j t

# ð24Þ

2fj x0 x2j j 0 ð1 2f Þ sin x t þ cos x0j t ; j j x0j xj ð25Þ

where all the quantities fj ; xj and x depend actually on the parameter h: 0 j

qffiffiffiffiffiffiffiffiffiffiffiffiffi ðiÞ kj ðhÞ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x0j ¼ xj 1 f2j ðhÞ:

xj ¼

ð26Þ ð27Þ

The damping ratios fj ðhÞ are, most advantageously, not specified by the an explicit damping matrix DðhÞ, but by modal damping ratios with a moderate increase with frequency xj . The damping ratios might be defined directly as function of uncertain quantities, e.g. fj ðhÞ ¼ cj expð aj hk Þ with given value for cj and aj . Suppose, the critical responses of interest, e.g. stresses, strains, accelerations, displacements, etc., are comprised in the vector yðt; hÞ. Each component yi ðt; hÞ of the vector must fulfill certain conditions in order to be regarded as reliable. It will be assumed that these critical types of structural response can be represented by a linear combination of the displacement or acceleration re€ ðt; hÞ, sponse uðt; hÞ or u

yðt; hÞ ¼ Q ðhÞuðt; hÞ;

ð28Þ

where Q is a constant matrix, independent of time t, and a function of the structural parameters which might depend on h. Similarly, as the displacement response, the variability due to the random excitation can be cast into a Karhunen–Loève representation.

ðtÞ; ð18Þ

yðt; h; nÞ ¼ yð0Þ ðt; hÞ þ

n X

ni yðiÞ ðt; hÞ:

ð29Þ

i¼1

The impulse response function associated with F ðiÞ is then:

wðiÞ ðt; hÞ ¼ Q ðhÞUðhÞzðiÞ ðt; hÞ: ðiÞ yk ðt; hÞ

The critical response due to the excitation term f therefore the result of the convolution

ð30Þ ðiÞ

ðtÞ is

H.J. Pradlwarter, G.I. Schuëller / Computers and Structures 88 (2010) 74–86

Velocity impulse response

mass 2

0.5

mass 1

0 −0.5

displacement [m]

mass 1 mass 2

velocity [m/s]

Displacements impulse response 0.2

mass 1 mass 2

0.1 0 −0.1

−1 0

−0.2

time t

4 6 time [s]

Fig. 1. Impulse response function.

ðiÞ

yk ðt; hÞ ¼

Z 0

ðiÞ

h ðsÞwk ðt s; hÞ ds:

ð31Þ

Numerically, this integral is most efﬁciently computed by using Fast Fourier Transforms, ðiÞ

ðiÞ

h ðtÞ ! FFT ! h ðxÞ;

ð32Þ

ðiÞ ðiÞ wk ðtÞ ! FFT ! wk ð Þ; ðiÞ ðiÞ ðiÞ yk ð Þ ¼ h ð Þwk ð Þ; ðiÞ ðiÞ yk ð Þ ! IFFT ! yk ðtÞ;

ð33Þ

x x

pi;f ðhÞ ¼ P½maxfyi ðt; hÞg P b i 06t6T

[ min fyi ðt; hÞg 6 bi 06t6T Z qðnÞdn ¼

where FFT denotes the discrete Fast Fourier Transform and IFFT its inverse. For the efficiency of the presented approach, it should be stressed that a single modal FE analysis is sufficient to compute the Gaussian distributed response. The associated Karhunen–Loève representation (29) provides the basis for the reliability assessment conditional on the structural parameters h. 2.5. Conditional reliability 2.5.1. First excursion probability In this section, the conditional first excursion problem pf ðN; hÞ is discussed for random Gaussian excitation and for the case where a random realization of H assumes a certain set of deterministic parameters h. The first excursion probability is then defined as the probability that the critical response exceeds at least once within the considered time period ½0; T the threshold bi of the ith component of the critical response yðt; hÞ [8]. The threshold bi as shown might consist of a lower limit bi and an upper limit b i in Fig. 2 for a single critical response.

ð37Þ

g i ðh;nÞ60

maxfy ðt; h; nÞg; g i ðh; nÞ ¼ minfb i i 06t6T

ð38Þ

min fyi ðt; h; nÞg bi g:

ð34Þ ð35Þ

ð36Þ

06t6T

This conditional first excursion probability corresponds to the case of stochastic excitation, specified by the random vector N and its probability density function qðnÞ, subjected to a deterministic strucR tural system. In the next section, the integration pf ¼ pf ðhÞqðhÞ dh over the whole domain of the probability density function qðhÞ will be discussed in detail, leading to the unconditional first excursion problem pf . To simplify the notations and the reliability problem, each critical response will be considered separately. Hence, the subscript ‘‘i” is dropped in the following developments. For this case, where the critical response yðtÞ can be represented by a Karhunen–Loève representation, as given in (29), several efficient numerical procedures have been developed recently to calculate the first excursion probability (e.g. [2,13]). 2.5.2. Line sampling in dynamics In this section, a procedure, based on Line Sampling (LS), is presented. Line sampling [7,15] is particularly efficient if a so called important direction can be computed, which points towards the failure domain nearest to the origin (see Fig. 3). Most importantly, it is not required that the vector a (see Fig. 3) points exactly towards the center of this domain, nor are any assumptions made

Fig. 2. First Excursions of thresholds b and b.

H.J. Pradlwarter, G.I. SchuĂŤller / Computers and Structures 88 (2010) 74â&#x20AC;&#x201C;86

1 c Ă°jĂ&#x17E; If nÂ˝i Ă°cĂ&#x17E; pďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192; exp dc 2 2p 1 Ă°jĂ&#x17E; Ă°jĂ&#x17E; ; Âź U bÂ˝i Ăž U b Â˝i

Z Ă°jĂ&#x17E; pf n~Â˝i ; h Âź

Ăž1

Ă°43Ă&#x17E;

where UĂ° Ă&#x17E; denotes the cumulative standard normal distribution, Ă°jĂ&#x17E; nÂ˝i ; h denotes the If Ă° Ă&#x17E; is an indicator function of failure, and pf ~ failure probability conditioned on the jth randomly selected line Ă°jĂ&#x17E; Ă°jĂ&#x17E; nÂ˝i Ăž caÂ˝i and on the strucin the ith important direction nÂ˝i Ă°cĂ&#x17E; Âź ~ tural parameter set h. The safe domain lies within the bounds h i Ă°jĂ&#x17E; Ă°jĂ&#x17E; bÂ˝i ; b Â˝i . Since the superposition law is valid, the response for Ă°jĂ&#x17E; ?Ă°jĂ&#x17E; nÂ˝i Âź caÂ˝i Ăž nÂ˝i Ă°cĂ&#x17E; can be speciďŹ ed by the following function:

Ă°jĂ&#x17E; ?Ă°jĂ&#x17E; y t; c; nÂ˝i Âź cy t; aÂ˝i Ăž y t; nÂ˝i

y t; aÂ˝i Âź

Fig. 3. Line sampling in high dimensions.

n X

Ă°44Ă&#x17E;

aÂ˝i k yĂ°kĂ&#x17E; Â˝i Ă°t; hĂ&#x17E;

Ă°45Ă&#x17E;

n X ?Ă°jĂ&#x17E; ?Ă°jĂ&#x17E; Ă°kĂ&#x17E; Âź yĂ°0Ă&#x17E; Ă°t; hĂ&#x17E; Ăž nkÂ˝i yÂ˝i Ă°t; hĂ&#x17E;: y t; nÂ˝i

Ă°46Ă&#x17E;

kÂź1

regarding the shape of the limit state function gĂ°h; nĂ&#x17E; Âź 0 [7]. LS is robust, since unbiased reliability estimates are obtained, irrespectively of the important direction a, and any deviation from the optimal direction merely increases the variance of the estimate. The basic procedure of LS is for completeness shown in the Appendix A. For structures with the properties speciďŹ ed uniquely by h and which are excited by a Gaussian distributed loading process, the important direction is well known to be proportional to the response yĂ°jĂ&#x17E; Ă°t; hĂ&#x17E;. Let cĂ°t; hĂ&#x17E; be the standard deviation of the response yĂ°t; hĂ&#x17E;

c2 Ă°t; hĂ&#x17E; Âź VarÂ˝yĂ°t; hĂ&#x17E; Âź

n X Â˝yĂ°jĂ&#x17E; Ă°t; hĂ&#x17E; 2 :

Ă°39Ă&#x17E;

h i Ă°jĂ&#x17E; Ă°jĂ&#x17E; along the jth random line in the ith Then, the safe domain bÂ˝i ; b Â˝i important direction is deďŹ ned as Ă°jĂ&#x17E; bÂ˝i

Ă°jĂ&#x17E; b Â˝i

jÂź1

Then, the unit vector aĂ°t; hĂ&#x17E; pointing towards the important direction is speciďŹ ed by

aj Ă°t; hĂ&#x17E; Âź

yĂ°jĂ&#x17E; Ă°t; hĂ&#x17E; cĂ°t; hĂ&#x17E;

Ă°40Ă&#x17E;

It should be noted that the important direction is a continuous function of time t. Moreover, linearity implies that any excitation deďŹ ned by n and orthogonal to aĂ°t; hĂ&#x17E;, will have a zero response at time t. Since the important direction aĂ°t; hĂ&#x17E; varies within the time interval 0 6 t 6 T, and the reliability index bĂ°t; hĂ&#x17E; might be of similar size over a considerable portion of time, the efďŹ ciency and reliability of Line sampling can L be increased by sampling in several important directions aÂ˝i iÂź1 . The algorithm to determine these Â˝i L important directions a iÂź1 is presented in Appendix B. The procedure involves a transformation of the KL vectors yĂ°jĂ&#x17E; Ă°t; hĂ&#x17E; such L that aÂ˝i Âź ei iÂź1 , where ei are unit vectors of which the ith component contains the value one and zeros elsewhere. In LS, each point n in the standard normal space is decomposed into the one dimensional space caÂ˝i and the Ă°n 1Ă&#x17E; dimensional subspace n? Ă°aÂ˝i Ă&#x17E; orthogonal to the direction aÂ˝i ; i Âź 1; 2; . . . ; L, Â˝i

Â˝i

n Âź ca Ă°hĂ&#x17E; Ăž n Ă°a Ă&#x17E;:

Ă°41Ă&#x17E;

8 y t; n?Ă°jĂ&#x17E; <b Â˝i

H y t; aÂ˝i yĂ°t; aÂ˝i Ă&#x17E; 9 ?Ă°jĂ&#x17E; = b y t; nÂ˝i Â˝i Ăž H y t; a ; 0 6 t 6 T ; yĂ°t; aÂ˝i Ă&#x17E; 8 y t; n?Ă°jĂ&#x17E; <b

Â˝i Âź min H y t; aÂ˝i : yĂ°t; aÂ˝i Ă&#x17E; 9 ?Ă°jĂ&#x17E; = b y t; nÂ˝i Â˝i Ăž H y t; a ; 0 6 t 6 T Â˝i ; yĂ°t; a Ă&#x17E; Âź max

Ă°jĂ&#x17E;

; bÂ˝i < nk Ă°aÂ˝i Ă&#x17E; < b Â˝i

k â&#x20AC;&#x201C; i and 0 < k 6 L:

^Ă°jĂ&#x17E; p f Ă°hĂ&#x17E; Âź

L X

Ă°jĂ&#x17E;

pf Â˝i Ă°hĂ&#x17E;

Ă°50Ă&#x17E;

iÂź1 Ă°jĂ&#x17E;

pf Â˝i Ă°hĂ&#x17E; Âź ^f Ă°hĂ&#x17E; Âź p

Ni 1 X Ă°kĂ&#x17E; p Ă°~n ; hĂ&#x17E; Ni kÂź1 f Â˝i

N 1X ^Ă°jĂ&#x17E; Ă°hĂ&#x17E; p N jÂź1 f

~nĂ°jĂ&#x17E; Âź nĂ°jĂ&#x17E; Ă°aÂ˝i T nĂ°jĂ&#x17E; Ă&#x17E;aÂ˝i : Â˝i

2.6. Design point for stochastic structural systems

Ă°jĂ&#x17E;

Ă°49Ă&#x17E;

For the very unlikely cases in which the above condition is not satisďŹ ed, the line will be ignored and another line is generated instead. The independent estimates pf Ă°hĂ°jĂ&#x17E; Ă&#x17E; for the failure probability allow to compute an unbiased estimate for the conditional failure probability

and the variance

For each independent random realization ~ nÂ˝i , the probability of failĂ°jĂ&#x17E; ure conditioned on the value ~ nÂ˝i and h can be determined by

Ă°48Ă&#x17E;

in which HĂ°yĂ&#x17E; Âź 1 for y P 0 and HĂ°yĂ&#x17E; Âź 0 for y < 0. To each of the important directions faÂ˝i gLiÂź1 , a disjunct failure domain is associated by imposing the condition

Ă°jĂ&#x17E; N In the following it is assumed that ~ n jÂź1 are independent sample ? Â˝i points of the subspace n Ă°a Ă&#x17E; drawn by direct Monte Carlo simulation. The random sample ~ nĂ°jĂ&#x17E; is generated by simulating ďŹ rst n independent standard normally distributed components of a vector nĂ°jĂ&#x17E; and then subtracting the component which points towards the direction of aÂ˝i ,

Ă°42Ă&#x17E;

Ă°47Ă&#x17E;

VarĂ°pf Ă°hĂ&#x17E; Âź

N 2 X 1 Ă°jĂ&#x17E; ^f Ă°hĂ&#x17E; ; pf Ă°hĂ&#x17E; p NĂ°N 1Ă&#x17E; jÂź1

Ă°51Ă&#x17E; Ă°52Ă&#x17E;

Ă°53Ă&#x17E;

which is the basis for deriving further conďŹ dence intervals.

In this section, the domain which contributes most to the unconditional total failure probability pf will be determined. The-

H.J. Pradlwarter, G.I. Schuëller / Computers and Structures 88 (2010) 74–86

oretically, the failure probability pf jh , conditioned on speciﬁc realizations h, need to be integrated over the whole domain of pH ðhÞ

@b2 ðn; hÞ ¼ 0; @hk

pf ¼

which leads to the solution

pf ðhÞpH ðhÞ dh:

ð54Þ

Since the number of uncertain structural parameters, characterized by the random quantities fH1 ; H2 ; . . . ; HS g, is in general large, numerical integration is not feasible. Basically the above integral can be estimated by Monte Carlo sampling

^f ¼ p

N 1 X p ðhðjÞ Þ; N j¼1 f

ð55Þ

h k ¼

where N independent realizations h are drawn from the distribution pH ðhÞ and the associated conditional failure probability pf ðhðjÞ Þ is determined as described in the previous section. Such an approach is only efficient in case the variability of pf ðhðjÞ Þ is small, i.e. might be within one order of magnitude. However, in case the structural variability is high, the conditional first excursion probabilities will vary over many orders of magnitude and direct Monte Carlo approaches will be very inefficient. Considering Eq. (54), it is not difficult to recognize that sampling in the neighborhood of the structural design point h , characterized by

pf ðh ÞpH ðh Þ P pf ðhÞpH ðhÞ;

ð56Þ

contributes most to the total (unconditional) failure probability. In order to identify this domain, composed of the standard normal variables fn; hg, the point fn ; h g with the minimal distance b to the failure surface is determined at the most critical time t 1 (see (B.8)). Since the uncertainty in the excitation and of the structural parameters, speciﬁed by n and h , respectively, are independent and therefore orthogonal in the standard normal space, this distance is speciﬁed according to Pythagoras by

b2 ðn; hÞ ¼ knk2 þ khk2 :

ð57Þ

Fig. 4 shows a sketch of the failure domain for this case, where bðn; hÞ can only be determined in the complete space ðn; hÞ. The threshold b is reached for 2

b knk2 ¼ 2 c ðt ; hÞ yð0Þ ðt ; hÞ; yð0Þ ðt ; hÞ b : b ¼ min½b

ð58Þ ð59Þ

The failure point ðh ; n Þ, with the highest probability density and deﬁned as the nearest failure point to the origin in standard normal space, is derived by imposing the necessary condition

3 ðt ; h Þ

ð60Þ

@ cðt ; h Þ : @hk

ð61Þ

An accurate solution of the above relation can only be obtained in an iterative manner, where convergence is reached for s > S with ðSÞ smaller than the tolerance, ðsþ1Þ

hk ðjÞ

k ¼ 1; 2; . . . ; K;

ðsÞ

¼ ð1 wÞhk

þw

@ cðt ; h ðsÞ Þ @hk c3 ðt ; h ðsÞ Þ b

ð62Þ

ðsþ1Þ ¼ kh ðsþ1Þ h ðsÞ k=kh ðsþ1Þ k;

ð63Þ

where w ¼ 0:5 leads to a stable and fast convergence. However, there is not much gain in evaluating h very accurately. A first or second step ðs ¼ 0; 1Þ is usually sufficiently accurate for the integration procedures as developed in the next section. The above iterative evaluation of the structural design point h requires a gradient computation which might hamper the efficiency of the approach in case the number of uncertain structural parameters is large. For such cases, a gradient estimation procedure – as shown in the Appendix C – might be used to improve the computational efficiency. 2.7. First excursion probability for stochastic systems In this section, the procedure to estimate the total unconditional first excursion probability by integrating over the complete space of uncertain structural parameters is shown (see Eq. (54)). The proposed approach is to limit the application of LS to the subspace of the structural parameters h. This in fact leads to robust results, even for quite large uncertainties of the structural parameters, since the important directions a½i ðhÞ are then always computed in the optimal directions. The procedure is outlined as follows: 1. Determine the design point h with acceptable accuracy and compute

bS ¼ kh k h aS ¼ bS

ð64Þ ð65Þ

where the index ‘‘S” denotes the subspace of the structural uncertainties. 2. Generate samples fh?ðjÞ gNj¼1 using direct MCS in the subspace of h, which are perpendicular to the vector h . 3. Compute for each parallel line for, say, ﬁve discrete points hðjÞ ðck Þ; k ¼ 1; . . . ; 5, as shown in Fig. 5

hðjÞ ðck Þ ¼ h?ðjÞ þ ck aS ; ck ¼ bS þ ðk 3ÞD;

ð66Þ k ¼ 1; 2; . . . ; 5

ð67Þ

D 0:6

ð68Þ ðjÞ

the associated conditional failure probability pf ðck Þ. ðjÞ 4. Estimate the conditional failure probability pf along each line,

1 ðjÞ pf ¼ pffiffiffiffiffiffiffi 2p

e c

2 =2

ðjÞ

pf ðcÞ dc:

ð69Þ ðjÞ

Fig. 4. Structural design point h .

with quadratic interpolation of the function lnðpf ðcÞÞ by using the available discrete values. This estimation will be discussed subsequently in more detail. 5. Estimate the mean and variance of the failure probability pf ðtÞ according to Eqs. (52) and (53).

H.J. Pradlwarter, G.I. Schuëller / Computers and Structures 88 (2010) 74–86 ðiÞ

due to random fluctuation of the estimates pf ðcl Þ. Experience showed that the results using either a linear or quadratic approximation are quite similar. However, a linear approximation is less ðiÞ sensitive to random fluctuations of the estimate pf ðcl Þ. Hence the integration over all points along the line is efficiently approximated, requiring only three to five finite element analyses per line. 3. Numerical example 3.1. General remarks

Fig. 5. Line sampling in the uncertain structural paramameter space.

ðiÞ

It should be noted that pf ðcÞ > 0 for 1 < c < 1, since the position along each line speciﬁes uniquely the properties of a strucðiÞ ture subjected to random excitation. Since pf ðcÞ is always a ðiÞ positive quantity, it is proposed to represent pf ðcÞ as the exponential of a linear or quadratic polynomial, ðiÞ

pf ðcÞ ¼ exp½a0 þ a1 c þ a2 c2 =2 ;

ð70Þ

where the coefﬁcients are obtained by solving a least square problem

h i ðiÞ a0 þ a1 cl þ a2 c2l =2 ¼ ln pf ðcl Þ ;

l ¼ 1; 2; . . . ; 5

ð71Þ

The advantage of this approximation is that the inﬁnite integral in Eq. (69) can be represented in closed form for a2 < 1

1 a21 ðiÞ : pf ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp a0 þ 2ð1 a2 Þ 1 a2

ð72Þ

h i ðiÞ A linear approximation of ln pf ðcÞ in the least square solution should be applied ða2 ¼ 0Þ, whenever a2 > 0:4, which might be

The method, as developed in Section 2, is now exemplified within this section. Usually, proposed methods have been demonstrated by applying them to exceptionally small academic type of examples, where the feasibility of the approach to real world problems remains an open issue, because computational difficulties and many issues of computational efficiency are avoided. Since reliability evaluations in structural dynamics are computationally very demanding, the computational efficiency of the proposed method is naturally in the focus of interest. However, to be of practical value, the efficiency of procedures should be demonstrated by applying them to models which reflect the complexity of relevant engineering structures. By choosing a realistic FE-model, modeling a 12-story building made of reinforced concrete, the requirement of practical applicability is shown. 3.2. Structural system 3.2.1. Geometry A twelve story building with an additional cellar floor made of reinforced concrete is considered. The FE-model consists of 4046 nodes and 5972 elements using shell and 3-D beam elements for modeling the girders, resulting in 24,276 degrees of freedom. Fig. 6 provides a view of the building showing the displacement field according to the first three mode shapes. The building is axis-symmetric and consists of 13 floors of 24:0 24:0 m. The foundation plate is 0.5 m thick and rests on soil modeled by elastic springs. Fig. 7 shows a plan view for all floors, with the exception that the cellar floor is surrounded by cellar walls of 0.3 m thickness. The height of each story is 4.0 m. The weight of the structure is carried by four groups of concrete walls forming a cross. The four groups of supporting concrete walls are connected by four girders of type g-1 the height of which is 1.2 m and a width identical to the

Fig. 6. First three mode shapes of twelve story building.

H.J. Pradlwarter, G.I. Schuëller / Computers and Structures 88 (2010) 74–86

lation coefficients in the order of 0.86. The Young’s moduli of the remaining primary reinforced concrete parts are assumed somewhat higher to be 3.2e+10 N/m2 with a coefficient of variation of 0.144. This relatively large value reflects, to a considerable extent, the lack of knowledge on the dynamic stiffness. It implies also a large correlation of the stiffness, which is reflected by a correlation coefficient in the order of 0.7. The live loads (masses) are represented by a function of five random variables per floor, which are assumed to be independent between the floors and correlated within the floors. The mean value is assumed to be 100 kg/m2 and the coefficient of variation of 0.224 with correlation coefficients of 0.8. 3.3. Dynamic excitation

Fig. 7. Floor plan of twelve story building supported by croncrete walls (w) and girders of type g-1 and type g-2. Critical strains are considered at positions p-1 and p-2.

walls. The thickness of the supporting walls varies with respect to the respective story. In the cellar floor and first and second floor the wall thickness is 0.6 m, the next two floors (third and fourth) 0.5 m, and 0.4 for the fifth until the twelfth floor. The concrete plate of 0.2 m thickness is supported also by girders of type g-2 of height 0.5 m and constant width of 0.3 m. Above the concrete plate, a floor construction with the weight of 150 kg/m2 have been assumed for all floors. 3.2.2. Uncertain structural properties Concrete and reinforced concrete are widely used in building constructions. Its material properties, however, are hard to specify by just a few characteristic quantities. The stiffness of reinforced concrete construction parts depends on many factors which are still difficult to predict and to control. Due to inhomogeneities and possible cracking, the stiffness under dynamic loading reveals uncertainties. Moreover, it might depend also on the loading history (see e.g. [3]). At the present state-of-the-art, quantitative models for reinforced concrete structures, which are both feasible and realistic (including bond-slip, cracking, etc.) are not used yet for ordinary structural analysis as they are still topics of active research. Uncertainties of the stiffness is partially the result of inherent random factors and to a large part by lack of knowledge of the actual mechanisms. A practical way to cover all these uncertainties is to take a single quantity, such as the Young’s modulus, to represent the uncertainties in the stiffness, with the understanding that this quantity should cover the uncertainty of the stiffness and not only of the physical elastic constant. All structural uncertainties are assumed to be represented as a function of 244 independent standard normal variables. The stiffness of the soil bedding is represented by 9 random variables and the modal damping ratio by a single random variable. The mean value of the stiffness of the soil is assumed to be 4.44e+08 N/m3 in the horizontal and 8.88e+08 N/m3 in the vertical direction, with a coefficient of variation of 0.224. 169 random variables do represent the Young’s modulus of various parts of the reinforced concrete structure. The mean value for the Young’s modulus for the foundation plate and cellar walls are assumed to be 2.8e+10 N/m2 and its variability is modeled by a coefficient of variation of 0.215 and corre-

In this numerical example, the dynamic excitation is due to earthquake ground motion in terms of ground accelerations. Future ground motions are highly uncertain, regarding its amplitudes, frequency content and durations, respectively. For physical reasons, the average velocity and acceleration must vanish, and the frequency content depends on the unknown source mechanism and is influenced substantially by local soil conditions. The intensity (acceleration amplitudes) depends on the distance to the earthquake source and the duration on the magnitude of the energy release. It is difficult to cover all these uncertainties by a credible excitation model. In engineering practice, the excitation model is selected such that the model covers uncertainties conditioned on the acceptable hazard. There are several options to derive a suitable realistic excitation. In the ideal case, many records from different events are available – which after a suitable scaling – are used further to establish the covariance matrix (see (8)) from which the Karhunen–Loève terms can be deduced in a straightforward manner. A further option is to establish the covariance matrix such that it is compatible to some response spectra requirements. In this work, an approach based on filtered white noise is used. To cover the unknown frequency content and random amplitudes of the acceleration, the model proposed in [4] is applied, where the acceleration is represented as the output of the response of a linear filter excited by white noise

aðtÞ ¼ X2g v 1 ðtÞ þ 2fg Xg v 2 ðtÞ X2f v 3 ðtÞ 2ff Xf v 4 ðtÞ;

ð73Þ

in which Xg represents the dominant frequency of the ground and Xf ensure that the spectrum of the acceleration tends to zero for frequencies approaching zero. The linear ﬁlter is described by the differential equation [4]

8 9 v1 > > > > > > = <

0 6 X2 d v2 g 6 ¼6 > 4 0 dt > v 3> > > ; : > 2

2fg Xg

X2f

9 3 8 9 8 0 > v1 > > > > > > > > > > > < = < 7 0 7 wðtÞ = v2 þ 7 1 5 > v3 > > > 0 > > > > > > ; : ; > : > 2ff Xf 0 v4 0

ð74Þ where wðtÞ denotes white noise. The intensity of wðtÞ can either be characterized by its intensity IðtÞ or by the constant (over frequency x, i.e. Sðx; tÞ ¼ S0 ðtÞ) spectral density S0 ðtÞ ¼ IðtÞ=ð2pÞ, which satisfy the following relation [8]

E½wðtÞwðt þ sÞ ¼ IðtÞdðsÞ ¼ 2pS0 ðtÞdðsÞ

ð75Þ

where dð Þ denotes Dirac’s delta function with the property dðsÞ ¼ 0 R for s – 0 and dðsÞds ¼ 1; > 0. Since white noise is uncorrelated for any s > 0, white noise excitation can be discretized by a sequence of independent impulses with zero mean and standard deviation rI ðkDtÞ or with linearly interpolated independent force amplitudes with standard deviation rF ðkDtÞ at subsequent time steps kDt

H.J. Pradlwarter, G.I. SchuĂŤller / Computers and Structures 88 (2010) 74â&#x20AC;&#x201C;86

qďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192; rI Ă°kDtĂ&#x17E; Âź IĂ°kDtĂ&#x17E;Dt rďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192;ďŹ&#x192; IĂ°kDtĂ&#x17E; : rF Ă°kDtĂ&#x17E; Âź Dt

Variance of horizontal acceleration

Ă°76Ă&#x17E; Ă°77Ă&#x17E; 1

The non-stationary properties of earthquake ground motion is described by a time varying white noise intensity. 0.8

for t 6 0 Ă°78Ă&#x17E;

2 4

for 0 < t < 2 for 2 6 t 6 8 for t P 8:

[m /s ]

8 0 > > > < t=2 IĂ°tĂ&#x17E; Âź I0 >1 > > : expĂ° 0:16Ă°t 8Ă&#x17E;Ă&#x17E;

0.4

The following constants are used in this work: I0 Âź 0:16 m2 =s3 ; Xg Âź 2p rad=s; fg Âź ff Âź 0:6, and xf Âź 0:1Xg . In a next step, the covariance matrix C Âź Â˝C ij of the acceleration is established with Dt Âź 0:01 s, and 0 < i; j 6 1600 and C ij Âź EÂ˝aĂ°iDtĂ&#x17E;aĂ°jDtĂ&#x17E;]. For this purpose, the impulse response function of the linear ďŹ lter is determined by the free motion of the ďŹ lter with zero initial displacement, and v_ 1 Ă°0Ă&#x17E; Âź v_ 3 Ă°0Ă&#x17E; Âź v_ 4 Ă°0Ă&#x17E; Âź 0 and v_ 2 Ă°0Ă&#x17E; Âź 1. To obtain the impulse response function of the acceleration aIRF Ă°tĂ&#x17E;, Eq. (73) is used. The result is shown in Fig. 8. To establish the covariance matrix, a single impulse at time kDt is considered ďŹ rst. The associated covariance is

C ij Ă°kĂ&#x17E; Âź Dt IĂ°kDtĂ&#x17E;

aIRF Ă°Ă°i kĂ&#x17E;DtĂ&#x17E;aIRF Ă°Ă°j kĂ&#x17E;DtĂ&#x17E;

for i P k and j P k

for i < k or j < k Ă°79Ă&#x17E;

0.2

0 0

C ij Ă°kĂ&#x17E;:

Fig. 9. Variance of horizontal acceleration over time.

