Symplectic superposition solution for the buckling problem of orthotropic rectangular plates with four clamped edges

Abstract

The main objective of this study is to uniformly solve the buckling problem of fully clamped (CCCC) orthotropic/isotropic rectangular plates with different thicknesses. The analysis uses the symplectic superposition method. This method describes the buckling problem of orthotropic rectangular moderately thick plates (RMTPs) in the Hamiltonian system for treatment in the symplectic space. First, the governing equations of RMTPs are represented by Hamiltonian canonical equations. Then, the original buckling problem of a CCCC rectangular moderately thick plate (RMTP) is divided into two sub-buckling problems. The variable separation method in the Hamiltonian system is used to calculate the general solutions of these two sub-buckling problems. The symplectic superposition solution of the original buckling problem is obtained by superimposing the general solutions of the two sub-buckling problems. Finally, the analysis results of the buckling load and modal shape of orthotropic rectangular plates under various thicknesses and aspect ratios are presented in numerical examples.

Introduction

RMTPs, as primary load-carrying members, play an important role in rigid pavements, bridges, ships, and other engineering structures. In practical applications, the buckling behavior of plates may lead to instability or even premature damage or failure of the plate structure. To optimize the structural design of plates and ensure the safety of the plate structure, the buckling load factor and modal vibration mode provide important reference values for plate designers. Therefore, the buckling problem of RMTPs has received research attention from many scholars. This type of problem can be studied using numerical and analytical methods. For example, R. Vescovini and L. Dozio [1] applied the Ritz method to obtain reference solutions for buckling and vibrating modes of variable stiffness moderately thick rectangular plates. A. Yiotis and J. Katsikadelis [2] solved the buckling problem of RMTPs on a biparametric elastic foundation based on the meshless method. O. Civalek [3] applied the discrete singular convolution method to obtain numerical solutions for the buckling problem of rectangular thick plates with clamped and simply supported boundary conditions. In addition to the above-mentioned methods, numerical methods such as the exact finite strip method [4], differential quadrature method [5], and finite difference method [6] can effectively solve the buckling problem of rectangular plates. Compared with the number of studies applying numerical methods, the number of studies applying analytical methods to investigate such problems is much smaller. Owing to the high order of the partial differential governing equations for RMTPs, it is very difficult to analytically solve the buckling problem of RMTPs under complex boundary conditions. Nevertheless, analytical solutions are often used as benchmark solutions to test the errors of numerical solutions, and they help to conduct parameter analysis and optimization in plate structure design quickly. Therefore, the use of analytical methods to study the solution of such problems is still necessary. Currently, by using traditional inverse or semi-inverse methods, scholars have achieved some good research results on solving the buckling problem of rectangular plates with fully simply supported or two opposite edges simply supported. For example, P. Bera et al. [7] and M. Abdollahi et al. [8] applied the Navier method to solve the buckling problem of orthotropic/isotropic rectangular plates with four simply supported edges and the buckling problem of thick functionally graded piezoelectric rectangular plates with four simply supported edges, respectively. H. Thai and D. Choi [9] obtained a set of Lévy-type buckling solutions of isotropic rectangular thick plates with two opposite sides simply supported. The buckling problem of rectangular plates without two parallel edges simply supported is difficult to solve using the traditional inverse or semi-inverse methods. Therefore, it is necessary to seek other analytical methods to solve these problems. For example, S. Ullah et al. [10] applied the finite integral method to solve the buckling problem of CCCC isotropic rectangular plates. I. Shufrinm and M. Eisenberger [11] applied the extended Kantorovich method to study the buckling problem of isotropic elastic rectangular thick plates under CCCC boundary conditions. S. Yu et al. [12] applied the superposition method to determine the critical buckling load of rectangular plates under five combinations of clamped and simply supported edges. In addition to the abovementioned analytical methods, the symplectic elasticity method [13] proposed by Professor Zhong Wanxie who introduces the traditional problem of solving problems based on the Lagrangian system in the Euclidean space into the Hamiltonian system based on elastic mechanics and then solves the problem in symplectic space, enabling many rectangular plate problems to be solved [14, 15, 16–17]. In recent years, based on the symplectic elastic mechanics method, the SSM has been proposed by Li Rui et al. [18]. This method not only inherits the advantages of the symplectic elasticity method, without the need to preset any type of trial function and strict step-by-step deduction during the solving process but also further expands the scope of solvable problems. Compared to the symplectic elasticity method, the SSM can solve plate and shell problems under more complex boundary conditions. At present, scholars have applied SSM to problems such as rectangular thin plates [19, 20, 21–22], RMTPs [23, 24], rectangular thick plates [25, 26, 27–28], nanoplates [29, 30], functionally graded plates [31, 32], and functionally graded shells [33]. In particular, there is little literature on the study of the buckling problem of CCCC rectangular plates by SSM. For example, B. Wang et al. [21] studied the buckling problem of CCCC isotropic rectangular thin plates. C. Zhou et al. [24] and S. Xiong et al. [25] studied the buckling problem of CCCC isotropic RMTPs based on the first-order shear deformation theory and the buckling problem of CCCC isotropic rectangular thick plates based on the third-order shear deformation principle, respectively. To the best of our knowledge, no literature exists that applies analytical methods to solve the buckling problem of orthotropic RMTPs without two parallel simply supported edges. Therefore, this study applies the SSM to solve the buckling problem of CCCC orthotropic RMTPs. In the first section of the article, we construct a Hamiltonian system based on the governing equations of orthotropic RMTPs. In the second section, we use the variable separation method and symplectic eigenfunction expansion method to solve the sub-buckling problems under two opposite simply supported conditions obtained from the original buckling problem decomposition in symplectic space. Then by superimposing the general solutions of the two sub-buckling problems, the buckling solution of the original orthotropic RMTP is obtained. In the third section, we applied the obtained buckling solution to provide the buckling loads and typical modal shapes of isotropic and orthotropic rectangular plates with different thicknesses in specific examples. We analyzed the convergence of the buckling load factors for orthotropic rectangular plates and further demonstrated the correctness and effectiveness of the obtained symplectic superposition solution by comparing the obtained buckling load factors of the isotropic rectangular plates with the results of the finite integral method and the finite element method.

