## Appendix A  
### Generalized Equation for the Moment of Inertia of the Area of \(n\)-Layered Corrugated Paperboard  
To represent the moment of inertia of the area for \(n\)-layered corrugated paperboard (the number of liner: \(n\)), it was expected to be useful to analyze surface properties of various corrugated paperboards and multi-layered corrugated structures. This study developed a generalized equation for the moment of inertia of the area of the \(n\)-layered corrugated paperboard further investigating the moment of inertia of area analysis methods for various corrugated paperboards used in Lee and Park [42]. A generalized equation for the moment of inertia of the area of the \(n\)-layered corrugated paperboard could be developed with an assumption that the corrugated paperboard components consisted of the same materials (where the width of the \(n\)-layered corrugated paperboard \(=\omega\), the thickness of the \(n\)th liner \(=t_{n}\), the thickness of the \(k\)th corrugating medium \(=t_{kk}\), height of the \(k\)th flute \(=h_{k}\), and the take-up factor of the \(k\)th flute \(=\alpha_{k}\)). * The neutral axis of CD cross-section of the \(n\)-layered corrugated paperboard: \[ \begin{array}{c}\overline{z}=\frac{1}{A}\int_{\lambda}z\,dA=\\ \frac{\sum_{n=1}^{n}t_{n}\left(\sum_{n=1}^{n-1}t_{n}+\frac{1}{2}t_{n}+\frac{1 }{2}t_{n}+\sum_{n=1}^{n-1}t_{n}\right)+\sum_{n=1}^{n-1}t_{nk}a_{k}\left(\sum_ {n=1}^{n-1}t_{n}+\sum_{k=1}^{n-2}t_{k}+\frac{1}{2}t_{k}\right)}{\sum_{n=1}^{n }t_{n}+\sum_{k=1}^{n-1}t_{nk}a_{k}}\end{array} \] (1)
* Moment of inertia of the area for the neutral axis of CD cross-section of the \(n\)-layered corrugated paperboard: \[ \begin{array}{c}I_{\overline{x}\overline{x}}=\sum_{n=1}^{n}\frac{\omega_{n} ^{3}}{12}+\sum_{n=1}^{n}\left(\overline{z}-\sum_{n=1}^{n-1}t_{n}-\frac{1}{2}t _{n}-\sum_{k=1}^{n-1}h_{k}\right)^{2}\omega t_{n}+\sum_{k=1}^{n-1}(I_{ \overline{x}\overline{x}})_{k}+\\ \sum_{k=1}^{n-1}\left(\overline{y}-\sum_{n=1}^{n-1}t_{n}-\sum_{k=1}^{n-2}h_{k} -\frac{1}{2}h_{k-1}\right)^{2}\omega t_{kk}a_{k}\end{array} \] (2)
* Moment of inertia of the area for the neutral axis of the CD cross-section of flute:for small area segment of flute, \(dA\cong t_{kk}ds\); for small length segment of flute, \(ds=\sqrt{1+\left(\frac{dz}{dx}\right)^{2}}dx\). If the shape of the flute is considered as a sinusoidal function, \(z=\frac{h_{k}}{2}\ \sin\frac{2\pi x}{\lambda_{k}}\rightarrow\frac{dz}{dx}=\frac{\pi h _{k}}{\lambda_{k}}\cos\frac{2\pi x}{\lambda_{k}}\); therefore, the moment of inertia of the area of the neutral axis of the flute can be expressed in Equation (A3) or (A4):  
\[\left(I_{\overline{xx}}\right)_{k}=\int y^{2}dA=\frac{4\omega}{\lambda_{k}} \Biggl{(}\int_{o}^{\frac{\lambda_{k}}{4}}\ \frac{t_{kk}h_{k}}{4}\sin^{2}\frac{2\pi x}{\lambda_{k}}\times\sqrt{1+\frac{ \pi^{2}h_{k}^{2}}{\lambda_{k}^{2}}\cos^{2}\frac{2\pi x}{\lambda_{k}}}dx\Biggr{)}\] (A3)  
\[\left(I_{\overline{xx}}\right)_{k}=\frac{\omega}{\lambda_{k}}\times\frac{\pi} {4}\left\{\left(\frac{h_{k}}{2}\right)^{3}\!\left(\frac{\lambda_{k}}{2}\right) -\left(\frac{h_{k}}{2}-t_{kk}\right)^{3}\!\left(\frac{\lambda_{k}}{2}-2t_{kk} \right)\right\}\] (A4)  
_Appendix A.2.