# Structural Analysis and Design of Reinforced Concrete Bridge Corbels
## 3 Discussion
### Strut-and-tie Model  
In this case, a 2D strut-and-tie model (Figure 5) was developed to catch the overall behavior (the structural response described with the parameters listed in Table 1). The model (Figure 5) can be described as follows: The compressive force transferred by the steel baseplate was first deviated toward the horizontal direction by the corbel's stirrups acting in tension. This force was equilibrated by the tension in the vertical branch of the stirrups themselves and by another concrete strut converging to the node (C) formed by the anchor and the upper (flexural) reinforcements of the deck (or slab). The compressive strut under the compressed portion of the steel plate was further deviated by the presence of the upper reinforcement of the deck, the tensioned anchor, and the vertical branch of the corbel stirrups (Node C- E -F). As shown in Figure 5, equilibrium conditions must be fulfilled for six nodes (A-F):  
* The node located close to the external edge of the concrete element under the compressed part of the plate (A). In this node, three elements of the model are converging: Two struts in compression and the top reinforcement of the curb in tension. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Geometry** & **Actions/Forces** & **Design** \\ \hline \(Z\), \(Z_{2}\), \(h_{1}\), \(H_{\text{corbel}}\), \(H_{\text{deck}}\) (see Figure 5) & T, \(V_{\text{Ed}}\), \(M_{\text{Ed}}\) & \\  & & Area of stirrups within 2 steel \\ \(Z_{1}=Z-h_{1}\cdot\tan(\theta_{1})\) & \(F_{1}=\frac{T}{\cos(\theta_{1})}\) & plate width \\  & & \(A_{T}=\frac{\max(H_{t},H)}{f_{\mu}}\) \\ \hline \multirow{2}{*}{\(c_{\text{s}}\) concrete cover} & \multirow{2}{*}{\(H=F_{1}\text{sen}(\theta_{1})+V_{\text{Ed}}\)} & Steel area at the top of the deck \\  & & \(A_{\text{s}}^{\prime}=\frac{T_{\mu}}{f_{\mu}}\) \\ \hline \(S_{1}=\)spacing between U-shaped rebars & & \\ \(S_{2}=\)spacing between the vertical branches of the U-shaped rebars & & Area of concrete under tension \\ \(X\) (see Figure 5)—determined as the half of the neutral axis at the ultimate limit state of the slab with a width B = \(3S_{1}\) & \(H_{\text{v}}=F\text{sen}(\theta)\) \\ \hline \(p=H_{\text{curb}}+H_{\text{deck}}-c_{\text{s}}-h_{1}-X-Z_{2}\cdot\tan(\theta_{ 3})\) & \(C=\frac{M_{\text{deck}}+V_{\text{deck}}H_{\text{deck}}}{H_{\text{deck}}-c_{\text{ s}}-\text{s}/2}\) \\ \hline \(\theta_{1}=10^{\circ}\) & \(T_{r}=C+V_{\text{Ed}}\) \\ \hline \(\theta=arctg\left(\frac{h_{1}}{Z_{2}}\right)\) & \(F_{3}=\frac{c_{\text{s}}}{\cos(\theta_{3})}\) \\  & & \(F_{2}=\frac{F_{1}\cos(\theta_{1})}{\text{sen}(\theta_{3})}\) \\ \hline \(\theta_{2}=arctg\left(\frac{P}{Z_{1}}\right)\) & \(T_{1}=\) & \\  & \(F_{2}\text{sen}(\theta_{2})-F_{3}\text{sen}(\theta_{3})\) \\ \hline \(\theta_{3}=arctg\left(\frac{H_{\text{s}}}{C}\right)\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Components of the strut-and-tie model of the corbel. Figure 5: Scheme of the 2D strut-and-tie model, geometrical quantities, and nodes. * The node located at the internal upper corner of the curb (B): Here the stirrups are bent at 90\({}^{\circ}\) and the two ties, which represent the vertical and horizontal branches of the stirrups, are equilibrated by an inclined concrete strut. * The node between the two struts in compression deviated by the effect of the tensile force in the upper reinforcement of the deck slab (C). * The node (D), placed where the compressed concrete strut coming from node B is equilibrated by the difference in vertical tension in the bar and the difference in horizontal tension in the upper reinforcement of the deck. * The node (E), placed at the bottom of the bar where the tension in the bar contributes to the deviation of the compressive force in the concrete struts. * The node (F), placed where the vertical branch of the corbel stirrups meets the two compressed struts coming, respectively, from node E and from the compressed lower chord of the deck. Table 1 describes the procedure to evaluate the main quantities of the strut-and-tie model.