Efficient mesh freee methods for plate analysis

These problems include bending, free vibration and buckling analyses of thin plates, large-amplitude free vibration of thin plates, free vibration of Mindlin plates, characteristic analy

Effective Mesh-free Methods for Plate Analysis

Wu Wenxin

NATIONAL UNIVERSITY OF SINGAPORE

2007

Effective Mesh-free Methods for Plate Analysis

Wu Wenxin (B Eng., Shanghai Jiaotong University)

A THESIS SUBMITTED

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF MECHANICAL ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2007

To Shu Rong and Jia Xi

I

I would like to express my deepest gratitude to my supervisors, Professor C Shu and

Professor C M Wang, for their invaluable guidance, encouragement and patience

throughout this study

My gratitude also extends to my wife and daughter for their support and

encouragement over my PhD candidature period

Finally, I wish to thank the National University of Singapore for providing me with a

research scholarship, which makes this study possible

Wu Wenxin

II

Acknowledgements I

Table of Contents II

Summary XI

List of Tables XIII

List of Figures XVII

Notations XXV

Abbreviations XXVIII

Chapter 1 Introduction 1

1.1 Background of Plate Analysis 1

1.1.1 Introduction to Plate Theories 1

1.1.2 Analytical Methods for Plate Analysis 4

1.1.3 Numerical Methods for Plate Analysis 6

1.2 Literature Review on Mesh-free Methods 8

1.2.1 Disadvantages of Traditional Numerical Methods 9

1.2.2 Concept of Mesh-free 11

1.2.3 Classification of Mesh-free Methods 13

1.2.3.1 A Particle Method: Smoothed Particle Hydrodynamics (SPH) 17

1.2.3.2 Mesh-free Methods of Integral Type 18

1.2.3.3 Mesh-free Methods of Non-Integral Type 20

1.2.4 Desirable Mesh-free Methods for Plate Analysis 24

1.3 Objectives of Thesis 25

III

Chapter 2 LSFD Method and LRBFDQ Method 29

2.1 Least Squares-based Finite Difference (LSFD) Method 29

2.1.1 Conventional Finite Difference Method (FDM) 29

2.1.2 Least Squares-based Finite Difference (LSFD) Method 30

2.1.2.1 Formulas for Derivative Discretization 30

2.1.2.2 Chain Rule for Discretization of Derivatives 37

2.1.2.3 LSFD Formulas in Local (n, t)-Coordinates at Boundary 39

2.1.2.4 Numerical Analysis of a Sample PDE Using LSFD 41

2.1.3 Function Value Problems and Eigenvalue Problems 45

2.1.4 Concluding Remarks 46

2.2 Local Radial Basis Function-based Differential Quadrature (LRBFDQ) Method 47

2.2.1 Radial Basis Functions (RBFs) and Interpolation Using RBFs 47

2.2.2 Traditional RBF-based Schemes for Solving PDEs 50

2.2.3 Local Radial Basis Function-based Differential Quadrature (LRBFDQ) Method 51

2.2.3.1 Formulas for Derivative Discretization 51

2.2.3.2 LRBFDQ Formulas in Local (n, t)-Coordinates at Boundary 54

2.2.3.3 Numerical Analysis of Sample PDEs Using LRBFDQ 55

2.2.4 Concluding Remarks 72

IV

3.1 TM Modes and TE Modes in Metallic Waveguides – Application of LSFD 74

3.1.1 Definition of Problem 74

3.1.2 Numerical Algorithm 75

3.1.2.1 Numerical Discretization by LSFD 75

3.1.2.2 Dealing with Singular Points on Boundary Γ 78

3.1.3 Results and Discussion 79

3.1.3.1 Rectangular Waveguide 79

3.1.3.2 Double-Ridged Waveguide 81

3.1.3.3 L-Shaped Waveguide 82

3.1.3.4 Single-Ridged Waveguide 83

3.1.3.5 Coaxial Rectangular Waveguide 85

3.1.3.6 Vaned Rectangular Waveguide 86

3.1.4 Concluding Remarks 87

3.2 Free Vibration of Uniform Membranes – Application of LRBFDQ 87

3.2.1 Definition of Problem 87

3.2.2 Numerical Discretization by LRBFDQ 88

3.2.3 Results and Discussion 89

3.2.3.1 Circular Membrane 90

3.2.3.2 Rectangular Membrane 92

3.2.3.3 Half Circle+Triangle Membrane 94

3.2.3.4 L-Shaped Membrane 96

3.2.3.5 Concave Membrane with High Concavity 98

3.2.4 Concluding Remarks 103

Chapter 4 Plate Theories and Numerical Implementation 105

4.1 Thin Plate Theory for Small Deflection Problems 105

4.1.1 Displacement Components 105

4.1.2 Strain-Displacement Relations 106

4.1.3 Stresses and Stress Resultants 107

4.1.4 Differential Equation for Transversely Loaded Plates 109

4.1.5 Differential Equation for Freely Vibrating Plates 113

4.1.6 Differential Equation for Buckling of Plates 114

4.1.6.1 Plates under Combined Transverse and In-Plane Loads 114

4.1.6.2 Buckling of Plates 117

4.1.7 Boundary Conditions 118

4.1.8 Numerical Implementation 120

4.1.8.1 Discretization of Governing Equations 120

4.1.8.2 Implementation of Boundary Conditions 123

4.2 Thin Plate Theory for Large Deflection Problems 133

