analysis of the deflections, vibrations ... - virginia tech

Analysis of the Deflections, Vibrations, and Stability of Leaning Arches

by

Aili Hou

Thesis submitted to the Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

IN

CIVIL ENGINEERING

APPROVED:

Raymond H. Plaut, Chairman

Siegfried M. Holzer

September 1996

Blacksburg, Virginia

Richard M. Barker

Keywords: Leaning arch, Deflection, Frequency, Vibration, Buckling

Analysis of the Deflections, Vibrations, and Stability of Leaning Arches

by

Aili Hou

Raymond H. Plaut, Chairman

Civil Engineering

(ABSTRACT)

In recent years, leaning arches have been used in frameworks for some tent structures.

Various people have studied the behavior of a single vertical arch; however, only a few

researchers have considered the three-dimensional behavior of arches and leaning arches.

The objective of this thesis is to analyze the three-dimensional nonlinear behavior of

leaning arches, particularly the load-deflection and load-frequency relationships, and to

provide a basis for future design guidelines.

In this study, vertical arches of different shapes and load combinations are analyzed in

order to compare with previous results given by other researchers. Then, the behavior of

single tilted arches with different tilt angles is considered. Finally, a leaning arch structure,

with two arches inclined to each other and joined at the top, is considered. The load-

displacement and load-frequency relationships, as well as some buckling modes, are

discussed and presented in both tabular and graphical formats.

Acknowledgments

I would like to express my sincere gratitude to my advisor, Dr. Raymond H. Plaut, for his

guidance, patience, understanding, and support throughout my work on this research. His

advice and expertise greatly enhanced the value of this study. I wish to express my special

thanks to Dr. Siegfried Holzer and Dr. Richard Barker for reviewing my thesis and being

members of my graduate committee.

This research was supported in part by the Anny Research Office under Grant No.

DAAH04-95-l-Ol 75.

I would like to thank my sister, Elizabeth B. Hess, and her family, for their love and

support throughout the years of my graduate study.

Finally, I would like to dedicate this thesis to my parents for their deepest love, support,

and encouragement throughout my life.

Acknowledgments 111

Table of Contents

Chapter 1. Introduction ........................................................................................... 1

Chapter 2. Literature Review ................................................................................... 6

2.1 In-plane buckling of arches .................................................................................... 6

2.2 Out-of-plane deformation .................................................................................... 9

2. 3 Leaning arches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.1 Steeves (1979) ........................................................................................... 11

2.3.2 Krainski (1988) .......................................................................................... 13

Chapter 3. Method of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 .1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 .2 Basic concepts of arches ....................................................................................... 17

3.2.1 Bifurcation and limit-load buckling .............................................................. 17

3 .2.1 Load-deformation relationship ..................................................................... 17

3.2.3 Geometric nonlinearity ............................................................................... 18

3.2.4 Out-of-plane instability of arches ................................................................ 19

3 .2. 5 Leaning arches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3. 3 Computer analysis ................................................................................................ 22

Table of Contents lV

3. 3 .1 Types of element .......................................................................................... 22

3.3.2 Element discretization ................................................................................. 22

Chapter 4. Results for Single Vertical Arches ......................................................... 25

4.1 Introduction ......................................................................................................... 25

4.2 In-plane buckling ................................................................................................. 26

4.2.1 Vertical concentrated load ......................................................................... 28

4.2.2 Vertical uniform load ................................................................................. 30

4.3 Out-of-plane buckling .......................................................................................... 32

4. 4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 5

Chapter 5. Results for Single Tilted Arches ............................................................ 36

5. 1 Arch with 10 degrees tilt angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 6

5. 1.1 Vertical concentrated load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 7

5.1.2 Vertical uniform load ................................................................................. 41

5.1.3 Half vertical uniform load .......................................................................... 43

5.2 Arch with 30 degrees tilt angle ........................................................................... 47

5.2.1 Vertical concentrated load ......................................................................... 47

5.2.2 Vertical uniform load .... . ... .. . ... .. ......... ... ... ... ....... .. . . .... . . .. .... .. ..... .... .. .... .... .. . 52

5.2.3 Half vertical uniform load ... .................... ... . ........... ... .... .... ........... ... . ..... ..... 54

5.3 Discussion .......................................................................................................... 59

Table of Contents v

Chapter 6. Results for Leaning Arches ................................................................... 60

6.1 Load types ........................................................................................................... 60

6.2 Leaning arches with 10 degrees tilt angle ............................................................. 64

6.2.1 Concentrated vertical load .......................................................................... 64

6.2.2 Vertical uniform load .................................................................................. 66

6.2.3 Half vertical uniform load ........................................................................... 71

6.2.4 Normal wind load ....................................................................................... 73

6.2.5 Angle load .................................................................................................. 78

6.2.6 Sideways load . . . . . . . . . . . .. .. . . . .. . . . .. . .. .. . .. . . . . .. . .. .. . . .. .. . . .. .. . . .. . .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . 80

6.3 Leaning arches with 20 degrees tilt angle ............................................................. 85

6.3.1 Concentrated vertical load ......................................................................... 85

6.3.2 Vertical uniform load ................................................................................ 87

6.3.2.1 Vertical uniform load acting on 120-element leaning arch

structure ..................................................................................... 87

6.3.2.2 Vertical uniform load applied to the modified leaning arch

structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.3.3 Half vertical uniform load .......................................................................... 96

6.3.3.1 Half vertical uniform load acting on 120-element leaning arch

structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.3.3.2 Half vertical uniform load applied to the modified leaning arch

structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Table of Contents Vt

6.3.4 Normal wind load ................................................................................... 105

6.3.5 Angle load ............................................................................................... 110

6.3.6 Sideways load .......................................................................................... 112

6.4 Leaning arches with 30 degrees tilt angle ........................................................... 117

6. 4. 1 Concentrated vertical load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 7

6.4.2 Vertical uniform load ............................................................................... 119

6.4.2.1 Vertical uniform load acting on 120-element leaning arch

structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.4.2.2 Vertical uniform load applied to the modified leaning arch

structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.4.3 Half vertical uniform load ......................................................................... 128

6. 4. 3. 1 Half vertical uniform load acting on 120-element leaning arch

structure ..................................................................................... 128

6. 4. 3. 2 Half vertical uniform load applied to the modified leaning arch

structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3

6.4.4 Normal wind load ..................................................................................... 138

6.4.5 Angle load ............................................................................................... 140

6.4.6 Sideways load ........................................................................................... 145

Chapter 7. Conclusions and Recommendations ................................................... 149

7. 1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Table of Contents vu

7.2 Recommendations for future study .................................................................... 151

References .............................................................................................................. 153

Vita ......................................................................................................................... 155

Table of Contents V111

List of Figures

Figure 1. 1 General concept of a leaning arch structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Figure 1.2 Example ofleaning arches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Figure 3. 1 Some possible load-displacement paths for symmetric arches . . . . . . . . . . . . . . . . . . 18

Figure 3 .2 Layout ofleaning arches ......................................................................... 21

Figure 3.3 Numbering of integration points for beam element ................................. 22

Figure 3. 4 In-plane vertical arch with concentrated load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Figure 4 .1 Vertical concentrated load vs. vertical displacement at the top ................ 28

Figure 4 .2 Vertical concentrated load vs. vibration frequencies ................................ 29

Figure 4.3 Vertical uniform load vs. vertical displacement at the top ........................ 30

Figure 4 .4 Vertical uniform load vs. vibration frequencies ........................................ 3 1

Figure 4.5 Vertical uniform load vs. vertical displacement at the top ........................ 33

Figure 4. 6 Vertical uniform load vs. vibration frequencies ........................................ 34

Figure 5. 1 Layout of single tilted arch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 7

Figure 5.2 Vertical concentrated load vs. vertical displacement at the top ................ 38

Figure 5.3 Vertical concentrated load vs. vibration frequencies ................................ 39

Figure 5. 4 Equilibrium shape (dashed line) and buckling mode (solid line) for arch with I 0 degrees tilt angle and concentrated load ........................ 40

Figure 5.5 Uniform load vs. vertical displacement at the top ................................... 42

List ofFigures lX

Figure 5.6 Uniform load vs. vibration frequencies ................................................... 43

Figure 5. 7 Equilibrium shape (dashed line) and buckling mode (solid line) for arch with 10 degrees tilt angle and uniform load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Figure 5. 8 Illustration of a half vertical uniform load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Figure 5.9 Half uniform load vs. displacement at the top ......................................... 46

Figure 5 .10 Half uniform load vs. vibration frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 7

Figure 5.11 Equilibrium shape (dashed line) and first vibration mode (solid line) for arch with 10 degrees tilt angle and half uniform load ........................ 48

Figure 5.12 Vertical concentrated load vs. vertical displacement at the top ............... 49

Figure 5.13 Vertical concentrated load vs. vibration frequencies ............................... 50

Figure 5. 14 Equilibrium shape (dashed line) and buckling mode (solid line) for arch with 10 degrees tilt angle and concentrated load . . . . . . . . . . . . . . . . . . . . . . . . 51

Figure 5 .15 Uniform load vs. vertical displacement at the top .................................... 53

Figure 5 .16 Uniform load vs. vibration frequencies ................................................... 54

Figure 5. 17 Equilibrium shape (dashed line) and buckling mode (solid line) for arch with 3 0 degrees tilt angle and uniform load . . .. . . . . . . . . .. .. . . . .. .. .. .. . .. . 5 5

Figure 5 .18 Half uniform load vs. displacement at the top ......................................... 56

Figure 5. 19 Half uniform load vs. vibration frequencies ............................................ 5 7

Figure 5. 20 Equilibrium shape (dashed line) and first vibration mode (solid line) for arch with 30 degrees tilt angle and half uniform load h = 35.77 lb/in ....... 58

Figure 6.1 Local axis definition for beam-type elements .......................................... 62

Figure 6.2 Concentrated load vs. vertical displacement at the top ........................... 65

Figure 6.3 Concentrated load vs. vibration frequencies ............................................ 66

List of Figures x

Figure 6. 4 Equilibrium shape (dashed line) and buckling mode (solid line) for leaning arches with 10 degrees tilt angle and concentrated load .......................... 67

Figure 6.5 Uniform load vs. vertical displacement at the top .................................... 68

Figure 6.6 Uniform load vs. vibration frequencies .................................................... 69

Figure 6. 7 Equilibrium shape (dashed line) and buckling mode (solid line) for leaning arches with 10 degrees tilt angle and uniform load .................................. 70

Figure 6.8 Half uniform load vs. displacement at the top .......................................... 72

Figure 6. 9 Half uniform load vs. vibration frequencies ............................................. 73

Figure 6. 10 Equilibrium shape (dashed line) and buckling mode (solid line) for leaning arches with 10 degrees tilt angle and half uniform load ........................... 7 4

Figure 6. 11 Normal wind load vs. displacement at the top ......................................... 7 5

Figure 6.12 Normal wind load vs. vibration frequencies ............................................. 76

Figure 6.13 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 10 degrees tilt angle and normal wind load n = 55.65 lb/in ......................................................................................... 77

Figure 6.14 Angle load vs. displacement at the top .................................................... 79

Figure 6.15 Angle load vs. vibration frequencies ....................................................... 80

Figure 6.16 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 10 degrees tilt angle and angle load a=5.954lb/in ......................................................................................... 81

Figure 6.17 Sideways load vs. displacement at the top ............................................... 82

Figure 6.18 Sideways load vs. vibration frequencies .................................................. 83

Figure 6. 19 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 10 degrees tilt angle and sideways load s = 38.87 lb/in ......................................................................................... 84

Figure 6.20 Concentrated load vs. vertical displacement at the top ............................ 86

List of Figures Xl

Figure 6.21 Concentrated load vs. vibration frequencies ............................................ 87

Figure 6.22 Equilibrium shape (dashed line) and buckling mode (solid line) for leaning arches with 20 degrees tilt angle and concentrated load .......................... 88



Figure 6.25 Equilibrium shape (dashed line) and buckling mode (solid line) for leaning arches with 20 degrees tilt angle and uniform load .................................. 91

Figure 6.26 Top view of the wavy leaning arch ......................................................... 93



Figure 6.29 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 20 degrees tilt angle and uniform load q = 49.28 lb/in ......................................................................................... 97

Figure 6.30 Half uniform load vs. displacement at the top .......................................... 98

Figure 6.31 Half uniform load vs. vibration frequencies ............................................. 99

Figure 6.32 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 20 degrees tilt angle and half uniform load h = 63.32 lb/in ....................................................................................... 100

Figure 6.33 Top view of the wavy leaning arch ........................................................ 102

Figure 6.34 Half uniform load vs. displacement at the top ........................................ 104

Figure 6.35 Half uniform load vs. vibration frequencies ........................................... 105

Figure 6.36 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 20 degrees tilt angle and half uniform load h = 46.963 lb/in ..................................................................................... 106

Figure 6. 3 7 Normal wind load vs. displacement at the top . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

List of Figures Xll

Figure 6.38 Normal wind load vs. vibration frequencies ........................................... 108

Figure 6.39 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 20 degrees tilt angle and normal wind load n = 56.07 lb/in ....................................................................................... 109

Figure 6.40 Angle load vs. displacement at the top .................................................. 111

Figure 6.41 Angle load vs. vibration frequencies ..................................................... 112

Figure 6.42 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 20 degrees tilt angle and angle load a= 26.94 lb/in ....................................................................................... 113

Figure 6.43 Sideways load vs. displacement at the top ............................................. 114

Figure 6.44 Sideways load vs. vibration frequencies ................................................ 115

Figure 6.45 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 20 degrees tilt angle and sideways load s = 42.66 lb/in ....................................................................................... 116

Figure 6.46 Concentrated load vs. vertical displacement at the top ........................... 117

Figure 6.47 Concentrated load vs. vibration frequencies .......................................... 119

Figure 6.48 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 3 0 degrees tilt angle and concentrated load P = 17.249 kips ..................................................................................... 120

Figure 6.49 Uniform load vs. vertical displacement at the top .................................. 121

Figure 6.50 Uniform load vs. vibration frequencies .................................................. 122

Figure 6.51 Equilibrium shape (dashed line) and buckling mode (solid line) for leaning arches with 30 degrees tilt angle and uniform load ................................ 123

Figure 6.52 Top view of wavy leaning arch ............................................................. 125

Figure 6.53 Uniform load vs. vertical displacement at the top ................................... 127

Figure 6.54 Uniform load vs. vibration frequencies .................................................. 128

List of Figures Xlll

Figure 6.55 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 30 degrees tilt angle and uniform load q = 40.269 lb/in ..................................................................................... 129



Figure 6.58 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 30 degrees tilt angle and half uniform load h = 54.73 lb/in ....................................................................................... 132

Figure 6.59 Top view of the wavy leaning arch ........................................................ 134



Figure 6.62 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 30 degrees tilt angle and half uniform load h = 43.2 lb/in ......................................................................................... 137

Figure 6.63 Normal wind load vs. displacement at the top ....................................... 139

Figure 6.64 Normal wind load vs. vibration frequencies .......................................... 140

Figure 6.65 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 30 degrees tilt angle and normal wind load n = 56.84 lb/in ..................................................................................... 141

Figure 6.66 Angle load vs. displacement at the top .................................................. 142

Figure 6.67 Angle load vs. vibration frequencies ..................................................... 143

Figure 6. 68 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 30 degrees tilt angle and angle load a= 46.50 lb/in ....................................................................................... 144

Figure 6.69 Sideways load vs. displacement at the top ............................................. 146

Figure 6.70 Sideways load vs. vibration frequencies ................................................ 147

List of Figures XIV

Figure 6.71 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 30 degrees tilt angle and sideways load s 43.09 lb/in ....................................................................................... 148

List of Figures xv

List of Tables

Table 2.1 Buckling loads for circular arches with vertical concentrated load at crown .................................................................................................... 7

Table 2.2 Buckling loads for circular arches with vertical load uniformly distributed along arch axis ( antisymmetrical modes) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Table 3. 1 Element discretization comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Table 4.1 Comparison of results for in-plane buckling of vertical arch ..................... 27

Table 4.2 Relationship between concentrated load and displacement at the top ....... 28

Table 4.3 Relationship between concentrated load and vibration frequencies ........... 29

Table 4.4 Relationship between uniform load and displacement at the top ............... 30

Table 4.5 Relationship between uniform load and vibration frequencies .................. 31



Table 4.8 Comparison of results for out-of-plane buckling of vertical arch .............. 35

Table 5 .1 Relationship between concentrated load and displacement at the top ....... 3 8

Table 5.2 Relationship between concentrated load and vibration frequencies .......... 39

Table 5.3 Relationship between uniform load and displacement at the top .............. 41

Table 5.4 Relationship between uniform load and vibration frequencies ................. 42

Table 5.5 Relationship between half uniform load and displacement at the top ....... 45

Table 5. 6 Relationship between half uniform load and vibration frequencies ......... 46

List of Tables XVl

Table 5. 7 Relationship between concentrated load and displacement at the top ....... 49




Table 5.11 Relationship between half uniform load and displacement at the top ........ 56

Table 5.12 Relationship between half uniform load and vibration frequencies ............ 57

Table 5.13 Summary of results for single tilted arch .................................................. 59

Table 6.1

Table 6.2

Table 6.3

Table 6.4

Table 6.5

Table 6.6

Table 6.7

Table 6.8

Table 6.9

Relationship between concentrated load and displacement at the top ....... 64

Relationship between concentrated load and vibration frequencies ........... 65

Relationship between uniform load and displacement at the top ............... 68

Relationship between uniform load and vibration frequencies .................. 69