Hamiltonian system

Based on the Mindlin first-order shear deformation theory, the governing equations for buckling of the orthotropic RMTPs are

1

$$\begin{array}{cc}\hfill \frac{\partial M_x}{\partial x}& +\frac{\partial M_{\mathit{xy}}}{\partial y}-Q_x=0,\hfill \\ \hfill \frac{\partial M_y}{\partial y}& +\frac{\partial M_{\mathit{xy}}}{\partial x}-Q_y=0,\hfill \\ \hfill \frac{\partial Q_x}{\partial x}& +\frac{\partial Q_y}{\partial y}+N_x\frac{{\partial }^{2}w}{\partial x^2}+N_y\frac{{\partial }^{2}w}{\partial y^2}=0,\hfill \\ \hfill \end{array}$$where the domain of the plate is $\left\{\left(\left(x,y,z\right)\right|0\le x\le a,0\le y\le b,-\frac{h}{2}\le z\le \frac{h}{2}\right\};$$N_x$ and $N_y$ are the membrane forces of the plate; $w$ is the transverse modal displacement of the plate; $M_x$ and $M_y$ are the bending moments of the plate, $M_{\mathit{xy}}$ is the twisting moment of the plate, $Q_x$ and $Q_y$ are the shear forces of the plate, they are given as

2

$$\begin{array}{cc}\hfill M_x& =-D_x\frac{\partial {\psi }_x}{\partial x}-D_1\frac{\partial {\psi }_y}{\partial y},M_y=-D_1\frac{\partial {\psi }_x}{\partial x}-D_y\frac{\partial {\psi }_y}{\partial y},M_{\mathit{xy}}=-D_{\mathit{xy}}\left(\frac{\partial {\psi }_x}{\partial y}+\frac{\partial {\psi }_y}{\partial x}\right),\hfill \\ \hfill Q_x& =C_{\mathit{xz}}\left(\frac{\partial w}{\partial x}-{\psi }_x\right),Q_y=C_{\mathit{yz}}\left(\frac{\partial w}{\partial y}-{\psi }_y\right),\hfill \\ \hfill \end{array}$$in Eq. (2), ${\psi }_x$ and ${\psi }_y$ are the rotating modal angles, $C_{\mathit{xz}}$ and $C_{\mathit{yz}}$ are the shear stiffness, $D_x$ and $D_y$ are the flexural stiffness, and $D_1$ and $D_{\mathit{xy}}$ are the effective torsional stiffness of the plate. These can be represented by Young’s moduli $E_x$, $E_y$, shear moduli $G_{\mathit{xy}}$, $G_{\mathit{xz}}$, $G_{\mathit{yz}}$, the shear correction coefficient $k$, thickness $h$, and Poisson’s ratios $v_x$, $v_y$ as

3

$$\begin{array}{cc}\hfill D_x& =\frac{E_x{h}^3}{12\left(1-v_x{v}_y\right)},D_y=\frac{E_y{h}^3}{12\left(1-v_y{v}_x\right)},D_1=v_x{D}_y=v_y{D}_x,\hfill \\ \hfill D_{\mathit{xy}}& =\frac{G_{\mathit{xy}}{h}^3}{12},C_{\mathit{xz}}=G_{\mathit{xz}}hk,C_{\mathit{yz}}=G_{\mathit{yz}}hk\hfill \\ \hfill \end{array}$$

We introduced auxiliary functions to obtain the Hamilton system of the governing Eq. (1) of the orthotropic RMTPs:

4

$$L_x=Q_x+N_x\frac{\partial w}{\partial x},L_y=Q_y+N_y\frac{\partial w}{\partial y}.$$

From Eqs. (1), (2), and (4), we can get:

5

$$\begin{array}{cc}\hfill \frac{\partial {\psi }_x}{\partial x}& =-\frac{1}{D_x}{M}_x-\frac{D_1}{D_x}\frac{\partial {\psi }_y}{\partial y},\hfill \\ \hfill \frac{\partial \left(-M_{\mathit{xy}}\right)}{\partial y}& =\frac{D_1}{D_x}\frac{\partial M_x}{\partial y}+\left(\frac{D_1^2}{D_x}-D_y\right)\frac{{\partial }^2{\psi }_y}{\partial y^2}+C_{\mathit{yz}}{\psi }_y-C_{\mathit{yz}}\frac{\partial w}{\partial y},\hfill \\ \hfill \frac{\partial L_x}{\partial x}& =C_{\mathit{yz}}\frac{\partial {\psi }_y}{\partial y}-\left(C_{\mathit{yz}}+N_y\right)\frac{{\partial }^{2}w}{\partial y^2},\hfill \\ \hfill \frac{\partial M_x}{\partial x}& =-\frac{C_{\mathit{xz}}{N}_x}{C_{\mathit{xz}}+N_x}{\psi }_x+\frac{\partial \left(-M_{\mathit{xy}}\right)}{\partial y}+\frac{C_{\mathit{xz}}}{C_{\mathit{xz}}+N_x}{L}_x,\hfill \\ \hfill \frac{\partial {\psi }_y}{\partial x}& =-\frac{\partial {\psi }_x}{\partial y}+\frac{1}{D_{\mathit{xy}}}\left(-M_{\mathit{xy}}\right),\hfill \\ \hfill \frac{\partial w}{\partial x}& =\frac{C_{\mathit{xz}}}{C_{\mathit{xz}}+N_x}{\psi }_x+\frac{1}{C_{\mathit{xz}}+N_x}{L}_x.\hfill \\ \hfill \end{array}$$

