research required to achieve a safe design for reinforced concrete members
6
.
Historically, at some stage ductile steel reinforced concrete, in Fig. 2 box 2a, was considered
to be a new technology and research was required to bring it to the level of a safe general
design for the observed or envisaged failure mechanisms. Where possible, sound structural
mechanics models were developed that simulated and quantified the failure mechanisms, box
5a, and as they were based on structural mechanics principles they required relatively little
testing to validate and calibrate. An example is the material capacities of RC members
subjected to flexural and axial loads and only for members with ductile steel reinforcement
and normal strength concrete; such that the crushing of concrete at an effective strain ε
c
always governs failure, that is, members always fail by concrete crushing. Further examples
are strut and tie modelling as it is based on equilibrium, and the buckling mechanism.
Where sound structural mechanics models were not available due to the incredible
complexity of reinforced concrete behaviour, a large amount of testing, that is through
experiments and/or numerical modelling, had to be done to develop empirical models, Fig. 2
box 6a: to temporarily fill in the knowledge gap; to ensure a safe design; and to allow the
rapid application of this new technology at the time. Examples are shown in box 6a: the
momentrotation (M/θ) of hinges through the use of momentcurvature (M/χ) and empirical
hinge lengths; moment redistribution which depends on the neutral axis depth factor k
u
; the
concrete component of the shear capacity V
c
; and the effect of concrete confinement ζ
lat
. As
can be imagined, the development of these empirical models required a very large amount of
testing as they could only be used within the bounds of their testing regimes, but their
development was absolutely essential in order to be able to develop a safe design.
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
38
Having converted reinforced concrete with ductile steel reinforcement and normal strength
concrete to a safe general design (as represented by boxes 1, 2a, 5a and 6a in Fig. 2), the
question is: what has to be done to bring in a new technology, such as the use of brittle FRP
reinforcement or fibre reinforced concrete in box 2b? The easiest part is to convert the
existing sound structural mechanics models, in box 5a, which were based on concrete
crushing, to allow for the other reinforcement failure mechanisms of fracture or debonding
7
as in box 5b, as these structural mechanics models are generic. However, the major problem
in developing a new technology to a point where it can be used safely for design is in the
testing required where sound structural mechanics models are not available (box 6b), as it is
very expensive and time consuming. Hence, the necessity for developing new structural
mechanics models to replace the existing empirical approaches as in box 6c, which is the
subject of this paper. This paper deals with what might be loosely termed as hinge regions
where large deformations occur through flexure and shear which lead to failure and where,
because of these large deformations, concrete confinement has the greatest effect.
The aim of this paper is to show how the peripheral research areas of shearfriction, partial
interaction and rigidbodydisplacements (in box 7 in Fig. 2) can be combined to form a
generic unified model for the behaviour of reinforced concrete beams at the ultimate limit
state. Some direct validation of aspects of the unified model are given in this paper, although,
most of this direct validation, where theoretical results are compared with tests on beams,
eccentrically loaded prisms, pull tests and confined concrete cylinders, are published
elsewhere
4,813
. However, it is not the aim of this paper to give the precise material properties
required for the analyses but, instead, to demonstrate how the unified model can be used to
explain behaviours that are known to occur in experimental tests but for which current
empirical models have difficulty explaining. Consequently, bringing a clearer insight into
reinforced concrete behaviours and further confidence in the validity of the approach. It may
also be worth noting that, unlike numerical models such as finite element models, because the
unified model is based on structural mechanics models this allows the development of closed
formed mathematical models of failure mechanisms
1114
which will aid in the development of
design rules. This new unified model has been presented in this paper in terms of a single
primary crack because the structural mechanics models have been developed for this
problem
1114
. It is realised that more than one primary flexural crack can dominate and
structural mechanics models
15
are being developed for this and numerical simulations based
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
39
on the concepts described in this paper have been developed for multiple cracks
10,16,17
but this
is the subject of further research.
The current empirical approaches or models used to quantify momentrotation (M/θ), the
concrete component of the shear capacity (V
c
), and the effect of lateral confinement (ζ
lat
) are
first described in this paper, as these behaviours are used to illustrated the generic unified
model. There already exists a great deal of very good research in what has been termed
peripheral approaches in box 7 in Fig. 2 as they are not often used directly in design. Theses
peripheral approaches are: shearfriction (SF) sometimes referred to as aggregate interlock
which quantifies the behaviour across a crack interface; partialinteraction (PI) where slip
occurs at an interface causing a discontinuity in the strain profile; and rigidbody
displacements (RBD) such as in the opening up or sliding of crack faces. The fundamental
principles behind these peripheral approaches are described and then they are combined to
form a new integrated failure mechanism that is the basis for the development of a unified
model for the momentrotation behaviour of RC hinges. It is then shown how this unified
M/θ model can be extended to allow for shear deformations and, interestingly, how shear
failure depends on flexure. Finally, it is shown how the unified M/θ model can be used to
quantify the effect of confinement in cylinder tests and more importantly in rectangular
flexural members.
Current empirical approaches
The empirical models mentioned in box 6a in Fig. 2 will now be described to explain the
need for structural mechanics models. Empirical models are frequently dimensionally
incorrect but this is certainly not of any major concern as empirical models are there to ensure
a safe design. Special attention will be paid in the following section to current moment
rotation approaches as it will be shown later that it is the moment rotation behaviour which
also controls the shear behaviour as well as that due to confinement.
Momentrotation
An example of a slab that was subjected to a blast load
18
and which failed in flexure is shown
in Fig. 3. The slab can be divided into the nonhinge region and the hinge region as in Fig. 4.
The nonhinge region generally encompasses most of the member and is associated with
narrow flexural cracks and where the concrete in compression is in its ascending or first
branch as in the stress profile ζ
nonhinge
. The analysis of this nonhinge region is
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
40
straightforward as standard methods of equilibrium and compatibility apply directly but it is
complex as partialinteraction, through reinforcement slip, and disturbed regions have to be
accommodated
10,16,17
. In contrast, the hinge region occurs: over a small length; is associated
with concrete softening in a wedge; and is also associated with wide cracks where the
rotation θ is concentrated and in particular where the permanent rotation is concentrated. It is
the moment rotation of the hinge region which is considered here and not that of the non
hinge region where standard procedures can be applied.
Close up views of typical hinges are shown in Fig. 5: the hinge in (a) was formed in a RC
slab made with steel fibre concrete which was subject to a blast load; and the hinge in (b) was
formed in a steel plated beam subject to monotonic load
1
. The threedimensional moment
rotation response of flexural members is required in design to quantify such things as the
column drift, moment redistribution and the ability to absorb the energy from dynamic loads.
A common approach is to use a fullinteraction sectional analysis to derive the moment
curvature relationship
7
and then to integrate this over a hinge length to get the moment
rotation relationship; this will be referred to as the moment curvature hinge length approach.
For ease of discussion, let us idealise the moment curvature relationship as bilinear
19
as in
Fig. 6. Let us assume that the crosssection of the continuous beam in Fig. 7(a) has the bi
linear relationship OAB in Fig. 6 which has a rising branch OA, with a peak moment M
A
which occurs at a curvature χ
A
, followed by a falling branch AB. On initially applying the
load to the beam in Fig. 7(a), all crosssections of the beam, that is the whole length of the
beam, follow the rising branch OA in Fig. 6 until the moment at the supports just reaches
M
A
at a curvature χ
A
as shown in Fig. 7(b). Hence there are only two points in the beam,
which are at the supports, where the moment M
A
has been achieved and these are referred to
as the hinge points in Fig. 7(b).
