Size effect on shear strength of reinforced concrete beams with tension reinforcement ratio

Abstract

It is well known that shear stress at peak of reinforced concrete beams decreases with increasing effective depth. Thus, several existing design codes and models have included various forms of formulas considering the size effect on shear strength of reinforced concrete beams; however, past experimental researches show that tension reinforcement ratio is also associated with the shear strength of reinforced concrete beams. To examine the effect of tension reinforcement ratio and effective depth on shear strength of reinforced concrete beams, this study has conducted experiments in which the effective depth (150, 300, 500, and 780 mm) and tension reinforcement ratio (1%, 2%, and 3%) are employed as variables. Besides, a formula for the shear strength considering both variables is proposed based on data samples collected from the present experiment and previous research. The proposed formula gives predictions comparable to the results of existing shear tests. Furthermore, rational predictions are made for effective depth of beams, compressive strength of concrete, shear span-to-depth ratio, and even tension reinforcement ratio exceeding 3%.

Keywords

compressive strength of concrete effective depth of beam reinforced concrete beam shear span-to-depth ratio shear strength size effect tension reinforcement ratio

Introduction

With the recent trends of super high-rise structures and extending span ranges, the size of structural members has expanded. While most research on reinforced concrete (RC) structures has designated the effective depth of beams as a variable, it has been shown that shear stress $(V / b_{w} d)$ clearly decreases with increasing effective depth in past experiments (Collins and Kuchma, 1999; Kani, 1966, 1977). In this respect, there has been renewed research interest in size effect. Size effect refers to the drop in average resistance that occurs with increasing dimensions, for instance, when a larger RC member fails to achieve a level of resistance proportionate to its size.

Size effect on shear strength of RC beams was first confirmed by Kani (1966, 1977) following the shear failure of RC members at a warehouse located on Wilkins Air Force Depot in Shelby, OH, USA, in August of 1955. It has since been experimentally and theoretically verified in various studies. Based on fracture mechanics, Bazant and Kim (1984) proposed a formula for shear strength, while taking into account the size effect arising from the tension reinforcement ratio, and the effective depth of beams. Bazant and Sun (1987) developed a shear strength formula that considers not only the size effect but also the maximum aggregate size. Later, Bazant and Kazemi (1991) carried out experiments on geometrically similar beams with the same tension reinforcement ratio, while increasing the effective depth from 20.6 to 330.2 mm. Podgorniak-Stanik (1998) performed large-scale experiments with the tension reinforcement ratio, concrete strength, and effective depth as variables. Collins and Kuchma (1999) found that shear failure occurs at lower shear stress for a larger effective depth due to the increased width of shear cracks and developed a formula considering aggregate size and crack width based on the modified compression field theory (Vecchio and Collins, 1986). The formula proposed by Zararis and Papadakis (2001) considers the size effect and tension reinforcement ratio using the neutral axis depth ratio. Muttoni and Ruiz (2008) predicted the shear strength of RC beams without shear reinforcement based on an estimation of the crack width in the critical shear region. Rao and Injaganeri (2011) carried out non-linear regression analysis using the results of previous experimental studies and predicted both the cracking and the ultimate shear strengths of RC beams without shear reinforcement. Sneed and Ramirez (2014) examined the influence of size effect using results of previous experimental research. They confirmed that the size effect is a function of differences in relative behavior at failure largely associated with lack of geometric scale in the cracking behavior.

As shown in Figure 1, previous experimental studies have demonstrated a decrease in shear strength with increasing effective depth (Ahmad et al., 1986; Bazant and Kazemi, 1991; Bazant and Kim, 1984; Bazant and Sun, 1987; Choi et al., 2010; Collins and Kuchma, 1999; Elzanty et al., 1986; Ghannoum, 1998; Kani, 1966, 1967; Kim and Park, 1996; Krefeld and Thurston, 1966; Leonhardt and Walther, 1964; Lubell et al., 2004; Mathey and Watstein, 1963; Mphonde and Frantz, 1984; Podgorniak-Stanik, 1998; Rajagopalan and Ferguson, 1968; Sneed and Ramirez, 2010; Walraven and Lehwalter, 1994), and the size effect has been considered in numerous formulas. However, the tension reinforcement ratios in past studies were mostly less than 3%, and those with ratios higher than 3% had an effective depth below 400 mm. It is thus necessary to further investigate how size effect changes when the tension reinforcement ratio exceeds 3%.

