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Table 1 Equilibrium and compatibility equations from GSVATM for RC beams (Bernardo et al. 2015a).

From: Generalized Softened Variable Angle Truss Model for PC Beams under Torsion

Plain truss analogy for a RC thin beam under shear:
\( R = \sqrt {C^{2} + T^{2} } \quad (1) \)
\( \upbeta = \arctan \left( {T/C} \right)\quad (2) \)
\( \upgamma = \alpha + \beta \quad (3) \)
\( C = \sigma_{2}^{c} t_{c} d_{v} \cos \upalpha \quad (4) \)
\( T = \sigma_{1}^{c} t_{c} d_{v} \sin \upalpha \quad (5) \)
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Space truss analogy for a RC box beam under torsion:
Equilibrium equations:
\( M_{T} = \frac{{2A_{o} R\sin \gamma }}{{d_{v} }}\quad (6) \)
\( t_{c} = \frac{{A_{sl} f_{sl} }}{{\sigma_{2}^{c} p_{o} }}\frac{\cos \beta }{\cos \alpha \cos \upgamma }{\text{ for }}\upgamma = \upalpha + \upbeta \le 90^{ \circ } \quad (7) \)
\( \upalpha = \arctan \left( {\frac{{\sqrt {F^{2} \left( {\tan \upbeta } \right)^{2} + F\left( {\tan \upbeta } \right)^{4} + F + \left( {\tan \upbeta } \right)^{2} } }}{{F\left( {\tan \upbeta } \right)^{2} + 1}}} \right){\text{ with }}F = \frac{{A_{st} f_{st} p_{o} }}{{A_{sl} f_{sl \, } s}}\quad (8) \)
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Compatibility equations:
\( \varepsilon_{st} = \left( {\frac{{A_{o}^{2} \sigma_{2 \, }^{c} \sin \gamma }}{{p_{o} M_{T \, } \cos \beta \tan \alpha \sin \alpha }} - \frac{1}{2}} \right)\varepsilon_{2s}^{c} \quad (9) \)
\( \varepsilon_{sl} = \left( {\frac{{A_{o}^{2} \sigma_{2 \, }^{c} \sin \upgamma }}{{p_{o} M_{T \, } \cos \upbeta \cot \upalpha \sin \upalpha }} - \frac{1}{2}} \right)\varepsilon_{2s}^{c} \quad (10) \)
\( \theta = \frac{{\varepsilon_{2s}^{c} }}{{2t_{c \, } \sin \upalpha \cos \upalpha }}\quad (11) \)
\( \varepsilon_{1s}^{c} = 2\varepsilon_{1}^{c} = 2\varepsilon_{sl} + 2\varepsilon_{st} + \varepsilon_{2s}^{c} \quad (12) \)