to my parents - mcgill universitydigitool.library.mcgill.ca/thesisfile43847.pdf · kani' s...
TRANSCRIPT
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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
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TO MY PARENTS
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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
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CHAPTER
NINE Conclusions A. Summary • B. Future Research
BIBLIOGRAPHY
Page
130 134
136
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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
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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
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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
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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
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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
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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.
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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
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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)
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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)
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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
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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 .
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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:
1 tensile strength = 100 + 12 compressive strength (f ~ ~ 5000 psi)
c constant (f c > 5000 psi)
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.
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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
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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
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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
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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
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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)
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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
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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
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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
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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:
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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.
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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 .
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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)
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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
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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
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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,
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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.
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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
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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.
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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
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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.
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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.
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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
the eventual section of failure.
~--T-···-·--·"-···-I--·~ =li_~~······-···--·---·-----··l·-··-·-·············-··-.. ~
(17;.);- li. - é7'
~._----_._-_ .... _ .... _ ....... _-_._._ .. _.i-._ .... __ ._ .. __ .-_ ..
--"----+------------------":...-;;:...-'---------_ ... _-
Figure 3. stress and Strain Diagrams at Eventual Sp.ction of Failure
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35.
From the geometry ·of the stl'ain diagram the following
relations are obtained:
d f:. c (fi - 1) .•• (3 .6)
and
(See Figure 3 for meani.ug' of symbols.)
From the parabolic stress strain curve with ultimate values
of f c and (;u:
.•• (3.8)
substituting (3.7) into (3.8) gives
••. (3.9)
where, \~ ).,.= longitudinal direct. stress at y from the extreme
compressive fiber
A-Y... ''''"' - h
Note that €. c is the strain, at some load, at the extreme compressive
fiber of the eventual seetion of failure.
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~6.
1. InternaI Compressive Force
Let C = total compressive force in concrete. C is made
up from the compressive forces of the web as well as the flange.
Thus,
C - b' ~"( <T.)" dy + (b -b' l( (lT'.j.,. dy ... (3.10) o 0 ..... -_._---' ... ... ..
web contribution flangf3 contribution
where, b ls the effective flange width, i.e., that width of the
flange which has an internaI compressive stress
equal to. the compressive stress in the web for
any one y.
Substituting equation (3.9) into equa tion (3.10) gi Ve s
c = f~ [b' th f 2'1' (1-13) - "," (1-p..V} dy
T (b -b't f 2't' (1 -r-+ '" ~(I-p..)' J d Y ] ... (3.11) o
integrating and simplifying:
... (3.12)
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37.
A further simplification can be made by negiecting the
terms ( th'*' - I:h2. ... 't' -- ~ .) ( ) _ Il in the R. H. S. term of equation 3.12
being small compared to Che factor 2,considering 'V.oe:::. t· and
~ L.. 1 for thin flanged T-beams.
Then: ... (3 .13)
2. InternaI Bending Moment.
The center of pressure of the web is given by
.•• (3.14)
Let Mi = internaI resisting bending moment and assuming that the t
center of p:essure of the flange is at y = 2' , then
Mr=b' (d- g) f"(U-xh' d~ +(b-b')(d- i) it.(~)y d~ ... (3.15) ,~ _____ o. 0 J
web contribution flange contribution
substituting (3.9) and (3.14) into (3.15), integrating and simpIifying:
Mi = b''fh2.{~ [4~ (3-'41)- (4-\V)] + 12-
+ f~ t 'fi (d - ~ ) (b - b') (2 - 'fi ) ••• (3.16)
where, d ~ = h·
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38.
3. InternaI Shear Force
Figure 4 shows the stress and strain diagrams of sections
A and D of a beam~ a distance dx apart. Section A is the failure section.
t h
___ L ____________________ _
__ _ ____ ------'-_-J..---_______ _ - -L-L..--.-l--Section A Se.ction B
Fi gure 4. Stress and Strain at Adjacent Sections
let the horizontal shear stress at depth y ={<r)l.'()y « Static equilibrium
of part of the section up to a depth y gives (neglecting the effect of the
flange on the shear stress distribution):
••• (3.17)
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39.
or .... (3.18)
now differentiating equatlon (3.9) gives:
Substituting equation(3.19) into equalion(3.18)and integrating gives:
Since the horizontal and verUcal shear stress at a point are equal, the
vertical shear stress at depth y' is also given by equation (3.20).
Therefore, if Vi ïs total shear force and assuming that tbe shear stress
in the flange is equal to that ln the web at the same depth y:
Vi .~~ \)' jlT~If)d 8 i-o ' .. - .~"--...-. -- ,,- ,.'
(b-bl)l~('T,cy) d~ ... (3.21) ~ o~_
web contribut.ion flange contribution
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Substituting (3.20) iuto (3.21) gives
Unknowns in (3.16) and (3.22) are:
'fi, h} ~~ and ~ , i.e.,4 equations are
needed for solution.
40.
... (3.22)
(i) As the steel is assumed to be yielding, the force in the steel at
the failure section is constant and thusC is constant:
ac = 0 3x
differentiating (3.13) and simplifying:
{"h L' ( '.p) 0 f b!l 2 6' h ( f) 't' 0 1 -::; fJ 'l( + . 11- '3" > ~ + 2, b - b l
..• (3.23)
••• (3.24)
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41.
(ii) Tensile foree jn st.eel = total compressive force in concrete,
from (3.13)
pb'd fy:: f(~[b'ho/(1-l)
+t·+-(2 ---'fi) (b -- br)] .,.(3.25)
(iii) from (3.16) and (3n22)
_ a ëf for symmetrieal loading
or
=?rcf (taking a.::; ~ , see Kani (35) ) 1 for dis- '. tributed
= 4.5 (see Laupa, Seiss and Newmark (12)) loading
3e Not-é. that 3d = 4.5 for l = 6' 9 d = 6" as in tests.
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4. Method of Solution
The u.nknowns in the expression for Mi (equation 3.16)
are 'V and h~ The problem is ta finà the lowest value of .'V whose
corresponding shear and direct: stresseS do not: violate the failure
criteria (equations 3.1, 3.3 and 3.5).
The following method of solu.tion i.s suggested:
42.
a. assume a value of 'V ~ preferably less than its value at
failure (a value between 0.2 and 0.3 is a good choice)
b. from equation (3.25) determine the value of h for the
selected value of 'V (aU beam geometr-y and material pro
perties should be known)
c. determine i; and ~~ from equations (3.24) and (3.26)
d. find t.he (f" and tj"y distributions trom equations (3.9)
and (3.20) respectively
e. compare the values obtained for q' l< and <T'ICV' to the failure
criteria (or a plot thereof) and adjust the initial value of
'fi. The same procedure is repeated until values of q-~
and 'T~" are obtained which just ·reach the failure value.
Note that the point of falluI'e in question is not necessarily
located at the extreme über. Whether this local failure
leads tü a fallure of the beam as a whole is not always certain.
f. calculat'e Mi from equation (3.16)
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43.
The equations required for the solution of the ultimate
strength of rectangular beams are (33) (malntaïning the original
numbering~ adding "a")
••• (3.16a)
••• (3.24a)
••• (3.25a)
••• (3.26a)
The method of solution of these equations as suggested
above is very Ume consuming and tedious unless a computer program
is written for their solution. Before this can be done, however, a
theoretical method should he developed, backed up by tests, to deter-
mine the effective wïdth of the flange. The effective width will not only
depend on the usual parameters such as f~, p~ t, etc., but also on the
boundary conditions of the flange. It will de pend on the extent to which
ficticious ïndividual st'.l'ips into which the tlange might be "cut" can
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44.
act as shaUow beams, carrying pàrt of imposed bending and shear
stresses. The problem then is to find the fl~Jlge width b over which the
bending and shear stresses are equal to these stresseS in the web for
any one y.
The failurecriteria were obtained from the literature and M 3E
their correctneSS is assumed, as, are the Vd = 4.5 or 8d values.
The theory presented in this chapter was not used to predict
the ultimate strength of the beams in this test program because the
author considers this theoretical analysis to be only a smaU step towards
the complete theoretical solution of the ultimate strength of reinforced
cOllcrete members. Test results were too scattered to come to a definite
conclusion regarding the applicability of this theory to predict the
ultimate strength of T-beams. It is the author' s belief that a major
breakthrough in the shear problem may occur if future research is
concentrated on the determination of a reliable failure criterion fOi'
concrete and appying this to different geometric shapes as was done for
rectangular beams by Jones and as was done above for T-beams. By
continuing the present trend in reinforced concrete research, that is,
by testing great numbers of beams under one or two concentrated loads,
and varying one or two variables, not much ground will eVer be gained
towards a solution of the problem. The more basic question: When
does concrete fail? , should first be answered.
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45.
CHAPTER FOUR
MECHANISMS OF FAILUR~
Although the many investigations into the shear problem of
reinforced concrete members have not led to a universally applicable
solution, much light has been shed on the different types of failure in
general, and the mechanisms of shear failure in particular.
In general two tyPes of failure may occur. One is a
flexural failure, the other a shear fallure.