Relationship between half uniform load and displacement at the top ........ 71

Relationship between half uniform load and vibration frequencies ............ 72

Relationship between normal wind load and displacement at the top ........ 75

Relationship between normal wind load and vibration frequencies ........... 76

Relationship between angle load and displacement at the top .................. 78

Table 6.10 Relationship between angle load and vibration frequencies ...................... 79

Table 6.11 Relationship between sideways load and displacement at the top ............. 82

Table 6.12 Relationship between sideways load and vibration frequencies ................ 83

Table 6.13 Relationship between concentrated load and displacement at the top ....... 85


List of Tables xvii





Table 6.19 Relationship between half uniform load and displacement at the top ........ 98

Table 6.20 Relationship between half uniform load and vibration frequencies ........... 99


Table 6.22 Relationship between half uniform load and vibration frequencies .......... 104

Table 6.23 Relationship between normal wind load and displacement at the top ....... 107

Table 6.24 Relationship between normal wind load and vibration frequencies ......... 108

Table 6.25 Relationship between angle load and displacement at the top ................. 110

Table 6.26 Relationship between angle load and vibration frequencies .................. 111

Table 6.27 Relationship between sideways load and displacement at the top .......... 114

Table 6.28 Relationship between sideways load and vibration frequencies .............. 115

Table 6.29 Relationship between concentrated load and displacement at the top ...... 118

Table 6.30 Relationship between concentrated load and vibration frequencies ........ 118

Table 6.31 Relationship between uniform load and displacement at the top ............ 121

Table 6.32 Relationship between uniform load and vibration frequencies ............... 122

Table 6.33 Relationship between uniform load and displacement at the top ............ 126

Table 6.34 Relationship between uniform load and vibration frequencies ............... 127

Table 6.35 Relationship between half uniform load and displacement at the top ...... 130

List of Tables X.Vlll

Table 6.36 Relationship between half uniform load and vibration frequencies .......... 131


Table 6.38 Relationship between half uniform load and vibration frequencies ........... 136

Table 6.39 Relationship between normal wind load and displacement at the top ....... 138

Table 6.40 Relationship between normal wind load and vibration frequencies ......... 139

Table 6.41 Relationship between angle load and displacement at the top ................. 142

Table 6.42 Relationship between angle load and vibration frequencies .................... 143

Table 6.43 Relationship between sideways load and displacement at the top ........... 145

Table 6.44 Relationship between sideways load and vibration frequencies ............... 146

Table 7.1 Summary ofresults for leaning arches ................................................... 150

List of Tables XIX

Chapter 1. Introduction

The study of nonlinear responses of nonshallow arches is of great interest in the design of

large tent structures for vehicles and aircraft. These tent-like maintenance shelters could

be structures supported by a framework of inflated tubes, acting much like arches and

beams. To study the behavior of such tent structures, the first step is to analyze the

framework. One possible component of such a framework consists of leaning arches, that

is, two arches connected together at the top. The objective of this study is to analyze the

three-dimensional behavior of leaning arches, particularly the load-deflection and load-

frequency relationships, and to provide a basis for future design guidelines.

Numerous studies on a single semi-circular arch have been presented over the past forty

years. One of the first applications was conducted by Langhaar, Boresi, and Carver

(1954). They calculated an approximate value of the critical downward point load P acting

on the crown of a two-hinged semi-circular arch of constant cross section. Since then,

various papers have been published directly relating to the subject of the in-plane behavior

of arches. In comparison, relatively few studies are available on the three-dimensional

behavior of arches and leaning arches. Therefore, a study considering the load-

displacement relationship of three-dimensional leaning arches is warranted.

Chapter 1. Introduction 1

The basic module for the leaning arch concept is formed by tipping two arches towards

each other and securing them together where they meet at the midspan point. The two

arches must be secured so that they can not move relative to each other, as this is essential

to the stability of the structure. The layout of leaning arches is illustrated in Figures 1.1

and 1.2.

This thesis considers a three-dimensional leaning arch structure, which consists of two

semi-circular arches connected together at the top and pinned at the ground. The structure

is discretized by the finite element method. A numerical investigation is conducted by

using ABAQUS (Hibbitt, Karlsson & Sorensen, 1994), a finite element analysis software

package. It is assumed that, on the basis of the Euler theory of buckling, the total weight

of the arch is negligible. A straight beam element is chosen for the analysis of the arch.

Geometric nonlinearity of the structure is considered ..

The following three steps are taken in this investigation. First, vertical arches of different

shapes and load combinations are analyzed. The buckling loads and displacements are

compared with previous results given by other researchers. Second, the behavior of single

tilted arches with different tilt angles is considered. Finally, a pair of leaning arches is

considered. The relationships between different load types and the top displacements, as

well as the relationships between different load types and natural frequencies, are


discussed and presented in both tabular and graphical formats. Also, some vibration and

buckling modes are shown.


Front View Side View

z

x

Top View

3D

Figure I. I General concept of a leaning arch structure

Chapter I. Introduction 4

n ::r ~ ~ -5" a §" ~ 5· ::s

VI

Figure 1.2 Example of leaning arches (Courtesy of Vertigo, Inc.)

Chapter 2. Literature Review

Compared to other areas of engineering study, relatively few papers have been published

on the subject of leaning arches. Steeves (1979) and Krainski (1988) are the only authors

who deal with the displacement behavior of leaning arch framing schemes; therefore, the

discussion of their works will be more detailed than those of the others.

2.1 In-plane buckling of arches

Many theoretical and experimental solutions for the elastic in-plane buckling of arches

have been reported. Among the first investigators concerned about the stability of arches,

Langhaar, Baresi, and Carver (1954) consider a hinged-hinged semi-circular arch under a

concentrated vertical load at the crown. By ignoring certain small terms in the strain

energy expression, they calculate the value of the critical load to be P = 6.54 E I I R2,

where EI is the bending stiffness and R is the radius. This result agrees fairly well with

their test solutions. Lind (1962) extends the theory to non-circular symmetric arches. He

also calculates the critical value of concentrated loads on circular arches for any

subtending angle. Dapeppo and Schmidt (1969) reconsider the previous problem on the

basis of the inextensional elastica theory. They compute the value of the critical load and

Chapter 2. Literature Review 6

the vertical deflection of the crown for semi-circular arches with opening angles from 1t

through 21t.

Austin and Ross (1976) present an extensive investigation of the in-plane elastic buckling

of arches which have prebuckling displacements. The solution of the exact theory is given

in their paper using a pseudo-critical load numerical procedure. They calculate the critical

loads and the corresponding reactions, maximum moments, and crown displacements for

two-hinged and fixed parabolic and circular arches of constant cross section subjected to

either a concentrated vertical load at the crown or a uniform vertical load along the arch

axis. Their solutions for circular arches are given in Table 2.1 and Table 2.2.

Table 2.1 Buckling loads for circular arches with vertical concentrated load at

crown

Two-Hinged Arch Fixed Arch Antisymmetrical Modes Symmetrical Modes

9 (deg) hi/L QL2/EI he I hi QL2/EI he/hi 50 0.1109 19.32 0.761 24.5 0.67 90 0.2071 30.6 0.738 38.6 0.65 120 0.2887 34.3 0.729 44.3 0.61 180 0.5000 23.5 0.805 41.8 0.53


Table 2.2 Buckling loads for circular arches with vertical load uniformly distributed

along arch axis (antisymmetrical modes)

Two-Hin2ed Arch Fixed Arch 9 (deg) hi/L qL3/EI he/ hi qL3/EI hc/h1

50 0.1109 31.2 0.994 64.8 0.992 90 0.2071 44.0 0.981 95.5 0.969 120 0.2887 42.8 0.968 99.8 0.938 180 0.5000 20.0 0.950 66.0 0.854

Notation: Q =concentrated load

q = distributed load

hi = initial rise of arch

he = height of arch at crown at instant of buckling

e = opening angle of the circular arch

L= span

I· L


2.2 Out-of-plane deformation

When a curved beam or arch is loaded in its plane, it may buckle by deflecting laterally out

of its plane and twisting. Papangelis and Trahair (1987) develop a flexural-torsional

buckling theory for circular arches of doubly symmetric cross section. Nonlinear

expressions for the axial and shear strains are derived for arches that deform in three-

dimensional space. In another one of their papers (1988), the extended theory for arches

of monosymmetric cross section is developed, based on doubly symmetric arches. The

out-of-plane buckling equation is derived in terms of nonlinear strain-displacement

relations for the axial and shear strain.

Wen and Lange (1981) discuss a beam initially curved in one plane but deformable in

three-dimensional space. Geometric nonlinearities are considered in the analysis and

eigenvalues are calculated to obtain the bifurcation buckling loads of arches. The curved

beam element they propose can be used to calculate the in-plane or out-of-plane buckling

loads of arches of different shapes. The basic concepts of their curved element are that

continuity of the slopes and curvatures along the curved axis is satisfied by using fourth-

order polynomials, and the displacement functions are approximated by cubic polynomials.

Geometric nonlinearities are considered by including the effect of rotations on the

longitudinal strains. Effects of warping are neglected. Based on a few numerical results,

the buckling loads agree with those of the classical linear theory of stability.


Another method for a three-dimensional space system is developed by Wen and Suhendro

(1992) using the principle of stationary potential energy and polynomial functions ..

Averaging the nonlinear part of the axial strain can improve the accuracy of the element.

The method of solution is based on fixed Lagrangian coordinates and the Newton-

Raphson procedure. Compared with the results of other methods, the out-of-plane load-

displacement behavior leads to results quite close to the lateral buckling load given by

Wen and Lange (1981) via an eigenvalue solution.

A curved beam element model for the three-dimensional nonlinear analysis of arches is

presented by Pi and Trahair (1996). The model includes higher-order curvatures which

make the order of bending strains consistent. Low-order polynomial interpolations are

used for all displacement fields. As a result, membrane locking problems are avoided. The

model is applied to several numerical examples. For arches with either pinned or fixed

ends subjected to vertical concentrated loads at the crowns, the flexural-torsional buckling

resistance of arches is significantly reduced due to large compression developed in the

arches. In comparison with the existing experimental and analytical results, the curved

beam model is very effective and efficient in terms of accuracy and number of elements

needed for convergence.


2.3 Leaning arches

A leaning arch structure consists of two arches which are inclined to each other and joined

at the top. The two arches must be anchored tightly at the top to prevent relative

movements, and thus improve stability.

2.3.1 Steeves (1979)

In his report, the use of pressure-stabilized structural elements in tentage support

structures is discussed. The report includes a description of the frame concepts, fabrication

of the structural elements, assembly of the prototype tents, and simulation snow load tests.

A tent with a length and width of 5.04 m, height of 2.12 m, radius of 2.44 m of a semi ..

circle, and angle of 53 degrees between the two arches is chosen for his work. The leaning

arch frame concept is selected based on its inherent stability in comparison with the other

frame concepts.

To demonstrate the stability of tent structures, a snow load is tested. The author assumes

that snow would not stay on a surface having a slope greater than 45 degrees. The

deflection of the midpoint of the arches is measured for each load increment. These


deflections are measured with tape measures whose smallest division is 1/16 of an inch.

The loading is increased until the point of collapse is reached.

Within his experiment, the overall deformation of the tentage structure is generally

observed. In the leaning arch concept, the rather large unsupported area at the center of

the tent between two sets of leaning arches undergoes quite a large deformation, on the

order of a meter or more, shortly before collapse. These deformations are so large that the

structure becomes unstable. The author recommends adding more structural support, such

as midspan beams and center arches, to that region as a modification of the tent design.

The experimental load-deformation curves are shown in Steeves' report. In general, the

curves fall within two main regions: in the first region the loads extend from zero up to the

wrinkling load, and the rate of deformation is relatively low; in the second region, from

wrinkling to collapse, deformation increases rapidly as the load is steadily increased. The

wrinkling loads are determined by observing the arches and noting when wrinkling occurs.

Compared to the wrinkling load and the collapse load of the other frame concepts

considered by Steeves, the leaning arch has the strongest load capacity, but it is necessary

to mention that the leaning arch has some characteristics in certain cases that are not well

understood. The leaning arch concept exhibits very similar behavior in the first region of

the load-deformation curves to the other arches. In the second region, however, the


leaning arch has much larger deformations than any of the other concepts. Steeves

observes that the deformation of the leaning arch during failure remains in the planes of

the arches until the point of collapse.

In conclusion, Steeves states that fabrication of a stable tent support structure using

pressure-stabilized structural elements is possible. With the aid of design frame concepts,

the operational snow load requirements can be satisfied. The results show that of all the

single module structures, the leaning arch gives the best load-carrying capacity. More

load-deformation research needs to be carried out to obtain more detailed information

about the leaning arch.

2.3.2 Krainski (1988)

Krainski investigates framing schemes using several pressure-stabilized leaning arches for

the support of the tent structure. He analyzes the stresses and displacements for various

arch diameters under snow load and wind load. The Nonlinear Finite Element Structural

Analysis (NONFESA) computer code is used for numerical analysis. Linearly elastic,

straight, three-dimensional beam types are used in the program. Based on the results

obtained from the various framing arrangements, general relative advantages and

disadvantages in terms of weight, deflections, and load are drawn.


The framing models are constructed in two tent sizes. First, a tent 18 feet wide by 22 feet

long, providing 400 square feet of floor area, is considered. Next, the floor area is about

300 square feet and the overall size is 18 feet by 17 feet. The geometry of the pressurized

arches is chosen in order to provide the greatest amount of floor area and headroom. A

leaning arch structure which consists of three pairs of arches of circular cross section is

considered. The angle of inclination of each pair of arches is 17. 5 and 13. 7 degrees for the

400 and 300 square feet structures, respectively. The finite element model consists of a

total of 95 nodal points to define the structure geometry, that is, 32 straight beam

elements are used for each pair of leaning arches.

Due to vanous environmental conditions, a combination of hinged (X-, Z-global

directions) and fixed end (Y-global direction) supports are chosen, where the X and Z

axes lie in the horizontal plane and the Y axis is vertical. For this support type,

displacements along the three global axes are set to zero. Free rotation is permitted about

the global X- and Z-axes, but not allowed about the Y-axis. The boundary conditions lead

to relatively conservative pressurized arch stresses and displacements.

Load conditions considered in the analysis include a snow load of I 0 psf and a wind load

of 30 mph, both directed perpendicular and parallel to the tent axis. In general, both loads

are prescribed as uniform or variable acting over either the entire structure or a portion of

it. Because the leaning arch is only the support frame, snow and wind load pressures


acting over the tent surface have to be decomposed into concentrated forces in the three

global coordinate directions at each nodal point.

According to the results of the author's study, an 11-in. diameter beam type in conjunction

with the arch frame leads to an acceptable design when inflation pressure is less than or

equal to 10.5 psi. In each load instance, the pressure, stresses, and deflections of the

leaning arch are less than those of the other arch arrangements. The leaning arch

arrangements are suitable for minimum weight-driven design and have an advantage in

terms of lower minimum required pressures.


Chapter 3. Method of Analysis

3.1 Assumptions

The purpose of this thesis is to analyze the load-displacement and load-frequency

relationships for three-dimensional leaning arches subject to various types ofloads.

Four basic assumptions are made in the analytical model. First, the effect of the structural

weight is neglected. Second, it is assumed that the material of the arches is steel with a

specific mass weight of 0.29 lb/in3, elastic modulus of 29000 ksi, and Poisson's ratio of

0.3. Third, linear elastic material and nonlinear geometry are considered in this study.

Fourth, the load is increased until one of the following occurs:

a. A bifurcation point is reached on the load-deflection curve.

b. A limit point is reached on the load- deflection curve.

c. The displacement in any of the three global directions exceeds 15 in.

Chapter 3. Method of Analysis 16

3.2 Basic concepts of arches

3.2.1 Bifurcation and limit-load buckling

There are two main categories of instability of structures. One is called bifurcation of

equilibrium, that is, the deformation of members suddenly changes from one mode into a

different pattern when the load reaches a critical value. The other one is instability that

occurs when the system reaches a maximum or limit load on a plot of load as a function of

displacement without any previous bifurcation.

3.2.2 Load-deformation relationship

The general deformational behavior of a symmetric arch under symmetric loading is

indicated in Figure 3.1. For a small load, the arch deflects symmetrically with a nonlinear

load-deflection curve. If an anti-symmetric mode does not first become dominant, the arch

eventually becomes unstable when the tangent to the load-deflection curve becomes

horizontal (Figure 3.1 (a)). Then the arch buckles in a symmetric mode, which is called a

snap-through phenomenon. However, the arch will buckle in an anti-symmetric

configuration if the corresponding critical load is less than the maximum load for

symmetric bending. Anti-symmetric buckling is a bifurcation phenomenon (Figure 3.l(b)).

The load-displacement plot (equilibrium path) for an unsymmetric arch or an


unsymmetrically loaded arch may also involve a critical load at either a limit point or a

bifurcation point.

3.2.3 Geometric nonlinearity

Geometric nonlinearities are considered in this thesis. Actual design loading on arches

usually produces both axial compression and bending moment on a cross section of the

arch. These internal forces cause a change in shape of the arch before buckling occurs.

This problem is called geometric nonlinearity which applies equilibrium on the deformed

shape and uses nonlinear strain-displacement relations.

Load Load Symmetric mode Limit load

Displacement Displacement

(a) (b)

Figure 3 .1 Some possible load-displacement paths for symmetric arches (a) Critical load at limit point (b) Critical load at bifurcation point


3.2.4 Out-of-plane instability of arches

When applied forces acting in the plane of a curved member reach a certain critical level, a

combination of twisting and lateral bending may cause the member to deform out of its

original plane. The critical load is influenced by the loads, the shape of the axis of the

member, the boundary conditions, and the flexural and torsional stiffness of the cross

sections.

