stress analysis of non-uniform guided composite structures

Stress Analysis of Non-Uniform Guided Composite Structures of Hybrid

Laminates

by

Md. Jamil Hossain

MASTER OF SCIENCE IN MECHANICAL ENGINEERING

Department of Mechanical Engineering

BANGLADESH UNIVERSITY OF ENGINEERING AND TECHNOLOGY

June 2016

iv

DEDICATION

______________________________________________

This thesis is dedicated to my parents.

v

ACKNOWLEDGEMENTS

______________________________________________

I would like to express my deepest gratitude to my supervisor Prof. Dr. Shaikh Reaz

Ahmed, Professor, Department of Mechanical Engineering, Bangladesh University of

Engineering and Technology, Dhaka for his support, guidance, inspiration,

constructive suggestions and close supervision throughout the entire period of my

graduate study. I am grateful to the Department of Mechanical Engineering for

providing me the required facilities for my thesis work. I would like to thank the

members of the board of examiners for their constructive comments and criticism.

I am deeply indebted to Mr. Partha Modak for his guidance throughout the thesis

work.

I want to express my deepest gratitude to my parents and siblings, for their

unparalleled love, dedication, and encouragement throughout my whole life. Without

their effort none of this would have been possible. They taught me how to learn, a gift

which I appreciate as much as any other. And they contributed to making me a better

person.

Finally, I am extremely grateful to my wife Umme Hani and brother Mohammed

Sajid Hossain for their countless efforts throughout the time of my thesis work.

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TABLE OF CONTENTS

______________________________________________ Item Page Title Page i Board of Examiners ii Declaration iii Dedication iv Acknowledgements v Table of Contents vi List of Figures ix List of Tables xiii Nomenclature xvi Abstract xvii Chapter 1 Introduction 1.1 Introduction 1 1.2 Literature review 6 1.3 Objectives 10 1.4 Outline of the methodology 10 1.5 Scope of the present research work 11 1.6 Significance of the present study 12 Chapter 2 Mathematical Background 2.1 Introduction 13 2.2 Stress at a point 13 2.3 Strains in terms of displacement and stress 15 components 2.4 Differential equations of equilibrium 16 2.5 Compatibility equations 17 2.6 Plane elasticity 18 2.6.1 Plane stress condition 18 2.6.2 Plane strain condition 19 2.6.3 Equilibrium equations and compatibility conditions for plane elasticity

20

2.7 Stress-strain relations for different types of materials

22

2.7.1 Anisotropic material 23 2.7.2 Monoclinic material 24 2.7.3 Orthotropic material 24 2.7.4 Isotropic material 25 2.8 Composite ply 27 2.8.1 Unidirectional composite ply 27

vii

2.8.2 Composite angle ply 29 2.9 Laminated composite material 30 2.9.1 Strain-displacement relations 31 2.9.2 Resultant laminate forces and moments 32 2.10 Hybrid laminates 33 2.11 Special cases of laminates 35 2.11.1 Symmetric laminates 35 2.11.2 Cross-ply laminates 36 2.11.3 Angle ply laminates 37 2.11.4 Balanced laminates 38 2.12 Available mathematical models of elasticity 39 2.12.1 Airy’s stress function formulation 39 2.12.2 Displacement parameter approach 41 2.13 Displacement potential formulation 42 2.13.1 Applicability of the formulation 46 2.13.2 Boundary conditions 47 2.13.3 Evaluation of stress components for individual ply

48

Chapter 3 Numerical Solution 50 3.1 Introduction 50 3.2 Discretization of the Computational Domain 50 3.3 Finite Difference Discretization of the Governing Equation

55

3.4 Finite Difference Discretization of Body Parameters

56

3.5 Management of boundary conditions at the corners

78

3.6 Placement of boundary conditions to Mesh points

83

3.7 Solution and Evaluation of ψ at the Internal and Boundary Mesh Points

84

3.8 Evaluation of Displacements, Strains and Stresses

86

3.9 Evaluation of Stress Components for Individual ply

92

3.10 Summary 92 Chapter 4 Analysis of Non-uniform Column of Hybrid Laminates

93

4.1 Introduction 93 4.2 Geometry, loading and material modelling of the composite column

93

4.3 Boundary conditions 96 4.4 Numerical modelling of the column 99 4.5 Results and Discussions 101 4.5.1 Determination of critical sections of the column

102

viii

4.5.2 Effect of laminate hybridization 103 4.5.3 Effect of eccentricity of applied loading 113 4.5.4 Effect of partial guides on the column behavior

116

4.6 Summary 120 Chapter 5 Analysis of Non-uniform Beam of Hybrid Laminates

121

5.1 Introduction 121 5.2 Geometry, loading and material of the Composite beam

5.3 Boundary conditions 122 5.4 Numerical modelling of the problem 127 5.5 Results and Discussion 127 5.5.1 Effect of aspect ratio on the elastic field 128 5.5.2 Effect of soft isotropic plies on the elastic field

134

5.5.3 Analysis of ply stresses 139 5.6 Summary 140 Chapter 6 Validation of the Computational Method 141 6.1 Introduction 141 6.2 Problem 1: A guided I-shaped hybrid laminated column subjected to eccentric loading

142

6.2.1 Comparison of results 144 6.3 Problem 2: A Uniform rectangular short sinking beam

150

6.3.1 Boundary conditions 150 6.3.2 Numerical modelling 151 6.3.3 Comparison of results 153 6.4 Salient features of the present computational scheme 157 6.5 Summary 159 Chapter 7 Conclusions 160 7.1 Conclusions 160 7.2 Recommendations for future works 161 References 163 Appendices A Flow chart of the Program 168

ix

LIST OF FIGURES

______________________________________________ No. Title Page 1.1 Mechanism of laminate formation (a) conventional laminate and (b)

hybrid laminate 2

1.2 Application of composites and hybrid composites in (a) an aircraft structure and (b) a helicopter rotor blade

4

2.1 Conventions of stress and displacement components of an

elementary cubic body 14

2.2 Stress components under plane stress conditions 19 2.3 Stresses on cubic element 23 2.4 Stress components on a plane of unidirectional fiber reinforced ply 28 2.5 Stress components on a plane of an angle ply 30 2.6 Relationship between displacements through the thickness of a

plate to mid-plane displacements and curvatures 31

2.7 Coordinate location of plies in a laminate. 32 2.8 Five ply hybrid laminate consisting of plies of two different fiber

materials in the same matrix 34

2.9 Components of displacements on a boundary segment 47 2.10 Components of stresses on a boundary segment 48 3.1 Different steps involved in the discretization of the domain of a

non-uniform body ABIJKLCDHGFE 51

3.2 Extreme nodal field for uniform geometry 53 3.3 A non-uniform geometry superimposed on the extreme nodal field. 53 3.4 Node numbering scheme of the extreme nodal field 54 3.5 Indicators 0 or 1 at each nodal point of the extreme field depending

on whether corresponding node is outside or inside the boundary 55

3.6 (a) Stencil for governing equation of general symmetric laminates (b) application of the governing equation stencil at internal points of the non-uniform structure

57

3.7 Indicators 1, 2, 3 or 4 at each nodal point depending on form on stencil of stress, strain and displacement components to be used in both stages pre- and post-processing

59

3.8 (a) Different forms of stencil for ux (b) application of the stencils at boundary and internal points of the non-uniform structure.

61

3.9 (a) Single form of stencil for uy (b) application of the stencils at boundary and internal points of the non-uniform structure.

62

3.10 (a) Different forms of stencil for σxx and σyy (b) application of the stencils at boundary and internal points of the non-uniform structure.

64

3.11 (a) Different forms of stencil for σxy (b) application of the stencils at 65

x

boundary and internal points of the non-uniform structure. 3.12 (a) Different forms of stencil for un or ut (b) application of the

stencils at different boundary points of the non-uniform geometry 70

3.13 (a) Different forms of stencil for σn or σt (b) application of the stencils at different boundary points of the non-uniform structure.

77

3.14 (a) Version 1 (b) version 2 with different forms and (c) version 3 with different forms of stencil for uy and application of the stencils at external corner points of the non-uniform structure.

82

3.15 Node numbering scheme applied to a non-uniform structure 84 3.16 (a) Different forms of stencil for εxx (b) application of the stencils at

different boundary and internal points of the non-uniform structure. 88

3.17 (a) Different forms of stencil for εyy (b) application of the stencils at different boundary and internal points of the non-uniform structure.

89

3.18 (a) Different forms of stencil for εxy (b) application of the stencils at boundary and internal points of the non-uniform structure.

91

4.1 Analytical model of the eccentrically loaded non-uniform

laminated column with partial guides. 94

4.2 Material modelling of hybrid laminate consisting of FRC-1 and FRC-2

95

4.3 3D views of: (a) hybrid of FRC-1 and FRC-2, (b) FRC-1 and (c) FRC-2 laminated columns

95

4.4 FDM Mesh network used to model I-shaped column 99 4.5 Developed extreme nodal field showing the involved and

uninvolved nodal points (1 and 0) for computation 100

4.6 Active field nodal points tagged with stencil indicating numbers (1, 2, 3, 4) for both stages of pre- and post-processing

101

4.7 Distribution of maximum principal laminate stress along the two opposing lateral surfaces of the column

102

4.8 Distribution of lateral stresses along the critical section b-b´ (y/L = 0.2) of identical plies of different laminates

105

4.9 Distribution of lateral stresses along the critical section e-e´ (y/L = 0.8) of identical plies of different laminates

106

4.10 Distribution of axial stresses along the critical section b-b´ (y/L = 0.2) of identical plies of different laminates

108

4.11 Distribution of axial stresses along the critical section e-e´ (y/L = 0.8) of identical plies of different laminates

109

4.12 Distribution of shear stresses along the critical section b-b´ (y/L = 0.2) of identical plies of different laminates

110

4.13 Distribution of shear stresses along the critical section e-e´ (y/L = 0.8) of identical plies of different laminates

111

4.14 I-shaped guided column subjected to axial loading on the top surface (a) uniform loading, (b) eccentric loading

113

4.15 Distribution of maximum principal stresses along the critical section (a) e-e´ (y/L = 0.8) and (b) b-b´ (y/L = 0.2) of hybrid laminated column subjected to both full and eccentric loading

114

4.16 Deformed shapes of hybrid laminated column subjected to (a) full loading and (b) eccentric loading

115

4.17 Eccentrically loaded I-shaped column (a) with and (b) without partial guides along the opposing upper lateral surfaces

116

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4.18 Distribution of maximum principal stress along the critical section EE´ (y/L = 0.8) of identical plies of both partially guided and unguided hybrid laminated column subjected to eccentric loading

118

4.19 Deformed shapes of eccentrically loaded I-shaped column (a) with and (b) without partial guides along the opposing upper lateral surfaces

119

5.1 Loading and geometry of a non-uniform sinking beam of laminated

composite 121

5.2 Material modelling of: (a) angle ply fiber reinforced composite (FRC) laminate, (b) angle ply hybrid laminate of fiber reinforced composite (FRC) and soft isotropic plies, (c) cross-ply fiber reinforced composite (FRC) laminate and (d) cross-ply hybrid laminate of fiber reinforced composite (FRC) and soft isotropic plies

123

5.3 3D views of the beam of: (a) fiber reinforced composite (FRC) laminate, (b) Hybrid laminate of FRC and isotropic ply

124

5.4 Distribution of overall laminate stresses at different sections of sinking beams with 12 isotropic plies in ±30° angle ply boron/epoxy laminate

129

5.5 Distribution of overall bending stresses at different sections of sinking beams with 12 isotropic plies in ±30° angle ply boron/epoxy laminate

130

5.6 Distribution of overall shear stresses at different sections of sinking beams with 12 isotropic plies in ±30° angle ply boron/epoxy laminate

132

5.7 Deformed shape of a ±30° angle ply boron/epoxy laminated sinking beam with various aspect ratios.

133

5.8 Distribution of overall laminate stresses at different sections of a I-shaped sinking beam (L/D = 4) with various isotropic plies in a ±30° angle ply boron/epoxy laminate

135

5.9 Distribution of overall laminate stresses at different sections of a I-shaped sinking beam (L/D = 4) with various isotropic plies in a ±55° angle ply boron/epoxy laminate

136

5.10 Distribution of overall laminate stresses at different sections of a I-shaped sinking beams of (L/D = 4) with various isotropic plies in a cross-ply boron/epoxy laminate

138

6.1 (a) Loading and geometry of the non-uniform hybrid laminated

composite column used for comparison with FEM solutions and (b) top portion of the column showing boundary nodes R, S and T and their physical conditions

143

6.2 (a) Geometry of the four noded isoparametric layered shell element and (b) Finite element modelling of the non-uniform laminated column using a commercial software

144

6.3 Comparison of stresses along different sections of boron/epoxy ply (θ = 75°) of hybrid laminated column subjected to eccentric loading

146

6.4 Comparison of stresses along different sections of boron/epoxy plies (θ = 30° and 75°) of hybrid laminated column subjected to eccentric loading

147

xii

6.5 Loading and geometry of the uniform sinking beam 150 6.6 Developed extreme nodal field showing the involved and

uninvolved nodal points (1 and 0) for computation 152

6.7 Active field nodal points tagged with stencil indicating numbers (1, 2, 3, 4) for both the stages of pre- and post-processing

153

6.8 Comparison of normalized axial displacement in a short sinking beam at y/L = 0.75

154

6.9 Comparison of normalized bending stress in a short sinking beam at y/L = 0

155

6.10 Comparison of normalized shear stress at various sections of a short sinking beam, L/D = 1.

156

xiii

LIST OF TABLES

______________________________________________ No. Title Page 4.1 Numerical modelling of the boundary conditions for different

boundary segments of the non-uniform laminated composite column

97

4.2 Numerical modelling of the boundary conditions for external and re-entrant corners of the non-uniform laminated composite column

98

4.3 Properties of unidirectional fiber-reinforced composite ply used to obtain the numerical results

102

4.4 Overall laminate stresses at the critical section e-e´( y/L = 0.8) 103 4.5 Comparison of critical ply stresses of the three different

laminates at the re-entrant corner d´ as a function of ply angle 112

4.6 Comparison of critical ply stresses of the three different laminates at the re-entrant corner d as a function of ply angle

112

5.1 Numerical modelling of the boundary conditions for different

boundary segments of the non-uniform laminated composite sinking beam

125

5.2 Numerical modelling of the boundary conditions for external and re-entrant corners of the non-uniform laminated composite beam

126

5.3 Properties of isotropic ply used to obtain the numerical results 127 5.4 Overall laminate stresses at the critical region of the laminated

composite I-shaped sinking beam 139

5.5 Comparison of critical ply stresses at the critical region of the θ = ±30° angle ply laminated composite I-shaped sinking beam

140

5.6 Comparison of ply stresses at the critical region of the cross-ply laminated composite sinking beam

140

6.1 Comparison of stresses and displacements at different points of

the boundary with known physical conditions of a hybrid laminated column subjected to eccentric loading

149

6.2 Numerical modelling of the boundary conditions for different boundary segments of the uniform sinking beam

151

6.3 Numerical modelling of the boundary conditions for different corners of the uniform sinking beam

151

6.4 Comparison of maximum normalized bending stress predictions with FEM, simple and modified theory estimates at various sections in a uniform short sinking beam

157

xiv

NOMENCLATURE

______________________________________________ x, y Rectangular co-ordinate L, D, h Dimensions of the laminated structures σ Stress σxx Normal stresses in x-direction σyy Normal stresses in y-direction σzz Normal stresses in z-direction σxy Shear stresses in xy planes σyz Shear stresses in yz planes σzx Shear stresses in zx planes σn Stresses in normal direction σt Stresses in tangential direction σ0 Maximum intensity of applied shear loading ux Displacement component along x-direction uy Displacement component along y-direction un Displacement component along normal direction ut Displacement component along tangential direction δ, δL Magnitude of applied shear displacement l, m Direction cosines θ Fiber orientation angle ε Strain εxx Normal strains in x-direction εyy Normal strains in y-direction εzz Normal strains in z-direction εxy Shear strains in xy plane εyz Shear strains in yz plane εzx Shear strains in zx plane E Elastic modulus Ef Elastic modulus of fiber material Em Elastic modulus of matrix material E1 Elastic modulus of composite ply along direction of fiber E2 Elastic modulus of composite ply along perpendicular direction of fiber ν Poisson ratio νf Major Poisson ratio of fiber material νm Major Poisson ratio of matrix material ν12 Major Poisson ratio of composite ply ν21 Minor Poisson ratio of composite ply hx Mesh length along x-direction ky Mesh length along y-direction [K] Coefficient matrix [C] Constant column matrix [S] Compliance matrix [Q] Stiffness matrix

xv

[Q ] Transformed reduced stiffness matrix [A] Extentional stiffness matrix [B] Coupling stiffness matrix [D] Bending stiffness matrix ψ Displacement potential function ϕ Airy’s stress function

xvi

ABSTRACT

The present research addresses a new computational study for the analysis of the

elastic field of both uniform as well as non-uniform guided structures of hybrid

laminated composites. Laminates composed of different ply materials of dissimilar

fiber orientations are considered for the present analysis. A displacement potential

based elasticity approach is used for the laminate, where the relevant displacement

components of plane elasticity are expressed in terms of a single scalar function. The

finite difference method is used to develop the single variable computational scheme,

which is capable of dealing with different ply materials as well as different fiber

orientations efficiently. The scheme is developed in such a way that it can handle

geometrical non-uniformity as well as mixed mode of physical conditions at the

surfaces of laminated structures.

The application of the computational scheme is demonstrated for a number of uniform

and non-uniform structural components, like beams and columns of hybrid laminated

composites. Balanced laminates composed of two different fiber reinforced composite

plies with various fiber orientations are considered for the non-uniform eccentrically

loaded column. On the other hand, laminates composed of fiber reinforced composite

plies and soft isotropic plies are considered for the non-uniform sinking beam

problem. The corresponding elastic fields of the overall laminate as well as individual

plies are analyzed mainly in the prospective of laminate hybridization. Both the fiber

materials and fiber orientation angles as well as geometrical aspect ratio of the

structural components are identified to play dominant roles in defining the design

stresses of the laminated structures.

Finally, in an attempt to verify the appropriateness as well as accuracy of the present

computational scheme, the present potential function solutions are compared with

available solutions obtained by standard computational method as well as those found

in the literature.

CHAPTER

Introduction

1.1 Introduction

With the development of industries such as the aeronautics, astronautics, national

defenses and nuclear energy, lightweight structures and materials like composites are

receiving more attention than conventional homogeneous materials. Composite

materials offer high stiffness to weight and strength to weight ratios when compared

with traditional metallic materials. Traditionally, composite materials were generally

costly which made them only attractive to very limited industries (e.g., the defense

industry). Advances in their manufacturing and new innovations have brought the cost

of these materials down and made them reasonably competitive. They have gained

more and more usage in the last three decades in the aerospace industry and have

recently been gaining more usage in the automotive industry. In automotive design,

they yield lighter structures which have positive impact on attributes like fuel

economy, emission and others. Applications of composite materials in the automotive

industry vary from thermo-plastics to fiber-reinforced structures. In particular, the use

of fiber-reinforced composites as the structural material is found to increase

extensively in almost all areas of structural applications mainly because of their

specific characteristics of light weight, high strength, stiffness, toughness, etc.,

compared to those of conventional materials. In most of the applications they are

found to be used as a laminate consisting of more than one ply bonded together

through their thickness.

Hybrid composites, on the other hand, usually contain more than one type of fiber in a

single matrix material as shown in Figure 1.1. In principle, several different fiber

types may be incorporated into a hybrid laminate, but it is more likely that a

combination of only two types of fibers would be most beneficial [1]. Hybrid

composite Materials have extensive engineering application where strength to weight

ratio, low cost and ease of fabrication are required. They provide combination of

properties such as tensile modulus, compressive strength and impact strength which

cannot be realized in composite materials. In recent times hybrid composites have

been established as highly efficient, high performance structural materials and their

1

CHAPTER 1 | INTRODUCTION

2

(a)

(b)

Figure 1.1: Mechanism of laminate formation (a) conventional laminate and (b) hybrid laminate

Fiber material 2

Fiber material 1

Fiber x

y


3

use is increasing rapidly. They are usually used when a combination of properties of

different types of fibers have to be achieved, or when longitudinal as well as lateral

mechanical performances are required. Combining two or more types of fiber in a

matrix to form a hybrid composite might create a material possessing the combined

advantages of the individual components and simultaneously mitigating their less

desirable qualities. Hybrid composites have unique features that can be used to meet

various design requirements in a more economical way than conventional composites.

This is because expensive fibers like graphite can be partially replaced by less

expensive fibers such as glass.

The aeronautical industry is dependent on materials with high specific properties.

Commercial aircraft applications are the most important uses of hybrid composites. In

cases where high moduli of elasticity values are less important, hybrid is the natural

option because of the low cost of material. Glass and carbon fiber reinforced hybrid

composites are the most desired materials in the aeronautical industry. Hybrid

laminates are also seen to be used in primary (load-carrying) structures of an aircraft

such as wings and fuselages. For example, carbon-fiber aluminum hybrid laminate is

used as a part of the primary structure of Airbus A380. Some of the applications of

composites and hybrid composites in the aeronautical industry are shown in Figure

1.2. The vast majority of marine structures such as ship hulls are constructed from

common carbon steels, which are obviously susceptible to corrosion, stress

concentration and reduced fatigue life. Hulls constructed out of reinforced polymer

hybrid composite materials, on the other hand, have many advantages over carbon

steel, including a much higher strength to-weight ratio, lower maintenance

requirement, and an ability to be formed into complex shapes. Hulls of hybrid

composites also offer a number of stealth benefits, high durability and increased

fatigue life. Hybrid of carbon and glass fibers in epoxy matrix is used for fabrication

of blades for wind power generation. On the other hand, these fibers reinforced in

plastic are used to form hybrids for the use in civil constructions like bridges.


4

(a)

(b)

Figure 1.2: Application of composites and hybrid composites in (a) an aircraft structure and (b) a helicopter rotor blade.


5

A guided structure is one where a boundary surface of the structure, in part or whole,

is allowed to move only along a certain direction. For example, a boundary surface

attached to rollers can be realized as a guided surface. In general, the geometry of

structures are not always of uniform rectangular shape. Any shape other than the

uniform rectangular shape may be called the non-uniform shape and structures with

non-uniform geometry are called here non-uniform structures.

The analysis of composite structures has now become a key subject in the field of

solid mechanics. In case of engineering problems, the elementary methods of strength

of materials are not adequate to provide sufficient and accurate information regarding

the elastic behavior of the corresponding body, especially if the body is of non-

uniform geometry. So, some more powerful methods are needed in the study of elastic

field. Structural analysis comprises the set of physical laws and mathematics required

to study and predict the behavior of structures. It is such an engineering artifact whose

integrity is judged largely based upon the ability to withstand loads on building,

bridge, ship, submarine, aircraft, etc. From theoretical perspective, the primary goal of

structural analysis is the computation of deformations, internal forces and stresses. In

practice, structural analysis can be viewed more abstractly as a method to derive the

engineering design process or to prove the soundness of a design without dependence

on directly testing it. Engineering disciplines which deal the matter are namely the

mechanics of materials (also known as strength of materials) and the theory of

elasticity. Again for the cases where the stress distribution in bodies with all

dimension of same order has to be investigated, neither strength of materials nor

theory of elasticity are adequate to furnish satisfactory information. For example, the

stresses in rollers and in balls of bearings can be found only by using the methods of

the theory of elasticity. So, to obtain satisfactory and reliable information of elastic

fields in engineering structures of practical applications, it is essential to adopt the

theory of elasticity. The equations of theory of elasticity are basically a system of

partial differential equations. Due to the nature of mathematics involved in solving

such equations, solutions have been only produced for relatively simple geometries.

For complex geometries, a modern computational facility is considered more suitable

for reliable solution.


6

1.2 Literature review

The elementary methods of strength of materials were the primary tools of the

practicing engineers for handling engineering problems of structural elements for

quite a long time. However, these methods are often inadequate to furnish satisfactory

information regarding local stresses near the loads and near the supports of the

structures. The elementary theory provides no means of investigating stresses in

region of sharp variation in cross section of beams, columns or shafts. Stresses in

screw threads, around various shapes of holes in structures, bear contact point on gear

teeth, rollers and balls of bearing, have all remained beyond the scope of elementary

theories. It is thus obvious that, for the designer of modern machines, recourse to

more powerful methods of the theory of elasticity is necessary.

Numerous occasions may arise in realistic engineering problems that might not be

dealt with any analytical techniques. In case where rigorous solution could not be

obtained, approximate methods have been developed. In other cases, where even

approximate methods could not be developed, solutions have been obtained by using

experimental methods. Photoelastic methods, soap-film methods, application of strain

gages, moiré fringe etc. are some of these experimental methods applied in the study

of stress concentration at points of sharp variation of cross-sectional dimension and at

sharp fillets of re-entrant corners. These results have considerably influenced the

modern design of machine parts and helped in many cases to improve the construction

by eliminating weak spots through which crack may initiate and thereby propagate.

As the elementary formulae of strength of materials are often not accurate enough, the

theory of elasticity has been found noteworthy in the solution of practical engineering

problems. The field of elasticity deals mainly with deformation parameters and stress

parameters for the solution of two dimensional problems since most of the three

dimensional problems may be resolved through a two dimensional one. If it still

remains beyond the extent of analytical studies, the problem has to be handled

experimentally as a particular case.

Even though the elasticity problems were formulated a long time ago, exact solutions

of practical problems are hardly available because of the inability of managing the

associated physical conditions in a justifiable manner. Analytical methods treat each

problem separately, for example, a beam and a shaft are analyzed as separate


7

problems. The famous Saint Venant’s principle is still applied and its merit is

evaluated in solving problems of solid mechanics [2-4], in which full boundary effects

could not be taken into account satisfactorily in the process of solution. Even now,

photo elastic studies are being carried out for classical problems like uniformly loaded

beams on two supports mainly because the boundary effects could not be taken into

account fully in the analytical method of solutions. Actually, the management of

boundary conditions and boundary shapes are two major obstacles to the reliable

solution of practical problems of all disciplines of engineering, specially that of solid

mechanics. Mixed-boundary-value problems are those in which the boundary

conditions are specified as a mixture of boundary restraint and boundary loading;

even the composition of the mixture may also vary from one segment of the boundary

to the other. Analytical methods of solution could not gain that much popularity in the

field of structural analysis, as most of the problems are of mixed-boundary-value type,

for example, guided or stiffened structures. Among the existing mathematical models

for plane problems of elasticity, Airy’s stress function approach [5] and the

displacement parameter approach [6] are noticeable. The shortcoming of the stress

function approach is that it accepts boundary conditions only in terms of loadings.

Boundary restraints specified in terms of the displacement components cannot be

satisfactorily imposed on the stress function. As most of the practical problems of

elasticity are of mixed boundary conditions, the approach fails to provide any explicit

understanding of the state of stresses at the critical regions of supports and loadings.

However, successful application of the stress function formulation in conjunction with

finite-difference method (FDM) has been reported for the solution of plane elastic

problems, where all the conditions on the boundary are prescribed in terms of stresses

[4]. Further, Conway and Ithaca [7] extended the stress function formulation in the

form of Fourier integrals to the case where the material is orthotropic and obtained

analytical solutions for a number of ideal problems. Again, the two displacement

parameter approach involves finding two functions simultaneously from two elliptic

partial differential equations, which is extremely difficult, especially when the

boundary conditions are specified as a mixture of restraints and loadings [8]. The

conventional mathematical models of elasticity are not adequate to handle the

problems of mixed boundary conditions. This necessitates the adoption of an

approach that can deal with both the non-uniform geometry and mixed boundary

conditions.


