Nonlinear Static Methods for Design/Assessment of 3D Structures R. Bento & R. Pinho (Eds.) 5-6 May 2008 Lisbon, Portugal
FORCE/TOURQUE PUSHOVER METHOD FOR PLAN IRREGULAR STRUCTURES M. Savoia*, B. Ferracuti *, M. Serpieri* *DISTART – Structural Engineering, University of Bologna, Italy e-mails: marco.savoia@mail.ing.unibo.it, barbara.ferracuti@mail.ing.unibo.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 on the floor level is proposed here. For predicting the behaviour of irregular structures in the worst configurations more than one pushover analyses have to be performed. The proposed method is validate 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 non linear analysis. The results confirm the effectiveness of the proposed procedure to predict the structural behaviour in the most dangerous configurations.
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 (Ferracuti et al. 2007). 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). The definition and assessment of 3D pushover methods are much more complex than in the case of 2D analyses for several reasons. First of all, the horizontal loading distribution both over the frame height and on the various floors must be defined in order to correctly take torsional effects into account. Secondly, parameters of structural response (maximum displacement and maximum rotation of roof floor, relative rotation/displacement between floors, etc) to be predicted by the nonlinear static analyses must be preliminary selected. The second point is fundamental to evaluate the accuracy of different methodologies. In the present work, a new pushover procedure for 3D plan-irregular RC structures is proposed. By this new approach, a set of force and torque distributions is selected to predict the worst configurations of the structures. In order to validate such technique, comparison between results from a series of incremental dynamic analyses and pushover analyses has been carried out, with an accurate selection of representative parameters of structural response. In particular, the comparisons have been made in terms of global parameters (capacity curves, horizontal displacement versus rotation of the center of mass of the roof) and local parameter as well as floor shears, interstorey drift, floor rotations. A very good agreement between results of proposed pushover and dynamic analyses has been found.
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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 maximum rotation and 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), has been proposed. First of all, rigid diaphragm constraint is introduced for the individual floors of the structure. Then, the force distribution proportional to the fundamental mode of the structure with the highest participant factor for the selected ground motion direction is obtained. 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 for components is then introduced, so defining a class of force distributions with variable translational and torsional configurations. Forces applied at 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, 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 (1) corresponds to the force distribution proportional to the selected fundamental mode.
3 CASE STUDIES Simple multi-storey RC frame structures with plans symmetric about the y-axis but unsymmetric about the x-axis are considered (see Figure 2). In particular, two different structures are studied, whose column cross-sections and eccentricities are shown in Figure 2. Elastic modulus of concrete is 25000 MPa, whilst the distributed mass was considered as equal to 6.60 kN/m2. For case n.1, the frame structure has a small eccentricity of the centre of rigidity with respect to the centre of mass, whereas for case n.2 a large eccentricity is considered. Periods of the first three vibration modes are also reported in Table 1, together with 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). Proposed Force Distribution Limit Force Distributions
Figure 1.Proposed FTP technique: force distribution at each floor and two limit cases.
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Figure 2. Geometry of case studies. Table 1: Column cross-sections, first three periods T and mass participation factor Γ of structures with small eccentricity (case n. 1) and large eccentricity (case n. 2).
Col. 1 Col. 2 Col. 3 Col. 4
Case n.1 Cross Section 30x25 30x25 25x25 25x25
Case n.2 Cross Section 25x25 25x25 25x25 25x25
ey/L T1-Γ1 T2-Γ2 T3-Γ3
0.133 0.97 s 0% 0.94 s 76.7% 0.80 s 7.5%
0.433 0.90 s 0% 0.86 s 45.3% 0.59 s 33.1%
4 COMPARISON OF RESULTS FROM DYNAMIC AND PUSHOVER ANALYSES 4.1
Linear range The aim of the present work is to validate the proposed pushover procedure by comparing results from static and dynamic analyses. In the present work the comparison has been made at the same limit state, corresponding to the achievement of ultimate strain in concrete core (εcu=0.35%). For the case studies, the pushover procedure proposed in Section 2, has been performed by selecting a set of values of weight coefficient β (0, 0.25,0.3, 0.5, 0.75, 0.9, 1.0). For both cases n.1-2, 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 the weight coefficient β, are reported in Figure (3-4)b. By increasing β, then reducing translational force component and increasing torsional force component, 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.
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The results from static analyses have been compared with results from linear dynamic analyses performed by adopting 12 artificial response spectrum-compatible time-histories with response spectrum from Eurocode 8 (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 (Vamvatsikos and Cornell, 2002). As an example, the response in terms of displacement versus rotation of CM3 obtained from one time history analysis (Ag1a) with scaling factor SF=169 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 Figure 5a - 6a. In particular, from dynamic analyses points corresponding to maximum values of displacement δx, rotation θ and concrete deformation εcu (the limit state condition) are indicated with markers on the hysteretic curve. For pushover analyses, only the ultimate values corresponding to the limit state condition are reported. It is worth to observe that the ultimate points from pushover analyses for different values of the coefficient β are located along the dashed line.
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(a) (b) Figure 3: Case n. 1 - (a) Maximum displacement and (b) maximum rotation of centre of mass (CM) of the three floors obtained by proposed pushover method with different values of the weight coefficient β. 0.20
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(a) (b) Figure 4: Case n. 2 - (a) Maximum displacement and (b) maximum rotation of centre of mass (CM) of the three floors obtained by proposed pushover method with different values of the weight coefficient β.
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Figure 5: Case n. 1 - 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. 0.04
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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. 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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Figura 7: Case n. 1 - (a) Base shear Vx in the direction of the seismic action vs top displacement of CM from static analysis compared with the maximum value of response parameters obtained from dynamic analyses. (b) Torque versus rotation.
