AN EFFICIENT LAYER-WISE FINITE ELEMENT MODEL FOR STATIC, VIBRATION AND BUCKLING ANALYSIS OF...

AN EFFICIENT LAYER-WISE FINITE ELEMENT MODEL FOR STATIC, VIBRATION AND BUCKLING ANALYSIS

OF COMPOSITES AND SANDWICH LAMINATES WITH INTER-LAMINAR IMPERFECTIONS

A thesis submitted for the award of the degree of

Doctor of Philosophy

by

Anupam Chakrabarti

DEPARTMENT OF OCEAN ENGINEERING AND NAVAL ARCHITECTURE INDIAN INSTITUTE OF TECHNOLOGY

KHARAGPUR - 721 302, INDIA November 2003

With dedication to

My parents

With Love to

My wife Rupa

And little daughter Ahana

Who endured all the sufferings silently

And looked forward to this day…..

CERTIFICATE

This is to certify that the thesis entitled ‘AN EFFICIENT LAYER-WISE FINITE ELEMENT MODEL FOR STATIC, VIBRATION AND BUCKLING ANALYSIS OF COMPOSITES AND SANDWICH LAMINATES WITH INTER-LAMINAR IMPERFECTIONS’ being submitted to the

Indian Institute of Technology, Kharagpur by Anupam Chakrabarti for the award of the

degree of Doctor of Philosophy is a record bonafide research work carried out by him under our

supervision and guidance, and Mr. Chakrabarti fulfills the requirements of the regulations of the

degree. The results embodied in this thesis have not been submitted to any other University or

Institute for the award of any degree or diploma. (Abdul Hamid Sheikh) Associate Professor Department of Ocean Engineering and Naval Architecture Indian Institute of Technology, Kharagpur-721302 INDIA (Sagar Kumar Sengupta) Professor and Head Department of Civil Engineering Jalpaiguri Government Engineering College, West Bengal-735102 INDIA

Acknowledgement

I am most grateful to my supervisors, Prof. A.H. Sheikh and Prof. S.K. Sengupta for their valuable advices, interests and encouragements throughout the period of the research work, right from the inception of the problem to the final preparation of the manuscript. It is due to their experience and timely suggestions that the work has taken its present shape.

I sincerely appreciate Prof. D. Sen, Head, Department of Ocean Engineering and Naval Architecture, IIT Kharagpur and Prof. S.K. Satsangi, former Head of the Department, for extending all necessary computational facilities of the department

I am highly indebted to the Directorate of Technical Education, Government of West Bengal for granting me the necessary leave to carry out the research work and the authorities of Jalpaiguri Government Engineering College, Jalpaiguri for allowing me to utilize all the required computational facilities of the Civil Engineering department.

Special thanks are due for my co-scholar Pijush Topdar for extending all sorts of helps without any hesitation during my tenure at IIT, Kharagpur. My other co-scholars in the Naval Computer Laboratory, Jahangir, Malabika, Ranadev, Joydeep, Sujoy, Anindya, Mihir … deserve special mention for making my stay a very pleasant and memorable one.

I am really grateful to my colleagues Prof. J. J. Mondal and Prof. M.K. Mukherjee, Department of Civil Engineering, Jalpaiguri Government

Acknowledgement

ii

Engineering College for extending tremendous mental support during my research work.

My sincerest gratitude goes to all my family members and relatives for their encouragements and co-operations shown during this period. It was a difficult proposition for me to undertake Ph.D. work away from home. However, I am indebted to my in-laws for not only taking care of my family in my absence but relieving me of all such responsibilities during my stay at IIT, Kharagpur.

I am also grateful to the faculty and staff of the Department of Ocean Engineering and Naval Architecture, IIT Kharagpur for their immense help throughout the tenure of my research. Special mention should be made of Mr. P. K. Ray, system administrator for his help and cooperation especially during the preparation of the manuscript. I am also thankful to Mr. C. Mukherjee for preparing some of the figures.

Finally, I am thankful to the authorities of Indian Institute of Technology, Kharagpur for providing me all necessary assistance in the form of research and guidance.

Indian Institute of Technology Kharagpur (A. Chakrabarti)

Contents

Preface ix Nomenclature xii List of Figures xvii List of Tables xx Chapter 1 Introduction 1-9 1.1 Introduction 1 1.2 Objective and scope of present investigation 7 Chapter 2 Review of Literature 10-39 2.1 Introduction 10 2.2 Shear deformation theories for laminated plates 12 2.2.1 Single layer theories 12 2.2.2 Layer-wise theories 14 2.2.2.1 Discrete layer theories 15 2.2.2.2 Refined shear deformation theories 16 2.3 Imperfections at the layer interfaces 18 2.4 Finite elements for laminated plates 21 2.4.1 Elements based on First order shear deformation theory (FSDT) 21 2.4.2 Elements based on Higher order shear deformation theory (HSDT) 23 2.4.3 Elements based on Discrete layer theories 25 2.4.4 Elements based on Refined shear deformation theories 26 2.5 Important analytical studies on laminated plates 29 2.5.1 Static analysis 29

Contents iv

2.5.2 Free vibration analysis 31 2.5.3 Buckling analysis 35 2.6 Motivation of the present study 39

Chapter 3 Mathematical Formulation and Computer Implementation 40-64 3.1 Introduction 40 3.2 Basic Assumptions 41 3.3 Governing equations for the proposed analyses 42 3.4 Stress strain relationship 42 3.5 Representation of displacement components as per the refined plate model 43 3.6 Finite element implementation 46 3.6.1 Proposed element 47 3.6.2 Element Stiffness Matrix and Load Vector 51 3.6.3 Element Mass Matrix 52 3.6.4 Element Geometric Stiffness Matrix 53 3.6.5 Numerical integration 55 3.6.6 Assembly of Element Matrices 56 3.6.7 Boundary conditions 56 3.7 Solution Techniques 57 3.7.1 Static analysis 57 3.7.2 Free Vibration and Buckling analysis 57 3.8 Computer implementation 59 3.8.1 Introduction 59 3.8.2 Application domain 59 3.8.3 Description of the computer program 60 3.8.3.1 Preprocessor 61 3.8.3.2 Processor 61 3.8.3.3 Postprocessor 64

Contents v

Chapter 4 Results and Discussions 65 – 156 4.1 Introduction 65 4.2 Patch test 67 4.3 Plates having perfect interfaces 69 4.3.1 Static Analysis 69 4.3.1.1 Analysis of laminated composite plates based on HSDT/FSDT 69 4.3.1.1.1 Simply supported square cross-ply laminated composite plate 69 4.3.1.1.2 Simply supported rectangular cross-ply laminated composite plate 71 4.3.1.1.3 Simply supported un-symmetric angle-ply laminate 74 4.3.1.1.4 Skew cross-ply laminate simply supported at the edges 74 4.3.1.2 Analysis of composites and sandwich laminates based on RHSDT 76 4.3.1.2.1 Simply supported cross-ply square laminate 76 4.3.1.2.2 Cylindrical bending of a cross-ply laminate 79 4.3.1.2.3 Simply supported cross-ply square laminate 80 4.3.1.2.4 Un-symmetric angle-ply square laminate 81 4.3.1.2.5 Skew cross-ply laminated plate simply supported at the edges 83 4.3.1.2.6 Square sandwich plate (f/c/f) subjected to uniform load 85 4.3.1.2.7 Double core (f/c/f/c/f) square sandwich plate subjected to uniform

load 87 4.3.1.2.8 Rectangular laminated sandwich plate simply supported at the edges 88 4.3.1.2.9 Double core square sandwich plate with laminated stiff sheets 91 4.3.1.2.10 Sandwich plate having angle-ply laminated stiff sheets

at the two faces 92 4.3.1.2.11 Skew laminated sandwich plate having different boundary

conditions 93 4.3.2 Free Vibration Analysis 94 4.3.2.1 Analysis of laminated composite plates based on HSDT/FSDT 94 4.3.2.1.1 Rectangular cross-ply and angle-ply laminates 94 4.3.2.1.2 Cross-ply square laminate having different boundary conditions at the edges 98

Contents vi

4.3.2.1.3 Triangular cross-ply and angle-ply laminates simply supported at the edges 99

4.3.2.1.4 Simply supported square laminate having concentrated mass at center 100

4.3.2.2 Analysis of composites and sandwich laminates based on RHSDT 102 4.3.2.2.1 Un-symmetric cross-ply and angle-ply laminates 102 4.3.2.2.2 Skew angle-ply (30/-30/30) laminate clamped at the edges 103 4.3.2.2.3 Simply supported square laminated plate having orthotropic layers 105 4.3.2.2.4 Square laminated sandwich plate simply supported at the four edges 106 4.3.2.2.5 Square laminated sandwich plate having different boundary

conditions 107 4.3.2.2.6 Simply supported triangular double core laminated sandwich plate 109 4.3.2.2.7 Skew angle-ply laminated sandwich plate simply supported at the

edges 110 4.3.3 Buckling Analysis 112 4.3.3.1 Analysis of laminated composite plates based on HSDT/FSDT 113 4.3.3.1.1 Simply supported cross-ply square laminate under uniaxial

compression 113 4.3.3.1.2 Skew cross-ply laminate having different boundary conditions 114 4.3.3.1.3 Effect of different parameters on buckling load of a square laminate 116 4.3.3.2 Analysis of composites and sandwich laminates based on RHSDT 118 4.3.3.2.1 Simply supported square angle-ply laminate under axial compression 118 4.3.3.2.2 Simply supported square plate having orthotropic layers 119 4.3.3.2.3 Rectangular sandwich plate with different boundary conditions 120 4.3.3.2.4 Simply supported multi-core rectangular sandwich plates 121 4.3.3.2.5 Simply supported rectangular sandwich plate with laminated face

sheets 122

Contents vii

4.3.3.2.6 Double core skew sandwich plate with laminated stiff sheets 124 4.4 Plates having imperfect interfaces 126 4.4.1 Static Analysis 127 4.4.1.1 Cylindrical bending of a cross-ply (0/90/0) laminate 127 4.4.1.2 Simply supported cross-ply (0/90/90/0) square laminate 127 4.4.1.3 Simply supported un-symmetric cross-ply (90/0/90/0) square laminate 130 4.4.1.4 Simply supported sandwich plate (f/c/f) having orthotropic face sheets 135 4.4.1.5 Laminated sandwich plate having different boundary conditions 136 4.4.1.6 Double core rectangular sandwich plate with laminated stiff sheets 137 4.4.1.7 Skew laminated sandwich plate simply supported at the four edges 138 4.4.2 Free Vibration Analysis 142 4.4.2.1 Simply supported cross-ply laminate having variable layer thickness 142 4.4.2.2 Initially stressed cross-ply laminate simply supported

at the edges 142 4.4.2.3 Simply supported square sandwich plate with laminated face sheets 143 4.4.2.4 Square sandwich plate with laminated face sheets having clamped edges 145 4.4.2.5 Double core simply supported triangular laminated sandwich plate 146 4.4.2.6 Initially stressed laminated sandwich plate simply supported at edges 148 4.4.3 Buckling Analysis 151 4.4.3.1 Simply supported cross-ply composite laminate having variable layer thickness 151 4.4.3.2 Simply supported double core rectangular laminated sandwich plate 151

Contents viii

4.4.3.3 Sandwich plate with different fibre orientations at laminated stiff sheets 153 4.4.3.4 Skew sandwich plate with laminated face sheets clamped at the edges 155 Chapter 5 Closure 157-160 5.1 Summary and conclusions 157 5.1.1 Summary 157 5.1.2 Conclusions 158 5.2 Scope of future research 159

References 161-175

Preface

Plate is a very common structural element in various activities of civil, mechanical, aerospace, marine and other engineering fields. The use of composite materials in laminated plates is gaining popularity due to its high strength/stiffness to weight ratio compared to the conventional materials. Laminated plates with different fibre orientations or having a sandwich construction with low strength core and high strength face sheets provide unique solution where weight minimization is one of the major concerns.

The most important feature of laminated plate is that it is comparatively weak in shear. This phenomenon is quite significant in case of sandwich plates due to wide variation of materials between core and face. As such the effect of shear deformation should be incorporated properly in the analysis of these layered structures. Development of an appropriate mathematical model for an accurate analysis of composites and sandwich laminates is one of the prime requirements to assess the strength and stability of these structures under different conditions. The refined higher shear deformation theory (RHSDT) has been proved to be most suitable for this purpose with reasonable economy and accuracy.

Moreover, the problem becomes further complex if inter-laminar imperfections are introduced in the plate.

The finite element method is proved to be the most powerful and versatile numerical tool for structural analysis due to its accuracy and generality. As a result, several commercial software packages based on this method have been developed for the analysis of structures including laminated plates. However, almost all of these software packages have the capability of modeling the

Preface x

transverse shear only up to a certain limit where the effect of inter-laminar imperfections cannot be included in the plate model.

In the exploitation of the plate model (RHSDT), it is realized that there is a genuine requirement for the development of new finite elements since very few elements have the capability of accommodating the plate theory (RHSDT) in an appropriate manner.

The present study is made on the development of an efficient and accurate modelling for the analysis of laminated composites and sandwich plates with imperfect layer interfaces based on an efficient layer-wise plate theory (RHSDT). A new triangular finite element is developed for the present purpose.

In this thesis, a detail investigation is made on static, vibration and buckling analysis of composites and sandwich laminates with imperfect layer interfaces while the case of perfect interface has been restored as a special one.

The content of the thesis is divided in five different chapters, which are as follows:

In Chapter 1, a general introduction along with the objective and scope of the present study is described.

Chapter 2 gives a review of the literature related to the scope of the present investigation. The developments of different shear deformation theories for the analysis of laminated composites and sandwich plates including inter-laminar imperfections are discussed, which is followed by the subsequent development of new finite elements for that purpose. The literature available on static, free vibration and buckling analysis of this structure is also included at the end of this chapter.

The detail formulation for the proposed analysis of laminated plate is presented in Chapter 3. The formulation is based on the refined higher order shear deformation plate theory. The effect of imperfection is incorporated in the

Preface xi

formulation by using a linear spring-layer model. The formulation of the six noded new triangular element developed for the present purpose has been described in detail. The formation of the element stiffness matrix, mass matrix and the geometric stiffness matrix are presented separately. The solution technique used for different analyses and its computer implementation are presented at the end of this chapter.

Chapter 4 deals with the application of the proposed formulations described in Chapter 3 by solving several numerical examples on static, free vibration and buckling of composites and sandwich laminates with or without inter-laminar imperfections covering different features such as convergence, shapes, boundary conditions, material properties, number of layers, fibre orientations, internal discontinuity and so on. Discussions are made on the results obtained in different problems. In many cases, the present results are compared with the published results. A large number of new results are presented for different types of analysis.

The summary and conclusions of the present study are presented in Chapter 5 where the scope of future research is also discussed.

A list of references is furnished at the end of the thesis.

Nomenclature

Most of the symbols are defined as they occur in the thesis. Some of the most common symbols, which are used repeatedly, are listed below.

a Dimension of the plate along x – axis [ ]B Strain-displacement matrix (derivative of shape

functions) b Dimension of the plate along y – axis Cr the ratio between the initial load and critical

buckling load c Core layer (each) in a sandwich plate [ ]D Rigidity matrix

1E Young’s modulus in the major principal material

direction of the lamina

2E Young’s modulus in the transverse material

direction of the lamina f Face/stiff layer in a sandwich plate

12G In-plane shear modulus

2313 ,GG Out-of-plane shear moduli

( )iH z z− , ( )1iH z z +− + Heavy side unit step functions h Overall thickness of the plate hc Thickness of the core in a sandwich plate hf Thickness of each face in a sandwich plate [ ]I Identity matrix

[ ]K Global stiffness matrix

[ ]GK Global geometric stiffness matrix

Nomenclature xiii

Kt A factor to define relative elastic moduli of the different layers of a laminated plate

[ ]k Element stiffness matrix

gk Element geometric stiffness matrix

[ ]M Global mass matrix

[ ]m Element mass matrix

[ ]N Shape function matrix

Nx Axial compression in the x-direction Nxy In-plane shear in the x-y plane Ny Axial compression in the y-direction [ ] [ ] [ ] [ ]1 2 3 4, , ,N N N N Displacement function vectors

ln Number of layers below the mid plane of the

laminated plate

un Number of layers above the mid plane of the

laminated plate

P Global load vector

p Element load vector k

Q

Rigidity matrix of the k-th lamina

q, q0 Intensity of transverse loading R Non-dimensional imperfection parameters R1 Non-dimensional imperfection parameters for upper

interfaces R2 Non-dimensional imperfection parameters for lower

interfaces

11 22 12 21, , ,k k k kR R R R Spring layer coefficients to define imperfection at

the interfaces

Nomenclature xiv

u In-plane displacement along x- direction at the mid plane of the plate

,u v In-plane displacement fields

v In-plane displacement along y- direction at the mid plane of the plate

w Transverse displacement at the mid plane of the plate

w,x , w,y First partial derivative of w with respect to x and y respectively

w Transverse displacement field x, y Cartesian co-ordinates/planes z Thickness co-ordinate/ Transverse direction

1 2 42, ,.....,α α α Unknown coefficients ixα , i

yα Change in slope at the i-th interface in the direction

- x or y ,k ku v Interfacial slips at the k-th layer interface along x

and y axes respectively

δ Global displacement vector

ε Strain vector at the mid plane of the plate

ε Strain vector at any point of the plate

Gε Geometric strain vector

,x yε ε In-plane normal strains

xγ Shear rotation about y- axis at the mid plane of the

plate

Nomenclature xv

xyγ In-plane shear strain

,xz yzγ γ Transverse shear strains

yγ Shear rotation about x- axis at the mid plane of the

plate λ Buckling load/ buckling load parameter

12ν In-plane major Poisson’s ratio

21ν In-plane minor Poisson’s ratio

θ Fiber orientation angle with respect to the principal material axis

xθ Rotation about y- axis at the mid plane of the plate

yθ Rotation about x- axis at the mid plane of the plate

ρ Mass density of the plate ρc Mass density of core layer in a sandwich plate ρf Mass density of face/stiff layer in a sandwich plate ρk Mass density of the k-th layer of the plate Ω Frequency parameter

σ Stress vector at the mid plane of the plate

σ Stress vector at any point of the plate

,x yσ σ In-plane normal stresses

xyτ In-plane shear stress

,xz yzτ τ Transverse shear stresses

,k kxz yzτ τ Transverse shear stresses at the k-th layer interface

ω Natural frequency ,ς η Transformed co-ordinates/planes

Nomenclature xvi

Subscript c Core f Face i, j Counters x x- direction y y- direction 1,2,6 Material axes system 13,23 Transverse directions Superscript i Layer/ Interface k Layer/ Interface nd Non-dimensionalised

Abbreviations CLPT Classical lamination plate theory COMPAG Computer program based on 3-D elasticity solution CPT Classical plate theory d.o.f Degrees of freedom FE Finite Element Fig. Figure FSDT First order Shear Deformation Theory GPa Giga Pascal HSDT Higher order Shear Deformation Theory PC Personal Computer RFSDT Refined First order Shear Deformation Theory RHSDT Refined Higher order Shear Deformation Theory 2-D Two Dimensional 3-D Three Dimensional

List of Figures Figures Page no.

3.1 Through thickness variation of In-plane displacement, u 44

3.2 A Typical element (a) before transformation (b) after transformation 473.3 Basic units of computer program 603.4 Flow chart for free vibration and buckling analysis 634.1 Different mesh arrangements for patch test 674.2 A rectangular plate having a mesh size of m x n 694.3 A skew plate having a mesh size of m x m 754.4 Convergence of central deflection of a composite plate with mesh

division 77

4.5 Variation of in-plane normal stress across the plate thickness 784.6 Variation of in-plane shear stress across the plate thickness 784.7 Variation of transverse shear stress across the plate thickness 794.8 Cross-section of the single core (f/c/f) sandwich plate 854.9 Cross-section of the double core (f/c/f/c/f) sandwich plate 874.10 Variation of in-plane normal stress across the depth of a sandwich

plate 89

4.11 Variation of in-plane shear stress across the depth of a sandwich plate

90

4.12 Variation of transverse shear stress across the depth of a sandwich plate

90

4.13 A triangular plate having a mesh division of m x m 1004.14 (a) Effect of skew angle and face ply orientations on fundamental 111

List of Figures xviii

frequency of a sandwich plate (h/a = 0.05) 4.14 (b) Effect of skew angle and face ply orientations on fundamental

frequency of a sandwich plate (h/a = 0.10) 111

4.14 (c) Effect of skew angle and face ply orientations on fundamental frequency of a sandwich plate (h/a = 0.15)

112

4.15 A rectangular plate subjected to uniaxial compression 1134.16 A skew plate subjected to uniaxial compression 1154.17 (a) Effect of different parameters on critical buckling load 1164.17 (b) Effect of different parameters on critical buckling load 1174.17 (c) Effect of different parameters on critical buckling load 1174.17 (d) Effect of different parameters on critical buckling load 1184.18 (a) Effect of skew angle and stiff sheets ply orientations on buckling

of a multi-core sandwich plate (h/a = 0.01) 124

4.18 (b) Effect of skew angle and stiff sheets ply orientations on buckling of a multi-core sandwich plate (h/a = 0.05)

125

4.18 (c) Effect of skew angle and stiff sheets ply orientations on buckling of a multi-core sandwich plate (h/a = 0.10)

125

4.18 (d) Effect of skew angle and stiff sheets ply orientations on buckling of a multi-core sandwich plate (h/a = 0.20)

126

4.19 (a) Variation of in-plane displacement across the depth of a laminate (h/a = 0.10)

131

4.19 (b) Variation of in-plane displacement across the depth of a laminate (h/a = 0.25)

131

4.20 (a) Variation of in-plane normal stress across the depth of a laminate (h/a = 0.10)

132

4.20 (b) Variation of in-plane normal stress across the depth of a laminate (h/a = 0.25)

132

4.21 (a) Variation of in-plane shear stress across the depth of a laminate (h/a = 0.10)

133

4.21 (b) Variation of in-plane shear stress across the depth of a laminate 133

List of Figures xix

(h/a = 0.25) 4.22 (a) Variation of transverse shear stress across the depth of a laminate

(h/a = 0.10) 134

4.22 (b) Variation of transverse shear stress across the depth of a laminate (h/a = 0.25)

134

4.23 (a) Effect of skew angle on buckling of a laminated sandwich plate having interlaminar imperfections (h/a = 0.05)

155

4.23 (b) Effect of skew angle on buckling of a laminated sandwich plate having interlaminar imperfections (h/a = 0.10)

156

List of Tables Table No. Page No.

4.1 Patch test for uniform moment, shear and twist 684.2 Deflection (wnd) at the center of a simply supported square laminated

composite plate (0/90/0) under uniformly distributed load 70

4.3 Deflection ( w ) and stresses ( 1σ , 2σ , 4σ , 5σ and 6σ ) of a simply

supported square laminate (0/90/0) under sinusoidal load

72

4.4 Deflection ( w ) and stresses ( 1σ , 2σ , 4σ , 5σ and 6σ ) of a simply

supported rectangular (b/a=3) cross-ply laminate (0/90/0) under sinusoidal load

73

4.5 Deflection (wnd) of a simply supported un-symmetric (45/-45/...) square laminate under uniformly distributed load

74

4.6 Deflection (wnd) and normal stresses ( nd1σ and nd

2σ ) at the center of

a simply supported cross-ply (0/90/0) skew laminate under uniform load

75

4.7 Central deflection (w*) of a simply supported cross-ply (0/90/90/0) laminate under cylindrical bending due to sinusoidal loading

80

4.8 In-plane displacement (u*) of a simply supported cross-ply (0/90/90/0) laminate under cylindrical bending due to distributed load of sinusoidal variation

81

4.9 Central deflection ( cw ) of a simply supported cross-ply (0/90/0) square laminate under sinusoidal loading

82

4.10 Deflection ( cw ) and stresses ( xσ , xzτ ) of a simply supported angle-

ply (-45/45….) square laminate under uniform loading

83

4.11 Deflection ( cw ) and stresses ( xσ , yzτ ) of a simply supported cross-

ply (0/90/0) skew laminate under uniform load

84

List of Tables xxi

4.12 Deflections (w) and stresses (σx, τxz) of a single core (f/c/f) square sandwich plate under uniformly distributed load

86

4.13 Deflections (w) and stresses (σx, τxz) of a simply supported double core (f/c/f/c/f) square sandwich plate under uniformly distributed load

87

4.14 Central deflection ( w ) of a simply supported rectangular sandwich plate with laminated facings (0/90/C/0/90) under sinusoidal load

88

4.15 Deflections and stresses of a simply supported double core square sandwich plate with laminated stiff sheets (0/90/0/C/0/90/0/C/0/90/0)

91

4.16 Deflections and stresses of a simply supported square sandwich plate with angle-ply laminated faces (θ/θ+90/C/θ/θ+90) under uniform load

92

4.17 Deflections and stresses of a skew sandwich plate with laminated faces (0/90/C/90/0) under uniformly distributed load with different boundary conditions

93

4.18 Frequency Parameters Ω = (ωa2/h)√(ρ/E2) of the two layer cross-ply (0/90) rectangular (a/b =1.0) laminate simply supported at all the edges

95

4.19 Frequency Parameters Ω = (ωa2/h)√(ρ/E2) of the four layer cross-ply (0/90/90/0) square laminate simply supported at all the edges

96

4.20 Frequency Parameters Ω = (ωa2/h)√(ρ/E2) of the four layer cross-ply (0/90/0/90) rectangular (a/b =1.5) laminate simply supported at the edges

96

4.21 Frequency Parameters Ω = (ωa2/h)√(ρ/E2) of the two layer angle-ply (45/-45) square laminate simply supported at all the edges

97

4.22 Frequency Parameters Ω = (ωa2/h)√(ρ/E2) of the six layer angle-ply (45/-45/45/-45/45/-45) rectangular (a/b =2.0) laminate simply supported at all the edges

98

4.23 Fundamental frequency parameters Ω = (ωa2/h)√(ρ/E2) of a cross-ply (0/90/0) square laminate having different boundary conditions at edges

99

4.24 Frequency parameters Ω = (ωa2/h)√(ρ/E2) of a triangular laminate simply supported at the three edges

101

List of Tables xxii

4.25 Frequency parameters Ω = (ωa2/h)√(ρ/E2) of a simply supported square laminate having a concentrated mass (Mc) at its center

102

4.26 Frequency parameters (Ω) of a simply supported un-symmetric square laminate

104

4.27 Frequency parameters Ω = ωa2√(ρh/D0) of an angle ply (30/-30/30) skew laminate for different skew angles (α)

105

4.28 Fundamental frequency parameters (Ω) of a simply supported square sandwich plate having orthotropic layers

106

4.29 Frequency parameters (Ω) of a simply supported square sandwich plate with laminated face sheets (0/90/../C/../90/0)

108

4.30 Frequency parameters (Ω) of a square sandwich plate with laminated face sheets (0/90/C/90/0) having different boundary conditions

109

4.31 Frequency parameters (Ω) of a double core triangular sandwich plate with laminated stiff sheets (0/90/C/0/90/C/0/90)

110

4.32 Buckling load parameters (λ =Nxb2/E2h3) of a simply supported cross-ply square laminate under uniaxial compression

114

4.33 Buckling load parameters (λ =Nxa2/E2h3) of a cross-ply (90/0/0/90) skew laminate under uniaxial compression

115

4.34 Simply supported square angle-ply laminate (45/-45/-45/45) under axial compression

119

4.35 Buckling load parameters (λ =12Nxb2/π2E11h2) of a square sandwich plate having orthotropic layers under uniaxial compression

120

4.36 Buckling stresses (σcr in N/mm2) of a rectangular sandwich plate having different boundary conditions

121

4.37 Buckling load parameter ((λ =Nxb2/Eh3) of a simply supported multi-core rectangular sandwich plate

122

4.38 Buckling load parameter ((λ =Nxb2/E2hc3) of a simply supported

rectangular sandwich plate (0/90/0/90/C/90/0/90/0) with cross-ply laminated face sheets

123

4.39 Buckling load parameter ((λ =Nxa2/E2hc3) of a simply supported

rectangular sandwich plate (θ/-θ/θ/-θ/C/-θ/θ/-θ/θ) with angle ply laminated face sheets

123

List of Tables xxiii

4.40 Central deflection ( cw ) of a cross-ply (0/90/0) laminate subjected to

a distributed load of sinusoidal variation (cylindrical bending)

128

4.41 Central deflection and stresses of a simply supported cross-ply (0/90/90/0) square laminate

129

4.42 Central deflection ( cw ) and stresses ( xσ , xyτ and xzτ ) of a simply supported un-symmetric cross-ply (90/0/90/0) square laminate

135

4.43 Central deflection and stresses of a simply supported single core (f/c/f) square sandwich plate subjected to sinusoidal variation of load

138

4.44 Central deflection and stresses of a laminated sandwich plate (0/90/0/C/0/90/0) having different boundary conditions

139

4.45 Double core laminated (-45/45/C/0/90/C/-45/45) rectangular sandwich plate subjected to uniformly distributed load

140

4.46 Skew laminated sandwich plate (0/90/C/0/90) simply supported at all the edges

141

4.47 Fundamental frequency parameters (Ω) of a simply supported square laminated plate having variable layer thickness

144

4.48 Fundamental frequency parameters (Ω) of a simply supported initially stressed cross ply square laminate (0/90/90/0)

145

4.49 Frequency parameters (Ω) of a simply supported square sandwich plate (0/90/../C/../90/0) with laminated face sheets

146

4.50 Frequency parameters (Ω) of a square sandwich plate with laminated face sheets having clamped edges

147

4.51 Frequency parameters (Ω) of a double core triangular sandwich plate (45/-45/C/0/90/C/45/-45) with laminated stiff sheets

148

4.52 Frequency parameters (Ω) of a simply supported square initially stressed sandwich plate with laminated face sheets (0/90/C/90/0)

150

4.53 Buckling load parameters (λ) of a square sandwich plate having variable layer thickness under uniaxial compression

152

4.54 Buckling load parameters (λ) of a simply supported double core rectangular sandwich plate with stiff laminated sheets (0/90/C/0/90/C/0/90)

153

List of Tables xxiv

4.55 Buckling load parameters (λ) of a square sandwich plate with different fiber orientations (θ/-θ/θ/-θ/C/-θ/θ/-θ/θ) at laminated stiff sheets

154

Chapter

ONE

INTRODUCTION

1.1 INTRODUCTION

The development of advanced composite materials (Fibre Reinforced Laminated Composites) is considered as one of the biggest technical revolutions in recent past. Due to the superior performance of these composite materials over conventional metallic materials, its use as structural material became popular in a wide range of applications all over the world. The improvement in the performance of advanced composites is primarily due to the use of the main load bearing material in fibre form, which is found to be stronger and stiffer than that used in bulk form. Composite material is preferred in many practical applications due to its high strength/stiffness to weight ratio and greater resistance to environmental degradation compared to steel. It has a specific advantage in naval vessels since magnetic signature generated in steel vessels can be totally avoided with the use of such material instead of steel. Again the coefficients of thermal expansion for many composites are much lower than those of metals. As a result of that composite structures show a better dimensional stability over a wide range of temperature variation. As the material possess high internal damping, it has better energy absorption capacity under dynamic condition, which is beneficial to

Introduction 2

reduce the transmission of noise, vibration and harshness. Moreover, a significant advantage of composite construction is its low maintenance cost.

The search for lighter, stronger and stiffer structures has encouraged the use of advanced composites and it is increasing day by day. The structural components made of composites are mostly found in the form of laminated plates and shells. A laminate has a layered construction, which consists of a number of lamina or ply stacked one over the others and bonded together to act as an integral structural element. As the individual layers may have any orientations, it is utilized in structural design to achieve the required strength and stiffness in a preferred direction. Due to all these merits, the laminated composite is gaining wide acceptance in various activities of structural engineering specifically in a situation where weight minimization is a major concern. In order to fulfill this requirement in more efficient manner, a sandwich construction having low strength core and high strength laminated stiff face sheets may be used. It gives substantially high strength and stiffness of the structure with reasonable low weight i.e., the specific strength and stiffness is remarkably increased. Moreover, a sandwich construction is preferred in a situation where there might be a requirement of higher thickness of the panel or it may be tapered or something else.

One of the important features of laminated composites is that it is weak in shear due to its low shear modulus compared to that of extensional rigidity. Thus the effect of shear deformation is quite significant and it should be considered in a proper manner in the analysis. This is much more severe and complex in case of sandwich constructions. Actually, the behavior of sandwich plate cannot be predicted by a simple plate theory like classical plate theory, as the core and face sheets deform in a different manner due to a wide variation of their material properties. In fact the analysis of sandwich plates was treated as a separate class of problem in earlier days (Allen 1969 and Plantema 1966). The typical feature of sandwich plate is that the variation of in-plane displacements across the thickness

Introduction 3

shows kink at the interface between the core and face sheets, which gives discontinuous transverse shear strain at these interfaces while transverse shear stresses at the interfaces are continuous. This phenomenon is also found in laminated composite plates for its layered construction but the discontinuity of transverse shear strain is not so prominent as that of sandwich plate. Actually, it depends on the difference in the values of transverse shear rigidity and thickness of adjacent layers, which is quite significant in sandwich plate compared to that of composite laminates. However, the effect of the discontinuity cannot be ignored in a multilayered thick composite laminate.

Considering this aspect in view and to represent some other features like transverse shear stress free conditions at the top and bottom surfaces of the plate in a better manner, a number of improved plate theories have been developed for satisfactory modeling of thick composite laminates. These plate theories can be broadly divided into two categories:

• Single layer plate theory

• Layer-wise plate theory.

In single layer theory, the deformation of the plate is expressed in terms of unknown parameters of reference plane i.e., a single plane, which is usually taken as the middle plane of the plate. In this group, the classical plate theory (CPT) was first developed where the effect of shear deformation is not considered. This is first considered in Reissner-Mindlin’s plate theory where the transverse shear strain is assumed to be uniform over the entire plate thickness. This plate theory is popularly known as first order shear deformation theory (FSDT, Reissner 1944, 1945, Mindlin 1951, Yang et al. 1966). A further improvement in this direction is responsible for the development of higher order shear deformation theory (HSDT, Hildebrand et al. 1949, Lo et al. 1977 a & b, Kant et. al 1982, Reddy 1984). It allows warping of plate cross-sections (normal to reference plane) to have higher order variation of transverse shear stresses/strains across the plate

Introduction 4

thickness. This is achieved by taking higher order through thickness variation of in-plane displacements. It has been subsequently realised that the increase in the order of variation of the in-plane displacements across the plate thickness is not sufficient enough to improve the performance of multi-layered thick laminate beyond a certain range. Actually, it gives a continuous variation of the transverse shear strain across the thickness, which gives discontinuity in the shear stress distribution at the layer interfaces due to different values of shear rigidity at the adjacent layers. It should be noted that the actual situation is just opposite i.e. the transverse shear strain is discontinuous and transverse shear stress is continuous at the layer interfaces.

In order to avoid the above disparity and to represent certain features in a better manner, the layer-wise theories are developed. The development of these plate theories were started with the discrete layer plate theory (Ambartsumyan 1970, Srinivias 1973, Toledano and Murakami 1987, Robbins and Reddy 1993) where unknown displacement components are taken at all the layer interfaces including top and bottom surfaces of the plate. With these unknowns at the different interfaces, the displacement components at any intermediate level are interpolated using piecewise linear functions. It gives a zigzag through thickness variation of in-plane displacements, which represent the desired shear strain discontinuity at the layer interfaces. The performance of this plate theory is good but it requires huge computational involvement, as the number of unknowns increases directly with the increase in the number of layers.

The above problem of discrete layer plate theories has been subsequently overcome by defining the unknowns at different interfaces in terms of those at the reference plane. This is achieved by satisfying the condition of transverse shear stress continuity at the layer interfaces (equating stresses of the adjacent layers at their interface). These plate theories may be defined as refined first order shear deformation theory (RFSDT, Di Scuiva 1984, Li and Liu 1995) where the unknowns are similar to those of the first order shear deformation theory (FSDT).

