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SIZE EFFECT ON SHEAR STRENGTH OF REINFORCED CONCRETE BEAMS
by
Wassim M. Ghannoum
November 1998
Department of Civil Engineering ancl Applied Mechanics
McGill University
Montréal, Canada
A thesis submitted to the Faculty of Graduate Studies
and Research in partial fulfilment of the requirements
for the degree of Master of Engineering
O Wassim M. Ghannoum, 1998
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0-6 12-506 1 O-X
Size Effect on Shear Strength of Reinforced Concrete Beams
Abstract
Given the great discord concerning the mechanisms that govern shear failure, the shear
behaviour of concrete bearn elements with no transverse reinforcement is investigated. The
variables introduced in the experimental program are member depth, amount of longitudinal
steel reinforcement and concrete strength. The effects of these variables on the shear stress at
failure of the concrete are investigated. Two geometrically similar series of beams of
different concrete strengths are compared. Beam heights in each of the series range from 90
mm to 960 mm and al1 the beams have a constant a/d ratio of 2.5.
Results show a strong "size-effect" in the behaviour of concrete beam or one-way slab
elements subjected to shear, where deeper members have smaller shear stresses at failure
than shallower ones. Increasing the amount of flexural reinforcement increases the shear
stress at failure while increasing the concrete compressive strength has little or no effect on
the diagonal shear resistance of concrete.
The AC1 Code equations for shear are found to be unconsewative for large elements
while the CSA Standard simplified shear design method yields conservative predictions
within the range of bearns tested. For the beams tested. with an a/d ratio of 2S1 the
combination of the modified compression field theory and a strut-and-tie analysis provides
more accurate predictions.
To My Parents
Effet de taille sur la résistance à I'effort tranchant de poutres en béton armé
Résumé
Etant donné le grand désaccord concernant les mécanismes qui régissent les ruptures en
cisaillement dans le béton, le comportement de poutres en béton, sans armatures à l'effort
tranchant. est étudié. Expérimentdement l'étude porte sur l'influence de la profondeur des
poutres. la quantité d'armature flexionnelle et la résistance du béton, sur la résistance en
cisaillement du béton. Deux séries de poutres géométriquement identiques, aux profondeurs
variant entre 90 mm et 960 mm et possédant différentes résistances de béton, sont
comparées. Toutes les poutres testées ont un rapport d d de 2.5.
Les résultats démontrent un important "effet de taille" par lequel les éléments les plus
profonds ont proportionnellement une moindre résistance en cisaillement que les éléments
les moins profonds. L'augmentation de la quantité d'amature flexionnelle augmente la
résistance à l'effort tranchant des poutres alors que l'augmentation de la résistance en
compression du béton n'a presque aucun effet.
Les équations du code américain AC1 donnant la résistance en cisaillement du béton ne
sont pas sécuritaires dans leurs prédictions pour les poutres de grande profondeur, alors que
les expressions simplifiées du code canadien CSA sont sécuritaires dans la gamme des
valeurs des paramètres testés. Pour les poutres testées, dont le rapport a/d est de 2.5, la
combinaison de la théorie modifiée des champs de compression avec l'analyse par la
méthode des bielles tendues et comprimées donne des prédictions plus précises.
Acknowledgements
The author would like to express his deepest gratitude to Professor Denis Mitchell for
his continued encouragement and knowledgeable advise throughout this research program.
Furthemore. the author would like to thank Dr. William Cook for his guidance and support.
The research was carried out in the Jamieson Stmctures Laboratory at McGilI
University. The author wishes to thank Ron Sheppard, Marek Przykorski, John Bartczak and
Damon Kiperchuk for their assistance in the laboratory. The author wouid also like to thank
Carla Ghannoum, Stuart Bristowe, Pierre Koch, Bryce Tupper, Emmet Poon, Kevin Li,
Pedro Da Silva and Robert Zsigo for their assistance.
The cornpletion of this project would not have been possible without the patience and
valuable help of the secretaries of the Civil Engineering Department. particularly Sandy
Shewchuk-Boyd, Lil ly Nardini. Ann Bless. and Donna Sears.
The financial assistance provided by the Natural Sciences and Engineering Research
Council of Canada WSERC) is gratefully acknowledged. As well. the author would like to
acknowledge Professor M. P. Collins and Evan Bentz for providing the software program
RESPONSE 2000 used in this thesis.
Finally. the author would like to thank his friends and family especially his parents,
for their constant support and encouragement during his years at McGill.
Wassim M. Ghannourn November, 1998
... III
Table of Contents
Abstract .............................. .. ................................................................................... i
. . Résumé ................................................................................................................................~l
Table of Contents ............................................................................................................... iv
List of Figures ..................................................................................................................... vi
. . List of Tables .....................................................................................................................~II
... ................................................................................................................... List of Symbols viii
1 Introduction and Literature Review ........................................................................... 1
...................................................................................................... 1.1 introduction 1 3 .................................................... 1 -2 Previous Size Effect Investigations ..............-
................................................................... 1.3 Shear and High-Strength Concrete 6 1 -4 Sliear Design Methods ...................................................................................... 7
........................................................... 1.4.1 AC1 Shear Design Procedure 7 1 .4.2 CSA Simplified Shear Design Procedure ......................................... 8 1.4.3 Modified Compression Field Theory ................................................ 9
1.5 Objectives of Research Program ....................................................................... 12
2 Experimental Program .................................................................................................. 13
2 . I Design and Details of the Beam Specimens ........................ .. ................ 13 .................................................................................... 2.2 Specimen identification 15
................................... 2.3 Material Properties ..... 16 2.3.1 Reinforcing Steel Properties ............................................................. 16 2.3 -2 Concrete Properties ........................................................................... 17
............................................................................................. 1.4 Testing Procedure 1 9 2.4.1 Test Setup and Loading Apparatus ................................................ 19 2.4.2 Instrumentation ..................................................................... 21
..................................................................... 3 Experimental Results and Cornparisons 24
3.1 Introduction ...................................................................................................... 24 .................... 3.2 General Behaviour ... ............................................................... 25
3.3 Normal-Strength Concrete Series ..................................................................... 25 .......................................................................... 3.4 High-Strength Concrete Series 35
3.5 Summary of kesults .......................................................................................... 44 3.6 interpretation and Comparison of Results ........................................................ 46
......................................................................................................... 4 Analysis of Results 49
4.1 AC1 Code Predictions ....................................................................................... 49 4.2 CSA Sirnplified Expressions ........................ .. ................................................ SI 4.3 Predictions Using the Mod ified Compression Field Theory and
........................................................................................ Strut-and-Tie Models 54
5 Conclusions and Rccommeadations ............................................................................. 63
References ........................................................................................................................... 65
0 Appendix A-Response 2000 Input and Output .............................................................. 67
List of Figures
Chapter 1 1.1
Chapter 2 2.1 2.2 2.3 2-4 2.5 3.6 2.7 2.8 2.9
Chapter 3 3.1 3.2 3 -3 3 -4 3 -5 3.6 3.7 3 -8 3 -9 3.10 3.1 1 3.12 3.13 3-14 3.15 3.16
Chapter 1 4.1
Relative strength (ultimate moment/flexural moment) vs . a/d ratio '7 (Kani 1967) .......................................................................................................-
Influence of member depth and aggregate size on shear stress at faiture for tests carried out by Shkya 1989, taken fiom Collins and Mitchell. 1997 ........ 5
Specirnen reinforcement details ........................................................................ 14 ................................... Typical tensile stress-strain curves for reinforcing steel 16
..................................... Typicai concrete compressive stress-strain responses 18 Concrete shrinkage readings ............................................................................. 19
............................................ Test setup and clamping of the failed weaker end 20 .............................. One-point loading arrangement used for some specimens 21
LVDT and concrete strain gauge locations ...................................................... 22 Typicai steel strain gauge locations (top view) ................................................ 22 Specimen strain target locations ........................................................................ 23
Test results for specimen N90 ............................................................................. 28 Test results for specimen N 1 55 .......................................................................... 29 Test results for specirnen N220 .......................................................................... 30
..................................... Test results for specirnen N350 ............................ ... 31 Test results for specimen N485 .......................................................................... 32
......................................................................... Test results for specimen N960 33 .................................................................... Normal-strength series afier failure 34
............................................................................. Test results for specimen H90 37 Test results for specimen H 1 55 .......................................................................... 38 Test results for specimen H220 .......................................................................... 39 Test results for specimen H350 .......................................................................... 40
......................................... ......................... Test results for specirnen H485 ... 41 Test results for specimen H960 .......................................................................... 42 High-strength series afier failure ........................................................................ 43 Shear stress versus specimen depth ................................................................... 44
................................ Nonnalised shear stress at failure versus specimen depth 46
Cornparison of predictions using the AC1 simplified expression with test results ................................................................................................... 50 Cornparison of predictions using the CSA simplified expressions with test results ................................................................................................... 52 Sectional model versus strut-and-tie model predictions for Kani's tests (Kani 1967), taken from Collins and Mitchell ................................................... 54
......................................... . Predictions for the normal-strength p= 1.2% series 58 ............................................ Predictions for the normal-strength, p=2% series 58 ............................................. Predictions for the high-strength, p= 1.2% series 59
................................................ Predictions for the high-strength, p=2% series 59
List of Tables
Chapter 2 2.1 2.2
................................................................................ Reinforcing steel properties 15 Concrete mix designs ............................ .... ................................................... 17
..................................................................... 2.3 Concrete properties for both series 18
Chapter 3 3.1 Summary of results .................... .. ................................................................... 45 3.2 Shear strength difference between p=2% and p=l.2% .................................... 47 3.3 Shear strength difference between the high-strength and normal-strength
concrete series ..................................................................................................... 48
Chapter 4 4 . I Comparison of predictions using the AC1 simplified expression
with test results ................................................................................................... 51 4.2 Comparison of predictions using the CSA simplified expressions
................................................................................................... with test results 53 ................... ............*...... 4.3 Modi fied compression field theory predictions ... 60
4.4 Strut-and-tiemodelpredictions .......................................................................... 61 4.5 Combined predictions of the modified compression field theory and
strut-and-tie mode1 .............................................................................................. 62
vii
List of Symbols
shear span area of concrete area of longitudinal steel reinforcement in tension zone overall width of specimen minimum effective web width within depth d distance frorn extreme compression fibre to centroid of tension reinforcement nominal diameter of reinforcing bars distance measured perpendicular to the neutral axis between the resultants of the tensile and compressive force due to flexure longitudinal steel reinforcement modulus of elasticity specified compressive strength of concrete limiting compressive stress in concrete stmt modulus of rupture of concrete splitting tensile stress of concrete ultimate strength of reinforcement yield strength of reinforcement overall thickness of specirnens moment at section moment at failure axial load at section crack spacing in the 8 direction shear force at section sliear stress resistance provided by concrete shear resistance of concrete shear force at failure average crack width ratio of average stress in rectangular compression block to the specified concrete strength tensile stress factor which accounts for the shear resistance of cracked concrete principal tensile strain in cracked concrete concrete strain at fcr tensile strain in tensile tie reinforcement yield strain of reinforcement ultimate strain of reinforcement longitudinal strain of flexural tension chord of rnember ratio of longitudinal tension reinforcement, AJbd angle of cracks with respect to the longitudinal direction smallest angle between compressive strut and adjoining tensile ties
Chapter 1
Introduction and Literature Review
1.1 Introduction
in spite of the numerous research efforts directed at the shear capacity of concrete, there
is still great discord conceming the mechanisms that govern shear in concrete. Proposed
theories Vary radically from the simple 45" truss mode1 to the very complex non-linear
fracture mechanics. Yet nearly al1 the resulting design procedures are empirical or semi-
empirical at best and are obtained by a regression fit thrciugh experimental results.
