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4-1981

Centre of Rigidity Concept and its Application tothe Static and Dynamic Analysis of Multi-StoryBuildingsDavid Ngai Shing Chan

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Recommended CitationChan, David Ngai Shing, "Centre of Rigidity Concept and its Application to the Static and Dynamic Analysis of Multi-Story Buildings"(1981). Open Access Dissertations and Theses. Paper 2856.

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CENTRE OF RIGIUITY CONCEPT AND ITS APPLICATION

TO ~HE STATIC AND DYNAMIC ANALYSIS

OF MULTI-STORY BUILDINGS

,..

by

\..Davi.d Ngal Shing Chan

B. Eng. (Civil)

A Thesis

Submitted to the School of Graduate Studies.

in Partial Fulfilment of the Requirements

for the Degree

Master of ~ngineering

McMaster University

April 1981

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MASTER OF ENG~RING (1981)(Civil Engineering andEngineering Mechanics)

McMASTER UNIVER£ITY~

Hamilton, Ontario

TITLE:jt

"

Centre of Rigidity Concept and i~s

Application to ,the Static and DynamicAnalysis of ~Ulti-Story,Buildings

AUTHOR:"I

"David Ngai Shing Chan,B.Eng. (McMaster University)

SUPERVISORS: Dr. W.K. Tso and Dr. A.C~ Heidebrecht

NUMBER OF PAGES: xv, 125

, ,

ABSTRACT

. The pbject of this investigation is to establish'-

a method of determining the locus of. centres of

rigidity of an asymmetric multi-story bui~ding

structure, which can be applied to determine, with

, acceptable' accuracy, thE?, she a r .for c esin the

load-resisti~g pa~els in a static or dynamic analysis.

Based on a more general definition for the

centre of rigidity within a particular story of w

multi-story building structure, a method is develC?ped

for the evaluation of the l~cus of centres of rigidity

of a multi-story building structure.' Other existing

methods are also discussed. It has ~een found that the

locus of centres of rigidity is not only a fupction of

the structural properties but also a function of the

distribu·tion of lateral forces acting on the m.ulti­

story building structure. It has also been shown that

for a building structure consisti~g of panels which are

dispropor.tional in stiffness, the traditional method·p

.. (i~ which only the structural elements in an isolated

story is considered for the determination of the centre~/ '

of rigidity within that story) do no~ give a satis-

~ory estimation of the locus of centres of rigidity.f.l\-

iii

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...

The appl i cabi 1 i ty of the locus 'Of cen tres of

rigidity' to the evaluation of the shear forces in the

load-resisting' panels in a static, analysis is

'investigated. - Based on the results of this inve.sti­

,gation, methods of determining the panel. shear forces

in a static analysis are recommended for various types

of building structures.

The applicability of the locus of centres of

rigidity to the evaluation of the paner shear forces in

a dyna~ic anafysis using a three degree~of-freedom

model is also studied. The results of this study show

that the distri~ of the panel shear ~orces. in a

frame buildin9 structure obtained by using the model,

agrees well with that obtained by using a three-.

dimensional ~ynamic frame program. The model can only'

. d h'" f th . t d f hprovl e a roug estlmatlon 0 e magnl u e 0 t e

panel shear forces. \

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ACKNOWLEDGEMENTSQ

The author wtshes to ex~ress his sincere

gratitude tc his research supervisors"Dr. W.K. T50 and, .

Dr. A.C. Heidebrecht, for their suggestions, guidance

and encouragement throughout the, course of this"

research work.

The author also wishes to thank Dr. W.K. Tso,

former Chairman and Dr. R.G. Drysdale, Chairman of the

Department of Civil Eng ineering, for making available

to him the scholarship and teaching assistantshi~.

Special thanks are due to Mr. E. Chu for his

invaluable advice on computer graphics, Miss K.A. 'Hugh

Sam for her useful suggestions on report-writing, and

·Ms. N.M. Sine ~f the Facufty of Engineering Word

Processing Centre for her skillful typing of this

thesis.

The. author wishes to dedicate this work to his

parents whose encouragement, -support and understanding

are gratefully appTeciated. Thank you, Mom and Dad.

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TABLE OF CONTENTS•

PageABSTRACT

iiiACKNOWLEDGEMENTS

TABLE OF CONTENTS

LIST OF FIGU~~!tIST OF TABLES

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vi

v,i i i

xv

CHAPTER 1

CHAPTER, 2

INTRODUCTION

1.,'1 General ,1.2 Literature Review1.3 Objective and Scope

-,STRUCTURAL MODELLI~G

1

137

102.1

2.2

Application of Meng's Model toa Multi-Story StructureStructural Modelling.

