The Discrete Rotation Behaviour of Reinforced
Concrete Beams under Shear Loading
by
Wade Doyle Lucas
B.E. Civil & Structural Engineering (Hons)
Thesis submitted for the degree of Doctor of
Philosophy at The University of Adelaide
(The School of Civil, Environmental and
Mining Engineering)
Australia
 March 2011 
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
i
Table of Contents
Table of Contents ................................................................................................................................................... i
Abstract ................................................................................................................................................................. ii
Sta
tement of Originality ...................................................................................................................................... iii
L
ist of Publications .............................................................................................................................................. iv
Journal Papers .................................................................................................................................................. iv
Conference Papers ............................................................................................................................................ iv
Acknowledgements .............................................................................................................................................. vi
Introduction & General Overview .................................................................................................................... vii
Chapter 1 – Background ...................................................................................................................................... 1
Introduction ....................................................................................................................................................... 1
List of Manuscripts ............................................................................................................................................ 1
Our Obsession with Curvature in Reinforced Concrete Modelling ................................................................... 3
A Generic Unified Reinforced Concrete Model ............................................................................................... 34
FRP Reinforced Concrete Beams – A Unified Approach Based On IC Theory ............................................... 75
Chapter 2 – The Shear Failure Mechanism ................................................................................................... 106
Introduction ................................................................................................................................................... 106
List of Manuscripts ........................................................................................................................................ 106
The Formulation of a Shear Resistance Mechanism for Inclined Cracks in RC Beams ................................ 107
Chapter 3 – The Shear Friction Mechanism .................................................................................................. 134
Introduction ................................................................................................................................................... 134
List of Manuscripts ........................................................................................................................................ 135
The Shear Friction Mechanism of Reinforced Concrete ................................................................................ 136
Shear Friction Behaviour in FRP Reinforced Concrete ................................................................................ 161
Chapter 4 – Numerical Model Development and Calibration ...................................................................... 182
Introduction ................................................................................................................................................... 182
List of Manuscripts ........................................................................................................................................ 182
Simulation of Shear Failure in RC beams without Stirrups ........................................................................... 183
Chapter 5 – Numerical Model Expansion and Validation ............................................................................ 212
Introduction ................................................................................................................................................... 212
List of Manuscripts ........................................................................................................................................ 212
The Failure Mechanism of RC Beams with Stirrups ...................................................................................... 213
Chapter 6 – Concluding Remarks ................................................................................................................... 241
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
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Abstract
This thesis presents a body of research into the behaviour of reinforced concrete (RC) beams
unde
r combined shear and flexural loading. This research is presented in the form of a series
of manuscripts that are either accepted, submitted or in preparation for journal publication.
C
urrently, many code approaches are built around the assumption that the shear and flexural
c
apacities of RC beams can be assessed separately despite acknowledging that there is an
interaction between the two. This is due to the fact that quantifying this interaction and
specifically the shear resistance of RC members has been found to be a very complex
problem as the inclined sliding planes along which failure occurs transcend both initially
cracked and uncracked planes.
This thesis introduces a mechanics based mechanism built around simulating the observed
physical behaviour. Developing a mechanism that simulates what is seen in practice provides
valuable insight into the complexities of the shear and flexural interaction. The mechanism
developed is built upon the well established research areas of rigid body displacement, shear
friction theory and partial interaction theory. Generic equations based on this mechanism are
derived for RC beams both with and without transverse reinforcement and implemented into
numerical models for RC members under both direct shear loads and combined flexural and
shear loading.
Comparison between the failure loads predicted by the developed numerical models and
empirically derived results show good agreement in magnitudes and more importantly
exhibits similar trends in behaviour. As a consequence, the numerical models are used to
conduct an indepth investigation into the variables that have a significant effect on the shear
resistance of RC beams and used to examine the physical behaviour behind the observed
trends. This knowledge is used to further advance and calibrate the derived numerical
models. In addition, the numerical models are used to demonstrate various advantages that
c
an be obtained through the use of structural mechanics based approaches.
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
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Statement of Originality
This work contains no material which has been accepted for the award of any other degree or
diploma in any university or other tertiary institution to Wade Lucas and, to the best of my
knowledge and belief, contains no material previously published or written by another
person, except where due reference has been made in text.
I give consent to this copy of my thesis when deposited in the University Library, being made
available for loan and photocopying, subject to the provisions of the Copyright Act 1968.
The author acknowledges that copyright of published works contained within this thesis (as
listed below) resides with the copyright holder(s) of those works.
I also give permission for the digital version of my thesis to be made available on the web,
via the University‟s digital research repository, the Library catalogue and also through web
search engines, unless permission has been granted by the University to restrict access for a
period of time.
________________________________________ ______________
Wade Doyle Lucas Date
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
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List of Publications
Journal Papers
Our Obsession with Curvature in Reinforced Concrete Modelling
Oehlers, D.J, Haskett, M., Mohamed Ali M.S., Lucas, W., and Muhamad, R.
Adv
ances in Structural Engineering 2011: v 14, n 3, p 391404.
A Generic Unified Reinforced Concrete Model
Oehlers, D.J., Mohamed Ali M.S., Griffith, M.C., Haskett, M., and Lucas, W.
Proc. ICE , Structures and Buildings 2010: accepted paper
FRP Reinforced Concrete Beams – A Unified Approach Based On IC Theory
Oehlers, D.J., Mohamed Ali M.S., Haskett, M., Lucas, W., Muhamad, R., and Visintin, P.
ASCE Composites for Construction 2010: accepted paper
The Formulation of a Shear Resistance Mechanism for Inclined Cracks in RC Beams
Lucas, W., Oehlers, D.J., Mohamed Ali, M.S.
ASCE Journal of Structural Engineering 2011: accepted paper
The Shear Friction Mechanism of Reinforced Concrete
Lucas, W., Oehlers, D.J., Mohamed Ali, M.S., Griffith, M.C.
T
ext in Manuscript
Shear Friction Behaviour in FRP Reinforced Concrete
Lucas, W., Oehlers, D.J., Mohamed Ali, M.S., Griffith, M.C.
Advances in Structural Engineering 2011: accepted paper
Simulation of Shear Failure in RC beams without Stirrups
Lucas, W., Oehlers, D.J., Mohamed Ali, M.S.
Engineering Structures 2011: submitted paper
The Failure Mechanism of RC Beams with Stirrups
Lucas, W., Oehlers, D.J., Mohamed Ali, M.S.
ASCE Journal of Structural Engineering 2011: submitted paper
C
onference Papers
Design of FRP Reinforced Concrete Beams against Shear Failure
Lucas, W., Oehlers, D.J., Mohamed Ali, M.S., Griffith, M.C.
Proceedings of the 9th International Symposium of the FiberReinforced Polymer
Reinforcement for Reinforced Concrete Structures (FRPRCS9) 2009, Sydney, Australia.
A Structural Mechanics Shear Capacity Model for FRP Plated RC Members
Lucas, W., Oehlers, D.J.
Proceedings of the AsiaPacific Conference on FRP in Structures 2009: pp. 117122.
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
v
FRP Design using Structural Mechanics Models
Oehlers, D.J., Haskett, M., Mohamed Ali, M.S., Lucas, W. and Muhamad, R.
Keynote Paper, Proceeding CICE 2010 – The 5
th
International Conference on FRP
Composites in Civil Engineering, Beijing, China, September 27
th
29
th
, pp. 3744.
