# Stress Analysis in Damaged Pipeline with Composite Coating
## 2 Materials and Methods
### Stress in the Pipeline with Wound Composite Coating
#### 2.2.2 Effective Characteristics of the Spiral-Wound Layer  
The specificity of the spiral-wound layer is that its macroscopic mechanical properties in the winding direction and in the transverse direction are different: so-called constructive orthotropy takes place. Let \(\alpha\) be the angle of winding inclination to the axis of the pipe, let \(L\) be the width of the bandage, and let \(b\) be the width of overlap at one edge (Figure 2). Let \(a=L-2b\) stand for the width of the nonoverlapped area. Let us introduce the main axes of orthotropy (\(I\), \(2\)): \(1\) in the winding direction, and \(2\) perpendicular to \(1\). Thus, we have an \((a+b)\) periodic structure in direction \(2\) consisting of an isotropic band with the width \(a\) and thickness \(h\), and an isotropic band with the width \(b\) and thickness \(2h\), which are in perfect contact. Let us denote \(E,\
u\) as the modulus of elasticity of the wound material.  
Let us carry out the procedure of averaging [PERSON]'s law over the period \(a+b\). We take [PERSON] 2005 into account regarding the idea that in direction \(I\), we have a parallel connection of elastic bands and sum up the stresses and, hence, the rigidity, according to  
Figure 2: Diagram of a pipe with spiral-wound composite coating: (**a**) longitudinal section of the wall of the pipe with the composite coating; (**b**) scheme of laying strips of fiberglass.  
Rice; in direction 2, we obtain a series connection of elastic bands with a summation of elongations and, hence, compliance, according to [PERSON].  
As a result, we obtain [PERSON]'s law for a constructive orthotropic material in orthotropy axes  
\[\left(\begin{array}{c}\sigma_{1}\\ \sigma_{2}\\ \tau_{12}\end{array}\right)=\left(\begin{array}{ccc}Q_{11}&Q_{12}&0\\ Q_{21}&Q_{22}&0\\ 0&0&Q_{66}\end{array}\right)\left(\begin{array}{c}\varepsilon_{1}\\ \varepsilon_{2}\\ \varepsilon_{12}\end{array}\right),\]  
where  
\[\begin{array}{c}Q_{11}=\frac{E}{1-\
u^{2}},\ Q_{12}=Q_{21}=\frac{vE}{1-\
u^{2 }}\frac{a+b}{a+b/2}\frac{h}{\langle h\rangle},\\ Q_{22}=\frac{E}{1-\
u^{2}}\frac{a+b}{a+b/2}\frac{h}{\langle h\rangle}\Big{(}1 -\
u^{2}+\
u^{2}\frac{a+b}{a+b/2}\frac{h}{\langle h\rangle}\Big{)},\ Q_{66}= \frac{E}{1+\
u}\frac{a+b}{a+b/2}\frac{h}{\langle h\rangle},\end{array} \tag{6}\]  
and \(\langle h\rangle=h(a+2b)/(a+b)\) is the average thickness of the spiral-wound layer.  
Taking into account the rules of transformation of the components of tensors at the rotation of the axes, for effective rigidities of a layer anisotropic in the coordinate system \((x,\varphi)\), we obtain [43; 44]  
\[\begin{array}{c}C_{11}=U_{1}+U_{2}\cos 2\alpha+U_{3}\cos 4\alpha,\ C_{22}=U_{1}-U_ {2}\cos 2\alpha+U_{3}\cos 4\alpha,\\ C_{12}=C_{21}=U_{4}-U_{3}\cos 4\alpha,\ C_{66}=U_{5}-U_{3}\cos 4\alpha,\\ C_{16}=C_{61}=-\frac{1}{2}U_{2}\sin 2\alpha-U_{3}\sin 4\alpha,\ C_{26}=C_{62}=- \frac{1}{2}U_{2}\sin 2\alpha+U_{3}\sin 4\alpha;\end{array} \tag{7}\]  
elastic coefficients \(U\) have the form  
\[\begin{array}{c}U_{1}=\frac{1}{8}(3Q_{11}+3Q_{22}+2Q_{12}+4Q_{66}),\ U_{2}= \frac{1}{2}(Q_{11}-Q_{22}),\\ U_{3}=\frac{1}{8}(Q_{11}+Q_{22}-2Q_{12}-4Q_{66}),\ U_{4}=\frac{1}{8}(Q_{11}+Q_{22 }+6Q_{12}-4Q_{66}),\\ U_{5}=\frac{1}{8}(Q_{11}+Q_{22}-2Q_{12}+4Q_{66});\end{array} \tag{8}\]  
\(\alpha\) is the winding angle measured counterclockwise from the positive direction of axis \(x\) to the positive direction of axis 1.