ROSE SCHOOL DISPLACEMENT-BASED DESIGN OF CONTINUOUS CONCRETE BRIDGES UNDER TRANSVERSE SEISMIC EXCITATION

Transcription

1 Istituto Universitario di Studi Superiori Università degli Studi di Pavia EUROPEAN SCHOOL FOR ADVANCED STUDIES IN REDUCTION OF SEISMIC RISK ROSE SCHOOL DISPLACEMENT-BASED DESIGN OF CONTINUOUS CONCRETE BRIDGES UNDER TRANSVERSE SEISMIC EXCITATION A Dissertation Submitted in Partial Fulfilment of the Requirements for the Master Degree in EARTHQUAKE ENGINEERING by JUAN CAMILO ORTIZ RESTREPO Supervisor: Prof. M.J.N. PRIESTLEY June, 6

2 The dissertation entitled Displacement-Based Design of Continuous Concrete Bridges Under Transverse Seismic Excitation, by Juan Camilo Ortiz Restrepo, has been approved in partial fulfilment of the requirements for the Master Degree in Earthquake Engineering. M.J.N. PRIESTLEY _ G.M. CALVI

3 Abstract ABSTRACT In this work a displacement-based design procedure for multi-span reinforced concrete bridge structures when subjected to seismic action in the transverse direction is presented. The procedure, initially proposed by Priestley [Priestley, 99], is reviewed and some improvements are implemented. The design methodology is then applied to different possible bridge configurations. The accuracy of the method in terms of reaching the target displacements under the design earthquake level is then assessed using inelastic time-history analysis. Discussion of the appropriate level of damping to be considered in the inelastic time-history analysis of this type of structures is provided based in a recent a recent work developed at the ROSE School on equivalent damping for displacement-based design applications [Grant et al., 4]. Dynamic amplification of the deck transverse moments is investigated and compared with analytical results using different variations of the modal superposition approach. What has been called the Effective Modal Superposition, is then proposed as an efficient method to account for higher mode effects on the deck transverse moment distributions. A comparison of the direct displacement-based design and the force-based design, also assessed with time history analysis, is carried out for the different bridges configurations. Results in terms of pier ductility demands, displacements, deck moments and longitudinal steel reinforcement ratios are presented and discussed. Finally, some analyses of a Rail Bridge configuration with lower deck transversal stiffness are presented to provide an idea of the scope and applicability of the design procedure under different conditions to those assumed for the initial designs. Keywords: bridges; performed-based seismic design; higher modes i

4 Acknowledgements ACKNOWLEDGEMENTS I would like to mainly thank Professor Nigel Priestley for his wise advice during the development of this work. Also thanks to Professor G.M.Calvi, director of the ROSE School, and Lorenza Petrini and Tim Sullivan, who were always available to help me and answer my questions. A exceptional thanks to Juan Camilo Alvarez who was all the time helping me to save time and make this work more proficient. Thanks to all my friends at ROSE School. I would principally like to thank Ana Beatriz, Juan Esteban, Juan Pablo, Carlos and Natalia, Jason and Nasha, Joao and Ana, Alex, Luca and Randolph for all the great times we shared. I would also like to thank my former employer in Colombia, Luis Gonzalo Mejía, for his wise advises, his example of life and his constant search to making me a better engineer and mainly a better person. This work and this Masters are entirely dedicated to my wife, Paulina, for her great love, support and company during this time in Italy. A special mention for my parents, Luis Javier and Gloria, and my brother, Alejandro, who have always be sustaining and encouraging me in every project of my life. ii

5 Index TABLE OF CONTENTS Page ABSTRACT...i ACKNOWLEDGEMENTS...ii TABLE OF CONTENTS...iii LIST OF FIGURES...vi LIST OF TABLES...x. INTRODUCTION.... WHY DISPLACEMENT-BASED DESIGN?.... SCOPE.... FUNDAMENTALS OF DIRECT DISPLACEMENT BASED DESIGN...4. DISPLACEMENT BASED DESIGN OF MULTI-SPAN BRIDGES...9. REGULAR AND IRREGULAR BRIDGES CONFIGURATIONS...9. DESIGN PROCEDURE..... Design Displaced Shape..... The Equivalent SDOF System System Design Displacement Equivalent System Damping Pier Yield Displacement Forces taken by Piers and Abutments Effective System Mass: Equivalent SDOF Design Required Columns Strength Additional notes.... APPLICATION TO DIFFERENT BRIDGE CONFIGURATIONS..... Bridge Information and Assumptions... iii

6 Index... Materials: Abutments: Bridge Deck: Piers and Cap Beam: Seismic Input Design Results Series of Regular Bridges Series 7: SMM Series 8: SML Series 9: SLL Series : SSM Series : SSL Series : MSL Series : SSMLL() Series 4: SSMLL() Series 5: SSLMS Series 6: MSLMS Series 7: LMSSM() Series 8: LMSSM() PERFORMANCE ASSESMENT USING TIME-HISTORY ANALYSIS MODELING ISSUES Hysteretic Rule Damping SPECTRUM-COMPATIBLE TIME HISTORIES DESIGN VERSUS TIME-HISTORY RESULTS Target Displacements and Deck Transverse Moments Series of Regular Bridges Series 7: SMM Series 8: SML Series 9: SLL Series : SSM Series : SSL Series : MSL Series : SSMLL() Series 4: SSMLL() Series 5: SSLMS...86 iv

