Structural analysis and deployable development of cable strut systems

STRUCTURAL ANALYSIS AND DEPLOYABLE
DEVELOPMENT OF CABLE STRUT SYSTEMS












SONG JIANHONG

NATIONAL UNIVERSITY OF SINGAPORE
2007




STRUCTURAL ANALYSIS AND DEPLOYABLE
DEVELOPMENT OF CABLE STRUT SYSTEMS











SONG JIANHONG
(B.Eng. Xi’an Jiao Tong University)

A THESIS SUBMITTED
FOR THE DEGREE OF DOCTER OF PHILOSOPHY
DEPARTMENT OF CIVIL ENGINEERING
NATIONAL UNIVERSITY OF SINGAPORE
2007


ii
ACKNOWLEDGEMENT


Firstly, I would like to thank my supervisors, Professor Koh Chan Ghee and Associate
Professor Liew Jet Yue, Richard for their invaluable advice, continuous guidance and
generous support throughout my graduate study at the department of civil engineering,
National University of Singapore.

Secondly, I would thank all my classmates, colleagues and friends who have helped me
in any way and to any extent since the beginning of my graduate study in January 2003.

Thirdly, I owe my thanks to my parents, wife and parents-in-law for their continuous
love, trust and support. They have been looking after my daughter since her birth in mid
2003, and thus I can concentrate my attention on the study. I would also say thanks to my
lovely daughter who gives me much courage and happiness.

Finally, scholarship and other financial assistances from the National University of
Singapore are gratefully acknowledged.

iii
TABLE OF CONTENTS

TITLE PAGE i

ACKNOWLEDGEMENT ii

TABLE OF CONTENTS iii

SUMMARY vi

NOMENCLATURE viii

LIST OF FIGURES xiv

LIST OF TABLES xxii

CHAPTER 1 INTRODUCTION 1

1.1 Introduction 1
1.2 Cable strut systems 2
1.2.1 Structural types 2
1.2.2 Tension cable strut systems 3
1.2.3 Free standing cable strut systems 4
1.2.3.1 Non-deployable structures 4
1.2.3.2 Deployable structures 6
1.3 Structural analyses of the two focused systems 12
1.3.1 Analysis types and methods 12
1.3.2 Simplified analysis of cable truss 16
1.3.3 Simplified analysis of cable-strut truss 19
1.4 Objectives and scope 23
1.5 Organization of thesis 24

CHAPTER 2 STATIC ANALYSIS OF RADIALLY ARRANGED CABLE TRUSS 27

2.1 Introduction 27
2.2 Initial configurations 27
2.3 Simplified solutions 30
2.3.1 Irvine’s solution 30
2.3.2 Improved solution 31
2.3.3 Solutions for strut force 35
2.4 Numerical verification 38
2.4.1 Finite element theory 38
2.4.2 Numerical verification 39
2.4.3 Nonlinear effect 48
2.4.4 Validity on other structural types 52
2.5 Effect of different parameters on structural behavior 54

iv
2.6 Summary 57

CHAPTER 3 FREE VIBRATION ANALYSIS OF RADIALLY ARRANGED CABLE
TRUSS
58

3.1. Introductions 58
3.2. Analytical free vibration solution 59
3.2.1. Irvine’s solution for cable truss with a cubic shape 59
3.2.2. Solution for cable truss with a parabolic shape 60
3.2.2.1. Single layer circular shallow membrane 60
3.2.2.2. Double layer shallow membrane 63
3.2.2.3. Solution for radially arranged cable truss 65
3.3 Numerical free vibration analysis 68
3.3.1 Finite element theory for free vibration analysis 68
3.3.2 Numerical verification 70
3.4 Summary 73

CHAPTER 4 EARTHQUAKE ANALYSIS OF RADIALLY ARRANGED CABLE
TRUSS 89

4.1 Introduction 89
4.2 Structural behavior study based on finite element analysis 90
4.2.1 Finite element analysis methods 90
4.2.2 Input parameters for numerical model 93
4.2.3 Structural behavior under earthquake 97
4.3 Proposed simplified procedure 104

