- Original article
- Open Access

# Concentrically Loaded Circular RC Columns Partially Confined with FRP

- Sahar Y. Ghanem
^{1}Email author and - Issam E. Harik
^{2}

**12**:52

https://doi.org/10.1186/s40069-018-0283-2

© The Author(s) 2018

**Received:**13 March 2017**Accepted:**28 May 2018**Published:**30 July 2018

## Abstract

Wrapping reinforced concrete (RC) columns with fiber reinforced polymer (FRP) composites is effective in increasing their capacity. The current state of the art concentrates primarily on fully wrapped columns and few studies dealt with partially wrapped ones. The objective herein is to evaluate the effectiveness of partial wraps (or strips) and to develop a confined concrete compressive stress–strain (*f*_{c} − *ε*_{c}) model that accounts for partial wrapping. Three-dimensional finite element (FE) models are generated to evaluate the influence of different parameters on the behavior of concentrically loaded RC circular columns that are partially and fully wrapped with FRP. The results indicated an increase in ductility as the number of FRP strips is increased, and revealed that longitudinal steel had little influence on the confined *f*_{c} − *ε*_{c} relationship. The proposed *f*_{c} − *ε*_{c} model, derived from the parametric study, accounts for the effect of partial and full confinement, the unconfined concrete strength \( f^{\prime}_{c} \), and yielding of transverse steel. Comparison of the results generated using the proposed model with FE and experimental results are in good agreement.

## Keywords

- reinforced concrete
- columns
- confinement
- stress–strain
- finite element
- ductility

## 1 Introduction

The application of fiber reinforced polymer (FRP) wraps to reinforced concrete (RC) columns is an established and efficient technique for enhancing the capacity of columns. Columns that are fully wrapped with FRP showed an increase in ductility, moment and ultimate compressive load capacity, ultimate deformability and energy absorption compared to unconfined columns (Mirmiran and Shahawy 1997; Toutanji 1999). Several studies focusing on fully wrapped FRP confined concrete columns have been carried out to generate models for predicting their behavior (Nanni and Bradford 1995; Lam and Teng 2003; El Fattah and Rasheed 2015). Research on columns partially wrapped with FRP sheets (or strips) is very limited (Saadatmanesh et al. 1994; Barros and Ferreira 2008; Wu et al. 2009). Most of the studies did not account for the influence of the existing steel reinforcement on the column’s behavior (Lam and Teng 2003), or simply estimate the total confinement pressure as the sum of the confinement pressure due to the external FRP jacketing and the confinement pressure due to the internal transverse steel reinforcement (Barros and Ferreira 2008; El Fattah and Rasheed 2015). Few models dealt with concrete confined by both FRP and transverse steel (Lee et al. 2010).

The focus of this paper is to better understand the influence of internal steel reinforcement and partial or full external FRP wrap/reinforcement on concrete confinement. A series of finite element (FE) models are developed to analyze the effect of the aforementioned parameters on the confined concrete column. FE models have been successfully used to simulate the behavior of RC beams (Hawileh et al. 2012; Hawileh et al. 2013) and columns (Mirmiran et al. 2000) wrapped by FRP sheets. The influence of partial wrapping on the increase in strength and ductility is evaluated. The results from the FE parametric analyses were used to derive a new confined concrete compressive stress–strain model for concentrically loaded RC circular columns that are partially and fully wrapped with FRP.

## 2 Finite Element Modeling

*f*

_{c}, is calculated by averaging stresses across the entire column cross section. Perfect bond is assumed between the FRP sheets, and FRP and concrete.

Material properties and FE elements for the control (or unwrapped) column and the eight columns in Fig. 2.

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 mm | 0.05 in | ||

\( \phi \) 10 | 71 mm | 0.11 in | |||

FRP | \( t_{f} \) | 0.15 mm | 0.0059 in | SHELL 181 | |

\( f_{fu} \) | 2848 MPa | 413 ksi | |||

\( E_{f} \) | 139 GPa | 20,160 ksi |

*l*

_{u}= 600 mm (23.62 in.) and diameter

*D*= 200 mm (7.9 in.). The 28 day compressive strength of unconfined concrete \( f^{\prime}_{c} \) = 16 MPa (2.32 ksi). The tests were carried out under displacement control at a rate of 1 mm/min. Figure 1 shows the FRP wrapping arrangements and columns cross section.

