Strength Evaluation of Shear Deficient RCC Beams Strengthened
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PRE-TENSIONED CONCRETE BEAMS UNDER UNIFORM LOAD
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PRE-TENSIONED CONCRETE BEAMS UNDER UNIFORM LOAD
· .
STRENGTH AND BEHAVIOUR OF PRE-TENSIONED CONCRETE BEAMS
SUBJECTED TO UNIFORMLY DISTRIBtrrED LOAD
by
Ciro R. Martoni
A thesis submitted to the Faculty of Graduate Studies
and Research in partial fulfiLment of the
requirements for the degree of
Master of Engineering
Department of Civil Engineering and Applied Mechanics
McGill University
Montreal, Canada.
® Ci ro R. :-!artvni 1971
August 1970
TO THE MEMORY OF
KY FATHER
STRENGTH AND BEHAVIOUR OF PRE-TENSIONED CONCRETE BEAMS
SUBJECTED TO UNIFORMLY DISTRIBUTED LOAD
Department of Civil Engineering and App1ied Mechanics
Ciro R. Martoni
ABSTRACT
- i -
M.Eng. August 1970
The object of this investigation was to study the strength and beha-
viour characteristics of simply supported pre-tensioned prestressed concrete
I-beams subjected to a uniform1y distributed load. The uniform load was
successfu1ly applied through two water-filled hoses placed directly on the top
of the test beam.
The parameters examined were the level of prestress and concrete
compressive strength. Duplication of the specimens demonstrated that the re-
producibility was excellent.
A previously developed flexure-shear strength theory was modified to
make it applicable for the analysis of beams under uniform load. Satisfactory
agreement was obtained between measured and computed ultimate capacities for
the specimens tested here and by another investigator.
In comparison to point-loaded beams failing in flexure-shear, simi-
lar beams under uniform load were found to possess substantially larger ulti-
mate load-carrying capacities.
- ii -
RESISTANCE ET COMPORTEMENr DE POurRES PRECONrRAINrES SOUS L'EFFET
D'UNE CHARGE UNIFORMEMENr REPARTIE
Département de Génie Civil et de Mécanique Appliquée
Ciro R. Martoni
RESUME
M.Eng. Août 1970
L'objet de cette étude est la détermination de la résistance et
l'analyse du comportement de poutres isostatiques de section en l en béton a
précontrainte initiale sous l'effet d'une charge uniformément répartie. La
charge uniforme fut appliquée avec succès par l'intermédiaire de deux boyaux
remplis d'eau et placés directement sur la face supérieure des poutres d'essai.
Les paramètres examinés furent l'intensité de la précontrainte et la
résistance à la compression du béton. La répétition des essais a déuiontré que
la reproductibilité des résultats était excellente.
Une théorie existante prédisant la résistance des poutres en flexion
composée fut modifiée pour la rendre appliquable au cas de poutres soumises à
une charge uniformément répartie. Les charges de rupture ainsi obtenues fu-
rent comparées à celles des essais, et a celles obtenues indépendamment par un
autre chercheur. Un accord satisfaisant fut noté.
Il est montré que des poutres identiques qui sont soumises à la rup-
ture par flexion composée supportent une charge supérieure lorsque celle-ci
est distribuée uniformément au lieu d'être concentrée.
- iii -
ACKNOWLEDGEMENTS
The writer wishes to acknowledge the following persons and organiza-
tions:
Prof. J.O. McCutcheon, Chairman, Department of Civil Engineering and
Applied Mechanics, who acted as research director, for his guidance and en
couragement throughout the project and constructive criticism during the wri
ting stages of this thesis;
Dr. M.A. Sheikh, Chief Civil Engineer, P.I.D.C. House, Pakistan, who
acted as research director in the early stages of this investigation, for
suggesting the problem;
Ors. M. Celebi, M.S. Mirza and J. Nemec for their helpful suggestions
gained through informaI seminars;
Prof. B. Gersovitz for his advice concerning the practical aspects
of the study;
Fellow graduate students for their moral support and Mr. B. Cockayne
and his staff for their technical assistance;
Canada Cement Co. and the Steel Co. of Canada who generously donated
the cement and prestressing strands, respectively;
:':_: ior.<~l I\esca~:ch Council of ·Canada who provided financial assistance
under Grant No. 282-01; and
Miss R. Then for her exce1lently typed transcription of this mate-
rial.
- iv -
TABlE OF CONTENT S
p~gë -N""o-:--
ABSTRACT. • . . . . . • . • . . • . . . . . . . . • • • . . • • • . • • . • . • • • . . . . . . . • • • • • • . • . • . • . . . . i
RESUME............................................................... ii
AC KNOWLEDGEMENT S. . . . • • • • • . . • • • • . • • • . • . . . . . . . . . . . . . . . . • . • • • • • • . . . • . . • . i i i
TABLE OF CONTENTS.................................................... iv
LIST OF TABLES........ . . • . • . • . . . . . . . • . . . . . • • . • • • • . . . . • • • • • . . . • . . . . . • • vii
LIST OF FIGURE S. • . . • • . • • . . . • . . • • . . . . • . . . . . . . . . . . . • • . . . • . . • • . • • . . • • . . . vii i
NŒATIONS............................................................ x
1. INI' RODUCT ION. . . • . • . . . . . . • . . . • . . • . . . • . . . . . • • . • . • • • . • . • . . • . . . . . . . . . 1
1 . 1 Gene ra 1. . . • . . • . . . . . • . . . . . . • . . . . . . . . • . • . . . . . . . . . . . . • • . . • . . . . . 1
1.2 Object and Scope............................................ 2
1.3 Theoretica1 Approach... ...•.....•••..•........•..•..•.•..... 3
2. REVIEW OF PREVIOUS WORK.......................................... 4
3. SPECIMENS AND TESTING METHOD..................................... 31
3. 1 Nomenc 1ature for Specimens..................... . . . . . . . . . . . . . 31
3.2 Description of Specimens.................................... 31
3.3 Test Bed and Loading Apparatus.............................. 32
3.4 Measurements and Test Procedure............................. 36
4. EXPERIMENTAL RESULTS AND OBSERVATIONS............... ............. 39
4.1 General..................................................... 39
4.2 Load-Deflection Relationships............................... 39
4.3 Crack Patterns.............................................. 45
4.4 Modes of Failure............... ............................. 56
4.5 Measured Concrete Strains.... ............................... 59
4.6 SUDlDary..................................................... 62
- v -
TABLE OF CONTENTS (Con~d)
Page No.
S. THEORETICALANALYSIS............................................. 64
5.1 MechanismofFailure ..•..•...........•........•............. 64
5.2 Factors Affecting the Flexure-Shear Strength of Prestressed Concrete Beams.............................................. 66
5.3 Strength of the Compression Zone............................ 66
5.4 Deformation Conditions..................... ..........•...•.. 75
5.5 Equilibrium Conditions...................................... 87
5.6 Determination of the Depth of Compression Block............. 89
5.7 Effective Moment-Shear Ratio................................ 91
5.8 SUlDlDary..................................................... 92
6. QUANTITATIVE EVALUATION OF TEST RESULTS.......................... 93
6.1 General..................................................... 93
6.2 Ulttmate Load Analysis of a Beam Under Uniform Load......... 94
6.3 Evaluation of McGill Tests.................. ................ 95
6.4 Evaluation of Tests by Kar (45, 46)............... .......... 97
6.5 Influence of Loading Arrangement on Load-Carrying Capacity.. 103
6.6 Comparison with ACI 318-63 (5).............................. 108
6.7 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . 109
7 • CONCLUS IONS. . . . . . . . . . . . • . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7. 1 Overall SUDlDary............................................. 110
7 . 2 Future Research............................................. 112
8. REFERENCES. . . . . . . . • . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . 114
- vi -
TABLE OF CONTENTS (Cont'd)
Page No.
APPENDIX A: MATERIALS, FABRICATION AND INSTRUMENrATION.............. 121
A.1 Materials................................................... 121
A.2 Beam Forms. • . . • • • • . • • • . • . . . . . . . . . . . . . . . • • • . • . • • . . • . . • • . • . . • . 123
A.3 Pres tres s ing Equipment............................ . • • • • . • • . . 124
A.4 Preparation Prior to Tensioning.. .. .... ........... ....... ... 126
A.S Tensioning Procedure...... .........................•.•...... 127
A.6 Cas t ing and Curing............. . • . . . . • . . . . . • . . • . . • • • . . . . . . . • 127
A.7 Release of Prestress........................................ 128
A.8 Ins t rument at ion. • . • • . . • . . . . . . . . . . . • . . . . . • . . • . . . • . . . • . . . . . . . • 130
APPENDIX B: LOADING APPARATUS.. . . . . . . . • • . . . • . . . • • • • . . • • • . . . • • • . . • . . • 133
Table No.
1
2
3
4
5
6
7
8
A.l
LIST OF TABLES
Title
Properties of Test Beams
Measured Total Loads (Kips)
McGill Tests - Measured and Computed Ultimate Capacities in Terms of Total Load on Beam
Properties of Beams Tested by Kar o (45, 46)
Tests by Kar (45, 46) - Measured and Computed Ulttmate Capacities in Terms of Total Load on Beam
McGill Tests - Comparison of Beam Ulttmate Capacities Under Point and Uniform Loading
Tests by Kar (45, 46) - Comparison of Beam Ulttmate Capacities Under Point and Uniform Loading
McGill Tests - Comparison With AC! 318-63 (5)
Properties of Concrete Mixes
- vii -
Page No.
35
40
98
99
102
105
106
108
122
Fig. No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
LIST OF FIGURES
TUle
Nominal Dimensions of Beams
Stress-Strain Relationship for 3/8 in. strand
Loading Arrangement Prior to Testing
Slip-Measuring Arrangement
Load Deflection Curves - Series l
Load Deflection Curves - Series II
Crack Pattern Beam Pl
Crack Pattern Beam PlA
Crack Pattern Beam P2
Crack Pattern Beam P2A
Crack Pattern Beam P3
Crack Pattern Beam P3A
Crack Pattern Beam P2-Cl
Crack Pattern Beam P2-ClA
Crack Pattern Beam P2-C3
Crack Pattern Beam P3-C3
Longitudinal Strain Distribution. Drawn for Beam P3A
Flexure - Shear Fai1ures
Stress Condition at Different Points Along the Beam Represented in the Form of Mohr's Stress Circ1es
Assumed Straight Line Mohr's Enve10pe
Determination of the Factors c, 9 fram Assumed Mohr's Envelope
plot of Interaction Curve of Concrete Strength Under BiAxial State of Stress (Eq. 5) for f If' = 0
z c
- viii -
Page No.
33
34
37
37
41
42
47
47
48
48
49
49
50
50
51
51
60
65
68
70
70
72
Fig. No.
23
24
25
26
27
28
29
30
31
32
A.l
A.2
A.3
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LIST OF FIGURES (Cont'd)
Title Page No.
Plot of Eq. 5 for Different Values of f If', and ~ = 0.14 Z c 73
Comparison of Various Interaction Equations for the Bi-Axial Strength of Concrete in Beams (f If' = 0, p. = 1/8) 76
Z c ,-
Disturbed Region Contributing to the Deformations of Steel and Concrete at the Critical Section 77
Schematic Representation of Strain Distribution at the Extreme Compression Fibre of Beam Shawn in Fig. 25 77
Equi',alent Compression Stress Block Used in This Thesis 88
Comparison of Various Compression Stress Blocks 88
General Stress-Strain Diagram of Steel 90
Determination of the Depth of Neutral Axis 90
Determination of the Position of the Failure Section and the Strength of a Simply-Supported Beam Under Uniform Load. Drawn for Beam P2-Cl 96
Stress-Strain Relationship for 0.276 in. wire Used by Kar (45, 46) 100
Reaction Frame 125
Prestressing and Casting Bed 125
Prestress-Releasing Mechanism 129
A
Ab
A s
a
b
b 1
bl
D
D 0
d
F se
Fsi FI
s
F su
fI' f 2 fI c
f cu
f cv
f pc
ft r
f sp
f' t
NOTATIONS
= gross cross-sectional area of the beam
= bearing area of the loading plate
= area of longitudinal reinforcement
length of shear span
= width of rectangular beam or width of compression flange of a flanged beam
thickness of web of a flanged beam
- x -
equivalent width of the beam at the level of the longitudinal reinforcement
diameter of tendon used
reference size of tendon taken as 1 in.
= effective depth to the centroid of the ·tensile reinforcement in a beam
total effective prestressing force
= initial prestressing force
total steel force at failure of beam
= ultimate tensile strength of the total tensile reinforcement in the beam
= principal stresses in a two-dimensional stress system
= compressive strength of standard 6 in. by 12 in. cylinders
= effective strength of the concrete in the compression zone of a beam at pure flexural failure = klk3f~
•
..
=
shear-compressive strength of concrete
compressive stress in the concrete at the centroid of the crosssection due to effective prestressing force
modulus of rupture strength from 6 in. by 6 in. by 24 in. control beams loaded at third points over a 18 in. span
splitting strengtb determined by indirect tension test
tensile strength of concrete
f s
f se
f' s
f su
f Y
f z
h
l s
K
k u
L
M u
v
v c
- xi -
= stress in steel at any stage of loading
= effective prestress in reinforcement
= stress in reinforcement at failure of beam
= ultimate tensile strength of longitudinal reinforcement
= initial prestress in tendons
= normal flexural stress at a point
= yield strength of longitudinal reinforcement
= normal vertical stress due ta applied load
= overall depth of a beam
= moment of inertia of tensile steel plus the concrete in tension surrounding and below the tensile steel
= factor taking into account the effect of effective moment-shear ratio in As
= modification of factor K for beams under uniformly distributed load
= parameters describing the properties of the concrete compressive stress black
= coefficient for bond qua lit y between main steel and concrete
= ratio of neutral axis depth at failure of the beam to effective depth
• ratio of neutral axis depth ta effective depth for pure flexural failure in under-reinforced beam
• length of beam span
= moment at any stage of loading
• resisting moment at failure
• resisting moment for a pure flexural failure
total shear at any stag~ of loading
shear force carried by the concrete in the compression zone
Vd • shearing force carried by dowel action
V • shear force corresponding to failure load u
- xii -
v = shear stress
v = shear stress acting in x-z plane xz
x = distance from support to the cross-section under investigation
0( = a/d-ratio at which and above which a b'eam with a point load at mid-span would fail in flexure
f'lf (= f' If' in this thesis) t cu t c
~c = deformation of the concrete at the extreme compression fibre of the critical section
~ s = deformation of the tension chord (steel) at the critical section
~su
=
max~ compressive concrete strain at the critical section at failure of the beam
(E:su - ESsi) = increase in steel strain, in an ordinary reinforced concrete beam, due to applied loading
= strain in steel when the strain in concrete at the steel level is zero
=
=
=
=
=
strain in steel at the failure of the beam V
degree of shear safeguard to the compression zone = 1 - -S V
ratio of stress in steel at any stage of loading to the ulttmate strength
roughness coefficient of longitudinal steel surface
f If (= f If' in this thesis) cv cu cv c
Other notation found in the text are defined where used.
- 1 -
1. INrRODUCT ION
1.1 General
The strength of concrete beams, both ordinary reinforced and pre
stressed, in flexure is quite weIl defined but their strength in resisting
shear or the combination of shear and flexure cannot be predicted with preci
sion. The factors inf~uencing behaviour and strength of concrete beams fail
ing in shear are numerous and complex. They include the proportions and
shape of the beam, the structural restraints and the interaction of the beam
with other components in the system, the amount and arrangement of tensile,
compressive, and transverse reinforcement, the degree of prestress, the load
distribution and loading history, the properties of the concrete and steel,
the concrete placement and curing and the environmental history (1). The
problem of shear is further complicated by the redistribution of internaI
forces which takes place with the formation of diagonal tension cracks.
Although an extremely large number of tests have been performed in
the study of shear strength, the problem still remains relatively unsolved.
Due to the complexity of the issue, investigators have frequently resorted to
empirical or semi-empirical expressions to predict strength in shear. While
these expressions usually agree quite well with the corresponding test results,
they are not applicable for general design use as they are related to or in
fluenced by the beam properties and loading conditions from which they were
derived. A rational solution involving aIl of the ab ove mentioned factors
would be ideal. However such a solution does not seem possible at the present
ttme due to the unknown and bigbly complicated interaction of the factors in
volved.
- 2 -
1.2 Object and Scope
Much expertmenta1 research has been performed on the shear strength
of concrete beams subjected to symmetrica11y p1aced isolated concentrated
loads in which the shear span, the distance fram the support to the nearest
load, is subjected to a constant shear force and a linear1y varying bending
moment. Although this 10ading arrangement is useful in determining the
effect of shear and in locating the critical cross-section, i.e. at the load
point where both shear and moment reach their max~ values simultaneously,
it is not representative of the loadings used in practical designs. With
distributed loading in the form of either a large number of closely spaced
point loads or a uniformly app1ied pressure, the shear force varies across
the span in a linear fashion and the bending moment distribution is parabolic.
The position of the critical section and hence the ultimate capacity are thus
unknown since the shear and moment maxima do not occur at the same section.
The object of this investigation is to study the strength and beha
viour characteristics of sLmply supported pre-tensioned prestressed concrete
I-beams subjected to a uniformly distributed load. In addition, a previously
proposed theory for the flexure-shear strength of concrete beams will be em
ployed to predict the u1timate capacities of the test beams.
The system most commonly used to simulate a uniformly distributed
load i8 multiple load points, the accuracy never being very great, unless a
large number of point loads are employed. In this study the applied load was
distributed as uniformly as possible with the aid of two water-filled hoses, a
technique previously used by Leonhardt and Walther (2).
Of the vast number of parameters that could possibly be investigated
to determine their influence, only two, concrete compressive strength and
level of prestress, are under study here. The concrete strength ranged fram
- 3 -
4640 psi to 7600 psi, while the effective prestress level ranged fram 71 ksi
to 127 ksi. The specimens were duplicated to evaluate the quality of reprodu
cibility.
1.3 Theoretical Approach
The semi-rational theory proposed by Sheikh (3, 4) to estimate the
flexure-shear capacity of prestressed concrete beams will be used to compute
the ultimate capacities of the beams reported here. In its proposed form th~
theory is capable of analysing simple and continuous prestressed concrete
beams subjected to isolated point loads. If required, the theory can be modi
fied to make it applicable to analyze simply supported beams under uniform
load. Prestressed concrete test beams under uniform load found in the litera
ture will also be analyzed. The theory in its present form is described
briefly but in sufficient detail in Chapter 5, along with possible modifica
tions. Only ref~rence to simply supported beams will be considered in this
thesis, i.e. the analysis of continuous beams will not be dealt with here.
The test beams reported here are also discussed with reference to the ACI 318-
63 (5) shear clause&. To bring out the differences, if any, between beams
loaded by uniform load and point loading, similar beams are analyzed under
both types of loading.
2 • REVIEW OF PREVIOUS WORK
A large number of the concepts used by recent researchers to
explain aspects of the behaviour of concrete beams failing in shear were
first described by the pioneers in this field.
- 4 -
It has long been recognized that the cause of failure is mainly
diagonal principal tensile stress and not horizontal shear as was originally
envisaged. The origin of the diagonal tension concept in reinforced concrete
beams is uncertain but a clear explanation of this was presented in 1899 by
Ritter (6). He also introduced the "truss analogy" in which a reinforced
concrete beam subject to shear forces is assumed to behave as a pin jointed
truss. In this analogy the compression zone of the beam acts as the com
pression chord of the truss, the longitudinal reinforcement as the tension
chord, the stirrups or bent-up bars as the web members in tension and the
concrete in between the compression zone and the longitudinal reinforcement
as the web members in compression. The truss analogy has been the basis of
shear reinforcement calculations in the codes of the West European countries.
It is still considered the basis for the CEB Code (7) and the British
Standard Code of Practice No. 114 (1957). The ACI-ASCE Committee 326 (8)
recommendation is also basically an empirical adaptation of that method.
In 1903 Morsch (6) presented an equation for the nominal shear
stress in reinforced concrete beams and showed that this stress could be
used as a measure of the diagonal tension stresses that appeared to cause
failure in shear. His equation for the nominal shear stress, v • V/bjd,
was based on the following assumptions: the strains varied linearly on a
transverse section of the beam, the stresses in both the concrete and the
steel remained in the elastic range and the concrete could resist the diago
nal tension stress inferred by the equation but could not resist any longi-
- 5 -
tudinal flexural tension stresses. For beams with web reinforcement the dia
gonal tension strength of the concrete was ignored and the ultimate shearing
stress was derived,- using the truss analogy, solely fram the strength of the
web reinforcement. In spite of its shortcomings the equation for the nominal
shear stress is still used today except for the minor change of omitting "j".
In 1907 Morsch (6) concluded that the internaI forces are redistri
buted when diagonal cracks form in a beam without shear reinforcement. The
compression zone and the dowelling action of the longitudinal steel were
thought to resist the applied shear force. However, the dowel action was not
considered to account for much of the shear resistance. To resist shear the
compression force must become inclined above the diagonal crack. Apart from
dowel action it was assumed that no shear could be transmitted across the
crack, an assumption apparently accepted without question by nearly aIl later
research workers.
Based on a number of tests of reinforced concrete beams, Talbot (6)
in 1909 concluded that the shear strength was dependent upon: the strength of
the concrete, the amount of flexural and shear reinforcements and the length
of the shear span. Stirrup stresses were found to be smaller than those in
ferred by the equations based on the truss analogy and from this it was de
duced that part of the shear was resisted by the compression zone of the beam.
lt was recommended that stirrups be dtmensioned for two-thirds of the exter
nal shear, the remaining one-third being carried by the concrete in the com
pression zone.
In 1916 Faber (9) demonstrated that a shear force could be resisted
by means of arch action. lt was assumed that the compression force could
become inclined between the load and support points and consequently the ten
sion force in the re1nforcement was constant between these two points. For
- 6 -
this arching action to occur the bond resistance of the reinforcement must be
overcome and extensive slip must occur between the concrete and steel.
Recent research work (e.g. Ref. 30) has shawn that this arching action can
take place only in short beams.
In 1927 Richart (10), noting that the stirrup stresses were consis
tently less than those predicted by the truss ana logy , proposed an expression
for the ultimate shearing strength which was composed of two terms. One term
accounted for the shear strength of the concrete and the other for the
strength of the shear reinforcement.
Prior to 1945 investigators tested reinforced concrete members for
diagonal tension cracking stress and stresses in the shear reinforcement as a
function of the applied load. The present general tendency is to attempt to
express quantitatively the influence of the different factors on the shear
strength of reinforced concrete members. However, owing to the complexity of
the problem frequent resort has been made to empirical interpretation of test
results.
Moretto (11) in 1945 reported that the percentage of longitudinal
steel, p, was an important parameter affecting the ultimate shearing strength.
He proposed an empirical expression for the ultimate shear strength in terms
of nominal stress which indicated the contributions of the concrete com
pressive strength and the amounts of longitudinal and shear reinforcements.
Altbough some previous investigators had indicated that the manner
of loading would affect the shear carrying capacity of a beam, Clark (12) in
1951 showed tbat the 10ading condition, expressed in terms of the ratio dIa
(effective depth/shear span), exhibited a considerable influence. For the
same concrete strength the resistance to fai1ure in diagonal tension was
found to increase as the loads were moved from midspan towards the supports.
