Barbara Ferracuti, Marco Savoia DISTART – Structural Engineering. Università di Bologna. Viale Risorgimento 2, 40136 Bologna.
Rui Pinho Department of Structural Mechanics, University of Pavia.
ANIDIS2009BOLOGNA
3D Pushover Analysis for Evaluating Torsional Effect of RC Structures
Keywords: 3D Pushover Analysis, Incremental Dynamic Analysis, RC frame structures. ABSTRACT In the present study, a new pushover procedure for the seismic static analysis of plan irregular 3D frame structures is proposed, based on the application of a set of horizontal force and torque distributions at each floor level. In order to predict the most severe configurations of an irregular structure subjected to an earthquake, more than one pushover analysis has to be performed. The proposed method is validated by a consistent comparison of results from static pushover and dynamic simulations 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 the nonlinear case. The results confirm that, in order to predict the structural behaviour in the most severe configurations, different force distributions must be defined.
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 pushover techniques by comparison with results from dynamic analyses (see Ferracuti et al. 2007, amongst many others). On the contrary, few methods to perform pushover analyses for irregular 3D frame structures have been proposed (Chopra and Goel, 2004; Fajfar et al., 2005; Moghadam and Tso, 2000; Penelis and Kappos, 2002). These methods are usually extensions to the 3D case of techniques proposed for planar structures, taking due consideration of the fact that the approaches prescribed in current design codes or guidelines proposals (e.g. ATC, 2005), involving the relatively straightforward application of force distributions proportional to the floor masses or linearly varying along the building height, are not able to take proper
account of the dynamic torsional amplification effect for plan-irregular structures. The methods for static pushover analysis of plan-irregular frame structures can perhaps be sub-divided into two main categories. The majority of them are based on a prescribed loading pattern that is defined a-priori according to the modal characteristics of the structure in the linear range. For instance, FEMA-440 (ATC, 2005) suggests the use of a loading pattern proportional to the fundamental vibration mode. Moghadam and Tso (2000) showed that this loading pattern gives results closer to dynamic simulation results with respect to mass proportional or linearly varying force distributions. Penelis and Kappos (2002) used a set of storey forces related with the elastic spectral modal displacements (combined via SRSS method). Also in this case, a single pushover analysis is performed. On the opposite end of the spectrum, the Modal Pushover Analysis (MPA), originally proposed for 2D frame structures (Chopra and Goel (2002)), has been extended to 3D structures by Chopra and Goel (2004). Independent pushover runs are performed, adopting load distributions proportional to various modes of the structure. Results in terms of capacity curves for
various modal shapes are transformed in a capacity curve for an equivalent SDOF. Seismic demands are separately evaluated for each system SDOF and a-posteriori combination of results is performed for storey shears, interstorey drifts, etc., typical by means of the SRSS method. Even though the a-posteriori mode combination introduces some approximations in the nonlinear range, the main advantage of MPA is to combine the effects of various modes occurring not simultaneously during the dynamic event, so being able to estimate the most severe configurations the structure may be subject to. Recently, a different set of approaches whereby different loading patterns, corresponding to varying worst-case configurations that must be predicted by static pushover analyses, have been recently proposed by Bosco et al. (2008) and Lucchini et al. (2008). In spatial analysis, the assessment of pushover methods through comparison with dynamic results is also more complex than for planar structures. In fact, the number of parameters to be predicted by the static analysis is much greater than in the planar case (maximum displacements, rotations, storey shears, storey torques, etc), and they typically occur in different time instants during the dynamic event. The fundamental point in 3D pushover analyses is the selection of the distribution of the horizontal forces along the frame height and over the individual floors. If the building floors are sufficiently rigid, the horizontal forces at the floor level can be divided into translational and torsional contributions. By analyzing the dynamic response of an irregular structure subjected to the 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 all the structural elements of the frame cannot really be defined. In the present work, a new pushover procedure for 3D RC structures, named Force/Torque pushover (FTP) is proposed; a set of force and torque distributions is selected, in order to predict the most severe configurations the structure may undergo during the earthquake. Three different worst-case performance conditions for the structure have been selected, corresponding to the attainment of i) the maximum displacement δmax, ii) the maximum rotation θmax, iii) the maximum strain in concrete core εcu. In the present work, the latter of these conditions has been chosen as the limit state, since it may represent the failure condition for a given structural element.
