Measurement of Moment Redistribution Effects in
Reinforced Concrete Beams
R.H. Scott
Reader in Engineering
School of Engineering
Durham University
South Road
Durham DH1 3LE, UK
[email protected]
R.T. Whittle
Consultant
Arup Research and
Development
13 Fitzroy Street
London W1T 4BQ, UK
[email protected]
A.R. Azizi
Procurement Adviser
Crown Agents
Banda, Aceh
Indonesia
[email protected]
ABSTRACT
Moment redistribution in reinforced concrete beams has traditionally been considered as an Ultimate Limit State (ULS)
phenomenon closely associated with considerations of reinforcement ductility. The work described here demonstrates that a
significant proportion of this redistribution will almost always occur at the Serviceability Limit State (SLS) because of the
mismatch between the flexural stiffnesses assumed when calculating moments for the ULS and those actually occurring at the
SLS due to variations in the reinforcement layout along the member and the influence of cracking. Tests on 37 two span
reinforced concrete beams are presented, parameters investigated being values of redistribution, beam depth, reinforcement
arrangements, concrete strength and the effect of brittle reinforcement. The results confirm that there is significant moment
redistribution at the SLS and also suggest that it should be possible to increase the permissible limits for redistribution beyond
those currently prescribed in design codes.
INTRODUCTION
Normal practice is to assume linear elastic behaviour when calculating the bending moment and shear force distributions in a
reinforced concrete structure. This assumption is reasonable at low levels of loading but it becomes increasingly invalid at
higher loads due to cracking and the development of plastic deformations. Design codes permit an elastic analysis to be used
at the Ultimate Limit State (ULS) but acknowledge the non-linear behaviour by allowing design engineers to redistribute
moments from one part of the structure to another subject to maintaining the rules of static equilibrium. Implicit in the current
use of moment redistribution is the assumption that sections possess sufficient ductility for the requisite plastic deformations to
occur. This view was challenged in 1987 by the work of Eligehausen and Langer [1] who investigated the consequences of a
finite limit to reinforcement ductility on the failure mode of a section. They demonstrated that reinforcement strain at the ULS
could be the controlling parameter in the more lightly reinforced types of members, such as slabs, and this prompted
considerable discussion concerning the ductility requirements for steel reinforcement.
The authors conducted an investigation which aimed to explore the nature of moment redistribution as load was increased on
a structure and thus provide some design guidance on the issues outlined above. To do this a series of two-span reinforced
concrete beams, some containing strain gauged reinforcement, was tested to investigate moment transfer between the central
support and the adjacent spans.
In the discussion which follows, the percentage moment redistribution quoted at any particular section along a beam was
calculated as follows:-
% Redistribu tion =
(Moment after redistribu tion - Moment before redistribu tion )
Moment before redistribu tion
x 100
The distribution of bending moments in a structure is heavily dependent on assumptions concerning flexural stiffness (EI). For
practical design purposes, EI is almost always based on the concrete section (the entire concrete cross section, ignoring the
reinforcement) and the resulting bending moment distribution then provides the baseline for the redistribution procedure. This
approach was adopted when calculating Moment before redistribution in the above expression. The Moment after
redistribution was determined from measurements of the applied loads and support reactions. For convenience, a reduction in
the centre support moment (i.e. redistribution from the support into the adjacent spans) was considered as being positive
whilst an increase in the centre support moment (i.e. redistribution from the spans back to the centre support) was considered
as being negative.
SPECIMEN DETAILS AND TEST PROCEDURE
A total of 37 two span beams, 5.2 m long overall and 300 mm wide, were tested to investigate moment redistribution effects
between the centre support and adjacent spans. Each span was 2435m long and loaded approximately at its mid-point. The
effects of the following parameters were investigated:•
•
•
•
•
•
Depth of section (150, 250 and 400 mm used)
Different values of design moment redistribution (see below).
Different percentages of tension steel at the centre support (and consequently in the spans).
Different arrangements of reinforcement (e.g. large bars versus small bars).
Different concrete strengths (i.e. use of high strength concrete for some specimens).
The effect of brittle reinforcement (i.e. glass fibre reinforced polymer (GFRP) bars).
