Assessment of Non-Linear Static Analysis of Irregular Bridges

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1 Assessment of Non-Linear Static Analysis of Irregular Bridges M JARA, JO NAVARRO, JM JARA AND BA OLMOS Civil Engineering Department University of Michoacán Avenida Francisco J. Múgica s/n, Ciudad Universitaria , Morelia, Michoacán MÉXICO mjarad10@gmail.com Abstract: - The non-linear static procedures for the seismic analysis has been the focus of extensive research, particularly for the analysis of bridges. Two relevant aspects for the success of these procedures are the reference point selection and the lateral load configuration applied to the deck. The effect of these parameters was assessed in this study by the analysis response of eight bridges with different level of transverse stiffness irregularity. A lumped plasticity and distributed plasticity analysis methods were developed for the response evaluation, considering six different combinations of the reference point and lateral load configuration. Based on the results, the reference point plays an important role in the displacement configuration, shear base distribution and characteristics or the capacity curve. Due to the sequential damage observed for increasing levels of seismic intensity, an adaptable load configuration is advisable for improving the accuracy of the analysis response of irregular bridges. Key-Words: - Non-linear static analysis, control point, load configuration, adaptable load 1 Introduction In recent years, the non-linear static analysis has been the focus of extensive research. Based on these studies different variations of the original methodology have been proposed, with the aim of improving the accuracy of the analysis while keeping their simplicity. The original non-linear static procedure was developed for the analysis of buildings; for this type of structures the selection of the node where the displacements are monitored (reference point), and the configuration of the incremental forces that are applied to the structure, are straightforward. By contrast, in the analysis of bridges, the selection of the reference point and the configuration of the applied load are two critical aspects that affect the success of the procedure, particularly for irregular bridges. The different procedures intend to extend the applicability of the original non-linear static analysis, to irregular bridges with significant higher mode effects. The influence of the reference point and load configuration on the accuracy of the non-linear static procedures is evaluated in this study by means of the analysis of eight bridges with different transverse stiffness irregularities. Three lateral load configurations were assessed: uniform, parabolic, and proportional to the first transverse mode. Besides that, two reference points were also considered for each lateral load applied to the structure: one is on the centre of mass of the deck, and the other at the position of the most flexible pier. 2 Non-linear static analysis The capacity curve is the relation of the base shear versus the displacement of a reference or control point, usually recommended at the center of mass of the deck. This curve is the first step in the static non-linear analysis procedures, and represents the capacity of the structure up to the collapse. In a second stage, the demand is estimated by means of an inelastic displacement spectrum through the capacity curve transformed for an equivalent single degree of freedom system. The force and displacement associated to yield and the post elastic stiffness of the equivalent system are needed for constructing the single degree of freedom capacity curve. Then, the accuracy of the non-linear static methods depends on the adequacy of the capacity curve obtained in the first stage of the analysis. 2.1 Non-linear static procedure for bridges The non-linear static analysis was originally proposed for buildings, with the reference point located in the center of mass of the upper level. This position of the control point has proved to be adequate for buildings without significant torsional eccentricities. Typically, the shape of the ISBN:

2 incremental lateral load is derived from the building s first natural period. However, in the case of bridges, the location of the reference point and the lateral load configuration must be investigated carefully, as the accuracy of the procedure is strongly dependent on the selection of these parameters. The Eurocode 8/2 [1] recommends to choose the center of mass of the deck as reference point for bridges; however, various authors [2-4] have demonstrated that the selection of this node drive to erroneous results, particularly for the transverse analysis of irregular bridges. Kappos and others [2] propose the maximum transverse displacement of the deck, as the reference point for the case of irregular bridges. With respect to the lateral load configuration, Isakovic and others [3, 4] propose the shape of the first transverse mode, in special for irregular bridges; other proposals are the uniform or the parabolic shapes. Fig. 1 shows the six possible combinations of the two parameters involved in the definition of the capacity curve: the two reference points and the three lateral load configurations. As it can be seen in Fig. 1 the six cases are the reference point in the center of mass combined with the uniform, parabolic and 1 