# Static Response of Double-Layered Pipes via a Perturbation Approach
## 3 Perturbation Analysis
\]  
\(c_{4}(v_{1}^{\prime}-\vartheta_{1})\) (one arbitrary constant is introduced); (2) substitution of that in Equation (15)\({}_{2}\) and double integration of both the sides of it to obtain \(\vartheta_{1}\) (two more arbitrary constants are introduced); (3) substitution of the obtained \(\vartheta_{1}\) in the integrated Equation (15)\({}_{1}\) and one more integration of both its sides, to evaluate \(v_{1}\) (one more arbitrary constant is introduced). The four arbitrary constants are evaluated imposing the four boundary conditions (b.c.) in Equation (15)\({}_{3,4}\). To continue the chain-solving procedure, Equation (19) is substituted to the right-hand side of Equation (16). The solution for variable \(u_{2}\) is directly obtained with two steps of integration of Equation (16)\({}_{1}\) and use of its b.c. :  
\[ \begin{split} u_{2}&=\frac{p_{v}^{2}s^{2}}{5040c_{ 7}^{2}}(-5s(42l^{4}-63l^{3}s+42l^{2}s^{2}-14 ls^{3}+2s^{4}))\\ &\quad+\frac{p_{v}^{2}s^{2}}{60c_{1}c_{4}c_{7}}(c_{1}-c_{4})(-15 l^{3}+20l^{2}s-10 ls^{2}+2s^{3})\end{split}  \tag{20}\]  
and for variable \(a_{g,2}\), after double integration of Equation (16)\({}_{4}\) and use of its b.c. :  
\[a_{g2}=\frac{p_{v}(c_{57}+c_{4}c_{11})}{c_{4}c_{7}c_{21}}\left[(s-l)+\text{ sech}\left(l\sqrt{\frac{c_{21}}{c_{25}}}\right)\left(l\cosh\left((l-s)\sqrt{ \frac{c_{21}}{c_{25}}}\right)-\sqrt{\frac{c_{21}}{c_{25}}}\sinh\left(s\sqrt{ \frac{c_{21}}{c_{25}}}\right)\right)\right] \tag{21}\]  
The solution to the coupled problem Equation (16)\({}_{2,3}\) in the variables \(a_{p,2}\) and \(a_{w,2}\) is written here in compact form, due to its large dimensions. In particular, its expression is evaluated with the method of variation of the constants: a vector of state variables is defined as \(\mathbf{y}_{2}=(a_{p,2},a_{w,2},a_{p,2}^{\prime},a_{w,2}^{\prime})^{T}\), where the superscript \(T\) indicates the transpose, and Equations (16)\({}_{2,3}\) are written in the state-space form as:  
\[\mathbf{y}_{2}^{\prime}(s)=\mathbf{A}\mathbf{y}_{2}(s)+\mathbf{f}_{2}(s) \tag{22}\]  
where:  
\[\mathbf{A}= \begin{pmatrix}0&0&1&0\\ 0&0&0&1\\ \frac{c_{22}}{c_{19}}&0&0&-\frac{c_{20}}{c_{19}}\\ 0&\frac{c_{26}}{c_{30}}&\frac{c_{20}}{c_{30}}&0\end{pmatrix} ,\qquad\mathbf{f}_ {2}(s)= \begin{pmatrix}0\\ 0\\ \frac{p_{v}^{2}c_{18}}{4c_{7}c_{19}}(l-s)^{4}\\ \frac{p_{v}^{2}c_{6}}{2c_{4}c_{7}c_{30}}(l-s)^{3}\end{pmatrix}  \tag{23}\]  
Then, its solution turns out to be:  
\[\mathbf{y}_{2}(s)=\mathbf{V}e^{\Lambda s}\mathbf{V}^{-1}\boldsymbol{\alpha}+ \int_{0}^{s}\mathbf{V}e^{\Lambda(s-\xi)}\mathbf{V}^{-1}\mathbf{F}_{2}(\xi)d\xi \tag{24}\]  
where:  
\[\Lambda= \begin{pmatrix}\lambda_{1}&0&0&0\\ 0&\lambda_{2}&0&0\\ 0&0&\lambda_{3}&0\\ 0&0&0&\lambda_{4}\end{pmatrix}  \tag{25}\]  
with \(\lambda_{i}\) eigenvalues of \(\mathbf{A}\), which are:  
\[\lambda_{i}=\pm\frac{\sqrt{c_{26}c_{30}c_{19}^{2}+c_{30}(c_{17}c_{30}-c_{20}^{ 2})c_{19}\pm\sqrt{c_{19}^{2}c_{30}^{2}(c_{20}^{4}-2(c_{19}c_{26}+c_{17}c_{30}) c_{20}^{2}+(c_{19}c_{26}-c_{17}c_{30}))^{2}}}}{\sqrt{2}c_{19}c_{30}} \tag{26}\]  
with \(i=1,\ldots 4\). \(\mathbf{V}\) is the matrix of the corresponding right eigenvectors, so that \(\mathbf{A}\mathbf{V}=\mathbf{V}\Lambda\) (the expression for the components of the eigenvectors is given in Appendix C); \(\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})^{T}\) are the column of the arbitrary constants, which are evaluated by imposing the relevant boundary conditions (16)\({}_{5,6}\), which are \(a_{p,2}(0)=0\), \(a_{w,2}(0)=0\), \(a_{p,2}^{\prime}(l)+\frac{c_{20}}{c_{19}}a_{w,2}(l)=0\), \(a_{w,2}(l)=0\), and that can be written as:\[\mathbf{B}_{0}\mathbf{y}_{2}(0)+\mathbf{B}_{l}\mathbf{y}_{2}(l)=\mathbf{0} \tag{27}\]  
where:  
\[\mathbf{B}_{0}= \begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&0&0\\ 0&0&0&0\end{pmatrix} ,\qquad\mathbf{B}_{l}= \begin{pmatrix}0&0&0&0\\ 0&0&0&0\\ 0&\frac{c_{20}}{c_{19}}&1&0\\ 0&1&0&0\end{pmatrix}