masters thesis: structural reliability of seismic design

Delft Center for Systems and Control

Structural Reliability of SeismicDesign Methodologies for ShearWalls and Distributors in RCBuildings - APPENDICES

W.B. van der Linde

Mas

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nce

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is

Photo cover: ©Weber Thompson, http://www.weberthompson.com/

Copyright ©All rights reserved.

Structural Reliability of Seismic DesignMethodologies for Shear Walls and

Distributors in RC Buildings -APPENDICES

Master of Science Thesis

For the degree of Master of Science in Structural Engineering at DelftUniversity of Technology

W.B. van der Linde

November 3, 2015

Faculty of Civil Engineering and Geosciences · Delft University of Technology

Delft University of TechnologyDepartment of Civil Engineering and Geosciences

Structural Reliability of Seismic Design of Shear Wallsand Distributors in RC Buildings - APPENDICES

by

W.B. van der Linde

Graduation Committee:Prof. Dr. A.V. Metrikine Delft University of TechnologyDr. ir. M.A.N. Hendriks Delft University of TechnologyDr. ir. C van der Veen Delft University of TechnologyProf. G.G. Deierlein Stanford UniversityN.J. Mathias, PE, SE Skidmore, Owings & MerrillJ. Zhang, SE Skidmore, Owings & Merrill

Contents

I Archetype structure and structural analysis model - APPENDICES 1

A Modelling of shear walls 3A-1 Steel stress-strain relationship . . . . . . . . . . . . . . . . . . . . . . . . . 3

A-1-1 Theoretical steel stress-strain relationships . . . . . . . . . . . . . . 3A-1-2 Steel stress-strain relationship in Perform 3D . . . . . . . . . . . . . 5

A-2 Concrete stress-strain relationship . . . . . . . . . . . . . . . . . . . . . . . 6A-2-1 Theoretical concrete stress-strain relationships . . . . . . . . . . . . 6A-2-2 Concrete stress-strain relationship in Perform 3D . . . . . . . . . . . 13

A-3 Shear stress-strain relationship . . . . . . . . . . . . . . . . . . . . . . . . . 19A-3-1 Theoretical shear force-deformation relationship . . . . . . . . . . . 19A-3-2 Shear-deformation relation in Perform 3D . . . . . . . . . . . . . . . 23

A-4 Other modelling assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 24A-4-1 Meshing of the shear wall . . . . . . . . . . . . . . . . . . . . . . . 24

B Modelling of coupling beams 27B-1 Coupling beams in archetype structure . . . . . . . . . . . . . . . . . . . . 27

B-1-1 Reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27B-2 Strength properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

B-2-1 Material strength properties . . . . . . . . . . . . . . . . . . . . . . 30B-2-2 Shear strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31B-2-3 Flexural strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

B-3 Perform 3D Coupling beam model . . . . . . . . . . . . . . . . . . . . . . . 34B-3-1 Elastic beam segments . . . . . . . . . . . . . . . . . . . . . . . . . 34B-3-2 Shear hinge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

i

ii

C Modelling of foundation 41C-1 Procedure for developing springs and dashpots . . . . . . . . . . . . . . . . . 41

C-1-1 Flexibility of shallow bearing footings . . . . . . . . . . . . . . . . . . 41C-1-2 Stiffness and damping of the foundation . . . . . . . . . . . . . . . 47

II Input - APPENDICES 55

D Hazard identification, target response spectra and selecting ground motionrecords 57D-1 Probabilistic Seismic Hazard Analysis . . . . . . . . . . . . . . . . . . . . . 57

E Input Code Design 59E-1 Input parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

E-1-1 Site input parameters . . . . . . . . . . . . . . . . . . . . . . . . . 59E-1-2 Building input parameters . . . . . . . . . . . . . . . . . . . . . . . 60E-1-3 Seismic Performance Factors . . . . . . . . . . . . . . . . . . . . . 60E-1-4 Design spectral acceleration parameters . . . . . . . . . . . . . . . . 62

E-2 Design Response Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 63E-2-1 Response spectrum input variables . . . . . . . . . . . . . . . . . . 66

E-3 Seismic Base Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66E-3-1 Seismic Response Coefficient . . . . . . . . . . . . . . . . . . . . . 67

E-4 Modal Response Spectrum Analysis . . . . . . . . . . . . . . . . . . . . . . 68E-4-1 Comparison Modal Analysis SAP2000 vs. Perform-3D . . . . . . . . . 71E-4-2 Vertical distribution of Seismic Forces . . . . . . . . . . . . . . . . . 72E-4-3 Transfer forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

F Input PEER/TBI Analysis 81F-1 Selected Ground Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . 85F-2 Results UHS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

F-2-1 Combined shear forces in diaphragms . . . . . . . . . . . . . . . . . 87

G Input Structural Reliability Analysis 89G-1 Results PSHA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

G-1-1 Ground Motion Prediction Equations . . . . . . . . . . . . . . . . . 89G-2 Selection of Ground Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . 91G-3 Selected Ground Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

ii

Contents

III Validation of analysis results - APPENDICES 97

H Overall structural behaviour 99H-1 Shear force over story height . . . . . . . . . . . . . . . . . . . . . . . . . . 99

H-1-1 Bending moment over story height . . . . . . . . . . . . . . . . . . . 101H-1-2 Interstory drift ratio over story height . . . . . . . . . . . . . . . . . 103

I Diaphragm shear stress distribution 105I-1 Usage ratio plots from NLTHA . . . . . . . . . . . . . . . . . . . . . . . . 105

I-1-1 NLTHAs with results similar to Case 2, 3, and 4 . . . . . . . . . . . 105

IV Design - APPENDICES 111

J Design and acceptance criteria 113J-1 Design loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

J-1-1 Specified loads on structure . . . . . . . . . . . . . . . . . . . . . . 113J-1-2 Load combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

J-2 Expected material properties . . . . . . . . . . . . . . . . . . . . . . . . . . 115J-2-1 Expected strength properties . . . . . . . . . . . . . . . . . . . . . 115J-2-2 Expected stiffness properties . . . . . . . . . . . . . . . . . . . . . . 116

J-3 Building Code acceptance criteria . . . . . . . . . . . . . . . . . . . . . . . 116

K Diaphragm Design 119K-1 Building Code demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

K-1-1 Diaphragm demand . . . . . . . . . . . . . . . . . . . . . . . . . . 119K-1-2 Collector demand . . . . . . . . . . . . . . . . . . . . . . . . . . . 120K-1-3 Design force from MRSA . . . . . . . . . . . . . . . . . . . . . . . 120

K-2 PEER/TBI demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121K-2-1 Design force from NLTHA . . . . . . . . . . . . . . . . . . . . . . . . 121

K-3 Shear capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122K-3-1 Shear strength provided by concrete for nonprestressed members . . . 122K-3-2 Shear strength provided by shear reinforcement . . . . . . . . . . . . 123K-3-3 Shear friction capacity . . . . . . . . . . . . . . . . . . . . . . . . . 124

K-4 Maximum shear capacity diaphragm . . . . . . . . . . . . . . . . . . . . . . 125K-5 Maximum collector capacity diaphragm . . . . . . . . . . . . . . . . . . . . 126K-6 Diaphragm transfer force design . . . . . . . . . . . . . . . . . . . . . . . . 127

K-6-1 Code design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127K-6-2 PEER/TBI design . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

iii

iv

L Shear wall design 131L-1 PEER/TBI demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

L-1-1 Design force from NLTHA . . . . . . . . . . . . . . . . . . . . . . . . 131L-2 Force-Based Response Parameters . . . . . . . . . . . . . . . . . . . . . . . 132

L-2-1 Shear strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132L-2-2 Axial Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

L-3 Deformation-Based Response Parameters . . . . . . . . . . . . . . . . . . . 133L-3-1 Flexural Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

V Structural reliability analysis - APPENDICES 135

M Structural reliability 137M-1 AD-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

M-1-1 Various hypothesized distributions . . . . . . . . . . . . . . . . . . . 137M-1-2 Lognormal distribution . . . . . . . . . . . . . . . . . . . . . . . . . 142

M-2 Probabilities of failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144M-3 Estimated Collapse fragility curves . . . . . . . . . . . . . . . . . . . . . . . . 151M-4 Estimated deaggregation of mean annual frequency of collapse . . . . . . . . 163

iv

Part I

Archetype structure and structuralanalysis model - APPENDICES

1

Appendix A

Modelling of shear walls

A-1 Steel stress-strain relationship

A-1-1 Theoretical steel stress-strain relationships

The reinforcing steel stress-strain relationship used by Orakcal and Wallace [Orakcal etal, 2004] is the nonlinear relation of Menegotto and Pinto [Menegotto and Pinto, 1973],as extended by Filippou et al. [Filippou et al., 1983] to include isotropic hardeningeffects. The model by Menegotto and Pinto accurately simulates the hysteric behaviourof reinforcement. Cyclic deterioration is incorporated in the model by the curvatureparameter R. The constitutive model is shown in Figure A-1. It is noted that the strainand stress pairs (εr, σr) and (ε0, σ0) are updated after each strain reversal.The effect of isotropic strain hardening can be important when modelling the cyclicbehaviour of reinforcing bars in RC elements. To improve the ability of the model toaccount for isotropic strain hardening, Filippou et al. [Filippou et al., 1983] proposed amodification to the original model by introducing a stress shift to the yield asymptote, asshown in Figure A-2.

A-1-1-1 Definition of Menegotto-Pinto model

Yielding pointThe yielding point is defined by the yield stress (fy,exp and yield strain εy. The expectedyield stress fy,exp was defined by Table J-6 as fy,exp = 1.17fy. For reinforcement steelASTM A706 Grade 60, fy,exp = 70.2 [ksi]. The modulus of elasticity for non-prestressed

3

4

Figure A-1: Constitutive model for steel as developed by [Menegotto and Pinto, 1973]

Parameter Valuefy,exp 70.2 [ksi]fu,exp 105.3 [ksi]

εy 0.0024 [-]εu 0.0629 [-]εfailure 0.200 [-]

Table A-1: Material parameters for reinforcement steel [Menegotto and Pinto, 1973]

reinforcement Es = 29000 [ksi], as calculated by Equation J-4. The yield strain εy iscalculated by Equation A-1.

εy = fy,exp

Es

= 0.00242 [-] (A-1)

Ultimate stress pointThe ultimate stress point is defined by the expected ultimate stress fu,exp and the strainat reaching the expected ultimate stress εu. The expected ultimate stress fu,exp wasdefined by Table J-6 as fu,exp = 1.5fy,exp. For Grade 60 reinforcement steel, the expectedultimate stress fu,exp = 105.3 [ksi]. A strain hardening ratio of αy = 2% was assumed[ACI 318-11, 2011], the strain at reaching the expected ultimate stress was calculated by

4

A-1 Steel stress-strain relationship

Figure A-2: Stress shift due to isotropic strain hardening [Filippou et al., 1983]

Equation A-2.

εu = εy + fu,exp − fy,exp

αy · Es

= 0.06294 [-] (A-2)

Failure strainThe failure strain of reinforcement steel εfailure is given in Equation A-3 [ACI 318-11, 2011].

εfailure = 0.200 [-] (A-3)

A-1-2 Steel stress-strain relationship in Perform 3D

The constitutive model for steel reinforcement in Perform 3D is typically bilinear ortrilinear. The trilinear material relation was employed in this study. The model can bemodified to include strain hardening and stiffness degradation on reverse loading. Cyclicdegradation of the reinforcement was included by specifying energy factors. The energyfactors represent the relationship between the area of the degraded hysteresis loop andthe area of the undegraded loop as a function of the maximum strain in a given hysteresisloop. The energy factors make the material backbone curve dependent on the loadinghistory by altering the curve with each load excursion. The energy factors are typicallycalibrated using test data.The reinforcement steel ASTM A706 Grade 60 is modelled using a symmetric trilinearstress-strain relationship. The steel stress-strain relationship is defined by three points,yielding, ultimate stress and failure, given in Equation A-4. The required input parameters

5

6

to define the inelastic steel stress-strain relationship are derived in this section and givenin Table A-1.

Point A: (FY, DY) = (fy,exp , εy )Point B: (FU, DU) = (fu,exp , εu )Point C: (FU, DX) = (fu,exp , εfailure )

(A-4)

A-1-2-1 Definition of Perform 3D steel stress-strain relationship

Reinforcement steel ASTM A706 Grade 60 is used for the concrete shear walls. Theexpected, symmetric, trilinear stress strain relationship, based on Menegotto and Pinto[Menegotto and Pinto, 1973], is shown in Figure A-3.

Figure A-3: Expected steel stress-strain relationship

The energy degradation parameters as verified by Orakcal et al. [Orakcal et al., 2006] for arectangular wall (RW2 test specimen) and adopted by Ghodsi and Ruiz [Ghodsi and Ruiz, 2009]are also used in this study. The cyclic degradation parameters are given in Table A-2.

A-2 Concrete stress-strain relationship

A-2-1 Theoretical concrete stress-strain relationships

Two different theoretical constitutive concrete models were considered to simulate thecyclic behaviour of both confined and unconfined concrete implemented in the wall

6


Strain Energy FactorDY 0.70

0.0025 0.680.004 0.640.006 0.62DX 0.60

Table A-2: Cyclic degradation parameters for reinforcing steel [Ghodsi and Ruiz, 2009]

model: the model by Yassin [Yassin, 1994] and the model by Chang and Mander[Chang and Mander, 1994]. Both models are capable of capturing the hysteretic be-haviour, progressive degradation of stiffness for increasing strain values, and the effectsof confinement and tension stiffening. The hysteretic constitutive model by Yassin isrelatively simple and commonly used. The primary shortcoming of the Yassin model isits inability to simulate gradual gap closure. Sudden crack closure results in pronouncedand overestimated pinching behaviour of the walls. The model by Chang and Mandersuccessfully simulates gradual crack closure, and is overall more comprehensive andgeneral [Orakcal et al., 2006]. The Chang and Mander model was used by Orakcal et al.[Orakcal et al., 2006] for modelling the cyclic stress strain behaviour of concrete.

Tension stiffening, the contribution of cracked concrete to the tensile resistance of RCmembers, was modelled by Orakcal et al. following the relations developed by Belarbiand Hsu [Belarbi and Hsu, 1994]. Tension stiffening contributes to the tensile resistanceof the member and plays an important role in reducing post-cracking deformations.

A-2-1-1 Definition of Modified Mander model

The modified Mander model is defined by Equation A-5, where the expected concretecompressive stress f ′c,exp is a function of the concrete compressive strain εc. Other inputvariables are the expected concrete compressive strength of the confined concrete f ′cc,exp,

7

8

r and x.

f ′c(εc) =f ′cc,exp · x · rr − 1 + xr

Where,

f ′cc,exp =f ′cc,exp

f ′c,exp

· f ′c,exp

r = Ec

Ec − Esec

x = εc

εcc

(A-5)

Expected compressive strength of confined concrete f ′cc,exp

The expected compressive strength of the confined concrete f ′cc,exp is calculated by multi-plying the expected compressive strength of unconfined concrete f ′c,exp and the confinedstrength ratio f ′cc,exp/f

′c,exp, derived from Figure A-4, as given in Equation A-6.

f ′cc,exp =f ′cc,exp

f ′c,exp

· f ′c,exp

= 13.73 [ksi]Where,f ′cc,exp

f ′c,exp

= 1.32 [-]

f ′c,exp = 1.3 · f ′c,nom

= 10.40 [ksi]

(A-6)

The input parameters to calculate the confined strength ratio f ′cc,exp/f′c,exp are the largest

confining stress ratio f ′lx,exp/f′c,exp and the smallest confining stress ratio f ′ly,exp/f

′c,exp. In

order to calculate the expected effective lateral confining stress in x- and y-direction,f ′lx,exp and f ′ly,exp respectively, the expected lateral confining stress in x- and y-direction,flx,exp and fly,exp, and the confinement effectiveness coefficient have to be determined.

The expected lateral confining stress in x- and y-direction are determined by EquationA-7, using the ratio of volume of transverse confined steel to volume of confined concretecore in x- and y-direction, ρsx and ρsy respectively, and the expected yield strength ofsteel fy,exp. In order to calculate the confinement effectiveness coefficient ke by EquationA-7, the clear horizontal spacing between centrelines of perimeter hoop bars bc and dc,clear vertical spacing between hoop bars s′, clear distance between longitudinal bars in x-and y-direction wix and wiy, and ratio of longitudinal reinforcement to the area of core of

8


Figure A-4: Confined strength determination from lateral confining stresses for rectangular sections

section ρcc need to be determined.

flx = ρsx · fy,exp

= 0.803 [ksi]fly = ρsy · fy,exp

= 0.433 [ksi]

ke = Ae

Acc

=

(1−

n∑i=1

(w′i)2

6bcdc

)(1− s′

2bc

)(1− s′

2dc

)

(1− ρcc)

=

(1− (nlx − 1)(w′ix)2 + (nly − 1)(w′iy)2

6bcdc

)(1− s′

2bc

)(1− s′

2dc

)

(1− ρcc)= 0.880

(A-7)

9

10

Where,bc = b− 2 · c− ds

= 129.875 [in]dc = d− 2 · c− ds

fy,exp = 1.17fy,Grade 60

= 70.2 [ksi]= 21.875 [in]

s′ = s− ds

= 3.38 [in]

w′ix = bc − ds − dl · nlx

nlx − 1= 5.24 [in]

w′iy = dc − ds − dl · nly

nly − 1= 19.5 [in]

ρcc = nl · π · d2l

4 · bc · dc

= 0.0093 [-]

ρsx = d2s · π · nlx

s · bc

= 0.0130 [-]

ρsy = d2s · π · nly

s · dc

= 0.0070 [-]

(A-8)

10


And,b = 24 [in]c = 0.75 [in]d = 132 [in]dl = 0.875 [in]ds = 0.625 [in]fy,Grade 60 = 60.0 [ksi]c = 0.625 [in]nl = 44 [-]nlx = 22 [-]nly = 2 [-]s = 4.00 [in]

(A-9)

Input variable xThe input variable x is a function of the strain εc and the strain at compressive strengthof confined concrete εcc, as given in Equation A-10.

x = εc

εcc

Where,

εcc = εc0

[1 + 5

(f ′cc,exp

f ′c,exp

− 1)]

= 0.0052 [-]And,εc0 = 0.0020 [-]f ′c,exp = 10.40 [ksi]f ′cc,exp = 13.73 [ksi]

(A-10)

Input variable rThe input variable r is determined by the modulus of elasticity of concrete Ec and thesecant modulus of confined concrete at peak stress Esec, as given in Equation A-5 and

11

12

A-11.

r = Ec

Ec − Esec

= 2.08 [-]Where,Ec = 40000

√f ′c,exp + 106

= 5079 [ksi]

Esec =f ′cc,exp

εcc

= 2640 [ksi]And,f ′c,exp = 10.40 [ksi]f ′cc,exp = 13.73 [ksi]εcc = 0.0052 [-]

. (A-11)

Other input variablesTo complete the definition of the modified Mander model, the expected modulus ofelasticity of concrete Ec,exp, the yield strain εc,y, residual compressive strain εcr andultimate compressive strain εcu.

