BUCKLING BEHAVIOR OF SYMMETRIC ARCHES

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Authors Qaqish, Samih Shaker, 1950-

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QAQISH, Samih Shaker, 1950-BUCKLING BEHAVIOR OF SYMMETRIC ARCHES.

The University of Arizona, Ph.D., 1977 Engineering, civil

University Microfilms International r Ann Arbor, Michigan 48106

BUCKLING BEHAVIOR OP SYMMETRIC ARCHES

by

Sairiih Shaker Qagish

A Dissertation Submitted to the Faculty of the

DEPARTMENT OF CIVIL ENGINEERING AND ENGINEERING MECHANICS

In Partial Fulfillment of the Requirements For the Degree of

DOCTOR OF PHILOSOPHY WITH A MAJOR IN CIVIL ENGINEERING

In the Graduate College

THE UNIVERSITY OF ARIZONA

19 7 7

THE UNIVERSITY OF ARIZONA.

GRADUATE COLLEGE

I hereby recommend that this dissertation prepared under my

direction by Samih Shaker Qaqish

entitled Buckling Behavior of Symmetric Arches

be accepted as fulfilling the dissertation requirement for the

degree of Doctor of Philosophy

Dissertation Director Date

"4 tf-p

As members of the Final Examination Committee, we certify

that we have read this dissertation and agree that it may be

presented for final defense.

;^L*/

Himm

U PS *-0-"— •

/)*» 77

AUv-?7

2a mi 7.a (vl^. fin

Final approval and acceptance of this dissertation is contingent on the candidate's adequate performance and defense thereof at the final oral examination.

STATEMENT BY AUTHOR

This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.

Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author,

SIGNED

ACKNOWLEDGMENTS

The author wishes to express his sincere gratitude

to Dr. Donald DaDeppo, major professor and dissertation

director, for his helpful suggestions, wise counsel,

guidance, and encouragement during the course of this

study.

Special thanks are extended to the committee

members for reviewing the manuscript.

The writer expresses his appreciation to the

National Science Foundation for their financial assistance

under Grant ENG-76-10904.

Finally, this work would not have been possible

without the love, encouragement, and patience of Shaker

and Martha, the author's parents and brothers.

iii

TABLE OF CONTENTS

Page

LIST OF ILLUSTRATIONS vi

LIST OF TABLES X

ABS T R A C T . . . . . . . x i i i

CHAPTER

1. INTRODUCTION 1

Historical Perspective .......... 6 Purpose and Scope ............ 8

2. LITERATURE REVIEW 13

Arches with Uniform Cross Section .... 14 Arches of Variable Cross Section 22

3. FORMULATION OF ARCH EQUATIONS 27

Geometry of Deformation ......... 27 Stress Resultants ...... 32 Equilibrium Equations 34

Boundary Conditions . 39 Nondimensional Equations . . 41

Arches with Variable Cross Section .... 43 Circular Arch 4 3 Parabolic and Catenary Arches .... 48

4. METHOD OF SOLUTION 49

Stability of Equilibrium . 68

5. DATA PRESENTATION AND ANALYSIS ........ 77

Circular Arches ...... 78 Parabolic Arches ........ 102 Catenary Arches ............. 109 Arches Under Nonsymmetrical Loading . . . 119

6. CONCLUSIONS AND RECOMMENDATIONS 132

Recommendations for Further Research . . . 135

iV

V

TABLE OF CONTENTS—Continued

Page

APPENDIX A. NOMENCLATURE 136

REFERENCES 140

LIST OF ILLUSTRATIONS

Figure Page

1. Load-deflection curve showing the limit point . 3

2. Snap-through buckling ............ 4

3. Load-deflection curve showing the bifurcation point ... 5

4. Bifurcation buckling . 5

5. Extensional circular arches .... 9

6. Circular, parabolic, and catenary arches ... 11

7. Critical centrally placed concentrated load versus subtending angle a 19

8. Load-deflection diagrams for an eccentrically loaded arch, 2a = 106.2602° . 19

9. An element of the arch rib in the deformed and undeformed configuration 28

10. Deformation of an element of the arch .... 30

11. Centroidal element in the deformed state with all the external forces and stress r e s u l t a n t s „ . . . . . . . . . . 3 5

12. Point A in the deformed and undeformed configuration ....... 37

13. Centroidal element in the deformed and undeformed configuration 40

14. Circular arch with, variable cross section . . 44

15. Arch composed of many segments . 63

vi

vii

LIST OF ILLUSTRATIONS—Continued

Figure Page

16, Load-deflection and determinant-deflection curves for inextensional, uniform parabolic arch with clamped end supports under downward point load at the crown 70

17, Load-deflection and determinant-deflection curves for inextensional parabolic arch of Hg/Hc equal to 3.0 and with clamped end supports under vertical distributed load 71

18, Parabolic arch 73

19, Catenary arch 76

20, Buckling loads versus Hc/Rg for uniform, extensional circular arches with clamped end supports 84

21, Buckling loads versus HC/Rq for extensional circular arches of Hs/Hc equal to 2.0 and with clamped end supports ......... • 85

22, Buckling loads versus Hc/Ro for extensional circular arches of Hs/Hc equal to 3.0 and with clamped end supports 86

23, Buckling loads versus HC/Rq for extensional circular arches of Hs/Hc equal to 4.0 and with clamped end supports under downward point load at the crown 87

24. Load-deflection and S55~deflection curves for extensional circular arches of Hs/Hc equal to 3.0 and with clamped end supports subjected to vertical distributed load over the arch axis 90

25. Load-deflection and determinant-deflection curves for extensional circular arch with clamped end supports subjected to concentrated load at the crown 92

viii

LIST OF ILLUSTRATIONS—Continued

Figure Page

26. Hc/R0 versus Hs/Hc for circular arches with clamped ends 94

27. Buckling loads versus Hs/Hc for extensional circular arches subjected to vertical dis t r i b u t e d l o a d o v e r t h e a r c h a x i s . . . . 98

28. Buckling loads versus Hs/Hc for inextensional circular arches under dovmward point load at the crown 99

29. Buckling loads versus Hs/Hc for inextensional parabolic arches under vertical distributed load over the arch axis .... 105

30. Buckling loads versus Hs/Hc for inextensional parabolic arches under downward point load at the crown 106

31. Load^Hc/Ro and deflection-Hc/RQ curves for extensional uniform catenary arches with clamped end supports subjected to dis­tributed load over the arch axis ...... Ill

32. Buckling loads versus Hs/Hc for extensional catenary arches under vertical dis­tributed load over the arch axis 114

33. Buckling loads versus Hs/Hc for inextensional catenary arches under downward point load at the crown 115

34. Buckling loads versus Hs/Hc for inextensional circular arches subjected to nonsymmetrical load with p equal to 0.2 122

35. Buckling loads versus Hs/Hc for inextensional parabolic arches subjected to non­symmetrical load with p equal to 0.2 . . . . 124

36. Buckling loads versus Hs/Hc for inextensional catenary arches subjected to nonsymmetrical load with p equal to 0.2 127

ix

LIST OF ILLUSTRATIONS—Continued

Figure Page

•37. Buckling loads versus Hs/Hc for inextensional parabolic arches with hinged end supports subjected to nonsymmetrical load with different ratios of p ..... 131

LIST OF TABLES

Table Page

1. Critical loads and corresponding vertical crown deflections for extensional, uniform circular arches of different ratios E^/Rq and with clamped end supports 79

2. Buckling loads and corresponding vertical crown deflections for different ratios Hc/Rq for extensional circular arches of Hs/Hc equal to 2.0 and with clamped end supports 80

3. Buckling loads and corresponding vertical crown deflections for different ratios HC/R0 F°R extensional circular arches of Hs/Hc equal to 3.0 and with clamped end supports 81

4. Buckling loads and corresponding vertical crown deflections for different ratios Hc/Ro f°r extensional circular arches of Hs/Hc equal to 4.0 and with clamped end supports subjected to downward point load at the crown 82

5. Values of HC/Rq for circular arches with clamped end and different ratios of Hs/Hc 93

6. Critical loads and corresponding vertical crown deflections for inextensional circular arches of different ratios Hs/Hc under vertical, downward, dis­tributed load over the arch axis 96

7. Critical loads and corresponding vertical crown deflections for inextensional circular arches of different ratios of Hs/Hc under vertical, downward, point load at the crown 97

x

xi

Table

• 8.

9.

10,

11.

12.

13.

14.

15.

LIST OF TABLES—Continued

Page

Buckling loads for inextensional, uniform circular arches 101

Critical loads and corresponding vertical crown deflections for inextensional parabolic arches of different ratios Hs/Hc under vertical, downward, distributed load over the arch axis 103

Critical loads and corresponding vertical crown deflections for inextensional parabolic arches of different ratios Hs/Hc under vertical, downward, point load at the crown 104

Comparison between the buckling loads computed in this study and the buckling loads obtained by Austin and Ross (2) for inextensional, uniform, parabolic arches . 108

Buckling loads and corresponding vertical crown deflections for extensional catenary arches of different ratios Hs/Hc and Hc/Ro with clamped end supports, subjected to distributed load over the arch axis 110

Buckling loads and corresponding vertical crown deflections for different ratios Hs/Hc for extensional catenary arches of Hc/RQ equal to 0.01, subjected to vertical, downward, distributed load over the arch axis . 112

Critical loads and corresponding vertical crown deflections for inextensional catenary arches of different ratios of Hs/Hc under vertical, downward, distributed load over the arch axis 113

Comparison of the buckling loads for uniform catenary arches subjected to downward, distributed load over the arch axis ..... ....... 118

xii

LIST OF TABLES—Continued

Table Page

16. Buckling loads and corresponding vertical crown deflections for inextensional circular arches of different ratios Hs/Hc subjected to nonsymmetrical load with p equal to 0.2 120

17. Buckling loads and corresponding vertical crown deflections for inextensional parabolic arches of different ratios Hs/Hc subjected to nonsymmetrical load with p equal to O . 2 . . • 1 2 3

18, Buckling loads and corresponding vertical crown deflections for inextensional catenary arches of different ratios Hs/Hc subjected to nonsymmetrical load with p equal to 0.2 126

19. Buckling loads and corresponding vertical crown deflections for inextensional parabolic arches of different ratios Hs/Hc with hinged end supports subjected to nonsymmetrical load with different ratios of p ...... . 129

ABSTRACT

The stability of inextensional, slender circular,

parabolic, and catenary arches with variable cross sections

at large deflections is treated in this analysis. Clamped

and hinged end supports, as well as vertically distributed

loads over the arch axis, vertical downward point loads

at the crown, and nonsymmetrical loads are considered. For

nonsymmetrical loads, a ratio p = 0.2 of the distributed

load over the arch axis to the distributed load on the

horizontal projection over one-half of the arch axis is

assigned. The ratio of the depth of section at the Hs .

abutment to the depth of section at the crown, 5—, is "C

given values in the range 1 to 4. In addition, parabolic

arches with hinged ends subjected to nonsymmetrical loads

with ratio p = 0.3 and p = 0.4 are analyzed. . Furthermore,

extensional nonuniform circular arches with clamped ends

under distributed load over the arch axis and concentrated

load at the crown have been discussed.

The derivation of the differential equations of the

arch is based upon the assumption that cross sections of

the undeformed state remain undeformed and plane, but not

necessarily normal to the reference axis during deformation.

The developed differential equations are exact in the sense

xiii

xiv

that no restriction is placed on the magnitudes of the

deformations.

The governing differential equations were differen­

tiated with respect to time to derive associated rate

equations. With the aid of the rate equations the original

differential equations are solved numerically in an

iterative step-by-step integration process employing the

Runge-Kutta method.

It was found that the increases in the total load

at buckling for all arches are almost directly proportional

Hs to the increase in the ratio ==—. Circular arches with 0.3

Hc height-to-span ratio can withstand a larger total load at

buckling than either catenary or parabolic arches.

Furthermore, catenary arches can carry greater loads

before buckling than parabolic arches. The increases in

the total loads at buckling for parabolic and catenary

arches are higher than the corresponding increases for Hs

circular arches for all ratios of 5-7. Clamped arches can Hc

withstand greater total loads before buckling than hinged

arches. For the nonsymmetrical case, the total loads at

buckling decrease as the ratio p increases.

For extensional circular arches with clamped ends

subjected to downward vertical concentrated load at the Hc

crown, the values 0.06, 0.05, and 0.067 of tT~ are considered. K0

Hs These correspond to =— equal to 1, 2, and 3 respectively. Hc

The effect of rib shortening can be neglected for all

XV

values of less than or equal to these values. The corre-R0

sponding .values of distributed load over the arch axis

are 0.07, 0.03, and 0.02. For distributed load, the ratio Hc Hs — decreases as the ratio =j— increases, while for downward R° "C Hs point load this ratio decreases up to equal to 2.0 and

c Hs

then increases up to ==— equal to 3.0. Hc

CHAPTER 1

INTRODUCTION

In the classical linear elastic theory of struc­

tures, deformations and rotations are assumed to be very

small. When these conditions are satisfied, the equilibrium

equations may be written with respect to the undeformed

configuration. Most designs for arches are based on the t

fact that the pressure line to the most significant

combination of dead and live loads coincides with the

centroidal axis of the arch. In these cases, linear

theory is adequate because only very small deformations are

experienced before buckling occurs. Several studies of

elastic stability were made of such arches by Hurlbrink

(25), Dischinger (14), Ramboll (3 9), and Timoshenko and

Gere (49). For a loading, when the funicular curve does

not coincide with the centroidal line, the classical theory

is incapable of giving a true picture of the behavior of

an arch since it neglects the interaction between axial

forces and bending moments due to geometry changes. In

such cases the arch functions by carrying significant

bending moments as well as axial forces. The change in

geometry, which can have an appreciable effect on the

stresses, is taken into account in deflection theories of

1

2

structures in which equations of equilibrium are formulated

with respect to the deformed geometry of the structure. An

essential feature of all deflection theories of structures

is that the governing differential equations are inherently

nonlinear. In structures where the deformations are so

large that they require analysis by deflection theories,

they are described as geometrically nonlinear structures.

Because of the nonlinearities it is virtually impossible to

obtain analytical solutions of the equations of even the

simplest deflection theories. Thus, in the literature one

finds many formulations of problems but few solutions.

The field of nonlinear analysis has progressed

rapidly in recent years. The analysis of geometrically

nonlinear structures has received considerable attention

during the last decade, primarily because such problems may

be solved using large scale digital computers.

The analysis and design of many metal elastic

structures are sensitive to the relationship between the

states of deformation and the external loads, especially

if this relationship is nonlinear. Large distortions may

be due to small changes in external loads, while the

material behaves according to Hooke's law. This type of

behavior is encountered when the members are slender and

when the axial loads in these members approach critical

values.

Typical load-deflection curves of thin elastic

arches are shown in Fig. 1. The point on the equilibrium

path at which the load is a relative maximum is called a

limit point (29).

LIMIT POINT

a < o

VERTICAL CROWN DEFLECTION

Fig« 1, Load-deflection curve showing the limit point.

The load corresponding to this limit point is

called a buckling load or a critical load, and the

corresponding buckling mode is referred to as a snap-

through mode. Figure 2 illustrates symmetrical snap-

through buckling.