4.2.1 Bending Equations of Plates with Large Deflections 133

4.2.2 Equations of Motion for Large-Amplitude Free Vibration of Thin Plates 136

4.2.3 Boundary Conditions 137

4.2.4 Numerical Implementation 137

4.2.4.1 Bending of Plates with Large Deflections 137

VI

4.3 Shear Deformable Plate Theory for Small Deflection Problems 143

4.3.1 Displacement Components 144

4.3.2 Strain-Displacement Relations 144

4.3.3 Stress Resultant-Displacement Relations 145

4.3.4 Governing Equations of Motion 146

4.3.5 Governing Equations for Bending 147

4.3.6 Governing Equations for Free Vibration 148

4.3.7 Boundary Conditions 149

4.3.8 Numerical Implementation Using LSFD Method 151

4.3.8.1 Discretization of Governing Equations 151

4.3.8.2 Implementation of Boundary Conditions 152

Chapter 5 LSFD for Thin Plate Vibration 158

5.1 Small-Amplitude Free Vibration of Thin Plates with Arbitrary Shapes 159

5.1.1 Simply Supported and Clamped Plates 160

5.1.1.1 Introduction 160

5.1.1.2 Results and Discussion 162

• Symmetric Trapezoidal Plates 162

• Symmetric Parabolic Trapezoidal Plates 162

• Rhombic Plates 164

• Sectorial Plates 165

• Circular and Elliptical Plates 167

VII

5.1.1.3 Concluding Remarks 170

5.1.2 Completely Free Plates 172

5.1.2.1 Introduction 172

5.1.2.2 Problem Definition and Numerical Solution 173

5.1.2.3 Results and discussion 174

• Frequencies of Circular and Elliptical Plates 175

• Frequencies of Lifting-Tab Shaped and 45o Right Triangular Plates 176

• Verification of Radii of Nodal Circles of the Circular Plate 178

• Verification of Natural Boundary Conditions 179

• Distributions of Mode Shapes and Modal Stress Resultants 183

• Peak Values of Modal Deflections and Modal Stress Resultants 191

5.1.2.4 Concluding Remarks 193

5.2 Large-Amplitude Free Vibration of Thin Plates with Arbitrary Shapes 194

5.2.1 Motivation and Literature Review 194

5.2.2 Results and discussion 196

5.2.2.1 Square Plates 197

5.2.2.2 Circular Plates 201

5.2.2.3 L-Shaped Plate 202

5.2.2.4 Square Plates with Semi-Circular Edge Cuts 204

5.2.3 Concluding Remarks 206

VIII

6.1 Simply Supported Rectangular Plates 209

6.1.1 Sinusoidal Load 209

6.1.1.1 Definition of Problem 209

6.1.1.2 Results and Discussion 210

6.1.2 Uniform Load 214

6.2 Continuous Rectangular Plate 217

6.2.1 Definition of Problem 217

6.2.2 Numerical Algorithm 218

6.2.3 Results and Discussion 222

6.3 Sectorial Plates 229

6.3.1 Problem Definition 229

6.3.2 Numerical Algorithm 230

6.3.3 Results and Discussion 231

Chapter 7 LSFD for Buckling of Highly Skewed Plates 235

7.1 Motivation and Literature Review 235

7.2 Problem Definition 237

7.3 Numerical Algorithm 238

7.4 Results and Discussion 241

7.4.1 Verification of the LSFD Method 241

7.4.1.1 Convergence Study of k-Values for Two SSSS Rhombic Plates with 45 θ = o and θ =80o 241

IX

Angles 243

7.4.2 LSFD Results for Rhombic Plates with Very Large Skew Angles 244

7.5 Concluding Remarks 249

Chapter 8 LSFD for Vibration of Mindlin Plates 250

8.1 Motivation and Literature Review 250

8.2 Problem Definition 252

8.3 Numerical Algorithm 253

8.4 Results and Discussion 254

8.4.1 Varification of LSFD Method 254

8.4.1.1 Comparison Study Using an Elliptical Plate 254

8.4.1.2 Comparison Study Using a Circular Plate 256

8.4.2 Solution of a Rectangular Plate That Models VLFS 259

8.4.2.1 Comparison in Accuracy of LSFD Solution with Ritz Solution 259

8.4.2.2 Comparison in Computational Cost of LSFD Solution with Ritz

Solution 267

8.5 Concluding Remarks 268

Chapter 9 Conclusions and Recommendations 270

9.1 Conclusions 270

9.2 Recommendations for Future Research 275

Appendix 293

List of Publications 300

XI

In recent years, mesh-free methods have been developed for solving partial differential equations (PDEs) effectively These methods are invented in order to overcome difficulties that may be faced when solving PDEs using traditional mesh-based numerical methods These difficulties include (1) modeling of complex domain shapes, (2) accurate approximation of high order derivatives, (3) capturing of steep variations of functions and derivatives, and (4) implementation of multiple boundary conditions and internal restraints These difficulties may be due to (a) the use of mesh along coordinate directions such as in FDM and DQM, or (b) the use of elements such as in FEM and FVM, or (c) the use of global trial functions such as in the Ritz method, or (d) weak form

