1 Quake Summit /08/2010 Coupled Axial-Shear-Flexure Interaction Hysteretic Model for Seismic Response Assessment of Bridges Shi-Yu Xu, Ph.D. Student Jian Zhang, Assistant Professor Department of Civil & Environmental Engineering University of California, Los Angeles Shi-Yu Xu, Ph.D. Student Jian Zhang, Assistant Professor Department of Civil & Environmental Engineering University of California, Los Angeles

2 Quake Summit 2010 Outline  Introduction  Motivation & Objectives  Shear-Flexure Interaction Under Constant Axial Load  Proposed Axial-Shear-Flexure Interaction (ASFI) Scheme  Primary Curves and Hysteretic Models Considering Combined Actions  Generation of Primary Curve Family  Stress Level Index & Two-stage Loading Approach  Model Verification  Static Cyclic Tests  Comparison with Fiber Section Model under Seismic Loadings  Limitations and Known Issues  Factors Affecting ASFI & Effects on Bridge Responses  Arrival Time of Vertical Ground Motion  Vertical-to-Horizontal PGA Ratio  Summary

3 Quake Summit 2010 Introduction Motivation  Bridge columns are subjected to combined actions of axial, shear and flexure forces due to structural and geometrical constraints (skewed, curved etc.) and the multi-directional earthquake input motions.  Axial load variation can directly impact the ultimate capacity, stiffness and hysteretic behavior of shear and flexure responses.  Accurate seismic demand assessment of bridges needs to realistically account for combined actions. Objectives  An efficient analytical scheme considering axial-shear- flexural interaction  Shear and flexural hysteretic models reflecting the effects of axial load variation and accumulated material damage (e.g. strength deterioration, stiffness degrading, and pinching behavior)

4 Quake Summit 2010 Axial-Shear-Flexural Interaction Significance of Non-linear Shear-Flexural Interaction (Ozcebe and Saatcioglu 1989)  Shear displacement can be significant -- even if a RC member is not governed by shear failure (as is the case in most of RC columns).  Inelastic shear behavior -- RC members with higher shear strength than flexural strength do not guarantee an elastic behavior in shear deformation. Coupling of Axial-Shear-Flexural Responses (ElMandooh and Ghobarah 2003)  Dynamic variation of axial force -- will cause significant change in the lateral hysteretic moment-curvature relationship and consequently the overall structural behavior in RC columns.

5 Quake Summit 2010 Axial-Shear-Flexure Interaction at Material Level MCFT f sx f sy f cx f cy fxfx fyfy v xy v cxy f c1 f c2 EquilibriumStrain CompatibilityConstitutive Law (Vecchio and Collins 1986) Modified Compression Field Theory γ τ φ M +

6 Quake Summit 2010 Derivation of Flexural and Shear Primary Curves Discretize RC member into small pieces. For each piece of RC element, estimate M-φ and τ-γ relationship by Modified Compression Field Theory (MCFT, Vecchio and Collins 1986). M M=V*h dy V N yiyi V MCFT γ τ γ τ … φ M φ M … + + F-UEL S-UEL SSI spring FNDN DECK S-UEL F-UEL Rigid Column Input the V-Δ s and M-θ curve to Shear-UEL & Flexural-UEL. ΔsΔs V S-UEL ΔmΔm M θ M F-UEL Integrate curvature and shear strain to get displacement. δ=Σ { φ i *dy*y i + γ i *dy } Flexural deformation Shear deformation = h *θ + Δ s

7 Quake Summit 2010 Shear-Flexure Interaction (SFI) under Constant Axial Load dy V N yiyi M M=V*h  Sections with different M/V ratio (level of shear-flexural interaction) demonstrate different mechanical properties and behaviors  Section with higher M/V ratio:  Larger moment capacity  Smaller shear capacity  Maximum moment capacity is bounded by pure bending case

8 Quake Summit 2010 Improved Hysteretic Rules for Shear & Flexural Springs Unloading & reloading stiffness depend on:  Primary curve (K elastic, Crack, & Yield)  Cracked? Yielded?  Shear force level  Max ductility experienced  Loading cycles at max ductility level  Axial load ratio B E F A C D I J K L M N O P Q R S T U V Shear Displacement Shear Force V cr VyVy maximum peak (Δ m,V m ) hardening reference point (Δ m,V’ m ) previous peak (Δ p,V p ) pinching reference point (Δ p,V’ p ) G H  Structural characteristics  Damage in the column  Loading history  Varying during earthquake !! (Ozcebe and Saatcioglu,1989) Xu and Zhang (2010 ), EESD

