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IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 1
R. J. ELLWANGER
nance of frames, when a value lesser than 0,6, possibly close
to 0,5, should be adopted.
These errors, if expressed in relation to
a
1
, are apparently small.
However, is must be remembered that the instability parameter
computation requires a square root extraction. Consequently, on
verifying the exemption of performing a second order analysis, the
error on determining the needed stiffness can become significant.
This work aims to research a way of defining the instability parameter
limit
a
1
for associations of rigid frames and walls/cores, variable with
their stiffness factors proportion. At first, the linear behavior formula-
tion for these associations is presented, followed by an analytical study
about the geometric nonlinear behavior of isolated walls/cores and rigid
frames and then of their association. This study is based on the simpli-
fied model presented in section 1.1, applying the criterion expressed by
inequality (1); the differential equations are solved by Galerkin method.
Right away, the formula found for the variable limit
a
1
is tested in a
series of examples of buildings braced by wall-frame associations. 88
tests are performed, varying the number of floors, of frame spans and
the proportion between the stiffness factors of frames and walls.
2. Linear analysis
2.1 Equivalence between bracing substructures
The substructures of the wall or core types distinguish themselves
by a high stiffness to shear, predominating flexural deflections.
They may be modeled by simple beams, fixed on the building sup-
port, behaving as columns. Figure 2-a shows a wall or core, mod-
eled by a cantilever bar of length
l
, subject to an uniformly distribut-
ed horizontal load of ratio
w
. Representing the material longitudinal
elasticity modulus, the constant section moment of inertia and the
bending moment function respectively by
E
,
J
and
M(x)
, the differ-
ential equation of motion may be expressed as:
(13)
2/)
(
)(
/
/
2
2
2
x w xM dx dJE dx ydJE
- =
=
=
l
f
Bending moments inducing tension on the bar left side are posi-
tive, the deflected concavity becoming turned to right (
f
(x)
is its
slope). Introducing the appropriate boundary conditions,
y(x)
and
the top horizontal displacement
D
H
are obtained:
(14)
ú
ú
û
ù
ê
ê
ë
é
- +÷
ø
ö
ç
è
æ -
=
1 4
1
24
)(
4
4
l
l
x
x
EJ
w xy
l
(15)
EJ
w y
H
8/
)(
4
l
l
= = D
In substructures of the rigid frame type, the deflections due to bend-
ing of the individual beam and column members are predominant.
When the frame is subject to horizontal loads, the global bending
moment is mainly carried to the columns as axial efforts, for which
a function of
E
Ci
I
C
, or of
E
CS
I
C
if equation (7) is used. Representing
by
A
s
and
A
s
’, respectively, the tensile and compressive longitudi-
nal reinforcements areas, the following expressions can be written:
- slabs:
(8)
CCS
CCi
IE
IE
EI
353 ,0
3,0 ) (
sec
=
=
- beams:
(9)
'
sec
471 ,0
4,0 ) (
s
s
CCS
CCi
A A
IE
IE
EI
¹ '
=
=
(10)
'
sec
588 ,0
5,0 ) (
s
s
CCS
CCi
A A
IE
IE
EI
= '
=
=
- columns:
(11)
C CS
CCi
IE
IE
EI
941 ,0
8,0 ) (
sec
=
=
Furthermore, when the bracing substructure is exclusively con-
stituted by beams and columns (rigid frame) and the “importance
factor” of the second order global efforts (
g
z
) is lesser than 1,3 (cor-
responding to a “bland” nonlinearity) it is allowed to consider the
stiffness of the rigid frame members as a whole, as follows:
(12)
C CS
CCi
IE
IE
EI
824 ,0
7,0 ) (
sec
=
=
1.3 Reasons and targets of the research
The ABNT [8] code represented an improvement in relation to
the preceding one, on establishing procedures for checking if
second order global effects are unnecessary to consider. Con-
cerning to the instability parameter for buildings with four or
more floors, it treated differently the various types of bracing
systems, on determining different values for the
a
1
limit. How-
ever, the prescription of a fixed limit (
a
1
= 0,6) for associations
of walls and/or cores with rigid frames is questionable. As the
relation between the stiffness factors of walls/cores and frames
can vary,
a
1
also can vary from 0,5 to 0,7. This can lead to two
types of errors:
n
on behalf of safety: in associations with predominance of walls/
cores, the code restricts
a
1
to 0,6, when a larger value, possibly
close to 0,7, could be adopted;
n
contrary to safety: in the case of associations with predomi-