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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.
vi
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
xi
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
CHAPTER 1 | INTRODUCTION
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.
CHAPTER 1 | INTRODUCTION
4
(a)
(b)
Figure 1.2: Application of composites and hybrid composites in (a) an aircraft structure and (b) a helicopter rotor blade.
CHAPTER 1 | INTRODUCTION
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.
CHAPTER 1 | INTRODUCTION
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
CHAPTER 1 | INTRODUCTION
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.
CHAPTER 1 | INTRODUCTION
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
CHAPTER 1 | INTRODUCTION
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
CHAPTER 1 | INTRODUCTION
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,
CHAPTER 1 | INTRODUCTION
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.
CHAPTER 1 | INTRODUCTION
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
CHAPTER 2 | MATHEMATICAL BACKGROUND
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
CHAPTER 2 | MATHEMATICAL BACKGROUND
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]:
CHAPTER 2 | MATHEMATICAL BACKGROUND
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)
CHAPTER 2 | MATHEMATICAL BACKGROUND
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.
CHAPTER 2 | MATHEMATICAL BACKGROUND
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
CHAPTER 2 | MATHEMATICAL BACKGROUND
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)
CHAPTER 2 | MATHEMATICAL BACKGROUND
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)
CHAPTER 2 | MATHEMATICAL BACKGROUND
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)
CHAPTER 2 | MATHEMATICAL BACKGROUND
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
CHAPTER 2 | MATHEMATICAL BACKGROUND
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
CHAPTER 2 | MATHEMATICAL BACKGROUND
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)
CHAPTER 2 | MATHEMATICAL BACKGROUND
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)
CHAPTER 2 | MATHEMATICAL BACKGROUND
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)
CHAPTER 2 | MATHEMATICAL BACKGROUND
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
CHAPTER 2 | MATHEMATICAL BACKGROUND
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.
CHAPTER 2 | MATHEMATICAL BACKGROUND
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
CHAPTER 2 | MATHEMATICAL BACKGROUND
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
CHAPTER 2 | MATHEMATICAL BACKGROUND
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
CHAPTER 2 | MATHEMATICAL BACKGROUND
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].
CHAPTER 2 | MATHEMATICAL BACKGROUND
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
CHAPTER 2 | MATHEMATICAL BACKGROUND
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)
CHAPTER 2 | MATHEMATICAL BACKGROUND
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)
CHAPTER 2 | MATHEMATICAL BACKGROUND
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
M B B B D D DM B B B D D D
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)
CHAPTER 2 | MATHEMATICAL BACKGROUND
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
M B B B D D DM B B B D D D
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)
CHAPTER 2 | MATHEMATICAL BACKGROUND
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],
CHAPTER 2 | MATHEMATICAL BACKGROUND
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)
CHAPTER 2 | MATHEMATICAL BACKGROUND
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
u uu uA A A Ah x x y y
(2.53 b)
16 66 66 261 y yx x
xy
u uu uA A A Ah x x y y
(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.
CHAPTER 2 | MATHEMATICAL BACKGROUND
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)
CHAPTER 2 | MATHEMATICAL BACKGROUND
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)
CHAPTER 2 | MATHEMATICAL BACKGROUND
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)
CHAPTER 2 | MATHEMATICAL BACKGROUND
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].
CHAPTER 2 | MATHEMATICAL BACKGROUND
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.
CHAPTER 2 | MATHEMATICAL BACKGROUND
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
CHAPTER 2 | MATHEMATICAL BACKGROUND
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
CHAPTER 2 | MATHEMATICAL BACKGROUND
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)
CHAPTER 3 | NUMERICAL ANALYSIS
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
CHAPTER 3 | NUMERICAL ANALYSIS
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.
CHAPTER 3 | NUMERICAL ANALYSIS
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.
CHAPTER 3 | NUMERICAL ANALYSIS
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)
CHAPTER 3 | NUMERICAL ANALYSIS
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)
CHAPTER 3 | NUMERICAL ANALYSIS
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)
CHAPTER 3 | NUMERICAL ANALYSIS
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)
CHAPTER 3 | NUMERICAL ANALYSIS
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)]
xu C i j i j i j i ji j i j i j i j
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)]
xu C i j i j i j i ji j i j i j i j
i j
(3.7)
CHAPTER 3 | NUMERICAL ANALYSIS
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)]
xu C i j i j i j i ji j i j i j i j
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)
CHAPTER 3 | NUMERICAL ANALYSIS
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.
CHAPTER 3 | NUMERICAL ANALYSIS
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.
CHAPTER 3 | NUMERICAL ANALYSIS
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
1 and 3 Central Forward
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
2 and 4 Central Backward
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)
CHAPTER 3 | NUMERICAL ANALYSIS
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.
CHAPTER 3 | NUMERICAL ANALYSIS
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.
