3D PUSHOVER METHOD FOR IRREGULAR RC STRUCTURES Barbara Ferracuti, Marco Savoia, University of Bologna, DISTART – Structural Engineering, Italy e-mails: marco.savoia@mail.ing.unibo.it, barbara.ferracuti@mail.ing.unibo.it Rui Pinho University of Pavia, Department of Structural Mechanics, email: rui.pinho@eucentre.it ABSTRACT In the present study, a pushover procedure for 3D frame structures is proposed. A new way to define a set of horizontal force and torque distributions to apply at the floor levels is proposed here. For predicting the behaviour of irregular structures in the worst configurations, more than one pushover analyses must be performed. The proposed method is validated by a consistent comparison of results from static and dynamic analyses in terms of different response parameters, such as displacements, rotations, floor shears and floor torques. Starting from the linear analysis, the procedure is subsequently extended to nonlinear cases. The results confirm the effectiveness of the proposed procedure to predict the structural behaviour in the most dangerous configurations. KEYWORDS 3D pushover analysis, incremental dynamic analysis, RC frame structures, in-plane irregularity, multi-storey buildings. 1 INTRODUCTION In the recent years, nonlinear static analyses received a great deal of research attention within the earthquake engineering community. Their main goal is to describe the nonlinear capacity of the structure when subjected to horizontal loading, with a reduced computational effort with respect to nonlinear dynamic analyses. For 2D frame structures, many studies have been performed in order to validate different
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pushover techniques by comparison with results from dynamic analyses ([1], amongst many others). On the contrary, few methods to perform pushover analyses for irregular 3D frame structures have been proposed [2], [3], [4], [5]. The definition and assessment of 3D pushover methods are much more complex than in the case of 2D analyses for several reasons. Indeed, the fundamental point in 3D pushover analyses is the selection of the distribution of the horizontal forces along the frame height and across the individual floors. At floor level, horizontal forces can be divided into translational and torsional contributions. By analyzing the dynamic response of an irregular structure subjected to earthquake excitation, it can be observed that maximum displacement and maximum rotation do not occur at the same time step. Therefore, a unique pushover force distribution giving the most severe conditions for the structure cannot be defined. In the present work, a new pushover procedure for 3D RC structures, called Force/Torque pushover (FTP) is proposed; a set of force and torque distributions is selected to predict the most severe configurations the structure may undergo during the earthquake. In order to validate such technique, the comparison between results from a series of incremental dynamic analyses and pushover analyses has been carried out, considering different representative parameters of the structural response. In particular, both global parameters (capacity curves, horizontal displacement versus rotation of the centre of mass of the roof) and local parameters such as floor shears, interstorey drifts, floor rotations have been considered. A good agreement between results with the proposed pushover method and dynamic analyses has been preliminarily found. 2 PROPOSED METHOD: FORCE/TORQUE PUSHOVER(FTP) The fundamental point in 3D pushover analyses is to define the distribution of horizontal force over the frame height and on the individual floors. At the floor level, horizontal forces can be divided into translational and torsional contributions. By analyzing the dynamic response of an irregular structure subjected to earthquake excitation, it can be observed maximum displacement and maximum rotation do not occur at the same timing step. On the contrary, in a static pushover analysis the maximum rotation and the maximum displacement occur at the same force level. Therefore, a unique pushover force distribution giving the worst conditions for the structure cannot be defined. In the present work, a new pushover procedure for 3D irregular RC frame structure, called Force/Torque Pushover (FTP), is proposed. First of all, a rigid diaphragm constraint is introduced for the individual floors of the structure. Then, the method is based on the two following assumptions: 1) For plan-irregular structures, 2 modes are dominant, and higher order modes can be neglected.
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2) Displacement and rotation profiles of first and second mode along the frame height are related as:
δ x 2 ( z ) ≅ δ x1 ( z ) , δ y 2 ( z ) ≅ δ y1 ( z ) ,
θ2 ( z ) ≅ K R θ1 ( z ) ,
(1)
The modes have been normalized by setting to unity the top displacement in the dominant direction (e.g., the direction of the earthquake motion). Eqn (1) states displacement profiles of first two modes are approximately proportional to each other along the frame height, and the same can be said for rotation components, being KR the scale coefficient. Therefore, the force distribution proportional only to the fundamental mode of the structure with the highest participant factor for the selected ground motion direction is considered. The floor force resultant of the i-th floor level is divided into lateral forces Fx ,i , Fy ,i and torque Ti with respect to the centre of mass. In the proposed procedure, a weight coefficient β for the two components is then introduced, so defining a class of force distributions with variable translational and torsional configurations. The forces applied at i-th floor are then written as (see Figure 1): ⎧ Fx,i = (1 − β) Fx ,i ⎪ ⎨ F y ,i = (1 − β) F y ,i ⎪ Ti = βTi ⎩
(2)
The β weight coefficient can vary from zero to one. Therefore, translational forces only and torque only are applied, for the two limit cases, β=0 and β=1, respectively, as depicted in Fig. 1. For β =0.5, the force system (2) corresponds to the force distribution proportional to the selected fundamental mode. Proposed Force Distribution
Limit Force Distributions
Fig. 1: Proposed FTP technique: force distribution at each floor and two limit cases.