## 3 Partial FRP Wraps or Strips

*l*

_{u}= 600 mm (23.62 in.) and diameter

*D*= 200 mm (7.9 in.). One column is unwrapped and is the baseline column. The other eight columns are presented in Fig. 2, one of the columns is fully wrapped (FW) and the remaining seven are partially wrapped with strips varying from one strip (

*N*

_{f}= 1) on column S1 to seven strips on column S7 (

*N*

_{f}= 7). Each strip has a width

*w*

_{f}= 40 mm (1.6 in.). For the fully wrapped column,

*w*

_{f}=

*l*

_{u}and

*N*

_{f}= 1. The full wrap and each strip has four layers of CFRP fabric (

*n*

_{f}= 4), and the thickness of each layer

*t*

_{f}= 0.15 mm (0.0059 in.). The FRP volumetric ratio (

*ρ*

_{f}) for each column is determined as follows

*f*

_{c}), axial strain (

*ε*

_{c}) and lateral strain (

*ε*

_{l}). In addition to the unwrapped column, each group contains the eight columns in Fig. 2, and Group 1 is the baseline group. In Groups 2–4, three different parameters are varied: the 28 day compressive strength of unconfined concrete \( f^{\prime}_{c} \), the transverse steel reinforcement ratio,

*ρ*

_{st}, and the longitudinal steel reinforcement ratio,

*ρ*

_{sl}, respectively (Eq. 2)

*V*

_{st}is the volume of transverse steel;

*V*

_{c}is the volume of concrete;

*A*

_{sl}is the total area of longitudinal steel; and

*A*

_{g}is the gross area of the column section.

Column groups used in the parametric study.

Group# | \( f^{\prime}_{c} \) | 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 | ||||||

| 20.68 | 3 | 4 | 0.011 | 140 | 5.50 | 0.004 |

| 55.16 | 8 | 4 | 0.011 | 140 | 5.50 | 0.004 |

| 20.68 | 3 | 4 | 0.011 | 80 | 3.15 | 0.0064 |

| 20.68 | 3 | 12 | 0.027 | 140 | 5.50 | 0.004 |

*f*

_{c}) versus concrete axial strain (

*ε*

_{c}) and concrete lateral strain (

*ε*

_{l}).

*µ*). \( f^{\prime}_{cc} \) is the ultimate confined concrete compressive stress;

*ε*

_{ccu}is the ultimate confined concrete axial strain corresponding to the ultimate confined concrete compressive stress; \( \varepsilon^{\prime}_{c} \) is the concrete axial strain at the unconfined concrete compressive strength \( \left( {f^{\prime}_{c} } \right) \). The numerical values of these terms are listed in Table 3 for all Groups. The derivation of the ductility factor is based on the one proposed by Cui and Sheikh (2010).

Stresses, strains and ductility factors for the unwrapped columns (UW) and columns in Groups 1–4 (1 MPa = 0.145 ksi).

Group# | Columns | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

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 |

*β*, are introduced in Eq. 3 and Fig. 4 to compare the fully wrapped (FW) columns to unwrapped columns (UW) in each group.

For Fig. 4, refer to Table 2 for information on Groups 1 to 4, and to Fig. 2 for *ρ*_{f} values

### 3.1 FRP Volumetric Ratio, *ρ*
_{f}

In all column groups, as the FRP volumetric ratio in Eq. 1 increases from *ρ*_{f} = 0.0 for the unwrapped column to *ρ*_{f}= 0.012 for the fully wrapped column (Fig. 3), and as expected, there is an increase in the ultimate confined concrete compressive stress \( \left( {f^{\prime}_{cc} } \right) \), the ultimate confined axial strain (*ε*_{ccu}), and the ultimate lateral strain of confined concrete (*ε*_{lu}). Figure 4 and Table 3 provide more detailed results that clearly show the influence of the number of strips in Groups 1–4 on the strength ratio \( \left( {{{f^{\prime}_{cc} } \mathord{\left/ {\vphantom {{f^{\prime}_{cc} } {f^{\prime}_{c} }}} \right. \kern-0pt} {f^{\prime}_{c} }}} \right) \), strain ratio \( \left( {{{\varepsilon_{ccu} } \mathord{\left/ {\vphantom {{\varepsilon_{ccu} } {\varepsilon^{\prime}_{c} }}} \right. \kern-0pt} {\varepsilon^{\prime}_{c} }}} \right) \), and ductility factor (*µ*). As the number of strips is increased, the aforementioned ratios also increase.