Twenty seven specimens were designed for +30% redistribution from the centre support into the adjacent spans (i.e. support
moment decreased by 30%), one specimen for +55% redistribution from the centre support into the adjacent spans (i.e.
support moment decreased by 55%), five specimens for 0% redistribution, three specimens for -30% redistribution from the
spans back to the centre support (i.e. support moment increased by 30%) and one specimen for -60% redistribution from the
spans back to the centre support (i.e. support moment increased by 60%).
The normal strength concrete (NSC) had an average compressive cube strength of 51.8MPa (standard deviation 7.9MPa) and
an average indirect cylinder strength of 3.0MPa (standard deviation 0.2MPa). The high strength concrete (HSC) had an
average compressive cube strength of 120.4MPa (standard deviation 10.0MPa) and an average indirect cylinder strength of
4.1MPa (standard deviation 0.5MPa). Typical stress-strain relationships for all the reinforcement types for strains up to 14 000
microstrain are shown in Figure 1.
800
700
STRESS, MPa
600
500
400
T8 Bar
T10 & T12 Bars
T16 Bar
T20 Bar
GFRP
300
200
E Steel = 200GPa
E GFRP = 50 GPa
100
0
0
2000
4000
6000
8000
10000
12000
STRAIN, microstrain
Fig 1 Reinforcement Stress-Strain Curves.
Some of the reinforcement in fifteen of the specimens was strain gauged in order to obtain detailed measurements of the
reinforcement strain distributions. Detailed measurement of reinforcement strain distributions is complicated by the criterion
that the selected measurement technique must not disrupt the bond behaviour around the perimeter of the bars. This
precludes mounting electric resistance strain gauges on the surface of the bars as the gauges and their associated wiring are
a source of considerable disturbance within the concrete. Consequently, the technique of installing strain gauges in a
machined duct running longitudinally through the centre of the reinforcement was adopted for this investigation. This was
pioneered at Cornell University over 40 years ago, but has since been considerably developed and improved by the author
and his colleagues [2].
Each strain gauged bar is made by milling two solid bars to a half round and then sawing a longitudinal groove down the
centre of each machined half to accommodate the strain gauges and their wiring. After installation of the gauges, the two
halves are glued together to give the appearance of a normal reinforcing bar, but with the lead wires from the gauges coming
out at the ends. A 4x4 mm duct is used in bars of 12 mm diameter and above but this is decreased when smaller bars are
gauged so that the reduction in bar cross-section does not become unacceptably large, the smallest diameter gauged to date
being 6 mm with a duct size of 2.5x2.5 mm. Up to 100 gauges (gauge length 3 mm) can be installed in the 4x4 mm duct
meaning that, because of the three wire system used for the strain gauge wiring, 150 wires emerge from each end of the bar.
The upper limit for the 2.5x2.5 mm duct is 30 gauges (gauge length 2 mm). In this investigation, the gauged bars contained up
to 51 gauges.
The test procedure was to load the beams incrementally until failure occurred. Applied loads, support reactions and
reinforcement strain gauge readings (when appropriate) were recorded at every load stage. Surface strains were measured
on one side face of all specimens, at selected load stages, using a Demec gauge and a grillage of Demec points.
MOMENT REDISTRIBUTION BEHAVIOUR
Space limitations preclude a full presentation of the test results so the following discussion will concentrate on highlighting the
main findings.
Specimens Designed for +30% Moment Redistribution from the Centre Support into the Adjacent Spans
Twenty six specimens were designed for +30% redistribution, one 400 mm deep, nine 250 mm deep and sixteen 150 mm
deep, using a range of tension steel percentages, bar diameters and concrete strengths. Both flexural and shear failures
occurred.
All specimens which exhibited flexural failures, characterised by an inability to maintain applied loads due to the excessive
deflections which occurred as a result of the mechanism action, achieved, or came very close to, the designed +30%
redistribution at the ULS. This is illustrated in Figure 2 which shows the relationship between percentage redistribution at the
centre support and experimental bending moment at the centre support over the full load history of five geometrically similar
specimens. These were all 150 mm deep and reinforced with two T12 bars (0.63%) over the centre support with two being
made with HSC, as indicated in the Figure.
