st
1 Civil and Environmental Engineering Student Conference
25-26 June 2012
Imperial College London
Shear Resistance of Axially Loaded Reinforced Concrete Sections
X.Y.WU
ABSTRACT
Unlike flexural failure mechanism which has been extensively researched, the behavior of reinforced
concrete member under combined shear and axial stress remains challenging as there is still strong
disagreement between different code provisions at the influence of axial stress on shear strength. Apart
from the difficulties at conducting the proper experiment which aims only to introduce pure shear, the
analytical theories was not very much developed until more recent works such as the Modified Compression
Field Theory. Through verification of the accuracy of MCFT by using available experiment data, it can be
seen that MCFT can give very satisfied results at predicting the membrane shear response while gives more
conservative results of shear response of beam. The aim of this research project is to use the MCFT as a
key along with other available test data from literature research to further examine the fitness of some major
code provisions at predicting the shear strength under axial loading.
Keywords: axial stress; shear; shear failure mechanism; code provisions
According to ACI the concrete strength is
calculated based on the subjected axial stress,
either compressive or tensile. Unlike BS8110 and
EC2 the concrete strength term with the axial
loading parameter is carried into the shear
reinforcement calculation and therefore the effect
of axial loading on member is involved.
1.
INTRODUCTION
The flexural failure mechanism and the associated
influence of axial stress on the flexural response of
reinforced concrete members have been carefully
studied. However the influence of axial stress on
the shear strength predicted by current major code
provisions remains relatively large disagreement.
Nevertheless it is widely believed that with
increase amount of axial tensile force, the shear
resistance and the ultimate shear capacity of the
reinforced member would be reduced. On the
other hand, when the member is subjected to axial
compressive force, the ultimate shear capacity
would be increased.
The CSA code on the shear design incorporates
the MCFT and therefore it considers the effect of
axial loading on shear strength by calculating the
strain in x-direction which involves the axial
loading parameter N. The major objective in this
research is to investigate how the different codes
accommodate the effect of axial loading in
combination of shear, especially using the MCFT
as a key to carry out verifications of the fitness of
predictions from those major codes in terms if
shear strength under axial loading.
Reviewing on the current major code provisions
for shear reinforcement design, the BS8110 which
is based on fixed inclination angle or 450 truss
model, contains empirical expressions to take the
effect of axial load into account where the design
shear force is smaller than the concrete strength.
BS8110 now has been replaced by EC2 and again
similar expressions which incorporate the axial
loading effect into the concrete strength prediction
can be found. Nevertheless, neither of the codes
has incorporated the influence of axial loading into
the web reinforcement design where the design
shear force exceeds the concrete strength.
2.
SHEAR RESISTING MECHANISMS
It can be argued that the bending moment on the
section can be resisted by beam action and arch
action. The beam action is referred as variation of
tensile force in the longitudinal reinforcement bars
times a constant internal lever arm z. And arch
action or strut-and-tie action refers to the constant
flexural force times the variable lever arm. The
two actions are geometrically incompatible and
therefore beam action has to be broken before
arch action can take place.
Back in 1950s the accident of shear failure of U.S
Air force warehouse beams triggered a major
effort of research led by ACI, the objective was to
develop an ultimate strength design method. The
resulting surge in shear research produced a shear
design proposal which is still the foundation of
current ACI provision.(Collins et al., 2008).
The shear span to effective depth ratio a/d plays a
significant role which decides the balance of arch
action to beam action. When shear span to depth
ratio is smaller than 2.5, it is reckoned that the
arch action dominates the beam strength and
governs the failure mode. Whereas if the a/d is
1
greater than 2.5, then beam action plays a more
vital role over the arch action. The effect of a/d
over the two actions are shown below (Kim, Kim
and White, 1999).
In order to further study the trend of effect of axial
load on shear capacity, one particular membrane
with constant and equal reinforcement ratio of
1.79% and concrete strength of 26.6 MPa is tested
in the program Membrane2000, and the results are
shown in Fig.3 above. As expected the ultimate
shear strength decreases with the increase of
applied axial tension. Notice that both the ultimate
shear failure stress (shown as red line) and shear
cracking stress (shown as blue line) follow almost
a linearly decrease pattern, where the shear stress
at just cracking is well below the ultimate shear
stress of the membrane.
