A Semianalytical Three‐Dimensional Elasticity Solution for Vibrations of Orthotropic Plates with Arbitrary Boundary Conditions

Abstract

A semianalytical three-dimensional (3D) elasticity solution for the vibration of the orthotropic plate is presented under arbitrary boundary conditions. Three-dimensional (3D) elasticity theory provides the theoretical support for the energy function of orthotropic plates. The orthotropic plates which have the arbitrary boundary condition are realized by the way of arranging three sets of linear springs at the edges. With the aim of eliminating the nonsmooth phenomenon at the edges, the admissible displacement function of an orthotropic plate is expressed with a modified Fourier series solution. Under this framework, a change that occurs on the boundary conditions only needs to modify the boundary parameters of the orthotropic plate, without the need for new derivation, thus greatly saving the modeling time. The convergence and accuracy of the proposed method are better than those of the published literature. Lastly, the new vibration results and parametric research of thick orthotropic plates as well as the geometric parameter are also presented.

Three-Dimensional Bending Analysis of Multi-Layered Orthotropic Plates by Two-Dimensional Numerical Model

“…The relations in Equation (13) provide definitions of D i tensors in the relation D one. However, the explicit definitions of the non-zero components of the tensors are as follows (17) Equation ( 11) is now applied to Equation (10). Moreover, the volume integrals are separated into the in-plane and transverse ones:…”

“…This type of analysis is limited only to thin plates due to the Kirchhoff-Love plate theory applied to derive the employed method. The vibration response of orthotropic composite plates was analysed by Zhang et al [16] with a new approach regarding the analytical solution, while 3D semi-analytical vibrations of thick orthotropic plates were investigated by Cui et al [17]. In this research, the modified Fourier series were applied to obtain admissible displacement functions while boundary conditions were generated by arranging sets of linear springs at the edges.…”

Three-Dimensional Vibration Analysis of a Functionally Graded Sandwich Rectangular Plate Resting on an Elastic Foundation Using a Semi-Analytical Method

“…The latter would cause a tedious formula deduction and reduce computational efficiency. In previous work, the author’s team proposed a new type of displacement admissible function by adding an auxiliary function for the boundary based on the traditional Fourier series [60,61]. In this way, the admissible displacement function and its derivative at the edges of the structure can be solved mechanically.…”

Dynamic Modeling and Analysis of Boundary Effects in Vibration Modes of Rectangular Plates with Periodic Boundary Constraints Based on the Variational Principle of Mixed Variables

“…Besides the dynamical modeling and dynamical properties of beam structures with complex boundary conditions [19], the dynamical modeling and dynamical properties of plate structures [20][21][22][23][24], shell structures [25][26][27][28][29][30][31], and other types of structures [32][33][34] with complex boundary conditions have also been investigated [35][36][37][38][39]. For example, Xue et al [20] developed and solved a dynamics model for medium-thick composite laminates with arbitrary boundary conditions based on Mindlin's theory, Hamilton's principle, a modified Fourier series method, and the spring technique, with parametric studies on the effects of several key parameters, such as thickness-to-width ratio, number of plies, and lay-up angle between two plies.…”