st mode load configurations, and the reference point located on the node of maximum displacement of the deck, assumed in this study as the node located on the most flexible pier of the bridge, combined with the same three lateral load configurations. In the following, the analysis of eight bridges with different distribution of the transverse stiffness along the longitudinal direction of the bridge were carried out for assessing the variations on the capacity curve when different decisions are taken about the reference point and lateral load configuration. CM CM CM (a) (b) (c) Fig. 1 Six combinations for the analysis assessment. Two control points: mass centre (CM) and taller pier (); and three lateral load configurations: (a) Uniform, (b) Parabolic, (c) 1st. Transverse mode 3 Assessment of the non-linear analysis method for bridges 3.1 Characteristics of the bridge models The prototype model is a four span continuous cast in place box-girder bridge, with four equal spans 30 meters long for a total length of 120 meters, three single-column bents, and with the ends of the bridge pinned. Fig. 2 shows an elevation view of the structure. 120 (m) 30 (m) 30 (m) 30 (m) 30 (m) 1 2 Fig. 2 Geometrical characteristics of the prototype bridge Table 1 shows the height and stiffness of the three piers of each bridge model. In spite of the same height of the column elements in bridges 1 to 3, the stiffness is different because the dimensions of the transverse section of each pier vary for each model. The purpose of modifying the transverse section in the first three bridges is to assess the influence of the shear or flexure type of failure of the columns. Table 1. Height and stiffness of piers Pier Properties Case P1 (m) P2 (m) (m) K1 (N/m) K2 (N/m) K3 (N/m) E11 4.8E E E12 4.5E E E E9 7.9 E E9 2.6 E8 7.9 E E8 2.6 E8 4.1 E E8 0.2 E9 4.1 E E8 2.6 E8 4.1 E E8 4.1 E8 4.1 E8 The deck was represented by a shell finiteelement model and the columns by beam elements. Rigid links connect the superstructure to the supporting columns in order to properly transmit vertical loads and bending actions as well. The piers are fixed at their bases. 3 ISBN:

3 Table 2 presents the stiffness ratio of the piers 1 and 2 with respect to the most flexible pier (number 3). The period of the first transverse mode and the corresponding mode number of the 3D bridge are shown as well. The last column shows the participating modal mass (PMM) as an index of the transverse irregularity of each model. For the regular bridge (model number 8), the PMM is 0.8 and the first transverse mode is the first global mode of the 3D bridge, while for the most irregular bridge (model number 1) PMM is 0.23 and the first transverse mode is the fifth global mode of the system. Table 2. Properties of bridges Case K1/ K2/ PMM T K3 K3 1t (s) Mode (%) Methods of analysis Each model was analyzed for the six conditions displayed in Fig. 1 by two types of analysis with different precision level. In the first method, the material non-linearity of concrete is incorporated into analysis using an idealized stress strain curves for confined concrete by means of Mander s model [5], and the lumped plasticity or plastic hinge analysis is applied. For characterizing the plastic hinge behavior, the effect of the axial force on each pier was taken into account. The plastic hinges were located at both ends of the columns and an elastic behavior is assumed for regions of the elements away from the plastic length. SAP 2000 was used for the pushover analysis. The second method is a distributed plasticity analysis that models the spread of the inelasticity behavior through the cross sections and along the length of the columns. This method considers a finite element model where the cracking and crushing failure modes of the reinforced concrete are accounted for. The criterion for failure of concrete is expressed by the five strength parameters Willam and Warnke failure surface. The concrete material was modeled with the element SOLID65, incorporated in the ANSYS 14.5 software, and the steel was represented by elements type BEAM 188 with bilinear isotropic hardening. The steel is connected in the corner nodes of the finite elements of the concrete model. Fig. 3 shows a detail of the steel reinforcement at the top of the pier. Fig. 3 Detail of the steel reinforcement modelled by the element type beam 188 in ANSYS 14.5 In the analysis the lateral load is incremented step by step until the collapse condition of the structure is reached. Fig. 4 displays the damage condition of the bridge before collapse estimated by the finite element analysis. Cracks are spread along the height of the piers 1 and 2 and crushing of concrete is beginning at the base of the same piers. Cracks are also observed at the base of the most flexible pier (number 3). Fig. 4 Damage distribution before collapse condition (Bridge 1) 3.3 Analysis results Results of the analysis of all bridge models are not shown because of lack of space, but the results of the two bridges, which represent the extreme cases of stiffness irregularity, are briefly summarized in the following. Bridge 1 is the most irregular model of the set, with two short piers and one tall pier varying in height from 2 to 20 meters. The participating modal mass in the transverse direction is 23%, which corresponds to the fifth mode of the structure. On the other side, Bridge 8 is the most ISBN:

4 regular bridge of the group with all piers of the same length (20 meters) and a participating modal mass of 80% in transverse direction, which is also the first mode of the structure. Results and conclusions for the rest of the models are between these two extreme cases. Fig. 5 presents the transverse displacement configuration of the deck previous to the collapse condition for bridge 1. The displacement pattern is similar for both chosen reference points and for the two analysis methods. However, the magnitude of displacements is very different, being major for the lumped plasticity models. The maximum displacement of the deck corresponds to a node located between piers 2 and 3, that is, in the third span, and a displacement reduction is observed at the location of pier 3 as a consequence of the stiffness provided by the element at this position of the deck. An important difference in the displacement pattern is observed in the first span (between abutment and pier); if the reference point is chosen in the center of mass of the deck, the displacement is significant greater than the displacement obtained with the reference point located in the more flexible pier. The displacements obtained by the finite element analysis, is closer to the displacement configuration derived from the lumped plasticity method if the reference point is chosen at the pier 3. Max displacement (m) ANSYS_Uniform CM_SAP2000_Uniform _SAP2000_Uniform Deformed shape_bridge Location along the longitudinal axis (m) Fig. 5 Displacement shape of the deck for bridge 1. Uniform load, finite element and lumped plasticity methods and reference point at the centre of mass (CM) and pier 3 () The first transverse mode configuration closely resembles the displacement patterns observed in Fig. 5 in spite of the uniform lateral load applied to the deck. The same pattern is obtained for the other irregular bridges (2 and 3), suggesting that the first transverse mode displacement distribution can be assumed in the case of irregular bridges. Fig. 6 presents the displacement configuration of the deck for the regular bridge (model 8). A parabolic displacement pattern is observed with maximum displacements at the center of the deck for the two chosen reference points and for both type of analysis. However, the magnitude of displacements estimated by the concentrated plasticity method is considerably higher than the values obtained from the finite element analysis. The same pattern is obtained for the other regular bridges (5 and 7). These results suggest that the parabolic displacement distribution can be assumed in the case of regular bridges, and the choice of the reference point is not as significant as in the case of irregular bridges ANSYS_Uniform 0.40 Deformed shape_bridge Location 30 along 60 the Max displacemen Fig. 6 Displacement shape for bridge 8. Uniform load, finite element and lumped plasticity methods and reference point at CM and Fig. 7 displays the pushover curves for bridge 1. The capacity curves were estimated by the distributed plasticity analysis, considering two reference points: at the center of mass of the deck (CM) and at the most flexible pier (). Even though the shear load capacity is similar for the two chosen reference points, important differences are observed in the initial stiffness, yield force and yield displacement. Shear Force (MN) CM_ANSYS_Uniform _ANSYS_Uniform Pushover curves_bridge Displacement (m) Fig. 7 Pushover curves obtained by the finite element analysis for uniform load and reference points located at the CM and the most flexible pier for bridge 1 Similar trends are observed in the eight bridges evaluated in this study, driving to the conclusion that the reference point has a great effect on the ISBN:

5 characteristics of the capacity curves, and therefore on the results of the nonlinear static procedures. However, as the bridge becomes more regular, the initial stiffness and yield point (displacement and force) are closer, as it can be seen in Fig. 8 which corresponds to the more regular structure (bridge 8). Shear Force (MN) CM_ANSYS_Uniform _ANSYS_Uniform 5 Pushover curves_bridge Displacement (m) Fig. 8 Pushover curves obtained by the finite element analysis for uniform load and reference points located at the CM and the most flexible pier for bridge 8 From the comparison of Figs. 7 and 8 it is evident that the bridge 8 reaches maximum displacement approximately ten times the maximum displacements of the bridge 1, as a consequence of the different demand distribution and shear/flexure ratio on short piers. 3.4 Effect of the seismic intensity level It is expected that stiffness irregularities cause concentration of damage in few elements and sequential deterioration of the piers stiffness. The sequential damage of piers may cause substantial deviations from the elastic analysis results, making the response of the bridge dependent on the seismic intensity level. In view of this response variation, some non-linear static procedures have proposed an adaptable lateral load method with the aim of taking into account the changes of force demand and displacement configuration with the intensity of the applied load. Figs. 9 and 10 show the normalized displacement, defined as the ratio of the maximum elastic displacement and the maximum ultimate displacement along the deck of the bridges 1 and 8 respectively. If the displacement configuration does not experienced