behaviour and modeling of deep beams with low shear span-to-depth ratios

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Behaviour and Modeling of Deep Beams with Low Shear Span-to-Depth Ratios

byZhen Yu Li

August, 2003

Department o f Civil Engineering and Applied Mechanics McGill University Montreal, Quebec

Canada

A thesis submitted to the Faculty o f Graduate Studies and Research in partial fulfillment o f the Requirements for the degree o f Master o f Engineering

Zhen Yu Li, 2003


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Behaviour and Modeling of Deep Beams

with Low Shear Span-to-Depth Ratios

ABSTRACT

The purpose o f this research program was to study the bahaviour o f full-scale deep beams

with realistic reinforcement details. In the overall research program, a total o f eight deep

beams were tested. A companion study by Li (2003) presents the results o f four o f these

beams. This research examines the other four beams, two without uniformly distributed

crack control reinforcement and two with distributed horizontal and vertical

reinforcement. The specimens dimensions were 2000 mm long and 400 mm thick, with

two specimens having heights o f 1160 mm and the other two heights o f 1840 mm. The

specimens were loaded with a central loading plate 300 mm long and 400 mm wide. The

end bearing plates were 250 mm long and 400 mm wide. All specimens contained seven

15M bars forming the main tension tie reinforcement.

The test results provided information on the influence o f the uniformly distributed

reinforcement and the crack and strain development up to failure. The ductility o f the

specimens containing only the main tension ties was limited due to the formation of

splitting cracks along the anchorages o f the main tension ties during the later stages of

testing. The uniformly distributed reinforcement provided additional tension ties that

increased the capacity and the ductility. Strut-and-tie models were developed to predict

the capacities. The FIP Recommendations (FIP 1996) were used to determine the

contributions o f the two major mechanisms, direct strut action and indirect strut action.

This approach gave very conservative strength predictions. More refined strut-and-tie

models were developed for the specimens with uniformly distributed reinforcement.

These refined models gave more accurate predictions o f the capacities o f the deep beams.

i


Comportements et modelisation de poutres profondes ayant un faible rapport portee I hauteur

RESUME

Le but de ce programme de recherche etait detudier le comportement de poutres

profondes pleines grandeurs ayant un detail darmature realiste. En tout, huit poutres

profondes furent testees dans le cadre de cette recherche. Quatre de ces resultats sont

presentes par Li (2003) dans une recherche similaire. La presente etude examine les

quatre autres specimens, done deux nont pas darmature uniformement distribute et

deux qui sont armes avec des aciers verticaux et horizontaux. Les specimens etaient

longs de 2000 mm et avaient une epaisseur de 400 mm et deux dentre eux etaient

hauts de 1160 mm et les deux derniers avaient une hauteur de 1840 mm. Les poutres

etaient chargees a Iaide dune plaque centrale de chargement mesurant 250 mm par

400 mm. Tous les specimens disposaient de sept barres dacier 15M formant Iarmature

de tension.

Les resultats ont permis dacquerir des informations sur in fluence de Iuniformite des

armatures et du developpement des fissures et des deformations avant rupture. La

ductilite des specimens ayant seulement Iarmature de tension etait limitee due a la

formation de lignes de rupture le long des ancrages de Iarmature de tension qui se

developpa vers la fin des essais. Le fait de placer des aciers uniformement distribues a

fournit plus de resistance en tension ce qui augmenta la capacite totale et la ductilite de

ces poutres. Des modeles de bielles-et-tirants furent developpes afin de predire les

capacites. Les recommandations du FIP (FIP 1996) ont ete utilisees pour determiner la

contribution de deux mecanismes m ajeures: Taction direct des bielles et Taction

indirecte des bielles. Cette approche donna des predictions tres conservatrices sur la

resistance. Des modeles raffines de bielles-et-tirants ont aussi ete developpes pour les

specimens ayant des aciers verticaux et horizontaux. Ces derniers ont donnes des

predictions plus pres de la realite concernant la resistance des poutres profondes.

ii


ACKNOWLEDGEMENTS

The author would like to express his gratitude to Professor Denis Mitchell for his skillful

guidance, encouragement and patience throughout this research programme. Thanks are

also given to Dr. William Cook for his invaluable support and assistance and for his

ability to keep things running so smoothly.

The completion o f this research would not have been possible without the patience and

valuable help o f the technical staff in the Jamieson Structures laboratory at McGill

University. The assistance of Ron Sheppard, Marek Przykorski, John Bartczak and

Damon Kiperchuk as well as the cheerful and enthusiastic aid o f Katherine Lai, Claudia

Correa, Ding Li and Jian Zhou is greatly appreciated. The French translation o f the

abstract by Felix A.Boudreaults is also greatly appreciated.

Gratitude is also extended to the following people who have aided towards the

completion o f this research: Professor Colin Rogers, Professor Yixin Shao, Ann Bless,

Sandy Shewchuk-Boyd, and Franca Della Rovere.

Finally the author would like to thank his wife, Ning Ning Liu for her moral support,

constant encouragement, understanding, endurance and love throughout his stay at

McGill University.

Zhen Yu Li

August 2003.

iii


TABLE OF CONTENTS

A bstract................................................................................................................................... i

R esum e.................................................................................................................................... ii

Acknowledgements............................................................................................................... iii

List of Figures ........................................................................................................................ vi

List of Tables ......................................................................................................................... ix

List of sym bols........................................................................................................................ x

Chapter 1 Introduction and Literature Review ................................................................. 1

1.1 Introduction...................................................................................................... 1

1.2 Disturbed R egions........................................................................................... 1

1.3 Previous Research on Strut-and-Tie Models .............................................2

1.4 FIP Recommendation and Refined Strut-and-Tie M odels........................ 7

1.5 Research Objectives......................................................................................... 8

Chapter 2 Description o f Test Specim ens.......................................................................... 18

2.1 Details o f Specim ens........................................................................................18

2.2 Material Properties...........................................................................................20

2.2.1 Concrete ................................................................................................... 20

2.2.2 Reinforcing S tee l..................................................................................... 21

2.3 Test Setup and Instrumentation...................................................................... 22

2.4 Testing Procedure ............................................................................................23

Chapter 3 Experimental Results ............................................................................. 35

3.1 Specimen B -3 N ................................................................................................ 35

3.2 Specimen B -3 S ................................................................................................. 45

3.3 Specimen B -4 N ................................................................................................ 57

3.4 Specimen B -4 S ................................................................................................. 6 6

iv


Chapter 4 Analyses and Comparison o f R esu lts.................................................................78

4.1 Simple Strut-and-Tie Models for Deep Beam B-3N & 4N ..................78

4.2 Predictions Using 1996 FIP Recommendations for Deep Beam B-3S

..............................................................................................................................79

4.3 Refined Strut-and-Tie Model for B-3S ....................................................80

4.4 Refined Strut-and-Tie Model for B-4S ....................................................82

Chapter 5 Conclusions ........................................................................................................... 89

References................................................................................................................................90

V


LIST OF FIGURES

1.1 Examples o f disturbed regions.................................................................................... 10

1.2 A simple strut-and-tie model for deep beams............................................................11

1.3 Compressive strength of diagonally cracked concrete, as a function of the

Principal tensile strain, s i ........................................................................................... 12

1.4 Compressive strength of strut, as a function of the angle o f crossing tension tie

........................................................................................................................................13

1.5 Use o f strut-and-tie model and sectional model for prediction of series o f beams.

........................................................................................................................................14

1. 6 Failure o f simply supported deep beams....................................................................15

1.7 Deep beam with transverse stirrups, tested by Uribe and Alcocer.........................16

1.8 Strut-and-tie model for deep beam tested by Uribe and Alcocer..........................17

2.1 Overall view o f specimens.......................................................................................... 24

2.2 Details o f Specimen B-3S............................................................................................25

2.3 Details o f Specimen B-4S........................................................................................... 26