8

Composites are relatively new candidates in the field of engineering and structural

materials. An experimental study of the nonlinear response and failure characteristics

of internally pressurized 4 to 16 ply thick graphite/epoxy cylindrical panels is carried

out by Richard and Eric [9]. Experimental study on impact resistance and ageing of

corrosion resistant steel/rubber/composite hybrid laminated structures has been

carried out Sarlin et al. [10-11]. The stainless steel/rubber/glass fiber reinforced

plastic hybrid laminates were manufactured and high velocity impact tests were

carried out [10] and the environmental resistance of the hybrid structures were tested

by exposure to hot, moist and hot/moist environments and after the ageing by peel

testing [11]. However, stress analysis of composite structures is mainly handled by

numerical methods. More specifically, the analysis and design of laminated structural

components has now been entirely dependent on finite element method (FEM)

packages, and the corresponding examples in the literature are quite extensive [12-

15]. Management of arbitrary boundary shapes of structures is out of the scope of

analytical methods. Although the adaptations of the FEM relieved us from our major

inability of managing non-uniform boundary shapes in numerical modelling, we are

aware of its large amount of computational work and lack of sophistication, especially

in predicting the stresses at the surfaces of structural components [16-17]. On the

other hand, displacement potential elasticity approach has been verified to be

successful for mixed boundary value stress problems of arbitrary-shaped elastic

bodies of isotropic materials, when used in conjunction with finite-difference method

(FDM) of solution [18-21]. However, arbitrary shaped composite laminated structures

are still in the scope of the FEM [15]. The uncertainties associated with the prediction

of surface stresses by the standard FEM have been pointed out by several researches

[16, 22-23]. On the other hand, the accuracy of FDM in reproducing the state of

stresses along the bounding surfaces has been repeatedly verified to be much higher

than that of finite element analysis [23-25]. In the research of Ranzi [26], the FDM

has also been identified to be an adequate numerical tool for describing the composite

behavior of beams, and the corresponding FDM solutions are shown to be more

accurate when compared with the usual eight-dof-FEM solutions, even with finer

discretization for the latter one.

The analysis of non-uniform composite structures of hybrid laminates has become a

key subject of recent interest in the field of structural mechanics. Yu et al. conducted


9

a research on hybrid fiber reinforced polymer (FRP)-concrete-steel hybrid beams of

tubular shape and found that they have a very ductile response when the compressive

concrete is confined by the FRP tube and the steel tube provides ductile longitudinal

reinforcement [27]. Benatta et. al. [28] mathematically proved that by varying the

fiber volume fraction within a symmetric laminated beam and combining two fiber

types to create a hybrid can offer desirable increases in axial and bending stiffness.

Using FEM, Badie et al. [29] analyzed the effect of fiber orientation angles and

stacking sequence on the torsional stiffness, natural frequency, buckling strength,

fatigue life and failure modes of hybrid carbon/glass fiber reinforced epoxy composite

tubes.

Attempt is made in the present thesis to obtain solutions of structural problems that

contain material complexity, geometrical complexity, involvement of a large number

of singular points, as well as complexity in the physical conditions at the surfaces.

The material considered is a hybrid laminate composed of different fiber reinforced

composite (FRC) ply materials of dissimilar fiber orientation as well as soft isotropic

ply materials. The structure with geometrical non-uniformity is considered where a

large number of singular points are involved. It is well known that more the number

of singular points, the more the possibility of deviation from the actual solution.

Guided structures, in general, make the boundary modelling complicated since the

associated boundary conditions are of mixed type. Complex loading cases are

considered, for example, eccentric loading for a column structure or shear

displacement for a beam structure. Accurate and reliable analysis of the elastic field of

non-uniform hybrid laminated composite structures is of great concern, as we are

constantly aware of the lack of sophistication and doubtful quality of conventional

computational solutions, especially around the surfaces as well as regions of

singularities. No serious attempt has been reported so far in the literature that can

provide a reliable analysis of stresses of guided laminated structures with non-uniform

geometry. Recently, a displacement-potential based elasticity approach has been

developed for the boundary value problems of anisotropic composite ply as well as

symmetric laminated composites [30-31], which has eventually opened up an

effective alternative avenue for stress analysis of composite structures. This thesis

extends the potential of finite-difference technique in conjunction with the

displacement-potential formulation of solid mechanics, to develop an efficient


10

computational scheme for reliable stress analysis of uniform/non-uniform guided

composite structures of hybrid laminates.

1.6 Objectives

The present study is an attempt to stress analysis of non-uniform composite structures

of hybrid laminates with mixed mode of physical conditions through an efficient and

effective computational scheme based on displacement potential elasticity approach.

The specific objectives of the present research work are as follows:

a) Displacement-potential based single variable modelling of mixed-boundary

value stress problems of hybrid laminates.

b) Development of an efficient computational scheme for the numerical solution

of elastic field of both uniform and non-uniform composite structures of

hybrid laminates.

c) Analysis of the effect of laminate hybridization on overall laminate as well as

individual ply stresses in non-uniform composite structures of cross-ply and

angle ply laminates.

d) Analysis of the effect of local guides on the elastic field of non-uniform

composite structures of hybrid laminates.

e) Verification of soundness and reliability of the single variable computational

approach by comparing the results with those of available computational

techniques or in the literature.

1.4 Outline of the methodology

The potential-function based elasticity formulation of laminated composites [31] is

extended to model the stress problems of hybrid laminates in terms of a single scalar

function. Based on the mathematical model, an efficient computational scheme is

developed for analyzing the elastic field of laminated structures of both uniform and

non-uniform geometries. Here, the finite-difference method (FDM) is used to

discretize the governing partial differential equation of equilibrium as well as the

equations associated with the prescribed physical conditions. An imaginary boundary,


11

exterior to the physical boundary of the non-uniform body, is realized for the sake of

discretization of the domain using a central difference approximation to the

equilibrium equation. A variable node numbering scheme is adopted here to discretize

the non-uniform computational domain using a rectangular mesh-network, in which

the active field nodal points are renumbered a number of times at different stages of

pre- and post-processing. Special cares have been taken for finite-difference

modelling of the external and re-entrant corners of the non-uniform geometry, which

are, in general, the points of singularity in the solution. The discrete values of the

function at the mesh points of the structure are obtained by solving the system of

algebraic equations resulting from the application of equilibrium and boundary

conditions at the appropriate nodal points of the computational domain. Finally, both

the overall laminate stresses and individual ply stresses of the hybrid laminate are

calculated from the nodal values of the potential function and the corresponding

reduced stiffness matrix of the overall laminate and individual plies.

1.5 Scope of the present research work

The computational scheme developed for non-uniform structures of composite

laminate basically converts the laminate into a representative single ply and obtains

solution of this representative ply considering it as a plane stress problem. The plane

elasticity solution is then extended to individual plies of the laminate. The scheme is

capable of handling both uniform and non-uniform geometry, which can however be

applied to all kinds of plies of isotropic materials, unidirectional fiber reinforced

composites as well as symmetric cross-ply and angle ply laminates, balanced

laminates and hybrid laminates. All kinds of boundary conditions, namely – Dirichlet,

Neumann and mixed mode of boundary conditions can be managed in terms of the

scalar function with equal sophistication. All possible sorts of loading cases, such as,

moment loading, distributed eccentric loading, shear displacement loading, etc. can

readily be accommodated.


12

1.6 Significance of the present study

The present research will lead to an effective alternative to reliable analysis of

structural components of hybrid laminates with non-uniform geometries. Results of

the present analysis are thus expected to provide a reliable design guide to non-

uniform composite structures, like beams and columns of hybrid laminates. This study

also throws challenges to conventional computational approaches, especially in

context of managing boundary conditions where the boundary conditions change from

one type to the other, i.e., the external and re-entrant corners, which are, in general,

the points of singularities.

CHAPTER

Mathematical Background

2.1 Introduction

The response of a solid body to external forces is influenced by its geometry as well

as the mechanical properties of the body. Here interest will be restricted to elastic

materials in which the deformation and stress disappear with the removal of the

external forces, provided that the external forces do not exceed a certain limit. In fact,

almost all engineering materials possess, to a certain extent, the property of elasticity.

Structural analysis necessitates the requirements to investigate the elastic field, i.e.,

state of stresses, strains and displacements, at any point due to given body forces and

given conditions at the boundary of the body.

2.2 Stress at a point

There are two kinds of external forces which may act on bodies. Forces distributed

over the surface of the body are called surface forces while forces distributed over the

volume of the body are called body forces. Now, let us take an infinitesimal cubic

element as shown in Figure 2.1, which is cut off from an elastic body with sides

parallel to the coordinate axes. The forces acting on each six faces may be resolved

into two components - one perpendicular to the plane of the face and the other parallel

to the face. The stress component acting perpendicular to the plane of the face is

called the normal stress and usually denoted by σ with a subscript (Example - σxx, σyy,

σzz) to indicate its direction of action. According to general convention, these normal

stresses are taken positive when producing tension and negative when producing

compression. In the same way, the stress components acting parallel to the face are

known as shear stresses and they can be resolved into two components parallel to two

in-plane coordinate axes and are indicated by the same notation with double subscript

- the first indicating the direction of the normal to the face and indicating of the

normal of the face and the second indicating the direction of the components of the

stress. On any side, the direction of positive shearing stress coincides with the positive

2

CHAPTER 2 | MATHEMATICAL BACKGROUND

14

direction of the axis if the outward normal on this side has the positive direction of the

corresponding axis. If the outward normal has a direction positive to positive axis, the

positive shearing stress will also have the opposite direction of the corresponding

axis.

Figure 2.1: Conventions of stress and displacement components of an elementary cubic body

Though the cubic element has six different faces, basically, it has three mutually

perpendicular faces and the rest of the faces are parallel to these mutually

perpendicular faces respectively. Thus, corresponding a cubic element with edges

parallel to the three axis of a Cartesian co-ordinate system, the state of stress of the six

sides of the element are described by three symbols σxx, σyy, σzz for normal stress and

six symbols σxy, σyz, σzx, σyx, σzy, σxz for shear stress. A consideration of the equilibrium

of the cubic element shows that, for two perpendicular sides of the cubic element, the

components of shearing stress perpendicular to the line of intersection of these sides

are equal. Mathematically stated, from consideration of equilibrium of moments about

three mutually perpendicular axes, it can be shown that

xy yx

x

z

y

x, ux

y, uy

z, uz

σyy

σyx

σyz

σxy

σxx

σxz

σzz

σzy

σzx

σzy

σzx

σzz

σzx

σyx

σxx

σyz

σyy

σyx


15

yz zy

zx xz

Thus the nine components of the stress are reduced to six. These six quantities σxx, σyy,

σzz, σxy, σyz, σzx are, therefore, sufficient to describe the stresses acting on the co-

ordinate planes through a point and these will be called the components of stresses at

the point.

2.3 Strains in terms of displacement and stress components

Due to the application of external forces, the elastic body deforms and the

deformations can be specified by assigning three elongations in three perpendicular

directions and three shear strains related to the same direction. These directions are

taken as the direction of the coordinate axis and the symbol ε is used to denote the

strain components with the same subscripts to this symbol as for the stress

components. If the components of displacements of particle in the body are specified

by ux, uy and uz parallel to the co-ordinate axes x, y and z respectively, then the

relations between the components of strain and the components of displacement are

given by Timoshenko and Goodier [5]

xxx

ux

(2.1 a)

yyy

uy

(2.1 b)

zzz

uz

(2.1 c)

yxxy

uuy x

(2.1 d)

y zyz

u uz y

(2.1 e)

x zzx

u uz x

(2.1 f)

It is also observed that, xy yx , yz zy and zx xz


16

The equations (2.1) are called the strain-displacement relations, since they define the

strain components in terms of the displacement components.

By the application of Hooke’s Law, that is, the linear relation between the stress and

strain components and the principle of superposition, which are both based on

experimental observation, the relation between the components of stress and the

components of strain are given by Timoshenko and Goodier [5]

1

xx xx yy zzE

(2.2a)

1

yy yy zz xxE (2.2b)

1

zz zz xx yyE

(2.2c)

2 1xy xyE

(2.2d)

2 1yz yzE

(2.2e)

2 1zx zxE

(2.2f)

where, E is modulus of elasticity or Young’s modulus, and ν is Poisson’s ratio.

2.4 Differential equations of equilibrium

In section 2.2, the stress at a point of an elastic body has been considered. Let

variation of the stress as we change the position of the point. Let us consider the

conditions of equilibrium of a small rectangular parallelepiped with the sides x , y

and z , (Figure 2.1). The components of the stresses acting on the sides of this small

element and their positive directions are indicated in the Figure 2.1. It has taken into

account the small changes of the components of the stress due to small increase x ,

y and z of the coordinates. The subscript of σ denotes the value of stress component

at the point in x, y, z directions.

If Fx, Fy, Fz denote the components of body force per unit volume of the element, then

the three equations of equilibrium are obtained by summing all the forces acting on

the element in x, y, z direction. The three equilibrium equations are as follows [5]:


17

0xyxx xzxF

x y z

(2.3a)

0yy yx yzyF

y x z

(2.3b)

0zyzxzzzF

z x y

(2.3c)

Equations (2.3) must be satisfied at all points throughout the volume of the body. The

stress components vary over the volume of the body, and near the boundary they must

be such as to be in equilibrium with the external forces on the boundary of the body,

so that external forces may be regarded as a continuation of the internal stress

distribution.

2.5 Compatibility equations

It should be noted that the six components of strain at each point are completely

determined by the three functions ux, uy and uz representing the components of

displacement. Hence the components of strain cannot be taken arbitrary as a function

of x, y and z. Now Equations (2.1) are differentiated twice and after simple

manipulation, the following set of differential equations are obtained [5].

2 22 2

2 2 ; 2yy xy yz xyxx xx zx

y x x y y z x x y z

(2.4a)

2 2 22

2 2 ; 2yy yz yy yz xyzxzz

z y y z z x y x y z

(2.4b)

2 22 2

2 2 ; 2 yz xyxx zx zxzz zz

x z z x x y z x y z

(2.4c)

These differential relations are called the Compatibility Conditions or Compatibility

Equations. If there are no body forces or if the body forces are constant, another form

of the compatibility equations can be rewritten as [5]

2 2

2 221 0; 1 0xx yzx y z

(2.5a)


18

2 2

2 221 0; 1 0yy xzy x z

(2.5b)

2 2

2 221 0; 1 0zz xyz x y

(2.5c)

where 2 2 2

22 2 2

xx yy zz

x y z

The solution of an elasticity problem must satisfy the equilibrium equations (2.3) and

the compatibility conditions (2.4) along with the prescribed boundary conditions.

2.6 Plane elasticity

Although the elastic analysis, in general, is of three dimensional form, it can be

analyzed using two dimensions on the consideration of symmetry of planes. For such

simplification there are two options:

(a) Plane stress condition and

(b) Plain strain condition

2.6.1 Plane stress condition

The plane stress condition is considered to be a state of stress in which the normal

stress σzz and the shear stresses σzx and σyz directed perpendicular to the plane are

assumed to be zero. Generally, members that are thin (those with a small z dimension

compared to in-plane x and y directions) and whose loads act only in the x-y plane can

be considered to be under plane stress. Thus a state of plane stress exists in a thin

object loaded in the plane of its largest dimensions. The non-zero stresses σxx, σyy and

σxy have been shown in Figure 2.2 lie in the x-y plane and are independent of z and

hence the functions of x and y only. A thin beam loaded in its plane and a spur gear

tooth are good examples of plane stress problem.


19

Figure 2.2: Stress components under plane stress conditions

For plane stress conditions,

0 0 0zz zx yz (2.6)

The stress-strain relations in case of plane stress condition are

1xx xx yyE

(2.7 a)

1yy yy xxE

(2.7 b)

2 1xy xyE

(2.7 c)

2.6.1 Plane strain condition

Plain strain condition is said to be a state of strain in which the strain normal to the x-y

plane, εzz and the shear strains εzx and εyz are assumed to be zero. The assumptions of

the plane strain condition are realistic for long bodies (saying in the z direction) with

constant cross-sectional area subjected to loads that act only in the x and/or y

directions and do not vary in the z direction. Also the other component of

displacement uz is zero all over the body. These conditions can be stated

mathematically, from equations (2.1) as

σyy

σxx

σxy

σyy

σxx

σxy

σxy

σxy


20

0zzz

uz

(2.8 a)

0y zzy

u uz y

(2.8 b)

0x zzx

u uz x

(2.8 c)

The above equations (2.8), in combinations with (2.2c), (2.2e) and (2.2f), show that

the stress components σzx = σzy = 0 and stress components can be determined from the

knowledge of σxx and σyy by the relation

zz xx yy (2.9)

Therefore, for both the circumstances, the problem ultimately reduces to the

determination of σxx, σyy and σxy as a function of x and y only.

The relations in case of plane strain condition are

21 1 1xx xx yyE

(2.10 a)

21 1 1yy yy xxE

(2.10 b)

2 1xy xyE

(2.10 c)

2.6.3 Equilibrium equations and compatibility conditions for plane elasticity

For the case of both plane stress and plane strain problems, the equilibrium equation

(2.3) reduces to [5]

0xyxxxF

x y

(2.11 a)

0yy xyyF

y x

(2.11 b)


21

In the above equations of equilibrium (Eq. (2.11) for plane problems of elasticity, the

body force are assumed, as they will be throughout this work, to be absent. The

equations (2.11) thus become

0xyxx

x y

(2.12 a)

0yy xy

y x

(2.12 b)

These equilibrium equations (2.12) are required to be solved for the case of a two-

dimensional problem. These two equations are not sufficient for the determination of

three stress components σxx, σyy and σxy. Thus, to evaluate these three dependent

variables a third equation is necessary. This third equation comes from the

consideration of the elastic deformation of the body. This additional equation ensures

continuity of deformation in the body which is known as compatibility equation for

the present case. In fact, it ensures compatibility of displacement ux and uy. The

mathematical formulation of this condition can be obtained from the strain

displacement relation. For two dimensional cases, these relations are

xxx

ux

(2.13 a)

yyy

uy

(2.13 b)

yxxy

uuy x

(2.13 c)

Differentiating the equation (2.13 a) twice with respect to y, the equation (2.13 b)

twice with respect to x and the equation (2.13 c) once with respect to x and once with

respect to y, the expression for condition of compatibility in term of strain is found as

follows

2 22

2 2y xyx

y x x y

(2.14)


22

This equation is known as the condition of compatibility for plane elasticity. To

express this compatibility equation in terms of stress components, the strain

components present in equation (2.14) have to be eliminated by their relations with

the stress components. This relations can be obtained from the equations (2.2a), (2.2b)

and (2.2d) by considering σzz = 0 in case of plane stress and σzz = ν(σxx + σyy) in case of

plane strain. One can obtain the compatibility equation in terms of stress components

as follows:

2 2

2 21

1xx yyX Y

x y x y

(2.15)

2.7 Stress-strain relations for different types of materials

The stress-strain relationship for a general material that is not linearly elastic is more

complicated. In the simplest approximation the relation between stress and strain is

taken to be linear. The generalized Hooke’s law relating stresses to strains can be

written in contracted notation as Kaw [32]

1 111 12 13 14 15 16

2 221 22 23 24 25 26

3 331 32 33 34 35 36

23 2341 42 43 44 45 46

51 52 53 54 55 5631 31

61 62 63 64 65 6612 12

C C C C C CC C C C C CC C C C C CC C C C C CC C C C C CC C C C C C

(2.16)

where σ1, σ2, σ3, σ23, σ31 and σ12 are the stress components of a three-dimensional cube

in 1, 2, 3 co-ordinates as shown in Figure 2.3. Now, Inverting equation (2.16), the

general strain-stress relationship for a three dimensional body in a 1 - 2 - 3 orthogonal

Cartesian coordinate system is

1 111 12 13 14 15 16

2 221 22 23 24 25 26

3 331 32 33 34 35 36

23 2341 42 43 44 45 46

51 52 53 54 55 5631 31

61 62 63 64 65 6612 12

S S S S S SS S S S S SS S S S S SS S S S S SS S S S S SS S S S S S

(2.17)


23

where, 1

ijS

C

is called the compliance matrix and

11 12 13 14 15 16

21 22 23 24 25 26

31 32 33 34 35 36

41 42 43 44 45 46

51 52 53 54 55 56

61 62 63 64 65 66

ij

S S S S S SS S S S S SS S S S S S

SS S S S S SS S S S S SS S S S S S

(2.18)

Figure 2.3: Stresses on cubic element

2.7.1 Anisotropic material

The material that has 21 independent elastic constants at a point is called an

anisotropic material [32]. Once these constants are found for a particular point, the

stress and strain relationship can be developed at that point. The stiffness matrix, Cij

has 36 constants in equation (2.16). However less than 36 of the constants can be

shown to actually be independent for elastic material when important characteristics

of the strain energy are considered. From the consideration of strain energy density, it

2

3

1

σ3

σ32

σ31

σ23

σ2

σ21

σ13

σ12

σ1


24

can be shown that Cij = Cji. Thus 36 constants of the stiffness matrix come down to 21

independent constants and the stiffness matrix turns to a symmetric matrix as follows

[32]

11 12 13 14 15 16

21 22 23 24 25 26

31 32 33 34 35 36

41 42 43 44 45 46

51 52 53 54 55 56

61 62 63 64 65 66

ij

C C C C C CC C C C C CC C C C C C

CC C C C C CC C C C C CC C C C C C

(2.19)

2.7.2 Monoclinic material

Material having symmetry with respect to one plane is referred to as monoclinic

materials. For such case of material, transformation of axis can be done and found that

C14 = C15 = C24 = C25 = C34 = C35 = C46 = C56 = 0 and then the number of independent

elastic constant becomes 13 only. So the stiffness matrix of equation (2.19) reduces to

[32]

11 12 13 16

12 22 23 26

13 23 33 36

44 45

45 55

16 26 36 66

0 00 00 0

0 0 0 00 0 0 0

0 0

ij

C C C CC C C CC C C C

CC CC C

C C C C

(2.20)

2.7.3 Orthotropic material

If a material has three mutually perpendicular planes of material symmetry, then it is

called orthotropic or orthogonally anisotropic or specially orthotropic material [32].

The stiffness matrix can be derived by starting from the stiffness matrix [Cij] of

monoclinic material. With two more planes of symmetry, it gives C16 = C26 = C36 =

C45 = 0. Thus, the stiffness matrix becomes


25

11 12 13

12 22 23

13 23 33

44

55

66

0 0 00 0 00 0 0

0 0 0 0 00 0 0 0 00 0 0 0 0

ij

C C CC C CC C C

CC

CC

(2.21)

Three mutually perpendicular planes of material symmetry also imply three mutually

perpendicular planes of elastic symmetry. Again an orthotropic material has at least

two orthogonal planes of symmetry, where material properties are independent of

directions within each plane. Example of an orthotropic material include a single ply

of continuous fiber composite arranged in rectangular array. The compliance matrix

(inverse of stiffness matrix) of orthotropic materials reduces to

11 12 13

12 22 23

13 23 33

44

55

66

0 0 00 0 00 0 0

0 0 0 0 00 0 0 0 00 0 0 0 0

ij

S S SS S SS S S

SS

SS

(2.22)

2.7.3 Isotropic material

If all planes in an orthotropic body are identical, it is an isotropic material, then the

stiffness matrix is given by [32]

11 12 12

12 11 12

12 12 11

11 12

11 12

11 12

0 0 00 0 00 0 0

0 0 0 0 02

0 0 0 0 02

0 0 0 0 02

ij

C C CC C CC C C

C CC

C C

C C

(2.23)


26

For a linear isotropic material in the three-dimensional stress state, stress-strain

relationships at a point in an x – y - z orthogonal system in matrix form are

0 0 01 1

0 0 02 2

0 0 03 3

0 0 0 0 023 23

0 0 0 0 031 31

0 0 0 0 0 1212

1

1

1

1

1

1

E E E

E E E

E E E

G

G

G

(2.24)

11 2 1 1 2 1 1 2 1

11 2 1 1 2 1 1 2 1

11 2 1 1 2 1 1 2 1

0 0 01 1

0 0 02 2

0 0 03 3

0 0 0 0 023 23

0 0 0 0 031 31

0 0 0 0 012 12

E E E

E E E

E E E

G

G

G

(2.25)

The shear modulus G is a function of two elastic constants, E and ν, as

2(1 )

EG

Relating equations (2.23) and (2.25), we find

111

1 2 1EC

(2.26 a)

12 1 2 1EC

(2.26 b)

11 12

2 2 1C C E G

(2.26 c)


27

2.8 Composite ply

2.8.1 Unidirectional composite ply

A unidirectional fiber reinforced composite ply shown in Figure 2.4 falls under the

orthotropic material category. If the ply is thin and does not carry any out-of-plane

loads, one can assume plane stress conditions for the ply. Therefore, taking Equation

(2.17) and (2.22) and assuming σ3 = 0, σ23 = 0 and σ31 = 0, then the strain-stress

relation for an orthotropic plane stress problem can be written as Kaw [32] and Jones

[33]

1 111 12

2 12 22 2

6612 12

00

0 0

S SS S

S

(2.27)

Inverting Eq. (2.27), gives stress-strain relationship as Kaw [32] and Jones [33]

1 111 12

2 12 22 2

6612 12

00

0 0

Q QQ Q

Q

(2.28)

where, [Qij] is called the reduced stiffness matrix, the elements of which are related to

the engineering constants as follows:

1 12 211 12

12 21 12 21

222 66 12

12 21

1 1

1

E EQ Q

EQ Q G

(2.29)


28

Figure 2.4: Stress components on a plane of unidirectional fiber reinforced composite ply

Again for elastic constant Qij of equation (2.29) the reciprocal relations can be

reduced as:

12 21

1 2E E

(2.30)

where,

E1 = longitudinal Young’s modulus (in direction 1)

E1 = longitudinal Young’s modulus (in direction 2)

ν12 = major Poisson’s ratio, where, the general Poisson’s ratio, νij is defined as

the ratio of the negative of the normal strain in direction j to the normal strain

in in direction i, when only normal load is applied in direction i

G12 = in-plane shear modulus (in plane 1 - 2)

The unidirectional ply is a specially orthotropic ply because normal stress applied in

the 1 - 2 direction do not result in shearing strains in the 1-2 plane because Q16 = Q26

= S16 = S26 = 0.

2

1

σ22

σ11

σ12

σ22

σ11

σ12

σ12

σ12

Matrix Fiber


29

2.8.2 Composite angle ply

The co-ordinate system used for showing an angle ply is given in Figure (2.5). The

axes in the 1 - 2 coordinate system are called the local axes or the material axes. The

direction 1 is called the longitudinal direction and it is parallel to the fibers and the

direction 2 is called the transverse direction and it is perpendicular to the fibers. The

angle between the two axes is denoted by an angle θ. The stress-strain relationship for

two dimensional angle ply is given by Kaw [32], Jones [33]

11 12 16

12 22 26

16 26 66

xx xx

yy yy

xy xy

Q Q QQ Q QQ Q Q

(2.31)

where ijQ are called the elements of the transformed reduced stiffness matrix which

are given by

4 2 2 411 11 12 66 22cos 2 2 sin cos sinQ Q Q Q Q (2.32 a)

2 2 4 412 11 22 66 124 sin cos sin cosQ Q Q Q Q (2.32 b)

4 2 2 422 11 12 66 22sin 2 2 sin cos cosQ Q Q Q Q (2.32 c)

3 316 11 12 66 22 12 662 cos sin 2 sin cosQ Q Q Q Q Q Q (2.32 d)

3 326 11 12 66 22 12 662 sin cos 2 cos sinQ Q Q Q Q Q Q (2.32 e)

2 2 4 466 11 22 12 66 662 2 sin cos sin cosQ Q Q Q Q Q (2.32 f)

Note that there are six different elements are in the Q matrix and it can be seen that

they are just function of the four stiffness elements, Q11, Q 12, Q22, Q66 and the

orientation angle of the fiber in ply, θ.


30

Figure 2.5: Stress components on a plane of an angle ply

2.9 Laminated composite material

A real structure, in general, do not consist of a single ply but a laminate consisting of

more than one ply of at least two different materials bonded together through their

thickness (Figure 2.6). Lamination is used to combine the best aspects of the

constituent layers and bonding material in order to achieve a more useful material.

Bonding plies together results in a compellingly large increase in bending resistance.

Other properties that can be emphasized by lamination are strength, stiffness, low

weight, corrosion resistance, wear resistance, beauty or attractiveness, thermal

insulation, acoustical insulation, etc.

σxx

σxy

σyy

σxy

σyy

σxx

σxy

σxy

Matrix Fiber

1-2: Local co-ordinate system

x-y: Global co-ordinate system

y

x

1

2

θ


31

(b) Five ply laminate

(a) Five plies of various fiber orientations (c) Cross-section and mid-plane of laminate

Figure 2.6: Relationship between displacements through the thickness of a plate to mid-plane displacements and curvatures.

2.9.1 Strain-displacement relations

Knowledge of the variation of stress and strain through the laminate thickness is

essential to the definition of the extensional and bending stiffness of a laminate. The

classical lamination theory is used to develop these relationships. The following

assumptions are made in the classical lamination theory Kaw [32].

(a) Each ply is orthotropic.

(b) Each ply is homogenous.