4.3.1 Formula for estimation of response using response spectra 104
4.3.2 Evaluation of the proposed simplified procedure 106
4.4 Summary 113

CHAPTER 5 NOVEL DEPLOYABLE CABLE STRUT SYSTEM 114

5.1 Introduction 114
5.2 Proposed cubic truss system 114
5.3 Structural behavior studies 130
5.3.1 Evaluation method 130
5.3.2 Comparison of structural behavior between different systems 133
5.3.3 Optimal study on novel cubic truss system 138
5.4 Summary 139

CHAPTER 6 ENHANCED DEPLOYABLE CUBIC TRUSS SYSTEM 140

6.1 Introduction 140
6.2 Enhanced cubic truss system 140
6.2.1. Type-A enhanced cubic truss system 140
6.2.2. Type-B enhanced cubic truss system 143


v
6.2.3. Type-C enhanced cubic truss system 152
6.3. Proposed deployable shelter 184
6.4 Summary 188

CHAPTER 7 STATIC AND DYNAMIC ANALYSIS OF THE NOVEL CUBIC TRUSS
SYSTEM 189

7.1 Introduction 189
7.2 Static analysis 190
7.2.1 Plate Analogies 190
7.2.1.1 Thin plate analogy 190
7.2.1.2 Thick plate analogy 193
7.2.1.3 Numerical verification 194
7.2.2 Novel method based on 2-D planar truss 195
7.2.2.1 Introduction 195
7.2.2.2 Procedure 196
7.2.2.2.1 Simplification of the 3-D space system to 2-D planar system 196
7.2.2.2.2 Analysis of the 2-D system 199
7.2.2.3 Numerical verification 208
7.3 Dynamic analysis 215
7.3.1 Free vibration analysis 215
7.3.1.1 Simplified solution 215
7.3.1.2 Numerical verification 217
7.3.2 Earthquake analysis 219
7.3.2.1 Simplified solution 219
7.3.2.2 Numerical verification 220
7.3.3 Blast analysis 228
7.3.3.1 Blast loading 228
7.3.3.2 Blast response analysis 229
7.3.3.2.1 Introduction 229
7.3.3.2.2 Elastic single degree freedom (SDOF) system 230
7.3.3.2.3 Elastic Multi-degree freedom (MDOF) system 235
7.3.3.2.4 Numerical verification 238
7.4 Summary 245

CHAPTER 8 CONCLUSIONS AND RECOMMENDATIONS 247

8.1 Conclusions 247
8.2 Recommendations for further work 249

REFERENCES 251

APPENDIX: SIMULATION MOVIES FOR TYPE-C ENHANCED DEPLOYABLE
CUBIC TRUSS SYSTEM
258




vi
SUMMARY
Simplified analysis and deployable development of cable strut systems are conducted in
this thesis. There are two objectives. The first one is to propose efficient simplified
analysis methods for the preliminary design of cable strut roofs under both static and
dynamic loads. The second one is to propose a novel deployable cable strut system which
has better structural behavior and simpler stabilizing procedure than existing systems.

Various types of cable strut systems are investigated and generally classified into two
categories: tension and free standing systems. For the first category, radially arranged
cable truss with parabolic shape is chosen for study; for the second category, a novel
deployable cable strut system is proposed and chosen for study.

Concerning radially arranged cable truss, improved simplified solution is proposed for
calculating static response by considering inner ring effect. It is more accurate than the
existing solution. An empirical formula for predicting natural vibration frequency and
mode sequence is proposed based on membrane analogy method. The predicted results
are much closer to the numerical solutions when compared with classical approach. A

hand calculation formula for estimating the maximum earthquake responses is proposed
based on many important findings. Numerical verification suggests that it can be adopted
in preliminary design.

A novel deployable cable strut system named as cubic truss system is proposed. It has
basic and enhanced forms. The basic system is suitable for small span and load condition,


vii
while the enhanced system is developed for large span and load condition. To verify the
deployment and stabilization of the two systems, a prototype model is built for basic
cubic system and a computer simulation is conducted for enhanced system. Comparison
on structural efficiency is made between the proposed and existing deployable cable strut
systems. It is demonstrated that the proposed system has both easier stabilization
procedure and higher structure efficiency than existing cable strut systems. The optimal
depth/span ratio and module width/span ratio of the proposed system are investigated and
found to agree with the previous published results for other cable strut systems. A rapidly
assembled shelter formed by five deployable cubic panels is proposed.

