American Journal of Innovative Research and Applied Sciences. ISSN 2429-5396 I www.american-jiras.com

65

| Nisrin Bahjat Aljubayli *|

Tishreen University | department of structural engineering | Lattakia | Syria |

| Received August 12, 2020 | | Accepted September 22, 2020 | | Published October 12, 2020 | | ID Article| Nisrin-Ref.1-ajira130920|

ABSTRACT

Background: Shear failure of reinforced concrete members is considered one of the major problems that may cause a structure to fail suddenly, without warning. the main variables affecting the shear strength of reinforced concrete beams without web

reinforcement are, concrete compressive strength , effective depth of the beam , longitudinal reinforcement steel ratio ,

Shear span to depth ratio . Most of the existing models to predict the shear strength of the concrete were experimentally derived from a regression analysis of test data, and each model has different limits and variables. Objectives: The object of this study is to propose a model for predicting the ultimate shear strength of reinforced concrete beams without web reinforcement,

which takes into account the main variables ( ) and agrees appropriately with the experimental results available in the

literature. Methods: the influence of the main variables affecting the shear strength was estimated through the statistical evaluation of 168 experimental results of reinforced concrete beams without web reinforcement collected from the literature. The database of experimental results of the tested beams covered a wide range of the studied variables. Based on the experimental database and using statistical evaluation, the proposed model was compared to other available models. Conclusions: the study showed that the proposed model accurately considers the influence of the main variables affecting the shear strength of reinforced concrete beams without web reinforcement, and predicts the shear strength more accurately than other studied models. Keywords: reinforced Concrete beam, shear strength, Shear span to depth ratio, statistical Analysis.

1. INTRODUCTION Concrete is a brittle material with little shear resistance, and the shear failure of concrete elements occurs suddenly

without warning. the shear failure of concrete beams is affected by several variables such as: Concrete compressive strength

, Shear span to depth ratio , Longitudinal reinforcement ratio ρ, Effective depth of the beam section ,

maximum aggregate diameter , the concrete density (lightweigte concrete or normal concrete), the nature of the

loading, The nature of the bearing, tensile strength of concrete and the fibers used. So the prediction of the shear

strength of concrete is an important and complex problem in structural design.

The shear strength of reinforced concrete beam without web reinforcement is determined by contribution of the following principal shear transfer mechanisms:

1. Shear resistance of uncracked concrete in the compression zone, 2. Dowel action of the longitudinal reinforcement,

and, 3. Interlocking action of aggregates on both sides of a crack line.

The sum of these three principal mechanisms is called the beam action. Another mechanism contributes to the shear resistance, is the contribution of the beam arch action, whose effectiveness increases with a decrease the shear span to depth ratio . As a result, the ultimate shear strength equals to the sum of the contributions of beam action

and arc action. The arch action governs the failure of the short beams ( <2.5), whereas the beam action controles

the failure of slender beams ( ) [1, 2, 3].

In general, when calculating the expected shear strength of reinforced concrete beams without web reinforcement,

the main variables affecting the shear strength are considered and [1, 2].

Current design codes recommend using empirical models to calculate the shear strength of concrete beams as: ACI318-08 [4], BS8110-97 [5], NBR 6118/07 [6] and The CEB-FIP Model Code [7]. Each model has different variables

and constraints, and moreover, all of these models ignore the beam arc action. Models proposed by ACI318-08 [4], BS8110-97 [5] and CEB-FIP model code [7] take into account the effect of concrete compressive strength and

longitudinal reinforcement, but the model proposed by NBR 6118 [6] takes into account the effect of concrete

compressive strength only and ignores the influence of other variables. These differences cause inconsistencies in the

predictions of shear strength of reinforced concrete beams without web reinforcement [3-6].

ORIGINAL ARTICLE

SHEAR STRENGTH PREDICTION OF REINFORCED CONCRETE

BEAMS WITHOUT WEB REINFORCEMENTS

*Corresponding author Author & Copyright Author © 2020: | Nisrin Bahjat Aljubayli *|. All Rights Reserved. All articles published in American Journal of Innovative Research and Applied Sciences are the property of Atlantic Center Research Sciences, and is protected by copyright laws CC-BY. See: http://creativecommons.org/licenses/by-nc/4.0/.

