A PARAMETRIC STUDY ON DYNAMIC ANALYSIS OF MULTI-STOREY BUILDING WITH

UNCERTAINTIES USING INTERVAL RESPONSE SPECTRUM ANALYSIS (IRSA)

MASTER OF ENGINEERING IN

CIVIL ENGINEERING (With Specialization in Structural Engineering)

By

VISHWANATHULA ANIL KUMAR (0109–11109)

DEPARTMENT OF CIVIL ENGINEERING UNIVERSITY COLLEGE OF ENGINEERING (A)

OSMANIA UNIVERSITY, HYDERABAD 2011

A PARAMETRIC STUDY ON DYNAMIC ANALYSIS OF MULTI-STOREY BUILDING WITH UNCERTAINTIES

USING INTERVAL RESPONSE SPECTRUM ANALYSIS (IRSA)

MASTER OF ENGINEERING IN

CIVIL ENGINEERING (STRUCTURAL ENGINEERING)

By

VISHWANATHULA ANIL KUMAR

(0109–11109)

DEPARTMENT OF CIVIL ENGINEERING UNIVERSITY COLLEGE OF ENGINEERING (AUTONOMOUS)

OSMANIA UNIVERSITY HYDERABAD – 500 007

AUGUST 2011

A PARAMETRIC STUDY ON DYNAMIC ANALYSIS OF MULTI-STOREY BUILDING WITH UNCERTAINTIES

USING INTERVAL RESPONSE SPECTRUM ANALYSIS (IRSA)

A Dissertation work submitted to Osmania University in partial fulfillment of the

Requirements for the Award of Degree of

MASTER OF ENGINEERING

IN

CIVIL ENGINEERING (STRUCTURAL ENGINEERING)

By

VISHWANATHULA ANIL KUMAR

(Roll No. 0109–11109)

Under the guidance of

Dr. R. RAMESH REDDY Professor of Civil Engineering

DEPARTMENT OF CIVIL ENGINEERING

UNIVERSITY COLLEGE OF ENGINEERING (AUTONOMOUS) OSMANIA UNIVERSITY HYDERABAD – 500 007

AUGUST 2011

ii

DEPARTMENT OF CIVIL ENGINEERING

UNIVERSITY COLLEGE OF ENGINEERING (A) OSMANIA UNIVERSITY

CERTIFICATE

This is to certify that the dissertation entitled, “A Parametric Study on Dynamic

Analysis of Multi-Storey Building with Uncertainties Using Interval Response

Spectrum Analysis (IRSA)” is carried out by Mr. VISHWANATHULA ANIL

KUMAR (Roll No. 0109 - 11109) as partial fulfillment of the requirements for the

award of “MASTER OF ENGINEERING” in Civil Engineering with specialization

in Structural Engineering from the Department of Civil Engineering, University

College of Engineering(A), Osmania University and that it is a bonafide work

executed by him during the academic year 2010 – 2011 under my guidance and

supervision.

The results embodied in this dissertation have not been submitted to any other

University or Institution for the award of any degree.

______________________ __________________________ Dr. R. Ramesh Reddy Prof. Ravande Kishore Professor and Supervisor Professor and Head Department of civil Engineering Department of civil Engineering University College of Engineering (A) University College of Engineering (A) Osmania University, Hyderabad. Osmania University, Hyderabad.

iii

DECLARATION

This is to certify that the work reported in the present thesis titled A Parametric

Study on Dynamic Analysis of Multi-Storey Building with Uncertainties Using

Interval Response Spectrum Analysis (IRSA) is a record of work done by me in

Department of Civil Engineering, University College of Engineering, Osmania

University.

No part of the thesis is copied from books/journals/internet and wherever the

portion is taken, the same has been duly referred in the text. The report is based on the

project work done entirely by me and not copied from any other source.

V. ANIL KUMAR

iv

ACKNOWLEDGEMENT

I feel honored in expressing my profound thanks, indebtedness and deep sense of

gratitude to my supervisor Dr. R. Ramesh Reddy, Professor, Department of Civil

Engineering, University College of Engineering (Autonomous), Osmania University,

Hyderabad from the core of my heart for his timely suggestions, generous help and

right direction throughout the course of present work in spite of his busy academic

schedule.

I would like to express my sincere thanks to Prof. Ravande Kishore, Head of the

Civil Engineering Department for the kind help in facilitating the departmental

infrastructural facilities.

I would like to express my sincere thanks to Dr. Kumar Molugaram, Chairman,

BOS, Civil Engineering, Osmania University, for his valuable support and advice for

preparation of this report as per the format and facilitating viva-voce.

Thanks are due to Dr. N Murali Krishna, Dr. V. Bhikshma, Mrs. K. L. Radhika

(Student Advisor), Dr. D. Rupesh Kumar and other staff members of the Department

and Library staff for their all out help and support during the thesis work.

I would like to thank my fellow graduate students for their support, feedback, and

friendship.

I have drawn upon heavily from various websites and numerous technical articles

authored by learned and eminent professionals, consultants and academicians

published in many journals. I am indebted to them all and the journals/books which I

referred to.

Last but not least I am indebted to my Parents and siblings without whom I cannot

reach up to this juncture. Finally, I wish to thank all those people who helped me

directly or indirectly in successful completion of my project in time.

V. ANIL KUMAR

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CONTENTS Page No. TITLE i CERTIFICATE ii DECLARATION iii ACKNOWLEDGEMENT iv CONTENTS v LIST OF TABLES vii LIST OF FIGURES viii LIST OF NOTATIONS AND ABBREVIATIONS xii ABSTRACT xvi CHAPTER - 1 INTRODUCTION 1 – 4 1.1 General 1 1.2 Need for the Present Study 3 1.3 Objective and Scope of Work 3 1.4 Summary 4 CHAPTER - 2 LITERATURE REVIEW 5- 66 2.1 Introduction 5 2.2 Interval Analysis and its Historical Background 5 2.3 Interval Finite Element Methods 7 2.4 Dynamic Analysis of Structures for Design under

Uncertainty 11

2.4.1 Probabilistic Methods 12 2.4.2 Fuzzy Set Based Methods 14 2.5 Interval Modal updating method and Application 15 2.6 Application of the FFEM in Structural Dynamics 18 2.7 Fundamentals of Structural Dynamics 19 2.7.1 Introduction 19 2.7.2 Characteristics of Dynamic Problem 20 2.7.3 Historical Background of Structural

Dynamics 20

2.7.4 The Equation of Motion 21 2.7.5 Free Vibration 22 2.7.6 Forced Vibration 24 2.7.7 Response History Analysis (RHA) 26 2.7.8 Response Spectrum Analysis (RSA) 26 2.7.9 Response Spectrum Analysis Summary 29 2.7.10 Limitations to RSA 30 2.8 Fundamentals of Uncertainty Analysis 30 2.8.1 Introduction 30 2.8.2 Causes and Sources of Uncertainty 31 2.8.3 Analysis of Uncertainty 32

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2.9 Interval Finite Element Methods (IFEM) 45 2.9.1 Introduction 45 2.9.2 Interval Approach 47 2.9.3 Interval Arithmetic 49 2.9.4 What is Interval Arithmetic and why it is

Considered? 50

2.9.5 Interval Finite Element Analysis 51 2.10 Interval Response Spectrum Analysis (IRSA) 54 2.10.1 Introduction 54 2.10.2 Interval Representation of Uncertainty 55 2.10.3 Bounds on Natural Frequencies and Mode

Shapes 58

2.10.4 Bounds on Natural Frequencies 60 2.10.5 Bounding the Mode Shapes 62 2.10.6 Bounding Dynamic Response 64 2.11 Summary 66 CHAPTER - 3 PROBLEM DESCRIPTION AND

METHODOLOGY 67 - 74

3.1 Introduction 67 3.2 Building Frame Data 67 3.3 Model Description 70 3.4 Methodology 73 3.5 Summary 74 CHAPTER - 4 RESULTS AND DISCUSSION 75 - 93 4.1 Introduction 75 4.2 Modal Participating Mass Ratios 75 4.3 Modal Participation Factors 78 4.4 Modal Periods and Frequencies 81 4.5 Application of IRSA Method to the Output Results 84 4.6 Discussion on observations 92 4.7 Summary 93 CHAPTER - 5 CONCUSIONS AND FUTURE SCOPE 94 - 95 5.1 Conclusions 94 5.2 Future Scope of Work 95 REFERENCES 96 - 104 APPENDIX - A 105 - 122

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LIST OF TABLES Sl. No Description Page No 3.1 Building Frame Data 67 4.1 Modal Participating Mass Ratios for Model 1 75 4.2 Modal Participating Mass Ratios for Model 2 76 4.3 Modal Participating Mass Ratios for Model 3 76 4.4 Modal Participating Mass Ratios for Model 4 76 4.5 Modal Participating Mass Ratios for Model 5 76 4.6 Modal Participating Mass Ratios for Model 6 77 4.7 Modal Participating Mass Ratios for Model 7 77 4.8 Modal Participating Mass Ratios for Model 8 77 4.9 Modal Participating Mass Ratios for Model 9 77 4.10 Modal Participation Factors for Model 1 78 4.11 Modal Participation Factors for Model 2 78 4.12 Modal Participation Factors for Model 3 78 4.13 Modal Participation Factors for Model 4 79 4.14 Modal Participation Factors for Model 5 79 4.15 Modal Participation Factors for Model 6 79 4.16 Modal Participation Factors for Model 7 80 4.17 Modal Participation Factors for Model 8 80 4.18 Modal Participation Factors for Model 9 80 4.19 Modal Periods and Frequencies for Model 1 81 4.20 Modal Periods and Frequencies for Model 2 81 4.21 Modal Periods and Frequencies for Model 3 81 4.22 Modal Periods and Frequencies for Model 4 82 4.23 Modal Periods and Frequencies for Model 5 82 4.24 Modal Periods and Frequencies for Model 6 82 4.25 Modal Periods and Frequencies for Model 7 83 4.26 Modal Periods and Frequencies for Model 8 83 4.27 Modal Periods and Frequencies for Model 9 83 4.28 Bounds on Eigen Values for Case 1 84 4.29 Bounds on Eigen Values for Case 2 84 4.30 Bounds on Eigen Values for Case 3 85 4.31 Bounds on Eigen Values for Case 4 85 4.32 Bounds on Eigen Values for Case 5 86 4.33 Bounds on Eigen Values for Case 6 86

viii  

LIST OF FIGURES Sl. No Description Page No

2.1 A generic response spectrum for an external excitation p(t) 27 2.2 A deterministic algebraic variable 33 2.3 Probability Density Function of a random quantity 34 2.4 Membership function of a fuzzy quantity 35 2.5 An interval quantity 36 2.6 An interval vector 38 2.7 Determination of corresponding to a for a generic response

spectrum 64

3.1 3D Model of the Structure 71 3.2 Plan of the Structural Model 71 3.3 Elevation of the Structural Model 71 3.4 Response Spectrum for the Structural Model as per IS1893(Part1):

2002 72

4.1 Variation of RU in Eigen Values with Mode No. for Interval Young’s Modulus and Normal Live Load for Case 1

87

4.2 Variation of RU in Eigen Values with Mode No. for Interval Live Load with Normal Young’s Modulus Case 2

87

4.3 Variation of RU in Eigen Values with Mode No. for Interval Young’s Modulus and Lower Bound Live Load for Case 3

88

4.4 Variation of RU in Eigen Values with Mode No. for Interval Young’s Modulus and Upper Bound Live Load for Case 4

88

4.5 Variation of RU in Eigen Values with Mode No. for Interval Live Load with Lower Bound Young’s Modulus for Case 5

89

4.6 Variation of RU in Eigen Values with Mode No. for Interval Live Load with Upper Bound Young’s Modulus for Case 6

89

4.7 Variation of RU in Eigen Values with Mode No. for Interval Young’s Modulus with Various Live Loads (summary of Figures 5.1, 5.3 & 5.4)

90

4.8 Variation of RU in Eigen Values with Mode No. for Interval Live Load and Various Young’s Modulus Values (summary of Figures 5.2, 5.5 & 5.6)

90

4.9 Variation of RU in Eigen Values with Variation in Live Load for Interval Young’s Modulus

91

4.10 Variation of RU in Eigen Values with Variation in Young’s Modulus for Interval Live Load

91

Mode Shapes for Model 2(E=24.75GPa, wl = 3.6kN/m2) A.1.1 3D Mode Shape 1for Structure Model 1 105 A. 1.2 3D Mode Shape 2for Structure Model 1 105 A.1.3 3D Mode Shape 3for Structure Model 1 105

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A.1.4 3D Mode Shape 4 for Structure Model 1 105 A.1.5 3D Mode Shape 5 for Structure Model 1 105 A.1.6 3D Mode Shape 6 for Structure Model 1 105 A.1.7 3D Mode Shape 7 for Structure Model 1 106 A.1.8 3D Mode Shape 8 for Structure Model 1 106 A.1.9 3D Mode Shape 9 for Structure Model 1 106 A.1.10 3D Mode Shape 10 for Structure Model 1 106 A.1.11 3D Mode Shape 11 for Structure Model 1 106 A.1.12 3D Mode Shape 12 for Structure Model 1 106 Mode Shapes for Model 2(E=24.75GPa, wl = 4.0kN/m2) A.2.1 3D Mode Shape 1 for Structure Model 2 107 A.2.2 3D Mode Shape 2 for Structure Model 2 107 A.2.3 3D Mode Shape 3 for Structure Model 2 107 A.2.4 3D Mode Shape 4 for Structure Model 2 107 A.2.5 3D Mode Shape 5 for Structure Model 2 107 A.2.6 3D Mode Shape 6 for Structure Model 2 107 A.2.7 3D Mode Shape 7 for Structure Model 2 108 A.2.8 3D Mode Shape 8 for Structure Model 2 108 A.2.9 3D Mode Shape 9 for Structure Model 2 108 A.2.10 3D Mode Shape 10 for Structure Model 2 108 A.2.11 3D Mode Shape 11 for Structure Model 2 108 A.2.12 3D Mode Shape 12 for Structure Model 2 108 Mode Shapes for Model 3(E=24.75GPa, wl = 4.4kN/m2) A.3.1 3D Mode Shape 1 for Structure Model 3 109 A.3.2 3D Mode Shape 2 for Structure Model 3 109 A.3.3 3D Mode Shape 3 for Structure Model 3 109 A.3.4 3D Mode Shape 4 for Structure Model 3 109 A.3.5 3D Mode Shape 5 for Structure Model 3 109 A.3.6 3D Mode Shape 6 for Structure Model 3 109 A.3.7 3D Mode Shape 7 for Structure Model 3 110 A.3.8 3D Mode Shape 8 for Structure Model 3 110 A.3.9 3D Mode Shape 9 for Structure Model 3 110 A.3.10 3D Mode Shape 10 for Structure Model 3 110 A.3.11 3D Mode Shape 11 for Structure Model 3 110 A.3.12 3D Mode Shape 12 for Structure Model 3 110 Mode Shapes for Model 4(E=25.00GPa, wl = 3.6kN/m2) A.4.1 3D Mode Shape 1 for Structure Model 4 111 A.4.2 3D Mode Shape 2 for Structure Model 4 111 A.4.3 3D Mode Shape 3 for Structure Model 4 111 A.4.4 3D Mode Shape 4 for Structure Model 4 111 A.4.5 3D Mode Shape 5 for Structure Model 4 111 A.4.6 3D Mode Shape 6 for Structure Model 4 111 A.4.7 3D Mode Shape 7 for Structure Model 4 112

x  

A.4.8 3D Mode Shape 8 for Structure Model 4 112 A.4.9 3D Mode Shape 9 for Structure Model 4 112 A.4.10 3D Mode Shape 10 for Structure Model 4 112 A.4.11 3D Mode Shape 11 for Structure Model 4 112 A.4.12 3D Mode Shape 12 for Structure Model 4 112 Mode Shapes for Model 5(E=25.00GPa, wl = 4.0kN/m2) A.5.1 3D Mode Shape 1 for Structure Model 5 113 A.5.2 3D Mode Shape 2 for Structure Model 5 113 A.5.3 3D Mode Shape 3 for Structure Model 5 113 A.5.4 3D Mode Shape 4 for Structure Model 5 113 A.5.5 3D Mode Shape 5 for Structure Model 5 113 A.5.6 3D Mode Shape 6 for Structure Model 5 113 A.5.7 3D Mode Shape 7 for Structure Model 5 114 A.5.8 3D Mode Shape 8 for Structure Model 5 114 A.5.9 3D Mode Shape 9 for Structure Model 5 114 A.5.10 3D Mode Shape 10 for Structure Model 5 114 A.5.11 3D Mode Shape 11 for Structure Model 5 114 A.5.12 3D Mode Shape 12 for Structure Model 5 114 Mode Shapes for Model 6(E=25.00GPa, wl = 4.4kN/m2) A.6.1 3D Mode Shape 1 for Structure Model 6 115 A.6.2 3D Mode Shape 2 for Structure Model 6 115 A.6.3 3D Mode Shape 3 for Structure Model 6 115 A.6.4 3D Mode Shape 4 for Structure Model 6 115 A.6.5 3D Mode Shape 5 for Structure Model 6 115 A.6.6 3D Mode Shape 6 for Structure Model 6 115 A.6.7 3D Mode Shape 7 for Structure Model 6 116 A.6.8 3D Mode Shape 8 for Structure Model 6 116 A.6.9 3D Mode Shape 9 for Structure Model 6 116 A.6.10 3D Mode Shape 10 for Structure Model 6 116 A.6.11 3D Mode Shape 11 for Structure Model 6 116 A.6.12 3D Mode Shape 12 for Structure Model 6 116 Mode Shapes for Model 7(E=25.25GPa, wl = 3.6kN/m2) A.7.1 3D Mode Shape 1 for Structure Model 7 A.7.2 3D Mode Shape 2 for Structure Model 7 117 A.7.3 3D Mode Shape 3 for Structure Model 7 117 A.7.4 3D Mode Shape 4 for Structure Model 7 117 A.7.5 3D Mode Shape 5 for Structure Model 7 117 A.7.6 3D Mode Shape 6 for Structure Model 7 117 A.7.7 3D Mode Shape 7 for Structure Model 7 118 A.7.8 3D Mode Shape 8 for Structure Model 7 118 A.7.9 3D Mode Shape 9 for Structure Model 7 118 A.7.10 3D Mode Shape 10 for Structure Model 7 118 A.7.11 3D Mode Shape 11 for Structure Model 7 118

xi  

A.7.12 3D Mode Shape 12 for Structure Model 7 118 Mode Shapes for Model 8(E=25.25GPa, wl = 4.0kN/m2) A.8.1 3D Mode Shape 1 for Structure Model 8 119 A.8.2 3D Mode Shape 2 for Structure Model 8 119 A.8.3 3D Mode Shape 3 for Structure Model 8 119 A.8.4 3D Mode Shape 4 for Structure Model 8 119 A.8.5 3D Mode Shape 5 for Structure Model 8 119 A.8.6 3D Mode Shape 6 for Structure Model 8 119 A.8.7 3D Mode Shape 7 for Structure Model 8 120 A.8.8 3D Mode Shape 8 for Structure Model 8 120 A.8.9 3D Mode Shape 9 for Structure Model 8 120 A.8.10 3D Mode Shape 10 for Structure Model 8 120 A.8.11 3D Mode Shape 11 for Structure Model 8 120 A.8.12 3D Mode Shape 12 for Structure Model 8 120 Mode Shapes for Model 9(E=25.25GPa, wl = 4.4kN/m2) A.9.1 3D Mode Shape 1 for Structure Model 9 121 A.9.2 3D Mode Shape 2 for Structure Model 9 121 A.9.3 3D Mode Shape 3 for Structure Model 9 121 A.9.4 3D Mode Shape 4 for Structure Model 9 121 A.9.5 3D Mode Shape 5 for Structure Model 9 121 A.9.6 3D Mode Shape 6 for Structure Model 9 121 A.9.7 3D Mode Shape 7 for Structure Model 9 122 A.9.8 3D Mode Shape 8 for Structure Model 9 122 A.9.9 3D Mode Shape 9 for Structure Model 9 122 A.9.10 3D Mode Shape 10 for Structure Model 9 122 A.9.11 3D Mode Shape 11 for Structure Model 9 122 A.9.12 3D Mode Shape 12 for Structure Model 9 122

xii  

LIST OF NOTATIONS AND ABBREVIATIONS A Cross-sectional area

A Ordinary subset

[A] Symmetric Matrix

[Â] Perturbed Symmetric Matrix

Aα Interval of Confidence of (α) cut

c Viscous damping

[C] Global damping matrix

n D Scaled modal coordinate

E Modulus of elasticity

Interval Young’s Modulus of Elasticity

EL Lower Bound Young’s Modulus of Elasticity

EU Upper Bound Young’s Modulus of Elasticity

E Referential set

[E] Perturbation matrix

f (x) Probability density function

Fx (a) Cumulative probability function

H Hilbert space

i Imaginary number ( −1)

[I ] Identity matrix

Kn Generalized modal stiffness

[K] Global stiffness matrix

[ ] Deterministic element stiffness contribution to the global Stiffness matrix

[ ] Interval global stiffness matrix

[K c] Central stiffness matrix

[ Ke] Stiffness matrix for a truss element

[Ki ] Element stiffness matrix

[ ] Radial stiffness matrix

[Li] Element Boolean connectivity matrix

[L] Matrix representation of [A] on χ with respect to the basis [X]

LL Live Load

Mn Generalized modal mass

[M] Global mass matrix

xiii  

[ ] Deterministic element mass contribution to the global mass matrix

[ ] Interval element mass matrix

[Me] Mass matrix for a truss element

[Mi] Element mass matrix

p(t) External excitation

[ p] Projection matrix

Pn (t) Generalized modal force

{P(t)} Vector of external excitation

[Pi] Projection matrix

[Q] Matrix of eigenvectors

R(t) Load effect

R(x) Rayleigh quotient

ℜ Real number domain

Rn static modal load effect

RU Relative Uncertainty

Vector of rigid body pseudo-static displaced shape

[T ] Linear operator in Sylvester’s equation

μ A(x) Characteristic function defining the ordinary subset (A)

u or y Displacement field

u or Velocity field

ü or Acceleration field

{U} Vector of nodal displacement

{ } Vector of nodal velocity

{Üt} Vector of nodal absolute acceleration motion

{Ü} Vector of nodal acceleration

Interval Live Load

wl Live Load

wlL Lower Bound Live Load

wlU Upper Bound Live Load

[Xi] Matrix for representation of a subspace

[ ] Matrix for representation of a perturbed subspace

y(t) Modal coordinate

Interval number

xiv  

α Level of presumption

εi Interval of [−1,1] for each element

η Test function

λ Eigen Value

ξn Modal damping ratio

ρ Mass density

φ Interpolation function

{ϕ} Mode shape

χ Invariant subspace

ω Natural circular frequency

Γ Domain of boundary conditions for truss elements

Γn Modal participation factor

[Λ] Diagonal matrix of Eigen Values

[Φ2] Matrix of complimentary eigenvectors to {φ1}

[Ω2] Diagonal matrix of other natural circular frequencies

FEM Finite Element Method

FFEM Fuzzy Finite Element Method

IFEM Interval Finite Element Method

IFFEM Interval Fuzzy Finite Element Method

IRSA Interval Response Spectrum Analysis

PDF Probability Density Function

RSA Response Spectrum Analysis

xv  

DEPARTMENT OF CIVIL ENGINEERING

UNIVERSITY COLLEGE OF ENGINEERING (A)

OSMANIA UNIVERSITY, HYDERABAD

M.E DISSERTATION EVALUATION SHEET

Name of the Candidate : VISHWANATHULA ANIL KUMAR

Roll No : 0109-11109

Specialization : STRUCTURAL ENGINEERING

Date of External Viva voce :

Grade :

Signature of the External Examiner :

Signature of Supervisor Signature of the Chairman, BoS

Signature of the HEAD

xvi  

ABSTRACT In recent years, the new developments in Uncertainty analysis in structural

engineering problems simplified the approaches of static and dynamic analyses of

civil engineering structures. It is difficult to predict certain action of loading and/or

characteristics of materials exactly in mathematical modeling which represents the

real world structure. There will be many uncertainties in the analysis problems when

we model the phenomena mathematically either manually or by simulation software.

The original representation of uncertainty in structural problems can be described

as, “as the age of the structure increases there will be a variation in material properties

and loads”. This makes the structure’s dynamic response behavior uncertain. This

appears as the basis of the thesis inspired by a doctoral thesis on dynamic analysis of

Structures with interval uncertainty introducing a new method called Interval

Response Spectrum Analysis (IRSA). Interval Response Spectrum Analysis (IRSA)

method is proposed by Dr. Mehdi Modarreszadeh and is the latest development in the

direction of dynamic analysis of Structures with multiple uncertainties. The method

solves for Interval values of young’s modulus of elasticity and live loads and gets

Modal Frequencies and Modal Eigen Values for a specific problem and applies

bounds on various end parameters there by determining the Relative Uncertainties

using perturbation techniques.