On a further downward deformation of the beam in Fig. 7(a), the peak moment at the
supports, M
A
in Fig. 7(b), must reduce as the curvature changes from χ
A
in Fig. 6 to
accommodate the deformation; as any change in curvature from χ
A
in Fig. 6 must result in a
reduced moment as χ
A
occurs at the peak moment. Let us assume that the support moment
reduces to M
1
in Fig. 6 resulting in the distribution of moment in Fig. 7(b) with a maximum
M
1
. For this to occur, the curvature at the hinge points increases along the falling branch AB
in Fig. 6 to χ
AB1
to accommodate the deformation, as these are the only points which initially
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
41
lay on the falling branch AB. In contrast, the remainder of the beam, that is not including the
hinge points, which was wholly on the rising branch OA remains in the rising branch OA,
so that the curvature adjacent to the hinge points reduces to χ
OA1
. Hence there is a step change
in the curvature shown as Δχ
B
in Fig. 6 and which occurs over a hinge of zero length as
shown in Fig. 7(c). This is of course an impossibility as first recognised by Barnard and
Johnson
20
and Wood
21
and which will be referred here as the zero hinge length problem
9,19
.
To overcome the zero hinge length problem, which is peculiar to the moment curvature
approach, and in order to allow a safe design, the hinge length has been derived empirically
22
27
with the exception of a more advanced technique developed by Fantilli et al
28
which still
uses a moment curvature approach but which bases the hinge length on the softening wedge
size L
soft
in Fig. 4. This approach of combining a twodimensional momentcurvature analysis
with an empirically derived hinge length is mathematically convenient and, no doubt, gives
good results within the population of test results from which it was derived. However, it can
give very large scatters
27
when used outside the population of test results from which they
were derived.
It is also worth noting that a moment curvature analysis is a twodimensional analysis which
is being used to simulate the threedimensional moment rotation behaviour. Take for example
the hinges in Fig. 5 where it can be seen that most of the rotation is concentrated in the
flexural crack face rotation. The flexural cracks can only widen if there is slip between the
reinforcement (that is the longitudinal reinforcing bars in (a) represented by broken lines or
the plate in (b)) and the concrete. If the bond between the reinforcement and concrete is
strong and stiff then for a given force in the reinforcement, the interface slip will be small and
consequently the cracks will be narrow and the rotation small. Conversely, weak bond will
lead to large slips and consequently large rotations. It can be seen that a twodimensional
moment curvature analysis simply cannot allow for this threedimensional behaviour. It is
also worth noting that twodimensional moment curvature analyses cannot allow for the
threedimensional behaviour of the compression wedges which can be seen in Figs. 3 and 5
and which are a significant component of the overall behaviour.
Further evidence of problems associated with the application of the moment curvature
approach are illustrated in Fig. 6. Take, for example, the case where the falling branch AB
becomes less steep as in AC. The step change in curvature Δχ
B
, that the hinge of zero length
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
42
theoretically has to accommodate, must increase from Δχ
B
to Δχ
C
until for a perfectly ductile
member, that is where the „falling branch‟ is horizontal as in AD, the step change required in
the curvature tends to infinity. This is simply not correct. To complete the discussion, it is
often perceived that the hinges in member sections with rising second branches, such as AE
in Fig. 6, are easier to determine or can be quantified. This is a false assumption as the hinge
length is now purely a property of the bending moment distribution, that is the hinge length is
the region of the beam where the moment lies in the region from A to E. As further proof of
the inadequacies of this approach, it can be seen that when the slope of AE tends to zero, that
is it tends to the perfect ductility response AD, then the hinge is of zero length and where A
E tends to the elastic condition AF then the hinge length tends to L/2. All of which is
nonsensical.
From the above discussions, it can be seen that the twodimensional moment curvature
analysis cannot be used to quantify through structural mechanics the hinge length and this has
been referred to as the zero hinge length problem. Therefore, the moment curvature hinge
length approach is not what may at first glance appear to be a generic structural mechanics
model but an empirical model with its associated restrictions. Hence, the need for a structural
mechanics model of the moment rotation.
Concrete component of the shear capacity
An example of shear failure of a beam without stirrups is shown in Fig. 8. The concrete
component of the shear capacity V
c
is one of the most intractable problems in reinforced
concrete and an extensive amount of experimental testing has been required to develop
empirical models to ensure a safe design. An example
29
is shown in the following equation
3/1
2
2000
4.1
bd
fA
bd
a
dd
V
cst
c
(1)
whe
re d and b are the effective depth and width of a beam, a is the shear span and A
st
the area
of the longitudinal reinforcing bars. The lack of understanding of the structural mechanics
behind this extremely complex problem of shear failure is reflected in the need for a size
effect as in the first parameter, a dimensionally incorrect stress component of the fourth
parameter and the use of lower bounds to represent the design strengths
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
43
It needs to be emphasised that there is nothing wrong with this empirical approach which is
essential in developing a safe design and also in identifying the parameters that control shear
failure. However because of the empirical nature of Eq.1, it would be very risky to apply it
directly for new technologies such as the use of high strength or high performance concrete
or the use of brittle FRP reinforcement as this lies outside the bounds of the tests from which
Eq. 1 was derived. It will be shown later that shear failure is simply another limit to the
moment rotation capacity of a beam.
Concrete confinement
For a long time, confinement of concrete in compression has been known to enhance the
ductility of reinforced concrete members and as far back as 1928 Richart
30
produced the
following empirical expression for the effect of hydrostatic confinement on the compressive
cylinder strength
latccc
ff 1.4
(2)
where f
cc
is the confined strength, f
c
the unconfined strength and ζ
lat
the hydrostatic
confinement. An extensive amount of testing has been done over the intervening eighty years
to refine this empirical model and to adapt it for FRP confinement as in Fig. 9(a). However,
columns rarely if ever are designed or fail in pure compression. Consequently, the greatest
difficulty has been to take this empirical research on confined concrete cylinders as in Fig.
9(a) and apply it to real structures where moment exists as in Fig. 9(b) and also to apply it to
rectangular sections as in Fig. 9(c) where confinement is not uniform. It will be shown that
the formation and sliding of the compression wedges, in Figs. 3 and 5, is a major contribution
to the dilation of concrete. This allows dilation to be included in the moment rotation
behaviour which provides a structural mechanics solution.
Peripheral approaches
The peripheral approaches, of rigid body displacement, shear friction and partial interaction
(box 7 in Fig. 2) that are needed to develop a structural mechanics model for the empirical
models in box 6a are described here. It is also shown how they can be combined to form a
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
44
generic integrated failure mechanism that can be used to model the seemingly disparate
behaviours of moment rotation, shear failure and the effect of confinement.
Rigid body displacements
The rigid body deformations due to flexure of the hinges in Fig. 5 are shown idealised in Fig.
10 for a reinforced concrete beam with both externally bonded plates and reinforcing bars.