### 3.2 Unconfined Concrete Compressive Strength, \( f^{\prime}_{c} \)

*ε*

_{ccu}) as the unconfined concrete compressive strength \( f^{\prime}_{c} \) is increased. All nine columns in Group 1 (Figs. 3a and 4a, and Table 3) have strength ratios \( \left( {{{f^{\prime}_{cc} } \mathord{\left/ {\vphantom {{f^{\prime}_{cc} } {f^{\prime}_{c} }}} \right. \kern-0pt} {f^{\prime}_{c} }}} \right) \), strain ratios \( \left( {{{\varepsilon_{ccu} } \mathord{\left/ {\vphantom {{\varepsilon_{ccu} } {\varepsilon^{\prime}_{c} }}} \right. \kern-0pt} {\varepsilon^{\prime}_{c} }}} \right) \), ductility factors (

*µ*), and efficiency factors,

*β*(Eq. 3), larger than the one for the corresponding columns in Group 2 (Figs. 3b and 4b, and Table 3). For column S1, the concrete stress–strain (

*f*

_{c}−

*ε*

_{c}) relationship is presented in Fig. 5a for Groups 1–4 to show the influence of the different parameters.

### 3.3 Transverse Steel Reinforcement Ratio, *ρ*
_{st}

The influence of *ρ*_{st} is studied by comparing the columns in Group 1 (Fig. 3a and Table 3), with *ρ*_{st} = 0.004, and Group 3 (Fig. 3c and Table 3) with *ρ*_{st} = 0.0064. Except for columns S5 and S6, the columns in Group 1 have a lower ultimate confined concrete compressive stress \( f^{\prime}_{cc} \). For columns S5 and S6 in Group 3, there is an overlap between the FRP and transverse steel leading to a decrease in the volume of confined concrete, and in turn to a lower \( f^{\prime}_{cc} \). The columns in Group 1 also have a higher ultimate confined concrete axial strain, *ε*_{ccu}, compared to the ones in Group 3 (Table 3). The ductility factor, *μ*, and the efficiency factors, *β* in Eq. 3, are reduced by increasing *ρ*_{st} (Table 3 and Fig. 4). The influence of increasing *ρ*_{st} on \( f^{\prime}_{cc} \) can be seen in Fig. 5b for the column with one strip (S1), in Fig. 6a for the column with four strips (S4), and in Fig. 6b for the fully wrapped column.

### 3.4 Longitudinal Steel Reinforcement Ratio, *ρ*
_{sl}

The effect of *ρ*_{sl} is studied by comparing the columns in Group 1 (*ρ*_{sl}= 0.011) and Group 4 (*ρ*_{sl}= 0.027) in Figs. 4a, d, 5b and 6 and Table 3. At ultimate conditions, the columns in Group 1 have a slightly lower ultimate confined concrete compressive stress and higher ultimate axial concrete strain (Table 3). The efficiency factors, *β* (Eq. 3), for Group 1 are slightly larger than the ones for Group 4 (Fig. 4a and 4d). In general, the change in the strength ratios \( \left( {{{f^{\prime}_{cc} } \mathord{\left/ {\vphantom {{f^{\prime}_{cc} } {f^{\prime}_{c} }}} \right. \kern-0pt} {f^{\prime}_{c} }}} \right) \), strain ratios \( \left( {{{\varepsilon_{ccu} } \mathord{\left/ {\vphantom {{\varepsilon_{ccu} } {\varepsilon^{\prime}_{c} }}} \right. \kern-0pt} {\varepsilon^{\prime}_{c} }}} \right) \), and ductility factors (*µ*), due to the increase in the longitudinal steel reinforcement ratio has a little influence on concrete confinement for the columns under consideration (Figs. 4a and d, 5b and 6). Figures 5 and 6 clearly show that the *f*_{c} − *ε*_{c} plots for Groups 1 and 4 are difficult to separate. Consequently, the contribution of the longitudinal steel is neglected in the derivation of the confined concrete stress–strain model in the following sections.

