H
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UNIVERSITY OF ILLINOIS AT URBANACHAMPAIGN
PRODUCTION NOTE
University of Illinois at
UrbanaChampaign Library
Largescale Digitization Project, 2007.
1111111
Strength in Shear of Reinforced
Concrete Beams
by
Armas Laupa
Chester P. Siess
Nathan M. Newmark
UNIVERSITY OF ILLINOIS BULLETIN
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A REPORT OF AN INVESTIGATION
Conducted by
THE ENGINEERING EXPERIMENT STATION
UNIVERSITY OF ILLINOIS
In Cooperation with
THE OHIO RIVER DIVISION LABORATORIES
CORPS OF ENGINEERS
UNITED STATES ARMY
Price: One Dollar
UNIVERSITY OF ILLINO IS BULLETIN
Volume 52, Number SS; March, 1955S. Published seven times each month by the University of
Illinois. Entered as secondclass matter December 11, 1912, at the post office at Urbana, Illinois,
under the Act of August 24, 1912. Office of Publication, 207 Administration Building, Urbana, Ill.
Strength in Shear of Reinforced
Concrete Beams
by
Armas Laupa
FORMERLY RESEARCH ASSOCIATE IN CIVIL ENGINEERING
Chester P. Siess
RESEARCH ASSOCIATE PROFESSOR OF CIVIL ENGINEERING
Nathan M. Newmark
RESEARCH PROFESSOR OF STRUCTURAL ENGINEERING
ENGINEERING EXPERIMENT STATION BULLETIN NO. 428
CONTENTS
I. INTRODUCTION 7
1. Introduction 7
2. Object and Scope of Investigation 7
3. Acknowledgment 7
4. Notation 8
II. SIMPLESPAN RECTANGULAR BEAMS WITHOUT WEB REIN
FORCEMENT AND UNDER ONE OR TWO SYMMETRICAL
CONCENTRATED LOADS 10
5. Review of Earlier Research 10
6. Derivation of Basic Empirical Equation 11
7. Test Data 13
8. Theoretical Interpretation of Basic Empirical Equation 20
9. Properties and Limitations of Basic Empirical Equation 23
III. SIMPLESPAN RECTANGULAR BEAMS WITH WEB REIN
FORCEMENT AND UNDER ONE OR TWO SYMMETRICAL
CONCENTRATED LOADS 24
10. General Considerations 24
11. Stirrups as Web Reinforcement 25
12. BentUp Bars as Web Reinforcement 31
13. Maximum Useful Amount of Web Reinforcement 34
IV. SIMPLESPAN TBEAMS UNDER ONE OR TWO SYMMETRICAL
CONCENTRATED LOADS 37
14. TBeams Without Web Reinforcement 37
15. TBeams With Web Reinforcement 42
V. RESTRAINED BEAMS UNDER SYMMETRICAL CONCEN
TRATED LOADS 45
16. Modes of Failure 45
17. Test Data on Restrained Beams 48
VI. BEAMS UNDER OTHER TYPES OF LOADING 56
18. Limitations of ShearCompression Failures 56
19. ShearProper 58
20. Transition Region and Flexural Failures 61
21. Beams Under Uniform Load 63
VII. SUMMARY AND CONCLUSIONS 68
22. General Summary and Discussion 68
23. Summary of Equations 71
VIII. BIBLIOGRAPHY 73
FIGURES
1. Evaluation of Eq. 17 for SimpleSpan Rectangular Beams Without
Web Reinforcement 18
2. Effect of Longitudinal Steel Percentage on Ratio of Measured to
Computed ShearMoment. SimpleSpan Rectangular Beams
Without Web Reinforcement 19
3. Relation Between kik3 and Concrete Strength 21
4. Internal Forces at Section of Diagonal Crack 25
5. Effect of Web Reinforcement on Shear Strength. SimpleSpan
Rectangular Beams with Stirrups 31
6. SimpleSpan Rectangular Beams Failing in Flexure. Beams
Reinforced with Stirrups 32
7. SimpleSpan Rectangular Beams Failing in Flexure. Beams
Reinforced with BentUp Bars 33
8. Relation Between Strength in Shear and Flexure as Function of
Reinforcement Percentage 35
9. Maximum Useful Amount of Web Reinforcement as Function of
Concrete Strength and Yield Strength of Reinforcement 35
10. Tests by Ferguson and Thompson. SimpleSpan TBeams Without
Web Reinforcement 42
11. Failure Moment as Function of Concrete Strength. SimpleSpan
TBeams Without Web Reinforcement 43
12. Effect of Web Reinforcement on Strength of SimpleSpan TBeams
Failing in Shear 44
13. Restrained Beam Under Symmetrical Concentrated Loads 45
14. Continuous Top and Bottom Reinforcement. Restrained Beam
With No Bond Failure 45
15. Continuous Top and Bottom Reinforcement. Bond Destroyed in
Restrained Beam With One Crack 46
16. Continuous Top and Bottom Reinforcement. Bond Destroyed in
Restrained Beam With Two Cracks 46
17. Straight Bars Cut Off Beyond Point of Contraflexure.
Restrained Beam 47
18. Stripping Type of Bond Failure. Restrained Beam 47
19. Restrained Beam With All Bars Bent Up 48
20. Restrained Beam With BentUp and Straight Bars 48
21. Typical Restrained Beam of Richart and Larsen 48
22. Restrained Beams of Moody 50
23. Beams of Moody, Series I, II, and IV. Restrained Beams Without
Web Reinforcement 52
24. Beams of Moody, Series VI. Restrained Beams Without
Web Reinforcement 53
25. Beams of Moody, Series I and IV. Restrained Beams With
Web Reinforcement 54
26. Shear Force V Versus a/d. Possible Modes of Shear Failure for
SimpleSpan Beams 57
27. Beams of Graf, Heft 80. ShearProper Type of Failures 59
28. Nominal Shearing Stress Ratio Versus x/D for Failures in
ShearProper 61
29. Beam 1026 of Bach and Graf, Heft 48 64
30. Beam 1025 of Bach and Graf, Heft 48 64
31. Beam 1031 of Bach and Graf, Heft 48 64
32. Beam 1032 of Bach and Graf, Heft 48 64
33. Ratio of Measured to Computed Failure Moment as Function of
M/Vd. TBeams of Heft 48 under Sixteen Concentrated Loads 65
34. Beams 60 of Bach and Graf, Heft 20 66
35. Beams 62 of Bach and Graf, Heft 20 66
TABLES
1. Range of Test Variables for SimpleSpan Rectangular Beams
Without Web Reinforcement and Under One or Two
Symmetrical Concentrated Loads 13
2. Tests by Richart, Series 1910. SimpleSpan Rectangular Beams
Without Web Reinforcement 14
3. Tests by Richart, Series 1911. SimpleSpan Rectangular Beams
Without Web Reinforcement 14
4. Tests by Richart, Series 1913. SimpleSpan Rectangular Beams
Without Web Reinforcement 14
5. Tests by Richart, Series 1917. SimpleSpan Rectangular Beams
Without Web Reinforcement 14
6. Tests by Richart, Series 1922. SimpleSpan Rectangular Beams
Without Web Reinforcement 15
7. Tests by Richart and Jensen, 1931. SimpleSpan Rectangular
Beams Without Web Reinforcement 15
8. Tests by Thompson, Hubbard, and Fehrer, 1938. SimpleSpan
Rectangular Beams Without Web Reinforcement 15
9. Tests by Moretto, 1945. SimpleSpan Rectangular Beams
Without Web Reinforcement 15
10. Tests by Clark, 1951. SimpleSpan Rectangular Beams Without
Web Reinforcement 16
11. Tests at M.I.T., 1951. SimpleSpan Rectangular Beams Without
Web Reinforcement 16
12. Tests by Gaston, 1952. SimpleSpan Rectangular Beams
Without Web Reinforcement 16
13. Tests by Laupa, 1953. SimpleSpan Rectangular Beams
Without Web Reinforcement 16
14. Tests by Moody, Series A, 1953. SimpleSpan Rectangular Beams
Without Web Reinforcement 17
15. Tests by Moody, Series B, 1953. SimpleSpan Rectangular Beams
Without Web Reinforcement 17
16. Tests by Moody, Series III, 1953. SimpleSpan Rectangular Beams
Without Web Reinforcement 17
17. Range of Test Variables for SimpleSpan Rectangular Beams With
Stirrups and Under One or Two Symmetrical Concentrated Loads 26
18. Tests by Richart, Series 1910. SimpleSpan Rectangular Beams
With Stirrups 26
19. Tests by Richart, Series 1913. SimpleSpan Rectangular Beams
With Stirrups 27
20. Tests by Richart, Series 1922. SimpleSpan Rectangular Beams
With Stirrups 27
21. Tests by Slater, Lord, and Zipprodt, 1926. SimpleSpan
Rectangular Beams With Stirrups 27
22. Tests by Slater and Lyse, 1930. SimpleSpan Rectangular
Beams With Stirrups 28
23. Tests by Thompson, Hubbard, and Fehrer, 1938. SimpleSpan
Rectangular Beams With Stirrups 28
24. Tests by Johnston and Cox, 1939. SimpleSpan Rectangular
Beams With Stirrups 28
25. Tests by Moretto, 1945. SimpleSpan Rectangular Beams
With Stirrups 29
26. Tests by Clark, 1951. SimpleSpan Rectangular Beams
With Stirrups 30
27. Tests by Gaston, 1952. SimpleSpan Rectangular Beams
With Stirrups 30
28. Tests by Moody, Series III, 1953. SimpleSpan Rectangular
Beams With Stirrups 31
TABLES (Concluded)
29. Tests by Richart, Series 1917. SimpleSpan Rectangular Beams
With BentUp Bars 33
30. Tests by Richart, Series 1911. SimpleSpan Rectangular Beams
With BentUp Bars 34
31. Amount of Web Reinforcement Required to Prevent Shear
Failures in Rectangular Beams. Normal ACI Beams
Without Compression Reinforcement 36
32. Range of Test Variables for SimpleSpan TBeams Under
Two Symmetrical Concentrated Loads 38
33. Tests by Bach and Graf, Heft 10, 1911. SimpleSpan TBeams
Under Two Symmetrical Concentrated Loads 38
34. Tests by Braune and Myers, 1917. SimpleSpan TBeams
Under Two Symmetrical Concentrated Loads 39
35. Tests by Richart, Series 1922. SimpleSpan TBeams Under
Two Symmetrical Concentrated Loads 39
36. Tests by Thompson and Ferguson, 1950. SimpleSpan TBeams
Under Two Symmetrical Concentrated Loads 40
37. Tests by Ferguson and Thompson, 1953. SimpleSpan TBeams
Under Two Symmetrical Concentrated Loads 40
38. Tests by Bach and Graf, Heft 12, 1911. SimpleSpan TBeams
With BentUp Bars Under Two Symmetrical Concentrated Loads 41
39. Tests by Graf, Heft 67, 1931. SimpleSpan TBeams Under
Two Symmetrical Concentrated Loads 41
40. Tests by Richart and Larsen, Series 1917. Restrained Beams
With BentUp Bars 49
41. Tests by Moody, Series I, 1953. Restrained Beams Without
Web Reinforcement 51
42. Tests by Moody, Series II and IV, 1953. Restrained Beams
Without Web Reinforcement 51
43. Tests by Moody, Series VI and V, 1953. Restrained Beams
Without Web Reinforcement 51
44. Tests by Moody, Series I, 1953. Restrained Beams With
Web Reinforcement 53
45. Tests by Moody, Series IV and II, 1953. Restrained Beams
With Web Reinforcement 54
46. Tests by Graf, Heft 80, 1935. ShearProper Type of Failures 60
47. Other ShearProper Type of Failures 61
48. Tests by Graf, Heft 67, Series II, 1931. SimpleSpan TBeams
Under One Unsymmetrical Concentrated Load 62
49. Tests by Graf, Heft 67, Series I, 1931. SimpleSpan TBeams
Under Three Concentrated Loads 63
50. Tests by Bach and Graf, Heft 48, 1921. SimpleSpan TBeams
Under Sixteen Equal Concentrated Loads 65
51. Tests by Bach and Graf, Heft 20, 1912. SimpleSpan TBeams
Under Eight Eaual Concentrated Loads
I. INTRODUCTION
I. Introduction
Reinforced concrete, like other structural ma
terials, has been the subject of extensive experi
mental and analytical research and the past sixty
years have witnessed a steady advance in knowl
edge of the behavior of reinforced concrete members
under static loads. With the aid of numerous tests,
a rather complete understanding has been obtained
of the ultimate strength of such members in pure
flexure and under pure axial compression. In
addition, there have been developed theories for
members subjected to combined flexure and axial
compression. However, no such extensive informa
tion is available for members subjected to com
binations of flexure and shear, or of flexure, com
pression and shear.
In previous research, major emphasis has been
placed on the evaluation of the contribution of web
reinforcement, and the shear strength of a rein
forced concrete member has been interpreted in the
light of a truss analogy. Experimental evidence,
however, has forced certain modifications of the
original truss analogy equation. The contribution
of the beam itself, without the benefit of web
reinforcement, has been taken into consideration.
Furthermore, it has been found that the moment
shear ratio affects the ultimate strength in shear.
These modifications, suggested by different authors,
have retained essentially the truss analogy relation
but have added new terms to account for effects
other than that of web reinforcement. All the
modified equations, however, have been derived
experimentally for each given series of test speci
mens and have usually failed to give good correla
tion with other test data, outside the range of test
variables for which the equations were derived.
Current design specifications have apparently
been based on certain minimum values obtained
from tests. Although these specifications yield
satisfactory or even conservative values in most
practical cases, test specimens have been reported
which failed in shear at a lower load than that
given by the usual "safe working stresses." This
indicates a definite need for a better understanding
of the phenomenon of shear failure and for a more
reliable set of design rules.
2. Object and Scope of Investigation
The object of this investigation was to review
and correlate the results of previous research in the
field of shear and diagonal tension, to determine the
modes and characteristics of shear failure of rein
forced concrete beams, and to establish a general
expression for the shear strength of reinforced
concrete beams under different loading conditions.
The investigation was limited to members subjected
to combinations of shear and flexure only.
More than one thousand tests of beams having
a wide range of physical properties and subjected
to different types of loading were studied. A basi
cally new empirical equation was derived for the
shear strength of simplespan rectangular beams
without web reinforcement and under one or two
symmetrical concentrated loads. It is shown herein
that the basic equation can be interpreted with the
aid of the conventional theory of compression fail
ures of reinforced concrete beams. This equation
was first presented in a previous technical report.("*
The basic empirical equation was extended to
include beams with web reinforcement, and the
amount of web reinforcement required to prevent
shear failures was determined. Furthermore, the
same equation was modified to apply to simple
span Tbeams and restrained beams under sym
metrical concentrated loads. It was found also that
the basic equation could be used to determine the
shear strength of a reinforced concrete beam under
uniform load and, possibly, under any type of
distributed loading.
3. Acknowledgment
The studies reported herein were made as a part
of a research program to establish by analysis and
by studies of the available test data criteria for the
structural design of reinforced concrete box culverts.
The work was carried out in the Structural Re
search Laboratory of the Department of Civil Engi
neering in the Engineering Experiment Station of
* Superscripts in parentheses refer to corresponding entries in the
Bibliography.
ILLINOIS ENGINEERING EXPERIMENT STATION
the University of Illinois in cooperation with Ohio
River Division Laboratories, Corps of Engineers,
U.S. Army, under Contract DA33017eng222.
This bulletin is based upon a thesis by A. Laupa
submitted in partial fulfillment of the requirements
for the degree of Doctor of Philosophy in Civil
Engineering in the Graduate College of the Uni
versity of Illinois in 1953. The thesis was written
under the direction of N. M. Newmark and C. P.
Siess.
4. Notation
The following notation is used:
a = distance from end support to concentrated
load in simplespan beams
a= angle of inclination of web reinforcement
with respect to axis of beam
A = given by Eq. 45
A = compressive area of concrete as determined
by "straight line" theory
A, = area of web reinforcement
b= width of rectangular beam or width of
flange of Tbeam
b'= width of web of Tbeam
C=internal compressive force in concrete;
(also various numerical coefficients as de
fined in text)
d = distance from centroid of tension reinforce
ment to compression face of beam
D= total depth of beam
e = thickness of flange of Tbeam
o,= ultimate compressive strain in concrete,
taken as 0.004
: = strain in steel at yield point
Ec= modulus of elasticity of concrete
E,= modulus of elasticity of reinforcing steel
f =a distance as shown on Fig. 13
fc = compressive stress in extreme fiber of con
crete, given by straight line theory
f' = compressive strength of 6 by 12in. con
crete cylinders
fe'= compressive strength of concrete cubes
fr = modulus of rupture
f,= stress in tension reinforcement
f = stress in compression reinforcement
f,= yield stress of tension reinforcement
f,'= yield stress of compression reinforcement
f, = stress in web reinforcement
fA = yield stress in web reinforcement
F = total force in web reinforcement, see Fig. 4
F= shape factor of Tbeams, given by Eq. 34
g=a distance as shown on Fig. 13
h=a distance as shown on Fig. 13
Icr = moment of inertia of "straight line" cracked
transformed section, either rectangular or
Tsection
IR = moment of inertia of uncracked rectangu
lar section having the same width as the
flange of an otherwise similar Tsection
IT = moment of inertia of uncracked Tsection
jd= internal moment arm
kd = depth of compression zone of concrete as
determined by "straight line" theory
k,d = depth of compression zone of concrete at
shear failure
C
ki= k3f,,bd' a parameter which determines
k3f' kmbd
the magnitude of the compressive force C.
It is the ratio of the average compressive
stress to the maximum compressive stress
in concrete
k2.= fraction of the depth of compression zone
which determines the position of the com
pressive force C in concrete
k3 = ratio of maximum compressive strength of
concrete in beam to compressive strength
of standard test cylinders
K = (sin a + cos a) sin a
L=span length of test beam
L'= total length of test beam
M= bending moment
M,= shearcompression moment of beam with
out web reinforcement, given by Eqs. 18,
35, 44
M,= shearcompression moment of beam with
web reinforcement, given by Eq. 28
E,
n= E= elastic modular ratio, taken as
10,000
5+ ±
n' = = plastic modular ratio
A.
= , where A,= area of tension reinforce
ment
P= bd,' where A,'= area of compression rein
forcement
po = given by Eq. 43
p = given by Eq. 47
P = total load on beam
P, =load which corresponds to M,
Bul.428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
P, = load which corresponds to M,w
q= f" = reinforcing index
qcr = value of q which determines the boundary
between initial flexural failure by crushing
of concrete and by yielding of reinforce
ment, given by Eq. 32
r = A for rectangular beams
bs sin a
w for Tbeams
b's sin a
s = spacing of web reinforcement along axis of
beam
td= distance between centroids of tension and
compression reinforcement
V
v= nominal shearing stress in concrete, bjd
V V
,bkd or b as defined in text
bkd = nominal shearing stress at ultimate loabD
v.= nominal shearing stress at ultimate load
ve=nominal shearing stress at ultimate load
for shearproper, given by Eq. 48
V= shearing force
Vo = critical magnitude of shear needed to cause
diagonal tension failures in the transition
region of a/d between shearcompression
and flexural failuressee Section 18 and
Fig. 26
x= clear distance between two load blocks
see Fig. 27a; also other distances as de
fined in text
The following notation was used in designating
modes of failure in the tables:
B= bond
C = flexural compression
Cr = crushing at hooks
DT= reported diagonal tension failures; most
beams failed in shear, a few in bond as
marked in the tables
S = shear
T = flexural tension
T  S = tension with sheartype final collapse
II. SIMPLESPAN RECTANGULAR BEAMS WITHOUT WEB REINFORCEMENT AND
UNDER ONE OR TWO SYMMETRICAL CONCENTRATED LOADS
5. Review of Earlier Research
In previous research, shear failures have been
treated conventionally as failures in diagonal ten
sion. Since the real value of diagonal tension stress
was generally difficult to determine, the unit shear
ing stress
V
v = bjd(1)
was considered as a measure of diagonal tension.
The effect of web reinforcement was taken into
account by considering a beam acting as a truss,
in which the top chord was formed by the com
pression zone of the concrete, the bottom chord
by the longitudinal reinforcement, the tension web
members by the web reinforcement, and the com
pression web members by the concrete in the web
of the beam. From these assumptions the following
equation was derived to represent stress in the web
reinforcement:
f = (2)
where the value of K depended on the angle of in
clination of web reinforcement.
It was realized that Eqs. 1 and 2 were approxi
mate in nature, and thus empirical data were used
to correlate the real behavior of test beams with
the above theoretical considerations. It was ob
served that measured stresses in the web reinforce
ment were, in general, considerably less than
predicted by Eq. 2; this discrepancy was attributed
to the fact that a portion of the total shear was
carried by the concrete. In 1927 Richart(2) modi
fied Eq. 2 from the truss analogy in the following
manner:
v = C + Krf,
where the constant C was found to vary from 90 to
200 psi, and it was stated that C probably depended
"upon the percentage of web reinforcement used
and also on the quality of the concrete."
More complete conclusions regarding the con
tribution of the concrete to resist shear had been
reached by Talbot some twenty years earlier. In
1909, Talbot3") reported that for beams without
web reinforcement the ultimate nominal shearing
stress v increases as the quality of concrete in
creases, as the amount of longitudinal reinforce
ment increases, and as the span length L decreases.
These conclusions, however, were apparently dis
regarded by most later investigators. Only in rela
tively recent years have new attempts been made
to evaluate in quantitative terms the contribution
of the various elements of a beam to its strength
in shear. In 1945 Moretto(4) presented the following
equation for the shearing strength of a simply
supported beam:
v = Krfy, + 0.10 f,' + 5000 p
This was essentially an extension of Eq. 3 sug
gested by Richart. In 1951 Clark(5) reported the
following formula:
v = 2500 \/7 + 0.12 f' (d/a) + 7000 p (5)
This equation was the first to account quantita
tively for all of the variables listed by Talbot in
1909 as influencing the shearing strength of rein
forced concrete beams.
In a previous report(1) attempts were made to
correlate the results of previous research and to
investigate the validity of Eqs. 4 and 5 as well as
other empirical equations in the following form:
v = Krf,ý + Cif' + C2p (d/a)
v = Krf. + [Cif/' + Cap] (d/a)
All these attempts to relate the nominal shearing
strength of simplespan reinforced concrete beams
to a function consisting of the truss analogy term
Krf,, and linear terms of fe' and p failed to give
good correlation with test results. Thus, all of the
Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
empirical equations which were derived for a cer
tain range of test variables were found to be not
applicable outside that particular range.
Since the introduction of the concept of the
truss analogy over 50 years ago, major emphasis
has been placed, in general, on the evaluation of
the contribution of web reinforcement to shear
strength. The contribution of the beam itself, with
out the benefit of any web reinforcement, has
remained a relatively unknown quantity. Further
more, any uncertainties with regard to the contri
bution of web reinforcement have reflected directly
on the contribution of the beam itself, thus render
ing both questionable.
The first problem, therefore, is the evaluation
of shear strength of a beam without web reinforce
ment. In the following section a general expression
for the shearing strength of such beams is derived.
6. Derivation of Basic Empirical Equation
After the formation of a diagonal tension crack,
a reinforced concrete beam which does not fail in
tension will fail either in the compression zone of
the concrete or in bond. Although the cause of these
two types of failure is different, their appearance
is often very nearly the same. When a beam fails
by the destruction of the compression zone, the
shear force which was previously carried by the
concrete is transferred to the level of the longi
tudinal reinforcement. This leads to splitting of the
concrete along the reinforcing bars. When a beam
fails in bond, however, slipping of the longitudinal
reinforcement produces cracking of the concrete
along the bars and effectively reduces the bending
resistance of the section. This causes a concentrated
angle change at the end of the diagonal crack and
leads to a premature destruction of the compression
zone of the concrete. Since the above phenomena
take place simultaneously, it has often been difficult
to determine the real cause of failure. In the early
tests in which plain bars without end anchorage
were used as tension reinforcement, bond failures
were frequently considered as diagonal tension,
that is, shear failures. In many of the more recent
experimental investigations, however, the possi
bility of bond failure has been eliminated by the
use of special types of end anchorage in addition
to deformed bars having good bond characteristics.
Splitting along the longitudinal reinforcement has
still been observed and sometimes even considered
as a primary mode of failure. This phenomenon,
however, is believed to be secondary to the failure
of the beam by destruction of the compression zone.
Failure by destruction of the compression zone
takes place, in general, under a concentrated load,
at the section of maximum moment and maximum
shear. The real cause of failure has not been gener
ally understood. It has been suggested that this
failure is the result of the principal stresses, com
pressive or tensile, or of the maximum shearing
stress. As has been mentioned, the conventional
theory, treating shear failures as diagonal tension
failures, considered the nominal shearing stress v
as a measure of diagonal tension. Previous research
has indicated that v is a function of the following
variables:
v= b = F p, f', d ' Krfw
All empirical equations suggested by different in
vestigators, however, have failed to give good cor
relation with all of the available test results.
Furthermore, the conventional theory pictures the
nominal shearing stress v as being distributed over
the entire crosssection of a beam, uniform from the
level of tension reinforcement to the neutral axis.
The formation of a diagonal crack, however, radi
cally changes the state of stress in a reinforced
concrete beam. Since there can be no transfer of
stress across a crack, the nominal shearing stress
cannot possibly be the criterion of shear failure
which occurs at loads greater than that causing first
crack, and the state of stress in the uncracked com
pression zone should be investigated in order to de
termine the probable cause of final failure.
A basic equation for the shear strength of a
simplespan rectangular beam without any form of
web reinforcement and under one or two symmetri
cal concentrated loads was derived by considering
the state of stress in the compression zone of the
concrete. It was first assumed that the total shear
ing force V is resisted solely by the compression
area of the concrete. For beams without compres
sive reinforcement the area of the compression zone
is given by kdb, where the quantity k,d refers to
the depth of the compression zone at shear failure.
Thus the average shearing stress is given by
v = V/kdb. It was further assumed that the ulti
mate shearing unit stress v. was a function of fe'.
Test results have shown that the shear capacity of
the compression zone decreases as the moment
shear ratio M/V increases. Since the ratio M/V
ILLINOIS ENGINEERING EXPERIMENT STATION
equals a for the beams considered, this effect has
usually been taken into consideration by the d/a
ratio, and there seems to be a linear relationship
between this ration and the shear capacity of the
beam. Since both the horizontal compressive
stresses and the vertical shearing stresses are as
sumed to be resisted by the same compressive area,
it seems more reasonable to consider the shear
compressive force ratio V/C rather than the M/V
ratio as influencing the ultimate load in shear. For
the type of beam under consideration it can be
written that V/C = jd/a. Thus the ultimate shear
ing stress v, can be expressed as follows:
V _ d
v kdb jd F, (fZ') (9)
If both sides of Eq. 9 are multiplied by the factor
aks/dfc', and the ratio Fi (fe') /f,' considered as a
new function F(fe'), Eq. 9 can be rewritten as:
Va M
bd2f = k,jF(f/') or bdf = kjF(f/') (10)
Equation 10 is in a form which suggests that the
criterion for shear failure is a limiting moment
rather than a limiting shearing stress. There is some
supporting evidence for this observation in previous
test results. Beams with no web reinforcement
tested by Clark(5' had the d/aratio as the only
variable; all these beams failed at a nearly constant
moment, although the total shear force at failure
depended upon the location of the loads on the
beams. In 1906, Moritz(6) reported a series of tests
on small mortor beams with the d/aratio as the
only variable, and his results again show that the
ultimate moment was nearly the same for all posi
tions of loads. Thus the socalled shear or diagonal
tension failures seem to be failures in compression,
the criterion of failure being a limiting average
compressive stress or a limiting total compressive
force in the compression zone of the concrete. This
type of failure differs from flexural compression
failures only because the compressive area is re
duced in depth as a result of diagonal tension
cracking.
In Eq. 10 there are two main unknowns: the
depth of the compression zone, kd, and the limit
ing average compressive stress, related to F(fc').
The quantity j can be considered as a constant,
since it does not vary over a large range.
The depth of the compression zone can be de
termined accurately for flexural failures, both in
tension and in compression, by considering statical
equilibrium and the strain relations involved. For
shear failures, however, no theoretical relationship
between the extent of diagonal tension cracking and
the physical properties of the beam has been found.
Consequently, th6 depth of the compression zone
must be determined empirically. From previous in
vestigations it can be shown qualitatively that k,
is a function of If' and p. Furthermore, this func
tion must be a complex one, since different empiri
cal equations considering v as a linear function of
fc' and p have failed to agree with test results. In
order to facilitate the empirical evaluation of k,, it
was deemed advantageous to consider the ratio
ks/k rather than k, alone. The value of k as de
termined by the straight line theory is also a
function of f,' and p. It was felt that there might
be some similarity between the functions repre
senting k, and k, so that the ratio k8/k might be
easier to evaluate than k, alone. It was considered
that if the ratio ks/k is either a constant or a func
tion of fC', Eq. 10 can be written as
M
bd2f,' = k F(f.')
and the unknown function F(fc') can be
directly from available test data. If this
done, the ratio k,/k must also depend
Eq. 11 must be rewritten as
M
bd2f = k F(f.', p)
(11)
evaluated
cannot be
on p and
(11a)
Equation 11 was derived for beams without
compression reinforcement. For beams with both
tension and compression reinforcement, Eq. 11 must
be modified to take into account the added effect
of the compression reinforcement. If it is assumed
that a beam fails before the compression reinforce
ment yields, an expression for the limiting moment
of shear failure can be derived by considering that
the presence of compression reinforcement increases
the compression area of the transformed section by
an amount equal to np'bd, the steel area trans
formed to concrete; thus:
Ac = bkd + np'bd = bd (k + np')
(12)
This modified compression area leads to the follow
ing equation which corresponds to Eq. 11 for beams
without compression reinforcement:
M
, = (k + up') F(f,') (13)
Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
The quantity k refers to the theoretical depth
of the compression zone as ordinarily determined
from transformed areas. For beams with tension
reinforcement only, the numerical value of k is
obtained from the wellknown equation
k = (pn)2 + 2pn  pn (14)
For beams with both tension and compression re
inforcement the following equation can be derived:
k = V [n (p + p')]2 + 2n (p + p'  p't)
n (p + p') (15)
where td is the distance between the centers of the
tension and compression reinforcement.
In all subsequent calculations the value of the
modular ratio n used in the above equations was
determined by Jensen's formula(7)
5 + 10,000 (16)
n=5+ (16)
f'1
which has been found to give reliable results.
7. Test Data
In order to determine the unknown function
.F(f,') in Eqs. 11 and 13, experimental results of
previous research were analyzed. Attention was
directed first only to simplespan rectangular beams
without web reinforcement and subjected to one or
two symmetrical concentrated loads. All known
tests of such beams were included in the analysis
except those of very early beams for which there
was some doubt about the compressive strength
of the concrete used.
A total of 125 beams from 15 different investi
gations were considered. These beams were tested
over a period of 43 years and had a wide variation
in their physical properties. Table 1 lists the differ
ent investigations, giving their entry numbers in
the Bibliography and the numbers of the tables in
which they are analyzed. This table also sum
marizes the range of test variables for the different
groups of beams.
