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This is to certify that the

dissertation entitled
Ultimate Strength of Arch Structures Under

Two- And Three-Dimensional Loading

i

I

I

i

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» I

presented by I

Abdulaziz Mahmood AlHamad \
has been accepted towards fulfillment _ _ p - (
of the requirements for i ' i ‘
Ph . D . degree in Civil Engineering i

 

W Leé/ I
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Major professor

Robert K. Wen

Date February 28, 1992

Mfllienn Afr -' ‘ ' '- ll"

rr ' ' ' ' 0-12771

 

 

LIBRARY
Michigan State
University

 

 

 

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MSU Is An Affirmative Action/Equal Opportunity Institution
c:\circ\duedue. may. 1

 

 

ULTIMATE STRENGTH OF ARCH STRUCTURES
UNDER TWO— AND THREE—DIMENSIONAL LOADING

By

Abdulaziz Mahmood AlHamad

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Civil and Environmental Engineering

1992

ABSTRACT
ULTIMATE STRENGTH OF ARCH STRUCTURES
UNDER TWO- AND THREE-DIMENSIONAL LOADING

By

Abdulaziz Mahmood AlHamad

The ultimate load of parabolic two-hinged and fixed arches under
vertical, longitudinal (in-plane horizontal) and lateral (normal to the plane of
the arch) loads was studied. Plastic, elastic-nonlinear, and combined (including
both plastic and geometric nonlinearities) ultimate loads were obtained under
vertical and longitudinal loads (VH-load) and vertical and lateral loads
(VS-load). Also, three-dimensional ultimate load surface was considered.

In the developed computer program, nonlinear straight beam finite
elements along with a plastic hinge model were used to model the arch
behavior. The element plastic resistance is traced through the three possible
stages of its end joints being both elastic, one plastic, and both plastic. Of four
yield functions implemented, the inverse parabolic function best fitted
experimental results. Nonlinear equations were solved using accelerated
modified Newton-Raphson method.

For a given ratio of longitudinal to total load, h, and in-plane slender-
ness ratio, L/ry, the calculated arch ultimate VH-load, P,(h), and other

responses vary significantly with the type of analysis (i.e., the presumed

 

 

IV

 

nonlinearity) and with the ratio of ultimate loads under longitudinal-only and
vertical-only loads, Pho/on- The ratio of total to vertical—only plastic ultimate
load, P,,,(h)/on,p, generally decreases with h and L/ry. For elastic analysis the
corresponding ratio, P,,c(h)/Pw,.,, increases with h for low L/ry but decreases for
higher values. Plastic interaction curves between vertical and longitudinal
components of the ultimate load are convex, while elastic ones are concave.

Behavior under combined analysis is similar to plastic analysis for low
L/ry, and to elastic analysis for high L/ry. For L/ry = 125 to 325, fixing arch
supports increases plastic ultimate load by 0—50%, elastic one by loo-560%,
and combined one by 0—120%. Thrust action in the arch decreases and bending
increases with h.

Behavior under VH-load is similar to that under vertical unsymmetric
load. In particular, the decrease of the "vertical symmetric" component of the
arch ultimate with h is similar to its decrease with the unsymmetry ratio.

The ultimate VS-load is insensitive to restraining the in-plane support
rotation, but is affected by the type of analysis and it generally decreases with
the ratio of lateral to total load and L/ry. Also, it is influenced by the governing
mode of failure (i.e., in—plane or lateral-torsional). This is also true for three-

dimensional ultimate load surface.

 

To My Father, Mahmood, . . .

the Memory of my Mother, Bakriyah, . . .

and to my Family, all of its members:
my wife, Sana,

and my children: Eiman, Osama, Amal, Arwa, and Mahmood.

ACKNOWLEDGEMENTS

I would like to thank professor Robert K. Wen, my thesis advisor, for
his intellectual inspiration, encouragement, and patience. The completion of
this work is deeply indebted to his support. My thanks are also to members of
the guidance committee: Dr. Parvis Soroushian, Dr. Nicholas Altiero, and Dr.
David Yen, and to Dr. William Saul, chairman of the Civil and Environmental
Engineering Department.

The financial support of King Faisal University, Dammam, Saudi
Arabia, for my graduate study is sincerely appreciated. The computing
resources provided by the College of Engineering A. H. Case Center for
Computer-Aided Engineering and the Center for Academic Computing at
Michigan State University are also acknowledged.

My personal appreciation go to three friends: my brother AbdulRazzaq
AlHamad, Khaled Al-Hamoudi and Khaled Medallah. Their personal touch was
invaluable. And last but not least, the support and endurance of my family here

and at home throughout my study is priceless.

TABLE OF CONTENTS

LIST OF TABLES ................................... x
LIST OF FIGURES .................................. xi
LIST OF SYMBOLS ................................. xvi
CHAPTER I
INTRODUCTION ................................... 1
1.1 GENERAL ................................... 1
1.2 OVERVIEW .................................. 2
1.2.1 Arch Behavior under Different Loads ............. 2
1.2.2 Plasticity Model ........................... 4
1.3 SUMMARY OF PRESENT STUDY ................... 5
1.4 LITERATURE REVIEW .......................... 6
1.4.1 In-Plane Behavior of Arches ................... 6
1.4.2 Out-of—Plane Behavior of Arches ................ 11
CHAPTER II
THEORETICAL MODELS FOR
NONLINEAR ELEMENTS ............................. 16
2.1 INTRODUCTION ............................... 16
2.2 ELASTIC-NONLINEAR RESISTANCE OF STRAIGHT BEAM
ELEMENT ................................... 17
2.2.1 Element Elastic-Geometric Stiffness Matrix .......... 17
2.2.2 Calculation of Element Elastic Resistance ........... 20
2.3 ELASTIC-PLASTIC RESISTANCE OF STRAIGHT BEAM
ELEMENT ................................... 21
2.3.1 Element Elastic-Plastic Stiffness Matrix ........... 23
2.3.1.1 Case of One Joint Yielding ............. 25
2.3.1.2 Case of Two Joints Yielding ............ 26
2.3.2 Yield Surface Functions: Types and Implementation . . . . 27
2.3.2.1 Types of Yield Surface functions ......... 27

vi

 

 

 

 

 

2.3.2.2 Computer Implementation of Yield Behavior . . 34

2.3.3 Calculation of Straight Beam Elastic-Plastic Resistance . . 37
2.3.3.1 Locating Force Penetration Point ......... 39

2.3.3.2 Keeping Force Point on Yield Surface ...... 41

2.3.3.3 Steps of Resistance Computation .......... 45

2.4 COMBINED ELASTIC AND PLASTIC NONLINEARITY ..... 53

CHAPTER III
COMPUTER PROGRAM:

SCOPE, IMPLEMENTATION, AND VALIDATION ............. 57
3.1 INTRODUCTION ............................... 57
3.2 PROGRAM SCOPE AND IMPLEMENTATION ............ 58

3.2.1 Scope and Types of Analysis ................. 7. . 58
3.2.1.1 Types of Analysis ................... 59
3.2.1.2 Assumptions of Nonlinear Analysis ........ 60'

3.2.2 Procedure of Incremental Analysis ............... 61
3.2.2.1 Incremental Analysis: Definition of the Ultimate

Load and Optimization of Solution ......... 61
3.2.2.2 Steps of Analysis During a Loading Increment . 64
3.2.3 Check for Convergence or Divergence of Solution ..... 71
3.2.3.1 Basic Tests of Solution Convergence and
Divergence ....................... 71
3.2.3.2 Criteria for Convergence .............. 76
3.2.3.3 Criteria for Divergence ............... 78
3.2.4 Data Input and Output for Analysis ............... 82
3.3 PROGRAM VALIDATION AND CHECK ............... 84

3.3.1 Geometric Nonlinearity Problems ................ 85

3.3.2 Plastic and Combined Nonlinearities Problems ........ 88

CHAPTER IV

PARAMETRIC STUDY OF ARCH BEHAVIOR:

I. IN-PLANE LOAD INTERACTION ...................... 94
4.1 GENERAL INTRODUCTION TO PARAMETRIC STUDY ..... 94

4.1.1 Basic Arch Model and Parameters ................ 95

4.1.2 Loading Distributions and Combinations ............ 98

4.2 IN-PLANE LOAD INTERACTION: INTRODUCTION ....... 103
4.3 BEHAVIOR OF TWO-HINGED ARCH UNDER VERTICAL AND

LONGITUDINAL LOADING ....................... 104

4.3.1 Variation of the Ultimate Load of Two—Hinged Arch with
Slenderness Ratio .......................... 104
4.3.1.1 Plastic Ultimate Load of Two-Hinged Arch . . . 105
4.3.1.2 Elastic—Nonlinear Ultimate Load of Two-Hinged

Arch ........................... 107

4.3.1.3 Combined Ultimate Load of Two-Hinged Arch . 110

 

 

4.3.2 Variation of The Ultimate Load of Two-Hinged Arch with

the Ratio of Longitudinal Load to Total Load ........ 114
4.3.3 Ultimate Load Interaction Curves of Two-Hinged Arch . . 116
4.3.4 Failure Modes and Response Curves of Two-Hinged Arch 119
4.3.5 Plastic Hinge Formation and Force Path at Quarter Point of
Two-Hinged arch .......................... 124
4.4 BEHAVIOR OF FIXED ARCH UNDER VERTICAL AND
LONGITUDINAL LOADING ....................... 127
4.4.1 Variation of the Ultimate Load of Fixed Arch with
Slenderness Ratio .......................... 127
4.4.1.1 Plastic Ultimate Load of Fixed Arch ....... 127
4.4.1.2 Elastic-Nonlinear Ultimate Load of Fixed Arch 130
4.4.1.3 Combined Ultimate Load of Fixed Arch ..... 134
4.4.2 Variation of The Ultimate Load of Fixed Arch with the Ratio
of Longitudinal to Total Load .................. 138
4.4.3 Ultimate Load Interaction Curves of Fixed Arch ....... 140
4.4.4 Failure Modes and Response Curves of Fixed Arch ..... 142
4.4.5 Plastic Hinge Formation and Force Path at Quarter Point of
Fixed Arch .............................. 146
4.5 ARCH ULTIMATE LOAD UNDER VERTICAL
UNSYMMETRICAL LOAD: COMPARISON WITH VERTICAL
AND LON GITUDINAL LOAD ...................... 149
4.5.1 Ultimate Load of Two-Hinged Arch Under Vertical
Unsymmetrical Loading ...................... 150
4.5.2 Comparison of The Vertical Symmetric Component of The
Ultimate Load of Two-Hinged and Fixed Arches for The
Two Loading Cases ......................... 153

CHAPTER V

PARAMETRIC STUDY OF ARCH BEHAVIOR:

II. THREE-DIMENSIONAL BEHAVIOR .................... 160
5.1 INTRODUCTION ............................... 160
5.2 EFFECT OF SUPPORT TYPE AND SECTION PROPERTIES ON ’

THE ARCH ULTIMATE LOAD ..................... 161
5.3 BEHAVIOR OF FIXED ARCH UNDER VERTICAL AND
LATERAL LOADS .............................. 163
5.3.1 Variation of the Ultimate Load of Fixed Arch with In-Plane
Slenderness Ratio .......................... 164
5.3.1.1 Plastic Ultimate Load of Fixed Arch ....... 164
5.3.1.2 Elastic Ultimate Load of Fixed Arch ....... 166
5.3.1.3 Combined Ultimate Load of Fixed Arch ..... 168
5.3.2 Ultimate Load Interaction Curves ................ 171
5.3.3 Failure Modes and Response Curves .............. 173
5.4 THREE-DIMENSIONAL ULTIMATE LOAD SURFACE ...... 178

viii '

 

 

5.4.1 Two-Dimensional Interactions Between The Ultimate Load

Components ............................. 181
CHAPTER VI
SUMMARY AND CONCLUSIONS ........................ 186
6.1 SUMMARY .................................. 186
6.1.1 Nonlinear Finite Element and Solutions ............ 186
6.1.2 In-Plane Arch Behavior ...................... 188
6.1.3 Out-of—Plane Arch Behavior ................... 191
6.2 CONCLUSIONS ............................... 192
6.3 RECOMMENDATIONS ........................... 194
APPENDIX A
METHODS OF SOLUTION OF
NONLINEAR F.E..M EQUATIONS ....................... 195
A.1 INTRODUCTION ............................... 195
A.2 DIRECT NEWTON—RAPHSON METHOD ............... 196
A.3 MODIFIED NEWTON-RAPHSON METHOD ............. 199
A.4 DISPLACEMENT-CONTROLLED ANALYSIS ............ 202
LIST OF REFERENCES ............................... 208
ix

 

 

LIST OF TABLES

TABLE ‘ PAGE

Table 3-1 Comparison of different yield functions with calculated results of
Argyris et a1 ........................................ 89

Table 3-2 Comparison of different yield functions with experimental results of
Shinke et al. .......................................

Table 4-1 Plastic hinge formation in hinged arch for different types of analysis
under different load combinations. ......................... 125

Table 4-2 Relative Increase of fixed arch ultimate load in comparison with two-
hinged arch for different types of analysis and values of longitudinal load ratio,
h. .............................................. 130

Table 4-3 Plastic hinge formation in fixed arch for different types of analysis
under different load combinations. ......................... 147

 

LIST OF FIGURES

FIGURE PAGE

Figure 2— 1 End joint displacements of straight beam element 1n three
dimensional space. ................................... 19

Figure 2-2 Different types of yield surface in two-dimensional space. . . . 29

Figure 2—3 Different types of yield surface in three-dimensional space and
locations of ridge and corner regions. ....................... 31

Figure 2-4 Definition of the yield surface and the force and plastic
displacement increment vectors for a two-component joint. .......... 35

Figure 2-5 Definition of the discontinuities along ridge and corner regions of
octahedral yield surface ................................. 36

Figure 2-6 Locating the force increment penetration point of the yield surface40
Figure 2-7 Iteration process to keep force point at yield surface ........ 42

Figure 2-8 Treatment of force increment extending across surface
discontinuity. ...................................... 44

Figure 2-9 Determination of the initial yield status of a joint at the beginning of
load increment. ..................................... 47

Figure 2-10 Calculation of the first stage elasto—plastic resistance of straight
beam during load increment. ............................. 49

Figure 2-11 Calculation of the second stage elasto-plastic resistance of straight
beam during load increment. ............................. 51

xi

 

 

Figure 2-12 Calculation of the third stage elasto-plastic resistance of straight
beam during load increment. ............................. 53

Figure 3-1 Iterative solution of nonlinear equations of equilibrium using the
Newton—Raphson iteration process. ......................... 65

Figure 3-2 Comparison of program with beam—column results for elastic-
nonlinear behavior of single-story, single-bay frame. .............. 86

Figure 3-3 Comparison of program with finite—element results for nonlinear-
elastic behavior of a space frame. .......................... 87

Figure 3-4 Behavior of a two-story, two-bay frame reported by Argyris, et. a1.
under plastic combined loading: response curves and failure mode. ..... 90

Figure 3—5 Ultimate load of single-rib arch under vertical unsymmetric loading:
comparison of different yield functions with experimental results of Shinke et.
a1 ............................................... 92

Figure 3—6 Response curves of single-rib arch under vertical loading:
comparison of inverse parabolic yield function with experimental results by
Shinke et. al.. ...................................... 93

Figure 4—1 Finite element model and section properties of the arch used for
parametric study. .................................... 96

Figure 4-2 Individual unit loading distributions in the vertical, longitudinal,
and lateral directions. ................................. 99

Figure 4-3 Applied loading combinations for different loading cases of the
parametric study. .................................... 100

Figure 4-4 Variation of ultimate load of two-hinged arch under vertical load (r
= 0.99) using different types of analysis, with in-plane Slenderness
ratio, L/ry. ........................................ 101

Figure 4-5 Variation with L/ry of the plastic ultimate load of two-hinged arch
under vertical and longitudinal loading. ...................... 106

Figure 4—6 Variation with L/ry of the elastic—nonlinear ultimate load of two— '
hinged arch under vertical and longitudinal loading. .............. 108

Figure 4—7 Variation with L/ry of the combined ultimate load of two-hinged
arch under vertical and longitudinal loading. ................... 111

xii

 

Figure 4-8 Elastic and combined ultimate loads of two-hinged arch under
vertical and longitudinal loading normalized to corresponding plastic ultimate
loads of same value of h. ............................... 113

Figure 4—9 Variation of the plastic, elastic, and combined ultimate loads of
two—hinged arch with h for different values of L/ry. ............... 115

Figure 4-10 Interaction curves between vertical and longitudinal loading of
two-hinged arch for different values of L/ry. ................... 117

Figure 4—11 Deflected shapes of two—hinged stiff arch (L/ry = 125) at ultimate
load for different types of analysis and values of h ................ 120

Figure 4-12 Deflected shapes of two—hinged medium arch (L/ry = 225) at
ultimate load for different types of analysis and values of h. ......... 121

Figure 4-13 Displacement response curves at quarter point of medium two-
hinged arch (L/ry = 225) for different values of h. ............... 123

Figure 4-14 Relation between axial force and bending moment at the left
quarter point of hinged-arch (L/ry = 225). .................... 126

Figure 4—15 Variation with L/ry of the plastic ultimate load of fixed arch under
combined vertical and longitudinal loading. .................... 128

Figure 4-16 Variation with L/ry of the elastic ultimate load of fixed arch under
vertical and longitudinal loading. .......................... 131

Figure 4-17 Variation with Mr, of the combined ultimate load of fixed arch
under vertical and longitudinal loading. ...................... 135

Figure 4—18 Elastic and combined ultimate loads of fixed arch under vertical
and longitudinal load normalized to corresponding plastic loads of the same
value of h. ........................................ 137

Figure 4-19 Variation of the plastic, elastic, and combined ultimate loads of
fixed arch with h for different values of L/ry. .................. 139

Figure 4-20 Interaction curve between vertical and longitudinal loading of fixed
arch for different values of L/ry ............................ 141

Figure 4-21 Deflected shapes of fixed stiff arch (L/ry = 125) at maximum load
for different types of analysis and values Of h. .................. 143

xiii

Figure 4-22 Deflected shapes of fixed medium arch (L/ry = 225) at maximum
load for different types of analysis and values of h. ............... 144

Figure 4-23 Response curves of fixed medium arch (L/ry = 225) for different
types of analysis and values of h. .......................... 145

Figure 4-24 Relation between axial force and bending moment at the left
quarter point of fixed arch (L/ry = 225). ..................... 149

Figure 4-25 Plastic, elastic-nonlinear, and combined ultimate loads of two—
hinged arch under unsymmetrical vertical loading. ............... 151

Figure 4-26 Elastic and combined ultimate loads of two—hinged arch under
vertical unsymmetrical loading normalized to corresponding plastic ultimate
loads of same value of r. ............................... 154

Figure 4-27 Variation of the vertical symmetric component of the ultimate load
of two—hinged arch in the presence of vertical unsymmetric or vertical and
longitudinal loads. ................................... 155

Figure 4-28 Variation of the vertical symmetric component of the ultimate load
of fixed arch in the presence of vertical unsymmetric or vertical and
longitudinal loads. ............................ . ....... 158

Figure 5-1 Effect of the type of arch supports and its relative lateral to in-plane
stiffness on its vertical and lateral ultimate loads. ................ 162

Figure 5- 2 Variation with L/r of the plastic ultimate load of fixed arch under
vertical and lateral loading .............................. 165

Figure 5-3 Variation with L/ry of the elastic ultimate load of fixed arch under
vertical and lateral loading. .............................. 167

Figure 5-4 Variation with L/ry of the combined ultimate load of fixed arch
under vertical and lateral loading. .......................... 169

Figure 5-5 Interaction curves between vertical and lateral load components for
fixed arch. ........................................ 172

Figure 5-6 Deflected shapes of fixed medium arch (L/ry = 225) at ultimate load
for different types of analysis and values of s. .................. 174

Figure 5-7 Response curves for fixed arch under vertical and lateral loading.177

xiv

Figure 5-8 Ultimate load surface of fixed arch (L/ry = 225). ......... 180
Figure 5-9 Projections of the ultimate load surface onto its principal planes. 182

Figure A-l Direct Newton-Raphson iteration process for solution of nonlinear
equations. ......................................... 198

Figure A-2 Modified Newton-Raphson iteration process for solution of
nonlinear equations. .................................. 200

Figure A-3 Displacement-controlled analysis iteration process for solution of
nonlinear equations. .................................. 204

 

 

D',LCA(11)

Dim

LIST OF SYMBOLS

Area of the arch or the element cross section.

Amplification factor by which the applied load (live +
impact) used in linear analysis is increased to account for
nonlinear effects.

Total width and total depth, respectively, of the box-type
cross-section of the arch.

A constant and a 2x2 constant matrix, used in elasto-plastic
stiffness matrix calculation when one joint (I) yields, and
both joint yield, respectively.

Total structure (global) displacement vector at the end of
iteration i of nonlinear solution for jth load increment.

Total structure displacement vector at the end of load-
controlled analysis (LCA), to be used in the displacement-
controlled analysis procedure.

Average value of displacement at d.o.f. I‘, per unit load
vector, P", at the end of load-controlled analysis (LCA), to
be used in the displacement-controlled analysis procedure.

Displacement vector corresponding to unit load vector, P",
solved for at the beginning of incrementj, for use in the
displacement-controlled analysis procedure.

Total structure (global) displacement vector at the
beginning of load increment j, obtained from the previous
increment.

 

 

 

 

DU)? DO)

an;

dDilv

an,"

dPilv

dPi“J

xvii
The portions (or subvectors) of displacement vector D that
correpond to translation and rotation degrees of freedom,
respectively, needed in calculations for convergence tests.

D is a generic term that may be replaced by any specific
displacement vector, e.g., Dmf or dD,“.

Incermental (or iterative) structure displacement vector
during iteration i of nonlinear solution for jth load
increment.

The incremental displacement vector corresponding to the
incremental load vector, dPi‘V, added to the structure
displacement vector during iteration i of load increment j of
the displacement-controlled analysis procedure so that

dDi = dD,“ + dDi‘V.

The incremental displacement vector coeesponding to the
unbalanced load vector, dP,“, added to the structure
displacement vector during iteration i of load increment j of
the displacement—controlled analysis procedure so that

dDi = dD,“ + dDi‘V.

Incermental structure displacement vector during iteration i
of nonlinear solution for jth load increment, obtained using
the modified Newton-Raphson method before applying the
acceleration procedure.

Incremental structure diaplacement vector during the first
iteration of solution for jth load increment, recalculated for
the purpose of verifying the solution accuracy.

Incremental load parameter obtained during iteration i of
load increment j of the displacement—controlled analysis
procedure.

The incremental external load vector, defined in terms of
the unit load vector, P“, applied to the structure during
iteration i of load incrementj of the displacement-
controlled analysis procedure.

Unbalanced (or residual) structure force vector at the end
of iteration i (i.e., at the beginning of iteration 1+1) of
solution for jth load increment.

 

 

xviii
dPMj = Incremental structure force vector at the beginning of the
first iteration of solution for jth load increment,

recalculated for the purpose of verifying the solution
accuracy.

dP“j = Residual structure force vector still unbalanced after
convergence at the end of solution for jth load increment.

dQ = Force increment vector at the element joints during a load
increment.

in = Force increment during a load increment at joint i of the
element.

dQs = Element joint force increment corresponding to stage

number s of elasto-plastic resistance computation during a
load incrment, s = 1, 2, or 3.

in‘,s, Qi“,s, Qi‘p,s _
= Tentative values of the force increment vector, dQ"’, -at
joint i of an element- used during stage 5 of elasto-plastic
resistance calculation to obtain the final value of dQ‘.

dq = Total end joint displacement increment vector of the
element during a load incrment.

dq1 (=dQ“), dq2 (=dq°"), and dq3 (=dqp")
= Portions of the element displacement increment during a
load increment that corresponds to the first, second, and
third stages of elasto-plastic resistance, with both joints
being elastic, one joint plastic, and both joints plastic,

respectively.

dqci, dqpi = Elastic and plastic components of the total displacement
increment at joint i of an element during a load incrment,
repectively.

dqm, dqci,s, dqpi,s
= Total displacement at joint i of an elemenet, and its elastic
and plastic components, respectively, corresponding to
stage s of an element elasto-plastic resistance computation
during a load incrment.

 

 

xix

dql, dqcl, dqpl
= Total element displaCement increment vector and its elastic
and plastics components corresponding to stage 5 of the
element elasto-plastic resistance computation during a load
incrment.

dq‘” = The portion of element displacement increment, dq, that is
still not accounted for (i.e. its resistance not calculated) yet
at the beginning of stage 5 of elasto~plastic resistance

calculation.
E = Elasticity modulus of the material.
EN, Eff-j = The residual and total energy (i.e., work done by the forces

through their corresponding displacements) at the end of
iteration i of load increment j.

f = Span rise of the arch measured as the vertical distance from
the arch support to its crown.

f1,i7 f2,i) f3,i) f4,i1 f5,
= Constants calculated during iteration i for use in the
' acceleration procedure used in conjunction with the

modified Newton-Raphson procedure.

 

Fi,.(h), Fr..(h), Fh,p(h)

= Combined, elastic, and plastic longitudinal load component
of fixed arch under longitudinal and vertical loading,
respectively, for a given ratio of h.

Fho) Fso: E.., = Ultimate loads that the fixed arch can carry under
longitudinal~only, lateral~only, and symmetric vertical-only
loads, respectively.

Flame, Fho,” Fho,p
= Combined, elastic, and plastic ultimate loads of fixed arch
under longitudinal-only load, respectively.

Fa,c(s)v Fs,e(s), 1250(8)
= Combined, elastic, and plastic lateral load component of
fixed. arch under lateral and vertical loading, respectively,
for a given ratio of s.

XX

Fl0,c7 Fro,” Fso,p
= Combined, elastic, and plastic ultimate loads of fixed arch

under lateral—only load, respectively.

F..(h). F.,.(h), F.,p(h)
= Combined, elastic, and plastic total load of fixed arch
under longitudinal and vertical loading, respectively, for a
given ratio of h.

Ft,c(r)! Ft,c(r)! Ft,p(r)
= Combined, elastic, and plastic total loads of fixed arch
under vertical unsymmetric loading, respectively, for a
given ratio of r.

Fl,c(s)) Ft,c(s)) Ft,p(s)
= Combined, elastic, and plastic total loads of fixed arch
under lateral and vertical loading, respectively, for a given

ratio of s.

Fv,.(h), Fv,.(h). Fv,p(h)
= Combined, elastic, and plastic vertical load components of
fixed arch under longitudinal and vertical loading,
respectively, for a given ratio of h.

Fv,.(r). Fv,.(r), Fv,p(r)

= Combined, elastic, and plastic symmetric vertical load
components of fixed arch under vertical unsymmetric
loading, respectively, for a given ratio of r.

FV,¢(S)1 Fv,e(s)a Fv,p(s)
= Combined, elastic, and plastic vertical load component of
fixed arch under longitudinal and vertical loading,
respectively, for a given ratio of s.

= Combined, elastic, and plastic ultimate loads of fixed arch
under symmetric vertical-only load, respectively.

f0 = The ultimate load parameter corresponding to the ultimate
load of the arch.

Her = Critical horizontal reaction at arch support.

 

IIz

y,

Iy,archa Iy,deck

y,comb

KJ'Kai

1.0 t,1

k1}

xxi

Ratio of longitudinal load to total load applied to the arch
during vertical and longitudinal loading, and three-
dimensional loading, respectively.

Moments of Inertia of the element cross section about the
y-axis and z-axis, respectively.

Moment of inertia of the arch section and the deck section,
respectively, of a stiffened arch about their respective local
y—axis.

Combined moment of inertia of a stiffened arch about y-
axis.

A general term that refers to the structure global stiffness
matrix.

Structure global elastic~gemoetric stiffness matrix at the
beginning of increment j, formulated without including
plastic effects.

Structure elastic-linear stiffness matrix.

A general term that refers to the structure tangential
stiffness matrix.

Structure tangential stiffness matrix at the beginning of
increment j, and at the end of iteration i (i.e., at the
beginning of iteration i+l) during solution for load
increment j.

Coeffecient used in formula for buckling of arch.

General term for the element stiffness matrix.

Generic term for the elastic stiffness matrix of the element
that may be replaced by either the elastic-linear or elastic-

nonlinear one.

The submatrix of the elastic stiffness matrix, k,, that relates

- the force vector at joint I to the displacement vector at joint

I.

 

k.(i,j)

keg,“

k.

P

k. k.

k111i kuZ

k k

cg: eg.n

L/r

Y

L/r

Z

N

xxii

The term of elastic stiffness matrix, k,, that relates ith
component of the force vector to jth component of the
displacement vector.

Average value of the elastic-nonlinear stiffness matrix of
the element during a load increment.

The elasto-plastic stiffness matrix of an element.

The geometric and plastic incremental (also called
reduction) matrices of the element elasto-plastic stiffness
matrix, kep, that represent the geometric and plastic effects,

respectively.

First and second order nonlinear—elastic incremental
stiffness matrices of the element, respectively.

Element elastic-nonlinear (also called elastic-geometric)
stiffness matrix.

Element elastic-linear stiffness matrix.
Element tangential stiffness matrix (that may include either
or both geometric and plastic nonlinearity effects,

whichever is present).

Elemenet tangential stiffness matrix at beginning of stage 5
of elasto-plastic resistance calculation.

Length of the arch measured along its axis.

General term that refers to arch or element length: length
of the horizontal projection of the arch, or the finite
element or frame span length.

In-plane slenderness ratio of the arch.

Out-of—plane slenderness ratio of the arch.

A general term for axial force in a structural member.

{N’ VX’ Vy’ waiMy: M2}

Force vector at a joint of the element.

 

 

NQP) NQP,c =

NMM

P’ Y.P’ 1’17

p103) =

Plv =

Pa): Pa) =

P; =

Ph,c(h) , Ph,c(h) : Ph.p(

P1107 Pso) on =

PM,“ PM,” Pho,p

Pill” P311: P =

V“

xxiii
The axial force and critical (buckling) axial force,

respectively, at the quarter point of the arch span due to
applied load.

Plastic (also called ultimate or limit) forces at a joint of the
element.

The load applied to the structure, expressed in its general
form as a function of the structure displaced position, D.

Unit load vector applied to the structure. The actual load,
P, applied to the structure is measured in terms of PIv and a
load parameter so that, AP = Af-P‘V.

The portions (or subvectors) of force vector P that
correpond to translation and rotation degrees of freedom,
respectively, needed in calculations for convergence tests.
P is a generic term that may be replaced by any specific
force vector, e.g., Pref or dPi“.

Total load vector applied to the structure at the end of
incrementj (i.e., at the beginning of incrementj+1).

h)

Combined, elastic, and plastic longitudinal load
components of two-hinged arch under longitudinal and
vertical loading, respectively, for a given ratio of h.

Ultimate loads that the two—hinged arch can carry under
longitudinal-only, lateral-only, and vertical—only
(symmetric) loads, respectively.

Combined, elastic, and plastic ultimate loads of two-hinged
arch under longitudinal—only load, respectively.

Longitudinal, lateral, and vertical (symmetric) unit total
loads, respectively, applied to the arch due to a unit
uniformly distributed load along the horizontal projection
of the arch in the corresponding direction.

 

 

xxiv

Ps,c(s)) Pl,6(s)7 Ps.p(s)
' = Combined, elastic, and plastic lateral load component of
two-hinged arch under lateral and vertical loading,
respectively, for a given ratio of 5.

P30,” P30,” Pump
= Combined, elastic, and plastic ultimate loads of two-hinged
arch under lateral—only load, respectively.

 

P1,.(h), P.,.(h), P.,p(h)
= Combined, elastic, and plastic total loads of two-hinged
arch under longitudinal and vertical loading, respectively,
for a given ratio of s.

Pt,c(r)’ Pt,c(r)) Pt,p(r)
= Combined, elastic, and plastic total loads of two-hinged
arch under vertical unsymmetric loading, respectively, for
a given ratio of r.

Pt,e(s)) Pt,e(s)’ Pt,p(s)
= Combined, elastic, and plastic total load of two—hinged arch
under lateral and vertical loading, respectively, for a given

 

ratio of s.

Pm(h), Pm(s) = Unit total loads applied to the arch as a combination of
vertical and longitudinal, and vertical and lateral loads,
respectively.

Pv,.(h), Pv,.(h), Pv,p(h)
= Combined, elastic, and plastic vertical load component of
two—hinged arch under longitudinal and vertical loading,
respectively, for a given ratio of h.

PV.C(r)’ Pv,c(r): Pv,p(r)
= Combined, elastic, and plastic symmetric vertical load
components of two-hinged arch under vertical unsymmetric
loading, respectively, for a given ratio of r.

PVy¢(s)’ Pv,e(s)a Pv,p(s)

= Combined, elastic, and plastic vertical load component of
two-hinged arch under lateral and vertical loading,
respectively, for a given ratio of s.

on,c) on,e: on,p

PDL —

phu) pun pvu

‘pi =

Combined, elastic, and plastic ultimate loads of two-hinged
arch under symmetric vertical-only load, respectively.

Symmetric (dead) load intensity applied to the arch full
span along its horizontal projection.

Longitudinal, lateral, and vertical (symmetric) unit
uniformly distributed loads applied to the arch along the
horizontal projection of the arch in the corresponding
direction.

Unsymmetric (live) load intensity applied to half of the
arch span along its horizontal projection.

Ultimate load intensity (or uniformly distributed load)
applied along the arch horizontal projection that causes
failure, calculated using elastic stability or nonlinear
analysis.

The load intensity (or uniformly distributed load) applied
along the arch horizontal projection that causes yielding in
the arch at its supports, calculated using linear analysis.

Total load intensity of the different load intensities (e.g.,
live, dead, and longotudinal) applied to the arch half of full
length span of its horizontal projection.

Element force vector at its joints.
The force vector at the joint i of an element.
Old and new (calculated)force vectors (or points) at the

beginning and end of stage 3 of elasto-plastic resistance
calculation.

Tentative values of the new force vector, Q,‘, used during
stage s of elasto-plastic resistance calculation to obtain the
final value of On“.