7 Index 4... Series 6: MSLMS Series 7: LMSSM() Series 8: LMSSM() Dynamic Amplification of Deck Transverse Moments COMPARISON OF WITH THE FORCE BASED DESIGN METHOD FORCE BASED DESIGN TIME HISTORY ANALYSIS FOR FORCE BASED DESIGNED BRIDGES Hysteretic rule Damping Spectrum-Compatible time histories RESULTS COMPARISON FOR REGULAR BRIDGE CONFIGURATIONS Series and Series Series Series Series RESULTS COMPARISON FOR IRREGULAR BRIDGE CONFIGURATIONS Series 7, 8 and Series, and Series, 4, 5 and Series 7 and RAIL BRIDGE PREVIOUS STUDY DIRECT DISPLACEMENT-BASED DESIGN OF A RAIL BRIDGE span Rail Bridges span Rail Bridges COMPARISON OF AND - PERFORMANCE ASSESMENT USING TIME- HISTORY ANALYSIS span Rail Bridges span Rail Bridges CONCLUSIONS...49 REFERENCES...5 v

8 Index LIST OF FIGURES Page Figure. - Effective Stiffness...4 Figure. Design Displacement Spectra...6 Figure. - Regular and Irregular Bridges... Figure. - Possible transverse displacement shapes for continuous bridges... Figure. Uniform beam simply supported on elastic springs... Figure. 4 Equivalent damping for deferent Takeda Thin degrading-stiffness models...6 Figure. 5 Caltrans displacement ARS curves. Soil C, M = 8.±.5 and.7g PGA, for different levels of damping...9 Figure. 6 Model of the equivalent elastic system under transverse response... Figure. 7 Flowchart for Direct Displacement-Based Design of MDOF-bridges... Figure. 8 Bridge Typical Transverse Section...4 Figure. 9 Series of 4-span and 6-span Regular Bridges (H = 7.5 m,. m,.5m and 5. m)...5 Figure. Series of 4-span Irregular Bridges (H = 7.5 m,. m,.5m and 5. m)...6 Figure. Series of 6-span Irregular Bridges (H = 7.5 m,. m,.5m and 5. m)...7 Figure. Extended Caltrans displacement ARS curve for soil profile C, M = 8.±.5 and.7g PGA...8 Figure. Interaction Diagrams for piers...9 Figure. 4 Design results for bridges of Series... Figure. 5 Design results for bridges of Series... Figure. 6 Design results for bridges of Series...4 Figure. 7 Design results for bridges of Series Figure. 8 Design results for bridges of Series vi

9 Index Figure. 9 Design results for bridges of Series Figure. Design results for bridges of Series 7: SMM....4 Figure. Design results for bridges of Series 8: SML....4 Figure. Design results for bridges of Series 9: SLL...4 Figure. Design results for bridges of Series : SSM...46 Figure. 4 Design results for bridges of Series : SSL...47 Figure. 5 Design results for bridges of Series : MSL Figure. 6 Design results for bridges of Series, H=7.5m, with strength redistribution.49 Figure. 7 Design results for bridges of Series : SSMLL()....5 Figure. 8 Design results for bridges of Series 4: SSMLL() Figure. 9 Design results for bridges of Series 5: SSLMS...55 Figure. Design results for bridges of Series 6: MSLMS...56 Figure. Design results for bridges of Series 7: LMSSM ()...57 Figure. Design results for bridges of Series 8: LMSSM ()...58 Figure 4. Typical simplified plan model of bridge used in time-history analysis Figure 4. Takeda degrading stiffness model...6 Figure 4. Artificial time histories and associated set of spectra for different damping levels....6 Figure 4. 4 Artificial time histories and associated set of spectra for different damping levels Figure 4. 5 Design Vs THA for bridges of Series...66 Figure 4. 6 Design Vs THA for bridges of Series...67 Figure 4. 7 Design Vs THA for bridges of Series...68 Figure 4. 8 Design Vs THA for bridges of Series Figure 4. 9 Design Vs THA for bridges of Series Figure 4. Design Vs THA for bridges of Series Figure 4. Design Vs THA for bridges of Series 7: SMM....7 Figure 4. Design Vs THA for bridges of Series 8: SML Figure 4. Design Vs THA for bridges of Series 9: SLL Figure 4. 4 Elastic and Inelastic properties for bridges of Series 8: SML Figure 4. 5 Design Vs THA for bridges of Series : SSM Figure 4. 6 Design Vs THA for bridges of Series : SSL...79 Figure 4. 7 Design Vs THA for bridges of Series : MSL....8 vii

10 Index Figure 4. 8 Elastic and Inelastic properties for bridges of Series : MSL....8 Figure 4. 9 Design Vs THA, Elastic and Inelastic properties for bridge of Series : SML, H=7.5 m, with strength redistribution...8 Figure 4. Design Vs THA for bridges of Series : SSMLL()...84 Figure 4. Design Vs THA for bridges of Series 4: SSMLL()...85 Figure 4. Elastic and Inelastic properties for bridges of Series 4: SSMLL()...86 Figure 4. Design Vs THA for bridges of Series 5: SSLMS...88 Figure 4. 4 Design Vs THA for bridges of Series 6: MSLMS Figure 4. 5 Design Vs THA for bridges of Series 7: LMSSM()...9 Figure 4. 6 Design Vs THA for bridges of Series 8: LMSSM()...9 Figure 4. 7 Elastic and Inelastic properties for bridges of Series 8 LMSSM()...9 Figure 4. 8 Deck Moments for Series Figure 4. 9 Deck Moments for Series Figure 4. Deck Moments for Series Figure 4. Deck Moments for Series : SSMLL() Figure 4. Deck Moments for Series 4: SSMLL().... Figure 4. Deck Moments for Series 5: SSLMS.... Figure 4. 4 Deck Moments for Series 6: MSLMS... Figure 4. 5 Deck Moments for Series 7: LMSSM().... Figure 4. 6 Deck Moments for Series 8: LMSSM()....4 Figure 5. Typical simplified plan model of bridge used in Force Based Design Analysis....6 Figure 5. Acceleration Spectrum for Soil Type C (M = )....7 Figure 5. Typical simplified plan model of bridge used in time-history analysis....8 Figure 5. 4 Comparison of, and THA for bridges of Series... Figure 5. 5 Comparison of, and THA for bridges of Series... Figure 5. 6 Comparison of, and THA for bridges of Series... Figure 5. 7 Comparison of, and THA for bridges of Series Figure 5. 8 Comparison of, and THA for bridges of Series Figure 5. 9 Comparison of, and THA for bridges of Series Figure 5. Comparison of, and THA for bridges of Series 7: SMM....8 Figure 5. Comparison of, and THA for bridges of Series 8: SML....9 Figure 5. Comparison of, and THA for bridges of Series 9: SLL.... viii

