THE SHEAR STRENGTH OF REINFORCED CONCRETE T-BEAMS by René Hakkenberg van GafLsbeek A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Master of ,Engineering. Department of. Civil Engineering and Applied Mechanics, McGill University, Montreal. @) René Hakkenberg van Gap.sbeek 1967 Novem be r 1966
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THE SHEAR STRENGTH
OF
REINFORCED CONCRETE T-BEAMS
by
René Hakkenberg van GafLsbeek
A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the
requirements for the degree of Master of ,Engineering.
Department of. Civil Engineering and Applied Mechanics,
McGill University, Montreal.
@) René Hakkenberg van Gap.sbeek 1967
Novem be r 1966
TO MY PARENTS
TABLE OF CONTENTS i
List of Figures List of Tables . Synopsis Acknowledgements . Nomenclature •
CHAPTER
ONE
TWO
THREE
FOUR
FIVE
SIX
SEVEN
EIGHT
Introduction A. The Problem B· Purpose of Investigation .
Historical Review
Theoretical Discussion A. Failure Criteria for Concrete • B. Ultimate Strength of T-beams •
Mechanisms of Failure •
Test Assemblies , A. The Test Beams • B. The Reinforcement C. The Formwork
Experimental Procedure.
Test Materials and Materials Testing A. Mortar . B. Steel
Results, Observations and Analysis A. Crack Formation. B. Failure Loads and Stresses C. Dial Gauge Results .' D. Strain Gauge ~esults . E. Deflection of Sponge Rubber F. Comparison of Results with
Existing' Theories .
Page i
iv v
vi vii
1 3
5
31 33
45
49 52 57
60
69 73
78 85 96
108 110
121
CHAPTER
NINE Conclusions A. Summary • B. Future Research
BIBLIOGRAPHY
Page
130 134
136
Figure No.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
LIST OF FIGURES
Kani' s "Concrete Teeth" .
Stress-Strain Curve of Concrete
Stress and Strain Diagrams at Eventual Section of Failure
Stress and Strain at Adjacent Sections .
Series 1 - Beam Dimensions, Shear and Bending Moment Diagrams •
Series II - Beam Dimensions, Shear and Bending Moment Diagrams •
Series III - Beam Dimensions, Shear and Bending Moment Diagrams .. •
Reinforcement Details •
Flange Reinforcing .. ..
The Formwork •
Beam Ready for Testing
Experimental Set-Up •
Steel Plates on Rubber for Rectangular Beams
Dial Gauge Set-Up •
Numbering and Location of Dial Gauges
Numbering and Location of Strain Gauges ..
Grading Curve for Sand.
•
Typical Stress-Btrain Curve for Concrete Mortar ..
i.·
Page
27
34
34
38
53
54.
55
56
58
58
58
62
66a
66a
67
68
72
75
Figure No.
19.
20.,
21.
22.
23.
24.
250
26.
27.
280
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
Typical Stress-8train Curve for #2 Bar.
Typical Mechanism of Failure for Series 1 and n with Typical Dimensions •
Crack Pattern of Beam III - 9 x 8/4 a
Crack Pattern of Beam III - 9 x 3/4 b
Crack Pattern of Beam III - NF a
Typical Diagonal Tension Crack for Series 1 and II
Bearn II - 15 x 1 a after Failure
Beam III - 9 x 3/4 b after Failure
C rack Pattern of Beam III - NF b
Load Deflection Curves
" " "
" " "
" " " •
" " "
" " "
" " "
" " "
Experimental and Theoretical Load-DefiectiQn Curves for Beam III - 9 x 1 b
Load-8train Curves
" " "
" " "
" " "
li.
Page
77
79
83
83
85
87
87
88
88
99
100
101
102
103
104
105
106
107
111
112
113
114
Ui.
Figure No. Page
41. Load Strain Curves 115
42. " " " 116
43. " " " 117
44. Load Deflection Curves for Rubber 118
45. " " " Il " 119
46. " " " " " 120
Table No.
1.
2.
3.
4.
5.
6.
7.
8.
9.
LIST OF TABLES
Sieve Analysis of Sand
Typical Strain Gauge Results of Two Test Cylinders for Determination of the Modulus of Elasticity and Poisson' s Ratio
Typical Steel l'roperties
Concrete and Bearn Data
Test Results •
Evaluated Results .
Typical Dial and Strain Gauge Resulta at Center Span
Comparison of Te~t Data with Theoretical Shear Moment Values. •
Comparison of Allowable Shear with Actual Shear Stress •
iVe
Page
71
74
76
90
92
94
109
125
127
v.
SYNOPSI~
Tests on eighteen T -beams q,nd six rectangular beams are
reported. The rectangular beams have dimensions equal to the web
of the T-beams and serve on a strength comparison basis. The flange
widths and thicknesses of the T-beams are varied to study the effect
of this variation on the shear strength. AlI beams are subjected to a
uniformly distributed load by a newly devised loading system. The
end conditions of the beams vary from fully fixed to simply supported.
A study is made of the cracking patterns, the initial diagonal
cracking loads and the ultimate loads. "'oad-deflection and load-strain
curves are presented. Results are compared to code requirements
and to some recent ~heories presented by other investigators.
Ultimate loads are fairly scattered and no definite trend is
found of increasing shear strength with increasing flange dimensions.
A theoretical derivation is presented of the ultimate strength of
reinforced concrete T-beams without stirrups.
ACKNOWLEDGEMENTS
The author wishes to express his appreciation and sincere
thanksfor the assistance given to him by the following:
vi.
Professor P. J. Harris, the author 's director of research.
Messrs. N. Ahmed and B. Cockayne of the Strength of
Materials Laboratory, for their assistance in the fabrication and testing
of the test specimens.
Miss Sheila Paulin who generously helped the author in
the editing ~d proof reading of this thesis.
Mr. J •. A. Pastega and Mr. H. Reichert for their moral
support.
The National Research Council of Canada, Grant No. A-2737,
who provided the funds for the prosecution of the research.
A
Av
a
b
b '
d
f ' c
fv
ft
j
1
M
Mu
r
v
v
w
NOMENCLATURE
area of cross-section
total area of web reinforcement within distance s
length of shear span
width of web of rectangular section or width of flange of T-beam
width of web of T-beam
distance from extreme compression fibre to centroid of tension reinforcement
compressive, uniaxial strength of 3 x 6 - inch cylinder at time of test .
tensile stress in web reinforcement
tensile stress at a point
ratio of distance between centroid of compr~ssion and centroid of tension to the depth, d
general term for length
bending mOlllent
bending moment at ultimate load
flexural moment, neglecting shear
ratio of area of stirrups and the product bs
external shear force
external shear force at ultimate load
nominal shear stress
load in kips per inch of beam
vii.
1.
CHAPTER ONE
INTRODUCTION
A. The Problem.
The problem of shear and diagonal tension in reinforced
concrete beams has been a major concern of engineers since the
inception of the use of concrete as a building material. Although an
extremely large number of tests has been performed over the last sixt Y
years, no decisive breakthrough has occurred. Such questions as what
failure mechanism will occur; what is the initial cracking load; what is
the ultimate load have yet to be answered accurately and reliably for
any set of conditions. The main reason for this lack of knowledge is
the large number of parameters known to influence the shear strength
of a retnforced concrete beam. These include the concrete cylinder
strength (f c), the grade and percentage of reinforcing steel, as well as
its arrangement and location, the cross-sectional shape (e.g.
rectangular, T-section, L-section) and its absolute dimensions (e.g. b, M
b', d), the Vd ratio, the type of web reinforcement (e.g. vertical or
inclined stirrups, bent-up bars), the type of loading (single or multi-
point loading, uniformly distributed, symmetrical or unsymmetrical
loading), the type of support (simply supported, semi- or fully fixed).