(a) A line straight and perpendicular to the middle surface remains straight and

perpendicular to the middle surface during deformation.

(b) The laminate is thin and is loaded only in its plane.

(c) Each ply is elastic.

(d) No slip occurs between the ply interfaces.

According to the classical lamination theory, the strain-displacement relations are

Fiber

x

y

z

h/2

Mid plane

h/2

Z

x


32

0

0

0

xxxx xx

yy yy yy

xy xyxy

z

(2.33)

where,

0

0

0

xx

yy

xy

and xx

yy

xy

are the mid-plane strain and the mid-plane curvature and z

is distance from the mid-plane to different layers in laminate through thickness in z

direction as shown in Figures 2.6 and 2.7.

Figure 2.7: Coordinate location of plies in a laminate.

2.9.2 Resultant laminate forces and moments

The mid-plane strains and plate curvature in equation (2.33) are the unknowns for

finding the ply strains and stresses. The stresses of individual ply can be integrated

through the laminate thickness to give resultant forces and moments. The forces and

moment applied to a laminate will be known, so the mid-plane strain and plate

curvatures can then be found. The relationship between applied loads and strain and

plate curvatures can be written as Kaw and Jones [32, 33] 0

0

0

11 12 16 11 12 16

12 22 26 12 22 26

16 26 66 16 26 66

xx

yy

xy

xx xx

yy yy

xy xy

N B B BN B B B

B B BN

A A AA A AA A A

(2.34 a)

1

2

3

k-1

k

k+1

n

h/2

h2

h1

h1

z

x

h/2 hn

hn-1

hk

hk-1


33

0

0

0

11 12 16 11 12 16

12 22 26 12 22 26

16 26 66 16 26 66

xxxx xx

yy yy yy

xy xyxy

M B B B D D DM B B B D D D

B B B D D DM

(2.34 b)

11

, 1,2,6n

ij ij k kk k

A Q h h i j

(2.35 a)

2 21

1

1 , 1,2,62

n

ij ij k kk k

B Q h h i j

(2.35 b)

3 31

1

1 , 1,2,63

n

ij ij k kk k

D Q h h i j

(2.35 c)

where,

Nx, Ny = normal force per unit length.

Nxy = shear force per unit length.

Mx, My = bending moments per unit length.

Mxy = twisting moments per unit length.

hk and hk-1 is the coordinate location of plies as shown in Figure 2.7 and n is the total

number of ply. The [A], [B] and [D] matrices are called the extensional, coupling, and

bending stiffness matrices respectively. The extensional stiffness matrix [A] relates

the resultant in-plane forces to the in-plane strains and the bending stiffness matrix

[D] relates the resultant bending moments to the plate curvatures. The coupling

stiffness matrix [B] couples the force and moment terms to the mid-plane strains and

mid-plane curvature.

2.10 Hybrid laminates

A hybrid laminate is a mixture of two or more fiber or matrix systems to form a

laminate. For example, graphite/epoxy plies are used with Kevlar-49/epoxy plies to

create wing-to-body fairings for the Boeing 757 and 767 [33]. When designing lighter

and more economic products, hybrid structures offer great advantages because they

enable tailoring the properties of a product in a way which is unattainable by any

material alone. The main four types of hybrid laminates follow Kaw [32].


34

Interply hybrid laminates contain plies made of two or more different material

systems as shown in Figure (2.8). Sometimes they can be a combination of

isotropic plies and unidirectional fiber reinforced composite ply [34].

Intraply hybrid composites consist of two or more different fibers used in the

same ply.

An interply–intraply hybrid consists of plies that have two or more different

fibers in the same ply and distinct composite systems in more than one ply.

Resin hybrid laminates combine two or more resins instead of combining two

or more fibers in a laminate. Generally, one resin is flexible and the other one

is rigid.

Figure 2.8: Five ply hybrid laminate consisting of plies of two different fiber materials in the same matrix

The extensional stiffness matrix [A], the coupling stiffness matrix [B] and the bending

stiffness matrix [D] are functions of the transformed reduced stiffness matrix Q and

thickness of each ply in a hybrid composite laminate. Eventually, matrices [A], [B]

and [C] are functions of E1, E2, ν12, G12 and θ of each ply constituting the laminated

structure.

Fiber material 2

Fiber material 1


35

2.11 Special cases of laminates

This section is devoted to those special cases of laminates for which the stiffnesses

[A], [B] and [C] take on certain simplified values as opposed to the general form in

equation (2.35). Based on angle, material and thickness of plies, the symmetry or

antisymmetry of a laminate may zero out some elements of the three stiffness

matrices. They are important to study because they may result in reducing or zeroing

out the coupling of forces and bending moments, normal and shear forces, or bending

and twisting moments.

2.11.1 Symmetric laminates

A laminate is called symmetric if the material, angle and thickness of plies are same

above and below the mid-plane. For symmetric laminates from the definition of [B]

matrix, it can be proved that [B] = 0. Thus equation (2.34) can be rewritten as

0

0

0

11 12 16

12 22 26

16 26 66

xxxx

yy yy

xy xy

NNN

A A AA A AA A A

(2.36 a)

11 12 16

12 22 26

16 26 66

xx xx

yy yy

xy xy

MMM

D D DD D DD D D

(2.36 b)

This shows that the force and moment terms are uncoupled. Thus, if a laminate is

subjected only to force, it will have zero mid-plane curvatures. Similarly, if it is

subjected only to moments, it will have zero mid-plane strains.

For symmetric laminated composite, the effect of curvature of the laminate under in

plane loading is usually neglected. So,

0xx

yy

xy

and Eq. (2.33) can be written as

0

0

0

xx

yy

xy

xx

yy

xy

(2.37)


36

2.11.2 Cross-ply laminates

A laminate is called a cross-ply laminate (also called laminates with specially

orthotropic layers) if only 0° and 90° plies are used to make a laminate. For cross-ply

laminates, A16 = A26 = B16 = B26 = D16 = D26 = 0; Equation (2.34) can be written as

0

0

0

11 12 11 12

12 22 12 22

66 66

0 00 0

0 0 0 0

xxxx xx

yy yy yy

xy xyxy

N B BN B B

BN

A AA A

A

(2.38 a)

0

0

0

11 12 11 12

12 22 12 22

66 66

0 00 0

0 0 0 0

xxxx xx

yy yy yy

xy xyxy

M B B D DM B B D D

B DM

(2.38 b)

In this case, uncoupling occurs between the normal and shear forces, as well as

between the bending and twisting moments.

If a cross-ply laminate is symmetric, then in addition to the preceding uncoupling, the

coupling matrix [B] = 0 and no coupling takes place between the force and moment

terms. Thus, equation (2.38) can be expressed for Symmetric Cross-Ply Laminate as

0

0

0

11 12

12 22

66

00

0 0

xxxx

yy yy

xy xy

NNN

A AA A

A

(2.39 a)

11 12

12 22

66

00

0 0

xx xx

yy yy

xy xy

MMM

D DD D

D

(2.39 b)

Equation (2.39 a) and (2.39 b) is called the force and moment equations for Symmetric

Cross-ply Laminate. Substituting the value of equation (2.37), equation (2.39)

becomes

11 12

12 22

66

00

0 0

xx

yy

xy

xx

yy

xy

NNN

A AA A

A

(2.40)


37

Equation (2.40) is called the Force equation of Symmetric Cross-Ply Laminate under

in plane loading.

2.11.3 Angle ply laminates

A laminate is called an angle ply laminate if it has the plies of the same material and

thickness and only oriented at +θ and - θ directions. If a laminate has an even number

of plies, then A16 = A26 = 0. However, if the number of plies is odd and it consists of

alternating θ and - θ plies, then it is symmetric, given [B] = 0 and A16, A26, D16 and

D26 also become small as the number of layers increases for same laminate thickness.

This behavior is similar to the symmetric cross-ply laminates. If an angle ply consists

of even number of plies, force equation (2.34) for the angle ply laminates can be

written as

0

0

0

11 12 11 12 16

12 22 12 22 26

66 16 26 66

00

0 0

xxxx xx

yy yy yy

xy xyxy

N B B BN B B B

B B BN

A AA A

A

(2.41 a)

0

0

0

11 12 16 11 12 16

12 22 26 12 22 26

16 26 66 16 26 66

xxxx xx

yy yy yy

xy xyxy


B B B D D DM

(2.41 b)

If an angle ply laminate is symmetric, the coupling matrix [B] = 0 and no coupling

takes place between the force and moment terms. Thus, equation (2.41) can be

expressed for Symmetric Angle Ply Laminates as

0

0

0

11 12

12 22

66

00

0 0

xxxx

yy yy

xy xy

NNN

A AA A

A

(2.42 a)

11 12 16

12 22 26

16 26 66

xx xx

yy yy

xy xy

MMM

D D DD D DD D D

(2.42 b)


38

Since laminate is symmetric, equation (2.37) is also valid for symmetric angle ply

laminates. Equation (2.42 a) can be rewritten as

11 12

12 22

66

00

0 0

xx

yy

xy

xx

yy

xy

NNN

A AA A

A

(2.43)

Above (2.43) is called force equation of the angle ply laminates under in plane

loading. It is not only applicable for even number angle plies but also for large

number odd plies of symmetric laminates. Above force equation (2.43) of angle ply

laminate has the exact same form as that of symmetric cross-ply laminate.

2.11.4 Balanced laminates

A laminate is balanced if layers at angles other than 0° and 90° occur only as plus and

minus pairs of +θ and - θ. The plus and minus pairs do not need to be adjacent to each

other, but the thickness and material of the plus and minus pairs need to be the same.

Here, the terms A16 = A26 = 0. An example of a balanced laminate is [30/40/ − 30/30/ −

30/ − 40]. Equation (2.34) can be written as

0

0

0

11 12 11 12 16

12 22 12 22 26

66 16 26 66

00

0 0

xxxx xx

yy yy yy

xy xyxy

N B B BN B B B

B B BN

A AA A

A

(2.44 a)

0

0

0

11 12 16 11 12 16

12 22 26 12 22 26

16 26 66 16 26 66

xxxx xx

yy yy yy

xy xyxy


B B B D D DM

(2.44 b)

If a balanced laminate is symmetric, the coupling matrix [B] = 0 and no coupling

takes place between the force and moment terms. Thus, equation (2.44) can be

expressed for Symmetric Balanced Laminates as 0

0

0

11 12

12 22

66

00

0 0

xxxx

yy yy

xy xy

NNN

A AA A

A

(2.45 a)


39

11 12

12 22

66

00

0 0

xx xx

yy yy

xy xy

MMM

D DD D

D

(2.45 b)

Since the laminate is symmetric, equation (2.37) is also valid for symmetric balanced

laminate. Equation (2.45 a) can then be rewritten as

11 12

12 22

66

00

0 0

xx xx

yy yy

xy xy

NNN

A AA A

A

(2.46)

Above force equation (2.46) of balanced laminate has the exact same form as that of

symmetric cross-ply laminate.

2.12 Available mathematical models of elasticity

In the existing mathematical models of elasticity, the two-dimensional problems are

formulated either in terms of three stress components or in terms of two-displacement

components. However, neither of the approaches is suitable for solving the practical

problems of elasticity. This is mainly because of the inability to deal with a large

number of variables in their numerical computation and also of the involvement of

mixed boundary conditions. The problem is severe in cases of non-uniform structures.

In fact, serious attempts have been started towards the solution of two-dimensional

practical problems of elasticity after the introduction of the finite-element method of

solution. It is noted that in the finite element method of solution, at least two variables

are used at each node of an element for solving two dimensional problems. The

corresponding computational work remains similar to that of the displacement

formulation of the problems.

2.12.1 Airy’s stress function formulation

The usual solution procedure of the two-dimensional elastic problems is to introduce a

function ϕ(x, y), known as the stress function or Airy’s stress function defined by

Timoshenko and Goodier [5],


40

2

2xx y

(2.47 a)

2

2yy x

(2.47 b)

2

xy x y

(2.47 c)

The compatibility condition from equation (2.14) is

2 22

2 2yy xyxx

y x x y

(2.48)

By inverse operation, equation (2.43) can be written as

11 12

12 22

66

00

0 0

xx xx

yy yy

xy xy

I II I h

I

(2.49)

The inverse operation of the compliance matrix I and the stiffness matrix A have a

relation of [I] = [A]-1.

The function ϕ(x, y) defined by the equations (2.47) must satisfy the equilibrium

equations (2.12) and the compatibility equation (2.48). By making use of equations

(2.12), (2.47) and (2.49), one can obtain

4 4 466

22 12 114 2 2 42 02

II I Ix x y y

(2.50)

Combining the equations (2.12), (2.47) and (2.49), expression of the strain

components in terms of stress function ϕ can be written as

2 2

11 122 2xxu I I hx y y

(2.51 a)

2 2

22 122 2yyv I I hy x y

(2.51 b)

2

66xyu v I hy x x y

(2.51 c)


41

The solution of elasticity problems following the Airy stress function approach

requires the solution of equation (2.50). However, this approach appears to be

efficient only for stress boundary conditions as one can readily apply the stress

boundary conditions by using equation (2.47). When boundary conditions are

prescribed in terms of displacement or constrains, it is quite difficult to directly apply

the boundary conditions as one requires integration of equation (2.51) before applying

the boundary conditions. Thus, this approach seems to be inconvenient for

displacement and mixed boundary conditions.

2.12.2 Displacement parameter approach

Making use of equilibrium equations (2.11) (neglecting body forces), strain-

displacement relations (2.13) and (2.36 a), the differential equation of equilibrium for

laminated composites in terms of displacement parameters, ux and uy are

2 2 22 2 2

11 16 66 16 12 66 262 2 2 22 0y y yx x x u u uu u uA A A A A A Ax x y y x x y y

(2.52 a)

2 2 22 2 2

16 12 66 26 66 26 222 2 2 22 0y y yx x x u u uu u uA A A A A A Ax x y y x x y y

(2.52 b)

Putting the strain values from equation (2.13), the three stress-displacement relations

for the plane stress problems, obtained from equation (2.36 a) and (2.37) are as

follows:

11 16 16 121 y yx x

xx

u uu uA A A Ah x x y y

(2.53 a)

12 26 26 221 y yx x

yy


(2.53 b)

16 66 66 261 y yx x

xy


(2.53 c)

Equation (2.52) represents two coupled second order elliptic partial differential

equations of equilibrium. It is quite difficult to obtain the solution satisfying these two

partial differential equations simultaneously especially with mixed conditions on the

boundaries, although this is suitable for applying displacement boundary conditions.


42

Further, equation (2.53) is the corresponding stress equation for displacement

parameter approach. This stress equation is directly related with displacement

parameters.

2.13 Displacement potential formulation

As discussed above, both the airy stress function approach and the displacement

parameter approach are not adequate to obtain numerical solution of elasticity

problems, particularly the mixed boundary value problems of non-uniform structures.

This necessitates a method for the solution of elasticity problems of uniform and non-

uniform structures of composite materials under any boundary conditions prescribed

in terms of either stress or displacement or any combination of these, i.e., mixed

boundary conditions. To realize this, a new function, called displacement potential

function ψ(x, y), defined in terms of the relevant displacement components of plane

elasticity, has been introduced for both uniform and non-uniform shaped structures

and different materials like isotropic [6, 18-21], orthotropic [17, 24-25], anisotropic

[30] and symmetric composite laminate [31].

The formulation derived earlier for the symmetric laminates [31] has been used for

the present numerical modelling of non-uniform hybrid balanced composite laminated

structure with mixed boundary conditions. For symmetric laminates subjected to in-

plane loadings only, one can neglect the effect of curvature, i.e, [κ] = 0. Thus equation

(2.33) can be reduced as 0

0

0

xx xx

yy yy

xy xy

(2.54)

If curvature effect is not present, then variation of strain is equal to the variation of

mid-plane strain. The stress–strain relations for the symmetric laminated composite

materials are expressed through the transformed material stiffness matrix from

equation (2.36 a) as follows

11 12 16

12 22 26

16 26 66

1xx x

yy y

xy xy

A A AA A A

hA A A

(2.55)


43

For different cases of symmetric cross ply, angle ply and balanced laminates, it is seen

that A16 = A26 = 0. Thus, the stress-strain relations for general symmetric laminates are

expressed through the stiffness matrix as follows:

11 12

12 22

66

01 0

0 0

xx x

yy y

xy xy

A AA A

hA

(2.56)

With reference to a rectangular coordinate system, in absence of body forces, the two

equilibrium equations for the solution of a general symmetric laminated composites

under plane stress condition, in terms of the displacement components ux and uy, are as

follows

22 2

11 12 66 662 2 0yx xuu uA A A Ax x y y

(2.57 a)

2 22

66 12 66 222 2 0y yxu uuA A A Ax x y y

(2.57 b)

In the present approach, the two-dimensional problem of elasticity is reduced to the

determination of a single function by using a scheme of reduction of unknowns. This

is done by expressing the displacement components in terms of a potential function of

space variables ψ (x, y) as follows:

2 2 2

1 2 32 2,xu x yx x y y

(2.58 a)

2 2 2

4 5 62 2,yu x yx x y y

(2.58 b)

Here, αi’s are unknown material constants. Combining Eqs. (2.57) and (2.58), one

obtains the equilibrium equations in terms of the function ψ (x, y), as follows:

4 4 4

1 11 2 11 4 12 66 2 66 6 12 664 3 3

4 4

1 66 3 11 5 12 66 3 662 2 4

{ ( )} { ( )}

{ ( )} 0

A A A A A A Ax x y x y

A A A A Ax y y

(2.59 a)


44

4 4 4

4 66 1 12 66 5 66 3 12 66 5 224 3 3

4 4

2 12 66 4 22 6 66 6 222 2 4

{ ( ) } { ( ) }

{ ( ) } 0

A A A A A A Ax x y x y

A A A A Ax y y

(2.59 b)

The constants, αi are determined in such a way that one of the equilibrium equations,

that is, Eq. (2.59 a), for example, is automatically satisfied under all circumstances.

This will happen when coefficients of all the derivatives present in Eq. (2.59 a) are

individually zero. That is, when

1 11 0A (2.60 a)

2 11 4 12 66( ) 0A A A (2.60 b)

1 66 3 11 5 12 66( ) 0A A A A (2.60 c)

2 66 6 12 66( ) 0A A A (2.60 d)

3 66 0A (2.60 e)

Thus, for ψ to be a solution of the stress problem, it has to satisfy Eq. (2.59 b) only.

However, the values of αi must be known in advance. There are five homogeneous

algebraic equations (Eq. (2.60)) for determining six unknown αi’s. An arbitrary value

is thus assigned to any one of these six unknowns and the remaining αi are solved

from Eq. (2.60). Assuming α2 = 1, the values of αi thus obtained, are as follows:

1 3 5 0

2 1

114

12 66

AA A

666

12 66

AA A

When the above values of αi are substituted in Eq. (2.59 b), the governing differential

equation for the solution of general symmetric laminated composites becomes

24 4 422 12 12 22

4 2 2 466 11 66 11 11

2 0A A A Ax A A A A x y A y

(2.61)


45

The two displacement parameters ux and uy are now expressed in terms of the

displacement potential function ψ (x, y) as

2

,xu x yx y

(2.62 a)

2 2

11 662 212 66

1,yu x y A AA A x y

(2.62 b)

And stress components in terms of displacement potential are

3 366

11 122 312 66

,xxAx y A A

h A A x y y

(2.63 a)

3 3

212 12 16 11 22 22 662 3

12 66

1,yy x y A A A A A A Ah A A x y y

(2.63 b)

3 366

11 123 212 66

,xyAx y A A

h A A x x y

(2.63 c)

Strain components in terms of the potential function are

3

2,xx x yx y

(2.64 a)

3 3

11 662 312 66

1,yy x y A AA A x y y

(2.64 b)

3 3

11 123 212 66

1,xy x y A AA A x x y

(2.64 c)

The distinguishing feature of the present approach is that all modes of boundary

conditions can be satisfied appropriately, whether they are specified in terms of

loading or physical restraints or any combination thereof, which is, however, not the

case for the standard stress function formulation. Formulating the problem of

elasticity in terms of the potential function ψ eventually reduces the computational

work by an amount of 87% [30].


46

2.13.1 Applicability of the formulation

The governing equation (2.61) and the body parameters (2.62), (2.63) and (2.64) of

the general symmetric laminates can readily be applied to the following cases of

laminated composites:

(a) Cross-ply laminates

The governing equation and body parameters mentioned above are applicable

for symmetric cross-ply laminates with any even or odd number of piles.

(b) Angle ply laminates

If an angle ply laminate has an even number of plies, then the elements of the

stiffness matrix A16 = A26 = 0. Again, when the laminate is symmetric, it is

known that given [B] = 0. This behavior is similar to the symmetric cross-ply

laminates. Further, when the number of plies is odd and the laminate consists

of alternating +θ and -θ plies, and if it is symmetric, then [B] = 0 and for a

large odd number of layers, A16 and A26 become closer to zero. This behavior

can be considered similar to the symmetric cross-ply laminates. So, the

governing equation (2.61) and body parameters (2.62), (2.63) and (2.64) for

symmetric angle ply laminates are applicable.

(c) Balanced laminates

A laminate is balanced if layers with fiber orientations other than 0° and 90°

occur only as plus and minus pairs of θ. The plus and minus pairs do not need

to be adjacent to each other, but the thickness and material of the plus and

minus pairs need to be the same. Thus an interplay hybrid laminate will be

balanced if each material system has plies of fiber orientations of plus and

minus pairs with equal thickness. Here, the terms A16 = A26 = 0 and if it is

symmetric, given [B] = 0. If the interplay hybrid laminate consists of pairs of

isotropic layers, the terms A16 and A26 will also be zero [33]. The governing

equation (2.61), body parameters (2.62), (2.63) and (2.64) are, thus, applicable

for hybrid or non-hybrid symmetric balanced laminates.


47

2.13.2 Boundary conditions

The boundary conditions at any arbitrary point on the boundary are known in terms of

the normal and tangential components of displacement, un and ut, and of stress, σn,

and σt. These four components are expressed in terms of σx, σy, σxy, ux, uy —the

components of stress and displacement with respect to the reference axes x and y of

the body. In Fig. 2.9, the positive normal direction on the boundary is outward,

positive tangential direction is anti-clockwise and the positive ϕ, the angle drawn from

positive x-axis to positive normal, is anti-clockwise. With these conventions, the

relations between interior and boundary components of displacement can be written

as [18]

n x yu u l u m (2.65 a)

t y xu u l u m (2.65 b)

Here, l and m are the direction cosines of the normal to the boundary.

Figure 2.9: Components of displacements on a boundary segment

ϕ

x

y n

t

un

ut ux

uy

Boundary segment


48

With reference to Fig. 2.10, the normal and tangential components of stress can be

written as

2 22n xx xy yyl lm m (2.66 a)

2 2( ) ( )t xy yy xxl m lm (2.66 b)

Figure 2.10: Components of stresses on a boundary segment

In order to solve the mixed boundary-value problems of non-uniform-shaped

structures using the present formulation, the boundary conditions need to be expressed

in terms of ψ which can be done by substituting the expressions (2.62) and (2.63) in

the above equations (2.65) and (2.66).

2.13.3 Evaluation of stress components for individual ply

After the solution of ψ, the stress, displacement and strain components of the overall

laminate are determined using equations (2.62), (2.63) and (2.64) respectively. For a

symmetric laminate, the distribution of strain and displacement components for all the

ϕ

σyy

t

σxy

n

σxx

σxy

Y

X

x

y


49

plies is identical. However, due to different materials and fiber orientations, the

stiffness of plies are different, which results in different stress distribution in

individual plies. The stress components of individual plies are calculated with the help

of strain distribution of overall laminate or global strain distribution by the following

equation

11 12 16

12 22 26

16 26 66 ii

xx xx

yy yy

xy xykk

Q Q QQ Q QQ Q Q

(2.67)

Where Q

is the transformed reduced stiffness matrix of the kth ply.

CHAPTER

Numerical Solution

3.1 Introduction

The mixed-boundary-value problems of non-uniform boundary shapes are generally

beyond the scope of analytical methods of solution. Rather, numerical solution of this

class of problems is the only plausible approach. Considering the relative advantage

of the finite-difference method, especially for the present displacement potential

function formulation, the governing differential equation of equilibrium and the

differential equations associated with the boundary conditions are discretized in terms

of the nodal values of ψ by the method of finite-difference. Finally the system of

linear algebraic equations resulting from the discretization of the governing equation

and the associated boundary conditions are solved for the discrete values of the

potential function at the nodal points of the domain concerned.

3.2 Discretization of the Computational Domain

In the present approach, the boundary of a non-uniform shaped two-dimensional body

under investigation is defined by the coordinates of points on the boundary line

ABIJKLCDHGFE enclosing the body with reference to a two-dimensional Cartesian

co-ordinate system, shown in Figure 3.1 a. As a typical non-uniform geometry, this

particular shape is taken only for explaining the boundary modelling scheme. The

area of interest is then enclosed by a rectangle, ABCD, of sides equal to the maximum

dimensions of the non-uniform shaped region along the direction of the coordinate

axes as shown in Figure 3.1 b. The rectangle, ABCD, is then divided into user defined

number of meshes with grid-lines parallel to the rectangular coordinate axes in such a

way that all boundary lines pass through the mesh points. The solid line represents the

physical boundary of the structure under consideration. It can be noted that whatever

the shape of the body is, it will always be enclosed by the rectangle, ABCD. In other

words, the maximum area that the body can occupy is ABCD, for instance, if the body

3

CHAPTER 3 | NUMERICAL ANALYSIS

51

Figure 3.1: Different steps involved in the discretization of the domain of a non-uniform body ABIJKLCDHGFE.

(a)

(b)

(c)


52

under consideration is a uniform rectangle, its boundaries will coincide with the sides

of rectangle ABCD.

The discretized form of governing differential equation is applied to each and every

nodal point inside the physical boundary. This leads to a set of algebraic equations for

the determination of nodal unknowns within the physical boundary. However, the

number of unknowns is greater than the number of equation available, which in turn

make the problem intractable. This is because of the fact that the application of the

governing equation to the interior points also involves the points on and outside the

boundary as shown in Figure 3.1 c. The line thus formed by connecting the

intermediate neighboring nodal points outside the physical boundary is called

imaginary boundary. The total number of nodal unknowns will be equal to the number

of interior nodal points together with those on the boundary and the exterior points

involved by the application of governing equation to the interior nodal points. To

make the problem tractable, it is necessary to generate more equations. Since there are

two conditions to be satisfied at an arbitrary point on the physical boundary of the

solid body, two finite difference expressions of the differential equations associated

with the boundary conditions are applied to the same point on the boundary. It leads

to the fact that two algebraic equations are assigned to a single point on the boundary.

The computer program is organized in such a fashion that out of these two equations,

one is used to evaluate the physical boundary point and the remaining one for the

corresponding point on the imaginary boundary. Thus, every mesh point of the

domain will have a single algebraic equation and the resulting system of algebraic

equations be solved by a suitable numerical method of solution. Imaginary boundaries

composed of field nodal points have, however, no physical existence and are included

only for the sake of computation. Once again, as a limiting case, if a uniform

rectangular body of size equal to the rectangle ABCD is considered, it will have an

imaginary boundary immediately beyond the boundary of the rectangle ABCD, on

four sides, as shown in Figure 3.2. The nodal points shown in this figure represent the

maximum possible nodal points that can be involved for geometry of any arbitrary

shape, since the area of the body will not exceed beyond that of ABCD. From this

point and onwards, the maximum possible nodal points that can be involved, as

shown in Figure 3.2, will be called the extreme nodal field. Now, if a non-uniform

shaped body is placed on the extreme nodal field, it will leave some nodal points


53

Figure 3.2: Extreme nodal field for uniform geometry.

unoccupied, as shown in Figure 3.3. Here, the occupied nodal points i.e. the internal

nodal points along with the physical boundary nodal points and imaginary nodal

points are called the active field nodal points.

Figure 3.3: A non-uniform geometry superimposed on the extreme nodal field.


54

A variable node numbering scheme is adopted here to discretize the non-uniform

computational domain using a rectangular mesh-network. Whatever the shape the

two-dimensional structure is, the extreme nodal field (Figure 3.2) represent the

maximum possible nodes involved. So, in order to make the computational scheme

general for both uniform and non-uniform geometry, all the nodes of the extreme

nodal field are numbered and this is done from left to right for each rows starting from

top and ending at the bottom (see Figure 3.4). The program generates the extreme

nodal field along with the corresponding node numbers according to the maximum

number of meshes along x- and y-axis, which is taken as input from the user. From the

extreme nodal field generated from the program, the geometry of the non-uniform

body is identified by assigning a number to each node within the enclosing rectangle,

which represents whether a particular node is inside or outside the physical boundary.