Figure 1.

Shear strength–effective depth relationships.

This study performed experiments to examine the size effect on shear strength of RC beams using the tension reinforcement ratio and effective depth of beams as variables. A formula for shear strength was proposed with consideration of the size effect and tension reinforcement ratio based on the results of previous experimental studies.

Experimental design

Materials

The design strength of concrete used in this study was 30 MPa, and the concrete mix design is shown in Table 1. Figure 2(a) shows the stress–strain relationships for the tested concrete cylinders, which were derived from the test results for three concrete cylinders with lateral and axial strain gauges conducted in accordance with the standard test method specified in ASTM C39/C39M (American Society for Testing and Materials, 2015b), and the compressive strength averaged 33.5 MPa on the test day. For the specimens to experience shear failure before flexural failure, the steel bars were used D10 (542.4 MPa, $A_{s} = 71.3 m m^{2}$ ), D16 (546.5 MPa, $A_{s} = 198.6 m m^{2}$ ), D19 (489.9 MPa, $A_{s} = 286.7 m m^{2}$ ), D22 (513.7 MPa, $A_{s} = 387.1 m m^{2}$ ), D25 (517.8 MPa, $A_{s} = 506.7 m m^{2}$ ), and D29 (455.4 MPa, $A_{s} = 642.4 m m^{2}$ ). D indicates the deformed steel bar. The reinforcing steel bars used in the specimens were subject to tension tests in accordance with ASTM A615/A615M (American Society for Testing and Materials, 2015a). The stress–strain relationships for reinforcing steel bars and their specifications are given in Figure 2(b) and Table 2, respectively.

Table 1.

Concrete mix design.

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

Figure 2.

Stress–strain relationships: (a) concrete and (b) reinforcing steel bars.

Table 2.

Physical characteristics of reinforcement.

Reinforcement		$f_{y} (MPa)$	$ε_{y}$	$E_{s} (GPa)$
Tension	D10	542.4	0.0031	177
	D16	546.5	0.0030	181
	D19	489.9	0.0025	192
	D22	513.7	0.0029	177
	D25	517.8	0.0029	179
	D29	455.4	0.0026	174

Specifications and installation of specimens

This study conducted shear tests with the tension reinforcement ratio and effective depth of beams as variables to examine the size effect on shear strength of RC beams over varying tension reinforcement ratios. All specimens had a width of 300 mm, and the effective depth values were 150, 300, 500, and 780 mm. The specimens had different tension reinforcement ratios of 0.95%, 2.2%, and 3.4%. The shear span-to-depth ratio was set as 2.5, at which the shear effect is dominant according to Kani. The beams were designed to have a height below 900 mm because longitudinal skin reinforcement is needed above 900 mm (ACI Committee 318, 2014). A summary of the specimens is presented in Table 3, and the details of the specimens are shown in Figure 3.

Table 3.

List of specimens.

Specimens	$f'_{c} (MPa)$	$b_{w} (mm)$	$h (mm)$	$d (mm)$	$a (mm)$	$a / d$	$ρ_{t} (%)$	Test results
Specimens	$f'_{c} (MPa)$	$b_{w} (mm)$	$h (mm)$	$d (mm)$	$a (mm)$	$a / d$	$ρ_{t} (%)$	$V_{c} (kN)$	$ν_{c} (MPa)$
R1-150	33.5	300	190	150	375	2.5	0.95	81.7	2.06
R1-300	33.5	300	350	300	750	2.5		102.8	1.14
R1-500	33.5	300	550	500	1250	2.5		138.6	0.92
R1-780	33.5	300	860	780	1950	2.5		187.6	0.80
R2-150	33.5	300	200	150	375	2.5	2.2	113.4	2.52
R2-300	33.5	300	350	300	750	2.5		106.7	1.19
R2-500	33.5	300	580	500	1250	2.5		168.2	1.12
R2-780	33.5	300	870	780	1950	2.5		201.2	0.86
R3-150	33.5	300	200	150	375	2.5	3.4	168.4	3.74
R3-300	33.5	300	380	300	750	2.5		143.6	1.60
R3-500	33.5	300	590	500	1250	2.5		188.0	1.25
R3-780	33.5	300	900	780	1950	2.5		245.1	1.05

Figure 3.

