M. VENKATA RAMA RAO - .: Mathematical Sciences : … thank my friends S.R.Ranganath, P.Surya Prakash and A. Radha Krishna and my colleagues Sri. M.Bhasker, Sri. B.Veeranna and Sri

ANALYSIS OF

CABLE-STAYED BRIDGES

BY

FUZZY - FINITE ELEMENT MODELLING

M. VENKATA RAMA RAO

DEPARTMENT OF CIVIL ENGINEERING,

UNIVERSITY COLLEGE OF ENGINEERING (AUTONOMOUS),

OSMANIA UNIVERSITY,

HYDERABAD - 500 007. A.P.

MARCH, 2004

To

My Beloved Late Father

ANALYSIS OF

CABLE-STAYED BRIDGES

BY

FUZZY - FINITE ELEMENT MODELLING

By

M.VENKATA RAMA RAO

A Thesis Submitted in Fulfilment of the requirements for the award of the Degree of

DOCTOR OF PHILOSOPHY

DEPARTMENT OF CIVIL ENGINEERING, UNIVERSITY COLLEGE OF ENGINEERING (AUTONOMOUS),

OSMANIA UNIVERSITY, HYDERABAD- 500 007, A.P.

MARCH, 2004

CERTIFICATE

This is to certify that the Thesis entitled “ANALYSIS OF CABLE-STAYED

BRIDGES BY FUZZY - FINITE ELEMENT MODELLING” being submitted by

Mr. M.Venkata Rama Rao for the award of the Degree of Doctor of Philosophy in

Civil Engineering is a record of bonafide work carried out by him under my guidance

and supervision.

This thesis or part there of, has not been submitted to any other University or

Institute for the award of any Degree or Diploma.

(Prof. R. Ramesh Reddy) Professor and Thesis Supervisor Department of Civil Engineering University College of Engineering

(Autonomous)Osmania University Hyderabad – 500 007 A.P.

(Prof. L.V.A.Sesha Sai) Head, Department of Civil Engineering University College of Engineering

(Autonomous)Osmania University

DECLARATION

I do hereby declare that the research work reported in this thesis has been

carried out by me at the University College of Engineering (Autonomous), Osmania

University, Hyderabad for Ph.D. degree in Civil Engineering and that this thesis,

neither in full nor in part, has been submitted for the award of any Degree or Diploma

to any other institution.

(M.V. Rama Rao)

Date: 16-03-2004

Place: Hyderabad.

ACKNOWLEDGEMNTS

I am deeply indebted to my thesis supervisor Prof. R. Ramesh Reddy, Principal,

University College of Engineering, Osmania University for his active encouragement,

indulgence, academic and emotional support. I am grateful to him for his guidance

and suggestions from time to time throughout the progress of the present work.

I thank Prof. L.V.A. Sesha Sayi, Head, Department of Civil Engineering, University

College of Engineering, Osmania University for allowing me to make use of the

facilities in the college. I am grateful to Prof.D.S.Prakash Rao, Professor, Department

of Civil Engineering, University College of Engineering, Osmania University for his

constant encouragement and support. I express my sincere thanks to

Prof. B.Srinivas Reddy, Chairman, Board of Studies (Civil Engineering), University

College of Engineering, Osmania University for his help and support.

I am thankful to Prof. S.K.Jain, Department of Civil Engineering, Indian Institute of

Technology, Kanpur for his valuable advice and help throughout the course of my

work and particularly for the help and support extended to me in my literature survey.

I am grateful to the contributions and e-mail correspondence by the primary

researchers in the area of Fuzzy-finite element analysis of structures: Prof. Rafi L.

Muhanna, Department of Civil Engineering, Georgia Institute of Technology,

Savannah, USA and Dr. Andrzej Pownuk, Chair of Theoretical Mechanics, Faculty of

Civil Engineering, Silesian University of Technology, Poland. I also thank

Prof. Christian Jansson, University of Hamburg, Germany for useful discussion by

e-mail on the methods of solving linear interval equations (Jansson’s Algorithm).

iv

I thank the Management and the Principal of Vasavi College of Engineering,

Hyderabad for the help and support extended to me through the staff development

programme. I am also thankful to Prof. P.Jagannadha Rao, Head, Department of Civil

Engineering, Vasavi College of Engineering for his help and support.

I would like to express gratitude to Prof. E.S.Rao profusely for his valuable advice

and support throughout the course of my work. I also thank him for his careful

review of the manuscript and his valuable suggestions.

I thank my friends S.R.Ranganath, P.Surya Prakash and A. Radha Krishna and my

colleagues Sri. M.Bhasker, Sri. B.Veeranna and Sri. K.V.Ramana Reddy for their

encouragement and support. I thank Mr. M.V.Suresh for his help in typing the

manuscript.

I am thankful to my mother Mrs. Vijaya Lakshmi, brother Dr. M.V.Krishna Rao and

my wife Prameela Rani for their help and support. I particularly appreciate the

support extended by my little daughter Deepika by way of cheering me up in the

midst of all my travails.

(M.V. Rama Rao)

v

ABSTRACT

Design of a structural system requires the performance of the system to be

guaranteed over its lifetime. It is essential that structures need to be modelled

accurately in order to predict their behaviour during their design life period.

Normally, various parameters of a structure that influence its behaviour, are identified

and are incorporated in a mathematical model of the structure. Classical finite element

analysis is performed using the mathematical model under the influence of the given

loads and forces are evaluated. However, uncertainties associated with the material

and geometric properties of a real life structure as well as the range of service loads

acting on it could not be accounted. Classical finite element analysis, despite its

advantages, is not well suited to handle such uncertainties in the structural parameters

and response quantities i.e. material and geometric properties, loads, displacements,

moments and forces. Thus there is a need to incorporate the effect of these

uncertainties in the mathematical model and analysis of structures in order to predict

structural behaviour in a reliable manner.

In the present study, multiple uncertainties refer to the concomitant presence

and independent variation of uncertainties of material property (E), live load and mass

density. These uncertainties are considered as fuzzy interval values.

Fuzzy finite element analysis has not been hitherto applied to the study of

complex structures in order to understand its structural response. There exists no

literature, which incorporates multiple uncertainties in the analysis and design of

complex structures including cable-stayed bridges. In the light of the above, it is felt

that there is a need to apply fuzzy finite element methodology to evaluate the effect of

multiple uncertainties on the structural response in order to have a realistic evaluation

vi

of its behaviour. Also, there is a need to study the effect of these uncertainties on the

membership functions of the structural response quantities.

In this thesis, a modified methodology incorporating the cumulative effect of

multiple uncertainties, is proposed, validated and discussed. The element-by-element

model as suggested by Muhanna handling single uncertainty is modified to handle

multiple uncertainties. A new procedure for post-processing in the presence of

multiple uncertainties is suggested in the thesis. This procedure is implemented to

post-process the solution in order to obtain axial forces, bending moments and shear

forces in the presence of multiple uncertainties. Response of different structures with

mass and material uncertainties subjected to fuzzy interval loading is evaluated. The

proposed methodology is employed on several simple structural problems as

preliminary case studies and the results are validated with the available literature. The

structures response of beams, trusses and frames under fuzzy interval loads is

evaluated and the structures are observed to behave uniquely (Appendix C).

As a major case study, a cable-stayed bridge is analysed using validated fuzzy

finite element analysis in order to evaluate the effect of multiple uncertainties on the

structural response. The nature of the fuzzy membership functions of the structural

response quantities in the presence of multiple uncertainties is investigated. The

membership functions of the structural response quantities for various combinations

of uncertainties are plotted. Trapezoidal membership functions are evident in the

presence of multiple uncertainties unlike triangular membership functions for single

uncertainty demonstrated by Muhanna’s method.

Sensitivity analysis is performed to evaluate the relative uncertainties of

structural response quantities such as displacements, rotations, forces and moments at

a given node as well as over a set of given nodes. The sensitivity (relative variation

vii

of uncertainty) of structural response quantities to unit change in variation of load and

material uncertainties is evaluated.

The present work demonstrates the effectiveness of fuzzy finite element model

in evaluating the structural response in the presence of multiple uncertainties over the

existing methods. The usefulness of a detailed sensitivity analysis in order to

understand the relative sensitivity of structural response is also highlighted.

viii

CONTENTS

Certificate iiDeclaration iiiAcknowledgements ivAbstract viContents ixNotation xiiiList of Tables xviList of Figures xixList of Tables in Appendices xxivList of Figures in Appendices xxvi Chapter 1 Introduction and Literature Review

1.1 Introduction 2

1.2 Literature Review 4

1.2.1 Uncertainty in Engineering Analysis and Design 4

1.2.2 Handling Uncertainty 6

1.2.2.1 Anti-optimisation 6

1.2.2.2 Probability - stochastic approach for uncertainty modelling 7

1.2.2.3 Modelling qualitative uncertainty using fuzzy logic 8

1.2.3 Interval algebra in structural analysis 10

1.2.4 Fuzzy finite element model of a structure 11

1.2.5 State of the art 12

1.3 Cable-Stayed Bridges 18

1.3.1 General 18

1.3.2 Structural characteristics 20

1.3.3 Loads 24

1.3.4 Idealisation of the structure 26

1.3.5 Methods of analysis 27

1.4 A Brief Review of The Past Work 28

1.5 Objectives of Present Study 31

1.6 Summary 31

Chapter 2 Methodology

2.1 Introduction 34

ix

2.2 Assembled Finite Element Model 34

2.2.1 Uncertainty of live load 35

2.2.2 Post-processing of solution 37

2.2.3 Uncertainty of mass density 38

2.3 Material Uncertainty 38

2.4 Sources of Overestimation 39

2.4.1 Evaluation of overall displacement vector 42

2.5 Concomitant Presence of Load and Material Uncertainties 43

2.5.1 Cumulative effect of material and load uncertainties (α and β) 45

2.5.2 Evaluation of {λ}cc from assembled FEA model 46

2.5.3 Solution of linear interval matrix equations 47

2.6 Post-processing of Solution 48

2.7 Concomitant Presence of Mass Density and Material Uncertainties 49

2.8 Concomitant Presence of Mass density and Material and Live Load

Uncertainties

50

2.9 Sensitivity Analysis 50

2.10 Summary 51

Chapter 3 Case studies

3.1 Introduction 54

3.2 Case Study 1 – Cable-Stayed Bridge with Uncertainties of Material

Property and Mass Density

57


Property and Live Load

59

3.4 Case Study 3–Cable-Stayed Bridge with Uncertainties of Material

Property, Live Load and Mass Density

61

3.5 Cumulative Effect of Multiple Uncertainties α, β and γ on Displacements

and Forces

62


3.7 Summary 64

Chapter 4 Results and discussion

4. Results and discussion 71

4.1 Case study 1 – Cable-Stayed Bridge with Uncertainties of Material


71

x

4.1.1 Effect of concomitant variation of α and γ on displacements and rotations 72

4.1.1.1 Uncertain horizontal displacement at node 2 73

4.1.1.2 Uncertain vertical displacement at node 3 73

4.1.1.3 Uncertain rotation at node 4 73

4.1.2 Effect of concomitant variation of α and γ on shear forces and bending

moments

74

4.1.2.1 Axial force in deck in element 9 75

4.1.2.2 Axial force (kN) in deck in element 12 75

4.1.2.3 Shear force (kN) in deck just to the left of node 3 76

4.1.2.4 Bending moment (kNm) in deck at node 4 76

4.1.2.5 Axial force (kN) in pylon in element 17 (at node 11) 77

4.1.2.6 Axial force (kN) in cable 2 (at node 2) 77


4.2 Case study 2 – Cable-Stayed Bridge with uncertainties of material

property and live load.

93

4.2.1 Effect of concomitant variation of α and β on displacements and rotations 93




4.2.2 Effect of concomitant variation of α and β on shear forces and bending

moments

95









Property, Live Load and Mass Density

113

4.3.1 Effect of concomitant variation of α and β on displacements and rotations 113



xi



moments

116









4.4.1 Sensitivity analysis of displacements and rotations at a given node 151

4.4.2 Sensitivity analysis of forces and moments at a given node 152

4.4.3 Sensitivity analysis of a given response quantity at different nodes 152

4.5 Discussion 159

Chapter 5 Conclusions and Recommendations for Future Work

5.1 Conclusions 161

5.2 Recommendations for Future Work 163

References 164

Appendices Appendix- ‘A’ Fuzzy Sets and Membership Functions 169

Appendix- ‘B’ Fuzzy Finite Element Model- Muhanna’s Approach 174

Appendix- ‘C’ Preliminary Case Studies 176

Bio-Data 197

Publications 198

xii

NOTATION

α : Material uncertainty (of Young’s modulus)

αmax : Maximum material uncertainty (of Young’s modulus)

β : Live load uncertainty

β max : Maximum live load uncertainty

βl, βu : Lower and upper bounds of β-cut on membership function for live load

γ : Mass density uncertainty

γ max : Maximum mass density uncertainty

{λ} : Vector of Lagrange multipliers

{λ} : Fuzzy interval vector of internal forces

{λc},{λ}cc : Crisp mid-point vector representing the mid value of {λ}

{λαβ} : Fuzzy interval vector of internal forces under the influence of multiple

uncertainties α and β

{λcβ},{λβ} : Fuzzy interval vector of internal forces under the influence of live load

uncertainty β alone

{λ} α c : Fuzzy interval vector of internal forces under the influence of material

uncertainty β alone

{λαβ}(e) : Fuzzy interval vector of internal forces under the influence of multiple

uncertainties α and β at element level

{λ}(e) : Overall internal force vector for the element

∏ : Potential energy of a structural system for an assembled model

∏* : Fuzzy potential energy of a structural system for an element-by element

model

AHb : Hull of the solution vector {U}

xiii

A1, A2 : Axial Forces at the ends of a plane frame element

[C] : A crisp constraint matrix

[C] : A crisp constraint matrix used in element by element model

[D]α : Fuzzy interval diagonal matrix of size n×n

E : Young’s modulus

Ec : Young’s modulus of concrete

Es : Young’s modulus of steel

{g}(e) : Fuzzy interval vector of element-end forces

[K̃] : Crisp (non-interval) stiffness matrix

[K̃] (e) : Crisp (non-interval) stiffness matrix at the element level

[K ] : Point interval stiffness matrix.

[Kα] : Fuzzy interval stiffness matrix at a level of uncertainty α

[L] : A Boolean connectivity matrix of size md×n containing 0s and 1s

m : Number of elements in the structure

[M ̃] : A non-interval matrix of size n × m, with each column containing

contributions of loads acting on a given element

M1, M2 : Bending Moments at the ends of a plane frame element

md : Number of degrees of freedom of an element

n : Kinematic indeterminacy for the structure

{P} : Crisp force vector

{P´} : Point load vector

{Pβ} : Fuzzy interval force vector corresponding to level of uncertainty β

{Pβ}(e) : Fuzzy interval force vector corresponding to level of uncertainty β at

element level

{Pγ} : Fuzzy interval force vector corresponding to level of uncertainty γ

xiv

{P} : Point interval load vector

{P}(e) : Mid-point element load vector.

{q} : Fuzzy interval element load vector

[R̃] : A deterministic singular matrix of size n×n.

[S] : A deterministic singular matrix of size n×n.

[T](e) : Transformation matrix for the element

{U} : Crisp displacement vector

{Uαβ} : Fuzzy interval displacement vector corresponding to levels of uncertainty

α and β

{Uαγ} : Fuzzy interval displacement vector corresponding to levels of uncertainty

α and γ

{Uβ} : Fuzzy interval displacement vector corresponding to level of uncertainty β

{U} : Overall fuzzy interval displacement vector obtained by superimposing

{Uαβ} for individual loads

{δU} : Error involved in displacement vector {Uαβ}

V1, V2 : Shear Forces at the ends of a plane frame element

{x} : Lower bound vector to solution hull AHb

{y} : Upper bound vector to solution hull AHb

xv

LIST OF TABLES

Table. 3.1 Properties of Cable Stayed Bridge 58

Table. 3.2 Cable stayed Bridge - Concomitant Variation of Rotation at node 4

(×10-3 radians) w.r.t α and γ

69

Table. 3.3 Cable stayed Bridge - Concomitant Variation of Axial Force in deck

in element 12 (kN) w.r.t α and β

69

Table. 4.1.1 Cable stayed Bridge - Concomitant Variation of Horizontal

displacement of node 2 (×10-4 metres) w.r.t α and γ

79

Table. 4.1.2 Cable stayed Bridge - Concomitant Variation of Vertical

displacement of node 3 (×10-2 metres) w.r.t α and γ

79

Table. 4.1.3 Cable stayed Bridge - Concomitant Variation of Rotation at node 4

(×10-3 radians) w.r.t α and γ

79

Table. 4.1.4 Cable stayed Bridge - Concomitant Variation of Axial Force (kN) in

deck in element 9 w.r.t α and γ

80


deck in element 12 w.r.t α and γ

80

Table. 4.1.6 Cable stayed Bridge - Concomitant Variation of Shear Force (kN) in

deck just to the left of node 3 w.r.t α and γ

80

Table. 4.1.7 Cable stayed Bridge - Concomitant Variation of Bending Moment

(kNm) in deck at node 4 w.r.t α and γ

81


pylon in element 17 w.r.t α and γ

81


cable 2 w.r.t α and γ

82


cable 3 w.r.t α and γ

82


displacement of node 2 (×10-4 metres) w.r.t α and β

99


displacement of node 3 (×10-2 metres) w.r.t α and β

99

xvi


(×10-3 radians) w.r.t α and β

99


deck in element 9 w.r.t α and β

100


deck in element 12 w.r.t α and β

100


deck just to the left of node 3 w.r.t α and β

100


(kNm) in deck at node 4 w.r.t α and β

101


pylon in element 17 w.r.t α and β

101


cable 2 w.r.t α and β

102


cable 3 w.r.t α and β

102


displacement of node 2 (×10-4 metres) w.r.t α and β at γ =1.0

124



124



124



125


(×10-3 radians) w.r.t α and β at γ =1.0

125


(×10-3 radians) w.r.t α and β at γ =0.8

125


deck in element 9 w.r.t α and β at γ =1.0

126



126

xvii



126



127


deck just to the left of node 3 w.r.t α and β at γ =1.0

127


deck just to the left of node 3 w.r.t α and β at γ =0.8

127


(kNm) in deck at node 4 w.r.t α and β at γ =1.0

128


(kNm) in deck at node 4 w.r.t α and β at γ =0.8

128


pylon in element 17 w.r.t α and β at γ =1.0

128


pylon in element 17 w.r.t α and β at γ =0.8

129


cable 2 w.r.t α and β at γ =1.0

129



129



130



130

xviii

LIST OF FIGURES

Fig. 1.1 Longitudinal Layout of Stays 21

Fig. 1.2 Positions of the Cables in Space 22

Fig. 1.3 Types of Towers 23

Fig. 1.4 Types of Cables 24

Fig. 2.1 Assembled and Unassembled Model of a Structure 39

Fig. 2.2 Avoiding Overestimation by Uncoupling of Loads 42

Fig. 2.3 Bounds on Hull Solution 48

Fig. 3.1 Membership function for Material Uncertainty 65

Fig. 3.2 Membership Function for Live Load Uncertainty 65

Fig. 3.3 Membership function for Mass Uncertainty 66

Fig. 3.4 Cable Stayed Bridge - Fan Configuration 66

Fig. 3.5 Cable Stayed Bridge - Membership function for vertical displacement

at node 3 at β=0.8

67

Fig. 3.6 Cable Stayed Bridge - Membership function for Axial Force in cable 3

at α=0.6

67

Fig. 3.7 Sensitivity Analysis for displacements (for dead load) at node 3 at

γ=1.0

68

Fig. 3.8 Sensitivity Analysis of Forces and Moments in Element 12 (at node 5)

at γ=1.0

68

Fig. 4.1.1 Membership Functions of horizontal displacement at node 2 at

(a) γ=1.0, (b) γ=0.8, (c) α=1.0, (d) α=0.6

83

Fig. 4.1.2 Membership Functions of vertical displacement at node 2 at (a) γ=1.0,

(b) γ=0.8, (c) α=1.0, (d) α=0.6

84

Fig. 4.1.3 Membership Functions of rotation at node 4 at (a) γ=1.0, (b) γ=0.8,

(c) α=1.0, (d) α=0.6

85

Fig. 4.1.4 Membership Functions of Axial Force in deck in element 9 at

(a) γ=1.0, (b) γ=0.8, (c) α=1.0, (d) α=0.6

86

Fig. 4.1.5 Membership Functions of Axial Force in deck in element 12 at

(a) γ=1.0, (b) γ=0.8, (c) α=1.0, (d) α=0.6

87

xix

Fig. 4.1.6

Membership Functions of Shear Force in deck to left of node 3 at (a)

γ=1.0, (b) γ=0.8, (c) α=1.0, (d) α=0.8

88

Fig. 4.1.7 Membership Functions of Bending Moment in deck at node 4 at (a)

γ=1.0, (b) γ=0.8, (c) α=1.0, (d) α=0.6

89

Fig. 4.1.8 Membership Functions of Axial Force in pylon in element 17 at (a)

γ=1.0, (b) γ=0.8, (c) α=1.0, (d) α=0.8

90

Fig. 4.1.9 Membership Functions of Axial Force in cable 2 at (a) γ=1.0, (b)

γ=0.8, (c) α=1.0, (d) α=0.6

91

Fig. 4.1.10 Membership Functions of Axial Force in cable 2 at (a) γ=1.0, at (a)

γ=1.0, (b) γ=0.8, (c) α=1.0, (d) α=0.6

92

Fig. 4.2.1 Membership Functions for horizontal displacement at node 2 at (a)

β=1.0, (b) β=0.8, (c) α=1.0, (d) α=0.8

103

Fig. 4.2.2 Membership Functions for vertical displacement at node 3 at (a) β=1.0,

(b) β=0.8, (c) α=1.0, (d) α=0.8

104

Fig. 4.2.3 Membership Functions for rotation at node 4 at (a) β=1.0, (b) β=0.8,

(c) α=1.0, (d) α=0.8

105

Fig. 4.2.4 Membership Functions for Axial Force in deck in element 9 at

(a) β=1.0, (b) β=0.8, (c) α=1.0, (d) α=0.8

106


(a) β=1.0, (b) β=0.8, (c) α=1.0, (d) α=0.8

107

Fig. 4.2.6 Membership Functions for Shear Force in deck to left of node 3 at

(a) β=1.0, (b) β=0.8, (c) α=1.0, (d) α=0.8

108

Fig. 4.2.7 Membership Functions for Bending Moment in deck at node 4 at

(a) β=1.0, (b) β=0.8, (c) α=1.0, (d) α=0.6

109

Fig. 4.2.8 Membership Functions for Axial Force in pylon in element 17 at

(a) β=1.0, (b) β=0.8, (c) α=1.0, (d) α=0.8

110

Fig. 4.2.9 Membership Functions for Axial Force in cable 2 at (a) β=1.0,

(b) β=0.8, (c) α=1.0, (d) α=0.6

111

Fig. 4.2.10 Membership Functions for Axial Force in cable 3 at (a) β=1.0,

(b) β=0.8, (c) α=1.0, (d) α=0.6

112

xx

Fig .4.3.1 Membership Functions for horizontal displacement at node 2 at

(a) β=1.0, γ =1.0, (b) β=0.8, γ =1.0, (c) α=1.0, γ =1.0 (d) α=0.6, γ =1.0

131

Fig. 4.3.2 Membership Functions for horizontal displacement at node 2 at

(a) β=1.0, γ =0.8, (b) β=0.8, γ =0.8, (c) α=1.0, γ =0.8 (d) α=0.6, γ =0.8

132


γ =1.0, (b) β=0.8, γ =1.0, (c) α=1.0, γ =1.0, (d) α=0.6, γ =1.0

133


γ =0.8, (b) β=0.8, γ =0.8, (c) α=1.0, γ =0.8, (d) α=0.6, γ =0.8

134

Fig. 4.3.5 Membership Functions for rotation at node 4 at (a) β=1.0, γ =1.0,

(b) β=0.8, γ =1.0, (c) α=1.0, γ =1.0, (d) α=0.6, γ =1.0

135

Fig. 4.3.6 Membership Functions for rotation at node 4 at (a) β=1.0, γ =0.8,

(b) β=0.8, γ =0.8, (c) α=1.0, γ =0.8 (d) α=0.6, γ =0.8

136


(a) β=1.0, γ =1.0, (b) β=0.8, γ =1.0,(c) α=1.0, γ =1.0, (d) α=0.6, γ =1.0

137


(a) β=1.0, γ =0.8, (b) β=0.8, γ =0.8, (c) α=1.0, γ =0.8 (d) α=0.6, γ =0.8

138


(a) β=1.0, γ =1.0, (b) β=0.8, γ =1.0,(c) α=1.0, γ =1.0, (d) α=0.6, γ =1.0

139


(a) β=1.0, γ =0.8, (b) β=0.8, γ =0.8, (c) α=1.0, γ =0.8 (d) α=0.6, γ =0.8

140


(a) β=1.0, γ =1.0, (b) β=0.8, γ =1.0,(c) α=1.0, γ =1.0, (d) α=0.6, γ =1.0

141

xxi


(a) β=1.0, γ =0.8, (b) β=0.8, γ =0.8, (c) α=1.0, γ =0.8 (d) α=0.6, γ =0.8

142

Fig. 4.3.13 Membership Functions for Bending moment at node 4 at (a) β=1.0,

γ =1.0, (b) β=0.8, γ =1.0,(c) α=1.0, γ =1.0, (d) α=0.6, γ =1.0

143

Fig. 4.3.14 Membership Functions for Bending moment at node 4 at (a) β=1.0,

γ =0.8, (b) β=0.8, γ =0.8, (c) α=1.0, γ =0.8 (d) α=0.6, γ =0.8

144


(a) β=1.0, γ =1.0, (b) β=0.8, γ =1.0, (c) α=1.0, γ =1.0, (d) α=0.6, γ =1.0

145


(a) β=1.0, γ =0.8, (b) β=0.8, γ =0.8, (c) α=1.0, γ =0.8 (d) α=0.6, γ =0.8

146

Fig. 4.3.17 Membership Functions for Axial Force in cable 2 at (a) β=1.0, γ =1.0,

(b) β=0.8, γ =1.0,(c) α=1.0, γ =1.0, (d) α=0.6, γ =1.0

147


(b) β=0.8, γ =0.8, (c) α=1.0, γ =0.8, (d) α=0.6, γ =0.8

148

Fig .4.3.19 Membership Functions for Axial Force in cable 3 at (a) β=1.0, γ =1.0,

(b) β=0.8, γ =1.0, (c) α=1.0, γ =1.0, (d) α=0.6, γ =1.0

149


(b) β=0.8, γ =0.8, (c) α=1.0, γ =0.8, (d) α=0.6, γ =0.8

150

Fig. 4.4.1 Sensitivity Analysis for displacements (for dead load) at node 3 at

γ=1.0

154

Fig. 4.4.2 Sensitivity Analysis for displacements (for dead load) at node 3 at

γ = 0.8

154

xxii

Fig. 4.4.3 Sensitivity of displacements at node 3 in deck at α=1.0

155

Fig. 4.4.4 Sensitivity Analysis of Forces and Moments in Element 12 (at node 5)

at γ = 1.0

155

Fig. 4.4.5 Sensitivity Analysis of Forces and Moments in Element 12 (at node 5)

at γ = 0.8

156

Fig. 4.4.6 Sensitivity Analysis for Bending Moment (due to dead load) in deck

slab at γ =1.0

156

Fig. 4.4.7 Sensitivity Analysis for Axial Force (due to live load) in deck slab at

β =1.0

157

Fig. 4.4.8 Sensitivity Analysis for Axial Force (due to live load) in deck slab at

β =0.8

157

Fig. 4.4.9 Sensitivity Analysis for Shear Force (due to live load) in deck slab at

β =1.0

158

xxiii

LIST OF TABLES IN APPENDICES

Table. C-1 Material and Geometric Properties of Fixed beam 177

Table. C-2 Fixed Beam – Concomitant Variation of mid-span displacement (×10-

3 metres) w.r.t. α and β

179

Table. C-3 Comparison of present results with Response Surface Approach at

Degree of Belief (α or β) = 0.0 (Mid-span displacements)

179

Table. C-4 Fixed Beam – Concomitant Variation of Fixed end moment (kNm)

w.r.t. α and β

179

Table. C-5 Fixed Beam - Concomitant Variation of Fixed end Shear Force (kN)

w.r.t. α and β

179

Table. C-6 Material and Geometric Properties of Propped Cantilever Beam 182

Table. C-7 Propped Cantilever Beam - Concomitant Variation of Mid-span

displacement (×10-3 metres) w.r.t α and β

184

Table. C-8 Propped Cantilever Beam - Concomitant Variation of Rotation at

propped end (×10-3 radians) w.r.t α and β

184

Table. C-9 Propped Cantilever Beam - Concomitant Variation of Fixed End

Moment (Nm) w.r.t α and β

184

Table. C-10 Propped Cantilever Beam - Concomitant Variation of Shear Force

(N) at fixed end w.r.t α and β

184

Table. C-11 Material and Geometric Properties of Plane Truss 188

Table. C-12 Plane Truss-Concomitant Variation of Horizontal displacement of

node 4 (×10-5 metres) w.r.t α and β

190

Table. C-13 Plane Truss-Concomitant Variation of Vertical displacement of node

2 (×10-5 metres) w.r.t α and β

190

Table. C-14 Plane Truss-Concomitant Variation of Axial Force (kN) in Member 4

w.r.t α and β

190

Table. C-15 Plane Truss-Concomitant Variation of Axial Force (kN) for Member

10 w.r.t α and β

190

Table. C-16 Material and Geometric Properties of Plane Frame 192

Table. C-17 Plane Frame-Concomitant Variation of Vertical displacement of node

3 (×10-5 metres) w.r.t α and β

194

xxiv

Table. C-18 Plane Frame - Concomitant Variation of Rotation at node 3 (×10-3

radians) w.r.t α and β

194

Table. C-19 Plane Frame-Concomitant Variation of Bending Moment (kNm) at

node 2 w.r.t α and β

194

Table. C-20 Plane Frame - Concomitant Variation of Axial Force (N) in member

1 w.r.t α and β

194

xxv

LIST OF FIGURES IN APPENDICES

Fig. A-1 Fuzzy Membership Function 170

Fig. C-1 Fixed beam with u.d.l. over entire span 180

Fig. C-2 Fixed Beam – Membership Function for Material Uncertainty 180

Fig. C-3 Fixed Beam – Membership Function for Load Uncertainty 180

Fig. C-4 Fixed Beam – Displacement (mm) at the mid-span under fuzzy static

load

180

Fig. C-5 Fixed Beam – Membership Function for Mid-span Displacement at

β=1.0

181

Fig. C-6 Fixed Beam – Membership Function for Mid-span Displacement at

α=0.5

181

Fig. C-7 Fixed Beam – Membership Function for Fixed-end moment at β=0.75 181

Fig. C-8 Fixed Beam – Membership Function for Fixed-end moment at α=1.0 181

Fig. C-9 Propped Cantilever Beam with central point load 185

Fig. C-10 Membership Function for Material Uncertainty 185

Fig. C-11 Membership Function for Load Uncertainty 185

Fig. C-12 Propped Cantilever Beam – Membership Function for Mid-span

Displacement at α=0.4

185

Fig. C-13 Propped Cantilever Beam – Membership Function for Fixed-end

moment at β=0.8

186

Fig. C-14 Propped Cantilever Beam – Membership Function for Shear Force at

β=0.8

186

Fig. C-15 Propped Cantilever – Variation of Shear Force w.r.t. alpha 187

Fig. C-16 Propped Cantilever – Variation of Bending Moment w.r.t. alpha 187

Fig. C-17 Plane Truss 191

Fig. C-18 Plane Truss – Membership Function for Vertical Displacement at node

2 at α=0.6

191

Fig. C-19 Plane Truss – Membership Function for Axial Force in Member 4 at

β=0.8

191

Fig. C-20 Plane Truss – Membership Function for Axial Force in Member 10 at

α=0.6

191

xxvi

xxvii

Fig. C-21 Plane Frame 195

Fig. C-22 Plane Frame – Membership Function for Horizontal Displacement at

node 2 at β=1.0

195

Fig. C-23 Plane Frame – Membership Function for Vertical Displacement at node

3 at β=0.8

195

Fig. C-24 Plane Frame – Membership Function for Rotation at node 3 at α=0.6 195

Fig. C-25 Plane Frame – Membership Function for Bending Moment at node 2 at

β=1.0

196

Fig. C-26 Plane Frame – Membership Function for Shear Force just to the left of

at node 3 at α=1.0

196

Fig. C-27 Plane Frame – Membership Function for Axial Force in Member 1 at

β=0.8

196

Fig. C-28 Plane Frame – Membership Function for Axial Force in Member 1 at

α=0.6

196

Chapter 1

Introduction

and

Literature Review

Chapter 1

1. INRODUCTION AND LITERATURE REVIEW

1.1 Introduction

Analysis and design of structures occupies an important place in the field of

Civil Engineering. Modern day structures are usually complex in geometry and are

made of a combination of several materials such as concrete and steel. Construction of

structures involves a huge investment in terms of materials, men, money and

expertise. An enormous loss is incurred in the event of structural failure or un-

serviceability of the structure owing to development of cracks or excessive

deflections. Further, certain special structures such as aircrafts, space vehicles,

missiles, nuclear power reactors and dams are of strategic importance. The failure of

such structures results in catastrophic and unpredictable consequences. Structures are

subjected to a combination of loads during their lifetime. In order to ensure that

structures do not fail during their intended design life period, proper analysis and

design are mandatory. Design of a structural system requires that the performance of

the system be guaranteed over its lifetime. Accordingly, the objective of structural

analysis is to predict the behaviour of the structure during its estimated life period

accurately.

Methods of structural analysis are undergoing changes over a period of time.

Traditionally, some of the methods of structural analysis such as slope deflection,

moment distribution, Kani’s method etc. were widely used. However, with the

introduction of high speed electronic computing, the methods of structural analysis

also underwent a sea change. Matrix methods of structural analysis came to be used

widely with a great degree of computational precision coupled with high speed.

2

Classical finite element analysis is presently the most popular mathematical

tool for the analysis of structures. Finite element analysis attempts to construct a

mathematical model in order to make an accurate prediction of structural behaviour.

Various parameters of a structure which influence its behaviour are normally

identified and used in constructing a mathematical model for the structure. The

parameters could be material and geometric properties of the structure, support

conditions and applied loads. The mathematical model constructed could be analysed

for the given structural configuration under the influence of the given loads. The

parameters used in generating the mathematical model are normally crisp and certain

in nature. It is presumed that the structural response of the mathematical model

closely corresponds to the behaviour of the actual structure. Any variation in the

response of the structure predicted using the mathematical model and the response of

the physical structure could be due to the following factors (Muhanna and Mullen,

1999):

1. Uncertainties involved in material properties.

2. Uncertainties in geometric properties

3. Uncertainty in service loads

4. Uncertainty in boundary conditions

The errors due to above uncertainties can neither be handled nor eliminated by

the use of classical finite element analysis. Therefore, there seems to be a lot of

uncertainty involved in the structural response by not considering the above

parameters in the analysis. Thus, there is a need to improve the existing mathematical

model of the structure in order to bring its behaviour as close as possible to the actual

behaviour of the physical structure. Thus uncertainty needs to be introduced in the

engineering analysis and design of structures to enhance the functionality and

dependability of the mathematical model of the structure.

3

The uncertainty introduced in the mathematical model of the structure needs to

be reflected in the method of analysis and its output as well. This requires the

redefinition and extension of the classical finite element model to a fuzzy finite

element model, which allows the use of fuzzy interval variables in order to account

for uncertainties in parameters.

Further, there is a need to evaluate the sensitivity of structural response to unit

variation of uncertain parameters about their respective mean values. A detailed

analysis of sensitivity of the structural response quantities helps the analyst in

evaluating the structural behaviour in a reliable manner.

1.2 Literature Review 1.2.1 Uncertainty in engineering analysis and design

Many real-world engineering systems are too complex and ill-defined to be

modelled by conventional deterministic procedures. They often contain information

and features which are vague, imprecise, qualitative, linguistic or incomplete (Rao

and Li Chen, 1997). Imprecision or approximation is often involved in the selection

of either design parameters or empirical formulation (Rao and Sawyer, 1995).

Generally, behaviour of engineering structures is assessed during the design stage

using deterministic values of structural parameters and applied loads. Thus, an

idealised model with deterministic values of structural parameters and loads is

formulated and utilized in the process of analysis and design.

One of the major difficulties a designer faces is that both the external demands

of the system and its manufacturing variations are not known exactly. In order to

overcome this uncertainty, the designer requires to provide wider allowance and

devise a conservative design for the system. As new analysis tools continue to be

4

developed, the predictive skills of the designer become sharper. The demands of the

market require more efficient designs and innovations (Muhanna and Mullen, 1999).

At different stages of designing a structure, various design parameters have uncertain

values especially during the conceptualisation of structural configuration.

Intermediate models, defined as assumed structural configurations with tentatively

defined materials and geometry, experience both numerical and qualitative

uncertainty of behaviour. Parameters affecting the performance of the system are

required to be incorporated in the analysis of structural designs subjected to

uncertainties.

Classical finite element analysis, despite its advantages, is not suited to handle

uncertainties in the design variables of the structural system i.e. displacements, loads,

bending moments and shear forces. All structures in reality possess physical and

geometric uncertainties due to physical imperfections, model inaccuracies and system

complications. Floating-point numbers are used to describe the physical quantities in

the analysis and the output generated is crisp and non-fuzzy in nature. Thus, idealised

classical finite element model for analysis and design does not truly represent the

degree of imprecision or uncertainty involved in the physical model of the problem.

Therefore, it is necessary to include these uncertainties in analysis and design in order

to evolve a refined and reliable model. Thus, there is a need for an uncertainty-based

model for analysis and design of engineering structures.

Treatment of material uncertainty is a puzzling issue. Dependency,

cancellation, and range of tolerance are among the main sources of overestimation in

interval arithmetic as they can lead to catastrophic results. (Muhanna and Mullen,

2001)

5

It may thus be surmised that any uncertainty introduced in the mathematical

model requires to be reflected in the analytical approach as well as in the final result.

This classical finite element model needs to be reformulated to include the concept of

uncertainty.

1.2.2 Handling uncertainty

Depending on the nature and extent of uncertainty involved in an engineering

system, three different approaches are used for its analysis. (Rao and Sawyer, 1995)

a) If only fragmentary information on the uncertain quantity is available, it is

possible to establish an upper bound on the maximum response of a system

using the anti-optimisation approach.

b) If the system parameters are treated as random variables with known

probability distributions, the performance or output of the system can be

determined using the theory of probability or random processes.

c) The other way of representing uncertainty in engineering calculations is

the use of possibility theory based on the theory of fuzzy sets. If the

system parameters are described as linguistic or imprecise terms, fuzzy

theory can be used to predict the response.

These three approaches are broadly outlined in the following paragraphs.

1.2.2.1.Anti-optimisation

In this approach (McWilliam, 2001), the least favourable response under the

imposed constraints is determined. A bounded uncertainty approach is proposed for

this purpose, which relies upon knowledge of the bounds of uncertainty. In practical

applications of this approach, it is usual to use (elliptical) convex sets to model the

6

uncertain phenomena; while more recent work has used interval sets. This method

takes some account of the interactions between uncertain parameters in the stiffness

matrix and force vector. As the uncertain parameters are varied between predefined

upper and lower bounds, the displacement of the structure in each degree of freedom

varies, according to the values of the uncertain parameters, between certain upper and

lower bounds. The problem of evaluating the maximum (or minimum)

displacements, subject to the uncertain parameters is investigated.

1.2.2.2.Probability - stochastic approach for uncertainty modelling

Probability theory has a long history that can be traced back to the work of

Pascal. The use of probability in the design of engineered components, however, is a

much more recent endeavour. The use of finite element methods for random variables

has been developed over the last two decades. Stochastic finite elements have been

developed to calculate the reliability of linear engineering systems by first-order or

second-order methods (FORM and SORM) and are applied to both static and transient

problems. In addition, Monte Carlo simulations as well as non-linear extensions to

stochastic finite elements have been developed (Muhanna and Mullen, 1999).

Although there is no suitable technique available for the analysis of all types

of imprecision, the stochastic finite element method can be used to handle uncertain

parameters that are described by probability distributions. The stochastic or

probabilistic finite element method was developed in the 1980s to account for

uncertainties in the geometry or material properties of the structure, as well as the

applied loads. The uncertain variables are spatially distributed over the region of the

structure and are modelled as random variables/stochastic fields with known

characteristics (Rao and Sawyer, 1995).

7

Underlying the work of probabilistic analysis of uncertain data is the existence

of a probability density function. In most of the analyses, the probability density

functions have small but non-zero values for all ranges of dependent values (e.g.

Guassian distributions). The interpretation of the probability density functions is such

that any outcome is possible, albeit with a very small chance. The design and

determination of the reliability of system using probability methods often use the

region of very low probabilities. In most cases, it is the region of the probability

density function that is the least accurately assessed (Muhanna and Mullen, 1999).

