Boundary Nonlinear Dynamic Analysis

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Vernon Trevor Shepherd
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2 Boundary Nonlinear Dynamic Analysis Damper type Nonlinear Link Base Isolator type Nonlinear Link Modal Nonlinear Analysis : Equivalent Dynamic Load

3 Damper type Nonlinear Link Visco-Elastic Damper (VED) Solid Type Fluid Type Hysteretic System (HYS) Metal Damper Friction Damper

4 Effectiveness of of Damper Displacement Response Spectrum Period Shift Damping Period Damping Increase Energy Dissipation by Damper Viscosity or Damper Plastic Behavior Building Damage Reduction

5 Equivalent Damping by by Damper Visco-Elastic Damper (VED) Force Equivalent Damping F Displacement Damper Hysteretic System (HYS) Force Area = A 0 Area = A 1 Displacement ξ d = A0 4π A 1

6 Visco-Elastic Damper (VED) d d c d k b d b ( & ) S f = kd + csign d d& = k d d d d d d b b f f d = d + d d b k d Input Variables Brace Damper c d : VED Damping Constant k d : VED Stiffness S : Damping Exponent k b : Bracing Stiffness

7 Viscous Damping Force Nonlinear (1.0<s) Linear (s=0.0) = csignd& d& ( ) d d d S Nonlinear (0.0<s<1.0) Damping Exponent (S) Velocity of Deformation

8 Hysteretic System (HYS) f d f f = r k d + (1 r) F z k S z& = 1 z { αsgn( dz & ) + β} d& F y y Force ( f ) Input Variables F y k r k Deformation (d) k : Initial Stiffness F y : Yield Strength r : Post Yield Strength Ratio α, β : Loop Shape Parameter S : Bi-linear Transition Parameter

9 1 5 Loop Shape Parameter (α, β) Internal variable (z) d Deformation(d) α=0.1, β=0.9 Internal variable (z) z d Deformation(d) α=0.25, β= Internal variable (z) d Deformation(d) α=0.5, β=-0.5 z Internal variable (z) d Deformation(d) α=0.9, β=-0.1

10 1.2 Internal variable (z) z s = 1.0 s = s = 10.0 s = Deformation(d) d Bilinear Transition Exponent (S)

11 Visco-Elastic Damper Solid Type Visco-elastic Damper Implementation

12 Visco-Elastic Damper Orifice Fluid Fluid Type Visco-elastic Damper Implementation

13 Hysteretic System Metal Damper : ADAS system (Added Damping and Added Stiffness)

14 Hysteretic System Metal Damper : TADAS system (Triangular ADAS)

15 Hysteretic System Metal Damper : Unbonded Brace system

16 Base Isolator type Nonlinear Link Lead Rubber Bearing (LRB) Friction Pendulum System (FPS)

17 Base Isolation :: Fundamentals Fixed Base Base Isolation

18 Effectiveness of of Base Isolation Pseudo-Acceleration Response Spectrum Period Shift Displacement Response Spectrum Period Shift Damping Damping Period Period Period Shift Isolator Flexibility after Yielding Force Reduction : Building Damage Reduction Isolator Displacement Increase Damping Increase Energy Dissipation by Isolator Plastic Behavior Isolator Displacement Reduction Building Damage Reduction

19 Lead Rubbr Bearing (LRB) Lead : Energy Dissipation Wind Resistance Deformation Recovery Rubber : Lateral Flexibility Steel Plate : Vertical Load Capacity

20 Composition of 3 Springs in LRB Input Data of Axial Spring k x : Elastic Stiffness Elastic Axial Spring Nonlinear Shear Spring (Hysteretic System) x k x Input Data of Shear Spring k y,k z : Initial Stiffness F y,y, F y,z : Yield Strength r y, r z : Post Yield Strength Ratio α, β : Loop Shape Parameter y z k y,f y,y, r y, α, β k z, F y,z,r z α, β