Details of specimens: (a) 150 series, (b) 300 series, (c) 500 series, and (d) 780 series.

Figure 4 shows the loading apparatus. As shown in Figure 4, two linear variable differential transformers (LVDTs) were installed on the bottom of the specimens to measure the deflection due to load. The three-point loading method was employed, and a 2000-kN universal testing machine (UTM) was used to apply load under displacement control. The load was applied until the shear capacity dropped below 80% of the maximum load.

Figure 4.

Test setup of specimen: (a) obverse side and (b) reverse side.

Result of experiment

General behavior

Figure 5 gives the shear force–displacement relationships obtained from the experiments. Most specimens, as shown in Figure 5, attained maximum capacity with the formation of diagonal tension cracks, and shear capacity dropped below 80% of the maximum capacity after diagonal tension cracks were observed. As presented in Figure 5 and Table 3, the maximum shear force of each specimen increased with the beam cross section, and the maximum shear force during failure rose with tension reinforcement ratio. When the effective depth increased from 150 to 780 mm, the maximum shear force increased by 130% for the R1 series $(ρ_{t} = 0.95 %)$ and by 46% for the R3 series $(ρ_{t} = 3.4 %)$ . In addition, when the tension reinforcement ratio $(ρ_{t})$ increased from 1% to 3.4%, the shear capacity increased by 105% and 31% for specimens with an effective depth of 150 and 780 mm, respectively. Figure 6 shows the strain in tension reinforcement measured from the strain gauge attached on tension reinforcement at the mid-span of specimens. As shown in Figure 6, all specimens failed in shear before yielding of tension reinforcements.

Figure 5.

Shear force–displacement relationships: (a) R1 series, (b) R2 series, and (c) R3 series.

Figure 6.

Strain history of tension reinforcement.

Crack patterns

Figure 7 shows the cracks formed in the R1 series $(ρ_{t} = 0.95 %)$ , R2 series $(ρ_{t} = 2.2 %)$ , and R3 $(ρ_{t} = 3.4 %)$ series. Cracks at flexural crack load $(V_{c})$ were first formed in the bottom center of the specimens, which is subject to the maximum moment. With increasing load, the flexural cracks progressed to the top center, or new flexural cracks were formed at the loading point. Diagonal tension cracks at ultimate load $(V_{c})$ were formed at about 45° angle to the axis of the beams, at a distance equivalent to the effective depth away from the loading point. Specimens with an effective depth of less than 300 mm experienced failure due to diagonal tension cracks cutting across loading points as loads were delivered directly to support points. On the other hand, specimens with an effective depth greater than 300 mm experienced failure due to diagonal tension cracks and shear bond cracks. The R3 series $(ρ_{t} = 3.4 %)$ , with relatively higher tension reinforcement ratios, exhibited shear failure due to shear bond cracks in the second and third steel bars on the tension side. In this study, bond cracks occurred along longitudinal reinforcement after the diagonal tension crack is named shear bond crack.

Figure 7.

Crack patterns in specimens after test: (a) R1-150, (b) R1-300, (c) R1-500, (d) R1-780, (e) R2-150, (f) R2-300, (g) R2-500, (h) R2-780, (i) R3-150, (j) R3-300, (k) R3-500, and (l) R3-780.

Effect of effective depth of beam on the shear strength of RC beams

The shear strength of RC beams is known to increase with the cross section, and this concept has been applied to structural designs under the building code requirements of ACI 318-14 (ACI Committee 318, 2014). However, as demonstrated in past research (Bazant and Kazemi, 1991; Bazant and Kim, 1984; Bazant and Sun, 1987; Collins and Kuchma, 1999; Ghannoum, 1998; Kani, 1966, 1967; Lubell et al., 2004; Sneed and Ramirez, 2010), shear strength of RC beams may decrease instead of staying proportionate to increasing cross section. Figure 8 shows the shear strength–effective depth relationship with varying tension reinforcement ratios. As shown in the test results summarized in Table 3, the shear strength of the R1-150, R1-780, R3-150, and R3-780 specimens with various effective depths increased to 2.06, 0.80, 3.74, and 1.05 MPa, respectively. When the effective depth was increased from 150 to 780 mm, the shear strength decreased, whereas the maximum shear strength decreased by up to approximately 61% and 72%, respectively. It can be confirmed from Figure 8 that the peak shear stress decreases as the effective depth of the specimens increases, which means that the shear capacity of an RC beam does not increase in proportion to its effective depth.