11 Index Figure 5. Comparison of, and THA for bridges of Series : SSM.... Figure 5. 4 Comparison of, and THA for bridges of Series : SSL... Figure 5. 5 Comparison of, and THA for bridges of Series : MSL.... Figure 5. 6 Comparison of, and THA for bridges of Series : SSMLL().5 Figure 5. 7 Comparison of, and THA for bridges of Series 4: SSMLL().6 Figure 5. 8 Comparison of, and THA for bridges of Series 5: SSLMS...7 Figure 5. 9 Comparison of, and THA for bridges of Series 6: MSLMS....8 Figure 5. Comparison of, and THA for bridges of Series 7: LMSSM()....9 Figure 5. Comparison of, and THA for bridges of Series 8: LMSSM().... Figure 6. Typical transverse section of Rail Bridge.... Figure 6. Design results for Rail Bridges of Series....4 Figure 6. Design results for Rail Bridges of Series 8: SML....5 Figure 6. 4 Design results for Rail Bridges of Series : MSL....6 Figure 6. 5 Design results for Rail Bridges of Series Figure 6. 6 Design results for Rail Bridges of Series 4: SSMLL....9 Figure 6. 7 Design results for Rail Bridges of Series 8: LMSSM....4 Figure 6. 8 Comparison of, and THA for Rail Bridges of Series....4 Figure 6. 9 Comparison of, and THA for Rail Bridges of Series 8: SML...4 Figure 6. Comparison of, and THA for Rail Bridges of Series : MSL..44 Figure 6. Comparison of, and THA for Rail Bridges of Series Figure 6. Comparison of, and THA for Rail Bridges of Series 4: SSMLL Figure 6. Comparison of, and THA for Rail Bridges of Series 8: LMSSM...48 ix

12 Index LIST OF TABLES Page Table. - Material Properties for Design.... Table. Substitute SDOF parameters for bridges of Series to 6... Table. Substitute SDOF parameters for bridges of Series 7 to...9 Table.4 Substitute SDOF parameters for bridge of Series, H=7.5m, with strength redistribution...49 Table.5 Substitute SDOF parameters for bridges of Series to Table 6. Substitute SDOF parameters for 4-span Rail Bridges.... Table 6. Substitute SDOF parameters for 6-span Rail Bridges....7 x

13 Chapter. Introduction. INTRODUCTION Seismic design is currently going through a transitional period. Most of the seismic codes to date utilize force-based seismic design, or what can also be called strength-based design procedures. However, it is now widely recognized that force and damage are poorly correlated and that strength has lesser importance when designing for earthquake loading than for other actions. These, together with other problems and inconsistencies with force-based design, [Priestley, ], have led to the development of more reliable seismic design methodologies under the framework of what has been termed Performance-Based Seismic Design (PBSD). PBSD represents basically the philosophy of designing a structure to perform within a predefined level of damage under a predefined level of earthquake intensity.. WHY DISPLACEMENT-BASED DESIGN? It is known that displacements correlate much better with damage than forces do. Hence, if the design objective is to control the damage under a given level of seismic excitation it is reasonable to attempt to design the structures using as input the desired displacements to be sustained under the design seismic intensity. One of the more rational and relevant approaches that has been developed over the past years is the Direct Displacement-Based Design, which characterizes the structure to be designed by a single degree of freedom representation of performance at peak displacement response. The objective is to design a structure which would achieve, rather than be bounded by, a given performance limit state under a given seismic intensity [Priestley, 99 and Priestley, ]. The method utilizes the Substitute Structure approach developed by Gulkan and Sozen [Gulkan and Sozen, 974] to model the inelastic structure as an equivalent elastic single-degree-of-freedom (SDOF) system. The concepts of the methodology will be presented first in this work and its application to multi-span bridge structures discussed in detail subsequently.