50
% REDISTRIBUTION
40
30
NSC
NSC
NSC
HSC
HSC
20
10
0
2
4
6
8
10
12
14
16
18
EXPERIMENTAL BENDING MOMENT, kNm
Fig 2 Moment Redistribution History for 150 mm Deep Beams
Data from the strain gauged reinforcement indicated that up to around 8 kNm (see Figure 2) strains in all the reinforcing bars
remained within the elastic range. This important observation showed that between 15 and 35% redistribution was developed
at the SLS i.e. before onset of yield occurred either at the centre support or in the adjacent spans. This elastic redistribution
was observed in all the specimens tested including, as will be shown below, those which failed in shear.
Figure 3, which shows the redistribution history for five specimens all 250 mm deep, reinforces the above point. These all had
nominally 0.9% tension steel over the centre support but, as Figure 3 indicates, two T20 bars were used in three specimens,
five T12 bars in one specimen whilst the fifth specimen used five 13 mm diameter GFRP bars (1%) for the tension
reinforcement over the centre support as well as GFRP bars in the spans. Also indicated is that two specimens were made
with HSC. Only the two HSC specimens failed in flexure. The three NSC specimens, which included the one using GFRP, all
failed in shear. Again, significant levels of elastic redistribution (10-30%) were developed at the SLS even when using the
brittle GFRP bars. Test results throughout the experimental programme indicated that concrete strength was not a significant
parameter so far as elastic behaviour was concerned, although specimens made with HSC achieved a higher failure moment
due to their greater moment of resistance.
40
% REDISTRIBUTION
30
20
2T20 : NSC
2T20 : HSC
2T20B : HSC
5T12B : NSC
GFRP : NSC
10
0
-10
0
10
20
30
40
50
60
70
80
EXPERIMENTAL BENDING MOMENT, kNm
Fig 3 Moment Redistribution History for 250 mm Deep Beams
None of the reinforcing bars, which were all of UK origin, failed in spite of the very high strain levels reached in some
instances. Thus reinforcement ductility was not an issue in this test programme.
Overall, the results indicated that total redistribution has two components. These are elastic redistribution, which occurs at the
SLS when the reinforcement is behaving elastically, and plastic redistribution, which occurs when the reinforcement yields.
Elastic redistribution is caused by crack development in the member and leads to a distribution of flexural stiffness (EI) along
the member which is inconsistent with the constant EI assumed in the concrete section approach. Total redistribution is not all
plastic, as is commonly assumed at present, and elastic redistribution can nearly always be expected to occur even though
none may have been anticipated in the design calculations. Even specimens designed for zero redistribution will develop
elastic redistribution at the SLS with plastic redistribution then being necessary to achieve the designed bending moment
redistribution at the ULS. The mismatch between the distribution of EI at the SLS and that at the ULS is fundamental to the
understanding of moment redistribution behaviour.
Specimens Designed for Moment Redistribution from the Spans to the Centre Support
Four specimens were tested as a preliminary investigation of moment redistribution from the spans to the centre support.
Three were for -30%, two of which were 150 mm deep with four T10 (0.81%) and four T12 (1.16%) bars over the centre
support respectively with the third being 250 mm deep with four T20 (1.90%) bars over the centre support. The fourth beam
was also 250 mm deep but had four T20 bars over the centre support to give -60% redistribution. The resulting behaviour is
shown in Figure 4.
It proved much more difficult to achieve the designed level of redistribution with these specimens than was the case with +30%
specimens. The contribution of elastic redistribution observed in the tests was much reduced and a flexural failure was
necessary if the specimens were to reach anywhere near their designed redistribution levels. Both the 250 mm deep
specimens failed in shear at low redistribution levels and the redistribution behaviour of even the two 150 mm specimens was
unconvincing even though full mechanisms were developed. This has implications for the design recommendations to be
considered shortly.
The practicability of a higher limit for permissible redistribution was investigated by testing a beam designed for +55%
redistribution. It was 150 mm deep with two T10 bars (0.44%) as tension steel over the centre support. The results, shown in
Figure 5, indicate that this specimen actually achieved +50% redistribution at the ULS plus the quite remarkable value of +35%
redistribution at the SLS.
