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BUCKLING BEHAVIOR OF SYMMETRIC ARCHES
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Authors Qaqish, Samih Shaker, 1950-
Publisher The University of Arizona.
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78-5782
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
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
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
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
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
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
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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