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108
IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 1
A variable limit for the instability parameter of wall-frame or core-frame bracing structures
the structure has a high stiffness. The horizontal deflections are
mostly caused by global shear. Therefore, the rigid frames may be
modeled as vertical bars extremely stiff to global bending, prevail-
ing shear distortions.
Figure 2-b shows a rigid plane frame subject to an uniformly dis-
tributed horizontal load of ratio
w
. It is modeled by a vertical bar
predominantly deformable by shear. As shown in the figure, the
deflected shape of this bar characterizes itself by a maximum
f
(
x
)
slope at basis and tending to zero at top, just the contrary that hap-
pens to the bar simulating the wall or core. This slope is related
with the differences between horizontal displacements at adjacent
floors. In their turn, these differences are proportional to the global
shear
Q(x)
. According to Stamato [9], the deflected shape for this
case is described by the following equation:
(16)
)
(
)(
)(
/
x w xQ x S dx dyS
- =
=
=
l
f
The proportionality factor
S
represents the system (plane frame) stiff-
ness to global shear; it corresponds to the
G A
/
c
factor of a bar
with shear deformation, where
G
,
A
and
c
are, respectively, the shear
modulus, the section area and the section shape factor. Solving equa-
tion (16),
y(x)
is obtained, leading to the top horizontal displacement:
(17)
S w y
H
2/
)(
2
l
l
= = D
The relations established in this section have as their purpose to
obtain the inertia of a bar equivalent to a given rigid plane frame.
The item 15.5.2 of ABNT [8] code, on dealing with the instability
parameter, establishes a methodology to determine the
E
CS
I
C
fac-
tor of a constant section column, equivalent to a given rigid plane
frame. According to this methodology, the above-mentioned stiff-
ness factor should be obtained computing initially the horizontal
displacement on the bracing structure (frame) top, under the hori-
zontal loading, which is just
D
H
given by (17). The next step is to
obtain the stiffness of an equivalent column with constant section
such that, under the same loading, undergoes the same top hori-
zontal displacement which, in this case, is
D
H
given by (15). This
implies in equality between the two expressions for
D
H
, resulting:
(18)
2
/
4
l
JE S
=
2.2 Association of rigid frames with walls and/or cores
This section presents the formulation of the linear response of
frame-wall/core structures, in order to that it have further on to
be used by Galerkin method in obtaining an approximate solution
for the nonlinear behavior of these structures. Figure 3-a shows
the simplified model of a bracing system composed by substruc-
tures of the frame and wall/core types. The model consists in a
wall (representing all system walls and cores) and a rigid frame
(representing all system frames) connected among themselves by
hinges (representing the floor slabs). An uniform distribution of rate
w
is admitted for the wind loads.
EJ
1
represents the stiffness of the
frames set, according to equation (18).
EJ
2
represents the stiffness
of the walls/cores set.
Figures 3-b and 3-c show the loads to that the wall and the frame
will respectively be subject. These loads consist in top concen-
trated forces (
Q
T
for the frame and –
Q
T
for the wall) and distributed
forces that can be decomposed in constant and variable (along
the height) parcels. The constant parcels (
w
1
for the frame and
w
2
for the wall) are such that
w
1
+
w
2
=
w
. The variable parcels (rate
(
u(x)
for the frame and –
u(x)
for the wall), jointly with the forces
Q
T
,
represent internal forces originated from the wall-frame interaction;
since they are connected by the hinges, the wall and the frame
Figure 2 – Bracing substructure equivalent bars
Wall or core
Plane frame
B
A