Prediction of shear strength of reinforced concrete beams without shear reinforcement considering bond action of longitudinal reinforcements

Abstract

The shear strength of concrete beams without shear reinforcement is determined by the shear strength of the concrete compressive zone, the shear force due to dowel action, and the shear force due to aggregate interlock. The existing models and formulas to predict shear strength of concrete beams based on geometric theories related to beam action and arch action that only consider shear resistance in relation to arch action while neglecting dowel action and aggregate interlock have been unable to accurately explain the shear resistance mechanism. This study proposes a more rational shear strength prediction formula that reflects the contribution of each component by linking the bond characteristics of longitudinal reinforcements based on the size effect and stress changes in longitudinal reinforcement to dowel action. The precision of the proposed formula was assessed by carrying out experiments with the shear-span-to-depth ratio (a/d = 2.0, 2.5, 3.0, and 4.0) as a variable. A ratio of shear bond failure distance to effective depth and the shear strength of concrete beams without shear reinforcement were compared to those obtained from the proposed prediction formula. As a result, it was confirmed that the proposed formula gives rational predictions for each shear contribution and exhibits a high accuracy compared to previous experimental results.

Keywords

aggregate interlock bond action of longitudinal reinforcement diagonal crack dowel action shear bond failure shear-span-to-depth ratio shear strength

Introduction

According to the Building Code Requirement for Structural Concrete (ACI Committee 318, 2014), the shear strength of reinforced concrete (RC) beams is the sum of the contribution from concrete V_c and the contribution of shear reinforcement V_s. A relatively accurate prediction equation has been provided for the contribution of shear reinforcement, which is based on the truss theory. However, the contribution of concrete is highly influenced by material properties, while brittle failure due to diagonal cracks, bond cracks, and concrete crushing failure prevents accurate predictions of failure strength and time of occurrence.

Research on RC beams without shear reinforcement was conducted by Kani (1966) following the 1955 shear failure at the Wilkins Air Force Depot Warehouse in Shelby, Ohio, USA. This shear failure stimulated further studies on the shear strength of RC beams and predictions on their behavior. Based on previous experimental results, empirical and semi-empirical formulas have been proposed for the design of RC beams. The variables included size effect, tension reinforcement ratio, and compressive strength of concrete in relation to effective depth and shear-span-to-depth ratio of the beams. These formulas varied in their accuracy of shear strength prediction due to differences in development background, as well as the type and range of design variables. On the other hand, prediction formulas for shear resistance developed from mechanical and geometric theories such as beam action and arch action (Kani, 1964; Park and Paulay, 1975) have considered only the shear resistance mechanism due to the arch action of concrete while neglecting the bond action caused by the dowel action of longitudinal reinforcement and aggregate interlock. As such, these theories are not effective in providing a clear description of the shear resistance mechanism in terms of the contribution of each component.

Zararis and Papadakis (2001) used the ratio of the neutral axis depth to the effective depth of beams and the splitting tensile strength of concrete to predict shear strength of RC beams at failure. They found that the shear strength of beams is not influenced by the aggregate interlock and dowel action of longitudinal reinforcement and that the concrete compression zone resists tensile cracks. Choi et al. (2007) presented a failure mechanism based on interactions between the flexural moment and shear strength of concrete beams. While Reineck (1991), Walraven (1987), and Hsu (1988) also conducted research on the shear resistance mechanism of beam members, few studies have taken into account the vertical resistance arising from the dowel action of longitudinal reinforcement, the shear resistance due to aggregate interlock, and the shear resistance at concrete struts.

This study developed a shear resistance mechanism based on modeling the relationship between the bond action of longitudinal reinforcements and dowel action and proposed a shear strength prediction formula that considers the dowel action, aggregate interlock, and shear resistance in the concrete compressive zone. Previous experimental studies found that the shear resistance of RC beams without shear reinforcements is dominated by arch action when shear-span-to-depth ratio (a/d) is less than 2.5 and by beam action when a/d is greater than 2.5 and is less than or equal to 5.6. To quantize the shear force due to dowel action considering bond action of longitudinal reinforcements in RC beams with a/d in the range of 2.0–4.0, the experiments were carried out with the shear-span-to-depth ratio (a/d = 2.0, 2.5, 3.0, and 4.0) as a variable. Then the analytical results were compared with previous experimental results such as the ratio of shear bond failure distance to effective depth and the shear strength of RC beams to assess the precision of the proposed formula.