(b)
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Moreover, points from dynamic analysis corresponding to 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 type of behaviour can be observed by comparison static results with all the results from 12 time histories analyses. These results are summarized in Figures 5b-6b, where all the maximum values of response parameters from 12 dynamic analyses have been reported 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. The same results from both dynamic and static analyses, but in terms of capacity curves (Base Shear – displacement and Torque – rotation) are depicted in Figure 7 . It is observed that dynamic results at time step corresponding to the achievement of maximum displacement and maximum rotation are always smaller than results from static analyses corresponding to the attainment of the limit state (εcu=0.35%). This is due to the fact that maximum displacement and maximum 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 for static and dynamic analyses, a scaling coefficient has been introduced:
Coef − δ x =
ε cu ; εc , max (tδ )
Coef - θ =
ε cu ε c , max (t θ )
(2)
where εcu is the ultimate strain equal to 0.35%, tδ and tθ are the timing steps of dynamic analyses 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,
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Figure 8: Displacement versus rotation of CM3 maximum with amplified values of dynamic results: (a) Case n.1; (b) Case n.2. respectively for case n.1, whilst for case n.2 they are 1.3 and 1.089, respectively. The results of linear dynamic analyses at time instant tδ and tθ are 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 8a, the maximum values of response parameters from 12 dynamic analyses are reported amplified by coefficients reported in eqn (2).
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Results, in terms of displacement-rotation couple, corresponding to the attainment of the maximum rotation or maximum displacement, are close to each other because of slight irregularity of the structure. The structure has a small eccentricity, therefore the torsional effects are less influent than the case n.2 and dynamic results are close to each other. In this case, pushover analyses with weight coefficients β=0.30 and 0.66 give results close to the mean value of failure conditions corresponding to the attainment of the maximum displacement and maximum rotation of the structure, respectively. For case n.2, results, in terms of displacement-rotation couple, corresponding to the attainment of the maximum rotation or maximum displacement, are very far to each others, due to the high irregularity of the second structure (see Figure2). It is possible to observe that the ultimate value of static analysis with β=0 ( Fx ,i = Fx ,i and Ti=0) is very close to the mean value of failure conditions corresponding to the attainment of maximum displacements 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 ) is very close to the mean value of failure conditions corresponding to the attainment of the maximum rotation. These results show that, for regular structure, the structural behaviour in the linear range can be predicted adopting a force distribution proportional to the first mode (β=0.5). On the contrary, for irregular structures, such as case n.2, force 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 step corresponding to maximum displacement and maximum rotation is performed. For case n.1, displacement in x direction, interstorey drift in x direction, floor shears in x direction and floor torques have been reported in Figure 9 a,b,c,d. Results from static analysis with β=0.66 are reported with red line, whereas results from 12 dynamic analyses are those corresponding to the time instant where maximum rotation θmax has been attained, together with their mean value (black line). A perfect match between results from static analysis and mean value of dynamic analyses can be observed. Displacement and force over the structure height has been correctly defined. 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 9e. Dynamic results are significant spread, nevertheless, static analysis (red points) predicts very well the mean value of dynamic results (black points). Moreover , good agreement between results can be observed by comparing results from static analysis with β=0.30 and from mean values of the 12 dynamic analyses at time instant corresponding to maximum displacement δmax (see Figure 10 a,b,c,d,e). A very close match between results from static and mean value of dynamic analyses can be 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.
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Dynamic Analyses corresponding to θmax and 9
9
Pushover A1a A1b A1c A2a A2b A2c A3a A3b A3c A4a A4b A4c Dyn (mean)
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Figure 9: Case n.1 - Pushover analysis with β = 0.9 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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Dynamic Analyses corresponding to δmax and 9
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Figure 10: Case n. 1- Linear static analysis with β = 0 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; (e) Displacement vs rotation of three floors.
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4.2
Non Linear range In order to investigate the efficiency of the proposed procedure in the nonlinear range, nonlinear material behaviors for concrete and steel bar have been introduced. For the case n. 2, some preliminary results in terms of displacement versus rotation and capacity curve giving base shear vs top displacement in x direction are reported in Figure 11. It is worth noting that, in the nonlinear range, the mean value of failure conditions corresponding to the attainment of the maximum displacement and maximum rotation of the structure are closer to each other with respect to the linear case. The values of coefficient β giving the force distributions for the FTP pushover analyses able to predict the mean values corresponding to the attainment of the maximum displacement and maximum rotation of the structure, are 0.13 and 0.55, respectively. 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. 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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Figure 11: Case n. 2 - Nonlinear dynamic and pushover analysis: (a) Displacement versus rotation; (b) Base shear – displacement capacity curve in x direction. 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
Linear 0.3 0.66 0.42
Case n.2
Non linear 0.18 0.50 0.25
Linear 0 0.90 0.67
Non linear 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 represent 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 11, 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 from FTP analysis with β=0.38 and non linear dynamic results corresponding to the attainment of εcu is reported in Figure 12. Good agreement in terms of displacements, interstorey drift, shear and torque along the height of the structure is shown.
(b)
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Dynamic Analyses corresponding to εcu and Static analysis with β=0.38 9
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Pushover A1a A1b A1c A2a A2b A2c A3a A3b A3c A4a A4b A4c Dyn (mean)
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2
0
Pushover A1a A1b A1c A2a A2b A2c A3a A3b A3c A4a A4b A4c Dyn (mean)
8
Height [m]
8
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 0.02 0 -0.03
(e) -0.025
-0.02
-0.015
θ [Rad]
-0.01
-0.005
0
Figure 12: Case n. 2 - Non linear static analysis with β = 0.38 and results from 12 linear dynamic analyses at time instant corresponding to the achievement of limit state εcu: (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.
M. Savoia et al.
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 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 non linear 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.
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