Figure.4 Shear
Compression
Capacity
of
Membrane
Under
Bi-axial
Figure 1. The influence of a/d on Ratio of Ta/Tb
3.
INFLUENCE OF AXIAL FORCE
Membrane Response
First, the program Membrane2000 is used to
implement test data from different researchers
including Collins’ work (Bentz, Vecchio & Collins,
2006).The program Membrane2000 is developed
from the MCFT and is used to study and
membrane behaviour of reinforced concrete
particularly under axial loading. Both bi-axial and
uni-axial loading tests are carried out and the
results are shown below.
Table 2.MCFT Verification via Membrane2000 on Bi-axial Loading
Loadin
Longitudinal
PV
Transverse
Concrete
vpre
g Ratio
Steel
Steel
strength
/ vexp
v:fx:fy
fc’
ρ x fyx
ρ y fxy
MPa
PV3
PV11
MPa
MPa
1:0:0
3.17
3.17
26.6
1.04
3.07
15.6
1.01
0.86
1:0:0
4.20
PV12
1:0:0
8.39
1.21
16
PV23
1:-0.39:0.39
9.27
9.27
20.5
PV25
1:-0.69:0.69
8.34
8.34
19.2
0.89
PV28
1:0.32:0.3
2
8.64
8.64
19
0.98
0.82
Again as expected the ultimate shear capacity
(shown as red line) increases together with the
shear cracking stress (shown as blue line) under
higher axially applied compressive force. However
the gradient is not linear compared to the axial
tension loading case. Notice the turning point
occurs at the loading ratio of v:f x(f y)-1:-1 for
ultimate shear capacity, whereas the shear
cracking stress keeps further increasing until it
intersecting with the failure stress under a higher
axial compression loading. Then the concrete
cracking and concrete crushing is said to occur
simultaneously which governs the failure mode
and ultimate shear capacity, providing sufficient
amount of reinforcements.
Table 5.MCFT Verification via Membrane2000 on Uniaxial Loading
Loadin
Longitudinal
PV
Transverse
Concrete
vpre
g Ratio
Steel
Steel
strength
/ vexp
v:fx
fc’
ρ x fyx
ρ y fxy
As can be seen from the results above, the
predictions of Membrane2000 are said to be
sufficiently accurate. Regarding to this particular
verification, the prediction results tend to
underestimate a little the shear strength under
axial compression.
MPa
PV5
Figure 3. Shear Capacity of Membrane Under Bi-axial Tension
MPa
1:0
4.60
3.17
15.6
1.0
PV11
1:0
4.21
3.07
19.1
0.9
P30
1:0
7.82
1.21
28.3
0.92
8.95
3.12
28.1
1.02
27.7
1.0
PP2
2
MPa
1:-0.38
PP3
1:-0.8
8.48
3.13
TP2
1:3
9.18
4.60
23.1
1.01
KP2
1:3
8.77
3.39
24.3
1.03
KP3
1:3
8.77
0
21
1.13
TP3
1:3
9.18
0
20.8
1.27
PB10
1:5.94
4.72
0
24
To illustrate the stress transfer process, one
particular loading stage (f y=0) which is near the
peak load position is selected to explain the details
of the membrane. The membrane shown here has
equal amount of reinforcement and a reasonable
yielding stress which are allow reinforcements to
yield. It is observed that initially the stress in the
transverse reinforcement is larger than stress in
longitudinal reinforcement and it will reach yielding
condition first which governs the failure mode and
ultimate strength of membrane. As increasing the
axial compression, the stress in y-direction is
transferred gradually to x-direction. This effect can
be observed from Figure.7(d) andFigure.7(e)
which represent the local stress of x- and yreinforcement at crack position. It can be seen that
the local transverse stress is just on the yielding
plateau and the gradient of longitudinal stress is
changed or flatten which indicates the stress is
transferred from transverse reinforcement or
weaker direction, to the longitudinal reinforcement
or stronger direction since it is still far below the
yielding stress. Notice that the local shear on crack
is just about to be developed (Fig.7(f)) which will
help transfer shear stress across the crack. Finally
Fig.7(c) indicates the concrete is about to crush at
the peak load accompanying with the yielding of
transverse reinforcement.