Nowhere is this lack of understanding more evident than in the shear design provisions
of the AC1 Code (AC1 committee 3 18-1995) which consists of 43 empirical equations for
different types of members and different loading conditions. Moreover, there is great
discrepancy between design codes of different countries. Many of these codes do not even
account for some basic and proven factors affecting the shear capacity of concrete members.
Of these factors, much confusion is expressed with regards to the effect of absolute member
size on the shear capacity of beam etements. On this subject, there is a lack of consensus in
the approach to the problem due to the limited amount of experiments dedicated to this
effect, especially when it cornes to high-strength concrete elements.
The focus of this research is to evaluate the "size effect" in nonnal and high-strength
concrete bearns without web rein forcement in order to better understand the mechanisms
involved. As well, the closely related subject of "amount of longitudinal steel" is
investigated as it has been shown to greatly affect the shear behaviour of concrete beam or
one-way slab elements.
1.2 Previous Size Effect Investigations
In 1 955, the Wilkins Air Force Depot warehouse in Shelby, Ohio, collapsed due to the
shear failure of 36 in. (914 mm) deep beams which did not contain any stirrups at the
location of failure (Collins and Kuchma, 1997 and Collins and Mitchell, 1997). These beams
had a longitudinal steel ratio of only 0.45%. They failed at a shear stress of only about 0.5
MPa whereas the AC1 Building Code of the time (AC1 Committee 3 18, 195 1) permitted an
allowable working stress of 0.62 MPa for the 20 MPa concrete used in the beams.
Experirnents conducted at the Portland Cernent Association (Elstner and Hognestad, 1957)
on 12 in. (305 mm) deep model beams indicated that the beams could resist about 1.0 MPa.
However, the application of an axial tension stress of about 1.4 MPa reduced the shear
capacity by about 50%. It was thus concluded that tensile stresses caused by thermal and
shrinkage rnovements were the reason for the beam failures.
O I 2 3 4 S 6 7 8 S
Figure 1.1: Relative strength (ultimate moment/flexural moment) vs. a/d ratio (Kani 1967)
Kani (1 966 and 1967) was amongst the first to investigate the effect of absolute member
size on concrete shear strength after the dramatic warehouse shear failures of 1955 (Collins
and Kuchma, 1997 and Collins and Mitchell, 1997). His work consisted of beams without
web reinforcement with varying mernber depths, d, longitudinal steel percentages, p, and
shear span-to-depth ratios, dd. He determined that member depth and steel percentage had a
great effect on shear strength and that there is a transition point at a/d=2.5 at which beams
are shear critical (Le. the value of the bending moment at failure was minimum)(see Fig.
1.1).
Kani found this value of a/d to be the transition point between failure modes and is the
same for different member sizes and steel ratios. Below an d d value of about 2.5 the test
beams developed arch action and had a considerable reserve of strength beyond the first
cracking point. For a/d values greater than 2.5 failure was sudden, brittle and in diagonal
tension soon atter the first diagonal cracks appeared. This transition point is more
ernphasised in test beams containing higher reinforcement ratios and almost disappears in
specimens with lower reinforcement ratios. In addition. Kani found a ctearly defined
envelope bounded by limiting values of p and a/d. Inside this envelope diagonal shear
failures are predicted to occur and outside of this envelope flexural failures are predicted to
occur. These conclusions regarding the influence of both p and a/d were similar for al1 beam
depths tested. Kani also looked at the effect of beam width and found no significant effect on
shear strength. Kani's work was summarised in the textbook "Kani on Shear in Reinforced
Concrete" (Kani et al. 1979).
More recently, Bazant and Kim (1984) derived a shear strength equation based on the
theory of fracture mechanics. This equation accounts for the size effect phenomenon as well
as the longitudinal steel ratio and incorporates the effect of aggregate size. This equation was
catibrated using 296 previous tests obtained from the literature and was compared with the
AC1 Code equations. It was noted afler the comparison that the practice used in the AC1
Code of designing for diagonal shear crack initiation rather than ultimate strength does not
yield a uniform safety margin when different beam sizes are considered. It was also found.
according to the new equation. that for very large specimen depths the factor of safety in the
AC1 Code almost disappears. However, no experimental evidence was available yet to
confinn that fact as al1 the tests performed up to that tirne were on relatively small
specimens. This equation was improved by Bazant and Sun (1987) to account for the
maximum aggregate size distinctly from the size effect phenomenon and was extended to
cover the influence of stimps. This formula was calibrated using a larger set of test data
consisting of 46 1 test results compiled from the literature.
Later on, Bazant and Kazemi (1 99 1) performed tests on geometrically similar beams
with a size range of 1 : 16 and having a constant a/d ratio of 3.0 and a constant longitudinal
steel ratio, p. Beams tested varied in depth from 1 inch (25 mm) to 1 6 inches (406 mm). The
main failure mode of the specimens tested was diagonal shear but the smallest specimen
failed in flexure. This study confirmed the size effect phenomenon and helped corroborate
the previously published formula. However, the deepest beam tested was relatively small and
the authors concluded that for beams larger than 16 inches (406 mm) additional reductions in
shear strength due to size effect were likely.
Kim and Park (1994) performed tests on beams with a higher than normal concrete
strength (53.7 MPa). Test variables were longitudinal steel ratio, p, shear span-to-depth ratio,
a/d, and effective depth. d. Beam heights varied from 170 mm to 1000 mm while the
longitudinal steel ratio varied from 0.01 to 0.049 and d d varied from 1.5 to 6.0. Their
findings were similar to Kani's from which it was concluded that the behaviour of the higher
strength concrete is similar to that of normal-strength concrete. However, since only one
concrete strength was investigated no general conclusions could be made with respect to
concrete strength and shear capacity.
Shioya ( 1 989) conducted a number of tests on large-scale beams in which the influence
of member depth and aggregate size on shear strength was investigated. In this study, lightly
reinforced concrete beams containing no transverse reinforcement were tested under a
uniformly distributed load. The beam depths in this experimental program ranged from 100
mm to 3000 mm. Shioya found that the shear stress at failure decreased as the member size
increased and as the aggregate size decreased. It is interesting to note that the beams tested
by Shioya contained about the saine amount of longitudinal reinforcement as the roof beams
of the Air Force warehouse which collapsed in 1955 (Collins and Kuchma, 1997 and Collins
and Mitchell, 1997). The warehouse bearns had an effective depth of 850 mm and failed at a
shear stress of about 0.1 OK MPa. This shear stress level corresponds with the failure shear
stress observed in beams having a depth of 1000 mm in the Shioya tests. It is important to
mention that there was a tendency for reduced shear stresses at failure even with tests
inc luding 3000 mm deep beams. Figure 1.2 illustrates the results obtained by Shioya.
(m) 0.5 1 .O 1 5 20 25 3.0
psi viits 4 . 1 1
1 m. (25 m l nu^ r w r g a t r sur
Figure 1.2: Influence o f member depth and aggregate size on shear stress at failure for tests
carried out by Shioya 1989. taken from Collins and Mitchell, 1997.
Stanik (1 998) perfomed tests on a wide range of beam specimens at the University o f
Toronto. The specirnens tested had varying depths, d, ranging from 125 mm to 1000 mm,
varying amounts of longitudinal steel ( 0.76% to 1.31%) a s well a s varying concrete
strengths, fé , ranging from 37 MPa to 99 MPa. The longitudinal reinforcement was
distributed in some specimens along the sides and some specimens contained the minimum
amount of transverse reinforcement recommended by the CSA Standard (CSA 1994). In the
series with longitudinal bars aiong the sides, a set o f wider beams was also tested. T h e
purpose was to evaluate the influence of the amount, a s well as the distribution o f the
longitudinal steel on the shear strength. Stanik found that the size effect is very pronounced
in IightIy reinforced deep members. Members containing the minimum amount o f transverse
rein forcement or side distributed steel performed better than their counterparts with only
bottom longitudinal reinforcing bars. Deep members with side distributed reinforcement
performed nearly as well a s the shallow members containing only bottom longitudinal
rein forcement. As well, the wider members containing side distributed steel were weaker
than the narrower ones with similar side distributed steel. Stanik concluded that the size
effect is more related to measures controlling crack widths and crack spacing rather than the
absolute depth of the member. Moreover, Stanik found very little gain in shear strength with
the use of higher concrete strengths. In fact, he found that the shear strengths of the beams
with 100 MPa concrete were only marginally greater than the shear strength of the 40 MPa
beams. Stanik used the modified compression field theory proposed by the CSA Standard
(CSA 1994) to predict the response of his test beams. He found good agreement between his
experirnental results and these predictions. He also proposed to use an effective aggregate
size of zero in the modified compression field theory method for the very high-strength
concretes in order to account for the insignificant gain in shear strength from the lower
concrete strengths. Stanik also performed a cornparison between his experimental results and
the AC1 Code (AC1 committee 3 18-1995) expressions. He found that the AC1 expressions
substantially overestimate the shear contribution of concrete, notably in the deeper members.
1.3 Shear and High-Strengtb Concrete
High-strength concrete is a more brittle material than normal-strength concrete. This
means that cracks that fonn in high-strength concrete will propagate more extensively than
in normal-strength concrete. Previous shear tests on high-strength concrete have shown a
significant difference between the failure planes of high-strength concrete and that of
iiormal-strength concrete. This is due to the fact that cracks tend to propagate through the
aggregates in the higher strength concretes rather than around the aggregates as in normal-
strength concrete. The result is a much smoother shear failure surface meaning that the shear
carried by aggregate interlock tends to decrease with increasing concrete strength.
Mphonde and Frantz (1984) tested concrete beams without shear reinforcernent with
varying a/d ratios from 0.0 15 to 0.036 and concrete strengths ranging from 2 1 to 103 MPa.
They conchded that the effect of concrete strength becomes more significant with smaller
a/d ratios and that failures became more sudden and explosive with greater concrete strength.
It was also found that there is a greater scatter in the results of specimens with small a/d
ratios due to the possible variations in the failure modes.