Figure 8.

Shear strength–effective depth relationship: (a) R1 series, (b) R2 series, and (c) R3 series.

Effect of tension reinforcement ratio on the shear strength of RC beams

Figure 8 shows that shear strength of RC beams increased as the tension reinforcement ratio increased. The shear strength of specimens with an effective depth of 150 mm rose by 45% when the tension reinforcement ratio increased from 1% to 3%, while that of specimens with an effective depth of 780 mm only increased by 25%.

Regardless of the tension reinforcement ratio, ACI 318-14 (ACI Committee 318, 2014) and Bazant and Kim (1984) overestimate shear strength of RC beams when the effective depth of beams exceeds 500 mm. The CEB-FIP MC90 model code (Comite Euro-International du Beton, 1990) makes fairly accurate predictions for the R1 series with a tension reinforcement ratio of 1% but becomes less effective with increasing effective depth when the tension reinforcement ratio is higher than 2%. On the other hand, Japan Society of Civil Engineers (JSCE, 2012) standard specifications were likely to underestimate when beams had a small effective depth, despite making fairly accurate predictions for the effect of effective depth. Most existing formulas, as shown in Figure 8, consider the effect of effective depth in the size effect but not the effect of tension reinforcement ratio. As such, it is necessary to develop a formula for shear strength that reflects size effect with consideration of the effective depth of beams, as well as tension reinforcement ratio.

Shear strength prediction of RC beams considering size effect

Shear strength prediction formula with consideration of size effect

This section proposes a shear strength prediction formula that takes the size effect into account, with the tension reinforcement ratio as a variable, based on 207 data samples (see Appendix 2) collected from the present experiment and previous research as mentioned in the introduction (see Figure 1). The data samples had an effective depth of 65–1199 mm, an a/d of 2.5–8.52, a concrete compressive strength of 18–97 MPa, and a tension reinforcement ratio of 0.004–0.066. The proposed prediction formula described the effect of each factor on shear strength.

The shear strength–effective depth relationship in Figure 9(a) shows that shear strength decreases with increasing effective depth. The amount of decrease lessened when the effective depth was greater than 400 mm. In this study, the effect of effective depth on shear strength is expressed as follows

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

(1)

where $k_{1}$ is the coefficient representing the effect of effective depth.

Figure 9.

Principle parameters in proposed model: (a) effective depth, (b) shear span-to-depth ratio, (c) compressive strength of concrete, and (d) tension reinforcement ratio.

The shear strength–shear span-to-depth ratio relationship in Figure 9(b) shows that shear strength decreases with increasing shear span-to-depth ratio. The amount of decrease lessened gradually when shear span-to-depth ratio was greater than 4. This study expressed the effect of shear span-to-depth ratio on shear strength as follows

v_{2} = k_{2} (0.5 + \frac{d}{a})

(2)

where $k_{2}$ is the coefficient representing the effect of shear span-to-depth ratio.

The shear strength–concrete compressive strength relationship is shown in Figure 9(c). The shear strength increases gradually in proportion to compressive strength of concrete to the power of one-third. Equation (3) describes the effect of compressive strength of concrete on shear strength

v_{3} = k_{3} {(f'_{c})}^{1 / 3}

(3)

where $k_{3}$ is the coefficient representing the effect of the compressive strength of concrete.

As shown in Figure 9(c), the effect of the compressive strength of concrete on shear strength converges to a constant when the compressive strength exceeds 60 MPa. A constant value is applied for such cases.

Figure 9(d) shows the shear strength–tension reinforcement ratio relationship. As shown in the figure, shear strength increases gradually with tension reinforcement ratio. This relationship is described in equation (4)

v_{4} = k_{4} {(ρ_{t})}^{0.4}

(4)

where $k_{4}$ is the coefficient representing the effect of the tension reinforcement ratio.