The expected modulus of elasticity of concrete Ec,exp is calculated by Equation J-5 andA-12.

Ec = 40000√f ′c,exp + 106 = 5079.2 · 103 [psi]

Where,f ′cc,exp = 10.80 · 103 [psi]

(A-12)

The yield strain εc,y is calculated by Equation A-13.

εc,y =f ′c,exp

Ec

= 0.0020 [-]

Where,f ′c,exp = 10.80 [ksi]Ec = 5079.2 [ksi]

(A-13)

12


The residual compressive strain εcr is determined by Equation A-14.

εcr = 0.004 + 1.4 · ρs · fy,exp · εsm

f ′cc,exp

= 0.0183 [-]Where,f ′cc,exp = 13.73 [ksi]fy,exp = 70.2 [ksi]εsm,Grade60 = 0.10 [-]ρs = ρx + ρy

= 0.020 [-]And,ρx = 0.0130 [-]ρy = 0.0070 [-]

. (A-14)

The expected residual compressive stress f ′cr,exp is calculated as the compressive stress asdefined by the modified Mander model at the residual compressive strain εcr and givenby Equation A-15.

f ′cr,exp = f ′c(εcr) = 6.79 [ksi]Where,

f ′c =f ′cc,exp · x · rr − 1 + xr

εcr = 0.0183 [-]

. (A-15)

The ultimate compressive strain εcu is defined by Equation A-16.εcu = 0.0500 [-] (A-16)

A-2-2 Concrete stress-strain relationship in Perform 3D

Concrete compressive stress-strain relationships are defined using multi-linear relations.The tensile strength is neglected for simplicity, which is a conservative assumption.The multi-linear stress-strain relationships are derived from material models such asthe model by Chang and Mander [Chang and Mander, 1994]. Orakcal and Wallace[Orakcal et al., 2006] used the hysteretic stress-strain rules defined by Chang and Manderto simulate the cyclic behaviour of both confined and unconfined concrete implementedin the wall model. In Figure A-5, it is shown that a close match between the multi-linearrelations and the modified Mander model can be obtained if the correct values are chosen.The parameters from Orakcal and Wallace were used in this study to simulate the cyclicbehaviour of the wall model.

13

14

Figure A-5: Expected stress-strain curve for confined concrete with specified concrete strength (f ′c)

of 8 ksi (Ghodsi and Ruiz, 2010)

A-2-2-1 Definition of Perform 3D confined concrete model

Concrete compressive stress-strain relationships in Perform 3D are defined by four linearsegments. To approximate the modified Mander model, the points given in EquationA-17 were defined.

Point A: (FY, DY) = (f ′c,exp , εc,y )Point B: (FU, DU) = (f ′cc,exp , 0.8 · εcc )Point C: (FU, DL) = (f ′cc,exp , 1.1 · εcc )Point D: (FR, DR) = (f ′cr,exp , εcr )Point E: (FR, DX) = (f ′cr,exp , εcu )

(A-17)

The compressive stress-strain relationship in Perform 3D for confined concrete with anominal strength of 8.0 [ksi] is defined by the parameters given in Table A-3.

Selected concrete stress-strain relationshipThe vertical fibre elements in the fibre model are defined by constitutive material modelsto represent the expected behaviour of the structural wall. The concrete stress-strainrelationship was based on the modified Mander model for confined concrete (Mander etal., 1988), as employed by Orakcal et al. [Orakcal et al., 2006] for modelling the cyclicstress-strain behaviour of concrete. The tensile strength of concrete is assumed to be zero.Concrete compressive stress-strain relationships in Perform 3D are defined by four linear

14


Parameter Valuef ′c,exp 10.40 [ksi]f ′cc,exp 13.73 [ksi]f ′cr,exp 6.79 [ksi]

εc,y 0.0020 [-]εcc 0.0052 [-]εcr 0.0183 [-]εcu 0.0500 [-]

Table A-3: Material parameters for confined concrete [Mander, J.B.; Priestley, M.J.N., Park, 1988]

segments. The modified Mander model for confined concrete with expected compressivestrength f ′c,exp = 10.4 [ksi] and the approximation in Perform 3D are given in Figure A-6.

15

16

Figure A-6: Expected stress-strain curve for confined concrete with expected concrete strength(f ′

c,exp) of 10.4 [ksi]

16


17

18

Definition of input parametersDefinition of input parameters:

Ac = Area of core of section within center lines of perimeter spiralAcc = Area of core within center lines of perimeter hoop bars

excluding area of longitudinal steelAe = Area of effectively confined concreteb = Wall lengthbc = Concrete core dimension to centre line of perimeter hoop in x-directionc = Concrete coverd = Wall widthdc = Concrete core dimension to centre line of perimeter hoop in y-directiondl = Diameter of longitudinal reinforcementds = Diameter of hoop barsEc = Expected modulus of elasticity concreteEsec = Secant modulus of confined concrete at peak stressfy,exp = Expected yield strength of steelflx,exp = Expected maximum lateral confining stress in x-directionfly,exp = Expected maximum lateral confining stress in y-directionf ′c,exp = Expected compressive strength of unconfined concretef ′cc,exp = Expected compressive strength of confined concretef ′lx,exp = Expected effective lateral confining stess of concrete in x-directionf ′ly,exp = Expected effective lateral confining stess of concrete in y-directionke = Confinement effectiveness coefficientnl = Number of longitudinal bars in cross-sectionnlx = Number of longitudinal bars per in x-directionnly = Number of longitudinal bars per in y-directions = Vertical spacing between hoop barss′ = Clear vertical spacing between hoop barsw′i = ith clear transverse spacing between adjacent longitudinal barsεco = Strain at cracking of concreteεcc = Strain at maximum compressive strength of confined concreteεcu = Ulitmate compressive strain, defined as strain at first hoop fractureρcc = Ratio of area of longitudinal steel to area of core of sectionρsx = Ratio of volume of transverse confined steel to volume of confined

concrete core in x-directionρsy = Ratio of volume of transverse confined steel to volume of confined

concrete core in y-direction18

A-3 Shear stress-strain relationship

(A-18)


A-3-1 Theoretical shear force-deformation relationship

Shear and flexural/axial behaviour are uncoupled in Perform 3D. As the study by Orakcalet al. [Orakcal et al., 2006] focussed on modelling the flexural response, linear elasticforce-deformation behaviour was adopted for shear. Slender RC shear walls are capacity-designed for shear, which means that shear does not control lateral strength or energydissipation. Therefore, elastic shear behaviour is typically assumed in RC shear walls,even when nonlinear flexural behaviour is anticipated [Wallace, 2007].The theoretical model by Wallace [Wallace, 2007] was adopted.

A-3-1-1 Definition of Wallace model

Definition of shear-deformation relationship The shear stress-strain relationship is definedby two points, the cracking point and point of ultimate shear stress. The required inputparameters to define the inelastic shear stress-strain relationship of concrete shear wallsare derived in this section.

Cracking pointThe expected shear strength at cracking Vcr,exp is defined by Equation A-19 [ACI 318-11, 2011].

Vcr,exp = 4 · Acv ·√f ′c,exp

1 + Pu

4√f ′c,expAweb

1/2

≤ 0.6Vn,exp

Where,f ′c,exp = 10.4 [ksi]Acv = Varying area per horizontal wall section [in2]Aweb = Varying per E-W or N-S direction [in2]Pu = Varying average vertical force per vertical section [kips]

(A-19)

Hence, the expected shear stress υcr,exp at cracking is defined by Equation A-20 [ACI 318-11, 2011].

υcr,exp = 4 ·√f ′c,exp

1 + Pu

4√f ′c,expAweb

1/2

≤ 0.6υn,exp (A-20)

19

20

The expected cracking strain γcr is defined by Equation A-21 [Wallace, 2007].

γcr = υcr,exp

Gc

Where,

Gc = Ec

2(1 + υ)= 2116.3 [ksi]

And,Ec = 5079.2 [ksi]υ = 0.2 [-]

(A-21)

Ultimate shear strengthThe nominal shear strength of walls is defined by Equation A-22 [ACI 318-11, 2011].

Vn,ACI318 = Acv

(αc

√f ′c,exp + ρtfy,exp

)

Where,fy,exp = 70.2 [ksi]αc = 2.0 for hw/lw ≥ 2.0 [-]ρt = Varying transverse reinforcement ratio per wall section [in2]And,hw = 5638 [in]lw = Varying wall length [in]

(A-22)

The upper limit of the nominal shear strength for walls sharing lateral load is given byEquation A-23.

Vn,max = Acv

(8√f ′c,exp

)(A-23)

Tests by Wood [Wood, 1989] and Orakcal et al. [Orakcal et al., 2009] found that theratio of the maximum shear force obtained in the test to the ACI 318-95 nominal shearstrength Vtest/Vn was 1.38 with a standard deviation of 0.34. This indicates that the ACIshear strength provides close to a lower-bound estimate of wall shear strength. Based onthese studies, the median shear strength of the tests is approximately Vtest = 1.5Vn,ACI .The results from various tests are summarized by Gogus and Wallace [Gogus, 2010], andshown in Figure A-7. The data clearly show that wall shear strength degrades withincreasing ductility. Oesterle et al. [Oesterle et al., 1984] suggest that the reduction in

20


drift capacity is related to increase contribution of inelastic shear deformations leading toweb crushing failures. The relation suggested is approximately a median value.

Figure A-7: Wall shear strength as impacted by flexural ductility

Equation A-22 does not consider the effect of axial load. Orakcal et al. reported that wallshear strength is sensitive to axial load, with Vtest/Vn,ACI values of approximately 1.5 forwalls tested with an axial load of Pu/Agf

′c = 0.05, and 1.75 for walls with an axial load

of Pu/Agf′c = 0.10. The expected (or median) shear strength is calculated according to

Equation A-24 [Orakcal et al., 2009].

Vcu,exp = 1.50 · Vn,ACI318 for Pu/Agf′c,exp ≤ 0.05

Vcu,exp = 1.75 · Vn,ACI318 for Pu/Agf′c,exp ≥ 0.10

Where,Vn,ACI318 = Varying shear strength per ACI 318-11 [ksi]

(A-24)

The ultimate shear stress υcu,exp is defined by Equation A-25 [ACI 318-11, 2011].

υcu,exp = Vn,ACI318

Vcu,exp

(αc

√f ′c,exp + ρtfy,exp)

Where,Vcu,exp = Varying expected shear strength [ksi]

(A-25)

The post-cracked shear stiffness is substantially less than 0.4Ec. For typical walls innew construction, the effective shear stiffness after yield Gc,y is approximately Gc/10

21

22

[ATC 72-1, 2010], and this stiffness is used to calculate γcu,exp, given υcu,exp.

γcu,exp = υcu,exp − υcr,exp

Gc,y

+ γcr,exp

Where,

Gc,y = Gc

10= 0.212 [ksi]

γcr,exp = Varying expected cracking strainυcr,exp = Varying expected shear stress at crackingυcu,exp = Varying expected ultimate shear stress

(A-26)

Other input variablesTo complete the definition of the shear stress-strain relation, the maximum shear strainγmax, the shear strain at failure, has to be defined. The maximum shear strain is definedby Equation A-27 [ACI 318-11, 2011]; [350 Mission, 2014].

γc,max = 0.015 [-] (A-27)

Definition of input parametersDefinition of input parameters:

Acv = Cross-sectional web area of wallAweb = Wall area in E-W or N-S directionρt = Transverse reinforcement ratiofy,exp = Expected yield strength of transverse reinforcementf ′c,exp = Expected compressive strength of concretehw = Wall heightlw = Wall length per horizontal wall sectionPu = Average vertical force per vertical wall sectionVn,ACI318 = Nominal shear strength according to ACI 318-11Vcu,exp = Expected shear strengthαc = To account for observed strength increase for low-aspect-ratio wallsµcr,exp = Expected shear stress at crackingµcu,exp = Expected ultimate shear stressγc,max = Maximum shear strainγcr = Expected cracking strainγcu,exp = Strain at ultimate shear stress

22


(A-28)

A-3-2 Shear-deformation relation in Perform 3D

Shear and flexural/axial behaviour are uncoupled in Perform 3D. The shear-deformationrelation can be defined elastically or inelastically. A comparison of elastic and inelasticshear force-deformation behaviour is given in Figure A-8.

Figure A-8: Shear stress-strain relationships

A-3-2-1 Definition of Perform 3D shear-deformation relationship

To represent the wall behaviour as realistic as possible, the inelastic shear-deformationrelation is adopted for the shear-deformation relation of confined concrete wall sections.According to Wallace [Wallace, 2007], shear force can be represented using two points,cracking and ultimate. The definition of the cracking and ultimate shear stress-strainsare given in Appendix A-3.

For the archetype structure, the inelastic shear stress-strain relationship is calculatedfor 5 vertical sections (Floors B4-01, 01-04, 04-13, 13-25, and 25-41) and 5 horizontalsections (North-wall-west, North-wall-east, West-wall-north, West-wall-mid, and West-wall-south). The shear stress-strain relationships are derived from recommendations by[ACI 318-11, 2011]; [Wallace, 2007]; [ATC 72-1, 2010]; [Gogus, 2010]; [Orakcal et al., 2009].

23

24

A-4 Other modelling assumptions

A-4-1 Meshing of the shear wall

A-4-1-1 Overall meshing recommendations

Several studies ([Orakcal et al., 2006]; [Salas, 2008]; [Wallace, 2012]) provide meshingrecommendations for concrete shear walls. These studies provide insight in the effect ofthe number of elements and fibres on the representation of the strain distribution alongthe cross-section was studied. Furthermore, the effect of the choice of inelastic versuselastic elements in the upper stories was subject of research in these studies.

The overall results were found to be relatively insensitive to the number of fibres andnumber of elements. Increasing the number of elements does not notably improve thecorrelation between analysis results and test results. Increasing the number of elementscan however be valuable to obtain more information on local behaviour, such as strainsor moment curvature response at a given location [Orakcal et al., 2006]. As noted earlier,a larger number of elements also diminishes the influence of parameter c on the predictedresults.

The local responses were found to be sensitive to variations in mesh size. Nonlineardeformations can concentrate in a single element if a fine mesh is combined with elastic-perfectly-plastic reinforcing steel behaviour. The correlation between the analysis resultsand the test behaviour was found to improve significantly when a moderate post-yieldhardening slope of about 3-5% was included in the steel reinforcement constitutive model[Salas, 2008]. The moderate post-yield hardening slope effectively eliminated the problemof concentration of nonlinear deformations in a single element [ATC 72-1, 2010].

Wallace [Wallace, 2012] found that allowing yield in the upper stories substantially reducesthe shear forces, especially within the hinge region, and the bending moments at upperlevels. Instead of modelling the regions of the wall outside the designated yielding zoneusing elastic elements, the nonlinear elements were therefore extended over the full wallheight, as the design allows for minor flexural yielding over the full height of the wall.

A-4-1-2 Element height in plastic hinge zone

Use of an element height equal to the plastic hinge length produces good correlationwith test results [ATC 72-1, 2010]. The element height in the estimated plastic hingezone should therefore be approximately equal to the anticipated plastic hinge length. InASCE 41, a hinge length equal to 0.5 times the wall length, but not more than the storyheight, is specified. The plastic-hinge length (lp) in the walls for analyses purposes may

24

A-4 Other modelling assumptions

be calculated from the maximum of the following formulas [Paulay and Priestley, 1992]:

lp = 0.2 · lw + 0.03 · hw

lp = 0.08hw + 0.15fydb

Where,lw = Wall lengthhw = Wall heightdb = Nominal diameter of rebar

(A-29)

The wall length depends on the stiffness of the coupling beams. If the wall has very stiffand strong coupling beams, the wall behaves like a solid walls, it is reasonable to base thehinge length on the total length of the wall. If the coupling beams are weak, the wallsbehave more independently, and it is more reasonable to use a hinge length based on theindividual pier lengths. Weak coupling beams result in a smaller hinge length, whichtends to be more conservative [ATC 72-1, 2010].The height of the finite element used to model the plastic hinge length shall not exceedthe plastic hinge length, lp, or the story height at the location of the critical section. Thefirst iteration to calculate the plastic hinge length assumed weak coupling beams andindependent piers, resulting in conservatively short hinge lengths. As shown in Table A-4,the story height at the location of the critical section governs the finite element height forall walls. Therefore, a second iteration is not necessary and the finite element heights areset equal to the story height at the base of the wall.

Material lp = 0.2lw + 0.03hwlp =

0.08hw + 0.15fydbStory height Finite element

heightEW-wall 164.3 424.5 150.0 150.0NS-wall 181.1 424.5 150.0 150.0

Table A-4: Plastic hinge length

A-4-1-3 Element length outside hinge zone

The element length outside the estimate yielding zone is chosen to be equal to the storyheight, as recommended by Powell [Powell, 2007]. The gage length for calculating thesteel strains is therefore also equal to the story height.

25

26

26

Appendix B

Modelling of coupling beams

B-1 Coupling beams in archetype structure

The considered archetype core walls use coupling beams to connect structural walls. Anexample of typical coupling beam configuration is given in Figure B-1. In the North andSouth (N-S) walls, one coupling beam at every level connects two wall panels. In theWest and East (E-W) walls, two coupling beams at every level connect three wall panels.