There were 111 of these beams which failed in
shear, 7 of them, however, yielding before failure.
The remaining 14 beams failed in bond, although
their mode of failure was reported as diagonal
tension. These beams are discussed later in this
section. Thirty beams were provided with both
tension and compression reinforcement; all other
beams were reinforced in tension only.
The test results for the different groups of
beams are analyzed in Tables 2 through 16. Both
the physical properties as reported by the investi
gators and the calculated quantities are given for
each individual beam. All dimensions are given in
inches and the compressive strength of concrete in
pounds per square inch. In most cases, concrete
strength was determined from tests on 6 by 12in.
standard cylinders. In a few cases, tests either on
cubes or on modulus of rupture beams were em
ployed; these cases are noted in the tables and the
concrete strength is reduced to that of a standard
cylinder by the formulas:
f,' = 0.75 fe,,' for cubes
and
f/ = 6.7 f, for modulus of rupture beams.
In order to evaluate the function F (fc'), the
quantity M/bd2fC'(k + np') was calculated for
each beam. For beams without compression rein
forcement the term (k + np') reduces to k. In
Table 1
Range of Test Variables for SimpleSpan Rectangular Beams Without
Web Reinforcement and Under One or Two Symmetrical Concentrated Loads
Test
Series
Richart
Series 1910
Series 1911
Series 1913
Series 1917
Series 1922
Richart and Jensen
Thompson, Hub
bard, and Fehrer
Moretto
Clark
M.I.T.
Gaston
Laupa
Moody
Series A
Series B
Series III
Total
Entry
in.
Bibl.
(2)
(8)
(9)
(4)
(5)
(10)
(11)
(1)
(12)
Table No. No.
No. of of
Beams S,TS
Fail.
psi
20302670
14902350
2180
4770
36964522
22304760
2570
35504640
31203800
31304880
40204750
21404690
8804570
17705970
25003620
8805970
p p' b d a a/d L No.
of
Loads
1.231.92
1.651.94
1.47
2.743.71
2.33
2.80
2.50
3.98
0.98
1.403.14
1.381.90
0.934.11
% % in. in. in.
8 10 24
8 10 24
8 15 40
8.1 10 48
8 21 36
8 21 32
8 12 20
0.50 5.5
8
p'=p 46.25
6
... . (6
0.802.37 ....
1.90 .
2.724.25 p'=0.5p
0.804.11
18.25
15.37
7
10.58
10.5
32
1836
30
36
4851
2.4
2.4
2.67
4.8
1.71
1.52
1.67
1.75
1.172.34
4.28
3.40
4.484.79
7 10.5 31.5 3
6 10.56 36 3.41
7 21 32 1.52
1.174.8
ads
ILLINOIS ENGINEERING EXPERIMENT STATION
Table 2
Tests by Richart, Series 1910. SimpleSpan Rectangular Beams Without Web Reinforcement
Beam P.'
psi
280.1 2670
280.2 2320
280.3 2030
Reference: (2)
Dimensions: b=8; d=10; a  24; a/d=2.4; L=72; L'= 78
Loading: 2 equal loads at Xpoints
Reinforcement: Plain round bars; fI,= 38,500 psi for Beam 280.3; not given for others
Concrete Strength: Tests on 6in. cubes; reduced to cyl. strength by /' 0.75 f.'
Age at Test: Around 60 days
Reported Ca
p Reinf. Anch. Ps.t Mode k Mt__
Bars of bd2f'k
% No., Size kips Fail.
1.23 5%" None 23.8 DT 0.380 0.352
1.92 5Y/s
18.8 DT
210 DT
0.378 0.322
0.456 0.340
Iculated
Ratio Mode
Mtt of
M. Fail.
0.78 B
0.69 B
0.71 B
Table 3
Tests by Richart, Series 1911. SimpleSpan Rectangular Beams Without Web Reinforcement
Reference: (2)
Dimensions: b= 8; d= 10; D= 12; a= 24; L=72; L'f78
Loading: 2 equal loads at hpoints
Reinforcement: Plain round bars; f,= 34,200 psi for Beam 293.3; not given for others
Concrete Strength: Tests on 6 by 8 by 40in. control beams; reduced to cyl. strength by f/' 6.7 f,
Age at Test: Around 60 days
Beam f/
psi
291.1 1690
291.2
291.3
294.1 1490
294.2
294.3
293.4 2350
293.5
293.6
293.1 2040
293.2
293.3
286.1 1660
286.2
286.3
286.5 2160
286.6
286.7
* Nuts tightened
t Nuts not tightened
Reported
p Reinf. Anch. Pe.st Mode
Bars of
% No., Size kips Fail.
1.65 3'/ Hooks 25.3 DT
22.5 DT
27.7 DT
15in. 25.0 DT
over 20.2 DT
S " hang 24.7 DT
Nuts* 27.4 DT
and 34.5 DT,T
Plates 19.3 DT
Nutst 20.0 DT
and 21.4 DT
Plates 24.8 DT
None 18.0 DT
17.6 DT
22.5 DT
1.94 5% " 17.4 DT
18.5 DT
22.1 DT
0
0
0
0
0
0
Calculated
k Mt_ Ratio Mode
bdf,'k M__t of
M. Fail.
.446 0.503 1.02 8
0.448 0.91 8
0.551 1.12 8
.457 0.551 1.10 S
0.445 0.89 S
0.544 1.08 S
.421 0.415 0.89 S
0.523 1.13 S
0.293 0.63 B
.431 0.341 0.71 B
0.365 0.76 B
0.423 0.88 B
.448 0.363 0.73 B
0.355 0.72 B
0.454 0.92 B
.452 0.267 0.56 B
0.284 0.60 B
0.340 0.72 B
Table 4
Tests by Richart, Series 1913. SimpleSpan Rectangular Beams Without Web Reinforcement
Reference: (2)
Dimensions: b=8; d=15; D= 17; a=40; a/d=2.67; L=120; L'=126
Loading: 2 equal loads at %points
Reinforcement: %in. plain round bars; I,= 36,300 psi
Concrete Strength: Tests on 6in. cubes; reduced to cyl. strength by f/,'=0.75 ft.'
Age at Test: 225 days
Mode
of
Fail.
Reported
f,' p Reinf. Anch. Ptws
Bars
psi % No., Size kips
2180 1.47 4Y' Hooks 24.9
Beam
16B20.1
16B20.2
16B1.1
16B1.2
16B2.1
1BTt2 0
16B20.1 3210
16B20.2 3210
16B1.1 2450
16B1.2 2670
16B2.1 2450
16B2.2 2450
Calculated
k Mtu, Ratio Mode
bdf'k Mt.1t of
M. Fail.
0.409 0.311 0.66 B
Table 5
Tests by Richart, Series 1917. SimpleSpan Rectangular Beams Without Web Reinforcement
Reference: (2)
Dimensions: b=8.1; d=10; D12; a48; a/d=4.8; L=114; L'=120
Loading: 2 equal loads
Reinforcement: Plain round bars; f,45,700 psi for %in. bars; f 40,600 psi for %in. bars
Age at Test: About 60 days
Reported Calculated
p Reinf. Anch. Pt.st Mode k 'MIf Ratio Mode
Bars of bd2f'k Mt.1 of
% No., Size kips Fail. M. Fail.
ANALYZED WITH ACTUAL CONCRETE STRENGTH IN COMPRESSION ZONE
3.71 5%. None 31.0 DT 0.523 0.368 1.04 S
" " 29.6 DT 0.523 0.352 0.99 S
3.69 Hooks 32.0 DT 0.522 0.381 1.07 S
" 28.8 DT 0.522 0.343 0.97 S
2.74 5i " 26.6 DT 0.472 0.350 0.99 8
" 29.5 DT 0.472 0.388 1.09 8
ANALYZED WITH CONCRETE STRENGTH USED IN LOWER PORTIONS OF BEAMS
.... ... ..... .... ... 0.531 0.540 1.28
.... ... ..... .... ... 0.531 0.516 1.21
.... ... ..... .... ... 0.550 0.701 1.52
.... ... ..... .... ... 0.543 0.586 1.30
.... ... ..... .... ... 0.499 0.660 1.43
.... ... ..... .... ... 0.499 0.732 1.59
Beam
301.1
Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
Table 6
Tests by Richart, Series 1922. SimpleSpan Rectangular Beams Without Web Reinforcement
Reference: (2)
Dimensions: b = 8 d  21; D= 24; a= 36; a/d = 1.71; L= 108; L' 120
Loading: 2 equal loads at %points
Reinforcement: 1in. corrugated round bars; f/,52,400 psi
Age at Test: About 60 days
Reported Calculated
p Reinf. Anch. Ptt Mode k Mtt Ratio Mode
Bars of bdf,'k M of
% No., Size kips Fail. M. Fail.
2.33 41 %8 None 149.4 B,DT 0.441 0.424 1.09 8
148.0 B,DT 0.446 0.458 1.13 S
Hooks 165.5 B,DT 0.435 0.429 1.17 8
" 126.0 B,DT 0.437 0.339 0.91 8
Table 7
Tests by Richart and Jensen, 1931. SimpleSpan Rectangular Beams Without Web Reinforcement
Reference: (8)
Includes only those beams which were made of concrete with natural sand and gravel aggregates
Dimensions: b=8; d=21; D=24; a=32; a/d=1.52; L=96; L'=108
Loading: 2 equal loads at hpoints
Reinforcement: 1in. plain round bars; f/= 37,600 psi
Age at Test: 28 days (moist cured 28 days)
Reported Calculated
p Reinf. Anch. Ps.t Mode k Mget Mt
Bars of bde'k M
% No., Size kips Fail.
2.8 61' Hooks 142.9 DT 0.463 0.294 0
159.7 DT 0.463 0.339 0
151.8 DT 0.467 0.344 0
134.1 DT 0.473 0.333 0
105.8 DT 0.510 0.422 0.
atio Mode
rte of
f. Fail.
.83 S
.94 S
.91 S
.84 8
90 8
89 8
Table 8
Tests by Thompson, Hubbard, and Fehrer, 1938. SimpleSpan Rectangular Beams Without Web Reinforcement
Reference: (9)
Dimensions: b=8; d= 12; a= 20; a/d= 1.67; L=60; L'=74 for Series I; L'= 86 for Series II
Loading: 2 equal loads of hpoints
Reinforcement: Four %in. round oldstyle deformed bars; f, = 36,000 psi
Concrete Strength: The average value of f/' reported for all beams
Age at Test: 28 days
Reported Calculated
Beam l' p Reinf. Anch. Ptus Mode k Mt.. Ratio Mode
Bars of bdWf.'k M1..* of
psi % No., Size kips Fail. M, Fail.
I B1 2570 2.5 4%' Hooks 84.0 DT 0.482 0.589 1.30 S
I B2 " " " " 88.0 DT 0.616 1.36 S
I B3 " 86.0 DT 0.603 1.33 S
II K1 13in. 88.0 DT 0.616 1.36 8
II K2 " over 84.0 DT 0.589 1.30 S
hang
Note: These beams were apparently tested without rollers at the beam supports. This may have restrained the horizontal movement of the beams
during loading. The ultimate loads of the beams are unusually high and correspond to their flexural capacities.
Table 9
Tests by Moretto, 1945. SimpleSpan Rectangular Beams Without Web Reinforcement
Reference: (4)
Dimensions: b=5.5; d=18.25; D=21; a=32; a/d1.75; L=96; L'120
Loading: 2 equal loads of hpoints
Tension Reinforcement: Four 1in. sq. deformed bars; f,=48,000 psi
Compression Reinforcement: Two hin. sq. deformed bars
End Anchorage: Hooks
Age at Test: 28 days
Reported Calculated
Beam f/ p p' t Pt..t Mode k k+np' Mt.& Ratio Mode
of bdf,'(k+np') Mt of
psi % % kips Fail. M. Fail.
1N1 3550 3.98 0.50 0.932 70.0 DT 0.516 0.556 0.310 0.76 8
1N2 3620 " 88.0 DT 0.514 0.553 0.383 0.94 S
2N1 4340 " 78.5 DT 0.502 0.538 0.293 0.78 8
2N2 4640 " " " 90.5 DT 0.502 0.537 0.318 0.88 S
Beam
221.1
221.2
222.1
222.2
Beam
1
2
3
4
5
6
11 6.5 DT
0.498 0.403 0.
ILLINOIS ENGINEERING EXPERIMENT STATION
Table 10
Tests by Clark, 1951. SimpleSpan Rectangular Beams Without Web Reinforcement
Reference: (5)
Dimensions: b=8; d=15.37; D=18; L=72
Loading: 2 equal loads at various positions
Reinforcement: 2 No. 7 deformed bars; f,= 53,710 psi
End Anchorage: % by 8in. steel plates jin. thick welded to the end of bars
Age of Test: 28 to 30 days; beams kept moist until the day prior to testing
fReported Calculated
Beam f/ p a a/d Ph.k
psi % in. kips
AO1 3120 0.98 36 2.34 40.0
2 3770 " 48.5
3 3435 " " 53.5
BO1 3420 0.98 30 1.95 54.4
2 3468 " 42.4
3 3410 " " " 57.6
CO1 3580 0.98 24 1.56 78.4
2 3405 " " " 79.9
3 3420 " " " 75.1
DO1 3750 0.98 18 1.17 99.6
2 3800 116.9
3 3765 " 100.4
Mode k MteM
of bdW'k
Fail.
DT
DT
T
DT
DT
DT
DT
T
DT
DT
T
DT
0.329
0.320
0.324
0.324
0.323
0.324
0.322
0.324
0.324
0.320
0.320
0.320
0.370
0.382
0.457
0.388
0.299
0.412
0.430
0.458
0.428
0.394
0.457
0.395
Ratio Mode
Mt, of
M. Fail.
0.86 S
0.96 S
1.10 TS
0.93 S
0.72 S
0.99 8
1.05 TS
1.10 TS
1.03 8
0.98 S
1.15 TS
0.99 S
Table 11
Tests at M.I.T., 1951. SimpleSpan Rectangular Beams Without Web Reinforcement
Reference: (10)
Dimensions: b= variable; d= 7; D=8; a=30; a/d= 4.28; L= 60; L'=65
Loading: One load at midspan
Reinforcement: Type of bars not given; f,=52,220 psi for /in.,
f,= 48,370 psi for hin.; f,= 46,240 psi for %in. bars
End Anchorage: Not given
Age at Test: 8 days
Reported Calculated
Beam f' p p' Reinf. t b Ps,,. Mode k k+np' Mtso Ratio Mode
Bars of bdf/'(k+np') MA.* of
psi % % No., Size in. kips Fail. M. Fail.
T2b 3580 1.40 1.40 2W" 0.857 4 10.0 S 0.327 0.436 0.392 0.96 S
c " " " " 10.0 8 ." 0.392 0.96 S
T3a 3470 3.14 3.14 2%'" 10.5 S 0.405 0.652 0.355 0.86 S
b " " " " 7.0 S " " 0.237 0.57 8
c " " " 8.5 S " " 0.288 0.70 S
T5a 3460 2.18 2.18 2%" 9.5 S 0.372 0.544 0.387 0.94 S
b " " " " 2 10.1 S " 0.412 1.00 S
c " " " " 10.3 S " 0.417 1.01 8
T6b 3130 1.40 1.40 2%" 7.6 S 0.331 0.446 0.417 0.97 8
c " " 8.2 S " " 0.450 1.05 S
T11b 4190 1.40 1.40 2%" 6.25 12.0 S 0.321 0.425 0.330 0.87 S
T12a 4880 2.18 2.18 2%5 5.75 15.8 8 0.360 0.514 0.334 0.95 S
b " 15.0 S " 0.318 0.91 8
c " 14.3 8 " " 0.303 0.87 S
Table 12
Tests by Gaston, 1952. SimpleSpan Rectangular Beams Without Web Reinforcement
Reference: (11)
Dimensions: b=6; d=10.58; D=12; a=36; a/d=3.40; L=108; L'=120
Loading: 2 equal loads at %points
Reinforcement: Deformed bars
Age at Test: Around 30 days
Reported
Beam /' p Reinf. f, Anch. Mt5t Mode
Bars of
psi % No., Size ksi kipin. Fail.
T2Ma 4320 1.38 2No. 6 47.7 None 332.3 S
T2Mb 4020 " " 48.3 Hooks 351.7 S
T2Mc 4470 1.90 2No. 7 46.8 None 450.2 8
Calculated
k Mitt Ratio Mode
bdf'k Mt** of
M. Fail.
0.359 0.319 0.85 S
0.363 0.359 0.92 8
0.405 0.377 1.02 S
Table 13
Tests by Laupa, 1953. SimpleSpan Rectangular Beams Without Web Reinforcement
Reference: (1)
Dimensions: b= 6; D = 12; L= 108; L= 120; distance a given from center of end support to edge of column stub
Loading: One load at center of 108in. span, applied through 6 by 12in. column stub, 6 in. high
Reinforcement: Deformed bars
End Anchorage: None, straight bars
Age at Test: Around 28 days
Reported Calculated
Beam /' p Reinf. f, d a a/d Pt.t Mode k MAtt Ratio Mode
Bars of bdf.'k M" of
psi % No., Size ksi in. in. kips Fail. M, Fail.
S2 3900 2.08 3No. 6 41.2 10.58 48 4.54 19.1 S 0.415 0.421 1.07 S
S3 4690 2.52 2No. 8 59.4 10.44 4.60 23.9 S 0.446 0.419 1.17 S
S4 4470 3.21 2No. 9 44.8 10.37 4.63 25.0 8 0.478 0.435 1.18 S
S5 4330 4.11 2No. 10 45.7 10.31 4.66 22.4 S 0.531 0.367 0.98 S
S11 2140 1.90 2No. 7 47.5 10.51 4.57 15.2 8 0.450 0.571 1.20 S
S13 3800 4.11 2No. 10 44.1 10.31 4.66 22.4 8 0.528 0.420 1.05 8
S1 3940 1.46 3No. 5 44.6 10.65 51 4.79 16.8 TS 0.361 0.443 1.13 TS
S9 2140 0.93 3No. 4 44.3 10.72 48 4.48 11.5 TS 0.344 0.543 1.15 TS
S10 2280 1.39 2No. 6 41.8 10.58 " 4.54 15.4 TS 0.396 0.608 1.30 TS
Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
Table 14
Tests by Moody, Series A, 1953. SimpleSpan Rectangular Beams Without Web Reinforcement
Reference: (12)
Dimensions: b=7; d= 10.310.8; D=12.0; a=31.5; a/d= 2.923.06; L= 63; L'=75
Loading: One load at midspan
Reinforcement: Intermediate grade deformed bars
End Anchorage: None, straight bars
Age at Test: About 28 days
Reported Ca
d p Reinf.
psi in. %
4400 10.30 2.17
4500 10.50 2.15
4500 10.55 2.22
4570 10.63 2.37
3065 10.50 1.62
3125 10.55 1.63
2785 10.63 1.60
2430 10.69 1.66
920 10.55 0.81
880 10.70 0.83
1000 10.75 0.80
980 10.80 0.82
Bars
No., Size
111
28
27; 16
46
18; 24
27
26; 15
45
17
25
34
24: 23
Mode k
of
Fail.
0.426
0.423
0.428
0.437
0.401
0.401
0.404
0.419
0.395
0.403
0.384
0.391
Table 15
Tests by Moody, Series B, 1953. SimpleSpan Rectangular Beams Without Web Reinforcement
Reference: (12)
Dimensions: b=6; d=10.56; D=12; a=36; a/d=3.41; L=108; L'=120
Loading: 2 equal loads at %points
Reinforcement: 2 No. 7 intermediate grade deformed bars
End Anchorage: None, straight bars
Age at Test: About 28 days
Reported
p
lculated
Mt..s
bdV'k
0.306
0.322
0.357
0.319
0.420
0.435
0.442
0.483
0.501
0.608
0.577
0.569
Calculated
Pts.t Mode k Mte Ratio Mode
of bd2f,'k Mt__ of
kips Fail. M. Fail.
26.0 S 0.397 0.334 1.01 S
16.0 S 0.441 0.408 0.89 S
23.5 8 0.414 0.413 1.03 S
19.8 S 0.447 0.540 1.15 S
23.4 S 0.395 0.362 0.98 S
15.8 S 0.445 0.421 0.90 S
23.0 S 0.395 0.354 0.96 S
14.0 S 0.465 0.462 0.94 S
24.0 S 0.393 0.278 0.92 S
22.0 S 0.418 0.412 1.00 S
27.0 S 0.395 0.336 1.05 S
21.2 S 0.424 0.464 1.06 S
25.0 S 0.396 0.313 0.97 S
19.4 S 0.421 0.382 0.90 S
23.0 S 0.397 0.291 0.89 S
17.0 S 0.446 0.437 0.94 8
Table 16
Tests by Moody, Series III, 1953. SimpleSpan Rectangular Beams Without Web Reinforcement
Reference: (12)
Dimensions: b=7; d= 21; D f24; a= 32; a/d= 1.52; L= 96; L'= 120
Loading: 2 equal loads at hpoints
Tension Reinforcement: Four deformed bars
Compression Reinforcement: Two deformed bars; t=0.91
Age at Test: 28 days
Reported Calculated
Beam f/ p p' f, Anch. P..es Mode k k+np' Mtat Ratio Mode
of bdf.(k+snp') M" of
psi % % ksi kips Fail. M. Fail.
24a 2580 2.72 1.36 45.7 Hooks 133 S 0.432 0.552 0.484 1.07 8
b 2990 " " " 136 S 0.424 0.538 0.439 1.01 8
25a 3530 3.46 1.73 45.4 120 8 0.456 0.582 0.303 0.74 S
b 2500 " " 130 S 0.455 0.610 0.442 0.97 8
26a 3140 4.25 2.13 43.8 189 S 0.485 0.659 0.473 1.10 S
b 2990 " " 178 8 0.488 0.665 0.464 1.07 S
27a 3100 2.72 1.36 45.7 None 156 8 0.433 0.545 0.479 1.11 S
b 3320 " " " 160 S 0.429 0.538 0.464 1.10 S
28a 3380 3.46 1.73 45.4 " 136 S 0.458 0.596 0.350 0.84 8
b 3250 " .... " 153 S 0.519 0.519 0.470 1.11 S
29a 3150 4.25 2.13 43.8 175 S 0.485 0.659 0.437 1.02 8
b 3620 " " " 196 S 0.480 0.645 0.435 1.07 8
Ratio
M..
0.82
0.88
0.97
0.88
0.97
1.01
0.99
1.05
0.95
1.14
1.10
1.08
Beam
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
f/
ILLINOIS ENGINEERING EXPERIMENT STATION
M te
bd 'f(A
Fig. 1. Evaluation of Eq. 17 for SimpleSpan
Fig. 1 the above quantity is plotted against fe'. It
is seen that the concrete strength varies from about
1000 to 6000 psi. Within these limits the function
F(fc') can be approximated by a linear equation:
F(fe') = 0.57  4.5' (17)
10,
where f,' is the compressive strength of a standard
cylinder in pounds per square inch. Substitution of
Eq. 17 into Eq. 13 yields an equation for moment,
subsequently called the shearcompression moment,
at which a simplespan reinforced concrete beam
without web reinforcement and under one or two
symmetrical concentrated loads fails in shear:
= (k + np') (0.57  4.5 f(18)
Five beams analyzed in Table 8V) show unusu
ally high ratios of Mtest to M,. Moreover, the ulti
mate loads corresponded to the flexural capacity
of the beams although there was no indication of
Rectangular Beams Without Web Reinforcement
flexural failure in either the measured deflections
or the crack patterns reported. The peculiar be
havior of these beams may be due to the fact that
apparently no rollers were provided at the beam
supports. This may have restrained the longitudinal
movement of the beams and contributed to their
high shear strengths. Data for these beams are not
shown in Fig. 1.
The agreement between test results and Eq. 18
is believed to be satisfactory. The average ratio
of M/MS for the 106 beams which failed in shear
and are shown in Fig. 1 is 0.986; the standard devi
ation is 0.119. The group of beams which were
loaded through a column stub at midspan(1) failed
at a somewhat higher load than that given by
Eq. 18. Whether the apparent increase in shear
strength was caused by the column stub or by the
use of a single concentrated load could not be de
termined from the available data. Six beams from
four different investigations failed at a considerably
lower load than predicted by Eq. 18. However, all
Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
these beams had companion specimens which failed
in close agreement with the predicted values.
In this connection, it must be pointed out that
all compression failures are sensitive to the com
pressive strength of the concrete at the section of
failure. The compressive strength reported for a
test beam is the average strength obtained from
control cylinders. Since even control cylinders can
vary widely in strength, it is not reasonable to ex
pect the concrete strength to be uniform throughout
a test beam. If the concrete strength at the section
of failure happens to be different from the average
strength of the control cylinders, the test beam may
fail at a load different from the predicted load. It
is believed that much of the scatter in test results
can be attributed to variation of the concrete
strength from the average value. This is especially
true in those cases for which only the average con
crete strength was reported for the entire test series
or for a group of companion specimens. Further
more, it may be of significance that the beams were
tested over a period of almost a half of a century,
and that the beams were both made and cured
under greatly different conditions.
No systematic difference can be detected be
tween beams reinforced in tension only and beams
reinforced both in tension and compression. If the
A
MA
latter group of beams is considered separately, the
average ratio of M/Ms for thirty such beams is
0.940 and the standard deviation 0.14. It is inter
esting to note, however, that five of the six beams
which fell considerably lower than the predicted
values were provided with compression reinforce
ment. This explains also why the average ratio for
this type of beam is somewhat lower than that for
all beams combined; if these beams are excluded,
the average ratio is 0.986 and the standard devia
tion 0.084.
Equation 18 was based on assumptions made in
deriving Eq. 11; that is, the ratio k,/k is a function
of fc' alone and does not depend on p. In order to
check this assumption and to investigate whether
Eq. 11a might not represent better the moment at
shear failure, the ratios M/M, are plotted against
p in Fig. 2. Although the steel percentages used in
the test beams vary over a large range of values,
no consistent relationship can be detected between
the ratio M/M, and p. Consequently, the ratio k,/k
does not seem to be influenced by p and Eq. 18 is
therefore assumed to be valid for beams with any
amount of longitudinal reinforcement.
Series 1917 of the beams tested by Richart
(Table 5) provides further data for a study of the
mechanism of shear failure. These beams were pro
Fig. 2. Effect of Longitudinal Steel Percentage on Ratio of Measured to Computed ShearMoment.
SimpleSpan Rectangular Beams Without Web Reinforcement
ILLINOIS ENGINEERING EXPERIMENT STATION
vided with a 4in. thick layer of highstrength con
crete at the top of each beam "as a precaution
against premature failure of the beam by crushing
of the concrete." Table 5 gives an analysis of these
beams using the reported values of the concrete
strength both in the compression zone and in the
lower portions of the beams. It is seen that the use
of the actual concrete strength for the compression
zone gives very good agreement with Eq. 18,
whereas the use of the concrete strength for the
lower portions of the beams results in differences
of as much as 59 percent between measured and
predicted values. Thus it is clearly evident that the
load at failure is controlled solely by the strength
of the compression zone of the concrete. The
strength of the remaining part of the concrete sec
tion does not greatly influence the shear strength
of a beam.
Fourteen beams, although reported as diagonal
tension failures, failed in bond. These were beams
tested by Richart; three from Series 1910 (Table 2),
ten from Series 1911 (Table 3), and one from Series
1913 (Table 4). A total of 18 beams of Series 1911
were without web reinforcement. These beams were
very nearly the same in every respect except for
the end anchorage of the tension reinforcement. All
beams with the longitudinal steel wellanchored
either by conventional hooks, by overhang, or by
an end plate tightened against the end of the
beams, failed in shear at a load in good agreement
with Eq. 18. All other beams, however, either with
unanchored straight bars or with end plates not
tightened, failed at a much lower load; this sug
gested bond failures. Some typical beams of this
group were checked for their bond strength by a
procedure suggested by Mylrea(13). Mylrea gives an
empirical relationship between the length of em
bedment of a plain round bar in a simplespan
beam and the cumulative bond stress the bar can
develop before bond failure. By using as the length
of embedment the distance from the end of the bar
to the 45deg diagonal crack, the cumulative bond
stress as given by Mylrea agreed closely with the
steel stress calculated from the load at failure. This
indicates that the ultimate bond resistance was
reached and that the beams failed in bond before
developing their ultimate shear capacity. The three
beams of Series 1910 with unanchored straight bars
also failed in bond. The only beam of Series 1913
for which concrete strength was reported was rein
forced with hooked plain bars. However, it failed
at a low load, and a photograph at failure indicated
a possible bond failure.
8. Theoretical Interpretation of Basic Empirical
Equation
a. Beams Reinforced in Tension Only. Equa
tion 18 where the quantity np' reduces to zero for
beams without compression reinforcement can be
interpreted in the light of the conventional theory
of compression failures of reinforced concrete
beams. The only modification is in the depth of the
compression zone. The following stress block is
assumed:
C = kk 3c'ksbd
k)
T
= Cd (1  k2k,)
= kjk3fs'kbd' (1  k2k,)
M
bd2fJ = kikak, (1  k2k,)
For beams failing in flexure, the parameter kAks
is the ratio between the average stress in the con
crete of a beam and the strength of a standard 6
by 12in. test cylinder in axial compression. This
parameter has been evaluated experimentally by
previous investigators. In Fig. 3 the values of kak3
as obtained by Gaston("1 and Billet(14 have been
plotted against fe'. There is considerable scatter in
the measured values as would be expected in an
investigation of this kind. A reasonable approxima
tion, however, can be obtained by a linear relation
ship between kiks and f,'. When f,' is within the
limits of 2000 and 6000 psi, kxk3 can be approxi
mated as follows:
10.8f3' 0 4.5f'
kik3 = 1.37  10, = 2.4 0.57 10)
Substitution of this function into Eq. 19 and the
use of k2 = 0.45 as in the case of beams failing in
flexure gives:
iff
., =2.4 0.57 'I ) k, (10.45k,.)
Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
2.0
1.6
1.2
k,k,
0.8
0.4
0
Fig. 3. Relation Between
Equation 21 is based on two conditions of static
equilibrium, C = T and M = Cd(1  k2ks), and
on a fully developed concrete stressblock having
a value of kks3 as given by Eq. 20. Consequently,
Eq. 21 constitutes a general expression for the
ultimate moment of a rectangular beam reinforced
in tension only and is valid for any mode of failure,
provided that the properties of a concrete stress
block at the instant of failure are represented by
Eq. 20.
The depth of the compression zone of concrete,
ksd, remains the only unknown in Eq. 21. This dis
tance can be determined with the aid of strain re
lations for beams which fail in flexure. Test obser
vations have shown that a beam fails in flexure
when the concrete crushes at a limiting strain of
about 0.004 and that the strain distribution over
the depth of the beam remains practically linear up
to the final failure. This information and the known
stressstrain relationship of the reinforcing steel
permit the calculation of the parameter k8 from
the condition of static equilibrium of the internal
forces of a section. After the value of k, has been
determined, the ultimate moment can be calculated
from Eq. 21.