Introduction 5

A further improvement over RFSDT was made possible by combining the concepts of RFSDT and HSDT to get an efficient plate theory, which may be defined as refined higher order shear deformation theory (RHSDT, Bhaskar and Varadan 1989, Di Scuiva 1992, and Cho and Parmerter 1993). It gives a piecewise parabolic variation of transverse shear strains across the thickness with discontinuity at the layer interfaces as desired in a layered plate. Moreover, it satisfies the transverse shear stress free condition at the top and bottom surfaces of the plate. Thus it is obvious that this refined plate theory would be more useful in the analysis of sandwich plates since the above features are more prominent in sandwich construction as mentioned earlier. Actually, it is supposed to provide an improved modeling of laminated sandwich plate compared to its conventional modeling technique. Keeping all these aspects in view, this plate theory (RHSDT) may be considered as most efficient in the group of layer-wise plate theories.

It has already been reported that the performance of this plate theory (RHSDT) is extremely good. This is demonstrated by solving some problem of laminates having simple geometry, loading, boundary condition and stacking sequence by this plate theory where the problem is mostly solved by analytical means. In this situation an attempt has been made in this study to develop a finite element for this plate model. In this context, it should be noted that this plate theory possesses one problem, which is found in its finite element implementation. Actually, this plate theory involves second order derivatives of transverse displacement in its strain components, which requires C1

continuity of transverse displacement at the element edges. In fact this is a well-known problem of plate finite element, which is also found in RFSDT, HSDT (Reddy 1984) and CPT. It is be noted that any C0

continuous element, which is quite popular, cannot be used/upgraded accordingly. In this situation, a simple option is to upgrade an existing C1

continuous element for thin plate (CPT) according to the requirement of any refined theory but the availability of such element is very few. Moreover, these elements are quite complex in most of the cases and having

Introduction 6

certain drawbacks in general. So there is a genuine requirement of suitable elements based on refined higher order shear deformation theory (RHSDT), which may be employed for the solution of laminated plate problem having any arbitrary loading, geometry, boundary condition, stacking sequence and other features.

In most of the studies the layers are assumed to be perfectly bonded. But there might be imperfections at the interfaces between different layers due to various reasons. The imperfections may be a shear slip between the layers or may be in the form of delamination. In this context some theories are proposed for modeling of these interfacial imperfections in efficient manners. Though the development of these theories is based on some micro-mechanic studies on imperfect laminated composites but these theories can also be applied in macro mechanical scale. The linear spring layer model (Lu and Liu 1992, Di Sciuva 1997, Cheng et al. 1996a, 1996b, 1997, 2000) is one of them, which is satisfactorily used in the prediction of structural response of imperfect laminates. So far, it is applied to composite laminates only and the problem is solved analytically. In these theories, the slip at any interface is defined in terms of transverse shear stresses at that interface.

For the analysis of the present structure, there might be a tendency of using a software package for structural analysis since such a package is commercially available in large numbers. In that case, one of the options is to model the composite or sandwich laminate with plate or shell elements. As these elements are based on FSDT in almost all the packages, it will not be adequate for proper prediction of structural response specifically in the case of sandwich plates. In such analysis, the major concern lies with the evaluation of stresses particularly the interlaminate stresses. Moreover, the effect of interfacial imperfections cannot be handled with FSDT. The other option is to model the structure by three-dimensional solid elements where a huge number of elements will be required to model a multi-layered laminates since each layer will require

Introduction 7

at least one element in the thickness direction and the element size in the other directions cannot be made large in order to maintain its aspect ratio. Thus it will require a huge computational effort, which may not be affordable in many practical applications.

In this situation, the only option lies with the use of refined higher order shear deformation theory (RHSDT) and it can be applied to a wide range of problems through finite element implementation.

1.2 OBJECTIVE AND SCOPE OF PRESENT INVESTIGATION

The objective of the present investigation is to accurately predict the static, vibration and buckling responses of laminated composites and sandwich plates with/without inter-laminar imperfections using

• Refined higher order shear deformation theory (RHSDT)

• Linear spring layer model

• A new triangular element

The concept of refined higher order shear deformation theory (RHSDT) proposed by different investigators is utilized in this study to represent the deformation of the structure in a convenient manner. To fulfill the C1 continuity requirement for transverse displacement at the element interfaces demanded by this plate theory, a new triangular element is developed. The element has three corner nodes and three mid side nodes with equal number of degrees of freedom at each node. The element formulation is based on displacement approach and the formulation is done so efficiently that the element is found to be free from shear locking problem and spurious modes. The formulation of RHSDT has the advantage that it can be easily reduced to that of RFSDT, HSDT and FSDT by simply dropping certain terms, as these plate theories are basically the subsets of RHSDT. In the present study, options are kept for all these plate theories (RHSDT, RFSDT, HSDT, FSDT) so that any one of them can be used. The

Introduction 8

element with its default option of RHSDT is applied to solve a wide range of problems of laminated composite plates and laminated sandwich plates with or without interfacial imperfections. Since the abovementioned inter-elemental continuity requirement for transverse displacement is also necessary in HSDT of Reddy (1984) belonging to single layer plate theories, the performance of the element is tested with the option of HSDT at the beginning. Results are also generated with the other options (FSDT and RFSDT) of the present model in order to show the performance of the different plate theories.

The effect of interfacial imperfection is incorporated in the formulation by using a linear spring layer model whereas the problem of perfect case may be treated as a special case by taking the degree of imperfection as zero.

The formulation is implemented by a computer code developed for this purpose to carryout the following analyses of composites and sandwich laminates with inter-laminar imperfections.

• Static analysis

• Vibration analysis

• Buckling analysis

In static analysis, the solution of the governing equation is carried out by factorizing the structural stiffness matrix using the technique of Cholesky’s decomposition. The eigen value problem found in the governing equation of vibration and buckling analysis are solved by the simultaneous iteration technique of Corr and Jennings (1976). In order to store the large size matrices (stiffness matrix, mass matrix and geometric stiffness matrix) in an efficient manner, they are kept in single array following the skyline storage technique. The computer code is written in FORTRAN 90 and it is executed in the Workstations and PCs available in the Computer Laboratory of Ocean Engineering and Naval Architecture Department (Indian Institute of Technology, Kharagpur).

Introduction 9

The computer code based on the efficient layer-wise finite element model is used to solve a large number of numerical examples covering different features such as plate geometry, loading, boundary condition, stacking sequence, thickness ratio, aspect ratio, degree of interfacial imperfections and so on. The results obtained in the form of deflection, stresses; natural frequencies for first few modes and buckling loads are presented in tables as well as in graphical manner. The convergence of results with mesh refinement is tested with at least one case in each category of problem. In many cases, the results obtained by the proposed model are validated with the help of published results. In addition to those, a large number of new results specifically for laminated sandwich plates are presented for future references. Though some results are available for simple cases (rectangular plate geometry, simply supported boundaries, cross-ply stacking etc.) of laminated sandwich plates having perfect interfaces but there is no published result for the case of imperfect sandwich laminates. For composite laminates with inter-laminar imperfections, the published results are for simple cases only.

Chapter

TWO

REVIEW OF LITERATURE

2.1 INTRODUCTION

For the proper design of laminated composite structure, one should predict its behavior as accurately as possible. For this purpose, the modeling of structural deformation should be done appropriately. In case of composites and sandwich laminates, the modeling of transverse shear stresses specifically at the layer interfaces (inter-laminar shear stresses) is quite challenging. Actually, the effect of shear deformation is found to be quite significant due to relative weakness of this material in shear compared to stretching. In addition to that the layered construction of the structure makes the situation more severe. Moreover, the inter-laminar imperfections introduce further complications in the modeling. In this context a number of plate theories have been proposed by different investigators.

By this time, the number of investigations carried out on different aspects of laminated composite structure is really quite vast and these are reported in many places in different forms. Thus it is an extremely difficult task to compile all of them in one place and it is rather not the intention of the present work. The objective here is to cite the important works, which are absolutely relevant to the theme of the present investigation. Again, a particular topic will be projected with

Review of literature 11

some representative references and all the works on that topic may not be referred in some cases.

It has been attempted in this chapter to make a review on the different plate theories developed for the analysis of laminated composite and sandwich plates, theories developed for imperfections at the layer interfaces, the finite elements developed for the implementation of these improved plate theories and some important analytical studies on static, vibration and buckling analyses of these structures. This is presented as follows.

1. Different shear deformation theories

2. Theories to include imperfect interfaces

3. Finite Elements for the present structure

4. Important Analytical studies on present structure

• Static Analysis

• Free Vibration Analysis

• Buckling Analysis.


2.2 SHEAR DEFORMATION THEORIES FOR LAMINATED PLATES

The plate theories developed so far may be broadly divided in to two groups as follows.

• Single layer theories

• Layer-wise theories

2.2.1 Single Layer Theories

Ashton and Whitney 1970 have made the first attempt to analyse laminated composite plates where the classical plate theory (Love-Kirchoff’s hypothesis) has been used to model the plate deformation. This is popularly known as classical lamination plate theory (CLPT). This theory does not include the effect of transverse shear in the formulation.

Reissner (1944, 1945) has first provided a consistent theory to include the effect of shear deformation. The basic assumption made by Reissner (1944,1945) gives a consistent representation of stress distribution across the thickness, which results in a through-thickness linear variation of in-plane displacements (i.e., two terms of the Taylor’s series expansion) and constant normal deflection across the plate thickness.

The same degree of approximation has been employed by Mindlin (1951) on kinematic assumptions of the displacement fields given by Reissner (1944, 1945) without introducing corresponding stress distribution assumptions and obtained the governing equations from a direct method. In Mindlin’s derivation it was necessary to introduce a correction factor in the shear stress resultants to account for the warping of the cross-section of the plate. The correction factor was evaluated by comparison with an exact elasticity solution. The effect of rotary inertia was also included, which plays a significant role in the vibration problems of thick plates.


Reissner and Stavsky (1961) have first considered the coupling between in-plane stretching and transverse bending for laminated plate. In their study, plates consisting of two orthotropic sheets of equal thickness, which are laminated in such a way that the material axis makes a positive angle with the x-axis in one sheet and a same negative angle with the other sheet.

Yang et al. (1966) have generalized the Reissner-Mindlin thick plate theory for isotropic plates to arbitrary laminated anisotropic plates. The transverse shear strains are taken as constant across the plate thickness. In this plate theory, the in-plane stretching is also considered. This is broadly known as first order shear deformation theory (FSDT) for laminated plates.

The origin of higher order shear deformation theories (HSDT) goes back to the work of Hildebrand et al. (1949) who made significant contributions by dispensing all the assumptions of Kirchoff. A Taylor series expansion having three terms has been used to represent the displacement vector. The minimum potential energy principle along with full elasticity matrix has been used in the derivation.

Lo et al. (1977a,b) presented a formal plate higher order theory based upon the principle of stationary potential energy resulting in eleven second order partial differential equations to determine the eleven functions assumed in the displacement model. Four terms of Taylor series expansion for in-plane displacements and three terms of Taylor series expansion for the transverse displacement are assumed. Kant (1982) derived the complete set of equations of isotropic version of the Lo et al. (1977a,b) theory. Extensive numerical results are presented with a proposed numerical integration technique. In this theory in-plane displacements are assumed to have a cubic variation across the plate thickness whereas the transverse deflection is kept constant throughout the thickness of the plate. Kant along with his co-workers extended this work for application to fibre reinforced composites and sandwich plates. This theory involves some additional unknowns


having no physical meaning and also the number of unknowns is more than that used in FSDT.

Reddy (1984) proposed a simple higher order theory for the analysis of laminated composite plates. The theory contains the same independent unknowns as in first order shear deformation (FSDT) and accounts for parabolic distribution of transverse shear strains through the thickness of the plate; and also there is no need to use shear correction factors in computing the shear stresses. The displacement field is chosen in such a manner so that the condition of zero transverse shear stresses at top and bottom surfaces of the plate is satisfied. This theory expands the in-plane displacements as cubic functions of thickness co-ordinates and the transverse deflection is assumed to be constant throughout the plate thickness.

Khedir and Reddy (1999) proposed a second order shear deformation theory for the analysis of laminated composite plates. A generalized Levy type solution in conjunction with the state space concept is used to analysis the free vibration behaviors of the cross ply and anti-symmetric angle ply laminated plates. Exact fundamental frequencies of cross ply plate strips are obtained for arbitrary boundary conditions. The exact analytical solutions are obtained for thick and moderately thick plates as well as for thin plates and plate strips.

2.2.2 Layer-wise Theories

The layer-wise theories may be further classified as follows.

• Discrete layer plate theory

• Refined plate theory


2.2.2.1 Discrete layer theories

The development of layer-wise plate theories for the analysis of laminated composites and sandwich plates initiated with the discrete layer theories where unknown displacement components are taken at all the layer interfaces including top and bottom surfaces of the plate.

Ambartsumyan (1970) was among the earliest to present a technique with transverse shear stress continuity conditions across the laminate interfaces. Based on the parabolic distribution for the transverse shear stresses in a composite layer, he presented a shear deformation theory for the composite analysis. Several other investigators refined this technique subsequently.

Srinivas (1973) presented a layer wise plate theory where the governing differential equations and boundary conditions are derived by following a variational approach. The displacements are assumed piecewise linear across the thickness. The effects of transverse shear deformations and rotary inertias are included. A procedure is proposed for obtaining the general solution of the above governing differential equations in the form of hyperbolic trigonometric series.

Reddy (1987) developed a layer wise laminated plate theory in which full account is given to all the parameters involved in a three dimensional (3-D) analysis. The displacement field for in-plane and out of plane components have a linear variation within each layer. The displacement field is assumed to be continuous through the laminate thickness. The transverse strain components, however, is not to be continuous at the interfaces, which leaves the possibility of transverse shear stress components becoming continuous at the layer interfaces of the two layers. This generalized layer wise theory accounts for all the six strain components in a kinematically correct manner. However, in this theory the number of unknowns increase directly with the increase in the number of layers.

Toledano and Murakami (1987) proposed a new higher order laminated plate theory based upon Reissner’s mixed variational principle in order to


improve the accuracy of the in-plane responses. A zigzag shaped C0 function and Legendre polynomials are assumed to approximate in-plane displacement distributions across the plate thickness. The condition of transverse shear stress continuity is satisfied by this theory. The accuracy of the theory is examined by solving problems of cylindrical bending of laminated composite plates.

Cho et al. (1991) proposed a higher order individual layer theory to analysis simply supported laminated rectangular plates. This theory approximates the in-plane and normal displacements by third and second order functions of thickness co-ordinates respectively. The theory satisfies the displacement compatibility and stress equilibrium conditions along the interfaces between adjacent layers.

Li and Liu (1995) presented a so-called refined zigzag theory for composite plates. The in-plane displacements for each composite layer are assumed to consist of terms up to third order. To have total number of degrees of freedom independent of the layer number only two coefficients of in-plane displacements are allowed to be layer dependent. The remaining two coefficients are layer independent variables. A higher order variation of transverse displacement was also assumed in this theory. This higher order theory is a compromising theory between the FSDT and the layer wise theory and holds both the advantages of numerical accuracy and computational efficiency.

2.2.2.2 Refined shear deformation theories Di Sciuva (1984) overcome the problem of discrete layer plate theories by

developing a theory for multi-layered anisotropic plates where the unknowns taken at all the interfaces are expressed in terms of those at the reference planes. This is achieved by satisfying the condition of transverse shear stress continuity at the layer interfaces, which shows a linear zigzag variation of in-plane displacement through the thickness while the transverse shear stress is constant all through the thickness of the plate. The out-of-plane displacement, however, is assumed to be constant through the thickness. A variational method was used to


formulate the governing equations. Closed form solutions are presented for a few special classes of problems of laminated composite plates. These types of plate theories are defined as refined first order shear deformation theories (RFSDT) where the number of unknowns is same as that found in FSDT.

Lu and Liu (1992) have given an inter-laminar shear stress continuity theory for both thin and thick composite laminates. The in-plane displacements show a piecewise linear zigzag variation. The inter-laminar shear stresses are calculated directly from the constitutive equations. The theory is applied to the problem of composite plates by obtaining an analytical closed form solution.

Bhaskar and Vardan (1989) combined the concepts of HSDT and RFSDT where a linearly varying piecewise displacement field is superposed on an overall higher order variation of in-plane displacements. The examples include a problem of symmetric laminate subjected to anti-symmetric loading about the mid-plane of the plate in addition to the normal symmetric loading cases. The results obtained at different locations throughout the thickness are quite closer to the exact 3-D solution. This type of theory is known as the refined higher order shear deformation theory (RHSDT) for the laminated plates.

Di Sciuva (1990) improved his previous linear zigzag theory (RFSDT) to include piece-wise cubic through thickness variation of the in-plane displacement so that the static condition of zero transverse shear stresses on the top and bottom surfaces of the plate is fulfilled along with the continuity of these stresses at the layer interfaces. In this improvement only five generalized displacement components are required to describe the kinematics of the laminates. Extensive numerical investigations performed by using this theory indicates that the proposed approach allows accurate determination of the thickness distribution of the inter-laminar stresses also in the cases of thick multi-layered plates.

A refined higher order shear deformation theory (RHSDT) was developed by Cho and Parmerter (1992) for the analysis of symmetric laminated composites. This theory has been further modified by Cho and Parmerter (1993) for general


lamination configurations. This is obtained by superposing a cubic varying overall displacement field on a zigzag linearly varying displacement field. The theory has the same number of unknowns as first order shear deformation theory (FSDT) and the number of unknowns is independent of the number of layers. The displacement field satisfies transverse shear stress continuity conditions at the interface between layers as well as transverse shear stress free surface conditions. Thus an artificial shear correction factor is not needed. To demonstrate the performance of the theory in comparison with 3-D elasticity solution and RFSDT, analytical solution is obtained and results are shown for cylindrical bending with sinusoidal distribution of transverse load for composite laminates.

Carrera (1998) formulated a refined multi-layered theory by using a mixed model for the analysis of sandwich panels. The computational efficiency is guaranteed by preserving C0 continuity as in the case of Reissner-Mindlin model. Mechanical accuracy is acquired by allowing a zigzag in-plane displacement field in the thickness direction and by fulfilling inter-laminar equilibrium at the interfaces between the core and faces for the transverse shear stress components. The resulting mixed model is reduced to the standard displacement formulation by writing a weak form of Hook’s law. The model (RMZC, i.e., Reissner-Mindlin Zigzag inter-laminar Continuity) is equivalent to RFSDT for sandwich plate.

2.3 IMPERFECTIONS AT THE LAYER INTERFACES

In all the plate theories discussed in the previous section a perfect interface between the layers of the laminated plate is assumed, which is characterized by continuous displacements and stresses at the interfaces. But the assumption of perfect interfaces has been proved to be inadequate for multi-layered plates in many cases. Chen and Jang (1995), Lai et al. (1997) have published some interesting findings on multi-layered anisotropic medium under the state of generalized plane deformation with inter-layer thermal contact resistance. The complete solution of the entire layered medium was obtained by introducing the thermo-mechanical boundary and layer interface conditions


including interlayer imperfect thermal contact conditions. It is observed that in case of imperfect interfaces there should be jumps in the displacement components at the interfaces whereas inter-laminar shear stresses would remain continuous from equilibrium point of view.

In view of the above, a few analytical studies have been reported on the analysis of laminated composite plates with imperfections at the layer interfaces. On the other hand no such study is reported for sandwich plates. Moreover, solution of the problem of imperfections in composite laminates or sandwich plates using a finite element model is not found in the literature.

Toledano and Murakami (1988) developed a shear deformable two-layer plate theory with built-in inter-layer slip. Based upon the principle of virtual work and Ressiner’s mixed variational principle, well-posed boundary value problems of the proposed theory are defined. The theory is tested by examining the problem of cylindrical bending of two-layer plates consisting of like material layers. It was observed by them that the interlayer slip might have a dramatic effect on the overall structural behavior and in-plane normal stresses, if the interface stiffness falls below a critical value.

However the above theory is unable to accurately model the effects of shear slip because of their relatively poor description of in-plane displacements and transverse shear stresses. In this respect Lu and Liu (1992) substantially improved the precision of modeling the shear slip between layers of composite laminates by using a layer wise displacement field. The theory satisfies the continuity conditions of inter-laminar shear stresses at the layer interfaces. Because of its layer wise nature the theory is modified to include shear slip conditions based on a linear shear slip theory.

Di Sciuva et al. (1997, 1999) developed simpler shear slip models for composite laminates by using the zigzag displacement fields (RHSDT) and reformulated them in such a manner so as to account for the interfacial bonding


conditions. But the application based on analytical solution is limited only to the static analysis of the cross-ply laminated plates.

Cheng et al. (1996a) used a linear spring layer model to investigate the phenomenon of imperfection at the layer interfaces where the displacement jumps in a particular interface are assumed to be proportional to the inter-laminar shear stresses at that interface. They have implemented (Cheng et al 1996b, Cheng 1997) this linear spring layer model successfully on a plate model based on RHSDT where the bottom of the plate is used as the reference plane. The imperfections are considered at all the layer interfaces. Closed form solutions are presented for simply supported rectangular cross ply laminated plates having different thickness ratios with sinusoidal distribution of the transverse load.

Cheng et al. (2000) investigated the behavior of pre-stressed composite laminates featuring inter-laminar imperfections. The combination of linear spring layer model with a RHSDT as described before (Cheng et al. 1996a) forms the basis of the plate model. The principle of virtual work is used to derive a boundary value formulation for the laminated composite plates, which is initially in pre-stress state. The problem of bending, free vibration and buckling of symmetric as well as un-symmetric cross-ply laminated rectangular composite plates are studied analytically for illustrative purpose.

Shu et al. (1999) at first developed a five degrees of freedom model for the analysis of cross ply laminates with perfect as well as weakly bonded layers and subsequently modified the model (Shu 2001) with a seven degrees of freedom model for the analysis of imperfect anti-symmetric angle ply laminated plates. The suggested displacement field contains both symmetric and anti-symmetric components at the plate middle plane. Based on the 3-D elasticity consideration a set of four shape functions is determined. Closed form solutions are obtained for simply supported rectangular laminated composite plates under sinusoidal variation of distributed transverse loading. A linear spring layer model is used with the consideration of linear shear slip law.


The concept of the linear spring layer model in the above plate model is actually incorporated at the ply level from some micro-mechanics based studies at reinforcement matrix level. These studies are due to Benveniste (1985) and Aboudi (1987), Achenbach and Zhu (1989), Benveniste and Dvorak (1990), Hasin (1990, 1991a, 1991b, 1993), Qu (1993a, b) and many others.

However, few attempts have been made to evaluate the effects of weak bonding at the ply level as evident from the review above.

2.4 FINITE ELEMENTS FOR LAMINATED PLATES

The development of finite elements based on the different plate theories is presented in this section.

2.4.1 Elements based on First order shear deformation theory (FSDT)

A large number of plate elements based on Reissner-Mindlin’s hypothesis for isotropic plate is available and any one of these elements can be applied to the analysis of laminated plates. Though all these elements are based on FSDT but some important elements are presented below.

Pryor (1970) developed a four noded rectangular element having seven degrees of freedom per node. Two in-plane extensions, three bending and two shear modes are considered as degrees of freedom per node. The normal rotations are assumed to be constant throughout the thickness. The normal rotations are expressed as sum of the middle surface slopes and the shear rotations. The element is nonconforming one and has the disadvantage of using excessive degrees of freedom.

Panda and Natarajan (1979) used a superparametric quadrilateral plate element for the analysis of arbitrary laminated anisotropic composite plates based on FSDT. The element has eight nodes where each node contains five usual degrees of freedom (i.e., three displacements and two rotations). It is similar to Ahmad’s (Ahmad et al. 1970) superparametric shell element but the normal


rotations through the thickness are assumed to be constant similar to Pryor (1970).

Mawenya and Davies (1974) used Ahmad’s (Ahmad et al. 1970) quadratic shell element with independent normal rotations for each layer. This approach is similar to that of layer wise theories and as a result the size of the problem increases considerably with the increase in the number of layers. The basic element used is eight noded superparametric.

Kabir (1992) presented a shear locking free isoparametric three-node triangular element based on FSDT. The degrees of freedom at each node are the three displacement components and two rotations about the two axes. The transverse displacement is kept constant throughout the depth.

Lee and Fun (1996) studied the finite element analysis of composite sandwich plates. The face plates of the sandwich structures are modeled based on the Mindlin’s plate theory. The displacement fields of the sandwich core material are linearly interpolated in terms of the displacements of the two faceplates. The finite element analysis is done using a nine noded isoparametric element with usual degrees of freedom. Static and free vibration analysis problems have been solved to investigate the effect of the transverse normal deformation of the core.

Sheikh et al. (2002) have proposed a high precision shear deformable triangular element for the analysis of laminated composite plates based on the element developed by Sengupta (1991) for isotropic plate. The interpolation function used to approximate transverse displacement is a quartic polynomial and it is one order higher than that used for in-plane displacements and bending rotations, which has made the element free from locking in shear.


2.4.2 Elements based on Higher order shear deformation theory (HSDT)

Kant et al. (1982) developed a C0 finite element to implement the higher order shear deformation theory proposed by the authors. The element is a nine noded Lagrangian quadrilateral element where each node has nine degrees of freedom. Kant and his co-workers extended this work to static and vibration analysis of fibre reinforced composites and sandwich plates for both symmetric and anti-symmetric laminates by developing similar elements. These elements can be easily derived since the plate theory requires C0 continuity of displacement components at the element interfaces. This is really an advantage of the plate theory but it involves some additional field variables and nodal degrees of freedom having no physical meaning. In practical analysis, such unknowns may not be preferred to avoid the confusion lies with the incorporation of boundary condition, coordinate transformation and other operations. Moreover, it bears shear locking problem, as it is based on isoparametric formulation.

Phan and Reddy (1985) presented a four noded element based on the simple higher order shear deformation theory developed by the senior author. The theory demands C1 continuity of the transverse displacement at the element interfaces and it has been satisfied by taking eight degrees of freedom at each node. The in-plane displacements and two rotations are interpolated over an element by linear interpolation functions. On the other hand, a set of conforming cubic Hermite interpolation functions has been used to represent the transverse deflection. The element is theoretically consistent but it has rectangular shape and contains higher order derivatives in the nodal degrees of freedom.

Ghosh and Dey (1992) presented a four noded rectangular element for the analysis of composite plates. The element has non-conforming shape functions for transverse displacement and conforming shape functions for in-plane displacements as well as for the average shear rotations. The in-plane displacements and two shear rotations have been interpolated by bi-linear shape functions and the transverse displacement is interpolated using a truncated cubic


polynomial. Each node of the element has seven degrees of freedom. The element is based on displacement model. The objective of the authors was to upgrade the existing twelve degrees of freedom rectangular plate element based on CPT for isotropic plate to HSDT for the analysis of composite plates. It has been found that the element is severely affected in modeling HSDT due to the non-conforming shape functions chosen for transverse displacement though the element can model CPT successfully.

Singh et al. (1995) developed a four noded C1 continuous rectangular element by taking in-plane displacements in addition to transverse displacement where these three displacement components have been interpolated with the shape functions for transverse displacement of the element proposed by Bogner et al. (1965). Thus the element has fourteen degrees of freedom at each node. The shape functions are based on Lagrangian expansion. This element contains higher order derivatives as nodal degrees of freedom even for the in-plane displacements.

Liu (1996) presented a conforming rectangular finite element formulation for the governing equations of composite multi-layered plates using higher order theory (Reddy 1984). The element has four nodes and there are eight degrees of freedom per node. The transverse displacement is described by a modified bi-cubic displacement function while the in-plane displacements and the shear rotations are interpolated quadratically. This element is more or less similar to that of Phan and Reddy (1985).

Shankara and Iyengar (1996) developed a C0 continuous finite element model having seven nodal degrees of freedom per node for the analysis of laminated composite plates using higher order shear deformation theory of Reddy (1984). The formulation has been done by displacement approach. The field variables i.e., the independent displacement components taken are more than those defined in the plate theory (Reddy 1984) to get the C0 conformity condition.


This treatment is somewhat similar to that of Kant et al. (1982) and the element possesses the troubles of isoparametric formulation such as shear locking.

Nayak et al. (2002) developed an element similar to that of Shankara and Iyengar (1996) where the concept of assumed strain proposed by Hinton and Huang (1986) is applied to make element free from shear locking problem. The concept of Hinton and Huang (1986) works fine with the FSDT but its performance in HSDT is yet to be tested. The element is applied to free vibration analysis of composite and sandwich plates.

2.4.3 Elements based on Discrete layer theories

Khatua and Cheung (1973a) formulated a finite element for the bending and vibration analysis of multi-layered sandwich beams and plates. Each layer of the sandwich structure may have individual orthotropic properties of its own and the bending rigidities of the stiff layers are taken into account while direct stresses in cores are neglected. The conditions of common shear angle for all cores, which has been used by several authors, is not assumed in the formulation. The element is rectangular in shape having four nodes. The transverse deflection is represented by a twelve term of polynomial expansion while the in-plane displacements assumed at every layer interface are represented by same bilinear polynomial functions.

Owen and Li (1987) presented a displacement based local finite element model by using a discrete layer refined theory for thick anisotropic laminated plates. The 3-D problem is reduced to a 2-D one by assuming piecewise linear variation of the in-plane displacements and constant values of transverse displacement across the thickness. The element used is a sixteen noded isoparametric brick element. To define geometry within the element eight noded serendipity quadrilateral element shape functions are used. The displacement field is defined by a Heterosis quadrilateral element with quadratic Lagrangian


interpolation for in-plane displacements consisting of serendipity shape functions plus a bubble function.

Reddy and Barbero (1989) presented a plate bending element based on the generalized laminate plate theory (GLPT) developed by the senior author. The generalized displacements defined by this layerwise theory are expressed over each element as a linear combination of the 2-D interpolation functions and nodal values corresponding to total number of nodes per element. Different quadrilateral elements are formed for symmetry and no-symmetry cases. The problems of cylindrical bending of cross-ply laminates and bending of simply supported plates are solved analytically as well as by using the finite element method to compare the deflection and stress results.

The above layer wise laminate theory is also used by Robbins and Reddy (1993) to develop a layerwise 2-D displacement based finite element model of laminated composite plates that assumes a piece wise continuous distribution of the transverse strain through the laminate thickness. The resulting layer wise finite element model is capable of computing inter-laminar stresses and other localized effects with the same level of accuracy as a conventional 3-D finite element model. The finite element model is implemented in a computer code that construct a layer wise finite element mesh by combining a 2-D finite element in-plane discretisation and a one-dimensional finite element discretisation through thickness. The 2-D finite elements for in-plane discretisation are defined by a group of isoparametric elements having different node numbers, which are based on either a Lagrangian or a serendipity family of shape functions. For discretisation through the thickness linear, quadratic or cubic one-dimensional Lagrangian finite elements are used.

2.4.4 Elements based on Refined shear deformation theories

A displacement based multi-layered anisotropic flat plate element is developed by Di Sciuva (1985), which includes the effects of the transverse shear


deformation as in RFSDT. The discrete element is four noded rectangular one where each node contains eight degrees of freedom, which include extension; bending and transverse shear deformation states. The formulation is based on an improved bi-dimensional transverse shear deformation plate theory, which satisfies the contact conditions for the displacements and transverse shear stresses at the interfaces between the layers. Linear functions are used for approximating the in-plane displacements and shear rotations; cubic polynomials are used for approximating the transverse displacement.

Di Sciuva (1993) formulated a displacement based general quadrilateral multi-layered plate element making use of a refined third order shear deformation plate theory (RHSDT) proposed earlier by the same author. The plate model makes use of a displacement field that fulfills the geometric and stress continuity conditions at the interfaces between the layers and requires only five generalized displacements to describe the kinematics of the laminate deformation. A four noded quadrilateral plate element formulated on the basis of the above theory has ten degrees of freedom (i.e., five generalized displacements, two total rotations, two curvatures and the twist) at each node. In formulating the plate element the Lagrangian linear interpolation function has been used for approximating in-plane displacements and shear rotations whereas Hermite sixth order polynomials are used to describe the transverse displacement.

Cho and Parmerter (1994) presented a three noded triangular element based on their RHSDT developed for the analysis of symmetric laminated composites. The element is based on the element developed by Specht (1988). This non-conforming element has five degrees of freedom (i.e. the transverse displacement, its two derivatives and two shear rotations) in each node. This element satisfies C1 continuity conditions at the nodes as desired by the plate theory. At the element interfaces between nodes, C1 continuity is satisfied on an average rather than a point wise sense. The shape functions and their derivatives are same as that of Specht (1988). Area co-ordinates are used in the description of


shape functions for the two shear rotations. Application of the element is done in the convergence study, prediction of different stress components and deflections in relation to the static analysis of symmetric cross-ply laminates.

Di Sciuva (1995) developed a three noded, fully conforming, multi-layered anisotropic plate element of arbitrary triangular shape based on a refined third order shear deformation theory (RHSDT) proposed by the author. The element incorporates ten nodal degrees of freedom namely the two in-plane displacements; two shear rotations, the transverse displacement and its first and second derivatives; thus giving a total of thirty degrees of freedom per element. The element is an upgraded version of the high precision triangular element developed by Cowper et al (1969). The transverse displacement is represented by a complete quartic polynomial with few extra quintic terms. For uniform computation of the involved integrals, a topological transformation of the geometry from the physical plane to the transformed plane is accomplished by using Lagrangian interpolation functions. Thus a subparametric formulation has been used for transverse displacement. The element is tested for convergence and then applied for the solution of static and vibration problems of simply supported square laminated composite plates.

Based on the kinematics of the third order plate model (RHSDT) described before, Icardi (1998) formulated an eight noded curvilinear element having fifty-six degrees of freedom. Nodal parameters are membrane displacements; transverse shear rotations, deflections, slopes and curvatures for corner nodes, membrane displacements and transverse shear rotations for mid-side nodes. The element has C2 continuity for transverse displacement. Tensor products of Hermite’s quintic polynomials are used as interpolation function for transverse displacement at the reference plane to assure C2 approximation for this quantity. A C0 approximation being required for in-plane displacements and rotations, serendipity interpolation functions are used for these quantities.


2.5 IMPORTANT ANALYTICAL STUDIES ON LAMINATED PLATES

2.5.1 Static analysis

Pagano (1969) initiated a pioneering work by presenting an exact 3-D elasticity solution for the static analysis of composite laminates in cylindrical bending. The loading is considered to have a sinusoidal distribution. The general class of problems treated, involves the geometric configurations of any number of isotropic or orthotropic layers bonded together. Because of its general nature the analysis technique is also capable to represent the behavior of sandwich plates in cylindrical bending.

In continuation to the above study, Pagano (1970) presented a 3-D elasticity solution for the static analysis of rectangular laminates with simply supported edges under sinusoidal loading. The lamination geometry treated consists of arbitrary number of layers, which can be isotropic or orthotropic with material symmetry axes parallel to the plate axes. The necessary continuity requirements at the interfaces are satisfied along with appropriate stress conditions at the top and bottom surfaces of the plate. Several example problems of cross-ply laminated composite plates are solved including a problem of sandwich plate.

Srinivas and Rao (1970) presented a unified analysis technique for static, vibration and buckling analysis of a class of thick laminates. A 3-D linear, small deformation theory of elasticity is developed for the analysis of simply supported thick orthotropic rectangular laminates. All the nine elastic constants of orthotropy are taken into account. The solution is formally exact and leads to simple infinite series for stresses and displacements in flexure, forced vibration and beam-column type problems and to closed form characteristics equation for free vibration and buckling problems. Some numerical results are presented for simply supported three layer cross-ply orthotropic laminated plates.

Whitney (1972) developed a procedure for accurately calculating the mechanical behavior of a thick laminated composite or sandwich plate of arbitrary stacking


sequence. The procedure is an extension of an existing laminated plate theory, which includes the effect of transverse shear deformation Accuracy of the approach, is ascertained by comparing solutions from the modified plate theory to exact elasticity theory. Results are given for the static analysis of orthotropic angle ply rectangular laminates and sandwich plates.