B-1-1 Reinforcement

The coupling beam layout is given in Figure B-2. Note the diagonal reinforcement "a",the horizontal beam reinforcement "b", and full section confinement reinforcement "c".In the archetype structure, three different diagonal reinforcement layouts are used. Thereinforcement layouts are given in Figure B-3.

B-1-1-1 Diagonal Reinforcement "a"

The coupling beams were designed using diagonal reinforcement. The diagonal reinforce-ment "a" works as a compression and tension strut and the diagonal bars are intended toprovide the entire shear and moment resistance of the beam. In compliance with ACI318-11, diagonal bars are placed approximately symmetrically in the beam cross section,in two or more layers. The diagonal reinforcement configuration was first proposed byPaulay and Binney [Paulay, T.; Binney, ]. Tests by Paulay and Binney have shown thatconfined diagonal reinforcement provided adequate resistance in deep coupling beams.

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28

Figure B-1: Coupling beams configuration

The use of diagonal reinforcement is recommended in coupling beams with aspect ratiosless than 4. Deep coupling beams may be controlled by shear and may be susceptible tostrength and stiffness deterioration under earthquake loading. The purpose of providingdiagonal reinforcement was to improve performance of deep coupling beams with respectto sliding shear failures at high shear stress levels. However, experiments have shownthat diagonally oriented reinforcement is effective only if the bars are placed with largeinclination. Therefore, the use of diagonal reinforcement is restricted to beams with aspectratio ln/h < 4.

It is noted that diagonally reinforcement coupling beams behave differently from beamswith conventional shear reinforcement. If the beam is to behave as intended, the tensiondiagonal must extend substantially, whereas the compression diagonal is stiffer and hassmaller deformation. Therefore the beam must extend as a whole to achieve the intendedbehaviour. If the axial extension is restrained, for example by the core walls, the behaviourof the beam may not be the same as observed in a test without axial constraint. Theusual practice is to assume that axial restraint does not have a substantial effect [ACI318-11], and this assumption is also adopted in this study. Furthermore, the embedmentof the diagonal bars into the wall is noted, not less than 1.25 times the developmentlength of fy in tension, in compliance with ACI 318-11 [ACI 318-11, 2011].

28

B-1 Coupling beams in archetype structure

Figure B-2: Coupling beam layout

Figure B-3: Coupling beam reinforcement layout

B-1-1-2 Horizontal beam reinforcement "b"

It is required per ACI 318-11 that each crosstie and each hoop leg shall engage a longitudinalbar of equal or larger diameter. The horizontal beam reinforcement "b" in the couplingbeams is designed accordingly.

B-1-1-3 Full section confinement reinforcement "c"

Use of transverse reinforcement around the diagonal bar groups is required to enhance thecompressive strength and deformation capacity of the diagonal truss members as well as tosuppress buckling of the diagonal bars. It was concluded by Naish [Naish et al., 2010] thatthe full section confinement option of ACI 318-11 provided equivalent, if not improved,performance compared to confinement around the diagonal per ACI 318-05. The beamsin the structure use full section confinement.The axial load and deformation demands during earthquake loading are not know with

29

30

sufficient accuracy to justify calculation of required transverse reinforcement as a functionof design earthquake demands. Instead, ACI 318-11 requires the total cross-sectional areaof rectangular hoop reinforcement, Ash, shall not be less than:

Ash ≥ 0.3sbcf′c

fyt

[(Ag

Ach

)− 1

]≥ 0.09sbcf

′c

fyt(B-1)

Where,

s = Center-to-center spacing of transverse reinforcementbc = Cross-sectional dimension of member core measured to the outside edges

of transverse reinforcement composing area Ash

fyt = Yield strength of transverse reinforcementAg = Gross area of concrete sectionAch = Cross-sectional area of a structural member measured to the outside edges

of transverse reinforcement(B-2)

The transverse reinforcement shall be provided for the entire beam cross section. Thelongitudinal spacing shall not exceed the smaller of 6 inch and six times the diameter ofthe diagonal bars. The spacing of crossties shall not exceed 8 inches both vertically andhorizontally in the plane of the beam cross section.

B-2 Strength properties

The strength of the coupling beams was determined using ACI 318-11. This sectionelaborates on the material strength properties, shear and flexural strength. For the designper ACI 318-11, the diagonally placed bars are intended to provide the entire shear andcorresponding moment strength of the beam.

B-2-1 Material strength properties

To characterize the expected performance as closely as possible, the strength propertiesof the component backbone curve should be based on expected material strengths. Theexpected material properties are taken from PEER TBI.

30


Material Expected strengthReinforcing Steel fy,exp = 1.17 fy

Concrete f ′c,exp = 1.3 f ′c

Table B-1: Coupling beam expected material properties

B-2-2 Shear strength

B-2-2-1 Expected nominal shear strength

The shear strength was determined using ACI 318-11. In compliance with ACI 318-11,each group of diagonal consists of a minimum of four bars provided in two or more layers.Furthermore, the diagonal bars are embedded into the wall not less than 1.25 timesthe development length for fy in tension. The nominal expected shear strength Vn,exp

for coupling beams reinforced with two intersecting groups of diagonally placed basedsymmetrical about the midspan is determined by Equation B-3 (ACI 318-11, Eq. 21-9).

Vn,exp = 2Avdfy,exp sin(α) ≤ 10√f ′c,expAcw (B-3)

Where,

α = Angle between the diagonal bars and the longitudinal axis of the coupling beamAvd = Total area of reinforcement in each group of diagonal bars

(B-4)

B-2-2-2 Impact of slab on strength

Naish [Naish et al., 2009] investigated the impact of the slab on strength of couplingbeams, by including the slab as part of the test specimen. The slab might restrain axialelongations and impact stiffness, strength and deformation capacity. The reports by Naishconcluded that if the beam acts compositely with the floor slab, this should be taken intoaccount when calculating the beam stiffness and strength. The impact of the slab on thebackbone load-deformation curve for full-scale beam models is shown in Figure B-4.The impact of the slab provides an increase in capacity by approximately 20%. Thereinforced concrete slab increases beam shear strength approximately 15-20%, addingpost-tensioning increases the beam shear strength another 10%. The strength increasewas directly related to the increase in beam moment strength. The results are summarisedin Table B-2.The coupling beams in the archetype structure are connected to reinforced concrete slabs.The expected average nominal strength Vy is therefore 1.3Vn,exp.

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32

Figure B-4: Impact of slab on shear strength

Slab and coupling beam Vy/Vn,exp Strength increaseNo slab 1.1 -Reinforced concrete slab 1.3 15-20 %Post-tensioned concrete slab 1.4 25-30 %

Table B-2: Impact of slab on shear strength

B-2-2-3 Expected ultimate and residual shear strength

The expected ultimate and residual shear strength are calculated according to recom-mendations by Naish [Naish et al., 2010]. The recommended ratio between the expectednominal shear strength including the impact of reinforced concrete slab Vy and the ex-pected ultimate shear strength Vu,exp and the ratio between the ultimate shear strengthVu,exp and residual shear strength Vr,exp are given in Equation B-5.

Expected ultimate shear strength: Vu,exp = 1.33Vy

Expected residual shear strength: Vr,exp = 0.25Vu,exp

(B-5)

B-2-2-4 Shear strength of coupling beams in archetype structure

The ultimate shear strength was calculated for all coupling beams in shear wall accordingto Equations B-3 and B-2-1, and Table B-2. The ultimate shear strengths of the couplingbeams are given in Table B-3. The location of the coupling beams in the shear walls isgiven in Figure REF.

32


Coupling Beam ID Shear strength Vu

N-S wall E-W wall1A 211.0 [kips] 177.3 [kips]2A - [kips] 177.3 [kips]3A 494.5 [kips] 415.5 [kips]4A 494.5 [kips] 415.5 [kips]5A - [kips] 533.9 [kips]6A 642.0 [kips] 533.9 [kips]7A 776.8 [kips] 646.0 [kips]

Table B-3: Shear strength of coupling beams

B-2-3 Flexural strength

To determine the flexural strength of the coupling beam, the relationship between compres-sive stress distribution and concrete strain is assumed to be rectangular [ACI 318-11, 2011].Concrete stress of 0.85 f ′c is assumed uniformly distributed over an equivalent compressionzone. The compression zone is bounded by the edges of the cross section and straight lineparallel to the neutral axis at a distance a from the fibre of maximum compressive strain.The distance a is calculated by:

a = β1 · c (B-6)

Where,

• Parameter β1 is a factor relating depth of equivalent rectangular compressive stressblock to neutral axis depth. For f ′c above 4000 psi, β1 is reduced linearly at a rateof 0.05 for each 1000 psi of strength in excess of 4000 psi, but β1 shall not be takenless than 0.65.

• Parameter c is the distance from the fibre of maximum compressive strain to theneutral axis. The distance is measured in a direction perpendicular to the neutralaxis and calculated using the moment equilibrium.

To calculate the moment equilibrium, the stress in the groups of diagonal bars andcompressive stress in the equivalent concrete compression zone are taken into account,the longitudinal bars are not included in the calculation of the flexural strength. Thecalculated flexural strengths of the coupling beams is given in Table B-4.

33

34

Coupling Beam ID Flexural strength Mn,diagonal

N-S wall E-W wall1A 6166 [kips-in] 6108 [kips-in]2A - [kips-in] 6289 [kips-in]3A 12253 [kips-in] 12180 [kips-in]4A 12536 [kips-in] 12455 [kips-in]5A - [kips-in] 15627 [kips-in]6A 16229 [kips-in] 16123 [kips-in]7A 18725 [kips-in] 18612 [kips-in]

Table B-4: Flexural strength of coupling beams

B-3 Perform 3D Coupling beam model

B-3-1 Elastic beam segments

B-3-1-1 Effective stiffness

The effective stiffness values for coupling beam tests are considerably lower than specifiedin reference codes and standard [ATC 72-1, 2010]. The effective elastic stiffness valuesfor test beams are determined to be ∼ 15% of the gross section stiffness. Slip and ex-tension of the flexural reinforcement has a large contribution (40%-50%) to the beamdeformation prior to yield, and Naish [Naish et al., 2009] therefore recommends to utilizethe slip/extension hinge model detailed in Supplement 1 in ASCE 41 to approximate theelastic stiffness of the coupling beam.

For coupling beams with aspect ratios ln/h ≥ 2.0, yield deformations are dominatedby flexure. For beams with aspect ratios ln/h ≤ 1.4, deformation due to flexure andshear are about equal, and nonlinear behaviour is dominated by shear deformation. Therecommended effective flexural, shear and axial stiffnesses, based on [ATC 72-1] are givenin Table B-5. For clear span-to-depth aspect ratios 1.4 < ln/h < 2.0, linear interpolationof the effective stiffness values is the recommended approach.

The effective stiffness properties of the N-S walls and E-W walls were determined using therecommendations from ATC 72-1 [ATC 72-1, 2010]. The effective shear area for a rectan-gular cross-section is calculated byAeff = 5/6·b·d [SAP2000 Analysis Reference Manual, 2009].The effective stiffness properties of the coupling beams in the N-S walls and E-W wallsare given in Table B-6.

34


Aspect ratio Flexural[EcIeff ] Shear [Gc,eff ] Axial [EcAeff ]

2.0 ≤ Ln/h ≤ 4.0 0.15 EcIg Gc,eff = 0.4 Ec EcAc

Ln/h ≤ 1.4 0.15 EcIg Gc,eff = 0.1 Ec EcAc

Table B-5: Effective stiffness properties of coupling beams [ATC 72-1]

Aspect ratio Flexural [EcIeff ] Shear [Gc,effAeff ] Axial [EcAeff ]Ln/h = 1.70 (N-S wall) 0.15 EcIg 0.25 EcAeff EcAc

Ln/h = 2.10 (E-W wall) 0.15 EcIg 0.40 EcAeff EcAc

Table B-6: Effective stiffness properties of the coupling beams in structure

The study by Naish [Naish et al., 2010] concluded that the parameters used for couplingbeam strength and stiffness will have substantial impact on coupling beam rotation, butdo not have a substantial impact on interstory drift, and variations do not greatly impactthe core wall shear forces.

B-3-2 Shear hinge

B-3-2-1 Backbone curve

The backbone curve characterizes the force-deformation (F-D) relationship, which measuresstrength against translational or rotational deformation. The backbone curve is expectedto capture the elastic behaviour, strain hardening and post-peak softening response. If nocyclic deterioration has occurred, the backbone curve is close to the monotonic backbonecurve, and is referred to as the initial backbone curve. A monotonic backbone curve isproduced when a load pattern is progressively applied such that the deformation parametercontinuously increases from zero to an ultimate condition. The monotonic backbone curveis shown in Figure B-5.The cyclic backbone curve encloses the forces and displacements under cyclic loading[Deierlein et al., 2010]. The force-deformation relationship is described by hysteresis loops.As the hysteresis loops develop, the profile of the peak values forms the cyclic envelope.The cyclic envelope varies depending on the cyclic loading history. Therefore, backbonecurves derived from cyclic envelopes are typically based on standard loading protocols.The cyclic backbone curve incorporates degradation in strength and stiffness. Strengthand stiffness deviate from their initial relationship once yielding occurs. Strength gainor loss is indicated by the strength level achieved. Although the strength may increaseinitially through hardening behaviour, strength will ultimately degrade through softeningbehaviour. Stiffness degradation is indicated by the decrease in slope upon load reversal.

35

36

The consequence of strength and stiffness degradation is the lack of stability in theload-deformation response. The backbone curve derived from the cyclic envelope willtherefore be less than the monotonic curve which would result from the same structurebeing subjected to monotonic loading. The hysteresis loop and cyclic envelope are shownin Figure B-7.

Figure B-5: Coupling beam monotonic backbone curve [Napier and Abell, 2014]

Figure B-6: Linearised coupling beam monotonic backbone curve [Napier and Abell, 2014]

Figure B-7: Coupling beam hysteresis loops and cyclic envelope [Napier and Abell, 2014]

The nonlinear curve may be idealized as a series of linear segments to provide a numerically-efficient formulation. When the general curve in Figure B-5 is compared to the idealizedcurve, shown in Figure B-6, it is evident that an exact formulation may be simplified withminimal compromise to accuracy. Perform 3D employs linearised cyclic backbone curvesto provide a simple, yet accurate method for estimating the overall force-deformationbehaviour. The input parameters given in Equation B-7 are required as input to representthe linearised cyclic backbone curve. The cyclic backbone curve as linearised by Perform

36


3D is given in Figure B-8.

θU = Plastic link beam rotation at the ultimate shear stressθL = Plastic link beam rotation at the onset of shear strength degradationθR = Plastic link beam rotation at the residual shear strengthθX = Plastic link beam rotation at analysis terminationFY = Shear yield strengthFU = Ultimate shear strengthFR = Residual shear capacity

(B-7)

Figure B-8: Linearized cyclic backbone curve in Perform 3D

Accurate characterization of the cyclic backbone curve and strength and stiffness rela-tionships is an important provision of nonlinear modelling [Powell, 2013]. When properlycalibrated, cyclic backbone curves enable the model to capture the deteriorated cyclicresponse of the component. Therefore, the linearized cyclic backbone curves of the couplingbeams in the N-S and E-W walls are derived from tests and reference projects with couplingbeams closely matching the archetype structure’s link beam configuration. The backbonecurve modelling parameters for coupling beams from test results and reference projects aregiven in Table B-7, for both the coupling beams in the N-S walls (ln/h = 1.70) and in theE-W walls (ln/h = 2.10). The values of the backbone curve modelling parameters for thecoupling beams in the N-S and E-W walls were derived from the modelling parameters inTable B-7 and recommendation by Naish. The values of these shear hinge cyclic backbonecurve parameters are given in link beam plastic chord rotations in Table B-8. It is notedthat the link beam chord rotation θi is not input or measured directly in Perform 3Dshear hinge model. Instead, a shear spring displacement which is equal to the link beamplastic chord rotation θi multiplied by link beam clear span ln is used. The input valuesof the displacement-based shear hinge in Perform 3D are given in Table B-9.Finally, a strength loss interaction factor must be specified. A strength loss interactionfactor of 0 implies that strength loss in one direction has no effect on the strength loss in

37

38

the other direction. On the other hand, an interaction factor of 1 implies that strengthloss in one direction causes an equivalent loss in the other direction. A strength lossinteraction factor of 0.25 was specified for all coupling beams, based on recommendationsby Naish.

Reference for N-S walls (aspect ratio ln/h = 1.70)Aspect ratio θU θL θR θX FY/FU FR/FU

1.4 ≤ ln/h < 1.9 [SOM (111 S Main St)] 0.014 0.025 0.045 0.20 0.82 0.50

ln/h = 1.50[SOM (350 Mission) -Galano and Vignoli,2000]

0.014 0.025 0.045 0.20 0.82 0.50

ln/h = 1.70 [MacKay-Lyons(UToronto, 2013)] 0.018 0.058 0.100 0.15 0.75 0.25

Reference for E-W walls (aspect ratio ln/h = 2.10)Aspect ratio θU θL θR θX FY/FU FR/FU

1.9 ≤ ln/h < 4.0 [SOM (111 S Main St)] 0.021 0.050 0.090 0.30 0.77 0.25

2.0 ≤ ln/h ≤ 4.0 [SOM (350 Mission) -Naish, 2010] 0.021 0.050 0.090 0.30 0.77 0.25

ln/h = 2.10 [MacKay-Lyons, 2013] 0.018 0.058 0.100 0.15 0.75 0.25ln/h = 2.40 [Naish, 2010] 0.075 0.08 0.10 0.13 0.96 0.30ln/h = 2.50 [Barney et al., 1980] 0.03 0.05 0.06 0.10 0.69 0.40

Table B-7: Cyclic degradation parameters from test results

Coupling beam θU θL θR θX FU/FY FR/FU

N-S wall(ln/h = 1.70)

[Galano andVignoli (2000)] 0.014 0.025 0.045 0.300 1.33 0.25

E-W wall(ln/h = 2.10) [Naish (2010)] 0.021 0.050 0.090 0.300 1.33 0.25

Table B-8: Values of cyclic backbone curve parameters of shear hinge (given in plastic link beamrotations)

B-3-2-2 Energy degradation parameters

The cyclic performance and the shape of link beam hysteresis loops are controlled inPerform 3D by varying the following energy degradation parameters which reflect hysteresis

38


Coupling beam DU DL DR DX FU/FY FR/FU


[Galano andVignoli (2000)] 0.71 1.27 2.29 10.20 1.33 0.25

E-W wall(ln/h = 2.10) [Naish (2010)] 1.32 3.15 5.67 18.90 1.33 0.25

Table B-9: Values of cyclic backbone curve parameters of shear hinge

loops relative areas. It is noteworthy that Perform 3D shear displacement hinge doesnot feature unloading stiffness degradation capability; instead, it only uses cyclic energydegradation which gives less control on the individual cyclic characteristics. The inputparameters to account for cyclic energy degradation are given in Equation B-8.

