# Effect of Prestress Levels and Jacking Methods on Friction Losses in Curved Prestressed Tendons
## 2 Theoretical Background of Friction Loss (Friction Losses in Prestressing Tendon)  
The force generated along the length of a prestressing tendon tensioned by a hydraulic jack in a PSC structure is not uniform because of the tension between the sheath tube and tendons. This phenomenon is caused by the friction losses. The friction losses consist of two elements: curvature and wobble.  
For more efficient load resistance, in PSC, the tendons are embedded in a curved rather than a straight shape. The curvature friction loss is caused by the intended angular change of the tendon profile. If the tendon direction changes by the angle \(da\) along the tendon length \(dx\), a vertical force \(N\) equal to \(2P\sin(d\alpha/2)\) is generated as shown in Figure 1. Furthermore, if the friction coefficient between the tendon and sheath tube is \(\mu\), the friction loss at the infinitesimal length \(dx\) is \(\mu N\). It is possible to replace \(2\sin(d\alpha/2)\) with \(d\alpha\) because the related angles are generally very small. Thus, the friction loss generated by the tendon's curvature becomes \(\mu Pd\alpha\).  
The prestress force is not decreased by the friction loss if the prestressing tendon and sheath tube are perfectly linear. However, the tendon and sheath in the actual construction cannot be perfectly linear. Furthermore, the sheath tube cannot avoid small wobbles during the actual construction, even if it is placed linearly, as shown in Figure 1. The same applies to tendons arranged in a curved line. Therefore, angular changes occur regardless of whether the tendons are arranged linearly or in a curved line. As a result, the prestress force introduced to the tendons is inevitably reduced. This type of loss is called the loss by the length effect or wobbling effect of the tendon. The size of this loss depends on the tendon type, size and type of the sheath, and rigidity. The wobble loss along the infinitesimal length \(dx\) of the tendon is expressed as \(kPdx\), where \(k\) is the empirically determined wobble coefficient.  
The total friction loss along the infinitesimal length \(dx\) is as follows:  
\[dP=\mu Pd\alpha+kPdx \tag{1}\]  
The change in the tendon force between points A and B in Figure 1 can be expressed as follows:  
\[\int_{P_{B}}^{P_{A}}\frac{dP}{P}=\mu\int_{0}^{\alpha}d\alpha+k\int_{0}^{x}dx \tag{2}\]  
when Equation (2) is integrated, it becomes  
\[P_{B}=P_{A}e^{-(\mu\alpha+kx)} \tag{3}\]  
Finally, the friction loss can be expressed as follows:  
\[P_{F}=P_{A}-P_{B}=P_{A}[1-e^{-(\mu\alpha+kx)}] \tag{4}\]  
Figure 1: Curvature and wobble friction losses.  
where \(P_{A}=\) tendon force at location A, \(P_{B}=\) tendon force at location B, \(P_{F}=\) friction loss in the prestressing tendon, \(\mu=\) friction coefficient, \(\alpha=\) total intended cumulative angle change between A and B in radians, \(k=\) wobble friction coefficient per meter of tendon, and \(x=\) tendon length between A and B in m.  
Figure 2 shows that the friction loss affects the prestress force of a tendon along the length of the beam. Figure 2a clearly indicates that the prestress force changes when the tendon is tensioned to 50% of its tensile strength at the left end A of the beam. A considerable friction loss occurs along the length of the beam when a tendon with an appropriate length is prestressed at only one end. Figure 2b shows the change in the prestress force when the tendon is tensioned to 50% of the tendon's tensile strength at both ends A and B of the beam. The friction loss decreases considerably when the tendon is prestressed at both ends of the beam. However, few studies have been conducted on friction losses. In particular, there are almost no reports of experiments on the effects of prestress levels and the methods for determining the friction losses. Therefore, a study on the friction losses is required to calculate the initial prestress force accurately, which is crucial for the design and analysis of PSC structures.