Pushover Seismic Analysis of Bridge Structures Bernardo Frère Departamento de Engenharia Civil, Arquitectura e Georrecursos, Instituto Superior Técnico, Technical University of Lisbon, Portugal October 2012 Abstract: This thesis belongs to the field of seismic analysis of bridge structures and intends to evaluate the use of static non linear analysis, also known as pushover. This work only deals with pushover analysis in the longitudinal direction of regular bridges. A plastic hinge model is developed to represent the non-linear behavior of structures and, therefore, obtain the capacity curve of bridge piers. The elastic stiffness of each pier is obtained thru the mean values of its material properties. However, the resistant moment is computed using the design values to limit the moment to its maximum allowed by Eurocode 2. The major methodologies for non linear static analysis of bridges are then presented. Of those, the Capacity Spectrum Method and the methodology suggested in EC8-2 are used. Finally, the results obtained with the above pushover methodologies are compared with those obtained with a non linear dynamic analysis. Both isolated piers and two pier frames results are analyzed. The influence of the structure s dynamic period as well as the influence of the structure s irregularity is evaluated. To close, the pushover analysis of two bridges is carried in their longitudinal direction. Keywords: pushover, displacement, plastic hinge, capacity, static 1. Introduction The Eurocode 8-2 proposes several methods for seismic analysis of bridge structures. However, modal elastic analysis using response spectra is the most common in practice of bridge design in Portugal and only in more complex cases a non-linear dynamic analysis is used. This method is computationally more challenging than the elastic analysis but accounts for the nonlinear behavior of the structure and is, therefore, considered the most precise. Trying to propose a method that is computationally less challenging but still accounts for the structure s non-linear behavior, the Eurocode 8-2 presents nonlinear static analysis, also known as pushover analysis. The objective of this dissertation is to present a procedure to 1
perform pushover analysis of regular bridges in the longitudinal direction and to compare its results with those obtained with a time history analysis. Therefore, the main goal of this work is to assess the nonlinear static analysis in terms of its feasibility and accuracy. 2. Non-Linear Analysis This section presents the procedure used to model the non-linearity of structural behavior. For both concrete and reinforcement steel, the relation between stress and strain is non-linear and, therefore, cross sections also have nonlinear behavior which causes non-linear behavior of the structure. In this work, a plastic hinge model was used. 2.1 Material s Constitutive Law A bilinear stress-strain curve is used for the steel reinforcement as presented in EN 1992-1-1 3.2.7(2). The steel is considered to belong to the C Class and, therefore, according to EN 1992-1-1, has a characteristic strain at maximum force. For concrete, the stress-strain curve presented in EN 1992-1-1 3.1.5(1) was used. This strain-stress relation is the one advised for non linear analysis. For both concrete and steel, both mean and design of material properties are used. This topic is explained in 2.2. 2.2 Moment-Curvature Relation The Moment-Curvature ( ) relation is computed using a program implemented in MATLAB program explained in [FRERE, 2012] and is then simplified into a bilinear diagram, as shown in Figure 1. Figure 1: Simplified bilinear diagram To obtain the bilinear diagram, both stress-strain relations with mean values and with design values are used. The cracked elastic stiffness of the cross section,, is calculated using mean values in order to be closer to the average elastic behavior of the cross section. However, as EN 1992-1-1 limits the resistant moment, both the maximum moment,, and the maximum curvature,, are determined using design values. The yielding point, ), was determined as presented in Figure 2: a post yielding stiffness,, of is considered and so is the fact that cannot be 2
exceeded. Therefore, the yielding moment determined with mean values for material properties,, is reduced to a value. (2.3) The length for the plastic hinge is defined in EN 1998-2 Annex E.3 and depends on the characteristic yield stress,, and bar diameter, : (2.4) Figure 2: Procedure do to obtain the yielding point ) 2.3 Moment-Rotation Relation and Capacity Curve The plastic hinge model presented in EN 1998-2 Annex E.3 is used to obtain the Moment-Rotation ( ) diagram, where is the chord rotation. The model used concentrates in a plastic hinge all the deformation, both elastic and plastic, of the pier. Using the bilinear curve, the relation for a pier of length can be determined with the following equations: (2.1) (2.2) Where is the plastic rotation at the pier s collapse and can be visualized in Figure 3. depends of the plastic hinge s length,, and is obtained as follows: Figure 3: Plastic hinge model The capacity curve of the pier, which is the relation between the applied force,, and the top displacement of the pier,, is obtained directly from the diagram dividing the moment by the pier s length and multiplying the curvature by that same length. 3. Methodologies for Pushover Analysis of Bridges Two MATLAB programs were developed to perform non-linear static 3