The plastic (ultimate) force of the ith component of the
force vector at an element joint.

 

 

qt, qp
qj, ch: qu

qoa (Imp, qn

R(D)

ry) rz

SI and S2

tf: tw

xxvi

General term for the element end (or joint) displacement
vector.

Elastic and plastic components of element joint
displacement vector, dq.

The total, elastic, and plastic displacement vector at joint j
of an element.

The old, mid-point, and new joint displacement vector of
the element at the beginning, mid-point, and end of a load
increment, respectively.

Structure resistance vector corresponding to the structure
displaced position, D.

Structure resistance at the beginning of increment j.

Ratio of symmetrical (e. g. dead) load intensity to the total
(symmetrical and unsymmetrical, e. g., dead and live) load
intensity applied to the arch.

Equivalent unsymmetry ratio when applying both vertical
unsymmetrical and longitudinal loads to the arch along with
vertical symmetrical load.

Radius of gyration of the arch section about the y-axis and
z-axis, respectively.

symbols (or names) for the two section proportions of the
arch that were used in the parametric study.

Ratio of lateral load to total load applied to the arch during
vertical and lateral loading, and three-dimensional loading,
respectively.

scaling factor applied to the specified load parameter, Afo,
to obtain a smaller load increment in the procedure for
adjusting the load increment size.

Flange and web thicknesses, respectively, of the box-type
cross-section of the arch.

 

 

 

F’ffi—T

xxvii

U = Total strain energy in the element produced by internal
forces.

U1, U2, U3 = Portions of the element total strain energy, U,
corresponding to the quadratic, cubic, and quartic terms of
U.

1
V = Gradient vector of the element, = {V } {0)
{0} {v2}

V‘(Q‘) = Gradient vector of the force vector, Q, at joint i of an
element. '

Voi, Vni = Gradient vectors calculated at the beginning and end of a
displacement stage at joint i during the process of elasto-
plastic resistance calcultion of an element.

V“D = Estimated gradient vector at joint 1 of an element during

' iteration j of the iteration process to return the force point
to the yield surface.

V, V. = Ratio of vertical load to total load applied to the arch
during vertical and longitudinal or vertical and lateral
loading, and three-dimensional loading, respectively.

01.. = Coeffecient used in formula for buckling of arch.

Olei = Scaling factor that scales the force increment vector, dQ‘,
at joint i of an element to the yield surface during a load
increment.

01:" = Scaling factor that scales the force increment vector, dQ“,
at joint i of an element to the yield surface during stage s
of elasto-plastic resistance computation of a load
increment.

3, 'Y = Constants in formulas for arch ultimate load.

ADij = Total displacement increment during increment j,

accumulated through the end of iteration i for nonlinear
solution.

 

 

 

xxviii

Specified value of the displacement at the controlling
d.o.f., I‘, used in the displacement-controlled analysis
procedure.

Load parameter increment that defines the applied load for
the increment in terms of the unit load vector, AP =
Af - P”.

Total increment load parameter at the end of iteration i of
load increment j that define the applied external load for
the increment in the displacement—controlled analysis
procedure, Afi = Af,_1 + de.

The initial value of the load parameter increment, Af,
specified in the data input.

Specified load parameter value used in determining the
value of displacement at the specified d.o.f., AD,,,(I‘) in the
displacement-controlled analysis procedure.

Total magnitude of the load parameter to be applied to the
structure using the specified load vector.

Length of the horizontal projection of each arch segment
represented by a straight beam finite element, = L/n,
where n = number of elements.

Unit longitudinal, lateral, and vertical concentrated loads
that represent the uniformly distributed load applied to an
arch node, in their respective directions.

Load increment applied to the structure during load
increment j.

Structure incremental resistance during incrementj,
calculated at the end of iteration i.

The difference between the unbalanced load vectors at the
beginning of iterations i-l and i, 5P,“ = dP,u -dPi_1, used in
the acceleration procedure used in conjunction with the
modified Newton~Raphson method.

F
6(X,y,Z)
6r
6P76D)€E
<I>
(I’ll; @ul
WQ‘)
¢0i7 éIii
45” 0i: ¢i
N
i #1: V1, an
06.2)
00
0y
0' 0'

ll!

:11

xxix

The d.o.f. that is specified to be used as the controlling
d.o.f. in the displacement—controlled analysis procedure.

Normal strain at point (x,y,z) of the element.

Tolerance limit for yield surface definition of a force
function at an element’s joint.

Tolerance limits for force, displacement, and energy
convergence tests, respectively.

Force function (or plastic potential function) of the force
vector at an element’s joint.

Lower and upper limits of the yield surface definition.
Force function of force vector Qi at joint i of an element.
Force functions of force vectors Q0i and Q,‘, at the
beginning and end of load (or a displacement stage)

increment, respectively, atjoint i of an element.

Rotation displacements about x-, y—, and z—axes at joint i of
an element.

Proportionality constant that relates the plastic
displacement vector to the gradient vector at the yielded
joint 1 of an element.

 

Translational displacements in the direction of x-, y-, and
z—zxis, respectively, at joint 1 of an element.

Normal stress at point (y,z) of the element section.

Allowable (or ultimate) stress permitted to occur in the
arch due to applied loads.

Yield stress of the material.

Normal stress at a section due to normal force and bending
moment, respectively.

xxx
1, E, 12,1, = Limits for energy terms divergence tests used for check of
iteration process divergence during nonlinear solution for a
load increment.
111., 12,? = Limits for force norms divergence tests used for check of

iteration process divergence during nonlinear solution for a
load increment.

 

CHAPTER I

INTRODUCTION

 

1._1 GENERAL

The objective of this thesis is to study the ultimate load carrying
capacity of arches. The prevailing trend in structural engineering is toward the
ultimate strength design, mainly because it is based on a more rational

approach to the design process that would place the structural material where it

 

is needed. In addition, it helps to provide accurate estimation of the structure
reliability under different loading conditions as exemplified by the recently
adopted load—resistance factor design (LRFD) method. In this regard, this study
represents a continuation of the efforts to study the ultimate strength of
structures under different load combinations.

The main advantage of arch structures is their ability to carry loads
through axial thrust (with little or no bending moments) when the funicular
curve of the applied load is close to or coincides with the longitudinal axis of
the arch. For parabolic arches, this load is the symmetrical uniform in-plane
load applied to the horizontal projection of the arch. While this property allows

the construction of long—span bridges with relatively small section depth to

j,

 

2

carry dead loads, the disadvantage of possible sudden failure due to buckling
under high axial load requires consideration its of stability.

In addition, when the arch is subjected to loads other than the vertical
symmetrical load —for example, vertical unsymmetrical load- its load carrying
capacity is significantly reduced. In this study, the effects of other types of
loading is considered, in particular the combination of vertical load with
longitudinal (horizontal in—plane) and lateral (transverse) loads are considered.

These loads usually occur due to earthquake or wind effects.
1_.2 OVERVIEW

1A Arch Behavior under Different Loads

In general, the behavior of arch structures under different combinations
of loading conditions and the interaction between possible different components
of the structure itself (for example, the arch ribs, the deck, and different
bracing systems) is quite complicated. Simplifying assumptions that eliminate
some of these variables —such as neglecting possible deck and/or bracing
systems- are usually deemed necessary to first study the behavior of the main
structural member, i.e., the arch.

The nonlinear behavior of arches under vertical in-plane loading was
extensively studied where both elastic and plastic nonlinearities were included.
It is well documented (for example Austin (2), and Kuranishi (16)) that the arch

total ultimate load decreases as the vertical load unsymmetry increases, and

 

 

3

that unsymmetric failure mode governs for both the symmetric and unsym-
metric loads. This was found to be true also for stiffened arches where the deck
serves as a stiffener to the arch in bending.

Another mode of failure that was investigated in the literature is the out-
of—plane elastic behavior of arches, specially the lateral-torsional failure of
arches under vertical loading, where it was found that a single-rib arch often
prematurely fails in this mode before it reaches its in-plane ultimate strength.
To overcome this problem, enough lateral stiffness may be provided by using
double-rib arches connected with lateral bracing system. Inelastic behavior of
braced double—arch systems was studied under wind loading, where the lateral
load was assumed to be a small proportion of the vertical load, and the bracing
requirements were the main objective.

The arch may be subjected to three types of loading. First, a vertical in—
plane loading due to dead and live loads is the primary type of loading. Longit—
udinal (horizontal in-plane) loading may result from wind load or more likely
from earthquake load. Lateral (transverse) loading also usually results from
wind or earthquake.

While the in—plane and out-of—plane arch behavior had been extensively
studied under different combinations of vertical and lateral loads, the inter-

action between the vertical and longitudinal loads had not been investigated
probably bacause of the assumption that the arch elastic stiffness is much

higher in the longitudinal direction. However, this assumption may not be true

 

4

when material nonlinearity is included. This is considered in this study. In
addition, limited studies in the past on the interaction between symmetric and
unsymmetric vertical and lateral loads were limited to two-dimensional studies.

In this study, a case involving the three loads is considered.

1.2.2 Plasticity Model

 

Early studies of arch behavior concentrated on elastic buckling of arches
and neglected plastic effects. Recent studies included both elastic and plastic
nonlinearities mainly using a "fiber model" to represent the arch. In this
model, the arch is divided into multiple segments along its longitudinal axis
with each segment being divided into small fibers that extend along the segment
length (i.e., the cross section of each segment is divided into small areas). The
stresses and strains in each one of these fibers are assumed to be uniform
throughout the fiber. By assuming a nonlinear stress—strain relation (e.g.,
elasto-plastic relation) for each fiber of the segment and assuming that "plane
sections" remain plane after deformation, the compatibility and equilibrium
conditions at the interfaces of each two adjoining segments are enforced. As a
result, the forces and stresses in the arch can be solved as a set of nonlinear
equations, and the Spread of'yielding in the arch may be traced with the
progress of loading.

Although the fiber model was successful and produced accurate results,
its use for complicated structures is cumbersome and may be prohibitively time

consuming. In this research, the classic plastic hinge approach was used to

 

 

 

5

model plastic effects with special treatment of discontinuities of the yield
surface. Comparison with experimental results produced good agreement, and

this model was used in obtaining the data for the parametric studies.

1_.3_ SUMMARY OF PRESENT STUDY

In chapter two, a detailed, three-stage procedure to implement the plastic
hinge model is described. Four different functions, namely spherical,
octahederal, parabolic, and inverse parabolic, were used to model the yield
surface at a plastic hinge. The gradient singularity at regions of discontinuity in
the yield surface was treated by using special patch functions. In chapter three,
the scope and implementation aspects of the computer program for nonlinear
solution are described, and numerical results are presented for comparison with
experimental and theoretical solutions reported in the literature. The inverse
parabolic function was shown to produce the best agreement.

In chapter four, the elastic, plastic, and combined (that includes both
nonlinearities) ultimate loads of single-rib parabolic arch with different support
conditions, under different combinations of symmetrical vertical load and

longitudinal load (uniformly distributed along the arch horizontal projection)
are studied. Comparison with the arch behavior under unsymmetrical vertical
loading revealed similarity between the two cases, specially for the combined
analysis. In addition, the interaction between the vertical and horizontal load
, components, the failure modes and response curves, the formation of plastic

hinges and the force path at the arch quarter point were also examined.

 

 

 

6

In chapter five, the arch plastic, elastic and combined ultimate. loads
under different combinations of vertical load and lateral load (also uniformly
distributed along the arch horizontal projection) were obtained, and the
interaction curves, failure modes, and response curves were also examined. In
addition, the three—dimensional combined ultimate load surface of the arch was

constructed to demonstrate the interaction between the three types of loading.

L4 LITERATURE REVIEW
Literature review of the in—plane and out—of—plane behavior of arches is
presented in the following sections. For each section, the review is divided into

studies that deal with elastic behavior and those that include plastic effects.
1.4.1 In—Plane Behavior of Arches

A) Elastic Behavior

In 1971, Austin (2), summarized the research done in the area of arch
behavior to that date. He reviewed the studies of linear buckling of arches
under symmetrical loading. These studies proposed a formula for calculating
the arch strength in terms of the axial thrust at the quarter point of the arch
, span, NQP, that is analogous to the formula for buckling load of axially loaded
column,

NQP,c = 4a,, EIyl‘L'2 = 4r2 EI,/(k,,L')2
where None is the critical axial load at the span quarter point that causes the

arch to buckle, ac, and kcr are coefficients that depend on rise—to—span ratio,

 

 

7
f/L, and type of arch support (2-hinged, 3-hinged, or fixed), L' is the arch
length along its longitudinal axis, L is the arch span length (the horizontal
projection of L'), Iy is the moment of inertia about the out-of—plane axis, and E
is the elasticity modulus. The arch critical load under unsymmetrical vertical
loading was reviewed, where the studies suggested an amplification factors, AP,
similar to that of beam-column to correct linear moments and deflections for
nonlinear effects, AF = 1/(1-NQP/NQP,,), where NQP is the actual thrust resulting
from the applied load. However, when the ratio of NQP/NQP,e is high, the author
recommended a deflection (i.e., nonlinear) analysis.

Studies suggested the above formulas may be applied to non-uniform
arches using an equivalent arch with properties equal to the average properties
of the actual arch, and to stiffened arches by using an equivalent moment of
inertia, 1).,eq = 1””, + Imm-

In 1976, Austin and Ross (3), studied the symmetrical and unsym-
metrical buckling modes of symmetrically loaded arches, using both classical
(linear eigenvalue) and nonlinear analysis. They found that the unsymmetrical
buckling load was always lower than symmetrical buckling load for both two-
hinged and fixed arches, and that the classical theory produces large errors in
predicting buckling loads for symmetrical modes, and in predicting moments
for all modes. They also found that a parabolic arch carries larger load than

circular or catenary arches.

 

 

 

st

811

8

In 1980, Maeda and Hayashi (24), reported on the elastic nonlinear
.nalysis and design aspects of long span arches and compared the obtained
esults with the limits specified by the Japanese specifications. Based on their
ubservations they suggested more precise formulas that take the load
:nsymmetry into account.

In 1982, Harrison (12), reported a study of the elastic response of
rarabolic arches under unsymmetrical loading that extends from one edge to a
raction of the length (varied from 0.3 to 1.0). He concluded that the ultimate
)ad intensity (i.e., uniformly distributed load) was minimum when about 70%
f the span was loaded, with reduction of 10% for 2-hinged arches and 13% for

Lxed arches compared with symmetric loading.

 

In 1984, Medallah (25), reported on the nonlinear response of deck-
:iffened arch subjected to constant vertical load and loaded till failure by
)ngitudinal loading. He found the amplification factor method mentioned
)ove adequate to predict the elastic nonlinear response from linear analysis.

In 1989, Dusseau and Wen (11), reported the results of linear dynamic
ralysis of deck-stiffened arches under unequal seismic support earthquake
otion in vertical, longitudinal and lateral directions. The study found that
ngitudinal motion creates high stresses in the arch itself and in the longitudi-
l1 bracing system between the arch and deck; they found that the sum of
resses due to both vertical and longitudinal motions could exceed the yield

ress of the material under a credible earthquake.

 

Im

31 Plastic Behavior

The behavior of arches when both geometric and plastic nonlinearities
[1'6 included was a subject of study since 1972. In the majority of these studies,
he fiber approach described earlier was used. A survey paper by Yabuki and

linnakota (46), 1984, covers different aspects of research on arch behavior up

 

0 that date.

In 1972, Kuranishi and Lu (16), reported a study of the ultimate load of
i-hinged steel arches under vertical loading. Using rectangular and sandwich
ections, they studied the variation of the ultimate load ratio, po/pp (defined as
he ratio of the computed ultimate load intensity, p,, to that which first causes

'ield in the arch using linear analysis, pp), with main parameters such as in—

 

ilane slenderness ratio L/ry, unsymmetry ratio, r (= pDL/(pDL+pLL), where pDL
s the dead load applied over the full arch span, and pLL is the live load applied
ver half the arch span), and span—to-depth ratio, f/L (see Figures (4-1) and
4-2)). They found that pO/pp decreases with the increase of r and Llry. In
ddition, po/p, becomes significantly less than one as r increases, which
1dicates that unsafe design would result from using the linear elastic analysis.
In 1973 and based on the above study, Kuranishi (17), proposed two
)rmulas for conventional structural design based on elastic linear and
onlinear analysis, that are based essentially on the sum of stresses due to

ending and axial loads.

10

Shinke, Zui, and Namita (35), 1975, studied arches under symmetrical
1nd unsymmetrical vertical loads, and the relation of p,,/pp with other main
)arameters, and arrived at conclusions similar to Kuranishi and Lu. In another
:tudy (36), 1977, they compared analytical results with experimental results,

1nd found good agreement.

 

Kuranishi (18), 1977, studied the ultimate load of fixed arches and found
hat their behavior is similar to that of two—hinged arches with same properties
tut they have larger load carrying capacity. Also, he noted that for stiff arches
with low slenderness ratio), as the force path (i.e. the axial force—bending
noment relation) at the arch span quarter point approaches the yield surface, it
akes a path parallel to the yield surface with direction toward the axial force
.xis so that the axial force increases while the bending moment decreases; for
lender arches, the force path moves toward the moment axis.

Komatsu and Shinke (14), 1977, proposed a design formula for the
n-plane instability criteria of a symmetrically loaded arch analogous to the
eam-column formula used in current specifications, where the ratio of
llowable stress, 0'0, to yield stress, 0)., is related to the slenderness ratio of the
rch. In this formula two relationships in two distinct regions were defined,
ne controlled by plastic behavior and the other by elastic behavior. In
:ldition, they extended this formula for the case of unsymmetrical loading by

1troducing a correction factor for the above formula.

11

Kuranishi and Yabuki (19), 1979, compared an analytical method to
:xperimental data and found good agreement. They also found that residual
:tresses may reduce the ultimate load of the arch by up to 20%, while section
>roportioning has much smaller effect (up to ;l-_5 %). To avoid plastic local
)uckling of the arch section components, they suggested that maximum strain at
he arch springing should not exceed 4 times the yield strain.

In 1984, the same authors proposed a design formula based on research
)f plastic behavior of two-hinged arches (22), and in 1987 they (along with Lu)
:xtended it to fixed arches (48). The formula uses two regions based on the
evel of axial load at the quarter point of the arch, similar to the suggestion of
tustin (2), and correlates axial load and bending moment obtained using linear
,nalysis.

Yabuki, Vinnakota, and Kuranishi (47), 1986, reported a study of the
rehavior of fixed arches. They found that the elastic theories (linear and

onlinear) overestimate the ultimate load of the arch.

A2 MW

The out-of—plane behavior of arches under vertical loading, or lateral-
)rsional buckling, was extensively studied. However, arch behavior under
1rge lateral loading received less attention. Studies on elastic and elasto-plastic

rch behavior are reviewed here.

 

12
A) Elastic Behavior

In 1968, I. Ojalvo and Newman, (26), described a theoretical formula—
tion of the linear elastic stability of curved twisted members that was used in
several studies of arch behavior, (27,38,39,40). M. Ojalvo, Demuts, and
Tokarz (27), 1969, studied the linear elastic stability of circular ring segments
under thrust or pull force applied to the ring chord (the line that connects the
two ends of the ring). They found that torsional rigidity has a slight effect on
thrust buckling, but it has a more pronounced effect on pull buckling.

Shukla and Ojalvo (38), 1971, studied the elastic stability of parabolic
arches under vertical and tilting loads. Tokarz (39), 1971, reported on experi-
mental tests of lateral-torsional buckling of circular and parabolic arches with
different crown and end support conditions under vertical and tilted loads. In
1972, Tokarz and Sandhu, (40), presented an analytical study of the elastic
buckling of parabolic arches to predict the obtained experimental results above,
and found fair to good agreement.

These studies found that the load direction profoundly affected the arch
buckling load. It is highest when the load tilts with the arch (called hanger-type
load) and lowest when the load tilts opposite to it (called column-type load),
with vertical load (with no tilting) in between. The differences between highest
and lowest loads reach up to 400%. Also, the studied arches were found to fail

in symmetric out-of—plane mode due to bending, with insignificant influence of

_____ww

13

the ratio of torsional rigidity to lateral bending rigidity on the buckling load for
small rise—to-span ratio (f/L s 0.3).

Sakimoto and Namita (31), 1971, studied the out-of—plane elastic
buckling of double—rib arches with bar transverse bracing under radial in-plane
loads, and found good agreement with experimental verification. Yabuki and
Kuranishi (44), 1973, studied the effect of warping on the elastic buckling of

double—rib circular arches under lateral loading, and found that the effect of

 

warping on the arch behavior increases as torsional rigidity and slenderness

ratio increase.

B) Plastic Behavior

Using the fiber model and incremental nonlinear analysis, Komatsu and
Sakimoto (15), 1977 , studied the elasto-plastic lateral-torsional ultimate load of
two—hinged single parabolic arches under vertical and hanger-tilted loads. When
plastic effects were included, they found the hanger-type ultimate load to be
100% larger than that of vertical load. This is relatively small when compared
with elastic analysis, where the difference is 200%. They also found that f/L
ratio does not affect the ultimate load.

Sakimoto and Komatsu (32), 1979, studied the effect of truss—type
transverse bracing on the ultimate load of double-rib arches under combination
of vertical and lateral loading. They found the system fails due to bracing
buckling for slender arches (L/ry > 180), while the ribs failure controls for

stiff arches; a simple formula to predict the arch ultimate strength based on

i

14.

linear analysis was suggested. Another study by the same authors (33), 1982,
on the behavior of parabolic double-rib arches with different bracing systems
under vertical loading found that failure is dominated by lateral bending mode
rather than torsional one in both deformation and moments.

Based on the above studies, Sakimoto and Komatsu (34), 1983, proposed
a formula to predict the arch ultimate strength due to lateral instability, similar
to that of beam-column formula, that takes into consideration the type of end
support (two—hinged or fixed), lateral rigidity, and extent of bracing along the

arch length.

 

Kuranishi and Yabuki (21), 1981, studied the ultimate load of double-rib
parabolic arches with lateral bracing under vertical and lateral loading, and
found the in—plane ultimate load begins to decrease significantly when the
applied lateral load exceeds 50% of the actual ultimate lateral load —calculated
from nonlinear elasto-plastic analysis—. Under “practical" lateral load due to
wind loading -defined below-, they found the in-plane load carrying capacity
decreases slightly (less than 10%) and exhibits in-plane failure mode when
sufficient lateral stiffness is provided; the "practical" wind load was assumed to
be 10% of the arch ultimate lateral load calculated from simple plastic analysis
‘ (using linear analysis).

Extension of this study by Yabuki, Vinnakota, and Kuranishi (45), 1983,
found the in—plane vertical load in presence of practical lateral load decreases

with the increase of in—plane slenderness ratio, L/ry, and rise—to—span ratio, f/L;

15

the decrease reaches a maximum of 15% for L/ry = 300 and f/L = 0.3. Design
criteria to account for the effect of lateral load on the in-plane ultimate load
and to specify the required out—of—plane stiffness to prevent lateral failure mode

were suggested by Kuranishi and Yabuki (23), 1984.

 

 

CHAPTER II
THEORETICAL MODELS FOR

NONLINEAR ELEMENTS

 

2; INTRODUCTION
The nonlinear finite element program prepared for this study uses

straight beam elements to model the arch. Both geometric and material

 

nonlinearity effects were incorporated into the finite element model, where
either or both nonlinearities may be included in the analysis.

The tangential stiffness matrix of the straight beam element consists of
the elastic linear stiffness matrix plus the contribution of the geometric (elastic-
nonlinear) and plastic nonlinearities in the form of incremental matrices
(30,28). For elastic-nonlinear analysis and simple—plastic analysis only the
corresponding incremental matrix is included. For combined nonlinear analysis
where both geometric and plastic effects are present both incremental matrices
are included.

The details of formulation of the element stiffness matrices and calcula-

tion of its resistance for different cases of nonlinearity are detailed in the

16

 

17

following sections. The computer implementation of the nonlinear solution
procedure of the structure equilibrium is presented in the next chapter.
_2._2 ELASTIC-NONLINEAR RESISTANCE OF STRAIGHT BEAM
ELEMENT
For elastic-nonlinear analysis, the structure resistance during a load
increment is assumed to be linear, and is evaluated as the average of resistance
during that increment,

AQ = keg” Aq, (2.1)
where AQ is the element incremental resistance (i.e., force) during the load
increment, kc“, is its "average" elastic-geometric stiffness matrix that repre-
sents the element stiffness during the increment, and Aq is its incremental
displacement vector at its end joints, obtained from the structure global displa-
cement vector (through transformation from the structure global axes to the
element local axes).

In the following two sections, the model used to calculate the element
stiffness matrix is presented first, and then the procedure to evaluate the

average stiffness matrix during a load increment is considered next.

2.2.1 Element Elastic-Geometric Stiffness Matrix

_

 

The model for elastic—geometric stiffness matrix used here is that
developed by Wen and Rahimzadeh (42), and derived using Lagrange
coordinates for small rotations (abbreviated as Lagrange-SR), where the global

coordinates are fixed (not updated) and the rotation of the element chord is

 

 

18

small (less than 15 degrees). It is based on the "average axial strain" model,
where the contribution of the transverse displacement to normal strain,
e(x,y,z), (the second term in the equation below) is averaged over the element
length,

L 2 2 2 2
e(x,y,z)=fl+lfl EX + £9 dx+ydv+zd0
(IX L02 (IX dX dxz dxz,

 

and the interpolation functions are linear for longitudinal and torsional

displacements and cubic for lateral ones,

“- = 111+(1J'2'P-1)x/Li
d) = ¢1+(¢2_¢1)x/L2
v = v, + 6,): + (—26, -62+360)x2/L + (e, +62—260)x3/L2,
(1) = (01- “fix +(21111+III2'3IIJ°)X2/L +(—‘i’1_‘i12+2‘ilo)x3/L2‘
In these equations, {#1, V1, (.11, d1, 0,, d1, ,uz, ..., 1,02} are the element

joint displacement vector, q, Figure (2-1), 00 = (uZ—ul)/L and 1/10 = (ml-w2)/L
are the chord rotations, and L is the element original length. The x—axis is the
local longitudinal axis of the straight beam element while y- and z-axes are the
principal axes of its section. The normal strain expression is used to compute

the element strain energy, U, over its volume, dV,

U =f lEede
volumc2

=Uz‘LU3+U4

where E is the elasticity modulus and U2, U3, and U4 are the quadratic, cubic

and quartic terms of strain energy, U, respectively. The elements of the tangent

stiffness matrix are derived from the energy express1on,

 

 

19

rotations @ x- -axis ’9 4’2
/(in y- 2 plane)

Deformed
Element

Undcformed Element
(original configuration)

tfl X

 

 

 

 

 

 

 

 

 

2 W2

Element projection on local x-z plane

Figure 21 End joint displacements of straight beam element in three dimensional
space.

 

 

 

 

 

20

62U

6‘],an

62U2 + 62U3 62U4
— — + _—
aq. aq, 6q. 6q,- aq. 8q,
= k..(i.j) + 19.16.13 + India).

k,(i,j) =

The resulting elastic-geometric stiffness matrix is composed of three
matrices,
mg=nnnnaumi _ 9%
where k0 is the linear elastic stiffness matrix, and knl and knz are the first order
and second order incremental stiffness matrices. The last two matrices

constitute the incremental geometric matrix,

[kg] = [k,,] + [km] (2.3)

The detailed derivation and stiffness matrix elaments can be found in

reference (42) .

a; Calculation of Element Elastic Resistance

The element elastic-geometric stiffness matrix at a given displaced
position is evaluated as a function of its elastic linear stiffness matrix, kc, and
its joint displacement vector at that position, q,

g=flgat an

The element new displacement vector at the end of a load increment, qn,
may be determined, qn = q0 + Aq, where qo is the displacement vector at the

beginning of the increment. Also, the increment mid—point displacement can be

 

21

found, qmp = q0 + Aq/2. The element average stiffness matrix during the
increment may be evaluated at the midpoint displacement position of the
increment,

1(.,,W = f( k0, qmp),

or alternatively may be taken as the average of the stiffness matrices evaluated

at beginning of the increment, km” and at the end of the increment, kc“,

k...” = file... + k...)-

The obtained numerical results show that both methods give very similar
results, but the first method is less calculation intensive and is the one used
here.

When plastic effects are present, only the elastic component of the
element total displacement vector contributes to the elastic-geometric stiffness;
i.e., the plastic component is neglected. Hence, the above equations are applied
with the element total displacement vector replaced by its elastic component as
explained later in this chapter.

L1 ELASTIC-PLASTIC RESISTANCE OF STRAIGHT BEAM

ELEMENT

For plastic analysis, the element stiffness, and hence its resistance,
change when a joint yields, and in theory the element resistance needs to be
evaluated separately for every stage of stiffness change (elastic, one, or two

joints yielded) during the increment.

 

 

 

22

While the stiffness matrix formulation for an elastic-perfectly plastic (or
elasto-plastic) straight beam element with yielded joint(s) is well established
(28, 30, 41), the approximations involved in its implementation differ widely,
(28). The differences arise mainly from the handling of the basic assumption
that for an elasto-plastic material the incremental force vector at a yielded joint
is tangent to the yield surface, which in general causes the new force point to
drift from the yield surface, hence violating the assumption that the force
function, (I), has to stay on the yield surface. To keep the force point on the
yield surface, several procedures were proposed, some involving iteration
procedure while others involve one—step approximate force correction; both
have been applied with the use of either the elastic or elastic-plastic stiffness
matrix of the element (28). All these procedures treat the yielding of each joint
separately when determining the elastic and plastic portions of the applied
displacement, neglecting the interaction effect of one joint yielding before the
other on the response of the element.

In the following section, a procedure is introduced that combines two
features. First, following an iterative process, the element new force point
would be made to stay on the yield surface in a manner more strictly in
conformance with the theory of plasticity. Second, the incremental displa-
cement can be large enough so that the element may undergo the process of
changing from a state of total elasticity to having one yielding joint and then to

having both joints yielding (43).

 

 

23

The basic derivation of the elastic-plastic stiffness matrix and the
assumptions associated with it, (30, 28), are summarized in next section for

reference.

git Element Elastic-Plastic Stiffness Matrix
When an element yields as it goes through a displacement increment, dq,
the displacement increment may be expressed as the sum of its elastic and
plastic components,
{dq} = {dqe} + {dqp}
{qu} {qu} {dqfil (2.5)
{and} = {dqfi + {dqj}
where dqc and dqp are the elastic and plastic displacement components,
respectively, and dq,i and qu‘ are their respective components at joint i. From
the assumption of elasto-plastic material, the plastic displacement increment
does not generate forces in the element and the element incremental nodal force
vector is related only to the elastic displacement increment and the elastic
stiffness matrix, kc,
{dQ} = [ke]{dqe}
{{dQl} IR.1 1] [kill {dq.li (2.6)
{szli _ 1k?) 1k?) {qu1
where k6 is a generic term, replaced by either kcg or k, (depending on whether

geometric effects are included or not), that is composed of four submatrices,

k}, that relates the joint forces at joint 1, dQ‘, to the joint displacements at joint

 

 

24

j, dqj. The yielding condition at joint i is defined by the force function, <19, that
defines the yield surface when,

<1>i({ Qi},{Q,‘}) = 1.0, ' (2-7)
where {Q‘} and {Qpij are the applied force vector and the limit (plastic) force
vector at joint i, respectively. When the force function satisfies this condition,
the joint is considered to have reached plastic or yield state. Moreover, for
elasto-plastic material the force function cannot exceed unity.

For a yielded joint, i, the normality rule states that the plastic defor—
mation vector is normal to the yield surface and orthogonal to the element force
increment,

{dqpiiT {in} = o (2.8)
For associated material, the force function is assumed to be the same as the
plastic potential function, and consequently the plastic displacement increment
at joint i is determined as follows,

{qu} = 1, {vi} (2.9)
where A, is the proportionality constant for joint i and Vi is the gradient vector

of the force point, Q‘, on the yield surface,

{Vl}= —a—q)—l , m=1,n
6Q;

. 6322‘ L‘P‘T
aQiaQ.‘ 60.1

(2.10)

 

 

 

 

25

where n is the number of force components that compose the yield function.

From (2.5) and (2.9),
{dqj} = {{dq‘} —A,{V‘}} (2.11)

Figure (2-4.a) shows the yield surface of a yielded joint with vectors

 

{dQ}, {dq}, and {V} in a two-component space. The derivation of the elastic-
plastic stiffness matrix depends on whether one or both joints have yielded, as

discussed below.

2.3.1.1 Case of One ,loint Yielding

For this case, the plastic deformations are present only at the yielded

 

 

joint (denoted as joint A), while the other joint (labeled joint B) is elastic with

{quB} = 0, and XI, = 0. The normality rule applies only at the yielded joint.
Substituting the expression for {dq,} using (2.11) into (2.6), then

expressions for {dQA} from (2.6) and {quA} from (2.9) into (2.8), and

observing that AA has arbitrary value, it can be determined that,

1
A =_
A C

{VAP [1%Al kf] {dq}

A
where,
CA = {vA}T [keM] {VA}
and k,“ are the submatrices of the elastic stiffness matrix defined before.
The elastic-plastic stiffness matrix is found by substituting expressions

for AA into (2.11), then expression for {dq,} from (2.11) into (2.6),

 

26

k.“ 1.33 i {VA}{VA}T[k;‘1k;‘2]
I— cA

[ ] (2.12)
REA keBB

rk.,1 =
l 0 l

where [I] is an identity matrix. The negative term consists of an N'2N non-
zero matrix that corresponds to the yielded joint, complemented with the N-2N

zero matrix that corresponds to the elastic joint, where N is the number of

degrees of freedom per joint.