Prediction of shear strength of RC beams in consideration of bond action of longitudinal reinforcements

Stress distribution of longitudinal reinforcements of RC beams with cracks

Figure 1 shows the tension force distribution of longitudinal reinforcements of RC beams at shear failure. As shown in Figure 1, the RC beams subjected to bending moment and shear force develop flexural cracks in the mid-bottom after reaching the cracking moment. The flexural cracks increase with load acting on the beams and progress to the m–m′ position, while flexural cracks at the m–m′ position do not occur in the range of x due to the behavior of concrete under multiaxial stress state rather than the tensile stress state. It is well known that the distance of flexural cracks and diagonal cracks from supporting points and the shear strength of RC beams are affected by the longitudinal reinforcement ratio and the shear-span-to-depth ratio (Park and Paulay, 1975; Wight and MacGregor, 2009). The last flexural crack occurring up to the m–m′ cross section develops into flexural-shear cracks, and the beams experience failure due to diagonal cracks and shear bond failures along the longitudinal reinforcements. As shown in Figure 1, the stress distribution of longitudinal reinforcement at shear failure is non-linear from the supporting point up to m–m′, where the greatest stress gradient is observed. The stress gradient of longitudinal reinforcements represents bond stress, and Figure 1 shows that the greatest bond stress is exhibited near the m–m′ cross section. The RC beams without shear reinforcement were assumed to have experienced shear bond failures, which were caused by an increase in local bond stress at x that takes place at the same time as the diagonal cracks develop at m–m′, followed by the rapid spreading of shear bond failures over the range of x, and ultimately, failure.

Figure 1.

Tensile stress distribution of longitudinal reinforcements at shear failure.

Figure 2 depicts the assumed stress distribution and bond stress of longitudinal reinforcements at shear failure. The difference between the actual bond stress acting on the longitudinal reinforcements and their stress gradient results in the distribution of Figure 2(a). The actual bond stress at x, as shown in Figure 2(a), is significant at the point of the last flexural crack, but the concept of average bond stress was applied to replace it with equivalence bond stress, represented by $τ_{c o}$ in Figure 2(b). The maximum bond stress is exhibited near m–m′, where the last flexural crack occurred, and bond stress is assumed to be effective for 1/3 of x, the range of shear bond failure. The sum of tension forces in longitudinal reinforcements acting in this range, represented by T₂, can be expressed in terms of the average bond stress ( $τ_{a v e}$ ) of longitudinal reinforcements and the distance x

T_{2} = τ_{a v e} \sum ψ x

(1)

Figure 2.

Assumed stress distribution and bond stress of longitudinal reinforcement at shear failure: (a) actual and equivalent bond stresses on longitudinal reinforcements at shear bond failure distance x and (b) a concept of average bond stress.

The sum of all tension forces acting over the range x is represented by T, which is calculated as the sum of tension force T₁ at 2x/3, the tension force T₃ at $a - x$ , and T₂

T = T_{1} + T_{2} + T_{3}

(2)

Here, x is the distance from the supporting point up to the final flexural crack or the range of shear bond failure, $\sum ψ$ is the sum of nominal parameters of longitudinal reinforcements ( $= n π d_{b}$ ), n is the number of longitudinal reinforcements, d_b is the diameter of longitudinal reinforcements, and $τ_{a v e}$ is the average bond stress over x at failure or $τ_{c o} / 3$ . $τ_{c o}$ , the bond strength due to concrete, as proposed by Fuji and Morita (1982), is given by equations (3) and (4)

τ_{c o} = (0.096 b_{i} + 0.134) \sqrt{{f'}_{c}}

(3)

b_{i} = \frac{b}{n d_{b}} - 1

(4)

Here, b_i is the coefficient representing the shape of bond cracks, b is the width of beams, and $f'_{c}$ is the compressive strength of concrete.

Shear bond failure distance x

RC beams with a shear-span-to-depth ratio greater than 2 can experience failure due to diagonal cracks and shear bond failure along the longitudinal reinforcement (Kim and White, 1991). As shown in Figure 2, the shear bond failure occurs over x, and this is influenced by the shear-span-to-depth ratio.

