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1. Introduction
In the design of reinforced concrete structures, it is common
practice to use simplified models which penalize the stiffness
of structural elements, in order to avoid non-linear material
analysis. A lot of research is dedicated to improve these sim-
plified models. However, it is hard to find research works ad-
dressing the precision or errors of the simplified models. The
objective of the present article is to investigate the precision of
simplified stiffness-reducing models recommended in Brazil-
ian code ABNT NBR6118:2003 [1] in the evaluation of hori-
zontal displacements of plane reinforced concrete frames. The
investigation is based on a comparison, for a set of represen-
tative frames, of the displacements obtained using simplified
models and rigorous physical (material) non-linear analysis.
This article also investigates the reliability, with respect to ser-
viceability limit states for horizontal displacements, of plane
frames representing usual reinforced concrete buildings. Reli-
ability analyses are performed using rigorous non-linear mate-
rial analysis and using the simplified models recommended
in ABNT NBR6118:2003 [1]. Geometrical non-linearities are
treated in a consistent way in all the analyses. Reliability anal-
yses performed herein consider uncertainties in loads and in
the structural strengths, as well as the uncertainties originated
in the use of the simplified stiffness reduction models.
Non-linear structural analyses are performed using a finite
element code developed by the authors (CORELHANO [3]).
Reliability analyses are performed using the StRAnD soft-
ware (BECK [4]).
2. Non-linear analyses in reinforced
concrete
2.1 Non-linear geometrical analysis
A formulation is considered based on second-order Piola Kirchhoff
tensors, developed by WEN & RAMIZADEH [5]. The deformation
tensor and deformation energy are given, respectively, by:
(1)
2
0
0
0
0
1
'
. ''
( ') .
L
x
u Y v
v dx
L
e = - +
ò
(2)
2
2
0
0
0
0
1
1
( '
. )
( ') .
.
2
L
V
U u Y v
v dx E dV
L
é
ù
=
- +
ê
ú
ë
û
ò
ò
where:
x
ε
: longitudinal strains;
0
u
and
0
v
: axial and transversal displacements;
Y
: distance from a given fiber to the sessions gravity center (C.G.);
L
: lenght of the element;
E
: Young´s modulus;
U
: internal strain energy.
Details of the formulation can be found in CORRÊA [6].
2.2 Rigorous material non-linear analysis
In this article, material non-linearities are considered by the meth-
od of layers, which allows independent constitutive models to be
considered for each layer. The element cross-section is divided in
steel and concrete slices, and the sum of the contribution of each
layer defines the behavior of the cross-section (Figure 1).
Properties of the cross-section (stiffness EA and EI
z
) are evaluated
from the sum of the contribution of each layer, at the integration
points at the extremes of each element:
(3)
.
i
i
EA E A
=
å
(4)
.
Z
i Z i
EI
E I
=
å
where:
A
i
: area of the i
th
layer;
E
i
: Young’s modulus of the i
th
layer;
Iz
i
: inertia of the i
th
layer w.r.t. Z axis.
For the compressed concrete, the constitutive model of KENT &
PARK[7] is adopted, following
Figure 2
. Segment AB of this model
is described by:
(5)
2
0
0
2 '
c
c
c
f
e e
s
e
e
é
ù
æ ö
ê
ú
=
- ç ÷
ê
ú
è ø
ë
û
85
IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 1
A. G. B. CORELHANO | M. R. S. CORRÊA | A. T. BECK
Figure 1 – Details of a 2D beam element
formed by layers of steel and concrete