The problem of shear is further complicated by the redistribution of
internaI forces after cracking has occurred.
2.
The standard code formula for the nominal shear stress:
v=~ bd
•.. (1.1)
does not take into account most of these parameters and therefore
appears to be inadequate. Moreover, it seems illogical that in the
ultimate strength design method the nominal shear stress is obtained
with the above formula which is based on elastic conditions and the
fiexural stress is obtained from formulae based on plastic conditions
in the concrete. .
The many investigations into the shear J,roblem that have
been carried out have led to numerous empirical or semi-€mpirical
formulae. These formulae usually agree quite well with the correS-
ponding test results but are not applicable for general use, as they
pertain only to one particular set of beam parameters and loading con-
ditions and do not permit a rational study of the various variables
involved. A theoretical solution involving all variables would be
ideal, but this would be very difficult to achieve due to the unknown
effect of the interaction of the large number of variables and also because
the failure criterion of concrete is not fully known.
3.
B.Purpose of Investigation.
One of the most commonly used geometric shapes in
reinforced concrete is a t1oor- or roof slab, cast integrally with a
supporting beam, the ensemble forming a T-section. Consultation of
building codes reveals that the formula for the nominal shear stress
of T-beams is the same a.s the one used for rectangular beams
(equation 1 .1), replacing the width of the rectangular section, b, by
the width of the web of the T-beam, b'. Thus for T-beams:
v v =b'd ... (1.2)
One notices that the capacity of the flange, if any, to resist
part of the total shear force, is neglected in this formula.
The purpose of this investigation is:
1. To determine whether the flange of a T-beam has an effect
on its shear strength and, if so, a quantitative determination thereof.
2. To provide information on initial flexural cracking-loads,
initial diagonal tension cracking-loads and ultimate loads of fully
fixed, restrained and simply supported, uniformly loaded, T-beams.
Such information is completely or almost completely lacking in the
presently available literature on concrete research.
3. To provide load-deflection information on such beams.
4. To compare the test results with building code requirements
and sorne recent work in this field.
4.
5. A theoretlcal discussion on the ultimate strength of such
beams.
6. A comparison of the test results for T-beams with those
of rectangular beams with dimensions equal to the web of the T-beam.
CHAPTER TWO
HISTORIC AL REVIEW
5.
Reinforced concrete members were used long before the basic
principles were understood or a. rational design theory was developed.
Patented design systems were used in those days, the design methods
of which Were not known to the public. Serious questioning of these
systems did not sta:rt until the late 1800' s. Two schools of thought
developed concerning the mechanism of shear failure in reinforced
concrete members. One school of thought considered horizontal shear,
as experïenced with web rivets in steel girders, to be the cause of shear
failures in reinforced concrete members. Concrete was thought to have
a low horizontal shearing strength and vertical stirrups were used as
shear-keys against high horizontal shear stresses.
The second school of thought considered diagonal tension
the caUSe of shear failure. Although it is not known who develo~d the
original idea, in 1899 W. Ritter plèsented the first report on diagonal
tension as a mechanism of shear failure of remforced concrete members.
He also presented the" truss analogy", in which a reinforced concrete
beam with stirrups is compared to a truss in which the concrete com
pression zone acts as the top chord, the longitudinal reinforcing as the
bottom chord, the stirrup:; or bent-up bars as the web members in
tension and the concrete in between the compression zone and the
6.
longitudinal reinforcing as the web members in compression. Four
assumptions were made in the development of the truss analogy:
1. The tension reinforcement carries only horizontal tensile
flexural stresses.
2. The concrete compression zone carries only horizontal
compressive flexural stresses.
3. The stïrrups or bent-up bars carry aU inclined or vertical
stresses.
4. A diagonal t.ension crack extends from the tension rein-
forcement up to a height equal to the effective moment arm jd.
The truss analogy forms the basis for several code design for mul ae 1
such as for the spacing of vertical stirrups:
. , .(2.1)
Neville and Taub (26) demonstrated that the truss analogy is not really
valide In plrticular, they showed that the use of sm aller stirrups at
smaller pitch resulted in a considerable increase in ult!mate load than A
when larger stirruIB at larger pitch but with equivalent ; were used.
Although discussion between the two schools of thought
continued, experimental testing showed concrete shear strength to be
considerably larger than Us tensile strength,which seemed to support
the conception of di.agonal tension.
7.
A major contribution to the undershmding of reinforced
concrete members in shear was made by Morsch between the years
1902 and 1910. He pointed out that shear failures are the result of
principal tensile stresses and that, even in a state of pure shear, with
equal horizontal and vertical shear stresses, equal tensUe stresses
exist on planes at 450 to the neutral axis. He also develoJl:d the equation
for the nominal shearing stress:
. . . (2.2)
The fact that this formula is still universally used today (with the minor
change of omitting "j") shows the enormous influence sorne of the early
pioneers have had on modern design practice. The importance of
equation (2.2) is such that a closer look 1s justified. At any IDint in an
isotropie, homogeneous concrete beam the principal tensUe stress cau
be related to the shear stress and flexural stress at that poïnt by the
equation:
... (2.3)
Although equation (2.3) is not an accurate failure criterion for concrete,
most texts on reinforced concrete adopt it, with an explanation of its
approximate applicability. It is then argued that in the region of high v,
ft is relatively small, and thus ft (max) or the diagonal tension stress is
8.
approximately equal to v. Therefore the nominal sheari.ng stress as
expressed by equation (2.2) can only be llsed as an indication of the
diagonal tension stress and not as a quantity equal to it
Research in diagonal tension was also carried out in the
United States. Diagonal tension was taken as the cause of shear failure,
the horizontal shear idea being rejected by most early investigators.
In 1906 A. N. Talbot~ from the University of llUnois,published the first
report on modes of shear failures of reinforced concrete beams.
These were described as a fffailure due to yielding of the tension steel,
compression of the concrete, shearing of concrete, bond or slip of
reinforcing bars, diagonal tension of concrete and sorne miscellaneous
methods such as splitting of the bars away from the concrete." The
earliest research on T-beams was also performed by A. N. Talbot (1) 1
in 1906. Nine tests were reported on 8 x 12-in. T-beams with 3 if -in.
flange thickness, 16-in. to 32-in. flange width and a 10..ft. span. Flange . width varied from two, to three, to four times the web width. AU beams
1 were loaded by symmetrical cO,ncentrated loads at the 3' -points. The
object was to test the effect of the different flange widths and also to
test the efficacy of vertical stirrups in resisting flange stresses.
Unfortunately an beams failed in flexure, after considerable yielding
.,.. Source: Hognestad (9)
9.
of the tenslle reinforcing. The author concluded that aU beams
'exhibited in a common way the characteristics of rectangular beams
falling in flexure." The vertical stirrups were thought to be very effec
tive as the nominal shearlng stress reached 605 psi in one case. But
since no diagonal tension fallures took place, the real effectiveness of
the stirrups was not determined. Talbot suggested that the maximum
strength of T-beams to resist flexural stresses may be calculated using
the common formulae used for rectangular beams, taking the enclosing
rectangle of the T-beam to be the equivalent rectangular beam. On
the other hand, the actual width of the web should bé used in calcu
lating the vertical shear and diagonal tension stresses.
