# Flexural Behavior of a 30-Meter Full-Scale Simply Supported Prestressed Concrete Box Girder
## 4 A Nonlinear Numerical Model for the Tested Box Girder
### Material Stress-Strain (\(\sigma\)-\(\varepsilon\)) Relationship  
(1) Concrete Stress-Strain Relationship  
The compression and tension constitutive model developed in references [55; 58] was adopted herein. The compression constitutive model is as follows:  
\[\begin{cases}\frac{\sigma_{c}}{f_{c}}=2x-x^{2}&x=\frac{\varepsilon_{c}}{ \varepsilon_{cp}}\leq 1\\ \frac{\sigma_{c}}{f_{c}}=1&x=\frac{\varepsilon_{c}}{\varepsilon_{cp}}>1\end{cases} \tag{2}\]  
where \(\sigma_{c}\) and \(\varepsilon_{c}\) are, respectively, the concrete compressive stress and strain, \(\varepsilon_{cp}\) is the compressive strain when the stress is \(f_{c}\) (\(\varepsilon_{cp}\) = 2000\(\mu\varepsilon\)), and \(\varepsilon_{cu}\) is the ultimate compressive strain of concrete (\(\varepsilon_{cu}\) = 3300\(\mu\varepsilon\)).  
The constitutive model of the uniaxial tension stress-strain is given as [55; 58]  
\[\begin{cases}\frac{\sigma_{t}}{f}=1.2x-0.2x^{6}&x=\frac{\varepsilon_{c}}{ \varepsilon_{lp}}\leq 1\\ \frac{\sigma_{t}}{f_{t}}=\frac{x}{0.312f_{t}^{2}(x-1)^{1/2}+x}&x=\frac{ \varepsilon_{c}}{\varepsilon_{lp}}>1\end{cases} \tag{3}\]  
\[\left\{\begin{array}{l}\varepsilon_{tp}=f_{t}/E_{c}\\ \varepsilon_{tu}=2\varepsilon_{tp}\end{array}\right. \tag{4}\]  
where \(\sigma_{t}\) and \(\varepsilon_{t}\) are, respectively, the concrete tensile stress and strain, and \(f_{t}\) and \(E_{c}\) are the tensile strength and elastic modulus of concrete, respectively.  
(2) Reinforcement Stress-Strain Relationship  
The following elastic hardening model was adopted for the tension and compression stress-strain relationship of the reinforcement [59; 60]:  
\[\begin{cases}\sigma_{s}=E_{s}\varepsilon&\varepsilon\leq\varepsilon_{y}\\ \sigma_{s}=f_{y}+E_{s2}(\varepsilon-\varepsilon_{y})&\varepsilon>\varepsilon_{ y}\end{cases} \tag{5}\]  
where \(\sigma_{s}\) and \(\varepsilon\) are the stress and strain of the reinforcement, and \(f_{y}\) and \(\varepsilon_{y}\) are the yield stress and strain of the reinforcement, respectively. \(E_{S}\) and \(E_{S2}\) (\(E_{S2}=0.01E_{S}\)) are the elastic modulus of the reinforcement before and after yielding, respectively.  
The tension stress-strain relationship of the prestressed reinforcement is shown as [59; 60]  
\[\varepsilon_{p}=\frac{\sigma_{p}}{E_{p}}+0.002\bigg{(}\frac{\sigma_{p}}{f_{0.2}}\bigg{)}^{13.5} \tag{6}\]  
where \(\varepsilon_{p}\) and \(\sigma_{p}\) are, respectively, the strain and stress of the prestressed reinforcement, \(E_{p}\) is the elastic modulus of the prestressed reinforcement, and \(f_{0.2}\) is the nominal yield stress of the prestressed reinforcement.  
(3) Unloading Stress-Strain Relationship  
The initial stiffness of the material is \(E_{0}\), as shown in Equation (7).  
\[E_{0}=\left.\frac{d\sigma}{d\varepsilon}\right|_{\varepsilon=0} \tag{7}\]  
When the strain increases to its maximum value, the unloading elastic modulus is \(E_{1}\), and \(E_{1}=0.6E_{0}\)[59; 60]. In other cases, the elastic modulus is calculated by linear interpolation with \(E_{0}\) and \(E_{1}\).