A node on and within the physical boundary will be characterized by the number 1

and those exterior to the physical boundary will be characterized by the number 0.

Figure 3.5 illustrates the nodal identification scheme by numbers 0 and 1 for the non-

uniform geometry described in Figure 3.3.

Figure 3.4: Node numbering scheme of the extreme nodal field.


55

1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Figure 3.5: Indicators 0 or 1 at each nodal point of the extreme field depending on whether corresponding node is outside or inside the boundary.

3.3 Finite Difference Discretization of the Governing Equation

The governing differential equation (2.61) for the general symmetric laminates used

to evaluate the function ψ at the interior mesh points, can be expressed in its

corresponding difference form when all the derivatives are replaced by corresponding

central difference expressions. The central difference expressions of the individual

fourth order derivatives of ψ present in the governing equation are as follows:

4

4 4,

2

1 { ( 2, ) 4 ( 1, ) 6 ( , ) 4 ( 1, )

( 2, )} ( )xi j

x

i j i j i j i jx h

i j O h

4

4 4,

2

1 { ( , 2) 4 ( , 1) 6 ( , ) 4 ( , 1)

( , 2)} ( )yi j

y

i j i j i j i jy k

i j O k

(3.1)

(3.2)


56

4

2 2 2 2,

2 2

1 [ ( 1, 1) ( 1, 1) ( 1, 1)

( 1, 1) 4 ( , ) 2{ ( 1, ) ( , 1)( , 1) ( 1, )}] ( , )

x yi j

x y

i j i j i jx y h k

i j i j i j i ji j i j O h k

Thus the finite difference form of the governing equation at a general mesh point (i, j)

can be obtained by replacing the derivatives of equation (2.61) with their difference

formulae as given by equation (3.1), (3.2) and (3.3). Assuming ψ to be the continuous

function at different mesh points, the equation in its finite difference form becomes

1[ ( 2, ) ( 2, )] (4 1 2 2){ ( 1, ) ( 1, )}(2 2 4 3)[ ( , 1) ( , 1)] (6 1 4 2 6 3) ( , )

2[ ( 1, 1) ( 1, 1) ( 1, 1) ( 1, 1)]3[ ( , 2) ( , 2)] 0

pk i j i j pk pk i j i jpk pk i j i j pk pk pk i j

pk i j i j i j i jpk i j i j

where

411x

pkh

2

22 22 122 2

66 11 66 11

2 12x y

A A ApkA A A A h k

224

11

13y

ApkA k

Here hx and ky are the distance between adjacent nodal points in x and y directions

respectively as shown in Figure 3.1. Therefore, equation (3.4) is the difference

approximation to the governing equation (2.61) of general symmetric laminates and

valid for all internal nodal points. Figure 3.6 illustrates the corresponding FD stencil

of equation (3.4) and its application for an arbitrary interior nodal point (i, j) on the

computational domain.

3.4 Finite Difference Discretization of Body Parameters

Considering an arbitrary point on the boundary, the body parameters associated with

the boundaries may be specified by any one of the four groups of boundary

conditions, namely (un, ut), (un, σt), (ut, σn), or (σn, σt). Therefore, there are always two

(3.3)

(3.4)


57

Figure 3.6: (a) Stencil for governing equation of general symmetric laminates (b) application of the governing equation stencil at internal points of the non-uniform

structure.

(a)

(b)


58

conditions to be satisfied at an arbitrary point on the boundary and these two

conditions are sufficient to provide two equations for the point. While discretizing the

differential equations associated with the body parameters, nodal points not included

by the application of the governing equation (2.61) at the interior nodal points are not

included in the process of their discretization. In our present computational scheme,

an attempt is made to develop only four sets of equations for each body parameter. As

the differential equations associated with the body parameters contain second- and

third-order derivatives of the function ψ, the use of central difference expressions at

the boundary ultimately leads to the inclusion of points other than active nodal points.

The derivatives of the boundary expressions are thus replaced by different

combinations of forward, backward and central difference formulae, keeping the local

truncation error of second order O (hx2) and O (ky

2). A body parameter discretized by

using a forward difference scheme in both x- and y-directions is referred to as

forward-forward combination of discretized form and so on. For every active nodal

point, the program automatically selects the most appropriate form of the formula or

stencil from the available four forms so that no nodal point other than the active field

nodal points is included. In order to do so, the user is required to assign each nodal

point within the enclosing rectangle a stencil indicating number (1, 2, 3, 4). A typical

arrangement can be seen in Figure 3.7. Imaginary nodes, however, follow the stencil

indicating number of the corresponding physical boundary node since the equation is

applied at the physical node even though the equation is used to evaluate ψ is at the

corresponding imaginary node.

Different forms of finite-difference expressions of the equations for body parameters

(2.62), (2.63) and (2.64) are expressed as follows:

2

( , )xu x yx y

Set X Y

1 Forward Forward

1[9 ( , ) 12{ ( , 1) ( 1, )} 16 ( 1, 1)3{ ( , 2) ( 2, )} 4{ ( 1, 2) ( 2, 1)}

( 2, 2)]

xu C i j i j i j i ji j i j i j i j

i j

(3.5)


59

1 1 1 1 1 1 1 1 1 1 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 0 0 0 0 0 0 0 0 0 0 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 0 0 0 0 0 0 0 0 1 1 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 4 4 0 0 0 0 0 0 0 0 0 0 0 0 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 4 4 0 0 0 0 0 0 0 0 0 0 0 0 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 4 4 0 0 0 0 0 0 0 0 0 0 0 0 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 4 4 0 0 0 0 0 0 0 0 0 0 0 0 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 4 4 0 0 0 0 0 0 0 0 0 0 0 0 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 4 4 0 0 0 0 0 0 0 0 0 0 0 0 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

Figure 3.7: Indicators 1, 2, 3 or 4 at each nodal point depending on form on stencil of

stress, strain and displacement components to be used in both stages pre- and post-processing.

Set X Y

2 Forward Backward

1[ 9 ( , ) 12{ ( , 1) ( 1, )} 16 ( 1, 1)3{ ( , 2) ( 2, )} 4{ ( 1, 2) ( 2, 1)}

( 2, 2)]


i j

(3.6)

Set X Y

3 Backward Forward

1[ 9 ( , ) 12{ ( , 1) ( 1, )} 16 ( 1, 1)3{ ( , 2) ( 2, )} 4{ ( 1, 2) ( 2, 1)}

( 2, 2)]


i j

(3.7)


60

Set X Y

4 Backward Backward

1[9 ( , ) 12{ ( , 1) ( 1, )} 16 ( 1, 1)3{ ( , 2) ( 2, )} 4{ ( 1, 2) ( 2, 1)}

( 2, 2)]


i j

(3.8)

The above four forms of FD stencils of ux are illustrated in Figure 3.8 for nodal points

interior and on the boundary of the solid body.

2 2

11 662 212 66

1( , )yu x y A AA A x y

Set X Y

1, 2, 3 and 4 Central Central

2 3

2 3

{ ( 1, ) ( 1, )} { ( , 1) ( , 1)}2( ) ( , )

yu C i j i j C i j i jC C i j

(3.9)

The single form of FD stencil of uy are illustrated in Figure 3.9 for nodal points

interior and on the boundary of the structure. It can be noted that a single central

difference approximation to the above displacement component would be sufficient

for all the nodal points of interest, which is because of the fact that the equation has

symmetry about both the x- and y-axis.

3 366

11 122 312 66

( , )( )xx

Ax y A Ah A A x y y

Set X Y

1 and 3 Central Forward

4 5 4 5 4 5

5 5 4

(6 10 ) ( , ) (8 12 ) ( , 1) (2 6 ) ( , 2)( , 3) 3 ( , 1) [ 3 ( 1, ) 3 ( 1, )

4 ( 1, 1) 4 ( 1, 1) ( 1, 2) ( 1, 2)]

xx C C i j C C i j C C i jC i j C i j C i j i j

i j i j i j i j

(3.10)


61

(a)

(b)

Figure 3.8: (a) Different forms of stencil for ux (b) application of the stencils at boundary and internal points of the non-uniform structure.


62

(a)

(b)

Figure 3.9: (a) Single form of stencil for uy (b) application of the stencils at boundary and internal points of the non-uniform structure.


63

Set X Y

2 and 4 Central Backward

4 5 4 5 4 5

5 5 4

(6 10 ) ( , ) (8 12 ) ( , 1) (2 6 ) ( , 2)( , 3) 3 ( , 1) [3 ( 1, ) 3 ( 1, )

4 ( 1, 1) 4 ( 1, 1) ( 1, 2) ( 1, 2)]

xx C C i j C C i j C C i jC i j C i j C i j i j

i j i j i j i j

(3.11)

3 3

212 12 66 11 22 22 662 3

12 66

1( , ) ( )( )yy x y A A A A A A A

h A A x y y

Set X Y


6 7 6 7 6 7

7 7 6

(6 10 ) ( , ) (8 12 ) ( , 1) (2 6 ) ( , 2)( , 3) 3 ( , 1) [ 3 ( 1, ) 3 ( 1, )

4 ( 1, 1) 4 ( 1, 1) ( 1, 2) ( 1, 2)]

yy C C i j C C i j C C i jC i j C i j C i j i j

i j i j i j i j

(3.12)

Set X Y


6 7 6 7 6 7

7 7 6

(6 10 ) ( , ) (8 12 ) ( , 1) (2 6 ) ( , 2)( , 3) 3 ( , 1) [3 ( 1, ) 3 ( 1, )

4 ( 1, 1) 4 ( 1, 1) ( 1, 2) ( 1, 2)]

yy C C i j C C i j C C i jC i j C i j C i j i j

i j i j i j i j

(3.13)

The above two forms of FD stencils of σxx and σyy are illustrated in Figure (3.10) for

nodal points interior and on the boundary of the structure.

3 366

12 112 312 66

( , )( )xy

Ax y A Ah A A x y x

Set X Y

1 and 2 Forward Central

8 9 8 9 8 9

8

9

(6 10 ) ( , ) (8 12 ) ( 1, ) (2 6 ) ( 2, )[ 3 ( , 1) 3 ( , 1) 4 ( 1, 1) 4 ( 1, 1)( 2, 1) ( 2, 1)] [3 ( 1, ) ( 3, )]

xy C C i j C C i j C C i jC i j i j i j i j

i j i j C i j i j

(3.14)


64

Set X Y

3 and 4 Backward Central

8 9 8 9 8 9

8

9

(6 10 ) ( , ) (8 12 ) ( 1, ) (2 6 ) ( 2, )[3 ( , 1) 3 ( , 1) 4 ( 1, 1) 4 ( 1, 1)( 2, 1) ( 2, 1)] [3 ( 1, ) ( 3, )]

xy C C i j C C i j C C i jC i j i j i j i j

i j i j C i j i j

(3.15)

The above two forms of FD stencils of σxy are illustrated in Figure (3.11) for nodal

points interior and on the boundary of the structure.

(a)

(b)

Figure 3.10: (a) Different forms of stencil for σxx and σyy (b) application of the stencils at boundary and internal points of the non-uniform structure.


65

(a)

(b)

Figure 3.11: (a) Different forms of stencil for σxy (b) application of the stencils at boundary and internal points of the non-uniform structure.


66

The boundary conditions at any arbitrary point on the physical boundary are usually

known in terms of the normal and tangential components of displacement, un and ut,

and of stress, σn, and σt. These four components are expressed in terms of σxx, σyy, σxy,

ux, uy (Eqs. 2.65 and 2.66). Different forms of finite-difference expressions of the

equations for boundary conditions are expressed as follows:

n x yu u l u m

Set

1

1

2 3

2 3

1 2 3

[9 ( , ) 12{ ( , 1) ( 1, )} 16 ( 1, 1)3{ ( , 2) ( 2, )} 4{ ( 1, 2) ( 2, 1)}

( 2, 2)][ { ( 1, ) ( 1, )} { ( , 1) ( , 1)}

2( ) ( , )]{9 2( ) } ( ,

n x yu u l u ml C i j i j i j i j

i j i j i j i ji j

m C i j i j C i j i jC C i j

C l C C m i

1 3

1 3 2

1 2 1

1 1 1

1

) ( 12 ) ( , 1)3 ( , 2) ( , 1) ( 1, )( 12 ) ( 1, ) 16 ( 1, 1)4 ( 1, 2) 3 ( 2, ) 4 ( 2, 1)

( 2, 2)

j C l C m i jC l i j C m i j C m i j

C l C m i j C l i jC l i j C l i j C l i j

C l i j

(3.16)

Set

2

1

2 3

2 3

1 2 3

[ 9 ( , ) 12{ ( , 1) ( 1, )} 16 ( 1, 1)3{ ( , 2) ( 2, )} 4{ ( 1, 2) ( 2, 1)}

( 2, 2)][ { ( 1, ) ( 1, )} { ( , 1) ( , 1)}

2( ) ( , )]{ 9 2( ) } (


i j i j i j i ji j


C l C C m

1 3

1 3 2

1 2 1

1 1 1

1

, ) (12 ) ( , 1)3 ( , 2) ( , 1) ( 1, )(12 ) ( 1, ) 16 ( 1, 1)4 ( 1, 2) 3 ( 2, ) 4 ( 2, 1)

( 2, 2)

i j C l C m i jC l i j C m i j C m i j


C l i j

(3.17)


67

Set

3

1

2 3

2 3

1 2 3

[ 9 ( , ) 12{ ( , 1) ( 1, )} 16 ( 1, 1)3{ ( , 2) ( 2, )} 4{ ( 1, 2) ( 2, 1)}

( 2, 2)][ { ( 1, ) ( 1, )} { ( , 1) ( , 1)}

2( ) ( , )]{ 9 2( ) } (


i j i j i j i ji j


C l C C m

1 3

1 3 2

1 2 1

1 1 1

1

, ) (12 ) ( , 1)3 ( , 2) ( , 1) ( 1, )(12 ) ( 1, ) 16 ( 1, 1)4 ( 1, 2) 3 ( 2, ) 4 ( 2, 1)

( 2, 2)

i j C l C m i jC l i j C m i j C m i j


C l i j

(3.18)

Set

4

1

2 3

2 3

1 2 3

[9 ( , ) 12{ ( , 1) ( 1, )} 16 ( 1, 1)3{ ( , 2) ( 2, )} 4{ ( 1, 2) ( 2, 1)}

( 2, 2)][ { ( 1, ) ( 1, )} { ( , 1) ( , 1)}

2( ) ( , )]{9 2( ) } ( ,


i j i j i j i ji j


C l C C m i

1 3

1 3 2

1 2 1

1 1 1

1

) ( 12 ) ( , 1)3 ( , 2) ( , 1) ( 1, )( 12 ) ( 1, ) 16 ( 1, 1)4 ( 1, 2) 3 ( 2, ) 4 ( 2, 1)

( 2, 2)

j C l C m i jC l i j C m i j C m i j


C l i j

(3.19)


68

t y xu u l u m

Set

1

2 3

2 3

1

1 2 3

[ { ( 1, ) ( 1, )} { ( , 1) ( , 1)}2( ) ( , )]

[9 ( , ) 12{ ( , 1) ( 1, )} 16 ( 1, 1)3{ ( , 2) ( 2, )} 4{ ( 1, 2) ( 2, 1)}

( 2, 2)]{ 9 2( ) } (

t y xu u l u ml C i j i j C i j i j

C C i jm C i j i j i j i j

i j i j i j i ji j

C m C C l i

1 3

1 3 2

1 2 1

1 1 1

1

, ) (12 ) ( , 1)3 ( , 2) ( , 1) ( 1, )(12 ) ( 1, ) 16 ( 1, 1)4 ( 1, 2) 3 ( 2, ) 4 ( 2, 1)

( 2, 2)

j C m C l i jC m i j C l i j C l i j

C m C l i j C m i jC m i j C m i j C m i j

C m i j

(3.20)

Set

2

2 3

2 3

1

1 2 3

[ { ( 1, ) ( 1, )} { ( , 1) ( , 1)}2( ) ( , )]

[ 9 ( , ) 12{ ( , 1) ( 1, )} 16 ( 1, 1)3{ ( , 2) ( 2, )} 4{ ( 1, 2) ( 2, 1)}

( 2, 2)]{9 2( ) } (



i j i j i j i ji j

C m C C l i

1 3

1 3 2

1 2 1

1 1 1

1

, ) ( 12 ) ( , 1)3 ( , 2) ( , 1) ( 1, )( 12 ) ( 1, ) 16 ( 1, 1)4 ( 1, 2) 3 ( 2, ) 4 ( 2, 1)

( 2, 2)



C m i j

(3.21)


69

Set

3

2 3

2 3

1

1 2 3

[ { ( 1, ) ( 1, )} { ( , 1) ( , 1)}2( ) ( , )]

[ 9 ( , ) 12{ ( , 1) ( 1, )} 16 ( 1, 1)3{ ( , 2) ( 2, )} 4{ ( 1, 2) ( 2, 1)}

( 2, 2)]{9 2( ) } (



i j i j i j i ji j

C m C C l i

1 3

1 3 2

1 2 1

1 1 1

1

, ) ( 12 ) ( , 1)3 ( , 2) ( , 1) ( 1, )( 12 ) ( 1, ) 16 ( 1, 1)4 ( 1, 2) 3 ( 2, ) 4 ( 2, 1)

( 2, 2)



C m i j

(3.22)

Set

4

2 3

2 3

1

1 2 3

[ { ( 1, ) ( 1, )} { ( , 1) ( , 1)}2( ) ( , )]

[9 ( , ) 12{ ( , 1) ( 1, )} 16 ( 1, 1)3{ ( , 2) ( 2, )} 4{ ( 1, 2) ( 2, 1)}

( 2, 2)]{ 9 2( ) } (



i j i j i j i ji j

C m C C l i

1 3

1 3 2

1 2 1

1 1 1

1

, ) (12 ) ( , 1)3 ( , 2) ( , 1) ( 1, )(12 ) ( 1, ) 16 ( 1, 1)4 ( 1, 2) 3 ( 2, ) 4 ( 2, 1)

( 2, 2)



C m i j

(3.23)

The corresponding four forms of FD stencils of un and ut are illustrated in Figure

(3.12) for nodal points on the boundary of the structure.


70

(a)

(b)

Figure 3.12: (a) Different forms of stencil for un or ut (b) application of the stencils at different boundary points of the non-uniform geometry.


71

2 22n xx xy yyl lm m

Set

1

2 2

24 5 4 5 4 5

5 5 4

8 9

2

[(6 10 ) ( , ) (8 12 ) ( , 1) (2 6 ) ( , 2)( , 3) 3 ( , 1) { 3 ( 1, ) 3 ( 1, )

4 ( 1, 1) 4 ( 1, 1) ( 1, 2) ( 1, 2)}]2 [(6 10 ) ( , ) (8

n xx xy yyl lm m

l C C i j C C i j C C i jC i j C i j C i j i j

i j i j i j i jlm C C i j C

8 9 8 9

8

92

6 7 6 7 6 7

7

12 ) ( 1, ) (2 6 ) ( 2, ){ 3 ( , 1) 3 ( , 1) 4 ( 1, 1) 4 ( 1, 1)( 2, 1) ( 2, 1)} {3 ( 1, ) ( 3, )}]

[(6 10 ) ( , ) (8 12 ) ( , 1) (2 6 ) ( , 2)( , 3) 3

C i j C C i jC i j i j i j i j

i j i j C i j i jm C C i j C C i j C C i jC i j C

7 6

2 24 5 6 7 8 9

2 24 5 6 7 8

2 24 5 6 7

( , 1) { 3 ( 1, ) 3 ( 1, )4 ( 1, 1) 4 ( 1, 1) ( 1, 2) ( 1, 2)}]

{(6 10 ) (6 10 ) 2(6 10 ) } ( , ){ (8 12 ) (8 12 ) 6 } ( , 1){(2 6 ) (2 6 ) } (

i j C i j i ji j i j i j i j

C C l C C m C C lm i jC C l C C m C lm i j

C C l C C m

2 25 7

2 2 2 25 7 8 4 6 92 2 2 2

4 6 4 62 2

4 6 8 92 2

4 6 8

, 2) ( ) ( , 3)( 3 3 6 ) ( , 1) ( 3 3 6 ) ( 1, )(4 4 ) ( 1, 1) ( ) ( 1, 2){ 3 3 2(8 12 ) } ( 1, )(4 4 8 ) ( 1, 1)

i j C l C m i jC l C m C lm i j C l C m C lm i j

C l C m i j C l C m i jC l C m C C lm i j

C l C m C lm i j

2 24 6

8 8 9 8

8 9

( ) ( 1, 2)8 ( 1, 1) 2(2 6 ) ( 2, ) 2 ( 2, 1)2 ( 2, 1) 2 ( 3, )

C l C m i jC lm i j C C lm i j C lm i jC lm i j C lm i j

(3.24)


72

Set

2

2 2

2 24 5 6 7 8 9

2 24 5 6 7 8

2 2 2 24 5 6 7 5 7

2 25 7 8 4

2

{ (6 10 ) (6 10 ) 2(6 10 ) } ( , ){(8 12 ) (8 12 ) 6 } ( , 1){ (2 6 ) (2 6 ) } ( , 2) ( ) ( , 3)(3 3 6 ) ( , 1) (3

n xx xy yyl lm m


C C l C C m i j C l C m i jC l C m C lm i j C

2 26 9

2 2 2 24 6 4 62 2

4 6 8 92 2 2 2

4 6 8 4 6

8 8 9 8

3 6 ) ( 1, )( 4 4 ) ( 1, 1) ( ) ( 1, 2){3 3 2(8 12 ) } ( 1, )( 4 4 8 ) ( 1, 1) ( ) ( 1, 2)8 ( 1, 1) 2(2 6 ) ( 2, ) 2

l C m C lm i jC l C m i j C l C m i j

C l C m C C lm i jC l C m C lm i j C l C m i j

C lm i j C C lm i j C

8 9

( 2, 1)2 ( 2, 1) 2 ( 3, )

lm i jC lm i j C lm i j

(3.25)

Set

3

2 2

2 24 5 6 7 8 9

2 24 5 6 7 8

2 2 2 24 5 6 7 5 7

2 25 7 8

2

{(6 10 ) (6 10 ) 2(6 10 ) } ( , ){ (8 12 ) (8 12 ) 6 } ( , 1){(2 6 ) (2 6 ) } ( , 2) ( ) ( , 3)( 3 3 6 ) ( , 1) ( 3

n xx xy yyl lm m


C C l C C m i j C l C m i jC l C m C lm i j

2 24 6 9

2 2 2 24 6 4 6

2 24 6 8 92 2 2 2

4 6 8 4 6

8 8 9

3 6 ) ( 1, )(4 4 ) ( 1, 1) ( ) ( 1, 2){ 3 3 2(8 12 ) } ( 1, )(4 4 8 ) ( 1, 1) ( ) ( 1, 2)8 ( 1, 1) 2(2 6 ) ( 2, )

C l C m C lm i jC l C m i j C l C m i j


C lm i j C C lm i j

8

8 9

2 ( 2, 1)2 ( 2, 1) 2 ( 3, )

C lm i jC lm i j C lm i j

(3.26)


73

Set

4

2 2

2 24 5 6 7 8 9

2 24 5 6 7 8

2 2 2 24 5 6 7 5 7

2 25 7 8 4

2

{ (6 10 ) (6 10 ) 2(6 10 ) } ( , ){(8 12 ) (8 12 ) 6 } ( , 1){ (2 6 ) (2 6 ) } ( , 2) ( ) ( , 3)(3 3 6 ) ( , 1) (3

n xx xy yyl lm m


C C l C C m i j C l C m i jC l C m C lm i j C

2 26 9

2 2 2 24 6 4 62 2

4 6 8 92 2 2 2

4 6 8 4 6

8 8 9 8

3 6 ) ( 1, )( 4 4 ) ( 1, 1) ( ) ( 1, 2){3 3 2(8 12 ) } ( 1, )( 4 4 8 ) ( 1, 1) ( ) ( 1, 2)8 ( 1, 1) 2(2 6 ) ( 2, ) 2

l C m C lm i jC l C m i j C l C m i j


C lm i j C C lm i j C

8 9

( 2, 1)2 ( 2, 1) 2 ( 3, )

lm i jC lm i j C lm i j

(3.27)


74

2 2( ) ( )t xy yy xxl m lm

Set

1

2 2

2 28 9 8 9 8 9

8

9

6 7

( ) ( )

( ) [(6 10 ) ( , ) (8 12 ) ( 1, ) (2 6 ) ( 2, ){ 3 ( , 1) 3 ( , 1) 4 ( 1, 1) 4 ( 1, 1)( 2, 1) ( 2, 1)} {3 ( 1, ) ( 3, )}]

[(6 10 ) (

t xy yy xxl m lm

l m C C i j C C i j C C i jC i j i j i j i j

i j i j C i j i jlm C C

6 7 6 7

7 7 6

4 5 4 5 4 5

5

, ) (8 12 ) ( , 1) (2 6 ) ( , 2)( , 3) 3 ( , 1) { 3 ( 1, ) 3 ( 1, )

4 ( 1, 1) 4 ( 1, 1) ( 1, 2) ( 1, 2)}][(6 10 ) ( , ) (8 12 ) ( , 1) (2 6 ) ( , 2)(

i j C C i j C C i jC i j C i j C i j i j

i j i j i j i jlm C C i j C C i j C C i jC i

5 4

2 28 9 6 7 4 5

2 28 6 7 4 5

6 7 4 5

, 3) 3 ( , 1) { 3 ( 1, ) 3 ( 1, )4 ( 1, 1) 4 ( 1, 1) ( 1, 2) ( 1, 2)}]

{( )(6 10 ) (6 10 6 10 ) } ( , ){ 3 ( ) (8 12 8 12 ) } ( , 1){(2 6 2 6 )

j C i j C i j i ji j i j i j i j

l m C C C C C C lm i jC l m C C C C lm i jC C C C

7 52 2

8 7 52 2

9 6 4 6 42 2

6 4 8 9 6 42 2

8 6

} ( , 2) ( ) ( , 3){ 3 ( ) (3 3 ) } ( , 1){ 3 ( ) (3 3 ) } ( 1, ) (4 4 ) ( 1, 1)( ) ( 1, 2) { (8 12 )( ) (3 3 ) } ( 1, ){4 ( ) (4 4

lm i j C C lm i jC l m C C lm i jC l m C C lm i j C C lm i j

C C lm i j C C l m C C lm i jC l m C C

4 6 42 2 2 2 2 2

8 8 9 82 2 2 2

8 9

) } ( 1, 1) ( ) ( 1, 2)4 ( ) ( 1, 1) (2 6 )( ) ( 2, ) ( ) ( 2, 1)

( ) ( 2, 1) ( ) ( 3, )

lm i j C C lm i jC l m i j C C l m i j C l m i j

C l m i j C l m i j

(3.28)


75

Set

2

2 2

2 28 9 6 7 4 5

2 28 6 7 4 5

6 7 4 5 7 52 2

8 7 5

( ) ( )

{( )(6 10 ) (6 10 6 10 ) } ( , ){ 3 ( ) (8 12 8 12 ) } ( , 1){ (2 6 2 6 ) } ( , 2) ( ) ( , 3){ 3 ( ) (3 3 ) } ( , 1)

t xy yy xxl m lm

l m C C C C C C lm i jC l m C C C C lm i jC C C C lm i j C C lm i j

C l m C C lm i j

2 29 6 4 6 4

2 26 4 8 9 6 4

2 28 6 4 6 4

2 28 8

{ 3 ( ) (3 3 ) } ( 1, ) (4 4 ) ( 1, 1)( ) ( 1, 2) { (8 12 )( ) (3 3 ) } ( 1, ){4 ( ) (4 4 ) } ( 1, 1) ( ) ( 1, 2)4 ( ) ( 1, 1) (2 6

C l m C C lm i j C C lm i jC C lm i j C C l m C C lm i j

C l m C C lm i j C C lm i jC l m i j C

2 2 2 29 8

2 2 2 28 9

)( ) ( 2, ) ( ) ( 2, 1)( ) ( 2, 1) ( ) ( 3, )

C l m i j C l m i jC l m i j C l m i j

(3.29)

Set

3

2 2

2 28 9 6 7 4 5

2 28 6 7 4 5

6 7 4 5 7 52 2

8 7 5

( ) ( )

{ (6 10 )( ) (6 10 6 10 ) } ( , ){3 ( ) (8 12 8 12 ) } ( , 1){(2 6 2 6 ) } ( , 2) ( ) ( , 3){3 ( ) (3 3 ) } ( , 1){

t xy yy xxl m lm

C C l m C C C C lm i jC l m C C C C lm i jC C C C lm i j C C lm i j

C l m C C lm i j

2 29 6 4 6 4

2 26 4 8 9 6 4

2 28 6 4 6 42 2

8 8 9

3 ( ) (3 3 ) } ( 1, ) (4 4 ) ( 1, 1)( ) ( 1, 2) {(8 12 )( ) (3 3 ) } ( 1, ){ 4 ( ) (4 4 ) } ( 1, 1) ( ) ( 1, 2)4 ( ) ( 1, 1) (2 6 )


C l m C C lm i j C C lm i jC l m i j C C

2 2 2 28

2 2 2 28 9

( ) ( 2, ) ( ) ( 2, 1)( ) ( 2, 1) ( ) ( 3, )

l m i j C l m i jC l m i j C l m i j

(3.30)


76

Set

4

2 2

2 28 9 6 7 4 5

2 28 6 7 4 5

6 7 4 5 7 52 2

8 7 5

( ) ( )

{ ( )(6 10 ) (6 10 6 10 ) } ( , ){3 ( ) (8 12 8 12 ) } ( , 1){ (2 6 2 6 ) } ( , 2) ( ) ( , 3){3 ( ) (3 3 ) } ( , 1)

t xy yy xxl m lm

l m C C C C C C lm i jC l m C C C C lm i j

C C C C lm i j C C lm i jC l m C C lm i j

2 29 6 4 6 4

2 26 4 8 9 6 4

2 28 6 4 6 42 2

8 8 9

{3 ( ) (3 3 ) } ( 1, ) (4 4 ) ( 1, 1)( ) ( 1, 2) {(8 12 )( ) (3 3 ) } ( 1, ){ 4 ( ) (4 4 ) } ( 1, 1) ( ) ( 1, 2)4 ( ) ( 1, 1) (2 6


C l m C C lm i j C C lm i jC l m i j C C

2 2 2 28

2 2 2 28 9

)( ) ( 2, ) ( ) ( 2, 1)( ) ( 2, 1) ( ) ( 3, )

l m i j C l m i jC l m i j C l m i j

(3.31)

The corresponding four forms of FD stencils of σn and σt are illustrated in Figure 3.13

for nodal points on the boundary of the structure.