Y = Energy degradation factor at yieldU = Energy degradation factor at ultimate deformationL = Energy degradation factor at the onset of loss of lateral load capacityR = Energy degradation factor at the residual lateral load capacityX = Energy degradation factor prior to analysis termination

(B-8)

The energy degradation factors were derived from tests on link beams closely matchingthe structure’s link beam configuration. For consistency, the energy degradation factorswere derived from the same tests as te the cyclic backbone curve parameters; Galano andVignoli (Specimen P07, ln/h = 1.50) for the N-S wall link beams, Naish (ln/h = 2.40) forthe E-W wall link beams. The energy degradation parameters from these tests are givenin Table B-10. The cyclic degradation parameters assigned to the coupling beams in theN-S and E-W walls are given in Table B-11.

39

40

EnergyDegradationParameter

Aspect ratio coupling beam [Reference]ln/h = 1.50 2.00 ≤ ln/h ≤ 4.00

[Galano and Vignoli] [Naish]Y 0.250 0.500U 0.225 0.450L 0.225 0.400R 0.150 0.350X 0.150 0.350

Table B-10: Cyclic degradation parameters - reference

Coupling beam Y U L R X


[Galano andVignoli (2000)] 0.250 0.225 0.225 0.150 0.150

E-W wall(ln/h = 2.10) [Naish (2010)] 0.500 0.450 0.400 0.350 0.350

Table B-11: Cyclic degradation parameters

40

Appendix C

Modelling of foundation

C-1 Procedure for developing springs and dashpots

C-1-1 Flexibility of shallow bearing footings

"The structural flexibility and strength of the footing shall be consistent with the soilbearing pressure distribution assumed with the foundation assessment. The flexibilityassessment shall consider whether soil-footing contact remains or uplift occurs" (ASCE41-13). For rectangular plates (with plan dimensions L and B, thickness t, and mechanicalproperties E and υ) on elastic support (for example mat foundations) subjected to a pointload in the center, the foundation may be considered as rigid where:

4ksv

5∑

m=1

5∑

n=1

sin2(m · π

2

)sin2

(n · π

2

)

[π4D

(m2

L2 + n2

B2

)]+ ksv

< 0.03 (C-1)

Where the constant representing flexibility plate, D, is:

D = Et3

12(1− υ)2 2.203E + 08 [kips inch] (C-2)

The Winkler springs stiffness in vertical direction, ksv, is:

ksv = 1.3 ·GBf · (1− υ) = 1.147E − 02 [kips/inch] (C-3)

The dimensional parameters are:

41

42

Dimensional parameters ValuePlan length Lp 2736 [in]Plan width Bp 2724 [in]

Length of footing Lf 2736 [in]Width of footing Bf 2724 [in]Thickness of footing t 72 [in]Depth of soil-footing interface Df 480 [in]

Area of footing Af 7.453E+06 [in2]

Table C-1: Dimensional parameters foundation

The structural parameters are:

Structural parameters ValueConcrete compressive strength f ′c 6.0 [ksi]Expected concrete compressivestrength f ′c 7.8 [ksi]

Poisson’s ratio υc 0.2 [-]Expected Young’s modulus Ec,exp 4533 [ksi]

Table C-2: Structural parameters foundation

The site parameters are:

42


Site parameters ValueTotal unit weight of soil[Lunne et al., 1997]

γt 120.9 [lb/ft3]γt 6.997 E-05 [kips/in3]

Specific weight of water γw 62.43 [lb/ft3]γw 3.61 E-05 [kips/in3]

Atmospheric pressure pa 0.0147 [ksi]Pore-water pressure at depth(Df −Bf/2) u 5.61E-02 [ksi]

Poisson’s ratio for sand υs 0.3 [-]Standard penetration test blowcount normalized N60 25.00 [blows/ft]

Cone resistance qc 90 [tsf]qc 1.25 [ksi]

Table C-3: Site parameters foundation

The soil parameters are:

Soil parameters ValueElevation ground surface Elsite 300 [ft]Elevation ground water level Elgwl 276 [ft]Depth water table zw 24 [ft]

zw 288 [in]

Table C-4: Soil parameters foundation

43

44

Figure C-1: Cone Penetration Test by Gregg - LA Federal Courthouse

Using ASCE 41-13, the mean effective stress averaged over the relevant region below thefooting, σ′mp may be obtained as the larger value of:

σ′mp,1 = 16

(0.52− 0.04Lf

Bf

)QGf

Af

σ′mp ≥ σ′vo = (γt(Df +Bf/2))− u(C-4)

Loading on the foundation is summarized in Table C-5.

44


Loading on foundation ValueExpecting bearing load onfooting due to gravity load Qgf 1.588 E+05 [kips]

Mean effective stress averagedover the relevant region belowthe foundation

σ′mp,1 1.704 E-03 [ksi]

Effective vertical stress atdepth (Df −Bf/2) σ′vo 7.273 E-02 [ksi]

Mean effective stress σ′mp 1.289 E-01 [ksi]

Table C-5: Loading on foundation

The standard penetration test blow count normalized for an effective stress of 1.0 [ton/ft2]confining pressure and corrected to an equivalent hammer energy efficiency of 60%, (N1)60,is calculated by:

(N1)60 = N60CN = 0.8175 [blows/ft

CN =√pa/σ′vo = 3.270E-02 [−]

(C-5)

Using ASCE 41-13, the initial shear modulus for sandy soils is determined by:

G0 = 435 3√

(N1)60√pa · σ′mp = 17.70 [kips] (C-6)

Assuming a constant shear wave velocity profile of 800 [ft/s] = 267 [m/s], and nooverburden pressure due to the weight of the structure, and a peak acceleration of 0.15[g], the effective shear wave velocity is:

Vs,avg = 267 · 0.95 = 253 [m/s] (C-7)

From Figure C-2, and Vs = 253 [m/s], the site class is determined to be Site Class D.

45

46

Figure C-2: NEHRP Site class definitions

Using information from USGS, the effective peak acceleration Sxs is determined to be Sxs

= 0.1439 [g]. The effective shear modulus ratio, G/G0 = 0.95 [-], was determined usingFigure C-3.

Figure C-3: Effective shear modulus ratio (G/G0)

The soil shear modulus G can now be calculated:

G ∼= (G/G0) ·G0 = 16.82 [ksi] (C-8)

The rigidity of the foundation mat can now be checked using Equation C-1:

4ksv

5∑

m=1

5∑

n=1

sin2(m · π

2

)sin2

(n · π

2

)

[π4D

(m2

L2 + n2

B2

)]+ ksv

< 0.03

1.45E-05 < 0.03

(C-9)

46


It is concluded that the foundation can be assumed rigid.

C-1-2 Stiffness and damping of the foundation

The static stiffnesses were calculated using the Pais and Kausel [Pais and Kausel, 1988]equations, which are most often used in practice [NEHRP Consultants Joint Venture, 2012].In these equations, B and L are defined as the overall foundation half-width B = 1362[in] and half-length L = 1368 [in]. The rigid footing is shown in Figure C-4.

Figure C-4: Rigid footing, axes should be oriented such that L ≥ B.

Degree of freedom Pais and Kausel (1988)

Translation along z-axis Kz,sur = GB

1− υ

[3.1(L

B

)0.75+ 1.6

]

Translation along y-axis Ky,sur = GB

2− υ

[6.8(L

B

)0.65+ 0.8

(L

B

)+ 1.6

]

Translation along x-axis Kx,sur = GB

2− υ

[6.8(L

B

)0.65+ 2.4

]

Torsion about z-axis Kzz,sur = GB3[4.25

(L

B

)2.45+ 4.06

]

Rocking about y-axis Kyy,sur = GB3

1− υ

[3.73

(L

B

)2.4+ 0.27

]

Rocking about x-axis Kxx,sur = GB3

1− υ

[3.2(L

B

)+ 0.8

]

Table C-6: Elastic solutions for static stiffness of rigid footings at the ground surface

47

48

Degree of freedom Embedment Correction factors

Translation along z-axis ηz =[1.0 +

(0.25 + 0.25

L/B

)(D

B

)0.8]

Translation along y-axis ηy =[1.0 +

(0.33 + 1.34

1 + L/B

)(D

B

)0.8]

Translation along x-axis ηx ≈ ηy

Torsion about z-axis ηzz =[1.0 +

(1.3 + 1.32

L/B

)(D

B

)0.9]

Rocking about y-axis ηyy =[1.0 + D

B+( 1.6

0.35 + (L/B)4

)(D

B

)2]

Rocking about x-axis ηyy =[1.0 + D

B+( 1.6

0.35 + L/B

)(D

B

)2]

Table C-7: Embedment Correction Factors for Static Stiffness of Rigid Footings

Degree of freedom Dynamic Stiffness Modifiers

Translation along z-axis αz = 1.0−

(0.4 + 0.2

L/B

)a2

0( 10

1 + 3(L/B − 1)

)+ a2

0

Translation along y-axis αy = 1.0Translation along x-axis αx = 1.0

Torsion about z-axis αzz = 1.0−

(0.33 + 0.03

√L/B − 1

)a2

0( 0.81 + 0.33(L/B − 1)

)+ a2

0

Rocking about y-axis αyy = 1.0−

0.55a20(

0.6 + 1.41 + (L/B)3

)+ a2

0

Rocking about x-axis αxx = 1.0−

(0.55 + 0.01

√L/B − 1

)a2

0(2.4− 0.4

(L/B)3

)+ a2

0

Table C-8: Dynamic stiffness modifiers for Static Stiffness of Rigid Footings

48


Degree of freedom Dynamic Stiffness Modifiers

Translation along z-axis βz =[

4 [ψ(L/B) + (D/B)(1 + L/B)](Kz,emb/GB)

] [a0

2αz

]

Translation along y-axis βy =[

4 [(L/B) + (D/B)(1 + ψL/B)](Ky,emb/GB)

] [a0

2αy

]

Translation along x-axis βx =[

4 [(L/B) + (D/B)(ψ + L/B)](Kx,emb/GB)

] [a0

2αx

]

Torsion about z-axis βzz =

(4/3)[3(L/B)(D/B) + ψ(L/B)3(D/B) + 3(L/B)2(D/B) + ψ(D/B) + (L/B)3 + (L/B)

]a2

0(Kzz,emb

GB3

)[( 1.41 + 3(L/B − 1)0.7

)+ a2

0

]

[a0

2αzz

]

Rocking about y-axis βyy =

(4/3)[(L/B)3(D/B) + ψ(D/B)3(L/B) + (D/B)3 + 3(D/B)(L/B)2 + ψ(L/B)3] a2

0(Kyy,emb

GB3

)[( 1.81 + 1.75(L/B − 1)

)+ a2

0

] +

(43

)(L

B+ ψ

)D

B

3

Kyy,emb

GB3

[a0

2αzz

]

Rocking about x-axis βxx =

(4/3)[(D/B) + (D/B)3 + ψ(L/B)(D/B)3 + 3(D/B)(L/B) + ψ(L/B)

]a2

0(Kxx,emb

GB3

)[( 1.81 + 1.75(L/B − 1)

)+ a2

0

] +

(43

)(ψL

B+ 1

)D

B

3

Kxx,emb

GB3

[a0

2αzz

]

Table C-9: Radiation damping ratios for Static Stiffness of Rigid Footings

Dynamic stiffness modifiers, αj, are related to the dimensionless frequency for footings,a0, which is calculated by:

a0 = ωB

Vs

ω = T1

2 · π

(C-10)

The stiffness coefficient kj and damping coefficient cj are determined by:

kj = Kj · αj · ηj

cj = 2 · βj · kj

ω

(C-11)

The soil hysteretic damping, βs, is taken as zero.

49

50

Degree of freedom Kj,sur ηj αj βj kj cj

Translationalong z-axis

1.323E+05

1.217E+00 9.992 E-01 7.408 E-02 1.609

E+05 3.038 E+04

Translationalong y-axis

1.067E+05

1.434E+00

1.000E+00 6.084 E-02 1.530

E+05 2.372 E+04

Translationalong x-axis

1.067E+05

1.434E+00

1.000E+00 6.082 E-02 1.529

E+05 2.371 E+04

Torsion aboutz-axis

3.049E+11

2.023E+00 9.945 E-01 2.636 E-04 6.132

E+11 4.119 E+08

Rocking abouty-axis

2.105E+11

1.498E+00 9.962 E-01 1.380 E-03 3.141

E+11 1.105 E+09

Rocking aboutx-axis

2.092E+11

1.499E+00 9.963 E-01 1.387 E-03 3.125

E+11 1.105 E+09

Table C-10: Springs and dashpots foundation

Distribution of vertical springs and dashpots around the foundation:

Stiffness intensity kz,i 2.159 E-02 [kips/inch/inch2]Damping intensity cz,i 4.076 E-03 [kips seconds/inch/inch2]Length end region (=B/6) lend 454 [in]End length ratio Re 0.33 [-]

Table C-11: Sum of vertical springs and dashpots and end regions

Figure C-5: ASCE 41-13 - Vertical modelling stiffness modelling of shallow bearing footings

50


Figure C-6: Middle and end regions of foundation

The increase in spring stiffness in the end regions, Rk, is calculated by:

Rk,yy =

(3kyy

4kizBL

3

)− (1−Re)3

1− (1−Re)3 = 4.053

Rk,xx =

(3kxx

4kizBL

3

)− (1−Re)3

1− (1−Re)3 = 4.029

(C-12)

The correct for overestimation of rotational damping, dashpot intensities over the fulllength and width of the foundation are scaled by a factor Rc, computed as:

Rc,yy =

(3cyy

4cizBL

3

)

Rk,yy

(1− (1−Re)3

)+ (1−Re)3 = 0.0187

Rc,xx =

(3cxx

4cizBL

3

)

Rk,xx

(1− (1−Re)3

)+ (1−Re)3 = 0.188

(C-13)

Stiffness intensities and dashpot intensities in the middle and end regions:

51

52

Area kz,i cz,i Rk,jj Rc,jj kz,jj cz,jj

[kips/in3] [kipss/in3] [-] [-] [kips/in3] [kips

s/in3]Yellow (mid)region 2.159E-02 4.076E-03 1.000E+00 1.878E-02 2.159E-02 7.652E-05

Orange (edge)region (yy) 2.159E-02 4.076E-03 4.053E+00 1.873E-02 8.748E-02 3.093E-04

Blue (edge)region (xx) 2.159E-02 4.076E-03 4.029E+00 1.882E-02 8.696E-02 3.091E-04

Green (corner)region (xx/yy) 2.159E-02 4.076E-03 4.041E+00 1.878E-02 8.722E-02 3.092E-04

Table C-12: Vertical stiffness and dashpot intensities in middle and end regions

The horizontal springs and dashpots at the base-slab level are evaluated as kj,sur = kj/ηj ,the dashpots at the base-slab level are evaluated as cj,sur = cj/ηj. The springs aredistributed in both horizontal directions uniformly around the perimeter of the foundation.The remaining lateral stiffness and damping from the embedment is distributed over theheight of the basement wall (i.e. the depth of the embedment). The remaining stiffness istaken as kj(1− 1/ηj), the remaining damping is taken as cj(1− 1/ηj).

Horizontal dashpots ki,j [kips/inch]ky,slab 1.067E+05ky,emb 4.626E+04kx,slab 1.067E+05kx,emb 4.624E+04

Table C-13: Sum of horizontal springs stiffnesses

Horizontal dashpot ci,j [kips-seconds/inch]cy,slab 1.655E+04cy,emb 7.174E+03cx,slab 1.654E+04cx,emb 7.169E+03

Table C-14: Sum of horizontal dashpots

52


The sum of the springs stiffnesses and dashpots in a given direction should match thetotal stiffness from the impedance function. Seismic excitation is applied to the bathtubusing the free-field motion or a foundation input motion derived considering kinematicinteraction effects.

53

54

54

Part II

Input - APPENDICES

55

Appendix D

Hazard identification, target responsespectra and selecting ground motion

records

D-1 Probabilistic Seismic Hazard Analysis

The probabilistic seismic hazard analysis (PSHA) is composed of the following steps:

1. Identification of all earthquake sources, both faults and areal regions, capable ofproducing damaging ground motions to a site.

2. Characterization of the distribution of earthquake magnitudes. The cumulativedistribution function is computed for the magnitudes of earthquakes that are largerthan some minimum magnitudemmin and smaller than the upper bound of earthquakemagnitudes in a region mmax due to the finite size of the source faults:

FM(m) = P (M ≤ m|mmin < M < mmax)

= Rate of earthquakes with mmin < M ≤ m

Rate of earthquakes with mmin < M ≤ mmax

(D-1)

The cumulative distribution function can be used to compute the probabilities ofoccurrences of a discrete set of magnitudes, where the probabilities associated withall magnitudes between mj and mj+1 are assigned to the discrete value mj:

P (M = mj) = FM(mj+1)− FM(mj) (D-2)

57

58

3. Characterization of the distribution of source-to-site distances associated with po-tential earthquakes. For a given earthquake source, it is assumed that earthquakeswill occur with equal probability at any location on the fault.