2.3.1.2 Case of Two ,Ioints Yielding
When two joints yield simultaneously, equations (2.8) and (2.9) may be

written in general matrix form,
1r 'r
{dqpi {0’ {ldQll} = {0} (2.13)
{o}T {dq:}T {dQZ} 0 ~ 4
and,
{dqp} = {VH1}
{dqfii {V1} {0) A. (2-14)
{dqj} — {0} {V2} 12 ,

In a similar procedure, substituting expressions for dq,l and dq,2 from

(2.11) into (2.6), then expressions for dQi from (2.6) and dq,,i from (2.9) into
(2.13), a set of two simultaneous equations is obtained that can be solved for
{X},

{A} = [c]"{V}T[k°]{dq}

where,

 

[c] = {V}T[ke]{V}
The elastic-plastic stiffness matrix is obtained by substituting

expressions for A, and A2 into (2.11) for both joints, and back into (2.6),

[k .p]=[k.[Il][ "001611 “W H (2'15)
The expressions for elastic-plastic stiffness matrix and plastic deforma-
tion are used in the calculation of the resistance and deformation of the straight

beam element as described later in this chapter.

L342 Yield Surface Functions: Types and Implementation

In this section the different types of yield surface functions used in this
study to represent the plastic behavior of straight beams and their definition are
discussed first, followed by discussion of the procedure of implementing these

definitions into a computer program.

‘ 2.3.2.1 Types of Yield Surface functions

 

The type of yield surface defining an element plastic behavior is
obviously an important factor in calculating the plastic load carrying capacity
of a structure. The closer it is to the actual behavior of the element the better
, the predicted results, but the more difficult to implement the calculation
procedure or to generalize its application. On the other hand, simply const-

_ ructed models may affect the accuracy of the obtained results, (1,28), but are
easier to implement into a computer program and can be used to represent a

wide range of structural sections.

28

An example of the first type is a multi-faceted yield surface that is
described by several facets in each single region of the surface (defined by
principal axes), (28), creating difficulty in computer implementation. Another
example is a single continuous (usually complex) equation depicting the yield
surface for a limited range of a certain structural section (e.g. an I-beam
section), (28). While this approach is preferable for computer implementation,
it is not always possible since it is difficult to generalize for wide range of

sections and sizes. These disadvantages are specially true when more than two

 

force components contribute to the yield surface (1, 8).
In this research, four types of yield surface functions are implemented
and compared for the problem in hand. These types are: 1) octahedral,
2) spherical, 3) inverse parabolic, and 4) parabolic, as shown in Figure (2—2)
for two-dimensional space. The octahedral and spherical functions are
considered as the lower and upper limits for the plastic behavior of the
‘ element, respectively, while the parabolic and inverse parabolic functions lie in
between. The inverse parabolic function more closely represents the interaction
between axial force and bending moment for steel sections, (e.g. ref. 7). In
‘ these functions, torsion and shear forces are not included since they do not
i have significant effect on the plastic behavior of the studied problem.
Except for the spherical function, all these functions have some disconti-
nuity of the gradient vector either along a line where two surfaces meet, called

l a ridge, or at a point along one of the principal axes when more than two

29

 
      
  
 
    
    

Parabolic

Spherical

Inverse
Parabolic

Octahedral

 

 

 

Figure 2-2 Different types of yield surface in two-dimensional space.

surfaces meet or when the function has a conical shape, called a corner. As a
result, the gradient vector cannot be properly evaluated if the force point lies in
a discontinuity region.

To overcome this problem, the original function is patched in the regions
of discontinuity with special functions that provide definite value and contin- i
, uous change of the gradient vector in each region of discontinuity, avoiding
singularity of the gradient and allowing proper evaluation of the direction of
the force increment tangent to the yield surface. If the force point lies in a
’ discontinuity region, the gradient is evaluated using the special function. The
1 force function itself, however, is always evaluated using the original function.
The patch functions used here do not provide an entirely smooth change

i of the gradient from one facet of the yield surface to the other, since there is

30

still some discontinuity between the batch function and the original yield
surface. However, the main purpose of the patch function is to provide an
intermediate or average value of the gradients of the original intersecting
surfaces in the direction perpendicular to the ridge, for example, while still
providing the same value of the gradient as the original surfaces in the
direction along the ridge.

Using the gradient of the patch function, the "initial“ direction of the
force increment -tangent to the surface of the patch function- is determined, for
example toward one of the original facets of the yield surface or in a direction
parallel to the ridge. The location of the new force point is found, either in the
region of the original yield surface or in the region of the patch function. If as
a result the new force point drifts from the yield surface, it is returned back to
it by repeating the calculation of the force increment using a new, improved
value of the gradient that is equal to the average of the gradients at the
beginning and end of the force increment, as described later in this chapter.

For a corner region the special function is always taken as the spherical
function, while for a ridge it is specific to each of the original functions. The
mathematical representation of the original yield surface functions and their
special patch functions are described below.

The spherical function, Figure (2-3.a), which has no discontinuity of the

gradient vector, is quadratic and is defined by the equation,

 

31

Corner along
N/Np N -axis

     

 

Ridges in

 

 

 

 

 

 

(a) spherical yield surface. (b) Octahedral yield surface.

Corner along

N‘ms Ridge in

Nle plane N=0

 

My/MYJ)

 

 

 

M7JMZ,p

(C) Inverse parabolic yield surface. (d) Parabolic yield surface.

Figure 2-3 Different types of yield surface in three-dimensional space and locations of
ridge and corner regions.

 

l
Ilifhlr ,

32
2 2 2
we in f“)
NP MVP M1119

The octahedral function, Figure (2-3.b), is defined by the equation,

&
Mm

The function is composed of plane facets, each passing through —and

 

 

M

Z

Mz’p

 

+ + (2.17)

 

NP

 

 

 

 

 

 

defined by- three points of limit force. Each pair of the limit forces of the

section (in, iM and iMz’P) is located on -and defines- one of the three

H”

 

principal axes. Each adjacent two of these facets intersect along a straight line
ridge that extends between the two limit force points common to the two facets.
The ridge lies in the plane defined by the principal axes of these two limit force
points. For three-component function, this plane is mathematically defined by
the equation,

Q, = 0,
where r denotes the third principal axis.

It is seen from the figure that four ridges intersect at each limit force
point, forming a corner at that point. Therefore, the corner is mathematically
defined by the equation,

Q. = Q...-
where Q”, is the limit force point along the c-axis (the axis which the corner

lies on).

 

 

 

33

The special function used to evaluate the gradient vector in a ridge

region for three-component octahedral function is,

Q 2 n
(I) = r +
r ( er) 1‘21

where n = 3, is the number of force components. As mentioned before, the

Q

Q),

 

, (2.18)

 

 

 

 

spherical function was used to represent all corners.
The inverse parabolic function, Figure (2-3.c), is defined by the

equation,

1/2
¢=£2+ £2. M22 ‘ (2.19)
NP MY’P MZ-P

For this function, there are only two discontinuity regions, both along

 

 

the N—axis, where the function has a conical form resulting in singularity of the
gradient at the two limit points, N = in. This is avoided by defining a corner
region along the N-axis, with the spherical function being used as the special
function.

The parabolic function, Figure (2-3.d), is defined by the equation,

+ [if . [ M2 )2 (2.20)
Mm Map

which also has only one discontinuity region in the form of a ridge in the plane

N

NP

 

 

 

N = 0, resulting in a circular ridge (elliptical when the equation is not in
normalized form) in that plane. The special function used here is the spherical

function.

 

34

2.3.2.2 Computer Implementation of Yield Behavior

The new yield status of an element joint, i, (whether it is elastic or

 

plastic) for a given force vector, Q, at the end of a load increment is
determined using the force function (i.e., the location of the force point in the
force function space), <I>‘(Qi), where three cases are possible, Figure (2—4.a).
First, if the force point is inside the yield surface (<I>i < 1), then it is elastic
with no plastic deformation. Second, the joint reaches the plasticitycondition
when the force point lies on the yield surface (<I>i = l), but the joint does not
necessarily go through plastic deformation. Finally, if the force point lies
outside the yield surface (éi > 1), then it has advanced outside the yield
surface and has gone through plastic deformation; this condition, however,
violates the elasto—plastic condition, and the force point needs to be returned
back to the yield surface so that <I>i = 1.

To determine the yield status of the joint for computer implementation,
the yield surface is represented by a range that is defined as follows,

ch, = 1+e, > <I>i(Q‘,Qpi) 2 1-6, = on, 931)

where <1)“ and <I>ul are the lower and upper limits that define the yield surface
range, respectively, and e, is the tolerance limit. For this study, a value of
64. = 0.001 was used.

The yield surface range divides the force function space into three
distinct regions that are analogous to the three possible case-s above, Figure

(2-4.b): if <I>i < <I>ul then the force point lies in the elastic range and the yield

 

 

 

35

Region Outside
Q2 Region Outside Q2 Yield Surface
Yield Surface

Yield surface Yield Surface

   
   

Elastic
dqp Range

    
 

Elastic

      
     
 

 

 

 

 

 

Range
Force Point, Q
Force dQ dQ
Point, Q Current Yield \
T Q1 11’ Q1
Qi,p Ql,p
(a) Theoretical definition. (b) Computer implementation.

Figure 2-4 Definition of the yield surface and the force and plastic displacement
increment vectors for a two—component joint.

status of the joint is elastic. When 4? lies in the yield surface range (i.e., on the
yield surface), then the joint has reached plasticity and its yield status is depen-
dent on the previous yield status of the joint at the beginning of the load incre-
ment (i.e., the plastic history of the joint). If 4)‘ > i)”, then it lies outside the
yield surface and the joint yield status is plastic.

The ridges and corners of the yield surface are defined for computer
implementation in a manner similar to the definition of the yield surface range,
as shown in Figure (2-5) for an octahedral yield surface. The force point is
considered to lie in a corner region of the yield surface along the k-axis (where
k may be any one of the principal axes) if the absolute value of the force
component along that axis, Qk, normalized to its plastic force, Qkyp, is larger

than a certain limit, taken here as l—2e,,

 

 

 

 

 

 

36

let

j Qk = New
' Qk = 1

1+8¢
= 1

¢

¢

 

 

 

 

1|

Corner along “'
Qk-axis

./'

-\.

./

,//

 

.rl

 

./

 

 

Ridge region
defined by ile < 8.,

 

Figure 2-5 Definition of the discontinuities along ridge and comer regions of octahedral

yield surface.

37

a

KP

> 1 "26° . (2-22)

 

 

Otherwise, the force point is considered to lie in a ridge region -defined by the
k-axis- if the absolute value of the normalized force component is smaller than

64>, Figure (2-5),

< 6,”, (2.23)

 

 

where Qk is the force component defining the ridge plane. Notice here that the
above two equations (specially the latter one) are applied when the force

function is in the yield surface range, <I>,,l > @(Q‘) 2 <1)“. Also, the presence of
a corner along each of the principal axes is checked first (before the ridge) and
its tolerance is taken twice as large to avoid the situation of a force point lying

in two ridge regions simultaneously.

2.3.3 Calculation of Straight Beam Elastic-Plastic Resistance

During a load increment the element displacement increment, Aq, is
assumed to be linear, and in general may be divided into three segments as

follows,

Aq = dq°° + dq‘P + dqpp (2.24)
= dql + dq2 + dq3

where dq“ is the segment of Aq with both joints elastic, and dq‘” and dqpp are
those with one and two joints yielding, respectively. The element resistance is

calculated in three stages each corresponding to one of the three displacement

 

 

38

segments, which are also numbered sequentially to correspond to their
respective stage numbers. For a load increment, any one or two of the three
segments may be null; however those present would take place in the order as
given above.

For each stage of resistance calculation, the process is essentially the
same and involves some or all of the following three steps. First, the initial
tangential stiffness matrix at the beginning of the stage is used to find the
initial force increment vector corresponding to the unprocessed (or unaccounted
for) displacement at that stage, which is the total displacement increment, Aq,
minus the displacement segments that belong to stages prior to the current one.
Second, the length of the displacement segment corresponding to current stage
is determined by locating the intersection point of the force vector with the
yield surface for each yielding joint, and taking the joint that yields first as the
governing one. Third, check for drift of the new force point from yield surface
is performed for yielded joints. If present, a better estimate of the gradient
vector is obtained using an iterative procedure, and the last three steps are
repeated to return the force point to the yield surface until the elasto—plastic
condition is satisfied.

These three steps are not present in all stages. The first displacement
stage involves only the first two steps, the third stage involves only last two

steps, while the second stage involves all three.

 

39

In the following sections, the procedure for locating the intersection
point of the force increment vector with the yield surface at a joint, also called
"the penetration point", and the method of returning the force point to the yield
surface are discussed in detail, followed by detailed steps of resistance

calculation.

2.3.3.1 Locating Force Penetration Point

The intersection point between the force increment vector and the yield
surface needs to be found during resistance computation. This mainly arises in
the resistance calculation process when determining the size of different
displacement segments.

At joint 1, when the old and tentative new force points, Q,8 and Q,"’, at
the beginning and end of a displacement stage, 3, lie in the elastic range and
outside the yield surface, respectively, the forCe increment vector needs to be
scaled back so that the new force point lies in the yield surface range. If either
the old or new force point lies in the yield surface range there is no need for
scaling and the scaling factor is equal to 0 or 1.

The tentative force increment vector, dQ'”, and new force point, Qn“,
during stage s are calculated, dQ"s = k,’dq“”, and Q,"5 = Q: + dQ‘“, where
k,8 = f(k,, V) is the tangent stiffness matrix at the beginning of the stage, k, is
the elastic stiffness matrix, V is the gradient vector at the yielding joint(s), and
dq‘” is the unprocessed displacement vector (not yet accounted for at the

beginning of that stage),

 

 

 

 

 

 

 

 

 

 

(0.“ .<I>.‘"‘ )
1 (0.1" 9;") i
._ ._ To =. 1 __ .
(Q.i".<I>.i“)
£1le =
al“dQ“-"
ii Q1“ do“ >Qi': _

 

Joint 1

Figure 2-6 Locating the force increment penetration point of the yield surface.

dq“'s = Aq - E dq“ (2.25)

.=1
The penetration point at joint 1 during displacement stage s is defined by
the scalar factor oz,"s that scales the tentative force increment vector, dQ“, to
the yield surface, Figure (2—6), so that its function satisfies the equation,
rim: + a? dQ“) = 1. (226)
The resulting scaled force and displacement vectors are dQs = oz,"‘dQ"’, and
dqs = acivsdq“, respectively.

i The solution of equation (2.26) depends on the type of yield surface used
to represent the element behavior. For spherical yield function the equation is a
quadratic one and its solution is the positive one of its two roots. For other
types of yield surface, the equation may be solved using an iteration method,

such as the modified bisection (Regula-Falsi) method used here, where the

 

 

41

solution is satisfied when the target value lies in the yield surface range instead

of being exactly equal to unity.

2.3.3.2 Keeping Force Point on Yield Surface

 

When a joint has yielded at the beginning of a displacement stage, 3, the
tentative force increment vector at that joint during that stage, in'”, is tangent
to the yield surface, and because the yield surface is convex in general, the new
force point, Q,” is necessarily not in the elastic range; it lies either on or out- _
side the yield surface. The last case violates the elasto-plastic condition, which
stipulates that the force point should stay on the yield surface, and therefore the
force point needs to be returned to the yield surface.

The procedure used here for force point return is an iterative procedure
that involves evaluating the tangent stiffness matrix, k,‘ = f(k,, V,), where V0
is the gradient vector at the old force point. The initial estimation of gradient
vector is that of the old force point at the beginning of the stage, Voi‘o) =
Vi(Qo’). If the resulting new force point, Q,“5 = Q,‘" + in'”, lies outside the
yield surface, then its projection to the yield surface along a radial line passing
through the origin, Qni‘P", is found, and the gradient used to recalculate k,8 for
the first iteration is taken as the average of the gradient at the old force point
and that at the projected point, V0“) = {V0i + Vni'P}/2. When the gradient is

calculated in normalized form, V,” = V:', and the process of projecting the

force to the yield surface is not needed.

 

 

 

 

 

42

 

 

 

 

 

   

(a) New force point outside (b) New force point inside
yield surface. elastic range.

Figure 2-7 Iteration process to keep force point at yield surface.

If the new force point is still not converged (i.e., not on the yield
surface) and lies outside the yield surface, the estimated value of VJ“) for

iteration j is calculated as follows,
VJ“) = (vio‘l) + V:(J“l))/2

where VO‘G'” and Vnifi'” are the gradients corresponding to old and new force
points during the previous iteration, Figure (2-7.a).

On the other hand, if the new force point is not converged and lies in the
elastic range, then the last obtained value of V‘,i is overestimated, and a new
value is calculated using the 01d vector used in the previous iteration, Void"),

and the old vector of the iteration jj that last resulted in a force point outside

the yield surface, Void”, Figure (2-7.b),

 

 

 

 

 

43
Void) = (Voitii) + Void—1))”.

If convergence is not achieved within a certain number of iterations (5
used here), the iteration process is stopped and the new force point at the end
of the last iteration is scaled back to the yield surface (i.e., projected on it)
along a radial line passing through the origin. The projected force point is
taken as the final new force point, Q,,' = Qn'P", and the force increment for the
stage, dQ' = Q; - Q0“. From experience, when convergence is not achieved

within a few iterations the reason is most likely related to the global structure

 

stability, such as the current load increment being too large and exceeds the
ultimate load of the structure.

When both joints of the element yield simultaneously, the iteration
process is performed independently at each joint, with the resulting gradient
vectors at each joint used to evaluate the tangent stiffness matrix. Similarly, the
process of scaling the new force point to the yield surface is performed (when
needed) only at non-convergent joint(s).

The process of projecting the tentative force point, Q”, to the yield
surface along a radial line passing through the origin is done using the
procedure described by equation (2.26) in the previous section, with Q,5
replaced by a null vector -representing the origin point- and dQ‘s replaced by
Q,"’, so that d)‘(Qn'P) = (Ili(oz,i-Qn"’) = 1. However, when this equation is
solved by the iteration procedure the assumption that Q,"5 lies outside the yield

surface may not be true, since it may also lie in the elastic range. To avoid this

   

Final gradient Q2
vector, V "'

Initial force
increment, de‘m

   
    
   
 
  

 

Intermediate gradient

intermediate force
vector, V( nun")

’ / increment, dQflnmm)

Initial gradient
vector, V “it

Final force incre— l
ment, dQ "‘1

 
 

Original (initial)
yield surface

 
  
 
    

Substitute (final)
yield surface

//
f O]

Figure 2-8 Treatment of force increment extending across surface discontinuity.

   

 

 

 

situation, the length of the tentative new force vector is doubled, Q,"8 =

2 - QB", until its function lie outside the yield surface, (@(Qn'f) > 1+e,). The
solution then proceeds normally and the obtained scaling factor, 043, is adjusted
for the additional multiplying factor, with the final value of a,‘ being larger
than unity. This technique guarantees stability of the iteration process.

For a yield surface function with discontinuous gradient, such as the
octahedral function, the force increment may jump from one facet (or region)
on the yield surface to another. In this case, the process of exactly following
the force increment path as it moves across different facets is tedious and
elaborate, since it involves finding the intersection between the force increment

vector and different facets of the yield surface, and each displacement stage

 

 

 

 

45

may have to be further subdivided into two (or more) substages that trace the
force path through each facet.

An approximation to this situation that simplifies the calculation proce-
dure is to approximate the actual force path by a single straight line path that
extends from the old force point to the new force point and passes through the
elastic range, Figure (2-8). The new force point is found so that it satisfies the
elasto-plastic condition using the same iterative procedure of returning the
force point on the yield surface described in this section. This in effect replaces
the original discontinuous yield surface by an imaginary substitute yield surface
that is defined by the final gradient vector, V,"‘, (i.e. the final gradient vector
is perpendicular to this surface) and the final force increment vector, de,

which forms a part of the surface.

2A; Steps of Resistance Computation .

The initial step in calculating the element resistance is to determine the
initial new yield status of the element joints (at the end of a load increment)
which in turn determines the stages of resistance that need to be considered,
and calculate the resistance for only those stages. After the resistance of those
stages are calculated, the total element resistance during. the increment, AQ, is
calculated as the sum of those of the stages. In general, AQ = dQ‘ + dQ2 +
dQ-3, where dQ‘, dQZ, and dQ3 are the element resistance for the first, second,
and third stages, respectively, and the final new force point of the element,

Qn = Q, + AQ, where Q, is the old force vector (at the beginning of the load

 

 

46

increment). In addition, plastic displacements are calculated using the
applicable equations given above.
The detailed process of determining the initial yield status and different

stages of resistance computations are discussed in the following sections.

A. Determining Element Initial Yield Status

The first step to determine the new yield status at each joint is to assume
that the element behaves elastically and calculate the tentative element force
increment,AQ' = k,-Aq, where Aq is the element displacement increment
vector obtained from the structure global incremental displacement vector. The
tentative new force point is found, Q,’ = Q, + AQ'. The old and new force
functions at each joint, i, are calculated, 4',‘ = d>i(Q,) and it," = <I>i(Q,'),
respectively. The initial new yield status at each joint is determined as follows:
1. If ti," > do,“ the initial new yield status of the joint is plastic regardless of
the old yield status, Figure (2-9.a), and 0 5 oz,‘ < l.
2. If <I>,i‘ < CI)“, then the joint is in the elastic range, Figure (2—9.b), and a,‘ =
1. Here, if the old yield status was plastic, the joint is considered to have
unloaded.
3. If (1)," lies within the yield range, then <1),i has to be checked to determine the
new yield status:
3.a. If <1): < <19“, then the new position is elastic, Figure. (2-9.c), and a,‘ = 1.
3.b. Otherwise, both old and new positions lie in the yield range (since old

position is not allowed to lie in the plastic range at the end of previous

 

 

 

 

 

 

47

 

 

 

 

 

 

 

 

 

 

Q1
(21) New force outside yield surface. 0’) New force in 3133th range.
Q2 Q2
Q3
Q0
Q1
(c) Old force in elastic range, new (d) Both new and old forces
force in yield surface range. in yield surface range.

Figure 2-9 Determination of the initial yield status of a joint at the beginning of load
increment.

 

 

 

 

48

incremént), with the following possibilities:
' - If <I>," > <I>,i and the old yield status is plastic, then new yield status is
plastic, shown as point (1) in Figure (2-9.d), and a: = 0.
— If m," S <I>,i with the old yield status being plastic or if the old yield
status is elastic, then the new yield status is elastic and or,i = l, as
represented by the other three points on the same figure.
With the yield status at the beginning and at the end of the increment
being found, it is possible to determine which stages of element resistance need
to be considered, and proceed to calculate the resistance for the ones that are

not null, as follows.

B. First Elasto-Plastic Stage

For the first displacement stage, the initial force increment is the
tentative force increment found before, dQ"l = AQ', and the unprocessed
displacement is the total increment displacement, dq‘“l = Aq. Also, the old and
new force points at the beginning and end of the stage are Q,1 = Q, and Q,"1 =
Q,', and the yield status at those points are those obtained before.

The scaling factors at each joint, a,’ and a}, are calculated, and the
smaller of these scaling factors, or," = minimum(a,l, 01,2), determines the joint
that yields first (joint A), Figure (2-10). The force and displacement increments
corresponding to this stage are calculated, dQl = oz,"-dQ"1, and dq1 =

01,“- dq“, respectively. The new force point at the end of this stage is Q,1 —

Qo + dQl-

 

 

 

 

49

   

 

 

Qn
Joint A Joint B

 

 

Figure 2—10 Calculation of the first stage elasto—plastic resistance of straight beam
during load increment.

If a,A = 0, the first-stage is null (not present), and dQ1 = O, dql = 0. In
addition, if this is true for both joints (i.e. oz,A = 01,3 = O), the second stage is
also null, and sz = O and dq2 = 0. On the other hand, if the new yield status
of both joints is elastic (i.e. a,‘ = l, i=l,2), the member behaves elastically
through the whole load increment and the second and third stages are null. In
this case, the final force increment and new force vectors at the end of the load
increment are equal to the tentative ones, AQ = dQl = AQ', and Q, = Q,1 =
Q,', respectively, and the process of resistance calculation ends here.

When 0 < ix,“ < 1, the new force function at joint B is calculated,

‘1’,” = <I>B(Q,‘), (< 49,), and used to check for the presence of second stage. If
é,” > <1)“, then the new force point at joint B is in the yield range and joint B
has also reached the yield status simultaneously with joint A. Therefore, the

second stage is null, i.e. sz = 0 and dq2 = O, and the calculation proceeds to

 

 

 

50

the third stage. Otherwise 4),“ s <1)“, i.e., joint B is elastic, and the second

stage is not null.

C. Second Elasto-Plastic Stage

At the beginning of this stage, the force point at joint A is on the yield
surface, joint B is elastic, and dq“-2 is not null. The unprocessed displacement
and the old force vectors are calculated, dq‘“2 >= Aq - dql, and Q,2 = ,‘, and
the resistance calculation proceeds as follows:

1. The tangential stiffness matrix, k}, is formulated with joint A plastic and
joint B elastic. The initial gradient at joint A is calculated as that of the old
force point, V,”"2 = V"(Q,2). I

2. The tentative new force point and its function at joint B are calculated,

Q,"2 = Q,2 + dQ"2, and <I>,B"2 = <l>B(Q,"2), where dQ"2 = ktzodq”.

2.a. If <1),B"2 > (1),“ then the new yield status ofjoint B is plastic, and the force
increment vector, dQ"2, needs to be scaled back to the yield surface at joint B,
Q,"'2 = Q,2 + dQ”'2 = Q,2 + a,B°dQ"2, where a,3 is the scaling factor, Figure
(2—11).

2.b. If, however, if” S <I>,, then joint B is still elastic and oz,B = l (i.e.
dQ"‘2 = dQ”2), and the scaling process is not needed.

3. The new force function at joint A is calculated, €l>,""*2 = <1>A(Q,"’2), and
checked for drift from the yield surface:

3.a. If <I>ll < <I),A"’2 3 <1)“, then the new force point is still on the yield surface

and satisfies the elasto—plastic condition and no iteration is needed.

 

 

 

51

 

 

 

 

 

Joint A Joint B

Figure 2-11 Calculation of the second stage elasto-plastic resistance of straight beam
during load increment.

3.b. On the other hand, if the new force point is not on the yield surface,
61>,A"~2 > <1), (or 4,an < <1)” during subsequent iterations), then it has to be
returned back to the yield surface. Following the iteration procedure described
before, a new estimate of V,"-2 is obtained and used to recalculate the stiffness
matrix, k,"2. Steps (2) and (3) are repeated until the new force point is returned
to the yield surface, as represented by the dashed line in the figure.

At the end of this stage, the final incremental and new force vectors for
the second stage are sz = dQ“2 and Q,2 = Q,"’2, respectively, and the
displacement vector, dq2 = oz,“ dqu'z.

If 01,3 < 1, then third stage is not null, and the old force and
unprocessed displacement vectors at the beginning of that stage are found,

Q,3 = Q,2 and dq3 = dq“-3 = dq‘“2 - dqz, respectively. Otherwise, if a,” = 1,

then third stage is null and resistance calculation ends here.

 

n

L l

 

 

52

D. Third Elastg-Plastic Stage

With the old force points at both joints on the yield surface and the third
displacement segment, dq3 = dq‘“3 not null, the third is present, Figure (2-12),
and its resistance is calculated as follows:

1. The gradient vectors corresponding to the old force points at both joints,

 

V,1'3 = V‘(Q,3) and V,2'3 = V2(Q,3), are calculated and used as the initial
gradients to evaluate the tangent stiffness matrix, k3.

2. The tangential stiffness matrix, k3, is formulated with both joints being
plastic and the tentative incremental and new force vectors are calculated,
dQ"3 = k? dq3, and Q,"3 = Q,3 + dQ"3, respectively.

3. The new force functions at both joints are calculated, é,“ = <I>‘(Q,"3), i =
1,2, and each joint is checked separately for drift from the yield surface:

3.a. If at joint i, <I>,l 2 <I>,i"3 > <I>", the new force point at that joint is on the
yield surface and satisfies the elasto-plastic condition, and no iteration is

needed at that joint.

 

3.b. On the other hand, if the force point lies outside the yield surface range
(i.e., <I>,i"3 > in, or it,“ < <1>u during subsequent iterations), then the elasto-
plastic condition is violated, and the force point needs to be returned back to
yield surface. A new estimate of the gradient vector is calculated (as described
before) only at the joint(s) that do not satisfy the elasto-plastic condition, and

Steps (2) and (3) are repeated until convergence is obtained.

 

53

 

 

 

 

 

 

Joint A Joint B

Figure 2-12 Calculation of the third stage elasto-plastic resistance of straight beam
during load increment.

After convergence, the final incremental and new force vectors for this
stage are calculated, dQ3 = dQ"3, and Q,3 = Q,"3, respectively. As stated
before, the element resistance for the load increment is AQ = dQl + dQ2 +

dQ3, and its new force vector, Q, = Q, + AQ.

M COMBINED ELASTIC AND PLASTIC NONLINEARITY

The procedure of combining both elastic and plastic nonlinearity effects
is basically the combination of both separate procedures discussed above. The
elastic-plastic (or tangential) stiffness matrix, equations (2.12) and (2.15), is
evaluated using the elastic stiffness matrix, k,. When geometric effects are not

present, k, is taken as the elastic-linear stiffness matrix, k, = k,. When

geometric effects are present, k, is taken as the elastic-geometric stiffness

 

 

 

54

matrix, k, = k,,. In this case, the elastic—plastic stiffness matrix may be written
as follows,

[12.1 = k + [kg] + ls], m,

o]

[1%] = -l1$e.1[{VH01“<V>leesll

where k, and k, are the geometric and plastic incremental (or reduction)

(2.27)

matrices, respectively.

The elastic-geometric stiffness matrix is evaluated using the elastic
portion of the midpoint displacement of the current increment —as discussed
before-,

keg = f(k°, qemp) (2.28)
where Aqml, = q,,, + Aq,/2, and Aq, = Aq - Aq,. However, Aq, is not known
beforehand since the plastic components of the increment displacement vector,
qu, is found only during the plastic resistance calculation, which in turn
requires prior knowledge of k,,.

This leads to a predictor-corrector iterative procedure where the element
resistance has to be evaluated twice during every iteration, i, of thelnonlinear
solution for the structure equilibrium. Initially, the elastic-geometric stiffness
matrix, k,,, is evaluated assuming the incremental plastic displacement to be
null, Aqmmm = O (i.e. Aq,(,,,) = Aq). Using this assumption, the element elasto—
plastic resistance, AQ, is calculated. If this initial Calculation produces plastic
displacements, Aqwm 7f 0, then the above assumption is not correct and k,g

needs to be reevaluated using only the improved or corrected value of the

 

 

 

55

elastic displacement increment, Aq,(,,,,) = Aq - Aq,(,,,,), and AQ is calculated
again for the same iteration.

The requirement that the element plastic resistance be calculated twice
every single iteration adds considerably to calculations, while on the other
hand, assuming the displacement increment to be all elastic (the first step in the
above procedure) produces an error that may add to a significant difference
when accumulated over many increments.

A procedure to avoid excessive calculations while minimizing error is
used here where the element plastic resistance is calculated only once during
every iteration for nonlinear solution of the current load increment). In this
procedure, the estimate of the plastic displacement at the beginning of an
iteration, i, is taken as that calculated during the previous iteration so that
Aq,‘D = Aq‘" — qua"). The elastic-geometric stiffness matrix is evaluated,

k,g(i) = f(k,, (q,+Aq,®/2)), and the elasto-plastic resistance and plastic
displacement vectors, AQ“) and qu‘D, are calculated. For the first iteration, the
plastic incremental displacement is assumed to be zero, Aq,” = Aq“).

The basis for this procedure is that when the iterative solution for the
increment is near convergence, the element displacements and forces do not
change significantly from one iteration to the next, and hence the element
plastic displacements produced during an iteration are close to those obtained
from the previous one. When the iteration for nonlinear solution of equilibrium

converges during a given iteration, this implies that the error —on the element

 

 

 

56

level- due to this procedure during that iteration produces errors small enough
—smaller that the allowable tolerance- to allow for convergence of the nonlinear
solution. In other words, the iteration for elasto-plastic resistance on the
element level is shifted to the iteration for convergence on the global structure

equilibrium level.

 

 

 

 

CHAPTER III
CONIPUTER PROGRAM:

SCOPE, INIPLENIENTATION, AND VALIDATION

fl INTRODUCTION

The computer program performs static nonlinear analysis of three-
dimensional structures with the flexibility of including either or both geometric
and material nonlinearities. For nonlinear solution, both direct Newton-
Raphson (abbreviated N-R) method and modified Newton-Raphson method with
acceleration scheme based on a modified Aitken’s method were implemented
with the latter being used to obtain numerical results throughout the study.

The structure ultimate load is defined more precisely using an automatic
load step adjustment procedure, and the response curve may be traced beyond
the ultimate load using a displacement-controlled analysis procedure that is
implemented in analogy with the modified Newton-Raphson method.

The convergence criteria for nonlinear solution involves checking the
norms of the residual force and displacement vectors with separate checks for
translation and rotation degrees of freedom, as well as the residual structure

internal energy. In addition, the solution process is checked to avoid diver—

57

 

 

 

58

gence and to optimize solution speed, and adjusted if necessary by means of
modifying the global stiffness matrix used to obtain the solution, adjusting the
load increment size, or switching the nonlinear solution method (from load-
controlled to displacement-controlled analysis). The accuracy of the member
resistance calculation is also verified by comparing the work of external (nodal)
forces and internal energy.

The structure and loading data input includes data input modules for
global analysis parameters, nodal points, structure members, basic load

vectors, and incremental loading parameters. The analysis output is obtained

 

for specified parts or all of the structure load and displacement vectors and
members joint forces and displacements. This output may be obtained for every
load increment or only at the ultimate load point. In addition, the load—
displacement response curves for selected degrees of freedom of the structure
may be separately obtained for plotting.

Detailed description and implementation of the above topics is covered

in this chapter.

M PROGRAM SCOPE AND INIPLEMENTATION

 

3.2.1 Scope and Types of Analysis
In this section, the different types of analysis are described, and their

assumptions and constraints are discussed in detail.

 

59
am; limes—ofAnilXS—E

Three types of arch ultimate loads were calculated for the parametric
study. Elastic-nonlinear ultimate load involves geometric effects only, while
plastic ultimate load involves plastic (or material) effects only. Combined
. ultimate load includes both geometric and material nonlinearities.