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
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Acknowledgements
I can honestly say that this thesis would not have been possible without the assistance of
Professor Deric Oehlers. His continual support and advice was always there when I needed it
most. I remember that at the very start of my candidature he told me he‟d get me through this
kicking and screaming if need be. I like to think that I only kicked a little.
I would also like to thank the other academics who have helped me in my research and were
able to point me in the right direction whenever an obstacle seemed insurmountable, in
particular Dr. Mohamed Ali and Professor Mike Griffith.
Finally, I would like to thank my family and my very tolerant partner for providing support
when it was needed, distraction when it was required and motivation when I lost mine.
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
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Introduction & General Overview
When dealing with the calculation of the failure load of reinforced concrete (RC) members,
many code approaches have found it convenient to assume a separation of shear and flexural
capacities, the use of highly ductile reinforcing elements and a good bond between the
reinforcing elements and the surrounding concrete. This has enabled the development of
simple yet conservative design equations. However, with the increasing number of options in
reinforcing materials, including some that exhibit little to no ductile behaviour, and new
techniques of applying reinforcing elements, these assumptions are becoming increasingly
invalid and hence the design equations are reaching the limit of their usefulness. This thesis
investigates a new, alternative method for calculating the failure load of RC beams, with an
emphasis on the shear sliding phenomenon associated with shear failure.
This thesis is a collection of manuscripts that are either in preparation, submitted or accepted
for publication in internationally recognised journals. Each of the Chapters 15 are titled
according to how they fit into the overall research objectives and take the following format:
an introduction explaining the aims of the chapter in terms of the overall research objectives;
a list of all the manuscripts presented in the chapter; and finally presentation of each
manuscript.
Chapter 1 provides the general background information about the current approaches to
designing RC beams and also details the alternative “unified reinforced concrete model” that
is currently being developed. This chapter discusses in detail the limitations involved in the
current approaches and the advantages that can be obtained from the alternative approach. As
the unified reinforced concrete model is an extensive topic, this thesis focuses solely on the
shea
r failure mechanism of RC beams.
Chapter 2 provides a detailed investigation of the shear failure mechanism of RC beams. It
examines the physical behaviour associated with the shear failure phenomenon and attempts
to replicate that behaviour using a structural mechanics model that simulates both flexural
rotation and shear sliding. Ultimately this chapter identifies two relationships that are critical
in the application of this mechanism: the partial interaction relationship that links the force in
a reinforcing bar intersecting a crack with the associated width of the crack; and the shear
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
viii
friction relationship that defines the ability of a sliding plane to resist shear stresses in terms
of the confining stresses and the lateral displacement.
Chapter 3 investigates the shear sliding phenomenon that forms one half of the shear failure
mechanism of RC beams. This is achieved through the development of a numerical model to
replicate an experimental series of reinforced concrete blocks under direct shear loading. By
doing this it was possible to both investigate the key factors involved in the shear sliding
phenomenon and identify potential relationships that could be implemented into the shear
failure mechanism of RC beams.
Chapter 4 implements the partialinteraction and shear friction relationships identified in
Chapter 3 into the shear failure mechanism of longitudinally reinforced RC beams. From this
analysis it was found that in the case of RC beams without stirrups it is the strength prior to
the onset of shear sliding that controls the critical failure load. This made it possible to further
improve the shear friction properties using a regression analysis of the predicted failure loads
of theoretical RC beams without stirrups.
Chapter 5 shows the completion of the research objectives by developing a numerical model
that predicts the failure load of RC beams with stirrups. This numerical model implements
the mechanics based shear friction mechanism introduced in Chapter 2 and the improved
shear friction relationships developed in Chapter 4. Importantly it was found that for RC
be
ams with stirrups the numerical model could predict both the failure load and failure
mechanism. The predicted failure mechanisms included shear sliding, concrete crushing and
reinforcement rupture.
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
1
Chapter 1 – Background
Introduction
This chapter provides the general background information for the research conducted
throughout this thesis.
The first paper “Our Obsession with Curvature in Reinforced Concrete Beam Modelling”
focuses on the current approaches to the design of reinforced concrete members and the
associated limitations. In particular this paper investigates the momentcurvature approach to
reinforced concrete design and its dependency on ductile reinforcement with a good bond
with the surrounding concrete. The paper also introduces the recently developed alternate
momentrotation approach that implements IC debonding theory to remove the dependency
on ductile reinforcement and good bond characteristics.
The second paper “A Generic Unified Reinforced Concrete Model” provides a more in depth
look at the momentrotation approach and its implementation into a discrete rotation model.
The research areas, upon which the discrete rotation model is built, partialinteraction and
shear friction theory are discussed in detail and used to show how the momentrotation
approach can simulate both the shear and flexural behaviour of RC beams. It is this
simulation of the shear behaviour of reinforced concrete beams that is the subject of this
thesis.
The third paper “FRP Reinforced Concrete Beams – A Unified Approach Based on IC
The
ory” shows how the previously discussed discrete rotation model can be applied to both
FRP and steel reinforcement. In particular it is shown that this model is fundamentally
generic and can be applied to any type of reinforcement and concrete.
L
ist of Manuscripts
Our Obsession with Curvature in Reinforced Concrete Beam Modelling
Oehlers, D.J, Haskett, M., Mohamed Ali M.S., Lucas, W., and Muhamad, R.
Advances in Structural Engineering 2011: v 14, n 3, p 391404.
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
2
A Generic Unified Reinforced Concrete Model
Oehlers, D.J., Mohamed Ali M.S., Griffith, M.C., Haskett, M., and Lucas, W.
Proc
. ICE , Structures and Buildings 2010: accepted paper
FRP Reinforced Concrete Beams – A Unified Approach Based On IC Theory
Oehlers, D.J., Mohamed Ali M.S., Haskett, M., Lucas, W., Muhamad, R., and Visintin, P.
ASCE Composites for Construction 2010: accepted paper
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
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STATEMENT OF AUTHORSHIP
Our Obsession with Curvature in Reinforced Concrete Modelling
Advances in Structural Engineering 2011: v 14, n 3, p 391404.
P
ROFESSOR DERIC J. OEHLERS
School of Civil, Environmental and Mining Engineering
The University of Adelaide
Prepared manuscript and supervised all research.
I hereby certify that the statement of contribution is accurate
Signed …………………………………………………… Date …………………
DR. MATTHEW HASKETT
School of Civil, Environmental and Mining Engineering
The University of Adelaide
Contributed to moment rotation, bond characteristics, moment redistribution and energy
absorption.
I
hereby certify that the statement of contribution is accurate
Signed …………………………………………………… Date …………………
DR. MOHAMED ALI M.S.
S
chool of Civil, Environmental and Mining Engineering
The University of Adelaide
Contributed to moment rotation and bond characteristics.
I
hereby certify that the statement of contribution is accurate
Signed …………………………………………………… Date …………………
WADE D. LUCAS
PhD Candidate
School of Civil, Environmental and Mining Engineering
The University of Adelaide
Prepared figures and performed all analyses involving nonlinear stress and strain profiles.
I hereby certify that the statement of contribution is accurate
Signed …………………………………………………… Date …………………
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
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RAHIMAH MUHAMAD
S
chool of Civil, Environmental and Mining Engineering
The University of Adelaide
Contributed to tension stiffening.