14 Chapter. Introduction. SCOPE The objectives of this project are to introduce possible improvements to the direct displacement-based design procedure for the design of multi-span bridges for regular and irregular bridges configurations, initially proposed by Priestley [Priestley, 99 and Priestley, ] and subsequently studied by Alvarez Botero [Alvarez Botero, 4]; and to assess the accuracy of the method in terms of reaching the target displacements under the design earthquake level. The latter is done by carrying out inelastic time-history analyses for a series of bridge structures designed using the direct displacement-based design methodology. Additionally, the issue of dynamic amplification of deck transverse moments is investigated and an effective method to consider this phenomenon for bridges designed using direct displacement-based design is proposed. A comparison between the direct-displacement based design,, and the force-based design,, is done. Finally, a parametric study of a Rail Bridge with a low deck transversal stiffness also form part of the investigation and is aimed to assess the applicability of the procedure under diverse design constrains. Chapter provides the basic concepts behind the direct displacement-based design procedure and its general application. Chapter deals with the application of the method to the specific case of multi-span reinforced concrete bridges with continuous deck, and flexible lateral supports at abutments. Important issues regarding the consideration of the sources of energy dissipation and the calculation of the system damping are discussed. An iterative design procedure is introduced. Design results for 7 different bridges are presented and discussed. Chapter 4 presents the results of the assessment of the method in terms of reaching the target displacements when the designs are subjected to spectrum-compatible acceleration time histories. Description of the models used is made and a brief discussion on the seismic input for the inelastic time-history analysis is presented. Higher-mode effects on deck transverse moments are investigated. Chapter 5 presents the comparison of the method with the current generally used code forcebased design method in terms of reaching the target displacements when the designs are subjected to spectrum-compatible acceleration time histories. Description of the force-base design models is made and a short discussion on the seismic input for the inelastic timehistory analysis is presented. Deck transverse moments are also investigated. Final design results for both methods, and, are presented in terms of pier diameter, design moments and longitudinal reinforcement ratios. Chapter 6 deals with the application of the method using a Rail Bridge with low deck transversal stiffness. The methodology is applied to 6 different bridges and then

15 Chapter. Introduction assessed with inelastic time-history analysis. Finally results for direct displacement-based design,, and force-based design,, are presented. Finally, some conclusions are presented in Chapter 7.

16 Chapter Fundamentals of Direct Displacement-Based Design. FUNDAMENTALS OF DIRECT DISPLACEMENT BASED DESIGN Direct Displacement-Based Design is an approach in which, contrary to current Force-Based Design practice, forces are obtained for a desired performance level and based on inelastic response of the system. The objective is to design a structure which would achieve, rather than be bounded by, a given performance limit state under a given seismic intensity [Priestley, ]. The procedure is based in the Substitute Structure approach developed by Gulkan and Sozen [Gulkan and Sozen, 974], which models the inelastic structure as an equivalent elastic single degree of freedom (SDOF) system. The SDOF is represented by an effective stiffness (See Figure.), mass and damping. The aim of the design procedure is to obtain the base shear from a given target displacement and the level of ductility that can be estimated from the structural and element geometries. Figure. - Effective Stiffness Since the substitute structure is elastic, its response to a particular ground motion, and hence the response of the actual structure, can be determined form the elastic response spectrum for the appropriate level of damping. For a SDOF system the design displacement, d, for the performance level under consideration, can be based either on material strain limits or codespecific drift limits. The yield displacement, y, can be estimated from simplified relations for the yield curvature, ϕ y, [Priestley, ] and the displacement ductility calculated as: 4

17 Chapter Fundamentals of Direct Displacement-Based Design µ d = (.) Equivalent viscous damping can then be estimated as the sum of elastic and hysteretic damping, using some relations depending of the displacement ductility, µ, and structure period T eff. y ξeff = ξe + ξhyst (.) The hysteretic component, ξ hyst, can be computed using the equation (.) [Grant et al, 5] which is depends of the equivalent period, T eq. ξ hyst = a b + d µ ( Teq + c) (.) Where a, b, c and d are constants values that depend of the hysteretic model assumed, and µ is the displacement ductility. For the Takeda Thin degrading-stiffness-hysteretic rule, which is commonly used to represent ductile reinforced concrete columns response, these values are a =.5, b =.64, c =.84, d = [Grant et. al., 5] The elastic component, ξ el, is assumed to be 5% of the critical damping but some correction factor must be applied for the assumption of initial-stiffness or tangent-stiffness damping (See deeper discussion in Grant et al, 4). The correction factor for the elastic component can then be computed using eq. (.4). φ κ = µ (.4) Where µ is the displacement ductility and φ depend on the hysteretic rule used and the elastic damping assumption. For the Takeda Thin degrading-stiffness-hysteretic rule, using tangentstiffness elastic damping, φ is equal to As equation (.) is period dependent, an iterative procedure should be implemented to obtain the hysteretic damping (See Grant et. al., 5 for detailed process). Alternatively, as the period dependency of equation (.) is generally insignificant for periods greater than. seconds using the Takeda Thin Model [Grant et al, 5], and as will be unusual for normal bridges to have effective periods less than. seconds, it will generally be conservative to ignore the period dependency in design, and the simplified equation (.5) can be used instead of equation (.). ξ eff µ = µπ (.5) Once the design displacement has been defined and the corresponding damping estimated from the expected ductility demand, the effective period at maximum response, Teff, can be 5

18 Chapter Fundamentals of Direct Displacement-Based Design read directly from the displacement spectrum, reduced for the corresponding level of damping, as shown in Figure.. The effective stiffness, Keff, of the equivalent SDOF can then be determined from the period equation of a SDOF oscillator: K eff 4π M eff = (.6) T eff Where M eff represent the effective mass of the structure participating in the fundamental mode of vibration. Having the effective stiffness, the design lateral force can be readily obtained using Equation (.7). VB Keff d = (.7) Figure. Design Displacement Spectra For a SDOF system the procedure ends here, the design lateral force is the corresponding base shear of the system, and adequate strength must be then provided. Capacity design procedures are used to ensure shear strength exceeds maximum possible shear correspondent to flexural over-strength in the plastic region. However, for a MDOF system, the next step in the design process is the distribution of the design lateral force, VB, throughout the structure and a subsequent structural analysis under the distributed seismic forces. When the design method is applied to a MDOF system, the main issues are the definition of the Substitute Structure and the determination of the design displacement. However, the substitute structure can be easily defined by assuming a displaced shape for the real structure. This displaced shape is that which corresponds to the inelastic first-mode at the design level of seismic excitation. Representing the displacement by the inelastic rather than the elastic first-mode shape is consistent with characterizing the structure by its secant stiffness to maximum response [Priestley et al, 6]. During the last years, research efforts have been 6

