## 2 Methods  
### The Total Potential Energy of Plane Stress Members  
It is possible to generate a structural system as a plate problem by considering triangular elements having three nodes i, j, and k, as shown in Figure 1. For the member that has a homogeneous and continuous material, linear displacement fields defined as u(x,y) and v(x,y) are, respectively, given in x and y directions in Equations (1) and (2) for a triangular element in the x-y plane. Translational displacements of a node symbolized as i are shown as u\({}_{\text{i}}\) and v\({}_{\text{i}}\). Figure 1: Triangular elements used in the generation of plate systems. \[\mathrm{u}(\mathrm{x},\mathrm{y}) =\mathrm{u_{i}}+\mathrm{C_{1}x}+\mathrm{C_{2}y} \tag{1}\] \[\mathrm{v}(\mathrm{x},\mathrm{y}) =\mathrm{v_{i}}+\mathrm{C_{3}x}+\mathrm{C_{4}y} \tag{2}\]  
Knowing the displacement fields including constants named \"C\({}_{1}\), C\({}_{2}\), C\({}_{3}\), and C\({}_{4}\),\" the strains can be calculated as shown in Equations (3)-(5). \[\varepsilon_{\mathrm{x}}=\frac{\partial\mathrm{u}}{\partial\mathrm{x}}= \mathrm{C_{1}} \tag{3}\]  
\[\varepsilon_{\mathrm{y}}=\frac{\partial\mathrm{v}}{\partial\mathrm{y}}= \mathrm{C_{4}} \tag{4}\]  
\[\gamma_{xy}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}= \mathrm{C_{2}}+\mathrm{C_{3}} \tag{5}\]  
The three different strains that are symbolized as \(\varepsilon_{\mathrm{x}}\), \(\varepsilon_{\mathrm{y}}\), and \(\gamma_{\mathrm{xy}}\), are the normal strain in the x-direction, normal strain in the y-direction, and shear strain, respectively. The nodal displacements are written as in Equations (6)-(8). \[\mathrm{u}(0,0)=\mathrm{u_{i}},\ \mathrm{v}(0,0)=\mathrm{v_{i}} \tag{6}\]  
\[\mathrm{u}(\mathrm{a_{j}},\mathrm{b_{j}})=\mathrm{u_{j}},\ \mathrm{v}(\mathrm{a_{j}}, \mathrm{b_{j}})=\mathrm{v_{j}} \tag{7}\]  
\[\mathrm{u}(\mathrm{a_{k}},\mathrm{b_{k}})=\mathrm{u_{k}},\ \mathrm{v}(\mathrm{a_{k}}, \mathrm{b_{k}})=\mathrm{v_{k}} \tag{8}\]  
After the displacements of each node are obtained, the displacement matrix form can be written as follows using Equations (1) and (2). \[\left[ \begin{array}{c}u_{j}\\ v_{j}\\ u_{k}\\ v_{k}\end{array} \right]=\left[ \begin{array}{cccc}a_{j}&b_{j}&0&0\\ 0&0&a_{j}&b_{j}\\ a_{k}&b_{k}&0&0\\ 0&0&a_{k}&b_{k}\end{array} \right]\left[ \begin{array}{c}C_{1}\\ C_{2}\\ C_{3}\\ C_{4}\end{array} \right]+\left[ \begin{array}{c}u_{i}\\ v_{i}\\ u_{i}\\ v_{i}\end{array} \right] \tag{9}\]  
In Equations (10)-(13), the constants C\({}_{1}\), C\({}_{2}\), C\({}_{3}\), and C\({}_{4}\) are formulated according to matrix calculations of Equation (9). \[\mathrm{C_{1}}=\frac{\mathrm{b_{k}}(\mathrm{u_{j}}-\mathrm{u_{i}})}{\mathrm{ a_{j}}\mathrm{b_{k}}-\mathrm{a_{k}}\mathrm{b_{j}}}+\frac{\mathrm{b_{j}}( \mathrm{u_{k}}-\mathrm{u_{i}})}{\mathrm{a_{k}}\mathrm{b_{j}}-\mathrm{a_{j}} \mathrm{b_{k}}} \tag{10}\]  
\[\mathrm{C_{2}}=\frac{\mathrm{a_{k}}(\mathrm{u_{j}}-\mathrm{u_{i}})}{\mathrm{ a_{k}}\mathrm{b_{j}}-\mathrm{a_{j}}\mathrm{b_{k}}}+\frac{\mathrm{a_{j}}( \mathrm{u_{k}}-\mathrm{u_{i}})}{\mathrm{a_{j}}\mathrm{b_{k}}-\mathrm{a_{k}} \mathrm{b_{j}}} \tag{11}\]  
\[\mathrm{C_{3}}=\frac{\mathrm{b_{k}}(\mathrm{v_{j}}-\mathrm{v_{i}})}{\mathrm{ a_{j}}\mathrm{b_{k}}-\mathrm{a_{k}}\mathrm{b_{j}}}+\frac{\mathrm{b_{j}}( \mathrm{v_{k}}-\mathrm{v_{i}})}{\mathrm{a_{k}}\mathrm{b_{j}}-\mathrm{a_{j}} \mathrm{b_{k}}} \tag{12}\]  
\[\mathrm{C_{4}}=\frac{\mathrm{a_{k}}(\mathrm{v_{j}}-\mathrm{v_{i}})}{\mathrm{ a_{k}}\mathrm{b_{j}}-\mathrm{a_{j}}\mathrm{b_{k}}}+\frac{\mathrm{a_{j}}( \mathrm{v_{k}}-\mathrm{v_{i}})}{\mathrm{a_{j}}\mathrm{b_{k}}-\mathrm{a_{k}} \mathrm{b_{j}}} \tag{13}\]  
For a body with two dimensions, the strain energy (e) is formulated in Equation (14). \[e=\int\limits_{\varepsilon=0}^{\varepsilon}\sigma d\varepsilon=\frac{1}{2}( \sigma_{x}\varepsilon_{x}+\sigma_{y}\varepsilon_{y}+\tau_{xy}\gamma_{xy}) \tag{14}\]  
For linear plane strain problems, the stresses are defined as Equations (15)-(17) that include two material constants, namely, elasticity modulus (E) and Poisson's ratio (v).