Shi-Yu Xu, Ph.D. Student Jian Zhang, Assistant Professor

1. 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

2. 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. 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. 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. Axial-Shear-Flexure Interaction at Material Level fy vxy fc1 fx fsx fc2 fcx vcxy fcy fsy Equilibrium Strain Compatibility Constitutive Law M τ + φ γ MCFT Modified Compression Field Theory (Vecchio and Collins 1986)

6. Derivation of Flexural and Shear Primary Curves N yi dy M M τ τ + γ γ φ φ MCFT … … + • Integrate curvature and shear strain to get displacement. DECK M V M=V*h S-UEL M δ=Σ { φi*dy*yi + γi*dy } Flexural deformation Shear deformation =h *θ+ Δs F-UEL S-UEL Δm Δs Rigid Column M V θ S-UEL F-UEL • Input the V-Δs and M-θ curve to Shear-UEL & Flexural-UEL. F-UEL FNDN SSI spring V • 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).

7. Shear-Flexure Interaction (SFI) under Constant Axial Load N V M=V*h yi dy M • 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. Improved Hysteretic Rules for Shear & Flexural Springs Shear Force I maximum peak (Δm,Vm) hardening reference point (Δm,V’m) G Vy F A V previous peak (Δp,Vp) O pinching reference point (Δp,V’p) H Vcr U E N P J M Q T B D K Shear Displacement C L R S Xu and Zhang (2010 ), EESD • Unloading & reloading stiffness depend on: • Primary curve (Kelastic, Crack, & Yield) • Cracked? Yielded? • Shear force level • Max ductility experienced • Loading cycles at max ductility level • Axial load ratio  Structural characteristics  Damage in the column  Loading history  Varying during earthquake !! (Ozcebe and Saatcioglu,1989)

9. 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. 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-328 H/D=3.0 Calderone-828 H/D=8.0

11. Normalization of Primary Curves (c) yield load (d) ultimate capacity

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

13. Stress Level Index & Two-stage Loading Approach 10% Equivalent stress level 0% -5% Equivalent stress level 10% -5% d d c c d d d c c c Δ1 Δmax Δy Δ1 Δy Δ1 Δy Δ1 Δmax Δmax Δmax Keep Δ, change N: 10%  -5% Keep N, change Δ: Δ1 Δ2 10% -5% Δ2 Δ1 Assumption: Effective stress level of a loaded column at fixed ductility is independent of axial load.

14. 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. Cyclic Test: Experimental Program – TP031 ~ TP034 TP-033 TP-034 Height Diameter =

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

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

18. Dynamic Validation with Fiber Section Model ABAQUS ASFI Model OpenSees Fiber 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.

19. Limitations and Known Issues V M Δs θ • 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. 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. Factors Affecting ASFI: Arrival Time of Vertical Ground Motion No significant correlation is found. (a) Horizontal: WN22 (Tp=0.488s); Vertical: WN22 (Tp=0.138s) (b) Horizontal: WN22 (Tp=0.488s); Vertical: NO4 (Tp=0.322s) (a) H: WN22; V: WN22 (b) H: WN22; V: NO4

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

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

24. 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. ACKNOWLEDGEMENT The research presented here was funded by National Science Foundation through the Network for Earthquake Engineering Simulation Research Program, grant CMMI-0530737, Joy Pauschke, program manager. Thank You! Thanks for your attention !

26. Analytical Models for RC Columns Elastic or rigid beam Linear or Nonlinear spring elements y fiber z y x z • 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

27. 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. 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. 10% 0%