A modified solution for the free vibration analysis of moderately thick orthotropic rectangular plates with general boundary conditions, internal line supports and resting on elastic foundation

Abstract

In this investigation, a unified solution procedure based on the first-order shear deformation theory is presented for the free vibration analysis of moderately thick orthotropic rectangular plates with general boundary restraints, internal line supports and resting on elastic foundation. Under the current framework, regardless of boundary conditions, each of the displacement and rotation components of the plates is described as a standard Fourier cosine series supplemented with some auxiliary functions introduced to eliminate any possible discontinuities of the original displacements and their derivatives throughout the entire plate area including the boundaries and then to effectively enhance the convergence of the results. All the unknown expansion coefficients are treated as the generalized coordinates and determined by using the Raleigh–Ritz method. The current method can be universally applied to a variety of boundary conditions including all classical boundaries and their combinations and arbitrary elastic restraints. The excellent accuracy and reliability of current solutions is demonstrated by numerical examples and comparisons with the results available in the literature. In addition, the current method can also predict the vibration characteristics of the plate with internal line supports and elastic foundation. Comprehensive studies on the effects of elastic restraint parameters, locations of line supports and foundation coefficients are also reported. New results for plates subjected to elastic boundary restraints, arbitrary internal line supports in both directions and resting on elastic foundations are presented, which may serve as benchmark solutions for future researches.

Appendix: Representative calculation for stiffness and mass matrices

To illuminate the particular expression of the mass and stiffness matrixes clearly and tersely, six new variables are defined as:

$$p = m * (N + 1) + n + 1,\;\;\;q = (k - 1) * (N + 1) + n + 1,\;\;\;r = (k - 1) * (M + 1) + m + 1$$

$$p^{\prime } = m^{\prime } * (N + 1) + n^{\prime } + 1,\quad q^{\prime } = (k^{\prime } - 1) * (N + 1) + n^{\prime } + 1,\quad r = (k^{\prime } - 1) * (M + 1) + m^{\prime } + 1$$

The first row elements of K and M, are given below:

$$\begin{aligned} \{ {\mathbf{K}}_{1 - 1} \}_{{p,p^{\prime}}} &= 2\kappa G_{xz} h\varUpsilon_{cc}^{11} \varXi_{cc}^{00} + 2\kappa G_{yz} h\varUpsilon_{cc}^{00} \varXi_{cc}^{11} + (k_{y0}^{w} + \varTheta^{yy} k_{y0}^{w} )\varUpsilon_{cc}^{00} + (k_{x0}^{w} + \varTheta^{xx} k_{x0}^{w} )\varXi_{cc}^{00} \hfill \\ & \quad+ K_{w} \varUpsilon_{cc}^{00} \varXi_{cc}^{00} + K_{s} (\varUpsilon_{cc}^{11} \varXi_{cc}^{00} + \varUpsilon_{cc}^{00} \varXi_{cc}^{11} ) + \varLambda^{x} \varXi_{cc}^{00} + \varLambda^{y} \varUpsilon_{cc}^{00} \hfill \\ \end{aligned}$$

(28)

$$\begin{aligned} \{ {\mathbf{K}}_{1 - 2} \}_{{q,p^{\prime}}} = 2\kappa G_{xz} h\varUpsilon_{cc}^{11} \varXi_{\zeta c}^{00} + 2\kappa G_{yz} h\varUpsilon_{cc}^{00} \varXi_{\zeta c}^{11} + (k_{y0}^{w} + \varTheta^{yy} k_{y0}^{w} )\varUpsilon_{\zeta c}^{00} + \hfill \\ \begin{array}{*{20}c} {} & {} & {} & {} \\ \end{array} + K_{w} \varUpsilon_{cc}^{00} \varXi_{\zeta c}^{00} + K_{s} (\varUpsilon_{cc}^{11} \varXi_{\zeta c}^{00} + \varUpsilon_{cc}^{00} \varXi_{\zeta c}^{11} ) + \varLambda^{xy} \varUpsilon_{cc}^{00} + \varLambda^{x} \varXi_{\zeta c}^{00} \hfill \\ \end{aligned}$$

(29)

$$\begin{aligned} \{ {\mathbf{K}}_{1 - 3} \}_{{r,p^{\prime}}} &= 2\kappa G_{xz} h\varUpsilon_{\zeta c}^{11} \varXi_{cc}^{00} + 2\kappa G_{yz} h\varUpsilon_{\zeta c}^{00} \varXi_{cc}^{11} + (k_{x0}^{w} + \varTheta^{xx} k_{x0}^{w} )\varXi_{\zeta c}^{00} \\ & \quad+ K_{w} \varUpsilon_{\zeta c}^{00} \varXi_{cc}^{00} + K_{s} (\varUpsilon_{\zeta c}^{11} \varXi_{cc}^{00} + \varUpsilon_{\zeta c}^{00} \varXi_{cc}^{11} ) + \varLambda^{yx} \varXi_{cc}^{00} + \varLambda^{y} \varUpsilon_{\zeta c}^{00} \hfill \\ \end{aligned}$$