In this work a brief review is made for various types of uncertainties present in

structural modeling and methods for solving such uncertainties. A general

introduction is provided about the basic mathematical equations of structural

dynamics. The object and scope of the work includes discussion of the preexisting

methods for solving multiple uncertainties in static and dynamic analysis and

applying IRSA method for a Five-storey RCC symmetric building using Response

Spectrum Analysis as per IS 1893 (Part 1): 2002 successfully. SAP2000, v14.1

software is used to perform the modal analysis for getting results. Then bounds are

applied on Eigen Values for interval analysis. Combinatorial and Monte-Carlo

simulations are not required as already proved by Dr. Mehdi Modarreszadeh.

xvii  

The method is successfully applied for a Three Dimensional Five Storey Building

Frame. Interval response spectrum analysis is performed considering the effects of

uncertainties in Young’s modulus and Live load in terms of the structure’s response

that includes modal participating mass ratios, modal participation factors, mode

shapes and modal frequencies. Interval Eigen Values are obtained from the analysis

results. Then bounds are applied on them as per the method presented in this work.

Relative Uncertainty (RU) is a measure of defining the degree of variation of

uncertainty of the results in terms of a standard or normal case as a base reference for

application to the other problems easily and successfully by reducing the effort of

calculations. The purpose of this study is to analytically determine Relative

Uncertainties (RU) of Eigen-value bounds of a five-storey RCC building frame with

interval Young’s modulus and interval Live load independently. It is found that with

less variation in Young’s modulus there is more variation in structural response as

compared to the variation in Live load triggering the need for a strict quality control

in material properties. The Relative Uncertainty curves for Interval young’s modulus

are parabolic in form where as linear and straight for interval loads with increasing

mode numbers.

It is observed from the IRSA method for the used problem; the interval

uncertainty of ±1% in Young’s modulus resulted in 0.009 to 0.01 Relative

Uncertainties (RUs) in the Modal Eigen Value results and is positive. The interval

uncertainty of ±10% in Floor Live Load resulted in -0.0099 to -0.004 Relative

Uncertainties (RUs) in the Modal Eigen Value results and is negative. The variation

of Relative Uncertainty (RU) is on positive side of y-axis for Interval Young’s

Modulus’s. This is due to increase in Eigen values in the modal analysis from Lower

bound Young’s Modulus to Upper Bound Young’s Modulus increasing the effect of

stiffness matrix of the structure on Eigen Solution.

The future scope of work is discussed at the end for extension of the Interval

Response Spectrum Analysis (IRSA) Method for other problems like non-linear

analysis.

Key words: Uncertainty, Dynamic Analysis, Interval Response Spectrum

1  

CHAPTER - 1 INTRODUCTION

1.1 General

Uncertainty is a situation in which a person does not have the quantitatively and

qualitatively appropriate information to describe, prescribe or predict

deterministically and numerically a system, its behavior or other characteristic. [H. J

Zimmerman (2001)].

The term uncertainty analysis in Structural Engineering refers to solving

unknowingness or otherwise randomness of determining certain parameters both

aelatory and epistemic in nature. The term “epistemic” uncertainty can be expressed

as a system comprising the physical quantities with intrinsic variability being

stochastic and random in nature. “aelatory” refers to uncertainty that can be reduced

with additional data or additional information or better modeling and better parameter

estimation. [Muhanna (2006) (65-70)] .

As opined by Kompella and Bernard (1993) “Even the most accurate Finite

Element (FE) model of a structural dynamic system is just is presentation of a

nominal structure. In this context, several similar structures taken out of the

production line will exhibit variations on all of these individual properties and this

dispersion will reflect in the global response of the structure. Small changes in the

physical properties of the structure can directly affect the results, and the prediction

error can increase with frequency”

Uncertainty has a range of sources; we consider five broad sources: time,

statistical limits, model limits, randomness, and human error. [W. M. Bulleit (2008)

(11)] The problems of computational engineering and science requires to determine

various complex processes which poses challenges to Engineers and Analysts with

presence of multiple spatial and/or temporal scales, or even discontinuities in the

solution by challenging their computer simulations. There are various advanced

numerical methods devised by various mathematicians and researchers to tackle these

problems such as Finite Element Methods, Fuzzy Logic etc. As expressed by R. L.

Muhanna and others “Most of these methods work on a traditional basis where no

2  

uncertainty considerations are present in the modeling or computation. However, the

need for numerical treatment of uncertainty becomes increasingly urgent. In many

cases a given problem can be solved efficiently and accurately for a given set of input

data (such as geometry, boundary conditions, material parameters, load parameters

etc.), but little can be said about how the solution depends on uncertainties in these

parameters.”

The uncertainties shall be included in the simulations and modeling of the analysis

processes in both manual and computer programs to overcome the difficulties in these

problems. For solving uncertainty in such problems there exist various methods at

present such as Monte-Carlo simulations, Dempster-Shafer (22) Methods, Evidence

Theory, Perturbation Techniques, Interval Methods, and Fuzzy FEM Etc.

When we choose a model for the analysis of a structural system, there is

uncertainty about whether the model you choose is adequate to predict the behavior of

the not-yet-built structure under not-yet-to-have-occurred load events. Some aspects

of the model are based on code criteria, but others are based on the structural

engineer’s professional judgment about what aspects of the structure need to be

included in the model and how rigorously the behavior of the structural elements must

be modeled. [W. M. Bulleit (2008) (11), ASCE].

To date, much research has been done towards improving the numerical models,

hence towards achieving full spectrum response for dynamic analysis. However,

deterministic models, such as the finite element (FE) and the boundary element (BE)

methods, have limitations at higher frequencies due to the necessity of refining the

mesh, whereas analytical methods, such as the spectral element (SEM) are less

sensitive to the higher wave-numbers.

In terms of wavelength, it is well known that to achieve sufficient agreement

between numerical and analytical solutions, 6 to 10 linear elements per wavelength

are necessary to keep the prediction error within acceptable limits [Desmet,

2002(23)]. Experience shows that, as the frequency range increases, the structure

becomes very sensitive to small uncertainties in material properties and geometrical

details. Additionally, the high order modes of vibration also tend to have more

complex spatial fluctuations. As a result, the size of the finite element model increases

3  

with frequency [Shorter, 1998(87)]. Therefore, in terms of a frequency response

function (FRF), the response of the structure is not adequately predicted, and it is

more convenient to predict the frequency averaged response in terms of the energy

[Lyon and DeJong, 1995].

1.2 Need for the present study

Various studies are made in the direction of solving uncertainties in Structural

engineering problems and are still in progress. There are new and many methods

(such as the works of Muhanna R. L., Mullen R. L.(65-69), Rama Rao M. V., Ramesh

Reddy R., Vandipitte, W. M. Bulliet(11), Mehdi Modarreszadeh(58), etc.) are evolved

in recent years in this direction for solving uncertainty problems in static analysis and

dynamic analysis.

To solve uncertainties in analysis problems, Interval Finite Element Method

(IFEM) was introduced by Muhanna and Mullen in 2001. The Fuzzy Finite Element

Method (FFEM) was introduced by Dr. M. V. Rama Rao and R. Ramesh Reddy in

2004 and applied for cable-stayed bridge. A new method called Interval Response

Spectrum Anlaysis (IRSA) was introduced by Mehdi Modarreszadeh in 2005 for

solving uncertainties in dynamic analysis problems of one and two-dimension systems

There is a need to study the uncertainty methods in dynamic analysis for solving

uncertainty problems to minimize the system response to the loads for optimized

design processes. This dissertation applies a method IRSA devised by Dr.

Mehdimodarres Zadeh(58) for application to civil engineering structures by applying

it for a 3D multi-storey building structure by applying uncertainty bounds on Young’s

Modulus of Elasticity (Stiffness) and Live Load (Load).

1.3 Objective and Scope of Work

• The objective of the work is to review existing methods in uncertainty analysis

and to use IRSA method for a Five Storey R. C. C. building frame using SAP

2000, v14.1 software package with uncertain Young’s modulus and Live loads

separately and bounds are applied on Eigen Values at final stage.

4  

• The scope of work includes finding Relative Uncertainties of Eigen Value bounds

for given structure from the results of Modal analysis and to find the variation of

Relative Uncertainties within the interval values of Young’s modulus and Live

loads independently.

1.4 Summary

This chapter gives introduction to uncertainty in structural engineering and briefs

the latest developments in the direction of the present work and gives the scope and

objectives of the present study.

5  

CHAPTER - 2 LITERATURE REVIEW

2.1 Introduction

The advancement in Fuzzy Finite Element Analysis simplified the structural

analysis process in recent years in the area of structural engineering for solving

uncertainty. Many methodologies were developed to solve uncertainties in analysis

and design processes with help of new mathematical algorithms and simulations using

software. This caused a need for study in new directions for improving the precision

and accuracy in analysis models. The mathematical modeling of structural

engineering phenomena changed significantly in the recent years. Various

methodologies and approaches were discussed in the literature review solving

uncertain parameters in analysis process. The need for fuzzy finite Element

approaches is reviewed. Various researchers introduced new concepts and

methodologies for analyzing structures with uncertainties effectively. Until now

Fuzzy Finite Element approach is used for solving static analysis of structures

subjected to multiple uncertainties, dynamic analysis with single uncertainty. A new

method called Interval Response Spectrum Analysis (IRSA) is introduced by Dr.

Mehdi Modarreszadeh for solving uncertainties in Dynamic Analysis and is the latest

method available. This method is reviewed and use of this method in present work is

enumerated.

2.2 Interval Analysis and its Historical Background

The concept of representation of an imprecise real number by its bounds is quite

old. In fact, Archimedes (287-212 B.C.) defined the irrational number (π) by an

interval 310/71<π < 31/7, which he found by approximating the circle with the inscribed

and circumscribed 96-side regular polygons. Early work in modern interval analysis

was performed by W. H. Young (1908), who introduced functions with values which

are bounded between extreme limits. The concept of operations with a set of multi-

valued numbers was introduced by R. C. Young (1931)(97), who developed a formal

algebra of multi-valued numbers. Also, the special case of multi-valued functions

with closed intervals was discussed by Dwyer (1951). The introduction of digital

computers in the 1950’s provided impetus for further interval analysis as discrete

representations of real numbers with associated truncation error.

6  

Interval mathematics was further developed by Sunaga (1958) (94) who

introduced the theory of interval algebra and its applications in numerical analysis.

Also, R. Moore (1966) (63) introduced interval analysis, interval vectors and interval

matrices as a set of techniques that provides error analyses for computational results.

Interval analysis provides a powerful set of tools with direct applicability to

important problems in scientific computing. Alefeld and Herzberger (1983) presented

an extensive treatment of interval linear and non-linear algebraic equations and

interval methods for systems of equations. Moreover, Neumaier (1990) (68)

investigated the methods for solution of interval systems of equations.

The concept of interval systems has been further developed in analysis of

structures with interval uncertainty. Muhanna and Mullen (1999)(65-71) developed

fuzzy finite element methods for solid mechanics problems. For the solution of

interval finite element method (IFEM) problems, Muhanna and Mullen (2001)

introduced an Element-by- Element interval finite element formulation, in which a

guaranteed enclosure for the solution of interval linear systems of equations was

achieved.

The research in interval Eigen-value problem began to emerge as its wide

applicability in science and engineering was realized. Dief (1991) presented a method

for computing interval Eigen-values of an interval matrix based on an assumption of

invariance properties of eigenvectors. Using Dief’s method, the lower Eigen-values

have a wider range of uncertainty than the exact results.

The concept of the interval Eigen-value problem has been developed in dynamics

of structures with uncertainty. Qiu(17), Chen(15,16,17) and Elishakoff (1995) (27-29)

have introduced a method to find the bounds on Eigen-values. In their work, the

perturbation of the Eigen-value is derived from pre and post multiplying the perturbed

matrix by the exact eigenvector which is inconsistent with matrix perturbation

theories.

However, since the presence of perturbation in the matrix results in perturbation

of both Eigen-values and eigenvectors, applying the unperturbed eigenvector to

determine the perturbation of the Eigen-value may lead to incorrect results. The

7  

second problem in Modarreszadeh’s work solves the problem cited by Qiu, Chen and

Elishokoff (1995) for exact bounds.

Qiu, Chen and Elishakoff (1996) have introduced an alternate method for

bounding the natural frequencies of a structural system. However, their results are

wider than sharp values because of a non-parametric formulation and the existence of

variation inside the matrices. Moreover, their definition of the concept of maximin

characterization appears to be inconsistent with the formal mathematical definitions.

The constraint induced subspaces in this concept are not completely arbitrary but they

should be orthogonal to arbitrary vectors [Bellman (1960) and Strang (1976)]. The

third problem in the Modarreszadeh’s work solves the problem cited by Qiu, Chen

and Elishokoff (1996) and compares the results.

As part of Mehdi Modarreszadeh’s work, he considered the method introduced by

Modares and Mullen (2001, 2005)(68-69) for the solution of the parametric interval

Eigen-value problem resulting from semi-discretization of structural dynamics which

determines the exact bounds of the natural frequencies of a structure.

2.3 Interval Finite Element Methods (IFEM)

It is unavoidable in analysis and design process some uncertain conditions in

aspects of modeling a structural problem such uncertainties in material properties,

geometry, loads and boundary conditions of the structure. It is very important to keep

in mind that these uncertainties must be deduced and incorporated for accurate

modeling of the solution for optimization and economizing the structure. The

problems must be checked for the real world phenomena by simulating them with

mathematical tools. There are so many theories and research works are made by

various people for solving such problems. This Literature review presents such

attempts by various researchers.

There are various ways in which the types of uncertainty might be classified. One

is to distinguish between “aleatory” (or stochastic) uncertainty and “epistemic”

uncertainty. The first refers to underlying, intrinsic variability’s of physical quantities,

and the latter refers to uncertainty which might be reduced with additional data or

information, or better modeling and better parameter estimation [Melchers (1999)].

Probability theory is the traditional approach to handle uncertainty.

8  

This approach requires sufficient statistical data to justify the assumed statistical

distributions. Analysts agree that, given sufficient statistical data, the probability

theory describes the stochastic uncertainty well. However, traditional probabilistic

modeling techniques cannot handle situations with incomplete or little information on

which to evaluate a probability, or when that information is nonspecific, ambiguous,

or conflicting [Walley (1991), Ferson and Ginzburg (2007)(85), Sentz and Ferson

(2002)]. Many generalized models of uncertainty have been developed to treat such

situations, including fuzzy sets and possibility theory [Zadeh (1978)](98), Dempster-

Shafer theory of evidence [Dempster (1967)(22), Shafer (1976)], random sets

[Kendall (1974)], probability bounds [Berleant (1993), Ferson and Ginzburg

(2007)(85), Ferson et al. (2003)], imprecise probabilities [Walley (1991)], convex

models [Ben-Haim and Elishakoff (1990) (27)], and others.

These generalized models of uncertainty have a variety of mathematical

descriptions. However, they are all closely connected with interval analysis [Moore

(1979) (63)], in which imprecision is described by an interval (or, more generally, a

set). For example, a fuzzy number can be viewed as a nested collection of intervals

corresponding to different levels of confidence (so-called α-cuts). Thus, the

mathematical analysis associated with fuzzy set theory can be performed as interval

analysis on different α-levels [Muhanna and Mullen (1995), Lodwick and Jamison

(2002)], and fuzzy arithmetic can be performed as interval arithmetic on α-cuts. A

Dempster-Shafer structure [Dempster (1967)(22), Shafer (1976)] with interval focal

elements can be viewed as a set of intervals with probability mass assignments, where

the computation is carried out using the interval focal sets. Probability bounds

analysis [Berleant (1993), Ferson and Ginzburg (1996), Ferson et al. (2003)] is a

combination of standard interval analysis and probability theory. Uncertain variables

are decomposed into a list of pairs of the form (interval, probability). In this sense,

interval arithmetic serves as the calculation tool for the generalized models of

uncertainty.

Recently, various generalized models of uncertainty have been applied within the

context of the finite element method to solve a partial differential equation with

uncertain parameters. Regardless what model is adopted, the proper interval solution

will represent the first requirement for any further rigorous formulation. Finite

9  

element method with interval valued parameters results in Interval Finite Element

Method (IFEM).

Different formulations of IFEM have been developed. The use of IFEM solution

techniques can be broadly classified into two groups, namely the optimization

approach and the non-optimization approach.

In the optimization approaches [Koyluoglu et al. (1995), Rao and Chen (1998),

Akpan et al. (2001), Möller et al. (2000)(62)], optimizations are performed to

compute the minimal and maximal structural responses when the uncertain parameters

are constrained to belong to intervals. This approach often encounters practical

difficulties. Firstly it requires efficient and robust optimization algorithm. In most

structural engineering problems, the objective function is nonlinear and complicated,

thus often only an approximate solution is achievable. Secondly, this approach is

computationally expensive. For each response quantity, two optimization problems

must be solved to find the lower and the upper bounds.

More recently, non-optimization approaches for the interval finite element

analysis have been developed in a number of papers. For linear elastic problems, this

approach leads to a system of linear interval equations, and then the solution is sought

using various methods developed for this purpose. The major difficulty associated

with this approach is the “dependency problem” [Moore (1979)(63), Neumaier

(1990)(72), Hansen (1992)(40), Muhanna and Mullen (2001)]. In general, dependency

problem arises when one or several interval variables occur more than once in an

interval expression. The dependency in interval arithmetic leads to an overestimation

of the solution. A straightforward replacement of the system parameters with interval

ones without taking care of the dependency problem is known as a naïve application

of interval arithmetic in finite element method (naïve IFEM). Usually such a use

results in meaninglessly wide and even catastrophic results [Muhanna and Mullen

(2001)].

In the non-optimization category, a number of developments can be presented. A

combinatorial approach (based on an exhaustive combination of the extreme values of

the interval parameters) was used in [Muhanna and Mullen (1995), Rao and Berke

(1997)]. This approach gives exact solution in special cases of linear elastic problems.

10  

However, it is computationally tedious and expensive, and is limited to the solutions

of small-scale problems.

A convex modeling and superposition approach was proposed to analyze load

uncertainty in [Pantelides and Ganzerli (2001) (75)], and exact solution was obtained.

However, the superposition is only applicable to load uncertainty. A combinatorial

approach was used in [Ganzerli and Pantelides (1999)] to treat interval modulus of

elasticity. A static displacement bounds analysis was developed in Chen et al.

(2002)(15,16) have developed using matrix perturbation theory. The first-order

perturbation was used and the second-order term was neglected. The result is

approximate and not guaranteed to contain the exact bounds.

The paper of McWilliam (2000)(56) proposed two methods for determining the

static displacement bounds of structures with interval parameters. The first method is

a modified version of perturbation analysis. The second method is based on the

assumption that the displacement surface is monotonic.

However, for the general case, the validity of monotonicity is difficult to verify. In

Dessombz et al. (2001)(25), an interval FEM was introduced in which the interval

parameters were factored out during the assembly process of the stiffness matrix.

Then an enhanced iterative algorithm from Rump (1983) was employed for solving

the linear interval equation. In the work, the overestimation control becomes more

difficult with the increase of the number of the interval parameters, which does not

lead to useful results for practical problems.

In [Muhanna and Mullen (1995), Mullen and Muhanna (1996), Mullen and

Muhanna (1999)], an interval-based fuzzy finite element has been developed for

treating uncertain loads in static structural problems. Load dependency was

eliminated, and the exact solution was obtained. Also, in [Muhanna and Mullen

(2001)], an interval FEM was developed based on an element-by-element technique

and Lagrange multiplier method. Uncertain modulus of elasticity was considered.

Most sources of overestimation were eliminated, and a sharp result for displacement

was obtained. However, this formulation can only handle uncertain modulus of

elasticity, and it cannot obtain the sharp enclosures for element internal forces.

11  

A new formulation for interval finite element analysis for linear static structural

problems is developed in the work of Muhanna et al. (2005). Material and load

uncertainties are handled simultaneously, and sharp enclosures on the system’s

displacement as well as the internal forces are obtained efficiently.

Recently, new advancements has been made in the area of interval FEM, e.g.,

[Corliss et al. (2004)(21), Popova et al. (2003)(77)], and the significant development

in [Neumaier and Pownuk (6,72)] where sharp results are achieved for linear truss

problems even with large uncertainty.

2.4 Dynamic Analysis of Structures for Design under Uncertainty

In recent years, an important research effort has been deployed in the mid-

frequency range problem in structural dynamics. In this regard, one of the most

important issues is the numerical simulation of dynamic systems taking into account

the influence of nondeterministic input parameters.

In order to describe how uncertainty can influence the engineering design process,

both the variability of physical artifacts and their interaction with environments have

to be taken into account. Many possible factors can lead to uncertainty, such as

variations in the measurement process, variation in the outer environment,

geometrical and material variation of the same product and so on [Battil et al., 2000].

According to a review paper by [Manohar and Gupta, 2002(54)], the sources of

uncertainties in dynamic structural engineering problems can be split up into four

categories: physical or inherent uncertainties, model uncertainties, estimations errors

and human errors. Some of the sources, such as inherent uncertainties, are beyond the

control of engineers. With regard to model uncertainty some assumptions, usually

done in the mathematical models based on simplifying assumptions, also lead to

model uncertainty. The study of estimation errors belongs to the science of statistics,

while human errors can arise at any stage of the design process.

Other authors, such as Keese (2003)(5) summarize that uncertainties can be

caused either by the intrinsic variability of physical quantities such as irregularities in

12  

material properties caused by the manufacturing process or simply by lack of

knowledge, which can be called epistemic uncertainty.

In the context of non-deterministic numerical modeling, different methods using

different approximate models of uncertainties have been proposed to deal with this

problem. Basically, some important methods include worst-case scenario, safety

factors, Taguchi methods, probabilistic and fuzzy set based methods. According to

Maglaras et al. (1997)(53), the worst-case scenario concept used to improve safety

factors in many cases lead to over design. Also, for Taguchi methods in general, the

concept is to find values of some parameters for which system performance is close to

the target values and is insensitive to uncertainties.

In [Maglaras et al. (1997)(53)], experimental comparison of probabilistic and

fuzzy set based methods was conducted. They state that a model with uncertainty and

enough statistical information available is better represented by stochastic description;

otherwise fuzzy theory is better suited.

In terms of stochastic approach, the finite element method for stochastic problems

applied to dynamic analysis is a fast growing area of research. Many authors have

discussed such an application in papers reviews and books, see for instance in [Keese

(2003)(5), Elishakoff and Ren (2003)(29), Ghanem and Spanos (2003), Manohar and

Gupta (2002)(54), Manohar and Ibrahim (1999) and Matthies et al. (1997)].

2.4.1 Probabilistic Methods

The idea of probabilistic methods is to include uncertainties of the input

parameters in the analysis. Basically, important aspects as a risk of failure, safety

factors or simply targets established in the industry are possible examples of the

application of probabilistic approach.

In terms of mathematical foundation, the concept of probability is defined as a

number assigned to events of a universal set [Chen (2000)]. Probability also satisfies

the three axioms

i. The probability of any single event occurring is greater or equal to zero.

13  

ii. The probability of the universal set is one, i.e., in case the universal set

includes all possible outcomes.

iii. The probability of the union of mutually exclusive events is equal to the sum of

the probabilities of these events. This is also called the additivity axiom.

In other words, considering the objective sense, the probability concept is the

relative frequency of occurrence of an event [Siddall (1983)]. In this context, one

important factor to add is that to give a confidence interval, probability must be

estimated considering a large number of observations. In this scenario, it is important

to stress the quote proposed by Freudenthal cited in [Moens and Vandepitte

(2004)(57)] that,”... ignorance of the cause of variation does not make such variation

random”. Besides that, the concept of probability is also defined in terms of a

subjective view, which in general is called Bayesian interpretation. The probability

concept is then defined as likelihood that, an event will occur [Savage (1972)]. On

one hand, it is important to add that with the Bayesian method, it is possible to add

objective information if it becomes available. On the other hand, when no information

is available, the Bayesian approach is just a subjective representation of real-life cases

[Moens and Vandepitte (2004)(61)].

Going further, in terms of assumptions, one assumption which is very often

applied in probability theory is that variables are uncorrelated. However, this is not

true or at least it cannot be defined when little information is available. This can lead

to very inaccurate models or can be unrealistic in terms of real applications. In the

same context, it is important to add that modeling errors should also be considered in

the problem formulation. However, in general, they are not taken into account.

Considering, for instance, an optimization problem, where little statistical information

is available, this can also lead to poor results as a whole.

In addition, in [Maglaras et al. (1997)(53)], the effect of the choice of probability

distribution is also discussed. According to results presented by Fox and Safie in

1992, it is concluded that probability of failure is very sensitive to the choice of

distribution.

14  

Therefore, in real life applications, to give all statistical information which is

necessary to satisfy the probability method assumptions is, in general, time

consuming or, in most cases, impracticable.

2.4.2 Fuzzy Set Based Methods

Fuzzy arithmetic has been proposed in the literature as a methodology that can be

very helpful to analyze systems with respect to uncertain model parameters. The

concept of fuzzy has been developed to deal with imprecision and also what is called

verbal information. In the context of engineering applications, a fuzzy model adopted

for a dynamic problem can help in determining the range of results or simply intervals

of confidence, considering a variability of materials, geometrical dimensions,

manufacturing process and so on.

Going one step further, Zadeh’s extension principle provides the fundamental

basis of fuzzy arithmetic. He states that real valued functions can be extended to

functions of fuzzy numbers [Zadeh (1965)(99)]. In this concept, also a fuzzy set can

be defined as a class with a continuum of grades of membership. This is the case that

an element belongs to a fuzzy set to a certain degree.

The main idea behind the concept of the fuzzy set is to model uncertainty

considering subjective information or simply vagueness. In terms of practical

applications, the fuzzy arithmetic based on Zadeh’s extension principle, such as LR-

fuzzy numbers according to Dubois and Prade (1980) and the standard fuzzy

arithmetic described in [Kaufmann and Gupta (1991)], lead to a serious drawback.

