OPTIMIZATION OF SHALLOW ARCHES AGAINST SNAP-THROUGH BUCKLING A THESIS ...

205 Pages·2007·21.55 MB·English

OPTIMIZATION OF SHALLOW ARCHES
AGAINST SNAP-THROUGH BUCKLING
A THESIS
Presented To
The Faculty of the Division of Graduate
Studies and Research
By
Hartley McMullin Caldwell, III
In Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy in the School of
Engineering Science and Mechanics
Georgia Institute of Technology
March, 1977
OPTIMIZATION OF SHALLOW ARCHES
AGAINST SNAP-THROUGH BUCKLING
Approved:
_ r, r =
U
George J. Simitses, Chairman
C. V. Smith
Jl
S. Atluri
Date Approved by ChairmanMAR 14 1977
This Dissertation is Dedicated
To
Thornwell, A Home and School for Children
Clinton, South Carolina
Ill
ACKNOWLEDGMENTS
Special appreciation is extended to Professor
George J. Simitses, the author's thesis advisor, for his
patience and encouragement during the course of this inves
tigation. The many enlightening discussions and remarkable
insights shared with th€> author will always be a constant
source of motivation. Thanks are also extended to Professors
C. V. Smith, S. Atluri, C. E. S. Ueng and R. L. Carlson who
served on the thesis committee and provided invaluable
comments and advice.
A special note of thanks to Ms. Brenda Smiley and
Ms. Barbara Geiger for an excellent job in typing the final
manuscript and to Ms. Sharon Flores for her assistance in
putting the early drafts into a readable form.
And finally, sincere appreciation is extended to my
family for whose encouragement and devotion I will be
forever indebted.
IV
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS . . . . . . .. iii
LIST OF TABLES vi
LIST OF ILLUSTRATIONS . vii
SUMMARY x
NOTATIONS , xiii
Chapter
I. INTRODUCTION 1
1.1 Statement of Problem
1.2 Historical Review
II. MATHEMATICAL FORMULATION 12
2.1 Buckling Analysis
2.2 Development, of Optimality Criteria
III. SOLUTION PROCEDURE 34
3.1 Energy Interpretation of the Optimality
Criterion
3.2 Optimization for Anti-Symmetric Modes
3.3 Optimization for Symmetric Modes
IV. RESULTS AND DISCUSSION 50
4.1 Analysis of Anti-Symmetric Optimal Designs
4.2 Analysis of Symmetric Optimal Designs
4.3 Determination of the Strongest Design
V. CONCLUSIONS AND RECOMMENDATIONS 68
5.1 Conclusions
5.2 Recommendations
Appendix
A. ANALYTIC APPROACH FOR UNIFORM ARCH BUCKLING . 91
V
Appendix Page
B. FINITE ELEMENT METHOD APPLIED TO BUCKLING
OF SHALLOW ARCHES 116
C. ANALYTIC APPROACH FOR ANTI-SYMMETRIC
OPTIMIZATION 129
D. DATA GENERATION AND PROGRAM LISTING 140
REFERENCES . . . .. 183
VITA 188
VI
LIST OF TABLES
Table Page
Al. Effect of Rotational Restraint on
the Critical Axial Stress, p, and
the Lowest Rise Parameter for Anti
symmetric Buckling. 107
vii
LIST OF ILLUSTRATIONS
Figure Page
1. Shallow Arch Geometry and Sign Convention . . .. 71
2. Anti-Symmetric Response of Optimum Shapes
for Various Values of n, 3 =°° 72
0
3. Anti-Symmetric Response of Optimum Designs
for Various Values of n, 3 = 10.0 73
0
4. Anti-Symmetric Response of Optimum Designs
for Various Values of n,B = 0 74
0
5a. Optimum Designs for Various Values of n.
Anti-Symmetric Buckling, 3 = °°. 75
5b. Optimum Designs for Various Values of n.
Anti-Symmetric Buckling, $ = 0 76
o
5c. Optimum Designs for Various Values of n.
Anti-Symmetric Buckling, 3 =10.0 77
6. Critical Load Versus Rise Parameter for
T ^ and I , 3 = «, n=2 78
opt u' 0 '
7. Critical Load Versus Rise Parameter for
^opt and V Bo= 10-0' n=2 78
8. Critical Load Versus Rise Parameter for
i . and I , 3 = 0, n=2 79
opt u 0
9a. Anti-Symmetric Response, Different Designs,
3 = °°, n=2 80
0
9b. Anti-Symmetric Response, Different Designs,
S =10, n=2 81
9c. Anti-Symmetric Response, Different Designs,
3 =0, n=2. . 82
o
.9d. Limit Point Load Versus Rise Parameter for
Symmetric Optimum Designs at e=6, 8, and 10,
3 =~, n=2. . 83
Q
Vlll
Figure Page
9e. Limit Point Load Versus Rise Parameter
for Symmetric Optimum Designs at e=6,
8, and 10, 3 =0, n=2 . 83
o
10. Symmetric Critical Response, Symmetric
Optimum Designs at e=6, 8, 10,3 =co, n=l,3 84
11. Variation in Symmetric Optimum Designs
with Rise Parameter, 3 = °° n=2 85
0
12. Variation in Symnietric Optimum Designs with
n, 3 =oo, e=10,0. 86
0
13. Variation in Symmetric Optimum Designs
with Boundary Conditions, e=6, n=2 87
14. Critical Response of I , 1 ,, and-I
3 =«», n=2. I . . . . ?. ?P* . . ? ?^ 88
o
15. Critical Response of I , 1 , , and i .
r J
& =10,n=2. . . . . . ?. ?Pf . . .6.°?t 89
0
A S
16. Critical Response of I , I , , and _i
B =0, n=2. . . . . . ? '. ?PV. . A0?* 90
o
Al. Critical Load Vs. Rise Parameter, Simply
Supported, Half Sine Load, Half Sine Arch 112
A2. Critical Load Vs. Rise Parameter, Clamped,
Half Sine Load, Half Sine Arch 112
A3. Critical Load Vs. Rise Parameter, Simply
Supported, Uniform Load, Parabolic Arch 113
A4. Effect of Rotational Restraint, Symmetric Response. 114
A5, Effect of Rotational Restraint, Anti-Symmetric
Response, Half Sine Load, Half Sine Arch 114
A6. Critical Load Vs. Rise Parameter, Simply
Supported, Uniform Load, Half Sine Arch 115
A7. Critical Load Vs, Rise Parameter, Clamped,
Uniform Load, Half Sine Arch 115
CI. Comparison of Analytic and Finite Element
Optimum Shapes, Anti-Symmetric Buckling,
3 =00, =2 " li9
n
0
ix
Figure Page
Dl. Relation Between Number of Iterations
to Convergence and Starting Value of EXPl . . . 148
D2. Flow Chart for Anti-Symmetric Optimization . . . 151
D3. Flow Chart for Symmetric Optimization 152