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3 CASE STUDIES
Two simple multi-storey RC frame structures with floor plans symmetric about the y-axis but asymmetric about the x-axis are considered (see Fig. 2). Column crosssections and eccentricities of the Centre of Rigidity (CR) with respect to the Centre of Mass (CM) are present, elastic modulus of concrete is 25000 MPa, whilst the distributed mass is 6.60 kN/m2. For case n. 1, the frame structure is slightly irregular, whilst case n. 2 features a significant eccentricity. The periods of the first three vibration modes are also reported in Fig. 2, together with the mass participation factors for a ground motion acting in x-direction. The two structures have been modelled by a fibre finite element code [6]. Case n.1
Case n.2
Cross Section Cross Section Col. 1
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ey/L 0.133 0.433 T1-Γ1 0.97 s 0% 0.90 s 0% T2-Γ2 0.94 s 76.7% 0.86 s 45.3% T3-Γ3 0.80 s 7.5% 0.59 s 33.1% Fig. 2: Geometry of case studies, column cross-section dimensions (cm), eccentricity, first three periods T and mass participation factor Γ of the two structures with small eccentricity (case n. 1) and large eccentricity (case n. 2). 4 COMPARISON OF RESULTS FROM DYNAMIC AND PUSHOVER
ANALYSES 4.1 Linear range
The aim of the present work is to validate the proposed FTP procedure through comparison with results from dynamic analyses. The comparison has been made at a given limit state, corresponding to the attainment of the ultimate strain in concrete core (εcu=0.35%) in a column. For the case studies, the pushover procedure proposed in Section 2, has been performed by selecting a set of values of weight coefficient β. For the two cases, two selected response parameters, i.e., the maximum displacement and the maximum rotation of the centre of mass (CM) of the three floors, adopting different values of the weight coefficient β, are reported in Figs. 3,4. By increasing β, i.e. reducing forces and increasing the torsional component, the displacement of CM decreases and rotation increases. It is interesting to observe that for case n. 2, by increas-
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ing β, displacement of CM decreases and rotation increases according to an almost linear law. The results obtained from the static analyses have been compared with results from linear dynamic analyses performed by adopting 12 artificial time-histories compatible with Eurocode 8 response spectrum (seismic zone 2 and type B ground). Artificial ground-motions have been preferred here over natural records in order to reduce the variability of the structural response and to obtain a homogeneous statistical sample. The artificial records have been scaled up to the achievement of the limit state (εcu=0.35%) for the structure under study. As an example, the response in terms of displacement versus rotation of CM3 obtained from one time history analysis (Ag1) with scaling factor SF=1.69 and SF=1.89 (corresponding to the attainment of the limit state), for case n. 1 and n. 2, respectively, is compared with results from pushover procedures in Figs. 5a, 6a. In particular, from the dynamic analyses, the points corresponding to the maximum values of displacement δx, rotation θ and concrete deformation εcu (the limit state 0.25
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Fig. 6: Case n. 2 - Displacement versus rotation of CM3: (a) results from time history Ag1 with SF=1.86 and maximum values from static analyses with different values of β; (b) maximum values of response parameters from 12 dynamic analyses, their mean values, and results from static analyses with different values of β.
