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

AC I Comm ittee 3 1 8, "Building Code Requirements for Reinforced Concrete ". Arnerican Concrete Institute. Detroit, 195 1.

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.

Bazant, Z.P., and Kazemi, M.T., "Size Eficr on Diagonal Shear Failure of Beams without Stirrups". AC1 Structural Joumal, v88 No.3, May-June 1991, p. 268-276.

Bazant, Z.P., and Kim, Jin-Keun, "Size ef/ect in Shear Failure of Longitudinally Reinforced Beams ". ACI Journal, v8 1, September-October 1984, p. 456-468.

Bazan t, 2. P., and Sun, Hsu-Hue i, "Size Eflect in Diagonal Shear Failure: Influence of Aggregate Size and Stirrups ". AC1 Materials Joumal, v84 No.4, July-August 1987, p. 259- 272.

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Collins. M.P., Mitchell, D., "Prestressed Concrere Sfructures ". Response Publications. Canada. 1997.766 pp.

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