Non-Linear Modeling of Reinforced Concrete Structures for Seismic Applications

Transcription

1 2/18/21 Non-Linear Modeling of Reinforced Concrete Structures for Seismic Applications Luis A. Montejo Assistant Professor Department of Engineering Science and Materials University of Puerto Rico at Mayaguez

2 Outline Justification Extended moment curvature analysis Finite Element Modeling: Lumped plasticity Distributed plasticity / fiber based Fiber based lumped plasticity) Non-linear static analysis (pushover) Non-linear time history analysis: Seismic Input Damping Application Example Conclusion

3 When do we need non-linear modeling? Design verification of very important and/or unusual structures.

4 When do we need non-linear modeling? Have you ever use a force-reduction factor (R) in your design?.7 W Tn acc celeration [g] A [g] F el =A [g]*w Δ Tn period [s] F el /R Δ 1* S d <=Δ limit

5 25 Fn (Mn) 2 When do we need non-linear modeling? Have you ever use a force-reduction factor (R) in your design? μ1 lateral force 15 1 first yield F Δ 1* S d ~Δ limit Δ εc φ displacement εy

6 Outline Justification Extended moment curvature analysis Finite Element Modeling: Lumped plasticity Distributed plasticity / fiber based Fiber based lumped plasticity Non-linear static analysis (pushover) Non-linear time history analysis: Seismic Input Damping Application Example Conclusion

7 Moment-Curvature Analysis εc φ εy Stress concrete : Confined Concrete : Unconfined Concrete Strain Stress [MPa] steel Strain

8 Moment-Curvature Analysis 6 Moment (k kn-m) Curvature(1/m)

9 From M-Φ to F-Δ P Δy Δp (M) L Lsp actual idealized Lp structure and moment φp φy distribution curvature profile displacements θp 2 Δ = φ L / 3 Δ = Δ y + φ L p p L φ φ φ > φ y y (Park and Priestley, 1988)

10 From M-Φ to F-Δ Force (k kn) Displacement(m)

11 Shear Capacity V = Vp + Vs + Vc V p D c = P P 2L > revised UCSD shear model (Kowalsky and Priestley, 2), illustration by Pablo Robalino

12 Shear Capacity V = Vp + Vs + Vc V s = A sx f yh H clb + s d h c cot( θ ) picture by Pablo Robalino

13 Shear Capacity V = Vp + Vs + Vc V s = αβγ f ' c A e α : aspect ratio β : longitudinal reinforcement γ : aggregate interlock

14 Shear Capacity 4 35 Force (k kn) shear failure Displacement(m)

15 lateral force [kn] Shear Capacity [in] μ6 μ4 μ8 μ8 μ6 μ [kips] displacement [mm]

16 Onset of buckling characteristic capacity buckling flexural tension strain steel tension strain growth strain column strain-ductility behavior curvature ductility Moyer and Kowalsky 23

17 lateral force [kn] Onset of buckling displacement [in] μ8 μ6 μ5 μ displacement [mm] μ4 μ5 μ6 μ8 : bar buckl. : theor. buckl [kips] lateral force

18 Outline Justification Extended moment curvature analysis Finite Element Modeling: Lumped plasticity Distributed plasticity / fiber based Fiber based lumped plasticity Non-linear static analysis (pushover) Non-linear time history analysis: Seismic Input Damping Application Example Conclusion

19 Finite element modeling of RC structures non-linear spring Spring moment - rotation Lumped Plasticity Model

20 Lumped plasticity models Disadvantages: -Axial force-moment interaction and axial-force stiffness interaction are separate from the element behavior -Need to use M-C analysis to find: Elastic and post-yield stiffness Non-linear axial force/moment interaction envelope

21 force-moment interaction and axial-force stiffness interaction +4% +2% +1% +5% % -5% T C C T

22 Outline Justification Extended moment curvature analysis Finite Element Modeling: Lumped plasticity Distributed plasticity / fiber based Fiber based lumped plasticity Non-linear static analysis (pushover) Non-linear time history analysis: Seismic Input Damping Application Example Conclusion