10

13

CHAPTER 3 LOCUS OF CENTRES OF RIGIDITY OF AMULTI-STORY BUILDING STRUCTURE 25

3.1 Existing Methods of Evaluation 25

3.1.1 Traditional Methoa 253.1.2 Improved Traditional 28

Method3.1.3 General Method Specified 29

in New Zealand StandardNZS 4203:1976

3.2 Proposed Exact ~ethod 303.3 Factors Affecting the Locus of 34

Centres of Rigidity ~

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CHAPTER,4

TABLE OF CONTENTS (cont'd)

STATIC ANALY,SIS OF A MULTI-STORYBUILDING STRUCTURE

53

4.1 Method of the Static Analysis 534.2 Panel Shear 'Forces by the . 57

S~ (3 tic A.n a1ysis .

CHAPTER 5 DYNAMIC ANALYSIS OF A MULTI-STORYBUILDING STRUCTURE .

85

5.1 Modified Meng's Model - A Three 85Degree-of-Freedom Dynamic Model

5.2 Panel Shear ForGes Obtained by 87U~ing the Modified Meng's Model

5.3 Comparison with~esults Obtained 98by Using the Static Analysis

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CHAPTER 6 CONCLUSIONS

REFEREN~ES

APPENDIX I MENG'S MATHE,ATICAL MODEL

vii

106

110

115

LIST OF FIGURES

Page'

FIGURE 2.1

FIGURE 2.2

Comparison of the Deflection Shapesof the Example Frame Described inTabl'e 2.1

Deformed·Interior Intermediate St6iyof a Frame Subjected to a P~int

Load at Top

12

14

FIGURE 2.3 beformed Beam Segm~nt, AB, $ubjectedto a Bending Moment at B

FIGURE 2.4' Deformed Interior Bottom Story of aFrame SUbjepted to a Point Loadat Top

14

18

FIGURE 3.1

FIGURE 3.2

FIGURE 3.3

Plan View of the ith Story of aMulti-Story Frame Building Structure

Force-Resisting Panels Connectedby Rigid Bars for the Determinationof the Locus of Centres of Rigidity

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Fl~ibility Coeffi,cients for the ith'Floor Level, Obtained by ~pplying

a Un i t Fo rceo r a ,Un i t Mom entatthe Reference Point of th~ ith FloorLevel

26

26

31

FIGURE 3..4

FIGURE 3.5

'"\IGURE 3.,6

The Lateral Force and Torsional 31Moment at the Reference Axis,or theCentre Qf Rigidity Within the ithS~ory

Sketch of the Ten-Story Building 37Structure with Set-Backs Referencedin Table 3.1

•Comparison of Loci of Centres of 38Rigidity Obtained by Various MethOQSof Evaluation for Case 1 in Table 3.1

viii

LIST OF FIGURES (cont'd)

FIGURE 3.7 Comparison of Loci of Centres of 39Rigidity Obtained by Various Methodsof Evaluation for Case 2 in Table 3.1

FIGURE 3.8 Comparison of Loci of Centres of 40Rigidity Obtained by Various Methodsof Evaluation for Case 3 in Table 3.1

FIGUR~ 3.9 Comparison of Loci of Centres of 41Rigi~i~y Obtained by Various ~ethods

of Evaluation for Case 4 in Table 3.1I '

FIGURE 3.10 Comparison of-Lcci of Ce~tres of 42Rigidity Obtained by Various Methodsof ,Evaluation for Case 5 in Table 3.1

FIGURE 3.11 C~mparison of Loci of Centres of 43Rigidity Obtained by Various,Methodsof Evaluation for Case 6 in Table 3.1

FIGURE 3.12 Comparison of ,Loci of Centres of 44Rigidity Obtained by Various Methodsof, Ev a 1 uat ion for Ca s e 7 in Ta b1e 3. 1

FIGURE 3.13 Comparison of Lo~i of Cent~es of 45Rigidity Obtained by Various Methodsof Evaluation for Case 8' in Table 3.1

51

50Comparison of Loci·of Centres ofRigidity by Using the New ZealandStandard (Extensible Columns Assumed)and Various Distributions of LateralForces for Case 6 in Table 3.1

Comparison of Loci of Centres of- Rigidity by Using the New'Zealand

Standard (Extensibl~ ~olumns Assumed)and Various Distributions of LateralForces 'for Case 7 in Table 3.1

\FIGURE 3.15

FI GURE 3.14

FIGURE 4.1 The Lateral Force and Torsional I

Moment at the Centre of RigidityWithin the ith Story

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ix

FIGURE 4.2

FIGURE 4.3

.LIST OF FIGURES (cont'd)

Comparison of Panel Bhear ForcesObtained by Various Methods ofEvaluation of Locus of Centres ofRigidity aQd Relative PanelStiffnesses for Case 1 in Tabie 3.1