As will be shown in Section 2, the method requires a coefficient (β) to be defined, representing the weight of translational and torsional force contributions. This weighting coefficient must be calibrated through and extensive numerical campaign, and depends on the structural response parameter to be predicted, as well as on the nonlinear behaviour of the structure. Evidently, the complete definition of the method requires a full numerical study to be performed, involving the analysis of a set of different plan irregular structures, in order to obtain/calibrate a series of regression expressions that yield the weight coefficient β as a function of the most important structural parameters (e.g. mass eccentricity, stiffness and strength centre positions, etc.). In the present paper, the weight coefficient β has been calibrated with reference to a simple three-floor frame structure with variable eccentricity. The comparison between the 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 the results with the proposed pushover method and dynamic analyses has been 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: For plan-irregular structures, 2 modes are dominant, and higher order modes can be neglected. Displacement and rotation profiles of first and second mode along the frame height are related as:
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 Figure 1. For β =0.5, the force system reported in Eq. (2) corresponds to the force distribution proportional to the selected fundamental mode.
δ x 2 ( z ) ≅ δ x1 ( z ) , δ y 2 ( z ) ≅ δ y1 ( z ) , θ 2 ( z ) ≅ K R θ1 ( z )
3
(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 ⎩
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 Figure 2). Column cross-sections and eccentricities of the Centre of Rigidity (CR) with respect to the Centre of Mass (CM) are reported in Figure 2; 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 Figure 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 (Seismosoft 2007).
(2)
Case n.1
Case n.2
Cross Section
Cross Section
Col. 1
30x25
60x25
Col. 2
30x25
60x25
Col. 3
25x25
25x25
Col. 4
25x25
25x25
Proposed Force Distribution
Limit Force Distributions
ey/L T1-Γ1 T2-Γ2 T3-Γ3
Figure 1. Proposed FTP technique: force distribution at each floor and two limit cases.
0.133 0.97 s 0.94 s 0.80 s
0% 76.7% 7.5%
0.433 0.90 s 0% 0.86 s 45.3% 0.59 s 33.1%
Figure 2. Geometry of case studies, column cross-section dimensions (cm), eccentricity ey, 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
0.20 CM3
CM2
CM1
0.15
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 increasing β, displacement of CM decreases and rotation increases according to an almost linear law.
δx [m]
Linear range
0.10
0.05
0.00 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
β
(a)
-0.035
CM3
CM2
CM1
-0.025
θ [rad]
4.1
-0.015
-0.005
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
β
(b) Figure 4. Case n. 2 - (a) Max. displacement and (b) max. rotation of centre of mass (CM) of the three floors obtained by proposed pushover method with different values of the weight coefficient β.
0.25 CM3
δx [m]
0.2
CM2
CM1
0.15 0.1 0.05 0 0
0.1
0.2
0.3
0.4
0.5
0.6
β
0.7
0.8
0.9
1
(a)
-0.04 -0.035 -0.03
CM3
CM2
CM1
θ [rad]
-0.025 -0.02
-0.015 -0.01 -0.005 0 0
0.1
0.2
0.3
0.4
0.5
β
0.6
0.7
0.8
0.9
1
(b) Figure 3. Case n. 1 - (a) Max. displacement and (b) max. rotation of centre of mass (CM) of the three floors obtained by proposed pushover method with different values of the weight coefficient β.
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 condition) are indicated with markers on the cyclic curve.