- 7 -
He found that after the yield stress was reached in one stirrup leg, the
stress increased in adjacent stirrups, indicating a redistribution of internal
stresses. An empirical expression was suggested for the ultimate shearing
resistance in terms of nominal shear stress. This expression included the
contributions of the ratio of longitudinal tension reinforcement, p, concrete
compressive strength, f~, ratio of effective depth to shear span, dia, and
ratio of shear reinforcement, r. Clark emphasized that his formula was in
tended to indicate the factors which must be considered rather than to be
used for general design purposes. The ACI-ASCE Committee 326, Shear and Dia
gonal Tension, (8) stated that "this investigation was a major step toward a
modern understanding of the ultimate strength of reinforced concrete beams in
shear".
In 1953, Ferguson and Thompson (13) reported tests on T-beams with
out stirrups in which the chief variables were the concrete strength and the
effect of extra web width over part of the beam depth. They found that the
0.03 f~ permissible unit shear allowed by the ACI Building Code then in
effect led to a greatly reduced factor of safety when high strength concrete
was used since the ratio Vult/f~ decreased for increasing values of f~. Test
results of the T-beams with "shoulders" indicated that the use of the minimum
web width in the calculation of the nominal shear stress ia overly conserva
tive. Hence, they suggested that the area of the 5houlders below the neutral
axis be added to "b'd" for use in the nominal shear formula. The authors
pointed out a large variation in shear strength for different a/d ratios.
The increased str€ngth for loads near the reaction was said to be the effect
of vertical compressive stresses over and near the reaction that cancel or
greatly reduce potential diagonal tension stresses. This was shawn to be 50
in a test beam where for a distance of "d" fram the support the depth of the
- 8 -
beam was reduced by fifty percent. The local reaction effect was enough to
offset the reduction in depth. The a/d ratio was 3.4. The authors questioned
the majority of previous work where the a/d ratios were equal to two or less
and the litt1e emphasis placed on that factor. Rence, it is possible that
too much reliance had been p1aced on tests that did not indicate truly depen
dable mintmum strengths.
Zwoyer and Siess (14) in 1954 were the first to put forward the con
cept of the "shear-compression" mode of failure. Failure is caused by the
rapid increase of the diagonal cracks resu1ting in a reduced compression area
and a concentration at the apex of the diagonal crack of the compressive
strains on the top fibre of the beam. This is followed by the crushing of the
concrete over the inclined cracks and adjacent to the loading point. The
destruction of the compression zone appears to be similar in Many ways to that
of the flexural failure of over-reinforced beams. Zwoyer and Siess deduced
that the shear-compression mode of failure occurs at some limiting moment
rather than at some ltmiting value of the shear itself. Based on this concept
an empirical formula was developed to predict the ultimate shear strength of
prestressed concrete beams, which was then extended to ordinary reinforced
concrete beams.
In 1954 Moody, Viest, Elstner and Rognestad (15) published a four
part report on restrained and simple reinforced concrete beams under concen
trated loads. Test results indicated that for beams with relatively large
M/Vd ratios, but not large enough to cause flexural failure. the initial dia
gonal tension cracking May produce an ~iate failure of the beam. This
mode of failure was called "diagonal tension". Within the ranges of variables
studied the diagonal tension strength, expressed in terms of the nominal shear
stress. was shawn to be pr1marily dependent on the compressive strength of
- 9 -
concrete and the ratio M/Vd. The equations formulated should not be used for
beams with M/Vd larger than 3.5. The ratio M/Vd was employed in the equation
rather than a/d because, although these ratios are stmilar for stmply sup
ported beams, they are different for end-restrained beams. Beams with rela
tively smaller M/Vd ratios were able to sustain loads greater than the ini
tial diagonal tension cracking load and finally failed by destruction of the
compression zone above the diagonal tension crack, i.e. by shear-compression.
The presence of the diagonal crack was said to be a necessary condition for
shear failure. The shear-compression capacity expressed as a moment was
shawn to be independent of the magnitude of the shear but to depend primarily
on the dimensions of the cross-section, amounts of longitudinal and web rein
forcements, compressive strength of concrete and the M/Vd ratio. Although
the basic equation was developed from conditions of statical equilibrium, it
contained several emperical parameters describing the stress distribution in
the concrete above the diagonal crack and the stress in the longitudinal
reinforcement. The equations presented apply only to statically determinate
beams loaded in such a manner that the maxÜDWm shear is constant over a
portion of the span and the maxÜDWm moment occurs at one or both ends of the
region of maxÜDWm shear. Thus, they are not applicable to beams loaded with
a distributed load.
In 1955 Laupa, Siess and Newmark (16) published an extensive analy
tical report. The object of this report was to review and correlate the
results of previous research in the field of shear and diagonal tension, to
determine the modes and characteristics of shear failure of reinforced con
crete beams and to establish a general expression for the shear strength of
reinforced concrete beams under different loading conditions. Having pointed
out that the conventional formula for nominal shearing stress cannot be a
- 10 -
true criterion of shear failure as no transfer of stresses occurs across
cracks, they assumed that the total shear force is resisted solely by the area
of concrete in the compression zone and that the criterion of fa.Uure is an
ultimate shearing stress related to the compressive strength of the concrete.
With these assumptions an expression was formulated which suggested that the
real criterion for shear failures was a limiting moment rather than an ulti
mate shear stress. This observation was supported by certain test results
reported in the literature. Beams tested by Clark and Moritz having the a/d
ratio as the only variable failed at a nearly constant moment, although the
total shear force at failure depended upon the location of the loads on the
beams. It was concluded that shear failures were actually a compression
phenomenon. Their studies indicated that the "shear" moment was influenced
mainly by the cross-sectional dimensions of the beam, the amount of longitudi
nal reinforcement and the concrete compressive strength but not by the a/d
ratio. However, the test beams analyzed had a/d-ratios which varied only
between 1.17 and 4.80. It was this variation in a/d-values that did not seem
to have any effect on the agreement between the test results and the predicted
values. The shear-moment equation included the empirical determination of the
depth of the compression zone and the average compressive stresses in it, both
of vhich vere concluded to be primarily a function of the concrete compressive
strength and on1y secondarily of the amount of longitudinal reinforcement.
This equation vas then modified to fit various types of beams and 10adings.
Laupa et al. briefly discussed beams under uniformly distributed
load. For simple-span beams under uniform load the value of M/Vd ranges from
zero at the support to infinity at midspan. The beam cannot fail in shear at
the section of maximum moment because there are no diagonal cracks at that
section. Hence, the critical section for a shear-compression failure 15 one
- Il -
at which the shear is large enough for a diagonal tension crack to form, yet
the moment is sufficiently large to cause crushing of the compression zone.
Based on a lÛDited number of tests, Laupa et al. concluded that the critical
section occurred in the region where M/Vd was approximately 4.5. This criti-
cal value of M/Vd was found by plotting along the length of the beam the ratio
of the actual moment at failure to the predicted moment and observing where
this ratio equalled unity.
In 1956 Bernaert and Siess (17) reported a study on the shear
strength of rectangular reinforced concrete beams without web reinforcement
under simulated uniform load using ten point loads. The inclined crack inter-
sected the tension steel at ~ distance of O.llL, on the average, from the
support. Variations in this distance could not be related to the variations
of the parameters under study and the scatter was believed to be completely
random. This section, at O.llL from the support, was taken as the location of
the critical section for diagonal tension cracking and used in the development
of an equation for the nominal shear stress at the diagonal cracking load.
The nominal shear stress was expressed in terms of p, fé and the L/d-ratio.
Other conditions being equal, it was found that the extent to which the beams
carried load after first diagonal tension cracking depended primarily on the
L/d-ratio, with the increase in load being greater for beams with lower L/d-,
ratios. A sharp increase in the steel stress took place at the section where
the diagonal tension crack crossed the reinforcement. The authors believed
this to be caused by the additional stresses developed in the steel at that
section due to dowelling action. By the use of electric resistance strain
gauges it vas found that the concrete below the diagonal tension crack
carried an appreciable amount of compression. However. the strain distribu-
tion below the crack remained essentially unchanged fram the cracking load to
- 12 -
failure. There was evidence for some of the beams that failure in shear
compression occurred after considerable yielding of the steel had taken place
and for some beams the strain hardening portion of the steel was reached.
However, at the diagonal cracking load no strain ab ove the yield point strain
was measured. Distances fram the support to the centres of the zones of
crushing again could not be related to the variables investigated. The aver
age value was 0.33L. A comparison was made between the ultfmate moment at the
section of failure for the beams of this investigation and the shear-moment
expression developed by Laupa (16) for beams under concentrated loads. Taking
the critical section at 0.33L, it was found that the value of the ratio of
failure moments was always greater than unit y and averaged out to 1.35. The
authors claimed that this was an indication that the shear moment for beams
under uniform load is higher than the shear moment for stmilar beams under
isolated concentrated loads. The M/Vd-ratio at 0.33L from the support for the
beams tested varied between 5.90 and 10.30 with the majority above 7.40. The
shear-moment expression developed by Laupa (16) for beams under isolated point
loads was independent of the M/Vd-ratio due to the fact that he only analyzed
beams in a narrow range of M/Vd-ratios, i.e., from 1.17 to 4.80. It is known
(34, 42) that above a certain value of M/Vd, say about 5, the shear-moment is
no longer independent of M/vd but increases with an increase of M/Vd. Hence,
the line of reasoning used by the authors to arrive at the conclusion that
beams under uniform load fail at higher shear-moments than stmilar beams under
iaolated point loada was not correct.
In a paper published in 1956 Ferguson (18) developed a hypothesis
stating that it is possible to describe an "unrestrained" diagonal tension
failure as a series of tension crack developments, eacb of wbicb can be ex
plained in terme of the conventional tbeory of combined stresses. The term
- 13 -
"unrestrained" 1& used to designate a failure generally removed from the
influence of local stresses around loads and reactions. Rather than depending
upon purely empirical relationships, Ferguson envisioned the constructive use,
at~least in research, of the theory of combined stresses in connection with
diagonal tension. The hypothesis visualized the following steps in the deve
lopment of an unrestrained failure: (a) an initial diagonal crack forming near
middepth and stopping within the compression area and in the tension area
somewhere near the longitudinal steel; (b) a somewhat flatter extension of
this crack in the compression zone at increasing loads; (c) a general cracking
in the zone around the steel, which may develop stmultaneously with (b);
(d) a sudden final failure by an extension of the tension crack to the top of
the beam or the shear-compression failure, with a secondary failure in split
ting and bond at the steel level. In some cases the splitting and bond dis
tress can develop into the prtmary failure. Two exploratory series of tests
were also reported. The first investigated the effect of extra or multiple
loads and suggested that higher shear strengths are available near supports.
The vertical compressive stresses under the exterior loads delayed the open
ing of the first diagonal crack. The second series indicated that ~ch of
the increased capacity associated with small a/d-ratios is 10st if the loads
are applied as shears over the depth of the beam or if the reactions are
taken out as shears. Without the vertical compression required to stop the
diagonal tension crack, tbere is notbing to stabilize the beam for a shear
compression failure.
Investigating the cracking shear strengtb of knee frames and stub
beams, Morrov and Viest (19) in 1957 deve10ped a seœitheoretica1 equation for
tbe unit cracking shear strengtb by assuming that a critical diagonal tension
crack formed wben the principal tensile stresses, due to both shear and
- 14 -
flexure, reaches the value of the modulus of rupture of the concrete set
equal to 9.5~. This equation was the first to give relatively accurate re-
sults for aIl normal values of fc, p and M/Vd. Also developed was an express-
ion for the shear-compression moment capacity which was quite similar in deri-
vation to that of Moody et al. (15). The results of this investigation indi-
cated that, contrary to some earlier studies, the shear-moment capacity of
beams without web reinforcement is not independent of the a/d-ratio but in-
creases with decreasing values of this ratio. The authors claimed that within
the limits of their investigation axial compression affected the shear and
diagonal tension strengths only insofar as it changed the conditions of stati-
cal equilibrium.
In 1957 Whitney (2) proposed an ultimate strength theory for shear
which was radically different from those then in use. He suggested that the
shear strength, expressed as nominal shear stress, dependa on the ultimate
flexural capacity of the member and on the ratio of effective depth to shear
span. The author urged the use of the diagonal cracking load as the ultimate
load for beams without shear reinforcement •
• Bresler and Pister (21) in 1958 and Guralnick (22) in 1959 advanced
the concept of shear failure, as a result of failure of the concrete in the
compression zone, a stage further. Assuming that the concrete ln the com-
pression zone failed under the combined action of the compression and shear-
lng forces, equations were derived to predict the strength of plain concrete
subjected simultaneously to compression and shear. These equatlons were
applied to the compression zone of the beam, the size of this zone being
taken equal to the size derived from standard flexural theory. lt vas assumed
that aIl the shear force was resisted by the compression zone.
- 15 -
Noting the problem of determining the size of the compression zone,
Walther (23, see also 24) in 1957 attempted to overcome this by considering
the compatibility requirements of concrete and steel deformations near a dia
gonal crack. He postulated that the defo.rmations in the region of a beam
containing a diagonal crack could be represented by a rotation occurring about
the end of this crack. lt was shawn how such a rotation could be related to
the depth of the neutral axis, the length of the crack, the pull out of the
reinforcement and the shortening of the extreme compression fibre of the beam.
By considering the postulated mode of deformation, the depth of the com
pression zone could be determined. Walther's work was of great significance
in that it high-lighted the compatibility problem that so many previous
workers had ignored.
In 1960 Brock (25) developed the hypothesis that the effect of
shear is simply to reduce the pure flexural capacity. He used this hypothesis
as a basis for formulating a method of predicting the ulttmate load and fail
ure mode under any type of loading. The author proposed that, despite the
appearance of failure, all failures which are not characterized by a reduction
of flexural strength be treated as flexural failures. True shear failures
always cause a reduction in the flexural capacity and are a function of the
reinforcement index, p/po, where Po is the proportion of reinforcement re
quired for balanced failure, and the value of M/Vd at the section of failure.
In Brock's method of analysis the actual moment capacity curve is first drawn
across the span. The actual moment capacity at a section is determined by
substituting the actual M/vd value at éhat particular section onto the curve
of a/d versus ulttmate moment for the appropriate p/po' The ultimate load
and the location of the failure section for a particular beam and loading
condition can be determined by supertmposing that bending moment diagram
- 16 -
which Just touches but does not cross the capacity curve. The objection to
this method is that curves of ulttmate moment, expressed as Mu/(fbbd2),
against a/d for particular values of p/Po have to be determined experimentally
and are not generally available to the designer.
In 1960 Diaz de Cossio and Siess (26) reported tests on rectangular
reinforced concrete stmply supported beams and frame members under several
different types of loading. These included: two-point loading, midspan con
centrated loads, simulated uniform load (multiple load points) and axial load.
The uniformly loaded beams discussed in this paper were the ones previously
reported by Bernaert and Siess (17). For beams under uniform load the in
clined portion of the diagonal crack intersected middepth always in the
neighborhood of 0.15 of the span from the support, and before final collapse
always became horizontal for a certain length in the compression zone. The
type of vertical loading was shawn to have a profound influence on the shear
carrying capacity of a member. Members with almost identical properties
failed under uniform load at loads much larger than under isolated concentra
ted loads. For those beams in which comparisons could be made, the ratio of
the capacity under uniform load to the capacity under symmetrical isolated
concentrated loads was always in the neighborhood of 1.5. For a simply
supported beam failing in flexure, the ratio is 2.0. The authors pointed out
that this again suggests that shear strength is to a certain degree dependent
on moment, or in other words, that there ls an interaction between moment and
shear in a member which affects its shear capacity. The transition point at
which beams under uniform load no longer susta1n load after the formation of
the first fully developed diagonal tension crack appeared to be at a value of
L/d - 12 or 13. For values of the L/d-ratio above this transition point the
beam failed simultaneously with tbe sudden appearance of the first fully
- li -
developed diagonal tension crack. The authors found that the effect of the
"slenderness" of the member whether expressed in terms of the a/d-ratio or
the L/d-ratio was most apparent. AlI other conditions being the same, the
less slender the beam, the larger the diagonal cracking load and the ultimate
shear capacity. Also, the less slender the member, the larger the ratio of
the ultimate shear capacity ta the diagonal cracking load. The presence of
axial load had a relatively small influence on the shear capacity of the mem
bers tested, with this influence being somewhat larger on the ultimate shear
capacity than on the diagonal cracking load. Due to the presence of the com
pressive axial la ad the inclined cracks formed closer ta midspan, were not as
high and developed at a slower rate than in the corresponding beams with no
axial load. An expression for the nominal shear stress at the critical sect
ion in terms of the diagonal cracking load was also presented. For stmply
supported beams under uniform load, the critical section was taken at a dis
tance of O.15L from the support, i.e. the point where the diagonal crack
causing failure crossed middepth.
Krefeld and Thurston (27) in 1962 published a report on an inves
tigation involving the testing of over two hundred stmply supported beams
with and without shear reinforcement under bath isolated point loads and
uniformly distributed loads (eight point loads). It was observed that after
the formation of the diagonal tension crack the magnitude of the shear re
sisted by the tension steel, due ta dowel action, and the concrete below was
appreciable even for beams without web reinforcement. This was also observed
by Mathey and Watstein (28) in 1958. Beams developed longitudinal cracks in
the concrete above the reinforcement due ta these dawel forces. The authors
suggested that, due ta the relative unreliability of the ultimate shear
compression strength, the load at which these longitudinal cracks formed
- 18 -
along the reinforcement due to dowel action be conservatively taken as a mea
sure of the ultimate shear strength for beams without web reinforcement. An
expression was derived to estimate this "critical" nominal shear intensity at
the critical shear section in terms of known beam properties and empirically
determined constants. Since the form of the equation is the same for both
concentrated and distributed loads, locations of the experimentally determined
critical sections are given. The locations of the critical sections were
found to be dependent on the a/d-ratio for beams under concentrated loads and
the L/d-ratio for beams under distributed loads. As the a/d- or L/d-ratio
decreased the proposed expression gave increasingly conservative results due
to the increasing reserve capacity beyond the critical cracking load, i.e. the
load at which longitudinal cracking along the reinforcement occurred. The
effect of stirrups in retarding failure by diagonal tension was visualized as
a means of augmenting the dowel resistance and delaying horizontal cracking
along the bars. After these cracks have been formed the stirrups provide
additional resistance against separation of the beam segments. The authors
suggested that the ultimate shear strength of beams with web reinforcement be
taken as the diagonal cracking shear strength of an identical beam without
web reinforcement plus the shear capacity of the web reinforcement.
In 1962 the ACI-ASCE Committee 326, Shear and Diagonal Tension (8),
using the average or nominal shear stress and the criterion that the critical
diagonal tension cracking represents the usable ultimate strength of beams
w1thout web reinforcement, reported on a systematic study of available data
which indicated that tbe shear capacity depends primarily on three variables,
viz., p, M/Vd and f~. Other variables have minor effects on shearing strength.
The equation proposed for the sbear strength of beams without web reinforce
ment represents only a sl1ght modification of the equation suggested by
- 19 -
Krefeld and Thurston (27). The difference lies in the values of the empirical
ly determined numerical coefficients. This difference can be accounted for in
part by the difference between using conset'vative values and mean or average
values for shear resistance and in part by using different expressions for the
locations of the critical shear sections. Committee 326 used conservative
values for the shear resistance to insure that beam design would be governed
by flexure rather than by shear. For beams with web reinforcement, the Commit
tee recommended the same procedure as Krefeld and Thurston (27). However,
slightly different expressions, based on the truss analogy, were proposed for
the shear capacity of the web reinforcement. It shou1d be noted that an upper
limit of the stirrup yie1d point stress of 60,000 psi was set due to the fact
that stirrups with very high yie1d limits May not be capable of deve10ping
these high stresses before fai1ure of the beam. The 1963 ACI Building Code
(5) provisions for u1timate shear strength are identica1 to those proposed by
ACI-ASCE Commit tee 326.
For intermediate values of the a/d-ratio, say greater than 3.0, in
clined cracks deve10p in the shear span as an extension of a f1exural crack
which progressively bends over unti1 the inclined crack is formed. Bath Moe
(29) in 1962 and Kàni (30) in 1964, amongst others, have idealized this
mechanism as the breaking off of a concrete "tooth" between two flexural
cracks. The concrete teeth behaving as cantilevers fixed at the compression
zone are acted on by the bond forces transmitted by the reinforcement. A
tooth was assumed to fail when the tensile stress at the root of the tooth
equa11ed the tensile strength of the concrete. Moe considered the 1ongest,
and hence weakest, tooth and assumed a certain amount of shear transfer across
the crack by aggregate interlock and dowel action. On the other hand, Kani
considered the average tooth rather than the longest and weakest one and
- 20 -
. .
neglected any shear transfer across the crack either by aggregate interlock or
dowel action.
In 1962 Leonhardt and Walther (2) published a report of a very ex-
haustive study of the shear strength of reinforced concrete beams of rectangu-
lar and T---cross-section under both isolated concentrated loads and uniformly
distributed load (using two water-filled hoses). Onlya few of the conclu-
sions arrived at will be mentioned here. In aIl of the tests, uniformly dis-
tributed loading was associated with a 20 to 40 percent higher shear strength,
if the largest shear force is adopted as the criterion. This was claimed to
be due to the effect of the bending moment which for concentrated loads is
larger at the shear failure section than for uniformly distributed loading and
to the pressures exerted by the loading, which strengthens the compression
zone. For beams under uniform load failing in shear the destruction of the
compression zone was located at a distance of between 2d to 3.5d from the
support. A tentative expression was proposed for the determination of the
M/Vd-value at the section of crushing in the compression zone. According to
this expression the critical M/Vd-value is solely dependent on the L/d-ratio.
The nominal shear stress at failure was found to decrease with an increase
in the L/d-ratio. However, the decrease in the shear strength was found to
be not very significant with an increase in the L/d ratio after a value of
approximate1y 12.
Sheikh (31) in reviewing the above work pointed out that the values
of M/Vd at the failure section or at the section of crushing are haphazard
and do not seem to bear any relation to L/d before the value of L/d = 12, and
after this value of L/d • 12 the value of M/Vd in these tests decreases gra-
dual1y vith an increase in the L/d-ratio and hence a decrease in the shear
carrying capacity of the beam. This is, now, in complete contrast to the
- 21 -
effect of the M/Vd-ratio on s~p1y supported beams under is01ated concentrated
10ads, where the shear strength increases with a decrease in the M/Vd-ratio.
Remarking that nothing conclusive cou1d be said on the basis of these few
tests, the author fe1t that for estimating the shear-carrying capacity of a
uniform1y 10aded beam, the tendency of assuming a critica1 section, sayat a
distance d fram the support, for taking into account the M/Vd effect cou1d be
somettmes dangerous.
Previous to about 1955 (32) numerous prestressed concrete beams had
been tested to determine their strength in flexure, but very few in shear.
However, between 1955 and the present, 1970, 1itera11y thousands of spec~ns
have actua11y been tested to de termine their strength in resisting shear pre
daminate1y or moment and shear, with or without web reinforcement. Unfortu
nate1y, due to the comp1exity of the prob1em and our inabi1ity to iso1ate the
variables in our exper~nts and analyses, it cannot be c1a~d that the pro
b1em has been solved, even with the wea1th of information now avai1ab1e.
At working 10ad the resistance of prestressed concrete to shear is
much more than that or ordinary reinforced concrete since the prestress
greatly reduces the principal tensi1e stresses and will usua11y prevent the
occurrence of shrinkage cracks which could conceivab1y destroy the shear re
sistance of reinforced concrete beams, especial1y near the point of contra
flexure. However, the resistance against shear is quite sûni1ar at ultimate
load for both reinforced and prestressed concrete, and the conditions may
even be worse in prestresaed concrete because the tendons may have a sma1ler
resistance to shear than the heavy reinforcing bars used in ordinary rein
forced concrete. Moreover, the cross-section of the beam is usual1y smaller
than that of a corresponding reinforced concrete beam.(33)
- 22 -
In 1958 Hicks (34) classified the type of shear failure to be ex
pected in terms of the a/d-ratio. However, it has been shawn that the failure
mode of beams different from his is not necessarily correctly predicted by his
data. In addition, the effect of the concrete strength was found to be inde
pendent of the a/d-ratio.
Evans and Hosny (35) in 1958 reported on a study of 93 shear fail
ures of prestressed concrete beams. The shear failures were classified under
three categories: shear-compression, diagonal crushing of the web and shearing
of the compression zone. The shear-compression mode of failure, the most
common failure for rectangular beams, was found to be s~ilar to that of ordi
nary reinforced concrete. In the second mode, diagonal crushing of the web
(for I-beams), the final rupture takes place after the destruction of bond in
the neighbourhood of the cracks and the formation of tension cracks on the
top fibre of the beam that lead to the crushing of the web at the toe of the
top tension cracks. The third mode of failure, shearing of the compression
zone, is usually expected for I-beams with web reinforcement. In this type of
failure the inclined cracks extena up to near the top fibre of the beam and
afterwards the remaining uncracked section shears suddenly. The different
anchorage systems or degree of bond did not seem to affect the mode of failure
as long as excessive loss of bond that would reduce the effect of prestressing
was prevented. The authors assumed that the penetration of the diagonal
cracks into the zone of pure flexure, the loss of bond in the neighbourhood of
the web or longitudinal reinforcement or the yielding ûf the web reinforcement
make the transfer of the shearing force only possible through the concrete
above the diagonal cracks at loads very near to failure. Based on this
assumption the authors suggested tbat one expression could be obtained to pre
dict the shear strength, regardless of the mode of failure. The ulttmate
- 23 -
shear strength vas discussed from tvo points of viev: the ultimate nominal
shear stress and the ultimate shear moment. Expressions for the nominal shear
stress at failure for both rectangular and I-sections vere given indicating
how it increaseJ vith an increase in ~ and the product pfse and a decrease
in the a/d-ratio. Since shear failure is considered primarily a failure in
diagonal tension, it vas considered reasonable to relate the calculated Vu to
the tensile strength of the concrete or to ~. It is believed that this is
the reason vhy aIl the early empirical expressions trying to relate Vu to fe
failed to give good results over a vide range of fe. The percent age increase
in the shear strength of rectangular sections vith the increase in pfse vas
found to be much higher than that of the I-sections. Because the Many beams
tested vith only one variable, the position of the loading points, failed at a
nearly constant moment, it vas suggested that the ultimate shear strength be
expressed as a ILmiting moment rather than a nominal shear stress. Due to the
non-linearity of strains an empirical coefficient vould have had to be intro-
duced to compute the strain in the concrete at the level of the reinforcement.
The authors claimed that this coefficient vould be dependent on the shape of
the cross-section, the prestressing force, the a/d-ratio and the bond charac-
teristics. More tests vere said to be required to evaluate aIl of these va-
riables. The expression for this coefficient proposed by Zvoyer and Siess
(14) vas not considered to be generally valid. Nonetheless, the ultimate
2 shear moment, Ms/bd, vas expressed directly in terms of f~ and pf se regard-
less of the strain relations. The increase of the ultimate shear strength of
be~ vith web reinforcement was found to be affected by both the properties
of the web reinforcement and the shear strength of the beam itself. An addi-
tive type of equation, similar to that recommended by Laupa (16) for ordinary
reinforced concrete beams, was found to agree reasonably weIl with test results.
- 24 -
An extensive amount of work has been carried out on the shear
strength of prestressed concrete members at the University of Illinois (36 -
40). This work has formed the basis for the shear clauses of the present ACI
code (318-63). For the purposes of establishing a design criterion, the dia
gonal cracking load has been considered to be the practical ulttmate load for
beams without web reinforcement. Diagonal tension cracks in prestressed con
crete beams have been categorized into two types according to the condition of
the beam before its formation. If the portion of the beam in which the diago
nal crack forms is uncracked, shear stresses dominate the principal tensile
stress and this type of crack has been called a "web-shear" crack. If the
diagonal crack is initiated by a crack in the extreme fiber related directly
to bending stresses, the interaction of bending and shear stresses is critical.
This type of crack has been called a "flexure-shear" crack. Because a web
shear crack is initiated before flexural cracks can develop in its vicinity,
the principal stresses computed in the web of an uncracked section will appro
ximate with reasonable accuracy the state of stress at initial cracking. For
prestressed I-beams with short shear spans and thin webs MacGregor (38) has
indicated that the max~ principal tensile stress along the potential diago
nal crack occurs at or close to the centroid. Since the combined stress ex
pression is cumbersome for the designer, the ACI code equation for the web
shear cracking load, Vcw ' is a simplification of that expression. Derived by
Mattock (41), the code equation is based on an assumed concrete tensile
strength of 3.5~ and the prestress at the centroid. The equation for Vcw
is thus a rational ratber than an empirical one. Attempts to predict analyti
cally the flexure-shear cracking load have not been fully successful so far.
This is due to the fact that the stress redistribution which follows the for
mation of initial flexural cracks greatly influences their shape and propaga-
- 25 -
tion and analyses which neglect factors influencing this redistribution cannot
adequately predict the behaviour which fol1ows it (1). Recourse to semiempiri
cal methods was thus necessitated. MacGregor (38) was perhaps the first to ex
press the flexure-shear cracking load as the sum of the load which causes a
flexural crack at a point in the shear span and an additional increment of
shear required to cause the diagonal crack after the flexural crack had deve
loped. This concept of the f1exure-shear cracking load was stmplified and re
fined for design purposes by Sozen and Hawkins (40). The shear corresponding
to the load causing the critical initial flexural crack was taken at a dis
tance of d/2 fram the load, or, more genera1ly, from the section unaer consi
deration. The increment of shear between the development of the initiating
flexural crack and the development of the diagonal crack was conservatively
taken as 0.6b'd~. The influence of the amount of longitudinal reinforce
ment on the propagation of the diagonal crack was neglected. The expression
proposed by Sozen and Hawkins is identical to Vci of the ACI code. The ex
pression for Vci is thus partly rational and part1y empirical. The shear, Vc '
carried by the concrete at diagonal cracking is taken as the lesser of Vcw and
Vci. Both web-shear and flexure-shear cracks occurred in the tests of beams
with draped reinforcement, although the properties of the beams tested were
such that most of them developed flexure-shear cracks (38). Draping the longi
tudinal reinforcement appeared to increase the diagonal cracking load in the
beams which developed web-shear cracks, and appeared to de crea se the diagonal
cracking load for the beams which developed flexure-shear cracks. In computing
shear stresses and principal stresses, only the net shear is considered.
Since the web-shear cracking load is a function of principal tensile stresses,
draping the longitudinal reinforcement should increase the web-shear cracking
load. With draped reinforcement. the bending moment for the initiating flexu-
- 26 -
raI crack is smaller than in a beam with straight tendons. Due to the close
relationship between flexural and diagonal cracking, the flexure-shear cracking
load is decreased correspondingly. MacGregor (38) found that the difference
between the ultimate shear and the shear corresponding to diagonal cracking was
by far the most significant variable governing the amount of web reinforcement
necessary to prevent a shear failure. He offered the following empirical ex
pression as the contribution of the stirrups: l.~fyd/s. The empirical co
efficient 1.1 presumably includes a term to relate the horizontal projection of
the diagonal crack to the effective depth and also allows for the variations in
the actual stress in the stirrups crossing the crack since some of the stirrups
near the apex of the diagonal crack May not be stressed to the yield point.
The ACI code dropped the coefficient 1.1 from the above expression. Although
the method of designing shear reinforcement is quite stmilar for both rein
forced and prestressed concrete, the code procedures for supplying the minimum
amount of web reinforcement differ. Whereas shear reinforcement can be omitted
in ordinary reinforced concrete, the ACI code requires a specified minimum
shear reinforcement in aIl prestressed beams unless it is shawn by tests that
the required ultimate flexural and shear capacity can be developed when web
reinforcement is omitted.
In 1960 Sethunarayanan (42) reported a study of the shear strength of
pre-tensioned I-beams without web reinforcement. The beams had rather thin
webs and failed mainly by distress in the web, either by crushing due to diago
nal compression or by splitting due to diagonal tension. The diagonal crushing
mode of failure was typical for the thin-webbed I-beams with a/d-ratios of
between 1.5 to 4.0, while the diagonal tension or splitting mode of failure
took place when a/d exceeded about 4. The author developed an expression for
the resistance of the beam against web crushing that was based on a truss
- 27 -
action in the shear span. An expression was also offered for the diagonal ten
sion resistance. It was found that the diagonal compression mode of failure
occurred at an approximately constant bending moment whereas the diagonal ten
sion failure occurred at an increasing bending moment when the a/d-ratio was
increased. The actual point of transition from diagonal compresssion to diago
nal ~ension failure appeared to depend mainly on the relation between the com
pressive and tensile strengths of the concrete and on the amount of prestress.
The transition from diagonal tension to flexural failure tended' to be governed
mainly by the amount of prestress.
In 1964 Wilby and Nazir (43) studied shear failure in uniformly
loaded grouted post-tensioned I-beams and tested on a short span (L/d = 5.4).
The .beams had rather thin webs and were without any shear reinforcement. They
failed mainly by the destruction of the web either by splitting along the line
of the diagonal tension crack or by the crushing of the concrete due to arch-
action. The authors found that the ultimate shear strength of a beam increa
ses appreciably with an increase in the concrete compressive strength. stmi
lar beams were tested under uniform load (eight point loads) and under two
point and mid-point loadings. The authors claimed that the beams tested under
uniform load failed at higher ultimate shear moments than the beams tested
under iso1ated point 10ads. The ratio of the u1timate shear moments for the
beams under the ~o systems of loading was said to be 1.25. From the test
data supp1ied by the authors it is not clear how they arrived at the last ~o
conclusions mentioned above. The authors found that the diagonal crack
crossed middepth at a distance of O.22L from the support, on the average.
Diaz de COlsio and Siels (26), testing ordinary reinforced concrete beams un
der uniform load, found that the diagonal crack crossed middepth a1ways in the
neighborhood of O.ISL from tbe support. This would seem to imply that the in-
- 28 -
troduction of prestressing forces the formation of the diagonal crack closer
to midspan.
Arthur (44) in 1965, studying pre-tensioned I-beams without web re
inforcement, showed that it was not possible, witbin the range of his tests
(a/d = 1.12 to 4.57), to predict the type of failtJre by reference only to the
a/d-ratio. Altbougb aIl failures involving flexure took place in the higher
part of the range of a/d-values, and aIl the simple diagonal crack failures in
the lower part, the two categories overlapped in the middle of the range, and
non-bending failures of the web-distortion type (i.e. when web tension formed
a series of multiple cracks in the sbear span), took place at every value of
a/d studied. The value of a/d at which the change fram one type of failure to
the other occurs was said to depend on the properties of the beam.
Kar (45, 46) reported a study of the shear strength of grouted post
tensioned beams under both isolated point loads and uniform load. The uniform
load was stmulated by means of ten point loads. The location of the critical
section for diagonal cracking was found to vary between 0.29L and 0.39L fram
the support while the location of the critical section for ultimate failure
was found to vary between 0.37L and 0.43L fram the support. The critical sec
tion for diagonal cracking was defined as the section at which the diagonal
crack has progressed in an inclined direction over a distance of d/2 measured
along the cable. The critical section for ultimate failure was defined as
the section of crushing in the compression zone. The cause of the variation
in the location of the critical section could not be traced to any possible
influence of the variables investigated. These variables included concrete
strength, L/d-ratio, pre stress level, size of beam and amount of steel. Semi
empirical expressions were proposed to esttœate the diagonal cracking and ul
timate loads. The same expressions were used for point and uniformly loaded
beams.
- 29 -
Due to the rather large amount of research that has been carried out
on the shear strength of concrete beams, only a few of the major contributions
have been discussed in this review. A number of items have not bean dealt
with here. An abundance of data on simply supported beams under one or two
symmetrically placed point loads is available in the literature. The fact
that isoleted point loading, although convenient for any theoretical analysis
due to its constant maximum shear force and bending moment, cannot be consi
dered a practical loading arrangement, does not seem to affect most investiga
tors.
Very little work has been done on the shear strength of beams under
uniform load, a practical loading arrangement. The main conclusions derived
from investigating the effect of uniform load were mentioned previously and
will be briefly summarized here. The consequence of varying the L/d-ratio for
beams under uniform load is s1milar to varying the a/d-ratio for beams under
isolated point loads. AlI other conditions being the same, the smaller the
L/d-ratio the larger the diagonal cracking and ultimate shear capacity. AIso,
the smaller the L/d-ratio, the larger the ratio of the ultimate shear capacity
to the diagonal cracking load. The location of the critical section of fail
ure, although defined differently by different investigators, was generally
found to vary randomly, say witbin ten percent of the beam span, and could not
be conclusively correlated to pertinent variables, such as concrete strength,
L/d-ratio, prestress level, size of beam and amount of steel. Krefeld and
Thurston (27), however, have stated that the location of the critical section
ls dependent on the L/d-ratio. The type of vertical loading has been shawn to
have a profound influence on the shear-carrying capacity of a member. Uni
formly distrlbuted loading was associated with a 20 to 40 percent higher shear
strength than s~ilar beama under lsolated point loads. Seme investigators
- 30 -
(17, 43) have stated that the shear moment for beams under uniform load is
higher than the shear moment for stmilar beams under isolated point loads.
The line of reasoning used to arrive at this conclusion was shawn above to be
incorrect.
Most of the work carried out on beams under uniform load has been
done on reinforced concrete beams. Little has been performed to determine the
influence of uniform load on the shear strength of prestressed concrete beams,
the type under study here.
- 31 -
3. SPECIMENS AND TESTING METHOD
3.1 Nomenclature for Spectmens
The beams were designated according to the 1eve1 of prestress.
Three levels of prestress were used in this study and these have been distin-
guished by using numeral 1, 2 or 3 after the 1etter P in the designation. To
distinguish the concrete strength for any particular specimen, the let ter C
and an appropriate numeral were added as in P2-Cl, to signify the intermediate
level of prestress and the l~est concrete strength. If there is no let ter C
in the designation, then C2 for the intermediate concrete strength is ta be
understood. The letter A was added as in P2-ClA ta differentiate the dupli-
cate fram the original beam.
The tests were grouped into two series, based on the number of varia-
bles studies, as presented in the following table:
Test Series
l
II
Variable
level of prestress
concrete strength
3.2 Opscription of Specimens
Beams
Pl, PlA, P2, PlA, P3, P3A, P3-C3
P2-Cl, P2-ClA, P2, PlA, P2-C3, P3-C3
Each test beam had an overall length of 10 ft. 6 in. and possessed a
doubly symmetrical I-shaped cross-section with the following nominal dtmen-
sions: overall depth of 12 L~., flange width of 6 in., flange thickness of 2
in., web thickness of 3 in. and an effective depth of 9.25 in. Rectangular
end-blocks 15 in. long were provided at the ends of each beam. A step was
formed at the top of the beam over each support ta terminate the loaded area
- 32 -
at the extremeties of the 9 ft. simple span. The nominal dimensions of the
beam are shawn in Fig. 1.
Since the beams were cast in steel forms there were negligible, if
any, variations between the different specimens. However, the web thickness
was actually 2.875 ius., with the other actual dimensions being stmilar to the
nominal ones.
AlI of the specimens were pre-tensioned with four 3/8 in. diameter
high tensile strength strands for a total steel area of 0.32 sq. in. The
strands were straight throughout the length of the test beam, i.e. they were
not draped. A stress-strain curve for the strand is shawn in Fig. 2. The
beams were without any transverse reinforcement throughout their length inclu
ding the end-blocks.
The average tensile and compressive concrete strengths
and the initial and effective prestress levels are listed in Table 1. Details
of the materials, fabrication and instrumentation of the test specimens can be
found in Appendix A. The prestress was released on the fourth day after cast
ing and the beams were usually tested on the eighth day after casting.
3.3 Test Bed and Loading Apparatus
A loading frame in existence prior to the initiation of this study
was slightly modified in order to be suitable. The frame in its original con
dition was essentially composed of two 14 ft. long steel beams of size l8WFS5
with transverse stiffeners every 4 ft. 6 in. resting on two transverse steel
beams of size l2WF27. Only one of the longitudinal steel beams was used. Due
to the position of an overhead crane used to hoist the test beams into place
and other permanently fixed items, the centres of the test set-up and loading
frame did not coincide; the test set-up was 10cated on the left-hand part of
the 10ading frame. For the first few tests the transverse stiffeners and
- 33 -
....
C'\
... ..- " \D
N
"' . .;t
c:: . .... c:: .... C""l . ,.... 0 \D 11"\ ,.... ft Il
< ~
~
l' j:Q
"" II~/l 6 = P 0 tI)
z 0 ~ tI)
0 Z 1 -C'\ M • • ~
Cl \D Il
:c- • • ,...l
~ t-4 X 0 z
l-9 IIZ .-1
II~/( (.!)
IIZ1 t-4
""
I 1
1
=il - ! - 1 0\ 1
1 ! ,
- 34 -
280r------,------------------------------------r-----~
1
240 I-----.+-----+_- ----1
200 ----------- -- ---.--------
~ 160 ~----+--------___,I_----------------------0 --CI)
~ -fi) fi)
~ 120 1--______ +#--__ ---4-__________________________ --0--- -------
Area = 0.08 sq. in.
80~----~~------------ f = 280 ksi su
E 6
= 29.2 x 10 psi
40 ~-___.f__---------~-----------------
0~----------------------------------------------------------4 0.2 0.4 0.6 0.8 1.0
srRAIN (percent)
FIG. 2 SIRESS-SIRAIN REIÂTIONSHIP FOR 3/8 INo SIRAND
1.2 1.4
- 35 -
TABLE 1
PROPERTIES OF TEST BEAHS
fI f sp f' f si f Mark c r se
psi psi psi ksi ksi
Pl 6650 600 630 90.0 72.0
PlA 6750 690 560 90.0 71.3
P2 6100 580 520 120.0 99.0
P2A 6600 515 590 120.0 97.3
P3 6200 650 600 150.0 127.0
P3A 6160 680 620 150.0 123.0
P2-C1 4640 520 450 120.0 98.5
P2-ClA 4890 590 500 120.0 98.5
P2-C3 7400 600 650 120.0 99.0
P3-C3 7600 600 670 150.0 127.3
- 36 -
transverse beams were left in their original positions. However, it was dis
covered that this lack of symmetry of the stiffeners and transverse beams with
respect to the test set-up affected the cracking pattern of the test beam.
This point will be discussed at a later time. For this reason, the transverse
stiffeners were removed and the right-hand transverse beam moved inwards in
order to locate the ~o transverse beams in positions symmetrical about the
centre of the test set-up.
The applied load was uniformly distributed along the top surface of
the test beam by means of two water-filled hoses. Load from two hydraulic
rams was distributed to the hoses through ~o layers of steel beams. A photo
graph of the loading arrangement Just prior to testing is shawn in Fig. 3.
Complete details of the loading apparatus can be found in Appendix B.
3.4 Measurements and Test Procedure
The load applied was measured by means of two aluminum dynamometers
placed between each ram and the bottom of the longitudinal steel beam of the
loading frame. These dynamometers are described in detail in Appendix A. The
dynamometer calibration was 200 micro in./in./lOOO lb. The dawnward deflect
ion of the concrete beam and upward deflection of the longitudinal steel beam
of the loading frame were measured at midspan with O.OOI-in. dial indicators.
Hence, the net deflection of the test beam was determined. Any possible slip
page of the strands relative to the concrete was measured by the placement of
a O.OOI-in. dial gauge as shawn in Fig. 4. Strains in the longitudinal rein
forcement at midspan and along tbe centreline of the top surface of the test
beam were measured by electrical resistance strain gauges. Water pressure in
the hoses was measured by means of a Marsh water gauge.
- 37 -
FIG. 3 LOADING ARRANGEMENT PRIOR TO TESTING
FIG. 4 SLIP-MEASURING ARRA.'iGEMENr
37
- 38 -
After the beam was placed in the test bed, the loading apparatus was
assembled as shawn in Fig. 3 and everything properly aligned and balanced.
Great care was taken to insure that the loading heads and base plates were
level at the beginning of the test since any amount of unlevelness would in
crea se as the test advanced. This would cause one of the hoses to be com
pressed to a higher degree than the other, in addition to other complications.
A small amount of load was applied and then removed before the zero readings
vere recorded.
Load was applied in increments of two kips up to flexural cracking
and then in increments of one kip until failure was achieved. Load was
applied rather slowly as the flexural, diagonal cracking and ulttmate loads
were anticipated. After each increment of load, the valve between the pump
and the rams was closed. Deflection readings, slippage readings, water pres
sure readings and strain measurements were taken and the cracks were marked
with the total load, expressed in kips, indicated at the point of furthermost
propagation of the crack. There was usually, and especially at high loads,
some drop-off in the load and some increase in deflection while cracks were
being marked and readings taken. These changes were noted prior to the appli
cation of the next load increment. This drop off was due to creep in the test
specimen since no leaks were detected in the hydraulic system and the shut-off
valve vas capable of vithstanding 10,000 psi without leaking. No attempt was
made to maintain the load cOnstant while readings were being taken. Loading
vas continued until failure of the beam. Each test required approxtmately
three hours and vas concluded vith the testing of the control specimens.
- 39 -
4. EXPERIMENrAL RESUUS AND OBSERVATIONS
4.1 General
In this chapter the experimental results obtained and the behaviour
observed will be reported and discussed. Listed in Table 2 are the flexural,
diagonal cracking and ultimate loads of the specimens tested. It should be
mentioned that beam P3-C3 although not originally planned was accidentally ob
tained when the wrong concrete mix was used. It nevertheless provided useful
data and will be included here. As mentioned in the previous chapter a 0.001-
inch dial gauge was attached to a strand protruding from the end-block, see
Fig. 4. This gauge was employed to determine any relative slip between the
steel and concrete. No slippage was observed tmplying th~t bond was satis
factory. The reproducibility of specimens was excellent.
4.2 Load-Deflection Relationships
The best method of studying beam behaviour is by means of load
deflection relationships taking into account the other phenomena observed du
ring testing. The load-deflection diagrams obtained from tests of beams fail
ing in shear differ from those of beams failing in flexure since for the for
mer, part of the ductility, if any, is contributed by the opening up of diago
nal cracks. Nevertheless, the curves are of value in evaluating and comparing
"load-worthiness".
The load-deflection curves have been grouped according to the two
main variables in this study, Le. level of prestress and concrete compressive
strength. They have aIl been plotted to the same scale for purposes of direct
comparison and can be found in Figs. 5 and 6. The points corresponding to the
observed diagonal cracking loads are indicated on each curve. In aIl curves
the deflections measured at midspan were plotted versus the total live load,
- 40 -
TABLE 2
MEASURED TOTAL LOADS (KIPS)
Mark Flexural Diagonal Ultimate Cracking Cracking
pl 19.0 33.3 41.0
PlA 18.5 30.0 40.0
P2 28.0 42.0 44.2
P2A 24.3 38.0 44.0
P3 28.0 42.8 48.0
P3A 32.0 43.0 49.2
P2-Cl 21.9 34.4 38.8
P2-ClA 23.0 35.6 39.8
P2-C3 26.0 42.0 46.4
P3-C3 32.0 45.9 53.5
t-------------, ~ .
i
- 41 -
··0 ___ . __ .l. ___ .
i 1
! : ~---~----~
1 :
+--------;-.-g..I5------lf---------.---------+----+-1 ~~-+_:_i-- ·_~U ~ 1 ! rd 1 i .... -t- ··t .-~--- 0 -- •• ----
-+-~'-----+--------.- ---"---"'---'- ~
1 i CI) +---i-----4-----+- -------- ----------.--.----.. -- .. - J -' - - .. -· .... -M-
f ~ t.)
I-----+-----+------l~-----------.--.-------+_- ----S;;rr.-
o -:t
(sdl~) ŒY01 ïVlDl
M -o . • c:: t-4 or4 -
Z S tj ~
co Cl · 0 Cl
Z < g.. CI) Cl .... X
\D · 0
-:t · o
o
...... II)
0-.... :oc: '-'
§ 0-1 g i-4
48
40
32
24·· -
--,...--i 1
P2-C3
t -1
i
1 1 - ! 1
i 1
--1- ------T---! 1
1
--+- -j 1
161 .. -. 1
1
1 1
1
1
r---T. l~;u'-1--- l r-+- i --; 1
8.- ___ )- ----i-- 1 _______ -+ _____ ~--_J~IG~_L 4-DEFLfCTIONj CURVEh - SErIES Ir
1
1 1 --"----- - -
l' 1 -f
1 1 1 1
! t i
1 1 1 -- .------.-- ---- -- ---t----- -~--~-+--i----I H' ~ -- - - +
0.2 0.4 0.6 0.8 1.0 -1. 2 1.4 MIDSPAN DEFLECTION (in.)
~ N
- 43 -
i.e. the weight of the test beam and loading apparatus were not included.
Two well-defined regions can be observed fram the load-deflection
curves corresponding to stages before and after flexural cracking. The first
stage of the curve is the elastic range of the beam in which the deflections
are almost directly proportional to the load except in the later parts of this
stage. The duration of this stage is a function of the tensile strength of
the concrete, the magnitude and eccentricity of the prestressing force, the
moment-shear relationship and the type and the location of the load, while the
slope of the curve is dependent upon the loading history, the modulus of elas
ticity of the concrete, the amount of longitudinal reinforcement, the moment
shear relationship and the shape of the concrete cross-section (38). This
stage is terminated by the formation of a flexural crack at the section of
maximum moment.
As expected, an increase in either concrete strength or level of pre
stress increased the beam resistance against flexural cracking. The load
deflection diagram deviates fram the linear portion prior to the flexural
cracking load reported in Table 2. This deviation could he due to the fact
that the tensile stress-strain curve for concrete is not linear up to failure
(38). In addition, observation of the flexural cracking load is a visual one
and hence any micro-cracking would not he noticed. This would delay the ob
servation of any flexural cracks.
With a knowledge of the properties of the steel and concrete, the
flexural cracking load and the amount of deflection up to this load can he cal
culated on the hasis of an uncracked section analysis. However, because of
the inherent variations in concrete properties, such computations cannot be
very accurate. Sozen (36) has shawn that the measured and computed flexural
cracking loads agree reasonably weIl for beams with high prestress levels.
- 44 -
The agreement becomes poorer with an increase in the relative contribution of
the modulus of rupture to the cracking load.
The second stage of the load-deflection curve is characterized by a
constantly changing rate of increase of deflection with respect to load. This
phenomenon is due to such factors as the formation of new flexural cracks
which decrease the stiffness of the beam, a continuing rise in the position of
the neutral axis as the cracks extend higher into the beam and inelastic ac
tion in the concrete above the cracks (38). An important development in this
stage was the formation of diagonal cracking. This was accompanied by a
slight drop in load, similar to the drops in load which took place between
loading increments. Since these drops were not critical they were omitted
from the load-deflection curves. No other significant changes in the curves
occurred due to diagonal tension cracking.
In this study the diagonal tension cracking load was defined as that
load at which a diagonal crack traverses the web of the beam and reaches the
upper flange-web junction. Although frequently reported by investigators, the
diagonal cracking load by its very nature is not a precise quantity. lt lacks
a generally accepted definition and is dependent on visual observation for its
determination.
With an increase in concrete strength or prestress level, the diago
nal cracking load increased. Prestressing contributes to the principal ten
sile stress in such a manner as to counteract the contribution fram the shear
force. The ulttmate loads also increased with an increase in concrete
strength or prestress level and for the range of effective prestres8 investi
gated here, 72 ksi to 127 ksi, the load carried beyond diagonal cracking
appeared to be proportionately amaller as the prestress level increased.
Thele reductions vere in no vay substantial and hence no definite conclusions
- 45 -
can be drawn here. However, Sozen (36), in testing beams under one and two
point loads over a larger effective prestress range, observed these reductions.
The ductility appears to increase as the concrete strength decreases,
cf. P2-C3 with P2-ClA, and as the level of prestress decreases, cf. P3 with Pl.
This is due to the fact that as the concrete strength or level of prestress
decreases, the diagonal tension cracks occur earlier and the "life" of the
beam is relatively longer after diagonal cracking. The normal deflection due
to bending is augmented by that due to the opening of diagonal tension cracks.
In comparing the load-deflection relationships obtained in this
study for beams under authentically uniformly distributed load with those re
ported in the literature for beams under isolated point loads, no essential
differences appear to exist.
4.3 Crack Patterns
The prtmary macrocracks observed in concrete beams, prestressed or
ordinarily reinforced, may be arbitrarily divided into three categories
according to the dominant influences on their formation: flexural cracks, web-
shear cracks and flexure-shear cracks. Usually the first cracks to form are
flexural cracks, resulting from tensile stresses caused by bending at the
• section of max~ moment. Web-shear cracks are diagonal or inclined cracks
that form in the web of a beam prior to the appearance of flexural cracks in
their vicinity. A flexure-shear crack is a diagonal or inclined crack that
forma as an extension of a p~eviously developed flexural crack or forma over
or beside such a flexural crack. In the formation of flexure-shear cracks,
the previously developed flexural cracks act as stress-raisers.
In addition to the prtmary cracks mentioned above, secondary cracks
often result due to slip between steel reinforcement and concrete, aggregate
interlocking and dowel action forces in the longitudinal bars transferring
- 46 -
shear across a crack.
Initially, a flexural crack extends into the beam perpendicular to
the longitudinal steel and continues to extend until the tension in the steel
and the internaI lever arm are sufficient to restore moment equilibrium. The
height to which the crack must extend for a given moment is prtmarily a func-
tion of the amount of longitudinal reinforcement, the prestressing force and
the effective depth. A comparison of beams Pl and P3, Figs. 7 and Il, res-
pectively, will clearly demonstrate the influence of the magnitude of the pre-
stressing force on the height of propagation of the flexural crack. In the
former, with an effective prestress of 72 ksi, the flexural crack extended to
a height of approximately 6 in., while in the latter, with an effective pre-
stress of 127 ksi, the flexural crack extended only to about 2 in. above the
beam soffit. The first flexural crack usually did not occur at midspan but
at a slight distance fram midspan because of a local weakness in the concrete.
However, due to the fact that the bending moment diagram is relatively fIat in
this region, the deviation in the flexural cracking moment fram that at mid-
span is negligible.
As the load increased the flexural cracks progressed upward and re-
mained essentially vertical in the region near midspan. In addition, flexural
cracks located closer to the support formed as the loading progressed. The
vertical cracks near midspan developed rapidly at first but at a later stage
in tbeir development tbeir propagation almost stopped.
Web-sbear cracks, as defined above, were not observed in any of the
specimens in this study. Tbese cracks usually form in beams witb high pre-
stress, thin webs and short spans where the principal tensile stresses in the
• web may exceed the tensile strength of the concrete before flexural cracks
occur in tbeir vicinity (38).
- 47 -
.... :5 Il. Il.
~ ~ = = ~ ~ ~ ~
5 5 Il. Il.
lII:' lII: c.J c.J
~ ~ c.J c.J
,... co
(.!) (.!) .... .... Ca. Ca.
- 47 -
< ..... :... :...
l: l: ~ ~ -::::::
Z ~
Z ..--~
t.:..: E-
t.:..:
E-~
~} E-
~ E-< :...
~ ~ , ' ..... <
u ~ ~ \.; .....
r-- X
,. ~ -'
~ :.:.. "
'"
\
"
- 48 -
N ~ Q.. Q..
~ ~ = = ~ i5 &'la M
~ S Q.. . Q..
~ ~
~ tJ
~ tJ tJ
0\ 0 .... (!) . .... (!) ra. ....
ra.
- 48 -
< N ('J
::... ::...
:L: :L:
~ ~ -Z Z ::.:: ::.:: c.:: :.:.: E- E-E- E-< :;:
~ ~
~ :...; < ::.::
:...; :...;
0"- 0
~ ...... -.. ...... :....
- 49 -
C"') < C"')
~ ~
~ ~ cc cc
~ ~ rd rd
5 5 ~ ~
~ ~ u
~ :! u U
.-4 N
.-4 ..... . .
C!I C!I .... .... "" ""
- 49 -
< c:' M
" ~ ~
~ tS ::::: ::::: Z Z 0::: 0::: ~ tz.J r t""'
1-< 1-< ;:; ~
~ ~ V u ~ ~ v u
.::: :.:..
- 50 -
.... ;S tJ tJ 1 1
("II ("II g,. g,.
~ ~ c:Q c:Q
~ ~ ral
5 5 g,. g,.
~ ~ tJ tJ
~ ~ tJ tJ
M ~ .... .-4
. . 0 0 .... .... la.. la..
......
::...
z :z:: ;-:.....
< U
1 C'l ::...
50 -
FIG. 15 CRACK PATTERN BEAH P2-Cl
FIG. 16 CRACK PATTERN BEAM Pl-Cl
VI ....
:'>1
M M G U
1 1 ~j M ::... ::...
:<::
~ < a z z ::.:: ct: -." :.:.: :....
~ :..... < ~ :..:: :..:: -- u -< ~ ::.:: -- u
" ,::;
~ .... '. -
- 52 -
Due to the increase in shear force and decrease in bending moment in
the direction towards the support, the flexural cracks located closer to the
support started to bend over tovards midspan and become flexure-shear cracks,
the angle of inclination with respect to the vertical being greater as the
crack was located closer to the support. In addition, the cracks deviated
from the vertical sooner, i.e. at lover depths, as they were located closer to
the support. An examination of the beam photographs, Figs. 7 to 16, will re
veal that relatively more flexure-shear cracks, which are potential diagonal
tension failure cracks, form in a member under uniform load than one subjected
to isolated point loads. Whereas only one diagonal crack usually forms in the
shear span for beams under point loads (e.g. 36) three and sometimes four dia
gonal cracks were observed on either side of midspan for beams under uniform
load.
The flexure-shear crack causing failure was generally the one clo
sest to the support. The variation in the location of this crack is believed
to be random and, in general, can be attributed to local weaknesses in the
concrete and the presence of previously developed cracks. This critical crack
cros8ed the steel level, on the average, at 0.2L from the support, the varia
tion being from 0.18 to 0.24L. It crossed middepth at 0.27L, with the loca
tion ranging from 0.25 to 0.29L. Finally, it cut through the top surface of
the beam at about O.44L, the range being from 0.42 to 0.47L. Other investiga
tors (17, 27, 43, 45, 46) while studying ordinarily reinforced or prestressed
concrete beams under uniform load, have shown that none of the variables which
affect shear-carrying capaclty, except posslbly for the span/depth ratio have
a significant influence on the location of the variously-defined critical sec
tion.
- 53 -
As Just mentioned the critical flexural cracks which initiated the
flexure-shear type of diagonal crack were generally those closest to the sup-
port. With further application of load they propagated vertically and then
proceeded with a gradually flatter slope. At the diagonal tension cracking
load there was usually no splitting along the longitudinal steel. However, at
the attainment of this load beams PlA and P2-ClA split along the steel towards
the support for a distance of about one inch fram tbe point where tbe crack
crossed the steel. In aIl beams up to the diagonal cracking load the crack
pattern was remarkably symmetrical with respect to midspan. After propagating
along the upper flange-web junction for about two or three inches the diagonal
crack suddenly extended downward approximately tangent to itself and in the
direction of the nearest support. These downward extensions either travelled
directly towards the support or intercepted the centroid of the longitudinal
steel at about 12 in. fram the support and then travelled along the steel to-
wards the support.
At the location where the downward extension was tangent to the dia-
gonal crack, the diagonal crack formed a rather large angle with the horizon-
tal. It is believed that transverse displacement across the diagonal crack
brought a considerable aggregate interlocking force into action and this is
the main cause of the flatter downward extension of the diagonal crack. A few
progressively flatter cracks were also formed until a stage was reached at
wbich the interlocking vas insufficient to produce further cracking. Due to
the large opening and fIat angle of the last crack produced the interlocking
across it vas negligible.(47) Except for beams P2, P2-C3 and P3-C3 in wbich
these dovavard extensions occurred only on the left-hand side, aIl beams deve-
Ioped these extensions on both left- and right-hand sides almost simultaneous-
Iy.
- 54 -
With further application of load beyond the load causing the down-
ward extensions a fair degree of inclined hair-l~ne cracking occurred in the
web where the downward extension of the diagonal crack crossed the longitudi-
nal steel. lt is not known if these cracks started in the web and moved down-
wards or formed at the steel level and moved upward. Shear transfer by means
of aggregate interlocking is believed to be primarily responsible for the de-
velopment of these secondary cracks if they formed in the web and moved down-
wards. However, if they formed at the steel level and moved upwards, they are
associated with either bond distress or dowel action of the reinforcement.
The main diagonal cracks were equally wide on either side of midspan and fail-
ure could have taken place on either side or both. Failure occurred soon
afterwards in a manner that in general seemed to be a stmultaneous collapse of
the compression and tension zones. This will be discussed in the section on
modes of failure. AlI beams failed only on one side of mi~span and the diago-
nal cracks on the other side closed sl1ghtly just after failure.
lt was previously mentioned in Section 3.3 that for the first few
tests tbe transverse stiffeners and transverse beams were left in their or1gi-
nal positions. As can be seen from a comparison of beams P2, P2-C3 and P3-C3
witb the otber beams, tbe lack of symmetry of the transverse stiffeners and .... beama about tbe centre of the test set-up affected tbe cracking pattern after
the diagonal tension cracking load. Whereas in most of the beams the cracking
pattern was remarkably symmetrical about midspan up to the point of fallure.
tbe tbree above-mentioned beams developed dowuward extensions of the diagonal
crack only on tbe left of midspan. apparently because the test set-up vas not
equally flexible on either side of tbe midspan of the test beam. However, the
ult~te capacittes did not seem to be affected as a comparison of the ulttm&te
cap4cities of tbe otber test beams revealed.
-.. 5~. -.
From an examination of the tests reported here and others found in
the literature, one could conclude that cracks assume a flatter slope in the
compression zone for beams under uniformly distributed load than for beams un
der isolated point loads. This can be attributed to a combination of numerous
effects. In a beam under uniform load, the shear force decreases towards mid
span while the bending moment increases. Rence, the normal compressive
stresses due to bending exhibit a greater influence than in point-loaded beams
on the value and especially the inclination of the principal tensile stresses.
The restraining effect offered by the applied loading due to its being dis tri
buted over the entire span rather than over a short distance removed fram the
critical flexure-shearcrack could also contribute to the flattening of the
diagonal crack. An inspection of the photographs published by Bernaert and
Siess (17) of their rectangular beama under uniform load would reveal this
flattening of the diagonal crack in the compression zone. In addition, as an
examination of the photographs presented here will reveal, the horizontal tra
vel of the inclined crack takes place at the upper flange-web junction. It
may be argued that the crack flattens out and travels along the flange-web
junction rather than penetrating into the flange because of the fact that the
stress level is higher in the web than in the wider flange.
In the case of a beam with web reinforcement the stirrups that are
effective against shear are the ones that are crossed by the diagonal crack,
the angle of inclination of the diagonal crack being generally assumed as 45
degreel to the horizontal. Due to the flattening of the diagonal crack the
number of Itirrups crolled in the case of a beam under uniform load would of
courIe be larger. Bence it vould seem that the web reinforcement could resist
more of the applied shear. AlI of this assumes that the stirrups have suffi
clent anchorage to prevent tbeir being pulled out.
- 56 -
Under uniform load the diagonal tension crack causing failure is 10-
cated closer to midspan in prestressed beams than in reinforced concrete
beams. For the beams tested in this study and those by Kar (45), which were
rectangular and grouted, the critical diagonal crack crossed the middepth at
0.27 to 0.3L fram the support. Bernaert and Siess (17), testing reinforced
concrete beams, reported that the critical diagonal crack crossed middepth in
the neighborhood of 0.15L. Although Krefeld and Thurston (27), who tested
reinforced concrete beams, did not state where the diagonal crack causing
failure crossed the middepth for their tests, it can be inferred tram their
data that it was much less than O.27L.
A possible explanation for this behaviour is as follows. It is weIl
known that the presence of precompression in the beam due to prestressing in
creases the beam's resistance to cracking. As the loading is increased the
precompression is overcome at sections progressively closer to the support and
hence flexural cracks form at these sections as soon as the tensile strength
of the concrete is overcome. A loading stage is reached at which the shear at
a section is sufficient to cause diagonal tension cracking, the diagonal crack
usually being an extension of a flexural crack. The sections closer to the
support, at this stage, are still compressed to a certain degree. In contrast,
for the case of ordinarily reinforced concrete beams, flexural cracks form ra
ther close to tbe support due to the absence of precompression. The flexural
crack closest to the support is the one that usually develops into the criti
cal diagonal tension crack.
4.4 Modes of Failure
Failure generally occurred in a manner that seemed to be a simulta
neous collapse of the compression and tension zones. Except for beams PlA and
- 57 -
and P2, aIl test beams failed in a stmilar manner in the compression zone.
The main diagonal crack which had been propagating along the upper flange -
web junction suddenly and violently turned upwards and caused failure of the
compression zone by rupturing or shearing along its line of propagation rather
than by crushing. The diagonal crack entered the top flange only when failure
w~s tmminent. The point of exit of the diagonal crack fram the top of the
beam varied between 0.42 and 0.47L fram the support. This variation seemed to
be randam and could not be conclusively correlated to the parameters investi
gated.
At first sight beam PlA would appear to have failed by crushing in
the reduced compression zone. However, this crushing took place when the beam
due to its rather large deflection touched and rotated about the wooden block
at midspan used to protect the deflection gauges. Beam P2 failed in the ten
sion zone prior to the destruction of the compression zone.
AlI of the beams failed in the tension zone in a manner that Neville
and Taub (48) have called "shear-tension". The widening of the lower portion
of the diagonal crack led to a deformation of the longitudinal steel; the two
parts of the beam to either side of the diagonal crack rotated relative to
each other approxtm&tely about the apex of the crack. This caused the rein
forcement in the crack to be pressed down so that the force in that part of
the reinforcement which crossed the diagonal crack was no longer horizontal.
The vertical component of tbis force, and tbe horizontal as weIl, caused split
ting of the beam along the steel, beginning from the bottom end of the diago
nal crack. Tbis splitting towards the support rrogressed as the loading in
creased. At the point of failure this splitting bad reached tbe support. For
soma of tbe specimens tbe cover below the reinforcement fell off in the vlcl
nit y of the support.
- 58 -
The failures of the test beams in this study also exbibited certain
features wbich are considered to be secondary and to be the effects rather
than the causes of failure. Due to the rotation of the portion of the beam
above the diagonal crack about its apex, this portion of the beam was broken
into two or more pieces by vertical cracks originating at the diagonal crack
and propagating to the top surface; this fracturing took place in aIl but
three of the beams. The final appearance of beams PlA and P2-ClA exhibits
characteristics of the failure of a two-hinged tied arch, i.e., the formation
of a tension crack at the top surface with its downward extension and the
crushing of the concrete in the web. This arching action was more pronounced
in beam PlA than in P2-ClA because beam PlA did not have an opportunity to
deflect freely and fail in the usual shearing mode as it touched and rotated
about the wooden block Just before failure. lt is believed that with the
destruction of bond up to the support at failure the tension in the steel be-
came constant between the support and the point where the critical diagonal
crack crossed the steel. This transformed the beam action to one of a "par-
tial" tied arch and since there did not seem to be any slippage of the steel
at the end-block buckling of the arch took place wben its thrust line cut the
diagonal crack. The transformation from beam action to that of a tied arch
was not complete as bond was still maintained in the central part of the
beam.{49, 50} As closely al could be' observed this arching took place stmul-
taneoualy with the destruction of the compression and tension zones.
Another crack propagation of interest took place in aIl beams except
P2, P2-Cl and Pl-Cl. Immediately prior to failure the diagonal cracks on ei
• ther lide of midspan were equally wide and failure could have taken place on
either or both aides. However, the diagonal crack always cut through the top
flange on one aide only and the other diagonal crack suddenly propagated down-
- 59 -
ward diagonally across tbe web. Tbis could be due to tbe release of a consi
derable amount of energy wbicb took place at failure or to tbe beams' collap
sing to tbe wooden block placed at midspan.
4.5 Measured Concrete Strains
Strains along tbe centreline of the top surface of the beams were
measured witb 20 mm. long electrical resistance strain gauges to obtain the
relationship between the strain distribution and the stage of development of
tbe crack pattern, and the compressive strain at the location of failure.
In a prismatic beam of elastic material the shape of the longitudi
nal strain distribution should be stmilar to that of the bending moment dis
tribution. However, the appearance of cracking introduces peaks and discon
tinuities into the strain distribution. In addition, due to the fact that
concrete is not an ideal material for the application of strain gauges, dis
continuities occur in the strain distribution prior to any cracking.
The variation of the shape of the longitudinal strain distribution
with load was essentially stmilar for aIl beams. For this reason, the re
Bults of only one beam, P3A, will be discussed here. Strain gauges were
placed at 3 in. intervals aIl along the span alang the centreline of the top
surface except for one foot near either support. Fig. 17 reveals that before
the flexural cracking load of 28 kips, the longitudinal strain distribution
followed tbe parabolic sbape of tbe bending moment diagram. At loads above
tbe flexural cracking load tbe strain distribution exhibited peaks and dis
tortions due to tbe presence of cracks. However, the strain pattern still
varied approxtmately as tbe moment diagram for loads up to diagonal cracking,
wbicb took place at 43 kips. Prior to tbe formation of diagonal cracks tbe
strain distribution on eitber side of midspan maintained tbe seme direction of
curvature, i.e., it remained concave downward.
1:1 H
~ ~
fWj
,i § ~ ~
~ -..,.. " -
~
o 8
~'~ _._-
~ ~-_ri~ ~
~ ~.~ <
'_._-~ ...
III - -Cl. ~ N (J\ CD
~ - -1 _. -_.-.. - ._. .--"'0 ~ III
< 1 --~.----••
~ ,-- Pl ,--:J -IL ~ ,
LONGITUDINAL STRAIN DISTRIBUTION ALONG f ON TOP OF BEAN (micro in./in.)
CD o o
-
~tb.-. • 4_ .~ ...
~ N
8 .... (J\
o o
--_ .. _-
i
N o o o
'11-._-- -_ .......... ~ .... ~ -\:: ~. ::.. ~ -------. ---..; ~:-=-.:: ::.::.:- ----'-~~~ -t> t .ft ~.~.-~.-
lJ --- ;::.. ._.~ ..
r1 /....-'1 __ f--- f----1
~ I~-f-....?"~ !
.L ~ 1 i
~ ___ I____ 1
1 1
!
1
--
N ~ o o
_.
------'-~.~ ------
. '
"
N OD o o
--~ ---
-~----
FIG. 17 LONGITUDINAL STRAIN DISTRIBUTION. ' DRAWN FOR BEAN P3A
W N o o
--- ---~ /--CD -A ---
1--- --
W 0\ o o
--.. --~ -------__ J
(J\
o
- 61 -
After the formation of the diagonal crack or cracks the strain dis
tribution varied with the propagation and development of the diagonal cracks.
The main compressive force in the concrete above the diagonal crack slopes
downward fram the section at the head of the crack in the general direction of
the support due to the onset of arch action or at least due to the disruption
of beam action. Consequently, the strains in the top surface away fram the
apex of the diagonal crack are reduced, even changing to tension as shown in
Fig. 17 and as evidenced by the small tensile cracks observed at the top of
the beam near the end block. Large strain concentrations take place at the
apices of the diagonal cracks because of the hinging of the fracture faces on
either side of the diagonal cracks about the apices. After the formation of
the diagonal crack or cracks the curvature of the strain distribution changed
to one which was convex downward. The concentration of strains near the
apices continued until failure.
In comparison to beams under point loads, the "peaking" or the con
centration of longitudinal strains along the top surface of the beam above
the apex of the diagonal crack spreads over a larger portion of the top sur
face of the beam. This is due to the restraining effect offered by the uni
formly distributed loading against the tendency of the upper portion of the
beam to buckle upward when acted upon by the inclined compressive thrust.
All otber things being equal, tbe reported value of tbe compressive
strain at tbe point of failure is dependent on tbe technique used to measure
lt and on tbe time at wbicb the last reading can be taken. For tbe beams
tested bere tbe maximum strain was usually obtained witbin six incbes of mid
span. correspouding to tbe point of exit of tbe diagonal crack fram tbe top
flange. Its value varied betveen 0.003100 in./in. for beam P2A and 0.006140
in./in. for beam P3.
- 62 -
4.6 SUDIIIary
The method employed to obtain a uniformly distributed load proved to
be very successful as seen from the parabolic distribution of longitudinal
strains. The reproducibility of spectmens was found to be excellent. Table 2
lists the flexural, diagonal cracking and ultimate loads, Figs. 5 and 6 show
the load-deflection curves, while Figs. 7 through to 16 are the photographs of
the test beams after failure. Relatively more f1exure-shear t}~e diagonal
cracks form in a beam under uniform load than one subjected to point loads.
Whereas only one diagonal crack usua1ly forme in the shear span for beams under
point loads, three and somettmes four diagonal cracks were observed on either
side of midspan for beams under uniform load. The diagonal crack causing fail
ure was generally the one c10sest to the support. The location of this criti
cal diagonal crack could not be conc1usively corre1ated to the parameters
studied. This crack crossed the steel level at 0.2L, on the average, from the
support, the variation being from 0.18 to 0.24L. It crossed middepth at 0.27L,
with the location ranging fram 0.25 to 0.29L. Finally, it cut through the top
surface of the beam at about O.44L, the range being from 0.42 to 0.47L.
Crack. assume a flatter slope in the compression zone for beams under uniform
load tban for beams under point loads. This fact wou1d increase the number of
stirrups that are effective in resisting shear because more stirrups would be
traversed. In camparison with ordinary reinforced concrete beams under uni
form load, for the case of prestressed concrete beams tested under stmilar
loading the diagonal crack causing failure is located c10ser to midspan. For
prestressed concrete beams tested here and elsewhere (45), the critica1 diago
nal crack crossed middepth in the vicinity of 0.27 to 0.3L fram the support.
For reinforced concrete, this crack crossed middepth in the neighborhood of
0.15L.(17, 27) Failure generally occurred in a manner tbat seemed to be a
- 63 -
stmultaneous collapse of the compression and tension zones. In comparison to
beams under point loads, the "peaking" or the concentration of longitudinal
strains along the top surface of the beam ab ove the apex of the diagonal crack
spreads over a larger portion of the top surface for beams under uniform load.
- 64 -
5. TBEORETlCAL ANALYSIS
5.1 Mechanism of Failure
The failure of concrete beams is generally classified under two
broad categories, viz. flexural and shear failure~. Flexural failures take
place in the case of rather slender beams, i.e. beams with relatively high mo
ment-sbear ratios, M/vd, or beams having low percentages of tensile reinforce
ment. Investigators are in general agreement as to the manifestations of this
type of failure and hence it will not be discussed further.
The term "shear faUure" is quite confusing because concrete as a
material does not exhibit a shear type failure. The cause of the failure in a
region where shear forces predominate is normally the principal tensile stress
es, referred to by many as a diagonal-tension failure. In cases where both
flexural and shear stresses are acting, the failure can Qccur either in the
form of the so-called shear-compression or the diagonal-tension failure. Al
though these two types of failure are generally dealt with under different
headings, they will be treated as one type here and stmply called flexure
shear failures. The two types of failure are considered to be the same except
that they exhibit different modes of rupture of the compression concrete in
the critical zone above a diagonal-tension crack or cracks. That is, both
types of failure are really only different demonstrations of a failure in the
compression zone. A diagonal-tension fallure, as shawn in Fig. l8(a), takes
place when a characteristic diagonal crack continues at a decreased slope
through the compression zone up to the comprepsion surface of the beam. On
the other hand, a shear-compression failure, see Fig. l8(b), takes place when
the concrete in the greatly reduced compression zone above the diagonal crack
crusbes in a manner stmilar to the flexural failure of an over-reinforced beam.
- 65 -
(a) DIAGONAL TENSION FAIWRE
(b) SBEAR-COHPRESSION FAILURE
FIG. 18 FIEXURE-SHEAR FAILURES
- 66 -
Numerous other modes of failure have been observed and classified by
various investigators, viz. shear proper (16), shear bond (25), web distress
(36), etc. lt is not intended to discuss these types of failure here.
5.2 Factors Affecting the Flexure-Shear Strength of Prestressed Concrete Beams
The strength of a concrete beam is a function of the strength of con
crete in the compression zone and the depth of the compression zone. The fac
tors affecting the strength of concrete and the depth of the compression zone,
thus, affect the load-carrying capacity of the beam.
The strength of a material under combined stresses is obviously a
function of the magnitude of these stresses. The strength of the compression
zone will be discussed in Section 5.3. The factors controlling the depth of
the compression zone can be enumerated as follows: (a) strength of concrete in
the compression zone, (b) deformation of the compression zone, and (c) defor
mation of the tension chord. The deformation conditions will be discussed in
Section 5.4.
To determine the internaI forces that occur at failure it is neces
sary to consider, in addition to the deformation conditions, the equilibrium
conditions wbich state that the borizontal compressive forces must be equal
to the borizontal tensile forces. The equilibrium conditions will be dis
cU88ed in Section 5.5.
5.3 Strengtb of tbe Compre8sion Zone
The strengtb of an element in the compression zone depends on the
normal flexure stre8s, fx ' tbe shearing stress, v, and tbe normal vertical
stre88, fz. The first ~o of these are related to one anotber Dy the confi
guration of loading 80 that their effect can he taken into account by the
effective moment-8hear ratio, M!Vcd. Here, Vc is the shearing force carried
- 67 -
by the concrete in the compression zone, so that the effective moment-shear ra
tio allows for the sbear carried by the longitudinal and web reinforcement.
The effect of vertical stresses, f z ' particularly for small values
of M/vd, on the strengtb of concrete beams has been noted by many investiga
tors (18, 51, 52). Under biaxial compression, the strengtn of concrete in the
compression zone increases but wben vertical loads are applied througb secon
dary beams, concrete in tbe compression zone is subjected to biaxial compres
sion-tension, and its strength is reduced. Tbus, the strength of concrete in
the compression zone depends on the vertical stress, tbe moment-shear ratio,
the amount of longitudinal reinforcement, the degree of shear reinforcement and
of course on the quality of the concrete.
Full consideration of the strength of concrete under a complexstate of
stress will not be made here and further discussion will be based on Mohr's
theory. Altbough this theory may not be applicable to concrete under triaxial
stress (53, 54), it has been found to be valid for biaxial stresses (22, 24,
55), this being the general state of stress in tbe compression zone of a con
crete beam. Mohr's circles for some typical elements at potential failure are
shown in Fig. 19. To generalize from sucb information it is necessary to know
tbe sbape of Mobr's envelope and to establisb tbe relation between normal and
shearing stresses wbicb, when acting togetber, cause failure to occur in the
compression zone.
Various suggestions about tbe shape of tbe envelope bave been made
in tbe past, including a straigbt line, a parabola and a cubic parabola.
Guralnick (22) used a straigbt line envelope based on pure tensile and compres
sive strengtb circles (circles C3 and C2, respectively, in Fig. 19). A simllar
approacb is adopted bere but, instead of pure tension, pure shear (circle Cl)
ls used since tbe stresses at failure in tbe majority of beams lie be~een
P 2
- 68 -
j x--=--_-+---_----r-' --.----88
z
(a)
(h) (c)
(e)
v , \
C4\ ,
(d)
Element B for M
larger Vd ratios
__ ~-+~~ __ ~--________________ ~ ____ ~ ______ ;--.-f (Tension)
f' t
f' c /
./ - ."...
(Compression)
Element B for M
small Vd ratios Element
A
FIG. 19 STRESS CONDITION AT DIFFERENT POINTS ALONG THE BEAH REPRESENIED IN THE FORH OF HOUR 1 S STRESS CIRCLES
- 69 -
pure shear and pure compression. It is appreciated, however, that for small
values of the moment-shear ratio, vertical stress, f z ' may cause flexural
stress, fx' on an element such as B (Fig.19) to fall somewhat beyond the pure
compression circle (see circle CS). The assumption of a straight line enve-
lope within this l~ited range of stresses, thus, seems justified.
Fig. 20(a) shows an element under a general bi-axial state of stress.
Mohr's circle with the envelope is shown in Fig. 20(b) for the element.
From Fig. 20(b) we can obtain:
----- (1)
It is weIl known that the major and minor principal stresses are:
- x z 2 J f - f 2 + ( 2 ) + vxz
fx + fz J fx fz 2 f 2 - } + 2
2 - ( 2 vxz
----- (2)
Substituting Eq. 2 into Eq. 1:
V/x 2 fz
) + 2 - %(fx + fz) sin 9 + c cos 9 2 vxz ----- (3)
Putting )9 • ft/f~ and from geometrical considerations from Fig. 21 we can
obtain:
sin 9
COB 9
c cos 9
• (l - 2')
• 2 Vft - Il 2'
• f' t
The value of ft is taken as the modulus of rupture of concrete to fit with the
aSBumed shape of tbe envelope. Sheikh (3) bas shawn that the use of the uni-
axial tensile strength would require a parabolic envelope. Substituting in
Eq. 3:
----- (4)
(a)
,
f z
v ·v xz
v
- 70 -
_-Assumed straight Une envelope
E~--~~-+~~~----------~~----------T-~----~
(b)
FIG. 20 ASSUHED STRAIGHr LINE MOHR' 5 ENVELOPE
Assumed straight line envelope
v
v xz
~~----~---+--~----~------------~~---f fc
FIG. 21 DEl'ERKIHATION OF THE FACTORS c, 9 FROM ASSUHED K>BR' 5 ENVELOPE
(compression)
whence
vxz - = f' c
r
- 71 -
(5)
Tbe interaction curves of Eq. 5 bave been p10tted in Figs. 22 and 23
for different values of fz/f~ and ~ • Tbese figures c1ear1y sbow tbat tbe
effect of fz/f~, even for sma11 ratios, cannot be neg1ected and sbou1d be ta-
ken into account when examining tbe test beams.
At tbe critica1 section tbe effective vertical stresses appear to be
contro11ed by tbe sbearing force, Vu. Hence it is suggested tbat the vertical
stress, fz, be taken as:
(6)
Eq. 6 is based on fz - Vu/Ab for M/Vd • O. The magnitude, and hence the
effect, of fz decreases witb an increase in the H/Vd-ratio. Walther (24) has
sbawn tbat tbe effect of vertical stresses is neg1igib1e for H/Vd greater tban
3. Eq. 6 gives fz = 0 for H/Vd ~ 4. The value of fz for M/Vd between 3 and
4 given by Eq. 6 will not be too large, and a continuous transition between
maxtmum and zero values of fz is acbieved. A second degree transition is
assumed.
In tbe case of beams, tbe ratio vxz/fx can be expressed in terma of
tbe important ratio H/Vd. To eva1uate tbis relationsbip, an equivalent com-
pression b10ck witb constant stresses fx and vxz is assumed; tbe value of tbe
lever arm is taken as 0.9 ttmes tbe effective depth. Then
fcv • • K K ----- (7)
0.24
0.20
0.16
0.08
0.04 ~------~-------------4-------4-----4~
o 0.2 0.4
f ~ f'
c
0.6 0.8 1.0
FIG. 22 PLOr OF INrERACIION CURVE OF CONCREIE STRENGTH UNDER BI-AXIAL STATE OF STRESS (Eq. 5) FOR f If' • 0 z c
- 72 -
1- U >\6.1 •
=1- u > \6.1
0.7~------~------~------T-------~----~
0.6r-------+-------+-------+-------~----__1
0.5~------+-------+-------~=-----+-----~
0.4
0.3
0.2~~~~~~----------__ ~------~------~
0.1r-----~~----~--------------_+--~~_4
o 0.2 0.4 f ~ f'
c
0.6 (l.8 . 1.0
- 73 -
FIG. 23 P1Dl' OF EQ. 5 FOR DIFFERENl' VAWES OF f If', and f3 • 0.14 z c
- 74 -
Also, Vc vxz
.. bkud ----- (8)
Therefore,
fcv fx M =- - .. vxz vxz 0.9 Vcd ----- (9)
or
fcv M - SI
0.9 (1 - "'l ) Vd vxz -----(10)
where
Vc .. 1 --
V
Walther (24) called the factor "l the "degree of shear safeguard" •
He used it to allow for the contribution of web reinforcement only, but in
thi8 the8is it is u8ed to take into account any part of the shearing force not
resisted by the compression zone of the concrete. This factor, thus, also
takes into account the effect of dowel action and the shear force resisted by
the vertical component of the force in a draped tendon in prestressed concrete
beama or inclined bars in ordinary reinforced concrete beams.
Substituting fx/vxz in terms of M/Vd in Eq. 5 the following equation
can be obtained after SOlDe simple transformation8:
2 2 fz 2 f 2 -,8 - (IJ - 2~ )F + (/& -!& )(f~) .. 0
c c (11)
where ~.. fcv/f~
Altbougb Eq. Il appears uuvieldy, it does not really pose &Dy serious
problema as far as its practical utility is concerned. Charts can be prepared
.. 75 -
for values of s-' with respect to M/Vd for different values of fJ ' ., and
fz/f~. The shear-compressive strength, fcv' can then be chosen from the
charts for the known values of 13 , M/Vd, "l and fz/f~. A comparison of the
proposed interaction curves (Eq. Il) for ~ = 1/8 with those of other investi
gators (22, 24, 56) is shawn in Fig. 24.
The interaction of shear and moment on the strength of the compres
sion zone was derived in a general manner independent of the type of loading.
No change is required here. However, the effect of vertical stresses, fz,
requires some discussion. According to the water pressure gauge inserted at
the end where the two hoses were interconnected the contact width per hose was
approxfmately 0.75 in. since the pressure was 300 psi for a total load of 50
kips. If the load was distributed over the entire top surface of the beam
instead of two strips of 0.75 in. width, the contact pressure, for the same
total load, would have been 75 psi, being quite small. Hence, vertical
stresses were thought to have negligible effect and were ignored. In Eq. Il,
aIl the terms involving fz were set equal to zero when analyzing beams under
uniform load.
5.4 Deformation Conditions
A major difficulty in determining ultfmate strength of a section
subjected to combined bending and shear is that, under this condition of load
ing, plane sections no longer remain plane. The disturbed region, which bas
to be considered in computing the deformation of the compression zone, as
shawn in Fig. 25, extends approxfmately between the load point, toward wbich
the diagonal crack is oriented (referred to as section 2), and a point some
vbat to tbe far side of the bottom of tbe diagonal crack (section 1). The
critical section is adjacent to tbe load point so that the deformation condi
tion at section 2 needs to be considered.
1. Oll--r-----:~~:::::=P=::::::::= ... ,...--I-___.,
~u 0.~--~--4_------~~~~~----_+------~---'" >
U IW
a "1=0
~ ~ 0.4~rl_--+_~~~----~----~------~--__,
Proposed
1---- Walther (24) o. 2 J-..4+--+--41I~-----+------l _. _. - Bay (56)
o
- - - Guralnik (22)
1 2 4 5
FIG. 24 COHPARISON OF VARIOUS INTERACTION EQUATIONS FOR THE BI-AXIAL STREN(il1I OF CONClŒrE IN BEAMS (f If' =0 0, $1 ·-"1/8) z c ,-
..
6
- 76 -
11'---". ___ a -------;1
Major diagonal tension crack
V disturbed region
1 2
FIG. 25 DISTURBED REGION CONTRIBUtING TO THE DEFORMATIONS OF STEEL AND CONCRETE AT THE CRITlCAL SECTION
FIG. 26 SCHEHATIC REPRESENtATION OF STRAIN DISTRIBUtION AT THE EXTREME COMPRESSION FIBRE OF BEAM SHOWN IN FIG. 25 (See references 3, 23, 57 and 58)
- 77 -
- 78 -
Strictly speaking, the total deformation at section 2 is the summa-
tion of the unit deformations over the entire shear span in which the section
is located. However, it has been shown (23, 57, 58) that deformations outside
the disturbed region 1-2 do not contribute much and can be neglected (see Fig.
26).
The deformation conditions for the region under consideration must
be such that the rotations based on the concrete deformation and the rotations
based on the steel deformation are equal. As most of the rotations taking
place at section 2 occur over a short distance they can be assumed as concen-
trated at section 2 (see Fig. 26). The above condition can therefore be rea-
sonably represented as:
...o.c - :. ~s
-----(12)
As is weIl known, shortening of the compression surface is concen-
trated over a short length near the upper end of the diagonal crack, the
effective length contributing to this deformation being an increasing function
of the depth of the compression zone (24, 58). The deformation of the com-
pression edge will thus be assumed as:
----- (13)
The value of Ecu is controlled by the ductility of the beam, which
in turn is controlled by the inelastic deformation of the steel. A beam
reaching its flexural capacity shows greater concrete strain at failure than a
beaœ failing in flexure-shear below its flexural capacity. Ductility of the
beam depends also on the effective moment-shear ratio, which governs the mode
of failure of the beam.
Measured values of E cu for beams failing in flexure are generally
reported in tbe range 0.0035 to 0.0040. Sheikh (3, 4) assumed a value of
- 79 -
0.0036 for beams having M/Vcd ) 6. Sheikh's tests for beams failing in fle
xure-shear indicated a linear relation between Ecu and the M/Vcd-ratio. Ig-
noring other possible influences on ~ cu' such as quality of concrete and qua-
lit Y of bond between steel and cOncrete, we can write for M/Vcd < 6:
Ecu = M
0.0006 V d c
-----(14)
The deformation of the tension chord mainly depends on: the magni-
tude of the steel strain 4S s due to the applied loading; the extent of the
disturbed region, taken into account by assuming~s to be proportional to
d(l - ku); the quality of bond between the steel and the surrounding concrete;
and the effective moment-shear ratio. The quality of bond affects the dis-
tribution of steel strain in the uncracked portion of the beam; poor bond leads
to a more uniform steel strain and hence a larger steel deformation.
The influence of M/Vcd arises from the development of cracks in the
beam. Flexure-shear cracks rise higher than flexural cracks because, for the
same applied moment, the principal tensile stress at the tip of the crack is
greater the greater the shear force, i.e. the lover the effective moment-
shear ratio. A higher crack means greater deformation of the longitudinal
reinforcement. Furthermore, the lover the effective moment-shear ratio (for
the s&me applied moment), the higher the bond stress and hence the larger the
relative movement between steel and concrete. Total deformation of the steel
is thus greater the lover the effective moment-shear ratio.
The deformation of the tension chord will therefore be put as:
-----(15)
The effect of M/Vd ia negligible for flexural failures (8), and,
since it has been observed (42, 48) that K/Vcd greater than 6 precipitatea a
flexural failure, the factor K for M/Vcd > 6 will be taken aa unity. For
- 80 -
H/v cd < 6 the factor K will be assumed to be:
-----(16)
While this expression has not been derived in a rational manner, it
is apparent from the discussion of the effect of H/Vcd on the deformation of
the tension chord that (a) K > 1 for H/Vcd < 6 and (b) a change in M/Vcd
results in a change in rate of development of diagonal cracks and also in a
change in bond stress. The steel deformation is thus accordingly affected.
The effect of Vcd/H on steel defonaation is greater than linear, and K has
n been assumed to be proportional to (Vcd/H) • On the basis of experimenta1
results, the value of n = 2 is used in further computations.
Substituting for K in Eq. 15:
-----(17)
Substituting Eqs. 13 and 17 in Eq. 12 and rearranging:
----- (18)
For flexural failure Eq. 18 can be stmplified to:
-----(19)
Eq. 18 thus provides a deformation condition which takes into
account both flexura1 and flexure-shear types of failure. This equation is
applicable to both reinforced and prestressed concrete members, since no dis-
tinction betveen the tvo was made in its development. For prestressed con-
crete, however, tbe equation can be vritten more conveniently as:
-----(20)
- 81 -
In considering the acceptability of Eqs. 18 and 19 it should be noted
that Walther (24) found an equation s~ilar to Eq. 19 to give better agreement
with test results for flexural failure than when the Navier-Bernoulli assump
tion is accepted. Furthermore, tests have shown that quality of bond between
steel and concrete affects the strain distribution and hence the deformation
of the steel, thus justifying introduction of the factor kb.
The value of kb for varying bond conditions, as proposed by Walther
(24), will be adopted in the absence of any better suggestion. For pre
tensioned beams this is:
•
The values of " are:
for plain bars and wires
for deformed wires
for efficiently deformed wires and strands
1.50
1.25
1.00 to 0.75
----- (21)
When the vertical loading is uniformly distributed the so-called
peaking of the strain distribution (see Fig. 26) is spread over a larger por
tion of the top surface. Hence a greater portion of the beam top contributes
to the total deformation at the section under consideration. In addition, the
horizontal projection of the diagonal crack is, in general, larger for a beam
under uniform load than for a beam under isolated concentrated loads. Thus
the disturbed region, see Fig. 25, is larger.
As discussed in Section 4.5 strain gauges were placed at close in
tervals along the top surface of the beam to obtain the longitudinal strain
distribution. At ulttm&te, or as close as possible, the strain distributions
vere plotted. Tbe areas under tbe strain diagrams vere obtained grapbically
and compared to Eq. 13, wbicb vas found to predict the area under the strain
diatribution diagram for beams under concentrated loads quite accurately. Tbe
- 82 -
areas under the diagrams for beams under uniform load were always larger than
twice the area given by Eq. 13 and usually in the neighborhood of three ttmes.
From this Ac was set conservatively at twice the value given by Eq. 13, Le.:
-----(22)
It was felt that Eq. 14 for the max~ compressive concrete strain
at the critical section of failure was not valid for beams under uniform load.
Firstly, it was obtained for beams under lsolated point loads and it is not
known if the relationship between "cu and M/Vcd would be the same for beams
under uniform loading. Secondly, at M/Vcd = 6, which for the beams reported
here takes place at about 2.70 ft. from the support if "i = 0.0, Ecu takes on
the value of 0.0036 which is taken as the crushing strain at flexural fallure.
This is not correct since the beam cannot possibly fail in flexure at Any 10-
cation other than midspan, a distance of 4.50 ft. from the support. For these
and possibly other reasons Eq. 14 was not used.
The strain ~cu is dependent on the extreme fibre stress, which is
fcv for a beam failing in flexure-shear. Assuming that the stress-strain dia-
gram for the concrete is a quadratic parabola, we have:
·cu • • 2
0.0036 Y' -----(23)
The value of 0.0036 was still adopted as the maxtmum compressive strain at
flexural failure. Eq. 23 is stmilar in form to that assumed by Walther. (24)
The factor K, Eq. l6,·which takes into account the effect of the
effective moment-shear ratio, requires modification. The effect of M/Vd ls
negliglble for the case of flexural failure and this was indicated by setting
K equal to unit y for this type of failure. However, whereas beams under point
loads migbt fail in flexure if M/Vcd ~ 6, beams under uniform load fail in
flexure only when tbe effective moment-sbear is equal to inf1nity, i.e. they
- 83 -
fail in flexure only at midspan. Rence, the effect of the effective moment-
shear ratio as formulated by Eq. 16 cannot be used when analyzing beams under
uniform load.
A method of determining the location of the failure section and hence
the ultimate load carrying capacity of the member is to assume, in turn, that
various sections along the span are the critical or failure sections and to
compute their flexure-shear capacity on that assumption. The proposed modifi-
cation of the factor K can be obtained using the following line of argument.
Define as x the distance fram the support to the cross-section under investiga-
tion. lt can then be said that x/d, at the assumed critical section, for a
beam under uniform load "corresponds" to a/d, at the known critical section, for
a beam under isolated point loads. This i8 only a correspondence between x/d
and a/d and not an equality. We are interested in a ratio between the two to
1 be able to formulate a quantity, defined as K , that describes the influence
of the effective moment-shear ratio on the deformation of the tension chord.
A beam with a point load at midspan will faU in flexure if (a/d) :> CI(, say.
The quantity 0( will be derived below. A beam under uniform load will fail in
flexure only at the midspan section. Assuming this correspondence between a/d
(equal to 0( to produce flexural failure) and x/d, the proposed modification
of K becOlDes:
KI 2 2
0( (~) • (x/d) .. x -----(24)
for ~ '1 0.0, ve have:
KI (~(l 2
• - "'l » x -----(25)
Sheikh (3,4) stated that a value of the exponent, equal to 2, was
chosen on the basis of exper1mental results. For a beam under uniform load,
a8 coœpared to one under point loads, the magnitude of M!Vd increases more
- 84 -
rapidly in the direction a~ay from the support, exhibiting the larger in--
fluence of moment as compared to shear force. Hence, the influence on the to-
tal deformation of the steel of the effective moment-shear ratio at the sec-
tion under investigation would tend to be smaller. Therefore, an exponent
less than 2 would be appropriate, say 1.3. However, to keep ~s compatible
with the increased value of ~c, a value of the exponent larger than say 1.5
should be used. Hence the value of 2 seems logical and will be used here.
The expression for 0(, the a/d-ratio at which and above which a beam
with a point load at midspan would fail in flexure, will now be derived. A
constant value for this ratio, for example, equal to 6 as chosen by Sheikh, is
not general and does not convey the influence of the physical properties of
the beam. The line of reasoning to be used here was outlined by Rangan (66)
while analyzing reinforced concrete beams.
The critical location for failure is at the level of the neutral
axis where only shear stress exists. This shear stress will result in a pair
of principal stresses which are compressive-tensile in nature and are equal in
magnitude to the shear stress. Concrete, being relatively weak in tension,
will fail whpn the principal tensi1e stress reaches its ultimate tensi1e
strength. Hence failure will take place when v = f~. The tensile strength
of concrete is here taken as the modulus of rupture, sÛDilar to the value used
when the shear-compres81ve strength of the compression zone was evaluated.
tained a8:
hence,
The bending moment at the critical section of failure can be ob-
M cr
k d u
k u
• A f d(l - -) 882
• 2(A f d - M )
8 8 cr A f s s
-----(26)
-----(27)
- 85 -
The shear carried by the concrete in the compression zone at the critical sec-
tion is denoted as Vc and so:
= = (1 - "'l )V -----(28)
Substituting for the value of kud and solving for v (= f~), we obtain:
fI t =
0.5V c M cr
bd(1 - A f d) s s
-----(29)
Based on the test results of Krefeld and Thurston (27), Rangan stated that ~
M = 0.833A f d cr s s -----(30)
Assuming that the flexural ultimate moment i8 0.9A f d, Eq. 29 can be manipulas y
ted to determine the value of the a/d-ratio at which the bending moment at the
critical section of failure, Mcr ' is equal to the flexural ultimate moment.
Rence, we obtain:
0<. = 2.7f p(l - "( )
y f'
t
-----(31)
The above expression for ~ does not bring out the differences be-
tween reinforced and prestressed concrete members or for that matter the
effect of different levels of precompression due to different levels of pre-
stress. From the theory of elasticity for stresses at a point it can be shown
that:
v • f 0.50
f (1 +~) t ft
-----(32)
where f - compressive stress due to prestress. The addition of prestress is pc
equivalent to increasing the tensile strength of the concrete. Although a
fair degree of the precompression in the concrete is overcome at the ultimate
load stage, it is generally observed that beams with higber levels of prestress
possess bigber u1ttmate sbear capacities, aIl other conditions being equa1.
- 86 -
For this reason, the following expression is offered for the "effective" ten-
sile strength of the concrete at failure:
v = f 0.125
f'(l+~) t f'
t -----(33)
While the adoption of the one-eighth exponent in Eq. 33 is arbi-
trary, it gives good agreement with experÜDental results. If so wished, Eq.
31, ignoring the effect of prestressing, can be conservatively used for pre-
stressed concrete beams. However, if use is made of Eq. 33, then Eq. 31
becomes:
0(= 2.7f p(1 - '71) y\
f'(l + f If , )0.125 t pc t
-----(34)
In the event that the longitudinal steel does not possess a defi-
nite yield point, as is the case for most steel used for prestressing, the
"0.2'7. offset" load or the more recent "1.0% extension" load (ASTM A4l6) can
be used for the yield stress f. If the modulus of rupture strength i6 not y
known it can be esttmated as 7.5~, being based upon the best knowledge pre-
sently available (67). The value of the precompression, f pc ' is based on the
overall cross-section of the beam and is equal to FIA. se
Making use of Eqs. 22 and 25, the value of k as given by the conu
sideration of deformation conditions can be expressed as:
-----(35)
-----(36)
- 87 -
s.s Eguilibrium Conditions
The actual stress distribution over the compression block is consi-
dered to be parabolic. Numerous investigators have suggested different equi-
valent stress blocks to stmplify the calculations. It must, however, be
pointed out that the differences in the calculated ult1mate moments resulting
from various reasonable choices are rather small. Fig. 27 shows the stress
block used here, leading to satisfactory results.
From Fig. 27(a) it can be seen that for an under-reinforced section
failing in pure flexure:
k = uf
F su bf d cu
The moment of resistance for ideal flexural failures is then:
k M f = F d(1 _ uf)
u su 2
-----(37)
-----(38)
For failures other than in pure flexure, i.e. for flexure-shear
failures, Fig. 27(b) represents the forces in the section. From statics we
have:
A fI = fl/fcu bk d s a u -----(39)
Writing f' 1: À fau' Eq. 39 becomes: a
k Fsu .2L • u f bd $V cu
-----(40a)
or, ku - k .~ uf "
-----(40b)
Eq. 40 (b) is val1d for rectangular sections and for T- and I-
section beama in wbich the neutral axis lies in the flange. If the neutral
axis is located below the f1an8e, equUibrium conditions bave to be estabHsbed
in the u8ual manner.
, ,
d -if
.
.. --F = A f su s su F' = A fI = A À f
S S 8 8· SU
(a) (b)
FIG. 27 EQUIVALENT COMPRESSION STRESS BLOCK USED IN THIS THESIS
(a) Actuai
-.. F'
S
(b) ACI
-
,
klkud
Il
F~ S
(c) Proposed
FIG. 28 COHPARISON OF VARIOUS COMPRESSION STRESS BLOCKS
- 88 -
- 89 -
Fig. 28 (a) shows the internaI forces at a section with the parabolic
stress distribution in the compression zone. The equivalent stress block
suggested by ACI 318-63 (5) is given in Fig. 28 (b); the equivalent stress
block used here is shawn in Fig. 28 (c). The major reason for using this
stress block is its simplicity as compared to the ACI stress block. Specifi-
cally, the factors kl and k3 generally occur as klk3 and it seems reasonable
to treat them as one factor. The factor klk3 in the present tests is taken as
unit y because the tests are considered to be comparable in size to a 6 in. by
12 in. standard control cylinder. In larger beams it may be somewhat smaller
because of a decrease in concrete strength with an increase in size of member.
5.6 Detenaination of the Deptb of Compression Block
In order to calculate tbe depth of the compression block, Eqs •. 20 or
36 and 40 (h) need to be solved stmultaneously. This can be done either by a
trial and error procedure suggested by Warwaruk et al. (59) or by a graphical
approacb proposed by Guyon (60). Needless to say, the value of k can also be u
obtained by digital computer with the aid of a simple programme. Guyon's
metbod will be discussed bere. Eq. 20 can be written as:
k u
Eq. 41 can be written witb reference to Fig. 29 as:
It u
-----(41)
-----(42)
where G i8 a constant for a particular section and g and gl are function
notations. ..
S~larly Eq. 36 can be written as:
It u • G 1
2 Eicu
-----(43)
l'
1.0 1.0
~
l: ~J--/ ::s co
IW • 1 -co 1 ~
ft
--<
o r .. 0 vL-__________ -L'L' __________ ~~ __________ ~
Csi ~su
Strain k
FIG. 29 GENERAL STRESS-STRAIN DIAGRAM OF STEEL
u
FIG. 30 DETERMINATION OF THE DEPTH OF NEUlRAL AX IS
\0 o
- 91 -
Eq. 42 or 43 can now be plotted as shawn in Fig. 30 for various va-
lues of E (or ~) and E . Eq. 40 (b) is also plotted in Fig. 30. The in-su cu
tersection of the ~o curves gives the required values of ~ andk. u
The moment of resistance of the section can now be calculated from:
k M • À F d(l - -l!.) u su 2
-----(44)
k or M = ~ f bk d2(l - 2 u) u cu u -----(45)
The preceding derivations have been made on the assumption that
there is no longitudinal steel in the compression zone but the method can be
extended in the usual manner to the case when steel is present (61).
5.7 Effective Moment-Shear Ratio
The determination of the effective moment-shear ratio involves a
knowledge of the term , ' the degree of shear force not resisted by the con
crete in the compression zone. That part of ~ which is due to the dowel
force, denoted by ~d' can be computed as (62):
Vd
0.7f~(dls)0.25blO.75 - - . v V -----(46)
On the basis of a large number of tests (27, 28, 63, 64) it appears
that the value of ~ d in reinforced concrete beams after diagonal cracking
ranges up to 0.6. Dowel action is effective in aIl cases wbere beam action is
present, but 1s reduced or absent wben splitting along tbe tension steel
occur8 and arch bebaviour develop8. Thi8 takes place only in simply-supported
beams (30) witb K!Vd < 2.5, and for such beams the value of "l has to be
modified. (3, 4)
No test data on dovel action in prestressed concrete beams are
available but, due to the lover stiffne8s of the thin wires and strands normal-
- 92 -
ly used, ~ d may not exceed approxtmately 0.2. In beama witb web reinforcement,
sbear carried by tbat reinforcement is relatively large (65) so tbat dowel
action can be neglected.
Tbe vertical component of draped longitudinal reinforcement resist-
ing shear can be easily calculated. This is also the case with web reinforce
o ment, it generally being assumed that the diagonal crack is incline& at 45 to
the horizontal.
5.8 Summary
In this chapter a procedure that had been previously derived (3, 4)
for the computation of the flexure-shear capacity of beama under isolated
point loads was described along with modifications to the procedure in order
to make it applicable for the analysis of beams under uniform load.
In order to compute the ulttmate moment capacity of a section the
following need to be calculated: value of ~ fram Eq. 11 and the value of k u
from Eqs. 40 (b) and 42 or 43. The ulttmate moment capacity can then be calcu-
lated fram Eqs. 44 or 45.
- 93 -
6. QUANtITATIVE EVAWATION OF TEST RESULTS
6.1 General
The validity of the method outlined in Chapter 5 for the analysis of
prestressed, concrete beams, both simply supported and continuous, under isola-.,' ..
ted point loads has already been shawn elsewhere (3, 4). In order to check
the validity of the modifications made in Chapter 5 for general use it is ne-
cessary to compare the esttmated load-carrying capacity of various type test
beams using these modifications with the actual failure load. To be ab~e to
achieve this goal it was deemed advisable to supplement the tests carried out
at McGill University with published exper1mental results. Unfortunately there
is very little data in the literature dealing with the flexure-shear strength
of prestressed concrete beams under uniform load.
Wilby and Nazir (43) reported on five grouted post-tensioned beams
under uniform load. However, sufficient information of the physical proper-
ties of the beams vas not given, making analysis of these beams impossible.
The only other data on prestressed concrete beams under uniform load that was
found in the literature vas that of Kar.(45, 46) In addition to other tests,
Kar tested Il grouted post-tensioned beams under uniform load. These speci-
mens will be analyzed in this chapter in addition to the McGi11 specimens.
In order to bring out the differences in load-carrying capacities of
beams under uniform load and beams subjected to isolated point loads, the
McGill and Kar beams will be analyzed under two-point loading according to the
method de.cribed in Chapter 5. In addition, the McGi11 Beams will be analyzed
according to the shear clauses of ACI 318-63 (5) for prestressed concrete,
the.e clau.es being based on test results of beams under isolated point loads.
Although the method outlined in Chapter 5 was said to be applicable
to both reinforced and prestres.ed concrete member. with suitable modification
- 94 -
of the bond coefficient, reinforced concrete members will not be analyzed here.
6.2 Ult~te Load Analysis of a Beam Under Uniform Load
The method of analysis of a stmple beam under uniform load to deter
mine the location of the weakest or critical section and hence the ult~te
lOad-carrying capacity will be briefly outlined here.
The required given quantities are as follows:
a) the span length, L, and cross-section dfmensions;
b) the concrete compressive and tensile strengths, f~'and ft, respectively;
c) the stress-strain curve of the longitudinal steel, including the values of
the yield and ult~te strengths, fy and f su' respectively;
d) the percent age of longitudinal reinforcement, p;
e) the effective prestress ~n the longitudinal steel, f se ' and the strain in
the steel whea the strain in concrete at the steel level is zero, e si;
f) the coefficient for bond quality between longitudinal steel and concrete,
kb, as given by Eq. 21; and
g) the degree of shear safeguard to the compression zone, , , (see section 5.7).
The value of ~, the a/d-ratio at which and above which a beam having
stmilar cross-section and steel properties as the beam under consideration
would fail in flexure with a point load at midspan, can now be computed by
means of Eq. 31 or Eq. 34. The quantity C( is dependent on the physical pro
perties of the cross-section and for the McGill beams was found to vary appro
ximately within the range of 6.0 to 9.0. For Karls beams 0( varied between
7.5 and 10.5.
The flexure-shear strength at various sections along the beam can be
calculated with the aid of the modifications proposed in Chapter 5. The ex
temal bending moment can then be drawn so that it Just touches the flexure
shear strength curve. The point of contact of these two curves will give the
- 95 -
position of the failure section. Knowing the position of the failure section
and the flexure-shear strength of this section the ultimate load-carrying capa
city of the beam can then be determined.
6.3 Evaluation of McGill Tests
The ten tests carried out in the course of this investigation were
analyzed by the procedure outlined in Section 6.2. AlI of the quantities re
quired to carry out the analysis were known except for the value of ~ , the de
gree of shear safeguard to the compression zone. For the McGill beams the
only contribution to ~ was from the dowel action of the longitudinal steel.
It was observed in the course of the testing of these beams that cracks along
the longitudinal steel occurred prior to failure. The extent of this longitu
dinal splitting varied from beam to beam. Nevertheless, the occurrance of
such splitting indicates that a certain amount of the dowel resistance of the
beam has been overcome. Thus, the computation of the dowel resistance at the
time of failure becomes quite indeterminate. For this reason ~ was neglected
in the analysis of these beams. In addition, it can be noted that the ex
pression proposed by Jones (62), Eq. 46, is only approximate. He neglected
the influence of axial tension in the reinforcement and did not consider the
relative transverse displacement of the bar ends across the diagonal crack.(l)
Sections, at 0.2 ft. intervals, were investigated between the sec
tion at 1.0 ft. from the support and midspan. Representative of the shape of
the flexure-shear capacity distribution alang the span for aIl of the beams
analyzed here, both fram McGill and Kar (45, 46), is the capacity distribution
shawn in Fig. 31 for beaœ P2-Cl. Due ta symmetry only one-half of the distri
bution i. shawn. The location of the critical section for beam P2-CI can be
seen to be located at 1.80 ft. fram the support; this corresponds ta a dis
tance of 0.2L fram the support. In the analysis of the MeGill tests, the
-. s:: .... 1 Cl. .... ~ -1
700 1 i 1
1 - f---- --A'
j ", i , / 1
" i ,
/ 1
i , 1 ; /
1 , i ~
600
! : ! ,
1
i FLElÜJRE-SuEAR / STRENGTH
,
~/ ,
1 1 1
1 1 7 ~ 1 1 ! i
1
, / i / 1
i ~
1
500
400
300
1 //'- 1
, i B.M. D.
i -1 1 y. 1
, 1 :1. ' : i
1 tr------:--1
j , '
1
, i
/: : ~ i
1 , --
1 /1 1
1
, 1
i .... ~ag;cIroAT roD'T' i 1 //' IrTroA T 1
1
~ I--t- SEct ION : 1 i . " .. Iii
1
'."<1 : 1
/ 1
1
, 1
i 1 1
1 i ,
200
V 1
1
,
1
,
1 i 1
.l.- I 1
fi 1 1
i 1
1 1
i
l i 1
100
/ 1 1
1
: 1 i !
1 : 1 ! 1
o 1 2 3 4
DISTANCE FROM LEFr-HAND SUPPORT (ft.)
FIG. 31 DETERMINATION OF THE POSITION OF THE FAIWRE SECTION AND THE STRENGTH OF A SIMPLY-SUPPORTED BEAM UNDER UNIFORM LOt\D. DRAWN FOR BEAM P2 -c 1
- 96 -
- 97 -
location of the critical section deviated only slightly from 0.2L from the
support, varying between 0.18L and 0.22L. Due to this very smail deviation,
the theoretical location of the critical section can be said to be independent
of variations in concrete strength and prestress level. lt is not known if the
location of the critical section would change if the L/d-ratio varied since the
L/d-ratio was kept constant for these beams.
Table 3 lists the measured and estimated ultfmate capacities for the
McGill beams. The estfmated ultimate capacities are those corresponding to
the moment at the section at 0.2L from the support. The average ratio of the
measured to the computed strengths is 1.04 with a standard deviation of 0.037.
Good agreement between the measured and computed strengths is, thus, observed.
6.4 Evaluation of Tests by Kar (45, 46)
Kar tested eleven grouted post-tensioned beams under sfmulated uni
form load. The beams were of rectangular cross-section and were without any
shear reinforcement. The uniform load was applied through five hydraulic
jacks, each one transmitting the load through a rail piece and two cast iron
loading blocks. Details of the properties of the beams are given in Table 4.
Although the stress-strain curve of the wire was not given, sufficient infor
mation of its properties were supplied to allow its drawing. lt is shown in
Fig. 32 along with pertinent information.
lt is not known to what degree longitudinal splitting along the
steel occurred jU8t prior to failure. For this reason the contribution of the
dowel resistance of the beam will be con8ervatively ignored. AlI of the
quantities required for the analyst8 are now known. However, according to
Leonhardt (68), the bond coefficient requires seme modification. Leonhardt
recommend8, for pre8tressed concrete beams vith grouted post-tensioned cab les
con8i8ting of individual vire8 or strands, the introduction of an effective
- 98 -
TABLE 3
- McGILL TESTS -
MEASURED AND COMPtrrED ULTIMATE CAPACITIES
IN TERMS OF TOIAL LOAD ON BEAM
Mark Meas. Camp. Meas. Kips Kips Camp.
Pl 41.0 36.8 1.11
PlA 40.0 36.7 1.09
P2 44.2 42.3 1.04
P2A 44.0 43.0 1.02
P3 48.0 48.8 0.98
P3A 49.2 47.9 1.03
P2-Cl 38.8 37.2 1.04
P2-ClA 39.8 38.6 1.03
P2-C3 46.4 44.8 1.03
P3-C3 53.5 53.0 1.01
Average 1.04 Standard Deviation 0.037 Coef. of Variation 3.61.
- 99 -
TABLE 4
PROPERTIES OF BEAMS TESTED BY KAR (45, 46)
Mark fI b h d A F L c s 2 se psi in. in. in. in. kips in.
A-U-2 5140 5 10 7 0.30 13.82 90
A-U-3 5600 5 10 7 0.30 23.57 90
A-U-4 5300 5 10 7 0.30 28.29 90
A-U-5 4160 5 10 7 0.24 22.70 90
A-U-6 4160 5 10 7 0.30 23.45 90
A-U-8 4040 5 10 7 0.30 18.57 90
A-U-ll 4900 5 10 7 0.30 28.29 90
B-U-1 5600 4 8 6 0.24 14.72 90
B-U-2 4900 4 8 6 0.24 18.59 90
B-U-4 5300 4 8 6 0.24 18.59 90
B-U-5 4600 4 8 6 0.24 15.0 90
Note - the prestressing wires were p1aced and grouted in a duct of diameter
1 3/8 in.
- 100 -
280T-------~------~------~----~------~------~
240 ~------+_------~------+_----__4
200 r-------+_------+-------+----~~~
160 ~------~----~----
! 1 l ' 1 ! 120 +--------~--------+------+------~-----.-- .. ----
Area = 0.06 sq. in.
80 +----------+------------, f = 236 ksi su
E 6 = 28 x 10 psi
40 r----.~----------------~--------------,_------~
o ~----------------------------------------------~ 0.2 0.4 0.6 0.8 1.0 1.2
STRAIN (percent)
FIG. 32 STRESS-STRAIN REIATIONSHIP FOR 0.276 IN. WIRE USED BY KAR (45, 46)
= 4 Vii' D
- 101 -
where n denotes the number of wires or strands in the cable, and D denotes the
actual diameter of the individual wire or of a strand. In Eq. 21 this value
Di can then be treated as though it were the diameter of the relevant indivi
dual bar.
Sections, at 0.2 ft. intervals, were investigated between the sec-
tion at 0.7 ft. fram the support and midspan. The general shape of the capa-
city distribution was as shawn in Fig. 31 for beam P2-Cl. Again the location
of the critical section deviated fram the section at 0.2L by only a very small
amount. The L/d-ratios for these beams were 12.9 and 15.0, as compared to
Il.7 for the McGill Beams. Either the location of the critical section is in-
dependent of the L/d-ratio as reported by some investigators (17, 45), or this
variation in the L/d-ratio fram Il.7 to 15.0 was not large enough to signifi-
cantly influence the location of the critical section. A few hypothetical
beams possessing the same cross-sectional properties but with varying L/d-
ratio were analyzed to determine the influence of this ratio on the location
of the critical section. The variation in the theoretical position of the cri-
tical section was found to be not too sensitive to changes in the L/d-ratio.
However, the trend observed was that the relative position of the critical
lection, expresled as a fraction of the span, occurred further from the sup-
port al the L/d-ratio decreased.
Table 5 lists the measured and esttmated ulttmate capacities for the
beams tested by Kar. Again the esttmated ulttmate capacities are those cor-
relponding to the moment at the critical section at 0.2L fram the support. In
beam B-U-2 grouting of tbe cable duct was said to be poor and this possibly
explains tbe premature failure of tbis beam. Excluding beam B-U-2, the average
ratio of the measured to tbe computed strengtbs is 1.07 with a standard devia-
- 102 -
TABLE 5
- TESTS BY KAR (45, 46) -
MEASURED AND COMPurED ULTIMATE CAPAClTIES
IN TERMS OF TOTAL LOAD ON BEAM
Mark Meas. Camp. Meas. Meas. Kips Kips Comp. Comp.(Kar)
A-U-2 20.4 17.3 1.18 0.88
A-U-3 28.9 25.2 1.15 1.07
A-U-4 26.9 28.1 0.96 0.93
A-U-5 21.9 23.4 0.94 0.95
A-U-6 26.1 23.6 1.10 1.04
A-U-8 24.1 20.0 1.20 1.05
A-U-ll 28.1 27.8 1.01 1.00
B-U-1 18.1 15.3 1.18 1.07
B-U-2 14.5 17 .1 0.85 0.83
B-U-4 17 .8 17 .5 1.02 1.01
B-U-5 14.1 14.7 0.96 0.89
Exc1uding B-U-2: Average 1.07 0.99 Standard Deviation 0.103 0.072 Coef. of Variation 9.61. 7.31.
- 103 -
tion of 0.103. Satisfactory agreement between the measured and the estimated
capacities is, therefore, obtained.
Also listed in Table 5 are the ratios of measured to estimated ulti
mate capacities as obtained by the method presented by Kar. At first sight it
would seem that better agreement was obtained by Kar's method than by the
method proposed here. However, it should be noted that his estimates are
slightly on the opttmistic side. In addition, his method is of a rather empi
rical nature and was based on the results of a number of tests, including
those reported here.
6.5 Influence of Loading Arrangement on Load-Carrying Capacity
In order to bring out the differences, if any, in flexure-shear load
carrying capacities of beams under uniform load and beams subjected to isola
ted point loads, it would be ideal to test stmilar beams under both loading
arrangements. Wilby and Nazir (43) have done this but unfortunately they have
given very little direct data as to the ultimate capacities of their beams.
As mentioned in Chapter 2, it ia not clear how they arrived at the conclusion
that beams loaded uniformly have a higher ultimate shear moment than beams
10aded with isolated point loads. The line of reasoning uaed by Bernaert and
Siess (17) to arrive at this same conclusion was shawn in Chapter 2 to be
incorrect.
Before one ia able to compare the ultimate load capacities, it must
be decided at what points along the span to place the point loads. Kani (69)
has suggested tbat a uniformly dlstrlbuted load compares best with a point
loading arrangement of ~ point loads at tbe quarter points. In tbis case,
for tbe s&me total load, tbe max~ moment, the reactlons, and tberefore, the
maxtmum sbear forces are identlcal. Hence, lt waa declded to place point loads
at the quarter points of tbe MeGlil and Kar beams and analyze these beams by
- 104 -
the method described in Chapter 5. This method was found to give very good
agreement with beams actually tested under isolated point loads. lt can be
noted that for the McGill Beams the M/Vcd-ratio, for a uniform load, at 0.2L
from the support is 3.11 whereas for the same beams with point loads at the
quarter points the value of this ratio is 2.92. For the beams tested by Kar,
the corresponding values of the M/Vcd-ratio are 3.43 and 3.23, respectively,
for series A and 4.00 and 3.76, respectively, for series B. Thus, it is seen
that the positioning of the point loads at the quarter points corresponds
quite weIl with the critical section at 0.2L fram the support for a beam under
uniform load, as far as the effective moment-shear ratio is concerned.
Listed in Tables 6 and 7 are the measured ultimate capacities for
the McGill and Kar beams under uniform load. Also listed are the estimated
ultimate capacities for these same beams with point loads at the quarter
points. lt can be seen that the differences in ultimate capacities are quite
substantial. The average value of the ratio of ultimate capacity under uni
form load to that under point load is 1.51 for the McGill beams and 1.46 for
the beams reported by Kar.
This increase in the ultimate load-carrying capacity is believed to
be due ma1nly to the following reason. In the case of a beam subjected to
isolated concentrated loads, the critical section for failure is at or near
the load points where both maxtmum shear and maxtmum bending moment take
place. But in the case of a beam under uniform load the maxtmum shear and
maxtmum moment do not take place at the same section. The location of the
critical diagonal tension crack occurs at some 10termediate point between the
support and midspan and not at the location of maxtmum shear, i.e. at the
support, as calculated 10 the usual manner. This is due to the fact that the
actual highest principal tensile stress does not occur at or near the support
- 105 -
TABLE 6
- McGILL TESTS -
COMPARISON OF BEAM ULXIMATE CAPACITIES
UNDER POINr AND UNIFORM LOADING
Meas. Camp. Uniform Mark (Uniform) (Point) Point Kips Kips
Pl 41.0 26.0 1.58
PlA 40.0 26.0 1.54
P2 44.2 29.4 1.51
P2A 44.0 29.7 1.48
P3 48.0 32.8 1.46
P3A 49.2 32.4 1.52
P2-C1 38.8 26.5 1.46
P2-ClA 39.8 27.4 1.46
P2-C3 46.4 30.6 1.52
P3-C3 53.5 34.7 1.54
Average 1.51
- 106 -
TABLE 7
- TESTS BY KAR (45, 46) -
COMPARISON OF BEAM ULTIMATE CAPACrrIES
UNDER POINr AND UNIFORM WADING
Meas. Camp. Uniform Mark (Uniform) (Point) Point K~ps Kips
A-U-2 20.4 13.6 1.50
A-U-3 28.9 17 .9 1.61
A-U-4 26.9 19.5 1.38
A-U-5 21.9 15.8 1.39
A-U-6 26.1 16.7 1.56
A-U-8 24.1 14.9 1.62
A-U-ll 28.1 19.2 1.46
B-U-1 18.1 12.2 1.48
8-U-2 14.5 13.0 1.12
8-U-4 17.8 13.3 1.34
B-U-5 14.1 11.7 1.21
Exc1uding B-U-2, Average 1.46
- 107 -
but at a section where the combined effects, on the principal tensile stress,
of the bending moment and the shear force are greatest.
From the above it follows that the critical section in a simply
supported beam under uniformly distributed load is subjected to less severe
conditions for the Bame maximum shear at the supports as compared to the cri
tical section in a simply supported beam under isolated concentrated loads.
Thus we can expect that the beams under uniform loading would show a higher
shear strength if, for example, the largest shear force is adopted as the
criterion.
It can be noted that, in general, there was no difference or very
little in the u1timate shear moments at the section located at 0.2L from the
support for beams under uniform load and at the point load for beams loaded at
the quarter points.
6.6 Comparison with ACI 318-63(5)
According to the ACI 318-63 Code, the shear at diagonal cracking in
prestressed concrete beams is taken as the lesser of the loads that cause fle
xure-shear or web-shear cracking. These terms have been defined in Chapters 2
and 4. In this study the diagonal cracking load has been defined as the load
at which a diagonal crack traverses the web of the beam and reaches the upper
f1ange-web junction. The diagonal cracking loads for the beams tested here
have been listed in Table 8 along with the esttmate of this load by the ACI
code. The diagonal cracking load for beams under uniform load is always
greater than the estimate by the code, the equations of which were based on
tests of beams under isolated point loads. The average ratio of measured to
esttmated is 1.30. exhibiting the influence of the type of loading.
Tbis increase in tbe diagonal cracking load can be expected because
of tbe re.son given in the previous section. However. due to the different
- 108 -
TABLE 8
- McGILL TESTS -
COMPARISON WITH ACI 318-63 (5)
Meas. Camp. Meas. (Diagonal Mark Cracking) (ACI 318-63) Camp.
Kips Kips
Pl 33.3 25.8 1.29
PlA 30.0 25.8 1.16
P2 42.0 29.4 1.43
PU 38.0 29.4 1.30
P3 42.8 33.6 1.28
P3A 43.0 33.6 1.28
P2-Cl 34.4 27.0 1.27
P2-ClA 35.6 27.0 1.32
P2-C3 42.0 30.4 1.38
P3-C3 45.9 35.4 1.29
Average 1.30
- 109 -
definitions given for the diagonal cracking load by various investigators, the
actual increase in the diagonal cracking load for beams under uniform load
cannot be definitely stated. The definition for the diagonal cracking load
adopted at the University of Illinois (36), whose work formed the basis of the
ACI shear equations, was that load at which the diagonal crack starts to af
fect the behaviour of the beam, i.e., when it severely distorts the strain dis
tribution over the depth of the beam, and/or when it triggers a chain of local
failures which leads to total or partial loss of beam action. One must admit
that this definition does leave some room for interpretation.
6.7 Summary
In order to check the validity of the modifications proposed in the
previous chapter, the estimated ultimate load-carrying capacities of the beams
tested here and of others found in the literature were compared with the actual
failure loads. Satisfactory agreement was obtained. The theoretical location
of the failure or critical section was found to vary very little from the sect
ion at 0.2L from the support. Bence, in lieu of analyzing numerous sections,
an accurate estimation of the failure load can be obtained by analyzing only
this section. In comparison to point-loaded beams failing in flexure-shear,
s~ilar beams under uniform load were found to possess substantially larger ul
timate load-carrying capacities. Placing the point loads at the quarter
points, the ratio of the ultimate capacities was of the order of 1.50. In com
paring the diagonal cracking loads for the beams tested here with the est~te
of that load by the ACI 318-63, it was found that the code gave consistently
conservative reBults. The ratio of the diagonal cracking loads was of the or
der of 1.30. Due to lack of expertmental data, prestressed concrete beams with
web or drapped reinforcements were Dot analyzed.
- 110 -
7 • CONCLUSIONS
7 • 1 Overall Summary
The object of this investigation was to study the strength and beha
viour characteristics of simp1y supported pre-tensioned prestressed concrete
I-beams subjected to a uniform1y distributed 10ad. The uniform 10ad was ap
p1ied through two water-fi11ed hoses p1aced direct1y on the top of the test
beam. This method proved to be very successfu1. The par~ters investigated
were the 1eve1 of prestress and the concrete compressive strength. The
effective prestress 1eve1 ranged from 71 ksi to 127 ksi, while the concrete
strength ranged from 4640 psi to 7600 psi. Duplication of the specimens
demonstrated that the reproducibility was excellent. Ten spec~ns in aIl
were tested. In or der to esttmate the ulttmate load-carrying capacities of
the test specimens, a previously developed flexure-shear strength theory was
modified to make it applicable for the ana1ysis of beams under uniform load.
Tests of another investigator were also analyzed in order to check the genera1
validity of the proposed modifications. The ulttmate load-carrying capacities
of similar beams under uniform and point loadings were compared. In addition,
the McGi11 tests were analyzed by the shear clauses of the ACI 318-63 Code for
prestressed concrete.
Major conclusions drawn from this investigation can be summarized as
follows:
1) Re1atively more flexure-sbear type diagonal cracks fo~ in a
beam under uniform 10ad than one subjected to isolated point
10ads. Whereas only one diagonal crack usua11y forms in tbe sbear span for
beama under point loads, tbree and sometimes four diagonal cracks were ob
served on eitber side of m1dspan for beams under uniform load.
- 111 -
2) The diagonal crack causing failure was generally the one closest
to the support. The location of this critical diagonal crack
could not be conclusively correlated to the parameters studied. This crack
crossed the steel level at 0.2L, on the average, from the support, the varia
tion being from 0.18 to 0.24L. It crossed middepth at 0.27L, with the loca
tion ranging from 0.25 to 0.29L. Finally, it cut through the top surface of
the beam at approxfmately 0.44L, the range being fram 0.42 to 0.47L.
3) Cracks assume a flatter slope in the compression zone for beams
under uniform load than for beams under isolated point loads.
This would increase the number of stirrups that are effective in resisting
shear, as more stirrups would be traversed.
4) In comparison to ordinary reinforced concrete beams under uni-
form load, for the case of prestressed concrete beams tested un
der s~ilar loading the diagonal crack causing failure is located closer to
midspan. For prestressed concrete beams tested here and elsewhere, the criti
cal diagonal crack crossed middepth in the vicinity of 0.27 to 0.3L from the
support. For reinforced concrete, this crack crossed middepth in the neigh
borhood of 0.15L.
5) In comparison to beams under point loads, the "peaking" or the
concentration of longitudinal strains alang the top surface of
the beam above the apex of the diagonal crack spreads over a larger portion of
that surface for beams under uniform load.
6) Modifications of a previously developed flexure-shear strength
theory vere checked for general validity against the tests per
formed here and those of another investigator. Satisfactory agreement was ob
tained in compartng esttmated and measured ulttmate loads.
- 112 -
7) The theoretical location of the failure or critical section was
found to vary very little from the section at 0.2L from the
support. Renee, in lieu of analyzing numerous sections, a rather accurate es
t~tion of the ult~te load can be obtained by analyzing only this section.
8) In comparison to point-Ioaded beams failing in flexure-shear,
beams similar to those tested here and by another investigator
under uniform load were found to possess substantially larger ult~te load
carrying capacities. Placing the point loads at the quarter points, the ratio
of the total ult~te load for a uniformly loaded beam to a similar but point
loaded beam was of the order of 1.50.
9) In comparing the diagonal cracking loads for the beams tested
here with the estimate of that load by the ACI 318-63 Code, it
was found that the code gave consistently conservative results. The ratio of
the measured diagonal cracking load to the diagonal cracking load given by the
code was of the order of 1.30.
7.2 Future Work
On the basis of the present investigation the following suggestions
for future work in this field can be recommended.
1) Due to time and space limitations, only two parameters were in
vestigated. tt is recommended that other parameters be studied,
especial1y the effect of different amounts of web reinforcement and span!
depth ratio.
2) Only prestressed concrete beams were analyzed. It is suggested
that reinforced concrete beams be analyzed to vertEy if the mo
difications made bere are equally valid for this type of beam, and, if not,
wbat cbanges are required.
- 113 -
3) A1though a very small number of reinforced concrete continuous
beams have been tested under uniform load, there is a complete
lack of information on prestressed continuous beams under uniform load. lt ia
recommended that these tests be carried out.
8. REFERENCES
The following abbreviations were used:
ACI - American Concrete Institute
ASCE - American Society of Civil Engineers
PCI - Prestressed Concrete Institute
1. Bresler, Band MacGregor, J.G.
2. Leonhardt, F. and Walther, R.
3. Sheikh, M.A.
4. Sheikh, M.A., de Paiva, H.A.R. and Neville, A.M.
S. ACI, Committee 318.
6. Hognestad, E.
7. Comité Européen du Béton.
8. Report of ACIASCE Committee 326.
9. Faber, O.
Review of Concrete Beams Failing in Shear.
Contributions to the Treatment of the Problems of Shear in Reinforced Concrete Construction.
Flexure-Shear Strength of Pretensioned Prestressed Concrete Beams.
Calculation of Flexure-Shear Strength of Prestressed Concrete Beams.
Builèing Code Requirements for Reinforced Concrete.
What Do We Know about Diagonal Tension and Web Reinforcement in Concrete?
Recommendations for an International Code of Practice for Reinforced Concrete.
Shear and Diagonal Tension.
Researches on Reinforced Concrete Beams, With New Formulae for Resistance to Shear.
- 114 -
ASCE Proceedings, Vol. 93, No. STl, Feb. 1967, pp 343-371.
Cement and Concrete Association (London), Translation No. Ill, 1962.
Ph.D. Thesis, Dept. of Civil Engineering, University of Calgary, May 1967.
PCI Journal, Vol. 13, No. l, Feb. 1968, pp. 68-85.
June 1963.
Bulletin No. 64, University of Illinois Engineering Experiment Station, 1952.
Translation published by ACI.
ACI Journal, Proceedings, Vol. 59, Jan., Feb., Mar. 1962.
Concrete and Constructional Engineering (London), Vol. Il, Nos. 5-11, May-Nov. 1916.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
Richart, F.E.
Moretto, O.
Clark, A.P.
Ferguson, P.M. and Thompson, J.N.
Zwoyer, E.M. and Siess, C.P.
Moody, K.G., Viest, I.M., Elstner, R.C. and Hognestad, E.
Laupa, A., Siess, C.P. and Newmark, N.M.
Bernaert, S. and Siess, C.P.
Ferguson, P.M.
Harrow, J. and Viest, I.M.
Whitney, C. S.
An Investigation of Web Stresses in Reinforced Concrete Beams.
An Investigation of the Strength of Welded Stirrups in Reinforced Concrete Beams.
Diagonal Tension in Reinforced Concrete Beams.
Diagonal Tension in T-Beams Without Stirrups.
Ulttmate Strength in Shear of Stmply-Supported Prestressed Concrete Beams Without Web Reinforcement.
Shear Strength of Reinforced Concrete Beams - 4 Part Series.
Strength in Shear of Reinforced Concrete Beams.
Strength in Shear of Rein-forced Concrete Beams Under Uniform Load.
Seme Lmplications of Recent Diagonal Tension Tests.
Shear Strength of Reinforced Concrete Frame Members Without Web Reinforcement.
Ultimate Sbear Strengtb of Reinforced Concrete Flat Slabs, Footings, Beams and Frame Members.
- 115 -
Bulletin 166, University of Illinois Engineering Expertment Station, June 1927.
ACI Journal, Proceedings, Vol. 42, No.5, Nov. 1945, pp.14l-l62.
ACI Journal, Proceedings, Vol. 48, No.4, Oct. 1951, pp.145-l56.
ACI Journal, Proceedings, Vol.49, No.7, Mar.1953, pp.665-675.
ACI Journal, Proceedings, Vol. 51, No.2, Oct. 1954, pp.18l-200.
ACI Journal, Proceedings, Vol.5l, Dec. 1954 to Mar 1955.
Bulletin No. 428, University of Illinois Engineering Experiment Stat ion, 1955.
Civil Engineering Studies, Structural Research Series No.120, University of Illinois, 1956.
ACI Journal, Proceed-inge, Vol. 53, No.2, Aug.1956, pp.157-172.
ACI Journal, Proceed-ing., Vol. 53, No.9, Mar.1957, pp.833-869.
ACI Journal, Proceed-ings, Vol. 54, No.4, Oct.1957, pp.265-298.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
Bresler, B. and Pister, K.S.
Guraln ik, S.
Walther, R.
Walther, R.
Brock, G.
Diaz De Cossio, R. and Siess, C.P.
Krefeld, W.J. and Thurston, C.W.
Watstein, D. and Mathey, R.G.
Mae, J.
Kani, G. N.J.
31. Sheikh, M.A.
32. Lin, T.Y.
Strength of Concrete Under Combined Stresses.
Shear Strength of Reinforced Concrete Beams.
The Ultimate Strength of Prestressed and Conventionally Reinforced Concrete Under the Combined Action of Moment and Shear.
Calculation of the Shear Strength of Reinforced and Prestressed Concrete Beams by the Shear Failure Theory.
Effect of Shear on Ultimate Strength of Rectangular Beams with Tensile Reinforcement.
Behaviour and Strength in Shear of Beams and Frames Without Web Reinforcement.
Studies of the Shear and Diagonal Tension Strength of Simply Supported Reinforced Concrete Beams.
Strains in Beams Raving Diagonal Cracks.
Discussion of Report by ACIASCE Committee 326.
The Riddle of Shear Failure and its Solution.
- 116 -
ACI Journal, Proceedings, Vol.30, No.3, Sept.1958, pp.32l-345.
ASCE Proceedings, Vol. 85, No.STl, Jan.1959, pp.1-42.
Report 223.17, Lehigh University, Fritz Engineering Laboratory, Oct. 1957.
Cement and Concrete Association (London), Translation No. 110, 1964.
ACI Journal, Proceedings, Vol.56, No.7, Jan.1960, pp.6l9-637.
ACI Journal, Proceedings, Vol. 56, No.8, Feb.1960, pp.697-735.
Columbia University, New York, June 1962.
ACI Journal, Proceedings, Vol.55, No.6, Dec.1958, pp.7l7-728.
ACI Journal, Proceedings, Vol. 59, No.9, Sept. 1962, pp.1335-1339.
ACI Journal, Proceedings, Vol.6l, No.4, April 1964, pp.44l-467.
Shearing Strength of Reinforced M.Sc. Dissertation, Concrete Beams with Web Rein- University of Leeds, forcement. 1965.
Design of Prestressed Concrete Structures.
Second Edition, John Wi ley and Sons, New York, 1963.
33. Abeles, P.W.
34. Hicks, A.B.
35.
36.
37.
38.
39.
40.
41.
42.
Evans, R.H. and Hosny, A.H.H.
Sozen, M.A., Zwoyer, E.M. and Siess, C.P.
Hernandez, G.
MacGregor, J. G. , Sozen, M.A. and Siess, C.P.
Olesen, S.O., Sozen, M.A. and Siess, C.P.
Sozen, M.A. and Hawkins, N.M.
ACI COIIIIlittee 318.
Setbumarayanan, R.
An Introduction to Prestressed Concrete.
The Influence of Shear Span and Concrete Strength Upon the Shear Resistance of a Pretensioned Prestressed Concrete Besm.
The Shear Strength of PostTensioned Prestressed Concrete Beams.
Strength in Shear of Beams Without Web Reinforcement.
Strength of Prestressed Concrete Besms With Web Reinforcement.
Strength and Behaviour of Prestressed Concrete Beams With Web Reinforcement.
Strength in Shear of Besms With Web Reinforcement.
Discussion of Report by ACIASCE Committee 326.
Commentary on Building Code Requirements for Reinforced Concrete.
UlttmAte Strengtb of Pretensioned 1 Beams in Combined Bending and Shear.
- 117 -
Vol.l, Concrete Publications, London, 1964.
Magazine of Concrete Research (London), Vol. 10, No.30, Nov.1958, pp.115-l22.
Proceedings, Third Congress of the Fédération Internationale de la Précontrainte, Berlin, 1958, pp.112-132.
Bulletin No.452, University of Illinois Engineering Experiment Station, 1959.
Civil Engineering Studies, Structural Research Series No.153, University of Illinois, 1958.
Civil Engineering Studies, Structural Research Series No. 201, University of Illinois, 1960.
Bulletin No. 493, University of Illinois Engineering Experiment Station, 1967.
ACI Journal, Proceedings, Vol. 59, No.9, Sept.1962, pp.1341-1347.
Special Publication No. 10, 1965, pp.81-83.
Magazine of Concrete Research (London), Vol. 12, No.35, July 1960, pp.83-90.
43. Wilby, C.B. and Nazir, C.P.
44. Arthur, P.D.
45. Kar, J.N.
46. Kar, J.N.
47. Placas, A.
4B. Neville, A.M. and Taub, J.
49. Lorentsen, M.
50. SWamy, R.N., Andriopoulos, A. and Adepegba, D.
51. Taub, J. and Neville, A.M.
52. Walther, R.
Shear Strength of Uniformly Loaded Prestressed Concrete Beams.
The Shear Strength of Pretensioned I Beams With Unreinforced Webs.
Diagonal Cracking in Prestressed Concrete Beams.
Shear Strength of Prestressed Concrete Beams Without Web Reinforcement.
Shear Strength of Reinforced Concrete Beams.
Resistance to Shear of Rein-forced Concrete Beams - 5 Part Series.
Theory for the Combined Action of Bending Moment and Shear in Reinforced and Prestressed Concrete Beams.
