Dynamic stiffness formulation and free vibration analysis of specially orthotropic Mindlin plates with arbitrary boundary conditions

Abstract

A novel dynamic stiffness formulation which includes the effects of shear deformation and rotatory inertia is proposed to carry out the free vibration analysis of thick rectangular orthotropic plates i.e. the formulation is based on an extension of the Mindlin theory to orthotropic plates in a dynamic stiffness context. The modified trigonometric basis is used to construct the general solution for the free vibration problem of the plate in series form, permitting the derivation of an infinite system of linear algebraic equations which connect the Fourier coefficients of the kinematic boundary conditions of its four edges.The boundary conditions are essentially the deflections and angles of rotations constituting the displacement vector and shear forces and bending moments constituting the force vector. The forcedisplacement relationship is then obtained by relating the two vectors via the dynamic stiffness matrix.Essentially the ensuing infinite system forms the fundamental basis of the paper, which defines the dynamic stiffness matrix for an orthotropic Mindlin plate in an exact sense. Numerical results are obtained by using the Wittrick-Williams algorithm as solution technique. The theory is first validated using published results and then further illustrated by a series of numerical examples. The paper concludes with significant conclusions.

Dynamic stiffness formulation for isotropic and orthotropic plates with point nodes

“…Recently, Wei et al [41] made a noteworthy contribution when they formulated the dynamic stiffness matrix for transverse and in-plane vibration of rectangular plates with arbitrary boundary conditions. Their work differs from the work described in [38,42,43] in that the choice of the trigonometric function to describe the series solution was somehow different, which yielded slightly different results. It should be noted that strictly speaking, for all the above approaches, the dynamic stiffness matrix in the exact formulation is an infinite matrix because the boundary values of the displacements and forces form an infinite system of equations.…”

“…( 1). The composition of the solution is based on the separation of variables technique for each of the four component cases of symmetry [38,42] which are described above by symmetric-symmetric (SS), symmetric-anti-symmetric (SA), anti-symmetric-symmetric (AS) and anti-symmetric-antisymmetric (AA), respectively. The complete general solution for the amplitude of the bending displacement W is given by the sum of all the four individual comments of the symmetry as follows…”

“…Following similar procedure as in [38,42] it is now possible to write the general solution of Eq. ( 1) in the form of infinite series for all four cases of symmetry as…”

Spectral dynamic stiffness theory for free vibration analysis of plate structures stiffened by beams with arbitrary cross-sections

“…In order to remove the above restrictions in the literature, many researchers have proposed different dynamic stiffness models in recent years [30][31][32][33][34][35][36][37][38][39] . Some of these models are applicable to plate elements with more general boundary conditions, and the elements can be assembled in two directions.…”

“…Some of these models are applicable to plate elements with more general boundary conditions, and the elements can be assembled in two directions. Amongst these contributions, Liu, Banerjee and their coauthors [30][31][32][33][34] proposed the spectral dynamic stiffness method (SDSM) for both the transverse [30][31][32][33] and inplane [34] vibration problem of the plate by combining the spectral method with the classical dynamic stiffness method for plate with classical boundary conditions (BCs) and non-classical BCs [35,36] . The method has also been extended to buckling analysis of plates [37,38] .…”