δ x,max
0.03
β=1
ε cu
β= 0
-0.01
β=0
0
-0.03
(a) -0.15
-0.1
-0.05
0
δx [m]
0.05
0.1
0.15
0.2
-0.04
0.25
β=0 β=0.25 β=0.30 β=0.50 β=0.66 β=0.75 β=0.90 β=1 ε cu static
0.035
β=1
0.03
-0.1
-0.05
0
δx [m]
0.05
0.1
0.15
0.2
0.025
θmax (mean) ε cu dyn ε cu (mean)
0.02
0.25
(a)
β=0 β=0.25 β=0.30 β=0.50 β=0.75 β=0.90 β=1 ε cu static
0.03
θmax
0.01
-0.15
0.035
δ x,max (mean)
0.015
-0.2
0.04
δ x,max
0.02
-0.25
β= 1
-0.2
θ [Rad]
-0.25
0.04
θ [Rad]
β=0 β=0.25 β=0.30 β=0.50 β=0.75 β=0.90 β=1
-0.02
-0.03
0.025
ε cu
-0.01
-0.02
-0.04
θmax
0.01
θ [Rad]
0
δ x,max
0.02
β=0 β=0.25 β=0.30 β=0.50 β=0.75 β=0.90 β=1
0.01
Dyn
0.03
θmax
0.02
θ [Rad]
0.04
Dyn
β=1
0.04
δ x,max δ x,max (mean)
0.015
θmax θmax (mean)
0.01
ε cu dyn ε cu (mean)
β=0
0.005
0.005
β=0 0 0
0.05
0.1
δx [m]
0.15
0.2
(b)
0 0
0.05
0.25
Figure 5. Case n. 1 - Displacement versus rotation of CM3: (a) results from time history Ag1 with SF=1.69 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.
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 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
0.1
δx [m]
0.15
0.2
0.25
(b)
Figure 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.
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 Figure 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.
β=0 β=0.25 β=0.30 β=0.50 β=0.66 β=0.75 β=0.90 β=1 ε cu static
0.04
0.035
β=0.9
β=1
0.03
θ [Rad]
0.025
β=0.66
δ x,max
0.02
δ x,max (mean) θmax
0.015
θmax (mean) ε cu dyn
0.01
ε cu (mean)
β=0.3
0.005
β=0
0 0
0.05
0.1
δ x [m]
0.15
0.2
(a)
β=0 β=0.25 β=0.30 β=0.50 β=0.75 β=0.90 β=1 ε cu static
0.04
β= 1
0.035
0.03
0.025
θ [Rad]
0.25
δ x,max δ x,max (mean)
0.02
θmax θmax (mean)
0.015
ε cu dyn 0.01
0.005
0 0
ε cu (mean)
β=0
0.05
0.1
δx [m]
0.15
0.2
0.25
(b)
Figure 7. Displacement versus rotation of CM3 with amplified values of dynamic results: (a) Case n. 1; (b) Case n. 2.
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 Figure2). 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 Figures 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 Figure 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 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 (Figures 9 a,b,c). A very close match between results from static and mean value of dynamic analyses is observed. 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 θmax and Static analysis with β=0.66
9
6 5 4 3
7 6 5 4 3
2
2
1
1
0 0
0.5
1
1.5
2
δx/H [%]
2.5
0
3
0.5
1
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2
2.5
3
3.5
Interstorey drift ratio δx/h [%]
4
(b)
9
Pushover A1a A1b A1c A2a A2b A2c A3a A3b A3c A4a A4b A4c Dyn (mean)
7 6 5 4 3
7 6 5 4 3
2
2
1
1
0
100
200
300
400
500
600
700
Storey Shear [kN]
Pushover A1a A1b A1c A2a A2b A2c A3a A3b A3c A4a A4b A4c Dyn (mean)
8
Height [m]
8
Height [m]
0
(a)
9
0
Pushover A1a A1b A1c A2a A2b A2c A3a A3b A3c A4a A4b A4c Dyn (mean)
8
Height [m]
7
Height [m]
9
Pushover A1a A1b A1c A2a A2b A2c A3a A3b A3c A4a A4b A4c Dyn (mean)
8
0
800
0
500
1000
1500
Storey Torque [kN⋅m]
(c)
(d)
0.25
0.2
|δ| [m]
0.15
0.1
Pushover A1a A1b A1c A2a A2b A2c A3a A3b A3c A4a A4b A4c Dyn (mean)
3rd floor 2nd floor
1st floor
(e)
0.05
0 -0.04
-0.035
-0.03
-0.025
-0.02
θ [Rad]
-0.015
-0.01
-0.005
0
Figure 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.