Such effect is well known in interval arithmetic as the effect of overestimation or

simply defined as dependency effect. In [Hanss (2002b)] the effect of overestimation

is discussed and simple examples are treated to show the major drawback in applying

standard fuzzy arithmetic.

In order to address the limitation described above, other alternatives in the

literature have been proposed. In this context, algorithms for fuzzy arithmetic based

on interval-based branch-and-bounds codes according to Hansen (1992)(40) and

point-based methods or just called constrained fuzzy arithmetic, such as proposed by

[Hanss (2002b), Dong and Shah (1987)] have been suggested.

15  

The well known overestimation effect in fuzzy arithmetic is discussed in the

literature and different approaches have been proposed. A non-overestimating

approach, also called fuzzy weighted averages (FWA) has been proposed by Dong

and Shah (1987) as an alternative to address problem. However, in its initial proposal,

only monotonic functions with respect to all of their fuzzy variables were used.

In Wood et al. (1992), an enhancement is achieved for the case of non-monotonic

function applications. According to Klimke (2006)(51), the proposed algorithm

requires an additional routine that should be used to locate the internal extrema. For

such kind of application, numerical and analytical approaches are used combined with

special algorithms that take into account the non-linear functions as input.

Additionally, a new theoretical framework proposed by Klir (1997) takes into

account dependencies of the fuzzy parameters. Such an approach in literature is called

constrained.

In this sense, the transformation method is proposed by Hanss (2002) as a

practical approach to evaluate fuzzy parameterized models in order to avoid any other

extra optimization routine such as suggested by Wood et al. (1992). The

transformation method is considered an advanced approach of the Vertex Method

proposed by Dong and Shah (1987).

The use of fuzzy set methods applied in engineering problems has been

extensively discussed in literature [Hanss (2004), Hanss (2004), Hanss et al. (2002),

Hanss (2002), Hanss (1999) and Klimke (2006)(51)].

Nunes et al. (2005)(84) and Arruda et al. (2004) presented the application of the

fuzzy set combined with the spectral element method. In both papers, the standard

transformation method in its reduced form proposed by Hanss (2002) is applied to

estimate frequency response function envelopes.

2.5 Interval Model updating method and Application

Finite element techniques are nowadays essential tools in engineering design of

civil and mechanical structures.

16  

The prediction of finite element model can be improved by using deterministic

finite element model updating approaches [J.E. Mottershead, M.I. Friswell (1993,

1995)(64, 37)]. These techniques are about improving and correcting invalid

assumptions by processing experimental results. However experimental data includes

variability due to different sources.

Experimental variability is supposed not to be inherent to the test structure itself,

but arises from other sources such as measurement noise, the use of sensors that affect

the measurement or signal processing that might introduce bias. Such variability is

reducible by increased information. Statistical techniques such as minimum variance

method [J.D. Collins, G.C. Hart, T.K. Hasselman, B. Kennedy (1974) (20), M.I.

Friswell (1989)(32)] have been implemented in deterministic finite element model

updating to treat reducible uncertainty in measured data. However, manufacturing and

material variability in structures is not reducible and needs to be considered as part of

the model. The problem of interval model updating in presence of irreducible

uncertain measured data (manufacturing and material variability) is considered in the

paper of Khodaparast et al...

Fonseca et al. [J.R. Fonseca (48,49), M.I. Friswell (32,33,59,64,78), J.E.

Mottershead (64), A.W. Lees (2008)(59)] proposed an optimization procedure for the

purpose of stochastic model updating based on the maximizing a likelihood function

and applied it to a cantilever beam with a point mass at an uncertain location. Hua et

al. [X.G. Hua, Y.Q. Ni, Z.Q. Chen, J.M. Ko (2008)] used perturbation theory in the

problem of test-structure variability. The predicted output mean values and the matrix

of predicted covariance’s were made to converge upon measured values and in so

doing the first two statistical moments of the uncertain updating parameters were

determined.

Khodaparast et al. [H. Haddad Khodaparast, J.E. Mottershead, M.I. Friswell

(2008) (34-37)] developed two perturbation methods for stochastic model updating.

The first method required only the first order sensitivity matrix and therefore was

computationally efficient compared with the method developed by Hua et al. [X.G.

Hua, Y.Q. Ni, Z.Q. Chen, J.M. Ko (2008)] which needed the calculation of the second

order sensitivity matrix.

17  

Khodaparast and Mottershead [H. Haddad Khodaparast, J.E. Mottershead (2008)

(36-39)], and also Govers and Link [Y. Govers, M. Link (2010)], proposed an

objective function for the purpose of stochastic model updating. The objective

function consisted of two parts:

1. The Euclidean norm of the difference between mean values of measured data

and analytical output vectors, and

2. The Frobenius norm of the difference between the covariance matrices of

measured data and analytical outputs.

The stochastic model updating methods have made use of probabilistic model for

updating so far. This usually needs large volumes of data with consequent high costs.

The interval model can be used as an alternative approach when those large quantities

of test data are not available.

In the article by H. Haddad Khodaparast, J.E. Mottershead, K.J. Badcock

(2010)(36) the problem interval model updating in the presence of irreducible

uncertain measured data is defined and solutions of the problem are made available in

two cases. In the first case, the problem is solved by using the parametric vertex

solution [Z. Qui, X. Wang, M.I. Friswell (2005)]. It is shown that the parameter

vertex solution can be only used when (i) the overall mass and stiffness matrices are

linear functions of the updating parameters, (ii) can be decomposed into non-negative-

definite substructure mass and stiffness matrices and (iii) the output data are the

Eigen-values of the dynamic system. Two recursive updating equations are developed

to update the bounds of an initial hypercube of updating parameters in this case.

However, the parameter vertex solution cannot be used when the output data include

the eigenvectors of the structural dynamic system and the system matrices are

nonlinear functions of the updating parameters. In this case, the problem is solved by

using a meta-model which acts as a surrogate for the full finite element model so that

the region of input data is mapped to the region of output data with parameters

obtained by regression analysis. The Kriging predictor is chosen as the meta-model in

the paper of Khodaparast et al. and is shown to be capable of predicting the region of

input and output parameter variations with very good accuracy, even in the difficult

case of close modes. The method is validated numerically by using a three degree of

freedom mass-spring system with close modes. The method is also applied to a frame

18  

structure with uncertain internal beams locations. It is shown that the updated bounds

are in good agreement with the known real bounds on the position of the beams [H.

Haddad Khodaparast, J.E.Mottershead, K.J. Badcock (2010)].

2.6 Application of the FFEM in Structural Dynamics

Through the years, scientists and researchers have provided us with procedures for

calculating specific dynamic features of any structure. For most problems that have

been analyzed, the solution starts from deterministic values for the quantization of the

necessary physical properties of the analyzed structure.

Results from a deterministic analysis often are successfully used as a

representation for the dynamic feature that is analyzed. Nevertheless, in some cases,

the deterministic analysis is not sufficient. If the deterministic supposition is not valid,

different concepts are applied to incorporate the uncertainties into the analysis. How

to deal with uncertainties in structural dynamics depends on the goal of the analysis.

One way to deal with the uncertainties is adapting initially uncertain features of a

model in order to make the analysis coincide with a known physical result. Updating

techniques have been developed in the area of structural dynamics (e.g. model

updating). This is a useful tool when the considered model is intended for further

analysis, which will be more reliable if the model is updated. From a designer’s

viewpoint, this approach is not useful since no reference data are available during the

design optimization phase. A designer is interested in the expected dynamical features

of the design. Restricting the evaluation of a design to one deterministic analysis in

most cases will not be sufficient, as shown by Guyader and Parizet (1997). In general,

when uncertainties exist on the definition of a model (which is nearly always the case)

the designer may only be interested in the resulting uncertainty on the analyzed

feature. Different techniques dealing with this problem have been proposed in the

paper of D. Moens at al. (ISMA23, Sept. 1998).

The perturbation method calculates the uncertainty on the result of an analysis by

making a linear approximation of the relations between inputs and outputs. If this

assumption is correct and the result is normally distributed, the perturbation method

gives exact results. Stochastic structural analysis is the alternative if the above

assumptions are not met. One type of stochastic analysis utilizes the Monte Carlo

19  

simulation technique. This method determines the mean value, variances and co-

variances of the results of the analysis from a number of samples. Unlike the

perturbation method, this method has no real restrictions. It has been proven very

useful in many cases. The major drawback is that the number of samples should be

sufficiently high in order to reach a reliable result. However, with growing

computational capacity of computer systems, this drawback is only restricting when

large models are analyzed. For those cases, reduction techniques are a useful tool for

limiting the size of an analysis, as shown by Teichert (1998) for the analysis of a large

spacecraft structure. However, during the design phase, a fast estimation of the

uncertainty of a dynamic feature could be preferred above the exact uncertainty. For

these cases, the Fuzzy Finite Element Method (FFEM) is proposed as an alternative

for the Monte Carlo simulation.

2.7 Fundamentals of Structural Dynamics

2.7.1 Introduction

As the term itself defines ‘dynamic’ loading is time-varying in magnitude,

direction and position. Two basic approaches are available for evaluating structural

response to dynamic loads: deterministic, non-deterministic. The choice of method

depends on the definition of loading. If the time-variation of loading is fully known

the loading is termed as prescribed dynamic loading and the analysis of response is

known as deterministic analysis. If the time-variation of dynamic loading is not

known fully but can be defined in a statistical sense the loading is termed random

dynamic loading and the analysis of such loading response is called nondeterministic

analysis.

Generally in problems of dynamic analysis, the structural response to dynamic

loading expressed basically in terms of displacements. Thus a deterministic analysis

leads directly to the displacement time-histories corresponding to prescribed dynamic

loading history. The second phase of the analysis defines the other related response

quantities like stresses, strains, internal forces. A nondeterministic analysis provides

only statistical information of the displacements resulting from the statistical defined

loading and the corresponding information of structural response then generated with

help of independent nondeterministic analysis procedures.

20  

There are two types of prescribed dynamic loadings viz. periodic and non-

periodic. The periodic loading exhibits same time variation successively for large

number of cycles (ex: sinusoidal variation termed as simple harmonic). Any periodic

loading can be represented by Fourier analysis as the sum of the series of simple

harmonic components. Non-periodic loadings may be either short-duration impulsive

loading (blast) or long duration general loading (earthquake).

2.7.2 Characteristics of Dynamic Problem

There are two differences that are to be noted by a dynamic problem. The first is

the time-varying nature of the dynamic problem. As both the loading and response

vary with time, it is evident that a dynamic problem does not have a single solution as

a static problem does and a succession of solutions must be established

corresponding to all times of interest in the response history. Thus a dynamic analysis

is more complex and time consuming than a static analysis. The second and more

fundamental distinction between static and dynamic problems is the displacements of

a structural system resulting from application of a dynamic loading depend not only

upon the loading but also upon the inertial forces in the structural system which

opposes the accelerations producing them. Thus corresponding internal moments and

shear forces must equilibrate not only the externally applied load bust also the inertial

forces resulting from the accelerations of the system.

2.7.3 Historical Background of Structural Dynamics

Modern theories of structural dynamics were introduced mostly in mid 20th

century. M. A. Biot (1932) introduced the concept of earthquake response spectra and

G. W. Housner (1941) was instrumental in the widespread acceptance of this concept

as a practical means of characterizing ground motions and their effects on structures.

N. M. Newmark (1952) introduced computational methods for structural dynamics

and earthquake engineering. In 1959, he developed a family of time-stepping methods

based on variation of acceleration over a time-step.

A. W. Anderson (1952) developed methods for considering the effects of lateral

forces on structures induced by earthquake and wind and C. T. Looney (1954) studied

the behavior of structures subjected to forced vibrations. Also, D. E. Hudson (1956)

21  

developed techniques for response spectrum analysis in engineering seismology. A.

Veletsos (1957) determined natural frequencies of continuous flexural members.

Moreover, he investigated the deformation of non-linear systems due to dynamic

loads. E. Rosenblueth (1959) introduced methods for combining modal responses and

characterizing earthquake analysis.

J. Biggs (1964) developed dynamic analyses for structures subjected to blast

loads. Moreover, numerical methods for dynamics of structures and modal analysis

were further developed by J. Penzien and R. W. Clough (1993).

The mathematical theory of dynamical systems is based on the qualitative theory

of ordinary differential equations the foundations of which were laid by Henri

Poincaré (1854–1912). An essential role in its development was also played by the

works of A. M. Lyapunov (1857–1918) and A. A. Andronov (1901–1952). At present

the theory of dynamical systems is an intensively developing branch of mathematics

which is closely connected to the theory of differential equations. (I. D. Cheushov,

ACTA 2002)(19).

2.7.4 The Equation of Motion

The equation of motion can be derived by using the d'Alembert's principle which

gives a concept that a mass develops an inertial force proportional to its acceleration

and opposing the acceleration of the mass.

The equation of motion of a dynamic system can be formulated by directly

expressing its equilibrium of all forces acting on the mass of the system using

d’Alembert’s Principle as:

fI(t) + fD(t) + fS(t) = p(t)… …. …. 2.1.1

Each of the forces represented on the left hand side of this equation is a function

of the displacement u(t) or one of its time derivatives. The positive sense of these

forces has been deliberately chosen to correspond with the negative-displacement

sense so that they oppose a positive applied loading.

In accordance with d'Alembert's principle, the inertial force is the product of the

mass and acceleration

22  

fI(t) = mü(t)… … … 2.1.2.a

Assuming a viscous damping mechanism, the damping force is the product of the

damping constant c and the velocity

fD(t) = c (t)… … … 2.1.2.b

Finally, the elastic force is the product of the spring stiffness and the displacement

fS(t) = ku(t)… … … 2.1.2.c

When Eqs. (2.1.2) are introduced into Eq. (2.1.1), the equation of motion for this

SDOF system is found to be

mü(t) + c (t) + ku(t) = p(t)… …. …. 2.1.3

Integrating over the domain, the equation of motion for vibration of a multiple

degree of freedom (DOF) system is defined as a linear system of ordinary differential

equations as:

[M]{Ü} + [C]{ } + [K]{U} = {P(t)}… … … 2.1.4

2.7.5 Free Vibration

The equilibrium equations for the free vibration of an undamped multiple degree

of freedom system are defined as a set of linear homogeneous second-order ordinary

differential equations as:

[M]{Ü} + [K]{U} = {0}… … … 2.1.5

Assuming a harmonic motion for the temporal displacement, ({U} = {φ} e-iωt) Eq.

(2.1.5) is transformed to a set of linear homogeneous algebraic equations as:

([K] – (ω2)[M]{φ} = {0} … … … 2.1.6

Or

([K] {φ} = (ω2)[M]{φ} … … … 2.1.7

23  

Eq. (2.1.6) is known as a generalized Eigen-value problem between the stiffness

and mass matrices of the system.

The values of (ω) are the natural circular frequencies and the vectors {φ} are the

corresponding mode shapes.

2.7.5.1 Solution to Eigen value Problem

For non-trivial solutions, the determinant of ([K] − (ω²) [M]) must be zero. This

leads to a scalar equation, known as the characteristic equation, whose roots are the

system’s natural circular frequencies of the system (ω).

Substituting each value of circular frequency in Eq. (2.1.6) yields a corresponding

eigenvector or mode shape, that is defined to an arbitrary multiplicative constant. The

modal matrix [{φ₁} ... {φn}] spans the N-dimensional linear vector space.

This means that the eigenvectors [{φ₁} ... {φn}] form a complete basis, i.e., any

vector such as the vector of dynamic response of a multiple degree of freedom

(MDOF) system, {U(t)}, can be expressed as a linear combination of the mode shapes:

… … … … 2.1.8

in which, the terms y (t) n are modal coordinates and therefore, {U(t)} is defined

in modal coordinate space, since the values of {φ } are independent of time for linear

systems, Eq. (2.7).

Furthermore, the temporal derivatives of total response can be expressed as:

… … … … 2.1.9

… … … … 2.1.10

Which are also defined in modal coordinate space.

24  

2.7.5.2 Orthogonality of Modes

Considering the generalized Eigen-value problem for the mth and nth circular

frequencies and corresponding mode shapes:

([K] – (ωm2)[M]{φm} = {0} … … … 2.1.11

([K] – (ωn2)[M]{φn} = {0} … … … 2.1.12

Pre-multiplying Eq. (4.11) and Eq. (4.12) by {φn}T and {φm}T , respectively:

{φn}T [K]{φm} – (ωm2) {φn}T [M]{φm} = {0} … … … 2.1.13

{φm}T [K]{φn} – (ωn2) {φm}T [M]{φn} = {0} … … … 2.1.14

Then, transposing Eq. (4.14) and invoking the symmetric property of the [K] and

[M] matrices yields:

{φn}T [K]{φm} – (ωn2) {φn}T [M]{φm} = {0} … … … 2.1.15

Subtracting Eq. (2.15) from Eq. (2.13) yields:

((ωm2 – (ωn

2)) {φn}T [M]{φm} = {0} … … … 2.1.16

For any (m ≠n), if (ωm2 ≠ (ωn

2) :

{φn}T [M]{φm} = {0} … … … 2.1.17

{φn}T [K]{φm} = {0} … … … 2.1.18

Eqs. (2.1.17,2.1.18) express the characteristic of “orthogonality” of mode shapes

with respect to mass and stiffness matrices, respectively.

2.7.6 Forced Vibration

The equation of motion for forced vibration of an un-damped MDOF system is

defined as:

… … … … 2.1.19

Expressing displacements and their time derivatives in modal coordinate space:

25  

… … … 2.1.20

Pre-multiplying each term in Eq. (2.20) by φ T:

… … … … 2.1.21

Invoking orthogonality, Eq. (2.1.20) is reduced to a set of N uncoupled modal

equations as:

… … … … 2.1.22

or:

… … … … 2.1.23

Where, M φ T M φ , K φ T K φ , P t φ T P t are

generalized modal mass, generalized modal stiffness and generalized modal force

respectively.

Dividing by modal mass Mn and adding the assumed modal damping ratio, (ξn),

Eq. (2.1.23) becomes:

2 … … … … 2.1.24

2.7.6.1 Proportional Excitation

If loading is proportional {P(t) = {P}p(t), meaning the applied forces have the

same time variation defined by p(t) (such as ground motion), Eq. (2.1.24) can be

expressed as:

2 … … … … 2.1.25

Defining a modal participation factor, , as:

Γφ T P

Mφ T P

φ T M φ … … … … 2.1.26

26  

Also defining a scaled generalized modal coordinate:

… … … 2.1.27

Eq. (4.24) is rewritten in terms of the scaled modal coordinate D t as:

2 … … … … 2.1.28

Therefore, using modal decomposition, the equation of motion for an N-DOF

system is uncoupled to N equations of motion of generalized single degree of freedom

(SDOF) systems.

2.7.7 Response History Analysis (RHA)

In response history analysis (RHA), N uncoupled SDOF modal equations, Eq.

(2.1.28), are solved for the modal coordinates (Dn(t)), and then, by superposing the

modal responses, the total displacement response of the system is obtained as:

D t Γ … … … … 2.1.29

in which the “time history” of the total response is obtained by the summation of

modal responses as products of time history of modal coordinates (Dn(t)), modal

participation factors (Гn), and modal displacements (mode shapes) {φn}. Moreover, the

time history of any load effect, R(t) , may be expressed as:

… … … … 2.1.30

In which, {Rn} is a static modal load effect.

2.7.8 Response Spectrum Analysis (RSA)

In response spectrum analysis (RSA), for each uncoupled generalized SDOF

modal equation, Eq. (2.1.28), the maximum modal coordinate (Dn,max) is obtained

using the response spectrum of the external excitation p(t) and assumed modal

damping ξn (Figure 2.1).

27  

Response spectra are found by obtaining the maximum dynamic amplification

(maximum ratio of dynamic to static responses) for a set of natural frequencies.

Figure 2.1: A generic response spectrum for an external excitation p(t)

Therefore, the modal response is obtained as:

, D , Γ … … … … 2.1.31

2.7.8.1 Superposition of modal maxima

The total response is obtained using superposition of modal maxima. The

superposition can be performed by summation of absolute values of modal responses.

U U ,

N

… … … … 2.1.32

This provides a conservative estimate of the maximum response. As an

approximation, the method of Square Root of Sum of Squares (SRSS) of modal

maxima can be used when natural frequencies are distinct (Rosenblueth 1959):

U U ,

N

… … … … 2.1.33

28  

Also, the method of complete quadratic combination (CQC) can be used.

2.7.8.2 Ground Excitation- Response Spectrum Analysis

The equation of motion for an undamped MDOF system subjected to ground

excitation (support motion) from an earthquake is:

0 … … … … 2.1.34

Where; {Üt} is the vector of absolute acceleration. The vector {U} is defined as the

relative displacement vector, defined as:

… … … … 2.1.35

Where; {r}(Ug) is the vector of rigid body pseudo-static displaced shape due to

horizontal ground motion. Substituting Eq. (2.1.35) in Eq. (2.1.34) yields:

… … … … 2.1.36

As before, solving the linear Eigen-value problem, defining the response in modal

coordinate space, uncoupling and adding assumed modal damping yields:

2 … … … … 2.1.37

Defining the modal participation factor, Гn, as:

… … … … 2.1.38

Also, defining the scaled generalized modal coordinate Dn(t) = yn (t)/Γn , Eq.

(2.1.36) may be rewritten in terms of the scaled modal coordinate (Dn(t)) as:

2 … … … … 2.1.39

Performing response spectrum analysis for ground excitation, for each uncoupled

generalized SDOF modal equation, Eq. (2.1.39), the maximum modal response is

obtained using earthquake response spectra such as the Newmark Blume Kapur

(NBK) design spectra.

29  

Therefore, the maximum modal coordinate is obtained as:

, , … … … … 2.1.40

The total response is obtained using superposition of modal maxima. The

superposition is performed by considering Square Root of Sum of Squares (SRSS) of

modal maxima:

, … … … … 2.1.41

2.7.9 Response Spectrum Analysis Summary

Response spectrum analysis to compute the dynamic response of a MDOF to

external forces and ground excitation can be summarized as a sequence of steps as:

1. Define the structural properties.

• Determine the stiffness matrix [K] and mass matrix [M].

• Assume the modal damping ratio ξn.

2. Perform a generalized Eigen-value problem between the stiffness and mass

matrices.

• Determine natural circular frequencies ωn.

• Determine mode shapes {φn}.

3. Compute the maximum modal response.

• Determine the maximum modal coordinate (Dn,max) using the

excitation response spectrum for the corresponding natural circular

frequency and modal damping ratio.

• Determine the modal participation factor Гn.

• Compute the maximum modal response as a product of maximum

modal coordinate, modal participation factor and mode shape.

4. Combine the contributions of all maximum modal responses to determine

the maximum total response using SRSS or other combination methods.

30  

2.7.10 Limitations to RSA

In the presence of uncertainty in the structure’s physical or geometrical

parameters, the deterministic structural dynamic analysis cannot be performed and

hence, a new method must be developed to incorporate an uncertainty analysis into

the conventional response spectrum analysis.

2.8 Fundamentals of Uncertainty Analysis

2.8.1 Introduction

The steady development of powerful computational hardware in recent years has

led to high-resolution finite element models of real-life engineering structural

systems. However, for high-fidelity and credible numerical models, a high resolution

in the numerical mesh is not enough. It is also required to quantify the uncertainties

and robustness associated with a numerical model. As a result, the quantification of

uncertainties plays a key role in establishing the credibility of a numerical model.

Uncertainties can be broadly divided into two categories. The first type is due to the

inherent variability in the system parameters, for example, different cars

manufactured from a single production line are not exactly the same. This type of

uncertainty is often referred to as aleatoric uncertainty. If enough samples are present,

it is possible to characterize the variability using well established statistical methods

and consequently the probability density functions (pdf) of the parameters can be

obtained. The second type of uncertainty is due to the lack of knowledge regarding a

system, often referred to as epistemic uncertainty. This kind of uncertainty generally

arises in the modeling of complex systems, for example, in the modeling of cabin

noise in helicopters. Due to its very nature, it is comparatively difficult to quantify or

model this type of uncertainty.

There are two broad approaches to quantify uncertainties in a model. The first is

the parametric approach and the second is the non-parametric approach. In the

parametric approach the uncertainties associated with the system parameters, such as

Young’s modulus, mass density, Poisson’s ratio, damping coefficient and geometric

parameters are quantified using statistical methods and propagated, for example, using

the stochastic finite element method [R. Ghanem, P. Spanos (1991), M. Kleiber, T.D.

31  

Hien (1992), I. Elishakoff, Y.J. Ren (2003]. This type of approach is suitable to

quantify aleatoric uncertainties. Epistemic uncertainty on the other hand does not

explicitly depend on the system parameters. For example, there can be un-quantified

errors associated with the equation of motion (linear or non-linear), in the damping

model (viscous or non-viscous), in the model of structural joints, and also in the

numerical methods (e.g, discretisation of displacement fields, truncation and round-

off errors, tolerances in the optimization and iterative algorithms, step-sizes in the

time-integration methods).

It is evident that the parametric approach is not suitable to quantify this type of

uncertainty. As a result non-parametric approaches [C. Soize (2006), S. Adhikari

(2006)] have been proposed for this purpose. [Sondipon Adhikari, Michael I.

