# Structural Behaviour of Concrete Deep Beams Reinforced with Aluminium Alloy Bars
## 2 Analysis of AA RC Beams
### FE Model
#### 2.2.1 Material Model  
The Hognestad parabola [36] is the default model in VecTor2 software [35] for determining concrete compression pre-peak (ascending branch) behaviour. It is also recommended for normal concrete strengths (i.e., less than 40 MPa). The stress-strain relation for the pre-peak behaviour of concrete in compression is given by Equation (1).  
\[f_{ci}=f_{p}\left\{2*\left\{\frac{\epsilon_{ci}}{\epsilon_{p}}\right\}-\left\{ \frac{\epsilon_{ci}}{\epsilon_{p}}\right\}^{2}\right\} \tag{1}\]  
where \(\mathrm{f_{ci}}\) (MPa) represents the concrete compressive stress at a given strain \(\epsilon_{ci}\) (mm/mm); \(\mathrm{f_{p}}\) (MPa) is the peak concrete cylinder compressive strength and \(\epsilon_{p}\) (mm/mm) is the corresponding strain.  
The stress-strain curve of a modified Park-Kent model [35], which was modified by [PERSON] et al. [37], was used to define the post-peak (descending branch) behaviour of concrete in compression.  
As can be seen in Figure 2, the descending branch linearly reaches a fifth of concrete compression stress. The relation between stress and strain for the post-peak behaviour of concrete in compression is given in Equation (2).  
\[f_{ci}=\left[f_{p}+Z_{m}*f_{p}\left(\epsilon_{ci}-\epsilon_{p}\right)\right] \text{ }or\ 0.2*f_{p} \tag{2}\]  
where Zm is given by Equation (3).  
\[Z_{m}=\frac{0.5}{\frac{3+0.29|f_{c}|}{145|f_{c}|-1000}*\left(\frac{\epsilon_{0 }}{-0.002}\right)+\left(\frac{|f_{m}|}{170}\right)^{0.9}+\epsilon_{p}} \tag{3}\]  
where fc is the concrete cylinder compressive strength, \(\epsilon_{0}\) is the concrete compressive strain corresponding to fc, and \(\mathrm{f_{lat}}\) is the sum of the principal stresses (i.e., \(\mathrm{f_{c1}}+\mathrm{f_{c2}}+\mathrm{f_{c3}}-\mathrm{f_{ci}}\leq 0\), i = 1 or 2) acting in a direction opposite to the one being considered [35].  
Figure 2: Post-peak concrete compression behaviour [35; 37].  
The default model of Model A in VecTor2 proposed by [PERSON] and [PERSON] [38] for concrete compression softening was used. The compression softening model is described in Equations (4)-(8).  
\[\beta_{d}=\frac{1}{1+C_{s}\ C_{d}}\leq 1 \tag{4}\]  
\[C_{d}=\left\{\begin{array}{rl}0&\text{if $r<0.28$}\\ 0.35(r-0.28)^{0.80}&\text{if $r>0.28$}\end{array}\right. \tag{5}\]  
\[r=\frac{\epsilon_{\text{c1}}}{\epsilon_{\text{c2}}}\ \leq 400 \tag{6}\]  
\[C_{s}=\left\{\begin{array}{rl}0&\text{if shear slip not considered}\\ \\ 0.55&\text{if shear slip considered}\end{array}\right. \tag{7}\]  
\[f_{p}=\beta_{d}f_{c}\ and\ \varepsilon_{pk}=\beta_{d}\varepsilon_{0} \tag{8}\]  
where \(\beta_{d}\) is the reduction factor for the concrete compressive strength (fc) and its corresponding strain (\(\epsilon_{0}\)). \(\epsilon_{\text{c1}}\) and \(\epsilon_{\text{c2}}\) are the principal tensile and compressive strains, respectively. \(C_{d}\) is the strain softening factor, and \(C_{s}\) is the shear slip factor.  
The default model of the bilinear tension softening model in VecTor2 software was used. This model was adapted from the CEB-FIP Model Code [39] by the software. The stress-strain model for 6 and 8 mm diameter AA bars had three phases. The first phase was linear-elastic behaviour up to the yield strength of 6 or 8 mm diameter AA bars, which was given in Section 2.1 (see Table 1). This was followed by the second phase, which consisted of a yield plateau until the start of nonlinear strain hardening. The third phase ended with the rupture of AA bars.