### 3.5 Strip Arrangement

*ρ*

_{f}= 0.003 and

*ρ*

_{f}= 0.006 (Eq. 1). One unwrapped (

*ρ*

_{f}= 0.0) and one fully wrapped column, and three columns wrapped with 1, 3, and 6 strips are compared in Fig. 7. All columns have the same cross section and material properties as the columns in Group 1 (Table 2). In order to maintain a constant FRP ratio,

*ρ*

_{f}, the thickness,

*t*

_{f}, and/or number of FRP sheets per strip,

*n*

_{f}, is varied to achieve different strip arrangements. Figure 7 clearly shows that, for the same volume of CFRP material bonded to the column, the fully wrapped is more effective in increasing the ultimate compressive stress and strain, and thus, ductility. The effectiveness is more pronounced when the CFRP volumetric ratio is increased to 0.006 in Fig. 7b. The increase in the number of strips, from 1 to 6, leads to an increase in the ultimate compressive stress and strain, and ductility. Although the fully wrapped column is more effective, in certain instances, a specific number of strips could satisfy the design requirements. This may be of interest when retrofitting columns that are not easily accessible (e.g. over a waterway) where the placement of strips maybe more economical than placing a full wrap.

## 4 Current FRP Confined Concrete Stress–Strain Models

Stress–strain models for FRP confined concrete.

