# Effect of Prestress Levels and Jacking Methods on Friction Losses in Curved Prestressed Tendons
## 4 Test Results and Discussion  
### Theoretical Friction Loss  
Table 5 lists the friction coefficients recommended by KSCE and KCI design codes [15; 16]. The KCI design code indicates that the curvature friction coefficient \(\mu\) depends on the tendon type and sheath surface characteristics, and varies between 0.05 and 0.30 per meter. The wobble friction coefficient \(k\) is a function of both the sheath strength and the curvature friction coefficient \(\mu\), and it varies between 0.0003 and 0.0066 per radian. Information on the prestressing system used is required to calculate the friction losses accurately because these friction coefficients may vary within the tolerance. In principle, the friction coefficients for the selected prestressing tendon must be obtained empirically. However, it is economically unreasonable and even not feasible to conduct experiments to acquire the friction coefficients of all the tendons used to design the PSC girders. Consequently, structural engineers determine the friction loss based on the friction coefficients presented in the design codes, and, typically, the simpler friction coefficients given in the KSCE design code are preferred to those in the KCI design code. Therefore, the theoretical friction loss was calculated using the curvature coefficient \(k\) = 0.005  
Figure 4: Prestressing systems (**a**) prestressing jack; (**b**) anchorage; (**c**) hydraulic pump.  
and wobble coefficient \(\mu\) = 0.25 based on the recommendation of the KSCE design code, considering that seven-wire strands and a galvanized metal sheath were used for the PSC girders.  
Table 3 lists the information about the longitudinal positions of the tendons in the girders depicted in Figure 3; the height of each tendon is presented according to the distance from midspan. Each tendon was assumed as parabolic in shape. The derived quadratic equations of each tendon and the corresponding first-order differential equations are outlined in Table 6.  
Table 7 lists the variations in the angle with the distance of the tendon from midspan, based on the differential equations in Table 6.  
Table 8 lists the theoretical friction losses of tendon 3 in the GBE girder and tendon 1 in the GOE girder at the midspan for the virtual prestress forces, based on the variation in angle as given in Table 7 and the friction loss coefficients (\(k\) = 0.005, \(\mu\) = 0.25) in the KSCE design code [15]. The theoretical friction losses were calculated by Equation (3). The values in this table reveal that the theoretical friction losses have a linear relationship with the prestress forces, which is already well known. Furthermore, there was a difference in the friction losses between tendon 1 in the GOE girder and tendon 3 in the GBE girder because the tendons had different profiles.  
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Tendon Number** & **Quadratic Equation** & **Differential Equation** & **Remark** \\ \hline
1 & \(y=0.00587564x^{2}\) & \(y^{\prime}=0.01175128x\) & \(y\): height from bottom of PSC \\
2 & \(y=0.00453342x^{2}\) & \(y^{\prime}=0.00906684x\) & \(y\): height from bottom of PSC \\
3 & \(y=0.00277116x^{2}\) & \(y^{\prime}=0.00554232x\) & \(y\): height to tendon. \\
4 & \(y=0.00091764x^{2}\) & \(y^{\prime}=0.00183528x\) & \(x\): distance from midspan \\ \hline \hline \end{tabular}
\end{table}
Table 6: Quadratic equations of each tendon and corresponding first-order differential equations.