A few cases of uncertainties, especially those involving descriptive and

linguistic variables as well as those based on scanty information, cannot be handled

satisfactorily by stochastic finite element approach. (Rao and Sawyer, 1995)

1.2.2.3. Modelling qualitative uncertainty using fuzzy logic

Until recently, probabilistic analysis was the only methodology available in

literature to handle uncertainties. Since available statistical information about

structural parameters and loads is scanty, there has been increased interest in the

application of models of uncertainty that need not depend upon such detailed

knowledge.

An alternative method for representing uncertainties in engineering

calculations is the use of possibility theory based on the theory of fuzzy sets

(Muhanna and Mullen, 1999). The fuzzy or imprecise information may be present in

the geometry, material properties, applied loads, or boundary conditions of a

structural system. In the traditional (deterministic) finite element approach, all the

parameters of the system are taken to be precisely known (Akpan et al, 2001). In

practice, however, there is always some degree of uncertainty associated with the

8

actual values for structural parameters and applied loads. In addition, the pattern of

live loads, applied on a structure during its lifetime, is not deterministic.

However, these uncertainties in the system properties are usually bounded

from above and below, and can be considered to be defined with in envelope bounds.

As a consequence of this, the response of the structure will always exhibit some

degree of uncertainty. Uncertainties will be introduced as interval values i.e., the

values are known to lie between two limits, but the exact values are unknown. The

associated subjective type of uncertainty is known as fuzziness and is usually present

in any engineering design process (Köylüoğlu, 1995).

A realistic or natural way of representing uncertainty in engineering problems

might be to consider the values of unknown variables, which are defined within

intervals that possess known bounds or limits (Muhanna and Mullen, 2001).

Thus, the problem is of determining conservative intervals for the response

quantities, displacements, internal forces, and stresses under uncertain loads in the

presence of material, geometric and load uncertainties.

Extensive research helped in the understanding of the behaviour of

imprecisely defined systems using fuzzy logic. Uncertainty in the input data as well

as in the behaviour of systems was explained by introducing vagueness in qualitative

terms in the definition of design variables of the problem. Use of fuzzy logic in order

to understand and model the behaviour of structural systems is of recent origin.

Concerted efforts were made since then to handle uncertainty in engineering problems

realistically by introducing fuzziness in material and geometric properties of structural

systems and also service loads to which the structures are exposed to during their

design life period. In order to understand the applicability of fuzzy logic to solve

structural engineering problems, it is necessary to understand the basic concepts of

9

fuzzy logic and fuzzy set theory. Some fundamentals of fuzzy set theory are

presented in Appendix A.

Interval numbers can be extracted from fuzzy variables by using a concept

known as α-cut approach. The concept of α-cut approach, properties of interval

numbers and their relation with fuzzy logic are described in Appendix-A

1.2.3 Interval Algebra in Structural Analysis

Though interval arithmetic was introduced by Moore (1966), the application

of interval concepts to structural analysis is more recent. Modelling with intervals

provides a link between design and analysis where uncertainty may be represented by

bounded sets of parameters.

Interval computation has become a significant computing tool with the

software packages developed in the past decade. Comparing with traditional floating-

point arithmetic, interval arithmetic has the following advantages:

1. Computational results obtained with interval arithmetic are reliable since there

is no loss of information.

2. Interval valued parameters, which appear frequently in real-world application,

can be directly used for computations with interval arithmetic.

3. New algorithms may be developed to solve some difficult problems with

interval analysis.

Interval algebra has been applied to develop several methods for the solution

of linear interval systems with a given interval vector on right hand side. Among

them is the inclusion theory developed by Gay (1982), Neumaier (1987, 1989, 1990),

Jansson (1990) and Rump (1990). In their theories, the exact hull for the interval

solution set of the linear interval system is found to be sharply bounded by outer and

inner estimates.

10

1.2.4 Fuzzy Finite Element Model of a Structure

The uncertainty model for analysis and design is constructed by combining the

concepts of fuzzy interval numbers and classical finite element analysis. In traditional

finite element methods, the development of the element stiffness matrices and load

vectors requires the evaluation of integrals over the domain or region of the element.

The resulting assembled finite element equations are to be solved using a suitable

technique such as Guassian Elimination or one of its variants (Rao and Sawyer,

1995).

In the case of a fuzzy system, the integrals of fuzzy quantities are to be

evaluated over fuzzy domains. Similarly, the methods of solving a system of

equations are to be redefined to handle fuzzy information. All of these theoretical and

computational aspects can be developed starting from the basic definitions of a fuzzy

quantity and fuzzy arithmetic operations. In particular, the system of linear interval

matrix equations that arise out of the fuzzy integrals is solved using several

algorithms (Hansen, 1965 and Jansson, 1990).

Fuzzy finite element analysis is of recent origin. The studies made in the field

of fuzzy finite element analysis by various researchers are set out in the succeeding

paragraphs to present the state of the art.

In the following section, a survey of the existing advances in uncertainty

modelling using fuzzy finite element analysis has been made. The limitations of the

existing methodologies are identified and the need for further improvement is

underlined. The causes and the nature of various uncertainties are also identified and

discussed.

11

1.2.5 State of the Art

Mid-nineties can be considered as the period when the main activities for

uncertainty treatment in the form of intervals have started in the area of

Computational Mechanics and when the efforts for development of interval finite

element methods have commenced.

Köylüoğlu, Cakmak and Nielsen (1995) have developed an interval approach

utilizing finite element method to deal with pattern loading and structural

uncertainties. In their work, structural and loading uncertainties, bounded by upper

and lower, were considered within a finite element formulation to determine

conservative bounds for the displacement and force response quantities.

Discretisation of a continuum with material uncertainties was illustrated using a linear

elastic beam. This yielded the elements of the stiffness matrix with uncertainties, to

be defined in bounded intervals. Then, the response quantities become uncertain, yet

bounded. The solutions for the system in the form of linear interval equations utilized

the triangle inequalities and linear programming. The results were conservative

within the bounds for the response quantities. Also, the problem of pattern-loading

was solved using this approach. Sample frames were worked out to illustrate the

theory and to examine the sharpness of the bounds in the studied examples, for which

exact solutions and Monte-Carlo simulations were also computed.

Köylüoğlu and Elishakoff (1998) made a comparison of stochastic and interval

finite elements as applied to the problem of shear frames with uncertain material

properties. In their approach, structural uncertainties were modelled using stochastic

and interval methods to quantify the uncertainties in the response quantities. A shear

frame of multi-degree of freedom was considered, and the bending rigidities of the

storey columns were taken as uncertain fields. The uncertain bending rigidity of each

12

beam-column element is modelled using two methods. One is a probabilistic method,

where the uncertain field is assumed to be a weakly homogenous stochastic field of

known first order and second order statistical information. The other method is a set-

theoretical interval method, where the uncertain field is assumed to be strictly

bounded from above and below. Galerkin finite elements with cubic deterministic

shape functions were used to descretise the uncertain field.

Rao, S.S. and Sawyer, J.P. (1995), Rao, S.S. and Berke L (1997), and Rao,

S.S. and Li Chen (1998) have developed different versions of interval based finite

element methods to account for uncertainties in engineering problems, these works

were mainly restricted to narrow intervals and approximate numerical results.

Rao, S.S. and Sawyer, J.P. (1995) developed a fuzzy finite element approach

for analysis of imprecisely defined systems. The development of methodology started

from the basic concepts of fuzzy numbers in fuzzy arithmetic and implemented

suitably defined concepts of fuzzy calculus and a solution to finite element equations.

Simple stress analysis problems involving vaguely defined geometry, material

properties, external loads, and boundary conditions were solved to establish and

illustrate the new procedure. The approach developed is applicable to systems that

are defined in linguistic terms as well as those that are described by incomplete

information. Solution vector was obtained using techniques of optimisation, in

particular, unconstrained minimisation of the design vector.

Rao, S.S. and Berke L. (1997) made a study of analysis of uncertain structural

systems using interval analysis. This work considered the modelling of uncertain

structural systems using interval analysis. By representing each uncertain input

parameter as an interval number, a static structural analysis problem was expressed in

the form of a system of linear interval equations. In addition to the direct and

13

Guassian-elimination based methods, a combinatorial approach (based on an

exhaustive combination of extreme values of the interval numbers) and an inequality-

base method are presented for finding the solution of interval equations. The range or

interval of the solution vector was found to increase with the increasing size of the

problem in all the methods. An interval-truncation approach is used to limit the

growth of intervals of response parameters so that realistic and accurate solutions

could be obtained in the presence of large amount of uncertainty. Numerical

examples were presented to illustrate the computational aspects of the methods and

also to indicate the importance of the truncation approach in practical problems. The

utility of interval methods in predicting the extreme values of response parameters of

structures is underlined.

A sincere effort was made by Rao, S.S. and Li Chen (1998) to develop a new

algorithm for solution of linear interval equations. The algorithm used search-based

operations with an accelerated step size in an attempt to find an optimum setting of

unknown vector components. In one of the examples it was observed that 64

operations were needed to obtain the interval solution, where 128 operations were

required by the combinatorial method (exact). This shows the inefficiency of the

algorithm especially in large size problems, besides the fact that the sharpness of the

results is limited to narrow intervals.

Tonon and Bernardini (1996 and 1999) have developed a mono and multi-

objective optimisation approach for structural uncertainty using fuzzy set and random

set theories. In their work, they utilised a fuzzy set to model the engineer’s judgment

about every objective function. Uncertainty was modelled by means of fuzzy

numbers and the meaning was clarified by random set and possibility theories.

14

Nakagiri and Yoshikawa (1996) have developed finite element interval

estimation by convex model. The first-order approximation was employed to estimate

the change of structural responses due to the change of structural parameters that are

assumed to exist in a convex hull. The Lagrange multiplier method was employed to

search for the bounds of the response.

Also, Nakagiri and Suzuki (1999) developed an interesting application of

interval mathematics. In their work, they investigated the load identification as one of

inverse problems of applied mechanics as well as the identification of material

constants and boundary conditions.

Abdel-Tawab and Noor (1999) used parameters with interval representation

for analysis of dynamic thermo-elasto-viscoplastic damage response. The number of

parameters was limited and a combinatorial solution was used for the interval system

to construct fuzzy membership of the response values.

Kulpa, Pownuk, and Skalna (1998), investigated possibilities to apply interval

methods in analysis of linear mechanical systems with parameter uncertainties. This

approach was applied to trusses and frames. The purpose of their study was to

investigate possibilities of and problems with application of interval methods in

(qualitative) analysis of linear mechanical systems with parameter uncertainties in

some truss structures and frames. Their study gave an introduction to interval

arithmetic and systems of linear interval equations, including an overview of basic

methods for finding interval estimates for the set of solutions of such systems. The

methods were further illustrated by several examples of practical problems, solved by

the hybrid system of analysis of mechanical structures. Finally, several general

problems using interval methods for analysis of such linear systems were identified,

with promising avenues for further research indicated as a result. The problems

15

discussed include estimation of inaccuracy of the algorithms (especially the

fundamental problem of matrix coefficient dependence), their computational

complexity, as well as inadequate development of methods for analysis of interval

systems with singular matrices.

Mullen and Muhanna (1999) introduced a new treatment of load uncertainties

in structural problems based on fuzzy set theory. A fuzzy-finite element method

(FFEM) for treating uncertain loads in static structural problems was developed.

Using this approach, the problem domain is first discretised, which resulted in a

system of fuzzy algebraic equations. An efficient algorithm for calculating

guaranteed inclusions for the solution of such fuzzy systems was implemented. In the

case of uncertain loading, the resulting system of equations is linear and only the right

hand side vector contains fuzzy values. Solutions due to all load combinations, as a

special case of load uncertainties can be obtained by a single computation, leading to

a set of possible displacement values corresponding to all loading cases. In

conventional analysis, due to combinatorial nature of the problem, calculating a

possible structural response for all possible loading combinations becomes

computationally intractable for typical structures. Examples show that the possible

interval values calculated using FFEM provide a sharp bound on possible nodal

displacements and element forces calculated by combinatorial computation of all

possible loading combinations. Thus the extreme values of element forces for all

loading combinations can be calculated using fuzzy finite element analysis.

Comparison of extreme values of element forces with typical design loading schemes

show that the current design procedures can lead to nonconservative predictions of

element forces. The method is thus applicable to static analysis of any structural

system that can be analysed using fuzzy-finite element methods.

16

Muhanna and Mullen (1999) have developed interval finite element analysis

procedure utilizing the concept of fuzzy sets through interval calculations. They have

also computed the response of different structural systems due to geometric and

loading uncertainties. Results were exact in the case of load uncertainty where all

types of dependency were eliminated in the new formulation. Exact bounds on

possible nodal displacements and forces were calculated by combinatorial calculation

of all loading patterns when computationally feasible.

Pownuk, A. (2001), introduced an interval global optimisation method. As

stated in this work, the interval algorithm guarantees that all stationary global

solutions have been found. According to him, existing optimisation methods usually

are not reliable or cannot use the non-differentiable, non-continuous objective

functions or constraints. An interval global optimisation method is very stable and

robust, universally applicable and fully reliable. The interval algorithm guarantees

that all stationary global solutions have been found. The convergence of this method

depends on choice of a good inclusion function. In his work, a method of

constructing the inclusion function was presented. This method was based on special

tests of monotonicity. The algorithm was applied for optimisation of mechanical

systems. The preceding results indicated that the presented algorithm was an

effective and efficient method of global optimisation. However, it would probably be

quite slow if many local minima have values of function differing very little from the

global value. The pure interval algorithm guaranteed that all stationary global

solutions (in the initial interval) had been found. The bounds on the solution(s) were

guaranteed to be correct. Error from all sources was accounted for. The algorithm

could solve the global optimisation problem also when the objective function is non-

differentiable or even not continuous.

17

A practical approach for analysing the structures with fuzzy parameters was

developed by U.O. Akpan et al (2001). The uncertainties in material, loading and

structural properties were represented by convex normal fuzzy sets. Vertex solution

methodology that was based on α-cut representation was used for the fuzzy analysis.

Response surface methodology and combinatorial optimisation were used to

determine the binary combinations of the fuzzy variables that resulted in fuzzy

responses at an α- cut level. These binary combinations of the fuzzy variables were

then used to obtain extreme responses to the finite element model.

More recently, Muhanna and Mullen (2001) have developed an interval finite

element method based on a new Element-By-Element (EBE) formulation for

uncertainty in solid and structural mechanics. The method allows the treatment of

uncertainty in the form of intervals (tolerances) in stiffness terms. This formulation

avoids most sources of overestimation and computes a very sharp hull for the solution

set of interval linear equations in the field of solid and structural mechanics, with

wide interval quantities.

1.3 Cable-Stayed Bridges

1.3.1 General

The concept and practical application of the cable-stayed bridge date back to

the 1600’s, when a Venetian engineer named Verantius built a bridge with several

diagonal chain-stays (Kavanagh, 1973). The modern cable-stayed bridge consists of a

superstructure of steel or reinforced concrete members supported at one or more

points by cables extending from one or more towers. The concept attracted to

engineers and builders for many centuries and experimentation and development

continued until its modern-day version evolved in 1950 in Germany. The renewal of

18

the cable-stayed system in modern bridge engineering was due to the tendency of

bridge engineers in Europe, primarily Germany, to obtain optimum structural

performance from material which was in short supply during the post-war years

(Troitsky, 1972).

During the past three decades, cable-stayed bridges have found wide

applications all over the world, especially in Western Europe and United States. In

particular, the cable-stayed girder type of design is fast gaining popularity among the

bridge designers, particularly for medium and long spans.

Cable-stayed bridge stands out as the most recent technological development

in bridge construction as demonstrated by several bridges existing all over the world,

built of different materials and techniques. The Stromsund Bridge, which was

constructed in Sweden in 1955 with a central span of 183 m is the world’s first cable-

stayed highway bridge. Subsequently, a number of cable-stayed bridges were

constructed all over the world in many countries. The Second Hoogly Bridge over the

river Ganga at Howrah is one of the longest bridges in the world with a span of 457.2

m, the Tatara Bridge in Japan being the longest with a span of 890 m. Efforts are on

to increase the span further beyond 1000 m. For medium spans of 100 - 300 m, cable-

stayed bridges are considered to be the most suitable system.

Due to their aesthetic appearance, efficient utilization of structural materials

and other notable advantages, cable-stayed bridges have gained popularity in recent

decades. This fact is due, on one hand, to the relatively small size of the substructures

required and on the other hand, to the advent of efficient construction techniques apart

from the rapid progress in the analysis and design of this type of bridges.

Wide and successful application of cable-stayed systems was realized only

recently, with the introduction of high-strength steel, orthotropic type decks,

19

development of welding techniques and progress in structural analysis. The

development and application of electronic computers opened up new and practically

unlimited possibilities for the exact solution of these highly statically indeterminate

systems and for precise static analysis of their three-dimensional performance.

The recent developments in design technology, material quality, and efficient

construction techniques in bridge engineering will enable construction of not only

longer but also lighter and slender bridges. Thus, very long span slender cable-stayed

bridges are being built, and the aim is to further increase the span length and use

shallower and more slender girders for future bridges. To achieve this, accurate

procedures need to be developed that can lead to a thorough understanding and a

realistic prediction of the structural response to not only wind and earthquake loads

but also traffic loads.

1.3.2 Structural characteristics

Cable-stayed bridges are constructed along a structural system, which

comprises an orthotropic deck and continuous girders, which are supported by stays

i.e. inclined cables passing over or attached to towers located at the main piers.

Modern cable-stayed bridges present a three-dimensional system consisting of

stiffening girders, transverse and longitudinal bracings, orthotropic-type deck and

supporting parts such as towers in compression and inclined cables in tension.

A multiple-stay bridge is a highly redundant system. The paths of the forces

are dictated to a great extent by the relative stiffnesses of the load-bearing elements –

the stays, the pylons and the deck.

Depending on the arrangement of longitudinal stay cables, the cable-stayed

bridges can be divided into four basic systems as shown in Fig. 1.1 (Walther, 1981).

20

i. Harp system

ii. Fan system

iii. Semi-Harp system

iv. Asymmetric system

(d) Asymmetric pattern

(b) Fan pattern

(a) Harp pattern

(c) Semi-harp pattern

Fig. 1.1 Longitudinal Layout of Stays (Courtesy: Walther, 1988)

With respect to the various positions in space, which may be adopted for

planes in which cable-stays are disposed, there are two basic arrangements:

21

i. Two plane systems

a) Two vertical plane systems

b) Two inclined plane systems

ii. Single plane systems

(b) Two- Inclined Plane System

(d) Single Plane System

(c) Single Plane System

(a) Two- Vertical Plane System

Fig. 1.2 Positions of Cables in Space (Courtesy: Troitsky, 1972)

22

The various possible types of tower construction are:

i. Trapezoidal portal frames

ii. Twin towers

iii. A-frames

iv. Single towers

Fig. 1.3 Types of Towers (Courtesy: Troitsky, 1972)

Most cable-stayed bridges have orthotropic decks, which differ from one

another only as far as the cross sections of the longitudinal ribs and the spacing of the

cross-girders depending upon physical constraints. The deck may be made of

different materials such as steel, concrete or prestressed concrete.

The cable-stays that are usually are of the following types:

i. Parallel-bar cables

ii. Parallel-wire cables

iii. Stranded cables

iv. Locked-coil cables

The choice of any of these types depends on the mechanical properties

required (modulus of elasticity, ultimate tensile strength, durability etc.) as well as on

structural and economic criteria (erection and design of the anchorages).

23

.3.3 Loads

wide disparities throughout the world concerning the loads on

(c) Stranded Cables (d) Locked Coil cables

(b) Parallel-wire cables (a) Parallel-bar cables

Fig. 1.4 Types of Cables (Courtesy: Walther, 1988)

1

There are

bridges. There are two considerations governing the loads on the bridges i.e. the load

carrying capacity and the service requirements.

24

For serviceability, it is only necessary to consider cases of loads likely to be

encountered. For major structures, it may be advisable to base the design loads on a

probabilistic traffic analysis. There is a wide range of statistical data available for the

estimation of probable loads. It is still necessary to take note of the fact that the

permanent loads themselves are often decisive, setting aside the phenomena of

vibration and physiological effects on the users (Walther, 1988).

In the case of load-carrying capacity, one is led inexorably to subjective

considerations. It is not, in fact, possible to define a theory based on probability,

however ambitious it may be, given the fact that the majority of serious accidents

causing structures to collapse are due to non-stochastic causes (for example, human

error). These unforeseeable risks must be covered, at least partly, by the margin of

safety. It matters little whether this safety margin is ensured indirectly by overall or

partial safety factors, provided that it reduces the risks to an acceptable level

(Walther, 1988).

For the serviceability limit state, the permanent loads require to be taken as the

actual loads, allowing normal tolerances for the materials used and for the method of

construction (e.g. concrete pre-cast or poured in-situ). Particular attention requires to

be paid to these loads. Agreement between calculations and actual practice, notably

in the deflection of the deck and the strengths of the stays, narrowly depends on the

precision of the self-weight and the permanent loads (surfacing and kerbs)

(Walther, 1988).

The loads to be considered when checking the ultimate limit state depend, in

the first instance on the safety philosophy. According to FIP recommendations, the

load carrying capacity is determined from design material strengths – characteristic

strength divided by partial safety factors, such as

25

γc = 1.50 for concrete and

γs = 1.15 for steel

The permanent loads (mean values) then require to be increased by partial

coefficients given as

γq = 1.35 (unfavourable effect) and

γq = 1.00 (favourable effect)

The distinction between favourable and unfavourable effects is fairly vague

and calls for clarification (Walther, 1988).

Following FIP recommendations, the factor of safety for variable loads (live

loads) is usually taken as γq = 1.5

1.3.4 Idealisation of the structure

Model simulation of a structure consists of idealizing it as a system of

appropriate members, which allow its behaviour to be analysed with sufficient

accuracy and with a reasonable amount of calculation.

Depending on the complexity of the structure and the stage of the design,

different models may be used. These may be plane or spatial systems, covering the

whole structure or only a part, and can comprise of a wide range of members. The

pylons generally can be represented by beam elements. The same can be said of the

deck if this actually behaves like beam. The cables can be represented by bars by

assigning them a very small bending inertia and an idealised modulus of elasticity,

which makes it possible to take into account the effects of cable sag. In this model the

cables are sufficiently tensioned under permanent loads. Thus any compression,

which is likely to arise under live loads, results only in a reduction of initial tension

(Walther, 1988).

26

The behaviour of cable-stayed bridges under the action of live loads is difficult

to predict by means of simple intuitive methods. Thus, it is advantageous during the

initial design stages to model the cable-stayed bridges as plane frames. Final

dimensioning can also be done on the basis of a plane frame model in case of

structures where pylons experience no transverse bending under dead weight plus live

loads due to traffic (Walther, 1988).

1.3.5 Methods of Analysis A cable-stayed bridge is highly statically indeterminate structure in which the

stiffening girder behaves as a continuous beam supported elastically at the points of

cable attachments. Except in the case of a very simple cable-stayed bridge, a

computer is necessary for the solution of this type of structure, its use being primarily

in analysis rather than in design application.

Computer programs are necessary to generate the influence diagrams for cable

forces, stiffening girder, bending moments and shear, and tower and pier reactions.

The computer is also required for the rapid solution of various parametric efforts and

loadings that have to be taken into account in achieving a reasonably efficient design.

Probably, the most important problems are the determination of the optimum section

of the stiffening girder section, cable configuration and size.

In a simplified approach to the solution, the structure is assumed to be a linear

elastic system, which may be analysed using the standard stiffness or flexibility

method. Several general computer programs are available which use this approach,

e.g. FRAN, STRESS, STRUDL (Troitsky, 1972).

As with a conventional structure, the analysis of a cable-stayed bridge consists

of several stages. The first involves calculations to give preliminary sizes to the deck,

27

pylons and stays. The general aim is to check the feasibility of the work, as well as to

estimate the quantities required for its completion. At this stage of the scheme, it is

generally sufficient to make use of simplified calculations, without taking into

consideration secondary influences or long term effects. In the second stage, final

calculations are prepared, determining the strength and deformations based on the

final dimensions (Walther, 1988).

1.4 A Brief Review of the Past Work

A review of the work done by the previous researchers reveals the following

points:

1. Modelling of uncertainty is a recent approach to structural analysis.

2. In particular three major approaches are used to model uncertainty i.e. anti-

optimisation, stochastic finite element methodology (based on the theory of

probability) and fuzzy finite element analysis.

3. Of the above three methods, the review of literature reveals that anti-

optimisation and stochastic finite element methodology are rather complicated

and thus require large numbers of computations. Further, only problems of

smaller magnitude are solved using these approaches, as is evident from the

literature review.

4. Fuzzy finite element analysis is more elegantly suited to handle uncertainty in

engineering problems. The work done by Muhanna and Mullen (1999,2001)

and Pownuk (1999) is a significant step in this direction.

5. In particular, Muhanna and Mullen integrated the concepts of fuzzy interval

algebra with classical finite element methodology, thereby evolving fuzzy

finite element methodology.

28

6. Making use of this methodology, the above researchers were able to study the

individual effects of various types of uncertainties namely, load uncertainty,

material uncertainty and geometric uncertainty in problems related to

mechanics. Simple cases such as propped cantilever beam, plane truss, and

plate problems were handled and fuzzy structural response was evaluated in

terms of fuzzy displacements and forces.

7. However, the above studies were confined to the studies of the individual

effects of uncertainties alone. But the realistic effect of multiple uncertainties

(simultaneous presence of more than one uncertainty in the system) was not

considered. The fuzzy response quantities were represented by triangular

membership functions. The effect of multiple uncertainties on the fuzzy

structural response and the corresponding representation has not been

researched. This necessitates the need to study the effect of multiple

uncertainties on the fuzzy response of structures in terms of the fuzzy

displacements, bending moments, axial forces and shear forces. Further, the

effect of the multiple uncertainties on the fuzzy membership functions of the

above response quantities needs to be probed.

8. The sensitivity of fuzzy structural response in the presence of multiple

uncertainties is also not considered in the earlier research. Thus there is a

need to take up this study. In particular, further research needed to evaluate

the individual sensitivities of various response quantities to multiple

uncertainties. This is required in order to evaluate the relative sensitivity of

the fuzzy structural response of all the response quantities to multiple

uncertainties.

29

A review of literature of the analysis of the cable-stayed bridges reveals the

following points:

1. A cable-stayed bridge is highly redundant structural system, requiring the use

of advanced methods of structural analysis for a proper evaluation of its

behaviour (Walther, 1988).

2. The service loads acting on the bridge are highly variable in nature. The

magnitude of the live loads cannot be evaluated exactly using existing

methodologies. Thus, there is always an amount of uncertainty involved in the

evaluation of loads. Currently, the problem of uncertainty of live load is

handled by various national codes by probabilistic approach (Köylüoğlu and

Elishakoff, 1998).

3. The material properties, in particular, the Young's modulus of the materials of

cable stays and concrete deck may also be uncertain in nature. Owing to the

property of visco-elasticity, these values tend to change during the service

period of the bridge. A maximum variation of 10% in the value of the

Young's modulus has been reported in the literature (Muhanna and Mullen,

2001).

4. In addition, uncertainty may be associated with the density of the materials

used in the construction of the bridge depending on the method of construction

(Walther, 1988). Further, during the lifetime of the bridge, the value of the

mass density may undergo variation owing to the effect of corrosion, etc.

There is no existing literature, which incorporates the effect of these uncertainties

in the analysis and design of cable-stayed bridges. This apart, fuzzy finite element

analysis has not been applied to the study of cable-stayed bridges in order to

understand its structural response. In the light of the above, it is felt that there is a

30

need to apply fuzzy finite element methodology to evaluate the effect of multiple

uncertainties on the structural response of cable-stayed bridges in order to have a

realistic evaluation of its behaviour.

1.5 Objectives of the Present Study

The objective of present study is to understand the response of structures in

the presence of multiple uncertainties using fuzzy finite element analysis. The nature

of the fuzzy membership functions of the structural response in the presence of

multiple uncertainties requires to be investigated. Further, sensitivity analysis needs

to be performed in order to evaluate the relative sensitivity of structural response in

the presence of multiple uncertainties. Accordingly, the present investigation is

carried out with the following objectives.

1. To generate a methodology to handle multiple uncertainties in fuzzy finite

element analysis of structures.

2. To validate the new methodology by conducting preliminary case studies.

3. To analyse a cable-stayed bridge using the validated methodology.

4. To incorporate sensitivity analysis in the methodology and evaluate the

relative sensitivity of structural response

1.6. Summary

The general scheme of organization of the present thesis is summarised below.

1. The analytical procedure for fuzzy finite element model of a structure to

handle multiple uncertainties is presented in Chapter 2. This chapter presents

a new approach to handle multiple uncertainties using an element-by-element

model. A new procedure suggested by the researcher is implemented to post-

31

32

process the solution in order to obtain axial forces, bending moments and

shear forces in the presence of multiple uncertainties.

2. The proposed method is tested on beams, frames and trusses and then

validated with the available literature (Appendix C). The validated new

approach is applied to a more complex study of a cable-stayed bridge in

Chapter 3.

3. The fuzzy-finite element model of the structure is analysed for three possible

combinations of uncertainties. The results of these case studies are presented

and discussed in Chapter 4.

4. Finally, the present work is summarised, conclusions are drawn and

recommendations are made for further research in Chapter 5.

Chapter 2

Methodology

Chapter 2

2. METHODOLOGY

2.1 Introduction

In this chapter, a methodology of handling multiple uncertainties using fuzzy

finite element approach is presented. The multiple uncertainties considered are

material property (E), live load and mass density respectively. These uncertainties are

designated as α,β and γ respectively.

A fuzzy-finite element model for tackling a single uncertainty was developed

by Muhanna (Muhanna and Mullen, 2001) (Appendix B). This model is suitably

modified to include the effect of multiple uncertainties. A new methodology is

developed to handle post-processing of the solution.

The following combinations of uncertainties are considered:

a) Uncertainty of live load (0≤β≤1 and α=γ=1).

b) Uncertainty of material property (E) (0≤α≤1 and β=γ=1).

c) Uncertainty of mass density (0≤γ≤1 and α=β=1).

d) Simultaneous presence of live load and material uncertainties (0≤α,β≤1).

e) Simultaneous presence of live load and mass density and material

uncertainties (0≤α,β,γ≤1).

The proposed fuzzy finite element model of the structure in the presence of the

above uncertainties is presented in the following sections.

2.2 Assembled Finite Element Model

The stiffness equations for an assembled finite element model of a structure

with reference to structural axes are

[K̃]{U} = {P} - (2.1)

34

where [K̃] is a crisp (non-interval) assembled stiffness matrix, {U} is the

global displacement vector and {P} global load vector.

Considering the case of a fuzzy structural system with a material uncertainty α

and subjected to a fuzzy interval live load at a level β, Eq. 2.1 is reformulated as

[Kα]{Uβ} = {Pβ} - (2.2)

where [Kα] is the fuzzy-interval stiffness matrix (assembled) at a level of material

uncertainty α, {Uαβ} is the fuzzy-interval displacement vector corresponding to

simultaneous variation of material and live load uncertainties at levels α and β

respectively and {Pβ} is the fuzzy-interval live load vector at the level of

uncertainty β.

The set of equations represented by Eq. 2.2 represents a system of linear

interval equations in the presence of live load (Muhanna and Mullen, 1999).

Uncertainty of mass density, if present in the system, gets reflected as the uncertainty

in the dead load at a level γ. In such a case, Eq. 2.2 is represented as

[Kα]{Uαγ} = {Pγ} - (2.3)

2.2.1 Uncertainty of live load

In this case, the structural system has crisp material property (Young’s

modulus) and is subjected to a fuzzy interval live load. Accordingly, Eq. 2.2 is

recast as

[K̃]{Uβ} = {Pβ} - (2.4)

Here, [K̃] is a crisp stiffness matrix.

35

Eq. 2.4 represents a system of linear interval equations. Solution of this system

of interval equations directly by Guassian elimination leads to overestimation.

Several methods are available in literature to avoid overestimation in the solution of

linear interval matrix equations (Hansen, 1965, Gay, 1982, Alefeld and Herzberger,

1983, Neumaier, 1987,1989 and 1990, Jansson, 1991, Muhanna and Mullen, 1999).

In particular, Muhanna (Muhanna and Mullen, 1999) handled the problem of

dependency by multiplying all the non-interval values first and performing operations

on fuzzy values as a final step. The procedure suggested by Muhanna (Muhanna and

Mullen, 1999) is modified as follows.

The interval body-force vector {P} is considered as the product of a non-fuzzy

matrix [M ̃] and an interval element live load vector {q} as shown below.

{P}n×1 = [M ̃] n×m {qβ} m×1 - (2.5)

where n is the kinematic indeterminacy for the structure and m is the number

of elements in the structure. The jth column of [M ̃] matrix contains the contribution to

element forces due to a unit live load on the jth element of the structure.

Substituting Eq. 2.5 in Eq. 2.4, the following equation is obtained

[K̃]{Uβ} = [M ̃] {qβ} - (2.6)

Thus, the fuzzy nodal displacement vector {Uβ} is obtained as:

{Uβ} n×1 = [ [K̃] -1n×n [M ̃] n×m] {qβ} m×1 - (2.7)

Thus all non-interval values are multiplied first, and the last multiplication

involves the fuzzy-interval quantities. If this order is not maintained, the resulting

interval solution will not be sharp (Muhanna and Mullen 1999).

In case the structure is acted upon by set of fuzzy point loads, the point load

vector {P´} n×1 is given by:

36

{Uβ} n×1= [K̃] -1n×n {P´β} n×1 - (2.8)

where Pí = [Píl , Píu] is the fuzzy point load applied at the ith degree of freedom.

2.2.2 Post-processing of solution

For post-processing of the solution, the following approach is proposed:

After solving for the fuzzy displacement vector {Uβ} n×1 (primary unknown),

the values of fuzzy axial force, fuzzy shear force and fuzzy bending moment

(secondary unknowns) are obtained by post-processing the solution at the element

level.

Considering the equation of equilibrium at the element level,

[K̃] (e) md×md [T] (e)

md×md {Uβ}(e) md×1 = {Pβ}(e)

md×1 + {g}(e) md×1 - (2.9)

where md is the number of degrees of freedom for the element, [T](e) is the

transformation matrix for the element and {g}(e) is the vector of element-end forces.

In case of a plane frame element, {g}(e) consists of {-A1,-V1,-M1,A2,V2,M2}T

where A1, A2, V1, V2, M1, M2 are the axial forces, shear forces and bending moments

at the first node and the second node of the element respectively.

Now, the displacements appropriate to a given element are extracted as

{Uβ}(e) md×1 = [L] md×n {Uβ} n×1 - (2.10)

where [L] md×n is a Boolean connectivity matrix containing 0s and 1s.

Substituting Eq. 2.10 and Eq. 2.7 in Eq. 2.9 for uniformly distributed loads,

[K̃](e) md×md [T](e)

md×md [L] md×n [K̃]-1n×n [M}n×m{qβ}m×1 ={Pβ}(e)

md×1+{g}(e) md×1 - (2.11)

Substituting Eq. 2.10 and Eq. 2.8 in Eq. 2.9 for point loads,

[K̃] (e) md×md [T] (e)

md×md [L] md×n [K̃]-1n×n {P′}n×1 = { P′β}(e)

md×1+{g} (e) md×1 - (2.12)

37

From Eq. (2.11) and Eq. (2.12), {g}(e) is computed and the fuzzy values of A1,

A2, V1, V2, M1, M2 are obtained for a given element.

2.2.3 Uncertainty of mass density

Uncertainty of mass density is processed in the same manner as load

uncertainty. In this case, the governing equation, given by Eq. 2.3, is modified as

[K̃]{Uγ} = {Pγ} - (2.13)

The displacement vector {Uγ} and the vector of internal forces {λγ} are obtained by an

appropriate modification of the procedure followed in the previous sections (section

2.2.1 and section 2.2.2).

2.3 Material Uncertainty

Considering the case of a structural system with a crisp load (β=1) and with

material uncertainty at a level α. Thus, Eq. 2.2 is rewritten as

[Kα]{Uα} = {P} - (2.14)

where [Kα] is the structure matrix with material uncertainty α for an assembled finite

element model.

The solution of the set of equations given by Eq. 2.14 for the assembled finite

element model using the methodology described in section 2.2.1 and section 2.2.2

leads to a conservative solution of a large width owing to the problem of dependency,

overestimation and element coupling (Appendix A).

An element-by-element model with a fuzzy-interval domain of displacements

was proposed by Muhanna (Muhanna and Mullen, 2001) to avoid overestimation

(Appendix B). In this model, the governing equations for assembled finite element

38

model given by Eq. 2.2 are modified in order to impose constraints on the

displacements of the classical finite element model. The element-by-element model

of the structure has the following characteristics (Fig. 2.1)

a) The set of elements of the model is dissembled. Each element has its own set

of nodes so that a given node belongs to only one element.

b) A set of additional constraints (in addition to the constraints that are imposed

at the boundary) is introduced to force unknowns associated with coincident

nodes to have identical values.

50kNm 50kN

100kN 100kN

25kN/m 4 m

25kN/m

3 m 3 m

(b) Unassembled Model (a) Assembled Model

Fig. 2.1 Assembled and Unassembled Models of a Structure

2.4 Sources of Overestimation

Several problems involving uncertain Young’s modulus and crisp live load

were solved by Muhanna using the approach outlined in Appendix B. Initially, an

attempt was made in the present study to utilise the above methodology to problems

involving multiple uncertainties (α and β). However, it was found that the application

of Muhanna’s methodology in the presence of multiple uncertainties resulted in an

39

overestimated solution. A detailed numerical investigation and error tracking led to

the following points of observation:

a) In case of structures with crisp stiffness properties (α=1), the problems of

dependency and overestimation due to element coupling of stiffness

contributions (as discussed in Appendix A) are completely eliminated, as per

the observations made by Muhanna (Muhanna and Mullen, 2001). Thus, in

the case of a structure with a crisp Young’s modulus and uncertain interval

load (α=1, 0≤β≤1), the assembled finite element model (section 2.2) and

Muhanna’s element-by-element model (Appendix B) are required to yield

exactly the same value of the displacement vector {U}. However,

investigations made in the present study could not validate this assumption by

numerical calculations. This suggests that there is a possible source of

overestimation due to coupling of elements of the interval load vector at the

elemental level itself due to contribution of various interval loads

simultaneously acting on each element.

In the case of classical finite element analysis using crisp loads, the

sum total of the fixed end reactions at a particular degree of freedom at the

element boundaries due to the contribution of each of the loads acting on a

given element (for a given load-case) is computed. The element load vector is

obtained by reversing the sign of the vector of fixed end forces for a given

element. But, it is observed in the present study that the application of this

procedure in the case of several interval loads acting on the elements causes

overestimation. This indicates the existence of load coupling in the same

manner as the element coupling described by Muhanna (Appendix A).

40

However, this problem was not encountered in the problems solved by

Muhanna, because crisp loads were used in his case studies.

b) Even in the case of single uncertain load acting on the structure, the result of

the assembled model did not exactly match the result of the element-by-

element model. Further numerical investigations pointed the source of error to

the approximation of the interval vector of internal forces {λ} by its mid-point

(crisp) vector {λc}. This is because in the case of a statically indeterminate

structures internal forces depend on both the structural stiffness and external

load applied. Thus in the presence of material uncertainty (at a level α) and

load uncertainty (at a level β), the internal forces are expected to exhibit the

combined effect of both uncertainties. However, in the presence of material

uncertainty (0≤α≤1 and β=1), it is observed that the use of the mid-point

vector of the interval vector {λ} i.e. {λc} in Eq. 7 of Appendix-B yields the

sharp solution.

The above sources of error are removed by incorporating the following

modifications in Muhanna’s method.

i) The contribution of the loads to the overall solution is kept separate

throughout the solution process in order to eliminate overestimation due to

coupling of load vector (Fig. 2.2).

ii) A new approximation to the vector of internal forces {λαβ} (under the

influence of multiple uncertainties α and β) is proposed. Thus the interval

vector {λαβ} is approximated by {λcβ} (The first subscript c indicates the

crisp value of material property i.e. α=1).

41

By incorporating the modifications mentioned in points (i) and (ii) above, a

sharp solution is obtained in the case of multiple uncertainties.

=

+ +

Fig. 2.2 Avoiding Overestimation by Uncoupling of Loads

2.4.1 Evaluation of overall displacement vector

The structure is analysed by keeping the uncertain loads acting on each of the

elements separate throughout the course of the solution in order to prevent load

coupling at the element level. The overall fuzzy displacement vector {U} which

represents combined effect of all joint loads and element loads is obtained by

superimposing, at the end of the solution process, the fuzzy-interval displacement

vectors {Uαβ} obtained for individual load cases. As brought out later in section 2.5,

a similar methodology is employed in post-processing as well to maintain the

sharpness of forces and moments.

