Force Based Design Fundamentals. Ian Buckle Director Center for Civil Engineering Earthquake Research University of Nevada, Reno

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1 Force Based Design Fundamentals Ian Buckle Director Center for Civil Engineering Earthquake Research University of Nevada, Reno

2 Learning Outcomes Explain difference between elastic forces, actual forces and Modified Design Forces Define a plastic hinge Explain Capacity Protection ti Philosophyh Describe the Component Capacity/Demand Retrofit Method (Method C) Describe how P-Δ effects are considered in design

3 Late eral For rce Force vs. Displacement F EQ Elastic Response F EQ Δ EQ Δ EQ Displacement - Δ Elastic Analysis & Response

4 Force vs. Displacement For moderate-large earthquakes, F EQ is a very large force (can be > structure weight) and it is uneconomical to design an ordinary bridge to resist this force elastically Inelastic behavior (damage) is therefore accepted, provided it does not cause collapse is ductile in nature (not brittle), and occurs in designated components (members) Exceptions are made for special structures where the higher cost of elastic (or almost elastic) behavior can be justified

5 Force vs. Displacement La ateral Fo orce F mechanism First-Order Rigid- Plastic Analysis F mechanism F mechanism Plastic hinge Displacement - Δ Plastic hinge First Order Rigid-Plastic analysis assumes all deformations take place at discrete regions, called plastic hinges, and that a sufficient number of plastic hinges have formed to form a mechanism in the pier/bent. First-Order Rigid Plastic Pier Response

6 Force vs. Displacement l Force Latera Fir rst-order Rig gid-plastic Re esponse Elastic Response Idealized Elasto-Plastic Response Displacement - Δ Idealized Elasto-Plastic Response

7 Force vs. Displacement La ateral Fo orce F max 1 2 Actual Response F EQ 3 4 Plastic Hinge Δ EQ Displacement - Δ Actual Response Milestones 1 - Pseudo-yield Point 2 Maximum Plastic Deformations 3 Onset of Collapse 4 - Collapse

8 Force vs. Displacement Elastic Response Δ EQ l Force F EQ Actual Response F EQ Latera Δ EQ Displacement - Δ Elastic vs. Actual Response

9 Force vs. Displacement l Force Latera Fir rst-order Rig gid-plastic Re esponse Elastic Response Idealized Elasto-Plastic Response Displacement - Δ At Actual lresponse Idealized Elasto-Plastic Response

10 Force vs. Displacement La ateral Fo orce Elastic Analysis Displacement - Δ First-Order Elastic-Plastic Second-Order Elastic-Plastic (slender column response shown- requirements are in place to limit this type of behavior) Elastic Analysis conducted using to linear methods First-order elasto-plastic behavior includes reduced pier/bent stiffness and strength with progressive plastic hinge formation. Second-order elasto-plastic behavior traces formation of plastic hinges and includes geometric non-linearities (e.g. P-Δ) Pier Response with Higher Order Effects

11 Force vs. Displacement Member strengths: Nominal strength, S n Design strength, S d = φ S n where φ is the strength th reduction factor <1.0 (see AASHTO LRFD Specifications) [Expected strength, S e = φ e S n where, in absence of mill certificates, φ e =1.2 for steel and 1.3 for concrete members] Over-strength, S o = φ o S n where φ o is the over-strength factor >1.0 ( )

12 Force vs. Displacement F elastic Latera al Force F overstrength F mechanism Design for lesser of F elastic or F overstrength Overstrength Factor Displacement - Δ F overstrength = φ o F mechanism where φ o = over-strength factor overstrength o mechanism o Over-strength Forces

13 Force vs. Displacement Later ral Forc ce μ = Δ max Δ yield Δ yield Δ max Displacement - Δ Displacement Ductility Factor - μ

14 Force vs. Displacement To calculate Δ max Nonlinear time history analysis using software such as SAP2000, SEISAB-NL, ADINA, ABAQUS Elastic analysis using either ULM, SMSA, MMSA, TH to find maximum elastic displacement Δ EQ and then assume either: Equal displacements, i.e. Δ max = Δ EQ or: Equal work done during elasto-plastic response as during elastic response, and solve for Δ max

15 Force vs. Displacement F elastic Define R = F elastic F mechanism (Response Modification Factor) rce Lat teral Fo F mechanism Δ yield F elastic R Δ max = Δ EQ Displacement - Δ Then: Δ max = Δ EQ and: μ = Δ max = R Δ yield (Displacement Ductility Factor) Relationship between ductility and R Equal Displacement Assumption