If, however, a linear distribution of strain over
the depth of the beam is assumed for shear failures,
the ultimate concrete strain of about 0.004 and the
fact that the steel strain must be below yielding,
S[(psi)
kik, and Concrete Strength
say below 0.0015, result in a value of k, at failure
greater than 0.7. This is incompatible with test
observations which show that the diagonal cracks
extend higher than the flexural cracks at failure.
Since the depth of the compression zone rarely ex
ceeds about 0.3  0.4d for flexural failures, the
value of k, must be much below 0.7 for shear fail
ures. This would be possible if the ultimate con
crete strain were, say, 10 times smaller than 0.004,
but such a small strain is not possible at the loca
tion of a diagonal crack. Actually, the presence of
the diagonal crack disrupts the normal distribution
of the steel strain along the tension reinforcement.
Since there can be no transfer of stress across the
diagonal crack, consideration of moment shows that
the steel stress must be the same both at a vertical
section through the upper end of the crack and at
the intersection of the reinforcing bars and the
crack. Thus, the steel strain must be practically
uniform over this distance. Furthermore, in order to
preserve the continuity of the beam, the total elon
gation of steel between these two sections must
have a geometrically corresponding, although not
numerically equal, total shortening of the top con
crete fiber over a much shorter distance at the
location of the diagonal crack. This requires a con
centration of concrete strain in that region. Conse
quently, the strain distribution over the depth of
the beam cannot be linear at the section of failure.
ILLINOIS ENGINEERING EXPERIMENT STATION
Since the concrete strain must be concentrated
at the location of the diagonal crack, it is likely
that the ultimate concrete strain is still about 0.004,
as in the case of flexural failures, and that the
concrete stressblock is fully developed. However,
the actual distribution of strain is unknown and
cannot be determined from the data available at
the present time. Consequently, no theoretical re
lationship can be written for the depth of the com
pression zone at shear failure. In order to interpret
test results and to determine a general expression
for shearcompression failures, either the value of
k,, or the magnitude of the steel stress, or some
relationship between the average strains in the re
inforcement and in the concrete must be determined
empirically. In this investigation it was chosen to
evaluate k, empirically. Equation 18 was obtained
to represent the shearcompression capacity of rec
tangular simplespan beams under one or two con
centrated loads.
A comparison between Eqs. 18 and 21 reveals
that both these equations have the same form.
Equating the two yields a relationship between
k, and k:
k, = k (22)
= 2.4 (1  0.45 k,)
from which:
k, = 1.11  V 1.23  0.926 k (23)
Since k remains usually within 0.2 and 0.5, Eq. 22
shows that k, is practically a constant fraction of k,
the depth of the compression zone computed by
the "straight line" theory. This finding shows
why the previous attempt to use k as a measure of
k, gave satisfactory results. However, since the
relationship between the two was determined em
pirically, it can only be speculated why these two
quantities are related.
Zwoyer used in his investigation(29) an empiri
cal relationship between the average values of the
concrete strain on the top surface of the beam and
at the level of the reinforcing steel. In addition,
the parameter kks3 was determined from data ob
tained in tests of prestressed concrete beams and
the same value was used subsequently for ordinary
reinforced concrete beams. The average ultimate
strain in the concrete was found to be 0.00385.
Moody used the parameter kiks3 as obtained from
flexural failures and evaluated the magnitude of the
steel stress from test results.(12) Two different ex
pressions were obtained for the steel stress; one for
simplespan beams and another for restrained rec
tangular reinforced concrete beams under sym
metrical concentrated loads.
b. Beams Reinforced in Both Tension and
Compression. Equation 19 was derived for beams
without compression reinforcement. For beams re
inforced in both tension and in compression it can
be modified as follows:
M = khk3fc'k.bd2 (1  k2k,) + f,' p'bd2t (24)
where td is the distance between the centers of the
tension and compression reinforcements, f/' is the
stress in the compression reinforcement, and p' is
the ratio of compression reinforcement.
Since the ultimate strain in the concrete is ap
proximately 0.0040 and the yield strain for rein
forcing bars is usually less than 0.0017, yielding of
the compression reinforcement precedes crushing
of the concrete in most flexural compression fail
ures. For shear compression failures, however, di
agonal cracks extend higher than the vertical
cracks caused by flexural tension. It is conceivable
that a beam can fail in shear either before or after
the compression reinforcement yields. Expressions
for the ultimate shear moment for both of these
cases are derived in the following paragraphs, and
the validity of these equations is determined with
the help of experimental data.
If it is first assumed that the compression rein
forcement has reached its yield stress f,' at shear
failure and that k, is still given by k, = 2.4 (1 k2k,)'
2.4 (k2k.)'
Eq. 24 for maximum shearmoment can be writ
ten as:
Ma= k (0.57  4.5f/) + n'p't
bd2f, ' k 10 ) npt
Since this equation assumes that the compression
reinforcement has yielded while the tension rein
forcement is still elastic, the elastic modular ratio
n is to be used for the tension reinforcement and
the plastic modular ratio n' = f'//f' for the com
pression reinforcement when computing the quan
tity k.
An expression for the maximum shear moment
for the second case, a beam failing in shear before
the compression reinforcement yields, was derived
previously:
bdf = (k + np') (0.57  4.5f)
I
Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
In this expression the elastic modular ratio n is
used for both tension and compression reinforce
ment in computing the quantity k.
Equations 25 and 18, based on different assump
tions, are greatly different. Equation 25 gives a
much higher ultimate moment than Eq. 18. In the
analysis of previous test data, thirty of the 106
beams under consideration were provided with
compression reinforcement. If these beams are con
sidered as having failed after the compression re
inforcement had yielded, the internal resisting
moment given by the compression reinforcement
acting at its yield stress is almost as large as and
in several cases even larger than the total external
moment. Thus, it must be concluded that these
beams failed in shear before the compression rein
forcement yielded. Furthermore, since the thirty
beams with compression reinforcement gave good
agreement with Eq. 18, this equation can be used to
take the effect of compression reinforcement into
consideration. According to Eq. 18 the shear
strength of a beam with compression reinforcement
is but little greater than that of the same beam
without; p' decreases the value of k while adding
the term np', so that the quantity (k + np') is but
little greater than the value of k for a beam with
out compression reinforcement.
9. Properties and Limitations of Basic Empirical
Equation
The basic empirical equation was derived for
simplespan rectangular beams without web rein
forcement and under one or two symmetrical con
centrated loads. Different variables have the
following effect on Eq. 18:
a. Ratio of a/d. Equation 18 considers shear
failures as compression failures. The load at failure
is determined by a limiting shearcompression mo
ment. In that sense, the ratio a/d loses its usual
meaning; that is, as affecting the shearing strength
of a beam. The quantity a relates the magnitude
of the applied load to the moment at failure, M =
Va, and the effective depth d affects both the lever
arm of the internal moment and the area of the
compression zone. For the beams analyzed, the
ratio a/d varied from 1.17 to 4.80. This variation
did not seem to have any effect on the agreement
between the test results and the predicted values.
It is conceivable, however, that as the ratio a/d
increases and the relative magnitude of the shearing
stresses decreases, a beam will either fail in shear
at a higher load than that given by Eq. 18 or, for
still higher values of a/d will fail in flexure. This
phenomenon is discussed further in Section 18. Con
versely, as the ratio a/d decreases to a very small
value, it is expected that the mode of failure
changes from shearcompression to shearproper.
This question is discussed in more detail in Sec
tion 19.
b. Tensile Reinforcement. The amount of ten
sile reinforcement affects the size of the compres
sion zone of the concrete. It was found empirically
that the moment at failure could be related to k
and that the actual depth of the compression area
was practically a constant proportion of k, or
k, = c
2.4 (1  k2k.)
c. Concrete Strength. The shear strength of a
beam is directly proportional to the following func
tion of fe': f,'(0.57  4.5 fc'/10)k. It is seen that
as f,/ increases, both the quantity (0.57  4.5 fc'/
10) which represents the effect of kfks, and the
value of k decrease. Thus the shear strength is not
a linear function of fe'. As an example, for a beam
with one percent tension reinforcement, an increase
of f,' from 2500 to 5000 psi increases the shear
strength 36 percent.
d. Compression Reinforcement. The contribu
tion of compression reinforcement to the shear
strength is rather small and can be included in the
analysis by considering p' in computing both the
elastic k and the transformed concrete area. This
procedure led to Eq. 18.
e. Column Stub. Beams which had a column
stub cast integrally with the beam at midspan
failed consistently at slightly higher loads than
beams without a column stub. These increases in
strength were somewhat larger for lower values of
concrete strength than for higher values of concrete
strength.
III. SIMPLESPAN RECTANGULAR BEAMS WITH WEB REINFORCEMENT AND
UNDER ONE OR TWO SYMMETRICAL CONCENTRATED LOADS
10. General Considerations
In Chapter II a rather simple expression was
derived for the shear strength of a simplespan
beam without web reinforcement. Here an attempt
is made to extend that expression to beams which
are provided with web reinforcement.
The contribution of web reinforcement to the
shear strength can be pictured in different ways.
As has been mentioned, the conventional theory
originally assumed that all shear was carried by
the web reinforcement. Later modifications of the
concept of truss analogy, prompted by experimental
evidence, allocated a certain proportion of the shear
to be resisted by the concrete. Essentially, even the
modified expressions for the shear strength implied
that the contribution of web reinforcement was
determined by the properties of the web reinforce
ment alone, as expressed by the term Krfy,, and
not influenced by the shear strength of the beam
without web reinforcement.
Another approach to the effect of web reinforce
ment is to consider that its contribution is de
termined by both the properties of the web
reinforcement and the shear strength of the beam
itself. The two alternatives are examined in more
detail in the following paragraphs.
Test observations show that, in general, web
reinforcement which crosses the main diagonal
crack yields before the beam fails in shear. Figure
4a shows a simplespan beam shortly before shear
failure. For convenience, only the main diagonal
crack is shown, although in reality numerous cracks
appear as the beam is being loaded. Figure 4b
shows the portion of the beam to the left of the
crack as a freebody diagram, and Fig. 4c shows
the approximate locations of the internal forces at
the assumed 45deg diagonal crack. The force F
is the resultant of all stirrup forces crossing the
crack. It has been projected down to the level of the
tension reinforcement and divided into horizontal
and vertical components. The other symbols have
their usual meanings.
One possible assumption is that the contribution
of web reinforcement is independent of the shear
strength of the same beam without web reinforce
ment. If this is true, then it must be possible to
determine the increase of the shear capacity of the
beam solely from the amount and physical prop
erties of the web reinforcement. The following cal
culation attempts this:
(1/s) (cot a + 1) jd = number of stirrups
crossed by crack
(1/s) (cot a + 1) jd Afy, = F = total tension
force in stirrups
(1/2) cos 2a (jd)2 brfy = moment given by
stirrups acting at their yield stress, about
point A
The moment due to all forces about A is then:
Va = Cjd  (1/2) b(jd)2 cos 2a rfJ
According to this equation the total internal
resisting moment is made up of two parts; the web
reinforcement resists directly a part of the applied
moment, the remainder being resisted by the com
pressive force C. The direct contribution of the web
reinforcement is influenced by the angle of inclina
tion of the stirrups. For vertical stirrups, cos 2a =
 1, and the moment of the stirrup forces is added
to Cid. As the angle a decreases, the direct contribu
tion of the web reinforcement decreases also. At
a = 45 deg, this contribution is zero. For a less
than 45 deg, cos 2a reverses its sign; this indicates
that the direct contribution is detrimental to the
shear strength of the beam since the moment of the
stirrup forces is subtracted from Cjd. The remain
ing part of the internal resisting moment is pro
vided by Cjd. For vertical stirrups the horizontal
component of the stirrup force F reduces to zero.
Consequently, the term Cjd is equal to that of the
beam without web reinforcement, given by Eq. 18.
As the value of a decreases, the horizontal com
ponent of F increases and, consequently, the value
of C increases. This increases the part of the in
ternal resisting moment provided by Cjd.
24
Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
The above assumptions regarding the effect of
web reinforcement can easily be checked for verti
cal stirrups where the magnitude of C is determined
by Eq. 18. For this case the preceding expression
for Va can be rewritten as follows:
Va = (1/2)b(jd)2 rfv, + bd2 f' k (0.57  4.5
fc'/10")
or
M = M, + (1/2) b (jd)2rf
In this equation all quantities can be determined
and the validity of the equation can be checked
against test results. This was done for Clark's and
Moretto's beams with vertical stirrups. It was
found, however, that the increase of the shear ca
pacity of the beams was much greater than the
direct contribution of the web reinforcement as
given by the above equation. Furthermore, the
difference between the two was consistently larger
than could be accounted for by inaccuracies in the
assumed locations of the internal forces, e.g., as
Fig. 4. Internal Forces at Section of Diagonal Crack
given by the 45deg crack in Fig. 4c. Consequently,
it was concluded that the foregoing assumptions
regarding the effect of web reinforcement were not
valid.
The above reasoning served one useful purpose.
It showed not only that the shear strength is af
fected by the internal forces in the stirrups but
also that the presence of web reinforcement changes
the location of the neutral axis. Web reinforcement
hinders the development of diagonal cracks; thus
a larger compression area is available to resist the
compressive stresses in the concrete. The combined
effects of web reinforcement on the shearcompres
sion capacity are (1) to contribute directly a por
tion of the internal resisting moment which can be
either beneficial or detrimental, depending on the
angle of inclination of the web bars, (2) to provide
a larger ultimate compression force through a larger
compression area in the concrete, and (3) to de
crease the moment arm of the larger compression
force through lowering the neutral axis of the beam.
It is also conceivable that the presence of web re
inforcement restricts the concentration of the com
pressive concrete strain in the region of the main
diagonal crack.
In estimating the total effect of web reinforce
ment, only the direct contribution of stirrup forces
can be determined rationally. However, even this
contribution depends on the assumed angle of in
clination between the main diagonal crack and the
axis of beam. The other two effects of web rein
forcement cannot be determined rationally. More
over, there is no theoretical basis for estimating
the effect of stirrups on restricting the concentra
tion of concrete strain in the region of the diagonal
crack. For these reasons it was considered desirable
to express the total effect of web reinforcement
empirically rather than to attempt to separate the
different effects. This is done in the following sec
tion by assuming that the shear strength of a beam
with web reinforcement is greater than that of the
same beam without web reinforcement by an
amount that is a function of the strength of the
unreinforced beam and the amount and yield
strength of the web reinforcement provided.
11. Stirrups as Web Reinforcement
The findings of the previous section suggest that
the shear strength of a beam with a reinforced web
is affected not only by the amount and properties
of web reinforcement but also by the shear strength
of the beam itself. Since the most important func
ILLINOIS ENGINEERING EXPERIMENT STATION
tion of web reinforcement appears to be its resist
ance to the extension and widening of diagonal
cracks, it is logical to assume that a given amount
of web reinforcement will increase the shear
strength of a beam in proportion to that of the
same beam without web reinforcement. Further
more, test results show that in most cases web
reinforcement yields before the beam fails in shear,
the latter indicating that both the amount of web
reinforcement and its yield strength influence the
load at failure.
All available test data on simplespan beams
with stirrups as web reinforcement were analyzed
in the light of the above assumptions. A total of
179 beams from 11 different investigations were
included; 87 of them failed in shear, 91 in flexure,
and one additional beam failed because of insuffi
cient anchorage of stirrups. Different groups of
beams are analyzed in Tables 18 through 28; Table
17 summarizes the range of test variables. In addi
tion to shear failures, it was found advantageous to
consider also beams which failed in flexure.
Several empirical expressions for the shear
strength of such beams were investigated. The most
consistent results were obtained by plotting the
ratio P/P, against the quantity rfw, where P is the
measured load and P, the load corresponding to the
shear capacity of the same beam without web rein
forcement, Eq. 18. Figure 5 shows such a plot for
the 87 beams which failed in shear. Satisfactory
agreement with test results was obtained with the
following linear equation:
PW/P, = 1 + 1
Table 17
Range of Test Variables for SimpleSpan Rectangular Beams
With Stirrups and Under One or Two Symmetrical Concentrated Loads
Test
Series
Richart(l)
Series 1910
Series 1913
Series 1922
Slater, Lord,
Zipprodt(W)
Slater, Lyse(1)
Table No. No. No.
No. of of of
Beams Shear Flex.
Fail. Fail.
f' d a/d
psi in.
20303570
13802180
36894124
30005960
12105060
10
15
21
32.75
16.9
4.1
12.2
npson, Hub 23 3 3 .. 2570 12
rd, FehrerO)
ston, Cox<") 24 20 10 10 3190 12
etto('> 25 40 26 14 23205060 18.25
19.50
k() 26 50 43 7 20006900 15.37
12.37
on(n" 27 9 .. 9 21205900 9.23
10.72
dy(1) 28 2 2 .. 3250;3680 21
179 87* 91
* One additional beam failed because of insufficient anchorage of stirrups.
t Assumed values.
Table 18
Tests by Richart, Series 1910. SimpleSpan Rectangular Beams With Stirrups
Reference: (2)
Dimensions: b=8; d=10; a=24; a/d=2.4; L72; L'=78
Loading: 2 equal loads at %points
Tension Reinforcement: Monolith, ovoid, and corrugated bars
Concrete Strength: Tests on 6in. cubes; reduced to cyl. strength by f/,= 0.75 f.'
Age at Test: From 60 to 70 days
Beam // Tension p
Reinf.
psi No., Size %
282.1 2420 2%" 1.40
2 3570 monolith
3 2410
281.1 2670 31iM' 1.56
2 2320 ovoid
3 2030
BOTH ST
281.5 2570 241s« 1.48
6 2570 and 1%'
7 2030 ovoid
284.1 2420 4%' 1.50
2 2560 corr.
3 2410
284.5 2570 4%' 1 50
6 2900 corr.
7 2030
* Bentup bars not included in web reinforces
f, Web
Reinf.
ksi
STIRRUPS AS WEB
%' round
37.7 loops
40.0 Ms6 round
loops
IRRUPS AND BENTUP BAB
37.6 iMe" round
loops and
63.3 fie' round
stirr. and
2%
"' sq.
stirr. and
64.8 2~%"
a r A. rf.,
deg % ksi psi
3 REINFORCEMENT
45 0.35 54.5 191
90 0.52 93.3 485
A USED AS WEB REINFORCEMENT
90 0.34* 99.4 339*
45 0.25* 63.7 159*
and " "
90
45 0.56* 55.6 311*
and " 5 "
90
Mode
of
Fail.
T
T
T
T
T
T
T
DT,B
T
T
T
DT
T
T
T
p' a
Thor
ba
John
More
Clar
Gast
Mood
Tota
%
1.401.56
1.47
2.33
2.332.50
2.14.7
2.50
0.390.87
.3.98
1.86
1.633.42
0.627.22
4.25
deg
45; 90
45
90
90
90; 20
90
90
90; 67.5;
45
90
90
90
%
0.35;0.52
0.171.39
1.381.40
0.230.88
0.420.85
0.36
0.10
0.281.12
0.341.22
0.281.83
0.52;0.95
ksi
54.5; 93.3
40t
39.642.9
70
73.4
38.2
45t
46.055.0
48.0
45t
44.0; 47.3
P.
Eq. 18
kips
29.3
36.1
29.3
32.4
29.8
27.4
31.0
30.9
26.9
30.1
31.1
30.0
31.1
33.4
27.0
Ratio
P.
1.09
0.89
1.15
1.24
1.22
1.34
1.32
1.22
1.49
1.76
1.58
1.58
1.75
1.50
1.88
Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
Table 19
Tests by Richart, Series 1913. SimpleSpan Rectangular Beams With Stirrups
Reference: (2)
Dimensions: b= 8; d=15; D= 17; a= 40; a/d=2.67; L  120; L'= 126
Loading: 2 equal loads at 4pointa
Tension Reinforcement: Four %in. plain round bars; p=0.0147; f,=36,300 psi
End Anchorage: Hooks
Web Reinforcement: Y, %, and Min. plain round bars; f,= 40,000 psi assumed
Concrete Strength: Tests on 6in. cubes; reduced to cyl. strength by f// =0.75 f4'
Age at Test: From 80 to 238 days
a r rf,. Pgt. Mode P.
of Eq. 18
deg % psi kips Fail. kips
45 0.17 68 38.7 DT 37.8
0.35 140 39.5 T 34.8
0.35 140 40.2 T 32.0
0.78 312 38.0 T 33.6
0.78 312 40.0 T 28.6
0.78 312 40.0 T 33.4
0.78 312 35.7 T 34.8
1.39 556 40.7 T 33.0
0.82 328 44.0 T 36.2
Table 20
Tests by Richart, Series 1922. SimpleSpan Rectangular Beams With Stirrups
Reference: (2)
Dimensions: b=8; d=21; D=24; a=36; a/d=1.71; L108; L'=120
Loading: 2 equal loads at spoints
Tension Reinforcement: Four 1%in. corrugated round bars; p=0.0233; /,f52,400 psi
End Anchorage: Hooks
Web Reinforcement: Plain round vertical stirrups
Age at Test: Around 60 days
Beam
303.1
304.1
2
305.1
2
306.1
2
307.2
308.1
Beam
223.1
2
224.1
2
225.1
2
229.1
2
BENTUP BARS AS WEB REINFORCEMENT
1%Y 0.96 52.4 503 223.4 T 1
ntup " " " 211.7 T 1
Table 21
Tests by Slater, Lord, and Zipprodt, 1926. SimpleSpan Rectangular Beams With Stirrups
Reference: (15)
Dimensions: a=57; L= 114; L'= 128; D= 36 (18 for Beam 61)
Loading: One load at midspan
Flexural Reinforcement: Equal tension and compression reinforcement; 1lin. round plain
bars; f,=about 55,000 psi; some bars, not known which, had much
lower yield strength
End Anchorage: Hooks
Web Reinforcement: % and %in. plain round vertical bars; f.= about 70,000 psi
Age at Test: About 60 days
d a/d p=p' t Stirr.
Size
in. % in.
32.75 1.74 2.50 0.901 3
2.48
2.48
16.9 3.37 2.33 0.867 %
where P,, is the shear strength of a beam with
stirrups, P,, that of the same beam without web
reinforcement, and fy, is expressed in pounds per
square inch.
It is seen that most beams fall within ± 15 per
cent of the value predicted by Eq. 26. Only 7 beams
failed at considerably lower load. All these beams
had a very small a/dratio, and for two of them,
tested by Moody,("2) it was reported that the stress
in the stirrups was but 83 and 67 percent of their
yield strength. It is likely that these beams did not
fail in shearcompression but in shearproper. This
mode of failure is discussed in more detail in Sec
tion 19.
r r/,. Pt«t
% psi kips
0.82 574 496.2
0.81 567 496.2
0.88 616 540.0
0.23 161 121.3
Mode
of
Fail.
T
T
T
DT
Ratio
Pt..s
P.
1.02
1.13
1.26
1.13
1.40
1.20
1.02
1.23
1.22
P.
;q. 18
kips
36.6
30.2
36.4
32.0
32.2
35.4
34.1
37.6
p kRatio
Eq. 18 PML
lkips P.
415.6 1.19
347.6 1.43
429.6 1.26
99.2 1.22
The average ratio between the load at failure
and that given by Eq. 26 is 1.017 for the 80 beams
which failed in shearcompression; the standard
deviation is 0.089. This agreement is somewhat
better than that obtained previously for beams
without web reinforcement.
As a further check on Eq. 26, the ratio P/P, is
plotted against rf,,, in Fig. 6 for beams which failed
in flexure, either in tension or in compression. It is
well known that beams which have been tested
to obtain information about their shear strength
have frequently failed in tension. Some of these
beams, however, were rather close to their shear
capacity at failure, as indicated by welldeveloped
a r A. rf,, Pitt Mode
of E
in. % ksi psi kips Fail.
STIRRUPS AS WEB REINFORCEMENT
4 1.38 42.9 592 212.5 T 1
S. 216.4 T 1
7 1.40 40.1 561 218.5 T 1
.. 216.0 T 1
11 1.39 39.6 550 227.3 T 1
S091 2 T 1
2
Be
Beam /,'
psi
43 4880
48 3000
50 5960
61 3600
Ratio
F*
P..
0.90
Ratio
P.
1.56
1.66
1.60
1.64
1.72
1.63
1.67
1.54
Ratio
P..
0.92
ILLINOIS ENGINEERING EXPERIMENT STATION
Table 22
Tests by Slater and Lyse, 1930. SimpleSpan Rectangular Beams With Stirrups
Beam f/
psi
IA 1210
B 1520
C 1450
2A 2530
B 2940
C 2910
3A 4020
B 4200
C 4000
4A 4670
B 4660
C 5060
6A 2490
B 2600
C 2670
7A 2800
B 2860
C 3200
8A 3020
B 2650
C 2600
9A 3120
B 2670
C 2900
10A 3040
B 2750
C 2660
10AA 3730
B 3900
Reference: (16)
Dimensions: a=36; L=114; L'=132; a/d2.95 to 8.78
Loading: Two equal loads
Tension Reinforcement: Railsteel bars; f, = from 59,300 to 63,000 psi
End Anchorage: Hooks
Web Reinforcement: %in. stirrups; f,w=73,400 psi
Age at Test: 28 days
d p a r rf,, Krf,. Pwie,
in. %
10.2 2.1
10.3 2.8
3.7
10.1 4.7
14.2 3 0
12.2 2.8
8.0 3.1
5.9 3.2
4.1 3.0
4.0
* Stirrups too short.
psi psi kips
627 33.0
32.0
36.2
47.4
40.0
46.8
58.6
64.8
66.6
74.5
71.1
79.0
92.5
106.6
92.0
619 69.3
63.9
71.3
309 135 25.8
31.7
33.4
317 139 15.3
18.6
16.6
315 138 6.8
7.7
6.2
9.8
10.3
" 9.0
Mode
of
Fail.
C
C
C
C
C
C
C
C
C
C
C
C
DT*
C
C
C
C
C
C
DT
C
C
C
C
C
C
C
C
C
C
P.
Eq. 18
kips
15.3
17.9
17.3
27.9
30.5
30.3
39.1
39.8
39.0
43.0
42.9
43.9
53.7
55.3
56.1
42.1
42.6
45.4
19.1
17.7
17.5
10.4
9.5
10.0
4.9
4.6
4.5
6.0
6.1
6.0
Ratio
P.
2.15
1.79
2.09
1.70
1.31
1.54
1.50
1.63
1.71
1.73
1.66
1.80
1.73
1.92
1.64
1.65
1.50
1.57
1.35
1.79
1.91
1.47
1.96
1.66
1.37
1.65
1.37
1.63
1.68
1.49
Table 23
Tests by Thompson, Hubbard, and Fehrer, 1938. SimpleSpan Rectangular Beams With Stirrups
Reference: (9)
Dimensions: b=8; d= 12; a=20; a/d= 1.67; L=60; L'= 74
Loading: 2 equal loads at 3points
Tension Reinforcement: Four %in. round oldstyle deformed bars; f,=36,000 psi
End Anchorage: Hooks
Web Reinforcement: %in. round vertical stirrups at 3.5 in.; r =0.0036; fI. =38,200 psi
Age at Test: 28 days
Beam f/ p rf,, Pt.t Mode P. Ratio Ratio
of Eq. 18 Pt__ Ptt
psi % psi kips Fail. kips P. P..
I CI 2570 2.5 136 97.0 S 65.0 1.49 1.17
2 " 88.0 S 1.35 1.06
3 " 98.0 S 1.51 1.18
Note: These beams, like those in Table 8, were tested without rollers at the beam supports. Their ultimate loads are significantly above their com
puted flexural strengths.
Table 24
Tests by Johnston and Cox, 1939. SimpleSpan Rectangular Beams With Stirrups
Reference: (17)
Dimensions: b=12, d  12; D= 13.3; a=36; a/d=3.00; L 108; L'= 120
Loading: Two equal loads at %points
Tension Reinforcement: Hard grade deformed and sq. twisted bars
End Anchorage: Hooked
Concrete Strength: Average concrete strength reported
Web Reinforcement: Vertical Yin. deformed stirrups at 8 in.; inter, grade; /, 45,000 psi assumed
Aee at Test: 28 days
%451
0.451
f, rf,. Pt, Mode P. Ratio Ratio
of Eq. 18 P. __
ksi psi kips Fail. kips P. P..
62.2 47 30.6 T 30.9 0.99 0.90
. . 30.9 T " 1.00 0.91
59.2 28.7 T 29.0 0.99 0.90
" 28.4 T " 0.98 0.90
60.3 28.3 T 29.6 0.96 0.88
" 28.2 T " 0.95 0.87
59.2 45.2 DT 38.9 1.16 1.06
45.3 T " 1.16 1.06
60.3 44.5 T 40.2 1.11 1.01
45.2 DT " 1.12 1.02
63.2 52.8 T 39.9 1.32 1.21
.. 52.9 DT " 1.32 1.21
58.6 46.0 DT 39.7 1.16 1.06
" 45.5 DT " 1.15 1.05
58.4 54.2 DT 40.7 1.33 1.22
54.3 T " 1.33 1.22
61.8 47.1 DT 39.5 1.19 1.09
45.7 DT " 1.16 1.06
64.4 42.8 DT 38.9 1.10 1.01
" 45.5 DT " 1.17 1.07
Beam
Al I
II
A2 I
II
A3 I
II
Bl I
II
B2 I
II
B3 I
II
I1I I
T1 I
II
T2 I
II
T3 I
II
Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
Table 25
Tests by Moretto, 1945. SimpleSpan Rectangular Beams With Stirrups
Reference: (4)
Dimensions: b= 5.5; d = 18.25; D= 21; a= 32; a/d= 1.75; L= 96; L'= 120
Loading: 2 equal loads at hpoints
Tension Reinforcement: Four 1in. square deformed bars; p =0.0398; f,= 48,000 psi
End Anchorage: Hooks
Compression Reinforcement: Two %in. square deformed bars; t=0.932; p'= 0.005
Web Reinforcement: 4in. plain bars, %in. and hin. deformed bars; s=6.5 in.
Age of Test: 28 days
r
%
0.28
",
0.615
11
1.1
"
Beam
IV Y 1
2
2V Y4 1
2
11 1
2
21 Y 1
2
1D Y 1
2
2D Y 1
2
IV % 1
2
2V % 1
2
11 % 1
2
21 Ya 1
2
LD % 1
2
2D % 1
2
lV / 1
2
2V~ 1
2
II ' 1
2
21 ) 1
2
1D q 1
2
2D M 1
2
laV Y 1
2
laV Y 1
2
0.28
0.615
in.; a= 32
46.0
52.0
'
rf,, Pes..t
kips
154 116.4
116.8
135.8
134.4
126.9
115.0
144.0
120.0
115.3
121.8
142.3
138.8
295 142.9
150.8
148.3
139.0
162.5
155.5
171.0
165.0
132.0
127.5
139.9
147.4
568 157.0
157.0
188.8
184.0
177.5
178.0
196.3
196.1
165.0
145.0
171.5
180.2
SERIES IA
in.; a/d= 1.64; p= 0.0186;
129 105.2
107.0
320 115.6
119.1
Mode
of
Fail.
