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115
IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 1
R. J. ELLWANGER
It can be observed that the difference
Y
(
l
) –
Y
(0) made the con-
stant
C
4
to vanish. Substituting
C
1
,
C
2
and
C
3
, respectively, by
equations (26), (27) and (82), transforms equation (83) into:
(84)
1
24 )1
)(
12 3( )1
)( 8 6(
96 2
4
2
4 2
4 3
3
1
5
1
2
+
--
-++
+
×
+ =
K
K
K
K
e
Ke
e K
e K K
KEJ
qwa
w M(0)
l
l
Introducing the expressions for
C
1
and
C
2
also in equation (79) and
substituting the formula of
a
1
obtained in such a manner into equa-
tion (84), leads to the expression of the bending moment at support
of the system composed by frames and walls/cores:
(85)
ú
û
ù
ê
ë
é
+-
++
+ -+
- -
- + +
+
+
=
K
K
K
K
K
K
K
Ke
eK
e K K
eK /q EJ
Ke
e K
e /K K
w M(0)
2
4 2
4 3
4 3 3
1
2
4 2
4
3
2
2)1 ( 2)1
)( 2 ( )1 ( )
16(
8 )1
)( 41( )1
)(3 8 2(
1
2
l
l
Applying the condition expressed by inequality (1) to this bending
moment, results:
(86)
2
41 1,1
2)1 ( 2 )1
)(
2 41
16(
8)1
)(
41( )1
)(3 8 2(
1
2
41
2
2
4 2
4
3
3
3
1
2
4 2
4
3
2
l
l
l
w,
Ke
eK
eK K q,/KEJ
Ke
e K
e /K K
w,
K
K
K
K
K
K
´ £
ú
ú
û
ù
ê
ê
ë
é
+-
++
- -
--
-++
+
+
In inequality (86), the factor
q
l
3
/
EJ
1
can be isolated, giving:
(87)
K
K
K
K
Ke ,
e K,
e K,
K,
eK /
EJ
q
2
4 2
4 3
4 3
1
3
624 )1 () 612 3( )1 () 68 36(
)1 ( )7 24(
- -
- + +
+
+
£
l
Calling
J
the sum of inertias of the walls/cores and frames assemblag-
es, and considering the preceding definition of
K
, it may be written:
(88)
2
2
1
2
1 1 2 1
)1 (
K KJ KJ J J J J
+
=
+ = + =
Isolating
J
1
:
(89)
)1 (
2
2
1
+
=
KJK J
I
C1
and
I
C2
are defined as the gross inertias, respectively, of the
frames and walls assemblages. Calling
I
C
the sum of
I
C1
and
I
C2
and
applying the relations (11) and (66), gives:
(90)
1
2
2
2
1
1
2
1
2
1
0625 1
5385 1
941 0 650
941 0 650
J
K
,
K ,
K ,
J
,
J
,
J
,
J
I
I
I
C C C
+
=
+ =
+ = + =
From (90), the factor
EJ
1
may be expressed by:
(91)
C CS
IE
,
K ,
K
EJ
0625 1
5385 1
2
2
1
+
=
Equation (87) can be rewritten, substituting
EJ
1
by expression (91),
q
l
by
N
k
(total vertical load) and
l
by
H
tot
(building height). After,
extracting the square root of both members, results:
(92)
1
α
IE
N
H
C CS
k
tot
£
´
where
(93)
] e 24,6 1)
)(e
12,6
(3 1)
)(e
8,6
,3
1,0625)[(6
(1,5385
1)
(e
(24/7)
α
2
4 2
4 3
2
4 5
1
K
K
K
K
K
K
K K
K
K
- -
- + +
+
+
+
=
In this manner, an expression for the limit
a
1
of the instability pa-
rameter, variable with
K
(relation between the reduced inertias of
frames and walls/cores), was obtained. However, in order to obtain
a
1
, it is more practical to deal with the gross inertias. Combining
equations (66) and (90), the following relation between
K
and
I
C1
/
I
C
(proportion between the frames gross inertia and the total one)
is obtained:
(94)
) /
1/() /
(
831 ,0
1
1
C C
C C
I I
I I
K
-
=
Thus, given any proportion
I
C1
/
I
C
,
K
is obtained applying equa-
tion (94); soon after,
a
1
is obtained applying equation (93). The
sequence of
a
1
values, presented in table 1 and graphically
represented in figures 7 and 8, shows a rough variation for
I
C1
/
I
C
close to 1 (predominance of frames) and a more smooth
one for
I
C1
/
I
C
close to 0 (predominance of walls). It can also be
observed that equation (93), at its domain ends, reproduces
equation (46) exactly, but presents a discrepancy of 1,8% for
bracing systems composed exclusively by rigid frames. The
Table 1 – Values of
a
, varying the I /I ratio
1
C1 C
I /I
C1 C
I /I
C1 C
I /I
C1 C
a
1
a
1
a
1
0 0,773
0,50
0,755
0,90 0,651
0,10 0,772
0,60 0,744
0,95 0,611
0,20 0,771
0,70
0,726
0,98 0,574
0,30 0,768
0,80 0,699
0,99 0,555
0,40 0,763
0,85 0,679
1,00 0,509