2.4 Details o f Specimen B-3N & 4 N ............................................................................... 27

2.5 Representative concrete compressive stress-strain curves......................................28

2.6 Measured concrete shrinkage strains......................................................................... 28

2.7 Stress-strain curves for the 10M bars.........................................................................29

2.8 Stress-strain curves for the 15M bars.........................................................................29

2.9 Specimen B-4S under the MTS testing machine.................................................... 30

2.10 Details o f bearing and loading devices...................................................................... 31

2.11 LVDT locations for specimen B-3S & 3N................................................................32

2.12 Strain gauge locations and crack measurement lines for specimen B-3N 32

2.13 Strain gauge locations and crack measurement lines for specimen B-3S 33

2.14 LVDT locations for specimen B-4S & 4N............................................................. 33

2.15 Strain gauge locations and crack measurement lines for specimen B-4N 34

2.16 Strain gauge locations and crack measurement lines for specimen B-4S 34

3.1 Load-deflection response o f Specimen B-3N .......................................................... 38

vi


3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

3.10

3.11

3.12

3.13

3.14

3.15

3.16

3.17

3.18

3.19

3.20

3.21

3.22

3.23

3.24

3.25

Strains in main tension tie o f Specimen B-3N, determined from strain readings

.............................................................................................................................................39

Longitudinal strains from LVDTs at the level o f main tension tie o f Specimen

B-3N..............................................................................................................................40

Calculated rosette strain responses in Specimen B-3N..........................................41

Cracking patterns o f Specimen B-3N at first yielding o f main tension tie 42

Cracking patterns o f Specimen B-3N at general yielding o f main tension tie .. 43

Cracking patterns o f Specimen B-3N at peak load.................................................44

Load-deflection response o f Specimen B-3S.......................................................... 48

Strains in main tension tie o f Specimen B-3S, determined from strain readings.

...................................................................................................................................... 49

Strains in vertical distributed reinforcement of specimen B-3S, determined from

strain readings.............................................................................................................. 50

Rosette strain responses in Specimen B-3S.............................................................51

Calculated rosette strain responses in Specimen B-3S...........................................52

Longitudinal strains from LVDTs at the level o f main tension tie o f Specimen

B-3S.......................................................... 53

Cracking patterns o f Specimen B-3S at first yielding o f main tension tie...........54

Cracking patterns o f Specimen B-3S at general yielding o f main tension t ie ... 55

Cracking patterns o f Specimen B-3S at peak load..................................................56

Load-deflection response o f Specimen B-4N..........................................................59

Strains in main tension tie o f Specimen B-4N, determined from strain readings.

60

Calculated rosette strain responses in Specimen B-4N..........................................61

Longitudinal strains from LVDTs at the level o f main tension tie of Specimen

B-4N............................................................................................................................... 62

Cracking patterns o f Specimen B-4N at first yielding of main tension tie 63

Cracking patterns o f Specimen B-4N at general yielding o f main tension tie... 64

Cracking patterns of Specimen B-4N at peak load.................................................65

Load-deflection response o f Specimen B-4S.......................................................... 70

Strains in main tension tie o f Specimen B-4S, determined from strain readings.

vii


...................................................................................................................................... 71

3.26 Strains in vertical distributed reinforcement o f specimen B-4S, determined from

strain readings........................................................................................................ 72

3.27 Calculated rosette strain responses in Specimen B-4S........................................... 73

3.28 Longitudinal strains from LVDTs at the level o f main tension tie o f Specimen

B-4S.........................................................................................................................74

3.29 Cracking patterns o f Specimen B-4S at first yielding o f main tension tie 75

3.30 Cracking patterns o f Specimen B-4S at general yielding of main tension tie ... 76

3.31 Cracking patterns o f Specimen B-4S at peak load................................................... 77

4.1 Simple strut-and-tie model for Specimen B-3N....................................................... 84

4.2 Simple strut-and-tie model for Specimen B-4N....................................................... 84

4.3 Strut-and-tie model for Specimen B-3S using FIP Recommendations (1996).

.......................................................................................................................................85

4.4 FIP model for Specimen B-3S assuming strain hardening and spreading of

yielding in main tension tie.................................................................................. 8 6

4.5 Refined strut-and-tie model for Specimen B-3S.....................................................87

4.6 Refined strut-and-tie model for Specimen B-3S assuming strain hardening and

spreading o f yielding in main tension tie........................................................... 8 8

4.7 Refined strut-and-tie model for Specimen B-4S.................................................... 8 8

viii


LIST OF TABLES

1.1 Effective stress levels in struts......................................................................................... 4

2.1 Concrete mix proportions................................................................................................. 20

2.2 Concrete properties.................................................................................................. 21

2.3 Reinforcing steel properties..............................................................................................22

3.1 Key load stages for Specimen B-3S.............................................................................37

3.2 Key load stages for Specimen B-3N .............................................................................47

3.3 Key load stages for Specimen B-4S.............................................................................58

3.4 Key load stages for Specimen B-4N .............................................................................67

4.1 Comparison between the prediction and testing results..............................................83

ix


LIST OF SYMBOLS

a shear span

aw effective length of vertical stirrups

As area o f reinforcing steel

b width o f beams

C forces in compression strut

c distance from extreme compression fiber to neutral axis

d distance from extreme compression fiber to centroid o f main tension reinforcement

db nominal diameter o f bar, wire or prestressing strand

Ec modulus o f elasticity o f concrete

Es modulus o f elasticity o f reinforcing steel

f 2max limiting compressive stress o f diagonally cracked concrete

f c concrete stress

f c ' specified compressive strength of concrete

f cr concrete cracking stress

f cu limiting compressive stress in concrete compression strut

f r modulus o f rupture o f concrete

f sp splitting tensile strength o f concrete

fuit Ultimate tensile strength o f reinforcement

f y specified yield strength o f nonprestressed reinforcement

f yt specified yield strength o f transverse reinforcement

h overall depth o f beams

ki reinforcing bar location factor in development length expression

k2 reinforcement coating factor in development length expression

k3 concrete density factor in development length expression

k4 bar size factor in development length expression

lb length o f bearing

Id development length o f reinforcement

Idb basic development length

Idh development length of standard hook in tension, measured from critical section to

x


outside end o f hook (straight embedment length between critical section and start of

hook plus radius o f bend and one bar diameter)

n number o f bars being developed along the potential plane o f bond splitting

p total applied load

s spacing o f reinforcement parallel to axis o f the member

T tension force in reinforcement

V shear force at section

z effective lever arm at section

sc compressive strain

gcr strain in concrete at cracking

Erupt rupture strain o f reinforcement

es strain in reinforcing steel

Esh strain o f reinforcement at strain hardening

ex horizontal tensile strain

y yield strain o f reinforcement

Et principal tensile strain

e2 principal compressive strain

0 angle o f compressive strut from horizontal direction

Chapter 1

Introduction and Literature Review

1.1 Introduction

Strut-and-tie models have become useful tools to design regions o f both reinforced and

prestressed concrete structures. It provides a simple tool for the analysis o f disturbed

regions. Strut-and-tie model design procedures were first codified in the Canadian

Standards Association Standard A23.3 in 1984 (CSA 1984). The United States has just

recently adopted this design method (American Concrete Institution Code, the year 2002).

The main advantage o f this method is that designers can visualize the flow o f stresses.

Traditional engineering beam theory is based on the assumption that plane sections may

remain plane, but it does model how the forces were introduced into the members. This

chapter first presents the definition and behaviour o f disturbed regions, and then provides

a brief historical review o f the development o f strut-and-tie models. Finally this chapter

presents recent developments o f the 1996 FIP Recommendation (FIP 1996) and the use of

refined strut-and-tie models. Information on the developments o f strut-and-tie models is

given in the publication Recent Approaches to Shear Design o f Structural Concrete

(ASCE-ACI 1998) and in the ACI Special Publication Experimental Verification of

Strut-and-Tie Models (ACI 2002).