Simplified analysis methods for truss systems are proposed based on studies on the novel
cubic truss system. For static analysis, plate analogy method is adopted by deriving the
equivalent stiffness expressions for the novel cubic truss system. A novel simplified
analysis method based on 2-D planar truss is proposed for the analysis of orthogonal truss
systems with aim to overcome the boundary limitation of the plate analogy method. Both
methods are verified by finite element method. For dynamic analysis, frequency formulae
to cover all common boundary conditions are established. A hand calculation formula
similar to that for cable truss is proposed for estimating earthquake response. Diagrams
for estimating maximum blast response under different frequencies and weights are
established based on Dynamic load factor (DLF) method. Numerical verification suggests
the proposed simplified methods for calculating frequency and dynamic responses can be

adopted in the preliminary design.




viii
NOMENCLATURE

a
ratio of radius of inner ring to radius of cable truss

1
a
beam spacing at x direction, or modular width

A
area

1, 2,

n
A
AA

areas for different members

b
ratio of axial stiffness of inner ring / axial stiffness of radial cable

1

b
beam spacing at y direction

r
b
ratio of axial stiffness of cable / axial stiffness of strut

c
spacing angle between radial cable

1, 2,
,
n
CCC C

coefficients used to calculate equivalent loads in Chapter 7


[
]
C
damping matrix

d
sag of cable


dx,dy,dz
derivative in x,y,z direction respectively

D
flexural stiffness

e
ratio of strut spacing to half span used in cable truss

e
s

ratio of strut spacing to the radial distance between inner and outer ring

e
r

aspect ratio

E
Yong’s modulus

1, 2,

n
E
EE

Yong’s modulus for different members

f
frequency

1, 2,

n
f
ff

frequency corresponding to 1
st
, 2
nd
,…nth mode

,,,abcd
f
fff

coefficients used to calculate displacement and reaction forces in
Chapter 7
s
f

sag ratio used in cable truss



ix
r
f

generalized force for mode r

F
pretension force of strut in cable truss

1
F
pretension force of strut not connected to inner ring in cable truss

2
F
pretension force of strut connected to inner ring in cable truss

{
}
F
force vector

g
gravity acceleration

h
additional horizontal pretension force in cable

h
′

dimensionless additional horizontal pretension force in cable

1
h

modular or structural height

H

horizontal pretension force in cable

1
,ii
coefficients

I
moment of inertia

j
coefficient

J
Bessel function of the first kind

k

coefficient used in Chapter 2 and 4

12
,kk
coefficients used in Chapter 7

3
k
coefficient used in Chapter 3

a
k
unit axial stiffness of truss member in Chapter 7

b
k
equivalent lateral stiffness of supporting beam in Chapter 7

c
k
equivalent lateral stiffness of supporting column in Chapter 7

H
k
coefficient used for calculating equivalent pretension force in Chapter 3

r
k
generalized mass associated with mode
r


s
k
equivalent lateral stiffness of supporting structure in Chapter 7



x

EA
k
coefficient used for calculating equivalent axial stiffness in Chapter 3

[
]
K
stiffness matrix

l
length

l
e

length of cable

L
span

m
mass per unit area

m
p

Maximum intensity of mass per unit length

r
m

generalized mass associated with mode
r


M
moment

[
]
M

mass matrix

n
the number of modules

s
n
the number of frame spans


u
p

uniformly distributed load per length

p
maximum intensity of a triangular load per unit length

P

concentrated force on node

r
P
generalized force associated with mode
r


R
p

pink blast pressure

q
uniformly distributed load per area

Q
shear force

r
coordinate in radial direction

r
′

dimensionless coordinate in radial direction

R
radius of outer ring in cable truss, radius of circle membrane

1
R

radius of inner ring in cable truss



xi
h
R

reaction force in horizontal direction

v
R

reaction force in vertical direction

w
R

load bearing capacity to weight ratio

c
R

load bearing capacity to cost ratio

S

unit shear stiffness

01, 02 1, 2, 6
, SS SS S
coefficients used to calculate member forces, displacement and reaction
forces in Chapter 7
d
S
displacement response spectra