of PDEs to be solved Most mesh-free methods are used for solving weak form of PDEs When using these methods, the implementation of multiple boundary conditions is still a difficulty since the natural boundary conditions are not enforced in the solution process

The earliest foundation for the development of two mesh-free methods, namely the least squares-based finite difference (LSFD) method and the local radial basis function-based differential quadrature (LRBFDQ) method, was laid by Ding et al (2004) and Shu

et al (2003) In this thesis, the development of these two methods is advanced further, i.e high order schemes of the LSFD method are introduced, the chain rule for discretization

of high order derivatives is proposed, the LSFD and LRBFDQ formulations in terms of local (n, t)-coordination system at boundary are developed, a practical approach for searching a proper value of the shape parameter in the LRBFDQ method is proposed By

XII

free approximation and the enforcement of all boundary conditions, the aforementioned unfavorable factors (a) to (d) have been successfully removed, and consequently the aforementioned difficulties (1) to (4) in solving PDEs have been overcome to a great degree by using these two methods

In this thesis, a wide range of plate problems and other engineering problems with various complexities have been solved These problems include bending, free vibration and buckling analyses of thin plates, large-amplitude free vibration of thin plates, free vibration of Mindlin plates, characteristic analysis of metallic waveguides, and free vibration of uniform membranes The complexities associating with these problems correspond actually to the difficulties (1) to (4) mentioned above in solving PDEs For example, the governing equations for plates in the classical thin plate theory are fourth-order PDEs; the plate shapes are arbitrary; there are two boundary conditions for a plate edge; and in the free edge region of a thin plate, the variation of stresses is very steep, etc These studies demonstrate the accuracy and versatility of the two mesh-free methods after the new developments in this thesis have been made

XIII

Table 2.1 Lower limitation of number of supporting points for different LSFD

schemes 44

Table 3.1 Analytical and LSFD cutoff wavenumbers of TE modes of rectangular waveguide: Case 1: N =2232, F-9) ( ; Case 2: N =2232, F-14) ( ; Case 3: 3341, F-9) N = ( 80

Table 3.2 Cutoff wavenumbers for the double-ridged waveguide 82

Table 3.3 Cutoff wavenumbers of the L-shaped waveguide 83

Table 3.4 Cutoff wavenumbers of the single-ridged waveguide 84

Table 3.5 Cutoff wavenumbers of the coaxial rectangular waveguide 85

Table 3.6 Cutoff wavenumbers of the vaned rectangular waveguide 86

Table 3.7 Comparison of wavenumbers of the circular membrane (R =1) obtained by LRBFDQ method, the exact method, Kang (1999), and FEM 91

Table 3.8 Comparison of wavenumbers of the rectangular membrane (1.2×0.9)

obtained by LRBFDQ method, the exact method, Kang (1999), and FEM 93

Table 3.9 Comparison of wavenumbers of the half circle+triangle membrane obtained by LRBFDQ method, Kang (1999), and FEM 95

Table 3.10 Comparison of wavenumbers of the L-shaped membrane obtained by LRBFDQ method, Kang (2000), and FEM 97

Table 3.11 Comparison of wavenumbers of the concavely shaped membrane obtained

by LRBFDQ method, Kang (2000), and FEM 99

Table 3.12 Comparison of wavenumbers of the multi-connected membrane obtained

by LRBFDQ method, Kang (2000), and FEM 102

Table 5.1 First six vibration frequencies of a symmetric trapezoidal plate 163

Table 5.2 First six vibration frequencies of the symmetric parabolic trapezoidal plate with various boundary conditions 164

Table 5.3 First six vibration frequencies of simply supported rhombic plates 165

Table 5.4 First six vibration frequencies of clamped rhombic plates 166

XIV

Table 5.6 First six vibration frequencies of clamped sectorial plates 168

Table 5.7 First six vibration frequencies of an eccentric sectorial plate 169

Table 5.8 First six vibration frequencies of clamped circular and elliptical plates 169

Table 5.9 First four vibration frequencies of an annular plate 171

Table 5.10 LSFD solution for the first six frequencies of completely free circular and elliptic plates 177

Table 5.11 LSFD solution for the first six frequencies of the completely free lifting-

tab shaped plate 178

Table 5.12 LSFD solution for the first six frequencies of the completely free 45° right triangular plate 178

Table 5.13 Radii of nodal circles ρ=r a for a completely free circular plate 179

Table 5.14 Verification of boundary conditions M = and n 0 V = of the completely n 0 free circular plate 182

