3.3.1 Prediction of Shear Strength
Several studies attempted to predict equations for the shear capacity of fiber RC beam to contribute to shear design method. Data analysis of several studies indicates that the behavior of SFRC beam remains a complex problem (Aoude et al.
2012; Foster
2010; Sharma
1986; Al-Ta’an and Al-Feel
1990; Swamy et al.
1993; Kuntia et al.
1999; Yakoub
2011; RILEM TC 162-TDF
2000b; AFGC
2013). All the proposed calculation methods for shear load capacity of beams were divided into the following three terms: (1) shear resistance of the member without shear reinforcement due to the compression zone, longitudinal reinforcement, aggregate interlock et al., (2) contribution of shear reinforcement, (3) contribution of steel fibers, as shown in Eq. (
1) as follows:
$$V_{d} = \, V_{c} + \, V_{s} + \, V_{fib}$$
(1)
where
Vd denotes the shear load capacity,
Vc denotes the shear resistance of the member without shear reinforcement,
Vs denotes the shear resistance of contribution of the shear reinforcement, and
Vfib denotes the shear resistance of contribution of steel fibers.
Generally, in order to compute the shear strength of the reinforced concrete, the shear resistance of the concrete matrix is computed by using the following equation: \(V_{c} = \emptyset_{c} \beta \sqrt {f^{\prime}_{c} } b_{w} d_{v}\), where bwdv denotes the effective shear area represented by the web width multiplied by the effective shear depth of beam. Additionally, the shear resistance of contribution of the shear reinforcement is based on the number of shear reinforcement.
Therefore, in this section, the ultimate shear strength obtained by the experimental investigation and the predicted value from design guidelines and that proposed by extant studies is compared (Table
9). These predictions of shear capacity for effect of fiber were mainly divided into the following two types: (1) separate from concrete contribution and accounted for by using fiber factor (Type 1), (2) included in concrete contribution and accounted for by using SFRC material properties in tension (Type 2). The predictions of shear strength by RILEM TC-TDF recommendation (RILEM TC 162-TDF
2000b) and Hassan et al. (Aoude et al.
2012) is a typical equation for each concept.
Table 9Summary of a typical predictions of shear strength.
| Vfib, SAK = kft(d/a)0.25 |
Al-Ta’an and Al-Feel ( 1990) | Vfib, AT = \(\frac{8.5}{9}\)kf(Vf\(\frac{{L_{f} }}{{d_{f} }}\))bwd |
| Vfib, SRN = 0.37τ(Vf\(\frac{{L_{f} }}{{d_{f} }}\))bwd |
| Vfib,KM= 0.25β(Vf\(\frac{{L_{f} }}{{d_{f} }}\))\(\sqrt {f_{c} }\)bwd |
| Vfib, FSJ = 0.9kfdfwbwdcotθ |
| Vfib, HA = 0.83FpNfibbwdvcotθ |
| Vfib,YHE=2.5β\(\sqrt {f_{c} }\)(1 + 0.70Vf\(\frac{{L_{f} }}{{d_{f} }}R_{g}\))\(\frac{d}{a}\) for \(\frac{a}{d}\)≤2.5 Vfib,YHE= β\(\sqrt {f_{c} }\)(1 + 0.70Vf\(\frac{{L_{f} }}{{d_{f} }}R_{g}\)) for \(\frac{a}{d}\)>2.5 |
| Vfib,RIL= kfklτfibbwd |
| Vfib,AC=(0.9bwdσRd,f)tanθ |
| Vfib,SN= 0.22voffcfh/[(a-x0)bwdw] |
The RILEM TC 162-TDF recommendation (Yakoub
2011) that is a very widely used code for SFRC, and it was proposed to predict the equation for shear resistance of contribution of the shear reinforcement (
Vfib,RIL). It is based on the use of more than 100 test data of SFRC beam to establish a predictive relation for
Vfib,RIL, and to verify against the possibility of occurrence of a failure state similar to that of plain concrete (Lucie
2000; Vandewalle
2000). The details of the equation are as follows:
$$V_{fib,R} = k_{f} k_{l} \tau_{fib} b_{w} d$$
(2)
$$k_{f} = 1 + n\left( {\frac{{h_{f} }}{{b_{w} }}} \right)\left( {\frac{{h_{f} }}{d}} \right) \le 1.5$$
(3)
$$n = \frac{{b_{f} - b_{w} }}{{h_{f} }} \;{\text{where}}\;n \le 3\;{\text{and}}\;\frac{{3b_{w} }}{{h_{f} }}$$
(4)
where
kf denotes the factor taking into account the contribution of the beam section,
hf denotes effective depth of the section,
bf denotes width of the flanges,
bw denotes width of the web,
kl is equal (1600-d)/1000 and greater than or equal to 1, and
τfib denotes the design value of the increase in shear strength due to steel fibers (0.12
feqk,3).
Aoude et al. (
2012) suggested a prediction equation of shear strength resistance with respect to the contribution of steel fibers (
Vfib,HA) that is related to the pullout strength of the fibers crossing cracking plane and fibers randomly oriented in three dimensions. Additionally, a crack inclination
θ of the SFRC beam was considered under the free body diagram. Thus, the
Vfib,HA was calculated as follows.
$$V_{fib,HA} = 0.83F_{P} N_{fib} b_{w} dcot\theta$$
(5)
$$F_{P} = \left( {\tau_{fib} \pi d_{f} \frac{{L_{f} }}{2}} \right) + \Delta P^{\prime}$$
(6)
$$N_{fib} = \frac{{v_{f} }}{{A_{f} }} \alpha \eta_{l}$$
(7)
where
FP denotes the pullout strength of hooked-end steel fiber,
τfib denotes the bond strength between fiber and concrete matrix,
df denotes the fiber diameter,
Lf denotes fiber length,
∆P′ denotes the effect of hooked-end steel fiber on pullout test,
Nfib denotes the fibers randomly oriented in three dimension,
vf denotes the volume fraction of fibers in the matrix,
Af denotes the cross-sectional area of the fiber,
ɑ and
ηl denotes the orientation factor and length factor of the fiber, respectively.