I hereby certify that the statement of contribution is accurate
Signed …………………………………………………… Date …………………
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
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Our obsession with curvature in RC beam modelling
Deric John Oehlers, Matthew Haskett, Mohamed Ali M.S.,
Wade Lucas and Rahimah Muhamad
Abstract
Much of the early research in reinforced concrete dealt with steel reinforcement that was both
ductile and had a very strong bond with the concrete. Hence partialinteraction, that is slip
between the reinforcement and concrete and subsequently debonding, has not been a major
issue. This has allowed researchers to develop the twodimensional fullinteraction moment
curvature approach to model the threedimensional behaviour of reinforced concrete. It is
shown in this paper that this twodimensional fullinteraction momentcurvature approach
relies on a large amount of empirical calibration to ensure a safe design. Furthermore, it is
shown that a threedimensional partialinteraction momentrotation approach can lead to
more advanced structural mechanics models of reinforced concrete behaviours and
subsequently better accuracy and more versatile models.
Keywords: reinforced concrete beams; ductility; rotation; tension stiffening; deflection;
hinges; and moment redistribution.
Introduction
Steel members can be considered to be homogenous so that linear strain profiles can be
applied and subsequently fullinteraction (FI) momentcurvature (M/χ) approaches can be
used. In contrast, the behaviour of reinforced concrete members, as illustrated in Fig. 1, is
incredibly complex: with undisturbed regions between cracked zones where FI M/χ
approaches can also be used; and disturbed regions at cracked zones which occur at both
serviceability and ultimate limit states and where linear strain profiles and subsequently FI
M/χ are only an approximation.
To help develop design rules for these complex structures, reinforced concrete members were
limited to the following three criteria. (1) Steel reinforcement with very large material
ductility, that is very high strain capacities, so that fracture of the reinforcement would not
occur or govern design. (2) Reinforcement with very strong and stiff bond with the concrete,
so that debonding was rarely an issue. Furthermore, the slip between the reinforcement and
concrete could be ignored that is FI, which is a continuity of strain between the reinforcement
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
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and concrete, as opposed to partialinteraction (PI), that is a discontinuity of strain between
the reinforcement and concrete (Newmark et al 1951; Oehlers and Bradford 1995; Yuan et al
2004) could be assumed. And (3) only failure of the concrete in compression at the ultimate
limit state, that is reinforcement debonding and fracture never governed design. In short,
reinforced concrete beams were originally restricted to ductile steel reinforcement with a
strong and stiff bond such that concrete compressive failure governed design. These three
restrictions tend to lend themselves to the FI M/χ approach which is a sectional (two
dimensional or 2D) analysis technique such as in the derivation of the curvature and the
strength of overreinforced members. However, this 2D FI M/χ approach has also been used
to model in general different threedimensional (3D) aspects of reinforced concrete behaviour
such as the formation of hinges where the 2D curvature is integrated over a hinge length and
member deflection where effective flexural rigidities from the 2D FI M/χ approach are used.
To allow the FI 2D M/χ approach to be used to model 3D reinforced concrete behaviours, the
models have been calibrated through an extensive amount of testing. As such, these models
being empirically based are restricted to the bounds of the experimental regimes from which
they were developed as this ensures a safe design.
In this paper, the appropriateness of using the FI M/χ approach in the disturbed regions in
reinforced concrete members is explored. It is suggested that a partialinteraction (PI)
momentrotation (M/θ) approach may be more appropriate for the disturbed regions as it
allows partialinteraction structural mechanics models to be developed; this may help in
reducing the amount of empirical calibration required in developing reinforced concrete
design models outside the present restriction of ductile steel, strong and stiff bond and
compression failure. The development of design rules is first discussed to explain the
importance of empirical models, such as the curvature based empirical models, in allowing
the rapid advancement and application of reinforced concrete. This is followed by a
description of the well established fullinteraction momentcurvature approach to emphasise
the fundamental principles on which it is based. A partialinteraction moment rotation, PI
M/θ, approach is then described which can actually model through structural mechanics the
discrete rotation at individual cracks and the results compared with the FI M/χ approach to
question the appropriateness of using the FI M/χ approach in disturbed regions. These two
approaches are then used to discuss current curvature based empirical techniques for
quantifying the: behaviour of reinforced concrete hinges; moment redistribution and the
ability to absorb energy; and tension stiffening.
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
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Development of a safe design
Structural engineering advances by continually bringing in new technologies or innovations.
A major aim of structural engineering research is to develop design rules for these new
technologies that cover all failure mechanisms and in a form that can be applied to a wide
range of structural types both safely and without further research. This is shown as Box 1 in
Fig. 2 and will be referred to here as current technology that can be applied to established
applications (Denton 2007). An example of current technology is reinforced concrete
structures with steel ribbed reinforcing bars made with ductile steel and which have a very
good bond with the concrete. Most structural engineers would agree that because of the huge
amount of research over the last sixty years this technology can be considered current
technology and can be applied with confidence to established applications such as standard
bridges and buildings under gravity and wind loads. Bringing in new technology such as
brittle reinforcing bars will require further research or applying the current technology to new
applications such as blast loads will also require further research
Seventy years ago steel reinforced concrete was a new technology as shown in Box 2 in Fig.
2. The question is, how was this new technology of reinforced concrete brought into the
market quickly and efficiently whilst maintaining a safe design that is all failure modes were
covered. To do this, the researcher resorted to available structural mechanics models as in
Box 4 which are the laws of nature. However because reinforced concrete is such an
incredibly complex structure, structural mechanics models were not available for all possible
failure mechanisms so these gaps in understanding were filled in by testing, that is empirical
models as in Box 5 were developed. Empirical models are absolutely essential as they allow
new technologies to be brought in rapidly.
Structural mechanics models, such as those listed in Box 4 in Fig. B, need some empirical
calibration mainly in determining the material properties to be used and they need some
empirical validation but the fundamental model itself is not changed empirically. Hence
structural mechanics models tend to have a wide application often well beyond the bounds for
which they were originally validated. The FI M/χ approach for homogenous structures in
Box. 4 is a structural mechanics model as it is based on a fundamental principle of a linear
strain profile, equilibrium and compatibility so it can be used in all types of homogenous
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
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8
structures. Strut and tie modeling can also be considered a structural mechanics model as it is
based on equilibrium and many aspects of buckling can be applied to any type of column.
In contrast to structural mechanics models, empirical models may be based in part on
structural mechanics principles which, however, fall short of providing the whole model to
simulate the mechanism being modelled. Hence, an important part of developing empirical
models is to develop the components of the model that are not covered by structural
mechanics principles being used and this is done through well planned experimental testing.
There are of course empirical models that are purely based on testing without even a
modicum of structural mechanics such as the well used concrete component of the shear
capacity V
c
(AS3600 1988). Because components of empirical models have to be determined
purely through testing, unlike structural mechanics models, empirical models do not have a
wide application but can only be used within the bounds of the tests from which they were
developed.
Examples of empirical models are given in Box. 5. The use of the 2D FI M/χ approach with
empirical hinge lengths to make it three dimensional is an empirical model as it is only
applicable within the range of tests from which the empirical hinge lengths were determined.
Moment redistribution based on the well known neutral axis depth factor k
u
which is obtained
from 2D FI M/χ analyses is also empirically based in order to convert the 2D FI M/χ k
u
approach to solve the 3D problem. And the empirical method of converting flexural rigidities
EI from the 2D FI M/χ approach to cope with nonhomogenous regions in 3D beams is also
an empirical approach as the results are only applicable within the bounds of the tests. The
concrete component of the shear capacity V
c
used in most national standards is an example of
a pure empirical model. It is also an example of how important empirical models are because
the shear behaviour of reinforced concrete is even more complex than its flexural behaviour
so this empirical model has allowed us to design RC members with safety just as long as they
are used within the bounds of the testing regime from which they were developed.
