Non-Linear Modeling of Reinforced Concrete Structures for Seismic Applications

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

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

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

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].6.5.4 A [g].3 2.2.1 F el =A [g]*w Δ 1 1 2 3 4 5 Tn period [s] F el /R Δ 1* S d <=Δ limit

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 5 1 2 3 4 5 6 7 8 displacement εy

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

Moment-Curvature Analysis εc φ εy Stress 6 5 4 3 2 1 concrete : Confined Concrete : Unconfined Concrete.1.2.3 Strain Stress [MPa] 6 4 2-2 -4-6 steel -.1 -.5.5.1 Strain

Moment-Curvature Analysis 6 Moment (k kn-m) 5 4 3 2 1.5.1.15 Curvature(1/m)

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)

From M-Φ to F-Δ Force (k kn) 4 35 3 25 2 15 1 5.5.1.15.2 Displacement(m)

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

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

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

Shear Capacity 4 35 Force (k kn) 3 25 2 15 1 shear failure 5.5.1.15 Displacement(m)

lateral force [kn] Shear Capacity 4 2-2 -4 [in] -2.4-1.6 -.8.8 1.6 2.4 μ6 μ4 μ8 μ8 μ6 μ4 9 45-45 -9 [kips] -6-4 -2 2 4 6 displacement [mm]

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

lateral force [kn] Onset of buckling displacement [in] -5.9-3.9-2. 2. 3.9 5.9 1 5-5 -1 μ8 μ6 μ5 μ4-15 -1-5 5 1 15 displacement [mm] μ4 μ5 μ6 μ8 : bar buckl. : theor. buckl. 2 1-1 -2 [kips] lateral force

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

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

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

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

Material constitutive relationships Stress [MPa] 3 2 1 Mander monotonic envelope Concrete2 4.4 2.9 1.5 [ksi] Unconfined concrete.5.1.15.2 Strain 3 Mander monotonic envelope 4.4 Confined concrete Stress [MPa] 2 1 Concrete2 2.9 1.5 [ksi].5.1.15.2 Strain

Material constitutive relationships ReinforcingSteel material (Mohle and Kunnath, 26), account for degrading strength and stiffness due to cyclic reversals. 75 5 Raynor monotonic envelope 19 73 Stress [MPa] 25-25 -5-75 -36 ReinforcingSteel -73 material -19 -.2.2.4.6 Strain 36 [ksi]

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

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

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)

Fiber based lumped plasticity model Normalized Force AB ξ [%] 1.5 -.5-1 4 3 2 1 -.5.5 Drift 1 2 3 cycle # Normalized Force Curvature ductility 1.5.2.4.6.8 3 2 Drift 1 : Experimental : Simulation.2.4.6.8 Drift

Non-linear Static Analysis (Pushover) 8 1798 Force [k kn] 6 4 : +2 C 2 : -4 C : serviceability : dam. control.2.4.6.8.1 Lateral drift 1348 899 449 [kips]

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

Non-linear Time History Analysis: Seismic input Seismic input: 1 PSA [%g].8.6.4.2 1 2 3 4 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)

Non-linear Time History Analysis: Seismic Input acceleration [g].6.4.2 T1 T2 displacement/g.25.5 1 1.5 2 2.5 3.2.15.1.5 Period [s] period shift.5 1 1.5 2 2.5 3 Period [s] error

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

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

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.

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

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

Application: Alaska DOT bridges (F, Δ) post-tensioning (2) hydr. jacks Cross beam 8#7 or 8#9 linear potentiometers #3@6mm 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

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

Application: Alaska DOT bridges MOMENT-CURVATURE ANALYSIS 25 [in].8 1.6 2.4 3.1 3.9 4.7 56.2 [in]. 2. 3.9 5.9 7.9 9.8 11.813.8 15.7 17.7 19.7 12 lateral force [kn N] 2 15 1 5 2 4 6 8 1 12 displacement [mm] TOP HINGE : measured : calculated : ε s = ε y : ε s =.15 : ε s =.6 44.9 33.7 22.5 11.2 [kips] lateral force [kn N] 9 6 3 : ε st =.28 5 1 15 2 25 3 35 4 45 5 displacement [mm] BOTTOM HINGE : measured : calculated : ε st = ε y : ε st =.8 22.2 134.8 67.4 [kips]

Application: Alaska DOT bridges FIBER MODEL CALIBRATION TOP HINGE norm malized force AB ξ [%] 1-1 4 2 -.5.5 drift 1 2 3 cycle # norm malized force curvature ductility 1.5.2.4.6 drift 4 2 : Experimental : Simulation.2.4.6 drift

Application: Alaska DOT bridges FIBER MODEL CALIBRATION BOTTOM HINGE norm malized force 1-1 -.5.5 drift norm malized force 1.5 Experimental e Simulation.1.2.3.4.5 drift AB ξ [%] 4 2 1 2 3 cycle # curvature ductility 1 5.1.2.3.4.5 drift

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

Application: Alaska DOT bridges M-C ANALYSIS Moment [kn- -m] 16 12 8 4 : top hinge : bottom hinge : first yied : serviceability : damage control 11796 8847 5898 2949 [kips-ft].2.4.6.8.1 φd

Application: Alaska DOT bridges lateral force [ kn] 12 1 8 6 4 2 damage control top hinge PUSHOVER RESULTS serviceability bottom hinge first yield bottom hinge serviceability top hinge first yield top hinge 2247 1798 1348 899 449 [kips].5.1.15 drift

Application: Alaska DOT bridges SEISMIC INPUT Pseudo Accelera ation [g] 1.2 1.8.6.4.2.5 1 1.5 2 Period [s] Displacemen nt/g.3 2.2.1.5 1 1.5 2 Period [s]

Application: Alaska DOT bridges.12.1 INCREMENTAL DYNAMIC ANALYSIS : Average : Eq. records damage control lateral drift.8.6.4.76g.2.2g serviceability first yield.5 1 1.5 peak ground acceleration [g]

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.

Available (FREE) tools Moment-curvature analysis: http://www.ecf.utoronto.ca/~bentz/home.shtml http://blogs.uprm.edu/montejo/ Nonlinear FEM (lumped and distributed plasticity): http://opensees.berkeley.edu/index.php http://www.seismosoft.com Generation of spectrum compatible earthquake records: contact the author http://blogs.uprm.edu/montejo/