19 Chapter Fundamentals of Direct Displacement-Based Design focused on the definition of design displaced shapes for different structural systems. The design displacement of the substitute structure depends also on the limiting displacement of the critical member, C, which in turn depends on the strain or code-drift limit for the performance level under consideration. For bridge structures, the critical member will normally be the shortest column. Having defined the displacement of the critical member and the design displacement shape the displacements of the individual masses can be obtained using Equation (.8). c i = φi (.8) φc Where φ is the design displaced shape, i.e. the fundamental inelastic mode shape. Having now the actual design displacement pattern, the system design displacement is computed using Equation (.9), which is based on the requirement that the work done by the equivalent SDOF system is equivalent to the work done by the MDOF force system, [Calvi, et al., 995]. = d ( mi i ) ( mi i) (.9) To fully define the equivalent SDOF system an effective mass needs to be computed. The effective mass, Meff, is defined as the mass participating in the fundamental inelastic mode of vibration. Being consistent with the work equivalence between the two systems, the effective mass can be obtained using Equation (.). M eff ( m ) ( m ) ( m ) = = i i i i d i i (.) The equivalent SDOF system is now fully defined. Using Equations (.6) and (.7) the total design lateral force is obtained. This shear force must be distributed as design forces to the various discretized masses of the structure, in order that the design moments for potential plastic hinges can be established. Assuming essentially sinusoidal response at peak response, the base shear should be distributed in proportion to mass and displacement at the discretized mass locations. Thus the design force at mass i is given by Equation (.), [Priestley et al, 6]. F i = V B ( mi i) ( m ) i i (.) The subsequent analysis under the distributed seismic forces is straightforward; however, careful consideration of member stiffnesses to be used in the analysis is required. In order to be compatible with the substitute structure concept, member stiffnesses should be representative of effective secant stiffnesses (See Figure.) at the design displacement response. 7

20 Chapter Fundamentals of Direct Displacement-Based Design Particulars of the Direct Displacement-Based Design approach and its application to several structural systems can be found in [Priestley et al, 6]. In the next chapter of this work application of the methodology to continuous RC bridges is presented in more detail. 8

21 Chapter Displacement Based Design of Multi-Span Bridges. DISPLACEMENT BASED DESIGN OF MULTI-SPAN BRIDGES Direct Displacement-Based Design is an approach in which, contrary to current Force-Based Design practice, forces are obtained for a desired performance level and based on inelastic response of the system. The objective is to design a structure which would achieve, rather than be bounded by, a given performance limit state under a given seismic intensity [Priestley, ]. The procedure is based in the Substitute Structure approach developed by Gulkan and Sozen [Gulkan and Sozen, 974], which models the inelastic structure as an equivalent elastic single degree of freedom (SDOF) system. The SDOF is represented by an effective stiffness (See Figure.), mass and damping. The aim of the design procedure is to obtain the base shear from a given target displacement and the level of ductility that can be estimated from the structural and element geometries.. REGULAR AND IRREGULAR BRIDGES CONFIGURATIONS As previous studies were done in bridges with regular configurations [Alvarez Botero, 4], in this dissertation a Regular Bridge will be defined as a bridge in which the structure center of mass, CM, coincides with the structure center of strength, CV. In this case the translational modes of vibration rule the seismic response and the rotational ones are not excited and consequently do not participate in the seismic response of the structure. An Irregular Bridges will be defined as a bridge in which the structure center of mass, CM, do not coincides with the structure center of strength, CV. In this case the seismic response is a combination of the translational and rotational modes of vibration. 9

22 Chapter Displacement Based Design of Multi-Span Bridges Figure. - Regular and Irregular Bridges. Certainly, as the method is based in the shape of the first inelastic mode shape, its efficiency will depend of the similarities between the fundamental elastic and inelastic mode shapes for both, Regular Bridge and Irregular Bridge. In the cases in which the first fundamental elastic and inelastic mode shapes are very different, care must be taken. Previous research [Alvarez Botero, 4] has shown that depending of the seismic level considered, the parabolic inelastic mode shape can or can not be developed, and the bridge maximum displacements, and consequently its behaviour, can be still dominated by the response in the elastic range.. DESIGN PROCEDURE The displacement-based design of multi-degree-of-freedom bridge structures is based on the concepts presented in Chapter. However, some specific issues must be considered carefully during the process. The design displacement shape is a function of the relative stiffness between columns, abutments and the deck. Resistance to transverse seismic excitation is mainly provided by bending of the bridge piers, which are designed to respond inelastically; and, if the abutments provide some restraint to transverse displacements, superstructure bending will also develop. In normal seismic design practice the bridge deck is required to remain elastic under the design level earthquake. As a consequence the seismic inertia forces