Foile (1970) presented an appropriate solution procedure for the static analysis of orthotropic rectangular sandwich plates under general boundary conditions. The shear deformation is considered in a similar manner as that of the FSDT. The governing equations for an orthotropic sandwich plate with the corresponding boundary conditions are derived by the variational principle of minimum potential energy. A numerical integration method for solving the equation is described and an explicit solution is presented for a clamped plate.

Monforton and Ibrahim (1975) studied the static analysis problem of rectangular sandwich plates constructed of an orthotropic core and unbalanced cross-ply face sheets. A double Fourier series approach is used for the analysis of simply supported sandwich plates under transverse loading. The results indicate that the effect of bending-membrane coupling depends mainly upon the relative thickness of the core and the faces; other factors include the shear stiffness of the core, the degree of anisotropy of the individual plies, the total number and layups of plies in the plate and the aspect ratio.

Noor and Burton (1990) proposed a three-dimensional elasticity solution for the stress problems of multi-layered anisotropic plates. The plate is assumed to have rectangular geometry and antisymmetric lamination with respective to the middle plane. A mixed formulation is used with the fundamental unknowns consisting of the six stress components and three displacement components of the plate. Each of the plate variables is decomposed into symmetric and antisymmetric components in the thickness direction and is expressed in terms of a double Fourier series in the Cartesian surface co-ordinates. Extensive numerical


studies are made to show the effects of variation in the lamination and geometric parameters of composite plates on various stress and strain components.

Chen and Cheung (1972) adopted the finite strip method for the static and free vibration analysis of multi-layered sandwich plates. The assumption of common shear angle for the cores as used by some previous investigators, are omitted. Several numerical examples of sandwich plates with various examples are presented. The loading is considered as uniform. The examples include 3-layer and 5-layer (multi-core) simply supported isotropic and orthotropic sandwich plates. The thickness ratio is considered in the range of thin plate and effect of aspect ratio is studied.

Cho and Averill (2000) presented a first order zigzag sub-laminate plate theory for the analysis of laminated composites and sandwich plates. A 3-D finite element plate model is chosen for the implementation of the theory. The in-plane displacement fields in each sub-laminate are assumed to be piecewise linear functions and vary in a zigzag fashion through the thickness of the sub-laminate. The transverse displacement field is assumed to vary linearly through the thickness. The finite element is developed with the topology of an eight-noded brick element, allowing the thickness of the plate to be discretised into several elements or sub-laminates. Each node has five engineering degrees of freedom (i.e., three translations and two rotations). However, the number of unknowns is dependent on the number of sub-laminates. Numerical performance of the current element is investigated for a composite armored vehicle and for a sandwich plate.

2.5.2 Free vibration analysis

Srinivas et al. (1970) developed a three-dimensional linear, small deformation theory of elasticity by direct method for the free vibration of simply supported thick rectangular plates. The solution is exact and involves determining a triple infinite sequence of eigenvalues from a double infinite set of closed form transcendental equations. The present analysis yields symmetric thickness modes,


which neither of the approximate theories can identify. The numerical results are given for three-ply laminates and are used to assess the accuracy of thin plate theory predictions for laminates. Extension to general lateral surface conditions and forced vibrations is also indicated.

Noor (1973) proposed a solution by three-dimensional elasticity theory for free vibration of simply supported multi-layered cross-ply laminates. A square plate having both symmetric and skew-symmetric laminates with respective to the middle plane of the plate are considered. The thickness of each laminate is assumed to be same. Solution of the three-dimensional elasticity theory is obtained using the higher order finite difference scheme. Two parameters are varied, namely the degree of orthotropy of the individual layers and the thickness ratio of the plate. The results are presented in graphical form for the fundamental frequencies and the associated mode shapes and modal stresses.

Khedir (1988) developed an analytical procedure to investigate the free vibration of anti-symmetric angle-ply laminated plates for various boundary conditions. The procedure based on a generalized Levy type solution considered in conjunction with the state space concept, enables one to solve exactly the equations governing the laminated anisotropic plate theory, which is a generalization of Mindlin’s theory for isotropic plates to laminated anisotropic plates and includes shear deformation and rotary inertia effects.

Kant and Mallikarjuna (1989) analysed free vibration of laminated composites and sandwich plates using the shear deformation theory developed by the senior author in conjunction with a C0 finite element formulation. The theory is based on a higher order displacement model and the three dimensional Hook’s law for plate material, which gives rise to more realistic representation of the cross-sectional deformation. A special mass lumping procedure is used in the dynamic equilibrium equations.


Nosier et al. (1993) performed free vibration of laminated plates using Reddy’s (Reddy 1987) layerwise theory. In this theory full account is taken for various 3-D effects. The elasticity equations are solved utilising the state space variables and the transfer matrix. A detail analysis has been carried out by uncoupling the Navier equations to study various mode shapes and natural frequencies of a homogeneous, transversely isotropic plate. Results are shown for symmetric and anti-symmetric cross ply laminates.

Wu and Chen (1994) used local higher order shear deformation theory to determine the natural frequencies and buckling loads of laminated composite plates. This theory accounts for the effects of transverse shear and normal deformation. In this theory the layer-by-layer or sub-laminate by sub-laminate in-plane displacement components are approximated by cubic polynomial expansion whereas the transverse displacement variation is considered as quadratic. The displacement continuity conditions between different layers are restored. The equations of motion based on this theory are obtained by using the Hamilton’s principle. The analytical solutions for the natural frequencies and buckling loads of simply supported rectangular cross-ply laminates are determined by applying the Fourier series expansion method.

Rais-Rohani and Marcellier (1999) presented an extension of the small deflection theory of the sandwich plates to the free vibration and buckling analysis of composite sandwich plates with all edges rigidly supported against transverse displacements and with different degrees of elastic restraint against rotations. The plate deflections and the transverse shear forces are represented by an independent set of functions that satisfy the essential boundary conditions for all and the natural boundary conditions for specific cases. The Rayleigh-Ritz method is used to solve for in-plane buckling loads and transverse natural frequencies through the solution of an eigenvalue problem.

Wang et al. (2000) studied the free vibration of skew sandwich plates composed of an orthotropic core and laminated facings. The p-Ritz method is


adopted for the analysis. Since no vibration solution is available for such skew sandwich plates, the validity, convergence and accuracy of the Ritz formulation are established by comparing with other results of vibration frequencies for various subset plate problems involving rectangular sandwich plates and skew laminated plates.

Matsunaga (2000, 2001) presented the vibration and stability analysis of cross-ply and angle-ply laminated composite plates according to a global higher order plate theory, which accounts for shear deformation, thickness change and rotary inertia. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional higher order theory for thick rectangular laminates subjected to in-plane stresses is derived through Hamilton’s principle. Several sets of truncated approximate theories are applied to the solution of the problem of a simply supported thick laminated plate.

Kant and Swaminathan (2001) presented an analytical solution for free vibration of laminated composite and sandwich plates based on a higher order refined theory developed by the first author. The theoretical model presented incorporates laminate deformations, which account for the effects of transverse shear deformation, transverse normal strain/stress and a non-linear variation of in-plane displacements with respect to the thickness co-ordinates. The equations of motion are obtained by Hamilton’s principle. Solutions are obtained in closed form using the Navier technique and by solving the eigenvalue equation. Using the theory some new results for sandwich plates are also presented.

Yuan and Dawe (2002) used spline finite strip method for predicting the natural frequencies and modes of vibration of rectangular sandwich plates with laminated faces. The faceplates are treated as being classically thin. The core is modelled as 3-D body. Finite strip stiffness and mass properties are based on a displacement field, which represents eight fundamental through thickness displacements as series of product of longitudinal B-spline functions and


crosswise Lagrangian or Hermitian polynomial shape functions. The solution procedure utilizes the efficient super strip concept in conjunction with the extended Strum sequence bisection approach.

Wang (1997) presented an approach to free vibration analysis of skew fibre-reinforced composite laminates based on first order shear deformation theory (FSDT). He proposed a B-spline Rayleigh Ritz method for the analysis. The composite laminates may have arbitrary layups, admitting the possibility of coupling between in-plane and out-of-plane behaviour and general anisotropy. In this approach displacement field consists of the three mid surface translational displacements and the two through thickness shear strains instead of the two rotations in order to avoid the shear-locking phenomenon.

Krishna Reddy and Palaninathan (1999) extended a general high precision triangular plate bending finite element (Rao et al. 1974) to the free vibration analysis of laminated skew plates by deriving the consistent mass matrix in explicit form. The boundary conditions on the skew edges are imposed through the transformed element matrices. The fundamental frequencies are obtained for simply supported and clamped antisymmetric angle ply skew laminates. In this analysis the effects of skew angles, fibre orientation angles, number of layers and stacking sequence on the fundamental frequency is studied.

2.5.3 Buckling analysis

Srinivas and Rao (1969) obtained a closed form characteristics equation for the buckling stress of thick rectangular plates. This analysis considers the effect of transverse shear stress and forms the basis of three dimensional elasticity analyses of buckling problems for thick ordinary and laminated plates. The effect of change in the thickness ratio is demonstrated to have considerable impact on the response of thick plates compared to that of thin plates.

Khatua and Cheung (1973b) applied the finite element method to the buckling analysis of multi-layered sandwich structures with n stiff layers and n-1


weak cores, in which each layer or core can assume different material properties so that the concept of common shear angle is no longer applicable. A rectangular and triangular element is used separately in the analysis and their stability matrices are developed. The elements are applied to solve several standard stability problems of multi-layered sandwich plates with uniaxial edge loading.

Chan and Foo (1977) presented a finite strip method for the stability analysis of rectangular multi-layered sandwich plates. The method permits freedom of in-plane displacements for all the stiff layers, thus excludes the assumption of common shear angle for all cores as adopted in some previous works. Numerical examples of various multi-layered sandwich plates with different boundary conditions, aspect ratios and under different edge loadings are considered.

Rao (1985) extended the small deflection theory of orthotropic sandwich plates developed by Libove and Batdorf for the analysis of highly anisotropic sandwich plates with thin faces. Buckling coefficients of a simply supported rectangular symmetric anisotropic sandwich plate under combined longitudinal compression and bending are evaluated using the Rayleigh Ritz Method. The results of various examples show that the multi-ply-faced orthotropic sandwich plate is stronger than a single-ply-faced orthotropic one and that the maximum longitudinal buckling strength occurs in plates with lower aspect ratios when the enforcing fibres are longitudinal and in plates of higher aspect ratios when the fibers are oriented at about 400 with respect to the longitudinal axis.

Heder (1991) obtained the uniaxial buckling loads of sandwich panels by analytical methods. Using an energy method with deflection functions satisfying the boundary conditions the derivation of buckling loads for different boundary conditions is made. The work is restricted to panels with isotropic face and core materials; and to panels with faces of equal thickness. For comparison of the analytical results a finite element is also developed, which consists of combination of a shell and solid element representing the faces and the core. The


shell element used has four nodes with six degrees of freedom per node. The core is represented by an eight noded solid element, which has three displacement degrees of freedom per node.

Krishna Reddy and Palaninathan (1995) extended a general high precision triangular plate bending finite element to the buckling analysis of laminated skew plates. This procedure involves the development of the transformation matrix between global and local degrees of freedom for nodes lying on the skew edges and suitable transformation of element matrices. New results are obtained for antisymmetric angle ply and cross ply laminated skew plates. In this analysis the critical buckling loads for different skew angles with various lamination parameters such as number of layers, fiber orientation angles, different boundary conditions and loadings are presented.

Chattopadhay and Gu (1996) presented an exact elasticity solution for the buckling analysis of a simply supported orthotropic plate whose behaviour is referred to as cylindrical bending. The general class of problem involving geometric configurations of any number of orthotropic layers bonded together and subjected to an in-plane compressive load is also analysed by assuming a uniform pre-buckling stress, which is equivalent to the membrane assumption used in the plate theories. The closed form expression for the displacements and stresses are derived and the nonlinear eigenvalue equations are presented, which are used to solve the critical loads.

Hadi and Mathews (1998) presented a method for calculating the critical buckling load of sandwich panels. Based on earlier zigzag models for through thickness displacements, the transverse shear deformation of the faces is included and thus allowing for the analysis of thick sandwich plates. The Rayleigh Ritz method has been used to form the governing equations for the closed form solution. Applying the present method to investigate the buckling behaviour of solid orthotropic and laminated plates it is necessary to divide the plate’s thickness into three components, namely upper face, core and the lower face. The


thickness of each component is arbitrary. Studies are carried out to investigate the effect of layup angles of the faces and the aspect ratio on the buckling of anisotropic symmetric angle ply sandwich panels under uniaxial, biaxial and shear loadings.

Gu and Chattopadhay (2000) presented a three-dimensional elasticity solution for the buckling of simply supported orthotropic and laminated composite plates. This is an extension of their previous work (Chattopadhay and Gu 1996) on buckling of orthotropic plates.

Yuan and Dawe (2001) presented a B-spline strip method for predicting the buckling stresses of rectangular sandwich plates. The core is represented as three-dimensional in which the in-plane displacements vary quadratically through the thickness while the out-of-plane displacement varies linearly. The faceplates may in general be composite laminates. The method is applied to predict the elastic buckling loads of various rectangular sandwich plates under uniaxial, biaxial and shear loadings for different aspect ratios, boundary conditions and others. Some examples are also included to study the effect of localised buckling of the wrinkling type.

Hu and Tzeng (2000) studied the elastic stability problem of laminated skew composite plates subjected to uniaxial in-plane compressive loads. The critical buckling load of the laminated skew plates are calculated by the bifurcation buckling analysis implemented in finite element programme ABAQUS. The plate model is based on the FSDT. The effects of skew angles, laminate layups, plate aspect ratio, plate thickness ratio, central circular cutouts and edge conditions on the buckling resistance of the laminated skew composite plates are presented through several numerical examples and the results are shown graphically.


2.6 MOTIVATION OF THE PRESENT STUDY

Based on the review made on the existing literature, it has been observed that there is no investigation on laminated sandwich plates with inter-laminar imperfection so far. Moreover, the studies carried out on perfect sandwich laminate and imperfect composite laminate are few. Again these are mostly restricted to simple plate problems since the solution is based on analytical technique. Interestingly, these categories of problems can be modeled by a layer-wise plate theory or by 3-D idealization. In addition to that, it is desirable to apply finite element technique for the solution of the problem so that a wide range of plate problem having practical complexity can be handled.

In order to keep the computational involvement within manageable range and to have reasonably accurate structural modeling, the refined higher order shear deformation theory (RHSDT) should be the best option according to the review presented above. Actually, this plate theory can predict with sufficient accuracy the structural responses including inter-laminar stresses without introducing any additional unknown into the formulation. However, this theory is applied to some simple problems of laminated plates only by this time. Thus the potential of this theory is not fully utilized and it can be achieved through a proper finite element implementation of it. Incidentally, the plate theory requires C1 continuity of transverse displacement at the element interfaces and there are very few elements having that capability. Moreover, these elements may not be attractive in all the cases due to the presence of higher order derivative in the nodal degrees of freedom and some other disadvantages. For the representation of imperfections at the layer interfaces, the linear spring layer model should be most suitable and it can be nicely combined with the refined higher order shear deformation theory (RHSDT).

The above observations have inspired the author of this thesis to undertake the present investigation.

Chapter

THREE

MATHEMATICAL FORMULATION AND COMPUTER IMPLEMENTATION

3.1 INTRODUCTION

The development of an appropriate mathematical model for accurate prediction of the behavior of composites and sandwich laminates has drawn a considerable amount of attention of researchers all over the world in last two decades.

It is understood from the literature review carried out in the previous chapter that the refined higher order shear deformation theory (RHSDT) will be most suitable for the analysis of laminated composites and sandwich plates. This is due to the fact that the theory satisfies the necessary conditions of inter-laminar stress continuity at the layer interfaces as well as the condition of zero transverse shear stresses at the top and bottom surfaces of the plate. However, the plate theory (RHSDT) has one problem and it is faced during its finite element implementation since the theory requires C1

continuity of the transverse displacement at the element edges. In fact this has restricted the use of such an elegant theory in the finite element analysis. Thus it is required to develop new finite elements for this purpose since availability of elements having the capability of fulfilling C1 continuity requirement of the plate theory is very few.

Mathematical formulation 41

As discussed in the last chapter, linear spring layer model will be suitable for accurate modeling of the laminated plates having inter-laminar imperfections. In this study finite element is implemented to refined higher order shear deformation theory (RHSDT) in combination with linear spring layer model and it has been used for the present analysis. In this chapter the basic assumptions of the formulation is first stated. Then the application of the proposed analysis in the static, vibration and buckling problems are summarized. Finally the mathematical formulation is explained in detail for laminated plates having inter-laminar imperfections, which is followed by a section where the computer implementation of the mathematical formulation is discussed.

3.2 BASIC ASSUMPTIONS

The formulation is based on the following assumptions:

• The middle plane of the plate is taken as reference plane.

• The laminated plates consist of a number of layers bonded together where each layer is treated as homogeneous and orthotropic.

• The bonding between the layers may be perfect or imperfect.

• The orthotropic layers may have any orientation with respect to the reference structural (plate) axes system.

• The materials used obey Hook’s law.

• The lateral deflection is small compared to the thickness of the plate.

• The normal stress in the transverse direction is small compared to other stress components and it is neglected.

• The in-plane displacements have cubic variations across the plate thickness with kinks at the layer interfaces according to RHSDT where the transverse displacement is constant over the entire thickness.

Mathematical formulation

42

3.3 GOVERNING EQUATIONS FOR THE PROPOSED ANALYSES

The present study consists of static, vibration and buckling analyses. The governing equations for these analyses are as follows.

• Static Analysis: [ ] K P∆ = (3.3.1)

• Free Vibration Analysis: [ ] [ ] 2K Mω∆ = ∆ (3.3.2)

• Buckling Analysis: [ ] [ ] GK Kλ∆ = ∆ (3.3.3)

where [ ]K , [ ]M , [ ]GK , P , ∆ , ω and λ are the stiffness matrix, mass matrix,

geometric stiffness matrix, nodal load vector, nodal displacement vector, natural frequency of vibration and buckling load factor of the structure respectively. The

derivation of the system matrices [ ]K , [ ]M , [ ]GK and P are presented in the

following sections.

3.4 STRESS-STRAIN RELATIONSHIP

The stress-strain relationship of an orthotropic layer/lamina (say k-th layer) having any fibre orientation with respect to structural axes system (x-y) may be expressed as

11 12 16

12 22 26

16 26 66

55 45

45 44

0 0

0 0

0 0

0 0 0

0 0 0

x x

y y

xy xy

xz xz

yz yz

Q Q Q

Q Q Q

Q Q Q

Q Q

Q Q

σ ε

σ ε

τ γ

τ γ

τ γ

=

or kQσ ε= (3.4.1)

where the rigidity matrix kQ can be formed with the material properties (E1,

E2, ν12, G12, G13, G23) and fibre orientation (θ) of the lamina as follows:

( )4 2 2 411 11 12 66 22cos 2 2 sin cos sinQ Q Q Q Qθ θ θ θ= + + +

( ) ( )2 2 4 4

12 11 22 66 124 sin cos cos sinQ Q Q Q Qθ θ θ θ= + − + +


( )4 2 2 4

22 22 12 66 11cos 2 2 sin cos sinQ Q Q QQ θ θ θ θ= + + +

( ) ( )3 316 11 12 66 12 22 662 sin cos 2 sin cosQ Q Q Q Q Q Qθ θ θ θ= − − + − +

( ) ( )3 326 11 12 66 12 22 662 sin cos 2 sin cosQ Q Q Q Q Q Qθ θ θ θ= − − + − +

( ) ( )2 2 4 466 11 22 12 66 662 2 sin cos cos sinQ Q Q Q Q Qθ θ θ θ= + − − + +

2 244 13 23cos sinQ G Gθ θ= + , ( )45 13 23 sin cosQ G G θ θ= −

2 255 23 13cos sinQ G Gθ θ= +

where, 111

12 211E

Qν ν

=−

, 12 212

12 211E

Qν

ν ν=

− , 2

2212 211

EQ

ν ν=

− and 66 12Q G= .

3.5 REPRESENTATION OF DISPLACEMENT COMPONENTS AS PER

THE REFINED PLATE MODEL

The variation of in-plane displacements (u , v ) across the plate thickness for general lamination configuration with imperfections at the layer interfaces as shown in Fig. 3.1 may be expressed as

( ) ( )1 11

lnix ii i

iu u z z u H z zα + +

== + − − ∆ − +∑

( ) ( ) 2 3

1

l u

l

n nix i i i x x

i n

z z u H z z z zα β η+

= +

+ − + ∆ − + +∑ (3.5.1)

( ) ( )1 11

lniy i i i

i

v v z z v H z zα + +=

= + − − ∆ − +∑

( ) ( ) 2 3

1

l u

l

n niy i i i y y

i n

z z v H z z z zα β η+

= +

− + ∆ − + ++ ∑ (3.5.2)

where ( )iH z z− and ( )1iH z z +− + are the unit step functions.

This is obtained by combing the concept of refined higher order shear deformation theory (RHSDT) with the linear spring-layer model for inter-laminar


44

imperfections in a convenient manner. The transverse displacement is assumed to be constant over the plate thickness i.e.,

w w= . (3.5.3)

Figure 3.1 Through Thickness Variation of In-plane Displacement, u

The imperfection at the k-th interface is characterized by the displacement jumps ku∆ (Fig. 3.1) and kv∆ , which may be expressed in terms of inter-laminar

shear stresses at that interface as

11 12k k k k

k xz yzu R Rτ τ∆ = + (3.5.4)

and 21 22k k k k

k xz yzv R Rτ τ∆ = + . (3.5.5)

11kR , 12

kR , 21kR and 22

kR in the above equations are the compliance coefficients of

the idealized linear spring layer at the k-th interface where kxzτ and k

yzτ are the

transverse shear stresses at that interface. Taking an adjacent layer of the k-th interface, k

xzτ and kyzτ may be expressed in terms of xzγ and yzγ (transverse shear

strains) of that layer at this interface using equation (3.4.1). Again, equations


(3.5.1)-(3.5.3) may be used to express xzγ = / /w x u z∂ ∂ + ∂ ∂ and yzγ

= / /w y v z∂ ∂ + ∂ ∂ where ku∆ and kv∆ will not fortunately appear and it will help

to express ku∆ and kv∆ in terms of other terms easily. Now utilizing the

transverse shear stress free condition at the plate top and bottom surfaces, βx, βy, ηx and ηy may be expressed in terms of other parameters appeared in equations (3.5.1) – (3.5.3) as

1

12

l un ni

x xih

β α+

=

= − ∑ ,1

12

l un ni

y yih

β α+

=

=− ∑ , 21 1

4 1 1,3 2 2

l l u

l

n n ni i

x x x xi i n

wh

η α α+

= = +

=− − +

∑ ∑

and 21 1

4 1 1,3 2 2

l l u

l

n n ni i

y y y yi i n

wh

η α α+

= = +

=− − +

∑ ∑ .

Finally, the condition of transverse shear stress continuity at the interfaces between the layers is imposed to express i

xα and iyα in terms of the quantities at

the reference plane as

( ) ( ) , ,ix xx x xy y xx x xy ya a b w b wα γ γ= + + +

( ) ( ) , ,iy yx x yy y yx x yy ya a b w b wα γ γ= + + +

where 1( , , )lnx x x x xw wγ θ α += − = + and 1( , , )ln

y y y y yw wγ θ α += − = + are the

transverse shear strains at the reference plane. The constants (axx, axy, bxx, byy …) found in the above equation are dependent on the material properties of the two layers adjacent to the i-th interface.

Interestingly, the refined higher order shear theory (RHSDT) can be easily reduced to RFSDT, HSDT and FSDT by dropping certain terms from the expressions of in-plane displacements obtained in RHSDT. Actually RHSDT may be considered as the most general one where RFSDT, HSDT and FSDT are basically its subsets and they may be obtained by omitting the following terms from equation (3.5.1) and (3.5.2).

• RFSDT - omitting βx, βy, ηx and ηy


46

• HSDT - omitting all ixα , i

yα except 1lnxα + , 1ln

yα +

• FSDT - omitting βx, βy, ηx, ηy and all ixα , i

yα except 1lnxα + , 1ln

yα +

Though the plate model presented in this section is for laminates having inter-laminar imperfections but it can be used for a perfect laminate by taking the

compliance coefficients ( 11kR , 12

kR , 21kR and 22

kR ) as zero. In this context, it should

be noted that the effect of imperfection can be taken into account with layer-wise plate models (Discrete layer theory, RFSDT and RHSDT) only.

3.6 FINITE ELEMENT IMPLEMENTATION

Regarding finite element implementation, it has already been mentioned that the present plate theory requires C1 continuity of transverse displacement, w since the strain components contain second order derivatives of w due to the presence of w,x, w,y in the expression of u , v . Unfortunately, this requirement cannot be satisfied easily and it has been proved that it cannot be achieved with the usual degrees of freedom (w, w,x, w,y) at the nodes. Actually this is a well-known problem of plate finite element, which has been identified at the beginning of finite element era. Though this problem, faced with CPT for the first time, has been avoided in Mindlin type plate elements (FSDT) according to isoparametric formulation but it reappeared in the improved plate theories for laminated plates such as HSDT (Reddy 1984) and all the theories in the group of RFSDT and RHSDT.

In this situation, one of the simple options is to upgrade an existing conforming plate element based on CPT, which fulfills the above continuity requirement of w. Such an attempt has been made by Di Sciuva (1993) where the conforming rectangular plate element of Bogner et al. (1965) has been upgraded according to his RHSDT. In a similar manner Di Sciuva and Icardi (1995) have upgraded the high precision triangular element of Cowper et al. (1969). Extending the concept of Bogner et al. (1965), Icardi (1998) has developed an eight noded


curvilinear element based on RHSDT (Di Sciuva 1992) where different degrees of freedom (DOF) are taken at the different nodes. In addition to the other complications, these elements (Di Sciuva 1993, Di Sciuva and Icardi 1995, Icardi 1998) contain higher order derivatives of w as nodal unknowns, which are not usually preferred in a practical analysis.

In order to avoid these problems, Cho and Parmerter (1994) have made an attempt where the nonconforming plate element (CPT) of Specht (1988) has been upgraded according to their RHSDT and applied it to the analysis of symmetric composite laminates. It is quite interesting that the element has successfully modeled RHSDT (Cho and Parmerter, 1993) though it (Specht, 1988) does not satisfy the inter-element continuity requirement. This may give an impression that any successful nonconforming plate element based on CPT can be upgraded in a similar manner but the success is not assured, which has already been experienced in some studies. In this situation an effort has been given to develop an element, as the requirement of an efficient and trouble free element for the solution of the present problem genuinely exists.

3.6.1 Proposed Element

The element may have any triangular shape as shown in Fig. 3.2(a). The arbitrary triangular geometry in Fig. 3.2(a) is mapped in a different plane (ζ-η) to have a regular shape as shown in Fig. 3.2(b).

Figure 3.2 A typical element (a) before transformation (b) after transformation


48

The relationship between these two axes system may be defined as

321)1( xxxx ηςης ++−−= , (3.6.1)

321)1( yyyy ηςης ++−−= , (3.6.2)

∆ς

2222 ycxba ++

= (3.6.3)

and ∆

η2

333 ycxba ++= , (3.6.4)

where 32321 xyyxa −= , 13132 xyyxa −= , 132 yyb −= , 312 xxc −= ,

21213 xyyxa −= , 213 yyb −= , 123 xxc −= , and 2/)( 321 aaa ++=∆ .

The element (Fig. 3.2) has three corner nodes and three mid-side nodes where each node contains u, v (in-plane displacements), w (transverse displacement), θx, θy (rotations of normal at the reference plane), γx =∂w/∂x-θx and γy = ∂w/∂y-θy (shear rotations at the reference plane) as the degrees of freedom. The formulation of the element is made taking u, v, w, γx and γy as the field variables where all the field variables except w are approximated by a complete quadratic polynomial having six unknowns. The transverse displacement w is approximated by a truncated quintic polynomial having eighteen unknowns. This is obtained by imposing cubic variation of normal slope along the three sides, which helped to express the three unknowns amongst the twenty-one unknowns of a complete quintic polynomial in terms of other unknowns.

Actually, the nodal unknowns adopted in the present element can give a quadric variation of normal slope along the three sides whereas a complete quintic polynomial demands a quartic variation of normal slope along the sides if it is chosen as displacement function for w. The difference in the order of variation of normal slope along the sides is reduced by the following treatment, which has helped to achieve the success of this element.


Considering all the terms of the complete quintic polynomial, the transverse displacement w may be expressed as

2 2 3 2 2 3 41 2 3 4 5 6 7 8 9 10 11

3 2 2 3 4 5 4 3 2 2 3 4 512 13 14 15 16 17 18 19 20 21

w c c c c c c c c c c c

c c c c c c c c c c

ς η ς ςη η ς ς η ςη η ς

ς η ς η ςη η ς ς η ς η ς η ςη η

= + + + + + + + + + + +

+ + + + + + + + +

where c1… c21 are the unknown coefficients.

To get a cubic variation of normal slope along the sides 1-5-3, 1-6-2 and 2-4-3 (Fig. 3.2b), the quartic terms found in the expression of normal slope along these sides obtained by taking derivatives of w, are accordingly set to zero as follows.

From side 1-5-3: c20 = 0

From side 1-6-2: c17 = 0

From side 2-4-3: c18 = -(5c16 + c19 + 5c21)

Using the above three conditions, twenty-one unknown coefficients in the expression of w above may be reduced to eighteen unknowns. With these, the field variables (independent displacement components at the reference plane) may be expressed as follows

[ ] uNu α1= , (3.6.5)

[ ] vNv α1= , (3.6.6)

[ ] wNw α2= , (3.6.7) [ ] xx N γαγ 1= (3.6.8)

and [ ] yy N γαγ 1= , (3.6.9)

where [ ]T

u 654321 ααααααα = , [ ]Tv 121110987 ααααααα = ,

[ ]13 14 15 16 30.... Twα α α α α α= , [ ]31 32 33 34 35 36

Txγα α α α α α α=

[ ]Ty 424140393837 αααααααγ = , [ ] 2 2

1 1N ς η ς ςη η= and


50

[ ] 2 2 3 2 2 3 4 3

2

2 2 3 4 5 3 2 2 3 3 2 5 3 2

1

( 5 ) ( ) ( 5 ) .

N ς η ς ςη η ς ς η ςη η ς ς η

ς η ςη η ς ς η ς η ς η η ς η

=

− − −

With the help of equations (3.6.3) - (3.6.4) and (3.6.7) - (3.6.9), ( ),x x xwθ γ= −

and ( ),y y ywθ γ= − may be expressed as

x x x

w w wx x x

ς ηθ γ γ

ς η∂ ∂ ∂ ∂ ∂

= − = + −∂ ∂ ∂ ∂ ∂

[ ] [ ] [ ] [ ] [ ] 1 12 2 1 3 12 2 w x w x

b cN N N N Nγ γα α α α

ς η∂ ∂

= + − = −∂ ∆ ∂ ∆

(3.6.10)

and y y y

w w wy y y

ς ηθ γ γ

ς η∂ ∂ ∂ ∂ ∂

= − = + −∂ ∂ ∂ ∂ ∂

[ ] [ ] [ ] [ ] [ ] 2 22 2 1 4 12 2 w y w y

b cN N N N Nγ γα α α α

ς η∂ ∂

= + − = −∂ ∆ ∂ ∆

(3.6.11).

Substituting the expressions of u, v, w, θx, θy, γx and γy as found in equations (3.6.5)-(3.6.11) at the six nodes of the element (Fig. 3.2), the coefficients (αi) in the displacement functions (3.6.5)-(3.6.11) can be expressed in terms of nodal unknowns as

[ ] αδ A= or [ ] δα 1−= A (3.6.12)

where [ ]62221111111 ...... yyxyxT wvuwvu γγγθθδ = ,

[ ]424321 ...... αααααα =T and the matrix [A] having an order of 42x42

can be formed with (xi,yi) and ( iς ,ηi) of the six nodes.

It should be noted that the order of approximation functions (x-y plane) for transverse displacement, slope/rotation of normal, bending moment and shear force is quintic, quartic, cubic and quardatic respectively in this element. Thus a proper hierarchy is maintained in this regard and this is achieved by the appropriate selection of displacement functions for the field variables. This is not


maintained in the elements based on refined plate theories (Di Sciuva 1993, Di Sciuva and Icardi 1995, Icardi 1998, Cho and Parmerter 1994).

3.6.2 Element Stiffness Matrix and Load Vector

The derivation of element stiffness matrix and nodal load vector required in static analysis is presented in this section.

The strain components ε at any point within the plate may be expressed

in terms of reference plane parameters using the expressions of ,u v and w as presented in equations (3.5.1) – (3.5.3) as

[ ] T

u vH

x yu v u w v wy x z x z y

ε ε∂ ∂

= =∂ ∂

∂ ∂ ∂ ∂ ∂ ∂ + + + ∂ ∂ ∂ ∂ ∂ ∂ , (3.6.13)

where

2 2 2

2 2

yxu v w w wx y x y x y x y

u vy x

γγε

∂∂ ∂ ∂ ∂ ∂ ∂=

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂+ ∂ ∂

y xx y

w wx y x y

γ γγ γ

∂ ∂ ∂ ∂

∂ ∂ ∂ ∂

(3.6.14)

and the matrix [H] having an order of (5 x 14) contains z and some constant quantities dependent on material properties, layer thickness and compliance parameters for inter-laminar imperfections.

The field variables as expressed in equations (3.6.5)-(3.6.9) may be substituted in the above equation to express ε in terms of α as

[ ] Bε α′= . (3.6.15)

where the matrix [ ]B′ contains [N1] and [N2] and their derivatives.

It (ε) may be further expressed in terms of nodal displacement vector δ with the help of equation (3.6.12) as

[ ] [ ] 1B Aε δ−′= or [ ] δε B= . (3.6.16)


52

Now the element stiffness matrix can be derived following the technique of energy minimization (available in any standard text on finite element method) in usual manner and it may be expressed using equations (3.4.1), (3.6.13) and (3.6.16) as

[ ] [ ] [ ] [ ][ ] [ ] [ ][ ]1

u ln n T TT i

i

k B H Q H B dxdydz B D B dxdy+

=

= = ∑ ∫ ∫ (3.6.17)

where the rigidity matrix [D] may be defined as

[ ] [ ] [ ]1

Tlun n

i

iD H Q H dz

=

+

= ∑ ∫ (3.6.18)

The element load vector can also be derived in that process and it may be expressed with the help of equations (3.6.7) and (3.6.12) as

[ ] [ ]T

p N q dx dy= ∫ (3.6.19)

where [ ] [ ] [ ] [ ] [ ] [ ] [ ] 10 0 2 0 0N N N N N N A − = , [N0] is a null vector of order

1x6 and q is intensity of distributed transverse load acts on the plate.

3.6.3 Element Mass Matrix A consistent mass matrix can be derived in a similar manner as that of stiffness matrix and they may be used in the vibration analysis. For free vibration problem, the acceleration at any point within the plate may be expressed in terms of reference plane parameters with the help of equations (3.5.1)–(3.5.3) as

[ ] 2

2 22 F ff f

u uv v

tw w

ω ω= − = −=

∂ = ∂

(3.6.20)

where the matrix [F] of order 3x7 contains z and some constant quantities like that of [H] and

T

x yw wf u v wx y

γ γ ∂ ∂

= ∂ ∂


It can finally be expressed in terms of nodal displacement vector δ with the help of equation (3.6.5)-(3.6.9) and (3.6.12) as

[ ] [ ][ ] [ ] 1f C C A Cα δ δ−′ ′= = = (3.6.21)

where the matrix [ ]C′ having an order of 7x42 contains [N1], [N2] and its

derivatives like that of [ ]B′ .

