1 Introduction
2 Finite Element Modeling
Material | Parameter | FE element | |||
---|---|---|---|---|---|
Concrete |
\( f^{\prime}_{c} \)
| 20.68 MPa | 3 ksi | SOLID 65 | |
55.16 MPa | 8 ksi | ||||
\( E_{c} \)
| 21.50 GPa | 3118 ksi | |||
35.13 GPa | 5095 ksi | ||||
Steel |
\( f_{y} \)
| 413.68 MPa | 60 ksi | LINK 180 | |
\( A_{s} \)
| \( \phi \) 6 | 32 mm2 | 0.05 in2 | ||
\( \phi \) 10 | 71 mm2 | 0.11 in2 | |||
FRP |
\( t_{f} \)
| 0.15 mm | 0.0059 in | SHELL 181 | |
\( f_{fu} \)
| 2848 MPa | 413 ksi | |||
\( E_{f} \)
| 139 GPa | 20,160 ksi |
3 Partial FRP Wraps or Strips
Group#a |
\( f^{\prime}_{c} \)
| Longitudinal steel \( \phi \) 10 mm (#3) | Transverse steel stirrups \( \phi \) 6 mm (#2) | ||||
---|---|---|---|---|---|---|---|
MPa | ksi | Number of bars |
\( \rho_{sl} \)
| Spacing |
\( \rho_{st} \)
| ||
mm | in | ||||||
1
| 20.68 | 3 | 4 | 0.011 | 140 | 5.50 | 0.004 |
2
| 55.16 | 8 | 4 | 0.011 | 140 | 5.50 | 0.004 |
3
| 20.68 | 3 | 4 | 0.011 | 80 | 3.15 | 0.0064 |
4
| 20.68 | 3 | 12 | 0.027 | 140 | 5.50 | 0.004 |
Group#a | Columnsa | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
UW | S1 | S2 | S3 | S4 | S5 | S6 | S7 | FW | ||
1 | \( f^{\prime}_{cc} \) (MPa) | 22.13 | 22.32 | 23.51 | 26.22 | 27.81 | 29.23 | 33.18 | 35.24 | 48.80 |
\( \varepsilon_{ccu} \)
| 0.0025 | 0.0068 | 0.0076 | 0.0102 | 0.0120 | 0.0126 | 0.0171 | 0.0195 | 0.0303 | |
\( \varepsilon_{lu} \)
| – | 0.003 | 0.0035 | 0.0055 | 0.0061 | 0.0070 | 0.0089 | 0.0102 | 0.0179 | |
\( f^{\prime}_{cc} /f^{\prime}_{c} \)
| 1.079 | 1.070 | 1.137 | 1.268 | 1.345 | 1.414 | 1.604 | 1.704 | 2.360 | |
\( \varepsilon_{ccu} /\varepsilon^{\prime}_{c} \)
| 1.266 | 3.390 | 3.808 | 5.088 | 5.995 | 6.321 | 8.562 | 9.741 | 15.166 | |
\( \mu \)
| 5.744 | 6.357 | 6.952 | 8.422 | 9.215 | 9.219 | 10.269 | 11.266 | 14.327 | |
2 | \( f^{\prime}_{cc} \) (MPa) | 55.20 | 55.22 | 57.52 | 28.35 | 59.86 | 62.35 | 63.39 | 68.06 | 78.27 |
\( \varepsilon_{ccu} \)
| 0.0028 | 0.0058 | 0.0065 | 0.0066 | 0.0074 | 0.0098 | 0.0104 | 0.0135 | 0.0154 | |
\( \varepsilon_{lu} \)
| – | 0.0021 | 0.0025 | 0.0029 | 0.0032 | 0.0044 | 0.0045 | 0.0068 | 0.0077 | |
\( f^{\prime}_{cc} /f^{\prime}_{c} \)
| 1.001 | 1.001 | 1.043 | 1.058 | 1.085 | 1.130 | 1.149 | 1.234 | 1.419 | |
\( \varepsilon_{ccu} /\varepsilon^{\prime}_{c} \)
| 1.083 | 2.267 | 2.534 | 2.580 | 2.903 | 3.840 | 4.062 | 5.273 | 6.036 | |
\( \mu \)
| 3.53 | 3.552 | 3.970 | 4.041 | 6.177 | 6.762 | 6.915 | 7.087 | 7.339 | |
3 | \( f^{\prime}_{cc} \) (MPa) | 23.89 | 24.55 | 24.78 | 26.65 | 28.80 | 28.83 | 32.04 | 35.99 | 49.79 |