23 Chapter Displacement Based Design of Multi-Span Bridges developed in the deck are taken by two different load paths, one portion is transmitted to the piers foundations by column inelastic bending and the remainder transmitted to the abutments by superstructure elastic bending. The portion of load carried by each of the two different load paths is unknown at the start of the design process and depends strongly on the relative effective-column and deck stiffnesses as well as on the degree of lateral restrain provided by the abutments. Since column stiffnesses are also unknown at the start of the design process, an iterative procedure is required. The design procedure presented here considers the discretization of the deck mass as lumped masses at the top of the piers and at the abutments. A portion of the column masses and the cap beam masses can also be lumped at the top, following the recommendations given in [Priestley, et al., 996]. The Direct Displacement-Based Design procedure for multi-degree-of-freedom bridge structures can be summarized in the following basic steps:. Determination of the design displaced shape.. Characterization and evaluation of the equivalent SDOF system.. Application of the displacement-based design approach to the SDOF system. 4. Determination of column required strengths and design... Design Displaced Shape A bridge structure composed by several columns connected to a superstructure of defined flexibility will deform in a manner that is influenced by variations in strength, stiffness and mass distribution. The transverse displaced shape will depend strongly on the relative column stiffness, and more considerably, on the degree of lateral restrain provided at the abutments. Figure. depicts two different bridge configurations and the possible transverse displaced shapes indicated for the different abutment conditions.

24 Chapter Displacement Based Design of Multi-Span Bridges (a) Uniform Height Piers (b) Irregular Height Piers Figure. - Possible transverse displacement shapes for continuous bridges Some flexibility will normally be considered at the abutments; hence the actual displaced shape will be in between the fully restrained and free profiles. Since the displaced shape depends on the relative effective stiffness of the piers, which are unknown at start of the design, some iteration may be required to determine the relative displacements between the abutments and the critical pier. Generally a parabolic displaced shape between abutments and piers can be initially assumed for design purposes. In this work, as in the Alvarez Botero study [Alvarez Botero, 4], the deck-only elastic first-mode shape has been used as a first approximation to the bridge displacement profile. Deck first-mode deformed shape can be obtained either by solving the eigen-problem for the deck or by using an approximate first mode shape function as the one shown in Equation (.) based on a half-sine wave loading shape [Alfawakhiri et al., ]. Where: φ ( x) sin = ( π x ) + π L + π B B (.) EI B = (.) K L A In Equation (.), E is the deck elastic modulus, I is the deck transverse moment of inertia, K A is the elastic abutment stiffness, and L is the total bridge length.

25 Chapter Displacement Based Design of Multi-Span Bridges Figure. Uniform beam simply supported on elastic springs Evaluation of the limiting displacements for each of the piers is also required in order to determine the target displacement pattern. The limiting displacements for the piers are function of the performance level under consideration. The final displacement pattern is likely to consist of only one or two of the piers reaching the limiting displacement. However, determination of the limiting displacements for all them is required in order to identify the critical column. Normally the shortest column governs the selection of the displacement pattern. For the purposes of this work, pier limiting displacements are based on a specified drift limit rather than on material strain limits, which can also be used. Once the critical member has been identified, the displaced shape is scaled in such a way that the critical member reaches its limiting displacement using Equation (.8). It is important to bear in mind that the critical member can change if the assumed displaced shape is not close enough to the actual fundamental inelastic mode of vibration for the given seismic intensity... The Equivalent SDOF System The required properties to characterize the MDOF system as an equivalent SDOF system are: the system design displacement, sys, the system equivalent damping, ξsys, and the system effective mass, Meff.... System Design Displacement Having the target displacement pattern the system design displacement is readily obtained using Equation (.9).... Equivalent System Damping The equivalent system damping can be obtained from a combination of the energy dissipated by the different mechanisms activated during the structure response to seismic excitation. Some approaches, [Kowalsky, ], suggest to make a weighted average of the damping

26 Chapter Displacement Based Design of Multi-Span Bridges based on the work done by the members at each degree of freedom. Another way that has been used to weight the member damping is based on the proportion of load taken by each of the piers and the abutments, as expressed in Equation (.), which gives a reasonable estimate of the actual system damping in a preliminary design. ξ sys = ( Vξ ) i V i i (.) However, the previous forms of calculating the system damping do not consider the contribution of the energy dissipated through elastic deck bending; besides, if high proportions of the load are transmitted to the abutments, their contribution to the system damping is over estimated. Energy dissipation at the abutments is activated if they displace, however, it is localized and should not have a large influence on the overall system damping, especially for relatively large bridges. Given the above reasons, it is suggested to calculate the system damping using Equation (.4), which explicitly considers the contribution from the elastic deck beam-action, the abutment displacements and inelastic pier behaviour. ξ sys ( ) ( ) ( ) VA FA + VA FA ξdeck + FA + FA ξa + Viξi piers = V n i (.4) Where F A and F A are the seismic forces applied to the degrees of freedom associated to the abutments, and are calculated using Equation (.5) and (.6); V A and V A represents the proportion of the lateral force that is taken by each abutment, while V i represent the shear force at the degree of freedom i. F F A A = V B = V B n m m n A A ( m ) i i A A ( m ) i i (.5) (.6) Where V B is the total lateral design force, obtained using Equation (.7), m A and m A are the masses associated to each abutment degree of freedom and A and A are the displacement of each abutment. The equivalent viscous damping for the individual column members, ξ i, is obtained from the relationship between displacement ductility and damping, developed for the Modified Takeda hysteretic rule using the Equation (.) [Grant et al., 5] which is reproduced forward in the Equation (.7) when the period dependency is considered. As was said previously, this generates an iterative process. For a detailed process see Grant et al., 5. 4

