Table 4 Constitutive models for reinforcing bar and concrete.

Reinforcing bar

\(\sigma_{s} = \left\{ {\begin{array}{*{20}c} {E_{s} \varepsilon_{s} } & {{\text{for}}\left| {\varepsilon_{s} } \right| \le \varepsilon_{yd} } \\ {f_{yd} + E_{h} \left( {\varepsilon_{s} - \varepsilon_{yd} < 1.25f_{yd} } \right)} & {{\text{for}}\left| {\varepsilon_{s} } \right| > \varepsilon_{yd} } \\ \end{array} } \right.\)

\(f_{yd} = f_{y} + 6\ln (10^{5} \left| {\dot{\varepsilon }} \right|/5) \le f_{y} + 6\ln (2 \times 10^{5} ){\text{ (CEB)}}\)

Confined concrete under compression

\(\sigma_{cc} = \left\{ {\begin{array}{*{20}c} {kf{\prime}_{cd} \left[ {\frac{{2\varepsilon_{c} }}{{\varepsilon_{cod} K}} + \left( {\left[ {\frac{{\varepsilon_{c} }}{{\varepsilon_{cod} K}} + } \right]} \right)^{2} } \right]} & {{\text{for }}\varepsilon_{c} \ge - \varepsilon_{cod} K} \\ {kf{\prime}_{cd} \left[ {1 + Z_{m} \left( {\varepsilon_{c} + \varepsilon_{cod} K} \right)} \right] \ge 0.2f{\prime}_{cd} } & {{\text{for }}\varepsilon_{c} < - \varepsilon_{cod} K} \\ \end{array} } \right.\)

\(K = 1 + \rho_{t} f_{yt} /f_{cd}^{'}\)

\(Z_{m} = \frac{0.5}{{\frac{{3 + 0.29f_{cd}^{'} }}{{145f_{cd}^{'} - 1000}} + \frac{3}{4}\rho_{t} \sqrt {\frac{{h_{0} }}{s}} - \varepsilon_{cod} K}}\)

\(f_{cd}^{'} = \left\{ {\begin{array}{*{20}c} {f{\prime}_{c} \left( {10^{5} \left| {\dot{\varepsilon }/3} \right|} \right)^{0.014} } & {{\text{for }}\dot{\varepsilon }_{c} {\text{ < 30/s }}} \\ {0.012f^{\prime}\left( {10^{5} \left| {\dot{\varepsilon }/3} \right|} \right)^{01/3} } & {{\text{for }}\dot{\varepsilon }_{c} \ge 3 0 / {\text{s }}} \\ \end{array} } \right.{ (}fib{ 2010) }\)

Unconfined concrete under compression

\(\sigma_{cu} = f_{cd}^{'} \left[ {\frac{{2\varepsilon_{c} }}{{\varepsilon_{cod} }} + \left( {\frac{{\varepsilon_{c} }}{{\varepsilon_{cod} }}} \right)^{2} } \right]{\text{ for }} - \varepsilon_{cu} \le \varepsilon_{c} \le 0\)

Concrete under tension

\(\sigma_{t} = E_{cd} \varepsilon_{c} ,0 < \varepsilon_{c} \le f_{td}^{'} /E_{cd}\)

\(f_{td} = \left\{ {\begin{array}{*{20}c} {f{\prime}_{c} \left( {10^{6} \left| {\dot{\varepsilon }} \right|} \right)^{0.018} } & {{\text{for }}\dot{\varepsilon }_{c} {\text{ < 10/s }}} \\ {0.0062f_{t} \left( {10^{6} \left| {\dot{\varepsilon }} \right|} \right)^{1/3} } & {{\text{for }}\dot{\varepsilon }_{c} \ge 1 0 / {\text{s }}} \\ \end{array} } \right.{ (}fib{ 2010) }\)

\(f_{t} = \left\{ {\begin{array}{*{20}c} {0.3f_{c}^{{'2/3}} \quad \quad \quad \quad \quad \quad \quad {\text{for }}f_{c}^{'} < 50{\text{MPa}}} \\ {2.12\ln [1 + 0.1(f_{c}^{'} + 8)]{\text{ }}\;\; {\text{for }}f_{c}^{'} \ge 50{\text{MPa}}} \\ \end{array} } \right.(fib{\text{ }}2010)\)