## 2 Theoretical Derivations  
### Calculation of Prestress Loss, \(\sigma_{st,l}\)  
The prestress loss persisted during and after the tensioning operation, and can be divided into two categories, namely, instantaneous loss and long-term loss [13; 14; 15]. Instantaneous loss, i.e., short-term loss during the tensioning operation, described the prestress losses caused by friction resistance  
Figure 1: Structural drawing of a prestressed concrete cylinder pipe (PCCP) strengthened with prestressed steel strands. between the surface of the pipe wall and the steel strands, the anchor deformation, the concrete elastic compression induced by stepwise tensioning operation, and cracks closures. Long-term prestress losses included prestress losses [13; 16], while taking into account the materials aging, including the effects of shrinkage and the creep losses of concrete, and the long-term relaxation losses of prestressed steel strands. Types of prestress loss of steel strands applied to PCCPs are illustrated in Figure 2. Since the reinforcement of PCCPs with external prestressed steel strands is a post-tensioning method, the impact of temperature can be removed from consideration when considering the reinforcement of PCCPs with external prestressed steel strands. 1.1 Calculation of Retraction Length, \(l_{re}\), and Its Corresponding Retraction Angle, \(\theta_{re}\)  
As far as we know, the stress distribution along the strand is nonlinear. The anchor influenced the prestressed steel strand within a certain length range due to the static friction caused by the retraction of the strand. This length is called the retraction length, \(l_{re}\). Strands within the retraction length showed a displacement opposite to the tension direction, which decreases the prestress. The movement trend is demarcated at point \(C\), and the stress is redistributed from \(ACB\) to \(A^{\prime}\,CB\) (Figure 3). The circumferential micro-segment of the prestressed steel strand is regarded as the research object, where the corresponding angle is \(d\theta\) (Figure 4). Assuming that the normal stress of the steel  
Figure 3: Distribution of strand stress caused by retraction. Figure 2: Types of prestress loss of steel strands applied to PCCPs. strand in the micro-segment is evenly distributed, a differential equation can be established according to the static equilibrium conditions:  
\[T\cdot sin\left(\frac{d\theta}{2}\right)+(T+dT)\cdot sin\left(\frac{d\theta}{2} \right)-dP=0 \tag{1}\]  
where \(T\) and \(P\) stand for the tension force and the normal pressure of the strand, respectively. Higher variables were omitted, taking \(\frac{d\theta}{2}=sin\left(\frac{d\theta}{2}\right)\). Equation (1) can be simplified to \(Td\theta=dP\). The equation describing the momentary balance for rotation around the center of curvature, \(O\), can be written as follows:  
\[r_{st}\cdot\mu dP+r_{st}dT=0 \tag{2}\]  
where \(T=T_{0}\) when \(\theta=0\), thus, \(T=T_{0}e^{-\mu\theta}\). \(r_{st}\), where \(r_{st}\) is the calculated radius of the strand wrapped outside the pipe (m) and \(\mu\) is the friction coefficient between the prestressed strands and the outer surface of the deteriorating pipe. The influencing factors of \(\mu\) mainly include the type of steel, the type of lubricating grease, the materials wrapped outside, and the quality control of the construction. Here, \(\mu\) ranges from 0.08 to 0.12, with a mean value of 0.1. The stress for an arbitrary cross section is calculated as per Equation (3):  
\[\sigma=\sigma_{st}e^{-\mu\theta} \tag{3}\]  
where \(\sigma_{st}\) is the tension stress of prestressed steel strands (N/mm\({}^{2}\)). \(\sigma_{st}=f_{st,t^{\prime}}\alpha\), in which \(\alpha\) is the control coefficient for the tension of the steel strands (N/mm\({}^{2}\)). Normally, this value ranges between 0 and 0.75 [17; 18; 19]. \(f_{st,t}\) is the nominal tensile strength of the prestressed strand (N/mm\({}^{2}\)).