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Table 4 Stress–strain models for FRP confined concrete.

From: Concentrically Loaded Circular RC Columns Partially Confined with FRP

Lam and Teng (2003) Pellegrino and Modena (2010)
\( - for \quad 0 \le \varepsilon_{c} \le \varepsilon_{t} , f_{c} = E_{c} \varepsilon_{c} - \frac{{\left( {E_{c} - E_{2} } \right)^{2} }}{{4 f^{\prime}_{c} }} \varepsilon_{c}^{2} \)
\( - for\quad \varepsilon_{t} \le \varepsilon_{c} , f_{c} = f^{\prime}_{c} + E_{2} \varepsilon_{c} \)
\( \varepsilon_{t} = \frac{{2 f^{\prime}_{c} }}{{E_{c} - E_{2} }} \); \( E_{2} = \frac{{f^{\prime}_{cc} - f^{\prime}_{c} }}{{\varepsilon_{ccu} }} \)
\( f_{cc}^{'} = f^{\prime}_{c} \left( {1 + 3.3 \frac{{f_{l} }}{{f^{\prime}_{c} }}} \right) \)
\( \varepsilon^{\prime}_{c} \left[ {1.75 + 12\left( {\frac{{f_{l} }}{{f^{\prime}_{c} }}} \right)\left( {\frac{{0.586.\varepsilon_{fu} }}{{\varepsilon^{\prime}_{c} }}} \right)^{0.45} } \right] \)
\( - for \quad 0 \le \varepsilon_{c} \le \varepsilon_{ccu} , f_{c} = \frac{{(E_{c} - E_{1} )\varepsilon_{c} }}{{\left[ {1 + \left( {\frac{{(E_{c} - E_{1} )\varepsilon_{c} }}{{f_{0} }}} \right)^{n} } \right]^{1/n} }} + E_{1} \varepsilon_{c} \)
\( n = 1 + \frac{1}{{({{E_{c} \varepsilon^{\prime}_{c} } \mathord{\left/ {\vphantom {{E_{c} \varepsilon^{\prime}_{c} } {f^{\prime}_{c} )}}} \right. \kern-0pt} {f^{\prime}_{c} )}} - 1}} \); \( f_{0} = f^{\prime}_{cc} - E_{1} \varepsilon_{ccu} \); \( E_{1} = \frac{{f^{\prime}_{cc} - f^{\prime}_{c} }}{{\varepsilon_{ccu} - \varepsilon^{\prime}_{c} }} \)
\( f_{cc}^{'} = f^{\prime}_{c} \left[ {1 + A\left( { \frac{{f_{l} }}{{f^{\prime}_{c} }}} \right)^{ - \alpha } \left( {\frac{{f_{l} }}{{f^{\prime}_{c} }}} \right)} \right] \)
\( \varepsilon_{ccu} = \varepsilon^{\prime}_{c} \left[ {2 + B\left( {\frac{{f_{l} }}{{f^{\prime}_{c} }}} \right)} \right] \)
\( A,B{\text{ and }}\alpha \, \) are coefficients defined in Tables 3 and 4, and 5 (Pellegrino and Modena 2010)
Lee et al. (2010) Proposed model (Ghanem and Harik)
\( \begin{array}{*{20}c} { - for} & {0 \le } \\ \end{array} \varepsilon_{c} \le \varepsilon^{\prime}_{c} { , }f_{c} = E_{c} \varepsilon_{c} + (f^{\prime}_{c} - E_{c} \varepsilon^{\prime}_{c} )\left( {\frac{{\varepsilon_{c} }}{{\varepsilon^{\prime}_{c} }}} \right)^{2} \)
\( \begin{array}{*{20}c} { - for} & {\varepsilon_{c}^{'} \le } \\ \end{array} \varepsilon_{c} \le \varepsilon_{c,s} { , }f_{c} = f^{\prime}_{c} + (f_{c,s} - f^{\prime}_{c} )\left( {\frac{{\varepsilon_{c} - \varepsilon^{\prime}_{c} }}{{\varepsilon_{c,s} - \varepsilon^{\prime}_{c} }}} \right)^{0.7} \)
\( \begin{array}{*{20}c} { - for} & {\varepsilon_{c,s}^{{}} \le } \\ \end{array} \varepsilon_{c} \le \varepsilon_{ccu} { , }f_{c} = f_{c,s} + (f^{\prime}_{cc} - f_{c,s} )\left( {\frac{{\varepsilon_{c} - \varepsilon_{c,s} }}{{\varepsilon_{ccu} - \varepsilon_{c,s} }}} \right)^{0.7} \)
\( \begin{array}{*{20}c} {\left. {\begin{array}{*{20}c} {\varepsilon_{c,s} = \varepsilon_{ccu} \left[ {0.85 + 0.03\left( {\frac{{f_{l,f,\hbox{max} } }}{{f_{l,s,\hbox{max} } }}} \right)} \right]} \\ {f_{c,s} = 0.95f^{\prime}_{cc} } \\ \end{array} } \right\}} & {f_{l,f,\hbox{max} } \ge f_{l,s,\hbox{max} } } \\ \end{array} \)