\( \varepsilon_{ccu} \)
| 0.0051 | 0.0066 | 0.0070 | 0.0083 | 0.0084 | 0.0106 | 0.0157 | 0.0173 | 0.0294 | |
\( \varepsilon_{lu} \)
| – | 0.0023 | 0.0028 | 0.0035 | 0.0044 | 0.0052 | 0.0077 | 0.0096 | 0.0178 | |
\( f^{\prime}_{cc} /f^{\prime}_{c} \)
| 1.155 | 1.187 | 1.198 | 1.289 | 1.393 | 1.394 | 1.549 | 1.740 | 2.408 | |
\( \varepsilon_{ccu} /\varepsilon^{\prime}_{c} \)
| 2.554 | 3.313 | 3.491 | 4.129 | 4.194 | 5.309 | 7.841 | 8.650 | 14.717 | |
\( \mu \)
| 5.674 | 5.796 | 5.813 | 6.446 | 6.983 | 7.315 | 9.271 | 9.765 | 13.426 | |
4 | \( f^{\prime}_{cc} \) (MPa) | 22.2 | 22.38 | 25.02 | 26.22 | 27.31 | 29.89 | 32.54 | 36.16 | 48.49 |
\( \varepsilon_{ccu} \)
| 0.0028 | 0.0066 | 0.0072 | 0.0096 | 0.0108 | 0.0122 | 0.0150 | 0.0193 | 0.0307 | |
\( \varepsilon_{lu} \)
| – | 0.0031 | 0.0033 | 0.0053 | 0.0056 | 0.0067 | 0.0088 | 0.0113 | 0.0187 | |
\( f^{\prime}_{cc} /f^{\prime}_{c} \)
| 1.074 | 1.082 | 1.210 | 1.268 | 1.320 | 1.445 | 1.573 | 1.749 | 2.345 | |
\( \varepsilon_{ccu} /\varepsilon^{\prime}_{c} \)
| 1.384 | 3.323 | 3.618 | 4.795 | 5.391 | 6.082 | 7.527 | 9.637 | 15.374 | |
\( \mu \)
| 5.972 | 6.205 | 6.231 | 7.918 | 8.415 | 9.206 | 10.026 | 10.697 | 14.155 |
3.1 FRP Volumetric Ratio, ρf
3.2 Unconfined Concrete Compressive Strength, \( f^{\prime}_{c} \)
3.3 Transverse Steel Reinforcement Ratio, ρst
3.4 Longitudinal Steel Reinforcement Ratio, ρsl
3.5 Strip Arrangement
4 Current FRP Confined Concrete Stress–Strain Models
Lam and Teng (2003) | Pellegrino and Modena (2010) |
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\( - for \quad 0 \le \varepsilon_{c} \le \varepsilon_{t} , f_{c} = E_{c} \varepsilon_{c} - \frac{{\left( {E_{c} - E_{2} } \right)^{2} }}{{4 f^{\prime}_{c} }} \varepsilon_{c}^{2} \)
\( - for\quad \varepsilon_{t} \le \varepsilon_{c} , f_{c} = f^{\prime}_{c} + E_{2} \varepsilon_{c} \)
\( \varepsilon_{t} = \frac{{2 f^{\prime}_{c} }}{{E_{c} - E_{2} }} \); \( E_{2} = \frac{{f^{\prime}_{cc} - f^{\prime}_{c} }}{{\varepsilon_{ccu} }} \)
\( f_{cc}^{'} = f^{\prime}_{c} \left( {1 + 3.3 \frac{{f_{l} }}{{f^{\prime}_{c} }}} \right) \)
\( \varepsilon^{\prime}_{c} \left[ {1.75 + 12\left( {\frac{{f_{l} }}{{f^{\prime}_{c} }}} \right)\left( {\frac{{0.586.\varepsilon_{fu} }}{{\varepsilon^{\prime}_{c} }}} \right)^{0.45} } \right] \)
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\( - for \quad 0 \le \varepsilon_{c} \le \varepsilon_{ccu} , f_{c} = \frac{{(E_{c} - E_{1} )\varepsilon_{c} }}{{\left[ {1 + \left( {\frac{{(E_{c} - E_{1} )\varepsilon_{c} }}{{f_{0} }}} \right)^{n} } \right]^{1/n} }} + E_{1} \varepsilon_{c} \)
