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IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 5
Punching strength of reinforced concrete flat slabs without shear reinforcement
the region of the slab at the ends of the columns, and was among
the first to check the stress concentration at the corners of square
columns. The author concluded that the stress concentration could
justify the fact that slabs supported on square columns presented
lower resistance than those supported on circular columns, in whi-
ch he observed a uniform distribution of stresses.
In rectangular columns, which are the most commonly used in buil-
dings, the concentration of stresses in the corners may be even
greater. Hawkins et al. [26] varied the ratio between the largest and
the smallest sides of the column (
c
max
/c
min
) from 2.0 to 4.3 and ob-
served that for ratios greater than two the nominal shear strength
decreases with increasing ratios between the column sides. This
research conducted by Hawkins is the basis of the recommenda-
tions of ACI for the consideration of the rectangularity index of co-
lumns (
μ
), which can reduce by more than a half the nominal shear
strength around rectangular columns.
OLIVEIRA et al. [27], analyzing slabs tested by Forssel and Holmberg
supported on a rectangular column with sides of 300 x 25 mm (
c
max
/
c
min
= 12)
observed that the punching resistance can be well estimated
using the recommendations of CEB-FIP MC90 [6], which does not
take into account the relationship
c
max
/
c
min
. OLIVEIRA et al. [27] believe
that this can be explained by the relationship
c
max
/
d
that for this specific
slab is around 2.88·
d
, value that may be considered small compared
to the usual cases. After conducting an experimental program with 16
slabs, Oliveira et al. [27] concluded that the relationship
c
max
/
d
may be
a better parameter than the relationship
c
max
/
c
min
for determining the
punching strength of slabs supported on rectangular columns and
proposed a correction factor
λ
to refine the recommendations for co-
des such as ACI 318 [7 ] and CEB-FIP MC90 [6].
3.4 Size-Effect
It is common to use scale factors in the definition of the dimension of
specimens used for experimental tests of concrete elements. This is
done in order to save material resources but mainly because testing
full-scale structural elements can be a difficulty in most laboratories.
For this reason, many of the tests carried out on flat slabs have
been made on specimens with reduced dimensions. Muttoni [3] sta-
tes that when the current formulation for estimating the punching
resistance of slabs presented by ACI was originally developed in the
1960’s, only tests in relatively small thickness slabs were available
and therefore, the influence of the size effect was not apparent. But
as the punching expressions are also normally used for the verifica-
tion of both thick slabs and footings, testing in experimental models
thicker have been carried out and this effect became evident.
The first ones that observed that the nominal shear strength could
vary in non-proportional way with the thickness of the slabs were
Graf [28] and Richart [29]. At the time these authors have propo-
sed formulas to describe this effect, but they are no longer used.
Subsequently, various expressions have been proposed. Regan
and Braestrup [23] and Broms [30] suggest that the reduction of
the nominal shear strength with increasing thickness of the ele-
ment (size effect) can be estimated by
(1/
d
)
1/3
. CEB-FIP MC90 [6]
and EUROCODE 2 [8] recommend that the size effect should be
estimated by
1+(200/
d
)
1/2
, however, Eurocode limits results of this
expression to the maximum of 2.0. The effect of this limitation is
to reduce the increase in estimates of punching resistance of flat
slabs with effective depth less than 200 mm by limiting the value of
ξ
. It is noteworthy that a solid experimental basis to justify this limi-
tation is not evident and thus a series of tests seeking to evaluate
the recommendation of Eurocode could be of interest.
Some experimental results that can aid understanding of the variation
of the nominal shear strength as a function of effective depth of the
slab come from tests made by Li [31] and Birkle [32]. Li [31] varied
the effective depth of his slabs from 100 mm to 500 mm. In slabs
with effective depth of 100 mm, 150 mm and 200 mm the flexural
reinforcement ratio used was 0.98%, 0.90% and 0.83% respectively.
For slabs with effective depth of 300 mm, 400 mm and 500 mm was
Figure � � �ariation of the nominal shear strength as a function of the effective depth of the slab