The crack faces are shown as straight lines for convenience to emphasise that the variation of
crack width over the crack height h
cr
is linear; this linear variation in the crack width occurs
irrespective of the shape of the crack face due to the rigid body displacement. Slip between
the reinforcements and the concrete, Δ
rebar
and Δ
plate
, allows the crack width to increase from
zero to h
rebar
and h
soft
, and the consequence of which is the rotation θ of the flexural crack
where most of the hinge rotation is now concentrated. The compression wedge, which has
been much studied through research on eccentrically loaded prisms
31
as in Fig. 11, also
exhibits rigid body displacements through the slip s
soft
and separation h
soft
shown in Fig. 10.
The rigid body deformation due to shear across the critical diagonal crack in Fig. 8 is shown
idealised in Fig. 12. The shear displacement s induces a separation of the crack faces h due to
aggregate interlock.
Shear Friction
There has been extensive research on shear friction, sometimes referred to as aggregate
interlock, which is the behaviour across a crack, such as the critical diagonal crack in Fig. 8
(idealised in Fig. 12) that is subject to a shear rigid body displacement. Mattock and
Hawkins
32
quantified the shear capacity v
u
in terms of the normal force across the crack ζ
n
nu
mcv
(3)
where c is the cohesive component and m the frictional component of the MohrCoulomb
failure plane as shown in Fig. 13 and which depends on whether the failure plane was
initially cracked, or initially uncracked but deformed through the formation of a herringbone
formation of cracks.
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
45
Walraven
33
studied the effect of discrete particles bearing on each other as in Fig. 12 and
quantified the relationship between the interface slip s, the crack widening h induced by s
through aggregate bearing, the interface normal force ζ
n
and the shear v
u
as shown in Fig. 14.
As an example, if an interface is displaced as shown in Fig. 12 by both a specific slip s =
0.7mm and crack width h = 0.4mm, then from Fig. 14 it can be seen that there is only one
combination of v
u
= 0.76 N/mm
2
and ζ
n
= 0.38 N/mm
2
that can occur. Mattock and Hawkins‟
and Walraven‟s research
32,33
fully describe the interaction between the shear behaviours s, h
and v
u
across a crack when subjected to given normal force ζ
n
.
Partial Interaction
Finally, the third peripheral research area is partialinteraction theory which can be defined as
the behaviour when there is slip across an interface s as this produces a step change in the
strain profile at the interface. Partialinteraction theory was first developed by Newmark,
Siess and Viest
34
for the study of composite steel and concrete beam behaviour
35,36
and very
recently in the study of intermediate crack debonding of adhesively bonded plates
37,38
.
Whenever a crack of any description intercepts reinforcement such as the reinforcing bar in
Fig. 5(a), or the externally bonded (EB) plate in Fig. 15, or the near surface mounted plate
(NSM) plate in Fig. 16, or the externally bonded plates used for shear strengthening in Fig.
17, then slip must occur across the interface between the concrete and the reinforcement to
allow the crack to widen. This is, therefore, a partial interaction problem and the solution to
this problem requires knowledge of the interface bond slip characteristics which can be
determined from experimental tests. Typical bond slip values are shown in Fig. 18 for a
ribbed reinforcing bar, a near surface mounted plate and an externally bonded plate, where η
is the interface shear stress and δ the interface slip. The relationship is often idealised in the
generic bilinear form shown (Fig. 18) where the important characteristics are the peak shear
stress η
max
and peak slip δ
max
.
From partial interaction theory and for reinforcement that has a unilinear stressstrain
relationship such as FRP or steel prior to yield, the structural mechanics relationship between
the force in the reinforcement P
reinf
, in Fig. 10, and the slip of the reinforcement at the crack
face Δ
reinf
is given by the following equations
11
which apply when the reinforcement is fully
anchored
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
46
max
inf
1
max
inf
1arccossin
re
per
re
L
P
(4)
inf1max
max
1
re
per
AE
L
(5)
where L
per
is the width of the failure plane perimeter which for round bars is the
circumference and for externally bonded plates the width of the plate, A
reinf
is the cross
sectional area of the reinforcement and E
1
is the modulus of the reinforcement material. For
reinforcement with a bilinear stressstrain relationship such as steel after yield in which the
initial elastic modulus E
1
changes to E
2
at a stress f
1
then the behaviour after f
1
is given by
1inf
max
1inf
2
max
inf
1arccossin fA
L
P
re
re
per
re
(6)
inf2max
max
2
re
per
AE
L
(7)
where Δ
1
is the slip at stress f
1
which can be derived from Eqs. 4 and 5. For reinforcement
that is not fully anchored, similar expressions which depend on the boundary conditions
14
are
available.
Integrated failure mechanism
It can now be seen that if there is a rigid body displacement across two crack faces as in Fig.
12, then the widening of the crack through shearfriction or aggregate interlock by h, which is
twice the slip of the reinforcing bar as shown in Fig. 10, will induce forces in the reinforcing
bars that can now be quantified through partialinteraction theory. Mattock and Hawkins‟ and
Walraven‟s research
32,33
depends on knowing the normal force ζ
n
in Fig. 12. Partial
interaction theory has completed their research by quantifying the normal stress induced by
the reinforcement crossing this shear interface and consequently the shear capacity which
depends on this normal stress. However, partialinteraction theory has also allowed Mattock
and Hawkins‟ and Walraven‟s research
32,33
to be used not only under shear displacements but
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
47
also under the much wider application of shear and flexural rigid body displacements as
shown in Fig. 19 which will be used in the following unified models for moment rotation,
shear and concrete confinement.
Of interest, it can be seen that the shear resisted directly by the dowel action of the
reinforcement crossing the crack is not addressed directly in the integrated failure mechanism
in Fig. 19. Dowel action is a recognised component of the shear resistance. However, any
additional shear resisted by dowel action will be offset by the reduction in the aggregate
interlock shear capacity due to the reduced axial capacity of the reinforcement after allowing
for the stresses due to the dowel action.
Moment rotation model
The rigid body rotations of the hinges in Figs. 3 and 5 have been idealised in Fig. 10. The
flexural hinge can be considered to have the three distinct components shown in Fig. 20: the
shearfriction concrete compression softening wedge of depth d
soft
which can resist a force
P
soft
and which through partial interaction slips a value s
soft
; the depth of concrete in the
ascending portion of its stressstrain relationship d
asc
which resists a force P
asc
which can be
obtained from standard procedures and where the peak strain is ε
pk
as shown; and the partial
interaction tensile zone of height h
cr
associated with rigidbodyrotation of the crack faces
and where the reinforcement forces are P
reinf
.
Let us first consider the tension zone in Fig. 20. For a given height of crack h
cr
and for a
given rotation θ, the slip in the reinforcement layers Δ
reinf
is fixed so that the force P
reinf
in
each reinforcement layer can be determined from Eqs. 47. The limits to this rotation occur
when either the reinforcements reach their slip at fracture which can also be obtained from
Eqs. 46 or when they debond which, as a lower bound, occurs at δ
max
in Fig. 18. What is still
required is the limit to this rotation due to concrete compression wedge failure and also the
extent of the compression zone, that is d
soft
plus d
asc
, as this fixes the crack height h
cr
which is
needed for determination of the rotation θ for a specific reinforcement slip Δ
reinf
.
The behaviour of the concrete wedges in Figs. 3, 5 and 11 and required in the analysis in Fig.