Various KHLâ&#x2C6;&#x2019;terms for horiz. acceleration 0.4 10â&#x2C6;&#x2019;th 30â&#x2C6;&#x2019;th 60â&#x2C6;&#x2019;th 80â&#x2C6;&#x2019;th

0.2

Ă°80Ă&#x17E; 2

kÂź0

[m/s ]

C ij Âź

time in [s]

Since the impulses at instants kDt are all independent, the coefďŹ cients C ij are given by the sum 1 600 X

0.6

The diagonal terms C ii Âź EÂ˝a2 Ă°tĂ&#x17E; specify the variance over the time t i Âź iDt and are shown in Fig. 9. Applying further the procedure as described in Section 2.1, the Karhunenâ&#x20AC;&#x201C;LoĂ¨ve functions aĂ°jĂ&#x17E; Ă°tĂ&#x17E; (see Eq. (12)) are computed. For illustration, some of these functions are shown in Fig. 10. Finally, it should be stated that also the vertical earthquake acceleration has been modeled. It turned out that for the structure under consideration this vertical acceleration has only a negligible effect on the critical response and hence it is not described further.

â&#x2C6;&#x2019;0.2

â&#x2C6;&#x2019;0.4 0

8 10 time in [s]

Fig. 10. Karhunenâ&#x20AC;&#x201C;LoĂ¨ve functions aĂ°jĂ&#x17E; Ă°tĂ&#x17E;; j Âź 10; 30; 60; 80 of the horizontal acceleration.

Impulse acceleration response of horizontal filter 8

3.4. Critical response 7 6 5

[m/s ]

4 3 2 1 0 â&#x2C6;&#x2019;1 â&#x2C6;&#x2019;2

4 time in [s]

Fig. 8. Acceleration impulse response function of ďŹ lter.

Under dynamic earthquake loading, two reinforced structural components are likely to exceed the critical strain. The strain in vertical direction of the walls in the basement, forming a cross, might be critical. These walls support the weight of the entire structure and are, in addition, highly strained by bending due to the horizontal earthquake accelerations. Since the structure is symmetric, and the horizontal accelerations are in both directions independent and identically distributed, it sufďŹ ces to consider a single position, indicated by p-1 in Fig. 7. Girders form the remaining critical parts, denoted as g-1 in Fig. 7, which connect separated walls. These girders transmit large shear forces from one group of walls to the counterpart, and contribute essentially to the bending stiffness of the twelve story building. The critical part is the curvature of the girder at the connection to the walls. The position is indicated by p-2 in Fig. 7. Table 1 provides an overview of the strains in these components, i.e. for all ďŹ&#x201A;oors. Accepting as limit state function a vertical

H.J. Pradlwarter, G.I. Schuëller / Computers and Structures 88 (2010) 74–86 Table 1 Critical strain due to static weights and maximum standard deviation due to horizontal earthquake motion. Floor

Width

Wall at pos. p-1

Girder g-1 at p-2

(–)

(m)

v. Strain static 10 6

v. Strain std. dev. 10 6

Curvature static 10 6

Curvature std. dev. 10 6

0 1 2 3 4 5 6 7 8 9 10 11 12

0.6 0.6 0.6 0.5 0.5 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4

143 110 100 108 97 107 93 80 66 53 39 25 18

266 228 181 170 134 125 80 73 60 46 33 20 12

9 27 27 32 32 41 41 42 42 42 42 42 34

281 466 548 643 654 704 675 620 544 457 367 289 187

strain of 0.0016 for the walls without having to expect serious damage and a maximum acceptable curvature of 0.004 in the girders g-1, the curvature in the girder are more likely to be exceeded. Hence, the reliability analysis will focus on the reliability of the girders for not exceeding the limit state function the curvature of 0.004. Taking into account the static solution of floor five,the limit state function has the lower and upper bounds are b ¼ 0:004þ ¼ 0:004 þ 000041 ¼ 0:004041, respec000041 ¼ 0:003959 and b tively. 3.5. Reliability of critical component 3.5.1. Estimation by direct monte carlo simulation Before evaluating the first excursion probability, the Karhunen– Loève terms (see Eqs. (29) and (35)) of the critical response, i.e. the curvature at girder g-1, are determined. These terms yðjÞ ðt; hÞ depend only on the set of uncertain structural parameters h and their evaluation requires a single FE analysis for each distinct set hðjÞ . Hence, the investigation of the variability of the response due to uncertain structural properties is the computationally expensive part. The efficiency of the procedure will therefore be governed by the required number of FE analyses; the larger the FE-model, the more important this number will be. Fig. 11 shows some of these critical response functions for the nominal solution h ¼ 0. The associated first excursion problem is determined by Line sampling as outlined in Section 2.5, where

N ¼ 4000 lines have been used. Note that the number N is selected such that the computation time is approximately one ﬁfth of the time for the FE analysis to compute the Karhunen–Loève terms. The failure probability assumes for the nominal system ðh ¼ 0Þ ^f ¼ 4:67 10 7 and a standard deviation of this estithe value p mate of rp^f ¼ 1:46 10 7 . However, it is well known that the reliability is quite sensitive with respect to the variations of structural properties. This sensitivity is shown by the results of Direct Monte Carlo simulation (see Eq. (55)) by randomly sampling over the uncertain structural parameters fhðjÞ g1600 j¼1 . Direct Monte Carlo sampling leads ^f ¼ 2:32 10 4 and the standard deviation of to the estimate p rpf ¼ 3:43 10 5 . Fig. 12 shows the histogram of all independent estimates. It is observed, that the estimate for the failure probability varies over many orders of magnitude. Hence, the procedure of Direct Monte Carlo Simulation cannot be used in an efﬁcient manner. 3.5.2. Critical domain of uncertain structural parameters The domain of the structural parameters which contributes most to the total failure probability has been derived in Section 2.6. Eq. (61) provides a quantitative guidance to determine this domain. Its solution, shown in Table 2, requires the gradient of the standard deviation due to the dynamic excitation with respect to the uncertain structural parameters. To avoid direct differentiation for all 244 uncertain structural parameters, the gradient estimation

KHL-terms of curvature of girder at p-2 0.0003

Histogram of 1600 DMC estimates p_f=10^{-k} 1

1 10 40 80

0.0002

0.1

0.0001 0.01

-0.0001

0.001

-0.0002 0

8 time t [s]

0.0001 0

k Fig. 11. Karhunen–Loève terms yðjÞ ðtÞ; j1; 10; 40; 80 of the curvature of girder g-1 at position p-2 in ﬂoor ﬁve for h ¼ 0.

Fig. 12. 1600 estimates by direct Monte Carlo of the stochastic structure.

H.J. Pradlwarter, G.I. Schuëller / Computers and Structures 88 (2010) 74–86

Table 2 The ﬁrst three most important components (k) of the structural design point h computed by three iterations s ¼ 1; 2; 3. s

c ðsÞ

0 0.00070

k 26 244 114

hk 0 0 0

1 0.00190

ð0Þ

2 0.00110

ð1Þ

3 0.00111

ð2Þ

hk 1.92 1.52 0.69

ð3Þ

hk 2.07 1.18 0.76

hk 2.05 1.17 0.75

Twenty inpendent estimates of pf by LS 0.001 0.0009 0.0008 0.0007 0.0006 0.0005

procedure is employed [12,11] to compute these components which significantly influence the standard deviation of the critical response. It was observed that the variability of cðt ; hÞ is governed by only three random variables, namely the numbers 26, 244 and 114. Number 26 controls the stiffness of the girder of all floors, number 244 controls the log-normally distributed damping ratio, and the number 114 the local stiffness deviation at the considered girder at floor 5. These three random variables cause 97% of the total variability of cðt ; hÞ, i.e. Eq. (C.4) lead to ¼ 0:03. The gradient estimation procedure has been used only in the first step, i.e. to identify the most important parameters and to compute the associated accuracy in terms of . This procedure required 44 FE analyses to arrive at the solution as shown. The iteration for s ¼ 1; 2 in Eq. (62) are then performed by finite differences of the three identified random variables. Table 2 summarizes the result of the failure point nearest to the origin in standard normal parameter space. 3.5.3. Line sampling within the uncertain structural parameter space The procedure outlined in Section 2.7 is applied to compute efficiently the total first excursion probability of the uncertain strucðjÞ tural system. Fig. 13 shows the five discrete values pf ðck Þ of the conditional failure probability at five points along 20 random lines by using bS ¼ 2:48 and D ¼ 0:6. The significant increase of the conditional failure probabilities in the direction of the design point demonstrates the high sensitivity of the failure probability with respect to the particular uncertain structural parameters. Fig. 14 shows twenty independent estimates of the total first excursion probability. To obtain these results 100 FE analyses (20 ^f ¼ 0:00046 5) have been carried out. The estimated mean is p with a standard deviation of rp^f ¼ 0:000040. When compared with results of Direct Monte Carlo sampling in Fig. 12 in the uncertain parameter space, one observes that these results are approximately by a factor of two larger. LS, however, explores the impor-

0.1

0.0004 0.0003 0.0002 0.0001 2

Fig. 14. Twenty independent estimates of the total ﬁrst excursion probability of the ^f ¼ 0:00046 with standard deviation uncertain structure lead to the mean p rp^f ¼ 0:000040.

tant domain systematically and is therefore more reliable. This can be also seen by the substantial smaller coefficient of variation of approximately 8% compared with 16% and a 16 times larger number of FE analyses when using Direct MCS. 4. Conclusions The developments as shown here allow to draw the following conclusions: 1. The reliability evaluation of linear structural systems (general FE-models) with uncertain structural parameters subjected to general Gaussian dynamic excitation is feasible. To the authors knowledge, the feasibility of a reliability evaluation in dynamics for a large FE-model and a large number of structural uncertainties is demonstrated the first time in a numerical example. 2. A novel procedure to identify the critical uncertain parameters has been introduced. 3. Efficiency is gained by performing modal analysis, impulse response functions combined with Fast Fourier Transforms (FFT) and a new Line Sampling approach which focuses on the important failure domain within the uncertain parameter space. 4. The presented approach is accurate and robust, also for cases where the uncertainties of the structural parameters are large. 5. The reliability of linear systems is quite sensitive to structural uncertainties – as demonstrated in the numerical example – and therefore must not be ignored.

Acknowledgement

0.01

This work has been supported by the Austrian Science Foundation (FWF) under contract number P19781-N13 (Simulation Strategies for FE Systems under Uncertainties), which is gratefully acknowledged by the authors.

0.001

0.0001

Appendix A. Line sampling in a single direction 1e-05 2

Fig. 13. Five discrete conditional failure probabilities along twenty lines in the direction of the structural design point using bS ¼ 2:48 and D ¼ 0:6.

Line sampling (LS) [7,15] has been developed for estimating low probabilities of failure in high dimensional reliability problems. LS is particularly efﬁcient if a so called important direction can be computed, which points towards the failure domain nearest to

H.J. Pradlwarter, G.I. SchuĂŤller / Computers and Structures 88 (2010) 74â&#x20AC;&#x201C;86

the origin (see Fig. 3). LS is robust, since unbiased reliability estimates are obtained, irrespectively of the important direction a, and any deviation from the optimal direction merely increases the variance of the estimate. Suppose the important direction a has been estimated by an appropriate procedure, e.g. gradient estimation, stochastic search algorithm, design point, etc. Letâ&#x20AC;&#x2122;s assume further that the reliability is computed in standard normal space of dimension n. Then, any point n in the n- dimensional standard normal space is represented by the projection on a

c Âź aT n

Ă°A:1Ă&#x17E;

c2Â˝i Ă°t; hĂ&#x17E; Âź

nĂž1 i X h

Ă°jĂ&#x17E;

yÂ˝i Ă°t; hĂ&#x17E;

Ă°B:6Ă&#x17E;

jÂź1

The important direction aÂ˝i for LS is selected at time t i ; aÂ˝i Âź aĂ°t i Ă&#x17E;, when the excursion probability assumes its highest value, or equivalently when the reliability index bi Ă°tĂ&#x17E; assumes it smallest value:

aÂ˝i Ă°hĂ&#x17E; Âź aĂ°t i ; hĂ&#x17E; Ă°B:7Ă&#x17E; " # " # Ă°0Ă&#x17E; Ă°0Ă&#x17E; Ă°0Ă&#x17E; Ă°0Ă&#x17E; b y Ă°t i Ă&#x17E; y Ă°t i Ă&#x17E; b b y Ă°t; hĂ&#x17E; y Ă°t; hĂ&#x17E; b 6 min ; min ; ; cÂ˝i Ă°t i Ă&#x17E; cÂ˝i Ă°t i Ă&#x17E; cÂ˝i Ă°t; hĂ&#x17E; cÂ˝i Ă°t; hĂ&#x17E; 06t6T

Ă°B:8Ă&#x17E;

and the Ă°n 1Ă&#x17E; dimensional space n? Ă°aĂ&#x17E;:

n Âź ca Ăž n? Ă°aĂ&#x17E;

Ă°A:2Ă&#x17E; f~ n?Ă°jĂ&#x17E; Ă°aĂ&#x17E;gNjÂź1

LS is performed by selecting random points in the space n? Ă°aĂ&#x17E; by applying direct Monte Carlo Simulation. Random lines are speciďŹ ed by the variable c and ~ n?Ă°jĂ&#x17E; Ă°aĂ&#x17E;

nĂ°jĂ&#x17E; Ă°cĂ&#x17E; Âź ca Ăž ~n?Ă°jĂ&#x17E; Ă°aĂ&#x17E;:

Ă°A:3Ă&#x17E;

j is determined, For each random line, the safe domain bj < c < b leading to an estimate for the failure probability Ă°jĂ&#x17E; j Ă&#x17E;; pf Âź UĂ°bj Ă&#x17E; Ăž UĂ° b

Ă°A:4Ă&#x17E;

where UĂ° Ă&#x17E; denotes the cumulative standard normal distribution. Finally, an unbiased estimate for the failure probability is obtained from the average

pf Âź

N 1 X Ă°jĂ&#x17E; p : N jÂź1 f

Ă°A:5Ă&#x17E;

Appendix B. Evaluation of important directions in linear dynamics To obtain these speciďŹ c important directions, aĂ°tĂ&#x17E; at discrete instances ft i gLiÂź1 are selected. Â˝i

a Ă°hĂ&#x17E; Âź

aĂ°t i ; hĂ&#x17E;

Ă°B:1Ă&#x17E;

The instance t i is determined in the Ă°n Ăž 1 iĂ&#x17E;-dimensional subspace nÂ˝i n, deďŹ ned as the subspace orthogonal to all previously selected important directions faÂ˝j gi 1 jÂź1 and deďŹ ning the initial condition nÂ˝1 Âź n Â˝j

Â˝i

ha ; n i Âź 0 8 j Âź 1; . . . ; i 1;

i Âź 2; 3; . . . ; L

Ă°B:3Ă&#x17E;

Â˝i

where R is a Ă°n Ăž 1 iĂ&#x17E; Ă°n iĂ&#x17E; dimensional orthonormal matrix orthogonal to aÂ˝i . This matrix is established by populating ďŹ rst all entries of RÂ˝i by independent random numbers and applying subsequently the well known Gram-Schmidt orthogonalization algorithm. The deterministic KL functions yĂ°jĂ&#x17E; Ă°t; hĂ&#x17E; need also to be transformed because of the change of orientation. DeďŹ ne

YÂ˝i Âź

Ă°1Ă&#x17E; Ă°2Ă&#x17E; Ă°nĂž1 iĂ&#x17E; Â˝yÂ˝i Ă°t; hĂ&#x17E;; yÂ˝i Ă°t; hĂ&#x17E;; . . . ; yÂ˝i Ă°t; hĂ&#x17E;

with the initial condition transformed by

Ă°jĂ&#x17E; yÂ˝1 Ă°t; hĂ&#x17E;

YÂ˝i Ă°t; hĂ&#x17E; Âź YÂ˝i 1 Ă°t; hĂ&#x17E; RÂ˝i ;

i Âź 2; 3; . . . ; L

2 Â˝i Ă°t; hĂ&#x17E;

In the following, the actual evaluation of the derivatives Ă°sĂ&#x17E; @ cĂ°t ; h Ă°sĂ&#x17E; Ă&#x17E;=@hk is discussed, since it might considerably affect the efďŹ ciency of the procedure. For the case where only very few uncertain structural quantities are speciďŹ ed, differentiation for all parameters fhk gKkÂź1 by ďŹ nite differences can be carried out in a straight forward manner, requiring additional K ďŹ nite element analyses (FEA). In general, however, the number K of uncertain structural quantities might be quite large, where K FEA would seriously reduce the efďŹ ciency of the procedure. Although, the number of uncertain structural properties might be large, only a limited number of uncertain parameters have usually a signiďŹ cant effect on the standard deviation cĂ°t ; hĂ&#x17E;. These quantities are sometimes known a priori and often they cannot be speciďŹ ed with sufďŹ cient conďŹ dence. Suppose an estimate where only I < K uncertain quantities signiďŹ cantly inďŹ&#x201A;uence the standard deviation cĂ°t ; hĂ°sĂ&#x17E; Ă&#x17E;. It is Ă°sĂ&#x17E; then proposed to compute ďŹ rst these derivatives @ cĂ°t ; hĂ°sĂ&#x17E; Ă&#x17E;=@hi2I by ďŹ nite difference. These directly evaluated derivatives are denoted as

ji2fIg Âź @ cĂ°t ; hĂ°sĂ&#x17E; Ă&#x17E;=@hĂ°sĂ&#x17E; i2fIg

Ă°C:1Ă&#x17E;

Since the set fIg is just an estimate, a check whether indeed no important quantities hjRI are missing is necessary. For this purpose, the standard deviation cĂ°t ; hĂ&#x17E; has to be computed for the set fcĂ°t ; hĂ°jĂ&#x17E; Ăž DĂ°kĂ&#x17E; gIkÂź0 , with the following properties:

DĂ°0Ă&#x17E; Âź 0; i DĂ°j>0Ă&#x17E; Âźd i

i Âź 1; 2; . . . ; K

Ă°C:2Ă&#x17E;

1 for 0 6 ui;j < 0:5

Ă°C:3Ă&#x17E;

Ăž1 for 0:5 6 ui;j 6 1

Ă°B:2Ă&#x17E;

where h ; i denotes the inner product. The Ă°n Ăž 1 iĂ&#x17E;-dimensional subspace nÂ˝i is computed by the linear transformation

nÂ˝i Âź nÂ˝i 1 RÂ˝i ;

Appendix C. Gradient estimation

Ă°B:4Ă&#x17E;

Ă°jĂ&#x17E;

Âź y Ă°t; hĂ&#x17E;, these terms are also

Ă°B:5Ă&#x17E;

The variance c of the critical response yĂ°t; hĂ&#x17E; in these subspace nÂ˝i is determined analogous as in Eq. (39)

with 0:01 6 d 6 0:1 and uniformly distributed independent random numbers 0 < ui;j < 1 drawn by a random number generator. The accuracy achieved by this procedure can be estimated by comparing the Euclidean norm of the vector b and the vector c,

Âź

kck kbk

Ă°C:4Ă&#x17E;

where the components of the vectors are deďŹ ned as follow:

bk Âź cĂ°t ; hĂ°sĂ&#x17E; Ăž DĂ°kĂ&#x17E; Ă&#x17E; cĂ°t ; hĂ°sĂ&#x17E; Ă&#x17E;; X c k Âź bk ji DĂ°kĂ&#x17E; i

k Âź 1; 2; . . . ; J

Ă°C:5Ă&#x17E; Ă°C:6Ă&#x17E;

i2fIg

In case < 0:2, the estimate might be regarded as sufďŹ ciently accurate, otherwise the estimate requires improvements. For an efďŹ cient procedure for improvements it is referred to [12,11]. An abbreviated version is shown in the following: Components hj , which are likely to have a great effect, can be Ă°kĂ&#x17E; recognized by the correlation of Dj and the component ck . It is suggested to determine for all components fj R fIgg the estimate

H.J. Pradlwarter, G.I. Schuëller / Computers and Structures 88 (2010) 74–86

j^ j ¼ P

ðkÞ k¼1 Dj c k

2 ¼ ðkÞ J k¼1 Dj

ðkÞ k¼1 Dj c k 2

ðC:7Þ

In a further step, the component m with the absolutely largest esti^ jRfIg j is selected and the derivative jm is computed by ^ m j P jj mate jj ﬁnite differences. The improvement is measured by after updating the vector c

ck #ck jm DðkÞ m :

ðC:8Þ

This procedure is repeated until a satisfactory accuracy is obtained. If the improvements are insigniﬁcant, the sample size J needs to be enlarged as shown in [11]. References [1] Adomian G. Stochastic Systems. New York: Academic Press; 1983. [2] Au SK, Beck JL. First excursion probabilities for linear systems by very efﬁcient importance sampling. Probabilistic Engineering Mechanics 2001;16:193–207. [3] Chryssanthopoulos MK, Dymiotis C, Kappos AJ. Probabilistic evaluation of behaviour factors in ec8-designed r/c frames. Engineering Structures 2000;22: 10281041. [4] Clough RW, Penzien J. Dynamics of Structures. Intern. Student ed. Auckland: McGraw-Hill; 1975.

[5] Ghanem R, Spanos P. Stochastic Finite Elements: A Spectral Approach. Springer-Verlag; 1991. [6] Karhunen K. Über lineare Methoden in der Wahrscheinlichkeitsrechnung. American Academy of Science Fennicade Series A 1947;37:3–79. [7] Koutsourelakis PS, Pradlwarter HJ, Schuëller GI. Reliability of structures in high dimensions, part I: algorithms and applications. Probabilistic Engineering Mechanics 2004;19(4):409–17. [8] Lin YK. Probabilistic Theory of Structural Dynamics. New York: McGraw-Hill, Inc., McGraw-Hill Company; 1967. [9] M. Loève, Fonctions aleatoires du second ordre, supplemeent to P. Levy. In Processus Stochastic et Mouvement Brownien. Gauthier Villars, Paris, 1948. [10] W.L. Oberkampf, S.M. DeLand, B.M. Rutherford, K.V. Diegert, K.F. Alvin, Estimation of total uncertainty in computational simulation. Technical Report, Sandia National Laboratories, Albuquerque, NM SAND2000-0824, USA, 2000. [11] Pellissetti MF, Pradlwarter HJ, Schuëller GI. Relative importance of uncertain structural parameters, part II: applications. Computational Mechanics 2007;40(4):637–49. [12] Pradlwarter HJ. Relative importance of uncertain structural parameters part I: algorithm. Computational Mechanics 2007;40(4):627–35. [13] Pradlwarter HJ, Schuëller GI. Excursion probability of non-linear systems. International Journal of Non-Linear Mechanics 2004;39(9):1447–52. [14] Schenk CA, Schuëller GI. Uncertainty Assessment of Large Finite Elements Systems. Berlin/Heidelberg/New York: Springer-Verlag; 2005. [15] Schuëller GI, Pradlwarter HJ, Koutsourelakis PS. A critical appraisal of reliability estimation procedures for high dimensions. Probabilistic Engineering Mechanics 2004;19(4):463–74. [16] Soong TT, Grigoriu M. Random Vibration of Mechanical and Structural Systems. New Jersey: Prentice-Hall; 1993.