Figure 3 shows the shear failure distance in relation to the shear-span-to-depth ratio a/d, and the ratio of effective depth x/d is based on 44 experimental results on RC beams with shear reinforcement (Alam and Hussein, 2013; Bentz and Buckley, 2005; Choi et al., 2010; Ghannoum, 1998; Jeong et al., 2014, 2016; Kani, 1967; Sneed and Ramirez, 2010; Walraven and Lehwalter, 1994). With an increasing shear-span-to-depth ratio, x/d tends to increase. When the shear-span-to-depth ratio rises to 4, x/d converges to 1.5. Based on previous research, this study proposed the distance from the supporting point up to the final flexural crack x using equation (5)

x = \frac{4 d (a / d - 1)}{3 {0.9 + {(a / 3 d)}^{2}}}

(5)

Here, x is the distance from the supporting point up to the final flexural crack with $x \leq 1.5 d$ , a/d is the shear-span-to-depth ratio, d is the effective depth of beams, and a is the shear span.

Figure 3.

Relation between shear-span-to-depth ratio and ratio of shear bond failure distance to effective depth.

Shear strength contribution of RC beams without shear reinforcement

As given in equation (6), the shear force V_c of RC beams without shear reinforcement was assumed to face resistance from the shear force V_cc in the concrete compressive zone, the shear force V_d due to the dowel action considering the bond action of longitudinal reinforcements, and the shear force V_ay due to aggregate interlock. The contributions of the shear strength prediction formula were derived using the sum T of longitudinal reinforcement, and the size effect correction coefficient k_d was proposed by reflecting on the influence of the size effect on each contribution

V_{c} = k_{d} (V_{c c} + V_{d} + V_{a y})

(6)

It is well known that while the arch action dominates the shear strength of concrete beams when shear-span-to-depth ratio (a/d) is less than 2.5, the beam action dominates that of concrete beams when a/d is greater than 2.5 and is less than or equal to 5.6. This study was conducted to quantize the shear force V_d in RC beams with a/d in the range of 2.0–4.0. Equation (6), thus, is limited to a/d in the range of 2.0–4.0. Additional research is needed to apply equation (6) to the case of $a / d < 2.0$ or $a / d > 4.0$ .

Shear force in the compressive zone of concrete

Figure 4 shows the flow of stress directly delivered to the load acting on a loading point through the concrete struts formed by an arch action. The neutral axis depth c of RC beams with cracks, which corresponds to the strain distribution of longitudinal reinforcement, may be assumed constant over the distance between the loading point and the point of the last flexural crack, or $a - x$ , and grows deeper toward the supporting point. The distance between the centers of longitudinal reinforcement and concrete j_d grows shorter toward the supporting point, which directly receives the load acting on the loading point through the concrete strut formed by the arch action. The shear force V_cc at the concrete compressive zone, which is the shear force delivered by $T_{1} + T_{3}$ at the angle $θ_{1}$ , is the normal component of the sum T of longitudinal reinforcement calculated by equation (2). As shown in Figure 2(a), $T_{1} + T_{3}$ was assumed to have a stress similar to T₂, thus allowing the shear force V_cc at the concrete compressive zone to be calculated using equation (7)

V_{c c} = (T_{1} + T_{3}) \tan θ_{1}

(7)

Here, $\tan θ_{1}$ can be expressed as $(h - c) / (x - s)$ , where h is the total member height and c is the neutral axis depth or $(d - j_{d}) / 3$ . $j_{d}$ ( $= 0.875 d$ ) is the distance between the center of forces. s is the distance determined by the angle $θ_{1}$ between the neutral axis depth c and the vertical plane of the concrete strut. Geometrically, it is calculated by

s = 0.5 [x - \sqrt{x^{2} - 4 (h c - c^{2})}]

(8)

Figure 4.

Shear resistance mechanism in concrete compressive zone by arch action.