t
time

,
rd
tt
rising time and time duration for blast loading

T
member force

,,,tbvd
TTTT
force in top strut, bottom strut, vertical strut and diagonal cable

1
T
tension force in radial cable

2

T
tension force in inner ring

{
}
{
}
{
}
,,uuu
 

acceleration vector, velocity vector and displacement vector

r
u

,
r
u

,
r
u

uncoupled acceleration vector, velocity vector and displacement vector
for mode r
u,v,w displacement in x,y,z direction

u



velocity in x direction

g
u



acceleration of the ground motion
s
t
u
displacement under equivalent static loading

1
s
t
u
displacement under equivalent static loading when fundamental
frequency is equal to 1
d
u
displacement under dynamic loading

,,uvw
′′ ′
dimensionless displacement in vertical z direction

V

vertical pretension force in cable



xii
W
1
, W
2
total weight of strut and cable respectively

x,y,z rectangular coordinates

()()()
,,
x
yz
′′′

derivative of x,y,z coordinate

α

coefficient used for
α
method

,
ϕ
β

,
γ

angle

11
,
α
β

Raleigh damping coefficients

ξ

damping ratio

r
ξ

damping ratio for mode r

ir
Φ
r-th mode shape vector at node i

[
]
r
Φ
r-th mode shape vector

λ

coefficient used in cable truss

r
γ

participation factor for mode r

ω

circular frequency

1, 2,

n
ω
ωω

circular frequency corresponding to 1
st
, 2
nd
,…nth mode

r
ω

circular frequency for mode r

ω

damped frequency

ω
′

dimensionless circular frequency

υ

poison’s ratio

ρ

density

θ

coordinate in radial direction

Δ
change of length

σ

stress

xiii
Superscript


0 initial state

1,2a,2b,p state “1”, “2a”, “2b” and “p” respectively

B
1
support condition where no horizontal reaction force is generated

B
2
,
3
support condition where reaction force is generated

e1 elastic support condition where horizontal reaction force is in
compression
e2 elastic support condition where horizontal reaction force is in tension

f 1 fixed support condition where horizontal reaction force is in
compression
f 2 fixed support condition where horizontal reaction force is in tension

k

thin plate solution

M
thick plate solution




Subscript


b
bottom

c
chord

d
diagonal

eq
equivalent

max
maximum

t
top

v

vertical

x
x direction

y
y direction

z
z direction



xiv
LIST OF FIGURES

Figure 1.1: Radial cable truss structure—Lev Zetlin’s cable roof over the auditorium in
the city of Utica, U.S.A. (Berger, 1996)
8

Figure 1.2 The cable dome by David Geiger (Robin, 1996) 8

Figure 1.3 The suspen-dome system. (Kitipornchaia, 2005) 9

Figure 1.4 An exhibition hall of Guangzhou international convention center, China
（Chen，2003） 9

Figure 1.5 A double layer single curvature tensegrity system. (Motro, 1990) 10

Figure 1.6 Novel cable-strut roof formed by modules (Liew et al, 2003) 10

Figure 1.7 Module of RP system (Vu et al, 2006) 11

Figure 1.8 Module of SP system (Wang and Li, 2003) 11

Figure 2.1 Radially arranged cable truss-concave type 29

Figure 2.2 Equivalent cable considering inner ring effect 34

Figure 2.3 Equilibrium of a unit cable segment 37

Figure 2.4 Modified stress-strain relationship for cable 38

Figure 2.5 Comparison of solutions for M3-6 45

Figure 2.6 Comparison of solutions for M3-9 45

Figure 2.7 Load displacement curve for M6-6 49

Figure 2.8 Load displacement curve for M6-9 49

Figure 2.9 Load displacement curve for M8-6 50

Figure 2.10 Load displacement curve for M8-9 50

Figure 2.11 Load displacement curve for M10-6 51

Figure 2.12 Load displacement curve for M10-9 51

Figure 2.13 Three types of cable truss (struts are only shown in half of span) 53

xv
Figure 2.14 Comparison of analytical solution with various types of cable truss 54

