Parametrically Dissipative Explicit Direct Integration Algorithms for Computational and Experimental Structural Dynamics

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Lehigh University

Lehigh Preserve

Theses and Dissertations

2016

Parametrically Dissipative Explicit Direct

Integration Algorithms for Computational and

Experimental Structural Dynamics

Chinmoy Kolay

Lehigh University

Follow this and additional works at:http://preserve.lehigh.edu/etd Part of theCivil and Environmental Engineering Commons

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Recommended Citation

Kolay, Chinmoy, "Parametrically Dissipative Explicit Direct Integration Algorithms for Computational and Experimental Structural Dynamics" (2016).Theses and Dissertations. 2667.

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Parametrically Dissipative Explicit Direct Integration Algorithms for

Computational and Experimental Structural Dynamics

by Chinmoy Kolay

Presented to the Graduate and Research Committee of Lehigh University

in Candidacy for the Degree of Doctor of Philosophy

in

Structural Engineering

Lehigh University

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c

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Approved and recommended for acceptance as a dissertation in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Date Dr. James M. Ricles Dissertation Advisor Accepted Date Committee Members: Dr. John L. Wilson Committee Chair Dr. Richard Sause Member Dr. Shamim N. Pakzad Member Dr. Oreste S. Bursi External Member

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Acknowledgements

I express my sincere gratitude to Professor James M. Ricles, my research advisor, for his valuable guidance, encouragements, and innumerable critical discussions during the course of this study. I would like thank my dissertation committee members Professor John L. Wilson (committee chair), Professor Richard Sause, Professor Shamim N. Pakzad, and Professor Oreste S. Bursi for their advice and guidance.

The research presented in this dissertation was conducted at the NEES@Lehigh Equip-ment Site and Engineering Research Center for Advanced Technology for Large Structural Systems (ATLSS), Department of Civil and Environmental Engineering, Lehigh University, Bethlehem, Pennsylvania.

Financial support from the Pennsylvania Department of Community and Economic Development through the Pennsylvania Infrastructure Technology Alliance and the National Science Foundation (NSF) is greatly appreciated. I sincerely acknowledge the financial support provided by the P.C. Rossin College of Engineering and Applied Science (RCEAS) fellowship, the Yen fellowship, and the Gibson fellowship through the Department of Civil and Environmental Engineering, Lehigh University.

I would like to thank the ATLSS laboratory technical staff for their contribution to the experimental part of this work, in particular Mr. Thomas Marullo, Mr. Darrick Fritchman, Mr. Edward Tomlinson, Mr. Carl Bowman, Mr. Peter Bryan, and Mr. Gary Novak.

I would like to thank Dr. Andreas Schellenberg for writing the OpenSees implementation

code for the KR-α method developed in this study, and Dr. Frank McKenna for providing

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developed in this study might not have been implemented into OpenSees.

I would like to express my gratitude to my parents for their unwavering support and encouragement. I wish I knew how to thank my six month old son Sourik for being so adorable. His continuous smile has been a great source of energy for writing my dissertation. Last but not the least, I would like to thank my wife Sumita for her love, support, and

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List of Tables

Table 2.1 Well known members of the Newmark method. . . 29

Table 2.2 Parameter values for OC method (Hilber and Hughes, 1978). . . 35

Table 3.1 Transfer function coefficients of the G-α method. . . 68

Table 3.2 Coefficients ofG0(z)of the G-α method. . . 72

Table 3.3 Transfer function coefficients of the proposed SE-α method. . . 78

Table 3.4 Transfer function coefficients of the proposed E-α method. . . 82

Table 4.1 Design criteria and parameter conditions for the KR-α and SSE-α-1 methods. . . 99

Table 4.2 Displacement and velocity at the first time step for the E-α, SE-α and G-α methods whenΩ→∞. . . 104

Table 4.3 Design criteria and parameter conditions for the MKR-α and SSE-α-2 methods. . . 106

Table 5.1 Coefficients ofG0(z)(see Figure 3.1) of the proposed SE-α method. . . 140

Table 5.2 Coefficients ofG0(z)(see Figure 3.1) of the proposed E-α method. . . . 141

Table 6.1 Summary of model-independent integration parameters for the single-parameter subfamilies of algorithms in terms ofρ_∞∗. . . 168

Table 6.2 Recommended procedure for selection ofρ_∞∗ for dynamic analysis using the proposed methods. . . 169

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Table 6.3 Nonlinear dynamic analysis procedure using the SSE-α-1 and SSE-α-2

methods. . . 173

Table 6.4 Nonlinear dynamic analysis procedure using the KR-α and MKR-α

methods. . . 176

Table 6.5 Errors (%) in roof displacement response of the shear frame in

Exam-ple 1 computed by the proposed methods with∆t=0.01 s andρ_∞∗ =1

for up to 2 s with respect to the exact solution. . . 178

Table 6.6 PE (%) andξ (%) for all three modes of the undamped shear frame in

Example 1. . . 181

Table 6.7 Computational time for collapse simulation of the six-story RC planar

frame. . . 198

Table 6.8 Computational time (s) for collapse simulation of the ten-story RC

building. . . 201

Table 8.1 NEE (%) for roof drift(θr), axial force(P)and moment(Mcol)at

first-story south side MRF column base, and moment at center of south

side roof RBS(M_RBS)calculated for two consecutive values ofρ∞with

MCE level ground motion. . . 267

Table 8.2 NRMSE (%) for roof drift(θr), axial force(P)and moment(Mcol)at

first-story south side MRF column base, and moment at center of south

side roof RBS(M_RBS)calculated for two consecutive values ofρ∞with

MCE level ground motion. . . 268

Table 9.1 Comparison of story drifts (%) from numerical simulations using the

DBE and MCE level ground motions where all flexibility-based

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Table 9.2 NEE (%) in roof displacement response under the MCE level ground motion from numerical simulations for a fixed number of element

itera-tions. . . 308

Table 9.3 NRMSE (%) in roof displacement response under the MCE level ground

motion from numerical simulations for a fixed number of element

itera-tions. . . 309

Table 9.4 Characterization test matrix showing the amplitude and frequency

com-binations used and the maximum velocities noted inside the table cells in mm/s (in./s). . . 315

Table 9.5 Identified nonlinear Maxwell damper model parameters from

character-ization tests. . . 318

Table 9.6 RTHS test matrix for investigation of stability and accuracy for the

KR-α and MKR-α methods. . . 339

Table 9.7 Story drifts (%) from RTHS using the MCE level ground motion with

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List of Figures

Figure 2.1 Variation of spectral radius (ρ) withΩfor various methods. . . 44

Figure 2.2 Variation of equivalent damping ratio

ξ

withΩfor various methods. 45

Figure 2.3 Variation of period error (PE) withΩfor various methods. . . 47

Figure 3.1 Block diagram representation of the G-αmethod for linear SDOF systems. 71