16 Force vs. Displacement F elastic Define R = F elastic F mechanism (Response Modification Factor) rce Lat teral Fo F mechanism Δ yield F elastic R Δ max = Δ EQ Displacement - Δ Then: Δ max = Δ EQ and: μ = Δ max = R Δ yield (Displacement Ductility Factor) Relationship between ductility and R Equal Displacement Assumption

17 Force vs. Displacement Define R = F elastic F mechanism (Response Modification Factor) Force Lateral F elastic F mechanism F elastic R Then: Δ max = (R Δ EQ + Δ yield ) / 2 and: μ = Δ max (R 2 + 1) / 2 Δ yield (Displacement Ductility Factor) Δ yield Δ EQ Δ max Displacement - Δ Relationship between ductility and R Equal Work Done Assumption

18 Force vs. Displacement Define R = F elastic F mechanism (Response Modification Factor) Force Lateral F elastic F mechanism F elastic R Then: Δ max = (R Δ EQ + Δ yield ) / 2 and: μ = Δ max (R 2 + 1) / 2 Δ yield (Displacement Ductility Factor) Δ yield Δ EQ Δ max Displacement - Δ Relationship between ductility and R Equal Work Done Assumption

19 Force vs. Displacement μ = R = 1 Late eral For rce μ = Δ max R Δ yield μ = R = 2 μ = R = 3 μ = R = 5 In ncreasin ng Ducti ility Δ yield Displacement - Δ Δ max Substructure Type Strength and Ductility Relationship

20 Force vs. Displacement Late eral For rce μ = Δ max R Δ yield Δ yield Displacement - Δ Δ max μ = R = 1 μ = R = 2 μ = R = 3 μ = R = 5 In ncreasin ng Ducti ility Substructure Type Strength and Ductility Relationship xibility Δ EQ sing flex higher Δ Increas period, NOTE: longer

21 Force vs. Displacement Late eral For rce μ = Δ max R Δ yield Δ yield Displacement - Δ Δ max μ = R = 1 μ = R = 2 μ = R = 3 μ = R = 5 In ncreasin ng Ducti ility Substructure Type Strength and Ductility Relationship xibility Δ EQ sing flex higher Δ Increas period, NOTE: longer

22 Force vs. Displacement Latera al Force F elastic F overstrength F mechanism = F design F R F elastic Effec ct of R Axial Force (Pn, M n ) ΦP n ΦM n ign P,M des (P n, M n ) w/ Over-strength PM P,M elastic Displacement - Δ Moment F elastic & P,M elastic force effect generated from Extreme Event 1 load combination Elastic vs. Modified Design vs. Over-strength Forces

23 Capacity Protection The principle of Capacity Protection Design is that yielding in one component (or member) will cap (or limit) the forces and moments in an adjacent component (member). i.e. reaching the elastic capacity (yield) of one member protects adjacent members from excessive forces

24 Capacity Protection Bridge Geometry F, Δ A B C D h 1 h 2 E F

25 Capacity Protection Bridge Geometry Bending Moment Diagram F, Δ A B C D F, Δ M p 2 4 h 1 h 2 M M p 1 V EB E F Two plastic hinges in BE, 3 V FC V EB max = 2 M p / h 1

26 Capacity Protection Bridge Geometry Bending Moment Diagram F, Δ A B C D F, Δ M p 2 4 h 1 h 2 M M p 1 V EB F 2 E 3 4 F Two plastic hinges in BE, V EB max = 2 M p / h 1 3 V FC 1 V FC F = V EB + V EB V FC Δ

27 Capacity Protection Bridge Geometry Bending Moment Diagram F, Δ A B C D F, Δ M p 2 4 h 1 h 2 M M p 1 V EB F E Two plastic hinges in EB, 3 V FC F V EB max =2M p /h Maximum shear and moment in V FC foundation at E is limited by yielding F = V EB + in EB, i.e. the foundation is capacity V EB V FC protected by column yield Δ

28 Capacity Protection Bridge Geometry Bending Moment Diagram F, Δ A B C D F, Δ M p 2 4 h 1 h 2 M M p 1 V EB E Two plastic hinges in EB, 3 V FC F F V EB max =2M p /h NOTE: For design M V p = φ M n, but to 1 FC calculate max. value of V EB, should F = V EB + use the overstrength moment, V EB V FC i.e. M p =φ o M n where (φ o > 1) Δ

29 Seismic Retrofit Process Three step process: Screening and Prioritization Detailed Evaluation Retrofit strategies, approaches and measures

30 Evaluation Methods FHWA Manual describes 5 basic methods for evaluation Methods vary in rigor from no-analysis analysis to nonlinear dynamic time history analysis Selection / application depends d on the Seismic Retrofit Category A D for the bid bridge (seismic i hazard and performance required), and its geometry (regularity).