Shear force due to dowel action considering bond action of longitudinal reinforcements

Figure 5 presents the shear resistance mechanism due to the dowel action that considers the bond action of longitudinal reinforcements. The RC beams without shear reinforcement were assumed to have experienced shear bond failures and ultimate shear failure due to the increase in local bond stress at x, which took place at the same time as the diagonal cracks developed. After the development of local bond cracks at the point of the last flexural crack, shear bond failure was assumed to have spread rapidly over the entire range of x. As described previously, the bond stress results from difference in stress acting on longitudinal reinforcements, and bond failure occurs when the bond stress on longitudinal reinforcements reaches the average bond strength $τ_{a v e}$ acting over x. As such, the bond stress in this range can be obtained using the concept of average bond stress, as shown in equation (1). The shear force V_d due to the dowel action considering the bond action of longitudinal reinforcements, as shown in Figure 5, is the shear force delivered due to the angle $θ_{2}$ formed between the tension force T₂ and x. This is given by equation (9)

V_{d} = T_{2} \tan θ_{2} = (τ_{a v e} \cdot \sum ψ x) \tan θ_{2}

(9)

Here, $\tan θ_{2}$ is $d / (a - x / 2)$ .

Figure 5.

Shear resistance mechanism due to dowel action of longitudinal reinforcement.

Shearing force due to aggregate interlock

Eurocode 2 (British Standards Institution (BSI), 2004) defines the maximum shear stress delivered due to aggregate interlock as the coulomb failure criteria ( $τ = C + μ σ_{c r}$ ) (Sagasteta and Vollum, 2011). Since $μ σ_{c r}$ ( $= ρ_{v} f_{v}$ ) is the contribution due to shear reinforcement at failure, this term was excluded from the proposed prediction formula for RC beams without shear reinforcement. In accordance with Eurocode 2, the shear force V_a at the crack surface was subject to the compressive strength of concrete as shown below

V_{a} = 0.4 (0.21 {f'_{c}}^{2 / 3} / γ_{c}) b \cdot d (f'_{c} < 50 MPa)

(10a)

V_{a} = 0.4 [1.48 \ln (1 + {f^{'}}_{c} / 10)] b \cdot d (50 MPa \leq {f^{'}}_{c} < 90 MPa)

(10b)

Here, $γ_{c}$ is the coefficient of concrete, or 1.5.

As shown in Figure 6, the shear force V_a forms the angle $θ_{3}$ in the horizontal direction of the member. V_ay, which is the normal component of V_a, is given by equation (11)

V_{a y} = V_{a} \sin θ_{3}

(11)

Here, $\sin θ_{3}$ can be expressed geometrically as $(d - c) / \sqrt{{(a - x)}^{2} + {(d - c)}^{2}}$ . a is the shear span, d is the effective depth of beams, and c is the neutral axis depth.

Figure 6.

Shear resistance mechanism due to aggregate interlock.

Influence of size effect

Several experimental studies on normal beams ( $a / d \geq 2.5$ ) reported that the shear strength of RC beams decreases with an increase in depth, and many of the researchers described this phenomenon as the size effect. Kani (1967) observed that about 40% of the shear strength of RC beams decreased when the effective depth was increased from 300 to 1200 mm. Walsh (1972) confirmed that a characteristic depth to observe the size effect on RC beams exists and it is about 225 mm. Bazant (1984) developed size effect law based on the dimensional analysis of strain energy release caused by macro crack growth. Jeong et al. (2016) also found that the rate of decrease in k_d with respect to the effective depth slows when the effective depth exceeds 400 mm.

To reflect the influence of size effect on each contribution, this study presented the size effect correction coefficient k_d based on 51 previous experimental results (Ahmad et al., 1986; Alam and Hussein, 2013; Bentz and Buckley, 2005; Choi et al., 2010; Ghannoum, 1998; Jeong et al., 2014, 2016; Kani, 1967; Lee et al., 2013; Mathey and Watstein, 1963; Sneed and Ramirez, 2010; Walraven and Lehwalter, 1994). Figure 7 shows that the shear strength tends to decrease with an increasing effective depth. By reflecting the decrease in shear strength with effective depth, the size effect correction coefficient k_d can be calculated using equation (12) proposed by Jeong et al. (2016)

k_{d} = 1.72 {(\frac{400}{d^{1.5}})}^{1 / 6}

(12)

Figure 7.

Size effect correction coefficient k_d.