Figure 3.2 Root-loci of the characteristic equation of the G-α method for linear

SDOF systems with no inherent damping and various values ofρ∞. . . 73

Figure 3.3 Proposed model-based methods and their single-parameter subfamilies

of algorithms and nondissipative algorithms. . . 75

Figure 3.4 Variation of model-independent parametersγ,β,αm, andαf withρ∞

for the proposed methods. . . 90

Figure 4.1 Variation ofβ withγ for the proposed and G-α methods. . . 107

Figure 4.2 Classification of the E-α and SE-α methods inαm–αf space. . . 109

Figure 4.3 Variation of spectral radius(ρ)withΩfor the proposed and G-α methods.114

Figure 4.4 Loci of the eigenvalues in the complex plane for the proposed methods. 115

Figure 4.5 Variation of relative period error (PE) withΩfor the proposed and G-α

methods. . . 118

Figure 4.6 Variation of equivalent damping ratio

ξ

withΩfor the proposed and

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Figure 4.7 Variation of normalized amplitude coefficient (A/Ae), phase error (θe−

θ), coefficient of spurious root (c3) andc3λ3withΩfor the proposed

and G-α methods whenx0=1 and∆tx˙0=0. . . 125

Figure 4.8 Variation of normalized amplitude coefficient (A/Ae), phase error (θe−

θ), coefficient of spurious root (c3) andc3λ3withΩfor the proposed

and G-α methods whenx0=0, and∆tx˙0=1. . . 126

Figure 4.9 Overshoot response of the KR-α and MKR-α methods whenx0=1

and∆tx˙0=0. . . 129

Figure 4.10 Overshoot response of the SSE-α-1 and SSE-α-2 methods whenx0=1

and∆tx˙0=0. . . 130

Figure 4.11 Overshoot response of the KR-α and MKR-α methods when x0=0

and∆tx˙0=1. . . 131

Figure 4.12 Overshoot response of the SSE-α-1 and SSE-α-2 methods whenx0=0

and∆tx˙0=1. . . 132

Figure 4.13 Numerical dynamic magnification factor (Rnum_d ) for the proposed and

G-α methods compared with the exact solution (Rd) forξ =0.1. . . 134

Figure 5.1 Approximation of the incremental restoring force∆rn₋αf. . . 143

Figure 5.2 Block diagram representation of the proposed SE-α and E-α methods

for nonlinear structural behavior. . . 143

Figure 5.3 Root-locus of the open-loop transfer functionH(z)G0(z)of the proposed

methods with∆t =0.01 s when applied to a nonlinear SDOF system

withm=1, initial frequencyω =10π rad/s and damping ratioξ =0. . 147

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withm=1, initial frequencyω =1000πrad/s and damping ratioξ =0. 149

withm=1, initial frequencyω=1000πrad/s and damping ratioξ =0.1.150

Figure 5.7 Variation of critical values of Ω (Ωcrit) with kt/k for the proposed

methods applied to nonlinear SDOF systems. . . 155

Figure 6.1 Comparison of roof (DOF-3) displacement response of the shear frame

in Example 1 subjected to initial conditionX1₀= (φ1+φ2)inches for

the proposed methods withρ_∞∗ =1 and∆t=0.01 s. . . 178

Figure 6.2 Roof (DOF-3) displacement response of the shear building in Example 1

with two initial conditionsX1₀=φ1+φ2andX02=φ1+φ2+0.5φ3and

various values ofρ_∞∗. . . 180

Figure 6.3 Example 2 – Two story linear nonclassically damped shear frame: (a)

input ground acceleration, (b) first floor displacement response, and (c) second floor displacement response. . . 182

Figure 6.4 Normalized responses of the Duffing softening oscillator in Example 3. 184

Figure 6.5 Normalized responses of the Duffing stiffening oscillator in Example 3. 185

Figure 6.6 FE model of the structure in Example 4 . . . 186

Figure 6.7 Response of the system in Example 4. . . 187

Figure 6.8 FE model of the two-story moment resisting frame (MRF) in Example 5. 188

Figure 6.9 Comparison of the seismic response of the frame in Example 5, PD

model: (a) horizontal displacement of floor-2; and, (b) hysteretic re-sponse of Element 1 (see Figure 6.8(a)). . . 190

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Figure 6.10 Comparison of the seismic response of the frame in Example 5 at Node 2

(see Figure 6.8(a)), PD model: (a) displacements (D); (b) velocities (V);

and (c) accelerations (A). . . 191

Figure 6.11 Comparison of the seismic response of the frame in Example 5, PD model—damping forces at Node 2. . . 192 Figure 6.12 Comparison of the seismic response of the frame in Example 5, NPD

model—Floor-2 horizontal displacement obtained for various values of

ρ_∞∗ using: (a) KR-α; and (b) SSE-α-1 methods. . . 193

Figure 6.13 Comparison of the seismic response of the frame in Example 5, NPD model—normalized moment at Node 2 of Element 1 obtained for

vari-ous values ofρ_∞∗ using: (a) KR-α; and (b) SSE-α-1 methods. . . 193

Figure 6.14 Comparison of the seismic response of the frame in Example 5, NPD model—damping forces at Node 2. . . 194 Figure 6.15 Layout of the six-story three bay planar RC frame for column removal

analysis. . . 197 Figure 6.16 Vertical displacement of node N2 at the top of column C2 following its

removal. . . 198 Figure 6.17 Vertical displacement of node N1 at the top of column C1 following its

removal. . . 199 Figure 6.18 Ten-story three dimensional model of the RC frame building for seismic

collapse simulation. . . 200 Figure 6.19 Deformed shape of the ten-story RC frame at selected time instants

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Figure 7.1 Real-time hybrid simulation: (a) an example structural system subjected to seismic excitation, and (b) a schematic representation of real-time

hybrid simulation for the example structural system. . . 209

Figure 7.2 Magnitude of the complex frequency response function of the ATS

compensator for J =3 ∆t= ₁₀₂₄3 s and three sets of values of the

coefficients. . . 217

Figure 7.3 Hydraulic actuator power curves. . . 220

Figure 7.4 Real-time integrated control architecture of the RTMD facility, ATLSS

Center at Lehigh University. . . 222

Figure 7.5 Implementation of the SE-α method for RTHS. . . 228

Figure 7.6 Simulink block diagram of the implemented SE-α method for RTHS. . 229

Figure 7.7 Timing of various blocks in Simulink model of the SE-α and E-α

methods whenJ=4, that is,∆t= ₁₀₂₄4 s. . . 231

Figure 7.8 Implementation of the E-α method for RTHS. . . 234

Figure 7.9 Simulink block diagram of the implemented E-α method for RTHS. . . 236