\( \begin{array}{*{20}c} {\left. {\begin{array}{*{20}c} {\varepsilon_{c,s} = 0.7\varepsilon_{ccu} } \\ {f_{c,s} = \left( {\frac{{\varepsilon_{c,s} }}{{\varepsilon_{ccu} }}} \right)^{0.4} f^{\prime}_{cc} } \\ \end{array} } \right\}} & {f_{l,f,\hbox{max} } < f_{l,s,\hbox{max} } } \\ \end{array} \)
\( k_{s} = \left\{ {\begin{array}{*{20}c} {2 - \frac{{f_{l,f,\hbox{max} } }}{{f_{l,s,\hbox{max} } }}} & {{\text{for }}f_{l,f,\hbox{max} } \le f_{l,s,\hbox{max} } } \\ 1 & {{\text{for }}f_{l,f,\hbox{max} } > f_{l,s,\hbox{max} } } \\ \end{array} } \right. \)
\( f_{cc}^{'} = f^{\prime}_{c} \left( {1 + 2 \frac{{f_{l} }}{{f^{\prime}_{c} }}} \right) \)
\( \varepsilon_{ccu}^{{}} = \varepsilon^{\prime}_{c} \left[ {1.75 + 5.25\left( {\frac{{f_{l,f,\hbox{max} } + k_{s} f_{l,s,\hbox{max} } }}{{f^{\prime}_{c} }}} \right)\left( {\frac{{\varepsilon_{fu} }}{{\varepsilon^{\prime}_{c} }}} \right)^{0.45} } \right] \)
\( - for ,\quad 0 \le \varepsilon_{c} \le \varepsilon_{c,s} , { }f_{c} = \frac{{(E_{c} - E_{1} )\varepsilon_{c} }}{{\left[ {1 + \left( {\frac{{(E_{c} - E_{1} )\varepsilon_{c} }}{{f_{0} }}} \right)^{n} } \right]^{1/n} }} + E_{1} \varepsilon_{c}^{m} \)
\( - \, for \, \varepsilon_{c,s} \le \varepsilon_{c} \le \varepsilon_{ccu} ,\,f_{c} = f_{c,s} + E_{2} (\varepsilon_{c} - \varepsilon_{c,s} ) \)
\( E_{1} = \frac{{f_{c,s} - f_{0} }}{{\varepsilon_{c,s} }};\,E_{2} = \frac{{f^{\prime}_{cc} - f_{c,s} }}{{\varepsilon_{ccu} - \varepsilon_{c,s} }} \)
\( n = 1 + \frac{1}{{({{E_{c} \varepsilon^{\prime}_{c} } \mathord{\left/ {\vphantom {{E_{c} \varepsilon^{\prime}_{c} } {f^{\prime}_{c} )}}} \right. \kern-0pt} {f^{\prime}_{c} )}} - 1}} \)
\( m = \left[ {\frac{1}{{\ln (\varepsilon_{c,s} )}}} \right]\left\{ {\ln \left[ {\frac{1}{{E_{1} }}\left( {f_{c,s} - \frac{{(E_{c} - E_{1} )\varepsilon_{c,s} }}{{\left\{ {1 + \left[ {\frac{{(E_{c} - E_{1} )\varepsilon_{c,s} }}{{f_{0} }}} \right]^{n} } \right\}^{1/n} }}} \right)} \right]} \right\} \)
\( f_{c,s} = \frac{{f_{core} A_{core} + f_{{\text{cov} er}} A_{{\text{cov} er}} }}{{A_{g} }} \)
\( \varepsilon_{c,s} = 0.85\varepsilon^{\prime}_{c} \left( {1 + 8\frac{{(f_{l,fy} + f^{\prime}_{l,s,\hbox{max} } )}}{{f^{\prime}_{c} }}} \right).\left\{ {\left[ {1 + 0.75\left( {\frac{{\varepsilon_{l,y} }}{{\varepsilon^{\prime}_{c} }}} \right)} \right]^{0.7} - \exp \left[ { - 7\left( {\frac{{\varepsilon_{l,y} }}{{\varepsilon^{\prime}_{c} }}} \right)} \right]} \right\} \)
\( f_{cc}^{'} = f^{\prime}_{c} \left[ {1 + 1.55\left( {\frac{{f_{l,f,\hbox{max} } }}{{f^{\prime}_{c} }}} \right)\left( {\frac{{N_{f} w_{f} }}{{l_{u} }}} \right)^{0.3} + 1.55\left( {\frac{{f_{l,s,\hbox{max} } }}{{f^{\prime}_{c} }}} \right)} \right] \)
\( \varepsilon_{ccu}^{{}} = \varepsilon^{\prime}_{c} \left[ {2.4 + 15\left( {\frac{{f_{l,f,\hbox{max} } }}{{f^{\prime}_{c} }}} \right)\left( {\frac{{N_{f} w_{f} }}{{l_{u} }}} \right)^{0.3} + 7.7\left( {\frac{{f_{l,s,\hbox{max} } }}{{f^{\prime}_{c} }}} \right)} \right] \)