3.2.S Leaning arches

The leaning arch structure consists of a pair of arches. These two arches face each other

and are joined at the midspan point. The two arches are secured at the top so that one can

not move relative to the other. Therefore, they will have the exact same coordinates,

displacements, and rotations at the apex. The basic components of the leaning arch

concerned in this study are two semi-circular arches of radius R = 100 in. and tilted at an

angle r from the vertical axis. A solid circular cross section of radius r = 1. 0 in. is

considered. An illustration of the model is shown in Figure 3.2.

The arch height is H = Rcosr and the distance between the supports at corresponding ends

of the two arches is 2Rsiny. The locations of the four supports are {x, y, z) = (0, 0, 0),


(2R, 0, 0), (0, 0, 2Rsiny), and (2R, 0, 2Rsiny). The supports are assumed to be pinned in

all directions and to have no deflections. The horizontal X-axis passes through the

supports of one of the arches and is parallel to the line passing through the supports of the

other arch. The Y-axis is vertical downward and the Z-axis is horizontal.


R= 100 in. x z

J yl

~ L = 200 in. ~ 1. 2Rsiny

·I Front View Side View

z

x

Top View

3D

Figure 3 .2 Layout of leaning arch


3.3 Computer analysis

3.3.1 Types of element

A finite element computer software package, ABAQUS (Hibbitt, Karlsson & Sorensen,

1994) is used for the numerical analysis. A three-dimensional beam element is selected for

the arches. There are two basic beam element types for space structures in ABAQUS. The

first one is B32 which is a 3-node quadratic beam element. Timoshenko beam theory is

used and transverse shear deformation is considered for B32. The second one is B33

which is a 2-node cubic beam element that uses Euler-Bernoulli beam theory, and

transverse shear deformation is ignored. Six degrees of freedom are active for each node:

displacements in the x, y, and z directions, and rotations about the x-axis, y-axis, and z-

axis. Numbering of integration points for the element is shown in Figure 3.3.

2 2

3 2 1 1

3-node quadratic element 2-node cubic element

•node x Gaussian point

Figure 3 .3 Numbering of integration points for beam element


Both B32 and B33 beam element types provide inter-element continuity of deflections and

their first derivatives at the nodes. This characteristic satisfies the continuity of deflections

and slopes of arches. It is not necessary to consider the effect of shear deformation

because the behavior of these thin arches is much more like that of slender beams.

Therefore, the B33 beam element type, which is a straight element, is selected for the

analysis of the arches considered here.

3.3.2 Element discretization

The accuracy of the ABAQUS results is checked for some element discretizations. For the

semi-circular arch shown in Figure 3.4, the critical load and corresponding vertical

displacement at the top are calculated and compared to those from the exact solution from

Austin and Ross (1976) in Table 3.1. Four cases of the element discretization are tested

and the error is presented for each case.

It is obvious that a finer arch discretization provides more accurate results. However, the

computational expense should be considered when deciding on the discretization to apply.

Even though an 80-element model provides closer results to the exact solution, a 60-

element model which yields less than I% error in Table 3 .1 is chosen due to consideration

for computer resources.


l

I~ L = 200 in.

Semi-circular arch of radius R = 100 in. Circular cross section of radius r = 1 in.

IH=lOOin.

Vertical concentrated load is applied at the crown of the arch

Figure 3 .4 In-plane vertical arch with concentrated load

Table 3.1 Element discretization comparison (by B23)

Exact 20 elements 40 elements 60 elements critical load 13.38 13.05 13.18 13.30

(kips) error - 2.47% 1.49% 0.60%

y-displacement 19.50 18.77 19.05 19.40 (in)

error - 3.74% 2.31% 0.51%

Chapter 3. Method of Analysis

80 elements 13.35

0.22% 19.54

0.20%

24

Chapter 4. Results for Single Vertical Arches

4.1 Introduction

This chapter presents the results for vertical arches. Vertical concentrated loads and

vertical, horizontally-uniform loads are considered. The purpose of this section is: (a) to

compare the results with those of previous studies, so as to further check the correctness

of our modeling of arches, element type, and programming algorithms, and (b) to examine

the displacement and frequency behavior. An arch which is initially curved in the vertical

plane is considered. In-plane and out-of-plane buckling are considered.

In general, the semi-circular arches are divided into 120 straight beam elements. However,

there are exceptional cases, such as whole and half vertical uniform loads for leaning

arches with tilted angles of 20 and 30 degrees (Chapter 6). For those special cases, the

leaning arches will be modeled with 600 straight beam elements, because the arches

contact each other at points other than the apex.

The following sections include load-deflection curves and load-frequency diagrams. The

first five natural frequencies are calculated. By convention, the value of the downward

vertical displacement at the apex is used throughout the analysis.

Chapter 4. Results for Single Vertical Arches 25

4.2 In-plane buckling

In-plane buckling means that the arch is initially defined in a vertical plane and all of its

deformations are restricted to that initial plane, that is, the arch does not have any out-of-

plane deformation. This is a two-dimensional analysis. Two cases with different load types

are considered. The displacement at the crown, buckling load, and natural frequencies are

calculated using ABAQUS.

The arch is semi-circular and has the following properties:

Span: L = 200 in.

l p

Height: H= 100 in.

Radius: R= 100 in.

Circular cross section: r 1.0 in.

Material: steel

Young's modulus: E = 29,000 ksi L ! 1 I I ! 1 q

Poisson's ratio: u=0.3

Total nodes: 61y

Element type: B23

Total elements: 60


Boundary conditions: Pinned at both ends

Two different load cases, vertical concentrated load and vertical horizontally-uniform

load, are considered. The critical loads for both cases are bifurcation loads. The first

vibration mode shapes are anti-symmetric. The results obtained from ABAQUS and those

from Austin and Ross (1976) are quite close to each other, as seen in Table 4.1

Table 4.1 Comparison of results for in-plane buckling of vertical arch

Concentrated load P Vertical uniform load q Critical load Y-disp. (in) Critical load Y-disp. (in)

(kips) (lb/in) ABAQUS 13.30 19.40 56.88 4.947

Exact theory 13.38 19.5 56.94 5 Error 0.6% 0.5% 0.1 % 0.5%

Results for the concentrated load are presented in section 4.2.1 (Tables 4.2 and 4.3, and

Figures 4.1 and 4.2), and results for the uniform load are presented in section 4.22 (Tables

4.4 and 4.5, and Figures 4.3 and 4.4). The critical load occurs when one of the vibration

frequencies reduces to zero. The plots of load versus displacement correspond to the basic

symmetric equilibrium shape, which becomes unstable at the critical load. The apex

vertical displacement for the bifurcating equilibrium path, corresponding to an

unsymmetric shape, is not determined or plotted in this study.


4.2.1 Vertical concentrated load

Table 4.2 Relationship between concentrated load and displacement at the top

Load (lb.) Disp. (in.) 492.9 0.4181 972.1 0.8391 1666 1.475 2654 2.441 4025 3.910 5891 6.150 8280 9.562 11155 14.67 13300 19.40 14298 21.90 17345 31.27 19882 41.94

Load vs. Displacement

20000 ~· 18000 16000 ~·

~ 14000 ,,,,,..

;.. 12000 /. Q. 10000 /. i .9 8000 /.

6000 ./· 4000 2000 .. ·' . o -r·

0 10 20 30 40 50

Vertical Displacement (in)

Figure 4 .1 Vertical concentrated load vs. vertical displacement at the top


Table 4.3 Relationship between concentrated load and vibration frequencies

load P frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 (lb) (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec) 0 22.291 68.031 137.31 224.04 333.02

1000 21.306 66.432 135.59 222.81 331.68 3000 19.273 63.136 132.05 220.15 328.82 5000 17.103 59.732 128.38 217.12 325.70 7000 14.757 56.271 124.60 213.59 322.27 9000 12.039 52.647 120.65 209.22 318.33 10000 10.500 50.845 118.86 206.72 316.19 11000 8.7502 49.060 116.68 203.97 313.94 12000 6.5770 47.182 114.62 200.80 311.43 13000 3.3130 45.335 112.59 197.33 308.78 13100 1.5835 45.089 112.18 197.01 308.70 13300 0.4442 44.957 112.04 196.74 308.50

Load vs. Vibration Frequency

14QQ0 T • • • 12000 • • . • ~ . • • 4

:a 10000 • • • ~ ~ • • • - 8000 \ \ \ a. • • 4 ,,

6000 \ \ \ ~

.9 • • 4

4000 \ \ \ • • 4

2000 \ \ \ • • ~

0 ;--+-• • 0 50 100 150 200 250 300 350

Vibration Frequency (rad/sec)

Figure 4.2 Vertical concentrated load vs. vibration frequencies


4.2.2 Vertical uniform load

Table 4.4 Relationship between uniform load and displacement at the top

q (lb/in) Y-disp.(in) 0 0

4.874 0.2975 9.51 0.597 16.05 1.050 25.04 1.736 36.98 2.781 52.08 4.377 56.88 4.947 70.17 6.819 90.60 10.550 112.40 16.160 134.60 24.390

uniform load vs. displacement

140

120 c ~ 100 "CJ 80 Ill ..2 e 60 .2 40 ·= = 20

0 0 5 10 15 20 25

y-displacem ent (in)

Figure 4.3 Vertical uniform load vs. vertical displacement at the top


Table 4.5 Relationship between uniform load and vibration frequencies

load q frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 (lb/in) (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec)

0 22.281 68.031 137.31 224.04 333.02 IO 20.196 65.751 134.77 221.57 330.57 20 17.880 63.411 132.16 219.06 328.06 30 15.234 61.016 129.47 216.49 325.48 40 12.045 58.568 126.70 213.87 322.83 50 7.6713 56.074 123.85 211.17 320.08 55 4.0085 54.815 122.39 209.79 318.68 56 2.7466 54.562 122.10 209.51 318.39

56.8 0.8559 54.361 121.86 209.29 318.16 56.88 0.2277 54.340 121.84 209.27 318.14

Uniform Load vs. Vibration Frequency

60 T •,, !! !

' 1 i - 50 • • ... c \ \ \ ! 40 • • ... ,, \ \ \ CIS .9 30 • • ... e \ \ \ .... ~ 20 • • ... c \ \ \ ::::>

10 • • " \ \ \

0 •-+-• ...

0 50 100 150 200 250 300 350

Vibration Frequency (rad/sec)

Figure 4.4 Vertical uniform load vs. vibration frequencies


4.3 Out-of-plane buckling

A parabolic arch which is initially defined in the vertical plane but deformable in three-

dimensional space is considered. The arch exhibits out-of-plane buckling behavior at the

bifurcation point. Note that the critical load is well below the limit load and that in-plane

deflections are negligible.

The arch has the following properties:

Span: L = 59 in.

Height: H = 11.8 in.

Cross section: 0.192 in. by 1.5 in.

Material: aluminum ! I I I I I lq

Young's modulus: E = 10,700 ksi

Poisson's ratio: u =0.3

Total nodes: 61

Element type: B33

Total elements: 60

Boundary conditions: fixed at both ends

A vertical, horizontally-uniform load is applied along arch.


Results are presented in Tables 4.6 and 4. 7, and Figures 4.5 and 4.6.


load q (lb/in) Y-disp. (10 ..J in) 0 0

0.2 0.1845 0.6 0.5536 0.8 0.7382 1 0.9227

1.2 1.1072 1.4 1.2918 1.6 1.4763 1.8 1.6609 1.82 1.6793

1.823 1.6821 1.9 1.7531

3.189 2.9425 5.238 4.8330 8.309 7.6670


10 c

8 '== ~ ,, 6 "' ,g E 4 a ·2 2 ::::J

0 0 2 4 6 8

Y·displacement (in)

Figure 4.5 Vertical uniform load vs. vertical displacement at the top


Table 4. 7 Relationship between uniform load and vibration frequencies

load q (lb/in)

0 0.2 0.6 0.8 l

1.2 1.4 1.6 1.8

1.82 1.823

frequency 1 frequency 2 frequency 3 frequency 4 (rad I sec) (rad I sec) (rad I sec) (rad I sec)

54.798 156.95 312.4 520.63 51.763 153.17 308.35 516.41 45.042 145.30 300.09 507.87 41.248 141.19 295.88 503.54 37.048 136.95 291.60 499.18 32.281 132.58 287.26 494.77 26.642 128.04 282.62 490.33 19.384 123.33 278.37 485.85 6.2963 118.43 273.82 481.32 2.4331 117.93 273.36 480.36 0.9271 117.85 273.29 480.80

load vs. vibration frequency

2T • c ~ @. 1.5 ,

i , .2 1 , E • ._ I a • '§ 0.5 \

• ' 0 ·~--A-+-~IE---+-~~-t-~~--(_~

0 200 400 600 800

vibration frequency (rad/sec)

Figure 4. 6 Vertical uniform load vs. vibration frequencies

Chapter 4. Results for Single Vertical Arches

frequency 5 (rad I sec)

780.89 776.56 767.81 763.41 758.97 754.51 750.02 745.50 740.96 740.50 740.43

34

The results for out-of-plane buckling of the vertical arch from ABAQUS are compared

with the theoretical solutions of Tokarz {1971) in Table 4.8. The agreement is quite good.

Table 4.8 Comparison of results for out-of-plane buckling of vertical arch

Critical load (lb/in) ABAQUS 1.823

Theoretical load 1.89 Error 3.54%

4.4 Discussion

The comparisons of results for the vertical arches demonstrate that the proposed arch

model using straight beam elements is suitable for the analysis of arches in a plane and in

space. The results of ABAQUS are good and reasonable for practical engineering

purposes. Therefore, the assumptions, the method of analysis, and the beam element type

{B33 for space) of ABAQUS are used for the investigation of the tilted arches and leaning

arches.


Chapter 5. Results for Single Tilted Arches

In this chapter, a single semi-circular arch whose initial plane is tilted with an angle y from

the global Y-axis is considered (See Figure 5 .1 ). The arch deforms in three-dimensional

space. There is no deflection in the horizontal z-direction at the top; in other words, the

middle point of the arch can not move out of a vertical plane parallel to the x-y plane.

Three load combinations are considered for the tilted arches:

a. A vertical concentrated load is applied at the top of the arch.

b. A vertical horizontally-uniform load is applied along the arch.

c. A vertical horizontally-uniform load is applied along half of the arch.

Section 5 .1 considers an arch with a tilt angle of 10 degrees, and section 5 .2 treats an arch

with a tilt angle of 30 degrees.

5.1 Arch with 10 degrees tilt angle

The properties are the same as in section 4.2.1 except that H = 98.48 in. and there is no z-

deflection at the top.

Chapter 5. Results for Single Tilted Arches 36

H

Front view Side view

Figure 5. 1 Layout of single tilted arch


A vertical concentrated load is applied at the crown of the tilted arch. Results are

presented in Tables 5.1 and 5.2, and Figures 5.2 to 5.4. Bifurcation buckling occurs on the

load-displacement equilibrium path. The bifurcation load is 8.42 kips, and the

corresponding vertical displacement at the top is 10.21 in. Anti-symmetric buckling occurs

under the critical load, and flexural deformation dominates in the equilibrium shape.

Twisting and sideways movement occur in the buckling mode (which is the first vibration


mode at the critical load). The equilibrium shape at the bifurcation load (dashed line) and

buckling mode (solid line) are shown in Figure 5. 4.


d p (lb) Y-disp.(in) 492.6 0.4307 970.9 0.8641 1662 1.549 2644 2.512 4012 4.020 5842 6.317 8173 9.795 8420 10.21 10927 14.93 13153 20.17 14975 25.45

load vs. displacement

15000

:c 10000 -5000

0 10 20 30

displacement at the top (in)

Figure 5.2 Vertical concentrated load vs. vertical displacement at the top


Table 5.2 Relationship between load and vibration frequencies

load P frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 (lb) (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec)

0 20.971 22.285 46.447 67.991 135.15 1000 19.640 21.356 45.494 66.419 133.78 2000 18.225 20.418 44.527 64.817 132.40 4000 15.073 18.521 42.565 61.552 129.59 6000 11.122 16.594 40.539 58.201 126.63 8000 4.8296 14.676 38.489 54.841 122.97 8400 1.1710 14.274 38.03 54.129 122.22 8420 0.6136 14.255 38.009 54.095 122.18


10000 T aooo·· ! "

@: 6000 \~ \ "C \ \ \ .2 4000 ·u ,

2000 •• ... .. ~

0 ··--~+-------------+-----------; 0 50 100 150


Figure 5. 3 Vertical concentrated load vs. vibration frequencies


(1 ::r .g !i VI

~ Cf.l E.. ...... Cf.l

O> ..., en ~· (i"

~ -g_ > ~ ~ Cf.l

~ 0

FRONT VIEW

SIDE VIEW

TOP VIEW

/.---·-·-·-·-·-·-·-·-·-·-·-·-·-· -·-·-·-·-·-·-·-·-·-·-·-·-·-·-........ )

3D

i \ \ \ \ ~

~-·-·-·--·-·-·--·-·-·-·-·-·-·,.,

Figure 5.4 Equilibrium shape (dashed line) and buckling mode (solid line) for arch with 10 degrees tilt angle and concentrated load

.,., \._:

·~


A vertical, horizontally-uniform load is applied along the arch. Results are presented in

Tables 5.3 and 5.4, and Figures 5.5 to 5.7. Bifurcation buckling occurs on the load-

displacement equilibrium path. The bifurcation load is 29.22 kips, and the corresponding

vertical displacement at the top is 2.342 in. Bending is the main deformation in the anti-

symmetric buckling mode, which involves twisting and sideways movement. The

equilibrium shape at the bifurcation load (dashed line) and the buckling mode (solid line)

are shown in Figure 5. 7.


load q (lb/in) Y-disp.(in) 0 0

9.324 0.640 17.42 1.270 27.70 2.192 29.22 2.342 39.86 3.518 52.89 5.368 55.41 5.803 57.71 6.227 58.90 6.458 60.60 6.806


uniform vertical load vs. displacement

c 70

~ 60

~ 50

0 2 4 6 8


Figure 5.5 Uniform load vs. vertical displacement at the top



0 20.971 22.285 46.447 67.991 135.15 10 16.936

i 20.576 43.982 65.706 132.65

20 11.667 18.036 41.370 63.373 130.08 29 1.7345 15.810 38.866 61.237 127.67

29.2 0.9687 15.768 38.821 61.195 127.63 29.22 0.1440 15.755 38.806 61.186 127.62


uniform load vs vibration frequency

30 -. • .... c 25 \\ \ ' ~ 20 •• .... "C \\ \ cu .2 15 E ... 10 •• & g

\\ \ c 5 ::::J

0 ·--· 0 50 100 150


Figure 5.6 Uniform load vs. vibration frequencies

5.1.3 Half vertical uniform load

A vertical, horizontally-uniform load is applied along half of the arch (see Figure 5.8).