4

Fig. 2. Snap-through buckling.

The point at which the primary equilibrium path

intersects a secondary path is called a bifurcation point

as shown in Fig. 3. The load at this point is also

referred to as a buckling or critical load, and for a

symmetrical arch under symmetrical load the corresponding

buckling mode shape is antisymmetrical as shown in Fig. 4.

The buckling illustrated in Fig. 4 is sometimes referred

to as sidesway buckling or bifurcation buckling.

Structural stability is one of the major considera­

tions in the design of light and thin structures. Generally

these structures are desirable, not only for the immediate

savings in material cost, but, more importantly, for the

savings in fabrication, construction, and service life of

the structure. Aerospace structures are the most obvious

example of the use of light and thin members. These

a <

BIFURCATION

VERTICAL CROWN DEFLECTION

Fig, 3. Load-deflection eurve showing the bifurcation point.

Fig. 4. Bifurcation buckling.

6

structures are designed to satisfy strength requirements

and are very slender. The buckling may occur at a stress

level less than the elastic limit of the material. Vacuum

vessels, when exposed to uniform external pressures, may

exhibit instability and collapse at a relatively low stress

if the ratio of the thickness of the shell to the diameter

is small (50).

Historical Perspective

The arch has been employed as a structural unit for

more than fifty centuries. The Assyrians, Chaldeans,

Hindus, and Chinese all used the arch. The Egyptians, who

may have gained their knowledge of the arch from the

Ethiopians, used arches primarily for architectural effect,

rather than as a structural unit.

In Italy, however, the Etruscans began to use the

true arch to a great extent. When the Romans conquered

them and ruled Italy and the ancient world, they spread not

only their policies, but also architectural influence.

Arched bridges and aqueducts were constructed throughout

her territories.

The Roman arch structures were usually built of

stone, brick, or stone shells with concrete filling.

However, others were built of stone laid without mortar.

Some of these arch structures still stand today as

monuments to Roman engineers.

With the decline of the Roman Empire, aesthetic and

intelligent bridge construction came to a halt. By the

twelfth century, religious orders known as "Fratres Pontes"

came into being. Although these groups of devout men were

dedicated to the construction and repair of bridges, un­

fortunately the structures of the middle ages lacked the

engineering skills of the Romans. Introduced at this time

was the semi-Gothic or pointed Ogival arch, probably

imported from Persia by the Crusaders.

The Renaissance brought a revival of art and

architecture. Many beautiful stone masonry arch bridges

were constructed, some using the flat elliptical arch,

which was a notable innovation of this era.

The modern period of arch bridge building saw its

beginning in the formation of the French "Departement des

Ponts et Chaussees" in 1716. This was followed by a series

of technological advances. The first cast-iron arch bridge

was constructed in 1716. Iron could be rolled into

. structural shapes by 1783. The use of steel in bridge

work followed, in 1828. Much later, the art of making

arch bridges from reinforced concrete was introduced.

The development of this type of construction is

credited to Joseph Monier of Paris, who built his first

arch bridge in 1867. In 1894 F. Von Emperger introduced

the "Melan" system (which employed rolled I beams for

8

reinforcing! into the United States and built the first

reinforced-concrete arch bridges of considerable span (34).

Thus the arch has seen many changes and improve­

ments, and for 50 centuries or more paralleled the develop­

ment of civilization. And parallel to this development of

the arch as a long span, relatively thin structure, there

developed a concern over the stability of these structures.

Purpose and Scope

In the literature, no attempt has been made to

study the stability of nonuniform arches at large deflec­

tions. Nor has the effect of the extensibility of the

centroidal axis on the buckling loads for nonuniform arches

been discussed.

This study is concerned with the finite deforma­

tions and buckling behavior of slender symmetrical arches

of variable cross section at large deflections. In addi­

tion, discussions will include the effect of rib shortening

on the buckling loads of arches of variable cross section,

the behavior of extensional circular arches of variable

cross section under downward point load at the crown and

distributed load over the arch axis is analyzed.

Extensional circular arches are studied with

clamped end supports and a height to span ratio of 0.3 as Hc shown in Fig, 5. Different ratios of =—, the thickness at K0

the crown to the radius of curvature of the undeformed

•xniniiii

1.0

Ca} Vertical uniform distributed load over the arch axis.

1.0

CbJ Downward point load at the crown.

Fig. 5. Extensional circular arches,

10

centroidal axis, are assigned such that the corresponding

Ho values of the thickness at the support to the radius at

R0 curvature of the centroidal axis, are in the range 0-0.2.

Hc The effect of variations of these ratios, , on the R0

buckling loads as well as the vertical crown deflections

are analyzed. A comparison between the values of the Hc

buckling load corresponding to different ratios of p^-,

H c and the buckling load corresponding to — equal to zero,

0 or inextensional deformation is also presented.

As shown in Fig. 6, slender and inextensional

arches of variable cross section under concentrated loads

at the crown, vertically distributed loads over the arch

axis, and nonsymmetrical loads are also investigated. The

nonsymmetrical load is composed of uniform, vertical,

distributed load over the arch axis, LD, and uniform,

vertical, distributed load on the horizontal projection

L over one-half of the arch axis, L. A ratio p = =— of 0.2 •L-D

is considered which is adequate for practical engineering

purposes.

Three types of arches are considered: Type I

arches are circular with a 0.3 height to span ratio which

corresponds to a 123,856° subtending angle. Type II arches

are parabolic. Type III arches are catenary. In

parabolic and catenary arches, the height to span ratio is

equal to that of circular arches.

0-3

4 1.0

Nonsymmetrical loading

0-3

1.0

Uniform distributed load

(a) Hinged supports

0.3

1.0

Downward point load at the crown

0.3

1.0

Nonsymmetrical loading

0.3

1.0

Uniform distributed load

(b) Clamped supports

0-3

1 0

Downward point load at the crown

Fig. 6. Circular, parabolic, and catenary arches. H

12

Catenary arches under vertical, uniform, dis­

tributed load over the arch axis are analyzed only as

extensional arches. There are no prebuckling deformations

of inextensional catenary arches under this type of loading.

All three types are studied with both clamped and

hinged end supports as shown in Fig. 6.

In all types, the ratios of the depth at the support Hs to the depth at the crown, =—, are given values in the Hc

range 1.0 to 4,0.

Furthermore, parabolic arches with hinged end

supports under two additional ratios of p, 0.3 and 0.4,

are considered.

Analysis and discussion of the results are

presented later.

i

CHAPTER 2

LITERATURE REVIEW

A considerable amount of work has been done on the

stability of arches. Surprisingly, little documentation

of the work has been done. Schmidt and DaDeppo (43) did a

survey of literature on large deflections of nonshallow

arches as well as the finite deflections of shallow arches.

Briefly, in the course of the literature review, it was

found that:

1. No research has been done on the stability of

arches of variable cross section at large deflec­

tion.

2. No extensive studies have been made to determine

arch properties for which the effects of rib

shortening on large deflection behavior are

negligible.

3. No studies have been made of the nonlinear

behavior of arches under nonsymmetric distributed

loads as are commonly employed in design.

The main purpose of this chapter is to give the

reader a comprehensive idea about the state of the art on

the stability of arches in general and at large deflections

13

14

in particular. A secondary objective is to establish

that the main results of this research are new.

Arches with Uniform Cross Section

Euler (16) was the first investigator to discuss

the nonlinear theory of curved members. Euler's theory is

suitable for calculating deflections of slender arches

where the extensional effect of the centroidal axis may be

neglected. Winkler and Bach (cited in 50, p. 419)

developed the engineering theory of stress and strain of

nonslender curved beams. The Winkler-Bach hypothesis

states that plane sections normal to the undeformed

centroidal line remain plane, unextended, and normal to

the undeformed centroidal line.

Levy (30) and Carrier (5) determined an implicit

closed form solution (in terms of elliptical integrals)

for the postbuckling behavior of inextensional circular

rings.

Fung and Kaplan (17) discussed the stability of

shallow arches against snap-through under static loading,

while Hoff and Bruce (20) studied this phenomenon under

dynamic loading.

Langhaar, Boresi, and Carver (28) developed an

approximate approach for calculating buckling loads of

symmetrical circular arches with large deflections. On

the basis of the Winkler-Bach assumption, an expression

for the total strain energy is formulated. By discarding

certain terms in this expression, the problem was reduced

to a mathematically linear problem. A two-hinged semi­

circular arch under a vertical, downward, concentrated

load at the crown was analyzed. The critical load obtained

2 was 6 . 5 4 E I / R q. This agreed with the results obtained by

Lind (31), Langhaar et al. (28) also performed tests on

semicircular arches subjected to downward point load at

the crown. The experimental and theoretical results

agreed fairly well.

Dinnik (13) calculated buckling loads of in-

extensional prismatic parabolic arches subjected to down­

ward, vertical, load uniformly distributed on the

horizontal projection, prismatic circular arches subjected

to normal load uniformly distributed along the arch axis,

and prismatic catenary arches subjected to downward,

vertical, distributed load over the arch axis. In those

three cases, the arches under the specified loading are in

pure compression and undergo no deformation prior to

buckling.

Wempner and Kesti (51) presented a complete set of

nonlinear equations for curved members based on Winkler-

Bach hypothesis.

Lind (31) extended the theory of (28) to parabolic

and catenary arches. The buckling loads for circular

arches with hinged end supports for different hight-to-span

16

ratios subjected to concentrated load at the crown are

presented. In addition, the stability of parabolic and

catenary arches with hinged ends under downward pointload

at the crown is included in Lind's analysis. It was found

that the buckling load is independent of the shape of the

arch for the height to span ratios.

Gjelsvik and Badner (18) used the energy approach

to analyze the mechanical behavior of a nonlinear struc­

tural system that undergoes snap buckling. A complete

theoretical and experimental analysis of shallow arches

under vertical concentrated load at the crown was presented.

Good agreement between these results and those obtained by

more thorough theoretical analyses was obtained.

Nordgren (35) analyzed finite symmetrical deflec­

tions of circular arches with clamped edges subjected to

downward point load at the crown.

Tadjbakhsh (45), Greenberg (19), and Huddleston

(23) added the effect of center line extensibility to the

problem of straight elastics.

Lo (32) and Lo and Conway (33) presented the

critical load of two-hinged symmetrical circular arches

under both end couples and concentrated load at the

crown. It was found that the critical loads for circular

arches subjected to end couples, computed according to

Euler (16) theory of inextensional elastica and the exten-

sional theory, are in agreement for the rise to span ratio

£ > 0.13 if = 6400, and for £ > 0.19 if = 1600. L X li X

For arches subjected to downward point load at the crown,

the differences between the critical loads computed accord­

ing to both the inextensional and extensional theories,

b AT 2

are large for > 0.3, if —^— = 1600, i.e., if the arch

is not very slender. In the above expressions and in this

study, A is the area, L is the span of the arch, h is the

height, and I is the moment of inertia of the cross

section.

Kammel C26) and Dym (15) studied the extensional

buckling problem of arches under constant directional

pressure, while the relationship between ring buckling and

that of semicircular rings had been discussed by Singer

and Babcok (42) under dead pressure.

Huddleston (23, 24) studied the buckling

phenomenon of prismatic two-hinged arches with several

height-to-span ratios under vertical concentrated load at

the crown. The problem was formulated as a two point

boundary-value problem consisting of six nonlinear first

order differential equations. The problem was solved by

using a shooting method which is presented by Huddleston

(22), No restrictions were placed on displacements or

thicknesses of arches. It was shown that the critical

load for a pinned circular, inextensional arch with

EX hieght-to-span ratio of 0.25 is Pr = 13.01 —Results r Ro

were also presented for two hinged parabolic arches.

DaDeppo and Schmidt (6, 7) used the exact inexten-

sional theory of deep arches to analyze two-hinged circular

arches subjected to concentrated load at the crown and

point load applied eccentrically near the crown. For

concentrated load at the crown, the critical values of the

loads were calculated for different subtending angles.

EI For a semicircular arch, a critical load of 5.8605 —~ was R0

calculated. This value is 11.6% lower than the values

obtained by both Langhaar et al. (28) and Lind (31). A

comparison between Lind's values and the values obtained

by DaDeppo and Schmidt (6) is shown in Fig. 7. It was

found that Lind's (31) results are fairly accurate for

arches that have subtending angles less than 90°. For

downward point load near the crown, four different rise-

to-span ratios were considered. Figure 8 shows the load

deflection curve for a height-to-span ratio corresponding

to 106.2602° subtending angle. In this figure, a is the

radius of the arch in the undeformed state, h^ is the

horizontal displacement, v^ is the vertical displacement,

and e is the eccentricity of the application of the load

from the crown in the undeformed state. "he stability of

hinged arches under distributed and combined loads was

also studied by DaDeppo and Schmidt (8, 101.

DaDeppo and Schmidt (9) studied the nonlinear

theory of in-plane flexure of slender elastic curved

members taking into account the effect of transverse shear

45

40

ss

so

25 Pa' £1

20

IS

10

5

0 -O 20' 40' SO' to' WO' 120' HO' ISO' ISO'

aC

Fig. 7. Critical centrally placed concentrated load versus subtending angle a.

hp/a

\ L2«/L *0 ! 1^—2«/L • 0.001 : I—2«/L • 0.005 j

I 2 t / L » 0 . 0 1 0 ,

Po« ez

OJO aoB h P /a t v p /a

0.06 0.02

Fig. 8. Load-deflection diagrams for an eccentrically loaded arch, 2a = 106.2602° — Height—to—span ratio of 0.25.

2 0

deformation and rotary inertia. The large deflections of

elastic arches and beams with shear deformation were also

discussed by DaDeppo and Schmidt (11). However, no

numerical results were produced.

Dym (15) studied the bifurcation buckling and post-

buckling behavior of steep, compressible, circular arches

with pinned and clamped edges under uniform constant

directional pressure. It was shown that pinned arches

demonstrate a change from unstable to stable behavior as a

semicircular arch is approached, while clamped arches

are unstable after bifurcation. Dym's analyses were based

on Koiter's (27) theory.

Hubka (21) studied the initial stages of post-

buckling of a circular ring subjected to hydrostatic

pressure by use of the quadratic form of Newton's method

which is discussed in detail by Thurston (48).

Rehfield (40) examined the postbuckling behavior

of circular rings for two types of loading: [1] external

hydrostatic pressure (the loading is always directed

normal to the centerline of the ring), and [2] uniform

radial loading (the loading is always directed parallel to

its original direction prior to buckling).

It was found that a ring under hydrostatic

pressure is unstable after buckling, while the uniform

radially loaded ring remains stable. The analysis developed

by Budiansky (3) and Budiansky and Hutchinson (4) on the

basis of the postbuckling theory of Koiter (27} was

employed.

DaDeppo and Schmidt C12) studied large deflections

and stability of hingeless circular arches under inter­

acting loads. Vertical concentrated load at the crown,

as well as the arches' dead weight, were investigated. It

was found that the buckling mode of nonshallow hingeless

arches depends upon the relative magnitudes of the point

load to the dead weight of the arch.

Large deflections and stability of circular,

slender, inextensional tied arches under vertical downward

point load were analyzed by Samara (41). Two types of

arches were considered:

1. Circular arches with hinged and roller end supports

elastically tied at the base.

