1 Fundamental Equations of Laminated Beams，Plates and Shells. 1
1.1 Three-Dimensional Elasticity Theory in Curvilinear Coordinates 1
1.2 Fundamental Equations of Thin Laminated Shells 3
1.2.1 Kinematic Relations 3
1.2.2 Stress-Strain Relations and Stress Resultants 5
1.2.3 Energy Functions 9
1.2.4 Governing Equations and Boundary Conditions 11
1.3 Fundamental Equations of Thick Laminated Shells 16
1.3.1 Kinematic Relations 17
1.3.2 Stress-Strain Relations and Stress Resultants 18
1.3.3 Energy Functions 23
1.3.4 Governing Equations and Boundary Conditions 26
1.4 Lamé Parameters for Plates and Shells. 29
2 Modified Fourier Series and Rayleigh-Ritz Method 37
2.1 Modified Fourier Series 38
2.1.1 Traditional Fourier Series Solutions. 39
2.1.2 One-Dimensional Modified Fourier Series Solutions 43
2.1.3 Two-Dimensional Modified Fourier Series Solutions 48
2.2 Strong Form Solution Procedure 53
2.3 Rayleigh-Ritz Method Weak Form Solution Procedure 58
3 Straight and Curved Beams 63
3.1 Fundamental Equations of Thin Laminated Beams 64
3.1.1 Kinematic Relations 64
3.1.2 Stress-Strain Relations and Stress Resultants 65
3.1.3 Energy Functions 66
3.1.4 Governing Equations and Boundary Conditions 68
3.2 Fundamental Equations of Thick Laminated Beams. 71
3.2.1 Kinematic Relations 71
3.2.2 Stress-Strain Relations and Stress Resultants 72
3.2.3 Energy Functions 73
3.2.4 Governing Equations and Boundary Conditions 74
3.3 Solution Procedures 76
3.3.1 Strong Form Solution Procedure 77
3.3.2 Weak Form Solution Procedure Rayleigh-Ritz Procedure 80
3.4 Laminated Beams with General Boundary Conditions 83
3.4.1 Convergence Studies and Result Verification 83
3.4.2 Effects of Shear Deformation and Rotary Inertia 85
3.4.3 Effects of the Deepness Term 1+zR. 87
3.4.4 Isotropic and Laminated Beams with General Boundary Conditions. 90
4 Plates 99
4.1 Fundamental Equations of Thin Laminated Rectangular Plates. 100
4.1.1 Kinematic Relations 100
4.1.2 Stress-Strain Relations and Stress Resultants 102
4.1.3 Energy Functions 103
4.1.4 Governing Equations and Boundary Conditions 105
4.2 Fundamental Equations of Thick Laminated Rectangular Plates. 107
4.2.1 Kinematic Relations 108
4.2.2 Stress-Strain Relations and Stress Resultants 108
4.2.3 Energy Functions 109
4.2.4 Governing Equations and Boundary Conditions 111
4.3 Vibration of Laminated Rectangular Plates 113
4.3.1 Convergence Studies and Result Verification 116
4.3.2 Laminated Rectangular Plates with Arbitrary Classical Boundary Conditions 117
4.3.3 Laminated Rectangular Plates with Elastic Boundary Conditions. 122
4.3.4 Laminated Rectangular Plates with Internal Line Supports. 125
4.4 Fundamental Equations of Laminated Sectorial， Annular and Circular Plates 131
4.4.1 Fundamental Equations of Thin Laminated Sectorial， Annular and Circular Plates 134
4.4.2 Fundamental Equations of Thick Laminated Sectorial， Annular and Circular Plates 137
4.5 Vibration of Laminated Sectorial， Annular and Circular Plates 139
4.5.1 Vibration of Laminated Annular and Circular Plates 139
4.5.2 Vibration of Laminated Sectorial Plates 144
5 Cylindrical Shells 153
5.1 Fundamental Equations of Thin Laminated Cylindrical Shells 156
5.1.1 Kinematic Relations 156
5.1.2 Stress-Strain Relations and Stress Resultants 157
5.1.3 Energy Functions 158
5.1.4 Governing Equations and Boundary Conditions 159
5.2 Fundamental Equations of Thick Laminated Cylindrical Shells 162
5.2.1 Kinematic Relations 162
5.2.2 Stress-Strain Relations and Stress Resultants 163
5.2.3 Energy Functions 165
5.2.4 Governing Equations and Boundary Conditions 167
5.3 Vibration of Laminated Closed Cylindrical Shells 169
5.3.1 Convergence Studies and Result Verification 172
5.3.2 Effects of Shear Deformation and Rotary Inertia 175
5.3.3 Laminated Closed Cylindrical Shells with General End Conditions 177
5.3.4 Laminated Closed Cylindrical Shells with Intermediate Ring Supports 184
5.4 Vibration of Laminated Open Cylindrical Shells 188
5.4.1 Convergence Studies and Result Verification 192
5.4.2 Laminated Open Cylindrical Shells with General End Conditions 193
6 Conical Shells. 199
6.1 Fundamental Equations of Thin Laminated Conical Shells 200
6.1.1 Kinematic Relations 201
6.1.2 Stress-Strain Relations and Stress Resultants 201
6.1.3 Energy Functions 202
6.1.4 Governing Equations and Boundary Conditions 202
6.2 Fundamental Equations of Thick Laminated Conical Shells 207
6.2.1 Kinematic Relations 208
6.2.2 Stress-Strain Relations and Stress Resultants 209
6.2.3 Energy Functions 210
6.2.4 Governing Equations and Boundary Conditions 211
6.3 Vibration of Laminated Closed Conical Shells 215
6.3.1 Convergence Studies and Result Verification 217
6.3.2 Laminated Closed Conical Shells with General Boundary Conditions. 219
6.4 Vibration of Laminated Open Conical Shells 225
6.4.1 Convergence Studies and Result Verification 227
6.4.2 Laminated Open Conical Shells with General Boundary Conditions. 228
7 Spherical Shells 235
7.1 Fundamental Equations of Thin Laminated
內容試閱：
Chapter 1
