Shear Strength of Reinforced Concrete Beams Under Uniformly Distributed Loads

ACI Structural Journal/November-December 2008 711

ACI Structural Journal, V. 105, No. 6, Nov.-Dec. 2008.MS No. S-2007-142 received April 20, 2007, and reviewed under Institute publication

policies. Copyright © 2008, American Concrete Institute. All rights reserved, including themaking of copies unless permission is obtained from the copyright proprietors. Pertinentdiscussion including author’s closure, if any, will be published in the September-October2009 ACI Structural Journal if the discussion is received by May 1, 2009.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

An analytical theory for shear resistance of reinforced concretebeams subjected to uniformly distributed loads is presented. Slenderbeams with a span length-to-depth ratio (l/d) greater than 10, aswell as deep beams in which l /d < 10, are examined; and simpleexpressions are derived for the ultimate shear capacity. The impact ofsize effect in the shear strength of slender beams is taken into account.A criterion for minimum shear reinforcement is also considered.

The derived formulas are verified by comparisons to well-grounded experimental data from the literature. Data wereobtained on slender beams and deep beams with various strengthsof concrete, longitudinal steel ratios, shear reinforcement ratios, l/d,and geometrical sizes.

The shear strength of beams, both slender and deep, under auniform load is found to be much higher than the shear strength ofbeams under a loading arrangement of two concentrated loads atthe quarter points.

Keywords: beam; reinforced concrete; shear; strength; uniform load.

INTRODUCTIONShear resistance of reinforced concrete beams has been

studied extensively over the last few decades. Nevertheless,the study of shear resistance of beams subjected to uniformlydistributed loads is limited.1-6 One of the reasons is thedifficulty to achieve a uniformly distributed load in experiments.Another reason is that the mechanism of shear failure wasdifficult to be found, as most of the research has concentratedon the simpler case of two-point loading. The shear behaviorof beams under a uniformly distributed load has been examinedin earlier studies5,7 to be essentially the same as the behaviorunder a point loading arrangement of two-point loads at thequarter points. Thus, the shear span of a beam with uniformload has been defined as a = /4, where is the span length ofbeam. This is probably why this particular type of loading isnot mentioned in the current provisions for shear in internationalcodes, such as the ACI 3188 or Eurocode 2.9 As a result,the shear strength of beams is calculated using the knownempirical formulas that apply to any type of loading. Such aconsideration, however, is not correct. Tests show thatthe shear strength of beams under a uniform load isconsiderably higher than the strength under a one- ortwo-point loading arrangement.

Theories have been proposed in previous works10-12 thatuse the internal forces at diagonal shear cracks13-14 todescribe the diagonal shear failure in slender beams as wellas the shear compression failure in deep beams underconcentrated loads. These theories determine:1) the ultimateshear capacity of slender beams with or without stirrupsunder concentrated loads; 2) the ultimate shear capacity ofdeep beams with or without stirrups under concentratedloads; 3) the impact of size effect and how it relates to diagonalshear failure; and 4) a criterion that the minimum amount of

shear reinforcement must satisfy to restrain the growth ofdiagonal cracking and prevent brittle failure.

In this study, the previously referred theories are adaptedfor the case of beams under uniformly distributed loads andthe following was determined:

1. The ultimate shear capacity of reinforced concreteslender beams under a uniform load, where the size effect istaken into account and a criterion for the minimum shearreinforcement is considered; and

2. The ultimate shear capacity of reinforced concrete deepbeams under a uniform load.

It is shown that the theoretical results can explain, in arigorous and consistent way, the experimentally observedbehavior of slender as well as of deep beams failing in shear.

RESEARCH SIGNIFICANCEReinforced concrete beams subjected to uniformly

distributed loads are the most common case in practice. Thestudy of shear resistance of these beams, however, is verylimited. The proposed theory explains the mechanism ofshear failure in slender and deep beams subjected touniform loading. The theory results in simple and easy-to-use expressions, which predict the ultimate shear forcewithin 4.5% and 4.3% of experimental observations ofslender and deep beams, respectively.

SHEAR STRENGTH OF BEAMS UNDER CONCENTRATED LOADS—AN OVERVIEW

Slender beams without shear reinforcement10

In slender beams (that is, beams with a shear span-to-depthratio [a/d] > 2.5) without shear reinforcement under two-point loading (or one-point loading at midspan), the criticalcrack, leading to collapse, typically involves two branches(Fig. l(a)). The first branch is a slightly inclined shear crack,the height of which is approximately that of the flexuralcracks. The second branch initiates from the tip of the firstbranch and propagates toward the load point crossing thecompression zone, with its line meeting the support point.Failure occurs by the formation of this second branch. Thesecond branch of the critical diagonal crack is caused by atype of splitting of concrete in the compressive zone. Thestress distribution along the line of splitting, however, is notsimilar to that occurring in the common split cylinder test(Fig. 2).

The theory10 results in a simple expression Vcr = (c/d)fctbwd,where bw is the width of the beam. The nominal shear stress

Title no. 105-S65

Shear Strength of Reinforced Concrete Beams under Uniformly Distributed Loadsby Prodromos D. Zararis and Ioannis P. Zararis

ACI Structural Journal/November-December 2008712

(vcr = Vcr/bwd) at the diagonal tension cracking (formationof the second branch of the critical crack) is a product of theratio of neutral axis depth c to effective depth d of the beamand the splitting tensile strength of concrete fct.

Moreover, the problem of size effect on the shear strength ofbeams reduces to a problem of size effect on the splitting tensilestrength of concrete. Introducing a correction factor to accountfor the size effect in slender beams, the expression becomes

(1)

where

(2)

For values of 1.2 – 0.2(a/d)d < 0.65, the correction factoris still taken as 0.65.

Taking into account that a = (a/d)d, the size effect inbeams appears to depend not only on the depth d, as iscommonly believed, but also on the a/d.

Vcr 1.2 0.2ad---d–⎝ ⎠

⎛ ⎞ cd--- fctbwd=

1.2 0.2ad---d 0.65 (d in m)≥–

The shear force Vcr in Eq. (1) represents the ultimate shearforce of a slender beam without shear reinforcementsubjected to one- or two-point loads acting at a distance a(shear span) from the support.

The depth c of the compression zone in Eq. (1) is given bythe positive root of the following equation11

(3)

where fc′ is the compressive strength of concrete, MPa; ρ isthe ratio of main tension reinforcement = As/bwd; ρ′ is theratio of compression reinforcement = As′/bwd; and d′ is theeffective depth to compression reinforcement.

Finally, the splitting tensile strength of concrete fct , whennot known from experiments, can be calculated from

fct = 0.30fc′2/3 (MPa) (4)

This expression, cited in Eurocode 2,9 fits well with theresults of the common split cylinder test. Moreover, usingthis expression in Eq. (1), the obtained predictions for theshear strength of beams without stirrups are in very goodagreement with the experimental results.10,11

Slender beams with stirrups11

The crack pattern of slender beams with stirrups is similarto that of beams without stirrups (Fig. 1). The critical crack,again as the beam without stirrups, typically involves twobranches, which are formed in the same region. It is rationalto consider that the cause of the formation of the secondbranch of the critical diagonal crack and the correspondingcracking load is identical in both cases. Up to the formationof the second branch of the critical crack, the effect of stirrupscan be considered negligible.

By the cracking of the second branch of the critical crack,the stirrups take action and the strength of the beamincreases. An essential condition for the shear failure ofbeams is the yielding of stirrups at the critical crack. Thiscondition, however, is not sufficient. Shear failure of aslender beam occurs only when the shear force Vd developed inthe longitudinal steel bars (Fig. 2) causes a horizontal splittingof the concrete cover along the longitudinal reinforcement(Fig. 1). This splitting results in the loss of the shear force Vdand, consequently, the failure of beam.

By analyzing the way that horizontal splitting occurs alongthe longitudinal reinforcement,11 the following simpleexpression of shear strength of slender beams is derived.This involves a superposition of the shear strength of beamswithout shear reinforcement and the shear strength providedby shear reinforcement.

(5)

where ρv is the ratio of shear reinforcement = Av/bws; Av isthe area of vertical stirrups within a distance s; fyv is the yieldstrength of shear reinforcement; and Vcr is the shear strengthat diagonal tension cracking, that is, shear strength of beamwithout stirrups, from Eq. (1).

Moreover, according to this analysis,11 to avoid anundesirable widening of the critical diagonal crack (as well

cd---⎝ ⎠

⎛ ⎞ 2600ρ ρ′+

fc′-------------- c

d--- 600ρ ρ′ d′ d⁄( )+

fc′-------------------------------–+ 0=

Vu Vcr 0.5 0.25ad---+⎝ ⎠

⎛ ⎞ ρv fyvbwd+=

ACI member Prodromos D. Zararis is a Professor of civil engineering at the AristotleUniversity of Thessaloniki, Thessaloniki, Greece. He received his MSc and DIC inconcrete structures and technology from the Imperial College of Science and Technology,London, UK, and his PhD from Aristotle University. His research interestsinclude the study of behavior of reinforced concrete structural elements undervarious loading conditions.

Ioannis P. Zararis is a Chartered Civil Engineer. He is a PhD candidate at the AristotleUniversity of Thessaloniki. His research interests include the study of behavior ofreinforced concrete structural elements under various loading conditions.

Fig. 1—Final crack pattern of slender test beams: (a)without stirrups2; and (b) with stirrups.15

Fig. 2—Distribution of normal stresses along line of secondbranch of critical diagonal crack (+ = tension, — =compression).


as that of the horizontal splitting crack), providing anincreased ductility and preventing a sudden shear failure, theratio ρv of shear reinforcement, in relation to the ratio ρ ofmain tension reinforcement, must satisfy (approximately)the following equation

ρ/ρv ≤ 1.75(a/d) (6)

Equation (6) constitutes the criterion for minimum shearreinforcement.

Deep beams12

In deep beams (that is, beams with a shear span depth ratioa/d < 2.5), the critical diagonal crack is governed by shearrather than by bending, because it initiates very close to thesupport (Fig. 3). In this case, an arch action exists. Especiallyfor 1.0 < a/d < 2.5 (approximately), the shear failure ismainly caused by concrete crushing in the compression zoneat the top of the critical diagonal crack (Fig. 3). This type offailure is known as shear compression failure.

A determining factor in shear compression failure of deepbeams is the depth cs of the compression zone above the criticaldiagonal crack, where failure occurs. This depth cs is muchsmaller than the depth c of the compression zone above thetip of the flexural cracks (Fig. 3).

According to analysis,12 taking into account the forcesacting on the critical diagonal crack (Fig. 4(a)), the depth csis given by the simple expression

(7)

where

R = 1 + (ρv/ρ)(a/d)2 (8)

Equation (7) shows that the depth cs is a portion of thedepth c above the flexural cracks, found by Eq. (3).

The stirrups yield upon higher loading after the formationof the critical diagonal crack. Then the shear force Vd of the

cs

d---- 1 0.27R a d⁄( )2+

1 R a d⁄( )2+----------------------------------------- c

d---=

longitudinal reinforcement (Fig. 4(a)) increases signifi-cantly, resulting in a horizontal cleavage of concrete coveralong the main reinforcement that eventually causes the lossof force Vd. After that, the normal and shear forces in theconcrete compression zone above the critical diagonal crackincrease excessively, eventually resulting in concretecrushing in this zone.

According to the analysis12 of the equilibrium of forcesacting on a free-body diagram of a deep beam at failure(Fig. 4(b)), the ultimate shear force is given by

(9)

which is valid for deep beams with and/or without webreinforcement.

SHEAR STRENGTH OF BEAMS UNDER UNIFORM LOAD

Slender beamsAs previously discussed, the shear failure of slender

beams is caused by a type of concrete splitting along the lineof the second branch of the critical diagonal crack. This splittingoccurs in the most diagonally compressed area. In beamsunder two-point loading (Fig. 1), this area is near a pointload; but in beams under a uniform load, this area is near thesupport reaction. The critical diagonal crack in slenderbeams under a uniform load always occurs near the supportand not near a quarter point of beam. This is clearly observedin the patterns of cracking of test beams (Fig. 5(a)). The criticaldiagonal crack also has two branches. The height of the firstbranch is approximately that of the flexural cracks, whereas

Vubwd

a d⁄----------

cs

d---- 1 0.5

cs

d----–⎝ ⎠

⎛ ⎞ fc′ 0.5ρv fyv 1cs

d----–⎝ ⎠

⎛ ⎞2 a

d---⎝ ⎠

⎛ ⎞ 2+=

Fig. 3—Final crack pattern of deep test beam: (a) front side;and (b) detail of critical diagonal crack.16

Fig. 4—Forces on free-body diagram of a deep beam: (a) byformation of critical diagonal crack: and (b) at failure.

714 ACI Structural Journal/November-December 2008

the line of the second branch is meeting the support point(Fig. 5(a) and (b)).

For slender beams under a uniform load, an ideal shearspan ai must be determined by analogy to point-loadedbeams. It is the distance from the support to the tip of the criticaldiagonal crack (Fig. 5(b)). The uniformly distributed load, qforce per unit length, over the ideal shear span, length ai,may be substituted by a statically equivalent loadingarrangement of two concentrated loads at the ends of theideal shear span, each having a value of 0.5qai (Fig. 5(c)).

Under this point loading arrangement, the shear force overthe ideal shear span has a constant value of 0.5q(l – ai),where is the span length of the beam. The stress state in thisarea is similar to that in a slender beam under two-pointloading (Fig. 2). Thus, the derived equations for the case ofslender beams under two-point loading are also valid forslender beams under a uniform load, where the total shearforce V in these equations is equal to 0.5q(l – ai).

Furthermore, taking into account that: 1) in slender beamsunder a uniform load, concrete splitting, which results in theformation of the second branch of the critical diagonal crack,possibly occurs at the most stressed area, that is, in the areaclosest to the support area; and 2) for a slender beam (a/d >

2.5), one can conclude that under a uniform load, the idealshear span to depth ratio is approximately

ai/d = 2.5 (10)

Substituting the force 0.5q( – ai) for the force V in Eq. (1)and (5) with Eq. (10), the uniform load qu under which theshear failure occurs in slender beams with or without stirrupscan be obtained by

(11)

where the size effect factor in this case is

(1.2 –0.5d) ≥ 0.65 (d in m) (12)

The depth c of the compression zone and the splittingtensile strength fct in Eq. (11) can be taken from Eq. (3) and(4), respectively.

Also, by Eq. (6) and (10), the criterion for minimum shearreinforcement is approximately

ρ/ρv ≤ 4.4 (13)

Deep (short) beamsFor a deep beam under a uniform load, an ideal shear span

ai = l/4 can be considered, as it has been suggested inprevious analytical studies.17 Then, taking into account thata/d < 2.5, it is shown that the span length to depth ratio (l/d)of a deep beam is less than 10. Experimental evidence1-3

indicates that behavior of test beams with approximately l/d > 10is different from that of beams with l/d < 10. In test beamswith l/d > 10, a failure occurs after a critical diagonal crackconsisting of two branches, as it is shown in Fig. 5, has beenformed (diagonal shear failure). By contrast, in test beamswith l/d < 10, a failure occurs due to concrete crushing in areduced compression zone above the tip of the critical diagonalcrack (shear compression failure). Failure occurs atapproximately the quarter points of the beam span (Fig. 6).

As in the case of deep beams under concentrated loads,and in the case of uniform load, a determining factor in shearcompression failure of deep beams is the depth cs of thecompression zone above the tip of the critical diagonal crack,where the concrete crushing occurs. Following theconsiderations made by a previous analysis of deep beamsunder two-point loading12 and the forces that act on thecritical diagonal crack (Fig. 7(a)), the depth cs is obtained

(14)

where

R = 1 + (ρv /ρ)(l/d)2/16 (15)

It can be seen that Eq. (14) can be obtained by substituting,in Eq. (7), the ratio a/d with the ratio l/(4d). Equation (14)shows that the depth cs is a portion of the depth c above theflexural cracks given by Eq. (3).

qu2bw

d--- 2.5–---------------- 1.2 0.5d–( )c

d--- fct 1.2ρv fyv+=

cs

d---- 1 0.27R l d⁄( )2 16⁄+

1 R l d⁄( )2 16⁄+-------------------------------------------------- c

d---=

Fig. 5—Slender beams under uniform load: (a) final crackpattern of test beam2; (b) ideal shear span determination;and (c) statically equivalent loading arrangement.