2. Circular arches with hinged end supports,

elastically tied at intermediate points.

Austin and Ross (2) presented buckling loads, as

well as the buckling modes of inextensional parabolic,

circular, and catenary arches with uniform cross section

for different height-to-span ratios. Parabolic and

circular arches subjected to vertical, downward, point

load at the crown, and vertical, downward, uniform, dis­

tributed load over the arch axis with clamped and hinged

end supports were analyzed. Buckling loads for parabolic

22

and catenary arches with clamped and hinged end supports

acting in pure axial compression are also presented.

Arches of Variable Cross Section

According to Parrae and Holland (37), the depth of

an arch rib at the two critical sections, namely the crown

* and abutment, is assigned for strength requirement. In

general, the variation in depth between these sections is

dictated by aesthetic consideration. In the literature a

number of different rules for variation in depth of

section or moment of inertia are prescribed.

Most work in the field of arches with variable

cross sections is not concerned with stability. The

exceptions are the two papers by Dinnik (13) and Austin (1).

Dinnik (13) presented the buckling loads for

circular and parabolic arches with variable cross section

acting in pure axial compression. For circular arches,

the variation of the moment of inertia along the arch

axis was according to the following formula:

Ii ft I = IQI1 - (1 - -i) C2.1)

U J.Q U

where: I = the moment of inertia at any given point,

1^ = the moment of inertia at the crown,

1^ = the moment of inertia at the support,

0 = the angle between the tangent and the

horizontal at any given point.

23

For parabolic arches, the cross-sectional moment of inertia

was assumed to vary along the arch axis following the law:

I = Iq Sec <j> (2.2)

where: <}> = the angle between the horizontal and the

tangent to the arch axis at any given point.

Austin CD presented the buckling loads of

parabolic arches with variable cross section subjected to

vertical downward load uniformly distributed on the

horizontal projection for different height to span ratios.

Two variations of the moment of inertia along the arch

axis were suggested:

1 = 1 S e c $

and

1 = 1 S e c 3 < f > c

where: I = moment of inertia at the crown, C

<p = the angle between the tangent to the centroidal

axis at any given point and the horizontal.

Taylor and Thompson (46, p . 476) and Sutherland

and Clifford (44, p. 289) utilized Whitney's method for

the variation of the cross section of nonprismatic

parabolic arches. Whitney's approach assumed that the

variation of the moment of inertia from the crown to the

abutment is either:

24

1. A straight line variation where the moment of

inertia at any given section is computed from:

I = I /II - tl-m) cos <P (2.3) X C li

2. A quadratic variation where the moment of inertia

at any given section is evaluated from:

4X^ Ix = Ic/I1 ~ t1_in) "l"J cos <f> (2.4)

I where: m =

rs cos *s

I = moment of inertia at the crown,

<J>s = the angle between the horizontal and the

tangent to the arch axis at the supports,

X = horizontal distance from the center line of

the parabolic arch to any given point on the

centroidal axis,

L = span of the arch.

Timoshenko and Young (50) assumed that the variation

of the moment of inertia of circular arches along the

centroidal axis is according to:

1 = 1 S e c < p ( 2 . 5 ) u

or

1 = 1 Sec3 <j> (2.6) c

where: I = moment of inertia at the crown. c

25

<f> = the angle between the horizontal and the

tangent to the arch axis at any given point.

Much work in the field of arch analysis was

performed by the Portland Cement Association (38). Circu­

lar and parabolic arches with variable cross section were

investigated and the results tabulated for stress resultants

at different points on the arch axis. All analyses were

based on the linear elastic theory. The variation in

depth of circular arches was according to Parme (36):

t = t (e - d cos <j)) (2.7)

where: t = thickness at the crown. c

t = thickness at any given point,

K - cos e = -= t— with t = thickness at supports,

1 - COS <j>A a tftr I

<}>_ = angle from crown to supports, and k = t /t , A CX

d = e - 1,

<|) = angle measured from crown to any point.

The variation in depth of parabolic arches was determined

by the following law (37):

= f cc + a " V Cf > 2 «•«)

where: t_ = thickness at the crown, c '

t = depth of section at supports,

26

x = distance from the crown to any intermediate

point,

a = half of the span.

This formula will also be adopted in this study for the

variation of the cross section of parabolic and catenary

arches.

Parme and Holland (37) presented design tables for

symmetrical parabolic arches of variable cross section

with different height to span ratios. The Equation (2.8)

was used for the variation of depth along the arch axis.

Different ratios of abutment to crown thickness with

various loading conditions were investigated. All results

were based on the linear elastic theory.

CHAPTER 3

FORMULATION OF ARCH EQUATIONS

The derivation of the governing differential equa­

tions of the arch are based on the assumptions that plane

cross sections of the undeformed state remain inextensional

and plane, but not necessarily normal to the centroidal

axis, during deformation. This allows the consideration

of transverse shear deformation. The basic geometric

variables are the current position and the stretch £ =

1 + £, rather than the displacement from the reference

configuration and strain £. As derived here, the nonlinear

differential equations which govern the mechanical behavior

of the arch are exact in the sense that no restrictions are

placed on the magnitudes of the deformations.

Geometry of Deformation

Figure 9 shows an element of the arch in the

undeformed state. Its length, central angle, and radius

of curvature of the centroidal line are dS, d<f>, and R

respectively. After deformation the element has gone

through a rotation g, and the new length, central angle,

and radius of curvature are dS*, d<f>*, and R* respectively.

27

28

d(D

Fig. 9. An element of the arch rib in the deformed and undeformed configuration.

29

The deformed length of the element is

dS* = (1 + ec) dS (3.1)

where: e = the strain at the centroidal axis. c

The central angle in the deformed state

d(j>* = dcf> + dB (3.2)

and the radius of curvature of the deformed centroidal

line is

K* = s-*-=al£- t3-3)

Substitution for d<J>* and dS* from Equations (3.1) and (3.2)

into Equation (3.3) yields

K* = — = (3 4) K R* (l+ec)ds* 1 ' '

The curvature in the undeformed state is

v — i. = (3 5) K ~ R dS* {'

By making use of Equations (3.5), (3.3), and (3.1),

Equation (3.4) becomes

K* = 5T = tirsp- «+(!'> or §fi=K+B' (3.6)

where prime denotes the derivative with respect to S.

From Fig. 10, the length of the element in the deformed

state at any distance z from the centroidal line can be

expressed as

d<P

Fig. 10. Deformation of an element of the arch.

31

ds* = (R*-z cos y) d(j>* - z sin y + z sin (y+dy) . (3.7) z

With the aid of the relation cos dy = 1 and sin dy = dy,

expansion of sin (y+dy) yields

dS* = (R* - z cos y) d<J>* + z cos y dy. (3.8) z

However, dS* can also be written as z

dS* = dS (1+e ) ' (3.9) z z «

and

dS = dS - zd<|>. C3.10) z

By making use of Equation (3.10), Equation (3.9) yields

dS* = dS - zd<f> + e (ds-zd<}>) . (3.11) z z

Equations (3.6), (3.8), and (3.11) may be combined to

obtain

1+e (K+g? - z cos y)(K + 3') + z cos y y'

= * - i + » - i 2 »

Rearrangement of the terms in Equation (3.12) yields an

expression for ez:

e, = A + I " (1+3'R-y'R) cos yj}. (3.13) Z i\""Z C i\

The effect of transverse shear deformation is very small

and can be neglected for practical engineering purposes,

32

especially if the arch is slender. When shear deformation

is negligible Equation (3.13) reduces to

S - R?i (6C " ZS'! <3-14)

or, in view of relation (3.5) as

s - rhc? (Ec - *»•>• <3-15)

Stress Resultants

The material is considered to be linearly elastic

and to obey the one dimensional form of Hooke's law,

a = Ee

where: a = the stress,

E = Young's modulus,

e = the strain.

The normal force and bending moment can be expressed as

N = f f a dA = // Ee dA (3.16)' A Z A Z

M = - // CT„ z dA = - // E z e_ dA. (3.17) A Z A Z

The value of &z from Equation (3.15) substituted into

Equations (3.16) and (3.17) yields

N - lec " rh* da - B" !l i=k5- (3-18) A A

M = -E Ie f f dA - // dA] (3.19) A A

33

2

but

Kz2 , 1 Kz , n + z and ,——— = •=——— + 1 1-Kz 1-Kz 1—Kz 1-Kz

and the reference line is taken to pass through the

centroid of the cross section and therefore,

f l i=s - K i t i=s - jr (3-20)

and

1 2 z2 f f dA = K // —— dA + A = A Cl+Z) (3.21) A A

where

A

Substitution of Equations (3.20) and (3.21) into Equations

C3.18) and (3.19) yields

N = EA (1+Z) ec - 3' (3.22)

EAZ , EAZ - | #— oo\ M = - ec + —2~ 3r. (3.23)

K

Expressions for ec and 31 in terms of M and N can be

obtained by solving Equations (3.22) and (3.23). Thus,

e = F,N + F~M C 1 2

34

(3.24)

3' = F2N + F3M (3.25)

where

F1 EA K (1+Z)

In terms of the stretch C = 1 + £c and the angle <p* = <J> + 3

Equations (3.24) and (3.25) become

Equilibrium Equations

Assume that distributed normal and tangential loads

and pt act along the reference line. Figure 11 shows

an element of the centroidal axis of the arch in the

deformed state with all the external forces and stress

resultants. Summation of forces in £n and directions

yields

^ i d<i* » „ dd>* — . d<b* PtdS* + Nr cos —^ Nil cos —j— - Qr sin —|—

C = 1 + F-jN + F2M (3.26)

»*= 1 + F 2N + F 3M (3.27)

QZ sin = 0

^ - j. , d<t>* „ . d<f>* _ d<b* PndS* + Nr sin —+ N& sin —• + Qr cos —|—

t QA cos = 0. (3.28)

35

d<P

Fig. 11. Centroidal element in the deformed state with all the external forces and stress resultants.

36

Assuming d<J>* is very small, then sin d<j>* = d<j>* and

cos d<j>* ~ 1, and Equations (3.28) become

Nr - N£ = - Pt dS* + (Qr + QZ)

Qr - Ql = - P ds* - (Nr + N5<) (3.29) n 2.

By neglecting small quantities of higher order, the

summation of moments at point 0 yields

Mr - MA = 2— (Qr + Q^) (3.30)

As dS* approaches zero, Equations (3.29) and (3.30) yield

d N _ = _ £ + Q

dS* *t u dS*

( 3"31 )

dM_ = _ Q dS* g*

Equations (3.31) are the differential equilibrium equations

of the arch rib.

To complete the formulation of the governing

equations, one needs relations between position variables

and deformation variables. Let the position of a point

on reference line be referred to a rectangular cartesian

coordinate system as shown in Fig. 12. Let Xq and Yq be

the coordinates of point A in the undeformed state and let

X and Y be the coordinates of the same point A in the

37

Yn

X

Fig. 12. Point A in the deformed and undeformed configura­tion.

38

deformed state, then the following geometric relationship

can be obtained:

gly = sin <J>* and g§* = cos <J>* (3.32)

Equations (3.31) and (3.32) can be written in

terms of S and £ as

dN d0^ - _ rp dS y dS ^t

| a + N g l = . t J n ( 3 . 3 3 )

s - - »

and

dX „ . , j. dS~ = sin 4>*

|| = C cos <J>*. (3.34)

At this point, the nonlinear differential equations

(3.26) , (3.27), (3.33), and (3.34) which govern the

mechanical behavior of the arch are assembled in one group

as

g = 5 sin **

g = ? oos •*

*

dN _ 0 £$1 = - rp dS w ds ^t

39

n

dM dS

Z, = 1 + FN + F2M

(3.35)

Boundary Conditions

Forces at the ends of a segment of an arch are

resolved into rectangular components as shown in Fig. 13.

These end forces at A' (X^,Y^) may be expressed as functions

of Q^, M^, and <j>* by writing the equations of

equilibrium for an element at A1. Thus,

cos <}>£ + sin <j>* = 0

P ^ sin (j)^ + cos cj>* = 0

M el + M

1 0 (3.36)

Similarly, at B' (X2,Y2) one obtains

Px2 ~ Q2 cos 2 ~ N2 sin ^2 = 0

M e2 M 2

0 (3.37)

40

undeformed reference state

deformed

state

>x

Fig. 13. Centroidal element in the deformed and undeformed configuration.

Nondimensional Equations

To put Equations 03.35), (3.36}, and (3.37) in

a form more suitable for computations, they are non-

dimensionalized. This is accomplished in two steps.

First, introduce the nondimensional variables

OR0 NE0 MR0 q " EI0 " = EI0 ' m " EI0 '

a 3 A 3 _ 2 PnR0 _ _ PtR0 „ _ PxlR0

Pn EIq ' Pt EIq ' Pxl EIq

— 2 — 2 — 2 p -.Rn M ,Rn P 0Rft yl 0 _ el 0 ^ x2 0

pyl EIq ' el EI0 ' Px2 EIQ '

Py2R0 ^e2R0 = X_ py2 EI0 ' e2 ~ EIQ ' RQ

y = I- , s = |- , k = , and a = | . R0 R0 Ro A0

where Aq, Iq, and RQ are the cross sectional area, the

moment of inertia, and the curvature of the centroidal

line respectively in the undeformed state at the crown.

Second, substitute the above defining relations into

Equations (3.35) to obtain

x' = £ sin <J)*

y' = C cos 0*

q' = -n<j>*' - cpn

where

n' = qtj)*' - Cpt

m* = - Cq

£ = 1 + C^n + C2m

<J)*' = C4 + C2n + C3n (3

Cn = 1

1 R2A ats) R0 0

C2 ~ kCl

r Aro AIo

3 ~ AI A0Ials)

C4 = k = cj)'

A = A (1+Z)

1 ~ r? ZA

j

pn = Pn(s/X/Yf <P*r CfXd)

Pt = Pt (s,x,y,<j>*,?,Ad)

Ad = distributed load intensity parameter.

By making use of the previous nondimensional quantities

Equations (3.36) and (3.37) become

pxl + ql COS 1 + nl sin *1 = 0

p ^ sin (f>^ + n1 cos <J>* = 0 '

M el + m 1 0 . 0 (3.39)

and

Px2 " q2 cos ^2 " n 2 s i n ^2 = 0

M L r\ e2 m 2 0 (3.40)

Arches with Variable Cross Section

Many formulas for the thickness variations of

arches have been suggested as mentioned in Chapter 2. The

depth at the two critical sections, the crown and the

abutment, are usually assigned to satisfy the strength

requirements. The variation in depth is usually ruled by

artistic considerations, rather than strength require­

ments. For circular arches, the approach developed in

this study is similar to the method used by the Portland

Cement Association (38), while the Parme and Holland (37)

approach is adopted for parabolic and catenary arches.

Circular Arch

Figure 14 shows a circular arch with variable

cross section. X, Y, R, X*, Y*, and as used in this

44

Fig. 14. Circular arch with variable cross section.

45

section are in reference to the undeformed state. The

extrados, intrados, and the centroidal line are defined as

circles with centers at different points.

The equation of the centroidal circle A is

2 2 2 X + Y = R

where: R = radius of the centroidal circle.

