结构振动：任意边界条件层合梁、板、壳结构的准确解法（英文版）

1 Fundamental Equations of Laminated Beams，Plates and Shells.
1.1 Three-Dimensional Elasticity Theory in Curvilinear Coordinates
1.2 Fundamental Equations of Thin Laminated Shells
1.2.1 Kinematic Relations
1.2.2 Stress-Strain Relations and Stress Resultants
1.2.3 Energy Functions
1.2.4 Governing Equations and Boundary Conditions
1.3 Fundamental Equations of Thick Laminated Shells
1.3.1 Kinematic Relations
1.3.2 Stress-Strain Relations and Stress Resultants
1.3.3 Energy Functions
1.3.4 Governing Equations and Boundary Conditions
1.4 Lamé Parameters for Plates and Shells.

2 Modified Fourier Series and Rayleigh-Ritz Method
2.1 Modified Fourier Series
2.1.1 Traditional Fourier Series Solutions.
2.1.2 One-Dimensional Modified Fourier Series Solutions
2.1.3 Two-Dimensional Modified Fourier Series Solutions
2.2 Strong Form Solution Procedure
2.3 Rayleigh-Ritz Method (Weak Form Solution Procedure)

3 Straight and Curved Beams
3.1 Fundamental Equations of Thin Laminated Beams
3.1.1 Kinematic Relations
3.1.2 Stress-Strain Relations and Stress Resultants
3.1.3 Energy Functions
3.1.4 Governing Equations and Boundary Conditions
3.2 Fundamental Equations of Thick Laminated Beams.
3.2.1 Kinematic Relations
3.2.2 Stress-Strain Relations and Stress Resultants
3.2.3 Energy Functions
3.2.4 Governing Equations and Boundary Conditions
3.3 Solution Procedures
3.3.1 Strong Form Solution Procedure
3.3.2 Weak Form Solution Procedure (Rayleigh-Ritz Procedure)
3.4 Laminated Beams with General Boundary Conditions
3.4.1 Convergence Studies and Result Verification
3.4.2 Effects of Shear Deformation and Rotary Inertia
3.4.3 Effects of the Deepness Term (1+z/R).
3.4.4 Isotropic and Laminated Beams with General Boundary Conditions.

4 Plates
4.1 Fundamental Equations of Thin Laminated Rectangular Plates.
4.1.1 Kinematic Relations
4.1.2 Stress-Strain Relations and Stress Resultants
4.1.3 Energy Functions
4.1.4 Governing Equations and Boundary Conditions
4.2 Fundamental Equations of Thick Laminated Rectangular Plates.
4.2.1 Kinematic Relations
4.2.2 Stress-Strain Relations and Stress Resultants
4.2.3 Energy Functions
4.2.4 Governing Equations and Boundary Conditions
4.3 Vibration of Laminated Rectangular Plates
4.3.1 Convergence Studies and Result Verification
4.3.2 Laminated Rectangular Plates with Arbitrary Classical Boundary Conditions
4.3.3 Laminated Rectangular Plates with Elastic Boundary Conditions.
4.3.4 Laminated Rectangular Plates with Internal Line Supports.
4.4 Fundamental Equations of Laminated Sectorial， Annular and Circular Plates
4.4.1 Fundamental Equations of Thin Laminated Sectorial， Annular and Circular Plates
4.4.2 Fundamental Equations of Thick Laminated Sectorial， Annular and Circular Plates
4.5 Vibration of Laminated Sectorial， Annular and Circular Plates
4.5.1 Vibration of Laminated Annular and Circular Plates
4.5.2 Vibration of Laminated Sectorial Plates

5 Cylindrical Shells
5.1 Fundamental Equations of Thin Laminated Cylindrical Shells
5.1.1 Kinematic Relations
5.1.2 Stress-Strain Relations and Stress Resultants
5.1.3 Energy Functions
5.1.4 Governing Equations and Boundary Conditions
5.2 Fundamental Equations of Thick Laminated Cylindrical Shells
5.2.1 Kinematic Relations
5.2.2 Stress-Strain Relations and Stress Resultants
5.2.3 Energy Functions
5.2.4 Governing Equations and Boundary Conditions
5.3 Vibration of Laminated Closed Cylindrical Shells
5.3.1 Convergence Studies and Result Verification
5.3.2 Effects of Shear Deformation and Rotary Inertia
5.3.3 Laminated Closed Cylindrical Shells with General End Conditions
5.3.4 Laminated Closed Cylindrical Shells with Intermediate Ring Supports
5.4 Vibration of Laminated Open Cylindrical Shells
5.4.1 Convergence Studies and Result Verification
5.4.2 Laminated Open Cylindrical Shells with General End Conditions

6 Conical Shells.
6.1 Fundamental Equations of Thin Laminated Conical Shells
6.1.1 Kinematic Relations
6.1.2 Stress-Strain Relations and Stress Resultants
6.1.3 Energy Functions
6.1.4 Governing Equations and Boundary Conditions
6.2 Fundamental Equations of Thick Laminated Conical Shells
6.2.1 Kinematic Relations
6.2.2 Stress-Strain Relations and Stress Resultants
6.2.3 Energy Functions
6.2.4 Governing Equations and Boundary Conditions
6.3 Vibration of Laminated Closed Conical Shells
6.3.1 Convergence Studies and Result Verification
6.3.2 Laminated Closed Conical Shells with General Boundary Conditions.
6.4 Vibration of Laminated Open Conical Shells
6.4.1 Convergence Studies and Result Verification
6.4.2 Laminated Open Conical Shells with General Boundary Conditions.

7 Spherical Shells
7.1 Fundamental Equations of Thin Laminated
……
8 Shallow Shells
References and Further Reading

　　Chapter 1
　　Fundamental Equations of Laminated Beams， Plates and Shells Beams， plates and shells are named according to their size or/and 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 z/Rα and z/Rβ can be neglected compared to unity in the strain-displacement relations （i.e.， z/Rα ? 1 and z/Rβ ? 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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