0.92
The membranes above are loaded in uniaxial
stress with variable amount of transverse steel
reinforcement.
The
predictions
from
Membrane2000 can be reckoned as sufficiently
accurate
enough
where
the
maximum
overestimation is about 1.27 and underestimation
about 0.9. If taking a closer look at the prediction
under axial tension, the results are a little
overestimated in this case. However, the overall
accuracy of Membrane is very satisfied.
Figure.6 Uni-axial Compression with Constant Reinforcement
Ratio Case
If further increasing the axial compressive stress,
at some point the longitudinal stress will finally
overtake the transverse stress (absolute value),
however concrete crushing would occur first and
limit the shear strength of membrane in this case.
Again to further study the trend of influence of
axial loading on shear strength capacity, one
particular membrane is selected with constant
longitudinal and transverse reinforcement ratio
while the applied axial compression is increased.
Comparing to the trend of bi-axial loading case,
the turning point of ultimate shear strength is
shifted further up to higher axial compression
loading ratio (in this case about fx/v=-2), rather
than fx/v=1:1. Also the intersection point of
cracking shear stress with the ultimate shear
strength is shifted further to higher load as well.
Further
analyzing
the
trend
by
using
Membrane2000 shows there is a shift of stress
transfer accompanied by failure mode change
happening with the increasing axial compression
force.
Figure.8 Effect of Transverse Reinforcement Ratio on Shear
Capacity
Figure.7 Details of Membrane at One Particular Loading Stage
To study the influence of transverse reinforcement
on the shear capacity, one membrane with 1.79%
longitudinal reinforcement is loaded in bi-axial and
uniaxial forces. As can be seen from Fig.8, the
uniaxial loading follows a more or less linear
pattern in both axial compression and axial tension
case, whereas the bi-axially loaded membrane is
subjected to the limit at v:f x(f y)=1:-1 in
compression and therefore has a earlier turning
point observed. As expected, with higher amount
3
of transverse reinforcement, the shear capacity is
increased.
As can be seen from Fig.10 for beams under
simply supported condition, the predictions from
Response2000 are more conservative or much
underestimated compared to the test data.
Especially in terms of axial compression, the
actual test result increases at a relatively large
gradient than the more flattened gradient of
prediction results. Therefore there exists a
difference at shear strength regarding to the
support condition.
Beam Response
In this research, experiment data from Bara’ test at
Imperial College (Bara, 1971) is used to
investigate the effect of axial loading on shear
strength of beam. In Bara’s test, two groups of
beams are tested under restrained condition and
simply supported condition. The geometry of the
two group beams are similar and the shear span to
effective depth ratio and amount of reinforcement
are varied.
Figure.11 Influence of Web Reinforcement on Shear Capacity
Figure.9 Effect of Axial Force on Ultimate Shear Capacity under
Restrained Support
The actual test data together with the predictions
from Response2000 which is based on MCFT are
shown in Fig.9 for beams under restrained support.
As can be seen the predictions are more
conservative both in axial compression and axial
tension loading cases. The trend from prediction
follows more or less a linear pattern, whereas the
actual test results show a jump at axial
compression loading envelope about v:fx=1:-4
ratio. The exceptional point of test data which is
loaded under axial compression but ended with
much lower shear capacity is explained as the
randomness or fluctuation observed in the test,
potentially due to poor cast of the RC beam.
However, the general trend can still be concluded
as the increasing axial compression has a
beneficial effect on the shear capacity whereas
axial tension does the reverse effect.
In the Bara’s experiment, it was concluded that the
effect of axial stress on the shear capacity was not
dependent on the member reinforcement ratio.
However the MCFT is more sensitive with the
reinforcement ratio parameters. Figure.11 shows
the results of prediction from Reponse2000, for a
beam loaded in constant axial tension and
constant longitudinal tension reinforcement of
1.46%. The shear reinforcement and compression
bars are varied to study the corresponding effect.