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Comparison of Panel Shear ForcesObtained by Various Methods ofEvaluation of Locus of Centres ofRigidity and Relative PanelStiffnesses for Case 2 in Table 3.1

58

5!l

~URE 4.4 Comparison of Panel Shea~ Forces-Obtained by Various Methods of

Evaluation_ Q.I_LoGUs _Qf .Centres ofRigidity and Relative PanelStiffnesses for Case 3 1n Table 3.1

600'

FIGURE 4.5 Comparison of Panel Shear Forces 61Obtained by Various Methods 'bf )Evaluation of Locus of Centres ofRigidity and Relative PanelStiffnesses for Case 4 in Table 3.1

FIGURE 4.6 Comparison of Panel Shear Forces 62Obtained by Various Methods ofEvaluation of Locus of Centres of •Rigidity and Relative Panel

~ Stiffnesses for Case 5 in Table 3.1

FIGURE 4.7 Comparison of Panel Shea'r Forces 63Obtained by Various Methods of&~luation of Locus of Centres ofFigidity and Relative PanelStiffnesses for Case 6 in Table 3.1

FIGURE 4.8 Comparison of Panel Shear Forces 64Obtained oy Various Methods ofEvaluation of Locus of Centres ofRigidity and Relative PanelStiffnesses for Case 7 in Table 3.1

FIGURE 4.9 Comparison of Panel Shear Forces 65. Obtained by Various,Methods of

Evaluation of Locus of Centres ofRigidity and Relative PanelStiffnesses Eor Case 8 in Table 3.1

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LIST OF FIGURES (cont'd)

FIGURE 4.10 Comparison of Panel Shear Forces 66Obtained by various Methods ofEvaluation of Locus of Centres ofRigiditt an~ Relative PanelStiffnegses for, Case 1 in T!-ble 3.1

FIGURE 4.11 Comparison, of Panel Sh-ear Forces 67Obtained by Va r ious 14ethods' ofEvaluation of Locus of Ce'ntres of

') Rigidity and Relative Panel"- Stiffnesses for Case 2 in Table 3.1, ,

----FI GURE ---4-;-:1 "2 Comparison of Panel Shear Forces 68

Obtained by various~~~~ds of -

Evaluation of ~ocus of Centres ofRig id i ty and Relative PanelStiff(lesses for Ca$e 3 in Table 3.1

of'

FIGURE 4.13 Comparison of Panel Shear Forces 69Obtained by Var ious Methods ofEvaluation of Lo..e-us of Centres ofRig id i ty and Relative P.anelStiffnesses for Case 4. in Table 3.1

~

FIGURE 4.14 Comparison of ,Panel Shear Forces 70Obtained by Various Methods ofEvaluation of Loc !-IS of Centres ofRig id i ty and Relative PanelStiffnesses for Case 5 in Table 3.1

FIGURE 4.15 Comparison of Panel Shear Forces 71Obtained by Various Methods ofEvaluation of Locus of Centres ofRigidity and Relative paneJStiffnesses for Case 6 in Table 3.1

<,

t FIGURE 4.16 Comparison o~ Pan-el Shear Fo rces 72Obtained by Various Methods of" .Eval ua t-i on of Locus of Centres ofRigidity and Relative PanelStiffnesses for Case 7 in Table 3.1

FIGURE 4.17 Comparison of Panel Shear Fo rces 73Obtained by 'Various Methods ofEvaluation of Locus of Cen.tres ofRigidity and Relat ive pan~l

Stiffnesses for Case 8 in'Table 3.1

xi

LIST OF FIGURES (cont'd)

FIGURE 4.18 Comparison of Fanel Shear Forces 79for the Case in which the Momentsof Inertia of the Columns in Panel 3Equal Two Times those 01 theCorresponding Columns in pa~el 1

FIGURE 4.19 Comparison of Panel Shear Forces 80for the Case in which the Momentsof Inertia of the Columns in Panel 3Equal ,Two Times thpse of theCorresponding Columns in.... Panel 1

FIGURE 4.20 Comparison 'of Panel Shear Forces 81Obtained by Various Methods ofEvaluating the Torsio~al MomentWithin a Story for Case 6 in* T~ble 3.1

•FIGURE 4.21 Comparison of Panel Shear ForcesObtained by Various Methods of

• Evaluating the Torsional MomentWit h ina Stor y for Ca s e 4' i. nTable 3.1

•83

FIGURE 5.1 Comparison of Panel Shear Fo~ces 88Obtained by 'Applying Various ~ethods "of Evaluation in the Modified Meng's"~10del and Normalized with Respect tothe Stor¥ Shear Force Obtain~d by ~he

Three-Dimensional Dynamic Frame Programin SAP IV for Case 1 in Table 3.f

'FIGURE 5.2 Comparison of panel Shear Forces 89Obtained .by Applying Various Methodsof "Evaluation in the ~odified Meng'sModel and Normalized with Respect tQthe Story Shear Force Obtaihed by the