The definition of the coefficients used in the above expressions is given below:

11

4 x y

Ch k

112 2

12 66( ) x

ACA A h

663 2

12 66( ) y

ACA A k

11 664 2

12 66( )2 x y

A ACh A A h k

12 665 3

12 66( )2 y

A ACh A A k

212 12 66 11 22

6 212 66( )2 x y

A A A A ACh A A h k

22 667 3

12 66( )2 y

A ACh A A k

12 668 2

12 66( )2 x y

A ACh A A h k

11 669 3

12 66( )2 x

A ACh A A h


77

(a)

(b)

Figure 3.13: (a) Different forms of stencil for σn or σt (b) application of the stencils at different boundary points of the non-uniform structure.


78

3.5 Management of boundary conditions at the corners

The most difficult part in the application of finite-difference method in solving the

partial differential equations arises from the boundary conditions, especially those at

the corner points. In general, the boundary conditions, when expressed in finite-

difference form, generate the most ill-conditioned algebraic equations and introduce

the maximum local truncation error when the boundary conditions are of mixed

nature. The magnitude of the difficulty is more seriously felt when re-entrant corners

are introduced. The computer program is developed in such a way that it has

sufficient generality and flexibility of being utilized in all the varieties of problems.

Special cares have been taken to model the boundary conditions associated with the

corner points of the geometry, which are, in general, the points of singularity. Each

external corner point is considered as the common point to the connecting two

boundaries, for which four conditions are available. In the present modeling scheme,

three out of the four boundary conditions associated with each external corner points

are satisfied and the remaining one is treated as redundant. It can be mentioned here

that, in case of conventional computational approaches, a maximum of two conditions

are taken into consideration to model the external corner nodes. A re-entrant corner

point is also considered as a common point for the adjacent two boundaries. Unlike to

the case of external corner points, for most of the cases, the boundary conditions

associated with the adjacent boundaries of the re-entrant corner points are of the same

kind. Therefore, each re-entrant corner point basically contains two boundary

conditions. The present finite difference solutions obtained on the basis of assigning

one out of the available two boundary conditions for each re-entrant corner points.

The neighboring mesh network of the re-entrant corner points suggests that identical

consideration would be necessary for one of the adjacent points of the re-entrant

corner points.

The present computer program is organized in such a fashion that the user can choose

any three out of the available four boundary conditions associated with each external

corner points. One of these conditions is used to evaluate ψ at the corner mesh point

itself and the remaining two conditions are used to evaluate ψ at the corresponding

imaginary mesh points of the corner point. The option to play with the

correspondence between mesh points and selected boundary conditions are also made


79

open for the users. The input for selection of external corner point boundary

conditions is to be maintained a specific order. This is done by defining a vector

named ‘corner’, for each external corner, with five elements – each element

representing the boundary conditions associated with ux, uy, σxx, σyy and σxy

respectively. The number assigned to each element indicates a particular

correspondence between mesh points and boundary conditions. A number of ‘0’

assigned to a certain element of ‘corner’ vector indicates that the corresponding

boundary condition cannot be used or selected. A number of ‘1’ indicates that the

corresponding boundary condition is used to evaluate ψ at the corner of the physical

boundary. Numbers of ‘2’ and ‘3’ indicate that the conditions are used to evaluate ψ at

the imaginary points top/bottom and left/right associated with the physical corner

point, respectively. As an example, the external corner point A of Figure 3.3 might

have conditions from two connecting surfaces AB and AD as (un, ut) and (σn, σt),

respectively. The conditions un and ut on AB turn out to be uy and ux respectively since

l = 0 and m = -1. Similarly, σn and σt on AD become σxx and σxy respectively since l = -

1 and m = 0. Now, among the available conditions ux, uy, σxx and σxy, the user has to

choose any three. It should be pointed out here that for each and every combination of

boundary conditions at the external corner points, the solution may not converge. In

case of divergence of the solution the user will be notified in the output screen while

the execution of the program. Then another trial may be executed by changing the

selection for boundary conditions at the external corners. For instance, one might

consider ux as redundant and use σxx, σxy and uy to evaluate ψ at the physical corner,

imaginary point above and left to the physical corner respectively. The boundary

condition σyy will not be available in this case. The ‘corner’ vector for this particular

external corner will be taken from the user as input as [0 3 1 0 2]. The default value 0

at the 4th element automatically deselects σyy. From experimenting numerically, it has

been investigated that very few combinations of boundary conditions selection and

correspondence between mesh points ensure convergence of the system.

In addition, to ensure convergence, three different versions of difference equations for

uy are used for external corners. Each version has two or four different forms or sets

as follows:


80

Version Set X Y

1 1, 2, 3 and 4 Central Central

2 3

2 22 3

{ ( 1, ) ( 1, )} { ( , 1) ( , 1)}

2( ) ( , ) ( , )y

x y

u C i j i j C i j i j

C C i j O h k

(3.32)

2 3

22 3

{ 2 ( 1, ) ( 2, )} { ( , 1) ( , 1)}

( 2 ) ( , ) ( , )y

x y


C C i j O h k

(3.33)

2 3

22 3

{ 2 ( 1, ) ( 2, )} { ( , 1) ( , 1)}

( 2 ) ( , ) ( , )y

x y


C C i j O h k

(3.34)

2 3

2 22 3

{ 5 ( 1, ) 4 ( 2, ) ( 3, )} { ( , 1) ( , 1)}

2( ) ( , ) ( , )y

x y

u C i j i j i j C i j i j

C C i j O h k

(3.35)

2 3

2 22 3

{ 5 ( 1, ) 4 ( 2, ) ( 3, )} { ( , 1) ( , 1)}

2( ) ( , ) ( , )y

x y

u C i j i j i j C i j i j

C C i j O h k

(3.36)

From numerical experimentation, version 2 has been identified the most proper for

convergence for structures of uniform geometry, while different versions from the

above three ensure convergence for different cases of non-uniform geometry. It

Version Set X Y

2 1 and 2 Forward O(hx) Central

Version Set X Y

2 3 and 4 Backward O(hx) Central

Version Set X Y

3 1 and 2 Forward O(hx2) Central

Version Set X Y

3 3 and 4 Backward O(hx2) Central


81

should be noted that the most proper version here is defined as the version for which

all the basic physical requirements are satisfied and the boundary conditions are

reproduced best after compiling and running the program. Since introducing re-

entrant corner points tend to make the system ill-conditioned, having three different

versions of uy for the external corner points and the flexibility for the user of the

program to choose among the three versions widens the possibility of convergence.

The above three versions and different forms of FD stencils of uy are illustrated in

Figure (3.14) for external corner nodal points of the body. It is to be noted that,

versions 2 and 3 cannot be used to evaluate ψ at imaginary points above or below the

physical corner point since the application of these stencils at the external corner

points do not involve any points above (set 1 and 2) or below (set 3 and 4) the

physical corner point and this, eventually, incorporates a zero diagonal element in the

corresponding row of the coefficient matrix.

The available conditions for a re-entrant corner point and connecting boundaries are

usually given in terms of σn and σt. At each re-entrant corner point and, in some cases,

at any one of the adjacent boundary points, only one out of the two available

conditions are to be used, the corresponding option has been made available to choose

either of the conditions for their application. For example, referring to Figure 3.3, the

re-entrant corners J and K and the adjacent boundary points J´ and K´ require only

one condition to be satisfied, as the associated imaginary nodal points are missing at

the re-entrant corner region. Now, the re-entrant corner point J can be considered a

point on either boundary segment JK or IJ. The value of σn becomes equal to σxx and

σyy if J is considered to be a point on JK and IJ respectively. The value of σt becomes

equal to σxy for either boundary segments. However, since J´ is a point on the

boundary segment JK, the available conditions for this point is σxx and σxy and it is

seen that σxx is the best choice for all applications. For point J the user has the freedom

to choose between σxx, σyy or σxy. From several trials of different problems of non-

uniform bodies, it is seen that choosing one condition from σxx or σyy at the re-entrant

corner is most preferable in terms of convergence of the solution. The external corner

points satisfy 75% of the available conditions while re-entrant corners and adjacent

points satisfy only 50% of the available conditions for which the system tends to

become difficult to solve. So special care should be taken in the selection of


82

(a)

(b)

(c)

Figure 3.14: (a) Version 1 (b) version 2 with different forms and (c) version 3 with different forms of stencil for uy and application of the stencils at external corner points

of the non-uniform structure.


83

conditions at the external and re-entrant corner regions and it might take several trials

for different combinations of boundary conditions and correspondence between mesh

points and selected boundary conditions.

3.6 Placement of Boundary Conditions to Mesh Points

There are two conditions to be satisfied at an arbitrary point on the physical boundary

of the body – a normal and a tangential component of body parameters. As a result,

two finite difference expressions of the differential equations associated with the

boundary conditions are applied to the same point on the boundary. Out of these two

equations, one is used to evaluate the function ψ at the point on the physical boundary

and the other one for the corresponding point on the imaginary boundary. In the case

of external corner points, three conditions are to be satisfied and the finite difference

expressions of the differential equations associated with these three conditions are

assigned to the same external corner point. One of these equations is used to evaluate

ψ at the point on the physical corner and the other two are used to evaluate ψ at the

corresponding two imaginary points. No imaginary points are associated with re-

entrant corner points and, in some cases, for one of the adjacent boundary points

which can be seen at points J, K, J´ and K´ in Figure 3.3. Node numbers of the

different boundary segments and external and re-entrant corners are identified from

the extreme nodal field generated from the program (see Figure 3.15). Appropriate

boundary conditions are applied at the boundary and corner nodes while ψ is

evaluated for each boundary and corresponding imaginary nodes. So only a single

algebraic equation corresponding to a single condition is used to evaluate the function

at each re-entrant corner and at any boundary point next to it. Governing equation is

applied for the evaluation of each internal nodal point.


84

Figure 3.15: Node numbering scheme applied to a non-uniform structure.

3.7 Solution and Evaluation of ψ at the Internal and Boundary

Mesh Points

For a uniform rectangular body, every nodal point of the extreme nodal field will have

a single algebraic equation tagged to it. The discrete values of the potential function,

ψ (x, y), at mesh points are solved from the system of algebraic equations resulting

from the discretization of the governing equation and the associated boundary

conditions. Attempt is made to organize the difference expressions in such a fashion

that the concerned point for which the equation is to be written must be included

together with other surrounding mesh points. Therefore, the coefficient matrix which

is, of course, a square matrix, generated from the system of algebraic equations, will

have all non-zero diagonal elements. This criterion must be satisfied by the coefficient

matrix; otherwise the solution will be impossible. For the problem at hand, the

important issue is to solve a system of linear algebraic equations, expressed in matrix

equation as

[ ]{ } { }K C (3.37)


85

Expressing explicitly, equation (3.37) becomes

11 12 13 1 1 1

21 22 23 2 2 2

31 32 33 3 3 3

1 2 3

p

p

p

p pp p p pp

K K K K CK K K K CK K K K C

CK K K K

(3.38)

where,

p – is the number of nodal points in the extreme nodal field.

[K] - is a known square matrix, called the co-efficient matrix.

{C} - is a given constant matrix (a column matrix), whose elements are zero

except for those correspond to points on or outside the boundary.

{ψ} - is an unknown column matrix the elements of which are to be

determined.

At the beginning of the program [K] matrix is created with p number of rows and p

number of columns with default zero elements. The constant column matrix {C} is

also created with p number of default zero elements. As algebraic equations are used

for each active field nodes, each corresponding row of the [K] matrix changes and

several zero elements are replaced by the coefficients of that particular equation. The

elements of the constant column matrix corresponding to the equations which have

non-zero constants at the right hand side will be replaced by the constants. It is

obvious that the matrix [K] is non-singular and hence a unique solution exists. In the

present problem, the number of unknowns in the system of equations is extremely

large, but they are only a few in each individual equation. The iterative method is

advocated for this kind of large sparse system of linear algebraic equation, but the

most unfortunate part in this method is that the technique is successful to very few

cases of co-efficient matrix. Considering this difficulty, the author decided to solve

the system of equation by Cholesky’s decomposition method. The requirement of the

storage space in the computer and also the computer time is relatively less in this

method. It has been verified that, up to a large number of equations, this elimination

method can produce promising results with the minimum truncation error.


86

For the case of non-uniform geometry as in Figure 3.3, not all nodes of the extreme

nodal field are involved. Some nodes are left out and ψ is not evaluated and no

equations are generated for those nodal points. But the program is developed in such a

way that each row of the co-efficient matrix corresponds to a particular node. So for p

number of total nodal points, involved and uninvolved nodes all together, the

coefficient matrix [K] will still have a size of p×p. Since no equations are generated

for uninvolved nodes, the corresponding rows of the coefficient matrix will have

default zero elements. Again, each column of the co-efficient matrix corresponds to

the involved nodal points of the stencils of each equation. So, for uninvolved nodes,

the corresponding columns of the co-efficient matrix will also have default zero

elements. For the same reason the corresponding element of the constant matrix will

also be zero by default. For example, in Figure 3.15, node no. 13 is left out of the

physical domain. So the 13th row and the 13th column of the co-efficient matrix will

have all-zero elements and so on. As such, the system of equations becomes singular

and thus unsolvable. To make the system solvable, the rows and columns

corresponding to the uninvolved nodal points are discarded. The corresponding

elements of the constant column matrix are also removed. The reduced coefficient

matrix will now have rows and columns equal to the number of active nodal points, p´

of the domain. Now the system becomes solvable and after the solution, the {ψ}

matrix will have number of elements equal to the number of active nodal points. In

order to trace the ψ value of each active nodal point according to the extreme nodal

field numbering scheme, the {ψ} matrix is expanded by inserting zeros at appropriate

positions corresponding to uninvolved nodal points. Thus, the {ψ} matrix will

eventually have number of elements equal to the total nodal points of the extreme

nodal field as shown in Figure (3.15). The zero values will represent the ψ values of

uninvolved nodes which are actually not evaluated and will not be used any further in

post-processing of the results. The non-zero values will represent the ψ values of the

active nodal points.

3.8 Evaluation of Displacements, Strains and Stresses

Using the known values of ψ, the displacement and stress components of the overall

laminate can be determined for all the nodal points on and inside the physical


87

boundary of the body using the stencils used for body parameters. To determine the

strain components of the laminate, the following difference equations for strain

components are used in four different forms. 3

2( , ) xxx

ux yx x y

Set X Y


1[ 3{ ( 1, ) ( 1, ) 4 ( 1, 1) ( 1, 1)}6 ( , ) 8 ( , 1) 2 ( , 2) { ( 1, 2) ( 1, 2)}]

xx L i j i j i j i ji j i j i j i j i j

(3.39)

Set X Y


1[3{ ( 1, ) ( 1, ) 4 ( 1, 1) ( 1, 1)}6 ( , ) 8 ( , 1) 2 ( , 2) { ( 1, 2) ( 1, 2)}]

xx L i j i j i j i ji j i j i j i j i j

(3.40)

The above two forms of FD stencils of εxx are illustrated in Figure (3.16) for nodal

points interior and on the boundary of the structure. 3 3

11 662 312 66

1( , ) yyy

ux y A A

y A A x y y

Set X Y


2 3 2 3 2 3

3 3 2

(6 10 ) ( , ) (8 12 ) ( , 1) (2 6 ) ( , 2)( , 3) 3 ( , 1) [ 3 ( 1, ) 3 ( 1, )

4 ( 1, 1) 4 ( 1, 1) ( 1, 2) ( 1, 2)]

yy L L i j L L i j L L i jL i j L i j L i j i j

i j i j i j i j

(3.41)

Set X Y


2 3 2 3 2 3

3 3 2

(6 10 ) ( , ) (8 12 ) ( , 1) (2 6 ) ( , 2)( , 3) 3 ( , 1) [3 ( 1, ) 3 ( 1, )

4 ( 1, 1) 4 ( 1, 1) ( 1, 2) ( 1, 2)]

yy L L i j L L i j L L i jL i j L i j L i j i j

i j i j i j i j

(3.42)

The above two forms of FD stencils of εyy are illustrated in Figure 3.17 for nodal

points interior of the boundary of the structure.


88

(a)

(b)

Figure 3.16: (a) Different forms of stencil for εxx (b) application of the stencils at different boundary and internal points of the non-uniform structure.


89

(a)

(b)

Figure 3.17: (a) Different forms of stencil for εyy (b) application of the stencils at different boundary and internal points of the non-uniform structure.


90

3 3

11 123 212 66

1( , ) yxxy

uux y A Ay x A A x x y

Set X Y

1 and 2 Forward Central

4 5 4 5 4 5

5 5 4

(6 10 ) ( , ) (8 12 ) ( 1, ) (2 6 ) ( 2, )( 3, ) 3 ( 1, ) [ 3 ( , 1) 3 ( , 1)

4 ( 1, 1) 4 ( 1, 1) ( 2, 1) ( 2, 1)]

xy L L i j L L i j L L i jL i j L i j L i j i j

i j i j i j i j

(3.43)

Set X Y

3 and 4 Backward Central

4 5 4 5 4 5

5 5 4

(6 10 ) ( , ) (8 12 ) ( 1, ) (2 6 ) ( 2, )( 3, ) 3 ( 1, ) [3 ( , 1) 3 ( , 1)

4 ( 1, 1) 4 ( 1, 1) ( 2, 1) ( 2, 1)]

xy L L i j L L i j L L i jL i j L i j L i j i j

i j i j i j i j

(3.44)

The above two forms of FD stencils of εxy are illustrated in Figure (3.18) for nodal

points interior and on the boundary of the structure.

The definition of the coefficients used in the above expressions is given below:

1 21

2 x y

Lh k

112 2

12 66

12 x y

ALA A h k

663 3

12 66( )2 y

ALA A k

124 2

12 66( )2 x y

ALA A h k

114 3

12 66( )2 x

ALA A h


91

(a)

(b)

Figure 3.18: (a) Different forms of stencil for εxy (b) application of the stencils at boundary and internal points of the non-uniform structure.


92

3.9 Evaluation of Stress Components for Individual ply

The previous section describes the method of evaluating displacement, stress and

strain components for overall laminate. Stress and displacement for individual plies of

the composite has been solved by the help of strain distribution. Strain distributions

for overall laminate or global strain distribution has been evaluated directly from the

ψ values in equation (3.38) of the mesh point. The distributions of strain components

are same for all plies of whole laminate. These strain components are directly

multiplied with the transformed reduced stiffness matrix of the each ply to evaluate

the different stress components at these plies by using equation (2.67).

3.10 Summary

In order to summarize the steps involved in the present computational scheme, the

overall programming philosophy is sequentially described in the flow diagram, given

in Appendix A.

CHAPTER

Analysis of Non-uniform Column of Hybrid Laminates

4.1 Introduction

The elastic field of an eccentrically loaded hybrid balanced laminated composite

column of non-uniform shape is analyzed under the influence of partial frictionless

guides. As an example of a non-uniform column structure, I-shaped column has been

chosen which includes both external and re-entrant corners. An efficient

computational algorithm is developed, in which a displacement-potential is introduced

to model the non-uniform shaped column of hybrid balanced laminated composite

with mixed boundary conditions. Solutions of stresses at different layers of the

laminated composite are obtained, some of which, especially those around the critical

regions of the non-uniform column are presented in a comparative fashion mainly in

the form of graphs and Tables. Results are found to be accurate and reasonable when

analyzed in light of basic principles of mechanics and given boundary conditions.

4.2 Geometry, loading and material of the composite column

The geometry and loading of the guided I-shaped laminated column under eccentric

loading is illustrated with reference to a Cartesian coordinate system, in Figure 4.1

(Case-I). The length, width and thickness of the column are represented by L, D and

h, respectively. The upper lateral edges of the column are assumed to be partially

guided, which is realized by assuming boundary segments e-f and e´-f´ to be guided

by frictionless guides such as rollers. These ends can only move along the y-direction.

The aspect ratio (L/D) of the column is kept 4 for the present analysis. Two different

fiber-reinforced composite (FRC) plies with the same matrix material and two

different fiber materials having dissimilar fiber stiffness are considered. Here, the

FRC ply having the fiber material with higher stiffness is designated as FRC-1 while

the one having lower fiber stiffness is designated as FRC-2. Both types of FRC plies

4

CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES

94

Figure 4.1: Analytical model of the eccentrically loaded non-uniform laminated column with partial guides.

are assumed to be linearly elastic throughout the present analysis. These two types of

FRC plies with various fiber orientations are considered to form the symmetric hybrid

laminate for the column. Along with the hybrid laminate, two additional identical

non-hybrid laminates of FRC-1 and FRC-2 are considered for the comparative

analysis. The laminates are assumed to be composed of twenty eight plies having an

overall thickness, h =14 mm. The length and width of the column are assumed to be L

= 400 cm and D = 100 cm respectively. The material modelling of the hybrid

laminated column together with its stacking sequence is shown in Figure 4.2, while

the non-hybrid laminates of FRC-1 and FRC-2 follow the identical sequence of fiber

c c´ b

d e

a

f

b´

d´ e´

f´

a´

D/2

0.6 L

0.2 L

0.2 L

x

y

D

D/4 D/4

Eccentric loading

Frictionless guides

σ0


95

orientations. The three-dimensional view of the three laminates is shown in Figure

4.3.

Figure 4.2: Material modelling of hybrid laminate consisting of FRC-1 and FRC-2.

Figure 4.3: 3D views of: (a) hybrid of FRC-1 and FRC-2, (b) FRC-1 and (c) FRC-2 laminated columns.

(a)

(b) (c)

[±45 / ±30/ ±45 / ±75

/ ±30

/ ±75

/ ±45

]

s

FRC-2 Laminate FRC-1 Laminate Hybrid Laminate

(FRC-1 and FRC-2)

FRC-1 ply

FRC-2 ply FRC-1 ply FRC-2 ply

(a)

x

y h

FRC-1, +45 FRC-1, -45

FRC-2, +75 FRC-2, -75

FRC-1, -45 FRC-1, +45

[±451 / ±301 / ±452 / ±751 / ±302 / ±752 / ±451]s 1: FRC-1; 2: FRC-2

Ply1, +45B


96

4.3 Boundary conditions

The physical conditions to be satisfied along the different boundary segments of the

column can be expressed mathematically with reference to Figure 4.1, as follows:

1. The bottom surface (boundary a-a´) is rigidly fixed. Both the normal and

tangential components of displacement are assumed to be zero here, that is,

un (x, 0) = 0 and ut (x, 0) = 0 [ 0 ≤ x ≤ D]

2. The top surface (boundary f-f´) of the column is the eccentrically loaded

boundary. The left half of the top surface is free from loading. So both the

normal and tangential components of stress are assumed to be zero here, that

is,

σn (x, L) = 0 and σt (x, L) = 0 [ 0 ≤ x < D/2]

And the loading of the right half of the top surface is realized through a

uniform distribution of normal compressive stress of the intensity of σ0 while

the associated tangential stress component is assumed to be zero, that is,

σn (x, L) = -σ0 and σt (x, L) = 0 [D/2 ≤ x ≤ D]

The magnitude of σ0 is chosen here arbitrarily as 3 MPa for the present

calculation.

3. The boundary segments e-f and e´-f´ are assumed to be guided by frictionless

guides. Thus displacement is restricted along the normal direction while the

associated tangential stress component is assumed to be zero, that is,

un (D, y) = 0 and σt (D, y) = 0 for segment e-f and

un (0, y) = 0 and σt (0, y) = 0 for segment e´-f´ [ 0.8L ≤ y ≤ L]

4. All other boundary segments are free from loading and restraints. Thus, the


is,

σn (D, y) = 0 and σt (D, y) = 0 for segment a-b and

σn (0, y) = 0 and σt (0, y) = 0 for segment a´-b´ [ 0 ≤ y ≤ 0.2L]

σn (0.75D, y) = 0 and σt (0.75D, y) = 0 for segment c-d and

σn (0.25D, y) = 0 and σt (0.25D, y) = 0 for segment c´-d´ [ 0.2L ≤ y ≤ 0.8L]

σn (x, 0.2L) = 0 and σt (x, 0.2L) = 0 for segment b-c and

σn (x, 0.8L) = 0 and σt (x, 0.8L) = 0 for segment d-e [0.75D ≤ x ≤ D]

σn (x, 0.2L) = 0 and σt (x, 0.2L) = 0 for segment b´-c´ and


97

σn (x, 0.8L) = 0 and σt (x, 0.8L) = 0 for segment d´-e´ [0 ≤ x ≤ 0.25D]

Numerical modelling of the boundary conditions are summarized in Table 4.1. It has

been observed that the choice of boundary conditions at the singularity points is

important. Accuracy depends on the combination of boundary conditions used at the

point of singularity. Table 4.2 illustrates the scheme for treating the boundary

conditions of the external and re-entrant corner points of the I-shaped column, which

are, in general, the points of singularity.

Table 4.1: Numerical modelling of the boundary conditions for different boundary segments of the non-uniform laminated composite column.