Figure 7.10 Alternative implementation of the E-α method for RTHS. . . 238

Figure 7.11 Variation of relative period error (PE) withΩfor the proposed methods

applied to a linear SDOF system withce_eq=ηcc,keeq=k, andξ =0.1. . 242

Figure 7.12 Variation of equivalent damping ratio ξ

withΩ for the proposed

methods applied to a linear SDOF systemce_eq=ηcc,keqe =k, andξ =0.1.243

Figure 7.13 Variation of relative period error (PE) withΩfor the proposed methods

applied to a linear SDOF systemce_eq=c,ke_eq=ηkk, andξ =0.1. . . 245 Figure 7.14 Variation of equivalent damping ratio

ξ

withΩ for the proposed

methods applied to a linear SDOF system withce_eq=c,ke_eq=ηkk, and

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Figure 8.1 Prototype building structure. . . 251

Figure 8.2 Elevation view of 0.6-scale MRF. . . 253

Figure 8.3 Elevation view of 0.6-scale DBF. . . 254

Figure 8.4 Configuration for RTHS showing the FE model of the analytical

sub-structure, a schematic of the experimental subsub-structure, and rigid floor diaphragms connecting them. . . 255

Figure 8.5 Experimental substructure. . . 257

Figure 8.6 Comparison of normalized maximum response for proportional and

nonproportional damping under the DBE and MCE level ground motions from numerical simulations using the AA algorithm. . . 262

Figure 8.7 Floor displacements under DBE and MCE level ground motions from

numerical simulations. . . 264

Figure 8.8 Normalized member force time histories under DBE and MCE level

ground motions from numerical simulations. . . 265

Figure 8.9 Variation of absolute maximum responses withρ∞under MCE level

ground motion from numerical simulations. . . 267

Figure 8.10 Synchronization subspace plots for target(xt)and measured(xm)floor

displacements from a typical RTHS using MCE level ground motion. . 270

Figure 8.11 Effects of different cases of extrapolation on maximum story drift from RTHS using DBE level ground motion. . . 271 Figure 8.12 Effects of perturbation inCe_eq,Ke_eq, and bothCe_eqandKe_eqon maximum

story drift from RTHS using DBE level ground motion. . . 272 Figure 8.13 Target floor displacement response under DBE and MCE level ground

motions from RTHS. . . 273 Figure 8.14 Normalized member force time histories in analytical substructure under

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Figure 8.15 Hysteretic response of members in analytical substructure under DBE and MCE level ground motions from RTHS. . . 275 Figure 9.1 Forces (s,D(x)) and deformations (v,d(x)) at the element and section

level for a two dimensional frame element in simply supported basic system. . . 282

Figure 9.2 Implementation of flexibility-based element state determination at time

step(n+1)for application to RTHS utilizing an explicit direct

integra-tion algorithm. . . 287

Figure 9.3 Two-story reinforced concrete prototype building with nonlinear viscous

dampers. . . 288

Figure 9.4 Design section details for the prototype test frame with a nonlinear

viscous damper in the second story. . . 290

Figure 9.5 Prototype test frame: (a) RTHS configuration showing FE model of

analytical substructure, (b) plastic hinge integration, and (c) fiber section discretization for flexibility-based elements. . . 291

Figure 9.6 Modified Kent and Park concrete material model. . . 295

Figure 9.7 Hysteretic concrete stress-strain relation. . . 295

Figure 9.8 Test setup for characterization tests and RTHS. . . 297

Figure 9.9 Comparison of floor displacement response from numerical simulations

using the MCE level ground motion where all flexibility-based elements

converged withEtol=10−16. . . 303

Figure 9.10 Comparison of moment-rotation response from numerical simulation using the MCE level motion where all flexibility-based elements

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Figure 9.11 Comparison of moment-curvature response from numerical simulation using the MCE level motion where all flexibility-based elements

con-verged withEtol=10−16. . . 304

Figure 9.12 Maximum number of element iterations required for the KR-α/MKR-α

method to satisfyEtol=10−16 withρ_∞∗ =1 for numerical simulation

under the MCE level ground motion. . . 305 Figure 9.13 Roof displacement response under the MCE level ground motion from

numerical simulations using the KR-α method with a fixed number of

element iterations. . . 306

Figure 9.14 Roof displacement response under the MCE level ground motion from

numerical simulations using the MKR-α method with a fixed number

of element iterations. . . 307 Figure 9.15 Energy increment at the end of maximum number of iterations reached

for the first story column element on the south side from numerical

simulations using the KR-α method withρ_∞∗=1. . . 309

Figure 9.16 Energy increment at the end of maximum number of iterations reached for the first story column element on the south side from numerical

simulations using the MKR-α method withρ_∞∗=1. . . 310

Figure 9.17 Normalized energy increment at the end of maximum number of itera-tions reached for the first story column element on the south side from

numerical simulations using the KR-α method withρ_∞∗ =1. . . 311

Figure 9.18 Normalized energy increment at the end of maximum number of itera-tions reached for the first story column element on the south side from

numerical simulations using the MKR-α method withρ_∞∗ =1. . . 311

Figure 9.19 Moment-rotation response under the MCE level ground motion from

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Figure 9.20 Moment-curvature response under the MCE level ground motion from

numerical simulation usingmaxIter=2 and CO = Yes. . . 313

Figure 9.21 Input actuator displacement profile for characterization tests. . . 314 Figure 9.22 Nonlinear Maxwell model for the experimental substructure (nonlinear

damper). . . 315 Figure 9.23 Simulink model for the solution of the nonlinear ordinary differential