21 f R P P k f Friction Pendulum System (FPS) μ P P Shear Spring in FPS Friction Material (Friction Coefficient = μ) Spherical Concave Surface (Radius of Curvature = R)

22 f P f P Initial Stiffness (= k ) Shear Stiffness of Slider & Connected Member Before Sliding Usually Column Stiffness (in which isolator is installed) Before Sliding f P = k d Before Sliding f Column Shear Stiffness (= k ) P P f = d + P µ z R f After Sliding

23 Composition of 3 Springs in FPS GAP type Axial Spring Input Data of Axial Spring k x : Elastic Stiffness O : Gap Opening = 0 (fixed) Input Data of Shear Spring Nonlinear Shear Spring (Friction Pendulum System) y x z R y k y μ fast, y μ slow, y r y k x k z R z μ fast, z μ slow, z r z k y, k z : Initial Shear Stiffness R y, R z : Radius of Curvature of Friction Surface μ fast,y, μ fast,z : Friction coefficient at Fast deformation μ slow,y, μ slow,z : Friction coefficient at Slow deformation r y, r z : Friction coefficient Transition Ratio

24 Implementation of of Base Isolation

25 Implementation of of Base Isolation

26 Implementation of of Base Isolation

27 Modal Nonlinear Analysis : Equivalent Dynamic Load Composite Spring Link Spring Composition Spring Location Equivalent Dynamic Load Mothod Effective Stiffness Equivalent Dynamic Load Choice of of Effective Stiffness Modal Nonlinear Analysis Dynamic Load

28 L Nonlinear Link :: Composite Spring Link c yi L k dx c yj L y z x joint i c zi L k dy c zj L k dz joint j k rx k ry k rz

29 NL-Link with Spring Location :: Shear Shear & Moment Couple Couple Similar Similar to to Beam Beam Element r i rigid d r =r j -r i k r k u rigid r j M j V j L j r j u j -u i d u L i r i V i M i L i L j L shear force diagram V V i V x V j bending moment diagram M M i M x M j Moment of Spring Depends on Spring Location

30 NL-Link without Spring Location :: Shear Shear & Moment Decouple r i k u k r rigid d r =r j -r i r j M j V j d u =u j -u i M i rigid V i L shear force diagram V V i V j bending moment diagram M M i M j

31 Effective Stiffness Nonlinear Spring Elastic Spring with Effective Stiffness Nonlinear Spring Elastic Spring with Effective Stiffness Nonlinear Analysis = + ( - ) Linear Analysis = Equation of Motion : ( ) ( ) Mu&& () t + Cu& () t + K + K u() t = B p() t + B f ()- t f () t S N P N L N

32 Equivalent Dynamic Load Method p p p f L f L f N f L f N f L f N p p f N f L f N fl f L f L

33 Modal Nonlinear Analysis p f N f L = + + d N d 1 d 2 d 3 Effective Stiffness Model Nonlinear Link Force : f N = function of d N (=d 1 +d 2 +d 3 )

34 Choice of of Effective Stiffness Non-zero Value Mode Shapes depends on on Effective Stiffness Unstable Stable by Effective Stiffness Structure with Isolator Type Nonlinear Link

35 Choice of of Effective Stiffness Zero Value Original Mode Shape Structure with Damper Type Nonlinear Link Stable without Effective Stiffness

36 Gravity Load W Scale (=0 to 1) Earthquake Load W Scale (=1) Dynamic Load For Nonlinear Analysis Important Case Case :: FPS FPS type type Isolator GA (g) t1 t2 Dynamic Load 1.0 t0 t1 Scale GA t2

Improving the earthquake performance of bridges using seismic isolation

Improving the earthquake performance of bridges using seismic isolation Ian Buckle Professor, University of Nevada Reno TRB Webinar February 10, 2016 Sponsored by TRB Committee AFF50: Seismic Design and