# Stress Analysis in Damaged Pipeline with Composite Coating
## 2 Materials and Methods
### Stress in the Pipeline with Wound Composite Coating
#### 2.2.3 Stresses in the Steel Pipeline with Composite Coating  
Let us consider a steel pipeline (layer 0) on which a layered coating of the following structure is applied: epoxy substrate (layer 1), spiral-wound fiberglass cloth (layer 2), and polyurethane (layer 3), with the parameters shown in Table 1.  
Let us write down the components of the rigidity matrix for each layer.  
For isotropic layers (\(i=0,\ 1,\ 3\)):  
\[\begin{array}{c}C_{11}^{(i)}=C_{22}^{(i)}=E_{i}/\left(1-\
u_{i}^{2}\right), \ C_{12}^{(i)}=C_{21}^{(i)}=v_{i}E_{i}/\left(1-\
u_{i}^{2}\right),\\ C_{66}^{(i)}=2G_{i}=E_{i}/(1+v_{i}),\ C_{16}^{(i)}=C_{61}^{(i)}=C_{26}^{(i)}=C_{6 2}^{(i)}=0.\end{array}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Layer** & & & & & **Winding** \\
**Number** & \(\mathbf{h_{i}}\)**, **mm** & **Material** & \(\mathbf{E_{i}}\)**, **MPa** & \(\mathbf{v_{i}}\) & **Options** \\  & & & & & **(Parameters)** \\ \hline
0 & 15.2 & steel & \(2.1\times 10^{5}\) & 0.3 & Isotropy \\ \hline
1 & 0.5 & epoxy & \(2\times 10^{4}\) & 0.4 & Isotropy \\ \hline
2 & 1.5 & fiberglass & \(5\times 10^{4}\) & 0.4 & \(\begin{array}{c}\alpha=60^{\circ},\\ a=b=L/3,\\ L=200\ \mathrm{mm}\end{array}\) \\ \hline
3 & 2.2 & polyurethane & 20 & 0.4 & Isotropy \\ \hline \hline \end{tabular}
\end{table}
Table 1: Geometric and physical parameters of layers [2; 11].  
For spiral-wound fiberglass (\(i=2\)), by assuming that in Formula (6), \(E=E_{2}\), \(\
u=v_{2}\), \(a=b=L/3\), we find  
\[Q_{11}=\frac{E_{2}}{1-v_{2}^{2}},\ Q_{12}=Q_{21}=\frac{v_{2}E_{2}}{1-v_{2}^{2}} \frac{8}{9},\] \[Q_{22}=\frac{E_{2}}{1-v_{2}^{2}}\frac{8}{9}\bigg{(}1-\frac{v_{2}^ {2}}{9}\bigg{)},\ Q_{66}=\frac{E_{2}}{1+v_{2}}\frac{8}{9},\ \langle h\rangle=1.5h_{2}.\]  
Therefore, using Formulas (7) and (8) at \(\alpha=60^{\circ}\), we can calculate the effective anisotropy coefficients \(C_{kl}^{(2)}\).  
Given these data, using Formula (5) for a given operating pressure \(p=10\) MPa and \(R=610\) mm, we can find stresses in the pipe and in each layer of the coating relating to the plane deformation and constrained torsion of the composition. In the common denominators of fractions (5) under \(h_{2}\), we mean \(\langle h\rangle=1.5h_{2}\)--i.e., the average thickness of the wound layer.  
In the case of coating a pipeline that is under a repair pressure \(p_{repair}\) less than the operating pressure, the resulting stresses in the package are calculated as the superposition of the stresses in the pipe without the coating caused by the repair pressure:  
\[\left(\begin{array}{c}\sigma_{x}^{(0)}\\ \sigma_{\varphi}^{(0)}\\ \tau_{x\varphi}^{(0)}\end{array}\right)=p_{repair}\frac{R}{h_{0}}\left( \begin{array}{c}v_{0}\\ 1\\ 0\end{array}\right)\]  
and stresses (5) in the pipe with the coating caused by the pressure \(p-p_{repair}\).  
Table 2 shows the results of the calculations of the stress tensor components for different values of repair pressures.  
Analysis of the obtained data shows that coating the structure under consideration reduces the maximum stress in the pipe by 4.5% and 2.3%, respectively, at \(p_{repair}=0\) and at \(p_{repair}=0.5p\). In the case of equal repair and operating pressures, the discharge of the pipe is obviously absent. The stresses in the epoxy and polyurethane layers are negligibly small. The main reinforcement of the structure is due to the wound fiberglass bandage. In this case, the structural anisotropy is almost non-evident (\(\tau_{x\varphi}^{(2)}\) is small). Additional analysis of the rigidity distribution of the composition shows that the integral rigidity of the considered layered coating \(\sum\limits_{i=1}^{3}C_{22}^{(i)}h_{i}\) in the direction \(\varphi\) is two orders of magnitude less than the integral rigidity of the pipe \(C_{22}^{(0)}h_{0}\). When considering the interaction of a defective pipe with a layered coating, the established features allow simulating it with an isotropic layer with a modulus of elasticity approximately 100 times smaller than the modulus of elasticity of the pipe material.