Table 5.15 Verification of boundary conditions M = and n 0 V = of the completely n 0 free elliptical plate 183

Table 5.16 Verification of boundary conditions M = and n 0 V = of the completely n 0 free lifting-tab shaped plate 183

Table 5.17 Verification of boundary conditions M = and n 0 V = of the completely n 0 free 45° right triangular plate 183

Table 5.18 Peak values and corresponding locations of modal displacements W ,

modal principal bending moments Mx′ and My′, maximum modal

twisting moments Mx y′′ ′′ and maximum modal shear forces Qx′′′ for completely free circular plate 191

Table 5.19 Peak values and corresponding locations of modal displacements W,

modal principal bending moments Mx′ and My′, maximum modal

twisting moments Mx y′′ ′′ and maximum modal shear forces Qx′′′ for completely free elliptical plate 192

XV

x ′ y ′

twisting moments Mx y′′ ′′ and maximum modal shear forces Qx′′′ for

completely free lifting-tab shaped plate 192Table 5.21 Peak values and corresponding locations of modal displacements W,

modal principal bending moments Mx′ and My′, maximum modal

twisting moments Mx y′′ ′′ and maximum modal shear forces Qx′′′ for

completely free 45° right triangular plate 193

Table 5.22 Linear frequency parameters (λL =ωLa2 ρh D) and period ratios

(TNL TL) of a simply supported square plate 198

Table 5.23 Linear frequency parameters (λL =ωLa2 ρh D) and period ratios

(TNL TL) of a clamped square plate 200

Table 5.24 Linear frequency parameters (λL =ωLa2 ρh D) and period ratios

(TNL TL) for the fundamental modes of a square plate with different

Table 5.26 Linear frequency parameters (λL =ωLa2 ρh D, a = ) and period ratios 2

(TNL T ) for the fundamental mode of the L-shaped plate 203L

Table 5.27 Linear frequency parameters (λL =ωLa2 ρh D) and period ratios

(TNL T ) of the simply supported square plate with edge cuts L

(2r a =0.4) 205

Table 5.28 Linear frequency parameters (λL =ωLa2 ρh D) and period ratios

(TNL T ) of the clamped square plate with edge cuts (L 2r a =0.4) 206

Table 6.1 Comparison between LSFD solution and exact solution to the simply

supported square plate under the sinusoidal lateral load 211Table 6.2 Comparison between LSFD solution and exact solution to the simply

supported square plate under the uniform lateral load 215

XVI

Table 6.4 The extreme values of deflections and stress resultants in the annular

sectorial plate 232Table 6.5 Comparison of bending results for SCSC annular sectorial plate (α =π 6,

very large skew angles (Case I) 244

Table 7.4 Convergent LSFD results for buckling factors k for rhombic plates with

very large skew angles (Case I, upper and/or lower edges are free) 247Table 7.5 Buckling factors k′ for rhombic plates (Case II) 248

Table 8.1 First six frequency parameters for an elliptical Mindlin plate with free

edge 255Table 8.2 First five frequency parameters for a circular Mindlin plate with free edge 257Table 8.3 Frequency parameters Ω and maximum values of stress resultants for a

rectangular Mindlin plate 261

XVII

Fig 2.1 A computational domain with an unstructured distribution of points 31

Fig 2.2 Local (n, t)-coordinate system 40

Figs 2.3 to 2.7 Log10 (error) vs Number of supporting points 44

Figs 2.8 to 2.13 Log10 (error) vs c (MQ) 56

Figs 2.14 to 2.19 Log10 (error) vs c (IMQ) 59

Figs 2.20 to 2.25 Log10 (error) vs c (IQ) 61

Figs 2.26 to 2.29 Log10 (error) vs c (TPS) 63

Figs 2.30 to 2.35 Log10 (error) vs c (Gaussian) 64

Fig 2.36 Perspective view of u 691 Fig 2.37 Perspective view of u 692 Fig 2.38 Perspective view of u 693 Fig 2.39 Perspective view of u 694 Fig 2.40 Perspective view of u 695 Fig 2.41 Log10 (error) vs c for Eq (2.57) (N=1004, m=16) 71

