## 2 Mechanical Analysis Model for TSWs
### Determining Coefficients for Deflection Expressions with Additional Terms  
The governing equation above is a non-homogeneous partial differential equation in terms of deflection. While a closed-form integration method exists, it becomes complex when accounting for the effects of both transverse and longitudinal loads [27; 28]. Hence, the superposition method is commonly used for thin plates with all edges supported [23; 29]. To further simplify the solution, the method of complementary terms can be applied. [PERSON] [30] introduced a deflection expression that includes supplementary terms, offering both ease of calculation and broad applicability. The general deflection expression is given as follows:  
\[\omega=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}b_{mn}\sin\frac{m\pi y}{L}\sin \frac{n\pi z}{h}+\sum_{n=1}^{\infty}F_{n}(y)\sin\frac{n\pi z}{h}+\sum_{m=1}^{ \infty}F_{m}(z)\sin\frac{m\pi y}{L} \tag{7}\]  
where  
\[F_{n}(y)=\frac{1}{6L}\bigg{(}\frac{A_{n}-B_{n}}{D}\bigg{)}y^{3}-\frac{1}{2} \frac{A_{n}}{D}y^{2}+\frac{L}{6D}(A_{n}+2B_{n})y \tag{8}\]  
\[F_{m}(z)=\frac{1}{6h}\bigg{[}\frac{C_{m}-D_{m}}{D}\bigg{]}z^{3}-\frac{1}{2} \frac{C_{m}}{D}z^{2}+\frac{h}{6D}(C_{m}+2D_{m})z \tag{9}\]In Formula (7), the term \(b_{mn}\) represents the Fourier coefficient of the deflection expression. According to the derivation results from the book Fourier Series Solutions in Structural Mechanics [30], the deflection coefficient \(b_{mn}\) as follows:  
\[b_{mn}=\frac{g_{n}+g_{m}+\frac{q_{mn}}{D}}{\left[\left(a_{\text{m}}^{2}+\beta_{n }^{2}\right)^{2}+a_{\text{m}}^{2}\frac{N_{y}}{D}+\beta_{n}^{2}\frac{N_{z}}{D} \right]} \tag{10}\]  
\[g_{n}=-\frac{2}{m\pi D}\bigg{[}\beta_{n}^{2}\bigg{(}2+\frac{\beta_{n}^{2}}{ \alpha_{\text{m}}^{2}}\bigg{)}+\frac{N_{y}}{D}+\frac{\beta_{n}^{2}}{\alpha_{ \text{m}}^{2}}\frac{N_{z}}{D}\bigg{]}\Big{[}A_{\text{n}}+(-1)^{\text{m}+1}B_{ \text{n}}\Big{]} \tag{11}\]  
\[g_{m}=-\frac{2}{n\pi D}\bigg{[}\alpha_{\text{m}}^{2}\bigg{(}2+\frac{\alpha_{ \text{m}}^{2}}{\beta_{n}^{2}}\bigg{)}+\frac{\alpha_{\text{m}}^{2}}{\beta_{n}^ {2}}\frac{N_{y}}{D}+\frac{N_{z}}{D}\bigg{]}\Big{[}C_{m}+(-1)^{\text{n}+1}D_{ \text{m}}\Big{]} \tag{12}\]  
where \(\alpha_{m}=\frac{m\pi}{L}\), \(\beta_{n}=\frac{m\pi}{h}\). The coefficients \(A_{\text{n}}\), \(B_{\text{n}}\), \(C_{\text{m}}\), \(D_{\text{m}}\) in Formulas (7) to (12) represent the sinusoidal steps of the normal bending moments on the left edge ((\(y=0\)), as shown in Figure 3), the right edge (\(y=L\)), the back edge (\(z=0\)), and the front edge (\(z=h\)) of the rectangular thin plate. The expression for the sinusoidal steps of the normal bending moments on these four edges is as follows:  
\[\left\{ \begin{array}{l}M_{y}(0,z)=\sum\limits_{n=1}^{\infty}A_{n}\sin\frac{ n\pi z}{h}\\ M_{y}(L,z)=\sum\limits_{n=1}^{\infty}B_{n}\sin\frac{n\pi z}{h}\\ M_{z}(y,0)=\sum\limits_{m=1}^{\infty}C_{m}\sin\frac{m\pi y}{L}\\ M_{z}(y,h)=\sum\limits_{m=1}^{\infty}D_{m}\sin\frac{m\pi y}{L}\\ \end{array} \right. \tag{13}\]  
The coefficients \(A_{\text{n}}\), \(B_{\text{n}}\), \(C_{\text{m}}\), \(D_{\text{m}}\) in Formulas (8) to (12) need to be determined based on the boundary supported conditions of the rectangular thin plate. These coefficients \(A_{\text{n}}\), \(B_{\text{n}}\), \(C_{\text{m}}\), \(D_{\text{m}}\) represent the sine function of the normal moments on the edges as outlined in Formula (13). This paper explores a rectangular thin plate fixed and supported on all four edges; Thus, the values of these coefficients \(A_{\text{n}}\), \(B_{\text{n}}\), \(C_{\text{m}}\), \(D_{\text{m}}\) must be determined through the boundary conditions of the edges. The four edges of the TSW are all connected to the original rock, and their boundary displacements are basically negligible and boundaries are fixed and supported by the surrounding rock, similar to a thin plate with four fixed and supported edges, The deflection and angular displacement around the perimeter of the rectangular thin plate are zero. Thus, TSW's boundary conditions are expressed as follows:  
\[