It is interesting to note that some of the early investi-
gators in the early 1900' s, such as Ritter, Morsch and Talbot, developed
most of the basic ideas, design formulae and fallure criteria that are
nowadays universally accepted. This proves the enormously complicated
task of arriving at a complete solution to the shear and diagonal
tension problem of reinforced concrete members.
In 1907 Withey * published two reports in which formula
(2.1) was introduced into tha American literature. He found that the
equation gave too high stirrup stresses and concluded that the concrete
of the compression zone carried considerable shear stresses. He also
if Source: Hognestad (9)
10.
pointed out the possible dowel action of the tension reinforcement.
In 1909 Talbot reported the results of tests in 188 concrete
beams and came to the following important conclusions:
"1. The nominal shearing strength v increases with cement
content.
2. v increases with the age of the concrete.
3. v increases with the amount of longitudinal reinforcing.
4. v increases with decreasing span of beam for the same
cross-section.
5. Bent-up bars were found to be most advantageous when
distributed over the region of high shear.
6. It was recommended that stirrups be dimensioned for
two-thirds of the external shear, the remaining one-third
being carried by the concrete in the compression zone."
By 1910 the horizontal shear viewlDint was virtually abandoned
in favour of the diagonal tension concept of shear fallures of reinforced
concrete beams.
In 1914 L. J. Johnson and J. R. Nichols (3) relDrted tests
on 28 T..beams. The webs were made first and only after several days
of hardening were the flanges poured. The object of the tests was to
check whether the joint between the web and flange was sufficiently
strong under load. Stirrups which extended from the web into the flange
11.
were used and the top surface of the web was roughened in sorne caseS
to determine any difference of behaviouf or slip between flange flnd
web. The author pointed out that if "sufficient" stirrups are provided
for the an\l!horage of tne ~ange, the beam can be considered as a
regular T-beam, and designed as such. The meaning of "sufficient" was
not defined and in the discussions following the report this weakness
is pointed out.
In 1915 J. Gilchrist (2) published a report in which he
pointed out that in the design method for reinforced concrete beams
(including T-beams) that was used at the time, the increase in the
ultimate shear strength of the web due to the }resence of stirrups is
added to the increase of ultimate strength due to bent..up bars. The
shear strength due to stirrups and bent-up bars are then added to the
strength Qf plain concrete. The author concluded that tests, made by the
German Reinforced Concrete Committee, revealed that this addition
is not correct as the· shear stren~th obtained by adding the three indi-
vidual strengths were too high, thus giving unsafe designs. He alao
cop.cluded that the shear strength due to stirrups is not directly propor ..
tio~al to the stirrup-area. An empirical formula was proposed as a
means of calculating the ultimate tot~l shear strength of the web of a
T-beam. Limiting values of ultimate shear stress were suggested for
three different a~ounts of web-reinforcement, for one particular f c .
12.
The same author published another report on T-beams in
1927 (4). Tests were done on T-beams to determine the vartation of
the limiting shear stress (as mentioned in his 1917 report) with
variations in f C. First a relationship was found between the compressive
and tensile strengths of concrete. This was found to be:
lt is the t~nsile strength of the concrete that should be compared to the
calculated shear stress, as the cracks that ultimately give rise to
failure of the beam are due to the tensile strength of concrete being
exceeded by the tensUe stresses associated with shear. The method
of determination of the tensile strength was not specified in the report.
Tlte author pointed out that it was clear from the tests' on T-beams,
that the cracking and ultimate shearing stresses are not solely a
{unction of the tensilestrength of concrete. Other conclusions were:
1. tensile strength is not directly proportioned to the shear
strength;
2. tensile reinforcement has an infl~ence on the shear strength; _ V _
3. derived equation: v = Da ; where v is the shearing stress
at the neutral axis;
4. neglecting the ten&ile strength of concrete when determining
d is suff~ciently accurate.
13.
In 1934 Mylrea (5) published a report entitled 'Tests of
Reinforced Concrete T-beams" in which the Scott system of reinforcing
was described. In this system many small tensile reinforcing bars were
used which were bent up at points where they were no longer needed
to carry the bending moment. They crossed the neutral axis at 450 and
whan the flange was reached the bars were bent transversely into the
flange where they were anchored with small anchor plates. A detailed
description was presented of tests on 5 T -beams, reinforced with the
Scott system. Nominal shearing stresses of up to 1200 psi were observed
much to the surpr ise of many investigators who did not realize that
in the Scott system the reinforcing bars are highly effective because they
follow the principal tensile stress trajectories of the web.
In the period from 1910 to about 1945 relatively few contri
butions were made towards the understanding of the shear problem.
After 1945, however, much effort was put into obtaining a rational
mathematical expression for the shear strength of reinforced concrete
beams.
ln 1951 A. P. Clark (8) tested beams of two cross-sections,
four span lengths and varying f c under different loading conditions. He
found tha~ after the yield stress was reached in one stirrup the stress
in adjacent stirrups would increase, indicating a redistribution of
internaI stresses. Resistance to failure inqreased as the loading points
14.
were shifted front the center towards the supports. An empirical
formula was suggested that agreed closely with the test results, but
genefal application was not recommended. The formula indicates that
the shear strength of beams varies with (1) ~he percentage of longitudin~l
reinforcement, (2) the square root of the percentage of web reinforce-
ment, and (3) the compressive concrete strength, multiplied by the ratio
of the effective depth and the shear span length.
Tests on 25 T-beams were reported by P. Ferguson and
J. N. Thompson (10) in 1953. One series of tests consisted of normal
T-beams, the other of T-beams with extra web width over part of the
beam depth. Concrete strength was also a variable. Higher values of
vult. were found for higher f c values but ~ decreased for increasing f ' c
values of f è and the code' s unit allowable working stress of 0.03 f è
was considered too high for high f è. The authors pointed out a large a a
variation in shear strength for different d ratios. For small d large
values for v were obtained which was attributed t9 compressive stresses
in the concrete near the support which greatly reduce diagonal tension
stresses in that area. The "shoulders" on the second series of beams
were found to increase the shear strength and it was suggested that the
area of the shoulders below the neutral axis be added to b'd for use in
the nominal shear formula.
In 1954 Moody, Viest, Elstner and Hognestad (14) publisbed
a four part report on reinforced concrete beams under different types
15.
of loading and supports, ranging from sim ply supported to fully fixed. a
Test results indicated that "the strength of beams with large ëi ratios
is governed by the load causing first cracking whereas the strength
of shorter beams is governed by the load causing the destruction of the
compression zone above the diagonal tension crack." A formula expressing
the ultimate shear load multiplied by the shear span (Vua) in beam
geometry-terms multiplied by f c , suggested that the ultimate strength
depended on the ultimate resisting moment (Vua = Mu) of the beam,
independent of the magnitude of the shear. This was later supported by
Brock (24) for a certain range of â- and t. The redistribution of
internaI stresses was also emphasized. Before cracking occurs the
internaI moment and shear are distributed along the beam in the same
way as the external moment and shear. Once a crack forms, however,
a redistribution of internaI stresses takes place, suddenly increasing
the stress in the steel at the location of the crack. Conclusions drawn
from the tests on restrained beams without web reinforcement were
identical to the ones from simply supported beams, namely that the
cracking load may be predicted on the basis of nominal shearing stress
and the ultimate load from the ultimate moment. Analytical expressions
were developed for the diagonal tension cracking strength and the
shear strength of simply supported and restrained reinforced concrete
beams, loaded by one or two concentrated loads. The authors pointed
16.
out that the exprf~ssions are only vaUd for beams with a constant
maximum shear force over a part of the span and a maximum moment
occurring at the loading point(s). The writer of this thesis has doubts
on the value of the equations as the one or two point concentrated loading
system can certainly not be considered as representing a generally
practical case. The expressions derived should be compared to
expressions for distributed loadings to determine the discrepmcies in
load capacities. In 1955 A. Laupa, C. P. Siess and N. M. Newmark (12)
published an extensive analytical report. The object was to study and
correlate test results of earlier investigations in the field of shear and
diagonal tension to determine modes and characteristics of shear
failures and to derive analytical expressions for the strength of rein-
forced concrete beams failing in shear under different loading conditions.