Geometric nonlinearity is that resulting from the change of the geomet-
rical configuration (deflection) of the structure under loading, that affects the
equilibrium of the structure. This type of analysis is called here "elastic—
nonlinear" or "elastic—geometric" analysis. The elastic-nonlinear ultimate load
of a structure is essentially equivalent to its buckling load under the same
loading conditions.

Plastic nonlinearity is that resulting from the permanent change, or
yielding, of the material response under high stress level. "Plastic" or "simple-
plastic" analysis as defined here includes only the material effects.

When the analysis includes both geometric and material nonlinearities, it
is called here "combined analysis". Combined analysis is the lower bound of
elastic-nonlinear and plastic ultimate loads, and is more representative of the
structure actual behavior. When combined analysis is dominated by one of the
two nonlinear effects, it produces results that are close to or the same as those
produced when only that effect is included. Nonetheless, plastic and elastic-
nonlinear ultimate loads are needed for comparison with known solutions in the

literature and specifications criteria based on these types of analysis.

 

 

 

' 60
3.2.1.2 Assumptions of Nonlinear Analysis

The nonlinear solution procedure is implemented using the assumption of
fixed coordinates, where reference is made to the original configuration of the
structure. Loads are applied at the nodal points.

The elements stiffness matrices are derived using the Lagrange-small
rotations (Lagrange-SR) coordinates, where the member generalized deflection
coordinates are measured from the original global coordinates. This derivation
is valid for small member chord rotations from the original structure
configuration, where 0, and it, are less than about 15 degrees (see Figure
(2-1)). In addition, the member end rotations relative to the deformed member
chord are assumed to be small, and the effect of torsional rotations on axial and
shear strains are neglected.

When plastic effects are included (in plastic and combined types of
analysis), the material nonlinearity considered here is based on the elastic-
perfectly plastic material behavior, neglecting strain hardening effects, residual
stresses, and the gradual spread of yielding across the member section. The

member yielding is represented using the plastic hinge model, as described in

 

chapter two, where the plastic deformations are assumed to be concentrated at
its ends by formation of plastic zones of zero length while the remainder of the

member behaves elastically.

 

61
3,2._2 Procedure of Incremental Analysis
In this section, the procedure of applying the incremental load and ',
optimization of the solution speed are considered, and then the detailed steps of

nonlinear solution during a load increment are described.

3.2.2.1 Incremental Analysis: Definition of the Ultimate
Load and Optimization of Solution

 

The structure ultimate load is defined as the maximum load it can carry
before failure. For the purpose of this study, failure is assumed when the
structure can no longer carry any additional applied loads. The structure
behavior beyond the point of ultimate load or when the structure goes into large
deflection -of the magnitude of the structure dimensions- is not of practical
importance, and is not considered here.

The structure ultimate load is obtained by incrementally loading the
structure up to its ultimate load. The ultimate load is reached when either the
solution for equilibrium becomes unstable due to divergence of the iteration
process when applying a new loading increment, or when the load parameter

decreases if displacement-controlled analysis is used.

Defining the ultimate load point

To define the ultimate load point more precisely, a procedure for
automatic adjustment of the load increment size, similar to that suggested by

Bathe and Cimento (4), was implemented.

 

 

 

62

In this procedure, the applied load for increment j, AP], is determined
using the initial load parameter, Af = Af,, where Af, is the load parameter
specified in the data input so that,

APj = Af . P“,
where P" is the unit load vector applied to the structure. If the nonlinear
solution process diverges during iteration, a smaller load is applied by sealing
down the load parameter so that Af = sfI ~Af,, where sfl is a scaling factor, and
the solution for the increment is repeated using the new load parameter. If the
solution is still diverging, the scaling process is successively repeated -for a
total of three times- using a smaller load parameter each time,
Af = sfIn - Afo ,
where sf,, is the scaling factor for the mth attempt. Scaling factors of 1.0 (for

the initial attempt), 0.50, 0.20, and 0.05 are used for m = O, l, 2, and 3,

 

respectively.

If the smallest load parameter still results in solution divergence, the
structure is assumed to have reached its ultimate load at the end of the last
converging load increment. On the other hand, if convergence is obtained for a
load parameter, it is used in the next load increment, as explained later.

Displacement-controlled analysis was implemented to trace the load
response curve beyond the limit of load—controlled analysis, and to define the

ultimate load point even more precisely. However, the added accuracy in

 

63

defining the ultimate load point was found negligible, with the improvement in

the value of ultimate load being typically less than 0.2%.

Optimization of Solution Speed

In addition to handling calculation divergence, the same scaling
procedure described above is used to handle slow convergence rate of the
solution. To avoid endless calculations for convergence during a load incre-
ment, the maximum number of iterations allowed during a load increment is
limited to 40 iterations. If this limit is reached, a smaller load increment is
used (by using the next smaller scaling factor) and the solution is repeated.

In addition, if the number of iterations needed for convergence during a
converging load increment exceeds a certain limit (taken here as 25 iterations),
then a smaller load increment is used for the following increment. By reducing
the nonlinearity handled in a single increment, the calculation time spent on
every increment is decreased and the total solution effort may be minimized.
This process may be repeated until the smallest factor is used.

To optimize the solution speed and avoid repeatedly encountering the
same likely divergence situations, the first scaled load parameter that results in
solution convergence is used for at least the next four load increments to
guarantee that at least the total magnitude of the last diverging load increment
is successfully applied to the structure. This process avoids likely divergence
situations due to, for example, unnecessarily large load increment near the

ultimate load, or due to steep change in the structure stiffness.

 

 

64

After four consecutive fast converging increments (i.e., each requiring
less than 25 iterations for convergence), a larger load parameter is used -by
using the next larger scaling factor— to avoid excessively small load increments.
This process may be repeated if necessary until the initial load parameter is

reached again.

3.2.2.2 Steps of Analysis During a Loading Increment

 

The steps of nonlinear solution procedure during a load increment are
described in this section for the modified Newton—Raphson method, which was
used for obtaining results of this study. Direct Newton—Raphson method was
implemented for comparison of convergence rate and check of the accuracy of
nonlinear solution. When the implementation of direct Newton-Raphson method

is different for a given step, the difference is mentioned as relevant.

 

The principles and implementation of the three nonlinear solution
methods of analysis is explained in detail in appendix A. A summary is

described here for completeness of discussion.

Theoretical Basis of Nonlinear Solution
The nonlinear equilibrium equation of the structure at the end of load
increment j requires that,
f(D) = Pj(D) — R3(D) = o,
where P5(D) is the applied load through load increment j, R5(D) is the structure

resistance at the end of that increment, and D is the (unknown) structure displa—

 

 

 

65

 

 

 

  
  

 

 

P
i dPI‘ii
// () dp: =
aP.“ (M) ” dP’J”
ml.
(0
I. — I
AP: = (H) AR, _ AR
dPo“ ARiH
ARI
ARI-I
, (D: .Ri (1).»
(0)
00' F dDi ——>l‘<— dl)i+l a! an D

 

 

 

 

AI)1 = :1 dDi ———4

Figure 3-1 Iterative solution of nonlinear equations of equilibrium using the Newton-
Raphson iteration process.

 

cement vector that satisfies the equation. By using Taylor’s series to expand the
above equation about the converged (known) solution point at the beginning of
load increment j, the direct Newton-Raphson equation for iteration i of the

nonlinear solution process may be written as, Figure (3-1),

APj - AR,j_1(Dij_1) = dPiu—‘i = K214 (Dij—l) ' dDij :
where APj is the applied load during increment j, AR,_,5(D,_lj) is the calculated
structure resistance at the end of iteration i-l as a function of DH", the total
structure displacement vector at the end of that iteration; dPi_l“-"(D,_15) is the
unbalanced force at the end of iteration i—l; KLHKDH’) is the structure tangen~

tial stiffness matrix at the beginning of iteration i-l, and dD,j is the unknown

 

 

 

 

 

66

incremental displacement vector during iteration i that is needed to attempt to
satisfy the equilibrium equation. The solution is accepted when equation (3.1)
is satisfied to within an assumed tolerance limit.

For modified Newton-Raphson method, Kw,” is replaced by the stiffness
matrix at the beginning of the increment, Km", to avoid the calculation-intensive
decomposition process for every iteration. An acceleration scheme is
implemented to compensate for the resulting decrease in the speed of solution
convergence.

The displacement vector through iteration i, Dij, may be written as Dij =

DJ"1 + AD"

I,

where D"1 is the displacement at beginning of current load incre—
ment, and AD,j is the accumulated incremental displacement during the current
load increment through iteration i, AD,j = AD,j + dDfl. The increment index,

j, may be omitted in the following discussion for simplicity.

Steps of Analvsis for a Load Increment

The typical solution for nonlinear equilibrium for a given load
increment, j, consists of the following steps:
1. Specifying the Initial Load Increment:

The analysis begins by specifying the initialload increment size, AP’, to
be applied for the increment, which is assumed to be the unbalanced load for

the first iteration of the increment, dP,“’j

APJ‘ = dpg‘J = Af'P‘V+dP"H, (3-3)

 

 

 

 

 

67

where Af is the load parameter that defines the actual load forces as a ratio of
the applied unit load vector, P", and dP'J'l is the residual load vector still
unbalanced after convergence at the end of the previous increment. The value
of Af is influenced by the convergence rate and its value during the previous
converged load increment, as discussed before.

2. Assembly and Test of the Structure Global Stiffness Matrix:

At the beginning of the increment, the structure tangential stiffness
matrix, Km, is formulated as a function of the structure displaced position at
the beginning of the increment (D,5 = D“), and decomposed using the
Cholesky decomposition method. During this process the stiffness matrix, K,_,,
is verified to be positive-definite, and if this is true its determinant is
calculated and the initial displacement vector, le, corresponding to the initial
applied load, dP,“, is obtained using equation (3.4) below.

If the determinant of ij is small compared with the determinant of the
linear-elastic stiffness matrix of the structure, the solution accuracy is verified
using the procedure described in the next section. If the solution is found
acceptable, the obtained displacement vector is accepted as that of the first
iteration. For modified N-R method, the decomposed stiffness matrix is used
during all following iterations. For direct N-R method, the stiffness matrix is
updated and decomposed for every iteration but is checked for accuracy of
solution only during the first iteration, since the iteration process itself acts as

an indirect check in the following iterations.

 

 

 

 

68

On the other hand, if the stiffness matrix is not positive-definite or the
solution was found unacceptable then the structure is considered either unstable
or close to that state, and the solution is considered to be diverging. To
overcome this situation, the stiffness matrix is formulated again using a
different assumption, as explained in step (5) below, and the solution is
repeated.

3. Calculation of the Structure Nonlinear Resistance:

The structure nonlinear resistance for the increment is calculated during
every iteration as follows,

3.a. The incremental global displacement vector for iteration i is obtained by

solving for the unbalanced load at the end of previous iteration,

de = K;‘.dP,‘:, for MNR, or,

1

db, = K11. dP,‘_‘, for DNR,
where dDi' and dDi are the incremental displacement vectors obtained using the
modified and direct Newton—Raphson methods, respectively. For accelerated
modified N-R method, dD,‘ is replaced by the modified (accelerated) vector,
dDi = f(dDH, dDi’), as that of current iteration. The total displacement vector
for the increment is calculated as, the sum of displacement vectors for all
iterations up to the current one, ADi = AD,l + dDi.
3.b. The resistance of individual elements corresponding to AD,- (transformed

into the element’s local coordinates), is calculated as described in chapter two,

and assembled into the incremental structure resistance, ARi, after being trans-

 

 

 

 

' 69
formed back into global coordinates. The new unbalanced (or residual) load
vector at the end of current iteration, dP,“, is found,
dP,“ = AP — ARi.

The vectors, dP,u and dDi, Figure (3—1), are used as the residual force
and displacement vectors in the process of convergence/divergence check
during that iteration.

4. Check for Convergence or Divergence of the Solution:

Check for convergence or divergence of the solution process is
performed using the convergence/divergence monitoring procedure described
later. Depending on the results, the solution procedure continues as follows,
4.a. If the solution process has converged, then the solution for the increment
is accepted and control is transferred to step (6) below.
4.b. If convergence has not been obtained yet but the solution is converging,
then the solution process continues by going to step (2) above for iteration i+l.
4.c. If the solution process is determined to be diverging, then control is
transferred to step (5) below.
5. Divergence Handling:

If the solution process is diverging, the load step is repeated with

modified assumptions and the solution process is repeated. These modifications

are implemented in the following order:

 

 

 

 

70

5.a. If the process diverges during the iteration process, the loading step size
is probably too large and the solution is repeated with a Smaller increment size
using the scaling procedure described before.
5.b. If solution still diverges for the smallest load step size or if the stiffness
matrix is unstable at beginning of the increment, then the structure stiffness
matrix is formulated without the plastic nonlinearity effects if present. If the
process still diverges in the presence of geometric effects alone or if plastic
nonlinearity is not present, the stiffness matrix is formulated using the elastic—
linear stiffness matrix, K,. This option is not applicable for direct Newton-
Raphson method.
5.c.‘ If the process is still diverging after the above two steps, then the load—
controlled analysis has reached its end. The ultimate load is taken as that at the
end of last converging increment. If displacement—controlled analysis is
required, the solution method is switched to displacement-controlled analysis
and the solution process is continued. Otherwise, the analysis is ended.
6. End of Increment:

After convergence of the load increment, the final step is to update the
structure and element quantities, such as displacements and forces, for use
during next increment. Also, output for printing and plotting of the required

data is performed. The analysis then proceeds to the next increment or is ended

if the ultimate load has been reached.

   

 

 

 

71

3.2.3 Check for Convergence or Divergence of Solution

The solution process for nonlinear equilibrium during the load increment
is monitored for convergence every iteration. If the convergence criteria is
satisfied, the solution is accepted and analysis proceeds to the next load
increment. At the same time, the solution process is checked for divergence to
avoid large errors in the obtained solution (i.e., false convergence), false or
premature termination of analysis, prevent needless calculations, or to end the
analysis. If divergence is encountered, the solution is repeated using different
assumptions, or the analysis is ended.

In this section, the different tests of convergence/divergence are
presented first, then their implementation as a criteria for convergence or

divergence is described.

3.2.3.1 Basic Tests of Solution Convergence and Divergence

The basic, or individual tests used to check the convergence and
divergence of the solution process are the force, displacement, and energy
tests. For the first two tests, the second order (Eucledian) norms of the residual
(unbalanced) force vector of iteration i, dPi“, and the residual (or incremental)
displacement vector of the iteration, dDi, are traced and compared with their

respective reference norms. In these tests, the translation and rotation degrees

of freedom are checked separately so that,

 

 

 

 

72

idPfiflllinffl s
"dram/1mg n s e... (35)
ldDfijlllllfofl s 6,,
idD8)||/I|D{§fl s

an ,
where de“ and Pu)ref in the first equation are the force residual and reference
vectors corresponding to translation degrees of freedom, and EP is the allowable
force tolerance; similarly subsequent equations represent the tests for rotation
forces (moments), translation displacements, and rotation displacements, with
61) being the displacement tolerance limit. A tolerance limit of 0.0001 is used
for 61, and 6D. The force or displacement test is considered convergent if both
translation and rotation parts indicate convergence.

The energy test measures the work done by the unbalanced force going

through the incremental displacement and compares it to the total work done

for the whole increment,

E,“ / E,” s e, (3-6)

where E,“ and E,” are the residual and reference energy terms, respectively, at
the end of iteration i, and GE is the tolerance limit for energy test, taken here as
0.03.

The residual force vector for iteration i is calculated as the difference
between the incremental applied force and structure resistance at the end of
current iteration, dP,“ = AP - ARi. The forces and moments in this vector

contribute to their respective translation and rotation norms,

 

 

 

 

. ..lll‘il . L
illpii-Killlrx :

 

 

73

 

n,
u 1
up u — — P“(k)2,
<1). n. \ .‘é
. 1 'T—j
"Pull = — P"(k) ,
‘2’ n2 \ 231

where n1 and n2 are the number of translation and rotation degrees of freedom
in the structure, respectively. The reference norms are taken as the largest
values of the total force norms throughout the analysis, evaluated after

convergence of each increment,

11>ng = max.(l|P(‘f)lel) j=1jj—1
“Pg?" =_ max.(nP{;,‘~'u) ’ ’ ’

where jj is the current increment, and the norms of P‘°f'j for increment j are
calculated similar to those of P“ above. However, minimum values of these
norms are calculated as a function of the plastic forces of the structure
individual members, with the plastic axial forces, Np, and plastic bending
moments, MM and MM, contributing to their respective norms,

. -- 1 ""
mmdeftnn) = — ‘ 2N,<k>2,
m1 k=l

m2
—1—\J 2 {(M.,(k) + man/212.
m2 k=l

 

min. (i Pg?jj u.)

where m1 and m2 are the number of members contributing to each norm,
respectively.
The residual displacement vector for iteration i is taken as the incremen-

tal displacement vector for that iteration, dD,, obtained from equation (3.2),

 

 

 

 

74

with its two norms calculated similar to those of residual force vector above.
The reference norms are taken as the largest values of the total displacement
norms throughout analysis, evaluated after convergence of each increment. For
the first increment, it is taken as the norm of displacement vector after first
iteration. The residual displacement vector for the first iteration of every
increment is found by solving for the residual force, dD2 = K,‘,‘1'dPl“.

For energy check during an iteration i, the residual energy, E,“, is
calculated as the amount of work done by the residual forces on their corres-
ponding residual displacements, which is the scalar product of the two vectors.
Similarly, the reference energy, E3“, is the work done by the total increment
resistance force, AR, (at the end of current iteration) on its corresponding
(accumulated) displacement, ADi,

E.“ = {dP?}T - {(11),},
E,” = {ARilT -{AD,}.

After convergence of the solution process, an additional test of the
solution accuracy is made by calculating the structure internal energy on the
element level through the actual force path (for example, through the member’s
different elasto—plastic stages) and comparing the result with the external work
done by the total increment resistance force, AR,, through its corresponding
increment displacement, ADi. At convergence, the difference should be small

enough (less than 1%).for the solution to be accepted. This test is particularly

 

 

 

 

 

75

useful to verify that the approximations used to calculate the member resistance

produced no significant errors.

Tests for Divergence

The divergence criteria uses the same tests for force convergence
described above, but with an upper limit that indicates divergence if the test
exceeds this limit in place of the tolerance limit of convergence. For force
divergence test,

naps.) u / u Pa? 11 2 11,, (3.7)
u mg, n / u ngn 2 cu.
where rm, is the limit divergence for the residual force test. The value of TH,
used here is 10 to allow for large changes during the first few iterations
without declaring false divergence.

Another test of force divergence is the rate of convergence over consec-
utive iterations. If the solution is converging, the residual forces and displa-
cements for a given iteration, i, should be smaller than their values during the
preceding iteration, i—l. Therefore, divergence of the iteration process may be

tested by comparing the residual force norms during the current iteration to

 

 

those during the preceding one. If the ratio exceeds a certain limit for both

types of degrees of freedom, this test indicates divergence,

fldPilzl) ll / H delilu) ll 2 172.1” (3.8)
lldPi‘Ez) ll / "dPilium ll 2 72,1”

 

   

 

 

76 '
where 72,1, is the limit of divergence for the consecutive residual force test. A
value of 12,? = 1.2 is used here. However, to avoid reporting false divergence
when the solution is near convergence, this test is applied only if its corres-
ponding residual force test in equation (3.7) indicates large norm of the
residual force, i.e., dP“/Pref > 0.03.

The above two tests are not applied to the displacement vectors, since
the new yielding of a member joint during an iteration, for example, causes the
residual displacement vector to become larger than that of the previous
iteration. Also, the acceleration scheme used in the modified N—R method
creates a similar situation where the displacement vector is artificially
increased from one iteration to the other. Consequently, the displacement
vectors are not directly included for divergence testing, but are indirectly
included in the energy test.

The energy divergence tests are similar to those of the force tests,

IIEI'l/IlEfdll 2 :13.
"Erin/"EL" 2 rm,

and’ (3.9)

where E,“ and E,“ are the residual and reference energy terms for increment i.

The values of 71,, and 72,13 are 10 and 1.2, respectively.

3.2.3.2 Criteria for Convergence
In general, a single convergence test is not always effective in producing
accurate and robust prediction of convergence for all stages of analysis. While

a stringent tolerance limit for the energy test may assure accurate prediction of

 

 

 

 

77

convergence, it may cause premature end of the analysis for small load
increments by forcing very small absolute error tolerance and possibly
producing false divergence or needless calculations before convergence. On the

other hand, a relaxed tolerance limit may permit unacceptable large errors in

 

the solution for large load increments. The same may be said about individual
force or displacement tests, where the softening or hardening of the structure
stiffness may cause similar problems of inconsistency.

To provide an adequate check for solution convergence during all stages

of the analysis, the above three individual tests are combined to form an

 

adaptable criteria for convergence check. In this criteria, all three tests are .
calculated for every iteration, and the solution is accepted if all three tests are ‘
satisfied. The rationale for this criteria lies in the different reference norms
used for each of the convergence tests.
The reference value of the energy test is the only one that is directly
related to the increment magnitude, while those of the force and displacement
tests generally increase with the progress of analysis. For the first few load
increments of the analysis, the energy test is usually satisfied well before the
force and displacement tests. These two tests together, however, provide
sufficient control of convergence by keeping errors small.
As the analysis approaches the ultimate load, the structure stiffness
decreases and the displacement (and possibly the force) reference norms

become larger, leading to more relaxed convergence check. This results in an

78

increase in the absolute error magnitudes allowed in the solution for the same
tolerance limits. In addition, the load increment size may become smaller,
resulting in even larger relative errors compared with the increment forces and

displacements. The energy test imposes an upper limit on the allowable errors

 

in the solution by directly relating the errors to the increment forces and

 

displacements. As a result, the energy test may become the governing test of
convergence during the final stages of analysis.

In addition, for modified N-R method the structure stiffness matrix used
at the beginning of. the increment may be different from the actual tangential

stiffness matrix, as discussed before. For example, the linear-elastic stiffness

 

matrix may be used in equation (3.4) in an analysis that involves plastic effects.
This condition may result in very small displacement increments, dD,, for the
first few iterations, which may lead to false convergence. The acceleration
procedure usually corrects this situation afterward. To avoid this possibility, a
more stringent tolerance limit for energy test is used, 6,, = 0.001 during the

first four iterations.

3.2.3.3 Criteria for Divergence

Solution divergence may occur due to different reasons during the
analysis. Divergence may occur because of the structure instability as it
approaches the ultimate load, or due to sudden change in the structure stiffness
during a single load increment. Also, the nonlinear solution procedure may

cause instability or divergence of the solution, or may produce unacceptable

 

 

 

79

slow convergence rate. The procedures to identify and handle these situations

of divergence are discussed in this section.

A. Structure Unstable at the Beginning of Load Increment

As the analysis approaches the ultimate load, the structure becomes close
to instability and its stiffness close to being ill-conditioned or singular. This
condition may produce significant, unacceptable errors in the obtained solution.
To avoid this possibility, the solution is checked for accuracy every time the
structure tangential stiffness matrix at the beginning of a load increment, K,,,
is updated.

The proximity to the ultimate load is determined by the ratio of the
determinant of the structure stiffness matrix at the beginning of the load
increment, Det(Km), to that of the elastic—linear stiffness matrix at the
beginning of the analysis, Det(K,). If the ratio of Det(K,_,)/Det(K,) < 0.001,
then the solution is checked for accuracy, otherwise it is accepted and the
nonlinear solution procedure proceeds.

The solution accuracy is checked for instability using a simple method,
(13). This is done by recalculating the force and displacement vectors, dP,‘,“
and dD1,,, using double precision,

o .

(11):: = Ko.le’
dD = K'l-dP“

LC 0 cm ‘

dD, = K,'1'dP“

 

80

The difference between the calculated and original vectors is taken as a
measure of the stability of the solution. The errors in the force vector, (SP, is
found using the first order norm (i.e., the largest absolute value of the vector
elements) of the difference between the calculated and original vectors, dP,,,“ -
dP,“, and the original vector, dP,“. The error in displacement vector, 6D, is

calculated similarly so that,

5p = ndp:—ap3,j,i,/ udPS‘ul.
5D H le _ dD1,c"1 / '1le "1'

If the error in either the force or displacement is larger than a certain
tolerance limit, taken here as 0.001, then the stiffness matrix is considered
close to being singular or ill-conditioned, and the solution is considered to be
divergent. On the other hand, if both tests are satisfied, the solution is accepted
and the iteration procedure proceeds.

To handle divergence due to ill-conditioned structure stiffness matrix, it
is formulated again without the plastic nonlinearity if present, i.e., formulated
using the elastic-nonlinear stiffness matrix and the above check is repeated. If
it is still ill-conditioned using elastic-nonlinear stiffness matrix, it is formulated

again using the elastic—linear stiffness matrix.

B. Divergence of Solution During Iteration
Divergence of the iteration process may be due to the fact that
equilibrium is not possible under the applied load increment because it exceeds

the structure ultimate load, or due to instability of the solution procedure due

 

81

to severe nonlinearity. In both cases, the unbalanced load vector produces
unproportionally large displacements, that in turn produce large unbalanced
forces for the following iteration. For the first case, this cycle continues and
ends with divergence of the solution. For severe nonlinearity, however, the
iteration process may momentarily produce large residual forces and
displacements during a given iteration, but the solution may stabilize as the
iteration process continues and eventually converge.

To provide a flexible divergence criteria that detects real divergence but
at the same time allows for the possibility of temporary large divergence, the
divergence is monitored over consecutive iterations in addition to the current
iteration. For a given iteration, i, the four divergence tests described before are
checked, and the number of tests that indicate divergence is found. In addition,
the number of diverging tests during the previous iteration, i-l, is taken into
consideration; if at least two tests were diverging during that iteration, the
number of diverging tests for the current iteration is increased by one.

The divergence criteria assumes divergence of the iteration process if
four of the five tests indicate divergence, otherwise the iteration process is
continued. During the first four iterations where the solution and the accele—
ration procedure usually produces large oscillations, divergence is assumed
only if all five tests indicate divergence.

In addition to testing for divergence, a maximum limit on the number of

iterations for the increment is applied so the iteration process is stopped if

 

82

number of iterations exceeds 40 iterations. To handle divergence of the itera-
tion process, a smaller load increment is found using the scaling procedure

discussed before, and the solution process is repeated.

3.2.4 Data Input and Output for Analysis

The data input for the computer program is composed of four main
groups, 1) Main control parameters of the structure, loads, and type of
analysis, 2) description of the structure geometry and its elements and their

properties, 3) Data to describe the load vectors and their combinations, and 4)

 

Data for incremental loading and analysis parameters, and output of results.

The program control parameters specify the total number of structure
members, number of element groups, number of structure nodal points, number
of load vectors, and type of analysis. Nonlinear analysis parameters include the
maximum number of load increments allowed throughout the analysis, the
method of analysis (direct or modified N-R method), whether or not to switch
to displacement-controlled analysis and the necessary information for that, and
whether response curves are to be output.

Data input for the structure nodal points include their three—dimensional
coordinates and restraint conditions. When the structure includes an arch as
part of its geometry, the program can generate the y-coordinates of the arch
nodal points given the type of arch (parabolic or circular), its span rise at the
crown, and the coordinates of the end points. This procedure along with other

basic input and output, and linear analysis routines was obtained from a

83

program written by Ralph Dusseau along with professor Robert Wen at
Michigan state university, (10).

Data for each element group consists of three parts, namely the element
group control parameters data, data for element property groups, and data for
individual elements. The program includes three types of elements: truss
element, straight beam element, and curved beam element, but only the straight
beam element was used in this study. Data for control parameters specifies the
number of elements in the group, number of element property groups and
material properties (modulus of elasticity, yield stress, and Poison’s ratio).
Data for each element property group defines the section properties for that
group; i.e., the area, moments of inertia, torsional constant, and ultimate
plastic forces and moments of the section. Data for each element includes its
two end joint nodal point numbers, and for straight or curved beams, the nodal
point that defines its local y-axis in space. Also its section properties are
determined by its property group number.

The program can accept up to six different "basic" load vectors, such as
dead loads, live loads, or lateral loads. For a basic load vector, loading data
entries for each loaded nodal point of that vector is input in the form of six
load components, three forces and three moments, specified in terms of the
structure global coordinate system. The program can also generate "combined“
load vectors that are composed, or constructed, as combinations of the basic

load vectors. These combinations are specified in the input data.

 

 

84

Data for incremental loading specify the load vector number to be
applied, the total number of increments this vector is to be applied, NINC, the
initial load increment parameter, Af,, the maximum number of iterations
allowed before declaring divergence of the load increment, and the tolerance
limits for the force, energy, and displacements convergence tests. This load
vector is then applied to the structure for a total load parameter of Af,,“, =
NINC ' Af,, regardless of the actual number of load increments needed to apply
this load parameter, or until the ultimate load is reached. This accounts for the
adjustment procedure of the load increment size described before.

Output of the calculated results is controlled for incremental analysis by
specifying its frequency; for example, results are output every increment, every
few increments, or at the ultimate load only. Also, output for only selected

nodal points and elements may be specified.

Q PROGRAM VALIDATION AND CHECK

The computer program was checked against various past studies, both
experimental and analytical, reported in the literature with either or both
geometric and material nonlinearities included. Plane and space structures, and
building-type frame and arch-type structures were compared. In general, good
agreement was found between the obtained results and those reported. The

details of these examples are given in the following sections.

 

 

85

3_._3_._1, Geometric Nonlinearity Problems

When only geometric nonlinearity is present, the program produced very
close results to those obtained using beam—column theory for so called "small-
displacement" type problems (with displacement less than 1% of the structure
length). The straight beam element for nonlinear elastic behavior used here is
that developed by Wen and Rahimzadeh, (42), but the tolerance limits used
here is smaller. As would be expected, the current program produced very
close results for the same problems.

Solution of the one—story single-bay frame shown in Figure (3-2.a), is
given in Figure (3-2.b), along with the solution based on beam-column theory,
as reported in ref. (42) by Wen and Rahimzadeh. In both cases, one element
per member was used with load step of 250 kips. It is seen that the load-
displacement response and the critical load of the current solution is very close
to the beam-column solution. The obtained critical load was 4710 kips, while
the theoretical value was 4750 kips.

Figure (3—3.a) shows the geometry, loading, and properties of the space
arch frame reported by Wen and Lange in ref. (42). The reported lateral load-
displacement solutions using finite element analysis with updated and fixed
coordinates are shown in Figure (3-3.b), along with current solution. The
current solution is in close agreement with both solutions, and forms a lower
asymptotic curve to both solutions, where it is closer to the fixed coordinates

solution at the initial stage, and becomes closer to the updated coordinates

 

 

 

 

 

 

86
P P
.001 P V V
r C
2’;
52‘.
g A = 29.99 cm2
0 (11.77 in)
«a
3 4
m 1 = 12,907 cm
11 (3101.1 m“)
v—l
. E = 2.11*106 kg/crnz
L7 L (30,000 ksi)‘
I : : w

 

i7 L = 304.8 cm (120") Z

(21) Dimensions, properties, and loading.

 

   

 

 

8.0 —
E
"A 6.0 —
a.
n —.
E
g 4.0 —
23
an _
'2
3 2.0 — ___ Beam-column
(Wen & Rahimzadeh)
_ _ Program
0.0 I I I I I I l —l
0.0 2.0 4.0 6.0 8.0

Normalized Horizontal Deflection at c ( = u,*103/L)

(b) Response curves.

Figure 3-2 Comparison of program with beam—column results for elastic-nonlinear
behavior of single-story, single-bay frame.

 

 

 

40.0'

    

 

 

 

309'
[ 69.282' 61.436' 69.282'
L = 200'

Section 1 Section 2 Section I: All Longitudinal Members
A ft2 0 500 0 100 Section 2: Two Transverse Members
1,, n“ 0.400 0.050 6
1 ft4 0 133 0 050 E = 4'3”“) “f
2’ . .
GI, k-ft 4.15*105 1.66*105

 

 

 

 

 

(a) Dimensions, properties, and loading.

   

__ Rahimzadeh, updated

Laod Parameter ( = PLZ/EIy(1))

 

 

J _ _ Rahimzadeh, fixed

05 L _ Program

0.0 . t . t - t - a
0.0 1.0 2.0 3.0 4.0

Normalized Lateral Displacement at D ( = wD*103/L)

(b) Response curves.

Figure 3-3 Comparison of program with finite—element results for nonlinear-elastic
behavior of a space frame.

 

 

 

88

solution after the snap stage of the response. This might be attributed to the

more stringent tolerance limits of convergence used in current solution.

Li; Plastic and Combined Nonlinearities Problems

The two-story, two—bay frame reported by Argyris et. a1. as example 5.5
in ref. (1) is shown in Figure (3-4.a). Using the inverse parabolic function,
Argyris reported that the plastic ultimate load (using simple plastic analysis) is
22.4 kp, and the combined (geometric and plastic) ultimate load is 21.5 kp.

Table (3-1) shows the calculated ultimate loads using different load
functions, and the differences from those of ref. (1). It is seen that the inverse
parabolic and spherical functions produce very close results to those of
Argyris, with differences of less than 3%, while the other two functions
generate larger differences. The simple—plastic and combined nonlinear
response curves obtained by the program for the spherical function are shown
in Figure (3-4.b). Also shown in the same figure is the sequence of plastic-
hinge formation for the combined analysis. The sequence of plastic hinge
formation for both spherical function and Argyris solutions is very similar.

Shinke, et. al., (36), reported the results of experimental tests of in-
plane ultimate load of single-rib parabolic arches with rectangular cross—
sections under unsymmetric loading. Using 16 elements along the arch length
with equal horizontal projections, the load carrying capacity of these arches
were calculated using different types of yield surface functions. Spherical,

triangular, parabolic, and inverse parabolic functions were used. The results

 

 

89

Table 3-1 Comparison of different yield functions with calculated results of Argyris

 

 

 

 

 

 

 

et a1.
Function Type Ultimate Load, kp.
Plastic Combined
Analysis Analysis

Argyris: P 22.40 21.50

Inverse Parabolic

Triangular P 20.75 18.59
error, % ~7.37 -l3.53

Spherical P 22.56 21.22
error, % 0.71 -l.30

Inverse Parabolic P 22.43 20.90
error, % 0.13 -2.79

Parabolic P 21.70 19.74
error, % ' -3.l3 ~8.l9

 

 

 

 

 

 

are shown in Table (3-2), and plotted in Figure (3-5).