Equation (9.38). . . 318 Figure 9.24 Comparison of characterization test data with model prediction for input

displacement amplitude of 25.4 mm (1 in.). . . 319

Figure 9.25 Comparison of characterization test data with model prediction for input

displacement amplitude of 76.2 mm (3 in.). . . 319

Figure 9.26 Determination of equivalent damping(C_eq)and stiffness(K_eq)

coeffi-cients for the experimental substructure. . . 321 Figure 9.27 Variation ofK_eq(ωe)andCeq(ωe)with harmonic excitation frequencyωe

for the equivalent Kelvin-Voigt model corresponding to the nonlinear Maxwell damper model parameters in the first row of Table 9.5. . . 323 Figure 9.28 Identification of initial values of ATS compensator coefficients from

predefined BLWN test. . . 325 Figure 9.29 Verification of identified initial values of ATS compensator coefficients

from predefined BLWN test. . . 327

Figure 9.30 Synchronization subspace plot for target(xt) and measured(xm)

dis-placements from predefined BLWN test. . . 328 Figure 9.31 Verification of identified initial values of ATS compensator coefficients

from RTHS using BLWN, KR-α method withρ_∞∗ =0.5, ωe =ω1 for

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dis-placements from RTHS using BLWN, KR-α method with ρ_∞∗ =0.5,

andω_e=ω1forCeq(ωe)andKeq(ωe), andmaxIter=1 for all flexibility-based elements.. . . 331

dis-placements from RTHS using the MCE level ground motion with

ρ_∞∗ =0.25, ωe =ω1 for Ceq(ωe), and Keq(ωe) and maxIter=2 for all flexibility-based elements. . . 332 Figure 9.34 Damper displacement from RTHS using the MCE level ground motion

withρ_∞∗ =0.50,ωe =ω1 forCeq(ωe)andKeq(ωe), andmaxIter=2 for all flexibility-based elements. . . 333 Figure 9.35 Variation of β∆t2Keq(ωe), γ∆tCeq(ωe), and γ∆tCeq(ωe) +β∆t

2_K

eq(ωe) withωe corresponding to the variation ofKeq(ω)e andCeq(ωe)presented

in Figure 9.27. . . 337

Figure 9.36 Damper force-displacement hysteresis from RTHS using various values

ofωe, andρ∞∗ for the KR-α and MKR-α methods withmaxIter=2 for

all flexibility-based elements. . . 340 Figure 9.37 Normalized energy increment at the end of maximum number of

itera-tions reached for the first story column element on the south side from RTHS usingρ_∞∗ =0.75 andωe=0 forCeq(ω)e andKeq(ωe). . . 342

Figure 9.38 Comparison of roof displacement time history from RTHS usingρ_∞∗ =0.75

andωe=0 forCeq(ωe)andKeq(ωe)with offline simulation using the mea-sured damper force and allowing flexibility-based elements to converge

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List of Symbols

a A vector of the ATS adaptive coefficients

a₀,a₁,a₂ Adaptive coefficients of the ATS compensator

a_K Stiffness proportionality constant in Rayleigh damping formulation

aM Mass proportionality constant in Rayleigh damping formulation

avq Displacement transformation matrix from the global coordinate to the basic

system

b(x) Force interpolation function

c Damping coefficient of an SDOF system

ce_eq Estimated equivalent damping coefficient of the experimental substructure

in numerical hybrid simulation

ce₁,ce₂ Constants of exact free vibration solution

c0₁, . . . ,c0₃ Constants of discrete solution for free vibration

c₁, . . . ,c₃ Constants of discrete solution for free vibration

d A model-based parameter used to express the local truncation error for the

E-α and SE-α methods in a compact form

dα

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d_b Diameter of the longitudinal reinforcement bars in meters

d₀New, . . . ,d₃New Characteristic equation coefficients of the proposed methods

d₀0, . . . ,d₃0 Denominator coefficients of open-loop transfer functionG0(z)

d₀, . . . ,d₃ Denominator coefficients of a discrete transfer function

d(x) Vector of section deformations of a flexibility-based element

dr(x) Vector of section residual deformations of a flexibility-based element

f(t) Excitation force applied to an SDOF system in continuous time

fC Force in the dashpot in the nonlinear Maxwell damper model

f_D Damper force

˙

fD Time derivative of the damper force fD

f_De Experimentally measured damper force

f_Dp Predicted damper force using the nonlinear Maxwell model

f_D_thr Damper force at the small threshold velcity ˙u_C_thr

fK Force in the spring in the nonlinear Maxwell damper model

f_c Concrete stress

f_c0 Concrete compressive cylinder strength (MPa)

f_n Undamped natural frequency of thenth mode

f(x) Section tangent flexibility matrix of a flexibility-based element

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g Acceleration due to gravity

ηc Damping coefficient factor

ηk Stiffness coefficient factor

h0 Width of concrete core measured to outside of stirrups

i Unit imaginary number equal to√₋1

j Element level iteration index in the state determination of a flexibility-based

element

k (a) Stiffness coefficient of a linear SDOF system

(b) Initial linear elastic stiffness of a nonlinear SDOF system

k1,k2 Stiffness coefficients of a Duffing oscillator

ke_eq Estimated equivalent initial elastic stiffness coefficient of the experimental

substructure in numerical hybrid simulation

kt Tangent stiffness of an SDOF system

l Difference between the weighted excitation force and the restoring force

l_p Effective plastic hinge length in meters

l_pI,l_pJ Plastic hinge lengths at endIandJ of a flexibility-based element

m Mass of an SDOF system

maxIter Maximum number of element level iterations for the state determination of a flexibility-based element

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n_{f be} Number of flexibility-based elements in the analytical substructure

n0₀, . . . ,n0₃ Numerator coefficients of open-loop transfer functionG0(z)

n0, . . . ,n3 Numerator coefficients of a discrete transfer function

q Displacement vector of a flexibility-based element in the global coordinate

system

q An user defined parameter for the ATS data window selection

r (a) Magnitude of complex conjugate roots and is equal to√σ2+ε2

(b) Restoring force of a nonlinear SDOF system

u(t) Displacement of an SDOF system in continuous time

˙

u(t) Velocity of an SDOF system in continuous time

¨

u(t) Acceleration of an SDOF system in continuous time

s Vector of element forces of a flexibility-based element in the basic system

s Complex variable used in Laplace transform

s_h Center to center spacing of stirrups or hoops

t Continuous time variable

t_[k] Time variable corresponding to time indexkand is equal tokδt

t_n Time variable corresponding to time indexnand is equal ton∆t

t_n+(j)₁ Time variable corresponding to the jth substep of the(n+1)th _{time step}