Computers and Structures 88 (2010) 87–94

Contents lists available at ScienceDirect

Computers and Structures journal homepage: www.elsevier.com/locate/compstruc

A ﬁnite element procedure for multiscale wave equations with application to plasma waves Haruhiko Kohno a,*, Klaus-Jürgen Bathe b, John C. Wright a a b

Plasma Science and Fusion Center, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA

a r t i c l e

i n f o

Article history: Received 8 April 2009 Accepted 1 May 2009 Available online 2 June 2009 Keywords: Waves Multi-scale Spectral method Finite elements Plasma Radio frequencies

a b s t r a c t A ﬁnite element wave-packet procedure is presented to solve problems of wave propagation in multiscale behavior. The proposed scheme combines the advantages of the ﬁnite element and spectral methods. The basic formulation is presented, and the capabilities of the procedure are demonstrated through the solution of some illustrative problems, including a problem that characterizes the mode-conversion behavior in plasmas. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Although much research effort has been spent on solving wave propagation problems, the accurate solution of many such problems is frequently still difficult, in particular when multiscale behavior is involved. Of course, in general, a numerical solution needs to be employed [1,2]. However, when wave numbers vary in magnitude over the domain, the wave numbers may be very large in certain regions, in particular where there is resonance, requiring a fine mesh to capture this fine-scale variation accurately. In this case, if conventional numerical methods are used, even though most of the domain should not require a fine mesh, we still have to provide the fine discretization for the entire domain. The reason is that the waves will travel throughout the domain and we cannot predict precisely prior to the analysis where resonance will occur. Indeed, frequently, it is the objective of the analysis to detect the regions of resonance. A specific example in mind is the solution of waves in plasmas. Applying radio-frequency waves in order to raise plasma temperatures is an important subject of research for nuclear fusion. Much effort has been directed to uncover the mechanisms of electromagnetic wave propagations in plasmas (see Refs. [3,4] and the references therein) and computer programs to solve wave propagations in plasmas have been developed [5–7]. Since in these * Corresponding author. E-mail address: kono@mit.edu (H. Kohno). 0045-7949/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2009.05.001

numerical solutions also the phenomenon of mode conversion needs to be addressed, the usual numerical techniques to solve wave propagation problems are not efficient. To solve wave propagation problems accurately, the spectral method [8] or spectral finite element method have been used [9–11] and good results have been obtained in certain analyses. However, these methods can be computationally expensive, and more importantly, the methods show intrinsic difficulties in satisfying the boundary conditions for arbitrary-shaped domains. Since in many wave propagation analyses, the domain considered is geometrically complex, the available spectral techniques may not be effective. Another possibly more efficient approach is to utilize basic interpolation functions that are enriched with waves. This means in essence to construct special interpolation functions that are more amenable to capture the desired response. This approach is rather natural to increase the effectiveness of the finite element method for the solution of specific problems, and has been pursued for a long time, like for example (that is, not giving an exhaustive list of references) in the analysis of wave propagations [12–14], global local solutions [15,16], piping analyses [17], the development of beam elements [18], and in fluid flow analyses [19,20]. Such methods have lately also been referred to as partition of unity methods or extended finite element methods, see for example [21– 24]. In addition, recently, discontinuous Galerkin methods [25] and related techniques have been researched for the solution of wave propagation problems, but these techniques are computationally very expensive to use.

H. Kohno et al. / Computers and Structures 88 (2010) 87–94

Whenever such a problem-specific method is proposed, the generality for a specific class of problems and effectiveness are crucial. For plasma wave problem solutions, Pletzer et al. proposed a wave-packet approach using the Gabor functions as envelopes [26]. Although this method has several good features, five parameters need to be selected, where it is difficult to find near optimal choices. Also, since the values of the Gabor functions are nonzero in the entire calculation domain, a cutoff value has to be defined. Furthermore, it is difficult to incorporate general boundary conditions. Our objective in this paper is to present a finite element scheme in which basic finite element interpolations are used enriched with wave packets. The method is quite simple and is based on the standard finite element method [1] and spectral method [8], but does not have the above-mentioned disadvantages. It turns out that the resulting interpolation functions have the same structure as those proposed in References [12,13] but can be applied to a much broader range of problems. Specifically, the procedure can also be used to solve a range of plasma wave propagation problems, for example in which mode conversion occurs. In these cases, waves with dramatically different wavelengths can exist in localized regions, which are determined by sophisticated plasma models considering kinetic effects. An important point is that the governing equations corresponding to the kinetic models include integrals, since the dielectric tensor is evaluated by integrating over the whole of velocity space and past particle trajectory time. For that reason, the methods referenced above [12–14,21,22] cannot directly be used to solve such plasma wave problems, because they use solutions of some specific differential equations. Our approach utilizes classical finite element interpolations with spectral enrichments, and can be applied to the equations including integrals as well as general differential equations. The combined interpolation technique can be used to easily satisfy Dirichlet boundary conditions and solve for many different wave numbers in one solution. We first present the numerical procedure in detail and then give the solutions of some test problems, including a problem modelling wave behavior in plasmas. We show that the proposed finite element method gives more accurate results than the conventional finite element method for wave propagation problems. While we only consider one-dimensional linear problems, there is considerable intrinsic potential of the method to be effective for multidimensional and even nonlinear solutions.

2. Finite element wave-packet approach The method proposed here is based on three important features: the technique can be thought of as using the interpolations of the traditional finite element method enriched by waves, the resultant global coefficient matrix is sparse as in finite element methods, and the boundary conditions are easily incorporated. The purpose of this section is to describe each feature in detail.

The basis of the proposed scheme is a weak form of the weighted residual method [1]. Consider a general one-dimensional ordinary differential equation written as L[u] + f(x) = 0, where L is ^ be an approximate numeran ordinary differential operator. Let u ^ is determined such that the ical solution. The numerical solution u following integral equation is satisﬁed:

^ þ f ðxÞÞdX þ hðxÞðL½u

^ ÞdC ¼ 0; hðxÞðB½u B½u

2.2. Linear, quadratic and Hermitian wave-packet interpolation functions The interpolation functions are constructed by multiplying sinusoidal functions by well-known ﬁnite element interpolation ^ and the weight function functions. First, the numerical solution u h are expressed using the linear or quadratic wave-packet interpolation functions g(i,j) as follows:

^ ðxÞ ¼ g ði;jÞ ðxÞuði;jÞ ; u

hðxÞ ¼ g ði0 ;j0 Þ ðxÞhði0 ;j0 Þ ;

ð2Þ

where the superscript * denotes the complex conjugate; u(i,j), hði0 ;j0 Þ are nodal complex variables in the coordinate-frequency space 0 identified by the global node number i (i ) and the harmonic number 0 j (j ). Here the summation convention applies to the subscripts i and j. Since our methods utilize a finite element interpolation function as an envelope function, the value of the envelope function is one at some nodal point xk and zero at every xj (j – k). This allows the functions g(i,j) to be defined locally as follows. For the linear case:

g ða;jÞ ðnÞ ¼

1 Dx ð1 þ na nÞ exp i2pmj xe þ n : 2 2

ð3Þ

For the quadratic case:

g ða;jÞ ðnÞ ¼

na n Dx ð1 þ na nÞ þ ð1 n2a Þð1 n2 Þ exp i2pmj xe þ n ; 2 2 ð4Þ

where i, xe, Dx and n are the imaginary unit, the x-coordinate at the center of an element, the length of an element and the coordinate variable in the calculation space ( 1 6 n 6 1), respectively; the physical space is then related to the calculation space by x = xe + (Dx/2)n. The subscript a denotes the local node number, and the values of na are n1,2 = 1, 1 for the linear case and n1,2,3 = 1, 1, 0 for the quadratic case, respectively. The wave numbers 2pmj are determined by mj = jm, where m is the fundamental frequency and j is an integer in the range (NF 1)/2 6 j 6 (NF 1)/2 with the cutoff number of harmonics NF. Here NF P 1 is an odd integer. The schematic proﬁle of a linear wave-packet interpolation function is shown in Fig. 1. As we will see in numerical examples in Section 4, the quadratic wave-packet interpolation is actually more effective. Another possibly more efﬁcient wave-packet approach can be established by employing Hermitian cubic beam functions [1] where then the nodal values and also the derivative values at the nodes are used. This makes the expressions for the numerical solution and the weight function slightly different from Eq. (2):

~ ði;jÞ ; ^ ðxÞ ¼ g ði;jÞ ðxÞu u

~ 0 0: hðxÞ ¼ g ði0 ;j0 Þ ðxÞh ði ;j Þ

ð5Þ

Here the Hermitian wave-packet interpolation functions comprise two different expressions:

2.1. Foundation of the numerical method

solution and the weight function are given by the same type of interpolation functions, which are formulated next.

g ði;jÞ ¼ g 1ði;jÞ ¼

g 2ðk;jÞ ;

~ ði;jÞ ¼ uði;jÞ u ¼ u0ðk;jÞ

for 1 6 i 6 Nx ; for Nx þ 1 6 i 6 2Nx 1

Piecewise-linear envelope

Waves

ð1Þ

where h(x) is a weight function, B is an operator for the boundary term, X and C denote the calculation domain and its boundary, respectively. Using the standard Galerkin approach, the numerical

Fig. 1. Schematic diagram of a linear wave-packet interpolation function.

ð6Þ

H. Kohno et al. / Computers and Structures 88 (2010) 87–94

~ ), where N is the total number of nodes, and (same applies to h ði;jÞ x the value of the subscript k is related to the value of i by k = i Nx. In a similar way to the linear and quadratic wave-packet interpolations, these functions in Eq. (6) can be written locally as follows:

1 Dx ðn þ na Þ2 ð na n þ 2Þ exp i2pmj xe þ n ; 4 2 D x D x g 2ða;jÞ ðnÞ ¼ ðn þ na Þ2 ðn na Þ exp i2pmj xe þ n ; 8 2

g 1ða;jÞ ðnÞ ¼

ð7Þ

where n1,2 = 1, 1. The real-valued proﬁles of the Hermitian wavepacket interpolation functions are shown in Fig. 2. For a real-valued solution, we can easily derive the following restrictions from Eqs. (2) and (5):

uða;jÞ ¼ u ða; jÞ ; u0ða;jÞ ¼ u0 ða; jÞ ;

ð8Þ

where the equation involving derivatives is of course only considered for the Hermitian wave-packet interpolation functions. These relations reduce the number of unknowns to half and consequently, the size of the global matrix to a quarter. Using Eq. (8), for example, we can modify the linear wave-packet interpolation functions as follows:

^ ðxÞ ¼ g aða;0Þ uða;0Þ þ u

ðNFX 1Þ=2 h

ðRÞ

ðIÞ

g bða;jÞ uða;jÞ þ g cða;jÞ uða;jÞ

i ð9Þ

j¼1

~ ða;mÞ ; ¼ g ða;mÞ u with

1 ð1 þ na nÞ; 2 Dx ¼ ð1 þ na nÞ cos 2pmj xe þ n ; 2 Dx ¼ ð1 þ na nÞ sin 2pmj xe þ n 2

g aða;0Þ ¼ g bða;jÞ g cða;jÞ

g ða;mÞ ¼ g aða;0Þ ¼

~ ða;mÞ ¼ uða;0Þ u

g bða;jÞ ;

¼ g cða;kÞ

ðRÞ uða;jÞ ðIÞ

¼ uða;kÞ

ð10Þ

for m ¼ 0;

2.3. Imposing the boundary conditions

for 1 6 m 6 ðNF 1Þ=2; for ðNF 1Þ=2 þ 1 6 m 6 N F 1; ð11Þ

ðRÞ uða;jÞ ,

other by j = m, k = m (NF 1)/2. Of course, if we consider a general plasma wave, the numerical solution is always complex, and hence Eqs. (8)–(11) are not applicable. An interesting observation is that for j = 0 all the wave-packet interpolation functions given in Eqs. (3), (4), and (7) reduce to the usual finite element interpolation functions as a result of mj = jm = 0. Thus, for NF = 1, the present interpolation scheme consists only of the conventional finite element interpolation functions, and indeed the present wave-packet approach is identical to the conventional finite element method when NF = 1 (see Section 2.3). We will see that this property leads to the straightforward treatment of the boundary conditions. The present scheme results in a relatively low computational cost since the global matrix is sparse. This sparsity is due to the local interpolation of wave packets. As an example, we show the distribution of the global matrix elements for the case of using the Hermitian functions in Fig. 3, where the nonzero regions are block-diagonalized with a regular bandwidth of 3NF. As an illustration, consider a one-dimensional sine-wave problem described by u00 + a2u = 0 in the range 0 6 x 6 1 subject to the boundary conditions u(0) = 0, u0 (1) = a. Here a is a constant with cos a = 1. The exact solution for this problem is then given by u = sin (ax). Fig. 4a shows a numerical solution obtained by the linear finite element wave-packet approach for a = 4p, m = 0.5, Nx = 2 and NF = 9. As seen, with only one element used, we obtain virtually the exact analytical results. This is the desired result since the method is based on the Fourier decomposition technique so that any smooth function should be reproduced by the combination of sinusoidal waves with different wave numbers regardless of the value of Nx. Fig. 4b is a semi-log plot of the error norm, which R R ^ Þ2 dx= u2 dx 1=2 , as a function of NF. We is defined by k k ½ ðu u notice that the error decreases logarithmically with the number of harmonics for NF P 5. Due to this feature, the present wave-packet approach can yield more accurate results compared to the conventional finite element method by orders of magnitude.

ðIÞ uða;jÞ

where are the real and imaginary parts of u(a,j), respectively, and the subscripts j, k and m in Eq. (11) are related to one an-

An important feature of the present method is the ease of imposing the boundary conditions. Consider a one-dimensional problem governed by a second-order differential equation. When imposing the Dirichlet boundary condition, we choose a weight function whose value is forced to be zero at the boundary in the

0.05

1.2

g1(1,j)

1.0

g1(2,j)

0.04

0.8

0.03

0.6 0.02

0.4

0.01

Re(g)

0.2

Re(g)

g2(1,j)

0.0 -0.2

0.00 -0.01

-0.4

-0.02

g2(2,j)

-0.6 -0.03

-0.8

-0.04

-1.0 -1.2 -1.0

-0.5

0.0

(a)

0.5

1.0

-0.05 -1.0

-0.5

0.0

0.5

1.0

(b)

Fig. 2. Proﬁles of the Hermitian wave-packet interpolation functions together with their envelope functions for Dx = 0.1 and mj = 100: (a) plot of Re½g 1ða;jÞ vs. n; and (b) plot of Re½g 2ða;jÞ vs. n.

H. Kohno et al. / Computers and Structures 88 (2010) 87–94

^ ðxÞ. Thus, for example, if we intend to exactly satisfy ical solution u the Dirichlet boundary condition at the boundary x = xb (the righthand side boundary), the following equality must be satisﬁed:

^ ðxb Þ ¼ g ði;jÞ ðxb Þuði;jÞ ¼ ub u

ðfor the linear=quadratic caseÞ;

~ ði;jÞ ¼ ub ^ ðxb Þ ¼ g ði;jÞ ðxb Þu u

ðfor the Hermitian caseÞ

ð12Þ

For ub being real, Eq. (12) leads to

8 i ðN FP 1Þ=2 h > ðRÞ ðIÞ > > > < j¼ ðN 1Þ=2 cosð2pmj xb ÞuðNx ;jÞ sinð2pmj xb ÞuðNx ;jÞ ¼ ub ; F

> > > > :

ðN FP 1Þ=2

j¼ ðNF 1Þ=2

i ðRÞ ðIÞ sinð2pmj xb ÞuðNx ;jÞ þ cosð2pmj xb ÞuðNx ;jÞ ¼ 0;

ð13Þ

where we note that Eq. (13) does not lead to a unique solution for NF > 1. However, the following choice always satisﬁes the boundary condition for any m and xb: Fig. 3. An example of the structure of the global matrix for the analysis using the Hermitian ﬁnite element wave-packet method.

ðRÞ

uðNx ;jÞ ¼ ub

for j ¼ 0;

¼ 0 for j – 0; ðIÞ uðNx ;jÞ

1.5

u (Numerical solution)

¼ 0 for any j:

This corresponds to the concept of imposing the exact boundary value in the conventional finite element component (j = 0). Note that Eq. (14) is consistent with the statement in Section 2.2; the present scheme reduces to the conventional finite element method for NF = 1. On the other hand, the proposed method only approximately satisfies the Neumann boundary conditions, again as in the conventional finite element method. For the linear or quadratic wavepacket approach, the value of the weight function at the boundary can be arbitrary. The boundary term in the discretized equation is calculated in the same way as in standard finite element methods. 0 For the Hermitian wave-packet approach, we specify hði;jÞ ¼ 0 at the Neumann boundary and choose the boundary nodal values in a similar way to the Dirichlet boundary condition as follows:

1.0

0.5

0.0

-0.5

-1.0

-1.5 0.0

0.2

0.4

0.6

0.8

1.0

0ðRÞ

uðNx ;jÞ ¼ u0b

(a)

for j ¼ 0;

¼ 0 for j – 0; 0ðIÞ uðNx ;jÞ

1e+1

ð15Þ

¼ 0 for any j:

Here we assume that the Neumann boundary condition is imposed at x = xb. In general, the above choice does not exactly satisfy the Neumann boundary condition because

1e+0 1e-1

! 1 2 dg ða¼2;jÞ ðnÞ 0 ^ du 2 dg ða¼2;jÞ ðnÞ ¼ uða¼2;jÞ þ uða¼2;jÞ dn dn dx x¼xb Dx n¼1;x¼xb 1 2 dg ð2;jÞ ðnÞ ¼ þ u0b uð2;jÞ Dx dn

1e-2

Norm of error

ð14Þ

1e-3 1e-4 1e-5

ð16Þ

n¼1;x¼xb

1e-6

In general, the first term on the right-hand side is nonzero so that ^ =dxjx¼x –u0b . For NF = 1, the scheme reduces to the conventional du b Hermitian finite element method, and then the Neumann boundary condition is exactly satisfied.

1e-7 1e-8 1e-9 0

(b) Fig. 4. The numerical results obtained by the linear ﬁnite element wave-packet method for m = 0.5, Nx = 2: (a) the calculated wave for NF = 9; (b) the norm of error as a function of NF.

same way as in the conventional Galerkin ﬁnite element methods. But an important point to notice is that the interpolated nodal val~ ði;jÞ ) are not identical to the nodal values of the numerues u(i,j) (or u

3. A required condition in the fundamental frequency In the present scheme, we need to specify three numerical parameters: Nx, NF and m. Here we derive one required condition for a proper choice of m related to the value of Nx. First of all, an important point is that every integral in the locally discretized equations can be written in the following form:

I¼

X n¼0

! n

Cnn

expðan þ bÞdn;

ð17Þ

H. Kohno et al. / Computers and Structures 88 (2010) 87–94 2

where

d u b ¼ i2pðmj mj0 Þxe :

ð18Þ

Here n P 0 takes integer values, and Cn are the coefﬁcients determined depending on the differential equations considered. Now let

FðnÞ ¼

nn expðan þ bÞdn:

ð19Þ

P Then Eq. (17) is simply expressed by I ¼ n¼0 C n FðnÞ. Consider ﬁrst 0 the case of mj – mj0 (i.e., j – j ). For n P 1 one can rewrite Eq. (19) as follows:

FðnÞ ¼

n 1 n n Fðn 1Þ: expðan þ bÞ a a 1

Fð0Þ ¼

eanþb dn ¼

1 aþb ðe e aþb Þ: a

nn dn ¼

1 ½1 ð 1Þnþ1 : nþ1

0 6 x 6 2;

ð23Þ

where a2 ¼ a2I for 0 6 x < 1 and a2 ¼ a2II for 1 < x 6 2. We assume that sin aI = sin aII = 0 and cos aI = cos aII subject to the boundary conditions u(0) = 0 and u0 (2) = aII. The exact solution is then uI = (aII/aI)sin (aIx) in the range 0 6 x < 1 and uII = sin (aIIx) in 1 < x 6 2. Here we consider two cases: aI = 8p, aII = 4p in case 1 and aI = 64p, aII = 8p in case 2. The discretized equation for Eq. (23) is:

dg ða;j0 Þ dg ðb;jÞ

ð21Þ

Thus, using Eqs. (20) and (21) we obtain F(n) for any value of n 0 through successive calculations. For mj ¼ mj0 ðj ¼ j Þ, the integral in Eq. (19) is easily solved as follows:

FðnÞ ¼

þ a2 u ¼ 0;

2 g ða;j0 Þ g ðb;jÞ

du dx uðb;jÞ g ða;j0 Þ dx

ð20Þ

For n = 0 we have:

¼ 0: Neumann boundary

ð24Þ As for the parameters used in the numerical scheme, we set the number of envelope positions (i.e., nodes), the cutoff number of harmonics and the fundamental frequency to Nx = 9, NF = 5, m = 1.8 (Nx = 21, NF = 11, m = 6.0) for the linear, quadratic wave-packet methods and Nx = 5, NF = 5, m = 1.5 (Nx = 11, NF = 11, m = 6.0) for the Hermitian wave-packet method in case 1 (case 2). The proﬁles of the numerical solutions obtained by the Hermitian wave-packet method are shown in Fig. 5. Fig. 6 shows the

ð22Þ

These analytical expressions are desirable since we do not need to apply any numerical integration to the integral shown in Eq. (17); consequently, the computation of each term is fast without a numerical error due to numerical integration. Now, using Eqs. (20)–(22), consider the following two important limits: |a| ? 1 and |a| ? 0. Assume that a given differential equation is discretized by properly choosing finite element wavepacket interpolation functions. For |a| ? 1, we find that jIj¼j0 j=jIj–j0 j ! 1 and jIj¼j0 j ! 1 for j – 0 in a non-sparse block (i, i0 ), where jIj¼j0 j and jIj – j0 j are the integrals obtained by adding up all the discretized derivative terms for j = j0 and j – j0 , respectively, expressed in the form of Eq. (17). On the other hand, for |a| ? 0, we find that jIj – j0 j=jIj¼j0 j ! 1 and jIj – j0 j ! 1 in a nonsparse block (i, i0 ). Of course, the numerical solutions for these cases do not make any sense. Therefore, a required condition should be |a| 1, i.e., mDx 1, for which the magnitude of every term in Eq. (17) is about like in the conventional finite element discretization (j = j0 = 0). The physical interpretation of this constraint is that the waves in the wave packet should have at least one wavelength in an element (see Fig. 1).

1.5

1.0

u (Numerical solution)

a ¼ ipðmj mj0 ÞDx;

0.5

0.0

-0.5

-1.0

-1.5 0.0

0.5

1.0

1.5

2.0

1.5

2.0

(a) 1.5

4. Numerical examples

4.1. Wave propagation through different media Consider the wave propagation problem through different media described by the following equation:

u (Numerical solution)

1.0

In this section, we illustrate the performance of the finite element wave-packet approach using three test problems. First, we solve a wave propagation through different media, then we solve the problem described by the Airy-type equation, whose exact solutions are available for comparison with the numerical results. Finally we solve a more difficult problem which models the mode-conversion behavior of the radio-frequency waves in plasmas described by the Wasow equation. We take the last two examples from Ref. [26]. In all solutions we use uniform meshes and when we compare solution accuracies with the accuracy obtained using the conventional finite element method we employ the fact that the solutions are real and use the same number of unknowns (see Section 2.2).

0.5

0.0

-0.5

-1.0

-1.5 0.0

0.5

1.0

(b) Fig. 5. Numerical solutions of the wave propagation problem through different media: (a) u = 0.5 sin (8px) in 0 6 x < 1 and u = sin (4px) in 1 < x 6 2 (case 1); (b) u = 0.125 sin (64px) in 0 6 x < 1 and u = sin (8px) in 1 < x 6 2 (case 2).

H. Kohno et al. / Computers and Structures 88 (2010) 87–94 0.08

0.08 Linear wave-packet (10 x error) Quadratic wave-packet (100 x error) Hermitian wave-packet (100 x error)

0.04

0.02

0.00

-0.02

-0.04

-0.06

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

-0.08 0.0

2.0

0.4

0.6

0.8

1.0

(a)

1.2

1.4

1.6

1.8

2.0

1.2

1.4

1.6

1.8

2.0

0.08 Linear wave-packet (10 x error) Quadratic wave-packet (1000 x error) Hermitian wave-packet (1000 x error)

0.06

0.02

u - error

0.04

0.00

-0.02

-0.04

-0.06

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

-0.08 0.0

2.0

0.2

0.4

0.6

0.8

1.0

(b)

Fig. 6. Comparison of the numerical error for the wave propagation problem through different media among the three different wave-packet methods: (a) case 1; and (b) case 2.

^ u) for the linear, quadratic comparison of the numerical error (u and Hermitian wave-packet approaches. As seen, the error is considerably smaller if we use higher-order envelope functions, although the difference between the quadratic and Hermitian wave packets is small. Fig. 7 shows the comparison of the numerical error between the present wave-packet method and the conventional finite element method with Nx = 25 in case 1 and Nx = 121 in case 2, both of which utilize the Hermitian interpolation functions. We see that the numerical results obtained using the Hermitian wave-packet method are several orders of magnitude more accurate than the results obtained using the standard finite element method. Especially, the result in Fig. 7b demonstrates that a sufficient number of harmonics yields rapid convergence for a smooth function as for the standard Fourier series (see Fig. 4b). 4.2. Airy-type equation

Fig. 7. Comparison of the numerical error for the wave propagation problem through different media between the ﬁnite element wave-packet method and the conventional ﬁnite element method: (a) case 1; and (b) case 2.

value as in Ref. [26]), and the corresponding boundary conditions are given by u0 (0) = 8.3239 and u0 (1) = 9.8696 10 5. Fig. 8 shows the proﬁle of the corresponding exact solution. The fundamental frequency, the numbers of envelope positions and Fourier modes 0.6

0.4

0.2

Hermitian FEM (0.1 x error) Hermitian wave-packet (1000 x error)

0.06

0.04

-0.08 0.0

0.2

0.08

u - error

0.00

-0.02

-0.08 0.0

Hermitian FEM Hermitian wave-packet (100 x error)

0.06

u - error

0.06

0.0

-0.2

Second, the methods are applied to the following second-order differential equation:

-0.4

d u 2

þ a2 ð1 2xÞu ¼ 0;

0 6 x 6 1;

ð25Þ

whose exact solution is described by the Airy function: u = Ai[(a/2)2/3(2x 1)]. Here the coefﬁcient a is ﬁxed at 21p/2 (the same

-0.6 0.0

0.2

0.4

0.6

0.8

x Fig. 8. Exact solution of the Airy-type equation for a = 21p/2.

