# A New Proposal for the Shear Strength Prediction of Beams Longitudinally Reinforced with Fiber-Reinforced Polymer Bars

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Available Models and Design Recommendations

_{c}) consists of: the strength resulting from aggregate interlock, the shear strength of the concrete in the compression zone, the dowel action of the longitudinal reinforcement and residual tensile strength of concrete across the crack. The stiffness of composite rebars is much lower than that of steel reinforcement. In comparison with a steel-RC member, after cracking, the distance from the compressed fibres to the neutral axis in a concrete member longitudinally reinforced with FRP bars is smaller (the compression region of the cross section is reduced). This is due to the lower axial stiffness of FRP reinforcement. Since the compression zone extent is smaller, the shear strength of the concrete in the compression zone is also smaller [5,7]. Moreover, the crack width is larger in the case of FRP reinforcement. Hence, the component associated with aggregate interlock is smaller [5,7]. The low transverse stiffness of FRP rebars significantly reduces the component stemming from dowel action [5,7]. As a result, at the same longitudinal reinforcement area, the concrete member reinforced with FRP bars has a lower shear strength than the corresponding steel-RC member [13,14].

## 3. Experimental Database

## 4. Verification of the Available Models and Design Recommendations

_{n}was calculated without reduction factors taken into account. One should note that the values of strength V

_{test}were obtained for the short-term loading of the beams. This way of loading precludes any study of the long-term processes taking place in concrete and in FRP rebars and of the effect of an aggressive environment.

_{test}/V

_{n}, parameter X (inverse of regression curve slope), coefficient of variation (COV), mean absolute percentage error (MAPE) and the percentage of support zones with overestimated strength. The results of the comparative analyses of procedures included in the design recommendations ACI 440.1R-15 [5], Canadian CSA-S806-12 [8], Japanese JSCE-97 [9] and Italian CNR-DT 203/2006 [10] are presented in Table 3. The results of the comparative analyses of the algorithms proposed by Tottori et al. [15], Michaluk et al. [16], Deitz et al. [17], El-Sayed et al. [18], Wegian et al. [19], Nehdi et al. [20], Hoult et al. [21], Razaqpur et al. [22], Alam [23], Kara [24], Kurth [25], Jang et al. [26], Lignola et al. [27], Valivonis et al. [28], Thomas et al. [29] and Hamid et al. [30] are presented in Table 4.

_{test}< V

_{n}), which adversely affects the level of safety. The load capacities of more than 77% of the analysed support zones were overestimated when the procedure was used. In addition, a high value of the coefficient of variation and the mean absolute percentage error were obtained, which indicates a relatively large spread of results and an inappropriate adjustment of the model. Taking into account the criterion of conservativeness and the scatter of results, the best-fit model is included in the Canadian CSA S806-12 standard [8]. Using the procedures given in [8], the smallest values of the coefficient of variation and the mean absolute percentage error were obtained. Nevertheless, a relatively high percentage of support zones with an overestimated load capacity, with the parameter X of 1.47, indicates that the model is not adjusted properly.

_{test}/V

_{n}, and the value of parameter X. In addition, using the procedures proposed by Nehdi et al. [20], Razaqpur et al. [22] and Kurth [25], a relatively low value of the coefficient of variation and the mean absolute percentage error were obtained.

## 5. Proposed Model for Estimating the Shear Capacity of Support Zones Reinforced Longitudinally with FRP Bars without Shear Reinforcement

^{2}= 0.87, which is a measure of the model’s fit. In addition, the smallest coefficient of variation and the mean absolute percentage error were obtained, which indicates the smallest distribution of results among the described models. The percentage of beams with overestimated strength is 52.90%, which is a result of obtaining the average value of the ratio of the experimental and theoretical shear force V

_{test}/V

_{n}close to 1.00.

_{test}/V

_{n}, for the particular ranges. The distribution is close to the normal distribution, and the lower endpoint of the 95% confidence interval of the average value is 0.977.

_{test}, to the corresponding analytical, V

_{n}. Figure 2 shows the value of V

_{test}/V

_{n}in relation to shear slenderness, a/d, compressive concrete strength, f

_{c}, the axial stiffness of longitudinal reinforcement, ρ

_{f}E

_{fl}and the effective depth of cross section, d.

