1 Introduction
Steel fiber reinforced concrete (SFRC) is a concrete, where steel fiber was added. The benefit of steel fiber includes shear reinforcement, blast-resistant, precast, slope stabilization, repairs, etc. (Banthia & Trottier,
1994). The steel fiber can improve the concrete abrasive resistance (Mansouri et al.,
2020). It also plays a role in determining the failure mode of reinforced concrete beams with steel fiber (RSF beams) after fire (Antonius et al.,
2020). Moreover, it can be recycled for a more environmentally friendly application (Qin & Kaewunruen,
2022). In addition, it will act as crack arrestors similar to that of aggregates (Swamy & Mangat,
1974). This will increase the tensile resistance of concrete by controlling crack opening. The ductility will increase significantly, since steel fiber will continue to transfer forces across the cracks (Marti et al.,
1999). The addition of steel fiber increases the shear capacity and tends to change the failure mode from brittle to ductile (Khuntia et al.,
1999). Furthermore, there were also applications with other materials, for example, an ultra-high-performance fibre-reinforced concrete (UHPFRC) made with hybrid fibers (steel and polypropylene) (Nuaklong et al.,
2020).
For the shear failure of reinforced concrete (RC) beams, there are four failure modes called diagonal tension, shearing, shear compression, and web crushing (Placas & Regan,
1971). To estimate the shear capacity of RC beams, most codes of practice use semi-empirical relations based on the available experimental results (Tan et al.,
1993). It has been shown that the randomly dispersed and oriented steel fibers in concrete can enhance the mechanical properties of RC beams (Birincioglu et al.,
2022).
There are beam action and arch action in the shear resisting mechanisms of RC beams (Gunawan et al.,
2020). In the case of RC short beams, the final failure will be caused by a dowel failure or crushing of the compression zone (shear compression) (MacGregor,
1997). In such a case, the arch action is dominant, the compressive strut is the important component for the estimation of the shear capacity. By being able to calculate the compression force in the compressive strut, it is considered that the shear capacity can be estimated. The compression force can be calculated using the width of compressive strut and the compressive stress occurring in the strut.
Digital image correlation (DIC) can capture the displacement distribution that occurs on the surface of the object. It is able to memorize the pattern of the black and white on the object at the original stage in the photo image. The pattern can change after the object is deformed and DIC can analyze the differences between the changed pattern and the original pattern. Then, the strain can be calculated with the displacement field that contains displacements at many points in the object. For the preparation of DIC applied to the loading test of RC beams, the surface of beam specimens must be sprayed in black and white to create an original pattern for the analysis.
Normally, the image analysis and DIC were used to find the crack width and strain distribution. Their previous researches in the structural concrete field are summarized below.
Jongvivatsakul et. al. (
2011) applied the image analysis (using the grid of colored stickers as a pattern on the concrete surface) to obtain the crack width in RSF slender beams (
a/
d = 2.8) and proposed the prediction method for the shear capacity. Lee et. al. (
2019) used the image analysis to investigate the cracking and fracture stages in RC beams. Moreover, Zarrinpour and Chao (
2017) used DIC to investigate the crack width and strain distribution to calculate the contributions of compression zone, bridging effect, and dowel action in detail to the shear capacity of RSF slender beams. There was also an application of DIC to monitor the diagonal crack width in high strength (around 60–65 MPa) RC and RSF short beams. The results showed that the steel fiber delayed the appearance of the diagonal crack (Tahenni et al.,
2016). DIC was used for the crack propagation analysis, such as crack width and strain contours in synthetic, steel, hybrid fiber reinforced concrete beams (Bhosale & Prakash,
2020). DIC for crack measurement of RC beams can be verified by comparing the visual and DIC results (Destrebecq et al.,
2011). Furthermore, there was a combination of fiber-optic sensors and DIC for damage detection, localization in structural health monitoring (Morgese et al.,
2021).
However, in the case of RSF short beams, the method to estimate the width of compressive strut has not been developed so far. This research attempted to estimate the shear capacity of RSF short beams. To achieve this, the focus of this research is the compressive strut which is the main contribution in the shear capacity of RSF short beams. If the force in the compressive strut can be calculated, the shear capacity can be estimated. The force in the compressive strut is strongly related to the width of the compressive strut and the compressive stress. The compressive stress can be calculated using the compressive strength, while the width of compressive strut needs to be investigated.
