# Structural Behaviour of Concrete Deep Beams Reinforced with Aluminium Alloy Bars
## 4 Design Models  
When designing RC deep beams where arc action predominates and beam action does not apply, one can use a strut-and-tie model. This section aims to evaluate the analytical design model [56] in the published literature, together with ACI 318 [57] predictions. ACI 318 [57] provides design guidelines for deep beams where traditional beam theory does not apply, since the acting load is in a direct transfer to the support through direct compression. The shear strength of the deep beam is calculated by Equation (10) according to the ACI 318 design code [57]. \[V_{u}=F_{\text{ns}}*sin\dot{\mathcal{O}} \tag{10}\]  
where Fns, the compressive strength of a strut, is given in Equation (10). \(\dot{\mathcal{O}}\) is the angle of the diagonal strut (assumed to be 25\({}^{\circ}\)). The strut angle assumption is 25\({}^{\circ}\) based on the ACI 318 design code [57]. The accurate predictions were obtained with this assumption  
\[F_{\text{ns}}=f_{ce}*A_{cs} \tag{11}\]  
where Acs is the cross-section area of at the end of strut. fce can be calculated by Equation (12). \[f_{ce}=0.85*\beta_{c}*\beta_{s}*f_{c} \tag{12}\]  
The analytical model, considering transverse reinforcement to predict the shear strength capacity of deep beams, was developed by [PERSON] and [PERSON] [56]. The predicted shear strength, which is limited by less than 0.3 \({}^{\ast}\) fc, is considered with Equation (13). \[v_{u}=0.5*f_{c}*\left|K*\left(w_{h}+0.03\right)+\sqrt{K^{2}*\left(w_{h}+0.03 \right)^{2}+4*\left(w_{h}+0.03\right)*\left(w_{v}+0.03\right)}\right| \tag{13}\]  
\[w_{h}=\frac{p_{h}*f_{yh}}{f_{c}}\leq 0.26\ w_{v}=\frac{p_{v}*f_{yv}}{f_{c}} \leq 0.12 \tag{14}\]  
where ph and pv are the horizontal and transverse reinforcement ratios, respectively. fyh and fyv are the yield strengths of horizontal and transverse reinforcement, respectively. Figure 18: Comparison of transverse reinforcement ratio (%) [53; 54]. \[K=\left\{ \begin{array}{ll}\frac{2\pi d_{D}}{\hbar}&0<\frac{\text{a}}{\text{H}} \leq 0.5\\ \frac{d_{D}}{\hbar}\left[\frac{\text{h}}{\text{d}}\left(\frac{4}{3}-\frac{2\pi a }{3\hbar}\right)\right]&0.5<\frac{\text{a}}{\text{H}}\leq 2\\ 0&\frac{\text{a}}{\text{H}}\geq 2\end{array} \right. \tag{15}\]  
where dv is the effective shear depth, which can be calculated to be d - d\({}^{\prime}\), and d\({}^{\prime}\) is the distance between the centre of compression reinforcement and top of the beam. The numerically tested AA RC beams, which are identical to the beams that were physically tested by [PERSON] and [PERSON] [4], are given in Table 3. The beams reported in Table 3 failed in terms of shear. The shear force capacities predicted by both the design model of ACI 318 [57] and the design equation proposed by [PERSON] and [PERSON] [56] were given in Table 4. The reproduced beams by the FE model had shear force capacities in a range from 55.6 kN to 176.7 kN, with different physical and mechanical properties, as shown in Table 4. The design code of ACI 318 [57] predicted the shear force capacities of beams with an error of 3%. The standard deviation was 0.19. It should be noted that a safety factor of 0.75 was considered as recommended by ACI 318 [57]. Similarly, the proposed design model by [PERSON] and [PERSON] [56] predicted well with an error of 3% but better accuracy with a standard deviation of 0.12. The design model developed by [PERSON] and [PERSON] [56] for beams reinforced with steel bars offers a more balanced approach and can be adapted to AA RC beams. The 3% error remained consistent across the specimens, with a/d ratios between 1.14 and 2.44, tension reinforcement ratios from 2.15% to 5.99%, and transverse reinforcement ratios from 0.12% to 0.84%. Further studies are recommended to verify the accuracy for other AA deep beam configurations. Both models were originally developed for RC deep beams.