9 Quake Summit 2010 Outline  Introduction  Motivation & Objectives  Shear-Flexure Interaction Under Constant Axial Load  Proposed Axial-Shear-Flexure Interaction (ASFI) Scheme  Primary Curves and Hysteretic Models Considering Combined Actions  Generation of Primary Curve Family  Stress Level Index & Two-stage Loading Approach  Model Verification  Static Cyclic Tests  Comparison with Fiber Section Model under Seismic Loadings  Limitations and Known Issues  Factors Affecting ASFI & Effects on Bridge Responses  Arrival Time of Vertical Ground Motion  Vertical-to-Horizontal PGA Ratio  Summary

10 Quake Summit 2010 Effects of Axial Load Variation on Total Primary Curves Ultimate capacity and stiffness increase with compressive axial load level. Yielding displacement is almost fixed, regardless of applied axial load. Cracking point is getting smaller as axial force decreasing, implying the column being relatively easy to be cracked. Kunnath et al. H/D=4.5 Calderone-828 H/D=8.0 Calderone-328 H/D=3.0

11 Quake Summit 2010 Normalization of Primary Curves (c) yield load (d) ultimate capacity

12 Quake Summit 2010 Generation of Primary Curve Family (i) 0  crack: straight line (ii) crack  yield: interpolation (iii) yield  ultimate: interpolation (iv) ultimate  failure: constant residual strength ratio n% primary curve (predicted) I% initial primary curve (given) n% critical points, predicted from equations loading deflection I% critical points, on initial primary curve a a a bb b iiiiiiiv Objective:Generating the primary curves related to various axial load levels from a given primary curve subject to an initial axial load

13 Quake Summit 2010 Stress Level Index & Two-stage Loading Approach Equivalent stress level Equivalent stress level -5% ΔyΔy Δ1Δ1 d c Δ max 0% ΔyΔy Δ1Δ1 d c Δ max 10% d c ΔyΔy Δ1Δ1 Δ max Keep Δ, change N: 10%  -5%Keep N, change Δ : Δ 1  Δ 2 10% -5% Δ1Δ1 Δ2Δ2 10% c d -5% Δ1Δ1 c d Δ max Assumption: Effective stress level of a loaded column at fixed ductility is independent of axial load.

14 Quake Summit 2010 Outline  Introduction  Motivation & Objectives  Shear-Flexure Interaction Under Constant Axial Load  Proposed Axial-Shear-Flexure Interaction (ASFI) Scheme  Primary Curves and Hysteretic Models Considering Combined Actions  Generation of Primary Curve Family  Stress Level Index & Two-stage Loading Approach  Model Verification  Static Cyclic Tests  Comparison with Fiber Section Model under Seismic Loadings  Limitations and Known Issues  Factors Affecting ASFI & Effects on Bridge Responses  Arrival Time of Vertical Ground Motion  Vertical-to-Horizontal PGA Ratio  Summary

15 Quake Summit 2010 Cyclic Test: Experimental Program – TP031 ~ TP034 TP-033TP-034 Height Diameter =

16 Quake Summit 2010 Verification of Primary Curve Prediction TP-032 Sakai and Kawashima H/D=3.375 TP-031 Sakai and Kawashima H/D=3.375  TP-031 TP-032  Given the primary curve of TP-031, predicts the response of TP-032. Given the primary curve of TP-032, predicts the response of TP-031.