Using the above equations (3.6.20)-(3.6.21), the consistent mass matrix of an element can be derived and it may be expressed as

[ ] [ ] [ ] [ ][ ] [ ] [ ][ ]1

u ln nT T T

ii

m C F F C dxdydz C L C dxdyρ+

=

= =∑ ∫ ∫ (3.6.22)

where ρi is the mass density of the i-th layer and the matrix [L] is

[ ] [ ] [ ]1

u ln nT

ii

L F F dzρ+

=

= ∑ ∫ (3.6.23)

3.6.4 Element Geometric Stiffness Matrix

The geometric stiffness matrix required in buckling analysis can be derived in a similar manner as that of stiffness matrix. For this purpose, the nonlinear strain vector/geometric strain vector may be expressed as

[ ]

2 2 2

2 2 2

1 1 1

2 2 2

1 1 1

2 2 2

12G G

w u vx x x

w u v Ay y y

w w u u v vx y x y x y

ε θ

+ +

= + +

+ +

∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

(3.6.24)


54

where [ ]

0 0 0

0 0 0G

w u vx x x

w u vAy y y

w w u u v vy x y x y x

∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂

and T

w w u u v vx y x y x y

∂ ∂ ∂ ∂ ∂ ∂= ∂ ∂ ∂ ∂ ∂ ∂

θ

With the help of equations (3.5.1)-(3.5.3) and (3.6.16), the above equation may be expressed as

[ ] [ ][ ] G GH H Bθ ε δ= = , (3.6.25)

where the matrix [HG] of order 6x14 contains z and some constant quantities like that of [H] and [F].

Using the above equation, the geometric stiffness matrix [kg] of an element can be derived and it may be expressed as

[ ] [ ] [ ] [ ]1

u l

g

n nT T i

G Gi

k B H S H B dxdydz+

=

= ∑ ∫

= [ ] [ ] [ ]TB G B dxdy∫ (3.6.26)

where

[ ] [ ] [ ]1

u ln nT i

G Gi

G H S H dz+

=

= ∑ ∫ (3.6.27)

and the stress matrix [Si] may be expressed in terms of in-plane stresses of the i-th layer as


0 0 0 00 0 0 0

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

x xy

xy y

x xyi

xy y

x xy

xy y

S

σ τ

τ σ

σ τ

τ σ

σ τ

τ σ

=

. (3.6.28)

3.6.5 Numerical Integration

In the previous sections, it is found that integrations are involved in the expression of element stiffness matrix, mass matrix, geometric stiffness matrix and nodal load vector. These integrations are carried out numerically following the Gauss quadrature integration rule. As the element has a triangular shape, the integration in x-y plane found in equations (3.6.17), (3.6.19), (3.6.22) and (3.6.26) are to performed in a triangular domain. This is carried out following an efficient Gaussian quadrature rule proposed by Dunavant (1985) for triangles where the Gauss points (integration points) are taken symmetrically. The one-dimensional integration in the thickness direction found in equations (3.6.18), (3.6.23) and (3.6.27) is carried out by the usual Gaussian quadrature rule. The number of integration points is based on the precision required in an integration, which is detected by the order of polynomials in that integration. In some examples, it has been found that the distributed transverse load has a sinusoidal variation. As the polynomial expression for sinusoidal function contains infinite number of terms having increasing orders, the integration for the evaluation of load vector due to such a loading should be carried out with infinite number of integration points according to the rule. However, it is found that a higher number of integration points give a satisfactory level of accuracy. In the present analysis, these integrations are carried out with 12 points integration rule, which is obtained through a convergence study with respect to integration points.


56

3.6.6 Assembly of Element Matrices

Depending on the type of analysis, the stiffness matrix [k], mass matrix [m], geometric stiffness matrix [kg] and/or nodal load vector p are computed for all the elements and combined/assembled together to form the overall stiffness matrix [K], mass matrix [M], geometric stiffness matrix [KG] and/or nodal load vector P of the whole structure. The skyline storage technique is used to keep these large size matrices [K], [M] and [KG] in single array where a considerable amount of storage space in core memory is saved in an efficient manner. This has been implemented systematically in the computer code developed in the present study.

3.6.7 Boundary conditions

In the present investigation, it is required to satisfy the kinematic boundary conditions only since the formulation of the present element is based on displacement approach. The boundary conditions are incorporated through penalty function approach where the diagonal terms/elements of the overall stiffness matrix [K] corresponding to the restrained degrees of freedom are multiplied with a very high value (1015). The restrained degrees of freedom for different types of boundary conditions such as clamped, simply supported and symmetric conditions are as follows. Boundary line parallel to x-axis

Simply supported condition: u = w = θx = γx = 0

Clamped condition: u = v = w = θx = θy = γx = 0

Symmetric condition: v = θy = γy = 0

Boundary line parallel to y-axis

Simply supported condition: v = w = θy = γy = 0

Clamped condition: u = v = w = θx = θy = γy = 0

Symmetric condition: u = θx = γx = 0.

For a free edge, none of the degrees of freedom are restrained.


3.7 SOLUTION TECHNIQUES

In the preceding sections the finite element formulation for the laminated plates are explained. In this section, the solution procedures for static, free vibration and buckling analyses are presented separately.

3.7.1 Static analysis

The scheme adopted for static analysis eliminates zeros within the band of the stiffness matrix beyond the last non-zero value and reduces the storage

requirement. The static equilibrium equation [ ] K Pδ = is solved by the

Cholesky decomposition procedure according to the algorithm presented by Corr and Jennings (1976), which uses a skyline storage scheme. The deflection components δ at any point of the plate can be calculated by solving the static equilibrium equations as discussed above. Once these displacements at the

reference plane is known, the strain components, ε at any point of the plate can

be calculated by using the strain displacement relationship using equations

(3.6.13) and (3.6.16). Now the stress, σ at any point of the layered plate may

be calculated by multiplying the strain vector, ε with the matrix, k

Q

of the k-

th layer. For the calculation of transverse shear stresses the respective equilibrium equations are used.

3.7.2 Free Vibration and Buckling analysis

In this case the governing equations (3.3.2) and (3.3.3) are solved by, the simultaneous iteration technique of Corr and Jennings (1976) for the computation of eigen values and eigen vectors. In this method, [ ]K is positive definite and can

be decomposed into Cholesky factors as


58

[ ] [ ] [ ]TTLK = , (3.7.1)

where [ ]L is the lower triangular matrix. Using equations (3.3.2) and (3.3.3), the

equation is rewritten for the free vibration and buckling analyses as:

[ ] [ ] [ ] [ ] [ ] δω

δ TTT LLLML 21 1

=−− (3.7.2)

[ ] [ ] [ ] [ ] [ ] δλ

δ TTTG LLLKL 11 =−− . (3.7.3)

The Equations (3.3.2) and (3.3.3) represent standard eigen value problem and these have been solved to extract the eigen values and the eigen vectors. In

these equations 2

1ω

and λ1 are the eigen values. Thus, the eigen values

corresponding to the lowest natural frequencies and the buckling loads are obtained using the simultaneous iteration technique. The methodology is explained as follows:

(i) Set a trial vector [ ]U and ortho-normalize.

(ii) Back substitute [ ] [ ] [ ]UXL = (iii) Multiply [ ] [ ] [ ]XMY = or [ ] [ ] [ ]XKY G=

(iv) Forward substitute [ ] [ ] [ ]YVL T =

(v) Form [ ] [ ] [ ]VUB T=

(vi) Construct [ ]T so that 1=ijt and ( )[ ]2

2

ijiiijij

ijij bbsbb

bt

−+−

−= ,

where s is the sign of ( )ijii bb −

(vii) Multiply [ ] [ ] [ ]TVW =

(viii) Perform Schmidt ortho-normalization to derive [ ]U

(ix) Check tolerance [ ] [ ]UU − (x) If not satisfactory, go to step (ii)


3.8 COMPUTER IMPLEMENTATION

3.8.1 Introduction

A new computer code is to be developed for the computer implementation of these mathematical formulations based on RHSDT. A number of general-purpose commercial software packages such as ANSYS, NISA and NASTRAN are available in the market. They are mostly based on FSDT and their modules cannot be readily modified to suit an individual designer’s needs. Hence, an attempt has been made to develop a new numerical tool using the finite element method with a flexibility to incorporate modifications whenever desired. The mathematical formulation presented in the previous chapter for the analysis of laminated plates has been used to generate a set of computer codes using the FORTRAN 90 language. Efficient tools such as an automatic mesh generator and the skyline storage scheme are incorporated in the computer code.

3.8.2 Application domain

The computer code developed for the analysis of laminated plates has been shown in the Flow Chart in Fig. 3.3. It includes a wide spectrum of application domain. Various boundaries, loading conditions and shapes have been incorporated in the computer code to make it a generalized one. The analyses modules can solve the following types of problems:

1. Static analysis: to evaluate displacements, stresses and strains at various points of the laminated plates

2. Free vibration analysis: to determine the natural frequencies of the laminated plates along with the corresponding mode shapes.

3. Buckling analysis: to assess the elastic buckling load of the laminated plates and the corresponding buckled shape.


60

START

PRE-PROCESSORReads input datafor laminated plateReads BoundaryconditionsGeneratesautomatic meshfor laminated plate

PROCESSORGenerates element matrices for laminated plateAssembles the matrices to global matrixIncorporates the boundary conditionsSolves the equations for desired analysis :

1. Static analysis2. Free vibration analysis3. Buckling analysis

END

POST-PROCESSOREchos input dataPrints the output

Figure 3.3 Basic units of the computer program

3.8.3 Description of the computer program

The computer code developed using the finite element method involves three basic steps in terms of the computational procedure:

• Preprocessor

• Processor

• Postprocessor


3.8.3.1 Preprocessor

The preprocessor is instrumental in reading the necessary details of the given structure such as geometry, boundary conditions, material properties, loading configurations and their magnitudes, lamination details. The desired data as

described above for the laminated plates are read in a subroutine (input) written for this purpose.

3.8.3.2 Processor

Based on the mathematical formulation presented in previous sections, the processor unit of the computer code performs the following tasks:

1. Generation of elastic stiffness, mass and geometric stiffness matrices of the laminated plate element.

2. Assembly of element matrices using the skyline storage scheme

3. Imposition of boundary conditions

4. Solution of algebraic equations for static analysis to obtain nodal displacements and stress resultants for the laminated plate.

5. Extraction of eigen values and eigen vectors for the free vibration and buckling analyses using the simultaneous iteration technique.

The various operations used in this processor unit of the computer code are mostly developed independently for the present problem. However, some already developed subroutines are utilized in different stages of analyses, which are presented briefly below

Subroutine assem

This subroutine assembles the element matrices (stiffness, mass and geometric) to generate the overall matrices in a single array using a variable bandwidth profile storage form. The scheme involves storing only those elements from the first non-zero element to the leading diagonal. The storage locations of the leading


62

diagonal elements of the matrix are stored in a separate integer pointer vector. The storage position of any stored element ( )jiaij > in the global loading matrix

is given by k=s (i)-i+j, where s is the pointer vector. An off-diagonal term is allocated a space when k>s (i-j). A fully populated symmetric matrix can be stored in a triangular form with s (i) = i (i+1)/2.

Subroutine solve

This routine is used to solve the simultaneous algebraic equations generated in the

process of analysis. The equilibrium equation is in the form of [ ] K Pδ = ,

where [ ]K is the global stiffness matrix, P is the global load vector obtained due

to the assembly of element load matrix and ` δ is the nodal unknown vector to be

obtained. The program involves subroutine for forward elimination (forsol) and

back substitution (backsol) for the solution.

Subroutine r8usiv

This Subroutine r8usiv (Figure 3.4) is used for the eigen value solution on the basis of the simultaneous iteration technique of Corr and Jennings (1976). Free vibration and buckling analyses are carried out using the subroutine. The input data to the subroutine are the global elastic stiffness matrix, the global mass matrix, the overall geometric stiffness matrix and the corresponding pointer vectors. Eigen values and eigen vectors are extracted from this routine as the outputs. The subroutine requires three arrays u, v and w of size (n, m) where n is the total degrees of freedom and m is a value higher than the number of modes. The numerical value of m has been considered as 1.5 times the number of modes in the present program. The tolerance value has been set to 10-7 and the maximum number of iterations to 50. The initial trial vectors are generated from a random number generator.


Elastic stiffness matrixmass Matrix

Geometric stiffness matrix

RETURN

END

call r8upre

call r8ufor

call r8ufort

call r8ubac

call r8uerr

call r8udec

call r8uran

Callr8usiv

Computation ofNatural frequencies/

buckling load/mode shapes

call r8ured

Figure 3.4 Flow chart for free vibration and buckling analysis


64

A number of following subroutines are called in turn for the execution of r8usiv:

Subroutine reduce : Decompose the stiffness matrix into upper and lower triangular matrix

Subroutine random : Generate random initial eigenvectors

Subroutine orthog : Orthogonalise the trial vector by Schmidt decomposition

Subroutine backsub : Back substitution in linear equation solution

Subroutine premult : Premultiply a matrix by a vector

Subroutine forsub : Forward substitution in linear equation solution

Subroutine error : Estimate vector error in successive trials.

3.8.3.3 Postprocessor

In this final stage of programming, all the input data are echoed to check their

accuracy. A subroutine (output) sets the desired output data in the form of displacements stresses, strains, eigen values etc. depending on the type of analysis carried out. Then the results are stored in separate files for presentation in the form of graphs or tables.

Chapter

FOUR

RESULTS AND DISCUSSIONS

4.1 INTRODUCTION

In this chapter a large number of numerical examples on composites and sandwich laminates covering wide range of features are solved by the proposed finite element plate model. The results obtained are presented in the form of tables and figures. For the purpose of validation of the proposed model, a large number of these results are compared with the published results. In some cases numerical results are generated from some available analytical solutions by separate computer programs developed for this purpose.

It has been found in the literature that so far there is no result on sandwich laminates having inter-laminar imperfections. For the case of composite laminates, there are very few studies with imperfect interfaces where some simple cases have been handled as the problem has been solved analytically. The results are mostly found for simply supported rectangular cross-ply laminates under distributed load of sinusoidal variation. For perfect laminates, the available results are relatively more but the results based on refined plate theories are found again for simple plate problems only. Actually, the available results for relatively complex cases of laminated plates having different shapes, boundary conditions,

Results and Discussions

66

stacking sequence, loading and some other parameters are based on single layer plate theory and it is mostly with FSDT.

In order to study the performance of the proposed finite element model in complex plate problems having arbitrary features as mentioned above, a number of such available problems are solved by the single layer options (HSDT, FSDT) of the present plate model at the beginning. Some relatively simple plate problems are also solved in a similar manner to have some initial confidence about the plate model. In this case, emphasis has been given to HSDT because it is not only superior than FSDT but also requires C1 continuity of transverse displacement at the element interfaces like RHSDT and this particular aspect has been tested in this process. For the analysis with FSDT, the shear correction factors used are evaluated using the concept of Whitney (1973). Finally, the finite element model with its default option of RHSDT is applied to the analysis of composite and sandwich laminates with perfect as well as imperfect interfaces. In some cases, parametric studies are also made. A large number of new results are generated in this whole process, which should be useful for future references.

The results are presented according to the following two broad categories of plate problems in separate sections:

1. Plates having perfect interfaces

2. Plates having imperfect interfaces.

Again there are three separate divisions for static, vibration and buckling analyses under each of these two broad categories. The results are presented in tabular as well as graphical form where the results are given in the form of deflections and stresses in static analysis, natural frequency of vibration for first few lower modes in free vibration analysis and buckling load parameter in buckling analysis.

Before solving the different numerical examples, the eigen value analysis of the stiffness matrix of a single element having different configurations is carried


67

out and it is found that the element is free from any spurious modes. Also the performance of the element in patch test for uniform bending, shear and twisting is found to be satisfactory, which is shown below.

4.2 PATCH TEST

The patch test for uniform bending, shear and twist is carried out with a square plate (Fig. 4.1a) for three different types of patches (Fig. 4.1b–4.1d) taking thickness ratio (h/a) of the plate as 0.1, 0.001 and 0.00001.

(a)

(b)

(c)

(d)

Figure 4.1 Different Mesh arrangements for patch test

For the case of uniform moment, the plate (Fig. 4.1a) is clamped at x = 0 and subjected to uniformly distributed edge moment (m) per unit length at x = a. The boundary conditions imposed are: u = v = w = θx = θy = γy = 0 at the nodes along fixed edge (x = 0), θy = 0 at the nodes along the free edge (x = a) and θy = γy = 0 at the nodes along the boundaries at y = 0 and y = a. The same boundary


68

conditions are taken for the case of uniform shear where the free edge (x = a) is subjected to uniformly distributed transverse load (q) instead of m. The values of deflection obtained at the central node of the free edge (x = a) for both the cases are presented with the analytical results in Table 4.1.

For the case of uniform twist, the plate is supported (w = 0) at three corners while the fourth corner is free and subjected to a transverse load (P). The values of deflection obtained at the loaded corner are presented with the analytical results in the same table.

The table shows that the results obtained by the present element exactly match with the analytical results for a regular mesh (Fig. 4.1b). Though a little variation between results is found in irregular meshes (Fig. 4.1c and Fig. 4.1d) but it is not so prominent.

Table 4.1 Patch test for uniform moment, shear and twist

Mesh arrangement References h/a b c d

Edge displacement (Eh3w/ma2) due to uniform moment 0.1 5.460 5.485 5.485 0.001 5.460 5.485 5.485

Present

0.00001 5.460 5.485 5.485 Analytical All 5.460 Edge displacement (Ghw/qa) due to uniform shear Present 0.1 141.000 141.372 141.372 Analytical 141.000 Present 0.001 1.410x106 1.413x106 1.413x106 Analytical 1.410x106

Present 0.00001 1.410x1010 1.412x1010 1.412x1010 Analytical 1.410x1010 Displacement (Eh3w/Pa2) at the free corner due to uniform twist

0.1 7.800 7.813 7.813 0.001 7.800 7.813 7.813

Present element

0.00001 7.800 7.813 7.813 Analytical All 7.800


69

4.3 PLATES HAVING PERFECT INTERFACES

The problems on composite and sandwich laminates with perfect interfaces are presented in this section.

4.3.1 Static Analysis

In this section static analysis is carried out primarily with the option of HSDT and FSDT at the beginning and then by RHSDT for the remaining portion. In some cases, results based on FSDT, HSDT and RFSDT are also presented with those obtained by RHSDT to the show the relative performance of the different plate theories.

4.3.1.1 Analysis of laminated composite plates based on HSDT/FSDT

4.3.1.1.1 Simply supported square cross-ply laminated composite plate

The problem of a three ply (0/90/0) square (a = b) laminate (Fig. 4.2) subjected to uniformly distributed load, is studied for different thickness ratios (h/a) ranging from 0.01 to 0.5.

Figure 4.2 A rectangular plate having a mesh size of mxn

The plate is analysed with different mesh divisions (full plate) and the deflection obtained at the plate center is presented with the analytical solution of Reddy (1984) in Table 4.2, which shows that the present results have an excellent agreement with the analytical solutions (Reddy 1984) for both HSDT and FSDT.


70

The convergence of present results with mesh refinement for different cases is also shown in the same table (Table 4.2), which is found to be very good. Ghosh and Dey (1992) have proposed an element where an existing non-conforming plate element based on CPT has been upgraded as per the requirement of HSDT (Reddy 1984). The results obtained by the element (Ghosh and Dey 1992) are also presented in Table 4.2, which clearly shows the failure of their element. Though the individual layers possess different orientations but they have equal thickness and material property (E1/E2 = 25, G12 = G13 =0.5E2, G23 = 0.2E2, ν12 = 0.25 and ν13 = 0.01), which is also used for all other examples in this section (4.3.1.1).

Table 4.2 Deflection (wnd)a at the center of a simply supported square laminated composite plate (0/90/0) under uniformly distributed load

Thickness ratio (h/a) References 0.50 0.25 0.10 0.05 0.02 0.01

HSDT 7.7640 2.9214 1.1113 0.7975 0.7048 0.6913 Present (4x4)b

FSDT 7.6940 2.6764 1.0440 0.7788 0.7017 0.6905 HSDT 7.7660 2.9163 1.0997 0.7861 0.6938 0.6804 Present (6x6) FSDT 7.7088 2.6675 1.0321 0.7674 0.6907 0.6796 HSDT 7.7680 2.9134 1.0956 0.7819 0.6897 0.6763 Present (8x8) FSDT 7.7080 2.6642 1.0278 0.7632 0.6866 0.6755 HSDT 7.7673 2.9111 1.0926 0.7787 0.6865 0.6732 Present (12x12) FSDT 7.7074 2.6617 1.0246 0.7600 0.6835 0.6724 HSDT 7.7670 2.9103 1.0910 0.7763 0.6854 0.6720 Present (16x16) FSDT 7.7068 2.6608 1.0235 0.7588 0.6823 0.6713 HSDT 7.7670 2.9093 1.0910 0.7763 0.6841 0.6708 Present (20x20) FSDT 7.7068 2.6608 1.0235 0.7588 0.6813 0.6707 HSDT 7.7670 2.9093 1.0910 0.7763 0.6841 0.6708 Present (24x24) FSDT 7.7068 2.6608 1.0235 0.7588 0.6813 0.6707 HSDT 7.7671 2.9091 1.0900 0.7760 0.6838 0.6705 Reddy (1984)

FSDT 7.7062 2.6596 1.0219 0.7573 0.6807 0.6697 Ghosh and Dey (1992) HSDT - - 0.9650 0.7572 - 0.6823

a3

2

4100nd wh E

wqa

=

(also for other tables) b Quantities in parenthesis indicate mesh divisions


71

4.3.1.1.2 Simply supported rectangular cross-ply laminated composite plate

A rectangular laminate (Fig. 4.2) subjected to a load having a distribution of ( ) ( )byaxqyxq /sin/sin),( ππ= is considered in this example. The stacking

sequence and boundary conditions are identical to those in the previous example.

The study is made for two different aspect ratios (b/a = 1.0 and 3.0) taking thickness ratio (h/a) as 0.5, 0.1 and 0.01. In all the cases the analysis is done with three different mesh divisions and the deflection and stress components obtained at the important locations are presented with the analytical solution of Reddy (1984) in Table 4.3 and Table 4.4. There is an excellent agreement between the results obtained from the two sources. To assess the improvement of HSDT over FSDT the three dimensional elasticity solution of Pagano (1970) is also presented in these tables. In addition to the above results finite element solution of Panda and Natarajan (1979, FSDT) and Mawenya (1974) are included in Table 4.3 (b/a = 1.0). Some difference is found for the plate having h/a = 0.1. This may be due to inadequate mesh size taken by Panda and Natarajan (1979). Moreover, a large number of Gauss points are required to calculate the load vector accurately for a load having a sinusoidal variation. They (Panda and Natarajan 1979) may have taken a lower order integration scheme with less number of Gauss points for the calculation of the stiffness matrix. Unfortunately, these aspects are not mentioned in their paper (Panda and Natarajan 1979). As the analysis of Mawenya (1974) is based on multi-rotation finite element formulation, his results should be comparable with 3-D elasticity solution of Pagano (1970) rather than FSDT and HSDT. Unfortunately, the results obtained by Mawenya (1974) does not compare well with the 3-D elasticity solution.

In Table 4.4 (b/a = 3.0) the results based on classical plate theory are also included to show the variation of results obtained in different plate theories.


72

Table 4.3 Deflection ( w ) and stresses ( 1σ , 2σ , 4σ , 5σ and 6σ )a of a simply supported square laminate (0/90/0) under sinusoidal load h/a References w 1σ 2σ 4σ 5σ 6σ

HSDT 1.9232 0.7588 0.5079 0.2022 0.1832 0.0499Present (12x12) FSDT 1.7773 0.5132 0.4860 0.1440 0.1575 0.0372HSDT 1.9230 0.7500 0.5080 0.2023 0.1831 0.0499Present (16x16) FSDT 1.7770 0.4430 0.4843 0.1440 0.1569 0.0371HSDT 1.9230 0.7500 0.5080 0.2023 0.1831 0.0499Present (20x20) FSDT 1.7770 0.4430 0.4843 0.1440 0.1569 0.0371HSDT 1.9220 0.7345 - - 0.1832 - Reddy (1984) FSDT 1.7760 0.4369 - - 0.1562 -

0.25

Pagano (1970) 2.0059 0.7550 - - 0.2170 - HSDT 0.7142 0.5885 0.2790 0.2430 0.1007 0.0280Present (12x12) FSDT 0.6711 0.5282 0.2658 0.1614 0.0920 0.0254HSDT 0.7140 0.5806 0.2722 0.2437 0.1015 0.0279Present (16x16) FSDT 0.6700 0.5219 0.2582 0.1623 0.0918 0.0254HSDT 0.7140 0.5806 0.2722 0.2437 0.1015 0.0279Present (20x20) FSDT 0.6700 0.5219 0.2582 0.1623 0.0918 0.0254HSDT 0.7130 0.5684 - - 0.1033 - Reddy (1984) FSDT 0.6690 0.5172 - - 0.0915 -

Pagano (1970) 0.7405 0.5900 - - 0.1230 - Panda and Natarajan (1979)

FSDT 0.6274 0.5320 - - - 0.0250

0.10

Mawenya (1974) 0.8813 0.5420 - - - 0.0292HSDT 0.4360 0.5570 0.1905 0.2367 0.0816 0.0216Present (12x12) FSDT 0.4355 0.5565 0.1904 0.1546 0.0728 0.0216HSDT 0.4350 0.5496 0.1828 0.2401 0.0749 0.0215Present (16x16) FSDT 0.4350 0.5490 0.1825 0.1568 0.0709 0.0202HSDT 0.4350 0.5496 0.1828 0.2401 0.0749 0.0215Present (20x20) FSDT 0.4350 0.5490 0.1825 0.1568 0.0709 0.0202HSDT 0.4340 0.5390 - - 0.0750 - Reddy (1984) FSDT 0.4340 0.5384 - - 0.0703 -

Pagano (1970) 0.4368 0.5520 - 0.0938 0.0214Panda and Natarajan (1979)

FSDT 0.4346 0.5660 - - - 0.2230

0.01

Mawenya (1974) 0.4398 0.5510 - - - 0.0219

a ,2 2

nd a bw w=

; 2

11 1 1 2

, , ,2 2 2

nd nda b h h

qa

σσ σ σ= =

; 2

22 1 2 2

, , ,2 2 6

nd nda b h h

qa

σσ σ σ= =

;

2

44 4 4 2

0, , 0 ,2

nd ndb h

qa

σσ σ σ= =

;2

55 5 5 2

, 0, 0 ,2

nd nda h

qa

σσ σ σ=

= ;

2

66 6 6 2

0, 0, ,2

nd ndh h

qa

σσ σ σ= =

(also used in other tables in section 4.3.1.1)


73

Table 4.4 Deflection ( w ) and stresses ( 1σ , 2σ , 4σ , 5σ and 6σ ) of a simply supported rectangular (b/a=3) cross-ply laminate (0/90/0) under sinusoidal load

h/a References w 1σ 2σ 4σ 5σ 6σ HSDT 2.6558 1.1256 0.1553 0.2722 0.0217 0.0262 Present (8x8) FSDT 2.3712 0.6738 0.1518 0.1823 0.0182 0.0190 HSDT 2.6460 1.0820 0.1277 0.2723 0.0291 0.0264 Present (12x12) FSDT 2.3683 0.6405 0.1293 0.1852 0.0249 0.0196 HSDT 2.6437 1.0650 0.1209 0.2723 0.0320 0.0264 Present (16x16) FSDT 2.3650 0.6285 0.1137 0.1863 0.0274 0.0199 HSDT 2.6411 1.0356 0.1028 0.2724 0.0348 0.0263 Reddy (1984) FSDT 2.3626 0.6130 0.0934 0.1879 0.0308 0.0205

0.25

Pagano (1970) 2.8200 1.1000 0.1190 0.3870 0.0334 0.0281 HSDT 0.8762 0.7742 0.0981 0.2832 0.0045 0.0118 Present (8x8) FSDT 0.8175 0.6905 0.0986 0.1797 0.0017 0.0103 HSDT 0.8674 0.7321 0.0677 0.2847 0.0061 0.0117 Present (12x12) FSDT 0.8175 0.6531 0.0738 0.1842 0.0074 0.0103 HSDT 0.8649 0.7164 0.0383 0.2851 0.0106 0.0117 Present (16x16) FSDT 0.8013 0.6398 0.0367 0.1861 0.0110 0.0103 HSDT 0.8622 0.6924 0.0398 0.2859 0.0170 0.0115 Reddy (1984) FSDT 0.8030 0.6214 0.0375 0.1894 0.0159 0.0105

0.10


0.05


0.01

Pagano (1970) 0.5080 0.6240 0.0253 0.4390 0.0108 0.0083 CLPTa (Allen 1969) 0.5030 0.6230 0.0252 - - -

a Classical Lamination Plate Theory


74

4.3.1.1.3 Simply supported un-symmetric angle-ply laminate

The problem of an angle ply (45/-45/...) square laminate is studied by taking number of layers as two and ten where thickness ratio is 0.1 in both cases. The plate (Fig. 4.2) is simply supported and subjected to uniformly distributed load. The deflections obtained along y = b/2 is presented with that of Reddy (1989) in Table 4.5. The agreement between the results is good in all the cases.

Table 4.5 Deflection (wnd) of a simply supported un-symmetric square angle ply laminate (45/-45/...) under uniformly distributed loading

x/a Number of layers

References 0.500 0.375 0.250 0.125

HSDT 1.237 1.158 0.917 0.519 Present (4x4) FSDT 1.274 1.194 0.946 0.537 HSDT 1.242 1.161 0.917 0.520 Present (8x8) FSDT 1.279 1.196 0.946 0.538 HSDT 1.242 1.161 0.917 0.520 Present (12x12) FSDT 1.279 1.196 0.946 0.538

2

Reddy (1989) FSDT 1.281 1.196 0.948 0.545 HSDT 0.632 0.593 0.476 0.276 Present 4x4) FSDT 0.631 0.593 0.475 0.276 HSDT 0.632 0.593 0.475 0.276 Present (8x8) FSDT 0.631 0.592 0.475 0.276 HSDT 0.632 0.593 0.474 0.275 Present (12x12) FSDT 0.631 0.592 0.474 0.275

10

Reddy (1989) FSDT 0.631 0.591 0.474 0.274

4.3.1.1.4 Skew cross-ply laminate simply supported at the edges

The problem of a skew laminate (0/90/0) as shown in Figure 4.3 is studied for different skew angles (300, 450, and 600) taking h/a as 0.01 and 0.1. The plate is subjected to uniformly distributed load and layers of the plate are of equal thickness. The plate is modeled by the proposed element using the mesh arrangement as shown in Fig. 4.3, which shows that the elements possess a shape other than right angle triangle. As the sides BC and AD are inclined to global axis system (x-y), necessary transformation is made to express the degrees of freedom of the nodes on these two sides along x′-y′ (Fig. 4.3). The transformation is essential for simply supported boundaries, which is done in element level.


75

Figure 4.3 A skew plate having a mesh of mxm

In all the cases the analysis is performed with mesh size: 16x16. The deflections and principal stresses obtained in the present analysis at the plate center are presented in Table 4.6 with those of Kabir (1995) and Haldar (2002) based on FSDT.

Table 4.6 Deflection (wnd) and normal stresses ( nd1σ and nd

2σ ) at the center of a simply supported cross-ply (0/90/0) skew laminate under uniform load

h/a Skew angle (α)

References wnd nd1σ nd

2σ

HSDT 0.5452 0.6444 0.2629 Present FSDT 0.5447 0.6439 0.2626

300

Haldar (2002) FSDT 0.5458 0.6348 - HSDT 0.3631 0.4421 0.3007 Present FSDT 0.3628 0.4417 0.3006

450


Haldar (2002) FSDT 0.1455 0.1956 -

0.01

600

Kabir (1995) FSDT 0.1520 0.2322 - Present HSDT 0.8621 0.6617 0.3709 FSDT 0.8199 0.6098 0.3527

300

Haldar (2002) FSDT 0.8193 0.6005 - Present HSDT 0.5707 0.4543 0.3786 FSDT 0.5512 0.4143 0.3684

450


Haldar (2002) FSDT 0.2455 0.1758 -

0.10

600

Kabir (1995) FSDT 0.2600 0.0852 -


76

Kabir (1995) has used an element developed by him for the analysis where only the transverse displacement is restrained along the simply supported edges. Haldar (2002) has analysed the plate using the high precision element of Sengupta (1991) where he (Haldar 2002) has made all necessary treatments for the simply supported edges as done in the present analysis. Table 4.6 shows that the present results agreed better with those of Haldar (2002) compared to Kabir (1995) as expected.

4.3.1.2 Analysis of composites and sandwich laminates based on RHSDT

In the previous section, the results based on HSDT show definite improvement over that of FSDT, but it requires further improvement specifically for the stresses evaluation to have a comparable value with 3-D elasticity solution (Pagano 1970). This improvement is shown in this section by solving some problems of composites and sandwich laminates with the proposed element having the option of RHSDT. For the purpose of comparison, a separate computer program (COMPAG) based on 3-D elasticity solution (Pagano 1970) is developed to generate numerical results as required.

4.3.1.2.1 Simply supported cross-ply square laminate

The problem of a cross-ply (0/90/90/0) laminated plate (Fig. 4.2, a = b) subjected

to a distributed load of intensity ( ) ( )0 sin / sin /=q q x a y bπ π is taken in this

example. In this laminate, each ply has same thickness (h/4) and material properties (E1/E2 = 25, G12 = G13 = 0.5E2, G23 = 0.2E2, ν12 = 0.25). The full plate is analysed with the proposed element with mesh size (Fig. 4.2) of 4x4, 8×8, 12×12, 16×16, 20x20, 24×24 and 28×28 taking thickness ratio (h/a) of 0.1 and

0.2. The non-dimensional values of central deflection ( )3 42 0100 /w wE h q a=

obtained in the present analysis are plotted with the 3-D elasticity solution (Pagano 1970) in Figure 4.4. It is quite encouraging that the present results have


77

excellent agreement with the 3-D elasticity solution. The figure (Fig. 4.4) also shows that the convergence of the results with mesh refinement is very good.

4 8 12 16 20 24 280.720

0.725

0.730

0.735

0.740

0.745

0.750

0.755

0.760 Thickness ratio (h/a) = 0.10

Present 3-D elasticity (Pagano 1970)

Cen

tral d

efle

ctio

n

Mesh division

4 8 12 16 20 24 281.30

1.35

1.40

1.45

1.50

1.55

1.60Thickness ratio (h/a) = 0.20

Present 3-D elasticity (Pagano 1970)

Cen

tral d

efle

ctio

n

Mesh division

Figure 4.4 Convergence of central deflection of a composite plate with mesh division

Now the through thickness variation of non-dimensional stress components at some important points obtained in the present analysis (mesh size: 20x20, h/a = 0.2) are plotted with the 3-D elasticity results (Pagano 1970) using COMPAG in

Figures 4.5 – 4.7. The variation of in-plane normal stress ( )2 20/x xh q aσ σ= at the

plate center obtained by RHSDT, RFSDT, HSDT and FSDT are presented in Figure 4.5, which shows that the 3-D elasticity solution is closest to RHSDT. The potential of the proposed element based on RHSDT is clearly reflected by the order of accuracy exhibited in the prediction of stress distribution. Based on the above observations, it has been decided that proposed element will be used with its default option (RHSDT) in the subsequent analysis. The in-plane shear stress

( )2 20/xy xyh q aτ τ= at a corner of the plate and transverse shear stress

( )0/xz xzh q aτ τ= at the center of a side (x = 0, y = b/2: Fig. 4.2) obtained by

RHSDT are plotted with the 3-D elasticity solution in Figures 4.6 - 4.7 where


78

excellent agreement between the results are found like the other stress components.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.00 0.15 0.30 0.45 0.60

Composite plate (0/90/90/0)h/a=0.20

3-D Elasticity (Pagano) Present (RHSDT) Present (RFSDT) Present (HSDT) Present (FSDT)

Normalised in-plane normal stress

Nor

mal

ised

dep

th

Figure 4.5 Variation of in-plane normal stress across the plate thickness

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

-0.045 -0.030 -0.015 0.000 0.015 0.030 0.045


3-D Elasticity Present (RHSDT)

Normalised in-plane shear stress

Nor

mal

ised

dep

th

Figure 4.6 Variation of in-plane shear stress across the plate thickness


79

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.00 0.05 0.10 0.15 0.20 0.25 0.30


3-D Elasticity (Pagano) Present (RHSDT)

Normalised transverse Shear stress

Nor

mal

ised

dep

th

Figure 4.7 Variation of transverse shear stress across the plate thickness

4.3.1.2.2 Cylindrical bending of a cross-ply laminate

This plate problem has been chosen by Cho and Parmerter (1993) to show the performance of their plate theory (RHSDT). The problem has been solved analytically and some numerical results for this laminate (0/90/90/0) having thickness ratios (h/a) of 0.01, 0.02 and 0.04 have been given. The same problem is taken in the present study to assess the performance of the proposed element. For this purpose, a rectangular plate (Figure 4.2, b/a = 3), simply supported at the longer sides (x = 0 and x = a) is taken. The plate is subjected to a distributed load

of intensity ( )0 sin /q q x aπ= . All the layers of the laminate possess same

thickness and material properties (E1 = 172 GPa, E2 = 6.9 GPa, G12 = G13 = 3.4 GPa, G23 = 1.4 GPa, ν12 = ν13 =ν23 =0.25), which is maintained for all the subsequent examples of this section (4.3.1.2). The plate is analysed with the present element taking a number of mesh sizes (4×4, 6×6, 8×8, 12×12, 16×16, 20×20 and 24×24). The non-dimensional deflection values obtained at the plate


80

center are presented in Table 4.7 along with the analytical solution of Cho and Parmerter (1993) and 3-D elasticity solution of Pagano (1970).