42

In the case of structures with uncertainty of load (Young’s modulus being

crisp), it was found that the results obtained using assembled finite element model and

the corresponding element-by-element model matched exactly. This is because of the

absence of load coupling. The detailed methodology is outlined in section 2.5.

2.5 Concomitant Presence of Load and Material Uncertainties

Following Muhanna’s methodology, the fuzzy potential energy ∏* of a

structural system for an element-by element model is represented as

∏* = ½ {Uαβ} T [Kα]{Uαβ}–{Uαβ} T {Pβ} +{λαβ} T {[C̃]{Uαβ}-{0}} - (2.15)

where {Uαβ} is the fuzzy-interval displacement vector, {Pβ} is the fuzzy-interval load

vector, [C ̃] is a constraint matrix , [Kα] is the fuzzy-interval stiffness matrix and {λαβ}

is the fuzzy-interval vector of internal forces.

Using Rayleigh-Ritz approach, the following equations are obtained at a given

level of material uncertainty α and load uncertainty β as

[K]α {U}αβ + [C]T{λ}αβ = {P}β and - (2.16)

[C]{U}αβ = {0} - (2.17)

The interval stiffness matrix [K]α is a symmetric indefinite square matrix

owing to the dissembled state of elements in the EBE model (Muhanna and Mullen,

2001). If the unknowns of each element are numbered consecutively, then the

stiffness coefficients of [K]α will be clustered in the form of a square diagonal blocks

each of size equal to the number of unknowns per element. The size of [K]α is n×n.

The interval vector of Lagrange multipliers {λ}αβ represents the vector of internal

forces that are exposed due to the dissembled nature of elements in the EBE model.

The value of {λ}αβ depends on only applied loads and boundary conditions in the case

43

of statically determinate structures. But in the case of statically indeterminate

structures, the value of {λ}αβ depends on structural stiffness, applied loads and

imposed constraints on the boundary.

Eq. 2.16 is introduced as

[D]α[S]{U}αβ = {P}β -[C]T{λ}αβ - (2.18)

where [D]α is a diagonal matrix of size n×n containing interval terms corresponding to

material uncertainty at level α and [S] is a deterministic singular matrix of size n×n..

If Eq. 2.17 is multiplied on either side by [D]α[C]T and the result is added to

Eq. 2.18, the following equations are obtained:

[D]α[[S]+[C]T[C]] = {{P}β -[C]T{λ}αβ} - (2.19)

[D]α[R̃]{U}αβ = {{P}β -[C]T{λ}αβ} - (2.20)

{U}αβ = [R̃]-1 [D]α-1 {{P}β -[C]T{λ}αβ} - (2.21)

If the interval vector {λ}αβ is determined exactly, then the solution of Eq. 2.21

represents the exact hull of interval system of equations given by Eq. 2.2

In the case of statically determinate structures, the vector of internal forces

depends only on applied loads and boundary conditions and does not depend on

structural stiffness. Accordingly, the internal force vector is independent of material

uncertainty α and is represented as {λ}cβ. But, in the case of statically indeterminate

structures, {λ}αβ depends on uncertain structural stiffness(corresponding to material

uncertainty at a level α) in addition to uncertain loads(with an uncertainty level β) and

boundary conditions(crisp). Therefore the value of {λ}αβ cannot be determined

exactly. Thus, the procedure described by Muhanna (Muhanna and Mullen, 2001) is

suitably modified to consider the cumulative effect of uncertainties of material

property and load.

44

2.5.1 Cumulative Effect of Material and Load Uncertainties (α and β)

At a given level of uncertainty β, the interval load vector {P}β and vector of

internal forces is expressed as

{P}β = [βl, βu] {P} - (2.22)

and {λ}α β ≈ [βl, βu] {λ} α c - (2.23)

where βl and βu are the lower and upper bounds of the normalised variable for interval

vectors {P} β or {λ}αβ. The exact value of {λ}αc is not known and is approximated

with {λ}cc.

Thus,

{λ}cc ≈ {λ}αc - (2.24)

Substituting Eq. 2.24 in Eq. 2.23 leads to

{λ}α β ≈ [βl, βu] {λ}cc - (2.25)

Substituting the approximate value of {λ}αβ in Eq. 2.21, the following approximate

matrix equation is obtained

[D]α[R̃]{U}αβ = {{P}β -[βl, βu] [C]T{λ}cc} - (2.26)

The above equation is written as

{U}αβ = [R̃]-1[M]{δ}α - (2.27)

where {δ}α is an interval vector of size n×1 containing interval material properties of

n elements taken from the diagonal entries of [D]α. The solution of this equation

yields an exact solution in the presence of load uncertainty and a sharp solution if

material uncertainty is also present.

In order to solve Eq. 2.27, it is required to compute the value of the mid-point

internal force vector {λ}cc. This is done by considering the assembled finite element

45

model of the structure with deterministic value of structural stiffness subjected to

fuzzy interval loading at a level of load uncertainty β. The detailed procedure is

outlined in the succeeding section 2.5.2.

2.5.2 Evaluation of {λ}cc from Assembled FEA Model

{λ}cc is obtained using the following procedure:

An equivalent assembled Finite element model is obtained from the set of

unassembled discrete elements in the EBE model by assembling elements that are

kept unassembled hitherto. In this process, each set of hitherto coincident nodes of

the BE model is collated and treated as a single node common to all the coincident

elements. Owing to this process of assembling the elements, the internal force vector

{λαβ} vanishes at the inter-element boundary. Therefore, the structural stiffness is

taken as crisp (or equivalently, point interval) and only load uncertainty is allowed.

Accordingly, substituting α=1 and {λαβ} = {0} in Eq. 2.16, and assembling all

the elements leads to the following equation at the global level:

[K ]{U}cβ = {P}β - (2.28)

where [K ] is a point-interval stiffness matrix.

Replacing the point interval stiffness matrix [K ] by its equivalent floating-

point stiffness matrix [K ̃] leads to the following equation.

[K̃]{Ucβ} = {Pβ} - (2.29)

Here, [K̃] is the fixed-point (crisp) stiffness matrix.

As explained in section 2.2.1, the system of interval equations given by Eq.

2.29 cannot be solved directly because of the problem of overestimation of solution

vector owing to expansion of intervals during algebraic operations (dependency

46

problem). Hence, the equations are solved by using the procedure mentioned in

section 2.2.1.

The resulting interval vector of displacements {Ucβ} is further post-processed

as per the procedure outlined in section 2.2.2 and the vector of element-end forces

{λ}cβ(e) is calculated element-wise. The vector {λ}cβ

(e) is obtained from the following

equations for body forces and surface tractions respectively (by modifying

Eq. 2.11 and Eq. 2.12 ) .

{λ}cβ(e)

md×1=- {Pβ}(e) md×1+ [K̃] (e)

md×md [T] (e) md×md [L] md×n1[K] -1n×n1 [M ̃] n1×m {qβ}m×1 - (2.30)

and

{λ}cβ(e)

md×1= -{Pβ}(e) md×1+ [K̃] (e)

md×md [T] (e) md×md [L] md×n1 [K] -1n1×n1 {P′}m×1 - (2.31)

The vector of internal forces {λcβ} is obtained from the calculation of vector

{λcβ}(e) for all elements.

2.5.3 Solution of linear interval matrix equations

Eq. 2.27 represents a system of linear interval matrix equations which

approximate the value of the displacement vector {U}αβ. This is owing to the

approximation made in the computation of the interval vector {λ}αβ. Following

Muhanna’s approach (Muhanna and Mullen, 2001) and using Eq. 2.21 and Eq. 2.26,

the amount of error involved {δU} is expressed as

{δU} =[R̃]-1 [D]α-1 {{P}β -[C]T{λ}αβ}- [R̃]-1 [D]α-1 {{P}β -[βl, βu] [C]T{λ}cc} - (2.32)

A sharp solution to Eq. 2.27 is obtained by minimising the value of {δU} using

the principles of theory of inclusion developed by a number of researchers (Gay,

1982, Neumaier, 1987,1989 and Rump, 1990). The theory of inclusion aims at

obtaining an optimum interval enclosure for the solution of the given interval system.

47

Solution of these equations using the inclusion theory results in an optimal enclosure

AHb known as the hull of the solution. The lower bound vector {x} and the upper

bound vector {y} enclose the hull of the solution as

{y} ⊆ AHb ⊆ {x} - (2.33)

This concept is depicted schematically in Fig. 2.3 below.

xx

AHb

yy

Fig. 2.3 Bounds on Hull Solution

(Muhanna and Mullen, 2001)

Making use of Jansson’s algorithm (Jansson 1991, Muhanna 2001), the lower

bound vector {x} and the upper bound vector {y} enclosing hull of the solution are

obtained. The detailed procedure is outlined below:

In the present study, {Uαβ} is taken as the mean value of the inner bound

vector {y} and outer bound vector {x}.

2.6 Post-processing of solution

The [M] matrix in Eq. 2.27 has n rows and m columns. The matrix contains

terms of load vector at a level of uncertainty β, arranged element-wise in m columns.

The [M] matrix is related to the corresponding point-interval form [M ] as

[M] = [βl, βu] [M ] - (2.34)

Using Eq. 2.34, Eq. 2.27 is recast as

{U}αβ = [βl, βu] [R̃]-1[M ] {δ}α - (2.35)

48

Eq. 2.9 is reformulated as

[Kα] (e) md×md [T] (e)

md×md {Uαβ}(e) md×1 = {Pβ}(e)

md×1 + {λαβ}(e) md×1 - (2.36)

Now, the displacements appropriate to a given element is extracted by reintroducing

Eq. 2.14 as

{Uαβ}(e) md×1 = [L] md×n {Uαβ} n×1 - (2.37)

Here {Uαβ}(e) contains displacements appropriate to the given element with reference

to the global axes.

Substituting Eq. 2.35 in Eq. 2.36,

{Uαβ}(e) md×1 = [βl, βu] [L] md×n [R̃]-1[M]{δ}α - (2.38)

Substituting Eq. 2.38 in Eq. 2.36,

{λαβ}(e) md×1 =[βl, βu] [Kα](e)

md×md[T](e)md×md[L]md×n[R̃]-1[M]{δ}α- {Pβ}(e)

md×1 - (2.39)

Eq. 2.39 can be further simplified as

{λαβ}(e) md×1 = [βl, βu]([Kα] (e)

md×md [T](e)md×md[L] md×n[R̃]-1[M]{δ}α–{P}(e)

md×1) - (2.40)

where is {P}(e) is the mid-point element load vector.

As explained in the section 2.5.2, the vectors {λαβ}(e) obtained for individual

load cases are superimposed to obtain the overall internal force vector for the element

{λ}(e), which represents the combined effect of all loads acting on the structure.

2.7 Concomitant Presence of Mass Density and Material Uncertainties

A methodology similar to the one that was adopted in the previous case

(simultaneous presence of load and material uncertainties) is adopted in this case also.

Accordingly, the set of equations givens by Eq. 2.15 through Eq. 2.40 are modified by

replacing the subscript β by γ.

49

2.8 Concomitant Presence of Mass Density, Material Uncertainty and Live

Load Uncertainty

In this case the results are obtained by superimposing the following cases:

i) Structural response (displacement vector and vector of internal forces) in

the presence of dead load with material and mass density uncertainty.

ii) Structural response in the presence of live load with material and live load

uncertainty.

2.9 Sensitivity analysis

Sensitivity may be defined as the percentage variation of a structural response

quantity (displacement or force) about its mean value with reference to a unit

percentage variation of the uncertainty in the characteristics of the structure (material

property, live load and mass density) about their respective mean values.

In order to perform the sensitivity analysis, the material property, live load and

mass density are expressed as normalised variables. The structural response

quantities (Forces and moments) are also normalised. Normalisation refers to the

process of dividing the lower and upper values of a fuzzy interval number by its mid

value.

Considering an interval [a,b], the mid-point of the interval is given by

mid = 0.5 (a + b). The corresponding normalised interval is [φ,η] where φ and η

represent a/mid and b/mid respectively. Obviously, the mid-value of this normalised

interval µ is equal to 1 where as φ and η are less than 1 and greater than 1

respectively. A normalised interval [0.98,1.02] for a fuzzy response quantity

50

indicates that the normalised triplet (φ,µ, η) equals (0.98,1.00,1.02) and also that the

variation of the given response quantity about its mean value is ±2 percent.

Sensitivity graphs can be plotted for each response quantity by taking the

percentage variation of characteristic values of the structure on the x-axis and the

corresponding percentage variation of structural response quantities (about their

respective mean values) on the y-axis. A larger value of sensitivity for a given

response quantity indicates that the level of uncertainty associated with that particular

response quantity is quite significant. This information may help the designer to

adopt an appropriate design strategy.

2.10 Summary

An overview of the methodology discussed in this chapter is outlined below:

i) Analysis of structure by fuzzy finite element modelling is discussed.

ii) Various types of uncertainties present in the structural properties are

identified and discussed. The corresponding membership functions are

explained.

iii) Existing methodologies are discussed and the inapplicability of existing

methodology in the presence of multiple uncertainties α,β, and γ is

brought out. A modified methodology, which incorporates the cumulative

effect of multiple uncertainties, is proposed and is discussed.

iv) The concept of sensitivity analysis to ascertain the sensitivity of structural

response to uncertainty of structural parameters is introduced.

The modified methodology proposed in the current chapter is validated by

conducting certain preliminary case studies (Appendix C). After due validation, the

51

52

proposed methodology is applied to obtain the response of a cable-stayed bridge as

explained in Chapter 3.

Chapter 3

Case Studies

Chapter 3

3. CASE STUDIES

3.1 Introduction

A modified fuzzy finite element methodology is developed in Chapter 2 to

evaluate the cumulative effect of multiple uncertainties. To illustrate the applicability

and validity of the proposed methodology, several simple problems are solved and the

results are compared with the published work of previous researchers, wherever

available. The results of these problems are presented in Appendix-C.

After validation of the methodology by application to preliminary problems,

the proposed methodology is applied to the behaviour of a cable-stayed bridge under

the influence of multiple uncertainties.

In the present study, multiple uncertainties refer to the simultaneous presence

and the independent variation of uncertainties of material property (E), live load and

mass density. These uncertainties are considered as fuzzy interval values. The levels

of uncertainty associated with these quantities are denoted by α, β and γ respectively.

The fuzzy interval quantities are represented at a specified level of presumption (α,β

or γ) of the respective membership functions by a process known as the α-cut

approach. The concept of fuzzy interval numbers, fuzzy membership functions and

the concept of α-cut are explained in Appendix-A. By making an α-cut on the

membership function for a fuzzy quantity, a fuzzy interval variable in the form a

closed interval [a, b] (a≤x≤b) is obtained. In order to avoid confusion, the cuts made

on the membership functions of material property (E), live load and mass density are

denoted as α-cut, β-cut and γ-cut respectively. The uncertainties and the

corresponding membership functions are explained in the following paragraphs.

54

Fig. 3.1 shows the membership function of uncertainty of material property

(Young’s modulus E). The membership function for material uncertainty shows the

normalized values of uncertain Young’s modulus, taking the crisp value as 1.0. The

values of these uncertainties are shown on the y-axis at various levels of α ranging

from α=0.0 to α=1.0 in steps of 0.2. A maximum of ±5% variation about the mean

value is taken to represent maximum uncertainty (Muhanna and Mullen, 2001).

The membership function is triangular in nature with a central peak

corresponding to the mean value and the width of the interval corresponding to

various α-cut levels is equally distributed about the mean (crisp) value. This is

because of the uncertainty in the value of Young’s modulus arising out of minor

fluctuations in the manufacturing process. Thus the Young’s modulus of a given

sample has an equal chance of being above or below the mean value. Thus, an equal

variation about the mean value of Young’s modulus is expected.

At the level α=0.0, the α-cut on the membership function yields a fuzzy

interval number [0.95,1.05]. Similarly, for other values of α also, the corresponding

α-cut values of membership function is extracted. For example, at α=0.8 (a variation

of ±1% about the mean value), the normalized α-cut value is [0.99,1.01]. At α=1.0

(corresponding to 0 percent variation about the mean value), the normalized α-cut

value is a fuzzy point interval number [1.0,1.0]. This corresponds to total certainty.

Fig. 3.2 depicts membership function for load uncertainty. The membership

function for load uncertainty shows the normalized values of uncertain load, taking

the mean (crisp) value as 1.0. The load, which is being referred to, is a service load

and is subject to wide fluctuation about its mean value. In fact, load variation as

much as 100% about the mean value has been reported in the literature (Mullen and

55

Muhanna 1999, Muhanna and Mullen, 1999). Thus, ±100% variation about the mean

value has been chosen as maximum uncertainty, corresponding to a β value of 0.0.

In Fig. 3.2, the membership function shown to represent load uncertainty is

triangle with a central peak (corresponding to the crisp load). This is because the

service load at any instant has an equal chance of falling below or going above the

nominal (mean) value of the load. Fig.2.2 depicts the β-cut levels corresponding to

various levels of β ranging from β=0.0(total uncertainty) to β=1.0 (total certainty) in

steps of 0.2. For example at β=0.4, the normalized β-cut value is [0.4, 1.6] and at a

level of β = 0.8 the corresponding β cut value is [0.8, 1.2].

Fig. 3.3 presents membership function for mass uncertainty. The membership

function for mass uncertainty shows the normalized values of uncertain mass, taking

the mean (crisp) value as 1.0. The mass, which is being referred to, is the nominal

mass and is subject to fluctuation about its mean value owing to the uncertainty

associated with mass density of the materials during the manufacturing process. In

addition, corrosion may affect the mass of the structure during its design life. This

fact can also be taken into consideration during analysis by introducing uncertainty in

mass. Therefore, ±5% variation about the mean value has been chosen as maximum

uncertainty, corresponding to a γ value of 0.0. (Mass uncertainty is denoted by γ

whereas material uncertainty is denoted by α). In Fig. 3.3, the membership function

shown to represent mass uncertainty is triangle with a central peak. This is because

the mass density has an equal chance of falling below or going above the nominal

(mean) value of the load.

In Fig. 3.3 the γ-cut levels corresponding to various levels of γ ranging from

γ=0.0 (total uncertainty) to γ=1.0 (total certainty) in steps of 0.2 are presented. For

example at γ=0.4, the normalized γ-cut value is [0.970,1.030] and at a level of γ = 0.8

56

the corresponding γ cut value is [0.990,1.010]. Just as in the case of membership

function for material uncertainty, the membership function for mass uncertainty also

is a central peak triangle with the peak corresponding to the crisp mass density.

The uncertainty of mass density causes uncertainty of dead load (owing to

self-weight). This is because dead load intensity is the product of mass density and

cross-sectional area for a prismatic element. Thus, the uncertainty associated with

dead load is the same as the uncertainty due to mass density. Thus membership

function adopted for mass density together with the corresponding γ-cut can be

applied in the case of dead load as well.

The effect of multiple uncertainties on the structural response of cable-stayed

bridges is to be evaluated for the following cases.

1. Structural response to dead load in the presence of uncertainty of mass density

and uncertainty of material property (Young's modulus).

2. Structural response to fuzzy live load in the presence of uncertain material

property (Young's modulus).

3. Structural response to a combination of dead load and fuzzy live load in the

presence of uncertainty of mass density and uncertainty of material property

(Young's modulus).

3.2 Case Study 1 – Cable-Stayed Bridge with Uncertainties of Material Property and Mass Density

The case study considered here is a cable-stayed bridge with fan configuration

of cable-stays, shown in Fig. 3.4. This problem is adopted from the configuration of

Canal du Centre Bridge at Obourg, Belgium (Walther, 1988). This bridge is a

pedestrian bridge with two parallel planes of cables arranged in fan configuration.

The bridge is symmetric about the longitudinal axis. The properties of the bridge are

57

mentioned in Table 3.1. Owing to the symmetry of the bridge deck about the

longitudinal axis of the bridge, only one half of the bridge along with a single plane of

cables is used for analysis.

Table 3.1 Properties of Cable Stayed Bridge Nature of the bridge Pedestrian foot bridge across Canal du Centre, Belgium Total Span 2 × 67.00 = 134.00 metres Concrete Desk Slab (girder) Section Double T Material Precast PSC Overall width 1.8 metres Depth 0.6 metres Flange thickness 0.20 metres Web thickness 0.3 metres Pylon A-shaped – Steel Section Double armed Rectangular Pylon, each arm 0.60×0.80 m Height 20 metres above deck (30 metres total height) Load Live load 3.6 kN/m all over the span(4.0 kN/m2 intensity)Cables Stranded cables each with 37 strands of 12.7 mm φ Mass density (Concrete) 2500 kg/m3 Mass Density (Steel) 7850 kg/m3 Young’s Modulus (for concrete) 30 GPa Young’s Modulus (for steel) 200 GPa Material Uncertainty (αmax) 10 percent (±5% about mean value of E) Load Uncertainty (βmax) 200 percent (±100% about mean value of Load) Mass Uncertainty (γmax) 10 percent (±5% about mean value of density)

The structure is modelled using the methodology developed in Chapter 2. The

structural elements belonging to bridge deck and the pylon are idealized as plane

frame elements while cables are modelled as bar elements. The structure is hinged at

nodes 1 and 9 on the bridge deck and is fixed at node 14 at the bottom of the pylon.

A roller support is provided at node 5, at the point where the deck slab passes over the

cross-arm of the pylon.

In this case study, the cable-stayed bridge described above is subjected to the

action of a uniformly distributed dead load (owing to the self-weight).The dead load is

computed on the basis of mass density of steel and concrete as provided in Table 3.1.

58

Also, the bridge is subjected to simultaneous uncertainties in material property E at a

level α and mass density at a level γ respectively.

The structural response of the cable-stayed bridge is evaluated. Various

combinations of α and γ ranging from total certainty (α=1, γ=1) to total uncertainty

(α=0, γ=0) are considered. Fuzzy finite element model of the structure is used to

obtain fuzzy interval values of displacements and rotations. These results are further

post-processed to obtain fuzzy interval values of shear force and bending moment at

various nodes. The values of structural response (displacement and forces) obtained

at various levels of uncertainty are used to construct membership functions of

structural response.

The uncertainty associated with displacements and forces is a function of the

cumulative effect of uncertainties associated with material property and mass density

(α and γ respectively). Thus, the cumulative effect of the simultaneous variation of

material and mass density uncertainties (α and γ) on displacements and forces is

investigated and the results are shown in the form of tables and figures (for

membership functions). The tables and the membership functions represent the

cumulative effect of material property and mass density uncertainties on the values of

the displacements (horizontal, vertical and rotation), and forces (axial and shear forces

and bending moments).

3.3 Case Study 2 – Cable-Stayed Bridge with Uncertainties of Material Property and Live Load

In the second case study, a cable-stayed bridge (shown in Fig. 3.4) with the

configuration given in Table 3.1 is taken up for analysis. The bridge is subjected to

59

uncertain live load due to traffic at a level of uncertainty β. Uncertainty is also

introduced in the Young’s modulus of the materials (Steel and Concrete) at a level α.

As mentioned in Table 3.1 a live load intensity of 4.0 kN/m2 (for a pedestrian

bridge) is adopted to act all over the deck slab. For half the width of the slab in the

longitudinal direction (½ × 1.8 metres = 0.9 metres), the equivalent uniformly

distributed live load works out to be 0.9 × 4.0 = 3.6 kN/m2. The maximum

uncertainty for load βmax is taken as ±100% about mean value of load as is reported in

the literature (Muhanna and Mullen, 1999).

As in case study 1, a maximum of ±5% variation about the mean value is

taken to represent material uncertainty. The membership function of material

uncertainty plotted in Fig. 3.1 is adopted in the case study as well.

The cable-stayed bridge (considered in Fig. 3.4 with properties mentioned in

Table 3.1) is subjected to simultaneous uncertainties in material property E at a level

α and load at a level β respectively. The structural response of the cable-stayed

bridge is evaluated. Various combinations of α and β ranging from total certainty

(α=1, β=1) to total uncertainty (α=0, β=0) are considered. Fuzzy finite element

model of the structure is used to obtain fuzzy interval values of displacements and

rotations. These results are further post-processed to obtain fuzzy interval values of

shear force and bending moment at various nodes. The values of structural response

(displacement and forces) obtained at various levels of uncertainty are used to

construct membership functions of structural response.


cumulative effect of uncertainties associated with material property and live load (α

and β respectively). Thus, the cumulative effect of the simultaneous variation of

60

material and load uncertainties (α and β) on displacements and forces is investigated

and the results are shown in the form of tables and figures (for membership

functions). The tables and the membership functions represent the cumulative effect

of material property and load uncertainties on the values of the displacements

(horizontal, vertical and rotation), and forces (axial and shear forces and bending

moments).

3.4 Case Study 3 – Cable-Stayed Bridge with Uncertainties of Material Property, Live Load and Mass Density

Here the Cable stayed Bridge with fan configuration of cable-stays (taken in

case study 3.1 and case study 3.2) subjected to a combination of dead load and live

load is considered. The Cable-stayed Bridge is subjected to a uniformly distributed

live load (owing to traffic loading). The dead load uncertainty arises out of

uncertainty in mass density (γ). The cable-stayed bridge is subjected to uncertainties

in material property (α), load (β) and mass density (γ) respectively.

Varies combinations of α and β ranging from total certainty (α=β=1) to total

uncertainty (α=β=0) are considered at specified values of γ. The fuzzy finite element

model of the structure that incorporates the simultaneous variation of these

uncertainties is used to obtain fuzzy interval values of displacement and rotations.

These results post-processed to obtain fuzzy interval values of axial force, shear force

and bending moment at various nodes at specified values of α, β and γ.


cumulative effect of uncertainties associated with material, load and mass density (α,

β, γ respectively). Thus, the cumulative effect of the simultaneous variation of

material and load uncertainties (α and β) on displacements and forces is investigated

61

at a specific level of mass uncertainty γ and the results are shown in the form of tables

and figures (for membership functions). The tables and the membership functions

represent the combined effect of material, live load and mass uncertainties on the

values of the displacements (horizontal, vertical and rotation), and forces (axial and

shear forces and bending moments).

3.5 Cumulative effect of multiple uncertainties α, β and γ on displacements and forces

Table 3.2 and Table 3.3 present the sample output of the case studies

explained in section 3.1, section 3.2 and section 3.3 for various combinations of α,β

and γ respectively. Table 3.2 represents the variation of uncertain rotation (×10-3

radians) at node 4 for various combinations of material and mass density uncertainties

(α and γ). The value of the rotation shown at the top left corner of the table

corresponds to the combination α=γ=1. The value of the rotation shown at the bottom

right corner of the table corresponds to the combination α=γ=0. The uncertain values

of rotation in the first row of the table correspond to the presence of mass density

uncertainty alone (0≤γ≤1 and α=1). The uncertain values of rotation in the first

column of the table correspond only to the presence of material uncertainty (0≤α≤1

and γ=1).

Table 3.3 represents the variation of uncertain axial force in deck (kN) in

element 12 for various combinations of material and live load uncertainties (α and β).

The value of the axial force shown at the top left corner of the table corresponds to the

combination α=β=1. The value of the axial force shown at the bottom right corner of

the table corresponds to the combination α=β=0. The uncertain values of axial force

in the first row of the table correspond to the presence of live load uncertainty

62

(0≤β≤1 and α=1), while the uncertain values of axial force in the first column of the

table correspond to the presence of material uncertainty alone (0≤α≤1 and β=1).

Fig. 3.5 represents the membership function for variation of vertical

displacement at node 3 at beta=0.8 at various levels of α (0≤α≤1) in steps of 0.2.

Fig. 3.6 represents the membership function for axial force (kN) in cable 3 (at node 2)

at α=0.6 at various levels of β (0≤β≤1) in steps of 0.2. These results show that the

membership functions are trapezoidal in the presence of multiple uncertainties.

The detailed discussion of the results obtained for case studies 1,2 and 3 are

presented in sections 4.1, 4.2 and 4.3 respectively.

3.6 Sensitivity analysis

Sensitivity analysis is performed in order to ascertain the following:

1. Sensitivity of structural response at a given node. Sensitivity of horizontal and

vertical displacements and rotations together with axial force, shear force and

bending moment to variation of material property (E) is evaluated.

2. Sensitivity of a given structural response across a set of nodes is evaluated.

Fig. 3.7 depicts the sensitivity analysis of displacements of the deck at node 3

under the action of dead load at γ=1.0 with respect to percentage variation of material

property (E) about its mean value. Fig. 3.8 depicts the sensitivity analysis of forces

and moments of the deck at node 5 under the action of dead load at γ=1.0 with respect

to percentage variation of material property (E) about its mean value.

The results and the detailed discussion of this investigation are presented and

discussed in section 4.4.

63

3.7 Summary

The contents of the current chapter are summarised as follows:

a) The methodology developed in chapter 2, after being validated by way of

preliminary examples (Appendix C), is applied to the study of a cable-stayed

bridge.

b) Three case studies are considered, involving combinations of α,β and γ.

Results are tabulated to show the simultaneous variation of the response

quantities to α,β or γ as the case may be.

c) Finally sensitivity analysis is performed to evaluate the relative sensitivity of

response quantities.

64

Fig. 3.1 Membership Function for Material Uncertainty

0.950

0.960

0.970

0.980

0.990

1.0001.000

1.010

1.020

1.030

1.040

1.0500

0.2

0.4

0.6

0.8

1

0.950 0.960 0.970 0.980 0.990 1.000 1.010 1.020 1.030 1.040 1.050Normalised Young's Modulus

alph

a

Fig. 3.2 Membership Function for Live Load Uncertainty

0

0.2

0.4

0.6

0.8

11

1.2

1.4

1.6

1.8

20

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Normalised load

beta

65

Fig. 3.3 Membership Function for Mass Uncertainty

0.950

0.960

0.970

0.980

0.990

1.0001.000

1.010

1.020

1.030

1.040

1.0500

0.2

0.4

0.6

0.8

1

0.950 0.960 0.970 0.980 0.990 1.000 1.010 1.020 1.030 1.040 1.050Normalised mass density

gam

ma

Fig. 3.4 Cable Stayed Bridge – Fan Configuration

7 3 1

2 4 6 8 5

66

Fig. 3.5 Cable stayed bridge- Membership Function for vertical displacement at node 3 at beta=0.8

-3.263E-02

-3.164E-02

-3.066E-02

-2.968E-02

-2.871E-02

-2.775E-02 -1.730E-03

-8.500E-04

3.000E-05

9.100E-04

1.790E-03

2.670E-030

0.2

0.4

0.6

0.8

1

-3.50E-02 -3.00E-02 -2.50E-02 -2.00E-02 -1.50E-02 -1.00E-02 -5.00E-03 0.00E+00 5.00E-03Vertical Displacement (m)

alph

a

Fig. 3.6 Cable stayed Bridge - Membership Function for Axial Force in cale 3 at alpha=0.6

-30.58

-0.45

29.68

59.81

89.94

139.54

172.29

205.05

237.80

270.55

303.31

120.06

0

0.2

0.4

0.6

0.8

1

-50.00 0.00 50.00 100.00 150.00 200.00 250.00 300.00Axial Force (kN)

beta

67

Fig. 3.7 Sensitivity Analysis for displacements (for dead load) at node 3 at gamma=1.0

0.000

3.021

6.054

9.122

12.214

15.330

0.000

6.089

12.148

18.237

24.296

30.355

0.000

5.502

11.084

16.667

22.290

27.953

0.000

5.000

10.000

15.000

20.000

25.000

30.000

35.000

0.0 1.0 2.0 3.0 4.0 5.0

Percentage Variation of E about mean

Perc

enta

ge V

aria

tion

of d

ispl

acem

ent a

bout

mea

n

Horizontal DisplacementVertical displacementRotation

Fig 3.8 Sensitivity Analysis of Forces and Moments in Element 12 (at node 5) at gamma=1.0

0.00

3.05

6.03

8.93

11.75

14.51

0.00

2.07

4.17

6.29

8.45

10.64

0.00

3.34

6.65

9.92

13.17

16.40

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

0.0 1.0 2.0 3.0 4.0 5.0Percentage Variation of E about mean

Perc

enta

ge V

aria

tion

of F

orce

/mom

ent a

bout

mea

n

Axial ForceShear Force Bending Moment

68

69

Table. 3.2 Cable-Stayed Bridge – Concomitant Variation of Rotation at node 4 (×10-3 radians) w.r.t α and γ

γ→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [0.846,0.846] [0.766,0.925] [0.686,1.005] [0.606,1.085] [0.526,1.165] [0.447,1.244]

0.8 [0.731,0.960] [0.652,1.040] [0.572,1.120] [0.492,1.200] [0.413,1.280] [0.333,1.360]

0.6 [0.617,1.075] [0.537,1.155] [0.458,1.235] [0.378,1.315] [0.299,1.396] [0.219,1.476]

0.4 [0.503,1.190] [0.423,1.270] [0.344,1.351] [0.264,1.431] [0.185,1.512] [0.106,1.592]

0.2 [0.388,1.305] [0.309,1.386] [0.230,1.467] [0.151,1.548] [0.071,1.628] [-0.008,1.709]

0.0 [0.274,1.421] [0.195,1.502] [0.116,1.583] [0.036,1.664] [-0.043,1.745] [-0.122,1.826]

Table. 3.3 Cable-Stayed Bridge – Concomitant Variation of Axial Force in deck in element 12 (kN) w.r.t α and β

β→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [-128.14,-128.14] [-153.76,-102.51] [-179.39,-76.88] [-205.02,-51.25] [-230.65,-25.63] [-256.27,0.00]

0.8 [-132.15,-124.23] [-158.58,-99.38] [-185.01,-74.54] [-211.44,-49.69] [-237.87,-24.85] [-264.30,0.00]

0.6 [-136.27,-120.42] [-163.53,-96.34] [-190.78,-72.25] [-218.04,-48.17] [-245.29,-24.08] [-272.55,0.00]

0.4 [-140.51,-116.71] [-168.61,-93.37] [-196.72,-70.03] [-224.82,-46.68] [-252.92,-23.34] [-281.02,0.00]

0.2 [-144.87,-113.09] [-173.84,-90.47] [-202.82,-67.85] [-231.79,-45.24] [-260.76,-22.62] [-289.74,0.00]

0.0 [-149.35,-109.55] [-179.22,-87.64] [-209.09,-65.73] [-238.97,-43.82] [-268.84,-21.91] [-298.71,0.00]

Chapter 4

Results and Discussion

Chapter 4

4. RESULTS AND DISCUSSION

The present study involves the study of the cumulative effect of uncertainties

in material property (E), live load and mass density (α,β and γ) on the structural

response of the cable-stayed bridge. These uncertainties are allowed to vary

independent of each other in the range 0 through 1. The modified methodology

presented in chapter 2 is applied to EBE model of the structure in order to obtain the

structural response. The structural response is analysed in terms of fuzzy values of

displacements and rotations at various levels of uncertainty. The horizontal, vertical

displacements and rotations thus obtained are further post-processed using the

methodology described in chapter 2 and fuzzy values of axial forces and shear forces

and bending moments are evaluated. The results obtained are tabulated and

membership functions for the structural response quantities are constructed for

various combinations of α,β and γ. Sensitivity analysis is performed in order to assess

the relative variation of structural response to the corresponding change in the values

of material, load and mass density uncertainties.

4.1. Case Study 1 – Cable-Stayed Bridge with Uncertainties of Material


The Cable-stayed Bridge shown in Fig. 3.4 is subjected to a uniformly

distributed dead load (owing to the self-weight). Uncertainty is introduced in the

Young’s modulus of the materials (steel and concrete) and also in mass density.

The structural response of the cable-stayed bridge is evaluated for various

combinations of α and γ ranging from total certainty (α=1, γ=1) to total uncertainty

(α=0, γ=0). The results are presented in Table 4.1.1 through Table 4.1.10. The

following common characteristics can be observed in all the tables:

71

a) The values of structural response in the first row correspond respectively

to the presence of mass density uncertainty alone (0≤γ≤1 and α=1) and

first column corresponds to the presence of material uncertainty alone

(0≤α≤1 and γ=1). It is observed that the width of the interval increases

along and across each table.

b) The structural response (horizontal or vertical displacement or rotation)

shown at the top left corner of the tables (corresponding to the

combination α=γ=1) is represented by a normalised interval [1.0,1.0] or

[-1.0,-10]. An interval [a, b] is normalised by dividing its lower and upper

bounds by its mid-value 0.5*(a+b). In general, a normalised interval

[1-ε1, 1+ε2] indicates that lower bound and upper bound variations of the

structural response about its nominal (mean) value are ε1 and ε2

respectively.

c) Fuzzy membership functions (a), (b),(c) and (d) are constructed from

intervals to represent the uncertain structural response for various

combinations of α and γ, row-wise and column-wise in all the cases as

described below.

4.1.1 Effect of concomitant variation of α and γ on displacements and rotations.

Table 4.1.1, Table 4.1.2 and Table 4.1.3 show the concomitant variation of

horizontal and vertical displacements and rotations at nodes 2,3 and 4 respectively, for

various combinations of α and γ.

72

4.1.1.1 Uncertain horizontal displacement at node 2

Fig. 4.1.1 (a) and (b) represent the membership functions for horizontal

displacement at node 2 (m) at γ=1.0 and γ=0.8. The uncertain horizontal

displacements at α=0.0 and α=1.0 correspond to normalised values [0.884,1.121],

[1.0,1.0] in Fig. 4.1.1(a) and [0.868,1.137], [0.984,1.016] in Fig. 4.1.1 (b). Fig. 4.1.1

(c) and (d) represent the membership functions for horizontal displacement at node 2

(m) at α=1.0 and α=0.6. The uncertain horizontal displacements at γ=0.0 and γ=1.0

correspond to normalised values [0.919,1.081], [1.0,1.0] in Fig. 4.1.1(c) and

[0.873,1.129], [0.953,1.047] in Fig. 4.1.1 (d).

4.1.1.2 Uncertain vertical displacement at node 3

Fig. 4.1.2 (a) and (b) represent the membership functions for vertical

displacement at node 3 (m) at γ=1.0 and γ=0.8. In these figures, the uncertain vertical

displacements at α=0.0 and α=1.0 correspond to normalised values [-1.309,-0.696] ,

[-1.0,-1.0] in Fig. 4.1.2 (a) and [-1.353,-0.653] , [-1.044,-0.957] in Fig. 4.1.2(b). Fig.

4.1.2 (c) and (d) represent the membership functions for vertical displacement at node

3 (m) at α=1.0 and α=0.6. The uncertain vertical displacements at γ=0.0 and γ=1.0

correspond to normalised values [-1.218,-0.782] ,[-1.000,-1.000] in Fig. 4.1.2 (c) and

[-1.342,-0.662] , [-1.123,-0.879] in Fig. 4.1.2 (d).

4.1.1.3 Uncertain rotation at node 4

Fig. 4.1.3 (a) and (b) represent the membership functions for rotation (radians)

at node 4 at γ=1.0 and γ=0.8. The uncertain rotations at α=0.0 and α=1.0 correspond

to normalised values [0.324,1.680], [1.0,1.0] in Fig. 4.1.3 (a) and [0.230,1.775],

[0.905,1.093] in Fig. 4.1.3 (b). Fig. 4.1.3 (c) and (d) represent the membership

73

functions for rotation at node 4 at α=1.0 and α=0.6. The uncertain rotations at γ=0.0

and γ=1.0 correspond to normalised values [0.528,1.470], [1.000,1.000] in Fig. 4.1.3

(c) and [0.259,1.745] , [0.729,1.271] in Fig. 4.1.3 (d).

The normalised values of uncertain horizontal displacement in the first row of

the Table 4.1.1 correspond to [1.0,1.0], [0.984,1.016], [0.967,1.032], [0.951,1.049],

[0.935,1.065] and [0.919,1.081]. Similarly, the normalised values in the first column

of the Table 4.1.1 correspond to [1.000,1.000], [0.976,1.024], [0.953,1.047],

[0.930,1.072], [0.907,1.096], [0.884,1.121] respectively. From the above values, it is

observed that the variation of horizontal displacement about its nominal value is more

in the case of material uncertainty (12.1%) compared to mass density uncertainty

(8.1%). In the case of vertical displacement (Table 4.1.2) the corresponding variations

about its nominal value are 30.9% for material uncertainty and 21.8% for mass

uncertainty. In the case of rotation (Table 4.1.3), the corresponding variations about

its nominal value are 68.0% for material uncertainty and 47.0% for mass uncertainty.

4.1.2 Effect of concomitant variation of α and γ on shear forces and bending

moments

Table 4.1.4 through Table 4.1.10 show the concomitant variation of axial

forces, shear forces and bending moments for various combinations of α and γ. Table

4.1.4 and Table 4.1.5 represent the variation of axial force (kN) in element 9 and

element 12 respectively. Table 4.1.6 represents the variation of shear force in the

deck just to the left of node 3. Table 4.1.7 represents the variation of bending moment

in the deck at node 4. Table 4.1.8 represents the variation of axial force (kN) in pylon

in element 17 (at node 2). Table 4.1.9 and Table 4.1.10 represent the variation of

74

axial force (kN) in cable 2 (at node 2) and cable 3 (at node 3) respectively. In the

above tables, tensile force and sagging moment are considered positive.