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In previous research, various models were developed based on theoretical concepts and experimental results such as

Zsutty's model (1968), Mphonde and Frantz’s model (1984), and Ahmed’s model (2018) [1-3]. Each model has different limits and variables, for example, Zsutty's model considers the main variables affecting shear strength

including beam arc action, but Mphonde and Frantz’s model takes into account the effect of concrete compressive

strength only [1]. According to Ghaffar (2010) and Ahmed (2018), the models available in the literature differ in terms of the effect of the variables and their limitations, and only a few of them can well correlate the shear strength with

the independent variables (

) [1-3]. The main objective of this study is to propose a model to predict the

ultimate shear strength of reinforced concrete beams without web reinforcement. This model considers the main

variables , and presents more accurate predictations than other models available in the literature.

2. MATERIALS AND METHODS

In this study, the calibration approach applied by Kang and Kim (2011) was used to develop a model providing accurate predictions [8]. Experimental data for the tested beams were collected from the literature in order to

evaluate proposed and available shear strength models in predicting the ultimate shear strength of reinforced concrete beams without web reinforcement. Excel software was used to perform a statistical evaluation of the /

ratio, where the / ratio represents the experimental ultimate shear strength to shear strength calculated based on

the shear strength models for each tested beam. The mean, coefficient of variation (COV), minimum, maximum, and

coefficient of correlation (R) are good statistical values for evaluating the effectiveness of the proposed and available models in predicting shear strength. The mean value close to one achieves economical and conservative design, while

the lowest coefficient of variation is a good statistical indicator of consistent accuracy. The minimum and maximum

values are taken into consideration to achieve the safety factor and economical design respectively. According to Kang and Kim (2011), the slope of the linear regression line of / ratios can be used as a statistical index for evaluation

the sensitivity of / to each variable [8,9].

2.1 Shear strength models available:

The ACI 318-08 Building Code recommends the detailed empirical model to calculate shear strength of reinforced

concrete beams without shear reinforcement subjected to shear and flexure loadings, the formula is shown in Eq.(1) [4]:

( √

) √

Where: : factored shear force at the section considered; =factored moment occurring

simultaneously with at the section under evaluation. It should be noted that

for a simply supported beam

with a single point load or two symmetrical loads. for normal concrete for lightweight

concrete.

The BS8110 code provided a model to predict the shear strength of reinforced concrete members without web reinforcement shown in Eq.(2) [5]:

⁄ ⁄ ⁄ ⁄ ⁄

Where: : compressive cube strength of concrete, , ⁄ .

The CEB-FIP Model Code suggested an empirical model considering for the effect of size illustrated in Eq.(3) [6,7]:

(

)

[ √

]

The Brazilian code NBR 6118/07 proposed a model given by Eq.(4) [6]:

⁄

Zsutty (1968) suggested empirical models considering the beam arch action, as shown in Eq.(5), Eq.(6) [10].

(

)

⁄

(

)

⁄

(

)

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2.2 Database of experimental results

A total of 168 test results of reinforced concrete beams without web reinforcement were collected from the literature

[11, 12, 13, 14, 15]. All these beams were simply supported and failed in Shear. These collected beams cover wide range of studied variables. The concrete compressive strength

ranged from 12.2 to 80 MPa, longitudinal

reinforcement ratio ranged from 0.72% ≤ ≤9.42% and the shear span to depth ratio ranged from 0.81≤

≤9.74. The effective depth of beams ranged from 85 mm to 1150 mm. The detailed information about these beams

are shown in the table (1).

Table 1: The table presents the experimental results database adopted in this study.

Reference No, of beams

(mm) (Mpa) (%)

