Geology Reference
In-Depth Information
the first phase elastic optimization is to minimize
the concrete material cost of the structure as ex-
pressed explicitly in terms of design variables as
Based on the modal response of a building,
which is computed by commonly available engi-
neering software, the n th modal elastic displace-
ment at the j th floor level, u j ( ) , can be expressed
explicitly by the principle of the virtual work in
Box 1.where L i is the length of member i ; E and
G are the axial and shear elastic material moduli;
A X , A Y , and A Z are the axial and shear areas for the
cross-section; I X , I Y , and I Z are the torsional and
flexural moments of inertia for the cross-section;
F X ( ) , F Y ( ) , F Z ( ) , M X ( ) , M Y ( ) , and M Z ( ) are the
nth modal element internal forces and moments;
fXj, fYj, fZj, mXj, mYj, and mZj are the virtual
element forces and moments due to a unit of
virtual load applied to the building at the location
corresponding to the story displacement, uj.
Considering rectangular concrete elements
with the width ( B i ) and depth ( D i ) taken as
design variables and expressing the cross section
properties in terms of B i and D i , the modal dis-
placement Eq. (3) can be simplified as
N
i
Minimize: concrete cost f
=
w B D
1
c
ci
i
i
i
=
1
(1)
where w ci is the unit cost coefficient of concrete
for member i .
The intent of the elastic drift design is to ensure
that a building remains operational or serviceable
under the action of minor earthquakes. In checking
the seismic drift response of a building, an elastic
analysis procedure can be employed and the j =
1, 2,…, N j inter-story drift should comply with
the following requirement:
u
h
j
ψ
(
j
=
1 2
,
, ...,
N
)
(2)
j
j
j
where u j is the elastic inter-story drift of the jth
story; h j is the jth story height and ψ j is the
specified inter-story drift ratio limit for the jth
story.
Elastic linear response spectrum analysis
method, widely used in modern building codes
such as UBC (1997) and GB5011-2001 (2001),
is adopted in this study. This method eliminates
the time variable and provides a relatively simple
method for determining the maximum structural
responses in which individual modal responses
are first calculated and the maximum responses
are then obtained by combination rules.
( )
n
( )
n
( )
n
F f
E
F f
+
F f
1
L
i
X
Xj
Y
Yj
Z
Zj
+
dx
+
B D
5
G
/
6
0
i
i
i
( )
n
N i
M m
E
1
L
i
( )
n
i
Z
Zj
u B D
(
,
)
=
dx
+
j
i
i
B D
3
/
12
0
=
1
i
i
i
M m
G
( )
n
M m
E
( )
n
1
L i
X
Xj
Y
Yj
+
dx
3
B D
β
/
12
0
i
i
i
(4)
β denotes the torsional coefficient that depends
on the ratio value of depth to width of the element
i .For typical rectangular sections, it can be ap-
proximately set to 0.2.
Once the modal story displacement is formu-
lated explicitly, the maximum value of the inter-
Box 1.
Li
( )
n
( )
n
( )
n
( )
n
(
n
)
( )
n
N i
F f
EA
F f
GA
F f
GA
M m
GI
M
m
EI
M m
EI
= 1
(3)
( )
n
X
Xj
Y
Yj
Z
Zj
X
Xj
Y
Yj
Z
Zj
u
=
+
+
+
+
+
dx
j
i
X
Y
Z
X
Y
Z
o
i
 
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