# NONLINEAR_ANALYSIS_OF_A_LARGE_BRIDGE_WITH_ISOLATION_BEARINGS.pdf

NONLINEAR ANALYSIS OF A LARGE BRIDGE WITH ISOLATION BEARINGS Robyn M. Mutobe, P.E. SC Solutions, Inc., Santa Clara, California Thomas R. Cooper, P.E. Parsons Brinckerhoff Quade & Douglas, Inc., Sacramento, California ABSTRACT As a part of the California Toll Bridge Seismic Retrofit Program, a global nonlinear time history analysis of the Benicia-Martinez Bridge was conducted using ADINA. A key component of the retrofit strategy was the implementation of friction pendulum bearings. Proper application of the frictional contact surface and the simulation of the restoring force of the bearing were critical to the evaluation of the proposed seismic retrofit. Various local models of the bearing system were developed to study its response and sensitivity to the modeling parameters. Responses of the ADINA friction pendulum bearing representation were compared to those of other nonlinear codes. The behavior predicted by the system of elements used in ADINA for the friction pendulum bearing produced results that matched very closely with other programs. Other important issues for analyzing the Benicia-Martinez Bridge are also discussed. 1.0 INTRODUCTION The Benicia-Martinez Bridge spans the Carquinez Straits connecting the cities of Benicia and Martinez on Interstate Highway 680 between the counties of Solano and Contra Costa in California. Under a state mandate after the 1989 Loma Prieta earthquake, the existing Benicia- Nonlinear Analysis of a Large Bridge with Isolation Bearings Mutobe & Cooper 2 Martinez Bridge was slated for seismic retrofit improvements under the California Toll Bridge Seismic Retrofit Program. The Benicia-Martinez Bridge represents a lifeline structure to the Bay Area. It is a critical route for traffic and commerce between the San Francisco Bay Area and the Sacramento Valley. Therefore the State of California specified in performance criteria for the retrofit design that immediately following a maximum credible earthquake event, the structure should be operational and open to the public. Implicit in this performance specification is that the structure should not collapse, and that the life safety of its users should be ensured. The design criteria for the retrofit design of the structure were therefore based on the desired level of serviceability [1]. In terms of performance, the analysis and subsequent retrofit design of the Benicia-Martinez Bridge differs from other recent efforts associated with the California Toll Bridge Seismic Retrofit program on at least two counts. First, to meet the criteria of remaining operational after a seismic event, the retrofit design specification required that the bridge have little or no damage following the maximum credible earthquake. Secondly, the retrofit design relied heavily on the use of friction pendulum bearings – a relatively untested bearing for application to large bridge structures. To assess the retrofit design, the use of non- linear analysis was required to account for the bearing’s inherent nonlinear behavior. While such bearings have been used on a smaller scale to retrofit and provide isolation systems for buildings, the friction pendulum bearings have rarely been used on a large scale such as required for the retrofit of a major bridge structure. Nonlinear Analysis of a Large Bridge with Isolation Bearings Mutobe & Cooper 3 Material presented in this paper will describe the global model of the bridge, and in particular, the modeling of the friction pendulum bearings using the ADINA contact surface element, and the verification of the friction pendulum bearing simulation. 2.0 DESCRIPTION OF THE STRUCTURE AND SITE SEISMICITY The main span structure of the Benicia-Martinez Bridge consists of 11 spans totaling 4894 feet in length (Figure 2.1). The superstructure is composed of a steel truss, which connects to a system of floor beams and stringer beams to a concrete deck. The substructure consists of multi- celled concrete box piers, which are founded on caissons. The input ground motions, foundation damping and stiffness matrices were provided by the geotechnical consultant for the project. The ground motions were developed for each pier in each of the three orthogonal directions. The displacement time histories were used as input to the global ADINA model. The motions were filtered through the foundation damping and stiffness matrices. The Green Valley Event was used as input to assess the adequacy of the retrofit design [2]. 