Keywords: reinforced concrete, interaction curves, size effect, limit states, concrete in compression, ductility, stress-strain relationship.
In order to evaluate the progressive damage produced by crushing of compressed
concrete, a numerical model has been proposed in [
1]. With this model, it was possible
to reproduce the post-peak branch in the moment curvature

relationship of
RC beams in bending. The

diagrams obtained, also applied to elements in bending
and compression, seem to be close to the curves obtained by testing RC beams
with and without stirrups [
2]. Moreover, the stress

of compressed concrete is a
function of the strain

and of the extension

of the compressed zone, according
to the size effect theory proposed by Hillerborg [
3]. However, in the case of
beams in bending, the stress

obtained with the proposed model seems to depend
also on the cross-sectional curvature

.
All this aspects cannot be computed by means of Eurocode 2 [4], where the size
effect on the structural response of compressed concrete is not taken into account. In
the Eurocode 2 the concrete in compression is modelled by the parabola-rectangle
diagram. Thus in a generic cross-section the maximum moment is reached only
when one of the materials attains its limit strain.
The aim of this paper is to highlight how Eurocode 2 is not able to reproduce the
structural effects produced by the post peak branch of
. The influence of the
mechanical behaviour of compressed concrete on the interaction curves of RC cross-sections
is investigated by adopting different approaches. For a given value of the
normal force
, the maximum bending moments obtained by the proposed model are
compared to the ones of Eurocode 2 [4]. Both the approaches are applied to similar
cross-sections, whose dimensions are obtained by scaling the size factor
. In each
case, there is a deterministic evaluation of the maximum bending moment, so the partial
factors for materials are not applied. The non-dimensional bending moment
,
normal force
and mechanical reinforcement ratio
are introduced in order to compare
the numerical results. Varying the dimensions of the cross-section, the proposed
model provides different stress distributions in the compressed concrete and, consequently,
more than one interaction
curves. These curves, obtained for symmetrically
reinforced cross-sections and for asymmetrically reinforced ones (no
reinforcement in the compressed zone), with size factors
and
, are compared
with the
curves computed by assuming the parabola rectangle stress-strain
diagram for concrete in compression [4]. For a given value of
, a reduction of the
bending moment
with the increase of
is obtained. This phenomenon is particularly
evident for
, in the section symmetrically reinforced and in the case of
high mechanical reinforcement ratio.
From a practical point of view, it is also interesting to evaluate the rotation
of a
beam portion having a length equal to the height
of its cross-section. If the value of
the curvature at maximum bending moment is considered in the evaluation of
, the
effect of the inelastic behaviour of the beam are also included in the rotation. The
curves obtained in case of asymmetrically reinforced concrete sections
and
for different size factors
, show an increase of
when
decreases. Similarly, for a
given value of
, there is an increase of
with the decrease of
. When
and
, the rotation
can be
times lower than
. This difference considerably
decreases with the increase of
and
, and is particularly evident in beams with
lower
and
. This scaling behaviour cannot be reproduced with the classical
approach proposed in the Eurocode 2 [4].
In conclusion, with the proposed model for compressed concrete, the softening
branch of the stress strain relationship, and its effects on moment curvature diagrams,
can be defined. The post-peak branch of the
diagram clearly shows a size effect,
which remarkably affects the cross-sectional strength of a RC beam and its corresponding
rotation
. The decrease of
, observed when
increases, can be explained
as a reduced capability of larger structures to bear plastic deformations. Since size-effect
is not currently considered into Eurocode 2, it is desirable that future code
requirements will be take into account these phenomena.
- 1
- A.P. Fantilli, D. Ferretti, I. Iori, P. Vallini, "A Mechanical Model for the Fail-ure of Compressed Concrete in R/C Beams", Journal of Structural Engineering, ASCE, 128(5), 637-645, 2002. doi:10.1061/(ASCE)0733-9445(2002)128:5(637)
- 2
- A.P. Fantilli, I. Iori, P. Vallini, "A Mechanical Model for the Confined Com-pressed Concrete of RC Beams", 1st fib Congress: Concrete Structures in the 21st century, October 13-19 2002, Osaka, Japan.
- 3
- A. Hillerborg, "Fracture mechanics concepts applied to moment capacity and rotational capacity of reinforced concrete beams", Engineering Fracture Mechanics, 35(1/2/3), 233-240, 1990. doi:10.1016/0013-7944(90)90201-Q
- 4
- European Committee for Standardization, "Eurocode 2: Design of concrete structures- Part 1: general rules and rules for buildings", Brussels, 2001.
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