Swamy and Bahia (1985) [11] 1 207 44.6 4 4.5

Rosenbusch and Teutsch (2003) [11] 3 260-285 41.6-45.7 1.81-3.56 2.5-4.04

Ding et al., (2011) [11] 1 263 32.6 2.8 3

De Hanai and Holanda (2008) [11] 10 85-155 23.1-75.3 1.54 -3.44 1.94 -3.55

Cucchiara et al., (2004) [11] 1 219 41.2 1.91 2-2.8

Roberts and Ho (1982) [11] 4 200 32.5 2.37 0.81-2.41

Batson et al., (1972) [11] 3 127 35.1 3.12 4.8

Niyogi and Dwarakanathan (1985) [11] 1 125 32.5 0.8 3.2

Narayanan and Darwish (1987) [11] 2 114 52-53 3.13 2

Kearsley and Mostert (2004) [11] 1 209 44.8 1.08 2.87

Murty and Venkatacharyulu (1987) [11] 1 180 22.2 1.26 3

Narayanan and Darwish (1987)[11] 6 130 30.5-50 2 2,-3

Kaushiket al., (1987) [11] 1 131 21 1.72 2.34

Sharma (1986) [11] 2 263 42.3-43.2 1.34 1.9

Shinet al., (1994) [11] 7 175 80 4.71-9.42 2-6

Ayad et al., (2008) [12] 3 193 25.14 -27.2 2.4-3.07 2.5

Huang (2011) [13] 3 350 32.4 0.72 1.5-3.5

Chao (2009) [2] 12 170 23.1-53.8 1.33 1.5-3

Attaullah et al., (2012) [14] 21 260 50 1-2 3-6

Dassow (2014) [15] 3 541 25.2-30.1 1.02 2.5

Klein (2015) [15] 4 0.98-1.05 23.9-31.1 3-3.01 609.6-1219.2

Krefeld and Thurston (1966) [15] 78 238-483 12.2-39 0.8-5.09 2.34-9.74

3. RESULTS AND DISCUSSION Based on experimental database, a comparison of the available models was made. the ratio / of the measured

ultimate shear strength to shear strength calculated based on the available models Eq.(1), Eq.(2), Eq.(3), Eq.(4),

Eq.(5), Eq.(6) was determined for each beam of 168 tested beams. The mean, coefficient of variation (COV), minimum, maximum and coefficient of correlation R values of the ratios / , were calculated and listed as shown in

the Table 2.

Table 2: The table presents statistical evaluation of the proposed and available shear strength models. (Comparison of / values for tested beams)

model /

Eq.(1)

/

Eq.(2)

/

Eq.(3)

/

Eq.(4)

/

Eq.(5) Eq.(6)

/

Eq.(8) Eq.(9)

Average 1.66 1.23 1.47 1.35 1.19 1.48

Maximum 6.22 5.72 4.27 6.83 2.72 2.89

Minimum 0.55 0.54 0.78 0.38 0.51 0.78

COV % 50.9 55.9 38.7 55.4 37.1 23.8

R 0.90 0.90 0.92 0.9 0.92 0.95

No. of unsafe value, /

19 92 10 47 71 8

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Several observations can be made from the results of statistical evaluation of the / ratio cited in Table 2:

- All studied models, the ACI318 Eq.(1), BS8110 Eq.(2), CEB-FIP Eq.(3), NBR6118 Eq.(4), and Zsutty's models Eq.(5), Eq.(6) are conservative, as the mean value ( / ), But they have different safety level, for example Eq.(1) has

large safety margin of 66% on average. BS8110 code Eq.(1), and Zsutty,s models Eq.(5), Eq.(6) have the lowest values of mean ,which are 1.23 and 1.19 respectively. These lower values of mean are due to unsafe values of / ,

which are 92, 71 respectively. These values (points), which have / , can be seen in Fig. 1, Fig. 2, Fig. 3, Fig.

4.

- ACI318 Eq.(1), BS8110 Eq.(2), and NBR6118 Eq.(4) models have close values of coefficient of variation. The CEB-

FIP Eq.(3), and Zsutty's models Eq.(5), Eq.(6) have the lowest coefficient of variation for the studied models which are 38.7 , 37.1 respectively.

- The CEB-FIP Eq.(3), and Zsutty's models Eq.(5), Eq.(6) has the highest value of coefficient of correlation 0.92, and this means that there is a good match between the experimental values and expected values compared to other

models, but Zsutty's models Eq.(5), Eq.(6) provided unsafe predictions, as the number of unsafe values is 71.

For further comparison between the studied models, the sensitivity of these models to each independent variable was

established. The distributions of the / ratios versus main variables , were plotted as shown in Fig.1,

Fig.2, Fig.3 and Fig.4. It should be noted that conservative design occurs when points fall above the unit line.