20 can be derived from Mattock and Hawkins‟ shearfriction theory
32
represented in Fig. 13
and Eq. 3. For a given depth of wedge d
soft
in Fig. 21 and from shear friction theory
8
, the
force the wedge can resist is given by the following equation
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
48
sincossin
cossincos
m
mc
dwP
lat
softbsoft
(8)
where w
b
is the width of the wedge which is generally the width of the beam, ζ
lat
is the lateral
confinement which can be induced by the FRP wrap, as in Fig. 21, or by the stirrups, m and c
are the shear friction material properties shown in Fig. 13 and Eq. 3 and α is the angle of the
weakest plane in Fig. 21 which is given by
1arctan
2
mm
(9)
The difference in the concrete compressive strain at the wedge interface which is the peak
ascending branch strain ε
pk
minus the strain in the softening wedge ε
soft
is the slip strain
across the wedge interface which when integrated over the length of the hinge L
soft
as in Eq. 7
gives the interface slip which has a limit s
slide
that is a material property which can be
determined from tests
13,32
.
slidesoftsoftpksoft
sLs
(10)
It is this latter limit s
slide
that limits the depth of the softening wedge and by so doing restricts
the rotation.
Thus the three limits to the rotation, shown on the left of Fig. 22(a) are: sliding of the wedge
when s
soft
equals s
slide
; fracture of the reinforcement that is when the reinforcement slip Δ
reaches the slip at reinforcement fracture given by Eqs. 47; or debonding of the
reinforcement when the slip is at least δ
max
in Fig. 18. To determine which of these limits
comes first, let us consider the bilinear strain profile in Fig. 22(b) where there is a linear
variation in strain in the compression ascending region of depth d
asc
and zero tensile strain in
the cracked region of height h
cr
; this is a generally accepted strain profile at the ultimate limit
state when the flexural cracks are closely spaced. For a given width of concrete L, it can be
seen that the bilinear strain profile can be converted to a bilinear rigid body displacement as
shown and, importantly, the rotation within the compression ascending zone θ is simply the
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
49
crack rotation θ. It is this relationship that allows the interaction between the wedge
deformation and crack rotation
12
as shown in Fig. 22(c).
The moment rotation analysis
12
is depicted in Fig. 22(c). Importantly, it is based on a linear
rigid body displacement ABC as opposed to the conventional linear strain profile. Starting
with a small rotation θ, the rigid body displacement ABC is moved up or down whilst
maintaining θ until the axial forces P, from Eqs. 49, sum to zero after which the moment can
be taken for that specific rotation. The rotation is gradually increased to get the moment
rotation curve bearing in mind that a limit occurs when either the wedge reaches its sliding
capacity s
slide
or a layer of reinforcement either debonds or fractures. A typical analysis
12
is
compared with test results in Fig. 23 where in this case the rotation was limited by fracture of
the reinforcing bar which is, therefore, the ductility limit at M
cap
/θ
cap
.
Moment redistribution
As an example of the application of the moment rotation model, it is applied here to moment
redistribution to explain the difficulty with current empirical approaches. A two span
continuous beam is shown in Fig. 24 where it can be seen that virtually all of the permanent
rotation is concentrated in single cracks within the hinges. Hence, it would be expected that it
is these crack rotations which will control most of the moment redistribution.
Let us define the moment redistribution factor K
MR
as the moment redistributed as a
proportion of the moment if there were no redistribution; hence, 100K
MR
is the commonly
used percentage moment redistribution. Let us consider the case of a continuous or encastre
beam of span L and flexural rigidity (EI)
cr
as shown in Fig. 25 that is subjected to a uniformly
distributed load. Hence before the hinges are formed that is in the elastic range, the hogging
or negative maximum moments are twice the maximum sagging or positive moment. The
moment redistribution capacity of the beam
39
for redistribution from the hogging (negative)
to sagging (positive) region, K
MR
, is given by
crcapcap
crcap
MR
EILM
EI
K
2
2
(11)
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
50
where the rotation at the supports is θ
cap
when the moment at the supports is M
cap
; M
cap
and
θ
cap
are the limits to the moment rotation which can be derived from the analyses described
above and an example of which is shown in Fig. 23.
National standards usually quantify the moment redistribution K
MR
using the neutral axis
depth factor k
u
. Examples of these empirical approaches from four national standards are
shown in Fig. 26. In general, they have a bilinear variation characterised by large differences
between empirical models. Results from the unified moment rotation approach are also
shown in Fig. 26 for beams of varying depth. These also exhibit a „bilinear‟ variation where
the first part of the curve at lower values of k
u
are governed by bar fracture and the second
part by wedge sliding and between these curves there is a discontinuity. It can be seen that the
variations from the national standards are just one part of a family of curves and, hence, just
using the neutral axis factor k
u
will never provide an accurate analysis and which explains the
large variations between the empirical models from national standards.
Shear deformation
The momentrotation model described above and illustrated in Fig. 22 for a vertical crack can
just as easily be applied to inclined cracks as in Fig. 27(a) where a moment P
fl
d induces a
force in the reinforcement of P
fl
which in turn requires bar slips to produce a crack width of
h
fl
at the soffit. The same inclined crack is now only subjected to shear deformations as in
Fig. 27(b) which, through aggregate interlock, opens the crack by h
sh
. This induces axial
tensile forces in the bars of P
sh
but more importantly an equal but opposite axial compressive
force whose resultant is at the same position (although shown slightly offset in Fig. 27(b) for
clarity) and which provides the normal interface stress ζ
n
that is essential for resisting the
shear as shown in Figs. 1214.
The
flexural and shear deformations in Figs. 27(a) and (b) have been combined in Fig. 28. It
can be seen that the shear deformation increases the crack width by h
sh
and subsequently
increases the force in the reinforcement from P
fl
to P
fl
+P
sh
, but the resultant tensile force
remains at P
fl
because of the compressive force P
sh
across the interface so that the moment
remains at P
fl
d. From Walraven‟s research
33
illustrated in Fig. 14, the concrete component of
the shear capacity across the inclined plane (V
c
)
β
increases with P
sh
but reduces with the
crack width h, that is to say there are two opposing effects on the shear capacity. Hence the
shear capacity is limited not only by the remaining strength of the reinforcement after
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
51
allowing for flexure, which emphasises the importance of strain hardening, but also by the
crack width which depends on the bond and bar properties as illustrated in Fig. 18 and Eqs. 4
7. It can now be seen how flexure and shear interact no matter what the inclination of the
crack and that neither the shear capacity nor the flexural capacities are independent. It can
also be seen that shear failure can be envisaged as simply the fourth limit to the moment
rotation behaviour after wedge sliding, and reinforcement debonding or fracture.
It is also worth noting that vertical stirrups or externally bonded plates as in Fig. 17 and
shown as the transverse reinforcement in Fig. 29 can be treated the same way in this model as
the longitudinal reinforcement, as the forces in the vertical or transverse reinforcement, (P
fl
)
st
and (P
sh
)
st
also depend on the crack width due to flexure and shear and there is also an
interface compressive force (P
sh
)
st
. However, there is a subtle difference in their effects.