## 6. Conclusions

^{2}= 0.87. A satisfactory level of value of COV = 22.50% and MAPE = 18.62% was obtained. The proposed model appropriately takes into account the influence of particular parameters (such as shear slenderness, the compressive strength of concrete, the stiffness of the longitudinal reinforcement and the effective depth of cross section) on the shear capacity of support zones of beams longitudinally reinforced with composite bars. Based on the verification analysis, a satisfactory level of model fit was obtained—the best among the available proposals.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Distribution of values of the ratio of the shear resistance attained experimentally, V

_{test}, to the corresponding analytical, V

_{n}for particular ranges.

**Figure 2.**The ratio of the experimental and theoretical shear force V

_{test}/V

_{n}as a function of: (

**a**) shear slenderness, a/d; (

**b**) the compressive strength of concrete, f

_{c}; (

**c**) the stiffness of the longitudinal reinforcement, ρ

_{f}E

_{fl}; (

**d**) the effective depth of cross section, d.

**Table 1.**Algorithms for determining the shear resistance of the shear zones longitudinally reinforced with fibre-reinforced polymer (FRP) bars.

Algorithm | Shear Strength of Member without Shear Reinforcement | |
---|---|---|

Tottori et al. [15] | ${V}_{\mathrm{c}}=0.2{\left(100{\rho}_{\mathrm{f}}{f}_{\mathrm{c}}\frac{{E}_{\mathrm{fl}}}{{E}_{\mathrm{s}}}\right)}^{1/3}{\left(\frac{d}{1000}\right)}^{-1/4}\left(0.75+\frac{1.4}{\frac{a}{d}}\right){b}_{\mathrm{w}}d$ | (1) |

JSCE-97 [9] | ${V}_{\mathrm{c}}={\beta}_{\mathrm{d}}{\beta}_{\mathrm{p}}{\beta}_{\mathrm{n}}{f}_{\mathrm{vcd}}{b}_{\mathrm{w}}d$ ${\beta}_{\mathrm{d}}=\sqrt[4]{\frac{1}{d}}\le 1.5;d\mathrm{in}\text{}\left(\mathrm{m}\right);\text{}{\beta}_{\mathrm{p}}=\sqrt[3]{100{\rho}_{\mathrm{f}}\frac{{E}_{\mathrm{fl}}}{{E}_{\mathrm{s}}}}\le 1.5;$ ${\beta}_{\mathrm{n}}=1.0\mathrm{when}\text{}\mathrm{there}\text{}\mathrm{is}\text{}\mathrm{no}\text{}\mathrm{axial}\text{}\mathrm{force};{f}_{\mathrm{vcd}}=0.2\sqrt[3]{{f}_{\mathrm{c}}}\le 0.72\mathrm{MPa}$ | (2) |

Michaluk et al. [16] | ${V}_{\mathrm{c}}=\frac{{E}_{\mathrm{fl}}}{{E}_{\mathrm{s}}}\left(\frac{1}{6}\sqrt{{f}_{\mathrm{c}}}{b}_{\mathrm{w}}d\right)$ | (3) |

Deitz et al. [17] | ${V}_{\mathrm{c}}=3\frac{{E}_{\mathrm{fl}}}{{E}_{\mathrm{s}}}\left(\frac{1}{6}\sqrt{{f}_{\mathrm{c}}}{b}_{\mathrm{w}}d\right)$ | (4) |

El-Sayed et al. [18] | ${V}_{\mathrm{c}}={\left(\frac{{\rho}_{\mathrm{f}}{E}_{\mathrm{fl}}}{90{\beta}_{1}{f}_{\mathrm{c}}}\right)}^{1/3}\left(\frac{\sqrt{{f}_{\mathrm{c}}}{b}_{\mathrm{w}}d}{6}\right)\le \frac{\sqrt{{f}_{\mathrm{c}}}{b}_{\mathrm{w}}d}{6}$ ${\beta}_{1}=\{\begin{array}{ccc}0.85& \mathrm{for}& {f}_{\mathrm{c}}\le 28\mathrm{Mpa}\\ 0.85-0.05\frac{{f}_{\mathrm{c}}-28}{7}& \mathrm{for}& {f}_{\mathrm{c}}=28\xf756\mathrm{Mpa}\\ 0.65& \mathrm{for}& {f}_{\mathrm{c}}\ge 56\mathrm{Mpa}\end{array}$ | (5) |