Dynamic Analyses corresponding to δmax and Static analysis with β=0.30 9
9
Pushover A1a A1b A1c A2a A2b A2c A3a A3b A3c A4a A4b A4c Dyn (mean)
7
Height [m]
6 5 4 3
7 6 5 4 3
2
2
1
1
0 0
0.5
1
1.5
δx/H [%]
2
2.5
Pushover A1a A1b A1c A2a A2b A2c A3a A3b A3c A4a A4b A4c Dyn (mean)
8
Height [m]
8
0 0
3
0.5
1
1.5
2
2.5
3
Interstorey drift ratio δx/h [%]
(a)
3.5
4
(b)
0.25
0.2
|δ| [m]
0.15
0.1
Pushover A1a A1b A1c A2a A2b A2c A3a A3b A3c A4a A4b A4c Dyn (mean)
3rd floor
2nd floor
1st floor
0.05
0 -0.04
-0.035
-0.03
-0.025
-0.02
θ [Rad]
-0.015
-0.01
-0.005
0
(c)
Figure 9. Case n. 1- Linear FTP analysis with β = 0.30 and results from 12 linear dynamic analyses at the time instant corresponding to maximum displacement: (a) Displacement in x direction; (b) interstorey drift in x direction; (c) Displacement vs rotation of three floors.
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 Figure 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 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. 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 Figure 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. The comparison between results along the building height from FTP analysis with β=0.38 and non linear dynamic
results corresponding to the attainment of the limit state εc,max= εcu is reported in Figure 11. Good agreement in terms of displacements, interstorey drift, shear and torque along the height of the structure is shown. 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
0.025
β=0.55
0.02
θ [Rad]
β= 1
β=0.38 0.015
β=0.13
δ x,max
0.01
δ x,max (mean) θmax 0.005
θmax (mean) ε cu dyn
β= 0 0 0
0.05
0.1
ε cu (mean)
δx [m]
0.15
0.2
0.25
(a)
200
β=0 β=0.25 β=0.30 β=0.38 β=0.50 β=0.75 β=0.90 β=1 ε cu static
180
Base Shear Vx [kN]
160
0 β=
140 120 100
δ x,max
80
δ x,max (mean) θmax
60
θmax (mean)
40
ε cu dyn ε cu (mean)
20 0 0
β=1
0.05
0.1
0.15
0.2
0.25
(b) Figure 10. Case n. 2 - Nonlinear dynamics and FTP analysis results: (a) Displacement versus rotation; (b) Base shear – displacement capacity curve in x direction. δx [m]
Table 2. Values of coefficient β in FTP analyses giving the closest results to the mean values of dynamic results corresponding to the attainment of the maximum displacement δmax, maximum rotation θmax and maximum strain in concrete core εcu . Case n. 1 β−δmax β−θmax β-εcu
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 and other geometrical/mechanical parameters. The work is therefore on-going.
Linear 0.3 0.66 0.42
Nonlinear 0.15 0.50 0.25
Case n. 2 Linear 0 0.90 0.67
Nonlinear 0.13 0.55 0.38
CONCLUSIONS A new procedure, called Force/Torque Pushover (FTP) analysis, to select storey force distributions for 3D pushover analysis of planirregular RC frame structures has been proposed here. According to the proposed procedure, the most severe configurations in terms of maximum top displacement or maximum rotation are
ACKNOWLEDGEMENTS The 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 ATC 2005. Improvement of nonlinear static seismic analysis procedures, FEMA 440 Report, Applied Technology Council, Redwood City, CA. Bosco M. Ghersi A., Marino E.M. 2008. Eccentricities for the evaluation of the seismic response of asymmetric buildings by nonlinear static analysis, Fifth European Workshop on the Seismic Behaviour of Irregular and Complex Structures, September 16-17 2008, Catania, Italy. Chopra A. K., Goel R. K. 2002. A modal pushover analysis procedure for estimating seismic demands for buildings. Earthquake Engineering & Structural Dynamics, Vol. 31(3), 561-582. Chopra A. K., Goel R. K., 2004. A modal pushover analysis procedure to estimate seismic demands for unsymmetric-plan buildings, Earthquake Engineering and Structural Dynamics, 33, 903-927.