From the normalisation of interval values of axial force in Table 4.1.4, it is

observed that variation of axial force in deck is more in the case of material

uncertainty (upper bound 20.3%) compared to mass density uncertainty (upper

bound 8.1%). Similar behaviour is observed from all the other uncertain

forces/moments presented in Table 4.1.5 through Table 4.1.10.

4.1.2.1 Axial Force in deck in element 9

Fig. 4.1.4 (a) and (b) represent the membership functions for axial force in

deck in element 9 (kN) at γ=1.0 and γ=0.8. The uncertain axial forces at α=0.0 and

α=1.0 correspond to normalised values [0.811,1.203], [1.000,1.000] in Fig. 4.1.4 (a)

and [0.795,1.221], [0.984,1.016] in Fig. 4.1.4 (b). Fig. 4.1.4 (c) and (d) represent the

membership functions for axial force in deck in element 9 (kN) at α=1.0 and α=0.6.

The uncertain axial forces at γ=0.0 and γ=1.0 correspond to normalised values

[0.919,1.081], [1.0,1.0] in Fig. 4.1.4 (c) and [0.844,1.162], [0.923,1.079] in

Fig. 4.1.4(d).

4.1.2.2 Axial force (kN) in deck in element 12

Fig. 4.1.5 (a) and (b) represent the membership functions for axial force (kN)

in deck in element 12 at γ=1.0 and γ=0.8. The uncertain axial forces at α=0.0 and

α=1.0 correspond to normalised values [-1.166,-0.855], [-1.0,-1.0] in Fig. 4.1.5(a)

and [-1.177,-0.846], [-1.01,-0.99] in Fig. 4.1.5 (b). Fig. 4.1.5 (c) and (d) represent the

membership functions for axial force in deck in element 9 at α=1.0 and α=0.6. The

uncertain axial forces at γ=0.0 and γ=1.0 correspond to normalised values

75

[-1.05,-0.95], [-1.0,-1.0] in Fig. 4.1.5(c) and [-1.117,-0.89], [-1.06,-0.94] in

Fig. 4.1.5 (d).

4.1.2.3 Shear force (kN) in deck just to the left of node 3

Fig. 4.1.6 (a) and (b) represent the membership functions for shear force just

to the left of node 3 (kN) at γ=1.0 and γ=0.8. The uncertain shear forces at α=0.0 and

α=1.0 correspond to normalised values [-1.104,-0.903] , [-1.00,-1.00] in Fig. 4.1.6 (a)

and [-1.116,-0.891] , [-1.012,-0.988] in Fig. 4.1.6 (b). Fig. 4.1.6 (c) and (d) represent

the membership functions for shear force in deck in element 9 (kN) at α=1.0 and

α=0.6. The uncertain shear forces at γ=0.0 and γ=1.0 correspond to normalised values

[-1.062,-0.938], [-1.000,-1.000] in Fig. 4.1.6 (c) and [-1.103, -0.9] , [-1.041,-0.961] in

Fig. 4.1.6 (d).

4.1.2.4 Bending moment (kNm) in deck at node 4

Fig. 4.1.7 (a) and (b) represent the membership functions for bending moment

(kNm) in deck at node 4 at γ=1.0 and γ=0.8. The uncertain bending moments at

α=0.0 and α=1.0 correspond to normalised values [-1.189,-0.818],

[-1.000,-1.000] in Fig. 4.1.7 (a) and [-1.206,-0.802] , [-1.016,-0.984] in Fig. 4.1.7 (b).

Fig. 4.1.7 (c) and (d) represent the membership functions for bending moment at node

4 at α=1.0 and α=0.6. The uncertain bending moments at γ=0.0 and γ=1.0 correspond

to normalised values [-1.081,-0.919], [-1.000,-1.000] in Fig. 4.1.7 (c) and

[-1.157,-0.847] , [-1.075,-0.927] in Fig. 4.1.7 (d).

76

4.1.2.5 Axial force (kN) in pylon in element 17 (at node 11)

Fig. 4.1.8 (a) and (b) represent the membership functions axial force (kN) in

pylon in element 17 (at node 11) at γ=1.0 and γ=0.8. The uncertain axial forces at

α=0.0 and α=1.0 correspond to normalised values [-1.115,-0.900] , [-1.000,-1.000] in

Fig. 4.1.8 (a) and [-1.126,-0.891] , [-1.010,-0.990] in Fig. 4.1.8 (b) . Fig. 4.1.7 (c) and

(d) represent the membership functions for axial force in deck in element 9 (kN) at

α=1.0 and α=0.8. The uncertain axial forces at γ=0.0 and γ=1.0 correspond to

normalised values [-1.050,-0.950] , [-1.00,-1.00] in Fig. 4.1.8 (c) and

[-1.096,-0.910] , [-1.044,-0.958] in Fig. 4.1.8 (d).

4.1.2.6 Axial force (kN) in cable 2 (at node 2)


cable 2 (at node 2) at γ=1.0 and γ=0.8. The uncertain axial forces at α=0.0 and α=1.0

correspond to normalised values [0.823,1.195] and [1.000,1.000] Fig. 4.1.9 (a) and

[0.812,1.208], [0.988,1.012] in Fig. 4.1.9 (b). Fig. 4.1.9 (c) and (d) represent the

membership functions for axial force in cable 2 (at node 2) at α=1.0 and α=0.6. The

uncertain axial forces at γ=0.0 and γ=1.0 correspond to normalised values [0.94,1.06] ,

[1.0,1.0] in Fig. 4.1.9 (c) and [0.87,1.138] , [0.928,1.075] in Fig. 4.1.9 (d) .



cable 3 (at node 3) at γ=1.0 and γ=0.8. The uncertain axial forces at α=0.0 and α=1.0

correspond to normalised values [0.822,1.196], [1.0,1.0] in Fig. 4.1.10 (a) and

[0.812,1.209], [0.989,1.011] in Fig. 4.1.10 (b). Fig. 4.1.10 (c) and (d) represent the

membership functions for axial force in cable 3 at α=1.0 and α=0.6. The uncertain

77

axial forces at γ=0.0 and γ=1.0 correspond to normalised values [0.944,1.056],

[1.0,1.0] in Fig. 4.1.10 (c) and [0.874,1.135], [0.927,1.076] in Fig. 4.1.10 (d).

From all the above results, it is observed that membership functions

represented by figures (a) and (c) of Fig. 4.1.1 through Fig. 4.1.10 are triangular

because of the presence of a single uncertainty (α or γ).

Also, it is observed that membership functions represented by figures (c) and

(d) of Fig. 4.1.1 through Fig. 4.1.10 are trapezoidal in the presence of multiple

uncertainties (α and γ).

Further, the variation with reference to material uncertainty is found to be

more in comparison with the variation with reference to mass uncertainty. This is

because the material uncertainty causes uncertainty in the stiffness matrix [Kα] in Eq.

2.3 where as uncertainty of mass density causes uncertainty in the right hand side

vector {Pγ} of the same equation.

78

Table. 4.1.1 Cable stayed Bridge - Concomitant Variation of Horizontal displacement of node 2 (×10-4 metres) w.r.t α and γ

γ→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [7.169,7.169] [7.052,7.285] [6.936,7.401] [6.820,7.517] [6.704,7.633] [6.588,7.749]

0.8 [7.000,7.338] [6.885,7.455] [6.769,7.572] [6.654,7.689] [6.538,7.805] [6.422,7.922]

0.6 [6.833,7.509] [6.718,7.627] [6.603,7.744] [6.488,7.862] [6.373,7.979] [6.258,8.097]

0.4 [6.668,7.682] [6.553,7.801] [6.438,7.919] [6.324,8.037] [6.209,8.155] [6.094,8.273]

0.2 [6.503,7.857] [6.389,7.976] [6.275,8.095] [6.161,8.213] [6.047,8.332] [5.932,8.451]

0.0 [6.340,8.034] [6.226,8.153] [6.112,8.273] [5.999,8.392] [5.885,8.512] [5.771,8.631]

Table. 4.1.2 Cable stayed Bridge - Concomitant Variation of Vertical displacement of node 3 (×10-2 metres) w.r.t α and γ γ→

α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [-3.268, -3.268] [-3.411, -3.126] [-3.553, -2.984] [-3.696, -2.841] [-3.838, -2.699] [-3.980, -2.556]

0.8 [-3.468, -3.069] [-3.611, -2.927] [-3.754, -2.785] [-3.897, -2.643] [-4.040, -2.501] [-4.183, -2.359]

0.6 [-3.669, -2.871] [-3.812, -2.729] [-3.956, -2.587] [-4.099, -2.445] [-4.243, -2.304] [-4.386, -2.162]

0.4 [-3.871, -2.672] [-4.015, -2.531] [-4.159, -2.389] [-4.303, -2.248] [-4.447, -2.107] [-4.590, -1.965]

0.2 [-4.073, -2.474] [-4.218, -2.333] [-4.362, -2.192] [-4.507, -2.051] [-4.652, -1.910] [-4.796, -1.768]

0.0 [-4.277, -2.276] [-4.422, -2.135] [-4.567, -1.995] [-4.713, -1.854] [-4.858, -1.713] [-5.002, -1.572]

Table. 4.1.3 Cable stayed Bridge - Concomitant Variation of Rotation at node 4 (×10-3 radians) w.r.t α and γ

γ→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [0.846,0.846] [0.766,0.925] [0.686,1.005] [0.606,1.085] [0.526,1.165] [0.447,1.244]

0.8 [0.731,0.960] [0.652,1.040] [0.572,1.120] [0.492,1.200] [0.413,1.280] [0.333,1.360]

0.6 [0.617,1.075] [0.537,1.155] [0.458,1.235] [0.378,1.315] [0.299,1.396] [0.219,1.476]

0.4 [0.503,1.190] [0.423,1.270] [0.344,1.351] [0.264,1.431] [0.185,1.512] [0.106,1.592]

0.2 [0.388,1.305] [0.309,1.386] [0.230,1.467] [0.151,1.548] [0.071,1.628] [-0.008,1.709]

0.0 [0.274,1.421] [0.195,1.502] [0.116,1.583] [0.036,1.664] [-0.043,1.745] [-0.122,1.826]

79

Table. 4.1.4 Cable stayed Bridge – Concomitant Variation of Axial Force in deck in element 9 (kN) w.r.t α and γ

γ→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [385.17,385.17] [378.93,391.41] [372.69,397.65] [366.45,403.90] [360.21,410.14] [353.96,416.37]

0.8 [370.23,400.33] [364.06,406.65] [357.88,412.96] [351.71,419.27] [345.54,425.59] [339.36,431.89]

0.6 [355.48,415.73] [349.37,422.12] [343.26,428.51] [337.15,434.90] [331.04,441.29] [324.93,447.67]

0.4 [340.92,431.38] [334.87,437.85] [328.82,444.32] [322.77,450.79] [316.72,457.26] [310.67,463.73]

0.2 [326.52,447.30] [320.53,453.86] [314.53,460.41] [308.54,466.97] [302.54,473.53] [296.54,480.08]

0.0 [312.27,463.52] [306.33,470.17] [300.38,476.81] [294.44,483.46] [288.50,490.10] [282.55,496.74]

Table. 4.1.5 Cable stayed Bridge – Concomitant Variation of Axial Force in deck in element 12 (kN) w.r.t α and γ

γ→ α↓

1.0 0.8 0.6 0.4 0.2 0.0

1.0 [-278.72,-278.72] [-281.51,-275.93] [-284.30,-273.15] [-287.09,-270.36] [-289.88,-267.57] [-292.67,-264.78]

0.8 [-287.46,-270.22] [-290.34,-267.51] [-293.21,-264.81] [-296.09,-262.11] [-298.96,-259.40] [-301.84,-256.69]

0.6 [-296.43,-261.93] [-299.40,-259.31] [-302.36,-256.69] [-305.33,-254.06] [-308.30,-251.44] [-311.26,-248.82]

0.4 [-305.66,-253.85] [-308.71,-251.31] [-311.77,-248.77] [-314.83,-246.23] [-317.89,-243.69] [-320.94,-241.14]

0.2 [-315.14,-245.97] [-318.29,-243.50] [-321.45,-241.04] [-324.60,-238.58] [-327.75,-236.12] [-330.90,-233.66]

0.0 [-324.90,-238.28] [-328.15,-235.89] [-331.40,-233.51] [-334.65,-231.12] [-337.90,-228.74] [-341.15,-226.35]

Table. 4.1.6 Cable stayed Bridge – Concomitant Variation of Shear Force in deck just to the left of node 3 (kN) w.r.t α and γ

γ→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [-66.2,-66.2] [-67.0,-65.4] [-67.8,-64.6] [-68.6,-63.8] [-69.5,-63.0] [-70.3,-62.1]

0.8 [-67.5,-64.9] [-68.4,-64.1] [-69.2,-63.3] [-70.0,-62.5] [-70.8,-61.7] [-71.6,-60.9]

0.6 [-68.9,-63.6] [-69.7,-62.8] [-70.6,-62.0] [-71.4,-61.2] [-72.2,-60.4] [-73.0,-59.6]

0.4 [-70.3,-62.3] [-71.1,-61.5] [-71.9,-60.7] [-72.8,-59.9] [-73.6,-59.2] [-74.5,-58.4]

0.2 [-71.7,-61.0] [-72.5,-60.3] [-73.4,-59.5] [-74.2,-58.7] [-75.1,-57.9] [-75.9,-57.1]

0.0 [-73.1,-59.8] [-73.9,-59.0] [-74.8,-58.2] [-75.7,-57.5] [-76.5,-56.7] [-77.4,-55.9]

80

Table. 4.1.7 Cable stayed Bridge – Concomitant Variation of Bending Moment in deck at node 4 (kNm) w.r.t α and γ

γ→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [-172.2,-172.2] [-175.0,-169.4] [-177.8,-166.7] [-180.5,-163.9] [-183.3,-161.1] [-186.1,-158.3]

0.8 [-178.6,-165.9] [-181.4,-163.1] [-184.2,-160.4] [-187.0,-157.6] [-189.8,-154.9] [-192.6,-152.1]

0.6 [-185.1,-159.6] [-187.9,-156.8] [-190.7,-154.1] [-193.5,-151.4] [-196.4,-148.6] [-199.2,-145.9]

0.4 [-191.6,-153.3] [-194.4,-150.6] [-197.3,-147.9] [-200.1,-145.2] [-203.0,-142.4] [-205.8,-139.7]

0.2 [-198.2,-147.0] [-201.0,-144.3] [-203.9,-141.7] [-206.8,-139.0] [-209.7,-136.3] [-212.6,-133.6]

0.0 [-204.8,-140.8] [-207.7,-138.1] [-210.6,-135.5] [-213.5,-132.8] [-216.5,-130.1] [-219.4,-127.4]

Table. 4.1.8 Cable stayed Bridge – Concomitant Variation of Axial Force in pylon in element 17 (at node 11) (kN) w.r.t α and γ

γ→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [-901.8,-901.8] [-910.9,-892.8] [-919.9,-883.8] [-928.9,-874.7] [-937.9,-865.7] [-947.0,-856.6]

0.8 [-921.4,-882.8] [-930.6,-874.0] [-939.8,-865.1] [-949.1,-856.3] [-958.3,-847.4] [-967.5,-838.6]

0.6 [-941.5,-864.3] [-951.0,-855.7] [-960.4,-847.0] [-969.8,-838.4] [-979.3,-829.7] [-988.7,-821.0]

0.4 [-962.3,-846.4] [-971.9,-837.9] [-981.6,-829.4] [-991.2,-820.9] [-1000.9,-812.5] [-1010.5,-804.0]

0.2 [-983.7,-828.9] [-993.6,-820.6] [-1003.4,-812.3] [-1013.3,-804.0] [-1023.1,-795.7] [-1032.9,-787.4]

0.0 [-1005.8,-811.9] [-1015.8,-803.8] [-1025.9,-795.7] [-1036.0,-787.5] [-1046.0,-779.4] [-1056.1,-771.2]

81

82

Table. 4.1.9 Cable stayed Bridge – Concomitant Variation of Axial Force (kN) in cable 2 (at node 2) w.r.t α and γ

γ→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [286.1,286.1] [282.7,289.6] [279.2,293.0] [275.8,296.5] [272.4,299.9] [268.9,303.3]

0.8 [275.7,296.8] [272.3,300.3] [268.9,303.9] [265.5,307.4] [262.2,310.9] [258.8,314.4]

0.6 [265.4,307.7] [262.1,311.3] [258.8,314.9] [255.4,318.5] [252.1,322.1] [248.8,325.7]

0.4 [255.3,318.8] [252.0,322.5] [248.8,326.2] [245.5,329.9] [242.3,333.5] [239.0,337.2]

0.2 [245.3,330.2] [242.1,333.9] [238.9,337.7] [235.8,341.5] [232.6,345.2] [229.4,348.9]

0.0 [235.5,341.8] [232.4,345.6] [229.3,349.5] [226.1,353.3] [223.0,357.2] [219.9,361.0]

Table. 4.1.10 Cable stayed Bridge – Concomitant Variation of Axial Force (kN) in cable 3 (at node 3) w.r.t α and γ

γ→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [343.9,343.9] [340.0,347.7] [336.1,351.6] [332.3,355.4] [328.4,359.3] [324.5,363.1]

0.8 [331.2,356.8] [327.4,360.7] [323.7,364.7] [319.9,368.6] [316.1,372.6] [312.4,376.5]

0.6 [318.8,370.0] [315.1,374.0] [311.4,378.1] [307.8,382.1] [304.1,386.2] [300.4,390.2]

0.4 [306.6,383.4] [303.0,387.6] [299.4,391.8] [295.8,395.9] [292.2,400.1] [288.6,404.2]

0.2 [294.6,397.2] [291.1,401.5] [287.6,405.7] [284.1,410.0] [280.6,414.3] [277.1,418.5]

0.0 [282.8,411.4] [279.4,415.7] [276.0,420.1] [272.6,424.5] [269.1,428.9] [265.7,433.2]

83

(b) gamma=0.8

6.226E-04

6.389E-04

6.553E-04

6.718E-04

6.885E-04

7.285E-04

7.455E-04

7.627E-04

7.801E-04

7.976E-04

8.153E-04

7.052E-04

0

0.2

0.4

0.6

0.8

1

6.2E-04 6.4E-04 6.6E-04 6.8E-04 7.0E-04 7.2E-04 7.4E-04 7.6E-04 7.8E-04 8.0E-04 8.2E-04Horizontal Displacement(m)

alph

a

(a) gamma=1.0

6.340E-04

6.503E-04

6.668E-04

6.833E-04

7.000E-04

7.169E-047.169E-04

7.338E-04

7.509E-04

7.682E-04

7.857E-04

8.034E-040

0.2

0.4

0.6

0.8

1

6.34E-04 6.54E-04 6.74E-04 6.94E-04 7.14E-04 7.34E-04 7.54E-04 7.74E-04 7.94E-04 8.14E-04Horizontal Displacement(m)

alph

a

(c) alpha=1.0

6.588E-04

6.704E-04

6.820E-04

6.936E-04

7.052E-04

7.169E-047.169E-04

7.285E-04

7.401E-04

7.517E-04

7.633E-04

7.749E-040

0.2

0.4

0.6

0.8

1

6.6E-04 6.8E-04 7.0E-04 7.2E-04 7.4E-04 7.6E-04Horizontal Displacement (m)

gam

ma

(d) alpha=0.6

6.258E-04

6.373E-04

6.488E-04

6.603E-04

6.718E-04

7.509E-04

7.627E-04

7.744E-04

7.862E-04

7.979E-04

8.097E-04

6.833E-04

0

0.2

0.4

0.6

0.8

1

6.250E-04 6.500E-04 6.750E-04 7.000E-04 7.250E-04 7.500E-04 7.750E-04 8.000E-04 8.250E-04Horizontal Displacement (m)

gam

ma

Fig. 4.1.1 Membership Functions of Horizontal Displacement at node 2

(a) gamma=1.0

-4.277E-02

-4.073E-02

-3.871E-02

-3.669E-02

-3.468E-02

-3.268E-02-3.268E-02

-3.069E-02

-2.871E-02

-2.672E-02

-2.474E-02

-2.276E-020

0.2

0.4

0.6

0.8

1

-4.50E-02 -4.25E-02 -4.00E-02 -3.75E-02 -3.50E-02 -3.25E-02 -3.00E-02 -2.75E-02 -2.50E-02 -2.25E-02 -2.00E-02

Vertical Displacement (m)

alph

a

gamma=0.8

-4.422E-02

-4.218E-02

-4.015E-02

-3.812E-02

-3.611E-02

-3.126E-02

-2.927E-02

-2.729E-02

-2.531E-02

-2.333E-02

-2.135E-02

-3.411E-02

0

0.2

0.4

0.6

0.8

1

-4.50E-02 -4.25E-02 -4.00E-02 -3.75E-02 -3.50E-02 -3.25E-02 -3.00E-02 -2.75E-02 -2.50E-02 -2.25E-02 -2.00E-02Vertical Displacement (m)

alph

a

(c) alpha=1.0

-3.980E-02

-3.838E-02

-3.696E-02

-3.553E-02

-3.411E-02

-3.268E-02-3.268E-02

-3.126E-02

-2.984E-02

-2.841E-02

-2.699E-02

-2.556E-020

0.2

0.4

0.6

0.8

1

-4.000E-02 -3.800E-02 -3.600E-02 -3.400E-02 -3.200E-02 -3.000E-02 -2.800E-02 -2.600E-02Vertical Displacement (m)

gam

ma

(d) alpha=0.6

-4.386E-02

-4.243E-02

-4.099E-02

-3.956E-02

-3.812E-02

-2.871E-02

-2.729E-02

-2.587E-02

-2.445E-02

-2.304E-02

-2.162E-02

-3.669E-02

0

0.2

0.4

0.6

0.8

1

-4.50E-02 -4.25E-02 -4.00E-02 -3.75E-02 -3.50E-02 -3.25E-02 -3.00E-02 -2.75E-02 -2.50E-02 -2.25E-02 -2.00E-02Vertical Displacement (m)

gam

ma

84

Fig. 4.1.2 Membership Functions of Vertical Displacement at node 3

85

(a) gamma=1.0

2.740E-04

3.884E-04

5.027E-04

6.170E-04

7.312E-04

8.455E-048.455E-04

9.600E-04

1.075E-03

1.190E-03

1.305E-03

1.421E-030

0.2

0.4

0.6

0.8

1

2.74E-04 4.74E-04 6.74E-04 8.74E-04 1.07E-03 1.27E-03Rotation(radians)

alph

a

(b) gamma=0.8

1.948E-04

3.091E-04

4.233E-04

5.374E-04

6.516E-04

9.253E-04

1.040E-03

1.155E-03

1.270E-03

1.386E-03

1.502E-03

7.657E-04

0

0.2

0.4

0.6

0.8

1

1.90E-04 3.90E-04 5.90E-04 7.90E-04 9.90E-04 1.19E-03 1.39E-03Rotation(radians)

alph

a

(c) alpha=1.0

4.465E-04

5.263E-04

6.061E-04

6.859E-04

7.657E-04

8.455E-048.455E-04

9.253E-04

1.005E-03

1.085E-03

1.165E-03

1.244E-030

0.2

0.4

0.6

0.8

1

4.40E-04 5.15E-04 5.90E-04 6.65E-04 7.40E-04 8.15E-04 8.90E-04 9.65E-04 1.04E-03 1.12E-03 1.19E-03Rotation(radians)

gam

ma

(d) alpha=0.6

2.192E-04

2.988E-04

3.784E-04

4.579E-04

5.374E-04

1.075E-03

1.155E-03

1.235E-03

1.315E-03

1.396E-03

1.476E-03

6.170E-04

0

0.2

0.4

0.6

0.8

1

2.00E-04 4.00E-04 6.00E-04 8.00E-04 1.00E-03 1.20E-03 1.40E-03Rotation(radians)

gam

ma

Fig. 4.1.3 Membership Functions of Rotation at node 4

86

(a) gamma=1.0

312.3

326.5

340.9

355.5

370.2

385.2385.2

400.3

415.7

431.4

447.3

463.50

0.2

0.4

0.6

0.8

1

310 330 350 370 390 410 430 450 470Axial Force (kN)

alph

a

(b) gamma=0.8

306.3

320.5

334.9

349.4

364.1

391.4

406.7

422.1

437.9

453.9

470.2

378.9

0

0.2

0.4

0.6

0.8

1

305 325 345 365 385 405 425 445 465Axial Force (kN)

alph

a

(c) alpha=1.0

353.97

360.21

366.45

372.70

378.94

385.18385.18

391.42

397.66

403.90

410.14

416.380

0.2

0.4

0.6

0.8

1

353 363 373 383 393 403 413Axial Force (kN)

gam

ma

(d) alpha=0.6

324.9

331.0

337.2

343.3

349.4

355.5 415.7

422.1

428.5

434.9

441.3

447.70

0.2

0.4

0.6

0.8

1

320 340 360 380 400 420 440 460Axial Force (kN)

gam

ma

Fig. 4.1.4 Membership Functions of Axial Force in deck in element 9

87

(a) gamma=1.0

-324.9

-315.1

-305.7

-296.4

-287.5

-278.7-278.7

-270.2

-261.9

-253.8

-246.0

-238.30

0.2

0.4

0.6

0.8

1

-325 -315 -305 -295 -285 -275 -265 -255 -245 -235Axial Force (kN)

alph

a

(b) gamma=0.8

-331.4

-321.4

-311.8

-302.4

-293.2

-273.1

-264.8

-256.7

-248.8

-241.0

-233.5

-284.3

0

0.2

0.4

0.6

0.8

1

-335 -320 -305 -290 -275 -260 -245 -230Axial Force (kN)

alph

a

(c) alpha=1.0

-292.7

-289.9

-287.1

-284.3

-281.5

-278.7-278.7

-275.9

-273.1

-270.4

-267.6

-264.80

0.2

0.4

0.6

0.8

1

-293.0 -288.0 -283.0 -278.0 -273.0 -268.0 -263.0Axial Force (kN)

gam

ma

(d) alpha=0.6

-311.3

-308.3

-305.3

-302.4

-299.4

-296.4 -261.9

-259.3

-256.7

-254.1

-251.4

-248.80

0.2

0.4

0.6

0.8

1

-315 -305 -295 -285 -275 -265 -255 -245Axial Force (kN)

gam

ma

Fig. 4.1.5 Membership Functions of Axial Force in deck in element 12

88

(a) gamma=1.0

-73.100

-71.700

-70.300

-68.900

-67.500

-66.200-66.200

-64.900

-63.600

-62.300

-61.000

-59.8000

0.2

0.4

0.6

0.8

1

-74.000 -72.000 -70.000 -68.000 -66.000 -64.000 -62.000 -60.000Shear Force (kN)

alph

a

(b) gamma=0.8

-73.9

-72.5

-71.1

-69.7

-68.4

-65.4

-64.1

-62.8

-61.5

-60.3

-59.0

-67.0

0

0.2

0.4

0.6

0.8

1

-74.000 -72.000 -70.000 -68.000 -66.000 -64.000 -62.000 -60.000Shear Force(kN)

alph

a

(c) alpha=1.0

-70.3

-69.5

-68.6

-67.8

-67.0

-66.2-66.2

-65.4

-64.6

-63.8

-63.0

-62.10

0.2

0.4

0.6

0.8

1

-70.5 -69.5 -68.5 -67.5 -66.5 -65.5 -64.5 -63.5 -62.5 -61.5Shear Force (kN)

gam

ma

(d) alpha=0.8

-73.0

-72.2

-71.4

-70.6

-69.7

-68.9 -63.6

-62.8

-62.0

-61.2

-60.4

-59.60

0.2

0.4

0.6

0.8

1

-73.0 -71.0 -69.0 -67.0 -65.0 -63.0 -61.0 -59.0Shear Force(kN)

gam

ma

Fig. 4.1.6 Membership Functions of Shear Force in deck to left of node 3

89

(a) gamma=1.0

-204.8

-198.2

-191.6

-185.1

-178.6

-172.2-172.2

-165.9

-159.6

-153.3

-147.0

-140.80

0.2

0.4

0.6

0.8

1

-205.00 -195.00 -185.00 -175.00 -165.00 -155.00 -145.00Bending Moment (kNm)

alph

a

(b) gamma=0.8

-207.7

-201.0

-194.4

-187.9

-181.4

-175.0 -169.4

-163.1

-156.8

-150.6

-144.3

-138.10

0.2

0.4

0.6

0.8

1

-210.0 -200.0 -190.0 -180.0 -170.0 -160.0 -150.0 -140.0 -130.0Bending Moment (kNm)

alph

a

(c) alpha=1.0

-186.1

-183.3

-180.5

-177.8

-175.0

-172.2-172.2

-169.4

-166.7

-163.9

-161.1

-158.30

0.2

0.4

0.6

0.8

1

-187 -182 -177 -172 -167 -162 -157Bending Moment (kNm)

gam

ma

(d) alpha=0.6

-199.2

-196.4

-193.5

-190.7

-187.9

-185.1 -159.6

-156.8

-154.1

-151.4

-148.6

-145.90

0.2

0.4

0.6

0.8

1

-200 -190 -180 -170 -160 -150Bending Moment (kNm)

gam

ma

Fig. 4.1.7 Membership Functions of Bending Moment in deck at node 4

90

(a) gamma=1.0

-1005.8

-983.7

-962.3

-941.5

-921.4

-901.8-901.8

-882.8

-864.3

-846.4

-828.9

-811.90

0.2

0.4

0.6

0.8

1

-1010.0 -990.0 -970.0 -950.0 -930.0 -910.0 -890.0 -870.0 -850.0 -830.0 -810.0Axial Force (kN)

alph

a

(b) gamma=0.8

-1015.8

-993.6

-971.9

-951.0

-930.6

-910.9 -892.8

-874.0

-855.7

-837.9

-820.6

-803.80

0.2

0.4

0.6

0.8

1

-1015.0 -990.0 -965.0 -940.0 -915.0 -890.0 -865.0 -840.0 -815.0Axial Force (kN)

alph

a

(c) alpha=1.0

-947.0

-937.9

-928.9

-919.9

-910.9

-901.8-901.8

-892.8

-883.8

-874.7

-865.7

-856.60

0.2

0.4

0.6

0.8

1

-950.0 -940.0 -930.0 -920.0 -910.0 -900.0 -890.0 -880.0 -870.0 -860.0 -850.0Axial Force (kN)

gam

ma

(d) alpha=0.8

-988.7

-979.3

-969.8

-960.4

-951.0

-941.5 -864.3

-855.7

-847.0

-838.4

-829.7

-821.00

0.2

0.4

0.6

0.8

1

-1000.0 -980.0 -960.0 -940.0 -920.0 -900.0 -880.0 -860.0 -840.0 -820.0Axial Force (kN)

gam

ma

Fig. 4.1.8 Membership Functions of Axial Force in pylon in element 17

91

(a) gamma=1.0

282.8

294.6

306.6

318.8

331.2

343.9343.9

356.8

370.0

383.4

397.2

411.40

0.2

0.4

0.6

0.8

1

282.0 302.0 322.0 342.0 362.0 382.0 402.0Axial Force (kN)

alph

a

(b) gamma=0.8

279.4

291.1

303.0

315.1

327.4

340.0 347.7

360.7

374.0

387.6

401.5

415.70

0.2

0.4

0.6

0.8

1

279.0 299.0 319.0 339.0 359.0 379.0 399.0Axial Force (kN)

alph

a

(c) alpha=1.0

300.4

304.1

307.8

311.4

315.1

318.8 370.0

374.0

378.1

382.1

386.2

390.20

0.2

0.4

0.6

0.8

1

300.0 310.0 320.0 330.0 340.0 350.0 360.0 370.0 380.0 390.0Axial Force (kN)

gam

ma

(d) alpha =0.6

324.5

328.4

332.3

336.1

340.0

343.9343.9

347.7

351.6

355.4

359.3

363.10

0.2

0.4

0.6

0.8

1

324.5 329.5 334.5 339.5 344.5 349.5 354.5 359.5Axial Force (kN)

gam

ma

Fig. 4.1.9 Membership Functions of Axial Force in cable 2

92

(a) gamma=1.0

235.5

245.3

255.3

265.4

275.7

286.1286.1

296.8

307.7

318.8

330.2

341.80

0.2

0.4

0.6

0.8

1

235.0 255.0 275.0 295.0 315.0 335.0Axial Force (kN)

alph

a

(b) gamma=0.8

232.4

242.1

252

262.1

272.3

282.7 289.6

300.3

311.3

322.5

333.9

345.60

0.2

0.4

0.6

0.8

1

230.0 250.0 270.0 290.0 310.0 330.0 350.0Axial Force (kN)

alph

a

(c) alpha=1.0

268.9

272.4

275.8

279.2

282.7

286.1286.1

289.6

293.0

296.5

299.9

303.30

0.2

0.4

0.6

0.8

1

268.0 274.0 280.0 286.0 292.0 298.0 304.0Axial Force (kN)

gam

ma

(d) alpha=0.6

248.8

252.1

255.4

258.8

262.1

265.4 307.7

311.3

314.9

318.5

322.1

325.70

0.2

0.4

0.6

0.8

1

248.0 258.0 268.0 278.0 288.0 298.0 308.0 318.0Axial Force (kN)

gam

ma

Fig. 4.1.10 Membership Functions of Axial Force in cable 3

4.2. Case Study 2 – Cable-Stayed Bridge with Uncertainties of Material

Property and Live load

The Cable-stayed Bridge shown in Fig. 3.4 is subjected to a uniformly

distributed live load (owing to service loads). Uncertainty is introduced in the

Young’s modulus of the materials (steel and concrete) and also in live load. The

structural response of the cable-stayed bridge is evaluated for various combinations of

α and β ranging from total certainty (α=1, β=1) to total uncertainty (α=0, β=0). The

results are presented in Table 4.2.1 through Table 4.2.10.

The following common characteristics can be observed in all the tables:

1

2

The uncertain values of structural response in the first row of correspond to the

presence of live load uncertainty alone (0≤β≤1 and α=1). Further, the uncertain

values of structural response in the first column correspond to the presence of

material uncertainty alone (0≤α≤1 and β=1).

Fuzzy membership functions (a), (b), (c) and (d) are plotted as described in the

previous case study (Section 4.2).

4.2.1 Effect of concomitant variation of α and β on displacements and rotations.


horizontal and vertical displacements and rotations at nodes 2,3 and 4 respectively, for

various combinations of α and β.



displacement at node 2 (m) at β=1.0 and β=0.8. The uncertain horizontal


93

[1.0,1.0] in Fig. 4.2.1(a) and [0.566,1.456], [0.675,1.325] in Fig. 4.2.1 (b). Fig. 4.2.1

(c) and (d) represent the membership functions for horizontal displacement at node 2

(m) at α=1.0 and α=0.8. The uncertain horizontal displacements at β=0.0 and β=1.0

correspond to normalised values [-0.625,2.625], [1.0,1.0] in Fig. 4.2.1 (c) and

[-0.641,2.658], [0.976,1.024] in Fig. 4.2.1 (d).



displacement at node 3 (m) at β=1.0 and β=0.8. In these figures, the uncertain

vertical displacements at α=0.0 and α=1.0 correspond to normalised values

[-1.314,-0.691] , [-1.0,-1.0] in Fig. 4.2.2 (a) and [-2.214,0.181] , [-1.883,-0.117] in

Fig. 4.2.2(b). Fig. 4.2.2 (c) and (d) represent the membership functions for vertical

displacement at node 3 (m) at α=1.0 and α=0.8. The uncertain vertical displacements

at β=0.0 and β=1.0 correspond to normalised values [-5.413,3.413], [-1.000,-1.000]

in Fig. 4.2.2(c) and [-5.491,3.464] , [-1.062,-0.938] in Fig. 4.2.2(d).


Fig. 4.2.3 (a) and (b) represent the membership functions for rotation at node 4

at β=1.0 and β=0.8. The uncertain rotations at α=0.0 and α=1.0 correspond to

normalised values [0.302,1.701], [1.0,1.0] in Fig. 4.2.3 (a) and [-1.648,3.693],

[-0.963,2.963] in Fig. 4.2.3 (b). Fig. 4.2.3 (c) and (d) represent the membership

functions for rotation at node 4 (radians) at α=1.0 and α=0.8. The uncertain rotations

at β=0.0 and β=1.0 correspond to normalised values [-8.816,10.816], [1.000,1.000] in

Fig. 4.2.3 (c) and [-8.942,10.982] , [0.861,1.139] in Fig. 4.2.3 (d).

94

The normalised values of uncertain horizontal displacement in the first row of

the Table 4.2.1 correspond to [1.00,1.00], [0.675,1.325], [0.350,1.650], [0.025,1.975],

[-0.300,2.300] and [-0.625,2.625] respectively. Similarly, the normalised values in

the first column of the Table 4.2.1 correspond to [1.000,1.000], [0.976,1.024],

[0.953,1.048], [0.930,1.072], [0.907,1.096] and [0.884,1.121] respectively. From the

above values, it is observed that the variation of horizontal displacement is less in the

case of material uncertainty (upper bound 12.1%) compared to live load uncertainty

(upper bound 162.5%). A similar behaviour is observed in the case of vertical

displacement and rotation as well. This is because the material uncertainty adopted

(±5%) is less compared to live load uncertainty (±100%).


moments

Table 4.2.4 through Table 4.2.10 show the concomitant variation of axial

forces, shear forces and bending moments for various combinations of α and β. Table

4.2.4 and Table 4.2.5 represent the variation of axial force (kN) in element 9 and

element 12 respectively. Table 4.2.6 represents the variation of shear force in the

deck just to the left of node 3. Table 4.2.7 represents the variation of bending moment

in the deck at node 4. Table 4.2.8 represents the variation of axial force (kN) in pylon

in element 17 (at node 2). Table 4.2.9 and Table 4.2.10 represent the variation of

axial force (kN) in cable 2 (at node 2) and cable 3 (at node 3) respectively. In the

above tables, tensile force and sagging moment are considered positive.

4.2.2.1 Axial Force (kN) in deck in element 9

From the normalisation of interval values of axial force in Table 4.2.4, it is

observed that the variation of axial force in deck is less in the case of material

95

uncertainty (upper bound 20.4%) compared to live load uncertainty (upper bound

162.5%). Similar behaviour is observed from all the other uncertain forces/moments

presented in Table 4.2.5 through Table 4.2.10.

Fig. 4.2.4 (a) and (b) represent the membership functions for axial force in

deck in element 9 (kN) at β=1.0 and β=0.8. The uncertain axial forces at α=0.0 and

α=1.0 correspond to normalised values [0.810,1.204], [1.000,1.000] in Fig. 4.2.4 (a)


membership functions for axial force in deck in element 9 (kN) at α=1.0 and α=0.8.

The uncertain axial forces at β=0.0 and β=1.0 correspond to normalised values

[-0.625,2.625], [1.0,1.0] in Fig. 4.2.4 (c) and [-0.646,2.683], [0.961,1.039] in Fig.

4.2.4 (d).


Fig. 4.2.5 (a) and (b) represent the membership functions for axial force (kN) in deck

in element 12 at β=1.0 and β=0.8. The uncertain axial forces at α=0.0 and α=1.0

correspond to normalised values [-1.166,-0.855] , [-1.000,-1.000] in Fig. 4.2.5 (a)

and [-1.399,-0.684] , [-1.200,-0.800] Fig. 4.2.5 (b) . Fig. 4.2.5 (c) and (d) represent

the membership functions for axial force in deck in element 9 (kN) at α=1.0 and

α=0.8. The uncertain axial forces at β=0.0 and β=1.0 correspond to normalised

values [-2.001,0.000], [-1.0, -1.0] in Fig. 4.2.5 (c) and [-2.063,0.0], [-1.032,-0.970] in

Fig. 4.2.5 (d) .


Fig. 4.2.6 (a) and (b) represent the membership functions for shear force just

to the left of node 3 (kN) at β=1.0 and β=0.8. The uncertain shear forces at α=0.0 and

96

α=1.0 correspond to normalised values [-1.102,-0.905] , [-1.0,-1.0] in Fig. 4.2.6 (a)


the membership functions for shear force in deck in element 9 (kN) at α=1.0 and

α=0.8. The uncertain shear forces at γ=0.0 and γ=1.0 correspond to normalised values

[-2.218,0.218], [-1.000,-1.000] in Fig. 4.2.6 (c) and [-2.251,0.225] , [-1.02,-0.980] in

Fig. 4.2.6 (d).