Wegian et al. [19] | ${V}_{\mathrm{c}}=2{\left({f}_{\mathrm{c}}\frac{{\rho}_{\mathrm{f}}{E}_{\mathrm{fl}}}{{E}_{\mathrm{s}}}\frac{d}{a}\right)}^{1/3}{b}_{\mathrm{w}}d$ | (6) |

CNR DT 203/2006 [10] | ${V}_{\mathrm{c}}=1.3{\left(\frac{{E}_{\mathrm{fl}}}{{E}_{\mathrm{s}}}\right)}^{\frac{1}{2}}{\tau}_{\mathrm{Rd}}{k}_{\mathrm{d}}\left(1.2+40{\rho}_{\mathrm{f}}\right){b}_{\mathrm{w}}d\le {V}_{\mathrm{Rd},\mathrm{max}}=0.5{\upsilon}_{1}{f}_{\mathrm{c}}{b}_{\mathrm{w}}\xb70.9d$ ${\tau}_{\mathrm{Rd}}=0.25{f}_{\mathrm{ct}}$; ${k}_{\mathrm{d}}=1.6-d\left[m\right]\ge 1.0$; $1.3{\left(\frac{{E}_{\mathrm{fl}}}{{E}_{\mathrm{s}}}\right)}^{1/2}\le 1.0$ ${\upsilon}_{1}=\{\begin{array}{ccc}0.6& \mathrm{for}& {f}_{\mathrm{c}}\le 60\mathrm{Mpa}\\ 0.9-\frac{{f}_{\mathrm{c}}}{200}\ge 0.5& \mathrm{for}& {f}_{\mathrm{c}}60\mathrm{Mpa}\end{array}$ | (7) |

Nehdi et al. [20] | ${V}_{\mathrm{c}}=2.1{\left(\frac{{f}_{\mathrm{c}}{\rho}_{\mathrm{f}}d}{a}\frac{{E}_{\mathrm{fl}}}{{E}_{\mathrm{s}}}\right)}^{0.3}{b}_{\mathrm{w}}d\xb7\frac{2.5d}{a}$ $\frac{2.5d}{a}\ge 1.0$ | (8) |

Hoult et al. [21] | ${V}_{\mathrm{c}}=\beta \sqrt{{f}_{\mathrm{c}}}{b}_{\mathrm{w}}\xb70.9d$ $\beta =\frac{0.3}{0.5+{\left(1000{\epsilon}_{\mathrm{x}}+0.15\right)}^{0.7}}\frac{1300}{1000+{s}_{\mathrm{ze}}}$ ${\epsilon}_{\mathrm{x}}=\frac{\frac{M}{0.9d}+V}{2{E}_{\mathrm{fl}}{A}_{\mathrm{f}}}$; ${s}_{\mathrm{ze}}=\frac{31.5d}{16+{a}_{\mathrm{g}}}\ge 0.77d$; ${a}_{\mathrm{g}}=\{\begin{array}{ccc}{a}_{\mathrm{g}}& \mathrm{for}& {f}_{\mathrm{c}}<60\mathrm{Mpa}\\ {a}_{\mathrm{g}}-\frac{{a}_{\mathrm{g}}}{10}\left({f}_{\mathrm{c}}-60\right)& \mathrm{for}& 60\le {f}_{\mathrm{c}}70\mathrm{Mpa}\\ 0& \mathrm{for}& {f}_{\mathrm{c}}\ge 70\mathrm{Mpa}\end{array}$ | (9) |

Razaqpur et al. [22] | ${V}_{\mathrm{c}}=0.045{k}_{\mathrm{m}}{k}_{\mathrm{a}}{k}_{\mathrm{r}}\sqrt[3]{{f}_{\mathrm{c}}}{b}_{\mathrm{w}}d$ ${k}_{\mathrm{m}}={\left(\frac{Vd}{M}\right)}^{1/2}$; ${k}_{\mathrm{r}}=1+{\left({\rho}_{\mathrm{f}}{E}_{\mathrm{fl}}\right)}^{1/3}$; ${k}_{\mathrm{a}}=\{\begin{array}{ccc}1.0& \mathrm{for}& \frac{M}{Vd}\ge 2.5\\ \frac{2.5Vd}{M}& \mathrm{for}& \frac{M}{Vd}<2.5\end{array}$ | (10) |