1.2 Disturbed Regions

Regions o f concrete members in which the traditional engineering beam theory is

appropriate (in other words, the plane section remains plane and the shear stress can be

assumed to be uniform over the nominal shear area) are sometimes referred to as B-

regions (where B represents beam or Bernoulli). Their internal state o f stress complies

with the Bernoulli hypothesis and satisfies equilibrium with the sectional forces (bending

and torsional moments, shear and axial force). On the other hand, the regions adjacent to

concentrated loads, supports or abrupt changes in cross section are so-called Disturbed

1


Regions or D-regions (where D represents discontinuity, disturbance or detail etc.). Their

strain distribution is significantly non-linear due to a complex internal flow of stresses.

Several examples o f disturbed regions are shown in Fig 1.1 where dashed lines represent

the flow of compressive stresses and solid lines represent the tensile ties. D-regions are

indicated by shaded areas. Figure 1.1 also shows a deep beam subjected to concentrated

loading. Because o f the complex flow o f stresses from the top plate to the bottom plates,

the entire deep beam is a disturbed region.

1.3 Previous Research on Strut-and-Tie Models

In 1899, Ritter suggested truss models to analyze and design reinforced concrete beams.

In the early 1920s, Morsch introduced truss models for torsion analysis. These early truss

models consist o f compression chords, tension chords and diagonal compressive struts,

assumed to be inclined at 45 to the longitudinal direction. These truss models established

the basis o f code development in Europe and North America for design o f conventional

reinforced concrete beams.

Truss models have gained increased popularity in the last two decades for the design o f

disturbed regions. Strut-and-tie models are the most appropriate method for the design of

disturbed regions. The essential steps in design using strut-and-tie models are to visualize

the flow o f internal stress and to establish properly equilibrated models. Experience is

necessary to determine the more efficient strut-and-tie models for different situations.

Under most circumstances, for any given structures, many strut-and-tie models may

feasible so that there is not a unique solution. Schlaich and Shafer (1984) and Schlaich et

al. (1997) suggested choosing a strut-and-tie model after carrying out an elastic analysis.

They recommend that the angle o f the compression diagonals be within 15 o f the

angles o f the resultant o f the compressive stresses obtained from a linear elastic analysis

when choosing the geometry o f the ideal truss model.

2


In developing a simple strut-and-tie model it is necessary to first idealize the flow of

internal forces. In disturbed regions, high unidirectional compressive stresses may be

modeled as compressive struts, and tension ties are used to idealize the principal tension

reinforcement. For a deep beam, the loads are assumed to be transferred to the supports

by compressive concrete struts, requiring a tension tie between supports to satisfy

equilibrium. The internal resisting mechanism can be represented by a strut-and-tie

model, as shown in Fig 1.2.

There has been a significant amount o f research to investigate the limiting stresses in

concrete compressive struts and the influence of anchorage details on the geometry o f

these struts. Thurliman et al. (1983) and Marti (1985) draw the conclusion that the

compressive stress in the struts be not more than 0.60 f c', and Ramirez and Green (1991)

suggested the limiting compressive stress o f 2 . 4 9 (in MPa units). Schlaich et al.

(1987) and MacGregor (1997) proposed the effective stress level based upon different

conditions o f struts, shown in Table 1.1.

Vecchio and Collins (1986) suggested an equation for limiting compressive stresses for

the modified compression field theory that considered the strain softening of diagonally

cracked concrete (see Fig 1.3). The limiting compressive stress,/}:, is calculated as:

f cf j n tax0 . 8 + 170s, f c (1-1)

where: f c' = concrete compressive strength.

i = principal tensile strain where:

= ex +{sx + s2)cot20 ( 1.2)

where: ex = horizontal tensile strain,

2 = principal compressive strain,

6 = angle between the principal compressive strain and horizontal direction.

3


Table 1.1 Effective stress levels in stmts (Schlaich et al. 1987 and

MacGregor 1997)

Conditions of StrutEffective

Stress Level

Proposed

by

Undisturbed and uniaxial state of compressive stress that

may exist for prismatic struts0.80 f c'

Schlaich et

al.

(1987)

Tensile strains and /or reinforcement perpendicular to the

axis o f the strut may cause cracking parallel to the strut

with normal crack width

0 . 6 8 f c'

Tensile strains and /or reinforcement at skew angles to the

axis o f the strut may cause skew cracking with normal

crack width

0.51 f c

Skew cracks with extraordinary crack width (expected if

modeling o f the struts departs significantly from the theory

o f elasticitys flow o f internal stresses)

0.34 f c

Uncracked uniaxially stressed struts o f fields 1 . 0 v2/ c'(a)

MacGregor

(1997)

Struts cracked longitudinally due to bottle-shaped stress

fields with sufficient transverse reinforcement0.80 v2/ c'(a)

Struts cracked longitudinally due to bottle-shaped stress

fields without transverse reinforcement0.65 v2/ c'(a)

Struts in cracked zone with transverse tensions from

transverse reinforcement0.60 v^/c'(a)

(a) where v 2 =0.55 + 1.25

4


The CSA Standard A23.3 Design o f Concrete Structures for Buildings (1984, 1994),

Ontario Highway Bridge Design Code (CSA 1991) the Canadian Highway Bridge Design

Code (CHBDC 2000) and the American Association o f State Highway and

Transportation Officials (AASHTO 1993) have adopted the strut-and-tie methods

developed by Collins and Mitchell (1986, 1987). The expressions for the limiting

concrete compressive stress in the struts are given below:

Where: f cu =

f c =

l =

where: 6 2 -

Bs =

Figure 1.4 shows the variation o f the compressive strength f cu as a function o f the angle,

6S, between the strut and the tension tie crossing the strut (Collins and Mitchell 1987).

The Canadian Standards Association, Design of Concrete Structures for Buildings

(CSA 1984) provides the following limitations for the compressive stresses in the nodal

zones o f strut-and-tie models (Collins and Mitchell 1986):

0 .850,// for nodes bounded by compressive struts and bearing areas only (CCC nodes).

0.75a /c for nodes with only one direction tension tie is anchored (CCT nodes).

0.600 /c ' for nodes where tension ties are anchored in more than one direction (CTT

nodes).

5

fcu = ----- ^ 0-85 f'c (1.3)Jcu 0.8 + 170^

limiting compressive stress in the strut,

concrete cylinder strength,

principal tensile strain, where.

i = f s + f 2) cot 20 s (1-4)

principal compressive strain in the strut, taken as 0 .0 0 2 .

strain in the tension tie crossing the strut.

smallest angle between the strut and the tension tie crossing the strut.


Figure 1.5 shows the shear strength o f a simply supported reinforced concrete beam under

two point loading, as a function o f shear span-to-depth ratio, a/d. The beams in this series

had been tested by Kani in the 1960s and were published by Kani et al in 1979. The

tested beams contained only horizontal main tension tie without distributed

reinforcement. It can be easily visualized that the sectional model is appropriate when the

shear span-to-depth ratio is 2.5 or higher. The same amount o f tensile reinforcement and

different size o f bearing plates for each beam were used. This figure shows that the strut-

and-tie model provides more accurate predictions for shear span-to-depth ratios, a/d o f

less than about 2.5.

Numerous studies have investigated the stress distributions in deep members as a function

o f the shear span-to-depth ratio, a/d. For example, the size o f the bearing plates may

affect the principal stresses significantly and is very critical in the immediate vicinity o f

supports and the anchorage conditions o f the tensile reinforcement is another important

aspect for the design o f deep beams. Leonhardt and Walther (1966) carried out

experiments on simply supported deep beams at University o f Stuttgart. The applied loads

were introduced from either the top surface or a bottom ledge o f the specimen to

investigate top and bottom loading effects. When a uniformly distributed load was

applied to the top surface o f the beam (see Fig 1.6 (a)) the load path consisted mainly o f

compressive stresses fanning into the supports. A minimum reinforcement ratio o f 0.2 %

in both directions was concluded to be adequate (Park and Paulay 1975). When the load

was applied through a bottom ledge o f the beam (see Fig 1. 6 (b)), the total applied load

was transferred by means o f vertical stirrups into the compressive area o f the beam.