It was pointed out that the conventional formula for the nominal shearing V
stress, v = bjd , cannot he a true criterion of shear failure as no
transfer of stresses across cracks occurs and only the compression
area above the cracks should be used to expresS the nominal shear
stress at failure. Dowel action by the longitudinal reinforcing was
neglected. It was assumed that the ultimate unit shearing stress was a
function of f ~. Also, since no expressions for the depth of the com
pression zone are available for shear failures, thià depth has to be
determined empirically. The factor ks , which multiplied by d gives the
17.
depth of the compression zone, is found to be a non-linear function of
f c and p~ Using the above assumptions the following equation was obtained:
M = kF(f c) ... (2.4)
where, k refers to the theoretical depth of the compression
zone as ordinarily obtained from the transformed
steel area.
For beams with longitudinal steel reinforcement only:
k = - pn , .. (2.5)
also F(f~) refers to some' function of f C' related to
the limiting average compressive stress.
M From a plot pf bdafck versus f é it was found that for f è between
1000 and 6000 psi, F(f é) can be approximated by the linear equation:
= 0.57 _ 4.5f é 105
,., (2.6)
Substituting equation (2.6) into equation(2.4) yieldsan equation for the
moment, called the shear compression moment, at which a reinforced
concrete beam, with tension reinforcement only will fail:
2 4.5f é M = bd f ~ k(0.57 - ) 10 5
. . . (2.7)
18.
No assumptions were made regarding the effect of shear-span to depth
ratio or the ultimate shear strength and therefore equation (2.7) seems
to be applicable to beams loaded by concentrated loads as well as
uniformly distributed loads. This formula is subsequently expanded
to include the effects of compression reinforcing and/or web reinforcing.
Equation (2.7) which applies to rectangular beams was
modified to be applicable to T-beams. It was thought that the effect of
the geometrical shape of the beam is primarily dete"rmined by its moment
of inertia. At the instant of failure the ratio of the moment of inertia
of a T -beam and a rectangular beam is unknown. This ratio was
approximated by the ratio of the average values of lof the uncracked
~nd fully cracked state. Thus the shape factor is~
where,
. " .(2.8)
It and Ir refer to the uncracked T- and rectangular
sections respectively and 1er refers to the fully
cracked section of either a rectangular- or T-beam
(with equal b) as both have nearly the same 1 in the
fully cracked state and is obtained from the "straight
line "cracked and transformed section with k given
by equation (2.5).
Replacing the compression area bkd as obtained by the conventional
straight line theory by Ac in equation (2.4) and substituting we obtain
an equation for the moment at failure of T-beams without web
reinforcement.
Thus: M 4.5f C
A df ' = 0.57 - 105 c cee
19.
.•• (2.9)
It is the opinion of the writer of this thesis that the shape factor as
defined by equation (2.8) is a very arbitrary choice. No attempt was
made to justify its use,except that in most caSeS equation (2.9) agrees
fairly well with the test results. In sorne series of tests, however,
consistently lower shear strengths were found than pTedicted. The author
concludes that "this discrepancy could mean either that the sha~ factor
is fundamentally incorrect or that there are some other considerations
besides the effect of moment of inertia which determine the compressive
strain in the concrete." No attempt was made in the report to define
the effective flange width b. The discrepancies between the test results
and equation (2.6) occur mainly when b is large and this seems to point
to the fact that only some part of b acts as an effective compressive
concrete area. Thus equation (2.6) cannot he used for practical design
purposes unless some means is found to determine the effective flange
width. The report then continues to discuss beams under distributed
loadingo The difficulty with distributed loading is that unlike single or
two-point symmetrical loading where the critical section occurs where
both the bending moment· and the shear force are maximum, namely
20.
under the loading point, the critical section is unknown. Under M
distributed loading the Vd ranges from zero to infinity from the support
to the center of the beam. Equation (2.9) was assumed to be appicable M M
for values of Vd between certain limits. The value of va- which limits
the region of critical diagonal cracking capable of producing shear com-
pression failures was then determined empirically. This was done by
plotting along the length of the beams the ratio of the actual moment at
failure and the moment as obtained from equat~on (2.9). It is recalled
that when this ratio reached the value of one for a beam under concentrated
loads, a shear failure would occur: From the plot it was found that the M~~ M
ratio of M 1 equalled one at a value of ver about equal to 4.5. Thus ca c.
it appeared that equation (2.9) is applicable to T-beams under uniformly
distributed loading if the section at which the moment is calculated is M
that at which va is about 4.5.
E. 1. Brown (15) compared the strength of longitudinally
reinforced concrete T-beams under combined direct shear and torsion
to the strength under shear alone. He showed that reinforced concrete
under combined shear and torsion does not adhere to any particular
failure criterion, but the maximum stress criter~on, which is the
generally accepted failure criterion, was assumed applicable. This D
criterion offered no explanation for the 50% average increase in strength
under combined shear and torsion after cracking of the beams had
21.
occurred. Diagonal cracks formed under combined stresses at an average
of 66% of the ultimate load. Maximum rotation seemed to be a more
important factor determining the strength than maximum stress when
torsion is large. Plastic theory seems ta give more accurate results
in predicting the cracking load than does the more widely accepted
elastic theory.
A report entitled "Diagonal Tension Strength of Reinforced
Concrete T-beams with Varying Shear Span" was published in 1956 by
AI-Alusi (19). Emphasis,was laid on the variation of the shear span
and percentage of longitudinal reinforcement on the mode of failure, the a
'cracking and ultimate strengths. For values of CI between 4.0 and 8.0, v n; at first diagonal cracking and at failure were found to be essen-
tially constant and thus the actual ultimate moment increased in direct a
proportion to d. ~ a a
V- decreased with increasing a for CI between cr
2.0 and 4.0. The percentage of longitudinal reinforcement seemed to a
have no significant effect on the cracking and ultimate loads for d
between 4.0 and 8.0. The presence of comtression reinforcement in the
web or mesh reinforcement in the flange seemed to have no effect on
either cracking or ultimate loads. Whether failure was by diagonal a
tension or moment, was determined by the value of ëi and the ~rcentage
of longitudinal reinforcement.
Morrow, in his discussion of AI-Alusi' s report, suggests
the formula:
22.
4 0.15 + ~M~---
(-) + 10 V cr npd
for the cracking load of a T-beam, provided they are considered equivalent
to a rectangular section of dimensions equal to the web sizeof the T-beam.
Whitney, also in a discussion of the same paper suggests:
where,
Mu \ Id vcr = 50 + 0.26 (i1l V il ••• (2.11)
Mu is the ultimate moment call1city per inch of width.
Whitney also considers the scope of AI-Alusi' s tests too limited to accept
his conclusions without further research.
G. Brock (24) published an extensive study in 1960 in an
attempt to predict the mode of failure and the ultimate load of a rein-
forced concrete rectangular beam without web reinforcement and under
any type of loading. 'rhe author pointed out that the ultimate call1city of
a beam under a uniform bending moment can be established by well-
known design formulae. '.lJhen the beam was also subjected to shear
forces,the ultimate capacity might be lower than that predicted for
bending moment al one and the beam is said to fail in shear. Thus the
hypothesis was developed that the effect of shear is sim ply to reduce the
capacity of the beam in pure flexure. He distinguished between two
modes of shear failure which were called diagonal tension and shear bond.