CHAPTER 3 | NUMERICAL ANALYSIS
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( ) } (
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
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 C m i j C l i jC l i j C l i j C l i j
C l i j
(3.17)
CHAPTER 3 | NUMERICAL ANALYSIS
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( ) } (
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
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 C m i j C l i jC l i j C l i j C l 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( ) } ( ,
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
(3.19)
CHAPTER 3 | NUMERICAL ANALYSIS
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( ) } (
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
(3.21)
CHAPTER 3 | NUMERICAL ANALYSIS
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( ) } (
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
(3.22)
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( ) } (
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
(3.23)
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.
CHAPTER 3 | NUMERICAL ANALYSIS
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.
CHAPTER 3 | NUMERICAL ANALYSIS
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)
CHAPTER 3 | NUMERICAL ANALYSIS
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 C C lm i jC C l C C m C lm i j
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 C C lm i jC C l C C m C lm i j
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 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
8
8 9
2 ( 2, 1)2 ( 2, 1) 2 ( 3, )
C lm i jC lm i j C lm i j
(3.26)
CHAPTER 3 | NUMERICAL ANALYSIS
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 C C lm i jC C l C C m C lm i j
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
(3.27)
CHAPTER 3 | NUMERICAL ANALYSIS
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)
CHAPTER 3 | NUMERICAL ANALYSIS
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 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 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)
CHAPTER 3 | NUMERICAL ANALYSIS
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 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 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
CHAPTER 3 | NUMERICAL ANALYSIS
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.
CHAPTER 3 | NUMERICAL ANALYSIS
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
CHAPTER 3 | NUMERICAL ANALYSIS
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:
CHAPTER 3 | NUMERICAL ANALYSIS
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
u C i j i j C i j i j
C C i j O h k
(3.33)
2 3
22 3
{ 2 ( 1, ) ( 2, )} { ( , 1) ( , 1)}
( 2 ) ( , ) ( , )y
x y
u C i j i j C i j i j
C C i j O h k
(3.34)
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
CHAPTER 3 | NUMERICAL ANALYSIS
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
CHAPTER 3 | NUMERICAL ANALYSIS
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.
CHAPTER 3 | NUMERICAL ANALYSIS
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.
CHAPTER 3 | NUMERICAL ANALYSIS
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)
CHAPTER 3 | NUMERICAL ANALYSIS
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.
CHAPTER 3 | NUMERICAL ANALYSIS
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
CHAPTER 3 | NUMERICAL ANALYSIS
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 and 3 Central Forward
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
2 and 4 Central Backward
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
1 and 3 Central Forward
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 and 4 Central Backward
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.
CHAPTER 3 | NUMERICAL ANALYSIS
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.
CHAPTER 3 | NUMERICAL ANALYSIS
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.
CHAPTER 3 | NUMERICAL ANALYSIS
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
CHAPTER 3 | NUMERICAL ANALYSIS
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.
CHAPTER 3 | NUMERICAL ANALYSIS
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
CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES
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
CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES
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
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 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
CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES
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
CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES
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
Correspondence between mesh points and given boundary conditions
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
uy, σxx, σxy 2 [0 3 1 0 2] σxx = 0 σxy = 0 uy = 0
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
σxx, σyy, σxy [0 0 1 3 2] σxx = 0 σyy = 0 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
σxx, σyy, σxy [0 0 1 3 2] σxx = 0 σyy = 0 σxy = 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 ---
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
CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES
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
CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES
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.
CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES
101
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.
CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES
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
CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES
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
CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES
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.
CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES
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.
CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES
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
boron/epoxy and glass/epoxy plies.
CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES
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.
CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES
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
boron/epoxy and glass/epoxy plies.
CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES
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
boron/epoxy and glass/epoxy plies.
CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES
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
boron/epoxy and glass/epoxy plies.
CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES
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
boron/epoxy and glass/epoxy plies.
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
CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES
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
CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES
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
CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES
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
CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES
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
Original shapeDeformed shape
CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES
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
CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES
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.
CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES
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.
CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES
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
Original shapeDeformed shape
x-coordinate (m)
0.00 0.05 0.10
Original shapeDeformed shape
CHAPTER 4 | ANALYSIS OF NON-UNIFORM COLUMN OF HYBRID LAMINATES
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
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 a-b and
σn (0, y) = 0 and σt (0, y) = 0 for segment a´-b´ [ 0 ≤ y ≤ 0.2L]
CHAPTER 5 | ANALYSIS OF NON-UNIFORM BEAM OF HYBRID LAMINATES
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
CHAPTER 5 | ANALYSIS OF NON-UNIFORM BEAM OF HYBRID LAMINATES
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)
CHAPTER 5 | ANALYSIS OF NON-UNIFORM BEAM OF HYBRID LAMINATES
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 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´ 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
CHAPTER 5 | ANALYSIS OF NON-UNIFORM BEAM OF HYBRID LAMINATES
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
Parameters from two sides
Used boundary
Parameters
Form of uy
if used
Elements of ‘corner’ vector
Correspondence between mesh points and given boundary conditions
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´ 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
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 σn = σxx ; σt = σxy un = uy ; ut = -ux
uy, σxx, σxy 2 [1 3 2 0 0] ux = δ σxx = 0 uy = 0
uy, σxx, σxy 2 [0 3 1 0 2] σxx = 0 σxy = 0 uy = 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
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
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 ---
CHAPTER 5 | ANALYSIS OF NON-UNIFORM BEAM OF HYBRID LAMINATES
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].