The ability to resist shear across an interface has been shown by Mattock and Hawkins
32
and
Walraven
33
to depend on the compressive force normal to the interface, as shown in Figs. 13
and 14. Hence the interface compressive forces P
sh
in Fig. 29 have been resolved about the
slope of the interface which is at an angle β as shown in Fig. 30. An inclined FRP NSM plate
as in Fig. 31 is also shown in Fig. 30 where it is assumed to be perpendicular to the crack
face. The compressive interface forces (P
sh
)
st
cosβ + (P
sh
)
NSM
+ (P
sh
)
long
sinβ provide the
interface normal stress ζ
n
. Hence, depending on the variation of ζ
n
and the crack width h and
from Walraven‟s research
33
in Fig. 14, the concrete component of the shear strength (V
c
)
β
can
be determined. It can be seen that reinforcement that is placed perpendicular to the critical
diagonal crack, such as the NSM plate, are the most efficient in increasing the concrete
component of the shear capacity (V
c
)
β
. However, (V
c
)
β
must also provide the inclined forces
(P
sh
)
st
sinβ and (P
sh
)
long
cosβ to maintain equilibrium with the tensile forces in the
reinforcement. Hence, the concrete component of the shear capacity available to resist shear
is reduced by [(P
sh
)
long
cosβ  (P
sh
)
st
sinβ)]. Hence, from this integrated failure mechanism, it
can be seen that the longitudinal reinforcement induces a component of force that reduces the
concrete shear capacity and the vertical reinforcement induces a component that is beneficial
which explains the subtle difference in their effects.
Effect of spandepth ratio on shear
The concrete component of the shear capacity of a beam is known to increase with reduced
shear spans as can be seen in the second parameter 2d/a in Eq. 1. This increase in strength has
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
52
often been associated with arching action, however, this increase in shear capacity can also
be explained with this unified model. The concrete component of the shear capacity in
empirical models is usually determined from tests on beams with short shear spans, as in Fig.
32, to try to ensure shear failure precedes flexural failure. In these tests and because of the
short shear spans, a critical diagonal crack usually extends from the applied load to the
support as shown. This beam with this critical diagonal crack was analysed using a numerical
model based on the unified model and the shear spans were varied as shown to determine the
effect of the spandepth ratio.
For a fixed angle β in Fig. 32, the analysis consisted of first imposing a flexural crack
deformation of a maximum width h
fl
in Fig. 32 which is shown as OA at h
fl
= 0.06 mm in
Fig. 33 for an analysis in which the inclined crack β is at 45
o
. This crack width h
fl
fixes P
fl
(Eqs. 47) and consequently the applied moment. The shear crack width h
sh
in Fig. 32 was
then gradually increased, that is h
total
in Fig. 33, which imposed the interface force P
sh
in Fig.
32 and for each increment of h
sh
the shear capacity V
c
in Fig. 33 was determined to get the
variation BCD. From Walraven‟s research
33
, The shear capacity is proportional to P
sh
and
inversely proportional to h
total
. Hence, initially the shear capacity increases along BC as the
effect of increasing the interface force P
sh
dominates but after a while along CD the effect of
the crack width h
total
dominates, even though the interface force is increasing, to produce a
decline in strength.
As the flexural crack h
fl
in Fig. 32 is reduced to say h
fl
= 0.24 mm in Fig. 33 that is OE, the
force due to flexure in the reinforcement reduces and, hence, a greater proportion of the
reinforcement‟s strength is available to resist shear and, as would be expected, the shear
capacity increases as can be seen by comparing FGH with BCD. A further reduction in the
flexural crack width to h
fl
= 0.06 mm gives JKL The shear capacities K, G and C in Fig. 33
are the capacities that are available at specific applied moments and in this case reducing
moments from K to C. The results are plotted in Fig. 34 as the line MN. The results for crack
angles β of 40
o
and 35
o
are also shown as lines PQ and RS.
Figure 34 shows the variation of the shear capacity available with increasing moment. The
ratio between the applied shear and moment depends on the beam geometry and these are
shown as the loading lines OT at a/d = 1 (45
o)
, OU at a/d = 1.3 (40
o
) and OV at a/d = 1.6
(35
o
). Where these loading lines intercept the capacities gives the shear to cause failure, that
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
53
is V
c
at points W, X and Y. It can be seen that steep inclinations of crack such as at β = 45
o
with strengths MN are stronger than shallow inclinations such as at β = 35
o
with strengths at
RS and this is one explanation of the so called arching action. However, just as important,
the loading line is steeper for steep inclinations of crack which by itself would produce higher
shea
r capacities. Hence, there are two effects that produce the arching action. It can be seen
that the unified model can by itself explain the so called arching effect bringing a further
insight into shear behaviour.
Concrete confinement
It has been shown that the shearfriction analysis in Fig. 21 can be applied to the two
dimensional analysis of wedges in beams or prisms under flexure, such as in Figs. 3, 5 and
11. However, wedges also form in the three dimensional problem of confined cylinders and
rectangular prisms under pure compression, as in Figs. 9(a) and (c). In the following section,
shearfriction theory developed by Mattock and Hawkins
32
as in Fig. 13 will be applied to the
analysis of confined concrete cylinders as in Fig. 9(a) as further evidence of the usefulness of
the shearfriction component of the integrated failure mechanism. However, the analysis of
confined cylinders under pure compression is fairly academic as virtually all structures are
designed to take moment as in Fig. 9(b) and are often not circular. To provide a solution, it
will be shown that the unified moment rotation model can also be used to simulate dilation
and consequently the effect of confinement due to stirrups as well as for FRP wrap and for
the analysis of rectangular sections.
Pure compression members
The stressstrain relationship of hydrostatically confined concrete has the typical shapes
40,41
OAEDC in Fig. 35 which increases in magnitude with the lateral confinement (ζ
lat
)
n
. The
branch OA can be considered to be a material property with the peak value at the start of
softening of ζ
start
at A which depends on the amount of hydrostatic confinement. It is
hypothesised that: after the peak stress ζ
start
is reached, wedges start to form; and the wedges
are fully formed when the strength stabilises at the residual strength ζ
soft
. A typical variation
of the stressstrain relationship for FRP confined concrete is shown as the broken line E
1
E
2

E
3
. For example, at point E
2
the FRP fractures so the confinement stress is known and equal
to that for the hydrostatic variation OA
2
E
2
D
2
that is (ζ
lat
)
2
; that is the FRP stressstrain
relation is just the one point E
2
of this variation. Hence, the axial stress at FRP fracture
(ζ
fract
)
2
will be expected to be equal to or greater than the residual strength ζ
soft
.
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
54
The circumferential wedge in Fig. 36 encompasses the whole of the truncated cone and when
the depth of the wedge d
soft
is equal to the radius of the cylinder d
prism
/2, that is the failure
plane is shown as de, then the behaviour of the horizontal plane is fully governed by shear
friction so that the strength is ζ
soft
. The shearfriction theory has been applied to cylinders
13
.
When the wedge in Fig. 36, which encompasses the whole of the interior truncated cone,
occupies the whole of the horizontal truncated plane, then the stress at which this occurs is
the residual strength ζ
soft
of the cylinder and it is given by
sincossin
cossincos
m
mc
lat
soft
(12)
which is virtually the same equation as for the wedge in a rectangular section given by Eq. 8
and where the wedge angle α is given by Eq. 9.