DT
DT
DT
DT
DT
DT
DT
DT
DT
DT
DT
DT
DT
DT
DT
DT
DT
DT
DT
DT
DT
DT
DT
DT
C
C
C
C
C
C
C
C
C
C
C
C
p'=0.0047; t= 0.936
DT
DT
T,DT
T,DT
diagonal cracks. Furthermore, some beams tested
to provide data on their flexural strength can also
be utilized to obtain information about their shear
capacity. Figure 6 is used with the following cri
terion in mind: a beam will fail either in flexure
or in shear, whichever capacity is reached first.
Beams which failed in shear were used to derive
Eq. 26 for their shear strength. Beams which failed
in flexure, however, must fall below the line repre
senting their strength in shear in Fig. 6. If they
fall above, Eq. 26 cannot be correct; if it is correct,
the beams should have failed in shear rather than
in flexure since their shear capacity was smaller
than their flexural capacity.
Figure 6 indicates that Eq. 26 is a reliable
expression for shearcompression strength; all
beams with a few exceptions fall below the line
representing this equation. The flexural capacity of
the beams was reached at different ratios of P/P,,
Eq. 26 being the limit. Only 4 of the 91 beams fall
substantially above this limit. Two of these beams
were tested by Johnston and Cox17 ) and only the
average concrete strength was reported for 20
beams; it is likely, therefore, that the actual value
of fc' for the individual beams was greater than the
average and that P, for this strength would be in
creased sufficiently to bring the ratio P/Ps into
agreement with other test results. Two other beams
in this category were tested by Slater and Lyse.(16
One of the beams had one companion specimen
which failed in shear and another which failed at
a much lower load. Both companion specimens of
the other beam failed at a considerably lower load.
Figures 5 and 6 can also be used to determine
the relative effectiveness of different angles of in
clination and the yield strength of stirrups. Most
of the beams considered in the analysis had vertical
stirrups; there were, however, beams with stirrups
inclined at 67.5, 45 and 20 deg. The effect of differ
ent angles of inclination was taken into considera
tion in plotting Figs. 5 and 6 by computing the
b=5.5 in.; d= 19.5
P.
Eq. 18
kips
96.4
85.3
103.5
104.9
97.8
92.8
104.4
102.7
90.0
92.9
90.0
92.5
81.2
91.2
100.0
97.6
95.4
90.0
99.9
99.2
83.9
81.0
95.2
93.0
94.8
92.9
105.6
102.6
85.8
86.3
98.3
100.8
86.2
72.9
89.5
93.8
82.5
80.0
79.7
78.6
Beam A/ p a
psi % in.
Al1 3575 3.10 36
2 3430
3 3395
4 3590
B11 3388 3.10 30
2 3680
3 3435
4 3380
5 3570
B21 3370 3.10 30
2 3820
3 3615
B61 6110 3.10 30
C11 3720 2.07 24
2 3820
3 3475
4 4210
C21 3430 2.07 24
2 3625
3 3500
4 3910
C31 2040 2.07 24
2 2000
3 2020
C41 3550 3.10 24
C62 6560 3.10 24
3 6480
4 6900
D11 3800 1.63 18
2 3790
3 3560
D21 3480 1.63 18
2 3755
3 3595
4 3550
D31 4090 2.44 18
D41 3350 1.63 18
D16 4010 3.42 24
7 4060
8 4030
E12 4375 3.42 25
D26 4280 3.42 30
7 4120
8 3790
D41 3970 3.42 30
2 3720
3 3200
D51 4020 3.42 30
2 4210
3 3930
* Considered tension failure.
ILLINOIS ENGINEERING EXPERIMENT STATION
Table 26
Tests by Clark, 1951. SimpleSpan Rectangular Beams With Stirrups
Reference: (5)
Loading: 2 equal symmetrical loads at various positions on beam
Tension Reinforcement: Deformed bars
End Anchorage: A by 8in. plates jin. thick welded to the end of bars
Web Reinforcement: %in. vertical deformed bars; f,_= 48,020 psi
Age at Test: 28 to 30 days; beams kept moist until the day prior to testing
a/d a r rf,. Pt,,t Mode P. Ratio
of Eq. 18 Pt«t
in. % psi kips Fail. kips P,
8 by 18in. Beams; Span=72 in.; d= 15.37; f,=46,500 psi
2.54 7.2 0.38 182 100.0 DT 75.9 1.32
94.0 DT 74.5 1.26
100.0 DT 74.0 1.35
110.0 DT 76.0 1.44
1.95 7.5 0.37 178 125.4 DT 88.9 1.41
115.4 DT 92.3 1.25
128.1 DT 89.2 1.43
120.6 DT 88.6 1.36
108.6 DT 91.0 1.19
1.95 3.75 0.73 351 135.4 DT 88.4 1.53
144.9 DT 94.0 1.54
150.6 DT 91.5 1.64
1.95 7.5 0.37 178 170.6 DT 106.3 1.60
1.56 8.0 0.34 163 124.9 DT 100.9 1.24
139.9 DT 101.9 1.37
110.6 DT 97.6 1.13
128.6 DT 106.3 1.21
1.56 4.0 0.69 331 130.4 DT 97.0 1.34
. 4 135.4 DT 99.6 1.36
145.6 T 97.8 1.49
129.6 DT 100.5 1.29
1.56 8.0 0.34 163 100.6 DT 71.6 1.40
90.1 DT 70.9 1.27
84.6 DT 71.3 1.19
1.56 8.0 0.34 163 139.1 DT 113.5 1.23
1.56 8.0 0.34 163 190.6 DT 132.2 1.44
195.6 DT 132.1 1.48
192.7 DT 130.4 1.48
8 by 18in. Beams; Span=72 in.; d= 15.37 in.; f,= 48,630 psi
1.17 6.0 0.46 221 135.4 DT 124.1 1.09
160.4 T 121.0 1.32
115.4 DT 120.4 0.96
1.17 4.5 0.61 293 130.4 DT 119.0 1.10
. 4 140.4 DT 123.4 1.14
150.4 T 120.9 1.24
150.6 T 120.3 1.25
1.17 3.0 0.92 442 177.6 DT 148.5 1.20
1.17 2.25 1.22 586 140.4 DT 116.8 1.20
6 by 15in. Beams; Span 96 in.; d= 12.37; f, 46,500 psi
1.94 8.0 0.46 221 78,6 DT 60.3 1.30
80.6 DT 60.5 1.33
83.6 DT 60.4 1.38
6 by 15in. Beam; Span= 115 in.; d= 12.37 in.; f,= 46,500 psi
2.02 5.0 0.73 351 99.7 DT 60.0 1.66
6 by 15in. Beams; Span= 120 in.; d  12.37 in.; f,= 46,500 psi
2.43 6.0 0.61 293 75.7 DT 49.5 1.53
70.7 DT 48.8 1.45
75.7 DT 46.9 1.61
2.43 7.5 0.49 235 75.7 DT 46.0 1.58
70.7 DT 46.6 1.52
74.2 DT 43.2 1.72
2.43 10.0 0.37 178 65.7 DT 48.2 1.36
70.7 DT 49.3 1.44
70.7 DT 47.8 1.48
Table 27
Tests by Gaston, 1952. SimpleSpan Rectangular Beams With Stirrups
Reference: (11)
Dimensions: b = 6; D= 12; a= 36; a/d= 3.36 to 3.90; L= 108; L'= 120
Loading: 2 equal loads at Apoints
Tension Reinforcement: Deformed bars
End Anchorage: None, straight bars
Web Reinforcement: Y and %in. vertical deformed stirrups; fw= 45,000 psi assumed
Age at Test: Around 30 days
Beam // d p f, r rf,. MtAt Mode M. Ratio
of Eq. 18 Mt__t
psi in. % ksi % psi kipft Fail. kipft M.
T1Lb 2520 10.72 0.62 46.0 0.28 126 20.2 T 18.7 1.08
T2La 2120 10.65 0.97 40.4 0.42 189 24.2 T 20.0 1.21
T4Lb 2810 10.44 2.52 43.3 0.92 414 47.8 T 32.4 1.48
T5L 2500 10.37 3.22 40.2 0.92 414 53.9 T 32.3 1.67
T11L 2900 9.23 7.22 45.3 1.83 824 67.6 C 35.4 1.91
T1Ha 5880 10.58 1.38 44.2 1.05 473 35.1 T 34.9 1.01
T2H 5400 10.44 2.52 45.6 1.05 473 53.9 T 42.3 1.27
T3H 5920 9.52 4.20 43.2 1.83 824 67.7 T 42.4 1.60
T5H 5900 9.23 7.22 40.6 1.83 824 86.3 T 46.8 1.85
Ratio
Ptt
P..
0.97
0.92
0.99
1.06
1.04
0.92
1.05
1.00
0.88
0.90
0.90
0.96
1.18
0.94
1.03
0.85
0.91
0.81
0.82
0.90
0.78
1.06
0.96
0.90
0.93
1.09
1.12
1.12
0.76
0.92
0.67
0.69
0.72
0.78
0.79
0.64
0.55
0.90
0.92
0.96
0.98
0.96
0.91
1.02
1.07
1.03
1.17
1.00
1.06
1.09
Ratio
Pt
0.83
0.78
0.84
0.91
0.88
0.79
0.89
0.84
0.75
0.95
0.98
1.03
1.06
0.95
1.05
0.85
0.96
1.00*
1.03*
1.11*
0.98*
0.87
0.78
0.73
0.86
0.94
0.97
0.95
0.91
1.08*
0.78
0.89
0.94
1.01*
1.02*
0.83
0.96
0.81
0.82
0.86
1.04
0.96
0.90
0.98
0.97
0.92
1.02
0.84
0.90
0.91
Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
Table 28
Tests by Moody, Series III, 1953. SimpleSpan Rectangular Beams With Stirrups
Reference: (12)
Dimensions: b 7; d= 21; D= 24; a=32; a/d= 1.52; L= 96; L'= 120
Loading: 2 equal loads at 4points
Flexural Reinforcement: No. 11 deformed bars; f,= 43,800 psi; t=0.914
End Anchorage: Hooks
Web Reinforcement: Vertical stirrups
Age at Test: 28 days
Beam f/ p p' Web s
Reinf.
psi % % in.
30 3680 4.25 2.13 No. 3 6
31 3250 " No. 4
r f.
% ksi
0.52 47.3
0.95 44.0
r/fw Pte.t Mode P.
of Eq. 18
psi kips Fail. kips
246 215 S 179.5
418 228 S 169.8
Ratio Ratio
Pte.t Pte.t
P, P..
1.20 0.80
1.34 0.73
ratio of web reinforcement from the conventional
expression:
A, (27)
r bs sin a (27)
The conventional theory considers that the param
eter Krf,,, is the measure of shear strength. Since
the concept of the truss analogy is disregarded by
the present analysis, there is no justification for
employing the quantity K. Furthermore, while the
variation in K is rather small for a between 45 and
90 deg, for smaller values of a the coefficient K de
creases rapidly. For beams of Slater and Lyse
Fig. 5. Effect of Web Reinforcement on Shear Strength.
SimpleSpan Rectangular Beams with Stirrups
which had stirrups inclined at 20 deg, K is equal
to 0.44. The use of this low value of K would shift
these beams considerably to the left in Fig. 6.
Consequently, the beams would lie above the shear
strength line. Since the beams failed in flexure, the
use of Krfy, rather than rfJ, is not justified.
The yield strength of the web reinforcement
varied from about 44,000 to 73,400 psi for beams
which failed in shear and from about 40,000 to
93,000 psi for beams which failed in flexure. The
majority of the beams, however, had their yield
strength between 45,000 and 55,000 psi. This varia
tion is perhaps not large enough to bring out the
effect of yield strength. However, the beams re
ported by Slater and Lyse were reinforced with
stirrups of relatively high yield strength, f,,, =
73,400 psi. If the ratio P/P, were plotted against r
alone, these beams would again fall above the shear
strength line determined from other test results.
This indicates that the quantity rfw is a more cor
rect measure of shear strength than the ratio r
alone. It seems reasonable to believe that stirrups
with higher yield strength offer greater resistance
to the extension and widening of the diagonal
cracks than stirrups of low yield strength.
12. BentUp Bars as Web Reinforcement
Relatively few simplespan beams with bentup
bars as web reinforcement have been tested to
determine their strength in shear. The only source
of experimental data is the beams tested by
Richart,(2 but practically all these beams failed in
tension.
Series 1917 included 32 beams with hooked
bentup bars. The variables were the amount,
angle of inclination, and spacing of the web bars.
The main body of the beams was made of concrete
from 2450 to 3770 psi; at the top center of each
beam, however, there was a 4in. deep zone of
higher strength concrete, f/' = 4770 psi. The beams
were tested twice: they were first loaded to yielding
P,
ILLINOIS ENGINEERING EXPERIMENT STATION
2.4
2.2
2.C
1.6
1.6
Plest
Ps
1.4
1.2
/.C
0.6
Fig. 6. SimpleSpan Rectangular Beams Fail
with loads placed 48 in. from the end supports, and
they were then retested with loads 36 in. from the
supports. All beams failed in tension.
In order to obtain some indirect information
about the shear strength of these beams, some of
the beams with the smallest ratio of web reinforce
ment are analyzed in Table 29. The shear capacity
of the beams was calculated by using the steel
percentage p at the critical section, that is, under
the concentrated load, to determine the value of k.
Their P/P,ratios are plotted against rf,, in Fig. 7.
This figure shows that four beams with rf,, equal
to 210 psi were very close to shear failures, pro
vided that Eq. 26 holds true for beams reinforced
with bentup bars. Photographs taken of these
beams after failure show welldeveloped diagonal
cracks. In all probability the beams were very
close to their shear capacity.
Two beams of Series 1922 were also provided
with bentup bars as web reinforcement. These
beams are analyzed in Table 20 and shown in Fig.
ing in Flexure. Beams Reinforced with Stirrups
7. Both beams failed in tension and, as seen in the
figure, lie below their strength in shear as given
by Eq. 26.
Three beams of Series 1911 had one longitudinal
bar bent up at a rather small angle in order to
reinforce the entire shear span, 24 in. long. These
beams are reported to have failed in diagonal ten
sion. Table 30 analyzes the beams by using s = a
in Eq. 27 to calculate their ratio of web reinforce
ment. Undoubtedly, this procedure is approximate,
and these beams fall somewhat low in Fig. 7. How
ever, a sketch of one of the beams after failure
shows extensive cracking at the end hooks of the
reinforcement and indicates a possible failure in the
anchorage.
With the help of Fig. 7 and more numerous
tests on Tbeams which are analyzed later, it was
concluded that the contribution of bentup bars to
the shear strength of a beam is the same as that of
stirrups. Consequently, Eq. 26 can be used in both
cases.
Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
Table 29
Tests by Richart, Series 1917. SimpleSpan Rectangular Beams With BentUp Bars
Reference: (2)
Dimensions: b=8.1; d=10; D=12; a=48; a/d=4.8; L=114; L'=120
Loading: 2 equal loads
Tension Reinforcement: Plain round bars, hooked; f,= 37,500 to 45,700 psi
Web Reinforcement: Bentup bars, hooked
Concrete Strength: f// = 4770 psi for a zone 4in. deep and 54in. long at top center of each beam;
f/= 3040 to 3770 for the remainder
Age at Test: About 60 days
a r rf,«
deg % psi
45 0.56 210
45 0.56
45 0.80 320
45 0.80 370
45 3.28 1490
28 1.96 900
45 1.29 590
11 " 11
* Distance from load to first bentup bar.
Some additional information about the effective
ness of bentup bars is available from the tests of
Series 1917. One of the variables investigated was
the distance from the load point to the first bentup
P
7
Pt«t Mode
of
kips Fail.
40.2 T
40.0 T
42.2 T
40.8 T
40.8 T
40.0 T
41.9 T
41.5 T
36.3 T
31.5 T
45.5 T, DT
40.9 T
37.7 T
41.1 T
P.
Eq. 18
kips
30.3
30.3
30.3
30.3
30.1
30.1
30.3
30.3
30.0
30.3
30.2
30.2
30.2
30.2
bar. This distance varied from 9.6 to 16.8 in., or
up to 1.68 times the effective depth of the beams.
The analysis of some of these beams was included
in Table 29. It is seen that even these beams failed
rfFm (psi)
Fig. 7. SimpleSpan Rectangular Beams Failing in Flexure. Beams Reinforced With BentUp Bars
Beam
16B6.1
6.2
16B7.1
7.2
16B8. 1
8.2
16B9.1
9.2
16B10. 1
10.2
16B18.1
18.2
16B19.1
19.2
ILLINOIS ENGINEERING EXPERIMENT STATION
in tension, although a considerable part of the
shear span in the immediate region of maximum
moment was without any direct web reinforcement.
The highest P/P,ratio at failure was 1.51. Conse
quently, wellanchored bentup bars, although not
covering the entire shear span, appear to be bene
ficial in resisting the development of diagonal
cracks. This phenomenon was also observed for
Table 30
Tests by Richart, Series 1911
SimpleSpan Rectangular Beams With BentUp Bars
Reference: (2)
Dimensions: b 8; d0; D  12; a24; a/d2.4; L 72; L'=78
Loading: 2 equal loads at Apoints
Tension Reinforcement: Three tin. plain round bars, hooked; p = 0.0165;
f,=about 38,000 psi
Web Reinforcement: One oin. round bar bent up; a about 27 deg
Concrete Strength: Tests on 6 by 8 by 40in. control beams; reduced to
cylinder strength by f/' = 6.7 f,
Age at Test: Around 60 days
Beam f/ r* rf,. Pt.t Mode P. Ratio Ratio
of Eq. 18 Pte..t Pte.t
psi % psi kips Fail. kips p
292.1 1760 0.50 190 30.7 DT 25.5 1.20 0.87
2 " " " 28.9 DT 25.5 1.14 0.83
3 29.8 DT 25.5 1.17 0.85
*r computed as r= A sin 270.
beams of Series 1910 which had vertical and
diagonal stirrups supplemented by bentup bars.
It is seen in Table 18 that the addition of only one
layer of bentup bars, not covering the entire shear
span, increased the shear strength of the beams
sufficiently to permit a tension failure.
13. Maximum Useful Amount of Web Reinforcement
Excluding bond failure, a reinforced concrete
beam can fail either in flexure or in shear. Flexural
failures can be initiated either by yielding of ten
sion reinforcement or by crushing of concrete on
the top of the beam, depending on the physical
properties of the beam. Since the flexural capacity
of a beam can be determined accurately, the pur
pose of this analysis is to find the amount of web
reinforcement necessary to force a beam to fail in
flexure rather than in shear.
Expressions for the shear capacity of a simple
span rectangular beam under one or two sym
metrical concentrated loads were derived previ
ously. Equation 26 can be rewritten as:
M 1 + 2rf (28)
M8 = 10,
where Ms is given by Eq. 18.
Expressions for the flexural capacity of a beam
are taken from a previous technical report.(") The
ultimate flexural moment is given as:
Mf = pf. k2 pf.
bd f,' f' kk f, ' )
(29)
When a beam fails in tension, the yield stress f, is
substituted for f, in Eq. 29. For compression fail
ures, the steel stress f, is below its yield strength;
it can be determined from the following equation:
= _ E .ekk3f/ (1 ~uE)2 1
f. = \  p 2 EUE)  uE
(30)
Whether the stress in the tension reinforcement at
failure is below or at its yield stress is determined
by the following criterion. The reinforcing index q
is defined as:
q = ,
c
(31)
The critical value of q is given by
kik3
qcr= 
1 Cy
(32)
If q > qcr, the steel stress at failure is below its
yield stress and the beam fails in compression. If
q = Qcr, the beam fails by crushing of concrete as
soon as the tension reinforcement yields. If q < qer,
the steel stress at failure is either at or above its
yield stress and the beam fails initially by yielding
of the tension reinforcement. Another critical value
of q can be utilized to determine whether or not the
steel stress reaches work hardening at failure; this,
however, is an unnecessary refinement in the pres
ent analysis.
The following numerical values are used in the
above equations:
k2 = 0.45
kik3 = 2.4 (0.57  4.5f,)
10,
(20)
Eu = 0.004
E, = 30,000,000 psi
The behavior of beams with different values of
the reinforcing index is shown by Fig. 8. To facili
tate the presentation of expressions for shear
strength, the quantity M/bd2f/ for the ultimate
moment is plotted against the parameter p/f,'
Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
rather than q. The curves are drawn for f, =45,000
psi, fc' = 3,000 psi, and fy,, = 45,000 psi. If the
beams have sufficient web reinforcement to fail in
flexure, the ultimate moment is determined by
curves 1 and 2. For p/fi' < (P/fc') cr the beams
fail in tension according to curve 1, obtained from
Eq. 29 by substituting f, =fI. At (p/fc')r = 1.69
X 10" in.2/lb, computed from Eq. 32, the mode
of failure changes from tension to compression. For
p/fc > (p/fc') c, the ultimate moment is given by
curve 2, computed by Eq. 29 with steel stresses ob
tained from Eq. 30.
If, however, no web reinforcement is provided,
the maximum load is governed by curves 1 and 3.
Curve 3 represents the shear strength of a beam
without web reinforcement, given by Eq. 18. The
intersection between these two curves determines
the transition between tension and shear failures.
When some web reinforcement is provided, the
shear strength increases according to Eq. 28 and
the transition between the two types of failures
takes place at a larger value of p/fc. Curve 4
shows this for r = 0.005.
If it is desired that the beams fail in flexure for
any value of p/If', the shear strength must be
Fig. 8. Relation Between Strength in Shear and Flexure
as Function of Reinforcement Percentage
greater than the flexural strength for the entire
range of ultimate moment, curve 1 for tension and
curve 2 for compression failures. It is seen that a
shear strength curve 5 passing through the inter
section between curves 1 and 2 satisfies this con
dition. Computations based on the value of of per,
which corresponds to qcr obtained from Eq. 32,
show that the corresponding ratio of web reinforce
ment is 0.011 for the variables under consideration.
f" (ps/I
Fig. 9. Maximum Useful Amount of Web Reinforcement as
Function of Concrete Strength and Yield
Strength of Reinforcement
Thus r = 0.011 corresponds to the maximum useful
amount of web reinforcement for the values of fC',
/y, and fyw used in the above example. This limit
was calculated for other combinations of f//, If, and
fyw and is shown in Fig. 9 graphically.
The maximum useful amount of web reinforce
ment does not depend on the percentage of tension
reinforcement. It forces a beam of any amount of
tension reinforcement to fail in flexure, either in
tension or in compression. However, for any value
of p except that at the transition between tension
and compression failures, this maximum useful
amount is more than sufficient to insure flexural
failures; see curve 5 of Fig. 8. In practice, most
beams are designed to fail in tension if loaded to
destruction. These beams would fall considerably
to the left of the transition point and, consequently,
would require much less web reinforcement to pre
vent shear failures.
Table 31 shows an analysis of simplespan
beams designed according to the present ACI Code
balanced design requirements and loaded with
concentrated loads. This analysis considers rec
p/f' (10sin?1//
ILLINOIS ENGINEERING EXPERIMENT STATION
Table 31
Amount of Web Reinforcement Required to
Prevent Shear Failures in Rectangular Beams
Normal ACI Beams Without Compression Reinforcement
p//f.
105
in.'/lb
0.46
0.45
0.45
0.46
n
Eq. 16
10.0
9.0
8.3
7.7
k M.
Eq. 14 bdf//
Eq. 18
0.345 0.166
0.351 0.160
0.376 0.164
0.398 0.160
// q Mr Ratio r/,. r(%)
bdf/3 Mf/Al. psi computed for fA.(ksi) =
Eq. 31 Eq. 29 Eq. 26 40 45 50
f,=40,000 psi
2000 0.182 0.169 1.02 11 0.03 0.02 0.02
2500 0.181 0.167 1.04 22 0.06 0.05 0.04
3000 0.181 0.167 2 11 0.03 0.02 0.02
3750 0.184 0.168 1.05 25 0.06 0.06 0.05
fy = 45,000 psi
2000 0.205 0.188 1.14 69 0.17 0.15 0.14
2500 0.203 0.186 1.16 81 0.20 0.18 0.16
3000 0.204 0.186 1.14 69 0.17 0.15 0.14
3750 0.207 0.187 1.17 84 0.21 0.19 0.17
f, = 50,000 psi
2000 0.228 0.207 1.25 126 0.32 0.28 0.25
2500 0.226 0.205 1.28 140 0.35 0.31 0.28
3000 0.227 0.205 1.25 126 0.32 0.28 0.25
3750 0.229 0.205 1.28 141 0.35 0.31 0.28
* Steel percentages as given by ACI Code balanced design requirements
for f.= 20,000 psi and f. = 0.45 f/.
tangular beams reinforced in tension only with the
steel percentages corresponding to allowable
stresses of f, equal to 20,000 psi and fc equal to
0.45 fe'. The amount of web reinforcement neces
sary to prevent shear failures has been calculated
with the aid of Eq. 28 for several values of f, and
fy,. It is seen that as f, increases and fy decreases,
the amount of web reinforcement necessary to en
sure flexural failures increases. For f, equal to 50,
000 psi and f/w equal to 40,000 psi, about 0.35
percent web reinforcement is required, while for
both f, and fe, equal to 45,000 psi, about 0.20 per
cent will be sufficient.
IV. SIMPLESPAN TBEAMS UNDER ONE OR TWO SYMMETRICAL
CONCENTRATED LOADS
14. TBeams Without Web Reinforcement
The basic empirical equation 18 was derived for
simplespan rectangular beams. It has been shown
that this equation can be interpreted by means of
the conventional theory of compression failures, as
modified by diagonal tension cracking, and that the
failure criterion is the ultimate compressive strain
in the concrete.
The above concept of shear failures as shear
compression failures was extended to include T
beams. Since the momentrotation relationship of
a Tbeam differs from that of a rectangular beam,
a correction must be made to take into considera
tion the effect of the shape of the beam on the com
pressive strain in the concrete. But since the
distribution of the concrete strain had not been de
termined previously, the exact form of the shape
factor cannot be established. If a linear strain dis
tribution is assumed, strain in any fiber is given by
M
= El y
where y is the distance from the neutral axis to the
fiber under consideration. Comparing a Tsection
with a rectangular section of the same width as the
flange in the Tsection, the following relationship
can be written if the ultimate strain in the concrete
is the same in both cases:
MT = MR IcTYR (33)
where the subscripts R and T refer to rectangular
and Tsections, respectively, Ic refers to the mo
ment of inertia of a section transformed to concrete,
and y refers to the distance from the neutral axis
to the top fiber in the concrete, all quantities taken
at the instant of failure. If the strain distribution
were linear and all quantities could be determined,
the above expression would give the relationship
between shear moments of a Tsection and a rec
tangular section of the same width. However, the
formation of a diagonal crack produces a non
linear strain distribution. The stress in the tension
reinforcement is approximately uniform from the
lower end of the crack to a vertical section through
the uper end of the crack. These conditions affect
also the distribution of concrete strain at the top of
the beam, causing a certain concentration of strain
at the end of the diagonal crack. Furthermore, since
the section cracks progressively as load is applied,
the exact values of I and y cannot be determined.
Consequently, Eq. 33 cannot be applicable.
An approximate shape factor was derived by
assuming that the effect of shape of a beam is de
termined primarily by its moment of inertia. In an
uncracked state, the moment of inertia of a Tbeam
is considerably smaller than that of a similar rec
tangular beam. After extensive cracking, the value
of I of a section transformed to concrete is very
nearly the same in both cases. At the instant of
failure, the relationship between the two is un
known; it was approximated by the ratio of the
average values of I of the uncracked and the fully
cracked state. Thus, the shape factor takes the fol
lowing form:
IT + Icr
IR + Icr
where IR and IT refer to the uncracked rectangular
and Tsections, respectively, and Icr refers to the
"straight line" cracked transformed section of either
a rectangular or a Tsection since both have very
nearly the same moment of inertia.
The above shape factor makes it possible to
modify Eq. 18 for rectangular beams so that it
applies to Tbeams. The compressive area Ac of a
Tsection as determined by the conventional
"straight line" theory is substituted for bkd and
the equation is rewritten as follows:
dF M. 0.57 104.5
AodfF, 10s
The validity of Eqs. 34 and 35 must be deter
mined with the help of test results. All available
data on Tbeams under one or two symmetrical
ILLINOIS ENGINEERING EXPERIMENT STATION
concentrated loads were analyzed; the range of test
variables is summarized in Table 32 and the physi
cal properties and calculated quantities of indi
vidual beams are given in Tables 33 through 39. All
units are given in inches and pounds. The width of
the flange is indicated by b, that of the web by b',
and the thickness of the flange by e. Other symbols
have their usual meaning. Some beams were rein
forced with straight unanchored bars and failed in
bond; these beams are not included in the analysis.
Beams without web reinforcement are consid
ered first. Ferguson and Thompson(20, 21) have re
ported tests on beams of a number of different
shapes. Some of the beams were provided with
shoulders; that is, the width of the upper part of
the web was greater than that of the lower part.
These beams are analyzed in Tables 36 and 37, and
the quantity M/Acdf/'F is plotted against f/' in
Fig 10. It is seen that in most cases Eq. 35 gives
reasonable agreement with the test results. How
Table 32
Range of Test Variables for SimpleSpan TBeams. Under Two Symmetrical
Table No. No.
No. of of
Beams Shear
Fail.
I' A. Reinf.
in
Flange
Concentrated Loads
b b d
psi in.' in.
BEAMS WITHOUT WEB REINFORCEMENT
25102690 3.9 None 19.7
1700 1.56 Yes 42
3570;3610 3.91 None 20
25403500 0.88 None 19; 22
39606580 1.58 None 17
BEAMS WITH WEB REINFORCEMENT
2650 3.9 None 19.7
2580 3.9 None 19.7
1700 2.34;3.51 Yes 42
37994346 3.91 None 20
13701540 6.77 Yes 53.2
a/d F,
in. in. in.
13.9
10.9
21
4.57
8.25
13.9
13.4
10.010.9
21
21.3
3.9
4.25
6
1.5;2.13
1.5
3.9
3.9
4.25
6
3.9
2.83
3.30
1.71
4.06.22
3.39
2.83
2.94
3.33.6
1.71
2.77
0.815
0.55
0.76
0.580.65
0.650.75
0.82
0.80
0.59;0.62
0.76
0.63
Table 33
Tests by Bach and Graf, Heft 10, 1911. SimpleSpan TBeams Under Two Symmetrical Concentrated Loads
Reference: (18)
Dimensions: b=19.7; b'=7.9; Df15.7; d=13.9; e=3.9; a=39.4; a/d=2.83; L118.1; L'=133.9
Loading: 2 equal loads at %points
Tension Reinforcement: Two 1.57in. plain round bars; A.=,3.90 in.'; f,= 43,600 psi
Anchorage: Hooks
Web Reinforcement: Plain round vertical stirrups
Reinforcement in Flange: None
Concrete Strength: Average f/.'3530 psi; '//0.75 .f' =2650 psi; variation from 8 to +12 percent
Age at Test: About 45 days
Number of Beams: 3 companion specimens in each group; 2 in groups c and d
BEAMS WITHOUT WEB REINFORCEMENT
Group Size 8
W.R.
in. in.
b 0.51 3.35
c 0.51 5.51
d 0.28 5.51
8 0.39 7.87
9 0.28 7.87
10 0.20 7.87
11 0.39 5.91
12 0.28 5.91
13 0.20 5.91
15 0.39 3.94
16 0.28 3.94
17 0.20 3.94
18 0.20 1.97
14 0.39 5.91
19 0.28 5.91
20 0.28 5.91
21 .79by .08 5.91
22 .79by .08 5.91
23 0.28 5.91
Ps..t A. Fe
Muse Ratio
0.815
Adf'Ft
0.418
0.354
0.398
0.408
0.412
BEAMS WITH WEB REINFORCEMENT
f,. rf,, Pt.t F, P,
Eq. 35
ksi psi kips kips
37.8 580 94.4 0.82 60.6
38.6 370 86.0
40.2 109 77.2
41.0 160 80.0 "
43.8 83 72.0
48.2 48 65.8
41.0 213 82.9
43.8 114 79.4
48.2 63 72.4
41.0 324 94.1
43.8 171 88.2
48.2 96 80.1
48.2 188 89.3
41.0 213 82.2
43.8 114 71.9
43.8 114 74.4
57.5 305 87.5
52.2 141 77.2
43.8 114 67.6
M.