4. Prediction of the resulting probability distribution of ground motion intensity bya prediction model, as a function of predictor variables such as the earthquake’smagnitude and distance. Prediction models take the following general form:

ln IM = ln IM(M,R, θ) + σ(M,R, θ) · ε (D-3)

where ln IM is the natural log of the ground motion intensity of interest (i.e. spectralacceleration). The predicted mean ln IM(M,R, θ) and standard deviation σ(M,R, θ)are functions of the earthquake’s magnitude (M), distance (R) and other parameters(generically referred to as θ). The standard normal random variable ε represents theobserved variability in ln IM ; positive values of ε produce larger-than-average valuesof ln IM . For a discrete set of intensity measures, the probability of equalizing acertain intensity measure is:

P (IM = xj) = P (IM > xj)− P (IM > xj+1) (D-4)

5. Combination of uncertainties in earthquake magnitudes, source-to-site distances andground motion intensity, using the total probability theorem:

λ(IM > x) =nsources∑

i=1λ(Mi > mmin)

mmax∫

mmin

rmax∫

0

P (IM > x|m, r)fM,R(m, r)

(D-5)

The discretized PSHA equation is:

λ(IM > x) =nsources∑

i=1λ(Mi > mmin)

nM∑

j=1

nR∑

k=1P (IM > x|mj, rk)

· P (Mi = mj)P (Ri = rk)(D-6)

The PSHA equations give the rate of exceeding IM levels at varying intensity usingthe knowledge about rates of occurrence of earthquakes, the possible magnitudesand distances of those earthquakes, and the distribution of ground shaking intensitydue to those earthquakes.The conditional distribution of magnitude M and distance R given IM > x is:

λ(IM > x,M = m,R = r) =nsources∑

i=1λ(Mi > mmin)P (IM > x|mj, rk)

· P (Mi = m)P (Ri = r)(D-7)

58

Appendix E

Input Code Design

E-1 Input parameters

E-1-1 Site input parameters

The response spectrum is site-specific and the site latitude and longitude need to bespecified to determine the spectral response acceleration parameters. The site location isspecified in Table E-1.

Site latitude 34.050Site longitude -118.250

Table E-1: Site location

E-1-1-1 Risk Category

The Risk Category of the archetype structure is determined as Risk Category II, accordingto ASCE 7-10 Table 1.5-1.

E-1-1-2 Importance Factor

The Importance Factor Ie accounts for the degree of risk to human life, health and welfareassociated with damage to property or loss of use or functionality. According to ASCE7-10 Table 1.5-2, the Importance Factor Ie was determined Ie = 1.0.

59

60

E-1-1-3 Site Classification

With ASCE 7-10 Table 20.3-1 Site Classification and the average shear wave velocity Vs,30= 233 [m/s] from Appendix C, the Site Class is determined to be Site Class D Stiff soil.

E-1-2 Building input parameters

The building height and effective seismic weight W , as determined by ASCE 7-10 Section12.7.2, are input parameters for the modal response spectrum analysis and given inEquation E-1. The seismic weight was calculated by Perform 3D.

hn = 4918 [in]W = 1.0(DL+ SDL) = 91, 000 [kips] (E-1)

E-1-3 Seismic Performance Factors

The calculated forces on structures through linear methods of analysis are much greaterthan the specified design forces. The code relies upon ductility, arising due to inelasticmaterial behaviour, and overstrength, the additional strength in structures above thedesign strength, to account for this difference. Current building codes apply seismicperformance factors to estimate strength and deformation demands on lateral force-resisting systems that are designed using linear methods of analysis but are responding inthe nonlinear range.

The seismic performance factors include the importance factor (Ie), the response modifica-tion factor (R-factor), the system overstrength factor (Ωo), and deflection amplificationfactor (Cd). The importance factor Ie depends on the functional purpose of the building,characterized by hazardous consequences of its failure, post-earthquake functional needs,historical value or economic importance. The importance factor is intended to provide fora lower inelastic demand on a structure which should result in lower levels of structuraland non-structural damage. The response modification factor R is the representative ofthe inherent overstrength and global ductility of the seismic-force-resisting system. Alower R-factor implies a less ductile system.

Properly detailed structures also contain elements which are not capable of safely resistingthe seismic demands through inelastic behaviour. These elements must be designed withsufficient strength to resist the seismic demands in the elastic range. The overstrengthfactor Ωo approximates the overstrength in typical systems to account for increased forcedemands in elements that are intended to remain elastic.

60

E-1 Input parameters

Seismic Performance Factor Value Comments/Reference

Occupancy Category IIASCE 7-10 Table 1.5-1; Occupancycategory II includes the majority ofbuildings constructed under IBC

Seismic Importance Factor Ie 1.0 ASCE 7-10 Table 1.5-2

Seismic force-resisting system

Building Frame,Special

ReinforcedConcrete Shear

Walls

Building Frame System is a structuralsystem with essentially complete spaceframe providing support for verticalloads, seismic force-resistance providedby shear walls Table 12.2-1

Response modification factor R 6 Special structural wall designedto comply with seismicrequirements in 21.1 of ACI318Section 2.2

Overstrength factor Ωo 2.5Deflection amplification factor Cd 5

Seismic Design Category D IBC Table 1613.5.6; Occupancy Cate-gory II and SDS=1.00≥0.50

Redundancy factor ρ 1.0

Table E-2: Seismic Performance Factors

E-1-3-1 Structural system selection [ASCE 7-10, 2010]

Different seismic force-resisting systems are permitted to be used to resist seismic forcesalong each of the two orthogonal axes of the strucure. Where different systems areused, the respective R, Cd, and Ωo coefficients shall apply to each system. The selectedstructural system is Building Frame, Special Reinforced Concrete Shear Walls, theResponse Modification Factor R, Overstrength factor Ωo and Deflection AmplificationFactor Cd are given in Equation E-2.

R = 6Ωo = 2.5Cd = 5

(E-2)

E-1-3-2 Seismic Design Category

The Seismic Design Category was determined according to ASCE 7-10 Table 11.6-1 andTable 11.6.2; Seismic Design Category D was selected.

61

62

E-1-3-3 Redundancy Factor

From ASCE 7-10 Section 12.3.4.2 condition b and Seismic Design Category D, theredundancy factor ρ was determined ρ = 1.0.

E-1-4 Design spectral acceleration parameters

From ASCE 7-10 Figure 22-1 and 22-2 the spectral response acceleration parameters forshort periods and 1 [sec] periods, SS and S1 respectively, are derived. Site coefficientsFa and Fv are derived from ASCE 7-10 Table 11.4-1 and 11.4-2. The MCER spectralresponse acceleration parameter for short periods SMS and at 1 [sec] (SM1), adjusted forSite Class effects, is determined by Equation E-3.

SMS = FaSS

= 2.402 [g]SM1 = FvS1

= 1.264 [g]Where,Fa = 1.000 [-]Fv = 1.500 [-]SS = 2.402 [g]S1 = 0.843 [g]

(E-3)

Design earthquake spectral response acceleration parameter at short period, SDS and at 1[sec] SD1, shall be determined by Equation E-4.

SDS = 23SMS

= 1.601 [g]

SD1 = 23SM1

= 0.843 [g]

(E-4)

E-1-4-1 Design Spectral Periods

To define the design response spectrum, the approximate fundamental period of thestructure T , T0, TS and the long-period transition period TL have to be determined. Theapproximate fundamental period of the structure T is defined in Equation E-5.

T = 4.90 [sec] (E-5)

62

E-2 Design Response Spectrum

The periods T0 and TS are determined by Equation E-6.

T0 = 0.2SD1

SDS

= 0.105 [sec]

TS = SD1

SDS

= 0.527 [sec]

(E-6)

The long-period transition period TL is derived from ASCE 7-10 Figure 22-12 and definedin Equation E-7.

TL = 8 [sec] (E-7)


The design response spectrum curve is developed according to ASCE 7-10 Figure 11.4-1,as shown in Figure E-1.

Figure E-1: Code Design Response Spectrum (ASCE 7-10, Figure 11.4-1)

1. For periods less than T0 the design spectral response acceleration, Sa,shall be takenas:

Sa = SDS

(0.4 + 0.6 T

T0

)(E-8)

2. For periods greater than or equal to T0 and less than or equal to TS, the designspectral response acceleration, Sa, shall be equal to SDS.

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64

3. For periods greater than TS, and less than or equal to TL, the design spectral responseacceleration, Sa, shall be taken as:

Sa = SD1

T(E-9)

4. For periods greater than TL, Sa shall be taken as:

Sa = SD1 · TL

T 2(E-10)

The design response spectrum is given in Figure E-2.

Figure E-2: Design Response Spectrum

The MCER response spectrum is determined by multiplying the design response spectrumby 1.5 and given in Figure E-3.

64


Figure E-3: MCER Response Spectrum

65

66

E-2-1 Response spectrum input variables

Fa = Short-period site coefficient at 0.2 [sec]-periodFv = Long-period site coefficient at 1.0 [sec]-periodSDS = Design spectral response acceleration parameter at short periodsSD1 = Design spectral response acceleration parameter at a period of 1 [sec]SS = Mapped MCER spectral response acceleration parameter

at short periodsS1 = Mapped MCER spectral response acceleration parameter

at a period of 1 [sec]SMS = MCER, 5 percent damped, spectral response acceleration

parameter at short periods adjusted for site class effectsSM1 = MCER, 5 percent damped, spectral response acceleration

parameter at a period of 1 [sec] adjusted for site class effectsT = Fundamental period of the structureTL = Long-period transition period

(E-11)

E-3 Seismic Base Shear

The seismic base shear V in a given direction is determined by Equation E-12, accordingto ASCE 7-10 Equation 12.8-1.

V = Cs ·W= 6406 [kips]

Where,Cs = 0.0704 [-]W = 91000 [kips]

(E-12)

66

E-3 Seismic Base Shear

E-3-1 Seismic Response Coefficient

The seismic response coefficient Cs is determined by Equation E-13, according to ASCE7-10 Equation 12.8-2.

Cs1 = SDS(R

Ie

)

= 0.267 [-]Where,SDS = 1.601 [sec]R = 6 [-]Ie = 1.0 [-]

(E-13)

The value of Cs computed in accordance with Equation E-13 must be smaller than thevalue Cs2 computed in Equation E-14, according to ASCE 7-10 Equation 12.8-4.

Cs2 = SD1

T(R

Ie

) for T ≤ TL

= 0.0551 [-]

Cs2 = SD1TL

T 2(R

Ie

) for T > TL

Where,SD1 = 0.843 [sec]T = 2.55 [sec]

< TL

TL = 8.0 [sec]

(E-14)

The value of Cs shall not be less than the values Cs3 and Cs4 computed in Equation E-15,according to ASCE 7-10 Equations 12.8-5 & 12.8-6.

Cs3 = 0.044 · SDS · Ie ≥ 0.01= 0.0704 [-]

Cs4 = 0.5 · S1(R

Ie

) [-] For S1 > 0.6 [g]

= 0.0703 [-]Where,S1 = 0.843

(E-15)

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68

From Equations E-13, E-14, and E-15 follows that Cs4 governs and Cs4 = Cs = 0.0704.

E-3-1-1 Period Determination

The fundamental period of the structure T in the direction under consideration wasdetermined by Perform 3D, and must not exceed the product of the coefficient for upperlimit on calculated period (Cu) from ASCE 7-10 Table 12.8.2.1 and the approximatefundamental period Ta.The coefficient for upper limit on calculated period Cu = 1.4 for SD1 ≤ 0.4 accordingto ASCE 7-10 Table 12.8.2.1. The approximate fundamental period Ta is calculated byEquation E-16, according to ASCE 7-10 Equation 12.8-7, for structure type all otherstructural systems.

Ta = Cthxn

= 1.818 [sec]Ct = 0.02 [-]hn = 409.833 [feet]x = 0.75 [-]

(E-16)

The fundamental period in two directions and the upper limit of the fundamental periodare given in Equation E-17.

TNS = 4.43 [sec]TEW = 3.44 [sec]Tmax = Cu · Ta

= 2.55 [sec]

(E-17)

From Equation E-17 follows that the fundamental period of the structure T = Tmax = 2.55[sec] in both directions.

E-4 Modal Response Spectrum Analysis

An analysis was conducted by Perform 3D to determine the natural modes of vibrationfor the structure, according to ASCE 7-10 Section 12.9. From Table E-4 and E-5 itis found that a total of 35 modes must be included to obtain a combined modal massparticipation factor of at least 90 percent of the actual mass in each of the orthogonalhorizontal direction of response considered by the model.Think about whether this should even be included in an appendix, or whetherit can be ignored completely

68


1. Find the structural characteristics of the building. Compute the stiffness matrix K,the damping matrix C, and the mass matrix M.

The equation of motion for a damped system is given by:

Mx+ Cx+ Kx = F (t) = −mxg (E-18)

2. Find the eigenvalues and eigenvectors of the structure.

The undamped eigenvalues and eigenvectors of the MDOF system are found from thecharacteristic equation. The eigenvalues λi are given by ω2m

k. For a 3x3 matrix, 3

eigenfrequencies are found. The eigenvectors Xi are found by implementing the eigenvaluesinto the algebraic equation.

(−ω2iM +K) ·

X11X21X31

= (−ω2

iM +K)X1 (E-19)

The eigenmatrix E is given by:

E =[X1 X2 X3

](E-20)

3. Find the modal participation factors

Using equation E-21, the modal participation factors can be determined. In the event ofan earthquake, there is mass participating in the forcing function and mass participatingin inertia effects. The modal participation factor Γ is a comparison of these two masses.

ui + c∗iim∗ii

ui + ω2i ui = F ∗i (t)

m∗ii(E-21)

where F ∗i (t) = xTi F , c∗ii = xT

i Cxi, m∗ii = xTi Mxi.

The earthquake forcing function is defined as:

F = − a ·M · e (E-22)

where M is the mass matrix, a the ground acceleration and e is a vector with ei = 1.0 forall i.

The modal participation factor is given by:

Γk = ETi Me

ETME= Σmi · xk,i

Σmi · x2k,i

(E-23)

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70

4. Calculate the static equivalent nodal force

The static equivalent nodal force is given by:

Fki = Sa · Γk ·mi · φk,i (E-24)

5. Combine the modes

Several methods can be used for obtaining the total peak response quantity for a MDOFsystem. The most commonly used methods are the Absolute Sum (ABSSUM) method,the Square Root of Sum of Squares (SRSS) method, and the Complete QuadraticCombination (CQC) method.

The ABSSUM method assumes that all modal peaks occur at the same time and the peakresponses of all the modes are added. This method provides a conservative estimate ofthe total peak response quantity and provides an upper bound value of the total response.In the SRSS method, the maximum response is obtained by square root of sum of squareof response in each mode of vibration. The SRSS method of combining maximum modalresponses is fundamentally sounds where the modal frequencies are well separated, butyields poor results where frequencies of major contributing modes are very close together.The CQC method is better capable of combining maximum modal responses where closelyspaced modes have significant cross-correlation of translational and torsional response.The CQC method is therefore chosen to combine the modes.

The CQC method is a procedure where the maximum response from all modes is calculatedusing the correlation coefficient αij:

rmax =√√√√

n∑

i=1

n∑

j=1riαijrj (E-25)

where ri and rj are maximum responses in the ith and jth mode, respectively and αij isthe correlation coefficient given by:

αij = 8 · (ξiξj)12 · (ξi + βξj) · β

23

(1− β2)2 + 4ξiξjβ(1 + β2) + 4(ξ2i + ξ2

j )β2 (E-26)

where ξi and ξj are damping ratio in ith and jth modes of vibration, respectively and β isgiven by:

β = ωi

ωj

(ωj > ωi) (E-27)

The range of coefficient, αij is 0 < αij < 1 and αii = αjj =1.

70


E-4-1 Comparison Modal Analysis SAP2000 vs. Perform-3D

Mode Perform 3D SAP2000 Ratio SAP/Perform1 4.93 5.23 106%2 3.91 4.16 106%3 2.67 2.72 102%4 1.02 1.06 104%5 0.94 0.98 104%6 0.88 0.88 100%7 0.48 0.48 100%8 0.47 0.48 102%9 0.43 0.45 105%10 0.32 0.31 97%11 0.30 - -12 0.26 - -13 0.23 - -14 0.21 - -15 0.18 - -

Table E-3: Comparison of modal analysis results SAP2000 vs. Perform-3D

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72

E-4-2 Vertical distribution of Seismic Forces

The lateral seismic force Fx induced at any level is determined by Equation E-28, accordingto ASCE 7-10 Equation 12.8-11.

Fx = Cvx · VWhere,Cvx = Vertical distribution factor

= wxhkx

n∑i=1

wihki

And,wi, wx = Portion of W located at Level i or xhi, hx = Height from the base to Level i or xk = Exponent related to structure period

= 2

(E-28)

E-4-2-1 Diaphragm Design Force

Diaphragms shall be designed for both the shear and bending stresses resulting fromdesign forces. At diaphragm discontinuities, such as openings and re-entrant corners, thedesign shall assure that the dissipation or transfer of edge (chord) forces combined withother forces in the diaphragm is within shear and tension capacity of the diaphragm.Floor and roof diaphragms shall be designed to resist design seismic forces from thestructural analysis, but shall not be less than that determined in accordance with EquationE-29, according to ASCE 7-10 Equation 12.10-1.

Fpx =

x∑i=1

Fi

x∑i=1

wi

· wpx (E-29)

The force determined from Equation E-29 shall not be less than Fpx,min as calculated byEquation E-30, , according to ASCE 7-10 Equation 12.10-3, and shall not be larger thanFpx,max as calculated by Equation E-31, according to ASCE 7-10 Equation 12.10-3.

Fpx = 0.2 · SDSIewpx (E-30)

Fpx = 0.4 · SDSIewpx (E-31)

72


Where the diaphragm is required to transfer design seismic force from the vertical resistingelements above the diaphragm to other vertical resisting elements below the diaphragmdue to offsets in the placement of elements or to changes in relative lateral stiffness in thevertical elements, these forces shall be added to those determined by Equation E-29. Theredundancy factor ρ applies to the design of diaphragms in structures assigned to SDCD, E, or F. The redundancy factor is 1.0 for inertial forces calculated in accordance withEquation E-29. For transfer forces, the redundancy factor shall be the same as that usedfor the structure, ρ = 1.0.