Arch Action and Bond in Con-crete Shear Failures.
Shear Strength of Reinforced Concrete Beams Loaded Through Framed-in Cross Beams.
Critical Appraisal of MDmentShear Ratio.
- 118 -
Civil Engineering and Public Works Review, Vol. 59, No.693, April 1964, pp.457-46l.
Magazine of Concrete Research (London), Vol. 17, No.53, Dec.1965, pp.199-210.
ASCE Proceedings, Vol. 94, No.SII, Jan.196B, pp.83-l09.
Magazine of Concrete Research (London), Vol. 21, No.68, Sept. 1969, pp.159-170.
Ph.D. Thesis, Imperial College of Science and Technology, London, 1969.
ACI Journal, Proceedings, Vol.57, Aug-Dec. 1960.
ACI Journal, Proceedding, Vo1.62, No.4, April 1965, pp.403-420.
ASCE Proceedings, Vol. 96, No.SI6, June 1970, pp. 1069 -1091.
International Association for Bridge and Structural Engineering, Sixth Congress, Stockholm, 1960, Preliminary Publication, pp.77-84.
International Association for Bridge and Structural Engineering, Seventh Congress, Rio de Janeiro, 1964, Preltminary Publication, pp.76l-776.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
Richart, F. E. , Brandtzaeg, A.
Bresler, B. and Pister, K.S.
Cowan, H.J.
Bay, H.
Hawkins, N. M. , Sozen, M.A. and Siess, C.P.
Bruce, R.N.
Warwaruk, J., Sozen, M.A. and Siess, C.P.
Guyon, Y.
British Standard Institution.
Jones, R.
A Study of the Failure of Concrete Under Combined Compressive Stresses.
Failure of Plain Concrete Under Combined Stresses.
The Strength of Plain, Reinforced and Prestressed Concrete Under the Action of Combined Stresses, with Particular Reference to the Combined Bending and Torsion of Rectangular Sections.
Beigung und Querkraft betm Verbundquerschnitt.
Behaviour of Continuous Prestressed Concrete Beams.
An Experimental Study of the Action of Web Reinforcement in Prestressed Concrete Beams.
Strength and Behaviour in Flexure of Prestressed Concrete Beams.
The Strength of Statically Indeterminate Prestressed Concrete Structures.
The Structural Use of Prestressed Concrete in Buildings.
The Ultimate Strength of Reinforced Concrete Beams in Shear.
- 119 -
Bulletin No. 185, University of Illinois Engineering Expertment Station, 1928.
ASCE Transactions, Vol. 122, 1957, pp.1049-l059.
Magazine of Concrete Research (London), Vol.5, No.14, Dec.1953, pp.75-86.
Beton-und Stahlbetonbau (Berlin-Wilmersdorf), Vol.57, No.4, April 1962, pp.79-85.
International Symposium on Flexural Mechanics of Reinforced Concrete, ACI Special Publication No. 12, 1965, pp.259-294.
Ph.D. Thesis, University of Illinois, 1962.
Bulletin No. 464, University of Illinois Engineering Experiment Station, 1962.
Proceedings, Symposium on the Strength of Concrete Structures, Cement and Concrete Association (London), 1958, pp.305-376.
British Standard Code of Practice CP 115: 1959.
Magazine of Concrete Research (London), Vol. 8, No.23, Aug.1956, pp.69-84.
63. Acharya, D.N. and Kemp, K.O.
64. Armishaw, J.W., Bunni, N.G. and Neville, A.M.
65. Leonhardt, F.
66. Rang an , B.V.
67. PCI
68. Leonhardt, F.
69. Kani, G.N.J.
Significance of Dowe1 Forces on the Shear Fai1ure of Rectangu1ar Reinforced Concrete Beams Without Web Reinforcement.
Distribution of Shear in Rectangu1ar Besms (in two parts).
Reducing the Shear Reinforcement in Reinforced Concrete Beams and Slabs.
Shear Strength of Reinforced Concrete Besms With Uniform1y Distributed Loads.
Fundamenta1s of Prestressed Concrete Design.
Prestressed Concrete Design and Construction.
Basic Facts Concerning Shear Fai1ure.
- 120 -
ACI Journal, Proceedings, Vo1.62, No.10, Oct.1965, pp.1265-1279.
Concrete and Constructiona1 Engineering (London), Vo1.41, No.4, April 1966, pp.119-130; No.5, May 1966, pp.157-161,183.
Magazine of Concrete Research (London), Vol. 17, No.53, Dec.1965, pp.187-198.
The Indian Concrete Journal, Vo1.43, No.1, Jan.1969, pp.17-24.
Second Edition, PCI, 1964.
Second Edition, Ernst and Son, Berlin, 1964.
ACI Journal, Proceedings, Vo1.63, No.6, June 1966, pp.675-691.
- 121 -
APPENDIX A: MATERIALS, FABRICATION AND INSTRUMENl'ATION
A.l Materials
(a) Concrete
Type III high early strength Portland Cement manufactured by the
Canada Cement Co. was used for aIl test specimens. The coarse aggregate, con
sisting of crushed stone, had a maximum size of % in. The fine aggregate,
100% passing ASTM sieve no. 8, had a fineness modulus of 3.26.
The concrete mixes were designed by the trial-batch method. Three
batches were used in each beam with each batch making up approximately one
third of the beam depth. Table A.l lists the proportions of the concrete bat
ches used in each beam along with the average s lump , average compressive
strength, average tensile strength as determined by the modulus of rupture and
splitting strengths and age at the time of test. Proportions are in terms of
oven-dry weights. An open pan type mixer, with a capacity of 3\ cu.ft. was
used.
The compressive strength of the concrete was determined from tests
on standard 6 in. by 12 in. control cylinders. The cylinders were capped with
a thin layer of high-strength plaster of paris. Three cylinders were made
from each batch for a total of nine.
The tensile strength in splitting was determined from 3 in. by 6 in.
cylinders using 1/8 in. thick by tin. wide fibre board strips. The modulus
of rupture strength was obtained fram tests on 6 in. by 6 in. by 24 in. beams
tested on an 18 in. span using one-third-point loading.
(b) Prestressing Reinforcement
Stress-relieved seven vire high tensile strength strand of 3/8 in.
diameter manufactured by the Steel Co. of Canada was used for aIl the speci-
- 122 -
TABLE A.1
" PROPElUms OF CONCRETE MIXES
Mark fI f fI Cement: Sand: Water/ Slump Age at c sp r Grave 1 Cement test
psi psi psi in. days
Pl 6650 600 630 1: 1.69: 1.42 0.44 1. 75 8
PlA 6750 690 560 " " 1.5 8
P2 6100 580 520 " " 3 11
P2A 6600 515 590 " " 1.5 8
P3 6200 650 600 " " 1. 75 8
P3A 6160 680 620 " " 3 8
P2-Cl 4640 520 450 1: 2.58: 1.91 0.58 1.5 8
P2-ClA 4890 590 500 " " 2 8
P2-C3 7400 600 650 1: 1.24: 1.21 0.34 1 18
P3-C3 7600 600 670 " " 0.75 9
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mens. The stress-strain curve of the strand, along with relevent properties,
is shawn in Fig. 2. This curve is the average of results obtained from nu
merous samples of the strand cut from different portions of the coil. Strains
were measured with electrical resistance strain gauges and with a 24 in. ex
tensometer, stmilar results being given by both.
Prior to the fabrication of the beam, the cut lengths of strand
were wipeJ with a piece of cloth dipped in a weak solution of hydrochloric
acid and then rusted by storing in a moist room for approximately two weeks.
On their removal from the moist room, the rusted strands were cleaned with a
wire brush to remove aIl loose rust. In this manner a slightly pitted surface
which tmproved the bond characteristics was produced.
A.2 Beam Forms
The beam forma were composed of a bottom form, two side forms,
wood en filler sections and two end pieces. The bottom form was a 15 in. by
3.52 in. by 21 ft. channel (note: aIl channels are American Standard Channels)
with holes of ; in. diameter at 12 in. centres located at a distance of 1\ in.
from both flange edges to accommodate the fastening devices for the side forma.
Each of the two side forms was a 12 in. by 2.94 in. by 11 ft. channel with
holes in the upper flange of 3/4 in. diameter at approximately 2 ft. centres
to accommodate the external vibrator. Wooden filler sections (8 in. by 1.5
in. by 8 ft. with a 26.50 bevel at the edges) were centred and permanently
bolted to tbe back of the side forma. In addition, wood en filler blocks were
emp10yed to provide a step at the top of the beam to terminate the loaded area.
The ~o end pieces were 6 in. by 1.92 in. by 12 in. channe1s. AlI sections
and wooden pieces were given three coats of marine fibreglass paint. AlI sec
tions were assembled using bo1ts.
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A.3 Prestressing Eguipment
A11 strands were tensioned in a rectangu1ar prestressing bed 23 ft.
6 in. by 3 ft. 3 in. The frame of the bed consisted of two longitudinal pipes
and two reaction b10cks bo1ted together for ease of dismant1ing. The two
pipes were 6 in. in diameter and 21 ft. 9 in. long with 8 in. by 9 in. by ~ in.
plates a1igned and we1ded perpendicu1ar to the ends. The reaction b10cks were
fabricated fram two 10 ~n. by 2.89 in. by 3 ft. 3 in. channe1s set back to
back 2 in. apart and he1d in place at the ends by four ~ in. we1ded plates.
A reaction frame, as shown in Fig. A.1, was required at the end of
the beam to separate the prestressing strands and a110w for the placement of
adjusting bo1ts, dynamometers, grips and prestressing ram. The frame was con
structed fram one 9 in. by 3 in. by 1 in. plate and two 10 in. by 7 in. by
1~ in. plates, the three plates being interconnected by eight 1 1/8 in. diame
ter rods. An additiona1 plate of size 7 in. by 6 in. by 1t in. was p1aced be
hind the prestressing ram. Four 7/16 in. diameter ho1es were dri11ed through
each plate at the appropriate locations. The plates and rods were assemb1ed
using bo1ts. At the other end of the prestressing bed a reaction frame was
not required since the strands were a110wed to f1are out horizonta11y fram the
end piece of the beam form to the reaction b10ck. A 5 in. square plate 1 in.
thick wac used. Plates were a1so positioned tmmediate1y outside of the beam
end pieces to insure that the tensioned strands wou1d not cut into the end
pieces due to their being f1ared out.
Supreme Reusab1e grips of 3/8 in. size were used. To tension the
strands a 30 ton Hein-Werner hollow hydrau1ic ram operated by a Black~k pump
was emp1oyed. A photograph of the entire setup can be seen in Fig.-A.2.
- 125 -
FIG. A.1 REACTION FRAME
FIG. A.2 PRESTRESSING AND CASTING BEO
- 125 -
. ',':. i :' : ~.:" ( . . i \': :: .. '.. . .. ,', ..
-~. :- 121;
A.4 Preparation Prior to Tensioning
Upon their remova1 from the moist room, the strands were a110wed to
dry for one day and then c1eaned with a wire brush to elfminate any loose rust.
The strands were then fed through the reaction blocks, form end pieces and re-
action frame which included the adjusting nuts and bolts (1 in. diameter,
type A325),dynamometers, grips and prestressing rame In addition, a ~ in.
thick plate was placed on either side of each dynamometer to insure that its
entire cross-section would be loaded, i.e. to duplicate the condition under
which the dynamometer was calibrated. The adjusting nut and boIt was employed
on each strand to make fine adjustments in the final tension in each strand.
This was found to be necessary since the initial sag in each strand differed
and hence for the same elongation the final prestress would be dl:ferent.
Electrical resistance strain gauges were installed and waterproofed
on one wire of each strand at the midspan of the beam. The zero strain read-
ings for aIl strain gauges and prestressing dynamometers were recorded and a
small amount of tension was induced into the strands through the prestressing
ram to remove any noticeable sag.
AlI beam forms were cleaned and coated with a layer of light form
grease. Prior to the positioning and assembly of the end pieces and side
forms to each other and to the base channel, the strands were cleaned with
acetone to remove any dirt and grease that might affect the quality of bond.
To prevent bulging of the forms at the centre of the beam during casting, a
cross-strap was bolted in place on top of the forms. The cross-strap was
raised about one inch above the top of the form to facilitate trowelling of
the beam. The strands could now be tensioned.
- 127 -
A.5 Tensioning Procedure
All beams were pre-tensioned the day before casting. The prestress
ing force in each strand was measured by the dynamometers and by the strain
gauges on the strands. In general these forces differed by approximately five
percent at the most. However, the procedure adopted was ta consider the force
as given by the dynamometer ta be the more accurate and ta use the strain
differences as given by the strain gauges on the strands ta compute the pre
stress lasses.
With the aid of the pressure gauge on the pump, the strands were ten
sioned simultaneously ta approximately eightly percent of the final prestress
ing force. Then with the help of the ad)usting nuts and bolts three or four
trials were required ta tension the strands ta the desired level of prestress.
The procedure for making the fine adjustments with the adjusting nuts and
bolts was as follows. The ram was pumped until the forward grips which acted
against the dynamometers and adjusting bolts became unloaded. The tension in
the strands at this point was taken by the rear grips behind the ram. The nut
was then easily advanced along the bolt if an increase in prestress was de
sired or retracted if a decrease in prestress was desired. The ram was then
released and the forward grips were required ta carry the tension in the
strands.
A.6 Casting and Curing
Before casting all strand exits in the beam form were filled with
putty. Three batches of approximately three cu. ft. each were required for
each beam and the corresponding control specimens. From each batch three 6 in.
by 12 in. standard cylinders, one 3 in. by 6 in. cylinder and one 6 in. by
6 in. by 24 in. modulus of rupture beam were cast according ta ASTM specifica
tions. The dry ingredients were mixed for three minutes before the addition
- 128 -
of water and then mixed for another five minutes. The concrete was placed in
the forme in three approximately equal horizontal layers and each layer, as it
was placed, was vibrated with a Branford external vibrator.
Two ta three hours after casting the top surface of the beam was
trowelled smooth. The beam and control specimens were then covered with poly
ethylene. Twenty-four hours after casting the side forms were removed from
the beam as were also the forms for the control specimens. The beam and con
trol specimens were then covered with continuously wet burlap and polyethylene
up ta the fourth day after casting at which time the prestress was released.
The burlap and polyethylene were then removed ta allow the surface ta dry for
twenty-four hours before the application of strain gauges on the concrete
surface.
A.7 Release of Prestress
On the fourth day after casting the prestress was released. The
method of releasing the prestress must be graduaI and uniform for aIl the
strands. A dynamic release of prestress would disrupt the bond in the end
block of the beam and hence increase the length required for transmission of
prestress to the concrete. If the prestress is not released simultaneously in
aIl the strands the elastic recovery of the prestressing frame would tend to
put increased force on the strands that had not been released.
Burning of the tendons with an oxy-acetylene torch was attempted but
found ta be rather dangerous and unsatisfactory. In view of this, the use of
two shims of size 6 in. by 2 in. by l1j in. thick, as shawn in Fig. A.3, was
employed for the releasing mechanism. The hydraulic ram was placed between
the first and second plates of the reaction frame and acted against the reac
tion black through the releasing device shawn in Fig. A.3. This releasing de
vice was composed of a 4 in. by 5 in. by 1 in. plate and four 1 1/8 in. diame-
- 129 -
-,/
FIG. A. 3 PRESTRESS - RELEAS ING MECHANISM
- 129 -
~. l C;. :\. 3 ? RE 5T RE 5 5 - RE UA 5 e.:c ~ŒCH.;~: l 5~·~
- 130 -
ter rods 4 in. long. The ram was pumped until the shtms just became free and
fell to the ground. Release of the prestress was gradually and uniformly
carried out by a slow return of the ram. In aIl beams the steel strain read
ings were taken both before and after the release of the prestress to deter
mine the instantaneous prestress losses.
A.8 Instrumentation
(a) Dynamometers
Dynamometers were used for two purposes in this study to measure the
prestressing force induced into the strands prior to casting and to determine
the load applied to the beam during the test. Both were stmilar in design and
material (Alcan aluminum 6061), varying only in external dimensions. The pre
stressing dynamometers were l~ in. in diameter and 2~ in. long. A hole of
7/16 in. diameter was bored down the centre of the cylinder to allow it to be
placed around the strand. The dynamometers used during the test were l~ in.
in diameter and 2~ in. long.
Each dynamometer was composed of the aluminum cylinder and four
strain gauges (BLH type FAP-2S-l2Sl3). The strain gauges were placed at the
middepth of the cylinder and positioned along the circumference at 90 degree
intervals. Two diametrically opposite gauges were positioned longitudinally
and the other two transversely. The gauges were interconnected in the form of
a four-arm Wheatstone bridge. This arrangement effectively cancels out any
possible strain changes due to bending, torsion or temperature variations.
The dynamometers were connected to a standard four-pole rotary switch to make
the"measurement of their outputs more convenient.
- 131 -
(b) Strain Gauges on Strand
Other than the electrical resistance strain gauges used for the dyna
mometers, aIl strain gauges employed in this study were polyester gauges manu
factured ~y the Tokyo Sokki Kenkyujo Co. Ltd. Type PS-20 were used on the
strands. The nominal gauge 1ength and width of these gauges were 20 mm. and 1
mm. respective1y, the base dimensions being 30 mm. by 2 mm. The principal ad
vantages of and reasons for using these gauges were their narrow width and
f1exibility. The gauges were applied on one wire of each strand at the midspan
of each beam in the usua1 fashion using Eastman 910 adhesive and were water
proofed using GW-S Waterproofing manufactured by Automation Industries, Inc.
This waterproofing agent formed a protective surface which was smooth and fle
xible. The lead wires were brought out of the concrete beam individua11y
through the midd1e of the top surface and hence away from the two hoses.
Three-wire lead circuits were used for a11 of the gauges placed on the strands
and concrete surface. The desensitization error caused by lead-wire resis
tance was found to be negligible. A dummy gauge for temperature compensation
was mounted on an unstressed length of strand and then cast into a 3 in. by
6 in. concrete cylinder wh en the beam was cast.
(c) Strain Gauges on Concrete
PL-20 polyester gauges were used to determine the long ~_tudina1
strain distribution along the centre 1ine of the top surface of the beam.
These gauges had a nominal gauge 1ength and width of 20 mm. and 3 mm., res
pectively. They were installed in the standard manner recommended by the manu
facturers using their P-2 adhesive. The gauges were then coated with GW-1
Waterproofing Compound manufactured by Automation Industries, Inc. to prevent
variations in strain readings due to air temperature fluctuations.
- 132 -
(d) Def1ectometers
Deflections of the concrete beams and the supporting steel beam in
the 10ading frame were measured with Starrett spring dial gauges provided with
magnetic bases, reading 0.001 in. between the smallest divisions. The load
deflection relationships for deflections measured at midspan are given in Figs.
Sand 6. These dial gauges were also used to measure any possible slip of the
strand relative to the concrete. The arrangement is shown in Fig. 4.
(e) Strain Indicators
For aIl the dynamometers and strain gauges on the strands while in
the casting bed and prior to testing, the strains were measured by means of a
BLH Model l20C Portable Digital Strain Indicator. The outputs of the strain
gauges on both steel and concrete during the testing of the beam were obtained
using a Budd Mode1 Datran l Digital Strain Indicator with a strain gauge
switch and balance unit and a Victor Digitamatic print-out device.
- 133 -
APPENDIX B: LOADING APPARATUS
Each beam support was composed of a 12 in. long by 2 in. diameter
steel roller and a 12 in. long 6WF25 with stiffened web. The roller at the
left-hand support was fixed by means of welding while the roller at the right
hand support was free to rotate and move horizontally. If both supports would
have been fixed against horizontal movement the test beam would have acted as
a two-hinged arch and its load-carrying capacity would have been greater than
that of a simply supported beam.
As previously mentioned, two water-filleJ hoses were employed to ob
tain a uniformly distributed load. The hoses used were Gacord Type 43 manu
factured by George Angus Co., England. Their composition consisted of a
smooth neoprene rubber tube of 3/32 in. thickness reinforced with high tensile
rayon fabric which was spirally wound. Added to this was a yellow natural
rubber cover of approximately 1/8 in. thickness. The internaI diameter was
l~ in. and the external, 1 3/4 in. This hose was designed for a working
pressure of 250 psi and a burst of 1200 psi. The two hoses were sealed sepa
rately at one end and interconnected at the other in order to insure equal
water pressure in both. At the end where the hoses were interconnected provi
sion was made for the installation of a Marsh water-pressure gauge with a dial
which ranged from zero to 600 psi. A strip of 2 in. by ~ in. fine gum rubber
was installed beneath each hose leaving a gap of one inch width along the cen
treline of the top surface for the application of electrical resistance strain
gauges. This was done to prevent contact between the hoses and the strain
gauges.
The hoses were prevented from moving laterally by the longitudinal
placement of four 81 18.4 sections each of length 2 ft. 2~ in. with their webs
horizontal. Plates of ~ in. thickness were welded to the webs to slightly in-
- 134 -
crease their bending stiffness. The four I-sections were separated by approxi
mately ~ in. to prevent their contact as the test beam deflected. These 1-
sections did not co-operate longitudinally in compression since the hoses had
a high degree of shear deformability in the longitudinal direction. At the
centre of each of the horizontal I-sections were placed two steel blocks of
size 3 in. by 2 in. by 6 in. in length. Above these blocks were placed two 4
ft. long beams of size 6WF25.
The loading system was completed by the addition of two loading
heads, two base plates and eight tension rods. Each loading head was fabrica
ted from two 5 in. by 2 in. by 13 in. long channels set flange tip to flange
tip, with an 8 in. by 13 in. by 1 iu. thick plate above and an 8 in. by 13 in.
by ~ in. thick plate below. Both plates had l~ in. diameter holes drilled
through them to allow the passage of the tension rods. The plates were welded
flush to the channel flanges. A 3 in. diameter half-round was welded to the
bottom plate to maintain the loading head in a vertical position when the
upper distributing would rotate to follow the test beam's deflection profile.
To further insure that the two loading heads would remain vertical and in po
sition they were interconnected by the bolting of two 2 in. by 2 in. by ~ in.
angles. Strain gauges placed on the angles indicated that very little thrust
was transmitted through them from the loading heads tmplying the lack of
application of any substantial horizontal force to the loading heads. The two
base plates were placed below each of the two hydraulic rams and were of size
8 in. by 13 in. by l~ in. Holes of l~ in. diameter were drilled at the appro
priate locations to permit the passage of the tension rods. The tension rods
were threaded for their entire length and were l in. in diameter. Each rod .
was composed of two 3 ft. long pieces coupled together.
- 135 -
Load was applied with the aid of two 30 ton Enerpac hydraulic rams
connected to a single pump through a tee and valve system in order that both
rama could be operated simultaneously. Because of the nature of the hydraulic
system, the load on each ram was the same at any one ttme, regard1ess of its
extension, except for neg1igible differences due to friction. The 10ad
applied was determined by an aluminum dynamometer placed between each ram and
the bottom of the longitudinal steel beam of the 10ading frame. These dynamo
meters are described in detai1 in Appendix A.
Steel channe1s of size 4 in. by 1.72 in. by 5 ft. were p1aced in a
vertical position on either side of the test beam and were bo1ted to the
longitudinal steel beam of the loading frame. These channels were used purely
for purposes of safety. A gap of approximately ~ in. existed between each
side of the horizontal I-sections and the vertical channe1s. During the test
the horizontal I-sections remained pretty weIl in position and did not act
against the vertical safety channe1s. A photograph of the test bed and 10ad
ing apparatus immediately prior to testing can be seen in Fig. 3.