The equation of the intrados is

(3.41)

where: R, = radiu

The equation of

angle <f> with the x,.

Y = - X coC

(.3.42)

\s centers of the

.es.

A and making an

( 3 . 4 3 )

This line intersects the centroidal axis at point c, and

the intrados circle at point b. The coordinates of

point c are

and

X = - R sin <£

Y = R cos $

( 3 . 4 4 )

( 3 . 4 5 )

The coordinates of point b are

1/2-, X* = J s;*-n 2$ - 2 sin <j> (R^~a s;*-n ^ (3.46)

45

section are in reference to the undeformed state. The

extrados, intrados, and the centroidal line are defined as

circles with centers at different points.

The equation of the centroidal circle A is

X2 + Y2 = R2 (3.41)

where: R = radius of the centroidal circle.

The equation of the intrados is

X2 + (Y-a-jJ2 = R2 (3.42)

where: R^ = radius of the intrados circle,

a^ = vertical distance between the centers of the

centroidal and intrados circles.

The equation of any line passing through A and making an

angle <J> with the vertical axis, is

Y = - X cot <}) (3.43)

This line intersects the centroidal axis at point c, and

the intrados circle at point b. The coordinates of

point c are

X = - R sin <p C 3 . 4 4 )

and

Y = R cos <j> (3.45)

The coordinates of point b are

X* = | I-a1 sin 2<J> - 2 sin (f> (R2-a2 sin 24>)1/2] (3-46>

46

and

2 2 2 2 '^2 Y* = a^ cos <J> + cos <j> CR -3! s*n ^ (3.47)

The depth of the section at any given point may be obtained

from the relation,

H2 = 4 I (X-X*) 2 + (Y-Y*) 2J . (3.48)

By substituting the values of X, Y, X*, and Y* from

Equations (3.44), (3.45), (3.46), and (3.47) into Equation

(3.48), and by rearranging the terms, an expression for H

is obtained.

2 2 2 H = 2 JR + R^ + a^ cos 2<j) - 2 R a^ cos <}>

2 2 2 1/2 1/2

+ 2 (R-l - ax sin <|>) ^ cos <f> - R)J ( 3 , 4 9 )

H Q

where: a^ = R - R^ - y-

H = the thickness at the crown. c

R^ can be expressed as a function of R2, the depth

of the section at the crown Hc, and the depth of the

section at the support H . From Fig. 14 b

H R^ sin <f>^ = R sin <J>2 - s n $2

H s R^ cos <)>^ = R cos $2 ~ Y~ cos 2 " al (3.50)

47

and

R2 = CRX sin (J^)2 + (R1 cos (f^)2 (3.51)

t

Substitution of the values R^ sin <j>^ and R£ cos <f>^ from

Equations (3.50) into Equation (3.51), and rearrangement

of the terms yields

2R2 (l-cos$2) + j(H2+H2) -R (Hs+Hc) +Rcos<J)2 (Hc+Hs) Ri = 1 2R Ccoscf^-l)-I^cos^+H^

H Ht ^ scos4)9 — (3.52)

To obtain an expression for the depth of the section in

nondimensional form, let H = r|R, Hs = n-^R, Hc = HqR, and

R^ = anc substitute these values into Equations (3.4 9)

and (3.52). Thus,

2 2 2 2 2 ^/2 n = 2Il+ri2+a^cos2(J)-2a^cos(J)+2 (ri2~aisin 4")

1/2 CajCostf.-DJ (3.53)

•

where

12 (l-cosc|>2) + -(ti2+t1q) - (n j+n ..Q.) +cos4>2 ^no+t1i^

n2 icos<j>2 (2-n1)+n0-2J

nln0 - 2—

\

48

Parabolic and Catenary Arches

In this study, the depth of the section at any

given point will be computed according to the formula

suggested by Parme and Holland (37).

- 2

H = Hc + (Hs-Hc) (*) (3.54) a

where: H = depth of section at an intermediate point,

Hc = depth of section at crown,

Hg = depth of section at abutments,

x = distance from the crown to any intermediate

point,

a = half of the span.

Let H = nH and H = . Substitution of these values of c s 1 c

H and H into Equation (3.54) yields the nondimensional s

depth,

- 2

11 = l + Cn-L-1) (|) C3.55)

CHAPTER 4

METHOD OF SOLUTION

Nonlinear, first order, differential equations

(3.3 8) govern the mechanical behavior of arches. These

equations will be solved numerically because it is not

possible to obtain an analytical solution. These equations

may be differentiated with respect to time to obtain rate

equations which are linear with respect to the rates. The

rate equations can be numerically integrated first over arc

length to obtain the relationship of the rate of loading to

the rate of displacement, and then with respect to time to

obtain the load-displacement history of an arch.

Consider that all dependent variables, including

load, are functions of time as well as arc length that vary

quite slowly with time such that the arch undergoes quasi

static deformation. For quasi static loading and deforma­

tion, the character of instantaneous stiffness matrix is

related to stability of equilibrium. The instantaneous

stiffness matrix can be obtained through integration of

the rate equations. The instantaneous stiffness coeffi­

cients are the forces at the ends of a segment needed to

cause a, unit rate of displacement at the nodes. When

49

50

dealing with rate equations only, the increment in dis­

placements or loads in the integration with respect to time

must be very small to obtain accurate numerical solutions.

This analysis is based on both the rate equations and the

original differential equations. The results obtained in

this study are exact in the sense that the instantaneous

stiffness matrix is evaluated in the true equilibrium

position which satisfies the given conditions at the nodes

and the original differential equations for all segments.

Increment size is limited only to extent that the Newton-

Raphson method converges.

The nondimensional rate equations are obtained by

differentiating Equations (3.3 8) with respect to time.

These equations are

x' = £ sin <j>* + C 4>* cos (J)*

y' = C cos (f>* - C sip cj>*

q» = - n<p*' - n<j>*' - Cpn - CPn

n' = q<f>*' + q4>*' - bt - ZPt m' = ?q - ?q.

£ = C^n + C2m $*' = C2n + C3m (4.1)

where

Pn = 3p a * + 3x

3p *n 3y

y + 3p

3<|>* 4>* 3p

•

e 3p

+ 3 Ad Ad (4. 2)

*xj.

fi­ll

3Pt .

3x X +

3pt

3y y +

8pt 3 4»*

<j>* 9pt 3?

•

X. + 9Pt + 3 Ad Ad (4. 3)

51

3 ,3x. _ 3 ,8x. 91 l3iT' ~ 9s 9t

y. = 9_ C9Z) = 1_ (iZ) Y 91 l9s; 9s 9t

A i _ ^ f^S.) etc q at e3s] '

Similarly, differentiation of boundary conditions (3.39)

and (3.40) with respect to time yields

Pxl = - cos <j>* - nx sin <J>* + 4*^ sin <J>*

- cos <j)J

pyl = q1 sin cf>| - n^ cos «J>* + 4)*^ cos 0*

+ sin <J>*

= ~ m 2 C 4 . 4 )

and

Px2 = q2 cos 2 + n2 S n ^2 ~ ^2q2 S'*'n ^2

+ 4>2n2 cos ^2

• • • • p^2 ~ 2 S"*"n ^2 + n2 COS 2 ~ ^2^2 COS 2

- <}>2n2 s -n $2

Me2 = iti2 C4.5)

Equations (4.1) to (4.5) are linear with respect to the

rates.

Consider a segment of the reference line in which

the stress resultants and their derivatives with respect

to arc length are continuous. Let

s = undeformed nondimensional arc length,

s^ = s at "1" end of the segment,

S2 = s at "2" end of the segment,

such that

Si < s < s2

and let

Ix,y, <j>* ,q, n,m] s_.s = I*1,y1,«|)J,q1»nlfm1J

Ix,y,<J)*,q,n,mJ s=s = Ix2 ,y2, <J>* ,q2 ,n2 ,m2J

The solutions of Equations (3.39), (3.40), and .

(3.41). can be expressed as functions of s with x^, y^,

<j>*, q^, n^, m^, and Ad = Xd(t) as parameters. Thus,

x = x (s; x1,y1,c|)J,q1,n1,m1; Xd)

y = y (s; ,. . 7 ^d)

(}>* = (j>* (s; ... ; Xd)

q = q (s; .. . ; Xd)

53

n = n (s; ..

m = m(s;

; Xd)

; Xd)

and

Pxl

Pyl

Mel

PX2

Py2

M e2

~ pxi(xi'yi'<f>i'qi'ni'ini

Pyl C* *

Px2(--

Py2 *

Me2(. •

At s = s2 Equations C4.6) yield

x2 = x Cs; x1,y1,<j)J,q1,nlfm1

y2 = y(s;

<{>*=(})* (s; ...

q2 = q(s; ...

n2 = n (s; ...

m2 ~~ m (s} ...

Xd)

Xd)

Xd)

Xd)

Xd)

Xd)

Xd)

Xd)

Xd)

Xd)

Xdl

Xd)

(4.6)

(4.7)

(4.8)

The solutions of Equations (4.1) satisfying the

one-point boundary value problem

y = Y]/ 4>* = <3 = q-jy n = nlf and m = m1

C 4.9)

x = x 1'

54

at s = are

3x • , 3x • , 3x 1* . 3x A j. -X x = 35^ X1 + 3^ yl + *i + 9q^ ql ^ nl

+ 3HJ" ml + 3Ad

y = x1 + ... + g^- Ad

« . 3 d ) * * . . 3 ( j i * ? , <j> - x1 + . . . + Ad

a = l3_ i + + ^SL Xd q 3x1 1 *" 3 Ad

• _ 3n • , , 3n s , n - xx + .. . + 3Xd Ad

i = |*L_ i + ... + Ad (4.10) 3x^ 1 3Ad

At s = s^, Equations (4.10) yield Equations (4.9). At

s = ^2' Equations C4.10) become

x2 = ullxl + U12^1 + u13^1 + U14ql + U15nl

+ u16m1 + u17Ad

y2 - + u22y1 + u23i* + u244x + u25n1

+ u26m1 + u27Xd

55

*2 " u31xl + u32^ + + u34ql + u35nl

+ + u37*a

<*2 = u41xl + "42^1 + u43*l + u44ql + u45nl

+ u46ml + u47Xd

"2 = u51xl + u52^1 + u53^1 + u54ql + u55nl

+ u^mj + u57Xd

m2 = u61xl + u62^1 + u63^1 + u64ql + u65nl

+ Uggin^ + Ug7Ad (4.11)

where

3x 3x 3x „ _ 3x U , , = / U , ~ = ^ r - — / U , 0 = 7 T " T T f U , lll " 3x7' 12 ~ 3y ' 13 ~ 3<J>*' 14 3q '

3x 3x j _ 3x U15 " 3n2' U16 ~ Sn^' and u17 3Ad

- <*£ '21 3x Uo, - *zr-r

2

The force rates pxl, pyl, Mel, Px2, py2, and Me2

can be expressed as linear functions of x^, y^, q^,

n^, and Differentiating equations (4.7) with respect

to time, yields

56

3pxl 3pxl • 9pxl 8pxl • 3vn Y1 3<j>* 3q, ql xl ~ X1 ' 9yx *1 ' rl " Bq^

^Pxl . 3pxl • 9pxl * + 3^ nl + 3HT- ml + 3Ad" Xd

p. 3pyl • 8pyl

yl 3x, X1 + + 3Ad

9Mel • 8Mel • M. i = ,, x, + ... + " Ad el 3x^ 1 3Ad

- Px2 * , 3px2 ? , px2 3xx X1 3Ad X

- Py2 • . i 3py2 py2 ~ 3x, X1 3Ad

9Me2 • SMe2 • Me2 ~ Sx - X1 + ••• + 3Ad Ad C4.12)

The solutions of the governing differential equa­

tions and the rate equations are expressed as functions of

x t ^x' x' 1' **1' ^1 ^1' 1' x' x' 1' 1

respectively. In this study the Runge-Kutta method is

used to generate these solutions numerically.

An important step in the numerical analysis and

adaptation to arches with concentrated loads is to find q^,

n f and m^ such that when s = s2 in Equations (4.6.1)

57

through (4.6.3) the calculated values x(s2), y(' an<

<J>* (s2) are equal to prescribed values of x2, y2* an< ^*2

respectively. Assume that q-[a / ' an( mia are

approximations of the true values q-^, n^, and m-^, which are

consistent with x2, y2, an< ^2' t ien x2^ ' 2^ ' anc 2^

corresponding to q-[a r n±a ' an< m;[a ma written as

Ca) , (a) (a) (a) x2 = x Cs2fx1,y1,(|)*,q1 /n1 ,1^ ,-Xd)

y2aJ = Y Cs2,x1,y1,(j)*fq;[a) ,n.[a) ,m.[a) ;Ad)

$*U) = <j>* Cs2 ,x1 ,ylf (j)* ,q a) /n a) ,m|a) ; Ad) C4.13)

The Newton-Raphson method is applied to obtain improved

values of q-[a , n|a / mia to t le true values q^, n f and

respectively. Subtracting Equations (4.13) from

Equations (4.8.1) through (4.8.3) and expanding the

difference in Taylor's Series about q-[a , n{a r an< m;[a

and considering only the linear terms one gets

, . 9x_ 9x 9x_ x 2 - = 9 ^ 6 q l + a H ^ 5 n l + 3 i ^ 6 m l

Cal 3y2 s , 2 . . 8y2 . y2 ~ *2 = 3^ Sl! + 3^- + 3^ «">x

9<J>5 9^5 3<t>*

$2 $2 = 9ql" 6ql + 3E7 6nl + 3mT 6ml C4.14)

58

where 6q^ = q^ - q-[a t Sn^^ = , and 611^ =

are small corrections to be applied to q-j^ ' ' anc

to obtain improved approximations q-£a+"^ = + q^#

n a+"^ = and m a+ = + <5m^ to q^, n^, and

m^. This procedure can be repeated to obtain any degree of

accuracy desired. The differential coefficients in

(a) Equations C4.14) are evaluated at s = s2 with q^ = q^ ,

n^ = n a , and m^ = m a . These coefficients are identical

to the coefficients of q^, , and m1 appearing in Equa­

tions (4.11.1) through (4.11.3). Therefore, Equations

C4.14) can be written as

x2 " x2al = u14Sql + u15Snl + u165ml

y2 - y2(a) = u24Sqi + u25 n1 + "26^

• * - 4>*tal = u346qi + u356n1 + 61^ (4.15)

A forward integration scheme such as the Runge-

Kutta method may be used to calculate the entries needed in

Equations (4.13) and (4.15) for prescribed values of x^,

Yl, <j>*, q^, n^, and m^. With q^, n^, and m^ consistent

with the values for x2, y2, $2' the differential coeffi­

cients in Equations (4.10), (4.11), and (4.12), that are

also consistent with the required values for x2, y2# an<

<p£r are obtained i The calculation of the rates q^, n^.