Verification of the shear strength prediction formula with consideration of bond action of longitudinal reinforcement

Experiment design and details

This study assessed the precision of the proposed formula by conducting experiments with the shear-span-to-depth ratio as a variable. The compressive strength of concrete was set as 30 MPa, and the tension reinforcement ratio was set as 1.43% using three D-19 steel bars to attain shear failure before flexural failure. Table 1 and Figure 8(a) show the concrete mix design and the average stress–strain relationship for three concrete cylinders tested in accordance with the standard test method specified in ASTM C39/C39M, respectively. The compressive strength of concrete cylinders averaged 34 MPa on the test day. Table 2 and Figure 8(b) show the physical characteristics and the average stress–strain relationship for three D-19 steel bars tested in accordance with ASTM A615/A615M, respectively. To measure the strain of longitudinal reinforcement due to span, strain gauges were attached at regular intervals, as shown in Figure 9. All specimens were simply supported beams subjected to three-point loading, as shown in Figure 9, and the deflection of specimens was measured using two LVDTs installed at the bottom of loading points.

Table 1.

Concrete mix design.

Design strength (MPa)	W/C (%)	Unit weight (kg/m³)
		Water	Cement	Aggregate		Air entraining agent
				Fine	Coarse
30	51.7	177	301	936	844	2.39

Figure 8.

Stress–strain relationship of materials: (a) concrete and (b) D-19 steel bars.

Table 2.

Physical characteristics of steel bar.

Steel bar	Diameter (mm)	Area (mm²)	$f_{y}$ (MPa)	$ε_{y}$	$E_{s}$ (GPa)
D-19	19.1	286.5	476.5	0.0034	140.15

Figure 9.

Specimen details: (a) S2.0, (b) S2.5, (c) S3.0, and (d) S4.0.

Behavior assessment

All specimens reached shear failure before the yield of longitudinal reinforcement (see Figure 13). Figure 10 shows the shear force–deflection relationship for all specimens. The shear force and stiffness of all specimens decreased with an increasing shear-span-to-depth ratio, and the deflection at maximum shear force increased with the shear-span-to-depth ratio. The maximum shear force was 105.2 kN for S2.0 specimen and there was a similar force of 107.2 kN for S2.5 specimen. The S3.0 and S4.0 specimens, with a shear-span-to-depth ratio of 3.0 and 4.0, respectively, were significantly influenced by the bending moment with an increasing shear-span-to-depth ratio. Their maximum shear force was 70.5 and 62.7 kN, respectively. The maximum shear force of each specimen decreased by approximately 60% when the shear-span-to-depth ratio increased from 2 to 4. Table 3 presents the comparison of the design shear strength calculated by the proposed formula and ACI 318-14 (see Table 4) with the test results. ACI 318-14 is likely to underestimate the test results; however, the proposed prediction formula provides relatively accurate predictions for the specimens with $a / d < 3.0$ . In the case of $a / d \geq 3.0$ , on the other hand, ACI 318-14 provides more accurate predictions. This is because the proposed prediction formula employs the average bond stress $τ_{a v e}$ of longitudinal reinforcement to estimate the shear force due to dowel action, while ACI 318-14 directly considers bending moment to estimate the shear force due to dowel action and aggregate interlock.

Figure 10.

Shear force–deflection relationship.

Table 3.

Comparison of analytical results with test results.

Specimen	Variables					Analytical results		Test results
	${f_{c}}^{'}$	$b$	$d$	$a$	$ρ_{t}$	$V_{p r o}$	$V_{A C I}$	$V_{t e s t}$	$V_{p r o} / V_{t e s t}$	$V_{A C I} / V_{t e s t}$
	(MPa)	(mm)	(mm)	(mm)	(%)	(kN)	(kN)	(kN)	–	–
S2.0	34.0	200	300	600	1.43	117.5	71.1	105.2	1.12	0.68
S2.5	34.0	200	300	750	1.43	98.5	66.1	107.2	1.01	0.62
S3.0	34.0	200	300	900	1.43	89.8	63.5	70.5	1.27	0.90
S4.0	34.0	200	300	1,200	1.43	76.5	61.0	62.7	1.22	0.97

Table 4.

Existing prediction models for shear strength of beams without shear reinforcement.