Boundary segment Given and used

boundary conditions

Correspondence between mesh points and given boundary conditions

Mesh point on the physical boundary

Mesh points on the imaginary boundary

a-a´ un = -uy = 0 ut = ux = 0 ux = 0 uy = 0

f-f´(left half) (Case-I)

σn = σyy = 0 σt = -σxy = 0 σxy = 0 σyy = 0

f-f´(right half) (Case-I)

σn = σyy = σ0 σt = -σxy = 0 σxy = 0 σyy = σ0

f-f´(Case-II) σn = σyy = σ0 σt = -σxy = 0 σxy = 0 σyy = σ0

a-b σn = σxx = 0 σt = σxy = 0 σxx = 0 σxy = 0

a´-b´ σn = σxx = 0 σt = σxy = 0 σxx = 0 σxy = 0

c-d σn = σxx = 0 σt = σxy = 0 σxx = 0 σxy = 0

c´-d´ σn = σxx = 0 σt = σxy = 0 σxx = 0 σxy = 0

b-c σn = σyy = 0 σt = -σxy = 0 σxy = 0 σyy = 0

b´-c´ σn = σyy = 0 σt = -σxy = 0 σxy = 0 σyy = 0

d-e σn = σyy = 0 σt = -σxy = 0 σxy = 0 σyy = 0

d´-e´ σn = σyy = 0 σt = -σxy = 0 σxy = 0 σyy = 0

e-f (Case-I) un = ux = 0 σt = σxy = 0 ux = 0 σxy = 0

e-f (Case-III) σn = σxx = 0 σt = σxy = 0 σxx = 0 σxy = 0

e´-f´ (Case-I) un = ux = 0 σt = σxy = 0 ux = 0 σxy = 0

e´-f´ (Case-III) σn = σxx = 0 σt = σxy = 0 σxx = 0 σxy = 0

Case-I: Hybrid laminated I-shaped column with partial guides subjected to eccentric loading Case-II: Hybrid laminated I-shaped column with partial guides subjected to full loading Case-III: Hybrid laminated I-shaped column without partial guides subjected to eccentric loading


98

Table 4.2: Numerical modelling of the boundary conditions for external and re-entrant corners of the non-uniform laminated composite column. (see Figure 4.1)

Corner Point

Available boundary

Parameters from two sides

Used boundary

Parameters

Form of uy

if used

Elements of ‘corner’ vector


Mesh point on the

physical boundary

Mesh points on the imaginary

boundary

a´ σn = σxx ; σt = σxy un = -uy ; ut = ux

uy, σxx, σxy 2 [0 3 1 0 2] σxx = 0 σxy = 0 uy = 0

f´ (Case-I) un = -ux ; σt = σxy σn = σyy ; σt = -σxy

ux, σyy, σxy 2 [1 0 0 3 2] ux = 0 σyy = 0 σxy = 0

f´ (Case-II) un = -ux ; σt = σxy σn = σyy ; σt = -σxy

ux, σyy, σxy 2 [1 0 0 3 2] ux = 0 σyy = σ0 σxy = 0

f (Case-III) σn = σxx; σt = σxy σn = σyy ; σt = -σxy

σxx, σyy, σxy 2 [0 0 1 3 2] σxx = 0 σyy = 0 σxy = 0

a σn = σxx ; σt = σxy un = -uy ; ut = ux


f un = ux ; σt = σxy σn = σyy ; σt = -σxy

ux, σyy, σxy 2 [1 0 0 3 2] ux = 0 σyy = σ0 σxy = 0

f (Case-III) σn = σxx; σt = σxy σn = σyy ; σt = -σxy

σxx, σyy, σxy 2 [0 0 1 3 2] σxx = 0 σyy = σ0 σxy = 0

b´ σn = σxx ; σt = σxy σn = σyy ; σt = -σxy

σxx, σyy, σxy [0 0 1 3 2] σxx = 0 σyy = 0 σxy = 0

e´ un = -ux ; σt = σxy σn = σyy ; σt = -σxy

ux, σyy, σxy [1 0 0 3 2] ux = 0 σyy = 0 σxy = 0

e´ (Case-III) σn = σxx; σt = σxy σn = σyy ; σt = -σxy


b σn = σxx ; σt = σxy σn = σyy ; σt = -σxy


e un = ux ; σt = σxy σn = σyy ; σt = -σxy


e (Case-III) σn = σxx; σt = σxy σn = σyy ; σt = -σxy


c´ σn = σxx or σyy ; σt = σxy

σxx [0 0 1 0 0] σxx = 0 ---

c1´ (adjacent to c´) σn = σxx ; σt = σxy σxx [0 0 1 0 0] σxx = 0 ---

d´ σn = σxx or σyy ; σt = σxy

σxx [0 0 1 0 0] σxx = 0 ---

d1´ (adjacent to d´) σn = σxx ; σt = σxy σxx [0 0 1 0 0] σxx = 0 ---

c σn = σxx or σyy ; σt = σxy

σxx [0 0 1 0 0] σxx = 0 ---

c1 (adjacent to c) σn = σxx ; σt = σxy σxx [0 0 1 0 0] σxx = 0 ---

d σn = σxx or σyy ; σt = σxy

σxx [0 0 1 0 0] σxx = 0 ---

d1 (adjacent to

d) σn = σxx ; σt = σxy σxx [0 0 1 0 0] σxx = 0 ---

Case-I: Hybrid laminated I-shaped column with partial guides subjected to eccentric loading Case-II: Hybrid laminated I-shaped column with partial guides subjected to full loading Case-III: Hybrid laminated I-shaped column without partial guides subjected to eccentric loading


99

4.4 Numerical modelling of the column

In order to discretize the computational domain, the I-shaped structure is placed

within a rectangle of length L and width D as described in chapter 3. Then the domain

is divided into 48 meshes along the x-axis and 80 meshes along the y-axis. By doing

this, the maximum number of physical nodal points becomes m = 49 along x-axis and

n = 81 along y-axis. The size of each mesh is hx = D/48 and ky = L/80 along each

coordinate axes respectively. The whole mesh network including the uninvolved

meshes are shown in Figure 4.4 together with node numbers obtained by the node

numbering scheme described in chapter 3. Figure 4.5 shows the indicators of either 1

or 0 at each node of the extreme nodal field depending on whether the nodes are on

and inside the physical boundary or not. Figure 4.6 shows the stencil indicators of 1,

2, 3 or 4 at each node inside and on the physical boundary depending on which form

from the available four forms of stencil are to be used for applying boundary

condition in the pre-processing stage and evaluation of stress and displacement

components in the post-processing stage.

Figure 4.4: FDM Mesh network used to model I-shaped column


100

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Figure 4.5: Developed extreme nodal field showing the involved and uninvolved nodal points (1 and 0) for computation.


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1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

Figure 4.6: Active field nodal points tagged with stencil indicating numbers (1, 2, 3, 4) for both stages of pre- and post-processing.

4.5 Results and Discussion

The present scheme is capable of handling any materials. However, to generate the

present numerical results, two fiber reinforced composite plies namely - boron/epoxy

(FRC-1) and glass/epoxy (FRC-2) are considered. The effective mechanical properties

of the composite plies along with their constituents are listed in Table 4.3. The results

of the present investigation are presented mainly in the form of stress distributions at

critical sections of the laminated column, particularly in the form of Tables and

graphs. In all cases, stresses are normalized with respect to the maximum intensity of

the applied loading, σ0.


102

Table 4.3: Properties of unidirectional fiber-reinforced composite ply used to obtain the numerical results.

Material Property Composite

Glass/epoxy Boron/epoxy

Fiber Ef (GPa) 85.52 413.69 νf 0.22 0.20

Matrix Em (GPa) 3.45 3.45 νm 0.35 0.35

Composite

E1 (GPa) 38.6 204.0 E2 (GPa) 8.27 18.5 G12 (GPa) 4.14 5.59

ν12 0.26 0.23 4.5.1 Determination of critical sections of the column

In order to determine the critical sections in terms of stresses, the guided I-shaped

hybrid balanced laminated column is analyzed to determine the maximum principal

stress for the overall laminate. Figure 4.7 shows the distribution of normalized value

of maximum principal stress along the surfaces abcdef and a´b´c´d´e´f´. Total number

of nodal points along both surfaces are 105 each. From the figure, one can easily

determine that lateral sections BB´ (y/L = 0.2) and FF´ (y/L = 0.8) assume the

maximum magnitudes of stresses. In other words, these two sections are identified as

the critical sections of the non-uniform shaped eccentrically loaded column in terms

of stresses.

Figure 4.7: Distribution of the maximum principal laminate stress along the two opposing lateral surfaces of the column.

(p)max0

-25 -20 -15 -10 -5 0 5

Nod

al p

ositi

on a

long

surf

ace

a´b´

c´d´

e´f´

15

30

45

60

75

90

105

(p)max0

-45-40-15-10-505

Nod

al p

ositi

on a

long

surf

ace

abcd

ef

15

30

45

60

75

90

105(a) (b)

-40.45-20.26

-8.11-12.85

b

e d

c

a

f

b

d

c

e

f

a

σ0


103

4.5.2 Effect of laminate hybridization

In this section, the overall laminate stresses as well as individual ply stresses are

analyzed mainly in the perspective of laminate hybridization. This is done by taking

three different balanced laminates, namely - non-hybrid laminate of boron/epoxy

(FRC-1), non-hybrid laminate of glass/epoxy (FRC-2) and hybrid laminate of

boron/epoxy and glass/epoxy, and the corresponding stress fields are analyzed in a

comparative fashion.

Overall laminate stresses

Table 4.4 lists the overall laminate stresses at the critical section e-e´ (y/L = 0.8) of the

column for the case of boron/epoxy laminate, glass/epoxy laminate as well as the

hybrid laminate of boron/epoxy and glass/epoxy considered. Among the stress

components, the axial stress component is found to play the most dominant role in

defining the overall state of stresses. It would be worth mentioning that the maximum

magnitudes of all the components of stress and maximum principal stress for the

hybrid laminate are almost identical with those of non-hybrid boron/epoxy laminate

and non-hybrid glass/epoxy laminate. In other words, the effect of hybridizing

boron/epoxy and glass/epoxy plies for a guided non-uniform eccentrically loaded

column is almost negligible when analyzed in the perspective of overall laminate

stresses.

Table 4.4: Overall laminate stresses at the critical section e-e´( y/L = 0.8).

Laminate σxx/σ0 σyy/σ0 σxy/σ0 (σp)max/σ0 d´ d d´ d d´ d d´ d Hybrid laminate 0 0 -20.04 -39.96 2.09 -4.42 -20.26 -40.45 Boron/epoxy laminate 0 0 -21.24 -40.33 2.16 -4.32 -21.45 -40.78 Glass/epoxy laminate 0 0 -21.70 -42.44 1.66 -3.57 -21.83 -42.74

Individual ply stresses

In order to determine the effect of hybridization in the perspective of individual ply

stresses, identical plies from hybrid laminates as well as non-hybrid laminates are

analyzed in a comparative fashion. Plies of three different fiber orientations are


104

considered here, θ = 30°, 45° and 75°. Stresses are observed at two critical sections

y/L = 0.2 and 0.8.

(a) Distributions of the lateral stress component

Figure 4.8 shows the distribution of normalized lateral stress along the section y/L =

0.2 for individual plies of hybrid and non-hybrid laminates as a function of fiber

orientation of the plies. The distributions of lateral stresses are found to be affected by

both the fiber orientation and hybridization of ply materials in terms of magnitude as

well as nature of variation. The stress developed in the boron/epoxy ply is much

higher when it is in the hybrid laminate of boron/epoxy and glass/epoxy, than that in

the non-hybrid boron/epoxy laminate (see Figures 4.8 a, c and e). Boron/epoxy ply

has a higher fiber stiffness than glass/epoxy ply. Hybridizing boron/epoxy with plies

with lower fiber stiffness tend to soften the laminate. It has been seen earlier that the

overall laminate stresses for both hybrid and non-hybrid laminates are almost the

same. Now in order to maintain the equilibrium of stresses, the stresses in a

boron/epoxy ply in a hybrid laminate increase than that in a non-hybrid laminate of its

own. In other words, equilibrium is maintained according to the stiffness of the fiber.

However, an opposite phenomenon is observed when we consider the case of

glass/epoxy ply (see Figures 4.8 b, d and f). The re-entrant corner c assumes higher

stress than c´ for θ = 30° plies, however, the opposite is observed for θ = 45° and 75°

plies. Both plies have higher magnitude of stress for θ = 45° fiber orientation since

this particular case tends to make the ply stiff. The maximum value of lateral stress

developed in the boron/epoxy ply, θ = 45°, increases by 58.5 % when hybridized with

glass/epoxy plies to form the laminate (Figure 4.8 c). On the other hand, the lateral

stresses at the critical region in an identical glass/epoxy ply are found to decrease

69.33 % when hybridized with boron/epoxy plies (Figure 4.8 d), which, in turn,

reveals that the glass/epoxy (lower fiber stiffness) plies are more severely affected by

hybridization than the boron/epoxy (higher fiber stiffness) plies. This effect of

hybridization follow similar trend for both the boron/epoxy and glass/epoxy plies,

when the ply angle increases and decreases from 45° even though the magnitude of

stresses are comparatively lower.


105

Figure 4.8: Distribution of lateral stresses along the critical section b-b´ (y/L = 0.2) of identical plies of different laminates. The hybrid laminate is composed of

boron/epoxy and glass/epoxy plies.


106

Figure 4.9 shows the distribution of normalized lateral stress along the section y/L =

0.8. Stresses are higher than those at section y/L = 0.2 and for all fiber orientations the

re-entrant corner d assumes higher stress compared to d´. Similar to section y/L = 0.2,

the lateral stress is maximum for plies with 45° fiber orientation.

Figure 4.9: Distribution of lateral stresses along the critical section e-e´ (y/L = 0.8) of identical plies of different laminates. The hybrid laminate is composed of



107

(b) Distributions of the axial stress component

The distribution of normalized axial stress along the sections y/L = 0.2 and 0.8 are

shown in Figures 4.10 and 4.11 respectively. As mentioned earlier, the axial

component of stress is the most prominent. At section y/L = 0.2, the re-entrant corner

c´ assumes higher stress than c, however, at section y/L = 0.8, the re-entrant corner d

assumes higher stress than d´. Similar to the lateral stress component, the magnitude

of axial stress is higher at section y/L = 0.8 than 0.2. In addition, the magnitude of

stress increases as fiber orientation increases from 30° to 75°. Similar to lateral stress,

the axial stress developed in the boron/epoxy ply is much higher when it is in the

hybrid laminate, than that in the non-hybrid boron/epoxy laminate while the opposite

scenario is seen for the case of glass/epoxy ply. The maximum value of axial stress

developed in the boron/epoxy ply, θ = 75° at section y/L = 0.8, increases by 51.4 %

due to hybridization with glass/epoxy plies while the axial stresses at the same critical

region in an identical glass/epoxy ply are found to decrease by 61.2 % due to

hybridization with boron/epoxy plies.


108

Figure 4.10: Distribution of axial stresses along the critical section b-b´ (y/L = 0.2) of identical plies of different laminates. The hybrid laminate is composed of



109

Figure 4.11: Distribution of axial stresses along the critical section e-e´ (y/L = 0.8) of identical plies of different laminates. The hybrid laminate is composed of



110

(c) Distributions of the shear stress component

The distributions of shear stresses at the critical sections, y/L = 0.2 and 0.8 are shown

in Figures 4.12 and 4.13. Likewise the cases of lateral and axial stresses, the

glass/epoxy plies are found to experience greater effect of hybridization in terms of

shear stress than boron/epoxy plies. Similar to the axial stress, the effect associated

with the shear stress is also found to increase with the increase of ply angle from 30°

to 75°.

Figure 4.12: Distribution of shear stresses along the critical section b-b´ (y/L = 0.2) of identical plies of different laminates. The hybrid laminate is composed of



111

Figure 4.13: Distribution of shear stresses along the critical section e-e´ (y/L = 0.8) of identical plies of different laminates. The hybrid laminate is composed of


Observing the details of the stress fields it can be concluded that the stress level in an

individual ply of a laminate can be well controlled by hybridizing the plies of

appropriate fiber stiffness and orientation. The maximum magnitudes of stresses of

individual plies of different laminates for three different fiber orientations θ = 30°,

45° and 75° are summarized in Tables 4.5 and 4.6. For all the ply angles considered,

laminate hybridization causes the increase in the magnitude of maximum stresses for


112

higher stiffness plies (boron/epoxy), while an opposite characteristic is noticed for the

lower stiffness plies (glass/epoxy). Therefore, a careful analysis of ply stresses instead

of overall laminate stresses would be essential for reliable analysis of failure of hybrid

laminated non-uniform columns of the present type.

Table 4.5: Comparison of critical ply stresses of the three different laminates at the re-entrant corner d´ as a function of ply angle

Ply θ = 30º θ = 45º θ = 75º

σxx/σ0 σyy/σ0 σxy/σ0 (σp)max/σ0 σxx/σ0 σyy/σ0 σxy/σ0 (σp)max/σ0 σxx/σ0 σyy/σ0 σxy/σ0 (σp)max/σ0 Boron/epoxy (in non-hybrid) -9.32 -3.59 -4.96 -12.14 -6.61 -8.47 -6.03 -13.32 -1.91 -53.52 -13.02 -56.60

Boron/epoxy (in hybrid) -13.4 -5.20 -7.12 -17.44 -9.59 -12.45 -8.74 -19.34 -2.79 -77.3 -18.81 -81.76

Glass/epoxy (in non-hybrid) -6.76 -9.29 -3.17 -9.49 -5.15 -15.08 -4.30 -15.95 -2.18 -39.93 -6.63 -41.01

Glass/epoxy (in hybrid) -2.70 -3.03 -2.45 -3.37 -1.94 -5.12 -1.47 -5.22 -0.86 -15.16 -2.47 -15.54

Table 4.6: Comparison of critical ply stresses of the three different laminates at the re-entrant corner d as a function of ply angle

Ply θ = 30º θ = 45º θ = 75º σxx/σ0 σyy/σ0 σxy/σ0 (σp)max/σ0 σxx/σ0 σyy/σ0 σxy/σ0 (σp)max/σ0 σxx/σ0 σyy/σ0 σxy/σ0 (σp)max/σ0

Boron/epoxy (in non-hybrid) -10.72 -5.91 -4.78 -13.27 -14.99 -25.9/ -18.18 -39.43/ -3.63 -110.92 -28.43 -117.99

Boron/epoxy (in hybrid) -16.36 -9.09 -7.31 -20.27 -22.96 -39.40 -27.65 -60.03 -5.61 -167.94 -43.01 -178.63

Glass/epoxy (in non-hybrid) -8.93 -21.07 -3.61 -21.58 -8.30 -35.39 -15.33 -42.23/ -3.97 -83.24 -17.41 -86.78

Glass/epoxy (in hybrid) -3.71 -7.21 -1.00 -7.21 -2.66 -12.62 -4.88 -14.33 -1.50 -32.33 -6.74 -33.65


113

4.5.3 Effect of eccentricity of applied loading

In order to check the effect of eccentricity of loading on the stresses, the results of a

guided I-shaped hybrid laminated column subjected to eccentric loading are compared

with those of a column subjected to full axial loading. The results of the latter one is

obtained by applying a uniformly distributed normal loading of intensity of σ0 on the

top surface of the I-shaped guided hybrid laminated column as in Figure 4.14 (Case-

II). The boundary conditions and corner modelling of this problem are almost the

same as Case-I except for the top surface f-f´ and external corner f´. The required

changes are given below:

The top surface (boundary f-f´) of the column is the fully loaded boundary. The

loading is realized through a uniform distribution of normal stress of the intensity of

σ0 while the associated tangential stress component is zero, that is,

σn (x, L) = σ0 and σt (x, L) = 0 [0 ≤ x ≤ D]

(a) (b)

Figure 4.14: I-shaped guided column subjected to axial loading on the top surface (a) uniform loading, (b) eccentric loading.

For the external corner f´ the required change is shown in Table 4.2.

σ0

Eccentric loading

b

d

c

e

a

f

b

d

c

e

f

a

σ0

Full loading

b

d

c

e

a

f

b

d´

c

e

f

a


114

Now the results for both type of loading conditions (Case I and II) are analyzed in

terms of maximum principal stress of the overall laminates. Figure 4.15 shows the

distribution of stresses along the critical sections y/L = 0.2 and 0.8 for both cases of

loadings. It was seen earlier that at section y/L = 0.8, the re-entrant corner d assumes

maximum stress while at section y/L = 0.2, the re-entrant corner c´ assumes maximum

stress and can be seen once again in Figure 4.15. But in case of full loading, at section

y/L = 0.8 both re-entrant points d and d´ are equally stressed (Figure 4.15 a) since the

loading is symmetric about the axis of the column. Similarly at section y/L = 0.2, re-

entrant corners c and c´ are also equally stressed (Figure 4.15 b). Thus, it can be

explained that the unequal magnitudes of stress at re-entrant corners of the same

lateral sections is due to the eccentricity of the loading. Full axial loading not only

produces the same amount of stresses at the re-entrant corners of the same lateral

section, but also increases their magnitudes. However, although the amount of loading

is doubled, the magnitude of maximum stress does not increase proportionately. A

100% increase in load increases the maximum stress by only 41.5%.

Figure 4.15: Distribution of maximum principal stresses along the critical section (a) e-e´ (y/L = 0.8) and (b) b-b´ (y/L = 0.2) of hybrid laminated column subjected to both

full and eccentric loading.

x/D0.0 0.2 0.4 0.6 0.8 1.0

( p

) max

0

-60-50-40-30-20-10

010

x/D0.0 0.2 0.4 0.6 0.8 1.0

( p

) max

0

-20

-15

-10

-5

0

5

Full LoadingEccentric Loading

(a)

(b)

y/L = 0.2

y/L = 0.8


115

The deformed shapes of the non-uniform hybrid laminated column subjected to both

full and eccentric loading are shown in Figure 4.16.

(a) (b)

Figure 4.16: Deformed shapes of hybrid laminated column subjected to (a) full loading and (b) eccentric loading. (Magnification factor along x-axis: 4000, y-axis:

5000)

x-coordinate (m)

0.00 0.05 0.10

Original shapeDeformed shape

x-coordinate (m)

0.00 0.05 0.10

y-co

ordi

nate

(m)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40



116

4.5.4 Effect of partial guides on the column behavior

In order to check the effect of partial guides on the stresses, the results of the guided I-

shaped hybrid laminated column subjected to eccentric loading are compared with

those of a column without partial guides as in Figure 4.17 (Case-III). The results of

the latter one is obtained by changing the boundary conditions of Case-I at the

segments e-f and e´-f´ only and keeping other conditions as it is. The required changes

are given below:

The boundary segments e-f and e´-f´ are free from loading and restraints. Thus, the

normal and tangential components of stress are assumed to be zero here, that is,

σn (D, y) = 0 and σt (D, y) = 0 for segment e-f and

σn (0, y) = 0 and σt (0, y) = 0 for segment e´-f´ [ 0.8L ≤ y ≤ L]

For the external corner e, f, e´ and f´ the required changes are shown in Table 4.2.

(a) (b)

Figure 4.17: Eccentrically loaded I-shaped column (a) with and (b) without partial guides along the opposing upper lateral surfaces.

Now the results of the eccentrically loaded hybrid laminated column with and without

partial guides (Case I and III) are analyzed in terms of maximum principal stress of

b

d

c

e

a

f

b

d

c

e

f

a

σ0

Eccentric loading

Free from guides

b

d

c

e

a

f

b

d

c

e

f

a

σ0

Eccentric loading

Guides


117

identical plies. Distribution of maximum principal stress along the critical section e-e´

(y/L = 0.8) is shown in Figure 4.18. For both cases and ply materials the maximum

principal stress in a ply increases when fiber orientation increases from θ = 30° to 75°.

However, partial guides make the magnitude of stress increase even more in most

regions of the section. But at the re-entrant corner d, the stress decreases for c and 45°

plies while increases for 75° plies. Even though the re-entrant corner d assumes higher

stress than d´, partial guides do not change the magnitude of stress at d significantly

compared to the un-guided column. However, the change of magnitude of stress is

noticeable away from d and maximum at d´. This change of stress due to the use of

partial guides is most prominent for θ = 30° and decreases from 30° to 75° even

though the magnitude of stress increases. It is also noticeable that the effect of partial

guides is same for both ply materials boron/epoxy and glass/epoxy. In other words,

the effect of partial guides does not depend of ply materials, rather it is found to be a

function of fiber orientation only.


118

Figure 4.18: Distribution of maximum principal stress along the critical section e-e´ (y/L = 0.8) of identical plies of both partially guided and unguided hybrid laminated

column subjected to eccentric loading.


119

The deformed shapes of the eccentrically loaded non-uniform hybrid laminated

column with and without partial guides along the opposing upper lateral surfaces are

shown in Figure 4.19.

(a) (b)

Figure 4.19: Deform shapes of eccentrically loaded I-shaped column (a) with and (b) without partial guides along the opposing upper lateral surfaces. (Magnification factor

along x-axis: 4000, y-axis: 5000)

x-coordinate (m)

0.00 0.05 0.10

y-co

ordi

nate

(m)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40


x-coordinate (m)

0.00 0.05 0.10



120

4.6 Summary

The stress field of a guided non-uniform hybrid balanced laminated composite

column subjected to eccentric loading has been investigated using an efficient

computational algorithm based on a scalar potential of displacement components. No

appropriate analytical approach is available in the literature that can provide explicit

information about the actual stresses, since in general analytical approaches are

inadequate to take into account the effect of geometrical non-uniformity of structures.

Even the standard computational approaches have been verified to be inadequate to

predict the actual state of stresses accurately (see chapter 6), especially at the

bounding surfaces, whether it is subjected to restraints or external loading or even free

from loading.

The stress fields of the overall laminate as well as individual plies of non-uniform

laminated composite column have been analyzed mainly in the perspective of

laminate hybridization. The effect of hybridization on the overall laminate stress is

found to be nearly insignificant. However, the same is identified to be quite prominent

in case of individual ply stresses, especially around the re-entrant corners of the non-

uniform structure. Moreover, this effect of hybridization on ply stresses is further

found to be influenced significantly by the orientation angles of fibers in individual

plies of the laminate.

The stress fields of the non-uniform laminated composite column have also been

analyzed to investigate the effect of eccentricity of the applied loading and the effect

of partial guides. The effect of these issues on the elastic behavior of the laminated

non-uniform structure is found to be dependent on ply material as well as ply fiber

orientation of the hybrid laminate.

CHAPTER

Analysis of Non-uniform Beam of Hybrid Laminates

5.1 Introduction

The elastic field of a fixed non-uniform hybrid balanced laminated composite beam

with a sinking support is analyzed. Built-in beams subjected to shear displacement is

known as sinking beams [34]. The researches on sinking beam reported in the

literature are very few and none in case of non-uniform geometry. Using the efficient

computational algorithm developed, solution of elastic field of the laminated

composite beam are obtained, some of which, especially those around the critical

regions of the non-uniform beam are presented in a comparative fashion mainly in the

form of graphs and Tables.

5.2 Geometry, loading and material of the composite beam

As an example of a non-uniform beam structure, I-shape has been chosen once again.

With reference to a Cartesian-coordinate system, the geometry and loading of the I-

shaped laminated beam is illustrated in Figure (5.1). The length, width and thickness

Figure 5.1: Loading and geometry of a non-uniform sinking beam of laminated composite.

5

a b

c d

e f

a b e f

c d

Sinking support

x

y

D

0.6 L

δ

0.2 L 0.2 L

D/4

D/4

CHAPTER 5 | ANALYSIS OF NON-UNIFORM BEAM OF HYBRID LAMINATES

122

of the beam are represented by L, D and h, respectively. The left lateral edge of the

beam is attached to a fixed support while the right lateral edge is attached to a sinking

support. This is realized by assuming the right lateral edge is subjected to a shear

displacement of δ along the x-direction while the movement along the y-direction is

restrained as shown in Figure (5.1). The aspect ratio (L/D) of the beam considered for

the present analysis are 2, 2.5, 3, 4 and 4.5. Fiber reinforced composite (FRC) plies

with various fiber orientations are considered to form different non-hybrid symmetric

angle ply and cross-ply laminates. A varying number of soft isotropic plies are

inserted within the FRC plies to form the hybrid balanced laminates. The varying

numbers of soft isotropic plies used to constitute different laminates are four, eight

and twelve. All laminates are assumed to be composed of twenty eight plies having an

overall thickness, h =14 mm. The width of the beam is assumed to be D = 100 cm and

the length L is varied according to the beam aspect ratio. The material modelling of

the laminated beams together with their stacking sequence are shown in Figure 5.2.

The three-dimensional view of the laminates are shown in Figure 5.3.

5.3 Boundary conditions

The physical conditions to be satisfied along the different boundary segments of the

beam can be expressed mathematically as follows:

1. The left lateral surface (boundary a-a´) is rigidly fixed. Both the normal and

tangential components of displacement are assumed to be zero here, that is,

un (x, 0) = 0 and ut (x, 0) = 0 [ 0 ≤ x ≤ D ]

2. The right lateral surface (boundary f-f´) of the beam is attached to a sinking

support which is displaced by an amount δ along x-axis while movement along

y-axis is restrained. So the normal and tangential components of displacement

are as follows,

un (x, L) = 0 and ut (x, L) = -δ [ 0 ≤ x ≤ D ]

3. All other boundary segments are free from loading and restraints. Thus, the


is,

σn (D, y) = 0 and σt (D, y) = 0 for segment a-b and

σn (0, y) = 0 and σt (0, y) = 0 for segment a´-b´ [ 0 ≤ y ≤ 0.2L]


123

(a) (b)

(c) (d)

Figure 5.2: Material modelling of: (a) angle ply fiber reinforced composite (FRC)

laminate, (b) angle ply hybrid laminate of fiber reinforced composite (FRC) and soft isotropic plies, (c) cross-ply fiber reinforced composite (FRC) laminate and (d) cross-

ply hybrid laminate of fiber reinforced composite (FRC) and soft isotropic plies.

