New analytic buckling solutions of moderately thick clamped rectangular plates by a straightforward finite integral transform method

Abstract

A first endeavor is made in this paper to explore new analytic buckling solutions of moderately thick rectangular plates by a straightforward double finite integral transform method, with focus on typical non-Lévy-type fully clamped plates that are not easy to solve in a rigorous way by the other analytic methods. Solving the governing higher-order partial differential equations with prescribed boundary conditions is elegantly reduced to processing four sets of simultaneous linear equations, the existence of nonzero solutions of which determines the buckling loads and associated mode shapes. Both numerical and graphical results confirm the validity and accuracy of the developed method and solutions by favorable comparison with the literature and finite element analysis. The succinct but effective technique presented in this study can provide an easy-to-implement theoretical tool to seek more analytic solutions of complex boundary value problems.

Appendices

Appendix A: Definition of double finite integral transforms

The following double finite integral transforms are used for W, $\varphi _x$ and $\varphi _y$:

$$\begin{aligned} W^{\mathrm{ss}}= & {} \mathop \int \limits _0^a {\mathop \int \limits _0^b {W\left( {x,y} \right) \sin (\alpha _m x)\sin (\beta _n y)\hbox {d}x\hbox {d}y} } \\ \varphi _x^{\mathrm{cs}}= & {} \mathop \int \limits _0^a {\mathop \int \limits _0^b {\varphi _x \left( {x,y} \right) } } \cos (\alpha _m x)\sin (\beta _n y)\hbox {d}x\hbox {d}y \\ \varphi _y^{\mathrm{sc}}= & {} \mathop \int \limits _0^a {\mathop \int \limits _0^b {\varphi _y \left( {x,y} \right) } } \sin (\alpha _m x)\cos (\beta _n y)\hbox {d}x\hbox {d}y \\ \end{aligned}$$

where the superscripts “SS”, “CS” and “SC” represent the double sine, cosine-sine and sine-cosine integral transforms with respect to variables x and y.

Appendix B: Matrix elements in Eq. (12)

$$\begin{aligned} T_{11}= & {} \frac{C\left[ {C+D\left( {\alpha _m^2 +\beta _n^2 } \right) } \right] }{\left[ {C+D\left( {\alpha _m^2 +\beta _n^2 } \right) } \right] \left[ {\alpha _m^2 \left( {C+N_x } \right) +\beta _n^2 \left( {C+N_y } \right) } \right] -C^{2}\left( {\alpha _m^2 +\beta _n^2 } \right) },\\ T_{12}= & {} -\frac{CD\alpha _m }{CN_x \alpha _m^2 +D\alpha _m^4 \left( {C+N_x } \right) +\beta _n^2 \left[ {CN_y +D\alpha _m^2 \left( {2C+N_x +N_y } \right) } \right] +D\beta _n^4 \left( {C+N_y } \right) },\\ T_{13}= & {} -\frac{CD\beta _n }{CN_x \alpha _m^2 +D\alpha _m^4 \left( {C+N_x } \right) +\beta _n^2 \left[ {CN_y +D\left( {2C+N_x +N_y } \right) \alpha _m^2 } \right] +D\beta _n^4 \left( {C+N_y } \right) },\\ T_{21}= & {} -\frac{C}{D}T_{12} ,\\ T_{22}= & {} -\frac{D\left\{ {\begin{array}{l} \alpha _m^2 \left( {C+N_x } \right) \left[ {2C+D\alpha _m^2 \left( {1-\mu } \right) } \right] \\ +\beta _n^2 \left\{ {2CN_y +D\alpha _m^2 \left[ {C\left( {3-\mu } \right) +2N_x +N_y \left( {1-\mu } \right) } \right] } \right\} +2D\beta _n^4 \left( {C+N_y } \right) \\ \end{array}} \right\} }{\left[ {2C+D\left( {1-\mu } \right) \left( {\alpha _m^2 +\beta _n^2 } \right) } \right] \left\{ {\begin{array}{l} CN_x \alpha _m^2 +D\alpha _m^4 \left( {C+N_x } \right) \\ +\beta _n^2 \left[ {CN_y +D\alpha _m^2 \left( {2C+N_x +N_y } \right) } \right] +D\beta _n^4 \left( {C+N_y } \right) \\ \end{array}} \right\} },\\ T_{23}= & {} -\frac{D\alpha _m \beta _n \left\{ {2C^{2}-D\left( {1+\mu } \right) \left[ {\alpha _m^2 \left( {C+N_x } \right) +\beta _n^2 \left( {C+N_y } \right) } \right] } \right\} }{\left[ {2C+D\left( {1-\mu } \right) \left( {\alpha _m^2 +\beta _n^2 } \right) } \right] \left\{ {\begin{array}{l} CN_x \alpha _m^2 +D\alpha _m^4 \left( {C+N_x } \right) \\ +\beta _n^2 \left[ {CN_y +D\alpha _m^2 \left( {2C+N_x +N_y } \right) } \right] +D\beta _n^4 \left( {C+N_y } \right) \\ \end{array}} \right\} },\\ T_{31}= & {} -\frac{C}{D}T_{13} ,\\ T_{32}= & {} -T_{23} ,\\ T_{33}= & {} -\frac{D\left\{ {\begin{array}{l} 2CN_x \alpha _m^2 +2D\alpha _m^4 \left( {C+N_x } \right) \\ +\beta _n^2 \left\{ {2C\left( {C+N_y } \right) +D\alpha _m^2 \left[ {C\left( {3-\mu } \right) +N_x \left( {1-\mu } \right) +2N_y } \right] } \right\} +D\beta _n^4 \left( {1-\mu } \right) \left( {C+N_y } \right) \\ \end{array}} \right\} }{\left[ {2C+D\left( {1-\mu } \right) \left( {\alpha _m^2 +\beta _n^2 } \right) } \right] \left\{ {\begin{array}{l} CN_x \alpha _m^2 +D\alpha _m^4 \left( {C+N_x } \right) \\ +\beta _n^2 \left[ {CN_y +D\alpha _m^2 \left( {2C+N_x +N_y } \right) } \right] +D\beta _n^4 \left( {C+N_y } \right) \\ \end{array}} \right\} }. \end{aligned}$$