0.93
0.79
0.88
0.89
0.92
Ratio Ratio
P1 Pt1
P. P,,
1.56 0.73
1.42 0.82
1.27 1.04
1.32 1.00
1.19 1.03
1.08 0.99
1.37 0.96
1.31 1.06
1.19 1.05
1.55 0.94
1.45 1.08
1.32 1.10
1.47 1.07
1.36 0.96
1.19 0.97
1.23 1.00
1.44 0.90
1.27 0.99
1.12 0.92
Mode
of
Fail.
S,B
S,B
S,B
S,B
S,B
Mode
of
Fail.
B,S?
B,S?
S
S
S
S
S
8
S
S
S
S
S
S
8
S
S
Test
Series
Bach, Graf
Heft 10(1)
Braune
Myers(")
Richart
Ser. 1922(2)
Thompson
Ferguson(20)
Ferguson
Thompson(21)
Bach, Graf
Heft 100)'
Heft 12(22)
Braune
Myers(u)
Richart
Ser. 1922(2)
Graf
Heft 67(<)
Beam
e330
331
7441
442
444
kips
57.3
48.5
52.9
52.9
57.3
r
%
1.56
0.95
0.27
0.39
0.19
0.10
0.52
0.26
0.13
0.79
0.39
0.20
0.39
0.52
0.26
0.26
0.53
0.39
0.26
I
Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
ever, two series with the largest number of beams,
Series A and B, indicate consistently lower shear
strengths than those given by Eq. 35. This discrep
ancy could mean either that the shape factor given
by Eq. 34 is fundamentally incorrect or that there
are some other considerations besides the effect of
the moment of inertia which determine the com
pressive strain in the concrete. It is noticed that the
beams reported by Ferguson and Thompson have,
in general, very wide and thin flanges. It is known
that in such beams parts of the flanges at some
distance from the web do not resist their full share
of the bending moment. This phenomenon is dis
cussed in some detail for an elastic medium by
Timoshenko.241 It can be seen in Fig. 10 and
Tables 36 and 37 that beams which fall the greatest
percentage below the predicted values have very
large d/e and b/b'ratios; that is, the depth of the
beam is large relative to the thickness of the flange
and the width of the flange is large relative to that
of the web. The beams of Series A, B, C, and D
had the same d/eratio, 5.5, while b/b'ratio was
equal to 4.25 for Series A and 2.43 for Series D. The
two beams of Series D failed at a load rather close
to the predicted values, 85 and 93 percent, respec
tively, while the beams of Series A reached only
about 70 percent of their predicted capacity at fail
ure. The beams of Series B and C were provided
with shoulders of the same width as the web width
of the beams of Series D. The depth of the shoulder,
e", was 4 in. for Series B and 7 in. for Series C.
The addition of shoulders reduces the unsupported
width of the flange and in that sense, should have
the same effect as a decrease in the b/b'ratio.
Table 34
Tests by Braune and Myers, 1917. SimpleSpan TBeams Under Two Symmetrical Concentrated Loads
Reference: (19)
Dimensions: b=42; b'=6; D=12; e=4.25; a=36; L=108; L' 120
Loading: Two equal loads at Mpoints
Tension Reinforcement: 5%in. square twisted bars; f,= 40,320 and 66,350 psi for the two bars tested
Reinforcement in Flange: Four %in. round long. bars; Hin. square transverse bars at 8 in.; all beams except I
Web Reinforcement: hin. square plain bars and bentup bars
Concrete Strength: Average f,,'= 2270 psi; f'= 0.75 fe'= 1700 psi
Age at Test: 90 davys
BEAMS WITHOUT WEB REINFORCEMENT
Beam p* d a/d Ptest Mode
of
% in. kips Fail.
I1 0.34 10.9 3.30 33.4 S
2 " " 28.8 S
BEAMS WITH WEB REINFORCEMENT
Beam pt d a/d Web a r rf,.
Reinf.
% in. deg. % psi
II1 0.51 10.9 3.30 Stirr. 90 0.52 350
2 11" " " I '?
III1 0.51 10.9 3.30 3Bup ? ?
2 " " Bars
IV1 0.91 10.0 3.60 Stirr. 90 2.36 1580
2 "" +Bup 45
V1 0.91 10.0 3.60 Stirr. 90 2.82 1870
2 " 1 " +Bup 45
* Bars not hooked.
t Bars hooked.
Ft M 'tF
Asdfe'F,
Ptýt
kips
92.0
86.0
97.4
95.4
129.6
139.2
139.2
139.2
Mode
of
Fail.
T
T
T
T
T
T
T
T
0.562
0.484
F* P.
Eq. 35
kips
0.59 39.1
0.59 39.1
0.62 43.7
0.62 43.7
Ratio
M.
1.14
0.98
Ratio
Pte.
P.
2.36
2.20
2.49
2.44
2.97
3.19
3.19
3.19
Table 35
Tests by Richart, Series 1922. SimpleSpan TBeams Under Two Symmetrical Concentrated Loads
Reference: (2)
Dimensions: b=20; b'=8; D=f24; d=21; e=6; a= 36; a/d=1.71; L= 108; L'=120
Loading: 2 equal loads at Wpoints
Tension Reinforcement: Four 1%in. corrugated round bars; p= 0.0093; f,= 52,400 psi
Anchorage: Hooks
Web Reinforcement: Plain round vertical stirrups
Reinforcement in Flange: None
Age at Test: About 60 days
BEAMS WITHOUT WEB REINFORCEMENT
Pts. Mode A, Fs
of
kips Fail. in.2
180.3 DT 125.0 0.76
167.2 DT 125.4
BEAMS WITH WEB REINFORCEMENT
s r /,, r/,» Pte.s
in. % ksi psi kips
4 1.38 42.9 592 259.5
. ... 245.5
7 1.40 40.1 561 258.5
265.8
11 1.39 39.6 550 261.4
257.2
Mtt
A4df,'Ft
0.451
0.421
Mode
of
Fail.
T
T
T
T
T
T
Ratio Mode
M. " of
M, Fail.
1.10 S
1.03 S
Fs p, Ratio
Eq. 35 ___
kips P.
172.7 1.50
178.1 1.38
167.5 1.54
178.3 1.49
173.0 1.51
Beam
2210.1
2210.2
Beam
226.1
2
227.1
2
228.1
2
71 4.8 1.47
ILLINOIS ENGINEERING EXPERIMENT STATION
Although the beams of Series B failed at but
slightly higher loads than those of Series A, the two
beams of Series C reached 75 and 84 percent, re
spectively, of their predicted strength at failure.
These results show that for the same value of d/e,
the agreement between the measured and calculated
loads improves as the ratio b/b' decreases, and thus
addition of shoulders does have partially the same
effect as that of decreasing the b/b'ratio. Further
more, deeper shoulders have a greater effect on the
increase of the shear strength than shallower shoul
Beam
N1
2
3
G4
5
6
L1
2
3
HB2
5
8
KB1
4
7
ders. This is apparently related to the formation
and propagation of cracks in the tension zone of
the concrete. However, since the shape factor of
Eq. 34 was primarily intended for ordinary T
beams without shoulders, it is not expected that it
would apply equally well for more complex shapes
of Tbeams. The remaining beams had rather large
b/b'ratios, varying from 4.47 to 5.18, while the
d/eratio varied from 2.11 to 4.67. All these beams
except those of Series N failed at a load in good
agreement with Eq. 35. The beams of Series N
Table 36
Tests by Thompson and Ferguson, 1950. SimpleSpan TBeams Under Two Symmetrical Concentrated Loads
Reference: (20)
Dimensions: a=28; L=84; L'=96
Loading: Two equal loads at %points
Tension Reinforcement: Two %in. round deformed bars, inter, grade; A,= 0.88 in.2
End Anchorage: Welded anchorage plate
Web Reinforcement: None
Reinforcement in Flange: None
Shoulders: Width = b; depth from top of beam= e'= 0
Age at Test: 28 days
Series HB and KB: Beams with Btile considered in analysis; comp. strength of Btile= 4160 psi;
%in. layer of tile included in the overall dimensions of beams
Mode of Failure: All beams failed in shear
f.' Ptt A, F, M__
Adfc'F,
psi kips in.'
b=19; b'=4.25; b"=0; d=7; D=7.5; e=1.5; e"=0; a/d=4.0; d/e=4.67; b/b'=4.47
3000 10.68 30.67 0.65 0.357
2990 10.76 30.67 0.361
2540 9.66 30.97 0.378
b= 22; b'= 4.25; bV= 0; d=4.5; D= 5.5; e= 1.5; e'=0; a/d6.22; d/e =3.00; b/b'= 5.18
3320 6.30 30.99 0.58 0.326
3150 7.10 31.28 0.383
3170 7.90 31.19 0.425
b=19; b'=4.25; b"=7; d=6.25; D=7.5; e= 1.5; e= 3.5; a/d=4.48; d/e=4.17; b/b'= 4.47
3150 12.30 30.95 0.61 0.463
3280 13.40 30.81 0.487
3220 12.30 30.88 0.454
b=22;b'= 4.25; Vb=0; d=4.5; D=5.5; e=2.13; e"=0; a/d=6.22; d/e=2.11; b/b'=5.18
3270 9.14 31.09 0.60 0.470
3150 9.14 31.19 " 0.487
3020 8.90 31.48 0.489
b=19; b'= 4.25; b"= 8.25; d= 6.25; D=7.5; e=2.13; e'=4.13; a/d= 4.48; d/e=2.93; b/b'= 4.47
3340 13.78 34.44 0.62 0.435
3350 12.25 34.44 0.385
3500 14.76 34.20 0.447
Ratio
Mtet
M,
0.82
0.83
0.83
0.78
0.90
1.00
1.08
1.15
1.07
1.11
1.13
1.12
1.03
0.92
1.08
Table 37
Tests by Ferguson and Thompson, 1953. SimpleSpan TBeams Under Two Symmetrical Concentrated Loads
Reference: (21)
Dimensions: b=17; d=8.25; D=9.5; e=1.5; a= 28; a/d=3.39; L=64; L'=72
Loading: Two equal loads at Jpoints
Tension Reinforcement: Two No. 8 deformed bars, rail steel; A.= 1.58 in.'
End Anchorage: Welded steel block at each end
Web Reinforcement: None
Reinforcement in Flange: None
Age at Test: Around 28 days
Shoulders: Width = b", depth from top of beam = e
Mode of Failure: All beams failed in shear
I' b' b" el d/e b/b' Ptes. A, Ft Mt
Beam
A 1
2
3
4
5
6
D 1
2
B 1
2
3
4
5
C 1
2
kips
13.06
12.12
15.12
14.22
15.22
16.00
21.90
23.40
15.94
14.20
17.72
19.72
17.22
19.74
17.44
in.'
31.38
31.54
31.10
31.10
30.74
30.58
35.16
35.37
34.81
35.02
34.60
34.39
34.46
35.02
35.02
est Ratio
M.
0.254 0.68
0.256 0.65
0.252 0.74
0.238 0.70
0.198 0.72
0.246 0.78
0.312 0.85
0.350 0.93
0.231 0.69
0.218 0.62
0.231 0.75
0.238 0.83
0.220 0.73
0.295 0.84
0.261 0.75
Bul.428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
had the largest d/eratio, 4.67, and failed at a
somewhat lower load than that predicted. In con
clusion, the above findings suggest that the shape
factor of Eq. 34 is applicable whenever beams with
abnormally high d/e' and b/b'ratios are excluded.
In the above comparison, the beams of Series
HB and KB were of composite tileconcrete con
struction. One 5/sin. thickness of tile was included
in the overall dimensions of the beams in calcu
lating the shear strength of the beams. These beams
were made with Btype tile which had but slightly
higher compressive strength than that of the con
crete used. It was seen that these beams failed
at about the predicted load. Beams made with tiles
of higher concrete strength, not included in the
analyses, failed at a somewhat higher load; the
high strength tiles seemed to have acted as a form
of web reinforcement in increasing the load at
failure.
Figure 11 shows the above beams, except Series
A and B, together with results from other investi
gations. The beams of Series A and B were ex
cluded because of the simultaneous high ratios of
d/e and b/b', and as discussed previously, the
shoulders of the beams of Series B were not deep
enough to increase their shear strength. It is seen
Table 38
Tests by Bach and Graf, Heft 12, 1911. SimpleSpan TBeams With BentUp Bars Under Two Symmetrical Concentrated Loads
Group Pt.es Ratio
Ptest
kips P,
25 76.0 1.34
29 92.6 1.63
50 81.9 1.44
31 83.8 1.48
34 86.7 1.53
33 92.6 1.63
47 105.8 1.86
45 86.9 1.53
46 95.2 1.68
36 101.4 1.79
38 109.1 1.92
48 100.7 1.77
49 107.3 1.89
43 90.0 1.58
44 99.9 1.76
40 100.3 1.77
42 104.3 1.84
* Cr = crushing at hooks.
Reference: (22)
Dimensions: b= 19.7; b'V=7.9; D=15.7; d =around 13.4; e=3.9; a= 39.4; a/d=2.94; L=118.1; L'=133.9
Loading: 2 equal loads at %points
Tension Reinforcement: From 4 to 7 plain round bars; A,=about 3.9 in.2; f,= about 47,000 psi
Anchorage: Only hooked bars included
Web Reinforcement: Bentup bars
Reinforcement in Flange: None
Concrete Strength: Average f.' =3440 psi; /' =0.75; f1' = 2580 psi; variation from 8.3 to +7.0 percent
Age at Test: Around 45 days
Number of Beams: Three companion specimens in each group
Calculated Quantities: Ft=0.80; P.=56.8 kips (from Eq. 35)
No. of
Bup
Layers
1
2
3
5
a No. and Area of
Bentup Bars
deg No.sq in.
18 21.91
45 31.79
31.77
45 21.78;21.78
21.78;21.78
21.78;21.78
21.78;21.78
30 21.78;21.78
21.78;21.78
45 10.89;20.88;20.88
10.89;20.88;10.88
10.89;20.88;20.88
10.89;20.88;10.95
30 10.89;20.88;20.88
10.89;20.88;10.95
45 Five times 10.54
Five times 10.54
No. and Area
Straight Bars
No.sq in.
21.92
32.10
32.00
10.35
10.35
10.35
10.35
10.35
10.35
11.25
21.25
11.25
21.18
11.25
21.18
11.25
21.25
Spacing Between
Bends, From
Load (in.)
0
10.8
14.8
2.025.6
027.6
2.025.6
3.917.3
2.010.8
2.021.7
012.89.8
012.88.5
3.910.87.9
3.910.86.5
2.014.49.9
2.014.010.0
1.28.38.15.98.5
1.08.58.15.98.5
Beam P'
psi
III 61 1540
2
III 71
2
III 81
2
Table 39
Tests by Graf, Heft 67, 1931. SimpleSpan TBeams Under Two Symmetrical Concentrated Loads
Reference: (23)
Dimensions: b=53.2; b'= 9.9; D= 23.6; d=21.3; e=3.9; a=59.1; a/d=2.77; L= 177.2; L'= 205
Loading: 2 equal loads at Hpoints
Tension Reinforcement: Ten 0.866in. plain round bars; A.= 6.77 in.2; f,= 46,000 psi; all bars hooked
Reinforcement in Flange: Four 0.28in. long. plain round bars; 0.28in. transverse bars at 4.5 in.,
under loads at 2.5 in.; f, = 48,000 psi
Web Reinforcement: Five long. bars bent up at 45 deg, s=about 10.2 in.; 0.28in. vert. stirrups at 7.1 in.
Concrete Strength: Tests on 7.9in. cubes; reduced to cyl. strength by f,'=0.75 f..'
Age at Test: Around 30 days
Type of r rf,. Pt.es Mode F, P. Ratio Ratio Ratio
Bentup of Eq. 35 Pt Ptt Ptt
Bars % psi kips Fail. kips P. P.. PI
148370 c
1370 7
III 91 1410 k.
2*
* Cr = Crushing at hooks.
1.33 638 231 T
220 T
170 SCr*
165 SCr
209 S,Cr
176 S,Cr
209 S,Cr
182 S,Cr
0.63 85 2.72 1.20 1.11
85 2.59 1.14
82 2.07 0.91
82 2.01 0.88
76 2.75 1.20 0.94
76 2.32 1.02 '
78 2.68 1.18 0.96
78 2.33 1.02
r r,.
% psi
1.15 540
1.15 540
1.15 540
1.50 710
3.52 1650
1.90 890
1.24 580
1.24 580
1.08 510
1.08 510
1.37 640
1.34 630
1.02 480
1.02 480
Mode*
of
Fail.
Cr
Cr
Cr
Cr
Cr
Cr
T
Cr
Cr
T, Cr
T
T, Cr
T
Cr
Cr
T, Cr
T
ILLINOIS ENGINEERING EXPERIMENT STATION
0.6
0.5
0.4
0.3
UMest
0.2
0.I
0
c, (psi)
Fig. 10. Tests by Ferguson and Thompson. Si
that when beams with abnormally large d/e and
b/b'ratios are excluded, Eq. 35 gives satisfactory
agreement with test results. In some beams of Bach
and Graf, there is some doubt about the primary
mode of failure; heavy cracking at the end hooks of
of the tension reinforcement indicated possible
anchorage failure. This might explain why one of
these beams is somewhat low. Beams of Richart and
of Braune and Myers show good agreement with
Eq. 35. Although the beams of Braune and Myers
had a very high b/b'ratio, 7.0, the d/eratio was
rather small, 2.56, and the beams failed according
to Eq. 35.
It is concluded that the shear strength of
simplespan Tbeams without web reinforcement as
normally used in construction can be predicted by
Eq. 35 where the shape factor is computed by
Eq. 34. Beams with abnormally high d/e' and b/b'
ratios are outside the scope of Eq. 35, their shear
strength is lower because the effective width of such
flanges is reduced. No attempt was made, however,
to determine an expression for the effective flange
width. Moreover, Tbeams of such dimensions are
not permitted by the present ACI Code require
impleSpan TBeams Without Web Reinforcement
ments for isolated beams.* In the following section,
it is shown that the use of transverse reinforcement
in the flange effectively counteracts the reduction in
the effective width of the flange and thereby in
creases the scope of Eq. 35. This phenomenon was
also observed for the beams of Braune and Myers
in the present comparison.
15. TBeams With Web Reinforcement
Tbeams considered in this section are analyzed
in Tables 33, 34, 35, 38, and 39. A summary of the
test variables is included in Table 32. The ratio
P/P,, where P, was obtained from Eq. 35, was cal
culated for each beam. This ratio is plotted against
the parameter rf,, in Fig. 12 for beams which failed
in shear. The ratio of web reinforcement was com
puted with respect to the width of the web.
Heft 10 by Bach and Graf reports tests on 81
beams.0s8 The beams were tested in 28 groups, 25
groups of three and 3 groups of two companion
specimens. All beams were reinforced with two ten
sion bars. One group of 3 beams had 2.82 sq in. of
tension reinforcement and failed in tension. The
*Building Code Requirements for Reinforced Concrete (ACI 31851),
American Concrete Institute, Detroit, 1951.
Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
remaining beams were provided with about 3.9 sq
in. of tension steel and failed either in shear or in
bond. Beams with straight unhooked bars failed
at a lower load than similar beams with hooked
bars, apparently in bond. Beams with hooked bars
failed mostly in shear; these beams are analyzed
in Table 33. Figure 12 shows that most of the
beams give good correlation with Eq. 26, originally
derived for rectangular beams. Only two groups of
beams failed at a somewhat lower load than that
predicted. However, photographs of beams after
failure show rather extensive cracking at the end
hooks of the tension reinforcement. Beams with
larger amounts of web reinforcement resisted a
higher load at failure and showed more marked
cracking. It is possible that the two groups with
the highest amount of web reinforcement failed
in bond through excessive bending of the anchorage
hooks.
Heft 67 by Graf reports tests on 8 Tbeams
under two symmetrical concentrated loads.(23)
These beams were provided with transverse rein
forcement in the flanges, and although the flanges
were rather thin and wide, no reduction was noticed
0.6
o0
0.
Mte
A .df
0.
'4
3
st
Ft
2
n
0.57
/05
Z
£
*
in the effective flange width. All beams were rein
forced with the same amount of web reinforcement;
the only variable was the arrangement of bentup
bars. Four different groups of two beams were in
vestigated; the test results are given in Table 39.
Beams of Group 6 were reinforced with regular
bentup bars, the horizontal part of the bends being
carried over the transverse reinforcement in the
flanges. This arrangement of web reinforcement was
the most effective one; the beams failed in tension,
and as seen in Fig. 12, the load at failure was about
20 percent higher than that predicted for shear.
Beams of Group 8 were reinforced with "brought
back" bentup bars; all longitudinal bars were first
taken to the end of the beam, bent up there and
then bent down at the desired spacing to serve as
web reinforcement. The beams failed in shear at a
load slightly higher than the predicted load; some
crushing of concrete was observed at the end hooks
of the "broughtback" bars. Beams of Group 9
were provided with conventional bentup bars ex
cept that the bends had no horizontal extension at
the top of the beam. This type of web reinforce
ment was about as effective as that of Group 8.
/000 2000 300 f (ps) 4000 5000
Fig. 17. Failure Moment as Function of Concrete Strength. SimpleSpan TBeams Without Web Reinforcement
bU600
0
00 +15%
0 0
0
0
0 0
0
+ 15%
o ^ * 
0
15%
* Bach and Graf, Heft /0
* Braune and Myers
o Richart, Series 1922
o Ferguson and Thompson
except Series A and B
ILLINOIS ENGINEERING EXPERIMENT STATION
2.8
2.6
2.4
2.2
2.0
Ptest
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
500 600 700
Tension failure
/
+15 %
2rf,
1/+ //
03
* Bach and Gra/ Heft /0, 3 beams each
o Graf, Heft 67, 2 beams each
Pw/P. = 1 ±+ ,10
10
(26)
Fig. 12. Effect of Web Reinforcement on Strength of
SimpleSpan TBeams Failing in Shear
Beams of Group 7 were reinforced with loose,
"floating" type of inclined bars, hooked on both
ends. These beams failed at a somewhat lower load
than that predicted, indicating that this type of
web reinforcement was not fully effective.
Tbeams tested by Richart were provided with
vertical stirrups.(2) As seen in Table 35, all these
beams failed in tension. Beams tested by Braune
and Myers had both vertical stirrups and bentup
where P, is determined from Eq. 35 and r from the
following equation:
As
b's sin a
(27a)
For bentup bars, there is some danger of a prema
ture failure because of cracking and crushing of
concrete at the hooks. This failure can be prevented
by using sufficiently large hooks and, especially, by
using transverse reinforcement in the flanges of the
beam.
0 /00 200 300 400
rfy, (psi)
bars as web reinforcement."(9 These beams failed
in tension also; see Table 34. However, the web re
inforcement was sufficiently effective to permit high
ratios of P/P, at failure, the highest ratio being
3.19.
Beams reported by Bach and Graf in Heft 12
were reinforced with bentup bars.(22) A total of 87
beams were tested; the tension reinforcement con
sisted of from 4 to 7 bars of the same total area,
and no transverse reinforcement was used in the
flanges. All beams with unhooked longitudinal bars
failed at a lower load than similar beams with
hooked bars, evidently in bond. Beams with hooked
longitudinal and bentup bars are analyzed in
Table 38. It is seen that despite the large ratios of
web reinforcement only a few beams failed in ten
sion. Some other beams might have had yielding of
the lower layer of the tension reinforcement at fail
ure. In most beams, failure was initiated by exces
sive cracking and crushing of the concrete at the
hooks. The most effective arrangements of bentup
bars can be found from Table 38.
It was concluded that the shear strength of
simplespan Tbeams with web reinforcement can
be determined by the same expression as that for
rectangular beams:
V. RESTRAINED BEAMS UNDER SYMMETRICAL CONCENTRATED LOADS
16. Modes of Failure*
Simplespan beams under concentrated loads
fail at the location of an applied load, at the sec
tion of maximum shear and maximum moment.
Shear stresses combined with flexural stresses are
instrumental in producing a main diagonal crack;
after this crack has formed, the beam fails in
compression.
In restrained beams, shear and moment con
ditions permit, in general, the formation of three
main diagonal cracks as shown in Fig. 13. The
A
P, I
2/ 1
'p,  p?
ID
4h
17Z
Possible Location of Main Diagonal Cracks
r
P
Shear Diagram
Moment Diogram
Fig. 13. Restrained Beam Under Symmetrical Concenlrated Loads
beam may fail at any of these three cracks, depend
ing on the magnitude of shear and moment at the
section under consideration and on the arrange
ment of both longitudinal and web reinforcement.
Although the static moment is the same on both
sides of section A, the magnitude of shear can be
different in spans f and g. The crack at the section
of greater shear forms first, and for small shear
ratios it is even conceivable that the beam fails at
that section before the other crack has formed. Al
* The modes of failure of restrained beams, and particularly the for
mation of two main diagonal cracks in the region in which the moment
changes sign, were first described by E. Hognestad in an unpublished
report, "Shear Failures in Concrete Beams," Department of Theoretical
and Applied Mechanics, University of Illinois, 1951.
though span g has constant shear, the moments can
be different at sections A and B. Depending mainly
on the relative magnitudes of moment, either one
or two cracks form. The crack at the larger moment
develops first and the beam fails, in general, at that
crack.
Various modes of failure are discussed below.
Special emphasis is placed on the arrangement of
reinforcement. It is assumed that span f has suffi
cient reinforcement so that the beam fails in span g.
a. Continuous Top and Bottom Reinforcement.
A freebody diagram for this arrangement of longi
tudinal reinforcement is shown in Fig. 14. It is as
sumed first that only one diagonal crack forms
before failure. Figure 14 shows crack 2 and assumes
that shear is resisted exclusively by the compres
A B
Fig. 14. Continuous Top and Bottom Reinforcements.
Restrained Beam With No Bond Failure
sion area of the concrete. The top longitudinal
reinforcement is in tension at crack 2 and in com
pression at section B. If there is no possibility for
bond failure between these two sections, e.g., if span
g is long relative to the effective depth of the beam
and bars of good bond characteristics are used, a
shear failure similar to that in simplespan beams
is expected to take place. Thus, Eq. 18 can be em
ployed directly to determine the shear strength of
such beams without web reinforcement and Eqs. 18
and 28 to determine that of such beams with web
reinforcement. If two cracks are present and bond
failure does not occur between them, the mode of
failure is unchanged and the shear capacity of the
beam can be determined by the same equations at
the section of maximum moment.
I
I
I
I
f"
ILLINOIS ENGINEERING EXPERIMENT STATION
If, however, bond is destroyed between the re
inforcing bars and the concrete in the middle por
tion of span g, Eqs. 18 and 28 no longer represent
the shear strength of the beam. Bond failures are
likely to take place when span g is relatively short.
Then only a small distance separates the diagonal
crack from either section A or B, and a change in
stress from tension to compression in the reinforce
ment must take place over this length. If a bond
failure results from the high bond stresses in this
region, both the top and bottom reinforcing bars
are in tension as shown in Fig. 15 for one crack and
in Fig. 16 for two cracks at failure. For simplicity,
it is assumed that the whole tensile force, TA or TB,
is carried through the middle portion of the beam.
This redistribution of internal forces is very un
favorable to the shear capacity, and the beam fails
at a much lower load than it would if no "com
pressive" reinforcement was provided.
An approximate expression for the shear
strength of a beam with both top and bottom re
inforcement in tension can be derived as follows:
C = kik3kdbfc'
C = Ta + TB
7,
t V
T,
Section B
Fig. 15. Continuous Top and Bottom Reinforcement. Bond
Destroyed in Restrained Beam With One Crack
Fig. 16. Continuous Top and Bottom Reinforcement. Bond
Destroyed in Restrained Beam With Two Cracks
It is further assumed that the factor k, as derived
for simplespan beams and given by Eq. 22 remains
valid for restrained beams. The quantity k is de
termined again by the "straight line" theory. This
can be done as follows:
C = (1/2) bkdfc
TB = pbdn k f
M. = Cd (1  k2k,)  TAtd
(37)
From Eqs. 36, 39, 40, and 41:
Equation 37 determines the moment at shear
failure. However, there are two unknowns, k, and
TA, which must be evaluated before the shear mo
ment M, can be expressed quantitatively. If the
tensile force TA is determined by assuming that the
moment arm is the same for both sections A and B,
the following relationship can be written:
k = V/ (pn)2 + 2pon  pon
where
p= p(1+ MA
(43)
The shear moment as given by Eq. 37 can now
be rewritten as:
TA/TB = MA/MB
(38)
From Eqs. 36 and 38:
where
C
TA =M 
MB
MX "+1
(39)
= Ak (0.57 4.5f'
bd2f ' =5 10, )
A = 1 
(M + 1) (1  kwk,)
(44)
(45)
Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
k is given by Eq. 42
k2 is taken as 0.45
k, is given by Eq. 23.
Equation 44 determines the shear strength of a
restrained beam which fails at section B after bond
has been destroyed from this section to crack 2 so
that both the top and bottom reinforcing bars are
in tension. This equation can be used for any sec
tion provided that the subscript B refers to the
section under consideration and the subscript A to
the adjacent section from which the tensile force
TA is carried through to the section B. It was de
rived by assuming that the longitudinal reinforce
ment was continuous throughout the entire length
of the beam and that the whole tensile force at one
section was carried through to the other section.
This is a conservative estimate, since it is likely
that in some cases a part of the tensile force is re
sisted by partial bond which can exist even though
the reinforcing bar might be slipping in the entire
region from section A to B. If the actual ratio be
tween the top and bottom tensile forces can be de
termined for the section at failure, the actual ratio
Ts/TA should be substituted for MB/MA in Eqs. 43
and 45.