1.0

H. Kohno et al. / Computers and Structures 88 (2010) 87–94

are m = 2.0, Nx = 9 for the linear, quadratic cases, Nx = 5 for the Hermitian case, and NF = 5. Fig. 9 gives the numerical error for the three different wavepacket approaches. As before, the errors obtained using the higher-order wave-packet interpolations are much smaller than the error obtained using the linear interpolation. Also, it is observed that the higher-order ﬁnite element wave-packet methods are comparable in accuracy with the Gabor element method developed by Pletzer et al. [26]. Fig. 10 shows the comparison of the numerical error between the present wave-packet method and the conventional ﬁnite element method (with Nx = 25), both of which utilize the Hermitian interpolation functions. Again, it is observed that the numerical result using the Hermitian wave-packet method is much more accurate; note that the error-scale differs by two orders of magnitude.

0.08 0.06 0.04

u - error

0.02 0.00 -0.02 -0.04 Linear wave-packet (1000 x error) Quadratic wave-packet (10000 x error) Hermitian wave-packet (10000 x error)

-0.06 -0.08 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

4.3. Wasow equation

x Fig. 9. Comparison of the numerical error for the Airy-type equation among the three different wave-packet methods.

Lastly, we consider the numerical solution of the Wasow equation, which models the mode conversion effects of radio-frequency waves in plasmas. The equation considered here is given by

(

0.08

0.02

u - error

þ k ½1 0:5ðx 0:5Þ

)

2 2

þ k ½1 160ðx 0:5Þ u þ au ¼ 0;

0 6 x 6 1;

0.04

0.00 -0.02 -0.04

Hermitian FEM (100 x error) Hermitian wave-packet (10000 x error)

-0.06 -0.08 0.0

dx 0.06

)(

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

x Fig. 10. Comparison of the numerical error for the Airy-type equation between the ﬁnite element wave-packet method and the conventional ﬁnite element method.

ð26Þ

where k2 = 2 103 and a = 8 106 subject to the boundary conditions u(0) = 0, u(1) = 1 and u0 (0) = u0 (1) = 0 (the same boundary conditions as in Ref. [26]). Eq. (26) implies the formation of multiscale waves with different wave numbers by a factor of 320. Here a comparison is made between the finite element wave-packet method and the conventional finite element method, both utilizing the Hermitian interpolation functions which can be straightforwardly applied to this fourth-order differential equation. As numerical parameters, we choose m = 10.5, Nx = 10 and NF = 11. Since an analytical solution to this problem is not available, we first calculate the problem with a very fine mesh using the Hermitian interpolation functions, and utilize the obtained result as a ‘‘quasi-exact” solution. Fig. 11 shows the numerical solution obtained with 1000 elements. We see that the fast and slow waves are coupled on the left half of the domain (see Fig. 11b), while only the slow wave having a shorter wavelength is evanescent on the

25 2

20 1

5 -1

0 -2

-5

-10 0.0

0.2

0.4

0.6

0.8

1.0

-3 0.0

0.1

0.2

0.3

(a)

(b)

0.4

Fig. 11. Numerical solution of the Wasow equation: (a) macroscopic oscillation; and (b) ﬁne scale oscillation.

0.5

H. Kohno et al. / Computers and Structures 88 (2010) 87–94

tics of these more complex waves. In further research the method should be applied to and tested in two- and three-dimensional solutions with nonuniform meshes. Also, a mathematical convergence analysis should be pursued to identify the rate and order of convergence, and the optimal value of fundamental frequency.

0.15

0.10

u - error

0.05

Acknowledgements 0.00

We would like to thank Prof. Jeffrey Freidberg and Dr. Paul Bonoli of M.I.T. for their valuable comments on this work for the application to plasmas. This work was supported in part by DoE Contract No. DE-FG02-99ER54525.

-0.05

-0.10

References

Hermitian FEM (0.1 x error) Hermitian wave-packet -0.15 0.0

0.2

0.4

0.6

0.8

1.0

x Fig. 12. Comparison of the numerical error for the Wasow equation between the ﬁnite element wave-packet method and the conventional ﬁnite element method.

right half. This is also confirmed in Eq. (26); although the sign of r1 = k2[1 0.5(x 0.5)] is always positive in the entire domain, the sign of r2 = k2[1 160(x 0.5)] changes from positive to negative at x = 0.5. The former corresponds to propagation of the fast wave at every point, whereas the latter corresponds to evanescence of the slow wave on the right half of the domain. The mixing of these very different waves makes it more difficult to accurately solve the Wasow equation compared to the equations in the previous problems. ^ uquasi exact ) between A comparison of the numerical error (u the finite element wave-packet method and the conventional finite element method is shown in Fig. 12. Again, the present wave-packet approach gives a more accurate numerical result compared to the conventional finite element solution. 5. Conclusions We presented in this paper a finite element wave-packet method for the analysis of waves through media, and solved some illustrative problems. The method is in particular directed to solve waves in plasmas accurately with a reasonable computational cost. The key idea is to enrich the usual finite element interpolations with wave packets. We see that this approach results into some favorable features drawing from both, conventional finite element and spectral methods. First, the interpolation functions are locally defined in the same way as in the conventional finite element methods, which is effective for programming. Second, this local definition results in the formation of a sparse global matrix. Third, all the integrals in the discretized equation are analytically solved, yielding simple expressions (of course, numerical integration could be used and probably has to be used for wave equations of higher dimensions). Fourth, the boundary conditions are easily incorporated in the discretized equation. In fact, the Dirichlet and Neumann boundary conditions are treated in a similar way as in the conventional finite element methods. Fifth, using the wave packets can give more accurate results than using the corresponding conventional finite element methods under the same computational costs. Plasma wave equations can be far more complex than the onedimensional equations we solved here, but the one-dimensional equations/solutions exhibit some of the fundamental characteris-

[1] Bathe KJ. Finite element procedures. Prentice-Hall; 1996. [2] Cohen GC. Higher-order numerical methods for transient wave equations. Springer; 2001. [3] Stix TH. Waves in plasmas. American Institute of Physics; 1992. [4] Freidberg JP. Plasma physics and fusion energy. Cambridge University Press; 2007. [5] Brambilla M, Krucken T. Numerical simulation of ion cyclotron heating of hot tokamak plasmas. Nucl Fusion 1988;28:1813–33. [6] Jaeger EF, Berry LA, D’Azevedo E, Batchelor DB, Carter MD. All-orders spectral calculation of radio-frequency heating in two-dimensional toroidal plasmas. Phys Plasmas 2001;8:1573–83. [7] Wright JC, Bonoli PT, Brambilla M, Meo F, D’Azevedo E, Batchelor DB, et al. Full wave simulations of fast wave mode conversion and lower hybrid wave propagation in tokamaks. Phys Plasmas 2004;11:2473–9. [8] Karniadakis GE, Sherwin S. Spectral/hp element methods for computational fluid dynamics. 2nd ed. Oxford University Press; 2005. [9] Gopalakrishnan S, Chakraborty A, Mahapatra DR. Spectral finite element method. Springer-Verlag; 2008. [10] Beris AN, Armstrong RC, Brown RA. Spectral/finite-element calculations of the flow of a Maxwell fluid between eccentric rotating cylinders. J Non-Newtonian Fluid Mech 1987;22:129–67. [11] Steppeler J. A Galerkin finite element-spectral weather forecast model in hybrid coordinates. Comput Math Appl 1988;16:23–30. [12] Astley RJ. Wave envelope and infinite elements for acoustical radiation. Int J Numer Methods Fluids 1983;3:507–26. [13] Bettess P. A simple wave envelope element example. Commun Appl Numer Methods 1987;3:77–80. [14] Bettess P, Chadwick E. Wave envelope examples for progressive waves. Int J Numer Methods Eng 1995;38:2487–508. [15] Avanessian V, Dong SB, Muki R. Forced asymmetric vibrations of an axisymmetric body in contact with an elastic half-space – a global–local finite element approach. J Sound Vib 1987;114:45–56. [16] Belytschko T, Lu YY. Global-local finite element-spectral-boundary element techniques for failure analysis. Comput Struct 1990;37:133–40. [17] Bathe KJ, Almeida C. A simple and effective pipe elbow element – linear analysis. J Appl Mech 1980;47:93–100. [18] Dvorkin EN, Celentano D, Cuitino A, Gioia G. A Vlasov beam element. Comput Struct 1989;33:187–96. [19] Kohno H, Bathe KJ. A flow-condition-based interpolation finite element procedure for triangular grids. Int J Numer Methods Fluids 2006;51: 673–99. [20] Banijamali B, Bathe KJ. The CIP method embedded in finite element discretizations of incompressible flows. Int J Numer Methods Eng 2007;71:66–80. [21] Melenk JM, Babuška I. The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 1996;139: 289–314. [22] Babuška I, Melenk JM. The partition of unity method. Int J Numer Methods Eng 1997;40:727–58. [23] Sukumar N, Moes N, Moran B, Belytschko T. Extended finite element method for three-dimensional crack modelling. Int J Numer Methods Eng 2000;48:1549–70. [24] Fries TP, Belytschko T. The intrinsic XFEM: a method for arbitrary discontinuities without additional unknowns. Int J Numer Methods Eng 2006;68:1358–85. [25] Bathe KJ, editor. Computational fluid and solid mechanics. Proceedings of the third MIT conference on computational fluid and solid mechanics 2005. Amsterdam: Elsevier; 2005. [26] Pletzer A, Phillips CK, Smithe DN. Gabor wave packet method to solve plasma wave equations. Proceedings of the 15th topical conference on radio frequency power in plasmas, vol. 694; 2003. p. 503–6.

Computers and Structures 88 (2010) 95–104

Contents lists available at ScienceDirect

Computers and Structures journal homepage: www.elsevier.com/locate/compstruc

A constraint Jacobian based approach for static analysis of pantograph masts B.P. Nagaraj a,1, R. Pandiyan a,2, Ashitava Ghosal b,* a b

ISRO Satellite Centre, Bangalore 560 017, India Dept. of Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India

a r t i c l e

i n f o

Article history: Received 10 April 2009 Accepted 21 September 2009

Keywords: Pantograph masts Deployable structures Null-space Jacobian Stiffness matrix

a b s t r a c t This paper presents a constraint Jacobian matrix based approach to obtain the stiffness matrix of widely used deployable pantograph masts with scissor-like elements (SLE). The stiffness matrix is obtained in symbolic form and the results obtained agree with those obtained with the force and displacement methods available in literature. Additional advantages of this approach are that the mobility of a mast can be evaluated, redundant links and joints in the mast can be identiﬁed and practical masts with revolute joints can be analysed. Simulations for a hexagonal mast and an assembly with four hexagonal masts is presented as illustrations. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Deployable structures can be stored in a compact conﬁguration and are designed to expand into stable structures capable of carrying loads after deployment. In their general form, they are made up of a large number of straight bars (links) connected by revolute joints and with one or more cables used for deployment or increasing the stiffness of the deployed structure (see, for example, [1,2]). Initially, the whole assembly of bars can be stowed in a compact manner and, when required, can be unfolded into a predeﬁned large-span, load bearing structural form by simple actuation of one or more cables. This characteristic feature makes them eminently suitable for a wide spectrum of applications, ranging from temporary structures that can be used for various purpose in ground to the large structures in aerospace industry. Deployable/ collapsible mast are often used for space applications since in their collapsed form they can be easily carried as a spacecraft payload and expanded in orbit to a desired size. Many deployable systems use the pantograph mechanism or scissor-like elements (SLE’s). Typically, an SLE has a pair of equal length bars connected to each other at an intermediate point with a revolute joint. The joint allows the bars to rotate freely about an axis perpendicular to their common plane. Several SLE’s are connected to each other in order to form units which in plan view appear as regular polygons with their sides and radii being the SLE’s. Several such polygons, in turn,

* Corresponding author. Tel.: +91 80 2293 2956; fax: +91 80 2360 0648. E-mail address: asitava@mecheng.iisc.ernet.in (A. Ghosal). 1 Spacecraft Mechanisms Group. 2 Flight Dynamics Group. 0045-7949/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2009.09.007

are linked in appropriate arrangements leading to deployable structures that are either flat or curved in their final deployed configurations. The assembly is a mechanism with one degree of freedom from the stowed/folded configuration till the end of deployment. The deployment is through an active cable and after deployment the assembly is a pre-tensioned structure. Active cables control the deployment and pre-stress the pantograph and passive cables are pre-tensioned in the fully deployed configuration. These cables have the function of increasing the stiffness in the fully deployed configuration [3]. 1.1. Kinematics and mobility The kinematics of multi-body mechanical systems can be studied by use of relative coordinates [4], reference point coordinates as used in the commercial software ADAMS [5] or Cartesian coordinates (also called natural/basic coordinates) [6]. In Refs. [7,8], Garcia and co-workers have used Cartesian coordinates to obtain the constraints equations for different types of joints and for kinematic analysis of mechanisms. Typical pantograph masts are overconstrained mechanisms according to Grübler–Kutzbach criteria, and in Ref. [9], Cartesian coordinates have been used to study the kinematics and mobility of deployable pantograph masts – the authors use the derivative of the constraint equations and develop an algorithm to obtain redundant link and joints in over-constrained deployable masts, perform kinematic analysis and obtain global degrees of freedom. The key advantage of Cartesian coordinates is that the constraint equations are quadratic (as opposed to transcendental equations for relative coordinates), and, hence their derivatives are linear. As shown in [9], these features allows easier

B.P. Nagaraj et al. / Computers and Structures 88 (2010) 95–104

manipulation and simplification of expressions in a computer algebra system to obtain symbolic expressions and closed-form solutions for the kinematics of pantograph masts. A disadvantage of Cartesian coordinates is that the number of variables is typically larger and tends to be (on average) in between relative coordinates and reference point coordinates. However, for analysis of pantograph masts, the number is not too large and could be handled without much difficulty in the computer algebra system, Mathematica, used in this work. The masts in their deployed configuration become pre-tensioned structures. For pre-stressed structures with pin jointed bars, the necessary condition for the structure to be statically and kinematically determinate is given by the Maxwell’s rule

3j b c ¼ 0

ð1Þ

where, j is the number of joints, b is the number of bars or links and c is the number of kinematic constraints. Calladine [10] generalized the Maxwell’s rule as

s¼b r m ¼ 3j c r 3j b c ¼ m s

ð2Þ

where, m is the number of internal mechanisms, s is the number of states of self-stress, and r is the rank of the equilibrium matrix. This equation is referred to as the extended Maxwell’s rule. The values m and s depends on the number of bars and joints, topology of the connection and on the geometry of the frame work. The numerical values of the vectors describing s and m, for a given system, can be determined from the singular value decomposition (SVD) of the equilibrium matrix. The concept of using a Jacobian matrix to evaluate the mobility was first presented by Freudenstein [11] for an over-constrained mechanism. Later, the first and higher order derivatives of constraint equations has been used for under constrained structural systems to evaluate mobility and state of self-stress by Kuznetsov [12,13] 1.2. Structural matrix The mechanism at the end of deployment becomes a pre-tensioned structure and the structural matrices are useful for evaluating the stiffness/displacement of the SLE masts in the deployed configuration. In literature, researchers have used various methods for formulating the structural matrix for an SLE. These are termed as force method [14], displacement method [15] and equivalent continuum model [16]. We describe each of these methods in brief below. 1.2.1. Force method In the force method, as used by Kwan and Pellegrino [14], the SLE is discretised into four beam elements. The equilibrium, compatability and flexibility matrices are derived for a typical beam element in a local coordinate system using shear force and bending moment relationships. These equations are transformed to the global coordinate system by using the rotation matrices and are assembled for the four beam elements, which make up the SLE. The equilibrium matrix is reduced in size by matrix partitioning and by setting the end moments to zero [18]. In this approach one can evaluate the number of self-stress states and the number of infinitesimal mechanisms of the given system by using singular value decomposition (SVD) of the equilibrium matrix [19]. 1.2.2. Displacement method The displacement method is used by Shan [15] to formulate stiffness matrix for the SLE. In his approach, each link of the SLE is called an uniplet. One uniplet of the SLE is modeled as an assem-

bly of two beam elements with mid node at the pivot point of SLE. The stiffness matrix was partitioned to have the translation terms and rotational terms in order. The final reduced stiffness matrix is obtained by condensing and removing the rotational degrees of freedom of all the three nodes. In Ref. [20], the authors have formulated the stiffness matrix for two uniplets, called as a duplet, by using the stiffness matrix of the uniplet developed above. Matrix partitioning is used to get the reduced stiffness matrix which condenses the translational degrees of freedom of the pivot node. 1.2.3. Equivalent continuum model This approach was used to predict the stiffness characteristics of deployable flat slabs when they are subjected to normal loads [16,17]. In this method, the SLE is considered as an equivalent uniform beams that deflects identically to the given loading as that of an SLE. The flat large deployable structure is substituted with an equivalent grid of uniform beams running in particular directions The beams are rigidly connected to each other. This arrangement is reduced to an equivalent orthotropic plate of constant thickness and stiffness matrix is obtained. The results predicted by this method are approximate unlike above methods and hence can only be used for initial design phase which reduces the computational time. In an exact finite element modeling the storage space requirements are large for large number of SLE units due to the complicated pivotal connections and hinged connections that require more than one nodal point to be described accurately. The equivalent approach can significantly reduce the computational effort during preliminary design stage. 1.2.4. Comparison of existing methods The force method gives the additional information about the states of self-stress and infinitesimal mechanisms. The displacement method or equivalent continuum model does not give this information. The force method uses two matrix reductions which reduces the matrix of dimension 18 14 to 12 8 in the first step. Further in the second step the matrix dimension is reduced from 12 8 to 10 6, to obtain the final reduced equilibrium matrix. The displacement method has a stiffness matrix of dimension 18 18 for the two assembled beam elements with six degrees of freedom at each node. By condensing the rotational degrees of freedom at all the nodes the matrix dimension reduces to 9 9. The reduced matrix has only translational degrees of freedom at each node. The equivalent continuum approach is useful for very large repetitive structures. However, this method does not give the accurate results when compared to other two methods and, hence, can be used only for initial design phase to reduce computational time. As mentioned earlier, at the end of deployment we get a structure capable of bearing loads, and in this paper, we extend the approach in [9] to the static analysis of deployable pantograph masts. We present a new approach to formulate the structural matrices for a typical SLE using Cartesian coordinates, the kinematic equations of the SLE/pantograph element, and the constraint Jacobian matrix. These matrices are derived by using the symbolic computation software Mathematica [21]. The results of formulations obtained by this approach matches exactly with the results of force and displacement based methods. Our approach has the advantages of the force method in evaluating the states of self-stress and infinitesimal mechanisms. However, in our approach, the final reduced equilibrium matrix can be obtained in a single step unlike in the force and displacement methods. In addition, the constraint equations of the links and joints are useful in studying the kinematics behavior of pantograph masts during deployment, in evaluating the redundancy in the links/joints of these over-constrained systems, and in obtaining the final degrees of freedom of the deployable masts. In literature the successive SLE joint connection

B.P. Nagaraj et al. / Computers and Structures 88 (2010) 95–104

are assumed to be spherical joints. In a practical pantograph mast, two revolute joints with intersecting axis are used. In this work, we have used the revolute joint constraints for the SLE connected to the successive SLEs by revolute joints. This paper is organized as follows: In Section 2, we present a brief description of the deployable masts considered for the analysis. and present the constraint equations for the links, joints and the SLE with the Jacobian matrix. In Section 3, we present the mathematical approach for the evaluation of stiffness matrix for the SLE and the detailed equations are presented in an Appendix A. In Section 4, we present the stiffness matrix for the cables used in pantograph masts. In Section 5, we present the additional constraints and the stiffness matrix due to revolute joints. In Section 6, we illustrate our approach by using a planar stacked mast and three-dimensional SLE based masts. Finally, in Section 7, we present the conclusions. 2. Kinematic description of the SLE masts In this section a brief description of the SLE masts and formulation of the constraint equations are presented for the sake of completeness (see, [9], for details). In the next section, we use these equations to derive stiffness matrices. The simplest planar SLE is shown in Fig. 1. The revolute joint in the middle connects the two links of equal length. The assembly has one degree of freedom. Fig. 2 shows a two-dimensional stacked SLE mast [3]. This consists of four SLEs stacked one above the other. The deployment angle b can vary continuously from b ¼ 0 , when the assembly is fully folded, i.e. lying flat on its base and all links are collinear, to b ¼ 45 which corresponds to the fully deployed configuration. This has eight passive cables connecting the adjacent joints of the SLEs. The cables are taut in the fully deployed configuration and slack at all other configurations. One active cable which is firmly connected to the joint 3 of SLE mast, runs over a pulley at joint 4, zig-zags down the SLE following the route shown in the figure (it runs over a pulley at each kink) and, after passing over a pulley at joint 1, is connected to the motorised drum located below the base. This mast remains stress free during folding. It can be deployed simply by turning the drum below the base and thus winding in the active cable. When the passive cable is taut the deployment is complete. At this stage the active cable is wound in little more to set up a state of self-stress in the system. Usually it is desirable that all the passive cables be in a state of pre-tension while the structure is operational to avoid the possibility of some of them might going slack when the mast is subjected to the action of external loads; it is easiest to aim for uniform state of pre-stress

Fig. 2. Stacked planar SLE mast – (a) fully deployed and (b) partially deployed.

in all cables. The uniform pre-stress can be obtained by introducing the second active cable [3]. The triangular SLE mast can be created with three SLE’s. The stacked triangular SLE mast [2] is shown in Fig. 3. This has twelve passive cables and an active cable. The active cable is ﬁrmly attached to joint 5. The double loops are connected at the intermediate joints as shown in Fig. 3. This pre-tensions uniformly all the cables in the mast. A drum is used to wind the active cable. The function of passive cables are (a) for termination of deployment, (b) increasing the stiffness of fully deployed structure, and (c) setup a state of pre-stress in the fully deployed structure resulting in pre-tensioning of all passive cables. An active cable is such that its length reduces monotonically as the structure deploys. The functions of active cable is to (a) control the deployment process, (b) setup a state of pre-stress in a fully deployed structure resulting in pre-tensioning the whole system, and (c) elimination of backlash at all joints. More than one active cable is often introduced in some structure. In practice it is advisable to have no less than two active cables to ensure minimum level of redundancy should an active cable fail. However it is impractical to introduce many active cables in the structure because different cables may require independent winding mechanisms and control units. A structure with passive and active cables remains essentially stress free in folded/partially folded conﬁgurations and is pre-stressed in the fully deployed state. These structures have high stiffness when fully deployed. 2.1. Formulation of constraints

Fig. 1. Basic planar module of SLE.

In this section we derive the kinematic constraint equations for the SLE. We will use the Cartesian/natural coordinates [6] to model the SLE. The natural or Cartesian coordinates are deﬁned at the locations of the joints and unit vectors along the joint axis to deﬁne the motion of the link completely. In the natural coordinate system the constraint equations originate in the form of rigid constraints of links and joint constraints. Consider a SLE shown in Fig. 1. This is considered as an assembly of two links 1–2 and 3–4 with a pivot p. The link 1–2 with pivot p is considered as an assembly of two link segments 1–p and p–2

B.P. Nagaraj et al. / Computers and Structures 88 (2010) 95–104

Fig. 3. Stacked triangular SLE mast – (a) full mast, (b) passive cables, and (c) active cable.

with lengths l1 and l2 , respectively. Likewise, the link 3–4 with pivot p is considered as an assembly of two link segments 3–p and p–4 with lengths l3 and l4 , respectively. 2.1.1. Rigid link constraint A rigid link is characterized by a constant distance between two natural coordinates i and j. This is given by

rij rij ¼ L2ij

ð3Þ

where rij ¼ ½ðX i X j Þ; ðY i Y j Þ; ðZ i Z j Þ T with ðX i ; Y i ; Z i Þ; ðX j ; Y j ; Z j Þ are the natural coordinates of i, j, respectively, and Lij is the distance between i and j. Using this equation for the SLE of Fig. 1 we get the following systems of equations for the four segments: 2

ðX p X 1 Þ2 þ ðY p Y 1 Þ2 þ ðZ p Z 1 Þ2 l1 ¼ 0 2

ðX 2 X p Þ2 þ ðY 2 Y p Þ2 þ ðZ 2 Z p Þ2 l2 ¼ 0 2

ðX 3 X p Þ2 þ ðY 3 Y p Þ2 þ ðZ 3 Z p Þ2 l3 ¼ 0 2

ðX p X 4 Þ2 þ ðY p Y 4 Þ2 þ ðZ p Z 4 Þ2 l4 ¼ 0

ð4Þ

2.1.2. Constraint for SLE Referring to Fig. 1, the node p is a pivot, the link segments 1 p and p 2 of link 1 2 are aligned at pivot. Hence, the cross-product of the two adjacent link segments 1 p and p 2 is given by

r1p rp2 l1 l2 sin /1 ¼ 0

ð5Þ

here, /1 is the angle between the two link segments (equal to 0 degrees for a pantograph mast). Similarly the constraint equation for the link 34 is given by

r3p rp4 l3 l4 sin /2 ¼ 0

ð6Þ

where, /2 the angle between the two link segments (0 degree in our case). Eqs. (5) and (6) along with rigid link constraint equations (Eq. (4)) of the four link segments 1p, p2, 3p and p4 form the complete set of equations for a single SLE. 2.1.3. System constraint equations The rigid constraint equations and joint constraints of SLE can be written together as

fj ðX 1 ; Y 1 ; Z 1 ; X 2 ; . . . ; Z n Þ ¼ 0 for j ¼ 1; . . . ; nc

ð7Þ

where, nc represents the total number of constraint equations including rigid link and joint constraints of SLE, and nt ¼ 3n is the total number of Cartesian coordinates of the system. The derivative of the constraint equations give the Jacobian matrix and can be symbolically written as

½J dX ¼ 0

ð8Þ

Since, Eq. (8) is homogeneous, one can obtain a non-null dX if the dimension of the null-space of ½J nc nt is at least one. The existence of the null-space implies that the mechanism possess a degree of freedom along the corresponding dX [6]. The null-space of [J] can be obtained numerically. The dimension of the null-space is the degree of freedom/ mobility of the deployable system. The deployable systems will have large number of links and joints arranged in a repetitive pattern. Using the above equations one can evaluate the possible change in degree of freedom of the deployable system with addition of each link/joints and also can identify the redundant links/ joints in the system [9]. The deployable systems at the end of deployment lock and the cables attached to the successive joints get pre-stressed there by reducing the mechanism to a structure. Using the null-space dimension of the Jacobian matrix one can evaluate the minimum number of cables required to reduce the mechanism to structure. 3. Stiffness matrix for the SLE In this section we present a method to evaluate the stiffness matrix for SLE from the constraint Jacobian matrix discussed in the previous section. The SLE is considered to have constant cross sectional area and uniform material properties. The cross section of the SLE remains plane and perpendicular to the longitudinal axis during deformation. The longitudinal axis which lies within the neutral surface does not experience any change in length. The SLE beam is long and slender and the transverse shear and rotary inertia effects are negligible. These assumptions allows the use of Euler–Bernoulli beam theory. 3.1. Stiffness matrix from length constraints From the length constraint equations, the elongation in the structural members, dL, can be related to the system displacements, dX, as

