## 2 Theoretical Derivations
### Calculation of Area of Prestressed Steel Strands
#### 2.2.1 Stress of PCCPs under Combined Loads  
The combined loads acting on the pipe include the vertical earth pressure at the top of the pipe, \(F_{sw,k}\), the lateral earth pressure, \(F_{ep,k}\), the ground pile load, the weight of the pipe, \(G_{1k}\), the weight of fluid in the pipe, \(G_{wk}\), and the variable load. The values of \(F_{sw,k}\) and \(F_{ep,k}\) are calculated according to [PERSON]'s theory [17] and [PERSON]'s earth pressure theory [26], respectively. The variable load can be regarded as the ground stacking load, and its standard value is defined as \(q_{mk}=10\) kN/m. The weight of the pipe can be written as per Equation (23):  
\[G_{1k}=\pi r_{G}(D_{i}+h_{c})h_{c} \tag{23}\]  
where \(D_{i}\) is the inner diameter of the pipe (m), \(h_{c}\) is the thickness of concrete core (m), and \(r_{G}\) is the gravity density of the pipe (kN/m\({}^{3}\)). The weight of fluid in the pipe, \(G_{wk}\), can be calculated by Equation (24):  
\[G_{wk}=\frac{r_{W}\pi D_{i}^{2}}{4} \tag{24}\]  
where \(r_{W}\) is the gravity density of the fluid in the pipe (kN/m\({}^{3}\)). 2.2 Calculation of the Area of Prestressed Strands, Considering the Concrete Core Compression of Ultimate Limit States  
According to Chinese specifications [27; 28], the design requirements for the calculation of ultimate limit states under the external soil load, weight of pipe, weight of fluid, and other variable loads are detailed. The design value of the maximum bending moment of the pipe at the spring-line, \(M_{max}^{l}\), can be calculated using Equation (25). The value of \(M_{max}^{l}\) is negative, indicating that the outer surface of the concrete core is subjected to tension. The absolute value can be taken when the formula is substituted for reinforcement. The design value of the maximum axial tension of the pipe at the spring-line, \(N^{l}\), is written as per Equation (26):  
\[M_{max}^{l}=\gamma_{0}\tau\Big{[}k_{vm}\Big{(}\gamma_{G3}F_{svk}+\psi_{c} \gamma_{Q2}q_{vk}D_{1}\Big{)}+k_{hm}\gamma_{G4}F_{epk}D_{1}+k_{vm}\gamma_{G2}G_ {vk}+k_{gm}\gamma_{G1}G_{1k}\Big{]} \tag{25}\]  
\[N^{l}=\gamma_{0}\Big{[}\psi_{c}\gamma_{Q1}P_{d}\tau\times 10^{-3}-0.5\Big{(}F_{ svk}+\psi_{c}q_{vk}D_{1}\Big{)}\Big{]} \tag{26}\]  
where \(\gamma_{0}\) is the factor of importance. It varies with the structure and the layout of the pipes. The value of \(\gamma_{0}\) is generally 1.1. For side-by-side pipelines, the \(\gamma_{0}\) value should be taken as 1.0, in particular. Moreover, the value of \(\gamma_{0}\) should also be taken as 1.0 for pipes with storage facilities or those which are used for drainage. \(r\) is the calculated radius of the pipe (\(m\)), and \(k_{hm}\), \(k_{hm}\), \(k_{gm}\) represent the bending moment coefficient of the bending moment at the spring-line of the pipe under the vertical earth pressure, lateral earth pressure, the weight of fluid inside the pipe, and the weight of the pipe, respectively. These factors were determined according to Appendix E [27]. The \(k_{gm}\) of the arc-shaped soil bedding can be adopted according to the data of the bedding angle of \(20^{\circ}\). \(\gamma_{Gi}\) and \(\gamma_{Qj}\) are the partial coefficients under the permanent load i and the variable load j. \(\psi_{c}\) is the combination coefficient of the variable loads and usually takes the value of 0.9. \(D_{1}\) is the outer diameter of the pipe (m). \(P_{d}\) is the designed water pressure (N/mm\({}^{2}\)). The area of the prestressed strands of ultimate limit states should be calculated by Equation (27):  
\[A_{st}\geq\frac{\lambda_{y}}{f_{pyk}}\bigg{(}N^{l}+\frac{M_{max}^{l}}{d_{0}}- A_{sc}f_{yy}^{\prime}\bigg{)} \tag{27}\]  
where \(\lambda_{y}\) is the comprehensive adjustment factor of the PCCP, \(f_{pyk}\) is the design strength of prestressed strands (N/mm\({}^{2}\)), \(d_{0}\) is the distance from the prestressed strand to the center of gravity of the pipe(m), \(A_{sc}\) is the area of the cylinder per unit length (m\({}^{2}\)/m), and \(f_{yy}^{\prime}\) is the design strength of the cylinder (N/mm\({}^{2}\)).