## 3 Numerical Example
### FEM Configuration
By including the loading combinations both static and dynamic [43], CDP containing four constitutive parameters [44] to define the yield surface in ABAQUS. Here, for assigning these parameters, the data from [PERSON] et al. [23] were used. The illustration of the tension and compression behavior can be expressed either by the stress-strain curve for compression and also in crack length investigation in tension softening. The stress-strain curve is drawn by two adopted expressions as followed by EN1992-1-1 up to nominal ultimate strain (Figure 7a), and the rest can be defined by the sinusoidal expression (Figure 7b). According to EN1992-1-1, the mathematical expression up to the ultimate strain for the relationship between the compressive stress, \(\sigma_{c}\), and shortening strain, \(\varepsilon_{c}\), is determined by the following equation:  
\[\sigma_{c}=f_{cm}\left[\frac{k\eta-\eta^{2}}{1+(k-2)\eta}\right],\;\varepsilon _{c}\geq\varepsilon_{c\omega 1} \tag{13}\]  
where the concrete compressive stress is expressed by \(\sigma_{c}\), \(\eta=\frac{\varepsilon_{c}}{\varepsilon_{c1}}\), the concrete compressive strain is denoted by \(\varepsilon_{c}\), \(\varepsilon_{c1}\) is the peak stress compressive strain in the concrete is defined by \(f_{c}\) and \(k=\frac{1.05\varepsilon_{cm}|\varepsilon_{c1}|}{f_{cm}}\). The nominal ultimate strain \(\varepsilon_{c\omega 1}=0.0035\) were adopted from EN1992-1-1 for all aged concrete (0-75 years). Therefore, the extension of the stress-strain curve was made with the sinusoidal part between points using the following equation:  
\[\sigma_{c}=f_{cm}\left[\frac{1}{\beta}-\frac{sin\left(\mu^{\alpha_{LD}}\alpha_ {tE}\frac{\pi}{2}\right)}{\beta sin\left(\alpha_{tE}\frac{\pi}{2}\right)}+ \frac{\mu}{\alpha}\right],\;\varepsilon_{c\omega 1}<\varepsilon_{c}\leq \varepsilon_{c\omega 1} \tag{14}\]  
[PERSON] et al. [45] well explained the parameters for the calculation of this sinusoidal part. The stress-strain behavior of our selected CGD is determined using the above expressions and pictured in Figure 8a for the four (0, 25, 50 and 75 years) different aging conditions. Figure 8b shows the tension stiffening behavior considered the crack opening ratio by adopting the method of concrete exponential tension softening model [46]. The maximum tensile crack starts from \(\sigma_{t}\), which follows the equation explained in [PERSON] et al. [47] and is also used in [PERSON] et al. [23]. Figure 7: Stress–strain of concrete diagram of concrete: (**a**) up to ultimate condition; (**b**) with extended sinusoidal part. The equation for defining the tensile softening behavior is expressed by the following exponential function:  
\[\frac{\sigma_{1}}{f_{t}}=f(w)-\frac{w}{w_{c}}f(w_{c}) \tag{15}\]  
\[f(w)\ =\left[1+\left(\frac{c_{1}w}{w_{c}}\right)^{3}\right]\exp\left(-\frac{c_{2} w}{w_{c}}\right) \tag{16}\]  
where the displacement for crack opening is denoted by \(w\), when the stress cannot be transferred, then, the notation is defined by \(w_{c}\), and \(c_{1}\) and \(c_{2}\) are material constants for normal concrete. Figure 8 shows that by increasing the age, the compression and tensile performance of the concrete are decreased. As the hygro-chemo-mechanical model affects the modulus of elasticity with time, the tensile strength is reduced, and this tensile stress for each year has been adopted for the limit state of this study.