Therefore, a vertical stirrups amount must be provided to satisfy the force requirement as

well as to control cracks. For this test series the thickness of the deep beams was only 200

mm and small diameter bars with unusual anchorage details were used for the main

tension ties.

In order to design disturbed regions more accurately, elastic finite element analysis may

be used to determine the flow of stresses inside the concrete member prior to cracking,

however it is not appropriate to predict for the cracked concrete member due to the

6


significant redistribution o f stresses after cracking. Non-linear finite element analysis can

be used to predict the full response including the post-cracking response o f reinforced

concrete members. The computer program, FIELDS, was developed (Cook and Mitchell

1988) using two-dimensional non-linear finite elements and the compression field theory

(Vecchio and Collins 1986). Program FIELDS, along with a series o f tests on disturbed

regions (Cook and Mitchell 1988) was used to provide additional guidance during the

development o f the strut-and-tie design provisions o f the 1984 CSA Standard (CSA

1984).

1.4 FIP Recommendations and Refined Strut-and-Tie Models

Design approaches using strut-and-tie models have been specified in the CSA Standards

(1984, 1994) and in Appendix A of the ACI Code (2002). While these codes do not

provide specific guidelines on suitable strut-and-tie models for different situations, the

FIP Recommendations (FIP 1996) provide such guidance. For deep beams, the

Recommendations assume that the load is transferred from the loading plate to supports

by both a direct strut mechanism and an indirect strut mechanism. The direct strut

mechanism means that part o f the load is transferred to the support directly through an

inclined strut, while the indirect strut mechanism assumes that the remainder is carried by

stirrups in a truss with two inclined struts at each beam end. In accordance with the 1996

FIP Recommendations, the part of the total load transferred by indirect strut mechanism is

based on the shear span-to-intemal lever arm ratio, a/z, as given by 1/3 (2a/z-l). For using

this equation, the shear span is taken as the distance between the centres o f the loading

and support bearing plates.

Uribe and Alcocer (Mitchell et al 2001) carried out an experiment on a deep beam

containing transverse reinforcement, using design approach o f the 1996 FIP

Recommendations to predict the maximum load. Figure 1.7 shows Specimen MT that is

simply supported on two bearing plates. The specimen had vertical stirrups placed over

the bearing nodal zone on one end with the other without this reinforcement in order to

investigate the effect o f confinement along the bar anchorage. The beam was intentionally

7


designed to avoid a flexure failure o f the main tension tie so that it became possible to

impose large shear force demands. The strut-and-tie model was established in accordance

with the 1996 FIP Recommendations. It was assumed that stirrup yielding controlled the

failure mode. From the testing results, yielding was recorded in nearly all of the stirrups

at the peak load. This strut-and-tie model, shown in Fig. 1.8, gave a conservative

prediction because the contribution of horizontally distributed reinforcement was not

considered.

The simple strut-and-tie model is based upon the assumption that the compressive strut

may be represented by straight lines from the loading bearing plate to the support bearing

plates directly, and it neglects the contribution o f any uniformly distributed

reinforcement. As a result, this simply strut-and-tie model usually gives conservative

capacity predictions. A more refined model was developed so as to provide a more

accurate estimate o f the failure load. The refined strut-and-tie model accounts for not only

the main tension tie reinforcement but also the uniformly distributed reinforcement

normally provided for crack control. Mitchell et al. (2001) adopted refined strut-and-tie

model to predict the capacities of deep beams tested by Leonhardt and Walther (1966).

This refined model utilized the additional horizontal reinforcement in the tension zone

and provided more accurate predicted capacities compared with the simple strut-and-tie

model. This refined model gave conservative predictions. The CSA Standard A23.3-94

and the 1996 FIP Recommendations all require that the uniformly distributed

reinforcement should be provided for crack control at service load levels.

1.5 Research Objectives

This research program is part o f a comprehensive study conducted at McGill University

to investigate the behaviour of deep beams with various shear span-to-depth ratios and to

model the load transferring system using strut-and-tie models. This thesis reports on four

o f a total o f eight full-scale deep beams that were constructed and tested under

concentrated loading. Li (2003) reported on the other four specimens.

8


The objectives o f this research programme are:

1 . to study the complete behaviour of full-scale reinforced concrete deep beams,

2 . to compare the predicted responses using simple strut-and-tie models, strut-and-tie

models using 1996 FIP Recommendations and refined strut-and-tie models,

3. to investigate the crushing concrete stress and the role o f anchorage o f the main

tension tie on the behaviour,

4. to investigate the influence of crack control reinforcement for various span-to-

depth ratios.

9


Deep Beam r----- ----------

Corbel

Beam with opening

y

iFooting

Beam with Dapped Ends

Figure 1.1 Examples of disturbed regions

(Adapted from CAC Handbook (CAC 1995))

10


Tension tie

Figure 1.2 A simple strut-and-tie model for deep beams

(Adapted from on Collins and Mitchell (1986) and CAC (1995)

li


r^2niux

K 5

(a) Average concrete compressive stress,^, from strains e, and e2

1.2

0.8

~ 0.6

0.4

0.2

(b) Reduction in compressive strength with increasing values of e,

Figure 1.3 Compressive strength of diagonally cracked concrete,

as a function of the principal tensile strain, 6i

(Taken from Vecchio and Collins 1986)

12


Figure 1.4 Compressive strength of strut, as a function of the angle of

crossing tension tie (Collins and Mitchell 1986)

13


0 .2 5 -

0 .2 0 -

0.15

bdf:

0 . 1 0 -

0.05

o 152 x 76 x 9.5 mm plate 152 x 152 x 25 mm plate

-tj> 152 x 229 x 51 mm plate

24 in (610 mm)

/ / = 27.2 MPa fy = 372 MPa

max. agg. = 19 mm d = 538 mm b = 155 mm

A, = 2277 mm2

76 .7 4 ,75 -

strut-and-tie model sectional model

Figure 1.5 Use of strut-and-tie model and sectional model for prediction of

series of beams (taken from Collins and Mitchell 1991)

14


(a) Load was introduced at top surface

(b) Load was introduced at bottom ledge

Figure 1.6 Failure of simply supported deep beams (Leonhardt and Walther

1966).

15


'4 No.6 0 200 {No. 4 stirrups) -2 No. 8 fi No. 4 250

-U No. 4 .30

200305

250

250

305220

100 jTHTjr.7 0 l 4 ( J U N ' 8

*2 No.5 No. -5 No.

1200

^ - 2 NO. 8-

^ 6 No. 4

-350

A - A

.4 No. 8

-N o . 4

A No- 8 2 No. 8 + 3 No. 8

- 5 No. 8

.4 No. B-

- 6 No. 4

U Jo . 4 -

3 No. {

- 5 No. J

4 No. 8 .

, - 6 No. 4-

L350

C - C'

- 5 No. 8-

D im cnsions in mm

Figure 1.7 Deep beam with transverse stirrups, tested by Uribe and

Alcocer (Mitchell et al. 2001)

16


.*00.

119

m

1400

(a) Strut-and-tie model

140

: : 55409 kN

1110

409 kN 35

396 kN

(b) Direct strut mechanism

762 kN

MH646,5 kN 646.5 h i" \

211

/ \ftm / p

646.5: kN

908

762 kN

(c) Indirect truss mechanism

Figure 1.8 Strut-and-tie model for deep beam

tested by Uribe and Alcocer (Mitchell et al. 2001)

17


Chapter 2

Description of Test Specimens

Eight full-scale deep beams were constructed and tested in order to study their complete

responses as part of a testing program. This thesis reports on four o f these beams, the

other four beams are reported by Ding Li (2003). Their dimensions have been chosen in

order to provide experimental evidence of the change in response as the beams become

deeper. These deep beams were designed using the strut-and-tie approach of the CSA

Standard A23.3-94 (CSA 1994) and the 1996 FIP Recommendations (FIP 1996). The bar

size and spacing o f the uniformly distributed horizontal and vertical reinforcement was

chosen to satisfy the provisions o f Clause 11.5.5 o f CSA A23.3-94, that states that the

ratio o f reinforcement area to gross concrete area shall not less than 0.002 in each

direction in order to satisfy the minimum reinforcement ratio requirements for crack

control.