Results are presented in Tables 5.5 and 5.6, and Figures 5.9 to 5.11. Displacements of the

crown in the x-direction and y-direction are shown. The load is increased until the vertical

displacement reaches 15 in. No buckling occurs in this load range. The arch bends and

twists as the load is increased. The equilibrium shape (dashed line) and first vibration

mode (solid line) at the load of 47.31 lb/in are shown in Figure 5.11.


(j

5 ~ ""l Vt

~ Cl.I = -~ O' ""l

CIJ s· ~ 0 -3 -· ~ 0..

> (i ::r' ('I) Cl.I

t

FRONT VIEW

// /

'/ 'i v

~

,,, ...... · ,,.,,,.. ,,,,..,,, ...... -·-·;;;..·

SIDE VIEW

TOP VIEW

-·-·-·-·-·-·-·-·-·-·-·-·-·-....... ,,

3D

/ /

/ r/

/

,,,,..,,,.,,,.,,,,. ..... : .... · /

.---·-·-·--·-·-·-·-·-·-·--..... ,.,.,_

Figure 5.7 Equilibrium shape (dashed line) and buckling mode (solid line) for arch with 10 degrees tilt angle and uniform load

'..,\ \

:\.

h ! I 1 1 I l

x

Front view

Figure 5. 8 Illustration of a half vertical uniform load

Table 5.5 Relationship between half uniform load and displacement at the top

load h (lb/in) X-disp. (in) Y-disp.(in) 0 0 0

9.114 -0.9923 0.324 16.21 -1.862 0.645 24.02 -2.934 1.156 29.19 -3.707 1.715 32.76 -4.265 2.344 35.38 -4.679 3.061 38.23 -5.124 4.288 40.38 -5.448 5.722 42.13 -5.679 7.368 43.63 -5.895 9.217 44.97 -6.055 11.26 46.19 -6.181 13.47 47.31 -6.274 15.83


c ::::: J:l :::.. ,, ns .2 'ii u t: Cl> > E ~ c :::s

!:!::: ns .c

half uniform load vs. displacement

x-disp. y-disp. 50

\ 45 --·-·--· . .~· •• 40 ...........

\ 35 .~"" \. 30 ,;

\25 ' ~o I 1'~ • 1~-J s\-f

t--~~-;--~~o-•~~-+-~~-+-~~--1-~~--;

-10 -5 0 5 10 15 20


Figure 5. 9 Half uniform load vs. displacement at the top

Table 5.6 Relationship between half uniform load and vibration frequencies

load h frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 (lb/in) (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec)

0 20.971 22.285 46.447 67.991 135.15 10 18.994 21.288 45.231 66.853 133.89 20 16.656 20.242 43.974 65.692 132.54 30 13.825 19.171 42.637 64.440 130.93 40 11.025 18.310 40.981 62.527 127.86 45 10.827 18.273 39.647 60.386 124.22

47.31 11.119 18.416 38.803 58.910 121.72


half uniform load vs. vibration frequency 50

• • ~ 45 • ii \ :? 40 I I • • ...

g_ 35 \ I \ "C cu 30 • • ... .2 \\ I E 25 .... ~ 20 •• ... c \\ I ::s 15 ,._ 'iij .s::. 10 •• .

5 \\ I 0 ·•--+-· 0 20 40 60 80 100 120 140


Figure 5. 10 Half vertical uniform load vs. vibration frequencies

5.2 Arch with 30 degrees tilt angle

The properties are the same as in section 4.2 except that H 86.60 in. and there is no z-

deflection at the top.


A vertical concentrated load is applied at the crown of the arch. Results are presented in



(j ::r ~ CD" """ v.

~ tl'l s:: fi' CP """ en s·

(IQ (j)

;] -8.. > ~ ::r ('I) tl'l

.a::.. 00

FRONT VIEW

...,..._...-·-·-......................... , ,.,,,· ·-....,

./ .,., // .,.,

/ ., / ' I '

I ' I ' I .,

f ·,,_ ! ' ! '·,

SIDE VIEW /'.

I i i ; I i i i \ \ \ \ \ \ \ \ \ ~ ........... ,,,,..,,,..·"'·

///

\ \ i i ; i i

i //

TOP VIEW

\ --- ,,,,..,,. .. . ,,,..·""' '·-·-·-·-·_ ................... .

,,,.. ,,,..

3D

,,,..,,,.. ............. -·-·-·-·~·-·-·-·-·-·- ........ / -

/ -/ ~ / '

/ ' I ' I .

I I I i i i

Figure 5.11 Equilibrium shape (dashed line) and first vibration mode (solid line) for arch with 10 degrees tilt angle and half uniform load h = 47.31 lb/in


vertical displacement at the top is I 0. 99 in. Anti-symmetric buckling is observed at the

critical load, and flexural deformation is the main action in the equilibrium shape. Twisting

and sideways movement also occur in the buckling mode. The equilibrium shape (dashed

line) at the bifurcation load and the buckling mode (solid line) are shown in Figure 5. 14.

Table 5. 7 Relationship between concentrated load and displacement at the top

load P(lb) Y-disp.(in) 0 0

489.6 0.5564 959.9 1.115 1630 1.957 2565 3.226 3821 5.141 5424 8.016 6800 10.99 7314 12.26 8188 14.68 9268 18.26

Load vs. Displacement

10000 i 8000 .---·-------· ::c- .-

- 6000 .~ ~ 4000 . /.~

2000 t _.,,.·"'· 0 ~ ~

0 5 10 15 20

y-displacement at the top (in)

Figure 5 .12 Vertical concentrated load vs. vertical displacement at the top



loadP frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 (lb) (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec) 0 20.973 22.287 42.510 67.543 135.15

1000 19.372 21.230 41.184 65.808 133.58 2000 17.612 20.147 39.796 64.023 131.97 3000 15.682 19.048 38.358 62.213 130.32 4000 13.573 17.955 36.896 60.420 128.67 5000 10.810 16.778 35.254 58.480 126.87 6000 7.2267 15.613 33.566 56.570 124.84 6800 0.4111 14.671 32.123 55.015 123.05


sooo T .. • .&

- 6000 '•, ii • ::9 • • l - \' ' '"C 4000 •• .&

~ ~- l .Q " ' 2000 ~~ ~ ·• ... II f 0 ·~-,&-f-~---~~->-~~---F----

0 50 100 150


Figure 5. 13 Vertical concentrated load vs. vibration frequencies


(j ::r .§ -(I) .... Vt

~ E.. f.rl" ~ ""I

00 ~· (;"

::i ~ CJ,..

J ti.)

Vt -

FRONT VIEW TOP VIEW

.,,,....,,....-·-·-·-·-·-·-·-·-· -. ..._ ................... , .,

/. i

i i

/

,,,,..,,, /

SIDE VIEW

............ '.,. ' \

\ \ /

/ /

/

3D

,,,....,,....-·-·-·-·-·-·-·-·-·--·

Figure 5.14 Equilibrium shape (dashed line) and buckling mode (solid line) for arch with 30 degrees tilt angle and concentrated load

'·

"""""'· ........ _

'· ....... ,

' ., ., ., ., \

.,

\ \ \ \

\ \ I i


A vertical, horizontally-uniform load is applied along the arch. Results are presented in



vertical displacement at the top is 4. 843 in. Bending is the main deformation in the anti-

symmetric buckling mode. Twisting and sideways movement are also exhibited in the

buckling mode. The equilibrium shape (dashed line) at the bifurcation load and the

buckling mode (solid line) are shown in Figure 5 .17.


load q (lb/in) Y-disp.(in) 0 0

12.63 1.728 22.36 3.480 26.69 4.481 26.90 4.526 27.21 4.613 27.68 4.732 28.10 4.843 28.35 4.910 28.85 5.045 30.27 5.446 32.29 6.057 35.00 6.978 38.50 8.376 42.68 10.50 45.88 12.67 46.55 13.22 47.48 14.06 47.97 14.53 48.21 14.79


50

c ! 40 'Cl

~ 30 Cii 0 ~ ~ 20 E .... .e ·2 ::::J

Uniform Vertical Load vs. Displacement

0 5 10 15

displacement in y-direction at the top (in)

Figure 5 .15 Uniform load vs. vertical displacement at the top



0 20.973 22.287 42.51 67.543 135.15 10 16.616 998 39.002 64.9 132.12 20 10.929 17.446 35.08 62.153 8.69 28 1.3222 15.128 31.527 59.858 125.49

28.1 0.5648 15.096 31.48 59.828 125.45


uniform load vs. vibration frequency

30 T • c • • :.::::: 25 \\ \ .c :::::.. 20 "C •• • ca \\ \ .2 15 E 10 "- •• • ~ 5 \\ \ c :I

0 ·--· 0 50 100 150


Figure 5 .16 Uniform load vs. vibration frequencies


A vertical, horizontally-uniform load is applied along half of the arch (see Figure 5.8).

Results are presented in Tables 5.11 and 5.12, and Figures 5.18 to 5.20. Displacements of

the crown in the x-direction and y-direction are shown. The load is increased until the

vertical crown displacement reaches 15 in. No buckling occurs in this load range. The arch

bends and twists as the load is increased. The equilibrium shape (dashed line) and first

vibration mode (solid line) at the load of35.77 lb/in are shown in Figure 5.20.


Ci

I .... Vl

~ r/.l E.. -r/.l O> .... r./1 s·

(1Q n-::i ff 0..

~ ::r t'D r/.l

Vl Vl

FRONT VIEW TOP VIEW

,,,,,,.,,,,,, ....... -·-·-·-·-· ·-·-·-·-·-·-·-................... .

//

//

/.,,,,,.· / '·,., .,

.,\ ·-·-·-·-·-·-·-·-·-·-·-.......

. ............ , /

I i ;

SIDE VIEW

\ \ \

\1

3D

/ /

/ /

I i i i ; ; i i i \ \ \

/

,;t,' /,;

/

Figure 5.17 Equilibrium shape (dashed line) and buckling mode (solid line) for arch with 30 degrees tilt angle and uniform load

...... ............. ,.,

., .,, \ \ \

.,\ \ \ i

\


load h (lb/in) X-disp.(in) Y-disp.(in) 0 0 0

12.53 -1.129 1.124 19.30 -1.681 2.460 23.52 -1.947 3.933 26.55 -2.074 5.548 28.98 -2.114 7.369 31.04 -2.090 9.454 32.84 -2.013 11.84 34.42 -1.884 14.55 35.77 -1.708 17.57

half load vs. displacement

-c X-disp. 40 1 :*.35 .. ~ ;30 i ,, ~ !2s

e '~ .e 1\5 '2 • ::J ..... 'ii .c

-5 0

• . / ........

/ •

5

,,,,,,,,,,,..

Y-disp.

---· ·---· ---·--

10 15



Chapter 5. Results for Single Tilted Arches

20

56



0 20.973 22.287 42.510 67.543 135.15 10 18.732 21.169 40.846 66.154 133.57 20 15.848 20.019 39.286 64.381 131.46 30 12.323 18.977 37.857 61.253 127.59

35.77 10.138 18.558 37.096 57.645 122.81

half uniform load vs. vibration frequency

40 ! 35 • • •

~ 30 \ ' \ • • • :::. \\ \ i;, 25 <'C .2 e 20 •• • J2 15 \\ \ ·c: ::s ~ 10 •• • <'C

\\ \ .c

5

0 ;. • 0 20 40 60 80 100 120 140


Figure 5. 19 Half uniform load vs. vibration frequencies


(j

J Vl

~ (1>

E. ...... VJ

~ -; en ~· n--i -· -ft 0.

~ ~ VJ

Vl 00

FRONT VIEW

SIDE VIEW

/ i

i

/

,... /

i \ i i i i i i i i ; i

i i i / . / I . . / , . ; .,,/' . ;'

! ......... I .-·,,...·

-·-·-·-·-·.,,,.,,.~..,..,..

TOP VIEW

3D

/ /

/ I

I i

i i i ; i i

//

,,,,.,,., ..... /

....... ......... -·-·-·-·-·--.........................

Figure 5.20 Equilibrium shape (dashed line) and first vibration mode (solid tine)

............ ,

for arch with 30 degrees tilt angle and half uniform load h = 35.77 lb/in

., ., .. ,

''\

5.3 Discussion

Table 5.13 Summary of results for single tilted arch

load pattern

vertical concentrated critical load I bifurcation load (kips) load (P) I limit load (kips)


vertical uniform load critical load I bifurcation load (lb/in) (q) I limit load (lb/in)


half vertical uniform critical load load (h)

bifurcation load (lb/in) limit load (lb/in)

angle 10 deg. 30 deg.

8.42 6.8 - -

10.21 10.9s9

29.22 28.1 - -

2.342 4.843

Table 5.13 shows that the tilted arches exhibit bifurcation buckling for the cases in which

the arch is loaded symmetrically. For the larger tilt angle, the buckling load is lower and

the vertical displacement of the crown at the buckling load is higher than for the smaller

tilt angle. The vertical arch has a much higher buckling load and corresponding vertical

displacement at the crown.


Chapter 6. Results for Leaning Arches

In this chapter, a leaning arch structure which consists of two semi-circular arches

connected together at the crown with tilt angle y from the global Y-axis is considered (see

Figure 3.2). The arches are constrained to have equal deflections and rotations at the

crown. Sections 6.2, 6.3, and 6.4, respectively, consider leaning arches with tilt angles of

I 0 degrees, 20 degrees, and 30 degrees.

6.1 Load types

Six load types are considered for analysis of the load-displacement relationship and load-

frequency relationship of the leaning arches.

a. A vertical concentrated load (P) is applied at the top of the leaning arches.

p

l

Chapter 6. Results for Leaning Arches 60

b. A vertical, horizontally-uniform load ( q) is applied to both arches.

I 1 I 1 I 1 q

!

c. A half vertical, horizontally-uniform load (h) is applied to both arches.

h 111111 l

d. A normal wind load (n) is applied perpendicular to the axes of both arches (varying

from n to -n).

n -n

Front View


The orientation of a beam cross-section is defined by ABAQUS in terms of a local, right-

handed, ( t, n 1, n2) axis system, where t is the tangent to the axis of the element, positive

in the direction from the first to the second node of the element, and n 1 and n2 are basis

vectors that define the local 1- and 2-directions of the cross-section. The local axis

definition for the beam element is shown in Figure 6.1. Based on the local axis definition,

normal wind loads (n) are perpendicular to the axis of the arch in space; in other words,

there are no components of normal wind loads along the axes of the arches.

2

"' / I

/ I

I I ,

, I

# ,

-.... --- ---

nl

__________ ....... ~

--- ----------------_, ..

Figure 6.1 Local axis definition for beam-type elements


1

62

e. A uniform horizontal load (a) is applied at 45 degrees in the x-z plane only on one arch;

that is, the components along the x and z directions are equal, and there is no component

along the y direction.

z

a

l x x

y Front View Top View

f. A sideways uniform horizontal load (s) is applied in the x-direction to half of both

arches.

z

s

x x

Front View Top View


6.2 Leaning arches with 10 degrees tilt angle

The properties are the same as in section 4.2 except that H = 98.48 in.