Fig. 4.2.7 (a) and (b) represent the membership functions for bending moment

(kNm) in deck at node 4 at γ=1.0 and γ=0.8. The uncertain bending moments at


Fig. 4.2.7 (a) and [-1.512,-0.522] , [-1.313,-0.686] in Fig. 4.2.7 (b). Fig. 4.2.7 (c) and

(d) represent the membership functions for bending moment at node 4 at α=1.0 and

α=0.6. The uncertain bending moments at β=0.0 and β=1.0 correspond to normalised

values [-2.567,0.569], [-1.000,-1.000] in Fig. 4.2.7 (c) and [-2.668,0.613] ,

[-1.072,-0.929] in Fig. 4.2.7 (d).



pylon in element 17 (at node 11) at β=1.0 and β=0.8. The uncertain axial forces at


Fig. 4.2.8 (a) and [-1.342,-0.718] , [-1.200,-0.800] in Fig. 4.2.8 (b) . Fig. 4.1.7 (c) and

(d) represent the membership functions for axial force in deck in element 9 (kN) at

α=1.0 and α=0.8. The uncertain axial forces at β=0.0 and β=1.0 correspond to

97

normalised values [-2.00,0.00], [-1.000,-1.00] Fig. 4.2.8 (c) and [-2.044,0.00] ,

[-1.022,-0.978] in Fig. 4.2.8 (d).



cable 2 (at node 2) at β=1.0 and β=0.8. The uncertain axial forces at α=0.0 and α=1.0

correspond to normalised values [0.820,1.198] , [1.000,1.000] in Fig. 4.2.9 (a) and

[0.620,1.454] , [0.775,1.225] in Fig. 4.2.9 (b). Fig. 4.2.9 (c) and (d) represent the


axial forces at β=0.0 and β=1.0 correspond to normalised values [-0.127,2.127],

[1.0,1.0] in Fig. 4.2.9(c) and [-0.148,2.26], [0.926,1.077] in Fig. 4.2.9 (d).



cable 3 (at node 3) at β=1.0 and β=0.8. The uncertain axial forces at α=0.0 and α=1.0

correspond to normalised values [0.820,1.198] , [1.000,1.000] in Fig. 4.2.10 (a) and

[0.601,1.468] , [0.758,1.242] in Fig. 4.2.10 (b). Fig. 4.2.10 (c) and (d) represent the


axial forces at β=0.0 and β=1.0 correspond to normalised values [-0.211,2.211],

[1.0,1.0] in Fig. 4.2.10 (c) and [-0.236,2.34], [0.926,1.077] in Fig. 4.2.10 (d).

From all the above results, it is observed that membership functions

represented by figures (a) and (c) of Fig. 4.2.1 through Fig. 4.2.10 are triangular

because of the presence of a single uncertainty (α or β). Also, it is observed that

membership functions represented by figures (c) and (d) of Fig. 4.2.1 through Fig.

4.2.10 are trapezoidal in the presence of multiple uncertainties (α and β).

98

Table. 4.2.1 Cable stayed Bridge – Concomitant Variation of Horizontal displacement of node 2 (×10-4 metres) w.r.t α and β

β→ α↓

1.0 0.8 0.6 0.4 0.2 0.0

1.0 [3.275,3.275] [2.211,4.340] [1.147,5.404] [0.082,6.469] [-0.982,7.533] [-2.047,8.598]

0.8 [3.198,3.353] [2.139,4.424] [1.079,5.494] [0.019,6.564] [-1.040,7.634] [-2.100,8.705]

0.6 [3.122,3.432] [2.067,4.508] [1.012,5.584] [-0.043,6.661] [-1.098,7.737] [-2.153,8.813]

0.4 [3.046,3.511] [1.995,4.593] [0.945,5.676] [-0.011,6.758] [-1.156,7.841] [-2.206,8.923]

0.2 [2.970,3.591] [1.924,4.680] [0.879,5.768] [-0.167,6.857] [-1.214,7.946] [-2.260,9.034]

0.0 [2.896,3.672] [1.854,4.767] [0.812,5.862] [-0.229,6.957] [-1.271,8.052] [-2.313,9.147]

Table. 4.2.2 Cable stayed Bridge – Concomitant Variation of Vertical displacement of node 3 (×10-2 metres) w.r.t α and β

β→ α↓

1.0 0.8 0.6 0.4 0.2 0.0

1.0 [-1.474,-1.474] [-2.775,-0.173] [-4.076,1.128] [-5.377,2.429] [-6.678,3.730] [-7.979,5.031]

0.8 [-1.566,-1.382] [-2.871,-0.085] [-4.177,1.213] [-5.483,2.511] [-6.788,3.808] [-8.094,5.106]

0.6 [-1.658,-1.291] [-2.968,0.003] [-4.279,1.298] [-5.589,2.592] [-6.900,3.887] [-8.210,5.181]

0.4 [-1.750,-1.200] [-3.066,0.091] [-4.381,1.383] [-5.697,2.674] [-7.012,3.966] [-8.328,5.257]

0.2 [-1.843,-1.109] [-3.164,0.179] [-4.484,1.468] [-5.805,2.756] [-7.125,4.045] [-8.446,5.333]

0.0 [-1.937,-1.018] [-3.263,0.267] [-4.588,1.553] [-5.914,2.839] [-7.240,4.124] [-8.566,5.410]

Table. 4.2.3 Cable stayed Bridge – Concomitant Variation of Rotation at node 4 (×10-3 radians) w.r.t α and β

β→ α↓

1.0 0.8 0.6 0.4 0.2 0.0

1.0 [0.381,0.381] [-0.367,1.129] [-1.115,1.877] [-1.863,2.625] [-2.611,3.373] [-3.359,4.121]

0.8 [0.328,0.434] [-0.419,1.184] [-1.166,1.934] [-1.913,2.684] [-2.660,3.434] [-3.407,4.184]

0.6 [0.275,0.487] [-0.471,1.239] [-1.217,1.992] [-1.963,2.744] [-2.709,3.496] [-3.454,4.248]

0.4 [0.221,0.541] [-0.523,1.295] [-1.268,2.049] [-2.013,2.803] [-2.758,3.558] [-3.502,4.312]

0.2 [0.168,0.595] [-0.576,1.351] [-1.319,2.107] [-2.063,2.864] [-2.807,3.620] [-3.551,4.377]

0.0 [0.115,0.648] [-0.628,1.407] [-1.371,2.166] [-2.114,2.924] [-2.856,3.683] [-3.599,4.442]

99

Table. 4.2.4 Cable stayed Bridge – Concomitant Variation

of Axial Force in deck in element 9 (kN) w.r.t α and β β→

α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [175.99,175.99] [118.80,233.19] [61.61,290.38] [4.42,347.57] [-52.77,404.77] [-109.96,461.94]

0.8 [169.15,182.94] [112.57,240.79] [56.00,298.65] [-0.57,356.49] [-57.15,414.35] [-113.72,472.19]

0.6 [162.39,189.99] [106.39,248.55] [50.41,307.09] [-5.59,365.64] [-61.58,424.19] [-117.57,482.74]

0.4 [155.72,197.17] [100.27,256.46] [44.82,315.74] [-10.63,375.03] [-66.08,434.32] [-121.53,493.60]

0.2 [149.12,204.46] [94.17,264.53] [39.23,324.59] [-15.71,384.67] [-70.66,444.73] [-125.60,504.80]

0.0 [142.58,211.89] [88.11,272.78] [33.64,333.67] [-20.84,394.56] [-75.31,455.46] [-129.78,516.35]

Table. 4.2.5 Cable stayed Bridge – Concomitant Variation of Axial Force in deck in element 12 (kN) w.r.t α and β

β→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [-128.14,-128.14] [-153.76,-102.51] [-179.39,-76.88] [-205.02,-51.25] [-230.65,-25.63] [-256.27,0.00]

0.8 [-132.15,-124.23] [-158.58,-99.38] [-185.01,-74.54] [-211.44,-49.69] [-237.87,-24.85] [-264.30,0.00]

0.6 [-136.27,-120.42] [-163.53,-96.34] [-190.78,-72.25] [-218.04,-48.17] [-245.29,-24.08] [-272.55,0.00]

0.4 [-140.51,-116.71] [-168.61,-93.37] [-196.72,-70.03] [-224.82,-46.68] [-252.92,-23.34] [-281.02,0.00]

0.2 [-144.87,-113.09] [-173.84,-90.47] [-202.82,-67.85] [-231.79,-45.24] [-260.76,-22.62] [-289.74,0.00]

0.0 [-149.35,-109.55] [-179.22,-87.64] [-209.09,-65.73] [-238.97,-43.82] [-268.84,-21.91] [-298.71,0.00]

Table. 4.2.6 Cable stayed Bridge – Concomitant Variation of Shear Force in deck just to the left of node 3 (kN) w.r.t α and β

β→ α↓

1.0 0.8 0.6 0.4 0.2 0.0

1.0 [-31.64,-31.64] [-39.34,-23.93] [-47.05,-16.22] [-54.76,-8.51] [-62.47,-0.8] [-70.18,6.91]

0.8 [-32.26,-31.02] [-40.05,-23.39] [-47.85,-15.77] [-55.64,-8.14] [-63.43,-0.51] [-71.23,7.11]

0.6 [-32.89,-30.41] [-40.78,-22.86] [-48.66,-15.31] [-56.54,-7.77] [-64.42,-0.22] [-72.3,7.33]

0.4 [-33.54,-29.81] [-41.51,-22.34] [-49.48,-14.87] [-57.46,-7.39] [-65.43,0.07] [-73.39,7.55]

0.2 [-34.19,-29.21] [-42.26,-21.81] [-50.33,-14.42] [-58.39,-7.02] [-66.46,0.38] [-74.52,7.77]

0.0 [-34.86,-28.62] [-43.03,-21.29] [-51.19,-13.97] [-59.35,-6.65] [-67.51,0.68] [-75.67,8.01]

100

Table. 4.2.7 Cable stayed Bridge – Concomitant Variation of Bending Moment in deck at node 4 (kNm) w.r.t α and γ

β→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [-83.2,-83.2] [-109.2,-57.1] [-135.3,-31.0] [-161.4,-4.9] [-187.5,21.2] [-213.6,47.3]

0.8 [-86.1,-80.2] [-112.5,-54.3] [-138.8,-28.5] [-165.1,-2.6] [-191.4,23.2] [-217.8,49.1]

0.6 [-89.2,-77.3] [-115.7,-51.6] [-142.3,-26.0] [-168.9,-0.3] [-195.5,25.3] [-222.0,51.0]

0.4 [-92.2,-74.3] [-119.0,-48.9] [-145.9,-23.4] [-172.7,2.0] [-199.6,27.5] [-226.4,52.9]

0.2 [-95.3,-71.4] [-122.4,-46.2] [-149.5,-20.9] [-176.6,4.4] [-203.8,29.6] [-230.9,54.9]

0.0 [-98.4,-68.5] [-125.8,-43.4] [-153.2,-18.3] [-180.6,6.7] [-208.0,31.8] [-235.5,56.9]

Table. 4.2.8 Cable stayed Bridge – Concomitant Variation of Axial Force in pylon in element 17 (kN) w.r.t α and β

β→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [-362.39,-362.39] [-434.87,-289.91] [-507.34,-217.43] [-579.82,-144.96] [-652.30,-72.48] [-724.78,0.00]

0.8 [-370.46,-354.55] [-444.55,-283.64] [-518.64,-212.73] [-592.73,-141.82] [-666.82,-70.91] [-740.91,0.00]

0.6 [-378.77,-346.94] [-454.52,-277.55] [-530.27,-208.16] [-606.03,-138.77] [-681.78,-69.39] [-757.54,0.00]

0.4 [-387.33,-339.53] [-464.79,-271.63] [-542.26,-203.72] [-619.73,-135.81] [-697.19,-67.91] [-774.66,0.00]

0.2 [-396.15,-332.34] [-475.38,-265.87] [-554.62,-199.40] [-633.85,-132.93] [-713.08,-66.47] [-792.31,0.00]

0.0 [-405.25,-325.34] [-486.30,-260.27] [-567.35,-195.20] [-648.40,-130.13] [-729.45,-65.07] [-810.50,0.00]

101

102

Table. 4.2.9 Cable stayed Bridge – Concomitant Variation of Axial Force in cable 2 (kN) w.r.t α and β

β→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [155.28,155.28] [120.28,190.27] [85.29,225.27] [50.29,260.26] [15.30,295.26] [-19.69,330.25]

0.8 [149.49,161.18] [115.34,197.04] [81.18,232.89] [47.02,268.75] [12.85,304.61] [-21.30,340.47]

0.6 [143.83,167.20] [110.47,203.95] [77.11,240.71] [43.76,277.46] [10.40,314.22] [-22.95,350.97]

0.4 [138.26,173.35] [105.67,211.03] [73.09,248.72] [40.51,286.39] [7.93,324.08] [-24.64,361.76]

0.2 [132.78,179.64] [100.95,218.29] [69.12,256.95] [37.28,295.60] [5.45,334.26] [-26.37,372.91]

0.0 [127.39,186.09] [96.28,225.81] [65.17,265.53] [34.07,305.25] [2.96,344.96] [-28.15,384.68]

Table. 4.2.10 Cable stayed Bridge – Concomitant Variation of Axial Force in cable 3 (kN) w.r.t α and β

β→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [129.62,129.62] [98.23,161.01] [66.84,192.40] [35.45,223.79] [4.06,255.18] [-27.33,286.57]

0.8 [124.80,134.53] [94.05,166.59] [63.31,198.65] [32.56,230.71] [1.81,262.76] [-28.93,294.82]

0.6 [120.06,139.54] [89.94,172.29] [59.81,205.05] [29.68,237.80] [-0.45,270.55] [-30.58,303.31]

0.4 [115.41,144.65] [85.87,178.13] [56.34,211.61] [26.80,245.08] [-2.73,278.56] [-32.26,312.04]

0.2 [110.83,149.88] [81.86,184.10] [52.89,218.33] [23.93,252.56] [-5.03,286.78] [-33.99,321.01]

0.0 [106.32,155.22] [77.90,190.22] [49.48,225.23] [21.06,260.24] [-7.35,295.25] [-35.77,330.25]

103

(a) beta=1.0

2.896E-04

2.970E-04

3.046E-04

3.122E-04

3.198E-04

3.275E-043.275E-04

3.353E-04

3.432E-04

3.511E-04

3.591E-04

3.672E-040

0.2

0.4

0.6

0.8

1

2.8E-04 2.9E-04 3.0E-04 3.1E-04 3.2E-04 3.3E-04 3.4E-04 3.5E-04 3.6E-04 3.7E-04Horizontal Displacement (m)

alph

a

(b) beta=0.8

1.854E-04

1.924E-04

1.995E-04

2.067E-04

2.139E-04

2.211E-04 4.340E-04

4.424E-04

4.508E-04

4.593E-04

4.680E-04

4.767E-040

0.2

0.4

0.6

0.8

1

1.75E-04 2.25E-04 2.75E-04 3.25E-04 3.75E-04 4.25E-04 4.75E-04Horizontal Displacement (m)

alph

a

(c) alpha=1.0

-2.047E-04

-9.820E-05

8.200E-06

1.147E-04

2.211E-04

3.275E-043.275E-04

4.340E-04

5.404E-04

6.469E-04

7.533E-04

8.598E-040

0.2

0.4

0.6

0.8

1

-3.000E-04 -1.000E-04 1.000E-04 3.000E-04 5.000E-04 7.000E-04 9.000E-04Horizontal Displacement (m)

beta

(d) alpha=0.8

-2.100E-04

-1.040E-04

1.900E-06

1.079E-04

2.139E-04

3.353E-04

4.424E-04

5.494E-04

6.564E-04

7.634E-04

8.705E-04

3.198E-04

0

0.2

0.4

0.6

0.8

1

-3.000E-04 -1.000E-04 1.000E-04 3.000E-04 5.000E-04 7.000E-04 9.000E-04Horizontal Displacement (m)

beta

Fig. 4.2.1 Membership Functions for horizontal displacement at node 2

104

(a) beta=1.0

-1.937E-02

-1.843E-02

-1.750E-02

-1.658E-02

-1.566E-02

-1.474E-02-1.474E-02

-1.382E-02

-1.291E-02

-1.200E-02

-1.109E-02

-1.018E-020

0.2

0.4

0.6

0.8

1

-2.000E-02 -1.850E-02 -1.700E-02 -1.550E-02 -1.400E-02 -1.250E-02 -1.100E-02Vertical Displacement (m)

alph

a

(c) alpha=1.0

-7.979E-02

-6.678E-02

-5.377E-02

-4.076E-02

-2.775E-02

-1.474E-02-1.474E-02

-1.730E-03

1.128E-02

2.429E-02

3.730E-02

5.031E-020

0.2

0.4

0.6

0.8

1

-8.00E-02 -6.50E-02 -5.00E-02 -3.50E-02 -2.00E-02 -5.00E-03 1.00E-02 2.50E-02 4.00E-02Vertical Displacement (m)

beta

(d) alpha=0.8

-8.094E-02

-6.788E-02

-5.483E-02

-4.177E-02

-2.871E-02

-1.382E-02

-8.500E-04

1.213E-02

2.511E-02

3.808E-02

5.106E-02

-1.566E-02

0

0.2

0.4

0.6

0.8

1

-8.094E-02 -6.594E-02 -5.094E-02 -3.594E-02 -2.094E-02 -5.940E-03 9.060E-03 2.406E-02 3.906E-02Vertical Displacement (m)

beta

(b) beta=0.8

-3.263E-02

-3.164E-02

-3.066E-02

-2.968E-02

-2.871E-02

-2.775E-02 -1.730E-03

-8.500E-04

3.000E-05

9.100E-04

1.790E-03

2.670E-030

0.2

0.4

0.6

0.8

1

-3.50E-02 -3.00E-02 -2.50E-02 -2.00E-02 -1.50E-02 -1.00E-02 -5.00E-03 0.00E+00 5.00E-03Vertical Displacement (m)

alph

a

Fig. 4.2.2 Membership Functions for vertical displacement at node 3

105

(a) beta=1.0

1.150E-04

1.680E-04

2.210E-04

2.750E-04

3.280E-04

3.810E-043.810E-04

4.340E-04

4.870E-04

5.410E-04

5.950E-04

6.480E-040

0.2

0.4

0.6

0.8

1

1.150E-04 1.950E-04 2.750E-04 3.550E-04 4.350E-04 5.150E-04 5.950E-04Rotation (radians)

alph

a

(b) beta=0.8

-6.280E-04

-5.760E-04

-5.230E-04

-4.710E-04

-4.190E-04

-3.670E-04 1.129E-03

1.184E-03

1.239E-03

1.295E-03

1.351E-03

1.407E-030

0.2

0.4

0.6

0.8

1

-6.280E-04 -3.280E-04 -2.800E-05 2.720E-04 5.720E-04 8.720E-04 1.172E-03Rotation (radians)

alph

a

(c) alpha=1.0

-3.359E-03

-2.611E-03

-1.863E-03

-1.115E-03

-3.670E-04

3.810E-043.810E-04

1.129E-03

1.877E-03

2.625E-03

3.373E-03

4.121E-030

0.2

0.4

0.6

0.8

1

-4.000E-03 -3.000E-03 -2.000E-03 -1.000E-03 0.000E+00 1.000E-03 2.000E-03 3.000E-03 4.000E-03 5.000E-03Rotation (radians)

beta

(d) alpha=0.8

-3.407E-03

-2.660E-03

-1.913E-03

-1.166E-03

-4.190E-04

4.340E-04

1.184E-03

1.934E-03

2.684E-03

3.434E-03

4.184E-03

3.280E-04

0

0.2

0.4

0.6

0.8

1

-4.00E-03 -3.00E-03 -2.00E-03 -1.00E-03 0.00E+00 1.00E-03 2.00E-03 3.00E-03 4.00E-03 5.00E-03Rotation (radians)

beta

Fig. 4.2.3 Membership Functions for rotation at node 4

106

(a) beta=1.0

142.58

149.12

155.72

162.39

169.15

175.99175.99

182.94

189.99

197.17

204.46

211.890

0.2

0.4

0.6

0.8

1

140.00 151.00 162.00 173.00 184.00 195.00 206.00Axial Force (kN)

alph

a

(b) beta=0.8

88.11

94.17

100.27

106.39

112.57

118.80 233.19

240.79

248.55

256.46

264.53

272.780

0.2

0.4

0.6

0.8

1

80.00 105.00 130.00 155.00 180.00 205.00 230.00 255.00 280.00Axial Force (kN)

alph

a

(c) alpha=1.0

-109.96

-52.77

4.42

61.61

118.80

175.99175.99

233.19

290.38

347.57

404.77

461.940

0.2

0.4

0.6

0.8

1

-110.00 -35.00 40.00 115.00 190.00 265.00 340.00 415.00Axial Force (kN)

beta

(d) alpha=0.8

-113.72

-57.15

-0.57

56.00

112.57

182.94

240.79

298.65

356.49

414.35

472.19

169.15

0

0.2

0.4

0.6

0.8

1

-115.00 -40.00 35.00 110.00 185.00 260.00 335.00 410.00Axial Force (kN)

beta

Fig. 4.2.4 Membership Functions for Axial Force in deck in element 9

107

(a) beta=1.0

-149.35

-144.87

-140.51

-136.27

-132.15

-128.14-128.14

-124.23

-120.42

-116.71

-113.09

-109.550

0.2

0.4

0.6

0.8

1

-150 -144 -138 -132 -126 -120 -114Axial Force (kN)

alph

a

(b) beta=0.8

-179.22

-173.84

-168.61

-163.53

-158.58

-153.76 -102.51

-99.38

-96.34

-93.37

-90.47

-87.640

0.2

0.4

0.6

0.8

1

-180 -168 -156 -144 -132 -120 -108 -96Axial Force (kN)

alph

a

(c) alpha=1.0

-256.27

-230.65

-205.02

-179.39

-153.76

-128.14-128.14

-102.51

-76.88

-51.25

-25.63

00

0.2

0.4

0.6

0.8

1

-257 -219 -181 -143 -105 -67 -29Axial Force (kN)

beta

(d) alpha=0.8

-264.30

-237.87

-211.44

-185.01

-158.58

-124.23

-99.38

-74.54

-49.69

-24.85

0.00

-132.15

0

0.2

0.4

0.6

0.8

1

-264.30 -234.30 -204.30 -174.30 -144.30 -114.30 -84.30 -54.30 -24.30Axial Force (kN)

beta

Fig. 4.2.5 Membership Functions for Axial Force in deck in element 12

108

(a) beta=1.0

-34.86

-34.19

-33.54

-32.89

-32.26

-31.64-31.64

-31.02

-30.41

-29.81

-29.21

-28.620

0.2

0.4

0.6

0.8

1

-35.00 -34.00 -33.00 -32.00 -31.00 -30.00 -29.00 -28.00Shear Force (kN)

alph

a

(b) beta=0.8

-43.03

-42.26

-41.51

-40.78

-40.05

-39.34 -23.93

-23.39

-22.86

-22.34

-21.81

-21.290

0.2

0.4

0.6

0.8

1

-45.00 -41.25 -37.50 -33.75 -30.00 -26.25 -22.50Shear Force (kN)

alph

a

(c) alpha=1.0

-70.18

-62.47

-54.76

-47.05

-39.34

-31.64-31.64

-23.93

-16.22

-8.51

-0.80

6.910

0.2

0.4

0.6

0.8

1

-70.18 -60.18 -50.18 -40.18 -30.18 -20.18 -10.18 -0.18Shear Force (kN)

beta

(d) alpha=0.8

-71.23

-63.43

-55.64

-47.85

-40.05

-31.02

-23.39

-15.77

-8.14

-0.51

7.11

-32.26

0

0.2

0.4

0.6

0.8

1

-72.00 -62.00 -52.00 -42.00 -32.00 -22.00 -12.00 -2.00 8.00Shear Force (kN)

beat

Fig. 4.2.6 Membership Functions for Shear Force in deck to left of node 3

109

(a) beta=1.0

-98.4

-95.3

-92.2

-89.2

-86.1

-83.2-83.2

-80.2

-77.3

-74.3

-71.4

-68.50

0.2

0.4

0.6

0.8

1


alph

a

(b) beta=0.8

-125.8

-122.4

-119.0

-115.7

-112.5

-109.2 -57.1

-54.3

-51.6

-48.9

-46.2

-43.40

0.2

0.4

0.6

0.8

1

-130.0 -120.0 -110.0 -100.0 -90.0 -80.0 -70.0 -60.0 -50.0 -40.0Bending Moment (kN)

alph

a

(c) alpha=1.0

-213.6

-187.5

-161.4

-135.3

-109.2

-83.2-83.2

-57.1

-31.0

-4.9

21.2

47.30

0.2

0.4

0.6

0.8

1

-215.0 -190.0 -165.0 -140.0 -115.0 -90.0 -65.0 -40.0 -15.0 10.0 35.0Bending Moment (kN)

beta

(d) alpha=0.6

-222.0

-195.5

-168.9

-142.3

-115.7

-77.3

-51.6

-26.0

-0.3

25.3

51.0

-89.2

0

0.2

0.4

0.6

0.8

1

-225.0 -175.0 -125.0 -75.0 -25.0 25.0 75.0Bending Moment (kNm)

beta

Fig. 4.2.7 Membership Functions for Bending Moment in deck at node 4

110

(a) beta=1.0

-405.25

-396.15

-387.33

-378.77

-370.46

-362.39-362.39

-354.55

-346.94

-339.53

-332.34

-325.340

0.2

0.4

0.6

0.8

1

-410 -400 -390 -380 -370 -360 -350 -340 -330Axial Force (kN)

alph

a

(b) beta=0.8

-486.3

-475.38

-464.79

-454.52

-444.55

-434.87 -289.91

-283.64

-277.55

-271.63

-265.87

-260.270

0.2

0.4

0.6

0.8

1

-495 -470 -445 -420 -395 -370 -345 -320 -295 -270Axial Force (kN)

alph

a

(c) alpha=1.0

-724.78

-652.3

-579.82

-507.34

-434.87

-362.39-362.39

-289.91

-217.43

-144.96

-72.48

00

0.2

0.4

0.6

0.8

1

-800 -710 -620 -530 -440 -350 -260 -170 -80 10Axial Force (kN)

beta

(d) alpha=0.8

-740.91

-666.82

-592.73

-518.64

-444.55

-354.55

-283.64

-212.73

-141.82

-70.91

0

-370.46

0

0.2

0.4

0.6

0.8

1

-825 -725 -625 -525 -425 -325 -225 -125 -25Axial Force (kN)

beta

Fig. 4.2.8 Membership Functions for Axial Force in pylon in element 17

111

(a) beta=1.0

127.39

132.78

138.26

143.83

149.49

155.28155.28

161.18

167.20

173.35

179.64

186.090

0.2

0.4

0.6

0.8

1

125.00 135.00 145.00 155.00 165.00 175.00 185.00Axial Force (kN)

alph

a

(b) beta=0.8

96.28

100.95

105.67

110.47

115.34

120.28 190.27

197.04

203.95

211.03

218.29

225.810

0.2

0.4

0.6

0.8

1

90.00 110.00 130.00 150.00 170.00 190.00 210.00 230.00Axial Force (kN)

alph

a

(c) alpha=1.0

-19.69

15.30

50.29

85.29

120.28

155.28155.28

190.27

225.27

260.26

295.26

330.250

0.2

0.4

0.6

0.8

1

-50.00 0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.00Axial Force (kN)

beta

(d) alpha=0.6

-22.95

10.40

43.76

77.11

110.47

167.20

203.95

240.71

277.46

314.22

350.97

143.83

0

0.2

0.4

0.6

0.8

1

-50.00 0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.00Axial Force (kN)

beta

Fig. 4.2.9 Membership Functions for Axial Force in cable 2

112

(a) beta=1.0

106.32

110.83

115.41

120.06

124.8

129.62129.62

134.53

139.54

144.65

149.88

155.220

0.2

0.4

0.6

0.8

1

100 108 116 124 132 140 148 156Axial Force (kN)

alph

a

(b) beta=0.8

77.90

81.86

85.87

89.94

94.05

98.23 161.01

166.59

172.29

178.13

184.10

190.220

0.2

0.4

0.6

0.8

1

75.00 90.00 105.00 120.00 135.00 150.00 165.00 180.00 195.00Axial Force (kN)

alph

a

(c) alpha=1.0

-27.33

4.06

35.45

66.84

98.23

129.62129.62

161.01

192.40

223.79

255.18

286.570

0.2

0.4

0.6

0.8

1

-60.00 -20.00 20.00 60.00 100.00 140.00 180.00 220.00 260.00 300.00Axial Force (kN)

beta

(d) alpha=0.6

-30.58

-0.45

29.68

59.81

89.94

139.54

172.29

205.05

237.80

270.55

303.31

120.06

0

0.2

0.4

0.6

0.8

1

-50.00 0.00 50.00 100.00 150.00 200.00 250.00 300.00Axial Force (kN)

beta

Fig. 4.2.10 Membership Functions for Axial Force in cable 3

4.3 Case Study 3 – Cable-Stayed Bridge with Uncertainties of Material Property, Live Load and Mass Density

The Cable-stayed Bridge shown in Fig. 3.4 is subjected to a combination of

dead load and uniformly distributed live load (owing to service loads). Uncertainty is

introduced in the Young’s modulus of the materials (Steel and Concrete), in live load

and also in mass density (α,β and γ). The structural response of the cable-stayed

bridge is evaluated in the concomitant presence of these uncertainties. Various

combinations of α and β ranging from total certainty (α=1, β=1) to total uncertainty

(α=0, β=0) are considered at γ=1.0 (crisp) and γ=0.8 (±1% variation) respectively.

Correspondingly, the uncertainty of mass density γ is reflected in the uncertainty of

dead load (self weight).

4.3.1 Effect of concomitant variation of α and β on displacements and rotations.


horizontal and vertical displacements and rotations at nodes 2, 3 and 4 respectively,

for various combinations of α and β at γ=1.0. Also, Table 4.3.2, Table 4.3.4 and

Table 4.3.6 show the concomitant variation of horizontal and vertical displacements

and rotations at nodes 2, 3 and 4 respectively, for various combinations of α and β at

γ=0.8.

In Table 4.3.1, the uncertain values of horizontal displacement in the first row

correspond to the normalised values [1.000,1.000], [0.898,1.102], [0.797,1.204],

[0.694,1.307], [0.593,1.408], [0.490,1.511] respectively. Similarly, the normalised

values of horizontal displacement in the first column are [1.000,1.000], [0.977,1.024],

[0.954,1.048], [0.930,1.072], [0.907,1.097], [0.885,1.122] respectively. From the

above values, it is observed that the variation of horizontal displacement is less in the

113

case of material uncertainty (upper bound 12.2%) compared to live load uncertainty

(upper bound 151.1%). A similar behaviour is observed in the case of vertical

displacement and rotation as well. This is because the material uncertainty adopted

(±5%) is less compared to live load uncertainty (±100%).


Table 4.3.1 and Table 4.3.2 represent the variation of uncertain horizontal

displacement (×10-4m) at node 2 for various combinations of material and live load

uncertainties (α and β) at γ=1.0 and γ=0.8 respectively.


displacement at node 2 (m) at β=1.0,γ=1.0 and β=0.8,γ=1.0. The uncertain horizontal


[1.0,1.0] in Fig. 4.3.1 (a) and [0.784,1.226], [0.898,1.102] in Fig. 4.3.1 (b).

Fig. 4.3.1 (c) and (d) represent the membership functions for horizontal displacement

at node 2 (m) at α=1.0,γ=1.0 and α=0.6,γ=1.0. The uncertain horizontal displacements

at β=0.0 and β=1.0 correspond to normalised values [0.490,1.511], [1.0,1.0] in

Fig. 4.3.1 (c) and [0.448,1.563], [0.954,1.048] in Fig. 4.3.1 (d).


displacement at node 2 (m) at β=1.0,γ=0.8 and β=0.8,γ=0.8. The uncertain horizontal



Fig. 4.3.2 (c) and (d) represent the membership functions for horizontal displacement

at node 2 (m) at α=1.0,γ=0.8 and α=0.6,γ=0.8. The uncertain horizontal displacements

at β=0.0 and β=1.0 correspond to normalised values [0.479,1.521], [0.989,1.011] in

Fig. 4.3.2 (c) and [0.437,1.575], [0.943,1.059] in Fig. 4.3.2 (d).

114



displacement at node 3 (m) at β=1.0,γ=1.0 and β=0.8,γ=1.0. The uncertain vertical


[-1.0,-1.0] in Fig. 4.3.3 (a) and [-1.591,-0.424] , [-1.274,-0.726] in Fig. 4.3.3 (b).

Fig. 4.3.3 (c) and (d) represent the membership functions for vertical displacement at

node 3 (m) at α=1.0,γ=1.0 and α=0.6,γ=1.0. The uncertain vertical displacements at

β=0.0 and β=1.0 correspond to normalised values [-2.373,0.371], [-1.0,-1.0] in

Fig. 4.3.3 (c ) and [-2.506,0.487] , [-1.124,-0.878] in Fig. 4.3.3 (d).


displacement at node 3 (m) at β=1.0,γ=0.8 and β=0.8,γ=0.8. The uncertain vertical


[-1.030,-0.970] in Fig. 4.3.4 (a) and [-1.620,-0.395] ,[-1.304,-0.694] in Fig. 4.3.4 (b).

Fig. 4.3.4 (c) and (d) represent the membership functions for vertical displacement at

node 3 (m) at α=1.0,γ=0.8 and α=0.6,γ=0.8. The uncertain vertical displacements at

β=0.0 and β=1.0 correspond to normalised values [-2.403,0.401], [-1.030,-0.970] in

Fig. 4.3.4 (c) and [-2.536,0.517], [-1.154,-0.848] in Fig. 4.3.4 (d).


Table 4.3.5 and Table 4.3.6 represent the variation of uncertain rotation

(×10-3 radians) at node 4 for various combinations of material and live load


Fig. 4.3.5 (a) and (b) represent the membership functions for uncertain rotation

at node 4 at β=1.0,γ=1.0 and β=0.8,γ=1.0. The for uncertain rotations at node 4 at

α=0.0 and α=1.0 correspond to normalised [0.317,1.686] , [1.0,1.0] in Fig. 4.3.5 (a)

115

and [-0.289,2.305] , [0.390,1.610] in Fig. 4.3.5 (b). Fig. 4.3.5 (c) and (d) represent the

membership functions for uncertain rotation at node 4 at α=1.0,γ=1.0 and

α=0.6,γ=1.0. In these figures, the uncertain vertical displacements at β=0.0 and β=1.0

correspond to normalised values [-2.048,4.048] , [1.0,1.0] in Fig. 4.3.5 (c) and

[-2.312,4.338] , [0.727,1.273] in Fig. 4.3.5 (d).

Fig. 4.3.6 (a) and (b) represent the membership functions for uncertain rotation

at node 4 at β=1.0,γ=0.8 and β=0.8,γ=0.8. The uncertain rotations at node 4 at α=0.0

and α=1.0 correspond to normalised values [0.253,1.752] , [0.935,1.064] in

Fig. 4.3.6 (a) and [-0.353,2.371] , [0.325,1.674] in Fig. 4.3.6 (b). Fig. 4.3.6 (c) and (d)

represent the membership functions for uncertain rotation at node 4 at α=1.0,γ=0.8

and α=0.6,γ=0.8. The uncertain vertical displacements at β=0.0 and β=1.0 correspond

to normalised values [-2.113,4.112], [0.935,1.064] in Fig. 4.3.6 (c) and

[-2.377,4.403] , [0.662,1.338] in Fig. 4.3.6 (d).

4.3.2 Effect of concomitant variation of α and β on shear forces and bending moments

Table 4.3.7 through Table 4.3.10 represent the variation of axial force (kN) in

element 9 and element 12 respectively. Table 4.3.11 and Table 4.3.12 represent the

variation of shear force in the deck just to the left of node 3 at γ=1.0 and γ=0.8

respectively. Table 4.3.13 and Table 4.3.14 represent the variation bending moment

in the deck at node 4 at γ=1.0 and γ=0.8 respectively. Table 4.3.15 and Table 4.3.16

represent the variation of axial force (kN) in pylon in element 17 (at node 2) at γ=1.0

and γ=0.8 respectively. Table 4.3.17 and Table 4.3.18 represent the variation of axial

force (kN) in cable 2(at node 2) at γ=1.0 and γ=0.8 respectively. Table 4.3.19 and

116

Table 4.3.20 represent the variation of axial force (kN) in cable 2(at node 2) at γ=1.0

and γ=0.8 respectively.

4.3.2.1 Axial Force (kN) in deck in element 9

In Table 4.3.7, the uncertain values of axial force in element 9 in the first row

correspond to normalised values [1.0,1.0], [0.898,1.102], [0.796,1.204],

[0.694,1.306], [0.592,1.408] and [0.490,1.509] respectively. Similarly, the uncertain

values of axial force in element 9 in the first column correspond to normalised values

[1.000,1.000], [0.961,1.039], [0.923,1.079], [0.885,1.120], [0.848,1.161] and

[0.810,1.204] respectively. From the above values, it is observed that the variation of

axial force in element 9 is less in the case of material uncertainty (upper bound

31.0%) compared to live load uncertainty (upper bound 137.1%). Similar behaviour

is observed in the case of other forces/moments as well.

Fig. 4.3.7 (a) and (b) represent the membership functions for uncertain axial

force (kN) in element 9 at β=1.0,γ=1.0 and β=0.8,γ=1.0. The axial forces at α=0.0

and α=1.0 correspond to normalised values [0.810,1.204], [1.0,1.0] in Fig. 4.3.7 (a)


membership functions for axial force (kN) in element 9 at α=1.0,γ=1.0 and

α=0.6,γ=1.0. The uncertain axial forces at β=0.0 and β=1.0 correspond to normalised

values [0.490,1.509], [1.0,1.0] in Fig. 4.3.7 (c) and [0.424,1.601], [0.923,1.079] in

Fig. 4.3.7 (d).



and α=1.0 correspond to normalised values [0.8,1.215], [0.989,1.011] in Fig. 4.3.8 (a)


membership functions for axial force (kN) in element 9 at α=1.0,γ=0.8 and

117


values [0.479,1.521] , [0.989,1.011] in Fig. 4.3.8 (c) and [0.413,1.612] , [0.912,1.091]

in Fig. 4.3.8 (d).




and α=1.0 correspond to normalised values [-1.165,-0.855],[-1.0,-1.0] in Fig. 4.3.9 (a)


the membership functions for axial force (kN) in element 12 at α=1.0,γ=1.0 and


values [-1.314,-0.685] , [-1.00,-1.00] in Fig. 4.3.9 (c) and [-1.398,-0.643] ,

[-1.063,-0.940] in Fig. 4.3.9 (d).



and α=1.0 correspond to normalised values [-1.173,-0.849] ,[-1.007,-0.993] in

Fig. 4.3.10 (a) and [-1.247,-0.795] , [-1.070,-0.930] in Fig. 4.3.10 (b). Fig. 4.3.10 (c)

and (d) represent the membership functions for axial force (kN) in element 12 at

α=1.0,γ=0.8 and α=0.6,γ=0.8. The uncertain axial forces at β=0.0 and β=1.0

correspond to normalised values [-1.321,-0.678] , [-1.007,-0.993] in Fig. 4.3.10(c) and

[-1.405,-0.637], [-1.071,-0.933] in Fig. 4.3.10 (d).


Fig. 4.3.11 (a) and (b) represent the membership functions for the uncertain

shear force (kN) in deck just to the left of node 3 at β=1.0,γ=1.0 and β=0.8,γ=1.0. The

118

uncertain shear forces at α=0.0 and α=1.0 correspond to normalised values

[-1.104,-0.904] , [-1.0,-1.0] in Fig. 4.3.11 (a) and [-1.187,-0.829] and [-1.079,-0.921]

in Fig. 4.3.11 (b). Fig. 4.3.11 (c) and (d) represent the membership functions for

shear force (kN) in deck just to the left of node 3 at α=1.0,γ=1.0 and α=0.6,γ=1.0. The

uncertain shear forces at β=0.0 and β=1.0 correspond to normalised values

[-1.395,-0.606] ,[-1.0,-1.0] in Fig. 4.3.11 (c) and [-1.444,-0.576] , [-1.041,-0.961] in

Fig. 4.3.11 (d ).


shear force (kN) in deck just to the left of node 3 at β=1.0,γ=0.8 and β=0.8,γ=0.8.