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u_C Deformation of the dashpot in the nonlinear Maxwell damper model

uD Total damper deformation in the nonlinear Maxwell damper model

uK Deformation of the spring in the nonlinear Maxwell damper model

˙

u_C Velocity of the dashpot in the nonlinear Maxwell damper model

˙

u_D Damper velocity

˙

u_K Velocity of the spring in the nonlinear Maxwell damper model

(ust)₀ Maximum static deformation

v Deformation vector of a flexibility-based element in the basic system

vr Vector of element residual deformations of a flexibility-based element

x Displacement of an SDOF system in discrete time

xc A vector of compensated displacements used in identification of the ATS

adaptive coefficients

xc Compensated displacement

˙

x Velocity of an SDOF system in discrete time

¨

x Acceleration of an SDOF system in discrete time

bx¨ Weighted acceleration for the proposed methods

xh Homogeneous solution of displacement difference equation

xm A rectangular matrix of measured displacements, velocities, and

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xm Actuator/specimen measured displacement

xp Particular solution of displacement difference equation

xt Actuator/specimen target displacement

z Complex variable

z₁, . . . ,z₃ Roots of the characteristic equation of an algorithm

A (a) Amplitude of discrete free vibration solution

(b) Cross sectional area

A (a) Amplification matrix

(b) A model-based parameter in the implementation of the E-α method for

RTHS

Ae Amplitude of exact free vibration solution

A₁ 1₂ of trace of amplification matrixA

A2 Sum of principal minors of amplification matrixA

A₃ Determinant of amplification matrixA

B (a) A matrix used in linearized stability analysis

RTHS

C (a) Damping matrix

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Cele₂ Tangent stiffness proportional damping matrix of the flexibility-based

ele-mentele

CD Damper coefficient

CE Damping matrix for explicit group of elements in the OS methods

CI Damping matrix for implicit group of elements in the OS methods

Ca_ID Inherent damping matrix for the analytical substructure

Ce_ID Inherent damping matrix for the experimental substructure

CIP Damping matrix for determination of integration parameters in an RTHS

CO Cary over unbalanced section forces and apply the corrections

C1 Damping matrix of the analytical substructure excluding contribution of the

flexibility-based elements

Ce Damping matrix of the experimental substructure used for restoring force

extrapolation

Ca_eq Equivalent damping matrix of the analytical substructure

Ce_eq Equivalent damping matrix of the experimental substructure

C_eq(ωe) Frequency(ω)e dependent equivalent damping coefficient of the Maxwell

model

C_lin Linearized dashpot coefficient in the nonlinear Maxwell damper model

Cs The seismic response coefficient according to ASCE 7

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C∗_j Modal damping coefficient for the jth mode

D A model-based parameter used to present the amplification matrix of the

SE-α method in a compact form

D (a) A matrix used in linearized stability analysis

RTHS

DU(x) Vector of unbalanced section forces of a flexibility-based element

D(x) Vector of section forces of a flexibility-based element

E (a) Modulous of elasticity

(b) Total mechanical energy of an SDOF system under free vibration

EI Energy increment in the state determination of a flexibility-based element

Etol User defined tolerance on the normalized energy increment in the state

determination of a flexibility-based element

F (a) Excitation force vector in discrete time

(b) Tangent flexibility matrix of flexibility-based element

FD(s) Laplace transfrom of damper force fD(t)

FE Excitation force vector for explicit group of elements in the OS methods

FI Excitation force vector for implicit group of elements in the OS methods

FID Inherent damping force

b

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Fob j Objective function in the damper parameter identification using the PSO algorithm

F(s) Laplace transform of input excitation f(t)

F(t) Excitation force vector in continuous time

F_y Yield strength of steel material

F(z) z-transforms of f_n+₁

G₁(z) Open-loop discrete transfer function for nonlinear SDOF systems

G_ATS(iω)_e Complex frequency response function of the ATS compensator

GATS(z) Discrete transfer function of the ATS compensator

G_CLNL(z) Closed-loop discrete transfer function of the proposed methods applied to

nonlinear systems

G0(z) Open-loop discrete transfer function

Gs(s) Continuous transfer function of an SDOF system

Gs(iωe) Complex frequency response function of an SDOF system

G(z) Discrete transfer function of an integration algorithm

G_z(z) Identical toG(z). The subscriptzis introduced to distinguish it fromG_s(s)

in Chapter 4

H₁(z) Discrete transfer function

H₂(z) Discrete transfer function

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I Moment of inertia

I Identity matrix

J An integer number of substeps in RTHS

K (a) Stiffness matrix of a linear system

(b) Initial linear elastic stiffness matrix of a nonlinear system (c) Tangent stiffness matrix of a flexibility-based element

KD Coefficient of the spring in the nonlinear Maxwell damper model

KE Stiffness matrix for explicit group of elements in the OS methods

KI Stiffness matrix for implicit group of elements in the OS methods

KI Initial elastic stiffness matrix

KIP Stiffness matrix for determination of integration parameters in an RTHS

Ka_I Initial elastic stiffness matrix of the analytical substructure

Ka_I∗ (a) Initial elastic stiffness matrix of the analytical substructure excluding the

contribution of the elements undergoing significant inelastic deformations in RTHS of the steel building

(b) Initial elastic stiffness matrix of the analytical substructure excluding the flexibility-based elements in RTHS of the RC building

K∗_I Initial elastic stiffness matrix of a nonlinear system excluding the

contribu-tion of the elements undergoing significant inelastic deformacontribu-tions

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Ka_Rayleigh Stiffness matrix of the analytical substructure for constructing Rayleigh damping matrix

Ke_Rayleigh Stiffness matrix of the experimental substructure for constructing Rayleigh damping matrix

K_S(ωe) Frequency(ω)e dependent storage modulus of the Maxwell model

KT Tangent stiffness matrix

Ka_T Tangent stiffness matrix of the analytical substructure

Kc A factor which accounts for the strength increase in concrete due to

con-finement

Ke Stiffness matrix of the experimental substructure used for restoring force

extrapolation

Ke_eq Equivalent initial elastic stiffness matrix of the experimental substructure

K_eq(ωe) Frequency(ωe)dependent equivalent stiffness coefficient of the Maxwell

model

K∗ Modal stiffness matrix

K∗(iω_e) Complex dynamic modulus of the Maxwell model

K∗_j Modal stiffness coefficient for the jthmode

L Length of a flexibility-based element

L Laplace transform operator

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M Moment

M (a) Mass matrix

(b) Analytically defined mass matrix in an RTHS

ME Mass matrix for explicit group of elements in the OS methods

MI Mass matrix for implicit group of elements in the OS methods

MIP Mass matrix for determination of integration parameters in an RTHS

Me Mass matrix of the experimental substructure

b

M1 A system matrix calculated only once in a numerical simulation and RTHS

using the proposed methods

b

M2 A system matrix calculated only once in a numerical simulation and RTHS

using the proposed methods

Mj jth floor seismic mass lumped on the lean-on column

Mp Plastic moment of a wide-flange section

M∗ Modal mass matrix

M∗_j Modal mass coefficient for the jth mode

N Total number of time steps

NEE Normalized energy error

NEI Normalized energy increment in the state determination of a

flexibility-based element

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P Axial force

PE Relative period error

Pj jth floor axial force on the lean-on column

P_y Member yield strength

R Restoring force vector

Ra Analytically determined restoring force vector

R_d Dynamic magnification factor

Rnum_d Numerical dynamic magnification factor

Re Experimentally measured restoring force vector

Rm Measured restoring force vector

S_D₁ Design spectral response acceleration parameter at a period of 1 s

S_DS Design spectral response acceleration parameter in the short period range

T Natural period of an SDOF system

T₀ Equal to 0.2SD1

SDS

T_S Equal to SD1

SDS

T_a The approximate fundamental period according to ASCE 7

T Apparent natural period of an SDOF system

T_l (a)lth coefficient of the lth derivative ofu(t)in the local truncation error

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(b) Period of thelth mode

T_min Period of the highest mode in an MDOF system

U(t) Displacement vector in continuous time

U_D(s) Laplace transfrom of damper displacementu_D(t)