The residual strength in Eq. 12 varies according to the amount of confinement ζ
lat
. Applying
typical values for the Mohr Coulomb properties in Fig. 13 of m = 0.8 and c = 0.17f
c
, Eq. 12
converts to
latcsoft
f 3.471.0
(13)
which is a remarkably similar form to Richart‟s
30
(1928) empirical equation for the strength
of confined concrete in Eq. 2 although it should be remembered that Richart‟s equation
30
measures the peak strengths f
cc
shown as ζ
start
in Fig. 35, whereas, the shearfriction
expression in Eq. 11 is a measure of the residual strengths ζ
soft
in Fig. 35.
The theoretical shearfriction residual strengths from Eq. 12 have been compared
13
with
tests
4043
in which the residual strength and confinement were measured directly and the
results are shown in Fig. 37 for both hydrostatically restrained concrete and steel spirally
confined concrete and show good correlation. They have also been compared with FRP
confined concrete
13
in Fig. 38 where, as to be expected, they have a strength that is slightly
greater than ζ
soft
.
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
55
The results in Figs. 37 and 38 are good if not very good bearing in mind the large scatter
associated with confinement research and also bearing in mind that a single value of m and c
was used and research has shown some variation as shown in Fig. 13. It would, therefore,
appear that shearfriction provides a structural mechanics model for confined concrete in
cylinders and gives further confidence in the procedure outlined in Fig. 21. It is now a
question of: how do we apply the research on confined concrete under uniform compression
to the real problem of beams or columns that have moment? This will be dealt with in the
following section.
Compression members subjected to flexure
The analysis in Fig. 22 allows the moment rotation behaviour and limits to be determined.
Even though the wedge moves upwards as a consequence of the interface sliding s
soft
in Fig.
22(a), it was not necessary to include this movement as the analysis was primarily concerned
with unconfined concrete. However, sliding across the wedge interface causes an upward or
dilation movement of the wedge as shown in Fig. 39. The upwards movement consists of
two components. If the wedge interface was smooth, then sliding across the interface at an
angle α would cause an upward movement h
α
, as shown, which depends purely on the
geometry of the wedge. However, aggregate interlock also causes an upwards movement h
a
which can be derived from Walraven‟s research
33
in Fig. 14. Hence, the wedge by itself
causes a dilation with two components h
a
and h
α
as well as the usual dilation due to the elastic
mate
rial Poisson effect.
The moment rotation analysis in Fig. 22 is shown again in Fig. 40 to illustrate the effect of
confinement from both stirrups and FRP wrap. To do the analysis
44
an additional iterative
procedure, to that described for Fig. 22, has to be included. For a fixed depth of wedge d
soft
in Fig. 40(b), guess the confinement ζ
lat
. For the depth of wedge d
soft
, the force in the wedge
P
soft
from Eq. 8 induces a slip s
soft
from Eq. 10. This slip s
soft
produces an upward movement
h
prism
equal to h
a
+h
α
. This is in effect a crack width in Fig. 40. If the plate is unbonded, then
the strain in the plate due to the movement h
prism
gives the stress and consequently the force
in the wrap P
wrap
. If the wrap is adhesively bonded, then the Eqs.4 to 7 can be used to
determine P
wrap
and it can be shown that this is better at confinement at the early stages of the
crack development. The same approach can be used to derive the force in the stirrups P
stirrup
.
The confinement ζ
lat
can be derived from P
wrap
and P
stirrup
. If this is not equal to the initial
guess of ζ
lat
then it will be necessary to iterate towards a solution. The rest of the procedure
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
56
for deriving the moment rotation is unchanged. Hence the unified moment rotation model can
also provide a structural mechanics model for confinement.
An example of a moment rotation analysis
44
is shown in Fig. 41 for a reinforced concrete
beam with 1% steel and in which the rotation was limited by wedge sliding. The upper curve
is for a 900 mm deep beam and the lower for a 600 mm deep beam, point a is the rotation
when the steel yielded, point b for a beam without stirrups or wrap, point c for a beam with
stirrups, and point d for a beam with both stirrups and an FRP wrap. The moment rotation
analysis reflects what is known to occur in tests that is deep beams are less ductile than
shallow beams, stirrups increase the ductility, and the wrap does not enhance the strength but
further enhances the ductility; which is further evidence of the validity of the generic unified
reinforced concrete model.
Conclusions
It has been shown that at present empirical models are used to quantify the momentrotation,
concrete component of the shear capacity and the effect of lateral confinement. Because of
the incredible complexity of reinforced concrete behaviour, these empirical models have been
essential in plugging the structural mechanics gaps in our understanding to allow a safe
design and the introduction of reinforced concrete when it was a new technology. However,
these empirical models, by their very nature, can only be used within the bounds from which
they were obtained and are, consequently, of limited help in deriving more accurate
reinforced concrete design procedures or in allowing new technology such as the use of high
strength concrete, fibre reinforced concrete, fibre reinforced polymer reinforcement or in new
applications such as blast loading.
A generic integrated failure mechanism has been developed that combines the well
established research areas of shear friction, partial interaction and rigid body displacement to
produce a reinforced concrete failure mechanism which can be used for: (1) any type of
reinforcement material such as brittle FRP, or ductile or brittle steel or whatever; (2)
numerous shapes of reinforcement such as circular ribbed bars or flat externally bonded
plates or rectangular near surface mounted plates; and for (3) both shear and flexural
deformations and their combinations.
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
57
This generic integrated failure mechanism has been developed into a structural mechanics
moment rotation model which has upper limits due to the concrete wedge sliding or
reinforcement fracture or debonding. It is also shown that this generic moment rotation model
can also be used to quantify the effects of shear and consequently the shear capacity and also
the effects of confinement due to stirrups or wraps and, hence, is in effect a generic unified
reinforced concrete model.
It is shown that this model can explain many phenomenon that are known to occur but have
eluded structural mechanics solutions such as: the reason the neutral axis depth factor will
never truly quantify the moment redistribution because the approach is part of a family of
curves; the shear capacity depends on the flexural forces and why transverse reinforcement is
better at enhancing the concrete component of the shear capacity than longitudinal
reinforcement; how compression wedges that form in flexural members as well as in
compression members are not an illusion but a shear friction mechanism that can be
quantified allowing the residual strength of confined concrete to be quantified; and how
stirrups increase the flexural ductility and FRP wrap does not increase the flexural strength
but increases the ductility.
Ac
knowledgements
This research was supported by Australian Research Council Discovery Grants “Ductile
retrofit of concrete frames subjected to static and earthquake loading.” and “Development of
innovative fibre reinforced polymer plating techniques to retrofit existing reinforced concrete
structures”.