23 Finite element modeling of RC structures Longitudinal steel fibers A A Unconfined concrete cover fibers Confined concrete core fibers Section A-A Distributed Plasticity Model Material stress - strain

24 Material constitutive relationships Stress [MPa] Mander monotonic envelope Concrete [ksi] Unconfined concrete Strain 3 Mander monotonic envelope 4.4 Confined concrete Stress [MPa] 2 1 Concrete [ksi] Strain

25 Material constitutive relationships ReinforcingSteel material (Mohle and Kunnath, 26), account for degrading strength and stiffness due to cyclic reversals Raynor monotonic envelope Stress [MPa] ReinforcingSteel -73 material Strain 36 [ksi]

26 Distributed plasticity models Advantages: -No prior M-C analysis required -No need to define hysteretic response (it s defined by the material models) -The influence of axial load is directly modeled -Post-peak strength reduction factor resulting from material strain-softening or failure can be directly modeled. Disadvantages: -Shear strength and shear deformations still under development -Time consuming -Strain localization

27 Strain localization problem Force based Displacement based 5 IP 3 E Ba ase shear 3 IP 8 IP 5 E 2 E Base curvature 8 IP 5 IP 3 IP 2 E 5 E 3 E Displacement at top Displacement at top

28 Outline Justification Extended moment curvature analysis Finite Element Modeling: Lumped plasticity Distributed plasticity / fiber based Fiber based lumped plasticity Non-linear static analysis (pushover) Non-linear time history analysis: Seismic Input Damping Application Example Conclusion

29 Fiber based lumped plasticity model Force-based-fiber sections E, A, I node i Lpi Linear Elastic L Lpj node j BeamWithHinges element (Scott and Fenves, 26)

30 Fiber based lumped plasticity model Normalized Force AB ξ [%] Drift cycle # Normalized Force Curvature ductility Drift 1 : Experimental : Simulation Drift

31 Outline Justification Extended moment curvature analysis Finite Element Modeling: Lumped plasticity Distributed plasticity / fiber based Fiber based lumped plasticity Non-linear static analysis (pushover) Non-linear time history analysis: Seismic Input Damping Application Example Conclusion

32 Non-linear Static Analysis (Pushover) Force [k kn] 6 4 : +2 C 2 : -4 C : serviceability : dam. control Lateral drift [kips]

33 Non-linear Static Analysis (Pushover) Disadvantages: Higher mode effects are missed With an unidirectional push the hysteretic characteristics of fthe structure t can not tbe evaluated If force controlled: tends to become unstable after the peak force is reached If displacement controlled: how do you specified the displacement vector in a multistory building potential soft-storey building mechanisms can be inhibited

34 Outline Justification Extended moment curvature analysis Finite Element Modeling: Lumped plasticity Distributed plasticity / fiber based Fiber based lumped plasticity Non-linear static analysis (pushover) Non-linear time history analysis: Seismic Input Damping Application Example Conclusion

35 Non-linear Time History Analysis: Seismic input Seismic input: 1 PSA [%g] T (s) Real records (Historic) Artificial records: Full artificial (Simqke Gasparini and Vanmarke, 1976) Modified historic records: Using Fourier: Wes Rascal (Silva and Lee, 1987) Using CWT: ArtifQuakeLet (Suarez and Montejo, 23) Using Wavelets: rspmatch (Hancock et al, 25)

36 Non-linear Time History Analysis: Seismic Input acceleration [g] T1 T2 displacement/g Period [s] period shift Period [s] error

37 Outline Justification Extended moment curvature analysis Finite Element Modeling: Lumped plasticity Distributed plasticity / fiber based Fiber based lumped plasticity Non-linear static analysis (pushover) Non-linear time history analysis: Seismic Input Damping Application Example Conclusion

38 Non-linear Time History Analysis: Damping ξ = ξ + ξ hyst el Damping = hysteretic + elastic (viscous) Hysteretic damping:

39 Non-linear Time History Analysis: Damping Elastic damping: represents damping not captured by the hysteretic model: Hysteretic damping on the elastic range Foundation compliance and non-linearity Radiation damping Interaction I t ti between structural t an non-structural t members

40 Non-linear Time History Analysis: Damping Elastic damping: mx && + cx& + kx = mx & c = 2 mωξ = 2ξ mk What values of k and ξ are appropriate? Traditional lumped plasticity model: 5% concrete, 2% steel Fiber model: very low -2% Initial stiffness based viscous damping may result in inelastic damping forces that are unrealistically high. Use tangent-stiffness viscous damping.