The uncertain shear forces at α=0.0 and α=1.0 correspond to normalised values

[-1.112,-0.896], [-1.008,-0.992] in Fig. 4.3.12(a) and [-1.195,-0.821],

[-1.087,-0.913] in Fig. 4.3.12 (b). Fig. 4.3.12 (c) and (d) represent the membership

functions for shear force (kN) in deck just to the left of node 3 at α=1.0,γ=0.8 and

α=0.6,γ=0.8. The uncertain shear forces at β=0.0 and β=1.0 correspond to normalised

values [-1.403,-0.598] , [-1.008,-0.992] in Fig. 4.3.12 (c) and [-1.452,-0.567] ,

[-1.049,-0.953] in Fig. 4.3.12 (d).



bending moment (kNm) in deck at node 4 at β=1.0,γ=1.0 and β=0.8,γ=1.0. The

uncertain bending moments at α=0.0 and α=1.0 correspond to normalised values

[-1.187,-0.819], [-1.0,-1.0] in Fig. 4.3.13 (a) and [-1.294,-0.721],[-1.102,-0.898] in

Fig. 4.3.13 (b). Fig. 4.3.13 (c) and (d) represent the membership functions for bending

moment in deck at node 4 at α=1.0,γ=1.0 and α=0.6,γ=1.0. The uncertain bending

119

moments at β=0.0 and β=1.0 correspond to normalised values [-1.511,-0.489],

[-1.0,-1.0] in Fig. 4.3.13 (c) and [-1.594,-0.425] ,[-1.074,-0.928] in Fig. 4.3.13 (d).


bending moment (kNm) in deck at node 4 at β=1.0,γ=0.8 and β=0.8,γ=0.8. The

uncertain bending moments at α=0.0 and α=1.0 correspond to normalised values

[-1.199,-0.809] ,[-1.011,-0.989] in Fig. 4.3.14 (a) and [-1.306,-0.711] and

[-1.113,-0.887] in Fig. 4.3.14 (b). Fig. 4.3.14 (c) and (d) represent the membership

functions for bending moment in deck at node 4 at α=1.0,γ=0.8 and α=0.6,γ=0.8. The

uncertain bending moments at β=0.0 and β=1.0 correspond to normalised values

[-1.522,-0.478] , [-1.011,-0.989] in Fig. 4.3.14 (c) and [-1.605,-0.414] ,[-1.085,-0.917]

in Fig. 4.3.14 (d).



axial force (kN) in pylon in element 17 at β=1.0,γ=1.0 and β=0.8,γ=1.0. The

uncertain axial forces at α=0.0 and α=1.0 correspond to normalised values

[-1.116,-0.900] , [-1.0,-1.0] in Fig. 4.3.15 (a) and [-1.180,-0.848] , [-1.058,-0.943] in

Fig. 4.3.15 (b). Fig. 4.3.15 (c) and (d) represent the membership functions for axial

force (kN) in pylon in element 17 at α=1.0,γ=1.0 and α=0.6,γ=1.0. The uncertain

axial forces at β=0.0 and β=1.0 correspond to normalised values [-1.287,-0.713],

[-1.0,-1.0] in Fig. 4.3.15 (c) and [-1.344,-0.684] , [-1.045,-0.958] in Fig. 4.3.15 (d).


axial force (kN) in pylon in element 17 at β=1.0,γ=0.8 and β=0.8,γ=0.8. The uncertain

axial forces at α=0.0 and α=1.0 correspond to normalised values [-1.124,-0.893] ,

[-1.007,-0.993] in Fig. 4.3.16 (a) and [-1.188,-0.842] , [-1.065,-0.936] in

120

Fig. 4.3.16 (a). Fig. 4.3.16 (c) and (d) represent the membership functions for axial

force (kN) in pylon in element 17 at α=1.0,γ=0.8 and α=0.6,γ=0.8. The uncertain

axial forces at β=0.0 and β=1.0 correspond to normalised values [-1.294, -0.706],

[-1.007, -0.993] in Fig. 4.3.16 (c) and [-1.352, -0.677], [-1.052, -0.951] in

Fig. 4.3.16 (d).



axial force (kN) in cable 2 (at node 2) at β=1.0,γ=1.0 and β=0.8,γ=1.0. The uncertain

axial forces at α=0.0 and α=1.0 correspond to normalised values [0.822,1.196],

[1.000,1.000] in Fig. 4.3.17 (a) and [0.752,1.286] , [0.921,1.079] in Fig. 4.3.17 (a).

Fig. 4.3.17 (c) and (d) represent the membership functions for axial force (kN) in

cable 2 (at node 2) at α=1.0,γ=1.0 and α=0.6,γ=1.0. The uncertain axial forces at

β=0.0 and β=1.0 correspond to normalised values [0.604,1.396], [1.0,1.0] in

Fig. 4.3.17 (c) and [0.549,1.492] , [0.927,1.076] in Fig. 4.3.17 (d).




[0.993,1.009] in Fig. 4.3.18 (a) and [0.745,1.295] , [0.914,1.088] in Fig. 4.3.18 (b).



β=0.0 and β=1.0 correspond to normalised values [0.596,1.405] , [0.993,1.009] in

Fig. 4.3.18 (c) and [0.542,1.501] , [0.920,1.085] in Fig. 4.3.18 (d).

121


Table 4.3.19 and Table 4.3.20 represent the variation of uncertain axial force

(kN) in cable 3 (at node 3) for various combinations of material and live load





[1.0,1.0] in Fig. 4.3.19 (a) and [0.762,1.271] , [0.934,1.066] in Fig. 4.3.19 (b).




Fig. 4.3.19 (c) and [0.609,1.422] [0.927,1.076] in Fig. 4.3.19 (d).








Fig. 4.3.20 (c )and [0.601,1.430] , [0.919,1.085] in Fig. 4.3.20 (d).

It is observed that membership functions (a) and (c) depicted in Fig. 4.3.1,

Fig. 4.3.3, Fig. 4.3.5, Fig. 4.3.7, Fig. 4.3.9, Fig. 4.3.11, Fig. 4.3.13, Fig. 4.3.15,

Fig. 4.3.17, Fig. 4.3.19 are triangular owing to the presence of a single uncertainty

(α or β).

122

It is further observed that membership functions (b) and (d) depicted in

Fig. 4.3.1, Fig. 4.3.3, Fig. 4.3.5, Fig. 4.3.7, Fig. 4.3.9, Fig. 4.3.11, Fig. 4.3.13,

Fig. 4.3.15, Fig. 4.3.17, Fig. 4.3.19 are trapezoidal in the presence of multiple

uncertainties (α and β).

It is also observed that membership functions (a),(b),(c),(d) depicted in Fig.

4.3.2, Fig. 4.3.4, Fig. 4.3.6, Fig. 4.3.8, Fig. 4.3.10, Fig. 4.3.12, Fig. 4.3.14, Fig. 4.3.16,

Fig. 4.3.18, Fig. 4.3.20 are trapezoidal in the presence of multiple uncertainties (α and

β and γ).

123

Table. 4.3.1 Cable stayed Bridge - Concomitant Variation of Horizontal displacement of node 2 (×10-4 metres) w.r.t α and β at γ=1.0

β→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [10.44,10.44] [9.38,11.51] [8.32,12.57] [7.25,13.64] [6.19,14.70] [5.12,15.77]

0.8 [10.20,10.69] [9.14,11.76] [8.08,12.83] [7.02,13.90] [5.96,14.97] [4.90,16.04]

0.6 [9.96,10.94] [8.90,12.02] [7.85,13.09] [6.79,14.17] [5.74,15.25] [4.68,16.32]

0.4 [9.71,11.19] [8.66,12.28] [7.61,13.36] [6.66,14.44] [5.51,15.52] [4.46,16.61]

0.2 [9.47,11.45] [8.43,12.54] [7.38,13.63] [6.34,14.71] [5.29,15.80] [4.24,16.89]

0.0 [9.24,11.71] [8.19,12.80] [7.15,13.90] [6.11,14.99] [5.07,16.09] [4.03,17.18]

Table. 4.3.2 Cable stayed Bridge - Concomitant Variation of Horizontal displacement of node 2 (×10-4 metres) w.r.t α and β at γ=0.8

β→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [10.33, 10.56] [9.263,11.625] [8.199,12.689] [7.134,13.754] [6.070,14.818] [5.005,15.883]

0.8 [10.08,10.808] [9.024,11.879] [7.964,12.949] [6.904,14.019] [5.845,15.089] [4.785,16.160]

0.6 [9.840,11.059] [8.785,12.135] [7.730,13.211] [6.675,14.288] [5.620,15.364] [4.565,16.440]

0.4 [9.599,11.312] [8.548,12.394] [7.498,13.477] [6.542,14.559] [5.397,15.642] [4.347,16.724]

0.2 [9.359,11.567] [8.313,12.656] [7.268,13.744] [6.222,14.833] [5.175,15.922] [4.129,17.010]

0.0 [9.122,11.825] [8.080,12.920] [7.038,14.015] [5.997,15.110] [4.955,16.205] [3.913,17.300]

Table. 4.3.3 Cable stayed Bridge - Concomitant Variation of Vertical displacement of node 3 (×10-2 metres) w.r.t α and β at γ=1.0

β→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [-4.74,-4.74] [-6.04,-3.44] [-7.34,-2.14] [-8.64,-0.84] [-9.95,0.46] [-11.25,1.76]

0.8 [-5.03,-4.45] [-6.34,-3.15] [-7.64,-1.86] [-8.95,-0.56] [-10.26,0.74] [-11.56,2.04]

0.6 [-5.33,-4.16] [-6.64,-2.87] [-7.95,-1.57] [-9.26,-0.28] [-10.57,1.02] [-11.88,2.31]

0.4 [-5.62,-3.87] [-6.94,-2.58] [-8.25,-1.29] [-9.57,0.00] [-10.88,1.29] [-12.19,2.58]

0.2 [-5.92,-3.58] [-7.24,-2.29] [-8.56,-1.01] [-9.88,0.28] [-11.19,1.57] [-12.52,2.86]

0.0 [-6.21,-3.29] [-7.54,-2.01] [-8.86,-0.72] [-10.19,0.56] -11.52,1.85] -12.84,3.13]

124

Table. 4.3.4 Cable stayed Bridge - Concomitant Variation

of Vertical displacement of node 3 (×10-2 metres) w.r.t α and β at γ=0.8 β→

α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [-4.88,-4.60] [-6.18,-3.29] [-7.49,-1.99] [-8.79,-0.69] [-10.09,0.60] [-11.39,1.90]

0.8 [-5.18,-4.31] [-6.48,-3.01] [-7.79,-1.71] [-9.09,-0.42] [-10.39,0.88] [-11.70,2.18]

0.6 [-5.47,-4.02] [-6.78,-2.73] [-8.09,-1.43] [-9.40,-0.14] [-10.71,1.16] [-12.02,2.45]

0.4 [-5.76,-3.73] [-7.08,-2.44] [-8.39,-1.15] [-9.71,0.14] [-11.03,1.43] [-12.34,2.73]

0.2 [-6.06,-3.44] [-7.38,-2.15] [-8.70,-0.86] [-10.02,0.42] [-11.34,1.71] [-12.66,3.00]

0.0 [-6.36,-3.15] [-7.68,-1.87] [-9.01,-0.58] [-10.34,0.70] [-11.66,1.99] [-12.99,3.27]

Table. 4.3.5 Cable stayed Bridge - Concomitant Variation of Rotation at node 4 (×10-3 radians) w.r.t α and β at γ=1.0

β→ α↓

1.0 0.8 0.6 0.4 0.2 0.0

1.0 [1.227,1.227] [0.479,1.975] [-0.269,2.723] [-1.017,3.471] [-1.765,4.219] [-2.513,4.967]

0.8 [1.059,1.394] [0.312,2.144] [-0.435,2.894] [-1.182,3.644] [-1.929,4.394] [-2.676,5.144]

0.6 [0.892,1.562] [0.146,2.314] [-0.600,3.067] [-1.346,3.819] [-2.092,4.571] [-2.837,5.323]

0.4 [0.724,1.731] [-0.02,2.485] [-0.765,3.239] [-1.51,3.993] [-2.255,4.748] [-2.999,5.502]

0.2 [0.556,1.900] [-0.188,2.656] [-0.931,3.412] [-1.675,4.169] [-2.419,4.925] [-3.163,5.682]

0.0 [0.389,2.069] [-0.354,2.828] [-1.097,3.587] [-1.84,4.345] [-2.582,5.104] [-3.325,5.863]

Table. 4.3.6 Cable stayed Bridge - Concomitant Variation of Rotation at node 4 (×10-3 radians) w.r.t α and β at γ=0.8

β→ α↓

1.0 0.8 0.6 0.4 0.2 0.0

1.0 [1.147,1.306] [0.399,2.054] [-0.349,2.802] [-1.097,3.550] [-0.999,5.144] [-2.593,5.046]

0.8 [0.980,1.474] [0.233,2.224] [-0.514,2.974] [-1.261,3.724] [-1.277,5.434] [-2.755,5.224]

0.6 [0.812,1.642] [0.066,2.394] [-0.680,3.147] [-1.426,3.899] [-1.555,5.726] [-2.917,5.403]

0.4 [0.644,1.811] [-0.100,2.565] [-0.845,3.319] [-1.590,4.073] [-1.832,6.018] [-3.079,5.582]

0.2 [0.477,1.981] [-0.267,2.737] [-1.010,3.493] [-1.754,4.250] [-2.110,6.311] [-3.242,5.763]

0.0 [0.310,2.150] [-0.433,2.909] [-1.176,3.668] [-1.919,4.426] [-2.387,6.606] [-3.404,5.944]

125


of Axial Force in deck in element 9 (kN) w.r.t α and β at γ=1.0 β→

α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [561.16,561.16] [503.97,618.36] [446.78,675.55] [389.59,732.74] [332.40,789.94] [275.21,847.11]

0.8 [539.38,583.27] [482.80,641.12] [426.23,698.98] [369.66,756.82] [313.08,814.68] [256.51,872.52]

0.6 [517.87,605.72] [461.87,664.28] [405.89,722.82] [349.89,781.37] [293.90,839.92] [237.91,898.47]

0.4 [496.64,628.55] [441.19,687.84] [385.74,747.12] [330.29,806.41] [274.84,865.70] [219.39,924.98]

0.2 [475.64,651.76] [420.69,711.83] [365.75,771.89] [310.81,831.97] [255.86,892.03] [200.92,952.10]

0.0 [454.85,675.41] [400.38,736.30] [345.91,797.19] [291.43,858.08] [236.96,918.98] [182.49,979.87]

Table. 4.3.8 Cable stayed Bridge - Concomitant Variation of Axial Force in deck in element 9 (kN) w.r.t α and β at γ=0.8

β→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [554.92,567.40] [497.73,624.60] [440.54,681.79] [383.35,738.98] [326.16,796.18] [268.97,853.35]

0.8 [533.21,589.59] [476.63,647.44] [420.06,705.30] [363.49,763.14] [306.91,821.00] [250.34,878.84]

0.6 [511.76,612.11] [455.76,670.67] [399.78,729.21] [343.78,787.76] [287.79,846.31] [231.80,904.86]

0.4 [490.59,635.02] [435.14,694.31] [379.69,753.59] [324.24,812.88] [268.79,872.17] [213.34,931.45]

0.2 [469.65,658.32] [414.70,718.39] [359.76,778.45] [304.82,838.53] [249.87,898.59] [194.93,958.66]

0.0 [448.91,682.06] [394.44,742.95] [339.97,803.84] [285.49,864.73] [231.02,925.63] [176.55,986.52]

Table. 4.3.9 Cable stayed Bridge - Concomitant Variation of Axial Force in deck in element 12 (kN) w.r.t α and β at γ=1.0

β→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [-406.9,-406.9] [-432.5,-381.2] [-458.1,-355.6] [-483.7,-330.0] [-509.4,-304.4] [-535.0,-278.7]

0.8 [-419.6,-394.5] [-446.0,-369.6] [-472.5,-344.8] [-498.9,-319.9] [-525.3,-295.1] [-551.8,-270.2]

0.6 [-432.7,-382.4] [-460.0,-358.3] [-487.2,-334.2] [-514.5,-310.1] [-541.7,-286.0] [-569.0,-261.9]

0.4 [-446.2,-370.6] [-474.3,-347.2] [-502.4,-323.9] [-530.5,-300.5] [-558.6,-277.2] [-586.7,-253.9]

0.2 [-460.0,-359.1] [-489.0,-336.4] [-518.0,-313.8] [-546.9,-291.2] [-575.9,-268.6] [-604.9,-246.0]

0.0 [-474.3,-347.8] [-504.1,-325.9] [-534.0,-304.0] [-563.9,-282.1] [-593.7,-260.2] [-623.6,-238.3]

126


of Axial Force in deck in element 12 (kN) w.r.t α and β at γ=0.8 β→

α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [-409.7,-404.1] [-435.3,-378.4] [-460.9,-352.8] [-486.5,-327.2] [-512.2,-301.6] [-537.8,-275.9]

0.8 [-422.5,-391.7] [-448.9,-366.9] [-475.4,-342.1] [-501.8,-317.2] [-528.2,-292.4] [-554.6,-267.5]

0.6 [-435.7,-379.7] [-462.9,-355.7] [-490.2,-331.6] [-517.4,-307.5] [-544.7,-283.4] [-572.0,-259.3]

0.4 [-449.2,-368.0] [-477.3,-344.7] [-505.4,-321.3] [-533.5,-298.0] [-561.6,-274.7] [-589.7,-251.3]

0.2 [-463.2,-356.6] [-492.1,-334.0] [-521.1,-311.4] [-550.1,-288.7] [-579.1,-266.1] [-608.0,-243.5]

0.0 [-477.5,-345.4] [-507.4,-323.5] [-537.2,-301.6] [-567.1,-279.7] [-597.0,-257.8] [-626.9,-235.9]

Table. 4.3.11 Cable stayed Bridge - Concomitant Variation of Shear Force in deck just to the left of node 3 (kN) w.r.t α and β at γ=1.0

β→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [-97.8,-97.8] [-105.5, -90.1] [-113.3, -82.4] [-121.0, -74.7] [-128.7, -67.0] [-136.4, -59.3]

0.8 [-99.8,-95.9] [-107.6, -88.3] [-115.4, -80.7] [-123.1, -73.0] [-130.9, -65.4] [-138.7, -57.8]

0.6 [-101.8, -94.0] [-109.7, -86.5] [-117.6, -78.9] [-125.4, -71.4] [-133.3, -63.8] [-141.2, -56.3]

0.4 [-103.8, -92.1] [-111.8, -84.6] [-119.8, -77.2] [-127.8, -69.7] [-135.7, -62.2] [-143.7, -54.8]

0.2 [-105.9, -90.2] [-114.0, -82.8] [-122.0, -75.4] [-130.1, -68.0] [-138.2, -60.6] [-146.2, -53.2]

0.0 [-108.0, -88.4] [-116.1, -81.1] [-124.3, -73.8] [-132.5, -66.5] [-140.6, -59.1] [-148.8, -51.8]

Table. 4.3.12 Cable stayed Bridge - Concomitant Variation of Shear Force in deck just to the left of node 3 (kN) w.r.t α and β at γ=0.8

β→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [-98.6,-97.0] [-106.3, -89.3] [-114.1, -81.6] [-121.8, -73.9] [-129.5, -66.2] [-137.2, -58.5]

0.8 [-100.7, -95.1] [-108.5, -87.5] [-116.3, -79.9] [-124.0, -72.2] [-131.8, -64.6] [-139.6, -57.0]

0.6 [-102.6, -93.2] [-110.5, -85.7] [-118.4, -78.1] [-126.2, -70.6] [-134.1, -63.0] [-142.0, -55.5]

0.4 [-104.6, -91.3] [-112.6, -83.8] [-120.6, -76.4] [-128.6, -68.9] [-136.5, -61.4] [-144.5, -54.0]

0.2 [-106.7, -89.5] [-114.8, -82.1] [-122.8, -74.7] [-130.9, -67.3] [-139.0, -59.9] [-147.0, -52.5]

0.0 [-108.8, -87.6] [-116.9, -80.3] [-125.1, -73.0] [-133.3, -65.7] [-141.4, -58.3] [-149.6, -51.0]

127


of Bending Moment in deck at node 4 (kNm) w.r.t α andβ at γ=1.0 β→

α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [-255.4, -255.4] [-281.4, -229.3] [-307.5, -203.2] [-333.6, -177.1] [-359.7, -151.0] [-385.8, -124.9]

0.8 [-264.7, -246.1] [-291.1, -220.2] [-317.4, -194.4] [-343.7, -168.5] [-370.0, -142.7] [-396.4, -116.8]

0.6 [-274.3, -236.9] [-300.8, -211.2] [-327.4, -185.6] [-354.0, -159.9] [-380.6, -134.3] [-407.1, -108.6]

0.4 [-283.8, -227.6] [-310.6, -202.2] [-337.5, -176.7] [-364.3, -151.3] [-391.2, -125.8] [-418.0, -100.4]

0.2 [-293.5, -218.4] [-320.6, -193.2] [-347.7, -167.9] [-374.8, -142.6] [-402.0, -117.4] [-429.1, -92.1]

0.0 [-303.2 -209.3] [-330.6, -184.2] [-358.0, -159.1] [-385.4, -134.1] [-412.8, -109.0] [-440.3, -83.9]

Table. 4.3.14 Cable stayed Bridge - Concomitant Variation of Bending Moment in deck at node 4 (kNm) w.r.t α andβ at γ=0.8

β→ α↓

1.0 0.8 0.6 0.4 0.2 0.0

1.0 [-258.2, -252.6] [-284.2, -226.5] [-310.3, -200.4] [-336.4, -174.3] [-362.5, -148.2] [-388.6, -122.1]

0.8 [-267.5, -243.3] [-293.9, -217.4] [-320.2, -191.6] [-346.5, -165.7] [-372.8, -139.9] [-399.2, -114.0]

0.6 [-277.1, -234.1] [-303.6, -208.4] [-330.2, -182.8] [-356.8, -157.1] [-383.4, -131.5] [-409.9, -105.8]

0.4 [-286.6, -224.9] [-313.4, -199.5] [-340.3, -174.0] [-367.1, -148.6] [-394.0, -123.1] [-420.8, -97.70]

0.2 [-296.3, -215.7] [-323.4, -190.5] [-350.5, -165.2] [-377.6, -139.9] [-404.8, -114.7] [-431.9, -89.40]

0.0 [-306.1, -206.6] [-333.5, -181.5] [-360.9, -156.4] [-388.3, -131.4] [-415.7, -106.3] [-443.2, -81.20]

Table. 4.3.15 Cable stayed Bridge - Concomitant Variation of Axial Force in pylon in element 17 (at node 11) (kN) w.r.t α and β at γ=1.0

β→ α↓

1.0 0.8 0.6 0.4 0.2 0.0

1.0 [-1264.2, -1264.2] [-1336.7, -1191.7] [-1409.1, -1119.2] [-1481.6, -1046.8] [-1554.1, -974.3] [-1626.6, -901.8]

0.8 [-1291.9, -1237.4] [-1366.0, -1166.4] [-1440.0, -1095.5] [-1514.1, -1024.6] [-1588.2, -953.7] [-1662.3, -882.8]

0.6 [-1320.3, -1211.2] [-1396.0, -1141.9] [-1471.8, -1072.5] [-1547.5, -1003.1] [-1623.3, -933.7] [-1699.0, -864.3]

0.4 [-1349.6, -1185.9] [-1427.1, -1118.0] [-1504.6, -1050.1] [-1582.0, -982.2] [-1659.5, -914.3] [-1737.0, -846.4]

0.2 [-1379.9, -1161.2] [-1459.1, -1094.8] [-1538.3, -1028.3] [-1617.6, -961.8] [-1696.8, -895.4] [-1776.0, -828.9]

0.0 [-1411.1, -1137.2] [-1492.1, -1072.2] [-1573.2, -1007.1] [-1654.2, -942.0] [-1735.3, -877.0] [-1816.3, -811.9]

128


of Axial Force in pylon in element 17 (at node 11) (kN) w.r.t α and β at γ=0.8 β→

α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [-1273.3,-1255.19] [-1345.77,-1182.71] [-1418.24,-1110.23] [-1490.7,-1037.76] [-1563.2, -965.28] [-1635.68, -892.8]

0.8 [-1301.06,-1228.55] [-1375.15,-1157.64] [-1449.24,-1086.73] [-1523.3,-1015.82] [-1597.4,-944.91] [-1671.51, -874.0]

0.6 [-1329.77,-1202.64] [-1405.52,-1133.25] [-1481.27,-1063.86] [-1557.03,-994.47] [-1632.7,-925.09] [-1708.54,-855.7]

0.4 [-1359.23,-1177.43] [-1436.69,-1109.53] [-1514.16,-1041.62] [-1591.63,-973.71] [-1669.0,-905.81] [-1746.56, -837.9]

0.2 [-1389.75,-1152.94] [-1468.98,-1086.47] [-1548.22,-1020.0] [-1627.45,-953.53] [-1706.6,-887.07] [-1785.91, -820.6]

0.0 [-1421.05,-1129.14] [-1502.1-1064.07] [-1583.15,-999.0] [-1664.2 ,-933.93] [-1745.2,-868.87] [-1826.3, -803.8]

Table. 4.3.17 Cable stayed Bridge - Concomitant Variation Of Axial Force (kN) in cable 2 (at node 2) w.r.t α and β at γ=1.0

β→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [441.4,441.4] [406.4,476.4] [371.4,511.4] [336.4,546.4] [301.4,581.4] [266.4,616.4]

0.8 [425.2,458.0] [391.0,493.8] [356.9,529.7] [322.7,565.6] [288.6,601.4] [254.4,637.3]

0.6 [409.2,474.9] [375.9,511.7] [342.5,548.4] [309.2,585.2] [275.8,621.9] [242.5,658.7]

0.4 [393.6,492.2] [361.0,529.8] [328.4,567.5] [295.8,605.2] [263.2,642.9] [230.7,680.6]

0.2 [378.1,509.8] [346.3,548.5] [314.4,587.2] [282.6,625.8] [250.8,664.5] [218.9,703.1]

0.0 [362.9,527.9] [331.8,567.6] [300.7,607.3] [269.6,647.1] [238.5,686.8] [207.4,726.5]

Table. 4.3.18 Cable stayed Bridge - Concomitant Variation Of Axial Force (kN) in cable 2 (at node 2) w.r.t α and β at γ=0.8

β→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [438.0,444.9] [403.0,479.9] [368.0,514.9] [333.0,549.9] [298.0,584.9] [263.0,619.9]

0.8 [421.8,461.5] [387.6,497.3] [353.5,533.2] [319.3,569.1] [285.2,604.9] [251.0,640.8]

0.6 [405.9,478.5] [372.6,515.3] [339.2,552.0] [305.9,588.8] [272.5,625.5] [239.2,662.3]

0.4 [390.3,495.9] [357.7,533.5] [325.1,571.2] [292.5,608.9] [259.9,646.6] [227.4,684.3]

0.2 [374.9,513.5] [343.1,552.2] [311.2,590.9] [279.4,629.5] [247.6,668.2] [215.7,706.8]

0.0 [359.8,531.7] [328.7,571.4] [297.6,611.1] [266.5,650.9] [235.4,690.6] [204.3,730.3]

129

130

Table. 4.3.19 Cable stayed Bridge - Concomitant Variation of Axial Force (kN) in cable 3 (at node 3) w.r.t α and β at γ=1.0

β→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [473.5,473.5] [442.1,504.9] [410.7,536.3] [379.4,567.7] [348.0,599.1] [316.6,630.5]

0.8 [456.0,491.3] [425.3,523.4] [394.5,555.5] [363.8,587.5] [333.0,619.6] [302.3,651.6]

0.6 [438.9,509.5] [408.7,542.3] [378.6,575.1] [348.5,607.8] [318.4,640.6] [288.2,673.3]

0.4 [422.0,528.1] [392.5,561.5] [362.9,595.0] [333.4,628.5] [303.9,662.0] [274.3,695.4]

0.2 [405.4,547.1] [376.5,581.3] [347.5,615.5] [318.5,649.8] [289.6,684.0] [260.6,718.2]

0.0 [389.1,566.6] [360.7,601.6] [332.3,636.6] [303.9,671.6] [275.5,706.7] [247.0,741.7]

Table. 4.3.20 Cable stayed Bridge - Concomitant Variation of Axial Force (kN) in cable 3 (at node 3) w.r.t α and β at γ=0.8

β→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [469.60,477.32] [438.23,508.71] [406.84,540.10] [375.45,571.49] [344.06,602.88] [312.67,634.27]

0.8 [452.20,495.23] [421.45,527.29] [390.71,559.35] [359.96,591.41] [329.21,623.46] [298.47,655.52]

0.6 [435.16,513.54] [405.04,546.29] [374.91,579.05] [344.78,611.80] [314.65,644.55] [284.52,677.31]

0.4 [418.41,532.25] [388.87,565.73] [359.34,599.21] [329.80,632.68] [300.27,666.16] [270.74,699.64]

0.2 [401.93,551.38] [372.96,585.60] [343.99,619.83] [315.03,654.06] [286.07,688.28] [257.11,722.51]

0.0 [385.72,570.92] [357.30,605.92] [328.88,640.93] [300.46,675.94] [272.05,710.95] [243.63,745.95]

131

(a) beta=1.0 , gamma=1.0

9.236E-04

9.473E-04

9.714E-04

9.955E-04

1.020E-03

1.044E-03

1.069E-03

1.094E-03

1.119E-03

1.145E-03

1.171E-03

1.044E-03

0

0.2

0.4

0.6

0.8

1


alph

a


8.194E-04

8.427E-04

8.663E-04

8.900E-04

9.139E-04

9.380E-04 1.151E-03

1.176E-03

1.202E-03

1.228E-03

1.254E-03

1.280E-030

0.2

0.4

0.6

0.8

1


alph

a

(c) alpha=1.0 , gamma=1.0

5.122E-04

6.187E-04

7.251E-04

8.316E-04

9.380E-04

1.044E-031.044E-03

1.151E-03

1.257E-03

1.364E-03

1.470E-03

1.577E-030

0.2

0.4

0.6

0.8

1


beta

(d) alpha=0.6 , gamma=1.0

4.680E-04

5.735E-04

6.790E-04

7.845E-04

8.900E-04

1.094E-03

1.202E-03

1.309E-03

1.417E-03

1.525E-03

1.632E-03

9.955E-04

0

0.2

0.4

0.6

0.8

1


beta

Fig. 4.3.1 Membership Functions for horizontal displacement at node 2 at γ=1.0

132


9.122E-04

9.359E-04

9.599E-04

9.840E-04

1.008E-03

1.056E-03

1.081E-03

1.106E-03

1.131E-03

1.157E-03

1.183E-03

1.033E-03

0

0.2

0.4

0.6

0.8

1


alph

a

(b) beta=0.8 and gamma=0.8

8.080E-04

8.313E-04

8.548E-04

8.785E-04

9.024E-04

9.263E-04 1.163E-03

1.188E-03

1.214E-03

1.239E-03

1.266E-03

1.292E-030

0.2

0.4

0.6

0.8

1


alph

a

(c) alpha=1.0 and gamma=0.8

5.005E-04

6.070E-04

7.134E-04

8.199E-04

9.263E-04

1.056E-03

1.163E-03

1.269E-03

1.375E-03

1.482E-03

1.588E-03

1.033E-03

0

0.2

0.4

0.6

0.8

1


beta

(d) alpha=0.6 and gamma=0.8

4.565E-04

5.620E-04

6.675E-04

7.730E-04

8.785E-04

1.106E-03

1.214E-03

1.321E-03

1.429E-03

1.536E-03

1.644E-03

9.840E-04

0

0.2

0.4

0.6

0.8

1


beta

Fig. 4.3.2 Membership Functions for horizontal displacement at node 2 at γ=0.8

133


-6.21E-02

-5.92E-02

-5.62E-02

-5.33E-02

-5.03E-02

-4.74E-02-4.74E-02

-4.45E-02

-4.16E-02

-3.87E-02

-3.58E-02

-3.29E-020

0.2

0.4

0.6

0.8

1

-6.30E-02 -5.95E-02 -5.60E-02 -5.25E-02 -4.90E-02 -4.55E-02 -4.20E-02 -3.85E-02 -3.50E-02Vertical Displacement (m)

alph

a

(b) beta=0.8 , gamma=1.0

-7.54E-02

-7.24E-02

-6.94E-02

-6.64E-02

-6.34E-02

-3.44E-02

-3.15E-02

-2.87E-02

-2.58E-02

-2.29E-02

-2.01E-02

-6.04E-02

0

0.2

0.4

0.6

0.8

1


alph

a


-1.13E-01

-9.95E-02

-8.64E-02

-7.34E-02

-6.04E-02

-4.74E-02-4.74E-02

-3.44E-02

-2.14E-02

-8.40E-03

4.60E-03

1.76E-020

0.2

0.4

0.6

0.8

1

-1.13E-01 -9.80E-02 -8.30E-02 -6.80E-02 -5.30E-02 -3.80E-02 -2.30E-02 -8.00E-03 7.00E-03Vertical Displacement (m)

beta


-1.19E-01

-1.06E-01

-9.26E-02

-7.95E-02

-6.64E-02

-4.16E-02

-2.87E-02

-1.57E-02

-2.80E-03

1.02E-02

2.31E-02

-5.33E-02

0

0.2

0.4

0.6

0.8

1

-1.19E-01 -1.04E-01 -8.90E-02 -7.40E-02 -5.90E-02 -4.40E-02 -2.90E-02 -1.40E-02 1.00E-03 1.60E-02Vertical Displacement (m)

beta

Fig. 4.3.3 Membership Functions for vertical displacement at node 3 at γ=1.0

134


-6.36E-02

-6.06E-02

-5.76E-02

-5.47E-02

-5.18E-02

-4.60E-02

-4.31E-02

-4.02E-02

-3.73E-02

-3.44E-02

-3.15E-02

-4.88E-02

0

0.2

0.4

0.6

0.8

1


alph

a


-7.68E-02

-7.38E-02

-7.08E-02

-6.78E-02

-6.48E-02

-6.18E-02 -3.29E-02

-3.01E-02

-2.73E-02

-2.44E-02

-2.15E-02

-1.87E-020

0.2

0.4

0.6

0.8

1

-7.68E-02 -6.93E-02 -6.18E-02 -5.43E-02 -4.68E-02 -3.93E-02 -3.18E-02 -2.43E-02Vertical Displacement (m)

alph

a


-1.14E-01

-1.01E-01

-8.79E-02

-7.49E-02

-6.18E-02

-4.60E-02

-3.29E-02

-1.99E-02

-6.90E-03

6.00E-03

1.90E-02

-4.88E-02

0

0.2

0.4

0.6

0.8

1

-1.14E-01 -9.90E-02 -8.40E-02 -6.90E-02 -5.40E-02 -3.90E-02 -2.40E-02 -9.00E-03 6.00E-03Vertical Displacement (m)

beta


-1.20E-01

-1.07E-01

-9.40E-02

-8.09E-02

-6.78E-02 -2.73E-02

-1.43E-02

-1.40E-03

1.16E-02

2.45E-02

-5.47E-02-4.02E-02

0

0.2

0.4

0.6

0.8

1

-1.20E-01 -1.03E-01 -8.50E-02 -6.75E-02 -5.00E-02 -3.25E-02 -1.50E-02 2.50E-03 2.00E-02Vertical Displacement (m)

beta

Fig. 4.3.4 Membership Functions for vertical displacement at node 3 at γ=0.8

135


3.890E-04

5.560E-04

7.240E-04

8.920E-04

1.059E-03

1.227E-031.227E-03

1.394E-03

1.562E-03

1.731E-03

1.900E-03

2.069E-030

0.2

0.4

0.6

0.8

1

3.000E-04 5.500E-04 8.000E-04 1.050E-03 1.300E-03 1.550E-03 1.800E-03 2.050E-03Rotation (radians)

alph

a


-3.540E-04

-1.880E-04

-2.000E-05

1.460E-04

3.120E-04

4.790E-04 1.975E-03

2.144E-03

2.314E-03

2.485E-03

2.656E-03

2.828E-030

0.2

0.4

0.6

0.8

1

-5.000E-04 0.000E+00 5.000E-04 1.000E-03 1.500E-03 2.000E-03 2.500E-03 3.000E-03Rotation (radians)

alph

a


-2.513E-03

-1.765E-03

-1.017E-03

-2.690E-04

4.790E-04

1.227E-031.227E-03

1.975E-03

2.723E-03

3.471E-03

4.219E-03

4.967E-030

0.2

0.4

0.6

0.8

1


beta


-2.837E-03

-2.092E-03

-1.346E-03

-6.000E-04

1.460E-04

1.562E-03

2.314E-03

3.067E-03

3.819E-03

4.571E-03

5.323E-03

8.920E-04

0

0.2

0.4

0.6

0.8

1


beta

Fig. 4.3.5 Membership Functions for rotation at node 4 at γ=1.0

136


3.100E-04

4.770E-04

6.440E-04

8.120E-04

9.800E-04

1.306E-03

1.474E-03

1.642E-03

1.811E-03

1.981E-03

2.150E-03

1.147E-03

0

0.2

0.4

0.6

0.8

1

2.500E-04 5.500E-04 8.500E-04 1.150E-03 1.450E-03 1.750E-03 2.050E-03Rotation (radians)

alph

a


-4.330E-04

-2.670E-04

-1.000E-04

6.600E-05

2.330E-04

3.990E-04 2.054E-03

2.224E-03

2.394E-03

2.565E-03

2.737E-03

2.909E-030

0.2

0.4

0.6

0.8

1

-5.000E-04 0.000E+00 5.000E-04 1.000E-03 1.500E-03 2.000E-03 2.500E-03 3.000E-03Rotation (radians)

alph

a


-2.593E-03

-1.845E-03

-1.097E-03

-3.490E-04

3.990E-04

1.306E-03

2.054E-03

2.802E-03

3.550E-03

4.298E-03

5.046E-03

1.147E-03

0

0.2

0.4

0.6

0.8

1


beta


-2.917E-03

-2.172E-03

-1.426E-03

-6.800E-04

6.600E-05

1.642E-03

2.394E-03

3.147E-03

3.899E-03

4.651E-03

5.403E-03

8.120E-04

0

0.2

0.4

0.6

0.8

1

-3.500E-03 -2.250E-03 -1.000E-03 2.500E-04 1.500E-03 2.750E-03 4.000E-03 5.250E-03Rotation (radians)

beta

Fig. 4.3.6 Membership Functions for rotation at node 4 at γ=0.8

137

(a) beta= 1.0,gamma=1.0

454.85

475.64

496.64

517.87

539.38

561.16561.16

583.27

605.72

628.55

651.76

675.410

0.2

0.4

0.6

0.8

1

425 460 495 530 565 600 635 670Axial force (kN)

alph

a

(b) beta= 0.8,gamma=1.0

400.38

420.69

441.19

461.87

482.8

503.97 618.36

641.12

664.28

687.84

711.83

736.30

0.2

0.4

0.6

0.8

1

375 425 475 525 575 625 675 725Axial Force (kN)

alph

a

(c) alpha= 1.0,gamma=1.0

275.21

332.4

389.59

446.78

503.97

561.16561.16

618.36

675.55

732.74

789.94

847.110

0.2

0.4

0.6

0.8

1

250 330 410 490 570 650 730 810Axial Force (kN)

beta

(d) alpha= 0.6,gamma=1.0

237.91

293.9

349.89

405.89

461.87

605.72

664.28

722.82

781.37

839.92

898.47

517.87

0

0.2

0.4

0.6

0.8

1

200 275 350 425 500 575 650 725 800 875Axial Force (kN)

beta

Fig. 4.3.7 Membership Functions for Axial Force in deck in element 9 at γ=1.0

138

(a) beta= 1.0,gamma=0.8

448.91

469.65

490.59

511.76

533.21

567.4

589.59

612.11

635.02

658.32

682.06

554.92

0

0.2

0.4

0.6

0.8

1

425.0 460.0 495.0 530.0 565.0 600.0 635.0 670.0Axial Force (kN)

alph

a

(b) beta= 0.8,gamma=0.8

394.44

414.7

435.14

455.76

476.63

497.73 624.6

647.44

670.67

694.31

718.39

742.950

0.2

0.4

0.6

0.8

1

375 420 465 510 555 600 645 690 735Axial Force (kN)

alph

a

(c) alpha= 1.0,gamma=0.8

268.97

326.16

383.35

440.54

497.73

567.4

624.6

681.79

738.98

796.18

853.35

554.92

0

0.2

0.4

0.6

0.8

1

250 325 400 475 550 625 700 775 850Axial Force (kN)

beta

(d) alpha= 0.6,gamma=0.8

231.8

287.79

343.78

399.78

455.76

511.76 612.11

670.67

729.21

787.76

846.31

904.860

0.2

0.4

0.6

0.8

1

200 275 350 425 500 575 650 725 800 875Axial Force (kN)

beta


139

(a) beta=1.0,gamma=1.0

-474.3

-460.0

-446.2

-432.7

-419.6

-406.9-406.9

-394.5

-382.4

-370.6

-359.1

-347.80

0.2

0.4

0.6

0.8

1

-485.0 -465.0 -445.0 -425.0 -405.0 -385.0 -365.0 -345.0 -325.0Axial Force (kN)

alph

a

(b) beta=0.8,gamma=1.0

-504.1

-489.0

-474.3

-460.0

-446.0

-432.5 -381.2

-369.6

-358.3

-347.2

-336.4

-325.90

0.2

0.4

0.6

0.8

1

-525.0 -500.0 -475.0 -450.0 -425.0 -400.0 -375.0 -350.0 -325.0 -300.0Axial Force (kN)

alph

a

(c) alpha=1.0,gamma=1.0

-535.0

-509.4

-483.7

-458.1

-432.5

-406.9-406.9

-381.2

-355.6

-330.0

-304.4

-278.70

0.2

0.4

0.6

0.8

1

-550.0 -515.0 -480.0 -445.0 -410.0 -375.0 -340.0 -305.0Axial Force (kN)

beta

(d) alpha=0.6,gamma=1.0

-569.0

-541.7

-514.5

-487.2

-460.0

-432.7 -382.4

-358.3

-334.2

-310.1

-286.0

-261.90

0.2

0.4

0.6

0.8

1

-600.0 -550.0 -500.0 -450.0 -400.0 -350.0 -300.0 -250.0Axial Force (kN)

beta


140


-477.50

-463.16

-449.22

-435.67

-422.49

-404.07

-391.74

-379.73

-368.02

-356.59

-345.44

-409.70

0

0.2

0.4

0.6

0.8

1

-480.00 -460.00 -440.00 -420.00 -400.00 -380.00 -360.00 -340.00Axial Force (kN)

alph

a


-507.37

-492.13

-477.32

-462.93

-448.92

-435.27 -378.44

-366.89

-355.65

-344.68

-333.97

-323.530

0.2

0.4

0.6

0.8

1

-525.00 -500.00 -475.00 -450.00 -425.00 -400.00 -375.00 -350.00 -325.00 -300.00Axial Force (kN)

alph

a


-537.78

-512.16

-486.53

-460.90

-435.27

-404.07

-378.44

-352.81

-327.18

-301.56

-275.93

-409.70

0

0.2

0.4

0.6

0.8

1

-550.00 -505.00 -460.00 -415.00 -370.00 -325.00 -280.00Axial Force (kN)

beta


-571.95

-544.69

-517.44

-490.18

-462.93

-435.67 -379.73

-355.65

-331.56

-307.48

-283.39

-259.310

0.25

0.5

0.75

1

-575.00 -535.00 -495.00 -455.00 -415.00 -375.00 -335.00 -295.00 -255.00Axial Force (kN)

beta


141


-108.0

-105.9

-103.8

-101.8

-99.8

-97.8-97.8

-95.9

-94.0

-92.1

-90.2

-88.40

0.2

0.4

0.6

0.8

1

-109.0 -106.0 -103.0 -100.0 -97.0 -94.0 -91.0 -88.0Shear Force (kN)

alph

a


-116.1

-114.0

-111.8

-109.7

-107.6

-105.5 -90.1

-88.3

-86.5

-84.6

-82.8

-81.10

0.2

0.4

0.6

0.8

1

-118.0 -113.0 -108.0 -103.0 -98.0 -93.0 -88.0 -83.0Shear Force (kN)

alph

a


-136.4

-128.7

-121.0

-113.3

-105.5

-97.8-97.8

-90.1

-82.4

-74.7

-67.0

-59.30

0.2

0.4

0.6

0.8

1

-140.0 -129.0 -118.0 -107.0 -96.0 -85.0 -74.0 -63.0Shear Force (kN)

beta


-141.2

-133.3

-125.4

-117.6

-109.7

-101.8 -94.0

-86.5

-78.9

-71.4

-63.8

-56.30

0.2

0.4

0.6

0.8

1

-145.0 -135.0 -125.0 -115.0 -105.0 -95.0 -85.0 -75.0 -65.0 -55.0Shear Force (kN)

beta

Fig. 4.3.11 Membership Functions for Shear Force in deck to left of node 3 at γ=1.0