2.1 Details of Specimens

Deep beam specimens (B-1S & IN, B-2S & 2N, B-3S & 3N and B-4S & 4N) were cast

with normal-strength concrete having an assumed design concrete compressive strength,

f c , of 35 MPa. These specimens have the same overall length o f 2 m and the same

thickness o f 400 mm. The depths o f deep beams are 520 mm for B-1S & IN, 810 mm for

B-2S & 2N, 1160 mm for B-3S & 3N and 1840 mm for B-4S & 4N respectively. The

complete test series o f eight beams is shown in Fig. 2.1.

The main tensile reinforcement on the bottom of each deep beam was identical,

consisting o f 7-15M bars in a single layer. The longitudinal reinforcement was anchored

with 90-degree standard hooks to achieve adequate development length. The specimens

contained 9-10M two-legged stirrups spaced at 219 mm resulting in a reinforcement ratio

o f 0.225% in the vertical direction. All the standard hooks and bends conformed to

Clause 12.2 o f CSA Standard A23.1 (CSA 2001).

18


The tension development length, o f the 15M reinforcing is determined as:

ld = 0 . 4 5 k l k 2k 3k 4 ^ j = d b = 0 .45x1 .0 x 1 .0 x 1 .0 x 0 .8x-^22rxl6 = 389mm (2.1)

v / c' V35

lhb = 100db l { f i = 1QJ ^ 16 = 270mm (2.2)

The corresponding side concrete cover is 60 mm and the net cover on the bar extension

beyond the 90 hooks is 50 mm and hence the basic development length, kb, is multiplied

by the modification factor 0.7 in accordance with Clause 12.5.3b..

lhb = 270x0 .7 = 189mm (2.3)

In accordance with Table N. 12.5.2 (CAC 1995) the distance from the point o f tangency

o f the hook to the end o f the hook is equal the inside bend radius plus the bar diameter,

db, or 98mm.

A minimum area of reinforcement o f 0.002Ag must be provided in each direction. Using

10M stirrups, Av=200 mm , the required spacing o f transverse reinforcement, in

accordance with the requirements o f Clause 11.5.5, is:

s < -----^22-----= 250mm, and shall not exceed 300 mm (2.4)0.002x400 v 7

Along the beam length o f 2 m, 9-10M stirrups are required to fulfill the minimum crack

control requirement. In order to arrange the stirrups uniformly, the spacing o f the

transverse reinforcement was chosen to be 219 mm.

Over the depth o f the beam, the spacing o f the horizontal reinforcement is 262 mm for

specimen B-3S and 247 mm for specimen B-4S. These pairs o f horizontal bars had 90

degree bend hooks at their ends. The steel was placed such that the free end extensions of

the hooks were lap spliced over a length of 230 mm through the thickness o f the beam.

The overall specimen details are shown in Figs 2.2 to 2.4.

19


2.2 Material Properties 2.2.1 Concrete

The eight beams were cast with ready-mix concrete. The specified concrete strength was

30 MPa with a water to cement ratio (w/c) of 0.47 and a maximum aggregate size of

Table 2.1 Concrete mix proportions

Components Quantity (kg/m3) Volume (L)

Cement 340 108.14

Fine aggregate 787 290.74

Coarse aggregate 20 mm 472 175.35

Coarse aggregate 14 mm 575 214.32

Water 160 160

Total 2334 998.55

Admixtures (ml /100 kg)

Water reducing agent 313 1.06

Air entraining agent 56 0.19

Retarding agent 95 0.32

Slump 150 mm

Air content 6.0 %

Water / cement ratio 0.47

Density 2334 kg / m3

20 mm. The slump and air content measurements were taken upon delivery and are given

in Table 2.1. The test specimens were covered with wet burlap and plastic sheeting a few

hours after casting, and were kept moist during the first 10 days. The control cylinders

and flexural beams were stripped o f their formwork and cured in 100% humidity

condition 24 hours after casting. The average compressive s tre n g th ,^ ', was determined

from the results o f testing 6 standard, 150mm diameter by 300 mm long, concrete

cylinders. Representative compressive stress-strain curves for the concrete are shown in

20


Fig 2.5. The average modulus o f rup ture,/-, derives from 6 flexural beam tests, 3 in a wet

surface condition and another 3 in a dry surface condition. The 150 x 150 x 400 mm sized

beams were subjected to third-point loading over a span of 300 mm. In addition, three

Brazilian split cylinder specimens, 150 mm diameter by 300 mm long cylinders, were

tested to provide the splitting tensile strength, f sp. The average values o f the measured

concrete properties are given in Table 2.2. The concrete compressive strength of the six

cylinders varied between 38.4 and 40.0 MPa. Shrinkage strains o f the concrete over time

were determined from standard shrinkage specimens measuring 3 x 3 x 10 in. One

shrinkage specimen was air dried, while the other was cured in 100% humidity condition.

The shrinkage strains are shown in Fig 2.6.

Table 2.2 Concrete properties

/ ' (MPa)

average

(std. dev.)

/ ( MPa) average

in wet condition

(std. dev.)

/(M P a )

average

in dry condition

(std. dev.)

fsp (MPa)

average

(std. dev.)

38.6 5.91 4.34 3.67

(1.072) (0.327) (0.261) (0.076)

2.2.2 Reinforcing Steel

Steel reinforcement consisted o f 10M and 15M deformed bars with a specified grade o f

400 MPa. Three tensile coupons were tested for each bar size. An extensometer with

gauge length o f 50.8 mm was used to determine the strains during testing. The properties

o f the reinforcing bars are summarized in Table 2.3. The 10M and 15M reinforcing steel

exhibit a distinct yield plateau. The reinforcing steel must conform to CSA Standard

G30.18 and be o f weldable grade. Stress-strain curves for the two different bar sizes are

shown in Figs 2.7 and 2.8. The modulus o f elasticity for all reinforcing steel has been

regarded as 200 GPa for both design and analysis purposes.

21


Table 2.3 Reinforcing steel properties

Bar

Description

f y, MPa

average

(std. dev)

Sy

Esh

average

(std.dev)

f uit, MPa

average

(std.dev)

Erupt

average

(std.dev)

10M459.8

(11.72)0.00230

0.0195

(0.00245)

616

(16.60)

0.0208

(0.0016)

15M455.8

(9.30)0.00228

0.0192

(0.00042)

578

(4.787)

0.297

(0.0396)

2.3 Test Setup and Instrumentation

The deep beams were installed under the 11,400 kN capacity MTS universal testing

machine (see Fig 2.9). Figure 2.10 shows the bearing details used for specimens. The

deep beams were simply supported on the laboratory strong floor. The bearing plates

were 25 mm thick and were 250 x 400 mm. The bearing plates rested on a rocker, having

a radius o f 250 mm, and in turn, rested on two 152 mm diameter rollers placed between

two 76 mm thick steel rectangular plates. These support conditions permitted elongation

o f the beams and rotation at the ends. Monotonic load was transferred through the

spherical seat o f the testing machine to the top loading plate at midspan o f the beams. The

size of the 35 mm thick top loading plate was 300 x 400 mm. High-strength capping

compound was placed at the interface between the beam and bearing and loading plates.

Vertical displacements o f the beams at the two supports and at mid-span were measured

by three Linear Voltage Differential Transducers (LVDTs). The corrected central

deflection o f each beam was calculated by subtracting the average reading o f the LVDTs

at the two supports from the LVDT reading at mid-span. Five LVDTs were placed

horizontally at the level o f the centroid o f the longitudinal tension reinforcement, between

the centers o f two bottom bearing plates. The LVDTs were attached to short lengths of

threaded rod that were grouted into the concrete. These connecting rods were placed at

22


350 mm on centres, such that the average strain could be determined over this gauge

length. In addition, six more LVDTs were centralized symmetrically at the intersection

o f the beam mid-height and centerlines o f net shear spans to form rosettes with 350 mm

gauge lengths. Figures 2.11 to 2.16 show the layout o f LVDT locations.