27 Chapter Displacement Based Design of Multi-Span Bridges φ ξi = ξe + ξ hyst = µ ξ i el + a b + (.7) d µ i ( Teq + c) Where the values of a, b, c, d and φ were previously defined in Chapter. ξ i can also be obtained with in the simplified Equation (.5) reproduced forward as Equation (.8) [Priestley et al., 6], when the structure actual period is greater than. second. µ i ξi = (.8) µ π Values of column displacement ductility, µ, are obtained by dividing the displacements from the target displacement pattern by the respective yield displacements. At this point it important to note that in previous work developed by Alvarez Botero [Alvarez Botero, 4], Equation (.9) proposed by Kowalsky [Kowalsky, ] was used to compute the equivalent viscous damping. ( r) i µ r µ i i ξi =.5 + (.9) π Where r is the ratio of post-elastic stiffness to the elastic stiffness, normally between. and.5 for concrete members. A value of r =. was used in the work done by Alvarez Botero, and will be used herein when defining the Time History Analysis post-yielding parameters. In Figure.4 the Equations (.7), (.8) and (.9) are plotted. It can be seen that using Equation (.9) results in greater equivalent damping values than using Equations (.7) and (.8) for equivalent periods greater than.5 seconds. Therefore in the results of the present work, generally when any configuration bridge studied in Alvarez Botero dissertation is analyzed, for a given ductility displacement, lesser equivalent damping values will be obtained resulting at the end of the process in bigger values of base shear, V B, and corresponding flexural strengths in piers. i 5

28 Chapter Displacement Based Design of Multi-Span Bridges ξ eq - Equivalent Damping µ - Displacement Ductility Eq. (.7) - Grant et al - Teq =.5 s Eq. (.7) - Grant et al - Teq =.5 s Eq. (.7) - Grant et al - Teq =.75 s Eq. (.7) - Grant et al - Teq =. s Eq. (.7) - Grant et al - Teq =.5 s Eq. (.7) - Grant et al - Teq =. s Eq. (.7) - Grant et al - Teq =. s Eq. (.7) - Grant et al - Teq = 4. s Eq. (.7) - Grant et al - Teq = 5. s Eq. (.8) - Priestley Simplified Eq. (.9) - Alvarez Botero, 4 Figure. 4 Equivalent damping for deferent Takeda Thin degrading-stiffness models... Pier Yield Displacement The yield displacement, y, for a cantilever pier is given by Equation (.), where φ y is the yield curvature and H e is the effective pier height, that considers yield penetration, [Priestley et al., 996], and can be estimated using Equation (.). He y = φy (.) H = H + L (.) e sp In Equation (.) L sp is the strain penetration length and is given by Equation (.), where f y is the longitudinal bar yield stress and d bl is the longitudinal reinforcement bar diameter. L =. fd (.) sp y bl The yield curvature for circular columns, φ y, can be estimated using Equation (.), [Priestley, ], where ε y is the longitudinal bar yield strain, and D is the column diameter..5ε φ y y = (.) D 6

29 Chapter Displacement Based Design of Multi-Span Bridges...4 Forces taken by Piers and Abutments Note that at this stage of the design, the forces taken by each pier and by the abutments are unknown. However, the actual values of the forces are not required and only the relative proportion is needed in order to weight the damping contributions. An initial assumption of the proportion of the total seismic force carried by superstructure bending, SS, and transmitted to the abutments, has to be made, and the remaining distributed among the piers. Seismic Force Carried by Pier Bending Since the relative strength of the piers is a design choice, a practical alternative would be to provide the same longitudinal steel ratio and column diameter and hence the same flexural strength to all the piers. This selection introduces a convenient simplification for the calculation of the column design forces. By doing so, it is found that, if all piers achieve a displacement ductility of at least one, the lateral force resisted by a column is approximately inversely proportional to the pier height, H. F (.4) H That is, if F C (%) is the proportion of the total seismic force carried by pier bending, the proportion resisted by a column is given by Equation (.5), where SDF is the shear distribution factor and is calculated using Equation (.6). V (%) SDF F ( ) = % (.5) i i C SDF i Hi = (.6) H i For the case in which some of the piers remain elastic, i.e. have a displacement ductility demand less than one, the secant stiffness at maximum response is the secant stiffness at yield displacement, that is, the cracked stiffness, and Equation (.6) must be modified. In such a case, the pier force is proportional to the fraction of the yield displacement over the column height for the elastic columns, as shown in Equation (.7). Where µ <. F µ (.7) H 7

30 Chapter Displacement Based Design of Multi-Span Bridges Seismic Force Carried by the Abutments For Regular Bridges, in which the torsional modes do not participate in the seismic response, each abutment will take half of the seismic force carried by superstructure bending, SS. In the case of Irregular Bridges, the seismic force taken by each abutment can be computed using Equations (.4) and (.5) based in the displacements of each abutment. As the initial displacement assumption is based in the first deck mode shape, clearly the shear distribution factor, SDF will be equal for both abutments at the initial stage, but along the iteration process will be changed according to each updated displacement pattern. SDF SDF A A = SS (.8) A + A A A = SS (.9) A + A...5 Effective System Mass: The effective system mass M eff is defined as the mass participating in the fundamental inelastic mode of vibration under the design earthquake level, and can be obtained using Equation (.), which is rewritten below as Equation (.). M eff = ( mi i) (.) d.. Equivalent SDOF Design Having characterized the equivalent SDOF system the effective period of the Substitute Structure is obtained by entering the displacement spectrum, for the appropriate level of damping, with the system design displacement (See Figure.5). The elastic displacement spectrum for the required level of damping can be obtained from the 5% damping spectrum for normal accelerograms measured at least km from the fault rupture, using Equation (.), from EC8, where T,5 is the response displacement for 5% damping. Figure.5 shows a displacement response spectra set from the Caltrans, Seismic Design Criteria, soil profile C, magnitude 8.±.5,.7g PGA [Caltrans, ] reduced for different levels of damping using Equation (.). 5 + ξ T, ξ = T,5.5 (.) 8