With respect to the hooked-end steel fibers used in this study, the effect of the hook (
∆P′) is examined on the pullout test by debonding of the fiber that is the mechanical effect of the hook in the straightening process of the fiber on the pullout from the matrix (Aoude et al.
2012; Alwan et al.
1999). This parameter is calculated as follows:
$$\Delta P^{\prime} = \frac{3.05}{{{ \cos }\left( {45^\circ \times \pi /180^\circ } \right)}}\left( {f_{y} \frac{{\pi (d_{f} /2)^{2} }}{6}} \right)$$
(8)
where
fy denotes the fiber yield strength.
In this study, the average bond strength (
τfib) was obtained as 6.16 MPa, which is based on single fiber pullout test results (Fig.
5b). The fiber orientation factor (
ɑ) and length factor (
ηl) were suggested by using 3/8 and 0.5, respectively (Foster
2001; Aoude
2007).
3.3.2 Comparison of Shear Strength
Table
10 compares the shear strength value between experimental and prediction equations. The HSC beams without steel fiber and shear reinforcement exhibited 226.0 kN shear strength (Table
8), which is approximately the pure shear strength of the concrete beam. In the HSC beams with shear reinforcement, the shear resistance capacity of shear reinforcement (
Vs, TR) was calculated by Eq. (
1), and the values of specimens with increases in shear reinforcement ratios were 163.4 kN, 267.7 kN, and 311.8 kN, respectively. Therefore, the shear resistance capacity increases with increases in the shear reinforcement ratios, which were approximately 63.8% and 90.8% higher than those of
Vs, 0.16 (minimum shear reinforcement). Additionally, the shear resistance capacity of steel fibers in the specimen with steel fibers is 185.1 kN, which approximately increases shear resistance by 13.2% when compared with the HSC beam with minimum shear reinforcement (
Vs, 0.16). Therefore, given the test results, the HSC use of fiber volume fraction of 0.75% can replace the minimum shear reinforcement through the use of hooked steel fibers as recommended by the ACI 318-19 Code.
Table 10Summary of predictions of the shear resistance of contribution of steel fibers.
Type 0 |
Vfib, exp | – | 185.1 | 100 | Fiber effect is: • calculated by experimental value |
Vs, 0.16 | 163.4 | – | 88.3 |
Vs, 0.21 | 267.7 | – | 144.6 |
Vs, 0.32 | 311.8 | – | 168.4 |
Type 1 |
Vfib, AT | – | 71.7 | 38.7 | Fiber effect is: • separate from concrete contribution • accounted for by using fiber factor |
Vfib, SRN | – | 97.2 | 52.5 |
Vfib, KM | – | 135.0 | 72.9 |
Vfib, HA | – | 205.3 | 110.9 |
Vfib, YHE | – | 154.5 | 83.5 |
Type 2 |
Vfib, SAK | – | 93.2 | 50.4 | Fiber effect is: • included in concrete contribution • accounted for by using SFRC material properties in tension |
Vfib, FSJ | – | 83.9 | 45.3 |
Vfib, RIL | – | 53.1 | 28.7 |
Vfib, AC | – | 81.4 | 44.0 |
Vfib,SN | – | 58.9 | 31.8 |
A comparison of the predicted shear strength using the proposed Type 1 equation and the experimental value, due to consider only a factor influencing the factor reflecting the fiber shape (k) or specific pullout tests on the FRC (τ), the predicted shear strength through equation Vfib, AT and Vfib, SRN resulted approximately 38.7–52.5% lower than Vfib, exp. The predicted shear strength through equation Vfib, KM and Vfib, YHE were close to the experimental values although they also below 16.5% of the suitable value when compared with Vfib, exp. It is based on considering an additional factor, namely the factor of fiber shape and concrete type (strength). Additionally, the equation Vfib, HA considered more than five factors in the FRC, namely fiber orientation factor, fiber length factor, bond-shear strength between fiber and matrix, the contribution of the hook to fiber pull out strength, and angle of inclination of the shear crack. Hence, the predicted value of Vfib, HA only varied by 10.9% and was very closer to Vfib, exp.
Furthermore, when compared to the predicted shear strength that used the proposed Type 2 equation and experimental value, the following are obtained: the predicted shear strength equations were mainly accounted for by using steel fiber reinforcement concrete material properties in tension, which was based on the direct tensile or bending test. Therefore, the predicted shear strength values were less more than 50% when compared with the actual experimental value (Vfib, exp). Proving, once again, the predict shear strength using Type 2(accounted for by using material properties by tensile test) may lead to high data error.
Given the fore-mentioned equations, the Vfib.HA exhibited a more closes predicted shear strength value with a 10.9% difference in experimental values. A modification factor of 0.83 was employed in the equation as used by extant studies in normal strength concrete, and a size effect may play a role in reducing the shear resistance of test beams. Thus, modification factor and size effect should be determined for the optimal prediction of an equation to design the shear strength of the HSC with steel fibers in a future study.