Figure 2.15 Effects of sag ratio 55

Figure 2.16 Effects of ratio a 55

Figure 2.17 Effects of ratio b 55

Figure 2.18 Effects of spacing angle c 56

Figure 2.19 Effects of ratio e
s
56

Figure 2.20 Effects of prestress 56

Figure 3.1 Single layer circular membrane 61

Figure 3.2 Radially arranged cable truss 64

Figure 3.3 Curve fitting for coefficient
H
k 67

Figure 3.4 Curve fitting for coefficient
3

k 68

Figure 3.5 The first mode in rotational direction, highest participation in z-rotation 86

Figure 3.6 The second mode in θ direction, highest participation in x-direction 86

Figure 3.7 The first symmetrical mode in z direction, highest participation in z-direction

87

Figure 3.8 The second symmetrical mode in z direction 87

Figure 3.9 The first anti-symmetrical mode in z direction 88

Figure 3.10 The second anti-symmetrical mode in z direction 88

Figure 4.1 Radially arranged cable truss 94

Figure 4.2 Displacement spectra corresponding to vertical component of 1987 New
Zealand Earthquake acceleration history
95

Figure 4.3 Displacement spectra corresponding to vertical component of 1994 Northridge
Earthquake acceleration history
96

Figure 4.4 Displacement spectra corresponding to vertical component of 1940 Imperial
Valley Earthquake acceleration history
96

xvi
Figure 4.5 Relative vertical displacement at different time points for a piece of cable in
M8-9 under Northridge earthquake
100

Figure 4.6 Relative displacement history in radial direction for the node in the middle of
radial cable in M8-9 under Northridge earthquake
101

Figure 4.7 Relative displacement history in angular direction for the node in the middle
of radial cable in M8-9 under Northridge earthquake,
101

Figure 4.8 Relative displacement history in vertical direction for the node in the middle
of radial cable in M8-9 under Northridge earthquake
102

Figure 4.9 Earthquake induced stress for radial cable near the support in M8-9 under
Northridge earthquake
102

Figure 4.10 Earthquake induced stress for inner ring in M8-9 under Northridge
earthquake
103

Figure 4.11 Earthquake induced stress for the strut element near the support in M8-9
under Northridge earthquake
103

Figure 4.12 Comparison of the first mode shape for M2-6 107

Figure 4.13 Comparison of the first

mode shape for M2-9 108

Figure 4.14 Comparison of the first mode shape for M8-9 108

Figure 4.15 Comparison of the first mode shape for M8-9 108

Figure 5.1 A basic module in deployed state 118

Figure 5.2 A basic module in folded state 119

Figure 5.3 A basic module in partially deployed state 120

Figure 5.4 A Type-1 enhanced module in deployed state 121

Figure 5.5 A Type-1 enhanced module in partially deployed state 122

Figure 5.6 A Type-2 enhanced module with 2 stabilizing cables S1 and S2 in deployed
state (the four diagonal cables D1-D4 are not shown here)
123

Figure 5.7 A Type-2 enhanced module with 2 stabilizing cables S1 and S2 in partially
deployed state (the four diagonal cables D1-D4 are not shown here).
124

Figure 5.8 Perspective view of a panel comprising 3x3 modules 125

xvii

Figure 5.9 Plan view of a slab system comprising 3x3 modules 126

Figure 5.10 Elevation view of a slab system comprising 3x3 modules 126

Figure 5.11 Close-up view of a top joint 127

Figure 5.12 Close-up view of a bottom joint 128

Figure 5.13 A prototype modal formed by 2x2 modules in a folded state 129

Figure 5.14 A prototype modal formed by 2x2 modules in a partially deployed state 129

Figure 5.15 A prototype modal formed by 2x2 modules in a deployed state 130

Figure 5.16 Load displacement curve for different systems 133

Figure 5.17 A typical numerical model for cubic truss system 136

Figure 5.18 Structure efficiency under different depth /span ratio 138

Figure 5.19 Structure efficiency under different grid width /span ratio 139

Figure 6.1 Area with high shear force for panel supported along the 4 perimeters 141

