## 2 The GSVATM  
For the sake of this article, this section summarizes the GSVATM model for RC rectangular beams. The main equations, as well as the solution procedure, are also presented here. Additional information about the model can be found in [1]. The first equations (Equations (1-5), see Table 1) are derived based on a plain truss used to model a RC thin beam element under a shear force \(V\) (Figure 2a). This loading induces a shear flow \(q\) in the cross section of the thin beam. This beam element is used to model the walls of the equivalent RC box beam element under torsion \(M_{T}\) illustrated in (Figure 2b), and from which Equations (6)-(11) are derived (see Table 1). When compared to the VATM [2,3], the GSVATM incorporates an additional feature which is a concrete tie with a tensile force \(T\) in the perpendicular direction to the concrete strut (see Figure 2a). This last one carries a compressive force \(C\) and makes an angle \(\alpha\) to the longitudinal axis of the beam. This angle is equal to the angle of cracks (see Figure 2a). The concrete tie aims to simulate the tensile concrete in the referred direction. In Figure 2a, \(R\) is the resulting force (Equation (1)) with an angle \(\beta\) (Equation (2)) to the force \(C\) and with an angle \(\gamma\) (Equation (3)) to the longitudinal axis of the beam. Forces \(C\) (Equation (4)) and \(T\) (Equation (5)) are, respectively, the resultants of the compressive and tensile stress fields in concrete (\(\sigma_{2}^{c}\) and \(\sigma_{1}^{c}\), respectively). Parameter \(d_{v}\) is the distance between the centers of longitudinal bars and parameter \(t_{c}\) is the width of the thin cross section (see Figure 2a). This latter is equal to the width of the concrete strut and tie. In the box beam element, the torsional moment \(M_{T}\) and the shear flow \(q\) are related by using the classical [PERSON]'s thin tube theory for thin tubes. Three equilibrium equations (Equations (6)-(8)) are derived to calculate the following: the torsional moment \(M_{T}\), the effective thickness \(t_{c}\) and the angle \(\alpha\). If \(\gamma=\alpha+\beta>90^{\circ}\), Equation (7) is multiplied by the factor (\(-1\)). In the previous equations, the other parameters are the following: \(A\) and \(p\) are, respectively, the area enclosed and the perimeter of the center line of the shear flow \(q\) which is assumed to coincide with the center line of the walls (\(A=(x-t_{c})(y-t_{c})\), with \(x\) and \(y\) the minimum and maximum outer dimensions of the cross section, and \(p=2(x-t_{c})+2(y-t_{c})\)); \(A_{sl}\) is the area of the longitudinal steel reinforcement; \(A_{st}\) is the area of one bar of the transverse steel reinforcement (closed stirrups); \(s\) is the spacing of the stirrups in the longitudinal direction; \(f_{sl}\) and \(f_{st}\) are the stresses in the longitudinal and transverse steel reinforcement, respectively. Figure 2: Beam elements to derive the equations for the generalized softened variable angle truss-model (GSVATM): (**a**) thin beam, (**b**) box beam. For the box beam element, three compatibility equations (Equations (9-11)) are also derived to calculate the following: the strains in the transverse and longitudinal steel reinforcement, \(\varepsilon_{st}\) and \(\varepsilon_{sl}\) respectively, and the angular deformation per unit length (twist) \(\theta\). Equation (12) relates the strains. In the previous equations, the other parameters are: \(\varepsilon_{2s}^{c}\) is the maximum compressive strain which occurs at the outer fiber of the concrete strut; \(\varepsilon_{1}^{c}\) and \(\varepsilon_{2}^{c}\) are the average strains in the concrete strut and tie, respectively. The distribution of the strains among the thickness of the struts and ties accounts for the bending of the walls which induces a strain gradient (Figure 3). A smeared and softened stress (\(\sigma\))-strain (\(\varepsilon\)) relationship is used for the compressive concrete in the strut.