Alam [23] | ${V}_{\mathrm{c}}=\frac{0.2\lambda}{{\left(\frac{a}{d}\right)}^{2/3}}{\left(\frac{{\rho}_{\mathrm{f}}{E}_{\mathrm{fl}}}{d}\right)}^{1/3}\sqrt{{f}_{\mathrm{c}}}{b}_{\mathrm{w}}d$ $\frac{0.1\lambda d}{a}\sqrt{{f}_{\mathrm{c}}}{b}_{\mathrm{w}}d\le {V}_{\mathrm{c}}\le 0.2\lambda \sqrt{{f}_{\mathrm{c}}}{b}_{\mathrm{w}}d$ | (11) |

Kara [24] | ${V}_{\mathrm{c}}=0.997{b}_{\mathrm{w}}d{\left(6.837\sqrt[3]{\frac{d}{a}}{f}_{\mathrm{c}}{\rho}_{\mathrm{f}}\frac{{E}_{\mathrm{fl}}}{{E}_{\mathrm{s}}}\right)}^{1/3}$ | (12) |

CSA S806-12 [8] | ${V}_{\mathrm{c}}=0.05\lambda {k}_{\mathrm{m}}{k}_{\mathrm{a}}{k}_{\mathrm{s}}{k}_{\mathrm{r}}\sqrt[3]{{f}_{\mathrm{c}}}{b}_{\mathrm{w}}{d}_{\mathrm{v}}$ ${k}_{\mathrm{m}}=\sqrt{\frac{Vd}{M}}\le 1.0$; ${k}_{\mathrm{a}}=\frac{2.5Vd}{M}$; $1.0\le {k}_{\mathrm{a}}\le 2.5$; ${k}_{\mathrm{s}}=\frac{750}{450+d}\le 1.0$; ${k}_{\mathrm{r}}=1+{\left({\rho}_{\mathrm{f}}{E}_{\mathrm{fl}}\right)}^{1/3}$; ${d}_{\mathrm{v}}=\mathrm{max}\left(0.9d;0.72h\right)$; $0.11\sqrt{{f}_{\mathrm{c}}}{b}_{\mathrm{w}}{d}_{\mathrm{v}}\le {V}_{\mathrm{c}}\le 0.22\sqrt{{f}_{\mathrm{c}}}{b}_{\mathrm{w}}{d}_{\mathrm{v}};{f}_{\mathrm{c}}60\mathrm{MPa}$ | (13) |

Kurth [25] | ${V}_{\mathrm{c}}=\beta \frac{1}{313}\kappa {\left(100{\rho}_{\mathrm{f}}{E}_{\mathrm{fl}}{f}_{\mathrm{c}}\right)}^{1/3}{b}_{\mathrm{w}}d$ $\beta =3\frac{d}{a}\ge 1.0$; $\kappa =1+\sqrt{\frac{200}{d}}\le 2.0$; $d$ in (mm) | (14) |

Jang et al. [26] | ${V}_{\mathrm{c}}=\frac{1}{6}{\beta}_{f}\sqrt{{f}_{\mathrm{c}}}{b}_{\mathrm{w}}d$ ${\beta}_{f}=0.716+0.466\frac{{E}_{\mathrm{fl}}}{{E}_{\mathrm{s}}}-0.095\frac{a}{d}+32.101{\rho}_{\mathrm{f}}$ | (15) |

Lignola et al. [27] | ${V}_{\mathrm{c}}=1.65{\left(\frac{{E}_{\mathrm{fl}}}{{E}_{\mathrm{s}}}\right)}^{0.6}{C}_{\mathrm{Rd},\mathrm{c}}k{\left(100{\rho}_{\mathrm{f}}{f}_{\mathrm{c}}\right)}^{1/3}{b}_{\mathrm{w}}d$ $k=1+\sqrt{\frac{200}{d}}\le 2.0$; $d$ in (mm); ${C}_{\mathrm{Rd},\mathrm{c}}=\{\begin{array}{c}0.18\mathrm{for}\mathrm{normal}\mathrm{concrete}\\ 0.12\mathrm{for}\mathrm{lightweight}\mathrm{concrete}\end{array}$ | (16) |