The first crack in span g will form at the section
of maximum moment. If a beam fails in shear after
only this crack has formed as shown in Fig. 15,
redistribution of the internal forces due to bond
failure has taken place at section B, whereas the
bottom longitudinal reinforcement is still in com
pression at section A. Although both the top and
bottom reinforcing bars are in tension at section B,
diagonal cracking has not reduced the compression
area, and the beam cannot fail in shear at that sec
tion. Consequently, section A is the critical section,
and the shear strength of the beam is determined
by Eq. 18 at the section of maximum moment. If
two cracks are present at failure and full redistri
bution of the internal forces has taken place as
shown in Fig. 16, Eq. 44 is applicable at section A
as well as at section B. The shear strength of the
beam is determined by Eq. 44 at the section of
maximum moment. Under certain conditions it is
conceivable that despite the formation of two
cracks only partial bond failure and redistribution
of the internal forces has taken place. This may be
the case if, for example, the moment at section A
is much greater than the moment at section B.
Then the bond stresses are much higher in the top
reinforcement than in the bottom reinforcement,
and local bond failure may take place only along
the top longitudinal bars. The shear capacity of
the beam is given by Eq. 44 at section B and by
Eq. 18 at section A, the section of maximum
moment. The beam fails at the section of the small
est shear strength. However, since the conditions
for partial redistribution of the internal forces
cannot be determined in advance, it is more conser
vative to assume full bond failure and full redistri
bution whenever two cracks are present at failure.
The validity of Eq. 44 is checked against test
results in Section 17.
b. Straight Bars Cut Off Beyond the Theoreti
cal Point of Contraflexure. A diagram for this ar
rangement of longitudinal reinforcement is shown
in Fig. 17. When the length of embedment, both
x and y, is sufficient to prevent a bond failure, it is
expected that the shear strength of a restrained
beam can be determined by Eqs. 18 and 28. How
ever, when the length of anchorage is small or re
inforcing bars of poor bond characteristics are used,
the failure may be a sudden stripping out of the
reinforcement and a complete destruction of the
beam. Failures of this type have been reported by
Richart and Larson(25' and by Moody.(12) Figure 18
shows a sketch of a beam in this category after
failure.
c. Beams With All Bars Bent Up. Figure 19
shows this arrangement of longitudinal reinforce
ment which appears to be an effective one since it
prevents any possibility of bond failures and uses
the bentup bars as web reinforcement. When the
bars are bent up at some distance from the support,
P,+ . cs  n r  dP o
Fig. 17. Straight Bars Cut Off Beyond Point of
Contraflexure. Restrained Beam
Fig. 18. Stripping Type of Bond Failure. Restrained Beam
ILLINOIS ENGINEERING EXPERIMENT STATION
it seems advisable to use a few stirrups between
the first bend and the load point. The shear
strength of such beams is determined by Eqs.
18 and 28. While such an arrangement of reinforce
ment is very effective, care must be taken with the
design and fabrication of bends. Richart and Lar
son 25) observed frequent crushing of the concrete
at the bends after yielding of reinforcement.
d. Beams With Both BentUp and Straight
Longitudinal Bars. A diagram of such a beam is
shown in Fig. 20. This type is similar to that dis
cussed in paragraph c. When bond failures are pre
vented, shear capacity is given by Eqs. 18 and 28.
I'
I I I
II I
I I II
Ii
1 P  g
Fig. 19. Restrained Beam With All Bars BentUp
Pr i
Fig. 20. Restrained Beam With BentUp and Straight Bars
When, however, numerous bars are left straight, a
premature bond failure similar to that discussed
in paragraph b is possible.
17. Test Data on Restrained Beams
The only tests on restrained beams reported in
the literature are those by Richart and Larson(25)
and by Moody.<12) These tests are analyzed and the
validity of previously derived equations checked
in the following paragraphs.
a. Tests Reported by Richart and Larson.
Richart and Larson reported tests on 59 beams, 17
in Series 1911 and 42 in Series 1917. Beams of
Series 1911 failed either in tension or in bond, and
the concrete strength was not recorded for all
beams. Thus, very little information is available
about the shear strength of these beams, and they
are not included in the present analysis.
Beams of Series 1917 were designed to investi
gate the effect of various arrangements of bentup
bars in span g. The type of beam is shown in
Fig. 21, and Table 40 gives the arrangement of
reinforcement for each individual beam. All beams
Shear I I P/4 I
MM, 8P
Momenl
Fig. 21. Typical Restrained Beam of Richart and Larsen
had eight 5/8in. round plain bars over the sup
port A. The overhanging portions of the beam,
span f, were heavily reinforced so as to produce
failures in span g. Most beams of Series 1917 failed
in tension. There are, however, a few beams which
throw some light on the modes of shear failure as
discussed under paragraphs b, c, and d in the
previous section.
Beams 380 represent beams with straight longi
tudinal bars cut off beyond the point of contra
flexure. From the description and photographs of
failure it appears that these beams failed in bond
by stripping off the concrete above the bars at
failure as shown in Fig. 17. This premature bond
failure cannot be predicted by any of the shear
strength equations of this report; it is a matter of
bond characteristics of the reinforcing bars.
The rest of the beams were of the types dis
cussed under paragraphs c and d in Section 16,
with some or all of the longitudinal bars bent down
in span g. Beams 388, 389, and 400, which had four
of the eight bars at support A bent down in one
layer, appear to have failed in bond after yielding
of the reinforcement. They correspond to a bond
failure of the type discussed in Section 16, para
graph b. Beams which had four or more bars bent
down in two or three layers failed in tension with
out any tendency for stripping of the concrete at
the straight bars. However, crushing of the concrete
inside the bends was frequently the cause of final
failure. Furthermore, diagonal cracks were ob
served to intersect the reinforcement at the bends.
These two phenomena were often responsible for
a sudden sheartype final collapse of the beams.
This occurred, however, well after the yielding of
reinforcement. The ratios P/Pf in Table 40 were
computed by Eq. 29, using f1= f, and kl2/kkc3 = 0.5
I
Bul.428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
Beam f,'
psi
380.1 3060
2 3665
400.1 3158
2 3165
382.1 3315
2 2748
386.1 2870
2 3525
391.1 2892
2 3495
392.1 2818
2 2795
383.1 3082
2 2950
385.1 2985
2 3362
387.1 3398
2 2965
388.1 3260
2 2970
389.1 3210
2 3102
390.1 2905
2 2735
393.1 3155
2 2325
394.1 3145
2 3355
395.1 3120
2 3015
384.1 3080
2 3442
399.1 3352
2 2810
396.1 3410
2 3362
397.1 3235
2 2682
398.1 2682
2 2990
381.1 3070
2 3385
Table 40
Tests by Richart and Larsen, Series 1917. Restrained Beams With BentUp Bars
Reference: (25)
Dimensions: b= 8; d= 15
Spans: f32 in.; g  48 in.; h =48 in.; L = 216 in.
Loading: P i P/4; Psi P/4; MA SP in.k; MB=4P in.k
Long. Reinforcement: Eight %in. round plain bars at support, from 4 to 8 bars at midspan
(See Fig. 21), f,=about 37,600 psi; p=about 1.95%
Web Reinforcement: %in. round plain vertical stirrups; f~.=45,000 psi
Age at Test: 60 days
Bentup Bara
No. of Bars a
Total Layers deg
4 i 22"
5 2 22
6 3 32.5
6 3 32.5
6 3 32.5
6 3 32.5
5 2 32.5
4 2 32.5
4 1 32.5
4 1 32.5
8 4 45
6 3 45
6 3 45
6 3 45
6 3 45
5 2 45
6 2 45
6 2 45
6 2 45
6 2 45
Stirrups
No. as
in.
2 4
5 12
5 16
kips
102.8
104.0
151.0
149.7
175.5
183.7
188.2
188.0
187.8
172.0
146.4
176.4
183.2
181.5
176.3
190.0
182.3
168.4
173.8
143.2
174.0
169.0
181.2
186.0
164.0
170.0
172.3
185.4
180.6
167.0
176.7
178.6
176.1
184.9
182.6
165.0
192.5
179.6
168.0
173.1
165.0
124.1
Mode
of
Fail.
DT
DT
DT
DT
T,DT
T,Cr,DT
T,Cr
T,Cr
T
T,DT
DT
T,DT
T
T,Cr,DT
T,Cr
T
T,Cr,DT
T,DT
T,B,Cr
B,DT
T,DT
T
T,Cr
T,Cr
T,Cr
T,DT
T,Cr
T,Cr,DT
T,Cr,DT
T,Cr,DT
T,Cr,DT
T,Cr,DT
T,Cr,DT
T,Cr
T,Cr
T,Cr
T,Cr
T,Cr
DT
T,DT
Cr,DT
DT
P.
Eq. 18
kips
127.3
139.6
129.5
129.8
133.2
120.0
123.1
136.9
123.7
136.4
121.9
121.2
128.0
125.3
125.4
133.5
133.9
124.8
131.2
125.0
131.4
128.8
124.9
121.0
129.4
108.5
129.6
134.1
129.2
126.6
128.7
136.3
134.0
122.1
135.3
134.1
132.1
119.2
117.9
125.6
127.7
134.3
in the calculation of the flexural capacity P,. Since
these ratios are greater than one, the reinforcement
was presumably stressed in the workhardening
region at final failure.
The main variables intended to be investigated
were the angle of inclination and the number and
spacing of bends. Even the largest spacing of bent
up bars gave a value of r which was sufficient to
prevent shear failures. A few shear failures were
obtained, however, when the first bar was located
so far from support A that a diagonal crack could
form without intersecting any inclined bars. Such
failure was observed in Beam 381.2, where the first
bend was 24 in. from the support. As seen in Table
40, this beam failed before yielding and at a ratio
P/Ps equal to 0.92. Thus, the load at failure was
governed by Eq. 18. The companion specimen failed
in tension at a higher load, however. Beams 398,397,
and 396 were similar to Beams 381 except that they
were provided with vertical stirrups as additional
web reinforcement. All of these beams failed in
tension, although the final failure of Beam 398.1
was a sudden break, called diagonal tension by
Richart and Larson.
In conclusion, it can be said only that the be
havior and strength of the restrained beams with
bentup bars in these tests is not inconsistent with
the behavior of simplespan beams as predicted by
Eqs. 18 and 28. Bond failures are outside the scope
of these equations; beams must be designed so that
the possibility of destroying the bond is eliminated.
Care must be taken in the design of bends to avoid
crushing of the concrete inside the bends in the
reinforcing bars.
b. Tests Reported by Moody. Moody reports
tests on 96 restrained beams in five series.(12) The
dimensions of the beams and the arrangement of
reinforcement and loads are shown in Fig. 22. All
beams were provided with equal amounts of top
and bottom longitudinal reinforcement, four bars
placed in two layers. In all but three beams
the four top bars and the two lower bottom bars
were continuous throughout the total length of the
beam; the other two bottom bars were cut off 4 in.
Ratio
Ptet
P.
0.81
0.75
1.17
1.15
1.32
1.53
1.53
1.37
1.52
1.26
1.20
1.45
1.43
1.45
1.41
1.42
1.36
1.35
1.32
1.15
1.32
1.31
1.45
1.53
1.27
1.56
1.33
1.38
1.40
1.32
1.37
1.31
1.31
1.51
1.35
1.23
1.46
1.50
1.42
1.38
1.29
0.92
Ratio
Pts..t
P1
0.71
0.70
1.04
1.03
1.19
1.27
1.31
1.27
1.31
1.17
1.02
1.23
1.26
1.26
1.23
1.31
1.26
1.12
1.20
1.04
1.19
1.16
1.24
1.26
1.06
1.23
1.14
1.26
1.24
1.15
1.20
1.28
1.22
1.29
1.24
1.12
1.20
1.27
1.19
1.19
1.14
0.88
ILLINOIS ENGINEERING EXPERIMENT STATION
B
7
T i
LLQ
0frf~vI f
Series II//
I7
LLI?
0
Fig. 22. Restrained Beams of Moody
from the supports. In the remaining three beams
the longitudinal reinforcement was cut off at the
supports and the inner load points in accordance
with the present ACI Code. The test variables in
cluded the percentage of longitudinal reinforcement,
the concrete strength, the dimensions of the beams,
and the magnitude of moments and shear as re
flected by different arrangements of loads. Sixty
one beams were tested without web reinforcement,
29 with vertical stirrups, and 6 with 45deg
stirrups.
Beams with no web reinforcement are analyzed
in Tables 41, 42, and 43. The beams of Series I, II,
and IV had continuous longitudinal reinforcement
and equal moments at sections A and B. From
crack patterns and strain measurements recorded
for Beam I2c, it was observed that bond was de
stroyed in span g so that both the top and bottom
reinforcing bars were stressed in tension. The beams
failed after developing, in general, two main diag
onal cracks between sections A and B. Conse
quently, Eq. 44 should apply at both these sections.
In order to apply Eq. 44, the weakest critical sec
tion must be determined first. Everything else re
maining the same, the shear capacity of a section
is determined by the square of its effective depth.
This distance was always 0.25 in. larger at section
B than that at section A, indicating that A was
the critical section. However, it is recalled that
there were some differences in the arrangement of
the longitudinal reinforcement at these sections.
This might have a larger effect on Eq. 44 than the
small difference in the values of d. From the con
dition of equal moments and entirely continuous top
reinforcement it can be concluded that TA = TB
at section B. This assumes that the total tensile
force is carried through from section A to section B
so that Eq. 44 can be used with TA/TB 1 at
section B. At section A, however, only half of the
bottom reinforcement is continuous. After bond is
destroyed, it is likely that the stress in the continu
ous bars is increased relative to its magnitude be
fore bond failure. In order to transmit the total
force TA to section B, the cutoff bars must be com
4P/14
_T
6.86P
_L
" 5.33P \ \P/3
SERIES V
T_
15
Kl
0
Q  >
ri
c
Q
' kQ
Series ki
J
*»f
#i
4)
Bul.428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
Table 41
Tests by Moody, Series I, 1953
Restrained Beams Without Web Reinforcement
Reference: (12)
Critical Section: Inner Loadpoint
Calculation of Ultimate Moment: Eq. 44; TA/TB=1; p.=2p
Loading: See Fig. 22; MB 5.33P
Dimensions: bo= ; a= 12; t= 0.729; g
Beam f/' p Ptest k
kips
1950 Series
77 0.623
87 0.615
75 0.673
95 0.664
94 0.641
89 0.715
101 0.696
100 0.587
89 0.591
120 0.631
110 0.631
115 0.672
120 0.685
115 0.578
100 0.580
145 0.629
110 0.630
130 0.664
130 0.670
1952 Series
90 0.405
89 0.486
99 0.550
105 0.469
109 0.541
107 0.467
105 0.534
118 0.466
128 0.527
133 0.572
130 0.628
140 0.659
Table 42
Tests by Moody, Series II and IV, 1953
Restrained Beams Without Web Reinforcement
Reference: (12)
Critical Section: Inner load point
Calculation of Ultimate Moment: Eq. 44; TA/TB= 1; p.= 2p
Loading: See Fig. 22
=I 2.67 Beam /' p Patt k A M.t.t
A M.t Ratio b/fkAA
bdf/'kA Mt^t psi % kips
M. Series II; b=7; d. 21; t 0.845; MAf=5.33P; g/d =1.52
17a 2650 2.15 188 0.572 0.518 0.414
0.578 0.451 0.98 b 3000 170 0.561 0.519 0.337
0.579 0.462 1.04 18a 2170 2.72 220 0.626 0.510 0.523
0.571 0.436 0.94 b 2700 " 180 0.609 0.513 0.364
0.572 0.484 1.08 19a 3030 3.46 241 0.642 0.508 0.422
0.575 0.356 0.89 b 3240 " 219 0.637 0.508 0.361
0.565 0.509 1.09 20a 2890 4.25 235 0.679 0.502 0.412
0.568 0.455 1.04 b 2960 249 0.678 0.502 0.427
0.582 0.359 0.95 IIa 3820 0.54 130 0.322 0.547 0.324
0.581 0.340 0.88 b 3720 0.84 145 0.395 0.540 0.316
0.577 0.429 1.11 c 4040 1.20 168 0.446 0.534 0.302
0.577 0.396 1.02 d 3440 1.63 210 0.506 0.527 0.396
0.571 0.349 0.96 Series IV; b=7; d= 12; t0.729; MB 6.86P; g/d= 4.0
0.583 0.377 1.06 IVg 3390 0.95 63 9.419 0.601 0.502
0.583 0.313 0.91 h 3750 1.47 70 0.483 0.595 0.442
0.577 0.441 1.25 i 3490 2.10 68 0.548 0.587 0.412
0.577 0. 41 0.95 3600 2.86 83 0.598 0.581 0.451
0.577 0.3 1.05 k 3630 3.76 88 0.644 0.576 0.444
0.572 0.350 1.05 1 3920 4.76 81 0.679 0.571 0.363
0.572 0.386 1.07
0.603 0.440 1.18
0.591 0.462 1.12
0.587 0.488 1.16
0.596 0.409 1.16
0.588 0.469 1.18
0.596 0.399 1.17
0.589 0.402 1.08
0.596 0.429 1.28
0.590 0.431 1.26
0.584 0.353 1.17
0.578 0.388 1.11
0.574 0. 330 1.09
pletely inactive and the continuous bars must resist
twice their former stress. However, stress measure
ments in Beam I2c showed that although the stress
increased in the continuous bars, it never reached
more than about 120 percent of its former value.
This indicates, using the proper subscripts, that the
ratio TA/TB is less than one at section A. The
smaller is this ratio, the larger is the factor A in
Eq. 44 and, consequently, the shear strength of
the beam. Thus, section B must be considered as
the critical section for Eq. 44, using MA/MB
TA/TB = 1.
Beams of Series I, II, and IV are analyzed in
Tables 41 and 42, and the quantity M/bd2Jc'kA is
plotted against fc' in Fig. 23. It is seen that, in
general, test results give satisfactory agreement
with Eq. 44. Thus, it appears that the assumptions
made in deriving this equation are essentially cor
rect and that this equation can be used to deter
mine the shear strength of restrained beams with
continuous reinforcement whenever the shear fail
ure takes place subsequent to destruction of bond.
This type of failure is still a primary shear failure
since the destruction of bond in the high bond stress
region does not in itself constitute failure of the
beam; it only causes a redistribution of the internal
Ratio
M.
0.92
0.78
1.11
0.77
0.97
0.85
0.94
0.98
0.81
0.78
0.78
0.95
1.20
1.10
1.00
1.10
1.09
0.92
Table 43
Tests by Moody, Series VI and V, 1953
Restrained Beams Without Web Reinforcement
Reference: (12)
Critical Section: Support
Calculation of Ultimate Moment: Eq. 18
Loading: See Fig. 22
Beam Pc' p Pt.t k k+np' Mt,. Ratio
bdf.'(kc+np') M..
psi % kips M.
Series VI; b=7; d=11.75 t=0.817; p'0.5p; MA=6.4P; g/d2.73
Via 4090 0.95 77 0.300 0.335 0.372 0.96
b 4160 1.47 129 0.351 0.406 0.505 1.31
c 3580 2.10 110 0.401 0.483 0.421 1.03
d 3900 2.86 118 0.435 0.543 0.369 0.93
e 4120 3.76 128 0.467 0.606 0.339 0.88
f 5570 2.10 140 0.383 0.455 0.365 1.14
g 5530 2.86 130 0.422 0.519 0.300 0.94
h 5300 3.76 155 0.457 0.586 0.330 0.99
i 6020 4.76 146 0.483 0.641 0.251 0.84
Series V; b = 7; d= 11.75; p'=0; MA =5.33P; g/d=2.73
Vb 3770 1.47 64.0 0.379 ..... 0.247 0.62
d 3600 2.86 76.5 0.484 ..... 0.242 0.59
f 3380 4.76 74.5 0.574 ..... 0.212 0.51
forces, so that the new combination of the tensile
forces at a certain section requires a larger com
pressive force than before. For Moody's beams the
new compressive force is about twice as large as
that before the destruction of bond. The greatly in
creased compressive force leads to a lower shear
strength since the capacity of the compressive zone
of the beam is but little greater than that for
simplespan beams. Thus the factor A of Eq. 44
can be considered as a reduction factor for re
strained beams which fail after local bond failure.
In deriving Eq. 44 it was assumed that the
entire tensile force at once section is transmitted to
the adjacent section. This assumption is, in general,
a conservative estimate since some of the tensile
force is probably resisted by partial bond. Tables
41 and 42 show that the ratio Mtest/M, increases
as the g/dratio increases or as the size of the rein
ILLINOIS ENGINEERING EXPERIMENT STATION
Fig. 23. Beams of Moody, Series I, II, and IV. Restrained Beams Without Web Reinforcement
forcing bars, as indicated by the percentage of
reinforcement, decreases. In both cases the relative
importance of partial bond is more pronounced and,
consequently, not all of the tensile force is trans
mitted from one section to the adjacent section.
The true ratio TA/TB is thus smaller than that ob
tained from the bending moments and the shear
strength of the beams is thereby increased. How
ever, as seen in Tables 41 and 42, the increase in
the shear capacity was rather small even for the
largest value of g/d and the smallest size of rein
forcing bar used in the tests. Furthermore, beams
of Series II which had the smallest value of g/d
fall even somewhat low in Fig. 23.
The limits of applicability of Eq. 44 cannot be
determined from Moody's tests. For the beams
shown in Fig. 23, the g/dratio varied from 1.52 to
4.0. All of these beams failed after redistribution
of the internal forces. Thus, it appears that local
bond failures are possible with g/dratios larger
than four.
Beams of Series V are analyzed in Table 43.
These beams had their longitudinal reinforcement
cut off at the supports and the inner load points.
Failure took place by a sudden stripping out of the
longitudinal reinforcement as discussed in Section
16, paragraph b. The beams were analyzed by Eq.
18 with the support as the critical section, and the
load at final bond failure was found to be about
onehalf the theoretical shear capacity.
Beams of Series VI had unequal bending mo
ments, MA being twice Ms. The beams failed, in
general, after developing only one main diagonal
crack at section A. Thus it is likely that local bond
failure had taken place only in the top reinforce
ment, so that both the top and bottom bars were
in tension at section B, whereas the bottom bars
were still in compression at section A. This possi
bility was illustrated in Fig. 15. Consequently, the
shear strength of these beams should be governed
by Eq. 18 at section A, and the beams are analyzed
in Table 43 accordingly. The quantity M/bd2fc
(k + np') is plotted against f,' in Fig. 24, and it is
seen that there is good agreement between the
measured and calculated moments.
Beams with web reinforcement are analyzed in
Tables 44 and 45. For beams of Series I and IV
the quantity M/Ms is plotted against rfw, in Fig.
Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
'c (ps//
Fig. 24. Beams of Moody, Series VI. Restrained Beams Without Web Reinforcement
25. Beams in both these series had equal moments simplespan beams. Beams with vertical stirrups
at sections A and B, and the shear moment M, was were tested in two different groups, 6 beams in 1950
calculated by Eq. 44. Figure 25 shows that the and 10 beams in 1952. Beams of 1950 had the
beams of Series I which had 45deg stirrups give closed ends of the stirrups placed toward the lower
very good agreement with Eq. 28, derived for face of the beam, whereas the beams of 1952 had
Beam // p
psi %
lOa 3070 4.76
b 2810
lla 3560
b 3180
12a 4000
b 3220
Is 3470 4.76
t 3700
u 3740 2.86
v 3580
w 4210
x 3830
y 4790 4.76
z 4850
a 5070
P 5130
13a 3460 4.76
b 2860
14a 3510
b 3600
15b 3710
16a 3610 4.76
b 3240
Table 44
Tests by Moody, Series I, 1953. Restrained Beams With Web Reinforcement
Reference: (12)
Critical Section: Inner load point
Calculation of Ultimate Moment: Eqs. 44, 28; TA/TB = 1; po=2p
Loading: See Fig. 22; MBs= 5.33P
Web Reinforcement: Stirrups of inter, grade deformed bars
Dimensions: b= 7; d= 12; t=0.729; g/d=2.67
WEB REINFORCEMENT Ptet k A
Size s r f/, rf.,
No. in. % ksi psi kips
Vertical Stirrups; 1950 Series
3 6 0.52 47.3 246 163 0.694 0.568
138 0.700 0.568
4 0.95 44.0 418 190 0.685 0.570
174 0.692 0.569
5 1.47 41.2 606 190 0.678 0.571
159 0.691 0.569
Vertical Stirrups; 1952 Series
5 5 1.72 47.6 819 220 0.686 0.570
4 2.14 47.6 1019 240 0.682 0.570
3 6 0.52 53.8 280 160 0.596 0.582
3 4.5 0.70 53.8 377 170 0.599 0.581
4 6 0.95 45.8 435 180 0.589 0.582
5 6 1.47 47.6 700 217 0.594 0.582
3 6 0.52 53.8 280 220 0.669 0.578
3 4.5 0.70 53.8 377 222 0.668 0.578
4 6 0.95 45.8 435 260 0.666 0.573
5 6 1.47 47.6 700 279 0.665 0.573
45deg Inclined Stirrups
3 6 0.74 47.3 350 185 0.687 0.570
170 0.699 0.568
4 1.35 44.0 594 250 0.686 0.570
240 0.684 0.570
5 2.09 41.2 861 304 0.683 0.570
TBeams; Vertical Stirrups; b = 23, b'=7, d  11.75, e=4; t=0.839
Critical Section: Support; Eqs. 44, 28; TA/TB=0.5; p.=0.0714
5 6 1.47 41.2 606 271 0.636 0.675
" 282 0.643 0.675
M.
Eq. 44
kin.
527
500
574
539
609
542
566
586
525
514
554
531
661
664
667
668
566
505
570
577
587
Ratio
M.
1.65
1.47
1.76
1.72
1.66
1.57
2.07
2.18
1.62
1.76
1.73
2.18
1.77
1.78
2.08
2.23
1.74
1.79
2.33
2.22
2.76
611 2.36 1.07
576 2.61 1.18
ILLINOIS ENGINEERING EXPERIMENT STATION
Table 45
Tests by Moody, Series IV and II, 1953. Restrained Beams With Web Reinforcement
Beam ft
psi
IVm 2860
n 3710
o 3420
21a 3560
b 3640
22a 3000
b 2710
23a 3230
b 3160
IHe 3420
f 3330
the closed ends always in the compression zone.
Beams of 1950 give good agreement with Eq. 28
except for the two beams with the largest amount
of web reinforcement. These beams were observed
to split longitudinally along the reinforcement and
the stirrups did not reach yielding at failure. It is
possible that longitudinal splitting destroyed the
anchorage of stirrups so that they were unable to
develop their full effectiveness. In the beams of
1952, longitudinal splitting took place on a more
3.5
3.0
2.5
pn
0.5
oC
0 200 400
bUU
rfyw(psi)
800
/OUU /200
Fig. 25. Beams of Moody, Series I and IV. Restrained
Beams With Web Reinforcement
A M. Ratio
Eq. 44 Mtet
kin. M.
.568 504 1.99 1.28
.570 586 2.32 1.24
.570 562 2.66 1.11
516 1373 1.20 0.80
516 1387 1.09 0.73
514 1247 1.28 0.70
513 1169 1.32 0.72
515 1301 1.23 0.56
514 1284 1.45 0.66
516 1344 1.55 0.62
515 1323 1.37 0.48
restricted scale, and then only in the region where
the stirrups were openended. This suggests that
longitudinal splitting will not occur if the rein
forcing bars are tied together in the transverse
direction. Beams of 1952 give good agreement with
Eq. 28 except for two beams with very high values
of rfy,. It is noticed, however, that at the value
of rfy, at which the beams of 1950 fell below the
predicted ultimate moment, the beams of 1952 still
agree with Eq. 28. Thus, the anchorage of stirrups
was more effective in Series 1952 than in Series
1950; only with very high values of rfy, did the
vertical stirrups not develop their full strength at
failure.
Among the beams of Series I two beams were
provided with a 23 by 4in. flange. Since the
flange area increases the compression area of the
concrete at section B, the beams are analyzed for
section A as the critical section in Table 44. In line
with the previous discussion about the effect of
cutting off onehalf of the bottom bars at the sup
ports, the ratio TA/TB must be less than one at that
section. The beams were analyzed with TA/TB
equal to 0.5, or only onehalf of the tensile force
at the inner load point transmitted to the section
at the support. The use of this ratio gave satis
factory agreement with Eq. 28, the loads at failure
being 7 and 18 percent more than the predicted
loads. If the ratio TA/TB had been taken larger
than 0.5, the calculated ultimate moment would
have been still smaller.
Three beams of Series IV were provided with
vertical stirrups. These beams were included in
Table 45 and Fig. 25. The test moments were found
to be from 11 to 28 percent larger than the calcu
lated moments. Beams of Series IV had the largest
g/dratio used in these tests, g/d = 4. Since the
beams without web reinforcement in this series had
Reference: (12)
Critical Section: Inner Load Point
Calculation of Ultimate Moment: Eqs. 44, 28; TA/TB= 1; p.=2p
Loading: See Fig. 22
Web Reinforcement: Vertical stirrups of inter, grade deformed bars
WEB REINFORCEMENT Ptet k
Size a r f/ rf,,,
No. in. % ksi pali kips
Series IV; b 6= 7; d = 12; t = 0.729; MB = 6.86P; g/d=4.0
3 6 0.52 53.8 280 146 0.698 0
4 " 0.95 45.8 435 198 0.682 0.
5 " 1.47 47.6 700 218 0.687 0
Series II; b =7; d = 21; t =0.845; MsB=5.8P; g/d= 1.52
3 6 0.52 47.3 246 310 0. 590 0.
283 0.589 0.
4 " 0.95 44.0 418 300 0.602 0.
290 0.608 0.
5 1.47 41.2 606 300 0.596 0.
350 0.598 0.
5 1.72 43.5 748 390 0. 593 0.
4 2.14 " 931 340 0.595 0.
* A5%
/5%
0 /
Series I
* Vertical Stirrups, 1950
o Vertical Stirrups, 1952
A 45degree Stirrups
* TBeams, Vertical Stirrups
Series IV
a Vertical Stirrups
/a
V y
* ^^
/ 0
0 >
/
^
Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
only slightly larger shear capacities than given by
Eq. 44, the addition of web reinforcement appears
to have restricted the development of diagonal
cracks so that the relative importance of partial
bond was increased. The shear capacity of the
beams was thereby increased also.
Beams of Series II had the smallest g/dratio
of all beams, g/d = 1.52. It is seen in Table 45 that
these beams failed at a considerably lower load than
that given by Eq. 44. Since the beams failed at
about 30 percent greater loads than similar beams
without web reinforcement, an increase in the
amount of web reinforcement apparently did not
produce a corresponding increase in shear strength.
These beams appear to have failed in shearproper,
and they are analyzed accordingly in Section 19.
VI. BEAMS UNDER OTHER TYPES OF LOADING
1 8. Limitations of ShearCompression Failures
Equation 18, the basic equation for shear
strength, was derived for simplespan rectangular
beams without web reinforcement and under one or
two symmetrical concentrated loads. This equation
considers shear failures as compression failures.
Shearing stresses together with flexural tension
stresses are combined in the principal tension
stresses and produce a diagonal crack which ex
tends higher than the flexural tension cracks. After
this crack has formed, final failure takes place by
crushing of the concrete in the reduced compression
area.