B.P. Nagaraj et al. / Computers and Structures 88 (2010) 95–104

where the Jacobian matrix, ½Jm , can be obtained from Eq. (4) (see Appendix) and dX; dL are given by

d/2 ¼ ½d/2x ; d/2y ; d/2z T are the rotations in the global coordinate system. The transformation matrix relating the global and local coordinate system is given by

dX ¼ ½dX 1 ; dY 1 ; dZ 1 ; dX 2 ; dY 2 ; dZ 2 ; dX 3 ; dY 3 ; dZ 3 ; dX 4 ;

d/01 ¼ ½R d/1

½Jm dX ¼ dL

ð9Þ

dY 4 ; dZ 4 ; dX p ; dY p ; dZ p dL ¼ ½dl1 ; dl2 ; dl3 ; dl4

d/02 ¼ ½R d/2

If the elongation dL are elastic, the member forces, dT, can be expressed with a diagonal matrix of member stiffnesses as given below:

½Sm dL ¼ dT

ð10Þ

where, the member stiffness matrix ½Sm for the length segments of SLE is given by

1 E1

A2 E2 l2

A3 E3 l3

6 6 6 0 6 ½Sm ¼ 6 6 0 6 4 0

7 7 0 7 7 7 0 7 7 5

ð11Þ

ð16Þ

where,

2 6 6 ½RW ¼ 6 6 4

Cx Cz pffiffiffiffiffiffiffiffiffiffi 0 pffiffiffiffiffiffiffiffiffiffi 2 2 2 2 C x þC z

C x þC z

Cx z p C ffiffiffiffiffiffiffiffiffiffi 0 pffiffiffiffiffiffiffiffiffiffi 2 2 2 2

3 7 7 7 7 5

C x þC z

0 2

0 0

6 ½RH ¼ 6 4 0 cos H 0

sin H

1 3

7 sin H 7 5 cos H X X

Y Y

where, for the nodes i and j with length L, C x ¼ i L j ; C y ¼ i L j and Z Z C z ¼ i L j , and H is the angle from one of the principal axis of cross section of the SLE beam. Using Eqs. (14) and (15) we get

d/02 ¼ ½R ½J34 dX34

dF ¼ ½dF 1x ; dF 1y ; dF 1z ; dF 2x ; dF 2y ; dF 2z ; dF 3x ; dF 3y ; dF 3z ; dF 4x ; dF 4y ; dF 4z ; dF px ; dF py ; dF pz T with the right-hand side denoting the load components at nodes. To be statically determinate, the load must be in the column space of the equilibrium matrix, in which case it is the equilibrium load. Substituting the Eqs. (9) and (10) in Eq. (11), we get

ð12Þ

The above equation can be written as

½Km dX ¼ dF

½R ¼ ½RH ½RU ½RW

d/01 ¼ ½R ½J12 dX12

where dF is given by

½Jm T ½Sm ½Jm dX ¼ dF

where and are the rotations in the local coordinate system and [R] is the transformation matrix relating the local and global coordinate systems [22]. The transformation matrix [R] is given by

C x þC z

A4 E4 l4

½Jm T dT ¼ dF

d/02

3 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 2x þ C 2z Cy 0 7 6 7 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 ½RU ¼ 6 2 2 7 6 C y C 0 þ C x z 5 4

In the above equation E is the Young’s modulus, A the cross sectional area the diagonal elements correspond to the axial stiffness due to elongation of the link segments, and dT ¼ ½dT 1 ; dT 2 ; dT 3 ; dT 4 T are the forces in the link segments. The equilibrium matrix for the reference conﬁguration can be written in terms of the transpose of the Jacobian matrix and we can write

ð13Þ

where, ½Km ¼ ½Jm ½Sm ½Jm is the elastic stiffness matrix of four length segments of the SLE. 3.2. Stiffness matrix due to bending The rotations d/ in the structural members can be obtained from the cross-product Eqs. (5) and (6) of SLE. The rotations are the constraint variations related to the system displacements dX and in terms of the Jacobian matrix [J] are given as

½J12 dX12 ¼ d/1 ð14Þ

The detailed matrix is given in Appendix. In the above equation dX12 is the vector ½dX 1 ; dY 1 ; dZ 1 ; dX 2 ; dY 2 ; dZ 2 ; dX p ; dY p ; dZ p T and dX34 is the vector ½dX 3 ; dY 3 ; dZ 3 ; dX 4 ; dY 4 ; dZ 4 ; dX p ; dY p ; dZ p T . These are the displacements of the link 1–p–2 and 3–p–4, respectively. Finally, d/1 ¼ ½d/1x ; d/1y ; d/1z T and

ð17Þ

Considering the bending deformation of the links and neglecting torsion the above equations can be written as

d/001 ¼ ½J1 dX12 d/002 ¼ ½J2 dX34

ð18Þ

where, d/001 ¼ ½d/01y ; d/01z T and d/002 ¼ ½d/02y ; d/02z T . Combining the above equations, we can write

d/00 ¼ ½Jn dX 00

½J34 dX34 ¼ d/2

ð15Þ

d/01

ð19Þ ½d/001 ; d/002 T .

where, d/ ¼ ments is given by

The relation between the forces and mo-

dF ¼ ½JTn dM00

ð20Þ

where, dM00 ¼ ½dM012y ; dM012z ; dM034y ; dM 034z T . If the rotations d/00 are elastic, the member moments dM00 can be expressed with a diagonal matrix of member stiffnesses as given below:

½Sn d/00 ¼ dM00

ð21Þ

where, the member stiffness matrix ½Sn for the SLE is given by

2 3E1 Iz l1 þl2

6 6 0 6 ½Sn ¼ 6 6 6 0 4 0

3E1 Iy l1 þl2

3E2 Iz l3 þl4

7 0 7 7 7 7 0 7 5

3E2 Iy l3 þl4

In the above equation E is the Young’s modulus, Iz and Iy are the second moment of area of cross section about Z and Y axes, respectively, and the diagonal elements correspond to the bending stiffness of the links 1–p–2 and 3–p–4 about Z and Y axis.

100

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By using a similar procedure as given in previous section, the ﬁnal equations can be obtained by substituting the Eqs. (19) and (21) in (20). We get

½Jn T ½Sn ½Jn dX ¼ dF

ð22Þ

and this equation can be written as

½Kn dX ¼ dF

ð23Þ

where, ½Kn ¼ ½Jn T ½Sn ½Jn is the elastic stiffness matrix for the SLE. By combining the stiffness matrix due to length constraint Eq. (13) and SLE constraint Eq. (23), we get

½Ks dX ¼ dF

ð24Þ

Fig. 4. Typical truss element with coordinates.

where, ½Ks ¼ ½Km þ ½Kn is the elastic stiffness matrix due to length and SLE constraints 3.3. Rank of stiffness matrix The stiffness matrix is given by

½Ks ¼ ½Js T ½Ss ½Js ¼ ½Js T ð½S s T ½S s Þ½Js ¼ ð½Js ½S s Þ T ð½Js ½S s Þ

ð25Þ

½S s

where is a diagonal matrix whose diagonal elements are square root of the diagonal elements of [Ss ] and [Js ] is the Jacobian matrix of a single SLE. Since for a regular bar frame works all elements from ½Ss are positive, the rank of ½Js ½S s takes it value from the rank of ½Js . Furthermore

rankð½Ks Þ ¼ rankðð½Js ½Ss ÞT ð½Js ½Ss ÞÞ ¼ rankð½Js ½Ss Þ ¼ rankð½Js Þ

ð26Þ

3.4. Comparison with other methods In Ref. [15], the displacement method was used to derive the stiffness matrix. The link 1–2 is also called as an uniplet in the reference paper. By using the ﬁrst two length constraint equations in (4) of the link 1–p–2 and the SLE constraint Eq. (5) we can formulate the Jacobian matrix. The stiffness matrix for the uniplet can be computed by using Eq. (24). It can be observed that the stiffness matrix, obtained by our method, matches exactly with the matrix formulated in Ref. [15] by the displacement method. In Ref. [14], the authors have used force method to arrive at the stiffness matrix. Using the length constraint Eq. (4) and SLE constraint Eqs. (5) and (6) we can formulate the Jacobian matrix as described in the previous section. By using the coordinate system of Ref. [14] and making the substitutions in Jacobian matrix Eq. (8), we can observe that matrix obtained by our method is same as the matrix shown in Eq. (37) of Ref. [14] obtained by the force method. It can be observed that the transpose of this matrix relates the forces and the moments of SLE.

6 6 rs 6 6 6 rt Ac Ec 6 6 ½Kc ¼ lc 6 6 r2 6 6 6 rs 4 rt X X

t 2

Y Y

7 st 7 7 7 27 t 7 7 7 rt 7 7 7 st 7 5 t2

Z Z

where, r ¼ i lc j ; s ¼ i lc j and t ¼ i lc j . In the above equation cross sectional area is denoted by Ac , Young’s modulus is denoted by Ec and the bar/cable length is denoted by lc . By combining the stiffness matrix of SLE elements and the cable we can write

½K dX ¼ dF where the total stiffness matrix [K] is given by ½Ks þ ½Kc . 5. Revolute joint constraints

The two adjacent SLE’s are connected by revolute joints as shown in Fig. 5. This enforces additional constraints of the form

4. Equation for the cable As already described earlier, cables are added in the masts to enhance their stiffness. These cables are slack in the stowed conﬁguration and are taut at the end of deployment. A cable can be assumed to be bar in the taut conﬁguration. The stiffness matrix for the bar may be found in many textbooks and is described below for completeness. For the bar shown in Fig. 4 connecting the joints i and j, the stiffness matrix is given by

ð27Þ

Fig. 5. A spherical joint replaced by two revolute joints.

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rip Um Lip cosða1 Þ ¼ 0 rjq Un Ljq cosða2 Þ ¼ 0

ð28Þ

where the unit vectors Um and Un are along the revolute joint axis as shown in Fig. 5. The angles a1 and a2 are the angles between the unit vectors and r. In our approach, Lagrange multiplier is used to enforce these constraints on the stiffness matrix equations presented in the previous section. 5.1. Lagrange multiplier method For a steady state discrete linear system with potential energy functional P expressed by

1 2

P ¼ UT ½K U UT F

ð29Þ Fig. 6. Hexagonal SLE mast in the deployed conﬁguration.

where, U is the displacement of the structural system, [K] is the stiffness matrix and F is the external load, the equilibrium equations can be found for the condition which make the variation of P stationary. We get T

dP ¼ dU ð½K U FÞ ¼ 0

ð30Þ

with respect to the admissible virtual displacements dU. Since dU is arbitrary the above equation yields

½K U ¼ F

ð31Þ

The constraint equations due to revolute joints can be written in the general form as

/ðUÞ ¼ ½C U D ¼ 0

ð32Þ

where ½C p q is the constraint matrix, p is the number of constraint equations, q is the number of variables and D is a vector of constants. In the Lagrange multipliers method the potential function is appended with the revolute joint constraints and we get

P ¼ P þ kT /ðUÞ

ð33Þ

where, k ¼ ½k1 ; . . . ; kp are the Lagrange multipliers. The stationary of this functional P is

dP ¼ dP þ dUT ðCT kÞ þ dkT ðCU DÞ ¼ 0

ð34Þ

For arbitrary dk and dU the above equation gives

U k

F D

ð35Þ

The advantage of Lagrange multiplier method is that the constraints are satisﬁed exactly but this is at the expense of larger set of equations. This method also gives the magnitudes of constraint forces since the Lagrange multipliers can be obtained by solving Eq. (35). 6. Results and discussion In this section, the degree of freedom and the redundancy of the joints/links are ﬁrst obtained for a hexagonal mast. The stiffness of the mast is then evaluated and the variation in stiffness with addition of cables is presented. We also present the stiffness evaluation for an assembly of four hexagonal masts. 6.1. Degree of freedom and redundancy evaluation The hexagonal mast built out of SLE’s, is presented in Fig. 6. The mast has six SLEs. Each SLE has 4 rigid link constraints and 6 SLE

Table 1 Input data: coordinates of joints for hexagonal mast. Joints

X coordinate (mm)

Y coordinate (mm)

Z coordinate (mm)

Joint Joint Joint Joint Joint Joint Joint Joint Joint Joint Joint Joint

0.0 500.0 1500.0 2000.0 1500.0 500.0 0.0 500.0 1500.0 2000.0 1500.0 500.0

0.0 866.0254 866.0254 0.0 866.0254 866.0254 0.0 866.0254 866.0254 0.0 866.0254 866.0254

0.0 0.0 0.0 0.0 0.0 0.0 700.0 700.0 700.0 700.0 700.0 700.0

1 2 3 4 5 6 7 8 9 10 11 12

Table 2 [J] matrix details for hexagonal mast. Contents

Size of [J]

Nullspace

+SLE +SLE +SLE +SLE +SLE +SLE

(20, 39) (28, 42) (36, 45) (44, 48) (52, 51) (60, 54)

21 18 15 12 10 10

1 2 3 4 5

(62, 54) (64, 54) (66, 54) (68, 54) (70, 54)

8 6 5 4 4

+FACE 6

(72, 54)

+Boundary conditions ðX 1 ¼ Y 1 ¼ Z 1 ¼ 0Þ

(75,54)

Mechanism

+Cable 1–2

(76, 54)

Structure

1 2 3 4 5 6

+FACE +FACE +FACE +FACE +FACE

Remarks

SLE – 6 is redundant

Revolute joints are redundant Revolute joints are redundant

constraint equations at the pivot point. Fixed boundary conditions are used at joint 1. The coordinates of joints of the mast are presented in Table 1. The results of null-space analysis of the constraint Jacobian matrix are presented in Table 2. It is observed from the table that the dimension of null-space reduces on adding each SLE and the null-space does not change for the last SLE. Hence, the last SLE is redundant. The above analysis assumes spherical joints

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70 Transverse to mast Along the mast

Stiffness (N/mm)

50 40 30 20 10 0 −10

Angle of deployment (deg) Fig. 8. Axial and lateral stiffness during deployment.

Fig. 7. Stacked SLE units of Ref. [14].

for the points connected by the adjacent SLEs. The revolute joint constraints are added further for each face and null-space is evaluated. It can be observed from the table that the null-space reduces for addition of revolute joints on each face. The null-space does not change for the revolute joints added for the face 5 and face 6. Hence these joints are redundant. By adding the boundary condition the mast will be a single degree of freedom system. It can be observed that the last SLE and the revolute joints on the face 5 and face 6 are redundant from the kinematic point of view. In order to reduce this mast to a structure a cable can be added at any two successive joints. The simulation is further continued by adding a cable between the joint 1 and joint 2. The null-space dimension of the Jacobian matrix is zero indicating that the mast a structure.

Fig. 9. Hexagonal SLE mast in the deployed conﬁguration with cables.

Table 3 Variation of stiffness with addition of cables for hexagonal mast.

6.2. Stiffness evaluation In this section the stiffness matrix (see Eq. (24)) developed in previous section for the SLE is used along with the Lagrange multiplier (see Eq. (35)) to evaluate the stiffness of the mast. Fig. 7 shows the two-dimensional straight deployable structure consisting of four pantograph units presented in Ref. [14] and an active cable zig-zagging across the pantograph. A constant tension spring keeps the active cable pre-tensioned in all configurations. The structure is deployed from nearly flat b ¼ 1:0 to the configuration shown in Fig. 7, b ¼ 45 , by shortening gradually the active cable. The cables have AE ¼ 1:5 105 N and the pantograph units have AE ¼ 3:5 106 N and EIz ¼ 9:6 107 N mm2 . The length of arm is 1000 mm. The tip stiffness of the assembly as it deploys is evaluated by applying two forces of 0.5 N to top joints in X and Y directions. Fig. 8 presents the axial and lateral stiffness of the system. These results matches with those presented in literature [14]. Fig. 9 shows a hexagonal mast in the deployed configuration. This mast has six SLEs and the six cables in the top, six cables in

Mast Mast Mast Mast

with with with with

top or bottom cables only vertical cables top and bottom cables all cables

Stiffness in X direction (N/mm)

Stiffness in Y direction (N/mm)

Stiffness in Z direction (N/mm)

24.27 27.88 28.61 31.73

53.24 30.48 98.36 138.72

11.32 7.39 18.33 23.85

the bottom and six vertical cables are connected as shown in the ﬁgure. These cables are slack during deployment and becomes taut at the end of deployment. The SLE and cables have the cross sectional area, A, of 138:23 mm2 and 1:0 mm2 , respectively. the Young’s Modulus E of the SLE and cables are 70000.0 N/mm2 and 63000.0 N/mm2, respectively. The second moment of inertia Iz , of SLE is 8432.0 mm4. An unit load is applied at joint 10. The stiffness of the mast due to these loads were found to be 31.73 N/ mm,138.72 N/mm and 23.85 N/mm in X, Y and Z direction, respectively. In order to study the sensitivity of the mast stiffness with cables the simulations were carried out by adding top, bottom and vertical cables individually and in combinations. The results

B.P. Nagaraj et al. / Computers and Structures 88 (2010) 95–104

103

Fig. 10. Assembly of four hexagonal masts in the deployed conﬁguration with cables.

Table 4 Input data: coordinates of joints for assembled hexagonal mast. Joints

X coordinate (mm)

Y coordinate (mm)

Z coordinate (mm)

Joint Joint Joint Joint Joint Joint Joint Joint Joint Joint

0.0 500.0 1500.0 2000.0 2000.0 3000.0 3500.0 3000.0 4500.0 5000.0

0.0 866.0254 866.0254 0.0 1732.10 1732.10 866.0254 0.0 866.0254 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

1 2 3 4 13 14 15 16 27 28

Table 5 Variation of stiffness with addition of cables for assembled hexagonal mast.

Mast Mast Mast Mast

with with with with

top or bottom cables only vertical cables top and bottom cables all cables

Stiffness in X direction (N/mm)

Stiffness in Y direction (N/mm)

Stiffness in Z direction (N/mm)

32.01 40.46 65.44 114.23

104.31 81.17 175.42 326.64

17.56 10.28 27.25 39.26

are presented in Table 3. It can be observed that the stiffness does not increase signiﬁcantly when either top, bottom or vertical cables are used individually. The stiffness increases by more than 50% when top and bottom cables are used together. The stiffness further increases by additional 30% or more when all three cables, namely top, bottom and vertical, are used together. The stiffness in Y direction is found to be higher than in the other two directions. Fig. 10 shows a deployed mast consisting of four hexagonal masts. This mast has nineteen SLEs and the nineteen cables in the top, nineteen cables in the bottom and sixteen vertical cables are connected as shown in the ﬁgure. Fixed boundary conditions are used at joint 1. The coordinates of the bottom joints of the mast are presented in Table 4. The other coordinates are symmetrical. The top coordinates are located at 700 mm along Z axis. The cables are slack during deployment and becomes taut at the end of deployment. The geometrical and material properties of the SLE

and cables are same as in the example of the single hexagonal mast presented earlier. An unit load is applied at joint 31. The stiffness of the mast due to these loads were found to be 114.23 N/ mm,326.64 N/mm and 39.26 N/mm in X, Y and Z direction, respectively. In order to study the sensitivity of the mast stiffness with cables the simulations were carried out by adding top, bottom and vertical cables individually and in combinations. The results are presented in Table 5. It can be observed that the stiffness does not increase signiﬁcantly when either top, bottom or vertical cables are used individually. The stiffness increases by more than 55% when top and bottom cables are used together. The stiffness further increases by additional 45% or more when all three cables, namely top, bottom and vertical, are used together. The stiffness in Y direction is found to be higher than in the other two directions.

7. Conclusions In this paper, Cartesian coordinates and symbolic computations have been used for kinematic and static analysis of threedimensional deployable SLE masts. The mobility of the masts were evaluated from the dimension of null-space of the Jacobian matrix formed by the derivative of the constraint equations. The stiffness matrix for the SLE was obtained from the constraint Jacobian. The stiffness matrix obtained by our approach is same as those obtained with the force and displacement methods of literature. The main advantage of the constraint Jacobian based approach are (a) ease of obtaining the stiffness matrices, (b) determination of mobility and the redundant joints/links of the mast, and (c) ease of incorporating revolute joint constraints by using Lagrange multipliers. The stiffness due to cables, an integral part of deployable masts, are also considered. The constraint Jacobian approach was used for the analysis of a hexagonal mast and a assembled hexagonal mast, and the stiffness of the masts in different directions were obtained. The approach presented in this paper can be extended to masts of different shapes and to stacked masts.

Appendix A. Stiffness matrix for the SLE The matrices ½Jm and ½Jmn associated with the stiffness matrix for SLE are given by

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1 X p

B l1 B 0 B ½Jm ¼ B B B 0 @ 0

6 Zn Zp ½Jmn ¼ 6 4 lm ln

Y p Y n lm ln

Y 1 Y p l1

Z 1 Z p l1

X p X 1 l1

Y p Y 1 l1

Z p Z 1 l1

X 2 X p l2

Y 2 Y p l2

Z 2 Z p l2

X p X 2 l2

Y p Y 2 l2

Z p Z 2 l2

X 3 X p l3

Y 3 Y p l3

Z 3 Z p l3

X p X 3 l3

Y p Y 3 l3

X 4 X p l4

Y 4 Y p l4

Z 4 Z p l4

X p X 4 l4

Y p Y 4 l4

Z p Z n lm ln

Y n Y p lm ln X p X n lm ln

0 X n X p lm ln

Z m Z p lm ln

Z p Z m lm ln Y m Y p lm ln

0 X p X m lm ln

Y p Y m lm ln X m X p lm ln

Z n Z m ln lm

Y m Y n lm ln

Z m Z n lm ln

X n X m lm ln

Y n Y m lm ln

X m X n lm ln

References [1] Kwan AKS, Pellegrino S. A cable rigidized 3D pantograph. In: Proceedings of 4th European symposium on mechanisms and tribology, France, September 1989, ESA-S P-299; March 1990. [2] You Z, Pellegrino S. Cable stiffened pantographic deployable structures, Part 1: triangular mast. AIAA J 1996;34(4):813–20. [3] Kwan AKS, Pellegrino S. Active and passive elements in deployable/retractable masts. Int J Space Struct 1993;8(1–2):29–40. [4] Hartenberg RS, Denavit J. Kinematic synthesis of linkages. McGraw-Hill; 1965. [5] MSC Adams users manual. USA: MSC Software Corporation. [6] Garcia De Jalon J, Bayo E. Kinematic and dynamic simulation of multi-body systems: the real time challenge. Springer-Verlag; 1994. [7] Garcia De Jalon J, Angel Serna Migual. Computer methods for kinematic analysis of lower pair mechanism -1 Velocities and acceleration. Mech Mach Theory 1982;17(6):303–97. [8] Garcia De Jalon J, Unda J, Avello A. Natural coordinates of computer analysis of multi body systems. Comput Methods Appl Mech Eng 1986;56:309–27. [9] Nagaraj BP, Pandiyan R, Ghosal A. Kinematics of pantograph masts. Mech Mach Theory 2009;44(4):822–34. [10] Calladine CR. Buckminister Fuller’s Tensigrity structure and Clerk Maxwell’s rule for the construction of stiff frames. Int J Solids Struct 1978;14:161–72. [11] Freudenstein F. On the verity of motions generated by mechanism. J Eng Ind, Trans ASME 1962;84:156–60.

C C C C Z p Z 3 C C l3 A Z p Z 4 l4

3 7 7 5

[12] Kuznetsov EN. Under constrained structural systems. Int J Solids Struct 1988;24(2):153–63. [13] Kuznetsov EN. Orthogonal load resolution and statical-kinematic stiffness matrix. Int J Solids Struct 1997;34(28):3657–72. [14] Kwan AKS, Pellegrino S. Matrix formulation of macro-elements for deployable structures. Comput Struct 1994;50(2):237–54. [15] Shan W. Computer analysis of foldable structures. Comput Struct 1992;42(6):903–12. [16] Gantes CJ. Deployable structures: analysis and design. 1st ed. Boston: WIT Press; 2001. [17] Gantes CJ, Conner JJ, Logcher RD. Equivalent continuum model for deployable ﬂat lattice structures. J Aerospace Eng, ASCE 1994;7:72–91. [18] Pellegrino S, Kwan AKS, Van Heerden TF. Reduction of equilibrium, compatibility and ﬂexibility matrices in the force method. Int J Numer Methods Eng 1992;35:1219–36. [19] Pellegrino S. Structural computations with the singular value decomposition of the equilibrium matrix. Int J Solids Struct 1993;30(21):3025–35. [20] Kaveh A, Davaran A. Analysis of pantograph foldable structures. Comput Struct 1996;59:131–40. [21] Wolfram S. Mathematica: a system for doing mathematics by computer. 2nd ed. Addition Wesley Publishing Co.; 1991. [22] Krishnamoorthy CS. Finite element analysis: theory and programming. New Delhi: Tata McGraw-Hill Publishing Company Limited; 1991.