20
4T20B : NSC
5T20 (-60%) : NSC
4t12 : NSC
4T10 : NSC
% REDISTRIBUTION
10
0
-10
-20
-30
0
10
20
30
40
50
60
70
EXPERIMENTAL BENDING MOMENT, kNm
Fig 4 Moment Redistribution History for Beams with -30 and -60% Redistribution
% REDISTRIBUTION
60
40
20
0
-20
0
2
4
6
8
10
12
14
EXPERIMENTAL BENDING MOMENT, kNm
Fig 5 Moment redistribution history for beam with 55% redistribution
IMPLICATIONS FOR DESIGN
The test results have shown that total moment redistribution has both elastic and plastic components with the former making a
significant contribution to the whole. Elastic redistribution is a SLS phenomenon which occurs due to the mismatch between
the assumption, for purposes of analysis, of a constant EI (flexural stiffness) value along the member and the actual EI values
which are a consequence of the designed reinforcement layout.
Moment redistribution is permitted from supports into adjacent spans and from spans back to the supports and the load history
associated with each type of redistribution may be summarised as follows:Support to Span
•
Before any cracking, as the load is increased more moment will be attracted to the support. The first flexural crack will
form over the support, reducing the stiffness of the beam over the support. This will lead to immediate moment
redistribution into the span since the span will now have become stiffer than the support.
•
Cracks will develop in the span. The span will still be stiffer than the support but to a reduced degree. There will not be a
great reduction in span stiffness until the number and width of cracks has increased. There may be a slight reduction in
the amount of redistribution when the first span crack forms but as the load is increased a fairly stable situation is quickly
established while the crack pattern is being fully developed. This redistribution is all elastic.
•
The reinforcement over the support will yield at a load less than that to cause the span reinforcement to yield. This
causes plastic redistribution into the span. As the load is increased further the span reinforcement will yield, a mechanism
will develop and the bending moment distribution in the beam will then correspond to that anticipated in the design
calculations.
•
Over the entire load history the redistribution is from the support into the span. Thus the redistribution is always in the
direction which was assumed in the design calculations. The contribution of elastic redistribution will be significant and
the failure will be highly ductile.
Span to Support
•
Initially, behaviour is the same as that for support to span redistribution with the first flexural crack forming over the
support, the consequent reduction in stiffness leading to immediate moment redistribution into the span. This initial
redistribution is thus in the opposite direction to that assumed in the design calculations. The term reverse redistribution
may be coined to describe this situation where, due to cracking, moment redistribution at the SLS is from the supports into
the spans but, at the ULS moment redistribution back to the supports is required to achieve the designed bending moment
diagram at failure. It most obviously occurs in beams designed from the outset for redistribution back to the supports but
it can also occur in less obvious situations, such as beams designed for zero redistribution.
•
Moment can only be transferred back to the support if the span stiffness becomes significantly smaller than that at the
support. Also, a significant amount of rotation must occur in the span. This process is started when the span cracks but,
if the beam is loaded with a udl, the span bending moment distribution will be relatively flat in contrast to the well defined
peak at the support. Thus, before significant redistribution can occur, several cracks in the span must form, each extra
crack resulting from a further increase in load and there will be a significant increase in load before the number and size of
the cracks have increased sufficiently. Consequently, span cracking will not have such an immediate effect on
redistribution to the support as the localised support cracking described above had on the span. The design redistribution
is likely to take place at higher level of load and cause a more sudden collapse mechanism.
•
Initially, span rotation will be achieved by an increase in the number and width of cracks in the span as load is increased
giving elastic redistribution back to the support. In due course plastic redistribution will occur when the reinforcement
yields and, ultimately, a mechanism will form leading to the bending moment distribution anticipated at the design stage
being achieved. However, whilst the hinge at the support will be well defined, yield in the span will be spread over a much
greater length than that at the support. Thus the hinge in the span will be much less well defined than that at the support.
•
With span to support redistribution the direction of the redistribution has to reverse during the load history of the beam.
Mobilising rotation in the span is much more crucial than is the case with support to span redistribution but this is hindered
by the way in which the span cracks. The load at which hinging action in the span is fully developed will be much closer
to the failure load than is the case with support to span redistribution. Also, the hinge at the support will form after that in
the span. Less warning of failure can thus be expected when moments are redistributed back to the support than when
they are distributed into the span i.e. the failure will be more brittle.