While the triangular yield function produced values closer to the
experimental values on the lower side with error range of -l.3% to —‘6.7%, the
inverse parabolic function produced more consistent results with errors between
9.3% to 10.4%. The spherical function produced larger errors on the higher
end, with range of 14.8% to 19%. Since the plastic hinge model does not
account for gradual yielding in the member, it tends to over-estimate the actual
load carrying capacity of the structure because progressive yielding in its

members tends to decrease its load carrying capacity by softening its stiffness,

 

 

 

90

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P/2 P P P P/2
2p Columns Beams
.7 C A. m2 0.0192 0.0118
Hz 1) p P P/2 5 I, m4 .511=*10'3 .292*10‘3
2P i " NP, kp 73.2*103 42.0*103
V Mp,kp-rn 480*103 295*103
E
O.
I E = 2.10*1010
”727 ”777 /7777 v = 0.30
L 4 @ 3.0 m J
(a) Dimensions, properties, and loading.
25.
_| Simple-plastic analysis 10 11
.1
E“ \ 8 9
M 20. — 456 7 a 9 10,11
5 ' 4,5 6-7
3': - 3 3 Combined analysis
91
11 15. - 1.2 1,2
,_, _
0
3'3
E 10. —
(U
H
m
m _
2?:
o 5' _
D—]
0- . 1 ' 1 . 1 . ‘1
0 5 10 15 20

Horizontal Displacement at C, cm.

 

 

 

(b) Response curves and sequence of plastic hinge formation.

Figure 3-4 Behavior of a two-story, two-bay frame reported by Argyris, et. al. under
plastic combined loading: response curves and failure mode.

 

 

91

Table 3-2 Comparison of different yield functions with experimental results of Shinke
et al.

 

 

 

 

 

 

 

 

 

 

 

Function Type .Live to Dead Load Ratio, r’ = (l-r)/r
0.05 0.20 0.50

Experimental P 24.00 17.60 12.80
Results

Triangular P 23.70 16.74 11.99

error, % -l.25 -4.89 -6.33

Spherical P 28.57 20.89 14.76

I error, % 19.04 18.69 15.31

Inverse P 26.24 19.43 14.06

Pammhc error, % 9.33 10.40 9.84

Parabolic P 27.17 19.18 13.47

error, % 13.21 8.98 5.23

 

 

 

 

which in turn produces larger displacement thataccelerate its failure.
Consequently, it is reasonable to expect the obtained results to be on the higher
side of the actual capacity of the arch. Therefore, the triangular yield function,
although produces the smallest errors, is not the most accurate representative of
the arch behavior.

The inverse parabolic yield function, on the other hand, produced more
consistent error percentage. At the same time, it more closely represents the
interaction curve between axial force and bending moment for steel sections,
and therefore can be assumed to produce the most reasonable results. Using the

fiber model that accounts for gradual yielding and initial imperfection,

 

 

 

92

 

llllLlj

f=l7cm

 

 

L L=102crn J

l“

 

 

 

 

 

7,
2
A = 1.492 cm Pp = 4.335 kg T = PDL/(PLL+PDL)
4 a
r = 0.7913 cm M, = 2.767 kg-cm r = (1-r)/r = PLI/PDL
E = 2.06*106 kg/cm2 0‘, = 2,940 kg/cmz P = (PDL + 0.5pLL)L
(a) Dimensions, properties, and loading.
30. —l
7 U\ H Experimental tests
_ V \
a _ A:\ \ \ v. ...v Parabolic function
(\I\ 25. ._ \\ \
1.1 \\ \ A‘ —A Inverse parabolic function
9" ‘ \ \\ \
n - \\\\ \ \ [3' “Cl Spherical function
v - \ \\ \
\ Octahederal function
g 20. _ \ G- '"O
9 1
E
at
H _
a
94 _
E 15. —
o
,4 _
10. I I 1 I l I 1 l 1 1 Ifi
0.0 0.2 0.4 0.6

Ratio of uniform live to dead load, r’ = (l-r)/r

(b) Comparison of different yield functions with experimental tests.

Figure 3-5 Ultimate load of single-rib arch under vertical unsymmetric loading:
comparison of different yield functions with experimental results of Shinke et. al..

 

93

   

 

 

30. j
C: 1 ,e ’0‘ ~o
NE ‘ /°/(Y
E 20. — f0
H _
v _ y? r’ = 0.05
3;: f
o ' J .—
§ ' I /9— '0 ’— 6~ ‘0
g: 10. — I’
'3 ‘ ‘P
o - l
1..) Experimental,
(Shinke et. al.)
_ Program
0. 5, r ‘Ii 17 I If f *I—fi
0.000 0.003

Normalized Maximum Vertical Displacement, vmax/L

Figure 3—6 Response curves of single-rib arch under vertical loading: comparison of
inverse parabolic yield function with experimental results by Shinke et. al..

Kuranishi and Yabuki (19), reported errors up to 8%. Figure (3-6) shows the
response curves produced by this function as compared to the experimental
curves reported by Shinke et. al., where the agreement is better as the ratio of
live to dead load, r’ = (l—r)/r, increases. This may be attributed to neglecting

the effect gradual yielding in the arch.

 

 

CHAPTER IV
PARANIETRIC STUDY OF ARCH BEHAVIOR:

I. IN—PLANE LOAD INTERACTION

_4._1 GENERAL INTRODUCTION TO PARAMETRIC STUDY

The presence of longitudinal or lateral loads on the arch -due to

 

earthquake or wind loading, for example- results in reduction of the arch
vertical ultimate load. As this reduction becomes significant, it requires
consideration in the design process.

To study the effect of different load combinations on the arch ultimate
load, the parametric study was divided into two parts. The first part covers in-
plane behavior where interaction between longitudinal (in—plane horizontal) and
vertical loading is investigated and reported in this chapter. The obtained
ultimate load were compared with the arch ultimate loads under vertical unsym-
metric loading. The second part, reported in next chapter, involves lateral
(transverse) loading and consists of two portions where the interaction between
lateral and vertical loading was studied first. In addition, the ultimate load
surface (in three—dimensions) of a representative arch was constructed to

explore the effect of combining all three types of loading.

 

   

 

 

95

The main parametric study variables include the loading combination,
support type (two-hinged or fixed), type of analysis (plastic, elastic nonlinear,
or combined), and the slenderness ratio. In addition to the arch total ultimate
load, the study examined the vertical component of the ultimate load, the
interaction curves between different loads for different types of analysis,
failure modes and response curves, plastic hinge formation patterns, and the
force path at the arch quarter point. For lateral loading, the effect of arch

section proportions on ultimate load was also investigated.

 

The basic arch model and parameters, and the load patterns and

combinations used in the study are described first in the next two sections.

M Basic Arch Model and Parameters

The parametric study was carried out using a parabolic steel arch with
rise to span ratio, f/L = 0.15. The arch was modeled using the straight beam
finite element. Sixteen elements with equal horizontal projection, as shown in
Figure (4—1.a), were used to model the arch. A preliminary investigation of the
effect of the number of elements on the ultimate load produced insignificant
improvement (less than 0.5%) when the arch was modeled using 32 elements.
The material properties used are those typical of steel, with the ratio of
elasticity modulus to yield stress, E/oy = 805.5. ~

Two section proportions were chosen for this study, Figure (4-l.b). The
first section, denoted S1, was used in the first part for in-plane loading

(vertical and longitudinal load interaction), with dimensions representative of a

 

 

 

96

 

 

 

- r-1
0
.2 In
"‘ v-4
‘5' 9-:
o‘.
909° 1
8° 73° l arch length, L l
0
"k ‘ = 16 segments: @ AL = L/16 each A

(a) Arch finite element model.

 

 

 

 

 

 

 

 

 

Section 81 Section 82
D/B 2 1
D/tf 50 100
Bit“, 100 100 '
Iy/AZ 4.085 4.125 f I
ry/rz 2 1
Z Z

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l

B B
Section SI Section 82

(b) Dimensions and proportions of arch sections.

Figure 4-1 Finite element model and section properties of the arch used for parametric
study.

   

 

97

typical cross section of a box-type single arch. This section has proportions of
depth to width ratio, D/B = 2, depth to flange thickness ratio, D/tf = 50, and
depth to web thickness, D/tw = 100. The second section, denoted 82, was used
in studies involving lateral loading and has symmetric geometric properties
about its two axes of symmetry to serve as a standard section, with D/B = l,
D/tf = 100, and D/tw = 100.

The above three ratios define (or result in) one nondimensional variable
for the in-plane study, Iy/AZ. For out—of—plane study, an additional non-
dimensional variable is defined by these ratios, i.e., the ratio of in-plane to the
out-of—plane radius of gyration, ry/rz. The values of Iy/A2 for the two sections
are 4.085 and 4.125, respectively, and the corresponding ratios of ry/r, are 1.0
and 2.0. The resulting in—plane properties of the two sections (A, 1,, N,, and
M”) for the same in—plane slenderness ratio, L/ry, are similar enough (with
difference of less than 1%), that the in-plane section properties are considered
constant for the two sections in the study of the arch behavior under lateral
loading. For in-plane loading, the results obtained from the two sections were
virtually identical.

For in-plane study, the arch slenderness ratio, L/ry, was varied from 75
to 375 with increments of 25. For the study involving lateral load, L/ry

increments of 50 over the same range were used along with an extra point at

L/ry = 450.

 

 

 

98
4;; Loading Distributions and Combinations

The distributions of different loads used in the study are shown in Figure
(4—2). For vertical load, the probability of unsymmetric live load occurrence
simultaneously with an earthquake is very small. Therefore it was taken in this
study as uniformly distributed load along the horizontal projection of the arch
with slight unsymmetry (r = 0.99). The slight unsymmetry is assumed here to
produce-unsymmetric deflection mode under vertical—only load throughout the
loading process and avoid the sudden shift from symmetric mode to unsym—
metric one at failure if a perfectly symmetric load was used.

The unit vertical load, PW, is defined as the total load applied to the arch
using a unit uniformly distributed vertical load (with intensity of one force unit
per unit horizontal length of the arch, pm), so that P,u = L -p,, (neglecting the
effect of the slight unsymmetry). The uniformly distributed loads are lumped
into concentrated loads that are applied through the arch nodes as shown in
Figure (4-3) for this and other loading cases.

The arch ultimate vertical load, P,,, is expressed in terms of Pv, and the
load parameter at ultimate load, f,, so that, P,, = f,vP,,, and it may be
expressed in non-dimensional form, P.,,-Lz/E'Iy = p,,~L3/Eoly. Figure (4—4)
shows the change with L/ry of the ultimate load of two-hinged arch under
symmetric vertical loading using plastic, elastic, and combined types of
analysis. It is seen from the figure that the elastic ultimate load is essentially

constant when expressed in non-dimensional form.

 

 

99

Unit vertical concentrated
load applied to each node,

MEI—111111 INIPWI T1 1 IWE

Vertical load

 

 

 

APhu = Phu'AL
“I: :L :l l ’l :l \l ’l l
I ’l \l ’l :J l 4L I

 

Longitudinal (horizontal in-plane) load

Psu AL

z/m;//////

Psu

 

Lateral (transverse) load

v-AP,u

   

1:63P“ - or.
)Z/A arch span length = L

Figure 4-2 Individual unit loading distributions in the vertical, longitudinal, and lateral
directions.

 

 

 

 

 

 

100

 

0.99 v-APvu
. / \
v-AP“, "
\H \//
h'APhn
‘— Arch span length = L A
applied unit load vector, Pm = v.P,,, + h-Phu
v + h = l
(a) Vertical and longitudinal loading.
0.99 v-APV,
//
v-AP‘,u \
\\\\ ‘

\//...

S'APsu

applied unit load vector, Pm = v-Pvu + s-P“
v + s = l

(b) Vertical and lateral loading.

 

 

applied unit load vector,
Pl}, = v’-P,,.0 + h’-Ph., + s’-P,,,

v’+h’+s’=1

(0) Vertical, longitudinal, and lateral loading.

Figure 4-3 Applied loading combinations for different loading cases of the parametric
study.

 

 

 

 

    
  
 

 

 

 

101
0.2 '
H
LT-l
\
”.4
g 0.1 ~-
Q‘ plastic
analysis
elastic
analysis
\ combined analysis
0.0 ' ' ' ' i - - . - i . . . , j , fii
50 150 250 350

In-plane slenderness ratio, L/ry

Figure 4—4 Variation of ultimate load of two-hinged arch under vertical load (r = 0.99)
using different types of analysis, with in-plane slenderness ratio, L/ry.

Longitudinal (in-plane horizontal) forces on arch structures result mainly
from earthquake loading where dynamic effects generate longitudinal and
vertical inertial forces in the arch. For static analysis, the longitudinal forces
may be assumed to be proportional to the distribution of the structure mass
along the arch horizontal length. This assumption is justified because this
distribution represents the worst case loading by producing unsymmetric failure
mode, which is the first modal shape of the arch. In addition, since the arch is
a relatively simple structures -as compared to multi—story buildings, for
example— this mode is dominant compared to higher modes.

Lateral loading was also assumed to be uniformly distributed along the

arch horizontal length, since wind loads produce this distribution. Also,

 

. (Inert Ill-Ii

 

102

earthquake lateral inertial forces of the same distribution produce the worst
case lateral loading. Similar to vertical load, the unit longitudinal and lateral
loads are expressed in terms of their respective uniformly distributed loads as
P,, = Loph, and P'“. = L-p,,. Notice that the value (i.e., the magnitude) of P,,,
P,,, and P,,, is the same although their directions are different.

When applying a combination of two or all of the three basic loads'to the
arch, for example vertical and longitudinal loading, the applied unit total load,
P,,, is applied as a proportional load with the sum of the scaling factors being
equal to unity so that,

P, =v-Pw+h-P,,

= (l-h)-P,,, + h-P,,,
where (v oPN) is the vertical component of P,,, defined as a proportion of the
unit vertical load, PW, (defined above) using the scaling factor v. Similarly,
(h 'P,,) is the longitudinal component of Pm, defined in terms of\P,, by the
scaling factor h. Also, v + h = 1. For proportional loading, the ratio of the
longitudinal component to the total load is constant throughout the loading
process, i.e., h-Phul P, = h = constant. In other words, the unit total load,
Pm, always has the same value for all values of h and v, since the absolute sum
of its components, v -P,,, and h -P,, is always constant.

Similarly, the unit total load due to combined vertical and lateral loads
is,

P,, = s-P,u + V°Pv,,,

 

 

103

where s is the ratio that defines lateral (or transverse) load. Also, 5 + v = 1.
Three-dimensional loading was used to construct the ultimate load
surface in three-dimensional space using combined analysis. For this purpose it

was more suitable to form the unit total load in terms of the arch combined
ultimate loads under vertical-only, longitudinal-only, or lateral-only loads, i.e.,
P,,,,, P,,,,, and P,,,,, respectively, so that,

P,,' = v"P,,,,, + h"P,,,, + s"P,,,,,
where v', h', and s' are the respective scaling factors corresponding to vertical,
longitudinal, and lateral ultimate loads. In addition, v' + h' + s' = 1. Note that

P,,' does not always have constant value.

L2 IN-PLANE LOAD INTERACTION: INTRODUCTION

The in-plane interaction between vertical and longitudinal loads and its
effect on the arch behavior is presented in the rest of this chapter. The main
objective is to study the arch ultimate load and how it is affected by different
variables, such as the arch slenderness ratio, type of analysis, and the longit-
udinal load ratio. Other aspects of arch behavior, like the interaction curves
between vertical and longitudinal components of the load, failure modes, and
tracing the force path at the quarter point of the arch with the progress of
loading are also studied.

Both types of arch end support conditions, two-hinged and fixed, are
considered. The behavior of two-hinged arch is presented first, followed by the

behavior of fixed arch. To study the effect of longitudinal load on the arch

 

 

104

behavior, the ratio of longitudinal load to total load, h, is varied from 0.0
(vertical load only) to 1.0 (longitudinal load only), with intermediate values of
0.0625, 0.25, 0.50, and 0.75. Values for other variables were discussed in the
previous section.

Finally, the arch ultimate load under unsymmetric vertical loading was
studied and the obtained results were compared with the arch ultimate load
under vertical and longitudinal loads. In particular, the effect on the arch
capacity to carry vertical symmetric load (e. g., dead load) due to the presence
of vertical unsymmetric load or longitudinal load is studied and compared for
the two cases. The vertical load unsymmetry ratio, r, is varied from 0.0 (arch
loaded on half span only) to 0.99 (practically symmetric vertical load), with
intermediate values of 0.33, 0.50, and 0.88.

iii BEHAVIOR OF TWO- HINGED ARCH UNDER VERTICAL

AND LONGITUDINAL LOADING

The ultimate load and behavior of two-hinged arch under different types
of analysis are presented in this section.

4.3.1 Variation of the Ultimate Load of Two-Hinged Arch with
Slenderness Rgtjp

 

The ultimate load of two-hinged arch is obtained for the three types of
analysis, namely plastic analysis where only plastic effects are included, elastic
nonlinear analysis where only elastic nonlinearity (or geometric nonlinearity) is

included, and combined analysis where both plastic and elastic nonlinearities

 

 

 

 

105

are included. The last type represents the lower limit of the first two, while the
AASHTO design formula is based on the elastic analysis. In the following,

discussion of the three types of analysis is presented.

4.3.1.1 Plastic Ultimate Load of Two-Hinged Arch

 

The total plastic ultimate load of parabolic arch under a vertical
symmetric uniformly distributed load, P,,,,, is obtained for reference based on
linear analysis as the load that causes the stress at the arch support to‘reach the

yield stress,

 

This formula applies to both cases of two-hinged and fixed arch, since the arch
fails due to thrust action at the support in both cases.
Figure (4-5.a) shows the change with L/ry of the arch total plastic

ultimate load, P normalized to the vertical-only plastic ultimate load, P,,,,.

1.1»
From this figure it is seen that the presence of longitudinal load generally
reduces the arch ultimate load from its capacity under vertical-only load. The
exception is for low values of h and low values of the slenderness ratio, L/ry,
where it is slightly higher, because the longitudinal component of the ultimate
load has negligible effect on reducing the vertical load carrying capacity of the
very stiff arch.

For a given value of h, the arch normalized ultimate load, I’m/Pm”

decreases as L/ry increases. The relationship is nearly linear (i.e. has constant

 

106

h = 0.0 (Vertical load only)

 

 

000 r "f fir ' 44—. .—
. r I I | r I [ v I I l [

50 150 250 350

 

Slenderness ratio, L/ry

(a) Total load normalized to PVC,

1 = 0.0 (Vertical load only)

1.00 l

0.25

 

P v,p / P vo,p

0.50 t

j 0.50

0.75

 

 

 

 

0.00 ' ‘ I I i r r r r i I r I 1 i
50 ' 150 250 350

Slenderness ratio, L/ry

(b) Vertical component normalized to Pymp.

Figure 4-5 Variation with L/r, of the plastic ultimate load of two-hinged arch under

vertical and longitudinal loading.

 

107

rate of decrease) along the L/ry axis for small values of h (= 0.0625). For
larger values of h, the relationship changes to nonlinear with the rate of
decrease being largest for low values of Ur), then stabilizing to a constant rate
as L/ry increases. In general, the plastic ultimate load becomes more sensitive
to the presence of longitudinal load (i.e., its decrease from P,,, is larger) as
L/ry increases. The ratio of the longitudinal-only to vertical-only ultimate
loads, Pup/Pup, ranges from 60% (for L/ry = 75) to 12% (for L/ry = 375).

The vertical component of the total plastic ultimate load, PM,
normalized to Pm, is shown in Figure (4-5.b), where the trends are similar to
those of the total ultimate load but with less degree of nonlinearity along the
L/ry axis. Since the arch capacity under longitudinal-only load is smaller than
its capacity under vertical-only load, the arch ultimate load becomes controlled
by its longitudinal component as h increases; i.e., the arch reaches its ultimate
load when its longitudinal component (PM) approaches its capacity (P,,,,).
Consequently, as Pup/PW, decreases, the ultimate load and therefore its
vertical component decrease for the same value of h.

4.3.1.2 mstic-Nonlinear—Ultimate Load of Two-Hingd
rch

 

>

 

Figure (4-6.a) shows the arch elastic ultimate load normalized to P,,,,. It
is seen that the relationship is similar for all values of the longitudinal load
ratio, h, and all curves are grouped in a narrow band. However, the nonlinear-

ity of the curves along the L/ry axis increases with the increase of h, with the

 

 

 

 

108

 

 

 

 

 

 

j b — 0.0
1.00 7’ """""""""" \— ------------------------------------------------
e-
,f l
\ ' \ h = 0.0
tical
3‘ _ intermediate values, \ (ver
9* 0'50 h = .0625, .25, .50, .75 \ load “I”
i
‘ h = 1.0 (Longit-
l udinal load only) \ a
0.00 # fl r a— ’fi .— fi . r . fi— . . f—: , L
50 150 250 350
Slenderness ratio, L/ry .
(a) Total load normalized to Pv0.p'
0.50 0% 4%!
o '0
1 00 _ “2K h = 0.0 (Vertical load only)
° l
o i M625 N 0.0625
9' i\
9-1 I \ 0.25
\ .
3: J I 0.50
“* !
0.50 ‘ l 0.75
i h = 1.0 -
. I
l
i I L/ry = 185,
l i/Pm(h=0) = P,,(h=1)
0.001 :t....:.,_
50 150 250 350

Slenderness ratio, L/ry

(b) Total load normalized to PW,

Figure 4-6 Variation with L/ry of the elastic-nonlinear ultimate load of two-hinged arch
under vertical and longitudinal loading.

 

 

 

 

 

 

109
h = 0.0 (Vertical load only)
1.00 if
0. - \\
O
> . M
C“ 0.0625
\
0.
D: .
0.50
h = 0.75
0.00 f— . ' . I9 r r~ . . : fl . . fi : , fi_
50 ’ 150 250 350

Slenderness ratio, L/ry

(0) Vertical component normalized to Pym.

Figure 4_-6 continued.

curves shifting down and to the right at the same time. In other words, the
ultimate load'for longitudinal-only load, P1,”, is higher than that for vertical-
only load, PM” for stiff arches. But as L/iry increases, PM,e decreases by a
faster rate to become lower than Pm. for slender arches. These trends are
shown more clearly in Figure (4-6.b) where the same results are normalized to
the elastic ultimate vertical-only load, Pym. The two curves of h = 0.0 and h
1.0 intersect (i.e., PM, = Pym) at L/ry = 185.

The vertical component of the ultimate load, PM, is shown in Figure
(4-6.c) normalized to Pym. It is seen that for most of the studied range of L/ry,
the rate of reduction of Pm with h is largest for small values of h. Along the

L/ry axis, the reduction (for constant h) is smallest for low L/ry, and becomes

 

 

110

larger as L/ry increases. These trends are similar to those of plastic analysis,
Figure (4-5.b), but the reductions are smaller than those of plastic analysis,
i.e., the ratio of PhMIPW,c is always larger than the corresponding ratio of

PWJPW,p for the same L/ry.

@432 - We LoaLd of Two-Hinged Arch

The arch combined ultimate load (where both plastic and elastic-
nonlinear effects are included) is shown in Figure (4-7.a), normalized to PM,
The combined ultimate load basically represents the lower envelope of the
individual plastic and elastic-nonlinear curves, where it becomes asymptotic to
the elastic curve for high slenderness ratio, L/ry, (L/ry > L/ry)c, = 165, which
is defined as the value of L/ry where plastic and elastic ultimate vertical-only
loads are equal) whereas it approaches the plastic curve for lower L/ry. I

Fig (4-7.b) shows the total combined ultimate load normalized to the
combined ultimate load under vertical-only load, PW, In this figure, the
reduction in combined ultimate load due to the presence of longitudinal load
(for a given h) is largest for the region around the critical slenderness ratio,
L/ry,,,. The formation of this plunge in the curves is due to the transient shift of
the combined curves from following the plastic curves to following the elastic
curves (which are higher, i.e., exhibit less reduction than the correponding

plastic ones for the same values of “h and L/ry) in the region around L/r,)c,.

 

 

 

111

h = 0.0,

 

 

     
   
    

 

 

\/ Elastic analysis
100 ._ ------------ X .....................................................
a. \ h = 0.0
6 (Vertical
a: load only)
\ l
as} .
0.50 ' :
l i
l
l
l
l
g my," = 165
0_00 j fi_r f Fl. . .fi %- . . .fi%. fi—
50 150 250 350

Slenderness ratio,”L/ry
(a) Total load normalized to onp.

 

 

 

 

 

J /— h = 0.0 (Vertical load only)
1.00 -- -----------------------------------------------------------------
0.0625
g
9
a.
\ .
o. 1
9-4
0.50 ~-
l
0.00 ‘ l l jfi

 

50 150 250 350

Slenderness ratio, L/ry

(b) Total load normalized to on’c.

Figure 4-7 Variation with Mr, of the combined ultimate load of two-hinged arch under
vertical and longitudinal loading.

 

 

 

 

 

 

112

 

 

 

 

 

 

 

 

h = 0.0 (Vertical load only)
-_ r——.
1.0 ‘
D4 . ' ——--
\ 0.0625
:- .
9-1 i 0.25
0.5 '-
0.50
\ l1 = 0.75
0.0 . .— . r— l - . . fl : . . fl ,— : , f
50 lSO 250 ‘ 350

Slenderness ratio, L/ry

(c) Vertical component normalized to PW”.

Figure 4—7 continued.

Figure (4-7.c) shows the vertical component of the total combined
ultimate load normalized to PM” which follows the same trends as the total
load, but with less fluctuation. This figure represents the actual reduction in the
arch vertical ultimate load due to presence of longitudinal load. For example,
for h = 0.25 (where longitudinal load = 0.33 of vertical load), the arch
vertical load capacity ranges from 50% to 75% of its capacity under vertical-
only load, Pym, while for h = 0.50 (where longitudinal load = vertical load)
the arch capacity ranges from 25% to 47% of Pym.

To more clearly show the effect of combining both geometric and elastic
nonlinearities on the arch ultimate load, the elastic and combined loads for each

value of h, P,,,(h) and P,,,(h), were normalized to the plastic load for that value,

 

 

 

     
 

 

 

1.00 "
5
q.
a?
3
x.
a: 0.50 '-
____ Elastic analysis 0-25
—— Combined analysis
0.00 - - - . l . . . . l . . . . l r f
50 150 250 350

Slenderness ratio, L/ry

Figure 4-8 Elastic and combined ultimate loads of two-hinged arch under vertical and

longitudinal loading normalized to corresponding plastic ultimate loads of same value

of h.
P,_P(h) in Figure (4-8). From this figure it is seen that in the range where plastic
load governs, the combined ultimate load has the smallest reduction from
plastic ultimate load for h = 0.0 and 1.0, while it is most significant for h =
0.25. The latter ratio is where the direction of the total load resultant on the
left half of the arch span is approximately perpendicular to the chord extending
from the arch support to its crown; such loading case would be the most effec-
tive in producing unsymmetric deflection mode.

For high L/ry the curves follow the elastic behavior where the reduction

is largest for h = 0.0 and smallest for h = 1.0. In addition, as h increases, the

point of intersection of the elastic curves with the plastic curves (represented

 

 

 

114

by the horizontal dashed line) shifts to the right along the L/r, axis, in effect

extending the range of significance for combined effect.

4.3.2 Variation of The Ultimjte Load of Two-Hinged Arch with
the Rjaflo of Longitudinal Load to Total Load

 

The effect of the change in the ratio of longitudinal load to total load, h,
on the arch total load carrying capacity is shown in Figure (4-9). The effect of
the value of h on the "vertical component" of the arch load carrying capacity is
presented later in this chapter when the arch behavior under unsymmetric
vertical load is discussed.

Figure (4-9.a) shows the variation with h of the plastic ultimate load,
PW (normalized to Fwy) for different values of L/ry. For L/ry = 75, the
relation begins with a slight increase in ultimate load to 108% of PM, as h
increases from 0.0 to 0.25, then decreases to a low of 60% of PM, as
h increases from 0.25 to 1.0. For higher L/ry, the protrusion disappears and the
ultimate load decreases with h for its entire range to a minimum of 12% for
h = 1.0 and L/ry = 375. However, the rate of decrease of ultimate load
becomes smaller as L/ry increases.

Figure (4-9.b) shows the variation with h of the elastic ultimate load,
Pm, normalized to PW” for different values of L/ry. For L/ry = 75 , the
ultimate load increases with h for its entire range, with accelerating rate of
increase as h increases to reach 400% of Pm. for h = 1.0. As L/ry increases,

the sharp increase of individual curves with h rapidly changes to a decrease

 

 

Pt,p/on,p

 

115

 

 

 

 

 

 

 

1.0 -
- L/ry = 75
i 125
0 5 T 175
_ 225
' L/ — 375 £275
- r)’ — 325
0-0 l I 1 fir 4% I I r r *I' fig
0.00 0.50 1.00
Horizontal load ratio, h
(a) Plastic analysis.
2.0 ~-
i L/ry = 75
1 125
I75
1-0 } !,————/ /225
/275
- L/ry = 375 \ 325
0.0 l I I I I I I l I I I
0.00 0.50 1.00
Horizontal load ratio, h
(b) Elastic—nonlinear analysis.
1.0 -

 

 

l (L/ry = 75)

  

~_ ~.
. ~‘

0.50
Horizontal load ratio, h
(c) Combined analysis.

Figure 4-9 Variation of the plastic, elastic, and combined ultimate loads of two-hinged
arch with h for different values of L/ry.

 

_ ' 116

over the entire range of h, with the decrease reaching 50% for L/ry = 375. The
sharp increase with h may be related to the ratio of the arch longitudinal to
vertical stiffness, which is higher than unity for low L/ry and less than unity for
high L/ry. For a given value of h, the rate of decrease of the ultimate load
becomes smaller as L/ry increases.

Figure (4-9.c) shows the variation with h of the combined ultimate load,
Pm, normalized to PW, for different values of L/ry, where the curves exhibit the
same phenomenon of being similar to plastic curves for lower L/ry and then
shifting to the elastic curves as L/ry increases. The first three curves decrease
gradually following plastic ones, then the second two (marked by dashed lines)
increase marking the region of transition. Finally, the last two curves (marked
by dash-dotted lines) decrease again similar to elastic curves. It is noteworthy
that most of the curves are banded closely together; for h = 0.5 (where vertical
and longitudinal loads are equal), five of the seven curves are enclosed by the

limits of 65% and 55%.

_—

4.3.3 Ultimate 1M Interaction Curves of Two-Hinged Arch

Figure (4-10) shows three sets of interaction curves between the vertical
and longitudinal components of the total load for different values of slenderness
ratio (L/ry = 125, 225, and 325). For each set the vertical and longitudinal
components of the ultimate load from plastic, elastic, and combined analyses,
P,“x and PM, are normalized first to the plastic vertical-only and longitudinal-

only ultimate loads, Pm, and PM” respectively (figures on left), then to their

 

 

    
     
     

 

 

 

 

 

 

 

 

 

 

 

 

' 117
:k h 25 1.00 -
n.- l.50 --\ ‘0‘ _1
o .
a: \ I’ A;
\ \
:- i \ '1
0.. . I > X =' c
1.00 _ l \ ‘ = c ‘3‘ \ ‘.< (combined)
. 'x=p\ It 0,50» \ ./‘
‘ .’ (plastic) \ (° 1“ ‘9 \,/ ~
‘ \ \ ‘3 \
. ‘ \ 33/ \
050 -- ' x = c \\ - 4/‘/ —\
‘ . (combined) \ . ‘_°/' ", ”f \
‘ .l \ \ '/ (c ast c)
‘l \ . ./
(I \\ /'/
0.00 vaevrsw 0.00 ++ a
0.00 4.00 8.00 0-00 0.50 1.00
Pun: / Pho,p Phat / Phoat
(a) stiff arch, L/ry = 125. h =o,25
1.00 l
q.
0
ti:
\
u.
>
n.
0.50
0.00 . .
0.50 LOO
Ph.x / Pho,p Ph.x / Pho,x
(b) medium arch, L/ry = 225.
1.00
a.»
O
K.
\
x.
>
a.
0.50
0.00

 

 

Pin: I Ph0,x

Pt»: I Pho,p
(c) slender arch, I_./ry = 325.

Figure 4-10 Interaction curves between vertical and longitudinal loading of two-hinged
arch for different values of L/ry.

 

 

118

corresponding vertical-only and longitudinal-only ultimate loads, Pm, and PM.,(
(figure on right). The suffix "x" represents type of analysis where x = p for
plastic (for example, Pw/on‘p), e for elastic, or c for combined.

The figures on left show the relative values of elastic and combined
ultimate loads compared to plastic ultimate loads for different values of L/ry.
For a given value of L/ry, of the ratio of elastic ultimate load to plastic one for
vertical-only load, PVM/PVM, is always smaller than the corresponding ratio for
longitudinal-only loading, Pkg/Pb”. This is also obvious in Figure (4-8). The
effect of this difference on the interaction curves for combined analysis (which
is basically a lower envelope for the other two curves) is evident for L/ry =
225 where elastic behavior controls the vertical-only loading while plastic
behavior controls longitudinal-only loading, resulting in a significant reduction
in the combined ultimate load for intermediate values of h compared to either
plastic or elastic ones.

The figures on right show that the shape of the interaction curve for
plastic and elastic analyses are essentially independent of L/r,: convex for
plastic analysis and concave for elastic analysis. Nonetheless, as L/ry increases,
the location on the interaction curve of the point corresponding to a given value
of h shifts from the vertical axis toward the horizontal axis. For example, for
h = 0.25, the value of Pym/Pmp decreases while PIMP/Pm,p increases with L/ry, as
shown in the figure.This is attributed to the decrease in the ratio of Pump/Pm,p as

L/ry increases. This behavior is also true for elastic and combined analysis. For

 

 

 

119

combined analysis, the interaction curves shift from convex form for low L/ry
values, where it is closer to that of plastic analysis, to concave form -similar to

the elastic curve- for higher values of L/ry.