The following matrix equation can then be obtained from Eq. (5):

6

$$\stackrel{·}{Z}=HZ$$where ${\mathit{Z}=({\psi }_x,-M_{\mathit{xy}},L_x,M_x,{\psi }_y,w)}^T,$$\dot{\mathit{Z}}$ represents the partial derivative of $\mathit{Z}$ concerning $x$,$$\mathbf{H}=\left(\begin{array}{cccccc}0& 0& 0& -\frac{1}{D_x}& -\frac{D_1}{D_x}\frac{\partial }{\partial y}& 0\\ 0& 0& 0& \frac{D_1}{D_x}\frac{\partial }{\partial y}& \left(\frac{D_1^2}{D_x}-D_y\right)\frac{{\partial }^2}{\partial y^2}+C_{\mathit{yz}}& -C_{\mathit{yz}}\frac{\partial }{\partial y}\\ 0& 0& 0& 0& C_{\mathit{yz}}\frac{\partial }{\partial y}& -\left(C_{\mathit{yz}}+N_y\right)\frac{{\partial }^2}{\partial y^2}\\ -\frac{C_{\mathit{xz}}{N}_x}{C_{\mathit{xz}}+N_x}& \frac{\partial }{\partial y}& \frac{C_{\mathit{xz}}}{C_{\mathit{xz}}+N_x}& 0& 0& 0\\ -\frac{\partial }{\partial y}& \frac{1}{D_{\mathit{xy}}}& 0& 0& 0& 0\\ \frac{C_{\mathit{xz}}}{C_{\mathit{xz}}+N_x}& 0& \frac{1}{C_{\mathit{xz}}+N_x}& 0& 0& 0\end{array}\right).$$

It can be verified through a calculation that $\mathbf{H}$ satisfies the relationship $⟨\mathit{u},\mathbf{H}\mathit{v}⟩=⟨\mathit{v},\mathbf{H}\mathit{u}⟩$, where $\mathit{u}$ and $\mathit{v}$ are the simply supported full-state vectors [13]. Therefore, Eq. (6) is the Hamiltonian system of the governing Eq. (1) for orthotropic RMTPs.

Analytic solutions of the buckling problems by the SSM

As shown in Fig. 1, the parallelograms all represent RMTPs with a length of a and a width of b on the xy coordinate plane. Figure 1a shows the buckling problem of a CCCC orthotropic rectangular plate. The buckling problem of the CCCC rectangular plate is divided into two sub-buckling problems with opposite simply supported edges by analyzing its boundary conditions, as shown in Fig. 1b and c:

[See PDF for image]

Fig. 1

Symplectic superposition structure of the CCCC rectangular plate

Subproblem (i): The orthotropic RMTPs are simply supported in the y direction, and satisfied the following boundary requirements in the x direction (Fig. 1b):

7

$${\left({\psi }_y\right|}_{x=0,a}=0,{\left(w\right|}_{x=0,a}=0,M_x(a,y)=\sum _{n=1}^{\infty }{F}_{n}sin\left({\alpha }_{n}y\right),M_x(0,y)=\sum _{n=1}^{\infty }{E}_{n}sin\left({\alpha }_{n}y\right)$$

Subproblem (ii): The orthotropic RMTPs are simply supported in the x direction, and satisfied the following boundary requirements in the y direction (Fig. 1c):

8

$${\left({\psi }_x\right|}_{y=0,b}=0,{\left(w\right|}_{y=0,b}=0,M_y(x,0)=\sum _{n=1}^{\infty }{G}_{n}sin\left({\beta }_{n}x\right),M_y(x,b)=\sum _{n=1}^{\infty }{H}_{n}sin\left({\beta }_{n}x\right)$$

Both subproblems (i) and (ii) can be solved using the Hamiltonian system’s variable separation method. Then, by superimposing the general solutions of these two sub-buckling problems, the buckling solution of the original problem presented in Fig. 1a can be obtained.

Solutions to sub-buckling problems

To study the subproblem (i) shown in Fig. 1b, the variable separation in the symplectic framework is first applied to solve the Hamiltonian system (6), make $\mathit{Z}=\mathit{X}\left(x\right)Y\left(y\right)$, and substitute it into Eq. (6). We obtain the following two equations:

9

$$\mathbf{H}\mathbf{Y}(y)=\mu \mathbf{Y}(y),$$

10

$$\frac{dX(x)}{\mathit{dx}}=\mu X(x)$$where $\mu$ is the eigenvalue, $\mathit{Y}\left(y\right)={\left(Y_1\left(y\right),Y_2\left(y\right),Y_3\left(y\right),Y_4\left(y\right),Y_5\left(y\right),Y_6\left(y\right)\right)}^T$ is the corresponding eigenvector. Equation (9) can be rewritten as follows:

11

$$\left(\mathbf{H}-\mu \mathbf{I}\right)\mathbf{Y}(y)=0.$$

As shown in Fig. 1b, the rectangular plate in subproblem (i) satisfies the simply supported conditions in the y direction:

12

$${\left(w\right|}_{y=0,b}=0,{\left(M_y\right|}_{y=0,b}=0,{\left({\psi }_x\right|}_{y=0,b}=0.$$

According to Eq. (11) and the simply supported conditions (12), two sets of eigenvalues and associated eigenvectors can be obtained:

(1) When $n=0$, the eigenvalues ${\mu }_1=\sqrt{\frac{C_{\mathit{yz}}}{D_{\mathit{xy}}}},{\mu }_2=-\sqrt{\frac{C_{\mathit{yz}}}{D_{\mathit{xy}}}}.$ can be obtained, and the associated eigenvectors ${\mathbf{Y}}_l(y)(l=1,2)$ can be obtained as follows:

13

$${\mathbf{Y}}_l(y)=\left\{0,\frac{D_{\mathit{xy}}{\mu }_l^2\left(D_x{\mu }_l^2\left(C_{\mathit{xz}}+N_x\right)-C_{\mathit{xz}}{N}_x\right)}{C_{\mathit{xz}}{C}_{\mathit{yz}}+\left(D_1+D_{\mathit{xy}}\right){\mu }_l^2\left(C_{\mathit{xz}}+N_x\right)},0,\right){\left(0,\frac{D_x\left(C_{\mathit{xz}}+N_x\right){\mu }_l^4-C_{\mathit{xz}}{N}_x{\mu }_l^2}{C_{\mathit{xz}}{C}_{\mathit{yz}}{\mu }_l+\left(D_1+D_{\mathit{xy}}\right)\left(C_{\mathit{xz}}+N_x\right){\mu }_l^3},0\right\}}^T$$

(2) When $n\ne 0$, the eigenvalues ${\mu }_{n2}=-{\mu }_{n1}$, ${\mu }_{n4}=-{\mu }_{n3}$ and ${\mu }_{n6}=-{\mu }_{n5}$$(n=1,2,3,...)$ can be obtained, and the associated eigenvectors ${\mathbf{Y}}_{\mathit{ni}}(y)(n=1,2,3,...;i=1,2,3,4,5,6)$ can be obtained as follows:

14

$$\begin{array}{c}{\mathbf{Y}}_{\mathit{ni}}(y)=\left\{sin\left({\alpha }_{n}y\right),,\frac{C_{\mathit{xz}}{D}_{\mathit{xy}}\left(C_{\mathit{yz}}{\alpha }_n^2+{\eta }_{\mathit{ni}}\right)-{\sigma }_{\mathit{ni}}}{{\alpha }_{\mathit{ni}}{Q}_{\mathit{ni}}},cos,\left({\alpha }_{n}y\right),,,\frac{\left(\left(C_{\mathit{xz}}+N_x\right)\left((,D_1+D_{\mathit{xy}}\right){\eta }_{\mathit{ni}}+C_{\mathit{yz}}{\tau }_{\mathit{ni}}\right)+N_x{\tilde{\text{n}}}_{\mathit{ni}}}{{\tilde{\text{n}}}_{\mathit{ni}}},sin,\left({\alpha }_{n}y\right),,\right)\\ {\left(\frac{{\sigma }_{\mathit{ni}}+C_{\mathit{xz}}\left(D_1{\eta }_{\mathit{ni}}-C_{\mathit{yz}}{D}_x{\mu }_{\mathit{ni}}^2\right)}{{\mu }_{\mathit{ni}}{Q}_{\mathit{ni}}}sin\left({\alpha }_{n}y\right),\frac{{\eta }_{\mathit{ni}}\left(C_{\mathit{xz}}+{\tau }_{\mathit{ni}}\right)-C_{\mathit{xz}}{\tau }_{\mathit{ni}}{\mu }_{\mathit{ni}}^2}{{\alpha }_n{\mu }_{\mathit{ni}}{Q}_{\mathit{ni}}}cos\left({\alpha }_{n}y\right),\frac{\left(D_1+D_{\mathit{xy}}\right){\eta }_{\mathit{ni}}+C_{\mathit{yz}}{\tau }_{\mathit{ni}}+{\tilde{\text{n}}}_{\mathit{ni}}}{{\mu }_{\mathit{ni}}{\widetilde{\mathit{textn}}}_{\mathit{ni}}}sin\left({\alpha }_{n}y\right)\right\}}^T\\ \end{array}$$where ${\alpha }_n=\frac{n\pi }{b},{\tau }_{\mathit{ni}}=D_{\mathit{xy}}{\alpha }_n^2-D_x{\mu }_{\mathit{ni}}^2,{\eta }_{\mathit{ni}}=\left(C_{\mathit{yz}}+N_y\right){\alpha }_n^2-N_x{\mu }_{\mathit{ni}}^2,{\sigma }_{\mathit{ni}}=D_{\mathit{xy}}\left(D_1{\alpha }_n^2+D_x{\mu }_{\mathit{ni}}^2\right)\left({\eta }_{\mathit{ni}}-\right)\left(C_{\mathit{xz}}{\mu }_{\mathit{ni}}^2\right),$${\varrho }_{\mathit{ni}}=C_{\mathit{xz}}\left(C_{\mathit{yz}}+\left(D_1+D_{\mathit{xy}}\right){\mu }_{\mathit{ni}}^2\right)-\left(D_1+D_{\mathit{xy}}\right){\eta }_{\mathit{ni}}$

Based on the expansion method of the eigenfunctions and the solution of the differential Eq. (10), the general solution of the Hamiltonian system (6) under the condition of simply supported (12) can be assumed as:

15

$$\mathbf{Z}(x,y)=\sum _{l=1}^2{f}_l{e}^{{\mu }_{l}x}{\mathbf{Y}}_l(y)+\sum _{n=1}^{\infty }\sum _{i=1}^6{f}_{\mathit{ni}}{e}^{{\mu }_{\mathit{ni}}x}{\mathbf{Y}}_{\mathit{ni}}(y),$$where $f_l(l=1,2)$ and $f_{\mathit{ni}}(n=1,2,3,...;i=1,2,3,4,5,6)$ are undetermined coefficients. Extract the expressions for the fourth, fifth, and sixth components of the general solution (15), which are the expressions for the bending moment $M_x$, rotation angle ${\psi }_y$, and transverse modal displacement $w$ corresponding to the orthotropic RMTP under the simple support conditions (12). Then, by combining the boundary conditions (7) that the rectangular plate needs to satisfy on the $x=0$ and $x=a$ sides in subproblem (i), the undetermined coefficients $f_1=f_2=0$ and the undetermined coefficients $f_{\mathit{ni}}(n=1,2,3,...;i=1,2,3,4,5,6)$ represented by the undetermined constants $E_n,F_n$ can be obtained. Further, the solutions of subproblem (i) are obtained as:

16

$${\psi }_x^{(i)}(x,y)=\sum _{n=1}^{\infty }\sum _{i=1}^{6}sin\left({\alpha }_{n}y\right)f_{\mathit{ni}}{e}^{x{\mu }_{\mathit{ni}}}$$

17

$${\psi }_y^{(i)}(x,y)=\sum _{n=1}^{\infty }\sum _{i=1}^6\frac{cos\left({\alpha }_{n}y\right)\left(f_{\mathit{ni}}{e}^{x{\mu }_{\mathit{ni}}}\left({\eta }_{\mathit{ni}}{\tau }_{\mathit{ni}}+C_{\mathit{xz}}\left({\eta }_{\mathit{ni}}-{\tau }_{\mathit{ni}}{\mu }_{\mathit{ni}}^2\right)\right)\right)}{{\alpha }_n{\varrho }_{\mathit{ni}}{\mu }_{\mathit{ni}}},$$

18

$$w^{(i)}(x,y)=\sum _{n=1}^{\infty }\sum _{i=1}^6\frac{sin\left({\alpha }_{n}y\right)\left(f_{\mathit{ni}}{e}^{x{\mu }_{\mathit{ni}}}\left(\left(D_1+D_{\mathit{xy}}\right){\eta }_{\mathit{ni}}+C_{\mathit{yz}}{\tau }_{\mathit{ni}}+{\varrho }_{\mathit{ni}}\right)\right)}{{\varrho }_{\mathit{ni}}{\mu }_{\mathit{ni}}}$$

As shown in Fig. 1, subproblem (i) (Fig. 1b) and subproblem (ii) (Fig. 1c) have symmetry. Therefore, by variable substitution from the solution of subproblem (i), we can directly obtain the solution of subproblem (ii). Specifically, replacing $a,b,x,y,E_n,F_n,N_x,C_{\mathit{xz}},C_{\mathit{yz}},D_x,D_y,{\alpha }_n,f_l,f_{\mathit{ni}},{\tau }_{\mathit{ni}},{\eta }_{\mathit{ni}},{\sigma }_{\mathit{ni}},{\varrho }_{\mathit{ni}}$ in subproblem (i) with $b,a,y,x,G_n,H_n,N_y,C_{\mathit{yz}},C_{\mathit{xz}},D_y,D_x,{\beta }_n,g_l,g_{\mathit{ni}},$${\tilde{\tau }}_{\mathit{ni}}$, ${\tilde{\eta }}_{\mathit{ni}}$, ${\tilde{\sigma }}_{\mathit{ni}}$, ${\tilde{\varrho }}_{\mathit{ni}}$, respectively. The eigenvalues ${\xi }_l$ and ${\xi }_{\mathit{ni}}$, eigenvectors ${\mathbf{X}}_l(x)(l=1,2)$ and ${\mathbf{X}}_{\mathit{ni}}(x)(n=1,2,3,...;i=1,2,3,4,5,6)$, and general solutions represented by rotation angles ${\psi }_x$, ${\psi }_y$ and transverse modal displacement $w$ of subproblem (ii) can be obtained:

19

$${\psi }_y^{(ii)}(x,y)=\sum _{n=1}^{\infty }\sum _{i=1}^{6}sin\left({\beta }_{n}x\right)g_{\mathit{ni}}{e}^{y{\xi }_{\mathit{ni}}}$$

20

$${\psi }_x^{(ii)}(x,y)=\sum _{n=1}^{\infty }\sum _{i=1}^6\frac{cos\left({\beta }_{n}x\right)\left(g_{\mathit{ni}}{e}^{y{\xi }_{\mathit{ni}}}\left({\tilde{\eta }}_{\mathit{ni}}{\tilde{\tau }}_{\mathit{ni}}+C_{\mathit{yz}}\left({\tilde{\eta }}_{\mathit{ni}}-{\tilde{\tau }}_{\mathit{ni}}{\xi }_{\mathit{ni}}^2\right)\right)\right)}{{\beta }_n{\tilde{\varrho }}_{\mathit{ni}}{\xi }_{\mathit{ni}}}$$

21

$$w^{(ii)}(x,y)=\sum _{n=1}^{\infty }\sum _{i=1}^6\frac{sin\left({\beta }_{n}x\right)\left(g_{\mathit{ni}}{e}^{y{\xi }_{\mathit{ni}}}\left(\left(D_1+D_{\mathit{xy}}\right){\tilde{\eta }}_{\mathit{ni}}+C_{\mathit{xz}}{\tilde{\tau }}_{\mathit{ni}}+{\tilde{\varrho }}_{\mathit{ni}}\right)\right)}{{\tilde{\varrho }}_{\mathit{ni}}{\xi }_{\mathit{ni}}}$$

Symplectic superposition solution

After deriving the solutions for subproblem (i) and subproblem (ii), we use the superposition method to solve the original buckling problem. To ensure the equivalence of the superposition of the original CCCC buckling problem and the two subproblems, it is also necessary to ensure that the summation of the rotation angles of the two sub-buckling problems is zero on all four edges.