[ ±θ / ±θ / ±θ / ±θ /±θ / ±θ / ±θ ]s

Ply1, +45B

Angle ply (θ = 30°, 55°)

4 isotropic plies: [ ±θF / ±θF / ±θF / ±θF /±θF / ±θF / I / I ]s

Ply1, +45B 8 isotropic plies: [ ±θF / ±θF / ±θF / I / I /±θF / ±θF / I / I ]s

Ply1, +45B 12 isotropic plies: [ I / I / ±θF / ±θF / I / I /±θF / ±θF / I / I ]s

Ply1, +45B

h h

FRC, +θ FRC, -θ

Isotropic ply

[ 0 / 90 / 0 / 90 / 0 / 90 / 0 / 90 / 0 / 90 / 0 / 90 / 0 / 90 ]s

Ply1, +45B

Cross-ply

4 isotropic plies: [ 0F / 90F / 0F / 90F / 0F / 90F / 0F / 90F / 0F / 90F / 0F / 90F / I / I ]s 8 isotropic plies: [ 0F / 90F / 0F / 90F / 0F / 90F / I / I / 0F / 90F / 0F / 90F / I / I ]s 12 isotropic plies: [ I / I / 0F / 90F / 0F / 90F / I / I/ 0F / 90F / 0F / 90F / I / I ]s

Ply1, +45B

h

FRC, θ = 90°

Isotropic ply

FRC, θ = 0°

h


124

(a)

(b)

Figure 5.3: 3D views of the beam of: (a) fiber reinforced composite (FRC) laminate, (b) Hybrid laminate of FRC and isotropic ply.

FRC ply

FRC ply Isotropic ply

FRC laminate

Hybrid Laminate (FRC and Isotropic ply)


125

σn (0.75D, y) = 0 and σt (0.75D, y) = 0 for segment c-d and

σn (0.25D, y) = 0 and σt (0.25D, y) = 0 for segment c´-d´ [ 0.2L ≤ y ≤ 0.8L]

σn (D, y) = 0 and σt (D, y) = 0 for segment e-f and

σn (0, y) = 0 and σt (0, y) = 0 for segment e´-f´ [ 0.8L ≤ y ≤ L]

σn (x, 0.2L) = 0 and σt (x, 0.2L) = 0 for segment b-c and

σn (x, 0.8L) = 0 and σt (x, 0.8L) = 0 for segment d-e [0.75D ≤ x ≤ D]

σn (x, 0.2L) = 0 and σt (x, 0.2L) = 0 for segment b´-c´ and

σn (x, 0.8L) = 0 and σt (x, 0.8L) = 0 for segment d´-e´ [0 ≤ x ≤ 0.25D]

Numerical modelling of the boundary conditions have been summarized in Table 5.1.

Table 5.2 illustrates the scheme for treating the boundary conditions of the external

and re-entrant corner points of the I-shaped beam, which are, in general, the points of

singularity.

Table 5.1: Numerical modelling of the boundary conditions for different boundary segments of the non-uniform laminated composite sinking beam.


boundary conditions




a-a´ un = -uy = 0 ut = ux = 0 ux = 0 uy = 0

f-f´ un = uy = 0 ut = -ux = -δ ux = δ uy = 0

a-b σn = σxx = 0 σt = σxy = 0 σxx = 0 σxy = 0

a´-b´ σn = σxx = 0 σt = σxy = 0 σxx = 0 σxy = 0

c-d σn = σxx = 0 σt = σxy = 0 σxx = 0 σxy = 0

c´-d´ σn = σxx = 0 σt = σxy = 0 σxx = 0 σxy = 0

b-c σn = σyy = 0 σt = -σxy = 0 σxy = 0 σyy = 0

b´-c´ σn = σyy = 0 σt = -σxy = 0 σxy = 0 σyy = 0

d-e σn = σyy = 0 σt = -σxy = 0 σxy = 0 σyy = 0

d´-e´ σn = σyy = 0 σt = -σxy = 0 σxy = 0 σyy = 0

e-f σn = σxx = 0 σt = σxy = 0 σxx = 0 σxy = 0

e´-f´ σn = σxx = 0 σt = σxy = 0 σxx = 0 σxy = 0


126

Table 5.2: Numerical modelling of the boundary conditions for external and re-entrant corners of the non-uniform laminated composite beam.

Corner Point

Available boundary


Used boundary

Parameters

Form of uy

if used



Mesh point on the

physical boundary

Mesh points on the

imaginary boundary

a´ σn = σxx ; σt = σxy un = -uy ; ut = ux


f´ σn = σxx ; σt = σxy un = uy ; ut = -ux

ux, uy, σxx 2

[1 3 2 0 0] ux = δ σxx = 0 uy = 0

uy, σxx, σxy [0 3 1 0 2] σxx = 0 σxy = 0 uy = 0



f σn = σxx ; σt = σxy un = uy ; ut = -ux

uy, σxx, σxy 2 [1 3 2 0 0] ux = δ σxx = 0 uy = 0


b´ σn = σxx ; σt = σxy σn = σyy ; σt = -σxy


e´ un = -ux ; σt = σxy σn = σyy ; σt = -σxy


b σn = σxx ; σt = σxy σn = σyy ; σt = -σxy


e un = ux ; σt = σxy σn = σyy ; σt = -σxy


c´ σn = σxx or σyy ; σt = σxy

σxx [0 0 1 0 0] σxx = 0 --- σyy [0 0 0 1 0] σyy = 0 ---

c1´ (adjacent to c´) σn = σxx ; σt = σxy σxx [0 0 1 0 0] σxx = 0 ---

d´ σn = σxx or σyy ; σt = σxy

σxx [0 0 1 0 0] σxx = 0 --- σyy [0 0 0 1 0] σyy = 0 ---

d1´ (adjacent to d´) σn = σxx ; σt = σxy σxx [0 0 1 0 0] σxx = 0 ---

c σn = σxx or σyy ; σt = σxy

σxx [0 0 1 0 0] σxx = 0 --- σyy [0 0 0 1 0] σyy = 0 ---

c1 (adjacent to c) σn = σxx ; σt = σxy σxx [0 0 1 0 0] σxx = 0 ---

d σn = σxx or σyy ; σt = σxy

σxx [0 0 1 0 0] σxx = 0 --- σyy [0 0 0 1 0] σyy = 0 ---

d1 (adjacent to d) σn = σxx ; σt = σxy σxx [0 0 1 0 0] σxx = 0 ---


127

5.4 Numerical modelling of the problem

Discretization of the computational domain of the I-shaped beam is done in the same

process previously done for the I-shaped column in chapter 4. As mentioned earlier,

for different aspect ratios, only the value of L is varied while keeping the value of D is

kept constant. For all aspect ratios, the mesh network in Figure 4.4 is applicable.

Thus, for different values of L, the value of ky varies.

5.5 Results and Discussion

The elastic field of non-uniform hybrid laminated sinking beams are analyzed in the

perspective of aspect ratio and the number of isotropic plies that constitute the

laminates. Both FRC and isotropic plies are assumed to be linearly elastic throughout

the analysis. In order to generate results, the FRC material is taken as boron/epoxy

and the soft isotropic ply material is assumed to be rubber. The effective mechanical

properties of rubber are listed in Table 5.3. It is worth mentioning that a large number

of researches in the literature, especially those on the stress analysis of rubber based

components, are found to consider rubber as a linear elastic material [36-39].

Moreover, direct experiments in the laboratory showed [39] that the stress-strain

relation for a truck tire rubber was linear for the low strain range (0 ~ 0.25). Wang et

al. [37] reported that the FEM results based on linear elastic behavior of rubber were

in substantial agreement with the corresponding experimental results.

Table 5.3: Properties of isotropic ply used to obtain the numerical results.

Property Symbol Rubber Elastic modulus E (GPa) 0.05 Shear modulus G12 (GPa) 0.02 Major Poisson’s ratio υ12 0.49

The results of the present investigation are presented mainly for the critical sections of

the laminated beam, particularly in the form of tables and graphs. In all cases, stresses

are normalized with respect to the maximum bending stress of a uniform rectangular

(L×D) sinking beam of unidirectional boron/epoxy ply based on simple theory which

is found to be 3E1Dδ/L2 [34].


128

5.5.1 Effect of aspect ratio on the elastic field

In order to investigate the effect of aspect ratio (L/D) on the stresses and deformed

shape, sinking I-shaped beams of various aspect ratios such as 2, 2.5, 3, 4 and 4.5 are

taken. Material of the beams taken is symmetric θ = ±30° angle ply boron/epoxy

balanced laminate with twelve isotropic plies with stacking sequence of [ I / I / ±30B /

±30B / I / I /±30B / ±30B / I / I ]s. In this section, the effect of aspect ratio is analyzed in

the perspective of overall laminate stresses.

Lateral stresses

Figure 5.4 shows the distribution of normalized lateral stress of the overall laminate

along different lateral sections as a function of beam aspect ratio. The distributions of

lateral stresses are found to be affected by the aspect ratio in terms of magnitude as

well as nature of variation. The magnitude of lateral stresses are quite small at section

y/L = 1 (see Figure 5.4 c). On the contrary, stresses are high at sections y/L = 0.2 and

0.8 (Figures 5.4 a and b), especially around the re-entrant corners. At both sections

y/L = 0.2 and 0.8 the normalized lateral stress is higher for aspect ratio of 2 while the

magnitude decreases as aspect ratio increases from 2 to 4.5. This is because

decreasing the aspect ratio means shortening the beam and the same magnitude of

applied shear displacement induces higher amount of end moments for lower beam

aspect ratios. This, in turn, increases the stress. At other lateral sections the effect of

aspect ratio on lateral stress is negligible.

Bending stresses

Distributions of the overall laminate bending stresses at different sections of the

laminated beam are shown in Figure 5.5. Among the stress components, the bending

stress component is found to play the most dominant role in defining the state of

stresses. The re-entrant corners assume the maximum bending stress (see Figures 5.5

a and b). Similar to the case of lateral stress, the normalized bending stress decreases

as the beam becomes slender. It is obvious that for lower aspect ratio the magnitude of

bending stress will be higher for the same shear displacement. As mentioned earlier,


129

Figure 5.4: Distribution of overall laminate stresses at different sections of sinking beams with 12 isotropic plies in ±30° angle ply boron/epoxy laminate.

0.0 0.2 0.4 0.6 0.8 1.0

xx .(

L2 /3E 1D

)

-10-8-6-4-202468

10y/L = 0.2

0.0 0.2 0.4 0.6 0.8 1.0

xx .(

L2 /3E 1D

)

-20-15-10-505

1015

L/D = 2 L/D = 2.5L/D = 3L/D = 4L/D = 4.5

y/L = 0.8

x/D0.0 0.2 0.4 0.6 0.8 1.0

xx .(

L2 /3E 1D

)

-0.8-0.6-0.4-0.20.00.20.40.6

y/L = 1(c)

(b)

(a)


130

Figure 5.5: Distribution of overall bending stresses at different sections of sinking beams with 12 isotropic plies in ±30° angle ply boron/epoxy laminate.

0.0 0.2 0.4 0.6 0.8 1.0

yy .(

L2 /3E 1D

)

-15

-10

-5

0

5

10

15y/L = 0.2

0.0 0.2 0.4 0.6 0.8 1.0

yy .(

L2 /3E 1D

)

-30

-20

-10

0

10

20

L/D = 2L/D = 2.5L/D = 3L/D = 4L/D = 4.5

y/L = 0.8

x/D0.0 0.2 0.4 0.6 0.8 1.0

yy .(

L2 /3E 1D

)

-0.8-0.6-0.4-0.20.00.20.40.6

y/L = 1(c)

(b)

(a)


131

stresses are normalized with a factor whose magnitude is equal to the maximum

bending stress of a uniform unidirectional boron/epoxy ply (3E1Dδ/L2) which

increases as aspect ratio decreases. However, as the beam becomes shorter and the

increasing magnitude of the bending stress is normalized with an increasing factor, it

is found that the normalized value increases instead of remaining the same. This is

because the present solutions are based on elasticity theory while the normalizing

factor is based on simple theory which is not adequate for short beams.

Shear stresses

Figure 5.6 shows the distribution of normalized shear stress of the overall laminate

along different lateral sections of the I-shaped beam. Similar to the other stress

components, sections y/L = 0.2 and 0.8 assume maximum shear stress. However,

unlike the lateral and bending stresses, the distributions of shear stresses are not anti-

symmetric. The effect of aspect ratio follow a similar trend for all stress components.

Deformed shape

Figure 5.7 shows the deformed shape of the non-uniform sinking beam of symmetric

θ = ±30° angle ply boron/epoxy laminate as a function of beam aspect ratio. For both

cases of beam aspect ratio, it can be seen that the boundary conditions specified in

terms of displacements are well reproduced in the deformed shapes. Comparing the

deformed shapes for both aspect ratios it can be determined that the deformed shapes

of the sinking non-uniform beams are dependent on the magnitude of applied shear

displacement rather than beam aspect ratio. The deformed shapes of θ = ±30° angle

ply boron/epoxy laminate with any number of isotropic plies are similar to those

shown in Figure 5.7. In other words, the nature of the deformed shapes has no effect

of isotropic ply numbers.

From the distribution of the elastic field of the overall laminate, it is found that aspect

ratio plays a vital role in defining the state of stresses of a non-uniform laminated

composite sinking beam, especially around the critical regions. Magnitudes of stresses

decrease as aspect ratio increases. Around the critical regions, this effect is most

prominent.


132

Figure 5.6: Distribution of overall shear stresses at different sections of sinking beams with 12 isotropic plies in ±30° angle ply boron/epoxy laminate.

0.0 0.2 0.4 0.6 0.8 1.0

xy .(

L2 /3E 1D

)

-1.5-1.0-0.50.00.51.01.52.02.5

L/D = 2L/D = 2.5L/D = 3L/D = 4L/D = 4.5

y/L = 0.2

0.0 0.2 0.4 0.6 0.8 1.0

xy .(

L2 /3E 1D

)

-2

-1

0

1

2

3

4

y/L = 0.8

x/D0.0 0.2 0.4 0.6 0.8 1.0

xy .(

L2 /3E 1D

)

-0.4-0.20.00.20.40.60.81.01.2

y/L = 1

(c)

(b)

(a)


133

Figure 5.7: Deformed shape of a ±30° angle ply boron/epoxy laminated sinking beam with various aspect ratios. (Magnification factor along x-axis and y-axis: 20)

y-coordinate (m)

0.00 0.05 0.10 0.15 0.20 0.25 0.30

x-co

ordi

nate

(m)

-0.020.000.020.040.060.080.100.120.14

L/D = 3

(a)

y-coordinate (m)

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

x-co

ordi

nate

(m)

-0.020.000.020.040.060.080.100.120.14

L/D = 4


(b)


134

5.5.2 Effect of soft isotropic plies on the elastic field

For the purpose of investigating the effect of soft isotropic ply numbers on stresses,

the overall maximum principal stresses of laminates consisting of plies of similar fiber

orientations and various number of soft isotropic plies as shown in Figure 5.2 are

presented in a comparative fashion. The investigation has been carried out for a particular

value of the aspect ratio 4.

Stress field of a θ = ±30° angle ply laminate

The overall maximum principal stresses of symmetric θ = ±30° angle ply balanced

laminates of boron/epoxy with various number of soft isotropic plies are observed at

different lateral sections of the beam as shown in Figure 5.8. The stacking sequence of

the laminates considered are as follows:

Laminate with no isotropic ply: [±30B / ±30B / ±30B /±30B / ±30B / ±30B/ ±30B]s

Laminate with 8 isotropic plies: [±30B / ±30B / ±30B / I / I /±30B / ±30B / I / I]s

Laminate with 12 isotropic plies: [I / I / ±30B / ±30B / I / I /±30B / ±30B / I / I ]s

As seen earlier, the stresses are critical at sections y/L = 0.2 and 0.8 (Figures 5.8 a and

b) irrespective of isotropic ply numbers. At section y/L = 1, the effect of isotropic ply

numbers on stresses is insignificant (Figure 5.8 c). However, the effect of isotropic

ply numbers is prominent at the critical sections although the nature if variation is

almost the same. At section y/L = 0.8, a wide region around re-entrant corner d is

highly stressed compared to d´. As the number of isotropic plies increase from none to

twelve, the magnitude of stress decreases drastically. This is due to the fact that

increasing soft isotropic plies in a laminate decreases the overall stiffness of that

laminate. Eventually the magnitude of maximum stress decreases.

Stress field of a θ = ±55° angle ply laminate

Figure 5.9 shows the overall maximum principal stresses of symmetric θ = ±55° angle

ply laminates of boron/epoxy with various number of isotropic plies at different


135

Figure 5.8: Distribution of overall laminate stresses at different sections of an I-shaped sinking beam (L/D = 4) with various isotropic plies in a ±30° angle ply

boron/epoxy laminate.

0.0 0.2 0.4 0.6 0.8 1.0

( p

) max .(

L2 /3E 1D

)

-2.7

-1.8

-0.9

0.0

0.9

1.8

2.7y/L = 0.2

0.0 0.2 0.4 0.6 0.8 1.0

( p

) max .(

L2 /3E 1D

)

-4

-2

0

2

4

6

8y/L = 0.8

x/D0.0 0.2 0.4 0.6 0.8 1.0

( p

) max .(

L2 /3E 1D

)

-0.5

0.0

0.5

1.0

1.5

2.0

2.5y/L = 1(c)

(b)

(a)

no isotropic ply 8 isotropic plies12 isotropic plies


136

Figure 5.9: Distribution of overall laminate stresses at different sections of an I-shaped sinking beam (L/D = 4) with various isotropic plies in a ±55° angle ply


0.0 0.2 0.4 0.6 0.8 1.0

( p

) max .(

L2 /3E 1D

)

-6-4-202468

1012

y/L = 0.2

0.0 0.2 0.4 0.6 0.8 1.0

( p

) max .(

L2 /3E 1D

)

-10-505

1015202530

y/L = 0.8

x/D0.0 0.2 0.4 0.6 0.8 1.0

( p

) max .(

L2 /3E 1D

)

-1.0

-0.5

0.0

0.5

1.0

1.5y/L = 1(c)

(b)

(a)



137

lateral sections of the beam. The stacking sequence of the laminates considered are as

follows:

Laminate with no isotropic ply: [±55B / ±55B / ±55B / ±55B / ±55B / ±55B / ±55B]s

Laminate with 8 isotropic plies: [±55B / ±55B / ±55B / I / I /±55B / ±55B / I / I]s

Laminate with 12 isotropic plies: [ I / I / ±55B / ±55B / I / I /±55B / ±55B / I / I ]s

At the critical sections (Figures 5.9 a and c) the magnitude of stresses are higher

compared to θ = ±30° laminates. At section y/L = 0.8, stresses around the re-entrant

corner d are higher than that of d´. On the other hand, at section y/L = 0.2, stresses

around the re-entrant corner c´ are higher than that of c. The effect of isotropic ply

numbers on the stresses is comparatively lower than that of θ = ±30° laminates.

Stress field of a cross-ply laminate

Figure 5.10 shows the overall maximum principal stresses of symmetric cross ply

laminates of boron/epoxy with various number of isotropic plies at different lateral

sections of the beam. The stacking sequence of the laminates considered are as

follows:

Laminate with no isotropic ply: [0B/90B/0B/90B/0B/90B/0B/90B/0B/90B/0B/90B/0B/90B] s

Laminate with 8 isotropic plies: [0B/90B/0B/90B/0B/90B/I/I/0B/90B/0B/90B/I/I] s

Laminate with 12 isotropic plies: [I/I/0B/90B/0B/90B/I/I/0B/90B/0B/90B/I/I] s

It is found that the overall laminate stresses of cross-ply laminates are the most

prominent. Moreover, the region only a single re-entrant corner assumes a rise of

stress at each critical section and the maximum stress is more localized than those of

the angle ply laminates (Figures 10 a and b). Effect of isotropic ply numbers on

stresses are significant and the stresses follow a regular pattern with increasing

isotropic ply numbers as seen for the θ = ±30° laminates.

It is seen that beam aspect ratio and isotropic ply numbers play a vital role in defining

the overall laminate stresses and this is further influenced significantly by the ply

fiber orientations. The maximum stresses of laminated beams of various aspect ratio,

isotropic ply numbers and ply angles are summarized in Table 5.4. It can be observed


138

that, the magnitude of maximum stress decreases individually with increasing beam

aspect ratio and increasing isotropic ply numbers. The effects are most significant in

case of cross-ply laminates.

Figure 5.10: Distribution of overall laminate stresses at different sections of an I-shaped sinking beams of (L/D = 4) with various isotropic plies in a cross-ply


0.0 0.2 0.4 0.6 0.8 1.0

( p

) max .(

L2 /3E 1D

)

-200

20406080

100120

y/L = 0.2

0.0 0.2 0.4 0.6 0.8 1.0

( p

) max .(

L2 /3E 1D

)

-200

20406080

100120140160

y/L = 0.8

x/D0.0 0.2 0.4 0.6 0.8 1.0

( p

) max .(

L2 /3E 1D

)

-0.20.00.20.40.60.81.01.2

y/L = 1

(c)

(b)

(a)



139

Table 5.4: Overall laminate stresses at the critical region of the laminated composite I-shaped sinking beam.

5.5.3 Analysis of ply stresses

In this section, the individual ply stresses are analyzed in the perspective of ply fiber

orientation, aspect ratio and number of isotropic plies. The maximum stresses in

individual plies of the laminates are summarized in the form of Tables. Table 5.5

shows the individual ply stresses of a θ = ±30° angle-ply boron/epoxy laminate with

various number of isotropic plies. Results are shown for boron/epoxy plies with fiber

orientation, θ = 30° and -30° and isotropic ply for different aspect ratios. Table 5.6

details the maximum stresses of boron/epoxy plies with fiber orientation, θ = 0° and

90° along with isotropic plies in the cross-ply laminates of various aspect ratios. The

effect of aspect ratio and isotropic ply numbers on individual ply stresses follow

similar trend as that of overall laminates. However, the magnitude of stresses in plies

of cross-ply laminates are higher compared to the overall cross-ply laminate stresses.

For all cases, the magnitude of stresses decrease individually as aspect ratio or

isotropic ply numbers increase.

Number of isotropic

plies L/D

θ = ±30° laminate θ = ±55° laminate Cross-ply laminate

σxx σyy σxy σxx σyy σxy σxx σyy σxy

0 3 10.0191 12.9317 4.0001 23.1347 83.1959 14.0553 7.2747 210.8277 1.6572 4 4.9017 6.3914 2.2429 6.5240 23.3237 4.7086 5.6727 142.7037 1.8975

4 3 5.5731 7.1856 2.2591 18.3676 66.0836 11.2553 6.9768 204.3462 1.6450 4 4.3337 5.6487 1.9799 11.1724 39.6318 7.2787 5.7494 171.4380 2.2682

8 3 4.4583 5.7632 1.8411 15.7772 56.7663 9.5877 4.8195 132.6081 1.0807 4 3.5516 4.6278 1.6052 6.1544 21.9669 4.3799 4.6644 141.0196 1.8693

12 3 4.0726 5.2540 1.6408 10.3964 37.4514 6.4672 4.6502 135.1846 1.0908 4 2.6543 3.4623 1.2170 5.1051 18.2362 3.5891 3.1847 85.4281 1.1421


140

Table 5.5: Comparison of maximum ply stresses of the θ = ±30° angle ply laminated composite I-shaped sinking beam.

Table 5.6: Comparison of maximum ply stresses of the cross-ply laminated composite sinking beam.

5.6 Summary

The stresses and deformed shapes of the overall laminate as well as individual ply

stresses of a non-uniform laminated composite beam have been analyzed mainly in

the perspective of ply fiber orientation, beam aspect ratio and number of isotropic ply

constituting part of the laminates considered. The effect of aspect ratio and number of

isotropic plies on the overall laminate stress is found to be significant, especially

around the re-entrant corners of the non-uniform structure. Moreover, the same is

identified to be quite prominent in case of individual ply stresses. In addition, the

effect of these issues on overall laminate stresses as well as individual ply stresses is

further found to be influenced significantly by the orientation angles of fibers in

individual plies of a laminate.

Number of isotropic

plies L/D

θ = +30° ply θ = -30° ply Isotropic ply


0 3 9.3944 14.8016 4.5586 11.6771 11.0619 -6.0630 4 4.8238 7.4398 2.4623 5.5815 5.3429 -2.9081

4 3 6.1411 9.6099 2.9896 7.5410 7.1460 -3.9101 0.0050 0.0310 -0.0004 4 4.9726 7.6658 2.5381 5.7544 5.5064 -2.9981 0.0039 0.0244 -0.0002

8 3 5.9408 9.2612 2.9005 7.2074 6.8518 -3.7313 0.0048 0.0298 -0.0003 4 4.8625 7.5197 2.4757 5.6775 5.4190 -2.4737 0.0039 0.0240 -0.0002

12 3 6.7114 10.5108 3.2642 8.2738 7.8272 -4.2922 0.0055 0.0340 -0.0004 4 4.5755 7.0374 2.3372 5.2768 5.0470 -2.5254 0.0036 0.0224 -0.0002

Number of isotropic

plies L/D

θ = 0° ply θ = 90° ply Isotropic ply


0 3 12.9102 33.7141 1.6572 6.4495 374.9440 1.6572 4 9.9192 23.5696 1.8975 4.6262 261.8379 1.8975

4 3 14.4219 39.3185 1.9180 7.4511 437.4434 1.9180 0.0629 0.1376 0.0069 4 11.6328 32.9836 2.6447 6.2356 366.9999 2.6447 0.0527 0.1154 0.0095

8 3 11.9767 30.6297 1.5108 5.8817 340.5872 1.5108 0.0493 0.1073 0.0054 4 11.3445 3.7212 2.6133 6.1334 362.2145 2.6133 0.0520 0.1139 0.0093

12 3 14.4126 39.0041 1.9038 7.3950 433.9373 1.9038 0.0624 0.1365 0.0068 4 9.7010 24.6658 1.9933 4.7644 274.2031 1.9933 0.0398 0.0864 0.0071

CHAPTER

Validation of the Computational Method

6.1 Introduction

Analytical solutions of elasticity problems are obtained case by case in an individual

fashion for separate modes of boundary conditions. Moreover, non-uniform boundary

shape of a structure and the associated mixed mode of boundary conditions are the

major obstacles in obtaining reliable analytical solutions to these problems. The only

plausible option to obtain the solution of non-uniform hybrid laminated composite

structures is through numerical methods. The necessity of the management of

boundary shape has led to the invention of the finite element method, a widely used

computational technique especially for structural analysis. It may be added here that

the finite element prediction of nodal stresses involves an interpolation based

approach together with an averaging scheme to refine the values of stresses. This

problem of approximation becomes more serious when we need to predict stress at the

surfaces of the structures. This indirect method of solution may be realized to be an

inadequate one to predict the actual state of stresses in the critical regions of the

structure. Unfortunately, this critical regions for the case of structural elements are

invariably found to be on the surfaces, which includes external and re-entrant corners

as well. The finite difference solutions are, on the other hand, obtained directly from

the set of algebraic equations derived from the Taylor’s series expansions, free from

any post-processing. Finite difference method has the capability to reproduce

accurately the boundary conditions imposed on the structure of the problem at hand,

although the management of boundary shape is rather a bit involved. Surveying the

literature, it is evident that the analysis of a hybrid laminated composite structure

subjected to mixed mode of boundary conditions, together with geometrical non-

uniformity has never been attempted in the past with Finite difference method.

However, this limitation has been removed through the present investigation.

Moreover the present computational scheme is developed in such a way that it can

handle uniform as well as non-uniform shaped structures of hybrid laminated

composites with both uniform as well as mixed type boundary conditions in an

6

CHAPTER 6 | VALIDATION OF THE COMPUTATIONAL METHOD

142

efficient manner using displacement potential method (DPM) in conjunction with

finite difference method (FDM).

