# Static Response of Double-Layered Pipes via a Perturbation Approach
## 3 Perturbation Analysis  
The nonlinear boundary value problem (8)-(10) is addressed via a perturbation method, so as to get an asymptotic expression for the solution. The cases of applied distributed transverse load \(p_{v}\) and tip load \(P_{v}\) are considered, whereas the other loads are assumed as null, i.e.,  
\[p_{u}=c_{\vartheta}=q_{p}=q_{g}=q_{w}=0\quad\text{and}\quad P_{u}=C_{ \vartheta}=Q_{p}=Q_{w}=0 \tag{11}\]  
Furthermore, similar but not equal geometric and mechanic features of the two plies are assumed, i.e., Young modulus \(E_{i}\simeq E_{e}\), transverse elastic modulus \(G_{i}\simeq G_{e}\) and thickness \(h_{i}\simeq h_{e}\). The dependent variables are series-expanded after introducing the small positive parameter, \(\varepsilon\ll 1\), as:\[u =\epsilon^{2}u_{2}+\epsilon^{4}u_{4}, \tag{12}\] \[v =\epsilon v_{1}+\epsilon^{3}v_{3},\] \[\vartheta =\epsilon\vartheta_{1}+\epsilon^{3}\vartheta_{3},\] \[a_{p} =\epsilon^{2}a_{p,2}+\epsilon^{4}a_{p,4}\] \[a_{w} =\epsilon^{2}a_{w,2}+\epsilon^{4}a_{w,4}\] \[a_{g} =\epsilon^{2}a_{g,2}+\epsilon^{4}a_{g,4}\]  
where the odd and even natures of the dominant part of \(v,\vartheta\) and of \(u,a_{p},a_{w}\) are exploited, respectively, in case of transverse load application; more specifically, the perturbation scheme is based on the idea that the loads directly trigger the global bending problem (variables \(v,\vartheta\)), at the first step; then, in turn, at the second step, the bending problem triggers the axial and local distortion problems (variables \(u\) and \(a_{p},a_{w},a_{g}\)) by the nonlinear coupling; finally, higher order corrections to the global bending and local distortion problems, respectively, are evaluated. Since the coefficients \(c_{5}\), \(c_{11},c_{12},c_{13},c_{14}\) tend to zero in case of exactly equal internal and external piles (\(E_{i}=E_{e}\), \(G_{i}=G_{e}\), \(h_{i}=h_{e}\)), then they are scaled at order \(\epsilon\), as:  
\[c_{5}=\epsilon c_{5},\,c_{11}=\epsilon c_{11},\,c_{12}=\epsilon c_{12},\,c_{13 }=\epsilon c_{13},\,c_{14}=\epsilon c_{14} \tag{13}\]  
giving rise to vanishing linear coupling between the variables \(a_{g}\) and \(v,\vartheta\), and suggesting for the former to start at the second order (Equation (12)\({}_{6}\)). Moreover, the applied loads are considered of order \(\epsilon\), so that they appear at the the leading order, i.e.,  
\[p_{v}\rightarrow\epsilon p_{v},\qquad P_{v}\to eP_{v} \tag{14}\]  
Under the aforementioned assumptions, after substituting Equations (12)-(14) in Equations (8)-(10) and letting to zero the terms at the same powers of \(\epsilon\), the following perturbation equations and boundary conditions are obtained, at order \(\epsilon\):  
\[ \begin{cases}c_{4}(v_{1}^{\prime\prime}-\theta_{1}^{\prime})+p_{v}=0\\ c_{7}\theta_{1}^{\prime\prime}+c_{4}(v_{1}^{\prime}-\theta_{1})=0\\ \end{cases}  \tag{15}\] \[ \begin{cases}v_{1}(0)=\theta_{1}(0)=0\\ c_{4}(v_{1}^{\prime}(I)-\theta_{1}(I))=P_{v}\\ c_{7}\theta_{1}^{\prime}(I)=0\end{cases} \]  
at order \(\epsilon^{2}\):  
\[ \begin{cases}c_{11}u_{2}^{\prime\prime}=(c_{1}-2c_{4})\vartheta_{1}\theta_{1 }^{\prime}+(c_{4}-c_{1})(v_{1}^{\prime}\theta_{1})^{\prime}\\ c_{19}a_{p,2}^{\prime\prime}-c_{17}a_{p,2}+c_{20}a_{w,2}^{\prime}=c_{18}( \theta_{1}^{\prime})^{2}\\ c_{30}a_{w,2}^{\prime\prime}-c_{26}a_{w,2}-c_{20}a_{p,2}^{\prime}=c_{6}\theta _{1}^{\prime}(v_{1}^{\prime}-\theta_{1})\\ c_{25}a_{g,2}^{\prime\prime}-c_{21}a_{g,2}=-c_{5}(v_{1}^{\prime}-\theta_{1}) -c_{11}\theta_{1}^{\prime\prime}\\ \end{cases}  \tag{16}\] \[ \begin{cases}u_{2}(0)=a_{p,2}(0)=a_{w,2}(0)=a_{g,2}(0)=0\\ c_{1}u_{2}^{\prime}(l)=\left(\frac{c_{1}}{2}-c_{4}\right)\theta_{1}(l)^{2}+(c_ {4}-c_{1})v_{1}^{\prime}(l)\vartheta_{1}(l)\\ c_{19}a_{p,2}^{\prime}(l)+c_{20}a_{w,2}(l)=0\\ c_{30}a_{w,2}^{\prime}(l)=0\\ c_{25}a_{g,2}^{\prime}(l)=-c_{11}\theta_{1}^{\prime}(l)\end{cases}