At edge $x=0$ and edge $x=a$, the sum of the rotations ${\psi }_x$ of the two subproblems is zero, i.e., $({\psi }_x^{(i)}+{\left({\psi }_x^{(ii)})\right|}_{x=0}=0$ and $({\psi }_x^{(i)}+{\left({\psi }_x^{(ii)})\right|}_{x=a}=0$, which yields

22

$$\sum _{i=1}^6{f}_{\mathit{ji}}+\sum _{n=1}^{\infty }\sum _{i=1}^6\frac{2j\pi g_{\mathit{ni}}\left(1-\text{cos}\left(j\pi \right)e^{b{\xi }_{\mathit{ni}}}\right)\left(\left(C_{\mathit{yz}}+{\tilde{\tau }}_{\mathit{ni}}\right){\tilde{\eta }}_{\mathit{ni}}-C_{\mathit{yz}}{\tilde{\tau }}_{\mathit{ni}}{\xi }_{\mathit{ni}}^2\right)}{{\beta }_n{\tilde{\varrho }}_{\mathit{ni}}{\xi }_{\mathit{ni}}\left(b^2{\xi }_{\mathit{ni}}^2+j^2{\pi }^2\right)}=0$$and

23

$$\sum _{i=1}^6{f}_{\mathit{ji}}{e}^{a{\mu }_{\mathit{ni}}}+\sum _{n=1}^{\infty }\sum _{i=1}^6\frac{2j\pi \text{cos}\left(a{\beta }_n\right)g_{\mathit{ni}}\left(1-\text{cos}\left(j\pi \right)e^{b{\xi }_{\mathit{ni}}}\right)\left(\left(C_{\mathit{yz}}+{\tilde{\tau }}_{\mathit{ni}}\right){\tilde{\eta }}_{\mathit{ni}}-C_{\mathit{yz}}{\tilde{\tau }}_{\mathit{ni}}{\xi }_{\mathit{ni}}^2\right)}{{\beta }_n{\tilde{\varrho }}_{\mathit{ni}}{\xi }_{\mathit{ni}}\left(b^2{\xi }_{\mathit{ni}}^2+j^2{\pi }^2\right)}=0$$$j=1,2,3,...$.

At edge $y=0$ and edge $y=b$, the sum of the rotations ${\psi }_y$ of the two subproblems is zero, i.e., $({\psi }_y^{(i)}+{\left({\psi }_y^{(ii)})\right|}_{y=0}=0$ and $({\psi }_y^{(i)}+{\left({\psi }_y^{(ii)})\right|}_{y=b}=0$, which yields

24

$$\sum _{i=1}^6{g}_{\mathit{ji}}+\sum _{n=1}^{\infty }\sum _{i=1}^6\frac{2j\pi f_{\mathit{ni}}(1-cos\left(j\pi \right)e^{a{\mu }_{\mathit{ni}}})((C_{\mathit{xz}}+{\tau }_{\mathit{ni}}){\eta }_{\mathit{ni}}-C_{\mathit{xz}}{\tau }_{\mathit{ni}}{\mu }_{\mathit{ni}}^2)}{{\alpha }_n{\varrho }_{\mathit{ni}}{\mu }_{\mathit{ni}}(a^2{\mu }_{\mathit{ni}}^2+j^2{\pi }^2)}=0,$$and

25

$$\sum _{i=1}^6{g}_{\mathit{ji}}{e}^{b{\xi }_{\mathit{ni}}}+\sum _{n=1}^{\infty }\sum _{i=1}^6\frac{2j\pi cos\left(b{\alpha }_n\right)f_{\mathit{ni}}(1-cos\left(j\pi \right)e^{a{\mu }_{\mathit{ni}}})((C_{\mathit{xz}}+{\tau }_{\mathit{ni}}){\eta }_{\mathit{ni}}-C_{\mathit{xz}}{\tau }_{\mathit{ni}}{\mu }_{\mathit{ni}}^2)}{{\alpha }_n{\varrho }_{\mathit{ni}}{\mu }_{\mathit{ni}}(a^2{\mu }_{\mathit{ni}}^2+j^2{\pi }^2)}=0$$$j=1,2,3,...$.