important changes on the normalized displacements, it means that the elastic and inelastic configurations are similar for all intensity levels; on the contrary, if significant differences are observed, it means that the displacement configuration is strongly dependent on the seismic intensity level and the adaptable nonlinear static procedures are needed for improving the accuracy of the analysis results. The irregular bridge (Fig. 9) shows important differences between the elastic and inelastic displacements, leading to the conclusion that an adaptable load configuration is preferable. On the contrary, the regular bridge (Fig. 10) gives similar curves, making unnecessary the use of an adaptable load configuration. Normalized displacement (m/m) ANSYS_Ultimate_Uniform ANSYS_Elastic_CM_Uniform SAP2000_Ultimate_CM_Uniform SAP2000_Elastic_CM_Uniform 0.20 Normalized deformed shape _Bridge Location along the longitudinal axis (m) Fig. 9 Normalized deformed shape for bridge 1. Uniform load and control point located at the CM and Pier 3 Normalized displacement (m/m) ANSY_Ultimate_Uniform Ansys_Elastic_CM_Uniform SAP2000_Ultimate_CM_Uniform SAP2000_Elastic_CM_Uniform 0.20 Normalized deformed shape_bridge Location along the longitudinal axis (m) Fig. 10 Normalized deformed shape for bridge 8. Uniform load and control point located at the CM and Pier 3 Reference point plays an important role with respect to the shear base distribution as it can be seen in Fig. 11. If a uniform lateral load configuration is applied to the deck, shear force demand on the shortest pier is 8 MN, whereas the force is only 1.69 MN if the lateral load pattern is like the first transverse mode. In contrast, when the lateral load applied to the deck is proportional to the first mode, shear force on pier 3 is higher than shear force on the same pier obtained with the uniform distribution of the lateral load. If the parabolic distribution of load is considered, the maximum force demand occurs at the central pier, whereas the maximum force occurs in pier 1 if the uniform load is adopted. According to these observations, is complicated to identify a trend in the shear base distribution, making it difficult to define a conservative approach or the most accurate method. For the regular bridge, the reference point has little significance on the base shear distribution, and ISBN:

6 the magnitude of shear forces is approximately the same for all loads (Fig. 12). Shear force (MN) st. Mode Parabolic Uniform P1 P2 Pier Fig. 11 Shear base distribution for the irregular bridge and different load configurations Shear force (MN) st. Mode Parabolic Uniform P1 P2 Pier Fig. 11 Shear base distribution for the regular bridge and different load configurations 4 Conclusion The reference point where the displacements are monitored, and the lateral load configuration, are critical for the success of the non-linear static analysis, particularly for irregular bridges. The influence of these parameters was assessed by the study of eight bridges with different transverse stiffness distribution. A lumped plasticity method and a finite element analysis with distributed plasticity were developed for the evaluation of the bridge response. Based on the displacement configuration of the deck, estimated by both analysis methods, it is recommendable to apply a lateral load proportional to the first transverse mode and to choose the node of maximum displacement as reference point. For regular bridges, the analysis results suggest a parabolic distribution of the lateral load, and the position of the reference point seems to be irrelevant. The capacity curves are also sensitive to the reference point selection for irregular bridges. Important differences were founded in the initial stiffness and yield point, essential parameters on the non-linear static analysis results. However, as the bridges become more regular the capacity curves tend to be more similar. Analysis results for irregular bridges show significant changes between the elastic and ultimate displacements configurations, leading to the conclusion that an adaptable load configuration pattern is pertinent for the success of non-linear static procedures. In regular bridges the use of an adaptable load configuration seems to be unnecessary. Reference point plays an important role on the shear base distribution for irregular bridges. It is difficult to try to identify a trend on the shear base distribution, making it difficult to define a conservative method. This is not the case for regular structures. References: [1] Eurocode 8. Design of structures for earthquake resistance. Part 2: Bridges, European Committee for Standarization (CEN), Bruxelles, [2] A. Kappos, M. Saiidi, M. Aydinoglu, and T. Isakovic, Seismic Design and Assessment of Bridges. Inelastic Methods of Analysis and Case Studies. Springer Science, [3] T. Isakovic and M. Fischinger, Higher mode effects in simplified inelastic seismic analysis of single column bent viaducts. Earthquake Engineering and Structural Dynamics, Vol.35, 2006, pp [4] T. Isakovic, M.P. Lazaro and M. Fischinger, Applicability of pushover methods for the seismic analysis of single column bent viaducts. Earthquake Engineering and Structural Dynamics, Vol. 37, 2008, pp [5] J. B. Mander, M.J.N. Priestley and R. Park, Theoretical stress-strain model for confined concrete. Journal of Structural Engineering. ASCE. Vol. 114, 1988, pp ISBN: