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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.
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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
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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
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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
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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
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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
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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
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...................................................................................................................................... 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
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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
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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
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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
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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
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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
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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
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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.
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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
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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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
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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
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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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
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Figure 1.4 Compressive strength of strut, as a function of the angle of
crossing tension tie (Collins and Mitchell 1986)
13
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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
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(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
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'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
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.*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
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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
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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
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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
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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
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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
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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
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Notes: dimensions in mm
Figure 2.1 Overall views of specimens
24
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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
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1160
4002000
300
9-No.lO stirrups (vertical distributed
reinforcem ent) s = 219 mm
7-No. 15 tension reinforcement
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
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1840
7-No. 15 tension reinforcemelnt
No. 10 rebar
Section A-A
Specimen B-3N
7-No. 15 tension reinforcement
No. 10 rebar
B-Specimen B-4N
Section B-B
Figure 2.4 Details of Specimens of B-3N and B-4N
27
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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
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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
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Figure 2.9 Specimen B-4S under the MTS testing machine
30
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_660_
____55S_
300 X 400
25
10276
152
76
250 X 400
Figure 2.10 Details of loading and bearing devices
31
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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
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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
Figure 2.13 Strain gauge locations and crack measurement lines for
Specimen B-3S
WST CV EST* 350 ^
Note: dimensions in mm typ.
Figure 2.14 LVDT locations for Specimen B-4N & 4S
33
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, 2 1 9 ^ 2 1 9 ,
Figure 2.15 Strain gauge locations and crack measurement lines for
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
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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
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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 permission of the copyright owner. Further reproduction prohibited without permission.
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
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Reproduced
with perm
ission of the
copyright ow
ner. Further
reproduction prohibited
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 3.6 Cracking patterns o f Specimen B-3N at general yielding
of the main tension tie
(Total applied load of 1787 kN)
43
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 3.7 Cracking patterns of Specimen B-3N at peak load
(Total applied load of 2020 kN)
44
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
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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.
Figure 3.9 shows the applied load vs. horizontal strains measured in the bottom
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 3.2 Key load stages for Specimen B-3S
Load
stage
Applied load
(kN)
Relative
displacement
(mm)
Notes
0 18 0 Initial seating
1 700 0.78 First flexural crack
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
9 2254.8 3.5 General yielding of the system
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced
with perm
ission of the
copyright ow
ner. Further
reproduction prohibited
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced
with perm
ission of the
copyright ow
ner. Further
reproduction prohibited
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
reproduction prohibited
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 3.16 Crack patterns of Specimen B-3S at peak load
(Total applied load of 2580 kN)
56
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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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
1 1632 1.44 First flexural crack
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
7 2843.6 3.24 General yielding of the system
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
Figure 3.18 shows the applied load vs. horizontal strains measured in the bottom
reinforcing steel of the main tension tie determined from strain gauges.
58
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
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Reproduced
with perm
ission of the
copyright ow
ner. Further
reproduction prohibited
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
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ission of the
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ner. Further
reproduction prohibited
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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