## 4 Calculation for Axial Compressive Bearing Capacity
### Steel Plate Resistance
#### 4.1.2 Local Buckling Analysis of Stiffened Steel Plate  
For the steel plate with stiffening ribs, the stiffening coefficient \(\kappa\) represents the effect of axial rigidity and torsional rigidity on the improvement of the local strength, which is defined as the following equation [9]:  
\[\kappa=A_{1}/A \tag{12}\]  
where \(A_{1}\) is the cross-sectional area of the stiffening ribs, and \(A\) is the cross-sectional area of the steel plate.  
Since the stiffening coefficient \(\kappa\) of specimen HGW-2 is 0.125, the axial rigidity and torsional rigidity of specimen HGW-2 are neglected, without considering the initial inflect of the steel plate [13]. Besides, the boundary condition of the steel plate with stiffening ribs was assumed to be simply supported on four edges, the critical buckling load was calculated by the energy method, and the deflection surface function was adopted by Equation (7). For the stiffening ribs, the strain energy and the potential energy of the external force can be calculated using the following equations [11]:  
\[U=EI\int_{0}^{a}{[(\frac{\partial^{2}w}{\partial x^{2}})^{2}_{y=b/4}+(\frac{ \partial^{2}w}{\partial x^{2}})^{2}_{y=b/2}+(\frac{\partial^{2}w}{\partial x^ {2}})^{2}_{y=3b/4}]}\mathrm{d}x \tag{13}\]  
\[V=-AN_{x}\int_{0}^{a}{[(\frac{\partial^{2}w}{\partial x^{2}})^{2}_{y=b/4}+( \frac{\partial^{2}w}{\partial x^{2}})^{2}_{y=b/2}+(\frac{\partial^{2}w}{ \partial x^{2}})^{2}_{y=3b/4}]}\mathrm{d}x \tag{14}\]  
where \(A\) and \(I\) denote the cross-sectional area and the inertia moment of one rib, respectively.  
The total potential energy of the stiffening ribs was the sum of the strain energy and the potential energy of the external force, which was obtained after Equation (7) had been incorporated into Equation (15), with a series of integration:  
\[\Pi=U+V=(EI-AN_{x})\int_{0}^{a}{[(\frac{\partial^{2}w}{\partial x^{2}})^{2}_{ y=b/4}+(\frac{\partial^{2}w}{\partial x^{2}})^{2}_{y=b/2}+(\frac{\partial^{2}w}{ \partial x^{2}})^{2}_{y=3b/4}]}\mathrm{d}x \tag{15}\]  
\[\Pi=\frac{(EI-AN_{x})\pi^{4}}{a}\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}a_{mn}^ {2}m^{4} \tag{16}\]  
The total potential energy of the walling system is the sum of the two steel materials, including the steel plate and stiffening ribs. The potential energies of the stiffening ribs and steel plate should be superimposed in the calculation of the total potential energy, presenting the potential energy of the external force. Therefore, the following equation can be obtained through the superposition of Equations (8) and (16):  
\[\Pi=\frac{\pi^{4}abD}{8}\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}a_ {mn}^{2}(\frac{m^{2}}{a^{2}}+\frac{n^{2}}{b^{2}})^{2}-\frac{\pi^{2}bN_{x}}{8a} \sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}a_{mn}^{2}m^{2}+\frac{( EI-AN_{x})\pi^{4}}{a}\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}a_{mn}^{2}m^{4} \tag{17}\]In view of the single half-wave of the uniform pressure plate in the transverse direction [12] (i.e., \(n=1\)), supposed \(m=1\) and \(\frac{\partial\Pi}{\partial u_{11}}=0\) were established, the coefficient determinant was viewed as zero, and the critical local buckling stress of the steel plate with the stiffening ribs was obtained.  
\[N_{x}=\frac{8a}{b+8A\pi^{2}}[\frac{\pi^{2}D(a^{2}+b^{2})^{2}}{a^{2}b^{3}}+EI \pi^{2}] \tag{18}\]  
\[\sigma_{\sigma}=\frac{8a}{(b+8A\pi^{2})t}[\frac{\pi^{2}D(a^{2}+b^{2})^{2}}{a^{ 2}b^{3}}+EI\pi^{2}] \tag{19}\]