Figure 6.2 Cable-enhanced shear module in deployed state 142

Figure 6.3 Cable-enhanced shear module in partially deployed state 143

Figure 6.4 Strut-enhanced shear module in deployed state 145

Figure 6.5 Strut-enhanced shear module in partially deployed state 146

Figure 6.6 Joint detail 1 for strut-enhanced shear module in deployed state 147

Figure 6.7 Joint detail 2 for strut-enhanced shear module in partially deployed state 148

Figure 6.8 Joint detail 3 for strut-enhanced shear module in partially deployed state 149

Figure 6.9 A cubic truss system formed by 10x10 modules and enhanced by strut
enhanced shear modules-perspective view
150

Figure 6.10 A cubic truss system formed by 10x10 modules and enhanced by strut
enhanced shear modules -plan view
151



xviii
Figure 6.11 A cubic truss system formed by 10x10 modules and enhanced by strut
enhanced shear modules -elevation view
151

Figure 6.12 A cubic truss system formed by 10x10 modules and enhanced by strut
enhanced shear modules - Close-up view
152

Figure 6.13 A set of sliding struts in Type-C enhanced cubic truss system 156

Figure 6.14 Bar A in Figure 6.13 157

Figure 6.15 Close-up view of sliding bolt a in Bar A 158

Figure 6.16 Close-up view of sliding bolt b in Bar A 158

Figure 6.17 Bar B in Figure 6.13 159

Figure 6.18 Bar C in Figure 6.13 160

Figure 6.19 Hinged struts in Type-C enhanced cubic truss system 161

Figure 6.20 Perspective view of a basic module in Type-C enhanced cubic truss system in
deployed state
162

Figure 6.21 Elevation view of a basic module in Type-C enhanced cubic truss system in
deployed state
163

Figure 6.22 Plan view of a basic module in Type-C enhanced cubic truss system in
deployed state
164

Figure 6.23 One pair of sliding struts in deployed state corresponding to Figure 6.20. 165

Figure 6.24 Close-up view of the intersection of a pair of sliding struts in Figures 6.23166

Figure 6.25 Close-up view of the connection between Bar B at the intersection shown in
Figure 6.24
167

Figure 6.26 Close-up view of the intersection between two pairs of the sliding struts in
Figure 6.20
168

Figure 6.27 Close-up view of the top joint in deployed state 169

Figure 6.28 Close-up view of the bottom joint in deployed state 170

Figure 6.29 Perspective view of a basic module in Type-C enhanced cubic truss system in
partially deployed state
171



xix
Figure 6.30 Elevation view of a basic module in Type-C enhanced cubic truss system in
partially deployed state
172

Figure 6.31 Plan view of a basic module in Type-C enhanced cubic truss system in
partially deployed state
173

Figure 6.32 One pair of sliding struts in partially deployed state corresponding to Figure
6.29

174

Figure 6.33 Perspective view of a basic module in Type-C enhanced cubic truss system
in compact state 1
175

Figure 6.34 Elevation view of a basic module in Type-C enhanced cubic truss system in
compact state 1
176

Figure 6.35 Plan view of a basic module in Type-C enhanced cubic truss system in
compact state 1
177

Figure 6.36 One pair of sliding strut in compact state 1 corresponding to Figure 6.33 . 178

Figure 6.37 Perspective view of a basic module in Type-C enhanced cubic truss system in
compact state 2
179

Figure 6.38 Elevation view of a basic module in Type-C enhanced cubic truss system in
compact state 2
180

Figure 6.39 Plan view of a basic module in Type-C enhanced cubic truss system in
compact state 2
181

Figure 6.40 One pair of sliding struts in compact state 2 corresponding to Figure 6.37 182

Figure 6.41 A Type-1 enhanced module in Type-C enhanced cubic truss system in
deployed state
183

Figure 6.42 A Type-1 enhanced module in Type-C enhanced cubic truss system in
partially deployed state
184