The
structural mechanics models and empirical models in Boxes 4 and 5 in Fig. 2 have been
very well calibrated and validated over a length of time and can be used with confidence over
the range of the testing regimes from which they were both validated and calibrated. This
testing regime encompassed normal concrete, ductile steel and good bond which ensured that
concrete crushing only controlled failure. If we wish now to bring in innovation as in Box 3,
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
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such as the use of brittle steel or nonductile FRP reinforcement, then we can adapt the
structural mechanics models in Box 4 which should not be a problem if they are truly laws of
nature. The problem is in repeating all the testing in Box 5 to widen the scope of the
empirical models to cope with the wider bounds and this can be both expensive and time
consuming. The alternative is to develop structural mechanics models to replace the empirical
models in Box 5 which will be described in the following section, but first let us revisit the
standard 2D FI M/χ approach which is the basis for many of our structural mechanics models.
Traditional curvature approach
A traditional 2D FI M/χ analysis (Oehlers 2007) is shown in Fig. 3 for a crosssection of a
reinforced concrete beam with any distribution of reinforcement and any crosssectional
shape as in Fig. 3(a). The analysis is shown for what might be referred to as standard beams,
that is for beams with: ductile reinforcement material in which the fracture strain is very large
so that the possibility of achieving the fracture strain is unlikely and can be ignored; and with
very good bond as associated with ribbed reinforcement such that the possibility of achieving
the debonding strain can also be ignored. In this case, the only material failure is the concrete
crushing strain ε
c
as shown in Fig. 3(b). This point is, therefore, known at failure so ε
c
can be
used as the pivotal point for a linear strain profile as shown in Fig. 3(c). Essential to this
analysis is the material stressstrain (ζε) relationship so that for each possible strain
distribution in Fig. 3(c) can be derived the stress profile in Fig. 3(d) and subsequently the
force profile in Fig. 3(e). The distribution of strain in Fig. 3(c) in which the force distribution
in Fig. 3(e) is in equilibrium, i.e. in this case sums to zero, is the correct distribution at failure
from which can be determined the moment capacity M
cap
, the curvature at the moment
capacity χ
cap
and the depth of the neutral axis at failure (k
u
d)
cap
and consequently the height of
the flexural crack h
crcap
.
Because the FI M/χ analysis is a structural mechanics model, it can in theory be applied to
members other than the standard reinforced concrete section such as that shown in Fig. 4. For
example, if brittle steel reinforcement is being used then bar fracture at a strain of ε
fbar
in
Fig. 4(b) may occur before concrete crushing. This approach can also be applied to other
types of reinforcement such as the adhesively bonded steel side plates that may fracture at ε
f
sp
or debond at ε
dsp
or the externally bonded FRP plates which may fracture at ε
feb
or debond
at ε
deb
and which is used as the pivotal point in Fig. 4(c). The FI M/χ analysis is very
versatile because for a given curvature χ in Fig. 4(c) the neutral axis position can be varied
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
10
until equilibrium is achieved so that the FI M/χ can be plotted from initial loading to fracture.
Consequently whilst the material remains elastic, it can be used to determine the flexural
rigidity EI for cracked and uncracked sections and the position of the neutral axis or crack
height h
crel
. And at the ultimate limit it can be used to determine M
cap
, χ
cap
, (k
u
d)
cap
and h
crcap
.
The 2D FI M/χ sectional analysis is such a versatile tool that it is not only used in undisturbed
homog
enous regions but also in developing empirical rules in 3D disturbed regions at cracks.
It depends on Bernoulli‟s fundamental principle of a single continuous linear strain profile as
shown in Fig. 4(c), that is fullinteraction, so it does not allow for partialinteraction slip
between the reinforcement and the concrete (Yuan et al 2004; Oehlers et al 2005; Mohamed
Ali M.S. 2008b). It assumes a linear strain profile not only prior to cracking but also after
cracking. And it depends on being able to convert the strains to stresses and, hence, it is
essential to know the material stressstrain properties or at least equivalent values than can be
used in the analysis. The appropriateness of these assumptions are now considered by first
looking at the behaviour of the different regions of a reinforced concrete beam and in
particular the discrete rotation at an individual crack.
Reinforced concrete beam behaviour
A deformed reinforced concrete beam is shown in Fig. 5(a). Prior to cracking, or in regions
well away from cracks, from Bernoulli‟s fundamental principle, plane sections such as lines
AA remain plane so that there is a linear distribution of strain ε
u
within these regions at a
curvature χ
u
as shown. When a crack forms, there is a sudden increase in deflection, or step
change in deflection, due to the widening or rotation of the crack 2θ; stresses build up in the
reinforcement through slip between the reinforcement and the concrete as the region goes
from an uncracked or undisturbed state to a cracked or disturbed state. The distribution of
strain in the immediate vicinity of the crack is disturbed by the crack as represented by the
strain profile ε
d
where adjacent to the crack the tensile strain may reduce or parts could be
ignored as shown. Hence the curvature at the vicinity of a crack is disturbed. The deformation
of the beam is, therefore, due to two components: the continuous variation in curvature as
represented by Fig. 5(b) which can be integrated along the length of the beam to derive the
continuity of rotation and deflection δ
χ
; and the deformation due to the discrete rotation at
each crack as shown in Fig. 5(b) that is δ
θ
.
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
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11
As well as the deformation of the beam, the strength of the beam is also affected by the
discrete rotations. In the undisturbed region in Fig. 5(b), both the strength and deformation of
the beam depends on the strain profile as in ε
u
in Fig. 5(a). However in the disturbed regions
of a crack, the deformation depends on the discrete rotation and the strength depends on the
distribution of strain between the opposite faces of an individual crack and it is a question of
what is this distribution and how does it affect both the stiffness and strength.
Discrete crack rotation
Each individual crack in Fig. 1 can be idealised as two individual crack faces of height h
cr
as
in Fig. 6(b), that through rigid body displacements rotate apart by 2θ such that there is a
linear variation in crack widths from zero width at the neutral axis, or apex of the crack, to
w
sof
at the soffit as in Fig. 6(c) (Oehlers et al 2005; Oehlers et al 2008; Oehlers et al 2009;
Haskett et al 2009a; Haskett et al 2009b). It may be worth noting that adjacent to the
re
inforcement in Fig. 6(c) the crack width w is equal to 2Δ as shown where Δ is the slip of the
reinforcement relative to the concrete at the crack face. This slip Δ at the crack face is due to
the slip δ between the embedded reinforcement and the concrete that is along the anchorage
zone of the reinforcement. Furthermore, the slip in the anchorage zone δ is affected by the
strain in the reinforcement in this anchorage zone which depends on whether the
reinforcement has yielded or not in the anchorage zone. Hence widening of the crack is not
due to straining of the reinforcement between crack faces as this would require infinite strains
as the crack width is initially of zero length when once formed.
Hence, the crack width can only widen through slip Δ at the crack face between the
reinforcement and the concrete that is partialinteraction. This slip is essential in widening the
crack as the formation of a crack is independent of the slip but once it has formed, and even if
the steel has yielded, crack widening without slip requires infinite strains in the reinforcement
which is an impossibility. Hence, the reinforcement on either side of the crack pulls out by Δ
r
such that
rr
h
(1)
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
12
where h
r
is the distance from the reinforcement to the neutral axis, Δ
r
is both the slip of the
reinforcement at the crack face and half the crack width at the level of the reinforcement, and
the discrete rotation at a crack, that is the step change in rotation along the beam, is 2θ.