23.
Diagonal tension cracks run roughly from the point of support to the
loading point, while shear bond is characterized by a breaking away of
the concrete at the level of the longitudinal reinforcemento Shear bond
failures were thought to be a transition from diagonal tension to flexural p
failures and their occurrence for a certain Po' de pend on f è and fy.
When both these values are high there is a greater chance for a shear
bond fallure to occur. Shear failures reduce the flexural capacity, and p M
are a function of the reinforcement index Po and the value of Vd M a
(note Vd : d for two concentrated loads) at the section of fallure.
To predict the critical section from known shear and bending moment
diagrams two assumptions Were made:
1. "The potential capa city for r~sisting moment at any section M
of the beam depends on the value of Vd at that section. This capacity Mu a
can be found from the curve of f 'bd i against d for the appropriate c
p value of p;-."
2. "T he critical section of the beam will be that at which the
actual bending moment first reaches its capacity value and the ultimate
load will be that load which produced the capacity morpent at the
critical section."
The actual capacity moment of the beam at any section is found by M
substituting the actual Vd value at that particular section onto the
curve of ~ against ultimate moment for that particular :a .
24.
Values of this moment capacity curve are plotted along the beam. To
determine the ultimate load for a particular capacity curve one has to
superimpose the bending moment diagram which Just touches the
capacity curve. The point where the two curveS touch will be the critical
section. The drawback of this system is that curveS of ultimate moment
against à for a particular:a are not generally available in p'actice.
Two series of tests were run on the shear strength of
restrained reinforced concrete beams under concentrated loading by
J. Bower and 1. M. Viest (25). In the Ursi: series the principal variable
was the ratio of the maximum positive to the maximum negative moment M Vcr
in the shear span. Plots of the +-M ratio against bd m show that
the moment ratio has no effect on the magnitude of the shear at initial
diagonal tension cracking. The sarne seemed to be true for the shear
at ultimate load, but this was not shown conclusively as the variation
in ultimate load was considerable (up to 30% from the average). In the
second series of tests the main variable was the ratio of the maximum M
moment and the external shear multiplied by the effective depth (Vd) •
It appeared that for the beams tested the shear-moment capa city was M
inde pendent of ver. The following expression was suggested:
bd Vfc .•• (2.12)
where, A and B are empirical constants,
V (-) the shear-rnoment ratio al the section of initial M c
diagonal cracking~
The authors assume that the section of initial diagonal cracking is located
at a distapce d ~way from the section of maximum moment, but no closer
than half-way to the other end of the shear span. Then,
(M.) = a - d V c
The method of least squares was used to determine the values of A and . Vc M m
B of equahon (2.12) from a plot of bd V f é v.s. (V) c' ïid' Values of A: 1.917 and B :: 2725 were found. It was ~tressed that the
capacity of beams aft.er initial diagonal tension cracking is unreliable
and that therefore the initial diagonal t~nsion cracking load should be
taken as the ultimate design load for beams without web reinforcement.
"Sorne Factors in the Shear Strength of Reinforced Concrete
Beams", a report published in 1960 by A. M. Neville (27) , indicated
that with the increasing USe of ultimate strength design which is based
on the pastic theory it is illogical to use the generally accepted shear
design procedure which is based on the elastic theory and an uncracked
section. Unlike Brock, Neville distinguishes between four different types
of shear failure consequent upon the forming of a diagonal tension crack.
Two of these are similar to the oneS suggested by Brock although they
26.
are named differentlyo The third type, which occurs when the upper
end of the diagonal tension crack exlends at a continuously decreasing
slope over the fulliength of the beam, thereby separating the beam
into 2 parts, is not discussed by Brock. The fourth type of failure in
shear occurs when a shear bond failure is prevented by action of the
stirrups which restrain the downward movement of the longitudinal
reinforcement, resulting in a yielding of the longitudinal reinforcement.
The test program consisted of beams of 3 different cross sections,
rectangular, T- and L-shapeso All beams were simpy supported and
loaded by 2-point concentrated loads. Shear span to depth ratios were
varied from 1.63 to 2.30 and to 3.41. Curves of deflection versus load
show a greater stiffness for T-beams than L- and rectangular beams,
which is to be expected due to their larger moment of inertia. Cracking
and ultimate loads are also generally higher for T-beams, "the increase
being greater the more favourable the conditions for shear failure."
Beams with compression reinforcement showed a change in
the mode of failure and the cracking pattern, but the ultimate strength
appeared to he unaltered. No attempt was made to develop an analytical
expression for the shear strength of T-beams as the flange size was
unchanged throughout the test program, leaving the actual flange-effect
unknown.
Several papers were pUblished by Dr. G. J. Kani (31), from
the University of Toronto, in which he noted that there is no such material
27.
constant as·"shear strength" in concrete, because tensile cracks will
always introduce flexural'fa'ilure before shear fallure can be reached.
Also, be,cause the shear stress is only one component of the total stress
field, it cannot be considercd as a true measure of the stress causing
diagonal tension fallure. A completely new concept of the mechanism
of diagonal tension was developed from which the diagonal compression
idea was derived. When the load on a beam is increased flexural cracks
form as is shown in Figure 1 â.
(Cl ) (b)
Figure 1: Kani' s "concrete teeth"
Two adjacent cracks separate concrete blocks which were called
"concrete teeth." Figure 1 b shows a free body diagram of one of the
teeth. The T and T + AT forces are due to the longitudinal reinforcing,
28.
resulting in a shear force of ë:. T at the root of the tooth. Because of
the A T forces, the force in the longitudinal reinforcement can vary
along the be am according to the bending moment. The teeth act as
unreinforced (if no stirrups) concrete cantilevers acted upon by a force
AT. If under increasing load one of the cantilevers breaks, the
A T force it carried is now taken over by the remaining teeth untH
aU teeth have br<*en off. Once that happens the beam is transformed
into a tied arch with a constant force T in the longitudinal reinforcement.
The resultant compressive force acts in a straight Une from the support
to the loading point (under 1- or 2-point loading). If sorne teeth are
still resisting load the compressive force acts in a curved line with
decreasing slope from support to loading point. The arch being in com-
pression gave rise to the term "diagonal compression."
In a second paper Kani (35) put forward the idea that the
maximum bending momentat failure, Mu, is a much better indication
of diagonal failure than the maximum shear stress at failure. Reasons
suggested are (quoting):
"1. The upper value of the flexural strength, Mfl, which depends on few parameters necessitates only a simple calculation.
2. The lowest values of Mu for all beams tested were in the vicinity of 0.50 Mn. Thus, all values of Mu range between 50 and 100 percent of Mfl instead of the 1500 ~rcent variation in VU evident ;:rom test results studied from the Uterature.
::s. The prevention of premature failure by the formation of a diagonal crack is the very problem of "shear failure." When we obtain a diagonal failure at 70 percent of the flexural failure load, this means that we are just 30 percent short of our goal, i.e. the full flexural capacity of the cross-section.
4. The purpose of the web reinforcement is to increase the strength of the beam to 100 percent of Mn. Thlls a result of Mu = 0.70 Mn for a beam without web reinforcement expresses the requirement: a web reinforcement which increases the capa city by 30 percent of Mn."