Elzanaty et al. (1986) looked at the problem of shear in high-strength concrete and
observed a smoother failure plane in the higher strength concrete specimens. Their study was
performed on a total of 18 beams with concrete strengths, f l , ranging from 2 1 to 83 MPa.
Apart from concrete strength, test variables included p and the shear span-to-depth ratio, dd.
The conclusions drawn from these tests were that the shear strength increased with
increasing fi but less than that predicted using the AC1 Code equations. These equations
predict an increase in shear strength in proportion to K. Elzanaty et al. concluded that an
increase in the steel ratio led to an increase in the shear capacity of the specimens regardless
of concrete strength.
Ahmad et al. (1986) studied the effects of the a/d ratio and longitudinal steel percentage
on the shear capacity of bezms without web reinforcement. For their tests, the concrete
strength was maintained as constant as possible with f i in the range of 63 to 70 MPa.
Findings were similar to previous experiments with a transition in the failure mode at an a/d
ratio of approximately 2.5. The envelope involving limits on a/d and p which separates shear
faiiures from flexural failures was found to be similar to the envelope for the normal-strength
concrete. However, more longitudinal steel was required to prevent flexural failures. Ahmad
et al. found that the shear capacity was proportional to fi0 3.
1.4 Shear Design Methods
1.1.1 AC1 Shear Design Procedure
The AC1 Code (AC1 Committee 3 1 8- 1995) shear design equations for non-prestressed
reinforced concrete beams were derived in 1962 based on tests involving relatively small
(d,,, = 340 mm) and rather heavily reinforced (p,,, = 2.2%) beams and do not recognise the
size effect on the shear performance. These equations are:
In lieu of equation [1.1], the AC1 Code allows the foilowing simplet equation to be
used:
V, = 0.1 66$7b,d with l?% mm] 11.21
These equations for predicting the shear strength of concrete beam elements are based
on the shear causing significant diagonal cracking. At the time these expressions were
derived, the ACI-ASCE Committee 326 on shear and diagonal tension (ACI-ASCE
Committee 326, 1962) concluded that for mentbers without stirrups, any increase in shear
capacity beyond the shear which caused significant diagonal cracking was unpredictable due
to the great variation in failure mechanisms and should thus be ignored. Bazant et al. (1984,
1987, 199 1) criticised this assumption since the diagonal cracking load is not proportional to
the ultimate load, and hence a uniform factor of safety against failure is not provided.
1 A.2 CSA Simplified Shear Design Procedure
The simplified expression in the 1994 CSA Standard (CSA 1994) for the evaluation of
the contribution of the concrete, V,. to the shear capacity are given below:
a) For sections having either the minimum amount of transverse reinforcement required in
the Standard (CSA 1994), or an effective depth, d. not exceeding 300 mm :
Where @, is the material resistance factor for concrete, equal to 0.60.
The factor of "0.2" in the above equation was artificially increased from that
corresponding to the nominal value of 0.166 to account for the low value of 4,. Hence the
nominal resistance can be written as:
b) For sections with effective depths greater than 300 mm and with less transverse
reinforcement than the minimum required :
v, = ( 260 )d,,/F&d not Iess than 0.10(~fib,d m. mm] [I.j] 1000 + d
Similarly the nominal resistance can be written as:
not less than 0833Jf;;b,d IN, mm] [ 1.61
As can be seen from Equation [1.5], the CSA Concrete Standard (CSA 1994) includes a
term to account for the size effect in its simplified shear design expression but does not take
account of the reinforcing steel ratio, p. This shows the concern of this code regarding the
size effect phenomenon. However the linear nature of the t em added to the shear equation
cannot account for the complexity of the problem. More research is needed to adjust this
equation to account for higher concrete strengths and amount of longitudinal reinforcement,
p. Sornç limitations on the distribution of the longitudinal reinforcement may also be
requited.
1.4.3 Modified Compression Field Theory
In lieu of the latter simplified shear design equations, the CSA Standard (CSA 1994)
proposes a more rational method of approach to the shear design "problem" based more on
fundamental principles than on empirical equations. This method treats the stress-strain
charxteristics of the cracked concrete using average stresses and strains in the concrete and
utilises equilibrium and compatibility of strains. The crack pattern is also idealised as a series
of parallel cracks occurring at an angle 8 to the longitudinal direction. The theory considers
that the shear strength of concrete at a crack location is dependent on the width of the crack
as well as the maximum aggregate size used (Le., it looks at the crack roughness). This
method accounts for the strain softening of the diagonally cracked concrete in compression
and also accounts for the tensile stresses in the cracked concrete (Vecchio and Collins 1982).
The modified compression field theory is explained in detail by Collins and Mitchell (1997)
and by Collins et al. (1996) and yields the following design equations for predicting the
concrete contribution to the shear strength:
Where:
b, = Minimum effective web width within the depth of the section
d, = Distance measured perpendicular to the neutral axis between the resuitants of the tensile
and compressive force due to flexure, may be taken as O.9d for non-prestressed concrete
mem bers
p = Tensi le stress factor which accounts for the shear resistance of cracked concrete
033 cot 8 0.1 8 P = 24w l+J500E, 0 3 +
a + I6
Where w is the average crack width which is taken as:
W here:
E, = Principal tensile strain in cracked concrete
s, = Crack spacing in the 0 direction
and
E l = Ex + v (tan 8 + cot 0K0.8 + [ l . l O ]
For the case of a non-prestressed beam with bottom chord reinforcement the
longitudinal strain of the flexural tension chord can be taken as
M / d, + OS(N + V cot 8) Ex =
Es As
Where:
M = Moment at section
N = Axial load at section (positive in tension)
V = Shear force at section
Es = Modulus of elasticity of longitudinal steel reinforcernent
A, = Area of Iongitudinal steel reinforcement in tension zone
In order to simpli@ the design procedure using the modified compression field theory, a
set of tables and plots was developed (CSA 1994, Collins et al. 1996) with corresponding
values of 0, B, E, and v/f& Applying the method requires an iterative process where a value
of E, is assumed and the corresponding 8 value is calculated from which another value of E,
is obtained. The process is repeated until the values of E, converge.
The method has given very accurate predictions of the shear response in beam elements
(Collins et al. 1996, Vecchio and Collins 1988) especially when member size is involved.
The method's predictions will be cornpared with the test results of this research.
The cornputer program "RESPONSE" has k e n developed at the University of Toronto
by Felber (see Collins and Mitchell 1997). This program uses a "plane-section'' analysis
technique and uses the modified compression field theory for shear. It performs sectional
analyses using the stress-strain relationships for the diagonally cracked concrete and the
complete stress-strain relationship for the steel reinforcement. The analysis accounts for the
sectional properties as wel I as combined loading conditions (moment, shear and axial load),
and provides the response of a section up to and beyond failure. A later, widows-based.
version of the program called "RESPONSE 2000" is currently under development at the
University of Toronto by Collins and Bentz (Collins and Bentz 1998). This version allows
more flexibility in the definition of sections, perforrns a "dual-section" analysis (Vecchio and
Collins 1982) and provides full graphical output of stresses and strains at key stages of
loading. A beta version of this program will be used in this research program to predict the
response of the elements tested according to the modified compression field theory.
1.5 Objectives of Research Program
The objective of this research program is to investigate a number of issues related to the
"size effect" in shear. An experimental program was planned to investigate the following:
1 ) The reduction in shear stress at failure as the size of beams or one-way slabs
increases.
2) The influence of concrete strength on the shear stress at failure.
3) The effect of amount of longitudinal steel reinforcement on the shear stress at
fai l ure.
4) The combined effects of the three variables p, d and fl on the shear stress at failure
in beams or one-way slabs.
A comparison will also be performed between the test results and predictions given by
current shear design methods. The purpose being to evaluate the different approaches and
theories supporting these methods.
Chapter 2
Experimental Program
2.1 Design and Details of the Beam Specimens
Twelve beam specimens were constructed and tested in the Structures Laboratory of the
Department of Civil Engineering at McGill University. The specimens were cast in two
batches of different concrete strengths producing two sets of geometrically identical
specimens. Concrete strengths were 35 MPa for the "normal-strength" specimens and 60
MPa for the "high-strength" specimens. The specimens al1 had a width of 400 mm and their
heights varied from 90 mm to 960 mm.
The specimens were designed according to the modified compression field theory to fail
in shear rather than bending. No transverse reinforcement was provided for any of the
specimens. The shear span-to-depth ratio, a/d, of al1 specimens was kept constant at a/d=2.5,
in order to produce a shear critical specimen (Kani 1966, 1967, 1979). AI1 of the beams were
subjected to a two-point loading arrangement as shown in Fig. 2.1.
Each specimen had two different iongitudinal steel ratios. The north end of the bearns
contained a larger number of reinforcing bars giving p=2% while the south end of the beams
had a smaller number of bars with p=1.2% (see Fig. 2.1). This was achieved by lapping the
bottom flexurat reinforcement in the region between the two loading points. Sumcient
overIap was provided to ensure adequate steel development and thus provide suficient
flexural strength to produce shear failures in the high p section of the specimens. The
purpose here was to perform two tests per specimen. After the failure of the weaker south
end, external stirrups were clamped ont0 the failure region to enable further loading and
enable a shear failure on the strong end of the beams.
The longitudinal steel was evenly distributed along the width of the specimens leaving a
40 mm clear cover on each side. Bottom covers were chosen according to the 1994 CSA
Standard (CSA 1994) for slabs subjected to interior exposure. The CSA Standard requires a
South North
North Shear Span Reinforcement Details ( ~ 2 % )
O Bars O Bars
1 O\# 30 Bars 5 4 30 Bars 5 h 20 an
NOTE: All dimensions in mm All specimens 400 mm wide Bars are evenly spread All side avers are 40 mm
South Shear Span Reinforcement Details (p=1.2%)
L/ 6 30 Ban
NOTE: Dimensions similar to sechion for north shear span
3 ) 10 Bars
Figure 2.1: Specimen reinforcement details
minimum clear cover of 20 mm but the clear cover must at least equal the bar diameter, d,.
Hence for specimens with No. 20 bars and smaller, the 20 mm cover was used and for larger
diameter bars, a cover equal to d, was used. It is noted that the AC1 Code (AC1 Committee
3 18, 1995) requires a constant minimum cover of 20 mm for bars up to and including No. 35
bars. All specimens had one layer of bottom steel except for the 960 mm deep beams which
had two layers (see Fig. 2.1). Two top bars were added in the three largest specimens. AI1
reinforcement details are shown in Fig. 2.1.
2.2 Specimen Identification
A total of twelve specimens were divided into two series: the normal-strength series (N),
and the high-strength series (H). Both series have similar geometry and steel reinforcement.
Specimens were numbered according to their absolute height (in mm) as follows: N90,
N 1 55, N220, N350, N485, N960, H90, H 155, H220, H350, H485 and H960.
A further notation will be used in this report to distinguish the strong end from the weak
end of the specimens. The added notations wilt be (w) for the weak end and (s) for the strong
end. This notation will be added to the end of the specimen name.