As shown in equations (1) to (4), the factors influencing shear strength were effective depth, shear span-to-depth ratio, concrete compressive strength, and tension reinforcement ratio. These effects were combined for the shear strength prediction formula presented in equation (5)

V_{c} = 3.6 {(\frac{400}{d^{1.5}})}^{1 / 6} (0.5 + \frac{d}{a}) f'_{c}^{1 / 3} ρ_{t}^{0.4} b_{w} d

(5)

Comparison of shear strength prediction formula

This section assesses the accuracy of the proposed formula through comparisons with ACI 318-14, CEB-FIP MC90, JSCE (2012), and Bazant and Kim for 207 data samples without shear reinforcement. The formulas used in the comparison are given in Table 4, and comparisons between the experiment and analysis are presented in Figure 10.

Table 4.

The existing equations to predict ultimate shear strength of beams without shear reinforcement.

Code or investigator	Predictive equation, Unit : SI	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 $f'_{c}$
JSCE (2012)	$V_{c} = 0.009 [{(0.961 ρ_{t} f'_{c})}^{1 / 3} {(d / 100)}^{- 1 / 4}] b_{w} d$	d, $ρ_{t}$ , $a / d$ , and $f'_{c}$
Bazant and Kim	$V_{c} = \frac{10 \sqrt[3]{ρ_{t}}}{\sqrt{1 + d / 25 d_{a}}} [0.083 \sqrt{f'_{c} + 20.69 \sqrt{ρ_{t} / {(a / d)}^{5}}}] b_{w} d$	d, $ρ_{t}$ , $a / d$ , $f'_{c}$ , and $d_{a}$
Proposed	$V_{c} = 3.6 {(\frac{400}{d^{1.5}})}^{1 / 6} (0.5 + \frac{d}{a}) f'_{c}^{1 / 3} ρ^{0.4} b_{w} d$	d, $ρ_{t}$ , $a / d$ , and $f'_{c}$

Figure 10.

Comparison of test results with analytical results: (a) ACI 318-14, (b) CEB-FIP MC90, (c) JSCE (2012), (d) Bazant and Kim, and (e) proposed model.

As shown in Figure 10(a), ACI 318-14 predicted an average of 1.18 and a coefficient of variation of 27.9%. ACI 318-14 does not consider the size effect and overestimates shear strength of RC beams if the beams have an effective depth greater than 400 mm. There was a tendency to overestimate test results when the tension reinforcement ratio was less than 2% and to underestimate when the ratio was more than 2%.

Under CEB-FIP MC90, shear strength is inversely proportional to shear span-to-depth ratio to the power of one-third and proportionate to concrete compressive strength to the power of one-third. The size effect of effective depth was taken into consideration. As shown in Figure 10(b), CEB-FIP MC90 provided relatively accurate predictions with an average of 1.09 and a coefficient of variation of 17.7%. However, the test results were underestimated when the effective depth was less than 200 mm. In addition, there was a tendency to underestimate results with increasing tension reinforcement ratio.

Under JSCE (2012), shear strength is proportionate to compressive strength to the power of one-third and inversely proportional to effective depth to the power of one-quarter. Compressive strength and cross section are considered to have less significant effects compared to other standards. As shown in Figure 10(c), JSCE (2012) predicted an average of 1.16 and a coefficient of variation of 18.6%. There was a tendency to underestimate most test results.

Bazant and Kim proposed a formula considering the size effect based on non-linear fracture mechanics. Their formula also takes into account the effect of steel ratio and maximum aggregate size. As shown in Figure 10(d), Bazant and Kim predicted an average of 0.96 and a coefficient of variation of 18.0%. However, there was a tendency to overestimate results with increasing effective depth.

The formula proposed in this study, considering size effect, was found to be highly effective. As shown in Figure 10(e), it gave an average of 1.02 and a coefficient of variation of 14.6%. Moreover, the proposed method was effective even when the tension reinforcement ratio of beams exceeded 3%.

Conclusion

This study proposed a new formula for shear strength prediction based on test results on the size effect with varying tension reinforcement ratios. The following conclusions were derived:

The experiments showed a decrease in shear strength with increasing effective depth. The shear strength decreased by 61% and 72% when the tension reinforcement ratios were 1% and 3%, respectively. The amount of decrease grew larger with increasing tension reinforcement ratio.

