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78
IBRACON Structures and Materials Journal • 2013 • vol. 6 • nº 1
Considerations about the determination of
g
z
coefficient
(16)
3,2
3
3
3
3
3
3
3,2
3
3
3,2
B
M
M M
M
M M
1
B
M
M
1
1
B
Adding up
M
1
,
M
2
and
M
3,
equations (8), (11) and (14), and
∆M
1
,
∆M
2
and
∆M
3
, equations (9), (12) and (15) gives:
(17)
M
1
+ M
2
+ M
3
= F
h1d
L + 2F
h2d
L + 3F
h3d
L
(18)
M
1
+
M
2
+
M
3
= P
1d
u
1
+ P
2d
u
2
+ P
3d
u
3
Comparing equations (17) and (18) with equations (6) and (7) we
can write:
(19)
M
1,tot,d
= M
1
+ M
2
+ M
3
(20)
M
tot,d
=
M
1
+
M
2
+
M
3
By substituting equations (19) and (20) in equation (1), the
g
z
coef-
ficient becomes defined as:
(21)
)
( )
( )
(
3
3
2
2
1
1
3
2
1
M M M M M M
MMM
z
Inverting equation (21) gives:
(22)
3
2
1
3
3
2
2
1
1
)
( )
( )
( 1
MMM
M M M M M M
z
(7)
M
tot,d
= P
1d
u
1
+ P
2d
u
2
+ P
3d
u
3
The
B
2
coefficient, given by equation (5), shows distinct values for
each storey of the structure. Thus, referring to the
B
2
coefficient of
storey
i
as
B
2,i
and the parts
(L . ΣH
Sd
)
and
(∆
0h
. ΣN
Sd
)
as
M
i
and
∆M
i
,
respectively, we get:
n
1st storey:
(8)
(9)
(10)
1,2
1
1
1
1
1
1
1,2
1
1
1,2
1
1
1
B
M M M
M
M M
B
M
M
B
n
2nd storey:
(11)
M
2
= L
(F
h2d
+ F
h3d
) = F
h2d
L + F
h3d
L
(12)
(13)
2,2
2
2
2
2
2
2
2,2
2
2
2,2
B
M
M M
M
M M
1
B
M
M
1
1
B
n
3rd storey:
(14)
M
3
= L
(F
h3d
) = F
h3d
L
(15)
M
3
= (u
3
– u
2
)
(P
3d
) = P
3d
u
3
– P
3d
u
2