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
58
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The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
61
Notation
A
st
= crosssectional area of longitudinal reinforcement
A
reinf
= crosssectional area of reinforcement
a = length of shear span
CDC = critical diagonal crack
b
v
= width of beam
c = cohesive component of Mohr Coulomb failure plane
d = effective depth of beam
d
asc
= depth of compression concrete strain hardening
d
prism
= cylinder diameter
d
soft
= depth of wedge
E = modulus
EB = externally bonded plate
(EI)
cr
= flexural rigidity of cracked member
FRP = fibre reinforced polymer
f
c
= unconfined compressive cylinder strength
f
cc
= confined compressive cylinder strength
HSC = high strength concrete
h = crack width
h
a
= crack opening due to aggregate interlock
h
cr
= height of flexural crack
h
soft
= transverse movement of compression wedge
h
α
= crack opening due to sliding along smooth wedge
IC = intermediate crack
K
MR
= moment redistribution factor
k
u
= neutral axis depth factor
L = length of prism; length of beam
L
per
= perimeter length of failure plane
L
soft
= length of wedge; length of hinge
M = moment
m = frictional component of Mohr Coulomb failure plane
M
cap
= hinge moment at limit of rotation
MD = moment distribution
NSM = near surface mounted plate
P = axial force in reinforcement; longitudinal force
PI = partial interaction
P
soft
= maximum longitudinal force the wedge can resist
RBD = rigid body displacement
SF = shear friction
s = slip across interface
s
slide
= sliding capacity of interface
s
soft
= slip across wedge interface
V
c
= concrete component of the shear capacity
(V
c
)
β
= shear capacity of concrete along inclined plane
V
R
= shear force at support
v
u
= interface shear stress; interface shear capacity
w
b
= width of the wedge
α = slope of weakest wedge
β = angle of inclined crack
χ = curvature
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
62
δ = interface slip
δ
max
= maximum interface slip that resists shear
Δ = slip of reinforcement at crack face
Δχ = step change in curvature
ε = strain; strain profile
ε
c
= concrete crushing strain
ε
pk
= concrete strain at maximum strength
ε
soft
= longitudinal strain in compression wedge
θ = rotation
θ
cap
= hinge rotation at limit of rotation
ζ = stress; stress profile
ζ
fract
= confinement stress at FRP fracture
ζ
lat
= confinement stress
ζ
n
= compressive stress normal to crack interface
ζ
soft
= stress in softening wedge; residual strength
ζ
start
= stress at commencement of wedge formation; peak strength
η = interface shear stress
η
max
= maximum interface shear capacity
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
63
Figures:
Current Technology
normal strength concrete
with ductile steel rebars
New Technology
FRP, HSC,
brittle steel rebars
Established Applications
for standard bridges and
buildings under seismic
and monotonic loads
New Applications
such as resistance
to blast loads
safe general
design
[for all failure
mechanisms]
Innovation
Innovation
Innovation
research
research
research
box 1:box 2:
box 3:box 4:
Fig. 1 Innovation analysis tool
safe general design
of
failure mechanisms
research
research
Current
Technology
New
Technology
Innovation
Established
Applications
ductile
steelreinforced
concrete
Failure mechanisms
box 5b:
Eg. Adapt steel RC
struct/mech model
for brittle FRP fracture
and debond
box 6b: Repeat testing
program in box 6a
box 6c: Develop generic
struct/mech models
M/
V
c
,
lat
box 7 : A unified reinforced concrete model
Peripheral approaches: shearfriction (SF);
partialinteraction (PI); rigidbodydisplacements (RBD)
FRP reinforcement
HSC, fibre concrete,
brittle steel rebars
box 1:
box 2a:box 2b:
box 5a:
Struct/Mech models
Eg.:
Flex/Axial capacities for
ductile rebars (
c
).
.
Strut/Tie modelling.
Buckling.
box 6a:
Empirical models
Experimental and numerical modelling
Eg.: M/M/ and hinge length L
soft
moment redistribution  k
u
Concrete component of shear capacityV
c
Effect of concrete confinement
lat
Failure mechanisms
Fig. 2 Types of research to develop a safe design
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
64
Fig.3 Slab subjected to blast load
maximum
moment
point of
contraflexure
nonhinge region
d
nonhinge
point of
contraflexure
hinge
L
soft
<f
c
first branch
first branch
second branch
nonhinge regionhinge region
f
c
Fig.4 Hinge and nonhinge regions
(a) RC beam (b) plated beam
Fig.5 Hinge rotations
M
A
M
1
A
falling branch
O
rising
branch
M
A
B
C
B
C
D
D
infinity
E
F
L
soft
0
L
soft
= f(MD)
L
soft
= L/2
AB1
OA1
Fig.6 Moment curvature of RC member
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
65
M
(b) moment
M
A
(c) curvature
hinge points
A
M
1
(a) beam
hinge points
AB1
OA1
Fig.7 Continuous beam
Fig.8 Shear failure of beam without stirrups
(a) cylinder (b) column (c) rectangle
Fig.9 Confined concrete
rebar
rigid
body
rotation
concrete
softening
zone
crack
face
rigid body rotation
h
soft
wedge
s
soft
h
rebar
reinforcing
bar
externally
bonded plate
h
plate
plate
P
reinfrebar
P
reinfplate
h
cr
Fig.10 Rigid body deformations due to flexure
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
66
Fig.11 RBD in eccentrically loaded prism
s
s
h
n
n
v
u
v
u
v
u
n
n
v
u
h
h
Fig.12 Rigid body deformation due to shear
uncracked plane
v
u
0
n
cracked plane
c
un
c
cr
m
1
m
2
m
3
Fig.13 Mattock and Hawkins‟ approach
v
u
n
s
h = 1.0 mm
h = 1.0 mm
h = 0.4 mm
h = 0.4 mm
h = 0.1 mm
h = 0.1 mm
0
1.0
N/mm
2
0.5
N/mm
2
0.5
N/mm
2
2mm
s = 0.7mm
0.76
0.38
Fig.14 Walraven‟s approach
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
67
Fig.15 IC debonding of EB plate across flexural crack
NSM
CFRP
side
face
strip
s
Centre support
Fig.16 IC debonding of NSM side plate
critical
diagonal crack
IC debonding
Fig.17 IC debonding of EB plate across CDC
EB
NSM
deformed reinforcing bar
0 0.17mm 5 mm 15 mm
max
= 15 MPa
max
max
max
max
= 10 MPa
max
= 6 MPa
rising branch
falling branch
Fig.18 Bondstress/slip characteristics
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
68
v
u
s
P
reinf
rigid body
shear and
flexural
displacement
n
P
reinf
n
v
u
n
h
2
reinf
= h
Fig.19 Integrated failure mechanism
primary
flexural
crack
compression
wedge
s
soft
wedge
plane
f
c
at
pk
concrete
stress
s
soft
h
soft
h
reinf
P
reinfrebar
tension zone
compression
wedge or
softening zone
concrete
hardening
zone
h
cr
d
asc
d
soft
reinf
= h
rebar/2
P
reinfplate
plate
= h
plate/2
P
soft
P
asc
Fig.20 Idealised hinge components
FRP confinement force
stirrup
confinement
force
interface
compressive
force
V
u
n
P
soft
s
soft
d
soft
L
soft
latFRP
latstirrups
Fig.21 Shearfriction behaviour of concrete wedge
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
69
reinfrebar
d
soft
d
asc
P
asc
P
soft
P
reinfrebar
s
soft
s
soft
f
c
at
pk
concrete
stress
h
soft
h
reinf
P
reinfrebar
h
cr
profile
h
cr
(a) RBD in hinge (b) RBD interaction (c) RBD analysis
L
soft
rigid body
displacement
RBD
profile
A
B
C
reinfrebar
d
asc
wedge
sliding
limit
reinf.