Friswell1, and Kuldeep P. Lonkar (2007)]

2.8.2 Causes and Sources of Uncertainty

The uncertainties that a structural engineer encounters during a design come from

a range of sources. The following five sources of uncertainty cover the vast majority

of examples.

• Time: There is uncertainty in predicting the future (e.g., how much snow load

will our structure experience?) and uncertainty in knowing the past (e.g., what

was the concrete strength in the old building we need to renovate?);

• Statistical limits: We never can get enough data. (I took some cores from the

old building and tested the concrete. Do these test values truly represent the

concrete strength?);

• Model limits: The structural model used in the analysis and design leaves out

or simplifies many aspects of the structure, and it is possible that the model is

not conceptually correct;

• Randomness: The structural properties (e.g., modulus of elasticity, concrete

strength_ are not a single number but vary over some range. The properties are

random variables);

• Human error: It is possible that an error was made during the design or the

construction.

32  

None of these five causes of uncertainty separate uncertainties cleanly into

aleatory or epistemic. Generally there are aspects of both in each of the five causes.

But, the five causes allow us to categorize the uncertainties that we will encounter

subsequently, and whether an uncertainty is aleatory or epistemic will allow us to

focus on which uncertainties can be reduced through probabilistic techniques and

which must be dealt with in other ways.

The sources of uncertainty in modeling analysis problem may be listed as follows:

1. Uncertainty in Geometry: Actual geometry may differ from that specified

2. Uncertainty in Material Properties: Actual material properties may be different

from that specified. Uncertainty in stiffness, mass

3. Uncertainty in loading: Actual loads may differ from those assumed; Actual

loads may be distributed in a manner different from assumed.

4. Uncertainty in Boundary Conditions: The existing boundary conditions of the

structural system may vary with time due to age of the structure or with external

effects over time.

5. Uncertainty in mathematical modeling of dynamic problem

6. The assumptions and simplifications inherent in any analysis may result in

calculated load effects – moments, shears, etc., - different from those that, in

fact act in the structure

7. The actual structural behavior may differ from that assumed, owing to imperfect

knowledge

2.8.3 Analysis of Uncertainty

There are basically two methods to solve the problems of Uncertainty analysis,

viz.:

• Deterministic Analysis

• Non-deterministic Analysis

These methods are discussed in preceding sub-sections.

33  

2.8.3.1 Deterministic analysis

In deterministic analysis of physical systems, defining the system’s characteristics

as point quantities, using conventional deterministic algebraic values, is sufficient to

model the system and perform the analysis (Figure 2.2).

a

x = a

Figure 2.2: A deterministic algebraic variable

In order to perform uncertainty analysis on a physical system, the uncertainty

present in the system’s physical characteristics must be fully mathematically

quantified.

2.8.3.2 Non-Deterministic analyses

Presently, these are methods to consider uncertainties in non-deterministic

structural analysis are:

i. Stochastic analysis

ii. Fuzzy analysis

iii. Interval analysis

iv. Monte Carlo simulation Method

v. Perturbation Method

These methods were briefly discussed in preceding subheadings.

i. Stochastic Analysis

In stochastic analysis, the theory of probability which was developed based on

aleatory uncertainty. Extensions have been made such as “degree of belief”

probability on subjective probability which includes epistemic effects.

The stochastic approach to uncertain problems is to model the structural

parameters as random quantities [Pascal 1654]. Therefore, all information about the

structural parameters is provided by the probability density functions. This probability

density function is then used to determine an estimate of the system’s behavior.

34  

Random Variable

A random quantity, used in stochastic analysis, is defined by a deterministic

function that yields the probability of existence of the random variable in a given

subset of the real space (See Figure 2.3), (See Equation 2.2.1):

Figure 2.3: Probability Density Function of a random quantity

∞… … … 2.2.1

In which, Fx(a) is cumulative probability distribution function evaluated for

random variable (a) and f (x) is the corresponding probability density function.

ii. Fuzzy Analysis

In fuzzy analysis, the theory of possibility for fuzzy sets is used which assumes

epistemic uncertainty.

The fuzzy approach to the uncertain problems is to model the structural

parameters as fuzzy quantities [Lotfi-zadeh (1965)]. In conventional set theories,

either an element belongs or doesn’t belong to set. However, fuzzy sets have a

membership function that allows for “partial membership” in the set. Using this

method, structural parameters are quantified by fuzzy sets. Following fuzzifying the

parameters, structural analysis is performed using fuzzy operations.

35  

Fuzzy Subset

Considering E as a referential set inℜ, an ordinary subset A of the referential set is

defined by its characteristic function μA(x) as:

x E : (x) 0,1 μA … … … 2.2.2.a

This exhibits whether or not, an element of E belongs to the ordinary subset A. For

the same referential set E, a fuzzy subset A is defined by its characteristic function,

membership function μA(x), as:

x E : (x) 0,1 μA … … … 2.2.2.b

A fuzzy number is defined by its membership function whose domain is ℜ while

its range is bounded between [0, 1]. The domain of the membership function is known

as the interval of confidence and the range is known as the level of presumption.

Therefore, each level of presumption α (α-cut membership, α [0, 1]) has a unique

interval of confidence [Aa = aa, ba ], which is a monotonic decreasing function of α

(see Figure 2.4), (Eqs. (2.2.2.c, 2.2.2.d)):

, 0,1 , … … … … … 2.2.2.c

Or

, 0,1 , , , … … … … … 2.2.2.d

Figure 2.4: Membership function of a fuzzy quantity

36  

iii. Interval Analysis

In interval analysis, the theory of convex (interval) sets is used which assumes

epistemic or aleatory uncertainties (such as Dempster-Shafer bounds that are

epistemic bounds on aleatory probability functions).

The interval approach to the uncertain problems is to model the structural

parameters as interval quantities. In this method, uncertainty in the elements is viewed

by a closed set-representation of element parameters that can vary within intervals

between extreme values. Then, structural analysis is performed using interval

operations.

Interval (Convex) Number

A real interval is a closed set defined by extreme values as (Figure 2.5):

, | … … … … … 2.3.1

a b

,

Figure 2.5: An interval quantity

One interpretation of an interval number is a random variable whose probability

density function is unknown but non-zero only in the range of interval.

Another interpretation of an interval number includes intervals of confidence for α

-cuts of fuzzy sets. This interval representation transforms the point values in the

deterministic system to inclusive set values in the system with bounded uncertainty.

Interval Arithmetic Operations

Interval arithmetic is a computational tool that can be used to represent

uncertainty as:

1. A set of probability density functions (pdf).

2. In Dempster-Shafer models for epistemic probability.

3. α - cuts in fuzzy sets.

37  

In this work, the symbol (~) represents an interval quantity.

Considering X a, b and Y c, d as two interval numbers, the basic interval

arithmetic operations are:

Addition:

, … … … … … 2.3.2

Subtraction:

, … … … … … 2.3.3

Multiplication by scalar:

, , , … … … … … 2.3.4

Multiplication:

, , , , , , , … … … … … 2.3.5

Properties of Interval Multiplication:

Associative:

… … … … … 2.3.6

Commutative:

… … … … … 2.3.7

Distributive:

… … … … … 2.3.8

Therefore, the distributive property of interval multiplication is weaker than that

in conventional algebra and it is one possible cause of loss of sharpness in interval

operations.

Division:

,1

,1

, 0 , … … … … … 2.3.9

38  

Interval Vector (2-D):

,, … … … … … 2.3.10

Which; represents a “box” in 2-D space as the enclosure (Figure 2.6).

Figure 2.6: An interval vector

Transformation of Interval to Perturbation

Perturbation methods often use small change in a parameter, ε. To express interval

problems in terms of perturbation, an interval perturbation, ε = [−1, 1], is introduced

so that a general interval is written as summation of center and radial values.

Considering Z l, u as an interval number, the median and radius can be

defined as:

2 … … … … … 2.3.11

2 … … … … … 2.3.12

So, Zcan be redefined as:

… … … … … 2.3.13

39  

Where; the interval number is shown as its median subjected to a perturbation of

radius by which, the result encompasses the range of the interval between the extreme

values.

Functional Dependency of Interval Operations

Considering X 2,2 and Y 2,2 as two independent interval numbers,

the functional dependent interval multiplication results in:

0,4

In contrast, the functional independent interval multiplication results in:

4,4

Sharpness Considerations in Engineering

In interval operations, the functional dependency of intervals must be considered

in order to attain sharper results. In fact, the issue of sharpness and overestimation in

interval bounds is the key limitation in the application of interval methods. Naïve

implementation of interval arithmetic algorithms (substituting interval operations for

their scalar equivalence) will yield bounds that are not useful for engineering design.

Therefore, there is a need to develop algorithms to calculate sharp or nearly sharp

bounds to the underlying set theoretic interval problems.

For instance, the calculation of exact sharp bounds to the interval system of

equations resulting from linear static analysis using the finite element method has

been proved to be computationally combinatorial problem. However, even the 2n

combinations of upper and lower bounds do not always yield the bounds.

In problems with narrow intervals associated with truncation errors, the naïve

implementation of interval arithmetic will yield acceptable bounds. However, for

wider intervals representing uncertainty in parameters, the naïve method will

overestimate the bounds by several orders of magnitude.

40  

Successful applications of the interval method in the linear static problem have

required the development of new algorithms that are computationally feasible yet still

provide nearly sharp bounds [Muhanna and Mullen (2003)].

iv. The Monte Carlo Simulation Method

Monte Carlo methods (or Monte Carlo experiments) are a class of computational

algorithms that rely on repeated random sampling to compute their results. Monte

Carlo methods are often used in simulating physical and mathematical systems.

Because of their reliance on repeated computation of random or pseudo-random

numbers, these methods are most suited to calculation by a computer and tend to be

used when it is infeasible or impossible to compute an exact result with a

deterministic algorithm. This method is also used to complement the theoretical

derivations.

Monte Carlo simulation methods are especially useful in studying systems with a

large number of coupled degrees of freedom, such as fluids, disordered materials,

strongly coupled solids, and cellular structures (see cellular Potts model). More

broadly, Monte Carlo methods are useful for modeling phenomena with significant

uncertainty in inputs, such as the calculation of risk in business. These methods are

also widely used in mathematics: a classic use is for the evaluation of definite

integrals, particularly multidimensional integrals with complicated boundary

conditions. It is a widely successful method in risk analysis when compared with

alternative methods or human intuition. When Monte Carlo simulations have been

applied in space exploration and oil exploration, actual observations of failures, cost

overruns and schedule overruns are routinely better predicted by the simulations than

by human intuition or alternative "soft" methods.

One of its major advantages is that accurate solutions can be obtained for

problems whose deterministic solution is known. Since it is completely general this

method is frequently used to calibrate and validate other methods. The main

disadvantage is that it is time consuming, but among all numerical methods that rely

on n-point evaluations in a d-dimensional space to produce an approximate solution,

the Monte Carlo method has an absolute error estimation that decreases as n−½, while

in the absence of an exploitable special structure all others methods have an error

41  

estimation that decreases as n−1/d at best. Since each realization is independent of the

others Monte Carlo simulations can be easily parallelized.

In practical engineering problems, most real models possess some kind of

uncertainty, such as in domain geometry, material properties or even loads which are

not known for certain. In fact, another possibility is that uncertainty can also be

considered due to a lack of knowledge.

The amount of research in this area and also the interest in new methodologies has

been increased. One possibility is to adopt updating techniques, where a conventional

FE model is used with measurement data. In such an approach, the FE models are

compared with experimental data and the simulation results must be matched to the

measurements. This technique allows to increase the accuracy of the FE prediction

with respect to one particular realization of the structure, but does not improve the

representation of the statistics of the predicted response [Shorter (1998)].

Another possible way to describe the uncertainty parameters may through the use

of approximate methods. For instance, in a Monte-Carlo simulation, the global

structure is described in terms of sample individual structures, which can be used to

achieve the statistical information responses. Monte Carlo simulation is considered a

stochastic technique, when used to solve mathematical problems in general. The word

stochastic means that it uses random numbers and probability statistics to obtain

results. Monte Carlo methods were originally developed for the Manhattan Project

during World War II. However, this technique has found application in many fields,

such as stock market, forecasting, biology, etc.

The term "Monte Carlo method" was coined in the 1940s by physicists working

on nuclear weapon projects in the Los Alamos National Laboratory, after the famous

Monte Carlo casino, a gambling venue based on random-number generation.

Regarding the name Monte Carlo, it originated from the city Monte Carlo in

Monaco, whose main attractions are casinos and gambling. In a fashion similar to

gambling, Monte Carlo simulations use a random selection process which is repeated

many times to create multiple scenarios for the proposed problem. Each time a value

is randomly selected, it forms one possible scenario and solution to the problem.

Taking all scenarios together gives a range of possible solutions, some of which are

42  

more probable, and others less so. It is clear that for many scenarios, say 10, 000 or

more, the average solution will give an approximate answer to the problem. Naturally,

to improve the accuracy of this answer, more scenarios should be used.

In terms of practical application, unfortunately, due to the number of members to

be sampled and the number of samples to perform the statistical responses, the use of

the Monte Carlo approach can be prohibitive. For more details on Monte Carlo

simulation, please refer to Huber (1999) and Kleiber and Hien (1992).

Thus, the Monte Carlo approach is usually associated with methods to reduce the

computational burden of the FE dynamic model, such as model reduction methods

and the response surface methods.

v. Perturbation Method

The idea behind the perturbation method is a simple one. Faced with a problem

that we cannot solve exactly, but that is close (in some sense) to an auxiliary problem

that we can solve exactly, a good approximate solution to the original problem should

be close (in a related sense) to the exact solution of the auxiliary problem.

The simple pendulum is a excellent paradigm for studying the nonlinear behavior

of non-equilibrium systems. Moreover, a sequence of coupled pendula will provide a

natural setting for introducing the Sine Gordon nonlinear Partial Differential Equation

(PDE), an integrable nonlinear PDE, from which another integrable pde, the

Nonlinear Schr¨odinger (NLS) equation, can be derived as a small amplitude

approximation.

These provide a nice illustration of how the techniques of weakly nonlinear

analysis are developed systematically.

Exact solutions are available to the simple pendulum problem for comparison. We

will see that the nonlinear dependence of the frequency of oscillation of the pendulum

on its amplitude of oscillation is a crucial signature of nonlinear behavior. In fact, this

naturally suggests two weakly nonlinear paradigms: the anharmonic oscillator, a

weakly nonlinear oscillator and the Van der Pol oscillator, an essentially linear

oscillator with a nonlinear damping. These two models will illustrate the use of

43  

singular perturbation methods to derive uniformly valid perturbation corrections to the

basic oscillator amplitude and frequency. The principal idea is to expand the oscillator

amplitude in an asymptotic series, allowing for sufficient flexibility to avoid

unbounded (secular) growth of the correction term to the amplitude at each order in

the perturbation expansion. We will first see how a regular perturbation expansion

lacks this flexibility.

Singular perturbation expansions are extremely powerful analytic tools for

studying a whole class of nonlinear problems. They will form the basis for deriving

many of the hierarchy of soliton equations (the Nonlinear Schrödinger equation

(NLS), in particular) and universal order parameter equations (of Complex Ginzburg-

Landau type) valid near a bifurcation point in spatially extended systems. As these

perturbation theories involve expansions in a small parameter, one might be left with

the impression that they are of limited utility. Remarkably, in many instances the

results prove accurate even for values of the parameter approaching

The perturbation method, which is equivalent to a lower-order Taylor expansion,

has been widely used for its tractability and computational time-saving. It expresses

the structural matrices and response in terms of a lower-order polynomial function

with respect to the parameters centered at the mean values, i.e., makes an

approximation of the response surface. Introducing the following notation

, , … , … … … … … 2.4.1

, , … , … … … … … 2.4.2

, , … , … … … … … 2.4.3

For any variable Φ which depends of the parameters x1, x2, …, xn then the

stiffness and matrices are expanded as

12

… … … … 2.4.4

12

… … … … 2.4.5

44  

Where, . In the same manner, for the Eigen-value problem

0 … … … … 2.4.6

The resulting Eigen-values λ and eigenvectors φ are expanded as

12

… … … … 2.4.7

12

… … … … 2.4.8

The mean and variances of the Eigen-values are

12

, … … … … 2.4.9

, … … … … 2.4.10

It can be shown that λI is given by

… … … … 2.4.11

Where λ0 and φ0 are obtained by the mean eiganvalue problem

0 … … … … 2.4.12

The natural frequencies can be expanded as

√

12

128 … … 2.4.13

Therefore considering up to the first order term, their mean and standard deviation

can be obtained from equations (5.4.9) and (5.4.10) as

… … … … 2.4.14

45  

1

2… … … … 2.4.15

It is required that the random variables involved in the analysis do not deviate

much from their expected values. If the coefficient of variation is not very small the

solution cannot ever be improved by using a finer mesh. Of greater importance than

the magnitude of variability of the original random variables is how appropriate the

response surface is. The response quantity should be chosen in such a way that these

quantities are not highly nonlinear in respect to the random variables, e.g., a

frequency response function instead of a response in time domain.

If the number of random variables is large, such as in problems involving random

fields, or if a high-order expansion is used, then the calculation effort becomes

prohibitive.

2.9 Interval Finite Element Methods (IFEM)

2.9.1 Introduction

There are various ways in which the types of uncertainty might be classified. One

is to distinguish between “aleatory” (or stochastic) uncertainty and “epistemic”

uncertainty. The first refers to underlying, intrinsic variabilities of physical quantities,

and the latter refers to uncertainty which might be reduced with additional data or

information, or better modeling and better parameter estimation [Melchers (1999)].

Probability theory is the traditional approach to handle uncertainty. This approach

requires sufficient statistical data to justify the assumed statistical distributions.

Analysts agree that, given sufficient statistical data, the probability theory describes

the stochastic uncertainty well.

However, traditional probabilistic modeling techniques cannot handle situations

with incomplete or little information on which to evaluate a probability, or when that

information is nonspecific, ambiguous, or conflicting [Walley (1991), Ferson and

Ginzburg (1996), Sentz and Ferson, (2002)]. Many generalized models of uncertainty

have been developed to treat such situations, including fuzzy sets and possibility

theory [Zadeh (1978)], Dempster-Shafer theory of evidence [Dempster, (1967),

Shafer (1976)], random sets [Kendall (1974)], probability bounds [Berleant (1993),

46  

Ferson and Ginzburg (1996), Ferson et al. (2003)], imprecise probabilities [Walley

(1991)], convex models [Ben-Haim and Elishakoff (1990)]9, and others.

Recently, various generalized models of uncertainty have been applied within the

context of the finite element method to solve a partial differential equation with

uncertain parameters. Regardless what model is adopted, the proper interval solution

will represent the first requirement for any further rigorous formulation. Finite

element method with interval valued parameters results in Interval Finite Element

Method (IFEM). Different formulations of IFEM have been developed. The use of

IFEM solution techniques can be broadly classified into two groups, namely the

optimization approach and the non-optimization approach.

More recently, non-optimization approaches for the interval finite element

analysis have been developed in a number of papers. For linear elastic problems, this

approach leads to a system of linear interval equations, and then the solution is sought

using various methods developed for this purpose. The major difficulty associated

with this approach is the “dependency problem” [Moore (1979), Neumaier (1990),

Hansen (1992), Muhanna and Mullen (2001)]. In general, dependency problem arises

when one or several interval variables occur more than once in an interval expression.

The dependency in interval arithmetic leads to an overestimation of the solution. A

straightforward replacement of the system parameters with interval ones without

taking care of the dependency problem is known as a naive application of interval

arithmetic in finite element method (naïve IFEM). Usually such a use results in

meaninglessly wide and even catastrophic results [Muhanna and Mullen (2001).

In the non-optimization category, a number of developments can be presented. A

combinatorial approach (based on an exhaustive combination of the extreme values of

the interval parameters) was used in [Muhanna and Mullen (1995), Rao and Berke

(1997)]. This approach gives exact solution in special cases of linear elastic problems.

However, it is computationally tedious and expensive, and is limited to the solutions

of small-scale problems.

A new formulation for interval finite element analysis for linear static structural

problems is developed in the work of [Muhanna et al. (2005)]. Material and load

47  

uncertainties are handled simultaneously, and sharp enclosures on the system’s

displacement as well as the internal forces are obtained efficiently.

Recently, new advancements has been made in the area of interval FEM, e.g.,

Corliss et al. (2004), Popova et al. (2003), and the significant development in

[Neumaier and Pownuk (5)] where sharp results are achieved for linear truss

problems even with large uncertainty.

2.9.2 Interval approach

The interval approach to the uncertain problems is to model the structural

parameters as interval quantities. In this method, uncertainty in the elements is viewed

by a closed set-representation of element parameters that can vary within intervals

between extreme values. Then, structural analysis is performed using interval

operations.

In many cases, only a range of possible values for a non-deterministic quantity is

available, but no further information about which values are more likely to occur is

available. In this situation, a uniform distribution is often used in probability theory.

As such, the lack of knowledge is filled by subjective information assumed by the

analyst, expressed in the form of uniform distribution.

In the interval approach, an uncertain quantity is assumed to be unknown but

bounded, and it has lower and upper bounds (endpoints) without assigning a

probability structure. Therefore, the interval representation of an uncertain quantity x

is given by

; … … …2.5.1

Where x and x are the lower and upper bounds of the uncertain quantity

respectively. The interval approach concept can be directly attributed to Moore.

Moore's purpose for introducing intervals was to provide a treatment of rounding

errors and truncation errors in finite precision arithmetic. For example, in a computer

with five significant digits, the number √2 can be represented using an interval √2

[1.4142, 1.4143]. Thus the rounding error is enclosed in the interval. Besides

bounding the effects of rounding errors, interval arithmetic can probe the behavior of

48  

functions efficiently and reliably over whole sets of arguments at once. By its nature,

interval arithmetic yields rigorous enclosures for the range of operations and

functions. The results are intervals in which the exact results must lie. This

characteristic has made interval arithmetic useful in scientific computing, including

• Bounding effects of rounding errors and truncation errors Moore

• Bounding the range of functions Hansen

• Bounding the error term in Taylor's theorem Neumaier

• Bounding the results of Monte Carlo simulations in reliability analysis

• Computing rigorous bounds on the solution of ordinary and partial differential

equations

• Global optimization - Hansen

• Solving nonlinear system

• Construct rigorous bounds around an approximate solution, in which an actual

solution must lie

• Exhaustively search a region to find all roots of a nonlinear system

Interval arithmetic has been successfully applied to many applications. These

include reliable modeling and optimization for chemical engineering, estimating the

performance of financial trading systems, calculating validated reliability bounds,

computer-assisted proofs in mathematical physics, solving interval constraints in

computer-aided design, existence verification and construction of robust controlle,

and others. More information about applications for interval arithmetic is found in

Muhanna and Mullen.

If sufficient probabilistic information is available, the use of the interval approach

to describe uncertainties only gives the bounds of the responses, and the additional

likelihood information about the responses is lost. In this situation, probabilistic

approach is generally preferred. However, since interval analysis usually requires

much less computation than a probability analysis such as Monte Carlo simulation, it

might still be valuable to use the interval approach to perform a rapid analysis of the

response ranges. Another practical application of the interval approach is in the study

49  

of the sensitivity of the system behavior with respect to changes in input parameters.

The system behavior over the entire interval range of parameter variation can be

studied by interval analysis.

2.9.3 Interval Arithmetic

Early use of interval representation is associated with the treatment of truncation

errors in numerical calculations. For example, in a computational system with four

decimal digits accuracy, the number 4.1231 would be represented as an interval

[4.123, 4.124]. This approach allows the range of errors introduced by round-off

errors to be precisely determined. Moore (1966), instead of computing a numerical

approximation using limited-precision arithmetic, proceeded to construct intervals

known in advance to contain the desired results. Several authors have bound rounding

errors using intervals [Dwyer (1951), Sunaga (1958)]. However, Moore extended the

use of interval analysis to bind the effects of errors from different sources, including

approximation errors and errors in data.

Interval arithmetic was developed as an effective tool to obtain bounds on

rounding and approximation errors. It is still to be seen how effective this tool can be

when the range of the number is due to physical uncertainties instead of rounding

errors.

A number of software libraries and extensions to programming language have

been developed to implement interval calculations using computers [Blecher etal

(1987), Kullisch (1987)]. Additionally scientific calculators that are capable of

dealing with interval arithmetic operations in addition to normal arithmetic have been

developed

Definitions of real intervals and operations with intervals can be found in a

number of references [Hansen (1965), Moor (1966), Alefeld and Herzberger (1983),

Neumaier (1990)]. The fundamental concepts of interval arithmetic that has been seen

in engineering applications [Mullen and Muhanna (1999)

An interval number is a closed set in R that includes the possible range of an

unknown real number, where R denotes the set of real numbers. A real interval is a set

of the form

50  

, |

Based on the above definitions, interval arithmetic is defined on sets of intervals,

rather than on sets of real numbers. Interval mathematics can be considered a

generalization of real numbers mathematics. Overestimation is a major drawback in

interval computations. One reason is that only some of the algebraic laws, valid for

real numbers, remain valid for intervals; other laws hold only in a weaker form

[Neumaier (1990), pp. 19-21]. There are two general rules for the algebraic properties

of interval operations.

2.9.4 What is Interval Arithmetic and why is it Considered?

Interval arithmetic is an arithmetic defined on sets of intervals, rather than sets of

real numbers. A form of interval arithmetic perhaps first appeared in 1924. Modern

development of interval arithmetic began with R. E. Moore’s dissertation.

In this report, boldface will denote intervals, lower case will denote scalar

quantities, and upper case will denote vectors and matrices. Brackets “[ ]” will delimit

intervals while parentheses “( )” will delimit vectors and matrices. Underscores will

denote lower bounds of intervals and over scores will denote upper bounds of

intervals. Corresponding lower case letters will denote components of vectors. The set

of real intervals will be denoted by . Interval vectors will also be called boxes. If

x x, x and y y, y then the four elementary operations for idealized interval

arithmetic obey

| , , , … … … 2.5.2

Thus, the image of each of the four basic interval operations is the exact range of

the corresponding real operation. Although Equation (2.5.2) characterizes these

operations mathematically, interval arithmetic’s usefulness is due to the operational

definitions. For example,

, , … … … 2.5.3

, , … … … 2.5.4

, , , , , , , , … … … 2.5.5

51  

1 , 1 0 0 … … … 2.5.6

1 … … … 2.5.7

The ranges of the four elementary interval arithmetic operations are exactly the

ranges of the corresponding real operations. If such operations are composed, bounds

on the ranges of real functions can be obtained. For example, if

1 , … … … 2.5.8

Then

0,1 0,1 0,1 1 0,1 1,0 1,0

The power of interval arithmetic lies in its implementation on computers. In

particular, outwardly rounded interval arithmetic allows rigorous enclosures for the

ranges of operations and functions. This makes a qualitative difference in scientific

computations, since the results are now intervals in which the exact result must lie. It

also enables use of computations for automated theorem proving.

2.9.5 Interval Finite Element Analysis

Considering all the structural parameters as an interval number, a system of

interval equations can be formulated in general as

k.q=p … … … 2.5.9

Or in the following explicit form:

, , … , … ,, , … , … ,

. . … … . . . … … .

, , … , … , . . … … .

, , … , … ,

,,..,.,

,,..,.,

... 2.5.10

For the case of interval loads, the stiffness matrix k is the conventional

deterministic linear stiffness. The loading vector p will be interval quantity. The

element generalized forces and the generalized displacements will be linear

52  

transformations of the interval quantities. In conventional finite-element formulations,

the nodal load is given by

… … … 2.5.11

Where pc = vector of concentrated load; and pb = nodal load contribution from an

element and has the form

∑ … … … 2.5.12

Where L = Boolean connectivity matrix, b(x) =applied Traction; and Ni= shape

function for node i. Also note that pb itself can be broken in terms of element

generalized nodal loads pi .

… … … 2.5.13

While analyzing a structure for load patterns and load combinations, only the

function b(x) (the magnitude of the load) is allowed to be an interval. To correctly

evaluate inclusive interval values for pi , attention must be paid to the sign of the

terms Ni, as whenever Ni is positive, upper limit of interval need to be integrated

however whenever Ni change sign to negative, the lower limit must be integrated.

As mentioned previously some of the conventional laws hold weakly in interval

algebra, care has to be given to the order of multiplication as otherwise it will have a

strong influence on the width of resulting intervals. One of the challenges that have to

be faced in interval algebra will be controlling the width of the interval. One way to

control width effectively will be delaying the use of interval values as much as

possible.

It is important to see that some of the conventional characteristics of various

analysis parameters still have to be satisfied. As an example, shape function Ni(x) if

selected as a polynomial, automatically satisfies number of requirement of finite

element for convergence, compatibility, rigid body motion and stability. Additionally

it is practical to choose a loading function b(x) on element m in terms of an nth order

polynomial:

53  

… … … 2.5.14

The element coefficient Amj for each term of the polynomial n on element m can

be written in matrix form as Fi with the dimension of (k × 1), where k is the number

of polynomial coefficients.

.

.

.

… … … 2.5.15

for i=1,2…m, where m = number of elements; for the whole system F can be

expressed as

.

.

.

… … … 2.5.16

Note that the dimension of F is [(m × k) × 1]

The pb vector now takes following form

… … … 2.5.17

With the dimension of (ndof × 1), where ndof is number of degrees of freedom in

the system, and where

… … … … … 2.5.18

With the dimension of [ndof × (m × k)]. The matrix Mi can be written as

… … … … 2.5.19

Note that the dimension of Mi is (ndof × k). The expression for Qi may be given

as

54  

1,2,3 … . … … … 2.5.20

And the dimension of Qi is (ndof × ndofel). ndofel is element’s number of degrees of

freedom.

These expressions have both real and interval numbers embedded in them. As

such all non-interval values are multiplied first and the last multiplication involves the

interval quantities. In this process, width of resulting interval is reduced to the

minimum possible value. Since the formulation of interval finite analysis is already in

place, the next step will be to see how uncertainty in the presence of live load can be

handled using such formulation.

2.10 Interval Response Spectrum Analysis (IRSA)

2.10.1 Introduction

This method was introduced by Mehdi Modarreszadeh in 2005 for solving

uncertainty in Dynamic Analysis with interval uncertainty.

First, IRSA method defines the uncertainty in the system’s parameters as closed

intervals; therefore, the imprecise property can vary within the intervals between

extreme values called bounds. Then, having the uncertain parameters represented by

interval variables for each element, the interval global stiffness and mass matrices of

MDOF system are assembled. This assemblage is performed such that the element

physical characteristics and the matrix mathematical properties are preserved.

Then an interval generalized Eigen-value problem between the interval stiffness

and mass matrices is established. From this interval Eigen-value problem, two

solutions of interest are obtained:

1. Bounds on variation of circular natural frequencies (Interval natural

frequencies)

2. Bounds on directional deviation of mode shapes (Interval mode shapes)

Then, the interval modal coordinate and the maximum modal coordinate are

determined using the excitation response spectrum evaluated for the corresponding

55  

interval of natural circular frequency and assumed modal damping ratio. Then, the

interval modal participation factor is computed. Dependency or independency of

variations in interval modal participation factor is considered. Following this, the

maximum modal response is computed as a maximum of the product of the maximum

modal coordinate, the interval modal participation factor and the interval mode shape.

Finally, the contributions of all maximum modal responses are combined to

determine the maximum total response using SRSS or other combination methods.

2.10.2 Interval Representation of Uncertainty

The presence of uncertainty in a structure’s physical or geometrical property can

be depicted by a closed interval. Considering q as a structure’s uncertain parameter:

, … … … 2.6.1

In which, l and u are the lower and upper bounds of the uncertain parameter,

respectively.

2.10.2.1 Interval Stiffness Matrix

The structure’s deterministic global stiffness matrix can be viewed as a linear

summation of the element contributions to the global stiffness matrix,

… … … 2.6.2

where, [Li] is the element Boolean connectivity matrix and [Ki] is the element

stiffness matrix in the global coordinate system (a geometric second-order tensor

transformation may be required from the element local coordinates to the structure’s

global coordinates).

Considering the presence of uncertainty in the stiffness characteristics, the

nondeterministic element stiffness matrix is expressed as:

, … … … 2.6.3

56  

in which [li , ui] is an interval number that pre-multiplies the deterministic element

stiffness matrix.

Considering the variation as a multiplier outside of the stiffness matrix preserves

the element physical characteristics such as real natural frequencies and rigid body

modes as well as stiffness matrix properties such as symmetry and positive semi-

definiteness. In terms of the physics of the system, this means that the stiffness within

each element is unknown but bounded and has a unique value that can independently

vary from the stiffness of other elements.

This parametric form must be used to preserve sharp interval bounds. The

uncertainty in each element’s stiffness is assumed to be independent. For a

substructure with an overall interval uncertainty, Eqs. (2.6.2,2.6.3) are used to

assemble the substructure’s stiffness matrix.

For coupled elements, matrix decompositions can be used. For instance, in a

beam-column, if functional independent values of axial and bending properties are

uncertain, the axial and bending components can be additively decomposed as:

, , … … … 2.6.4

Likewise, for continuum problems with functional independent uncertain

properties at integration points, the contribution of each integration point can be

assembled independently.

2.10.2.2 Interval Global Stiffness Matrix

The structure’s global stiffness matrix in the presence of any uncertainty is the

linear summation of the contributions of non-deterministic interval element stiffness

matrices:

, … … … 2.6.5

or:

57  

, , … … … 2.6.6

In which K is the deterministic element stiffness contribution to the global

stiffness matrix.

2.10.2.3 Interval Mass Matrix

Similarly, the structure’s deterministic global mass matrix is viewed as a linear

summation of the element contributions to the global mass matrix as:

… … … 2.6.7

Where, M is the element stiffness matrix in the global coordinate system.

Considering the presence of uncertainty in the mass properties, the

nondeterministic element mass matrix is:

, … … … 2.6.8

In which l , u is an interval number that pre-multiplies the deterministic element

mass matrix. Considering the variation as a multiplier outside of the mass matrix

preserves the element physical properties. Analogous to the interval stiffness matrix,

this procedure preserves the physical and mathematical characteristics of the mass

matrix.

The structure’s global mass matrix in the presence of any uncertainty is the linear

summation of the contributions of non-deterministic interval element mass matrices:

, … … … 2.6.9

or:

, , … … … 2.6.10

In which is the deterministic element mass contribution to the global mass

matrix.

58  

2.10.3 Bounds on Natural Frequencies and Mode Shapes

2.10.3.1 Interval Eigen-value Problem

The Eigen-value problems for matrices containing interval values are known as

the interval Eigen-value problems. Therefore, if [Ã] is an interval real matrix

(Ã ℜnxn) and [A] is a member of the interval matrix (A Ã) or in terms of components

(aij ãij), the interval Eigen-value problem is shown as:

([A] − λ[I ]){x} = 0, (A Ã) … … … 2.6.11

2.10.3.2 Solution for Eigen-values

The solution of interest to the real interval Eigen-value problem for bounds on

each Eigen-value is defined as an inclusive set of real values λ such that for any

member of the interval matrix, the Eigen-value solution to the problem is a member of

the solution set. Therefore, the solution to the interval Eigen-value problem for each

Eigen-value can be mathematically expressed as:

, | Ã 0 … … … 2.6.12

2.10.3.3 Solution for Eigenvectors

The solution of interest to the real interval Eigen-value problem for bounds on

each eigenvector is defined as an inclusive set of real values of vector x such that for

any member of the interval matrix, the eigenvector solution to the problem is a

member of the solution set. Thus, the solution to the interval Eigen-value problem for

each eigenvector is:

| Ã, 0 … … … 2.6.13

2.10.3.4 Interval Eigen-value Problem for Structural Dynamics

For dynamics problems, the interval generalized Eigen-value problem between the

interval stiffness and mass matrices can be set up by substituting the interval global

stiffness and mass matrices, Eq. (2.6.6, 2.6.10), into Eq. (2.1.7). Therefore, the

nondeterministic interval Eigen-value problem is obtained as:

59  

, , … … … 2.6.14

Hence, determination of bounds on natural frequencies in the presence of

uncertainty can be mathematically interpreted as performing an interval Eigen-value

problem on the interval-set-represented non-deterministic stiffness and mass matrices.

Two solutions of interest are:

ω : Interval natural frequencies or bounds on variation of circular natural

frequencies.

φ : Interval mode shapes or bounds on directional deviation of mode shapes.

While the element mass matrix contribution can also have interval uncertainty, in

this work only problems with interval stiffness properties are addressed. However, for

functional independent variations for both mass and stiffness matrices, the extension

of the proposed work is straightforward.

2.10.3.5 Transformation of Interval to Perturbation in Eigen-value Problem

The interval Eigen-value problem for a structure’s with stiffness properties

expressed as interval values is:

, … … … 2.6.15

This interval Eigen-value problem can be transformed to a pseudo-deterministic

Eigen-value problem subjected to a matrix perturbation. Introducing the central

and radial (perturbation) stiffness matrices as:

2 … … … 2.6.16

2 , 1,1 … … … 2.6.17

60  

Using Eqs. (6.6,6.7), the non-deterministic interval eigenpair problem, Eq.(6.5),

becomes:

… … … 2.6.18

Hence, the determination of bounds on natural frequencies and bounds on mode

shapes of a system in the presence of uncertainty in the stiffness properties is

mathematically interpreted as an Eigen-value problem on a central stiffness matrix ([

Kc]) that is subjected to a radial perturbation stiffness matrix . This

perturbation is in fact, a linear summation of non-negative definite deterministic

element stiffness contribution matrices that are scaled with bounded real numbers .

2.10.4 Bounding the Natural Frequencies

2.10.4.1 Eigen-value Perturbation Considerations

A real symmetric matrix subjected to an arbitrary perturbation can produce

complex conjugate Eigen-values and therefore, the bounds on Eigen-values are then

in the complex domain. However, since the stiffness and mass matrices governing the

structural behavior are symmetric, the natural frequencies of the structure are always

real. To retain correct physical results, constraints must be imposed on the non-

deterministic Eigen-value problem. These constraints are intrinsically present in the

non-deterministic eigenpair problem. These constraints result in a radial perturbation

matrix which is a linear combination of non-negative definite matrices that are

scaled by bounded real numbers. Therefore, this characteristic of the radial

perturbation matrix must be considered in the development of any scheme to bound

the natural frequencies.

2.10.4.2 Determination of Eigen-value Bounds (Interval Natural Frequencies)

Using the concepts of minimum and maximin characterizations of Eigen-values

for symmetric matrices, the solution to the generalized interval Eigen-value problem

for the vibration of a structure with uncertainty in the stiffness characteristics,

Eq. (2.6.18), is shown as:

For the first Eigen-value:

61  

… … 2.6.19

For the next Eigen-values:

, ,…,

, ,…,

… … 2.6.20

Substituting and expanding the right-hand side terms of Eqs. (2.6.19,2.6.20):

∑ ∑ … … 2.6.21

Since the matrix is non-negative definite, the term is non-

negative. Therefore, based on the monotonic behavior of Eigen-values for symmetric

matrices, the upper bounds on the Eigen-values in Eqs. (2.6.19, 2.6.20) are obtained

by considering maximum values of interval coefficients of uncertainty

1,1 , 1 , for all elements in the radial perturbation matrix. Similarly,

the lower bounds on the Eigen-values are obtained by considering minimum values of

those coefficients, 1 , for all elements in the radial perturbation matrix.

Also, it can be observed that any other element stiffness selected from the interval set

will yield Eigen-values between the upper and lower bounds.

Hence, the bounds on the Eigen-values of the perturbed matrix are obtained as:

m 2 2

… … 2.6.22

62  

2 2

… … 2.6.23

Therefore, the deterministic Eigen-value problems corresponding to the maximum

and minimum natural frequencies are obtained as:

… … … 2.6.24

… … … 2.6.25

This means that in the presence of any interval uncertainty in the stiffness of

structural elements, the exact upper bounds of natural frequencies are obtained by

using the upper values of stiffness for all elements in a deterministic generalized

Eigen-value problem. Similarly, the exact lower bounds of natural frequencies are

obtained by using the lower values of stiffness for all elements in another

deterministic generalized Eigen-value problem.

2.10.5 Bounding the Mode Shapes

2.10.5.1 Determination of Eigenvector Bounds (Interval Mode Shapes)

The perturbed generalized Eigen-value problem for structural dynamics, Eq.

(2.6.18) can be transformed to a perturbed classic Eigen-pair problem as:

… … 2.6.26

Hence, the symmetric perturbation matrix is:

… … 2.6.27

Substituting for radial stiffness, , Eq. (2.6.17), in Eq. (2.6.27), the error matrix

becomes:

63  

2 … … 2.6.28

Using the obtained error matrix in eigenvector perturbation equation for the first

Eigenvector, perturbed Eigen vector yield the dynamic perturbed mode shape as:

2 … 2.6.29

In which, {φ1} is the first mode shape, (ω1) is the first natural circular frequency,

[Φ2] is the matrix of remaining mode shapes and [Ω2] is the diagonal matrix of

remaining natural circular frequencies obtained from the unperturbed eigenvalue

problem.

Moreover, Eq. (2.6.29) can be written as:

… … 2.6.30

In which:

, i = 1,2, … ... ,n.

Simplifying Eq. (7.30),the interval mode shape is:

… … 2.6.31

For the other mode shapes, the same procedure can be used.

64  

2.10.6 Bounding Dynamic Response

2.10.6.1 Maximum Modal Coordinate

The interval modal coordinate, is determined using the excitation response

spectrum evaluated for the corresponding interval of natural circular frequency,

and assumed modal damping ratio (Figure 2.7).

Figure 2.7: Determination of corresponding to a for a generic response

spectrum

Having the interval modal coordinate, the maximum (upperbound) modal

coordinate Dn,max is determined as:

, … … … 2.6.32

2.10.6.2 Interval Modal Participation Factor

If excitation is proportional, the interval modal participation factor is obtained as:

… … … 2.6.33

2.10.6.3 Maximum Modal Response

The maximum modal response is determined as the maximum as the maximum of

the product of the maximum modal coordinate, the interval modal participation factor

and the interval mode shape as:

65  

, . Γ … … … 2.6.34

To achieve sharper results, functional dependency of intervals in the

multiplicative terms must be considered. Maximum modal response, Eq. (2.6.34), is

expanded using the definitions of the interval mode shapes and the interval modal

participation factor, Eqs. (2.6.31), (2.6.33) as:

,

. ∑ ∑

∑ ∑ …

∑ ∑ ∑ ∑

… 2.6.35

Thus, considering the dependency of the intervals of uncertainty for each element,

(εi), the sharper results for maximum modal response are considered.

2.10.6.4 Maximum Total Responses

Finally, the contributions of all maximum modal responses are combined to

determine the maximum total response using SRSS or other combination methods.

, … … … 2.6.36

2.10.6.5 Summary

The interval response spectrum analysis (IRSA) is summarized as following:

1. Define the uncertain physical or geometrical characteristics with closed

intervals.

• Determine the interval stiffness matrix and interval mass matrix .

• Assume modal damping ratio ζn.

2. Perform an interval Eigen-value problem between the interval stiffness and

interval mass matrices.

66  

• Determine the bounds on natural frequencies (interval natural

frequencies).

• Determine the bounds on mode shapes (interval mode shapes).

3. Compute the maximum modal response.

• Determine the interval modal coordinates and the maximum modal

coordinate Dn,max using the excitation response spectrum for the bounds of

corresponding natural circular frequency and assumed modal damping ratio.

• Determine the interval modal participation factor Γ .

• Compute the maximum modal response as the product of the maximum modal

coordinate, the interval modal participation factor and the interval mode shape.

4. Combine the contributions of all modal responses to determine the maximum

total response using SRSS or the other combination methods. 

2.11 Summary

This chapter provides the historical background of Structural dynamics and

Interval Analysis. Various literature references are cited for understanding. The basic

principles and detailed discussion about the basics of Structural Dynamics and

Uncertainty Analysis are presented. The need for Interval Response Spectrum

Analysis is discussed. A detailed procedure of interval response spectrum analysis is

given.

67  

CHAPTER - 3 PROBLEM DESCRIPTION AND METHODOLOGY

3.1 Introduction

There is lot of developments in the field of Structural Engineering for solving

uncertainty problems by various authors. Dr. Rama Rao M. V. developed a new

method called Interval Fuzzy Finite Element Method (IFFEM) for solving multiple

uncertainties for a cable stayed bridge successfully. Dr. Mehdi Modarreszadeh has

developed a new method called Interval Response Spectrum Analysis (IRSA) for

solving uncertainties in Dynamic Analysis of one-dimensional elements and two-

dimensional frames successfully. This work takes both methods into consideration

and applies IRSA method to a three-dimensional Multi-storey Frame Model with five

floors.

3.2 Building Frame Data

Table 3.1: Building Frame Data

Sl.

No. Description Information Notes

1 Building Location Shillong

2 No. of Floors Five Floors

3 Type of Building Special Moment Resisting Frame-

SMRF (RCC)

4 Length of the

Building = 4 x 5.0m = 20.0m

5 Width of the

Building = 4 x 3.0m = 12.0m

6 Height of the

Building = 5.0 x 3.5m =17.5m

7 Bay length along

X- direction 5.0m

8 Bay length along 3.0m

68  

Y- direction

9 Height of Each

Floor 3.5m

10 Beam Size (all) 0.30m X 0.60m

11 Column Size (all) 0.60m X 0.45m

12 Thickness of Slab 0.20m

13

Young’s Modulus

of Elasticity for

Concrete, 5000√fck

Interval É Interval

Young’s

Modulus of

Elasticity

EL EU

0.99x25GPa

=24.75GPa

1.01x25GPa

=25.25GPa

14 Load Data

i. Dead Load due

to Self weight of

slab

0.20 X 1 X 1 X 25 = 5kN/m2

ii. Dead Load Due

to Floor

Finishes

1.5kN/ m2

iii. Live load on

Floors

wlL wl

U Interval

Live Load (0.90x4.0kN/m2)

3.6kN/m2

(1.10x4.0kN/m2)

4. 4kN/m2

iv. Live Load on

Roof 1.5kN/m2

15 Seismic Load Data as per IS 1893-2002

a. Seismic Zone V IS 1893-

2002

b.Zone Factor, Z 0.36 (Table 2) IS 1893-

2002

c. Soil Condition Medium Stiff

IS 1893-

2002

69  

d. Building is

Supported on Isolated Foundation

IS 1893-

2002

e. Importance

Factor, I 1.0 (Table 6)

IS 1893-

2002

f. Response

Reduction

Factor, R

5.0 (SMRF) IS 1893-

2002

g. The frame will

detailed as per IS 13920 -1993

IS 1893-

2002

h. Modal Damping

coefficient, ζ

0.05 (5% for RCC Structures)

IS 1893-

2002

i. Dead Load

Considered for

Seismic

Calculation

(Lumped)

100% IS 1893-

2002

j. Live Load

Considered for

Seismic

Calculation

(Lumped)

50%

Table 8

IS 1893-

2002

70  

3.3 Model Description

A five storey symmetrical reinforced concrete structure is selected. It has a

module typical of residential buildings. The first level of the selected building is a

parking area servicing the occupants. The building is comprised of a reinforced

concrete structural frame with infill masonry walls. The columns in all selected

models are assumed fixed at the base for simplicity since the foundation influence is

not the focus of the present study. For the purpose of this presentation the uncertain

live load is taken to be (3.6kN/m2, 4. 4kN/m2), the floor finish load is taken as

1.5kN/m2 .Wind loading is not considered because it has no bearing on the intended

context. The IS1893 (Part 1): 2002 response spectrum (Figure 3.4) with 5% damping

ratio is adopted in the study. The unit weights for concrete and masonry are taken as

25kN/m3 and 20kN/m3 respectively. The uncertain elastic modulus of concrete is

taken as (0.99, 1.01) x 25GPa. The Poisson's ratio for both concrete and masonry is

taken as 0.2. The total height of the building is 17.5 meters comprised of five identical

floors. The length of the building is 20 meters while the width is 12 meters. The

general layout is kept as regular as possible in order to focus an undistracted attention

on the effect of the infill wall distribution. The numerical models are built using SAP

2000 version 14.1. The live load contribution to the seismic mass is estimated at 50%

(as per Table 8 of IS 1893(Part 1): 2002 in addition to the contribution of the full dead

load of the structure.

In constructing the various numerical models, all columns are assumed having a

square cross section of 0.60 m x 0.45 m; solid slabs are modeled as shell elements of

0.20 m thickness sitting on continuous drop beams of 0.30 m x 0.60 m section. Beams

and columns are modeled as frame elements. The frame element is a two-node (each

having six degrees of freedom) element using a three dimensional beam column

formulation which includes the effects of biaxial bending, torsion, axial deformation

and biaxial shear deformations. Slabs are modeled as shell elements. The shell

element is a four-node formulation (each having six degrees of freedom) that

combines separate membrane and plate-bending behavior. The membrane behavior

uses an isoperimetric formulation that includes translational in-plane stiffness

components and a rotational stiffness component in the direction normal to the plane

of the element. The plate bending behavior includes two-way, out-of-plane, plate

71  

rotational stiffness components as well as a translational stiffness component in the

direction normal to the plane of the element. The slab shell plates are divided into

parts of size 1/10 of the span in each direction to meet the convergence criteria. The

slab is modeled as a rigid diaphragm. The stairs are modeled as part of the building

roof or floor system. Furthermore, only elements of prime significance to structural

behavior are modeled. Since the design is not the objective of the present discussion,

un-cracked sections are specified. The construction material is assumed isotropic and

linear. Figure 2 shows the general layout plan of the building used in the study. A set

of 12 eigenvectors are requested. Appropriate meshing of all shell elements was

generated to assure solution convergence. (See Figure 3.1, 3.2 and 3.3)

Figure 3.1: 3D Model of the Structure

Figure 3.2: Plan of the Structural Model Figure 3.3: Elevation of the Structural Model

72  

Figure 3.4: Response Spectrum for the Structural Model as per IS1893 (Part1):2002

In the following study on nine different models are numerically investigated; they

vary in elastic modulus and in the live load. Both the long and the short directions are

as such included. The models are described as follows:

1. Model 1: Bare frame with diaphragm floor slab having concrete elastic modulus

of 24.75GPa, Live Load Intensity of 3.6kN/m2.

2. Model 2: Bare frame with diaphragm floor slab having concrete elastic modulus

of 24.75GPa, Live Load Intensity of 4.0kN/m2.

3. Model 3: Bare frame with diaphragm floor slab having concrete elastic modulus

of 24.75GPa, Live Load Intensity of 4.4kN/m2.

4. Model 4: Bare frame with diaphragm floor slab having concrete elastic modulus

of 25.00GPa, Live Load Intensity of 3.6kN/m2.

5. Model 5: Bare frame with diaphragm floor slab having concrete elastic modulus

of 25.00GPa, Live Load Intensity of 4.0kN/m2.

73  

6. Model 6: Bare frame with diaphragm floor slab having concrete elastic modulus

of 25.00GPa, Live Load Intensity of 4.4kN/m2.

7. Model 7: Bare frame with diaphragm floor slab having concrete elastic modulus

of 25.25GPa, Live Load Intensity of 3.6kN/m2.

8. Model 8: Bare frame with diaphragm floor slab having concrete elastic modulus

of 25.25GPa, Live Load Intensity of 4.0kN/m2.

9. Model 9: Bare frame with diaphragm floor slab having concrete elastic modulus

of 25.25GPa, Live Load Intensity of 4.4kN/m2.

3.4 Methodology

The Finite Element Modeling is performed using the software package SAP 2000.

Response spectrum method of analysis based on the modal superposition is performed

by using the design spectrum specified in IS 1893:2002 for all the models. Modal

analysis types can be chosen between Eigenvector or Ritz vector. Here Ritz vector can

provide a better basis when used for Response spectrum as Ritz vector analysis seeks

to find modes that are excited by a particular loading. For modal combination

Complete Quadratic Combination option (CQC) is selected. Square Root of Sum of

Squares option as directional combination and a scale factor of 9.806, that multiplies

each acceleration load which has units of acceleration, and should be consistent with

the length units in use is selected. It is assumed that the buildings are situated in Zone-

V of India on Medium stiff soil. Analysis is done using SAP 2000 for bare frame

model with floor/roof slabs using Response spectrum method. The slabs are modeled

as thin shell elements and analysis by Response spectrum method is done using SAP

2000. The dead load of infill walls is applied as uniformly distributed frame loads.

Dead and Live Loads are applied as per input data.

Initially the structure is modeled for response spectrum analysis with bounds on

Young’s Modulus of Elasticity using naïve interval analysis (R. L. Muhanna), then for

bounds on Live loads. The results of Eigen values are obtained from Response

Spectrum analysis as per IS 1893(Part 1): 2002. The Resulted Eigen Values are solved

for Bounds on Eigen Values using the IRSA method introduced by Prof. Mehdi

Modarreszadeh in his Doctoral Thesis submitted to Case Western Reserve University,

Cleveland, Ohio. The relative uncertainty of the structure is calculated for further

74  

discussion. The tabulated results were observed for bounds on the following

uncertainty combinations.

Case 1: Interval Young’s Modulus with Normal Live Load ( = [0.99, 1.01]

x25GPa, Live Load, wl =4.0kN/m2).

Case 2: Interval Live Load with Normal Young’s Modulus ( = [0.90, 1.10]

x4.0kN/m2, E= 25.00GPa).

Case 3: Interval Young’s Modulus with Lower Bound Live Load ( = [0.99, 1.01]

x25GPa, Live Load, wlL =3.6kN/m2).

Case 4: Interval Young’s Modulus with Upper Bound Live Load ( = [0.99, 1.01]

x25GPa, Live Load, wlU =4.4kN/m2).

Case 5: Interval Live Load with Lower Bound Young’s Modulus (w = [0.90,

1.10] x4.0kN/m2, E= 24.75GPa).

Case 6: Interval Live Load with Upper Bound Young’s Modulus (w = [0.90, 1.10]

x4.0kN/m2, E= 25.25GPa).

The above cases determine the relative uncertainty of the Eigen Values in terms of

Relative Uncertainties. The results were tabulated and discussed in the next chapter.

3.5 Summary

In this chapter the problem considered for the present study is described. The model

description and the software used are discussed. The methodology used for the work

is elaborated.

75  

CHAPTER - 4 RESULTS AND DISCUSSION

4.1 Introduction

Based on the work done and analyzing the structure by varying the Young’s

Modulus of Elasticity and Live Loads separately in nine combinations of models the

results thus obtained from Modal Analysis of the Multi-storey Frame Structure in

SAP2000 v14.1 the results were tabulated for further discussions.

The bounds on Final Eigen Values obtained from the analysis were applied as per

the IRSA method. Bounds are applied on the analysis results for Eigen Values of

various combinations of Load and Young’s Modulus uncertainties. Averaging the

Lower Bound and Upper Bound will give Central Values, deducting the Central

Values from Upper Bound will give Radial Values. The ratio of Radial values to

Central Values gives the Relative Uncertainties of Various Eigen Values.

The Relative Uncertainty gives the deviation of the solution results from the

conventional analysis with the present work. Relative uncertainties also give the idea

about the sharpness of the solution. The observations are discussed at the end of the

chapter for drawing relevant conclusion of this work. The observations were tabulated

below.

4.2 Modal Participating Mass Ratios

Table 4.1: Modal Participating Mass Ratios for Model 1

TABLE:  Modal Participating Mass RatiosOutputCaseStepType StepNum Period UX UY UZ SumUX SumUY SumUZ RX RY RZ SumRX SumRY SumRZ

Text Text Unitless Sec Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless UnitlessMODAL Mode 1 0.088951 0 0 0.70084 0 0 0.70084 5.45E‐18 2.57E‐18 0 5.45E‐18 2.57E‐18 0MODAL Mode 2 0.083176 0 0 2.26E‐19 0 0 0.70084 5.07E‐17 0.60199 0 5.61E‐17 0.60199 0MODAL Mode 3 0.079829 0 0 2.82E‐17 0 0 0.70084 0.66417 9.49E‐17 0 0.66417 0.60199 0MODAL Mode 4 0.074808 0 0 3.9E‐16 0 0 0.70084 2.94E‐17 4.76E‐15 0 0.66417 0.60199 0MODAL Mode 5 0.07405 0 0 0.06633 0 0 0.76717 3.3E‐15 1.42E‐15 0 0.66417 0.60199 0MODAL Mode 6 0.066961 0 0 8.33E‐16 0 0 0.76717 0.06086 4.5E‐13 0 0.72504 0.60199 0MODAL Mode 7 0.06669 0 0 0.05869 0 0 0.82587 1.9E‐13 1.51E‐14 0 0.72504 0.60199 0MODAL Mode 8 0.063344 0 0 4.01E‐15 0 0 0.82587 3.88E‐13 0.1002 0 0.72504 0.70218 0MODAL Mode 9 0.062751 0 0 2.79E‐15 0 0 0.82587 4.81E‐13 0.07166 0 0.72504 0.77385 0MODAL Mode 10 0.058893 0 0 2.34E‐15 0 0 0.82587 3.26E‐15 0.01398 0 0.72504 0.78782 0MODAL Mode 11 0.057761 0 0 1.21E‐13 0 0 0.82587 1.23E‐13 7.5E‐13 0 0.72504 0.78782 0MODAL Mode 12 0.057644 0 0 0.12349 0 0 0.94935 2.83E‐14 1.29E‐14 0 0.72504 0.78782 0

76  

Table 4.2: Modal Participating Mass Ratios for Model 2

Table 4.3: Modal Participating Mass Ratios for Model 3

Table 4.4: Modal Participating Mass Ratios for Model 4

Table 4.5: Modal Participating Mass Ratios for Model 5

TABLE:  Modal Participating Mass RatiosOutputCaseStepType StepNum Period UX UY UZ SumUX SumUY SumUZ RX RY RZ SumRX SumRY SumRZ

Text Text Unitless Sec Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless UnitlessMODAL Mode 1 0.089257 0 0 0.70131 0 0 0.70131 5.53E‐18 2.32E‐18 0 5.53E‐18 2.32E‐18 0MODAL Mode 2 0.083454 0 0 3.95E‐19 0 0 0.70131 4.49E‐17 0.60272 0 5.04E‐17 0.60272 0MODAL Mode 3 0.080094 0 0 2.89E‐17 0 0 0.70131 0.66456 6.61E‐17 0 0.66456 0.60272 0MODAL Mode 4 0.075051 0 0 4.4E‐16 0 0 0.70131 3.56E‐17 4.08E‐15 0 0.66456 0.60272 0MODAL Mode 5 0.074283 0 0 0.0664 0 0 0.7677 3.78E‐15 1.13E‐15 0 0.66456 0.60272 0MODAL Mode 6 0.067167 0 0 1.48E‐15 0 0 0.7677 0.06091 3.69E‐13 0 0.72547 0.60272 0MODAL Mode 7 0.066895 0 0 0.05868 0 0 0.82639 1.84E‐13 3.17E‐14 0 0.72547 0.60272 0MODAL Mode 8 0.063487 0 0 7.44E‐16 0 0 0.82639 7.7E‐14 0.09953 0 0.72547 0.70225 0MODAL Mode 9 0.06294 0 0 9.32E‐15 0 0 0.82639 4.76E‐13 0.07415 0 0.72547 0.77639 0MODAL Mode 10 0.059186 0 0 1.55E‐14 0 0 0.82639 2.56E‐16 0.01185 0 0.72547 0.78824 0MODAL Mode 11 0.057893 0 0 2.43E‐13 0 0 0.82639 3.96E‐14 5.07E‐13 0 0.72547 0.78824 0MODAL Mode 12 0.057838 0 0 0.12313 0 0 0.94951 2.08E‐14 1.4E‐13 0 0.72547 0.78824 0

TABLE:  Modal Participating Mass RatiosOutputCaseStepType StepNum Period UX UY UZ SumUX SumUY SumUZ RX RY RZ SumRX SumRY SumRZ

Text Text Unitless Sec Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless UnitlessMODAL Mode 1 0.089563 0 0 0.70176 0 0 0.70176 5.62E‐18 2.1E‐18 0 5.62E‐18 2.1E‐18 0MODAL Mode 2 0.083733 0 0 6.23E‐19 0 0 0.70176 3.93E‐17 0.60343 0 4.5E‐17 0.60343 0MODAL Mode 3 0.080359 0 0 2.94E‐17 0 0 0.70176 0.66493 4.45E‐17 0 0.66493 0.60343 0MODAL Mode 4 0.075294 0 0 4.93E‐16 0 0 0.70176 4.22E‐17 3.45E‐15 0 0.66493 0.60343 0MODAL Mode 5 0.074516 0 0 0.06645 0 0 0.76821 4.21E‐15 8.97E‐16 0 0.66493 0.60343 0MODAL Mode 6 0.067375 0 0 2.33E‐15 0 0 0.76821 0.06095 2.91E‐13 0 0.72588 0.60343 0MODAL Mode 7 0.067099 0 0 0.05867 0 0 0.82689 1.74E‐13 5.5E‐14 0 0.72588 0.60343 0MODAL Mode 8 0.063636 0 0 4.17E‐16 0 0 0.82689 7.3E‐14 0.09857 0 0.72588 0.702 0MODAL Mode 9 0.06313 0 0 5.71E‐16 0 0 0.82689 6.61E‐14 0.07681 0 0.72588 0.77881 0MODAL Mode 10 0.05948 0 0 2.32E‐14 0 0 0.82689 2.59E‐15 0.00992 0 0.72588 0.78873 0MODAL Mode 11 0.058032 0 0 3.7E‐13 0 0 0.82689 2.26E‐16 1.01E‐13 0 0.72588 0.78873 0MODAL Mode 12 0.058034 0 0 0.12278 0 0 0.94967 1.33E‐14 3.83E‐14 0 0.72588 0.78873 0

TABLE:  Modal Participating Mass RatiosOutputCaseStepType StepNum Period UX UY UZ SumUX SumUY SumUZ RX RY RZ SumRX SumRY SumRZ

Text Text Unitless Sec Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless UnitlessMODAL Mode 1 0.088505 0 0 0.70084 0 0 0.70084 5.45E‐18 2.57E‐18 0 5.45E‐18 2.57E‐18 0MODAL Mode 2 0.082759 0 0 2.26E‐19 0 0 0.70084 5.07E‐17 0.60199 0 5.61E‐17 0.60199 0MODAL Mode 3 0.079429 0 0 2.82E‐17 0 0 0.70084 0.66417 9.49E‐17 0 0.66417 0.60199 0MODAL Mode 4 0.074434 0 0 3.9E‐16 0 0 0.70084 2.94E‐17 4.76E‐15 0 0.66417 0.60199 0MODAL Mode 5 0.073679 0 0 0.06633 0 0 0.76717 3.3E‐15 1.42E‐15 0 0.66417 0.60199 0MODAL Mode 6 0.066626 0 0 8.33E‐16 0 0 0.76717 0.06086 4.5E‐13 0 0.72504 0.60199 0MODAL Mode 7 0.066356 0 0 0.05869 0 0 0.82587 1.9E‐13 1.51E‐14 0 0.72504 0.60199 0MODAL Mode 8 0.063026 0 0 4.01E‐15 0 0 0.82587 3.88E‐13 0.1002 0 0.72504 0.70218 0MODAL Mode 9 0.062437 0 0 2.79E‐15 0 0 0.82587 4.81E‐13 0.07166 0 0.72504 0.77385 0MODAL Mode 10 0.058598 0 0 2.34E‐15 0 0 0.82587 3.26E‐15 0.01398 0 0.72504 0.78782 0MODAL Mode 11 0.057471 0 0 1.21E‐13 0 0 0.82587 1.23E‐13 7.5E‐13 0 0.72504 0.78782 0MODAL Mode 12 0.057355 0 0 0.12349 0 0 0.94935 2.83E‐14 1.29E‐14 0 0.72504 0.78782 0

TABLE:  Modal Participating Mass RatiosOutputCaseStepType StepNum Period UX UY UZ SumUX SumUY SumUZ RX RY RZ SumRX SumRY SumRZ

Text Text Unitless Sec Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless UnitlessMODAL Mode 1 0.08881 0 0 0.70131 0 0 0.70131 5.53E‐18 2.32E‐18 0 5.53E‐18 2.32E‐18 0MODAL Mode 2 0.083036 0 0 3.95E‐19 0 0 0.70131 4.49E‐17 0.60272 0 5.04E‐17 0.60272 0MODAL Mode 3 0.079693 0 0 2.89E‐17 0 0 0.70131 0.66456 6.61E‐17 0 0.66456 0.60272 0MODAL Mode 4 0.074675 0 0 4.4E‐16 0 0 0.70131 3.56E‐17 4.08E‐15 0 0.66456 0.60272 0MODAL Mode 5 0.07391 0 0 0.0664 0 0 0.7677 3.78E‐15 1.13E‐15 0 0.66456 0.60272 0MODAL Mode 6 0.066831 0 0 1.48E‐15 0 0 0.7677 0.06091 3.69E‐13 0 0.72547 0.60272 0MODAL Mode 7 0.066559 0 0 0.05868 0 0 0.82639 1.84E‐13 3.17E‐14 0 0.72547 0.60272 0MODAL Mode 8 0.063169 0 0 7.44E‐16 0 0 0.82639 7.7E‐14 0.09953 0 0.72547 0.70225 0MODAL Mode 9 0.062625 0 0 9.32E‐15 0 0 0.82639 4.76E‐13 0.07415 0 0.72547 0.77639 0MODAL Mode 10 0.058889 0 0 1.55E‐14 0 0 0.82639 2.56E‐16 0.01185 0 0.72547 0.78824 0MODAL Mode 11 0.057603 0 0 2.43E‐13 0 0 0.82639 3.96E‐14 5.07E‐13 0 0.72547 0.78824 0MODAL Mode 12 0.057548 0 0 0.12313 0 0 0.94951 2.08E‐14 1.4E‐13 0 0.72547 0.78824 0

77  

Table 4.6: Modal Participating Mass Ratios for Model 6

Table 4.7: Modal Participating Mass Ratios for Model 7

Table 4.8: Modal Participating Mass Ratios for Model 8

Table 4.9: Modal Participating Mass Ratios for Model 9

TABLE:  Modal Participating Mass RatiosOutputCaseStepType StepNum Period UX UY UZ SumUX SumUY SumUZ RX RY RZ SumRX SumRY SumRZ

Text Text Unitless Sec Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless UnitlessMODAL Mode 1 0.089114 0 0 0.70176 0 0 0.70176 5.63E‐18 2.1E‐18 0 5.63E‐18 2.1E‐18 0MODAL Mode 2 0.083313 0 0 6.23E‐19 0 0 0.70176 3.93E‐17 0.60343 0 4.5E‐17 0.60343 0MODAL Mode 3 0.079956 0 0 2.94E‐17 0 0 0.70176 0.66493 4.45E‐17 0 0.66493 0.60343 0MODAL Mode 4 0.074917 0 0 4.93E‐16 0 0 0.70176 4.22E‐17 3.45E‐15 0 0.66493 0.60343 0MODAL Mode 5 0.074143 0 0 0.06645 0 0 0.76821 4.21E‐15 8.97E‐16 0 0.66493 0.60343 0MODAL Mode 6 0.067037 0 0 2.33E‐15 0 0 0.76821 0.06095 2.91E‐13 0 0.72588 0.60343 0MODAL Mode 7 0.066763 0 0 0.05867 0 0 0.82689 1.74E‐13 5.5E‐14 0 0.72588 0.60343 0MODAL Mode 8 0.063317 0 0 4.17E‐16 0 0 0.82689 7.3E‐14 0.09857 0 0.72588 0.702 0MODAL Mode 9 0.062814 0 0 5.71E‐16 0 0 0.82689 6.61E‐14 0.07681 0 0.72588 0.77881 0MODAL Mode 10 0.059182 0 0 2.32E‐14 0 0 0.82689 2.59E‐15 0.00992 0 0.72588 0.78873 0MODAL Mode 11 0.057741 0 0 3.7E‐13 0 0 0.82689 2.26E‐16 1.01E‐13 0 0.72588 0.78873 0MODAL Mode 12 0.057743 0 0 0.12278 0 0 0.94967 1.33E‐14 3.83E‐14 0 0.72588 0.78873 0

TABLE:  Modal Participating Mass RatiosOutputCaseStepType StepNum Period UX UY UZ SumUX SumUY SumUZ RX RY RZ SumRX SumRY SumRZ

Text Text Unitless Sec Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless UnitlessMODAL Mode 1 0.088066 0 0 0.70084 0 0 0.70084 5.45E‐18 2.57E‐18 0 5.45E‐18 2.57E‐18 0MODAL Mode 2 0.082348 0 0 2.26E‐19 0 0 0.70084 5.07E‐17 0.60199 0 5.61E‐17 0.60199 0MODAL Mode 3 0.079035 0 0 2.82E‐17 0 0 0.70084 0.66417 9.49E‐17 0 0.66417 0.60199 0MODAL Mode 4 0.074064 0 0 3.9E‐16 0 0 0.70084 2.94E‐17 4.76E‐15 0 0.66417 0.60199 0MODAL Mode 5 0.073314 0 0 0.06633 0 0 0.76717 3.3E‐15 1.42E‐15 0 0.66417 0.60199 0MODAL Mode 6 0.066295 0 0 8.33E‐16 0 0 0.76717 0.06086 4.5E‐13 0 0.72504 0.60199 0MODAL Mode 7 0.066027 0 0 0.05869 0 0 0.82587 1.9E‐13 1.51E‐14 0 0.72504 0.60199 0MODAL Mode 8 0.062713 0 0 4.01E‐15 0 0 0.82587 3.88E‐13 0.1002 0 0.72504 0.70218 0MODAL Mode 9 0.062127 0 0 2.79E‐15 0 0 0.82587 4.81E‐13 0.07166 0 0.72504 0.77385 0MODAL Mode 10 0.058307 0 0 2.34E‐15 0 0 0.82587 3.26E‐15 0.01398 0 0.72504 0.78782 0MODAL Mode 11 0.057186 0 0 1.21E‐13 0 0 0.82587 1.23E‐13 7.5E‐13 0 0.72504 0.78782 0

TABLE:  Modal Participating Mass RatiosOutputCaseStepType StepNum Period UX UY UZ SumUX SumUY SumUZ RX RY RZ SumRX SumRY SumRZ

Text Text Unitless Sec Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless UnitlessMODAL Mode 1 0.088369 0 0 0.70131 0 0 0.70131 5.53E‐18 2.32E‐18 0 5.53E‐18 2.32E‐18 0MODAL Mode 2 0.082624 0 0 3.95E‐19 0 0 0.70131 4.49E‐17 0.60272 0 5.04E‐17 0.60272 0MODAL Mode 3 0.079297 0 0 2.89E‐17 0 0 0.70131 0.66456 6.61E‐17 0 0.66456 0.60272 0MODAL Mode 4 0.074304 0 0 4.4E‐16 0 0 0.70131 3.56E‐17 4.08E‐15 0 0.66456 0.60272 0MODAL Mode 5 0.073544 0 0 0.0664 0 0 0.7677 3.78E‐15 1.13E‐15 0 0.66456 0.60272 0MODAL Mode 6 0.066499 0 0 1.48E‐15 0 0 0.7677 0.06091 3.69E‐13 0 0.72547 0.60272 0MODAL Mode 7 0.066229 0 0 0.05868 0 0 0.82639 1.84E‐13 3.17E‐14 0 0.72547 0.60272 0MODAL Mode 8 0.062856 0 0 7.44E‐16 0 0 0.82639 7.7E‐14 0.09953 0 0.72547 0.70225 0MODAL Mode 9 0.062314 0 0 9.32E‐15 0 0 0.82639 4.76E‐13 0.07415 0 0.72547 0.77639 0MODAL Mode 10 0.058597 0 0 1.55E‐14 0 0 0.82639 2.56E‐16 0.01185 0 0.72547 0.78824 0MODAL Mode 11 0.057317 0 0 2.43E‐13 0 0 0.82639 3.96E‐14 5.07E‐13 0 0.72547 0.78824 0MODAL Mode 12 0.057262 0 0 0.12313 0 0 0.94951 2.08E‐14 1.4E‐13 0 0.72547 0.78824 0

TABLE:  Modal Participating Mass RatiosOutputCaseStepType StepNum Period UX UY UZ SumUX SumUY SumUZ RX RY RZ SumRX SumRY SumRZ

Text Text Unitless Sec Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless Unitless UnitlessMODAL Mode 1 0.088672 0 0 0.70176 0 0 0.70176 5.62E‐18 2.1E‐18 0 5.62E‐18 2.1E‐18 0MODAL Mode 2 0.0829 0 0 6.23E‐19 0 0 0.70176 3.93E‐17 0.60343 0 4.5E‐17 0.60343 0MODAL Mode 3 0.079559 0 0 2.94E‐17 0 0 0.70176 0.66493 4.45E‐17 0 0.66493 0.60343 0MODAL Mode 4 0.074545 0 0 4.93E‐16 0 0 0.70176 4.22E‐17 3.45E‐15 0 0.66493 0.60343 0MODAL Mode 5 0.073775 0 0 0.06645 0 0 0.76821 4.21E‐15 8.97E‐16 0 0.66493 0.60343 0MODAL Mode 6 0.066704 0 0 2.33E‐15 0 0 0.76821 0.06095 2.91E‐13 0 0.72588 0.60343 0MODAL Mode 7 0.066432 0 0 0.05867 0 0 0.82689 1.74E‐13 5.5E‐14 0 0.72588 0.60343 0MODAL Mode 8 0.063003 0 0 4.17E‐16 0 0 0.82689 7.3E‐14 0.09857 0 0.72588 0.702 0MODAL Mode 9 0.062502 0 0 5.71E‐16 0 0 0.82689 6.61E‐14 0.07681 0 0.72588 0.77881 0MODAL Mode 10 0.058888 0 0 2.32E‐14 0 0 0.82689 2.59E‐15 0.00992 0 0.72588 0.78873 0MODAL Mode 11 0.057454 0 0 3.7E‐13 0 0 0.82689 2.26E‐16 1.01E‐13 0 0.72588 0.78873 0MODAL Mode 12 0.057457 0 0 0.12278 0 0 0.94967 1.33E‐14 3.83E‐14 0 0.72588 0.78873 0

78  

4.3 Modal Participation Factors

Table 4.10: Modal Participation Factors for Model 1

Table 4.11: Modal Participation Factors for Model 2

Table 4.12: Modal Participation Factors for Model 3

TABLE:  Modal Participation FactorsOutputCase StepType StepNum Period UX UY UZ RX RY RZ ModalMass ModalStiff

Text Text Unitless Sec KN‐s2 KN‐s2 KN‐s2 KN‐m‐s2 KN‐m‐s2 KN‐m‐s2 KN‐m‐s2 KN‐mMODAL Mode 1 0.088951 0 0 44.767885 ‐4.687E‐07 ‐5.289E‐07 0 1 4989.47974MODAL Mode 2 0.083176 0 0 ‐2.542E‐08 0.00000143 ‐256.19187 0 1 5706.41149MODAL Mode 3 0.079829 0 0 ‐2.84E‐07 163.703476 ‐3.216E‐06 0 1 6194.9255MODAL Mode 4 0.074808 0 0 1.056E‐06 1.089E‐06 0.000023 0 1 7054.36514MODAL Mode 5 0.07405 0 0 ‐13.772602 ‐0.000012 ‐0.000012 0 1 7199.54788MODAL Mode 6 0.066961 0 0 ‐1.543E‐06 49.555859 0.000221 0 1 8804.67853MODAL Mode 7 0.06669 0 0 ‐12.955443 ‐0.000088 0.000041 0 1 8876.38262MODAL Mode 8 0.063344 0 0 ‐3.385E‐06 ‐0.000125 104.520112 0 1 9839.04286MODAL Mode 9 0.062751 0 0 2.824E‐06 ‐0.000139 88.394182 0 1 10025.64229MODAL Mode 10 0.058893 0 0 ‐2.586E‐06 ‐0.000011 39.036992 0 1 11382.17379MODAL Mode 11 0.057761 0 0 ‐0.000019 0.00007 0.000286 0 1 11833.0132MODAL Mode 12 0.057644 0 0 18.79185 ‐0.000034 0.000037 0 1 11880.8686

TABLE:  Modal Participation FactorsOutputCase StepType StepNum Period UX UY UZ RX RY RZ ModalMass ModalStiff

Text Text Unitless Sec KN‐s2 KN‐s2 KN‐s2 KN‐m‐s2 KN‐m‐s2 KN‐m‐s2 KN‐m‐s2 KN‐mMODAL Mode 1 0.089257 0 0 44.935762 ‐4.738E‐07 ‐5.042E‐07 0 1 4955.33694MODAL Mode 2 0.083454 0 0 ‐3.374E‐08 1.349E‐06 ‐257.11415 0 1 5668.40359MODAL Mode 3 0.080094 0 0 2.885E‐07 ‐164.22734 2.693E‐06 0 1 6154.02438MODAL Mode 4 0.075051 0 0 ‐1.126E‐06 ‐1.202E‐06 ‐0.000021 0 1 7008.8072MODAL Mode 5 0.074283 0 0 ‐13.826324 ‐0.000012 ‐0.000011 0 1 7154.58322MODAL Mode 6 0.067167 0 0 ‐2.064E‐06 49.720201 0.000201 0 1 8750.71453MODAL Mode 7 0.066895 0 0 12.998731 0.000086 ‐0.000059 0 1 8822.24029MODAL Mode 8 0.063487 0 0 1.463E‐06 0.000056 ‐104.48443 0 1 9794.5651MODAL Mode 9 0.06294 0 0 ‐5.18E‐06 ‐0.000139 90.180404 0 1 9965.50947MODAL Mode 10 0.059186 0 0 ‐6.677E‐06 3.221E‐06 36.052403 0 1 11270.04239MODAL Mode 11 0.057893 0 0 ‐0.000026 0.00004 0.000236 0 1 11778.84738MODAL Mode 12 0.057838 0 0 18.828598 ‐0.000029 0.000124 0 1 11801.41148

TABLE:  Modal Participation FactorsOutputCase StepType StepNum Period UX UY UZ RX RY RZ ModalMass ModalStiff

Text Text Unitless Sec KN‐s2 KN‐s2 KN‐s2 KN‐m‐s2 KN‐m‐s2 KN‐m‐s2 KN‐m‐s2 KN‐mMODAL Mode 1 0.089563 0 0 45.102865 ‐4.791E‐07 ‐4.819E‐07 0 1 4921.57445MODAL Mode 2 0.083733 0 0 ‐4.251E‐08 1.267E‐06 ‐258.03138 0 1 5630.7587MODAL Mode 3 0.080359 0 0 2.918E‐07 ‐164.74852 2.215E‐06 0 1 6113.52588MODAL Mode 4 0.075294 0 0 1.195E‐06 1.313E‐06 0.000019 0 1 6963.61887MODAL Mode 5 0.074516 0 0 13.879088 0.000013 9.946E‐06 0 1 7109.82558MODAL Mode 6 0.067375 0 0 ‐2.599E‐06 49.880182 0.000179 0 1 8696.89594MODAL Mode 7 0.067099 0 0 ‐13.041695 ‐0.000084 0.000078 0 1 8768.49094MODAL Mode 8 0.063636 0 0 1.099E‐06 0.000055 ‐104.28791 0 1 9748.74079MODAL Mode 9 0.06313 0 0 ‐1.287E‐06 0.000052 ‐92.057569 0 1 9905.718MODAL Mode 10 0.05948 0 0 8.198E‐06 ‐0.00001 ‐33.091948 0 1 11158.87267MODAL Mode 11 0.058032 0 0 ‐0.000033 3.038E‐06 0.000106 0 1 11722.66557MODAL Mode 12 0.058034 0 0 18.865883 ‐0.000023 0.000065 0 1 11721.75012

79  

Table 4.13: Modal Participation Factors for Model 4

Table 4.14: Modal Participation Factors for Model 5

Table 4.15: Modal Participation Factors for Model 6

TABLE:  Modal Participation FactorsOutputCase StepType StepNum Period UX UY UZ RX RY RZ ModalMass ModalStiff

Text Text Unitless Sec KN‐s2 KN‐s2 KN‐s2 KN‐m‐s2 KN‐m‐s2 KN‐m‐s2 KN‐m‐s2 KN‐mMODAL Mode 1 0.088505 0 0 44.767885 ‐4.687E‐07 ‐5.289E‐07 0 1 5039.87853MODAL Mode 2 0.082759 0 0 ‐2.542E‐08 0.00000143 ‐256.19187 0 1 5764.05201MODAL Mode 3 0.079429 0 0 ‐2.84E‐07 163.703476 ‐3.216E‐06 0 1 6257.5005MODAL Mode 4 0.074434 0 0 1.056E‐06 1.089E‐06 0.000023 0 1 7125.62135MODAL Mode 5 0.073679 0 0 ‐13.772602 ‐0.000012 ‐0.000012 0 1 7272.27059MODAL Mode 6 0.066626 0 0 ‐1.543E‐06 49.555859 0.000221 0 1 8893.61468MODAL Mode 7 0.066356 0 0 ‐12.955443 ‐0.000088 0.000041 0 1 8966.04305MODAL Mode 8 0.063026 0 0 ‐3.385E‐06 ‐0.000125 104.520112 0 1 9938.42714MODAL Mode 9 0.062437 0 0 2.824E‐06 ‐0.000139 88.394182 0 1 10126.91141MODAL Mode 10 0.058598 0 0 ‐2.586E‐06 ‐0.000011 39.036992 0 1 11497.14524MODAL Mode 11 0.057471 0 0 ‐0.000019 0.00007 0.000286 0 1 11952.53859MODAL Mode 12 0.057355 0 0 18.79185 ‐0.000034 0.000037 0 1 12000.87737

TABLE:  Modal Participation FactorsOutputCase StepType StepNum Period UX UY UZ RX RY RZ ModalMass ModalStiff

Text Text Unitless Sec KN‐s2 KN‐s2 KN‐s2 KN‐m‐s2 KN‐m‐s2 KN‐m‐s2 KN‐m‐s2 KN‐mMODAL Mode 1 0.08881 0 0 44.935762 ‐4.738E‐07 ‐5.041E‐07 0 1 5005.39085MODAL Mode 2 0.083036 0 0 ‐3.374E‐08 1.349E‐06 ‐257.11415 0 1 5725.66019MODAL Mode 3 0.079693 0 0 2.885E‐07 ‐164.22734 2.693E‐06 0 1 6216.18625MODAL Mode 4 0.074675 0 0 ‐1.126E‐06 ‐1.202E‐06 ‐0.000021 0 1 7079.60323MODAL Mode 5 0.07391 0 0 ‐13.826324 ‐0.000012 ‐0.000011 0 1 7226.85174MODAL Mode 6 0.066831 0 0 ‐2.064E‐06 49.720201 0.000201 0 1 8839.10559MODAL Mode 7 0.066559 0 0 12.998731 0.000086 ‐0.000059 0 1 8911.35383MODAL Mode 8 0.063169 0 0 1.463E‐06 0.000056 ‐104.48443 0 1 9893.5001MODAL Mode 9 0.062625 0 0 ‐5.18E‐06 ‐0.000139 90.180404 0 1 10066.17118MODAL Mode 10 0.058889 0 0 ‐6.677E‐06 3.221E‐06 36.052403 0 1 11383.8812MODAL Mode 11 0.057603 0 0 ‐0.000026 0.00004 0.000236 0 1 11897.82564MODAL Mode 12 0.057548 0 0 18.828598 ‐0.000029 0.000124 0 1 11920.61766

TABLE:  Modal Participation FactorsOutputCase StepType StepNum Period UX UY UZ RX RY RZ ModalMass ModalStiff

Text Text Unitless Sec KN‐s2 KN‐s2 KN‐s2 KN‐m‐s2 KN‐m‐s2 KN‐m‐s2 KN‐m‐s2 KN‐mMODAL Mode 1 0.089114 0 0 45.102865 ‐4.792E‐07 ‐4.818E‐07 0 1 4971.28732MODAL Mode 2 0.083313 0 0 ‐4.25E‐08 1.267E‐06 ‐258.03138 0 1 5687.63505MODAL Mode 3 0.079956 0 0 2.917E‐07 ‐164.74852 2.215E‐06 0 1 6175.27866MODAL Mode 4 0.074917 0 0 1.195E‐06 1.313E‐06 0.000019 0 1 7033.95845MODAL Mode 5 0.074143 0 0 13.879088 0.000013 9.946E‐06 0 1 7181.642MODAL Mode 6 0.067037 0 0 ‐2.599E‐06 49.880182 0.000179 0 1 8784.74338MODAL Mode 7 0.066763 0 0 ‐13.041695 ‐0.000084 0.000078 0 1 8857.06155MODAL Mode 8 0.063317 0 0 1.099E‐06 0.000055 ‐104.28791 0 1 9847.21292MODAL Mode 9 0.062814 0 0 ‐1.287E‐06 0.000052 ‐92.057569 0 1 10005.77576MODAL Mode 10 0.059182 0 0 8.198E‐06 ‐0.00001 ‐33.091948 0 1 11271.58856MODAL Mode 11 0.057741 0 0 ‐0.000033 3.038E‐06 0.000106 0 1 11841.07633MODAL Mode 12 0.057743 0 0 18.865883 ‐0.000023 0.000065 0 1 11840.15163

80  

Table 4.16: Modal Participation Factors for Model 7

Table 4.17: Modal Participation Factors for Model 8

Table 4.18: Modal Participation Factors for Model 9

TABLE:  Modal Participation FactorsOutputCase StepType StepNum Period UX UY UZ RX RY RZ ModalMass ModalStiff

Text Text Unitless Sec KN‐s2 KN‐s2 KN‐s2 KN‐m‐s2 KN‐m‐s2 KN‐m‐s2 KN‐m‐s2 KN‐mMODAL Mode 1 0.088066 0 0 44.767885 ‐4.687E‐07 ‐5.288E‐07 0 1 5090.27731MODAL Mode 2 0.082348 0 0 ‐2.542E‐08 0.00000143 ‐256.19187 0 1 5821.69253MODAL Mode 3 0.079035 0 0 ‐2.84E‐07 163.703476 ‐3.216E‐06 0 1 6320.07551MODAL Mode 4 0.074064 0 0 1.056E‐06 1.089E‐06 0.000023 0 1 7196.87756MODAL Mode 5 0.073314 0 0 ‐13.772602 ‐0.000012 ‐0.000012 0 1 7344.9933MODAL Mode 6 0.066295 0 0 ‐1.543E‐06 49.555859 0.000221 0 1 8982.55083MODAL Mode 7 0.066027 0 0 ‐12.955443 ‐0.000088 0.000041 0 1 9055.70348MODAL Mode 8 0.062713 0 0 ‐3.385E‐06 ‐0.000125 104.520112 0 1 10037.81141MODAL Mode 9 0.062127 0 0 2.824E‐06 ‐0.000139 88.394182 0 1 10228.18052MODAL Mode 10 0.058307 0 0 ‐2.586E‐06 ‐0.000011 39.036992 0 1 11612.11669MODAL Mode 11 0.057186 0 0 ‐0.000019 0.00007 0.000286 0 1 12072.06398MODAL Mode 12 0.057071 0 0 18.79185 ‐0.000034 0.000037 0 1 12120.88615

TABLE:  Modal Participation FactorsOutputCase StepType StepNum Period UX UY UZ RX RY RZ ModalMass ModalStiff

Text Text Unitless Sec KN‐s2 KN‐s2 KN‐s2 KN‐m‐s2 KN‐m‐s2 KN‐m‐s2 KN‐m‐s2 KN‐mMODAL Mode 1 0.088369 0 0 44.935762 ‐4.738E‐07 ‐5.041E‐07 0 1 5055.44476MODAL Mode 2 0.082624 0 0 ‐3.373E‐08 1.349E‐06 ‐257.11415 0 1 5782.91679MODAL Mode 3 0.079297 0 0 2.885E‐07 ‐164.22734 2.693E‐06 0 1 6278.34811MODAL Mode 4 0.074304 0 0 ‐1.126E‐06 ‐1.201E‐06 ‐0.000021 0 1 7150.39926MODAL Mode 5 0.073544 0 0 ‐13.826324 ‐0.000012 ‐0.000011 0 1 7299.12026MODAL Mode 6 0.066499 0 0 ‐2.064E‐06 49.720201 0.000201 0 1 8927.49664MODAL Mode 7 0.066229 0 0 12.998731 0.000086 ‐0.000059 0 1 9000.46737MODAL Mode 8 0.062856 0 0 1.463E‐06 0.000056 ‐104.48443 0 1 9992.4351MODAL Mode 9 0.062314 0 0 ‐5.18E‐06 ‐0.000139 90.180404 0 1 10166.83289MODAL Mode 10 0.058597 0 0 ‐6.677E‐06 3.221E‐06 36.052403 0 1 11497.72001MODAL Mode 11 0.057317 0 0 ‐0.000026 0.00004 0.000236 0 1 12016.80389MODAL Mode 12 0.057262 0 0 18.828598 ‐0.000029 0.000124 0 1 12039.82383

TABLE:  Modal Participation FactorsOutputCase StepType StepNum Period UX UY UZ RX RY RZ ModalMass ModalStiff

Text Text Unitless Sec KN‐s2 KN‐s2 KN‐s2 KN‐m‐s2 KN‐m‐s2 KN‐m‐s2 KN‐m‐s2 KN‐mMODAL Mode 1 0.088672 0 0 45.102865 ‐4.792E‐07 ‐4.818E‐07 0 1 5021.00019MODAL Mode 2 0.0829 0 0 ‐4.25E‐08 1.267E‐06 ‐258.03138 0 1 5744.5114MODAL Mode 3 0.079559 0 0 2.917E‐07 ‐164.74852 2.215E‐06 0 1 6237.03145MODAL Mode 4 0.074545 0 0 1.195E‐06 1.313E‐06 0.000019 0 1 7104.29803MODAL Mode 5 0.073775 0 0 13.879088 0.000013 9.946E‐06 0 1 7253.45842MODAL Mode 6 0.066704 0 0 ‐2.599E‐06 49.880182 0.000179 0 1 8872.59081MODAL Mode 7 0.066432 0 0 ‐13.041695 ‐0.000084 0.000078 0 1 8945.63217MODAL Mode 8 0.063003 0 0 1.099E‐06 0.000055 ‐104.28791 0 1 9945.68505MODAL Mode 9 0.062502 0 0 ‐1.287E‐06 0.000052 ‐92.057569 0 1 10105.83352MODAL Mode 10 0.058888 0 0 8.198E‐06 ‐0.00001 ‐33.091948 0 1 11384.30444MODAL Mode 11 0.057454 0 0 ‐0.000033 3.038E‐06 0.000106 0 1 11959.4871MODAL Mode 12 0.057457 0 0 18.865883 ‐0.000023 0.000065 0 1 11958.55315

81  

4.4 Modal Periods and Frequencies

Table 4.19: Modal Periods and Frequencies for Model 1

TABLE: Modal Periods And Frequencies OutputCase StepType StepNum Period Frequency CircFreq Eigenvalue

Text Text Unitless Sec Cyc/sec rad/sec rad2/sec2 MODAL Mode 1 0.088951 11.242 70.636 4989.5 MODAL Mode 2 0.083176 12.023 75.541 5706.4 MODAL Mode 3 0.079829 12.527 78.708 6194.9 MODAL Mode 4 0.074808 13.367 83.99 7054.4 MODAL Mode 5 0.07405 13.504 84.85 7199.5 MODAL Mode 6 0.066961 14.934 93.833 8804.7 MODAL Mode 7 0.06669 14.995 94.215 8876.4 MODAL Mode 8 0.063344 15.787 99.192 9839 MODAL Mode 9 0.062751 15.936 100.13 10026 MODAL Mode 10 0.058893 16.98 106.69 11382 MODAL Mode 11 0.057761 17.313 108.78 11833 MODAL Mode 12 0.057644 17.348 109 11881

Table 4.20: Modal Periods and Frequencies for Model 2

TABLE: Modal Periods And Frequencies OutputCase StepType StepNum Period Frequency CircFreq Eigenvalue

Text Text Unitless Sec Cyc/sec rad/sec rad2/sec2 MODAL Mode 1 0.089257 11.204 70.394 4955.3 MODAL Mode 2 0.083454 11.983 75.289 5668.4 MODAL Mode 3 0.080094 12.485 78.448 6154 MODAL Mode 4 0.075051 13.324 83.719 7008.8 MODAL Mode 5 0.074283 13.462 84.585 7154.6 MODAL Mode 6 0.067167 14.888 93.545 8750.7 MODAL Mode 7 0.066895 14.949 93.927 8822.2 MODAL Mode 8 0.063487 15.751 98.967 9794.6 MODAL Mode 9 0.06294 15.888 99.827 9965.5 MODAL Mode 10 0.059186 16.896 106.16 11270 MODAL Mode 11 0.057893 17.273 108.53 11779 MODAL Mode 12 0.057838 17.29 108.63 11801

Table 4.21: Modal Periods and Frequencies for Model 3

TABLE: Modal Periods And Frequencies OutputCase StepType StepNum Period Frequency CircFreq Eigenvalue

Text Text Unitless Sec Cyc/sec rad/sec rad2/sec2 MODAL Mode 1 0.089563 11.165 70.154 4921.6 MODAL Mode 2 0.083733 11.943 75.038 5630.8 MODAL Mode 3 0.080359 12.444 78.189 6113.5 MODAL Mode 4 0.075294 13.281 83.448 6963.6 MODAL Mode 5 0.074516 13.42 84.32 7109.8 MODAL Mode 6 0.067375 14.842 93.257 8696.9 MODAL Mode 7 0.067099 14.903 93.64 8768.5 MODAL Mode 8 0.063636 15.714 98.736 9748.7 MODAL Mode 9 0.06313 15.84 99.527 9905.7 MODAL Mode 10 0.05948 16.812 105.64 11159 MODAL Mode 11 0.058032 17.232 108.27 11723 MODAL Mode 12 0.058034 17.231 108.27 11722

82  

Table 4.22: Modal Periods and Frequencies for Model 4

TABLE: Modal Periods And Frequencies OutputCase StepType StepNum Period Frequency CircFreq Eigenvalue

Text Text Unitless Sec Cyc/sec rad/sec rad2/sec2 MODAL Mode 1 0.088505 11.299 70.992 5039.9 MODAL Mode 2 0.082759 12.083 75.921 5764.1 MODAL Mode 3 0.079429 12.59 79.104 6257.5 MODAL Mode 4 0.074434 13.435 84.413 7125.6 MODAL Mode 5 0.073679 13.572 85.278 7272.3 MODAL Mode 6 0.066626 15.009 94.306 8893.6 MODAL Mode 7 0.066356 15.07 94.689 8966 MODAL Mode 8 0.063026 15.866 99.692 9938.4 MODAL Mode 9 0.062437 16.016 100.63 10127 MODAL Mode 10 0.058598 17.065 107.22 11497 MODAL Mode 11 0.057471 17.4 109.33 11953 MODAL Mode 12 0.057355 17.435 109.55 12001

Table 4.23: Modal Periods and Frequencies for Model 5

TABLE: Modal Periods And Frequencies OutputCase StepType StepNum Period Frequency CircFreq Eigenvalue

Text Text Unitless Sec Cyc/sec rad/sec rad2/sec2 MODAL Mode 1 0.08881 11.26 70.749 5005.4 MODAL Mode 2 0.083036 12.043 75.668 5725.7 MODAL Mode 3 0.079693 12.548 78.843 6216.2 MODAL Mode 4 0.074675 13.391 84.14 7079.6 MODAL Mode 5 0.07391 13.53 85.011 7226.9 MODAL Mode 6 0.066831 14.963 94.017 8839.1 MODAL Mode 7 0.066559 15.024 94.4 8911.4 MODAL Mode 8 0.063169 15.831 99.466 9893.5 MODAL Mode 9 0.062625 15.968 100.33 10066 MODAL Mode 10 0.058889 16.981 106.7 11384 MODAL Mode 11 0.057603 17.36 109.08 11898 MODAL Mode 12 0.057548 17.377 109.18 11921

Table 4.24: Modal Periods and Frequencies for Model 6

TABLE: Modal Periods And Frequencies OutputCase StepType StepNum Period Frequency CircFreq Eigenvalue

Text Text Unitless Sec Cyc/sec rad/sec rad2/sec2 MODAL Mode 1 0.089114 11.222 70.507 4971.3 MODAL Mode 2 0.083313 12.003 75.416 5687.6 MODAL Mode 3 0.079956 12.507 78.583 6175.3 MODAL Mode 4 0.074917 13.348 83.869 7034 MODAL Mode 5 0.074143 13.488 84.745 7181.6 MODAL Mode 6 0.067037 14.917 93.727 8784.7 MODAL Mode 7 0.066763 14.978 94.112 8857.1 MODAL Mode 8 0.063317 15.793 99.233 9847.2 MODAL Mode 9 0.062814 15.92 100.03 10006 MODAL Mode 10 0.059182 16.897 106.17 11272 MODAL Mode 11 0.057741 17.319 108.82 11841 MODAL Mode 12 0.057743 17.318 108.81 11840

83  

Table 4.25: Modal Periods and Frequencies for Model 7

TABLE: Modal Periods And Frequencies OutputCase StepType StepNum Period Frequency CircFreq Eigenvalue

Text Text Unitless Sec Cyc/sec rad/sec rad2/sec2 MODAL Mode 1 0.088066 11.355 71.346 5090.3 MODAL Mode 2 0.082348 12.144 76.3 5821.7 MODAL Mode 3 0.079035 12.653 79.499 6320.1 MODAL Mode 4 0.074064 13.502 84.834 7196.9 MODAL Mode 5 0.073314 13.64 85.703 7345 MODAL Mode 6 0.066295 15.084 94.776 8982.6 MODAL Mode 7 0.066027 15.145 95.161 9055.7 MODAL Mode 8 0.062713 15.946 100.19 10038 MODAL Mode 9 0.062127 16.096 101.13 10228 MODAL Mode 10 0.058307 17.15 107.76 11612 MODAL Mode 11 0.057186 17.487 109.87 12072 MODAL Mode 12 0.057071 17.522 110.09 12121

Table 4.26: Modal Periods and Frequencies for Model 8

TABLE: Modal Periods And Frequencies OutputCase StepType StepNum Period Frequency CircFreq Eigenvalue

Text Text Unitless Sec Cyc/sec rad/sec rad2/sec2 MODAL Mode 1 0.088369 11.316 71.102 5055.4 MODAL Mode 2 0.082624 12.103 76.045 5782.9 MODAL Mode 3 0.079297 12.611 79.236 6278.3 MODAL Mode 4 0.074304 13.458 84.56 7150.4 MODAL Mode 5 0.073544 13.597 85.435 7299.1 MODAL Mode 6 0.066499 15.038 94.485 8927.5 MODAL Mode 7 0.066229 15.099 94.871 9000.5 MODAL Mode 8 0.062856 15.909 99.962 9992.4 MODAL Mode 9 0.062314 16.048 100.83 10167 MODAL Mode 10 0.058597 17.066 107.23 11498 MODAL Mode 11 0.057317 17.447 109.62 12017 MODAL Mode 12 0.057262 17.463 109.73 12040

Table 4.27: Modal Periods and Frequencies for Model 9

TABLE: Modal Periods And Frequencies OutputCase StepType StepNum Period Frequency CircFreq Eigenvalue

Text Text Unitless Sec Cyc/sec rad/sec rad2/sec2 MODAL Mode 1 0.088672 11.278 70.859 5021 MODAL Mode 2 0.0829 12.063 75.793 5744.5 MODAL Mode 3 0.079559 12.569 78.975 6237 MODAL Mode 4 0.074545 13.415 84.287 7104.3 MODAL Mode 5 0.073775 13.555 85.167 7253.5 MODAL Mode 6 0.066704 14.992 94.194 8872.6 MODAL Mode 7 0.066432 15.053 94.581 8945.6 MODAL Mode 8 0.063003 15.872 99.728 9945.7 MODAL Mode 9 0.062502 15.999 100.53 10106 MODAL Mode 10 0.058888 16.981 106.7 11384 MODAL Mode 11 0.057454 17.405 109.36 11959 MODAL Mode 12 0.057457 17.404 109.36 11959

84  

4.5 Application of IRSA Method to the Output Results

Case 1: Interval Young’s Modulus with Normal Live Load (E = [0.99, 1.01] x25GPa,

Live Load, wl =4.0kN/m2).

Table 4.28: Bounds on Eigen Values for Case 1

Mode Lower Bound

Upper Bound

Central Values

Radial Values

Relative Uncertainty(RU)

Mid Values

(L) rad2/sec2 (U) rad2/sec2 C = (L+U)/2 R = U-C R/C 1 4955.30 5055.40 5005.35 50.05 0.009999 5005.402 5668.40 5782.90 5725.65 57.25 0.009999 5725.703 6154.00 6278.30 6216.15 62.15 0.009998 6216.204 7008.80 7150.40 7079.60 70.80 0.010001 7079.605 7154.60 7299.10 7226.85 72.25 0.009997 7226.906 8750.70 8927.50 8839.10 88.40 0.010001 8839.107 8822.20 9000.50 8911.35 89.15 0.010004 8911.408 9794.60 9992.40 9893.50 98.90 0.009996 9893.509 9965.50 10167.00 10066.25 100.75 0.010009 10066.00

10 11270.00 11498.00 11384.00 114.00 0.010014 11384.0011 11779.00 12017.00 11898.00 119.00 0.010002 11898.0012 11801.00 12040.00 11920.50 119.50 0.010025 11921.00

Case 2: Interval Live Load with Normal Young’s Modulus (w = [0.90, 1.10]

x4.0kN/m2, E= 25.00GPa).