condition) are indicated with markers on the cyclic curve. For pushover analyses, only the ultimate values corresponding to the limit state condition are reported. The ultimate points from FTP, obtained for different values of the coefficient β are located along the dashed line between limit cases β =0 and β =1. Moreover, the points from dynamic analysis corresponding to the maximum values of response parameters are inside the area described by the two limit cases (β =0 and β =1) and the line of ultimate points from static analyses. The same behaviour can be observed when static results are compared with all the results from 12 time histories analyses. These results are summarized in Figs. 5b,6b, where all the maximum values of response parameters from 12 dynamic analyses have been
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compared with results from linear static analyses (red points corresponding to limit state conditions). It is worth noting that mean values (black markers) of dynamic results corresponding to maximum displacement, rotation and the limit state condition are inside the area delimited by β =0 and β =1 lines. It is observed that dynamic results corresponding to maximum displacement and rotation occur at different time steps with respect to that corresponding to the achievement of the limit state (εcu =0.35%). In order to perform a consistent comparison between static and dynamic analyses, scaling coefficients have been then introduced: Coef − δ x =
εcu ; εc , max (tδ )
Coef - θ =
εcu εc , max (tθ )
(3)
where tδ and tθ are the timing steps of dynamic analysis corresponding to maximum displacement and maximum rotation of third floor, respectively, εc,max(tδ) and εc,max(tθ) are maximum concrete strains corresponding to timing steps tδ and tθ, respectively. For the 12 dynamic analyses the mean values of 12 coefficients Coef − δ x and Coef - θ are 1.10 and 1.27, respectively for case n. 1, whilst for case n. 2 they are 1.3 and 1.09, respectively. The results of linear dynamic analyses at time instant tδ and tθ are then multiplied by the above coefficients in order to compare the results from the static and dynamic analyses in the same conditions (the attainment of the ultimate limit state). In Fig. 7a, the maximum values of those response parameters from 12 dynamic analyses are reported and compared with the failure states predicted by FTP for different values of β. For case n. 1, the results, in terms of displacement-rotation couples corresponding to the attainment of the maximum rotation or maximum displacement, are quite close to each other, due to the relatively small irregularity of the structure (small eccentricity). Indeed, for this case pushover analyses with weight coefficients β=0.30 and 0.66 give results close to the mean values of failure conditions, corresponding to the attainment of the maximum displacement and maximum rotation of the structure, respectively, whereas β=0.42 corresponds to the mean value of dynamic states when the limit state is attained.
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For case n. 2, displacement-rotation couples corresponding to the attainment of the maximum rotation or the maximum displacement are very distant from each other, due to the larger eccentricity of the centre of mass of the second structure (see Fig.2). In this case, the static analysis with β=0 ( Fx,i = Fx,i and Ti=0) turns to be very close to the mean value of failure conditions corresponding to the attainment of maximum displacement from the 12 dynamic analyses; analogously, the ultimate condition obtained from static analysis with β=0.9 (i.e., Fx,i = 0.1 ⋅ Fx,i and Ti = 0.9 ⋅ Ti ) coincides with the mean value of failure conditions corresponding to the attainment of the maximum rotation. These results show that, for regular structure, the most severe conditions in the linear range can be predicted adopting force/torque distributions quite close to the first mode distribution (β=0.3 and 0.6 instead of β=0.5). On the contrary, for irregular structures, such as case n. 2, force/torque distributions similar to the limit distributions β=0 and β=1 are required. In the following a comprehensive comparison of results from static and dynamic analyses at time steps corresponding to maximum displacement and maximum rotation is performed. For case n. 1, displacements, interstorey drifts, floor shears in x direction, and floor torques have been reported in Fig. 8 a,b,c,d. Results from static analysis with β=0.66 are reported with red line, whereas the mean value of results from 12 dynamic analyses, corresponding to the time instant where maximum rotation θmax has been attained, is depicted as black line. A good matching between results can be observed. Displacements and forces over the structure’s height have been correctly predicted. In order to compare the deformed configurations of the frame structure under seismic action and predicted by static analysis, displacement versus rotation of the three floors is reported in Fig. 8e. Although dynamic results are significantly spread, the static analysis (red points) predicts very well the mean value of dynamic results (black points). Moreover, good agreement between the results can be observed by comparing results from
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static analysis with β=0.30 and the mean values of the 12 dynamic analyses at the time instant corresponding to the max displacement δmax (Fig. 9 a,b,c). A very close match between results from static and mean value of dynamic analyses is observed. Dynamic Analyses corresponding to θmax and Static analysis with β=0.66 9
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Fig. 8: Case n. 1 - Linear FTP analysis with β = 0.66 and results from 12 dynamic analyses at time instant corresponding to maximum rotation: (a) Displacement in x direction; (b) interstorey drift in x direction; (c) Storey shear in x direction; (d) Storey torque; (e) Displacement versus rotation of three floors.
