View
7
Download
0
Category
Preview:
Citation preview
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
3.3 Case Study 2 – Cable-Stayed Bridge with Uncertainties of Material
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
3.6 Sensitivity Analysis 63
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
Property and Mass Density
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
4.1.2.7 Axial force (kN) in cable 3 (at node 3) 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
4.2.1.1 Uncertain horizontal displacement at node 2 93
4.2.1.2 Uncertain vertical displacement at node 3 94
4.2.1.3 Uncertain rotation at node 4 94
4.2.2 Effect of concomitant variation of α and β on shear forces and bending
moments
95
4.2.2.1 Axial force (kN) in deck in element 9 95
4.2.2.2 Axial force (kN) in deck in element 12 96
4.2.2.3 Shear force (kN) in deck just to the left of node 3 96
4.2.2.4 Bending moment (kNm) in deck at node 4 97
4.2.2.5 Axial force (kN) in pylon in element 17 (at node 11) 97
4.2.2.6 Axial force (kN) in cable 2 (at node 2) 98
4.2.2.7 Axial force (kN) in cable 3 (at node 3) 98
4.3 Case Study 3 – Cable-Stayed Bridge with Uncertainties of Material
Property, Live Load and Mass Density
113
4.3.1 Effect of concomitant variation of α and β on displacements and rotations 113
4.3.1.1 Uncertain horizontal displacement at node 2 114
4.3.1.2 Uncertain vertical displacement at node 3 115
xi
4.3.1.3 Uncertain rotation at node 4 115
4.3.2 Effect of concomitant variation of α and β on shear forces and bending
moments
116
4.3.2.1 Axial force (kN) in deck in element 9 117
4.3.2.2 Axial force (kN) in deck in element 12 118
4.3.2.3 Shear force (kN) in deck just to the left of node 3 118
4.3.2.4 Bending moment (kNm) in deck at node 4 119
4.3.2.5 Axial force (kN) in pylon in element 17 (at node 11) 120
4.3.2.6 Axial force (kN) in cable 2 (at node 2) 121
4.3.2.7 Axial force (kN) in cable 3 (at node 3) 122
4.4 Sensitivity Analysis 151
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
Table. 4.1.5 Cable stayed Bridge - Concomitant Variation of Axial Force (kN) in
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
Table. 4.1.8 Cable stayed Bridge - Concomitant Variation of Axial Force (kN) in
pylon in element 17 w.r.t α and γ
81
Table. 4.1.9 Cable stayed Bridge - Concomitant Variation of Axial Force (kN) in
cable 2 w.r.t α and γ
82
Table. 4.1.10 Cable stayed Bridge - Concomitant Variation of Axial Force (kN) in
cable 3 w.r.t α and γ
82
Table. 4.2.1 Cable stayed Bridge - Concomitant Variation of Horizontal
displacement of node 2 (×10-4 metres) w.r.t α and β
99
Table. 4.2.2 Cable stayed Bridge - Concomitant Variation of Vertical
displacement of node 3 (×10-2 metres) w.r.t α and β
99
xvi
Table. 4.2.3 Cable stayed Bridge - Concomitant Variation of Rotation at node 4
(×10-3 radians) w.r.t α and β
99
Table. 4.2.4 Cable stayed Bridge - Concomitant Variation of Axial Force (kN) in
deck in element 9 w.r.t α and β
100
Table. 4.2.5 Cable stayed Bridge - Concomitant Variation of Axial Force (kN) in
deck in element 12 w.r.t α and β
100
Table. 4.2.6 Cable stayed Bridge - Concomitant Variation of Shear Force (kN) in
deck just to the left of node 3 w.r.t α and β
100
Table. 4.2.7 Cable stayed Bridge - Concomitant Variation of Bending Moment
(kNm) in deck at node 4 w.r.t α and β
101
Table. 4.2.8 Cable stayed Bridge - Concomitant Variation of Axial Force (kN) in
pylon in element 17 w.r.t α and β
101
Table. 4.2.9 Cable stayed Bridge - Concomitant Variation of Axial Force (kN) in
cable 2 w.r.t α and β
102
Table. 4.2.10 Cable stayed Bridge - Concomitant Variation of Axial Force (kN) in
cable 3 w.r.t α and β
102
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
124
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
124
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
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
125
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
125
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
125
Table. 4.3.7 Cable stayed Bridge - Concomitant Variation of Axial Force (kN) in
deck in element 9 w.r.t α and β at γ =1.0
126
Table. 4.3.8 Cable stayed Bridge - Concomitant Variation of Axial Force (kN) in
deck in element 9 w.r.t α and β at γ =0.8
126
xvii
Table. 4.3.9 Cable stayed Bridge - Concomitant Variation of Axial Force (kN) in
deck in element 12 w.r.t α and β at γ =1.0
126
Table. 4.3.10 Cable stayed Bridge - Concomitant Variation of Axial Force (kN) in
deck in element 12 w.r.t α and β at γ =0.8
127
Table. 4.3.11 Cable stayed Bridge - Concomitant Variation of Shear Force (kN) in
deck just to the left of node 3 w.r.t α and β at γ =1.0
127
Table. 4.3.12 Cable stayed Bridge - Concomitant Variation of Shear Force (kN) in
deck just to the left of node 3 w.r.t α and β at γ =0.8
127
Table. 4.3.13 Cable stayed Bridge - Concomitant Variation of Bending Moment
(kNm) in deck at node 4 w.r.t α and β at γ =1.0
128
Table. 4.3.14 Cable stayed Bridge - Concomitant Variation of Bending Moment
(kNm) in deck at node 4 w.r.t α and β at γ =0.8
128
Table. 4.3.15 Cable stayed Bridge - Concomitant Variation of Axial Force (kN) in
pylon in element 17 w.r.t α and β at γ =1.0
128
Table. 4.3.16 Cable stayed Bridge - Concomitant Variation of Axial Force (kN) in
pylon in element 17 w.r.t α and β at γ =0.8
129
Table. 4.3.17 Cable stayed Bridge - Concomitant Variation of Axial Force (kN) in
cable 2 w.r.t α and β at γ =1.0
129
Table. 4.3.18 Cable stayed Bridge - Concomitant Variation of Axial Force (kN) in
cable 2 w.r.t α and β at γ =0.8
129
Table. 4.3.19 Cable stayed Bridge - Concomitant Variation of Axial Force (kN) in
cable 3 w.r.t α and β at γ =1.0
130
Table. 4.3.20 Cable stayed Bridge - Concomitant Variation of Axial Force (kN) in
cable 3 w.r.t α and β at γ =0.8
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
Fig. 4.2.5 Membership Functions for Axial Force in deck in element 12 at
(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
Fig. 4.3.3 Membership Functions for vertical displacement at node 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
133
Fig. 4.3.4 Membership Functions for vertical displacement at node 3 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
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
Fig. 4.3.7 Membership Functions for Axial Force in deck in element 9 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
137
Fig. 4.3.8 Membership Functions for Axial Force in deck in element 9 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
138
Fig. 4.3.9 Membership Functions for Axial Force in deck in element 12 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
139
Fig. 4.3.10 Membership Functions for Axial Force in deck in element 12 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
140
Fig. 4.3.11 Membership Functions for Shear Force in deck to left of node 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
141
xxi
Fig. 4.3.12 Membership Functions for Shear Force in deck to left of node 3 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
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
Fig. 4.3.15 Membership Functions for Axial Force in pylon in element 17 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
145
Fig. 4.3.16 Membership Functions for Axial Force in pylon in element 17 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
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
Fig. 4.3.18 Membership Functions for Axial Force in cable 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
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
Fig. 4.3.20 Membership Functions for Axial Force in cable 3 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
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 P´i = [P´il , P´iu] 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.
The uncertainty associated with displacements and forces is a function of the
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.
The uncertainty associated with displacements and forces is a function of the
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
Property and Mass Density
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)
Fig. 4.1.9 (a) and (b) represent the membership functions axial force (kN) in
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) .
4.1.2.7 Axial force (kN) in cable 3 (at node 3)
Fig. 4.1.10 (a) and (b) represent the membership functions axial force (kN) in
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.
Table 4.2.1, Table 4.2.2 and Table 4.2.3 show the concomitant variation of
horizontal and vertical displacements and rotations at nodes 2,3 and 4 respectively, for
various combinations of α and β.
4.2.1.1 Uncertain horizontal displacement at node 2
Fig. 4.2.1 (a) and (b) represent the membership functions for horizontal
displacement at node 2 (m) at β=1.0 and β=0.8. The uncertain horizontal
displacements at α=0.0 and α=1.0 correspond to normalised values [0.884,1.121],
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).
4.2.1.2 Uncertain vertical displacement at node 3
Fig. 4.2.2 (a) and (b) represent the membership functions for vertical
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).
4.2.1.3 Uncertain rotation at node 4
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%).
4.2.2 Effect of concomitant variation of α and β on shear forces and bending
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)
and [0.501,1.550], [0.675,1.325] in Fig. 4.2.4 (b). Fig. 4.2.4 (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
[-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).
4.2.2.2 Axial force (kN) in deck in element 12
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) .
4.2.2.3 Shear force (kN) in deck just to the left of node 3
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)
and [-1.360,-0.673] , [-1.243,-0.756] in Fig. 4.2.6 (b). Fig. 4.2.6 (c) and (d) represent
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).
4.2.2.4 Bending moment (kNm) in deck at node 4
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
α=0.0 and α=1.0 correspond to normalised values [-1.183,-0.823] , [-1.000,-1.000] in
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).
4.2.2.5 Axial force (kN) in pylon in element 17 (at node 11)
Fig. 4.2.8 (a) and (b) represent the membership functions axial force (kN) in
pylon in element 17 (at node 11) at β=1.0 and β=0.8. The uncertain axial forces at
α=0.0 and α=1.0 correspond to normalised values [-1.118,-0.898] , [-1.000,-1.000] in
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).
4.2.2.6 Axial force (kN) in cable 2 (at node 2)
Fig. 4.2.9 (a) and (b) represent the membership functions axial force (kN) in
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
membership functions for axial force in cable 2 at α=1.0 and α=0.6. The uncertain
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).
4.2.2.7 Axial force (kN) in cable 3 (at node 3)
Fig. 4.2.10 (a) and (b) represent the membership functions axial force (kN) in
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
membership functions for axial force in cable 3 at α=1.0 and α=0.6. The uncertain
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
-100.0 -95.0 -90.0 -85.0 -80.0 -75.0 -70.0Bending Moment (kNm)
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.
Table 4.3.1, Table 4.3.3 and Table 4.3.5 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=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%).
4.3.1.1 Uncertain horizontal displacement at node 2
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.
Fig. 4.3.1 (a) and (b) represent the membership functions for horizontal
displacement at node 2 (m) at β=1.0,γ=1.0 and β=0.8,γ=1.0. The uncertain horizontal
displacements at α=0.0 and α=1.0 correspond to normalised values [0.885,1.122],
[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).
Fig. 4.3.2 (a) and (b) represent the membership functions for horizontal
displacement at node 2 (m) at β=1.0,γ=0.8 and β=0.8,γ=0.8. The uncertain horizontal
displacements at α=0.0 and α=1.0 correspond to normalised values [0.874,1.133],
[0.989,1.011] in Fig. 4.3.2 (a) and [0.774,1.238], [0.887,1.114] in Fig. 4.3.2 (b).
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
4.3.1.2 Uncertain vertical displacement at node 3
Fig. 4.3.3 (a) and (b) represent the membership functions for vertical
displacement at node 3 (m) at β=1.0,γ=1.0 and β=0.8,γ=1.0. The uncertain vertical
displacements at α=0.0 and α=1.0 correspond to normalised values [-1.310,-0.694] ,
[-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).
Fig. 4.3.4 (a) and (b) represent the membership functions for vertical
displacement at node 3 (m) at β=1.0,γ=0.8 and β=0.8,γ=0.8. The uncertain vertical
displacements at α=0.0 and α=1.0 correspond to normalised values [-1.342,-0.665] ,
[-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).
4.3.1.3 Uncertain rotation at node 4
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
uncertainties (α and β) at γ=1.0 and γ=0.8 respectively.
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)
and [0.713,1.312], [0.898,1.102] in Fig. 4.3.7 (b). Fig. 4.3.7 (c) and (d) represent the
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).
Fig. 4.3.8 (a) and (b) represent the membership functions for uncertain axial
force (kN) in element 9 at β=1.0,γ=0.8 and β=0.8,γ=0.8. The axial forces at α=0.0
and α=1.0 correspond to normalised values [0.8,1.215], [0.989,1.011] in Fig. 4.3.8 (a)
and [0.703,1.324], [0.887,1.113] in Fig. 4.3.8 (b). Fig. 4.3.9 (c) and (d) represent the
membership functions for axial force (kN) in element 9 at α=1.0,γ=0.8 and
117
α=0.6,γ=0.8. The uncertain axial forces at β=0.0 and β=1.0 correspond to normalised
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).
4.3.2.2 Axial force (kN) in deck in element 12
Fig. 4.3.9 (a) and (b) represent the membership functions for uncertain axial
force (kN) in element 12 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 [-1.165,-0.855],[-1.0,-1.0] in Fig. 4.3.9 (a)
and [-1.239,-0.801] , [-1.063,-0.937] in Fig. 4.3.9 (b). Fig. 4.3.9 (c) and (d) represent
the membership functions for axial force (kN) in element 12 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 [-1.314,-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).
Fig. 4.3.10 (a) and (b) represent the membership functions for uncertain axial
force (kN) in element 12 at β=1.0,γ=0.8 and β=0.8,γ=0.8. The axial forces at α=0.0
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).
4.3.2.3 Shear force (kN) in deck just to the left of node 3
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 ).
Fig. 4.3.12 (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=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).
4.3.2.4 Bending moment (kNm) in deck at node 4
Fig. 4.3.13 (a) and (b) represent the membership functions for the uncertain
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).
Fig. 4.3.14 (a) and (b) represent the membership functions for the uncertain
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).
4.3.2.5 Axial force (kN) in pylon in element 17 (at node 11)
Fig. 4.3.15 (a) and (b) represent the membership functions for the uncertain
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).
Fig. 4.3.16 (a) and (b) represent the membership functions for the uncertain
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).
4.3.2.6 Axial force (kN) in cable 2 (at node 2)
Fig. 4.3.17 (a) and (b) represent the membership functions for the uncertain
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).
Fig. 4.3.18 (a) and (b) represent the membership functions for the uncertain
axial force (kN) in cable 2 (at node 2) 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 [0.816,1.205],
[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).
Fig. 4.3.18 (c) and (d) represent the membership functions for axial force (kN) in
cable 2 (at node 2) 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 [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
4.3.2.7 Axial force (kN) in cable 3 (at node 3)
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
uncertainties (α and β) at γ=1.0 and γ=0.8 respectively.
Fig. 4.3.19 (a) and (b) represent the membership functions for the uncertain
axial force (kN) in cable 3 (at node 3) 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 [0.822,1.197],
[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 (d) represent the membership functions for axial force (kN) in
cable 3 (at node 3) 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 [0.669,1.332], [1.0,1.0] in
Fig. 4.3.19 (c) and [0.609,1.422] [0.927,1.076] in Fig. 4.3.19 (d).
Fig. 4.3.20 (a) and (b) represent the membership functions for the uncertain
axial force (kN) in cable 3 (at node 3) 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 [0.815,1.206],
[0.992,1.008] in Fig. 4.3.20 (a) and [0.755,1.280], [0.926,1.074] in Fig. 4.3.20 (b).
Fig. 4.3.20 (c) and (d) represent the membership functions for axial force (kN) in
cable 3 (at node 3) 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 [0.660,1.340], [0.992,1.008] in
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
Table. 4.3.7 Cable stayed Bridge - Concomitant Variation
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
Table. 4.3.10 Cable stayed Bridge - Concomitant Variation
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
Table. 4.3.13 Cable stayed Bridge - Concomitant Variation
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
Table. 4.3.16 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=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
9.000E-04 9.500E-04 1.000E-03 1.050E-03 1.100E-03 1.150E-03 1.200E-03Horizontal Displacement (m)
alph
a
(a) beta=0.8 , gamma=1.0
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
8.000E-04 8.750E-04 9.500E-04 1.025E-03 1.100E-03 1.175E-03 1.250E-03Horizontal Displacement (m)
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
5.000E-04 6.250E-04 7.500E-04 8.750E-04 1.000E-03 1.125E-03 1.250E-03 1.375E-03 1.500E-03Horizontal Displacement (m)
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
4.000E-04 5.500E-04 7.000E-04 8.500E-04 1.000E-03 1.150E-03 1.300E-03 1.450E-03 1.600E-03Horizontal Displacement (m)
beta
Fig. 4.3.1 Membership Functions for horizontal displacement at node 2 at γ=1.0
132
(a) beta=1.0 , gamma=0.8
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
9.000E-04 9.500E-04 1.000E-03 1.050E-03 1.100E-03 1.150E-03 1.200E-03Horizontal Displacement (m)
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
8.000E-04 8.750E-04 9.500E-04 1.025E-03 1.100E-03 1.175E-03 1.250E-03Horizontal Displacement (m)
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
5.000E-04 6.250E-04 7.500E-04 8.750E-04 1.000E-03 1.125E-03 1.250E-03 1.375E-03 1.500E-03Horizontal Displacement (m)
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
4.000E-04 5.500E-04 7.000E-04 8.500E-04 1.000E-03 1.150E-03 1.300E-03 1.450E-03 1.600E-03Horizontal Displacement (m)
beta
Fig. 4.3.2 Membership Functions for horizontal displacement at node 2 at γ=0.8
133
(a) beta=1.0 , gamma=1.0
-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
-7.55E-02 -6.90E-02 -6.25E-02 -5.60E-02 -4.95E-02 -4.30E-02 -3.65E-02 -3.00E-02 -2.35E-02Vertical Displacement (m)
alph
a
(c) alpha=1.0 , gamma=1.0
-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
(d) alpha=0.6 , gamma=1.0
-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
(a) beta=1.0 , gamma=0.8
-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
-6.36E-02 -5.96E-02 -5.56E-02 -5.16E-02 -4.76E-02 -4.36E-02 -3.96E-02 -3.56E-02 -3.16E-02Vertical Displacement (m)
alph
a
(b) beta=0.8 , gamma=0.8
-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
(c) alpha=1.0 , gamma=0.8
-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
(d) alpha=0.6 , gamma=0.8
-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
(a) beta=1.0 , gamma=1.0
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
(b) beta=0.8 , gamma=1.0
-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
(c) alpha=1.0 , gamma=1.0
-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
-3.000E-03 -1.750E-03 -5.000E-04 7.500E-04 2.000E-03 3.250E-03 4.500E-03Rotation (radians)
beta
(d) alpha=0.6 , gamma=1.0
-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
-3.000E-03 -1.750E-03 -5.000E-04 7.500E-04 2.000E-03 3.250E-03 4.500E-03Rotation (radians)
beta
Fig. 4.3.5 Membership Functions for rotation at node 4 at γ=1.0
136
(a) beta=1.0 , gamma=0.8
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
(b) beta=0.8 , gamma=0.8
-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
(c) alpha=1.0 , gamma=0.8
-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
-3.000E-03 -1.750E-03 -5.000E-04 7.500E-04 2.000E-03 3.250E-03 4.500E-03Rotation (radians)
beta
(d) alpha=0.6 , gamma=0.8
-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
Fig. 4.3.8 Membership Functions for Axial Force in deck in element 9 at γ=0.8
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
Fig. 4.3.9 Membership Functions for Axial Force in deck in element 12 at γ=1.0
140
(a) beta=1.0,gamma=0.8
-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
(b) beta=0.8,gamma=0.8
-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
(c) alpha=1.0,gamma=0.8
-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
(d) alpha=0.6,gamma=0.8
-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
Fig. 4.3.10 Membership Functions for Axial Force in deck in element 12 at γ=0.8
141
(a) beta=1.0,gamma=1.0
-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
(b) beta=0.8,gamma=1.0
-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
(c) alpha=1.0,gamma=1.0