Code or researcher	Prediction model (SI unit)	Variables
ACI 318-14	$V_{c} = (0.16 \sqrt{{f'}_{c}} + 17.6 ρ_{t} \frac{V_{u} d}{M_{u}}) b_{w} d$	$ρ_{t}$ , $a / d$ , and $f'_{c}$
CEB-FIP MC90	$V_{c} = (0.15 {(3 d / a)}^{1 / 3} ς {(100 ρ_{t} {f'}_{c})}^{1 / 3}) b_{w} d$	$d$ , $ρ_{t}$ , $a / d$ and $f'_{c}$
Bazant and Kim	$V_{c} = \frac{10 \sqrt[3]{ρ}}{\sqrt{1 + d / 25 d_{a}}} [0.083 \sqrt{{f'}_{c} + 20.69 \sqrt{ρ / {(a / d)}^{5}}}] b_{w} d$	$d$ , $ρ_{t}$ , $a / d$ , $f'_{c}$ , and $d_{a}$
Zsutty	$V_{c} = 2.17 {({f'}_{c} ρ_{t} \frac{d}{a})}^{1 / 3} b_{w} d (a / d \geq 2.5)$	$ρ_{t}$ , $a / d$ , and $f'_{c}$
	$V_{c} = 2.17 (2.5 \frac{d}{a}) {({f'}_{c} ρ_{t} \frac{d}{a})}^{1 / 3} b_{w} d (a / d < 2.5)$

Figure 11 shows the cracks on the specimens failed due to the spread of diagonal cracks and shear bond failure. All specimens developed flexural cracks on the mid-bottom of the member with the largest bending moment, and flexural cracks spread toward the upper area and the supporting point with an increasing load. The flexural cracks that developed toward the supporting point are in the form of flexural-shear cracks and shear cracks. Shear failure is reached when the diagonal cracks and shear bond failure spread toward the supporting point. As shown in Figure 11, x increases in proportion to the shear-span-to-depth ratio. Figure 12 compares the results of the experiments on x to previous experimental results based on the ratio calculated by equation (5). By applying equation (5), which is derived from the existing experimental results (Fujii and Morita, 1982; Ghannoum, 1998; Hsu, 1988; Jeong et al., 2014, 2016; Kani, 1964, 1966, 1967; Kim and White, 1991), the average shear bond failure distance and coefficient of variation were found to be 0.93% and 17.9%, respectively. This demonstrates that the proposed equation to predict x is fairly accurate and reasonable.

Figure 11.

Crack pattern at failure: (a) S2.0, (b) S2.5, (c) S3.0, and (d) S4.0.

Figure 12.

Comparison between test and analytical results for shear bond failure distance.

Strain distribution of experimental section

Figure 13 shows the strain distribution with varying load of longitudinal reinforcement within the experimental section using measurements obtained from strain gauges. The strain of all specimens increased linearly toward the loading point during initial loading. The S2.0 and S2.5 specimens experienced high strain at the edges and had nearly constant stress distributions up to the loading point after exhibiting high stress gradients near the diagonal cracks. The constant stress distribution for all spans can be attributed to the dominant influence of arch action over beam action due to the relatively small shear-span-to-depth ratio. The S3.0 and S4.0 specimens had almost no strain at the edges, but high stress gradients near diagonal cracks. Previous experimental studies have concluded that a major change in behavior occurs at a shear-span-to-depth ratio, $a / d$ , of about 2.0–2.5. In the case of $a / d < 2.5$ , arch action dominates the strength of RC beams, and the tension force in longitudinal reinforcements remains constant over the length of the shear span. In the case of $a / d \geq 2.5$ , on the other hand, beam action is more dominant than arch action, and the tension force in longitudinal reinforcements comes close to zero near the supporting points (Wight and MacGregor, 2009). The strain distribution in Figure 13 presents that the failure mode of S2.0 and S2.5 specimens was determined by arch action while that of S3.0 and S4.0 specimens was determined by beam action.

Figure 13.

Strain distribution with varying load of longitudinal reinforcement: (a) S2.0, (b) S2.5, (c) S3.0, and (d) S4.0.