Electrical resistance strain gauges with a gauge length o f 5 mm were also used to detect

the tensile strain in the reinforcing bars. Figures 2.10, 2.11, 2.13 and 2.14 show the

positions o f electrical resistance strain gauges glued to the reinforcement prior to casting.

Eight gauges were situated on the surface o f the innermost bottom tension reinforcement

and another six (exclusive from B-3S & 4S) gauges were glued to the vertical distributed

reinforcement. Gauges LI and L8 were positioned at the inner edges of the bearing

plates.

2.4 Testing Procedure

The specimens were initially loaded to properly seat the bearing and loading plates.

Wedges used to prevent the movement o f the rollers were removed just before testing.

The experimental loading was controlled via displacement at an initial rate o f 0.1

mm/min. After general yielding o f each specimen, the testing rate was increased to 0.15

mm/min. The rate was further increased by 0.05 mm/min at later stages o f loading. For

taking measurements at load stages, the deflection was held constant while the crack

widths were measured and the crack patterns were sketched and photographed. The crack

widths were measured with a crack comparator at locations where the cracks crossed the

main tension reinforcement and where the cracks crossed the horizontal line at mid

height o f each beam. After yielding occurred along the overall length o f the main tension

reinforcement, load stages were recorded at increments o f 2 to 3 mm of the midspan

displacement. For each stage, the selected tensile strain, ss , was measured via the strain

gauges on the reinforcing steel and the strains were calculated from the LVDT readings.

After the peak load was reached, the loading was continued until the beam could only

resist 75% of the peak load.

23


Notes: dimensions in mm

Figure 2.1 Overall views of specimens

24


4002000300

9-No. 10 stirrups (vertical distributed

reinforcement) s = 219m m

7-No. 15 tension reinforcement

Section B-B

4-No. 10 double stirrups (horizontal distributed reinforcement)

s = 262 mm

Section A-A

Figure 2.2 Details of Specimen B-3S

25


1160

4002000

300

9-No.lO stirrups (vertical distributed

reinforcem ent) s = 219 mm


Section B-B

7-No. 10 double stirrups (horizontal distributed reinforcement)

s = 247 mm

Section A-A

Figure 2.3 Details of Specimen B-4S

26


1840

7-No. 15 tension reinforcemelnt

No. 10 rebar

Section A-A

Specimen B-3N


No. 10 rebar

B-Specimen B-4N

Section B-B

Figure 2.4 Details of Specimens of B-3N and B-4N

27


42 -40 - 38 36 - 34 - 32 30 - 28 - 26

= 22

Cylinder No.2Cylinder No.114 -

12 Cylinder No.3

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 0.0045 0.0050strain

Figure 2.5 Representative concrete compressive stress-strain curves

0.08

0.07 -

0.06 -Shrinkage (air-dried)

0.05 -

0.04ou>rcx.- 0.03

0.02 -

0.01

Shrinkage (mot:tst-cured)

- 0.010 20 40 60 80 100 120

days

Fig 2.6 Measured concrete shrinkage strains

28


650

600 -

550 -

500 -

450

400 -

S 350 -

8 300

250 -

200 -

150

100

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28strain

Fig 2.7 Stress-strain curves for 10M bars

650

600 -

550 -

500 -

450 -

400 -

S 350

8 300

250

200

150 -

100 -

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36

strain

Fig 2.8 Stress-strain curves for 15M bars

29


Figure 2.9 Specimen B-4S under the MTS testing machine

30


_660_

____55S_

300 X 400

25

10276

152

76

250 X 400

Figure 2.10 Details of loading and bearing devices

31


900__________ 550

WD*

W H

wv

ED

EH

EV

H5 H4 H3 H2 HI

WST CV EST4 350 ^

Note: dimensions in mm typ.

Figure 2.11 LVDT locations for Specimen B-3S & 3N

L8 L7 L6 L5 L4 L3 L2 LI

Figure 2.12 Strain gauge locations and crack measurement lines for

Specimen B-3N

32


ooo>n

S6 S5 S4 S3 S2 SIL 8 ' L7 ' L6 ' L5 1.4 ' 1.3 ' 1.2 ' LI

m i 219 < 219^ 219 219 ,20Q

ON


Specimen B-3S

WST CV EST* 350 ^

Note: dimensions in mm typ.

Figure 2.14 LVDT locations for Specimen B-4N & 4S

33


, 2 1 9 ^ 2 1 9 ,


Specimen B-4N

S6 S5 S4 L 8 ' L7 ' L6 ' L5

S3 S2 SI L4 ' L3 ' L2 ' LI

o

Chapter 3

Experimental Results

This chapter describes the experimental response of each specimen. This thesis reports

on four o f the eight beams forming a larger testing program carried out at McGill

University. The other four beams are reported by Ding Li (2003). The load-deflection

response is described in terms o f the total concentrated loading on the top o f the beam,

which is twice o f the applied shear force on each shear span.

3.1 Specimen B-3N

Specimen B-3N is 1160 mm deep and contains only the main tension tie reinforcement.

The first flexural hairline crack occurred at a load of 734 kN. The first flexural-shear

crack formed in the west shear span at an applied load o f 961 kN. The counterpart on the

east side occurred at a measured load o f 962 kN. These three major cracks dominated the

cracking pattern o f Specimen B-3N. At load stage 5, the two diagonal cracks extended to

the inner edge of the bottom bearing plates at an applied load o f 1326 kN. First yielding

occurred when the load reached 1584 kN at a deflection of 2.74 mm and all o f the strain

gauges yielded at a load of 1787 kN, at a deflection o f 3.10 mm. When the applied load

approached 2020 kN, a shear crack occurred suddenly on the east end. This major

cracking was accompanied by splitting of the full-depth diagonal strut which formed from

the comer o f the top loading plate to the middle of the bottom bearing plate

approximately, lead to brittle failure. The maximum deflection was 12.86 mm. The

measured width o f the crack was 6.0 mm resulting in a dramatic increase o f the east

diagonal (ED) LVDT reading located on the back of the specimen, from 0.14 mm to 8.15

mm. The width of the other diagonal crack at the west end was 5 mm and the main

flexural crack width was 10 mm. The strain gauge reading, L3, indicated that the

35


horizontal reinforcement most likely experienced strain hardening and no signs o f any

concrete crushing were apparent. It is noted that as the failure took place, the diagonal

cracks delineating the strut extended and followed the hook geometry. It is likely that loss

of anchorage occurred during failure.

The applied load versus relative midspan displacement of the beam is given in Fig 3.1.

The key load stages, peak load and relevant displacements are given in Table 3.1.

Figure 3.2 shows the applied load vs. horizontal strains measured in the bottom

reinforcing steel o f the main tension tie, determined from the strain gauges. It can be seen

that the strains are approximately the same throughout the overall length o f the main

tension tie (gauges LI to L8) as expected. At first yielding, the strains in the bottom

reinforcement determined from the strain gauges were 2063, 2281, 2219, 2214, 2241,

2201, 2227 and 1951 micro-strain for gauges LI to L8, respectively, at a total applied

load of 1595 kN. At general yielding, the strains were 2470, 2733, 2571, 2589, 2602,

2562, 2574 and 2284 micro-strain for gauges LI to L8, respectively, at a load o f 1806 kN.

Figure 3.3 shows the applied load vs. average horizontal strain determined from the

LVDTs readings at the level o f the main tension tie.

The results obtained from the rosettes mounted on the back o f the specimen, including the

principal strains, shear strains and the principal angle calculated using Mohrs circle of

strain are indicated in Fig 3.4. At stage 6 (first yield), the corresponding principal degrees

were 54.2 degrees and 53.6 degrees on west side and east side, respectively. At stage 7

(general yield), the corresponding principal degrees were 54.2 degrees and 54.9 degrees

on west side and east side, respectively.

36


Figures 3.5 to 3.7 show the development o f cracks in Specimen B-3N, including the

general yield and peak load stages.