analysis: one according to the methodology presented in EN 1998-2 for pushover analysis of bridges and another according to the methodology presented in ATC-40 ([APPLIED TECNHOLOGY COUNCIL, 1996]), the Capacity Spectrum Method (CSM). 3.1 Bridge Capacity Curve Both pushover methodologies used require the bridge capacity curve to be previously determined. Regular straight bridges can be considered a one degree of freedom system when analyzed in the longitudinal direction. Therefore, a bridge can be analyzed as a frame with a rigid beam. To compute its capacity curve for the longitudinal direction, it is first necessary to determine the curve of each resisting pier with the procedure presented in section 2, as presented in Figure 4. Figure 4: Capacity curve of a two pier frame 3.2 Methodology presented in EN 1998-2 This methodology is based in the equal displacement hypothesis that both a linear and a non-linear system have the same displacement when submitted to an earthquake. Response spectra are used to quantify the seismic action and the target displacement,, can then be obtained with: (3.1) Where is the acceleration obtained using a response spectrum, is the mass of the system and is its stiffness. The displacement obtained can then be compared with the capacity curve of the structure. 3.3 Capacity Spectrum Method The Capacity Spectrum Method is an iterative method to determine the target displacement and its procedure is presented in further detail in [FRERE, 2012]. The target displacement is the displacement that corresponds to the intersection of the structure s capacity spectrum and a reduced response spectrum. The response spectrum is in the ADRS (Acceleration Displacement Response Spectrum) format and is reduced in order to take into consideration both viscous and hysteretic damping. 4. Pushover Analysis Applications Results of non-linear static analysis, both using the EC8-2 methodology and the Capacity Spectrum Method, are compared 4
with those obtained with a time-history analysis. The following structures are analyzed: Isolated piers with different lengths in order to assess period of vibration s influence; Two pier frames with different periods of vibration and different relations between its two piers lengths; Real bridge structures based on the model presented in [ARRIAGA E CUNHA, 2011 ]. The section properties, both in its geometry and reinforcement, are kept the same in each of the previous analysis. 4.1 Analysis Procedure For each analysis the seismic action is adjusted to obtain a chosen value for the force ductility coefficient,, defined as follows: (4.1) where: is the elastic force due to the seismic action is the force when the first yielding occurs For pushover analysis the response spectrum is adjusted to obtain the chosen value for. For non-linear dynamic analysis this adjustment is done by a scale factor applied on each accelerogram and the results of this analysis is the average of the results obtained with 5 accelerograms. In each analysis the relative difference between the displacement determined with a pushover analysis and the one obtained with a time-history analysis is determined. Therefore, the relative is calculated for each analysis: (4.2) 4.2 Analysis of isolated piers Isolated piers with different periods of vibration were studied. Independently of the period of vibration, for the displacement calculated with both pushover methods are the same whereas it is not true for higher values of. Figure 5 shows the influence of the period of vibration in both pushover methodologies for. For lower periods, normally under, the displacement obtained was bigger using CSM than using EC8-2 s methodology, whereas the opposite happens for longer periods. For longer periods, Figure 5 shows that the target displacement calculated with the Capacity Spectrum Method becomes smaller than the one determined with a dynamic analysis, typically over. 5
Error Error 90% 70% 50% 30% 10% -10% 0,4 0,8 1,2 1,6 2 2,4 2,8-30% Figure 5: value for different periods of vibration. Isolated pier. 4.3 Analysis of two pier frames Two pier frames such as the one presented in Figure 6 are analyzed, Different values for both and are chosen in order to study the influence of the period of vibration and the structure s irregularity. T [s] x=3 EC8-2 x=3 CSM Figure 7 shows the progress of the relative difference between the displacement obtained with pushover methods and those obtained with a dynamic analysis, for different values of. Conclusions about the influence of the period of vibration are the same than for an isolated pier: for shorter periods (tipically under ), CSM s target displacement is greater than the one calculated with Eurocode s procedure; and the relative for the target displacement determined with CSM becomes negative for periods over. Comparing both pushover methodologies, Eurocode s procedure is less approximate and safer for periods over. Figure 6: Model for two pier frames 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% -10% 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5-20% -30% -40% T [s] β=1,2 EC8-2 β=1,5 EC8-2 β=2,0 EC8-2 β=1,2 CSM β=1,5 CSM β=2,0 CSM Figure 7: Progress of value of with the period of vibration for different values of. 6