ACI 440.1R-15 [5] | ${V}_{\mathrm{c}}=\frac{2}{5}k\sqrt{{f}_{\mathrm{c}}}{b}_{\mathrm{w}}d$ $k=\sqrt{2{\rho}_{\mathrm{f}}{n}_{\mathrm{f}}+{\left({\rho}_{\mathrm{f}}{n}_{\mathrm{f}}\right)}^{2}}-{\rho}_{\mathrm{f}}{n}_{\mathrm{f}};{n}_{\mathrm{f}}=\frac{{E}_{\mathrm{fl}}}{{E}_{\mathrm{c}}}$ | (17) |

Valivonis et al. [28] | ${V}_{\mathrm{c}}=\frac{2{\phi}_{\mathrm{f}}{f}_{\mathrm{ct}}{b}_{\mathrm{w}}{d}^{2}}{a}\ge 0.45{\phi}_{\mathrm{f}}{f}_{\mathrm{ct}}{b}_{\mathrm{w}}d$ ${\phi}_{\mathrm{f}}=0.4{\left(\frac{{E}_{\mathrm{fl}}}{{E}_{\mathrm{s}}}\right)}^{{\rho}_{\mathrm{f}}}$; $a\le 3.33d$ | (18) |

Thomas et al. [29] | ${V}_{\mathrm{c}}={k}_{1}{k}_{2}{\tau}_{\mathrm{c}}{b}_{\mathrm{w}}d$ ${\tau}_{\mathrm{c}}=\frac{0.85\sqrt{{f}_{\mathrm{c}}}\left(\sqrt{1+5\beta}-1\right)}{6\beta};\beta =\frac{{f}_{\mathrm{c}}}{45.55{p}_{\mathrm{t}}};$ ${k}_{1}=\{\begin{array}{ccc}2.2\frac{d}{a}+0.12& \mathrm{for}& \frac{a}{d}\le 2.5\\ 1.0& \mathrm{for}& \frac{a}{d}>2.5\end{array};$ ${k}_{2}=\{\begin{array}{ccc}\frac{750}{450+d}& \mathrm{for}& d>300\mathrm{mm}\\ 1.0& \mathrm{for}& d\le 300\mathrm{mm}\end{array};{p}_{\mathrm{t}}={\rho}_{\mathrm{f}}\frac{{E}_{\mathrm{fl}}}{{E}_{\mathrm{s}}}$ | (19) |

Hamid et al. [30] | ${V}_{\mathrm{c}}={f}_{\mathrm{c}}{b}_{\mathrm{w}}d\left[0.00203{\left({\rho}_{\mathrm{f}}{E}_{\mathrm{fl}}{f}_{\mathrm{c}}\right)}^{1/3}+0.153\frac{d}{a}\right]$ | (20) |

_{f}—longitudinal FRP reinforcement ratio of beam; f

_{c}—compressive strength of concrete (MPa); f

_{ct}—tensile strength of concrete (MPa); E

_{fl}—elastic modulus of FRP rebars (MPa); E

_{s}—elastic modulus of steel rebars (MPa); E

_{c}—elasticity modulus of concrete (MPa); d—effective depth of cross section (mm); a—length of the shear zone—distance of concentrated force from the support (mm); a/—shear slenderness; b

_{w}—web width (mm); λ—modification factor related to density of concrete; V—shear force (N); M—bending moment (Nmm); a

_{g}—maximum size of coarse aggregate [mm]; h—height of cross section (mm); p

_{t}—equivalent longitudinal FRP reinforcement ratio of beam regarding to steel.

Number of Support Zones | 310 | ||||
---|---|---|---|---|---|

Properties | Min | Max | Average | COV^{1} (%) | |

b_{w} | (mm) | 89 | 1000 | 251 | 68 |

h | (mm) | 100 | 1000 | 318 | 51 |

d | (mm) | 73 | 937 | 270 | 54 |

a | (mm) | 200 | 3055 | 907 | 53 |

a/d | (–) | 0.8 | 12.5 | 3.7 | 43 |

f_{c} | (MPa) | 20 | 93 | 44 | 39 |

ρ_{f} | (%) | 0.12 | 11.57 | 1.35 | 134 |

E_{fl} | (MPa) | 29,400 | 192,000 | 73,408 | 59 |

Longitudinal reinforcement material | (–) | AFRP, BFRP, CFRP, GFRP | |||

V_{test} | (N) | 9000 | 291,300 | 62,490 | 85 |

^{1}Coefficient of variation.

**Table 3.**Comparison of experimental (V

_{test}) and theoretical (V

_{n}) values of the concrete shear strength for available design recommendations. Results according to Equations (2), (7), (13) and (17).