Specimens with an effective depth less than 300 mm experienced failure due to diagonal tension cracks across loading points regardless of the tension reinforcement ratio. On the other hand, specimens with an effective depth greater than 500 mm experienced failure due to diagonal tension cracks and shear bond cracks, while those with a tension reinforcement ratio higher than 3% exhibited shear failure due to shear bond cracks in the second and third steel bars on the tension side.

For beams with an effective depth less than 400 mm, ACI 318-14, CEB-FIP MC90, and JSCE (2012) codes were likely to underestimate existing test results and to underestimate with increasing tension reinforcement ratio. The formula by Bazant and Kim provided relatively accurate predictions for each variable but had a tendency to overestimate with increasing effective depth.

The formula proposed in this study was found to be highly effective, giving an average of 1.02 and a coefficient of variation of 14.6%. Rational predictions were made for effective depth of beams, compressive strength of concrete, shear span-to-depth ratio, and even tension reinforcement ratios exceeding 3%.

Footnotes

Appendix 1

Appendix 2

Details of 207 data sample from literature.

Authors	Data numbers	$d (mm)$	$a / d$	$f'_{c} (MPa)$	$d_{a} (mm)$	$ρ_{t}$
Kani (1967)	33	132–1097	2.5–08.00	20.3–30.8	19	2.6–2.9
Ahmad et al. (1986)	11	184–208	2.30–4.00	64–67	12.7	2.25–6.64
Bahl (Bazant and Sun, 1987)	6	297–1198.9	3.00	232–29.7	25	0.6–1.3
Elzanty et al. (1986)	9	273.1	4.00	20.7–79.3	13	1.2–3.3
Choi et al. (2010)	5	360	2.50–3.25	24.7	25	0.8–1.6
Walraven and Lehwalter (1994)	3	125–720	3.00	34.2–34.8	16	0.7–0.8
Krefeld and Thurston (1966)	46	237.7–455.7	2.89–8.52	18.3–38.4	25	1.3–5.1
Kuchma (Lubell et al., 2004)	1	919.5	5.00	50.0	25	1.0
Kulkarni and Shah (Lubell et al., 2004)	1	152	3.50	45	25	1.4
Leonhardt (1964)	13	140–600	3.00–5.00	28.9–38.3	30	1.3–2.1
Sneed and Ramirez (2010)	7	233–822	3.00	65–74.8	25	1.2–1.3
Mathey and Watstein (1963)	9	402.8	2.84–3.78	23.5–30.5	25	0.47–2.5
Collins and Kuchma (1999)	6	110–925	3.00	36–39	10	0.8–1.0
Mphonde and Frantz (1984)	7	298.5	2.50–3.60	22.4–88.4	9.5	3.4
Rajagopalan and Ferguson (1968)	7	259–267	3.83–4.27	23.7–36.6	13	0.4–1.7
Scholz (Bazant and Sun, 1987)	1	362	3.00	96.8	25	1.9
Stanik (Lubell et al., 2004)	3	226.1–924.6	2.90–3.00	37.0	25	0.8–0.9
Taylor (Lubell et al., 2004)	3	233.7–929.6	3.00	25.3–28.8	25	1.4
Ghannoum (1998)	23	65–960	2.5	34.2–58.6	25	1.2–2.0
Xie et al. (Bazant and Sun, 1987)	1	215.9	3.0	39.7	25	2.1
This study	12	150–780	2.5	33.5	25	0.95–3.54
Total	207	65–880	2.3–8.52	18.3–96.8	9.5–30	0.4–6.64

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 work was supported by the research grant of the Kongju National University in 2015 (2015-0524-01).

References

ACI Committee 318 (2014) Building Code Requirements for Structural Concrete (ACI 318-14) and Commentary. Farmington Hills, MI: American Concrete Institute.

Ahmad

Khaloo

Poveda

(1986) Shear capacity of reinforced high strength concrete beams. ACI Journal Proceedings 83(2): 297–305.

American Society for Testing and Materials (2015a) Standard Specification for Deformed and Plain Carbon-Steel Bars for Concrete Reinforcement (A615/A615M-15a). West Conshohocken, PA: American Society for Testing and Materials.