31 Chapter Displacement Based Design of Multi-Span Bridges Spectral Period [s] 5% % 5% % % Figure. 5 Caltrans displacement ARS curves. Soil C, M = 8.±.5 and.7g PGA, for different levels of damping The effective stiffness at maximum response is then obtained using Equation (.), and the SDOF base shear force using Equation (.). K eff 4π M eff = (.) T eff VB Keff d = (.)..4 Required Columns Strength The total lateral seismic force must be now distributed to the discretized masses by means of Equation (.), rewritten below as Equation (.4), and analysis carried out. F i = V B ( mi i) ( m ) i i (.4) In order to be compatible with the Substitute Structure concept, member stiffness should be representative of effective secant stiffness, and can be evaluated from the target displacements pattern and the shear forces carried by each pier as: V i K s = (.5) i i 9

32 Chapter Displacement Based Design of Multi-Span Bridges The shear forces at each degree of freedom can be obtained using Equation (.6), where SDF refers to the shear distribution factors from Equation (.6), (.8) and (.9). V i = (.6) SDFi V i An elastic analysis of the equivalent elastic system must now be carried out under the vector of distributed seismic forces F i, and using the member effective stiffnesses, K si, from Equation (.5). Refer to Figure.6, which shows the simplified model that requires to be analyzed for the case of a four span bridge. Figure. 6 Model of the equivalent elastic system under transverse response From the equivalent elastic analysis a revised displaced shape of the bridge is obtained as well as a revised proportion of load carried by superstructure bending. Iteration may be required if the revised displaced shape is not close enough to the initial assumption. Using the updated displaced shape and percentage of load carried by superstructure bending, SS, obtained summing the reactions at the abutments and then dividing it by the base shear force, V B, the process is repeated until convergence of the displacements pattern is reached. Convergence is normally obtained with little iteration, and the final flexural strength to be provided to the individual columns is calculated from the final values of the vector of shear forces carried by each pier, Vi.

33 Chapter Displacement Based Design of Multi-Span Bridges..5 Additional notes The iterative procedure is illustrated in Figure.7; it can be easily implemented in Matlab, Math-Cad, Excel or any other programming software. Note that the convergence criterion is based on the displacement pattern; therefore a good initial assumption will reduce the required number of iterations. Application of the procedure using hand calculations is also likely to be done. Good initial approximations to the displaced shape can be achieved using the simply supported beam model depicted in Figure., and Equation (.). The method requires initial assumptions for the displaced shape and the proportion of load carried by superstructure bending. As pointed out before, a parabolic shape can generally be assumed. Initial displaced shape can be based on the deck first-mode displaced shape and an initial approximation to the proportion of load carried by superstructure bending, SS, can be obtained by applying the vector of displacements to the beam model shown in Figure..

34 Chapter Displacement Based Design of Multi-Span Bridges Figure. 7 Flowchart for Direct Displacement-Based Design of MDOF-bridges.

35 Chapter Displacement Based Design of Multi-Span Bridges. APPLICATION TO DIFFERENT BRIDGE CONFIGURATIONS The Direct Displacement-Based Design approach, as presented in Section. and summarized in Figure.7 was implemented in a Matlab subroutine and applied to eighteen different series of bridge structures, which are shown in Figures.9 to.. Figure.9 shows the six Regular Bridges (4-span and 6-span) studied in Alvarez Botero dissertation [Alvarez Botero, 4], which are re-computed herein with the new damping equation (.7) [Grant et al., 5]. Figures. and. shows twelve different series of Irregular Bridge structures studied in the present dissertation. Single column bents support the superstructure of the bridges. Pier heights were varied for each series, making H = {7.5 m;. m;.5 m and 5. m}, resulting in 7 different bridge designs. A single design limit state was considered and is represented by an arbitrarily chosen drift limit of 4%... Bridge Information and Assumptions... Materials: Concrete and reinforcing steel properties used for design purposes are presented in Table.. Table. - Material Properties for Design.... Abutments: Abutments are usually designed and detailed for service loads and are checked for seismic performance. Normally, equivalent linear springs are used in structural models to simulate the restrains of the superstructure provided by the abutments. The selection of equivalent springs must reflect the dynamic behaviour of the soil behind the abutment, the structural components of the abutment and their interaction with the soil. Substantial nonlinear behaviour can be

36 Chapter Displacement Based Design of Multi-Span Bridges expected as some of the elements constituting the abutments may be subjected to significant yielding, [Maroney and Chai, 994]. However, for all the cases that have been included in this work, the assumption of abutments responding elastically has been made. Nonlinear characterization of the abutments can be obtained from a pushover analysis and the results incorporated to the design procedure without significant difficulties. Abutment stiffness was then chosen to be K A = 75 kn/m and an arbitrary value of 8% damping was associated to their response. A limiting displacement of mm is also specified for the abutments as an additional design restriction.... Bridge Deck: Current design practice intends to avoid inelastic action in the bridge deck; therefore it is considered to respond elastically. The superstructure engages the substructure elements in the transverse direction with shear keys and inelastic action is intended to concentrate at the bottom of the piers. Representative dimensions of a two-lane bridge deck are used. A typical transverse section of the bridges is depicted in Figure.8, the deck transverse-moment of inertia is I yy = 44 m 4 and its torsional stiffness has been ignored. The distributed weight of the bridge deck, including asphalt, is taken as W deck = 75 kn/m. Figure. 8 Bridge Typical Transverse Section 4