Flexural failure usually occurs in long-span beams, or
beams with a low percentage of main reinforcement. After application
of a load to such a beam, vertical cracks appear in the region of
maximum moment. As the length of th~se flexural cracks increases
under increasing load, the compression area of the concrete above the
cracks decreases until a crushing fallure occurs. This failure may
or may not have been preceded by yielding of the tansile reinforce
ment, depending on its percentage and the con crete strength. Typical
flexural failures occur after large deflections and much yielding have
taken place. The effect of shear on this kind of' failure is negligible.
When failure of a beam occurs at a lower load than its
flexural capacity indicates, shear failure is said to have taken place.
Beams of average length, that have a normal ~rcentage of tensile
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46.
reinforcement (1.5 to 3%), usually fail in shear. These beams behave
similarly under low loads as des(;ribed above. As the load is increased,
however, an inclined crack, the diagonal tension crack, forms roughly
from the support to the loading pointo Since the tensile stresses due
to bending decrease when approaching the neutral axis, the crack in
this area is mainly due to a condition of pure shear and will cross the
neutral axis at approximately 450 • The diagonal tension crack
usuaUy includes the inclined tops of the flexural cracks. In sorne cases
more than one diagonal tension crack may form in a shear sPln.
Further increase of load is often possible to as much as twice or three
Umes the original diagonal tension cracking load. After the diagonal
tension crack has formed different failure mp.chanisms are possible.
Consulting the literature on t.his subject one finds much disagreement
among investigators as to the exact mechanisms of failure. Difference
in nomenclature for essentially the same tyPe of failure is widespread
and often confusing.
The writer considers A. M. Neville' s concept (27) of
failure mechanism s in shear the most appropriate and in aggreement
with the test results. Neville distinguishes between four modes of
failure after the initial diagonal crack has formed.
The first mode is called shear-compression failure and is
said to occur when the upward extension of the diagonal tension crack
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reduces the compressive concrete area to such an extent that crushing
oc(:,urs due to the bending and shear stresses. When the cross-
sectional are a of the compressive zone is so large that crushing cannot
occur, as ïs the caSe in 1'-beams, the diagonal tension crack may
extend horizontally towards the support at the level of the main longi
tudinal reinfor"cement. For this to happen the diagonal tension crack
has to widen and the splH:ting of the concrete along the reïnforcement
is due to dowel action of the reinforcing bars. This dowel force
constitutes a part of the applied shear force which is commonly neglected
in theoretical considerations of the shear problem. W. J. Kreffeld and
c. W. Thurston (32) point out that this force is considerable in many
caseS and thus should be inc1uded in any theoretical formulae. This
makes the problem indeterminate and compatibllity of deflection should
be used in addition to the equîlibrium equations of statics. Dowel action
cracking can be greatly reduced by the USe of web reinforcement such
as stirrups. The name of shear-tension fallure was suggested by Taub.
When sufficient web reinforcement. is used to prevent dowel action
cracking, fallure usually occurs by yielding of the tension steel in t.he
zone of maximum bending moment.. In t.his third mode~ fallure occurs
by extension of the diagonal tension crack through the compressive
zone of the concrete. The fourth mode occurs when the diagonal tension
crack ls continuous, with a decreasing slope towards the center of the
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48.
beam, thereby splitting the beam into a top and boUom half. Ferguson (16)
explained this mechanism in terms of principal stresses in the com
pression zone of the concret:e. It is still not always possible to predict:
exactly which mode of fallure will occur in a particular case, nor has
it been established whether the mode of failure affects the ultimate
load, or, if it does 1 to w hat extent.
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49.
CHAPTER FlVE
A. The Test Beams
As stated before, the object. of the test series was ta
determine the effect of the flange on the shear strength of reinforced
concrete T-beamso To ensure meaningful and generally applicable
results? pI'actical conditions were simulated as closely as possibleo
This was done by using:
la test beams of typically practical dimensions? c:oncrete
strength and reinforcing percentage
2. truly uniformly distributed loading
3. end conditions as found in practice
4. column stubs ta obtain a practical stress distribution in
the beams near the supports ,.
The only variable between different beams was to be the
flange width and thickness and, of course, unavoi~able differences in f è.
The original plan called for a total of 14 beams. Each
beam was to have a companion beam, exactly similar in aU respects,
as a check on the reliability of the results. In case of dissîmilarit:y
between companion beams, a third similar beam would be made as an
additional check. The beams were to have a column t.o column span
of 23' - 8" and a web cross-section of 14" x 28". Overhangs were
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50.
to be provided to simulate fixed end moments at the columns under a
continuous loado The overhangs were to he 9 ' - 8" long. As beams
of this size would be impossible to handle in the laboratory it was 1
decided that models at '4 scale would be used. Thus, the resulting 1
cross-section of the web was 3 2' "x 7"? column to column span of
5' - 11'' and overhangs of 2' - 5".
Two different widths of flanges were used ; one of 3' ? the
other of 5' (9" and 15" respectively in model form)u This was thought
to be in the range of the average practical distance from the beam to the
point of inflection of a typical slab. Wnh eac.h width, three different 3
slab thicknesses were used: 3", 4" and 5" (model slab of 4''', 1.0"
1 and 1 4' "). Thus the effects of flange width as well as tbickness were
to be studied.
Two rectangular beams.,dimensionally similar to the web
of the T-beams,were to be made for comparison purposes of the mode
of failure, and the cracking and ult.imate strengths.
In aU cases, one of the companion beams was strain gauged
(see Chapter Six for details). 1 1
All beams Were provided with column slubs of 3 2''' x 3 2" "
cross-section and 5" long.
After the testing of eight fixed-end beams (2 rectangular, 1 1 3
2 of 9" x l '4 " flange, 2 of 15" x l '4 "flange and 2 of 9" x '4" flange)
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510
this series of tests was discontinued as no t.rend in either the cracking
or ultîmale strength was discoveredo As this was attributed to the large
negative moment over the columns, it was decided to have a second
series of test beams with ends rest.rained ïn such a way that the negative
moment over the column was equal to the positive moment at center
span. Eight beams were tested in this series (2 rectangular? 2 of 9" x 1" 3 1
flange, 2 of 15 Il x 4" Il flange and 2 of 15" x l '4 "flange)o A third
series ~ also of eight beams (2 rectangular? 2 of 9" xl" flange? 2 of 3 1
9" x '4" flange, 2 of 9" x l '4 " flallge) weI'e tested under sim ply
supported conditions.
The three series are numbered l, II, and III respectively.
Each beam is identified by Us series number 1 Us flange dimensions and
an " a" or " b", referring to companion beams, the "a" beam being
strain gauged and the" b " beam being similar.,but without strain gaugeS. 3
Thus beam II - 15 x 4' a is a beam test.ed with equal positive and 3
negative moment (series II), a flange size of 15" x 4' " and equipped
with strain gauges. The rectangular beams are identified by the symb~l
NF (no flange). Bearn III - NF a is a rectangular, strain gauged beam
of Series nI. AU beams of Series II Wel'e of the same overall length as
the beams of Series l, the difference in moment comïng from the düference
of loading length over the overhang. Beams of Series III had overall
length of 6' - 8"? with a simïlar column to column span length of
...
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52.
1 5' - Il'' as the other series and an overhang of 4 2''' on each end for
securing of the longitudinal reinforcement to pre vent bond failureo
Figures 5 to 7 give the dimensional details of aU beams, and their shear
and bending moment diagrams •
.8. The Reinforcement
Beams of Series 1 and II ha~ similar reinforcement. Five
#~ bars were used in the negative moment tension area, below (note
beams are tested in "upside down" position) the columns. Three #2 bars
were used at center span, of which two were bentï-.upand one extended
to the columns. AU #2 bars were plain and, to pre vent bond failure,
hooks were provided at the ends. The reinforcing percentage was .61%
at center span and 1.14% OVer the columns. (See Figure 8 for reinforcing
details.)
. Beams of Series lIT were reinforced by 5-#3 deformed bars,
aU positioned in one plane. Although the spacing of these bars was
smaUer than allowed by the National Building Code, this was thought
to have a negligible effect as the spacing was considerably larger than
the largest aggregate size. In this series the percentage of reinforcing
was 2.51% (see Figure 8). 1
Reinforcement of the flange consisted of 8" diameter steel
rods, spanning across th~ flange, perpendicular to the beam. The
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53.
29~----~----------- 71 11 ______ ~-- 29/1
Il
î 3.5 i
---.~,...----
511
1
,sECT/oN A-A
3SS W-
Figure 5, Series 1: Bearn Dimensions, Sh ear and Bending Moment Diagrams
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54.
" '-------1-------- 129 --------'~II----~ .......
----- 71" ------...,.....dO--- 29-"-~-I
l {,7S 7 Il r t.:: /.00
L - - J. /'1$
\-c- sr' ~~1 ~l fo4---/5" ~rlo~
Fl.gure 6, Series II: Bearn Dimension, Shear and Bending Moment Diagrams
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55.