142


-108.8

-106.7

-104.6

-102.6

-100.7

-98.6 -97.0

-95.1

-93.2

-91.3

-89.5

-87.60

0.2

0.4

0.6

0.8

1

-110.0 -107.0 -104.0 -101.0 -98.0 -95.0 -92.0 -89.0 -86.0Shear Force (kN)

alph

a


-116.9

-114.8

-112.6

-110.5

-108.5

-106.3 -89.3

-87.5

-85.7

-83.8

-82.1

-80.30

0.2

0.4

0.6

0.8

1

-120.0 -115.0 -110.0 -105.0 -100.0 -95.0 -90.0 -85.0 -80.0Shear Force (kN)

alph

a


-137.2

-129.5

-121.8

-114.1

-106.3

-97.0

-89.3

-81.6

-73.9

-66.2

-58.5

-98.6

0

0.2

0.4

0.6

0.8

1

-140.0 -129.0 -118.0 -107.0 -96.0 -85.0 -74.0 -63.0Shear Force (kN)

beta


-142.0

-134.1

-126.2

-118.4

-110.5

-102.6 -93.2

-85.7

-78.1

-70.6

-63.0

-55.50

0.2

0.4

0.6

0.8

1

-150.0 -137.0 -124.0 -111.0 -98.0 -85.0 -72.0 -59.0Shear Force (kN)

beta

Fig. 4.3.12 Membership Functions for Shear Force in deck to left of node 3 at γ=0.8

143


-303.2

-293.5

-283.8

-274.3

-264.7

-255.4-255.4

-246.1

-236.9

-227.6

-218.4

-209.30

0.2

0.4

0.6

0.8

1


beta


-330.6

-320.6

-310.6

-300.8

-291.1

-281.4 -229.3

-220.2

-211.2

-202.2

-193.2

-184.20

0.2

0.4

0.6

0.8

1


alph

a


-385.8

-359.7

-333.6

-307.5

-281.4

-255.4-255.4

-229.3

-203.2

-177.1

-151.0

-124.90

0.2

0.4

0.6

0.8

1


beta


-407.1

-380.6

-354.0

-327.4

-300.8

-274.3 -236.9

-211.2

-185.6

-159.9

-134.3

-108.60

0.2

0.4

0.6

0.8

1


beta

Fig. 4.3.13 Membership Functions for Bending Moment at node 4 at γ=1.0

144


-306.1

-296.3

-286.6

-277.1

-267.5

-252.6

-243.3

-234.1

-224.9

-215.7

-206.6

-258.2

0

0.2

0.4

0.6

0.8

1

-310.0 -296.0 -282.0 -268.0 -254.0 -240.0 -226.0 -212.0Bending Moment (kNm)

alph

a


-333.5

-323.4

-313.4

-303.6

-293.9

-284.2 -226.5

-217.4

-208.4

-199.5

-190.5

-181.50

0.2

0.4

0.6

0.8

1


alph

a


-388.6

-362.5

-336.4

-310.3

-284.2

-252.6

-226.5

-200.4

-174.3

-148.2

-122.1

-258.2

0

0.2

0.4

0.6

0.8

1


beta


-409.9

-383.4

-356.8

-330.2

-303.6

-277.1 -234.1

-208.4

-182.8

-157.1

-131.5

-105.80

0.2

0.4

0.6

0.8

1


beta

Fig. 4.3.14 Membership Functions for Bending Moment at node 4 at γ=0.8

145


-1411.1

-1379.9

-1349.6

-1320.3

-1291.9

-1264.2-1264.2

-1237.4

-1211.2

-1185.9

-1161.2

-1137.20

0.2

0.4

0.6

0.8

1

-1420.0 -1382.0 -1344.0 -1306.0 -1268.0 -1230.0 -1192.0 -1154.0Axial Force (kN)

alph

a


-1492.1

-1459.1

-1427.1

-1396.0

-1366.0

-1336.7 -1191.7

-1166.4

-1141.9

-1118.0

-1094.8

-1072.20

0.2

0.4

0.6

0.8

1

-1500.0 -1450.0 -1400.0 -1350.0 -1300.0 -1250.0 -1200.0 -1150.0 -1100.0 -1050.0Axial Force (kN)

alph

a


-1626.6

-1554.1

-1481.6

-1409.1

-1336.7

-1264.2-1264.2

-1191.7

-1119.2

-1046.8

-974.3

-901.80

0.2

0.4

0.6

0.8

1

-1650.0 -1550.0 -1450.0 -1350.0 -1250.0 -1150.0 -1050.0 -950.0Axial Force (kN)

beta


-864.3

-933.7

-1003.1

-1072.5

-1141.9

-1211.2-1320.3

-1396.0

-1471.8

-1547.5

-1623.3

-1699.00

0.2

0.4

0.6

0.8

1

-1750.0 -1645.0 -1540.0 -1435.0 -1330.0 -1225.0 -1120.0 -1015.0 -910.0Axial Force (kN)

beta

Fig. 4.3.15 Membership Functions for Axial Force in pylon in element 17 at γ=1.0

146


-1421.05

-1389.75

-1359.23

-1329.77

-1301.06

-1273.3-1255.19

-1228.55

-1202.64

-1177.43

-1152.94

-1129.140

0.2

0.4

0.6

0.8

1

-1430 -1390 -1350 -1310 -1270 -1230 -1190 -1150Axial Force(kN)

alph

a


-1502.1

-1468.98

-1436.69

-1405.52

-1375.15

-1345.77 -1182.71

-1157.64

-1133.25

-1109.53

-1086.47

-1064.070

0.2

0.4

0.6

0.8

1

-1525 -1465 -1405 -1345 -1285 -1225 -1165 -1105Axial Force (kN)

alph

a


-1635.68

-1563.2

-1490.72

-1418.24

-1345.77

-1255.19

-1182.71

-1110.23

-1037.76

-965.28

-892.8

-1273.3

0

0.2

0.4

0.6

0.8

1

-1650 -1550 -1450 -1350 -1250 -1150 -1050 -950 -850Axial Force (kN)

beta


-1708.54

-1632.78

-1557.03

-1481.27

-1405.52

-1329.77 -1202.64

-1133.25

-1063.86

-994.47

-925.09

-855.70

0.2

0.4

0.6

0.8

1

-1750 -1640 -1530 -1420 -1310 -1200 -1090 -980 -870Axial Force(kN)

beta

Fig. 4.3.16 Membership Functions for Axial Force in pylon in element 17

147


362.9

378.1

393.6

409.2

425.2

441.4441.4

458.0

474.9

492.2

509.8

527.90

0.2

0.4

0.6

0.8

1

350.0 370.0 390.0 410.0 430.0 450.0 470.0 490.0 510.0 530.0 550.0Axial Force (kN)

alph

a


331.8

346.3

361.0

375.9

391.0

406.4 476.4

493.8

511.7

529.8

548.5

567.60

0.2

0.4

0.6

0.8

1

325.0 355.0 385.0 415.0 445.0 475.0 505.0 535.0 565.0Axial Force (kN)

alph

a


266.4

301.4

336.4

371.4

406.4

441.4441.4

476.4

511.4

546.4

581.4

616.40

0.2

0.4

0.6

0.8

1

250.0 290.0 330.0 370.0 410.0 450.0 490.0 530.0 570.0 610.0Axial Force (kN)

beta


242.5

275.8

309.2

342.5

375.9

409.2 474.9

511.7

548.4

585.2

621.9

658.70

0.2

0.4

0.6

0.8

1

225.0 275.0 325.0 375.0 425.0 475.0 525.0 575.0 625.0 675.0Axial Force (kN)

beta

Fig. 4.3.17 Membership Functions for Axial Force in cable 2 at γ=1.0

148


359.8

374.9

390.3

405.9

421.8

444.9

461.5

478.5

495.9

513.5

531.7

438.0

0

0.2

0.4

0.6

0.8

1

350.0 375.0 400.0 425.0 450.0 475.0 500.0 525.0Axial Force (kN)

alph

a


328.7

343.1

357.7

372.6

387.6

403.0 479.9

497.3

515.3

533.5

552.2

571.40

0.2

0.4

0.6

0.8

1

310.0 340.0 370.0 400.0 430.0 460.0 490.0 520.0 550.0 580.0Axial Force (kN)

alph

a


263.0

298.0

333.0

368.0

403.0

444.9

479.9

514.9

549.9

584.9

619.9

438.0

0

0.2

0.4

0.6

0.8

1

255.0 300.0 345.0 390.0 435.0 480.0 525.0 570.0 615.0Axial Force (kN)

beta


239.2

272.5

305.9

339.2

372.6

405.9 478.5

515.3

552.0

588.8

625.5

662.30

0.2

0.4

0.6

0.8

1

200.0 250.0 300.0 350.0 400.0 450.0 500.0 550.0 600.0 650.0 700.0Axial Force (kN)

beta


149


389.1

405.4

422.0

438.9

456.0

473.5473.5

491.3

509.5

528.1

547.1

566.60

0.2

0.4

0.6

0.8

1

375.0 400.0 425.0 450.0 475.0 500.0 525.0 550.0 575.0Axial Force (kN)

alph

a


360.7

376.5

392.5

408.7

425.3

442.1 504.9

523.4

542.3

561.5

581.3

601.60

0.2

0.4

0.6

0.8

1

350.0 380.0 410.0 440.0 470.0 500.0 530.0 560.0 590.0Axial Force (kN)

alph

a


316.6

348.0

379.4

410.7

442.1

473.5473.5

504.9

536.3

567.7

599.1

630.50

0.2

0.4

0.6

0.8

1

300.0 350.0 400.0 450.0 500.0 550.0 600.0 650.0Axial Force (kN)

beta


288.2

318.4

348.5

378.6

408.7

438.9 509.5

542.3

575.1

607.8

640.6

673.30

0.2

0.4

0.6

0.8

1

280.0 330.0 380.0 430.0 480.0 530.0 580.0 630.0 680.0Axial Force (kN)

beta


150


385.72

401.93

418.41

435.16

452.2

477.32

495.23

513.54

532.25

551.38

570.92

469.6

0

0.2

0.4

0.6

0.8

1

380 405 430 455 480 505 530 555Axial Force (kN)

alph

a


357.3

372.96

388.87

405.04

421.45

438.23 508.71

527.29

546.29

565.73

585.6

605.920

0.2

0.4

0.6

0.8

1

350 380 410 440 470 500 530 560 590Axial Force (kN)

alph

a


312.67

344.06

375.45

406.84

438.23

477.32

508.71

540.1

571.49

602.88

634.27

469.6

0

0.2

0.4

0.6

0.8

1

300 350 400 450 500 550 600Axial Force (kN)

beta


284.52

314.65

344.78

374.91

405.04

435.16 513.54

546.29

579.05

611.8

644.55

677.310

0.2

0.4

0.6

0.8

1

270 315 360 405 450 495 540 585 630 675Axial Force (kN)

beta


4.4 Sensitivity Analysis

As discussed in section 2.9 and section 3.6, sensitivity analysis aims at

analysing the relative variation of structural response to a given variation of structural

characteristics. Sensitivity analysis is performed to study

a) Sensitivity of structural response quantities at a given node.

b) Sensitivity of a given structural response quantity (displacements or forces)

across a set of nodes.

In the present study, sensitivity of structural response quantities

i.e. displacements, rotations, forces and moments is performed and the relative

sensitivity is ascertained. The results are presented and discussed in the following

sections.

4.4.1 Sensitivity analysis of displacements and rotations at a given node

Fig. 4.4.1 depicts the sensitivity analysis of displacements of the deck under

the action of dead load at γ=1.0 with respect to percentage variation of material

property (E) about its mean value. In this figure, the relative sensitivity of horizontal

displacement, vertical displacement and rotation are studied. It is observed from this

plot that that the slopes of the plots of horizontal displacement, vertical displacement

and rotation are 3.054, 6.071,5.591 respectively. Fig. 4.4.2 depicts the sensitivity

analysis of displacements of the deck under the action of dead load at γ=0.8 with

respect to percentage variation of material property (E) about its mean value. It is

observed from this plot that that the slopes of the plots of horizontal displacement,

vertical displacement and rotation are 3.519,6.934 and 6.456 respectively.

Fig. 4.4.3 depicts the sensitivity analysis of displacements of the deck under the action

of live load at α=1.0 with respect to percentage variation of live load about its mean

151

value. It is observed from this plot that that the slopes of the plots of horizontal

displacement, vertical displacement and rotation are 2.23, 4.413 and 4.372

respectively. Thus it is observed that in all these cases, vertical displacement has the

greatest sensitivity and horizontal displacement has the lowest sensitivity.

4.4.2 Sensitivity analysis of forces and moments at a given node

Fig. 4.4.4 depicts the sensitivity analysis of forces and moments of the deck


property (E) about its mean value. In this figure, the relative sensitivity of axial force,

shear force and bending moment are studied. It is observed from this plot that that the

slopes of the plots of axial force, shear force and bending moment are 2.902, 2.218

and 3.28 respectively.

Fig. 4.4.5 depicts the sensitivity analysis of forces and moments of the deck


property (E) about its mean value. It is observed from this plot that that the slopes of

the plots of axial force, shear force and bending moment are 3.074, 2.41 and 3.566

respectively. Thus it is observed that in all these cases, bending moment has the

greatest sensitivity and shear force has the lowest sensitivity. Further, the sensitivities

of all the above quantities are more at γ=0.8 compared to γ=1.0 because the

corresponding slopes of axial force, shear force and bending moment are more in case

of γ=0.8 than in the case of γ=1.0.

4.4.3 Sensitivity analysis of a given response quantity at different nodes

Fig. 4.4.6 depicts the sensitivity analysis of bending moment at nodes 2,3,4

and 5 on the deck slab under the action of dead load at γ=1.0 with respect to

152

percentage variation of material property (E) about its mean value. It is observed

from this plot that that the slopes of the plots of bending moment at these nodes are

16.208, 3.046, 3.280 and 3.648 respectively. Thus it is observed that the sensitivity of

bending moment varies along the span of the deck slab. The sensitivity at the node 2

is the highest because the bending moment increases from a value of zero at node 1 to

a non-zero value at node 2 along the deck slab.

Fig. 4.4.7 and Fig. 4.4.8 depict the sensitivity analysis of axial force at nodes

2, 4 and 5 on the deck slab under the action of live load, at β=1.0 and β=0.8

respectively, with respect to percentage variation of material property (E) about its

mean value. It is observed from Fig. 4.4.7 that that the slopes of the plots of axial

force at these nodes are 4.068, 4.904 and 2.902 respectively. Thus it is observed that

the sensitivity of axial force varies along the span of the deck slab in this case. The

sensitivity at the node 4 is the highest and the sensitivity at node 5 is the lowest.

It is observed from Fig. 4.4.8 that that the slopes of the plots of axial force at

the nodes 2, 4 and 5 on the deck slab are 4.498, 2.246 and 2.592 respectively. Thus it

is observed that the sensitivity of axial force varies along the span of the deck slab in

this case. The sensitivity at the node 2 is the highest and the sensitivity at node 4 is

the lowest.

Fig. 4.4.9 depicts the sensitivity analysis of shear force at nodes 1,3, 4 and 5

on the deck slab under the action of live load, at β=1.0, with respect to percentage

variation of material property (E) about its mean value. It is observed from Fig. 4.4.9

that that the slopes of the plots of shear force at these nodes are 1.698, 2.012, 2.396

and 2.128 respectively. Thus it is observed that the sensitivity of shear force varies

along the span of the deck slab in this case. The sensitivity at the node 4 is the highest

and the sensitivity at node 1 is the lowest.

153

Fig. 4.4.1 Sensitivity Analysis for displacements (for dead load) at node 3 at gamma=1.0

0.000

3.021

6.054

9.122

12.214

15.330

0.000

6.089

12.148

18.237

24.296

30.355

0.000

5.502

11.084

16.667

22.290

27.953

0.000

5.000

10.000

15.000

20.000

25.000

30.000

35.000

0.0 1.0 2.0 3.0 4.0 5.0


Perc

enta

ge V

aria

tion

of d

ispl

acem

ent a

bout

mea

n

Horizontal DisplacementVertical displacementRotation

Fig. 4.4.2 Sensitivity Analysis for displacements (for dead load) at node 3 at gamma=0.8

2.227

5.248

8.293

11.373

14.465

17.593

4.345

10.435

16.493

22.552

28.611

4.248

9.790

15.372

20.955

26.618

32.282

0.0

5.0

10.0

15.0

20.0

25.0

30.0

0.0 1.0 2.0 3.0 4.0 5.0


Perc

enta

ge V

aria

tion

of d

ispl

acem

ent a

bout

mea

n

Horizontal displacementVertical displacementRotation

35.0 34.670

154

Fig. 4.4.3 Sensitivity of displacements at node 3 in deck at alpha=1.0

0.00

44.62

89.24

133.86

178.58

223.07

0.00

88.27

176.53

264.79

353.05

0.00

87.38

174.84

262.31

349.69

441.32437.15

0.00

50.00

100.00

150.00

200.00

250.00

300.00

350.00

400.00

450.00

0.0 20.0 40.0 60.0 80.0 100.0

Percentage variation of live load about mean

Perc

enta

ge v

aria

tion

of d

ispl

acem

ent a

bout

mea

n

Horizontal displacementVertical displacementRotation

Fig 4.4.4 Sensitivity Analysis of Forces and Moments in Element 12 (at node 5) at gamma=1.0

0.00

3.05

6.03

8.93

11.75

14.51

0.00

2.07

4.17

6.29

8.45

10.64

0.00

3.34

6.65

9.92

13.17

16.40

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

0.0 1.0 2.0 3.0 4.0 5.0Percentage Variation of E about mean

Perc

enta

ge V

aria

tion

of F

orce

/mom

ent a

bout

mea

n

Axial ForceShear Force Bending Moment

155

Fig 4.4.5 Sensitivity Analysis of Forces and Moments in Element 12 (at node 5)at gamma=0.8

1.00

4.02

6.97

9.84

12.64

15.37

1.34

3.43

5.54

7.67

9.84

12.05

1.51

4.83

8.12

11.38

14.62

17.83

0

0

0

0

0

.0

.0

.0

.0

.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0


Axial ForceShear ForceBending Moment

0.

2.

4.

6.

8.

10

12

14

16

18

Perc

enta

ge V

aria

tion

of F

orce

/mom

ent a

bout

mea

n

156

Fig 4.4.6 Sensitivity Analysis for Bending Moment (due to dead Load) in deck slab at gamma=1

0.00

15.92

31.97

48.15

64.50

81.04

0.003.12

6.209.24

12.25

0.003.69

7.3511.00

14.62

0.003.34

6.659.92

13.17

15.2318.24

16.40

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

0.0 1.0 2.0 3.0 4.0 5.0

Percentage variation of E about mean

Pere

cnta

ge v

aria

tion

of B

endi

ng m

omen

t abo

ut m

ean

Node 2Node 3Node 4Node 5

156

157

Fig 4.4.7 Sensitivity Analysis for Axial Force (due to Live Load) in deck slab at beta=1.0

0.00

3.94

7.93

12.00

16.13

20.34

0.00

5.04

10.01

14.91

19.74

24.52

0.00

3.05

6.03

8.93

11.75

14.51

0.0

5.0

10.0

15.0

20.0

25.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0


Perc

enta

ge v

aria

tion

of a

xial

forc

e ab

out m

ean

Node 2Node 4Node 5

Fig. 4.4.8 Sensitivity Analysis of Axial Force due to live load in deck slab at beta=0.8

32.50

36.82

41.22

45.72

50.30

24.3026.46

28.6630.91

33.19

24.3727.01

29.6232.21

34.78

54.99

35.53

37.33

0.00

10.00

20.00

30.00

40.00

50.00

60.00

0.0 1.0 2.0 3.0 4.0 5.0


Perc

enta

ge v

aria

tion

of S

hear

For

ce a

bout

mea

n

node 2node 4node 5

he following points are observed from the results obtained:

1. Muhanna’s methodology for handling single uncertainty is modified in the present

study to handle multiple uncertainties. As described in Chapter 2, overestimation

is eliminated and sharp bounds to the solution are obtained.

2. The validated methodology is employed to evaluate the effect of multiple

uncertainties on the structural response of cable-stayed bridges and the results are

tabulated. The following common characteristics are be observed in all the tables:

a) It is observed that the width of the interval increases along and across each

table.

a) In each table, the values of structural response in the first column

correspond to the presence of material uncertainty alone (0≤α≤1) and

those in the first row correspond to the presence of uncertainty of mass

density (or live load) alone (0≤β≤1 or 0≤γ≤1).

4.5 Discussion

Fig 4.4.9 Sensitivity Analysis for Shear Force (due to Live Load) in deck slab at beta=1.0

0.00

1.65

3.32

5.01

6.74

8.49

0.00

1.95

3.93

5.93

7.98

10.06

0.00

2.35

4.72

7.11

9.53

0.00

2.07

4.17

6.29

8.45

10.64

11.98

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0.0 1.0 2.0 3.0 4.0 5.0


Pere

cnta

ge v

aria

tion

of S

hear

For

ce a

bout

mea

n

Node 1Node 3Node 4Node 5

14.0

T

158

159

b) The variation with reference to material uncertainty is found to be more in

comparison with the variation with reference to mass uncertainty.

c) The variation with reference to load uncertainty is found to be more in

comparison with the variation with reference to material uncertainty.

. Membership functions are plotted for various combinations of α,β and γ in the

case of horizontal, vertical displacements and rotations as well as forces and

moments. Triangular and trapezoidal membership functions are obtained under the

influence of single uncertainty (α or β, α or γ) and multiple uncertainties (α and γ,

α and β, α and β and γ) respectively. The present methodology is shown to be a

marked improvement over the earlier methodology developed by Muhanna.

. Sensitivity analysis is performed to evaluate the relative uncertainties of structural

ities viz. horizontal and vertical displacements, rotations, forces

en nodes. The following

ity.

d) Sensitivity of structural response is found to vary from node to node along the

The

succeeding

3

4

response quant

and moments at a given node as well as over a set of giv

points are observed from the sensitivity analysis:

a) Structural response is found to be more sensitive to material uncertainty in

comparison with uncertainties of live load and mass dens

b) Vertical displacement is found to have the greatest sensitivity while the

horizontal displacement has the lowest sensitivity.

c) Bending moment is found to have the greatest sensitivity while the shear force

has the lowest sensitivity.

span of the deck slab.

conclusions and recommendations for future work are addressed in the

chapter.

Chapter 5

Conclusions

and

Recommendations for future work

Chapter 5

5. CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK

5.1 Conclusions

On the basis of the work carried out on cable-stayed bridges using fuzzy-finite

element analysis, the following are conclusions arrived at:

1. In the present study, a new methodology to evaluate structural response in the

presence of multiple uncertainties is developed. For this purpose, Muhanna’s

methodology for handling single uncertainty is modified. Sharp bounds to the

solution vector in the presence of multiple uncertainties were established by

identifying and eliminating other sources of overestimation not considered so

far in the earlier studies. In the present study, overestimation of the structural

response is eliminated and a sharp enclosure to the solution vector is obtained

by

a) uncoupling of load vector by keeping the load contributions separate

throughout the solution process.

b) developing a new approximation to the vector of internal forces.

c) developing a new approach for post-processing the solution.

2. The applicability of new methodology to the detailed evaluation of the effect

of multiple uncertainties on the structural response of cable-stayed bridges is

demonstrated in the present work.

3. Sensitivity analysis is a new and useful concept proposed in the present study.

By performing sensitivity analysis it is observed that:

161

a) Structural response is more sensitive to variation in material property

(Young’s modulus) in comparison to variation in load. This indicates

that even a smaller variation in Young’s modulus may cause the actual

structural behaviour to be significantly different from the structural

behaviour predicted by the designer. This brings out the need for a

stricter quality control so that the variation in material properties from

their respective mean values is minimised.

b) As the age of the structure advances, Young’s modulus, mass density

and live load may change from their nominal values, thus leading to

uncertainties in these structural parameters as well as structural

response. Thus, at the design stage itself, present methodology can be

used to establish in advance, bounds to the structural response at

different stages of service period of the structure.

c) It is observed that different elements of the structure exhibit different

levels of sensitivity of structural response. Further it is revealed that

the sensitivity of the same response quantity changes from node to

node. Thus it is essential to make a detailed evaluation of the structural

response in order to get a complete understanding of the structural

behaviour.

Finally it is concluded that the fuzzy-finite element analysis is a near-realistic

approach for modelling of multiple uncertainties of structural behaviour in

comparison to the classical finite element approach.

162

163

5.2

Recommendations for Future Work

1. The scope of multiple uncertainties can be extended to include uncertainties

in geometry and support conditions.

2. The present methodology can be further advanced to the analysis of

two-dimensional and three-dimensional structures.

3. The methodology can be extended to dynamic analysis of structures. It is

also appropriate to evaluate fuzzy natural frequencies and mode shapes.

4. Uncertainty needs to be introduced in design philosophy of national codes to

enable the designer to design a structure at specified levels of uncertainty.

REFERENCES

Abdel-Tawab, K. Noor, A. K., (1999). "A Fuzzy-set Analysis for a Dynamic Thermo-

elasto-viscoplastic Damage Response," Computers and Structures Vol. 70, pp 91-107.

Akpan, U.O., Koko, T.S., Orisamolu, I.R., Gallant, B.K., (2001), "Practical fuzzy

Finite element analysis of structures", Finite Elements in Analysis and Design 38

(2001) 93}111. (Elsevier Publishing).

Bernardini, A. and Tonon, F. (1996). "A combined fuzzy and random-set approach to

the multi-objective optimization of uncertain systems," Proc, 7th ASCE EMD/STD

Joint Specialty Conference on Probabilistic Mechanics and Structural Reliability,

WPI, MA, August 7-9, 314-317.

Gay, D. M. (1982). "Solving Interval Linear Equations," SIAM Journal on Numerical

Analysis, Vol. 19, 4, pp. 858-870.

Hansen.E., (1965). "Interval Arithmetic in matrix computations", SIAM J. Numer.

Anal I(2), pp 308-320.

Jansson.C., (1991) "Interval Linear Systems with symmetric matrices, skew-

symmetric matrices, and dependencies in the right hand side.", Computing, 46,

265-274.

Kavanagh, T.C., Discussion of "Historical Developments of Cable-Stayed Bridges"

by Podolony and Fleming, Journal of the Structural Division, ASCE, Vol.99, No. ST

7, Proc. Paper 9826, July 1973.

Köyluoglu, U., Cakmak, A.S., and Nielsen, S. R. K. (1995). "Interval Algebra to Deal

with Pattern Loading and Structural Uncertainties, " Journal of Engineering

Mechanics, November, 1149-1157.

164

Köyluoglu, U. and Elishakoff, I. (1998). "A Comparison of Stochastic and Interval

Finite Elements Applied to Shear Frames with Uncertain Stiffness Properties",

Computers and Structures, Vol. 67, No. 1-3, pp.91-98.

Kulpa Z., Pownuk A., Skalna I., (1998) Analysis of linear mechanical structures with

uncertainties by means of interval methods. Computer Assisted Mechanics and

Engineering Sciences, vol. 5, pp.443-477.

McWilliam, Stewart.,(2001), "Anti-optimisation of uncertain structures using interval

analysis", Computers and Structures 79 (2001) 421-430.

Moore, R.E.(1966). Interval Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J.

Muhanna, R. L. and Mullen, R. L. (1999). "Formulation of Fuzzy Finite Element

Methods for Mechanics Problems," Computer-Aided Civil and Infrastructure

Engineering (previously Microcomputers in Civil Engineering), Vol.14, pp107-117.

Mullen, R. L. and Muhanna, R. L (1999). "Bounds of Structural Response for All

Possible Loadings," Journal of Structural Engineering, ASCE, Vol. 125, No. 1, pp

98-106.

Muhanna, R. L. and Mullen, R. L., (2000). “Sharp Enclosure for Material Uncertainty

in Solid and Structural Mechanics-Interval Based Approach”, Proc., 8th ASCE

Specialty Conference on Probabilistic Mechanics and Structural Reliability,

University of Notre Dame, Notre Dame, Indiana, July, 24-26.

Muhanna, R. L. and Mullen, R. L., (2001). " Uncertainty in Mechanics Problems -

Interval-Based Approach, "Journal of Engineering Mechanics, ASCE, Vol. 127,

No. 6, pp 557-566.

165

Nakagiri, S. and Yoshikawa, N. (1996). "Finite Element Interval Estimation by

Convex Model, " Proc., 7th ASCE EMD/STD Joint Specialty Conference on

Probabilistic Mechanics and Structural Reliability, WPI, MA, August 7-9, 278-281.

Nakagiri, S. and Suzuki, K. (1999). "Finite Element Interval analysis of external loads

identified by displacement input uncertainty," Comput. Methods Appl. Mech. Engrg.

168, pp. 63-72.

Neumaier, A. (1987). "Overestimation in Linear Interval Equations," SIAM Journal

on Numerical Analysis, Vol. 24, 1, pp. 207-214.

Neumaier, A. (1989). "Rigorous Sensitivity Analysis for Parameter-Dependent

Systems of Equations, "Journal of Mathematical Analysis and Applications, Vol. 144,

pp. 16-25.

Neumaier, A.(1990). Interval methods for systems of equations, Cambridge

University Press.

Pownuk A., (2001). New inclusion functions in interval global optimization of

engineering structures, European Conference on Computational Mechanics, Cracow,

26 - 29 June 2001, pp.460-461.

Pownuk A., (1999), “Optimization of mechanical structures using interval analysis”,

Computer Assisted Mechanics and Engineering Sciences, Polish Academy of

Sciences.

Rao, S. S., Sawyer, P. (1995). "Fuzzy Finite Element Approach for Analysis of

Imprecisely Defined Systems, " AIAA Journal, Vol. 33, No. 12, pp 2364-2370.

Rao, S. S., Berke, L. (1997). "Analysis of Uncertain Structural Systems Using

Interval Analysis, " AIAA Journal, Vol. 35, No. 4, pp 727-735.

166

167

Rao, S. S., Li Chen, (1998). "Numerical Solution of Fuzzy Linear Equations in

Engineering Analysis, " Int. J. Numer. Meth. Engrg. Vol. 43, pp 391-408.

Rump, S. M. (1990). " Rigorous Sensitivity Analysis for Systems of Linear and

Nonlinear Equations, "Mathematics of Computations, Vol. 54, 190, pp. 721-736.

Tonon, F. and Bernardini, A. (1999). "Multi-objective Optimization of Uncertain

Structures Through Fuzzy Set and Random Set Theory, "Computer-Aided Civil and

Infrastructure Engineering (previously Microcomputers in Civil Engineering),

Vol.14.

Troitsky. M.S. DSC; “Cable-Stayed Bridges: Theory and Design”, Crosby Lockwood

Staples, London, 1972.

Walther, Rene, (1988), "Cable Stayed Bridges", Thomas Telford, London.

Zadeh, L .A. (1965), “Fuzzy Sets”, Information and Control, 8, 338-353.

Appendices

APPENDIX-A

FUZZY SETS AND INTERVAL NUMBERS

1. Introduction

Fuzzy set theory was first developed by Zadeh (Zadeh, 1965) to model

uncertainties in qualitative terms. A fuzzy set A is a superset of an ordinary set A.

Thus the fuzzy set theory encompasses and extends the scope of classical set theory.

The difference between an ordinary subset A and a fuzzy subset A can be explained

as follows:

Let E be a referential set in R. An ordinary subset A of this referential set is

defined by its characteristic function, ∀x ∈ E

µA(x) ∈ [0,1]

which shows that an element of E belongs to or does not belong to A, according to the

value of the characteristic function (1 or 0).

For the same referential set E, a fuzzy subset A will be defined by its

characteristic function, the membership function µA(x) , which takes its values in the

interval [0,1].

There are several methods for the representation of fuzzy numbers. In one

such representation, the possible range of a value is used to represent the set

membership function. Thus a fuzzy set A can be represented by

A = {x : (x,µA(x))/ x∈E, 0≤µA(x)≤1}

where A represents the set of x each with membership function µA(x). Alternately,

A can be represented in the form

A = 0/0 + 0.1/1 + 0.3/2 + 0.8/3 + 1.0/4 + 0.7/5 + 0.3/6 + 0/7

where 0,0.1,0.2,0.3 etc. represent the membership function (or in other words, the

degree of belongingness to the given fuzzy s et ) of the values 0,1,2 etc. respectively.

169

Fig. A-1 Fuzzy Membership Function

0.950

0.960

0.970

0.980

0.990

1.0001.000

1.010

1.020

1.030

1.040

1.0500

0.2

0.4

0.6

0.8

1

0.950 0.970 0.990 1.010 1.030 1.050Variation about the mean value

Mem

bers

hip

Valu

e (a

lpha

)

In general, for discrete fuzzy sets, the number of elements in the fuzzy sets

could become quite large. Thus, in numerical computations, it is convenient to

express fuzzy numbers as sets of lower and upper bounds of a finite number of α-cut

subsets.

α-Cut Representation

At a level of α from the x- axis a cut is made (α-cut) to extract an ordered pair

in a closed interval form [xl, xu]. The α-cut can be taken anywhere ranging from α = 0

(total uncertainty) to α = 1(total certainty). An interval is a closed set in R that

includes the possible range of number. An interval number is represented as an

ordered pair [xl, xu] where xl ≤ xu. In case xl = xu, the interval is called a fuzzy- point

interval. e.g. [a, a]

The membership functions are usually constructed in terms of an interval of

confidence at several levels of presumption. The maximum level of presumption is

170

considered to be 1 and the minimum level of presumption is considered to be 0. The

level of presumption α , α∈[0,1], gives an interval of confidence Aα , defined as

Aα = {x∈R, µA(x)≥α}.

where Aα which is a monotonically decreasing function of α , that is

(α1<α2) ⇒ (Aα2⊂Aα1)

or

(α1<α2) ⇒ [a1α2,a2

α2] ⊂[a1α1,a2

α1] for every α1,α2 ∈[0,1]

The fuzzy numbers thus defined are known as intervals. Thus in the

investigation of fuzzy set methods applied to continuum mechanics, interval

representation is used. (Interval of confidence at a specific level of presumption

i.e. α-cut). Properties of fuzzy interval numbers are given below.

Interval Algebra

Interval analysis involves the definition, properties and the operations on

interval numbers and can be found in a number of sources (Alefeld and

Herzberger, 1983 and Jansson, 1991) . However, some of the properties of interval

numbers are presented below:

A = [a1, a2] ⇒ A = {a∈ R | a1≤ a≤ a2} and B = [b1, b2]

A+B = [a1+ b1, a2+ b2] ---- Addition

-A = [-a2, -a1] -- Additive Inverse

A-B = A + (-1). B = [a1- b2, a2- b1] ---- Subtraction

1/A = [1/ a1, 1/ a2] if 0∉ [a1, a2]

A*B = [min (a1b1, a1b2 ,a2b1,a2b2), max (a1b1, a1b2 ,a2b1,a2b2)]

A/B = A* (1/B) --- Division

171

Validity of laws of algebra Commutativity

A+B = B+A --- Addition

A× B = B×A --- Multiplication

Associativity

A+ (B+C) = (A+B) +C --- Addition

A× (B×C) = (A×B)×C --- Multiplication

Distributive law is not valid for interval numbers. Thus,

A× (B+C) ≠A×B + A×C Instead, the property of sub-distributivity is applicable as shown below:

Sub-Distributivity A× (B+C) ⊆ A×B + A×C Problems of Dependency and Overestimation

Failure of distributive law is a frequent source of expansion of intervals and

causes overestimation of result. Further, when two interval numbers, which represent

the same physical quantity are operated upon, obtaining a sharp enclosure for the

solution is a difficult task. This phenomenon is known dependency problem. The

work of Muhanna and Mullen primarily focuses on devising procedures to avoid

dependency to achieve a sharp enclosure for the solution.

This is because of failure of some algebraic laws that are valid in real

arithmetic. An overestimated solution encloses the solution hull but the width of the

solution obtained is very large, there by rendering the result very conservative,

meaningless and inapplicable in practical situations. The problems of dependency and

overestimation arise owing to the following reasons (Muhanna and Mullen, 2001):

1. Failure of distributive law in interval algebra. (Only sub-distributivity is valid

in interval algebra).

172

2. When one or several variables occur more than once in an interval expression.

In case of two or more interval variables representing the same physical

quantity, interval algebra treats them as separate variables having their own

range of variation. Thus each individual occurrence of the same physical

quantity i.e. uncertain Young’s modulus, if present as a multiplying factor for

all the individual elements of stiffness matrix, will be treated as an

independent entity by interval algebra. Thus, in order to avoid overestimation,

the uncertain Young’s modulus needs to be factored out from the element

stiffness matrix and be presented as a single multiplier for the entire stiffness

matrix

3. Another source of overestimation is the element coupling after the domain

assemblage is completed. In an assembled finite element model with uncertain

stiffness properties (arising out of uncertainty of material property i.e.

Young’s modulus), some elements have common nodes and the elements of

individual stiffness contributions (independent or not, depending on the nature

of the problem). These stiffness contributions are added at the corresponding

common degrees of freedom.

To keep the interval operations sharp and to avoid overestimation, it is

necessary to trace the contributions and their involvement in subsequent formulated

expression throughout the whole computational process. To overcome overestimation

due to coupling, an element-by-element procedure is proposed by Muhanna

(Muhanna and Mullen, 2001) in which elements are kept disassembled throughout the

course of solution. By doing so, the coupling that usually occurs in the conventional

finite element formulation is delayed. The Lagrange multiplier method imposes the

necessary constraints to ensure compatibility and equilibrium.

173

APPENDIX- B

FUZZY FINITE ELEMENT MODEL- MUHANNA’S APPROACH

Consider the potential energy Π as

Π = {U}T[K]{U} – {U}T{P} - (1)

Considering a set of m linearly independent constraints being imposed on the

solution in the form of constraint equations expressed as

[C] {U} = {V} - (2)

Here, [C] is a crisp constraint matrix. Using the Lagrange multiplier method,

the right hand side of Eq. 1 is modified using Eq. 2 to obtain,

∏* = ½ [U] T [K̃]{U}–{U} T {P} +{λ} T {[C̃]{U}-{V}} - (3)

Using Rayleigh- Ritz approach, stationarity of ∏* is invoked as

δ∏* = 0 - (4)

This leads to the deterministic (non-interval) system of equations represented by

[K̃] {U}+[C] T {λ} = {P} - (5)

Here, [K̃], {U} and {λ} are deterministic in nature. Also, {λ} is the vector of

Lagrange multipliers.