Figure 6.43 Shelter formed by 5 panels 185

Figure 6.44 Perspective view of the assembled shelter 185

Figure 6.45 Top view of the assembled house 186

Figure 6.46 Front view of the assembled house 186

Figure 6.47 Connection detail between adjacent roof and walls (viewed from outside) 187


xx

Figure 6.48 Connection detail between adjacent roof and walls (viewed from inside) 187

Figure 6.49 Connection detail between two adjacent walls 188

Figure 7.1 Shear deformation of a novel cubic module 194

Figure 7.2 Simplified orthogonal truss system 199

Figure 7.3 Simplified 2-D planar truss model under applied load and support condition 1

(named as state "P")
199

Figure 7.4 Simplified 2-D planar truss model under center unit virtual force 1 and support
condition 1 (named as state "1")
202

Figure 7.5 Simplified 2-D planar truss model under real load and support condition 2. 202

Figure 7.6 Simplified 2-D planar truss model under unit horizontal compression force 1
and support condition 1 (named as state “2a”)
203

Figure 7.7 Simplified 2-D planar truss model under unit horizontal tensile force 1 and
support condition 1 (named as state “2b”)
203

Figure 7.8 Simplified 2-D model under applied load and support condition 3 (named as
state "e1")
208

Figure 7.9 The first symmetrical mode in vertical direction 218

Figure 7.10 Displacement history for central top node 226

Figure 7.11 Stress history for central top strut 226

Figure 7.12 Stress history for central bottom strut 227

Figure 7.13 Stress history for boundary vertical strut 227

Figure 7.14 Stress history for boundary cable 228

Figure 7.15 Idealized blast loading Type-1 232

Figure 7.16 Idealized blast loading Type-2 233

Figure 7.17 DLF for blast loading Type-1 233

Figure 7.18 DLF for load Type-2 234



xxi
Figure 7.19 DLF for load Type-2 ( /
rd
ctt
=
) 234

Figure 7.20 The normalized maximum dynamic response under the ratio of /
dn
tt 237

Figure 7.21 The normalized maximum dynamic response under the ratio of
1
/
n
mm 238

Figure 7.22 Displacement history for central top node 242

Figure 7.23 Reaction force history at the middle support joint 243

Figure 7.24 Stress history for central top strut 243

Figure 7.25 Stress history for central bottom strut 244

Figure 7.26 Stress history for boundary vertical strut 244

Figure 7.27 Stress history for boundary cable 245

xxii
LIST OF TABLES

Table 2.1 Numerical model under different parameters 43

Table 2.2 Comparison of Irvine’s and improved solutions 44

Table 2.3 Comparison of strut force between derived and numerical solution for M6-6
(under the loading of 0.5kN/m
2
) 46

Table 2.4 Comparison of strut force between derived and numerical solution for M8-9
(under the loading of 0.5kN/m
2
) 46

Table 2.5 Comparison of strut force between derived and numerical solution for M10-6
(under the loading of 0.5kN/m

2
) 46

Table 2.6 Comparison of strut force between derived and numerical solution for M6-6
(under the loading of 10kN/m
2
) 47

Table 2.7 Comparison of strut force between derived and numerical solution for M8-9
(under the loading of 5kN/m
2
) 47

Table 2.8 Comparison of strut force between derived and numerical solution for M10-6
(under the loading of 5kN/m
2
) 47

Table 3.1 Model parameters 70

Table 3.2 Comparison of the frequency of 1
st
symmetrical mode in model M0 between
Irvine’s, proposed and numerical solution
74

Table 3.3 Comparison of the frequency of 1
st
anti-symmetrical mode in model M0
between Irvine’s, proposed and numerical solution

74

Table 3.4 Comparison of the mode sequence in model M0 between Irvine’s, proposed
and numerical solution
74

Table 3.5 Comparison of the frequency of 1
st
symmetrical mode in model M2 between
Irvine’s, proposed and numerical solution
75

Table 3.6 Comparison of the frequency of 1
st
anti-symmetrical mode in model M2
between Irvine’s, proposed and numerical solution
75

Table 3.7 Comparison of the mode sequence in model M2 between Irvine’s, proposed
and numerical solution
75

