# Experimental Analysis on Pre-Stress Friction Loss of Crushed Limestone Sand Concrete Beams
## 3 Test Result and Analysis  
### Measured Pre-Stress Loss  
Based on the project of Malinghe Bridge, the forces of active and passive ends at different stress degrees were measured and the \(\varepsilon_{i}\) is calculated by Equations (6)-(8), as shown in Figure 11. According to the theoretical basis above, there is a positive correlation between \(\varepsilon_{i}\) and the pre-stress friction loss \(\sigma_{l2}\). The standard deviation of \(\varepsilon_{i}\) in Figure 11 is 0.0311, 0.0253, 0.0646, 0.0118 and 0.0267. Despite certain discreteness, the pre-stress friction loss at each stress degree is generally stable. It should be mentioned that certain friction occurs in the 19-wire straight tendons during construction and installation, so the pre-stress friction loss at a low stress level is larger in Figure 11d. In most cases, the pre-stress friction loss in specimens with curve duct is relatively larger than that in straight duct. Table 5 shows that pre-stress friction loss directly increases in each duct. The large normal pressure on the curved part increases the additional friction. The pre-stress friction loss is in the range of 2-10% in straight duct and 8-20% in curve duct. As the number of tendons increases, the proportion of pre-stress friction loss first increases, but it subsequently decreases because that the increasing trend of total initial tension have exceeded the friction loss. The pre-stress loss is generally 4-16% in single-wire specimens, while 10-40% in multi-wire specimens. This illustrates that both duct arrangement and number of tendons have a great influence on pre-stress friction loss. The average value in each case is shown in Figure 12.  
Based on the project of Wujiang Bridge, the \(\epsilon_{i}\) at different stress degree is calculated as shown in Figure 13. The standard deviation of \(\epsilon_{i}\) in Figure 13a-c is 0.2046, 0.1224 and 0.0477, indicating that pre-stress loss in metal bellow and plastic bellow is small and stable. The pre-stress friction loss for each case is shown in Table 6. The value in specimen with rubber hose is the largest and nearly reaches 40%, as shown in Figure 14. Mainly due to being flexible, the rubber hose can easily become deformed during construction and concrete pouring. The metal bellow and plastic bellow have a relatively larger stiffness and smoother contact surface, allowing for the quality and precision to be strictly controlled. The duct-forming materials also significantly influence the friction loss. Figure 12: Average pre-stress losses: (**a**) straight duct; and (**b**) curve duct. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Wires**} & \multicolumn{4}{c}{**Straight Tendons**} & \multicolumn{4}{c}{**Curve Tendons**} \\ \cline{2-7}  & \(\mathbf{N_{a}/k}\)N & \(\mathbf{N_{b}/k}\)N & **Pre-Stress Loss** & \(\mathbf{N_{a}/k}\)N & \(\mathbf{N_{b}/k}\)N & **Pre-Stress Loss** \\ \hline \multirow{4}{*}{\(1\)} & 214.43 & 204.64 & 4.57\% & 217.63 & 190.93 & 12.27\% \\  & 215.26 & 208.25 & 3.26\% & 215.36 & 187.11 & 13.12\% \\  & 216.29 & 200.82 & 7.15\% & 214.02 & 178.76 & 16.48\% \\ \hline \multirow{4}{*}{\(3\)} & 585.98 & 538.02 & 8.18\% & 523.97 & 461.60 & 11.90\% \\  & 590.98 & 536.36 & 9.24\% & 530.58 & 467.45 & 11.90\% \\  & 601.00 & 543.80 & 9.52\% & 542.98 & 489.98 & 9.76\% \\ \hline \multirow{4}{*}{\(12\)} & 2327.21 & 2185.95 & 6.07\% & 2312.40 & 1835.56 & 20.62\% \\  & 2348.91 & 2214.88 & 5.71\% & 2258.68 & 1846.41 & 18.25\% \\  & 2337.23 & 2205.79 & 5.62\% & 2300.00 & 1823.87 & 20.70\% \\ \hline \multirow{4}{*}{\(19\)} & 3705.43 & 3465.15 & 6.48\% & 3548.84 & 3256.06 & 8.25\% \\  & 3713.18 & 3462.12 & 6.76\% & 3556.59 & 3265.15 & 8.19\% \\  & 3644.96 & 3357.58 & 7.88\% & 3499.22 & 3309.09 & 5.43\% \\ \hline \multirow{4}{*}{\(22\)} & 3996.90 & 3883.33 & 2.84\% & 4144.19 & 3650.00 & 11.92\% \\  & 4043.41 & 3939.39 & 2.57\% & 4167.44 & 3757.58 & 9.83\% \\ \cline{1-1}  & 4124.03 & 4012.12 & 2.71\% & 4192.25 & 3724.24 & 11.16\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Measured pre-stress loss.