˙

U(t) Velocity vector in continuous time

¨

U(t) Acceleration vector in continuous time

U(s) Laplace transform of output displacementu(t)

Ve Velocity vector of the ramp generator

X Displacement vector in discrete time

Xa Displacement vector associated with the analytical substructure DOFs

X_a(z) z-transforms of ¨x_n+₁

Xc(z) z-transform of the compensated displacementxc_[k]

˙

X Velocity vector in discrete time

˙

Xa Velocity vector associated with the analytical substructure DOFs

¨

X Acceleration vector in discrete time

b¨

X Weighted acceleration vector for the proposed methods

˙

Xe Velocity vector associated with the experimental substructure DOFs

b˙

X Velocity-like vector for the E-α method

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X∗ Modal coordinates associated with displacement ˙

X∗ Modal coordinates associated with velocity

¨

X∗ Modal coordinates associated with acceleration

e

X Predictor displacement vector

e˙

X Predictor velocity vector

Xt(z) z-transform of the target displacementxt_[k]

Xv(z) z-transforms of ˙xn+1

X(z) z-transform ofxn+1

Y A vector of numerical state variables in the recurrance relationship

Z Strain softening slope in the stress-strain envelope of concrete material

Z z-transform operator

α Velocity exponent in the force-velocity relationship of the nonlinear viscous

damper

α

αα0, . . . ,ααα3 Model-based integration parameter matrices

α0, . . . ,α3 Model-based scalar integration parameters

αf Integration parameter for dissipative methods

αm Integration parameter for dissipative methods

α

αα∗₁, . . . ,ααα∗₃ Model-based integration parameters of the proposed methods in modal

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α₁∗_j, . . . ,α₃∗_j Model-based integration parameters of the proposed methods for the jth

mode

β Integration parameter

γ Integration parameter

δ (a) An integration parameter equal toγ−2β in the CSE method

(b) Indicates small variation in displacement, velocity, and acceleration

δt Servo-controller sampling period

ε Imaginary part of a complex root

ε0 Concrete strain at the maximum stress

εc Concrete strain

εp Strain corresponding to complete unloading in the concrete stress-strain

hysteretic behavior

εr Strain corresponding to the point on the concrete stress-strain envelope

curve at which unloading starts

η (a) A parameter used to compare with the ratio k_kt for stability of a nonlinear

SDOF system

(b) Post-yield stiffness ratio

θ (a) Collocation parameter

(b) Phase of discrete free vibration solution

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θr Roof drift

θ∗ Collocation parameter of the OC method

κ Stability amplification parameter of the CSE method

λj jth eigvenvalue of the amplification matrixA

λ1,2 Complex conjugate eigenvalues of the amplification matrixA(principal

roots)

λ₁∞_,₂ Principal rootsλ1,2in the limitΩ→∞

λ3 Real eigenvalue of the amplification matrixA(spurious real root)

λ₃0 Spurious rootλ3in the limitΩ→0

λ₃∞ Spurious real rootλ3in the limitΩ→∞

ξ Inherent damping ratio of an SDOF system

ξ Equivalent damping ratio of an SDOF system

ξ_c Equivalent damping ratio corresponding toξc

ξc Inherent damping ratio corresponding to frequencyωc

ξ_∞ Equivalent damping ratio in the limitΩ→∞

ξn Modal damping ratio for thenth mode

ρ Spectral radius of amplification matrixA

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ρs Ratio of the volume of hoop reinforcement to the volume of concrete core measured to outside of stirrups

ρ_∞∗ User defined free parameter related toρ∞; a value of which provides the

same high-frequency dissipation(ξ_∞)in all of the proposed methods

σ Real part of a complex root

τ (a) Local truncation error

(b) Relaxation time in the Maxwell damper model

φ Phase of the complex frequency response functionGs(iωe)

φnum Phase of the discrete transfer functionGz(iω)e

φj Eigenvector of the jth mode

ω (a) Natural frequency of a linear SDOF system

(b) Initial linear elastic natural frequency of a nonlinear SDOF system

ωIP Frequency corresponding to stiffness coefficientkeeq

ω1 Initial linear elastic undamped fundamental frequency

ω Apparent natural frequency of an SDOF system

ωc Frequency of the highest contributing mode of interest

ωD Damped natural frequency

ωD Damped apparent natural frequency

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ωn Natural frequency of thenthmode

ωs Frequency above which modes can be considered as spurious

e

ω Frequency of a single harmonic excitation

∆ Denotes increment

∆Fαf(z) z-transform of∆fn−αf

∆L(z) z-transform of∆ln

∆Xαf(z) z-transform of∆xn−αf

∆crit Critical value of∆t for stability

∆t Integration time step size

Φ

ΦΦ Modal matrix

Ω Product of natural frequencyω and time step∆t

ΩIP A value ofΩfor determination of model-based integration parameters

Ω Product ofω and∆t

Ωbif Value ofΩat which complex conjugate eigenvalues bifurcte and become

real

Ωc Value ofΩcorresponding toωc

Ωcrit Critical value ofΩfor stability

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Ωmax Value of Ω corresponding to the highest mode frequency (ωmax) of an

MDOF system

Ωs Value ofΩcorresponding toωs

ξIP Damping ratio corresponding to damping coefficientceeqand requencyωIP

[·] Jump or undivided forward difference operator used in Chapter 5

h·i Mean value operator used in Chapter 5

≺ B_≺A(i.e.,A₋B0) meansA₋Bis positive definite

AB(i.e.,A₋B0) meansA₋Bis positive definite

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Abstract

Dynamic response of linear and nonlinear structural systems subjected to any arbitrary excitation is often determined by solving the equations of motion using a direct integration algorithm. Numerous direct integration algorithms have been developed in the past, which are generally classified as either explicit or implicit. Explicit algorithms are generally only conditionally stable, whereas implicit algorithms can provide unconditional stability. Implicit algorithms that are unconditionally stable and have some form of numerical dis-sipation are preferred for inertial problems where only a small number of low-frequency modes dominate the response. Nevertheless, implicit algorithms require an iterative solution procedure for nonlinear systems and can be computationally intense.

