Valutazione e riduzione della vulnerabilità sismica di edifici esistenti in c.a. Roma, 29-30 maggio 2008
SEISMIC ANALYSIS OF IRREGULAR RC STRUCTURES BY 3D FORCE/ TORQUE PUSHOVER METHOD
Barbara Ferracuti a, Marco Savoia b, Rui Pinho c, Maurizio Serpieri d a
b
DISTART – Struct. Engineering, University of Bologna, barbara.ferracuti@unibo.it DISTART – Structural Engineering, University of Bologna, marco.savoia@mail.ing.unibo.it c Department of Structural Mechanics, University of Pavia, rui.pinho@eucentre.it d DISTART – Structural Engineering, University of Bologna, maurizio.serpieri@ unibo.it
ABSTRACT In the present study, a new pushover procedure for 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 the effectiveness of the proposed procedure to predict the structural behaviour in the most severe configurations. KEYWORDS 3D Pushover Analysis, Incremental Dynamic Analysis, RC frame structures. 1 INTRODUCTION The main goal of nonlinear static analysis is to describe the nonlinear capacity of the structure when subjected to earthquake ground motion, with a reduced computational effort with respect to nonlinear dynamic analyses. For 2D frame structures, a large number of studies have been performed in order to validate different pushover techniques by comparison with results from dynamic analyses (e.g. Ferracuti et al. 2007, amongst many others). On the contrary, few pushover methods for irregular 3D frame structures have been proposed (Moghadam and Tso, 2000; Penelis and Kappos, 2002; Chopra and Goel, 2004; Fajfar et al., 2005). The definition and assessment of 3D pushover methods is however much more complex than in the 2D case. Indeed, the fundamental point in 3D pushover analyses is the selection of the distribution of 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
Topic: MND FC IRREG MIX TAMP SCALE NODI BIAX PREFAB
step. Therefore, a unique pushover force distribution giving the most severe conditions for the structure cannot really be defined. In the present work, a new pushover procedure for 3D RC structures, called Force/Torque pushover (FTP) is thus 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 relatively good agreement between results with the proposed pushover method and dynamic analyses has been preliminarily found. 2 PROPOSED METHOD: FORCE/TORQUE PUSHOVER (FTP)
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. Therefore, a unique pushover force distribution giving the most severe conditions for the structure cannot be defined. In the present study, Force/Torque Pushover has been proposed. To start with, a rigid diaphragm constraint is introduced at each individual floors of the structure. Then, the force distribution proportional to the fundamental mode of the structure with the highest participation factor for the selected ground motion direction is obtained. The floor force resultant at the i-th floor level is divided into lateral forces Fx ,i , Fy ,i and torque Ti with respect to the centre of mass. A coefficient β weighting the two components is then introduced, so defining a class of force distributions with variable force and torsional resultants. Forces applied at the i-th floor are then written as (see Figure 1):
⎧ Fx,i = (1 − β) Fx ,i ⎪ ⎨ Fy ,i = (1 − β) Fy ,i ⎪ Ti = β Ti ⎩
(1)
The β weight coefficient can vary from zero to one. Therefore, forces-only or alternatively torque-only are applied for the two limit cases, β=0 and β=1, respectively (see Figure 1). For β = 0.5, the force system (1) corresponds to the force distribution proportional to the selected fundamental mode.
FTP Force Distribution
Two Limit Force Distributions
Figure 1. Proposed pushover technique: force distribution at each floor and two limit cases.
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 Figure 2). Column cross-sections and eccentricities of 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 Figure 2, together with mass participation factors for a ground motion acting in x-direction. The structures have been modelled using the fibre finite element code Seismosoft (2007). Case n.1
Case n.2
Cross Section Cross Section Col. 1
30x25
60x25
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60x25
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ey/L T1-Γ1 T2-Γ2 T3-Γ3
0.133 0.433 0.97 s 0% 0.90 s 0% 0.94 s 76.7% 0.86 s 45.3% 0.80 s 7.5% 0.59 s 33.1%
Figure 2. Geometry of two case studies, column cross-sections, periods T and mass participation factors Γ for an earthquake acting in x direction.
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 achievement of 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 case n.1, two selected response parameters, i.e. maximum displacement and maximum rotation of the centre of mass (CM) of the three floors, obtained for different values of β, are reported in Figures 3a,b. By increasing β, i.e. reducing forces and increasing torsional component, displacement of CM decreases and rotation increases. The results from static analyses have been compared with those from linear dynamic analyses 0.25
-0.04 CM3
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β (b) (a) Figure 3. Case n.1: (a) Maximum displacement and (b) maximum rotation of centre of mass (CM) at failure of the three floors obtained by the proposed FTP method with different values of the weight coefficient β.
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
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Figure 4. Case n.1: Displacement versus rotation of CM3: (a) results from time history Ag1a with SF=1.69 and failure states from static analyses for different values of β; (b) maximum values of response parameters from 12 dynamic analyses, their mean values (bold symbols), and results from static analyses with different values of β.