\( n = 1 + \frac{1}{{({{E_{c} \varepsilon^{\prime}_{c} } \mathord{\left/ {\vphantom {{E_{c} \varepsilon^{\prime}_{c} } {f^{\prime}_{c} )}}} \right. \kern-0pt} {f^{\prime}_{c} )}} - 1}} \); \( f_{0} = f^{\prime}_{cc} - E_{1} \varepsilon_{ccu} \); \( E_{1} = \frac{{f^{\prime}_{cc} - f^{\prime}_{c} }}{{\varepsilon_{ccu} - \varepsilon^{\prime}_{c} }} \)
\( f_{cc}^{'} = f^{\prime}_{c} \left[ {1 + A\left( { \frac{{f_{l} }}{{f^{\prime}_{c} }}} \right)^{ - \alpha } \left( {\frac{{f_{l} }}{{f^{\prime}_{c} }}} \right)} \right] \)
\( \varepsilon_{ccu} = \varepsilon^{\prime}_{c} \left[ {2 + B\left( {\frac{{f_{l} }}{{f^{\prime}_{c} }}} \right)} \right] \)
\( A,B{\text{ and }}\alpha \, \) are coefficients defined in Tables 3 and 4, and 5 (Pellegrino and Modena 2010) |
Lee et al. (2010) | Proposed model (Ghanem and Harik) |
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\( \begin{array}{*{20}c} { - for} & {0 \le } \\ \end{array} \varepsilon_{c} \le \varepsilon^{\prime}_{c} { , }f_{c} = E_{c} \varepsilon_{c} + (f^{\prime}_{c} - E_{c} \varepsilon^{\prime}_{c} )\left( {\frac{{\varepsilon_{c} }}{{\varepsilon^{\prime}_{c} }}} \right)^{2} \)
\( \begin{array}{*{20}c} { - for} & {\varepsilon_{c}^{'} \le } \\ \end{array} \varepsilon_{c} \le \varepsilon_{c,s} { , }f_{c} = f^{\prime}_{c} + (f_{c,s} - f^{\prime}_{c} )\left( {\frac{{\varepsilon_{c} - \varepsilon^{\prime}_{c} }}{{\varepsilon_{c,s} - \varepsilon^{\prime}_{c} }}} \right)^{0.7} \)
\( \begin{array}{*{20}c} { - for} & {\varepsilon_{c,s}^{{}} \le } \\ \end{array} \varepsilon_{c} \le \varepsilon_{ccu} { , }f_{c} = f_{c,s} + (f^{\prime}_{cc} - f_{c,s} )\left( {\frac{{\varepsilon_{c} - \varepsilon_{c,s} }}{{\varepsilon_{ccu} - \varepsilon_{c,s} }}} \right)^{0.7} \)
\( \begin{array}{*{20}c} {\left. {\begin{array}{*{20}c} {\varepsilon_{c,s} = \varepsilon_{ccu} \left[ {0.85 + 0.03\left( {\frac{{f_{l,f,\hbox{max} } }}{{f_{l,s,\hbox{max} } }}} \right)} \right]} \\ {f_{c,s} = 0.95f^{\prime}_{cc} } \\ \end{array} } \right\}} & {f_{l,f,\hbox{max} } \ge f_{l,s,\hbox{max} } } \\ \end{array} \)
\( \begin{array}{*{20}c} {\left. {\begin{array}{*{20}c} {\varepsilon_{c,s} = 0.7\varepsilon_{ccu} } \\ {f_{c,s} = \left( {\frac{{\varepsilon_{c,s} }}{{\varepsilon_{ccu} }}} \right)^{0.4} f^{\prime}_{cc} } \\ \end{array} } \right\}} & {f_{l,f,\hbox{max} } < f_{l,s,\hbox{max} } } \\ \end{array} \)
\( k_{s} = \left\{ {\begin{array}{*{20}c} {2 - \frac{{f_{l,f,\hbox{max} } }}{{f_{l,s,\hbox{max} } }}} & {{\text{for }}f_{l,f,\hbox{max} } \le f_{l,s,\hbox{max} } } \\ 1 & {{\text{for }}f_{l,f,\hbox{max} } > f_{l,s,\hbox{max} } } \\ \end{array} } \right. \)