3.0 PERFORMANCE CRITERIA AND RETROFIT STRATEGY The retrofit strategy implemented for the Benicia-Martinez Bridge was developed based on the performance criteria set forth by the State of California. The criteria required that the bridge remain serviceable to emergency vehicles and the public immediately following a major seismic event. This criterion implied that damage to the structure should be kept to a minimum, such that the bridge would not pose any safety or access issues to its users following a major earthquake. Nonlinear Analysis of a Large Bridge with Isolation Bearings Mutobe & Cooper 4 The basic principle behind the subsequent retrofit strategy was that of isolating the superstructure from the substructure to minimize the damage to the important load carrying elements in the superstructure and allow the substructure to undergo large displacements during a strong seismic event independent of the superstructure. By uncoupling the superstructure from the piers, designers sought to reduce the levels of force in the superstructure elements, aiding tremendously in reducing structural damage in the bridge superstructure. To affect this isolation of the superstructure, the existing rocker bearings were replaced with friction pendulum bearings implemented between the main span superstructure truss and the top of the supporting concrete piers. Twenty-two friction pendulum bearings total were designed for the main span structure, two per pier. Additionally, the capacities of the foundations were substantially increased through the addition of caissons and enlargement of footings. The piers were also strengthened and made more ductile. Individual truss members and connections were strengthened to ensure that the main truss members remained linear throughout the earthquake loading. In particular, regions near the truss expansion hinges were strengthened. 4.0 STRUCTURAL ANALYSIS The ADINA finite element program was specified by the State of California for the California Toll Bridge Structure Retrofit Program, because it permits the user to evaluate important nonlinear and dynamic behavior for bridge structures. A full nonlinear multi-support time history analysis was conducted for 20.47 seconds of the Green Valley maximum credible earthquake (MCE). The direct integration time history analysis Nonlinear Analysis of a Large Bridge with Isolation Bearings Mutobe & Cooper 5 was carried out for 2047 time points at a maximum time step interval of .01 seconds. The global analytical model contains approximately 4500 nodes and 16,358 degrees-of-freedom. There are over 5300 discrete members defined in 230 element groups. Figures 4.1 and 4.2 show the global ADINA retrofit model. The global analysis process was refined such that changes in geometry could be implemented quickly, and the model could be exercised expeditiously. Batch processes were developed and used repeatedly for each new global analysis and the associated post-processing. Similarly, the local models used to evaluate some of the retrofit sub-systems were queued such that analysis and post-processing could be completed overnight. Throughout the project, local models were developed to study, verify, and validate various modeling assumptions and procedures. Such studies were conducted for the friction pendulum bearing assemblage before implementation into the global model. 5.0 DESCRIPTION OF FRICTION PENDULUM BEARING The friction pendulum bearing acts to isolate the structure through its unique geometry. An articulated slider on one side of the bearing transfers normal forces to the other side of the bearing, which is a concave, spherical surface with a special liner material (Figure 5.1). The liner material has known frictional coefficients, and the friction coefficient between the articulated slider and the spherical bearing surface can be varied slightly depending on the structural design requirements. Under non-isolated periods of vibration, lateral movement is resisted through Coulomb friction forces in the bearing given by the following equation: Nonlinear Analysis of a Large Bridge with Isolation Bearings Mutobe & Cooper 6 F friction = µ s N (5.1) where µ s is the static coefficient of friction between the articulated slider and the spherical bearing surface, and N is the normal force of the articulated slider on the bearing surface. However, when the friction force is exceeded, the slider moves and the structure responds at its isolated frequency. Because of the spherical geometry of the bearing surface and vertical loads resulting from seismic motion, the normal force is not constant. (The equations contained herein are taken from [3] and [4].) Also, when the friction force is exceeded the lateral displacement of the bearing is resisted by a “restoring” force, which results from the spherical geometry of the bearing surface. This tends to push the slider back toward the center of the bearing. The restoring force, is thus, highly non-linear, given by the following equation: F= R N U+µNsgn( U & ) (5.2) where N is the normal force given by Equation 5.3, R is the radius of curvature of the spherical bearing surface, U is the lateral displacement, µ is the sliding (dynamic) coefficient of friction, and U & is the velocity. The normal force for a vertically rigid structure also varies as given in the following equation: N=W ++ g U W dN 1 && (5.3) Nonlinear Analysis of a Large Bridge with Isolation Bearings Mutobe & Cooper 7 where W is the weight, dN is the additional normal force due to the spherical geometry of the bearing surface, and U && is the vertical acceleration, and g is the acceleration due to gravity. 