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111
IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 1
R. J. ELLWANGER
4.1 Substructures of shear wall or shear core types
In the case of bracing systems formed exclusively by shear walls
and/or shear cores, the differential equation of motion will be ob-
tained equalizing
2
2
/
dx ydEJ
to
M(x)
given by (31):
(33)
[
]
)()
( )( )(
2/ )
(
/
2
2
2
xyx
xY Yq
x w dx ydJE
- - -
+
- =
l
l
l
Deriving it in relation to
x
and considering that the rotations are
given by
f
(
x
) =
dy
/
dx
, changes equation (33) into:
(34)
0 )
(
)()
(
2
2
= - +
- +
x w x x q dx dJE
l
l
f
f
An approximate solution for equation (34) can be obtained
through the Galerkin method. Assuming that this solution is
proportional to
f
(
x
) due exclusively to first order effects, it may
be written:
(35)
ú
ú
û
ù
ê
ê
ë
é
÷
ø
ö
ç
è
æ - - =
=
3
1
11
1 1
)(
)(
l
x
a x a x
j
f
where
)(
1
x
ϕ
was obtained deriving equation (14) in relation to
x
and suppressing the constant that would remain in evidence. Ap-
plying equation (29) with
n
= 1, leads successively to:
(36)
(
)
0 )(
)(
)( )(
0
1
11
0
1
=
×
=
×
ò
ò
l
l
dxx
x aL
dx x
L
j j
jf
(37)
( )
( )
ò
=
ú
ú
û
ù
ê
ê
ë
é
÷
ø
ö
ç
è
æ - - ×
ïþ
ï
ý
ü
ïî
ï
í
ì
- +
ú
ú
û
ù
ê
ê
ë
é
÷
ø
ö
ç
è
æ - -
- +÷
ø
ö
ç
è
æ -
l
l
l
l
l
l
l
0
3
3
1
2
1
0
1 1
1 1
1
6
dx x
x w x
ax q
x EJa
Performing the integration and isolating
a
1
, results:
(38)
3
3
1
3
24
4
l
l
q EJ
w
a
-
=
On substituting (38) into (35), the approximate solution is obtained:
(39)
ú
ú
û
ù
ê
ê
ë
é
÷
ø
ö
ç
è
æ - -
-
=
3
3
3
1 1
3
24
4
)(
l
l
l
x
q EJ
w
x
f
Integrating (39) in relation to
x
and applying the condition of zero
displacement at support, leads to the displacements function. Inte-
grating again, leads to:
(40)
C x x
x
q EJ
w
xY
+
ú
ú
û
ù
ê
ê
ë
é
- +÷
ø
ö
ç
è
æ - -
-
=
l
l
l
l
l
2
5
3
4
2
1
5
3
24
)(
where
C
is the integration constant. The bending moment at sup-
port can be obtained on applying equations (32) and (40):
(41)
)
5(8
2
2
)0(
3
5
2
l
l
l
q EJ
qw
w M
-
+ =
Equation (41) can be transformed successively into:
(42)
3
3
2
3
3
2
8
5/
8
2
)
5(8
4
1
2
)0(
l
l
l
l
l
l
q EJ
q EJ
w
q EJ
q
w M
-
-
×
=÷÷
ø
ö
çç
è
æ
-
+ ×
=
The condition of that, in the ultimate limit state (loads multiplied by
1,4), the second order effects may not exceed the first order effects
in more than 10% (inequality (1)), is applied to the support bending
moment, obtaining:
(43)
2
1,4 1,1
1,4
8
/5 1,4
8
2
1,4
2
3
3
2
l
l
l
l
w
q
EJ
q
EJ
w
´ £
-
-
´
The terms
w
l
2
, multiplying both sides of the inequality, do vanish.
Performing the required algebraic transformations, results:
(44)
6349 ,0
/
3
£
EJ q
l
Since a wall or core has a behavior equivalent to the one of a
column, the physical nonlinearity may be considered adopting for
EJ
the expression 0,941
E
CS
I
C
, according to equation (11). On the
other hand, remembering that
q
l
is the total vertical load
N
k
and
l
is
the total height
H
tot
, inequality (44) becomes:
(45)
5974 0,
/
CCS
k
2
tot
£
´
IEN H
Extracting the square root of both members:
(46)
773 0,
/
C CS
k
tot
£
´
IEN H