___ 6b /1 ---------;:l .....
i s"
7/1 {
b.7S
l f. = 1.00 1.25
F-=-9---"-I~1 ,:SEc..TloN A -A
Figure 7, Series TIl: Bearn Dimensions, Shear and Bending Moment Diagrams
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~ /28" la /
L 45" • ,/ "'"' 45 N .... ;/ 7
/
L /{ 38' i3j/ / / ~ 38" 7/
r-f!."" fs, ;/ if ~ ÂLL&4RS #2 ~w/ /
Z 7/" - ----------~~-
R~/N;:l:}RCE.M4:Nr RNe Se,e;.êS Jo ~ .IL
~ ~. ;/ L/ , e ~' , ~ 3'" #~. /' /
r-. G\~" .a4.e.s-#3 OEFQlfiMED
/<.EINJ:'ORCC/tIfPV r F.o..e SL.:..eIG:.$:or
Figure 8. Reinforcement Details
4 #2 PLAINIl "LCtJ6
3STIReLlPS {Â IIPWIRE.I
COil.lMN RéINj:'O~ceN'lEN T
al ~ .
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object of the reinforcing was to prevent excessive deflection or cracking
of the flange. The rods were provided with hooks to prevent slipping. 113 1
They were spaced at l "2 ", 2" and 2 2 " for the 4 ", 1.0" and 1 4' "
flange thicknesses respectively. The rods were held in place by glueing 1
them with epoxy glue to two 8' " wires, running from end to end, parallel
to the beamo The epoxy glue was found to be sufficiently strong to hold
the flange reinforcement in position during the pouring of the concreteo
Figure 9 gives the details of the flange reinforcement.
Figure 8 also gives the details of the reinforcement of the
column stubs, consisting of four #2 bars, extending from the column into 1
the web, and three ties, made of 8''' wire.
C, Formwork
To he able to pour two beams simult~neously; two forms
were built, as shown in Figure 10. In the design of the formwork an
attempt was made to obtain a configuration that was easily put together 3
and dismounted. The main component was i" thick B. C. fir plywood.
To provide space for the column stubs the whole assembly had to he
lifted off the ground by awroximately 5 ". To accomplish this, the
assembly was supported on four pine beams, running over the full
length. To pre vent bulging of the plywood after the pouring of concrete,
wooden blocks were provided at 21" intervals along the beam. These
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58.
Figure 9.
Figure 10. Flange Reinforcing
The Forrnwork
Figure 11. Bearn Ready for Testing
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59.
blocks also supported the plywood on which the flanges were poured.
The concret.e forming the flange was contained along the edge by two
plywood strips of equal thickness to the required flange thickness and
nailed, one on each side of the web, at a distance apart. equal to the required
flange width. To prevent the sticking of concrete to the wood and to
facilita te the stripping operation, the forms Were oiled (Shell SAE 30) 3
about 24 hours before placing the concreteo Steel chairs of 4''' height
were used to Secure the longitudinal reinforcement in p>sitiono After
pouring the flange reinforcement was inserted to such a depth that the
concrete would just cover the rodso As the depth of the flange re in-
forcement was not critical, chairs Were not used.
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60.
CHAPTER SIX
EXPERIMENTAL PROCEDURE
Although third point loading is used in the vast majority of
tests described in the literature, this loading condition does not reflect
practice adequat.ely and any formulae derived are thus not generally
applicable. Uniformly distributed loading is by far the most corn mon in
practice and a method was devised to obtain a simple continuous loading
system.
The most commonly used approximation to a continuous
loading method is a multi-point load system, the accuracy never being
very great, unless a large number of point~oads are usedo Water or
sand pressureS have been used? but these are bothersome, comIiicated
and time consuming methods.
The loading system used in this investigation consisted of
a 3 "thick layer of Neoprene medium soft sponge rubber, lying on a level
floor under the testing machine. The T-beam was then put on the rubber
layer with its flange down and the web and column stubs pointing upwards.
Thus, the beams Were tested in an"upside down" position, the advantage
being that the mode of failure and the crack pattern in the web could
easily be recorded and studied. Figure 11 shows a beam ready for
testing. Steel plates were fixed on the column stubs with plaster of Paris
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61.
to ensure an even pressure distribution. To prevent eccentricity of
load, rollers were used hetween the plates and the loading beamo The
loading beam was a 6' ~ 10" long 14 WF-48 beam whlch was sufficiently
stiff to prevenl excessive deflection. A ball and soekel bearing was
used bet,ween the machine head and the loading beam~ again to prevent
eccentricity of loadlng'. AU tests were carrled out with a 400,000 Ibo
Baldwin~Tate~EmtH'Y hydraul1c testlng machlne1) Noo 512895 9 equlpped
with a Tate-=Emery load indicatol' 0 F1gure 12 shows the experimental
set-up.
As the sponge rubber was not commerclally avallable in
3" thickness, four layers of !" each were placed on top of each other.
A concrete slab 9 spanning between two parallel supporting
beams, has two points of inflection. If a free body diagram is drawn
of one beam and that part of the slab up to the point of inflection on
each side of the beam? the effect of the rest of the slab can be replaced
by a concentrated shear force, acting at the points of inflection. To
1
simulate this sheal' force in the .investigation, the sponge rubber was
1 1 made '2" wider than the flange width? thus providing a 4" overhang
of rubber on each side of the flange. This can be compared to a rigid
foundation footing on an elastïc soil ~ ha ving theoretically an infinite
1 . pressure at the edges. The '4" overhang was thought to Il'oVlde a
shear force in the' correct range of size. A î" overhang, which was
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62.
1
i i
-l''--r "-. • i
r"
.. ~! ~ 0.-
:r ..., CL>
'CIJ ...... CI1 .., s:: (1)
.S ~
~ ~ ~
K ~ lU ......t
~ (1) ~
Sn • ..-c r..
r'
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63.
t. ned imtially? provided sUlh a large force at lhe flange-edge that t.he
flange cracked al the web.
During loading the beam was forced down onto the rubber '1
wtnch exerled an even pressure on the beam provided the deflection of
the rubbl..:;r was equal for aH points along the beamo Two kinds
of defledions resuUf':do The fU'st was the deflection of the total beam
with respect to the !Joor" or the amount the sponge rubber was com-
pressed. The second was the deflection of the beam itself with respect
to the columnso The defle:ction of the beam itself was so smaU compared
to the defleclion of the beam as a whole with res{p-ct to the floor (less
than 1%) that the inaccuracy in the uniformity of the distributed loading,
brought about by the slight differences of compression of the rubber, was
neglected. In fact, dial gauge measurements of the deflection of the
beam with respect. to the l'loor at different points along the beam showed
negligible differenees.
Some difhculty was encountered with the rectangular beams.
Due to the small area in contact with the rubber as compared to the
[langed beams, the possibility existed of the beams punching through
3 the rubbero 1'0 preven!. this from happening,steel plates CS" x 2 If X 8")
were used between the rectangular beam and the rubber to provide a
large!' area to compress the rubbero This set-up was used in the
testing Di aIl r-ectangular beams and proved very successful (see
Figure 13),
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64.
Measurements of deflecUons and st.rains were taken at
2000 lbs. load intervalso As most beams failed in the neighbourhood
of 30 kips, this provided enough points for the plotting of graphs.
Whitewash was applied to the beams to facilitate the
detection of cracks. Black marking {)Emcils were used tü mark cracks
on the beam and the loads al which they formed. During' loading the
beams were carefully watched to determine the initial tlexural cracking
load and the initial diagonal tension cracking loado In many cases the
diagonal tension crack was an extension of an existing tlexural crack
and in such cases it was hard to give an exact value for the initial
diagonal tension cracking load.
Deflection Measurement.
"St.arrett" sproing dial gauges, provided with magnetic bases
and reading 0.001" belween the smallest divisions, were used for the
measurement of beam defleetions.
The compression of the rubber was measured by gauges
whose bases rested on the floor and whose measuring arms rested on
top of the webo The object was to obtain load-deflection curves for the
rubber and also to compare the deflection of the rubher at several points
along the beamo The latter values should be equal for any particular
load to ensure a uniformly distributed load on the beamo
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65.
The defleetion of the beams lhemselves was measured by
dial gauges with thei.r" measuring arm resting on top of the web and
1 1 3 their bases support€;d by a 2 2''' x 2 '2" x 8"" L" ,whieh was clamped
to the eolumns as shown lU Figure 140 Al" xl" xl" steel spaeer
was used belween t.he angle and the beam al the loeation of the clamps
to prevent: friet.lOllo When the beam as a whole deflected downward onto
the ru bb:e Y' " the angle and ils gauges defleded with il' by an equal amount
and thus the gauges measured only the deflection of the beam itself with
respect to the columns. Figure 15 gives the numbering and location of
aH dial gauges. Load-deflection curves were drawn as a means of
observing the redistribution of internaI stresses after cracking of the
concrete, and of comparing the detlections of beams with different
l'lange dimensions. Test: results are discussed elsewhere.
Str'ain MeasuremenL
In Series land U, eleetrieal strain gauges were applied to
both the reinforcing steel and the concrete. In Series III strain gauges
were applied to the steel only ..
The gauges measuring the strain in the reinforcing were
a pplied to the steel before the concreting of the beams. This course of
action was followed 1:0 reduce the possibility of damaging the beams,
should they be applied after pouring. This procedure was also lime
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saving. The type of gauges used was Budd C 6·~12L lt was learned
from experience Hl t.he laboratory that !,wo layers 01 wate:rproofing were
necessal'y to fully protect the gauges from damage during the placing
of the concrete" Budd GW -1 was used for the first layer of water -
proofing and Bndd GW -5 for the second. Twenty four hours or more
were required for t:he wat.erpr00fing ta seL In the third series.,deformed
bars were used and the deformatîons were removed by grinding the
bars uutU a smooth surface resulted for appUcation of the gauges. The
loss of bond area due to the waterproofing on the bars was neglected.
