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1. Introduction
The manufacturing components and industrial conditions involved in
producing precast concrete result in numerous advantages, including
facilities for quality control, low wastage of materials, and use of high-
strength concrete. However, when compared to concrete structures
cast on site, the connections between precast components compli-
cates evaluation of structural behavior(s), a situation of particular im-
portance when it comes to global stability analyses of structures with
multiple floors.
In general, the Physical Non-Linearity (PNL) and Geometric Non-
Linearity (GNL) are taken into account when analyzing the global
stability of concrete structures. One of the predominant ways in
which PNL is considered in concrete structures is through reduc-
ing stiffness in structural elements, as recommended by NBR
6118:2003 [1]. If the beam-to-column connections are perfectly
rigid, which is usually the case in concrete structures cast on site,
the indications from NBR-6118:2003[1] can be applied. However,
the indications from NBR 6118:2003 do not apply when the beam-
to-column connections are not perfectly rigid. Few studies in the
technical literature regarding precast concrete structures consider
conditions of semi-rigid connections. For the case of a pinned
beam-to-column connection, Hogeslag [2] recommends a stiffness
reduction coefficient (in columns) of 1/3.
Until the last decade of the twentieth century, connections in precast
concrete structures were either pinned or rigid. A program in the Euro-
pean Community called COST C1 (“Control of the Semi-Rigid Behav-
ior of Civil Engineering Structural Connections”) developed between
1991 and 1998 increased research in this regard for precast concrete
structures [3-8]. Since then, the possibility of developing relatively
simple beam-to-column connections that have semi-rigid behavior
has been investigated in relation to evaluations of global stability for
precast concrete structures with multiple floors.
The present study evaluates the reduction of stiffness in structural ele-
ments for typical situations involving precast concrete buildings with
multiple floors. The study is focused on the case of plane frames with
semi-rigid beam-to-column connections and columns embedded in
the foundation. The structural arrangements analyzed correspond to
the usual modulation(s) and load(s) in multi-storey precast concrete
framed structures. The study is performed using the M x N x 1/r dia-
grams for columns and beams, based on NBR 6118:2003 [1]. The
GNL is considered in a non-approximated way according to the finite
element model, which was constructed using ANSYS
®
software [9].
The values for the stiffness reduction coefficients obtained are com-
pared with the values recommended by standards and codes found
in the technical literature.
2. Values for reduction coefficients
in the technical literature
Taking PNL into account, the reduction of stiffness can be defined
as follows:
(1)
c ci
IE EI
a=
sec
where
α
represents the stiffness reduction coefficient;
EI
sec
represents the secant stiffness.
In this section, five recommendations from different codes, stan-
dards, and association committees are presented. It should be not-
ed that only some of these recommendations are directed towards
precast concrete structures. In principle, the first three pertain to
concrete structures cast on site, while the last two pertain to pre-
cast concrete.
a) Current Brazilian standard for structural concrete – NBR
6118:2003 [1]
The NBR 6118:2003 [1] allows for an approximate consideration of
physical non-linearity to analyze global stability in reticulated struc-
tures with at least four floors. For this, the secant stiffness (EI)
sec
, is
defined for each element as follows:
(2)
c ci
IE
EI
3.0
sec
=
Slabs:
(3)
c ci
IE
EI
4.0
sec
=
for A’
s
≠
A
s
Beams:
(4)
c ci
IE
EI
5.0
sec
=
for A’
s
=
A
s
(5)
c ci
IE
EI
8.0
sec
=
Columns:
where
I
c
is the moment of inertia of concrete, including, when appropriate,
collaborating flange;
E
ci
is the initial tangent modulus of elasticity;
A
s
is the cross-sectional area of tension longitudinal reinforcement;
A’
s
is the cross-sectional area of compression longitudinal rein-
forcement.
When the bracing frame is composed exclusively of beams and
columns and
γ
z
is less than 1.3, the beams and columns stiffness
can be calculated by the following equation:
(6)
EI
sec
= 0.7E
ci
I
The coefficient
γ
z
is determined from the results of a first-order lin-
ear analysis, where it is taken into account the structure’s tipping
moment and the moment caused by displacement of vertical loads.
M. C. MARIN | M. K. EL DEBS
317
IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 3