™1' 2' m2f **2' Pxl' etc' terms °f load rate Ad and

59

the end displacement rates may be expressed in matrix

form. Let

, • • r" " * _r,"- " i*iT * T t - T 'T, ~ ' z2 2'^2 2 ' —

• r' * " • r" * * *T r*T -T. = Iq1,n1,m1J , r0 = Iq9fn9,xn9J , r = [r, ,r0] l2 2 ' 2 ~ 1' 2-

Pl = ^Pxl'Pyl'"^' P2 = IPX2'Py2'"e2jT'

9 m 9 m m m

P = IPi'P2J

u ll

u,, u

u„, u

u_, u

u. 11 12 13

u. 21 22 23

u. 31 32 33

, U 12

u, . u

u„. u

u. 14 15 16

24 "25 U26

U34 U35 U36

, U

u 17

13 u 27

u 37

U 21

u,., u

uri u

u^_ u

41 "42 u43

ur 51 "52 "53

61 "62 U63

' U22

U,, U-c U , 44 45 46

u54 U55 u56

u64 u65 u66

u 47

, U 23

u 57

u 67

lll

COS <J>|

sin tpJ

0

sin J

cos (jjJ

0

0

0

-1

A22

cos <f>* sin <|>* 0

- sin <J>* cos <j>* 0

0 0 1

B 11

0 0

0 0

0 0

Cq^ sin <j>* - n^ cos <}>£)

Cq^ cos + n^ sin <}>*)

0

B 22

0 0

0 0

0 0

C- q2 sin <j>* + n2 cos

C- q2 cos <J>i£ - n2 sin 4>*1

0

3p xl ax.

3p

d y .

xl 3p xl 3<j>*

3p

3x. xl 3p

W-xl 3p xl

8 + J

3p

3xi yi

3 p. yi 305

s =

3M

3xn el

3 p.

3x. x2

3M el

3pv x2 3*5

3p,

3x. Z2

3M

"3x-e2

3p XL 3 + S

3M e2 3 <}>*

61

and

T f 9pxl 3pyl 3Mel 3px2 3py2 3Me2 C " l^Ad 3Xd 3Ad 3Ad 3Xd 3Ad

Then, Equations (.4.4), (4.5), and (4.11) can be written as

pi = BnJi + An^i (4-16>

= B22z2 + A22*2 C4.17)

z2 = Ullzl + U12rl + *dU13

l2 ~ U21zl + u22^1 + JdU23 (4-18)

Expressions for and can be obtained by

solving Equations (4.18). This yields

ll = Vll*l + V12*2 + dV13

*2 = V21^1 + V22Z2 + *dV23 (4.19)

where

V11 = ~ U12U11' V12 = U12' V13 = *" U12U13'

V21 = U21 ~ U22U12U11' V22 = U22U12' and

V23 = U23 " U22U12U13*

Substitution of Equations (4,19) into Equations (4.16) and

C4,17) yields

62

Pi = tBll+AllVll)Zl + A11V12Z2 + *dAllV13

^2 = A22V21Z1 + B22+A22V22 Z2 + + *dA22V23 (4.20)

Let

S11 ~ B11 + A11V11 S12 A11V12

S21 " A22V21

C1 = A11V13 and

S22 B22 + A22V22

C2 1 + A22V23

Then Equations (4.20) can be written as

Pi S11 S12 V C1 ss = + Ad

P2 S21 S22 *2 C2

Equations (4.21) may be rewritten as

p = Sz + AdC

where

S = S11 S12

S21 S22

, and C = IC, C_]

(4.21)

(4.22)

S is called the instantaneous or tangential stiffness

matrix of a segment of the arch.

Consider an arch composed of several segments

rigidly connected at nodes such that the tangent to the

reference line is continuous everywhere as shown in

Fig. 15. Assume that concentrated loads act only at the

63

Fig. 15. Arch composed of many segments.

nodes. By considering the equilibrium of node (j) which

connects segments Cj-1) and j, the following equations can

be obtained:

px^ px2(j-l} + pxl(j)

py py2Cj-l) + Pyl(jl

M = M 0 , . , . e e2(j-l) el (j ) (4.23)

In which the superscript in parentheses indicates the node

3nd the subscript in parentheses indicates the connecting

segments.

64

Let

PlCj) Ipxl'pyl'MelJ (j)

P2 Cj) = IPx2'Py2'Me2J (j)

-T j (1) CD M CD (2) C2) (2) , . P = IPX rPy fM0 ',px ,py ,Me , ... J (4.24)

Then according to Equations C4.23), p^^ can be expressed

as

pCjl = P2Cj-ll + P1 Cj) (4.25)

and

pT = IPi(i)'P2Cl) + pl(2)'p2(2) + P1C3) + J (4.26)

Differentiating Equation (4.25) with respect to time yields

pT = IplCH'P2Cl) + P1(2)'P2(2) + pl(3)' + ] (4*27)

Let z be a column position vector of the node

points with elements such, that

- IXU>,y

... J C4.28)

Then

y-Cj) -Cj) i*(j)-i = r* v .£* ] l x , Y , 9 J l x 2 Cj-1) ,y2 Cj-1) 2 Cj-1)

= -rxicj)^icj)^icj)J (4,29)

65

The force rate p in Equation C4.27) can be expressed in

terms of the displacement rates z and Ad by making use of

Equations C4.28). and £4.22) respectively. This yields

p = Sz + AdC (4.30)

where

s =

Sll(ll S12(1)

21 CI)

(s22 CD

+S11C2)}

21C21

, o

S12 (2) °

CS 22(2 )

+S11C3)}

. S21C3)

12(3)

CS22(3)

+S11(4)»

(S22(n-l)

+S . . ) 11 (n)

12 (n)

S21(n) S22(n)

is the instantaneous stiffness matrix for the whole arch

and

= ICltl) C2C1)+C1(2) C2(2)+Cl(3) C2(n)]

The node position variables, deformations, and

stress resultants used in computing S and C are evaluated

in the true equilibrium position which satisfies the given

conditions a,t the nodes and the differential equations for

66

all segments. The column position vector for the node

points z is composed of specified and unknown position

elements. The elements of the force vector p corresponding

to prescribed position variables are reactive forces and

those corresponding to the unknown position variables are

prescribed forces. To facilitate the discussion, let zu

represent the unknown position elements and zg represent

- (a) the specified position elements. If zu is an approximate

solution for the unknown position elements, the Newton-

Raphson method can be employed to obtain an improved

approximation z a+ "^ which corresponds to the prescribed

- - (a) z and prescribed loads. The force vector, p , which is s

required to satisfy all conditions in the problem with

position an z , can also be computed. Let Sz 3 be

a small correction to be applied to z a' such that

SUW + = zu is the correct position. Consider p as

_ _ -(a) a function of the elements of zu, expand p - p in a

Taylor's series about z a and retain only the terms linear

in Sz^ to obtain

p _ p t a ) = S p C a ) = sC a ) S z C a ) ( 4 . 3 1 )

_ (a) In Equations ( 4 . 3 1 ) the elements of <5z

corresponding to the elements of the position vector whose

values are prescribed must satisfy the condition <5z a = 0

and the elements 6pu of p - corresponding to the

position elements zu are known. Therefore, contained in

67

Equations (4.31) is a subset of equations of the form

SpCa) = SCa) S5Ca) (4.32) Ml U U

fa.) If the determinant of is nonzero, Equation- (4.32)

may be solved for the corrections,

The step-by-step analysis of finite deformations

of an arch under prescribed loading, with the aid of the

rate equations, is outlined in the following discussion.

The equilibrium position, stress resultants, and reactions

consistent with the applied loads and the prescribed dis­

placements zg, are assumed to be known. Then, the

matrices V^, v 2' anc V13 "*"n E(3uati°ns (4.19), the segment

stiffness matrix S in Equation (4.22), and the total

stiffness matrix for the arch S in Equation (4.30) can be

calculated. After a small interval of time p, z , and Ad s • •

change by the known increments 6p = p5t, 6z = z St, and s s * —

SAd = AdSt, Accordingly, if Su is nonsingular Sz^ can be

computed from

6pu = SuS5u + <5AdCu (4.33)

Having 6z^ and 5zg, one can next calculate Sr^ for each

segment of the arch. The displacement zu + 5z^, stress

resultants r^ + Sr^, and reactions pg + <5ps corresponding

to the prescribed loading and displacements pu + ^p^,

Ad + 6Ad, z + Sz represent an approximation to a new true s s

equilibrium state. Then the Newton-Raphson method can be

68

employed to obtain improved values to whatever degree of

accuracy is desired. The new equilibrium state then serves

as the starting point for another increment in loading.

Stability of Equilibrium

Equation (4.30) is basic for any discussion of

stability. Let the subset of these equations corresponding

to the prescribed load rates and unknown displacement rates

be written as

pu = Su*u + W=u t4-30)

Equation (4.30) .can be written as

K - S A ( 4 - 3 4 )

where

Fu = Pu - WCu

Equations (4.34) represent the rate equations of

equilibrium for continued quasi-static deformation. An

essential condition for continued quasi-static deformation

from an equilibrium configuration under arbitrary loading

rate F is that S be nonsingular. When S„ becomes u u 3 u A

singular, there will be a load rate vector Fu such that

Equation (4.3 4) has no solutions. This implies true

dynamic behavior governed by equations of motions. In

this case, the equilibrium is critical and the correspond­

ing load is said to be a critical load.

69

For conservative systems, the equilibrium is

stable if S is positive definite. As the load on the

structure increases, the equilibrium becomes critical if

S transits from being positive definite to being positive-

semi definite. Thompson (47) called this the primary

critical equilibrium state. In the general case, the

primary critical equilibrium state is associated with a

simple maximum on a load deflection curve as shown in

Fig. 16. At this point, the structure exhibits a transi­

tion from a stable to an unstable equilibrium state. The

corresponding buckling mode is called the snap through

buckling mode. In snap through buckling, Equations (3.34) • A

have a solution with respect to zu, only if F = 0.

Figure 16 illustrates the snap through buckling for an

inextensional, uniform, parabolic arch with clamped end

supports subjected to downward point load at the crown.

In special cases, the primary critical equilibrium

state is associated with a point where a secondary

equilibrium path intersects the primary equilibrium path.

Such points are called bifurcation points as shown in

Fig. 17, At these points Equations (4.34) have two solu-

tions z and at least one of these is associated with u A

nonzero f . Figure 17 shows the bifurcation buckling for

an inextensional, uniform, parabolic arch with clamped

end supports subjected to vertical downward distributed

loa,d oyer the arch axis.

70

70

60 - 12

I-50 10

LLlg CM O

017 <cu LOAD

a: 30

DETERMINANT 10

0.3 0.1 0.2

Y/R

Fig. 16. Load-deflection and determinant-deflection curves for inextensional, uniform parabolic arch with clamped end supports under downward point load at the crown.

71

o

X

H Z < z

100 90 80 70

60 50 40 30 20

10

0

-10

-20

IU O

CC LU |— -30

-40 -50 -60

-70 -80

-90 -100

50

40

30

20

J° UJ o o 0 GC

-10

-20

-30

-40

-50

LOAD

BIFURCATION POINT

DETERMINANT

0 02 0.03

X I R

0.01

Fig. 17. Load-deflection and determinant-deflection curves for inextensional parabolic arch of Hs/Hc equal to 3.0 and with clamped end supports under vertical distributed load.

In this study, all arches are symmetric and com­

posed of two segments. Thus the instantaneous stiffness

matrix has 9 rows and 9 columns. The buckling mode of

symmetric arches with clamped end supports, subjected to

symmetric loading, can easily be determined by examining

the instantaneous stiffness matrix which corresponds to

the critical load. After applying the boundary condi­

tions, the instantaneous stiffness matrix is reduced to

S = u

'44

54

'64

45

55

65

46

56

66

For symmetrical snap buckling, the coefficient = 0.

For all other cases the load versus the vertical crown

deflection, the load-deflection curve, was plotted and

the buckling mode was determined from examining that plot

at the critical load.

A general purpose computer program was written to

implement the basic numerical methods described in the

preceding section. The input data for the program

requires Xq, y^, <J>, and k as functions of s. This

section outlines how the quantities are computed. All

arches studied in this analysis are composed of two

identical segments. The number of points across each

segment of the arch used for numerical work varied from

73

51-91. These points are equally spaced on the arc length

of each segment. The input data needed for numerical work

is evaluated for circular, parabolic, and catenary arches.

Circular arches have constant curvature and do not require

discussion.

As shown in Fig. 18, the equation of parabolic

arch with respect to rectangular coordinates is

y - h = - a tx-k) 2 (4.35)

where

a = and k = L/2 kZ

yf

(Ml)

y— h=—a(x— kt2

•» X

L/2 L /2

Fig. 18. Parabolic arch.

74

The arc length to any point on the arch axis can be

expressed as

0 / Z 2 - / / i + ( f e SQCx) = / / 1 + (j£> dx (4.36)

Differentiating Equation (4,35) with respect to x and

substitue for in Equation (4.36) one obtains

X0 2 2 1/2 SnCx) = / (1 + 4a (x-k ) ) dx (4.37) U 0

Integration of Equation (4.37) yields

SQ(x) = /R + | (l+4a2k2) + |_ in (VR+2ax-2ak)

1 2 2 r- ~ In ((l+4a k ) - 2ak) (4.38)

where

1/2 -/R = I tl+4a2k2) - 8a2kx + 4a2x2]

For specific values of Sq (x) , Equation (4.38) can be

solved for the corresponding values of x by using the

Newton-Raphson method. Then y, k, (f> can also be evaluated

for the corresponding values of x. The curvature of the

parabolic arch is evaluated from:

K = f f = 23 - 3/2 C4-39> Il+4az (x-k) 2

75

The equation of catenary arches with respect to

x and y coordinates as shown in Fig. 19 is:

y = h + e II - cosh ^J (4.40) ° /

In Equation C4.40) e can be evaluated by using the Newton-

Raphson method and the condition y(0) = 0. The arc length

to any point on the arch axis may be written as

X0 , 1/2 S (x) = / II + sinh lX dx (4.41) 0 o e

Integrating Equation (4.41) yields

Sn(x) = alsinh C^r-^0 - sinh (- ^-)J C4.42) u a, »

the Newton-Raphson method may be employed to find the

value of x corresponding to a specific value of Sq. The

values of y, k, and (f> that correspond to x can also be

evaluated. The curvature in the undeformed state k is

computed from

K = as ,2 .x-k. C4-43) a cosh C—r—)

a

76

y= h+e (1 - cosh I2SzJL>) w

L/2 L /2

Fig. 19, Catenary arch.

CHAPTER 5

DATA PRESENTATION AND ANALYSIS

In most studies variation in height to span ratio

has been given primary attention. Shape of arches and

variation in depth of section has been given little or no

attention. In view of this, the many variables involved,

and the necessity of keeping the work within reasonable

limits, this study was confined to arches of a fixed rise-

to-span ratio of 0.3, This is a practical ratio for

engineering purposes.

The ratio of the depth at the support to the depth Hs

at the crown, is assigned values in the range 1 to 4. A Hs c depth ratio rz— of four seems to be the upper limit of the Hc

practical values for this ratio. For extensional arches, Hc different values of the compressibility parameter are

Hs R° assigned for each ratio of such that the corresponding

H c H values of are 0, 0.05-0.08, 0.1, and 0.2. equal to

R0 R0 zero corresponds to inextensional deformation. Clamped and

hinged end supports, concentrated load at the crown,

distributed load over the arch axis, and nonsymmetrical

load are considered.