American Society for Testing and Materials (2015b) Standard Test Method for Compressive Strength of Cylindrical Concrete Specimens (C39/C39M-15a). West Conshohocken, PA: American Society for Testing and Materials.

Bazant

Kazemi

(1991) Size effect on diagonal shear failure of beams without stirrups. ACI Structural Journal 88(3): 268–276.

Bazant

Kim

(1984) Size effect in shear failure of longitudinally reinforced beams. ACI Journal Proceedings 81(5): 456–468.

Bazant

Sun

(1987) Size effect in diagonal shear failure: influence of aggregate size and stirrups. ACI Materials Journal 84(4): 456–468.

Choi

Cho

. (2010) Experimental study on the shear strength of recycled aggregate concrete. Beams Magazine of Concrete Research 62(2): 103–114.

Collins

Kuchma

(1999) How safe our large, lightly reinforced concrete beams, slabs, and footings? ACI Structural Journal 96(4): 482–490.

10.

Comite Euro-International du Beton (1990) CEB-FIP Model Code 1990. London: Thomas Telford.

11.

Elzanty

Nilson

Slate

(1986) Shear capacity of reinforced concrete beams using high-strength concrete. ACI Structural Journal 83(2): 290–296.

12.

Ghannoum

(1998) Size effect on shear strength of reinforced concrete beams. Master’s Thesis, Department of Civil Engineering and Applied Mechanics, McGill University Montréal, Montréal, QC, Canada.

13.

Japan Society of Civil Engineers (JSCE) (2012) Standard Specification for Design and Construction of Concrete Structures, Design. JSCE. Available at: http://www.jsce-int.org/system/files/JGC15_Standard_Specifications_Design_1.0.pdf

14.

Kani

GNJ

(1966) Basic facts concerning shear failure. ACI Journal Proceedings 63(6): 675–692.

15.

Kani

GNJ

(1967) How safe are our large reinforced concrete beams? ACI Journal Proceedings 64(3): 128–141.

16.

Kim

Park

(1996) Prediction of shear strength of reinforced concrete beams without web reinforcement. ACI Materials Journal 93(3): 213–222.

17.

Krefeld

Thurston

(1966) Studies of the shear and diagonal tension strength of simply supported reinforced concrete beams. ACI Journal Proceedings 63(4): 451–476.

18.

Leonhardt

Walther

(1964) The Stuttgart Shear Tests. London: Cement and Concrete Association.

19.

Lubell

Sherwood

Bentz

. (2004) Safe shear design of large, wide beams. Concrete International 103(6): 794–802.

20.

Mathey

Watstein

(1963) Shear strength of beams without web reinforcement containing deformed bars of different yield strengths. ACI Structural Journal 60(2): 183–207.

21.

Mphonde

Frantz

(1984) Shear tests of high-and low-strength concrete beams without stirrups. ACI Structural Journal 81(4): 350–357.

22.

Muttoni

Ruiz

(2008) Shear strength of members without transverse reinforcement as function of critical shear crack width. ACI Structural Journal 105(2): 163–172.

23.

Podgorniak-Stanik

(1998) The influence of concrete strength, distribution of longitudinal reinforcement, amount of transverse reinforcement and member size on shear strength of reinforced concrete members. MASc Thesis, Toronto University, Toronto, ON, Canada, 771 pp.

24.

Rajagopalan

Ferguson

(1968) Exploratory shear tests emphasizing percentage of longitudinal steel. ACI Journal Proceedings 65(8): 634–638.

25.

Rao

Injaganeri

(2011) Evaluation of size dependent design shear strength of reinforced concrete beams without web reinforcement. Indian Academy of Sciences 36(3): 393–410.

26.

Sneed

Ramirez

(2014) Influence of Cracking on Behavior and Shear Strength of Reinforced Concrete Beams. ACI Structural Journal 111(1): 157–166.

27.

Vecchio

Collins

(1986) The modified compression-field theory for reinforced concrete elements subjected to shear. ACI Structural Journal 83(2): 219–231.

28.

Walraven

Lehwalter

(1994) Size effects in short beams loaded in shear. ACI Structural Journal 91(5): 585–593.

29.

Zararis

Papadakis

(2001) Diagonal shear failure and size effect in RC beams without web reinforcement. Journal of Structural Engineering 127(7): 733–742.