## 8 Bending Analysis
 \tag{120}\]  
For the bending analysis, we considered that:  
\[W=V_{0}+pV_{1}+p^{2}V_{2},\;\theta=\eta_{0}+p\eta_{1}+p^{2}\eta_{2},\;q=\xi_{0 }+p\xi_{1}+p^{2}\xi_{2} \tag{121}\]  
Inserting Equation (121) into (119) and (120) and identifying the coefficients of \(p^{i}\), \(i=0\), \(1\), \(2\) in the same manner as the above section, we can see that Equations (105) and (110) were unchanged, such that the solutions:  
\[V_{0}=\overline{A}_{0}sin\pi x,\;\;\;\;\;\;\eta_{0}=\overline{B}_{0}cos\pi x \tag{122}\]  
\[V_{1}=\overline{A}_{1}sin2\pi x,\;\;\;\;\;\;\eta_{1}=\overline{B}_{1}cos2\pi x \tag{123}\]  
\[V_{3}=\overline{A}_{2}sin3\pi x,\;\;\;\;\;\;\eta_{2}=\overline{B}_{2}cos3\pi x \tag{124}\]  
led to the conclusions:  
\[\overline{B}_{0}=\pi\overline{A}_{0},\;\;\;\;\;\;\overline{B}_{1}=\pi\overline{ A}_{1},\;\;\;\;\;\;\overline{B}_{2}=\pi\overline{A}_{2}, \tag{125}\]  
The Equation (105) can be rewritten as:  
\[(b_{11}-d_{11})\Big{(}V_{0}^{IV}-l^{2}V_{0}^{VI}\Big{)}+(\eta_{0}+K+K_{1})\Big{(} V_{0}-l^{2}V_{0}^{\prime\prime}\Big{)}=0 \tag{126}\]  
It follows that:  
\[\xi_{0}=\pi^{4}(b_{11}-d_{11})-K-K_{1} \tag{127}\]  
After some calculations, similarly to the previous section, from the coefficients of \(p^{1}\) and \(p^{2}\), for the values of \(V_{0}\), \(V_{1}\), \(V_{3}\), \(\eta_{0}\), \(\eta_{1}\)\[ \begin{array}{c}\xi_{1}=-\frac{\pi^{4}b_{11}\pi_{0}^{2}(146+231(\pi\omega n)^{2})}{3 5(1+\pi^{2}l^{2})}-\frac{\pi^{4}d_{11}\pi_{0}^{2}\big{(}56+148(\pi\omega n)^{2} \big{)}}{35(1+\pi^{2}l^{2})}+\frac{\pi^{4}e_{11}\pi_{0}^{2}\big{(}14+173(\pi \omega n)^{2}\big{)}}{35(1+\pi^{2}l^{2})}\\ +\frac{3\pi^{2}\lambda^{2}a_{11}(8-3(\pi\omega n)^{2}\big{)}\overline{A}_{0}}{ 35(1+\pi^{2}l^{2})}-\frac{(K+K_{1})\overline{A}_{1}\left(1+(4\pi\omega n)^{2} \right)}{\overline{A}_{0}\pi^{3}(1+2\pi l^{2})}-\frac{3(K_{3}+2K)\left(1+2(\pi \omega n)^{2}\right)\overline{A}_{0}^{2}}{4(1+\pi^{2}l^{2})}\end{array}  \tag{128}\]  
\[ \begin{array}{c}\xi_{2}=\frac{b_{11}\pi^{4}\overline{A}_{0}^{2}A_{1}}{3(1+ \pi^{2}l^{2})}\left[\frac{323}{35}-\frac{871}{505}\pi^{4}\overline{A}_{0}^{2} -l^{2}\left(\frac{362}{103}-\frac{1372}{105}\pi^{2}\overline{A}_{0}^{2}\right) \right]-\frac{\pi^{4}d_{11}\overline{A}_{0}\overline{A}_{1}}{3(1+\pi^{2}l^{2} )}\left(\frac{371}{105}-\frac{581}{105}\pi^{2}l^{2}\right)\\ -\frac{(K+K_{1})\overline{A}_{2}\left(1+(3\pi\omega n)^{2}\right)}{A_{0}\pi^{3 }\left(1+(\pi\omega n)^{2}\right)}\end{array}  \tag{129}\]  
Finally, from Equations (122)-(129), it holds that:  
\[W(x)=\overline{A}_{0}sin\pi x+\overline{A}_{1}sin2\pi x+\overline{A}_{2}sin3\pi x \tag{130}\]  
\[\theta(x)=\pi\big{(}\overline{A}_{0}cos\pi x+\overline{A}_{1}cos2\pi x+ \overline{A}_{2}cos3\pi x\big{)} \tag{131}\]  
\[ \begin{array}{c}q=\pi^{4}(b_{11}-d_{11})-K-K_{1}-\frac{\pi^{4}b_{11}\pi_{0}^ {2}\big{(}146+231(\pi\omega)^{2}\big{)}}{35(1+\pi^{2}l^{2})}-\frac{\pi^{4}d_{1 1}\pi_{0}^{2}\big{(}56+148(\pi\omega n)^{2}\big{)}}{35(1+\pi^{2}l^{2})}+\frac {\pi^{4}e_{11}\pi_{0}^{2}\big{(}14+173(\pi\omega n)^{2}\big{)}}{35(1+\pi^{2}l ^{2})}+\\ \frac{3\pi^{2}\lambda^{2}a_{11}(8-3(\pi\omega n)^{2})\overline{A}_{0}}{35(1+ \pi^{2}l^{2})}-\frac{(K+K_{1})\overline{A}_{1}\left(1+(4\pi\omega n)^{2} \right)}{\overline{A}_{0}\pi^{3}\left(1+\pi^{2}l^{2}\right)}-\frac{3(K_{3}+2K) \left(1+2(\pi\omega)^{2}\right)\overline{A}_{0}^{2}}{4(1+\pi^{2}l^{2})}+\frac {b_{11}\pi^{4}\overline{A}_{0}^{2}A_{1}}{3(1+\pi^{2}l^{2})}\left[\frac{323}{35 }-\frac{871}{505}\pi^{4}\overline{A}_{0}^{2}-\\ l^{2}\left(\frac{362}{103}-\frac{1372}{105}\pi^{2}\overline{A}_{0}^{2}\right) \right]-\frac{\pi^{4}d_{11}\pi_{0}\overline{A}_{1}}{3(1+\pi^{2}l^{2})}\left( \frac{371}{105}-\frac{581}{105}\pi^{2}l^{2}\right)-\frac{(K+K_{1})\overline{A }_{2}\left(1+(3\pi\omega n)^{2}\right)}{A_{0}\pi^{3}\left(1+(\pi\omega n)^{2} \right)}\end{array}  \tag{132}\]  
in which \(\overline{A}_{0}\), \(\overline{A}_{1}\), and \(\overline{A}_{2}\) were determined optimally as in the previous section.