Fundamental Equations of Laminated Beams， Plates and Shells Beams， plates and shells are named according to their size orand shape features. Shells have all the features of plates except an additional one-curvature Leissa 1969， 1973. Therefore， the plates， on the other hand， can be viewed as special cases of shells having no curvature. Beams are one-dimensional counterparts of plates straight beams or shells curved beams with one dimension relatively greater in comparison to the other two dimensions. This chapter introduces the fundamental equations including kinematic relations， stress-strain relations and stress resultants， energy functions， governing equations and boundary conditions of laminated shells in the framework of the classical shell theory CST and the shear deformation shell theory SDST without proofs due to the fact that they have been well established. The corresponding equations of laminated beams and plates are specialized from the shell ones.
1.1 Three-Dimensional Elasticity Theory in Curvilinear Coordinates Consider a three-dimensional 3D shell segment with total thickness h as shown in Fig. 1.1， a 3D orthogonal coordinate system α， β and z located on the middle surface is used to describe the geometry dimensions and deformations of the shell， in which co-ordinates along the meridional， circumferential and normal directions are represented by α， β and z， respectively. Rα and Rβ are the mean radii of curvature in the α and β directions on the middle surface z = 0. U， V and W separately indicate the displacement variations of the shell in the α， β and z directions. The strain-displacement relations of the three-dimensional theory of elasticity in orthogonal curvilinear coordinate system are Leissa 1973; Soedel 2004; Carrera et al. 2011:
where the quantities A and B are the Lamé parameters of the shell. They are determined by the shell characteristics and the selected orthogonal coordinate system. The detail definitions of them are given in Sect. 1.4. The lengths in the α and β directions of the shell segment at distance dz from the shell middle surface are see Fig. 1.1:
The above equations contain the fundamental strain-displacement relations of a 3D body in curvilinear coordinate system. They are specialized to those of CST and FSDT by introducing several assumptions and simplifications.
1.2 Fundamental Equations of Thin Laminated Shells According to Eq. 1.1， it can be seen that the 3D strain-displacement equations of a shell are rather complicated when written in curvilinear coordinate system. Typically， researchers simplify the 3D shell equations into the 2D ones by making certain assumptions to eliminate the coordinate in the thickness direction. Based on different assumptions and simplifications， various sub-category classical theories of thin shells were developed， such as the Reissner-Naghdi’s linear shell theory， Donner-Mushtari’s theory， Flügge’s theory， Sanders’ theory and Goldenveizer- Novozhilov’s theory， etc. In this book， we focus on shells composed of arbitrary numbers of composite layers which are bonded together rigidly. When the total thickness of a laminated shell is less than 0.05 of the wavelength of the deformation mode or radius of curvature， the classical theories of thin shells originally developed for single-layered isotropic shells can be readily extended to the laminated ones. Leissa 1973 showed that most thin shell theories yield similar results. In this section， the fundamental equations of the Reissner-Naghdi’s linear shell theory are extended to thin laminated shells due to that it offers the simplest， the most accurate and consistent equations for laminated thin shells Qatu 2004.
1.2.1 Kinematic Relations
In the classical theory of thin shells， the four assumptions made by Love 1944 are universally accepted to be valid for a first approximation shell theory Rao 2007:
1. The thickness of the shell is small compared with the other dimensions.
2. Strains and displacements are sufficiently small so that the quantities of secondand higher-order magnitude in the strain-displacement relations may be neglected in comparison with the first-order terms.
3. The transverse normal stress is small compared with the other normal stress components and may be neglected.
4. Normals to the undeformed middle surface remain straight and normal to the deformed middle surface and suffer no extension.
The first assumption defines that the shell is thin enough so that the deepness terms zRα and zRβ can be neglected compared to unity in the strain-displacement relations i.e.， zRα ? 1 and zRβ ? 1. The second assumption ensures that the differential equations will be linear. The fourth assumption is also known as Kirchhoff’s hypothesis. This assumption leads to zero transverse shear strains and zero transverse normal strain. Taking these assumptions into consideration， the 3D strain-displacement relations of shells in orthogonal curvilin
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