Bond stress distribution of experimental section

Figure 14 shows the bond stress distribution for the shear span to depth with a longitudinal reinforcement strain at maximum strength calculated by equation (13). The bond stress at diagonal cracks becomes zero due to the loss of bond strength between concrete and longitudinal reinforcements. The bond stress at the same shear-span-to-depth ratio mostly acts at 1/3 of x, thus supporting the proposed shear resistance model that gives consideration to the bond action of longitudinal reinforcements

τ_{b o n d} = \frac{(ε_{2} - ε_{1}) E_{s} A_{s t}}{\sum ψ \cdot l_{d}}

(13)

Here, $ε_{1}$ and $ε_{2}$ are the strains of the longitudinal reinforcement having the distance $l_{d}$ , $E_{s}$ is the modulus of the elasticity of steel, $A_{s t}$ is the cross-sectional area of the longitudinal reinforcements, and $l_{d}$ is the length for the bond stress calculation.

Figure 14.

Bond stress distribution of longitudinal reinforcement at maximum strength; (a) S2.0, (b) S2.5, (c) S3.0, and (d) S4.0.

Comparison with previous experimental results

This study compared the results obtained from the proposed shear strength prediction formula to previous experimental results, including ACI 318-14 (ACI Committee 318, 2014), CEB-FIP MC90 (Comite Euro-International du Beton, 1990), Bazant and Kim (1984), and Zsutty (1971). The experimental data used in the comparison were limited to research containing reinforcement details. The effectiveness and accuracy of the proposed formula were verified based on a total of 55 experimental results, including this study. Table 4 shows the existing prediction models to estimate shear strength of beams without shear reinforcement. As shown in Figure 15, ACI 318-14 gave an average of 1.25 and a coefficient of variation of 20.6%. ACI 318-14 was unreliable and more likely to underestimate the results for an effective depth of 400 mm and a tension reinforcement ratio higher than 2%. CEB-FIP MC90 gave an average of 1.17 and a coefficient of variation of 17.4% and was likely to overestimate the results with an increasing effective depth and tension reinforcement ratio. Bazant and Kim (1984) and Zsutty (1971) gave an average distance of 0.98 and 1.11, respectively, and a coefficient of variation of 23.6% and 22.3%, respectively. They were more inclined to overestimate results with an increasing effective depth and tension reinforcement ratio. The proposed shear strength prediction formula, which takes into account the bond action of longitudinal reinforcements, provided reasonable results for all variables with an average of 0.96 and a coefficient of variation of 14.5%. The discrepancy between test and analytical results obtained from the existing prediction models may be summarized in the following terms: (1) those are empirical or semi-empirical models considering the shear force V_d only due to the dowel action and (2) those retain large safety factors due to uncertainties in the design process or experimental results.

Figure 15.

Comparison of analytical and previous experimental results for shear strength of reinforced concrete beams: (a) proposed, (b) ACI318-14, (c) CEB-FIP MC90, (d) Bazant and Kim, and (e) Zsutty.

Figure 16 is a graph of the relationship between $(T_{1} + T_{3}) / T_{2}$ , calculated by equation (1), and the shear-span-to-depth ratio. The graph shows that $T_{1} + T_{3}$ increases more significantly than $T_{2}$ with an increasing shear-span-to-depth ratio. It is shown, however, that $T_{2}$ and $T_{1} + T_{3}$ have similar stress levels.

Figure 16.

Relationship between $(T_{1} + T_{3}) / T_{2}$ and shear-span-to-depth ratio.

Distribution of each shear contribution

Table 5 shows the rates of $V_{c c}$ , $V_{d}$ , and $V_{a y}$ calculated by the proposed prediction formula for $V_{c}$ obtained from this study and previous experimental results. Figure 17 compares the shear resistance contribution of the proposed shear strength prediction formula to that of previous experimental results. As shown in Table 5 and Figure 17, the shear contribution of concrete $V_{c c}$ in the compressive zone was predicted to be 42.7%–50.6% of the total shear force $V_{c}$ and the shear force $V_{d}$ due to the bond action of longitudinal reinforcements to be 25.0%–29.4% of the total shear force $V_{c}$ . The shear force $V_{a y}$ due to aggregate interlock, which only accounts for 18.6%–32.5%, tends to decrease with an increasing shear-span-to-depth ratio. As described previously, x increases with the shear-span-to-depth ratio, but the rate of increase drops as the shear-span-to-depth ratio approaches 4, leading to a slight decrease in the shear contribution of the concrete compressive zone. When the shear-span-to-depth ratio increases from 2 to 4, the shear force $V_{d}$ due to bond action increases slightly with x. On the other hand, the shearing force $V_{a y}$ due to aggregate interlock decreases with an increasing shear-span-to-depth ratio due to the smaller angle between the crack surfaces.