6.2.1 Concentrated vertical load

A vertical concentrated load is applied at the crown of the leaning arch. Results are

presented in Tables 6.1 and 6.2, and Figures 6.2 to 6.4. Bifurcation buckling occurs on the

load-displacement equilibrium path. The bifurcation load is 6.075 kips, and the

corresponding vertical displacement under the load is 2. 92 in. Sideways movement

dominates in the buckling mode; that is, the two arches move together in the z-direction

under the critical load, and there is not much bending or twisting movement. The




load P (lb.) Y-disp. (in) 49 .. 3 0.2147 985 0.4301 1704 0.7542 2754 1.243 4266 1.981 6075 2.920 6399 3.099 9334 4.797 13221 7.370 - - - - - 11.27 23809 17.03



25000 _____.,. :c 20000 ------::::. 15000 ~· D. ~· ~ 10000 ~· .2 5000 ............. ••• o.~,~~~-+-~~~~r--~~~-t--~~--1

0 5 10 15 20

vertical displacement at the top (in)

Figure 6.2 Concentrated load vs. vertical displacement at the top


load P (lb) frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec)

0 9.8276 22.279 22.415 46.447 46.494 1000 8.9867 21.793 21.815 45.97 46.022 2000 8.0588 21.161 21.350 45.492 45.550 3000 7.0036 20.508 20.881 45.009 45.072 4000 5.7574 19.842 20.411 44.524 44.595 5000 4.1475 19.157 19.939 44.037 44.115 6000 1.0978 18.453 19.465 43.548 43.633 6070 0.28873 18.403 19.431 43.514 43.599 6075 0.05687 18.399 19.429 43.511 43.597



0 10 20 30 40 50


Figure 6.3 Concentrated load vs. vibration frequencies


A vertical, horizontally-uniform load is applied along both arches. Results are presented in


displacement equilibrium path. The bifurcation load is 16. 409 lb/in, and the corresponding

vertical displacement at the top is 1.186 in. The main buckling movement of the leaning

arch is sideways again. The equilibrium shape at the bifurcation load (dashed line) and the

buckling mode (solid line) are shown in Figure 6.7


Cl ::r ~ ~ ?' ~ C/l c= ;::;' C/l

6> "'1

~ § ~·

> a ::r (')) C/l

°' -....J

FRONT VIEW

SIDE VIEW A I \ I \ I \ I I I I I I I

Figure 6.4

TOP VIEW

·""· ___ :::::::::::::=:=·-·--·-·--·-·-·=::::::::::::::::::: __ ·-·-.

3D

Equilibrium shape (dashed line) and buckling mode (solid line) for leaning arches with 10 degrees tilt angle and concentrated load


load q (lb I in) Y-disp. (in) 0 0

13.53 0.9571 16.41 1.186 24.29 1.886 37.36 3.218 47.12 4.477 54.63 5.663 54.97 5.723 55.48 5.813 56.22 5.947 57.29 6.147 58.85 6.443


- 60 c ..... ' 50 ~ ... <• .a - /. c:r 40 "C 30 /. ca .2 E 20 /. .... ....

10 / • .E ·2 :J 0 •

0 2 4 6 8




Table 6.4 Relation between uniform load and vibration frequencies


0 9.82760 22.279 22.415 46.447 46.494 5 8.19668 20.695 21.286 45.231 45.291 10 6.14420 18.839 20.250 43.981 44.055 13 4.48160 17.646 19.606 43.213 43.298 15 2.88180 16.811 19.166 42.694 42.786 16 1.55390 16.379 18.943 42.432 42.528

1.64 0.24156 16.203 18.852 42.327 42.424 16.409 0.07307 16.199 18.850 42.324 42.422

unifonn load vs. vibration frequency

0 10 20 30 40 50

vibraton frequency (rad/sec)



(")

s l ?' ~ ~ ii" O> "'1

i er OQ

~ ~ (/}

-....J 0

FRONT VIEW

SIDE VIEW A I \ I \ I ' I I I I I I

Figure 6.7

TOP VIEW

./"_::::::::=:==,=-·--·--·-·-·--·-·='='=:=:::::::::_._ ....

3D

Equilibrium shape (dashed line) and buckling mode (solid line) for leaning arches with 10 degrees tilt angle and uniform load


A vertical, horizontally-uniform load is applied along half of both arches (see page 61).

Results are presented in Tables 6.5 and 6.6, and Figures 6.8 to 6.10. Bifurcation buckling

occurs on the load-displacement equilibrium path. The bifurcation load is 29.28 lb/in, and

the corresponding horizontal and vertical displacements at the top are 4. 03 5 in. and 1. 18

in., respectively. The arches twist and move out-of-plane at the critical load for both the

equilibrium shape and the buckling mode. The deformed equilibrium shape is symmetric

with respect to a plane parallel to the x-y plane and passing through the crown. The




load h (lb/in) X-disp. (in) Y-disp. (in) 0 0 0

9.307 -1.059 0.3177 17.36 -2.122 0.6285 27.52 -3.720 1.090 29.28 -4.035 1.180 39.42 -6.119 1.792 52.11 -9.682 2.943 60.81 -13.11 4.286 67.07 -16.35 5.889 68.36 -17.12 6.333 69.55 -17.86 5.796 71.20 -18.95 7.528 73.39 -20.49 8.713 76.12 -22.56 10.70 78.27 -24.28 12.92 79.91 -25.58 15.35



X disp Y-disp. ........ - 80 ·-· ·-., ...... -•.• 70 .....

•+ L~

'· 60 ./ "-. I ." 50 i • 40 •

\.30 J ·'3~ • 1 c}..~

\J 1--~~----jf--~~--+~~---lcl-·~~~-t-~~~~

-30 -20 -10 0 10 20




load h frequency 1 frequency 2 frequency 3 frequen _ .. -requency 5 (lb/in) (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec)

0 9.8276 22.279 22.415 46.447 46.494 10 8.1324 20.702 21.28 44.678 45.813 20 5.7653 18.897 20.229 42.783 45.126 29 1.0308 17.184 19.221 40.962 44.465

29.2 0.56789 17.145 19.198 40.920 I 44.450 29.28 0.16208 17.130 19.189 40.904 44.443



30 +, .... c 25 \ \\ ~ ,, 20 • .... "' \ \\ .2 E 15 ._ a ·2 10 . ..... :::s \ \ -ii 5 .c

0 , ... 0 10 20 30 40 50


Figure 6.9 Half uniform load vs. vibration frequencies

6.2.4 Normal wind load

A normal wind load is applied perpendicular to the axes of both arches, varying from n to

-n (see page 61). Results are presented in Tables 6.7 and 6.8, and Figures 6.11 to 6.13.

Displacements in the x-direction and y-direction are shown. The load is increased until the

horizontal displacement reaches 15 in. No buckling occurs in this load range. The arches

twist and move along the x-axis as the load is increased. The deformed equilibrium shape

is symmetric with respect to a plane parallel to the x-y plane and passing through the

crown. The equilibrium shape (dashed line) and first vibration mode (solid line) at n=SS.65

lb/in are shown in Figure 6.13.


("'.) t:T' ~ ~ ..... ?" i;1 Cl.I c: a S' .....

[ ~·

> a ~ Cl.I

......J ~

FRONT VIEW

SIDE VIEW

TOP VIEW

p= '·-...... ·--- . =====·-·-·-·--·-=·c•=•=:::::::::-/

3D

Figure 6.10 Equilibrium shape (dashed line) and buckling mode (solid line) for leaning arches with I 0 degrees tilt angle and half uniform load

·-.... .......... ,_

Table 6. 7 Relationship between normal wind load and displacement at the top

load n (lb/in) X-disp. (in) Y -disp. (in) 2.5 0.7759 0.007675 5 1.552 0.030818

8.759 2.714 0.094531 13.78 4.210 0.2336 18.83 5.803 0.4344 23.91 7.338 0.6967 29.04 8.863 1.020 34.22 10.38 1.405 39.46 11.88 1.851 44.77 13.36 2.353 50.16 14.83 2.923 55.65 16.28 3.549

normal wind load vs. displacement

c 60 y-disp. x-disp.

:.::::: 50 ..c ::::.. "'C 40 cu ..2 30 "'C c 'i 20 ca 10 e 0 0 c

0 5 10 15 20


Figure 6 .11 Normal wind load vs. displacement at the top


Table 6.8 Relationship between normal wind load and vibration frequencies

load n frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 lb/in rad I sec rad I sec rad I sec rad I sec rad I sec

0 9.8276 22.279 22.415 46.447 46.494 5 9.8382 22.287 22.412 46.450 46.495

13.78 9.9101 22.335 22.392 0 46.497 23.91 10.087 22.341 22.446 46.504 34.22 10.390 22.263 22.614 46.471 46.534 44.77 10.860 22.152 22.830 46.448 46.582 55.65 11.571 22.016 23.113 46.401 46.649

normal wind load vs. vibration frequency

60 c 50 ~ "D 40 CIJ .2 "D 30 c 'i Ci 20 E .... 0 10 c

0 0 10 20 30 40 50


Figure 6.12 Normal wind load vs. vibration frequencies


('j =-~ ~ ""i

?" ~ E. ..+ tll

~ .... t'"'4 § ~·

> a g-tll

-...J -...J

FRONT VIEW

SIDE VIEW 1'· In\ I .. \ ;w ; i\ \ ii \ \ i i \ i i .

i i ii . i ! ; ! i ! .

I I . i /; ii . I !; I.

i/. l

TOP VIEW

________ :=•=========·-·--·-·-·-·-·~·========-·.>

3D

Figure 6.13 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 10 degrees tilt angle and normal wind load n = 55.65 lb/in

6.2.5 Angle load

A uniform horizontal load is applied at 45 degrees in the x-z plane only on one arch (see

page 63). Results are presented in Tables 6.9 and 6.10, and Figures 6.14 to 6.16.

Displacements in the x-direction, y-direction, and z-direction are shown. The load is

increased until the sideways horizontal displacement exceeds 15 in. No buckling occurs in

this load range. The arches twist and move along the z-axis as the load is increased. The

equilibrium shape (dashed line) and first vibration mode (solid line) at the load of 5.954

lb/in are shown in Figure 6 .16.

Table 6.9 Relationship between angle load and displacement at the top

load a (lb/in) X-disp. (in) Y-disp. (in) Z-disp. (in) 0 0 0 0

0.5015 0.2074 0.016709 1.354 1.006 0.4164 0.057407 2.707 1.769 0.732 0.1635 4.737 2.590 1.070 0.3362 6.899 3.420 1.409 0.5704 9.058 4.257 1.746 0.8659 11.21 5.102 2.081 1.222 13.36 5.954 2.412 1.640 15.50


6 c 5 :.::::: :e 4 -"'C 3 ca .2

2 G) r;,

1 c ca 0

0

angle load vs displacement y-disp.

x-disp.

5 10 15

displacement atthe top (in)

Figure 6. 14 Angle load vs. displacement at the top

z-disp.

20

Table 6.10 Relationship between angle load and vibration frequencies

load a frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 (lb/in) (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec)

0 9.8276 22.279 22.415 46.447 46.494 1 9.8613 22.228 22.524 46.352 46.553 2 9.8835 22.146 22.671 46.235 46.639 3 9.8948 22.058 22.831 46.112 46.738 4 9.8969 21.965 23.004 45.981 46.854 5 9.8837 21.860 23.183 45.839 46.979

5.954 9.8634 21.754 23.368 45.696 47.116


6

:E 5 -~4 "C .2 3 .!!! 2 C> ; 1 I

0

angle load vs. vibration frequency

10 20 30 40


Figure 6.15 Angle load vs. vibration frequencies

6.2.6 Sideways load

50

A uniform horizontal load is applied in the x-direction to half of both arches (see page 63).

Results are presented in Tables 6.11 and 6.12, and Figures 6.17 to 6.19. Displacements in

the x-direction and y-displacement at the top are shown. The load is increased until the

horizontal displacement exceeds 15 in. No buckling occurs in this load range. The arches

bend and twist as the load is increased. The equilibrium shape (dashed line) and first

vibration mode (solid line) at the load of 3 8. 87 lb/in are shown in Figure 6 .19.


('j ::r ~ ~ ?' ~ l'/.I a V1' ~ .., f; § :r

(JQ

~ ~ l'/.I

00 -

FRONT VIEW

SIDE VIEW

I /,

I I

I I

I

A ,1 ti

I i I I

I i

TOP VIEW

·-·-=:~~~-::...-::...-:.:..--:-.:.::::..~-·= -.-:-..:.~..:.-:-..:~..:.:::.:~:-.-·-· ,,,,.. .......

/ '

30

Figure 6.16 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 10 degrees tilt angle and angle load a = 5. 954 lb/in

Table 6.11 Relationship between sideways load and displacement at the top

load s (lb/in) X-disp. (in) Y-disp. (in) 0.5 0.2066 0.020153 1 0.4131 0.041576

1.75 0.723 0.076075 2.874 1.188 0.1331 4.562 1.885 0.2304 7.070 2.918 0.4007 9.580 3.949 0.6020 12.10 4.979 0.8341 14.64 6.007 1.097 17.21 7.032 1.390 19.29 8.054 1.713 22.40 9.072 2.067 25.04 10.09 2.450 27.72 11.10 2.862 30.44 12.10 3.304 33.20 13.10 3.775 36.01 14.09 4.274 38.87 15.08 4.802

sideways load vs. displacement

40 35 30

,, 25 C'CI

.Si! 20 ~ 15 ~ Cl> 10 ,,

5 u; 0

0 5 10 15


Figure 6.17 Sideways load vs. displacement at the top


Table 6.12 Relationship between sideways load and vibration frequencies

load s (lb/in)

0 10 20 25 30

38.87


9.8276 22.279 22.415 46.447 9.8798 22.429 23.502 46.006 10.113 22.766 24.710 45.697 10.294 22.996 25.354 45.602 10.510 23.260 26.017 45.540 10.980 23.798 27.242 45.515

sideways load vs. vibration frequency

40 ~ 35 :§. 30 i 25 .2 20 .,, ~ 15 ~ 10

"C ·u; 5 Q-+-~--c>----~-t--~-+-~~---+-~----+-~--1

0 10 20 30 40 50 60 vibration frequency (rad/sec)

Figure 6.18 Sideways load vs. vibration frequencies



46.494 47.794 49.218 49.964 50.726 52.117

83

(") =-~ & ::;t'

~ c:: 6f 8'> "'1

i ~·

~ [

00 ~

FRONT VIEW

SIDE VIEW ·t Ii\\ 'd I; \\

!; \ \ I; \ \ I; \ \ I; .

I; !; I.

I! ., !; / . . , ~ ~ /;

TOP VIEW

~ ·-·--. -- 7 ______ :::::=:===,=·-·-·-·- ~ /-·-- ·-·-·-·-·~,=:=:=:::::::: __ .>

3D

Figure 6.19 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 10 degrees tilt angle and sideways loads= 38.87 lb/in


The two arches have the same properties as those in section 4.2 except that H = 93.97 in.

6.3.1 Concentrated vertical load

A vertical concentrated load is applied at the crown of the leaning arch. Results are

presented in Tables 6.13 and 6.14, and Figures 6.20 to 6.22. Bifurcation buckling occurs

on the load-displacement equilibrium path. The bifurcation load is 17.201 kips, and the

corresponding vertical displacement at the top is 11. 96 in. Sideways movement is the main

action in the buckling mode; that is, the two arches move together in the z-direction. The

equilibrium shape at the bifurcation load (dashed line) and the buckling mode are shown in

Figure 6.22.


load P (lb) Y-disp. (in) 495.8 0.2351 983.1 0.470 1699 0.825 2739 1.359 4227 2.164 6318 3.379 9157 5.217 12853 7.991 17201 11.96 17375 12.14 19519 14.47 22299 17.96


25000

20000 -::9 15000 --g 10000 .2

5000

0


5 10 15 20

Y-cfisplacement (in)

Figure 6.20 Concentrated load vs. vertical displacement at the top


load P (lb) frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec)

0 19.003 22.26 26.057 45.027 45.258 1000 18.418 21.773 25.471 44.482 44.733 2000 17.82 21.284 24.881 43.935 44.207 4000 16.566 20.294 ~ 42.814 43.132 6000 15.229 19.294 22.406 41.671 42.043 8000 13.788 18.288 21.115 40.509 40.943 10000 12.187 17.269 19.762 39.308 39.814 12000 10.339 16.224 18.304 38.049 38.635 16000 5.0038 14.142 15.18 35.437 36.229 17000 2.4904 13.699 I 14.513 34.84 35.712 17200 0.19697 13.491 14.086 34.582 35.444 17201 0.13511 13.491 14.085 34.581 35.443



20000

g 15000

CL. 10000 "C ra .2

5000

0 0 10 20 30 40 50


Figure 6.21 Concentrated load vs. vibration frequencies


6.3.2.1 Vertical uniform load acting on 120-element leaning arch structure

A vertical uniform load is applied along both arches. Each arch is divided into 60 elements

as before. Results are presented in Tables 6.15 and 6.16, and Figures 6.23 to 6.25.

Bifurcation buckling occurs on the load-displacement equilibrium path. The bifurcation

load is 54.22 lb/in, and the corresponding vertical displacement at the top is 9.254 in. The

two arches bend and twist for both the deformed shape and the buckling mode, They

contact each other at points other than the apex under the critical load. The equilibrium

shape at buckling (dashed line) and the buckling mode (solid line) are shown in Figure

6.25.


('j ::r .§ ~ '"1

?"-~ CIJ c: lt" ~ '"1

r 0 ~ s·

(JQ

> ~ ::r ~ r.n

00 00

FRONT VIEW

SIDE VIEW

/;

I I

I

I

TOP VIEW

/" ,,.,

-·:::::::::=::'"=·-·-·-·-·-·-·-·-·-·-·=•=::::::::::_._''·,·~ _, . ..-..·--·

/\ 3D I \

I ' I

I

Figure 6.22 Equilibrium shape (dashed line) and buckling mode (solid line) for leaning arches with 20 degrees tilt angle and concentrated load

Table 6.15 Relationship between uniform load and vertical displacement at the top

70 c 60 :a - 50 i 40 .2 E 30 ... 20 g c: :J

0

load q (lb/in) Y-disp. (in) 0 0

4.778 0.4037 9.144 0.8049 15.03 1.402 22.59 2.29 31.73 3.606 41.88 5.547 .;.: 8.404 ..,,.,,,,,

54.22 9.254 58.55 11.3 62.91 14.25


5 10

Y-displacement at the top (in)



15

89



0 19.003 22.26 26.057 45.027 45.258 10 17.079 2 23.302 42.163 42.493 20 14.974 17.785 20.451 39.06 39.524 30 12.628 15.074 17.577 35.633 36.283 35 11.358 13.520 16.234 33.777 34.553 40 9.9937 11.723 15 31.773 32.705 50 6.5633 7.089 13.572 27.243 28.665 54 1.4603 5.9961 13.745 25.15 26.898

54.2 0.47195 5.9587 13.786 25.047 26.816 54.22 0.19744 5.9554 13.791 26.037 26.808





(') ::r ~ ft ..... ?" ?;' t/l c: ii' ~ .....

i ~·

~ ::r ('I> t/l

'° -

FRONT VIEW TOP VIEW

,,,,,·""'' ,,,.,,.. ,.,,,...-·-·-·-·-·-·-·-·

·-........................ * .......