Fig 2.42 Log10 (error) vs c for various PDEs (N=1004, u=u4) 71

Fig 3.1 Singular points on boundary Γ 78

Fig 3.2 A rectangular waveguide 78

Fig 3.3 A double-ridged waveguide 81

Fig 3.4 An L-shaped waveguide 81

Fig 3.5 A single-ridged waveguide 84

Fig 3.6 A coaxial waveguide 84

XVIII

Fig 3.8 Mode shapes of the circular membrane 91

Fig 3.9 Mode shapes of the rectangular membrane 93

Fig 3.10 Geometry of the half circle+triangle membrane 94

Fig 3.11 Mode shapes of the half circle+triangle membrane 95

Fig 3.12 Geometry of the L-shaped membrane 97

Fig 3.13 Mode shapes of the L-shaped membrane 97

Fig 3.14 Geometry of the membrane with high concavity 98

Fig 3.15 Mode shapes of the concavely shaped membrane 99

Fig 3.16 Geometry of the multi-connected membrane 102

Fig 3.17 Mode shapes of the multi-connected membrane 103

Fig 4.1 Part of a plate before and after deflection 106

Fig 4.2 Midplane of a plate element with positive stress resultants and load 110

Fig 4.3 Forces in the midplane of a plate element 115

Fig 4.4 Rotations of the normal planes 144

Fig 4.5 Local n-s coordinates at plate edge 150

Fig 5.1 A symmetric trapezoidal plate 161

Fig 5.2 A symmetric, parabolic, trapezoidal plate 161

Fig 5.3 A rhombic plate 161

Fig 5.4 An eccentric, circular, sectorial plate 161

Fig 5.5 An elliptical/circular plate 161

Fig 5.6 An annular plate 161

Fig 5.7 (a) Lifting-tab shaped plate (a=2b); (b) 45° right triangular plate 175

XIX

Fig 5.9 Verification of boundary conditions Mn =0, Vn =0 for completely free

elliptical plate (a b =2) vibrating in 4th mode 181

Fig 5.10 Verification of boundary conditions Mn =0, Vn =0 for completely free

lifting-tab shaped plate vibrating in 4th mode 181

Fig 5.11 Verification of boundary conditions Mn =0, Vn =0 for completely free

45° right triangular plate vibrating in 4th mode 182Fig 5.12 Modal deflections W for circular plate vibrating in 4th mode 184

Fig 5.13 1st principal modal bending moments Mx′ for circular plate vibrating in

Fig 5.18 1st principal modal bending moments Mx′ for elliptical plate (a b =2)

vibrating in 4th mode 186Fig 5.19 2nd principal modal bending moments My′ for elliptical plate (a b =2)

vibrating in 4th mode 186Fig 5.20 Maximum modal twisting moments Mx y′′ ′′ for elliptical plate (a b =2)

vibrating in 4th mode 187

Fig 5.21 Maximum modal shear forces Qx′′′ for elliptical plate (a b =2) vibrating

in 4th mode 187Fig 5.22 Modal deflections W for lifting-tab shaped plate vibrating in 4th mode 187

XX

Fig 5.24 2nd principal modal bending moments My′ for lifting-tab shaped plate

vibrating in 4th mode 188Fig 5.25 Maximum modal twisting moments Mx y′′ ′′ for lifting-tab shaped plate

vibrating in 4th mode 188

Fig 5.26 Maximum modal shear forces Qx′′′ for lifting-tab shaped plate vibrating in

4th mode 189Fig 5.27 Modal deflections W for 45° right triangular plate vibrating in 4th mode 189

Fig 5.28 1st principal modal bending moments Mx′ for 45° right triangular plate

vibrating in 4th mode 189Fig 5.29 2nd principal modal bending moments My′ for 45° right triangular plate

simply supported square plate under the sinusoidal load q q0sin xsin y

a b

Fig 6.3 Numerical bending moment Mx (left) and analytical bending moment M 0

(right) of the simply supported square plate under the sinusoidal load

Fig 6.5 Numerical shear force Qx (left) and analytical shear force Q 0 (right) of the

simply supported square plate under the sinusoidal load q q0sin xsin y

a b

Fig 6.6 Numerical shear force Qy (left) and analytical shear force Q 0 (right) of the

simply supported square plate under the sinusoidal load q q0sin xsin y

Fig 6.9 Numerical concentrated reaction force Rf (left) and analytical concentrated

reaction force Rf0 (right) of the simply supported square plate under the

sinusoidal load q q0sin xsin y

a b

= 214

Fig 6.10 Numerical deflection w (left) and numerical bending moment Mx (right)

of the simply supported square plate under the uniform load

2

0 10000N/m

q=q = 216

Fig 6.11 Numerical bending moment My (left) and numerical shear force Qx (right)

of the simply supported square plate under the uniform load

2

0 10000N/m

q=q = 216

Fig 6.12 Numerical shear force Qy (left) and numerical effective shear force Vx

(right) of the simply supported square plate under the uniform load

2

0 10000N/m

q=q = 216

XXII

reaction force Rf (right) of the simply supported square plate under the

uniform load q=q0=10000N/m2 217Fig 6.14 A continuous rectangular plate with edge and internal simple supports 217Fig 6.15 Deflections w of the continuous rectangular plate loaded in region 1