Attempt is made in this chapter to verify the soundness and accuracy of the solutions

obtained through the proposed computational methods by comparing with those of

conventional computational methods which mainly include the finite element method.

Problems of both uniform and non-uniform geometries are considered for the purpose

of verification. Finite element solutions are obtained from commercial softwares and

published results from the literature.

6.2 Problem 1: A guided I-shaped hybrid laminated column

subjected to eccentric loading

A 16-ply hybrid balanced laminate with a stacking sequence of [±75B / ±30G/ ±75G/

±45B]s is considered for the guided I-shaped eccentrically loaded column as shown in

Figure 6.1 a, for the comparison of present solutions with those of finite element

solutions. The aspect ratio of the column considered is taken as 4. Considering plies

of equal thickness, the overall thickness of the laminate is assumed to be 8 mm. The

boundary conditions, corner modelling as well as the overall numerical modelling are

the same for the present solutions as described as Case-I in Chapter 4. The finite

element solutions for this problem is obtained from a commercial software using a

total of 5600 four-noded, isoparametric layered shell finite elements, where maximum

of 80 and 100 elements are used along x- and y-directions, respectively. The shell

element is suitable for analyzing thin to moderately-thick structures. It is a four-noded

element with six degrees of freedom at each node: translations in the x, y, and z

directions, and rotations about the x, y, and z-axes. It is well-suited for linear, large

rotation, and/or large strain nonlinear applications. It can be used for layered

applications for modelling composite shells or sandwich construction. The accuracy

in modelling composite shells is governed by the first-order shear-deformation theory.

The geometry, node locations, and the element coordinate system for this element and

the finite element modelling of the non-uniform laminated column structure are

shown in Figure 6.2.


143

Figure 6.1: (a) Loading and geometry of the non-uniform hybrid laminated composite column used for comparison with FEM solutions and (b) top portion of the column

showing boundary nodes R, S and T and their physical conditions.

σyy

= -σ0; σ

xy = 0

ux = 0;

σxy

= 0

σyy

= 0; σxy

= 0

R S

T

Eccentric loading

Frictionless guides

σ0

0.6 L

0.2 L

0.2 L

x

y

D

D/4 D/4

(a) (b)


144

Figure 6.2: (a) Geometry of the four noded isoparametric layered shell element and (b) Finite element modelling of the non-uniform laminated column using a

commercial software.

6.2.1 Comparison of results

The displacement-potential based finite difference and finite element methods are

completely different in terms of their mathematical modelling as well as solution

processes. The results of the two solution methods for all stress components are

presented in the same graph so that one can readily compare the magnitude and the

nature of variation of the solutions. Individual ply stresses are compared for the two

(b) (a)


145

solutions. Stresses are normalized with respect to the magnitude of applied loading σ0

and distributions are shown at the loaded as well as other critical sections. Figure 6.3

shows the distribution of different stress components of boron/epoxy ply with fiber

orientation of 75° along different lateral sections. For all the stress components, the

two solutions are found to be in good agreement with each other, showing some

discrepancy mainly near the external corners and, particularly near the re-entrant

corner regions of the non-uniform columns, which are, in general, points of

singularity. Glass/epoxy plies of both 30° and 75° are also considered for the

comparison and results are presented in Figure 6.4. Solutions of glass/epoxy plies are

also in good agreement with the finite element solutions in terms of nature of

variation of the distribution, although the re-entrant corner regions are exceptions.

One of the reasons for discrepancy between the solutions is due to the fact that the

corner modelling schemes of these two methods are different. In case of the present

FDM solutions, a total of three conditions out of the available four are satisfied

appropriately at each of the external corner points, while one out of the available two

are satisfied at each re-entrant corner, which is, however, not the case for FEM

modelling. In the present computational scheme, each corner points are considered as

the common point to the connecting two boundaries while FEM considers them as

points of either one of the boundaries. Another reason for the discrepancy between the

solutions is due to the type of element selected in the finite element modelling. Using

a different type of element, for example layered solid element, the discrepancies

might have been different.

It would be worth mentioning here that the present computational scheme has been

developed in such a flexible fashion that it can handle any combination/type of

boundary conditions, especially for the points of singularity, which is, however, in

general, out of the scope of available commercial FEM softwares. Another advantage

of the present method is that it has the capability to reproduce the actual state of

boundary conditions imposed on the structure. This, however, cannot be verified from

the solutions of individual plies, since the boundary conditions are imposed on the

overall laminate and not on individual plies. The present computational program is

developed in such a way that it can obtain the stress field for all the individual plies as

well as the overall laminate independently. According to the conditions of

equilibrium, the average of the stresses of individual plies of equal thickness must be


146

Figure 6.3: Comparison of stresses along different sections of boron/epoxy ply (θ = 75°) of hybrid laminated column subjected to eccentric loading.

x/D0.0 0.2 0.4 0.6 0.8 1.0

xx

-7-6-5-4-3-2-101

DPMFEM

y/L = 0.8

x/D0.0 0.2 0.4 0.6 0.8 1.0

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

DPMFEM

y/L = 0.2

x/D0.0 0.2 0.4 0.6 0.8 1.0

y

-5

-4

-3

-2

-1

0

1

DPMFEM

y/L = 1

x/D0.0 0.2 0.4 0.6 0.8 1.0

-30-25-20-15-10-505

DPMFEM

y/L = 0.2

x/D0.0 0.2 0.4 0.6 0.8 1.0

xy

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

DPMFEM

y/L = 1

x/D0.0 0.2 0.4 0.6 0.8 1.0

-8

-6

-4

-2

0

2

DPMFEM

y/L = 0.2

(c) (d)

(a) (b)

(e) (f)


147

Figure 6.4: Comparison of stresses along different sections of boron/epoxy plies (θ = 30° and 75°) of hybrid laminated column subjected to eccentric loading.

equal to the overall laminate stresses at the corresponding location, which is the case

in the present solutions. On the other hand, most of the commercially available FEM

softwares do not directly show the overall laminate stress. To obtain the stress field of

the overall laminate from these softwares, one has to find the average of stresses for

all plies constituting the laminate. FEM results of some nodal points on the boundary

x/D0.0 0.2 0.4 0.6 0.8 1.0

xx

-0.30-0.25-0.20-0.15-0.10-0.050.000.050.10

DPMFEM

y/L = 1

x/D0.0 0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

1.5

DPMFEM

y/L = 0.2

x/D0.0 0.2 0.4 0.6 0.8 1.0

yy

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

DPMFEM

y/L = 1

x/D0.0 0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

DPMFEM

y/L = 0.2

x/D0.0 0.2 0.4 0.6 0.8 1.0

xy

-8

-6

-4

-2

0

2

DPMFEM

y/L = 0.8

x/D0.0 0.2 0.4 0.6 0.8 1.0

-0.6

-0.4

-0.2

0.0

0.2

0.4

DPMFEM

y/L = 0.2

(d)(c)

(f)(e)

(b)(a)


148

for each ply of the laminate are obtained and then averaged and compared to the FDM

solution of the overall laminate at the corresponding points which are summarized in

Table 6.1. The results are verified with the known physical conditions. The results of

all stress components are shown for nodal points R, S and T which are the middle

point of the unloaded segment of the top surface, the middle point of the loaded

segment of the top surface and the middle point of the right guided surface

respectively Figure 6.1 b. The corner modelling scheme are different for both solution

methods. The stress distribution at the top surface depends on the conditions applied

at the corners. Thus, it is probable that distributions might defer from each other. But

is seen that the present solutions conform to the known physical conditions imposed

to the boundary while the average of FEM solutions deviate from the known

condition as well as present FDM solution. In other words, the FEM solution does not

satisfy equilibrium conditions properly. For instance, the average of FEM prediction

of axial stresses at node R is found to be 0.2609 MPa. But it is known that the axial

stress of the overall laminate must be zero since no axial loading is present at that

particular region. There is a quite deviation between FEM and FDM predictions. For

loaded boundaries the deviation from equilibrium becomes severe especially for the

axial stress components. However, a different selection of element type for the finite

element modelling might decrease these deviations. On the other hand, the lateral

displacement, ux at node T predicted by both FE and FD solutions are also shown in

Table 6.1. It is seen that both solutions conform to the actual physical state. The

comparative analysis verifies the present displacement potential computational

solutions of hybrid laminated composite column to be highly reliable and are founded

on sound philosophy.


149

Table 6.1: Comparison of stresses and displacements at different points of the boundary with known physical conditions of a hybrid laminated column subjected to

eccentric loading.

Node R Node S Node T Ply no. σxx σyy σxy σxx σyy σxy σxx σyy σxy ux

1 0.0063 -1.2381 -0.2637 -2.2342 -6.3295 -1.2165 -1.6523 -5.7213 -1.1295 2 0.0524 -1.0771 0.2195 -2.1831 -6.0814 1.1675 -1.5917 -5.5102 1.0715 3 0.6740 0.2193 0.1790 -1.0443 -0.5555 -0.3232 -1.1852 -0.2231 -0.0816 4 0.6964 0.2326 -0.1945 -1.0195 -0.5449 0.3060 -1.1558 -0.2056 0.0611 5 0.1391 -0.1286 -0.0108 -0.4200 -0.7501 -0.1331 -0.1906 -0.6692 -0.1068 6 0.1448 -0.1138 0.0005 -0.4137 -0.7337 0.1217 -0.1830 -0.6497 0.0932 7 2.3429 1.9928 1.2601 -5.7426 -5.5754 -3.3307 -2.1946 -2.3760 -1.3546 8 2.5504 2.2000 -1.4038 -5.5132 -5.3459 3.1716 -1.9229 -2.1043 1.1662 9 2.5504 2.2000 -1.4038 -5.5132 -5.3459 3.1716 -1.9229 -2.1043 1.1662

10 2.3429 1.9928 1.2601 -5.7426 -5.5754 -3.3307 -2.1946 -2.3760 -1.3546 11 0.1448 -0.1138 0.0005 -0.4137 -0.7337 0.1217 -0.1830 -0.6497 0.0932 12 0.1391 -0.1286 -0.0108 -0.4200 -0.7501 -0.1331 -0.1906 -0.6692 -0.1068 13 0.6964 0.2326 -0.1945 -1.0195 -0.5449 0.3060 -1.1558 -0.2056 0.0611 14 0.6740 0.2193 0.1790 -1.0443 -0.5555 -0.3232 -1.1852 -0.2231 -0.0816 15 0.0524 -1.0771 0.2195 -2.1831 -6.0814 1.1675 -1.5917 -5.5102 1.0715 16 0.0063 -1.2381 -0.2637 -2.2342 -6.3295 -1.2165 -1.6523 -5.7213 -1.1295

FEM (Average) 0.8258 0.2609 -0.0109 -2.3213 -3.2396 -0.0296 -1.2595 -2.1824 -0.0351 0.0000

FDM (Overall) 0.6690 0.0000 0.0000 -1.4792 -3.0000 0.0000 -0.6105 -2.0558 0.0000 0.0000

Known values ---- 0.0000 0.0000 ---- -3.0000 0.0000 ---- ---- 0.0000 0.0000


150

6.3 Problem 2: A Uniform rectangular short sinking beam

In this section, attempt is made to verify results obtained from the displacement

potential method with those of published finite element results available in the

literature. Hardy et. al. [34] obtained finite element solution of built-in uniform

rectangular short beams with sinking support known as the sinking beam as shown in

Figure 6.5 using the standard facilities available in the PAFEC suite of programs. A

regular mesh of 200 8-noded plane stress, isoparametric elements (20×10) was used.

The material used for the short beam is steel with modulus of elasticity, E = 209 GPa

and major Poisson’s ratio, ν12 = 0.3. The same problem is solved using the present

computational approach for a single ply with necessary modification for isotropic

materials as done in Chap 5 for isotropic ply. Both FEM and DPM solutions of

stresses and displacements are compared for the sake of verification of the present

scheme.

Figure 6.5: Loading and geometry of the uniform sinking beam.

6.3.1 Boundary conditions

The left lateral end is fixed while the right lateral end is subjected to a shear

displacement of δL along the negative x-direction. The boundary conditions and

corner modelling the problem are described in Tables 6.2 and 6.3, respectively.

δL

x

y

D

L

a cF

b d


151

Table 6.2: Numerical modelling of the boundary conditions for different boundary segments of the uniform sinking beam.


boundary conditions




a-b un = -uy = 0 ut = ux = 0 ux = 0 uy = 0

c-d un = uy = 0 ut = -ux = δL ux = δL uy = 0

a-c σn = σxx = 0 σt = σxy = 0 σxx = 0 σxy = 0

b-d σn = σxx = 0 σt = σxy = 0 σxx = 0 σxy = 0

Table 6.3: Numerical modelling of the boundary conditions for different corners of the uniform sinking beam.

Corner Point

Available boundary


Used boundary

Parameters

Form of uy

if used



Mesh point on the

physical boundary

Mesh points on the

imaginary boundary



d σn = σxx ; σt = σxy un = uy ; ut = -ux


b σn = σxx ; σt = σxy un = -uy ; ut = ux


c σn = σxx ; σt = σxy un = uy ; ut = -ux


6.3.2 Numerical modelling

A 53×43 mesh network is used for discretizing the computational domain for the

sinking beam. As mentioned in Chap 3, there will be no uninvolved nodal points in

the extreme nodal field. Each nodal point is assigned a number of 1 indicating all

nodes are within the physical boundary as shown in Figure 6.6. Again, each node is

assigned numbers 1 through 4 indicating the form of stress and displacement stencils


152

to be used in the application of boundary conditions as well as for the calculation of

body parameters, which is illustrated in Figure 6.7.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Figure 6.6: Developed extreme nodal field showing the involved and uninvolved nodal points (1 and 0) for computation.


153

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

Figure 6.7: Active field nodal points tagged with stencil indicating numbers (1, 2, 3, 4) for both the stages of pre- and post-processing.

6.3.3 Comparison of results

Stresses are normalized with respect to the maximum bending stress obtained from

elementary theory which is 3EDδL/L2. Displacements are normalized with respect to

the magnitude of applied shear displacement δL. Figure 6.8 shows the comparison of

axial displacements as a function of beam aspect ratio at section y/L = 0.75. Observing

the results, one can verify that there is no discrepancy between the solutions of DPM

and FEM. The distribution curves of both the solutions are found to overlap each

other.


154

Figure 6.8: Comparison of normalized axial displacement in a short sinking beam at y/L = 0.75.

Figure 6.9 shows the distributions of bending stresses at the fixed end (y/L = 0) as a

function of beam aspect ratio. Both solutions are in great conformity with slight

exceptions at the corners of the support. This is due to the difference in corner

modelling in the two computational methods. However, the present solutions are

conservative in terms of critical bending stresses. However, at sections faintly away

from the support such as at y/L = 0.05, bending stresses for both solutions are found to

be in strong agreement without the slightest discrepancy.

uy/L

-0.2 -0.1 0.0 0.1 0.2

x/D

0.0

0.2

0.4

0.6

0.8

1.0

y/L = 0.75

L/D = 0.5

uy/L

-0.2 -0.1 0.0 0.1 0.2

y/L = 0.75

L/D = 1

uy/L

-0.2 -0.1 0.0 0.1 0.2

x/D

0.0

0.2

0.4

0.6

0.8

1.0

y/L = 0.75

L/D = 1.5

uy/L

-0.2 -0.1 0.0 0.1 0.2

y/L = 0.75

L/D = 2

(c) (d)

(a) (b)

DPMFEM

DPMFEM

DPMFEM

DPMFEM


155

Figure 6.9: Comparison of normalized bending stress in a short sinking beam at y/L = 0.

Figure 6.10 shows the comparison of shear stresses at various sections for a beam

aspect ratio of 1. Like the case bending stresses, both solutions of shear stresses are in

quite good agreement with slight exceptions around the points of singularity. From

Figure 6.10 it is seen that FDM solutions of shear stresses shows a normalized value

of 0 at the top and bottom surface, which appropriately reflects the associated given

physical condition at the corner point (see Table 6.3). But this is not the case in the

FEM solutions. FEM considers the corner points as the extreme points of the fixed

support only, and accordingly the restrained boundary conditions are only satisfied.

Away from the support, the discrepancy in shear stress solutions reduce. If the

concerned section is far enough from the supporting region, for example at y/L = 0.25

(Figure 6.10 c), the solutions of both methods overlap.

x/D0.0 0.2 0.4 0.6 0.8 1.0

yy .(

L2 /3ED

L)

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

DPMFEM

y/L = 0L/D = 0.5

x/D0.0 0.2 0.4 0.6 0.8 1.0

DPMFEM

y/L = 0L/D = 1

x/D0.0 0.2 0.4 0.6 0.8 1.0

yy .(

L2 /3ED

L)

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

DPMFEM

y/L = 0L/D = 1.5

x/D0.0 0.2 0.4 0.6 0.8 1.0

DPMFEM

y/L = 0L/D = 2

(c) (d)

(a) (b)


156

Figure 6.10: Comparison of normalized shear stress at various sections of a short sinking beam, L/D = 1.

Now, the bending stresses predictions of the present scheme are compared with those

of FEM and theoretical predictions found in reference [34]. Here, the theoretical

predictions at y/L = 0.05 are obtained from simple bending theory and modified

theories based on shear deformation and strain energy. Using simple theory, bending

stresses at y/L = 0 are also obtained. Table 6.4 summarizes the comparison of the

present solutions with those of FE, simple theory and modified theory solutions at

sections y/L = 0 and 0.05. The simple theory does not take the beam aspect ratio in to

account. So the predictions vary depending on the section concerned, irrespective of

beam aspect ratio. Comparing the results for DPM and FE solutions at the fixed end

(y/L = 0) it is seen that the present solutions are conservative in terms of critical

bending stresses. However, at section y/L = 0.05, the FE solutions are conservative.

Still, the present solutions predict safer design stresses since bending stress is critical

x/D0.0 0.2 0.4 0.6 0.8 1.0

xy .(

L2 /3ED

L)

-0.05

0.00

0.05

0.10

0.15

0.20

DPMFEM

(a) y/L = 0L/D = 1

x/D

0.0 0.2 0.4 0.6 0.8 1.0

DPMFEM

y/L = 0.1L/D = 1

x/D0.0 0.2 0.4 0.6 0.8 1.0

DPMFEM

y/L = 0.5L/D = 1

x/D0.0 0.2 0.4 0.6 0.8 1.0

xy .(

L2 /3ED

L)

-0.05

0.00

0.05

0.10

0.15

0.20

DPMFEM

y/L = 0.25L/D = 1

(b)

(c) (d)


157

at the supporting end. The modified theories underestimates the state of bending

stresses.

Table 6.4: Comparison of maximum normalized bending stress predictions with FEM, simple and modified theory estimates at various sections in a uniform short

sinking beam [34].

Maximum normalized bending stress

y/L = 0 y/L = 0.05

L/D DPM FEM Simple theory DPM FEM Simple

theory Modified theories

* † 0.5 0.53 0.23 1.00 0.15 0.18 0.90 0.05 0.07 1.0 0.83 0.51 1.00 0.31 0.35 0.90 0.18 0.22 1.5 1.04 0.74 1.00 0.44 0.48 0.90 0.33 0.38 2.0 1.16 0.90 1.00 0.54 0.57 0.90 0.46 0.51

* Modified theory based on shear deformation, shear coefficient = 1.5. † Modified theory based on strain energy, shear coefficient = 1.2.

6.4 Salient features of the present computational scheme

In this section, the salient features of the present computational technique of solving

problems of non-uniform laminated structures is discussed in comparison to the

existing commercial softwares.

The existing commercial softwares are based on the finite element method, which is

an indirect method of solution. In order to predict the behavior of the solution,

variations of parameters are assumed to be simple like polynomials of limited order.

Thus the solution is constrained to behave in the assumed mode, rather than the one it

would adopt naturally. Therefore, the existing softwares are likely to produce poor

results if the assumed variations do not satisfy the actual behaviour exactly. On the

other hand, the present computational scheme uses the finite difference technique

which does not presuppose any variation in its application and therefore produces

more attractive results as they are free from the shortcomings mentioned above. It

produces direct solution, unlike the existing softwares.

The FEM obtain solution at non-nodal points and uses extrapolation and averaging to

predict solution at nodal points. Meanwhile, the present computational scheme uses

finite difference method where primary solution ψ is obtained directly at the nodal


158

points without the aid of extrapolation. Subsequently, the stresses and displacements

at nodal points are calculated from the ψ solution. Moreover, the existing methods do

not consider singularity in its computation. An averaged result is predicted at the

external and re-entrant corners of a structure, which are the points of singularity. But

the present scheme is capable of handling singularity and the results at the external

and re-entrant corners are obtained directly satisfying maximum possible physical

conditions.

The present scheme permits the reduction of variables to be evaluated at each nodal

point of the domain, from three components of displacement to a single scalar

function. This reduction of unknowns at each nodal point ultimately reduces the total

number of algebraic equations to be solved. Hence, the computational work is reduced

drastically which makes the present computational scheme very efficient. Moreover,

the present scheme solves the elastic field of the laminate by converting it to an

equivalent single ply and then the solution is extended to each plies of the laminate.

This in turn reduces the extent of computation even more compared to the existing

methods since the existing methods obtain solutions for each ply of the whole

laminate individually. The present scheme is capable of producing both stresses of

overall laminate as well as individual plies independently. The overall laminate

stresses and global strains are obtained from the ψ solution, while individual ply

stresses are calculated analytically by multiplying the global strain field with the

reduced stiffness matrix of the corresponding ply. On the other hand, the existing

softwares are only capable of obtaining stresses of individual plies. The overall

laminate stresses have to be calculated from averaging stresses of individual plies.

The present computational method is limited to problems of plane elasticity.

However, the scheme can be extended to include three-dimensional analysis using the

displacement potential function formulation for three-dimensional analysis [35].

However, solving a problem of plane elasticity, the present scheme requires the input

of materials properties only for the relevant axes and plane. On the other hand, the

existing methods solve all type of problems as three-dimensional ones and require

input of material properties along all three axes and all three orthogonal planes. In

short, the present scheme takes less information as input than existing ones without

hampering the accuracy. Moreover, reliability of the solutions obtained by FEM


159

depend on the proper selection of element type, whereas the present scheme does not

bear this type of difficulty.

In the existing methods, the nodal points are not numbered in a uniform manner.

Thus, reading the solutions is rather troublesome because the results read from the

tabular form have to be matched with the nodal point number. Another way to obtain

the solution is to click on each node and read the results manually. For a large number

of nodes, this becomes a very inefficient procedure. However, this is not the case in

the present computational scheme. Results of the overall laminate as well as

individual plies can be easily shown at all the nodes as well as at any section of the

structure as both tabular and graphical form.

6.5 Summary

The stresses and displacements predicted by the present computational scheme are in

quite good agreement with the finite element solutions with slight exceptions around

the external or re-entrant corners of the boundary, which are points of singularity.

However, the present solutions are found to be conservative in terms of critical

stresses, which, in turn, verifies the adequateness and soundness of the present

computational approach in providing safe and economic design guides for both

uniform and non-uniform structures.

CHAPTER

Conclusions

7.1 Conclusions

The central objective of this thesis is to analyze the elastic field of guided non-

uniform composite structures of hybrid laminates. A single variable computational

method based on displacement potential modelling is developed which is well capable

of handling non-uniform geometry of the body and can manage all possible modes of

boundary conditions, whether they are prescribed in terms of stresses or constraints or

any combination of them. Moreover, the three dimensional laminates are analyzed

here as a plane stress problem and then solutions are extended from the laminate mid-

plane to any other plies of interest. The scheme is demonstrated through the solution

of a number of problems of hybrid laminated structures of non-uniform geometry.

The main conclusions are summarized as follows:

The potential-function based elasticity formulation of laminated composites is

extended for the elastic analysis of hybrid balanced laminates in terms of a

single scalar function.

A general finite difference numerical scheme is developed for the management

of both uniform and non-uniform geometry of the laminated composite

structures. A variable node numbering scheme is adopted here to discretize the

non-uniform computational domain using a rectangular mesh-network, in

which the active field nodal points are renumbered a number of times at

different stages of pre- and post-processing. Moreover, a flexible numerical

modelling scheme is introduced at the external and re-entrant corners of the

non-uniform geometry, which are, in general, the points of singularity in the

solution.

Based on the present numerical scheme, an investigation of the stress field of a

guided non-uniform hybrid balanced laminated composite column subjected to

eccentric loading has been carried out mainly in the perspective of laminate

hybridization. The effect of hybridization in terms of the overall laminate

stress is found to be nearly insignificant. However, the same is identified to be

7

CHAPTER 7 | CONCLUSIONS

161

quite prominent when analyzed in the perspective of individual ply stresses,

especially around the re-entrant corners of the non-uniform column.

Moreover, this effect of hybridization on ply stresses is further found to be

influenced significantly by the orientation angles of fibers in individual plies

of the laminate. The investigation also shows that stresses are affected by both

eccentricity of loading and partial guides, which is further influenced by the

ply fiber orientation.

Secondly, the elastic field of a non-uniform sinking beam of hybrid balanced

laminated composite (FRC and soft isotropic ply) is analyzed mainly in the

perspective of beam aspect ratio and number of soft isotropic plies forming

part of the laminate. Both aspect ratio and number of isotropic plies affect the

overall laminate stresses as well as individual ply stresses. Fiber orientation of

the FRC ply is, once again, found to be an influencing factor for defining the

state of stresses. Observing the details of the stress fields it can be concluded

that the stress level in an individual ply of a laminate can be well controlled by

hybridizing the plies of appropriate fiber stiffness and orientation.

Finally, the present numerical solutions are discussed in light of comparison

with these of conventional solutions and it is found that, the elastic field is in

good conformity with the corresponding FEM solutions with slight exceptions,

particularly around the critical regions. In all cases, the present solutions are,

however, found to be conservative in terms of critical stresses, which, in turn,

verifies the adequateness of the present computational approach in providing

safe and economic design guides for hybrid laminated structures.

7.2 Recommendations for future works

The present research will lead to an effective alternative approach for reliable analysis

of structural components of hybrid laminates with non-uniform geometries. Results of

the present analysis are also expected to provide a reliable design guide to non-

uniform composite laminated structures. In this connection it is recommended that the

scheme can be modified to incorporate the following works for further investigations:

CHAPTER 7 | CONCLUSIONS

162

The present numerical technique can be modified for handling arbitrary geometry of

structures. It can be further modified for managing inner boundary conditions for

structures with inner circular hole etc.

163

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APPENDIX A | FLOW DIAGRAM OF THE COMPUTER PROGRAM

168

Appendix A

Flow diagram of the computer program

Input: Geometry indicators 1 and 0

Stencil form indicators 1,2,3 and 4

Initializes coefficient matrix [K] with size p×pconstant column matrix {C} with size p×1

Extreme nodal field generation

with node numbers from 1 to p

Compute mesh lengthy ℎ𝑥, 𝑘𝑦

Create

Reduced stiffness matrix for each ply

transfomed reduced stiffness matrix for each ply

Extensional stiffness matrix of the laminate

Input: stacking sequence of laminate

properties θ, E1,E2, V12,G12 of each ply

Input: maximum length (y-axis) L

maximum width (x-axis) D

maximum nodes along x axis mi

maximum nodes along y axis nj

Start


169

Identification of inner

Non-uniform geometry external corner nodes

Identification of

Re-entrant corner nodes

Identification of boundary segments

Input: Choice of outer external corner boundary conditions and

Correspondence between mesh points and boundary conditions by

Defining ‘corner’ matrix for each external corner

Non-

Uniform geometry Input: choice of inner external corner

boundary conditions and correspondence between mesh points and boundary

conditions by defining ‘corner’ matrix for each inner external corner

Identification of external corner node

Y

N

Y

N


170

Expansion by inserting

Zero elements

Input: choice of re-entrant

Corner boundary conditions

Application of boundary conditions at boundary segments and corner points

for physical and imaginary nodal points

Generation of matrix [K] with size p×p

Constant column matrix {C} with size p×1

Non uniform geometry Generation of Reduced coefficient matrix

[K] of size p´×p´

Reduced constant column matrix {C} of size

p´×1

Solution by Choleskey’s Solution by Choleskey’s

Triangular Decomposition Triangular Decomposition

Creation of solution matrix Creation of solution matrix

{ψ} with size p×1 {ψ} with size p´×1

N

Y


171

Evaluation of body parameters of overall laminate

And write into file

Evaluation of stresses of individual ply by multiplying

Global strain with transformed reduced stiffness matrix

Of each and every ply and write into file

End

stress analysis of non-uniform guided composite structures

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