\( f_{cc}^{'} = f^{\prime}_{c} \left( {1 + 2 \frac{{f_{l} }}{{f^{\prime}_{c} }}} \right) \)
\( \varepsilon_{ccu}^{{}} = \varepsilon^{\prime}_{c} \left[ {1.75 + 5.25\left( {\frac{{f_{l,f,\hbox{max} } + k_{s} f_{l,s,\hbox{max} } }}{{f^{\prime}_{c} }}} \right)\left( {\frac{{\varepsilon_{fu} }}{{\varepsilon^{\prime}_{c} }}} \right)^{0.45} } \right] \)
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\( - for ,\quad 0 \le \varepsilon_{c} \le \varepsilon_{c,s} , { }f_{c} = \frac{{(E_{c} - E_{1} )\varepsilon_{c} }}{{\left[ {1 + \left( {\frac{{(E_{c} - E_{1} )\varepsilon_{c} }}{{f_{0} }}} \right)^{n} } \right]^{1/n} }} + E_{1} \varepsilon_{c}^{m} \)
\( - \, for \, \varepsilon_{c,s} \le \varepsilon_{c} \le \varepsilon_{ccu} ,\,f_{c} = f_{c,s} + E_{2} (\varepsilon_{c} - \varepsilon_{c,s} ) \)
\( E_{1} = \frac{{f_{c,s} - f_{0} }}{{\varepsilon_{c,s} }};\,E_{2} = \frac{{f^{\prime}_{cc} - f_{c,s} }}{{\varepsilon_{ccu} - \varepsilon_{c,s} }} \)
\( n = 1 + \frac{1}{{({{E_{c} \varepsilon^{\prime}_{c} } \mathord{\left/ {\vphantom {{E_{c} \varepsilon^{\prime}_{c} } {f^{\prime}_{c} )}}} \right. \kern-0pt} {f^{\prime}_{c} )}} - 1}} \)
\( m = \left[ {\frac{1}{{\ln (\varepsilon_{c,s} )}}} \right]\left\{ {\ln \left[ {\frac{1}{{E_{1} }}\left( {f_{c,s} - \frac{{(E_{c} - E_{1} )\varepsilon_{c,s} }}{{\left\{ {1 + \left[ {\frac{{(E_{c} - E_{1} )\varepsilon_{c,s} }}{{f_{0} }}} \right]^{n} } \right\}^{1/n} }}} \right)} \right]} \right\} \)
\( f_{c,s} = \frac{{f_{core} A_{core} + f_{{\text{cov} er}} A_{{\text{cov} er}} }}{{A_{g} }} \)
\( \varepsilon_{c,s} = 0.85\varepsilon^{\prime}_{c} \left( {1 + 8\frac{{(f_{l,fy} + f^{\prime}_{l,s,\hbox{max} } )}}{{f^{\prime}_{c} }}} \right).\left\{ {\left[ {1 + 0.75\left( {\frac{{\varepsilon_{l,y} }}{{\varepsilon^{\prime}_{c} }}} \right)} \right]^{0.7} - \exp \left[ { - 7\left( {\frac{{\varepsilon_{l,y} }}{{\varepsilon^{\prime}_{c} }}} \right)} \right]} \right\} \)
\( f_{cc}^{'} = f^{\prime}_{c} \left[ {1 + 1.55\left( {\frac{{f_{l,f,\hbox{max} } }}{{f^{\prime}_{c} }}} \right)\left( {\frac{{N_{f} w_{f} }}{{l_{u} }}} \right)^{0.3} + 1.55\left( {\frac{{f_{l,s,\hbox{max} } }}{{f^{\prime}_{c} }}} \right)} \right] \)
\( \varepsilon_{ccu}^{{}} = \varepsilon^{\prime}_{c} \left[ {2.4 + 15\left( {\frac{{f_{l,f,\hbox{max} } }}{{f^{\prime}_{c} }}} \right)\left( {\frac{{N_{f} w_{f} }}{{l_{u} }}} \right)^{0.3} + 7.7\left( {\frac{{f_{l,s,\hbox{max} } }}{{f^{\prime}_{c} }}} \right)} \right] \)
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