# Calculation of Reasonable Tension Value for Longitudinal Connecting Reinforcement of CRTSII
## 5 Check and Calculation in Operating Period
### Calculation of Reasonable Tension Value  
The pretension loss value was determined by the temperature difference between the concrete and the reinforcement when the concrete and reinforcement deformed together. The pretension decreases (Equation (11)) when the temperature difference is \(\Delta t\). When the reinforced concrete deforms together,  
\[N_{los}=\Delta t\alpha E_{s}A_{s,all}. \tag{11}\]  
The prestress of concrete \(N_{pre}=N_{los}\). At this time, the stress of the connecting reinforcement is  
\[\sigma_{s,c}=(N+N_{ten}-N_{los})/A_{s,all}. \tag{12}\]The strain of the connecting reinforcement is  
\[\varepsilon_{s,a}=(N-N_{prc})/E_{s}/A_{s,all}. \tag{13}\]  
From Equations (10), (11), and (13), it can be seen that the crack width is only related to the value of the prestress generated at the wide joint, while the prestress value is related to the temperature difference \(\Delta t\) inside the reinforced concrete. Figure 17 shows the relationship between the crack width and the temperature difference value between the concrete and the reinforcement. It can be seen that reinforcement and concrete temperature difference value should not be less than 29 \({}^{\circ}\)C, to guarantee the crack width is less than 0.5 mm. It can be seen from Equations (10) and (12) that the stress of reinforcement is related to the tension force and the internal temperature difference between concrete and reinforcement. Figure 18 shows the three-dimensional diagram of stress, tension of reinforcement, and temperature difference. The stress of reinforcement is directly proportional to tension force and inversely proportional to temperature difference. According to the checked results of crack width, the temperature difference value should be greater than 29 \({}^{\circ}\)C. The three-dimensional figure is mapped to the tension in the Figure 18, which is a flat surface temperature difference value under the condition of guaranteeing that the crack width and the reinforcement stress meet the design requirements of tension and temperature range, as shown in Figure 19 (shaded area). The maximum tensile force should not exceed 234 kN under the condition that the crack width meets the design requirements, and maximum tension force linearly increases with the rising of the reinforced concrete temperature difference. The maximum tension force should not exceed 317 kN when the reinforced concrete temperature difference value is 50 \({}^{\circ}\)C.  
Before the casting of wide-seam concrete, the track slab is in an eccentric compression state after the longitudinal reinforcing bar is stretched, due to the existence of the narrow joint. In order to ensure the track slab has a symmetry stress station after the casting of wide-seam concrete, the temperature increment value of the precast reinforcement should be determined according to different tensile values. The tensile force at this point is  
\[N_{ten}=2N_{los}=2\Delta t\alpha E_{s}A_{s,all}. \tag{14}\]  
Figure 17: The relationship between crack width and temperature difference between concrete and reinforcement. Figure 20 shows the relation between the tensile force value and temperature increment value of reinforcement under the condition of ensuring that the stress of reinforcement and the crack width of concrete meet the requirements and, at the same time, ensuring the tension force value of the track slab under the symmetry pressure condition. It can be seen that tension force range is 229-240 kN, and the temperature difference range is 29-30.3 \({}^{\circ}\)C, to meet the design requirements, and tensile force does not cause interface failure between the track slab and the mortar layer. Since the range of tension force values and the temperature difference of reinforcement are relatively small, this suggests that the tension force value should be 230 kN, and the temperature difference of reinforcement should be 30 \({}^{\circ}\)C for construction applications. Figure 19: Temperature and tension force range to guarantee appropriate reinforced stress and crack width.