Fig. 6—Final crack pattern of deep test beams under uniformload: (a) without stirrups2; and (b) with stirrups.3


Shear compression failure in a deep beam under auniformly distributed load occurs similar to failure in deepbeams under two-point loading.12 The stirrups yield uponhigher loading after the formation of a critical diagonalcrack. Subsequently, an increase in loading results in asignificant increase of the shear force Vd of the longitudinalreinforcement (Fig. 7(a)). This results in a horizontalcleavage of the concrete cover along the main reinforcement,which eventually causes the loss of force Vd. Then, thenormal and shear forces in the concrete compression zoneabove the critical diagonal crack increase excessively,eventually resulting in concrete crushing in this zone.

The forces acting on a free-body diagram of a deep beamat failure are shown in Fig. 7(b). Concrete crushing occurswith a combination of the concrete forces C and Vc in thecompression zone. Considering that the failure occurs whenthe concrete compressive stress exceeds the concretestrength fc′ over the entire depth cs of the compression zone,the concrete compressive force at failure is C = csbw fc′. Also,the stirrup force at failure is Vs = ρv fyvbwd(1 – cs/d)tanϕ.Considering that for deep beams, tanϕ ≈ ai/d, and T = C,from the equilibrium of moments of forces acting on thefree-body diagram of Fig. 7(b) at the point of application offorce C, the ultimate uniform load of the beam can be obtained

(16)

Equation (16) is valid for deep beams with and/or withoutweb reinforcement subjected to a uniform load.

EXPERIMENTAL VERIFICATION AND DISCUSSIONThe theory presented is applied for the prediction of the

ultimate shear strength of reinforced concrete beamssubjected to uniformly distributed load. Tables 1 and 2 showcomparisons of the theoretical results according to theproposed formulas, (Eq. (11) and (16)), with 45 test resultsof slender beams (l/d > 10) and 60 of deep beams (l/d < 10),respectively, extracted from the literature. Tables 1 and 2include the test results of Bernaert and Siess,1 Leonhardt andWalther,2 Rüsch et al.,3 Krefeld and Thurston,4 and Iguro etal.5 The seven sets of test data in Tables 1 and 2 have beenobtained on slender beams as well as deep beams withvarious strengths of concrete, longitudinal steel ratios, shearreinforcement ratios, l/d, and geometrical sizes. All testsbeams had adequate longitudinal reinforcement so thatflexural failure can be avoided. The experimental and thetheoretical results are in very good agreement for all tests.

Predicted shear capacities according to a point loadingarrangement of two-point loads at the quarter points are alsopresented in Tables 1 and 2 to compare the ultimate shearstrength of beams under a point loading arrangement toexperimental results from a uniformly distributed load. Forthe predictions under the two-point loading arrangement,Eq. (5) for slender beams and Eq. (9) for deep beams areused. The corresponding shear capacities are shown inColumn (11) of Tables 1 and 2 for slender and deep beams,respectively. It must be pointed out that these equationspredict with accuracy the shear strength of test beams underconcentrated loads.10-12

Moreover, the predicted shear capacities according to ACI318-028 are also presented in Tables 1 and 2. In preparing thevalues of Tables 1 and 2, the expression of code is used

qu8bw

0.75 l d⁄( )2--------------------------

cs

d---- 1 0.5

cs

d----–⎝ ⎠

⎛ ⎞ fc ′ 0.5ρv fyv 1cs

d----–⎝ ⎠

⎛ ⎞2

l

d---⎝ ⎠⎛ ⎞

2

16⁄+=

without including any factor of safety. Thus, for ACI Codepredictions (in SI units)

(17)

The splitting tensile strength fct of concrete used in Eq. (11)and Eq. (5) to predict the shear capacity of slender beamsunder a uniform load and under a two-point loading,respectively, is shown in Column (3) of Table 1. For the lastset of tests in Table 1,5 fct is taken from the experiments,whereas in the other two sets, fct is calculated from Eq. (4) inthe absence of experimental data.

By comparison of the observed shear capacities with thepredictions according to a two-point loading arrangement inTables 1 and 2, it can be seen that replacement of the uniformload with two-point loads at the quarter points (that is, takena shear span a = l/4) is a very conservative approximation. Itis shown that the shear strength of both slender and deepbeams under a uniform load is 40 to 50% and, in some cases,70%, higher than the shear strength of beams under a loadingarrangement of two-point loads at the quarter points.

Moreover, the ACI Code predictions for shear strength ofbeams under a uniform load are inaccurate. As shown inTables 1 and 2, as well as in Fig. 8 and 9, the ratio ofobserved shear capacity to the calculated ACI shear capacitygoes up to 3.0 for slender beams and 6.5 for short beams. InFig. 9, it is shown how the ACI predictions deviate whenl/d is decreased. The ACI 318-02 expression for shearstrength (Eq. (17)) seems to be unrealistic.

The ACI 318 Code does not address the size effect inestimating the shear strength of beams. As shown from thetests of Iguro et al.5 in Table 1, and also in Fig. 10, the ratio

Vu 0.166 fc′ ρv fyv+( )bwd=

Fig. 7—Forces on free-body diagram of a deep beam underuniform load: (a) by formation of critical diagonal crack;and (b) at failure.