Gauges were applied in triplets to check the reliability and accuracy
of the readingso Figure 16 gives the locatïon of aH strain gauges.
Baldwin·-Lima-Hamilton SR 4 strain gauges were used on
the concrete. Again~ three gauges were used~ side by side, in each
location? as a check on reliability of the readings. Terminal strips
were used for aU gauges to lessen the possibility of damage.
Strajn readings of the steel as well as the con crete were
read from a Baldwin-Lima-Hamilton Model 120 strain indicator.
AH gauges were appUed by standard methods as recom
mended by the manufacturerso
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66a
Figure 13. Steel Plates on Rubber for
Rectangular Beams
Figure 14. Dial Gauge Set-Up
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i 1
fi " ,.,
1 '.3 2.i 1l2l. J{ li L TI.)
~ 1" ,- 1 29/1 1 /7.l J7~ --; 1 _~
:/il
SE.!2IC.s l ~ JI
SE.R/ES .JlL
f---oe 29· ---1
::/J7
! 1 !
l " 1 1 3/1 2 2 -)(22 ~a L TO
suPPol2r 0. G.6UG€;:'
LEGE.rJD
()) .êVA' t;~aES MGASI./Iè/NIG .{)E/:"-
• '-ECrll>N6 OF ~ Nf IT.SEL.':
® D/~ ~GeS M,EjJS~~/M!'2 /).éF--
lecT/OII -o,c _ &;:"tlM A-S il v-/J./;ji. L
w...<2. T. ,cLüoR
i'igure 15. Number-ing and Location of Dial Gauges.
~ -J .
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~.
1--21.
1 CDNo2E r~ G.AuGES
L Nf) 1f),1/ .p /~ --T---~
/
3TseL ~A(Jt;,E.s No 1,.2 fI..3
~--;ç- =~-.;;;;~
" ---1
-=- ~ -.:z:> 1-" == = _ ~_ ~_~~_. ~
S7,EEL ~LJUc;,ES
1\0 4.s~ 6. COllal'lErE &;&.IGe,s
No 7, S ~ 9
-SEI2IE.s .z i ..Ir
zr
Figure 16. Numberingand Location of Strain Gauges
0-
f"
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69.
CHAPTER SEVEN
T EST MA TERIALS AND MATERIALS TESTING
A. Mortar
To be able t.o exercise strict quality control and because
of the fact that fairly small quantities of concrete-mortar were needed
at a Ume, the mixing of the mortar was done in the laboratory. About
350 lbs. of mortar were needed for the placing of two beams, the exact
weight depending on the paI'tlcular flange size. Due to the limited
capacity of the concrete mixer it was decided to split the total required
amount of mortar into three batches of 140 lbs. each, which left a
sufficient quantity for the making of test cylinders.
Cement used was Type 1 normal Portland cement, obtained
from the Canada Cement Company. The mix was designed for a com
pressive cylinder strength of 3000 psi. As no special conditions had
to be met, no admixtures were added to the mixe
Sand used was graded crushed quartz, obtained from the
Dominion Industrial Mineral Corporation. Five different grades were
mixed (using the supplier' s designation): 10%-#6, 25%-#10, 25%-#24,
25%-#35 and 15%-#70 (aU percentages are by weight). Results of a
sieve analysis performed on each grade of sand are shown in Table 1.
The grading curve of the sand is shown in Figure 17, along with the
requirements of A .S. T .M. designation C33 -61 T.
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70.
The water-cement ratio was 0.75 t.o 1.0 and the cement to
sand ratio was LO to 3.5. Thus a typical batch consisted of:
cement
sand
water
# 6 # 10 #24 #35 # 70
9.4 23.7 23.7 2307 14.1
total sand
lbs. " " " "
94.6 "
20.0 "
TOTAL 141.2 "
Curing of the beams consisted of covering the beams with
burlap and thoroughly soaking them with water several Urnes daily.
The beams were then covered with sheets of polyethelene to ensure
continuously moist conditions. After cu ring for one week the beams
were stripped of their forms and left in the atmosphere of the laboratory.
A minimum of two 3" x 6" test cylinders were made
from each batch of mortar in accordance .with ASTM specifications
C-192. This size of test cylinder was considered appropriate consi-1
dering the web dimensions of 3 2''' x 7" and flange thicknesses in the
range of 1 ". The curing method for the cylinders was exactly the same
as that used for the beams. AU cylinders were capJEd and tested for
their compressive strength within 24 hours of the corresponding beam
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Table l
SIEVE ANALYSIS OF SAND
Canadian 1 SAND
U .S. #6 #10 #24 #35 Standard Standard 10% 25% 25% 25% Screens Screens Percentage Retained:
No • 4 No. 4 .7 6 6 6.9 8 8 1.9
10 12 .3 5.6 14 16 .2 12.2 .2 20 20 5.5 6.7 28 30 1.4 12.2 .4 35 40 .2 5.2 4.5 48 50 .4 11.3 65 70 7.2
100 100 1.4 150 140 PAN .1 .3 .2
#70 15%
.1 1.8 5.5 3.9 3.7
Total Total % Accumulative Retained 1 Retained
.7 .7 6.9 7.6 1.9 9.5 5.9 15.4
12.6 28.0 12.2 40.2 14.0 54.2 9.9 64.1
11.8 75.9 9.0 84.9 6.9 91.8 3.9 95.7 4.3 100.0
-~ .... __ .- - ~.- --
%
-.J ..... .
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72.
--
1,.,. ~
1 .... ~ 1 , l'
"
'.
1" 1"-
i(1
1"0..
Figure 17. Grading Curve for Sand
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73.
test. The averages of the compressive strengths of the cylinders are
given in Table 4 for each beamo Four cylinders, picked at random,
were strain gauged with two gauges placed vertically and two gauges
placed horizontally at opplsite sides of the cylinder at about mid-height.
The vertical gauges served to obtain the stress-strain relationship,
and the horizontal gauges to obtain Poisson' s ratio of the mortar.
Typical results of the gauges are given in Table 2 and Figure 18 shows
a typical stress-strain curve for the mortar.
B. Steel
Three sizes of steel reinforcing were used in this inves-1
tigation: No. 3 deformed, No. 2 plain and 8" diameter wire for flange
reinforcement~ The purpose of the flange reinforcement was to prevent
excessive flange deflection (with respect to the web) or cracking at
the web. No tests were performed to determine the steel wire properties.
The No. 2 and No. 3 bars were made of billet steel of
intermediate structural grade conforming to ASTM requirements - A 15.
The bars were 20 feet long and were eut to length and bent to shape in
the laboratory. A wooden jig held the bars in position while 3" long, 1 a" diameter wires were spot welded transversely across the bars to
ensure the correct spacing. Spot welds were made at sections of low
bending moment to prevent any change in steel properties at a critical
section.
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Load Ki12s
0 2 4 6 8
10 12 14 16 18 20 22 24 26
0 2 4 6 8
10 12 14 16 18 20 22
Table 2
TYPICAL STRAIN GAUGE RESULTS OFTWO 'l'EST CYLINDERS FOR DETERMINATION OF
MODUL US OF ELASTICITY AND POISSON' S RATIO
Av. Axial Av. Hor. St.ress Strain Strain Poisson's
74.
Psi Micro-in/in Micro-in/in Ratio Ec 6
Psi x 10 Cylinder 1
00 00 00 283 60 17 0.284 4.71 566 152 25 0.164 3.73 849 240 29 0.121 3.54
1132 312 35 0.112 3.64 1415 407 39 0.096 3048 1698 461 51 00111 3.68 1981 552 67 0.121 4.29 2264 675 81 0 .. 120 3.36 2547 737 92 0.126 3.46 2830 791 110 0.139 3.58 3 113 889 183 0.206 3.50 3 396 1002 261 0.256 3.33 3 679 1167 372 0.319 ~.15
C ylinder 2 00 00 00
283 35 12 0.330 8.08 566 121 25 0.206 4.68 849 211 27 0.138 4.02
1132 297 32 0.108 3.81 1415 383 39 0.102 3.64 1698 467 52 0.111 3.68 1981 538 61 00f13 3033 2264 682 79 0.116 3.32 2547 821 93 0.113 3.10 2830 915 102 0.111 3.09 3113 1132 152 0.134 2.75
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75.
1/"
V
. ~ 1/
1
LI
,
11
,.,
Figure 18. Typical Stress-Strai.n Curve of Concrete Mortar
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76.
Three 2..feet long coupons were taken at random from each
of the two bar sizesn Two strain gauges were applied to each bar al
opposïte sides at midaheight. Table 3 gives the steel properties and
Figure 19 shows a typical stress--strain curve for the steel ..
Table 3
TYPICAL STEEL PROPERTIES
St rain at Modulus of Yield Ultimate Failure ElasticitY6 Stress Stress
Cou n No. A~2% psi x 10 ksi ksi
No. 2 Reinforcing
1 14.. 8 26.6 42.0 68.0 2 12.8 29.0 44 .. 7 72.0 3 16.3 30.1 43.2 71.7
No. 3 Reinforcing
1 16.2 30.6 45.2 65.3 2 16.7 29.4 43.7 67.2 3 13.9 30.1 40.9 71..9
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77.