Table 2.1: Reinforcing steel properties
Specimen
Designation
N 90
H 90
N, Hl55
N, H220
N, H350
N, H48S
N, H960
Steel
Location
Bottom
Bottom
Bottom
Bottom
Top
Bottom
Top
Bottom
TOP
Bottom
Size Designation
No. 1 O Grade 400
No. t O Grade 500
No. 15
No. 20
No. I O Grade 400
No. 25
No. 10 Grade 400
Na30
No. IS
No.30
f~
( M m
477
648
444
433
477
436
477
385
444
385
Area
(mm2)
100
100
200
300
100
500
100
700
200
700
&Y
( O h )
0.32
0.52
0.29
0.22
0.32
0.22
0.32
0.18
0.58
&sb
(%)
0.40
0.55
0.50
0.94
0.40
0.74
0.40
0.88
1.00
fa
<MP@ 670
672
667
686
670
675
670
637
667
637 0.18 0.88
2.3 Material Properties
2.3.1 Reinforcing Steel Properties Table 2.1 summarises the material properties of the deformed steel reinforcement used
in the construction of the specimens. All of the reinforcement used was Grade 400 except for
the reinforcement of the smallest specimen in the high-strength series which was Grade 500.
The values reported in Table 2.1 are the averages of three simples per bar sue chosen
randomly amongst the bars used. Figure 2.2 shows typical tensile stress-strain responses of
the reinforcing bars-
No. 10 1
No. 20
No. 15 Md
No. 25 Ml01
Figure 2.2: Typical tensile stress-strain curves for the reinforcing steel
16
2.3.2 Concrete Properties The concrete used to construct the specimens was provided by a local ready-rnix plant.
Table 2.2 summarises the mix designs provided by that plant. Following the cast, the normal-
strength concrete was moist cured for three days and the high-strength concrete for seven
days.
Table 2.2: Concrete mix designs I
30 MPa 70 MPa
1 Strength 1 Strength
I 1 Concrete 1 Concrete I 1
1 cernent (Type IO), kg/rn3 1 355 1 480' I I 1 fine aggregates (sand) . kg/m3 1 790 1 803
Coane aggregates, kg/m5 1040 1059
I 1 total water-, kg/m3 1 178 1 135
water-cernent ratio 0.50 0.28
I water-reducing agent, ml/mJ 1110 1502
I
air-entraining agent, ml/m5 I 180 -
1 1
1 air content, % 1 8.8 1 -
* Includes the water in admixtures
Table 2.3 summarises the material properties OF the concrete used in both the normal-
strength and high-strength concrete series. Compression, split-cylinder and four-point-
loading flexural beam tests were conducted to determine the mean values of the concrete
compressive strength f: . its associated strain cc'. the splitting tensile stress f,,, and the
modulus of rupture, f,. Standard cylinders, 150 mm in diameter and 300 mm long, were used
for the compression and split-cylinder tests. The flexural beams were 150 mm by 150 mm by
600 mm long. Four-point loading was applied over a span of 450 mm. Tests were conducted
once at the beginning of series testing and once at the end to observe variations in the
concrete material properties during testing. It was observed however that no significant
variations in the material properties occurred from one testing date to another. The values
shown in Table 2.3 are average values from three samples tested at the beginning of a series
and three samples tested at the end of a series.
O 0.001 0.002 0.003 0.004 0.005 0.006 0.007
Stnin (rnmlmm)
Table 2.3: Concrete properties for both series
Figure 2.3: Typical concrete compressive stress-strain responses
Series
Normal-Strength Concrete
std. Deviation
High-Strength Concrete
std. deviation
fi
34.19
0.49
58.55
2.56
EC'
x 10-~
4.09
0.258
3.98
0.268
fw m a )
3 .O8
0.17
3.49
O. 17
fr
(MPd
4.89
0.33
4.67
0.25
Figure 2.3 shows typical compressive stress-strain responses for both concrete strengths
and Fig. 2.4 shows the shrinkage readings taken on standard 75 mm by 75 mm by 280 mm
long shrinkage specimens. The values used to plot Fig. 2.4 are average values from two
shrinkage specirnens for each cast.
- NmalStrength Concrete
- . - - . . High-Strength Concrete
20 40 60 80 100 1 20 Time (days)
Figure 2.4: Concrete shrinkage readings
2.4 Testing Procedure
2.4.1 Test Setup and Loading Apparatus All specimens were tested under the MTS universal testing machine in the Structures
Laboratory of the Department of Civil Engineering at McGill University. The specimens
were supported on a pair of rollers, a rocker and a bearing plate at each support (see Figs. 2.5
and 2.7). A two-point loading scheme was used to apply loading to the specimens. The
distance separating the two Ioading points was constant for al1 the specimens at 500 mm. The
shear span separating the loading points from the supports was equal on both ends of the
specimens creating a zero shear region between the two loading points. The load transfer
from the MTS machine to the specimens was through a ball-bearing joint, a steel spreader
beam, a pair of rollen and a set of bearing plates that were grouted to the tops of the
specimens. The bearing plates at the supports and at the loading points for al1 the specimens
were one inch thick steel plates covering the entire width of the specirnen over a length of
100 mm,
Figure 2.5: Test setup and clamping of the failed weaker end
The loading was applied monotonically with load, deflection and strain values being
recorded at small increments of loading. At key load stages, the crack pattern and crack
widths were recorded. After the failure of the weaker side, extemal HSS clamps and threaded
rods were used to clamp the failed section (see Fig. 2.5), then the specimen was reloaded
until the stronger side failed. For the smaller specimens, it was sometimes impossible to
clamp the failed weaker side. ln these instances, the supports were moved to create a single-
point loading scheme (see Fig. 2.6). This was possible since the shear spans of these smaller
specimens was smaller than the 500 mm distance separating the two initial loading points.
Figure 2.6: One-point loading arrangement used for some specimens
2.4.2 Instrumentation The load values applied to the specimens were obtained from the MTS machine's load
cell. The deflections were monitored with a linear voltage differential transformer (LVDT) at
both loading points. Additional LVDTs were placed at both supports in order to monitor their
movement (see Fig. 2.7). Concrete strains were measured at the back face of the specimens
using LVDT rosettes centred at the middle of the shear spans (see Fig. 2.7). Strain targets
were glued at the front face of the specimens at the same location and in the same
arrangement as the rosettes (see Fig. 2.9). These targets were used to detemine the concrete
strains at the front face and to provide cornparison between the back and front sides in order
to veriQ that no torsion was induced in the specimens. The reading of the target strains was
pedonned using a 203 mm or 102 mm gauge length mechanical extensometer.
Figure 2.7: LVDT and concrete strain gauge locations
a
Electrical resistance strain gauges were glued to the reinforcement bars of the bottom
steel as shown in Fig. 2.8. For the 960 mm high specimens with two layers of bottom of
reinforcement, the strain gauges were placed on the bottom layer. Two additional strain
eauges were giued to the concrete surface just below both loading points (see Fig. 2.7). The C
strain readings obtained from the concrete strain gauges combined with those of the steel
gauges enabled the calculation of the curvature of the specimens at the maximum moment
and shear locations.
South
North - t--7 South
Sn 1 c h J North 1 m
\ I bncn(.-C.ugir r
Figure 2.8: Typical steel strain gauge locations (top view)
i I
Figure 2.9: Specimen strain target locations
Chapter 3
Experimental Results and Cornparisons
3.1 Introduction
In this chapter, the observed behaviour of the 12 beam specimens is presented. Among
the experimental results recorded were longitudinal steel strains, concrete strains and crack
widths at load stage intervals. These results are presented in Figs. 3.1 through 3.6 and Figs.
3.8 through 3.1 3. Figs. 3.7 and 3.14 are photographs of the normal-strength and high-strength
concrete specimens afier they have been tested.
A figure is given for each test beam. The response of the weak-end (p=1.2%) and
strong-end (p=2%) is plotted in each graph for comparison purposes. Each figure contains
the following:
a) A graph ploning the maximum applied moment versus the maximum flexural crack
width.
b) A drawing showing the Iocation of the instrumentation.
c) A graph plotting the maximum applied moment versus the longitudinal steel strain,
measured on the reinforcing bars below the loading points.
d) A graph plotting the applied shear versus the
was calculated from the strains in the rosettes
e) A drawing showing the failure crack pattern
tested.
principal concrete tensile strain which
placed in the center of the shear spans.
of the test beam after both ends were
As described in Chapter 2, the weak-end was reinforced with stirrup clamps afier it failed
to permit further testing of the strong-end of each beam eiement.
3.2 General Bebaviour
The general behaviour of the four largest bearns was quite similar. First, the flexural
cracks initiated in the pure bending region. With further increase of Ioad new flexural cracks
formed in the shear spans and curved toward the loading points. The failure in these
specimen was always sudden and in diagonal tension shortly after diagonal shear cracks
appeared. it was noted that the ultimate shear capacity of these beam elements was only
slightly higher than the load which caused diagonal cracking. It is for this reason that no
diagonal tension cracks could be measured prior to failure.
As for the smaller sizes, the crack development was similar to that of the other
specimens except where flexural yielding occured. This produced a different failure
mechanism which will be discussed in detail for each specimen.
3.3 Normal-Streogth Coocrete Series
$necben N90; In both the weak and strong ends of this specimen, the longitudinal
steel yielded before failure occurred due to the unexpectedly high shear resistance of the
specimen (Fig. 3.1 c)). Unfortunately, the strain gauge on the weak end was lost shortly afier
yield. First cracking is observed in Fig. 3.1 c) at an approximate moment of 2.0 kN.m which
corresponds to a modulus of rupture, f,, of 3.70 MPa. The principal tensile strain in the
concrete, obtained from the rosettes, changed very little prior to yielding of the longitudinal
steel. After y ielding, it Increased significantly (Fig. 3.1 d)) denoting large shear cracks before
failure. The failure mode of both ends of the specimen was a combination of flexural
yielding and shear. The longitudinal steel yielded first and as it elongated, it increased the
shear crack size until a shear failure occurred. Failure shears were 41.1 kN for the end with
p=I -2% and 74.5 kN for the end with p=2.0%.
Specben N155; The weak end of this beam failed in a similar fashion as the N90
specimen, with the steel yielding prior to shear failure (Fig. 3.2 c)). Unfortunately, the major
failure crack on the weak end was outside of the rosette coverage as well as k i n g outside of
the strain targets (see Fig. 3.2 e)). As for the strong end. it failed in shear just prior to the
yielding of the steel (Fig. 3.2 c)). It is to be noted for the strong end of this specimen, that
afier the first significant load drop at a shear force of 109.8 IrN, the specimen developed arch
action and was able to reach a shear force of 134.5 kN. The principal tensile strains recorded
for the strong end denote large shear cracks before failure (Fig. 3.2 d)). Faiture shears were
82.5 kN for the end with p=1.2% and 109.8 kN for the end with p=2.0%.
m e n N22& First cracking in this specimen occurred at an approximate moment of
12.5 kN.m (Fig. 3.3 c)) which corresponds to a modulus of rupture, f, of 3.87 MPa. Principal
tensile strains remained small until about 90% of the failure load. At this load level these
strains increased more significantly as very small shear cracks fonned causing the load to
drop off (Fig. 3.3 d)). Failure of both ends was in shear and in a brittle fashion without any
yielding of the longitudinal steel (Fig. 3.3 c)). Failure shears were 100.6 kN for the end with
p=1.2% and 1 19.7 kN for the end with p=2-0%.