(a) 3D view; (b) w at y=0.5m 224

Fig 6.16 Bending moments Mx of the continuous rectangular plate loaded in

region 1 (a) 3D view; (b) Mx at y=0.5m 225

Fig 6.17 Bending moments My of the continuous rectangular plate loaded in

region 1 (a) 3D view; (b) My at y=0.5m 225

Fig 6.18 Twisting moments Mxy of the continuous rectangular plate loaded in

region 1 (a) 3D view; (b) Mxy at y=0 and y=1m 226

Fig 6.19 Transverse shear forces Qx of the continuous rectangular plate loaded in

region 1 (a) 3D view; (b) Qx at y=0.5m 226

Fig 6.20 Transverse shear forces Qy of the continuous rectangular plate loaded in

region 1 227

Fig 6.21 Effective shear forces Vx along edges x=0 and x=3m of the continuous

rectangular plate loaded in region 1 (a) 3D view; (b) V at x x=0; (c) V x

at x=3m 227Fig 6.22 Effective shear forces Vy along edges y=0 and y=1m of the continuous

rectangular plate loaded in region 1 (a) 3D view; (b) V at y y=0 228

Fig 6.23 Concentrated reaction forces Rf at four corners (0, 0), (0, 1), (3, 0) and

(3, 1) of the continuous rectangular plate loaded in region 1 228Fig 6.24 An annular sectorial plate (a) Geometry of the plate; (b) Point distribution 230Fig 6.25 Deflections w of the annular sectorial plate (a) 3D view; (b) w at θ =15o 231Fig 6.26 Bending moments Mr of the annular sectorial plate (a) 3D view; (b) Mr at

15

θ = o 232Fig 6.27 Bending moments Mt of the annular sectorial plate (a) 3D view; (b) Mt at

15

θ = o 233

XXIII

Fig 6.29 Shear forces Qr of the annular sectorial plate (a) 3D view; (b) Qr at θ =15o

233Fig 6.30 Shear forces Qt of the annular sectorial plate (a) 3D view; (b) Qt at θ =0o 234Fig 7.1 Parallelogram plates under in-plane loads 236Fig 7.2 Convergence of buckling factor k of SSSS rhombic plate with θ =45o 242Fig 7.3 Convergence of buckling factor k of SSSS rhombic plate with θ =80o 242Fig 8.1a Modal twisting moment for the 4th mode of vibration of a rectangular plate

with free edges obtained by Ritz method 252Fig 8.1b Modal shear force for the 4th mode of vibration of a rectangular plate with

free edges obtained by Ritz method 252Fig 8.2 4th mode shape (n=2,s=1) of circular plate 257Fig 8.3 Modal stress-resultants of 4th mode (n=2,s=1) of circular plate 258Fig 8.4 Mode shape w for 4th mode corresponding to Ω =1.3557 261Fig 8.5 Bending moment M for 4th mode 262xFig 8.6 Bending moment My for 4th mode 262Fig 8.7 Twisting moment Mxy for 4th mode 262Fig 8.8 Shear force Q for 4th mode 263xFig 8.9 Shear force Qy for 4th mode 263Fig 8.10 Mode shape w for 5th mode corresponding to Ω =3.3033 263Fig 8.11 Bending moment M for 5th mode 264xFig 8.12 Bending moment My for 5th mode 264

XXIV

Fig 8.14 Shear force Q for 5th mode 265xFig 8.15 Shear force Qy for 5th mode 265Fig 8.16 Mode shape w for 6th mode corresponding to Ω =3.7325 265Fig 8.17 Mode shape w for 7th mode corresponding to Ω =6.8270 266Fig A1 -x y coordinate system and -n t coordinate system 299

XXV

c shape parameter in RBFs

D flexural rigidity of plate

E modulus of elasticity

(F-M) LSFD formula for derivative approximation in rectangular coordinates

(F-Ma) LSFD formula for derivative approximation in local (n, t)-coordinates at

boundary

G shear modulus of elasticity

h average point distance, plate thickness

i, j, k etc index of a point (or node)

ij index of the jth supporting point of the point i

ijk index of the kth supporting point of the point ij

m number of supporting points of a point

M twisting moment per unit distance on n plane at boundary

(n t) or (n s) normal and tangential coordinates at boundary

N, Ni, Nb total number of points, number of interior points, number of boundary

points, respectively, in a domain

Q shear force per unit distance on n plane at boundary

q lateral load per unit area

U, V, W modal deflections of a plate or membrane in x, y, and z directions

u, v, w displacements in x, y, and z directions

x, xj coordinates of a general point and the point j respectively

(x yi, i) rectangular coordinates of the point i

Γ boundary of a domain

ψ , ψs rotations of the n and s planes at boundary

Ω problem domain, frequency parameter of a plate

ω natural circular frequency of a plate or membrane

XXVIII

BEM boundary element method

DEM diffuse element method

DQM differential quadrature method

EFG element-free Galerkin method

FDM finite difference method

FEM finite element method

FPM finite point method

FVM finite volume method

GFD generalized finite difference method

IMQ inverse multiquadrics

IQ inverse quadratics

LBIE local boundary integral equation method

LRBFDQ local radial basis function-based differential quadrature method

LSCM least-squares collocation meshless method

LSFD least squares-based finite difference method

MLPG meshless local Petrov-Galerkin method

MLS moving least-squares

MLSDQ moving least-squares differential quadrature method

MLSRK moving least-square reproducing kernel method

MQ multiquadric

MQM multiquadrics method

MQRBF multiquadric radial basis function

XXIX

PDE partial differential equation

PU partition of unity

PUFEM partition of unity finite element method

RBF radial basis function

RKPM reproducing kernel particle methods

SPH smoothed particle hydrodynamics method

TPS thin plate spline

VLFS very large floating structure

Chapter 1

Introduction

1.1 Background of Plate Analysis

1.1.1 Introduction to Plate Theories

Plates are flat structural elements having thicknesses much smaller than the other