41 Non-linear Time History Analysis: Damping Petrini et al. 28

42 Non-linear Time History Analysis: Damping Petrini et al. 28

43 Outline Justification Extended moment curvature analysis Finite Element Modeling: Lumped plasticity Distributed plasticity / fiber based Fiber based lumped plasticity Non-linear static analysis (pushover) Non-linear time history analysis: Seismic Input Damping Application Example Conclusion

44 Application: Alaska DOT bridges Cap Beam Super Structure Columnn/Pile

45 Application: Alaska DOT bridges (F, Δ) post-tensioning (2) hydr. jacks Cross beam 8#7 or 8#9 linear potentiometers Cross beam (2) load cells plane stub string potentiometer 457mm OD steel tube 1651 mm base plate support block strong floor TOP HINGE BOTTOM HINGE pin specimen L d actuator 1 actuator 2 pin API-5L x52 OD: 24 in Thick.: 1 2 in pin fixed string pot. Δ 12#7 ASTM A76 pin

46 Application: Alaska DOT bridges BOTTOM HINGE TOP HINGE picture by Lennie Gonzales

47 Application: Alaska DOT bridges MOMENT-CURVATURE ANALYSIS 25 [in] [in] lateral force [kn N] displacement [mm] TOP HINGE : measured : calculated : ε s = ε y : ε s =.15 : ε s = [kips] lateral force [kn N] : ε st = displacement [mm] BOTTOM HINGE : measured : calculated : ε st = ε y : ε st = [kips]

48 Application: Alaska DOT bridges FIBER MODEL CALIBRATION TOP HINGE norm malized force AB ξ [%] drift cycle # norm malized force curvature ductility drift 4 2 : Experimental : Simulation drift

49 Application: Alaska DOT bridges FIBER MODEL CALIBRATION BOTTOM HINGE norm malized force drift norm malized force 1.5 Experimental e Simulation drift AB ξ [%] cycle # curvature ductility drift

50 Application: Alaska DOT bridges FINITE ELEMENT MODEL BeamWithHinges Linear elastic p-y springs Distributed plasticity

51 Application: Alaska DOT bridges M-C ANALYSIS Moment [kn- -m] : top hinge : bottom hinge : first yied : serviceability : damage control [kips-ft] φd

52 Application: Alaska DOT bridges lateral force [ kn] damage control top hinge PUSHOVER RESULTS serviceability bottom hinge first yield bottom hinge serviceability top hinge first yield top hinge [kips] drift

53 Application: Alaska DOT bridges SEISMIC INPUT Pseudo Accelera ation [g] Period [s] Displacemen nt/g Period [s]

54 Application: Alaska DOT bridges.12.1 INCREMENTAL DYNAMIC ANALYSIS : Average : Eq. records damage control lateral drift g.2.2g serviceability first yield peak ground acceleration [g]

55 Outline Justification Extended moment curvature analysis Finite Element Modeling: Lumped plasticity Distributed plasticity / fiber based Fiber based lumped plasticity Non-linear static analysis (pushover) Non-linear time history analysis: Seismic Input Damping Application Example Conclusion

56 Conclusion It was shown that non-linear analyses provide us with valuable information regarding the seismic behavior of RC structures otherwise impossible to obtain through conventional linear analyses. It is expected that, with the available computational tools, non-linear analyses become more popular in the design office environment.

57 Available (FREE) tools Moment-curvature analysis: Nonlinear FEM (lumped and distributed plasticity): Generation of spectrum compatible earthquake records: contact the author