As can be seen, the compression bars do not
significantly affect the shear strength of the beam
as the amount of transverse reinforcement does.
This can be explained as for this particular beam
test, the shear failure or yielding of stirrups
governs the failure strength rather than flexural
failure. The ultimate shear capacity increases with
about factor of 2.3 at Rq%=0.6% compared to
Rq%=0.1% based on the predictions which is a
significant increase in shear resistance. However
more test data are required to further conclude the
trend although MCFT tends to give more
conservative predictions.
Figure.10 Effect of Axial Force on Ultimate Shear Capacity under
Simply Supported Condition
Figure.12 Influence of Axial Tension on Shear Capacity with
Constant Reinforcement Ratio
4
Fig.12 shows the trend of influence of axial tension
on shear caoacity with constant reinforcement
ratio. As can be seen the member response
predicted follows a more linear pattern whereas
the sectional response decreases more rapidly
after applying over about 6 MPa axial tensile
stress in this case.
trend. Comparing the three predictions, EC2
intersects with MCFT under axial tension case and
is much conservative comparing ot MCFT under
axial compression case.
Figure 15 Shear Strength Prediction of EC2 under Axial Loading
Compared to Test Data
Figure.13 Influence of Axial Compression on Shear Capacity with
Constant Reinforcement Ratio
The prediction of members shown above have
same reinforcement ratio, for prediction of EC2
with a/d=1.95, the depth of beam is increased
while the shear span ‘a’ is kept as the same as
case of a/d=2.8 members. As can be seen, the
prediction of EC2 tends to give a much more
conservative estimating for members with
a/d=1.95 while overestimated for members with
a/d=2.8 assuming the longitudinal reinforcement is
sufficient enough to prevent flexural failure. The
prediction
from
Response2000
is
also
conservative in both cases of a/d. Also note that
Response2000 does not give very different
predictions in terms of member response, under
this particular situation.
In comparison, shear capacity predicted by
Response2000 under axial compression follows a
more linear pattern in terms of both sectional
response and member response although there is
a small jump of member response occurring at
about 6 MPa axial compression stress.
The difference can be explained as member
response takes the dowel action and arch action
into account which the sectional response does
not. The dowel action takes considerable amount
of shear resistance especially the shear
reinforcement is not sufficiently large. Therefore
member response predicted is generally greater
than the sectional response.
4.
CONCLUSION
After verifications of experiment data, it can be
seen that MCFT can give a very satisfied
prediction at membrane response under axial
loading. In terms of beam response, it tends to
give a conservative prediction and the effect of
support condition is not clearly reflected from the
MCFT comparing to the test data. Regarding to
EC2 prediction which is based on the variable
inclination angle truss model, it tends to give a
much more conservative estimating for members
with short a/d ratio while overestimating for
members with larger a/d ratio under axial loading
cases. However since the limited test data used,
more comparisons are required to be carried out
before making a more solid conclusion.
EC2 Prediction
Figure 14 Influence of Stirrup on Shear Capacity predicted by
Different Codes under Axial Compression
5. ACKNOWLEDGEMENTS
As can be seen,
all
predictions are
underestimating comparing to the actual test data,
however since there is only one test point
therefore more data are required to conclude the
Dr Robert L Vollum
5
6. REFERENCES
Bara, H. C., 1971. Investigation of the Effect of
Axial Loads on The Shear Strength of
Reinforced
Concrete Beams. Ph. D. Imperial College
London
Bentz, E. C., Vecchio, F. J. & Collins, M. P. (2006)
Simplified modified compression field theory for
calculating shear strength of reinforced concrete
elements. ACI Structural Journal. 103 (4), 614624
Collins, M. P., Bentz, E. C., Sherwood, E. G. & Xie,
L. (2008). An adequate theory for the shear
strength of reinforced concrete structures.
Magazine of Concrete Research. 60 (9), 635-50.
Kim, D., Kim, W and White, R.N., 1999. Arch
Action in Reinforced Concrete Beams: a rational
prediction of shear strength. ACI Structural
Journal, Volume.96, No.4, July-August 1999
6