After expanding the coefficients $f_{\mathit{ni}}$ and $g_{\mathit{ni}}$$(n=1,2,3,...;i=1,2,3,4,5,6)$ in Eqs. (22)–(25) into undetermined constants $E_n,F_n$, $G_n$, and $H_n$, Eqs. (22)–(25) can be transformed into linear equations for undetermined constants $E_n,F_n$, $G_n$, and $H_n$$(n=1,2,3...)$. If $E_n,F_n$, $G_n$, and $H_n$ are all equal to zero, the rectangular plate will not undergo buckling deformation. Therefore, to obtain the nonzero solutions of these constants, the buckling load value of the RMTP with four clamped edges can be obtained by making the determinant for the homogeneous system of Eqs. (22)–(25) equal to zero. In actual calculations, we need to truncate the infinite Eqs. (22)–(25) and then substitute the obtained buckling load values into Eqs. (22)–(25) to calculate the undetermined constants $E_n,F_n$, $G_n$, and $H_n$$(n=1,2,3...)$. By substituting the obtained numerical values into the general solutions $w^{(i)}(x,y)$ and $w^{(ii)}(x,y)$ of the two sub-buckling problems, the series expansion solution of the transverse modal displacement $w(x,y)$ of the original problem can be obtained. The results are obtained using the software Wolfram Mathematica 13.3 online version.

26

$$w(x,y)=w^{(i)}(x,y)+w^{(ii)}(x,y)$$

Remark: The relevant symbol operation results are obtained by programming and calculating with Wolfram Mathematica 13.3 software.

Numerical examples

Example 1

The buckling load factors of CCCC isotropic plates with different thicknesses under two ratios of length to width ($b\left./\phantom{ba}\right)a$) when $N_x\left./\phantom{N_x{N}_y}\right)N_y=1$ and $1.5$ are calculated separately by the sum of the first 60 terms ($N=15$) of the solution (26). The material properties were taken as $v_x=v_y=0.3$, $k=5\left./\phantom{56}\right)6$, $D_y=D_x=D$, and $G_{\mathit{xy}}=G_{\mathit{xz}}=G_{\mathit{yz}}=4.2D\left./\phantom{4.2D{h}^3}\right)h^3$. The first five buckling load factors are given in Tables 1 and 2, where the results reported in [10] and [24] (FEM) are added for comparison.

Table 1. Buckling load factors $-N_y{a}^2\left./\phantom{-N_y{a}^{2}D}\right)D$ for CCCC isotropic plate ($N_x\left./\phantom{N_x{N}_y}\right)N_y=1$)

$b\left./\phantom{ba}\right)a$	$h\left./\phantom{ha}\right)a$		$\mathit{Mode}$
			First	Second	Third	Fourth	Fifth
1	0.05	Present	50.202	85.673	85.673	115.93	137.17
		Ref [24]	50.168	85.600	85.600	115.71	137.04
	0.1	Present	44.886	71.114	71.114	90.898	103.60
		Ref [24]	44.711	70.813	70.813	90.142	103.14
	0.2	Present	31.977	42.896	42.896	49.816	53.115
		Ref [24]	31.610	42.451	42.451	48.990	52.572
2	0.05	Present	37.564	41.040	52.287	69.455	77.227
		Ref [24]	37.557	41.019	52.251	69.405	77.215
	0.1	Present	34.498	37.199	46.406	59.714	65.247
		Ref [24]	34.453	37.077	46.222	59.482	65.185
	0.2	Present	26.113	27.410	32.515	38.767	40.427
		Ref [24]	25.938	27.054	32.127	38.379	40.280

Table 2. Buckling load factors $-N_y{a}^2\left./\phantom{-N_y{a}^{2}D}\right)D$ for CCCC isotropic plate (Nx/Ny = 1.5)

$b\left./\phantom{ba}\right)a$	$h\left./\phantom{ha}\right)a$		$\mathit{Mode}$
			First	Second	Third	Fourth	Fifth
1	0.05	Present	40.060	40.060	61.206	77.530	92.545
		Ref [24]	40.064	40.064	61.210	77.535	92.551
		Ref [10]	40.033	61.151	77.468	92.369	99.146
	0.1	Present	35.771	50.792	64.203	72.460	75.042
		Ref [24]	35.773	50.795	64.207	72.465	75.046
		Ref [10]	35.632	50.565	63.946	71.862	74.760
	0.2	Present	25.355	30.633	38.304	38.504	39.565
		Ref [24]	25.357	30.634	38.306	38.505	39.566
		Ref [10]	25.068	30.311	37.913	38.228	38.954
2	0.05	Present	26.839	32.611	44.635	52.466	58.244
		Ref [24]	26.841	32.614	44.639	52.470	58.248
		Ref [10]	26.834	32.596	44.608	52.458	58.211
	0.1	Present	24.673	29.586	39.610	44.339	48.678
		Ref [24]	24.674	29.588	39.613	44.341	48.682
		Ref [10]	24.673	29.497	39.467	44.296	48.524
	0.2	Present	18.738	21.840	27.495	27.663	29.714
		Ref [24]	18.739	21.841	27.496	27.664	29.715
		Ref [10]	18.634	21.587	27.356	27.403	29.426

Example 2

The buckling load factors of CCCC orthotropic plates with different thicknesses under two ratios of length to width ($b\left./\phantom{ba}\right)a$) when $N_x\left./\phantom{N_x{N}_y}\right)N_y=1$ and $1.5$ are calculated separately by the sum of the first 60 terms ($N=15$) of the solution (26). The material properties are taken as $v_x=0.25$, $v_y=0.025$, $k=5\left./\phantom{56}\right)6$, $E_x=10E_y$, $G_{\mathit{xy}}=0.5E_y$, $G_{\mathit{xz}}=0.5E_y$, and $G_{\mathit{yz}}=0.2E_y$ [34]. The first six buckling load factors are given in Tables 3 and 4. In addition, we plotted the first six modal shapes of the CCCC orthotropic square plate with $b\left./\phantom{ba}\right)a=1,$$N_x\left./\phantom{N_x{N}_y}\right)N_y=1$ and $h\left./\phantom{ha}\right)a=0.1$ in Fig. 2. To reveal the rate of convergence of Eq. (26) for calculating the buckling load factors of rectangular plates with different thicknesses, we conducted convergence analysis on the first and sixth buckling load factors obtained at $b\left./\phantom{ba}\right)a=2$ and $N_x\left./\phantom{N_x{N}_y}\right)N_y=1.5$. The analysis showed that the accuracy of the five significant figures can be guaranteed when $N=15$, as shown in Fig. 3. Finally, Fig. 4 shows the buckling load factors of the CCCC orthotropic plates with different width–thickness ratios $h\left./\phantom{ha}\right)a$.