CHAPTER 5 | ANALYSIS OF NON-UNIFORM BEAM OF HYBRID LAMINATES
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,
CHAPTER 5 | ANALYSIS OF NON-UNIFORM BEAM OF HYBRID LAMINATES
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)
CHAPTER 5 | ANALYSIS OF NON-UNIFORM BEAM OF HYBRID LAMINATES
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)
CHAPTER 5 | ANALYSIS OF NON-UNIFORM BEAM OF HYBRID LAMINATES
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.
CHAPTER 5 | ANALYSIS OF NON-UNIFORM BEAM OF HYBRID LAMINATES
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)
CHAPTER 5 | ANALYSIS OF NON-UNIFORM BEAM OF HYBRID LAMINATES
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
Original shapeDeformed shape
(b)
CHAPTER 5 | ANALYSIS OF NON-UNIFORM BEAM OF HYBRID LAMINATES
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
CHAPTER 5 | ANALYSIS OF NON-UNIFORM BEAM OF HYBRID LAMINATES
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
CHAPTER 5 | ANALYSIS OF NON-UNIFORM BEAM OF HYBRID LAMINATES
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
boron/epoxy laminate.
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)
no isotropic ply 8 isotropic plies12 isotropic plies
CHAPTER 5 | ANALYSIS OF NON-UNIFORM BEAM OF HYBRID LAMINATES
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
CHAPTER 5 | ANALYSIS OF NON-UNIFORM BEAM OF HYBRID LAMINATES
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
boron/epoxy laminate.
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)
no isotropic ply 8 isotropic plies12 isotropic plies
CHAPTER 5 | ANALYSIS OF NON-UNIFORM BEAM OF HYBRID LAMINATES
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
CHAPTER 5 | ANALYSIS OF NON-UNIFORM BEAM OF HYBRID LAMINATES
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
σxx σyy σxy σxx σyy σxy σxx σyy σxy
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
σxx σyy σxy σxx σyy σxy σxx σyy σxy
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.
CHAPTER 6 | VALIDATION OF THE COMPUTATIONAL METHOD
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)
CHAPTER 6 | VALIDATION OF THE COMPUTATIONAL METHOD
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)
CHAPTER 6 | VALIDATION OF THE COMPUTATIONAL METHOD
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
CHAPTER 6 | VALIDATION OF THE COMPUTATIONAL METHOD
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)
CHAPTER 6 | VALIDATION OF THE COMPUTATIONAL METHOD
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)
CHAPTER 6 | VALIDATION OF THE COMPUTATIONAL METHOD
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.
CHAPTER 6 | VALIDATION OF THE COMPUTATIONAL METHOD
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
CHAPTER 6 | VALIDATION OF THE COMPUTATIONAL METHOD
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
CHAPTER 6 | VALIDATION OF THE COMPUTATIONAL METHOD
151
Table 6.2: Numerical modelling of the boundary conditions for different boundary segments of the uniform sinking beam.
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-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
Parameters from two sides
Used boundary
Parameters
Form of uy
if used
Elements of ‘corner’ vector
Correspondence between mesh points and given boundary conditions
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
d σn = σxx ; σt = σxy un = uy ; ut = -ux
uy, σxx, σxy 2 [0 3 1 0 2] σxx = 0 σxy = 0 uy = 0
b σn = σxx ; σt = σxy un = -uy ; ut = ux
uy, σxx, σxy 2 [0 3 1 0 2] σxx = 0 σxy = 0 uy = 0
c σn = σxx ; σt = σxy un = uy ; ut = -ux
uy, σxx, σxy 2 [0 3 1 0 2] σxx = 0 σxy = 0 uy = 0
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
CHAPTER 6 | VALIDATION OF THE COMPUTATIONAL METHOD
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.
CHAPTER 6 | VALIDATION OF THE COMPUTATIONAL METHOD
153
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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.
CHAPTER 6 | VALIDATION OF THE COMPUTATIONAL METHOD
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
CHAPTER 6 | VALIDATION OF THE COMPUTATIONAL METHOD
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)
CHAPTER 6 | VALIDATION OF THE COMPUTATIONAL METHOD
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)
CHAPTER 6 | VALIDATION OF THE COMPUTATIONAL METHOD
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
CHAPTER 6 | VALIDATION OF THE COMPUTATIONAL METHOD
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
CHAPTER 6 | VALIDATION OF THE COMPUTATIONAL METHOD
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
APPENDIX A | FLOW DIAGRAM OF THE COMPUTER PROGRAM
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
APPENDIX A | FLOW DIAGRAM OF THE COMPUTER PROGRAM
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
APPENDIX A | FLOW DIAGRAM OF THE COMPUTER PROGRAM
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
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