The relationship between the force in the bar within the crack, P
r
in Fig. 6(b), and the crack
face slip Δ
r
can be derived from well established partialinteraction theory first derived from
composite steel and concrete beams (Newmark et al 1951; Oehlers and Bradford 1995) and
further developed to simulate reinforcing bars pulling out of concrete (Wu et al 2002; Yuan et
al 2004) and in particular FRP reinforcement adhesively bonded to concrete (Mohamed Ali
M.S. et al 2008b). Central to partialinteraction theory is the bondslip (ηδ) relationship
which is determined through pulltests (Eligehausen et al 1983; Seracino et al 2007; Haskett
et al 2009b, De Lorenzis et al 2001, Nakaba et al 2001, Yoshizawa et al 2000). This bondslip
relationship has a nonlinear form as shown in Fig. 7 with a peak shear stress of η
f
that occurs
at a slip δ
1
and a peak slip δ
f
beyond which shear is no longer transferred, that is η = 0. These
nonlinear bondslip relationships can be idealised to help to form structural mechanics
models such as: the linearascending relationship at stiffness k
el
which is useful at
serviceability; the bilinear variation which is a simple approximation of the nonlinear
variation with the peak shear η
f
at δ
1
and a peak slip at δ
f
; and the lineardescending which is
useful in dealing with the ultimate limit state of debonding.
Partialinteraction theory has been used to develop numerical models for the PΔ relationship
(Mohamed Ali M.S. 2008b; Haskett et al 2009b) as well as closed form structural mechanics
models for specific idealised relationships in Fig. 7 (Muhamad et al 2010a). For example, the
following equation for the force in the reinforcement at a crack P
r
uses the linearascending
bondslip relationship in Fig. 7 of constant stiffness k
el
for reinforcement of crosssectional
area A that is still in its linear elastic material phase that is the modulus remains at the elastic
modulus E.
perrelrr
LEAkP
(2)
where (EA)
r
is the axial rigidity of the reinforcement and L
per
is the width of the debonding
failure plane which for circular bars can be taken as the circumference. Substituting Eq. 1
into Eq. 2 and rearranging gives the strain in the reinforcement as
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
13
r
elper
cr
crr
r
EA
kL
h
wh
2
(3)
which is the variation in reinforcement strain ε
r
for reinforcement at any position along the
crack height h
r
and for a specific width of crack at the soffit w
sof
as shown in Fig. 6(d). It can
be seen how partialinteraction theory can be used to derive the variation in reinforcement
strain along the height of a crack which is analogous to the linear strain profile used in FI M/χ
analyses.
It can be deduced from Eq. 3 that for a specific type of reinforcement, such that the modulus
E
r
, crosssectional area A
r
, failure perimeter L
per
and bond stiffness k
el
remain constant and
also for a specific crack configuration of height h
cr
with a linear variation in crack width w
cr
,
the variation in reinforcement strain ε
r
through the height of the crack is linear. For example,
when the width of the crack at the soffit is just sufficient to cause yield, w
sofy
, then the
variation in reinforcement strain is linear as shown by line OA in Fig. 6(d). Widening the
crack will cause yield which starts where the width of the crack exceeds w
sofy
such as at point
B; OB is linear and BC where yielding has occurred can be linear depending on the strain
hardening properties of the reinforcement. For steel with high ductility which can
accommodate very large crack widths such as 10w
sofy
in Fig. 6(d), the variation in the elastic
strain OD only occupies a small region so that the strain distribution is dominated by DE.
It can be seen that for a constant type of reinforcement: whilst the reinforcement remains
elastic, a linear strain profile as in the M/χ analyses in Figs. 3 and 4 is appropriate; and for
ductile steel reinforcement at the ultimate limit of failure a linear strain profile is also
appropriate; but in between as might occur with brittle reinforcement, a linear strain profile
may not be appropriate and this is explored further in the following section.
Strain profiles
The appropriateness of using Bernoulli‟s linear strain profile in Figs. 3 and 4 for cracked
regions is studied in this section: for various types of reinforcement ranging from steel
reinforcing bars to carbon FRP plates in which the elastic modulus E
el
is assumed to be
200GPa; for various types of bondslip properties whose values are summarised in Fig. 8; and
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
14
for a flexural crack height h
cr
in Fig. 6(b) of 500 mm that represents a reinforced concrete
beam and for beams with a concrete strength of 30 MPa.
The variation in the strain profile for a 16 mm ribbed steel reinforcing bar is shown in Fig. 9.
The reinforcement is assumed to have an elastic modulus of 200 GPa and a strain hardening
modulus of 4 GPa and to yield at 450 MPa. The bond properties are shown as OAB
(Haskett et al 2009b) in Fig. 8 where the rising component OA, which controls much of the
behaviour, is commonly accepted and used (CEB Model Code 90). When the crack width at
the soffit w
sof
is 0.67 mm, then the reinforcement at the soffit is about to yield. The parabolic
shape of the variation in the strain profile has the same shape as that of the bond
characteristic OA in Fig. 8. As the crack widens, the strain variation moves from a uni
parabolic to a biparabolic which approaches uniparabolic at very high crack widths and
strains. Because of the parabolic variation in the strain profile, a linear strain profile may be
considered safe as this would underestimate the strains between extremities of the parabolic
variations. Hence reiterating the conclusions from the linear strain profiles in the previous
section, for a beam with one type of steel reinforcement a linear strain profile is appropriate
whilst the reinforcement remains elastic or has reached very large strains but not appropriate
for intermediate conditions.
From Eq. 3 for the linear material and bond characteristics, it can be seen that the cross
sectional area of the bar A
r
and consequently the diameter affects the variation in strain. The
influence of bar diameter is shown in Fig. 10 for a crack width at the soffit of w
sof
= 0.53 mm
which from Fig. 9 is sufficiently small to prevent yield which occurs at a crack width greater
than 0.67mm. It can be seen in Fig. 10 that the diameter of the bar can substantially change
the strain even when all the other parameters including the bond characteristics are assumed
to be the same. When the soffit crack width was increased to 0.85 mm in Fig. 11, yielding
occurred causing an even greater difference in the strain distributions with bar diameter.
The variation in strain for smooth steel reinforcing bars, ribbed steel reinforcing bars,
externally bonded (EB) FRP plates and near surface mounted (NSM) FRP plates with the
bondslip properties in Fig. 8 are shown in Fig. 12 for a crack width (w
sof
) of 0.74 mm, which
allowed some yield in the ribbed steel reinforcement; in effect the remaining variables in Eq.
3 are now being varied. The bond for the smooth steel bar was too weak to allow the smooth
steel bar to yield and thus an approximately linear strain profile, the ribbed steel bars yielded
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
15
at point A, the EB plate reached its maximum bond strength capacity at B after which gradual
debonding accommodated the increase in the crack width, and the NSM plate had good bond
which allowed it to reach large strains. It can clearly be seen that at no depth h
r
can a single
strain be used to represent the existing strains and, therefore, a linear strain profile is totally
inappropriate. In summary, a linear strain profile for a single type of reinforcement that has
great ductility and good bond may be appropriate, but when the reinforcement material and
geometric properties are varied then a linear strain profile is totally inappropriate.
Tension stiffening
Tension stiffening is the effective increase in the stiffness of the tension reinforcement due to
its bond with the surrounding concrete. As such, it controls crack spacing (that is the position
where discrete rotation occurs as in Fig. 5(c)), crack widths (that is the amount of discrete
rotation at an individual crack), and consequently deflections due to the discrete rotation. In
studying tension stiffening, it is common practice (CEBFIB 1985, Eurocode 2 1991, Chang
and Sung 1996, David et al 2008, Gilbert 2007, Marti et al 1998) as a first step in the
understanding and quantification to idealise the tension region in Fig. 5(a) as an axially
concentrically loaded steel reinforced prism as in Fig. 13.
