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
![$ (M-\mu)$](cc03/img103.gif)
relationship of
RC beams in bending. The
![$ M-\mu$](cc03/img104.gif)
diagrams obtained, also applied to elements in bending
and compression, seem to be close to the curves obtained by testing RC beams
with and without stirrups [
2]. Moreover, the stress
![$ \sigma_c$](cc03/img105.gif)
of compressed concrete is a
function of the strain
![$ \varepsilon_c$](cc03/img106.gif)
and of the extension
![$ y_{c,max}$](cc03/img107.gif)
of the compressed zone, according
to the size effect theory proposed by Hillerborg [
3]. However, in the case of
beams in bending, the stress
![$ \sigma_c$](cc03/img105.gif)
obtained with the proposed model seems to depend
also on the cross-sectional curvature
![$ \mu$](cc03/img108.gif)
.
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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