Table 4.29: Bounds on Eigen Values for Case 2

Mode Lower Bound

Upper Bound

Central Values

Radial Values

Relative Uncertainty (RU)

Mid Values

(L) rad2/sec2 (U) rad2/sec2 C = (L+U)/2 R = U-C R/C 1 5039.90 4971.30 5005.60 -34.30 -0.006852 5005.402 5764.10 5687.60 5725.85 -38.25 -0.006680 5725.703 6257.50 6175.30 6216.40 -41.10 -0.006612 6216.204 7125.60 7034.00 7079.80 -45.80 -0.006469 7079.605 7272.30 7181.60 7226.95 -45.35 -0.006275 7226.906 8893.60 8784.70 8839.15 -54.45 -0.006160 8839.107 8966.00 8857.10 8911.55 -54.45 -0.006110 8911.408 9938.40 9847.20 9892.80 -45.60 -0.004609 9893.509 10127.00 10006.00 10066.50 -60.50 -0.006010 10066.00

10 11497.00 11272.00 11384.50 -112.50 -0.009882 11384.0011 11953.00 11841.00 11897.00 -56.00 -0.004707 11898.0012 12001.00 11840.00 11920.50 -80.50 -0.006753 11921.00

85  

Case 3: Interval Young’s Modulus with Lower Bound Live Load (E = [0.99, 1.01]

x25GPa, Live Load, wlL =3.6kN/m2).

Table 4.30: Bounds on Eigen Values for Case 3

Mode Lower Bound

Upper Bound

Central Values

Radial Values

Relative Uncertainty (RU)

Mid Values

(L) rad2/sec2 (U) rad2/sec2 C = (L+U)/2 R = U-C R/C 1 4989.50 5090.30 5039.90 50.40 0.010000 5039.902 5706.40 5821.70 5764.05 57.65 0.010002 5764.103 6194.90 6320.10 6257.50 62.60 0.010004 6257.504 7054.40 7196.90 7125.65 71.25 0.009999 7125.605 7199.50 7345.00 7272.25 72.75 0.010004 7272.306 8804.70 8982.60 8893.65 88.95 0.010002 8893.607 8876.40 9055.70 8966.05 89.65 0.009999 8966.008 9839.00 10038.00 9938.50 99.50 0.010012 9938.409 10026.00 10228.00 10127.00 101.00 0.009973 10127.00

10 11382.00 11612.00 11497.00 115.00 0.010003 11497.0011 11833.00 12072.00 11952.50 119.50 0.009998 11953.0012 11881.00 12121.00 12001.00 120.00 0.009999 12001.00

Case 4: Interval Young’s Modulus with Upper Bound Live Load (E = [0.99, 1.01]

x25GPa, Live Load, wlU =4.4kN/m2).

Table 4.31: Bounds on Eigen Values for Case 4

Mode Lower Bound

Upper Bound

Central Values

Radial Values

Relative Uncertainty (RU)

Mid Values

(L) rad2/sec2 (U) rad2/sec2 C = (L+U)/2 R = U-C R/C 1 4921.60 5021.00 4971.30 49.70 0.009997 4971.302 5630.80 5744.50 5687.65 56.85 0.009995 5687.603 6113.50 6237.00 6175.25 61.75 0.010000 6175.304 6963.60 7104.30 7033.95 70.35 0.010001 7034.005 7109.80 7253.50 7181.65 71.85 0.010005 7181.606 8696.90 8872.60 8784.75 87.85 0.010000 8784.707 8768.50 8945.60 8857.05 88.55 0.009998 8857.108 9748.70 9945.70 9847.20 98.50 0.010003 9847.209 9905.70 10106.00 10005.85 100.15 0.010009 10006.00

10 11159.00 11384.00 11271.50 112.50 0.009981 11272.0011 11723.00 11959.00 11841.00 118.00 0.009965 11841.0012 11722.00 11959.00 11840.50 118.50 0.010008 11840.00

86  

Case 5: Interval Live Load with Lower Bound Young’s Modulus (w = [0.90, 1.10]

x4.0kN/m2, EL= 24.75GPa).

Table 4.32: Bounds on Eigen Values for Case 5

Mode Lower Bound

Upper Bound

Central Values

Radial Values

Relative Uncertainty (RU)

Mid Values

(L) rad2/sec2 (U) rad2/sec2 C = (L+U)/2 R = U-C R/C 1 4989.50 4921.60 4955.55 -33.95 -0.006851 4955.302 5706.40 5630.80 5668.60 -37.80 -0.006668 5668.403 6194.90 6113.50 6154.20 -40.70 -0.006613 6154.004 7054.40 6963.60 7009.00 -45.40 -0.006477 7008.805 7199.50 7109.80 7154.65 -44.85 -0.006269 7154.606 8804.70 8696.90 8750.80 -53.90 -0.006159 8750.707 8876.40 8768.50 8822.45 -53.95 -0.006115 8822.208 9839.00 9748.70 9793.85 -45.15 -0.004610 9794.609 10026.00 9905.70 9965.85 -60.15 -0.006036 9965.50

10 11382.00 11159.00 11270.50 -111.50 -0.009893 11270.0011 11833.00 11723.00 11778.00 -55.00 -0.004670 11779.0012 11881.00 11722.00 11801.50 -79.50 -0.006736 11801.00

Case 6: Interval Live Load with Upper Bound Young’s Modulus (w = [0.90, 1.10]

x4.0kN/m2, EU= 25.25GPa).

Table 4.33: Bounds on Eigen Values for Case 6

Mode Lower Bound

Upper Bound

Central Values

Radial Values

Relative Uncertainty (RU)

Mid Values

(L) rad2/sec2 (U) rad2/sec2 C = (L+U)/2 R = U-C R/C 1 5090.30 5021.00 5055.65 -34.65 -0.006854 5055.402 5821.70 5744.50 5783.10 -38.60 -0.006675 5782.903 6320.10 6237.00 6278.55 -41.55 -0.006618 6278.304 7196.90 7104.30 7150.60 -46.30 -0.006475 7150.405 7345.00 7253.50 7299.25 -45.75 -0.006268 7299.106 8982.60 8872.60 8927.60 -55.00 -0.006161 8927.507 9055.70 8945.60 9000.65 -55.05 -0.006116 9000.508 10038.00 9945.70 9991.85 -46.15 -0.004619 9992.409 10228.00 10106.00 10167.00 -61.00 -0.006000 10167.00

10 11612.00 11384.00 11498.00 -114.00 -0.009915 11498.0011 12072.00 11959.00 12015.50 -56.50 -0.004702 12017.0012 12121.00 11959.00 12040.00 -81.00 -0.006728 12040.00

87  

Figure 4.1: Variation of RU in Eigen Values with Mode No. for Interval Young’s Modulus and Normal Live Load for Case 1

Figure 4.2: Variation of RU in Eigen Values with Mode No. for Interval Live Load with Normal Young’s Modulus Case 2

0.009995

0.010000

0.010005

0.010010

0.010015

0.010020

0.010025

0.010030

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Rel

ativ

e U

ncer

tain

ty (R

U)

Mode No.

Interval Young's Modulus

‐0.012

‐0.010

‐0.008

‐0.006

‐0.004

‐0.002

0.000

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Rel

ativ

e U

ncer

tain

ty (R

U)

Mode No.

Interval Live Load

88  

Figure 4.3: Variation of RU in Eigen Values with Mode No. for Interval Young’s Modulus and Lower Bound Live Load for Case 3

Figure 4.4: Variation of RU in Eigen Values with Mode No. for Interval Young’s Modulus and Upper Bound Live Load for Case 4

0.009970

0.009975

0.009980

0.009985

0.009990

0.009995

0.010000

0.010005

0.010010

0.010015

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Rel

ativ

e U

ncer

tain

ty (R

U)

Mode No.

Interval Young's Modulus

0.0099600.0099650.0099700.0099750.0099800.0099850.0099900.0099950.0100000.0100050.0100100.010015

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Rel

ativ

e un

cert

aint

y (R

U)

Mode No.

Interval Young's Modulus

89  

Figure 4.5: Variation of RU in Eigen Values with Mode No. for Interval Live Load with Lower Bound Young’s Modulus for Case 5

Figure 4.6: Variation of RU in Eigen Values with Mode No. for Interval Live Load with Upper Bound Young’s Modulus for Case 6

‐0.012000

‐0.010000

‐0.008000

‐0.006000

‐0.004000

‐0.002000

0.000000

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Rel

ativ

e U

ncer

tain

ty (R

U)

Mode No.

Interval Live Load

‐0.012000

‐0.010000

‐0.008000

‐0.006000

‐0.004000

‐0.002000

0.000000

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Rel

ativ

e U

ncer

tain

ty (R

U)

Mode No.

Interval Live Load

90  

Figure 4.7: Variation of RU in Eigen Values with Mode No. for Interval Young’s Modulus with Various Live Loads (Summary of Figures 4.1, 4.3 & 4.4)

Figure 4.8: Variation of RU in Eigen Values with Mode No. for Interval Live Load and Various Young’s Modulus Values (Summary of Figures 4.2, 4.5 & 4.6)

0.009960

0.009970

0.009980

0.009990

0.010000

0.010010

0.010020

0.010030

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Rel

ativ

e U

ncer

tain

ty (R

U)

Mode No.

Lower Bound Live Load

Normal Live Load

Upper Bound Live Load

‐0.0100

‐0.0090

‐0.0080

‐0.0070

‐0.0060

‐0.0050

‐0.0040

1 2 3 4 5 6 7 8 9 10 11 12

Rel

ativ

e U

ncer

tain

ty (R

U)

Mode No.

Lower Bound Young's Modulus

Normal Young's Modulus

Upper Bound Young's Modulus

91  

Figure 4.9: Variation of RU in Eigen Values with Variation in Live Load for Interval Young’s Modulus ( = [0.99, 1.01] x25GPa)

Figure 4.10: Variation of RU in Eigen Values with Variation in Young’s Modulus for Interval Live Load ( = [0.90, 1.10] x4.0kN/m2)

0.00996

0.00997

0.00998

0.00999

0.01

0.01001

0.01002

0.01003

3.00 3.50 4.00 4.50 5.00

Rel

ativ

e U

ncer

tain

ty (R

U)

Live Load Intensity, wl (kN/m2)

Mode1

Mode2

Mode 3

Mode 4

Mode 5

Mode 6

Mode 7

Mode 8

Mode 9

Mode 10

Mode 11

Mode 12

‐0.01100

‐0.01000

‐0.00900

‐0.00800

‐0.00700

‐0.00600

‐0.00500

‐0.00400

24.70 24.80 24.90 25.00 25.10 25.20 25.30

Rel

ativ

e U

ncer

tain

ty (R

U)

Young's Modulus, E (GPa)

Mode1

Mode2

Mode 3

Mode 4

Mode 5

Mode 6

Mode 7

Mode 8

Mode 9

Mode 10

Mode 11

Mode 12

92  

4.6 Discussion on observations

From the results it is observed that

• In Figures 4.1, 4.3 and 4.4 the trend line for Relative Uncertainty (RU) is on

positive side of y-axis. This is due to increase in Eigen values in the modal

analysis from Lower bound Young’s Modulus to Upper Bound Young’s Modulus

increasing the effect of stiffness matrix of the structure on Eigen Solution.

• In Figures 4.2, 4.5 and 4.6 the trend line for Relative Uncertainty (RU) is on

negative side of y-axis. This is due to reduction in Eigen values in the modal

analysis from Lower bound Live Load to Upper Bound Live Load by decreasing

the effect of stiffness matrix due to increase in Live load on the Eigen Solution.

• It is observed from all Figures that the RU is varying abruptly with increase in the

mode number which represents that there is drastic change of RU due to variation

in participation of modal mass with increase in number of modes.

• In Figure 4.9 and 4.10, it can be observed that the variations of RU in mode

numbers 3 to 6 are close and nearly same. This is due to effective utilization of

participation of modal mass in these modes.

• It is observed in the Figure 4.9 the variations of RU in case of Interval Young’s

Modulus are parabolic and this is due to variation in interval stiffness’s and

variation in participation of modal masses in the modal analysis.

• It is also observed that the variation of RU in case of Interval Live Load is not

much affected by the modal mass participation as Figure 4.10 is clearly indicating

nearly same variation and the lines are straight.

• There is an abrupt change of RU when we go to higher number of modes and this

may be due to complexity involved in the modal analysis as the number of

iterations performed by the computer increases relatively.

• The trend lines for all Figures 4.1 to 4.6 are non linear (variation of Relative

Uncertainties (RU)). Whereas the representation is triangular member ship values

in FFEM proposed by Dr. M. V. Rama Rao and Dr. Ramesh Reddy R. for analysis

of a cable-stayed bridge.

• Sensitivity analysis is not needed as the RU itself is a measure of the sharpness of

the results. Where as in FFEM (Rama Rao M. V. and Ramesh Reddy R.) a

sensitivity analysis is needed to check the sharpness of the membership values and

93  

is not needed due to the use of perturbed Eigen vectors in the final equations of

Mr. Mehdi Modarreszadeh.

• In the presence of any interval uncertainty in the characteristics of structural

elements, the proposed method of interval response spectrum analysis (IRSA) is

capable to obtain the nearly sharp bounds on the structure’s dynamic response.

• The Interval Response Spectrum Analysis can be applied for Three-Dimensional

analyses also with much accuracy in the analysis results for uncertain parameters

in the analysis model.

4.7 Summary

In this chapter the results of the Modal Analysis are tabulated and the Interval

Response Spectrum Analysis (IRSA) is applied using perturbation principles of IRSA.

The bounds on Eigen Values are solved for Relative Uncertainties RU’s) and the

observation were presented.

94  

CHAPTER - 5 CONCLUSIONS AND FUTURE SCOPE

5.1 Conclusions

• The Interval Response Spectrum Analysis (IRSA) Method is the latest

development in Dynamic Analysis of Structures with Uncertainties, and is

successfully applied to the Three-Dimensional Multi-Storey Building Structure for

finding Relative Uncertainty in Eigen Value Bounds.

• The interval uncertainty of ±1% in Young’s modulus resulted in 0.009 to 0.01

Relative Uncertainties (RUs) in the Modal Eigen Value results and is positive.

The interval uncertainty of ±10% in Floor Live Load resulted in -0.0099 to -0.004

relative uncertainties (RUs) in the Modal Eigen Value results and is negative.

• The variation of Relative Uncertainty (RU) is on positive side of y-axis for

Interval Young’s Modulus’s. This is due to increase in Eigen values in the modal

analysis from Lower bound Young’s Modulus to Upper Bound Young’s Modulus

increasing the effect of stiffness matrix of the structure on Eigen Solution.

• The variation of Relative Uncertainty (RU) is on negative side of y-axis for

Interval Live Loads. This is due to reduction in Eigen values in the modal analysis

from Lower bound Live Load to Upper Bound Live Load by decreasing the effect

of stiffness matrix due to increase in Live load on Eigen Solution.

• It is observed from all Figures that the RU is varying abruptly with increase in the

mode number which represents that there is drastic change of RU due to variation

in participation of modal mass with increase in number of modes.

• For both cases of Interval Young’s Modulus and Interval Live Load, it can be

observed that the variations of RU in mode numbers 3 to 6 are close and nearly

same. This is due to effective utilization of participation of modal mass in these

modes.

• The variations of RU in case of Interval Young’s Modulus are parabolic and this

is due to variation in interval stiffness’s and variation in participation of modal

masses in the modal analysis.

95  

• The variation of RU in case of Interval Live Load is not much affected by the

modal mass participation as there is a clear indication that there is nearly same

variation and the lines are straight.

• There is an abrupt change of RU when we go to higher number of modes and this

may be due to complexity involved in the modal analysis as the number of

iterations performed by the computer increases relatively.

5.2 Future Scope of Work

• The IRSA method can be further extended to the individual elements of the

structural model by extending the programming capabilities of the Software.

• The IRSA method can be extended further to the uncertainty problems for

dynamic analysis of shell elements to reduce the error in analysis.

• As the IRSA method is proved successful in determining linear dynamic response

of the Three-Dimensional structures due to seismic loading there is more scope for

future expansion of the method to non-linear analysis.

96  

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APPENDIX - A SOFTWARE MODEL ANALYSIS MODE SHAPE OUTPUT

Mode Shapes for Model 1(E=24.75GPa, wl = 3.6kN/m2):

Figure A.1.1: 3D Mode Shape 1for Structure Model 1 Figure A.1.2: 3D Mode Shape 2for Structure Model 1

Figure A.1.3: 3D Mode Shape 3for Structure Model 1 Figure A.1.4: 3D Mode Shape 4 for Structure Model 1

Figure A.1.5: 3D Mode Shape 5 for Structure Model 1 Figure A.1.6: 3D Mode Shape 6 for Structure Model 1

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Figure A.1.7: 3D Mode Shape 7 for Structure Model 1 Figure A.1.8: 3D Mode Shape 8 for Structure Model 1

Figure A.1.9: 3D Mode Shape 9 for Structure Model 1 Figure A.1.10: 3D Mode Shape 10 for Structure Model 1

Figure A.1.11: 3D Mode Shape 11 for Structure Model 1 Figure A.1.12: 3D Mode Shape 12 for Structure Model 1

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Mode Shapes for Model 2(E=24.75GPa, wl = 4.0kN/m2):

Figure A.2.1: 3D Mode Shape 1 for Structure Model 2 Figure A.2.2: 3D Mode Shape 2 for Structure Model 2

Figure A.2.3: 3D Mode Shape 3 for Structure Model 2 Figure A.2.4: 3D Mode Shape 4 for Structure Model 2

Figure A.2.5: 3D Mode Shape 5 for Structure Model 2 Figure A.2.6: 3D Mode Shape 6 for Structure Model 2

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Figure A.2.7: 3D Mode Shape 7 for Structure Model 2 Figure A.2.8: 3D Mode Shape 8 for Structure Model 2

Figure A.2.9: 3D Mode Shape 9 for Structure Model 2 Figure A.2.10: 3D Mode Shape 10 for Structure Model 2

Figure A.2.11: 3D Mode Shape 11 for Structure Model 2 Figure A.2.12: 3D Mode Shape 12 for Structure Model 2

109  

Mode Shapes for Model 3(E=24.75GPa, wl = 4.4kN/m2):

Figure A.3.1: 3D Mode Shape 1 for Structure Model 3 Figure A.3.2: 3D Mode Shape 2 for Structure Model 3

Figure A.3.3: 3D Mode Shape 3 for Structure Model 3 Figure A.3.4: 3D Mode Shape 4 for Structure Model 3

Figure A.3.5: 3D Mode Shape 5 for Structure Model 3 Figure A.3.6: 3D Mode Shape 6 for Structure Model 3

110  

Figure A.3.7: 3D Mode Shape 7 for Structure Model 3 Figure A.3.8: 3D Mode Shape 8 for Structure Model 3

Figure A.3.9: 3D Mode Shape 9 for Structure Model 3 Figure A.3.10: 3D Mode Shape 10 for Structure Model 3

Figure A.3.11: 3D Mode Shape 11 for Structure Model 3 Figure A.3.12: 3D Mode Shape 12 for Structure Model 3

111  

Mode Shapes for Model 4(E=25.00GPa, wl = 3.6kN/m2):

Figure A.4.1: 3D Mode Shape 1 for Structure Model 4 Figure A.4.2: 3D Mode Shape 2 for Structure Model 4

Figure A.4.3: 3D Mode Shape 3 for Structure Model 4 Figure A.4.4: 3D Mode Shape 4 for Structure Model 4

Figure A.4.5: 3D Mode Shape 5 for Structure Model 4 Figure A.4.6: 3D Mode Shape 6 for Structure Model 4

112  

Figure A.4.7: 3D Mode Shape 7 for Structure Model 4 Figure A.4.8: 3D Mode Shape 8 for Structure Model 4

Figure A.4.9: 3D Mode Shape 9 for Structure Model 4 Figure A.4.10: 3D Mode Shape 10 for Structure Model 4

Figure A.4.11: 3D Mode Shape 11 for Structure Model 4 Figure A.4.12: 3D Mode Shape 12 for Structure Model 4

113  

Mode Shapes for Model 5(E=25.00GPa, wl = 4.0kN/m2):

Figure A.5.1: 3D Mode Shape 1 for Structure Model 5 Figure A.5.2: 3D Mode Shape 2 for Structure Model 5

Figure A.5.3: 3D Mode Shape 3 for Structure Model 5 Figure A.5.4: 3D Mode Shape 4 for Structure Model 5

Figure A.5.5: 3D Mode Shape 5 for Structure Model 5 Figure A.5.6: 3D Mode Shape 6 for Structure Model 5

114  

Figure A.5.7: 3D Mode Shape 7 for Structure Model 5 Figure A.5.8: 3D Mode Shape 8 for Structure Model 5

Figure A.5.9: 3D Mode Shape 9 for Structure Model 5 Figure A.5.10: 3D Mode Shape 10 for Structure Model 5

Figure A.5.11: 3D Mode Shape 11 for Structure Model 5 Figure A.5.12: 3D Mode Shape 12 for Structure Model 5

115  

Mode Shapes for Model 6(E=25.00GPa, wl = 4.4kN/m2):

Figure A.6.1: 3D Mode Shape 1 for Structure Model 6 Figure A.6.2: 3D Mode Shape 2 for Structure Model 6

Figure A.6.3: 3D Mode Shape 3 for Structure Model 6 Figure A.6.4: 3D Mode Shape 4 for Structure Model 6

Figure A.6.5: 3D Mode Shape 5 for Structure Model 6 Figure A.6.6: 3D Mode Shape 6 for Structure Model 6

116  

Figure A.6.7: 3D Mode Shape 7 for Structure Model 6 Figure A.6.8: 3D Mode Shape 8 for Structure Model 6

Figure A.6.9: 3D Mode Shape 9 for Structure Model 6 Figure A.6.10: 3D Mode Shape 10 for Structure Model 6

Figure A.6.11: 3D Mode Shape 11 for Structure Model 6 Figure A.6.12: 3D Mode Shape 12 for Structure Model 6

117  

Mode Shapes for Model 7(E=25.25GPa, wl = 3.6kN/m2):

Figure A.7.1: 3D Mode Shape 1 for Structure Model 7 Figure A.7.2: 3D Mode Shape 2 for Structure Model 7

Figure A.7.3: 3D Mode Shape 3 for Structure Model 7 Figure A.7.4: 3D Mode Shape 4 for Structure Model 7

Figure A.7.5: 3D Mode Shape 5 for Structure Model 7 Figure A.7.6: 3D Mode Shape 6 for Structure Model 7

118  

Figure A.7.7: 3D Mode Shape 7 for Structure Model 7 Figure A.7.8: 3D Mode Shape 8 for Structure Model 7

Figure A.7.9: 3D Mode Shape 9 for Structure Model 7 Figure A.7.10: 3D Mode Shape 10 for Structure Model 7

Figure A.7.11: 3D Mode Shape 11 for Structure Model 7 Figure A.7.12: 3D Mode Shape 12 for Structure Model 7

119  

Mode Shapes for Model 8(E=25.25GPa, wl = 4.0kN/m2):

Figure A.8.1: 3D Mode Shape 1 for Structure Model 8 Figure A.8.2: 3D Mode Shape 2 for Structure Model 8

Figure A.8.3: 3D Mode Shape 3 for Structure Model 8 Figure A.8.4: 3D Mode Shape 4 for Structure Model 8

Figure A.8.5: 3D Mode Shape 5 for Structure Model 8 Figure A.8.6: 3D Mode Shape 6 for Structure Model 8

120  

Figure A.8.7: 3D Mode Shape 7 for Structure Model 8 Figure A.8.8: 3D Mode Shape 8 for Structure Model 8

Figure A.8.9: 3D Mode Shape 9 for Structure Model 8 Figure A.8.10: 3D Mode Shape 10 for Structure Model 8

Figure A.8.11: 3D Mode Shape 11 for Structure Model 8 Figure A.8.12: 3D Mode Shape 12 for Structure Model 8

121  

Mode Shapes for Model 9(E=25.25GPa, wl = 4.4kN/m2):

Figure A.9.1: 3D Mode Shape 1 for Structure Model 9 Figure A.9.2: 3D Mode Shape 2 for Structure Model 9

Figure A.9.3: 3D Mode Shape 3 for Structure Model 9 Figure A.9.4: 3D Mode Shape 4 for Structure Model 9

Figure A.9.5: 3D Mode Shape 5 for Structure Model 9 Figure A.9.6: 3D Mode Shape 6 for Structure Model 9

122  

Figure A.9.7: 3D Mode Shape 7 for Structure Model 9 Figure A.9.8: 3D Mode Shape 8 for Structure Model 9

Figure A.9.9: 3D Mode Shape 9 for Structure Model 9 Figure A.9.10: 3D Mode Shape 10 for Structure Model 9

Figure A.9.11: 3D Mode Shape 11 for Structure Model 9 Figure A.9.12: 3D Mode Shape 12 for Structure Model 9