Computers and Structures 88 (2010) 105–119

Contents lists available at ScienceDirect

Computers and Structures journal homepage: www.elsevier.com/locate/compstruc

Vibration modes and natural frequencies of saddle form cable nets Isabella Vassilopoulou a,*, Charis J. Gantes b,1 a b

Laboratory of Metal Structures, School of Civil Engineering, National Technical University of Athens, 12, Irinis Avenue, 15121 Pefki, Greece Laboratory of Metal Structures, School of Civil Engineering, National Technical University of Athens, 9 Heroon Polytechniou Street, GR-15780 Zografou, Athens, Greece

a r t i c l e

i n f o

Article history: Received 10 April 2009 Accepted 9 July 2009 Available online 7 August 2009 Keywords: Cable net Vibration modes Modal transition Deformable edge ring

a b s t r a c t The objective of this paper is to investigate the dynamic behaviour of cable networks, in terms of their natural frequencies and the corresponding vibration modes. A multi-degree-of-freedom cable net model is assumed, having circular plan view and the shape of a hyperbolic paraboloid surface. The cable supports are considered either rigid or ﬂexible, thus accounting for the deformability of the edge ring. On the basis of numerical analyses, empirical formulae are proposed for the estimation of the linear natural frequencies, taking into account the mechanical and geometrical characteristics of the cable net and the ring, expressed in the form of appropriate non-dimensional parameters. The sequence of the symmetric and antisymmetric modes of the network and the occurrence of modal transition can be predicted in relation to one of these parameters, in analogy to single cables. The differences between a network with rigid cable supports and one with boundary ring, concerning the eigenmodes and the corresponding eigenfrequencies are identiﬁed. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Many cable net roofs have been designed and constructed since the completion of the ﬁrst saddle form cable network, being the roof of the North Carolina State Fair Arena at Raleigh, in 1953, which opened a new chapter in architecture and engineering. This type of structures constitutes the most competitive solution for covering large spans, not only from aesthetic, but also from structural and economical point of view; such tensile structures provide appealing architectural shapes, while the loads are carried to the supports developing only tension in the members, leading thus to the best exploitation of the material. Saddle form cable networks usually have a rectangular, rhomboid, circular or elliptical plan and the shape of their surface is that of a hyperbolic paraboloid, where the two opposite boundary edges are higher than the other two. In this way, convex and concave curvatures develop in two perpendicular directions, respectively. The cables are anchored to a contour ring, usually made of prestressed concrete, having a closed box cross-section. Cable nets belong to the family of tension structures characterized by geometrically nonlinear behaviour. The system becomes stiffer as the deformation of the cables increases, as long as they remain in tension. The stiffness of these structures is obtained by the pretension of the cables and by the opposite curvatures of * Corresponding author. Tel.: +30 210 6141055; fax: +30 210 7723442. E-mail addresses: isabella@central.ntua.gr (I. Vassilopoulou), chgantes@central.ntua.gr (C.J. Gantes). 1 Tel.: +30 210 7723440; fax: +30 210 7723442. 0045-7949/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2009.07.002

the net. The pretension must be high enough to ensure the avoidance of cable slackening under any loading combination, because in that case the net becomes soft and may undergo large deformations. On the other hand, opposite curvatures are necessary to enable pretension, and to provide bearing capacity for loads directed both downwards, such as snow and upwards, such as wind suction. Moreover, nets of low curvatures are not very stiff and thus, unsuitable for large spans. The continuously increasing interest of engineers in cable structures has led to a growing demand to understand and, if possible, to predict their behaviour. The response of cables under static loads is quite well known, but their complicated dynamic response is still under investigation. Because of their lightness, they are vulnerable to dynamic excitation, especially due to wind action, which may lead to large amplitude oscillations, overstressing of cables and causing fatigue problems at the cable anchorages. In addition, because of their geometrical nonlinearity, cables are vulnerable to many kinds of resonance during their wind-induced vibrations. In undamped or lightly damped linear or nonlinear systems, when the loading frequency equals the eigenfrequency, even a weak excitation may lead to unbounded vibrations, with a continuously increasing amplitude, in which case the system is said to be in the well-known state of fundamental or primary resonance. In nonlinear systems, secondary resonances may also emerge: when the loading frequency X may be expressed as X = (1/n) x, where n is an integer, whereas x the eigenfrequency, superharmonic resonance may occur and when the loading frequency is larger than the frequency of the system, related to it with the expression X = n x, where n is again an integer, phenomena of subharmonic resonance

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Nomenclature 1A 1S 1SS 2A A Ar b D E Er f g

first antisymmetric mode about x or y-axis first symmetric mode of the cable net (cable net with rigid or flexible supports) first symmetric mode of the system (cable net with flexible boundary ring) first antisymmetric mode about x and y-axes cable cross-sectional area edge ring cross-sectional area ring cross-section width cable diameter elastic modulus of cable material elastic modulus of ring material maximum initial sag of the cables gravitational constant

are possible. In both cases, the system responds in such a way that the free oscillation term does not decay to zero, in spite of presence of damping and in contrast to the linear solution. Especially subharmonic oscillations, that occur for excitations with frequencies much larger than the natural frequency of a system, can have potentially catastrophic effects. Moreover, when two or more linearized natural frequencies, which depend on the geometrical and mechanical characteristics of the structure, are related with the expression n1 x1 þ n2 x2 þ þ nn xn 0; where n1, . . ., nn positive or negative integers, the corresponding modes may be strongly nonlinearly coupled, leading to internal resonances. If an internal resonance occurs in a system, energy imparted initially to one of the modes, will be continuously exchanged among all the modes involved in that resonance [1]. All these phenomena, related to the linear natural frequencies as well as the loading frequency, render unpredictable the dynamic response of a nonlinear system. The vibration modes and natural frequencies of individual cables exhibit interesting phenomena that have been studied by many researchers. First, Pugsley [2] gave some semi-empirical expressions for the first three in-plane modes of a sagged suspended chain, which could represent a hanging inextensible cable without pretension. Ahmadi-Kashani [3] used numerical methods to calculate the frequencies of an inclined hanging inextensible cable and compared the above semi-empirical formulae of Pugsley with the numerical results. New approximate formulae were given, for the first four in-plane and out of plane natural frequencies applicable to a wide range of sag/span ratios and inclination angles. Irvine and Caughey [4] focused their attention on the in-plane and out-of-plane vibrations of a sagged suspended cable, and derived specific formulae for the corresponding elastic natural frequencies of the symmetric and antisymmetric modes, comparing them with experimental results. In their study they considered the cable as extensible, with a sag-to-span ratio up to 1/8, and introduced the parameter k2, which involves the cable’s geometry and elasticity and governs the symmetric in-plane modes. When this parameter is very large, the cable may be considered inextensible, and when it is very small, the cable profile approaches that of a taut string. They also verified the existence of crossover points, at which modal transition occurs, for specific values of this parameter, which means that for the first crossover point the frequencies of the first symmetric and the first antisymmetric in-plane modes are equal, and the system is characterized by 1:1 internal resonance between these two modes. Rega and Luongo [5] noticed crossover points for the inextensible suspended cable with movable supports, for a wide range of values of the sag-to-span ratio. Rega et al. [6] studied the influence of quadratic and cubic nonlinearities of a simple sagged cable in the modes and the modal crossovers, using a sim-

Ir L N T0 x y z b

c k2

q qr x X

ring moment of inertia diameter of circular cable net plan view number of cables in each direction cable initial pretension x-coordinate of the cable net y-coordinate of the cable net z-coordinate of the cable net non-dimensional frequency of the cable net non-dimensional parameter for the ring non-dimensional parameter for the cable net cable unit weight ring unit weight natural frequency loading frequency

pler expression for the parameter k2, concluding that an infinite number of crossover points exists, instead of the single one of the linear theory. In [7,8] inclined cables are explored; it is noted that their mode shapes are totally different from those of horizontal cables, and the phenomenon of frequency avoidance is identified, meaning that, while in frequency crossover two natural frequencies become close, in frequency avoidance they always remain apart and never coincide. Hybrid modes are also detected, i.e. a mixture of symmetric and antisymmetric shapes, which are as important as the symmetric ones, regarding the amplification of the cable tension. Gambhir and Batchelor [9] investigated the influence of various parameters, such as the cable cross-sectional area, the initial pretension, the sag-to-span ratio, as well as the surface curvature, on the natural frequencies of 3D cable nets. Talvik [10] noticed that, in a cable network with an elliptical flexible contour ring, the first vibration mode involves mostly the contour ring, while the next four modes are determined only by cable net deformations. In [11] an attempt to calculate the natural frequencies of a cable cross with analytical methods is presented. The system consists of two crossing cables with equal spans and sag-to-span ratio. Each segment is modelled as one or two elements. The mass is considered either lumped or consistent. The formulae of the first three eigenfrequencies are derived, considering all three components of displacement unconstrained. The horizontal vibrations have equal frequencies, which, for sag-to-span ratios less than 1/8, are larger than the frequency referring to the vertical vibration, taking thus the place of the second and third eigenfrequencies of the system. A crossover point is noted for sag-to-span ratios larger than 1/8 and the frequency of the vertical vibration becomes the third eigenfrequency of the system. Seeley et al. [12] examined the natural frequencies and mode shapes of a circular concave cable network consisting of circular and radial cables. The sag f of the net was obtained by the static loading and the range of sag-to-span ratio was between 1/9 and 1/15. They derived an approximate formula of the fundamental circular frequency of the net, involving only the sag and the sag-to-span ratio, similar to the one given by Pugsley for the catenary, mentioned above. They concluded that the first natural frequency of the concave cable net is close to the average of the uncoupled in-plane and out-plane fundamental frequencies of an individual cable with the same sag/span ratio. They also noticed that, only the higher order frequencies depend on the extensibility of the network, expressed by a parameter in terms of the elastic modulus of the cable material, the cable cross-sectional area, the number of the radial cables, the diameter of the network and the uniformly distributed dead load. Buchholdt [13] suggested

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to calculate the eigenfrequencies of the deformed structure under the permanent loads and the wind load produced by the mean wind velocity, and then add the response due to the fluctuating component. He reported measured frequencies of a saddle-shaped net roof, with a circular plan of 125 m diameter, for the first seven modes having frequencies from 0.74 Hz to 1.12 Hz. Several computerized methods of analysis and other numerical techniques were developed to calculate the linear natural frequencies and nonlinear static or dynamic response of cable networks and membranes, by solving the governing equations of motion [14–19]. This paper deals with saddle form cable networks, with a circular plan view. The cable ends are considered either fixed or anchored to a boundary ring. Their behaviour under static loads depends mainly on the geometry of their boundaries, on the curvatures, on the stiffness of the system, which is determined by the cable axial stiffness and the ring flexural stiffness, and on the level of cable pretension. Furthermore, their dynamic response depends also on the mass of the system, consisting of the cable mass, the ring mass and the mass of the roof’s cladding, on the structural damping, as well as on the amplitude and the frequency of the excitation. A first attempt to provide insight to some very interesting aspects of the dynamic response of saddle form suspended roofs with fixed cable ends, was presented by the authors in [20]. Conducting parametric modal analyses of the cable nets with fixed cable ends, several similarities with the dynamic behaviour of a simple sagged suspended cable were revealed, with respect to their eigenmodes and eigenfrequencies. The introduction of a parameter k2 for saddle form cable nets, similar to the one for simple cables, makes possible the prediction of the modes sequence and of the crossover points at which modal transition occurs. Empirical formulae were proposed for the natural frequencies’ estimation of the first symmetric and antisymmetric modes. In [21] some similarities and differences between a network with rigid cable supports and one with the cables anchored to a flexible edge ring were highlighted. The dynamic behaviour of these two systems is further explored and compared in the present work, regarding their natural frequencies and vibration modes. Crossover points at which modal transitions occur are also detected for both systems. Semi-empirical formulae for the estimation the frequencies of the first vibration modes are provided and proposed to be used at a preliminary design stage. Solving analytically three dimensional cable structures turned out to be practically impossible, due to

107

their complex nonlinearity; thus numerical analyses have been conducted. If the main loading frequencies are known, e.g. the frequency spectrum of the wind action, the natural frequencies of a system provide important information on the nature of the response to dynamic loads and on the potential danger of nonlinear dynamic response and resonance phenomena. For preliminary structural design and initial selection of form as well as cable cross-sections and pretensions, instead of setting up multi-degree of freedom numerical models in order to calculate the natural frequencies of a cable net system, an empirical formula is preferable. Knowing the frequencies of the system and the parameters that inﬂuence them, it is possible to improve the design of such structure, avoiding internal or secondary resonances, protecting the cable net from oscillations of large amplitude and the cables from fatigue symptoms at their anchorage points.

2. Cable net model and assumptions The model adopted for this work is a three-dimensional cable net, with the geometry of a hyperbolic paraboloid surface and a circular plan view of diameter L (Fig. 1). The net consists of N cables in each direction, arranged in a grid of equal distances. The sag of the longest main and secondary cables is equal to f which is also considered as the sag of the roof. All cables have a circular cross-section with diameter D and area A and their material is assumed inﬁnitely linearly elastic with Young-modulus E; they are modelled by truss elements, that can sustain only tension, with initial strain equal to T0/EA, where T0 is the cable pretension. The cable unit weight is equal to q. A lumped mass matrix is used for the analysis. Each part of a cable between two adjacent net intersection points is modelled with one truss element. All three translational degrees of freedom are considered free for all internal nodes of the net. For the cable net with ﬁxed cable ends, the supports at all cable ends are considered as pinned. Without loss of generality, the edge ring, if considered, has a square box cross-section of width b, wall thickness b/10, with cross-sectional area Ar, moment of inertia Ir, unit weight qr and elastic modulus Er (Fig. 2). The z-displacement of the ring’s nodes is restrained. The displacement in the x-direction is not permitted for the two nodes of the ring with coordinate x = 0, and respectively, the y-displacement is not permitted for the two ring nodes with coordinate y = 0, in order to avoid rigid body

Fig. 1. Geometry of the net with rigid supports.

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Fig. 2. Geometry of the net with ﬂexible edge ring.

motion. Thus, the radial deformation of the ring is allowed, but not the overall rotation about the z global axis. The net is uniformly prestressed and no structural damping is taken into account in the present work. Linear modal analyses are performed to calculate the eigenmodes and eigenfrequencies. For the calculation of the natural frequencies of the system, the geometry and stiffness of the equilibrium state under prestressing are considered. All analyses have been carried out with the ﬁnite element software ADINA [22,23], which was validated for analyses of this type by comparisons with analytical results of the equation of motion for several cases of single cables as well as for a singledegree-of-freedom cable net, having the shape of a cross. For all these cases complete agreement of analytical and numerical results was observed.

k2 ¼

3. Cable net with rigid supports 3.1. The vibration modes In order to investigate the vibration modes of a cable net with rigid supports, parametric analyses were performed for a large number of cable net models with different geometrical and mechanical characteristics, eight of which are presented here, with the characteristics listed in Table 1 and the cross-sectional diameter D of all cables varying between 10 mm and 60 mm. The cable net’s eigenmodes can be distinguished in symmetric and antisymmetric ones. The former ones consist of symmetric vertical components and antisymmetric horizontal components about both horizontal axes x and y, while the latter ones consist of antisymmetric vertical components and symmetric horizontal components with respect to one, or to both horizontal axes. In case Table 1 Characteristics of the cable nets with rigid supports. Cases

N L (m) f/L T0 (kN) E (GPa) q (kN/m3)

two modes are antisymmetric with reference to x or y axis, they have similar shapes and equal eigenfrequencies. In this work, the first four modes are thoroughly examined, which are the first symmetric mode of the net, denoted as 1S, the first similar antisymmetric modes about x or y axis, respectively, which are treated as one mode, denoted, both of them, as 1A and the first antisymmetric mode about both horizontal axes, which is denoted as 2A (Fig. 3). Examining the natural modes of the cable nets, frequency crossovers and modal transitions, similar to those described for the simple suspended cable in Section 1, were also observed. To describe these phenomena, a parameter k2 is herein introduced for cable nets, similar to that for a simple cable, proposed by Rega et al. [6], which is given by the following expression:

25 100 1/20 400 165 100

25 100 1/20 600 165 100

25 100 1/35 400 165 80

25 100 1/35 400 165 100

25 100 1/20 400 165 80

25 100 1/20 400 148.5 90

25 50 1/20 400 165 100

35 50 1/20 400 165 100

2 EA f T0 L

ð1Þ

The modes of the cable net are found to depend on the parameter k2, in a similar manner as for the simple cable. This has been verified by means of numerical analyses for all cable nets examined. More specifically: (a) for k2 6 0.80 the first eigenmode of the system is the 1S mode, while the second and third eigenmodes are the 1A modes. The fourth eigenmode is the 2A mode. The 1S mode has a natural frequency smaller than the one of the 1A modes, which in turn, is smaller than the frequency of the 2A mode, that is x1S < x1A < x2A. For k2 = 0.80 the first three eigenmodes have equal natural frequencies, which means x1S = x1A, accounting for the first crossover point. The sequence of the first four eigenmodes is shown in Fig. 4. (b) for 0.80 < k2 6 1.00 the natural frequencies of the first two eigenmodes, which are the 1A modes, are equal and smaller than that of the 1S mode, which is the third cable net eigenmode, followed by the 2A mode. This means x1A < x1S < x2A. For k2 = 1.00 the natural frequencies of the 3rd eigenmode – which is the 1S mode – and the fourth eigenmode – which is the 2A mode – are equal, that is x1S = x2A (second crossover point). The first four eigenmodes have the sequence shown in Fig. 5. (c) for 1.00 < k2 6 1.17 (Fig. 6) a transition between the 3rd and 4th eigenmodes occurs. Thus, the 1S mode becomes the 4th cable net eigenmode, while the 1A modes remain

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Fig. 3. The ﬁrst four vibration modes of a cable net with rigid supports.

Fig. 4. The ﬁrst four eigenmodes of a cable net with rigid supports for k2 6 0.80.

Fig. 5. The ﬁrst four eigenmodes of a cable net with rigid supports for 0.80 < k2 6 1.00.

Fig. 6. The ﬁrst four eigenmodes of a cable net with rigid supports for 1.00 < k2 6 1.17.

Fig. 7. The ﬁrst four eigenmodes of a cable net with rigid supports for 1.17 < k2.

the first two eigenmodes. This means x1A < x2A < x1S. For k2 = 1.17 the natural frequencies of the 1st, 2nd and 3rd eigenmodes are equal, that is x1A = x2A (third crossover point). (d) for 1.17 < k2 (Fig. 7) a transition between the 3rd and the first two eigenmodes occurs. The 2A mode becomes the first mode of the system. The 1A modes become second and third, while the 1S remains the fourth mode. This means x2A < x1A < x1S. The above limits of k2 refer to the first four eigenmodes of a cable net with rigid supports. Transitions among higher modes also occur for different values of k2.

3.2. The natural frequencies In this section parametric analyses are presented for all cases of Table 1, in order to examine the relation between the natural frequencies and the characteristics of the cable net, and especially the parameter k2. Different values are given to this parameter by changing the cable cross-sectional area between 10 mm and 60 mm, keeping the characteristics of Table 1 constant. The 1S, 1A and 2A modes are examined again. The analysis results are shown in the charts of Fig. 8, where the parameter k2 is plotted pffiffiffiffiffiffiffiffiffiffiffiffi on the horizontal axis and the normalised frequency x= ðg=LÞ on the vertical axis for all eight cable nets of Table 1, where g is the gravitational constant considered equal to 10 m/s2.

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60 30 0

Case 2

60 30 0

3 2 λ

Case 3 ω /(g/L)0.5

ω/(g/L)0.5

Case 1 ω/(g/L)0.5

ω/(g/L)

0.5

60 30

3 λ2

Case 4

60 30

3 λ2

Case 6

Case 5

0.5

ω/(g/L)

ω/(g/L)0.5

30 0

3 λ2

Case 7

ω/(g/L)0.5

60 30

3 2 λ

Case 8

60 30 0

0 0

3 λ2

1S 1A 2A Fig. 8. Normalised natural frequencies of a cable net with rigid supports vs. k2.

From the charts some important remarks can be elicited: as the parameter k2 increases, indicating that the net becomes stiffer, the natural frequencies decrease. the natural frequencies and eigenmodes do not depend on the number of the cables in each direction (cases 7, 8). if k2, q, E, L and f/L are kept constant, changing the level of pretension does not affect the natural frequencies (cases 1, 2). keeping k2, q, E, T0 and L constant, the natural frequencies decrease as the sag-to-span ratio f/L decreases (cases 1, 4 and 3, 5).

keeping k2, q, E, T0 and f/L constant, the natural frequencies increase as L decreases (cases 1, 7). on the other hand, if k2, T0 and f/L remain constant, the pffiffiffiffi natural frequencies increase with respect to 1= q (cases 1, 5). moreover, if k2, T0 and f/L remain constant, the natural pffiffiffiffiffiffiffiffiffiffiffiffi frequencies do not change if the ratio ðE=qÞ remains the same (cases 1, 6). finally, the frequency crossovers occur at the same values of the parameter k2 for all cable nets, as already pointed out in Section 3.1.

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15 β

case 1 case 2 case 3 case 4 case 5 case 6 case 7 case 8

(1S)

case 1 case 2 case 3 case 4 case 5 case 6 case 7 case 8

(1A)

15 β

10 5

0 0

(2A)

case 1 case 2 case 3 case 4 case 5 case 6 case 7 case 8

15 β

10 5 0 0

3 2 λ

Fig. 9. Parameter b for the 1S, 1A and 2A modes of a cable net with rigid supports vs. k2.

3.3. Empirical formulae A new non-dimensional parameter b is introduced, in order to include all the above information in one chart. This parameter represents the non-dimensional cable net frequencies

b¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Lx ½q=ðEg Þ f =L

ð2Þ

In the charts of Fig. 9, plotting on the horizontal axis the parameter k2 and on the vertical one the parameter b for each one of the frequencies of the net, it is noted that each natural frequency follows the same curve for all cable nets. Based on the results of the previous modal analyses, it is possible to produce approximate mathematical formulae estimating the natural frequencies of the cable nets. According to the above charts, there is a relation between the two non-dimensional parameters b and k2, for each of the three modes, which can be expressed as follows:

k2 bn ¼ 7n ) vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffis pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#n u 2 " ffi Lxe ½q=ðEgÞ EA f f u Eg n T 0 L 2 n t ¼ 7 ) xe ¼ 7 ð Þ T0 L L EA f ðf =LÞ qL2 ð3Þ where the subscript e denotes that this is an empirical expression of the eigenfrequencies, n = n1S = 3 for the 1S mode, n = n1A = 2.5 for the 1A modes and n = n2A = 2 for the 2A mode. Thus, Eq. (3) becomes, for the three modes respectively:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffiv u u" 2 # f u Eg u T0 L 3 t t x1S;e ¼ 7 L EA f qL2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffi v u" 2 # u T0 L f Eg 2:5 x1A;e ¼ 7 ð 2 Þ t L EA f qL ffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ffi 7 gT 0 x2A;e ¼ L qA

ð4Þ

ð5Þ

The error of the above formulae is calculated for all the results obtained from the parametric modal analyses performed for each case, as the ratio (xn xe)/xn where xn and xe are the net’s frequencies calculated by numerical methods and by the empirical formulae, respectively. The mean value and the standard deviation of the error are calculated and tabulated in Table 2 for each case. The accuracy of the empirical formulae is also illustrated in Fig. 10, and it is considered as very satisfactory for preliminary design purposes.

4. Boundary ring In order to extend the above investigation into the natural frequencies of a cable net anchored to a deformable edge ring, at ﬁrst, the ring itself is examined, without the cables. Conducting a parametric linear modal analysis, the ﬁrst natural frequency of such a structure is calculated, considering four different geometries of the boundary, given in Table 3. The ring modulus of elasticity Er varies between 30 GPa, 34 GPa, 37 GPa and 39 GPa, accounting for the concrete categories B25, B35, B45 and B55, respectively, according to the DIN codes. The ring’s cross-section has the shape of a square box, as given in Fig. 2, with width b taking the values b = 5.00 m, b = 6.50 m and b = 8.00 m for the models with diameter L = 100 m, while for the model with diameter L = 50 m the width b varies between b = 2.00 m, b = 3.50 m and b = 5.00 m.

Table 2 Error mean value (MV) and standard deviation (SD) of the empirical formulae of cable nets’ natural frequencies xe.

1S mode 1A mode

ð6Þ

Error (xn xe)/xn

Cases

2A mode

MV SD MV SD MV SD

1 (%)

2 (%)

3 (%)

4 (%)

5 (%)

6 (%)

7 (%)

8 (%)

5.4 3.4 3.1 3.5 3.0 1.6

4.5 4.9 2.4 3.6 3.5 0.5

0.9 7.9 1.1 2.9 4.0 0

5.4 3.4 3.1 3.5 3.5 0

5.4 4.3 3.4 3.9 3.4 0

5.6 3.6 3.4 3.5 3.5 0

5.7 3.6 3.4 3.5 3.7 0

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Case 1-8 (1S)

ω 1A,e

ω 1S,e

Case 1-8 (1A)

0 0

ω 1S,n 20

ω 1A,n

Case 1-8 (2A)

ω 2A,e

15 10 5 0 0

ω 2A,n

Fig. 10. Empirical formulae for natural frequencies ﬁts numerical data.

The ﬁrst vibration mode is characterized by an in-plane breathing motion of the ring (Fig. 11). The results of the natural frequencies are plotted in Fig. 12 where on the horizontal axis the non-dimensional parameter c is represented, deﬁned as:

c¼

Er I r

ð7Þ

qr Ar L3

andpon the vertical axis the non-dimensional eigenfrequency ffiffiffiffiffiffiffiffiffiffiffi ffi xr = ðg=LÞ: From the chart, one can conclude that the relation between the non-dimensional natural frequency of the edge ring without cables and the parameter c is the same for all cases. In [24] the natural frequency of any mode of vibration is given, concerning the flexural vibration of a plane circular ring:

xr ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Er Ir g i ð1 i Þ2 4

q r Ar R

i þ1

¼4

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Er Ir g i ð1 i Þ2 4

qr Ar L

i þ1

ð8aÞ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi Er I r g

qr Ar L4

5. Cable net with boundary ring 5.1. The natural modes A large number of models are also examined, considering both components of the suspended roof, the cable net and the deformable edge ring. It is concluded that in case the ﬂexibility of the boundary ring is taken into account, among the eigenmodes of

Table 3 Characteristics of the edge ring. Cases

When i = 1, xr = 0 and the ring moves as a rigid body. For i = 2, the ring performs the fundamental mode of ﬂexural vibration and Eq. (8a) becomes:

xr ¼ 10:73

mode is not taken into account, because it cannot affect the vibration of the net, as will be shown next.

L (m) f/L qr (kN/m3)

100 1/20 25

100 1/20 35

100 1/35 25

50 1/20 25

ð8bÞ

Although, the above formula refers to a plane ring, it can also be used for the boundary ring of a hyperbolic paraboloid roof. The error of the above formula is calculated for all the results obtained from the parametric modal analyses performed for each case, as the ratio (xrn xre)/xrn, where xrn and xre are the ring’s frequency calculated by numerical methods and by Eq. (8b), respectively. The mean value and the standard deviation of the error are calculated and tabulated in Table 4 for each case. The accuracy of the formula is illustrated in Fig. 13, and is considered as sufficient for all practical purposes. When i = 3, the calculated frequency corresponds to the ring’s second mode, which is the first antisymmetric vibration mode, being 2.8 times larger than the first eigenfrequency of the ring. This

Fig. 11. Ring’s ﬁrst vibration mode (in-plane mode).

