## 2 Mechanical Analysis Model for TSWs
### Thin Plate Differential Equilibrium Equation  
Figure 3 shows the force distribution on a rectangular thin plate with all four edges supported, subject to transverse loads, longitudinal loads, and in-plane localized lateral forces (including shear). This generalized model supports broader theoretical derivation beyond the specific conditions in Figure 2.  
The purpose of using the general thin plate model for analysis and research is to make the theoretical derivation have a wider applicability; if the transverse and longitudinal loads are small, the bending effect for the thin plate can be ignored. In this case, the stress superposition effect brought by the longitudinal and transverse loads on the thin plate can be disregarded; however, when the longitudinal and transverse loads reach a certain level, the effect of the longitudinal and transverse loads on the bending effect of the thin plate must be considered. Due to the localized lateral force applied to the Z-Y plane, the value  
Figure 3: Rectangular plate subjected to edge forces in-plane and localized lateral forces. The in-plane forces are uniformly distributed in the midplane, while the localized lateral forces vary as a function of y and z.  
of which varies with different positions of the thin plate, it is expressed as a function with reference to y and z. Thus, the governing differential equation of the thin plate is as follows:  
\[D\
abla^{4}w-\left(N_{y}\frac{\partial^{2}w}{\partial y^{2}}+N_{z}\frac{ \partial^{2}w}{\partial z^{2}}+2N_{yz}\frac{\partial^{2}w}{\partial y\partial z }\right)=q(y,z) \tag{1}\]  
\[D=EW_{b}^{3}/12\Big{(}1-\mu^{2}\Big{)} \tag{2}\]  
where \(D\) is the flexural rigidity, \(w\) is the deflection, \(E\) is [PERSON]'s modulus, \(\mu\) is [PERSON]'s ratio, \(W_{b}\) is the thickness of the thin plate, \(N_{y}\), \(N_{\mathrm{z}}\), \(N_{\mathrm{yz}}\) are in-plane forces per unit width along the edge of the thin plate (\(N_{\mathrm{yz}}\) is the in-plane shear force, and \(N_{y}\) and \(N_{\mathrm{z}}\) are normal forces), \(q(y,z)\) is the localized lateral force on the plate per unit area.  
Because the in-plane shear forces \(N_{\mathrm{yz}}\) at the boundaries of the thin plate are zero [17], the influence of in-plane shear forces on the bending behavior of the thin plate can be ignored; so, the Formula (1) can be simplified as follows:  
\[D\
abla^{4}w-\left(N_{y}\frac{\partial^{2}w}{\partial y^{2}}+N_{z}\frac{ \partial^{2}w}{\partial z^{2}}\right)=q(y,z) \tag{3}\]  
\[q(y,z)=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}q_{mn}\sin\frac{m\pi y}{L}\sin \frac{n\pi z}{h} \tag{4}\]  
For linear loads localized laterally and varying with the z-direction, the Fourier coefficients \(q_{mn}\) in Formula (3) are determined based on Formula (a) on page 111 of Theory of plates and shells [23]. Consequently, the Fourier coefficients \(q_{mn}\) in Formula (3) are calculated as follows:  
\[q_{mn}=\frac{4q}{Lh}\int_{y_{0}-a/2}^{y_{0}+a/2}\int_{z_{0}-h/2}^{z_{0}+h/2} \sin\frac{m\pi y}{L}\sin\frac{n\pi z}{h}dydz \tag{5}\]  
\[q=q_{0}-kz \tag{6}\]  
where \(q\) is the linear load, \(L\) is the stope length, \(h\) is the stope height, \((y_{0},z_{0})\) is coordinates of the localized lateral load center, \(a\) is width of inter-chamber pillar, \(q_{0}\) is maximum value of lateral load in thin plate, \(k\) is lateral load per unit.