N350; First cracking in this specimen occurred at an approximate moment of
40 kN.m (Fig. 3.4 c)) which corresponds to a modulus of rupture, f,, of 4.90 MPa. Principal
tensile strains rernained small until about 90% of the failure load. At this load level these
strains increased more significantly as very small shear cracks formed causing the load to
drop off (Fig. 3.4 dj). Failure of both ends was in shear and in a brinle fashion without any
yielding of the longitudinal steel (Fig. 3.4 c)). Failure shears were 152.6 kN for the end with
p= 1 -2% and 173.1 kN for the end with p=2.0%.
S~ecimen N485; First cracking in this specimen occurred at an approximate moment of
62 kN.m (Fig. 3.5 c)) which correspondsto a modulus of rupture, f, of 3.95 MPa. Principal
tensile strains remained small until about 90% of the failure load. At this load level these
strains increased more significantly as very srnall shear cracks fonned causing the ioad to
drop off (Fig. 3.5 d)). Failure of both ends was in shear and in a brittle fashion without any
yielding of the longitudinal steel (Fig. 3.5 c)). Failure shears were 178.9 kN for the end with
p= 1.2% and 206.7 kN for the end with p=2.0%.
en N960; First cracking in this specimen occurred a t an approximate moment of
220 kN.m (Fig. 3.6 c)) which corresponds to a modulus of rupture, f, of 3.58 MPa. Principal
tensile strains remained small until very small shear cracks formed causing the load to drop
off (Fig. 3.6 d)). Failure of both ends was in shear and in a brittle fashion without any
yielding of the longitudinal steel (Fig. 3.6 c)). Failure shears were 340.5 khi for the end with
p=1.2% and 360.0 CcN for the end with p=2.0%.
Maximum Fkrunl Crack Width (mm)
Moment a. Mw. Fkxural Crack Wdtti
O 0.m2 0.004 0.006 0.008
E* (mm/mm) Moment vs. Longitudinal Steel Strain
Section 2-2 Section 1-1
O 0.002 0.004 0.006 0.W8 0.01 0.012
E~ (mmlmm) Shear vs. Principal Tensik Strain
Failure Crack Pattern for Specimen N90
e)
Figure 3.1: Test results for specimen N90
O 0.5 1 1.5 2 2.5 Maximum Fbxural Crack Width (mm)
Moment W. Max. Fkwural Crack Widai
1
O 0002 0004 0006 0008 O 01
r, (mmlmm) Moment vs. Longitudinal Sîeel Strain
Section 2-2 Section 1-1
O 0.001 0.002 0.003 0.004 0.005
E, (mmlmm) Shear vs. Principal Tensik Strain
Failure Crack Pattern for Specimen Ni 55
el
Figuro 3.2: Test results for specimen N 1 55
O 0.05 0.1 0.15 0.2 0.25 O 3 Maximum Fkxuril Crack W a (mm)
Moment vs. Max. Flaxurai Crack Width
I O 0.0005 O 001 0.0015 0 002
E* (mrnlmm) Moment vs. Longitudinal Steel Strain
Section 2-2 Section 1-1
O 0.001 0.002 0.003 0.004 D.W5 0.006
E, (mrnlmm) Shear vs. Principal Tensiie Strain
Failure Crack Pattern for Specimen N220
e)
Figure 33: Test results for specimen N220
O 0.05 0.1 0.15 0.2 0.25 0.3 Maximum Fkxural Crack Width (mm)
Moment vs. Max. Fkxural Cracâ Wdth
O 0.0005 0.001 00015 0.002 0.0025
E, (mmlmm) Moment vs. Longitudinal Steel Strain
Section 2-2 Section 1-1
O 0.005 0.01 0.015
E, (mmlmm) Shear vs. Prinapal Tensik Strain
Faiture Crack Pattern for Specimen N350
e)
Figure 3.4: Test results for specimen N350
O 0.0005 0.001 0 0015 O 0.005 0.01 0.015 0.02
Es (mmlmm) ~1 (mwmm) Moment vs. Longitudinal Steel Stmin Shear vs. Principal Tensik Strain
300 -
Failure Crack Pattern for Speamen N485
e)
250
Figure 3.5: Test results for specirnen N485
..-- . h, l ,L
2 Es2 Es 1 1
Section 2-2 Section 1-1
1
O 0.05 0.1 0.15 0.2 0.25 0.3 O O O O D
Maximum Fkxunl C m k Mdth (mm) 0 p=2% p=1.2%
Moment vs. Max. Fkxurol Cnck Width
q=2% . - -.
2 €1-1
O O. 1 0.2 0.3 0.4 Maximum Fkxural Cmck Wdüt (mm)
Moment vs. Max f kxural Crack Width
O 0.0002 OOOM O0006 0.0008 0001
E, (mmlmm) Moment vs. Longitudinal Steel Strain
Section 2-2
. - Es1 1 E l - 1
Section 1-1
O 0.001 0002 0.063 0.004 0.005 O 0 0 6
E, (mmlmm) Shear vs. Principal Tensik Stmin
Failure Crack Patîem for Specimen N960
e)
Figure 3.6: Test results for specimen N960
Figure 3.7: Normal-strength series after failure
34
3.4 High-Strength Concrete Series
-90; In the weak-end of this specimen, the longitudinal steel yielded before
failure occurred due to the unexpectedly high shear resistance of the specimen (Fig. 3.8 c)).
First cracking is observed in Fig. 3.8 c) at an approximate moment of 3.0 kN.m which
corresponds to a modulus of rupture, f,., of 5.56 MPa. At the weak-end, the principal tensile
strain in the concrete, obtained from the rosettes, changed very little prior to yielding of the
longitudinal steel. Afier yielding, it increased significantly (Fig. 3.8 d)) denoting large shear
cracks before failure. The failure mode of the weak-end was a combination of fiexural
yielding and shear. The longitudinal steel yielded first and as it elongated, the shear cracks
increased until a shear failure occurred. As for the strong-end, it failed in shear prior to the
yielding of the steel (Fig. 3.8 c)). The principal tensile strains recorded for the strong end
denote large shear cracks before failure (Fig. 3.8 d)). Failure shears were 50.8 kN for the end
with p=1.2% and 76.0 kN for the end with p=2.0%.
Specirnedl55: First cracking in this specimen occurred at an approxirnate moment of
12.0 kN.m (Fig. 3.9 c)) which corresponds to a modulus of rupture. f, of 7.49 MPa- Principal
tensile strains remained small until very small shear cracks formed causing the load to drop
off (Fig. 3.9 d)). Failure of both ends was in shear and in a brittle fashion without any
yielding of the longitudinal steel (Fig. 3.9 c)). Failure shears were 74.7 kN for the end with
p=1.2% and 102.9 kN for the end with p=2.0%.
S ~ e c i m e n First cracking in this specimen occurred at an approximate moment of
13.0 kN.m (Fig. 3.10 c)) which corresponds to a rnodulus of rupture, f,., of 4.03 MPa.
Principal tensile strains remained smalI until very small shear cracks formed causing the load
to drop off (Fig. 3.10 d)). Failure of both ends was in shear and in a brittle fashion without
any yielding of the longitudinal steel (Fig. 3.1 O c)). Failure shears were 102.8 kN for the end
with p= 1.2% and 132.3 kN for the end with p=2.0%.
-50; First cracking in this specimen occurred at an approximate moment of
40.0 kN.m (Fig. 3.1 1 c)) which corresponds to a modulus of rupture, f,, of 4.90 MPa.
Principal tensile strains remained small until very small shear cracks formed causing the load
to drop off (Fig. 3.1 I d)). Faiiure of both ends was in shear and in a brittle fashion without
any yielding of the longitudinal steel (Fig. 3.1 1 c)). Failure shears were 15 1.8 kN for the end
with p= 1.2% and 1 84.2 kN for the end with p=2.0%.
485; First cracking in this specirnen occurred at an approxirnate moment of
64.0 kN.m (Fig. 3.12 c)) which corresponds to a modulus of rupture, f, of 4.08 MPa.
Principal tensile strains remained small until very smalt shear cracks formed causing the load
to drop off (Fig. 3.12 d)). Failure of both ends was due to diagonal tension cracking, with the
shear failure forming in a brittle fashion, without any yielding of the longitudinal steel (Fig.
3.12 c)). Failure shears were 189.8 kN for the end with p=1.2% and 190.3 kN for the end
with p=2.0%.
H960; First cracking in this specimen occurred at an approximate moment of
260 kN.m (Fig. 3.13 c)) which corresponds to a moduhs of rupture, f,, of 4.23 MPa.