Mu a Kani subsequently Iiots Mn against d for one particular
fe ~ Comparison of these plots for different f c reveals that the shear
strength of rectangular reinforced concrete beams is independent of
the con crete compressive strength for the range of f ~ = 2500 to 5000 r ... 1 and
p = 0.50 to 2.80 percent. The percentage of longitudinal reinforcement Mu
is found ta have a large influence on Mfl ~ For p = 2.80 a value for M ~ of 0.50 is found while for the same concrete strength with '''~IMu p = 0.50 a value of 1.00 is obtained for Mn ~ The author suggests a
design procedure by expressing the ultimate strength of a beam by:
••• (2.14)
where r is a reduction factor which varies between 0.50 and 1.00 and a M
depends on the values of p and d or Vd. Values of r should he obtained
from tables or by formula. The problem of a good des ign would then
be ta determine the type and quantity of web reinforcement required to
increase -r to as close as possible to 1.
It bec orneS abundantly clear after reviewing the literature
on the shear problem in reinforced concrete members that notwith-
standing the large number of tests and their resulting emprical and
30.
semi-empirical design formulae, very little generally applicable
material has been developed. An abundance of results on sim ply
supported beams, under two-point loading is available. The fact that
simply supported beams hardly ever occur in practice and that two-
point concentrated loading, although convenient for any theoretical
discussion due to its constant maximum shear force and bending moment,
can certaiIÙy not be considered a practical loading arrangement, does
not Seem to occur to the investigators. It has, therefore, been the
object in the tests of this thesis to approach practical conditions as
closely as possible in beam geomp.try, material properties, loading and
end conditions. It is the author' s belief that if this had been done
consistently by aU investigators concerned, a far better understanding
of shear failure in general would have resulted than is now the case.
31.
CHAPTER THHEJ1:
THgORETICAL DISCUSSION
A. Failure Criterïa for ConcreLe
A fallure criterion of a mater'iai is an attempt. at an answer
to the question~ When does the materïal fa il ? Structural memhers
are usually subjec.t to a eomplex state of stresses. In concrete this is
further complicated by the non-homogeneity and non-isotropy of the
material and the preSence of steel reinforcement. Although the rein
forcement is assumed to have an effect on the mechanism of failure,
i.e., the prog'ress from initïal local failure to final collapse of the
member, it is generally assumed that the initial local failure in con
crete is the same for plain and reinfol'ced concrete. Thus, if the
failure criterion for plain concrete were known, the initiallacal failure
condition could be predicted for concrete members of any shape or
percentage of reinforcing.
Theories of failure have been praposed, such as the
maximum prilleiple stress theory (Rankine's theory), the maximum
shearing' stress theory (Coulomb's theory), the maximum strain theory
(Sto Venant' s theory) and others, but none af these seem to give reliable
resuUs far cancrete failures. Sorne emplrical failure criteria for
concrete have been developed by Mahr, Bresler and Pister, McHenry
32,.
and Karni, and Wastlund. These empirical criteria, which were
developed from test results should only be appied to structural mem-
bers that have a similar stress distribution to that encountered in those
tests. One such criterion is that of Bresler and Pister (20), which was
developed under uniform shear and compressive stresses and should
only be used under similar conditions. This criterion expresses the
relationship between the normal stress and shear stress at a point at
which failure occurs. ':'--
where,
h = 0.1 [0.62+7.86 (4) - 8.46 (~)J j ... (3.1)
\f - longitudinal direct stress at failure
'T = shear stress at failure
f c = nominal compressive strength of 6 12-in.
cylinders.
L. L. Jones (33) used Mohr's criterion with a parabolic
envelope to the stress circles in bis theoretical solution of ultimate
strength of rectangular concrete beams:
(a) When ••• (3.2)
failure at a point occurs when
'" Coefficients used are from the conservative "straight line" theory.
33 •
... (3.3)
(b) When ••• (3.4)
failure occurs at a point when
H.(3.5)
where, Q"L = numerical value of the ultimate uniaxial tensUe
stress
~ = 1 ~ f
fT = shear stress Xy
'rD = f ~ ( V(O(,;. ex.) - 0(, )
~ = longitudinal direct stress
B'. Ultimate Strength of T4>eams '1<
The following assumptions were made in the analysis below:
1. plane sections remain plane
2. longitudinal tensUe stresses in concrete are neglected
3. at failure the longitudinal reinforcement is yielding
4. no web reinforcement
'II" L. L. Jones' (33) work on rectangular beams i8 'modified and e;xtended to be applicable to T-beams under uniformly distributed load.
34.
5. The stress-strain curve of the concrete is parabolic with .
u ltimate values f ~ and Eu as shown in Figure 2.
po,ro/.)o{a
Figure 2. Stress-Strain 'Curve of Concrete
Figure 3 shows the assumed stress and strain diagrams at
ultimate loado If the load were again increased, much yielding would
occur in the steel a t the location of the cracko Deflectioll. would be
excessive and the load could no longer be considered to be uniformly
distributedo A further increase in load u.sually caused a diagonal
tension crack in the other shear span to widen and fail at loads somewhat
below the original failure loado In some other cases no initial diagonal
tension crack could be observed before the actual failure and the form-
ation of this crack and the failure of the beam took pace simultaneously.
Vi and Vu are the external shear forces at a distance d from
the center of the column.
Table 6 gives shear stresses at initial and ultimate loads. V· V
To take into account the variation in f é, columns of TI- and it are c c
included. As the modulus of rupture of concrete is more accurately
described as being proportional to
~ are included. Vi1
Vi VfJ , columns showing Vfi and c
Comparison of values of Table 6 reveal some interesting
results .. The increase in nominal shearing stress from the initial
diagonal tension cracking load to the ultimate load, taken as an average
for the six flanged beams per series, amounts to 40% in Series I,to 7C1fo
in Series n, .but is only 5% in Series ni. This indicates a considerable
"reserve capacity "beyond the initial diagonal cracking load for Series 1
and n, which i8 an important factor in ultimate load design. The low
Table 4
CONC~ETE"ANO "BEAM DATA
Beam No. f ' c vrr c b' d t bd bdf:' bd Vi! 1 -NF a 3180 56.4 3.65 6~25 22.81 72,535 1286 r-NFb 81.80 46.4 3.62 6.25 22.63 71,963 1275
1 - 9 x 3/4 a 4070 63.8 3.64 6.25 .82 22.75 92,592 1451 1 - 9 x 3/4b 4070 63.8 3.73 6.25 .86 23.31 94,872 1487 1 - 9 x 1-1/4 a 3510 59.4 3.54 6.25 1.30 22.13 77,676 1315 1 ~.g x"I-1/4 b 3510 59.4 3.55 6.25 1.30 22.19 77,887 1318 1 -15 x 1-1/4 a 3700 60.8 3.60 6.25 1.34 22.50 83,250 1368 1 - 15 x 1-1/4 b 3700 60.8 3.55 6.25 1.30 22.19 82,103 1349
TI -NF a 2700 52.0 3.80 6.25 23.75 64 125 , ... 1235 TI -NF b 2700 52.0 3.80 6.25 23.75 65,800 1267
TI -9 x 1 a 3250 57.0 3.56 6.25 1.05 22.25 72,400 1268 il-9rl b 3250 57.0 3.68 6.25 -1.02 23.00 74,750 1311 n -15 x 3/4 a 3013 54.9 3.71 6.25 .83 23.19 69,871 1273 il ~15 x 3/4 b 3013 54a9 3.63 6.25 .80 22.69 68,364 1246 TI -15 x 1 a 3100 56.4 3.65 6.25 1.10 22.81 70,711 1268 TI -15 x 1 b 3100 56.4 3.65 6.25 1.10 22.81 70,711 1268
(cont'd.)