Table 3.1 Key load stages for Specimen B-3N

Load

stage

Total applied

load (kN)

Midspan

displacement

(mm)

Notes

0 73.6 0 Initial seating

1 734 0.84 First flexural crack

3 961 1.40 First flexural-shear crack on west end

4 962 1.65Second flexural-shear crack on east

end

5 1326 2.30Diagonal crack propagating to comer

o f bearing plate on west end

6 1584 2.74 First yielding o f main tension tie

7 1787 3.10

General yielding o f main tension tie,

followed by a relatively flat load-

displacement response

8 1902.4 3.51 General yielding of the system

11 2020 12.86Peak load, splitting o f concrete stmt

and loss o f anchorage at hooks

37


tota

l app

lied

load

(KN)

2200

2000general yielding of the beai

1800leneral yielding of main tension

1600first yielding of main tension tie1400

1200

1000

800

600first flexural crack

400

200

0 1 2 3 4 6 7 1 12 14midspan deflection (mm)

Figure 3.1 Load-deflection response of Specimen B-3N


Reproduced

with perm

ission of the

copyright ow

ner. Further

reproduction prohibited

without

permission.

2500

2000

i 1500 o .21 1000 (0*

(a) First yield (total applied load o f 1584 klSi;

(b) General yield (total applied load of 1787 kN)

(c) Peak load (total applied load of 2020 kN)

Figure 3.3 Longitudinal strains from LVDTs at the level of

main tension tie of Specimen B-3N (mm/mmxlO'3)

40


Reproduced

with perm

ission of the

copyright ow

ner. Further


without

permission.

2500

W E S T

EA STo 1500

1000

500 -

0 0.0001 0.0002 0.0003 0.0004 0.0005strain

(a) maximum principal strains, et

2500

W E S T2000 -

EA ST

o 1500 -

1000 -

500 -

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012strain

2500

2000W E ST

1500

1000EA ST

500

-0.0008 -0.0006 - 0.0002-0.0004 0strain

(b) minimum principal strains, e2

2500

W E S T2000z

oreo 1500o0 EA STo.a0 1000o 500

0 20 40 60 80 100 120

(c) shear strain, yxy

degrees

(d) principal angle, 0 2

Figure 3.4 Calculated rosette strain responses in Specimen B-3N

41

Figure 3.5 Cracking patterns o f Specimen B-3N at first yielding

of the main tension tie

(Total applied load of 1584 kN)

42


Figure 3.6 Cracking patterns o f Specimen B-3N at general yielding

of the main tension tie


43


Figure 3.7 Cracking patterns of Specimen B-3N at peak load


44


3.2 Specimen B-3S

Specimen B-3S had an overall depth of 1160 mm and contained not only the

horizontal tension tie but also the bidirectional distributed reinforcement. First flexural

cracking of specimen B-3S occurred on the bottom face of the beam, close to

midspan, at an applied load of 700 kN. The first hairline diagonal crack formed from

the inner edge of the bearing at the east end support at load stage 4, at a load of 1312

kN. At stage 5, at an applied load of 1463 kN, a major diagonal shear crack initiated

from the west end support. This crack had a width of 0.2 mm.

First yielding of the main tension tie occurred at gauge L5 at an applied load of 1977

kN, at a deflection of 2.82 mm. General yielding of the main tensile reinforcement

occurred at stage 11 with an applied load of 2411 kN, at a deflection of 5.78 mm. At

the completion of the first four stages, flexural cracks had formed at the locations of

the transverse reinforcement and thus resulted in a uniform crack spacing of 220 mm.

Up to a load of 2411 kN, these flexural cracks became wider and extended, with no

significant new cracks forming. These cracks increased in width from 0.2 mm to 1.75

mm at the level of the centroid of the bottom reinforcement. At loads of 2430 and

2450 kN, two inclined cracks initiated from the comers of the top bearing plate and

abmptly penetrated over the full height of the beam to the middle of the bottom

bearing plates on the west and east sides. The width of four of the flexural cracks in

the midspan region of the beam reached widths of 4 mm indicating that the main

tension tie reinforcement had probably experienced strain hardening. The load

capacity continued to increase as the stirrups picked up some of the shear. Crushing of

the concrete immediately under the loading plate occurred at stage 14 corresponding

to the applied load of 2450 kN. The load continued to increase to the peak load of

2580 kN with a deflection of 26.68 mm. The load capacity of post-peak stage

decreased to 1963 kN, that is 76% of the peak load, at a deflection of 40.4 mm.

45


The applied load versus relative displacement of the beam is shown in Fig 3.8. The

load stages, peak load and displacement are given in Table 3.2. It should be noted that

there was no abrupt spalling and crushing of the concrete.


reinforcing steel of the main tension tie, determined from the strain gauges.

Figure 3.10 shows the applied load vs. vertical strain measured from the gauges on the

vertical, uniformly distributed reinforcement. There was a significant increase in the

tensile force in the stirrups after general yielding occurred. Prior to the peak load,

more than half of the vertical closed stirrups approached yielding. Gauges S4 and S6

were damaged at an early stage in the loading and hence the strains at these locations

could not be reported.

Figure 3.11 shows the responses of rosettes mounted at the back of specimen. The

principal strains, shear strains and principal angles determined from the rosettes

mounted on the back of the specimen, are shown in Fig 3.12. The principal angle is

defined from the horizontal direction and denotes the direction of the minimum

principal strain, in the other words, the maximum compressive strain. At stage 7 (first

yield), the principal angles were 59.2 degrees on the west end and 59.5 degrees on the

east end, respectively. At stage 11 (general yield), the principal angles were 63.8

degrees on west and 57.5 degrees on east, respectively. At stage 20 (peak load), the

principal angles were 57.7 degrees on west and 56.2 degrees on east, respectively

Figure 3.13 shows the total applied load vs. horizontal strain determined from the

LVDT readings at the level of main reinforcement on the west and east sides,

respectively.

Photographs showing the development of cracks in Specimen B-3S are given in Figs.

3.14 to 3.16, including at first yielding, general yielding and at the peak load stage.

46


Table 3.2 Key load stages for Specimen B-3S

Load

stage

Applied load

(kN)

Relative

displacement

(mm)

Notes

0 18 0 Initial seating


4 1312 1.78 0.1 mm wide First shear crack on east

5 1463 1.98 0.2 mm wide Shear crack on west

7 1977 2.82 First yielding of the main tension tie


11 2411 5.78General yielding of the main tension

tie

12 2430 6.35 First strut crack on west

13 2450 6.92 Second strut crack on east

14 2450 11.4Concrete crushing underneath the

loading plate

15 2488 14.7 Concrete cover spalling

20 2580 26.68 Peak load

21 2542 27.65Load dropped 40 kN after crack and

was reloaded to 2570 kN

25 1963 40.40 76% of peak load, end of testing


2800 2600 2400 2200

2000 ^ 1800 o 1600

1400 "g-^OO 1000 % 800

600 400 200

general yielding of main tension tie general yielding of the beam ]

first yielding of main tension tie

first flexural crack

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42midspan deflection (mm)

Figure 3.8 Load-deflection response o f Specim en B-3S

48


Reproduced

with perm

ission of the

copyright ow

ner. Further


without

permission.

3000

_ 2500 - zJX.S' 2000 -CBo

J 1500 - af 1000 -n

500

0 1000 2000 3000 4000m icrostrain

3000

2500 -z

v 2000 -ao| 1500 5. 1000

500 - L6 L3

0 1000 2000 3000 4000microstrain

3000yield

2500 -

o 2 0 0 0 -

1500Q .Q .

1000

500 -

0 1000 2000 3000 4000 5000m icrostrain

3000yiejld

2500 -

o 2 0 0 0 -

1500

1000L4500 -

0 1000 2000 3000 4000 5000 6000m icrostrain

Figure 3.9 Strains in m ain tension tie o f Specim en B -3S, determ ined from strain readings

49

3000 yield

2500

-o 2000

IS 1500 a

1000

500

1000 2000 3000 4000 5000-1000m icrostra in

3000

2500

o 2000rao

1500Q.