L.
1
Figure 19. Typical Stress-Strain Curve for #2 Bar
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CHAPTER EIGHT
RESULTS, OBSERVATIONS AND ANALYSIS
Ao Crack Formation
Series 1
78.
Except for a uniform compression of the sponge rubber,
no visible change took place in the experirnental set-up until a load of
about 10 kips was reached and tlexuI'al cracks st.arted forming. These
formed in the area of highest tensile stresses, in the extreme fïbers
of web and flange, just below the columns. These cracks formed at
equal or nearly equal loads at each end of the beam. Due to the test
arrangement, with the beams tested in an "upside-down" position,
they could only be observed after having reached a depth equal to the
thickness of .the flange. The cracks appeared in the flange perpen
dicular to the web and under increasing load more and more would
appear, spreading at intervals of 2 to 3 inc.hes, for a distance of about
15 inches each side of the column. No flexural cracks were observed
in the concrete tension zone of the beams al. mid-span. The initial
diagonal tension crack normally started off as a flexural crack which
would curve towards the intersection point of web and column. These
cracks crossed the neutral axis at an angle of about 450• In sorne
cases, the initial diagonal tension craek formed at about the neutral
..
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79.
axïs and extended upwards and downwards undèr increasing loado At
the ultimate load, the diagonal crack would open up and a failure
mechanism as shown in Figure 20 occurred which can he considered
to be a shear tension failure.
CANTILEveR .é.AlO t-
Figure 20. Typical Mechanism of Failure for Series 1 and II with Typical Dimensions
Dowel action of the longitudinal steel on the flange side forced a split
between flange and web which formed instantaneously at failure, thus
creating an " intermediate" piece of flange, cracked on each side but
held in place by the reinforcing steel. Close observation of the crack
which formed in between the flange and web showed clearly that the
crack formed in the plane of the reinforcing bars, which were embedded
inside the web at about mid'-depth of the flange. On each side of the
.. • • _ •• - • - ... __ , .~ •••• n. __ ._
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80.
outside bars this "plane of cracking" turned up towards t.he intersection
of the flange and web~ where the crack became externaUy visible. The
addition of web reinforcement in the form of stirrups which penetrate
as deeply into the flange as possible would have acted as a Unk between
the t1ange, and web and wou.ld have prevented the cracking of the flange
away from the web. This would have greatly increased the ultimate
load. Stirrups should therefore always be used in T-beams,even when
the desïgn theory does not requïre them.
Usually a failure as shown in Figure 20 occurred in one end
of the span, the other end also showing a diagonal crack which had not
yet opened up or failed. Bearn 1-15 x 1 i a failed on both sides 3
sîmultaneously. Bearn 1-9 x 4' failed on one side but. had a similar
diagonal tension crack at the opposïte end of the beam, in the 'canti-
lever" part. The rect.angular bearns had the same failure mechanism
as the T-beams, wïth a diagonal tension crack extending into a split of
the concrete along the longitudinal reinforcing, instead of the flange.
Although ït may apJEar that the beams in this series do not
act as T-beams as the flange is in tension due to the negative moment
below the columns, this is not necessarily true~ because the region of
tension is short (see the bending moment diagram, Figure 5). Thus the
actual critical section might well have been in the compression region
of the fla.nge. Also,the effect of dowel action of the flange could not
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81.
have been anticipated before the actual tests were performed. The point
of inflection of the beams occurred at about 16" from the column?
resulting in tension in the region of the crack close st to the column of
the "intermediate pie ce " and compression in the crack furthest from the
column. No crushing of concrete was observed in the compression
zone of the region of negative moment, where the diagonal tension crack
reached the column. This is believed to be due to the high biaxial
stresses (flexural stresses due t.o bending and compressive stresses
due to the reaction of the support). The compressive stresses due to
the support tend to decrease the effect of the fle.xural stresses and most
likely prevented crushing of the concrete in that area.
Series TI
Beams of Series II showed much the same crack formation
under load as those of Series 1. The main differences were a diagonal
tension crack which formed at a slightly shallower slOIB, thereby
extending further towards the center of the beam and the fact that due
to a larger bending moment at center span, flexural cracks formed not
only as described above, but also in the tension zone at the center of
the beam. These formed at a somewhat higher load than the flexural
cracks below the columns. They appeared over a length of about two
feet at intervals from three to six inches. The rectangular beams
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82.
3 showed more severe center cracking .. Beam n -15 x 4 b failed first
in a cantilever en d and only after increased load failed in shear tension
in the center span in the same baU of the beamo
Although with concentrated loading often several diagonal
cracks form in the shear span, of which one opens up at failure, all
beams in Series 1 and II had only one diagonal crack per shear span.
(The shear span is defined as that part of the beam extending from the
point of maximum shear to the point of zero shear or in this case from the
column to cen ter span.)
Series nI
In Series III, with simply supported beams, a different
crack formation was observed, t he main diff-erence being that in
the absence of a negative bending moment at the columns no flexural
cracking took place in that area. Flexural cracks appeared in the
tension zone in the center of the beam at approximately the same load
as the other series. Several slightly different crack patterns formed.
The simplest, as shown in Figure 21 which formed in beam III -3
9 x 4" a is a fairly straight diagonal crack, at about 300 to the longi-
tudinal axis. Note that rather than entering into the flange, the crack
followed the flange-web intersection for several inches, the flange
remaining completely uncracked.
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"If, -"",,, '''''''~ -.,
3 Figure 21. Crack Pattern of Bearn III - 9 x 4" ao
3
83.
Bearn TIl - 9 x 4' b showed, a different crack pattern as shown in
Figure 22"
e" ID"
--------)
3 Figure 22~: Crack Pattern of Bearn III - 9 x '4 b.
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84.
Two diagonal cracks formed in one shear span. The diagonal crack
closest to the column was thought to he the faUure crack, with' cracking
along the reinforcing bars, which ext.ended to the column. The second
diagonal crack was believed to form due to high shear stresses in the
reinforcing steel, in the region of the first diagonal crack. Due to these
high shear stresses a large force existed on the remaining part of the ..
beam, instantaneously causing the second diagonal crack. The crack
along the reinforcing steel continued on beyond the second crack for
about six inches towards the center of the beam. Although both
diagonal cracks occurred simultaneously, the one furthest away from
the co~umn was thought to form as a consequence of the one closest to
the column. Severe flange cracking was observed underneath the latter
crack while only a hairline crack formed in the flange underneath the
former diagonal crack. Beam TIl - 9 x 1 a had a crack pattern on
one side of the span as ls shown in Figure 21. When an attempt was
made to increase the load a crack pattern as shown in Figure 22 formed
on the other side of the span at a load of 3 kips below the first failure
load. AH other beams in this series failed with either one of the above
patterns or combinat.ions thereof.
The rectangular beams of this series showed slightly dif-
ferent crack patterns. Figure 23 shows the pattern of beam TIl - NF a.
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85.
1.:3 Il :/'
Figure 23. Crack Pattern of Bearn In - NF a.
Bearn III - NF b had a similar crack pattern except the second diagonal
crack, furthest from the column, did not occur.
It is clear that aU beams failed in diagonal tension which
is mainly due to the particular longitudinal steel percentages selected.
Figures 24 to 27 are photographs, iUustrating several
crack patterns.
Bo Failure Loads and Stresses
Tables 4, 5 and 6 give beam and con crete details, and the
evaluated data for shearing forces and stresses. The beam dimensions
of Table 4 refer to the average values, obtained at. a minimum of five
points along the beam. The initial flexural cracking load, P F'
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86.
Table 5, refers to the load at which flexural hairlïne cracks first became
visible. The initial diagonal tension cracking load, Pi, refers to the
load at which a diagonal tension .crack was first observed in the web.
At Urnes this load was hard to pinpoint due to diffïculty in observing il,
or due to dïffïculty in distinguishing it from flexural cracks. Values
for Pi should therefore be considered approximate. Pu is defined as
the ultimate load9 which occurred when the diagonal tension crack opened
. up, simultaneously with a considerable decrease in the load carrying
capacity of the beam. This load was always clearly defined, which is
assumed to be due to the method of loading. In the conventionalloading
method, with beams on rigid support, loaded by a hydraulic testing
machine, the diagonal tension crack, once it has formed, will slowly open
up under inc.reasing load, without a sharp drop in load carrying
capacity. This makes it at Umes difficult to pin-point the ultimate
capacity of the beam. Such inaccurate measures have been used as
defining Pu as that load at which the diagonal tension crack opens up a 1
certain amount, say 8'''4 Utilizing the sponge rubber loading system,
however, there was no d iffi cult Y in determining Pu. In. most cases the
initial diagonal tension crack would appear in the form of a hairline
crack. Under increasing load the diagonal tension crack would suddenly 1
open up, usually to about 4' ft, a breaking sound could be heard, and the
load would drop at least five kips. This load was considered to be the
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87
Figure 24. Typical Diagonal Tension Crack for Series l and II
Figure 25. Beam II -15 x 1 a after FaillJre
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88
Figure 26. Bearn III - 9 x 3/4 b after Failure
!,,', . ... ~ . ... ..,. .
