## 2 The GSVATM
As justified in [4], for the concrete in compression, the \(\sigma-\varepsilon\) relationship proposed by [PERSON] and [PERSON] [31] (Equations (13) and (14) in Table 2) with softening factor \(\beta_{*}=\beta_{\sigma}=\beta_{\varepsilon}\) for both the peak stress and corresponding strain proposed by [PERSON] and [PERSON] in 1998 [32] (Equations (15-18) in Table 2) is used. \begin{table}
\begin{tabular}{l l l l} \hline Equations for the plain truss (RC thin beam under shear): & \(\beta=\arctan(T/C)\) & (2) & \(\gamma=\alpha+\beta\) \\ \(C=\sigma_{2^{\prime}t}^{t}d_{v}\cos\alpha\) & (4) & \(T=\sigma_{1^{\prime}t}^{t}d_{v}\sin\alpha\) & (5) \\ \hline Equations for the space truss (RC box beam under torsion): & Equilibrium equations: & \\ \(M_{T}=\dfrac{2.4_{b}R\sin\gamma}{d_{v}}\) & (6) & \(t_{c}=\dfrac{A_{b}f_{d_{v}}}{\sigma_{2}^{c}p_{\phi}}\cos\alpha\cos\gamma\) & for \(\gamma=\alpha+\beta\leq 90^{\circ}\) \\ \(\alpha=\arctan\left(\dfrac{\sqrt{F^{2}\left(\tan\beta\right)^{2}+F\left(\tan \beta\right)^{4}+F+\left(\tan\beta\right)^{2}}}{F\left(\tan\beta\right)^{2}+1} \right)\) & with \(F=\dfrac{A_{b}f_{d_{v}}p_{\phi}}{A_{b}f_{d_{v}}s}\) \\ \hline Compatibility equations: & \\ \(\theta=\dfrac{\varepsilon_{2s}^{c}}{2t_{c}}\sin\alpha\cos\alpha\) & (11) & \(\varepsilon_{2s}^{c}=2\varepsilon_{2s}^{c}+2\varepsilon_{2s}^{c}+2\varepsilon_{2 s}^{c}\) & (12) \\ \hline \end{tabular}
\end{table} 
Table 1: Equations for GSVATM (reinforced concrete (RC) beams). Figure 3: Stress and strain gradients. Additionally, a smeared and stiffened \(\sigma-\varepsilon\) relationship is used for the tensile concrete in the tie and also for the tensile steel bars. For the tensile concrete, the \(\sigma-\varepsilon\) relationship from [PERSON] and [PERSON] in 1994 [33] and modified by [PERSON] and [PERSON] in 2009 [34] (Equations (19-23) in Table 2) is used. For the steel bars in tension, as justified in [4], the \(\sigma-\varepsilon\) relationship from [PERSON] and [PERSON] in 1994 [33] (Equations (30-32) in Table 2) is used. In the previous equations, the meaning of the parameters are as follows: \({f^{\prime}}_{c}\) is the average compressive strength of concrete and \(\varepsilon_{0}\) is the corresponding strain; \(\rho_{l}\) is the longitudinal reinforcement ratio (\(\rho_{l}=A_{sl}/A_{c}\), with \(A_{c}=xy\) ); \(\rho_{l}\) is the transverse reinforcement ratio (\(\rho_{l}=A_{sl}u/A_{c}s\), with \(u=2x+2y\)); \(f_{ly}\) and \(f_{ly}\) are the stresses corresponding to the yielding of the longitudinal and transverse reinforcement, respectively; \(E_{c}\) is the [PERSON]'s modulus for concrete; \(f_{cr}\) is the tensile concrete strength and \(\varepsilon_{cr}\) is the corresponding strain; \(f_{s}\) and \(\varepsilon_{s}\) are the stress and strain in the steel reinforcement, respectively; \(E_{s}\) is the [PERSON]'s modulus for steel; \(f_{y}\) is the stress corresponding to the yielding of the steel reinforcement; and \(\rho\) is the reinforcement ratio. The stress \(\sigma_{2}^{c}\) represents the average stress in the concrete strut (Equation (24) in Table 2) and accounts for the strain gradient (Figure 3). The stress \(\sigma_{1}^{c}\) represents the average stress in the concrete tie (Equation (27) in Table 2) and also accounts for the strain gradient (Figure 3). Parameters \(k_{2}^{c}\) (Equations (25) and (26)) and \(k_{1}^{c}\) (Equations (28) and (29)) represent the average compressive and tensile stresses, respectively. These parameters are computed from integration of Equations (13), (14) and (19), (20), respectively. The solution procedure of the GSVATM to calculate the \(M_{T}-\theta\) curve accounts for the nonlinearity of the equations and is based on an iterative trial-and-error technique.