7.7 Earthquake Spectrum Analysis of Structures Having MDF System
287
From Eqs. (3.147) and (3.153), we get
S
(r )
a = p
2
d r S
(r )
d
(7.56)
Combining Eqs. (7.55) and (7.56), we get
P
(r )
i
= m i φ
(r )
i B r S
(r )
a
(7.57)
IS Code 1893 has specified the equation for the load acting at each floor level in
the same form as Eq. (7.57).
The maximum storey shear at any level i in the rth mode then is
V
(r )
i
=
n
j=i
P j = B r S
(r )
a
n
j=i
m j φ
(r )
j
(7.58)
where n represents the topmost storey and i is the storey under consideration.
Though the expressions of Eqs. (7.54a–7.54b) and (7.58) appear to be different,
they are in fact identical. This is proved as follows.
Consider a three-storeyed shear building of Fig. 7.9. The equations of motion for
the masses in free vibration for the rth mode can be written as
− p
2
r
⎡
⎣
m 1 0 0
0 m 2 0
0 0 m 3
⎤
⎦
⎧
⎨
⎩
φ
(r )
1
φ
(r )
2
φ
(r )
3
⎫
⎬
⎭
+
⎡
⎣
k 1 + k 2 −k 2
0
−k 2 k 2 + k 3 −k 3
0
−k 3 k 3
⎤
⎦
⎧
⎨
⎩
φ
(r )
1
φ
(r )
2
φ
(r )
3
⎫
⎬
⎭
=
⎧
⎨
⎩
0
0
0
⎫
⎬
⎭
(7.59)
Fig. 7.9 Shear binding
287
From Eqs. (3.147) and (3.153), we get
S
(r )
a = p
2
d r S
(r )
d
(7.56)
Combining Eqs. (7.55) and (7.56), we get
P
(r )
i
= m i φ
(r )
i B r S
(r )
a
(7.57)
IS Code 1893 has specified the equation for the load acting at each floor level in
the same form as Eq. (7.57).
The maximum storey shear at any level i in the rth mode then is
V
(r )
i
=
n
j=i
P j = B r S
(r )
a
n
j=i
m j φ
(r )
j
(7.58)
where n represents the topmost storey and i is the storey under consideration.
Though the expressions of Eqs. (7.54a–7.54b) and (7.58) appear to be different,
they are in fact identical. This is proved as follows.
Consider a three-storeyed shear building of Fig. 7.9. The equations of motion for
the masses in free vibration for the rth mode can be written as
− p
2
r
⎡
⎣
m 1 0 0
0 m 2 0
0 0 m 3
⎤
⎦
⎧
⎨
⎩
φ
(r )
1
φ
(r )
2
φ
(r )
3
⎫
⎬
⎭
+
⎡
⎣
k 1 + k 2 −k 2
0
−k 2 k 2 + k 3 −k 3
0
−k 3 k 3
⎤
⎦
⎧
⎨
⎩
φ
(r )
1
φ
(r )
2
φ
(r )
3
⎫
⎬
⎭
=
⎧
⎨
⎩
0
0
0
⎫
⎬
⎭
(7.59)
Fig. 7.9 Shear binding