6.0 FRICTION PENDULUM BEARING ADINA MODELING AND ASSUMPTIONS The friction pendulum bearings were implemented in the global bridge model as a system of ADINA node-to-node frictional contact surface elements and linear springs (Figure 6.1). The contact surfaces implemented were flat. Because much of the behavior of the bearings is tied directly to the geometry of the system, several assumptions concerning the restoring force were addressed to elicit a more accurate representation of the bearing behavior. The surfaces were made arbitrarily large compared to the actual bearing surface. This was done to ensure that the lateral displacements during the earthquake did not exceed the limits of the contact surface element and cause the analysis to abort prematurely. The limits of each bearing were checked against the displacement trajectories as a post-processing issue after the time history run was complete. If needed, the bearings were then re-sized according to appropriate safety factors and maximum displacements recorded in the time history analysis. The node-to- node contact surface algorithm was invoked. The static coefficient of friction used for the contact surface elements was 6% for the final global analysis. The dynamic (velocity- dependent) coefficient of friction could not be implemented. The target node in the bottom of the bearing and the contactor node at the top of the bearing were coincident and at the center of the contact surface. Because the frictional contact elements were flat surfaces, the lateral restoring forces associated with the spherical geometry of the bearing dish were implemented via linear springs between the target node and contactor node. Nonlinear Analysis of a Large Bridge with Isolation Bearings Mutobe & Cooper 8 Equations 5.2 and 5.3 were assumed to be linear per verification and testing discussed at length in [3] and [4]. The normal force (Eqn. 5.3) was assumed to be constant and equal to the weight of the structure on the bearing. The contributions of additional normal force from the spherical bearing surface geometry and the vertical accelerations were ignored for the normal force as they were time- and displacement-dependent variables and could not efficiently be incorporated in the normal force calculation. Equation 5.2 was further linearized as the contribution of sliding friction, which was velocity-dependent, was also ignored in the calculation of the restoring force. Hence, the restoring force was a linear quantity given by the following equation: F= R W U (6.1) where W is the weight of the structure on the bearing, R is the radius of curvature of the spherical surface of the bearing, and U is the lateral displacement of the bearing. The stiffness of the linear springs used at each bearing in the model was, therefore, R W . As the friction pendulum bearings implemented in the retrofit design were quite large, the radius of curvature specified for the spherical surfaces was also large leading to relatively flat bearing surfaces. The bearing geometry worked in favor of the linear restoring force stiffness and flat bearing surface assumptions. Lastly, a “dummy” linear spring was implemented between the contactor node and target node in the vertical degree of freedom (global z-direction of the model). The spring was necessary to provide numerical stability to the system. The stiffness of the spring was made small in comparison to the lateral springs, 10% of the stiffness of the lateral springs. The forces in this Nonlinear Analysis of a Large Bridge with Isolation Bearings Mutobe & Cooper 9 spring were monitored as a post-processing step to ensure that the forces in the springs did not exceed the tolerances set for the bearings at any time during the earthquake motion. 7.0 LOCAL MODEL STUDIES As part of an on-going effort to verify and validate modeling issues for the global analysis model, local models were created and numerous studies were conducted to investigate assumptions and procedures, which were subsequently incorporated in the global analysis bridge model. These local studies played a very important role in the overall analysis effort. Many different issues were studied for the Benicia-Martinez Bridge. However, proper behavior of the friction pendulum bearings was a key issue for the designers and was examined in three different studies presented here. Because adequate documentation and research had been conducted to verify the properties of the friction pendulum bearings, the assumptions outlined in the previous section regarding the linear bearing properties were thought to be adequate and usable for the global bridge model. The model was instead tested against similar models in two other finite element codes to ensure that the ADINA modeling was accurately capturing the expected behavior. The other