Table 5.

Comparison of rates of shear resistance contribution of total shear force.

$a / d$	$V_{a n a} / V_{t e s t}$ (this study)			$V_{a n a} / V_{t e s t}$ (previous experimental results)
	$V_{c c, a n a} / V_{c, t e s t}$	$V_{d, a n a} / V_{c, t e s t}$	$V_{a y, a n a} / V_{c, t e s t}$	$V_{c c, a n a} / V_{c, t e s t}$	$V_{d, a n a} / V_{c, t e s t}$	$V_{a y, a n a} / V_{c, t e s t}$
2.0	54.4	27.5	17.7	50.6	23.6	32.5
2.5	39.7	27.4	14.8	48.6	25.0	29.1
3.0	57.2	38.1	18.2	46.5	26.5	25.5
4.0	63.0	32.2	13.6	42.7	29.4	18.6

Figure 17.

Comparison of analytical and previous experimental results for each shear resistance contribution.

Figure 18 presents the results of parametric analysis using the proposed model, ACI 318-14 (2014), CEB-FIP MC90 (1990), Bazant and Kim (1984), and Zsutty (1971). The effective depth, compressive strength of concrete, and longitudinal reinforcement ratio were used as variables in the analysis. The average and the coefficient of variation of the analytical results are similar to those of the analytical results with shear-span-to-depth ratio as a variable shown in Figure 15. The analytical results showed that the proposed model provides relatively reasonable values for effective depth compared to the others. This is because the size effect correction coefficient $k_{d}$ based on previous experimental results was considered in the contributions of the proposed shear strength prediction formula. Moreover, the results presented that the proposed model exhibits a high accuracy in predicting the shear strength of RC beams over a wide range of compressive strength of concrete and longitudinal reinforcement ratio. It is considered that an accuracy of each shear strength contribution was improved by applying the shear bond failure distance x obtained from previous experimental results to calculate the values of $T_{2}$ , $T_{1} + T_{3}$ , $θ_{1}$ , $θ_{2}$ , and $θ_{3}$ included in the proposed prediction formula.

Figure 18.

Results of parametric analysis with effective depth, compressive strength of concrete, and longitudinal reinforcement ratio as variables: (a) proposed model, (b) ACI 318-14 (2014), (c) CEB-FIP MC90 (1990), (d) Bazant and Kim (1984), and (e) Zsutty (1971).

Conclusion

This study proposed a shear resistance mechanism based on modeling the relationship between the bond action of longitudinal reinforcements and dowel action and presented a shear strength prediction formula that considers the dowel action, aggregate interlock, and shear resistance in the concrete compressive zone. By comparing the proposed formula with previous experimental results, the following conclusions were derived:

The distance from the supporting point up to the final flexural crack x was proposed to develop a shear resistance model based on the bond action of longitudinal reinforcements. The average distance and coefficient of variation obtained from the experiments were 0.93% and 17.9%, respectively. It was confirmed that the proposed equation to predict x is fairly accurate and reasonable.

The shear contribution of concrete V_cc in the compressive zone was predicted to be 47%–57% of the total shear force V_c. The shear force V_d due to the bond action of the longitudinal reinforcements was 25%–30% of the total shear force V_c and increased slightly when the shear-span-to-depth ratio increased from 2.0 to 4.0. The shear force V_ay due to aggregate interlock, which only accounted for 18%–30%, tended to decrease with an increasing shear-span-to-depth ratio.

A comparison of the proposed formula with the existing shear strength prediction formulas showed that ACI 318-14 was likely to underestimate a majority of previous experimental results by approximately 25%. The predictions by CEB-FIP MC90 and Bazant and Kim were unreliable for an effective depth of 400 mm and a tension reinforcement ratio higher than 2%. The proposed shear strength prediction formula, which takes into account the bond action of longitudinal reinforcements, provided reasonable predictions for previous experimental results with an average of 0.96 and a coefficient of variation of 14.5%.

Footnotes

Appendix 1 Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2014R1A1A2005431) and by the International Science and Business Belt Program through the Ministry of Science, ICT and Future Planning (2016K000298). This work (2015R1A2A2A01003397) was also supported by Mid-career Researcher Program through NRF grant funded by the MEST.

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