/ /

I /. I I i

/

SIDE VIEW

....... , ., '·,·,,\

\ \ \ i

3D

...... -,,,,.,,. .-'!::lf·.Jll" /..,,

fit::".-;:::::::.·=··=·=:::::.:::::-:.::..,..,.,:::::::::::·-._. __ ., .,

.,\ /.:,.·

/,,:"/ /' \

,, I. i \ \ \ \

Figure 6.25 Equilibrium shape (dashed line) and buckling mode (solid line) for leaning arches with 20 degrees tilt angle and uniform load

\ \

6.3.2.2 Vertical uniform load applied to the modified leaning arch structure

The diagram of the equilibrium shape and buckling mode (Figure 6.25) indicates that the

two circular arches cross each other, and this situation needs to be avoided. To correct

this problem, the points where the arches meet (except the apex) have to be determined,

and constraints have to be added to those points so that the arches can move together at

those points. Because the displacements in the middle part of the leaning arch are a major

concern for this case, 3 00 straight beam elements are used on each arch in order to get

more accurate results.

The two semi-circular arches are initially connected at the middle node 151. When the

vertical uniform load is increased to 31. 63 lb/in, the two arches start to contact each other

at nodes 150 and 152. The two arches have to be connected to have the same deflections

and rotations at nodes 150, 151, and 152 of both arches in the next load stage. When the

load reaches 36.83 lb/in, the arches meet at two more sets of nodes: nodes 143 and 144,

and nodes 158 and 159. Therefore, these four additional nodes of both arches are

constrained together for the next load stage. Then, these two arches contact each other at

nodes 137, 138 and 164, 165 when the load reaches 42.58 lb/in. As described previously,

the arches will then have more constraints at nodes 137, 138, 164, and 165. The number

of contact points of the two arches increases as the load is increased. As a result, the

deformation in the middle part of the arches becomes wavy (see Figure 6.26). Results are


presented in Tables 6.17 and 6.18, and Figures 6.27 and 6.29. The vibration frequencies

have sudden jumps when constraints are added. The equilibrium shape (dashed line) and

first vibration mode (solid line) at the load of 49.28 lb/in are shown in Figure 6.29.

151 152 151 150

load stage 1 : q = 3 0 lb/in load stage 2: q = 3 5 lb/in

1152151150

1152151150

load stage 3: q = 40 lb/in load stage 4: q = 45 lb/in

Figure 6.26 Top view of the wavy leaning arch structure



load Q (lb/in) Y-disp. (in) increment q (lb/in) increment Y-disp. (in) 0 0

4.775 0.4036 9.135 0.7047

15 1.4 22.54 2.284 31.63 3.591 32.63 3.7139 1 0.1229 33.63 3.8391 2 0.2481 34.63 3.9669 3 0.3759 35.63 4.0973 4 0.5063 36.63 4.2304 5 0.6394 36.83 4.2574 5.2 0.6664

37.3247 4.316533 0.4947 0.059133 37.8203 4.376 0.9903 0.1186 38.552 4.465 1.722 0.2076 39.632 4.5983 2.802 0.3409 41.21 4.7981 4.38 0.5407

41.572 4.8446 4.742 0.5872 41.93 4.8911 5.1 0.6337

42.287 4.9376 5.457 0.6802 42.58 4.9761 5.75 0.7187

43.0752 5.034823 0.4952 0.058723 43.5714 5.0939 0.9914 0.1178 44.305 5.1823 1.725 0.2062 45.388 5.3147 2.808 0.3386 46.977 5.5132 4.397 0.5371 49.28 5.8106 6.7 0.8345



0 2 3 4 5 6

vertical displacement at the top (in)




0 19.003 22.26 26.057 45.027 45.258 10 17.079 20.137 23.302 42.163 42.493 20 14.974 17.785 20.451 39.06 39.524 30 12.628 15.074 17.577 35.633 36.283

31.63 12.122 14.403 16.689 34.893 35.59 32.63 18.207 21.475 26.748 41.901 42.853 34.63 17.83 21.066 26.271 41.275 Hi= 36.83 17.408 20.606 25.746 40.573

37.8203 20.415 24.431 29.09 45.93 39.632 20.083 21.062 29.688 45.345 46.744 41.572 19.722 20.658 28.257 44.72 46.157 42.58 19.533 20.445 28.033 44.392 45.849

42.5714 21.391 22.459 31.631 49.546 51.478 45.388 21.021 22.128 31.247 48.962 50.916 46.977 20.691 21.835 30.912 48.445 50.438 49.28 20.202 21.404 30.425 47.685 49.721



50 45

c 40 '.a 35 ;- 30 .2 25 E 20 ... .e 15 c :::J 10

5 0

0 10 20 30 40 50 60




6.3.3.1 Half vertical uniform load action on 120-element leaning arch

A half vertical uniform load is applied along the x-axis to both arches (see page 61).


the x-direction and y-direction are shown. The load is increased until the horizontal

displacement at the top exceeds 15 in. No buckling occurs in this load range. The main

movements of the leaning arch are bending and twisting, and the two semi-circular arches

contact each other in the loading area of both arches when a certain load is


n I (I)

""' ?'-~ tll c: ii" O> ""' ~ § ~·

> (=! g' tll

'..() -...J

FRONT VIEW

/ i ; i

/ /

/ /

/

.,,,..,,,.. ,..,.. ....... -·-·-·-·-·-·-·-·

SIDE VIEW

---..................... .....................

....... , .,., ., \ \

\ \ 1 i

TOP VIEW

3D

1.l Ii ,,

ii I ,. i '· i \ \ \ \

,,_:::::::::::::::::::::::·-·-·-. "'·-....., .,.,

'\

Figure 6.29 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 20 degrees tilt angle and uniform load q = 49.28 lb/in

reached. The equilibrium shape (dashed line) and first vibration mode (solid line) at the

load of 63.32 lb/in are shown in Figure 6.32.


load h (lb/in) X-disp. (in) Y-disp. (in) 0 0 0

4.805 -0.5583 0.2026 9.238 -1.115 0.4057 15.27 -1.949 0.7126 23.11 -3.194 1.181 32.66 -5.044 1.916 43.32 -7.755 3.126 50.85 -10.32 4.5 56.32 -12.71 6.0079 60.34 -14.86 7.892 63.32 -16.73 9.952

1- load vs. displacement

X-disp 70 Y-disp.

-20 -15 -10 -5 0 5 10

dispalcement at the top (in)

Figure 6.30 Half uniform load vs. displacement at the top




0 19.003 22.26 26.057 45.027 45.258 10 18.086 21.213 24.708 43.036 44.436 20 17.213 20.093 23.393 40.771 43.672 30 16.461 18.875 22.234 38.275 42.857 40 15.88 17.5 21.609 35.461 41.955 45 15.589 16.717 21.827 33.895 41.455 50 15.203 15.845 22.778 32.231 40.92 55 14.615 14.729 24.866 30.328 40.291 60 13.21 13.788 28.287 28.746 39.571

63.32 11.674 13.047 32.839 26.887 38.999


70 I ., ... 60 : •• c ..... • i::;: ?-. @. 50 \\

'C •• llS ' \ .2 40 • • " e \ \ \ .e 30 • • " ·c: \ \ \ ::i - 20 . • ... ta \ \ \ .c

10 . • ...

\ \ \ 0 •+--•-... 0 10 20 30 40 50




(") ::r ~ ~ ?' ~ ~ -fij" 8'> '"1

i ~-

~ ~

-0 0

FRONT VIEW

/ /

/

/I i ! I i i \ \

,,... .,,.,,..,,, ........

SIDE VIEW

·-·-·-·-·-·-...... . ........... ..............

.................. ., ...... ,

....... , ........ ,

........ , ·,.,

.,.,_

TOP VIEW

3D

----·-·-.. ---·-· --·-------·-·-·-·-- ~ ., .,_

'\ \ \1

--..... ==::::.:::.:::::::::::::::::::::.:-::::-.,.-·-· --- .................. ,,..... -.... , . . .... ,_

Figure 6.32 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 20 degrees tilt angle and half uniform load h = 63.22 lb/in

6.3.3.2 Half vertical uniform load applied on the modified leaning arch structure

The physically unrealizable situation shown in Figure 6.32 must be avoided. In order to

simulate the assembly of real arches, the points where the arches meet have to be found,

and constraints have to be added to those points so that the arches can move together at

those points. To obtain more accuracy for the middle of the leaning arches, 300 straight

beam elements are used for every arch. They are numbered from right to left in the front

view, so that node 301 is at x = 0 on each arch and the load is applied from node 1 to

node 151.

The two semi-circular arches are initially connected at the middle node 151. When the half

vertical uniform load reaches 35.53 lb/in, the two arches start to contact each other at

node 150. They are connected at nodes 150 and 151 of both arches in the next load stage.

When the load reaches 41. 3 3 lb/in, the arches meet at nodes 146 and 14 7. Therefore, these

two additional nodes of both arches are constrained together for the next load stage. Then

the two arches contact each other at nodes 143 and 144 when the load reaches 42.53 lb/in.

As a result, the middle part of the arches is represented by a wavy curve as illustrated in

Figure 6.33. The number of contact nodes of the two arches increases as the load is

increased. Results are presented in Tables 6.21 and 6.22, and Figures 6.34 to 6.36. The

equilibrium shape (dashed line) and first vibration mode at the load of 46. 963 lb/in are

shown in Figure 6.36.


load stage 1: h = 30 lb/in

load stage 3: h = 4 2 lb/in

150 151


I l14sl ~





load h (lb/in) X-disp (in) Y-disp (in) Iner. h (lb/in) Iner. X-disp (in) incr. Y-disp f 0 0 0

4.802 -0.558 0.1937 9.227 -1.115 0.4059 15.25 -1.946 0.7116 23.05 -3.162 1.174 32.53 -5.015 1.905 33.53 -5.324 1.996 34.53 -5.459 2.09 35.53 -5.689 2.187

36.0227 '-5.77179 2.225956 0.4927 -0.08279 0.038956 36.5174 -5.8492 2.264677 0.9874 -0.1602 0.077677 37.009 -5.9289 2.3033 1.479 -0.2399 0.1163 37.493 -6.0087 2.3423 1.963 -0.3197 0.1553 38.21 -6.1283 2.4009 2.68 -0.4393 0.2139

39.268 .3073 2.4893 3.738 -0.6183 0.3023 40.814 -6.575 2.6227 5.284 -0.886 0.4357 41.33 -6.666 2.6685 5.8 -0.977 0.4815

41.8228 -6.75082 2.710454 0.4928 -0.084822 0.041954 42.3176 -6.83553 2.752159 0.9876 -0.16953 0.083659 42.53 -6.8716 2.7701 1.2 -0.2056 0.1016

43.0228 -6.95765 2.811964 0.4928 -0.086047 0.041864 43.1469 -6.9791 2.822378 0.6169 -0.1075 0.052278 43.2706 -7.0005 2.832783 0.7406 -0.1289 0.062683 43.3942 -7.0219 2.843184 0.8642 -0.1503 0.073084 43.4251 -7.0273 2.845783 0.8951 -0.1557 0.075683 43.4713 -7.0353 2.849684 0.9413 -0.1637 0.079584 43.541 -7.0474 2.85553 l.Oll -0.1758 0.08543 43.645 -7.0654 2.864302 1.115 -0.1938 0.094202 43.749 -7.0835 2.8731 1.219 -0.2119 0.103 43.853 -7.1015 2.8818 1.323 -0.2299 0.1117 44.007 7.1286 2.8950 1.477 -0.2570 0.1249 44.237 -7.1693 2.9149 1.707 -0.2977 0.1448 44.581 -7.2302 2.9447 2.051 -0.3586 0.1746 45.092 -7.3215 2.9896 2.562 -0.4499 0.2195 45.849 -7.4584 3.0572 3.319 -0.5868 0.2871 46.963 -7.6646 3.1710 4.433 -0.7930 0.4009



x-disp. 50 y-disp.

c :.:::: .c ::::. "Cl cu £ .... a; .r:.

-8 -6 -4 -2 0 2 4





0 19.003 22.26 26.057 45.027 45.258 10 18.086 21.213 24.708 43.036 44.436 20 17.213 20.093 23.393 40.771 43.672 30 16.461 18.875 22.234 38.275 42.857

35.53 16.018 18.048 21.566 36.644 42.354 36.5174 18.277 21.922 27.687 42.342 45.023 37.493 18.201 21.825 27.615 42.273 44.8 39.268 18.062 21.644 27.487 42.143 44.38 41.33 17.9 21.431 27.352 41.991 43.886

41.8228 18.427 21.97 29.351 41.956 47.673 42.3176 18.387 21.921 29.326 41.921 47.556 42.530 18.37 21.9 29.315 41.906 47.505

43.4713 18.531 21.939 30.483 41.838 49.688 45.092 18.401 21.781 30.424 41.724 49.303 45.849 18.336 21.702 30.399 41.667 49.111 46.963 18.244 21.59 30.366 41.586 48.836


1 ~50 45 ~ ' i . . .......... - ~ ~ ~-.5 40 • • .t. - • • .t. @.. 35 ·-·-... ---~ 30 \ \ \

~25 \\\

g2o •\ •\ "'\ § 15 lt::

jg 1~ \ \ \ 0 -+-------1-----•+--•-.t.--;-----+--------;


0 10 20 30 40 50


Figure 6. 3 5 Half uniform load vs. vibration frequencies



-n (see page 61). Results are presented in Tables 6.23 and 6.24, and Figures 6.37 to .39.

Displacements in the x-direction and y-direction are shown. The load is increased until the

horizontal displacement reaches 15 in. No buckling occurs in this load range. The arches

bend and twist as the load is increased. The equilibrium shape (dashed line) and first

vibration mode (solid line) at the load of 56.07 lb/in are shown in Figure 6.39.


(i ::r ~ ct ..... ?'-~ {/l

E.. -{/l

O> .... b § ~-

~ if {/l

-0

°'

FRONT VIEW TOP VIEW

___ .:::::::::::::=,-·-·-·-·-·-·-·-·-·-·=,=:::: _____ ·~ . ,,,...

·"'·

SIDE VIEW 3D

Figure 6.36 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 20 degrees tilt angle and half uniform load h = 46.963 lb/in

Table 6.23 Relationship between normal wind load and displacement at the top

load n (lb/in) X-disp (in) Y-diso (in) 0 0 0

2.5 0.772 0.007799 5 1.544 0.031644

8.761 2.701 0.097496 13.79 4.24 0.2414 18.85 5.773 0.4494 23.95 7.298 0.7212 29.10 8.812 1.057 34.32 10.31 1.455 39.61 11.8 1.917 44.99 13.27 2.441 50.47 14.72 3.027 56.07 16.15 3.674


c 60 y-disp. x-disp.

i.:: @. 50 "C

40 C'G .2 "C 30 c ·~ 20 c; E 10 0 c 0

0 5 10 15 20


Figure 6.37 Normal wind load vs. displacement at the top



load n (lb/in)

0 5

13.79 23.95 34.32 44.99 50.47 56.07


19.003 22.26 26.057 45.027 19.032 22.268 26.075 45.038 19.156 22.315 26.175 45.083 19.454 22.418 26.417 45.147 19.893 22.573 26.852 45.123 20.432 22.776 27.589 45.034 20.716 22.894 28.124 44.972 20.995 23.023 28.808 44.900

normal wind load vs. vibration frequency

I 50

~ 40

-g 30 'i 1\i 20 E ... g 10

o+-~~--+~~---4~H--A-~+--~~-+-~--------1

0 10 20 30 40 50





45.258 45.261 45.257 45.264 45.383 45.584 45.705 45.840

108

Q ~ !i' ?' S' ~ f1' ~ .....

i ~· > a ; r.n

-0 \0

FRONT VIEW TOP VIEW

.-...·-·-·-·_::::=:=:-·-·-·-·-·==:=:::::::::·-·-· ~ ....................... ,

....... '·

SIDE VIEW 3D

Figure 6.39 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 20 degrees tilt angle and normal wind load n 56.07 lb/in

6.3.5 Angle load



Displacements in x-, y-, and z-direction are shown. The load is increased until sideways

horizontal deformation reaches 15 in. The arches twist and move along z-direction as load

is increased. The equilibrium shape (dash line) and first vibration mode (solid line) at the

load of26.94 lb/in are shown in Figure 6.42.


load a (lb/in) X-disp (in) Y-disp (in) Z-disp (in) 0.5 0.2047 0.011528 0.3289 1 0.4099 0.026658 0.6576

1.758 0.7188 0.056117 1.150 2.897 1.184 0.1156 1.887 4.430 1.808 0.2234 2.868 5.982 2.433 0.3640 3.846 7.556 3.060 0.5376 4.820 9.155 3.688 0.7442 5.790 10.78 4.314 0.9839 6.757 12.44 4.938 1.257 7.719 14.12 5.558 1.563 8.676 15.84 6.174 1.903 9.629 17.60 6.785 2.275 10.58 19.39 7.389 2.680 11.52 21.22 7.985 3.118 12.46 23.09 8.573 3.588 13.39 25.00 9.150 4.089 14.31 26.94 9.717 4.622 15.24


angle load vs. displacement

30 y-disp. x-disp.