$\raisebox{1ex}{${V}_{\mathrm{test}}$}\!\left/ \!\raisebox{-1ex}{${V}_{n}$}\right.$ | ACI 440.1R-15 [5] | CSA S806-12 [8] |

Arithmetic mean | 2.20 | 1.18 |

Parameter X (inverse of regression curve slope) | 2.20 | 1.47 |

Coefficient of variation (COV) (%) | 63.38 | 43.18 |

Mean absolute percentage error (MAPE) (%) | 45.88 | 20.61 |

Percentage of beams with overestimated strength (%) | 0.97 | 39.03 |

Theoretical concrete shear strength versus experimental concrete shear strength | ||

$\raisebox{1ex}{${V}_{\mathrm{test}}$}\!\left/ \!\raisebox{-1ex}{${V}_{n}$}\right.$ | JSCE-97 [9] | CNR-DT 203/2006 [10] |

Arithmetic mean | 1.61 | 0.89 |

Parameter X (inverse of regression curve slope) | 2.03 | 1.38 |

Coefficient of variation (COV) (%) | 62.86 | 62.39 |

Mean absolute percentage error (MAPE) (%) | 28.29 | 56.32 |

Percentage of beams with overestimated strength (%) | 10.32 | 77.74 |

Theoretical concrete shear strength versus experimental concrete shear strength |

**Table 4.**Comparison of experimental (V

_{test}) and theoretical (V

_{n}) values of the concrete shear strength for algorithms proposed by other authors. Results according to Equations (1), (3)–(6), (8)–(12), (14)–(16) and (18)–(20).

$\raisebox{1ex}{${V}_{\mathrm{test}}$}\!\left/ \!\raisebox{-1ex}{${V}_{n}$}\right.$ | Tottori et al. [15] | Michaluk et al. [16] |

Arithmetic mean | 1.20 | 3.26 |

Parameter X (inverse of regression curve slope) | 1.57 | 3.87 |

Coefficient of variation (COV) (%) | 36.98 | 73.26 |

Mean absolute percentage error (MAPE) (%) | 19.38 | 59.09 |

Percentage of beams with overestimated strength (%) | 30.00 | 10.97 |

Theoretical concrete shear strength versus experimental concrete shear strength | ||

$\raisebox{1ex}{${V}_{\mathrm{test}}$}\!\left/ \!\raisebox{-1ex}{${V}_{n}$}\right.$ | Deitz et al. [17] | El-Sayed et al. [18] |

Arithmetic mean | 1.09 | 0.97 |

Parameter X (inverse of regression curve slope) | 1.29 | 1.09 |

Coefficient of variation (COV) (%) | 73.26 | 72.67 |

Mean absolute percentage error (MAPE) (%) | 64.15 | 49.49 |

Percentage of beams with overestimated strength (%) | 51.29 | 75.48 |

Theoretical concrete shear strength versus experimental concrete shear strength | ||

$\raisebox{1ex}{${V}_{\mathrm{test}}$}\!\left/ \!\raisebox{-1ex}{${V}_{n}$}\right.$ | Wegian et al. [19] | Nehdi et al. [20] |

Arithmetic mean | 1.44 | 1.12 |

Parameter X (inverse of regression curve slope) | 1.52 | 1.13 |

Coefficient of variation (COV) (%) | 43.55 | 25.50 |

Mean absolute percentage error (MAPE) (%) | 26.32 | 18.96 |

Percentage of beams with overestimated strength (%) | 12.58 | 34.52 |

Theoretical concrete shear strength versus experimental concrete shear strength | ||

$\raisebox{1ex}{${V}_{\mathrm{test}}$}\!\left/ \!\raisebox{-1ex}{${V}_{n}$}\right.$ | Hoult et al. [21] | Razaqpur et al. [22]. |

Arithmetic mean | 1.88 | 1.09 |

Parameter X (inverse of regression curve slope) | 2.21 | 1.04 |

Coefficient of variation (COV) (%) | 91.29 | 24.59 |

Mean absolute percentage error (MAPE) (%) | 34.15 | 18.97 |

Percentage of beams with overestimated strength (%) | 13.23 | 39.97 |

Theoretical concrete shear strength versus experimental concrete shear strength | ||