Dynamic Analyses corresponding to εcu and Static analysis with β=0.38 9
9
Pushover A1a A1b A1c A2a A2b A2c A3a A3b A3c A4a A4b A4c Dyn (mean)
7
Height [m]
6 5 4 3
7 6
4 3 2
1
1
0
0.5
1
1.5
2
δx/H [%]
2.5
0
3
6 5 4 3
4 3
1
150
200
2.5
3
(b) Pushover A1a A1b A1c A2a A2b A2c A3a A3b A3c A4a A4b A4c Dyn (mean)
5
2
100
2
6
1
0
250
Storey Shear [kN]
1.5
7
2
50
1
Interstorey drift ratio δ x/h [%]
8
Height [m]
7
0
0.5
9
Pushover A1a A1b A1c A2a A2b A2c A3a A3b A3c A4a A4b A4c Dyn (mean)
8
0
0
(a)
9
Height [m]
5
2
0
Pushover A1a A1b A1c A2a A2b A2c A3a A3b A3c A4a A4b A4c Dyn (mean)
8
Height [m]
8
(c)
0
100
200
300
400
Storey Torque [kN⋅m]
500
600
(d)
0.2 0.18 0.16 0.14
|δ| [m]
0.12 0.1 0.08 0.06
Pushover A1a A1b A1c A2a A2b A2c A3a A3b A3c A4a A4b A4c Dyn (mean)
3rd floor 2nd floor 1st floor
0.04
(e)
0.02 0 -0.03
-0.025
-0.02
-0.015
θ [Rad]
-0.01
-0.005
0
Figure 11: Case n. 2 - Non linear static analysis with β = 0.38 and results from 12 linear dynamic analyses at the time instant corresponding to the attainment of the limit state εcu: (a) Displacement in the x-direction, (b) interstorey drift in the x-direction, (c) Storey shear in the xdirection, (d) Storey torque, (e) Displacement versus rotation of three floors.
Eurocode 8 2003. Design for structures for earthquakes resistance – Part 1 – General rules, seismic actions and rules for buildings, Final Draft – prEN 1998-1. Fajfar P., Marusic D., Perus I., 2005. Torsional effects in the pushover-based seismic analysis of buildings, Journal of Earthquake Engineering, 9(6), 831-854. Ferracuti B., Pinho R., Savoia M., Francia R. 2007. Validation of nonlinear pushover analyses by Statistical Incremental Dynamic Analysis (S-IDA), in Proceedings COMPDYN, 13-16 June 2007. Lucchini A., Monti G., Kunnath S. 2008. Investigation on the inelastic torsional response of asymmetric-plan buildings, Fifth European Workshop on the Seismic Behaviour of Irregular and Complex Structures, September 16-17 2008, Catania, Italy. Marusic D., Fajfar P., 2005. On the inelastic response of asymmetric buildings under bi-axial excitation, Earthquake Engineering and Structural Dynamics, 34, 943-963.
Moghadam A.S., Tso W.K., 2000. Pushover analysis for asymmetric and set-back multistorey buildings, in Proceedings, WCEE, 12th, Upper Hutt, paper 1093. Penelis G. G., Kappos A. J., 2002. 3D Pushover analysis: The issue of torsion, in Proceedings, European Conference on Earthquake Engineering, 12th, London, paper 015. SeismoSoft 2007. SeismoStruct - A computer program for static and dynamic nonlinear analysis of framed structures, Available from URL: http://www.seismosoft.com. Stathopoulos K. G., Anagnostopoulos S. A., 2005. Inelastic torsion of multi-storey buildings under earthquake excitations, Earthquake Engineering and Structural Dynamics, 34, 1449-1465. Vamvatsikos D. and Cornell C. A. 2002. Incremental Dynamic Analysis, Earthquake Engineering and Structural Dynamics, 31, 491-514.