Table 1—Comparison of experimental and theoretical results of slender beams

Beam no.(1)

fc′, MPa(2)

fct, MPa(3)

b, cm(4)

d, cm(5)

l/d(6)

ρ, %(7)

Experimental Vu, kN

(8)

Theory Eq. (11) Theory Eq. (5) ACI Code

Vu , kN(9)

Vu,exp /Vu,th(10)

Vu, kN(11)

Vu,exp /Vu,th(12)

Vu, kN(13)

Vu,exp /Vu,th(14)

Leonhardt and Walter2

14-1 31.8 3.01 19.0 27.3 11.0 2.04 107.0 98.1 1.090 74.9 1.428 48.6 2.201

14-2 31.8 3.01 19.0 27.3 11.0 2.04 107.4 98.1 1.094 74.9 1.434 48.6 2.210

15-1 33.6 3.12 19.0 27.2 14.7 2.05 95.4 92.9 1.027 72.5 1.316 49.7 1.920

15-2 33.6 3.12 18.9 27.3 14.6 2.05 101.5 92.9 1.092 72.5 1.400 49.7 2.042

16-1 33.1 3.09 19.0 27.3 18.3 2.04 96.3 89.0 1.081 68.7 1.402 49.5 1.945

16-2 33.1 3.09 18.9 27.4 18.2 2.05 95.8 89.3 1.073 68.7 1.394 49.5 1.935

17-1 31.1 2.97 18.9 27.3 22.0 2.05 87.5 84.6 1.034 63.6 1.376 47.8 1.830

17-2 31.1 2.97 18.9 27.4 21.9 2.04 87.0 85.0 1.023 58.1 1.497 47.9 1.816

Mean 1.064 1.406 1.987

Standard deviation 0.029 0.048 0.142

Krefeld and Thurston4

1CU 19.0 2.13 15.2 25.6 11.9 1.00 48.5 48.0 1.010 36.8 1.318 28.2 1.720

2 CU 20.8 2.27 15.2 25.4 12.0 1.32 54.3 54.2 1.001 41.8 1.300 29.2 1.860

3 CU 20.5 2.25 15.2 25.6 11.9 1.99 71.6 62.5 1.145 48.0 1.492 29.2 2.452

4 CU 20.6 2.25 15.2 25.4 12.0 2.63 79.0 68.0 1.162 52.1 1.516 29.1 2.715

5 CU 20.4 2.24 15.2 25.2 12.1 3.35 82.7 72.0 1.148 55.3 1.495 28.7 2.881

6 CU 20.6 2.25 15.2 25.1 12.2 4.30 77.9 76.3 1.021 58.8 1.325 28.7 2.714

3 EU 17.6 2.03 15.2 25.6 14.3 1.99 61.8 56.8 1.088 44.2 1.398 27.1 2.280

4 EU 20.2 2.22 15.2 25.4 14.4 2.63 72.9 64.5 1.130 50.3 1.149 28.8 2.531

5 EU 19.3 2.16 15.2 25.2 14.5 3.35 77.4 67.5 1.146 52.6 1.471 27.9 2.774

6 EU 20.1 2.22 15.2 25.0 14.6 4.30 68.5 72.4 0.946 56.7 1.208 28.3 2.420

3 GU 22.6 2.40 15.2 25.6 16.7 1.99 58.0 60.0 0.966 46.8 1.239 30.7 1.889

4 GU 22.1 2.36 15.2 25.4 16.8 2.63 70.7 64.5 1.096 50.3 1.405 30.1 2.349

5 GU 21.3 2.30 15.2 25.2 16.9 3.35 65.9 67.9 0.970 53.1 1.241 29.3 2.249

6 GU 21.2 2.30 15.2 25.0 17.0 4.30 75.2 72.1 1.043 56.2 1.338 29.0 2.593

4JU 22.2 2.37 15.2 25.4 19.2 2.63 57.0 63.3 0.900 48.9 1.165 30.2 1.887

5JU 21.5 2.32 15.2 25.2 19.3 3.35 66.3 66.8 0.992 51.7 1.282 29.5 2.247

4CU8 32.3 3.04 15.2 25.4 12.0 2.63 97.0 79.2 1.224 61.0 1.590 36.4 2.665

5CU8 32.3 3.04 15.2 25.2 12.1 3.35 95.0 85.1 1.116 65.6 1.448 36.1 2.632

6CU8 36.8 3.32 15.2 25.1 12.2 4.30 107.6 95.6 1.125 73.9 1.456 38.4 2.802

3CU9 12.2 1.59 15.2 25.5 12.2 1.99 59.0 52.0 1.134 39.7 1.486 22.5 2.622

4CU9 17.1 1.99 15.2 25.4 12.0 2.63 70.7 63.2 1.118 48.7 1.452 26.5 2.668

5CU9 14.7 1.80 15.2 25.3 12.1 3.35 71.6 63.0 1.136 48.6 1.473 24.5 2.922

6CU9 13.7 1.72 15.2 25.0 12.2 4.30 71.6 64.1 1.117 49.4 1.119 23.4 3.606

3EU9 15.2 1.84 15.2 25.6 14.3 1.99 54.8 54.0 1.070 41.9 1.379 25.2 2.294

4EU9 14.3 1.77 15.2 25.4 14.4 2.36 50.3 56.4 0.891 44.0 1.143 24.2 2.078

5EU9 15.1 1.83 15.2 25.3 14.5 3.36 64.0 61.0 1.049 47.7 1.342 24.8 2.580

6EU9 12.8 1.64 15.2 25.1 14.6 4.30 61.8 59.6 1.036 46.6 1.326 22.7 2.722

3GU9 13.5 1.70 15.2 25.6 16.7 1.99 47.6 50.0 0.952 38.9 1.224 23.7 2.008

4GU9 11.6 1.54 15.2 25.4 16.8 2.63 44.1 50.2 0.878 39.2 1.125 21.8 2.023

5GU9 11.2 1.50 15.2 25.2 16.9 3.35 48.5 52.1 0.936 40.7 1.192 21.3 2.277

Mean 1.051 1.357 2.430


Iguro et al.5

1 20.6 1.85 15.8 10.0 12.0 0.80 17.6 16.2 1.086 12.7 1.386 11.9 1.480

2 19.7 1.87 15.8 20.0 12.0 0.80 35.4 32.0 1.106 24.6 1.439 23.3 1.519

3 21.1 1.81 30.0 60.0 12.0 0.40 111.2 105.5 1.054 78.0 1.426 137.2 0.810

4 27.2 2.05 50.0 100.0 12.0 0.40 237.0 232.0 1.021 170.5 1.390 432.9 0.547

5 21.9 2.23 50.0 100.0 12.0 0.40 264.0 276.0 0.956 203.5 1.297 388.4 0.680

6 28.5 2.73 100.0 200.0 12.0 0.40 924.0 1116.0 0.828 890.8 1.037 1772.4 0.521

7 24.3 2.19 150.0 300.0 12.0 0.40 1890.0 2104.0 0.898 1721.2 1.098 3682.3 0.513

Mean 0.993 1.296 0.867


Total of all 45 test data

Mean 1.044 1.356 2.109

Standard deviation 0.087 0.123 0.650Notes: 1 cm = 0.394 in.; 1 MPa = 145 psi; and 1 kN = 0.225 kips.


Table 2—Comparison of experimental and theoretical results of deep beams

Beam no.(1)

fc′, MPa(2)

b, cm (3)

d, cm(4)

l/d(5)

ρ, %(6)

ρv fyv, MPa(7)

Experimental Vu, kN

(8)


Vu, kN(9)

Vu,exp /Vu,th(10)

Vu, kN(11)

Vu,exp /Vu,th(12)

Vu, kN(13)

Vu,exp /Vu,th(14)

