## 2 Theoretical Derivations
### Calculation of Prestress Loss, \(\sigma_{st,l}\)
#### 2.1.2 Prestress Loss Caused by Friction Resistance, \(\sigma_{\Pi}\)  
The prestress loss caused by the friction resistance, \(\sigma_{\Pi}\), can be calculated based on the consideration of two parts, namely, the bending loss and the deviation loss. The radial pressing force, \(\sigma_{r}\), is produced between the strand and the pipe wall by prestressed strands, thereby resulting in extrusion friction. The bending loss accounted for a large proportion of the total friction loss.  
Based on the assumption of a rigid body, we hypothesized that the pressure between the strand and the pipe wall would be uniformly distributed [20], and that elastic deformation would occur when the two elastic bodies were pressed into contact with each other. The stress between the contact surfaces is ellipsoidal, and its value can be related to the radius of curvature and the elastic modulus of the contact object. It is not accurate enough to consider the contact stress as uniformly distributed under normal contact pressure due to the large tensile force of the prestressed steel strands.  
The scope of the bending loss can be related to the retraction length, \(l_{re}\). We can assume that the normal stress between the strands and the pipe wall would be distributed as a trigonometric function [21], as illustrated in Figure 5 and Equation (10).  
\[p_{(\alpha)}=p_{0}\cos^{2}\biggl{(}\frac{\pi}{\theta}\alpha\biggr{)} \tag{10}\]  
where \(\cos^{2}\biggl{(}\frac{\pi}{\theta}\alpha\biggr{)}=\left\{\begin{array}{c} \begin{array}{c}0\ \alpha=\frac{\theta}{2}\\ 1\ \alpha=0\\ 0\ \alpha=-\frac{\theta}{2}\end{array}\right.\)  
Figure 5: Distribution of the normal stress of the pipe wall excluding the friction.  
A balance of forces in the z-direction can be established by Equation (11). From Equation (11), we derived Equation (12). Therefore, the normal stress can be calculated using Equation (13):  
\[T\cdot\sin\!\left(\frac{\theta}{2}\right)+\left(T+dT\right)\cdot\sin\!\left( \frac{\theta}{2}\right)+2\int_{0}^{\frac{\theta}{2}}p_{(\alpha)}\cdot\cos\, add=0 \tag{11}\]  
\[p_{0}=\frac{\sin\frac{\theta}{2}}{\int_{0}^{\frac{\theta}{2}}\cos^{2}\!\left( \frac{\pi}{\theta}\alpha\right)\cos\,ad\alpha}\times\frac{T}{R} \tag{12}\]  
where \(\int_{0}^{\frac{\theta}{2}}\cos^{2}\!\left(\frac{\pi}{\theta}\alpha\right)\cos \,ad\alpha=\left(\cos^{2}\!\left(\frac{\pi}{\theta}\alpha\right)\sin\alpha \right)\!\!\int_{0}^{\frac{\theta}{2}}\sin\alpha\sin\!\left(\frac{2\pi}{ \theta}\alpha\right)\!\!d\alpha=\frac{2\left(\frac{\pi}{\theta}\right)^{2} \sin\frac{\theta}{2}}{\left(\frac{\pi}{\theta}\right)^{2}-1}\).  
\[p_{(\alpha)}=\left(2-\frac{\theta^{2}}{2\pi^{2}}\right)\cos^{2}\!\left(\frac {\pi}{\theta}\alpha\right)\!\frac{\sigma_{0}}{R} \tag{13}\]  
The prestress loss related to the bending loss, \(F\), during the tensioning operation is depicted in Equation (14).  
\[F=2\int_{0}^{\frac{\theta}{2}}\mu p_{(\theta)}dl=\mu\partial\sigma_{0}\!\left( 1-\frac{\theta^{2}}{4\pi^{2}}\right) \tag{14}\]  
The deviation loss stems from errors in pipe positioning and installation, which causes friction between the force rib and the pipe material, thereby forming contact friction. The deviation loss occupies a small proportion of the total friction loss. The correction coefficient, \(c_{1}\), is involved here, and the deviation loss is not separately calculated in this paper. As a result, the total prestress loss caused by the friction resistance can be calculated, as displayed in Equation (15).  
\[\sigma_{I1}=c_{1}F \tag{15}\]  
where \(c_{1}\) is the correction coefficient, accounting for the bending loss and the deviation loss, and is usually in the range of 1 to 1.3 [19].