77

78

Circular Arches

The buckling loads as well as the buckling modes

and the corresponding vertical crown deflections, for

extensional circular arches with clamped end supports are

illustrated in Tables 1, 2, 3, and 4. Vertical downward

distributed load over the arch axis and vertical concen­

trated load at the crown are considered. Four ratios of Hs rj— :1,2,3, and 4 for circular arches subjected to vertical c downward point load at the crown are assigned. As shown

Hs in Tables 1 through 4, three ratios of —:1,2, and 3 are Hc

considered for circular arches subjected to vertical dis­

tributed load over the arch axis.

The results displayed in Tables 1, 2, and 3 show

that extensional circular arches with clamped ends can

withstand a greater total load before buckling when the

arch is exposed to vertical downward distributed load over

the arch axis rather than a downward point load at the

crown. There is a better distribution of stresses along

the arch axis, as well as smaller prebuckling deformation,

when distributed load is used instead of a concentrated

load at the crown. Hs As the ratio increases, the buckling load Hc

increases. This is attributed to the fact that prebuckling

deformations become smaller as the arch becomes increasingly

stiff. For downward point load at the crown of an

inextensional arch, the buckling load increases by 55%,

Table 1. Critical loads and corresponding vertical crown deflections for extensional, uniform circular arches of different ratios H c / RQ and with clamped end supports.

Ratios of the thickness

at the crown to the radius of curvature

®</V

Downward point load at the crown

Distributed load over the arch axis

*VEI0 VR0 Buckling mode PnR0/EI0 VE0

Buckling mode

0 . 0 (inextensional) 14.336

0.05 14.214

0.1 13.845

0.2 12.434

0.2052 snap through 18.055

0.2090 snap through 17.949

0.2204 snap through 17.585

0.2656 snap through

0.357x10 ^ bifurcation

0.450x10 -1 bifurcation

.-1 0.748x10 bifurcation

Table 2. Buckling loads and corresponding vertical crown deflections for different ratios Hg/Rg for extensional circular arches of Hs/Hc equal to 2.0 and with clamped end supports.

Ratios of the thickness

at the crown to the radius of curvature (Hc/R0)

Downward point load at the crown

Distributed load over the arch axis

~ 2 pVEIo VR0 Buckling mode WEIo VR0

Buckling mode

0 . 0 (inextensional 22.148

0.03 22.015

0.05 21.851

0.1 21.138

0.1703 snap through 37.520 0.0365

0.1735 snap through 37.143 0.0449

0.1794 snap through 36.536 0.0608

0.2078 snap through

bifurcation

bifurcation

bifurcation

1

Table 3. Buckling loads and corresponding vertical crown deflections for different ratios Hc/Rg for extensional circular arches of Hs/Hc equal to 3.0 and with clamped end supports.

Ratios of the thickness

at the crown to the radius of curvature (Hc/R0)

Downward point load at the crown

Distributed load over the arch axis

Bucklina „ Buckling ^Rq/EIq Y2/R0 mode ^nR0/EI0 Y2/R0 mode

0.0 (inextens iona1) 28. 938 0. 1565 snap through 56. 855 0. 358x10 1 bifurcation

0.025 28. 801 0. 1647 snap through 55. 871 0. 457x10' •1 bifurcation

0.034 28. 772 0. 1609 snap through 55. 179 0. 561x10" •1 bifurcation

0.067 28. 590 0. 1940 snap through 49. 63 0. 953x10" •1 snap through

oo

Table 4. Buckling loads and corresponding vertical crown deflections for different ratios Hc/Rg for extensional circular arches of Hs/Hc equal to 4.0 and with clamped end supports subjected to downward point load at the crown.

Ratios of the thickness at the crown to the radius of curvature

CHc/R0)

Downward point load at the crown

Jro /eiO VR0 Buckling mode

0.0 (inextensional

0.02

0.025

0.05

35.295

35.232

35.301

stable

0.1598

0.1695

0.1812

snap through

snap through

snap through

Hs 100%, and 150% as the ratio =— varies from 1 to 2, 3, and 4 Hc

as shown in Tables 1 through 4 and Figures 20 through 23.

On the other hand, the corresponding buckling load for

distributed load over the arch axis increases by 110% and Hs 215% as the ratio varies from 1 to 2 and 3. This c

indicates that the increase in the buckling load for

distributed load is higher than for a downward point load Hs at the crown for the same ratios of 5-—. Hc

From Table 1 and Figure 20, the buckling loads for

uniform extensional circular arches decrease as the H

R

that as the arch becomes thicker, the effect of rib

Q compressibility parameter =— increases. This indicates

0

shortening becomes increasingly important. The effect

of rib shortening on the critical loads of circular arches

subjected to point load at the crown is greater than when

subjected to distributed load over the arch axis. This is

shown in Table 1 and Figure 20. The change in the buckling Hc load for different ratios of =— is smaller for circular R0

arches subjected to distributed load than downward point

load at the crown. As one might expect, the increase in

the vertical crown deflection for distributed load is Hc higher than for a downward point load for increasing

H 0 c .ratios. By studying Table 1, a value of equal to 0.06

for downward point load at the crown gives approximately

1% difference between the buckling loads computed according

25 25

DISTRIBUTED LOAD OVER THE ARCH AXIS

(RR3O/EI0) 20 20

n o

DOWNWARD POINT LOAD AT THE CROWN ( PRJ/EI0)

005 0.1 0.15 0.2

Fig. 20. Buckling loads versus HC/RQ for uniform, extensional circular arches with clamped end supports,

00

40

30

LU cn e CC < 0 =

20

10

40

30 -

LU CM Q

DC <CL

20

10

DISTRIBUTED LOAD OVER THE ARCH AXIS (^Rj j /E l 0 )

DOWNWARD POINT LOAD AT THE CROWN (PR5/EI0)

0.05 0.1 0.15

Hc /R0

Fig. 21. Buckling loads versus Hc/Rg for extensional circular arches of H /H.n equal to 2.0 and with clamped end supports. s

00 U1

86

60 60

50 50

DISTRIBUTED LOAD OVER THE ARCH AXIS

(RRj/Elo) 40 40

o

LLI

DOWNWARD POINT LOAD AT THE CROWN

(PR2/E I 0 )

20 20

0 1 005 0

Hc /R0

Fig, 2 2 , Buckling loads versus H c / RQ for extensional circular arches of Hs/Hc equal to 3.0 and with clamped end supports.

87

0.025

Hc /a

Fig. 23. Buckling loads versus Hc/Rg for extensional circular arches of Hs/Hc equal to 4.0 and with clamped end supports under downward point load at the crown.

88

to the extensional and inextensional theories. On the. Hc other hand, a ratio of — equal to 0.07 for distributed R0

load over the arch axis is within 1% deviation between

the buckling loads computed according to both the exten­

sional and inextensional theories. So, for circular

arches with uniform cross section, the deformation can be Hc considered as inextensional if is less than or equal to R0

these values. It is clear that this ratio for the same

arch is dependent on the type of loading.

The effect of rib shortening on the buckling loads Hs

for values of — greater than 1 is greater for distributed Hc

load than concentrated load.

Til a r ra ir r* V* 0 c s~\ -F „ Hs For extensional circular arches of 5— equal to 2.0

Hc c and with clamped end supports, values of =— equal to 0.03 R0

and 0.05 for distributed load over the arch axis and for

concentrated load at the crown are considered. The

buckling loads corresponding to those values deviate by

1.4% from the buckling loads computed according to the

inextensional theory. Therefore, the deformation may be Hc

considered as inextensional if =r— of the arch is less than R0

or equal to these values. As shown in Table 2 and Figure

21, the buckling load decreases as the compressibility Hc parameter increases for both downward point load at 0

the crown and distributed load over the arch axis. The

vertical crown deflections for distributed load increase

more rapidly than for downward point load for the same Hc ratios of . 0 Hs For extensional circular arches of 5— equal to 3.0

Hc c and with clamped end supports, values of 5— equal to 0.02 R0

and 0.067 for distributed load over the arch axis and

downward point load at the crown are considered. The

corresponding buckling loads deviate by 1.2% and 1.1%

respectively from the buckling loads computed according to

the inextensional theory. As shown in Table 3 and

Figure 22, the decrease in the buckling loads for downward

point load at the crown is smaller than the distributed Hc

load over the arch axis for different ratios of —. The R0

vertical crown deflections for distributed load increase

more than concentrated load at the crown.

For extensional circular arches subjected to

downward point load at the crown, the buckling mode is

snap through. Extensional circular arches subjected to

distributed load, buckle by sidesway, except for a single Hs Hc

case where equal to 3.0 and equal to 0.067 as shown Hc R0

in Table 3. The arch buckles by snap through. The load-

deflection and stiffness coefficient (S55)-deflection

curves for this case are shown in Fig. 24.

The determinant of the instantaneous stiffness

matrix is equal to zero if S55 is zero and the corresponding

load is the critical load.

16

1 4

12

10

8

6

4

S 2 0 z

0

-2

- 4

-6

-8

-10

Mg.

LOAD

0.08 3 . 0 6 0 . 0 9

ad-deflection and S55-deflection curves for extensional circular es of Hs/Hc equal to 3.0 and with clamped end supports subjected vertical distributed load over the arch axis.

91 Hs

For extensional circular arches of ==— equal to 4.0 Hc

and with clamped end supports subjected to downward point

load at the crown, the change in the buckling loads up to Hc =— equal to 0.025 is negligible as illustrated in Table 4 0 Hc and Figure 25. For =— equal to 0.05, the stiffness matrix

R0 remains positive definite throughout the load deflection

history. Figure 25 illustrates the load-deflection and

the determinant-deflection curves for this particular case.

As shown in Table 4, the vertical crown deflections

increase very slightly as the compressibility parameter Hc Hc =r— increases. The buckling modes -for values of =— up to R0 K0 0.025 are of the snap through type.

Hc Hs Values of for different ratios of tj— at which R0 Hc

the deformation of circular arches subjected to downward

point load at the crown and distributed load over the arch

axis may be considered as inextensional, are presented in Hc Hs

Table 5. Figure 26 shows a plot of versus .. For %c Hc

distributed load over the arch axis, =— decreases as the Hs 0

ratio ==— increases. For downward point load at the crown, Hc C Hs Hs =— decreases up to 5— equal to 2.0 and then increases for 5— R0 c c greater than 2.0 as shown in Fig. 26. This figure can be

Hc used to find the compressibility parameter =— corresponding H R°

to different — ratios of circular arches with clamped ends "c

at which the effect of rib shortening can be considered as

negligible.

92

4 0 80

7 0

6 0 - 3 0

5 0 • u j

<0. Z 4 0 20

LOAD

3 0

111

10 20

DETERMINANT

10

OL 0

0.2 0 0.1 0 . 3

X/R

Fig. 25. Load-deflection and determinant-deflection curves for extensional circular arch with clamped end supports subjected to concentrated load at the crown — The ratios Hs/Hc and Hc/Rq are 4 and 0.05 respectively.

93

Table 5. Values of H_/Rq for circular arches with clamped end and different ratios of Hs/Hc.

Downward point load at Distributed load over the crown the arch axis

H/H s c H /Rn c' 0 VR0 Deviations VR0 VR0 Deviations

1 0,06 0. 06 1% 0.07 0.07 1%

2 0. 05 0.1 1.4% 0.03 0. 06 1.4%

3 0,067 0.2 1.2% 0.02 0.06 1.2%

4 0. 025 0.1 — —

94

0.08 • i

Ho

R0 / 0.06 v EXTENSIONAl / -

- CONCENTRATED LOAD

\ INEXTENSIONAL

0.04 w *

w

DISTRIBUTED * » 0.02 LOAD

0 ' ' ' ' 0 1 2 3

Hs/Hc

Fig. 26. H^/Rq versus Hs/Hc for circular arches with clamped ends.

Tables 6 and 7 contain buckling loads, and

corresponding vertical crown deflections for inextensional Hs circular arches of different ratios of =7— with clamped and c

hinged end supports. Vertical, uniform, distributed load

over the arch axis and vertical, downward, concentrated

load at the crown are considered. W in these tables

represents the total distributed load the arch is

subjected to at buckling. As shown in Tables 6 and 7

and Figs. 27 and 28, the increase in the buckling load

is almost directly proportional to the increase in the Hs ratio —. The buckling loads for circular arches with c

clamped ends subjected to distributed load over the arch

axis increases by 110%, 215%, and 320%, while the

corresponding increase for hinged arches is 100%, 200%, Hs and 295% as the ratio 5— varies from 1 to 4. Therefore, c

the increase in the buckling loads for clamped arches is

slightly higher than for hinged arches. For inextensional

circular arches subjected to downward point load at the Hs crown, the buckling loads increase rapidly as the ratio ~— nc

increases as shown in Table 7 and Fig. 28. The buckling

loads for circular arches with clamped ends subjected to

downward point load at the crown increases by 55%, 100%,

and 150%, while for hinged arches, this increase is Hs

approximately 80%, 155%, and 200% as the ratios jj— vary

from 1 to 4. For inextensional circular arches subjected

to downward point load, the buckling loads for hinged end

Table 6. Critical loads and corresponding vertical crown deflections for inextensional circular arches of different ratios Hs/Hc under vertical, downward, distributed load over the arch axis.

Ratio of the

thickness at

the support

to the thick­ness at the

crown,

(VV

Clamped end supports Hinged end supports

R„Ro/EIo "V o VRo

Buckling mode R„Ro/EIo wrO/EIO VRO

Buckling mode

1

2

3

4

18.055 22.117

37,52 45.962

56.855 69.647

75,848 92.914

0.357x10 ^ Bifurcation

0.365x10 ^ Bifurcation

-1 0.358x10

0.350x10 -1

Bifurcation

Bifurcation

7.625 9.341 0.180x10 ^ Bifurcation

15.159 18.570 0.160x10 Bifurcation

22.671 27.772 0.143x10 ^ Bifurcation

30.145 36.928 0.128x10 ^ Bifurcation

Table 7. Critical loads and corresponding vertical crown deflections for inextensional circular arches of different ratios of Hs/Hc under vertical, downward, point load at the crown.

Ratio of the thick­ness at the support Clamped end supports Hinged end supports to the thickness at

the crown - Buckling ~ Buckling (W Pro/EI0 VR0 mode PRo/EI0 Y2/R0 m°de

1 14. 336 • 2052 Snap through 11. 036 0. 1429 Bifurcation

2 22. 148 0. 1703 Snap through 19. 695 0. 151 Bifurcation

3 28. 938 0. 1565 Snap through 27. 998 0. 1507. Bifurcation

4 35. 295 0. 1598 Snap through 36. 175 0. 1483 Bifurcation

98

80 -

60 •

111

r» ©

cc

<0.?

^ ° 4 0 CC

20

CLAMPED

HINGED

3 4 Hs/ Hc

Fig. 27. Buckling loads versus Hs/Hc for inextensional circular arches subjected to vertical dis­tributed load over the arch axis.

99

CLAMPED

HINGED

Fig. 28. Buckling loads versus Hs/Hc for inextensional circular arches under downward point load at the crown.

100

supports increase more rapidly than for clamped end Hs supports for the same 5— ratios. By examining Tables 6 and Hc

7, as expected the total load at buckling for circular

arches with clamped ends corresponding to distributed load

over the arch axis is higher than the downward point load

at the crown.

