# Flexural Behavior of a 30-Meter Full-Scale Simply Supported Prestressed Concrete Box Girder
## 4 A Nonlinear Numerical Model for the Tested Box Girder
### Discussion
#### 4.5.3 Prestressing Tendons  
The equilibrium destructive state of PC structures is that, when the prestressing tendon breaks, the concrete in the compression zone is crushed and damaged. In this failure mode, the ratio of prestressing tendons is called the equilibrium reinforcement ratio (\(\rho_{b}\)), which is defined as [55]  
\[\rho_{b}=\frac{\alpha f_{c}}{f_{u}}\left[\beta\left[\frac{\varepsilon_{cut}}{ \varepsilon_{cut}+\varepsilon_{p}}\right]+\left[\frac{b}{b_{w}}-1\right] \frac{t}{h_{0}}\right]-\frac{f_{y}}{f_{u}}\rho_{y} \tag{10}\]  
where \(\alpha\) and \(\beta\) are the coefficients of the equivalent rectangular stress diagram, and \(t\) is the flange thickness. The tested box girder is equivalent to an I-beam, and \(b_{w}\) and \(h_{0}\) represent the thickness of  
Figure 14: Ductility index values for different PPRs.  
Figure 13: Load–deflection curves for different degrees of prestress (PPRs).  
the web and effective height of the section, respectively. \(\rho_{y}\) is the ratio of reinforcement in the tensile zone of the bottom flange.  
The ratio of prestressing tendons (\(\rho_{p}\)) is defined as  
\[\rho_{p}=A_{p}/(b_{w}h_{0}). \tag{11}\]  
For the tested girder, the ratio of prestressing tendons is 0.934%, and the equilibrium reinforcement ratio is 1.446%. Girders with different ratios of prestressing tendon of 0.934%, 1.168%, 1.401%, 1.446%, 1.635%, and 1.868% are discussed. The load-deflection curve at the mid-span and the ductility index for each parameter are shown in Figures 15 and 16, respectively.  
It can be seen from Figures 15 and 16 that the ratio of prestressing tendons has an obvious influence on all of the control indices. The bearing capacity of the girder was observed to increase with increasing ratio of prestressing tendons, while the maximum deformation at the mid-span, the residual deformation, and ductility index decrease. For instance, as the ratio of prestressing tendons increases from 0.934% to 1.868%, the ultimate load increases from 1876.8 kN to 3706.2 kN, the maximum deformation at the mid-span decreases from 329.8 mm to 221.3 mm, the residual deformation decreases from 138.2 mm to 21.8 mm, and the ductility index of the component decreases from 1.83 to 1.32. With comprehensive consideration of the influence of various parameters, such as the bearing capacity,  
Figure 16: Ductility index values for different reinforcement ratios.  
Figure 15: Load–deflection curves for different reinforcement ratios.  
deformation, and ductility index of the structure, it is suggested that the ratio of prestressing tendons be increased to 1.401%. That is, the prestressing tendons should be increased from four strands to six strands in each duct of the tested girder. After the optimization of prestressing reinforcement, the ultimate load increases from 1876.8 kN to 2708.1 kN, the maximum deformation at the mid-span decreases from 329.8 mm to 269.2 mm, the residual deformation decreases from 136.2 mm to 65.4 mm, and the ductility index of the component decreases from 1.83 to 1.47.