25 z-disp.

c g_ 20 "O ~ 15 ..2 CD

"6> 10 c ~

5

0 0 5 10 15 20


Figure 6.40 Angle load vs. displacement at the top


load a frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 (lb/in) (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec)

0 19.003 22.26 26.057 45.027 45.258 5 19.195 22.313 26.516 44.49 45.541 10 19.289 22.387 27.072 43.968 46.017 15 19.28 22.443 27.707 43.539 46.603 20 19.174 22.45 28.386 43.188 47.279

26.94 18.895 22.353 29.325 42.794 48.332


I angle load vs. vibration frequency

30

- 25 c ::: g_ 20 "CJ ~ 15

~ 10 c cu 5

o+-~~---+-~~-U+-tf--ft--+-~~~t----C--~

0 10 20 30 40 50


Figure 6. 41 Angle load vs. vibration frequencies

6.3.6 Sideways load


Results are presented in Tables 6.27 and 6.28, and Figures 6.43 to 6.45. The horizontal

and vertical displacements at the top are shown. The load is increased until the horizontal

deformation exceeds 15 in. The arches twist and move along the direction of the load as

the load is increased. The equilibrium shape (dashed line) and first vibration mode (solid

line) at the load of 42.66 lb/in are shown in Figure 6.45.


(l

{ 0 .... 9' ~ E. ...... ti.I

& .... r § ~·

~ ~ 0 r.n

--w

FRONT VIEW TOP VIEW

,,,.. ............. :_-:.-:;_-:.;.-:-..;....,.n ... ~--~--

,,,...,,,..· "'.'.ii':,.·

SIDE VIEW 3D

Figure 6.42 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 20 degrees tilt angle and angle load a= 26.94 lb/in


load s (lb/in) X-disp (in) Y-disp (in) 0.4972 0.2044 0.021022

1 0.4087 0.043436 1.75 0.7153 0.079416

2.875 1.175 0.1389 4.564 1.865 0.2403 7.103 2.899 0.4201 10.13 4.123 0.6757 13.17 5.344 0.9773 16.24 6.562 1.325 19.35 7.775 1.717 22.49 8.982 2.154 25.69 10.18 2.636 28.94 11.37 3.162 32.26 12.56 3.731 35.64 13.73 4.343 39.11 14.9 4.996 42.66 16.05 5.692


~:~I ;e 35 -·. ; 30 .2 25 -~ 20 r ~ 15 ~ 10 ·u; 5

y-disp. x-disp.

o~~~~-+-~~~-+~~~--1~~~---i

0 5 10 15 20





loads {lb/in)

0 10 20 30 40

42.66


19.003 22.26 26.057 45.027 19.032 22.43 27.129 44.579 19.271 22.783 28.538 44.201 19.731 23.288 30.199 43.994 20.373 23.894 32.043 43.929 20.562 24.062 32.552 43.92


50 c ~ 40

-g 30 ..2 II)

~ 20 ~ ~ 10

0 -+-----~····-~-+---__,..~,....... 1------------.--.······-----+------<

0 10 20 30 40 50 60


Figure 6.44 Sideways load vs. vibration frequencies



45.258 46.467 47.885 49.399 50.982 51.41

115

f FRONT VIEW TOP VIEW ~

~ c: fl' ~ "'1

r-'

~ """'

.~:--·_. ___ ::::::=::=,,=·--·-·-·-·-·=,,::::::::::·--· -· -· ............. ' ~

i SIDE VIEW 3D

- Figure 6.45 Equilibrium shape (dashed line) and first vibration mode (solid line) -O'I for leaning arches with 20 degrees tilt angle and sideways loads= 42.66 lb/in


The properties are the same as in section 4.2 except that H = 86.60 in.


A concentrated vertical load is applied at the crown of the leaning arch structure. Results

are presented in Tables 6.29 and 6.30, and Figures 6.46 to 6.48. The load is increased until

the vertical displacement at the top exceeds 15 in. No buckling occurs in this load range.

Flexural deformation dominates in the equilibrium shape. Twisting movement occurs in

the first vibration mode. The equilibrium shape (dashed line) and first vibration mode

(solid line) at the load of 17.249 kips are shown in Figure 6.48.


18000 16000 14000

g: 12000 10000

"C

~ 8000 6000 4000 2000

0 0 5 10 15 20

vertical displacement at the top {in)

Figure 6. 46 Concentrated load vs. displacement at the top



load P (lb) Y-disp (in) 494.7 0.2749 979.6 0.5503 1688 0.9641 2711 1.586 4161 2.523 5607 3.525 6957 4.528 8220 5.532 9401 6.536 10507 7.539 11545 8.541 12518 9.541 13433 10.54 14292 11.53 15101 12.52 15861 13.51 16576 14.49 17249 15.47


load P frequency 1 frequency 2 (rad frequency 3 frequency 4 frequency 5 (lb) (rad I sec) I sec) (rad I sec) (rad I sec) (rad I sec) 0 22.223 27.028 30.643 42.563 43.247

1000 21.692 26.465 30.09 41.899 42.623 2000 21.16 25.9 29.502 41.229 41.996 4000 20.069 24.743 28.326 39.835 40.693 6000 18.962 23.566 27.141 38.394 39.354 8000 17.859 22.386 25.985 36.926 38.008 10000 16.668 21.117 24.707 35.294 36.499 12000 15.493 19.844 23.472 33.618 34.97 14000 14.309 18.56 22.298 31.859 33.393 16000 12.989 17.154 20.96 29.853 31.56 17000 12.32 16.44 20.317 28.779 30.591 17249 12.181 16.280 20.201 28.526 30.375



20000

- 15000 .a ;.. ,, 10000 ca .2

5000

0 0 10 20 30 40 50


Figure 6.47 Concentrated load vs. vibration frequency


6.4.2.1 Vertical uniform load acting on 120-element leaning arch structure

A vertical uniform load is applied along both arches. Each arch is divided into 60 elements

as before. Results are presented in Tables 6.31 and 6.32, and Figures 6.49 to 6.51.

Bifurcation buckling occurs on the load-displacement equilibrium path. The bifurcation

load is 46.34 lb/in, and the corresponding vertical displacement at the top is 12.85 in. The

main movement of the leaning arch is bending and twisting, and the two arches cross each

at a certain load. The equilibrium shape at buckling (dashed line) and the buckling mode

(solid line) are shown in Figure 6.51, but the result is not physically realizable.


n

! .... ?" ~ en s= --en O> .... ~ § :r

(JC)

~ go en

-N 0

FRONT VIEW

I

I i I \

/ /

/

,.. ............ -·-·-·-·-·-·-·-·-·-·-·-

SIDE VIEW

-·-·-·-·-·-..... ............ ,

.,\

\ \ I i i

TOP VIEW

30

..... ~· .,,. .... ;,./ /'1/ ( i ·-1. ( I I

' ' \ \ \

Figure 6.48 Equilibrium shape (dashed line) and buckling mode (solid line) for leaning arches with 30 degrees tilt angle and concentrated load P = 17.249 lb/in


load q (lb/in) Y-disp (in) 0 0

4.741 0.5693 9.004 1.141 14.64 2.002 21.7 3.306

29.11 5.064 34.86 6.85 39.36 8.666 42.93 10.51 43.7 10.98

44.77 11.69 45.34 12.09 45.64 12.31 46.09 12.65 46.34 12.85 46.7 13.15

47.02 13.42 47.56 13.85 47.77 14.1 48.15 14.46 48.68 15.02


50 c 45

~ 40 "O 35 tU ..2 30 E 25 .e ·c: 20 :s iii 15 u :e 10 cu > 5

0 0 5 10 15





load q frequency 1 frequency 2 frequency 3 frequency 4 lb/in rad I sec rad I sec rad I sec rad I sec

0 22.223 27.028 30.643 42.563 10 19.914 25.48 28.532 39.049 20 17.286 23.946 26.621 35.122 30 22.512 25.157 30.599 32 22.253 24.954 29.596 40 21.41 24.613 25.054 46 20.683 21.195 23.38

46.3 8 20.425 21.202 23.18 46.34 0.26037 20.39 21.203 23.153


50

-:§ 40 :§. ~ 30 .2 e 20 J2 ·2 10 ::I

0 10 20 30 40


Figure 6. 50 Uniform load vs. vibration frequencies


50

frequency 5 rad I sec

43.247 39.911 36.212 32.004 31.083 27.011 25.119 25.171 25.178

122

n ::r .§ !i ?' ~ en E. &;' O> .... ~ ~ JJ. ~ 0 go en

-N V..>

FRONT VIEW TOP VIEW

.......... / ......

..... ---·-·-·-·-·-·-· ·---............. , . ................... .,

// i i !

/

/ /

SIDE VIEW

·,·,·,., .,

\ \ \ i

3D

.~,.:::-_:::-..:.::-~~_:...~....:...-:-·-·-·-·.....J...-·

............. , .,

/

/.:f;f;;·

l ii u

Figure 6.51 Equilibrium shape (dashed line) and buckling mode (solid line) for leaning arches with 30 degrees tilt angle and uniform load

., ., \ \ \ \

6.4.2.2 Vertical uniform load applied to the modified leaning arch structure

The crossing problem happens again. The same method is used to determine the points

where the arches meet, and constraints have to be added to those points so that the arches

can move together at those points. Because the deflections in the middle part of the

leaning arches are mainly of concern for this case, 3 00 straight beam elements are used on

each arch in order to obtain more accurate results ..

The two semi-circular arches are initially connected at the middle node 151. When the

vertical uniform load reaches 31. 50 lb/in, the two arches contact each other at nodes 150

and 152. The two arches are connected at nodes 150, 151, and 152 of both arches in the

next load stage. When the load is 33.82 lb/in, the arches meet at two more sets of nodes:

nodes 146 and 147, and nodes 155 and 156. Therefore, these four additional nodes of

both arches are constrained together for the next load stage. Then these two arches

contact each other at nodes 143, 144 and 158, 159 when the load reaches 35.94 lb/in. As

described previously, the arches will have further constraints at nodes 143, 144, 158, and

159. The number of contact nodes of the two arches increases as the load is increased. As

a result, the displacement in the middle part of the arches will become wavy (see Figure

6.52). Results are presented in Tables 6.33 and 6.34, and Figures 6.53 to 6.55. The


lb/in are shown in Figure 6.55. No buckling occurs in the load range considered.


151

load stage I: q = 30 lb/in

1152151 150

/ 11551 ~

load stage 3: q = 3 5 lb/in

152151150

load stage 2: q = 3 3 lb/in

load stage 4: q 40 lb/in




load q (lb/in) Y-disp (in) Iner. q (lb/in) incr. Y-disp (in) 0 0

4.736 0.569 8.993 1.14 14.61 1.999

3.296 5.265

31.50 5.753 31.9957 5.8547 0.4957 0.1017 32.4869 5.9566 0.9869 0.2036 33.211 6.1094 1.711 0.3564 33.82 6.2404 2.32 0.4874

34.3158 6.33824 0.4958 0.09784 34.8075 6.4362 0.9875 0.1958 35.533 6.5833 1.713 0.3429 35.94 6.667 2.12 0.4266

36.4359 6.762517 0.4959 0.095517 36.928 6.8"~" 0.988 0.1912 37.655 7.0017 1.715 0.3347 38.721 7.2171 2.781 0.5501 40.269 7.5405 4.329 0.8735


uniform load vs. d isp lac em ent

45

40

c 35

~ 30 't:I 25 "' ..2 e 20 a 15 'E :::s

10

5

0 0 2 4 6 8

vertical displacement (in)



load q frequency 1 frequency 2 frequency 3 (rad frequency 4 frequency 5 (lb/in) (rad I sec) (rad I sec) I sec) (rad I sec) (rad I sec)

0 22.223 27.028 30.643 42.563 43.247 10 19.914 25.48 28.532 39.049 39.911 20 17.286 23.946 26.621 35.122 36.212

4.085 22.512 25.157 30.599 32.004 .324 22.195 24.456 29.549 30.973

32.4869 21.17 25.967 32.059 36.931 39.186 33.211 21.009 25.857 31.927 36.641 38.915 33.82 20.872 25.765 31.816 36.395 38.686

34.8075 21.167 27.711 34.027 38.29 41.23 35.533 21.005 27.6 33.899 38.337 40.954 35.94 20.914 27.537 33.828 38.172 40.799

36.928 21.167 29.069 35.677 39.901 42.812 37.655 21.004 28.955 35.551 39.608 42.532 38.721 20.763 28.787 35.368 39.172 42.116 40.269 20.4066 28.542 35.104 38.531 41.503



45

40 • • ~

35 .

c •. ~ 30 •

\ -g 25 ..2 e 20 . a 15 \ c :I

10 • 5 \ 0 •

0 10 20 30 40 50

vibration frequency {rad/sec)



6.4.3.1 Half vertical uniform load acting on 120-element leaning arch structure

A vertical, horizontally-uniform load is applied on half of both arches (see page 61 ).


the x-direction and y-direction are shown. The load is increased until the vertical crown

displacement reaches 15 in. No buckling occurs in this load range. The arches bend and

twist as the load is increased. The two arches contact each other at points other than the

crown when a certain load is reached. The equilibrium shape (dashed line) and first


(") t:J'" ~ FD .... ?' ~ C/l E.. ~

CP ""1

t'""'4 § s·

(IQ

~ C/l

-N

"°

FRONT VIEW

!/ I i i

/ /

/ /

,,,,,. ...... ,,,,,..

....... -·-·-·-·-·-·-·-·-·

SIDE VIEW

......................... ~ ....... . ........... ., .,

.,.,,\ \ \ \ i ;

TOP VIEW

3D

Figure 6.55 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 30 degrees tilt angle and uniform load q = 40.269 lb/in

vibration mode (solid line) at the load of 54.73 lb/in are shown in Figure 6.58, but this

situation is not physically realizable.


load h (lb/in) X-disp (in) Y-disp (in) 0 0 0

4.778 -0.6051 0.2869 9.132 -1.205 0.5795 14.97 -2.096 1.031 22.39 -3.408 1.743 30.28 -5.105 2.765 36.39 -6.719 3.881 41.14 -8.236 5.102 44.87 -9.642 6.439 47.79 -10.92 7.899 50.07 -12.05 9.485 51.82 -13.00 11.20 53.15 -13.78 13.03 54.10 -14.36 14.97 54.73 -14.74 17.01

half uniform load vs. displacement

x -dis p. 60 y-disp.

c ~ "C ca ..9 E ... .E ·c :::s -"i .s::;

-20 -10 0 10 20






0 22.223 27.028 30.643 42.563 43.247 10 21.081 26.268 29.646 40.226 42.161 20 19.818 25.454 29.046 37.347 41.253 30 18.348 24.388 29.385 34.036 40.28 35 17.457 23.714 30.128 32.167 39.743 40 16.367 22.933 31.476 30.139 39.148 45 14.857 22.032 33.632 27.944 38.468 50 12.349 20.946 36.952 25.716 37.63

54.73 5.4256 19.124 37.276 24.264 36.22


60

c 50 ! 'Cl 40 ca .2 e 30 g

20 c :I !: ca 10 .c

0 0 10 20 30 40 50




n s ~ ""1

?' ~ c.n E. -c.n Q> .....

i JJ. ~ ~ c.n

-w N

FRONT VIEW

.--·-·-·-·-·-·-·-...................... ...... .....................

...... ,

SIDE VIEW

.,., ., .,., .,., . ...... , .......

..............

TOP VIEW

3D

i i i i \ \ \ \ \: ·~

.-;.....----..--:-..:..-:-.:..-:::..-:-.:...--:::_-:-.:._--:-.:._ ... ~:.-.~, .. ,:::::-:--:r..."'· .............. ?--"" ........ ,

Figure 6.58 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 30 degrees tilt angle and half uniform load h = 54. 73 lb/in

.. ,.,,

6.4.3.2 Half vertical uniform load applied on the modified leaning arch

The crossing problem occurs for this load case, and the same method is used to find the

points where the arches meet and then add constraints to those points so that the arches

must move together at those points. Because deflections in the middle part of the leaning

arches are mainly of concern, 3 00 straight beam elements are used on each arch in order to

get more accurate results.