CQ' 0 .
Beam No; fé Vi1 rn-NF a 3000 54.8 rn-NF b 3550 '59.6
In - 9x 3/4 a 2940 54.2 rn -9 x 3/4 b 2940 54.2 rn-9x1 a 3000 54.8 rn -9 x 1 -b 3550 59.6 m -9 x 1-1/4 a 3330 57.7 rn -9 x 1-1/4 b 3330 57.7
Pf Pi Pu Initial Flexural Initial Diagonal Ultirnate Vi Vu Cracking Load Cracking Load Load at x = d at x = d
Bearn No. 1 kips kips kips kips kips
1 -NF a 8 14 29.5 3.18 6.70 1 -NF b 10 16 29.8 3.63 6.76
1 - 9 x 3/4 a 12 19 32.1 4.31 7.29 1 - 9 x 3/4 b 16 30 37.3 --6.81 ltï17 1 - 9 x 1-1/4 a 10 19 28.6 4.31 6.49 1 - 9 x 1-1/4 b 13 22 28.0 4.99 6.36 1 -15 x 1-1/4 a 13 25 28.0 5.68 6.36 1 - 15 x 1-1/4 b 12 29 41.6 6.58 9.45
TI-NF a 12 20 37.0 4.82 8.92 TI-NF b 12 18 52.4 4.34 11.74
TI-9xl a 10 20 32.2 4.82 7.76 ll-9xl b 10 19 32.1 4.58 7.74 il -15 x 3/4 a 8 18 32.5 4.03 7.83 II -15 x 3/4 b 10 16 31.2 3.86 7.52 ':D
II - 15 x 1 a 14 18 22.4 4.03 5.40 llo:) . TI -15 xl b 12 18 28.6 4.03 6.89
(cont'd.)
Table 5 (cont'd.)
.TEST.RESULTS
Pf Pi Pu ln itiaI Flexural Initial Diagonal Ultimate Cracking Load Cracking Load Load
Bearn No. ki ki ·ki
ID-NF a 10 16 17.3 rn-NF b 8 19 19.2
ID -9 x 3/4 a 12 16 16.4 rn -9 x 3/4 b 14 19 20.4 rn -9 x 1 a 10 16 16.0 ID -9 xl b 10 18 18.-0 ID -9 x 1-1/4 a 10 19 20.5 ID - 9 x 1-1/4 b 10 18 20.2
V· 1
at x = d ki
6.29 7.47
6.29 7.47 6.29 7.07 7.47 7.07
Vu
at x = d ki
6.80 7.55
6.45 8.02 6.29 7.07 8.06 7.94
co
'" .
Table 6
EVALUATED RESULTS
v· \T; Vu
1 \T; Vu = bd Vu VU V' - Da 1 Vu
1 - 1
Bearn No. psi. belf ' bdVf1 psi Ddfc IiIVfè Vi c
I-NF a 139.4 .0438 2.47 293.7 .0924 5.21 2.11 1 -NF b 160.4 .0504 2.85 298.7 .0939 5.30 1.86
1 - 9 x 3/4 a 189.4 .0465 2.97 32004 .0787 5.02 1.69 1 - 9 x 3/4 b 292.1 .0718 4.58 363.3 .0893 5.69 1.24 1 - 9 x 1-1/4 a 189.4 .0555 3.28 320.4 .0836 4.94 1.69 1 - 9 x 1-1/4 b 194.7 .0641 3.78 293.3 .0817 4.82 1.51 1 -15 x 1-1/4 a 252.4 .0682 4.15 282.7 .0764 4.64 1.12 1 -15 x 1-1/4 b 297.1 .0802 4.88 426.1 .1153 7.00 1.43
n -NF a 202.9 .0752 3.90 375.6 .1391 7.22 1.85 II -NF b 182.7 .0660 3.42 494.3 .1784 9.27 2.70
II-9xl a 216.6 .0666 3.80 348.0 .1072 6.12 1.61 II-9x1b 199.1 .0613 3.49 336.5 .1035 5.90 1.69 II -15 x 3/4 a 173.7 .0576 3.17 337.6 .1121 6.15 1.94 II -15 x 3/4 b 170.1 .0565 3.10 331.4 .1099 6.04 1.94 II -15 x 1 a 176.6 .0569 3.18 236.7 .0764 4.26 1.34 II -15 x 1 b 176.4 .0569 3.18 302.1 .0974 5.43 1.71
co ~ .
(cont'd.)
V· 1
Vi = bd V· o 1
Beam No. psi. bdf ' ~
rn-NF a 283.5 .0948 rn-NF b 341.4 .. 0964
m - 9 x 3/4 a 275.7 .0937 m -9 x 3/4 b 327.4 .1114 ID -9 x 1 a 283.4 .0945 rn -9 x 1 ob 309.0 .0910 m - 9 x 1-1/4 ~ 327.5 .0983 m - 9 x 1-1/4 b 305.7 .0918
m -9 x 3/4 a .222 -~ 139.88 108.1 8.4 .38 100.0 m -9 x 3/4 b .276 ..... 173.91 134.3 8.4 .38 100.0 m-9xla .216 - 136.10 105.0 8.3 .37 109.1 m - 9x 1 b .243 - 153.11 118.0 7.8 037 123.0 m -9 x 1-1/4 a .277 - 174.54 134.2 8.0 .35 98.2 m - 9 x 1-1/4 b .273 - 172.02 132.1 8.0 .35 98.2
M::
Mu
(9)
.58
.59
.71
.57
.80
.80
.56
.57
Qlal(.
)000&
1:'1:1 C» .
Table 9
COMPARISON OF ALLOWABLE SHEAR WITH ACTUAL SHEAR STRESS
VaU
Vd T Vult. f ' P M AC! code vaU vult. v.all c
Bearn psi. % cale. psi .. psi. psi. S.F.
1 -NF a 3180 1.14 .84 131.1 111.4 293.7 2.64 1 -NF b 3180 1.14 .84 lai.l 111.4 298.7 2.68
1 -9 x 3/4 a 4070 . 1.14 .84 145.1 123.3 320.4 2.59 1 .... 9 x 3/4 b 4070 1.14 .84 145.1 12303 363.3 2.94 1 - 9 x 1-1/4 a 3510 1.14 .84 136.8 116.3 320.4 2.75 1 ... 9 x·l-l/4 b 3510 1.14 .84 136.8 116.3 293.3 2.52 1 - 15 x 1-1/4 a 3700 1.14 .84 139.4 118.5 282.7 2.38 1 -15 x 1-1/4 b 3700 1.14 .84 139.4 118.5 426.1 3.59
m - 9 x 3/4 a 2940 2.51 .903 159.6 m -9 x 3/4 b 2940 2.51 .998 159.6 m -9 x 1 a 3000 2.51 .903 160.7 Ill",9x1b 3550 2.51 .903 169.8 m - 9 x 1 .. 1/4 a 3330 2.51 .903 166.2 m - 9 x 1-1/4 b 3330 2.51 .903 166.2
long constant shear regions of beams under concentrated loading. Also
the high shear regions of unüormly loaded beams occur close to the
support, where the large compressive stresses due to the support
tend to decrease the effect of shear and flexural stresses. Since there
is no allowance for extra shear strength of unüormly loaded beams
in the ACI code, it would ap{Ear that higher safety factors were obtained .
in this test program than had a concentra,ted loading system been used.
130.