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 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 (Ag1a) with scaling factor 1.69 (corresponding to the attainment of the limit state) is compared with results from FTP in Figure 4a. In particular, from dynamic analyses the points corresponding to maximum values of displacement δx, rotation θ and concrete deformation εcu (the limit state condition) are indicated with markers on the curve. For pushover analyses, only the ultimate values corresponding to the limit state condition are reported. The ultimate points from FTP for different values of the coefficient β are located along the dashed line between limit cases β =0 and β =1. Moreover, points from dynamic analyses corresponding to the maximum values of response parameters are inside the cone described by the two limit cases (β =0 and β =1) and the line of ultimate points obtained from static analyses. The same behavior can be observed when static results are compared with all the results from 12 time histories analyses. These results are summarized in Figure 4b, 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). 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
700
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θ [Rad] (a) (b) β=1 Figure 5. Case n.1: (a) Base shear in the direction of the seismic action Vx vs top displacement of CM from static analysis compared with the maximum value of response parameters obtained from dynamic analyses; (b) base torque versus rotation. δx [m]
The same results from both dynamic and static analyses, but in terms of capacity curves (Base Shear – displacement and Base Torque – rotation) are depicted in Figure 5. 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 =
ε cu ; ε c ,max (t δ )
Coef - θ =
ε cu ε c,max (t θ )
(2)
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 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. The results of linear dynamic analyses at time instant tδ and tθ are then multiplied by the above coefficients in order to compare results from static and dynamic analyses in the same conditions (the attainment of the ultimate limit state). In Figure 6, 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 because of the relatively reduced irregularity of the structure (due to its 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. 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 Figure 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 structures, 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,
θ [Rad]
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δx [m] (b) (a) Figure 6. Displacement versus rotation of CM3 with amplified values of dynamic results: (a) Case n.1, (b) Case n.2. δx [m]
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 7 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 Dynamic Analyses corresponding to θmax and FTP with β=0.66 9
9
Pushover A1a A1b A1c A2a A2b A2c A3a A3b A3c A4a A4b A4c Dyn (mean)
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Figure 7. 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 FTP analysis with β=0.30 9
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Figure 8. Case n.1- Linear FTP analysis with β = 0.30 and results from 12 linear dynamic analyses at time instant corresponding to maximum displacement: (a) Displacement in x direction; (b) interstorey drift in x direction; (c) Storey shear in x direction; (d) Storey torque.
structure’s height have been correctly predicted. In order to compare the deformed configurations of the frame structure under seismic action as predicted by static analysis, displacement versus rotation of the three floors is reported in Figure 7e. Although dynamic results are significantly spread, static analysis (red points) predicts very well the mean value of dynamic results (black points). Moreover, good agreement can be observed between results from static analysis with β=0.30 and mean values of the 12 dynamic analyses at time instant corresponding to maximum displacement δmax (see Figures 8 a,b,c,d). 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 bars have been introduced. For case n.2, preliminary results in terms of displacement versus rotation and capacity curves giving base shear vs top displacement in x direction are reported in Figure 9. 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. 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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0.025
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Figure 9. Case n.2 - Nonlinear dynamic and FTP analysis: (a) Displacement versus rotation; (b) Capacity curve in x direction.
5 CONCLUSIONS
A new procedure to select storey force distributions for 3D pushover analysis of irregular RC frame structures has been proposed here. The most severe configurations in terms of maximum top displacement or maximum rotation can be captured by a weighted distribution of force resultant and torque distributions. A preliminary parametric study to validate the proposed procedure has been performed in the linear range. Good agreement has been found between static and dynamic results in terms of displacement and rotation of the centre of mass of the top floor, interstorey drift, floor shear and floor torque. In the non linear range, to predict the most severe deformation states for an irregular structure at least two different force distributions must be selected, in order to capture the behaviour corresponding to maximum rotation and maximum displacement of the top floor. The optimum values of the distribution weight coefficient (β) depend on the degree of irregularity of the structure; further studies are thus needed to obtain realistic intervals of β. 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 Chopra A. K., Goel R. K., [2004]: “A modal pushover analysis procedure to estimate seismic demands for unsymmetric-plan buildings”, Earthquake Eng. and Struct.Dyn., 33, pp. 903-927. Fajfar P., Marusic D., Perus I., [2005]: “Torsional effects in the pushover-based seismic analysis of buildings”, Journal of Earthquake Engineering, Vol. 9(6), pp. 831-854. Ferracuti B., Savoia M., Pinho R., Francia R. [2007]: “Validation of non-linear pushover analyses by Statistical Incremental Dynamic Analysis (S-IDA)”, in Proceedings COMPDYN, 13-16 June 2007. 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" [online], URL: http://www.seismosoft.com.