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ω r/(g/L)0.5

Cases 1-4

60 case 1

case 2 20

case 3 case 4

0 0

γ Fig. 12. Normalised ﬁrst natural frequency of the ring without cables vs. non-dimensional parameter c.

Table 4 Error mean value (MV) and standard deviation (SD) of the formula of ring’s eigenfrequency xre. Cases

MV SD

5.2. The natural frequencies A sample of the above investigation is presented in this section, for the cable nets with characteristics given in Table 5. Parametric analyses are performed in order to evaluate the influence of the edge ring deformability to the net’s vibration modes, by varying the ring stiffness Er Ir and the ring mass qr Ar. The frequencies of this system are compared with those of the cable net with rigid supports. In Table 6 the parameter k2 and the frequencies are given for each net with rigid supports. Keeping the ring unit mass constant and equal to 25 kN/m3, the ring’s influence on the system’s frequency of the first symmetric mode (x1SS) is examined, by varying the elastic modulus Er and the ring cross-section width b as in Section 4, accounting for realistic values of the ring’s stiffness and crosssectional area. The variation of the non-dimensional first five natural frequencies of the system (including the double 1A frequency) with respect to the ring stiffness is given in the charts of Fig. 17. In these charts, x1S, x1A, x2A are the frequencies of 1S, 1A and 2A modes of the net, respectively, while x1SS is the frequency of the first symmetric mode of the system. The change of the combined system’s non-dimensional frequency x1SS of the first symmetric mode with respect to the ring stiffness is shown in the charts of Fig. 18. In these charts, the frequencies of the three systems are compared, the ring without the cables (xr), as calculated from Eq. (8b), the cable net without the ring (x1S) and the cable net with the ring (x1SS). For the first case, keeping the elastic modulus Er and the cross-section width b constant and equal to 37 GPa and 5.00 m respectively, the ring unit weight qr varies between 25 kN/m3 and 50 kN/m3. The variation of the non-dimensional frequency x1SS of the system with respect to the mass of the ring, is shown in Fig. 19.

Cases 1-4

ω re

the system there is also the in-plane mode of the ring, described in the previous section, which produces a symmetric vertical vibration of the cable net (Fig. 14). For common values of the ring’s stiffness and cable net stiffness in terms of pretension and cable cross-sectional area, the ring’s in-plane mode is the first eigenmode of the system and the corresponding frequency can be estimated by Eq. (8b), with negligible influence of the cable net. In this case, the following four eigenmodes are the same vibration modes of the cable net examined in Section 3.1 with negligible influence of the ring, and their frequencies can be expressed by Eqs. (4)–(6) with small errors. For very high values of the ring’s stiffness, its in-plane mode becomes of higher order, while the corresponding eigenfrequency still follows the law given by Eq. (8b). In this case the first four modes of the system are the vibration modes of the net examined previously, with frequencies that still follow Eqs. (4)–(6). Between these first four modes and the ring’s one, other vibration modes appear, most of them higher order net’s modes, but also hybrid ones, involving the ring and the net into the vibration. For illustration purposes, a cable net is considered with diameter L = 50 m, f/L = 1/20, T0 = 100 kN, D = 30 mm, q = 100 kN/m3, E = 165 GPa, while the ring’s characteristics are Er = 39 GPa, qr = 25 kN/m3, and the width b of the square box takes the values b = 2.00 m and b = 5.00 m. For the first structure the first mode is the ring in-plane mode and the vibration modes of the net follow, while for the second one, for which the ring is much stiffer than the cable net, the ring in-plane mode is the eighth mode of the system and the net modes are first. In Figs. 15 and 16 the first eight vibration modes and the deformed ring are shown for both structures, where one can distinguish the in-plane mode of the ring. For intermediate values of the ring’s stiffness, the symmetric vibration of the net and the in-plane one of the ring are not distinct; it is not possible to distinguish which mode represents a pure vibration of the net affecting also the ring and which one is mainly a vibration of the ring that produces a symmetric oscillation to the net. Consequently, in what follows, the 1st symmetric mode of the system is examined, whether this is produced mainly due to a net symmetric vibration or a ring in-plane one. The corresponding frequency will be named as x1SS.

20 10 0

Error (xrn xre)/xrn

1 (%)

2 (%)

3 (%)

4 (%)

0.2 0.1

1.3 0.1

0.1 0.3

ω rn

Fig. 13. Formula for the ﬁrst natural frequency of the ring without cables ﬁts numerical data.

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From the charts some important conclusions can be drawn: As the stiffness of the ring increases the frequency x1SS increases. If q, E, L, f/L and the stiffness of the ring are kept constant, an increase of the level of pretension increases slightly the frequency x1SS (cases 1, 2). Keeping q, E, T0, L, f/L and the ring stiffness constant, the frequency x1SS increases as the cable diameter D increases (cases 2, 3). Moreover, if k2, T0 and f/L remain constant, for the same levels of the ring’s stiffness, the frequency x1SS does not change, if the pffiffiffiffiffiffiffiffiffiffiffiffi ratio ðE=qÞ remains the same (cases 3, 6). Keeping q, E, T0, L and the ring stiffness constant, the frequency x1SS decreases as the sag-to-span ratio f/L decreases (cases 1, 5).

For the same levels of the ring’s stiffness and keeping f/L, T0, E, q constant, the frequency x1SS increases as L decreases (cases 1, 7). On the other hand, if the ring stiffness, E, T0, L, and f/L remain constant, the frequency x1SS increases slightly as the cable unit mass q decreases (cases 3, 4). The frequency x1SS does not depend on the number of cables in each direction (cases 7, 8). The frequency x1SS, for low levels of the ring’s stiffness, is the frequency of the in-plane mode of the ring and can be calculated using Eq. (8b), but as the stiffness increases the frequency diverges from the curve of the above equation and tends to become equal to the x1S of the net with rigid supports (Fig. 18). The presence of the edge ring changes the frequency of the 1S mode of the net (x1S), up to 36%, with respect to the one of the net with rigid supports. This occurs in case 5, in which the

Fig. 14. The first five eigenmodes of the cable net with the flexible edge ring.

Fig. 15. The ﬁrst eight eigenmodes of the cable net with ring cross-section width b = 2.00 m.

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first natural frequency x1SS of the combined system approaches the frequency of the 1S mode (x1S) of the net with rigid supports (Fig. 18) causing this change to the net’s frequency when the two components are taken into account (Fig. 17). In the other cases, instead, the change of the frequency of the 1S mode of the net (x1S), arises at only 11%. The frequency of the 2A mode (x2A) remains unchanged in presence of the edge ring (Fig. 17). The frequency of the 1A modes (x1A) does not change more than 3.4% due to the deformability of the edge ring (Fig. 17). This is the case because, for realistic values of ring flexural stiffness and cable axial stiffness, the antisymmetric vibration mode of the boundary ring is always much larger than the first four frequencies of the cable net. Thus, it cannot influence significantly the antisymmetric vibration mode of the net. If the stiffness of the ring is kept constant, and the mass of the edge ring increases, the frequency x1SS decreases (Fig. 19).

5.3. Empirical formulae Based on the aforementioned results of our investigation, it is concluded that the ﬁrst symmetric mode of a cable net with a ﬂexible boundary ring, depends on the ratio of the stiffness of the ring

Table 5 Characteristics of the cable nets with the ﬂexible edge ring. Cases

N L (m) f/L D (mm) T0 (kN) E (GPa) q (kN/m3)

25 100 1/20 40 400 165 100

25 100 1/20 40 600 165 100

25 100 1/20 60 600 165 100

25 100 1/20 60 600 165 80

25 100 1/35 40 400 165 100

25 100 1/20 63.2 600 148.5 90

25 50 1/20 40 400 165 100

35 50 1/20 40 400 165 100

and that of the cable net. If the ring is flexible enough with respect to the cable net, the first symmetric mode is the in-plane mode of the ring. On the other hand, when the ring is much stiffer than the cable net, it behaves as a rigid support to the cables, and its vibration mode is one of the higher order modes, while the first symmetric mode of the system is the one of the cable net. The stiffness ratio of the two components of such a system is expressed as the ratio of the natural frequencies of the two independent systems and constitutes the criterion that indicates whether the first symmetric mode of the system will be the in-plane mode of the ring or the first symmetric mode of the net.

Fig. 16. The ﬁrst eight eigenmodes of the cable net with ring cross-section width b = 5.00 m.

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Table 6 Eigenfrequencies of the nets with rigid supports. Cases 2

1.30 14.011 13.205 12.940

0.86 15.848 15.015 15.221

1.94 12.544 11.614 10.566

1.94 14.024 12.895 11.813

0.42 9.984 11.157 13.014

1.94 12.544 11.614 10.566

1.30 28.021 26.410 25.880

1.30 28.054 26.424 25.874

Case 1

ω/(g/L)0.5

Case 2

50 40 30 ω1S ω1Α ω2A ω1SS

20 10 0 0

40 30

ω1S ω1Α ω2A ω1SS

20 10 0

Case 4

60 50

ω/(g/L)0.5

Case 3

40 30 ω1S ω1Α ω2A ω1SS

20 10 0 0

40 30 ω1S ω1Α ω2A ω1SS

20 10 0

Case 5

Case 6

60 50

ω/(g/L)0.5

40 30 ω1S ω1Α ω2A ω1SS

20 10 0 0

40 30 ω1S ω1Α ω2A ω1SS

20 10 0 0

Case 7

Case 8

60 40

ω/(g/L)0.5

x1S (s 1) x1A (s 1) x2A (s 1)

ω/(g/L)0.5

ω1S ω1Α ω2A ω1SS

60 40 ω1S ω1Α ω2A ω1SS

20 0

0 0

Fig. 17. Normalised natural frequencies of the cable net with ring vs. c.

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Case 1

ω/(g/L) 0.5

50 40 30 ω1S-fixed ω1SS ωr-ring

20 10

40 30

10 0

Case 3

Case 4

ω/(g/L) 0.5

50 40 30 20

ω1S-fixed ω1SS ωr-ring

40 30 20

ω1S-fixed ω1SS ωr-ring

0 0

Case 5

Case 6

ω/(g/L) 0.5

ω1S-fixed ω1SS ωr-ring

30 ω1S-fixed ω1SS ωr-ring

20 10 0

30 20

ω1S-fixed ω1SS ωr-ring

Case 7

60 40 ω1S-fixed ω1SS ωr-ring

Case 8

ω /(g/L)0.5

ω/(g/L) 0.5

Case 2

60 40 ω1S-fixed ω1SS ωr-ring

20 0

0 0

Fig. 18. Normalised natural frequencies of the cable nets vs. c (x1S for the cable net with ﬁxed cable ends, x1SS for the ﬁrst symmetric mode of the cable net with the ring, xr for the ring with no cables).

From the charts of Fig. 18, it is evident that if the ring’s frequency quency of the combined system is close to the ring’s frequency and the first symmetric mode of the system is the in-plane mode of the ring. As the ring becomes stiffer and its frequency increases, the combined system’s first natural frequency approaches asymptotically the frequency (x1S) of the cable net with rigid supports and the first symmetric mode of the system is the symmetric mode of the net. Hence, if the ring’s frequency xr, according to Eq. (8b) is computed less than 65% of the x1S, then the frequency x1SS of the first symmetric mode of the cable net with edge ring can be evaluated from this equation. If, on the other hand, it results to more than

30 ω 1SS /(g/L)0.5

xr is less than, approximately, 65% of x1S, then the ﬁrst natural fre-

Case 1

20 10 0 200

300 400 ρrΑr [kN/m]

500

Fig. 19. Normalised natural frequency x1SS vs. qrAr.

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I. Vassilopoulou, C.J. Gantes / Computers and Structures 88 (2010) 105–119

Case 1

ω 1SS /(g/L)0.5

Case 2

40 30 20

50 40 30 20

0 0

Case 3

Case 4

ω 1SS /(g/L)0.5

40 30 20

50 40 30 20

ω 1SS /(g/L)0.5

Case 6

50 40 30 20

40 30

0 0

0.5

0 0

Case 8

ω 1SS /(g/L)

0.5

ω 1SS /(g/L)

Case 7

Case 5

60 40 n

Fig. 20. Numerical data (n) and empirical formula (e) for the frequency x1SS of the system’s ﬁrst symmetric mode vs. c.

65%, the frequency x1SS depends on the value of x1S and it may be approximated by:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi Er Ir g

If xr < 0:65 x1S ; then x1SS ¼ xr ¼ 10:73 " if xr P 0:65x1S ; then x1SS ¼ x1S

qr Ar L4

x1S 1 0:35 0:65 xr

ð9aÞ

Table 7 Error mean value (MV) and standard deviation (SD) of the empirical formula of the system’s frequency x1SS. Cases

2 # ð9bÞ

MV SD

Error (x1SS,n x1SS,e)/x1SS,n 1 (%)

2 (%)

3 (%)

4 (%)

5 (%)

6 (%)

7 (%)

8 (%)

2.3% 2.4%

1.0% 1.7%

7.5% 4.9%

6.0% 2.8%

1.2% 3.7%

7.5% 4.9%

3.6% 7.7%

5.6% 9.0%

I. Vassilopoulou, C.J. Gantes / Computers and Structures 88 (2010) 105–119

Knowledge of the natural frequencies of a nonlinear system and the relations between them provides us an important information about the existence of eventual internal resonances. It is possible to use the proposed formulae to calculate the natural frequencies of the cable net, with rigid or ﬂexible supports, at a preliminary design stage, in order to avoid 1:1 internal resonances between the ﬁrst vibration modes, which may lead to oscillations of large amplitude, with a continuous exchange of energy between the modes involved in the resonance.

Case 1-8 (ω 1SS )

119

ω 1SS,e

References

ω 1SS,n

Fig. 21. Empirical formula for the frequency x1SS ﬁts numerical data.

In Fig. 20, the frequency x1SS of the system’s first symmetric mode calculated by numerical methods and by the empirical formula is plotted vs. the non-dimensional parameter c. The error of the above formulae is calculated for all results obtained from the parametric modal analyses performed for each case, as the ratio (x1SS,n x1SS,e)/x1SS,n where x1SS,n and x1SS,e are the frequency x1SS calculated by numerical methods and by the empirical formula, respectively. The mean value and the standard deviation of the error are calculated and tabulated in Table 7 for each case. The accuracy of the empirical formulae is also shown in Fig. 21 for all eight cases of Table 5, and is evaluated as satisfactory for preliminary design purposes. Since the presence of the edge ring does not influence significantly the frequencies of the net’s antisymmetric vibration modes, the empirical formulae, given by Eqs. (5) and (6) can also be used for the case of a cable net with cables anchored to a deformable edge ring. 6. Conclusions The dynamic behaviour of cable nets having rigid supports, has been thoroughly investigated and many similarities with the case of a simple suspended cable have been observed. The introduction of a parameter k2 for cable nets makes possible the prediction of the modes appearance sequence and of the crossover points, at which modal transition occurs. The semi-empirical formulae provided in this work give satisfactory results and are suggested for predicting the frequencies of the first four vibration modes of the net. In case the deformability of the flexible contour ring is taken into account, the dynamic behaviour of the system becomes more complicated. The existence of the ring negligibly influences the antisymmetric modes of the cable net, but an in-plane mode of the ring produces a symmetric vertical vibration of the net, influencing significantly the frequency of the first symmetric mode of the net, with respect to the one of the cable net with fixed ends. Another semi-empirical formula is proposed for estimating the frequency of the system’s 1st symmetric mode, either produced by the symmetric vibration of the net, involving the ring, or by the in-plane mode of the ring, involving also the net.

[1] Vakakis A. Introduction in nonlinear dynamics. Lecture notes. Greece: National Technical University of Athens; 2002. [2] Pugsley AG. On the natural frequencies of suspension chains. Quart J Mech Appl Math 1949;II(4):412–8. [3] Ahmadi-Kashani K. Vibration of hanging cables. Comput Struct 1989;31(5):699–715. [4] Irvine HM, Caughey TK. The linear theory of free vibrations of a suspended cable. Proc Roy Soc London, Ser A Math Phys Sci 1974;341(1626):299–315. [5] Rega G, Luongo A. Natural vibrations of suspended cables with ﬂexible supports. Comput Struct 1979;12:65–75. [6] Rega G, Vestroni F, Benedettini F. Parametric analysis of large amplitude free vibrations of a suspended cable. Int J Solids Struct 1984;20(2):95–106. [7] Burgess JJ, Triantafyllou MS. The elastic frequencies of cables. J Sound Vib 1987;120(1):153–65. [8] Triantafyllou MS, Grinfogel L. Natural frequencies and modes of inclined cables. J Struct Eng 1986;112(1):139–48. [9] Gambhir ML, deV Batchelor B. Finite element study of the free vibration of 3D cable networks. Int J Solids Struct 1978;15:127–36. [10] Talvik I. Finite element modelling of cable networks with ﬂexible supports. Comput Struct 2001;79:2443–50. [11] Leonard JW. Tension structures, behavior and analysis. New York: McGrawHill, Inc.; 1988. [12] Seeley GR, Christiano P, Stefan H. Natural frequencies of circular cable networks. J Struct Div 1975;101(5):1171–7. [13] Buchholdt HA. An introduction to cable roof structures. London: Thomas Telford; 1999. [14] Porter Jr DS, Fowler DW. The analysis of nonlinear cable net systems and their supporting structures. Comput Struct 1973;3:1109–23. [15] Morris NF. Dynamic response of cable networks. J Struct Div 1974;100(10):2091–108. [16] Morris NF. Modal analysis of cable networks. J Struct Div 1975;101(1):97–108. [17] Swaddiwudhipong S, Wang CM, Liew KM, Lee SL. Optimal pretensioned forces for cable networks. Comput Struct 1989;33(6):1349–54. [18] Stefanou GD. Dynamic response of tension cable structures due to wind loads. Comput Struct 1992;43(2):365–72. [19] Tabarrok B, Oin Z. Dynamic analysis of tension structures. Comput Struct 1997;62(3):467–74. [20] Vassilopoulou I, Gantes CJ. Modal transition and dynamic nonlinear response of cable nets under fundamental resonance. In: Proceedings of the eighth HSTAM international congress on mechanics, vol. II, Patras, Greece; 2007, p. 787–94. [21] Vassilopoulou I, Gantes CJ. Vibration modes and dynamic response of saddle form cable nets under sinusoidal excitation. In: Proceedings of Euromech Colloquium 483, geometrically non-linear vibrations of structures, FEUP, Porto, Portugal; 2007. p. 129–32. [22] ADINA (Automatic Dynamic Incremental Nonlinear Analysis) v8.4. USA: ADINA R&D, Inc.; 2006. [23] ADINA (Automatic Dynamic Incremental Nonlinear Analysis) v8.4. Theory and modeling guide, ADINA solids and structures, vol. I. USA: ADINA R&D, Inc.; 2006. [24] Timoshenko S. Vibration problems in engineering. New York: D. Van Nostrand Company, Inc.; 1937.

Computers and Structures 88 (2010) 120–133

Contents lists available at ScienceDirect

Computers and Structures journal homepage: www.elsevier.com/locate/compstruc

Maximizing the fundamental eigenfrequency of geometrically nonlinear structures by topology optimization based on element connectivity parameterization Gil Ho Yoon School of Mechanical Engineering, Kyungpook National University, Republic of Korea

a r t i c l e

i n f o

Article history: Received 10 July 2009 Accepted 20 July 2009 Available online 13 August 2009 Keywords: Topology optimization Internal element connectivity parameterization method Modal analysis Nonlinear structure

a b s t r a c t This paper pertains to the use of topology optimization based on the internal element connectivity parameterization (I-ECP) method for nonlinear dynamic problems. When standard density-based topology optimization methods are used for nonlinear dynamic problems, they typically suffer from two main numerical difﬁculties, element instability and localized vibration modes. As an alterative approach, the I-ECP method is employed to avoid element instability and a new patch mass model in the I-ECP formulation is developed to control the problem of localized vibration modes. After the I-ECP based formulation is developed, the advantages of the proposed method are checked with several numerical examples. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Topology optimization methods are used for design purposes in civil and mechanical engineering [1–4]. In particular, layouts of linear dynamic structures, whose stiffness and mass matrices are independent on load and displacements, have been topologically optimized to reduce vibration or noise at target frequencies within pre-assigned design limits (see [5–16] and references therein for more details). In other words, by controlling the first eigenfrequency or some of the lowest eigenfrequencies of a linear structure, its dynamic characteristics could be improved indirectly. However, a review of current topology optimization methods for nonlinear dynamic problems highlights the difficulties related to unstable elements in geometrical nonlinear static analysis [2,17–24] and difficulties associated with the highly localized vibration modes in the modal analysis [12–16,25–29]. This study examines how these difficulties can be resolved and investigates the use of topology optimization for maximizing the fundamental eigenfrequency of geometrically nonlinear structures. In order to calculate eigenfrequencies of a geometrically nonlinear structure, it is necessary to perform a two-step numerical analysis; a modal analysis should be performed after a nonlinear static analysis. First, the deformation of a structure subject to a load at which nonlinear responses are detected in the structure should be computed using a standard nonlinear static solver commonly E-mail addresses: gilho.yoon@gmail.com, ghy@knu.ac.kr 0045-7949/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2009.07.006

with Newton–Raphson iteration. Second, we perform a modal analysis with the mass matrix and the tangent stiffness matrix computed at the last Newton–Raphson step of the first nonlinear static analysis [30,31]. In density-based topology optimization, flipped elements having negative Jacobian values – referred to as the unstable elements – inevitably appear around void areas simulated by elements with weak Young’s moduli during the Newton– Raphson iteration of the first nonlinear static analysis. Because these unstable elements make the Newton–Raphson iteration diverge, it is difficult to carry out topology optimization with the nonlinear static analysis. These difficulties have been overcome by developing numerical methods such as the modified Newton– Raphson iteration, the element removal and reintroduction method, and the displacement loading method [17,20,21]. In addition to the first difficulty of the unstable elements, the modal analysis for topology optimization problem is also suffering from highly localized vibrating modes in low-density regions having relatively low eigenvalues. Considering one or some of the lowest modes in topology optimization, these highly localized vibrating modes are troublesome. To resolve them, an optimization formulation maximizing localized buckling modes with a mesh independent filter have been proposed [12,26,27,29]. Alternatively several techniques involving mass density interpolation have also been proposed without changing optimization formulation [14,25]. In this research, to overcome the above mentioned difficulties, an alternative topology optimization approach called the internal element connectivity parameterization (I-ECP) method was

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G.H. Yoon / Computers and Structures 88 (2010) 120–133

employed. Unlike the standard density-based approach, this new method deﬁnes a structural layout by deﬁning connectivity among solid elements using zero-length links, as shown in Fig. 1 [19,32–

34]; it may be possible to interpret the element connectivity parameterization (ECP) method as a numerical method imposing weak continuity constraints among elements using the penalty

(a)

Weak links

Design variable Weak element Strong links

Solid element

SIMP approach (b)

Patch Design variable

I-ECP approach (c)

Fig. 1. Modeling by the density-based SIMP and the I-ECP (internal-element connectivity parameterization).

Fig. 2. Solution procedure for the nonlinear modal problem.

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G.H. Yoon / Computers and Structures 88 (2010) 120â&#x20AC;&#x201C;133

or the Lagrange multiplier (see related discussions in [20]). Because the material properties of solid elements do not change during optimization iterations, the unstable element of nonlinear structures can be eliminated. Depending on how solid elements

are connected with zero-length links, two ECP methods exist [32]. One is the external ECP (E-ECP) method in which solid elements are directly connected, and the other is the internal ECP (I-ECP) method, where the solid elements are indirectly connected

Fig. 3. Effects of the applied load on eigenfrequencies for a slender beam. (a) Problem deďŹ nition, (b) eigenmodes and eigenvalues for zero force, (c) eigenmodes and eigenvalues for nonzero force, and (d) curves of eigenfrequency versus applied load for the beam.

G.H. Yoon / Computers and Structures 88 (2010) 120–133

123

Fig. 4. e-th planar rectangular patch consisting of a plane ﬁnite element and four zero-length links.

through outer nodes. Because of the computational advantage in the latter, this study employs the I-ECP method [32,35]. For successful analysis and optimization, this study also develops a novel way of mass modeling formulation for the I-ECP method. So far, because of the ambiguity in the definition of the formulation of a mass matrix, the I-ECP method has not been applied to dynamic problems. It might have been obvious to define a mass matrix for a solid element inside a patch, but it was not apparent that a mass matrix could be defined for zero-length links; here ‘‘define” means calculating and assembling the mass matrix in the framework of the finite element (FE) procedure. Moreover, from several empirical test results discussed in Section 2.3, it appears that assembling the mass matrix of a solid element to the degrees of freedom of the solid element complicates the optimization process due to the presence of highly localized modes in patches

Fig. 5. Mass modelings using the I-ECP method. (a) A straightforward model (assigning the mass matrix at the solid element to the degrees of freedom of the inner nodes), and (b) the proposed patch mass matrix model (assigning the mass matrix to the degrees of freedom of the outer nodes).

[12–14,26–28]. To avoid the complication, an alternative approach in which the mass matrix of a solid element is assembled to the degrees of freedom of the outer nodes is presented in this study. Adopting the new mass modeling method proposed here and the mass interpolation functions proposed in [14,25], the difﬁculty arising from highly localized modes inside patches could be overcome and topology optimization could be carried out for nonlinear dynamic system. This paper is organized as follows. In Section 2, we provide an overview of the basic notations and governing equations for nonlinear modal analysis. We use the I-ECP method and study the mass modeling of the method. The parameterization of design variables and the sensitivity analysis of the I-ECP method are dealt with in Section 3. In Section 4, two numerical examples in which the fundamental eigenfrequency of two-dimensional structures is maximized are presented to show the potential of the proposed method in Section 4. In Section 5, some observations are made regarding designs of nonlinear dynamics system and future work is discussed.

Fig. 6. Solid element: (a) a ﬁnite element model, and (b) four eigenvalues and eigenmodes.