Table 4.7 Central deflection (w*)a of a simply supported cross-ply (0/90/90/0) laminate under cylindrical bending due to sinusoidal loading

Present element (RHSDT) H/a 4x4b 6x6 8x8 12x12 16x16 20x20 24x24

Pagano (1970)

Cho and Parmerter

(1993) 0.01 1.1555 1.0843 1.0747 1.0511 1.0450 1.0414 1.0414 1.0394 1.0348 0.02 1.2741 1.1926 1.1815 1.1555 1.1489 1.1449 1.1449 1.1438 1.1387 0.04 1.7321 1.6230 1.6020 1.5673 1.5580 1.5550 1.5550 1.5571 1.5504

a *

0

ww

w= , where 0

0 4 1 211 11

12( / 2 ) ( 7 )

q hw

h a Q Qπ=

+ is the deflection based on CLPT

b Indicates mesh division

Table 4.7 shows that the convergence of the present results with mesh refinement is very good. It also shows that the present results are in excellent agreement with those of Cho and Parmerter (1993) and Pagano (1970).

The through thickness variation of in-plane displacement component u at x = 0, y = b/2 is presented in Table 4.8 in a similar manner as presented by Cho and Parmerter (1993). The values of u at z = 0.1h, 0.2h, 0.25h, 0.3h, 0.4h and 0.5h obtained in the present analysis (mesh size: 20×20) are presented with those of Cho and Parmerter (1993) and Pagano (1970). The results based on Classical Lamination Plate Theory (CLPT) are also included in Table 4.8, which shows that the present results are sufficiently close to those of Cho and Parmerter (1993) and Pagano (1970).

4.3.1.2.3 Simply supported cross-ply square laminate

In order to show the performance of different plate theories a cross-ply (0/90/0) laminate (Figure 4.2, a = b) subjected to a distributed load of intensity

( ) ( )0 sin / sin /=q q x a y bπ π is considered in this example. The full plate is

analysed with the present element taking mesh size of 4×4, 6×6, 8×8, 12×12,


81

16x16 and 20×20. The analysis is carried out with all the options (RHSDT, RFSDT, HSDT and FSDT) of the proposed element taking thickness ratio of 0.25, 0.1, 0.05, 0.02 and 0.01. The values of central deflection

( )3 42 0100 /cw wE h q a= obtained are presented with those of Pagano (1970) in

Table 4.9. Table 4.9 shows that the agreement between the elasticity solution (Pagano 1970) and present results is best when RHSDT is used while it is worst for FSDT.

Table 4.8 In-plane displacement (u*)a of a simply supported cross-ply (0/90/90/0) laminate under cylindrical bending due to distributed load of sinusoidal variation

h/a z/h Present element (RHSDT)

Cho and Parmerter (1993)

Pagano (1970)

CLPT

0.10 -0.006235 -0.006195 -0.006235 -0.006285 0.20 -0.012471 -0.012401 -0.012471 -0.012566 0.25 -0.015589 -0.015513 -0.015589 -0.015708 0.30 -0.018004 -0.018714 -0.018000 -0.018850 0.40 -0.025262 -0.025142 -0.025252 -0.025133

0.01

0.50 -0.031761 -0.031616 -0.031752 -0.031416 0.10 -0.012008 -0.011864 -0.012011 -0.012566 0.20 -0.024037 -0.023820 -0.024031 -0.025133 0.25 -0.030075 -0.029862 -0.030047 -0.031416 0.30 -0.036798 -0.036613 -0.036798 -0.037699 0.40 -0.050560 -0.050336 -0.050531 -0.050266

0.02

0.50 -0.064721 -0.064430 -0.064662 -0.062832 0.10 -0.020392 -0.019558 -0.020442 -0.025133 0.20 -0.040931 -0.039850 -0.040951 -0.050266 0.25 -0.051331 -0.050502 -0.051251 -0.062832 0.30 -0.067471 -0.066783 -0.067342 -0.075398 0.40 -0.101511 -0.101092 -0.101253 -0.100531

0.04

0.50 -0.138722 -0.138341 -0.138372 -0.155664

a *

0

uu

w= , where 0

0 4 1 211 11

12( / 2 ) ( 7 )

q hw

h a Q Qπ=

+ is the deflection based on CLPT.

4.3.1.2.4 Un-symmetric angle-ply square laminate

The problem of a square laminate (Figure 4.2, a = b) subjected to uniformly distributed load of intensity q0 is studied for ply arrangement of –45/45 (two layer) and –45/45/-45/45 (four layer) taking h/a = 0.1 and 0.25. The whole plate is


82

analysed with the proposed element using mesh size of 12×12, 16×16 and 20×20. The values of central deflection cw , in-plane normal stress xσ (x = a/2, y = b/2

and z = h/2) and transverse shear stress xzτ (x = 0, y = b/2 and z = 0) obtained in

the present analysis are presented in Table 4.10.

Table 4.9 Central deflection ( cw ) of a simply supported cross-ply (0/90/0) square laminate under sinusoidal loading

Thickness ratio (h/a) References 0.25 0.10 0.05 0.02 0.01

Present Element (4x4)a RHSDT 1.9592 0.7714 0.5175 0.4612 0.4498 Present Element (6x6) RHSDT 1.9541 0.7616 0.5140 0.4526 0.4419 Present Element (8x8) RHSDT 1.9522 0.7583 0.5088 0.4495 0.4389 Present Element (12x12) RHSDT 1.9507 0.7560 0.5072 0.4471 0.4367 Present Element (16x16) RHSDT 1.9502 0.7522 0.5066 0.4462 0.4358 Present Element (20x20) RHSDT 1.9502 0.7522 0.5066 0.4462 0.4358 Present Element (20x20) RFSDT 1.8930 0.7202 0.5074 0.4446 0.4354 Present Element (20x20) HSDT 1.9230 0.7140 0.5052 0.4440 0.4350 Present Element (20x20) FSDT 1.7770 0.6740 0.4932 0.4421 0.4350 3-D Elasticity (Pagano 1970) 2.0059 0.7530 0.5164 0.4451 0.4347

a Indicates mesh size (applicable in all subsequent tables)

The non-dimensional representation of deflection and stresses are identical to those in the previous example and will be followed in the subsequent examples unless specified. Some of the results obtained by the present element based on RHSDT are compared with those of Topdar (2001) in Table 4.10, which shows that the agreement between them is very good. The results obtained by Topdar (2001) are based on an upgraded rectangular conforming element of Bogner et al. (1965) to implement the RHSDT.


83

Table 4.10 Deflection ( cw ) and stresses ( xσ , xzτ ) of a simply supported un-symmetric angle-ply (-45/45….) square laminate under uniform loading

Number of layers

h/a References cw xσ xzτ

Present Element (12x12) RHSDT 1.2516 0.3866 0.2004 Present Element (16x16) RHSDT 1.2531 0.3882 0.1984 Present Element (20x20) RHSDT 1.2531 0.3882 0.1984 Present Element (20x20) RFSDT 1.2133 0.3752 0.1362 Present Element (20x20) HSDT 1.2400 0.3622 0.4418 Present Element (20x20) FSDT 1.2410 0.3621 0.4419

0.10

Topdar (2001) RHSDT 1.2590 0.3860 0.2003 Present Element (12x12) RHSDT 2.0107 0.4590 0.1456 Present Element (16x16) RHSDT 2.0114 0.4608 0.1453 Present Element (20x20) RHSDT 2.0114 0.4608 0.1453 Present Element (20x20) RFSDT 1.8363 0.4032 0.1071 Present Element (20x20) HSDT 2.3346 0.4335 0.4106

2

0.25

Present Element (20x20) FSDT 2.6033 0.3472 0.4005 Present Element (12x12) RHSDT 0.8562 0.4282 0.2193 Present Element (16x16) RHSDT 0.8581 0.4289 0.2183 Present Element (20x20) RHSDT 0.8581 0.4289 0.2183 Present Element (20x20) RFSDT 0.8066 0.3680 0.1305 Present Element (20x20) HSDT 0.7015 0.2154 0.4810

0.10

Present Element (20x20) FSDT 0.6917 0.1968 0.3926 Present Element (12x12) RHSDT 2.2271 0.9239 0.1249 Present Element (16x16) RHSDT 2.2126 0.9187 0.1249 Present Element (20x20) RHSDT 2.2126 0.9187 0.1249 Present Element (20x20) RFSDT 2.1088 0.5700 0.0767 Present Element (20x20) HSDT 2.0311 0.4955 0.1811

4

0.25

Present Element (20x20) FSDT 2.0162 0.1929 0.1711

4.3.1.2.5 Skew cross-ply laminated plate simply supported at the edges

The problem of a cross-ply (0/90/0) skew laminate (Fig. 4.3) under uniformly distributed load of intensity q0 is studied in this example for skew angle α (Fig. 4.3) of 300, 450 and 600 taking h/a = 0.1 and 0.2. The whole plate is analysed with the proposed element using mesh size of 12x12, 16×16 and 20×20. The transformation for the degrees of freedom along the skew edge is carried out in a similar manner as explained in section 4.3.1.1.4. The values of deflection cw and in-plane normal stress xσ at the plate center and transverse shear stress


84

( )0/yz yzh q aτ τ= at the center of left edge AB obtained in the present analysis are presented in Table 4.11. For the purpose of validation, the results obtained with the FSDT option of the proposed element (mesh size: 20×20) are included in Table 4.11 along with some available finite element results (Haldar 2002, Kabir 1995) based on FSDT. Table 4.11 Deflection ( cw ) and stresses ( xσ , yzτ ) of a simply supported cross-ply (0/90/0) skew laminate under uniform load

h/a Skew Angle

References cw xσ yzτ

Present Element (12x12) RHSDT 0.8820 0.7188 0.2155 Present Element (16x16) RHSDT 0.8814 0.7125 0.2145 Present Element (20x20) RHSDT 0.8814 0.7125 0.2145 Present Element (20x20) FSDT 0.8201 0.6291 0.3026

300

Haldar (2002) FSDT 0.8193 - - Present Element (12x12) RHSDT 0.5738 0.4881 0.2125 Present Element (16x16) RHSDT 0.5742 0.4861 0.2119 Present Element (20x20) RHSDT 0.5742 0.4861 0.2119 Present Element (20x20) FSDT 0.5513 0.4234 0.3047

450

Haldar (2002) FSDT 0.5507 - - Present Element (12x12) RHSDT 0.2471 0.2194 0.1628 Present Element (16x16) RHSDT 0.2481 0.2201 0.1623 Present Element (20x20) RHSDT 0.2481 0.2201 0.1623 Present Element (20x20) FSDT 0.2455 0.1831 0.2467 Haldar (2002) FSDT 0.2455 - -

0.10

600

Kabir (1995) FSDT 0.2600 - - Present Element (12x12) RHSDT 1.6216 0.8598 0.2339 Present Element (16x16) RHSDT 1.6811 0.8549 0.2331 Present Element (20x20) RHSDT 1.6811 0.8549 0.2331

300

Present Element (20x20) FSDT 1.5604 0.5491 0.3345 Present Element (12x12) RHSDT 1.1756 0.5943 0.2017 Present Element (16x16) RHSDT 1.1790 0.5920 0.2013 Present Element (20x20) RHSDT 1.1790 0.5920 0.2013

450

Present Element (20x20) FSDT 1.0585 0.3640 0.3118 Present Element (12x12) RHSDT 0.5192 0.2904 0.1384 Present Element (16x16) RHSDT 0.5196 0.2906 0.1383 Present Element (20x20) RHSDT 0.5196 0.2906 0.1383

0.20

600

Present Element (20x20) FSDT 0.5158 0.1581 0.2368


85

4.3.1.2.6 Square sandwich plate (f/c/f) subjected to uniform load

This problem of simply supported sandwich plate (Fig. 4.2, a = b = 254 mm) having single core (f/c/f) construction (Fig. 4.8) has been studied by Khatua and Cheung (1973a) using a finite element model where the unknowns were taken at both the interfaces. Thus the resulting model of Khatua and Cheung (1973a) may be considered as a specific case of the discrete layer plate theory (Robbins and Reddy 1993).

The total thickness (h) of the plate is 20.4724 mm where thickness of each face layer is 0.7112mm. The plate subjected to uniformly distributed load of intensity 0.00688 N/mm2. The material properties used for the face layers are E11 = E22 = 68.8 GPa, G13 = G23 = 27.52 GPa and ν12 = 0.30. For the core, these are E11 = E22 = 6.88×10-12 GPa (a very small value), G13 = G23 = 0.2064 GPa and ν12 = 0.30. The problem has also been solved with different material properties keeping other parameters unchanged, which is referred as Case II where the earlier one is Case I. In Case II, the material properties of the face layers are: E11 = 68.8 GPa, E22 = 27.52 GPa, G12 = 12.9 GPa, G13 = G23 = 27.52 Gpa and ν12 = 0.30 whereas those of core are: E11 = 68.8×10-12 GPa, E22 = 27.52E-13 GPa, G12 = 12.9E-13 GPa, G13 = 0.2064 Gpa, G23 = 0.0826 Gpa and ν12 = 0.30. In the last case (Case III), the boundary condition at the four edges is changed to clamped where the other parameters identical to those of Case I except the value of transverse shear rigidity of the core, which is G13 = G23 = 0.3131 Gpa.

Figure 4.8 Cross-section of the single core (f/c/f) sandwich plate


86

A quarter of the plate is analysed with the proposed element with the options of RFSDT, HSDT and FSDT (mesh size: 8×8). The values of deflection w (x = a/2 and y = b/2), in-plane normal stress σx (x = a/2, y = b/2 and z = h/2) and transverse shear stress τxz (x = 0 and y = b/2) at the upper interface obtained in the present analysis are presented in Table 4.12. It shows that FSDT and HSDT have suffered much more in the present problem of sandwich plate as expected. The available finite element results (Khatua and Cheung 1973a) and analytical results (Azar 1968, Lockwood-Taylor 1968 and Foile 1970) are included in Table 4.12, which shows that the present results based on RHSDT/RFSDT are quite close to the published results. Table 4.12 Deflections (w) and stresses (σx, τxz) of a single core (f/c/f) square sandwich plate under uniformly distributed load

τxz (N/mm2)

Case References 100xw (mm.)

σx (N/mm2)

Upper Lower Present Element (4x4) RHSDT 1.8731 1.5998 0.0282 0.0282 Present Element (6x6) RHSDT 1.8734 1.5864 0.0283 0.0283 Present Element (8x8) RHSDT 1.8750 1.5817 0.0284 0.0284 Present Element (8x8) RFSDT 1.8510 1.5793 0.0285 0.0285 Present Element (8x8) HSDT 1.7233 1.5766 0.5730 0.0045 Present Element (8x8) FSDT 1.1899 1.5521 0.4406 0.0034 Khatua and Cheung (1973a) 1.8697 - - -

I

Series solution (Azar 1968) 1.8783 - - - Present Element (4x4) RHSDT 3.1118 2.5397 0.0324 0.0324 Present Element (6x6) RHSDT 3.1073 2.5077 0.0328 0.0328 Present Element (8x8) RHSDT 3.1056 2.4962 0.0329 0.0329 Present Element (8x8) RFSDT 3.0676 2.3294 0.0305 0.0305 Present Element (8x8) HSDT 2.9112 2.4551 0.3499 0.0056 Present Element (8x8) FSDT 2.1794 2.3294 0.4736 0.0076 Khatua and Cheung (1973a) 3.0810 - - -

II

Series solution (1968) 3.1140 - - - Present Element (4x4) RHSDT 0.9436 0.8934 0.0437 0.0437 Present Element (6x6) RHSDT 0.9483 0.8928 0.0500 0.0500 Present Element (8x8) RHSDT 0.9535 0.8916 0.0558 0.0558 Present Element (8x8) RFSDT 0.8951 0.8333 0.0460 0.0460 Present Element (8x8) HSDT 0.7779 0.8076 0.8976 0.0106 Present Element (8x8) FSDT 0.4466 0.7718 0.5086 0.0060 Khatua and Cheung (1973a) 0.8707 - - - Lockwood and Taylor (1968) 0.8865 - - -

III

Foile (1970) 0.8814 - - -


87

4.3.1.2.7 Double core (f/c/f/c/f) square sandwich plate subjected to uniform load

This plate has a double core sandwich construction (Figure 4.9) of total thickness h = 21.844 mm where thickness (hf) of each face is 0.508 mm. The plate size, material properties, loading and boundary conditions are identical to those used in Case II of the previous example but the full plate is analysed.

Figure 4.9 Cross-section of the double core (f/c/f/c/f) sandwich plate

The values of central deflection w, in-plane normal stress σx (x = a/2, y = b/2 and z = h/2) and transverse shear stress τxz (x = 0 and y = b/2) at the upper interface of the central stiff layer obtained in the present analysis are presented in Table 4.13. The deterioration in the performance of FSDT and HSDT is magnified further in this problem of multi-core sandwich plate. The FE results of Khatua and Cheung (1973a) and series solution of Azar (1968) included in Table 4.13 show that the performance of RHSDT and RFSDT is quite satisfactory.

Table 4.13 Deflections (w) and stresses (σx, τxz) of a simply supported double core (f/c/f/c/f) square sandwich plate under uniformly distributed load

τxz (N/mm2)

References 100xw (mm.)

σx (N/mm2)

Upper Lower Present Element (8x8) RHSDT 3.4061 2.9776 0.0385 0.0385 Present Element (12x12) RHSDT 3.4006 2.9639 0.0387 0.0387 Present Element (16x16) RHSDT 3.3973 2.9588 0.0388 0.0388 Present Element (16x16) RFSDT 3.3567 2.9461 0.0367 0.0367 Present Element (16x16) HSDT 1.7869 2.0715 0.0056 0.7491 Present Element (16x16) FSDT 1.7720 2.0645 0.0035 0.4633 Khatua and Cheung (1973a) 3.3833 - - - Series solution (Azar 1968) 3.3731 - - -


88

4.3.1.2.8 Rectangular laminated sandwich plate simply supported at the edges

The plate (0/90/C/0/90) is subjected to a distributed load of intensity

( ) ( )0 sin / sin /=q q x a y bπ π . It has a total thickness of h where the thickness of

the core is 0.8h and that of each ply in the face sheets is 0.05h. The plate is analysed with the proposed element taking aspect ratio b/a = 1 and 2, and thickness ratio h/a = 0.1 and 0.2 using mesh sizes (Fig. 4.2) of 4×4, 6×6, 8×8, 12×12, 16×16, 20×20, 24×24 and 28×28. The non-dimensional values of

deflection ( )3 40100 /w wEh q a= at the plate center obtained in the present

analysis are presented with those obtained from the 3-D elasticity solution (Pagano 1970) using COMPAG in Table 4.14, which shows that the agreement between the results is excellent. The table also shows that the convergence of the results with mesh refinement is very good.

Table 4.14 Central deflection ( w ) of a simply supported rectangular sandwich plate with laminated facings (0/90/C/0/90) under sinusoidal load

Aspect ratio (b/a) 1.0 2.0

References

h/a = 0.10 h/a = 0.20 h/a = 0.10 h/a=0.20 Present element (4x4)1 1.7465 4.2571 3.2843 7.4576 Present element (6x6) 1.7340 4.2498 3.2275 7.3963 Present element (8x8) 1.7297 4.2464 3.2095 7.3776 Present element (12x12) 1.7267 4.2437 3.1973 7.3653 Present element (16x16) 1.7257 4.2427 3.1931 7.3613 Present element (20x20) 1.7252 4.2422 3.1911 7.3593 Present element (24x24) 1.7249 4.2420 3.1902 7.3585 Present element (28x28)2 1.7249 4.2420 3.1902 7.3585 Present element (20x20)3 1.7131 4.1986 3.1700 7.2853 Present element (20x20)4 1.6738 3.9798 3.0957 6.9253 Present element (20x20)5 1.3334 2.6282 2.5147 4.7014 Elasticity solution (Pagano 1970)

1.7272 4.2447 3.1944 7.3707

1Quantities in the parenthesis indicate mesh division 2Refined Higher Order Shear Deformation Theory (RHSDT) 3Refined First Order Shear Deformation Theory (RFSDT) 4Higher Order Shear Deformation Theory (HSDT) 5First Order Shear Deformation Theory (FSDT)


89

In order to show the performance of RFSDT, HSDT and FSDT, the analysis has also been carried out with these options of the present element with a mesh size of 20x20 and the results obtained are included in Table 4.14, which shows the relative position of the different plate theories. As the performance of RHSDT is found to be the most superior, it has been used in the subsequent analyses in this section.

Now the through thickness variations of normalized in-plane normal stress ( )2 2

0/x xh q aσ σ= at the plate center, in-plane shear stress ( )2 20/xy xyh q aτ τ= at a

corner of the plate and transverse shear stress ( )0/xz xzh q aτ τ= at x = 0 and y = b/2 (Fig. 4.2) obtained in the present analysis (mesh size: 20x20, b/a = 1 and h/a = 0.2) are plotted with the three dimensional elasticity solution in Fig. 4.10 – 4.12. The figures show that the present results are in excellent agreement with the elasticity solution. For evaluation of the transverse shear stresses the present results are calculated from the equilibrium equations.

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Sandwich plate (0/90/C/0/90) h/a=0.20

Elasticity Present (RHSDT)

Normalised in-plane normal Stress

Nor

mal

ised

dep

th

Figure 4.10 Variation of in-plane normal stress across the depth of a sandwich plate


90

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

-0.06 -0.04 -0.02 0.00 0.02 0.04 0.06

Sandwich plate (0/90/C/0/90) h/a=0.20

Elasticity Present (RHSDT)

Normalised in-plane shear Stress

Nor

mal

ised

dep

th

Figure 4.11 Variation of in-plane shear stress across the depth of a sandwich plate

-0.50

-0.25

0.00

0.25

0.50

0.00 0.05 0.10 0.15 0.20

Sandwich plate(0/90/C/0/90) h/a = 0.20

Elasticity ( Pagano ) Present ( RHSDT )

Normalised Transverse Shear stress(τxz)

Norm

alis

ed d

epth

Figure 4.12 Variation of transverse shear stress across the depth of a sandwich plate


91

The material properties of the core are taken as E1 = E2 = 0.04E, G12 = 0.016E, G13 = G23 = 0.06E and ν12 = 0.25 while those of an orthotropic ply of the laminated face sheets are E1 = 25E, E2 = E, G12 = G13 = 0.5E, G23 = 0.2E and ν12 = 0.25. The same material properties are used for all the subsequent examples under this section (4.3.1.2).

4.3.1.2.9 Double core square sandwich plate with laminated stiff sheets

The square sandwich plate (0/90/0/C/0/90/0/C/0/90/0) is simply supported at its four sides and subjected to a distributed load of sinusoidal variation as in the previous example. In the present problem, the thickness of each core is taken as 0.4325h, while that of each play in the laminated stiff sheets is 0.015h. Taking h/a = 0.05, 0.1 and 0.2, the plate is analysed with the proposed element (RHSDT) with the mesh sizes: 16x16, 20x20 and 24x24 for the full plate. The non-dimensional values of central deflection w , in-plane normal stress xσ at x= a/2, y

= b/2, z = h/2 (Fig. 3), in-plane shear stress xyτ at x= 0, y =0, z=h/2, transverse

shear stress xzτ at x= 0, y =b/2, z = 0.0225h and ( )0/yz yzh q aτ τ= at x= a/2, y =0,

z = 0 obtained in the present analysis are presented with those obtained from the three dimensional elasticity solution (Pagano 1970) using COMPAG in Table 4.15. It shows that the present results are sufficiently close to the elasticity solution. Table 4.15 Deflections and stresses of a simply supported double core square sandwich plate with laminated stiff sheets (0/90/0/C/0/90/0/C/0/90/0)

h/a References w xσ xyτ xzτ yzτ

Present element (16x16) 5.0001 2.0567 -0.0970 0.1842 0.1452 Present element (20x20) 4.9965 2.0466 -0.0968 0.1855 0.1476 Present element (24x24) 4.9943 2.0409 -0.0967 0.1866 0.1487

0.20

Elasticity solution (Pagano 1970) 5.0469 2.0238 -0.0956 0.1864 0.1499 Present element (16x16) 2.5624 2.0846 -0.0892 0.1998 0.1332 Present element (20x20) 2.5594 2.0749 -0.0891 0.2005 0.1349 Present element (24x24) 2.5589 2.0697 -0.0890 0.2023 0.1359

0.10

Elasticity solution (Pagano 1970) 2.5700 2.0528 -0.0884 0.2014 0.1350 Present element (16x16) 1.9354 2.1445 -0.0864 0.2084 0.1273 Present element (20x20) 1.9325 2.1339 -0.0863 0.2090 0.1286 Present element (24x24) 1.9320 2.1284 -0.0862 0.2106 0.1293

0.05

Elasticity solution (Pagano 1970) 1.9348 2.1137 -0.0858 0.2101 0.1263


92

4.3.1.2.10 Sandwich plate having angle-ply laminated stiff sheets at the two faces

The square sandwich plate (θ/θ+90/C/θ/θ+90) is simply supported at its four edges and subjected to uniformly distributed load of intensity q0. The thickness distribution for the core and the individual ply in the laminated sheets is identical to that of the plate taken in the section 4.3.1.2.8. The whole plate is analysed with mesh size of 24x24 taking h/a = 0.05, 0.1 and 0.2, and θ = 00, 300 and 450. The values of w at the plate center, xσ at x= a/2, y = b/2, z=h/2 (Fig. 3), xyτ at x= 0, y

=0, z=h/2 and xzτ at x= 0, y =b/2, z=0.4h obtained in the present analysis

(RHSDT) are presented in Table 4.16. The present results corresponding to θ = 00 are compared with the 3-D elasticity results (Pagano 1970) using COMPAG, which gives validation of the present results up to a certain extent.

Table 4.16 Deflections and stresses of a simply supported square sandwich plate with angle-ply laminated faces (θ/θ+90/C/θ/θ+90) under uniform load

Fiber angle (θ)

h/a References w xσ xyτ xzτ

Present Element 6.2978 1.7792 0.1436 0.3482 0.20 Elasticity solution (Pagano 1970) 6.4208 1.7176 0.1442 0.3481 Present Element 2.6296 1.6249 0.1022 0.3612 0.10 Elasticity solution (Pagano 1970) 2.6384 1.6214 0.0975 0.3569 Present Element 1.7114 1.5988 0.0864 0.3732

00

0.05 Elasticity solution (Pagano 1970) 1.7116 1.5931 0.0834 0.3628

0.20 Present Element 5.9463 1.1165 1.0810 0.3762 0.10 Present Element 2.2237 0.8882 0.6672 0.3659

300

0.05 Present Element 1.2362 0.7653 0.5481 0.3603 0.20 Present Element 5.6079 0.5516 1.2139 0.2197 0.10 Present Element 1.9764 0.4444 0.7645 0.2203

450

0.05 Present Element 1.0615 0.4247 0.6362 0.2178


93

4.3.1.2.11 Skew laminated sandwich plate with two different boundary conditions

The skew sandwich plate (0/90/C/90/0) as shown in Figure 4.3 is analysed with the proposed element taking h/a = 0.1 and skew angle α = 00, 150, 300, 450 and 600. In the present study, the boundary condition at the four sides is taken as simply supported in one case while it is clamped in other case. The thickness distribution of the different layers is identical to that used in the previous example. The whole plate is analysed with mesh size (Fig. 4.3) of 24x24. The treatment with the skew edges as explained earlier is applied for simply supported boundaries. The values of w at the plate center, xσ at x = (a + b sinα)/2, y = (b

cosα)/2, z = h/2 (Fig. 4.3) and yzτ at x= a/2, y =0, z = 0.4h obtained in the

present analysis are presented in Table 4.17. For skew angle (α) of 00, the present results are validated with 3-D elasticity solution (Pagano 1970) using COMPAG, which shows a good correlation between them.

Table 4.17 Deflections and stresses of a skew sandwich plate with laminated faces (0/90/C/90/0) under uniformly distributed load with different boundary conditions

Boundary Condition

Skew angle References w xσ yzτ

Present Element 2.6235 1.6916 0.3562 00

3-D Elasticity (Pagano 1970) 2.6327 1.6890 0.3488 150 Present Element 2.3969 1.5402 0.2857 300 Present Element 1.8100 1.1397 0.1727 450 Present Element 1.1026 0.6532 0.1050

Simply supported

600 Present Element 0.5094 0.2817 0.1605 00 Present Element 1.4755 0.6054 0.3311

150 Present Element 1.3859 0.5705 0.2743 300 Present Element 1.1412 0.4754 0.1653 450 Present Element 0.7831 0.3396 0.1246

Clamped

600 Present Element 0.4031 0.1919 0.1641


94

4.3.2 Free vibration Analysis

Free vibration analysis of laminated composites and sandwich plates having perfect layer interfaces is carried out using the proposed finite element model initially by option of HSDT/FSDT for composites and then by RHSDT for composites as well as sandwich plates like static analysis. In some cases RFSDT, HSDT and FSDT are also applied with RHSDT to show their relative merits.


In this section the free vibration of perfect composite laminates is analysed with single layer theory (HSDT/FSDT) and the results obtained in the form of natural frequencies are compared with published results in many cases.

4.3.2.1.1 Rectangular cross-ply and angle-ply laminates

A rectangular laminate, simply supported at the four edges, is analysed with different mesh divisions taking thickness ratios (h/a) as 0.01, 0.10 and 0.20 etc. The convergence of the present element for the analysis of composites is reported in Table 4.18 by an example with ply arrangement of (0/90) for thickness ratios 0.1 and 0.2 with a/b=1.0. The study is also made in Table 4.19 with thickness ratio of 0.1 for ply arrangement of (0/90/90/0) to compare the performance of the present element with an upgraded finite element of Bogner et al (1965) using the higher order shear deformation theory (Reddy 1984). The similar study is done for ply arrangement of (0/90/0/90), (45/-45) and (45/-45/45/-45/45/-45/) where the aspect ratios (a/b) taken are 1.0, 1.5, 1.0 and 2.0 respectively (Fig. 4.2). The first six frequencies obtained in all the cases are presented with the analytical solution of Reddy and Phan (1985) in Table 4.18 to Table 4.22, which show that the present results have excellent agreement with the analytical ones (Reddy and Phan 1985). Though the individual layers have different fiber orientations, they possess same material properties and ply thickness. The material properties used are E1/E2 = 40, G12 = G13 = 0.6E2, G23 = 0.5E2, ν12 = 0.25.


95

Table 4.18 Frequency Parameters Ω = (ωa2/h)√(ρ/E2) of the two layer cross-ply (0/90) square laminate simply supported at all the edges

Mode number h/a References 1 2 3 4 5 6

Present (4x4) HSDT 10.668 26.649 26.675 37.263 49.644 49.650 FSDT 10.564 25.891 25.919 36.094 47.015 47.020 Present (6x6) HSDT 10.667 26.602 26.604 36.905 49.288 49.291 FSDT 10.564 25.849 25.850 35.750 46.707 46.709 Present (8x8) HSDT 10.667 26.596 26.601 36.844 49.228 49.234 FSDT 10.564 25.845 25.848 35.695 46.661 46.664 Present (12x12) HSDT 10.666 26.594 26.560 36.819 49.184 49.197 FSDT 10.564 25.842 25.844 35.678 46.637 46.644 Present (16x16) HSDT 10.665 26.590 26.594 36.801 49.132 49.155 FSDT 10.564 25.840 25.842 35.669 46.610 46.622 Present (20x20) HSDT 10.664 26.566 26.571 36.743 48.972 49.026 FSDT 10.564 25.830 25.833 35.634 46.519 46.547 Present (24x24) HSDT 10.664 26.566 26.571 36.743 48.972 49.026 FSDT 10.564 25.830 25.833 35.634 46.519 46.547 Reddy and Phan (1985)

HSDT 10.568 26.301 26.301 36.348 48.704 48.704

0.10

FSDT 10.469 25.565 25.565 33.227 46.164 46.164 Present (4x4) HSDT 9.210 19.853 19.892 26.978 33.156 33.172 FSDT 8.923 18.432 18.467 24.771 29.341 29.345 Present (6x6) HSDT 9.024 19.779 19.793 26.574 32.728 32.802 FSDT 8.919 18.389 18.395 24.463 29.103 29.125 Present (8x8) HSDT 9.198 19.717 19.741 26.419 32.450 32.598 FSDT 8.916 18.357 18.368 24.375 28.983 29.032 Present (12x12) HSDT 9.181 19.561 19.616 26.148 32.425 32.497 FSDT 8.906 18.251 18.295 24.225 28.912 28.934 Present (16x16) HSDT 9.156 19.522 19.545 26.111 32.398 32.405 FSDT 8.890 18.212 18.277 24.178 28.908 28.921 Present (20x20) HSDT 9.087 19.519 19.527 26.008 32.344 32.365 FSDT 8.844 18.203 18.213 24.110 28.906 28.911 Present (24x24) HSDT 9.087 19.519 19.527 26.008 32.344 32.365 FSDT 8.844 18.203 18.213 24.110 28.906 28.911 Reddy and Phan (1985)

HSDT 9.087 19.611 19.611 26.291 32.731 32.731

0.20

FSDT 8.807 18.199 18.199 24.150 28.917 28.917


96

Table 4.19 Frequency Parameters Ω = (ωa2/h)√(ρ/E2) of the four layer cross-ply (0/90/90/0) square laminate simply supported at all the edges

Mode number References 1 2 3 4 5 6

Present (4x4) HSDT 15.052 26.313 37.456 44.668 46.455 55.623 Bogner et al. (1965) (4x4)

HSDT 15.150 26.600 37.369 44.252 46.154 53.666


HSDT 15.108 26.452 37.086 43.232 44.307 56.325


HSDT 15.108 26.444 37.068 43.175 44.236 56.074


HSDT 15.107 26.442 37.063 43.162 44.223 56.023


HSDT 15.107 26.441 37.062 43.157 44.219 56.005


HSDT 15.107 26.441 37.061 43.154 44.127 55.996

Reddy and Phan (1985)

HSDT 15.398 26.347 38.460 43.334 44.199 56.153

Table 4.20 Frequency Parameters Ω = (ωa2/h)√(ρ/E2) of the four layer cross-ply (0/90/0/90) rectangular (a/b =1.5) laminate simply supported at the edges


Present (20x20) HSDT 13.230 24.730 47.374 49.063 52.550 68.472 FSDT 14.195 25.233 47.470 49.733 52.692 68.942 Reddy and Phan (1985) HSDT 13.241 24.758 47.462 49.122 52.717 68.740

0.01

FSDT 13.242 24.761 47.475 49.135 52.731 68.759 Present (20x20) HSDT 11.583 19.948 31.693 33.114 35.773 44.563 FSDT 12.293 20.393 32.027 33.730 36.083 45.076 Reddy and Phan (1985) HSDT 11.592 19.971 31.740 33.171 35.850 44.686

0.10

FSDT 11.641 20.081 31.968 33.390 36.066 44.918 Present (20x20) HSDT 8.971 14.095 19.710 20.839 22.557 27.314 FSDT 9.032 14.155 19.628 20.728 22.414 26.983 Reddy and Phan (1985) HSDT 8.911 14.163 19.936 21.029 22.767 27.596

0.20

FSDT 9.046 14.206 19.780 20.872 22.568 27.195


97

Table 4.21 Frequency Parameters Ω = (ωa2/h)√(ρ/E2) of the two layer angle-ply (45/-45) square laminate simply supported at all the edges

Mode number h/a References

1 2 3 4 5 6 Present (8x8) HSDT 14.620 33.717 33.723 58.331 62.596 62.597 FSDT 14.617 33.703 33.708 58.285 62.549 62.549 Present (12x12) HSDT 14.620 33.726 33.731 58.290 62.646 62.646 FSDT 14.617 33.711 33.716 58.244 62.598 62.599 Present (16x16) HSDT 14.621 33.733 33.736 58.286 62.634 62.634 FSDT 14.618 33.718 33.721 58.240 62.634 62.634 Reddy and Phan (1985) HSDT 14.621 33.745 33.745 58.289 62.749 62.749

0.01

FSDT 14.618 33.730 33.730 58.243 62.703 62.703 Present (8x8) HSDT 13.263 27.758 27.761 43.394 46.418 46.421 FSDT 13.043 26.919 26.921 41.370 44.445 44.445 Present (12x12) HSDT 13.262 27.756 27.760 43.334 46.414 46.417 FSDT 13.043 26.926 26.928 41.343 44.476 44.476 Present (16x16) HSDT 13.261 27.750 27.754 43.292 46.391 46.396 FSDT 13.044 26.930 26.930 41.340 44.496 44.496 Reddy and Phan (1985) HSDT 13.263 27.777 27.777 43.359 46.502 46.502

0.10


0.20

FSDT 10.299 18.732 18.732 26.150 28.381 28.381


98

Table 4.22 Frequency Parameters Ω = (ωa2/h)√(ρ/E2) of the six layer angle-ply (45/-45/45/-45/45/-45) rectangular (a/b =2.0) laminate simply supported at all the edges

Mode number h/a References

1 2 3 4 5 6 Present (8x8) HSDT 14.121 24.658 38.414 41.720 55.394 74.927 FSDT 14.122 24.660 38.419 41.725 55.989 74.945 Present (12x12) HSDT 14.130 24.697 38.555 41.854 55.854 75.118 FSDT 14.131 24.699 38.559 41.859 56.106 75.136 Present (16x16) HSDT 14.134 24.714 38.617 41.906 56.145 75.202 FSDT 14.134 24.716 38.622 41.911 56.155 75.220 Reddy and Phan (1985) HSDT 14.139 24.739 38.725 41.985 56.221 75.346

0.01


0.10


0.20

FSDT 9.372 12.786 16.901 19.099 21.362 24.722

4.3.2.1.2 Cross-ply laminate having different boundary conditions at the edges

The problem of a cross-ply (0/90/0) square laminate is studied for different combinations of boundary conditions at the four edges as shown in Table 4.23. The analysis is done with a mesh division of 20x20 taking thickness ratio (h/a) as 0.01, 0.05, 0.1 and 0.20 in all the cases. The fundamental frequencies obtained by the proposed element are presented in Table 4.23. Using third order shear deformation theory (HSDT), as well as FSDT Librescu et al. (1989) have solved this problem and presented results for h/a =0.1, which are included in Table 4.23.