E-4-3 Transfer forces

For the first three modes, the difference between the shear force in the walls just belowand above the ground floor diaphragm is almost equal to the shear force in the basementwalls. The force in the basement wall is the result of transfer force from the shear wallsto the basement walls through the podium diaphragm, and not by the inertial force ofthe podium diaphragm. For higher modes, the mass of the basement is activated andthe shear force in the basement walls is caused by both the transfer force from the shearwalls and mass inertia of the basement.The transfer force and inertial force components in the basement walls for contributingloads in H1- and H2-direction are given in Table E-6 and E-7, respectively.The individual transfer force components TFm1, TFm2, . . . TFmn and initial force compo-nents IFm1, IFm2, . . . IFmn are given in Table E-6 and E-7. The components are combinedby the square root of the sum of squares (SRSS) and given in Tables E-6 and E-7 fordirection H1 and H2. The residual combination component is also given, the combinedtransfer force, inertial force and combination component gives the total force, as shown inEquation E-32.

ForceBW =√F 2

m1 + F 2m2 + . . .+ F 2

mn

=√

(TFm1 + IFm1)2 + (TFm2 + IFm2)2 + . . .+ (TFmn + IFmn)2

=√TF 2

m1 + IF 2m1 + 2 · TFm1 · IFm1 + TF 2

m2 + IF 2m2+

2 · TFm2 · IFm2 + . . .+ TF 2mn + IF 2

mn + 2 · TFmnIFmn

=√√√√

n∑

i=1TF 2

mi +n∑

i=1IF 2

mi +n∑

i=12 · TFmi · IFmi

(E-32)

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74

Mode Period H1-Direction H2-Direction Cumulative CumulativeNo. (sec) for Mode for Mode H1-Direction H2-Direction1 4.434 0.4126 0.0000 0.4126 0.00002 3.439 0.0000 0.4284 0.4126 0.42843 2.423 0.0002 0.0001 0.4128 0.42854 0.947 0.1344 0.0030 0.5472 0.43155 0.879 0.0054 0.1298 0.5526 0.56136 0.802 0.0007 0.0002 0.5533 0.56157 0.465 0.0094 0.0076 0.5627 0.56918 0.442 0.0055 0.0262 0.5682 0.59539 0.411 0.0422 0.0154 0.6105 0.610810 0.292 0.0003 0.0004 0.6107 0.611211 0.284 0.0029 0.0289 0.6136 0.640112 0.246 0.0465 0.0060 0.6602 0.646113 0.213 0.0010 0.0022 0.6612 0.648314 0.205 0.0044 0.0402 0.6656 0.688515 0.179 0.0906 0.0270 0.7562 0.715516 0.169 0.0446 0.0938 0.8008 0.809317 0.165 0.0000 0.0127 0.8008 0.822018 0.155 0.0514 0.0084 0.8522 0.830419 0.150 0.0001 0.0000 0.8523 0.830420 0.149 0.0021 0.0234 0.8544 0.853821 0.134 0.0000 0.0002 0.8544 0.854022 0.124 0.0000 0.0001 0.8544 0.854123 0.121 0.0003 0.0000 0.8547 0.854124 0.119 0.0000 0.0000 0.8547 0.854125 0.116 0.0000 0.0000 0.8547 0.8541

Table E-4: Natural modes of vibration 1-25; modal mass participation in H1- and H2-direction

74


Mode Period H1-Direction H2-Direction Cumulative CumulativeNo. (sec) for Mode for Mode H1-Direction H2-Direction26 0.110 0.0000 0.0001 0.8547 0.854227 0.106 0.0008 0.0000 0.8555 0.854228 0.105 0.0432 0.0016 0.8987 0.855829 0.104 0.0031 0.0281 0.9018 0.883930 0.100 0.0000 0.0000 0.9019 0.883931 0.098 0.0010 0.0019 0.9028 0.885832 0.097 0.0244 0.0001 0.9272 0.885933 0.097 0.0050 0.0001 0.9322 0.886034 0.097 0.0000 0.0000 0.9322 0.886135 0.096 0.0002 0.0485 0.9324 0.934636 0.086 0.0000 0.0001 0.9324 0.934637 0.085 0.0006 0.0017 0.9329 0.936338 0.083 0.0047 0.0012 0.9376 0.937539 0.082 0.0035 0.0019 0.9411 0.939440 0.081 0.0002 0.0004 0.9413 0.939741 0.081 0.0001 0.0001 0.9415 0.939842 0.080 0.0003 0.0020 0.9417 0.941843 0.080 0.0004 0.0019 0.9421 0.943744 0.079 0.0011 0.0005 0.9432 0.944245 0.079 0.0000 0.0000 0.9432 0.944246 0.078 0.0000 0.0000 0.9432 0.944247 0.078 0.0014 0.0000 0.9446 0.944248 0.078 0.0001 0.0013 0.9448 0.945549 0.078 0.0003 0.0012 0.9451 0.946750 0.078 0.0000 0.0000 0.9451 0.9467

Table E-5: Natural modes of vibration 26-50; modal mass participation in H1- and H2-direction

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76

Force L1 Force B1 Transfer force Inertial ForceMode 1 8669 1841 6828 10Mode 4 13226 9025 4202 383Mode 7 10514 8364 2149 1399

Mode 11 7191 7104 87 3292

Table E-6: Distribution of transfer force and inertial force in diaphragm - Direction H1

Force L1 Force B1 Transfer force Inertial ForceMode 2 11601 5827 5774 69Mode 5 13704 9997 3707 463Mode 8 8588 6722 1865 1165

Mode 12 5398 4978 420 2133

Table E-7: Distribution of transfer force and inertial force in diaphragm - Direction H2

Total force Contributiontransfer force (SRSS)

Contribution inertialforce (SRSS)

Contributioncombinationcomponent (SRSS)

Mode 1-3 6837 6828 10 362Mode 1-6 8232 8017 384 1831Mode 1-9 8964 8300 1451 3061

Mode 1-12 9580 8300 3598 3153

Table E-8: Total force and transfer force component - Direction H1

Total force Contributiontransfer force (SRSS)

Contribution inertialforce (SRSS)

Contributioncombinationcomponent (SRSS)

Mode 1-3 5843 5774 69 894Mode 1-6 7179 6862 468 2058Mode 1-9 7792 7111 1255 2929

Mode 1-12 8200 7123 2475 3220

Table E-9: Total force and transfer force component - Direction H2

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E-4-3-1 MRSA for different number of modes

Shear Force L0 Transfer Force Basement Wall - H1 Basement Wall - H2Total - H1 8668 6828 6712 125Total - H2 11601 5774 142 5701North - H1 4101 2911 3361 63South - H1 3957 2877 3351 63West - H2 5587 2501 71 2848East - H2 5744 2774 71 2854

Table E-10: MRSA - first 3 modes



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78

Shear Force L0 Transfer Force Basement Wall - H1 Basement Wall - H2Total - H1 18990 6549 8801.4 163.093Total - H2 19903 6521 177.057 7615North - H1 9608.6 2978.3 4393.1 81.432South - H1 10014 2918.8 4408.3 81.661West - H2 10106 2907.5 88.426 3800.7East - H2 10363 3337.2 88.631 3814.3






78




79

80

80

Appendix F

Input PEER/TBI Analysis

Select models to generate target spectrum: PEER NGA-West2 Spectrum

Ground Motion Prediction Equation WeightAbrahamson-Silva-Kamai ’14 1.0Boore-Stewart-Seyhan-Atkinson ’14 1.0Campbell-Bozorgnia ’14 1.0

Table F-1: Ground Motion Predictions Equations for PEER NGA-West2 Spectrum

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82

Input variable NotesDamping ratio = 5 % Default value (only value available)Region = G/C Global/CaliforniaFault Type = SS Strike Slip

Magnitude = 7.19 [-] From USGS hazard deaggregation (alsoused for CS computation)

Rrup = 24.2 [km] Rrup = Rbar (from USGS hazarddeaggragation)

Rx = 24.2 [km] Rx = RbarRy0 = 24.2 [km] Default valueRjb = 24.2 [km] Rjb = Rbar

Ztor = 0 [km] Default value (also used for CScomputation)

Width = 10 [km] Default value

DIP = 90 [deg] Default value (also used for CScomputation)

Vs30 = 233 [m/s] From Chapter 8Vs30 type = M Measured

Z1.0 = 0.49 [km] Calculated default (based on Vs30)Z2.5 = 2.0 [km] Default (also used for CS computation)Zhyp = 9.0 [km] Calculated default (based on Vs30)

Hanging wall = No

Epsilon = 2.1752 [-] From USGS hazard deaggregation (alsoused for CS computation)

GMPE average = G [-] Geometric (default)

Table F-2: Input parameters to compute UHS

The uniform hazard spectrum is plotted in Figure F-1.

82

Figure F-1: Uniform Hazard Spectrum for site location

Secondary parameters to scale ground motions are M , R and local site conditions, suchas Vs,30. Search parameters were defined, as described in Table F-3.

Search parameters DefinitionFault Type All typesMagnitude (min, max) (6,8)Rjb (min, max) (10,30)Rrup (min, max) (10,30)Vs30 (min, max) (180,360)

Table F-3: Search parameters

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84

Search parameters DefinitionMaximum no. of records 7Initial Scale Factor (min, max) (0.5,3.5)

Table F-4: Additional characteristics parameters

Search parameters DefinitionSpectral Ordinate RotD50Damping Ratio 5%Suite Average Arithmetic

Table F-5: Suite

Search parameters DefinitionScaling method Minimize MSE

Table F-6: Suite

Computed weight mean squared error of record, and suite average, with respect to targetspectrum.

Search parameters DefinitionPeriod points 0.1, 1, 10Weights 1, 1, 1

Table F-7: Weight function

84

F-1 Selected Ground Motions

F-1 Selected Ground Motions

Recordnumber Earthquake

NGARecord

SequenceNumber

ScaleFactor

Timesteps

Totalnumber of

points

1 Superstition Hills,USA (1987) 721 2.680 0.005 8000

2 Superstition Hills,USA (1987) 728 3.266 0.005 8000

3 Loma Prieta, USA(1989) 776 2.513 0.005 11991

4 Landers, USA (1992) 900 3.262 0.020 2200

5 Kocaeli, Turkey(1999) 1158 2.156 0.005 5437

6 Darfield, NewZealand (2010) 6890 3.055 0.005 30000

7 Darfield, NewZealand (2010) 6953 2.835 0.005 10800

Table F-8: Suite of seven ground motions at MCE-level (RP = 2475 years; T1 = 4.90 [sec]),selected and scaled to match Uniform Hazard Spectrum

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86

F-2 Results UHS

Figure F-2: UHS - GM01 - Shear in podium diaphragm

Figure F-3: UHS - GM01 - Shear (2) in podium diaphragm

86

F-2 Results UHS

Figure F-4: UHS - GM01 - Deflection in Direction H1

F-2-1 Combined shear forces in diaphragms

Figure F-5: Diaphragm forces from Perform 3D - UHS-based ground motions 01-04

87

88

Figure F-6: Diaphragm forces from Perform 3D - UHS-based ground motions 05-07

Figure F-7: Diaphragm demands from Perform 3D

88

Appendix G

Input Structural Reliability Analysis

G-1 Results PSHA

Exceedanceprobability

ReturnPeriod[years]

Sa [g]Mean

MagnitudeM

Mean DistanceR0 [km] ε0 [-]

50 % in 30 years(SLE) 43 0.0321 7.15 56.7 0.10

20% in 50 years 224 0.0864 7.22 38.8 0.8210% in 50 years 475 0.1223 7.22 33.3 1.095% in 50 years 975 0.1625 7.21 28.9 1.332% in 50 years(MCE) 2475 0.2243 7.19 24.2 1.59

1% in 50 years 4975 0.2756 7.18 21.3 1.761% in 100 years 9950 0.3330 7.17 18.8 1.91

Table G-1: Results from PSHA for Conditional Spectrum (target spectrum)

G-1-1 Ground Motion Prediction Equations

Multiple Ground Motion Prediction Equations (GMPEs), formerly known as attenuationrelations, are typically used for PSHA computations. The USGS specify three models(Boore-Atkinson 2008, Campbell-Bozorgnia 2008, and Chiou-Youngs 2008) with equalweights. The percentage contribution for each GMPE is given in Table G-2.

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GMPE Percentage ContributionMean Hazard w/all GMPEs 100.0%Boore-Atkinson 2008 23.7 %Campbell-Bozorgnia 2008 55.7%Chiou-Youngs 2008 20.7%

Table G-2: Percentage contribution for each GMPE

G-1-1-1 Disaggregation results

Hazard disaggregation is required to compute the Conditional Spectrum (CS). The USGSprovides disaggregation calculations at the site. The parameters given in Table G-3 wereinput for the disaggregation analysis:

Input parameter ValueLatitude/Longitude 34.05/-118.25Exceedance probability 2% in 50 years (RP = 2475 yrs)Spectral Period T 5.0 [sec] (0.2 Hz)Average shear wave velocity V s30 233 [m/s]Run GMPE Deaggs Yes

Table G-3: Input parameters for disaggregation calculation for structure’s site

The disaggregation plot is given in Figure G-1.

90

G-2 Selection of Ground Motions

Figure G-1: Disaggregation results for LA site, 2% probability of exceedance in 50 years (MCE-level)

The disaggregation calculates the SA amplitude that is exceeded with a mean returnperiod of 2475 years. This SA amplitude, denoted as sa∗, will be the target value forCS computations. The mean distance, magnitude and ε0 values associated with thedisaggregation results are also used for CS computation. The disaggregation results aregiven in Table G-4.

2% in 50 yearsSpectral acceleration sa∗ [g] 0.22Mean Distance R [km] 24.2Magnitude M 7.19ε0-value 1.59

Table G-4: Disaggregation results LA site for exceedance probability of 2% in 50 years


The input parameters for computation of the target spectrum are given in Table G-5.

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92

Input parameter ValuePeriod of interest [sec] T1 4.90Magnitude of target scenario earthquake Mbar 7.19Distance corresponding to target scenario earthquake [km] Rbar 24.2ε-value for target scenario εbar 2.17Average shear wave velocity [m/s] Vs30 233Depth to top of coseismic rupture [km] Ztor 0Average dip of rupture place δ 90Rake Angle λ 180Depth to 2,5 km/s shear-wave velocity horizon Zvs 2Arbitrary component (three-dimensional structure) arb 0

Table G-5: Input parameters for target spectrum computation for the 2475-year return period

Notes Table G-5:

• The period of interest T1 is chosen equal to the fundamental period of the structure.• The ε-value corresponding to SA > sa∗ was computed by USGS using PSHA. Theεbar-value corresponding to SA = sa∗ was back-calculated with Equation G-1 usingGMPE input (Ztor, δ, λ, Vs30, Zvs, and arb) and USGS hazard and disaggregationinput (T1, Mbar, Rbar, and sa∗).

εbar(T1) = lnSa(T1)− µ(M,R, T1)σln Sa(T1) (G-1)

• For the three-dimensional structural model, the geometric mean of Sa is used asa target to select and scale two-component ground motions. Hence, the arbitrarycomponent is set arb = 0.

The input parameters for selection and scaling of ground motions are given in Table G-6.No further constraints were place on the ground motion selection other than limiting thescale factor to less than 3, with the primary selection focus being on the match of theground motion spectra to the target spectra.

92


Input parameter ValueNumber of ground motions to be selected nGm 61Spectra should be scaled before matching isScaled 1.0Maximum allowable scale factor maxScale 3Weights for error in mean and standard deviation Weights [1.0 1.5]Meta-variable of algorithm (Recommended value = 2) nLoop 2

Table G-6: Input parameters for ground motion selection for the 2475-year return period

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94

G-3 Selected Ground Motions


NGARecord

SequenceNumber

ScaleFactor

Timesteps

Totalnumber of

steps

1 Kobe, Japan (1995) 1120 2.71 0.010 40962 Chi-Chi, Taiwan (1999) 1180 1.15 0.004 375003 Chi-Chi, Taiwan (1999) 1240 2.23 0.004 375004 Chi-Chi, Taiwan (1999) 1492 0.89 0.005 180005 Chi-Chi, Taiwan (1999) 1538 2.31 0.005 180006 Mt. St. Elias, USA (1979) 1629 2.49 0.005 165947 Chi-Chi, Taiwan (1999) 1550 2.02 0.004 225008 Chi-Chi, Taiwan (1999) 1540 1.51 0.005 180009 Chi-Chi, Taiwan (1999) 1595 1.07 0.005 1180010 Chi-Chi, Taiwan (1999) 1552 2.98 0.004 3750011 Chi-Chi, Taiwan (1999) 1482 1.62 0.005 1800012 Chi-Chi, Taiwan (1999) 1531 1.72 0.005 1800013 Chi-Chi, Taiwan (1999) 1472 2.92 0.005 1800014 Kobe, Japan (1995) 1114 2.02 0.010 420015 Chi-Chi, Taiwan (1999) 1324 2.46 0.005 1800016 Chi-Chi, Taiwan (1999) 1537 1.78 0.005 1800017 Chi-Chi, Taiwan (1999) 1542 1.73 0.005 1800018 Chi-Chi, Taiwan (1999) 1554 2.65 0.004 3750019 Chi-Chi, Taiwan (1999) 1195 1.38 0.005 1800020 Chi-Chi, Taiwan (1999) 1233 2.51 0.005 18000

Table G-7: Selected set of 61 ground motions (part I) for probability of exceedance 2% in 50 years(RP = 2475 years)

