# Theoretical Study on Prestress Loss in Cross-Tensioned Concrete Pavement with BFRP Tendons
## 3 Theoretical Analysis of Prestress Loss of BFRP Tendons
### Results and Discussions
#### 3.2.1 Prestress Loss in the Prestressing Stage  
The prestress loss of BFRP tendons with different distribution angles (\(20^{\circ}\), \(25^{\circ}\), \(30^{\circ}\), \(35^{\circ}\), \(40^{\circ}\), and \(45^{\circ}\)) in the prestressing stage given the slab width of 3.0 m, 4.5 m, 9.0  
As described in Table 2, the theoretical prestress loss of BFRP tendons in the prestressing stage was between 7.8% and 10.6%, given the slab width of 3.0 m. The greater the distribution angle of the BFRP tendon, the higher the \(\sigma_{\rm 11}\) and the lower the \(\sigma_{\rm 12}\). When the distribution angle increased from 20\({}^{\circ}\) to 45\({}^{\circ}\), \(\sigma_{\rm 11}\) rose from 4.3% to 8.9% while \(\sigma_{\rm 12}\) reduced from 3.4% to 1.7%, and therefore the proportion of \(\sigma_{\rm 11}\) in the total prestress loss in the prestressing stage increased from 55.6% to 84.1%, implying that the main prestress loss of BFRP tendons in the prestressing stage given a 3.0 m wide slab was caused by anchorage deformation and prestress retraction (\(\sigma_{\rm 11}\)).  
It can be illustrated from Table 3 that the total prestress loss of the BFRP tendon in the prestressing stage was between 7.7% to 8.5%, given a 4.5 m wide prestressed concrete slab. With the increase of the distribution angle of the BFRP tendon, \(\sigma_{\rm 11}\) increased from 2.9% to 6.0% while \(\sigma_{\rm 12}\) experienced a drop of 2.6%, and the proportion of \(\sigma_{\rm 11}\) in the prestress loss in the prestressing stage increased from 36.0% to 70.3%, demonstrating that the main prestress loss was caused by the friction between the BFRP tendon and the surrounding casing pipe (\(\sigma_{\rm 12}\)) with the distribution angle less than 30\({}^{\circ}\). However, when the distribution angle was higher than 30\({}^{\circ}\), the main loss resulted from anchorage deformation and prestress retraction (\(\sigma_{\rm 11}\)).  
For a 9.0 m wide slab, the prestress loss in the prestressing stage decreased from 11.4% to 7.9% when the distribution angle of the BFRP tendon varied from 20\({}^{\circ}\) to 45\({}^{\circ}\) given a 1.6% increase in \(\sigma_{\rm 11}\) and a 5.0% decrease in \(\sigma_{\rm 12}\). It is shown in Table 4 that the main prestress loss was caused by the friction between the BFRP tendon and the surrounding casing pipe (\(\sigma_{\rm 12}\)), since the proportion of \(\sigma_{\rm 11}\) was always less than 38%, regardless of the distribution angle.  
Similar to the results for the slab width of 9.0 m, the main prestress loss given the slab width of 12.75 m was caused by the friction between the BFRP tendon and the surrounding casing pipe (\(\sigma_{\rm 12}\)), as shown in Table 5. The total prestress loss varied from 9.1% to 14.9%, with the variation of the distribution angle being between 20\({}^{\circ}\) and 45\({}^{\circ}\).  
In general, the prestress loss in the prestressing stage (\(\sigma_{\rm 11}\) and \(\sigma_{\rm 12}\)) varied with the distribution angle of the BFRP tendon and the slab width. Given the same slab width, \(\sigma_{\rm 11}\) increased and \(\sigma_{\rm 12}\) decreased when the distribution angle varied from 20\({}^{\circ}\) and 45\({}^{\circ}\), and the proportion of \(\sigma_{\rm 11}\) in the total prestress loss in the prestressing stage rose as well. However, given the same distribution angle, the \(\sigma_{\rm 11}\) decreased while \(\sigma_{\rm 12}\) significantly increased with the increase in slab width, and the proportion of \(\sigma_{\rm 11}\) went down gradually.