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) |
---|---|

\( \begin{array}{*{20}c} { - for} & {0 \le } \\ \end{array} \varepsilon_{c} \le \varepsilon^{\prime}_{c} { , }f_{c} = E_{c} \varepsilon_{c} + (f^{\prime}_{c} - E_{c} \varepsilon^{\prime}_{c} )\left( {\frac{{\varepsilon_{c} }}{{\varepsilon^{\prime}_{c} }}} \right)^{2} \) \( \begin{array}{*{20}c} { - for} & {\varepsilon_{c}^{'} \le } \\ \end{array} \varepsilon_{c} \le \varepsilon_{c,s} { , }f_{c} = f^{\prime}_{c} + (f_{c,s} - f^{\prime}_{c} )\left( {\frac{{\varepsilon_{c} - \varepsilon^{\prime}_{c} }}{{\varepsilon_{c,s} - \varepsilon^{\prime}_{c} }}} \right)^{0.7} \) \( \begin{array}{*{20}c} { - for} & {\varepsilon_{c,s}^{{}} \le } \\ \end{array} \varepsilon_{c} \le \varepsilon_{ccu} { , }f_{c} = f_{c,s} + (f^{\prime}_{cc} - f_{c,s} )\left( {\frac{{\varepsilon_{c} - \varepsilon_{c,s} }}{{\varepsilon_{ccu} - \varepsilon_{c,s} }}} \right)^{0.7} \) \( \begin{array}{*{20}c} {\left. {\begin{array}{*{20}c} {\varepsilon_{c,s} = \varepsilon_{ccu} \left[ {0.85 + 0.03\left( {\frac{{f_{l,f,\hbox{max} } }}{{f_{l,s,\hbox{max} } }}} \right)} \right]} \\ {f_{c,s} = 0.95f^{\prime}_{cc} } \\ \end{array} } \right\}} & {f_{l,f,\hbox{max} } \ge f_{l,s,\hbox{max} } } \\ \end{array} \) \( \begin{array}{*{20}c} {\left. {\begin{array}{*{20}c} {\varepsilon_{c,s} = 0.7\varepsilon_{ccu} } \\ {f_{c,s} = \left( {\frac{{\varepsilon_{c,s} }}{{\varepsilon_{ccu} }}} \right)^{0.4} f^{\prime}_{cc} } \\ \end{array} } \right\}} & {f_{l,f,\hbox{max} } < f_{l,s,\hbox{max} } } \\ \end{array} \) \( k_{s} = \left\{ {\begin{array}{*{20}c} {2 - \frac{{f_{l,f,\hbox{max} } }}{{f_{l,s,\hbox{max} } }}} & {{\text{for }}f_{l,f,\hbox{max} } \le f_{l,s,\hbox{max} } } \\ 1 & {{\text{for }}f_{l,f,\hbox{max} } > f_{l,s,\hbox{max} } } \\ \end{array} } \right. \) \( f_{cc}^{'} = f^{\prime}_{c} \left( {1 + 2 \frac{{f_{l} }}{{f^{\prime}_{c} }}} \right) \) \( \varepsilon_{ccu}^{{}} = \varepsilon^{\prime}_{c} \left[ {1.75 + 5.25\left( {\frac{{f_{l,f,\hbox{max} } + k_{s} f_{l,s,\hbox{max} } }}{{f^{\prime}_{c} }}} \right)\left( {\frac{{\varepsilon_{fu} }}{{\varepsilon^{\prime}_{c} }}} \right)^{0.45} } \right] \) | \( - for ,\quad 0 \le \varepsilon_{c} \le \varepsilon_{c,s} , { }f_{c} = \frac{{(E_{c} - E_{1} )\varepsilon_{c} }}{{\left[ {1 + \left( {\frac{{(E_{c} - E_{1} )\varepsilon_{c} }}{{f_{0} }}} \right)^{n} } \right]^{1/n} }} + E_{1} \varepsilon_{c}^{m} \) \( - \, for \, \varepsilon_{c,s} \le \varepsilon_{c} \le \varepsilon_{ccu} ,\,f_{c} = f_{c,s} + E_{2} (\varepsilon_{c} - \varepsilon_{c,s} ) \) \( E_{1} = \frac{{f_{c,s} - f_{0} }}{{\varepsilon_{c,s} }};\,E_{2} = \frac{{f^{\prime}_{cc} - f_{c,s} }}{{\varepsilon_{ccu} - \varepsilon_{c,s} }} \) \( n = 1 + \frac{1}{{({{E_{c} \varepsilon^{\prime}_{c} } \mathord{\left/ {\vphantom {{E_{c} \varepsilon^{\prime}_{c} } {f^{\prime}_{c} )}}} \right. \kern-0pt} {f^{\prime}_{c} )}} - 1}} \) \( m = \left[ {\frac{1}{{\ln (\varepsilon_{c,s} )}}} \right]\left\{ {\ln \left[ {\frac{1}{{E_{1} }}\left( {f_{c,s} - \frac{{(E_{c} - E_{1} )\varepsilon_{c,s} }}{{\left\{ {1 + \left[ {\frac{{(E_{c} - E_{1} )\varepsilon_{c,s} }}{{f_{0} }}} \right]^{n} } \right\}^{1/n} }}} \right)} \right]} \right\} \) \( f_{c,s} = \frac{{f_{core} A_{core} + f_{{\text{cov} er}} A_{{\text{cov} er}} }}{{A_{g} }} \) \( \varepsilon_{c,s} = 0.85\varepsilon^{\prime}_{c} \left( {1 + 8\frac{{(f_{l,fy} + f^{\prime}_{l,s,\hbox{max} } )}}{{f^{\prime}_{c} }}} \right).\left\{ {\left[ {1 + 0.75\left( {\frac{{\varepsilon_{l,y} }}{{\varepsilon^{\prime}_{c} }}} \right)} \right]^{0.7} - \exp \left[ { - 7\left( {\frac{{\varepsilon_{l,y} }}{{\varepsilon^{\prime}_{c} }}} \right)} \right]} \right\} \) \( f_{cc}^{'} = f^{\prime}_{c} \left[ {1 + 1.55\left( {\frac{{f_{l,f,\hbox{max} } }}{{f^{\prime}_{c} }}} \right)\left( {\frac{{N_{f} w_{f} }}{{l_{u} }}} \right)^{0.3} + 1.55\left( {\frac{{f_{l,s,\hbox{max} } }}{{f^{\prime}_{c} }}} \right)} \right] \) \( \varepsilon_{ccu}^{{}} = \varepsilon^{\prime}_{c} \left[ {2.4 + 15\left( {\frac{{f_{l,f,\hbox{max} } }}{{f^{\prime}_{c} }}} \right)\left( {\frac{{N_{f} w_{f} }}{{l_{u} }}} \right)^{0.3} + 7.7\left( {\frac{{f_{l,s,\hbox{max} } }}{{f^{\prime}_{c} }}} \right)} \right] \) |

Lam and Teng’s (2003) stress–strain model (Fig. 8a) accounts for FRP confinement only. Pellegrino and Modena (2010) proposed an analytical model (Fig. 8b) for partially wrapped columns. The model is based on Richard and Abbott’s model (1975) that accounts for steel reinforcement contribution to confinement in circular and rectangular columns. The partial wraps are accounted for by modifying the discontinuity coefficient used for transverse steel in Mander’s model (1988). The total lateral confining pressure, *f*_{l}, is derived by combining that of the transverse steel and FRP. Lee et al. (2010) introduced an empirical model for concrete confined with both steel spirals and FRP wraps (Fig. 8c). The model accounts for yielding of transverse steel and its contribution to the confining pressure.