For this load case, the two semi-circular arches are initially connected at the middle node

151. When the vertical uniform load reaches 3 5. 80 lb/in, the two arches begin to contact

each other at the nodes 150. The two arches are connected at nodes 150 and 151 of both

arches in the next load stage, so they have the same displacement and rotation at nodes

150 and 151. When the load reaches 37.55 lb/in, the arches meet at node 147. Therefore,

this additional node is constrained for the next load stage. Then, these two arches contact

each other at node 144 when the load reaches 39.70 lb/in. As described previously, the

arches will have a further constraint at node 144. The number of contact points of the two

arches increases as the load is increased. As a result, the deformation in the middle part of

the arches becomes wavy (see Figure 6.59). Results are presented in Tables 6.37 and 6.38,

and Figures 6.60 to 6.62. The equilibrium shape (dashed line) and first vibration mode

(solid line) at the load of 43.20 lb/in are shown in Figure 6.62. No buckling occurs in the

load range considered.


load stage 1 : h = 3 0 lb/in

I ~

load stage 3 : h = 3 9 lb/in

150151


I

[3£]

load stage 4: h = 4 2 lb/in




h (lb/in) X-disp (in) Y-disp (in) incr. h (lb/in) incr. X-disp (in) incr. Y-disp (in) 0 0 0

4.772 -0.6043 0.2866 9.118 -1.203 0.5788 14.93 -2.09 1.029 22.3 -3.391 1.734 30.94 -5.265 2.869 35.80 -6.551 3.759

36.2927 -6.64001 3.828042 0.4927 -0.089014 0.069042 36.788 -6.7277 3.8965 0.988 -0.1767 0.1375 37.274 -6.8157 3._ . 1.474 -0.2647 7 37.55 -6.8659 4.0055 1.75 -0.3149 0.2465

38.0427 -6.95709 4.074824 . 0.4927 -0.091185 0.069324 38.5386 -7.0471 4.1436 . 0.9886 -0.1812 0.1381 39.025 -7.1372 4.2131 1.475 -0.2713 0.2076 39.70 -7.2638 4.3115 2.15 -0.3979 0.306 40.70 -7.4515 4.4531 1 -0.1877 0.1416 41.20 -7.5472 4.526 1.5 -0.2834 0.2145 42.20 -7.7403 4.6749 2.5 -0.4765 0.3634 43.20 -7.9367 4.8288 3.5 -0.6729 0.5173


x-dsp. 50 y-dsp.

-8 -6 -4 -2 0 2 4 6

displacen&1t at the top (in)





0 22.223 27.028 30.643 42.563 43.247 10 21.081 26.268 29.646 40.226 42.161 20 19.818 25.454 29.046 37.347 41.253 30 18.348 24.388 29.385 3 ,')£ 40.28

35.80 17.158 23.367 30.03 3 . 11 39.614 36.788 21.779 24.76 34.702 39.529 40.424 37.274 21.726 24.71 34.716 39.461 40.297 37.55 21.697 24.682 34.725 39.42 40.228

38.5386 21.81 24.71 36.472 39.559 41.712 39.025 21.758 24.659 36.492 39.517 41.555 39.70 21.684 24.587 36.521 39.459 41.336 40.70 21.847 24.606 38.432 39.423 43.137 41.20 21.793 24.553 38.459 39.382 42.97 42.20 21.685 24.446 38.516 39.3 42.633 43.20 21.575 24.338 38.578 39.217 42.291


45 • • ~ ...

40 ; ; ... w

~ 35 ., .,-

i :~ \ \ ~ 20 •

~ 15 \ \

~ 1~ \ \ 0 -+------+---------+--• -•----1A-----+--'lllr>------l

0 10 20 30 40 50




("'.'.) ::r ~ ~ ?' ~ en E.. fir O' .....

i &. > (:! if en

-w .....:I

FRONT VIEW

,,.. /

/ /

/ i i i I i !

,,,.,,,.-·-·-·-·-·-·

SIDE VIEW

·-·-·-·-.............. .............

.............. , .,., .,.,

.,., . , '\

TOP VIEW

3D

, .. ~ /<;,,.·""

/ / / / .i /.

! I ,.,. 1· i '· ; i i \ \ \ \ \

I /

/

.. - .. --·--·-'·"""·

// ,,,,, ·-·-·-................ ......

.,., .. , \

..:..-:-.. ::::.:..-::.~::-·-·-· .... ·-·-·-·

...................... , ·,,

\ \

' \

Figure 6.62 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 30 degrees tilt angle and half uniform load h = 43 .2 lb/in



-n (see page 61). Results are presented in Tables 6.39 and 6.40, and Figures 6.63 to 6.65.

Displacements in the x-direction and y-direction are shown. The load is increased till the


twist and move horizontally as the load is increased. The equilibrium shape (dashed line)

and first vibration mode (solid line) at the load of 56.84 lb/in are shown in Figure 6.65.

Table 6.39 Relationship between normal wind load and displacement at the top

load n (lb/in) X-disp (in) Y-disp (in) 0 0 0

2.5 0.7644 0.008038 5 1.529 0.033224

8.765 2.674 0.1031 13.80 4.196 0.2563 18.88 5.712 0.4779 24.02 7.217 0.7675 29.22 8.710 1.125 34.51 10.19 1.549 39.89 11.64 2.040 45.40 13.08 2.597 51.04 14.49 3.220 56.84 15.88 3.906



y-disp. x-disp. - 60 c :.::::: .a 50 -,,

40 CG .2 ,, 30 c 'i 20 Ci E 10 '-0 c 0

0 5 10 15 20


Figure 6.63 Normal wind load vs. displacement at the top


load n frequency 1 frequency 2 frequency 3 frequency 4 (rad frequency 5 lb/in rad I sec rad I sec rad I sec I sec rad I sec

0 22.223 27.028 30.643 42.563 43.247 5 22.232 30.720 42.582 43.255

13.8 22.274 26.918 31.174 42.661 43.261 24.02 22.360 26.764 32.145 42.781 43.296 34.51 22.484 26.625 33.572 42.801 43.492 45.40 22.639 26.528 35.474 42.678 43.894 56.84 22.816 26.484 37.897 42.458 44.450


normal load vs. vibration frequency

60 c :a 50 =- 40 "CJ

~ 30 ca 20 E .... 0 10 c

0 0 10 20 30 40 50



6.4.5 Angle load



Displacements in the three global directions are shown. The load is increased till the


bend, twist, and move along the x- and z-directions as the load is increased. The


lb/in are shown in Figure 6. 68.


(".) ::r ~ ft .... ?"-~ ~ ~ ~ """'* i Jg.

~ 5 fl)

..... ~ .....

FRONT VIEW

// /

/

.///

/ /

, ..... ,,,.

SIDE VIEW

TOP VIEW

,,.,. ....... ---·-·-·-·-·-·-·-·-...... ,,.,.. ~

3D

Figure 6.65 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 30 degrees tilt angle and normal wind load n = 56.84 lb/in


load a (lb/in) X-disp (in) Y-disp (in) Z-disp (in) 0.5 0.2004 0.016427 0.1396 1 0.4009 0.035033 0.279

1.753 0.7019 0.067003 0.4878 2.883 1.154 0.1242 0.8001 4.586 1.832 0.2305 1.265 7.133 2.837 0.4336 1.951 9.717 3.841 0.6911 2.629 12.35 4.841 1.003 3.298 15.03 5.837 1.370 3.957 17.78 6.827 1.790 4.607 20.60 7.808 2.265 5.246 23.50 8.780 2.793 5.874 26.49 9.740 3.375 6.492 29.56 10.69 4.008 7.098 32.74 11.62 4.693 7.695 36.01 12.54 5.427 8.283 39.40 13.43 6.210 8.861 42.89 14.31 7.040 9.433 46.50 15.17 7.914 9.998

angle load vs. displacement

50 y-disp. z-disp. x-disp.

45

c 40

! 35 30 ,,

CV 25 .2 GI 20 c,

15 c ftS

10 5 0

0 4 8 12 16


Figure 6. 66 Angle load vs. displacement at the top



load a (lb/in)

0 10 20 30 40

46.5


22.223 27.028 30.643 42.563 22.433 27.155 32.005 41.582 22.782 26.752 33.659 41.147 23.123 26.187 34.771 41.516 23.328 25.652 35.23 42.323 23.329 25.397 35.336 42.836

angle load vs. vibration frequency

50 -c: :.::: 40 .c -"C 30 ca .2 20 CD

g> 10 ca

0 -1------t-----+--tl''lt----0 l---ft------1~~---I

0 10 20 30 40 50


Figure 6.67 Angle load vs. vibration frequencies



43.247 43.776 44.892 46.37

48.148 49.41

143

(1

f ""'t

?' ~ tlJ E.. -tfJ

O> "'1 ~ ('O

§ Er

(JQ

> ~ ::r ('O tlJ

-t

FRONT VIEW TOP VIEW

SIDE VIEW ~· ....... ·'"~ 3D

--......... ,

Figure 6.68 Equilibrium shape (dashed line) and first vibration mode (solid line)

....... ,., ·.\

\ \

\

for leaning arches with 30 degrees tilt angle and angle load a= 46.50 lb/in

6.4.6 Sideways load


Results are presented in Tables 6.43 and 6.44, and Figures 6.69 to 6. 71. Horizontal and

vertical displacements at the top are shown. The load is increased until the horizontal

deformation reaches 15 in. No buckling occurs in this load range. The arches twist and

move along the direction of the load as the load is increased. The equilibrium shape

(dashed line) and first vibration mode (solid line) at the load of 43.09 lb/in are shown in

Figure 6. 71.


load s (lb/in) X-disp (in) Y-disp (in) 0.4973 0.2004 0.022698

1 0.4008 0.046801 1.750 0.7013 0.08551 2.877 1.152 0.1494 4.568 1.828 0.2582 7.112 2.842 0.4508 10.45 4.161 0.7542 13.82 5.476 1.117 17.23 6.785 1.538 20.69 8.087 2.017 24.21 9.380 2.553 27.81 10.66 3.146 31.48 11.93 3.794 35.25 13.19 4.498 39.11 14.44 5.255 43.09 15.67 6.065


50 -c ~ 40 ::::. -g 30 ..2 ~ 20 ~ .g 10

·u;


y-disp. x-disp.

Ol::M"'"--~~-f-~~~-+~~~~+-~~~~

0 5 10 15 20




loads frequency 1 frequency 2 frequency 3 frequency 4 frequency 5 (lb/in) (rad I sec) (rad I sec) (rad I sec) (rad I sec) (rad I sec)

0 22.223 27.028 30.643 42.563 43.247 10 22.431 26.857 31.903 42.146 44.279 20 22.814 26.756 33.806 41.698 45.658 30 23.322 26.904 36.035 41.413 47.163 40 23.907 27.296 38.466 41.305 48.762

43.09 24.093 27.454 39.243 41.298 49.271



50 c :a 40 ::::::..

0 10 20 30 40 50


Figure 6. 70 Sideways load vs. vibration frequencies


9 ~ S' .... 9'i

~ = a CP """' i .... ~

~ [

-.i::.. 00

FRONT VIEW TOP VIEW

/ /

/ /

./· I

/ /

/

....... -·-·-·-·-·-·-·-·-·-·-,,,..,,,,. ·-,,-'/ ~

SIDE VIEW 3D

Figure 6. 71 Equilibrium shape (dashed line) and first vibration mode (solid line) for leaning arches with 30 degrees tilt angle and sideways loads= 43.09 lb/in

Chapter 7. Conclusions and Recommendations

7.1 Conclusions

A summary of the results for leaning arches is presented in Table 7 .1. The load values

designated by "exceed 15 in." refer to the first load obtained during the load steps used by

the computer program at which either the x-, y-, or z-displacement of the crown exceeded

15 in. Higher loads are not considered. Based on the results for leaning arches, some

conclusions can be drawn.

a. The comparison shows that the leaning arches buckle only under vertical load, such as

concentrated load, or whole or part of uniform load. Bifurcation occurs at all critical loads

obtained in this study.

b. For normal wind load and sideways load, no bifurcation load or limit point occurs on

the load-deformation equilibrium path. The displacement at the top of the arches increases

as the load is increased. Deformation in the x-direction is significant in these loading case.

c. The two arches contact each other at points other than the apex of the leaning arch

structure at a certain load for some load cases, such as whole and part of uniform vertical

load for leaning arches with 20 and 30 degrees of tilt angle. To handle this

Chapter 7. Conclusions and Recommendations 149

Table 7.1 Summary of results for leaning arches

load pattern angle 10 20 30

vertical concentrated critical load I bifurcation load (kips) 6.075 17.201 -load (P) I limit load (kips) - - -

load (exceed 15 in)(kips) - - 17.249 y-displacement at the top (in) 2.92 11.96 15.47

vertical uniform load critical load I bifurcation load (lb/in) 16.409 - -(q) I limit load (lb/in) - - -

load (exceed 5 or 7 in)(lb/in) - 49.28 40.27 y-displacement at the top (in) 1.186 5.810 7.541

half vertical uniform critical load I bifurcation load (lb/in) 29.28 - -load (h) I limit load (lb/in) - - -

load (exceed 7 in)(lb/in) - 46.96 43.20 x-displacement at the top (in) -4.035 -7.665 -7.937 y-displacement at the top (in) 1.18 3.171 4.829

normal wind load (n) critical load I bifurcation load (lb/in) - - -l limit load (lb/in) - - -

load (exceed 15 in)(lb/in) 55.65 56.07 56.84 x-displacement at the top (in) 16.28 16.15 15.88 y-displacement at the top (in) 3.549 3.674 3.906 z-displacement at the top (in) 0 0 0

angle load (a) critical load I bifurcation load (lb/in) - - -I limit load (lb/in) - - -

load (exceed 15 in)(lb/in) 5.954 26.94 46.5 x-displacement at the top (in) 2.412 9.717 15.17 y-displacement at the top (in) 1.64 4.622 7.914 z-displacement at the top (in) 15.5 15.24 9.998

sideways load ( s) critical load I bifurcation load (lb/in) - - -I limit load (lb/in) - - -

load (exceed 15 in)(lb/in) 38.87 42.66 43.09 x-displacement at the top (in) 15.08 16.05 15.67 y-displacement at the top (in) 4.802 5.692 6.065 z-displacement at the top (in) 0 0 0


phenomenon, extra constraints have to be added at the points where the two arches meet.

Thus, the two arches will have exactly the same displacements and rotations at these

points, and one arch cannot move relative to the other.

7 .2 Recommendations for Future Study

Research in any field usually creates as many questions as it does answers. This thesis

leaves open several areas where further study is warranted. Several of these areas are

discussed below.

Only semi-circular arches are considered in this study. It is necessary to consider different

shapes, such as parabolic shapes. Also, optimization of the shape and tilt angle for the

leaning arches should be investigated.

This study assumes that the supports of the arches are pinned. Results for other boundary

conditions should be obtained.

In this study, the two arches overlap each other for some load cases. To avoid this

crossing problem, constraints have to be added to the points where the two arches

contact, and they will have the same displacements and rotations in the three global


directions. Future study should consider a sliding contact situation for the two arches, that

is, only displacement in the z-direction is to be constrained at the points where the two

arches meet, so that one arch can move relative to the other in the x- and y-directions and

they will not be allowed to overlap.

Steel is used in this analysis. Future studies should consider other materials, including

composites, and optimize the properties of the materials for the leaning arches. With

regard to the potential use of leaning arches as part of the inflatable framework for large

tent structures, the behavior of inflated leaning arches should be investigated.


References

1. Austin, W. J., "In-plane Bending and Buckling of Arches", Journal of the Structural Division, ASCE, Vol. 97, No. ST5, May, 1971, pp. 1575-1591.

2. Austin, W. J. and Ross, T. J., "Elastic Buckling of Arches Under Symmetrical Loading", Journal of the Structural Division, ASCE, Vol. 102, No. ST5, May, 1976, pp. 1085-1095.

3. Cook, R. D., Malkus, D. S. and Plesha, M. E., Concepts and Applications of Finite Element Analysis, 3rd Edition, Wiley, New York, 1989.

4. DaDeppo, D. A. and Schmidt, R., "Nonlinear Analysis of Buckling and Postbuckling Behavior of Circular Arches", Journal of Applied Mathematics and Physics, Vol. 20, No. 6, 1969, pp. 847-857.

5. Galambos, T. V., ed., Guide to Stability Design Criteria for Metal Structures, 4th Edition, Wiley, New York, 1988.

6. Hibbitt, Karlsson, and Sorensen, Inc., ABAQUS, Pawtucket, Rhode Island, 1994.

7. Krainski, W. J., "Investigation of Alternative Framing Arrangements Using Pressure-Stabilized Beams for Battalion Aid Station Support", Technical Report TR-88-071, U. S. Army Natick Research and Development Laboratories, Natick, Massachusetts, September, 1988.

8. Langhaar, H. L., Boresi, A. P. and Carver, D. R., "Energy Theory of Buckling of Circular Elastic Rings and Arches", Proceedings of Second U.S. National Congress of Applied Mechanics, 1954, pp. 437-443.

9. Lind, N. C., "Elastic Buckling of Symmetrical Arches", University of Illinois Engineering Experiment Station Technical Report No. 3, 1962.

10. Papangelis, J. P. and Trahair, N. S., "Flexural-Torsional Buckling of Arches", Journal of Structural Engineering, Vol. 113, No. 4, April, 1987, pp. 889-906.

11. Pi, Y.-L. and Trahair, N. S., "Three-dimensional Nonlinear Analysis of Elastic Arches", Engineering Structures, Vol. 18, No. 1, January, 1996, pp. 49-63.

153

12. Steeves, E. C., "Fabrication and Testing of Pressurized Rib Tents", Technical Report TR-79-008, U. S. Army Natick Research and Development Laboratories, Natick, Massachusetts, April, 1979.

13. Tokarz, F. J., "Experimental Study of Lateral Buckling of Arches", Journal of the Structural Division, ASCE, Vol. 97, No. ST2, February, 1971, pp. 545-559.

14. Trahair, N. S. and Papangelis, J. P., "Flexural-Torsional Buckling of Mono symmetric Arches", Journal of Structural Engineering, Vol. 113, No. 10, October, 1987, pp. 2271-2288.

15. Wen, R. K. and Lange, J., "Curved Beam Element for Arch Buckling Analysis", Journal of the Structural Division, ASCE, Vol. 107, No. STl 1, November, 1981, pp. 2053-2069.

16. Wen, R. K. and Suhendro, B., "Nonlinear Curved-Beam Element for Arch Structures", Journal of Structural Engineering, Vol. 117, No. 11, November, 1991, pp. 3496-3515.

154

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