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G-3 Selected Ground Motions


NGARecord

SequenceNumber

ScaleFactor

Timesteps

Totalnumber of

steps

21 Chi-Chi, Taiwan (1999) 1238 1.15 0.004 3750022 Chi-Chi, Taiwan (1999) 1477 1.46 0.005 1800023 Chi-Chi, Taiwan (1999) 1311 2.81 0.004 4275024 Chi-Chi, Taiwan (1999) 1527 1.8 0.005 1800025 Chi-Chi, Taiwan (1999) 1490 2.02 0.005 1800026 Chi-Chi, Taiwan (1999) 1544 2.85 0.005 1800027 Chi-Chi, Taiwan (1999) 1337 2.96 0.005 1740128 Chi-Chi, Taiwan (1999) 1529 1.18 0.005 1800029 Chi-Chi, Taiwan (1999) 1244 0.94 0.005 1800030 Chi-Chi, Taiwan (1999) 1476 1.69 0.005 1800031 Chi-Chi, Taiwan (1999) 1331 1.63 0.004 2975032 Chi-Chi, Taiwan (1999) 1496 2.29 0.005 1800033 Chi-Chi, Taiwan (1999) 1514 2.76 0.005 1800034 Chi-Chi, Taiwan (1999) 1515 2.13 0.005 1800035 Chi-Chi, Taiwan (1999) 1343 1.04 0.004 3900036 Chi-Chi, Taiwan (1999) 1501 1.13 0.005 1800037 Chi-Chi, Taiwan (1999) 1239 2.56 0.004 3750038 Chi-Chi, Taiwan (1999) 1526 2.72 0.005 1800039 Chi-Chi, Taiwan (1999) 1553 1.42 0.004 3750040 Chi-Chi, Taiwan (1999) 1310 1.13 0.004 33750

Table G-8: Selected set of 61 ground motions (part II) for probability of exceedance 2% in 50 years(RP = 2475 years)

95

96


NGARecord

SequenceNumber

ScaleFactor

Timesteps

Totalnumber of

steps

41 Chi-Chi, Taiwan (1999) 1488 1.58 0.005 1800042 Chi-Chi, Taiwan (1999) 1505 0.63 0.005 1800043 Chi-Chi, Taiwan (1999) 1468 2.56 0.005 1800044 Chi-Chi, Taiwan (1999) 1336 2.09 0.004 3700045 Chi-Chi, Taiwan (1999) 1548 1.22 0.005 1800046 Chi-Chi, Taiwan (1999) 1502 1.85 0.005 1800047 Chi-Chi, Taiwan (1999) 1480 1.21 0.005 1800048 Chi-Chi, Taiwan (1999) 1483 1.59 0.005 1800049 Chi-Chi, Taiwan (1999) 1543 2.22 0.005 1800050 Chi-Chi, Taiwan (1999) 1516 2.41 0.005 1800051 Chi-Chi, Taiwan (1999) 1403 1.72 0.005 1060052 Chi-Chi, Taiwan (1999) 1494 2.24 0.005 1800053 Chi-Chi, Taiwan (1999) 1534 1.72 0.005 1800054 Chi-Chi, Taiwan (1999) 1506 1.16 0.005 1800055 Chi-Chi, Taiwan (1999) 1342 1.25 0.004 5075056 Chi-Chi, Taiwan (1999) 1246 0.78 0.004 3750057 Chi-Chi, Taiwan (1999) 2649 2.96 0.005 2480058 Chi-Chi, Taiwan (1999) 1539 2.51 0.005 1800059 Chi-Chi, Taiwan (1999) 1466 2.77 0.005 1800060 Chi-Chi, Taiwan (1999) 1535 1.35 0.005 1800061 Chi-Chi, Taiwan (1999) 1497 1.84 0.005 18000

Table G-9: Selected set of 61 ground motions (part III) for probability of exceedance 2% in 50 years(RP = 2475 years)

96

Part III

Validation of analysis results -APPENDICES

97

Appendix H

Overall structural behaviour

H-1 Shear force over story height

The individual results for shear force over the story height for RP975, RP2475, andRP9950 are given in Figures H-1, H-2, and H-3, respectively.

Figure H-1: Individual results for shear force over story height for IM=RP975

99

100



100


H-1-1 Bending moment over story height

The individual results for bending moment over the story height for RP975, RP2475, andRP9950 are given in Figures H-4, H-5, and H-6, respectively.

Figure H-4: Individual results for bending moment over story height for IM=RP975

101

102



102


H-1-2 Interstory drift ratio over story height

The individual results for interstory drift ratio in the North-East corner of the core overthe story height for RP975, RP2475, and RP9950 are given in Figures H-7, H-8, and H-9,respectively.

Figure H-7: Individual results for interstory drift ratio over story height for IM=RP975

103

104



104

Appendix I

Diaphragm shear stress distribution

I-1 Usage ratio plots from NLTHA

I-1-1 NLTHAs with results similar to Case 2, 3, and 4

I-1-1-1 Deformation (magnified 500x) and usage ratio plot

In Figure I-1, the force combination V1 and V2 works in the negative X-Y direction. InFigure I-2, the transfer forces V1 and V2 work in positive X-Y direction.

Figure I-1: RP9950 - GM20 - 44.6 [sec] - Deformation magnified 500x - Case 3 (−V1 & −V2)

105

106

Figure I-2: RP9950 - GM20 - 50.0 [sec] - Deformation magnified 500x - Case 3 (+V1 & +V2)

106


I-1-1-2 Other ground motion records with results similar to Case 2, 3, and 4

The usage ratio plots from other NLTHAs were also found to display principal loadingsimilar to static case 2, 3, and 4, such as RP9950-GM20. For some records, the loadcases were found in the two orthogonal directions, as shown in Figure I-3 for RP975-GM9.Other records resulted in usage ratios predominantly in one direction, as given in FigureI-4 for RP2475-GM13 and Figure I-5 for RP975-GM8.

Figure I-3: RP975 - GM9 - Case 2 (V1 & 0.50V2 ), Case 3 (V1 & V2), in both directions, Case 4(0.50V1 & V2) in two directions

Figure I-4: RP2475 - GM13 - Case 3 (V1 & V2), predominantly one direction

107

108

Figure I-5: RP975 - GM8 - Case 3 (V1 & V2), predominantly one direction

108


I-1-1-3 Deformation (magnified 500x) and usage ratio plot

In Figure I-6, the force V1 works in the positive X-direction. In Figure I-7, the transferforce V1 works in negative X direction.

Figure I-6: UHS - GM6 - 28.0 [sec] - Case 1 (+V1 only)

Figure I-7: UHS - GM6 - 31.5 [sec] - Case 1 (−V1 only)

109

110

I-1-1-4 Other ground motion records with results similar to Case 1 and 5

Other usage ratio plots from ground motion records were also found to display principalloading similar to Case 1 and Case 5, such as UHS-GM6. An example is UHS-GM1, asgiven in Figure I-8.

Figure I-8: UHS - GM1 - Case 1 (V1 only)

110

Part IV

Design - APPENDICES

111

Appendix J

Design and acceptance criteria

J-1 Design loads

J-1-1 Specified loads on structure

Material ValueNormalweight concrete 0.087 [lb/in3]

Table J-1: Self-weight (SW )

Occupancy or Use Value (L0)Parking Garage 40 [psf]Level 1 Retail (under tower footprint) 100 [psf]Level 1 Plaza (outside tower footprint) 100 [psf]Exit Areas (assumed inside core wall only) 40 [psf]Residential/Hotel (used for all elevatedlevels of the tower outside the core walls) 40 [psf]

Mechanical/Electrical 0 [psf]Roof 25 [psf]

Table J-2: Unreduced Live Loads (L0)

113

114

Occupancy or Use ValueParking Garage 3 [psf]Level 1 Retail (under tower footprint) 110 [psf]Level 1 Plaza (outside tower footprint) 350 [psf]Exit Areas (assumed inside core wall only) 28 [psf]Residential/Hotel (used for all elevatedlevels of the tower outside the core walls) 28 [psf]

Mechanical/Electrical 100 [psf]Roof 28 [psf]

Table J-3: Superimposed Dead Loads (SDL)

Occupancy or Use ValueExterior Cladding (Wall area) 15 [psf]

Table J-4: Cladding Load (CL)

J-1-2 Load combinations

ASCE 7-10 specifies two basic load combinations to account for earthquake loading,namely:

Load combination 5: 1.2D + 1.0E + L+ 0.2S + 1.6HLoad combination 7: 0.9D + 1.0E + 1.6H (J-1)

The load effects are specified in Table J-5.Combining the load effects from Table J-5 with the basic load combinations given inEquation J-1 gives the load combinations for strength design of reinforced concretediaphragms and collectors given in Equation J-2.

Load combination 5: (1.2 + 0.2Sds)D + ρ ·Qe + 0.2L0

Load combination 7: (0.9− 0.2Sds)D + ρ ·Qe

(J-2)

The seismic load combination with overstrength are also utilized for the collectors anddistributors. The load combinations with overstrength factor are given in Equation J-3.

Load combination 5 with overstrength: (1.2 + 0.2Sds)D + ΩoQe + L0

Load combination 7 with overstrength: (0.9− 0.2Sds)D + ΩoQe

(J-3)

For Modal Response Spectrum Analyses, 100% of the effects in one primary direction areto be combined with 30% of the effects in the other direction.

114

J-2 Expected material properties

Loading Value Comments

Earthquake load E Eh + EvLoad combination 5: gravity and seismic load effectsare additive

Eh − EvLoad combination 7: effects of seismic loads counter-act gravity

Horizontalseismic loadeffect

Eh ρ ·QeQe represent effects of horizontal seismic forces, ρ = 1;ASCE 7-10 Equation 12.4-3

Vertical seismicload effect Ev 0.2 · SDS ·D ASCE 7-10 Equation 12.4-4

Dead load D See weight calculation

Live load L 0.4 · L0

L shall not be taken less than 0.5L0 for members sup-porting one floor and 0.4L0 for members supportingtwo or more floors (ASCE 7-10, 4.7.2)

Snow load S 0 ASCE 7-10 Figure 7-1; Snow load is zero in LosAngeles

Load H 0Load due to lateral earth pressure, ground waterpressure, or pressure of bulk materials, assumed tobe zero

Table J-5: Load effects

J-2 Expected material properties

The structural wall model shall incorporate the expected (or median) material properties toreflect realistic estimates of material strengths and stiffnesses considering the anticipatedlevel of loading [LATBSDC, 2014]. The expected strength and stiffness properties wereused to define the constitutive steel, concrete and shear relations. Recommended valuesand formulas from literature were used to calculate the expected strength and stiffnessvalues.

J-2-1 Expected strength properties

The expected strength properties of reinforcing steel and concrete are given in Table J-6.

115

116

Material Strength Value

Reinforcing steel Yield strength fy,exp = 1.17 fy

Ultimate strength fu,exp = 1.50 fy,exp

Concrete Compressive strength f ′c,exp = 1.30 f ′c

Table J-6: Expected strength properties [LATBSDC, 2014]

J-2-2 Expected stiffness properties

The modulus of elasticity for non-prestressed reinforcement, Es, is calculated by EquationJ-4.

Es = 29000000 [psi] per ACI 318-11 (J-4)

The modulus of elasticity for normalweight concrete, Ec, is determined by Equation J-5.

Ec = 57000√f ′c [lb/in2] for f ′c ≤ 6000 psi per ACI 318-11

Ec = 40000√f ′c + 1 · 106 [lb/in2] for f ′c > 6000 psi per ACI 363R-92

(J-5)

The elastic shear modulus of concrete, Gc, is determined by Equation J-6.

Gc = Ec

2(1 + υ) Poisson’s ratio υ is equal to 0.20 (J-6)

J-3 Building Code acceptance criteria

In strength design, the margin of safety is provided by multiplying the service load bya load factor and the nominal strength by a strength reduction factor φ. The strengthreduction factor φ shall be taken as given in Table J-7. The basic requirement for strengthdesign is expressed in Equation J-7.

Design Strength ≥ Required Strengthφ · Nominal Strength ≥ U (J-7)

The required strength U is calculated using the load combinations from Equation 9-2and 9-3. The design strength provided by a member, its connections to other members,and its cross-sections shall be taken as the nominal strength, calculated according to therequirements and assumptions of ACI 318-11, multiplied by the strength reduction factor.The design strength can be expressed in terms of flexure, axial load, shear and torsion.The more local the parameter becomes (e.g. strain), the more sensitive the result typicallybecomes to the modelling. As an example, reinforcement strains are very sensitive to themodelling of bond between the reinforcement and the concrete and to the deterioration ofthe bond under cyclic loading. (ref: Miranda - Lecture)

116

J-3 Building Code acceptance criteria

Element Strength reduction factor Comments

Special structuralwall

φ for tension-controlledsections

0.90

Sections are tension-controlled if the nettensile strain the extreme tension steel, εt,is equal or great than 0.005 when the con-crete in compression reaches its assumedstrain limit of 0.003.

φ forcompression-controlledsections

0.65

Section are compression-controlled if thenet tensile strain in the extreme tensionsteel, εt is equal to or less than thecompression-controlled strain limit whenthe concrete in compression reaches its as-sumed strain limit of 0.003.

φ for shear 0.60

If the nominal shear strength of the mem-ber is less than the shear correspondingto the development of the nominal flexu-ral strength of the member. The nominalflexural strength shall be determined con-sidering the most critical factored axialloads and including E.

Diaphragms φ for shear 0.60φ for shear shall not exceed the minimumφ used for the vertical components of theprimary seismic-force-resisting system

Joints and diagonallyreinforced couplingbeams

φ for shear 0.85

Table J-7: Strength reduction factors

117

118

118

Appendix K

Diaphragm Design

K-1 Building Code demand

K-1-1 Diaphragm demand

Diaphragms shall be designed for both the shear and bending stresses resulting fromdesign forces. The diaphragms are to be designed for inertial forces determined as themaximum of Fx and Fpx.

The design seismic force Fx is the force from Modal Response Spectrum Analysis scaledby Ie/R:

Fx = rmax ·Ie

R=

√√√√n∑

i=1

n∑

j=1riαijrj (K-1)

The diaphragm design force Fpx is:

Fpx =

n∑i=x

Fi

n∑i=x

wi

· wpx

Fpx,min = 0.2SDS · Ie · wpx Minimum diaphragm design forceFpx,max = 0.4SDS · Ie · wpx Maximum diaphragm design force

(K-2)

Fi is the force at level i from Modal Response Spectrum Analysis.

119

120

Transfer forces are to be added to the calculated diaphragm design force Fpx. The forcesacting between a diaphragm and a vertical element are extracted from finite elementanalysis. For MRSA, the design value is obtained by determining the transfer force foreach vibration mode , and then combining the individual modal values using the CQCmethod.

K-1-2 Collector demand

Inertial collector design forces are the maximum of:

1. Forces resulting from application of Fx using the load combinations with overstrengthfactor Ωo.

2. Forces resulting from application of Fpx using the load combinations with over-strength factor Ωo.

3. Forces resulting from application of Fpx,min in the basic load combinations.

Transfer forces also need to be considered in the design of collectors. The transfer forcesfrom Fx come directly from the overall building analysis, and are subject to overstrengthfactor Ωo. For Fpx, the transfer forces are subject to the overstrength factor Ωo but notto redundancy factor ρ. The transfer forces need to be added for Fpx. For Fpx,min thetransfer forces are not subject to Ωo, but are subject to ρ. The maximum collector forcesneed not exceed Fpx,max in the basic load combinations, where the transfer forces areincluded and subject to ρ.

K-1-3 Design force from MRSA

Modal Response Parameters: the value for each force-related design parameter of interest,including story drifts, support forces, and individual member forces for each mode ofvibration shall be computed using properties of each mode and the defined responsespectra divided by R/Ie.

Fi,design = Ωo · Fpi = Ωo ·Fi,transfer

R/Ie

FH1,design = 3460 [kips]FH2,design = 2970 [kips]

(K-3)

Where,

120

K-2 PEER/TBI demand

FH1,transfer 8300 [kips]FH2,transfer 7150 [kips]R 6.0 [-]Ie 1.0 [-]Ωo 2.5 [-]

Table K-1: Code design force parameters

K-2 PEER/TBI demand

K-2-1 Design force from NLTHA

Seven ground motions were scaled to the UHS, the maximum transfer force from eachground motion was taken, the mean of all peak responses is multiplied by 1.5 to find thedesign transfer force. The peak transfer forces are given in Table K-2 and K-3.

Ground motion Total North wall South wall East wall West wallGM 1 Superstition Hills, USA (1987) 20033 8205 8437 1667 1723GM 2 Superstition Hills, USA (1987) 18784 8239 7354 1844 1791GM 3 Loma Prieta, USA (1989) 14703 6223 6113 1444 1359GM 4 Landers, USA (1992) 18400 8132 7851 2096 1827GM 5 Kocaeli, Turkey (1999) 14948 7268 5913 1336 1819GM 6 Darfield, New Zealand (2010) 19390 7341 7901 2572 1754GM 7 Darfield, New Zealand (2010) 19509 8307 8388 1964 2384

Average 17967 7673 7422 1846 1808Vmean * 1.5 26950 11510 11134 2769 2712

Table K-2: PEER/TBI design transfer force in diaphragm - Direction H1

121

122


Average 13488 781 764 7399 7660Vmean * 1.5 20232 1172 1146 11099 11490

Table K-3: PEER/TBI design transfer force in diaphragm - Direction H2

K-3 Shear capacity

K-3-1 Shear strength provided by concrete for nonprestressed members

For members subject to shear and flexure only, Equation K-4 is used to calculate theshear strength Vc.

Vc,DBE = φ · 2 ·√f ′c · bw · d

Vc,MCE = φ · 2 ·√f ′c,exp · bw · d

Where,φ = 0.75 [-] Per ACI 318-05f ′c = 5500 [psi] Nominal compressive strength concretef ′c,exp = 7150 [psi] Expected compressive strength concreteb = 12 [in] Web width (1 [ft])d = [in] Distance extreme compression fibre to centroid of longitudinal tension reinforcement

(K-4)

Parameter d shall be taken as the distance from the extreme compression fibre to centroidof the prestressed and nonprestressed longitudinal tension reinforcement, if any, but neednot be taken less than 0.80h.

122

K-3 Shear capacity

K-3-2 Shear strength provided by shear reinforcement

Shear reinforcement consisting of the following shall be permitted:

• Stirrups perpendicular to axis of member• Welded wire reinforcement with wires located perpendicular to axis of member• Spirals, circular ties, or hoops

For nonprestressed members, shear reinforcement shall be permitted to also consist of:

• Stirrups making an angle of 45 degrees or more with longitudinal tension reinforce-ment

• Longitudinal reinforcement with bent portion making an angle of 30 degrees or morewith the longitudinal reinforcement;

• Combinations of stirrups and bent longitudinal reinforcement

The values of fy and fyt used in design of shear reinforcement shall not exceed 60000 [psi].