Because explicit algorithms are non-iterative, they are preferred for hybrid simulation (HS) in earthquake engineering, an experimental method where the dynamic response of a structural system is simulated from coupled domains of physical and analytical substructures. Explicit algorithms are even more preferred for HS performed at the true time scale, known as real-time hybrid simulation (RTHS). For such simulations involving a large number of degrees of freedom, the need for unconditional stability and numerical dissipation within an explicit formulation is well recognized. Consequently, a new class of ‘model-based’ explicit methods evolved which can achieve unconditional stability through the use of model-based integration parameters. However, limited studies were conducted to assess the accuracy of model-based algorithms under nonlinear structural response. Furthermore, the studies on dissipative model-based algorithms and assessment of their efficacy in eliminating spurious participation of higher modes through actual tests are also limited.

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This research is focused on developing model-based algorithms for application to numer-ical simulation and RTHS of inertial problems. Two new families of model-based algorithms,

namely, the semi-explicit-α (SE-α) and explicit-α (E-α) methods, are developed where the

former uses an explicit displacement and implicit velocity formulation, and the latter uses explicit formulations for both displacement and velocity. These two methods are further analyzed and four single-parameter subfamilies of algorithms having second-order accuracy, unconditional stability, and controllable numerical dissipation with an optimal combination of high-frequency and low-frequency dissipation are developed. In particular, the

single-parameter semi-explicit-α-1 (SSE-α-1) and single-parameter semi-explicit-α-2 (SSE-α-2)

methods from the SE-α method, and Kolay-Ricles-α (KR-α) and modified-Kolay-Ricles-α

(MKR-α) methods from the E-α method are developed. Numerical characteristics of these

four methods are studied for free and force vibrations of linear systems and the advantages and limitations of these methods are presented. The results show that the controllable numerical dissipation provided by these method negligibly influences the low-frequency mode response while providing sufficient high-frequency dissipation to eliminate spurious

participation of higher modes. The analysis further show that the SSE-α-1 method possesses

the best numerical characteristics for linear systems compared with the other three methods.

When no numerical dissipation is used, the KR-α method shows some unusual tendency to

overshoot for higher modes which is however controlled with numerical dissipation. The

MKR-α method, which is designed to address this issue, further improves the overshoot

characteristics of the KR-α method. Stability characteristics of the proposed methods

applied to nonlinear systems are investigated using the concept of linearized stability and the necessary stability conditions are derived. The results show that a stiffness softening-type response is a necessary (may not be sufficient) condition for unconditional stability to be

achieved. The SSE-α-2 method compared with the SSE-α-1 method, and the MKR-α

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characteris-tics for nonlinear systems. The enhanced stability characterischaracteris-tics of the SSE-α-2 method is

achieved at the cost of increased overshoot for higher frequencies.

Efficient implementation procedures are presented for linear and nonlinear dynamic analysis using the proposed methods. Representative numerical examples of linear and nonlinear systems are presented to complement the analytical findings on the numerical

characteristics of the proposed methods. The results show that the SSE-α-1 method produces

large damping forces for inelastic seismic response analysis of frame structures, which lead to an inaccurate solution. The reason behind this is found to be associated with the

semi-explicit formulation of the method. The KR-α method, however, produces an accurate

solution for this type of problem. Application of the KR-α method for structural collapse

simulation is presented. The results indicate that the KR-α method is a computationally

efficient and accurate method for such applications.

Using the KR-α method, RTHS of a three-story 0.6-scale prototype steel building

with nonlinear elastomeric dampers are conducted with a ground motion scaled to the design basis and maximum considered earthquake hazard levels. The RTHS configuration consists of a moment resisting frame, gravity system, and seismic tributary masses modeled as the analytical substructure, and a damped-braced frame modeled as the experimental substructure. Inherent damping in the analytical substructure is defined using a form of nonproportional damping model. Through numerical simulation using an implicit algorithm it is found that the nonproportional damping model produces an accurate result that is comparable with that obtained using mass and tangent stiffness proportional damping. However, the nonproportional damping model when used with explicit integration algorithms can require a small time step to achieve the desired accuracy in an RTHS involving a structure with a large number of degrees of freedom. Restrictions on the minimum time step exist in an RTHS that are associated with the computational demand. Integrating the equations of motion in an RTHS with too large of a time step can result in spurious high-frequency

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oscillations in the member forces for elements of the structural model that undergo inelastic deformations. The problem is circumvented by introducing the controllable numerical

energy dissipation provided by the KR-α method. The results show that controllable

numerical energy dissipation can significantly eliminate spurious participation of higher modes and produce exceptional RTHS results.

Using the KR-α and MKR-α methods, RTHS of a two-story reinforced concrete (RC)

special moment resisting frame (SMRF) with a nonlinear viscous damper in the second story are conducted with a ground motion scaled to the maximum considered earthquake hazard level. The RC SMRF and the seismic masses are modeled analytically and the nonlinear viscous damper is modeled physically in the laboratory. To better model the complex hysteretic behavior of RC members, flexibility-based elements are considered. A new implementation scheme for the state determination of flexibility-based elements is developed based on a fixed-number of iterations for application to RTHS using explicit algorithms. The influence of unbalanced section forces which exist because of the limited number of iterations are studied numerically. The results show that the carrying over of the unbalanced section forces to the next integration time step and applying the necessary corrections can lead to an accurate solution with a small number of element level iterations. Inherent damping in the analytical substructure is modeled using a combination of mass, initial stiffness, and tangent stiffness proportional damping, where tangent stiffness is used for all flexibility-based elements. The equivalent stiffness and damping coefficients of the experimental substructure, which are required to determine the model-based integration parameters, are estimated based on a nonlinear Maxwell damper model and found to be frequency dependent. The parameters of the Maxwell model are identified from a suit of predefined sinusoidal characterization tests conducted at various excitation frequencies. The influence of the frequency dependency of the model-based parameters, which is due to the experimental substructure, on the stability and accuracy of RTHS results are investigated based on the test

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results. The test data shows that controllable numerical energy dissipation provided by the

KR-α and MKR-α methods plays an important role on the stability characteristics of an

RTHS. Accuracy of RTHS results with respect to this frequency dependency and numerical energy dissipation are assessed and found to be not sensitive. The influence of fixed-number of element iterations are assessed using RTHS results. The investigation shows that the proposed element implementation is efficient and accurate for application to RTHS.