In deriving Eq. 18, the unknown function F(fc')
was determined empirically. All available test data,
a total of 106 beams, were used in the analysis. The
ratio a/d which corresponds to the compressive
forceshear ratio in simplespan beams, C/V =
a/jd, varied from 1.17 to 4.80 for the beams con
sidered. This variation did not appear to have any
effect on the agreement between test results and the
values predicted by Eq. 18. Within these limits,
consequently, the shear strength of a beam was de
termined entirely by the physical properties of the
beam and was not a function of either the magni
tude of the shear or the momentshear ratio at the
section of failure. The beams failed at a limiting
moment Ms. This limiting moment was reached by
different combinations of V and a and the contri
bution of the shearing stresses was always large
enough to produce sufficient diagonal cracking
which is a prerequisite for this type of failure. The
addition of web reinforcement increased the limit
ing moment to Mw,. Otherwise, the mechanism of
failure remained the same as before.
As the ratio a/d increases, however, the ratio of
moment to shear at the section of maximum mo
ment increases, since M/V = a. Consequently, the
contribution of the shearing stress to the principal
tension stress decreases relative to the contribution
of the flexural stress at a given magnitude of mo
ment. Before cracking, the magnitude of the prin
cipal tension stress is determined by the magnitude
of the flexural tension, at the extreme tension fibers
of the beam, and by the magnitude of the shear
at the neutral axis. Between these two locations,
the magnitude of the principal tension stress is de
termined by the relative magnitudes of moment and
shear. As the magnitude of shear decreases at a
given value of moment, the trajectories of the
principal tension stresses become more and more
horizontal in the region of maximum moment. They
must still intersect the neutral axis at 45 deg, but
since the shear force is relatively small, the magni
tude of the principal tension stresses at that loca
tion is also relatively small. Since diagonal
cracking is the result of diagonal tension stress, the
cracks must start at the location of maximum stress
and progress first in an almost vertical direction.
Cracking, of course, alters the distribution of prin
cipal tension stress. Furthermore, their subsequent
distribution is beyond a theoretical analysis at the
present time. However, it is still likely that because
of small shear stresses, the cracks might remain
vertical or become but slightly inclined. Thus, for
large values of a/d, full diagonal cracking might
never develop and the beam might fail finally in
flexure rather than in shearcompression.
This behavior of beams with large ratios of a/d
can be observed from tests made by Johnson.30)
He tested a number of simplespan beams under
two concentrated loads to investigate the effect of
compression reinforcement. All beams were heavily
reinforced in tension, p = 0.046, and the ratio a/d
was equal to about 11. All beams but one failed in
flexural compression despite the fact that the mo
ment at failure was considerably larger than the
shearcompression moment given by Eq. 18. Fur
thermore, no diagonal cracking developed before
failure. It appears that at a/d = M/V equal to 11
the contribution of shear to the principal tension
stresses was too small to produce diagonal crack
ing. Consequently, the beams could not fail in shear
at the limiting shearcompression moment. The
beams continued to take load until their flexural
capacity was reached and they failed in flexure.
Figure 26 summarizes the above discussion and
presents a hypothesis for the limits of shearcom
Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
IUJLAIP
C/d
Fig. 26. Shear Force V Versus a/d. Possible Modes of
Shear Failure for SimpleSpan Beams
pression failures. Figure 26a shows onehalf of a
simplespan beam, loaded with two symetrically
placed concentrated loads; the loads Po, P1, P2, and
P3 represent several possible locations of the load
on the beam. Figure 26b shows the corresponding
moment diagrams. The ultimate flexural moment,
M, and the shearcompression moment Msw, are
also plotted on that figure. Both these moments are
determined only by the physical properties of the
beam crosssection. The corresponding values of
shear are plotted in Fig. 26c. These are obtained
from Eq. 28 for shearcompression failures and
from Eq. 29 for flexural failures.
For (a/d)1 < a/d < (a/d)c, it is assumed that
the beam fails in shearcompression. Load P1 repre
sents one possible load position in this range. As
the beam is loaded, the moment at the section
through load P1 increases. Finally, the limiting
moment Mw is reached and the beam fails in
shearcompression. The corresponding shear force
at failure is shown in Fig. 26c; it lies on a curve


determined from Eq. 28. Any other location of load
within the above limits would lead to the same re
sult, the limiting moment remaining constant while
the magnitude of shear decreases with increasing
ratio a/d as shown in Fig. 26c.
With respect to the limit of shearcompression
failures, the following hypothesis is advanced: a
certain value of a/d, (a/d)cr in Fig. 26, is assumed
to be the largest ratio of the internal compression
force to shear force at which the contribution of
the shearing stress is large enough to develop full
diagonal cracking. Thus, at the load location Po,
the limiting moment Msw and a critical magnitude
of shear, Vo, are reached simultaneously and the
beam fails in shearcompression.
As the ratio a/d increased beyond (a/d) c, there
is a transition region between shearcompression
and flexural failures. Considering load location P2
it is seen that as the beam is loaded, the limiting
moment Mw is reached at a load at which the cor
responding shear force V2 is below its critical mag
nitude Vo. As a consequence, full diagonal cracking
has not taken place and the beam cannot fail in
shear at that load. A further increase in load will
increase both the bending moment and shear at the
section P2. At a certain magnitude of load the shear
force reaches its critical value V. while the moment
has increased to a value of M2. Considering now
moment and shear conditions at a section closer to
the support, the section (a/d)c,, it is observed that
under the attained load both the shearcompression
moment and the critical magnitude of shear are
reached simultaneously at that section. This per
mits the formation of full diagonal cracking and
makes a shear failure possible. However, the cri
terion of failure is the attainment of the critical
magnitude of shear. A sudden formation of di
agonal cracks should occur as soon as this value
of shear is reached. In this sense, the resulting sud
den failure is not a shearcompression failure and
should be classified as a diagonal tension failure.
Furthermore, since the bending moment exceeds the
shearcompression moment anywhere between sec
tions P2 and (a/d) , while the shear force remains
at the constant value Vo, it is likely that the diag
onal tension failure can take place at any location
between these two sections.
This type of diagonal tension failure occurs
whenever the ratio a/d is between the values
(a/d)c, and (a/d),. The critical magnitude of the
shear force Vo is determined from the shearcom
ILLINOIS ENGINEERING EXPERIMENT STATION
pression moment M,. with the aid of the critical
ratio (a/d)c,, if the latter value can be uniquely
established. The higher limit of the transition re
gion, (a/d),, is not a constant ratio but a value
which depends on the relative magnitudes of M,,
and Mf. This is seen in Fig. 26c where the point of
intersection between the critical shear force Vo and
the shear curve corresponding to flexural failures
determines the value of (a/d) 2. Furthermore, a
constant shear force gives a linear relation be
tween moment and a/d in the range between
(a/d) cr and (a/d)2, as seen in Fig. 26b. The value
of (a/d)2 is then determined by the point of inter
section between the flexural moment M, and the
straight line through the origin in Fig. 26b which
passes through M,, at (a/d)c,.
For a/d > (a/d)2, the magnitude of the shear
force is never large enough to permit diagonal
cracking and the beam must fail in flexure. Con
sidering load location P3 in Fig. 26, it is seen that
as the load increases, the bending moment reaches
the flexural capacity of the beam before the shear
force reaches its critical magnitude Vo. As a con
sequence, diagonal cracking cannot develop, shear
failure is not possible, and the beam fails finally
in flexure.
In conclusion, ample test data is available to
justify the concept of shear failures as shearcom
pression failures for values of a/d below a certain
limit. Likewise, test data are available to show
that beams with high values of a/d fail in flexure
although their shearcompression moment has been
exceeded at the section of the maximum moment at
failure. Very little information is available, how
ever, for beams in the transition region between
the two types of failures. The above discussion is
presented as a possible explanation for the behavior
of beams in this region. It is realized that this
hypothesis is not supported by experimental evi
dence. If, however, the validity of this hypothesis
can be established by experiments and if the value
of the critical ratio (a/d)c, = (M/Vd)cr can be
uniquely determined, the behavior of a beam with
any value of a/d is fully described by a diagram
similar to Fig. 26. This diagram is determined by
Ms. and Mf, both of which depend on the physical
properties of the beam, and a critical ratio (a/d),r.
A few tests on Tbeams which fall in the transition
region between shearcompression and flexural fail
ures are discussed further in Section 20.
For very low values of a/d it is not expected
that a beam fails through beamaction. The mode
of failure seems to change from shearcompression
to what can be called shearproper; that is, actual
shearing off of the concrete. This type of failure is
discussed in the following section.
19. ShearProper
In the range of shearcompression failures, a
beam fails, after the formation of diagonal cracks,
in compression. However, as the ratio a/d decreases,
the mode of failure seems to change. With a con
centrated load close to a support, the cracks open
up near the load block in the tension zone of the
concrete and progress toward the other load block
in the compression zone. Since the load blocks are
but a short distance apart, the cracks are almost
vertical. The ultimate failure seems to take place
by the actual shearing off of the remaining concrete
in compression.
It is rather difficult to determine the true cri
terion of failure. Cracking of concrete is produced
by the principal tension stresses. As load on a beam
is increased, more cracks form and the existing
cracks both widen and extend higher. Consequently,
less and less concrete remains effective to resist
the complicated state of stress. Since the shear span
is short, the magnitude of the principal tension
stresses is also affected by the presence of compres
sive stresses. in the vicinity of the end reaction and
the concentrated load. These compressive stresses
will reduce the magnitude of the principal tension
stresses and will make them less inclined with the
axis of the beam. The closer is a load to a support,
the larger is the relative importance of the local
compressive stresses. Consequently, the tensile
stresses are smaller and it is expected that the
cracks will form and that the beam will fail at a
higher load than if the load were farther from the
support.
Some quantitative information on this type of
shear failure can be obtained from tests reported
by Graf in Heft 80.(26) A total of 26 beams were
tested, 21 small rectangular beams with the outside
dimensions and loading arrangement shown in
Fig. 27a and 5 large Tbeams as shown in Fig. 27b.
The variables included the size of the bearing block
for the concentrated load, the amount of longi
tudinal reinforcement, the amount and angle of
inclination of bentup bars, and to a minor extent
the compressive strength of concrete. In all tests
the distance x between the bearing blocks, Fig. 27a,
was either zero or a very small fraction of the depth
of the beams.
Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
An analysis of the test results shows that,
everything else remaining equal, the size of the
bearing block had no effect on the ultimate load.
This is true despite the fact that an increase in y
produced a larger moment under the load bearing
block, the load at failure being the same. It was
concluded, therefore, that the ultimate load depends
on the magnitude of the shear force V and the clear
the use of vertical stirrups, however, did not in
crease the ultimate load. Thus, there seems to be a
maximum value of a which limits the usefulness of
the bentup reinforcement.
The ultimate load increased as the concrete
strength increased. However, the range of f,' varied
generally only from 1500 to 2000 psi with but one
beam of about 3000 psi concrete strength.
The above observation suggested that the ulti
mate load could be expressed in terms of a nominal
shearing stress in the following form:
V
Vo =bD = C1 + C2f.' + C3Pt
where
= A (1 + sin a)
bD
C1, C2, C3 = numerical coefficients
"MIII 1
Fig. 27. Beams of Graf, Heft 80. ShearProper Type of Failures
shear span x rather than on the a/dratio. It also
appears that the size of the bearing area was suf
ficiently large in all cases to produce sheartype
failures; it is conceivable that local crushing of the
concrete can take place under the bearing block
when the bearing area is too small.
Some of the small beams were without any re
inforcement. The addition of longitudinal steel in
creased the ultimate load. Furthermore, it appears
that the use of longitudinal steel was equally ef
fective at any depth in the beam: in the bottom
half, at middepth, or in the top half of the beam.
The use of bentup bars was more effective than
the addition of longitudinal steel, and the effective
ness increased as the angle of inclination increased.
Judging from the load at failure, it seems that the
effectiveness of the inclined reinforcement increases
in proportion to the quantity (1 + sin a), at least
to the largest angle of inclination used in these tests,
a = 62.7 deg. Since the cracks were almost vertical,
and the quantity A,(1 + sin a) refers to the total
steel area crossing section AA, Fig. 27a. When
both horizontal and inclined reinforcement is used,
the reinforcement ratio Pt must be evaluated for
each part separately and the total value used in the
calculations.
This type of equation was checked against test
results. Reasonable agreement was found with the
following equation:
v, = 200 + 0.188f,' + 21,300 pt
where both v and f,' are expressed in pounds per
square inch. Since plain beams were included in the
analysis, the nominal unit shearing stress was de
termined for the gross section of the beams. For the
Tbeams of Heft 80 the value of v was calculated
by neglecting the flange area outside the web, since
the load was applied at a section in the end of the
beam where the flange was being tapered off to the
width of the web.
The physical properties of the beams and the
ratios v/v, are shown in Table 46, and in Fig. 28
the quantity v/vc is plotted against x/D, the ratio
of the clear distance between the load blocks to the
total depth of the beams. It is seen that Eq. 48 gives
satisfactory agreement with the test results; only
two plain concrete beams with the largest bearing
area fall more than 15 percent below the predicted
load and three beams are slightly more than 15
percent above. The five large Tbeams agree quite
well with Eq. 48.
&792
ILLINOIS ENGINEERING EXPERIMENT STATION
Table 46
Tests by Graf, Heft 80, 1935. ShearProper Type of Failures
(A) SMALL RECTANGULAR BEAMS
Reference: (26)
Dimensions: b=7.9 in.; D= 11.8 in.; See Fig. 27
Loading: See Fig. 27
Reinforcement: 0.39in. plain round bars; f,=49,000 psi; some bars bent as indicated below
Concrete Strength: Tests on 7.9in. cubes; f,'=0.75 fe,' assumed
Age at Test: 14 days
Horiz.
6
14
7
10
6
8
Arrangement of Reinforcement
No. of Bars
at AA
A,(1+sin a)
7 16
4 45
6 16
4 60
Physical Properties of Beams and Test Results
pe
Eq. 47
%
0.79
1.83
2.09
2.20
60.79
1.83
2.09
2.20
60.79
1.83
2.09
2.20
1.79
2.02
1.79
2.02
Group
a
b
d
e
f
g
Beam
la
b
c
d
e
2a
b
c
d
e
3a
b
c
d
e
4a
f
g
5a
f
g
Beam
1246
1247
1270
1271
1272
Eq. 48
psi
499
667
889
944
968
482
650
872
927
951
501
669
891
946
970
591
972
1021
563
944
993
(B) LARGE TBEAMS
Reference: (26)
Dimensions: b=49.2; b'=9.8; D=22.8; d=21.4; e=3.1; L=137.8; L'=161.4; x=2.0
Loading: One load 11.8 in. from end support; see Fig. 27
Tension Reinforcement: 0.63 and 0.71in. plain round bars, hooked; f,= 62,000 and 53,400 psi, resp.
Web Reinforcement: Bentup bars and 0.24in. round vert. stirrups at 7.9 in.
Reinforcement in Flange: Four 0.28in. round horiz. bars; 0.28in. round transverse bars at 4.9 in.
Concrete Strength: Tests on 7.9in. cubes, f/'=0.75 fI' assumed
Age at Test: From 12 to 23 days
Reinf. Bars a A,(1+sin a) pt Ptgt Vtmes Vtest
at AA Eq.47
Horiz. Bent deg in.' % kips kips psi
30.71 ...... .... 1.80 0.80 172 157 699
20.63
30.71 10.71 45 3.01 1.34 185 169 753
20.63 10.63
20.71 50.71 45 4.99 2.22 247 226 1004
10.63 10.63
30.71 30.71 62.7 4.59 2.04 231 212 942
20.63
30.71 30.71 62.7 4.59 2.04 296 271 1206
20.63
If
psi
1590
1500
1600
2080
1930
/f0'
psi
1560
1550
1780
1640
3040
Among previously analyzed test data there were
a few beams which failed at a lower load than
predicted by the shearcompression equations.
Those were the beams tested by Clark(5) which had
the shortest shearspan, and two simplespan beams
and eight restrained beams of Series II by
Moody;(12) all these beams had a very small a/d
ratio and were reinforced with vertical stirrups.
The beams for which strain readings were reported
failed in general before yielding of the web rein
forcement. These beams are reanalyzed in terms
of shearproper in Table 47.
The nominal shearing units stress vc as given
by Eq. 48 was computed for each beam, and the
ratio v/Vc is plotted against the parameter x/D
in Fig. 28. Some of Clark's beams failed in tension
and are not included in this comparison. Figure 28
shows that the ratio v/Vc decreases as x/D in
creases. Because Eq. 48 was entirely empirical by
nature and the number of tests was rather limited,
e.g., there were no beams in the range of x/D from
0.1 to 0.8, no attempt was made to write an expres
sion for the relationship between v/Vc and x/D
One possibility is shown by the dashed line in
Fig. 28.
Beams of Heft 80 with the load very close to
the supports showed no evidence that vertical stir
rups increased their shear strength. This is under
Tot. No.
of Reinf.
Bars
14
14
14
14
12
12
kips
88.2
110.2
154.3
154.3
154.3
88.2
125.7
167.6
165.3
185.2
77.2
132.3
158.7
165.3
176.4
88.2
209.4
231.5
77.2
183.0
207.2
Eq. 48
psi
664
777
1008
943
1206
pe
Eq. 47
0679
1.83
2.09
2.20
1.79
2.02
Ratio
Va
0.95
0.89
0.93
0.88
0.86
0.98
1.04
1.03
0.96
1.05
0.83
1.06
0.96
0.94
0.98
0.80
1.16
1.22
0.74
1.04
1.12
Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
1.4
1.2
0.8o
0.6
0.6
0.4
' 0
0 0.2 0.4 0.6 08 1.0 12
x/D
Fig. 28. Nominal Shearing Stress Ratio Versus x/D
for Failures in ShearProper
standable since the location of the load forced the
formation of almost vertical cracks. However, as
x/D increases in the region of shearproper, cracks
follow the edges of the bearing blocks and vertical
stirrups crossing the cracks produce a slight in
crease in the ultimate load. This is seen in Table 47
where for any value of x/D the ratio v/vc increases
somewhat as the ratio of web reinforcement in
creases. The beams fail, however, before the verti
cal web reinforcement yields. When the load is
removed sufficiently far from a support, a regular
shearcompression failure takes place.
The transition between shearcompression and
shearproper, point (a/d)1 in Fig. 26, seems to de
* Graf, Heft 80, rectangular beams
A Graf. Heft 80, T beams
o Clark, a =18"
* Moody, simp/espan beams, Series I//II
o Moody, restrained beams, Series II
Invest. Beam fI' b D p9
Eq. 47
psi in. in. %
Clark Dl1 3800 8 18 1.39
3 3560
D21 3480 1.39
2 3755
D31 4090 2.08
D41 3350 1.39
Moody 30 3680 7 24 5.57
Series 31 3250
III
Moody
Series
II
Table 47
Other ShearProper Type of Failures
a/d x x/D PLt f, at
Fail.
% of
in. kips fA.
1.17 14.5 0.81 135.4
115.4
130.4
140.4
177.6
140.4
1.52 24 1.00 215 83
228 67
Ratio
P.,
0.76
0.67
0.69
0.72
0.64
0.55
0.80
0.73
0.82
0.74
0.71
0.73
0.57
0.66
0.62
0.48
Ratio Vte.t vNe V. Ratio
Ptit Eq. 48 v"*t
kips psi psi
0.91 67.7 470 1210 0.39
0.78 57.7 401 1165 0.34
0.89 65.2 453 1150 0.39
0.94 70.2 487 1202 0.41
0.83 88.8 617 1412 0.44
0.96 70.2 487 1126 0.43
0.67 107.5 640 2081 0.31
0.72 114.0 679 2000 0.34
.... 103.3 615 1883 0.33
.... 94.3 561 1898 0.30
.... 100.0 595 1778 0.33
.... 96.7 576 1723 0.33
.... 100.0 595 1825 0.33
.... 116.7 695 1808 0.38
.... 130.0 774 1857 0.42
.... 113.3 674 1840 0.37
.4
8'
*
5~~   
3 a
pend both on the ratio x/D and the amount of web
reinforcement used. All beams of Table 47 had cor
responding test specimens without web reinforce
ment and these beams failed in shearcompression
in agreement with Eq. 18. Furthermore, Clark's
beams with 24in. shear span having x/D equal
to 1.14 and reinforced with vertical stirrups failed
in shearcompression. Thus the transition region
between the two types of failures seems to lie ap
proximately between x/D equal to 0.8 and 1.1,
increasing as the amount of vertical web reinforce
ment increases. The use of inclined web reinforce
ment, however, increases the ultimate load in
shearproper according to Eq. 48. Consequently,
whenever the clear shear span x approaches the
total depth of the beam, inclined web reinforce
ment should be used instead of vertical stirrups.
For restrained beams the distance x was con
sidered in the same way as for simplespan beams
 the clear distance between two load blocks. For
Series II of Moody's restrained beams this pro
cedure gave good results. It is seen in Fig. 28 that
both simplespan and restrained beams with the
same x/Dratio failed at about the same nominal
shearing stress. However, if the ratio x/D is con
sidered as a measure of principal tension stresses
and the extent of cracking, the use of x as defined
above is not strictly correct, since the magnitude
of flexural bending stresses for simplespan beams
is generally different from that for restrained
beams.
20. Transition Region and Flexural Failures
There is very little experimental data available
for beams with large ratios of a/d. The only tests
reported in the literature are those by Johnson,(30)
previously discussed in Section 18, and a few T
ILLINOIS ENGINEERING EXPERIMENT STATION
Table 48
Tests by Graf, Heft 67, Series II, 1931. SimpleSpan TBeams Under One Unsymmetrical Concentrated Load
Reference: (23)
Dimensions: b=49.2; b'=9.8; D=22.8; d=20.7; e=3.15; L=212.6; L'=240.2; a/d=2.09 for short
segment, 8.18 for long segment
Loading: One concentrated load 43.3 in. from support
Tension Reinforcement: Ten 0.87in. round plain bars at load, hooked; f,=about 46,000 psi
Reinforcement in Flange: Four 0.28in. long. bars; 0.28in. transverse bars at 4.9 in.; f,= about 48,000 psi
Web Reinforcement: Bentup bars and 0.28in. vertical stirrups
Concrete Strength: Tests on 7.9in. cubes; f/= 0.75 f,.'= 1370 psi
Age at Test: 27 to 35 days
Ft P.
Eq. 35
kips
0.64 51.6
Ratio
Pte.t
P.
3.20
3.20
2.56
2.35
SHORT SEGMENT
rf/. Pts.
psi p,;
650 1.39
1.39
1.11
Group Beam PteAt
kips
1 1203 165.3
1205 165.3
2 1204 132.3
1206 121.3
beams tested by Graf23) under one unsymmetrical
load or several concentrated loads.
Beams tested by Graf under one unsymmetri
cally placed concentrated load are reported as
Series II in Heft 67.(23) Four such beams were
tested; the two beams of Group 1 were reinforced
with bentup bars along the entire length of the
beams; the two beams of Group 2 had bentup bars
only in the short segment, whereas the long segment
was reinforced with a small amount of vertical
stirrups. The beams are analyzed in Table 48.
The ratio a/d was 2.09 for the short segment
and 8.18 for the long segment. The last value is
much larger than the range of a/d for which Eqs.
18, 35, and 26 were derived. It is likely that this
ratio corresponds either to the transition region
between flexural and shear failures or to the region
of flexural failures, Fig. 26. This observation is veri
fied by the test results. The two beams of Group 1
failed in tension at a load 2.18 times larger than the
shear strength of the long segment as given by
Eq. 26. The two beams of Group 2 failed in shear
and the load at failure was up to 2.44 times larger
than that given by Eq. 26 for the long segment. It
is interesting to note, however, that the beams
did not fail under the concentrated load at the
section of maximum moment but between the load
point and the end reaction in the long segment. The
final break took place about 68 in. from the sup
port for Beam 1026 and about 116 in. for Beam
1024. The magnitude of the moment at the actual
section of failure was 1.03 and 1.50, respectively,
times the shearcompression moment for the beams.
In both cases, it was reported that the failure was
sudden. Thus it appears that the ultimate load was
governed primarily by shear. Because of the long
shear span, the shearing stresses were relatively
small at the load which corresponded to the shear
compression moment, Ms, from Eqs. 35 and 28,
at the section of maximum moment in the more
lightly reinforced long segment. This load was less
than half the ultimate load. Photographs of the
beams show that at that load all cracks were prac
tically vertical. As the load increased, the magni
tude of the shearing stresses increased also and the
cracks started to incline. At a certain magnitude
of shear force, cracks were sufficiently inclined to
lead to a sudden shear failure. Since at that load
the moment was larger than the computed ultimate
shear moment over most of the beam, any random
occurrence of a diagonal crack could produce a
shear failure. This might be the reason that the
two beams failed at different sections.
Beams of Series I in Heft 67 were tested under
three equal and symmetrical concentrated loads.
Six Tbeams were tested in three groups: beams of
Group 1 had bentup bars along the entire length of
the beam; Groups 2 and 3 only between the end
supports and the first load. All tension reinforce
ment was carried through the two middle segments
of Group 2, whereas in Group 3 some of the bars
were cut off beyond the moment requirement and
hooked in the tension zone of the concrete. The
beams are analyzed in Table 49.
The quantity a/d has been used as a convenient
expression for the compressive forceshear ratio
C/V of simplespan beams. An equivalent expres
sion is given by a/d = M/Vd for other types of
loading. This ratio is 8.52 for the beams of Series
I, thus only a little greater than that for Series II.
As a consequence, these beams failed in a manner
similar to those of Series II. Beams of Group 1
failed in tension, beams of Group 2 in tension with
a sheartype final collapse, and those of Group 3
in shear at the section of the center load before
yielding of the tension reinforcement. The failure
of the last group of beams appears to have been
hastened by diagonal cracks which were initiated
at the hooks on the cutoff tension bars. The ratios
of the ultimate loads to the loads given by Eq. 26
LONG SEGMENT
rfy. Psest
psi pK.
230 2.18
" 2.18
20 2.44
.
Mode
of
Fail.
T
T
S
8
1. .k
2. 
Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
Table 49
Tests by Graf, Heft 67, Series I, 1931. SimpleSpan TBeams Under Three Concentrated Loads
Reference: (23)
Dimensions: b=49.2; b' =9.8; D=23.2; d=20.8; e 3.54; L=212.6; L'= 240.2; M/Vd  8.52 at midspan
Loading: Three equal and symmetrical concentrated loads, at midspan and at 35.4 in. from supports
Tension Reinforcement: Eleven 0.87in. round plain bars at midspan, hooked; f,= about 46,000 psi
Reinforcement in Flange: Four 0.28in. long. bars; 0.28in. transverse bars at 4.9 in; f, =about 48,000 psi
Web Reinforcement: Bentup bars and 0.24in. vertical stirrups
Concrete Strength: Tests on 7.9in. cubes; f//=0.75 f'= 1490 psi
Age at Test: 26 to 41 days
.... m.Be p... . F M AT FARST LOAD AT MIDSPAN
Eq. 35 rf/, Mttt AfMt rf, Mtt M.
kips in.kips psi M, M.. psi M. M,,
1 1197 209 0.64 2118 560 1.75 0.83 230 2.92 2.00
1200 218 " " " 1.83 0.86 " 3.04 2.08
2 1198 198 560 1.66 0.78 20 2.77 2.65
1201 209 1.75 0.83 " 2.92 2.79
3 1199 172 560 1.43 0.67 20 2.39 2.28
1202 187 " 1.56 0.74 2.60 2.49
are comparable to those of Series II since the
M/Vdratios are nearly the same in both cases.
From the results of these tests it is evident that
for high values of a/d = M/Vd a beam may fail
either in shear at a greater load than that given
by the shearcompression moment of Eq. 28 or in
flexure before developing any marked diagonal
cracking. In accordance with the hypothesis pre
sented in Section 18, it appears that the failure
criterion is a critical shear force Vo, determined
from the shearcompression moment with the aid
of a critical ratio (a/d)cr = (M/Vd)c,. If the
actual ratio a/d of a beam at the section of maxi
mum moment is larger than (a/d)c,, the beam fails
suddenly in what can be called diagonal tension as
soon as the shear force reaches its critical quantity
Vo. The corresponding moment at failure is larger
than the shearcompression moment M,,,. If, how
ever, the flexural capacity of the beam is reached
before the critical shear force V. is attained, the
beam fails in flexure.
The results of the above tests give the critical
value of a/d = M/Vd equal to 3.43.7. These val
ues are somewhat lower than the values of a/d at
which the rectangular simplespan beams under
one or two concentrated loads were still observed
to fail in shearcompression. The highest ratio a/d
was equal to about 4.8 in that case. However, the
present tests are too limited both in number and in
scope to provide a check on the validity of the
above hypothesis or to permit the setting of a
numerical value for the critical ratio (a/d)c,.
Furthermore, only Tbeams made of rather low
concrete strength, about 1000 psi, were tested.
This combination leads to very high ratios of
Pf/P, = MfM,, up to 2.79 as noted in Tables 48
and 49. For rectangular beams which are without
web reinforcement and which are made of more
normal concrete strength the flexural capacity
rarely exceeds that in shear by more than 5060
percent. This difference between the two types of
beams could also influence the mode of failure
which renders it impossible to draw any definite
conclusions as to the value of (a/d)cr from these
few test results.
21. Beams Under Uniform Load
It has been shown that within certain limits of
a/d = M/Vd the shear strength of a beam under
concentrated loads could be determined by Eqs. 18
and 28 for rectangular beams and by Eqs. 35 and
28 for Tbeams. Under this type of loading, the
beams tested failed at the section of maximum
moment and maximum shear, and the load at fail
ure was determined by the magnitude of moment.
As the value of M/Vd increased beyond these
limits, however, the actual shear strength was
found to be larger than that given by the above
equations. Furthermore, the location of failure was
not necessarily the section of maximum moment.
The upper limit of M/Vd for the applicability of
shearcompression equations and the shear strength
of a beam in the transition region between shear
and flexural failures could not be determined quan
titatively because of insufficient experimental data
for beams with high values of M/Vd.
For simplespan beams under uniform load the
value of M/Vd ranges from zero at the section of
no moment to infinity at the section of maximum
moment. The beam cannot fail in shear at the
section of maximum moment because there are no
diagonal cracks at that section. Consequently, if
a shear failure is to take place, it must occur at
a section where the value of M/Vd is such as to
permit diagonal cracking and where the moment
itself is sufficient to produce a shearcompression
failure. In the following paragraphs, the available
test data is analyzed in an attempt to find more
Mode
of
Fail.
Tat
Midspan
TS at
Midspan
S at
Midspan
ILLINOIS ENGINEERING EXPERIMENT STATION
quantitative information about the shear strength
of beams under uniform loading.
No tests could be found of beams under actual
uniform load. However, there are reports on tests
where uniform loading was simulated by a large
number of equal and equally spaced concentrated
loads. These beams were tested by Bach and Graf
in two series, one series under 16 equal loads as
reported in Heft 48,(27' and the other series under
8 equal loads as reported in Heft 20.(28)
Beams of Heft 48 were five simplespan T
beams loaded with sixteen equal concentrated loads.
The arrangement of loads and reinforcement is
shown in Figs. 29 through 32, and Table 50 gives the
physical properties and test results for these beams.