Principal tensile strains remained small until very small shear cracks formed causing the load
to drop off (Fig. 3. t 3 d)). Failure of both ends was in shear and in a brittle fashion without
any yielding of the longitudinal steel (Fig. 3.13 c)). Failure shears were 290.7 kN for the end
with p=I -2% and 3 1 1.4 kN for the end with p=2.0%. -
1 O 0.2 0.4 0.0 0.8 1 1 2
Maximum f kxunl Crack Width (mm)
Moment vs. Max. Fkxural Cradc Width
a)
O O O01 0.002 o. 003 E, (mmtmm)
Moment vs. Longitudinal Steel Strain
Section 2-2 Section 1-1
T l p=2% p=1.2%
O O002 0 . a 0.006 O O08 0.01
E, (mrnlmm) Shear vs. Principal Tensiie Strain
Failure Crack Pattern for Specimen Hg0
e)
Figure 3.8: Test results for specimen H90
O 0.05 0.1 0.15 0.2 0.25 0.3 Maximum Fkxural Crack Width (mm)
Moment vs. Max. Flexwal Crack WidFh
O 0.0002 O 0004 0.0006 0.0008 OM1
E, (mmlmm) Moment vs. Longitudinal Steel Strain
Section 2-2
rl p=2%
Section 1-1
O 0.002 0.004 0 . m 0.008 0.01 0.012
cl (mdmm) Shear vs. Principal Terisile Sûain
Failure Crack Pattern for Specimen Hl55
Figure 3.9: Test results for specimen H 1 55
O 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Maximum Fkxural Crack Wdth (mm)
Moment vs. Max Fkxural Ca& Wdth
O 00002 00004 00006 00008 0001 00012 OW14
E, (mmlmm) Moment vs. Longitudinal Steel Strain
Section 2-2 Section 1-1
n
C 0.01 0.02 0.03 O M 0.05 0.06
t, (mmlmm) Shear vs. Principal Tensiie Strain
Failure Crack Pattern for Specimen HZ20
Figure 3.10: Test results for specimen Hz20
Maximum Fkrunl Crack Wldth (mm)
Moment vs. Max. Fkxural Crack Wüith
a)
O 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014
E, (rnmlmm) Moment vs. Longitudinal Steel Strain
Section 2-2 Section 1-1
O 0.005 0.01 0.015 0.02
E, (rnmlmm) Shear vs. Principal Tensik Strain
Failure Crack Pattern for Specimen H350
Figure 3.11: Test results for specimen H350
Maximum Fkxunl Crack Width (mm)
Moment vs. Max. Flexurat Crack Width
a)
O 0.0002 0.0004 0.0006 0.0008
E. (mmlmm)
Moment vs. Longitudinal Steel Strain
Section 2-2 Section 1-1
O O.Mi5 0.01 0.015 0.02
E, (mmlmm) Shear vs. Principal Tende Strain
Failure Crack Pattern for Speamen H485
e)
Figure 3.12: Test results for specimen H485
. - Et-2 2 Es2 Es1 1 El-1
Section 2-2 Section 1-1
O O. 1 0.2 0.3 0.4 0.5 0.6 ..--. Maximum Fhxunl Crack Wdth (mm) ..--. U Moment vs. Max. Fkxural Crack W m p=2%
p=2%
O 0.0002 0 . m 0.0006 0.0008
E, (mdrnrn)
Moment vs. Longitudinal Steel Strain
cl
O 0.005 0.01 0.015 0.02 0.025
E, (mdmm) Shear vs. Principal Tensiie Strain
Failure Crack Pattern for Speamen Hg60
Figum 3.13: Test results for specimen H960
Figure 3.14: High-strength series after fai lure
43
3.5 Summary of Results
Fig. 3.15 shows the failure shear stress versus specimen depth for both the normal and
high-strength concrete series. The failure shear stress has k e n detemined by adding the
effective member self-weight and the weight of the loading apparatus to the applied Ioads. In
this figure it can be seen that, for the largest specimens, the high-strength concrete beam
actually has a smaller failure shear stress than the cornpanion normal-strength concrete
beam. Both series, regardless of the amount of reinforcement, had very comparable shear
strengths showing no significant gain in shear strength with increased concrete compressive
strength.
Figure 3.15: Shear stress versus specimen depth
Fig. 3.16 shows the variation of the nonnalised shear stress at failure with specimen
depth. This normalised shear stress is the shear stress Vhd divided by K. Table 3.1 summarises the test results for both series giving the failure shear, maximum
moment at failure, shear stress at failure, normalised shear stress at failure and the mode of
failure for each end of the beam elements. Values given in Table 3.1 include member self-
weight and the weight of the Ioading apparatus.
Table
Specimen
N90
NI55
N220
N350
N485
N960
H90
Hl55
H220
H350
H485
H960
Summary
P
(%)
1 -2
2.0
1.2
2.0
1.2
2.0
1.2
2.0
1.2
2.0
1.2
2.0
1.2
2 .O
1.2
2.0
1.2
2.0
1.2
2.0
1.2
2.0
1.2
2.0
3.1:
' fl (MPa)
34.2
34.2
34.2
34.2
34.2
34.2
58.6
58.6
58.6
58.6
58.6
58.6
of results
VmaX
(w 42.5
75.9
84.6
1 1 1.9
103.6
122.7
158.0
178.6
187.5
215.4
366.6
386.1
52.1
77.4
76.7
105
105.9
135.3
157.3
189.6
198.5
199.0
316.7
337.4
M m a x
(kN.m)
10.8
19.4
34.7
45.9
59.0
69.9
139.0
157.0
224.0
257.4
839.3
883.9
13.3
19.7
31.5
43.0
60.3
77.1
138.4
166.8
237.2
237.8
725.1
772.6
V M
(MPa)
1.63
2.92
1 -66
2.19
1.36
1.61
1.26
1.43
1 .O7
1 -22
1 .O5
1-10
2.00
2.98
1-50
2.06
1.39
1.78
1 -26
1.52
1.13
1.13
0.90
0.96
~ M q f :
0.28
0.50
0.28
0.38
0.23
0.28
0.22
0.24
0.18
0.2 1
0.18
0.19
0.26
0.3 9
0.20
0.27
0.18
0.23
0.16
0.20
0.15
0.15
O. 12
O. 13
Mode of Faiiure
Shear 1 Flexure
S hear 1 FIexure
Shear 1 Flexure
S hear
Shear
S hear
Shear
Shear
Shear
Shear
Shear
Shear
Shear l Flexure
Shear
Shear
S hear
Shear
Shear
Shear
Shear
Shear
S hear
S hear
Shear
Figure 3.16: Normalised shear stress at failure versus specimen depth
3.6 Interpretation and Cornparison of Results
In Figs. 3.15 and 3.16. the size effect on shear strength is evident. In both the normal
and high-strength concrete. regardless of the steel ratio, the deeper rnembers had lower shear
stresses at failure than the shallower ones. This effect is so pronounced that some of the
smailer specimens which have similar reinforcernent and geometries as the larger specimens,
failed in flexure rather than shear.
The important influence of the longitudinal steel ratio, p, on the shear stress at faiture is
also confmned as the bearns were consistently stronger at the end with p=2% than at the end
with p=I.2%- The effect of the longitudinal steel on the shear strength can be explained
through the aggregate interlock mechanism. In fact, a major component of shear strength in
concrete arises from the friction forces that develop across the diagonal shear cracks by
aggregate interlock. This component of shear strength is more significant if the cracks are
narrow. Thus higher percentages of longitudinal steel which reduce the shear crack widths,
would allow the concrete to resist more shear.
Table 3.2 shows the difference in strength between the ends with p=2% and the ends
with p=1.2%. This difference decreases as the specimen depth increases showing an
attenuation in the effect of p on shear strength, with increased specimen depth. This may be
due to the fact that the longitudinal steel has a limited zone of influence in controllifig the
width of diagonal cracks over the concrete cross section. Thus, the larger the specimen, the
smaller that zone of influence is with respect to the overall cross section.
This effect indirectly contributes to the size effect as welt. The srnaller specimens, k i n g
almost entirely in the zone of influence of the longitudinal steel, have their shear crack
widths controlled over most of their height whereas the larger specimens, whose cross-
section is only partially influenced by the steel, have their shear cracks controlled over only a
limited region.
The effect of concrete strength on shear capacity is summarised in Table 3.3. In this
table. a negative number implies that the high-strength specimen was weaker than the
normal-strength one. It can be seen from Table 3.3 that the normal-strength series and the
high-strength series were very close in ternis of shear strength, since the biggest difference
found in the table is 18.5%. This can also be observed in Fig. 3.15 where the N S and HS
curves are very close to each other. This is in great contradiction with current code equations
which predict a relationship of E b e t w e e n concrete shear strength and concrete
compressive strength.
Table 3.2: Shear strength difference between ends with ~'2% and ends with p=1.2%
I I ends with p=l.2% (in %)
Specimen Height Difference in shear strength between ends with p=2% and
(mm)
90
Normal-Strength Series
78.6
High-Strength Series
48.5
Table 3.3: Shear strength difference between the high-strength and normal-strength
concrete series
1 S p i m e n Height 1 Difference in shear strengîh between high-strength and
I normal-strengtb concrete series (in %)
1 L I
Difference = (V,, ., - V,, ,,) / (V,, ,,) * 100
Chapter 4
Analysis of Results
In this chapter, the experimental results are compared to the theoretical predictions
using the design expressions of the AC1 Code (AC1 committee 3 18, 1995) and the CSA
Standard (CSA 1 994). For the CSA Standard, the simplified expressions, the general method
based on the modified compression field theoty as well as the strut-and-tie approach are
investigated.
4.1 AC1 Code Predictions
The AC1 Code (AC1 Committee 3 18, 1995) shear design equations for predicting the
shear strength of concrete beam elements are:
Vd Vc = 0.1 5 8 K b w d + 1 7.24pw - bwd 5 0.291Jr;bwd IN, mm] M
In lieu of equation [4.1], The AC1 Code allows the following simpler equation to be
used:
V, = 0.1 66&b,d with & 5 J6g
Equation 14.11 when applied at a distance d from the support yielded normalised shear
stress ratios (V / bd c) in the range of 0.19 to 0.22. This equation gives normalised shear
stress ratios of 0.1 8 to 0.20 for sections located a distance d from the edge of the loading
plate, indicating that there is relatively little sensitivity to moment- These values are quite
unconservative when compared to the experimental results, especially for the high-strength
concrete beams. Hence. the more conservative Equation [4.2] is more appropriate for the
prediction of the diagonal shear response of beam or one-way slab elements.
Figure 4.1 compares Equation [4.2] with the experimental results obtained in terms of
the normalised shear stress ratio versus specimen depth. Results are also summarised in
Table 4.1.
*. \ -AC[ Code
Figure 4.1: Cornparison of predictions using the AC1 simplified expression with test results
As can be seen in Fig. 4.1 and from Table 4.1, the use of the AC1 simplified equation
[42 ] results in conservative predictions of the concrete shear strength in al1 but the three
largest high-strength specimens. It is noted that this expression does not provide a uniforrn
factor of safety against shear failure. These equations are based on the shear causing
signi ficant diagonal cracking rather than on an ultimate shear strength. However, since the
beam elements tested failed only at a slightly higher shear than that causing diagonal
cracking, the AC1 Code assumption is reasonable for these cases. This is only true in the case
of beams or one-way slabs without transverse shear reinforcement and therefore cannot be
generalised to elements with transverse reinforcement.
Table 4.1: Cornparison of predictions using the AC1 simplified expression with test results
Specirnen Normalised Sbear AC1 1 % Error 1 1 1 Stress st Fiilure 1 Siniplifid Eq. 1 1
Error = (normalised shear stress at failure - prediction) / normalised shear stress
Note: negative error numbers indicate unconse~ative predictions.
4.2 CSA Simplified Expressions
The simplified expressions of the 1994 CSA Standard (CSA 1994) for the evaluation of
the nominal shear capacity of concrete, V,, are taken from Chapter 1 as follows:
a) For sections having either the minimum amount of transverse reinforcement required in
the CSA Standard (CSA 1994), or an effective depth not exceeding 300 mm :
b) For sections with effective depths greater than 300 mm and with less transverse
reinforcement than the minimum required :
215 8 Vc = ( ) ~ ~ ; b . d not icss than 0.10&bWd [N, mm] 1000 + d
14-41
Note: to obtain the nominal shear strength, the original CSA equations were modified by
a factor of (0.16610.2) (see Chapter 1 for explanation).
Figure 4.2 compares the predictions of Equations 14.31 and 14-41 with the experimental
results obtained in terms of the norrnalised shear stress versus specimen depth. Results are
also summarised in Table 4.2.