Bernaest and Siess1

D-4 35.1 15.2 25.2 8.9 2.21 0 119.0 130.4 0.913 98.8 1.204 37.7 3.156

D-9 19.6 15.2 25.2 8.9 3.36 0 107.0 98.0 1.092 72.9 1.468 28.2 3.794

D-10 24.9 15.2 25.2 8.9 3.36 0 118.0 117.0 1.008 87.4 1.350 31.7 3.722

D-14 32.7 15.2 25.2 8.9 1.01 0 92.0 96.0 0.958 72.1 1.277 36.4 2.527

D-15 27.1 15.2 25.2 8.9 1.01 0 88.0 85.6 1.028 63.9 1.377 33.1 2.660

D-16 40.3 15.2 25.2 8.9 1.01 0 108.2 110.0 0.984 82.4 1.313 40.4 2.678

Mean 0.997 1.331 3.090


Leonhardt and Walther2

12-1 32.2 19.0 27.0 7.3 2.04 0 202.5 218.7 0.926 163.9 1.235 48.9 4.141

12-2 32.2 18.9 27.2 7.3 2.06 0 160.5 215.0 0.746 161.8 0.992 48.4 3.316

13-1 32.7 19.0 27.3 9.8 2.04 0 139.0 158.0 0.880 118.4 1.174 49.2 2.825

13-2 32.7 18.9 27.2 9.8 2.06 0 139.0 157.0 0.885 117.3 1.185 48.8 2.848

GA1 20.3 19.0 27.0 7.4 1.89 0 125.0 150.0 0.833 112.8 1.108 39.4 3.255

GA2 20.3 19.0 27.0 7.4 1.89 0 148.0 150.0 0.987 112.8 1.312 38.4 3.854

G6 26.6 19.0 27.0 7.4 2.47 0 200.5 196.9 1.018 147.3 1.361 43.9 4.567

Mean 0.896 1.195 3.544


Rusch et al.3

A1 22.1 10 12 7.2 3.77 0.31 60.0 48.2 1.244 35.9 1.668 13.1 4.580

A2 22.1 10 12 7.2 3.77 0.31 55.2 48.2 1.145 35.9 1.534 13.1 4.214

A3 21.1 10 12 7.2 3.77 0.31 60.4 47.0 1.285 34.7 1.740 12.9 4.682

A4 21.1 10 12 7.2 3.77 0.31 60.4 47.0 1.285 34.7 1.740 12.9 4.682

A5 23.6 10 12 7.2 3.77 0.31 57.6 51.0 1.129 37.7 1.527 13.4 4.300

A6 23.6 10 12 7.2 3.77 0.31 60.4 51.0 1.184 37.7 1.601 13.4 4.507

B1 22.4 10 20 7.2 3.75 0.31 98.8 81.5 1.212 62.0 1.594 21.9 4.511

B2 22.4 10 20 7.2 3.75 0.31 84.8 81.5 1.040 62.0 1.369 21.9 3.872

B3 22.4 10 20 7.2 3.75 0.31 88.8 81.5 1.089 62.0 1.433 21.9 4.055

B4 22.1 10 20 7.2 3.75 0.31 93.6 81.2 1.152 61.4 1.524 21.8 4.293

B5 22.1 10 20 7.2 3.75 0.31 87.2 81.2 1.073 61.4 1.420 21.8 4.000

B6 22.1 10 20 7.2 3.75 0.31 92.8 81.2 1.142 61.4 1.511 21.8 4.257

BO1 22.4 10 20 7.2 3.75 0 84.8 78.2 1.083 60.1 1.412 15.7 5.401

BO2 22.4 10 20 7.2 3.75 0 78.0 78.2 0.998 60.1 1.300 15.7 4.970

BO3 22.4 10 20 7.2 3.75 0 84.8 78.2 1.084 60.1 1.412 15.7 5.401

BO4 22.2 10 20 7.2 3.75 0 77.6 78.5 0.988 89.7 1.300 15.6 4.974

BO5 22.2 10 20 7.2 3.75 0 72.8 78.5 0.927 89.7 1.219 15.6 4.664

BO6 22.2 10 20 7.2 3.75 0 74.8 78.5 0.953 89.7 1.252 15.6 4.795

C1 24.1 10 28 7.2 3.77 0.31 136.8 120.0 1.136 89.5 1.528 31.5 4.343

C2 24.1 10 28 7.2 3.77 0.31 133.2 120.0 1.110 89.5 1.488 31.5 4.228

C3 22.9 10 28 7.2 3.77 0.31 130.0 116.2 1.118 86.3 1.506 30.9 4.207

C4 22.9 10 28 7.2 3.77 0.31 134.8 116.2 1.160 86.3 1.562 30.9 4.362

C5 23.9 10 28 7.2 3.77 0.31 130.0 119.6 1.087 89.1 1.459 31.4 4.140

C6 23.9 10 28 7.2 3.77 0.31 140.8 119.6 1.177 89.1 1.580 31.4 4.484

Mean 1.116 1.487 4.497



11A1 27.0 15.2 31.4 5.8 3.41 0 267.4 289.5 0.923 217.4 1.230 41.2 6.490

12A1 30.6 15.2 23.8 7.7 4.50 0 180.6 170.8 1.057 127.7 1.414 33.2 5.440

13A1 20.2 15.2 31.9 5.7 0.80 0 118.4 109.7 1.079 82.0 1.444 36.2 3.271