Table 3. Buckling load factors ${-N_y{a}^2\left./\phantom{-N_y{a}^{2}D}\right)D}_{11}$ for the CCCC orthotropic plate ($N_x\left./\phantom{N_x{N}_y}\right)N_y=1$)

$b\left./\phantom{ba}\right)a$	$h\left./\phantom{ha}\right)a$	$\mathit{Mode}$
		First	Second	Third	Fourth	Fifth	Sixth
1	0.05	14.048	15.109	20.479	21.048	26.883	28.734
	0.1	9.4330	9.7725	12.117	12.331	14.317	14.557
	0.2	4.1728	4.1762	4.4835	4.4983	4.6700	4.6843
2	0.05	12.642	12.713	14.455	14.565	16.870	17.873
	0.1	8.6181	8.6552	9.5689	9.6158	10.988	12.114
	0.2	4.1731	4.1755	4.3468	4.3489	4.4895	4.4913

Table 4. Buckling load factors ${-N_y{a}^2\left./\phantom{-N_y{a}^{2}D}\right)D}_{11}$ for the CCCC orthotropic plate ($N_x\left./\phantom{N_x{N}_y}\right)N_y=1.5$)

$b\left./\phantom{ba}\right)a$	$h\left./\phantom{ha}\right)a$	$\mathit{Mode}$
		First	Second	Third	Fourth	Fifth	Sixth
1	0.05	12.882	13.856	17.478	20.246	25.452	25.610
	0.1	8.7298	9.0907	11.184	11.669	13.598	13.654
	0.2	3.9314	3.9427	4.3549	4.3556	4.7323	4.8492
2	0.05	11.557	11.699	13.070	15.729	15.820	15.820
	0.1	7.9265	7.9865	8.7981	10.097	10.200	10.200
	0.2	3.7205	3.7231	3.9343	4.1637	4.1679	4.1679

[See PDF for image]

Fig. 2

The first six mode shapes of the CCCC orthotropic square plate with $N_x\left./\phantom{N_x{N}_y}\right)N_y=1$ and $h\left./\phantom{ha}\right)a=0.1$

[See PDF for image]

Fig. 3

Convergence analysis of the buckling load factors of CCCC orthotropic plates with different thicknesses

[See PDF for image]

Fig. 4

The buckling load factors of CCCC orthotropic plates with different width–thickness ratios $h\left./\phantom{ha}\right)a$

Conclusion

This study applies the SSM to solve the buckling problem of orthotropic RMTPs. First, the governing equations are formulated in the Hamiltonian system. Then, within the framework of the Hamilton system, without the need to preset any form of trial function and strict step-by-step deduction during the solving process, the buckling solution of CCCC orthotropic RMTPs is obtained by applying the separation variable method and symplectic eigenfunction expansion method. The obtained buckling solution can uniformly solve the buckling problems of isotropic and orthotropic rectangular plates with different thicknesses. In the examples, the buckling load factors of CCCC orthotropic/isotropic rectangular plates with different aspect ratios and width–thickness ratios are calculated by the symplectic superposition solution (26) and the results are obtained using the software Wolfram Mathematica 13.3. Then, we analyzed the convergence of the solution (26) in calculating some buckling loads of orthotropic rectangular plates and found that the convergence speed of this solution is fast and that the larger the width-to-thickness ratio $h\left./\phantom{ha}\right)a$, the faster the convergence speed of this solution. By analyzing the buckling load factors of CCCC orthotropic plates with different width–thickness ratios $h\left./\phantom{ha}\right)a$ (Fig. 4), we find that the larger the width–thickness ratios $h\left./\phantom{ha}\right)a$, the smaller the buckling load factor. By analyzing the data in Tables 3 and 4, we find that the larger the aspect ratio $b/a$, the smaller the buckling load factor. The larger the ratio of the membrane forces $N_x\left./\phantom{N_x{N}_y}\right)N_y$, the smaller the buckling load factor. In addition, because the SSM can effectively handle complex boundary-value problems of high-order partial differential equations, and the solution process of the SSM is general, we can apply the SSM to study the symplectic superposition solutions for buckling problems of rectangular plates with different thicknesses under more complex boundary conditions (such as clamped edges, free edges, and corner supports).

Acknowledgements

This work was supported by the National Natural Science Foundation of China (12362001 and 11862019) and the Natural Science Foundation of Inner Mongolia (2023MS01008).

Author Contribution

Z wrote the original draft; B provided the resources and software; B agrees to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved; Z and B made substantial contributions to the conception of the manuscript; Z and B made substantial contributions to the acquisition, and analysis of data in the manuscript; Z and B made a formal analysis; B and W revised for important intellectual content; B and W approved the version to be published.

Funding

National Natural Science Foundation of China (12362001 and 11862019), Natural Science Foundation of Inner Mongolia Autonomous Region,2023MS01008

Declarations

Conflict of interest

The authors declare no competing interests.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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