## 3 Solution Procedure  
### Static Buckling Analyses  
The static buckling solution can be obtained by dropping the damping terms and time-related terms. Equation (29) is reduced as:  
\[-P\Big{[}1-(e_{0}a)^{2}\
abla^{2}\Big{]}\frac{\partial^{2}w}{\partial x^{2}}-bB _{11}\Big{(}1-l^{2}\
abla^{2}\Big{)}\frac{\partial^{4}w}{\partial x^{4}}+ \frac{bA_{11}}{2L}\frac{\partial^{2}w}{\partial x^{2}}\int_{0}^{L}\bigg{(} \frac{\partial w}{\partial x}\bigg{)}^{2}dx=0 \tag{31}\]  
The lateral displacement is assumed in the form  
\[w=W_{\rm s}\sin\Big{(}\frac{\pi x}{L}\Big{)} \tag{32}\]Equation (31) can be written as \(\Gamma\Big{(}\frac{\partial^{2}w}{\partial x^{2}},\frac{\partial^{4}w}{\partial x ^{4}},\frac{\partial w}{\partial x}\Big{)}=0\), where \(\Gamma\Big{(}\frac{\partial^{2}w}{\partial x^{2}},\frac{\partial^{4}w}{ \partial x^{4}},\frac{\partial w}{\partial x}\Big{)}\) equals the left hand side of Equation (31). Then the [PERSON] method is applied as:  
\[\int_{0}^{L}\Gamma\Big{(}\frac{\partial^{2}w}{\partial x^{2}},\frac{\partial^{ 4}w}{\partial x^{4}},\frac{\partial w}{\partial x}\Big{)}\sin\Big{(}\frac{ \pi x}{L}\Big{)}\text{d}x=0 \tag{33}\]  
By substituting Equation (32) into Equation (33) and applying the integration, one obtains:  
\[P\Big{[}L^{2}+(e_{0}a)^{2}\pi^{2}\Big{]}-\pi^{2}bB_{11}\bigg{(}\frac{l^{2}\pi ^{2}+L^{2}}{L^{2}}\bigg{)}-\frac{\pi^{2}bL^{2}A_{11}}{4}W_{s}^{2}=0 \tag{34}\]  
In the prebuckling process the lateral displacement is zero. Solving the expression of \(P\) in Equation (34) with \(W_{s}=0\), the Eigen critical buckling force is obtained as:  
\[P_{cr}=\frac{\pi^{2}bB_{11}}{L^{2}}\cdot\frac{\pi^{2}l^{2}+L^{2}}{\pi^{2}(e_{0 }a)^{2}+L^{2}} \tag{35}\]  
In Equation (35), \(B_{11}\) is related to the porosity distribution, and whose expression is defined before, which can be solved as:  
\[B_{11} =\int_{-h_{f}/2}^{h_{f}/2}z^{2}C_{11}dz \tag{36}\] \[=\left\{\begin{array}{l}\frac{h_{f}^{3}}{12}C_{\max}(1-\chi_{0 }\alpha)\text{, for distribution 1}\\ \frac{h_{f}^{3}}{12}C_{\max}\Big{[}1-\frac{6h_{c}}{\pi h_{t}}\chi_{0}\sin \Big{(}\frac{\pi h_{t}}{2h_{c}}\Big{)}\Big{]}\text{, for distribution 2}\\ \frac{h_{f}^{3}}{12}C_{\max}\Big{[}1-\Big{(}\frac{6\sqrt{2}h_{c}}{\pi h_{t}}- \frac{96\sqrt{2}h_{c}^{2}}{h_{t}}\Big{)}\chi_{0}\sin\Big{(}\frac{\pi h_{t}}{4 h_{c}}\Big{)}\Big{]}\text{, for distribution 3}\end{array}\right.\]  
In the postbuckling process, the maximum vertical displacement can be obtained by solving the expression of \(W_{s}\) in Equation (34), i.e.:  
\[W_{s}=\frac{2\sqrt{L^{2}P\Big{(}\pi^{2}(e_{0}a)^{2}+L^{2}\Big{)}-\pi^{2}bB_{ 11}(\pi^{2}l^{2}+L^{2})}}{L^{2}\sqrt{bA_{11}}} \tag{37}\]  
where \(B_{11}\) is shown in Equation (36), and the other porosity related coefficient \(A_{11}\) is calculated as:  
\[A_{11}=\int_{-h_{f}/2}^{h_{f}/2}C_{11}dz=\left\{\begin{array}{l}h_{t}C_{\max }(1-\chi_{0}\alpha)\text{, for distribution 1}\\ h_{t}C_{\max}\Big{[}1-\frac{2h_{c}}{\pi h_{t}}\chi_{0}\sin\Big{(}\frac{\pi h_{ t}}{2h_{c}}\Big{)}\Big{]}\text{, for distribution 2}\\ h_{t}C_{\max}\Big{[}1-\frac{2\sqrt{2}h_{c}}{\pi h_{t}}\chi_{0}\sin\Big{(}\frac{\pi h _{t}}{4h_{c}}\Big{)}\Big{]}\text{, for distribution 3}\end{array}\right. \tag{38}\]