Eq. 5 is used to enforce multi-freedom constraints in assembled finite element

model to force the displacements of the slave nodes to match those of the master

nodes. The element-by-element model proposed by Muhanna (Muhanna and Mullen,

2001) is derived from a modification of constrained finite element methodology and

Eq. 1.

In the case of element-by-element model with uncertain material property (of

Young’s modulus), Eq. 2 takes the form

[C]{U}={0} - (6)

where [C] matrix is a deterministic in nature.

174

Using Eq. 6, Eq. 5 can be modified to in the case of element-by element

model as

[K] {U}+[C] T {λ} = {P} - (7)

where {P} is a point-interval load vector.

Using the relation [K] = [D] [S], Eq. 7 is reformulated as

[D][S]{U} = {P} -[C]T{λ} - (8)

where [K] is the interval stiffness matrix, [D] is an diagonal matrix containing the

value of uncertain Young’s modulus and [S ] is a crisp singular matrix, each of size

n×n. Also, {λ} = interval internal force vector of size n×1. Here, n is the product of

number of degrees of freedom per each element and the total number of elements in

the structure.

If Eq. 6 is multiplied on either side by [D][C]T and the result is added to

Eq. 8, and taking [R]=[S]+[C] T [C] the following equation is obtained

[D][R]{U}= {P}-[C]T {λ} - (9)

The solution of Eq. 9 is obtained by approximating the interval force vector

{λ} by its mid value {λc}. Eq. 9 is further modified as

{U} = [R]-1[M]{δ} - (10)

where [M] is an interval matrix of size n×m where m is the number of elements. Also,

{δ} is an interval vector of size m×1 and contains the inverse values of the Young’s

modulus for each of m elements. The solution of Eq. 10 represents the solution of

Eq. 9 by approximating the interval force vector of internal forces {λ} by its mid-point

vector {λc}. Eq. 10 is solved using Jansson’s algorithm (Jansson, 1991).

175

APPENDIX ‘C’

PRELIMINARY CASE STUDIES

1. Introduction

The methodology developed in Chapter 2 is illustrated by considering four

distinct structures as case studies in the following section. All structures considered

are statically indeterminate.

The following structures are taken up as case studies:

1. A fixed beam with a uniformly distributed load over the entire span

2. A propped Cantilever with a point load at the mid-span

3. A plane frame with point loads and moments acting at the joints and uniformly

distributed loads acting on the members.

4. A plane truss with point loads acting at the joints.

In all the above case studies, the simultaneous effect of material and live load

uncertainties is considered. All the case studies are modelled by using modified

Muhanna’s methodology suggested in Chapter 2. The displacements, shear forces and

bending moments obtained using the above approach are plotted with reference to

material and live load uncertainties. Each of the case studies is explained in detail

below:

2. Preliminary Case Study-1 -Fixed Beam

The first problem considered here is a fixed beam shown in Fig. C-1. This

problem was adapted from the work of Akpan (Akpan et al, 2001). The fixed beam

has the following properties (after conversion from U.S. Customary units) (Table.

C-1). The fixed beam is subjected to a uniformly distributed load over the entire

span. The uncertainty associated with Young’s modulus is denoted by α where as the

uncertainty associated with live load is denoted by β.

176

Table. C-1 Material and Geometric Properties of Fixed beam Span 10.16 metres (400 inches) Area of Cross section 0.07165 m2 (111.06 inch2) Moment of Inertia 1.2486×10-3 m4 (3000 inch4) Uniformly distributed load 70.076 kN/m (400lb/inch) Location of load Entire span Young’s Modulus (E) 206.9 GPa (30×10-6 psi) Material Uncertainty of E (αmax) 4 percent (±2% about mean value of E) Load Uncertainty (βmax) 20 percent (±10% about mean value of load)

Fig. C-2 and Fig. C-3 depict the membership functions of material and load

uncertainties respectively. The membership function for material uncertainty shows

the normalized values of uncertain Young’s modulus, taking the crisp values as 1.0.

The membership function for material uncertainty shows the normalized values of

live load, taking the crisp value as 1.0. Table. C-2 represents the simultaneous

variation of mid-span displacement for various combinations of α and β. It is

observed that the width of the interval increases along and across the table. The

variation with reference to material uncertainty is less compared to the variation with

reference to load uncertainty because more uncertainty (±10%) is associated with load

compared to Young's modulus (± 2%).

Table. C-3 shows a comparison of the results obtained by the present approach

with the results obtained by Akpan (Akpan et al, 2001) using the response surface

methodology. Fig. C-4 represents the membership function for mid-span displacement

(mm) for three cases of uncertainty (E and Load, Load only and E only). It is observed

that the present results compare very well with the results of Akpan et al for all the

three identical cases.

Table. C-4 and Table. C-5 represent the simultaneous variation of fixed end

moment and fixed end shear force for various combinations of α and β. The

membership function shown in Fig. C-5 reflects the variation of mid-span

177

displacement for various levels of material uncertainty α while the load is crisp

(β=1.0). Fig. C-6 depicts the variation of mid-span displacement at various levels of

load uncertainty β at α=0.5. Fig. C-7 and Fig.C-8 represent the membership functions

for fixed end moment at β=0.75 and α=1.0 respectively. In general it is observed that

membership functions are triangular in the presence of a single uncertainty and

trapezoidal in the presence of multiple uncertainties.

178

179

Table. C-2 Fixed Beam – Concomitant Variation of mid-span displacement (×10-3 metres)

w.r.t. α and β β→

α↓ 1.0 0.75 0.50 0.25 0.0

1.0 [-7.527,-7.527] [-7.715,-7.339] [-7.904,-7.151] [-8.092,-6.963] [-8.280,-6.774]

0.75 [-7.565,-7.490] [-7.754,-7.302] [-7.943,-7.115] [-8.132,-6.928] [-8.322,-6.741]

0.50 [-7.603,-7.453] [-7.793,-7.266] [-7.984,-7.080] [-8.174,-6.893] [-8.364,-6.707]

0.25 [-7.642,-7.416] [-7.833,-7.230] [-8.024,-7.045] [-8.215,-6.859] [-8.406,-6.674]

0.0 [-7.681,-7.379] [-7.873,-7.195] [-8.065,-7.010] [-8.257,-6.826] [-8.450,-6.641]

Table. C-3 Comparison of present results with Response Surface Approach at Degree of Belief (α or β) = 0.0 (Mid-span displacements)

Akpan (mm) Present (mm) Uncertainty Akpan (inch) Interval Normalised

interval Interval Normalised interval

Load and E [-0.332, -0.2615] [-8.433,-6.642] [-1.119, -0.881] [-8.450, -6.641] [-1.120,0.880]

Load only [-0.326,0.267] [-8.280,-6.782] [-1.099,-0.901] [-8.280, -6.774] [-1.100,-0.900]

E only [-0.303, -0.290] [-7.671,-7.366] [-1.02,0.980] [-7.681, -7.379] [-1.02,0.9799]

Table. C-4 Fixed Beam – Concomitant Variation of Fixed end moment (kNm) w.r.t. α and β

β→ α↓

1.0 0.75 0.50 0.25 0.0

1.0 [-602.80, -602.80] [-617.87, -587.73] [-632.94, -572.66] [-648.01, -557.59] [-663.08,-542.52]

0.75 [-608.68,-597.00] [-623.90, -582.08] [-639.12, -567.15] [-654.34, -552.23] [-669.55,-537.30]

0.50 [-614.66, -591.29] [-630.02, -576.51] [-645.39, -561.73] [-660.75, -546.94] [-676.12,-532.16]

0.25 [-620.72, -585.66] [-636.24, -571.02] [-651.75, -556.38] [-667.27, -541.74] [-682.79,-527.10]

0.0 [-626.87, -580.12] [-642.54, -565.61] [-658.21, -551.11] [-673.89, -536.61] [-689.56,522.10]

Table. C-5 Fixed Beam - Concomitant Variation of Fixed end Shear Force (kN) w.r.t. α and β

β→ α↓

1.0 0.75 0.50 0.25 0.0

1.0 [355.98,355.98] [347.08,364.88] [338.18,373.78] [329.28,382.68] [320.38,391.58]

0.75 [352.82,359.19] [344.00,368.17] [335.18,377.15] [326.36,386.13] [317.54,395.11]

0.50 [349.71,362.44] [340.96,371.50] [332.22,380.56] [323.48,389.62] [314.74,398.68]

0.25 [346.63,365.74] [337.97,374.88] [329.30,384.02] [320.64,393.17] [311.97,402.31]

0.0 [343.60,369.08] [335.01,378.31] [326.42,387.54] [317.83,396.77] [309.24,405.99]

180

Fig. C-2 Fixed Beam - Membership Function for Material Uncertainty

0.980

0.985

0.990

0.995

1.000

1.005

1.010

1.015

1.0200

0.25

0.5

0.75

1

0.980 0.985 0.990 0.995 1.000 1.005 1.010 1.015 1.020

Design Quanity

Mem

eber

ship

Val

ue (

alph

a)

Fig. C-3 Fixed Beam - Membership Function for Load Uncertainty

0.9

0.925

0.95

0.975

1

1.025

1.05

1.075

1.10

0.25

0.5

0.75

1

0.900 0.925 0.950 0.975 1.000 1.025 1.050 1.075 1.100Design Quanity

Mem

eber

ship

Val

ue (b

eta)

Fig. C-4 Fixed Beam - Displacement (mm) at the mid-span under fuzzy-static load

-7.379-6.641

-6.859

-7.080

-7.302

-8.215

-7.984

-7.754

-7.527

-8.280

-7.339

-7.151

-6.963

-6.774

-7.715

-7.904

-8.092

-7.603

-7.642

-7.681

-7.490

-7.453

-7.565

-7.416

0.00

0.25

0.50

0.75

1.00

-8.450 -8.250 -8.050 -7.850 -7.650 -7.450 -7.250 -7.050 -6.850 -6.650Displacement (mm)

Deg

ree

of b

elie

f (al

pha

or b

eta)

E and Load Load only E only

70.076 kN/m

10.16 m

Fig. C-1 Fixed beam with u.d.l over entire span

181

Fig. C-6 Fixed Beam - Membership Function for mid-span displacement at alpha=0.50

-8.364E-03

-8.174E-03

-7.984E-03

-7.793E-03 -7.266E-03

-7.080E-03

-6.893E-03

-6.707E-03

-7.453E-03-7.603E-03

0.00

0.25

0.50

0.75

1.00

-8.40E-03 -8.20E-03 -8.00E-03 -7.80E-03 -7.60E-03 -7.40E-03 -7.20E-03 -7.00E-03 -6.80E-03Displacement (m)

beta

Fig. C-5 Fixed Beam - Membership Function for mid-span displacement at beta=1.0

-7.681E-03

-7.642E-03

-7.603E-03

-7.565E-03

-7.527E-03-7.527E-03

-7.490E-03

-7.453E-03

-7.416E-03

-7.379E-030.00

0.25

0.50

0.75

1.00

-7.700E-03 -7.650E-03 -7.600E-03 -7.550E-03 -7.500E-03 -7.450E-03 -7.400E-03 -7.350E-03Displacement (m)

alph

a

Fig. C-8 Fixed Beam - Membership Function for Fixed End Moment at alpha=1.0

-663.08

-648.01

-632.94

-617.87

-602.80-602.80

-587.73

-572.66

-557.59

-542.520

0.25

0.5

0.75

1


beta

Fig. C-7 Fixed Beam - Membership Function for Fixed end moment at beta=0.75

-642.54

-636.24

-630.02

-623.90

-617.87 -587.73

-582.08

-576.51

-571.02

-565.610

0.25

0.5

0.75

1

-645.00 -635.00 -625.00 -615.00 -605.00 -595.00 -585.00 -575.00 -565.00Bending Moment(kNm)

alph

a

3. Preliminary Case Study-2 – Propped Cantilever

The second preliminary case study considered here is a propped cantilever

beam shown in Fig. C-9. This problem was adapted from the work of Muhanna

(Muhanna and Mullen, 2001). The beam has the following properties (Table C-6)

Table. C-6 Material and Geometric Properties of Propped Cantilever Beam Span 10 metres Area of Cross section 0.0086 m2 Moment of Inertia 100×10-6 m4 Load 10 kN Location of load Mid-span Young’s Modulus (E) 200 GPa Material Uncertainty of E (αmax) 10 percent (±5% about mean value of E) Load Uncertainty (βmax) 200 percent (±100% about mean value of load)

The propped cantilever shown above is subjected to a point load at the mid-

span. Fuzziness is introduced in the Young’s modulus and the applied load. Fig. C-10

and Fig. C-11 depict the membership functions of material and load uncertainties

respectively.

Table C-7 and Table C-8 represent the simultaneous variation of mid-span

displacement and propped end rotation respectively for various combinations of α and

β. Fig. C-12 depicts the variation of mid-span displacement at various levels of load

uncertainty β at α=0.4.

Table C-9 and Table C-10 indicate the results of the simultaneous variation of

bending moment and shear force respectively at fixed end for various combinations of

α and β. Fig. C-13 depicts the variation of fixed end bending moment at various

levels of material uncertainty α at β=0.8. Fig. C-14 depicts the variation of fixed end

shear force at various levels of material uncertainty α at β=0.8.

Fig. C-15 and Fig. C-16 show the variation of shear force (N) and bending

moment (kNm) along the span for various levels of α ranging from α=1.0 to α=0.0 at

β=1.0. The diagrams (shear force or bending moment) for the crisp load

(corresponding to α=1.0) are shown as thick black lines. The corresponding diagrams

182

for other values of α envelop the crisp diagrams from above and below. Therefore the

entire set of diagrams can be named as shear force and bending moment envelops.

The first column of Table C-7 represents the variation of mid-span displacement

corresponding to crisp load and uncertain material property (E) (β=1, 0≤α≤1). These

results are found to be in close agreement with the results given by Muhanna.

(Muhanna and Mullen, 2001).

183

184

Table. C-7 Propped Cantilever Beam Concomitant Variation of Mid-span displacement (×10-3 metres) w.r.t α and β

β→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [-4.557,-4.557] [-5.469,-3.646] [-6.380,-2.734] [-7.291,-1.823] [-8.203,-0.911] [-9.114,-0.0005]

0.8 [-4.603,-4.512] [-5.524,-3.609] [-6.445,-2.706] [-7.366,-1.804] [-8.287,-0.901] [-9.208, 0.0016]

0.6 [-4.650,-4.468] [-5.581,-3.573] [-6.512,-2.679] [-7.442,-1.785] [-8.373,-0.891] [-9.303, 0.0032]

0.4 [-4.698,-4.424] [-5.639,-3.538] [-6.579,-2.653] [-7.520,-1.767] [-8.460,-0.881] [-9.400, 0.0043]

0.2 [-4.747,-4.382] [-5.698,-3.504] [-6.648,-2.627] [-7.599,-1.749] [-8.549,-0.872] [-9.499, 0.0049]

0.0 [-4.797,-4.340] [-5.758,-3.471] [-6.718,-2.602] [-7.679,-1.733] [-8.639,-0.864] [-9.599, 0.0050]

Table. C-8 Propped Cantilever Beam Concomitant Variation of Rotation at propped end (×10-3 radians) w.r.t α and β β→

α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [1.562, 1.562] [1.250, 1.875] [0. 938, 2.187] [0. 625, 2.500] [0. 313, 2.812] [0.000, 3.125] 0.8 [1.547, 1.578] [1.237, 1.894] [0. 928, 2.210] [0. 618, 2.526] [0.309, 2.841] [0.000, 3.157] 0.6 [1.532, 1.594] [1.225, 1.913] [0. 917, 2.233] [0. 612, 2.552] [0.305, 2.871] [0.000, 3.190] 0.4 [1.517, 1.611] [1.213, 1.933] [0. 909, 2.256] [0. 606, 2.578] [0.302, 2.901] [0.000, 3.223] 0.2 [1.502, 1.628] [1.201, 1.953] [0. 901, 2.279] [0. 599, 2.605] [0. 299, 2.931 [0.000, 3.257] 0.0 [1.488, 1.645] [1.190, 1.974] [0. 892, 2.303] [0.594, 2.633] [0.296, 2.962] [0.000, 3.291]

Table. C-9 Propped Cantilever Beam

Concomitant Variation of Fixed End Moment (Nm) w.r.t α and β β→

α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [-18749,-18749] [-22499,-15000] [-26249,-11250] [-29999,-7500] [-33749,-3750] [-37497,-1.875]

0.8 [-19130,-18380] [-22956,-14704] [-26782,-11028] [-30608,-7352] [-34434,-3676] [-38258,-1.838]

0.6 [-19522,-18021] [-23427,-14417] [-27331,-10813] [-31236,-7209] [-35140,-3604] [-39043,-1.802]

0.4 [-19927,-17673] [-23913,-14138] [-27898,-10604] [-31883,-7069] [-35869,-3535] [-39852,-1.767]

0.2 [-20344,-17335] [-24413,-13868] [-28482,-10401] [-32551,-6934] [-36620,-3467] [-40687,-1.733]

0.0 [-20775,-17006] [-24930,-13605] [-29085,-10204] [-33240,-6803] [-37395,-3401] [-41548,-1.701]

Table. C-10 Propped Cantilever Beam Concomitant Variation of Shear Force (N) at fixed end w.r.t α and β

β→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [6875,6875] [5500,8250] [4125,9625] [2750,11000] [1375,12375] [0.687,13749] 0.8 [6739,7014] [5391,8417] [4044,9820] [2696,11223] [1348,12626] [0.674,14028] 0.6 [6608,7158] [5286,8590] [3965,10022] [2643,11453] [1322,12885] [0.661,14316] 0.4 [6480,7307] [5184,8768] [3888,10229] [2592,11691] [1296,13152] [0.648,14612] 0.2 [6356,7460] [5085,8952] [3814,10443] [2542,11935] [1271,13427] [0.636,14918] 0.0 [6236,7617] [4988,9141] [3741,10664] [2494,12188] [1247,13711] [0.624,15234]

185

Fig. C-10 Membership Function for Material Uncertainty

0.950

0.960

0.970

0.980

0.990

1.0001.000

1.010

1.020

1.030

1.040

1.0500

0.2

0.4

0.6

0.8

1

0.950 0.970 0.990 1.010 1.030 1.050Variation about the mean value

Mem

bers

hip

Valu

e (a

lpha

)

10 kN

10 m

Fig. C-9 Propped Cantilever Beam with central point load

Fig. C-11 Membership Function for Load Uncertainty

0

0.2

0.4

0.6

0.8

11

1.2

1.4

1.6

1.8

20

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0Variation of Load about mean value

Mem

bers

hip

Valu

e (b

eta)

Fig. C-12 Propped Cantilever- Membership Function for Mid-span displacement at alpha = 0.4

-9.400E-03

-8.460E-03

-7.520E-03

-6.579E-03

-5.639E-03 -3.538E-03

-2.653E-03

-1.767E-03

-8.810E-04

-4.698E-03 -4.424E-03

0

0.2

0.4

0.6

0.8

1

-1.0E-02 -9.0E-03 -8.0E-03 -7.0E-03 -6.0E-03 -5.0E-03 -4.0E-03 -3.0E-03 -2.0E-03 -1.0E-03 0.0E+00Mid-span displacement(metres)

beta

186

Fig. C-13 Propped Cantilever-Membership Function of Fixed End Bending Moment at beta =0.8

-24930

-24413

-23913

-23427

-22956

-22499 -15000

-14704

-14417

-14138

-13868

-136050

0.2

0.4

0.6

0.8

1

-25000 -23000 -21000 -19000 -17000 -15000 -13000Bending Moment (Nm) at Fixed End

alph

a

Fig. C-14 Propped Cantilever- Membership Function of Shear Force at Fixed End at beta =0.8

4988

5085

5184

5286

5391

5500 8250

8417

8590

8768

8952

91410

0.2

0.4

0.6

0.8

1

4900 5400 5900 6400 6900 7400 7900 8400 8900 9400Shear Force (N) at Fixed End

alph

a

187

Ben

ding

Mom

ent N

m

Span (m)

16500 14000 11500

9000 6500 4000 1500

Fig. C-16 Propped Cantilever - Variation of Bending Moment w.r.t alpha

alpha=0.0alpha=0.0alpha=0.2alpha=0.2alpha=0.4alpha=0.4alpha=0.6alpha=0.6alpha=0.8alpha=0.8

10.09.0alpha=1.0

8.0 7.0 6.05.04.03.02.0 1.0-3500 -1000

0.0

-6000 -8500

-11000 -13500 -16000 -18500 -21000

Span (m)

Fig. C-15 Propped Cantilever- Variation of Shear Force w.r.t. alpha

Shea

r For

ce (N

)

alpha=0.0alpha=0.0alpha=0.2alpha=0.2alpha=0.4alpha=0.4alpha=0.6alpha=0.6alpha=0.8alpha=0.8alpha=1.0

10.09.08.0 7.0 6.05.04.03.02.0 1.0

7300

6300

5300

4300

3300

2300

1300

300

0.0 -700

-1700

-2700

-3700

7617

6236

-3125-3462

-2834

6875

4. Preliminary CaseStudy-3 – Plane Truss

The third case study considered here is a two bay plane truss shown in

Fig. C-17. This problem is adopted from the work of Muhanna (Muhanna and

Mullen, 2001). The plane truss has the following properties (Table C-11)

Table. C-11 Material and Geometric Properties of Plane Truss

Area of Cross section of each member 0.01 m2 Span of each bay 10.0m Young’s Modulus (E) 200 GPa Material Uncertainty of E (αmax) ±5% about mean value of E Load Uncertainty (βmax) ±100% about mean value of load

The plane truss is subjected to point load acting downwards at node 2.

Uncertainty is introduced in the Young’s modulus and the applied loads. The material

and load uncertainties are given by membership functions shown in Fig. C-10 and

Fig. C-11.

Table C-12 and Table C-13 represent the simultaneous variation of horizontal

displacement at node 4 and vertical displacement at node 2 respectively for various

combinations of α and β. Fig. C-18 depicts the variation of vertical displacement at of

node 2 for various levels of load uncertainty β at α=0.6.

Table C-14 and Table C-15 represent the simultaneous variation of axial force

in member 4 and member 10 respectively for various combinations of α and β.

Fig. C-19 indicates the variation of axial force in member 4 at various levels of

material uncertainty α at β=0.8. Fig. C-20 depicts the variation of axial force in

member 10 at various levels of load uncertainty β at α=0.6.

188

In general it is observed that membership functions are triangular in the

presence of a single uncertainty and trapezoidal in the presence of multiple

uncertainties (Fig. C-18, Fig. C-19 and Fig. C-20).

The results presented in first column of Table C-12 and Table C-13 (horizontal

and vertical displacement) represents the presence of a single uncertainty (0≤α≤1 and

β=1). These results are found to be in close agreement with those given by Muhanna

(Muhanna and Mullen, 2001).

189

190

Table. C-12 Plane Truss-Concomitant Variation of Horizontal displacement of node 4 (×10-5 metres) w.r.t α and β

β→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [3.951,3.951] [3.161,4.741] [2.371,5.532] [1.580,6.322] [0.790,7.112] [0.000,7.902]

0.8 [3.845,4.058] [3.076,4.870] [2.306,5.682] [1.537,6.494] [0.768,7.306] [0.00,8.118]

0.6 [3.740,4.166] [2.991,4.999] [2.243,5.833] [1.494,6.667] [0.745,7.501] [-0.003,8.334]

0.4 [3.635,4.274] [2.907,5.130] [2.179,5.986] [1.451,6.841] [0.724,7.697] [-0.004,8.552]

0.2 [3.531,4.384] [2.824,5.262] [2.117,6.139] [1.409,7.017] [0.702,7.895] [-0.005,8.772]

0.0 [3.427,4.495] [2.741,5.395] [2.054,6.294] [1.368,7.194] [0.681,8.094] [-0.005,8.993]

Table. C-13 Plane Truss-Concomitant Variation of Vertical displacement of node 2 (×10-5 metres) w.r.t α and β

β→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [-19.93,-19.93] [-23.92,-15.95] [-27.91-11.96] [-31.89,-7.97] [-35.88,-3.99] [-39.87,0.00]

0.8 [-20.13,-19.74] [-24.16,-15.79] [-28.19,-11.84] [-32.22,-7.89] [-36.25,-3.94] [-40.28,0.01]

0.6 [-20.34,-19.54] [-24.41,-15.63] [-28.48,-11.72] [-32.55,-7.81] [-36.62,-3.89] [-40.69,0.02]

0.4 [-20.55,-19.35] [-24.66,-15.48] [-28.78,-11.60] [-32.89,-7.73] [-37.01,-3.85] [-41.12,0.02]

0.2 [-20.76,-19.17] [-24.92,-15.33] [-29.08,-11.49] [-33.24-7.65] [-37.39,-3.81] [-41.55,0.02]

0.0 [-20.98,-18.98] [-25.18,-15.18] [-29.39,-11.38] [-33.59-7.58] [-37.79,-3.78] [-41.99,0.02]

Table. C-14 Plane Truss- Concomitant Variation of Axial Force (kN) in Member 4 w.r.t α and β

β→ α↓

1.0 0.8 0.6 0.4 0.2 0.0

1.0 [-7.902,-7.902] [-9.483,-6.322] [-11.063,-4.741] [-12.64,-3.161] [-14.22,-1.580] [-15.805,0.0]

0.8 [-8.124,-7.687] [-9.748,-6.150] [-11.373,-4.612] [-12.99,-3.075] [-14.62,-1.537] [-16.247,0.0]

0.6 [-8.351,-7.477] [-10.02,-5.982] [-11.69,-4.486] [-13.36,-2.991] [-15.03,-1.495] [-16.702,0.0]

0.4 [-8.585,-7.273] [-10.30,-5.818] [-12.019,-4.364] [-13.74,-2.909] [-15.45,-1.455] [-17.170,0.0]

0.2 [-8.826,-7.074] [-10.59,-5.659] [-12.36,-4.244] [-14.12,-2.830] [-15.89,-1.415] [-17.652,0.0]

0.0 [-9.074,-6.880] [-10.89,-5.504] [-12.70,-4.128] [-14.52,-2.752] [-16.33,-1.376] [-18.148,0.0]

Table. C-15 Plane Truss-Concomitant Variation of Axial Force (kN) for Member 10 w.r.t α and β

β→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [8.834,8.834] [7.067,10.601] [5.301,12.368] [3.534,14.135] [1.767,15.902] [0.000,17.669]

0.8 [8.609,9.066] [6.887,10.879] [5.165,12.693] [3.444,14.506] [1.722,16.319] [0.000,18.132]

0.6 [8.389,9.305] [6.712,11.165] [5.034,13.026] [3.356,14.887] [1.678,16.748] [0.000,18.609]

0.4 [8.176,9.550] [6.541,11.460] [4.906,13.370] [3.270,15.280] [1.635,17.190] [0.000,19.100]

0.2 [7.968,9.802] [6.374,11.763] [4.781,13.723] [3.187,15.684] [1.594,17.644] [0.000,19.605]

0.0 [7.765,10.063] [6.212,12.075] [4.659,14.088] [3.106,16.100] [1.553,18.113] [0.000,20.125]

191

Fig. C-18 Plane Truss- Membership Function for vertical displacement of node 2 at alpha=0.6

-4.07E-04

-3.66E-04

-3.26E-04

-2.85E-04

-2.44E-04

-1.95E-04

-1.56E-04

-1.17E-04

-7.81E-05

-3.89E-05

-2.03E-04

0

0.2

0.4

0.6

0.8

1

-4.1E-04 -3.6E-04 -3.1E-04 -2.6E-04 -2.1E-04 -1.6E-04 -1.1E-04 -6.0E-05 -1.0E-05Vertical displacement (m)

beta

Fig. C-19 Plane Truss- Membership Function for axial force in Member 4 at beta=0.8

-10.89

-10.59

-10.3

-10.02

-9.748

-9.483 -6.322

-6.15

-5.982

-5.818

-5.659

-5.5040

0.2

0.4

0.6

0.8

1

-11.00 -10.50 -10.00 -9.50 -9.00 -8.50 -8.00 -7.50 -7.00 -6.50 -6.00 -5.50 -5.00Axial Force (kN)

alph

a

Fig. C-20 Plane Truss- Membership Function for Axial Force in Member 10 at alpha=0.6

0

1.678

3.356

5.034

6.712

9.305

11.165

13.026

14.887

16.748

18.609

8.389

0

0.2

0.4

0.6

0.8

1

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0Axial Force (kN)

beta

4 6 43 10

8 5 m 116 5 7

9 211 3

10 m 10 m

20 kN Fig. C-17 Plane Truss

5. Preliminary Case Study -4 –Plane Frame

The fourth case study considered here is a single bay-single storey portal

frame shown in Fig. C-21. The plane frame has the following properties (Table. C-16)

Table. C-16 Material and Geometric Properties of Plane Frame

Area of Cross section of each column 6.0e-2 m2 Moment of Inertia of each column 2.0e-4 m4 Area of Cross section of beam 7.5e-2 m2 Moment of Inertia of beam 3.91e-2 m4 Young’s Modulus (E) 200 GPa Material Uncertainty of E (αmax) ±5% about mean value of E Load Uncertainty (βmax) ±100% about mean value of load

The plane frame is subjected to the combination of loads as shown in

Fig. C-21. Uncertainty introduced in the Young’s modulus and the applied loads. The

membership functions shown in Fig. C-10 and Fig. C-11 are used to represent

material and load uncertainties in this case study also.

Table. C-17 and Table. C-18 represent the simultaneous variation of vertical

displacement and rotation respectively at node 3 for various combinations of α and β.

Fig. C-22 illustrates the variation of horizontal displacement at of node 2 various

levels of material uncertainty α at β=1.0. Fig. C-23 represents the variation of vertical

displacement at of node 3 various levels of material uncertainty α at β=0.8. Fig. C-24

represents the variation of rotation at of node 3 various levels of load uncertainty β at

α=0.6. Table. C-19 represents the simultaneous variation of bending moment at node

2 for various combinations of α and β. Table. C-20 represents the simultaneous

variation of axial force in member 1 for various combinations of α and β. Fig. C-25

illustrates the variation of bending moment at node 2 at various levels of material

uncertainty α at β=1.0. Fig. C-26 represents the variation of shear force just to the left

of node 3 at various levels of load uncertainty β at α=1.0.

Fig. C-27 depicts the variation of axial force in member 1 at various levels of

load uncertainty α at β=0.8. Fig. C-28 depicts the variation of axial force in member 1

at various levels of load uncertainty β at α=0.6.

192

6. Conclusions

In all the above case studies, uncertain structural response is evaluated in the

presence of multiple uncertainties. In all the cases, it is found that the uncertainty of

structural response (such as displacement, rotation, moment, shear force and axial

force) is found to vary simultaneously with the independent variation of material and

load uncertainties. Membership functions of uncertain structural response are found to

be triangular in the presence of a single uncertainty (Fig. C-5, Fig. C-8, Fig. C-22,

Fig. C-25 and Fig. C-26) and trapezoidal in the presence of multiple uncertainties

(Fig. C-6, Fig. C-7, Fig. C-12, Fig. C-13, Fig. C-14, Fig. C-18, Fig. C-19, Fig. C-20,

Fig. C-23 and Fig. C-24).

193

194

Table. C-17 Plane Frame-Concomitant Simultaneous Variation of Vertical displacement of node 3 (×10-5 metres) w.r.t α and β

β→ α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [1.912,1.912] [1.251,2.572] [0.591,3.232] [-0.069,3.892] [-0.729,4.553] [-1.390,5.213]

0.8 [1.874,1.949] [1.218,2.614] [0.562,3.279] [-0.095,3.944] [-0.751,4.609] [-1.407,5.274]

0.6 [1.838,1.987] [1.185,2.657] [0.533,3.327] [-0.119,3.996] [-0.772,4.666] [-1.424,5.336]

0.4 [1.801,2.025] [1.153,2.700] [0.504,3.375] [-0.145,4.049] [-0.793,4.724] [-1.442,5.398]

0.2 [1.765,2.064] [1.120,2.744] [0.476,3.423] [-0.169,4.103] [-0.814,4.782] [-1.459,5.462]

0.0 [1.729,2.104] [1.088,2.788] [0.447,3.473] [-0.194,4.157] [-0.835,4.841] [-1.476,5.526]

Table. C-18 Plane Frame-Concomitant Variation of Rotation at node 3 (×10-3 radians)

w.r.t α and β β→

α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [0.653,0.653] [0.369,0.938] [0.084,1.223] [-0.201,1.508] [-0.486,1.793] [-0.771,2.078]

0.8 [0.625,0.682] [0.343,0.969] [0.060,1.256] [-0.222,1.544] [-0.505,1.831] [-0.788,2.119]

0.6 [0.598,0.710] [0.317,0.999] [0.037,1.290] [-0.244,1.580] [-0.524,1.870] [-0.804,2.160]

0.4 [0.570,0.738] [0.292,1.031] [0.013,1.324] [-0.265,1.616] [-0.543,1.909] [-0.821,2.201]

0.2 [0.542,0.767] [0.266,1.062] [-0.010,1.358] [-0.286,1.653] [-0.562,1.948] [-0.838,2.243]

0.0 [0.514,0.796] [0.241,1.094] [-0.033,1.392] [-0.307,1.690] [-0.581,1.988] [-0.855,2.286]

Table. C-19 Plane Frame-Concomitant Variation of Bending Moment (kNm) at

node 2 w.r.t α and β β→

α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [-81.33,-81.33] [-106.34,-56.31] [-131.36,-31.30] [-156.37,-6.29] [-181.4,18.73] [-206.4,43.74]

0.8 [-77.97,-84.74] [-110.14,-53.33] [-135.54,-28.68] [-160.94,-4.04] [-186.3,20.61] [-211.7,45.26]

0.6 [-74.67,-88.20] [-114.01,-50.37] [-139.81,-26.08] [-165.62,-1.78] [-191.4,22.52] [-217.2,46.81]

0.4 [-71.41,-91.73] [-117.97,-47.45] [-144.19,-23.48] [-170.40,0.48] [-196.7,24.45] [-222.9,48.41]

0.2 [-68.20,-95.33] [-122.00,-44.55] [-148.68,-20.90] [-175.40,2.75] [-202.0,26.40] [-228.7,50.05]

0.0 [-65.02,-98.99] [-126.13,-41.67] [-153.28,-18.32] [-180.40,5.04] [-207.6,28.39] [-234.7,51.74]

Table. C-20 Plane Frame-Concomitant Variation of Axial Force (N) in member 1

w.r.t α and β β→

α↓ 1.0 0.8 0.6 0.4 0.2 0.0

1.0 [57.35,57.35] [37.54,77.16] [17.73,96.97] [-2.083,116.8] [-21.89,136.6] [-41.702,156.39]

0.8 [55.26,59.47] [35.69,79.52] [16.12,99.58] [-3.45,119.64] [-23.03,139.7] [-42.599,159.76]

0.6 [53.21,61.62] [33.87,81.94] [14.52,102.26] [-4.83,122.59] [-24.18,142.9] [-43.524,163.23]

0.4 [51.20,63.82] [32.06,84.42] [12.927,105.012] [-6.21,125.61] [-25.34,146.2] [-44.477,166.80]

0.2 [49.21,66.06] [30.27,86.95] [11.340,107.830] [-7.59,128.72] [-26.53,149.6] [-45.459,170.48]

0.0 [47.24,68.349] [28.50,89.54] [9.756,110.723] [-8.99,131.91] [-27.73,153.1] [-46.472,174.28]

195

Fig. C-22 Plane Frame- Membership Function for Horizontal Displacement at node 2 at beta=1.0

-5.097

-4.968

-4.84

-4.713

-4.587

-4.462-4.462

-4.338

-4.215

-4.092

-3.97

-3.8490

0.2

0.4

0.6

0.8

1

-5.2 -5.1 -5 -4.9 -4.8 -4.7 -4.6 -4.5 -4.4 -4.3 -4.2 -4.1 -4 -3.9 -3.8Horizontal Displacement(mm)

alph

a

Fig. C-23 Plane Frame- Membership Function for Vertical Displacement at node 3 at beta=0.8

1.088

1.12

1.153

1.185

1.218

1.251 2.572

2.614

2.657

2.7

2.744

2.7880

0.2

0.4

0.6

0.8

1

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8Vertical Displacement(X0.01mm)

alph

a

Fig. C-24 Plane Frame- Membership Function for rotation at node 3 at alpha=0.6

-0.804

-0.524

-0.244

0.037

0.317

0.710

0.999

1.290

1.580

1.870

2.160

0.598

0

0.2

0.4

0.6

0.8

1

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5rotation (X0.001 radians)

beta

Fig. C-21 Plane Frame

0 kN5

1

25 kN/m

100 kN

4 m

3 m

2 3

4

1

2

3

m

196

Fig. C-25 Plane Frame-Membership Function for Bending Moment at node 2 at beta=1.0

65.02

68.2

71.41

74.67

77.97

81.3381.33

84.74

88.2

91.73

95.33

98.990

0.2

0.4

0.6

0.8

1

65 70 75 80 85 90 95 100Bending Moment (kNm)

alph

a

Fig. C-26 Plane Frame- Membership Function for Shear Force just to left ofnode 3 at alpha=1.0

-156.39

-136.58

-116.78

-96.97

-77.16

-57.35-57.35

-37.54

-17.73

2.08

21.89

41.70

0.2

0.4

0.6

0.8

1

-160 -135 -110 -85 -60 -35 -10 15 40Shear Force (kN)

beta

Fig. C-27 Plane Frame- Membership Function for Axial Force in member 1 at beta=0.8

28.5

30.27

32.06

33.87

35.69

37.54 77.16

79.52

81.94

84.42

86.95

89.540

0.2

0.4

0.6

0.8

1

25 35 45 55 65 75 85Axial Force(kN)

alph

a

Fig. C-28 Plane Frame- Membership Function for Axial Force in member 1 at alpha=0.6

-43.524

-24.18

-4.83

14.52

33.87

61.62

81.94

102.26

122.59

142.9

163.227

53.21

0

0.2

0.4

0.6

0.8

1

-50 -25 0 25 50 75 100 125 150 175Axial Force (kN)

beta

BIO-DATA Mr. M.V.Rama Rao, the author of the present work, completed B.Tech in Civil

Engineering from JNT University in 1986 with distinction. He obtained M.Tech

degree in Structural Engineering from Indian Institute of Technology, Kanpur in

1988. He worked as a Junior Research Fellow for a brief period in a CSIR sponsored

project in the area of Structural Dynamics at I.I.T Kanpur.

As a student Mr. Rama Rao was an active participant in extra-curricular activities like

elocution, debate, essay writing and poetry. He won a number of merit certificates,

cups, medals and shields throughout the school and college studies. Mr. Rama Rao

worked as a software engineer in M/s IMI Engineering Ltd., Hyderabad during 1992.

His work included development and maintenance of finite element software related to

transmission line towers, multi-storeyed structures and bridges.

Mr. Rama Rao joined Vasavi College of Engineering, Hyderabad towards the end of

1992 as a Lecturer in Civil Engineering. Currently, he is working as a Senior

Assistant Professor in the same college. His areas of specialisation include Structural

Dynamics, Finite Element Analysis and Engineering Mechanics.

197

PUBLICATIONS

Rama Rao, M.V. and Ramesh Reddy, R. (2003). “Fuzzy Finite Element Analysis

of Structures with Uncertainty in Load and Material Properties”, Journal of

Structural Engineering, Structural Engineering Research Centre (SERC), Chennai

(accepted for publication).

Rama Rao, M.V. and Ramesh Reddy, R. (2003). “ Fuzzy Finite Element Analysis

of a Cable-Stayed Bridge”, Proc, 8th International Conference on Innovations in

Planning, Design and Construction Techniques in Bridge Engineering, IIBE,

Hyderabad, pp 321-331.


of a Plane Frame with Multiple Uncertainties.”, Proc, National Conference on

Emerging Trends in (ETSMC-2003), NIT, Rourkela, December, 01-02.

Rama Rao, M.V. and Ramesh Reddy, R. (2003). “Uncertainty in Engineering

Analysis and Design – An Overview.”, Proc, National Conference on Recent

Advances in Civil Engineering (RACE-2003), KITS Warangal, December 29-30.


of a Cable Stayed Bridge”, Proc, National Conference on Materials and

Structures (MAST-2004), NIT, Warangal, January 23-24, 214-218.

Rama Rao, M.V. and Ramesh Reddy, R. (2004). “Fuzzy Finite Element of a Plane

Truss with Multiple Uncertainties”, Proc, National Conference on Materials and

Structures (MAST-2004), NIT, Warangal, January 23-24, 250-254.

198

Documents

M. VENKATA RAMA RAO - .: Mathematical Sciences : … thank my friends S.R.Ranganath, P.Surya Prakash and A. Radha Krishna and my colleagues Sri. M.Bhasker, Sri. B.Veeranna and Sri