## 5 Proposed Confined *f*
_{c} − *ε*
_{c} Model

Richard and Abbott’s model (1975), a bilinear model that has been adopted and modified by others for concrete columns confined by FRP (Wu et al. 2009; Pellegrino and Modena 2010), is also adopted herein and modified based on the aforementioned parametric study. Since the contribution of longitudinal steel is minimal (Figs. 5 and 6), it is not considered in the model generation.

*f*

_{c}−

*ε*

_{c}model comprises a nonlinear portion for the strain range of 0 ≤

*ε*

_{c}≤

*ε*

_{c,s}and a linear portion for

*ε*

_{c,s}<

*ε*

_{c}≤

*ε*

_{ccu}(Fig. 8d).

*ε*

_{c,s}is the confined concrete axial strain at yielding of the transverse steel and

*ε*

_{ccu}is the ultimate confined concrete axial strain. The

*f*

_{c}−

*ε*

_{c}relationship is expressed as follows:

*f*

_{c}and

*ε*

_{c}are the concrete compressive stress and strain of FRP-confined concrete, respectively;

*f*

_{0}is the reference plastic stress at the intercept of the slope at yielding of transverse steel with the stress axis (Fig. 8d); and

*n*is a shape parameter in the transition zone, \( E_{1} \) is the slope of the stress strain curve at the yielding of transverse steel,

*E*

_{c}is the concrete modulus of elasticity and, for normal-weight concrete (ACI 2011), and

*E*

_{2}is the slope of the stress strain curve after yielding of transverse steel,

*f*

_{c,s}and

*ε*

_{c,s}are the compressive stress and strain in confined concrete at yielding of transverse steel. In Eq. (4),

*m*can be determined by setting the

*f*

_{c}(

*ε*

_{c,s}) =

*f*

_{c,s}at the point of yielding of transverse steel

*f*

_{0}is replaced by \( f^{\prime}_{c} \).

### 5.1 Ultimate Confined Concrete Stress and Strain, \( f^{\prime}_{cc} \) and *ε*
_{ccu}

*f*

_{l,f,max}), the maximum lateral confining pressure due to transverse steel only (

*f*

_{l,s,max}) and the ratio between the length of FRP wrap (

*N*

_{f}

*w*

_{f}) and the unbraced length of the column (

*l*

_{u}). Based on the regression analysis conducted on the data generated in the parametric study, the ultimate confined concrete stress \( f^{\prime}_{cc} \) and strain

*ε*

_{ccu}can be presented as follows

*f*

_{y}is the specified yield strength of non-prestressed steel reinforcement;

*A*

_{st}is the area of transverse steel;

*d*

_{s}is the concrete core diameter to center line of transverse steel;

*s*is the center to center spacing between transverse steel;

*ε*

_{fu}is the design rupture strain of FRP wrap.

### 5.2 Concrete Stress and Strain at Yielding of Transverse Steel, *f*
_{c,s} and *ε*
_{c,s}

*f*

_{c,s}and strain

*ε*

_{c,s}(Fig. 8d) is the transition between the nonlinear and linear stress–strain relationships. The increase in the compressive strength of concrete confined by the two materials can be derived by summing the increments of the compressive strength for each material (Lee et al. 2010). Consequently,

*f*

_{c,s}and

*ε*

_{c,s}can be determined by summing the strength of concrete due to FRP confinement and strength of concrete due to transverse steel at yielding of the transverse steel. Considering that the transverse steel yield occurred at a lateral strain

*ε*

_{l,y}, then

*ε*

_{l,y}is the confined concrete lateral strain at yielding of transverse steel and

*E*

_{s}is modulus of elasticity of the transverse steel. The strain in the confined concrete at yielding of the transverse steel may now be determined using the relationship introduced by Teng et al. (2007). for the lateral strain–axial strain relationship of FRP confined concrete

*f*

_{l,fy}is the lateral confining pressure exerted by FRP at yielding of transverse steel

*s′*is the clear spacing between the transverse steel (stirrups), and

*A*

_{core}is the column core area.

*A*

_{core}is the column cover area;

*f*

_{core}and

*f*

_{cover}are the compressive stresses of confined concrete for the column core and cover, respectively;

*f*

_{c,fy}is the component of confined concrete compressive stress at yielding of transverse steel due to FRP confinement only;

*f*

_{c,sy}is the component of the confined concrete compressive stress at yielding of transverse steel due to transverse steel confinement only.

