# Static Response of Double-Layered Pipes via a Perturbation Approach
## 2 Model Description
\begin{split} N&=\frac{\partial\phi}{\partial\varepsilon _{0}}=c_{1}\varepsilon_{0}+c_{2}\alpha_{w}^{2}+c_{3}\alpha_{g}^{2}\\ T&=\frac{\partial\phi}{\partial\gamma_{0}}=c_{4} \gamma_{0}+c_{5}\alpha_{g}+c_{6}\kappa_{0}\alpha_{w}\\ M&=\frac{\partial\phi}{\partial\kappa}=c_{7}\kappa_ {0}+c_{8}\kappa_{0}\alpha_{g}^{2}+c_{9}\kappa_{0}\alpha_{p}+c_{10}\kappa_{0} \alpha_{p}^{2}+c_{11}\beta_{g}+c_{12}\alpha_{p}\beta_{g}\\ &\qquad+c_{13}\alpha_{g}\beta_{p}+c_{14}\alpha_{w}\alpha_{g}+c_{ 15}\alpha_{w}^{2}+c_{16}\alpha_{w}\alpha_{g}+c_{6}\alpha_{w}\gamma_{0}\\ D_{p}&=\frac{\partial\phi}{\partial\kappa_{p}}=c_{ 17}\alpha_{p}+c_{10}\kappa_{0}^{2}\alpha_{p}+c_{16}\kappa_{0}\alpha_{w}\alpha_ {g}+c_{12}\kappa_{0}\beta_{g}+c_{18}\kappa_{0}^{2}\\ D_{w}&=\frac{\partial\phi}{\partial\alpha_{w}}=c_{ 26}\alpha_{w}+c_{20}\beta_{p}+c_{16}\kappa_{0}\alpha_{p}\alpha_{g}+c_{15} \kappa_{0}^{2}\alpha_{w}+c_{14}\kappa_{0}\alpha_{g}+c_{6}\kappa_{0}\gamma_{0} \\ &\qquad+c_{22}\alpha_{w}\alpha_{g}^{2}+c_{28}\alpha_{w}\beta_{w} +c_{29}\alpha_{w}\varepsilon_{0}+c_{27}\alpha_{w}^{3}+c_{3}\alpha_{g}\beta_{g }\\ D_{g}&=\frac{\partial\phi}{\partial\alpha_{g}}=c_{ 21}\alpha_{g}+c_{5}\gamma_{0}+c_{16}\kappa_{0}\alpha_{p}\alpha_{w}+c_{14} \kappa_{0}\alpha_{w}+c_{8}\kappa_{0}^{2}\alpha_{g}+c_{13}\kappa_{0}\beta_{p} \\ &\qquad+c_{22}\alpha_{w}^{2}\alpha_{g}+c_{3}\alpha_{w}\beta_{g}+c _{24}\alpha_{g}\varepsilon_{0}+c_{23}\alpha_{g}^{3}\\ B_{p}&=\frac{\partial\phi}{\partial\beta_{p}}=c_{ 19}\beta_{p}+c_{20}\alpha_{w}+c_{13}\kappa_{0}\alpha_{g}\\ B_{w}&=\frac{\partial\phi}{\partial\beta_{w}}=c_{ 30}\beta_{w}+c_{31}\alpha_{w}^{2}\\ B_{g}&=\frac{\partial\phi}{\partial\beta_{g}}=c_{ 25}\beta_{g}+c_{11}\kappa_{0}+c_{12}\kappa_{0}\alpha_{p}+c_{3}\alpha_{w}\alpha_ {g}\end{split}  \tag{7}\]  
Elastic coefficients \(c_{n}\), \(n=1,\ldots,31\) are defined in Appendix A. It is worth noting that the coefficients \(c_{n}\) depend on the longitudinal and transversal elastic moduli of both internal and external plies, namely, \(E_{i}\) and \(E_{e}\), \(G_{i}\) and \(G_{e}\), and on their thickness \(h_{i},h_{e}\); furthermore, coefficients \(c_{17},c_{21}\) depend also on the transversal elastic modulus of the adhesive \(G_{a}\) and its thickness \(h_{a}\). Equations (1), (2), (5)-(7) define the elastic problem for the homogeneous beam. Substitution of Equation (1) in (7), and then in (5), provides the following equilibrium equations in terms of displacement (where only the linear part is made explicit due to the large quantity of nonlinear terms):\[ \begin{split}& c_{1}u^{\prime\prime}+\mathcal{F}_{1}(u,v,\vartheta,a_{p},a _{g},a_{w})+p_{u}=0\\ & c_{4}(v^{\prime\prime}-\vartheta^{\prime})+c_{5}g_{g}^{\prime}+ \mathcal{F}_{2}(u,v,\vartheta,a_{p},a_{g},a_{w})+p_{v}=0\\ & c_{7}\vartheta^{\prime\prime}+c_{4}(v^{\prime}-\vartheta)+c_{ 1}a_{g}^{\prime\prime}+c_{5}g_{g}+\mathcal{F}_{3}(u,v,\vartheta,a_{p},a_{g},a _{w})+c=0\\ & c_{19}a_{p}^{\prime\prime}-c_{17}a_{p}+c_{20}a_{w}^{\prime}+ \mathcal{F}_{4}(u,v,\vartheta,a_{p},a_{g},a_{w})+q_{p}=0\\ & c_{30}a_{w}^{\prime\prime}-c_{26}a_{w}-c_{20}a_{p}^{\prime}+ \mathcal{F}_{5}(u,v,\vartheta,a_{p},a_{g},a_{w})+q_{w}=0\\ & c_{25}a_{g}^{\prime\prime}-c_{21}a_{g}+c_{5}(\vartheta-v^{ \prime})+c_{11}\vartheta^{\prime\prime}+\mathcal{F}_{6}(u,v,\vartheta,a_{p}, a_{g},a_{w})+q_{g}=0\end{split}  \tag{8}\]  
while the nonlinear terms, expanded up to the third order, are collected in the functions \(\mathcal{F}_{n}(u,v,\vartheta,a_{p},a_{g},a_{w})\), \(n=1,\ldots,6\), whose expressions are shown in Appendix B. A clamped-free beam is analyzed, leading to the following boundary conditions:  
\[ \begin{split} u(0)&=0\\ v(0)&=0\\ \vartheta(0)&=0\\ a_{p}(0)&=0\\ a_{w}(0)&=0\\ a_{g}(0)&=0\end{split}  \tag{9}\]  
and  
\[