Therefore, the objective of this study is to estimate the shear capacity of RSF short beams by investigating the width of compressive strut using DIC. The aim is to capture the contribution of compressive strut directly for the shear capacity of RSF short beams. To achieve this, DIC will be used to observe the minimum principal strain (Ɛ2) for the estimation of the width of compressive strut.
6 Calculation of Shear Capacity Using the Width of Compressive Strut from DIC
In this research, compressive strut and tension tie were assumed to bear only axial forces, as shown in Fig.
1. By considering the equilibrium of vertical forces at the supporting point, Eq.
1 can be obtained. Equation
2 describes the calculation of shear capacity using the force and angle of compressive strut:
$$V_{{{\text{DIC}}}} = D\sin \theta ,$$
(1)
$$V_{{{\text{DIC}}}} = f_{{\text{c}}}{^{\prime}} b_{{\text{w}}} kw\sin \left( {\cot^{ - 1} \left( {a/d} \right)} \right),$$
(2)
where
VDIC is the shear capacity calculated from DIC (N),
D is the force in compressive strut,
θ is the angle of compressive strut,
bw is the web thickness (mm),
k is the reduction factor regarding the compressive stress in cracked concrete and width of compressive strut at the peak load,
w is the width of compressive strut (mm).
If the full compressive strength and full width of strut were used for the calculation of the shear capacity (k = 1.00), it will be unconservative. By considering the failure and cracking of concrete at the peak load, cracked concrete at the peak load will not provide the full compressive strength.
According to Vecchio and Collins (
1986), cracked concrete subjected to high tensile strains in the direction perpendicular to the compression is softer and weaker in compression than concrete in a standard cylinder test. In addition, in this calculation, the width of compressive strut was determined from the distribution of
Ɛ2 and that the boundary of the strut was defined as where the
Ɛ2 reached zero.
However, judging from the calculation in Table
6, it was unconservative when
k = 1.00, giving the average accuracy (ratio between experimental and calculated values in each specimen) of 0.43. It is because the boundary of the strut is not reasonable, where
Ɛ2 reached zero. Moreover, the distribution of
Ɛ2 is not uniform in the section. However, the uniform distribution was assumed in this calculation. Hence, the compressive stress and the width of compressive strut should be reduced using a reduction factor from the estimation by Eq.
2. The accuracy was checked and the reduction factor,
k was proposed to be 0.43 in all the cases.
Table 6
Accuracy of shear capacity calculation by different equations
1 | 0.5 | 41.0 | 1.0 | 150 | 45 | 297.4 | 142 | 0.48 | 1.12 | 2.48 | 1.33 |
2 | 1.5 | 34 | 224.8 | 147a | 0.45 | 1.04 | 2.60 | 1.28 |
3 | 2.0 | 27 | 149.0 | 117 | 0.46 | 1.08 | 2.14 | 1.13 |
4 | 1.0 | 52.5 | 1.0 | 45 | 402.8 | 232 | 0.31 | 0.73 | 2.56 | 1.59 |
5 | 1.5 | 34 | 276.2 | 156 | 0.41 | 0.94 | 2.31 | 1.35 |
6 | 2.0 | 27 | 184.8 | 112 | 0.47 | 1.09 | 1.83 | 1.15 |
Average | 0.43 | 1.00 | 2.32 | 1.31 |
Coefficient of variation (CV) (%) | 14.8 | 14.8 | 12.7 | 12.9 |
To show the validity of the proposed method, the calculation of the shear capacity by the proposed method was compared with the existing methods. The existing methods were selected from the existing studies that proposed an equation for the shear capacity of RSF short beams. Equation
3 (Khuntia et al.,
1999) shows an existing equation for the calculation of the shear capacity which accounts the effect of steel fibers. This equation was chosen, because the effect of shear span ratio was included in the formulation. By considering the arch action of compressive strut, the equation is applicable for short beams:
$$V_{{{\text{Khu}}}} = \left[ {\left( {0.167\alpha + 0.25F} \right)\sqrt {f_{\text{c}}{^{\prime}} } } \right]b_{{\text{w}}} d,$$
(3)
where
VKhu is the shear capacity (N),
α is the arch action factor (
α = 2.5
d/
a < 3 for
a/
d < 2.5; 1 for
a/
d ≥ 2.5),
F is the fiber factor (
F = (
Lf/
Df)
pf df/100),
Lf is the length of steel fiber (mm),
Df is the diameter of steel fiber (mm),
pf is the volume fraction of steel fiber (%).
