Approximate Method for Transverse Response Analysis of Partially Isolated Bridges Gaetano Della Corte 1 ; Raffaele De Risi 2 ; and Luigi Di Sarno, M.ASCE 3 Abstract: Current analysis procedures for seismically isolated bridges frequently use an equivalent (approximate) linearization approach to represent the response of nonlinear isolation/energy dissipation devices. Linearization allows standard linear elastic analysis methods, e.g., the response spectrum method, to be conveniently used for design purposes. The linearization approach is by nature an iterative method implying the need to repeatedly correct and analyze a numerical finite-element model. A further simplification could be achieved using closed form equa- tions to represent (1) the structure displacement patterns and (2) the restoring forces from structural elements. The paper explores such a pos- sibility with reference to partially isolated continuous bridges, i.e., bridges with isolation devices at piers and pinned supports at abutments. The role and effect of higher modes of vibration on the system response are discussed, and an approximate method is proposed to account for such effects. An improvement of the classical Jacobsen’s approximation for the effective viscous damping ratio is also proposed using the results of response history analyses. The latter are carried out on two-dimensional numerical models of five case studies, generated from a real existing bridge supposed to be isolated with friction pendulum devices. Comparison of approximate predictions with response history analysis results is presented and discussed. Nonlinear dynamic analyses of a three-dimensional numerical model of the existing bridge were also carried out for comparison purposes. DOI: 10.1061/(ASCE)BE.1943-5592.0000473. © 2013 American Society of Civil Engineers. CE Database subject headings: Bridges; Design; Energy dissipation; Seismic effects. Author keywords: Bridges; Design; Energy dissipation; Isolation; Linearization; Seismic response. Introduction Seismic isolation and energy dissipation systems are currently used worldwide for either new or existing bridges. Numerous devices are available for efficient structural applications (Christopoulos and Filiatrault 2006), and design methods have been proposed and included in modern seismic codes [AASHTO 2010; European Committee for Standardization (CEN) 2005]. It is generally assumed to substitute the actual nonlinear isolation/energy dissipation device with a linear spring and dashpot, characterized by equivalent (or effective) stiffness and damping ratio. The equivalent linear multi- degree-of-freedom (MDOF) structure may be further reduced to an equivalent single-degree-of-freedom (SDOF) model to compute peak displacements and forces (CEN 2005). Design codes generally stipulate the use of numerical models requiring significant efforts of implementation and postprocessing of results. This situation is exacerbated in case of nonlinear seismic isolation systems because linearization is by nature an iterative process. Besides, there could be cases where the structure needs to be analyzed by multiple isolator properties (AASHTO 2010). The present paper discusses a simplified calculation procedure for partially isolated continuous bridges, i.e., those bridge systems isolated at intermediate supports and laterally restrained at abut- ments. The simplified analysis method uses the multimode push- over procedure (Chopra and Goel 2002) while considering the outcomes of more specific studies (Tsai 2008; Tubaldi and Dall’Asta 2011). The novel aspects presented herein are (1) an equation to calculate abutment reactions as a function of the (nondimensional) bridge parameters affecting the response; (2) the number of higher modes to be included in the analysis and their effects on both dis- placement and force demands; (3) an improvement of the classical Jacobsen’s approach for the effective damping ratio; (4) the inves- tigation of a larger number of both regular and nonregular sample bridges, using also a complete three-dimensional model (3D); and (5) the simplified predictions statistically tested using results of dynamic analyses. Simplified Seismic Response Prediction Outline of the Calculation Procedure The simplified analysis method assumes that deck displacements can be evaluated as the statistical combination of sinusoidal patterns as given by D n ¼ D max,n sin  np L x  n ¼ 1, 3, 5, ... , N (1) where L 5 bridge length; x 5 longitudinal abscissa; and N 5 number of (imposed sinusoidal) modes of vibration included in the analysis. Only odd modes of vibration are considered, because the anti- symmetric even modes are not activated during uniform transverse ground motions on a symmetric structure (null participating mass). 1 Assistant Professor, Dept. of Structures for Engineering and Architec- ture, Univ. of Naples “Federico II,” 80125 Naples, Italy (corresponding author). E-mail: gdellaco@unina.it 2 Research Fellow, Dept. of Structures for Engineering and Architecture, Univ. of Naples “Federico II,” 80125 Naples, Italy. E-mail: raffaele.derisi@ unina.it 3 Assistant Professor, Dept. of Engineering, Univ. of Sannio, 82100 Benevento, Italy. E-mail: ldisarno@unisannio.it Note. This manuscript was submitted on July 25, 2012; approved on January 30, 2013; published online on February 1, 2013. Discussion period open until April 1, 2014; separate discussions must be submitted for individual papers. This paper is part of the Journal of Bridge Engineering, Vol. 18, No. 11, November 1, 2013. ©ASCE, ISSN 1084-0702/2013/ 11-1121–1130/$25.00. JOURNAL OF BRIDGE ENGINEERING © ASCE / NOVEMBER 2013 / 1121 J. Bridge Eng. 2013.18:1121-1130. Downloaded from ascelibrary.org by DAPS LIBRARY on 10/16/13. Copyright ASCE. For personal use only; all rights reserved.