## 4 Calculation for Axial Compressive Bearing Capacity
### Steel Plate Resistance
#### 4.1.1 Local Buckling Analysis of Unstiffened Steel Plate  
Since the steel plate of specimen HGW-1 was surrounded by four neighboring studs, and the steel plate of specimen HGW-3 was divided into three narrow plate-strips by vertical separators, the simple support on four edges can be applied without a rotation constraint.  
The strain energy of the steel plate under slightly bending can be established using the following equation [11]:  
\[\begin{array}{ll}U&=\frac{1}{2}\int_{0}^{a}\int_{0}^{b}\left(M_{x}\frac{ \partial^{2}w}{\partial x^{2}}+M_{y}\frac{\partial^{2}w}{\partial y^{2}}+2M_{ xy}\frac{\partial^{2}w}{\partial x\partial y}\right)dxdy\\ &=\frac{D}{2}\int_{0}^{a}\int_{0}^{b}\left((\frac{\partial^{2}w}{\partial x^{2 }}+\frac{\partial^{2}w}{\partial y^{2}})^{2}-2(1-v)[\frac{\partial^{2}w}{ \partial x^{2}}\times\frac{\partial^{2}w}{\partial y^{2}}-(\frac{\partial^{2} w}{\partial x\partial y})^{2}]\right)dxdy\end{array} \tag{3}\]  
In which \(v\) denotes the Poisson ratio; \(\omega\) denotes the plane deflection at any point; \(M_{x}\), \(M_{y}\) denote the bending moment caused by plate bending; \(M_{xy}\) denotes the torque caused by plate bending; \(D\) denotes the bending rigidity of the steel plate, considering the Poisson ratio presented in the following equation:  
\[D=\frac{{E_{t}}^{3}}{12(1-v^{2})} \tag{4}\]  
In which \(E\) denotes the elastic modulus, and \(t_{1}\) denotes the thickness of steel plate.  
The potential energy of the external force can be shown in the following equation, from the straight state of the steel plate, to the bending state [11]:  
\[V=-\frac{1}{2}\int_{0}^{a}\int_{0}^{b}N_{x}(\frac{\partial w}{\partial x})^{2}dxdy \tag{5}\]  
According to the above equations, the total potential energy was the sum of the strain energy and the potential energy:  
\[\begin{array}{ll}\Pi=U+V=&\frac{D}{2}\int_{0}^{a}\int_{0}^{b}\left\{(\frac{ \partial^{2}w}{\partial x^{2}}+\frac{\partial^{2}w}{\partial y^{2}})^{2}-2(1 -v)[\frac{\partial^{2}w}{\partial x^{2}}\times\frac{\partial^{2}w}{\partial y ^{2}}-(\frac{\partial^{2}w}{\partial x\partial y})^{2}]\right\}dxdy\\ &-\frac{1}{2}\int_{0}^{a}\int_{0}^{b}N_{x}(\frac{\partial w}{\partial x})^{2}dxdy \end{array} \tag{6}\]  
Based on the small deflection theory, the equation for the critical local buckling was derived using the Rayleigh-Ritz method [11], in which a double trigonometric series was assumed as the bending surface function of the steel plate:  
\[w=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}a_{mm}\sin\frac{m\pi x}{a}\sin\frac{n \pi y}{b} \tag{7}\]  
where \(m\), \(n\) reflect the half wave number along the \(x\), \(y\) direction, after bending of the steel plate. Furthermore, Equation (7) conforms to the geometric boundary condition: \(w=0\) when \(x=0\) and \(x=a\); \(w=0\) when \(y=0\) and \(y=b\). So, Equation (7) was incorporated into Equation (6), and the following equation was obtained, after a series of integration:  
\[\Pi=\frac{\pi^{4}abD}{8}\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}a_{mn}^{2}(\frac {m^{2}}{a^{2}}+\frac{n^{2}}{b^{2}})^{2}-\frac{\pi^{2}bN_{x}}{8a}\sum_{m=1}^{ \infty}\sum_{n=1}^{\infty}a_{mm}^{2}m^{2} \tag{8}\]  
According to the principle of stationary potential energy, a group of linear algebraic equations, such as \(\frac{\partial\Pi}{\partial a_{11}}=0\), \(\frac{\partial\Pi}{\partial a_{12}}=0\), \(\frac{\partial\Pi}{\partial a_{mm}}=0\) were established, and the critical buckling equation was determined using the conditions of a non-zero solution:  
\[N_{x}=\frac{\pi^{2}D}{b^{2}}(\frac{mb}{a}+\frac{n^{2}a}{mb})^{2} \tag{9}\]Considering that the uniform pressure plate usually produced one half-wave in the transverse direction [12] (i.e., \(n=1\)), \(m=a/b\) can be obtained from \(\frac{\partial N_{x}}{\partial m}=0\) with a continuous variable of \(m\). The critical buckling local load (i.e., the minimum value of \(N_{x}\)) and the critical buckling stress were obtained as follows:  
\[N_{x}=\frac{4\pi^{2}D}{b^{2}} \tag{10}\]  
\[\sigma_{cr}=\frac{4\pi^{2}D}{b^{2}t} \tag{11}\]