# Calculation of Reasonable Tension Value for Longitudinal Connecting Reinforcement of CRTSII
## 5 Check and Calculation in Operating Period  
### Original Design Check and Calculation  
Three hundred kilometres of pre-pressure were applied to the concrete at the wide and narrow joint by applying 300 kN of longitudinal tension force to the longitudinal reinforcement between the track slabs in the design, and the track slab was considered as continuous in the longitudinal direction. The track slab was considered as the bar fixed at the beam end. Therefore, the strain of the track slab under the action of system temperature difference is  
\[\varepsilon_{\Delta t}=\alpha\Delta t. \tag{4}\]  
The shrinkage deformation, considered from 14 days to infinity of concrete, was carried out according to the specifications in [8]. The contraction strain is  
\[\varepsilon_{s}=-0.23. \tag{5}\]  
The stress of the reinforcement under the action of the live load was not taken into account. Only cooling and concrete shrinkage load were considered in the calculation because the longitudinal connecting reinforcement of the track slab was placed in the middle of the track slab. The stress of reinforcement is  
\[\sigma_{s}=E_{s}\cdot(\varepsilon_{\Delta t}+\varepsilon_{s}), \tag{6}\]  
where \(E_{s}\) is the elastic modulus of reinforcement.  
There were only six \(\Phi\) 20 connecting reinforcements in the joint of the track slabs. The stress of the reinforcement at the joint and the crack width was checked and calculated according to the tensile check result of the reinforcement inside the track slab. The axial force borne by the track slab was as Formula (7), under the combined action of cooling load and shrinkage load:  
\[N=\sigma_{s}\cdot A_{s}, \tag{7}\]  
where \(A_{s}\) is the cross-sectional area of a single connecting reinforcement.  
The stress of reinforcement caused by shrinkage and cooling of concrete, in the future, should be superimposed on the stress caused by the tension of the connecting reinforcement, because the tension step occurred before the concrete casting step. In addition, the prestress of the joint cannot play a role in limiting the crack width, due to the fact that the joint concrete is not perfused when the tension force is applied to the connecting reinforcement.  
The stress of the connecting reinforcement at the wide joint is  
\[\sigma_{s,c}=(N+N_{ten})/A_{s,all}, \tag{8}\]  
where \(A_{s,all}\) is the total cross-sectional area of six reinforcements and \(N_{ten}\) is the tension force.  
The average strain of reinforcement is  
\[\varepsilon_{s,a}=N/E_{s}/A_{s,all}. \tag{9}\]The crack width is checked and calculated according to the formula below:  
\[\omega_{cr}=l_{cr,\max}.(\epsilon_{s,a}-\epsilon_{c,a}), \tag{10}\]  
where \(\omega_{cr}\) is the calculated value of the crack width in mm, \(l_{cr,\max}\) is the maximum crack spacing in mm, \(\epsilon_{s,a}\) is the average strain of reinforcement, and \(\epsilon_{c,a}\) is the average strain of concrete.  
Take the distance (650 mm) between the false seam of track slab as \(l_{cr,\max}\). As the strain of concrete is much smaller than that of reinforcement, the average strain of concrete is considered to be 0.  
The limiting design value of reinforcement stress is 286 MPa, and the crack width is 0.5 mm according to the design specification [8]. The reinforcement stress was 577.6 MPa, and the crack width was 0.65 mm as calculated by Equations (8) and (10), and both the calculation results did not meet the design requirements.  
It can be seen from Equation (8) that the stress value of the connecting reinforcement can be reduced by reducing the tensile force of the connecting reinforcement. In order to reduce the crack width, the effective measures are to reduce the diameter of the connecting reinforcement or to apply prestress to the wide joint concrete.