4.3.4 Failure Modes and Response Curves of Two-Hinged Arch

The deflected shape of two-hinged arch at the ultimate load (or failure
modes) for different types of analysis and values of h are shown in Figure
(4-11) for L/ry = 125, and in Figure (4-12) for L/ry = 225. The deflection is
magnified by a factor of 5 for plastic and combined loading for clarity.

In both figures, the apparent symmetric (or almost symmetric) failure
modes of plastic curves under vertical loading (h = 0.0) indicate the arch
failure is due to thrust action at the support. This is confirmed by the curve
discontinuity near the left arch support due to plastic hinge formation. The only
other symmetric failure mode is that of stiff arch for combined analysis under
vertical loading, where geometric effects are negligible. For h > 0.0, the
failure modes are unsymmetric and smooth for all other cases. For elastic
curves, the deflection magnitudes show a clear gradual increase with h, but for
other types of analysis the curves are inseparable and overlapped. It is
important to mention here that for all studied cases, the apparent symmetric
failure modes at the ultimate load change to unsymmetric modes when the
analysis is continued past the ultimate load point (using displacement-controlled

analysis, as explained in chapter 3).

 

 

 

. 120

 

Deflection Magnification
Factor, DMF = 5

(a) Deflected shape under plastic analysis.

h = 0.50, 1.0
._— . , (overlapped)

  

 

DIVIF=5

(b) Deflected shape under combined analysis.

_.—-—~_—.—-_I—n

  
 

h = 0.25

  

h = 0.50

DMF=l

(c) Deflected shape under elastic analysis.

Figure 4-11 Deflected shapes of two-hinged stiff arch (L/ry = 125) at ultimate load for
different types of analysis and values of h.

 

 

 

 

 

 

 

121

h = 0.5, 1.0
(overlapped)

    

 

DMF=5

(a) Deflected shape under plastic analysis.

h = 0.25, 0.50, 1.00

/ (overlapped)

 

DMF = 5
(b) Deflected shape under combined analysis.

 

 

 

   

DMF=I

 

(c) Deflected shape under elastic analysis.

Figure 4—12 Deflected shapes of two—hinged medium arch (L/ry = 225) at ultimate load
for different types of analysis and values of h.

     

 

 

 

122

Comparison of the two figures reveals that the deflection at ultimate load
increases modestly with L/ry for plastic and combined analysis, where the arch
flexibility more than compensates for the lower ultimate load. For elastic
analysis, it decreases notably, since the ultimate load decreases more rapidly.
Within each figure, the deflection at ultimate load for elastic analysis is larger
than that of plastic or combined analysis, since geometric effects tend to
propagate deflections for elastic analysis while plastic hinge formation. forces
premature failure for the latter two cases.

Figure (4—13) shows the vertical and longitudinal displacement response
curves at the quarter point of a medium arch (L/ry = 225) for different values
of h, under combined analysis where it shows the influence of plastic and
geometric effects on displacement. As the arch is loaded, the response curves
for combined and elastic analysis are initially the same. If the plastic ultimate
load is significantly larger than the elastic one, the elastic and combined curves
follow the same path, as shown in Figures (4—l3.a). But if the plastic ultimate
load is close to or less than the elastic ultimate load, yielding occurs in the arch
(at some joint). This causes the combined curve to separate from the elastic
curve and follow a path similar or lower than that of plastic analysis, and reach
its combined ultimate load prematurely, Figure (4—13.b and c).

From the figure, the initial slope of the curves decreases drastically with
h; for example the initial slope under longitudinal-only load is about 5% that

for vertical-only load. The ratio of maximum vertical to longitudinal

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

123
30 .1- PD k.
plastic plastic
15 l"
elastic and combined elastic and combined
(overlapped) (overlapped)
horizontal displacement vertical displacement
-3 4.5 0 0 1.5
Displacement at quarter point, (x 10'3-L)
(a) Response under vertical load (h 7-== 0.0).
plastic P" k‘ plastic
elastic 10 ‘ elastic
bi 61
com ne \ combined
5 -
horizontal displacement i vertical displacement
-16 -8 ° 0 8
Displacement at quarter point, (x 10%-L)
(b) Response under vertical and horizontal load (h = 0.5).
: P,, k.
10 elastic
elastic
plastic __ plastic
combined 5 . \ combined
horizontal displacement vertical displacement
_20 -10 ° 0

IO
Displacement at quarter point, (x 10'3-1.)

(c) response under horizontal load (h = 1.0).

Figure 4-13 Displacement response curves at quarter point of medium two-hingedarch
(Ur, = 225) for different values of h. ‘

 

 

 

 

 

 

 

124

components of the plastic displacement is essentially the same for values of

h > 0.0 (about 1.6) but is much larger for h = 0.0 (about 3.3). The corres-
ponding ratios for elastic and combined displacements remain the same for all
values of h (about 2.3).

4.3.5 Plastic Hinge Formation and Force Path at Quarter Point
of Two-Hinged arch

Table (4—1) shows the order of plastic hinge formation with the progress
of loading in two—hinged arch (L/ry = 225) for different values of h due to
plastic and combined analysis. Figure (4—14) shows the force path at the left
quarter point (node #5, at the right end of member #4, see Figure (4—1)), where
the relation between axial force and bending moment is traced until failure.

For h = 0.0, the plastic force path follows a path that is close to the
N-axis, suggesting axial load—dominated behavior. In addition, the development
of plastic hinge near the left support causes moment release and further shift
toward the P—axis. On the other hand, the combined curve exhibits a significant
divergence from the plastic one and a shift toward the M—axis with increase of
loading, demonstrating the effect of elastic nonlinearity which generates
moment in the arch. Failure under combined analysis occurs without formation
of any plastic hinge in the arch.

With the increase of h from 0.0 to 0.25, the force path for plastic
analysis shifts notably toward the M—axis and the axial force at failure

decreases. Also, as h increases the position of the plastic hinge formation

 

 

 

. 125

Table 4-1 Plastic hinge formation in hinged arch for different types of analysis under
different load combinations.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Plastic Analysis Combined Analysis
h
Member: P at Yield Member: P at Yield
Joint Yield Status' Joint yield Status
PM, = 26.11 PM = 13.39
0.00
1:2 26.00 Y none - -
P”, = 24.59 PLc = 12.00
0.0625
15:1 24.40 Y 4:2 12.00 Y
3:2 24.58 Y
P”, = 16.74 PL, = 9.42
0.25
14:1 16.01 Y 14:1 9.42 Y
4:2 16.74 Y
5:1 16.74 Y
PLp = 10.26 Pm = 7.32
0.50
14:1 10.00 Y 14:1 7.32 Y
4:1 10.26 Y
P“, = 7.09 P... = 5.96
0.75
14:1 7.00 Y 14:1 5.96 Y
3:2 7.08 Y
4:1 7.08 Y
P”, = 5.35 P1,: = 5.01
1'00 3:2 5.34 Y 14:1 4.81 Y
14:1 5.34 Y 13:2 4.82 Y

 

 

 

 

 

 

 

* Y = joint yields, U = joint unloads.

moves along the arch axis from the left support toward the crown until it
reaches the quarter point for h = 0.25. In addition, another hinge forms near
the right quarter point briefly before the left hinge formation. For combined
analysis, the divergence of the force path from that of plastic analysis is less

significant since moment is generated in the arch in both cases. Only one

 

126

 

 

N/Np
1—
.! ------- Plastic
_ — Combined
g f h = 0.0
2 .'
Q .'
E :
u ': h = 0.0625 .—
:' ,x”\ h = 0.25
,-' x"/ h = 0.50
Negative Moment —> MY/MYm
O L l l l
0 h = 1.0 1
h = 0.75

Figure 4-14 Relation between axial force and bending moment at the left quarter point
of hinged-arch (L/ry = 225).
plastic hinge forms in the arch at the ultimate load causing instability of the

arch; this hinge forms at the left quarter'point for h = 0.0625 while it jumps to

the right side for h = 0.25.

 

127

As h increases beyond 0.25, the trend of decreasing axial force continues
where it gradually changes from compression to tension as h increases to 1.0.
Since the magnitude of axial force is small in this range, the difference between
plastic and combined curves becomes negligible. The position of the plastic
hinge that forms in the right half remains unchanged for both plastic and
combined cases, while that forming in the left, which forms during plastic
analysis only, slightly shifts back toward the support.
M BEHAVIOR OF FIXED ARCH UNDER VERTICAL AND
LONGITUDINAL LOADING
The behavior of fixed arch was studied using the same arch geometry,
section properties, and loading combinations. The effect of fixing the arch
supports increases the arch ultimate load, although the observed trends are
similar to those of two—hinged arch. In this section, the obtained results are
presented, with emphasis on the differences from those of two-hinged arch.

4.4.1 Variation of the Ultimate Load of Fixed Arch with
Slenderness Ratio

The plastic, elastic-nonlinear, and combined ultimate loads of the fixed
arch were obtained for the same values of L/ry and h used for two—hinged arch.

The results are presented in the following subsections.

4.4.1.1 Plastic Ultimate Load of Fixed Arch

 

Figure (4-15.a) shows the variation with L/ry of the plastic ultimate load

of fixed arch, F normalized to PW”, (the vertical-only ultimate load of two—

t,p:

 

 

128

1 g h = 0.0
(Vertical load only)

 

 

 

 

 

 

 

 

 

 

1.00 " \
n; - . 0.0625
8
D” _
a: _
I1: ..
0.50 "'
O.(x) I I l l I l I l I I I li‘r I fi
50 150 250 350
Slenderness ratio, L/ry
(a) Total load normalized to on.p (Pm,p = Fvo,p)-
- = 0.0 (Vertical load only)
1.00 " m
% _
m? _ 0.0625
\
D: _
1.1.> - 0.25
0.50 '
0.5
\\\\\\\\~UT‘~——jL92:_____________—__—___—————____
O_ml lil l l ##f If! I‘fl I 1+1 I
50 150 250 350

Slenderness ratio, L/ry

(b) Vertical component normalized to Fvo'p.

Figure 4—15 Variation with L/ry of the plastic ultimate load of fixed arch under combined
vertical and longitudinal loading.

 

129

hinged arch). From this figure, it is seen that the vertical—only ultimate load of
fixed arch is practically equal to that of two-hinged arch, i.e., Fm, z va. For
other curves of h- > 0.0, the general trend is similar to that of two—hinged arch,
Figure (4-5.a), with comparable degree of nonlinearity. However, the total
ultimate load of fixed arch for given values of h and L/ry is always larger than
the corresponding total ultimate load of two—hinged arch. Figure (4—15.b) shows
the vertical component of the ultimate load, Fm, normalized to Fun,» where the
same trends of total load are also true.

Table (4-2) presents the increase in the ultimate loads due to fixing the
arch supports in percentage form for selected values of h and L/ry for plastic,
elastic, and combined analysis. The values in the table also apply to the
comparison of the vertical components of the ultimate loads of‘fixed and two—
hinged arches since the loading is proportional and the scaling factors are the
same in both cases for each entry. From the table it is obvious that for constant
values of L/ry, the increase in plastic ultimate load due to fixing the arch
supports becomes larger as h increases. For stiff arches this increase is gradual
over the range of h, while for slender arches it is unevenly concentrated in the
range from h = 0.0 to 0.5.

Viewing the same data for constant values of h, the gain in plastic
ultimate load is largest for longitudinal-only loading, with the gain being
constant with L/ry for h = 0.0 (with no gain from the two-hinged case), and

h = 1.0 (where the gain ranges from 49% to 52%). For h = 0.25 and 0.50, the

 

 

130

Table 4-2 Relative Increase of fixed arch ultimate load in comparison with two—hinged
arch for different types of analysis and values of longitudinal load ratio, h.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Type of FLx(h)/P,1x(h), %
analySiS h L/r, = 125 L/r, = 225 L/ry = 325
0.00 0.00 0.19 0.56
Plastic 0.25 16.33 27.78 35.00
0.50 37.35 45.42 48.49
1.00 49.22 51.29 51.85
0.00 113.73 117.78 119.76
Elastic 0.25 137.56 154.31 163.32
nonlinear 0.50 161.83 187.86 201.31
1.00 561.84 317.46 304.69
0.00 0.78 76.62 120.71
combined 0.25 26.72 47.56 95.56
0.50 41.40 50.27 68.24
1.00 50.23 58.43 74.39

 

 

 

gain gradually increases with L/ry between the limits of the other above-

mentioned two cases.

4.4.1.2 Elastic-Nonlinear Ultimate Load of Fixed Arch

 

The elastic-nonlinear ultimate load of fixed arch, Fm is shown in Figure

(4-16.a) normalized to P

vo,pa

and Figure (4—16.b and c) normalized to PMC and

FR”, respectively. It is seen from the first figure that the main effect of fixing

the arch support is to significantly shift the critical slenderness ratio, L/rm, to

 

 

   
  

 

 

 

  
 

131

’21 _ \ h = .0,.0625

g- ‘\ (overlapped)

% 4.00 -- \

H, \

0.. ‘ ‘,

H \

O ‘ \

V \

a: - \

O

>

fie 2.00 -- h = 0.0 /\\

[If (two—hinged

' arch)
0.00 I I I I I I I I I I l I I l I __;
50 150 250 350
Slenderness ratio, L/ry
(a) Total load normalized to Pym.
4.00 --

Q _

S
0.. -
\ —

o h = 0.0 (Vertical
u: _ load only)

 

 

 

50 150 250 350

Slenderness ratio, L/ry

(b) Total load normalized to on’c.

Figure 4—16 Variation with L/ry of the elastic ultimate load of fixed arch under vertical
and longitudinal loading.

 

 

 

 

 

FV.C / FVO,6

1:t.e / Fvo,e

 

132

 
  
  

 

 

 

 

 

       
 

 

' = 1.00
0.5 075 h = 0.0
' ( tl al
0.25 .31 2.1.11
1.00 " \ \
_ h = 0.0625 \
________ h = 0.0 (two-hinged arch)
0.00 ‘l’ I I I I l I l I I I l l I F
50 150 250 350
Slenderness ratio, L/ry
(c) Total load normalized to Fvofi.
_ n = 0.0 (Vertical load only)
1'00 _ 0.0625
' 0.25
' 0.50
0.50 ‘7 ~~ — ~— __ _ -11: PP. (_“1’9' _________________________
_ hinged arch)
0.00 I l 1 Ir 1 I *1 I % l l l I ‘IWF
50 150 250 350

Slenderness ratio, L/ry

(d) Vertical component normalized to Fvofi.

Figure (4-16) continued.

 

 

 

133

the right (where Fm, = F",c at L/r = 240, compared with L/r = 165 for

it)" I)"

two-hinged arch), extending the range where plastic behavior dominates. Also,
the nonlinearity of the curves increases with h, with the difference between the
curves becoming more significant as L/ry decreases.

The second and third figures show that for the studied range of L/ry, the
lower envelope for the curves of total elastic ultimate load exhibits a gain of at
least 80% (for h = 0.50 and L/ry = 375) compared to PM”. For h = 0.0, the
/P

ratio of F is roughly constant with the gain in ultimate load about 114%

V0.6 VO,C

to 120%. As h increases, the curves shift mainly to the right along the L/ry
axis, with smaller shift down. The curves of h = 0.0 and h = 1.0. intersect
(i.e., Fhm = FVM) at L/ry = 310; in comparison, for two-hinged arch, PM. =
PM, at L/ry = 185.

The decrease in the vertical component of the elastic-nonlinear ultimate
load, Fm, Figure (4-16.d), exhibits similar trend to that of two-hinged arch,
but again the rate of decrease with both h and L/ry is smaller. For example,
vertical load component of fixed arch for h = 0.5 (where longitudinal load is
equal to the vertical load) is larger than the vertical-only ultimate load of two-
hinged arch, on,e2 for most of the L/ry range.

Table (4—2) shows that for constant values of h, the gain in elastic-
nonlinear ultimate load of fixed arch compared to corresponding two—hinged
case is at least 114% (for h = 0.0 and Mr, = 125), and reaches up to 560%

(for h = 1.0 and L/ry = 125). This drastic gain is attributed to the increased

 

 

 

134

nonlinearity of the curves of fixed arch with the increase of h, coupled with the
shift of all the curves of fixed arch along L/ry axis compared with those two-
hinged curves. Because of the combination of these two factors, the gain in
ultimate load with L/ry is not uniform, where it is constant for h = 0.0, and

increases for h = 0.25 and 0.50, but decreases for h = 1.0.

4.4.1.3 Combined Ultimate Load of Fixed Arch

 

The combined ultimate load of fixed arch is shown in Figure (4-17.a and
b) normalized to PVM, and PW» respectively. From the first figure, the
expansion of the plastic range in comparison with two-hinged arch is obvious.
Otherwise, the figure exhibits the same aspects of behavior as those of two-
hinged arch, but with smaller decrease of ultimate load for the same value of h
and L/ry.

The second figure shows the effect of fixing the arch supports on its
ultimate load as compared to two—hinged arch, Figure (4-7.b). Similar to plastic
analysis, the gain is modest for low L/ry, while it is significant for high L/ry,
coinciding with elastic analysis. The region of transition from the plastic- to
elastic—dominated regions may be roughly defined by the two values of the
critical slenderness ratio for two—hinged and fixed arch, L/ry)cr = 165 and 240.

Figure (4—17.c and (1) show the total and vertical component of the
combined ultimate loads, respectively, normalized to Fuc- These figures reflect
the change in the ultimate load and its vertical component of fixed arch due to

the presence of longitudinal load, where they show similar behavior to that of

 

 

135

 

 

    
    
 

 

 

 

 

A \\ h = 0.0, Elastic analysis
- k \/
1'00 h = 0.0, Plastic
q. ' analysis
9 1
CL:
\ ..
0.
m. _-
0.50 "
l
L/rym = 240
000 l I I *li I‘I I I F I n I‘li l I fl fi—
50 150 250 350
Slenderness ratio, L/ry
(a) Total load normalized to on.p-
2.00 '~
q —
9 -
D“ h = 0.0
\ ‘ (two—hinged
2. arch)
“-4
1.00 '—
1
0,00 I I I I I I I I I I I l . I I l 1—

 

 

50 150 250 350
Slenderness ratio, L/ry

(b) Total load normalized to on'c.

Figure 4—17 Variation with L/ry of the combined ultimate load of fixed arch under
vertical and longitudinal loading.

 

 

 

 

Ft.c / F vo.c

0.00

Fv.c / Fvo.c

0.00

136

 

 

 

 

 

  
 
  

 

50 150 250 350

Slenderness ratio, L/ry

(c) Total load normalized to FVM.
h =0.0 (vertical load only)
_ \ 0.0625 .
\\
_ \
\\ I1 = 0.0

l 0.25 \\/ (two-hinged)

 

50

 

‘\___

Slenderness ratio, L/ry

(d) Vertical component normalized to Fvofi.

Figure (4-17) continued.

 

   

 

 

 

 

 

 

 

137
__ \ \
2.00 \l\\\ \ \ \

- \ \ \ h = \
_ \\ \ \ 1.00 \

\ \ \ \ ‘
_ \ \
1 \\ \\ \ 0.75\\

= \\ \ \

3 “ § \ 0.25 \ \
LL. _ \ \ \ \\
\ \\ \

- 0.0625 \\ \ \
a \ \
V _
K.
in“? 1.00 ‘7
1
0.50 1
l _ _ _ _ Elastic
_ __ Combined
0_00 *I—I I I I r‘r I I I I I I I I I I
50 150 250 350

Slen derness ratio, L/ry

Figure 4-18 Elastic and combined ultimate loads of fixed arch under vertical and
longitudinal load normalized to corresponding plastic loads of the same value of h.

two-hinged arch, but with slower decrease with h and the middle plunge (due to
transition from plastic- to elastic-dominated behavior) being shifted to the
right.

Figure (4—18), shows the combined and elastic ultimate loads for a given
h normalized to the plastic ultimate load of the same h. Similar to two-hinged
case, it shows the most decrease in ultimate load -due to combining plastic and
geometric effects— is for h = 0.25 and 0.5. Measuring the decrease for a given

value of h as that at the point when F,c = F,_,,, it is seen that the decrease in the

 

 

138

case of fixed arch is slightly larger than that of two-hinged arch; for example,
the respective values for h = 0.25 are 40% and 30%.

The gain in combined ultimate load of fixed arch compared with that of
two-hinged arch is shown in Table (4—2). It is seen that for constant h the gain
in ultimate load always increases with L/ry, although the rate of this gain levels
out as h increases. This is attributed to the change from plastic- to elastic—
dominated behavior, which occurs at different stages for fixed and two-hinged
cases (i.e., L/ry,c, is different for fixed and two—hinged arches). Also, for each
case this change is not synchronized with L/ry (i.e., the value of L/ry where
F,',(h) = F,,,,(h) is not the same for all values of h, as clearly seen from Figure
(4-18)).

The combination of these two factors produces the fluctuating values of
the gain in ultimate load with h for constant L/ry. For example, the gain in
ultimate load for the same value of L/ry = 325 changes from 121% for h = 0.0
(where it is equal to that of elastic analysis), to 74% for h = 1.0 (which is
closer to that of plastic analysis). For h = 0.25, the gain is 95%, which is in
the middle between the corresponding plastic (35%) and elastic (163%) values.
For h = 0.5, the gain is 68%, which is the lowest value for L/ry = 325.

(Liz Variation of The Ultimate Load of Fixed Arch with the
Ratio of Longitudinal to Total Load
The variation with h of the plastic ultimate load of fixed arch is shown in

Figure (4-19.a) normalized to Film” where it is similar to that of two-hinged

 

 

 

 

139

 

 

 

 

 

 

  
   

 

 

L/ = 75
a 1.2 ’v
6
p
E 125
‘i‘
I-I-t 0.6 175
225
i 275
L/ry = 375 \ 325
0.0 ‘I 17 *I I I I ‘Ii I I k I
0.00 0.50 1.00
Horizontal load ratio, h
(a) Plastic analysis.
4.0 "
q _
LE 4
$0. _ L/ry = 75 125
.. 175
Ila 2.0 — 225
_i /
/ 275
— - ‘—=—* \ 325
‘1 L/ry = 375
0.0 l I 1 IT I I I fl I 1
0.00 0.50 1.00
Horizontal load ratio, h
(b) Elastic—nonlinear analysis.
1.2 —- =
o l (L/ry 75)
6 _
>
E _ 2 (125)
Q.
Li.” 0.6 -~ 7
_-:::::::::::::: ______ /(375)
i;
4 (225) \ 6
5 (275) (325)
0-0 I T l I I I r I I a
0.00 0.50 1.00

Horizontal load ratio, h

(0) Combined analysis.

Figure 4-19 Variation of the plastic, elastic, and combined ultimate loads of fixed arch
with h for different values of L/ry.

 

 

 

 

 

140

arch, but with slower decrease with h for all curves. In particular, for Ur, =
75 the increase with h of the normalized ultimate load continues from 0.0 to
0.50 where it peaks at 120% then decreases to 60% for h = 1.0. The lowest
ultimate load ratio of 20% occurs for h = 1.0 and L/ry = 375.

The elastic ultimate loads of fixed arch are shown in Figure (4-19.b)
normalized to Fm, The variation with h is similar to that of two-hinged arch,
but with slight shift up for all curves, where the ultimate load reaches a

minimum of 50% of Fm, for h = 1.0 and L/ry = 375.

 

The variation of combined ultimate load of fixed arch with h normalized
to Fm” Figure (4-19.c), shows a slight partial increase with h for L/ry = 75
-similar to plastic curves~ while it decreases with h for other curves of L/ry.
The rate of decrease gradually becomes smaller with L/ry until the ultimate load
bottoms out around L/ry = 275 and begins to increase again for the rest of the
range, indicating the shift from plastic to elastic behavior. The last stage of
decreasing again to coincide with elastic curves -exhibited by two—hinged arch—
is not present here, since it occurs outside the studied range of L/ry. Compared

to Figure (4-9), the figure also shows similar but wider banding of the curves.

 

fl; Ultimzfie Load IntLraLtion Curves of Fixed Arch

The general trends of the interaction curves for fixed arch, Figure
(4-20), are similar to those of two-hinged arch, where the plastic interaction
curves are convex, elastic curves are concave with higher nonlinearity for low

L/ry, and combined curves change with L/ry from the plastic to elastic ones.

 

 

 

 

141

 

 

 

 

 

 

 

 

  

     

 

 

4.00 r
a. I LIX)
or ‘1 .,
i. \ u:
a? \ \
\ a?
2.00 “ \\ 050
\
\
\
\
\
\
\
\
0.00 . . f4. . 0,00 . .
0.00 20.00 40.00 0.00 0.50 1.00
Plus I Fho.p Fh,x / Fho,x
(a) stiff arch, L/ry = 125.
1.00 l
1.00 K
a. a?
9 \
a. Is
N. X = e u“
ta? \(clast) 0.50 l- x = c \
0'50 \ - (combined) \\\\ \
x 3 P \ ‘
(plastic) \ x = e/\ \‘\
\ \\
x = c \ (elastic) \
“ (combined) \
\ L
0.00 - . 4 f . A. .A._: 0.00 . r .
0.00 3.00 1100 0.00 0.50 1.00
me / 13110.9 FILX I Fhom

(b) medium arch, L/ry = 225.

 

  

 

 

L I 9‘
l—‘U’ fér—Ii IfiiT’I ——T—'

1.00 2.00
Fh.x / Fho,p

0.00

 

(c) slender arch, L/ry = 325.

Figure 4-20 Interaction curve between vertical and longitudinal loading of fixed arch
for different values of L/ry.

 

 

 

 

 

142

For the first case of L/ry = 125, Figure (4-20.a), the curves to the right
in the figure (normalized to F,” and Fm?) show that the elastic ultimate load
for vertical-only loading is about 3.5 times its plastic ultimate load (compared
to 1.6 for hinged arch). For longitudinal-only loading the corresponding ratio is
36 (compared to 8 for hinged arch), with ten fold increase from that of vertical-
only loading.

The three figures on left show that plastic behavior controls the

combined behavior for h = 1.0 for the whole range of L/ry. For h = 0.0,

 

plastic behavior controls only the first case (L/ry = 125), while elastic behavior
controls the last case (L/ry = 325), with the intermediate case, (L/ry = 225),
influenced by both. As a result, the largest effect of combining plastic and

elastic effects for fixed arch is for the case of L/ry = 325.

w Mlure Modes and Response Curves of Fixed Arch

The deflected shape of fixed arch is shown in Figure (4—21) for L/ry =
125 (stiff arch), and in Figure (4-22) for L/ry = 225 (medium arch), for
different types of analysis and values of h. Fixing the arch supports results in
forcing symmetric failure mode (at the ultimate load) for h = 0.0 for both
plastic and combined analysis in both figures, and in preventing the large
rotation at the plastic hinges near the support. On the other hand, the plastic
curve for h = 0.25 and L/ry = 125 clearly exhibits bending-dominated

behavior, where large plastic deformations occur at the plastic hinges formed in

the two quarter—point regions before the arch reaches its ultimate load. This

 

 

 

143

  

h = 0.25

 

Deflection Magnification
Factor, DMF = 5

(a) Deflected shape under plastic analysis.

" .

 
  

    

Curves from top to bottom,
h = 0.0, 0.25, 1.00, and 0.50.

 

DMF=5

(b) Deflected shape under combined analysis.

 

    

h=0.50
DMF= 1

(c) Deflected shape under elastic analysis.

Figure 4-21 Deflected shapes of fixed stiff arch (L/ry = 125) at maximum load for
different types of analysis and values of h.

 

 

 

 

DMF=S

(a) Deflected shape under plastic analysis.

 

DNIF=5

(b) Deflected shape under combined analysis.

 

DMF=1

(c) Deflected shape under elastic analysis.

Figure 4-22 Deflected shapes of fixed medium arch (L/ry = 225) at maximum load for
different types of analysis and values of h.

 

 

145

 

elastic

combined

horizontal displacement

elastic

     
   

\ combined

vertical displ.

 

 

 

 

 

 

-3 -2 -1 00 1
Displacement at quarter point, (x 108-L)
(a) response under vertical load (h = 0.0).
30 T F,, k.
elastic I elastic
plastic 15 plastic
combined combined
horizontal displacement vertical displ.
-24 -16 -8 0 0 8
Displacement at quarter point, (x 10'3-L)
(b) response under vertical and horizontal load (h = 0.5).
I Fla k‘
40 - elastic
elastic -
lastic and
2 P
O — combined
plastic and combined _ (overlapped)
(overlapped)
horizontal displacement ‘ vertical displ.
I I I I I T I I I fi l 'I I I l 0 I 1* l I I
—30 -20 -l0 0 10

Displacement at quarter point, (x 10'3-L)

(c) response under horizontal load (h = 1.0).

 

Figure 4—23 Response curves of fixed medium arch (L/ry = 225) for different types of
analysis and values of h.

 

 

 

146

demonstrates the redundancy of fixed arch.

All other failure modes for all types of analysis are unsymmetric at
ultimate load, with essentially similar trends to those of two-hinged arch;
namely elastic deflections are larger than deflections of other types for same
L/ry, and deflection magnitude of plastic and combined curves increases and
that of elastic ones decrease with L/ry. However, the elastic deflection of fixed
stiff arch is much larger than that of two-hinged arch, due to the large increase
in ultimate load. Also, the plastic and combined curves for different values of h
within each group are more separated (i.e., increase gradually with h) than the
corresponding curves of two-hinged arch; this is particularly true for combined
curves of medium arch.

The longitudinal and vertical displacement response curves of fixed arch
(L/ry = 225) are shown in Figure (4-23), where they are comparable to those
of two—hinged arch with some differences. For example, the decrease in the
initial slope of the curves with h is similar to that of hinged arch but is slower.
Also, the ratio of maximum vertical to longitudinal displacements at quarter
point exhibit similar trends to those of hinged arch as they change with h, but
the ratios are higher.
w Plastic Hinge Formation and Force Path at Quarter Point
of Fixed Arch
Table (4-3) shows the order of plastic hinge formation in fixed arch

(L/ry = 225) for different values of h due to both plastic and combined

 

 

 

Table 4-3 Plastic hinge formation in fixed arch for different types of analysis under

different load combinations.

147

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Plastic Analysis Combined Analysis
h
Member: F at Yield Member: F at Yield
Joint Yield Action' Joint yield Action
F”, = 26.20 Fm = 23.65
0.00
1:1 23.10 Y 1:1 23.65 Y
16:2 24.30 Y
1:2 26.15 Y
16:2 26.15 U
F”, = 25.97 F,_c = 19.51
0.0625
1:1 21.66 Y 1:1 19.50 Y
16:2 24.40 Y
15:1 25.80 Y
4:2 25.97 Y
F”, = 21.39 FL. = 13.90
0.25
1:1 16.00 Y 1:1 13.35 Y
16:2 16.42 Y 16:2 13.64 Y
13:1 20.80 Y
14:1 21.36 Y
5:1 21.38 Y
13:1 21.39 U
F”, = 14.92 FLc = 11.00
0.50
121 10.01 Y 1:1 9.20 Y
16:2 10.03 Y 6:2 9.21 Y
13:1 14.41 Y 3:1 11.00 Y
5:1 14.92 Y
0.75 PM, = 10.67 F,’c = 9.22
1:1 6.84 Y 1:1 6.64 Y
16:2 6.85 Y 16:2 6.80 Y
13:1 10.42 Y 13:1 9.22 Y
4:2 10.66 Y
5:1 10.66 Y
1.00 FLp = 8.09 FLC = 7.95
1:1 5.21 Y 1:1} 5.21 Y
16:2 5.21 Y 16:2 5.22 Y
4:2 8.08 Y 13:1 7.80 Y
13:1 8.08 Y 12:2 7.83 Y
12:2 7.88 U
4:2 7.97 Y

 

* Y = joint yields, U = joint unloads.

 

 

 

 

 

 

148

analysis, and Figure (4~14) shows the force path at its left quarter point. From
the table, for all cases of plastic analysis the first and second plastic hinges
form at the left and right supports, respectively, effectively transforming the
fixed arch into a hinged one. Afterwards, inner plastic hinges form in a fashion
very similar to that of hinged arch, where a plastic hinge forms near the left
support for h = 0.0. Also, its position shifts toward the crown as h increases to
reach the quarter point at h = 0.25, then remain in the same region as h
increase to 1.0. On the right side, plastic hinges form near the right quarter
point for h = 0.0625 to 1.0.

For combined analysis, the same observations are essentially true, but
the arch may fail due to elastic nonlinearity before development of all the
plastic hinges. For h = 0.0, only one plastic hinge forms in the arch before
failure, but as h increases, more hinges develop in the arch.

Figure (4-24) shows the force path at the quarter-point of fixed arch
(L/ry = 225). Comparison of the figure with that of hinged arch, Figure (4-14),
reveals two main observations. First, all force path curves are shifted toward
the N—axis; this shift is more obvious for curves influenced by both axial force
and moment (h = 0.0625 and 0.25). Second, the formation of a plastic hinge in
the arch causes the force path to shift suddenly toward the M—axis, as evident

from the multi—segment curves of h = 0.25.

 

 

 

 

149

   
  
   
     

 

 

N/Np
1...
....... Plastic
— Combined
C:
.9.
§
0..
E
O
U
h = 0.50
Negative Moment —-> 1VIy/NIym
0 ' g ' L
0 h = 0.75 / 1
h = 1.0

Figure 4—24 Relation between axial force and bending moment at the left quarter point
of fixed arch (L/ry = 225).

4+5 ARCH ULTIMATE LOAD UNDER VERTICAL
UN SYMMETRICAL LOAD: COMPARISON WITH
VERTICAL AND LONGITUDINAL LOAD

The arch ultimate load under vertical unsymmetric loading was

extensively studied in the literature, and the different aspects of arch behavior

 

 

 

 

150

and design under this type of loading are well understood. By comparing the
load carrying capacity under vertical and longitudinal loads with the well
known case of vertical unsymmetric load, the process of accounting for the first
type of load may become easier.