For hinged arches, the buckling loads corresponding

to downward point load at the crown are higher than the

total buckling loads corresponding to distributed load over

the arch axis.

The buckling mode shape for all cases is bifurca­

tion, except for circular arches with clamped ends sub­

jected to downward point load at the crown. In this

instance, the buckling is of the snap through type. No

direct comparison between the buckling loads for uniform

circular arches presented in this study and the buckling

loads computed by Austin and Ross (2) can be made, since

the height to span ratio of 0.3 was not considered in

their analysis. Austin and Ross (2) present buckling loads

f for circular arches with height to span ratios, of

0,2887 and 0.3501, which correspond to subtending angles of

120 and 140 degrees respectively. A linear interpolation

between these yalues was carried out to estimate the

buckling loads corresponding to =0.3. As shown in

Table 8, these interpolated values deviate from the very

accurate values computed in this study.

Table 8. Buckling loads for inextensional, uniform circular arches.

Clamped end . supports Hinged end supports

Height Nondimensional Nondimensional Nondimensional Nondimensional to concentrated distributed concentrated distributed span load at the load over the load at the load over the ratio crown arch axis crown arch.axis

Reference Cf/L) (PR§/EI0) <Vo/EIo> (PR2/EIq) (PnVEIo)

Austin and 0.2887 14.225 18.160 11.014 7.788 Ross (2) 0.3 14.290 17.959 10.961 7.597

0.3501 14.578 17.068 10.725 6.751

This study 0.3 14.336 18.055 11.036 7.625

102

Parabolic Arches

Tables 9 and 10 illustrate the buckling loads as

well as the buckling modes and the corresponding vertical

crown deflections for parabolic arches of different depth Hs ratios of ==—. Clamped and hinged end supports as well as Hc

distributed load over the arch axis and downward point load

at the crown are considered. As shown in Figs. 29 and 30,

the buckling load increases with and is almost directly H£ K Hs proportional to the increase in the ratio . For 4c

inextensional arches with clamped ends subjected to dis­

tributed load over the arch axis, the buckling load

h K HS

increases by 140%, 290%, and 450% as the ratio 77— varies *c

from 1 to 4, While for hinged arches, the buckling load

increases by 115%, 235%, and 350% for the same ratios of Hs —. Thus, the increase in the buckling loads for Hc inextensional parabolic arches is higher for clamped than

Hs for hinged end supports for different ratios of The

"c increase of the buckling loads for parabolic arches with

clamped end supports, subjected to downward point load at

. Hs the crown, is 65%, 120%, and 170% as the ratio varies c

from 1 to 4, The corresponding increase in the buckling

loads for hinged arches is 95%, 190%, and 285%. Thus, for

inextensional parabolic arches subjected to downward point

load at the crown, the increase in the buckling load for

hinged end supports is higher than clamped end supports for

Table 9. Critical loads and corresponding vertical crown deflections for inextensional parabolic arches of different ratios Hs/Hc under vertical, downward, distributed load over the arch axis.

Ratio of the thickness at the support

to the Clamped end supports Hinged end supports thickness at the crown, , 2 Buckling 2 Buckling

<yc> gnVEI0 "VEI0 VR0 m°de gnVEI0 MR0/EI0 VR0 m°ae

1 7,927 9.544 0.110x10 Bifurcation 3.146 3.788 0.621x10 2 Bifurcation

• ^1 -2 2 18.815 22.653 0,121x10 Bifurcation 6.759 8.138 0.632x10 Bifurcation

-1 -2 3 30.976 37,295 0.121x10 Bifurcation 10.484 12.623 0.595x10 Bifurcation

-1 -2 4 43.907 52.864 0,118x10 Bifurcation 14.250 17.157 0.554x10 Bifurcation

Table 10. Critical loads and•corresponding vertical crown deflections for inextensional parabolic arches of different ratios Hs/Hc under vertical, downward, point load at the crown.

Ratio of the thick­ness at the support to the thickness at the crown (H /H ) s c

Clamped end supports Hinged end supports Ratio of the thick­ness at the support to the thickness at the crown (H /H ) s c Pr2

0/eiq VR0 Buckling mode PR0/EI0 VR0 '

Buckling mode

1 9.761 0.2449 Snap through 6.049 0.1058 Bifurcation

2 15.986 0.2079 Snap through 11.803 0.1256 Bifurcation

3 21.438 0.1905 Snap through 17.486 0.1328 Bifurcation

4 26.493 0.1832 Snap through 23.153 0.1355 Bifurcation

105

5 0

40

111

• J j S 3 0 <Q?

20 -

10 •

CLAMPED

HINGED

J I I L

4 5

Hs/Hc

Fig. 29. Buckling loads versus Hs/Hc for inextensional parabolic arches under vertical distributed load pver the arch axis.

106

CLAMPED

HINGED

Fig. 30. Buckling loads versus Hs/Hc for inextensional parabolic arches under downward point load at the crown.

107

Hs different ratios of ==—. It can also be concluded from Hc

Tables 9 and 10 that for hinged end supports, the total

load at buckling corresponding to downward point load at

the crown is greater than the total load at buckling

corresponding to distributed load over the arch axis. For

clamped end supports, the total loads at buckling cor­

responding to distributed load over the arch axis are

greater than the total loads at buckling corresponding to Hs

downward point load for different ratios of g—• A single ° Hs .

exception for clamped end supports is the case where is •c

equal to 1, the total loads at buckling, corresponding to

distributed load over the arch axis, are less than the

corresponding value of downward point load at the crown.

A comparison between the buckling loads for

uniform, inextensional parabolic arches computed in this

study and Austin and Ross (2) results is shown in Table 11.

The deviations are very small and clearly negligible for

practical engineering purposes.

The vertical crown deflections for parabolic

arches under distributed load over the arch axis with

both clamped and hinged end supports changed very slightly.

In contrast, clamped parabolic arches subjected to down­

ward point load at the crown have vertical crown deflec-Hs tions that decrease as the ratio 5— increases. While, c

hinged arches have vertical crown deflections that increase Hs as the ratio =— increases.. All the previous cases of Hc

Table 11. Comparison between the buckling loads computed in this study and the buckling loads obtained by Austin and Ross (2) for inextensional, uniform, parabolic arches.

Reference

Height to span ratio, Cf/L)

Clamped end supports

Nondimens ional concentrated load at the

crown, <PR2/EI0>

Nondimensional distributed load over the arch axis, < W E I 0 >

Hinged end supports

Nondimensional concentrated load at the

crown, (PR2/EI0)

Nondimensional distributed load over the arch axis,

<Vo/EIo'

Austin and Ross (2) 0.3 9.757 7.928 6.042 3.147

Buckling loads computed in this study 0.3 9.761 7.927 6.049 3.146

Deviations from the buckling loads computed in this study, % .401% 0.013% .12% 0.032%

109

parabolic arches buckle by sidesway except parabolic arches

with clamped ends subjected to downward point load at the

crown. These arches buckle by snap through.

Catenary Arches

Inextensional catenary arches subjected to downward

vertical distributed load over the arch axis are in pure

compression. Therefore, there are no prebuckling deforma­

tions for such arches. In this study, these arches are

treated as very thin arches (compressibility parameter Hc — = 0.01), that undergo extensional deformation. This 0 Hc value of is selected on the basis of results from a

R0 previous study by the author on catenary arches with a

similar type of loading. Table 12 and Fig. 31 show that the Hc buckling loads for uniform catenary arches for ratios of 5— R0

in the range 0.01 through 0.04 are almost identical. The Hc load for the lowest ratio of is selected as representa-0

tive of the buckling load for an inextensional catenary

arch.

Tables 13 and 14 present the buckling load as well

as the buckling mode and the corresponding vertical crown

deflection for catenary arches subjected to both distributed

load over the arch axis and downward point load at the

crown. As shown in Figs. 32 and 33, the buckling load for

catenary arches with clamped ends subjected to distributed

load over the arch axis increases by 135%, 360%, and 435% as

Table 12. Buckling loads and corresponding vertical crown deflections for extensional catenary arches of different ratios Hs/Hc and Hc/Rq with clamped end supports, subjected to distributed load over the arch axis.

Ratios of the thickness Ratios of the thickness at the support to the at the crown to the thickness at the crown, radius of curvature at , Buckling

<w the crown (Hc/Rg) PnVEI0 VR0 mode

1 0.01 10.418 0.274x10", Bifurcation 0.02 10.417 0.110x10"^ Bifurcation 0.03 10.417 0.247x10 Bifurcation 0.04 10.417 0.440x10 Bifurcation

2 0.01 24.399 0.601x10"^ Bifurcation 0.03 24.365 0.532x10 Bifurcation

Ill

12

10

8 LLJ n o DC

<Q?

LOAD

DEFLECTION

0.01 0.02 0 . 0 3 0 . 0 4 Hc/Ro

Fig. 31. Eoad-H^,/RQ and deflection-Hc/RQ curves for extensional uniform catenary arches with clamped end supports subjected to distributed load over the arch axis.

Table 13. Buckling loads and corresponding vertical crovm deflections for different ratios H§/Hc for extensional catenary arches of Hc/Rq equal to 0.01, subjected to vertical, downward, distributed load over the arch axis.

Ratio of the thickness at the support

to the Clamped end supports Hinged end supports thickness at the crown _ 2 Buckling ? Buckling

(W s c PnVEI0 WRo/EIo VE0 mode P R-/EI„ n 0 0 WRo/EIo VRo

mode

1 10.417 12.594 0. . 273xl0~ •3

Bifurcation 4.186 5.061 0 .837x10" •4

Bifurcation

2 24,395 29.494 0. 587xl0~ •3

Bifurcation 9.073 10.969 0 .140x10* •3

Bifurcation

3 39.672 47.963 0. ,945x10" •3

Bifurcation 14.112 17.061 0 .185x10" •3

Bifurcation

4 55.662 67.295 0. ,134x10" •2

Bifurcation 19.199 23.212 0 .222x10" •3

Bifurcation

Table 14. Critical loads and corresponding vertical crown deflections for inextensional catenary arches of different ratios of Hs/Hc under vertical, downward, distributed load over the arch axis.

Ratio of the thickness Clamped end supports Hinged end supports of the support to the

thickness at the 2 Buckling ^ 2 Buckling crown, CHg/Hc) §Rq/EIq Y2//R0 mode PRq/EIQ Y2//R0 mode

1 11. 129 0. 2321 Snap through 7. 344 0. 1151 Bifurcation

2 18. 243 0. 1964 Snap through 14. 377 0. 1341 Bifurcation

3 24, 533 0. 1813 Snap through 21. 316 0. 1397 Bifurcation

4 30. 437 0. 1785 Snap through 28. 235 0. 1409 Bifurcation

114

5 0

4 0 CLAMPED

3 0

cm q

HINGED 10

JL 1 2 3 4

Hs/Hc

Pig. 32. Buckling loads versus Hs/Hc for extensional catenary arches under vertical distributed load over the arch axis.

115

CLAMPED

HINGED

Fig. 33. Buckling loads versus Hs/Hc for inextensional catenary arches under downward point load at the crown.

116

Hs the ratio — varies from 1 to 4. For hinged arches this Hs increase is 120%, 240%, and 360% for the same =— ratios. Hc

This indicates that the increase in the total load at

buckling is higher for clamped end supports than hinged

end supports. For catenary arches with clamped ends

subjected to downward point load at the crown, the

increase in the buckling load is 65%, 120%, and 175%,

while for hinged end supports this increase is 100%, 190%, Hs

and 285% as the ratio of 5— varies from 1 to 2, 3, and 4. c

This also indicates that the percentage increase in the

total load at buckling is higher for hinged arches than

for clamped arches for downward point load at the crown.

By examining Tables 13 and 14 and Figs. 32 and 33, it can

be seen that the total loads at buckling for catenary

arches, with clamped end supports subjected to distributed

load over the arch axis, are greater than the buckling

loads.corresponding to downward point load at the crown.

For hinged end supports, the buckling loads corresponding

to downward point load at the crown are greater than the

total loads at buckling corresponding to distributed load

over the arch axis. As shown in Table 13, the vertical

crown deflections for catenary arches subjected to down­

ward distributed load are negligible and can be considered

as zero for practical engineering purposes. This is due to

the fact that these arches are in almost pure compression

and there are only very small prebuckling deformations.

117

The vertical crown deflection for inextensional catenary

arches subjected to downward point load at the crown with

clamped end supports is decreasing slightly as the ratio Hs ==• increases. For hinged arches the vertical crown ° Hs deflection increases slightly as the ratio — increases.

"c A comparison between the buckling loads obtained

in this study and the buckling loads obtained by Dinnik

(.13) and Austin and Ross (2) for uniform catenary arches

subjected to distributed load over the arch axis is

presented in Table 15. It can be seen that Dinnik's (13)

results deviate from the very accurate values computed in

this study by 2.5% for clamped arches and 2.9% for hinged

arches. Austin and Ross (2) values are in close agreement

with the results presented in this study, but for clamped

arches a very slight deviation in their values is noticed.

Austin and Ross (2) indicated that they were perhaps un­

certain of the accuracy of their calculations for catenary

arches. Previously published results by various authors

differ with each other and those obtained by Austin and

Ross (2). The analyses carried out in this study verify

that Austin and Ross (2) results for inextensional

catenary arches are accurate.

In all previous cases, catenary arches buckle

by sidesway, except the case where these arches are

subjected to downward point load at the crown and with

Table 15. Comparison of the buckling loads for uniform i catenary arches subjected to downward, distributed load over the arch axis.

pnR0/EI0

Deviations Deviations Height to in percentages in percentages

span ratio, from the value from the value Reference Cf/L) Clamped ends in this study Hinged ends in this study

Dinnik (13) 0,3 10.682 2.5% 4.064 2.9%

Austin and Ross (2) 0.3 10.419 0.02% 4.186 0.0%

This study 0.3 10.417 4.186

119

clamped end supports. The buckling mode is then snap

through.

From the previous discussion, one can see that

circular arches can withstand the highest total load at

buckling followed by the catenary and then the parabolic

arches. Also, the increase in the buckling load for all

types of arches is almost directly proportional to the Hs change in the ratio . Furthermore, the increase in the Hc

buckling loads for parabolic and catenary arches is greater

than the corresponding increase in the buckling loads for Hs circular arches as the ratio =— changes from 1 to 4. By Hc

studying Tables 6, 7, 9, 10, 12, and 13, it can easily be

seen that, as expected, clamped arches withstand greater

load at buckling than hinged arches. This phenomenon is

due to the fact that clamped arches have two additional

constraints over hinged arches.

Arches Under Nonsymmetrical Loading

Arches subjected to nonsymmetrical loads are also

considered in this study. Table 16 illustrates the

buckling loads as well as the buckling modes and the

vertical crown deflections for inextensional circular -

arches subjected to downward vertical distributed load Lp

over the arch axis, as well as vertical distributed load LL

on the horizontal projection over one-half of the arch. Lr

A basic load ratio of p = equal to 0.2 is considered, H)

Table 16. Buckling loads and corresponding vertical crovm deflections for inextensional circular arches of different ratios Hs/Hc subjected to nonsymmetrical load with p equal to 0.2.