17 Quake Summit 2010 Verification of Mapping between Different Axial Load Level TP-033 Sakai and Kawashima H/D=3.375 TP-034 Sakai and Kawashima H/D=3.375  TP-031 TP-032  TP-033 TP-034 Axial load decreasing Axial load increasing

18 Quake Summit 2010 Dynamic Validation with Fiber Section Model Proposed ASFI model in general produces larger displacement demand than the fiber section model. Vibration frequencies of the two models agree with each other indicating reasonable prediction on the tangent stiffness of the proposed ASFI model. Considering only the SFI can yield good prediction on the displacement demand. ABAQUS ASFI Model OpenSees Fiber Model

19 Quake Summit 2010 V ΔsΔs M θ Limitations and Known Issues Estimation on post-peak stiffness of primary curve family may not be adequate. May converge at an incorrect solution for systems with yielding platform. May converge at an inconsistent deformed configuration for softening systems. Use of full stiffness matrix can somehow improve the above-mentioned convergence issues, however, it is an asymmetric matrix which offsets most of the advantages.

20 Quake Summit 2010 Outline  Introduction  Motivation & Objectives  Shear-Flexure Interaction Under Constant Axial Load  Proposed Axial-Shear-Flexure Interaction (ASFI) Scheme  Primary Curves and Hysteretic Models Considering Combined Actions  Generation of Primary Curve Family  Stress Level Index & Two-stage Loading Approach  Model Verification  Static Cyclic Tests  Comparison with Fiber Section Model under Seismic Loadings  Limitations and Known Issues  Factors Affecting ASFI & Effects on Bridge Responses  Arrival Time of Vertical Ground Motion  Vertical-to-Horizontal PGA Ratio  Summary

21 Quake Summit 2010 Factors Affecting ASFI: Arrival Time of Vertical Ground Motion (a) H: WN22; V: WN22(b) H: WN22; V: NO4 (a) Horizontal: WN22 (Tp=0.488s); Vertical: WN22 (Tp=0.138s) (b) Horizontal: WN22 (Tp=0.488s); Vertical: NO4 (Tp=0.322s) No significant correlation is found.

22 Quake Summit 2010 Factors Affecting ASFI: Vertical-to-Horizontal PGA Ratio (a) H: WN22; V: WN22(b) H: WN22; V: NO4 (a) Horizontal: WN22 (Tp=0.488s); Vertical: WN22 (Tp=0.138s) (b) Horizontal: WN22 (Tp=0.488s); Vertical: NO4 (Tp=0.322s) t V peak – t H peak = -0.1s Larger PGA V /PGA H ratio tends to have larger influence on force demand. No significant correlation exists with drift demand.

23 Quake Summit 2010 Bridge Responses Considering ASFI Force v.s. total column drift (H/D=2.5) Considering axial variation does not change overall bridge responses much.

24 Quake Summit 2010 Summary Axial load considerably affects the lateral responses of RC columns. Primary curves of the same column under different axial loads can be predicted very well by applying the normalized primary curve and parameterized critical points. Mapping between loading branches corresponding to different axial load levels is made possible by breaking the step into two stages: constant deformation stage and constant loading stage. Model verification shows that the proposed method is able to capture the effects of axial load variation on the lateral responses of RC columns. Transient time analysis on individual bridge column and on prototype bridge system shows that considering axial load variation during earthquake events does not change the drift demand significantly.

25 Quake Summit 2010 ACKNOWLEDGEMENT Thanks for your attention ! The research presented here was funded by National Science Foundation through the Network for Earthquake Engineering Simulation Research Program, grant CMMI , Joy Pauschke, program manager. Thank You!

26 Quake Summit 2010 Analytical Models for RC Columns Plastic Hinge Models  Using equivalent springs to simulate shear and flexural responses of columns at the element level  Empirical and approximate  Difficult to couple together the axial, shear, and flexural responses  Numerical instability in the adopted hysteretic models may induce convergence problem Fiber Section Formulation  Controlling the element responses directly at the material level  Coupling the axial-flexural interaction  Rotation of principal axes in concrete (as large as ～ 30°) due to the existence of shear stress is not considered Elastic or rigid beam Linear or Nonlinear spring elements x y z fiber y z

27 Quake Summit 2010 Deficiencies of Current Numerical Models Deficiencies of Current Models  Non-linearity in shear deformation is not accounted for.  Material damage (strength deterioration and pinching) due to cyclic loading is not considered.  Axial-Shear-Flexural interaction is not captured.

28 Quake Summit 2010 Comparison of Primary Curve Family with Fiber Model Similar trends are observed except post-yield response. Fiber Section Model overestimates initial stiffness. Fiber Section Model underestimates axial load effects. 0% 10%