* 1000rc+->o

500

-1000

yield

1000 2000 m icrostra in

3000 4000

3000

2500

o 2000rao

| 1500 a

f 1000JSo

500

-1000

y eld

-

-7- P ~v ' U S3

-

)_C )_

1000 2000 m icrostra in

3000 4000

Figure 3.10 Strains in vertical distributed reinforcem ent o f Specim en

B-3S, determ ined from strain readings

50


3000EH

2500WH

2000

1500

1000

500

0.015 0.02-0.005 0 0.005 0.01strain

3000EV

2500WV

2000

1500

1000

500

- 0.001 0 0.001 0.002 0.003 0.004strain

3000

ED2500

z2000

o.2aara

1500

1000

500

0 0.005 0.01 0.015 0.02 0.025strain

Figure 3.11 R osette strain responses in Specim en B-3S


Reproduced

with perm

ission of the

copyright ow

ner. Further


without

permission.

3000EAST

2500 ---.W EST

o 2000

1! 1500 -

1000 -500 -

0 0.005 0.01 0.015 0.02 0.025s t r a i n

(a) maximum principal strains,

3000EAST

2500W EST

a 2000

1500 -

1000 -500

0 0.005 0.01 0.015 0.02 0.025 0.03s t r a i n

3000; EAST

2500

W EST 20001500

1000500

-0.006 -0.005 -0.004 -0.003 -0.002 -0.001 0 0.001s t r a i n

(b) minimum principal strains, 2

3000

2500WEST

D 2 0 0 0 -

1500 -EAST

1000 -500 -

0 20 40 60 80d e g r e e s

(c) shear strain, yxy (f) principal angle, 0 2

Figure 3.12 C alculated rosette strain responses in Specim en B-3S

52

Reproduced

with perm

ission of the

copyright ow

ner. Further


without

permission.

1.425 2.861 1.686 2.465 1.58210.623 7.971 7.79

2.023

(a) first yield (total applied load of 1977 kN) _____________

73.711 ......" 'i - if-'"-

n g g 38.286 37.62I "*3 ST,

(b) general yield (total applied load of 2411 kN)

84.844

44 901

27.853

8.006

(c) peak load (total applied load of 2580 kN) (d) after failure (total applied load of 1847 kN)

(Units: mm/mmxl0'3)

Figure 3.13 Longitudinal strains from LV D Ts at the level o f m ain tension tie o f Specim en B-3S

53

Figure 3.14 C rack patterns o f Specim en B-3S at first yielding

o f the m ain tension tie

(Total applied load o f 1977 kN)

54


Figure 3.15 Crack patterns of Specimen B-3S at general yielding

o f the m ain tension tie

(Total applied load o f 2411 kN)

55


Figure 3.16 Crack patterns of Specimen B-3S at peak load


56


3.3 Specimen B-4N

Specimen B-4N had an overall depth of 1840 mm and contained only the horizontal

tension tie reinforcement. Strain gauge L4 on the main reinforcement was accidentally

damaged during the formwork removal-and hence did not function during testing. The

load when the first flexural hairline crack occurred was 1632 kN and two more

flexural cracks appeared shortly afterwards. The first flexural-shear crack formed on

the east side at stage 3, at a load of 1839 kN. This crack propagated to the east support

at an applied load of 3155 kN. Symmetrical cracking occurred on the west side at

stage 4 (total load of 2144 kN). When the load approached 3000 kN, the flexure-shear

crack on the west side extended to the west support. The flexural cracks varied in

width between 0.2 mm and 0.75 mm due to non-uniform spacing of these cracks. First

yielding occurred at strain gauge L6 when the load reached 2424 kN at a midspan

deflection of 2.59 mm and all the strain gauges indicated yielding at a total load of

2808.8 kN at a midspan deflection of 2.98 mm. When the applied load approached

3247.2 kN (deflection of 11.36 mm), on the east side, an abrupt inclined crack formed

due to the splitting of the compressive struts. This splitting crack extended over the

full height of the beam, from the comer of top bearing plate to approximately the mid

of the bottom bearing plate, leading to a brittle failure. The splitting crack had

extended to the level of the main tension tie and then followed the profile of the hook.

The sudden failure was caused by loss of anchorage of the outer hooks on the east end

of the beam at the support. Even though this splitting crack extended into the east

rosette, the LYDT readings were lost due to the abrupt failure of the specimen. The

maximum crack width at the mid-height of the deep beam reached 10 mm before the

failure occurred. The cracks at the level of the main reinforcement varied in width

between 0.2 mm and 1.0 mm. According to the readings from the strain gauges and

the LVDTs, the main reinforcement most likely experienced strain hardening. No

signs of concrete crushing were apparent. The main reinforcement had yielded in

tension over the course of testing with the maximum tensile strain (strain gauge L4)

exceeding 10 times the yield strain just prior to failure. Gauges L5, L6 and L7 failed

57


due to the very high strains that were reached. Strain gauges L5, L6 and L7 reached 8,

9 and 4 times the yield strain, respectively, before failure of the gauges.

The total applied load versus relative midspan displacement of the beam is given in

Fig 3.17. The load stages, peak loads and corresponding displacements are given in

Table 3.3.

Table 3.3 Key load stages for Specimen B-4N

Load

stages

Applied load

(kN)

Relative

displacement

(mm)

Notes

0 48.8 0 Initial seating


3 1839 1.99First shear-flexural combined crack

on east

4 2144 2.32Second shear-flexural combined

crack on west

5 2424 2.59 First yielding of main reinforcement

6 2808.8 2.98General yielding of main

reinforcement


8 3202 9.06 Strain hardening

9 3106 9.47 Extremely wide crack, up to 10 mm

10 3247.2 11.36 Peak load and splitting of concrete


reinforcing steel of the main tension tie determined from strain gauges.

58


Figure 3.19 shows the strain responses, principal strains, shear strains and principal

angle calculated from the LVDT rosettes. At stage 6 (general yielding), the principal

angles were 72.2 degrees and 67.5 degrees on the west and east sides, respectively.

Figure 3.20 indicates the longitudinal strains measured from LVDTs at the level of

main reinforcement on several key stages and photos for development of cracks are

shown in Figs. 3.21 to 3.23.

3400 3200 3000 2800 2600

f 2400 ~ 2200 2000 1800 ! 1600 S 1400-s 1200

" 1000800600400200

\\g en e ra l yielding of the beam general yielding of main tension tie

first yielding of main tension tie

first flexural crack

0 2 4 10 126 8midspan deflection (mm)

Figure 3.17 Load-deflection response of Specimen B-4N


Reproduced

with perm

ission of the

copyright ow

ner. Further


without

permission.

3500 yield

3000

2500

2000

a 1500

2 1000

500

0 1000 2000 3000 4000m icrostrain

3500yield

3000

2500 -

2 2000 -

d. 1500 -

2 1000

500 -

1000 2000 3000 4000 5000 60000m icrostrain

3500

3000 -

2500 -

2000 -

a 1500 -

2 1000 -

500 -

0 1000 2000 3000 4000 5000 6000

3500

3000 -

2500 -

2000 -

a. 1500

2 1000 -

500 -

4000 50000 1000 2000 3000m icrostrain m icrostrain

Figure 3.18 Strains in m ain tension tie o f Specim en B-4N , determ ined from strain readings

60

Reproduced

with perm

ission of the

copyright ow

ner. Further


without

permission.

3500

3000 EAS2500

- 2000

=5. 1500

I 1000

500

0 2E-05 4E-05 6E-05 8E-05 0.0001 0.0001s t r a i n

(a) maximum principal strains, Ej

3500

3000 EASTi 2500

S 2000 -

o. 1500W EST

2 1000 -

500 -

0 0.0001 0.0002 0.0003 0.0004s t r a i n

(c) shear strain, yxy

3500

3

behaviour and modeling of deep beams with low shear span-to-depth ratios

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