Table 3.8 Comparison of the frequency of 1
st
symmetrical mode in model M2a between
Irvine’s, proposed and numerical solution
76


xxiii

Table 3.9 Comparison of the frequency of 1
st
anti-symmetrical mode in model M2a
between Irvine’s, proposed and numerical solution
76

Table 3.10 Comparison of the mode sequence in model M2a between Irvine’s, proposed
and numerical solution
76

Table 3.11 Comparison of the frequency of 1
st
symmetrical mode in model M2b between
Irvine’s, proposed and numerical solution
77

Table 3.12 Comparison of the frequency of 1
st
anti-symmetrical mode in model M2b
between Irvine’s, proposed and numerical solution
77

Table 3.13 Comparison of the mode sequence in model M2b between Irvine’s, proposed
and numerical solution
77

Table 3.14 Comparison of the frequency of 1
st
symmetrical mode in model M3 between

Irvine’s, proposed and numerical solution
78

Table 3.15 Comparison of the frequency of 1
st
anti-symmetrical mode in model M3
between Irvine’s, proposed and numerical solution
78

Table 3.16 Comparison of the mode sequence in model M3 between Irvine’s, proposed
and numerical solution
78

Table 3.17 Comparison of the frequency of 1
st
symmetrical mode in model M4 between
Irvine’s, proposed and numerical solution
79

Table 3.18 Comparison of the frequency of 1
st
anti-symmetrical mode in model M4
between Irvine’s, proposed and numerical solution
79

Table 3.19 Comparison of the mode sequence in model M4 between Irvine’s, proposed
and numerical solution
79

Table 3.20 Comparison of the frequency of 1

st
symmetrical mode in model M5 between
Irvine’s, proposed and numerical solution
80

Table 3.21 Comparison of the frequency of 1
st
anti-symmetrical mode in model M5
between Irvine’s, proposed and numerical solution
80

Table 3.22 Comparison of the mode sequence in model M5 between Irvine’s, proposed
and numerical solution
80

Table 3.23 Comparison of the frequency of 1
st
symmetrical mode in model M7 between
Irvine’s, proposed and numerical solution
81



xxiv
Table 3.24 Comparison of the frequency of 1
st
anti-symmetrical mode in model M7
between Irvine’s, proposed and numerical solution
81

Table 3.25 Comparison of the mode sequence in model M7 between Irvine’s, proposed
and numerical solution
81

Table 3.26 Comparison of the frequency of 1
st
symmetrical mode in model M8 between
Irvine’s, proposed and numerical solution
82

Table 3.27 Comparison of the frequency of 1
st
anti-symmetrical mode in model M8
between Irvine’s, proposed and numerical solution
82

Table 3.28 Comparison of the mode sequence in model M8 between Irvine’s, proposed
and numerical solution
82

Table 3.29 Comparison of the frequency of 1
st
symmetrical mode in model M8a between
Irvine’s, proposed and numerical solution
83

Table 3.30 Comparison of the frequency of 1
st
anti-symmetrical mode in model M8a
between Irvine’s, proposed and numerical solution

83

Table 3.31 Comparison of the mode sequence in model M8a between Irvine’s, proposed
and numerical solution
83

Table 3.32 Comparison of the frequency of 1
st
symmetrical mode in model M9 between
Irvine’s, proposed and numerical solution
84

Table 3.33 Comparison of the frequency of 1
st
anti-symmetrical mode in model M9
between Irvine’s, proposed and numerical solution
84

Table 3.34 Comparison of the mode sequence in model M9 between Irvine’s, proposed
and numerical solution 84

Table 3.35 Comparison of the frequency of 1
st
symmetrical mode in model M10a
between Irvine’s, proposed and numerical solution 85

Table 3.36 Comparison of the frequency of 1
st
anti-symmetrical mode in model M10a
between Irvine’s, proposed and numerical solution 85

Table 3.37 Comparison of the mode sequence in model M10a between Irvine’s, proposed
and numerical solution 85

Table 4.1 Earthquake records used for structural behavior study 94

Table 4.2 Comparison of maximum earthquake induced displacement for M2-6 99