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Chapter 1

Introduction

1.1

Motivation

Extreme events, both natural (e.g., earthquake, strong wind) and man-made (e.g., blast, explosions), continue to make civil infrastructure vulnerable to damage and demonstrate the fragility of our built environment. In order to prevent a catastrophe and make our civil infrastructure more resilient, the civil engineering community needs to improve their knowledge and understanding of the response of existing and new innovative structural sys-tems subjected to such extreme dynamic load events. Numerical analysis and experimental techniques to understand dynamic response of structures prove to be indispensable.

In the field of earthquake engineering, the hybrid simulation (HS) technique which combines both numerical and experimental methods has been introduced and developed over the past decades to reproduce seismic effects on structures. In an HS, a complete structural system is divided into coupled domains of analytical (numerical) and experimental substructures and the response of the hybrid system subjected to a seismic excitation is simulated based on an extended time scale. Due to the growing interest in rate dependent supplemental energy dissipation devices as an effective and efficient means of hazard mitigation, the HS technique has been extended to be performed in true time scale, which led to the real-time hybrid simulation (RTHS) method. During the past decades HS and RTHS have been shown to be viable alternative methods for simulating seismic response of civil infrastructure.

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the initial boundary value problem of second order hyperbolic partial differential equations (PDEs). These PDEs are first discretized into space either using finite element or finite differ-ence methods, which leads to the approximate initial value problem in structural dynamics. This initial value problem consists of a set of coupled second order ordinary differential equations, known as the semi-discrete equations of motion, and the corresponding initial conditions. Mathematical modeling of naturally discrete structural dynamic systems directly leads to the semi-discrete equations of motion. For linear and nonlinear systems subjected to arbitrarily varying excitations (e.g., earthquake loads), the solution of the equations of motion is conveniently obtained numerically in discrete time using a direct integration, also called time stepping, algorithm. In HS and RTHS, direct integration algorithms are also used to solve the temporally discretized equations of motion. This study is focused on the development of direct integration algorithms and their application to computational and experimental research (e.g., HS and RTHS) in structural dynamics. The experimental work presented in this study was conducted at the NEES@Lehigh Equipment Site, which has become currently the NHERI Lehigh Experimental Facility, located in the ATLSS Engineering Research Center of Lehigh University.

1.2

Problem Statement

Numerical methods for solution of the semi-discrete initial value problem in structural dynamics has a long history of development. Since the 1950s several direct integration algorithms (e.g., Houbolt, 1950; Newmark, 1959; Wilson, 1968; Hilber et al., 1977; Wood et al., 1980; Bazzi and Anderheggen, 1982; Chung and Hulbert, 1993; Chang, 2002; Chen and Ricles, 2008a) have been developed. These methods are generally classified as either

explicitorimplicit. The explicit algorithms are generally onlyconditionally stable, whereas

the implicit algorithms can possess unconditional stability. For inertial problems, also

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conditionally stable algorithms because in the former a relatively large time step can be em-ployed based on the desired accuracy for the participating low-frequency modes. Therefore, implicit algorithms that are unconditionally stable are well suited for such problems. In addition to unconditional stability, some form of numerical dissipation is useful and often required to reduce any spurious participation of high-frequency modes. Some examples of unconditionally stable implicit algorithms with controllable numerical dissipation include the methods developed by Hilber et al. (1977), Wood et al. (1980), and Chung and Hulbert (1993). However, the implicit algorithms can be computationally intense, although they re-quire a smaller number of time steps provided they are unconditionally stable. Furthermore, for highly nonlinear problems implicit algorithms often encounter convergence issues to satisfy equilibrium at the current time step associated with an iterative solution procedure. These convergence issues are often solved by resorting to a different nonlinear iterative scheme and/or reducing the time step size in a problem specific manner. For such problems researchers have also tried to use explicit algorithms to avoid nonlinear iterations despite being forced to use a smaller time step that is governed by the conditional stability limit of such algorithms. In the past decade, unconditionally stable explicit algorithms (e.g., Chang, 2002; Chen and Ricles, 2008a) have been developed. However, limited studies have been carried out on the application of these algorithms to highly nonlinear problems. Furthermore, these algorithms do not possess any numerical dissipation.

Due to the growing interest in performance assessment and fragility estimates of the civil infrastructure subjected to extreme events (e.g., earthquakes), researchers are using nonlinear dynamic finite element analysis technique more than ever. In the field of earth-quake engineering, numerical simulations of structures subjected to increasing magnitude of seismic hazards up to, or near, collapse are often used to assess collapse potential. This analysis technique, generally known as incremental dynamic analysis (IDA), often requires hundreds to thousands of nonlinear time history analysis to be carried out. Consequently, the

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huge computation effort required for such analysis, in addition to the aforesaid convergence issues have become important aspects to be considered. Due to the recent advances in com-putational power, e.g., super computers with multi-core processors and parallel computing capabilities, the computation time can be reduced significantly. Nevertheless, access to such super computing facilities and availability of parallel computing features in standard finite element programs can be limited. Therefore, studies need to be conducted to develop alternative numerical methods to solve such computationally intense problems in an efficient manner.

During late 1960s and early ‘70s, the HS technique was introduced to reproduce seismic effects on civil infrastructure in an effective and efficient manner. Subsequently, the concept of substructuring was introduced into the HS method, and it was also extended to RTHS method, as mentioned earlier. In HS and RTHS, a complete structural system is divided into two coupled domains consisting of analytical and experimental substructures, as mentioned above. The equations of motion for the complete hybrid system are solved using a step-by-step direct integration algorithm and the displacements are imposed on the experimental and analytical substructures. The restoring forces from the experimental substructure are then measured while a state determination is performed to compute the analytical substructure restoring forces. These combined restoring forces are then fed back to the equations of motion for determining the solution for the next time step. Explicit algorithms are generally preferred for HS and RTHS so as to avoid nonlinear iterations that can lead to undesired hysteresis due to loading and unloading of the experimental substructure during the iteration process. However, explicit algorithms are generally only conditionally stable requiring a small time step when used for multi-degree-of-freedom (MDOF) systems with a large number of degrees of freedom (DOFs). This is a severe limitation for RTHS because it imposes a restriction on the minimum time step that can exist due to the real-time computation nature of the simulation. Furthermore, explicit algorithms tend to excite

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the spurious higher modes due to a propagation of experimental errors. To address this, Nakashima et al. (1990) propo

Parametrically Dissipative Explicit Direct Integration Algorithms for Computational and Experimental Structural Dynamics

Lehigh University

Parametrically Dissipative Explicit Direct Integration Algorithms for

Acknowledgements

Table of Contents

List of Tables

List of Figures

List of Symbols

Abstract

Chapter 1

Problem Statement