99

As can be seen, the present results are in good agreement with those of Librescu et al. (1989). In the present problem the material properties used are E1/E2 = 25, G12 = G13 = 0.5E2, G23 = 0.2E2, ν12 = 0.25.

Table 4.23 Fundamental frequency parameters Ω = (ωa2/h)√(ρ/E2) of a cross-ply (0/90/0) square laminate having different boundary conditions at edges

Thickness ratio (h/a) Boundary conditions

References 0.20 0.10 0.05 0.01

Present element HSDT 3.473 3.790 3.884 3.916 FSDT 3.469 3.787 3.883 3.916 Librescu et al. (1989) HSDT - 3.790 - -

Two opposite edges simply supported and other edges free

FSDT - 3.788 - - Present element HSDT 9.076 13.856 18.783 22.490 FSDT 8.939 14.235 19.180 22.521 Librescu et al. (1989) HSDT - 13.815 - -

One edge clamped and other edges simply supported

FSDT 14.248 Present element HSDT 3.942 4.326 4.452 4.500 FSDT 3.916 4.319 4.450 4.500 Librescu et al. (1989) HSDT - 4.323 - -

One edge free and other edges simply supported

FSDT - 4.320 - - Present element HSDT 10.69

5 16.258 23.860 31.724

FSDT 9.478 16.367 24.468 31.797 Librescu et al. (1989) HSDT - 15.739 - -

Two opposite edges simply supported and other edges clamped

FSDT 9.476 16.354 24.426 31.724 Present element HSDT 5.104 6.103 6.635 6.873 FSDT 4.994 6.141 6.659 6.874 Librescu et al. (1989) HSDT - 6.095 - -

Two opposite edges simply supported, third edge free and forth edge clamped FSDT - 6.144 - -

4.3.2.1.3 Triangular cross-ply and angle-ply laminates simply supported at edges

The problem of a triangular laminate as shown in Figure 4.13 (a/b = 1.0) is studied for cross ply (0/90) and angle ply (45/-45) arrangements considering all the edges simply supported for both the cases. Assuming h/a = 0.1 and 0.2, the plate is analysed with the mesh arrangement shown in Figure 4.13. Similar to the case of skew laminate, the degrees of freedom of the nodes on the inclined edge


100

BC are expressed in a local axis system to incorporate the boundary conditions. The first six frequencies obtained with three different mesh divisions (Fig. 4.13) are presented in Table 4.24 with those of Haldar (2002) who have used a high precision triangular element based on FSDT to analyze the structure. The table shows the performance of the proposed element in case of triangular laminate. The material properties used are same as those used in the previous section (4.3.2.1.2).

Figure 4.13 A triangular plate having mesh division of mxm

4.3.2.1.4 Simply supported square laminate having concentrated mass at center

A simply supported square laminate having a concentrated mass at its center is analysed with a mesh division of 20x20 using h/a = 0.1. The study is made for cross ply (0/90) and angle ply (45/-45) arrangements with Mc/Mp = 0.5, 1.0 and 2.0 where Mp and Mc are the mass of the plate and the central concentrated mass, respectively. The first fix frequencies obtained in the present analysis are presented in Table 4.25 with those of Haldar (2002). These results are found to have good agreement, which indicates the performance of the proposed element in a problem having an internal discontinuity due to the concentrated mass. The


101

material properties used are E1 = 138GPa, E2 = 8.96GPa, G12 = 7.1GPa and ν12=0.3.

Table 4.24 Frequency parameters Ω = (ωa2/h)√(ρ/E2) of a triangular laminate simply supported at the three edges


Cross-ply (0/90) laminate having all the edges simply supported Present (8x8) HSDT 21.897 22.271 39.945 44.561 45.094 59.922 FSDT 21.326 22.271 38.143 43.218 44.562 56.083 Present (12x12) HSDT 21.893 22.251 39.966 44.507 45.079 59.910 FSDT 21.335 22.251 38.161 43.227 44.507 56.110 Present (16x16) HSDT 21.901 22.242 39.969 44.485 45.059 59.865 FSDT 21.339 22.242 38.173 43.273 44.486 56.137

0.10

Haldar (2002) FSDT 21.288 21.425 37.938 42.818 42.893 55.432 Present (8x8) HSDT 11.135 15.855 22.281 25.930 29.142 33.464 FSDT 11.135 14.902 22.281 23.562 26.737 32.061 Present (12x12) HSDT 11.125 15.026 22.142 22.256 24.060 27.640 FSDT 11.125 14.918 22.254 23.614 26.822 32.202 Present (16x16) HSDT 11.121 14.916 21.866 22.243 23.748 27.277 FSDT 11.121 14.925 22.243 23.041 26.864 32.276

0.20

Haldar (2002) FSDT 10.713 14.873 21.409 23.482 26.649 31.942 Angle-ply (45/-45) laminate having all the edges simply supported

Present (8x8) HSDT 23.940 38.244 50.238 56.222 63.306 67.891 FSDT 23.360 36.849 47.583 53.582 60.645 67.973 Present (12x12) HSDT 23.955 38.240 50.170 56.262 63.043 67.871 FSDT 23.383 36.900 47.593 53.704 60.601 68.001 Present (16x16) HSDT 23.954 38.202 50.093 56.227 62.823 67.890 FSDT 23.393 36.928 47.610 53.776 60.627 68.013

0.10

Haldar (2002) FSDT 22.338 36.421 46.481 50.787 60.488 62.379 Present (8x8) HSDT 16.647 18.892 24.910 28.540 31.158 34.831 FSDT 15.200 15.717 23.114 27.170 28.817 31.867 Present (12x12) HSDT 16.552 18.206 24.476 28.573 30.649 34.272 FSDT 15.200 15.718 23.171 27.174 28.817 31.540 Present (16x16) HSDT 16.406 17.688 23.955 28.580 30.026 33.548 FSDT 15.201 15.719 23.200 27.181 28.828 31.469

0.20

Haldar (2002) FSDT 15.136 15.146 23.048 26.708 28.286 30.244


102

Table 4.25 Frequency parameters Ω = (ωa2/h)√(ρ/E2) of a simply supported square laminate having a concentrated mass (Mc) at its center

Mode number Mc /Mp References 1 2 3 4 5 6

Cross-ply (0/90) laminate Present element HSDT 4.927 17.966 21.255 21.909 30.307 39.887 FSDT 4.822 16.623 21.390 21.391 29.549 38.200

0.50

Haldar (2002) FSDT 4.586 16.187 21.502 21.502 29.730 38.305 Present (16x16) HSDT 3.738 18.673 21.909 21.915 30.307 39.887 FSDT 3.643 17.114 21.390 21.391 29.549 38.200

1.0


2.0

Haldar (2002) FSDT 2.496 15.262 21.486 21.486 29.689 38.217 Angle-ply (45/-45) laminate

Present (16x16) HSDT 5.886 21.644 23.017 23.023 35.461 36.456 FSDT 5.695 20.096 22.391 22.393 33.990 35.421

0.50


1.0


2.0

Haldar (2002) FSDT 3.012 18.112 22.751 22.786 35.128 36.977


In this section, the RHSDT is applied to free vibration analysis of composite and sandwich laminates to demonstrate the capability of RHSDT in vibration analysis specifically for sandwich plates.

4.3.2.2.1 Un-symmetric cross-ply and angle-ply laminates

The free vibration analysis of a cross-ply (0/90/0/90) square laminate and an angle-ply (45/-45/45/-45) square laminate simply supported at edges is carried out in this example. Each ply has same thickness (h/4) and material properties (E1/E2 = 40, G12 = G13 = 0.6E2, G23 = 0.5E2, ν12 = 0.25) in its off axis system for both the laminates. The analysis is carried out with mesh sizes (full plate) of 4x4, 6x6, 8x8, 12x12, 16x16, 24x24 and 28x28 taking h/a = 0.1 and 0.2. The non-


103

dimensional frequency parameters Ω = (ωa2/h)√(ρ/E2) for the first six modes obtained in the present analysis are presented in Table 4.26. For the cross-ply laminate (h/a = 0.2), the fundamental frequency obtained in the present analysis is compared with the three-dimensional elasticity solution of Noor (1973) and the solution of Matsunaga (2000) based on higher order shear deformation theory where in-plane displacements are expressed with power series expansion. Noor (1973) has solved the problem by finite difference method while Matsunaga (2000) applied a series solution. It shows that the agreement between the results is very good where the present result is more close to the elasticity solution. For the angle-ply laminate (h/a = 0.1), the six frequency parameters obtained in the present analysis are compared with the results of Matsunaga (2001), which are obtained in a similar manner as that of Matsunaga (2000). It shows that the agreement between the results is similar to the earlier case. In one case, the results based on classical plate theory (CPT) are presented to show the relative improvement of the results obtained by the different plate theories.

4.3.2.2.2 Skew angle-ply (30/-30/30) laminate clamped at the edges

An angle-ply (30/-30/30) skew laminate (Fig. 4.3), clamped at all the sides, is analysed by the proposed element using the mesh arrangement (20x20) shown in Figure 4.3. The study is made for skew angle (α) of 200, 300 and 400 with thickness ratios (h/a) of 0.01 and 0.1. The first six frequencies obtained in the present analysis (RHSDT) along with the present results based on HSDT and FSDT are presented in Table 4.27. These results are compared with those of Han and Dickinson (1997) for h/a = 0.01 and Haldar (2002) for the other thickness ratios where both investigators have used FSDT to solved the problem. The results agreed well. The material properties used are E1 = 138GPa, E2 = 8.96GPa, G12 = 7.1GPa and ν12=0.3.


104

Table 4.26 Frequency parameters (Ω) of a simply supported un-symmetric square laminate

Mode Number h/a References

1 2 3 4 5 6 Cross-ply (0/90/0/90)

Present (4x4) RHSDT 14.639 31.989 32.045 42.512 51.750 51.756 Present (6x6) RHSDT 14.678 32.073 32.110 42.530 51.736 51.738 Present (8x8) RHSDT 14.694 32.117 32.141 42.574 51.769 51.770 Present (12x12) RHSDT 14.707 32.155 32.167 42.627 51.809 51.809 Present (16x16) RHSDT 14.712 32.170 32.178 42.651 51.827 51.827 Present (24x24) RHSDT 14.716 32.182 32.185 42.671 51.840 51.841

0.10

Present (28x28) RHSDT 14.716 32.184 32.187 42.675 51.844 51.844 Present (4x4) RHSDT 10.644 18.528 18.547 23.012 25.839 25.839 Present (6x6) RHSDT 10.657 18.525 18.526 22.811 25.700 25.704 Present (8x8) RHSDT 10.663 18.526 18.528 22.771 25.676 25.678 Present (12x12) RHSDT 10.668 18.529 18.529 22.762 25.670 25.671 Present (16x16) RHSDT 10.670 18.533 18.534 22.758 25.668 25.669 Present (24x24) RHSDT 10.671 18.535 18.535 22.754 25.668 25.668 Present (28x28) RHSDT 10.671 18.536 18.536 25.753 25.668 25.668 Matsunaga (2000)

HSDT 10.619 - - - - -

0.20

3-D elasticity (Noor 1973) 10.680 - - - - - Angle-ply (45/-45/45/-45)

Present (4x4) RHSDT 18.039 33.280 33.455 47.725 50.318 50.359 Present (6x6) RHSDT 18.015 33.290 33.463 47.425 50.362 50.417 Present (8x8) RHSDT 17.995 33.298 33.473 47.346 50.431 50.487 Present (12x12) RHSDT 17.978 33.300 33.472 47.298 50.482 50.515 Present (16x16) RHSDT 17.950 33.302 33.471 47.261 50.553 50.604 Present (24x24) RHSDT 17.934 33.300 33.464 47.237 50.584 50.634 Present (28x28) RHSDT 17.934 33.299 33.463 47.237 50.602 50.635 Matsunaga (2001)

HSDT 17.589 32.657 32.657 46.689 50.615 50.615

0.10

Classical Plate Theory 23.530 53.740 53.740 94.110 98.870 98.870 Present (4x4) RHSDT 11.877 18.801 19.002 24.135 25.388 25.503 Present (6x6) RHSDT 11.854 18.786 18.955 24.084 25.362 25.493 Present (8x8) RHSDT 11.837 18.780 18.941 24.016 25.334 25.454 Present (12x12) RHSDT 11.823 18.779 18.935 23.992 25.332 25.443 Present (16x16) RHSDT 11.818 18.778 18.930 23.964 25.330 25.439 Present (24x24) RHSDT 11.813 18.778 18.926 23.957 25.329 25.438

0.20

Present (28x28) RHSDT 11.813 18.778 18.925 23.956 25.329 25.438


105

Table 4.27 Frequency parameters Ω = ωa2√(ρh/D0)* of an angle ply (30/-30/30) skew laminate for different skew angles (α)

Mode number h/a α References

1 2 3 4 5 6 Present element RHSDT 19.582 35.212 43.350 56.063 63.457 77.800 Present element HSDT 19.583 35.214 43.352 56.066 63.462 77.805 Present element FSDT 19.584 35.215 43.356 56.072 63.463 77.816

200

Han et al. (1997) FSDT 19.690 35.490 43.810 56.650 64.280 - Present element RHSDT 20.521 40.098 42.623 63.211 69.954 75.183 HSDT 20.522 40.100 42.625 63.215 69.959 75.186 FSDT 20.523 40.101 42.630 63.222 69.958 75.206

300


0.01

400


200

Haldar (2002) FSDT 16.149 28.371 32.126 41.683 44.571 50.187 Present element RHSDT 16.806 29.969 31.020 42.335 46.530 48.311 HSDT 16.883 30.400 31.511 43.585 48.183 50.236 FSDT 16.909 30.272 31.644 43.323 47.690 50.321

300

Haldar (2002) FSDT 17.171 31.643 33.016 44.920 49.400 51.878 Present element RHSDT 19.128 32.680 36.216 45.323 52.574 55.520 HSDT 19.241 33.249 36.991 46.863 55.052 58.420 FSDT 19.276 33.246 37.012 46.521 54.822 58.070

0.10

400

Haldar (2002) FSDT 19.528 33.511 37.067 45.059 56.764 59.966 * D0 = E1h3/12(1-ν12ν21)

4.3.2.2.3 Simply supported square laminated plate having orthotropic layers

The problem of a plate (h/a = 0.1) having two orthotropic face layers of thickness 0.1h each and an orthotropic core layer of thickness 0.8h is studied for different values of material properties of the face layers expressed in terms of fraction/ratio (Rf) of those of core. In the present study, the above ratio (Rf) is varied from 1.0 to 15.0 (Table 4.28) where the elastic moduli of the face layers are obtained by


106

multiplying Rf with those of core, which are kept unchanged. The plate is analysed with the proposed element using mesh size (full plate) of 4x4, 8x8, 12x12, 16x16, 20x20 and 24x24 in all the cases. The fundamental frequency parameters Ω = ω√(ρh2/E11) obtained are presented in Table 4.28 with the three dimensional elasticity solution of Srinivas and Rao (1970) and RFSDT of Di Sciuva (1986). The results agreed well. The material properties used for the core are E22/E11 = 0.543, G12 /E11 = 0.2629, G13 /E11 = 0.1599, G23 /E11 = 0.2668, ν12 = 0.3.

Table 4.28 Fundamental frequency parameters (Ω) of a simply supported square laminated plate having orthotropic layers

Rf References 1 2 5 10 15

Present element (4x4) 4.7201 5.6810 7.6822 9.7592 11.1264 Present element (8x8) 4.7332 5.6912 7.6898 9.7653 11.1409 Present element (12x12) 4.7353 5.6937 7.6929 9.7686 11.1441 Present element (16x16) 4.7361 5.6946 7.6940 9.7699 11.1454 Present element (20x20) 4.7367 5.6953 7.6949 9.7708 11.1464 Present element (24x24) 4.7367 5.6953 7.6949 9.7708 11.1464 Srinivas and Rao (1970) 4.7419 5.7041 7.7148 9.8104 11.2034 Di Sciuva (1986) 4.7698 5.7244 7.7296 9.8222 11.2137

4.3.2.2.4 Square laminated sandwich plate simply supported at the four edges

The problem of a sandwich plate (0/90/../C/../90/0) having two laminated stiff layers at the two faces is taken in this example. In the stiff cross-ply layers the number of equal thickness ply is taken as 2 and 8, which are stacked symmetrically with respect to the core. The thickness of the central core is taken as 0.8h, while that of each stiff layer is 0.1h. The plate is analysed with the proposed element using mesh size (full plate) of 8x8, 12x12, 16x16 and 20x20 taking thickness ratios (h/a) of 0.05, 0.10 and 0.15. The non-dimensional frequency parameters Ω =100ωa√(ρc/E11) for the first six modes obtained in the present analysis are presented with the results of Wang et al. (2000) in Table 4.29. Wang et al. (2000) have taken two different angles of rotation for the face and the


107

core i.e., the usual modeling technique for sandwich plate (Plantema 1966, Allen 1969) and solved the problem analytically. This plate model gives linear variation of in-plane displacements across the thickness with kink at the interface between the core and face sheets and does not satisfy the inter-laminar shear stress continuity condition as well as shear stress free conditions at the top and bottom surfaces of the plate. Table 4.29 shows that the values of present results are somewhat lower that those of Wang et al. (2000), which is expected from the above observations. The material properties taken in off axis system for the core and a ply of the face sheets are as follows: Face: E11/E = 40.0, E22/E = 1.0, G12 /E = G13 /E = G23 /E = 1.0, ν12 = 0.25, ρf = ρ and Core: E/Ec = 11.945, Gc12 /Ec = Gc13

/Ec = 1.173/6.279, Gc23 /Ec = 2.415/6.279, νc12 = 0.0025 and ρ/ρc= 0.6818.

4.3.2.3.5 Square laminated sandwich plate having different boundary conditions

The square sandwich plate with two-ply laminated face sheets (0/90/C/90/0) studied in previous example is analysed with two different boundary conditions. The boundary conditions are CCCC i.e., all edges clamped and SCSC i.e., two opposite edges (parallel to y: Fig. 4.2) simply supported and the other edges clamped. Taking h/a = 0.05, 0.1 and 0.2, the analysis is carried out with mesh size of 12x12, 16x16, and 20x20 for the full plate. The first six frequency parameters

Ω =100ωa√(ρc/E11) obtained by the proposed element are presented in Table 4.30 with some available results (Wang et al. 2000) for the first four frequencies of the plate having thickness ratio (h/a) of 0.05 only. Table 4.30 shows that the agreement between the results is more or less similar as found in Table 4.29.


108

Table 4.29 Frequency parameters (Ω) of a simply supported square sandwich plate with laminated face sheets (0/90/../C/../90/0)

Mode Number h/a Layers in

each face

References

1 2 3 4 5 6

Present (8x8) 7.921 13.033 17.303 18.828 20.079 24.131 Present (12x12) 7.927 13.042 17.315 18.834 20.091 24.139 Present (16x16) 7.930 13.047 17.322 18.839 20.100 24.146 Present (20x20) 7.930 13.048 17.323 18.841 20.102 24.148

2

Wang et al. (2000) 8.029 14.858 16.984 21.111 - - Present (8x8) 7.877 13.407 16.856 19.697 19.964 24.546 Present (12x12) 7.883 13.415 16.867 19.701 19.972 24.526 Present (16x16) 7.886 13.420 16.873 19.705 19.980 24.528 Present (20x20) 7.886 13.420 16.875 19.706 19.982 24.529

0.05

8


2


0.10

8


2


0.15

8

Wang et al. (2000) 11.642 17.876 21.505 25.469 - -


109

Table 4.30 Frequency parameters (Ω) of a square sandwich plate with laminated face sheets (0/90/C/90/0) having different boundary conditions

Mode Number h/a Boundary conditions

References

1 2 3 4 5 6 Present (12x12) 8.640 13.590 17.611 19.537 20.477 24.753 Present (16x16) 8.630 13.555 17.605 19.479 20.440 24.671 Present (20x20) 8.625 13.540 17.602 19.453 20.424 24.637

SCSC

Wang et al. (2000)

9.098 15.441 17.442 21.840 - -

Present (12x12) 10.595 14.823 18.861 20.353 21.619 25.772 Present (16x16) 10.561 14.715 18.773 20.249 21.473 25.535 Present (20x20) 10.545 14.724 18.731 20.204 21.405 25.459

0.05

CCCC

Wang et al. (2000)

10.930 16.498 18.650 22.483 - -

Present (12x12) 10.438 15.054 19.148 20.430 21.697 25.337 Present (16x16) 10.431 15.053 19.140 20.393 21.662 25.260

SCSC


0.10

CCCC


SCSC


0.20

CCCC

Present (20x20) 11.869 15.683 19.487 20.075 21.130 23.668

4.3.2.2.6 Simply supported triangular double core laminated sandwich plate

The problem of a triangular sandwich plate (Fig. 4.13) having double core construction with two-ply laminated stiff sheet at the two faces as well as at the center (0/90/C/0/90/C/0/90) is studied in this example. The thickness of each core is taken as 0.425h, while that of each stiff laminated sheet having two equal thick ply is 0.05h. The material properties are same as those used in the previous example. Taking simply supported boundary condition at the three edges; the plate is analysed with the proposed element using mesh size (Fig. 4.13) of 20x20. The study has been made for aspect ratio (b/a) of 1.0, 1.5 and 2.0 (Fig.4.13) and


110

for thickness ratios (h/a) of 0.05, 0.10, 0.15 and 0.20. For all the cases, the first six frequency parameter Ω =100ωa√(ρc/E11) obtained by the proposed element are presented in Table 4.31.

Table 4.31 Frequency parameters (Ω) of a double core triangular sandwich plate with laminated stiff sheets (0/90/C/0/90/C/0/90)

Mode Number b/a h/a References

1 2 3 4 5 6 0.05 Present element 13.435 20.995 24.942 28.178 31.834 34.805 0.10 Present element 15.748 22.081 25.725 27.580 30.728 32.250 0.15 Present element 15.853 20.954 23.912 24.981 27.333 28.193

1.0

0.20 Present element 15.317 19.342 21.673 22.316 24.038 24.661 0.05 Present element 10.855 16.544 21.277 22.426 26.210 28.088 0.10 Present element 13.468 18.444 22.497 23.751 26.201 28.127 0.15 Present element 13.951 18.126 21.328 22.580 24.023 25.711

1.5

0.20 Present element 13.767 17.192 19.650 20.851 21.613 22.995 0.05 Present element 9.480 13.990 18.016 20.407 21.886 25.016 0.10 Present element 12.174 16.258 19.683 22.269 22.846 25.467 0.15 Present element 12.815 16.323 19.123 21.379 21.677 23.051

2.0

0.20 Present element 12.788 15.743 17.965 19.726 20.168 21.227

4.3.2.2.7 Skew angle-ply laminated sandwich plate simply supported at the edges

The problem of a simply supported skew sandwich plate (Fig. 4.3) having two-ply laminated face sheets (θ/-θ/C/θ/-θ) is studied for different values of θ (ply orientation) taking skew angle (α) of 00, 150, 300 and 450 (Fig. 4.3) and thickness ratio (h/a) of 0.05, 0.10 and 0.15. In this case, the thickness of the core is 0.8h, while that of each laminated face sheet having two equal thick ply is 0.1h. The material properties are same as those used in the previous example. The analysis is carried out with the proposed element using mesh size (Fig. 4.3) of 20x20 for all the cases. The frequency parameter Ω =100ωa√(ρc/E11) for the fundamental mode obtained in the present analysis is plotted against θ (ranging from 00 to 900) in Fig. 4.14 (a) – 4.14 (c).


111

0 15 30 45 60 75 906

7

8

9

10

11

12 h/a = 0.05

Skew Angle=00

Skew Angle=150

Skew Angle=300

Skew Angle=450

Freq

uenc

y pa

ram

eter

Fiber angles in degree

Figure 4.14 (a) Effect of skew angle and face ply orientations on fundamental frequency of a sandwich plate (h/a = 0.05)

0 15 30 45 60 75 90

8

9

10

11

12

13

14 h/a = 0.10

Skew angle = 00

Skew angle = 150

Skew angle = 300

Skew angle = 450

Freq

uenc

y pa

ram

eter


Figure 4.14 (b) Effect of skew angle and face ply orientations on fundamental

frequency of a sandwich plate (h/a = 0.10)


112

0 15 30 45 60 75 90

9

10

11

12

13

14 h/a = 0.15

Skew angle = 00

Skew angle = 150

Skew angle = 300

Skew angle = 450

Freq

uenc

y pa

ram

eter


Figure 4.14 (c) Effect of skew angle and face ply orientations on fundamental

frequency of a sandwich plate (h/a = 0.15)

4.3.3 Buckling Analysis

Buckling analysis of composites and sandwich laminates having perfect layer interfaces is carried out using the proposed finite element formulation. The results obtained in the form of buckling loads are compared with some published results in many cases. In case of an unsymmetrical laminate, if an axial load (e.g., Nx) is applied uniformly throughout the thickness, the resultant Nx will pass through the geometric centroid of the plate i.e., the middle plane of the plate. Due to different values of rigidity parameters of the individual layers in a particular direction (e.g., x), there will be an eccentricity in the structure and loading system. This will introduce bending moment at the edges once axial load will be applied. Thus transverse deformation will be produced from the beginning of the load history and it will not be a bifurcation-buckling problem. Change in distribution of axial load can eliminate the above-mentioned eccentricity. In that case, the intensity of


113

load at the different layers should be proportional to their rigidities. However, this proposition may not be attractive as its implementation will be difficult in practice. In spite of that, some problems of unsymmetrical laminates have been taken in this study. Actually, the intention here is to assess the performance of the proposed element and relative merits of different plate theories.


4.3.3.1.1 Simply supported cross-ply square laminate under uniaxial compression

The problem of a simply supported cross-ply square laminate (Fig. 4.15) subjected to uniaxial compression (Nx) is studied for (0/90) and (0/90/90/0) ply arrangements taking thickness ratio (h/a) of the laminate as 0.01, 0.02, 0.05, 0.10 and 0.20. The analysis is carried out on the basis of HSDT as well as FSDT using a number of mesh sizes in all the cases. The buckling load parameters obtained in the present analysis are presented with those obtained from the analytical solution of Reddy and Phan (1985) in Table 4.32. It shows that the results obtained with the proposed element have an excellent agreement with the analytical solution. The individual layers are oriented in different direction but they possess same thickness and material properties (E1/E2 = 40, G12 = G13 =0.6E2, G23 = 0.5E2, ν12 = 0.25).

Figure 4.15 A rectangular plate subjected to uniaxial compression


114

Table 4.32 Buckling load parameters (λ=Nxb2/E2h3) of a simply supported cross-ply square laminate under uniaxial compression

Thickness ratio (h/a) Ply orientation

References 0.01 0.02 0.05 0.10 0.20

Present element (4x4) HSDT 12.931 12.884 12.561 11.536 8.724 FSDT 12.940 12.885 12.515 11.353 8.270 Present element (6x6) HSDT 12.926 12.879 12.557 11.535 8.741 FSDT 12.928 12.873 12.504 11.343 8.269 Present element (8x8) HSDT 12.930 12.883 12.561 11.542 8.752 FSDT 12.930 12.875 12.506 11.345 8.271 Present element (12x12) HSDT 12.935 12.888 12.567 11.551 8.762 FSDT 12.934 12.879 12.510 11.349 8.275 Present element (16x16) HSDT 12.938 12.890 12.571 11.556 8.765 FSDT 12.936 12.881 12.512 11.350 8.276 Reddy and Phan (1985) HSDT 12.942 12.895 12.577 11.563 8.769

0/90

FSDT 12.939 12.884 12.515 11.353 8.277 Present element (4x4) HSDT 35.223 34.630 31.043 22.939 11.867 FSDT 35.249 34.667 31.142 23.141 11.543 Present element (6x6) HSDT 35.580 34.977 31.327 23.122 11.936 FSDT 35.592 35.001 31.416 23.295 11.545 Present element (8x8) HSDT 35.728 35.120 31.449 23.206 11.961 FSDT 35.736 35.141 31.532 23.359 11.537 Present element (12x12) HSDT 35.845 35.235 31.552 23.276 11.981 FSDT 35.851 35.254 31.625 23.409 11.507 Present element (16x16) HSDT 35.889 35.280 31.595 23.303 11.988 FSDT 35.894 35.297 31.659 23.428 11.512 Reddy and Phan (1985) HSDT 35.971 35.419 31.637 23.349 12.156

0/90/90/0

FSDT 35.955 35.356 31.707 23.471 11.525

4.3.3.1.2 Skew cross-ply laminate having different boundary conditions

The problems of a simply supported and a clamped cross ply (90/0/0/90) skew laminate subjected to uniaxial compression (Fig. 4.16) are studied for different skew angle (α). Taking h/a = 0.01, 0.02, 0.05, 0.10 and 0.25, the analysis is carried out with a mesh size of 20x20 and the results are presented in Table 4.33. Some of the present results are compared with those of Hu and Tzeng (2000), which shows that the agreement between the results is good. The coordinate transformation is to be done for the skew edges as explained before. In the present problem, the material properties used for the individual layers are E1 = 128GPa, E2 = 11GPa, ν12 = 0.25, G12 = G13 = 4.48GPa, G23 = 1.53GPa.


115

Figure 4.16 A skew plate subjected to uniaxial compression

Table 4.33 Buckling load parameters (λ=Nxa2/E2h3) of a cross-ply (90/0/0/90) skew laminate under uniaxial compression


Skew angle

References 0.01 0.02 0.05 0.10 0.25

Present element 37.418 35.817 27.758 15.437 5.945 00

Hu and Tzeng (2000) 37.272 - - - - Present element 38.798 37.059 28.215 15.620 5.967 100



Hu and Tzeng (2000) 49.000 - - - - Present element 52.907 50.357 38.791 18.654 6.340

Clamped

400

Hu and Tzeng (2000) 52.200 - - - - Present element 12.138 12.029 11.324 9.392 4.335 Reddy and Phan (1985) 12.169 12.072 11.428 9.605 4.583

00


Hu and Tzeng (2000) 13.000 - - - Present element 15.915 16.701 14.441 11.367 4.805 200


Hu and Tzeng (2000) 21.500 - - - - Present element 25.917 23.879 21.680 16.111 5.147

Simply supported

400

Hu and Tzeng (2000) 25.500 - - - -


116

4.3.3.1.3 Effect of different parameters on buckling load of a square laminate

In this example, the effect of different parameters on buckling load of a square laminated plate (Fig. 4.15) is studied. These are presented in Figure 4.17 (a) – 4.17 (d), where buckling load parameter verses thickness ratio (h/a) is shown for the different cases. The different parameters considered are ply orientation, boundary condition, material property, ratio of in-plane forces (Ny/Nx and Nxy/Nx), and thickness ratio (h/a). The boundary conditions considered are SSSS (simply supported at the four edges) and SSCC (simply supported at two opposite edges parallel to y and clamped at the other sides). The material properties used are as follows:

Material I: E1/E2 = 25, G12 = G13 =0.5E2, G23 = 0.2E2, ν12 = 0.25

Material II: E1/E2 = 40, G12 = G13 =0.6E2, G23 = 0.5E2, ν12 = 0.25.

0.00 0.05 0.10 0.15 0.20

5

10

15

20

25

30

Ply orientation : 45/-45 ( Material : I)Boundary : SSCC

Ny/Nx=0.0,Nxy/Nx=0.0 Ny/Nx=0.0,Nxy/Nx=0.5 Ny/Nx=0.0,Nxy/Nx=1.0 Ny/Nx=0.0,Nxy/Nx=2.0

Buck

ling

para

met

er(=

Nxa

2 /E2h

3 )

Thickness ratio, h/a

Figure 4.17 (a) Effect of different parameters on critical buckling load


117

0.00 0.05 0.10 0.15 0.202

4

6

8

10

12

14 Ply orientation : 45/-45 (Material : I)Boundary : SSCC


Buck

ling

para

met

er(=

NXa

2 /E2h

3 ) NY/NX=1.0,NXY/NX=0.0 NY/NX=1.0,NXY/NX=0.5 NY/NX=1.0,NXY/NX=1.0 NY/NX=1.0,NXY/NX=2.0

Figure 4.17 (b) Effect of different parameters on critical buckling load

0.00 0.05 0.10 0.15 0.20

6

12

18

24

30

36

Ply orientation : 0/90/0 ( Material : II)Boundary : SSSS

Ny/Nx=0.0,Nxy/Nx=0.0 Ny/Nx=0.0,Nxy/Nx=0.5 Ny/Nx=0.0,Nxy/Nx=1.0 Ny/Nx=0.0,Nxy/Nx=2.0

Buck

ling

para

met

er (N

xa2 /E

2h2 )


Figure 4.17 (c) Effect of different parameters on critical buckling load


118

0.00 0.05 0.10 0.15 0.202

4

6

8

10

12

14

Ply orientation : 0/90/0 (Material : II)Boundary : SSSS


Buck

ling

para

met

er (N

Xa2 /E

2h3 )

NY/NX=1.0,NXY/NX=0.0 NY/NX=1.0,NXY/NX=0.5 NY/NX=1.0,NXY/NX=1.0 NY/NX=1.0,NXY/NX=2.0

Figure 4.17 (d) Effect of different parameters on critical buckling load


4.3.3.2.1 Simply supported square angle-ply laminate under axial compression

In this example, an angle-ply square laminated plate (45/-45/-45/45) subjected to axial compressions (Ny /Nx = 0.0 and 1.0) is analysed with the proposed element (mesh size: 20x20). The analysis is carried out taking thickness ratio (h/a) of 0.01, 0.02, 0.05, 0.10 and 0.25. The buckling load parameters in the present analysis obtained by RHSDT, HSDT and FSDT are presented in Table 4.34, which shows the relative performance of the different theories. The material properties used for the individual layers are E1/E2 = 25, G12 = G13 =0.5E2, G23 = 0.2E2, ν12 = 0.25.