. Th~e~-Dimensional Dynami~ Frame Programin SAP 'IV for Case 2 in Table 3.1 '

FIGURE 5.3 Comparison of Panel Shear Fo~ces 90'Obtained by Applying Various Methods

'of Evaluation in the Modified Meng'sModel and Nor~alized with Respect tothe Story Shear Forc~ Obtained by theThree-Dimensional Dynamic Frame Programin SAP IV for Case 3 in Table 3.1

xii

FIGURE 5.. 4

FIGURE 5.5

FIGURE 5.6

fIGURE 5.7

FIGURE 5.8

,

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LIST OF FIGURES (cont'd)

-co~~arison of Panel Shear 'Forces 91Obtained by Applying Various Methodsof Eval~atlon in the-Modified Meng'sModel and Normalized with Respect tothe Story Shear Force Oatained by theThree-Dimensional Dynamic Frame Programin SAP IV ~r Case 4 in Table ~.l

Comparison of Panel Shear Forces 92Obtained by Applying Various Methodsof Evaluation in the Modified Meng'sModel and Normalized with Respect tothe Story Shear Force Obtained by theThree-Dimensional Dynamic Frqme Programin SAP IV for Case 5 in Table 3.1

Comparison of Panel Shear Forces' 93Obtained by Applying Various Methodsof Evaluation in the Modified Meng'sModel and Normalized with Respect tothe Story Shear Force Obtained by theThree-Dimensional Dynamic Frame .programin SAP IV for Case 6 in Table 3.1

Compa rison of Panel Shear Forces, 95Obtainea by Applying Various Methodsof .Evaluation in the Modified Meng'sModel and Normalized with Respect tothe Story Shear Force Obtained by theSame Method of Evaluation in theModified Meng's Model for Case 1 inTable 3.1

Comparison of Panel Shear Forces 96Obtained by Applying Various Meth~ds

of Evaluation in the ~odified Meng'sModel and Normalized with ~espect tothe Story Shear Force Obtained by theSame Method of Evaluation in the ­Modified Meng's Model for Case 6 inTable 3.1

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LIST OF FIGURES (cont'd)

FIGURE ~.9 Comparison of Panel Shear Forces 100Obtained. by Applying Various Methodsof Evaluation in t~e Static Analysisand Normalized with Respect to theStory Shear Forc€ Obtained by theSame Method of Evaluation in theStatic Analysis for Case 1 in Table 3.1

FIGURE 5.10 Comparison of Panel Shear Forces 101Obtained by Applying Various Methodsof Evaluation in the Static Analysisand Normalized with Respect to theStory Shear Force Obtained by theSame Method of Evaluation in .theStatic Analysis for Case 3 in Table 3.1

FIGURE 5.11 Comparison of Panel Shear Forces 102Obtained by Applying Various Methodsof Evaluation in the Static Analysisand Normalized with Respect to theStory Shear Force Obtained by theSame Method of Evaluation in theStatic Analysis for Case 6 in Table 3.1

FIGURE 5.12 Comparison of Panel Shear Forces 103Obtained by Applying Various Methodsof Evaluation in the Static Analysisand Normalized with Respect to theStory Shear Force Obtained by theSame Method of Evaluation in theStatic Analysis for Case 7 in Table 3.1

FIGURE I.l Flexural and Shear Springs 116

FIGURE I.2

FIGURE I.3

FIGURE I.4

FIGURE I.5

FIGURE I.6

Shear and Flexural Deformationsof Wall Structure

• •Shear and Flexural Deformatlonsof Frame Structure

Relationship of Geometry and Forcesof Multi-Degree-of-Freedom Unit

Spatial Spring Model

Geometric Relations of ArbitraryPoint and Centres of Mass andRigidity

xiv

118

118

120

123

124

LIST OF TABLES

Page

TABLE 2.1 Properties of the Example Frame '11 •

TAgLE 2.2 Applied Lateral Forces·for Different 21Values of r

TABLE 2.3 Deflection Characteristics of the 22Example Frame for Different Valuesof r (Cl. = 2/3 h Assumed)

TABLE 2.4 Deflection Characteristics of the 23Example Plane Warl for DifferentValues of r

TABLE 3.1 Member Propertie$ of Panel 3 of the 36Types of Building Structures Studied

TABLE 4.1 Recommended Methods Used in the Static 84An a1ysis for' the Eva 1ua t ion 0 f thePanel Shear Forces in Various Typesof Building Structures SUbjected toStatic Lateral Forces

TABLE 5.1 Recommended Methods used in 105Conjunction with the Modified Meng'sModel for the Evaluation of the PanelShear Forces in Various Types ofBuilding Structures Subjected ~o

Earthquake Excitation

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xv