14A1 22.8 15.2 24.3 7.5 1.05 0 89.4 95.0 0.941 71.2 1.255 29.3 3.051

15A1 19.2 15.2 31.6 5.8 1.34 0 154.8 174.0 0.890 133.4 1.161 34.9 4.435

16A1 21.1 15.2 24.0 7.6 1.77 0 105.4 102.9 1.024 75.2 1.401 27.8 3.791

1AU 21.9 15.2 25.6 9.5 0.99 0 64.5 67.3 0.958 50.4 1.280 30.2 2.136


Table 2 (cont.)—Comparison of experimental and theoretical results of deep beams

Beam no.(1)

fc′, MPa(2)

b, cm(3)

d, cm(4)

l/d(5)

ρ, %(6)

ρv fyv, MPa(7)

Experimental Vu, kN

(8)


Vu, kN(9)

Vu,exp /Vu,th(10)

Vu , kN(11)

Vu,exp /Vu,th(12)

Vu, kN(13)

Vu,exp /Vu,th(14)


2AU 21.2 15.2 25.4 9.6 1.32 0 76.1 71.2 1.068 53.3 1.428 29.5 2.580

3AU 22.7 15.2 25.5 9.5 1.99 0 93.0 86.2 1.078 64.3 1.445 30.7 3.030

4AU 17.9 15.2 25.4 9.6 2.63 0 90.7 77.0 1.170 57.5 1.577 27.1 3.347

5AU 20.6 15.2 25.2 9.6 3.35 0 108.0 90.0 1.200 67.2 1.608 28.9 3.737

3AAU 34.6 15.2 25.5 7.2 1.99 0 153.1 176.0 0.870 131.8 1.161 37.8 4.050

4AAU 36.4 15.2 25.4 7.2 2.63 0 183.3 196.0 0.935 146.7 1.249 29.7 4.736

5AAU 29.0 15.2 25.2 7.2 3.35 0 200.6 177.0 1.130 131.7 1.522 34.2 5.865

6AAU 64.4 15.2 25.1 7.3 4.30 0 214.9 209.6 1.025 156.9 1.369 37.1 5.792

4AU8 31.7 15.2 25.4 9.6 2.63 0 117.4 116.5 1.007 87.7 1.339 36.1 3.252

5AU8 31.7 15.2 25.2 9.7 3.35 0 132.6 123.1 1.077 91.7 4.446 35.8 3.704

6AU8 34.1 15.2 25.1 9.7 4.30 0 154.8 136.5 1.134 102.1 1.516 37.0 4.184

6A1 29.3 15.2 31.4 7.8 3.41 1.57 206.8 242.7 1.074 181.2 1.440 117.8 2.214

6B1 30.6 15.2 31.4 7.8 3.41 1.57 255.4 248.7 1.027 185.5 1.377 118.8 2.150

9A1 30.8 15.2 31.4 9.7 3.41 1.67 203.8 220.0 0.926 164.8 1.237 123.7 1.647

9B1 29.9 15.2 31.4 5.8 3.41 1.67 350.7 370.2 0.947 250.8 1.398 123.0 2.851

B-2 33.1 15.2 45.6 8.0 2.22 0.29 466.8 461.8 1.010 344.2 1.356 144.2 3.237

Mean 1.024 1.376 3.695


Total of all 60 test data

Mean 1.042 1.395 3.940

Standard deviation 0.112 0.154 0.985Notes: 1 cm = 0.394 in.; 1 MPa = 145 psi; and 1 kN = 0.225 kips.

Fig. 8—Effect of l/d on shear strength of slender beams withdifferent steel ratios.

Fig. 9—Effect of /d on shear strength of deep beams underuniform load and under two-point loading (steel ratio ρ =3.35%).

of observed shear capacity to calculated ACI shear capacitygoes down to 0.50 for beams with a depth d ≥ 1.0 m. It isworth noting that the existing number of tests on such largebeams under uniform loads is limited and more experimentalstudies could further justify the proposed theory.

The comparisons in Tables 1 and 2 and Fig. 8 though 10clearly demonstrate that limitations of the ACI Code, orthose resulting from a consideration of a two-point loadingarrangement, are effectively overcome by the proposedsimple formulas (Eq. (11) and Eq.(16)). These equationshave been derived via the presented theory, in which all the

factors affecting the shear strength of beams under a uniformload are rationally considered.