0.50
0 -45
0.40
:::: - 1994 CSA Code
0.25 _a, - > 0.20
0.15
o. 1 O
0.05
Figure 4.2: Comparison of predictions using the CSA simplified expressions with test resuIts
From Fig. 4.2 and Table 4.2 it'can be seen that these equations, which take into account
the size effect, result in quite consewative predictions for ail of the specimens except for the
485 mm deep beam in the high-strength concrete series. For this specimen the ptediction is
slightly unconservative by 1.8%.
By observing the trends in Fig. 4.2, the concem anses that the predictions of these
equations might not be conservative for the following cases:
i . Specimens with depths greater than about 1000 mm especially since there is a limit
of O. 1 on the normalised shear stress ratio given by Equation [4.4].
ii. Concrete strengths greater than about 60 MPa.
iii. Beams and one-way slabs with a reinforcement ratio, p, less than about 1.2%.
It can be noted as well, that these equations do not provide a uniform factor of safety
against diagonal shear failures, with very conservative predictions for srnall elements and
slightly unconservative predictions for larger elements (see Fig. 4.2).
Table 4.2: Comparison of predictions using the CSA simplified expressions with test results
Spccimen Normalised Shear CSA % Error
Stress at Failure Simplified Eq.
~ 1 . 2 % p=2.0% ~ 1 . 2 % p=2.0°h
N90 0.280 0.499 O. 166 40.6 1 66.75
N960 O. 1 79 0.189 0.1 15 35.78 39.03
H90 0.262 0.389 O. 1 66 36.65 57.35
Hl55 O. 197 0.269 O. 1 66 15.59 38.29
Hz20 O. 182 0.233 O. 166 8.75 28.63
H350 O. 164 0.198 O. 1 64 0.24 t 7.26
H485 0.147 0.148 O. 15 - 1 -79 - 1 -52
H960 0.1 18 O. 126 0.1 15 2.74 8.72
Error = (nonnalised shear stress at failure - prediction) / normalised shear stress
Note: negative error numbers indicate unconservative predictions.
4.3 Predictions Using the Modified Compression Field Theory and Strut-and-Tie
Models
The program Response 2 0 0 0 ~ developed at the University of Toronto by Michael P.
Collins and Evan C. Bentz (Collins and Bentz, 1998) was used to obtain predictions
according to the modified compression field theory. This program uses a sectional analysis
method which assumes that plane sections remain plane, combined with a dual-section
analysis and the modified compression field theory to determine the shear response.
strul and Ire mouel sectional modal
Figure 4.3: Sectional model versus strut-and-tie mode1 predictions for Kani's tests (Kani
1967), taken from Collins and Mitchell, 1997.
It is important to realise that for small shear span-to-depth ratios, a/d, sectional analysis
may not be appropriate. For small aid ratios, the applied load is close to the support and this
causes a disturbance in the flow of the stresses. There is a tendency for the forces to flow
from the point of application of the load, directly into the support reaction. This "strut
action" creates a "disturbed region" in which the assumptions of plane sections and of
uniformly distributed shear stresses are inappropriate.
Figure 4.3 compares test results for a series of k s m s tested by Kani (Kani, 1967) with
the predictions using a strut-and-tie model and the sectional analysis predictions obtained
using the modified compression field theory (Collins and Mitchell, 1997). From Fig. 4.3, it
can be seen that the sectional analysis method is more appropriate for aid ratios greater than
about 2.5. For a/d smaller than 2.5, a strut-and-tie analysis is more appropriate for this
particular series of tests.
For the predictions of the test results in this research program, the location chosen for
the sectional analysis was taken at a distance equal to the effective depth, d, from the edge of
the loading plate. This section is just outside of the disturbed region around the loading point
and is the most critical section for combined shear and moment effects. The measured
material properties were used for the predictions as well as the "as-built" cross-sectional
dimensions. The input and output values for each specimen are presented in Appendix A and
the results obtained from Response 2 0 0 0 ~ are summarised in Table 4.3 and Figs. 4.4 to 4.7.
The strut-and-tie model analysis was carried out for the series of beams tested. The
strut-and-tie model consisted of a direct strut going from the loading plate to the reaction
bearing plate. The bearings were sufficiently large to avoid crushing of the concrete at the
nodes in the strut-and-tie model. The failure mechanisms governing the strengths were
typically crushing of the compressive strut as it crosses the tension tie reinforcement. The
strut-and-tie design equations from the CSA Standard (CSA 1994) were used except that a
modification was made to the equation which gives the limiting compressive stress in the
compressive strut as a function of f: and the principal tensile strain, E,. The CSA Standard
expression limits the compressive strength in the strut to a value f,, as follows:
Where E, is calculated as
Where E, is the strain in the tension tie reinforcement which crosses the strut and 8, is
the smallest angle between the compressive strut and the adjoining tension tie. In order to
properly account for the influence of high-strength concrete Eq. [4.5] was changed to:
W here
The factor a, is the stress block factor giving the ratio of the average stress in the
rectangular compression block to the specified concrete strength. This stress block factor is
used in Clause 10 of the CSA Standard (CSA 1994) for the design for flexure. The CSA
Standard indicates that this stress block factor includes a reduction factor of 0.9 to account
for the difference between the in-place concrete strength and the strength of standard
concrete test cylinders. The introduction of this factor, a,, in Eq. 14-71 accounts for the
difference between the in-place strength and the cylinder strength and also accounts for the
presence of strain gradients across the compressive strut. Table 4.4 and Figs. 4.4 to 4.7 give
the shears corresponding to the strength predicted using the strut-and-tie model.
In comparing the predictions with the test results it is important to realise that the larger
of the two predictions made by the modified compression field theory and the strut-and-tie
model must be used (see Table 4 .9 .A~ can be seen in Table 4.5 and Figs. 4.4 and 4.5. the
predictions for the normal-strength concrete series with p=1.2% are slightly conservative but
very close to the actual experimental values, while the prediction for the normal-strength
concrete series with p=2% are more conservative, especially for the smaller specimens. The
predictions of the high-strength concrete series shown in Figs. 4.6 and 4.7 are very close to
the experimentally determined values, however some of the predictions are slightly
unconservative, particularly with the sectional analysis. While the smaller aggregate size of
10 mm was used in the predictions made with the modified compression field theory for the
high-strength concrete series, further reductions in the aggregate size could be made to
account for the fact that the diagonal cracks pass directly through the aggregates resulting in
a smoother failure surface. If this modification was to be made the predictions using the
modified compression field theory wouid be closer to the test results.
O 100 200 300 400 500 600 700 800 900
d (mm)
Figure 4.4: Predictions for the normal-strength, p=1.2% series
8 - - - Predicüon Envelope
../ Experimental
Figure 4.5: Predictions for the normal-strength. p=2% series
O 100 200 300 400 500 600 700 800 900
d (mm)
Figure 4.6: Predictions for the high-strength, p=1.2% series
*.. Experimental *. "
- - - Prediction Envelope
O 100 200 300 400 500 600 700 800 900
d (mm)
Figure 4.7: Predictions for the high-strength. p=2?40 series
Table 43: Modified compression field theory predictions
i
*:
b
Specimen
N90
NlSS
Nt20
N350
N485
N960
H90
Hl55
H220
H350
H485
H960
(vsxp- vp&
p (%)
1.2
2
1 -2
2
1.2
2
1.2
2
1 -2
7 - 1.2
2
1.2
2
1.2
2
1 -2
2
1.2
2
1.2
2
1.2
2
/ vexp *
Experimental Resu lts
va, (MPa)
1 -63
2.92
1 -66
2.19
1.36
1.61
1.26
1.43
1 .O7
1.22
1 -05
1-10
2.00
2.98
1.50
2.06
1.39
1.78
1.26
1.52
1.13
1.13
0.90
0.96
1 O0
Response 2000 Results
Ypred (MPa)
1.23
1.37
1.15
1.3 1
1.16
1.29
1 .O5
1.22
0.98
1.1 1
0.88
0.98
1.35
1.55
1.3 1
1 -49
1.33
1.53
1.20
1 -40
1.13
1.35
0.99
1.1 1
Difference between the predicted and
experimen ta1 resuits in %
24.7 1
52.96
30.73
40.39
14.73
20.34
16.94
14.37
7.76
9.03
15.92
11.51
32.44
47.80
12.9 1
27.8 1
4.58
14.0 1
4.23
7.4 1
-0.53
-19.38
-9.52
-1 5.58
Table 4.4: Strut-and-tie mode1 predictions
Specimen p Experimental Strut and Tie Difference betweeo (O/.) Results Predictions tbe prdicted and
var ( M W v P d (MW experimental results in % *
N90 1 -2 1.63 1.57 3.76
Table 4.5: Combined predictions of the modified compression field theory and the strut-and-
' ~ ~ e c i r n e n
N90
NI55
N220
N350
N485
N960
Hg0
Hl55
H220
H35O
H485
H960
*: (va,-
p (%)
1.2
2
1.2
2
1.2
2
1.2
2
1.2
2
1.2
2
1.2
2
1.2
2
1.2
2
1 -2
2
1 -2
2
1.2
2
vp,d) f
tie mode1
Experimen ta1 Resul ts
v,, (MPa)
1 .O3
2.92
1 -66
2.19
Corn bined Predictioos vw @Pa)
1.57
1.86
1.17
1.35
Difference between thepredictedand
experirnental results in % *
3 -76
36.36
29.20
38.34
1.36
1.61
1.26
1 -43
1 .O7
1.22
1 .O5
1.10
2.00
2.98
1.50
2.06
1.39
1.78
1.26
1.52
1.13
1.13
0.90
0.96
va, * 100
Prediction method
Stnit-and-Tie
Stmt-and-Tie
Strut-and-Tie
Strut-and-Tie
14.73
20.34
16.94
14.37
7.76
9.03
1 5.92
11.51
0.00
1 8.48
-9.13
6.19
-7.55
3.33
1.34
7.4 1
-0.53
- 19.38
-9.52
-1 5.58
1 .16
1 -29
1 .O5
1 -22
0.98
1.1 1
0.88
0.98
2.00
2.43
1 -64
1.93
1.50
1.72
1 -24
1.40
1.13
1.35
0.99
1.1 1
M. C. F. T.
M. C. F. T.
M. C. F. T.
M. C. F. T.
M. C. F. T.
M. C. F. T.
M. C. F. T.
M. C. F. T.
Strut-and-Tie
Strut-and-Tie
Strut-and-Tie
Strut-and-Tie
Strut-and-Tie
Strut-and-Tie
Stnit-and-Tie
M. C. F. T.
M. C. F. T.
M. C. F. T.
M. C. F. T.
M. C. F. T.
Chapter 5
Conclusions and Recommendations
The following conclusions were drawn from the results o f the experimental program on
the 1 2 bearn eiements:
1 ) The size effect is very evident in both the normal-strength and high-strength
concrete series. The shallower specimens were consistcntly able to resist higher
shear stresses than the deeper ones.