" .: .. _,. _..- -C")". ""'~-G'· ';<"""""'1""""1 - " \ Q ! 1 .', /,': ",:," , ,<l,J_ " . . ,.l.. 1
. . • : '",,~ ~ 1 . l .t~ .., . , .. \,..,.. , ., . , ... . ... ..... ~ .. ' : , .
Figure 27. Crack Pattern of Bearn III - NF b
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89.
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
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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 .
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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
1
Table 4 (cont'd.)
CONCRETE -AND BEAM DATA
b' d t bd
,3.55 6.25 22.19 3'.56 '6.25 21iSS
3.65 6.25 .80 22.81 3.65 6.25 .75 22.81 3.55 6.25 1.10 22.1~ 3.50 6.25 1.10 22.88 3.65 6.25 1.30 22.81 3.70 6.25 1.35 23.13
bdf' c
66,570 '77,656
67,061 67,061 66,570 77,656 75,957 76,990
bd~
1216 1304
1236 1236 1216 1304 1316 1334
(0 .....
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Table 5
TEST RESULTS
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.)
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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
'" .
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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.)
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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
Table 6 (cont'd.)
EVALUATED RESULTS
V· Vu 1 Vu =iii Vu
bdVf'c psi bdf ' c
5.17 306.4 .1021 5.72 345.1 .0972
5.09 282.7 .0962 6.04 351.6 .1195 5.17 283.4 .0945 5.42 309.0 0091"0 5.68 353.4 .1061 5.30 343.3 .1031
Vu
bdVTè 5.59 5.79
5.22 6049 5.17 5042 6.12 5.95
Vu v·
1008 1.01
1.03 1.07 LOO 1.00 1.09 1.12
(C C1I •
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96.
reserve capacity for the simply supported beams is merely an indica-
tion of Pi values which were nearly equal to those of Pu.
The same remarks are also true for the rectangular beams, Yu
for which the average values of vi were even larger.
Series II had a 27% higher average value of vi than Series l,
while the average value of 'v i of Series III was 60% higher than that of
Series II. On. the other hand, average values of 'vu are approximately
equal in aIl three series. In comparing results from Series III, one
should keep in mind the increase in the percentage of longitudinal
reinforcing from Series 1 and Il, which was necessitated to prevent
flexural failures. One beam,with Series 1 or II' s configuration,was
tested under sim ply supported conditions and failed in flexure, after
which the percentage of steel was increased for Series III.
c. Dial Gauge Results
The dial gauge results are prE:sented in the form of load
vs .. deflection curves. (See Figures 28 to 35.) The eight curves per
figure represent the deflection of the eight beams of a series. The two
L.H.S. curves correspond to the rectangular beams, the center curves
the strain~auged or "a" beams and the R.H.S. curves the "b" beams.
Thus deflections of aIl beams of a series can he easily compared.
Table 7 gives some typical load-deflection data. Although most curves
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97.
show roughly similar deflections, considerable düferences do exist in
sorne casesG Some of these differences can be attributed to the
variation of th~ moment of inertia due to the variable flange dimensions.
Others can only be attributed to non-intentional dimensional and concrete
strength variations.
Load deflection properties are thoroughly discussed in the
literature (7L Generally, the main conclusions reached can Ile briefly
sum marized as follows:
1. Before the stress at the extreme fiber reaches the modulus
of rupture of the concrete, the beam acts as a homogeneous member,
the full cross-section of which resists flexure and shear. The effect
of the tension reinforcement is usually neglected in deflection or stiffness
calculations.
2. After the load has been reached at which the extreme fiber
stress exceeds the modulus of rupture, the beam is assumed to be
cracked, the concret~ in tension is neglected and the section resisting
bending is the elastic, transformed section. This is usually true for
loads up to the "design load" which is about one half of the ultimate load.
3. For loads above the 'tlesign load" plastic conditions in the
concrete must be assumed in calculating deflections.
Figure 36 shows the experimental load-deflection curve of
beam In - 9 x 1 bo
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98.
The two straight lines are the theoretical deflections, the one with the
greater slope being based on the initial tangent Modulus (ET = 5.3 x
106 psi.) and the moment of inertia of the gross section, the other being 6
based on the secant modulus (ES = 3.0 x 10 psio) and the effective
moment of inertia, based on a cracked section. Until flexural cracking
starts, the experimental curve follows the former theoretical curve.
At this point a graduaI decrease of slope of the experimental curve takes
place, indicating that cracking has started and that ET is decreasing
towards ES. The moment of inertia varies along the beam from the
effective for cracked sections to the gross for uncracked sections. An
average moment of inertia of these two values would give most accurate
results for this region of the experimental curve, which lies in between
the two theoretical ones. Deflections depend to a large extent on the
amount of flexural cracking.
Large deflections result from yielding of the steel, which
was particularly evident for the cantileverends of Series 1 and il,
where deflections greatly increased during loading }Briods in which
extensive cracking and yielding occurred. Comparison of the deflec-
tions of Series 1 and Il shows 'generally larger deflections of the over-
hanging ends for Series l, but smaller deflections for points in the
main span, as expected from different end conditions.
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99.
\1 J.!I Io!" 1- "
t" Il
L
N J.,
~ rA
li
Il
LI. r7
Il
~
Il
Il
'" IL. Y
'II
Figure 28. Load Deflection Curves
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100.
f-f-
-
1 .. 1,. III
--
IArr lA
ri' V ln
" r-P- -Il 1/",
'Il 1/
III Il:
~
,
l7. 1""
~
Figur'e 29. Load Deflection Curves
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101.
if-,
If' ~
L..-
Il
- ~ 1/ 1"
Il ~
., IL
1
1
"" J
1,..
~~ f"" ~ 1'" ~ -
Figure 30. Load Deflection Curves
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102.
w r~
'1
r..-; l. -V 1'1'''''' l/
1- -J<:
t..;
"1'/
II'. lA
[;.00
~ 1:1
1/
Il ... ~ i0oi
~ r;
r-. IL
Ilj
:
~
1" IQ r" r" ~,t , .. R
lJ.j R..
Figure 31.0 Load Deflection Curves
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103.
1-
""" ~ ~~I] -
/
1(" 01/ 16U
II. IV
} ~
1 ~ Il
17
~ 1""'1"" -
Figure 32. Load Deflection Curves
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-
t:: IJ C-I.., II" ~ I~
.-
:,..-v
1/
Il L 11'
,-~ IL
" IL 17
1/ -tll ... IJ.~
l-"
IJ 1/' LI
7 IL
1 Il
.1 1.
1.
J
1" li 1
'lAI
1'"
.. -
.'fF 1 .... 1'" .
ti' ".1iL 1- 1111)(
Fïgure 330 Load Deflection Curves
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105.
-- -
-
1-. !oP 1/. 11'1 I~ I/IV. l-
ra.:
1
19, J ~
Il r.;4 li, li 1/
1.01
IJ I~
Il TJ /":II 1.7 17
Il
Il
Il Il
17 7
11/ Il
If
'8
1" l' 1""
P -rr lA iI'J
1
Figure 34. Load Deflection Curves
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106.
!....
if-' _.
;1 1
i?: , 1/ r:J
LI Il
rJi [;f
~
1/ 7 J
'fJ
1,1,
~
11
li rr
1""1 ~ II(
-11" ~,,(
Figure 350 Load Deflection Curves
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Lill/"
1 1
1"'"
1/
..
1-1-0 1/
r- I~
I/'r- [l' .. V ~ ,
V
i;'
t.;
~
~
V 1"1 ~ ~ t.;
r-
"'" il": ,~ rJ'LI.
Figure 360 Experimental and Theoretical D eflection Curves for Bearn III - 9 x 3/4 b
107.
L;
1-'
If.
-I~ 1/.
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108.
D. Strain Gauge Results
Much düficulty was encowltered with the strain gauges.
Although the gauges were applied in triplets in Series 1 and II and in
twins in Series TIl, results were very oUen erratic and unreliable.
Results are presented in the form of load vs. strain curves as shown
in Figures 37 to 43. Table 7 gives sorne typical load-strai,n data as
plotted in the graphs. Average values of the strain at a point were used
for the p~otting of the curveso In case readings from one gauge differed
considerably from the other two gauges at the same point, these
readings were omitted. If readings from aU three gauges' differed from
each other, the results were considered unreliable, and were not plotted.
Thus aU curves missing mean unreliable results from the gauges in
question. AU strain gauges appied to concrete were in compression
areas, those on steel in tension zones.
ln most curves abrupt changes of slope were encountered.
Changes of slope are generaUy associated with the formation of cracks
in the concrete. Once cracking occurs a redistribution of internaI
stresses takes place resulting in a higher strain in the reinforcing in
the region of the crack. Considerable stram differences were found
for different beams at the same load. This was believed to be only
partially due to the different flange thicknesses. Unreliability of the
strain results seems a factor also. Yielding of the longitudirut.l steel
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Table 7
TYPICAL DIAL AND STRAIN* GAUGE RESULTS AT CENTER SPAN
(Steel) (Steel) Dial Gauge Rdg. DiaI Gauge Rdg .. Strain Gauge Rdg. Strain Gauge Rdg.
