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114
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
Substituting (68) into (70) and re-arranging, results:
(71)
)(
)()
()
( )
()
(
2 1
2
1
2
2
2
x S x x
q q x
ww
dx
d EJ
f
f
f
+
- + - - + -=
l
l
Considering that
w
1
+
w
2
=
w
(total wind load),
q
1
+
q
2
=
q
(total
gravity load) and re-arranging again, leads to the differential equa-
tion describing the behavior of a system composed by rigid frames
and walls/cores, including the deflections influence:
(72)
0 )
(
)( )]
(
[
2
2
2
= - +
- - -
x w x x q S
dx
d EJ
l
l
f
f
In order to apply the Galerkin method to equation (72), it will be
assumed a solution given by a function
f(x)
multiplied by the linear
solution, expressed by equations (25), (26) and (27):
(73)
ú
û
ù
ê
ë
é
-
+
+
=
-
)
(
4
)(
)(
1
2
/ 2
2
/ 2
1
x
EJ
w
eC eCxf
x
Kx
Kx
l
l
l
l
f
Substituting (73) into (72), results:
(74)
+ú
û
ù
ê
ë
é
-
+
+
-
)
(
4
)(
1
2
/ 2
2
/ 2
1
2
x
EJ
w
eC eCx"f EJ
Kx
Kx
l
l
l
l
(
)
(
)
-
+
+ú
û
ù
ê
ë
é
-
-
-
-
l
l
l
l
l
l
l
/ 2
2
/ 2
1
2
2
2
1
2
/ 2
2
/ 2
1
2
)(
4
4
2 )(
2
Kx
Kx
Kx
Kx
eC eCxf EJ K
EJ
w
eC eCK x'f EJ
[
]
0 )
(
)
(
4
)
(
)(
1
2
/ 2
2
/ 2
1
= - +ú
û
ù
ê
ë
é
-
+
+
- -
-
x w x
EJ
w
eC eCx q Sxf
Kx
Kx
l
l
l
l
l
l
Assuming that
f(x)
is a constant function, leads the first and sec-
ond terms of equation (74) to be null, since they are multiplied by
the derivatives
)(
x'f
and
)(
x"f
. Furthermore, considering the
preceding definitions of
K
(section 2.2) and
S
(equation 18), it may
be written:
(75)
S
EJ
EJ K
=
=
2
1
2
2
2
4
4
l
l
Consequently, the third term of equation (74) cancels with some
parts of the fourth one, reducing the equation to:
(76)
[
]
0 )( 1 )
(
4
)(
1
2
/ 2
2
/ 2
1
=
- +ú
û
ù
ê
ë
é
-
+
+
-
xf
w x
EJ
w
eC eCxfq
Kx
Kx
l
l
l
l
The Galerkin method will be used in order to find a constant func-
tion
(x) f
that has to be a good approximation for
f(x)
appearing in
equation (76). According to (28), it may be written:
(77)
1 )(
)(
1
11
=
'
=
x
x a (x) f
j
j
For this case, equation (29) is applied in the following form:
(78)
[
]
{
}
0
) 1(
4)
(
0
1
1
2
2
2
2
1
1
=
-
+
ò
-
dx
- a
+ w
EJ x
w
e + C eCq a
Kx/
Kx/
l
l
l
l l
Performing the integration,
a
1
can be isolated, giving:
(79)
[
]
)1
(
)1 (
2
8
1
1
2
2
2
1
1
3
1
-
--
- -
=
-
K
K
eC
eC
Kw
q
EJ
q
a
l
Therefore, the approximate solution for equation (72) will be given by:
(80)
ú
û
ù
ê
ë
é
-
+
+
=
-
)
(
4
)(
1
2
/ 2
2
/ 2
1 1
x
EJ
w
eC eCa x
Kx
Kx
l
l
l
l
f
with
a
1
given by (79). Integrating twice leads to the primitive of the
displacements function:
(81)
(
)
ú
û
ù
ê
ë
é
+ +÷
ø
ö
ç
è
æ -
+
+
=
-
4
3
1
22
/ 2
2
/ 2
1 2
2
1
6 2
4
4
)(
C xC x
EJ
xw
eC eC
K
a xY
Kx
Kx
l
l
l
l
l
where
C
4
is an undetermined constant and
C
3
results from the con-
dition of zero displacement at support:
(82)
)
(eK
e )
K(e
EJ
w C
K
K
K
1
1
8
4 2
2
4
1
4
3
+
+ -
×
-
=
l
The bending moment at support is obtained, applying equation (32):
(83)
[
]
þ
ý
ü
î
í
ì
+
+ -
+-
+ =
-
l
l
l
l
3
1
5
2
2
2
1 2
2
1
2
12
)1
(
)1 (
4
2
C
EJ
w
eC
eC
K
qa w M(0)
K
K