Beam 1024 had no web reinforcement. It failed at
a very low load, the maximum moment at midspan
being only 68 percent of the shearcompression
moment as given by Eq. 35. A diagonal crack
formed at about the thirdpoint of the span shortly
before failure. Numerous longitudinal cracks ran
f P/16 P/16 P/16 P/16 P/16 P/16 P/16 P/16
from that crack toward the end support. It appears
that this beam failed in bond. Beams 1026, 1025,
and 1031 had almost identical arrangements of
bentup bars except that the size of the bars was
different, the area varying as 1.00:0.53:0.36. Beams
1026 and 1025 failed in tension and Beam 1031 in
shear. However Beam 1025 was rather close to its
shear strength at failure as indicated by marked
diagonal cracking all along the beam. Beam 1032
had only two bentup bars in the ends and failed
in shear.
Figures 29 through 32 show the arrangement of
loads and reinforcement and the main cracks at
failure. Furthermore, the actual ratio M/M, at
failure, where M, was computed by Eq. 35, and the
corresponding predicted ratio, 1 + 2rfK,/103 from
Eq. 28, are plotted along the beam for each indi
vidual beam. The shearcompression moment was
calculated for the section at midspan; the reduc
tion of the longitudinal steel area through bending
up bars at other sections was not taken into con
f P/16 P/16 P//6 P/16 P//6 P/16 P/16 P//6
Fig. 31. Beam 1031 of Bach and Graf, Heft 48
( P/16 P/16 P/16 P/16 P/16 P/16 P/16 P/16
Fig. 30. Beam 1025 of Bach and Graf, Heft 48
Fig. 29. Beam 1026 of Bach and Graf, Heft 48
[ P/16 P//16 P/16 P//16 P/16 P/16 P//6 P/16
Fig. 32. Beam 1032 of Bach and Graf, Heft, 48
Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
Table 50
Tests by Bach and Graf, Heft 48, 1921. SimpleSpan TBeams Under Sixteen Equal Concentrated Loads
Reference: (27)
Dimensions: b=47.2; b'=9.8; D= 27.6; d= 25.2; e=3.94; L=212.6; L' 244.1
Loading: 16 equal and symmetrical concentrated loads. See Figs. 2932
Tension Reinforcement: Round plain bars, hooked
Reinforcement in Flange: Two 0.28in. long. bars, 0.28in. transverse bars at 3.9 in.
Web Reinforcement: Bentup bars and 0.28in. round vertical stirrups
Concrete Strength: Tests on 7.9in. cubes; f/ = 0.75 f.'
Age at Test: 42 to 48 days
Beam /,' A. , Size of Pt.t Mt..e F, M. Ratio MM
Bentup Eq. 35 Mt1
Bars M,
psi in.2 ksi in. kips in.kips in.k in.k
1024 3230
1026 3250
1025 3050
1031 2750
1032 2750
5.79 50.5
5.73 50.5
5.78 51.2
5.63 49.8
5.87 50.5
None 105.8
0.98 262.3
0.71 264.6
0.59 211.6
0.98 202.8
sideration. The variation in 1 + 2rf,,w/103 was cal
culated using values of r at midheight of the
beams. If the relationship between the actual and
the predicted moment ratios is observed in these
figures, it is seen that the beams failed in shear
only when the ratio M/Ms approached the quan
tity 1 + 2rfv,/103 at about the fifth load point
from the end of the beam. It is recalled that a beam
under concentrated loads and in the shearcom
pression region of M/Vd would have failed in shear
as soon as the value of M/M, had exceeded that of
1 + 2rfyw/103 at the section of maximum moment.
Eq. 28. This difference between the two types o0

Beam 1031 Shear 
Beam /026 Tension
/675 8.18
/ 16.35 77S5
Beam 1031
Tension Capacity
Beam 1025 Tension
\ Beam 1032 Shear
. \
"..
5O10 3.43 2.32 1.49 0.83
4.57 2.90 /.79 097 03C
Q 8 7 6 5 4 J 2 I
Load Points
Fig. 33. Ratio of Measured to Computed Failure Moment as
Function of M/Vd. TBeams of Heft 48 under
Sixteen Concentrated Loads
2859
6973
7031
5625
5390
Ratio Mode
Mte i of
M! Fail.
0.57 4210 0.68 7031 0.41 B?
4210 1.65 6987 1.00 T
4140 1.69 7204 0.98 T
3750 1.50 6857 0.82 S
3793 1.42 7119 0.76 S
beams suggests that it might be possible to de
termine empirically the value of M/Vd which limits
the region of critical diagonal cracking capable of
producing shearcompression failures.
Figure 33 shows the ratio between the actual
moment at failure and the ultimate shearcompres
sion moment of Eq. 28 plotted along the beams.
The values of M/Vd at each side of the load points
are also marked in the figure. This figure shows
the effect of the M/Vdratio more clearly. Beam
1026, which failed in tension, has the ratio Mtest/
Ms. less than one at the fifth load point. Beam
1025, which failed in tension while being very close
to a shear failure, has the ratio just above one.
Beam 1031, which failed in shear, seems to have
failed just as the ratio exceeded one. The ultimate
flexural capacity of this beam is shown in the
figure also. It is seen that this load, if reached,
would have increased the ratio to considerably
higher than one. Finally, Beam 1032, which failed
in shear, has the ratio somewhat more than one,
1.16. However, Fig. 32 shows that in the case of
this beam there is some doubt as to what to con
sider as the value of 1 + 2rf,//103 at the fifth load.
The bentup bars do not cover that particular
section; their presence in the vicinity undoubtedly
offers some resistance to the formation of diagonal
cracks. This, in a sense, would mean an increase
in the value of rf,, which would bring the ratio
closer to one in Fig. 33.
Thus, it appears that the shearcompression
equations are applicable for the beams under con
sideration. However, the section at which the shear
moment is calculated is not that for maximum
moment but that at which the value of M/Vd is
equal to about 4.5, corresponding to the fifth load
point of the beams of Heft 48.
Heft 20 reports tests on 51 simplespan T
beams, tested in groups of three companion speci
L.8
1.6
1.4
1.2
Zo
e1t
Sw
08
0.6
0.4
0.2
'Vd
,q
M.
9 ,q 7 6 5 4 ,5 Z I
A//
ILLINOIS ENGINEERING EXPERIMENT STATION
mens. Sixteen groups of beams were loaded with
eight equal concentrated loads as shown in Figs. 34
and 35; one additional group had four loads omitted
on one half of the span. The physical properties of
the beams and the test results are given in Table 51.
The first four groups of beams were reinforced
with two 1.57in. plain round bars. The test vari
ables included the effect of anchoring of the longi
tudinal bars, either straight or hooked, and the
effect of web reinforcement which was provided by
vertical stirrups placed in accordance with the
shear diagram along the entire length of beam.
All these beams failed in bond as indicated by the
excessive end slip of the longitudinal bars which
was measured in most beams. Bond failure led to
longitudinal cracking along the reinforcing bars
and to the final opening of a diagonal crack, gen
erally between the first and the second load points.
Groups 55 and 56 were reinforced with four
1.10in. plain round bars, two of which were bent
P/B P/8
Fig. 34. Beams 60 of Bach and Graf, Heft 20
Fig. 35. Beams 62 of Bach and Graf, Heft 20
up at 13 deg. The ends of the bars were anchored
with small 90deg hooks. Beams of Group 55 had
no additional web reinforcement and failed in bond
by excessive slipping of the bars. Beams of Group
56 had additional vertical stirrups placed accord
ing to the shear diagram and failed in tension.
The remaining beams were reinforced with 6 or
7 round bars of different sizes. Two bars were
carried straight to the supports; the rest of the bars
were bent up at different locations. The middle
portion of the beams, not covered with bentup bars,
was reinforced with vertical stirrups. The beams
were tested in companion groups; in one the two
straight bars were left unhooked, in the other they
were hooked. All bentup bars were sufficiently
hooked in all beams. All beams with the straight
bars not hooked failed in bond by excessive end
slip. This led to the opening of a diagonal crack
at different locations in different beams. All beams
with hooked straight bars failed in tension with a
secondary crushing of the concrete at midspan.
Thus, no beams failed actually in shear. Some
indication of the shear strength of the beams can
be obtained, however, by analyzing the beams
which had the smallest number of bentup bars.
Figures 34 and 35 show beams of Groups 60 and 62
in this category. The arrangement of web reinforce
ment is shown together with the quantity 1 +
2rfw/103 and the ratio M/Ms along the beams.
The compressive forceshear conditions are repre
sented by the ratio M/Vd, given at both sides of
each load point. It is seen that the M/M.curve
intersects the web reinforcement curve near the
third load point, at about M/Vd equal to 5. Since
the beams failed in tension, the amount of web
reinforcement was sufficient to prevent a failure in
shear. Consequently, the critical value of M/Vd
for shear failures must be less than 5, which agrees
with the previous finding of about 4.5 for beams
of Heft 48.
From the results of the above two series of tests
it appears that the shear strength of beams under
uniform load can be represented by the shear
compression equations 18 and 28 for rectangular
beams and by Eqs. 35 and 28 for Tbeams. Since
there are no diagonal cracks in the region of maxi
mum moment and the inclination of cracks is very
small for high values of M/Vd, the beam cannot
fail in shear unless the bending moment is higher
than the shear strength as given by Eq. 28 at a
critical value of M/Vd. From the above results, the
critical value of M/Vd is set tentatively at about
Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
Table 51
Tests by Bach and Graf, Heft 20, 1912. SimpleSpan TBeams Under Eight Equal Concentrated Loads
Group No. of
Reinf.
Bars
51 2
52*
53
54*
55 4
56*
57 6
58
59
60
61 7
62
63
64*
65
66
Reference: (28)
Number of Beams: Three in each group
Dimensions: b=23.6; b'=7.9; D = 15.7; d= 13.6; e=3.94; L = 157.5; L'= 173.2
Loading: 8 equal and symmetrical concentrated loads. See Figs. 34, 35
Tension Reinforcement: Round plain bars, numerous sizes from 0.39 to 1.57 in. in diam; average /,=46,000 psi
Reinforcement in Flange: None
Web Reinforcement: Bentup bars and 0.28in. plain round vertical stirrups; f/, = 58,300 psi
Concrete Strength: Tests on 11.8in. cubes; f,'= 0.75 f.'= 2490 psi ± 7.7 percent
Age at Test: Around 45 days
Computed Quantities: Ft=0.77; M. 1259 in.k; P.=57.5 kips, average Mff=2135 in.k, Pf=97.5 kips
A. Anch. No. of a Pt.k Ratio Ratio
Long. BUp Pwt Ptmt
in.2 Bars Bars deg kips P. Pf
3.90 None .... 47.0 0.82 0.48
3.90 .. 67.6 1.17 0.69
3.90 Hooks .... 51.5 0.90 0.53
3.96 " 94.0 1.63 0.96
3.81 2 13 73.5 1.28 0.75
3.80 100.5 1.75 1.03
3.86 None 4 45 90.5 1.57 0.93
3.86 Hooks 95.5 1.66 0.98
3.91 None 86.1 1.50 0.88
3.91 Hooks 95.5 1.66 0.98
3.92 None 5 85.2 1.48 0.87
3.94 Hooks 99.6 1.73 1.02
3.90 None 90.3 1.57 0.93
3.90 Hooks 106.4 1.85 1.09
3.94 30 100.0 1.74 1.02
3.91 102.0 1.77 1.05
* Vertical stirrups along the entire span.
4.5. It is recalled, however, that the tests were far
from being conclusive and that only simplespan
Tbeams were tested. The validity of the above
concept of shear failures of beams under uniform
load and a more reliable value of the critical M/Vd
must be established by a more comprehensive test
program.
It appears, however, that the conventional
method of reinforcing simplespan beams under
uniform load against sheartype failures is incor
rect. Web reinforcement is placed to conform with
the shear diagram. This means that the amount of
web reinforcement in the region of the critical
value of M/Vd is smaller than that for lower values
of M/Vd. The above findings suggest, however, that
the web reinforcement should be placed at a uni
form spacing between the end reaction and the
region of critical M/Vd, say 4.5. Only beyond that
region should it be tapered off and reduced to zero
at midspan. If it is desired to prevent shear failures
altogether, the ultimate flexural and shear moments
must be calculated from the properties of the beam
by Eqs. 29 and 18 or 35. Then the ratio between
the ultimate flexural moment at the section of the
critical value of M/Vd and the shear moment of
Eq. 18 or 35 must be substituted into Eq. 28 in
order to find the necessary amount of web rein
forcement which would force the beam to fail in
tension at the section of the maximum moment
rather than in shear at the section of the critical
M/Vd.
Mode
of
Fail.
B
B
B
B
B
T
B
T
B
T
B
T
B
T
T
T
VII. SUMMARY AND CONCLUSIONS
22. General Summary and Discussion
A general expression for the shear strength of
reinforced concrete beams has been derived by con
sidering simplespan beams without web reinforce
ment. It was first assumed that the total shear
force is resisted solely by the compression area of
the concrete and that the criterion of failure is an
ultimate shearing unit stress, related to the com
pressive strength of the concrete. These assump
tions yielded an expression in a form which
suggested that the real criterion for shear failures
was a limiting moment rather than an ultimate
shearing stress. This observation was supported by
certain test results reported in the literature.(5,6) It
was concluded that shear failures were actually a
compression phenomenon. Shearcompression fail
ures differ from flexural compression failures only
because the compressive area of the concrete is re
duced by diagonal cracks which extend higher than
the flexural tension cracks at failure.
a. SimpleBeams Without Web Reinforcement.
Treating shear failures as compression failures and
assuming that the depth of the compression zone
was related to k as determined by the elastic
"straight line" theory, Eq. 18* was derived em
pirically to represent with good accuracy the shear
strength of rectangular simplespan beams without
web reinforcement and under one or two symmetri
cal concentrated loads
Equation 18 was based on the test results from
15 different investigations involving 106 beams
which failed in shear. These beams were tested
over a period of 43 years and had a wide variation
in their physical properties as summarized in
Table 1. The average ratio of measured to com
puted moments was 0.986 and the standard devia
tion 0.119. The agreement between the measured
and computed moments is shown graphically in
Fig. 2.
Equation 18 was also interpreted theoretically
in the light of the conventional theory of compres
sion failures of reinforced concrete beams. From
*The equations referred to in this section are summarized for con
venient reference in Section 23.
previous test results at the University of Illi
nois"1, 14) the value of kfk3 was approximated by
Eq. 20, and for this value of kks3 it was possible to
establish a relationship between k, and k, where k,
refers to the depth of the compression zone at shear
failures, Eq. 22. Since k remains usually within the
values of 0.2 and 0.5, Eq. 22 suggests that k, is
practically a constant fraction of k. This finding
explains why the previous attempt to use the value
of k as a measure of k, gave satisfactory agreement
with test results.
b. Web Reinforcement. The effect of web rein
forcement was investigated next. It was found that
the use of web reinforcement increased the shear
strength of a beam more than would be accounted
for by the internal forces in the stirrups. The total
contribution of web reinforcement was expressed
empirically by Eqs. 26 and 28.
These equations were based on the test results
for 80 beams. The average ratio between the meas
ured and calculated moments was 1.017 and the
standard deviation 0.089. The range of the physical
properties of the beams is summarized in Table 17
and the ratios of P/P, are shown graphically in Fig.
5. The equations were further checked by the help
of beams which had failed in flexure. It is seen in
Fig. 6 that although the flexural capacity of these
beams was reached at different ratios of P/P,, they
always failed at a load lower than their strength in
shear, given by Eq. 26.
Equations 26 and 28 were found to be applicable
for all angles of inclination and for different values
of yield strength of web reinforcement. It was
found also that there was no noticeable difference
between the effectiveness of bentup bars and stir
rups serving as web reinforcement.
Equations 26 and 28 show that a given amount
of web reinforcement will increase the shear
strength of a beam in proportion to its strength
without web reinforcement rather than by an
amount determined solely by the physical proper
ties of the web reinforcement. It appears that by
resisting the extension and widening of diagonal
cracks, the presence of web reinforcement increases
Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
the available compressive area of the concrete and,
conceivably, restricts the concentration of the com
pressive strain of concrete in the region of the main
diagonal crack.
The relationship between shearcompression and
flexural failures was discussed in Section 13. It was
found that the amount of web reinforcement neces
sary to prevent shear failures could be determined
for any beam by Eqs. 29, 18, and 28. Simplespan
rectangular beams reinforced in tension only and
designed according to the present ACI Code bal
anced design requirements were found to require
about 0.35 percent web reinforcement to ensure ten
sion failures. This assumed that the yield strength
of the tension reinforcement was 50,000 psi and that
of the web reinforcement 40,000 psi; it also as
sumed that the beams were loaded under one or
two symmetrical concentrated loads.
c. TBeams. Since the momentrotation rela
tionship of a Tbeam differs from that of a rec
tangular beam, Eq. 18 had to be modified to apply
for Tbeams. This was done by the use of a semi
rational shapefactor in the form of Eq. 34. Sub
stituting the compressive area Ac of a Tsection
as determined by the "straightline" theory for
bkd and using the shape factor of Eq. 34, Eq. 18
was rewritten as Eq. 35, applicable to Tbeams.
As seen from Fig. 11, Eq. 35 was found to give
satisfactory agreement with test results when
beams with abnormally large values of d/e and b/b'
were excluded. These beams had a lower shear
strength because the effective width of their flanges
was reduced. However, no attempt was made to de
termine an expression for the effective flange width.
Furthermore, it was found that the use of trans
verse reinforcement in the flange effectively counter
acted the reduction in the effective width and
thereby increased the scope of Eq. 35.
The shear strength of simplespan Tbeams with
web reinforcement could be determined from Eq. 28
which was derived for rectangular beams, but with
the value of r given by Eq. 27a. As seen in Fig. 12,
the agreement between the measured and calculated
quantities is satisfactory.
d. Restrained Beams. Simplespan beams under
one or two symmetrical concentrated loads develop
just one main diagonal crack under an applied
load and fail at that section. In restrained beams,
shear and moment conditions are such as to per
mit the formation of more than one main diagonal
crack. The beam may fail at any of these cracks,
depending on the magnitudes of moment and shear
and the arrangement of both longitudinal and web
reinforcement. It was found that whenever the pos
sibility of bond failures was precluded, the shear
strength of a restrained beam was determined by
the same equations as that of a simplespan beam,
Eqs. 18 and 28. The critical section was the section
of maximum moment.
When the longitudinal reinforcement was cut
off at some section, a sudden and complete bond
failure was possible by stripping out of the cutoff
reinforcement. This type of failure was outside the
scope of this investigation and was not examined in
more detail. Evidently, this is a question of bond
characteristics of the reinforcing bars and the
length of embedment from a diagonal crack to the
end of the bar.
Restrained beams with continuous top and bot
tom reinforcement may have another mode of fail
ure. Under certain conditions, e.g., when the dis
tance between a support and a load is short relative
to the effective depth of the beam, a local bond
failure may take place in the high bondstress
region between the sections of positive and nega
tive moments. As a result of local destruction of
bond, both the top and bottom longitudinal rein
forcement is in tension at a certain section. This
redistribution of the internal forces results in a
reduced shear strength of the beam. Assuming that
the entire tension force was transmitted from one
section to the adjacent section and that k, was
given by Eq. 22, Eq. 44 was derived to represent
the shear strength for this type of failure.
The validity of Eq. 44 was checked against test
results and satisfactory agreement was obtained.
Figure 23 shows the measured and calculated mo
ments graphically for all beams which failed after
a local bond failure. Most of the test specimens
show good agreement with Eq. 44; for some beams
a small increase in the shear strength was noticed
because of the effect of partial bond. This was dis
cussed in more detail in Section 17, paragraph b.
All beams shown in Fig. 23 had equal positive
and negative moments and developed, in general,
two main diagonal cracks before failure. This re
sulted in a full redistribution of the internal forces
and the shear strength of the beams was governed
by Eq. 44. For unequal positive and negative mo
ments, however, either one or two cracks may be
present at failure. Two cracks will produce, in
general, a full redistribution of the internal forces
ILLINOIS ENGINEERING EXPERIMENT STATION
and the shear strength of a beam will be given by
Eq. 44 at the section of maximum moment. One
crack will lead to a partial redistribution of the
internal forces, so that the shear strength will be
governed by Eq. 18 at the section of maximum
moment. Beams of Series VI by Moody had un
equal moments at sections A and B and failed at
section A after developing only one crack in span
g, Fig. 13. The beams were analyzed according
to Eq. 18 at that section and Fig. 24 shows that
good agreement was obtained between the measured
and the calculated moments.
From the available test data, it was not possible
to determine the limits of Eq. 44. The largest g/d
ratio for which test results were available was 4.0.
Since this ratio permitted a redistribution of the
internal forces, the limiting g/dratio must be
larger than four. Furthermore, it is apparent that
bond characteristics of the reinforcing bars have an
effect on the limiting value of g/d. The above re
sults were reported for beams reinforced with mod
ern deformed bars; plain bars undoubtedly are more
susceptible to local bond failures. Likewise, it was
not possible to determine the conditions under
which two cracks and, consequently, a full redis
tribution of the internal forces will occur for un
equal positive and negative moments. Until such
criteria can be established, the more conservative
condition of full redistribution should be assumed
in determining the shear strength of a restrained
beam.
It was found that the contribution of web re
inforcement could be determined in restrained
beams, as in simplespan rectangular and Tbeams,
by Eq. 28. Beams reinforced with 45deg stirrups
gave very good agreement with Eq. 28; beams pro
vided with vertical stirrups also agreed with this
equation except for two beams with the largest
values of rfw. It appears that in beams with rela
tively short shear span inclined stirrups are, in
general, more reliable than vertical stirrups. It is
conceivable that inclined stirrups have better an
chorage conditions whenever diagonal cracks are
forced to form in a restricted space and thereby can
develop their full effectiveness. Conversely, the an
chorage of vertical stirrups might be destroyed be
fore their full effectiveness is reached.
e. ShearFlexure Transition. All the shear
compression equations were derived and checked
for beams for which the a/dratio varied be
tween 1.17 and 4.80. The a/dratio represents
the compressive forceshear ratio for simplespan
beams under one or two concentrated loads; for
any other type of loading this ratio can be repre
sented by the equivalent ratio M/Vd. Within these
limits of M/Vd, the shear strength of a beam was
found to be determined solely by the physical prop
erties of the beam. It was not a function of either
the magnitude of the shear or the momentshear
ratio at failure.
However, as the M/Vdratio increases, the rela
tive importance of shear in connection with the di
agonal tension stresses decreases. Consequently, the
extent of diagonal cracking is less pronounced, and
it was found that the shear strength of such beams
was larger than that given by Eq. 28. It was also
noticed that the location of shear failure was not
necessarily the section of maximum moment. For
sufficiently large values of M/Vd the beams failed
in flexure rather than in shear. A hypothesis to
explain the behavior of beams with high values of
M/Vd was presented in Section 18. As seen in
Fig. 26, a critical value a/d = M/Vd was assumed
to represent the largest ratio of moment to shear
which would permit sufficient diagonal cracking for
shearcompression failures. Beams with higher than
the critical value of M/Vd did not crack sufficiently
and could not fail at the shearcompression mo
ment, Eq. 28. The criterion of failure for those
beams was a critical magnitude of shear force, Vo.
The magnitude of Vo was determined by the shear
compression moment, Msw, and the critical ratio
(M/Vd) cr. The resulting failure was classified as a
diagonal tension failure since the beams failed sud
denly as soon as the critical shear force was at
tained. The relative magnitudes of the flexural
ultimate and the shearcompression moments and
the value of (M/Vd) cr determined the upper limit
of the transition region, point (a/d), = (M/Vd)2
in Fig. 26. Beams with ratios M/Vd higher than
(M/Vd)2 failed in flexure since their flexural ca
pacity was reached prior to the attainment of the
critical shear force, Vo. The validity of this hy
pothesis and the critical value of M/Vd could not
be determined because of insufficient experimental
data.
f. ShearProper. Conversely, for very small
values of a/d the beams did not fail in shearcom
pression. The mode of failure appeared to be an
actual shearing off of the compression zone of the
concrete. This type of failure was tentatively called
shearproper. It was also found that the shear
Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS
strength of such beams depended on the x/Dratio
rather than a/dratio, where x denotes the clear
distance between the loadbearing blocks and D
the total depth of beam. For x/D equal to zero,
the shear strength of a beam could be related to a
nominal shearing stress. The entirely empirical ex
pression represented by Eq. 48 was found to give
satisfactory agreement with test results.
As the ratio x/D increased, the ratio between
the test and calculated shearing stresses decreased
as shown in Fig. 28. Since the number of tests was
limited, no expression could be determined for the
relationship between v/vc and x/D.
For small values of x/D the location of the
loadbearing blocks forced the formation of almost
vertical cracks and, consequently, vertical stirrups
were not found to contribute to the shear strength
of the beam. However, as x/D increased in the
region of shearproper, cracks followed the edges
of the bearing blocks and vertical stirrups crossing
the crack produced a slight increase in the shear
strength. The transition region between shear
proper and shearcompression was estimated to lie
approximately between x/D equal to 0.8 and 1.1,
increasing as the amount of vertical stirrups in
creased. Since the contribution of vertical stirrups
is very small, inclined stirrups should be used
whenever the x/Dratio approaches unity.
g. Uniform Loading. For simplespan beams
under uniform loading, the value of M/Vd ranges
from zero at the section of no moment to infinity
at the section of maximum moment. It is believed
that with certain modifications the shearcompres
sion equations, Eqs. 18, 35, and 28, could be used
to determine the shear strength of such beams.
Since there are no diagonal cracks in the region of
maximum moment and the inclination of cracks is
very small for high values of M/Vd, the beam can
not fail in shear unless the bending moment is
higher than the shear strength given by Eq. 28 at
a critical value of M/Vd. From test results studied,
this critical value of M/Vd was set tentatively at
about 4.5. However, since only a few Tbeams have
been tested under conditions which simulated uni
form loading, the validity of the above concept of
shear failures and a more reliable value of the
critical M/Vd must be established by a more com
prehensive test program.
It is conceivable that the same procedure can
be used for any type of beam under either uniform
or distributed loading to determine its strength in
shear. It involves only the determination of critical
sections for shear failures. Provided that the value
of M/Vd is in the region of shearcompression,
Eqs. 18 and 28 can be used directly at sections
where maximum shear and maximum moment co
incide. Where these maxima do not coincide, the
critical section at which the shearcompression
equations should be used is given by the critical
value of M/Vd.
Since sheartype failures result in a sudden and
complete destruction of a structure, they should be
avoided in actual construction. In order to deter
mine the amount of web reinforcement necessary
to ensure flexural failures, the flexural capacity of
the beam should be determined first and the cor
responding loading considered as applied loading.
Then, both the applied moment and shear moment
of Eqs. 18 and 35 should be determined for critical
sections of shear failure. The ratio between the two
substituted for M,w/M, in Eq. 28 will determine
the amount of web reinforcement required.
h. Indeterminate Structures. One additional
problem is confronted in statically indeterminate
structures whenever redistribution of moments near
the ultimate load is considered. In order to utilize
the full loadcarrying capacity of the structure, its
members must be so designed as to permit sufficient
rotation at the plastic hinges. Consequently, not
only primary but also secondary shear failures after
yielding of the reinforcement must be prevented.
This is a phase of the phenomenon of shear in re
inforced concrete which has received very little at
tention in the past.
23. Summary of Equations.
a. ShearProper. For x/D = 0 the shear
strength of a beam is determined by the following
expression:
V
VC = bD = 200 + 0.188f,' + 21,300 pt (48)
where
A, (1 + sin a)
Pt = bD
(47)
as x/D increases, the ratio v/vc decreases. The re
lationship between x/D and v/vc could not be de
termined, although some information is available
from Fig. 28. The transition region between shear
proper and shearcompression was estimated to lie
between x/D equal to 0.8 and 1.1, depending on
the amount of vertical stirrups. Otherwise, the
ILLINOIS ENGINEERING EXPERIMENT STATION
effect of vertical web reinforcement is neglected in
Eq. 47.
b. ShearCompression. In the shearcompres
sion range the shear strength of a beam without
web reinforcement and under concentrated loads is
given by the following equations for the maximum
shear moment, Ms:
For rectangular beams:
M. = 0.57  4.5f' (18)
bdf,' (k + np') 105
where k is given for beams reinforced in tension
only by
k = V (pn)2+ 2pn  pn (14)
and for beams reinforced in both tension and
compression by
k = V [n (p + p')]2 + 2n (p + p'  p't)
 n (p + p')
where
A MB+ 1) (1  kk,)
AMA
k = V (pon)2 + 2pn  pan
MA)
(45)
(42)
(43)
k. = 1.11  V/1.23  0.926 k (23)
k2 = 0.45
The contribution of web reinforcement is de
termined in all cases by the following expression for
the ratio of the maximum moment capacity Msw
of the beam with web reinforcement to the moment
capacity Ms of the same beam without web rein
forcement:
Mw + 2rfyw
M. 10,
where
and where n is given by
n=5± 10,000
n = 5For Tbeams:+
For Tbeams:
A
r = sw for rectangular beams
bs sin a
(16) and
r = 'A for Tbeams
b's sin a
(27a)
M, 4.5fo'
A ' 0.57  45f (35)
Acdf,'Ft 105
where
SIT±  Icr
Ft = IR + Icr (34)
For restrained beams: the shear strength is
given by Eq. 18, whenever bond failures are pre
vented, and by the following equation whenever
redistribution of internal forces has taken place as
a result of local bond failure in the high bond
stress region:
M = 0.57 4.5f' (44)
bd2f' kA 101
The upper limit of a/d = M/Vd for shearcom
pression failures could not be determined; the high
est value used in tests was 4.8. For high values of
M/Vd the shear strength is larger than that given
by the above equations.
c. Distributed Loading. At a section where
maximum moment and maximum shear coincide,
the shear strength of a beam under distributed
loading can be determined directly by the above
shearcompression equations, provided that the
value of M/Vd is in the range of applicability of
these equations. However, in regions of maximum
moment and no shear, the above equations should
be used at a section given by M/Vd equal to
about 4.5.
J
VIII. BIBLIOGRAPHY
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April 1953.
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4. Moretto, 0., "An Investigation of the Strength of
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nois, 1943.
8. Richart, F. E. and Jensen, V. P., "Tests of Plain and
Reinforced Concrete Made With Haydite Aggregates," Bul.
No. 237, Eng. Exp. Station, University of Illinois, 1931.
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"Concrete Beams With SheetSteel Web Plates," Civil
Engineering, Vol. 8, No. 12, Dec. 1938, pp. 815818; also:
Fehrer, J. N., "Tests of Concrete Beams With Steel Web
Plates," M.S. Thesis, Johns Hopkins University, 1937.
10. Galletly, G. D., Hosking, N. G., and Ofjord, A.,
"Behavior of Structural Elements Under Impulsive Loads
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Investigation of the LoadDeformation Characteristics of
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Civil Engineering Studies, Structural Research Series No.
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crete Beams Failing in Shear," Ph.D. Thesis, University of
Illinois, 1953.
13. Mylrea, T. D., "Bond and Anchorage," ACI Journal,
March 1948, Proc. Vol. 19, p. 521.
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