Since the concrete core is confined by transverse steel and FRP, the compressive concrete stress of the column core (*f*_{core}) can be calculated by adding the compressive concrete stress at yielding of steel due to steel confinement (*f*_{c,sy}) and FRP confinement (*f*_{c,fy}). Mander’s model (1988) is used to calculate *f*_{c,sy} using Eqs. (23–25). This model accounts for the effect of the compressive strength of unconfined concrete \( \left( {f^{\prime}_{c} } \right) \) on the column’s core. Teng model (2007) is used to calculate *f*_{c,fy} using Eqs. (27–29) and it includes the effect of \( f^{\prime}_{c} \) on *f*_{core}. Since both models include the effect of \( f^{\prime}_{c} \) on *f*_{core}, \( f^{\prime}_{c} \) is subtracted from Eq. (20) in order to avoid double counting its effect. The relationships between *f*_{c,sy} and \( f^{\prime}_{c} \), and *f*_{c,fy} and \( f^{\prime}_{c} \) are nonlinear. The nonlinearity leads to a complex model for the confined concrete stress–strain relationship. In order to simplify model, the assumption of a linear relationship generated results that are in very good agreement with experimental ones (Demers and Neale 1999; Rocca et al. 2006; Varma et al. 2009; Lee et al. 2010). Consequently, a linear relationship between *f*_{c,sy} and \( f^{\prime}_{c} \), and *f*_{c,fy} and \( f^{\prime}_{c} \) is adopted in the proposed model.

*r*

_{s}is a constant to account for the brittleness of concrete and is determined by Mander et al. (1988)

*ε*

_{ccu,s}are the peak compressive stress and strain, respectively, of confined concrete under the transverse steel confining pressure at yielding of transverse steel and can be calculated using the following equations (Mander et al. 1988).

*f*

_{c, fy}, can be expressed as follows

*r*

_{f}is a constant that accounts for the brittleness of concrete and can be calculated as Mander et al. (1988)

*ε*

_{ccu,f}are the peak compressive stress and strain, respectively, of FRP confined concrete at yielding of transverse steel. They can be determined using the following equations (Teng et al. 2007)

## 6 Comparison of Proposed Model with FE and Experimental Results

*f*

_{l}, combines the transverse steel and FRP pressures while the proposed model treats them independently.

## 7 Conclusions

This paper evaluated the effectiveness of partial wraps (or strips) and proposed an analytical model for describing the compressive behavior of RC columns that are partially and fully wrapped with FRP. Three-dimensional finite element (FE) models were generated to study the influence, on the behavior of the concentrically loaded columns, of the unconfined compressive strength \( \left( {f^{\prime}_{c} } \right) \), the number of strips (*N*_{f}), the FRP volumetric ratio (*ρ*_{f}), the transverse steel reinforcement ratio (*ρ*_{st}), and the longitudinal steel reinforcement ratio (*ρ*_{sl}). It should be noted that the columns wrapped with one, two, or three strips are not of practical interest and are used herein to illustrate the influence of partial wrapping as the analysis transitions from an unwrapped column to a partially wrapped column with one to seven strips, to a fully wrapped column.

- (1)
The influence of increasing the unconfined compressive strength \( \left( {f^{\prime}_{c} } \right) \) has a pronounced effect on the increase in the confined concrete compressive strength \( \left( {f^{\prime}_{cc} } \right) \)

- (2)
As the number of identical strips increases (or

*ρ*_{f}increases), the influence of the transverse steel confinement (*ρ*_{st}) decreases - (3)
The contribution of the longitudinal steel has little influence on the confined concrete stress–strain behavior

- (4)
The increase in the number of strips (

*N*_{f}= 1–7), while keeping the FRP volumetric ratio (*ρ*_{f}) constant, leads to an increase in the ultimate compressive stress, strain and ductility. This indicates that, for a specific*ρ*_{f}, it is more effective to fully wrap the column in order to increase the ultimate confined concrete compressive stress and axial strain.

## Notes

## Declarations

### Authors' contributions

SYG and IEH contributed to the design and implementation of the research, the analysis of the results and the writing of the manuscript. Both authors read and approved the final manuscript.

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## Authors’ Affiliations

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