fracture or
debonding
limits
P
reinfplate
reinfplate
P
reinfplate
reinfplate
L
Fig.22 Rigid body displacement of flexural hinge
0
20
40
60
80
0.00 0.01 0.02 0.03 0.04 0.05 0.06
Rotation (radians)
Moment (kNm)
Fracture (M
cap
,
cap
)
Experimental
Theoretical
Fig.23 Comparison with test results
Fig.24 Hinges in continuous beam
L
(EI)
cr
rotation capacity of hinges M
cap

cap
Fig.25 Moment redistribution from –ve region
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
70
Fig.26 Moment redistribution based on neutral axis depth
P
fl
P
fl
h
fl
P
fl
d
P
sh
h
sh
P
sh
P
sh
P
sh
(a) Flexural (b) Shear
Fig.27 Deformations across an inclined crack
h
fl
h
sh
P
fl
+P
sh
P
sh
P
fl
(V
c
)
= f(P
sh
, 1/h)
a
d
V
R
h
total
Fig.28 Superposition of flexural and shear deformations
h
fl
h
sh
(P
fl
+P
sh
)
st
(P
sh
)
long
P
fl
a
d
(P
fl
+P
sh
)
long
(P
sh
)
st
h
total V
R
Fig.29 Interaction between longitudinal and transverse reinforcement
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
71
(P
sh
)
st
(P
fl
+P
sh
)
long
(P
sh
)
long
(P
fl
+P
sh
)
st
(P
sh
)
long
sin
(P
sh
)
long
cos
(P
sh
)
st
cos
(P
sh
)
st
sin
(V
c
)
V
R
(P
fl
+P
sh
)
NSM
(P
sh
)
NSM
(P
fl
+P
sh
)
long
vertical stirrup
inclined NSM plate
longitudinal
reinforcement
Fig.30 Concrete component of the shear
Fig.31 Shear strengthening with inclined NSM plates
V
R
a
1
F
P
fl
P
sh
P
fl
+P
sh
d
(V
c
)
= f(P
sh
,1/h)
h
fl
h
sh
h
total
critical
diagonal
cracks
1
2
3
a
2
a
3
Fig.32 Critical diagonal crack for small shear spans
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
72
P
sh
dominates
h
total
dominates
I
K
L
h
fl
=0.06mm
J
0
G
h
fl
=0.24mm
E
H
F
A
C
D
h
fl
=0.42mm
B
β = 45°
0
10
20
30
40
50
0 0.5 1 1.5
Total Crack Width, h
total
(mm)
Concrete Component, V
c (kN)
Fig.33 Influence of crack width
40°
Q
P
S
R
35°
45°
N
M
T
a/d = 1.0
W
V
a/d = 1.6
Y
0
U
loading lines
X
a/d = 1.3
0
10
20
30
40
50
0 5 10 15 20 25 30 35 40
Applied Moment (kNm)
Shear Load (kN)
Fig.34 Influence of shear span
cylinder longitudinal strain ()
(
lat
)
1
(
lat
)
2
(
lat
)
3
(
start
)
3
(
start
)
2
(
start
)
1
(
soft
)
2
(
soft
)
3
(
soft
)
1
cylinder
longitudinal
stress ()
FRP fracture
hydrostatically
confined
concrete
FRP fracture
limit
shearfriction
sliding limit
fract
)
1
fract
)
2
fract
)
3
shearfriction
sliding
O
A
1
A
2
A
3
C
1
C
2
C
3
D
1
D
2
D
3
E
1
E
2
E
3
Fig.35 Residual strength
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
73
d
prism
d
soft
P
soft
L
soft
L
soft
s
slide
s
soft
a a
b
b
b
b
d
e
e e
e
upper
truncated
cone
lower
truncated
cone
horizontal
truncate
plane
softening
region
Fig. 36 Circumferential wedges
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.1 0.2 0.3 0.4 0.5 0.6
σlat/fc
σsoft test/theory
Fig. 37 Hydrostatically and spirally confined concrete
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.1 0.2 0.3 0.4 0.5
confinement pressure/ unconfined strength
strength at FRP farcture/ residual strength
Fig.38 FRP confined concrete
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
74
h
h
a
s
soft
d
soft
L
soft
original wedge interface
rough interface movement
smooth interface
movement
lat
P
soft
Fig.39 Dilation components in a beam
reinfbar
P
asc
P
soft
P
reinfbar
s
soft
h
a
+h
s
soft
lat
P
wrap
P
wrap
lat
P
stirrup
P
stirrup
(a) wedge restraints (b) wedge deformation (c) wedge confinement
h
a
+h
d
soft
Fig. 40 Confinement of flexural hinges
Fig. 41 Moment rotation analysis
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
75
STATEMENT OF AUTHORSHIP
FRP Reinforced Concrete Beams – A Unified Approach Based On
IC Theory
ASCE Composites for Construction 2010: accepted paper
PROFESSOR DERIC J. OEHLERS
School of Civil, Environmental and Mining Engineering
The University of Adelaide
Supervised research and wrote paper.
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 and supervised 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 redistribution and energy absorption
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
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
76
RAHIMAH MUHAMAD
S
chool of Civil, Environmental and Mining Engineering
The University of Adelaide
Contributed to crack spacings, widths and deflections.
I hereby certify that the statement of contribution is accurate
Signed …………………………………………………… Date …………………
PHILLIP VISINTIN
School of Civil, Environmental and Mining Engineering
The University of Adelaide
Contributed to confinement.
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
77
A
Oehlers, D.J., Mohamed Ali M.S., Haskett, M., Lucas, W., Muhamad, R., & Visintin, P.
(2011) FRP Reinforced Concrete Beams  A Unified Approach Based On IC Theory
ASCE Composites for Construction, v 15(3), pp. 293303
A
NOTE:
This publication is included on pages 77105 in the print copy
of the thesis held in the University of Adelaide Library.
A
It is also available online to authorised users at:
A
http://dx.doi.org/10.1061/(ASCE)CC.19435614.0000173
A
The Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
106
Chapter 2 – The Shear Failure Mechanism
Introduction
This chapter introduces the shear failure mechanism of RC beams that forms the basis of the
subsequent research conducted in this thesis. Importantly the shear failure mechanism
developed allows for the interaction between shear and flexural loading. This is achieved by
implementing a discrete displacement approach to simulate the observed behaviour
associated with both shear and flexure.
As thi
s is a structural mechanics based approach it can be used to derive generic equations for
the shear failure mechanism. This is achieved first for longitudinally reinforced beams and
then for beams with stirrups. From these generic equations the paper identifies two important
relationships that are required to apply this failure mechanism: the relationship between the
force in any reinforcing element and the widening of the sliding plane; and the relationship
linking the stresses confining the sliding plane and the shear stresses that the sliding plane
can transfer.
List of Manuscripts
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 Discrete Rotation Behaviour of Reinforced Concrete Beams under Shear Loading
Wade D. Lucas
107
STATEMENT OF AUTHORSHIP
The Formulation of a Shear Resistance Mechanism for Inclined
Cracks in RC Beams
ASCE Journal of Structural Engineering 2011: accepted paper
WADE D. LUCAS
PhD Candidate
School of Civil, Environmental and Mining Engineering
The University of Adelaide
Prepared manuscript, performed all analyses, and developed model and theory.
I hereby certify that the statement of contribution is accurate
Signed …………………………………………………… Date …………………
PROFESSOR DERIC J. OEHLERS
School of Civil, Environmental and Mining Engineering