Tiangui Ye

An analytical method for vibration analysis of cylindrical shells coupled with annular plate under general elastic boundary and coupling conditions

A unified solution for coupled cylindrical shell and annular plate systems with general boundary and coupling conditions is presented in the study by using a modified Fourier-Ritz method. Under the framework, regardless of the boundary and continuity conditions, each displacement for the cylindrical shell and the annular plate is invariantly expressed as the modified Fourier series composed of the standard Fourier series and auxiliary functions. The introduction of the auxiliary functions can not only remove the potential discontinuities at the junction and the extremes of the combination but also accelerate the convergence of the series expansion. All the expansion coefficients are determined by the Rayleigh-Ritz method as the generalized coordinates. The arbitrary axial position of the annular plate coupling with the cylindrical shell considered in the theoretical formulation makes the present method more general. The theoretical model established by present method can be conveniently applied to cylindrical shell-circular plate combinations just by varying the inner radius of the annular plate. The convergence and accuracy of present method are tested and validated by a number of numerical examples for coupled annular plate-cylindrical shell structures with various boundary restraints and general elastic coupling conditions. The effects of the axial position of the annular plate and elastic coupling conditions on the vibration behavior of the coupled system are also investigated. The power of present method compared to conventional finite element method is demonstrated with less computation cost. Some new results are presented to provide useful information for future researchers.

Vibration analysis and transient response of a functionally graded piezoelectric curved beam with general boundary conditions

The paper presents a unified solution for free and transient vibration analyses of a functionally graded piezoelectric curved beam with general boundary conditions within the framework of Timoshenko beam theory. The formulation is derived by means of the variational principle in conjunction with a modified Fourier series which consists of standard Fourier cosine series and supplemented functions. The mechanical and electrical properties of functionally graded piezoelectric materials (FGPMs) are assumed to vary continuously in the thickness direction and are estimated by Voigts rule of mixture. The convergence, accuracy and reliability of the present formulation are demonstrated by comparing the present solutions with those from the literature and finite element analysis. Numerous results for FGPM beams with different boundary conditions, geometrical parameters as well as material distributions are given. Moreover, forced vibration of the FGPM beams subjected to dynamic loads and general boundary conditions are also investigated.

Three-dimensional vibration analysis of functionally graded sandwich deep open spherical and cylindrical shells with general restraints

This paper presents three-dimensional (3D) vibration analysis of functionally graded (FG) sandwich deep open shells with general boundary restraints, including open spherical shells and the cylindrical ones. FG sandwich deep open shells composed of homogeneous cores and functionally graded material face sheets with material properties vary continuously through the thickness direction are considered in the present work. The 3D theory of elasticity in conjunction with an energy-based improved Fourier series method are combined to develop the theoretical formulation, in which each displacement of a deep open shell is approximated in terms of a triplicate product of the cosine Fourier series with the addition of certain supplementary terms introduced to remove the potential discontinuities associated with the original displacement and its relevant derivatives at the boundary faces. By using the present method, FG sandwich deep open shells with general boundary restraints, arbitrary geometry parameters, different material distributions and lamination schemes can be solved in a unified form. The accuracy and reliability of the present formulation are validated by comparisons with FEM solutions and those in the literature, and excellent agreements are obtained. Several 3D vibration results of FG sandwich deep open cylindrical and spherical shells with different dimensions in the meridional, circumferential and normal directions are presented for various types of boundary conditions and many representative lamination schemes, which may serve as benchmark solutions for future researchers in assessing two-dimensional approximate theories.

Three-Dimensional Vibration Analysis of Isotropic and Orthotropic Open Shells and Plates with Arbitrary Boundary Conditions

This paper presents elasticity solutions for the vibration analysis of isotropic and orthotropic open shells and plates with arbitrary boundary conditions, including spherical and cylindrical shells and rectangular plates. Vibration characteristics of the shells and plates have been obtained via a unified three-dimensional displacement-based energy formulation represented in the general shell coordinates, in which the displacement in each direction is expanded as a triplicate product of the cosine Fourier series with the addition of certain supplementary terms introduced to eliminate any possible jumps with the original displacement function and its relevant derivatives at the boundaries. All the expansion coefficients are then treated equally as independent generalized coordinates and determined by the Rayleigh-Ritz procedure. To validate the accuracy of the present method and the corresponding theoretical formulations, numerical cases have been compared against the results in the literature and those of 3D FE analysis, with excellent agreements obtained. The effects of boundary conditions, material parameters, and geometric dimensions on the frequencies are discussed as well. Finally, several 3D vibration results of isotropic and orthotropic open spherical and cylindrical shells and plates with different geometry dimensions are presented for various boundary conditions, which may be served as benchmark solutions for future researchers as well as structure designers in this field.

A Unified Accurate Solution for Three-dimensional Vibration Analysis of Functionally Graded Plates and Cylindrical Shells with General Boundary Conditions

Three-dimensional (3-D) vibration analysis of thick functionally graded plates and cylin‐ drical shells with arbitrary boundary conditions is presented in this chapter. The effective material properties of functionally graded structures vary continuously in the thickness direction according to the simple power-law distributions in terms of volume fraction of constituents and are estimated by Voigt’s rule of mixture. By using the artificial spring boundary technique, the general boundary conditions can be obtained by setting proper spring stiffness. All displacements of the functionally graded plates and shells are ex‐ panded in the form of the linear superposition of standard 3-D cosine series and several supplementary functions, which are introduced to remove potential discontinuity prob‐ lems with the original displacements along the edge. The Rayleigh-Ritz procedure is used to yield the accurate solutions. The convergence, accuracy and reliability of the current formulation are verified by numerical examples and by comparing the current results with those in published literature. Furthermore, the influence of the geometrical parame‐ ters and elastic foundation on the frequencies of rectangular plates and cylindrical shells is investigated.

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 in Vibration of Plates (NASA SP-160), US Government Printing Office, Washington, DC, pp. 1–353, 1969, Vibration Of Shells (NASA SP-288), US Government Printing Office, Washington, DC, pp. 1–428, 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.

Straight and Curved Beams

Beams, plates and shells are commonly utilized in engineering applications, and they are named according to their size or/and shape characteristics and different theories have been developed to study their structural behaviors. A beam is typically described as a structural component having one dimension relatively greater than the other dimensions. Specially, a beam can be referred to as a rod or bar when subjected to tension, a column when subjected to compression and a shaft when subjected to torsional loads (Qatu 2004). Beams are one of the most fundamental structural elements.

Modified Fourier Series and Rayleigh-Ritz Method

Although the governing equations and associated boundary equations for laminated beams, plates and shells presented in Chap. 1 show the possibility of seeking their exact solutions of vibration, however, it is commonly believed that very few exact solutions are possible for plate and shell vibration problems.