The first crack that forms in the beam in Fig. 13(a) is the left bound of the prism in Fig.
13(b). On the left of the prism, P
r
is the axial force in the reinforcement and Δ
r
is the slip at
the crack face. On the right of the prism there is a region of fullinteraction, that is the slip
strain ds/dx and slip s both tend to zero (ds/ds & s 0) and the axial force P
r
is resisted by
both the concrete and the reinforcement. The partialinteraction analysis, such as that used to
derive the P/Δ relationship in Eq. 2 (Muhamad et al 2010a), can be used to quantify the
behaviour of the prism in Fig. 13(b).
From partialinteraction theory (Muhamad et al 2010b) and for the linear ascending bond
slip relationship in Fig. 7 and for linear material properties which may be appropriate at
serviceability, the position S
pr
at which fullinteraction is first achieved (ds/ds & s 0) is
given by
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
16
c
r
rr
per
el
pr
EA
A
EA
L
k
S
1
2
(4)
where E
r
and A
r
are the reinforcement elastic modulus and area respectively and (EA)
c
is the
axial rigidity of the concrete prism. The axial tensile stress in the concrete prism is zero on
the left face in Fig. 13(b) and builds up to a maximum at the distance S
pr
where there is full
interaction. Hence, S
pr
is also the minimum spacing of the primary cracks should they occur.
Equation 4 is a structural mechanics model as the form of the equation does not rely on
empirical testing but only on the material properties such as k
el
, E
r
and the concrete modulus
E
c
that have to be obtained experimentally.
The primary crack spacing structural mechanics solution of Eq. 4 can be compared with the
following empirical solution from Eurocode 2 (1991) which has been rearranged to have the
same parameters as in Eq. 4 which is the reason why the variable A
r
could in theory be
cancelled out.
c
r
r
per
bond
pr
A
A
A
L
k
S
..
1
50
(5)
where k
bond
is a bond parameter that distinguishes between smooth and ribbed bars. Equation
5 is an empirical approach as the form of the equation was not determined from structural
mechanics but purely from the analysis of test result. Hence both the form and the material
properties have been obtained experimentally. Even though some of the important parameters
in Eq. 4 have been identified through testing in Eq. 5, it can be seen that Eq. 5 struggles to
reflect the true behaviour in Eq. 4 mainly because of the complexity of Eq. 4.
When a primary crack has formed at S
pr
, the partialinteraction problem is now of a
symmetrically loaded prism as in Fig. 13(c) where by symmetry the only boundary condition
is s = 0 at S
pr
/2 and this gives the following relationship at the crack face between the slip Δ
r
which is half the crack width w
cr
and the reinforcement load P
r
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
17
r
prr
r
EA
SP
2
)2tanh(
(6)
which can also be compared with the following Eurocode 2 approach (Eurocode 2 1991)
r
prr
r
EA
SP
2
(7)
where β is an average crack spacing parameter. Equation 7 has an identical form to the
structural mechanics solution in Eq. 6 probably because of the simplicity of Eq. 6. This
correlation is very good but the problem is that it depends on the crack spacing S
pr
in Eq. 5
which needs improving.
It can be seen that partialinteraction theory can be used to determine the position and onset
of both primary and secondary cracks. Furthermore as both Δ
r
and S
pr
are known, this can be
used to derive an effective strain which is often used in tension stiffening.
Having now derived structural mechanics models for the crack positions and crack widths,
these can be used in Fig. 5(c) to derive the change in deflection δ
θ
due to each crack. This can
be added to the curvature deflection in Fig. 5(b) to get the total deflection. This partial
interaction approach can be compared with the following current empirical curvature
approach (Branson 1965, AS3600)
3
s
cr
cruncrcref
M
M
IIII
(8)
which uses an effective moment of inertia I
ef
from the uncracked I
uncr
and cracked I
cr
values
that can be obtained from a 2D FI M/χ analysis as in Fig. 4. The form of this equation, such
as the exponent, has been derived purely empirically. It is correct when cracking has not
occurred as it gives the curvature deformation in Fig. 5(b) but it is based on the assumption
that a fullinteraction twodimensional analysis can be used to represent the discrete rotation
which is a threedimensional partialinteraction problem as it allows slip. For this reason, it is
suggested that the next step in accuracy may need the application of a partialinteraction
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
18
structural mechanics approach. Having considered serviceability, let us now consider the
ultimate limit state of which an essential component is the momentrotation relationship of
hinges.
Hinges and energy absorption
Let us consider the continuous beam in Fig. 14(a) where the loading conditions and
distribution of reinforcement is such that the first or primary hinges would form at the
negative or hogging regions and the secondary hinges would form in the positive or sagging
regions as in Fig. 14(b). As the beam is loaded, the rotation imposed on the primary hinges θ
χ
1
in Fig. 14(c) and that which the discrete rotation has to accommodate can be determined by
integrating the curvature along the undisturbed region. If the beam is loaded until the discrete
rotation limit at the secondary hinge θ
cap

2
is achieved as shown in Fig. 14(d), then the
rotation that the primary hinge has to accommodate is θ
χ1
+ θ
cap2
as shown. If the primary
hinge cannot accommodate this rotation then it fails before the rotation capacity of the
secondary hinge can be achieved. It can be seen that the rotation capacity of a beam and its
ability to deform and absorb energy depends on both the curvature rotation in the undisturbed
region θ
χ
and the discrete rotation and its limits θ
cap
in the disturbed regions.
At the ultimate limit state, much of the discrete rotation is concentrated about the first crack
as can be seen in Fig. 1. Let us consider the discrete rotation at a single crack as in Fig. 15
where failure can occur: either in the tension zone by debonding or fracture of the tension
reinforcement; or in the compression zone by concrete crushing through sliding of the wedge
(Haskett et al 2009a; Mohamed Ali M.S. 2010) or through shear failure (Lucas et al 2010),
although, these latter two limits are not dealt with in this paper which is only looking at the
discrete rotation due to the reinforcement slip at a flexural crack.
Numerical solutions (Oehlers et al 2005; Mohamed Ali M.S. et al 2008a) and closed form
solutions (Haskett et al 2009b) are available for the 3D PI M/θ analysis depicted in Fig. 6(a)
for the moment and discreterotation. At the ultimate limit state, the depth of the compression
zone can be derived from standard procedures such as for rectangular compression blocks
and gamma factors in Fig. 3(d) often used in national standards (AS3600 1988) which also
gives the position of the resultant compressive force from the tension reinforcement such as
d
1
in Fig. 6(b). Hence for a single layer of reinforcement, the moment at reinforcement
fracture M
f
is easily determined.
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
19
The rotation depends on the bondslip relationship such as those in Fig. 7. Based on the linear
descending bond slip characteristic in Fig. 7, the rotation at fracture (Haskett et al 2009b) is
given by
11
hh
shel
f
(9)
whe
re Δ
el
is the slip in the elastic range of the steel reinforcement to cause yield and Δ
sh
is the
slip in the strain hardening range to cause fracture, and where
ff
xb
1
f
1
x
δτ4
εd
11
h
δ
h
Δ
(10)
in which d
b
is the diameter of the reinforcement and for the elastic range Δ
x
is Δ
el
, f
x
is the
yield stress f
y
and ε
x
is the yield strain ε
y
, and for the strain hardening range Δ
x
is Δ
sh
, f
x
is the
increase in stress due to strain hardening f
f
f
y
and ε
x
is the increase in strain during strain
hardening ε
f 
ε
y
.
