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Table 2 \( \sigma{-} \varepsilon \) relationships for concrete and steel.

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

Concrete in compression (Belarbi and Hsu 1994 ; Zhang and Hsu 1998 ):
\( \sigma_{2}^{c} = \upbeta_{\sigma } f_{c}^{\prime } \left[ {2\left( {\frac{{\upvarepsilon_{2}^{c} }}{{\upbeta_{\upvarepsilon } \upvarepsilon_{o} }}} \right) - \left( {\frac{{\upvarepsilon_{2}^{c} }}{{\upbeta_{\upvarepsilon } \upvarepsilon_{o} }}} \right)^{2} } \right]{\text{ if }}\upvarepsilon_{2}^{c} \le \upbeta_{\upvarepsilon } \upvarepsilon_{o} \quad (13) \)  
\( \sigma_{2}^{c} = \upbeta_{\sigma } f_{c}^{\prime } \left[ {1 - \left( {\frac{{\upvarepsilon_{2}^{c} - \upbeta_{\upvarepsilon } \upvarepsilon_{o} }}{{2\upvarepsilon_{o} - \upbeta_{\upvarepsilon } \upvarepsilon_{o} }}} \right)^{2} } \right]{\text{ if }}\upvarepsilon_{2}^{c} > \upbeta_{\upvarepsilon } \upvarepsilon_{o} \quad (14) \)
\( \upbeta_{ * } = \upbeta_{\upsigma } = \upbeta_{\upvarepsilon } = \frac{{R\left( {f_{c}^{\prime } } \right)}}{{\sqrt {1 + \frac{{400\upvarepsilon_{1}^{c} }}{{\upeta^{\prime}}}} }}\quad (15) \)
\( \upeta = \frac{{\rho_{l} f_{ly} }}{{\rho_{t} f_{ty} \, }}\quad (16) \)
\( \left\{ {\begin{array}{*{20}l} {\upeta \le 1 \, \Rightarrow \, \upeta^{\prime} = \upeta } \\ {\upeta > 1 \, \Rightarrow \, \upeta^{\prime} = 1/\upeta } \\ \end{array} } \right.\quad (17) \)
\( R\left( {f_{c}^{\prime } } \right) = \frac{5.8}{{\sqrt {f_{c}^{\prime } \, \left( {\text{MPa}} \right)} \, }} \le 0.9\quad (18) \)
Concrete in tension (Jeng and Hsu 2009 ; Bernardo et al. 2013 ; Belarbi and Hsu 1994 ):
\( \upsigma_{1}^{c} = E_{c} \upvarepsilon_{1}^{c} {\text{ if }}\upvarepsilon_{1}^{c} \le \upvarepsilon_{cr} \quad (19) \)  
\( \sigma_{1}^{c} = f_{cr} \left( {\frac{{\upvarepsilon_{cr} }}{{\upvarepsilon_{2}^{c} }}} \right)^{0.4} {\text{ if }}\upvarepsilon_{1}^{c} > \upvarepsilon_{cr} \quad (20) \)  
\( k_{2}^{c} = \frac{{\upvarepsilon_{2s}^{c} }}{{\upbeta_{\upvarepsilon } \upvarepsilon_{o} }} - \frac{{(\upvarepsilon_{2s}^{c} )^{2} }}{{3(\upbeta_{\upvarepsilon } \upvarepsilon_{o} )^{2} }}\quad (21) \)  
\( E_{c} = 3875\;K\sqrt {f_{c}^{\prime } (\text{MPa} )} \quad (22) \)  
\( \varepsilon_{cr} = 0.00008\;K\quad K = 1.45{\text{ or }}1.24{\text{ (plain or hollow)}}\quad (23) \)  
Equations for stress in the concrete diagonal struts (Bernardo et al. 2015a ):
\( \sigma_{2}^{c} = k_{2}^{c} \upbeta_{\sigma } f_{c}^{\prime } \quad (24) \)
\( k_{2}^{c} = \frac{{\varepsilon_{2s}^{c} }}{{\upbeta_{\varepsilon } \varepsilon_{o} }} - \frac{{(\varepsilon_{2s}^{c} )^{2} }}{{3(\beta_{\varepsilon } \varepsilon_{o} )^{2} }}{\text{ if }}\varepsilon_{2s}^{c} \le \upbeta_{\varepsilon } \varepsilon_{o} \quad (25) \)
\( k_{2}^{c} = 1 - \frac{{\upbeta_{\varepsilon } \varepsilon_{o} }}{{3\varepsilon_{2s}^{c} }} - \frac{{(\varepsilon_{2s}^{c} - \upbeta_{\varepsilon } \varepsilon_{o} )^{3} }}{{3\varepsilon_{2s}^{c} (2\varepsilon_{o} - \upbeta_{\varepsilon } \varepsilon_{o} )^{2} }}{\text{ if }}\varepsilon_{2s}^{c} > \upbeta_{\varepsilon } \varepsilon_{o} \quad (26) \)
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Equations for stress in the concrete diagonal ties (Bernardo et al. 2015a ):
\( \sigma_{1}^{c} = k_{1}^{c} f_{cr} \quad (27) \)
\( k_{1}^{c} = \frac{{\varepsilon_{1s}^{c} }}{{2\varepsilon_{cr} }}{\text{ if }}\varepsilon_{1s}^{c} \le \varepsilon_{cr} \quad (28) \)
\( k_{1}^{c} = \frac{{\varepsilon_{cr} }}{{2\varepsilon_{1s}^{c} }} + \frac{{(\varepsilon_{cr} )^{0.4} }}{{0.6\varepsilon_{1s}^{c} }}[(\varepsilon_{1s}^{c} )^{0.6} - (\varepsilon_{cr} )^{0.6} ]{\text{ if }}\varepsilon_{1s}^{c} > \varepsilon_{cr} \quad (29) \)
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Non-prestress steel bars in tension (Belarbi and Hsu 1994 ):
\( f_{s} = \frac{{0.975E_{s} \varepsilon_{s} }}{{\left[ {1 + \left( {\frac{{1,1E_{s} \varepsilon_{s} }}{{f_{y} }}} \right)^{m} } \right]^{{\frac{1}{m}}} }} + 0.025E_{s} \varepsilon_{s} \quad (30) \)  
\( m = \frac{1}{9B - 0.2} \le 25\quad (31) \)  
\( B = \frac{1}{\rho }\left( {\frac{{f_{cr} }}{{f_{y} }}} \right)^{1.5} \quad (32) \)  
Prestress steel bars in tension (Hsu and Mo 1985b ):
\( f_{p} = E_{p} \varepsilon_{p} {\text{ if }}\varepsilon_{p} \le \varepsilon_{p0.1\% } = f_{p0.1\% } /E_{p} \quad (33) \)  
\( f_{p} = \frac{{E_{p} \varepsilon_{p} }}{{\left[ {1 + \left( {\frac{{E_{p} \varepsilon_{p} }}{{f_{pt} }}} \right)^{4.38} } \right]^{{\frac{1}{4.38}}} }}{\text{ if }}\varepsilon_{p} > \varepsilon_{p0.1\% } \quad (34) \)