(30)

$$\{ {\mathbf{K}}_{1 - 4} \}_{{p,p^{\prime}}} = 2\kappa G_{xz} h\varUpsilon_{cc}^{01} \varXi_{cc}^{00}$$

(31)

$$\{ {\mathbf{K}}_{1 - 5} \}_{{q,p^{\prime}}} = 2\kappa G_{xz} h\varUpsilon_{cc}^{01} \varXi_{\zeta c}^{00}$$

(32)

$$\{ {\mathbf{K}}_{1 - 6} \}_{{r,p^{\prime}}} = 2\kappa G_{xz} h\varUpsilon_{\zeta c}^{01} \varXi_{cc}^{00}$$

(33)

$$\{ {\mathbf{K}}_{1 - 7} \}_{{p,p^{\prime}}} = 2\kappa G_{yz} h\varUpsilon_{cc}^{00} \varXi_{cc}^{01}$$

(34)

$$\{ {\mathbf{K}}_{1 - 8} \}_{{q,p^{\prime}}} = 2\kappa G_{yz} h\varUpsilon_{cc}^{00} \varXi_{\zeta c}^{01}$$

(35)

$$\{ {\mathbf{K}}_{1 - 9} \}_{{r,p^{\prime}}} = 2\kappa G_{yz} h\varUpsilon_{\zeta c}^{00} \varXi_{cc}^{01}$$

(36)

$$\{ {\mathbf{M}}_{1 - 1} \}_{{p,p^{\prime}}} = \rho h\varUpsilon_{cc}^{00} \varXi_{cc}^{00} ,\;\{ {\mathbf{M}}_{1 - 2} \}_{{q,p^{\prime}}} = \rho h\varUpsilon_{cc}^{00} \varXi_{\zeta c}^{00} ,\;\{ {\mathbf{M}}_{1 - 3} \}_{{r,p^{\prime}}} = \rho h\varUpsilon_{\zeta c}^{00} \varXi_{cc}^{00} ,$$

(37)

$${\mathbf{M}}_{1 - 4} = {\mathbf{M}}_{1 - 5} = {\mathbf{M}}_{1 - 6} = {\mathbf{M}}_{1 - 7} = {\mathbf{M}}_{1 - 8} = {\mathbf{M}}_{1 - 9} = {\mathbf{0}}.$$

(38)

where

$$\varUpsilon_{cc}^{ef} = \int_{0}^{a} {\frac{{d^{e} \cos (\lambda_{am} x)}}{{dx^{e} }}} \frac{{d^{f} \cos (\lambda_{{am^{\prime}}} x)}}{{dx^{f} }}dx\varXi_{cc}^{ef} = \int_{0}^{b} {\frac{{d^{e} \cos (\lambda_{bn} y)}}{{dy^{e} }}} \frac{{d^{f} \cos (\lambda_{{bn^{\prime}}} y)}}{{dy^{f} }}dy$$

$$\varUpsilon_{\zeta c}^{ef} = \int_{0}^{a} {\frac{{d^{e} \zeta_{a}^{l} (x)}}{{dx^{e} }}} \frac{{d^{f} \cos (\lambda_{{am^{\prime } }} x)}}{{dx^{f} }}dx\varXi_{\zeta c}^{ef} = \int_{0}^{b} {\frac{{d^{e} \zeta_{b}^{l} (y)}}{{dy^{e} }}} \frac{{d^{f} \cos (\lambda_{{bn^{\prime } }} y)}}{{dy^{f} }}dy$$

$$\varTheta^{xx} = ( - 1)^{{m + m^{\prime}}} ,\;\varTheta^{yy} = ( - 1)^{{n + n^{\prime}}} ,\;\varLambda^{y} = \sum\limits_{i = 1}^{{N_{i} }} {sk_{yi}^{i} } cos(\lambda_{bn} y_{i} )^{2} ,\;\varLambda^{x} = \sum\limits_{j = 1}^{{M_{i} }} {sk_{xj}^{j} } cos(\lambda_{am} x_{j} )^{2}$$

$$\varLambda^{xy} = \sum\limits_{i = 1}^{{N_{i} }} {sk_{yi}^{i} } \zeta_{b}^{l} (y_{i} )cos(\lambda_{bn} y_{i} ),\;\varLambda^{yx} = \sum\limits_{i = 1}^{{M_{i} }} {sk_{xj}^{j} } \zeta_{a}^{l} (x_{i} )cos(\lambda_{am} x_{i} )$$

$$\lambda_{am} = \frac{m\pi }{a},\;\lambda_{bn} = \frac{n\pi }{b},\;\lambda_{{am^{\prime}}} = \frac{{m^{\prime}\pi }}{a},\;\lambda_{{bn^{\prime}}} = \frac{{n^{\prime}\pi }}{b}$$