Fig.1 shows that: The ACI-318 Eq.(4) and BS8110 Eq.(2) show a clear sensitivity to change , as a decrease in the

safety level was observed for beams with concrete compressive strength The shear strength predicted

by the CEB-FIP Eq.(3), Zsutty's models Eq.(5), and Eq.(6) were consistent with the experimental shear strength for beams with

. The BS8110 Eq.(2) and NBR6118 Eq.(4) overestimate the influence of

, whereas ACI-

318 Eq.(4) and CEB-FIP Eq.(3) provided sufficiently conservative results.

Fig.2 shows the significant effect of ( ) on the shear strength predicted by all studied models. The shear strength predictions of most models are scattered with (a/d) ratio, except for those of Zsutty's models Eq.(5), and Eq.(6), which takes into account the effect of beam arch action, but Zsutty's models Eq.(5),

and Eq.(6) are non-conservative for larg number of beams with . For all studied models, the level

of safety decreases with increasing the value (overerestimation). The ACI318 Eq.(1) and BS8110

Eq.(2) are non-conservative for significant number of beams with ( ), while NBR6118 Eq.(4) is

unsafe for large number of beams with ( ). This is due to the fact that NBR6118 Eq.(4) considers the concrete compressive strength

only.

y = 0.0132x + 1.1987

0

1

2

3

4

5

6

7

0 10 20 30 40 50 60 70 80 90

𝑣𝑢

/𝑣𝑛

f'c (MPa)

ACI 318-08

y = 0.0105x + 0.8676

0

1

2

3

4

5

6

7

0 10 20 30 40 50 60 70 80 90

𝑣𝑢

/𝑣𝑛

f'c (MPa)

BS8110

y = 0.0089x + 1.1543

0

1

2

3

4

5

6

7

0 10 20 30 40 50 60 70 80 90

𝑣𝑢

/𝑣𝑛

f'c (MPa)

CEB-FIP Mode

y = 0.0039x + 1.2117

0

1

2

3

4

5

6

7

0 10 20 30 40 50 60 70 80 90

𝑣𝑢

/𝑣𝑛

f'c (MPa)

NBR6118

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Figure 1: The figure presents the ratio / versus for the proposed and available models in this study.

Figure 2: The figure presents the ratio / versus a/d for the proposed and available models in this study.

Fig.3 shows that the shear strength predictions of most models are scattered with (ρ) ratio. Most models overestimate

the shear strength of the beams with ρ <2% (more risky). The ACI318 Eq.(1), CEB-FIP Eq.(3) are conservative for beams with ρ >1.5%. The BS8110 Eq.(2) and Zsutty's models Eq.(5), Eq.(6) give approximately constant values.

y = 0.0085x + 0.8963

0

1

2

3

4

5

6

7

0 10 20 30 40 50 60 70 80 90

𝑣𝑢

/𝑣𝑛

f'c (MPa)

Zustty

y = -0.0037x + 1.6066

0

1

2

3

4

5

6

7

0 10 20 30 40 50 60 70 80 90

𝑣𝑢

/𝑣𝑛

f'c (MPa)

The proposed model

y = -0.2009x + 2.5001

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8 9 10

𝑣𝑢

/𝑣𝑛

a/d

ACI 318-08

y = -0.2078x + 2.1046

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8 9 10

𝑣𝑢

/𝑣𝑛

a/d

BS8110

y = -0.1249x + 1.9891

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8 9 10

𝑣𝑢

/𝑣𝑛

a/d

CEB-FIP Mode

y = -0.167x + 2.0461

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8 9 10

𝑣𝑢

/𝑣𝑛

a/d

NBR6118

y = -0.0671x + 1.4723

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8 9 10

𝑣𝑢

/𝑣𝑛

a/d

Zustty

y = 0.0003x + 1.4774

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8 9 10

𝑣𝑢

/𝑣𝑛

a/d

The proposed model

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Fig.4 shows the effect of ( ) on the shear strength predicted by all the studied models. For all studied models, the

level of safety decreases with increasing the value . The CEB-FIP Eq.(3), and Zsutty's models Eq.(5), Eq.(6) are the

least sensitive to the changes in the effective depth, (the slope of the linear regression line is almost horizontal).