CHAPTER NINE
CONCLUSIONS
A. Summary
Three series of tests were performed with the prime
purpose of studying the effect of the fUmge of T-beams on tl1eir shear
strength. Each series consisted of six: T-beams and two rectangular
beams. The end conditions of the T-beams were either fully fixed,
restrained or simply supported for Series 1 to III respectively. AU
beams had exactly similar companion beams as a check on the va lid it y
of the results. Except for some unavoidable differences in f C' the
flange dimension was the only variable among companion beams of
each series. Beam dimensions and the experimental procedure were
designed in such a way to approximate practical conditions as closely
as possible. A method was devised to obtain a truly uniformly
distributed load. This was done by testing the T-beams, flange down,
on top of a 3" thick layer of medium soft sponge rubber. The beams
were loaded through the columns, thereby compressing the rubber
which exerted a uniformly distri buted load on the beams. This loading
system proved very successful.
131 •.
Test data included the measurement of deflections, steel
and concrete strains, the determination of the load causing initial
flexural cracking and initial diagonal tension cracking, as weIl as the
ultimate load, The findings can be summarized as follows:
1. Diagonal tension cracking loads and ultimate loads were
fairly scattered and no definite trend of increasing shear strength with
increasing flange Bize was observed. One should take into account,
however, that this research was the first of its kind and was therefo:re
too Umited in scope to generalize this conclusion and assume it to be
vaUd for any T-beam under any kjnd of loading and end condition.
2. The scatter of results complicated the issue and made it
düficult to draw definite conclusions. The compressive strength of the
beams, f C' was the average value obtained from six control cylinders.
It was not uncommon that the strength of one of the cylinders was as
much as 40% off the average~ It is therefore not reasonable to expect
a unüorm concrete strength throughout the test beams. If the concrete
strength at the section of fallure hapl2ned to be different from the
average value of the control cylinders, the cracking and ultimate load
would be different from the predicted loads. It is believed that much
of the scatter of the results can be attributed to non-uniformity of the
concrete strength of the test beams. As aU concreting was done in the
laboratory after carefully measuring the required amounts of cement,
132.
sand and water, no cause for the non-uniformity of the concrete
strength can he fmmd.
3. In Series 1 and II the flange in the column area was in
tension. One might expect therefore that the section did not act like a
T-section. This was by no means certain however, as the section of
failure of a beam is located at some distance away from the column,
towards the center span, thereby making it uncertain whether it occurs
in a tension or compression zone. In many cases the diagonal tension
crack was found to cross from the negative moment region into the
positive moment region.
4 •. T~e ultimate shear moment of the beams was calculated,
using the formulae presented recently in an extensive report by Laupa,
Siess and Newmark. Values of the calculated shear moment and the
experimental moment at ultimate load varied from .42 to 1.10, this
latter value being the only one larger than one. Most values were
below 0.80, indicating safe, but uneconomical design.
5. The nominal shear stress at ultimate load was calculated
and a comparison was made to the allowable shear stress as stipuVu
lated by the ACI Building Code. Values of vaU ranged from 2.0
to about 5.0. These values, or safety factors are in the expected
range for concrete structures.
133.
6. Load-deflection curves were presented of aU beams. Up
to the load at which initial flexural cracking took place, the deflections
agreed closely with the theoretical values, using the initial tangent
modulus and the moment of inertia based on the gross section. After
flexural cracking started, a decrease in the slope revealed a graduaI
change-over from the initial tangent modulus to the secant modulus
and from the moment of inertia based on the gross section to the
moment of inertia based on the" straight Une" transformed section.
7. In Series 1 and II the ultimate loads were very much
higher than the initial diagonal tension cracking loads, indicating a
large" reserve capacity". This was not the case for Series In, in which
the two loads closely coincided.
8. Cracking patterns and failure modes were very similar
for Series 1 and II, which were typical cases of diagonal tension failures.
Series nI also had diagonal tension failures but splitting along the
reinforcing bars was more pronounced and dowel action often caused
a second crack in the shear span, which formed at failure,and had the
appearance of a second diagonal tension crack.
9. Except for slight changes in the crack patterns, no definite
differences were encountered in the ultimate and cracking loads of
the T-beams and their corresponding rectangular beams.
10 •.. Even if the design procedure does not require them,
stirrups should always be used in T-beams to act as links between
web and flange.
134.
B. Future Research
As stated before, it is the author' s belief that the present
trend of research in reinforced concrete which consists of the testing
of large numbers of beams, mostly under concentrated loading and
simp.y supported end conditions wUI not lead to a solution of the shear
problem. Rather, the more basic and as yet unresolved questions
such as the foUowing ones should first be answered:
1. What is the true failure criterion for concrete?
2. Does the initial local fallure affect the final mode of
failure and/or ultimate load ?
3. Vice versa, does the mode of failure influence the initial
local failure and/or ultimate load?
" 4. What is the effect of the reinforcing on the initial local
failure, the mode of failure, and ultimate load?
5. What is the "reserve capacity" of a beam, once diagonal
tension cracking has started?
6. What is the effective width of the flange of a T-beam?
In trying to answer these points, one should at aU times
use practical conditions; a loading system such as developed in this
thesis is both simple and true to practice. A shrinkage prevention
mechanism should be incorporated in the test set-up and both short
135.
and long term loading should be used to determine the effects of creep.
Only when these very basic questions have been solved
can one proceed to more complicated shapes, such as complete
frameworl{s.
136.
BIBLIOGRAPHY
The following abbreviations were used:
A.S.C.E. - American Society of Civil Engineers A .C .1. - American Concrete Institute
1. Talbot, A. N.
2. Gilchrist, J.
3. Johnson, L. J. and Nichols, J. R.
4. Gilchrist, J.
Tests of Reinforced Concrete University of filinois, T-beams, series 1906. l!:ngineering Experiment
Station, Bulletin No. 12, Feb. '1907.
Reinforced Concrete T"; beams: Strength of Web in Shear.
Engineering (London), Vol. 100, Sept. 1915, pp. 293 -294.
Shearing Strength of Con- A.S.C.E. Transactions, struction Joints in Stems of Vol. 77, 1914, pp 1499-Reinforced Concrete T-beams 1522. as Shown by Tests.
Experiments on Shearing Strength of Reinforced Concrete Beams.
Engineering (London), Vol. 124, Oct. 1927, pp. 563 -566.
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Engineering News Record, Sept. 19, 1935.
Magazine of Concrete Research, No. 7, Aug. 19510
A.C.I. Journal, Prb.ceedings, Oct. 1951, Vol. 48, pp. 145-155.
~"
9. Hognestad, E.
10. Ferguson, P.M. and Thompson, J. N.
11. Hognestad, E.
12. Laupa, A., Siess, C. P., and Newmark, N. M,
1 3. Revesz, S.
14. Moody, K. G., Viest, 1. M., Eistner, R. C. and Hognestad, E.
What Do We Know about Diagonal Tension and Web Reinforcement in Concrete.
Diagonal Tension in T-beams without Stirrups.
Yield Line Theory for the Ultimate Flexural Strength of Reinforced Concrete Slabs.
The Shear Strength of Simple Span Reinforced Concrete Beams without Web Reinforcement.
Behavior of Composite T-beams with Prestressed and Unprestressed Reinforcement.
Shear Strength of Reinforced Concrete Beams.
137.
University of lllinois Engineering Experiment Station, Bulletin No. 64, 1952.
A. C .1. Journal, P:t'o~eedings, Volo 49, No. 7, March 1953, pp. 665-675.
A.C.I. Journal, Proceedings, Vol. 49, March 1953, pp. 637-664.
Civil Engineering Studies, Structural Research Series No. 52, University of llUnois, April 1953.