$\raisebox{1ex}{${V}_{\mathrm{test}}$}\!\left/ \!\raisebox{-1ex}{${V}_{n}$}\right.$ | Alam [23] | Kara [24] |

Arithmetic mean | 1.25 | 1.20 |

Parameter X (inverse of regression curve slope) | 1.70 | 1.27 |

Coefficient of variation (COV) (%) | 40.54 | 56.75 |

Mean absolute percentage error (MAPE) (%) | 21.89 | 22.05 |

Percentage of beams with overestimated strength (%) | 28.39 | 50.00 |

Theoretical concrete shear strength versus experimental concrete shear strength | ||

$\raisebox{1ex}{${V}_{\mathrm{test}}$}\!\left/ \!\raisebox{-1ex}{${V}_{n}$}\right.$ | Kurth [25] | Jang et al. [26] |

Arithmetic mean | 1.03 | 1.25 |

Parameter X (inverse of regression curve slope) | 1.26 | 1.13 |

Coefficient of variation (COV) (%) | 27.11 | 57.86 |

Mean absolute percentage error (MAPE) (%) | 19.09 | 33.80 |

Percentage of beams with overestimated strength (%) | 54.52 | 46.45 |

Theoretical concrete shear strength versus experimental concrete shear strength | ||

$\raisebox{1ex}{${V}_{\mathrm{test}}$}\!\left/ \!\raisebox{-1ex}{${V}_{n}$}\right.$ | Lignola et al. [27] | Valivonis et al. [28] |

Arithmetic mean | 1.07 | 0.98 |

Parameter X (inverse of regression curve slope) | 1.37 | 1.08 |

Coefficient of variation (COV) (%) | 64.64 | 39.97 |

Mean absolute percentage error (MAPE) (%) | 31.91 | 32.81 |

Percentage of beams with overestimated strength (%) | 64.84 | 65.48 |

Theoretical concrete shear strength versus experimental concrete shear strength | ||

$\raisebox{1ex}{${V}_{\mathrm{test}}$}\!\left/ \!\raisebox{-1ex}{${V}_{n}$}\right.$ | Thomas et al. [29] | Hamid et al. [30] |

Arithmetic mean | 1.06 | 1.38 |

Parameter X (inverse of regression curve slope) | 1.42 | 1.52 |

Coefficient of variation (COV) (%) | 33.86 | 40.66 |

Mean absolute percentage error (MAPE) (%) | 55.16 | 27.87 |

Percentage of beams with overestimated strength (%) | 21.18 | 23.87 |

Theoretical concrete shear strength versus experimental concrete shear strength |

**Table 5.**Comparison of experimental (V

_{test}) and theoretical (V

_{n}) values of the concrete shear strength for the proposed model.

$\raisebox{1ex}{${\mathit{V}}_{\mathbf{test}}$}\!\left/ \!\raisebox{-1ex}{${\mathit{V}}_{\mathit{n}}$}\right.$ | Proposed Model (21) |
---|---|

Arithmetic mean | 1.00 |

Parameter X (inverse of regression curve slope) | 1.03 (R^{2} = 0.87) |

Coefficient of variation (COV) (%) | 22.50 |

Mean absolute percentage error (MAPE) (%) | 18.62 |

Percentage of beams with overestimated strength (%) | 52.90 |

Theoretical concrete shear strength versus experimental concrete shear strength |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bywalski, C.; Drzazga, M.; Kamiński, M.; Kaźmierowski, M.
A New Proposal for the Shear Strength Prediction of Beams Longitudinally Reinforced with Fiber-Reinforced Polymer Bars. *Buildings* **2020**, *10*, 86.
https://doi.org/10.3390/buildings10050086

**AMA Style**

Bywalski C, Drzazga M, Kamiński M, Kaźmierowski M.
A New Proposal for the Shear Strength Prediction of Beams Longitudinally Reinforced with Fiber-Reinforced Polymer Bars. *Buildings*. 2020; 10(5):86.
https://doi.org/10.3390/buildings10050086

**Chicago/Turabian Style**

Bywalski, Czesław, Michał Drzazga, Mieczysław Kamiński, and Maciej Kaźmierowski.
2020. "A New Proposal for the Shear Strength Prediction of Beams Longitudinally Reinforced with Fiber-Reinforced Polymer Bars" *Buildings* 10, no. 5: 86.
https://doi.org/10.3390/buildings10050086