119

Table 4.34 Simply supported square angle-ply laminate (45/-45/-45/45) under axial compression


Ny/Nx References 0.01 0.02 0.05 0.10 0.25

Present (RHSDT) 31.314 30.483 26.184 16.623 4.994 Present (HSDT) 31.339 30.562 26.376 16.533 4.610

0.0

Present (FSDT) 31.393 30.749 27.119 17.656 5.123 Present (RHSDT) 15.679 15.266 13.141 9.175 3.374 Present (HSDT) 15.692 15.306 13.238 9.204 3.222

Simply supported

1.0


0.0


Clamped

1.0

Present (FSDT) 31.283 29.685 22.612 13.018 4.111

4.3.3.2.2 Simply supported square plate having orthotropic layers

In this example the laminated plate has two identical orthotropic face layers of thickness 0.1h each and an orthotropic layer of thickness 0.8h at the core where the elastic moduli of the face layers are expressed in terms of ratio (Rf) of those of the core. The material properties used for the core are E22/E11 = 0.543, G12/E11 = 0.2629, G13/E11 = 0.1599, G23/E11 = 0.2668, ν12 = 0.314. In the present study, the above ratio (Rf) is varied from 1.0 to 15.0 taking thickness ratio (h/a) of the plate as 0.05, 0.10 and 0.20. Under the action of uni-axial compression Nx, the plate is analysed with the proposed element using mesh sizes (Fig. 4.15) of 4x4, 8x8, 12x12, 16x16, 20x20 and 24x24 to show the convergence. The buckling load parameters obtained in the present analysis are presented in Table 4.35 with some available results, which include 3-D elasticity solution (Srinivas and Rao 1970), series solution based on RFSDT (Di Sciuva 1986) and thin plate theory. The table shows that the present results have better agreement with the 3-D elasticity solution of Srinivas and Rao (1970). In this problem, the distribution of Nx across plate thickness is varied in the different layers to have identical in-plane strain of all the layers, which is not so in other examples.


120

Table 4.35 Buckling load parameters (λ = 12Nxb2/π2E11h2) of a square sandwich plate having orthotropic layers under uniaxial compression

R h/a References 1 2 5 10 15

Present (4x4) 2.952 3.629 4.683 5.292 5.420 Present (8x8) 2.965 3.645 4.703 5.314 5.443 Present (12x12) 2.968 3.649 4.708 5.319 5.448 Present (16x16) 2.969 3.650 4.709 5.320 5.449 Present (20x20) 2.970 3.651 4.710 5.321 5.450 Present (24x24) 2.970 3.651 4.710 5.321 5.450 Srinivas and Rao (1970) 2.966 - - - -

0.05

Thin plate theory 3.039 - - - - Present (4x4) 2.756 3.313 4.026 4.184 4.001 Present (8x8) 2.768 3.326 4.041 4.197 4.013 Present (12x12) 2.771 3.330 4.044 4.200 4.016 Present (16x16) 2.772 3.331 4.046 4.201 4.017 Present (20x20) 2.772 3.331 4.046 4.202 4.017 Present (24x24) 2.772 3.331 4.046 4.202 4.017 Srinivas and Rao (1970) 2.770 3.330 4.046 4.200 4.037 Di Sciuva (1986) 2.843 3.410 4.136 4.283 4.106

0.10

Thin plate theory 3.039 3.768 4.984 5.852 6.283 Present (4x4) 2.204 2.490 2.476 1.888 1.492 Present (8x8) 2.207 2.498 2.483 1.918 1.479 Present (12x12) 2.208 2.499 2.485 1.887 1.478 Present (16x16) 2.208 2.500 2.486 1.888 1.478 Present (20x20) 2.209 2.501 2.486 1.888 1.478 Present (24x24) 2.209 2.501 2.486 1.888 1.478 Srinivas and Rao (1970) 2.210 - - - -

0.20

Thin plate theory 3.039 - - - -

4.3.3.2.3 Rectangular sandwich plate with different boundary conditions

This problem of ordinary sandwich plate is studied under three different boundary conditions. In the first case the plate is simply supported at the four edges (SSSS) and subjected to uni-axial compression Nx, while it carries Nx as well as Ny (= 0.5 Nx) with all the edges clamped (CCCC) in the second case. In the last case the plate is under the action of uniform in-plane shear Nxy, where two opposite edges are simple supported and other edges are clamped (SCSC). The study is made with four different values of aspect ratio (a/b = 0.5, 0.7, 1.0 and 2.0) taking b = 596.9mm (see Fig. 4.2). The thickness of each isotropic face layers (E = 65500.2N/mm2 and ν = 0.25) is hf = 0.5334mm while the core (Gxz = Gyz =


121

131.0N/mm2 with negligible in-plane rigidity) has a thickness of 4.597mm. The plate is analysed with the proposed element (mesh sizes of 12x12, 16x16 and 20x20 and the values of buckling stresses obtained are presented in Table 4.36.

Table 4.36 Buckling stresses (σcr in N/mm2) of a rectangular sandwich plate having different boundary conditions

Aspect ratio (a/b) Boundary condition References

0.5 0.7 1.0 2.0 Uniaxial compression (σcr = Nx/2hf)

Present (12x12) 73.096 54.828 49.305 49.184 Present (16x16) 73.161 54.853 49.319 49.247 Present (20x20) 73.192 54.865 49.326 49.278 Khatua and Cheung (1973b) 73.305 54.924 49.373 -

SSSS

Yuan and Dawe (2001) 73.227 54.877 49.334 - Biaxial compression (σcr = Nx/2hf)

Present (12x12) 169.13 110.74 80.72 66.55 Present (16x16) 169.57 110.90 80.79 66.72 Present (20x20) 169.79 110.98 80.83 66.81 Khatua and Cheung (1973b) 170.91 112.41 81.45 -

CCCC

Yuan and Dawe (2001) 170.11 111.15 80.95 - In-plane shear (σcr = Nxy/2hf)

Present (12x12) 255.67 171.40 134.03 108.65 Present (16x16) 256.09 171.47 134.07 109.03 Present (20x20) 256.29 171.50 134.07 109.19

SCSC

Yuan and Dawe (2001) 256.80 172.00 134.60 - Some of these results are compared with those of Khatua and Cheung (1973b) and Yuan and Dawe (2001), which show that the present results have very good agreement with the published results.

4.3.3.2.4 Simply supported multi-core rectangular sandwich plates

A number of soft layers (core) and stiff sheets are stacked alternately to form this multi-core sandwich plate where the two exterior face layers are always stiff sheets. The study is made for two-core sandwich plate having three stiff layers and three-core sandwich plate with four stiff layers. The thickness and Poisson’s ratio of each isotropic stiff layers (E = 206700 N/mm2) are hf = 0.635mm and ν = 0.25 for the two-core sandwich plate while those of the three-core sandwich plate are hf = 0.508mm and ν = 0.3. Each core (Gxz = Gyz = 68.9 N/mm2 with negligible in-plane rigidity) has thickness of 7.62 mm and 5.08mm for the two-core and


122

three-core sandwich plates respectively. In the present study the aspect ratios taken for the rectangular plate (Fig. 4.2) are b/a = 0.4, 0.7, 1.0 and 1.2 where b = 254 mm. Under the action of uni-axial compression (Nx), the plate is analysed with the proposed element using mesh sizes of 12x12, 16x16 and 20x20. The values of buckling load parameters obtained in the present analysis are presented with those of Khatua and Cheung (1973b) in Table 4.37, which shows that results obtained by the two sources are closed to each other.

Table 4.37 Buckling load parameter (λ = Nxb2/Eh3) of a simply supported multi-core rectangular sandwich plate

References Present

Number of Layers

Aspect ratio (b/a)

12x12 16x16 20x20

Khatua and Cheung (1973b)

0.4 1.295 1.296 1.297 1.301 0.7 0.764 0.764 0.764 0.766 1.0 0.690 0.691 0.691 0.693

5

1.2 0.719 0.719 0.719 0.726 0.4 1.207 1.208 1.209 1.213 0.7 0.708 0.709 0.709 0.711 1.0 0.639 0.640 0.640 0.642

7

1.2 0.665 0.665 0.665 0.668

4.3.3.2.5 Simply supported rectangular sandwich plate with laminated face sheets

The problem of a sandwich plate is studied by taking the face sheets as cross-ply laminates (0/90/0/90/C/90/0/90/0) as well as angle-ply laminates (θ/-θ/θ/-θ/C/-θ/θ/-θ/θ). In the laminated sheets, each play has same thickness and material properties in its off-axis system (E1=181GPa, E2=10.3GPa, G12=7.17GPa and ν12=0.28). The core (G13=0.146GPa and G23=0.0904GPa with negligible in-plane rigidities) has a thickness of hc = 10h/11. The plate having cross-ply laminated face sheets is subjected to bi-axial compression where the ratio of the in-plane forces are taken as Ny/Nx = 0.0, 1.0 and 2.0. For the other plate (θ/-θ/θ/-θ/C/-θ/θ/-θ/θ) under uni-axial compression (Nx), the fiber orientations in the laminated face sheets are varied (θ = 00, 150, 300 and 450). Taking a/b = 0.5, 1.0 and 2.0 (see Fig. 4.2) and hc/a = 0.05, 0.10 and 0.20, the analysis is carried out with the proposed


123

element using mesh size of 20x20 in all the cases. The values of buckling non-dimensional load parameters obtained by the proposed element are presented with some available results (Rais-Rohani and Marcellier 1999) in Table 4.38 and Table 4.39. The results of Rais-Rohani and Marcellier (1999) are found to be quite close to the present results.

Table 4.38 Buckling load parameter (λ = Nxb2/E2hc3) of a simply supported

rectangular sandwich plate (0/90/0/90/C/90/0/90/0) with cross-ply laminated face sheets

Aspect ratio (a/b) Ny/Nx hc/b References 0.5 1.0 2.0

Present element 11.148 5.839 5.830 0.01 Rais-Rohani and Marcellier (1999) 11.086 5.830 5.830

0.10 Present element 1.590 1.590 1.590

0.0

0.20 Present element 0.436 0.433 0.433 0.01 Present element 8.919 2.919 2.239 0.10 Present element 1.045 0.881 0.680

1.0


2.0

0.20 Present element 0.159 0.138 0.125

Table 4.39 Buckling load parameter (λ = Nxa2/E2hc

3) of a simply supported rectangular sandwich plate (θ/-θ/θ/-θ/C/-θ/θ/-θ/θ) with angle ply laminated face sheets

Face sheet fiber orientation (θ) hc/a a/b References 00 150 300 450

2.0 Present (20x20) 12.441 18.399 31.749 34.582 Present (20x20) 5.803 6.809 8.777 9.711 1.0 Rais-Rohani and Marcellier (1999)

- - - 9.702

Present (20x20) 4.485 4.376 3.937 3.091

0.01

0.5 Rais-Rohani and Marcellier (1999)

4.484 - - -

2.0 Present (20x20) 1.771 1.762 1.730 1.675 1.0 Present (20x20) 1.579 1.626 1.604 1.504

0.10

0.5 Present (20x20) 0.400 0.408 0.402 0.377 2.0 Present (20x20) 0.507 0.501 0.489 0.469 1.0 Present (20x20) 0.445 0.440 0.432 0.419

0.20

0.5 Present (20x20) 0.112 0.113 0.109 0.105


124

4.3.3.2.6 Double core skew sandwich plate with laminated stiff sheets

The problem studied in this example is skew sandwich plate (Fig. 4.3) having double core construction with laminated stiff sheets (45/-45/C/0/90/90/0/C/-45/45) where each ply in the laminated stiff sheets has same thickness and each core has thickness of hc = 5h/11. The material properties of these ply and core are same as those used in the previous example. Under the action of bi-axial compression (Ny/Nx = 0.0, 0.5, 1.0 and 2.0), the plate is analysed with the

proposed element using mesh size (see Fig. 4.3) of 20x20 taking skew angle (α) of 00, 150, 300, 450 and 600 (Fig. 4.3) and thickness ratio (h/a) of 0.01, 0.05, 0.10 and 0.15. The buckling load parameters (Nxb2/E2h3) obtained in the present

analysis are plotted against α in Figure 4.18 (a) – 4.18 (d).

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16Thickness ratio (hc/a) = 0.01

Nx/Ny = 0.0 Nx/Ny = 0.5 Nx/Ny = 1.0 Nx/Ny = 2.0

Buck

ling

load

par

amet

ers

Skew angles

Figure 4.18 (a) Effect of skew angle and stiff sheets ply orientations on buckling of a multi-core sandwich plate (h/a = 0.01)


125

0 10 20 30 40 50 600.0

0.5

1.0

1.5

2.0

2.5

3.0Thickness ratio (hc/a) = 0.05


Buck

ling

load

par

amet

ers

Skew angles

Figure 4.18 (b) Effect of skew angle and stiff sheets ply orientations on buckling of a multi-core sandwich plate (h/a = 0.05)

0 10 20 30 40 50 600.0

0.2

0.4

0.6

0.8

1.0Thickness ratio (hc/a) = 0.10


Buck

ling

load

par

amet

ers

Skew angles

Figure 4.18 (c) Effect of skew angle and stiff sheets ply orientations on buckling of a multi-core sandwich plate (h/a = 0.10)


126

0 10 20 30 40 50 600.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Thickness ratio (hc/a) = 0.20


Buck

ling

load

par

amet

ers

Skew angles

Figure 4.18 (d) Effect of skew angle and stiff sheets ply orientations on buckling of a multi-core sandwich plate (h/a = 0.20)

4.4 PLATES HAVING IMPERFECT INTERFACES

The behavior of composites and sandwich laminates having imperfect interfaces is studied by solving a number of numerical examples having different features using the proposed element with its full capability i.e., linear spring layer model is invoked with RHSDT. There is no published result on the problem of sandwich plates having imperfection at the layer interfaces. So the validation of the proposed approach is carried out with available results on sandwich plates without imperfections and ordinary composite plates with imperfections. Finally, the present element is applied to the analysis of imperfect sandwich laminates and a number of new results are generated for future references.


127

4.4.1 Static Analysis

4.4.1.1 Cylindrical bending of a cross-ply (0/90/0) laminate

Cheng et al. (1997) have solved the problem analytically taking imperfection at all the interfaces, which may not necessarily be so in the present approach. To assess the performance of the proposed FE model, it is used to solve this problem. The simply supported plate (Fig. 4.2, b/a = 3) is subjected to a distributed load of

intensity ( )0 sin /q q x aπ= . In this problem, all the layers are of equal thickness

and possess same material properties in their material axis system (E1 = 25E, E2 = E, G12 = G13 = 0.5E, G23 = 0.2E and ν12 = 0.25). The imperfections at the layer

interfaces are defined by the parameters: 11 22 /k kR R Rh E= = and 12 21 0.0k kR R= =

where the non-dimensional parameter R is varied from 0.0 to 0.6 (R = 0.0 for perfect interface). This is also valid for other examples of ordinary composite plates. Taking thickness ratio (h/a) of 0.01, 0.10 and 0.25, the plate is analysed by the present element using mesh sizes (Fig. 4.2) of 4×4, 8×8, 12×12, 16x16, 20x20, 24x24 and 28×28. The values of central deflection,

( )3 40100 /cw wEh q a= obtained by the proposed element are presented in Table

4.40 with the analytical solution of Cheng et al. (1997) for perfect as well as imperfect interfaces. For the case of perfect interfaces, the results based on three-dimensional elasticity solution of Pagano (1970) are also included in Table 4.40, which shows that the present results are in excellent agreement with those of Cheng et al. (1997) as well as Pagano (1970). The convergence of present results with mesh refinement is also very good.

4.4.1.2 Simply supported cross-ply (0/90/90/0) square laminate

The problem studied by Cheng et al. (1996b) is taken in this example. The plate is subjected to a distributed load of intensity ( ) ( )0 sin / sin /=q q x a y bπ π .


128

Table 4.40 Central deflection ( cw ) of a cross-ply (0/90/0) laminate subjected to a distributed load of sinusoidal variation (cylindrical bending)

Central deflection ( cw ) h/a References R = 0.0 R = 0.2 R = 0.4 R = 0.6

Present (4x4) 2.8496 3.4361 3.9719 4.4468 Present (8x8) 2.7706 3.3551 3.8950 4.3741 Present (12x12) 2.7618 3.3468 3.8872 4.3667 Present (16x16) 2.7593 3.3445 3.8850 4.3647 Present (20x20) 2.7583 3.3436 3.8841 4.3638 Present (24x24) 2.7575 3.3429 3.8833 4.3630 Present (28x28) 2.7575 3.3429 3.8833 4.3630 Cheng et al. (1997) 2.7567 3.3419 3.8825 4.3622

0.25

Pagano (1970) 2.820 - - - Present (4x4) 1.0077 1.1563 1.3207 1.4981 Present (8x8) 0.9334 1.0811 1.2450 1.4221 Present (12x12) 0.9248 1.0725 1.2365 1.4137 Present (16x16) 0.9224 1.0701 1.2341 1.4113 Present (20x20) 0.9214 1.0691 1.2331 1.4103 Present (24x24) 0.9205 1.0683 1.2323 1.4095 Present (28x28) 0.9205 1.0683 1.2323 1.4095 Cheng et al. (1997) 0.9197 1.0674 1.2314 1.4086

0.10


0.05


0.01

Pagano (1970) 0.508 - - -

The analysis is carried out by the proposed element using mesh sizes of 20×20 taking h/a = 0.25, 0.10 and 0.01 and R = 0.0, 0.2 and 0.6. The values of non-


129

dimensional central deflection cw ; transverse shear stress ( )0/xz xzh q aτ τ= at x =

0, y = b/2 and z = 0; transverse shear stress ( )0/yz yz h q aτ τ= at x = a/2, y = 0 and

z = 0.0; in-plane normal stress ( )2 20/x xh q aσ σ= at x = a/2, y = b/2 and z = h/2;

in-plane normal stress ( )2 20/y y h q aσ σ= at x = a/2, y = b/2 and z = h/4 and in-

plane shear stress ( )0/xy xyh q aτ τ= at x = 0, y = 0 and z = h/2 obtained are

presented in Table 4.41 with those of Cheng et al. (1996b) and Pagano (1970). Table 4.41 shows that the agreement between the results is excellent.

Table 4.41 Central deflections and stresses of a simply supported cross-ply (0/90/90/0) square laminate

h/a References cw xσ yσ xyτ xzτ yzτ

R=0 Present element 1.9065 0.7461 0.7044 0.0436 0.2079 0.3158 Cheng et al. (1996b) 1.9060 0.7368 0.7000 0.0434 0.2109 0.3148

0.25

Pagano (1970) 1.9540 0.7200 0.6630 0.0458 0.2190 0.2920 Present element 0.7364 0.5681 0.4106 0.0275 0.3006 0.2006 Cheng et al. (1996b) 0.7359 0.5611 0.4081 0.0274 0.3002 0.1995

0.10

Pagano (1970) 0.7430 0.5590 0.4010 0.0276 0.3010 0.1960 Present element 0.4351 0.5449 0.2727 0.0214 0.3399 0.1419 Cheng et al. (1996b) 0.4346 0.5389 0.2711 0.0214 0.3388 0.1390

0.01

Pagano (1970) 0.4385 0.5390 0.2710 0.0214 0.3390 0.1390 R=0.2

Present element 2.4816 0.8960 0.7815 0.0495 0.1950 0.2834 0.25 Cheng et al. (1996b) 2.4811 0.8850 0.7761 0.0494 0.1931 0.2818 Present element 0.8621 0.5893 0.4525 0.0296 0.2913 0.2121 0.10 Cheng et al. (1996b) 0.8615 0.5820 0.4498 0.0294 0.2887 0.2108 Present element 0.4367 0.5451 0.2734 0.0215 0.3404 0.1402 0.01 Cheng et al. (1996b) 0.4361 0.5390 0.2718 0.0214 0.3386 0.1392

R=0.6 Present element 3.6667 1.2147 0.9682 0.0627 0.1605 0.1905 0.25 Cheng et al. (1996b) 3.6662 1.2004 0.9609 0.0625 0.1582 0.1894 Present element 1.1637 0.6506 0.5375 0.0341 0.2686 0.2231 0.10 Cheng et al. (1996b) 1.1632 0.6426 0.5342 0.0339 0.2669 0.2213 Present element 0.4405 0.5458 0.2753 0.0216 0.3407 0.1412 0.01 Cheng et al. (1996b) 0.4400 0.5395 0.2731 0.0215 0.3381 0.1399


130

Now the through thickness variation of non-dimensional in-plane

displacement 2 0/E u q h at x = 0 and y = b/2; in-plane normal stress 0/x qσ at x =

a/2 and y = b/2; transverse shear stress 0/xz qτ at x = 0 and y = b/2 and in-plane

shear stress 0/xy qτ at x = 0 and y = 0 obtained in the present analysis (mesh size:

20x20) are plotted in the Figures 4.19 (a) - 4.22 (b) where R = 0.0, 0.2, 0.4, 0.6 and 0.9; and h/a = 0.25 and 0.10. In all these plots, the results based on elasticity solution of Pagano (1970) are included to compare the present results for the perfect cases (R = 0.0). The plots are found to follow a proper trend as expected.

4.4.1.3 Simply supported un-symmetric cross-ply (90/0/90/0) square laminate

The problem of an un-symmetric cross-ply (90/0/90/0) square laminate (Figure 4.2, a = b) with imperfections at all layer interfaces studied by Cheng and Kitipornchai (2000) is investigated by the proposed element. The plate is simply supported at the four sides and subjected to a distributed load of intensity

( ) ( )0 sin / sin /=q q x a y bπ π . The analysis is carried out with a mesh size of

20x20 taking R = 0.0, 0.2, 0.3, 0.4 and 0.6; and h/a = 0.1 and 0.25. The non-

dimensional values of central deflection cw , in-plane normal stress xσ (x = a/2, y

= b/2 and z = h/2), in-plane shear stress xyτ (x = 0, y = 0 and z = h/2) and

transverse shear stress xzτ (x = 0, y = b/2 and z = 0) obtained by the proposed

element are presented in Table 4.42. Some of the present results are compared with those of Cheng and Kitipornchai (2000) in Table 4.42, which show that the agreement between them is very good.


131

-0.500

-0.375

-0.250

-0.125

0.000

0.125

0.250

0.375

0.500

-8 -6 -4 -2 0 2 4 6 8

Composite plate (0/90/90/0)h/a = 0.10

Pagano(1970), R=0 Present, R=0.0 Present, R=0.2 Present, R=0.4 Present, R=0.6 Present, R=0.9

Normalised in-plane displacement

Nor

mal

ised

dep

th

Figure 4.19 (a) Variation of in-plane displacement across the depth of a laminate

(h/a = 0.10)

-0.500

-0.375

-0.250

-0.125

0.000

0.125

0.250

0.375

0.500

-1.0 -0.5 0.0 0.5 1.0



Normalised in-plane displacement

Nor

mal

ised

dep

th

Figure 4.19 (b) Variation of in-plane displacement across the depth of a laminate

(h/a = 0.25)


132

-0.500

-0.375

-0.250

-0.125

0.000

0.125

0.250

0.375

0.500

-70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70




Nor

mal

ised

dep

th

Figure 4.20 (a) Variation of in-plane normal stress across the depth of a laminate

(h/a = 0.10)

-0.500

-0.375

-0.250

-0.125

0.000

0.125

0.250

0.375

0.500

-25 -20 -15 -10 -5 0 5 10 15 20 25




Nor

mal

ised

dep

th

Figure 4.20 (b) Variation of in-plane normal stress across the depth of a laminate

(h/a = 0.25)


133

-0.500

-0.375

-0.250

-0.125

0.000

0.125

0.250

0.375

0.500

-4 -3 -2 -1 0 1 2 3 4




Nor

mal

ised

dep

th

Figure 4.21 (a) Variation of in-plane shear stress across the depth of a laminate

(h/a = 0.10)

-0.500

-0.375

-0.250

-0.125

0.000

0.125

0.250

0.375

0.500

-1.0 -0.5 0.0 0.5 1.0




Nor

mal

ised

dep

th

Figure 4.21 (b) Variation of in-plane shear stress across the depth of a laminate

(h/a = 0.25)


134

-0.500

-0.375

-0.250

-0.125

0.000

0.125

0.250

0.375

0.500

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0



Normalised transverse shear stress

Nor

mal

ised

dep

th

Figure 4.22 (a) Variation of transverse shear stress across the depth of a laminate

(h/a = 0.10)

-0.500

-0.375

-0.250

-0.125

0.000

0.125

0.250

0.375

0.500

0.00 0.25 0.50 0.75 1.00 1.25 1.50



Normalised transverse shear stress

Nor

mal

ised

dep

th

Figure 4.22 (b) Variation of transverse shear stress across the depth of a laminate

(h/a = 0.25)


135

Table 4.42 Central deflection ( cw ) and stresses ( xσ , xyτ and xzτ ) of a simply supported un-symmetric cross-ply (90/0/90/0) square laminate

R h/a References cw xσ xyτ xzτ

Present element 1.9488 0.7798 0.0438 0.2388 Cheng and Kitipornchai (2000) 1.9484 - - 0.2375

0.25

Pagano (1970) 1.9581 0.7444 0.0465 0.2325 Present element 0.7553 0.5409 0.0283 0.2740 Cheng and Kitipornchai (2000) 0.7550 - - 0.2730

0.0

0.10

Pagano (1970) 0.7624 0.5309 0.0291 0.2713 0.25 Present element 2.5364 0.9166 0.0522 0.2133 0.2 0.10 Present element 0.8773 0.5688 0.0300 0.2689

Present element 2.8325 0.9869 0.0564 0.1999 0.25 Cheng and Kitipornchai (2000) 2.8322 - - 0.1984 Present element 0.9460 0.5848 0.0310 0.2659

0.3

0.10 Cheng and Kitipornchai (2000) 0.9457 - - 0.2649

0.25 Present element 3.1235 1.0566 0.0607 0.1863 0.4 0.10 Present element 1.0195 0.6019 0.0321 0.2627

Present element 3.6756 1.1909 0.0688 0.1598 0.25 Cheng and Kitipornchai (2000) 3.6753 - - 0.1583 Present element 1.1791 0.6396 0.0344 0.2555

0.6

0.10 Cheng and Kitipornchai (2000) 1.1787 - - 0.2545 Present element 4.3976 1.3701 0.0795 0.1235 0.25 Cheng and Kitipornchai (2000) 4.3974 - - 0.1220 Present element 1.4446 0.7030 0.0382 0.2432

0.9

0.10 Cheng and Kitipornchai (2000) 1.4443 - - 0.2422

4.4.1.4 Simply supported sandwich plate (f/c/f) having orthotropic face sheets

A square sandwich plate (Fig. 4.2, a = b) simply supported at the four sides and

subjected to distributed load of intensity ( ) ( )0 sin / sin /=q q x a y bπ π is analysed

by the proposed element using mesh sizes of 16×16, 20×20 and 24×24 taking h/a = 0.01, 0.10 and 0.25. The plate has a central core (E1 = E2 = 0.04E, G12 = 0.016E,

G13 = G23 = 0.06E and ν12 = 0.25) of thickness 0.8h and an orthotropic stiff layer

(E1 = 25E, E2 = E, G12 = G13 = 0.5E, G23 = 0.2E and ν12 = 0.25) of thickness 0.1h at each face. The imperfections at the interfaces between core and stiff face layers

are defined by 11 22 /k kR R Rh E= = and 12 21 0.0k kR R= = where the value of R is


136

taken as 0.0, 0.3 and 0.6 in the present study. The values of non-dimensional

central deflection cw , in-plane normal stress xσ (x = a/2, y = b/2 and z = h/2), in-

plane shear stress xyτ (x = 0, y = 0 and z = h/2) and transverse shear stress xzτ (x

= 0, y = b/2 and z = 0.4h) obtained by the proposed element are presented in Table 4.43. For the perfect cases (R = 0.0), the present results are compared with those obtained from the elasticity solution of Pagano (1970) in Table 4.43, which shows the agreement between the results. The new results for the imperfect cases (R = 0.3 and 0.6) follow the expected trend of variation.

4.4.1.5 Laminated sandwich plate having different boundary conditions

The problem of a square (Fig. 4.2, b = a) sandwich plate (0/90/0/C/0/90/0) having stiff laminated face sheets (0/90/0) under uniformly distributed load of intensity q0 is studied for different boundary conditions and imperfections at the interfaces between core and face sheets. The core has a thickness of 0.85h while it is 0.025h for each ply in the face sheets. The material properties of the core and those of a ply in the face sheets are identical to those used in the previous example. The different boundary conditions taken are SSSS (all the sides simply supported), SCSC (simple supported at x = 0 and x = a; and clamped at the other sides) and CCCC (all the sides clamped). The imperfections are taken only at the interfaces

between core and face sheets where these are defined by 11 22 1 /k kR R R h E= = for

the upper interface, 11 22 2 /k kR R R h E= = for the lower interface and 12 21 0.0k kR R= =

for both these interfaces. The analysis is carried out with a mesh size of 20x20 taking h/a = 0.20 and different values of R1 and R2 to have a number of combinations for the imperfections (see Table 4.44). The values of non-

dimensional central deflection cw , in-plane normal stress xσ (x = a/2, y = b/2 and

z = h/2) and transverse shear stress xzτ (x = 0, y = b/2 and z = 0.425h) obtained in the present analysis are presented in Table 4.44. For SSSS boundary condition,


137

the present results corresponding to R1 = R2 = 0.0 (perfect cases) are compared with those obtained from the elasticity solution of Pagano (1970) in Table 4.44 which shows these results have very good agreement. The other results are presented for the first time.

4.4.1.6 Double core rectangular sandwich plate with laminated stiff sheets

The problem of a simply supported rectangular (Fig. 4.2) sandwich plate (-45/45/C/0/90/C/-45/45) subjected to uniformly distributed load of intensity q0 is studied in this example. It has a double core construction where thickness of each core is 0.44h. The stiff face sheets are angle ply laminates (-45/45) and the central stiff sheet is a cross ply laminate (0/90) where thickness of each ply in these laminates is 0.02h. The material properties of the core layers and those of a ply in these laminates are identical to those used in the previous example. In this case, the imperfections are taken only at the interfaces between the core layers and laminated sheets where the imperfections for all these interfaces are defined by

11 22 /k kR R Rh E= = and 12 21 0.0k kR R= = . The plate is analysed with the proposed

element (mesh size: 20x20) taking different values of thickness ratio (h/a), aspect ratio (b/a) and imperfection parameter (R). The values of non-dimensional central

deflection cw , in-plane normal stress xσ (x = a/2, y = b/2 and z = h/2) and

transverse shear stress xzτ (x = 0, y = b/2 and z = 0.02h) obtained by the proposed element are presented in Table 4.45.


138

Table 4.43 Central deflection and stresses of a simply supported single core (f/c/f) square sandwich plate subjected to sinusoidal variation of load

h/a References cw xσ xyτ xzτ

R = 0.0 Present (16x16) 7.6269 1.5949 0.1483 0.2153 Present (20x20) 7.6265 1.5841 0.1484 0.2280 Present (24x24) 7.6262 1.5775 0.1484 0.2322

0.25

Pagano (1970) 7.5960 1.5120 0.1481 0.2354 Present (16x16) 2.2022 1.1821 0.0718 0.2786 Present (20x20) 2.2015 1.1736 0.0717 0.2794 Present (24x24) 2.2011 1.1686 0.0716 0.2968

0.10

Pagano (1970) 2.2004 1.1520 0.0717 0.2974 Present (16x16) 0.8948 1.1226 0.0440 0.3092 Present (20x20) 0.8939 1.1143 0.0439 0.3116 Present (24x24) 0.8935 1.1095 0.0438 0.3259

0.01

Pagano (1970) 0.8924 1.0980 0.0437 0.3222 R = 0.3

Present (16x16) 8.0593 1.6337 0.1530 0.1291 Present (20x20) 8.0589 1.6227 0.1531 0.1318

0.25


0.10


0.01

Present (24x24) 0.8946 1.1096 0.0438 0.3259 R = 0.6

Present (16x16) 8.4989 1.6739 0.1576 0.0261 Present (20x20) 8.4985 1.6627 0.1577 0.0290

0.25


0.10


0.01

Present (24x24) 0.8957 1.1097 0.0439 0.3258

4.4.1.7 Skew laminated sandwich plate simply supported at the four edges

This problem of laminated sandwich plate (0/90/C/0/90) under uniformly distributed load of intensity q0 is studied for different skew angle α (Fig. 4.3). The thickness of the core is 0.8h and of each ply at the face sheets of the cross ply laminates (0/90) is 0.05h. The material properties are same as in the previous


139

example. The imperfections are also taken at the interfaces between core and laminated face sheets in the same manner. The analysis is carried out (mesh: 20×20) taking α = 00, 150, 300, 450 and 600, h/a = 0.1 and R = 0.0, 0.2, 0.4, 0.6 and 0.9. Transformations of axes along the skew edges are done as before. The values of central deflection cw , in-plane normal stress xσ (x = a/2, y = b/2 and z

= h/2) and transverse shear stress ( )0/yz yzh q aτ τ= (x = a/2, y = 0 and z = 0.4h)

obtained are presented in Table 4.46. The present results corresponding to R = 0.0 (perfect case) and α = 00 are compared with those obtained from the elasticity solution of Pagano (1970). These are found to be sufficiently close to each other.