length dimensions Plates have been used in many engineering structures, from airplanes

to ships, from offshore structures to structural buildings They form bulkheads, decks,

tanktops, bottoms, side panels, floors, deep girders, etc

The flexural properties and behavior of a plate depend greatly on its thickness in

comparison with its other dimensions Corresponding to this dependence, theories used in

plate analysis may be classified into two types: thin plate theories and shear deformable

(moderately thick) plate theories In plate analysis, an often applied criterion to define a

thin plate is that the ratio of the thickness to the smaller span length is less than 1/20

(Wang 2000a, Reddy 1999, Ugural 1999, Liew et al 1998)

For thin plates with small deflections, a very satisfactory approximate plate model has

been developed by making the following simplifying assumptions (Kirchhoff 1850,

Timoshenko and Woinowsky-Krieger 1959, Wang 2000a, Reddy 1999, Ugural 1999,

Liew et al 1998):

1 The deflection of the midsurface is small when compared to the thickness of the

plate The slope of the deflected surface is very small and the square of the slope

is a negligible quantity in comparison with unity

2 The midplane remains unstrained during bending

3 The stress normal to the midsurface, σz, can be neglected

4 Plane sections initially normal to the midsurface remain plane and normal to that

surface after bending, i.e the transverse shear strains γxz and γyz are neglected

5 For dynamic problems, the effect of rotary inertia is neglected

The above assumptions are known as the Kirchhoff hypotheses and the corresponding

theory is known as the classical thin plate theory or the Kirchhoff plate theory Using

these assumptions, the stress-resultants may be expressed in terms of deflection w (and its

derivatives) which is a function of the two coordinates in the plane of the plate This

function has to satisfy a fourth-order linear partial differential equation (PDE) and proper

boundary conditions The solution of this equation gives all necessary information for

calculating stresses at any point of the plate

For a thin plate with large deflection, the assumptions 1 and 2 of the Kirchhoff

hypotheses need to be modified, that is,

1 The deflection w is comparable with the thickness of the plate But the slope of

the deflected surface is still very small and the square of the slope is a negligible

quantity in comparison with unity

2 Unless the plate is bent into a developable surface, the large deflection results in

strains in the midplane, and the membrane forces must be taken into consideration

in deriving the differential equation of plate

Other assumptions of the Kirchhoff hypotheses remain unchanged In this way one

obtains nonlinear PDEs and the solution of the problem becomes more complicated The

transverse load on a thin plate with large deflection is balanced partly by the flexural

rigidity and partly by the membrane action of the plate Consequently, very thin plates

with negligible resistance to bending behave like membranes, and the governing

differential equations for membranes can be derived from the aforementioned nonlinear

PDEs by neglecting the flexural rigidity

For moderately thick plates, the effect of transverse shear deformation becomes

significant in bending and vibration problems The effect of rotary inertia should also be

considered for obtaining accurate frequencies and mode shapes associated with higher

modes of vibration If the Kirchhoff plate theory is used for solving moderately thick

plate problems, the vibrating frequencies and buckling loads are overpredicted while the

deflections are underpredicted This is because the theory neglects the effect of transverse

shear deformation, thereby making the plate “stiffer” Mindlin (1951) proposed a

first-order shear deformable plate theory to allow for the effects of transverse shear

deformation and rotary inertia In the Mindlin plate theory, the first three assumptions in

Kirchhoff hypotheses are maintained while the last two assumptions are modified as

follows:

4 Plane sections initially normal to the midsurface remain plane but not necessarily

normal to that surface after bending

5 For dynamic problems, the effect of rotary inertia is included

Based on the Mindlin assumptions, the stress-resultants are now expressed by deflection

w of the midsurface as well as the rotations ψ and x ψ (or y ψ and r ψ ) of the plane θsections The deflection w and rotations ψ and x ψ (or y ψ and r ψ ) are mutually θindependent functions of the two coordinates in the plane of the plate These three

functions have to satisfy three independent second order linear partial differential

equations and the boundary conditions Thus the solution of these simultaneous equations

gives all necessary information for calculating stresses at any point of the moderately

thick plate

1.1.2 Analytical Methods for Plate Analysis

The most ideal situation in solving the governing partial differential equation of plate

is to find an exact analytical solution However, analytical solutions are only possible for

very few simple types of loading, plate shapes and boundary conditions

Axisymmetrically loaded circular plates with homogeneous boundary conditions and

rectangular plates with sinusoidal loads and all edges simply supported are examples of

plate bending problems for which exact analytical solutions exist

There are basically two types of analytical methods for plates One type is the

equilibrium methods, and the other type is the energy methods These two techniques are

also referred to as the Newtonian and the Lagrangian approaches, respectively They are

equivalent to each other because the governing equations and the principle of minimum

total mechanical energy can be derived from each other

In the equilibrium methods, the governing PDEs are solved directly It is common to

attempt a solution by the inverse method This method relies on assumed solutions for w