In this section, the variation with L/ry of the ultimate load of two-hinged
arch under vertical unsymmetric load is presented for different values of the
vertical load unsymmetry ratio, r. This ratio is defined in terms of the
unsymmetric (e.g., live load) intensity, pm, which is applied only to one half
of the arch span, and the symmetric (e.g., dead load)’ intensity, pm, which is
applied to the full span length of the arch, so that r is the ratio of the
symmetric load intensity to the total (symmetric plus unsymmetric) load
intensity, r = pDL/p,, where pt = pDL+pLL. For this study, I is varied from 0.99
(practically symmetric load) to 0.0 (with no symmetric load), with intermediate
values of 0.88, 0.50, and 0.33.

Finally, comparison between the case of vertical unsymmetric load and
that of vertical and longitudinal loading is made by correlating the vertical
symmetric component of the ultimate load under the two cases. This was per—

formed for both two—hinged and fixed arches.

4.5.1 Ultimate Load of Two-Hinged Arch Under Vertical
Unsymmetrical Loading

The variation with L/ry of the plastic, elastic—nonlinear, and combined

ultimate loads of two-hinged arch under vertical unsymmetrical loading (for

 

 

151
2.00 --
\ \ \\\ Plastic
4:: ‘\ ‘ \ __ Elastic
g \ \ \ r = 0.5
9.4 \ \ ___ Combined
x. r = 0.0\ \\ r __ 099 ._.. Horizontal &
D: \ \ V/ _ ' vertical load
1.00 —

 

 

 

Slenderness ratio, L/ry

 

(a) Total loads normalized to on,p

 

1.00 a”

 

 

0.00 . e j . IL . r . . + fi fi . . ,_fi_
50 150 250 350
Slenderness ratio, L/ry

(b) Total loads normalized to respective vertical
loads, PW,“ (x = p, e, or c).

Figure 4-25 Plastic, elastic-nonlinear, and combined ultimate loads of two-hinged arch
under unsymmetrical vertical loading.

 

 

 

 

 

  

 

 

 

h = 0.0, r = 0.99 (Vertival load only)
1.0 --
_____ Plastic (r, h = .5)
2:. ___- Elastic (r, h = .5)
D: r = 0.88 — Combined
\
>< \ Vertical unsymm.
o \
> \\ 03d
9" 0.5 _, \\ Longitudinal and
\ vertical load (h = .5)
0_0 '. l l
50 150 250 350

Slenderness ratio, L/ry

(c) Vertical component normalized
to respective vertival loads.

Figure (4-25), continued.

r = 0.99, 0.5, and 0.0) and under vertical and longitudinal loading (for h =
0.50) are shown in Figure (4-25.a) normalized to on,p- Figure (4-25.b) shows
the total ultimate loads and Figure (4—25.c) shows the symmetric vertical
components, with the curves of each type of analysis normalized to their
corresponding vertical-only ultimate load, on’x, where x represents p (plastic),
e (elastic), and 0 (combined).

In the first two figures, it is seen that with the decrease of r, the plastic,
elastic, and combined ultimate loads significantly decrease from their
respective values under vertical-only load, on‘x. The decrease of the vertical
unsymmetric curves with r is for the whole range of L/ry, in contrast with

vertical and longitudinal load where, for example, the elastic curves of h = 0.5

exceed those of h = 0.0 at some point.

 

 

 

153

Comparing curves of unsymmetric vertical load for different types of
analysis, it is seen that the plastic load decreases with L/ry faster than elastic
load for constant r, with the combined curve shifting between the two. Figure
(4-25.c) shows that for r = 0.50 and h = 0.50, the decrease in vertical
component of ultimate load is comparable between the two loading cases for all
types of analysis, specially for combined analysis. The combined curves for
r = 0.33 and 0.88 are added for illustration of the rate of decrease with I.

Figure (4—26) shows the elastic and combined ultimate loads for r =
0.99, 0.5, and 0.0 normalized to their corresponding plastic ultimate loads of
the same value of r. Here also, the figure exhibits similar trends to those of

longitudinal loading, where the largest effect of combining elastic and plastic

 

nonlinear effects is for'r = 0.5.

4.5.2 Comparison of The Vertical Svmmetric Component of
The Ultimate Load of Two-Hinged and Fixed Arches for
The Two Loading Cases

 

Direct comparison of the two loading cases in terms of the arch total
ultimate load is not the proper measure of the reduction in the arch load
carrying capacity, since the loading directions and patterns are different.
Instead, the "vertical symmetric" component of the ultimate load (in both
cases) is a more accurate and representative measure, since it is the common
denominator and measures the arch capacity to carry dead load in the presence
of other types of load. The maximum vertical symmetric load that the arch

can carry occurs when r = 0.99 and h = 0.0, P,(r=0.99,h=0.0) = PW. The

 

 

 

 

 

 

 

100 \ \ \
-- ___§ \\ \\
1: . --_ \ \ \ \
V ‘\\““~\_‘ ‘\\ \ \
a: ‘\ “~2§‘ \
Q: ‘\~ \‘T‘\‘\‘ \
\‘\ \ \ ~-\ g \\
: \“\\\\ \\ I“ a‘}
C ‘\\a \\ \‘}§<= 0.0
X \\ \
a: 0.50 1i ~-~>\\ 05\\
‘\‘\. T: .
i __ Plastic \ \\
__ Elastic \\\\\
___ Combined r = 0-99 \‘
0.00 # . . : . . . . i . . . . M-
50 150 250 350

Slenderness ratio, L/ry

Figure 4-26 Elastic and combined ultimate loads of two-hinged arch under vertical

unsymmetrical loading normalized to corresponding plastic ultimate loads of same
value of r.

reduction with r and h is measured in comparison with this value.

For vertical and longitudinal load, the vertical component of the ultimate
load (which is symmetric by definition) for a given value of h, P,(h) =
v -P,(h) = (l-h) -P,(h). The value of h varies from 0.0 (vertical-only load) to
1.0 (with no vertical load).

For vertical unsymmetric load, the total ultimate load is P1 = pDLoL +
pLLcL/Z = ploL (r + (l-r)/2), while its vertical symmetric component, P,(r) =
L-pDL = p,-L-r. Expressing Pv in terms of Pt, we can write,

Pv(r) = P,(r)-2r/(l+r).

The unsymmetry ratio, r, varies from 0.99 (for practical vertical symmetric

load) to 0.0 (with no symmetric load).

 

 

 

 

 

 

 

 

 

 

 

 

 

155
1.00 0.50 0.00
1.00 4‘ ‘ 4* ' : I . I
j \ Vertical load unsymmetry ratio, r
‘1‘
g \ \ _____ Vertical unsym-
& ‘\ \ \\ metric load
2 \ \ i __ Longitudinal &
\a‘ ‘\ \\ \ vertical load
of x ‘\ o___° L/ry = 125
8 0.50 __ \\\ \\\ \\ E a L/ry = 225
D. \\ \\ \\ A—A L/Iy = 325
C; \\ \\ \\
> \\ \\ ‘
fi \\ °\ \‘x
C \\< \\\ \\\
‘13: \\B\\ \‘13\
> \ \ \\\
9" “ .jjs.~~ ‘\
Longitudinal load ratio, h \ :I“~~ZZ~:~
0.00 a . . - l . . -
0.00 0.50 1.00
(a) Plastic analysis.
1.00 0.50 ' 0.00
1.00 I I I I M I I I I
\ Vertical load unsymmetry ratio, r
v. ‘x
3 \ 9‘ _____ Vertical unsym- -
m * \ metric load
\ \ \
2 \\~\ Longitudinal & -
V0 \ \\ vertical load
a: \.\‘\\ °__e L/ry =125 '
‘5 0.50 —— \\\\‘e\ H L/ry = 225 _—
a \:\Q ‘\‘ A———A L/fy = 325
6 \ \‘ ‘\ _
> x \
O4 ‘ ~ \ ‘~\
\ . ~\ \\ .
3 ‘x‘\ t
o ‘\\\ \\ _
>‘ \ :I‘ ‘
‘34 ‘~~:~-
Longitudinal load ratio, h ‘ ‘N '
0.00 f . . f .L a . f . T
0.00 0.50 1.00

(b) Elastic analysis.

Figure 4—27 Variation of the vertical symmetric component of the ultimate load of two-
hinged arch in the presence of vertical unsymmetric or vertical and longitudinal loads.

 

 

 

156

1.00 0.50 0.00
1.00 ‘ ' ‘ ' l ' ' ' '
Vertical load unsymmetry ratio, r

 

\\ _____ Vertical unsym—
metric load

Longitudinal &

‘ \\ _ vertical load
‘ \ e——o' L/ry = 125
0.50 -- \\ ’B—El L/ry = 225 -_
‘ A__.__A L/ry = 325

Pv.c(r)/on,c 01' Pv.c(h)/on.c

 

 

Longitudinal load ratio, h

0.00 . - - . - .
0.00 0.50 1.00

 

(0) Combined analysis.

Figure (4—27), continued.

Figure (4—27.a, b, and 0) show the change in the vertical symmetric
component of the plastic, elastic-nonlinear, and combined ultimate loads of
two—hinged arch for the two loading cases. Similarly, Figure (4-28.a, b, and 0)
show the same curves for fixed arch. These figures show that the decrease with
(l-r) and with h of the vertical symmetric component, Pv, is essentially similar.

From Figure (4—27), both plastic and elastic curves of two—hinged arch
for both loading cases shift down (i.e., the curve nonlinearity increases with h
or (l-r)) as L/ry increases. However, the elastic curves exhibit less nonlinearity
than plastic ones, where the vertical and longitudinal elastic curve for L/ry =

125 is roughly linear, with the nonlinearity increasing with L/ry until the elastic

curve for L/ry = 325 become close to (i.e., coincides with) that of plastic curve

 

 

 

157

for L/ry = 125. This behavior explains the very little change in the
nonlinearity of combined curves, Figure (4—27.c), which are similar to plastic
curves for low L/ry and similar to elastic curves for high L/ry.

Comparing the curves of the two loading cases in each part of the figure,
it is seen that the plastic and combined curves are very close for a given value
of L/ry, while the difference is larger between the elastic curves, specially for
lower L/ry.

For fixed arch, Figure (4-28), the trends are similar to those of two—
hinged arch for plastic and elastic analysis,iwhile the combined curves show
larger change with L/ry. However, both plastic and elastic curves of fixed arch
exhibit less nonlinearity with h and (l-r) than their corresponding curves of
two—hinged arch. Also, for a given value of h or (l—r), the decrease of the
elastic curves from L/ry = 125 to 325 is larger than that of plastic curves. The
combination of these two factors clarify the larger change of combined curves
with L/ry. But more importantly, the curves of vertical unsymmetric load are
very similar to those of vertical and longitudinal ones for the same L/ry.

The nearly identical variation of the ultimate load vertical component,
PV, with (l-r) and h suggests that longitudinal loading on the arch may be
accounted for in the preliminary design process by transforming it into an
equivalent vertical unsymmetric-load. Also, if both vertical unsymmetric and
longitudinal loads are present simultaneously along with vertical symmetric

load, their effect on reducing the vertical symmetric load capacity may be

 

 

 

 

158

 

 

 

 

 

 

 

 

1.00 0.50 0.00
1.00 . ‘ - l
\G Vertical load unsymmetry ratio, r
e ‘ _
g x _____ Vertical unsym- _
E \\ metric load
2 \X \\ Longitudinal & -
\E‘ \ \ \\ vertical load
LI: \“ \\ , \\u 0—0 L/Ty = 125 -
‘5 0.50 -— \\ H L/ry = 225 -_
a x xx . \ , , L/ry = 325
g \\ \\ \x _
m \{ \\\ \\\
R \x \ \ _
C ‘~¢\ ‘\\
n- ‘\ ‘x ‘ _
>’ ‘x ‘ a
LL \4‘ \r:\
Longitudinal load ratio, h ‘~—t::\\
0.00 . . . i . - . -
0.00 0.50 1.00
(a) Plastic analysis.
l.00 0.50 0.00
1.00 I I I I : I I I I
Vertical load unsymmetry ratio, r

u \ _____ Vertical unsym-

6 \ metric load _
L7: ‘1 \\ Longitudinal &
E ‘\\ vertical load -
V0 ‘~\ \\ o o L/ry =125
LI:- ‘x‘fx B— E, L/ry = 225'

\ °~ L/r = 325

La 0.50 —- \\ \xa 5‘ y -—
o x \ ‘

o ‘ ‘(\\‘\ \\

. \‘ ‘Gx \\\

‘3 ‘wa. \ \

LL. ~,\~\ \
\ \\ \\
e \0-
0 ¢\:\\
>' ‘\:‘\
LL “11‘
Longitudinal load ratio, h “‘1:
0.00 - ' i ' ' '
0.00 050 1.00

(b) Elastic analysis.

Figure 4—28 Variation of the vertical symmetric component of the ultimate load of fixed
arch in the presence of vertical unsymmetric or vertical and longitudinal loads.

 

r—.——————————* ‘

159

1.00 0.50 0.00

 

    
     
     
  
    
  

Vertical load unsymmetry ratio, r

 

_____ Vertical unsym-
metric load

— Longitudinal &
vertical load
L/ry = 125

H L/ry = 225
L/ry = 325

  

    

   

,h

 

Longitudinal load ratio
0.00 - - - - . i . -
0.00 0.50 1.00

(0) Combined analysis.

Figure (4-28), continued.

 

accounted for by calculating an equivalent ratio of dead to total load, rcq =
pDL/pu where the total load intensity on the arch, pl = pDL + pLL + Pm where
ph is the longitudinal load intensity. For example, assume an arch with unsym—
metric live load intensity, pLL = 0.25 ~pDL and longitudinal load intensity, ph =
0.40-pDL. The load unsymmetry ratio, r = 1/(l+0.25) = 0.80, (l—r) = 0.20,
and the longitudinal load ratio, h = 0.40/(1 +0.40) = 0.29. If both effects are
added simultaneously, the total load intensity is pl = (l+0.25+0.40) -pDL =
1.65 0pm, and the equivalent load unsymmetry ratio, rcq = l/l.65 = 0.61, and
(l—rcq) = 0.39. Verification of this example for a two—hinged arch (L/ry = 225)
under combined analysis, produced a difference of less than 3% in the vertical

symmetric component of the combined ultimate load between the two cases.

 

CHAPTER V
PARAMETRIC STUDY OF ARCH BEHAVIOR:

II. THREE—DINIENSIONAL BEHAVIOR

Q INTRODUCTION

In this chapter, the arch behavior involving lateral loading is discussed.
The interaction between vertical and lateral loading was investigated first, and
then the three—dimensional ultimate load surface of an arch under combination
of vertical, longitudinal, and lateral loads was constructed.

In the first part, the behavior of fixed arch under combinations of

 

vertical and lateral loads was studied, where the total ultimate load and its
vertical component, the interaction curves between vertical and lateral load
components, and failure modes and response curves of the arch were obtained
using plastic, elastic, and combined analysis. In the second part, the three-
dimensional ultimate load surface was constructed using combined analysis, and
the interaction between each two of the three components was explored.

For analysis that involves lateral loading, the arch was always fixed in
the lateral direction; i.e., the rotations about the global X- and Y-axes were
restrained at the supports. This assumption is based on practical considerations

where the arch bridge usually consists of two ribs connected by lateral bracing,

 

 

161

and the whole system is restrained from rotation about the X- and Y-axes at the
supports. This is also supported by the preliminary analysis results performed
in the next section.
_5._2 EFFECT OF SUPPORT TYPE AND SECTION PROPERTIES

ON THE ARCH ULTIMATE LOAD

Figure (5—1) shows the effect on the ultimate load due to fixing the arch
support and varying its section properties in the presence of lateral load. For
this purpose, two sections were used, 51 and S2 (Figure (4-1)), with ratios of
the out—of—plane to in-plane slenderness ratios of the arch, (L/rZ)/(L/ry) = rylrz,
is 2.0 and 1.0, respectively. In Figure (5-l.a), the ultimate vertical loads of
S2-fixed arch (i.e., fixed arch with section S2) and SZ-hinged arch are plotted
against L/ry normalized to on,p(51)- In this figure, the ratio of the lateral load,
F,, to total load, F” s = F,/F,, are 0.0 and 0.001. The ultimate lateral-only
loads (3 = 1.0) of Sl—fixed, Sl-hinged, and S2-fixed arches are plotted in
Figure (5-1.b) against the lateral slenderness ratio, L/rz, normalized to
P,°_,,(SZ).

Note that the arch vertical-only ultimate load is the same for fixed and
two-hinged arches for both sections, i.e., F,°,p(Sl) = PVO,P(Sl) = on,p(82).

Under vertical—only load (3 = 0.0), the increase in the elastic in—plane
ultimate load due to _fixing the support is significant (up to 120%, see also
Table (4—2)). For the studied section proportions however, this increase

disappears almost entirely in the presence of even the smallest lateral load (3 =

 

 

l

h-d
O
O

u \\
Hinged Lateral ——-—\\\\
. \

Hinged In—Plane—fié
\\\

‘ Fixed Lateral ———\—\\\\

—.

Elastic, s 0.0
(in —plane buckling)

Elastic, s 0.00l
(lateral-tortional buckling)

Combined, s = 0.0

“ .\ (in-plane load)
\ Fixed arch
Hinged arch

.—.

Pt.x/on.p 01‘ Ft.x/on.p

 

0.00....:.....:....I....I
50 150 250

 

In-Plane Slenderness ratio, L/ry

(a) Vertical in-plane and lateral-torsional ultimate
loads normalized to Pymp.

        
     

Plastic
Elastic
Combined

 

1.00 -- .
‘ SZ-Fixcd arch

Sl arch (both

\ \ / hinged and fixed)
\

Pso,x/Pso.p(sz) 01' Fso.x/Pso,p(sz)

 

 

0.00-f..=...:...fi-
50

Slenderness ratio, L/rz(Sl) or L/rz(82)

(b) Lateral ultimate loads normalized to Pso,p(S2).

Figure 5-1 Effect of the type of arch supports and its relative lateral to in-plane stiffness
on its vertical and lateral ultimate loads.

 

 

163

0.001) where the failure mode changes from in—plane to lateral—torsional one
and the ultimate loads of hinged and fixed arches become almost equal, with
differences of less than 5%. Moreover, these ultimate loads are smaller than
—though close to- the vertical in-plane ultimate load of two-hinged arch, leading
to the conclusion that the restraint of the in—plane support rotation has negligi-
ble effect on the arch ultimate load if it fails in torsion.

Under lateral—only load (5 = 1.0), Figure (5-1.b), the plastic ultimate
loads of two—hinged and fixed arches of the same section are identical, PM, =

F since the arch is fixed in the lateral direction for both cases and the in-

50.1),

plane properties are irrelevant to lateral load when geometric effects are

neglected. The combined and elastic ultimate loads, however, are not identical

 

for the two types of support but are close enough -specially for combined
curves- to confirm the above conclusion. In addition, the figure shows the
effect of varying the ratio of lateral to in-plane slenderness ratios on the
plastic, elastic, and combined lateral ultimate loads where they are about
doubled for S2—fixed arch compared to Sl-fixed arch.
g BEHAVIOR OF FIXED ARCH UNDER VERTICAL AND
LATERAL LOADS
The interaction between vertical and lateral loading was studied for fixed

arch only, since the effect of support type on the arch ultimate load was found

 

insignificant. In addition to ultimate loads, the interaction curves between

lateral and vertical loading, the deflection modes, and the response curves of

 

 

164

the arch were also studied. The obtained results are discussed in the following
sections.
5.3—.1 Variation of the Ultimate Load of Fixed Arch with In-
Plane Slenderness Ratio
The plastic, elastic, and combined ultimate loads of fixed arch were
obtained for different values of the ratio of lateral to total applied load, 5 =
F,/F,. To adequately trace the change in ultimate load, the values of s = 0.0,

0.001, 0.01, 0.0625, 0.125, 0.25, 0.50, and 1.00 were used.

5.3.1.1 Plastic Ultimate Load of Fixed Arch

 

Figure (5-2.a) shows the variation with the in-plane slenderness ratio,

L/ry, of the total plastic ultimate load, F normalized to onm. Figure (5-2.b)

up,
shows the vertical component of the plastic ultimate load, Fm, also normalized
to Pym. From the first figure, it is seen that the ultimate load gradually
decreases with L/ry for constant 5. Also, the curves change from linear for
small values of s to nonlinear as 3 increases, with the nonlinearity (i.e., the
rate of change with L/ry) being higher in the low L/ry range. This behavior is
similar to that under vertical unsymmetric loading or vertical and longitudinal
loading.

The ratio of the plastic ultimate loads under lateral—only load (3 = 1.0)
to that under vertical—only load (5 = 0.0), Fwy/Fwy, decreases from 20% for

L/ry = 75 to 5% for Ur, = 450. It is notable that the plastic lateral~only

ultimate load is linearly dependent on the section out—of—plane properties, while

 

 

 

 

 

165

 

 

  
  

 

 

s = 0.0, 0.001 (overlapped)
1.00 - M
a: - s = 0.01
3
Ba
\ _
O: ..
i“ 0.50 ~—
0.00

 

 

50

Slenderness ratio, L/ry

(a) Total load normalized to Pymp.

 

 

  

 

s = .0, .001 (overlapped)

1.00 "
O... _ \
S s = 0.01
04
\
‘3:
> = 0'06
[L4 25

0.50 ‘

0.00 f I fl l I I l I l I 1 ‘li l l l—{

 

 

Slenderness ratio, L/ry

(a) Vertical component normalized to on‘p (= Fvom).

Figure 5—2 Variation with L/ry of the‘plastic ultimate load of fixed arch under vertical
and lateral loading

 

166

the vertical-only one is dependent on the' section in—plane properties, as
discussed before. Consequently, the choice of section properties has significant
effect on these ratios. In addition, while the proportions of section Sl were
chosen to reflect those of a typical box-arch bridge, those of 52 were chosen to
V have symmetric properties about its major axes. A practical arch bridge would
normally have larger lateral stiffness, and therfore larger ratios of Fwy/Fwy.
The vertical component of ultimate load, Figure (5-2.b), exhibits similar
trends to those of the total ultimate load but with less degree of nonlinearity for
all curves. Similar to the case of longitudinal and vertical loading, the vertical
load component always decreases with the lateral load ratio, 3. However, the
magnitude and rate of decrease also depends on the ratio of Fwy/Fwy, which
explains the observed behavior. In contrast, the interaction curves between the
vertical and lateral load components (discussed later) are less dependent on the
section properties. This discussion is also true for both elastic and combined

ultimate loads.

5.3—.12 Elastic Ultimate Load of Fixed Arch

The variation with L/ry of the total elastic-nonlinear ultimate load of
fixed arch, Fm is shown normalized to PM, in Figure (5—3.a), and ”to Fm: in
Figure (5-3.b). Its vertical component, F”, is shown in Figure (5-3.c)

normalized to F The curves in the first two figures show significant drop in

vo,c'

the ultimate load for s = 0.001 compared to that of s = 0.0 for the whole

range. As mentioned previously, this is due to the change from in-plane failure

 

 

 

167

    
 

 

 

 

 

 

 

2.00 p
‘2‘
O J
a?
\ 1.00" ~~~~~
ur' _ .9
. intermediate
values, s =
- .01, .0625,
.125, .25, .5 '
0.00 l I I I l —I—‘—I———i
50 250 450
Slenderness ratio, L/ry
(a) Total load normalized to onm'
0.50 i
q _ s = 0.001
3
LL
\ \ s = 0.01
3 5 = 0.0625
L“ 0.25 -- 8 = 0-25
.. 5 3 0.125
_ 8 § 0.50
~ s = 1.0
0.00 ' ' ' i ‘ f ' '
50 250 450

Slenderness ratio, L/ry
(b) Total load normalized to Fvofi.

Figure 5—3 Variation with L/ry of the elastic ultimate load of fixed arch under vertical and
lateral loading.

 

 

168

 

 

 

 

.. 0.50 J—
3. .
LL.
‘1‘ 0.25 +—
0.00 V I l i I I I fi
50 250 450

Slenderness ratio, L/ry

(c) Vertical component normalized to FVM.
Figure 5-3 continued.

mode to lateral one because the arch in—plane ultimate load is larger than its
lateral-torsional ultimate load (also called lateral-torsional buckling load).
From Figure (5-3.b), the curve of s = 0.001 shows a consistent decrease
of about 45% from the vertical load (Fvog) for the whole range of L/ry. Other
curves show similar consistent decrease except for very low L/ry, where they
tend to be more nonlinear with low L/ry. For L/ry = 75, the curves for s = 0.5
and 1.0 increase with s. The vertical component of the ultimate load, Figure
(5-3.c), shows consistent decrease for the whole range of L/ry (i.e., has less

nonlinearity) for all values of s.

5.3.1.3 Combined Ultimate Load of Fixed Arch

 

The variation with L/ry of the total combined ultimate load of fixed arch,

Fm, is shown in Figure (5~4.a) normalized to PM” and in Figure (5-4.b)

 

 

169

1.00 '

0.50 --

Ft.c / on,p

 

 

 

50 250 450
Slenderness ratio, L/ry

(a) Total load normalized to Pv0_p.

 

 

 

 

 

 

0.00 - a fi : . . , Mr,

Slenderness ratio, L/ry

(b) Total load normalized to Fvo’c.

Figure 5-4 Variation with L/ry of the combined ultimate load of fixed arch under vertical
and lateral loading.

 

170

 

1:“v.c / FVO,C

 

 

 

 

 

 

Slenderness ratio, L/ry

(c) Vertical component normalized to,Fvo‘c.
Figure 5-4 continued.

normalized to FVM. The vertical component, Fm, is shown in Figure (5-4.c)
also normalized to Fvo,c'

From the first figure, for low values of s (s 0.06), the combined curves
are asymptotic to both plastic and elastic curves, with a sharp decrease of the
ultimate load over a short interval of L/ry. For larger values of s, the transition
from plastic- to elastic-dominated behavior is smoother, where for low L/ry the
presence of geometric effects causes larger reduction in combined ultimate
loads compared to plastic ones. In addition, as L/ry increases, the reduction in
both plastic and elastic ultimate loads from their respective vertical ultimate
loads becomes comparable for constant 3, as illustrated in the example below.

Figures (5—4.b and c) show the transition from plastic—dominated to

elastic—dominated behavior in terms of the relative reduction of the ultimate

 

 

,C

 

171

load and its vertical component from the arch vertical ultimate load, FVW. At

L/ry z 250, the combined curves are predominated by the elastic behavior.
The above observations are illustrated by the following example. For

3 = 0.125 the reductions in the vertical load capacity of the arch due to plastic,

combined, and elsatic analysis (i.e., Fm/vap, Fv,c/Fvo,c7 and Fm/FVM) for

L/ry = 75 are 0.69, 0.58, and 0.37, respectively, and for L/ry = 250 they are

0.32, 0.20, and 0.23 for L/ry = 250. For L/ry = 450, these ratios are 0.19,

0.17, and 0.17, which are very similar for the three types of analysis.

m Ultimate Load Interaction Curves

The interaction curves between the vertical and lateral load components
of the ultimate load are shown in Figure (5-5) for three values of L/ry (125,
225, and 325), where they are presented similar to those of longitudinal and
vertical ones. The figures on the left show the relative values of the vertical
and lateral components of the plastic, elastic, and combined ultimate loads with
respect to their corresponding vertical-only and lateral—only plastic loads.
Similar to those of two-hinged arch under longitudinal and vertical loading, the
ratio of lateral elastic to plastic ultimate loads, Rog/Fwy,” is always larger than
the corresponding ratio of vertical loads, ch/va. Also, combining elastic and
plastic nonlinearity has the largest effect for L/ry = 225, with the plastic and
elastic behavior dominating for lower and higher L/ry, respectively.

In the figures on the right, every load component is normalized to its

corresponding ultimate load, for example, FV’JFW,e and Fae/Fm, and so on. It

 

 

 

 

    
 

 

 

 

 

 

 

 

 

 

 

 

 

1 60 1.00 1
a: _\ ’1
2 4., . 3
FL 0/ n.
\ Q‘/‘ \
u 4/ I:
>- 5/ In; Combined \
“I ' -»
0.80 l\\ 0'50 \(x = c) .
\ ' \
\ . \ \
\ \\ \
\\ Elastlc‘\\\ \\
\ (x = e) x
\
0.00 . \ : 0,00 . i . .
0.00 4.00 3.00 0.00 0.50 1.00
Fl'x / Flo-P F5.’ I 1:30.!
(a) Stiff arch, L/ry = 125.
1.00
a K.
g 2
m E
\ 5 I x.
:- 9/ u:
“‘ . 0.50
. \K\
0.00 . M. . . .L‘I 000
0.00 1.00 2.00 0.00 0.50 1.00
1:3,: / Fso.p Fs.x / Fso,x
(b) Medium arch, L/ry = 225.
L00
9.
o‘
>
D.
\
s.
>
LL
0.50
0.00 ' '
0.00 . .
FL! / F80.X

 

Fs,x / F004)
(c) Slender arch, L/ry = 325.

Figure 5-5 Interaction curves between vertical and lateral load components for fixed

arch.

 

 

 

173

is seen that the plastic interaction curves are convex, while the elastic and
combined curves are either linear or slightly concave for the range of s =
0.001 to 1.0. Also, the shapes of these curves are basically independent of
L/ry. The initial drop in the elastic ultimate load due to the change from
vertical in—plane failure mode for s = 0.0 to lateral-torsional failure mode for
s = 0.001, is consistent over the L/ry range, while it is gradual for combined

CUI'VCS.

5.3.3 Failure Modes and Response Curves

 

The vertical and horizontal projections of the deflected shapes of
medium arch (L/ry = 225) corresponding to the ultimate plastic, combined and
elastic loads are plotted in Figure (5—6) for s = 0.0, 0.001, 0.01, 0.0625, and
1.0. All deflections are magnified by a factor of 5 for clarity. It is well
established that the in-plane failure mode due to vertical-only load (3 = 0.0) is
always unsymmetric in the presence of geometric effects, although this

unsymmetry may not be well developed at the point of ultimate load, as in the

 

case of combined curves in Figure (5-6.b). However, tracing the deflection
through post—ultimate load always produces unsymmetric modes.

It is seen that in the presence of lateral load (i.e., for all values of s 2
0.001), the in-plane deflection mode is always symmetric with magnitudes that
increase slightly with s. On the other hand, the lateral deflections show distinct
increase in their magnitude with s. The only exceptions are the combined

curves of s = 0.01 and 0.06 that are almost identical, apparently due to

174

   
   

s = .0, .001, .01, .06
(overlapped)

In—plane mode (X-Y plane)
DMF

 

 

Out-of—plane mode (X-Z plane)
DMF =

(a) Deflections under plastic analysis.

 
     
 

s = .001, .01, .06
(overlapped)

In—plane mode (X-Y plane)
DMF = 5

 

 

 

s = .01, .06 s = 1-0
(overlapped)

Out-of-plane mode (X—Z plane)
DMF = 5
(b) Deflections under combined analysis.

Figure 5-6 Deflected shapes of fixed medium arch (L/ry = 225) at ultimate load for
different types of analysis and values of s.

 

 

 

175

  
 

 
   
 

s = .001, .01, .06

s = 0.0 (in order from top)

In-plane mode (X—Y plane)
DMF = 5

 

Out-of—plane mode (X-Z plane)
DMF = S

(c) Deflections under elastic analysis.

Figure 5—6 continued.

premature plastic failure.

The magnitudes of combined lateral deflections are smaller than the
corresponding plastic or elastic curves (with about 50% reduction) because of
premature failure due to combination of geometric and plastic effects, as
illustrated later. Also, the rotation at the supports is large for plastic analysis,
indicating yielding ofjoints at the supports. For elastic and combined analysis,
it is small indicating that the support did not yield yet at the maximum load (for
combined analysis), and the failure is governed by lateral instability of the arch
rather than local failure. I

The presence of geometric effects propagates lateral deflections, which
are substantially larger than corresponding in-plane ones for elastic and

combined curves. This becomes clear when comparing the lateral deflection

 

 

 

 

176

curves of s = 0.001 to those of s = 0.01 for the three types of analysis. For
example, the ratio of maximum lateral deflections at the crown for s = 0.001
and s = 0.01 is about 0.30 to 0.35 for elastic and combined analysis, while this
ratio is very small (< < 0.10) for plastic analysis.

The response curves at the quarter point of fixed arch (with L/ry = 225)
are shown in Figure (5—7) for s = 0.001, 0.01, and 0.06. Figure (5—7.a) clearly
shows the effect of geometric nonlinearity on the elastic and combined response
curves. As the load is initially applied, the lateral displacement (abbreviated
L.D.) is smaller than the in-plane displacement (IP.D.). Also, the curves of
elastic, plastic, and combined displacements are the same.

With the progress of loading, however, the lateral displacement
increases unproportionally -due to geometric effects— while the in-plane
displacement (IP.D.) increases steadily throughout the loading. This behavior
continues with accelerated rate until the magnitude of the lateral displacement
far exceeds that of in-plane displacement at failure. This is known as the snap—
through phenomenon, where the deflection changes from predominantly in-
plane mode to an out—of—plane one. For plastic analysis, the two displacements
increase proportionally throughout the loading process until failure.

The initial slope of the lateral displacement is more dependent on the
value of s where it significantly increases with 5, while that of in—plane

displacement is roughly the same for the three cases; the two become equal at s

 

= 0.01, Figure (5-7.b). The curves also show that the displacement magnitudes

   

1'77

   

Platic IP.D.

    
 
 
 

Platic L.D.