Table 4.44 Central deflection and stresses of a laminated sandwich plate (0/90/0/C/0/90/0) having different boundary conditions

Interfacial parameters

Boundary conditions References cw xσ xzτ

Present element 6.7444 2.0479 0.3484 SSSS Pagano (1970) 6.7576 1.9816 0.3514

SCSC Present element 5.6843 1.7504 0.3077

R1 = 0.0 R2 = 0.0

CCCC Present element 4.8696 0.9028 0.3296 SSSS Present element 6.8494 2.0579 0.3483 SCSC Present element 5.7863 1.7588 0.3076

R1 = 0.0 R2 = 0.2


R1 = 0.2 R2 = 0.2


R1 = 0.4 R2 = 0.2


R1 = 0.6 R2 = 0.2


R1 = 0.8 R2 = 0.2


R1 = 1.0 R2 = 0.2

CCCC Present element 5.4346 0.9416 0.4792


140

Table 4.45 Double core laminated (-45/45/C/0/90/C/-45/45) rectangular sandwich plate subjected to uniformly distributed load

Aspect ratio (b/a)

Thickness ratio (h/a)

References cw xσ xzτ

R = 0.0 0.25 Present element 8.8688 1.0070 0.1856 0.10 Present element 2.8050 0.9392 0.2613 0.01 Present element 1.6662 0.9457 0.2945


R = 0.9 0.25 Present element 12.3801 1.0559 0.1228 0.10 Present element 3.3684 0.9424 0.2494

1.0

0.01 Present element 1.6718 0.9456 0.2911 R = 0.0




1.5

0.01 Present element 3.4065 1.4552 0.3848 R = 0.0




2.0

0.01 Present element 5.0117 1.9200 0.4512


141

Table 4.46 Skew laminated sandwich plate (0/90/C/0/90) simply supported at all the edges

Skew Angle References cw xσ yzτ

R = 0.0 Present element 2.6300 1.6283 0.3600 00

Pagano (1970) 2.6357 1.6207 0.3521 150 Present element 2.3870 1.4718 0.2644 300 Present element 1.7706 1.0652 0.1477 450 Present element 1.0549 0.5886 0.0692 600 Present element 0.4787 0.2517 0.0320

R = 0.2 00 Present element 2.6903 1.6303 0.3591

150 Present element 2.4446 1.4748 0.2639 300 Present element 1.8196 1.0698 0.1468 450 Present element 1.0898 0.5931 0.0680 600 Present element 0.4976 0.2541 0.0314








142

4.4.2 Free Vibration Analysis

4.4.2.1 Simply supported cross-ply laminate having variable layer thickness

The vibration of the square plate (0/90/0) is studied for different degrees of imperfections at the layer interfaces taking thickness ratio (h/a) of the plate as 0.1. The imperfections taken at both the interfaces are characterized by

11 22 11/k kR R Rh E= = , where the non-dimensional parameter R is varied from 0.0 to

0.9. The values 12kR and 21

kR are taken as zero and it is followed in all the

subsequent examples. In this laminate the central layer has a thickness of 0.8h while that is 0.1h for both the outer layers. The elastic moduli taken for the thick layer are E22/E11 = 0.543, G12 /E11 = 0.2629, G13 /E11 = 0.1599, G23 /E11 = 0.2668 whereas these elastic moduli for the thin layers are varied and these are taken as a factor (Kt) of those of the central layer. The value of ν12 is taken as 0.3 for all the layers. The plate is analysed with the proposed element using mesh sizes of 8x8, 12x12, 16x16, 20x20 and 24x24 (full plate) in all the cases to assess convergence rate with respect to mesh density. The fundamental frequency parameters Ω = 100ω√(ρh2/E11) obtained in the present analysis are presented in Table 4.47 with the three dimensional elasticity solution of Srinivas and Rao (1970) and series solution of Di Sciuva (1993) based on RFSDT for perfect interfaces. The results obtained by Cheng et al (1997) are also included in Table 4.47 to compare the present results for imperfect interfaces. The table shows that the present results have very good agreement with the published results.

4.4.2.2 Initially stressed cross-ply laminate simply supported at the edges

The problem of an initially stressed symmetric (0/90/90/0) square laminate having imperfections at all the layer interfaces is studied in this example. The uniaxial in-plane initial stress is assumed in the x-direction. Taking Cr = 0.75, 0.50, 0.25, 0.0, -0.25, -0.50 and –0.75 (the factor Cr is taken as the ratio between the initial load and critical buckling load; Cr =0.0 corresponds to zero initial load), the analysis is carried out by the proposed element using h/a = 0.10 (mesh: 20×20, full plate).


143

All the layers are of equal thickness and possess same material properties in their material axis system (E1 = 40E, E2 = E, G12 = G13 = 0.6E, G23 = 0.5E and ν12 = 0.25). The imperfections at the layer interfaces are defined by the parameters:

11 22 /k kR R Rh E= = and 12 21 0.0k kR R= = where the non-dimensional parameter R is

varied from 0.0 to 0.6. The present results in the form of fundamental frequency parameter (Ω) for different situations are presented in Table 4.48. As there is no published results on the present problem, the present results corresponding to Cr = 0.0 and R = 0.0 for thickness ratio, h/a = 0.10 are compared in the same table with the published results of Wu et al. (1994) who used a local higher order theory and Cho et al. (1991) who used individual-layer plate theory for their analytical solutions. All other results in Table 4.48 based on the present element are presented for the first time.

4.4.2.3 Simply supported square sandwich plate with laminated face sheets

The vibration of the laminated sandwich plate (0/90/../C/../90/0) is studied taking imperfections at the interfaces between low strength core and high strength laminated face sheets. In the stiff face sheets the number of equal thickness ply in each laminate is taken as 2 and 8, which are stacked symmetrically with respect to the core. The thickness of the core (E/Ec = 11.945, Gc12 /Ec = Gc13 /Ec = 1.173/6.279, Gc23 /Ec = 2.415/ 6.279, νc12 = 0.0025 and ρ/ρc= 0.6818) is taken as 0.8h, while that of each face sheets (E11/E = 40.0, E22/E = 1.0, G12 /E = G13 /E = G23 /E = 1.0, ν12 = 0.25, ρf = ρ) is 0.1h. The imperfection is characterized by

11 22 /k kR R Rh E= = where R is varied as in the previous example. The plate is

analysed using mesh size of 20x20 (full plate) taking thickness ratio (h/a) of 0.01 and 0.10. The frequency parameters Ω = 100ωa√(ρc/E11) obtained in the present analysis for first six modes are presented in Table 4.49 with some results of Wang et al. (2000) for perfect cases. The results of Wang et al. (2000) are on the higher side compared to present results as observed and explained before. However, they follow a similar trend.


144

Table 4.47 Fundamental frequency parameters (Ω) of a simply supported square laminated plate having variable layer thickness

R References 0.00 0.20 0.40 0.60 0.90

Kt = 1 Present analysis (8x8) 4.7387 4.7321 4.7255 4.7188 4.7086 Present analysis (12x12) 4.7408 4.7343 4.7277 4.7209 4.7107 Present analysis (16x16) 4.7416 4.7351 4.7285 4.7217 4.7115 Present analysis (20x20) 4.7422 4.7357 4.7290 4.7223 4.7121 Present analysis (24x24) 4.7422 4.7357 4.7290 4.7223 4.7121 Di Sciuva (1993) 4.7698 - - - - Cheng et al. (1997) 4.7405 4.7340 4.7275 4.7209 - Srinivas and Rao (1970) 4.7419 - - - -





145

Table 4.48 Fundamental frequency parameters (Ω) of a simply supported initially stressed cross ply square laminate (0/90/90/0)

Cr References 0.75 0.50 0.25 0.00 -0.25 -0.50 -0.75

R = 0.0

Present (20x20) 7.559 10.690 13.093 15.118 16.903 18.516 20.000

Wu et al. (1994) - - - 15.069 - - -

Cho et al. (1991) - - - 15.066 - - -

R = 0.2

Present (20x20) 6.570 9.257 11.337 13.138 14.637 16.034 17.318

R = 0.6

Present (20x20) 5.279 7.193 8.810 10.510 11.373 12.459 13.457

4.4.2.4 Square sandwich plate with laminated face sheets having clamped edges

The vibration of the sandwich plate (0/90/C/90/0) having laminated face sheet of thickness 0.1h each (ply thickness 0.05h) is studied taking clamped boundary condition at the four edges (CCCC) in one case while it is simply supported at the two opposite edges and clamped at the remaining edges (SCSC) in the other case. The imperfection is taken in a similar manner as that of last example but the value of R is taken as 0.0, 0.9, 1.2 and 1.8. Taking h/a = 0.05, 0.10 and 0.20, the analysis is carried out with mesh size of 20x20 (full plate). The first six frequency

parameters Ω =100ωa√(ρc/E11) obtained by the proposed element are presented in Table 4.50 with some results obtained by Wang et al. (2000) for perfect cases. The agreement between the results is similar to that found in Table 4.49.


146

Table 4.49 Frequency parameters (Ω) of a simply supported square sandwich plate (0/90/../C/../90/0) with laminated face sheets

Mode number h/a R Na Reference 1 2 3 4 5 6

2 Present 2.2186 5.5928 6.3777 8.3599 10.8467 12.6391 0.0 8 Present 2.2175 5.7869 6.1724 8.3451 11.2865 12.6671 2 Present 2.2177 5.5854 6.3643 8.3467 10.8197 12.6089 0.2 8 Present 2.2165 5.7792 6.1702 8.3320 11.2584 12.6295 2 Present 2.2167 5.5780 6.3548 8.3334 10.7927 12.5787 0.4 8 Present 2.2155 5.7715 6.1611 8.3188 11.2303 12.5916 2 Present 2.2156 5.5705 6.3452 8.3201 10.7657 12.5484 0.6 8 Present 2.2145 5.7637 6.1519 8.3055 11.2022 12.5535 2 Present 2.2140 5.5591 6.3305 8.2992 10.7249 12.5019

0.01

0.9 8 Present 2.2130 5.7520 6.1379 8.2855 11.1601 12.4960

Present 10.1805 14.9134 19.6634 20.8018 22.4906 24.3906 2 Wang et al. (2000)

10.555 16.830 19.648 23.616 - -

Present 10.1198 15.4679 19.3421 22.4467 22.6176 24.3905

0.0

8 Wang et al. (2000)

10.704 16.823 20.476 24.240 - -

2 Present 10.0809 14.7856 19.4041 20.6373 22.2246 24.3906 0.2 8 Present 10.0237 15.3429 19.0960 22.2876 22.3650 24.3905 2 Present 9.9832 14.6600 19.1517 20.4759 21.9654 24.3905 0.4 8 Present 9.9293 15.2201 18.8566 22.1190 22.1314 24.3905 2 Present 9.8872 14.5366 18.9059 20.3175 21.7130 24.3905 0.6 8 Present 9.8366 15.0995 18.6235 21.8795 21.9782 24.3905 2 Present 9.7464 14.3556 18.5491 20.0849 21.3458 24.3905

0.10

0.9 8 Present 9.7006 14.9226 18.2853 21.5315 21.7536 24.3905

a Number of ply in each laminated face sheet

4.4.2.5 Double core simply supported triangular laminated sandwich plate

The vibration of the triangular sandwich plate (45/-45/C/0/90/C/45/-45) as shown Figure 4.13 is studied taking aspect ratio (b/a) of the plate as 1.0, 1.5 and 2.0. The thickness of each core is taken as 0.425h, while that of each laminated sheet is 0.05h (ply thickness = 0.025h). The material properties of a core and a ply in the stiff sheets are same as those used in the last example. The imperfections are

defined by 11 22 1 /k kR R R h E= = for the two upper interfaces and 11 22 2 /k kR R R h E= =

for the two lower interfaces. The values of R1 and R2 are taken to be same (i.e., R1 = R2 = R). With the mesh arrangement as shown in Figure 4.13, the plate is


147

analysed with mesh size of 20x20 taking thickness ratios (h/a) of the plate as 0.05, 0.10 and 0.20. The first six frequency parameter Ω =100ωa√(ρc/E11) obtained by the proposed element are presented in Table 4.51. The effect of imperfection is found in all the modes and it increases with plate thickness ratio as observed earlier.

Table 4.50 Frequency parameters (Ω) of a square sandwich plate with laminated face sheets (0/90/C/90/0) having clamped edges

Mode Number h/a R Boundary conditions

References

1 2 3 4 5 6 Present 8.648 13.646 17.741 19.793 20.694 24.391 SCSC Wang et al. (2000)

9.098 15.441 17.442 21.840 - -

Present 10.580 14.851 18.893 20.570 21.705 26.059

0.0

CCCC Wang et al. (2000)

10.930 16.498 18.650 22.483 - -

SCSC Present 8.401 13.244 16.919 19.188 19.818 24.216 0.9 CCCC Present 10.118 14.304 17.910 19.899 20.689 24.962 SCSC Present 8.321 13.115 16.660 18.996 19.542 23.960 1.2 CCCC Present 9.973 14.132 17.606 19.648 20.311 24.624 SCSC Present 8.1634 12.863 16.163 18.622 19.011 23.311

0.05

1.8 CCCC Present 9.695 13.801 17.030 19.222 19.776 23.969 SCSC Present 10.573 15.576 19.862 21.901 22.954 24.391 0.0 CCCC Present 11.713 16.307 20.753 22.387 23.778 28.430 SCSC Present 10.129 15.018 18.747 21.184 21.815 24.391 0.9 CCCC Present 11.122 15.649 19.582 21.604 22.584 27.180 SCSC Present 9.989 14.842 18.403 20.958 21.464 24.391 1.2 CCCC Present 10.941 15.446 19.225 21.361 22.219 26.797 SCSC Present 9.721 14.505 17.754 20.528 20.801 24.391

0.10

1.8 CCCC Present 10.599 15.060 18.555 20.898 21.533 26.076 SCSC Present 11.754 17.515 21.053 24.391 24.896 25.459 0.0 CCCC Present 12.660 18.128 22.649 25.896 26.319 32.457 SCSC Present 11.187 16.883 19.848 23.693 24.391 24.726 0.9 CCCC Present 12.022 17.444 21.389 25.055 25.126 31.184 SCSC Present 11.013 16.689 19.481 23.328 24.391 24.502 1.2 CCCC Present 11.829 17.236 21.012 24.677 24.893 30.803 SCSC Present 10.684 16.324 18.794 22.647 24.083 24.391

0.20

1.8 CCCC Present 11.470 16.848 20.312 23.977 24.457 30.101


148

Table 4.51 Frequency parameters (Ω) of a double core triangular sandwich plate (45/-45/C/0/90/C/45/-45) with laminated stiff sheets

Mode Number b/a R h/a References

1 2 3 4 5 6 0.05 Present 14.033 22.056 26.630 30.410 34.358 38.558 0.10 Present 16.975 25.082 29.543 33.226 37.424 41.260

0.0

0.20 Present 18.544 26.755 31.423 35.164 39.646 43.731 0.05 Present 13.185 20.494 24.525 28.058 31.453 35.440 0.10 Present 15.582 22.874 26.761 30.251 33.777 37.559

0.9


1.0

1.2


0.0


0.9


1.5

1.2


0.0


0.9


2.0

1.2

0.20 Present 12.687 16.939 20.809 22.825 24.709 27.627

4.4.2.6 Initially stressed laminated sandwich plate simply supported at edges

The problem of free vibration analysis of a simply supported square pre-stressed sandwich plate (0/90/C/90/0) having two laminated stiff layers at the two faces with imperfections only at the interfaces between the stiff face sheets and the core


149

is taken in this example. The stiff layers are cross-ply laminates where ply in each laminate is of equal thickness, which are stacked symmetrically with respect to the core. The thickness of the central core is taken as 0.8h, while that of each stiff layer is 0.1h. The plate is analysed with the proposed element using mesh size (full plate) of 20x20 taking thickness ratio (h/a) of 0.01, 0.10 and 0.20. The parameter Cr is taken as 0.50, 0.00 and –0.50 while the imperfection parameter (R) as defined before is varied from 0.0 to 0.60 (Table 4.52). The non-

dimensional frequency parameters Ω = 100ωa√(ρc/E11) for the first six modes obtained in the present analysis are presented with some results of Wang et al. (2000) in Table 4.52. Wang et al. (2000) have taken two different angles of rotation for the face and the core i.e., the usual modeling technique for sandwich plate and solved the problem analytically for perfect interfaces and zero pre-stress condition. This plate model gives linear variation of in-plane displacements across the thickness with kink at the interface between the core and face sheets. It does not satisfy the inter-laminar shear stress continuity condition as well as shear stress free condition at the top and bottom surfaces of the plate. Table 4.52 shows that the values of present results are somewhat lower that those of Wang et al. (2000) as expected. However, the present results are quite consistent in comparison with the results of Wang et al. (2000). The material properties for this example and all subsequent examples taken for the core and a ply of the face sheets are as follows.

Face: E11/E = 40.0, E22/E = 1.0, G12 /E = G13 /E = G23 /E = 1.0, ν12 = 0.25, ρf = ρ

Core: E/Ec = 11.945, Gc12 /Ec = Gc13 /Ec = 1.173/6.279, Gc23 /Ec = 2.415/6.279, νc12

= 0.0025 and ρ/ρc= 0.6818.


150

Table 4.52 Frequency parameters (Ω) of a simply supported square initially stressed sandwich plate with laminated face sheets (0/90/C/90/0)

Mode h/a Cr References 1 2 3 4 5 6

R = 0.0 +0.50 Present (20x20) 1.569 5.368 5.548 7.749 10.733 12.240 +0.00 Present (20x20) 2.219 5.593 6.378 8.360 10.847 12.639 -0.50 Present (20x20) 2.717 5.809 7.104 8.929 10.960 13.023


R = 0.6 +0.50 Present (20x20) 1.567 5.346 5.518 7.708 10.651 12.116 +0.00 Present (20x20) 2.216 5.571 6.345 8.320 10.766 12.548

0.01

-0.50 Present (20x20) 2.714 5.787 7.077 8.891 10.879 12.934 R = 0.0

+0.50 Present (20x20) 7.437 13.194 13.904 17.678 19.606 21.106 Present (20x20) 10.181 14.913 19.663 20.802 22.491 24.391 +0.00 Wang et al. [25] 10.555 16.830 19.648 23.616 - -

-0.50 Present (20x20) 12.328 16.454 21.933 24.083 24.391 24.391 R = 0.4

+0.50 Present (20x20) 7.336 13.003 13.542 17.294 19.324 20.491 +0.00 Present (20x20) 9.983 14.660 19.152 20.476 21.965 24.391 -0.50 Present (20x20) 12.063 16.148 21.567 23.456 24.391 24.391


0.10

-0.50 Present (20x20) 11.935 16.000 21.389 23.155 24.391 24.391 R = 0.0

+0.50 Present (20x20) 8.518 14.353 14.682 18.858 21.824 22.475 +0.00 Present (20x20) 11.245 16.122 20.764 23.026 23.900 24.391 -0.50 Present (20x20) 13.429 17.715 24.169 24.391 24.391 25.430



0.20

-0.50 Present (20x20) 12.944 17.195 23.596 24.391 24.391 24.419


151

4.4.3 Buckling Analysis

4.4.3.1 Simply supported cross-ply laminate having variable layer thickness

The problem is same as that of section 4.4.2.1. For buckling analysis, the plate is subjected to Nx where it has different distributions at the different layers to have identical in-plane strain in all the layers. The critical buckling load parameters (λ= 12Nxb2/π2E11h2) obtained are shown in Table 4.53 with the solution of Srinivas and Rao (1970) and Di Sciuva (1993) for perfect interfaces. The results obtained by Cheng et al (1996b) are also included in Table 4.53 to compare the present results for imperfect interfaces. The table shows that the agreement between the present results is very good with the published results.

4.4.3.2 Simply supported double core rectangular laminated sandwich plate

The buckling of a sandwich plate (0/90/C/0/90/C/0/90) having double cores with three laminated sheets is studied taking imperfections at the four interfaces between the cores and the stiff laminated sheets. The thickness of each core is 0.425h while that of each laminated sheet is 0.05h (i.e., ply thickness is 0.025h). The material properties of a core and a ply in the stiff sheets are same as those in

the last example. The imperfections are defined by 11 22 1 /k kR R R h E= = for the two

upper interfaces and 11 22 2 /k kR R R h E= = for the two lower interfaces where the

values of R1 and R2 are varied. Under the action of bi-axial compression (Ny/Nx = 0.0, 1.0 and 2.0), the plate is analysed with the proposed element (mesh size: 20x20) taking thickness ratio (h/b) of the plate as 0.01, 0.05 and 0.20; and aspect ratio (a/b) of 0.5, 1.0 and 2.0. The buckling load parameters (λ = Nxb2/Eh3) obtained are presented in Table 4.54. It is observed that the effect of imperfection is more in plates having higher thickness ratio and lower value of Ny/Nx.


152

Table 4.53 Buckling load parameters (λ) of a square sandwich plate having variable layer thickness under uniaxial compression

R References 0.00 0.20 0.40 0.60 0.90

Kt = 1 Present analysis (6x6) 2.7652 2.7554 2.7477 2.7399 2.7281 Present analysis (8x8) 2.7683 2.7586 2.7509 2.7431 2.7313 Present analysis (12x12) 2.7711 2.7611 2.7534 2.7456 2.7337 Present analysis (16x16) 2.7723 2.7621 2.7544 2.7465 2.7347 Present analysis (20x20) 2.7722 2.7625 2.7548 2.7470 2.7351 Present analysis (24x24) 2.7722 2.7625 2.7548 2.7470 2.7351 Di Sciuva (1993) 2.8432 - - - - Cheng et al. (1996b) 2.8077 2.7999 2.7919 2.7839 - Srinivas and Rao (1970) 2.770 - - - -

Kt = 5 Present analysis (6x6) 4.0372 3.9513 3.8664 3.7807 3.6514 Present analysis (8x8) 4.0412 3.9553 3.8703 3.7844 3.6548 Present analysis (12x12) 4.0441 3.9585 3.8734 3.7873 3.6575 Present analysis (16x16) 4.0462 3.9597 3.8745 3.7884 3.6585 Present analysis (20x20) 4.0461 3.6903 3.8750 3.7889 3.6590 Present analysis (24x24) 4.0461 3.6903 3.8750 3.7889 3.6590 Di Sciuva (1993) 4.1362 - - - - Cheng et al. (1996b) 4.1183 4.0318 3.9439 3.8554 - Srinivas and Rao (1970) 4.046 - - - -

Kt = 10 Present analysis (6x6) 4.1841 4.0172 3.8543 3.6942 3.4618 Present analysis (8x8) 4.1932 4.0260 3.8622 3.7014 3.4680 Present analysis (12x12) 4.1971 4.0295 3.8654 3.7043 3.4705 Present analysis (16x16) 4.2002 4.0324 3.8672 3.7066 3.4725 Present analysis (20x20) 4.2013 4.0332 3.8689 3.7075 3.4732 Present analysis (24x24) 4.2013 4.0332 3.8689 3.7075 3.4732 Di Sciuva (1993) 4.2022 4.0337 3.8693 3.7079 3.4736 Cheng et al. (1996b) 4.2833 - - - - Srinivas and Rao (1970) 4.2711 4.1006 3.9321 3.7670 -

Kt = 15 Present analysis (6x6) 4.0011 3.8069 3.5368 3.2455 2.8696 Present analysis (8x8) 4.0103 3.8138 3.5445 3.2514 2.8731 Present analysis (12x12) 4.0133 3.8167 3.5486 3.2546 2.8754 Present analysis (16x16) 4.0162 3.8189 3.5522 3.2575 2.8776 Present analysis (20x20) 4.0173 3.8197 3.5536 3.2587 2.8784 Present analysis (24x24) 4.0173 3.8197 3.5536 3.2587 2.8784 Di Sciuva (1993) 4.0172 3.8201 3.5543 3.2593 2.8789 Cheng et al. (1996b) 4.1063 - - - - Srinivas and Rao (1970) 4.0977 3.8743 3.6591 3.4535 -


153

Table 4.54 Buckling load parameters (λ) of a simply supported double core rectangular sandwich plate with stiff laminated sheets (0/90/C/0/90/C/0/90)

Aspect ratio (a/b) Ny/Nx h/b References 0.5 1.0 2.0

R1 = R2 = 0.0 0.01 Present 19.7397 10.1281 10.0853 0.05 Present 9.4745 6.9486 6.9312

0.0

0.20 Present 1.0327 1.0299 1.0314 0.01 Present 15.7920 5.0640 4.0427 0.05 Present 7.2017 3.4743 2.6399

1.0


2.0

0.20 Present 0.3189 0.2745 0.2505 R1 = 0.3, R2 = 0.9

0.01 Present 19.5573 10.1000 10.0573 0.05 Present 8.5383 6.6337 6.6176

0.0


1.0


2.0

0.20 Present 0.2981 0.2543 0.2286 R1 = 0.3, R2 = 1.8

0.01 Present 19.4120 10.0787 10.0360 0.05 Present 7.9296 6.4038 6.3888

0.0


1.0


2.0

0.20 Present 0.3198 0.2547 0.2210

4.4.3.3 Sandwich plate with different fiber orientations at laminated stiff sheets

The buckling of a simply supported square sandwich plate (θ/-θ/θ/-θ/C/-θ/θ/-θ/θ) under uniaxial compression (Nx) is studied for different fiber orientations (θ = 00, 150, 300 and 450) in the laminated face sheets. The core (G13 = 0.146GPa and G23

= 0.0904GPa, negligible in-plane rigidities) has a thickness of hc = 10h/11


154

whereas each ply in laminated face sheets has same thickness and material properties (E1=181GPa, E2=10.3GPa, G12=7.17GPa and ν12=0.28). The imperfection is taken at the interfaces between core and stiff face sheets and it is

defined by 11 22 2/k kR R Rh E= = where R is varied from 0.0 to 2.0. Taking hc/a =

0.01, 0.05, 0.10, 0.15 and 0.20, the analysis is carried out (mesh: 20x20) and the buckling load parameters (λ = Nxa2/E2hc

3) obtained are presented in Table 4.55. Amongst the different cases presented here, only one case with perfect interfaces has been investigated by Rais-Rohani and Marcellier (1999) and the result obtained by them is included in Table 4.55. It agreed well.

Table 4.55 Buckling load parameters (λ) of a square sandwich plate with different fiber orientations (θ/-θ/θ/-θ/C/-θ/θ/-θ/θ) at laminated stiff sheets

Face sheet fiber orientations hc/a R References 00 150 300 450

Present analysis 5.8025 6.8088 8.7773 9.7113 0.0 Rais-Rohani and Marcellier (1999) - - - 9.7024

0.9 Present analysis 5.7830 6.7952 8.7748 9.7104 1.2 Present analysis 5.7798 6.7916 8.7699 9.7096 1.5 Present analysis 5.7766 6.7878 8.7641 9.7078

0.01

2.0 Present analysis 5.7711 6.7816 8.7563 9.6981 0.0 Present analysis 3.4323 3.9842 4.7863 4.2999 0.9 Present analysis 3.3560 3.8959 4.6786 4.1693 1.2 Present analysis 3.3309 3.8668 4.6432 4.1269 1.5 Present analysis 3.3060 3.8380 4.6081 4.0849

0.05


0.10


0.15


0.20

2.0 Present analysis 0.3995 0.3953 0.3875 0.3754


155

4.4.3.4 Skew sandwich plate with laminated face sheets clamped at the edges

The buckling of the skew sandwich plate (-45/45/45/-45/C/-45/45/45/-45) having equal sides is studied for different values of skew angle (00, 150, 300, 450, 600and 750) taking thickness ratio (h/a) of 0.05 and 0.10. The material properties and thickness of core and a ply in the laminated face sheets are same as those taken in the previous example. The imperfection is also taken in a similar manner as that of previous example. Taking R = 0.0, 0.9, 1.2 and 1.8, the analysis is carried out with the proposed element using a mesh size of 20x20 for all the cases. The buckling load parameters (λ = Nxa2/E2hc

3) obtained in the present analysis are plotted against skew angles in Figure 4.23 (a) – 4.23 (b).

0 10 20 30 40 50 60 70 80

5.0

5.5

6.0

6.5

7.0

7.5

Thickness ratio (h/a) = 0.05

R = 0.0 R = 0.9 R = 1.2 R = 1.8

Buck

ling

load

par

amet

ers

Skew Angle

Fig. 4.23 (a) Effect of skew angle on buckling of a laminated sandwich plate having

inter-laminar imperfections (h/a = 0.05)


156

0 10 20 30 40 50 60 70 80

1.6

1.8

2.0

2.2

Thickness ratio (h/a) = 0.10

R = 0.0 R = 0.9 R = 1.2 R = 1.8

Buck

ling

load

par

amet

ers

Skew Angle

Figure 4.23 (b) Effect of skew angle on buckling of a laminated sandwich plate

having inter-laminar imperfections (h/a = 0.10)

Chapter

FIVE

CLOSURE

5.1 SUMMARY AND CONCLUSIONS

5.1.1 Summary

The investigation reported in the thesis may be summarized as follows: • An efficient layer-wise finite element plate model has been proposed

for the analysis of composites and sandwich laminates with inter-laminar imperfections.

• Refined higher order shear deformation theory (RHSDT) in combination with linear spring layer model is used in the proposed plate model.

• According to this plate model, the deformation of the plate can be expressed in terms of unknowns at the reference plane (i.e., plate middle plane) only.

• The proposed finite element model is also capable to produce results based on HSDT, RFSDT and FSDT. These results are used to compare the merit of different plate theories.

• For modeling of imperfect interfaces, slip/jumps in the in-plane displacements are taken at the layer interfaces to characterize these imperfections.

Closure

158

• A new triangular finite element has been developed where the formulation is done in such a manner that the problem lying with the satisfaction of inter-elemental continuity requirement of the plate theory is not manifested.

• The element has three corner nodes and three mid side nodes where each node contains seven physical quantities as degrees of freedom.

• The displacement approach is followed to do the formulation of this new element.

• The finite element model is applied to static, free vibration and buckling analysis of laminated composites and sandwich plates with or without inter-laminar imperfections. The present results are compared with the published results in many cases for validation.

• Some new problems such as skew plates, triangular plates, multi-core sandwich plates with laminated stiff layers etc. are framed and these problems are solved to generate new results.

5.1.2 Conclusions

Based on the present investigation, the following conclusions may be drawn: • The proposed finite element plate model has the capability of an

accurate modeling of laminated plates with/without inter-laminar imperfections.

• As the plate model retains unknowns at the reference plane only, it is computationally as efficient as any single layer plate theory with the abovementioned capability.

• The triangular element developed for the present purpose is quite elegant from the point of its use and computer implementation.

• The range of applicability of the proposed finite element model is demonstrated by analyzing plates of different boundary condition,

Closure

159

loading, geometry, thickness ratio, stacking sequence, aspect ratio and other parameters.

• For laminated sandwich plates, a large number of new results are generated, which should minimize the scarcity of available results on this problem.

• For imperfect sandwich laminates, the new results reported in the thesis should be really useful for future references, as there is no published result on this problem so far. Even in case of composite laminates with imperfect interfaces, the number of available results is not many and this paucity of result is brought down by presenting a number of new results.

• For the first time, the degree of imperfections is varied arbitrarily at the different layer interfaces to generate some new interesting results.

• The effect of imperfection is found to be more prominent in case of static problems compared to that in vibration and buckling.

• It is observed that the severest effect of imperfection is on inter-laminar shear stresses.

5.2 SCOPE OF FUTURE RESEARCH

The following are some of the possible areas of research where the present investigation can be extended in future:

• In the present investigation, the study has been confined to the linear analysis. The analysis with the effect of geometric and material non-linearity is a potential area of research.

• The present formulation can be extended to forced vibration analysis for steady state as well as transient response of composite and sandwich laminates with or without inter-laminar imperfections.

Closure

160

• It may also be extended to the random vibration analysis.

• Experimental investigations must be carried out to verify the accuracy and reliability of the proposed interface imperfection models.

• The present investigation can be extended to the dynamic instability analysis.

• It may also be extended to the post-buckling analysis.

• Environmental effects on composite materials are quite important. Change in temperature and moisture causes additional stresses and deformations in the structure and it may have complex variation. The present work can be extended to include such hygro-thermal effects.

• Fluid structure interaction may be a prospective research area if the laminated plate is used in ship and underwater structures.

• Effect of impact loading on laminated plates may be an interesting study extending the present finite element formulation.

• For sandwich laminates having very soft core, the deformation of the core in the transverse direction should be considered. The present study may be extended for taking into account the effect of core deformability.

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About the Author

Anupam Chakrabarti, the author, graduated in Civil Engineering with a first class degree from Bengal Engineering College, Howrah (India) in 1986. He started his professional career as a Design Engineer (Civil) in Consultants Combine Private Limited, Calcutta. In 1988, he joined the Development Consultants Limited (DCL), Calcutta as a Structural Engineer. Here he was responsible for the analysis and design of a few nuclear power plants of India. Later he went for further studies and subsequently obtained his master’s degree in 1991 with specialization in Structural Engineering from the Department of Civil Engineering, Bengal Engineering College. He worked for a brief stint in the analysis wing of the Engineers India Limited (EIL), New Delhi before moving over to the Public Works Department, West Bengal in the same year. Finally, in the year 1998 he shifted to academics to fulfill a long cherished dream as he joined Jalpaiguri Government Engineering College, Jalpaiguri as a faculty member in the Department of Civil Engineering. Recently he has been promoted to the post of Assistant Professor there.

The author has been carrying out his research in the field of Structural Engineering from January 2000 onwards in the Department of Ocean Engineering and Naval Architecture, Indian Institute of Technology, Kharagpur.

List of Publications:

International Journals:

Chakrabarti A. and Sheikh A.H. (2002). A new triangular element based on higher order shear deformation theory for flexural vibration of composite plates. International Journal of Structural Stability and Dynamics; 2(2): 163-184.

Sheikh A.H. and Chakrabarti A. (2003). A New Plate Bending Element Based on Higher Order Shear Deformation Theory for the Analysis of Composite Plates. Finite Elements in Analysis and Design; 39(9), 883-903.

Chakrabarti A. and Sheikh A.H. (2003). Buckling of laminated composite plates by a new element based on higher order shear deformation theory. Mechanics of Advanced Materials and Structures; 10 (4), 303-318.

Chakrabarti A. and Sheikh A.H. (2004). A New Triangular Element to model Inter-Laminar Shear Stress Continuous Plate Theory. International Journal for Numerical Methods in Engineering; 60(6), June (To be published).

Chakrabarti A. and Sheikh A.H. (2004). Vibration of laminate faced sandwich plate by a new refined element. Journal of Aerospace Engineering, ASCE (To be published).

Chakrabarti A. and Sheikh A.H. (2003). Analysis of Laminated Sandwich Plates Based on Inter-Laminar Shear Stress Continuous Plate Theory. Journal of Structural Engineering, ASCE (Communicated).

Chakrabarti A. and Sheikh A.H. (2003). Buckling of laminated sandwich plates by a new refined element. Journal of Engineering Mechanics, ASCE (Communicated).

Chakrabarti A. and Sheikh A.H. (2003). Behavior of laminated sandwich plates having interfacial imperfections by a new refined element. Computational Mechanics (Communicated).

Chakrabarti A., Topdar P. and Sheikh A.H. (2003). Vibration and buckling of laminated sandwich plates having interfacial imperfection. AIAA Journal (Communicated).

Chakrabarti Anupam, Topdar Pijush and Sheikh Abdul Hamid (2003). Vibration of pre-stressed laminated sandwich plates with interlaminar imperfections. Journal of Vibration and Acoustics, ASME (Communicated).

Chakrabarti Anupam and Sheikh Abdul Hamid (2003). Buckling of laminated composite plates subjected to partial edge compression. Composites Part B: Engineering (Communicated).

Chakrabarti Anupam and Sheikh Abdul Hamid (2003). Buckling of laminated sandwich plates subjected to partial edge compression. International Journal of Mechanical Sciences (Communicated).

Topdar P., Chakrabarti A. and Sheikh A. H. (2003). An efficient hybrid plate model for analysis and control of smart sandwich laminates. Computer Methods in Applied Mechanics and Engineering (Communicated).

National Journal:

Chakrabarti A., Sengupta S.K. and Sheikh A.H. (2003). Analysis of skew composite plates using a new triangular element based on higher order shear deformation theory, Journal of the Institution of Engineer, India. (To be published)

International conferences:

Chakrabarti A. and Sheikh A.H. (2001). A new plate bending element based on higher order shear deformation theory for the analysis of composite plates. International Conference on Theoretical, Applied, Computational and Experimental Mechanics (ICTACEM 2001); IIT Kharagpur, India.

Chakrabarti A. and Sheikh A.H. (2002). A new triangular element based on a layer wise zigzag theory for the analysis of composite panels having sandwich construction. International Conference on Ship and Ocean Technology (SHOT 2002), IIT Kharagpur, India.

Chakrabarti A., Sengupta S.K. and Sheikh A.H. (2003). Analysis of composite plates with interfacial slips by a new element based on

refined higher order plate theory, Structural Engineering Convention 2003, An International Meet, IIT Kharagpur, India. (Accepted).

National conferences:

Chakrabarti A., Sengupta S.K. and Sheikh A.H. (2002). A new plate bending element based on higher order shear deformation theory for the analysis of very thin to thick composite laminates. Proceedings of National Conference on Wave Mechanics and Vibration (WMVC 2002), Jalpaiguri, West Bengal, India.

Chakrabarti A., Sengupta S.K. and Sheikh A.H. (2003). Static and vibration analysis of composite plates based on a refined first order shear deformation theory using a new element. Proceedings of National Conference on Wave Mechanics and Vibration (WMVC 2003), Jalpaiguri, West Bengal, India.

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