•
Because of the reverse elastic redistribution the final required plastic redistribution from the span is greater than that
considered in the design. This could exceed the neutral depth limit for a ductile failure permitted by the design codes [3,4]
and the concrete would then start to crush in the span before the support reinforcement yielded, making the failure more
brittle.
Thus, to summarise further:•
Design redistribution from support to span: plastic contribution is less than the designed total redistribution. Provides a
ductile failure mechanism.
•
Design redistribution from span to support: plastic contribution is more than the designed total redistribution. May lead to
a brittle failure mechanism.
The work described in this paper indicates that there is no case for reducing the maximum permitted +30% limit to moment
redistribution from supports to the adjacent spans currently allowed in BS8110 [3] and EC2 [4]. Also, the limit of +20% given
in ACI 318 [5] would appear to be over-conservative. It seems reasonable that the upper limit can be increased beyond +30%
by taking into consideration the contribution of elastic redistribution and, with total redistribution being the sum of the elastic
and plastic components, it is suggested that the +30% limit (together with the current rules limiting neutral axis depth) now be
applied just to the plastic component. The absolute upper limit to total redistribution will depend on beam geometry and
loading arrangement but could be around +55%. However, crack width considerations and avoidance of reinforcement yield at
the SLS are likely to become limiting criteria. Nevertheless, there will frequently be scope for design engineers safely to
exceed the current +30% limit if they so wish.
Currently, there is no limit in the codes for redistribution from spans to the supports. The, admittedly limited, results from this
test programme suggest that this may be unwise. The results indicate that this type of redistribution does actually occur but
not necessarily to the extent assumed in the design calculations, except under very ideal circumstances. More research is
probably necessary but at this stage it seems appropriate to advise caution in this area.
CONCLUSIONS
1.
2.
3.
4.
5.
6.
An investigation of moment redistribution effects was undertaken comprising laboratory tests on 37 two span beams
supported by programmes of numerical modelling.
Total moment redistribution has two components. These are elastic redistribution, which occurs because of the nonuniformity of EI and plastic redistribution which occurs after yield of the tension steel. Elastic redistribution is caused
by a mismatch between the uniform flexural stiffness assumed and the stiffness values which actually occur due to
variations in the reinforcement layout along the member and the influence of cracking. The contribution of elastic
redistribution is significant even in members which fail in shear.
A consequence of elastic redistribution is that beams designed for zero redistribution will, in fact, undergo plastic
redistribution before the ULS is reached.
No failure of the reinforcing bars occurred in any of the tests which indicates that there was always sufficient ductility
in the UK reinforcement.
There was no evidence that the limits given in BS8110, EC2 and ACI 318 for support to span redistribution should be
reduced. The limit given in BS 8110 relating to the neutral axis depth is only related to plastic redistribution and 30%
total redistribution should always be possible. However, there is now a strong case for increasing this value although
it is not now possible to specify a unique upper limit since the contribution of elastic redistribution is dependent on
support conditions and loading arrangement. It should also be noted that, when current limits are exceeded, crack
widths and the possibility of yield at the SLS should be checked.
Designers should exercise caution in using span to support redistribution as the test results suggested that designed
levels of support to span redistribution are very difficult to achieve in practice.
ACKNOWLEDGEMENTS
Funding for this project was provided by the Engineering and Physical Sciences Research Council with additional support from
Arup Research and Development. The strain gauged beams were tested by Dr A.R. Azizi. In addition, the assistance of the
following is also gratefully acknowledged: Tarmac Topmix Ltd, Professor A.W. Beeby FREng, the technical staff of the
University of Durham and several generations of project students.
REFERENCES
1.
Eligehausen, R. and Langer P, Rotation capacity of plastic hinges and allowable degree of moment redistribution,
Stuttgart, 1987
2.
Scott, RH, Gilbert SG and Moss RM, Load Testing of Floor Slabs in a Full-Scale Reinforced Concrete Building,
Strain, Vol 38, No. 2, May 2002, pp 47-54
3.
BS8110: Structural Use of Concrete Part 1: 1997: Code of Practice for Design and Construction, British Standards
Institution, London.
4.
BS EN 1992-1-1 Eurocode 2: Design of Concrete Structures. Part 1-1: 2004 : General Rules and Rules for Buildings,
British Standards Institution, London.
5.
ACI 318: Building Code Requirements for Structural Concrete, American Concrete Institute, 2005