-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
(d) alpha=0.6,gamma=1.0
-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
(a) beta=1.0,gamma=0.8
-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
(b) beta=0.8,gamma=0.8
-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
(c) alpha=1.0,gamma=0.8
-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
(d) alpha=0.6,gamma=0.8
-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
(a) beta=1.0,gamma=1.0
-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
-310.0 -294.0 -278.0 -262.0 -246.0 -230.0 -214.0Bending Moment (kNm)
beta
(b) beta=0.8,gamma=1.0
-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
-340.0 -321.0 -302.0 -283.0 -264.0 -245.0 -226.0 -207.0 -188.0Bending Moment (kNm)
alph
a
(c) alpha=1.0,gamma=1.0
-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
-400.0 -365.0 -330.0 -295.0 -260.0 -225.0 -190.0 -155.0 -120.0Bending Moment (kNm)
beta
(d) alpha=0.6,gamma=1.0
-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
-415.0 -375.0 -335.0 -295.0 -255.0 -215.0 -175.0 -135.0 -95.0Bending Moment (kNm)
beta
Fig. 4.3.13 Membership Functions for Bending Moment at node 4 at γ=1.0
144
(a) beta=1.0,gamma=0.8
-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
(b) beta=0.8,gamma=0.8
-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
-340.0 -318.0 -296.0 -274.0 -252.0 -230.0 -208.0 -186.0Bending Moment (kNm)
alph
a
(c) alpha=1.0,gamma=0.8
-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
-400.0 -365.0 -330.0 -295.0 -260.0 -225.0 -190.0 -155.0 -120.0Bending Moment (kNm)
beta
(d) alpha=0.6,gamma=0.8
-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
-415.0 -375.0 -335.0 -295.0 -255.0 -215.0 -175.0 -135.0Bending Moment (kNm)
beta
Fig. 4.3.14 Membership Functions for Bending Moment at node 4 at γ=0.8
145
(a) beta=1.0,gamma=1.0
-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
(b) beta=0.8,gamma=1.0
-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
(c) alpha=1.0,gamma=1.0
-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
(d) alpha=0.6,gamma=1.0
-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
(a) beta=1.0,gamma=0.8
-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
(b) beta=0.8,gamma=0.8
-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
(c) alpha=1.0,gamma=0.8
-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
(d) alpha=0.6,gamma=0.8
-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
(a) beta=1.0,gamma=1.0
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
(b) beta=0.8,gamma=1.0
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
(c) alpha=1.0,gamma=1.0
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
(d) alpha=0.6,gamma=1.0
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
(a) beta=1.0,gamma=0.8
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
(b) beta=0.8,gamma=0.8
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
(c) alpha=1.0,gamma=0.8
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
(d) alpha=0.6,gamma=0.8
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
Fig. 4.3.18 Membership Functions for Axial Force in cable 2 at γ=0.8
149
(a) beta=1.0,gamma=1.0
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
(b) beta=0.8,gamma=1.0
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
(c) alpha=1.0,gamma=1.0
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
(d) alpha=0.6,gamma=1.0
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
Fig. 4.3.19 Membership Functions for Axial Force in cable 3 at γ=1.0
150
(a) beta=1.0,gamma=0.8
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
(b) beta=0.8,gamma=0.8
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
(c) alpha=1.0,gamma=0.8
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
(d) alpha=0.6,gamma=0.8
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
Fig. 4.3.20 Membership Functions for Axial Force in cable 3 at γ=0.8
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
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 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
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 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
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. 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
Percentage Variation of E about mean
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
Percentage Variation of E about mean
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
Percentage Variation of E about mean
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
Percentage variation of E about mean
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
Percentage variation of E about mean
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.
Recommended