df is the bond factor (
df = 1 for hooked or crimped steel fibers; 2/3 for plain or round steel fibers with normal concrete; 3/4 for hooked or crimped steel fibers with lightweight concrete).
Shahnewaz and Alam (
2014) found that the interactions between important parameters such as shear span ratio, aspect ratio, volume fraction of steel fiber are significant. Therefore, nonlinear relationships were proposed in the equations. A simplified equation for short beams is shown in the following equation:
$$\begin{aligned} V_{{{\text{Sha}}}} & = \left[ {0.2 + 0.034f_{\text{c}}{^{\prime}} + 19p_{{\text{w}}}^{0.087} {-}5.8\left( {{a \mathord{\left/ {\vphantom {a d}} \right. \kern-0pt} d}} \right)^{1/2} + 3.4p_{{\text{f}}}^{0.4} {-}800\left( {{{L_{{\text{f}}} } \mathord{\left/ {\vphantom {{L_{{\text{f}}} } {D_{{\text{f}}} }}} \right. \kern-0pt} {D_{{\text{f}}} }}} \right)^{ - 1.6} {-}12\left( {\left( {{a \mathord{\left/ {\vphantom {a d}} \right. \kern-0pt} d}} \right)p_{{\text{f}}} } \right)^{0.05} } \right. \\ & \quad \left. {{-}197\left( {\left( {{a \mathord{\left/ {\vphantom {a d}} \right. \kern-0pt} d}} \right)\left( {{{L_{{\text{f}}} } \mathord{\left/ {\vphantom {{L_{{\text{f}}} } {D_{{\text{f}}} }}} \right. \kern-0pt} {D_{{\text{f}}} }}} \right)} \right)^{ - 1.4} + 105\left( {p_{{\text{f}}} \left( {{{L_{{\text{f}}} } \mathord{\left/ {\vphantom {{L_{{\text{f}}} } {D_{{\text{f}}} }}} \right. \kern-0pt} {D_{{\text{f}}} }}} \right)} \right)^{ - 2.12} } \right]b_{{\text{w}}} d, \\ \end{aligned}$$
(4)
where
VSha is the shear capacity (N),
pw is the tensile reinforcement ratio (%),
pf is the volume fraction of steel fiber (%),
Lf is the length of steel fiber (mm),
Df is the diameter of steel fiber (mm).
Table
6 shows the accuracy of calculation of the shear capacity from the proposed equation and the existing equations.
Totally six RSF short beams were used for the estimation of the shear capacity. Beams were fabricated with the volume fraction of steel fiber from 0.5 to 1.0% and the shear span ratio ranging from 1.0 to 2.0. The accuracy of the proposed estimation method by DIC and other equations was compared. The proposed method by DIC estimated the shear capacity more accurately, while the variation was slightly higher compared to the others. Moreover, the accuracy of Eq.
3 was significantly lower than that of Eq.
4. By observing each equation, the calculation in Eq.
3 involves arch action factor, fiber factor, and compressive strength in a form of nonlinear equation, while Eq.
4 is also a nonlinear equation but includes an additional tensile reinforcement ratio and expresses a more complicated form. The experimental results in this research also shows that there is a correlation between the shear span ratio and the volume fraction of steel fiber. Therefore, a more complicated expression might be a solution for the calculation of the shear capacity. In addition, Eq.
4 might be more updated and based on a larger number of data.
