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110
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
3. The Galerkin method
In many engineering problems, as the ones that are presented
in the next sections, there arises the need to solve an equation of
the type
L(y)
= 0, where
L
is a differential operator, whose solu-
tion satisfies to homogeneous boundary conditions. The Galerkin
method consists in obtaining an approximate solution of the form:
(28)
å
=
=
n
i
x a
xy
1
i i
)(
)(
j
where
)(
i
x
ϕ
(
i
= 1, 2,...,
n
) are functions, previously chosen
and satisfying to the same boundary conditions; the
a
i
are coef-
ficients to be determined. The
n
functions
)(
i
x
ϕ
must be linearly
independent and belong to a system, represented by {
)(
i
x
ϕ
} (
i
= 1, 2,...,
n
) and endowed of the completeness property in the
solution domain. In order to
)(
xy
be the exact solution of the
given equation, it is necessary that
L
(
y
) be identically null. This
requirement, if
L
(
y
) is considered to be continuous, is equiva-
lent to the requirement of the orthogonality of the expression
L
(
y
) to all the functions
)(
i
x
ϕ
(
i
= 1, 2,...,
n
). However, having at
disposal only
n
constants
a
i
, only
n
orthogonality conditions can
be satisfied. Applying these conditions, the following system of
equations is obtained:
(29)
ò å
ò
=
=
× ÷
÷
ø
ö
ç
ç
è
æ
=
×
=
D
n
j
n
i
dxx
x a L
dxx
xyL
) ,...,2,1 (
0 )(
)(
)(
)) ((
i
1
j j
D
i
j j
j
The solution of this system (a linear one, in the case of a linear
operator
L
) provides the values of the coefficients
a
i
, from which
the approximate solution
)(
xy
is obtained. The proof of conver-
gence, as well as more detailed considerations about the Galerkin
method can be seen in Kantorovich and Krylov [10].
4. Exemption of the second order effects
consideration
The sections 4.1 and 4.2 present the formulation of the geometric
nonlinear behavior, respectively of wall/cores and rigid frames assem-
blages. For both cases, the limits
a
1
of the instability parameter are de-
duced, comparing them with the values prescribed by ABNT [8] code.
The section 4.3 does the same for the associations of these types of
substructures, obtaining an expression for the variable limit
a
1
, main
objective of this work.
Figure 4-a shows the deflected shape of a bar equivalent to a bracing
system, subject to uniformly distributed loads of rates
w
and
q
, respec-
tively in the horizontal and vertical directions;
q
is given by the sum of
the rates
p
and
v
of figure 1-b. Taking into account the bar deflections
(geometric nonlinearity) and representing by
Y
the primitive function of
the displacements
y(x)
, the bending moment will be given by:
(30)
x
x
dxy
yq
x w xM
x
)] ( )([
2/ )
(
)(
2
-
+
- =
ò
l
l
or
(31)
[
]
)()
( )( )(
2/)
(
)(
2
xyx
xY Yq
x w xM
- - -
+
- =
l
l
l
Considering that
y
(0) = 0, the bending moment at support will be
expressed by:
(32)
[
]
)0( )(
2
)0(
2
Y Yq
w M
-
+
=
l
l
Figure 4 – Deformations influence in the structure response
Bracing system equivalent bar
Shear deformation at the infinitesimal level
B
A