Ratios of

thickness at the support to the

thickness Clamped arches Hinged arches at the crown (H /H ) s c

* 3 , P R„/EI. n 0 0 WRo/EIo VRo

Buckling mode

/s 3 P R /EI nO 0 WRo/EIo

Y / R 2 0

Buckling mode

1 14,730 19.517 .657x10" •1

Snap through 6.567 8.701 0, .904xl0~ •1

Snap through

2 30.227 40.051 0.555xl0~ •1

Snap through 12.709 16.839 0 .720x10" •1

Snap through

3 45.899 60.816 0.479x10* •1

Snap through 18.809 24.922 0 .649xl0~ •1

Snap through

121

which would be realistic for practical engineering purposes.

As shewn in Fig. 34, the buckling load for circular arches

with clamped ends subjected to nonsymmetrical load with p Hs equal to 0,2 increases by 105% and 215% as the ratio 5— Hc

varies from 1 to 3. For hinged end supports, the

corresponding increase in the buckling loads is 90% and Hs 190% for the same ratios of . The total load at Hc

buckling for circular arches with clamped ends subjected

to nonsymmetrical load with p = 0.2 is less by 11.8%,

12.9%, and 12.6% than the total load at buckling Hs corresponding to p = 0 for — equal to 1, 2, and 3. In Hc

comparison, these values for hinged arches are 6.9%, 9.3%, Hs and 10.2% corresponding to the same ratios of . This Hc

indicates that hinged circular arches have deviations

between the total loads at buckling corresponding to

p = 0.2 and the total loads at buckling with p = 0 for Hs different ratios of 5— are less than for clamped circular Hc

arches.

Table 17 presents the buckling loads as well as the

buckling modes for parabolic arches subjected to non­

symmetrical load with p = 0.2. As shown in Fig. 35, the

increases in the total load at buckling for inextensional

parabolic arches with clamped end supports are 135% and

280%. For hinged arches this increase is 115% and 225% Hs as the ratio 5— varies from 1 to 2 to 3. This indicates c

that the increase in the buckling loads is almost directly

122

CLAMPED

HINGED

Fig. 34. Buckling loads versus Hs/Hc for inextensional circular arches subjected to nonsymmetrical load with p equal to 0.2.

Table 17. Buckling loads and corresponding vertical crown deflections for inextensional parabolic arches of different ratios Hs/Hc subjected to nonsymmetrical load with p equal to 0.2.

Ratios of thickness

at the support to the thickness Clamped arches Hinged arches

at tne crown ( H / H ) s c

A 3 p„VE1o

2 WR /EI 0 0

Y /R 2 0

Buckling mode

Vo/EIo WRo/EIo Y2/R0

Buckling mode

1 6.440 8.398 0. 508xl0~ •1

Snap through 2.653 3.460 0 .746x10* •1

Snap through

2 14.979 19.533 0. 411xl0~ •1

Snap through 5.593 7.293 0 .634x10" •1

Snap through

3 24,453 31.887 0. 361x10* •1

Snap through 8.620 11.240 0 .605x10 •1

Snap through

124

CLAMPED

HINGED

Fig. 35. Buckling loads versus Hs/Hc for inextensional parabolic arches subjected to nonsymmetrical load with p equal to 0.2.

125 Hs proportional to the increase in rr-. The increase in the *c

total load at buckling for clamped arches is higher than

for hinged arches. The total load at buckling for clamped

parabolic arches subjected to nonsymmetrical load with

p = 0.2 is less than the total load at buckling correspond-Hj

hT

Hs ing to p = 0 by 12%, 13.7%, and 14.5% as the ratio varies c

from 1 to 2 and 3. For hinged arches these values are Hs 8.66%, 7.9%, and 7.2% for the same ratios of 5—- From the Hc

previous discussion, one can conclude that the deviations

for clamped arches increase while the deviations for hinged Hs arches decrease as the ratio of tt— varies from 1 to 3. As Hc

shown in Table 17, the prebuckling deformations for hinged

arches are greater than for clamped arches.

Table 18 illustrates the buckling loads and the

corresponding vertical crown deflections for catenary arches

subjected to nonsymmetrical load with p = 0.2. As shown in

Fig. 36, the total loads at buckling for inextensional

catenary arches subjected to nonsymmetrical load with

p = 0,2 and with clamped end supports, increase 130% and Hs 275% as the ratio rr— varies from 1 to 2 to 3. The c

corresponding changes for hinged arches are 115% and 228% Hs for the same =— ratios. One can extract that the increase Hc

in the total loads at buckling for clamped end supports Hs are greater than for hinged end supports with the same jj—

ratios. The total loads at buckling for catenary arches

with clamped ends under nonsymmetrical load with p = 0.2

Table 18. Buckling loads and corresponding vertical crown deflections for inextensional catenary arches of different ratios Hs/Hc subjected to nonsymmetrical load with p equal to 0.2.

Ratios of thickness at the support to the

thickness at the crown

<W

Clamped arches

* 3 , 2, P R /EI WR /EI n 0' 0 0 0

Y / R 2' 0

Buckling mode

Hinged arches

P R /EI n 0 0

2, wrVET 0 0 Y„/R„ 2 0

Buckling mode

1

2

8.465 11.081 0.525x10 Snap through 3.55 4.647 0.775x10 Snap through

19.439 25.446 0.440x10 Snap through 7.532 9.859 0.647x10 ^ Snap through

3 31.403 41.106 0,392x10 ^ Snap through 11.628 15.221 0.609x10 Snap through

127

4 0

3 0

CLAMPED

10

HINGED

X

0 1 2 3

Hs/Hc

Fig. 36. Buckling loads versus Hs/Hc for inextensional catenary arches subjected to nonsymmetrical load with p equal to 0.2.

128

are less than the total loads at buckling corresponding to H

p = 0 by 12%, 13.7%, and 14.5% for — equal to 1, 2, and "c

3. The corresponding values for hinged arches are 8.66%, Hs 7.9%, and 7.2% for the same ratios of tj—. The deviations Hc

in the total loads at buckling are less for two-hinged

than for clamped arches.

By examining Tables 16, 17, and 18, the total

loads of buckling for circular arches subjected to non­

symmetrical load with p = 0.2 are greater than for both

catenary and parabolic arches. The total loads at

buckling for catenary arches are greater than parabolic

arches.

The buckling mode shape for all arches subjected

to nonsymmetrical load with p = 0.2 is the snap through.

The buckling loads as well as the buckling modes

for all arches subjected to nonsymmetrical load with

p = 0.2.were found by plotting the load-deflection and

the determinant-deflection curves. The load at which the

determinant of the stiffness matrix is zero is the

buckling load. This was done to save computer time since

the numerical search for the critical load in non­

symmetrical loading is a relatively slow process.

Buckling loads for inextensional parabolic arches

with hinged end supports under nonsymmetrical loads with

p ?=. 0.3 and 0,4 are also presented as shown in Table 19.

The total loads at buckling corresponding to p = 0.3

Table 19. Buckling loads and corresponding vertical crown deflections for inextensional parabolic arches of different ratios Hs/Hc with hinged end supports subjected to nonsymmetrical load with different ratios of p.

Ratios of thickness at the support to the

thickness p = 0.3 p = 0.4

at the 1 : crown _ 2 Buckling 2 Buckling

"W gnR0/EI0 "VEI0 Vo m°de gnVEI0 "VEI0 VR0 m°de

1 2. 484 3, 239 0. 888X10"1 Snap through 2. 338 3. 049 0. 996xl0~ •1

Snap through

2 5. 214 6. 799 0. 773xl0_1 Snap through 4. 893 6. 380 0. 883xl0_ -1

Snap through

3 8. 027 10. 467 0. 741xl0_1 Snap through 7. 525 9. 813 0. 847xl0~ •1

Snap through

130

increase by 110% and 225%, while this increase is 110% and Hs 225% for p = 0.4 as the ratio — varies from o to 2 to 3. c

Th.e decrease in the total loads at buckling for p = 0.3 is Hs

14.5%, 16.5%, and 17.1% in comparison with p = 0 for — Hc

equal to 1, 2, and 3. The corresponding values for p =

0.4 are 19.5%, 21.6%, and 22.3% for the same ratios of Hs

- Plots of the intensity of the uniform distributed Hc load over the arch axis at buckling for ratios of p equal

Hs 0, 0.2, 0.3, and 0.4 versus — is shown in Fig. 37. From Hc

Tables 9, 17, and 19 and Fig. 37, it is clear that the

higher the ratio of p, the lower the total load at

buckling the arch can withstand. This is due to the fact

that the higher the ratio of p, the greater the pre-

buckling deformations, the higher the effect of the

interaction between the bending moments and the axial

loads, and the lower the buckling loads.

The buckling loads as well as the buckling modes

for the tabulated results in Table 19 were obtained by

plotting the load-deflection and the determinant-deflection

curves.

The buckling mode for parabolic arches with

hinged ends under nonsymmetrical loads with p = 0.3 and

p = 0,4 is the snap through.

131

12

10 p=o.o -

8 P = 0-3

6

4

2

0 3 1 2 0

Hs(Hc

Fig. 37.. Buckling loads versus Hs/Hc for inextensional parabolic arches with hinged end supports subjected to nonsynunetrical load with different ratios of p.

CHAPTER 6

CONCLUSIONS AND RECOMMENDATIONS

The following conclusions are based on the analyses

of the results presented in this study:

1. The increase in the total load at buckling is

almost directly proportional to the increase in Hs the ratio, 5—, of depth of section at support to c

depth of section at crown.

2. Circular arches with 0.3 height-to-span ratio can

resist the highest total loads at buckling,

followed by the catenarian and then the parabolic

arches,

3. For all arches, the total load at buckling

corresponding to uniform vertical distributed

load over the arch axis is higher than the total

load at buckling corresponding to nonsymmetrical

load with load ratio p = 0.2.

4. For nonsymmetrical load, the higher the ratio p

the lower the total load at buckling.

5. The per cent increases in the total loads at

buckling for parabolic and catenarian arches are

higher than the corresponding increases in the

132

total loads at buckling for circular arches for the Hs ratios =— in the range 1 to 4. Hc

For the purpose of design, the total loads at

buckling for inextensional circular arches

subjected to downward point load at the crown,

distributed load over the arch axis, and non­

symmetrical load with p = 0.2 are presented in

Tables 6, 7, and 16. By choosing the appropriate

factor of safety, design loads can be directly

computed from the previous tables for ratios of Ho rr— in the range 1 to 4. Tables 9, 10, 17, 12, 13, "c and 18 can be used for the same purpose for

inextensional parabolic and catenarian arches. Hs The total loads at buckling for — between the Hc

values presented in the previously mentioned

tables can be obtained from Figs. 27, 28, 34, 29,

30, 35, 32, 33, and 36.

Design loads for inextensional parabolic arches

with hinged ends subjected to nonsymmetrical loads

with p equal to 0.3 and 0.4 can be obtained from Hs

Table 19 for — equal to 1, 2, and 3. Figure 37 "r-

Hs can be used for any value of 5— between the values Hc

presented in Table 19.

For extensional circular arches with clamped ends,

design loads corresponding to downward point load

at the crown and distributed load over the arch

134

axis can be computed from Tables 1, 2, 3, and 4 H~

for — equal to 1, 2, 3, and 4. Figures 20, 21, Hc

22, and 23 can be used to obtain the total loads

at buckling for any values between the values of Hs rr— mentioned in Tables 1, 2, 3, and 4. c

9. Arches with clamped ends can carry greater total

loads at buckling than arches with hinged ends.

10. For extensional circular arches with clamped ends

subjected to uniform distributed load over the Hs Hc arch axis, the higher rr— the lower =— (the Hc R0

compressibility parameter) at which the effect of

the extensibility of the centroidal line may be

neglected. For extensional circular arches with

clamped ends under downward point load at the Hc crown, at which the effect of rib shortening can ° «s

be considered as negligible, decreases up to rj— Jtl Hs equal to 2.0 and then increases up to ==— equal c

to 3,0,

11. The effect of rib shortening can be neglected for Hc all — less than 0.06, 0.05, and 0.067 for circular ° Hs

arches of 5— equal to 1, 2, and 3 and with clamped Hc

ends subjected to downward point load at the

crown.

12. The deformation can be considered as inextensional Hc for all less than or equal to 0.07, 0.03, and

0 H 0,02 for circular arches of ~ equal to 1, 2, and 3

135

and with clamped ends under vertical distributed

load over the arch axis.

Recommendations for Further Research

Investigate the stability at large deflections of

unsymmetrical arches with constant and variable cross

sections under combinations of loads, as well as different

boundary conditions and various rise-to-span ratios.

Examine the finite deformations and buckling

behavior at large deflections of arch-membrane. Arches

of constant and variable cross section with hinged or

clamped end supports subjected to various types of loading

can be considered.

«

APPENDIX A

NOMENCLATURE

cross sectional area

cross sectional area at the crown

ff 1 kz dA

A 1 - ¥• Ro

A A0

half the span

Young's modulus

depth of the section at the supports

depth of the section at the crown

moment of inertia of cross section about centroidal

axis

moment of inertia at the crown

2

!! Z kz A i - r

R0

span of the arch

distributed vertical load over the arch axis

distributed vertical load on the horizontal pro­

jection over one-half of the arch axis

moment internal bending couple

end couples of a segment

136

137

_ 2 M R

M = =— nondimensional end couples of a segment e EXQ

MR0 m = • nondimensional bending couple EIq

N = internal axial force, positive in tension

NR0 n = nondimensional axial force EI0

P = distributed normal load n P = P R^/EI_ nondimensional distributed normal load n n 0' 0

§£ = distributed tangential load

Pt = P^RQ/EIQ non.dimensional distributed tangential load

P = end forces of a segment in the x direction

_ 2 P = P RN /EI. nondimensional end forces of a segment x x 0' 0

in the x direction

P = end forces of a segment in the Y direction

_ 2 Py = P^RQ/EIQ nondimensional end forces of a segment

in the Y direction

Q = shear force on cross-section

2 g = QRQ/EIQ nondimensional shear force

R = initial radius of curvature of the centroidal line

R* = current radius of curvature

R^ = constant initial radius of intrados for circular

arches

R^ = constant initial radius of extrados for circular

arches

138

Rg = radius at the crown in the undeformed configuration

5 = position coordinate of an arch section referred to

the undeformed centroidal axis

s = S/Rq

S* = position coordinate of an arch section referred to

the deformed centroidal axis

X = abscissa of a point on the arch

X2 = horizontal current position at the crown

x = the distance from the crown to any intermediate

point on the centroidal axis

x = X/RQ

Y = ordinate of a point on the arch

y = VR0

Y2 = vertical current position at the crown

z = distance from centroidal axis to any point on the

section (Fig. 12)

z - K IH

6 = rotation of arch cross section

e = strain

K = initial curvature

k = KRq

K* = current curvature

a = normal stress

(j> = the angle between the horizontal and the tangent to

the centroidal axis at any given point

139

<f>^ = the angle between the horizontal and the tangent

to the centroidal axis at the supports

<j>* = the current angle between the vertical and the

tangent at the crown

£ = the stretch

• (dot) = !^r CI

' (prime) = f-^ C )

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