Load Bearn n -NF a Bearn ID - 9 x 1-1/4 a Bearn il -NF a Bearn m - 9 x 1-1/4 a
kips in. in. o 10-oAo lD. x . ID • -6/1 in. x 10 ln.
• 100 0.0 0.0 0.0 0.0 2 .007 .006 32 62 4 .012 .015 70 135 6 .017 .025 88 275 8 .022 ,,040 200 436
10 .030 .058 405 -594 12 .052 .077 598 749 14 .057 .096 650 892 16 .062 .111 782 1022 18 .074 .138 900 1148 20 .093 .150 1010 1270 22 .100 1208 24 .107 1376 26 .121 28 .155 30 .190
*" Readings are average of strains rneasured at that point.
... o ~
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110.
took place in sorne beams although it is believed that most yielding
occurred in the cantilever ends which had much larger deflections~
E. Deflection of Sponge Rubber
To obtain a uniformly distributed load on the beams, the
sponge rubber shou Id deflect equally at aU points along the beam. Dial
gauges, whose locations are shown in Figure 15, were used to measure
the deflection of the sponge rubber.
Figures 44 to 46 show typical load deflection curves of the
sponge rubber at düferent points along the beam. One typical curve
is drawn for each series. The curves start off in an alProximately
straight line, indicating linear elastic behaviour up to a load of about
10 kips, after which an up.yard curvature indicates a stüfening. Deflec
tions of the sponge rubber are in aU cases approximately equal along
each beam. A maximum difference of deflection between center span
and support occurs in Series Il of about 60 micro-inch, corresponding
to a variation of about one kip of machine load. This düference is
neglected in aU calculations.
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113.
-
h ~
1:'1 III LV ~
"'" 1
" 1'\
'\, 1 ....
" 1\1
1"'"
-1'1
1"1
~ 1\
l'iiô l'!r. II1II
"'" "
LI
" Il
.... C;
1'1"
r:;: r.. 1"
Ioo!\ Il
~ l-
l' "
10lI
Figure 390 Load Strain Curves
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If" 1\
" LI' 1\
1:;(
.fl'oJ 1"
l'
l'
Il Il
,
1\
1\'
RI.. 'to.."-~
- 1 .. 1" 1'::
-- - 1---
-1- -t---
Figure 410 Load Strain Curves
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1160
-,-~·~~~~~~~+++4~~+{~~~~~~t+~~rr++~~rr++Ti~rrTt~-HrtTt~lIltti
-~·_·II~~~~~~.~~~~++~~I~~~++~~~I\~4-~++~~4-~++~~rrtt~~rttt~-rGttt~~
1""
Ill"
I~I>
Figure 42e Load Strain Curves
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117.
"" 1"'0
, ~ ~
"-
50
1...\
,~
11'1 '" IJ' Uè
l-
I-
1
'~ 17
Figure 43. Lllad Strain C'i.lrves
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118.
u
II(
Vi
Fi gure 44. Load Deflection Curve for Rubber
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119.
-
....
" T- II( lA
I
IJ
li
"
la
~
.
1 -
~!'ll -
Figure 450 Load Deflection Curve for Rubber
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120.
ct'! IL ~ 1,,1:- ",.
l' lML
1
I~
ri
IJI:-
l\..
JI
-
~
",
L-'
V
IV 1-1-
Figure 460 Load Deflection Curve for Rubber
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121.
F. Comparison of Results with Existing Theories
The two most significant recent theories of shear failure
are those proposed by Moody, Viest, El~tner and Hognestad (14) and
by Laupa, Siess and Newmark*(12) (see Historical Review). In the
former report it is assumed that the shear sp~n to depth ratio has an
effect on the ultimate shear strength and thus the theory seems to he
limited to concentrated loading. The latter theory is applicable to
beams subjected to concentrated as weIl as distributed loads. The majQr
equations for calculating the ultimate shear moment, Ms, with the
L.S.N. theoryare for rectangular beams with tension reinforcement
orny, under concentrated loading:
where,
4.5f è Ms = bd2
f é k (0.57..,.. 10ij )
k = V (pn):L + 2pn - pn
fOr T-beams a shape factor oc is incorporated in the formula:
4.5f ~
where,
Ms = Acdf ~ oc (0.57 - 10 5 )
IT + 1er oC=
IR + 1er
* l{enceforth abbreviated L.S.N.
••• (8.1)
••• (8.2)
••• ( 8.3)
••• (8.4)
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122.
IT and IR refer to the uncracked section,
Icr refers to the "straight Une" cracked transformed
section of either a rectangular or T-beam since
both have nearly the same moment of inertia, and
Ac is the compressive concrete area, also determined
by the conventional straight-line theory.
At a section where maximum moment and maximum shear cOincide,
as is the case in Series 1 and II below the columns, the shear moment
of a beam under distributed loading can be directly determined by the
above equations. In regions of maximum moment and no shear, such
as at center span of Series nI, the above equations should be evaluated M
at the section where va = 4.5.
The applicability of the shape factor for T-beams seems
rather doubtful. In the report concerned, the authors simply assume
Us form, without any proof or explanation. Theoretical values, using
the shape factor, in sorne cases did not agree with the test results of
the report. Garulnick (18) applied the above formulae ta determine
"the shear moment capacity of 24 T-beams, omitting the shape factor.
The shape factor for the T-beams, tested by the author varied between
.60 and. 75. It was not used in the actual evaluation of the formq1ae,
in most cases giving more accurate results.
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123.
Columns 2 to 4, Table 8, give the experimental values of
the distributed load, and the maximum negative and positive :&.lloments
at ultimate loado The L.NoS. formulae are directly applicable to the
beams of Series 1 and II as the maximum moment and shear coincide. M
For Series III the formulae are applied for the section where Vd = 4.5
which is at X = 1806 ino (see colurnn 5). The values of the modular
ratio, n, column 6, are obtained from the formula:
••• {8.5)
Esteel This formula seemed more accurate than the use of n = =--
Econcr.
because of the fact that the Young's Moduli for steel and concrete
were not determined for each beam.
Column 8 gives the ultimate shear moment as calculated
by the t;aupa, Siess, Newmark theory, omitting the shape factoro The
ratio of calculated ultimate shear moment to the eXl2rimental moment
at ultimate load is given in column 9. Except for beam II - 15 x 1 a
aU ratios are smaller th an one, indicating safe design. In the majority
of cases the value ls below 0080, which i!l a practical case would be
considered an " over-designed "beam. The scatter of the test data
makes these results hard to interpret, but it should be emphasized
again that extreme care should be taken when using empirical formulae
to have exactly similar test and loading conditions as those under which
the formulae were derived.
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124.
The allowable unit shear stresses as stipulated by the ACI
Building Code No. 318-63 were calculated by the formula:
VaU = <P (1.9 VQ + 2500 Pw Vd ) M
••. (8.6)
and these values appear in Table 9, along with other data, such as the
concrete strength, th e percentage of longitudinal rei.nforcing and the Vd
value of M evaluated at a distance d from th e support:
for Series l, 1 Vd 12(2" - d)d M = 6 Id - 12
- 6d2. .•• (8.7)
for Series II, 1 Vd 48(2 - d)d M =
4(6 Id _12 - 6d") + 12
for Series III,
Vd (1 - 2d) M = (1 - d) eoo{8.9)
vult' also given in Table 9, is the nominal shear stress
at ultimate load, at a distance d from the columns. The last column
gives the ratio v:~!t.. These values or "Safety Factors" vary between
2.07 and 5.21. The safety factor seems higher for the restrained beams
than for the simply supported ones. It is anticipated that uniformly
loaded beams have higher shear strengths than beams loaded by one
or several large concentrated loads, due to the fact that uniformly
loaded beams have very short high shear regions in comparison to the
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Table B
COMPARISON OF TEST DATA WITH THEORETICAL SHEAR MOMENT VALUES
Test Results Calculated Results M· u
M!: Bearn M. Ms ... w (-)Mu (+)Mu at Vd ;: 4.5 n k L.S.N Theory Mu
kni in. kips in. kips in. kips in. kins (l) (2) (3) (4) (5) . ·(6)" . .. (7) (8) (9)
1 -NF a .229 96.18 48.09 - 8.1 .35 65.4 " .68 1 -NF b .231 97.04 48.57 - 8.1 .35 65.4 .67
1 - 9 x 3/4 a .249 104.60 52.30 - 7.4 .34 73.8 .71 1 - 9 x 3/4 b .289 121.41 60.71 - 7.4 .34 73.8 .61 1 - 9 x 1-1/4 a .222 93.26 46.63 - 7.8 .34 66.9 .72 1 - 9 x 1-1/4 b .217 91.16 45.58 - 7.8 .34 66.9 .73 1 -15 x 1 ... 1/4 a .217 91.16 45.58 - 7.7 .34 68.8 .75 1 -15 x 1-1/4 b .322 135.27 67.64 - 7.7 .34 68.8 .51
II -NF a .306 96.42 96.42 - 8.7 .35 58.1 .60 II -NF b .433 136.44 136.44 - 8.7 .35 58.1 .42
II-9xla .266 83.81 83.81 - 8.1 .35 .65 0 3 .77 II - 9 x-l b .266 83,,81 83.81 - 8.1 .35 65.3 .77 il -15 x 3/4 a .269 84.76 84.76 - 8.3 .35 6109 ,,73 il -15 x 3/4 b .258 81.30 81030 ...,. 8.3 .35 6109 .76 il -l5-x 1 a .185 57.74 57.74 - 8.2 .35 63.8 1010 il -15 x 1 b .236 74.36 74.36 - 8.2 .35 63.8 .86
Ma1t.