K-3-2-1 Spacing limits for shear reinforcement

Spacing of shear reinforcement placed perpendicular to axis of member shall not exceedd/2 in nonprestressed members, nor 24 [in]. Inclined stirrups and bent longitudinalreinforcement shall be spaced so that every 45-degree line, extending towards the reactionfrom mid-depth of member d/2 to longitudinal tension reinforcement, shall be crossed byat least one line of shear reinforcement.Where Vs exceeds 4

√f ′cbwd, maximum spacing given in the previous paragraph shall be

reduced by one-half.

K-3-2-2 Minimum shear reinforcement

A minimum area of shear reinforcement, Av,min, shall be provided in all reinforced flexuralmembers where Vu exceeds 0.5φVc,

K-3-2-3 Design of shear reinforcement

Where shear reinforcement perpendicular to axis of member is used:

Vs = Av · fyt · ds

(K-5)

123

124

Required area of shear reinforcement Av and its spacing s are computed by Equation K-6.

Av

s= (Vu − φVc)

φ · fyt · d(K-6)

K-3-3 Shear friction capacity

The relationship between shear-transfer strength and the reinforcement crossing the shearplane can be expressed in various ways. Equation K-7 gives a conservative prediction ofshear-transfer strength.

Vn = Avffyµ (K-7)

Where,

Avf = Area of shear reinforcementfy = Tensile strength of reinforcementµ = Coefficient of friction, normalweight concrete (λ = 1.0)

= 1.0λ

(K-8)

The coefficient of friction µ shall be taken as:

Concrete placed monolithically = 1.4λConcrete placed against hardened concete with surface intentionally roughened = 1.0λConcrete placed against hardened concrete not intentionally roughened = 0.6λConcrete anchored to as-rolled structural steel by headed studs or by reinforcing bars = 0.7λ

(K-9)

Other relations that give a closer estimate of shear-transfer strength can be used. Forexample, when shear-friction is perpendicular to the shear plane, the nominal shearstrength Vn is given by Equation K-10.

Vn = 0.8Avffy + AcK1 (K-10)

Where,

K1 = 400 [psi]Ac = Area of concrete section resisting shear transfer (K-11)

The first term represents the contribution of friction to shear-transfer resistance (0.8representing coefficient of friction). The second term represents the sum of the resistanceto shear of protrusions on the crack faces and the dowel action of the reinforcement.

124

K-4 Maximum shear capacity diaphragm

For normalweight concrete either placed monolithically or placed against hardened concretewith surface intentionally roughened, Vn shall not exceed the smallest of:

Vn ≤ 0.2f ′cAc

≤ (480 + 0.08f ′c)Ac

≤ 1600Ac

(K-12)

Where Ac is area of concrete section resisting shear transfer.Where concretes of different strengths are cast again each other, the value of f ′c used toevaluate Vn shall be that of the lower-strength concrete. The value of fy used for designof shear-friction reinforcement shall not exceed 60,000 [psi].When concrete is placed against previously hardened concrete, the interface for sheartransfer shall be clean and free of laitance (an accumulation of fine particles on the surfaceof fresh concrete due to an upward movement of water). If µ is assumed equal to 1.0φ,interface shall be roughened to a full amplitude of approximately 1/4 in.

K-4 Maximum shear capacity diaphragm

The shear capacity of the diaphragm is limited by the maximum shear force in thediaphragm and the maximum shear friction force in the connection between the shear walland diaphragm. The maximum shear friction capacity and shear capacity are determinedfor reinforcement #10-12.

Shear strength = Acv(2√f ′c + ρnfy)

vn = Acv(2√f ′c + ρnfy)

= 93.0 [kips/ft]vn,exp = Acv(2

√f ′c,exp + ρnfy,exp)

= 108.2 [kips/ft]

(K-13)

Where,

ρn (#10-12) 0.0094 [in2/in2]Acv (= b · d) 130.5 [in2]d (= h− c− 1/2φsl) 10.88 [in]b 12 [in]K1 (normalweight concrete) 400 [psi]


125

126

Shear friction strength = 0.8 · Avf · fy + AcK1

vn,sf = 0.8 · Avf · fy + AcK1

= 116.5 [kips/ft]vn,exp,sf = 0.8 · Avf · fy,exp + AcK1

= 126.5 [kips/ft]

(K-14)

Where,

Avf #10-121.23 [in2]

Ac b · h144.0 [in2]

h 12 [in]b 12 [in]K1 400 [psi]


In Code design, the overstrength factor is applied to connections to vertical elements ofthe seismic-force-resisting system, but not to the diaphragm as a whole. Therefore, theshear capacity for Code design is governed by the maximum shear friction force in theconnection between the shear wall and diaphragm. PEER/TBI design used unreducedforces from the nonlinear time history analysis in lieu of the application of overstrengthfactor Ωo. Therefore, the shear capacity is governed by the maximum shear force in thediaphragm. The maximum nominal and nominal expected shear capacity are given inEquation K-15.

vn,smax = 116.5 [kips/ft]vn,exp,smax = 108.2 [kips/ft] (K-15)

K-5 Maximum collector capacity diaphragm

Collector strength = fy · As ≤ f ′cAc

Vn,colmax = fy · As,n = 1425.5 [kips]Vn,exp,colmax = fy,exp · As,exp = 1984.9 [kips]

(K-16)

Where,

126

K-6 Diaphragm transfer force design

As,n 16-#1123.76 [in2]

As,exp 16-#1228.27 [in2]



There are several ways to design the diaphragm for transfer forces. The first boundarysolution is to utilize the full shear friction capacity and to transfer the remaining forcethrough the collectors. The other boundary solution is to maximize the collector forceand transfer the remaining force through shear friction, as recommended by NEHRP No.3. Both boundary solutions are explored for the Building Code and PEER/TBI design.

K-6-1 Code design

K-6-1-1 Boundary solution 1: Maximum shear friction force

Shear friction transverse reinforcement #10-12 is used across the entire wall section. Therequired collector reinforcement is calculated in this section. The length of the shear wallis given in Table K-7.

Wall N-S(Direction H1) lNS = 29.75 [ft]

Wall E-W(Direction H2) lEW = 39.50 [ft]

Table K-7: Length shear walls

Vn,NS = vn,smax · lNS + Vn,col,H1 ≤ VH1,design

φ

Vn,EW = vn,smax · lEW + Vn,col,H2 ≤ VH2,design

φ

(K-17)

127

128

Vn,col,H1 ≥ VH1,design

φ− vn,smax · lNS

≥ − 291.4 [kips]

Vn,col,H2 ≥ VH2,design

φ− vn,smax · lEW

≥ − 789.7 [kips]

(K-18)

A negative required collector force implies that transverse shear friction reinforcement issufficient to resist the total transfer force.

K-6-1-2 Boundary solution 2: Maximum collector force

Collector reinforcement 16-#11 is used at each end of the wall. The required shear frictionreinforcement is calculated over the length of the wall.

Vn,NS = vn,s · lNS + Vn,colmax,H1 ≤ VH1,design

φ

Vn,EW = vn,s · lEW + Vn,colmax,H2 ≤ VH2,design

φ

(K-19)

Vn,s,H1 ≥ VH1,design

φ− vn,colmaxH1

≥ − 376.0 [kips]

Vn,s,H2 ≥ VH2,design

φ− vn,colmax,H2

≥ 32.3 [kips]

(K-20)

A negative transverse shear force implies that the maximum collector reinforcement issufficient to resist the total transfer force and that transverse shear reinforcement is notrequired.

K-6-2 PEER/TBI design

K-6-2-1 Boundary solution 1: Maximum shear friction force

Shear friction transverse reinforcement #10-12 is used across the entire wall section. Therequired collector reinforcement is calculated in this section. The length of the shear wall

128


is given in Table K-7.

Vn,exp,NS = vn,exp,smax · lNS + Vn,exp,col,H1 ≤ VH1,design

φ

Vn,exp,EW = vn,exp,smax · lEW + Vn,exp,col,H2 ≤ VH2,design

φ

(K-21)

Vn,exp,col,H1 ≥ VH1,design

φ− vn,exp,smax · lNS

≥ 8103 [kips]

Vn,exp,col,H2 ≥ VH2,design

φ− vn,exp,smax · lEW

≥ 7020 [kips]

(K-22)

K-6-2-2 Boundary solution 2: Maximum collector force

Collector reinforcement 16-#12 is used at each end of the wall. The required shear frictionreinforcement is calculated over the length of the wall.

Vn,exp,NS = vn,exp,s · lNS + Vn,exp,colmax,H1 ≤ VH1,design

φ

Vn,exp,EW = vn,exp,s · lEW + Vn,exp,colmax,H2 ≤ VH2,design

φ

(K-23)

Vn,exp,s,H1 ≥ VH1,design

φ− vn,exp,colmaxH1

≥ 7352 [kips]

Vn,exp,s,H2 ≥ VH2,design

φ− vn,exp,colmax,H2

≥ 7325 [kips]

(K-24)

129

130

130

Appendix L

Shear wall design

L-1 PEER/TBI demand

L-1-1 Design force from NLTHA

Seven ground motion pairs were scaled to the UHS, the mean of the maximum shear forcefrom each ground motion is multiplied by β to find the design shear force. The peak shearforces are given in Table L-1 and L-2.


VUHS 15239 7853 7934 907 8791.5 · VUHS 22858 11779 11901 1361 13182.0 · VUHS 30477 15706 15868 1815 1758

Table L-1: Design shear force in shear wall - Direction H1

131

132


VUHS 15557 410 392 7627 86181.5 · VUHS 23335 615 588 11441 129272.0 · VUHS 31113 820 784 15255 17236

Table L-2: Design design shear force in shear wall - Direction H2

L-2 Force-Based Response Parameters

L-2-1 Shear strength

Nominal expected shear strength Vn,e of structural walls is calculated by Equation L-1from ACI 318-11 Equation 21-7.

Vn,e = Acv

(αcλ

√f ′c,exp + ρtfy,exp

)

≤ 8 · Acv

√f ′c,exp for all walls resisting a common lateral load

≤ 10 · Acv

√f ′c,exp for a single wall

(L-1)where αc is a factor dependent on the height to length ratio hw

lw, λ the factor related

to the concrete weight (i.e. λ = 1.0 for normal weight concrete), and ρt the transversereinforcement ratio.The factor αc is 3.0 for hw

lw≤ 1.5, 2.0 for hw

lw≥ 2.0, and varies linearly between 3.0 and 2.0

for hw

lwbetween 1.5 and 2.0.

ACI 318-11 for shear strength of walls does not consider the effect of axial load. Test resultshave indicated that wall shear strength increased, with Vtest/Vn values of approximatelyof 1.5 for walls with an axial load of Pu/Agf

′c = 0.05 and 1.75 for walls with an axial load

of Pu/Agf′c = 0.10. Furthermore, wall shear strength degrades with increasing ductility

demand.

132

L-3 Deformation-Based Response Parameters

L-2-2 Axial Strength

Nominal expected axial strength is calculated by:Pn,e ≤ Pn,max = 0.80[0.85f ′c,exp(Ag − Ast) + fy,expAst] (L-2)

L-3 Deformation-Based Response Parameters

L-3-1 Flexural Strength

The design of the shear wall subject to flexure and axial loads is based on stress and straincompatibility. Strain in reinforcement and concrete is assumed directly proportional tothe distance from the neutral axis.

Figure L-1: Strain distribution and net tensile strain (ACI 318-11 Fig. R10.3.3)

The stress distribution in concrete is calculated using the following assumptions:

• The compressive stress distribution is described by an equivalent rectangular distri-bution with an average stress of 0.85f ′c. The rectangle depth a = β1 · c, where c isthe distance from the fibre of maximum strain to the neutral axis and β1 = 0.85 forf ′c ≤ 4000 [psi].

• The maximum usable strain in compression is 0.003.• The tensile stress distribution is zero, since the tensile strength of concrete is neglected

in flexural calculations.

The stress in the steel reinforcement is calculated by:For εs ≤ εy : σs = AsEsεs

For εs ≥ εy : σs = Asfy

(L-3)

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134

134

Part V

Structural reliability analysis -APPENDICES

135

Appendix M

Structural reliability

M-1 AD-test

The null hypothesis at the 5% significance level is rejected for 5 out of 18 cases for theshear force distribution and 10 out of 18 cases for the transfer force distribution for thelognormal distribution. The null hypothesis at the 1% significance level is rejected at2 out of 18 cases for the shear force distribution and 7 out of 18 cases for the transferforce. The lognormal distribution performs best in the Anderson-Darling test and theload function S is therefore lognormal.

M-1-1 Various hypothesized distributions

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138

Hypothesized test results h

IM Walls Normal Exponential Extremevalue Lognormal Weibull

RP975All walls 1 1 1 0 1North wall 0 1 1 0 0South wall 1 1 1 0 1



RP975All walls 1 1 1 0 1East wall 1 1 1 0 1West wall 1 1 1 1 1



Summation 15 18 18 5 15

Table M-1: Shear force test result h

138

M-1 AD-test

Anderson-Darling test p-value


RP975All walls 0.0445 0.0005 0.0005 0.095 0.0130North wall 0.6043 0.0005 0.0007 0.0663 0.3135South wall 0.0031 0.0005 0.0005 0.2692 0.0020



RP975All walls 0.0013 0.0005 0.0005 0.1936 0.0005East wall 0.0015 0.0005 0.0005 0.0855 0.0005West wall 0.0019 0.0005 0.0005 0.0485 0.0005



Summation 1.3400 0.0090 0.0190 3.2634 0.6192

Table M-2: Shear force p-value

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140

Hypothesized test results h








Summation 10 18 16 10 10

Table M-3: Transfer force test result h

140

M-1 AD-test

Anderson-Darling test p-value








Summation 4.5074 0.009 0.316 2.6732 3.0402

Table M-4: Transfer force p-value

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142

M-1-2 Lognormal distribution

RP 975 RP 2475 RP 9950Walls p-value h-value p-value h-value p-value h-value

All walls 0.0950 0 0.4008 0 0.1727 0North Wall 0.0663 0 0.0750 0 0.5496 0South Wall 0.2692 0 0.2765 0 0.4501 0

Table M-5: Anderson-Darling test for hypothesized lognormal distribution - Shear Direction H1


All walls 0.1936 0 0.0104 1 0.1934 0East Wall 0.0855 0 0.0251 1 0.3010 0West Wall 0.0485 1 0.0344 1 0.0163 1

Table M-6: Anderson-Darling test for hypothesized lognormal distribution - Shear Direction H2


All walls 0.1635 0 0.0260 1 0.0005 1North Wall 0.0962 0 0.0354 1 0.0005 1South Wall 0.0811 0 0.0318 1 0.0311 1

Table M-7: Anderson-Darling test for hypothesized lognormal distribution - Transfer force DirectionH1

142

M-1 AD-test


All walls 0.0038 1 0.0022 1 0.2846 0East Wall 0.4733 0 0.0152 1 0.1374 0West Wall 0.5667 0 0.0005 1 0.7234 0

Table M-8: Anderson-Darling test for hypothesized lognormal distribution - Transfer force DirectionH2

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144

M-2 Probabilities of failure

Figure M-1: Probability of failure; Shear force in all walls - Direction H1

144


Figure M-2: Probability of failure; Shear force in north wall

Figure M-3: Probability of failure; Shear force in south wall

145

146

Figure M-4: Probability of failure; Shear force in all walls - Direction H2

Figure M-5: Probability of failure; Shear force in east wall

146


Figure M-6: Probability of failure; Shear force in west wall

Figure M-7: Probability of failure; Transfer force in all walls - Direction H1

147

148

Figure M-8: Probability of failure; Transfer force in north wall

Figure M-9: Probability of failure; Transfer force in south wall

148


Figure M-10: Probability of failure; Transfer force in all walls - Direction H2

Figure M-11: Probability of failure; Transfer force in east wall

149

150

Figure M-12: Probability of failure; Transfer force in west wall

150

M-3 Estimated Collapse fragility curves


Figure M-13: Collapse fragility curve; Shear force in all walls - Direction H1

151

152

Figure M-14: Collapse fragility curve; Shear force in north wall

152


Figure M-15: Collapse fragility curve; Shear force in south wall

153

154


154


Figure M-17: Collapse fragility curve; Shear force in east wall

155

156

Figure M-18: Collapse fragility curve; Shear force in west wall

156


Figure M-19: Collapse fragility curve; Transfer force in all walls - Direction H1

157

158

Figure M-20: Collapse fragility curve; Transfer force in north wall

158


Figure M-21: Collapse fragility curve; Transfer force in south wall

159

160


160


Figure M-23: Collapse fragility curve; Transfer force in east wall

161

162

Figure M-24: Collapse fragility curve; Transfer force in west wall

162

M-4 Estimated deaggregation of mean annual frequency of collapse


Figure M-25: Deaggregation of mean annual frequency of collapse; Shear force in all walls -Direction H1

163

164

Figure M-26: Deaggregation of mean annual frequency of collapse; Shear force in north walls

164


Figure M-27: Deaggregation of mean annual frequency of collapse; Shear force in south wall

165

166

Figure M-28: Deaggregation of mean annual frequency of collapse; Shear force in all walls -Direction H2

166


Figure M-29: Deaggregation of mean annual frequency of collapse; Shear force in east wall

167

168

Figure M-30: Deaggregation of mean annual frequency of collapse; Shear force in west wall

168


Figure M-31: Deaggregation of mean annual frequency of collapse; Transfer force in all walls -Direction H1

169

170

Figure M-32: Deaggregation of mean annual frequency of collapse; Shear force in north wall

170


Figure M-33: Deaggregation of mean annual frequency of collapse; Transfer force in south wall

171

172

Figure M-34: Deaggregation of mean annual frequency of collapse; Transfer force in all walls -Direction H2

172


Figure M-35: Deaggregation of mean annual frequency of collapse; Transfer force in east wall

173

174

Figure M-36: Deaggregation of mean annual frequency of collapse; Transfer force in west wall

174

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