# Stress Analysis in Damaged Pipeline with Composite Coating
## 2 Materials and Methods
### Stress in the Pipeline with Wound Composite Coating  
#### 2.2.1 Equilibrium of a Layered Anisotropic Cylindrical Shell  
Let us consider an infinite cylindrical shell which is made up of \(n+1\), where generally speaking, the anisotropic layers are perfectly connected to each other. It is necessary to develop a procedure for studying the stress-strain state of the shell caused by constant internal pressure, under given conditions of fixation at infinity. We introduce the cylindrical coordinates \((r,\varphi,x)\), by aligning the axis \(x\) with the axis of the pipe. Let \(R_{i}\) and \(h_{i}\) (\(i=\overline{0,n}\)) be the radii of the middle surfaces and the thickness of the layers, respectively. Then, the object under study occupies the area  
\[(r,\varphi,x)\in\left(R_{0}-\frac{h_{0}}{2},R_{n}+\frac{h_{n}}{2}\right)\times \left[0,2\pi\right]\times\left(-\infty,\infty\right)\]  
We considered uniform axisymmetric deformation and torsion of a multilayer cylinder along \(x\) within the classical momentless theory of shells [43]. The equilibrium equations that are integral through the shell thickness have the form  
\[\frac{dN_{x}}{dx}=0,\,N_{\varphi}=p\left(R_{0}-\frac{h_{0}}{2}\right),\,\frac {dN_{x\varphi}}{dx}=0, \tag{1}\]  
where  
\[N_{x}=\sum_{i=0}^{n}\sigma_{x}^{(i)}h_{i},\,\,N_{\varphi}=\sum_{i=0}^{n} \sigma_{\varphi}^{(i)}h_{i},\,\,N_{x\varphi}=\sum_{i=0}^{n}\tau_{x\varphi}^{( i)}h_{i} \tag{2}\]  
are the integral forces, \(\sigma_{x}^{(i)}\), \(\sigma_{\varphi}^{(i)}\), and \(\tau_{x\varphi}^{(i)}\) are the components of the stress tensor for the \(i\)th layer, and \(p\) is a given internal pressure. As the basic kinematic assumptions, we accepted the rigid normal hypothesis, which is uniform for the whole package:  
\[ \begin{array}{l}\varepsilon_{x}^{(i)}=\varepsilon_{x}=\text{const},\\ \varepsilon_{\varphi}^{(i)}=\varepsilon_{\varphi}=\text{const},\\ \varepsilon_{x\varphi}^{(i)}=\varepsilon_{x\varphi}=\text{const},\,i= \overline{0,n}.\end{array}  \tag{3}\]  
Expression (3) indicates a uniform distribution of deformations through thickness by assumption and is analogous to the compatibility equations for the composition. [PERSON]'s law for anisotropic layers is presented in the form  
\[\left( \begin{array}{c}\sigma_{x}^{(i)}\\ \sigma_{\varphi}^{(i)}\\ \tau_{x\varphi}^{(i)}\end{array} \right)=\left( \begin{array}{ccc}C_{11}^{(j )}&C_{12}^{(j)}&C_{16}^{(i)}\\ C_{21}^{(j)}&C_{22}^{(j)}&C_{26}^{(i)}\\ C_{61}^{(j)}&C_{62}^{(j)}&C_{66}^{(i)}\end{array} \right)\left( \begin{array}[] {c}\varepsilon_{x}\\ \varepsilon_{\varphi}\\ \varepsilon_{x\varphi}\end{array} \right), \tag{4}\]  
where \(C_{kl}^{(i)}\) represents the components of the symmetric stiffness tensor for the material of the \(i\)th layer. Relations (1)-(4) are a complete system of equations sufficient to determine the stress-strain state of a layered anisotropic cylindrical shell under pressure. While deriving them, we threw away values of \(h_{i}/R_{i}\) smaller than one, so we did not distinguish between the outer and inner radii of the thin layers by taking \(R_{i}\approx R\). Let us solve the formulated static problem for a shell fixed at infinity:  
\[\varepsilon_{x}(\pm\infty)=0,\,\,\varepsilon_{x\varphi}(\pm\infty)=0.\]  
Thus, we have a case of plane deformation and restricted torsion. As a result of relations (1)-(3), we get  
\[N_{x}=\sum\limits_{i=0}^{n}C_{12}^{(i)}h_{i}\,\varepsilon_{\varphi},\ N_{\varphi} =\sum\limits_{i=0}^{n}C_{22}^{(i)}h_{i}\,\varepsilon_{\varphi},\ N_{x\varphi}= \sum\limits_{i=0}^{n}C_{62}^{(i)}h_{i}\,\varepsilon_{\varphi},\ N_{\varphi}=pR.\]  
Then, \(\varepsilon_{x}=0,\varepsilon_{\varphi}=pR/\left(\sum\limits_{i=0}^{n}C_{22}^ {(i)}h_{i}\right)\), \(\varepsilon_{x\varphi}=0\).