The proposed method by DIC was able to grasp the width of compressive strut to be used for the estimation of the shear capacity of RSF short beams, while the other equations used the effect of related parameters only, without directly considering the contribution of the compressive strut. There is still a reduction factor in the proposed equation. However, DIC was used as the tool for investigating the compressive strut for the equation. This is the significant addition in the estimation of the shear capacity.
This shows that it is very important to study and reflect the effect of the applied load on the beam directly such as the strain distribution in the beams for the estimation of the shear capacity. According to MacGregor (
1997), after diagonal cracking or steel unbonding, the shear flow cannot be transmitted. The load after the interruption of shear flow was carried by arch action in the form of compressive strut which represented concrete compressive stress fields.
The compressive stress caused the concentration of the compressive strain. By capturing the strain distribution that reflected the effect of the applied load directly during the loading test, the contribution to the shear capacity was straightforwardly estimated. In this research, the contribution of the compressive strut of which width was evaluated by the strain distribution obtained from DIC and the compression force was calculated by the compressive stress in cracked concrete.
This research relies significantly on DIC for the estimation of the shear capacity of RSF short beams. However, DIC requires the knowledge and experience for the application. Moreover, there are limitations such as light sensitivity, preparation of measurement surface, loss of data points after spalling of concrete (Gencturk et al.,
2014).
In this research, the aim is to estimate the width of compressive strut by DIC. From the results of DIC, the width can be estimated. The ultimate strain in compression of concrete is generally considered to be 0.0035. The results from DIC around the compressive strut were less than and more than this ultimate strain. Nevertheless, the parabolic functions estimated from DIC can be used to estimate the width of compressive strut for the estimation of the shear capacity of RSF short beams. Moreover, the strain distribution at A, B and C was corresponding to the Saint–Venant’s Principle, the strain values decreased at B and C which are distanced from the loading and supporting points. Therefore, the results of DIC is reasonable, because they were corresponding to the theory of mechanics of structure.
The details and discussions regarding DIC will be described in the following. The subset size was 25 × 25 and the step was 12 × 12. The gage length was 30 mm in all cases. The virtual gage length was calculated using the gage length of 30 mm, the step, and the calibration factor for conversion between units of mm and pixels. Therefore, the strain filter size is varied in each case. The significant strain noise floor, such as due to imperfections of sprayed random pattern on specimens was not analyzed with DIC. Nevertheless, DIC in this research was conducted until near failure of specimens. The noise floor should be small compared to the strain due to loading tests. One of the causes of bias noise is the resolution of luminance differences between pixels. Furthermore, the bias noise also occurs from the movement of camera or light or the unexpected movement of a specimen unrelated to the loading test. In this research, the resolution was improved by accurately interpolating luminance values between pixels using bi-cubic interpolation. The deflections of specimens at the peak load ranged from 2.26 to 4.56 mm so movements of camera, light or specimens should be smaller than the deflection. Nevertheless, the typical pattern of bias such as stripe was not found and the results were able to be used for the estimation of the width of compressive strut for the estimation of the shear capacity of RSF short beams. Therefore, the bias noise was significantly smaller than that of the strain due to loading tests.
The gage length was fixed as 30 mm in this research. In addition, the maximum coarse aggregate size in this research was 20 mm. In concrete, when a crack occurred, if the crack passed through coarse aggregate of 20 mm size and the gage length was 20 mm, the measured strain will be incorrect, since the coarse aggregate is a hard material. Therefore, the application of a gage length larger than the maximum coarse aggregate size is preferred. According to Koohbor et. al. (
2017), a smaller virtual strain gage is good for the calculation of highly localized strain magnitudes, but will give a noisier strain distribution. On the contrary, a larger virtual strain gage will smooth the strain distribution but decrease the local strain information. In addition, according to Koohbor et. al. (
2017), the increase in the strain filter size reduces the noise, because the virtual strain gage is increased making the data points increase. In the case of the subset size, from Pan et. al. (
2008), the subset size is the size of area for tracking the displacement, it is critical to the accuracy of measured displacement. The subset size should be large enough for recognition of the distinctive pattern, but comes with larger error. The main correlation parameters such as subset, step and strain filter size can be optimized through sensitivity analyses for an accurate measurement (Koohbor et al.,
2017).
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.