# Experimental Analysis on Pre-Stress Friction Loss of Crushed Limestone Sand Concrete Beams
## 2 Experimental Program  
### Theoretical Basis  
Friction loss is conveniently considered in terms of the effect of length and curvature [27]. As shown in Figure 1, the pre-stressed tendons are first discrete, but they rearrange and squeeze against each other after tensioning. Consequently, certain friction occurs, which may resist the control stress. Furthermore, as shown in Figure 2, the location of tendon changes after tensioning. The effective length tends to be short and the local deviation cannot be ignored. Thus, the effective force should exclude the pre-stress loss caused by the friction and length deviation.  
According to the Code for Design of Concrete Structures of China (GB50010-2010), the pre-stress loss caused by the first source is defined in Equations (1) and (2):  
\[\sigma_{\mathrm{I2}}=\sigma_{\mathrm{con}}(1-\frac{1}{e^{\kappa x+\mu\theta}}) \tag{1}\]  
\[\theta=\sqrt{\alpha^{2}+\varphi^{2}} \tag{2}\]  
where \(\sigma_{\mathrm{I2}}\) is pre-stress loss due to friction; \(\kappa\) is the influence coefficient of the frictional resistance caused by local deviation in unit length of the duct; \(\mu\) is friction coefficient between the pre-stressed tendons and duct wall; \(\sigma_{\mathrm{con}}\) is the control stress under anchor; \(x\) is the projected length of duct; \(\theta\) is the angle between the active end and tangent of duct; \(\alpha\) is the angle between the tendon and the vertical axis; and \(\varphi\) is the angle between the tendon and the horizontal axis.  
For straight steel tendons, \(\theta\) can be regarded as 0 and \(\kappa\) is calculated as:  
\[\kappa=-\frac{\ln(1-\frac{\sigma_{\mathrm{I2}}}{\sigma_{\mathrm{con}}})}{x} \tag{3}\]  
Once \(\kappa\) is obtained, the \(\mu\) for curve steel tendons is calculated as Equation (4):  
\[\mu=\frac{-\left[\kappa x+\ln(1-\frac{\sigma_{\mathrm{I2}}}{\sigma_{\mathrm{ con}}})\right]}{\theta} \tag{4}\]  
The principle of least square method is adopted to reduce the errors and Equation (1) can be described as:  
\[\kappa x+\mu\theta-\ln(\frac{\sigma_{\mathrm{con}}}{\sigma_{\mathrm{con}}- \sigma_{\mathrm{I2}}})=0 \tag{5}\]  
Therefore, the \(\kappa\) and \(\mu\) can be obtained by at least two groups of data, while the certain error \(\Delta_{i}\) of \(i\)-th pre-stressed tendon can be calculated according to Equation (6). Based on the maximum principle of the least square method, the sum of squares of \(\Delta_{i}\) can be described as Equations (7) and (8):  
\[\varepsilon=\ln(\frac{\sigma_{\mathrm{con}}}{\sigma_{\mathrm{con}}-\sigma_{ \mathrm{I2}}}), \tag{6}\]  
\[\kappa x_{i}+\mu\theta_{i}-\varepsilon_{i}=\Delta_{i} \tag{7}\]  
Figure 1: The friction in pre-stressed tendons.  
Figure 2: Local deviation.  
\[\Omega=\sum\limits_{i=1}^{n}{{{\Delta_{i}}^{2}}}=\sum\limits_{i=1}^{n}{{(\kappa x_{ i}+\mu\theta_{i}-\varepsilon_{i})}^{2}} \tag{8}\]  
The error takes the minimum value when the partial derivatives are 0:  
\[\frac{\partial\Omega}{\partial\mu}=0\qquad\frac{\partial\Omega}{\partial\kappa}=0 \tag{9}\]  
The equations for \(\kappa\) and \(\mu\) are described as:  
\[\left\{\begin{array}{l}\mu\sum\limits_{i=1}^{n}{{\theta_{i}}^{2}}+\kappa \sum\limits_{i=1}^{n}{{\theta_{i}}l_{i}}=\sum\limits_{i=1}^{n}{{\varepsilon_{ i}}\theta_{i}}\\ \mu\sum\limits_{i=1}^{n}{{l_{i}}\theta_{i}}+\kappa\sum\limits_{i=1}^{n}{{l_{i} ^{2}}}=\sum\limits_{i=1}^{n}{{\varepsilon_{i}}l_{i}}\end{array}\right. \tag{10}\]  
where \(n\) is the quantity of ducts.  
Thus, the \(\kappa\) and \(\mu\) are precisely calculated after the data of \(l_{i}\) and \(\varepsilon_{i}\) are measured. The current specifications provide the suggested value of \(\kappa\) and \(\mu\) for normal concrete as shown in Tables 1 and 2[20]. These codes indicate that the coefficients depend on tendon type and duct material.