31 Evaluation Methods A and B:Default capacity method checks capacity of components due to non-seismic loads against specified minima no analysis is required (e.g. connections and support lengths) C: Capacity/demand method compares component capacities against force demands d from an elastic analysis - on a component-by-component basis

32 Evaluation Methods D: Capacity/spectrum method(s) use nonlinear capacity models for individual piers (or complete bridge), i.e. a pushover curve, and displacement demands from an elastic analysis E: Nonlinear dynamic method explicitly models nonlinear behavior of components in a time history analysis of complete bridge

33 Evaluation Method C Capacity / Demand Ratio Method Force demands are calculated by elastic methods, such as multi-modal spectral analysis method (i.e. without regard to any yielding that may occur). Elastic displacements also used Capacities are obtained from combination of theory (strength of materials) and engineering judgment, for each major component Gives good results for bridges that behave elastically, or nearly so

34 Evaluation Method C Five step procedure (FHWA 2006): 1. Determine applicability of the method 2. Determine capacity, Q ci for all components (i) using theory and engineering judgement (e.g. Appendix D, FHWA Manual, 2006) 3. Determine sum of non-seismic force and displacement demands, ΣQ NSi for all components for each load combination in LRFD design specification

35 Evaluation Method C Procedure continued: 4. Determine seismic demand, Q EQi on each component by an elastic method of analysis 5. For each component determine capacity/demand ratio from: r i = Q ci Σ Q NSi Q EQi

36 Evaluation Method C If r i > 1.0 component has adequate capacity If 0.5 < r i < 1.0 component may be acceptable without retrofitting, depending on other deficiencies in member, if any, and consequences of ffailure If 0.5 > r i component has inadequate capacity and retrofitting is indicated

37 Evaluation Method C Note that for each component, several c/d ratios (r i ) may need to be calculated. Example 1: Support lengths and bearings

38 Evaluation Method C Example 2: Columns Five (5) c/d ratios should be found for each column for the following: Anchorage length of longitudinal rebar, r ca Transverse confinement, r cc Splice length, r cs Shear force, r cv r confinement r splice Bending moment, r ec r anchorage

39 P-Δ Effects in Bridge Columns F, Δ h P F P K K pier pier h K pier F Δ A P Equilibrium in deformed state requires ΣM A = 0, i.e. F h + P Δ = (K pier Δ)h h Therefore F = (K pier P / h) Δ = K' pier Δ

40 P-Δ Effects in Elastic Bridge Columns Force F Elastic capacity curve for column, slope = K pier Elastic capacity curve with P-Δ included, slope = K' pier Loss in capacity at displacement Δ, due to P-Δ effect = PΔ / h Δ Displacement

41 P-Δ Effects in Yielding Bridge Columns Elasto-plastic capacity curve for column; initial slope = K pier, second slope = 0 Force F Elasto-plastic capacity curve with P-Δ included; initial slope = K' pier second slope = - P/h Loss in capacity at displacement Δ due to P-Δ effect = PΔ / h Δ yield Δ Displacement

42 P-Δ Effects in Yielding Bridge Columns Elasto-plastic capacity curve for column; initial slope = K pier, second slope = 0 Force F Elasto-plastic capacity curve with P-Δ included; initial slope = K' pier second slope = - P/h Loss in capacity at displacement Δ due to P-Δ effect = PΔ / h Δ yield Δ Displacement

43 P-Δ Effects in Yielding Bridge Columns AASHTO LRFD Specifications and FHWA Retrofit Manual require that if PΔ / h>025mp/h 0.25 h a refined analysis must be undertaken that explicitly includes nonlinear geometric effects (P-Δ) Encouraged to limit Δ such that Δ max < 0.25 Mp / P

44 Learning Outcomes Explain difference between elastic forces, actual forces and Modified Design Forces Define a plastic hinge Explain Capacity Protection ti Philosophyh Describe the Component Capacity/Demand Retrofit Method (Method C) Describe how P-Δ effects are considered in design

45 Thank you

Seismic Pushover Analysis Using AASHTO Guide Specifications for LRFD Seismic Bridge Design

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