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Therefore, the proposed FTP procedure is able to predict different structural dynamic configurations corresponding to maximum rotation and maximum displacement in terms of floor shear, floor torque, floor displacement and floor rotation. Dynamic Analyses corresponding to δmax and Static analysis with β=0.30 9
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4.2 Nonlinear range
In order to investigate the efficiency of the proposed procedure in the nonlinear range, nonlinear material models for concrete and steel bar have been introduced. For the case n. 2, preliminary results in terms of displacement versus rotation and capacity curve giving base shear vs top displacement in x direction are reported in Fig. 10. In the nonlinear range, mean value of failure conditions corresponding to the attainment of the maximum displacement and maximum rotation of the structure are closer to each other than in the linear case. The weight coefficients β giving the force distributions able to predict the mean values corresponding to the attainment of the δmax and θmax of the structure, are 0.13 and 0.55 instead of 0 and
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0.9 of the linear range. Therefore, in the nonlinear range, the interval of β coefficient for predicting the worst configurations of the structure is smaller than the case of linear analyses, as shown in Table 2 for the two case studies. This result seems to confirm that in the nonlinear range the effect of mass eccentricity tends to reduce. β=0 β=0.13 β=0.25 β=0.30 β=0.38 β=0.50 β=0.55 β=0.75 β=0.90 β=1 ε cu static
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Linear 0.3 0.66 0.42
Case n. 2 Nonlinear 0.15 0.50 0.25
Linear 0 0.90 0.67
Nonlinear 0.13 0.55 0.38
Three different severe conditions for the structure have been selected: i) maximum displacement δmax, ii) maximum rotation θmax, iii) maximum strain in concrete core εcu. In the present work, the last condition has been chosen as limit state, because it represents the failure condition for one structural element. It is worth noting that, for this limit state condition, dynamic simulations exhibit smaller values of maximum displacement δmax and maximum rotation θmax with respect static analysis. However, as shown in Fig. 10, the pushover curve with β=0.38 matches very well displacement and rotation corresponding to the mean value of the 12 dynamic results. Therefore, such force distribution predicts well the worst configuration of the structure. CONCLUSIONS
A new procedure, called Force/Torque Pushover (FTP) analysis, to select storey force distributions for 3D pushover analysis of plan-irregular RC frame structures
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has been proposed here. According to the proposed procedure, the most severe configurations in terms of maximum top displacement or maximum rotation are captured by a weighted distribution of force resultant and torque distributions. In order to validate the proposed procedure, two different cases (small and large stiffness eccentricities with respect to the center of mass) have been studied in the linear range. Good agreement has been found between static and dynamic results in terms of displacement and rotation of the center of mass of the top floor, interstorey drift, floor shear and floor torque. Moreover, the proposed technique has been adopted for RC structure in nonlinear range for both concrete and steel reinforcement. It is observed that, in order to predict the most severe deformation states for a plan - irregular structure different force distributions must be selected, in order to capture the behaviour corresponding to maximum rotation or alternatively the maximum displacement or the attainment of the limit state (ultimate stain in concrete core). The values of β coefficient required to capture those severe configurations depend on the degree of irregularity of the structure. Further studies are necessary to obtain intervals of β coefficient as a function of the center of mass eccentricity or other parameters. The work is therefore on-going. ACKNOWLEDGEMENTS
Financial support of the Italian Department of Civil Protection, through the two 2005-2008 framework programmes established with the Italian National Network of Earthquake Engineering University Laboratories (RELUIS, Task 7) and the European Centre for Training and Research in Earthquake Engineering (EUCENTRE), is gratefully acknowledged. REFERENCES
[1] Ferracuti B., Savoia M., Pinho R., Francia R., Validation of non-linear pushover analyses by Statistical Incremental Dynamic Analysis (S-IDA), in Proceedings COMPDYN, 13-16 June 2007. [2] Chopra A. K., Goel R. K., A modal pushover analysis procedure to estimate seismic demands for unsymmetric-plan buildings, Earthquake Eng, and Struct.Dyn., 2004, 33, pp. 903-927. [3] Fajfar P., Marusic D., Perus I., Torsional effects in the pushover-based seismic analysis of buildings, J. of Earthquake Eng., 2005, Vol. 9(6), pp. 831-854. [4] Moghadam A.S., Tso W.K., Pushover analysis for asymmetric and set-back multistorey buildings, in Proceedings, WCEE, 12th, 2000, Upper Hutt, paper 1093. [5] Penelis G. G., Kappos A. J., 3D Pushover analysis: The issue of torsion, in Proceedings, ECEE, 12th, 2002, London, paper 015. [6] SeismoSoft, SeismoStruct - A computer program for static and dynamic nonlinear analysis of framed structures, 2007 URL: http://www.seismosoft.com.