It can also be seen from Eq. 9 that yielding increases the slip and therefore rotation. It can be
seen from Eq. 10 that the discrete rotation is directly proportion to δ
f
which is a measure of
the ductility of the bond and inversely proportional to the depth of the reinforcement h
1
which
is proportional to the depth of the beam. Also from Eq. 10, it can be seen that increasing the
diameter of the bar, the stress capacity and the strain capacity also increases the rotation
indirectly. Conversely, increasing the fracture energy of the bondslip η
f
δ
f
reduces the rotation
which is the opposite effect of δ
f
so that the effect of the bond is not clearly defined.
Based on the linear ascending bondslip characteristics (Muhamad et al 2010) in Fig. 7, the
rotation at yield is
2
11
4 hk
fd
h
el
yyby
y
(11)
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
20
which clearly confirms the direct dependence of rotation on the bar diameter, and stress and
strain at yield and the inverse dependence on the beam depth which is proportional to h
1
and
bond stiffness.
Current hinge rotation philosophy is to use the 2D FI M/χ analysis from Figs. 3 or 4 to
determine the curvature at failure χ
f
and then to derive a hinge length L
hinge
empirically such
that the χ
f
L
hinge
gives the experimental rotation. Examples of empirical hinge lengths are
given in Table 1 (Baker 1956; Sawyer 1964; Corley 1966; Mattock 1967; Priestley and Park
1987; Haskett et al 2009b; Haskett et al 2009b; Panagiotakos and Fardis 2001). It can be seen
that the depth of the beam d can appear in either or both the numerator and denominator
suggesting that there is confusion with regard to its effect, whereas, the structural mechanics
model suggest an inverse relationship. The direct dependence on the length of the beam L or
that of the specific hogging or sagging region z may be a reflection that longer beams will
accommodate more cracks as in Fig. 1 which will allow more rotation, or that shorter beams
are more susceptible to shear failure which limits the rotation, or simply that the total rotation
that includes the curvature rotation has been attributed to the hinge. More recently (Priestly
and Park 1987; Panagiotakos and Fardis 2001) there has been a dependence on both L and the
diameter of the reinforcement d
b
which would suggest that the first parameter L accounts for
the curvature rotation and the second parameter d
b
accounts for the discrete rotation due to
bar slip which is in agreement with the discrete rotation model. Finally the empirical hinge
length approach does not consider the material ductility that is the fracture strain or stress
which is a major factor in the discrete rotation model.
Table 1. Empirically derived hinge lengths
Researcher Reference
Hinge length (
l
p
)
Hinge length variables
Baker (1956)
ddzk
4/1
/
span, depth
Sawyer (1964)
zd 075.025.0
span, depth
Corley (1966)
dzdd/2.05.0
span, depth
Mattock (1967)
zd 05.05.0
span, depth
Priestley and Park (1987)
yb
fdL 022.008.0
span, bar diameter
Panagiotakos and Fardis (2001)
yb
fdL 021.018.0
span, bar di
ameter
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
21
Moment redistribution
Moment redistribution is the ability of a hinge such as the ve hinge in Fig. 1 to hold a
moment and rotate while a require moment is achieved elsewhere. Let us consider the
continuous beam in Fig. 14 where discrete rotation hinges first form in the negative regions
with a moment capacity of M
1
and a rotation capacity of θ
cap1
as shown in Fig. 14(c). Hence
it is a question of finding the sagging moment M
sag
such that the variation of curvature in the
undisturbed region from M
sag
to M
1
causes a curvature rotation θ
χ1
rotation that is equal to
θ
cap1
. From this moment distribution can be determined the well known moment
redistribution factor K
MR
often given as a percentage which is the change in the hogging
moment from its elastic value as a proportion of its elastic value. The following moment
redistribution factor (Oehlers et al 2010; Haskett et al 2010) was obtained from structural
mechanics
LMEI
EI
K
MR
.2
2
(12)
where EI is the flexural rigidity of the undisturbed region which can be obtained from a FI
M/χ analysis as in Fig. 4 and M and θ is not only the moment and discreterotation
combination at failure but also any combination within the moment discreterotation response
of the hinge. It can be seen that structural mechanics can be used to quantify all aspects of
moment redistribution and, hence, this is a structural mechanics solution.
The empirical approach used frequently in national standards is based on the neutral axis
depth factor k
u
as defined in Fig. 3(d). From Fig. 3, it can be seen that the curvature is given
by
dk
u
c
(13)
If it is assumed that the hinge length is proportional to d as often assumed in Table 1, then the
rotation is given by
u
c
u
c
k
d
dk
d
(14)
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
22
that is it is inversely proportional to k
u
just as long as concrete crushing at ε
c
occurs. Based on
this assumption, the variation in K
MR
is then determined from tests results as shown for the
national standards in Fig. 16 (Oehlers et al 2010). It can be seen that not only is there a large
scatter of magnitudes but the variations also vary from linear, to bilinear to trilinear
although it could be said that in general there is a bilinear relationship. The variations from
the structural mechanics approach is also shown for different depths of beam. It can be seen
that this too is bilinear but that there is a family of curves that depend on the depth of the
beam. Hence the k
u
approach by itself will not be able to accurately quantify the moment
redistribution for ordinary reinforced concrete beams let alone those which do not fail by
concrete crushing but it is a safe lower bound.
Summary
The standard fullinteraction momentcurvature approach based on Bernoulli‟s linear strain
profile is appropriate for homogenous structures and in undisturbed regions. The question is
whether it should be used at all to quantify disturbed regions such as at cracks. A structural
mechanics rigidbodydisplacement partialinteraction model has been used to study the
strain distribution in reinforcement crossing a flexural crack. It is shown that a linear strain
profile is applicable when one type of reinforcement is used and when either the
reinforcement has not yielded or the reinforcement material is very ductile that it has a large
strain capacity. Hence the linear strain profile may be appropriate to current building
techniques that use ductile reinforcement with good bond characteristics but it is
inappropriate for brittle steel reinforcement. It is also shown that a linear strain profile is
totally inappropriate when the geometry or material properties of the reinforcement are varied
in a crosssection. It has also been shown that unlike fullinteraction momentcurvature
approaches that are used in a semiempirical approach to simulate and quantify disturbed
regions, a partialinteraction rigidbodydisplacement approach can quantify and explain the
mechanisms that control tension stiffening, deflection and moment redistribution.
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The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
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26
Figures
negative regionpositive region positive region
Fig. 1 Two span continuous beam
Failure mechanisms
safe general design
for all
failure mechanisms
research
research
Current Technology Innovation
Established
Applications
normal
steelreinforced
concrete
brittle steel
or brittle FRP
reinforcement
with brittle bond
Struct/Mech models
Momentcurvature (M/
Strut/Tie modelling.
Buckling.