CONCLUSIONSA theory has been presented about the shear strength of

reinforced concrete beams under a uniformly distributedload. The theory results in simple and easy-to-use expressionsfor the ultimate shear force (or ultimate uniform load) ofslender beams as well as deep beams under a uniform load.The impact of size effect in the shear strength of slenderbeams is taken into account.


The proposed theory accurately predicts capacitiescorresponding to the experimental observations of ultimateshear force of a well-grounded test series of slender beams aswell as deep beams under uniform load with various strengthsof concrete, longitudinal steel ratios, shear reinforcementratios, l/d, and geometrical sizes.

It is shown that the shear strength of beams, either slenderor deep, under a uniform load is much higher than the shearstrength of beams under a loading arrangement of twoconcentrated loads at the quarter points.

In this paper, it has been shown that the ACI Code predictionsfor the ultimate shear force of beams under a uniform loadare much lower for the cases of small size beams (eitherslender or deep), whereas for the larger beams, thepredictions are higher.

REFERENCES1. Bernaert, S., and Siess, O., “Strength in Shear of Reinforced Concrete

Beams under Uniform Load,” University of Illinois, Urbana, IL, June 1956.2. Leonhardt, F., and Walther, R., “Schubversuche an Einfeldrigen

Stahlbetonbalken mit und ohne Schubbewehrung zur Ermittlung derSchubtragfähigkeit und der Oberen Schubspannungsgrenze,” Deutcher

Ausschuss für Stahlbeton, Heft 151, W. Ernst u. Sohn, Berlin, Germany,1962, 68 pp. (in German)

3. Rüsch, H.; Haugli, F. R.; and Mayer, H., “Schubversuche an Stahlbeton—Rechteckbalken mit Gleischmässig Verteilter Belastung,” Deutscher Ausschussfür Stahlbeton, Heft 145, W. Ernst u. Sohn, Berlin, Germany, 1962, 30 pp.(in German)

4. Krefeld, W. J., and Thurston, C. W., “Studies of the Shear and DiagonalTension Strength of Simply Supported Reinforced Concrete Beams,” ACIJOURNAL, Proceedings V. 63, No. 4, Apr. 1966, pp. 451-476.

5. Iguro, M.; Shioya, T.; Nojiri, Y.; and Akiyama, H., “ExperimentalStudies on Shear Strength of Large Reinforced Concrete Beams underUniformly Distributed Load,” Japan Society of Civil Engineers (JSCE),Concrete Library, No. 5, Aug. 1985, pp. 137-154. (translation fromProceedings of JSCE, No. 345/V-1, Aug. 1984)

6. Shioya, T.; Iguro, M.; Nojiri, Y.; Akiyama, H.; and Okada, T., “ShearStrength of Large Reinforced Concrete Beams,” Fracture Mechanics:Application to Concrete, SP-118, V. C. Li and Z. P. Bažant, eds., AmericanConcrete Institute, Farmington Hills, MI, 1989, pp. 259-279.

7. Kani, G. N., J, “Basic Facts Concerning Shear Failure,” ACI JOURNAL,Proceedings V. 63, No. 6, June 1966, pp. 675-692.

8. ACI Committee 318, “Building Code Requirements for StructuralConcrete (ACI 318-02) and Commentary (318R-02),” American ConcreteInstitute, Farmington Hills, MI, 2002, 443 pp.

9. Eurocode No. 2, “Design of Concrete Structures, Part 1: GeneralRules and Rules for Buildings,” Commission of the European Communities,ENV 1992-1-1, Dec. 2004, 225 pp.

10. Zararis, P. D., and Papadakis, G., “Diagonal Shear Failure and SizeEffect in RC Beams without Web Reinforcement,” Journal of StructuralEngineering, ASCE, V. 127, No. 7, July 2001, pp. 733-742.

11. Zararis, P. D., “Shear Strength and Minimum Shear Reinforcementof Reinforced Concrete Slender Beams,” ACI Structural Journal, V. 100,No. 2, Mar.-Apr. 2003, pp. 203-214.

12. Zararis, P. D., “Shear Compression Failure in Reinforced ConcreteDeep Beams,” Journal of Structural Engineering, ASCE, V. 129, No. 4,Apr. 2003, pp. 544-553.

13. Zararis, P. D., “Failure Mechanisms in RC Plates Carrying In-PlaneForces,” Journal of Structural Engineering, ASCE, V. 114, No. 3, Mar.1988, pp. 553-574.

14. Zararis, P. D., “Aggregate Interlock and Steel Shear Forces in theAnalysis of RC Membrane Elements,” ACI Structural Journal, V. 94, No. 2,Mar.-Apr. 1997, pp. 159-170.

15. Placas, A., and Regan, P. E., “Shear Failure of Reinforced ConcreteBeams,” ACI JOURNAL, Proceedings V. 68, No. 10, Oct. 1971, pp. 763-773.

16. Mellis, S., and Tsalcatides, T., “Shear Strength of Beams underChanging Reversal Loading,” MSc dissertation, Department of CivilEngineering, Aristotle University of Thessaloniki, Thessaloniki, Greece,2001, 139 pp. (in Greek).

17. Kong, F. K.; Robins, P. J.; Singh, A.; and Sharp, G. R., “Shear Analysisand Design of Reinforced Concrete Deep Beams,” The Structural Engineer,V. 50, No. 10, Oct. 1972, pp. 405-409.

Fig. 10—Size effect on shear stress at failure of slenderbeams under uniform load (tests of Iguro et al.5).

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Shear Strength of Reinforced Concrete Beams Under Uniformly Distributed Loads

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