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G.H. Yoon / Computers and Structures 88 (2010) 120–133

displacements at time t + Dt of a generic point of a body by DU and U, the following update rules can be obtained:

2. Modal analysis of nonlinear structures using the I-ECP method

tþDt

2.1. Formulation of nonlinear vibration problem

The vibration motion of a structure with practical dynamic loads is represented by sinusoidally varying eigenmodes as well as static equilibrium displacements [30,31,36,37]. In the case of a linear structure, the static equilibrium displacements are zero and only the sinusoidal eigenmodes remain. Thus, the linear eigenvalue problems can be solved by using load-independent stiffness and mass matrices can be solved. In contrast, to calculate eigenvalues and their associated eigenmodes for a nonlinear structure, the modal analysis depicted in Fig. 2 should be used along with the tangent stiffness matrix computed for the deformed domain for given static loads and the mass matrix [30,31]. Consequently, eigenfrequencies of a geometrically nonlinear structure are dependent on current static displacements even for a structure made of linear elastic material. For nonlinear static analysis, the notations given in [30,31] were followed. By respectively denoting the updated displacements and

structure

UðkÞ ¼ tþDt Uðk 1Þ þ DUðkÞ ;

ðk 1Þ KT DUðkÞ

¼ Rð

tþDt

ðk 1Þ

Uð0Þ ¼ t U

ð1Þ

Þ ðsee ½19; 20 Þ

ð2Þ

tþDt

where the superscript (k) denotes the kth iteration step in the Newton–Raphson method. The incremental residual and tangent stiffðk 1Þ ness matrix are denoted by RðtþDt Uðk 1Þ Þ and t KT , respectively. After solving the nonlinear static equation (Eq. (2)), the following modal analysis should be carried out, where the displacementdependent tangent stiffness matrix tþDt KT and the displacementindependent mass matrices, M are used

ðtþDt KT x2 MÞU ¼ 0; UT MU ¼ I

ð3Þ

Here, x and U are eigenfrequencies and associated eigenmodes, respectively. The identity matrix is I is of the same size as the as the mass or stiffness matrix. In order to investigate how eigenvalues of nonlinear structures are affected signiﬁcantly by applied forces in practice, we considered the simple beam shown in Fig. 3a. The left side of the beam

Fig. 7. Comparison of mass models obtained using the ECP method ðlmax =kdiagonal ¼ 104 Þ. (a) A straight mass model, (b) a present patch mass model, and (c) modes of two methods.

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G.H. Yoon / Computers and Structures 88 (2010) 120–133

is clamped and a concentrated force is applied to the right side. The deformed shape of the beam determined by nonlinear static analysis in Fig. 3c and the associated eigenmodes obtained from the modal analysis of Eq. (3) indicate that the eigenfrequencies vary nonlinearly with the magnitude of the applied force; the first and second angular speeds at under the applied force of 0.003 N are increased approximately by 130% and 14%. However, the third angular speed is decreased by 8%. As observed in the figure, the eigenfrequencies are influenced significantly by the applied loads and associated deformations. Fig. 3c plots the eigenmodes of the undeformed structure; if the modes were plotted at the deformed structure, it would be difficult to compare them with those of linear analysis as shown in Fig. 2. Thus, later in the document, we present eigenmode plots of examples at undeformed structures.

2.2. Nonlinear static analysis using the ECP method The I-ECP method enables the realization of a layout different from that of the element density-based method. To underline the basic concepts of these two methods and some fundamental differences between them, consider the layout in Fig. 1a, which may be observed during topology optimization iterations. When the standard element density method is employed, design variables deﬁned for each element and their corresponding Young’s moduli are varied to realize the layout of Fig. 1a in terms of the potential energy. Although this method is robust and simple, it suffers from numerical instability called the unstable element with negative

Jacobian values in topology optimization for geometrically nonlinear structures [2,17–21,23,24]. As opposed to the density-based methods, the I-ECP method does not change the material properties of plane or cubic finite elements discretizing a domain during topology optimization as shown in Fig. 1c. To define a layout, all elements are disconnected from other elements and nodes at the same locations are connected using one-dimensional links. To reduce the computation time, the static condensation scheme, which condenses out for the degrees of freedom of inner nodes of the I-ECP patch from the global stiffness matrices, was proposed in [32]. A detail condensation implementation of the I-ECP method can be found in [35]. To formulate the I-ECP method for nonlinear static analysis, the eth patch shown in Fig. 4 is considered along with the assumption of geometrical nonlinearity. The nodes connecting elements are named the outer nodes and those defining plane elements are named the inner nodes. The displacements of the outer nodes ðkÞ ðkÞ and inner nodes are denoted by tþDt ue;out and tþDt ue;in in the kth iteration, respectively. The displacements are then updated as follows:

" tþDt

ðkÞ

ue;out

tþDt

ðkÞ

ue;in ðkÞ

" tþDt ¼

tþDt

ðk 1Þ

ue;out

ðk 1Þ

ue;in

" þ

DuðkÞ e;out DuðkÞ e;in

# ð4Þ

ðkÞ

where Due;out and Due;in denote the updated displacements for the outer nodes and the inner nodes, respectively, and are calculated by the following equation:

Fig. 8. Ratios of the condensed stiffness matrix to mass matrix with various mass interpolation functions for one element in Fig. 7 (SIMP penalty: 3). (a) The stiffness matrix behavior, (b) a linear mass interpolation function of Eq. (25), (c) a power mass interpolation function of Eq. (26), and (d) a C1 continuous mass interpolation function of Eq. (27) proposed in [14].

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G.H. Yoon / Computers and Structures 88 (2010) 120–133

kI;e ¼ le ðce ÞI8 8 (

ðwhere I8 8 is a 8 8identity matrixÞ

kI;e

0 0

#)"

0 t

DuðkÞ e;out

structure;ðk 1Þ

kT;e

DuðkÞ e;in

" ¼

ð5Þ ðk 1Þ

Re;out

ðk 1Þ

Re;in

ð6Þ The link stiffness of the eth patch, le, is a function of the design variable ce. The stiffness matrix and the residual force terms of the outer and the inner nodes of the eth patch are denoted by ðk 1Þ ðk 1Þ t structure;ðk 1Þ kT;e , Re;out and Re;in , respectively. ðk 1Þ ðk 1Þ For the I-ECP, Re;out and Re;in can be formulated as

# ðk 1Þ

Re;out

ðk 1Þ

Re;in 2 4

tþDt

tþDt ¼

link;ðk 1Þ 0 f e;out

tþDt link;ðk 1Þ f e;in 0

3 5¼

0 tþDt

kI;e kI;e

structure;ðk 1Þ 0fe

kI;e kI;e

" tþDt tþDt

2 4

tþDt link;ðk 1Þ 0 f e;out tþDt

link;ðk 1Þ

0 f e;in

3 ð7Þ

# ðk 1Þ

ue;out

ð8Þ

ðk 1Þ

ue;in

In Eq. (7), the externally applied force on the outer nodes and the internal force acting on the inner nodes are denoted by tþDt Re and tþDt structure;ðk 1Þ , respectively. Because the degrees of freedom of 0fe the inner nodes are independent of those of the other nodes, the static condensation scheme can be applied. t

ð9Þ ð10Þ

The global tangent matrix is assembled as t

ðk 1Þ

KCon ¼

Np X

ðk 1Þ

kCon;e

i structure kT;e x2 me ½/ ¼ |{z} 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |{z} 8 1 tþDt

ð11Þ

e¼1

ð14Þ

8 1

8 8

For the I-ECP patch of Fig. 7a:

kI;e

# x2

) "

ue;out

structure ue;in kI;e kI;e þ tþDt kT;e 0 me |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl} 16 16

1 ðk 1Þ structure;ðk 1Þ kCon;e ¼ ðkI;e kI;e kI;e þ tþDt kT;e kI;e Þ 1 structure;ðk 1Þ ðkÞ ðk 1Þ ðk 1Þ t ðk 1Þ kCon;e Due;out ¼ Re;out þ kI;e kI;e þ t kT;e Re;in

patch assembling the mass matrix of the solid element to the inner nodes is constructed in Fig. 7a. The modal analysis equations of the solid element and the patch are formulated as follows For the solid element of Fig. 6:

¼ |{z} 0 16 1

16 1

ð15Þ Here, u, ue,out, and ue,in are the eigenmodes of the solid element and the sub-vectors of the eigenmodes for the outer and the inner nodes of the patch in Fig. 7a, respectively. The stiffness matrix and the structure and me, mass matrix of the solid element are denoted by t kT;e respectively. In Eq. (15), note that the mass matrix of the solid element is assembled for the degrees of freedom of the inner nodes. structure is assembled for the Further, the tangent stiffness matrix t kT;e degrees of freedom of the inner nodes. Because four new nodes are added when constructing a patch in the I-ECP method, there are eight additional modes between the inner and the outer nodes in addition to the eight eigenmodes of the solid element. As seen in Fig. 7c, the ﬁrst four eigenvalues and eigenmodes of the outer nodes become almost identical with the high stiffness value of the links. In contrast to the case of links with a high stiffness value, highly localized vibrating modes inside the patch of Fig. 7a have been found as side effects when the stiffness value of links becomes small. In other words, with a small stiffness value, the fundamental eigenmode of the patch will be the motion observed between the solid element and links inside the patch. As this phenomenon

where Np is the total number of patches. Then the following system of equations is solved iteratively for DUðkÞ out , which is the global displacement vector for the outer nodes: t

ðk 1Þ

KCon DUðkÞ out ¼ RCon ðk 1Þ

where RCon ¼

Np X

ðk 1Þ

structure;ðk 1Þ

Re;out þ kI;e kI;e þ t kT;e

ðk 1Þ

ð12Þ

Re;in

e¼1

ð13Þ 2.3. Modeling of mass matrix in I-ECP method for modal analysis Eigenfrequencies of the nonlinear system that can be determined using the I-ECP method are also calculated by modal analysis (Eq. (3)). The tangent stiffness matrix can be easily computed by Eq. (2). But it is not clear how the mass matrix in Eq. (3) is constructed for the I-ECP method, which is one of the limitations of this method. The construction of the mass matrix is examined in the following sections. 2.3.1. Formulation 1: straightforward method for mass matrix modeling For dynamic analysis, assigning mass matrices to each solid elements is straightforward as shown in Fig. 5a. If the stiffness values of the links of a patch are sufﬁciently large, some eigenvalues of an I-ECP patch become equivalent to the eigenvalues of a solid element in principle. For example, consider the two-dimensional solid element in Fig. 6a under the assumption of small displacements. Because the degrees of freedom of the bottom nodes are clamped, there are 4 real and 4 inﬁnite eigenvalues in Fig. 6b. Here an I-ECP

Fig. 9. Cantilever problem in the case where the ends of the cantilever are clamped. (a) Problem deﬁnition, (b) a symmetric linear result with an evenly distributed initial guess, and (c) an unsymmetrical result with an unsymmetrical initial guess.

G.H. Yoon / Computers and Structures 88 (2010) 120–133

can be observed for any patch with a small stiffness value, many highly localized modes will appear. Furthermore, because the topology optimization formulation for dynamic structures usually involves the consideration of the fundamental eigenfrequency or some of the lowest eigenfrequencies, these localized modes of patches are not desirable. Consequently, although the proposed method, which assigns the mass matrices to solid elements is simple, it is not applicable to topology optimization. 2.3.2. Formulation 2: present mass matrix approach To resolve the above-mentioned problems related to the localized modes in patches, this paper presents a new concept assembling the mass matrices of solid elements in patches into the degrees of freedom of the outer nodes, as shown in Fig. 5b. Unlike the straightforward method, the solid element of the patch is modeled as massless, which makes the eigenmodes between the inner and the outer nodes numerically inﬁnite. To formulate this patch mass matrix approach, Eq. (15) is modiﬁed as below.

For the patch mass matrix : # (" kI;e kI;e kI;e

kI;e þ

tþDt

structure kT;e

ue;out ue;in

# ¼0

ð16Þ

127

This approach has the following features. (1) Compared to Eq. (15), the mass matrix of the solid element is simply assembled into the degrees of freedom of the outer nodes, which makes it simple to implement. (2) This simple modification of the mass matrix resolves the local modes observed between the inner and the outer nodes completely. For example, Fig. 7b and c shows the result of the reanalysis of the eigenmodes using this patch matrix approach, where four real eigenvalues and eight infinite eigenvalues are existing. (3) It is important to note that the local modes observed among patches, which correspond to local modes among void elements in element-density-based approaches, still exist. Therefore, links with very small stiffness values can produce highly localized vibrating modes which cause non-convergence in topology optimization. For the efficient construction of the present patch mass model, the following condensation scheme is used for the global stiffness matrix and mass matrix:

ðkI;e ue;out kI;e ue;in Þ x2 me ue;out ¼ 0

Fig. 10. Eigenfrequency histories for (a) a design with an initial uniform density, and (b) an optimized design with an unsymmetrical density distribution.

ð17Þ

128

G.H. Yoon / Computers and Structures 88 (2010) 120–133

structure kI;e ue;out þ kI;e þ tþDt kT;e ue;in ¼ 0 1 structure;ðk 1Þ t ðk 1Þ kCon;e ¼ kI;e kI;e kI;e þ tþDt kT;e kI;e t

KCon ¼

Np X

ðk 1Þ

kCon;e ; MCon ¼

e¼1

Np X

ð19Þ

Material model 2 : ( m0 ce ; ce > 0:1 me ¼ m0 ðc1 c6e þ c2 c7e Þ; ce 0:1; c1 ¼ 6 105 ; c2 ¼ 5 106

ð20Þ

e¼1

ð21Þ

3. Topology optimization formulation

For the sake of simplicity, we consider topology optimization to maximize only the fundamental eigenfrequency frequency of a geometrically nonlinear structure, which is deﬁned as c

For the topology optimization of dynamic problems, the values of two mechanical material properties, i.e., link stiffness and material density, are interpolated with respect to the design variable (c). In this paper, the following interpolation function is used for the link stiffness in Eq. (5) (see [32,35] for more detail):

a s structure kdiagonal

a ¼ lmax lmin ; b ¼ lmin cmin ce 1; cmin ¼ 0:001

! ð22Þ ð23Þ ð24Þ

where k is the number of degrees of freedom per node and s and n are penalties. A diagonal term of the linear stiffness matrix is destructure noted by kdiagonal . The upper and lower bounds of the stiffness of links are denoted by lmax and lmin, respectively. Numerical examples indicate that with sufﬁciently large and small values for lmax and lmin and appropriate penalties, similar results can be obtained (see [35] for the effects of these links in linear static structure cases and Section 4 for numerical examples in dynamic structure cases). structure structure Here, lmax and lmin are set to 104 kdiagonal and 10 4 kdiagonal , respectively. In all numerical examples, we assumed n = 3, k = 2, and s = 10 assumed that because optimal layouts similar to those obtained in the solid isotropic material with penalization (SIMP) method could be obtained. Fig. 8a shows the ratio of the condensed stiffness matrix to the normal stiffness. The mass matrix can be interpolated as follows:

me ¼ m0 ce

3.2. Topology optimization formulation

Max

3.1. Material Interpolation

s¼

ð26Þ

ð27Þ

½t KCon x2 MCon Uout ¼ 0

ce n þb 1 þ ð1 cne Þs

m0 ce ; ce > 0:1 m0 c6e ; ce 0:1

Material model 1 : me ¼

Finally, the following modal equation is solved for obtaining the outer nodes, Uout

le ¼ a

ð18Þ

s:t:

Min fxj g

j¼1;...;j Np X

q c v V

eð eÞ e e¼1 tþDt

Rð

ð28Þ

UÞ ¼ 0

ðtþDt KT ðtþDt U Þ x2 MÞU ¼ 0 where qe, ve and V* are the element density, element volume, and prescribed volume limit, respectively. The converged displacements obtained by solving the nonlinear static equation are denoted by t+Dt * U , and the number of candidate eigenfrequencies is denoted by J. To determine optimal topologies using a gradient-based optimizer, the sensitivity analysis of the jth eigenfrequency, xj, should be performed with respect to the design variable. In this study, the following sensitivity equation can be derived for the patch mass method (see [26–28,38]):

dxj 1 dle dme ¼ ðue;out ue;in ÞT ðue;out ue;in Þ x2j uTe;out ue;out dce 2xj dce dce ð29Þ

ð25Þ

where m0 is the nominal mass stiffness matrix. Numerical tests show that this interpolation function causes highly localized vibration modes inside I-ECP patches having links with low stiffness values [12,26–28].When the present patch mass method is used, it is observed that highly localized vibrating modes do not appear between the inner and outer nodes because the solid plane element inside an I-ECP patch only has stiffness and is massless, as shown in [16]. However, the localized modes continue to appear at the outer nodes, which are locally vibrating when the design variables have low values. That is because the ratio of the condensed stiffness matrix of the Eq. (19) to the mass matrix for the I-ECP patch becomes almost zero, similar to the case of the SIMP method; this results in the presence of localized vibration modes in patches with low link stiffness values (Fig. 8). (See [12,14,26–29] for localized vibration modes and sensitivity analysis of the SIMP method.) To suppress these local modes, some mass interpolation functions that have been veriﬁed in [12,14,25] are also tested here. By employing the following material interpolations, the locally vibration modes can be suppressed as expected

Fig. 11. Optimized results considering the geometrical nonlinearity. (a) A model with an arbitrary force, (b) with F = 10 N, and (c) with F = 20 N.

G.H. Yoon / Computers and Structures 88 (2010) 120–133

129

Fig. 12. Eigenfrequency histories for (a) a design with F = 10 N, and (b) a design with F = 20 N.

4. Topology optimization of nonlinear dynamic problems In order to illustrate the potential of the developed method, two-dimensional optimization problems with different loads are considered. Unless stated otherwise, uniform initial guesses of c satisfying given mass constraints are used. The dimension, material properties and magnitude of the loads are arbitrarily chosen to show the effectiveness of the present I-ECP method. For successful optimization, it is important to use a proper optimization algorithm such as sequential linear programming, sequential quadratic programming, and the method of moving asymptotes. In this study, the method of moving asymptotes is used as an optimization algorithm; however other optimization algorithms can also be used [39]. To get rid of checkerboard patterns inside the design domain, there are many regulation methods such as mesh-independent filtering, slope-constraint, density-filtering, and morphology filtering. In this paper, the mesh-independent sensitivity filtering is used (see Ref. in [40] for more details). 4.1. Example 1: cantilever beam A cantilever beam with clamped ends (Fig. 9a) is considered first. A finite element model of the design domain is constructed by using 5000 (200 25) patches containing bilinear Q4 elements. The volume considered is constrained to be less than 50% of the domain volume. Fig. 9b and c, and Fig. 10 show topological layouts

obtained by the proposed ECP method, and iteration histories. With a zero force (F = 0 N), the topological layout in Fig. 9b, which is comparable to the layouts presented in [14,15] using the SIMP method, could be obtained using a constant initial guess. Results of empirical tests revealed that another local optimum design could be obtained using an unsymmetrical initial guess in Fig. 9c.1 To consider the effect of large displacements on the optimal layouts, a concentrated point load is now applied at the center of the bottom surface of the beam in the downward direction (Fig. 11a). With sufficiently large loads (F = 10 N and 20 N) that elicit nonlinear responses in the beam, the unsymmetrical designs in Fig. 11b and c, whose eigenfrequency histories are shown in Fig. 12, could be obtained. Table 1 and Fig. 13 compare the eigenfrequencies of the four solutions with respect to the applied loads. From the analysis of the numerical results presented in Fig. 13, it is observed that the optimal layouts considering the geometrical nonlinearity in Fig. 11 are optimized for the given loads. At F = 10 N, the fundamental eigenfrequency of the nonlinear design 1 Up the best of our knowledge, the benchmark problem has been solved without considering a fixed mass by an evenly distributed initial design only [10,14]. Consequently, it appears that the first eigenfrequency, whose mode is the bending mode, is maximized. Therefore, from our numerical examples, we could obtain the unsymmetrical design shown in Fig. 9c with a randomly distributed initial guess. These empirical numerical tests imply that the optimization problem referred to has many local optima. Furthermore, eigenfrequencies and associated eigenmodes of a few initial iterations influence the solutions of the optimization problem.

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G.H. Yoon / Computers and Structures 88 (2010) 120–133

Table 1 Eigenfrequencies of optimized designs for the clamped beam. Design

Fig. 9b

Fig. 9c

Fig. 11b

Fig. 11c

Linear

223.06 (rad/s)

176.26 (rad/s)

256.99 (rad/s)

241.72 (rad/s)

Loading 1 (F = 10 N)

225.86 (rad/s)

189.92 (rad/s)

304.33 (rad/s)

297.24 (rad/s)

Loading 2 (F = 20 N)

222.63 (rad/s)

196.57 (rad/s)

322.41 (rad/s)

343.16 (rad/s)

Fig. 13. Load and eigenfrequency curves for the optimized results for the clamped beam.

Fig. 14. Numerical tests for examining the effects of link stiffness values. (The values in parentheses angular speed.)

of Fig. 11b, optimized for F = 10 N, are better than that of the design of Fig. 11c, optimized for F = 20 N, whereas the eigenfrequency of Fig. 11c is better than that of Fig. 11b at F = 20 N. Furthermore, a larger force (F = 20 N) produces a straight beam whose length is longer than the length of a straight beam of Fig. 11b. This example shows that a physically sound structure can be obtained using the present ECP method. Using Fig. 14 that shows numerical test results with different upper and lower bounds for the link stiffness values, similar local optimized results can be obtained in the case of both sufﬁciently large and sufﬁciently small link stiffness values (see [35]).

equivalent to those layouts obtained by solving the static compliance minimization problem (see Ref. [19] for more details). This might be explained by the fundamental structural eigenfrequency being proportional to the square root of the global stiffness. Fig. 16d plots the ﬁrst frequencies against the applied loads for each design. After exceeding near F = 0.5 N, the lower beam in

4.2. Example 2: beam structure with a point load or bending load In the second example, a beam with clamped boundary conditions is considered (Fig. 15a). The design domain is discretized by 200 30 patches. To avoid obtaining a trivial void structure, the densities of two elements at the loading point are ﬁxed as 20 kg/m3. The volume is also constrained to be less than 50% of the design domain. Fig. 15b shows an obtained linear design with a uniform initial density distribution for F = 0 N. It is found that this result is similar to the results obtained using the SIMP method too [14]. Subsequently, optimization problems with different loads were considered (Fig. 16). The layouts of the results maximizing the ﬁrst eigenfrequency in Fig. 16 are, in a certain sense,

Fig. 15. Beam structure with a point load. (a) Problem deﬁnition, and (b) result 1 (a linear optimized result).

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G.H. Yoon / Computers and Structures 88 (2010) 120–133

Fig. 16. Optimized results for point loading. (a) Problem deﬁnition, (b) result 2 (a result with a point load of 1.5 N), (c) result 3 (a result with a point load of 5 N), and (d) the curves of eigenfrequencies with respect to the applied point load.

Fig. 15b exhibits the buckling phenomenon. Consequently, the fundamental structural eigenfrequency of Fig. 15b decreases. In contrast, because the nonlinear designs of Fig. 16b and c can support larger forces without bucklings, their frequencies do not decrease at the given loads. It was observed that small density perturbations in the linear design of Fig. 15b can cause bucklings of the internal

Fig. 17. Optimized results for the bending loading case. (a) Problem deﬁnition, (b) result 2 (result for a bending moment of 1.25 Nm), (c) result 3 (a result for a bending moment of 2.5 Nm), and (d) the curves of eigenfrequencies with respect to the applied bending moment.

bar structure. Therefore, investigations on methods to control structural behaviors after post-buckling and eigenfrequency should be carried out in future. Table 2 compares the eigenfrequencies of the present designs.

Table 2 Eigenfrequencies of optimized designs for a cantilever beam subject to a point load. Design

Fig. 15b

Fig. 16b

Linear

8.43 (rad/s)

8.25 (rad/s)

Fig. 16c 7.74 (rad/s)

Shear (F = 1.5 N)

N/A (buckling of solid elements)

8.12 (rad/s)

7.77 (rad/s)

Shear (F = 5 N)

N/A (buckling of solid elements)

3.83 (rad/s)

7.87 (rad/s)

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G.H. Yoon / Computers and Structures 88 (2010) 120–133

Table 3 Eigenfrequencies of optimized designs for a cantilever beam subject to a bending moment. Design

Fig. 15b

Fig. 17b

Fig. 17c 6.81 (rad/s)

Linear

8.43 (rad/s)

6.61 (rad/s)

Bending (BM = 1.25 Nm)

N/A (buckling of solid elements)

8.51 (rad/s)

8.28 (rad/s)

Bending (BM = 2.5 Nm)

N/A (buckling of solid elements)

8.46 (rad/s)

8.99 (rad/s)

In addition to the point load, changes in the layout designs with the bending moments were investigated (Fig. 17). Similar to the case of the point loads, a large bending moment makes the leftend-lower beam prone to buckling as shown in Fig. 17d. Consequently, the ﬁrst structural eigenfrequency of the linear design of Fig. 15b is decreasing with respect to the magnitude of bending moment. Considering the geometrical nonlinearity, different topological layouts, which make the left-end parts stiffer and can support given larger moments, can be obtained in Fig. 17. The eigenfrequencies for these designs are given in Fig. 17d and Table 3. An investigation of the deformed shapes indicates that the curled left-end-upper beam of Fig. 17 becomes straight for the given bending moment, thereby increasing the overall stiffness. This example also shows that the present I-ECP method can be used to design for nonlinear dynamic structures.

5. Conclusions This paper pertains to topology optimization using the internal element connectivity parameterization (I-ECP) method in order to maximize the first eigenfrequency of a geometrically nonlinear structure. To calculate the eigenfrequencies of a nonlinear structure, modal analysis should be conducted after nonlinear static analysis using the Newton–Raphson method. However, in the framework of the standard density-based method, the modal and static analysis of nonlinear structures are hindered due to the instability of flipped elements among weak elements and the presence of locally vibrating modes. To the best of our knowledge, owing to these difficulties, there has not been any research on topology optimization in the context of nonlinear dynamic structures. To overcome the shortcomings of density-based methods, this paper proposes the use of the ECP method. This method was developed for compliance minimization problems involving large displacements, yet it was not applied to dynamic problems because of the ambiguity in the mass model. To examine whether distributed mass models can be constructed using the ECP method, this paper investigates the construction of two computational mass models. One mass model relates the mass matrix, formulated by the finite element method, to the degrees of freedom of a solid element inside a patch, while the other relates the same mass matrix to the degrees of the freedom of outer nodes forming a patch. Showing good agreement with each other for a solid element with a high stiffness value for links, the first mass model with a small stiffness value for links inherently has highly localized vibrating modes inside patches which can be troublesome on topology optimization. Therefore, this paper employs the second mass model (the patch mass modeling), which is relatively free from local modes, in preference to the direct method shown in an example with one element (Fig. 7). To illustrate the potential of the proposed approach, two numerical examples with sufficiently larger loads that elicit nonlinear responses from the structure considered are presented. For the sake of simplicity in optimization and sensitivity analysis, only the first eigenfrequency is considered. The optimized results show that it is possible to solve topology optimization problems by con-

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