Elatic and combined

 

 
 

 

 

 

 

 

 

 

.sé L.D. (overlapped)
is
u:
10 \
Elatic and IP.D. = ln-Plane Displacement
combined IP.D.
(overlapped) L.D. = Lateral Displacement
O I r l l I i l T I I %—
0 . l 2
Displacement at quarter point, (x 10'3-L)
(a) Response under predominantly vertical load, 3 = 0.001.
_ Platic IP.D.
' Platic L.D.
_ 20 T Elatic and l
*4 _ combined IP.D.
u; : / (overlapped) Elatic L.D.
10 -- \
- Combined L.D. (partially
— ovelapped with elastic)
0 . f . f .9 r . . . .-
0 5 10
Displacement at quarter point, (x 10-3-L)
(b) Response under vertical and lateral load, 5 = 0.01.
16 -— Platic IP.D.
_ _ Platic L.D.
.54
i: _
”j - latic IP.D
8 Elatic L.D.
_ Combined L.D.
‘ Combined IP.D.
0 I I r I f I fir fl M I Dr fil T l l] l [—‘l
0 5 10 15

Displacement at quarter point. (x 10‘3-L)

(c) Response under vertical and lateral load, 5 = 0.06.

Figure 5-7 Response curves for fixed arch under vertical and lateral loading.

 

 

 

178

at ultimate load significantly increase from s = 0.001 to s = 0.01, but increase
to a lesser extent from s = 0.01 to 0.0625. The combined response curve for

s = 0.0625 exhibits a sudden decrease after reaching the ultimate load, in
contrast with the smooth, horizontal curves for s s 0.01. This behavior
indicates failure due to plastic effects (i.e. , formation of plastic hinges), and
explains the overlapping of the deflection curves for s = 0.01 and 0.0625 in

Figure (5-6.b).

M THREE-DIMENSIONAL ULTIMATE LOAD SURFACE

 

The objective of this section is to study the interaction between the three
types of loading applied simultaneously to the arch under combined analysis,
and examine the change in relationship between any two of the three loads in
presence of the third. For this purpose, an arch with medium slenderness ratio,
L/ry = 225, with S2 section was used. V

The ultimate load surface was constructed by applying a load vector that

 

is a combination of the three principal ultimate loads,

P' = v'Fmc + h'Fho,c + s'ch,
where ch, Fho,c7 and Fso.c are the combined ultimate loads under vertical-only,
longitudinal—only, or lateral-only loading, respectively, also called the principal
ultimate loads, and v', h', and s' are their respective scaling factors so that,

v' + h' + s' = 1.

To obtain a uniform mesh of points on the surface, the value of v' was

varied from 0.0 to 1.0 with increments of 0.1. The individual values of h' and

179

s' were determined from the above relation so that h' + s' = l — v', and by
defining the ratio of s'/h'. Values of s'lh' = 0.0 (i.e., s' = 0.0), 0.33, 1.0, 3.0,
and on (i.e. h' = 0.0) were used. The resulting surface is defined by lines of
constant s'/h', called here longitudinal lines for the purpose of discussion.
These lines extend from the corner of the principal ultimate load on the vertical
axis (where v’ = l, h' = s' = 0) to the horizontal plane of longitudinal—lateral
loads (where v' = 0). These lines are intersected by lines of constant v', called
transverse lines.

In addition, a value of s' = 0.02 was used to produce an additional line
on the ultimate load surface that is parallel to the line of s’ = 0.0. This extra
line represents the interaction curve between vertical and longitudinal loading
in the presence of a small lateral force that would cause lateral failure mode
when it is the controlling mode. This provides insight into the arch behavior as

it fails in lateral mode.

 

The combined ultimate load surface is shown in Figure (5-8) in
normalized form, i.e., for each point on the surface the three ultimate load i
components are normalized to their respective principal ultimate loads. The
thick lines on the surface repreSent the interaction between each two of the
three loads. The line of s' = 0.0 represents the interaction between only
longitudinal and vertical loads, which is the same as the combined curve in
Figure (4-20.b), while the line of h' = 0.0 represents the interaction between

only vertical and lateral loads, and is the same as that of combined analysis in

180

I Vertical Load,
Fv,c/Fvo_c

     
 

)
V - vo,c + ‘- ho,c
+ s .FM,‘c

applied unit load vector,
P) = ’

 

  

v’+h’+s’=l

 

8'

‘\

  
 
 
 
 

Lateral
Load,

FS.C/FSO.C

/.

Longitudinal
Load,

Fh,c / Fho,c

Figure 5-8 Ultimate load surface of fixed arch (L/ry = 225).

181

Figure (5-5.b). The line of v' = 0.0 represents the interaction between
longitudinal and lateral loads, and is denoted as the surface base.

The shape of the ultimate load surface resembles a dome with the
vertical load axis as its axis of revolution. In the figure, the line of s' = 0.0 is
the only longitudinal line that extends smoothly for its entire range. All other
longitudinal lines exhibit sharp drop in the vertical component of the ultimate
load, Fm/FVM, from the vertical corner (where v' = 1.0) to the point of v' =
0.9. For this point, the ratio of FM/Fmc ranges from 0.55 (for s' = 0.02) to
0.52 (for s' = 1.0). This sudden drop in the arch vertical ultimate load is
attributed to the change of the mode of failure from in—plane to lateral one in
the presence of lateral load. From the point of v' = 0.9, these longitudinal
lines extend smoothly to the surface base (v' = 0.0), forming a convex—shaped

surface.

5.4.1 Two—Dimensional Interactions Between The Ultimate Load
Components

The projections of the ultimate load surface on its principal planes
produces the two—dimensional views shown in Figure (5-9.a, b, and c) for
vertical-longitudinal, vertical—lateral, and longitudinal-lateral views,
respectively.

The presence of lateral load has the effect of consistent reduction of the
arch ultimate load with the increase of s'/h', Figure (5—9.a). This is more

understood by defining the envelope of the arch ultimate load as the lateral-

 

 

182

  

1.0
applied unit load vector,
0 P’ = v’.FV°‘c + h’.Fho‘c
g + s’.Fso,c
u. v’ + h’ + s’ = I
\
O
>.
LL
0.5

 

 

 

 

0.0 0.5 l.0
Fh,c / Fho,c

(a) Relation between vertical and longitudinal components.

" v’ = .6, .7, .8. .9
(overlapped)
0.5 — .

    

 

 

 

 

 

Fs.c / Fso,c
(b) Relation between vertical and lateral components.

Figure 5—9 Projections of the ultimate load surface onto its principal planes.

183

 

F5.C / Fso.c

 

 

 

 

 

Fh,c / Fho,c

 

(0) Relation between longitudinal and lateral components.

Figure 5—9 continued.

torsional ultimate load, i.e., the line of s' = 0.02. This line represents the
practical limit on the arch capacity when it fails in lateral—torsional mode. The
lines of s’/h' = constant are basically parallel to this envelope for the whole
range of v' = 0.0 to 0.9, and the reduction in the arch ultimate load is
consistent as s'/h’ increases.

It is notable that the change of the failure mode from in—plane mode

(3' = 0.0) to lateral one (for s' 2 0.02), also changes the shape of the

184'
interaction curves of the ultimate load from a concave one to a convex-like one.
This means that an arch with its in-plane vertical ultimate load being smaller
than its lateral-torsional vertical ultimate load is more sensitive to the presence
of lateral loads; i.e., in the presence of a small lateral load, a longitudinal load
(with a given value of h', for example) would decrease the vertical load
capacity of such an arch more significantly than it would decrease an arch that
has a lateral—torsional vertical ultimate load smaller than its vertical in-plane
ultimate load.

The presence of longitudinal load has less consistent effect on the
relation between the vertical and lateral components of the ultimate load,
Figure (5-9.b). In this figure, the line of h' = 0.0 constitutes an upper envelope
of the other curves. However, the lines of s'/h' = constant are not parallel to
this line, but merge into it as v' increases. In other words, for a given v' =
constant, as the ratio of lateral load, 3', increases (i.e., the larger s'/h'), the
vertical load capacity decreases only slightly.

Analogy may be drawn between this case and the case of column
buckling under vertical load with small lateral perturbation load (which
corresponds to the lateral load component on the arch); the larger the lateral
load, the smoother the response curve and the smaller the obtained ultimate
load (as defined by a limit displacement) compared to the theoretical load.

The relation between longitudinal and lateral loads for constant v’,

Figure (5—9.c), changes gradually —but quickly— from convex for v' = 0.0 to

185

perfectly linear for v' = 0.6. For v' > 0.6, the relation continues to be linear
for the range of s' = 1.0 to s' = 0.02, which is the envelope for the arch
lateral—torsional behavior. For this range, the lines of v' = 0.6 to 0.9 are
overlapping, Figure (5—9.b), and the vertical component of the ultimate load
changes only slightly for all these curves. Consequently, it may be concluded
that the presence of longitudinal load has negligible effect on the arch behavior

as it fails in a lateral-torsional mode.

 

 

CHAPTER VI

SUIVIMARY AND CONCLUSIONS

Q , SUMMARY

The main goal of this research is the study of ultimate load capacity of
single parabolic arches under different loading combinations. In particular, the
interaction between vertical and longitudinal (horizontal in-plane) loading, and
between in—plane and lateral (transverse, i.e., normal to the plane of the arch)
loading. The developed computer program performs elastic, plastic, and
combined analysis (which considers geometric, material, and both

nonlinearities, respectively).

M Nonlinear Finite Element and Solutions

The arch was modeled using a straight beam finite element model. The
elastic nonlinear model used here is that developed by Wen and Rahimzadeh
(42), and based on the averaged axial strain model. The plastic behavior was
modeled using the classic elasto—plastic hinge model where the plasticity is
confined to the member ends, (28).

The member plastic resistance was calculated using a three—stage

procedure that traces the member resistance through the three possible stages of

186

 

 

187

having both end joints elastic, one plastic, and both plastic. For each stage
involving plasticity at a joint, the new force point -which is calculated using the
plastic stiffness matrix— was made to stay at the yield surface by iteratively
modifying the gradient vector at the beginning of that stage.

For combined analysis, the elastic-nonlinear stiffness matrix used in the
plastic hinge model is calculated as a function of the elastic portion of the
member displacement vector. But since it was not known beforehand, the value
at the previous iteration -on the structure level- was used.

Four types of functions were used to represent the yield surface, namely
spherical, triangular, parabolic, and inverse parabolic. The singularity of the
gradient in the regions of discontinuity in these functions, i.e., along ridges
and at corners of the yield surface, was avoided by using special substitute
functions that would produce a unique direction in the singularity region.

The nonlinear solution used the fixed coordinates (vis-a—vis moving
coordinates), and the elements stiffness matrices were derived using the
Lagrange-small rotations coordinates. The system of nonlinear equations on the
structure level was solved using a modified Newton—Raphson method with
acceleration scheme, (9). A displacement—controlled procedure, (6), was used
to trace the response beyond the ultimate load point when needed. In addition,
a simple procedure for load step adjustment was implemented to define the
ultimate load more precisely. The solution convergence was checked using

separate displacement, force and energy tests.

 

 

188

Comparisons with reported solutions in the literature seem to validate the
program and show good agreement between the current implementation of
nonlinear element models and those reported by others. Comparisons with
experimental results show that the inverse parabolic function gives the most

consistent results, and therefore was used in the parametric study.

6.1.2 In-Plane Arch Behavior
The plastic, elastic, and combined ultimate loads of both two-hinged and

fixed arches were studied under different combinations of vertical and

 

longitudinal loading.

The change with slenderness ratio, L/ry, in the total ultimate load from
that under vertical load only due to increasing the longitudinal load depends
significantly on the type of analysis (i.e., the presumed behavior of plastic,
elastic-nonlinear, or combined). The plastic ultimate load gradually decreases
with an increase in the ratio of longitudinal to total loads, h, except for very
low values of L/ry and h. The elastic ultimate load increases with h for lower
values of L/ry and decreases for higher values.

The combined ultimate load is similar to plastic ultimate load for lower
L/ry —where plastic effects control the arch behavior— while it is similar to
elastic load for higher L/ry. Also, the effect of combining both plastic and
geometric nonlinearities is maximum for h = 0.25, where the total load result—

ant is perpendicular to the chord extending from the arch support to its crown.

189

For two-hinged arch, the ratio of longitudinal to vertical plastic ultimate
loads, Pump/Pym, range from 60% for stiff arches to 15% for slender ones. For
fixed arch, the respective ratios are 90% and 20%. The corresponding ratios of
elastic ultimate loads, Pkg/Pm,” range from an increase of 400% to a decrease
of 45% for two—hinged arch, and from more than 800% to 90% for fixed arch.
The combined analysis ratios, Pug/PW,” for two—hinged arch decrease from
60% for stiff arches to 30% for medium arches, but the ratio increases to 45%
for slender arches. For fixed arch, these ratios are 90%, 30%, and 40%,
respectively. For a given value of h, the reduction in the arch capacity to carry
vertical load is strongly dependent on the ratio of the ultimate load under
vertical-only load to that under longitudinal-only load.

Compared to two-hinged arch, fixing the arch supports results in 100%
increase in the vertical-only elastic ultimate load and at least 300% increase in
longitudinal-only ultimate load. The vertical—only plastic load remains the same
because of failure due to pure axial load in both cases, while the longitudinal—
only plastic ultimate load increases. by 50%.

For both two-hinged and fixed arches, the interaction curves between
vertical and longitudinal components of the ultimate load are convex for plastic
analysis and concave for elastic analysis. Those of combined analysis change

from convex to concave as L/ry increases.

 

190

The arch capacity to carry vertical load (e.g. , dead load) due to the
presence of longitudinal load always decreases for all types of analysis.
However, this decrease is larger for plastic analysis than elastic analysis.

Comparison of the arch behavior under vertical symmetrical and
longitudinal load with its behavior under vertical unsymmetrical loading reveals
similar behavior. This is specially true with regard to the rate of decrease of
the vertical symmetric component in both cases. This similarity provides a
simple measure of the decrease in arch capacity to carry both live and
earthquake loads simultaneously with the dead load, for example.

The failure modes at ultimate load are unsymmetrical, and the deflection
magnitude at ultimate load increases with h for all types of analysis. The
magnitudes for combined analysis are smaller than those of elastic and plastic
types of analysis. As L/ry increases, the magnitude of elastic deflections at
ultimate load decreases, while that of plastic and combined analysis increases.

For two-hinged arch, the plastic hinge formation moves from near the ,
support toward the quarter point as h increases from 0.0 to 0.25, then stays in
that region as h increases further. For fixed arch, plastic hinges form at the
support first. Afterwards, the arch behaves similar to two-hinged arch. The
force path at the quarter point of the arch span shifts from predominantly
compression axial force for vertical load only to predominantly bending
moment as the longitudinal load ratio increases. This behavior is accelerated in

the presence of geometric effects, and due to fixing the arch supports.

 

191
M; mm

The lateral ultimate load is insensitive to fixing the arch against in-plane
rotation at the supports, while the section lateral properties affect it
proportionately. As a result, only fixed arch was studied.

Under vertical and lateral loading, the arch plastic ultimate load
decreases gradually as the ratio of lateral to vertical load, s, increases. The
elaStic ultimate load drops suddenly in the presence of lateral load due to
change from in-plane to lateral failure mode. The combined ultimate load shifts
from a gradual decrease to a sudden drop as L/ry increases. Since the in—plane
and out-of—plane ultimate loads are not directly related to each other, their
relative ratios are not meaningful, and the observed trends of ultimate load
cannot be generalized; instead, the reduction in the vertical component with s is
more meaningful.

The interaction curves are convex for plastic analysis, and slightly
concave for elastic analysis. Those of combined analysis are closer to elastic
curves for the whole range of L/ry. At the ultimate load, the lateral deflections
are distinctly larger than in—plane ones for all types of analysis, with the
combined curves having smaller magnitudes. The plastic hinge formation at
supports causes large rotations and contributes to the large deflections.

The combined ultimate load surface was constructed by applying the
three loads simultaneously, namely, vertical, longitudinal, and lateral loads.

The shape of the surface roughly resembles a dome with the vertical load axis

 

192

as its axis of revolution. However, the relative magnitude of the arch vertical
in—plane to the lateral-torsional ultimate loads is important in determining the
shape of the surface. For an arch with lateral—torsional ultimate load larger than
its in-plane vertical ultimate load, the ultimate load surface would have a
significantly different shape.

When the arch fails in lateral-torsional mode, as in the studied case, its
capacity to carry vertical load does not change significantly in the presence of
relatively small longitudinal load (up to 0.4 thM). Similarly, a small lateral

load (up to 0.2 ~F,,,,,) has negligible effects. This is due to the convex ultimate

 

load surface. The interaction curve between lateral and longitudinal loads is
convex for low vertical load ratio, and becomes linear as the magnitude of

vertical load component reaches the lateral-torsional ultimate load.

fl CONCLUSIONS

The main conclusions of this research are summarized in the following:
1. The plastic hinge model is versatile enough to represent the unique arch
behavior when. the axial force in the element is high. The function type used to
represent the yield surface in the plastic hinge model has significant effect on
the arch ultimate load. The inverse parabolic function produced values closer to
experimental results than other types of functions compared.
2. The presence of longitudinal loads on the arch significantly decreases the
arch capacity to carry vertical loads for all type of analysis, particularly plastic

analysis. The ratio of the arch ultimate load under longitudinal-only load to its

   

 

193

ultimate load under vertical-only load is an important factor in affecting the
rate of decrease. ‘

3. The combined analysis represents a lower bound for both elastic and
plastic types of analysis, and the arch behavior shifts from one that is
dominated by plastic behavior to one that is dominated by elastic behavior as
L/ry increases. The effects of combining both plastic and geometric
nonlinearities is highest when the ratio of longitudinal to total loads, h z 0.25,

when the direction of the total load resultant would be approximately

 

perpendicular to the chord extending between the arch support and its crown.
4. The presence of longitudinal load affects the arch behavior in many
aspects in a quite similar way to the presence of unsymmetrical gravity load
(e.g., live load), in particular, in the reduction of arch capacity to carry

vertical symmetrical load (e.g., dead load). This suggests using an equivalent

 

unsymmetrical load to represent the longitudinal load effects in a preliminary
design process.

5. When the arch lateral-torsional ultimate load is smaller than its in-plane
vertical ultimate load, the arch failure mode changes from in-plane to out-of-
plane mode. Moreover, the presence of lateral load on the arch causes signifi-
cant drop in the arch ultimate load.

6. The shape of the ultimate load surface is significantly related to the

failure mode. For out—of—plane failure mode, the surface shape due combined

analysis was found to convex-like.

   

 

194

§._3_ RECOMMENDATIONS

The behavior of arches under different combinations of loading is -,
broadly open for research. The following topics are some examples of topics
still in need of exploration.
1. The arch behavior under combinations of both vertical unsymmetrical
and longitudinal loads simultaneously with vertical symmetrical loads needs to
be studied, with the aim to develop a quantitative relationship between the arch
load carrying capacities for the two modes of loading. It is reasonable to expect

the results to support the current findings.

 

2. The study of arch bridges as a whole system need to be studied more
extensively. The extension of this study to include the effect of different
components of the arch is recommended. for example, the effect of the arch
deck and in—plane and out-of—plane bracing systems on the arch ultimate load
under longitudinal loads.

Also, expansion of the study of the interaction between vertical and
lateral loads and the ultimate load surface to include different ratios of lateral
to in—plane section properties and lateral bracing systems would produce useful
data for design.

3. Study of the dynamic effects of earthquake loading on the arch vertical
ultimate load capacity is important. In particular, estimation of the equivalent
static longitudinal load that represents the dynamic earthquake effects for

design purpose is needed.

APPENDIX

 

 

 

APPENDIX A
NIETHODS OF SOLUTION OF

NONLINEAR F...EM EQUATIONS

A; INTRODUCTION
The system of nonlinear equations generated by the finite element model

is solved here using procedures based on the Newton~Raphson iteration method.

 

The ultimate load of the structure is obtained by incrementally applying the unit
load vector to the structure until failure. This may be called load-controlled
analysis since the analysis is controlled by increasing the load parameter.

After the ultimate load point, more than one displacement path may be
possible for the same load decrement. For example, the displacement path may
follow an unloading path where the displacement direction is basically opposite
to that during loading, or may follow a post-ultimate load path where the
displacement path is a continuation of that during loading. The direction of
displacement is controlled by including it in some manner as a controlling
parameter in the analysis, and is called displacement—controlled analysis.

For this study, the load—controlled analysis procedure is the main

procedure used to obtain the ultimate load of the structure. The use of the

195

196

displacement-controlled procedure was limited to tracing the structure response
curves beyond the ultimate load and for comparison with previous analytical
methods.

Two methods for load—controlled nonlinear incremental analysis were
implemented: the direct Newton-Raphson (N—R) method, and the modified
Newton-Raphson method. The displacement-controlled analysis method is
implemented in parallel with the modified N—R method. The accelerated
modified Newton-Raphson method is used throughout the study for data
collection in association with displacement-controlled analysis when needed.

In this appendix, theoretical description and implementation of the three

methods is presented.

_._2 DIRECT NEWTON-RAPHSON METHOD

The nonlinear finite element equilibrium problem amounts to finding the
solution of the nonlinear equation, (for example, see ref. (5)),

am = P5(D) — Rer) = o,

where Pj(D) is the applied load through load increment j (assumed here to be
independent of D), Rj(D) the structure resistance at the end of increment j, and
D is the (unknown) structure displacement that satisfies the equation.

The direct Newton—Raphson method is a first order Taylor’s series
expansion of the nonlinear function, f(D), about the converged (known)

solution point at the beginning of increment j so that,

 

 

197

a

ij+dDj =ij+
<.. 1) (him)

1 an}.

In general, the derivative is the slope of the curve at point 0, and for structural
analysis problem it is the structure tangent stiffness matrix at that point, Km};
f(Doj + de) = Pj - R3, the equilibrium function at the end of increment j; and
f(Doj) = Poj - R°5(D°) = 0, the equilibrium function at the beginning of the
increment, Figure (A-1). The incremental form of the above equation is APj =
dPo‘“j z Km,j 'lej, where APj is the applied load for the increment at the
beginning of the solution, i.e., for the first iteration,

dry" = AP5 = Afj-Pl" + dP'J“, (A.1)
where AF is the load parameter that defines the actual loading forces as a ratio
of the applied unit load vector, P“, and dP"j'1 the residual force vector still
unbalanced after convergence at the end of previous increment. The increment
index, j, may be omitted in the following discussion for simplicity.

Since Taylor’s expansion was truncated, the obtained solution, le, is
only an approximation and the process needs to be repeated with the new point
(1) as the expansion point and so on until convergence is achieved. For
iteration i, the structure resistance at point (i) is calculated, ARi = AR(D,J',
iADi), where ADi = dD1 + ...+ dDi = AD,_l + dDi, and the new unbalanced
force vector at the end of iteration is found, dP,“ = AP — AR,. The incremental
equilibrium equation for iteration i may be written as,

f(Di) = 'AP — AR,(D°, AD,) = o, (A.2)

 

 

 

198

dpnh i A/

 

dP,“

 

AR,

D.‘.R.‘ Do
(0) /( < ))

"é— le 4% dDZ __9| one _>l an<'_ D

 

 

 

 

 

 

 

 

 

 

AD‘ = i an.

is].

Figure A-l Direct Newton—Raphson iteration process for solution of nonlinear
equations.

and the N ewton-Raphson iteration equation is,

K dDi = d Piil . (A’3)

Li—l
To avoid errors resulting from path—dependency, specially when plastic
effects are included, the increment resistance is calculated over the whole
increment for every iteration, AR, = AR(D°, AD,), rather than accumulating the
resistance increments for individual iterations, AR, = sum (dR,(D,, dD,), ...,
dRi(Di-la (1131))-
As discussed in chapter 3, if the solution process becomes unstable

during iteration, the step size is reduced and the increment is repeated. If it

 

 

 

199

becomes unstable for the smallest load increment, the analysis is either stopped

or switched to displacement-controlled procedure.

_.3 MODIFIED NEWTON-RAPHSON METHOD

In the direct Newton-Raphson method the structure tangent stiffness
matrix is evaluated for every iteration. For large number of degrees of
freedom, this consumes unproportionately large part of the computation time.
The modified Newton-Raphson method avoids this by using the tangent
stiffness matrix at the beginning of the increment, KM}, for all iterations during
the increment, Figure (A-2.a), so that,

Km dD,‘ = dP,“_ ,. (A.4)

This modification is based on the assumption that ij is a reasonable
estimate of the slope of the response curve at point i (i.e., Kai) for all
subsequent iterations. The disadvantage of this assumption is that the conver—
gence rate becomes slower and the expected benefit may be negated by the
increase in number of iterations required to obtain convergence. This is
specially true when the stiffness significantly changes during the increment,
due to new yielding of some member joints for example.

To overcome this drawback, an acceleration procedure reported by M.
A. Chrisfield, (9), and based on the secant approach was implemented to
accelerate convergence. In this procedure, the initial displacement vector

produced by standard modified Newton—Raphson method for a given iteration is

replaced by an improved estimate that is a function of the unbalanced force and

 

200

 

P idPn'fi i /
/7 (n-l) 4{
“I" WP;
/
(2)
(1)
AR AR'i
n—l n
AR2

AR,

 

i i
(0) ’(BO 9R0 (D0))

 

 

 

 

 

 

<— dD] ——>| dD2 ie 000

 

——>||<~dn,,
m————— AD‘ = i (ID, ___1
kl

(a) Standard modified Newton-Raphson method.

 

 

 

 

 

 

P dPii-l
(n-l) ( )—T dP:" =
n
dPl“ / 61132“ / , dPr‘aH
(2)
/ ...
/
1
API = ( ) AR:
v.5 AR _
dPo / AR2 " ‘
/ D
/ dDz
/ ARI
/
(0) (D: .R.‘. 03.))
/
<—— le He— dD2 ——>j

 

00' '—>i an
i D
AD‘ = 3:. an, ___)i

(b) Accelerated modified Newton—Raphson method.

 

Figure A-2 Modified Newton-Raphson iteration process for solution of nonlinear
equations.

201

displacement vectors of the current iteration and previous one.
In this procedure, Figure (A-2.b), the initial displacement vector, dD,',
for iteration i is found using equation (A.4) above, and the improved estimate

of the displacement vector, dB, is calculated,
dDI = flddDi—l + f2,1 dD,‘ I (A.5)
where f,,, and f2), are scalar factors defined as follows,

f2,i —f,,/r,,,
f1, = £21 (1 — gilt“) — 1,

where f3,,, f4,,, and f5), are the following scalar (inner) products,

tlj = dDgl ’ dPiu.2 s
t2,i = dDiT—l ' 5 Pilil :
t3, c113;T - 619:1, .

where 6P,_, is the difference between the unbalanced forces at the beginning of
the current and previous iterations, 6P,_, = dP,_,u - dP,_2“. The process is started
with f“, = 1, f“, = O, and dP,“ = AP.

Equation (A.5) above can be shown to be a form of secant relationship
by multiplying both sides of that equation by the difference between the

unbalanced forces, 6P,_,, which produces the following equation,
d1);r - 513,3, = dD,T_1- (11);.
For one degree-of—freedom system, this equation is described by the secant line

passing between points (0) and (l) in Figure (A-2).

 

 

202

When the change in stiffness during the increment and hence in response
is substantial, the acceleration procedure may cause false divergence of the
iteration process since it is based on the secant approach. To avoid this
instability, an upper limit is imposed on the norm of the accelerated
displacement vector so that,

IND," / |l dD,_1 II 3. maximum ratio = 5.

While slower convergence may occasionally result from this constraint, .
its stabilizing effect is generally advantageous.

In equation (A.4), the stiffness matrix at the beginning of the increment,
K,_,,, may produce unstable results. In this case it is reformulated with one or
both of the nonlinear effects being restricted. For example, when both plastic
and geometric nonlinearities are present the stiffness matrix is reformulated
using only geometric effects, Kg“. If it is still unstable or when only one effect
is present it is reformulated as the elastic-linear stiffness matrix, K,. If
divergence still occurs, it is an indication that ultimate load has been reached
and the analysis is either stopped or switched to displacement-controlled

analysis.

AA DISPLACEMENT-CONTROLLED ANALYSIS

The displacement—controlled analysis procedure used here is based on
controlling the displacement increment at a specified degree of freedom of a
selected structure node, as described by Batoz and Dhatt, (6). Instead of

specifying the direction and magnitude of the load parameter for load—

 

 

 

203

controlled increment and solving for the corresponding displacement vector, for
this procedure the direction and magnitude of the displacement at the specified
d.o.f., I‘, is specified and the nonlinear solution is sought for the rest of the
displacement vector and the load parameter (its magnitude and direction) that
satisfies the specified displacement.

In the following, the basic concept of this procedure is demonstrated by
explaining the steps through the first two iterations, Figure (A-3), then the
general iterative equations are presented. This procedure is implemented in
parallel with the modified Newton-Raphson method; i.e. , the structure stiffness
matrix is formulated and updated similar to that used for modified N-R method.
The initial steps of solution are as follows,

1. The solution begins by formulating the structure stiffness matrix at the
beginning of the increment and solving for the displacement vector, D‘V,
corresponding to the applied unit load vector, P".

The structure and solution instability is handled the same way as the
modified Newton-Raphson method. For direct Newton-Raphson method, the
stiffness would be updated at the beginning of every iteration.

2. The iteration process begins by assuming (or estimating) the displacement
vector for the first iteration, dD,, and the corresponding initial unbalanced
force vector, dPo“. The corresponding structure resistance and the unbalanced
force vectors are calculated, ARl = f(Do, dD,) and dP,“ = dPo“ — dR,,

respectively.

 

 

 

 

 

 

 

Pm
////
. - lv
dP,“(1‘) /./ // -dPI (F)
// dP“ r
/ 2 ( )
AP] =
“’3 a) MPG)
4132“ (r)= -le' (r)
ARI(F)
(0)
(0.5.11? (00))
AD‘(I‘) = AD,,,(1‘) = sold") DU“)

 

 

(a) Load—displacement Relation at d.o.f. F.

P /

 

 

 

 

 

 

 

 

 

 

/
dP1u /
// sz
= (I)
J v
“Po" A1321
AR, dDz 41132"
<———————é-
II
(D ‘.R i D do2
(0) / 0 0( 0))

 

 

(b) General load—displacement relation for the rest of the structure.

Figure A-3 Displacement-controlled analysis iteration process for solution of nonlinear
equations.

 

205

Batoz and Dhatt, (6), assume the value of dD1 to be zero for all entries
except for 1‘ where it is equal to the specified value, dD,(I‘) = AD,P(I‘), and the
initial iteration load parameter, Af1 = dfo = 0 (i.e., dP,“ = dPl'j“). An
alternative approach used here is to scale the displacement vector, D", and the
I unit load vector, P", so that the displacement at I‘ is equal to the specified
value, dD,(F) = AD,,,(I‘). The corresponding initial load parameter is, df0 =
AD,,,(I‘)/D"’(I‘), so that dP,“ = dfo-Plv + dP‘J‘l, and dD1 = dfo-D‘V. The load
parameter and displacement vector at the end of first iteration are, Af1 = dfo,

and AD1 = dD,.

 

The value of AD,,,(I‘) is estimated from its value at the end of the load—
controlled analysis under the same load vector. The average displacement per
unit load vector is calculated at the beginning of displacement-controlled
analysis, D""C"(I‘) = DLCA(I‘)/fLCA, where DLCA(I‘) and fLCA are the total
displacement at I‘ and total load parameter, respectively, at the end of load—
controlled analysis. The value of the specified displacement, AD,,,(I‘),
corresponding to a given value of the load parameter increment, Af,,,,, (speci—
fied in data input, for example) may be obtained, AD,,,(I‘) = Afm,,-D"LCA(I‘)
(note that Af,,, ¢ dfo). The final value of the load parameter after convergence
would be different from its initial value. The value of Af,,m is adjusted in case
of divergence during analysis similar to the load-controlled analysis.

3. For the second iteration, the unbalanced force vector at the beginning of the

iteration, dP,“, is applied to the structure, and the resulting displacement vector

 

 

 

206

is calculated, dDz“ = (K,,,,5)'1 'sz“, as shown by the dashed line in the figure.
This generates additional displacement at I‘, changing its specified value of
AD,p(I‘).

To keep AD,,,(I‘) constant, Figure (A-3.a), the change is offset by
applying a proportion of the unit load vector, dP,Iv = df1 'P", that produces a
displacement at 1‘, dDz'V(I‘), that is equal in magnitude but with an opposite
sign to dD2“(I‘) so that, dDz'V(I‘) = dfl oD‘V(I‘) = -dD2"(I‘). This is shown as the
dash-dotted line in the figure. The iterative load parameter, df,, is calculated
from this equation, df, = -dD2“(I‘)/D“’(I‘).

The incremental displacement vector for the iteration is calculated,
dD2 = dDz“ + dDz‘”. The values of displacements at other degrees of freedom
are modified as shown in Figure (A-3.b).

4. The total increment load parameter, Af2 = Af1 + df,, and the accumulated
increment displacement vector through the second iteration, AD2 = AD, + dD2
are obtained. The structure resistance and unbalanced force vectors at the end
of this iteration are evaluated as a function of ADz, and check for convergence

is performed.

The general steps of solution for iteration i may be summarized in the

following steps,

1. The displacement vector due to the unbalanced forces at the beginning of

the iteration, dP,,,“, is calculated, dD,“ = (K,,oj)‘1-dP,,,“.

 

 

 

 

 

207

The incremental load parameter for the iteration, df,,,, that keeps the
displacement at P constant is calculated, and the corresponding displacement
due to load vector, dD,“’, and incremental displacement for the iteration, dD,,

are found,

a.
rg-Ia
.1.

II

— dB.“ (1‘) / DMF).
(11).“ = df,_l.Dlv, (A.6)

l 1

do: + dD,“’.

a.
U
u

2. The total increment load parameter, Af,, the total applied force vector, AP,,

and total increment displacement, AD,, for iteration i are calculated,

Afi = Af,_l + de,
APi = A1>,_l + dr,_,~1>1", (A.7)
ADi = AD,_1 +dDi.

3. The structure resistance, AR, = f(Do, AD,), is calculated, and the
unbalanced force vector is obtained, dP,“ = AP, — AR,. Check for convergence

is performed, and the process is repeated until convergence is obtained.

 

 

 

 

 

 

 

 

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211

 

 

a. ' -

 

 

 

 

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212

 

 

 

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