Zhu Su

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.

Modified Fourier–Ritz Approximation for the Free Vibration Analysis of Laminated Functionally Graded Plates with Elastic Restraints

Free vibration analysis of moderately thick laminated functionally graded rectangular plates with elastic restraints is presented using the modified Fourier–Ritz method in conjunction with the first-order shear deformation plate theory. The material properties are assumed to change continuously through the lamina thickness according to a power-law distribution of volume fractions of the constituents. Each of the displacements and rotations of the laminated functionally graded plates, regardless of boundary conditions, is represented by a modified Fourier series which is constructed as the linear superposition of a standard Fourier cosine series and several closed-form auxiliary functions. The accuracy, convergence and reliability of the current solutions are demonstrated by numerical examples and comparison of the present results with those available in the literature. New results for free vibration of laminated functionally graded plates are presented, which may serve as benchmark solutions. The effects of the boundary conditions, volume fractions and lamina thickness ratios on the frequencies of the plates are 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.