Figure 3: The figure presents the ratio / versus ρ for the proposed and available models in this study.

y = 0.01x + 1.6336

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6 7 8 9 10

𝑣𝑢

/𝑣𝑛

ρ%

ACI 318-08

y = -0.0818x + 1.4398

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6 7 8 9 10

𝑣𝑢

/𝑣𝑛

ρ%

BS8110

y = -0.0522x + 1.5972

0123456789

0 1 2 3 4 5 6 7 8 9 10

𝑣𝑢

/𝑣𝑛

ρ%

CEB-FIP Mode

y = 0.0281x + 1.2759

0123456789

0 1 2 3 4 5 6 7 8 9 10

𝑣𝑢

/𝑣𝑛

ρ%

NBR6118

y = -0.0249x + 1.254

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6 7 8 9 10

𝑣𝑢

/𝑣𝑛

ρ%

Zustty

y = -0.0016x + 1.4825

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6 7 8 9 10

𝑣𝑢

/𝑣𝑛

ρ%

The proposed model

y = -0.0028x + 2.363

-1

0

1

2

3

4

5

6

7

8

9

0 200 400 600 800 1000 1200

𝑣𝑢

/𝑣𝑛

𝑑 mm

ACI 318-08

y = -0,0013x + 1,5611

-1

0

1

2

3

4

5

6

7

8

9

0 200 400 600 800 1000 1200

𝑣𝑢

/𝑣𝑛

𝑑 mm

BS8110

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Figure 4: The figure presents the ratio / versus for the proposed and available models in this study.

Based on the previous discussion, the following ultimate shear strength model is developed depending on the CEB-FIP

Eq.(3) and Zsutty's models Eq.(5), Eq.(6).

This model has the main variables affecting the shear strength of beams without shear reinforcement, and considers the arch action of the beam, which tends to occur when .

The proposed model has the following form:

[ (

)

] (

)

In this model, the term e takes into account the beam arc action. for ,

for .

for the following values of the constants , the model becomes appropriate

for the test results with the mean value (1.48), the coefficient of variation (23.8%) and the highest coefficient of

correlation of 0.95, as shown in table 2. The proposed model can be written as shown in Eq.(8), Eq.(9).

Where:

Fig 1, Fig 2, Fig 3, Fig 4 show that the proposed model shows less sensitivity to the change of the main variables

, as the safety level doesn’t changed with increasing each variable (the slope of the linear regression

line is nearly horizontal).

4. CONCLUSION

The most important conclusions can be summarized as follows:

1- A comparison of the available models shows that the ACI318 Eq.(1) and CEB-FIP Eq.(3) are the most conservative. Zsutty's models Eq.(5) and Eq.(6) are the most consistent. The NBR6118 Eq.(4) exhibits substantial scatter in

predicting shear strength of all tested beams, and was non-conservative for large number of slender beams (Unsafe

design for beams with > 2).

2- The CEB-FIP Eq.(3) appears to be the most accurate model for predicting the shear strength of beams without web

reinforcement, and the least sensitive to changes in the main variables that affect the shear strength, compared to other models.

3- The proposed model has been compared with the available models, the ACI318 Eq.(1), BS8110 Eq.(2), CEB-FIP Eq.(3), NBR6118 Eq.(4), and Zsutty's models Eq.(5) and Eq.(6). The comparison shows that the proposed model

y = -0,0014x + 1,8168

-1

0

1

2

3

4

5

6

7

8

9

0 200 400 600 800 1000 1200

𝑣𝑢

/𝑣𝑛

𝑑 mm

CEB-FIP Mode

y = -0,0022x + 1,8933

-1

0

1

2

3

4

5

6

7

8

9

0 200 400 600 800 1000 1200

𝑣𝑢

/𝑣𝑛

𝑑 mm

NBR6118

y = -0,0016x + 1,6033

-1

0

1

2

3

4

5

6

7

8

9

0 200 400 600 800 1000 1200𝑣𝑢

/𝑣𝑛

𝑑mm

Zustty

y = -0.0002x + 1.5274

-10123456789

0 200 400 600 800 1000 1200

𝑣𝑢

/𝑣𝑛

𝑑 mm

The proposed model

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accurately accounts for the effects of the main variables ) affecting the shear strength of beams without

web reinforcement. 4-The proposed model achieves good agreement with the test results compared to the available models with an

average value (1.48), coefficient of variation (23.8%), and the highest correlation coefficient of (0.95).

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Cite this article: Nisrin Bahjat Aljubayli. SHEAR STRENGTH PREDICTION OF REINFORCED CONCRETE BEAMS WITHOUT WEB REINFORCEMENTS. Am. J. innov. res. appl. sci. 2020; 11(3): 65-72.

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