(in case of thin plates) which satisfy the governing equation and the boundary conditions

Some cases may be treated by using polynomial expressions for w in x and y (or in r and

θ ), and undetermined coefficients In the inverse method, the Fourier series is often used

to get analytical solutions for the PDEs

One of the energy methods is the Ritz method, in which the principle of minimum

total mechanical energy is used to solve plate problems By using the Ritz method, a

series of trial functions for w is assumed in which each trial function is preset to satisfy at

least the geometric boundary conditions, and the undetermined coefficients in the series

are found by minimizing the total energy functional with respect to each of these

coefficients

Another energy method is the strain energy method in which the principle of virtual

work is used Similar to the Ritz method, this method involves the use of a series of trial

functions for w, in which the undetermined coefficients are found by equalizing the

virtual work δW done by the external forces p due to a virtual displacement δw to the increment of the strain energy δU due to the same virtual displacement, i.e δW =δU

The coefficients in the series for w can also be found by using the Galerkin method, which is an alternative procedure in applying the strain energy method

1.1.3 Numerical Methods for Plate Analysis

For most plate problems in practice, exact analytical solutions cannot be found

because of the difficulties in the mathematical treatment Therefore, numerical methods

have to be resorted in order to obtain approximate solutions to relatively complex plate

problems The aforementioned equilibrium and energy methods can both be used to solve

plate problems using numerical techniques, such as the finite element method (FEM), the

boundary element method (BEM), the finite difference method (FDM), the differential

quadrature method (DQM), the collocation method, the Galerkin method and the Ritz

method

The history of numerical methods for plate problems can be traced back to more than

one century ago The Rayleigh method (Rayleigh 1877) is based on the principle of

minimum total mechanical energy of a system assuming no energy dissipation By using

a trial function for the mode shapes, and assuming simple harmonic motion, the

equalization of the maximum potential energy and the maximum kinetic energy yields the

vibration frequencies The resulting frequency is an upper bound solution, unless an exact

eigenfunction of free vibration for the trial function is assumed Ritz (1909) generalized

the Rayleigh method by assuming the deflection w as a set of admissible trial functions in the plate domain As a result, a closer upper bound for the frequency could be derived by

minimizing the total mechanical energy with respect to each of the undetermined

coefficients of the trial functions Ritz used this method to get an approximate solution of

a completely free square plate for which no exact solution is possible Even in recent

decades, the Ritz method is still widely used in solving complicated plate problems with,

of course, the assistance of computers (Liew et al 1998; Wang et al 2000b, 2001)

The finite difference method (FDM) and the finite element method (FEM) are the two

among the most important numerical methods FDM is simple, versatile, suitable for

computer use, and accurate, provided that a relative fine mesh is used In FDM, the

derivative of a function with respect to a coordinate direction at a point is approximated

by a weighted sum of the function values at a set of nearby points on the line in that

direction In this way, the plate differential equations and the expressions defining the

boundary conditions are replaced by equivalent difference equations The solution of the

plate problem thus reduces to the simultaneous solution of a set of algebraic equations,

written for every nodal point within the plate domain

On the other hand, FEM has been proven to be a very powerful and versatile tool for

solving a plethora of plate problems This method was developed in the 1960s when the

increasing emphasis on numerical methods was generated due to the advent of computers

In FEM, the plate is discretized into a finite number of elements (usually triangular or

rectangular in shapes), connected at their nodes and along hypothetic inter-element

boundaries Instead of solving the governing differential equations, the weak form

equations are solved for solution The application of FEM has already been extended to

practical problems in most engineering fields and coded into many well known

commercial programs such as ABAQUS, NASTRAN and COSMOS

In addition to the Ritz method, FDM and FEM, the boundary element method (BEM)

is also a widely used numerical method in solving plate problems The main advantage of

BEM is its unique ability to provide a complete solution in terms of boundary values only,

with substantial savings in modeling effort (Rashed et al 1999, Ventsel 1997)

In DQM, very high order Lagrange functions are used to approximate solution

functions From this approximation, the derivative of the function with respect to a

coordinate direction at a point can be expressed as a weighted sum of the function values

at all discrete points on the line in that direction Therefore, the use of DQM usually gives

high accuracy for problems with rectangle-like domains (Shu 2000, Shu and Chew 1999,

Bert and Malik 1996)

1.2 Literature Review on Mesh-free Methods

We have seen that the abovementioned numerical methods are widely used in plate

problems However, there still exist some difficulties in analysis of plates when these

methods are used These difficulties may include the efficient modeling of complex plate

shapes, and the accurate solutions for stress resultants It is necessary for researchers to

continuously attempt to develop advanced numerical methods that will be able to

overcome these difficulties