Empirical models
Moment rotationM/ and empirical hinge lengths
Moment redistribution: neutral axis depth factor k
u
from M/
Deflections: effective moment of inertia I from M/
Concrete component of shear capacity V
c
New Technology
Box 1:Box 2:Box 3:
Box 4:Box 5:
Fig. 2 Structural engineering research
F
c
f
ail
cap
c
pivotal point
(b)failure strain (c)strain profile (d)stress profile (e)force profile
d
(a)
k
u
d)
cap
(k
u
d)
cap
neutral axis
h
crcap
Fig. 3. Traditional fullinteraction M/χ analysis
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
27
dbar
c
FRP plate
(b)
f
ail
cap
steel
side
plate
deb
feb
pivotal point
fbar
dsp
fsp
(a)
c
deb
(c)
(d)
Fig. 4 M/χ structural mechanics model
u
d
u
d
2
A
A
A
A
2
(a) deformation components
(b) curvature deformation
(c) discrete rotationdeformation
Fig. 5 Deformation in RC beam
1
1
2
2
P
1
P
2
P
comp
w
sof
h
1
d
1
(b) rigid body movement(a)
w
sof
/2
h
r
(c) crackwidths/slip
h
r
r
(d) reinf. strain
y
h
cr
y
w
sofy
A C E
B
D
Fig. 6. Discrete rotation at crack
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
28
f
f
linear ascending
linear descending
bilinear
nonlinear
k
el
Fig.7. Typical bondslip relationships
0
4
8
12
16
0 1 2 3
Slip δ (mm)
Bond Strength η (MPa)
EB
NSM
Smooth Steel
Ribbed Steel
Ribbed Steel: η
max
=
13.7MPa
δ
1
= 1.5mm
δ
max
= 15mm
α = 0.4
NSM: η
max
= 7.8MPa
δ
1
= 0.16mm
δ
max
= 1.6mm
α = 1
2.4mm x 10mm
EB: η
max
= 6.2MPa
δ
1
= 0.015mm
δ
max
= 0.15mm
α = 1
100mm x 1.2mm
Rising Branch: η = η
max
(δ/δ
1
)
α
δ
max
= 15mm
δ
max
=
15mm
Falling Branch: η = η
max
(δδ
max
)/(δ
1
δ
max
)
A
B
Smooth Steel:
η
max
= 1.6MPa
δ
1
= 0.1mm
δ
max
= 15mm
α = 0.5
Fig. 8. Bondslip properties
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
29
0
100
200
300
400
500
0.00 0.01 0.02 0.03 0.04
ε
r
hr (mm)
ε
y
5ε
y
15ε
y
0.67
0.88
2.22
w
sof
(mm)
Fig. 9 Strain profile for 16 mm ribbed bar
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
30
0
100
200
300
400
500
0.0000 0.0005 0.0010 0.0015 0.0020 0.0025
ε
r
hr (mm)
12mm
16mm
20mm
Fig. 10 Variation in bar diameter – strains prior to yield
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
31
0
100
200
300
400
500
0.000 0.004 0.008 0.012 0.016
ε
r
hr (mm)
12mm
16mm
20mm
yield
yield
yield
Fig. 11 Variation in bar diameter – strains after yielding
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
32
0
100
200
300
400
500
0.000 0.002 0.004 0.006 0.008
ε
r
hr (mm)
Smooth
Steel
(16mm)
EB
(100x1.2mm)
Ribbed Steel
(16mm)
NSM
(2.4x10mm)
A
B
Fig. 12 Strain profiles – all parameters varied
P
r
P
r
S
pr
ds/dx & s = 0
primarysecondary
b
b
(a) beam
first
(b) primary crack spacing
S
pr
/2
s = 0
(c) secondary crack
P
r
r
r
P
r
P
r
Fig. 13. Tension stiffening
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
33
primary hinge secondary
hinge
L/2 L/2
primary hingesecondary
hinge
cap2
cap
M
1
M
2
M
1
(a) beam
(b) hinges
(c) moment
redistribution
cap1
cap1
(d) energy
absorption
Fig. 14 Discrete and curvature rotation
Fig. 15 Discrete rotation at failure
Fig. 16 Moment redistribution results
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
34
STATEMENT OF AUTHORSHIP
A Generic Unified Reinforced Concrete Model
Proc. ICE , Structures and Buildings 2010: accepted paper
PROFESSOR DERIC J. OEHLERS
School of Civil, Environmental and Mining Engineering
The University of Adelaide
Prepared final manuscript and supervised development of model.
I
hereby certify that the statement of contribution is accurate
Signed …………………………………………………… Date …………………
DR. MOHAMED ALI M.S.
School of Ci
vil, Environmental and Mining Engineering
The University of Adelaide
Assisted in manuscript preparation and review
I hereby certify that the statement of contribution is accurate
Signed …………………………………………………… Date …………………
PROFESSOR MICHAEL C. GRIFFITH
School of Civil, Environmental and Mining Engineering
The University of Adelaide
Assisted in manuscript review
I hereby certify that the statement of contribution is accurate
Signed ……………………………………………….. Date ………………….
DR. MATTHEW HASKETT
School of Civil, Environmental and Mining Engineering
The University of Adelaide
Developed the flexural rigid body rotation model
I hereby certify that the statement of contribution is accurate
Signed …………………………………………………… Date …………………
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
35
WADE D. LUCAS
PhD Candidate
School of Civil, Environmental and Mining Engineering
The University of Adelaide
Developed the rigid body rotation model for combined shear and flexural loading
I hereby certify that the statement of contribution is accurate
Signed …………………………………………………… Date …………………
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
36
A generic unified reinforced concrete model
Oehlers
,
D.J., Mohamed Ali M.S., Griffith
M.C., Haskett
M. and Lucas W.
Abstract
The behaviour of reinforced concrete members with ductile steel reinforcing bars at the
ultimate limit state is extremely complex. Consequently, there has been a tendency for the
seemingly disparate research areas of flexure, shear and confinement to follow separate paths
in order to develop safe approaches to design. In this paper, it is shown how the already much
researched and established, but somewhat peripheral, areas of reinforced concrete research of
shearfriction, partialinteraction and rigidbodydisplacements can be combined to produce a
sing
le unified reinforced concrete model that simulates: the momentrotation of hinges and
their capacities; the shear deformation across critical diagonal cracks leading to failure; as
well as the effect of confinement on these behaviours. It is shown that this unified reinforced
concrete model is completely generic as it can be used to simulate a reinforced concrete
member with: any type of reinforcement material including brittle steel or fibre reinforced
polymer; various crosssectional shapes of reinforcement, that is, not only round bars but also
flat externally bonded or rectangular near surface mounted adhesively bonded plates; and any
type of concrete such as high strength or fibre reinforced concrete. Hence, this new model
should allow the development of more accurate and safe design procedures as well as
enabling more rapid development of new technologies.
Keywords: reinforced concrete; flexure; shear; confinement; hinge; ultimate limit state; FRP.
Introduction
The development of a unified reinforced concrete model stems from the research by the
authors on developing design rules for the new technology of fibre reinforced polymer (FRP)
and steel plating reinforced concrete and masonry structures
14
. The inclusion or practical
application of new technology is illustrated in the innovation analysis tool
5,6
in Fig. 1 which
depicts the role of research in taking innovative ideas into practice. It is suggested that the
main thrust of civil engineering research is to develop a safe general design for all failure
mechanisms, as shown in Fig. 1 box 1. This can then be classified as current technology for
use in established applications; an example of which is reinforced concrete with normal
strength concrete and ductile steel reinforcement which can be applied to standard bridges
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
37
and structures under monotonic and seismic loads. This safe general design is often codified,
but the emphasis in a national code or standard is to ensure that it is a safe design and not
necessarily the most advanced or efficient design.
It can be seen in Fig. 1 that applying the current technology in box 1 to new applications in
box 3, such as for the resistance to blast loads, will require further research. Furthermore,
applying new technology in box 2, such as the use of brittle FRP reinforcement, brittle steel
reinforcement or high strength concrete, to established applications will also require new
research. The latter scenario is taken one step further in Fig. 2 which depicts the types of
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