Erro The progress of the value of is the same for different values of, however, the effect of the structure s irregularity depends of the intensity of the seismic action. Figure 8 is similar than Figure 7 but presents the results for and shows that the differences of the value of are bigger for than for. 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% -10% 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5-20% -30% -40% T [s] β=1,2 EC8-2 β=1,5 EC8-2 β=2,0 EC8-2 β=1,2 CSM β=1,5 CSM β=2,0 CSM Figure 8: Progress of value of with the period of vibration for different values of. 4.4 Analysis of real bridges Based on the structure presented in Figure 9 two bridge models are analyzed. Piers lengths were altered in order to perform a pushover analysis for two different periods of vibration: model 1 with and model 2 with. Piers lengths are presented in Table 1. Piers Model 1 Model 2 e 10 20 e 15 25 e 20 30 25 35 Table 1: Piers' length for each bridge model Figure 9: Bridge model used for the analysis 7
Displacement [cm] Displacement [cm] Figures 10 and 11 present the results obtained for each of the two models, it shows the displacement determined for and and compare it with every pier s yielding and collapse displacement. The results are in accordance with the previously presented: for both displacements obtained with pushover methods are bigger 35 30 25 20 15 10 5 than those obtained with a time-history analysis; for the displacement obtained with Eurocode s methodology is almost the same than the one determined with a time history analysis - for and for - whereas the displacement calculated with the Capacity Spectrum Method is much lower - for and for. Yielding displacement collapse displacement x=3 ADNL x=3 EC8-2 x=3 CSM x=5 ADNL x=5 EC8-2 0 0 1 2 3 4 5 6 7 Pier Figure 10: Displacements. Model 1 ( ) x=5 CSM 120 110 100 90 80 70 60 50 40 30 20 10 0 1 2 3 4 5 6 7 Pier Figure 11: Displacements. Model 2 ( ) Yielding displacement Collapse displacement x=3 ADNL x=3 EC8-2 x=3 CSM x=5 ADNL x=5 EC8-2 x=5 CSM 8
Error 4.5 Influence of the postyielding stiffness For all previous analysis, the postyielding stiffness of a cross section,, was defined as 1% of the elastic stiffness,. Therefore, a new parameter,, is defined as follows: (4.3) and the influence of choosing bigger values for to perform a pushover analysis is evaluated. Figure 12 shows the meaning of changing the value while the behavior factor,, defined as, remains the same. Figure 12: Different values for The target displacement obtained with Eurocode s methodology is not affected by the value the parameter because only the elastic behavior is taken into account. Figure 13 presents the relative difference between the displacements obtained using the Capacity Spectrum Method with both and with the one obtained with. It shows that the displacements obtained are only slightly, generally under 10%, affected by the value of. 40% CSM k=0.05 30% 20% CSM k=0.1 10% 0% -10% 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5-20% T [s] Figure 13: Influence of the value of in determining the target displacement with CSM 5. Conclusions 5.1 Influence of the period of vibration For shorter periods of vibration, generally under, the target displacement obtained with the Capacity Spectrum Method is bigger than the one obtained with the methodology proposed by EC8-2. For longer periods, typically over, CSM determines smaller displacement than those obtained with a time-history analysis and the Eurocode s methodology becomes safer and more accurate. 5.2 Influence of the structure s irregularity Structure s irregularity has more influence as seismic action becomes more important. The relative differences 9
between the displacements obtained with both pushover methodologies were higher for more irregular structures and the difference increases with the value of the force ductility factor,. 5.3 Influence of the postyielding stiffness Choosing different post-yielding stiffness when creating the non-linear model to perform a pushover analysis has no effect when the target displacement is determined by the methodology proposed in EC8-2. However, the displacement calculated with CSM changes slightly, the difference is generally under 10%. References APPLIED TECHNOLOGY COUNCIL (1996). Seismic Evaluation and Retrofit of Concrete Buildings, Volume 1. California Seismic Safety Commission EN 1992-1-1 (2004). Eurocode 2: Design of Concrete Structures Part 1-1: General rules and rules for buildings. CEN EN 1998-2 (2004). Eurocode 8: Design of structures for earthquake resistance Part 2: Bridges. CEN. FRERE, B. (2012). Pushover Seismic Analysis of Bridge Structures. Master Thesis, IST Pushover analysis allows to determine the ductility demand of the structure and the sequence of plastic hinge formation. Therefore it allows better comprehension of the structure s behavior than an elastic analysis. However, its use might not be of any advantage if the objective of the analysis is only to determine the target displacement. Depending on the period of vibration the equal displacement hypothesis used in Eurocode s methodology might be a good estimate of the target displacement and, consequently, there is no need to account for the structure s non-linear behavior. 10