2) High-strength and normal-strength specimens of the same size and same
reinforcement ratios had almost equal shear stresses a t failure, showing no
significant gain in shear strength with increased concrete compressive strength. This
contradicts both the AC1 Code (AC1 Committee 3 1 8, 1995) and the CSA Standard
(CSA 1994) assumption that the concrete shear strength, V,, is proportional to the
square root of the concrete compressive strength, fl- 3) Increasing the amount o f longitudinal steel reinforcement increases the shear stress
a t failure in both the normal-strength and high-strength concrete series. The
influence of the longitudinal steel ratio, p, is found to attenuate with greater
specimen depth. This is due to the reduced effectiveness of the longitudinal steel in
controlling crack widths in the deeper elernents.
4) The AC1 Code (AC1 Committee 3 18, 1995) simplifted expression for the prediction
of the shear contribution o f concrete is highly unconservative for the deep, high-
strength concrete elements. This equation should include a term to account for
member size and a revision should be made to the term relating the concrete shear
strength to the concrete compressive strength.
5) The CSA Standard (CSA 1994) simpiified expressions for the prediction o f the shear
contribution o f the concrete are quite conservative in their predictions for al1 o f the
tested beam elements. These expressions should however be generalised to account
for the effect of the longitudinal steel ratio, p.
6) For these series of tests, with a/d of 2.5, the combined case of the modified
compression field theory and the strut-and-tie mode1 give much more realistic
predictions o f the shear capacity .
7) A modification to the compressive stress limit for the compressive struts gives much
more realistic predictions, particularly for the high-strength concrete beam elements.
8) The modified compression field theory accounts for important parameters such as
the size effect, the interaction with moment, the reinforcement ratio, p, and the
aggregate size.
It is hoped that the results obtained from this experimental program will help other
research efforts in better understanding the mechanisms affecting shear in concrete. It is
hoped as well that the experimental data obtained will be of use to researchers working
towards analytical models for the prediction of the shear response of concrete elements.
References
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AC 1 Cornmittee 3 1 8, "Building Code Requirements for Structural Concrete (AC1 3 18-95} and Commentary (AC1 318R-95) ". American Concrete Institute. Detroit, 1995, 369 pp.
ACI-ASCE Commitîee 326, "Shear and Diagonal Tension". AC1 Journal, v59, January- February-March 1962, p. 1-30,277-344 and 352-396.
A hrnad, S.H., Khaloo A.R., and Poveda A., "Shear Capacity of Reinforced High-Strength Concrete Beams ". AC1 Journal, v83, March-April 1986, p. 297-305.
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Collins, M.P., Mitchell, D.. Adebar, A., and Vecchio, F.J.. "A General Shear Design Method ". AC1 Structural Journal, v93 No. 1 , January-February 1996, p. 36-45.
CSA Comrnittee A23.3. "Design of Concrete Structures with fiplanatory Noies ". Canadian Standards Association. Rexdale, Dec. 1994.
Elstner, R.C., and Hognestad, E., "Laborutory Investigation of Rigid Frame Failure ". AC1 Journal, v53, January 1957, p.637-668.
E Izanaty , A. H ., Ni lson A.H ., and S late F.O., "Shear Capaciv of Reinforced Concrete Beams Using High-Sfrength Concrete ". AC1 Journal, v83, March-April 1 986, p. 290-296.
Kani, G.N.J ., "Basic Facts Concerning Shear Failure ". AC1 Journal, v63, June 1 966, p. 675- 692.
Kani, G.N.J., "How Sufe are our Large Reinforced Concrete Beams? ". AC1 Joumal, v64, March 1 967, p. 1 28- 14 1.
Kani, M., et al, "Kani on Shear in Reinforced Concrete ". University of Toronto Press, Toronto, Ont. 1979. 225 pp.
Ki m. Jin-Keun. and Park. Yon-Dong, "Shear Strength of Reinforced Hr'gh Strcngth Concrere Beams withour Web Reinforcement ". Magazine of Concrete Research, v46 No. 166, Marc h t 994, p. 7- 16.
Mphonde, A.G., and Frantz, G.C., "Shear Tests of High- and Low-Strength Concrete Beams rvithout Stirrups ". AC1 Journal, v8 1, July-August 1984, p. 350-357.
S h ioya, T., "Skar Properties of Large Reinforced Concrete Members '*- Special Report of Institute of Technology, Shimizu Corporation. No. 25, Feb. 1989.
Stanik. B., "The influence of Concrete Sfrength. Distribution of Longitudinal Reinforcement, Amounr of Transverse Reinforcement and Member Size on Shear Strcngrh of Reinforced Concrere Members ". Master's Thesis, Department of Civil Engineering, University of Toronto. Toronto, Ont.
Vecchio, F.J., and Collins, M.P., "The Response of Reinforced Concrete to In-Plane Shear and Normal Stresses ". Department of Civil Engineering, University of Toronto, March 1982. Publication No. 82-03, ISBN 0-7727-7029-8.332 pp.
Vecchio, F.J., and Collins, M.P., "Predicting the Response of Reinforced Concrete Beams Subjccted ro Shear Using the Mod~ped Compression Field Theory ". AC I Structural Journal, v85 No.3, May-June 1988, p. 258-268.
RESPONSE 2000~ Input and Output
Note: The program RESPONSE 2000~ version 0.7.5 (beta) was used.
67
Beam Cross Section
Crack Diaflram
/' 1 1 Principal Compressive stress
/ 1811 -- I l
...- - - - ------- -- Y (avg) = 1.21 mrnlm 2 . -
Axial Load = 0.0 k ~ - Moment:= 14.1 kNm Shear = 58.6 kN
Lon itudinal Strain
7l O
Shear Strain top
Shear on Crack
Transverse Strain top
Shear Stress
pot
Principal Tensile Stress
Response-2000 v 0.7.f NI 55s 1 998lIOl28 - 1:57 prn
Control : V-Gxy
I = 0.23 mmlm : 14.76 radlkm
y (avg) = 0.76 rnmlm --!!Y-_-- Axial Load = 0.0 kN
Crack Diaaram
Vinci~al Compressive Stress
Shear Strain
Y
Shear on Crack
Transverse Strain
Shear Stress
Principal Tensile Stre!
Moment:= 16.0 kNm Shear = 66.7 kN
Control : V-Gxy
y (avg) = 0.74 mmim -Y-. . ..- ------.- Axial Load = (5.0 k ~ - Moment:= 29.1 kNm Shear = 88.3 kN
I
I 1.2
Control : M q h i
, = 0.23 mmlm = 9.41 rad/km
Beam Cross Section
Crack Diagram
Vincipal Compressive Stress
-34.2 - .- . . -- .- -----
Strain Transverse Strain '
Shear Strain
[bot
bot
Shear on Crack. l0P
Shear Stress
Principal Tensile Stre
Control : M-Phi
9 = 7.77 radlkm l
Moment:= 32.3 kNm Shear = 97.7 kN
Bearn Cross Section
Crack Dianram
Longitudinal Strain
Shear Strain
Shear on Crack
rOP
Transverse Strain
pot
Shear Stress
Principal Tensile Strei
i"
OSE
Control: M-Phi 'Kr- --A
-- ~
= 0.07 mmlm 4.25 radlkm
y (avg) = 0.70 rnmlrn Gai ~ o a d y -0.3 kN Moment:= 79.1 kNm Shear = 152.9 kN
Beam Cross Section
Crack Diagram
Longitudinal Strain
Shear Strain
'rlncipal Compressive Stress
bol
Transverse Strain
Shear Stress
bot
Shear on Crack Principal Tenslle Stress
Gross Conc. Tranç(ln=16.49 -- ----- ---- .I__ _
220.6
5045.4
268
21 7
l883O. 1
oadina (N.M.V + dN.dM,dV)
0.00, 0.00, 0.00 + O,OO, 0.70, 1.00
Concrete Rebar k = 578 MPa
h = 1.83 MPa (aulo)
All units in millimetres Clear cover to reinforcment = 30 mm
1- 1 - .--
Control : V-üxy ------ I I I I I I I I I I
Control : Mah
y,(avg) = 2.39 mmlm Axial ~oad-=YoTkÏÜ-- Moment:= 121.9 kNm Shear = 172.9 kN
Beam Cross Section
Crack Diagram
'rinclpal Compressive Stress
Y bot Y
Shear Strain NP
Shear on Crack top
Transverse Strain OP
Shear Stress
Principal Tensile Stress
Control : Mahi
- I , = 0.12 mmlm %c # = 1.29 radlkm y (avg) = 0.56 mmlm Zia1 Load = - 0 K k ~ Moment:= 421.6 kNm Shear = 308.2 kN
Beam Cross Section Transverse Straln
Crack Diaaram Shear Strain Shear Stress
Incipal Compressive Stress
boi
Shear on Crack top
Principal Tensile Stre:
i
y (avg) = 1.53 mmlm Zia1 Load = OT~N Moment:= 4.9 kNm Shear = 35.2 kN
Beam Cross Section
Crack Diagrarn
q g i u d i n a l Strain
Shear Strain
6.85
1
bot
Shear on Crack
Transverse Strain ]top
Shear Stress
Principal Tensile Stress Ii
O 1 C .
: In, O , ! '3,
Control : V-Gxy
y (avg) = 0.67 mmtm . . - - - - - - Axial Load = -0.1 kN Moment:= 18.2 kNm Shear = 75.8 kN
Control : M-Phi
- 0.31 mmtm 1 -
Beam Cross Section
Crack Diagram
Vincl~al Compressive Stress
Strain
Shear Strain PP
Shear on Crack
Transverse Strain
Shear Stress
(bot
Principal Tensile Stress
/ 2.82
bot
OZZ
Controi : V-Gxy
Control : M-Phi
4 = 4.07 radlkrn y (avg) = 0.52 mmlm
- - - Axial Load = -0.4 kN Moment:= 91 .O kNm Shear = 175.5 kN
Beam Cross Section
Crack Diagram Shear Strain
'rlncipal Compressive Stress
--
1 bol
Shear on Crack
Transverse Strain ltop
Shear Stress
Principal Tensile Stress
Çontrol : iüi-Phi
- 0.09 mmlm 1 - = 2.61 rad/km
y (avg)= 0.51 mmlm wy- Axial Load = -0.4 kN Moment:= 168.0 kNm Shear = 237.5 kN
Beam Cross Section
--
Crack Diariram
Longitudinal Strain
---
bol
Shear Strain
lncipal Compressive Stress Shear on Crack
Transverse Strriin
Shear Stress
1-
Control : M-Phl--
Y (avg) = 0.40 mmlm - xy___" Axial ~ ~ ~ & = ~ o : ~ k ~ " - Moment:= 533.3 kNm Shear = 389.9 kN
Beam Cross Section
Crack Diagram
Longitudinal Strain
Shear Strain
Shear on Crack
Transverse Strain OP
Shear Stress
Principal Tensile Stress