.... t\:> c:TI .
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Table 8 (cont'd.)
COMPARISON OF TEST DATA WlTH THEORETICAL SHEAR MOMENT VALUES
Test P..esults Calculated Results
Bearn MMu Ms w (-)Mu (+)Mu at va ::-4.5 n k L.S.N. Theory
kpi in.kips in. kips in. kips in. kips (1) (2 ) (3) (4) (5) (6) (7) (8)
m-NFa .234 - 147.44 113.1 8.3 .48 85.1 , m-NFb .259 -- 163.20 126.2 7.8 .48 96.2
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» .
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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
li-NFa 2700 1.14/.92 .41 110.5 93.93 375.6 3.99 ll-NFb 2700 . 1.14/.92 .41 110.5 93.93 494.3 5.21
li-9xla 3250 1.14/.92 .41 120.0 102.0 348.0 3.41 II-1)xl b 3250 1.14/.92 .41 120.0 102.0 336.5 3.29 li ~15 x3/4 a 3013 1.14/.92 .41 116.0 98.6 337.6 3.42 li -15 x 3/4 b 3013 1.14/.92 .41 116.0 98.6 331.4 3.36 il.". 15·x 1 a 3100 1.14/.92 .41 118.9 101.1 236.7 2.34 II -15 xl b 3100 1.14/.92 .41 118.9 . 101.1 302.1 2.98 ......
tI:) -J .
(cont'd.)
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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 13i.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 .84 145.1 123.3 363.3 2.94 1 - 9 x 1-1/4 a 3510 1.14 .84 136.8 116.3 320.4 2.75 l '" 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
n-NFa 2700 1.14/.92 .41 110.5 93 0 93 375.6 3.99 il-NFb 2700 - 1.14/.92 .41 110.5 93.93 494.3 5.21
n ... 9xla 3250 1.14/.92 .41 120.0 102.0 348.0 3.41 II-~xl b 3250 1.14/.92 .41 120.0 102.0 336.5 3.29 II ~ 15 x 3/4 a 3013 1.14/.92 .41 116.0 98.6 337.6 3.42 il -15 x 3/4 b 3013 1.14/.92 .41 116.0 98.6 331.4 3.36 II '" 15 -x 1 a 3100 1.111/.92 .41 118.9 101.1 236.7 2.34 II -15 xl b 3100 1.1;1/'92 .41 118.9 . 101.1 302.1 2.98 .....
~
:..:J
(cont'd.)
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Table 9 (cont' d.)
Vd vaU JI)
f ' M c p ACI code Bearn psi. % cale. pd.
m -NFa 3000 2.51 .903 160.7 m-NFb 3550 2.51 .903 169~8
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
vaU Vult. psi. psi.
136.6 306.4 144.3 345.1
135.7 282.7 135.7 351.6 136.6 283.4 144.3 309.0 14L3 353.4 141.3 343.3
Vult.
vaU S.F.
2.24 2.39
2.08 2.59 2.07 2.14 2.50 2.42
.N foC
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129.
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.
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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.
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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,
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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.
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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.
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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
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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.
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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.
5. Mylrea, T. D. Tests of ReinfQrced Concrete A.C.I. Journal,Proceedings, T-beams. Vol. 30, No. U-12, May,
June 1934, pp. 448-464.
6. McCullough, C. B. Flexural Resistance of Shallow Concrete Beams.
7. Blackey, F. A. A Theory of Deflection of Reinforced Concrete Beams Under Short Term Loading.
8. Clark, A. P. and Diagonal Tensio~. in Rein ... forced Concrete Beams.
dis'cussion by Ferguson, P.M. and Moretto, O.
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.
~"
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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.
A.C .1. Journal, Proceedings, Vol. 49, No. 6, Feb. 1953, pp. 585-692.
Part 1 -Tests of Simple A.C.I. Journal, Proceedings, Beams. Vol. 51, No. 4, Dec. 1954,
pp. 317 -332.
Part II - Tests of Restrained Beams without Web Reinforcement.
Part III - Tests of Restrained Beams with Web Reinforcement.
Part IV - Analytical Studies.
A.C .1. Journal, Proceedings, Vol. 51, No. 5, Jan. 1955, pp. 417 -434.
A.C.I. Journal, Proceedings, Vol. 51, No. 6, Feb. 1955, pp. 525-639.
A.C.I. Journal, Proceedings, Vol. 51, No. 7, March 1955, pp. 697 -73 O.
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15. Brown, E. 1. and
di seussion by Ferguson, P. M.
1 6. Ferguson, P. M.
17. Jones, Ro
18. Garulnick, S. A.
19. AI-Alusi, A. F.
20. Bresler, B. and Pister, K. S.
21. McHenry, D. and Karni, J.
22. Rensaa, F. M.
23. Garulnick, S. A.
24. Brock, G.
Strength of Reinforced Concrete T-beams under Combined Direct Shear and Torsion.
Sorne Implications of Recent Diagonal Tension Tests.
The Ultimate Strength of Reinforced Concrete Beams in Shear.
An Investigation of High Strength Deformed Steel Bars for Concrete Reinforcement.
Diagonal Tension Strength of Reinforced Concrete T-beams with Varying Shear Span.
Strength of Concrete under Combined Stresses.
Strength of Concrete under Combined Tensile and Compressive Stress.
Shear, Diagonal Tension and Anchorage in Beams.
Shear Strength of Reinforced Concrete Beams.
Effect of Shear on Ultimate Strength of Rectangular Beams with Tensile Reinforcement.
138.
A.C.I. Journal, Proceedings, Vol. 51, No. 9, May 1955, pp. 889-902.
A.C.I. Journal, Proceedings, Vol. 53, No. 2, Aug. 1956, pp. 157-172.
Magazine of Concrete Research, VoL. 8, No. 23, August 1956.
Report No. TSR 4730-7146, School of Civil Engineering, Cornell University, July 195' 123 pp.
A.C.I. Journal, Proceedings, Vol. 53, No. 11, May 1957, pp. 1067-1077.
A.C.I. Journal, Proceedings, Vol. 55, No. 3, Sept. 1958, pp .. 321--345.
A.C.I. Journal, Proceedings, Vol. 54, No. 10, April 1958, pp. 829...a39.
A.C .1. Journal, Proceedings, Vol .. 55, No. 6, Dec. 1958, pp. 695-715.
A.S.C.E. Proceedings, Vol. 85, Jan. 1959, pp. 1-42.
A.C.I. Journal, Proceedings, Vol. 56, No. 7, Jan. 1960.
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25. Bower, J. E. and Viest,l. M.
26. Taub, J. and Neville, A. M.
27. Neville, 1. M.
28. Williams, D.
29. Faris, A. M.
30. Report of A.C.I. -A.S.C.E. Committee 326
3 1. 'K;lni, G. J.
32. Krefeld, W. J. and Thurston, C. W.
33. Jones, L. L.
139.
Shear Strength of Restrained A.C.I. Journal, Proceedings Concrete Beams without Web Vol. 57, No. 1, July 1960, Reinforcement. pp. 73 -98.
Resistance to Shear of Reinforced Concrete Beams, Part 1: Beams without Web Reinforcement.
Some Factors in the Shear Strength of Reinforced Concrete Beams.
Diagonal Tension Cracking of Reinforced Concrete Beams.
A. C .1. Journal, Proceedings Volo 57, No. 2, Aug. 1960, pp. 193 ";!20o
Structural Engineeer (London), Vol. 38, No. 7, July 1960, pp. 213 -223 .
M. Eng. Thesis, McGill University, 19610
Ultimate Shear in Reinforced M. Engo Thesis, COD.crete Beams with McGill University, 1962. Stirrups.
Shear and Diagonal Tension.
The Mechanism of So-called Shear Failure.
Contribution of Longitudinal Steel to Shear Resistance of Reinforced Concrete Beams.
A Theoretical Solution for the Ultimate Strength of Rectangular Reinforced Concrete Beams without Stirrups.
A.C.I. Journal, Proceedings, Vol. 59, 1962, pp. 3-30.
Transactions Engineering Institute of Canada, No. EIC '63-CN-5, April 1963.
A.C.I. Journal, Proceedings, Vol. 63, No. 3, March 1966, pp. 325-345.
Paper p-esented to the European Committee for Concrete at Wiesbaden, Cement and Concrete Association (London).
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34. Krefeld, W. J. and Thu,rston, C. W.
3 5. Kàni, G. J.
Studies of Shear and Diagonal Tension Strength of Simply Supported Reinforced Concrete Beams.
Bauic Facts Concerning Shear Failure.
140.
A.C.I. Journal, Proceedings Vol. 63, No. 4, April 1966, pp. 4514770
AoC.I. Journal) Proceedings: Vol. 63, No. 6, June 1966 1
ppo 675-692.