Civil Engineering Reference
In-Depth Information
L
L
⎡
⎤
⎡
⎤
()
()
Z
Z
c
0
k
0
⎢
⎥
⎢
⎥
ae
1
ae
1
2
0
2
0
c
and
k
=
⎢
⎥
=
⎢
⎥
(9.63)
ae
ae
n
n
L
L
⎢
⎥
⎢
⎥
(
)
(
)
Z
Z
0
c
0
k
⎢
ae
2
⎥
⎢
ae
2
⎥
2
0
2
0
⎣
⎦
⎣
⎦
n
n
where
and
are given in Eqs. 9.52 and 9.53.
c
k
ae
ae
0
0
On the other hand, if Eq. 9.60 is applicable (i.e. the concept of indicial functions is
adopted), then
t
⎧
⎫
L
L
⎪
⎡
⎤
⎪
()
(
)
( )
(
)
( )
∫
Q
t
=
c
Z
,
s
⋅
d
τ
+
k
Z
,
s
⋅
d
τ
d
τ
(9.64)
⎨
⎬
⎢
⎥
ae
ae
i
i
ae
i
i
i
n
0
0
2
2
⎣
⎦
⎪
⎪
⎩
⎭
0
n
and the corresponding load vector at the level of element
n
is given by
(
)
t
t
Zt
,
⎡
R
⎤
()
ae
1
(
)
( )
(
)
( )
⎦
∫
∫
R
t
=
=
c
ZZs
,
,
⋅
d
ττ
d
+
k
ZZs
,
,
⋅
d
ττ
d
⎢
⎥
ae
ae
12
n
ae
12
n
(
)
n
Zt
,
n
n
R
⎣
ae
2
n
0
0
(9.65)
where
L
L
⎡
⎤
⎡
⎤
(
)
(
)
c
Zs
,
0
k
Zs
,
0
⎢
⎥
⎢
⎥
ae
1
ae
1
2
0
2
0
and
c
=
⎢
⎥
k
=
⎢
⎥
ae
ae
n
L
n
L
⎢
⎥
⎢
⎥
(
)
(
)
0
c
Z
,
s
0
k
Z
,
s
ae
2
ae
2
⎢
⎥
⎢
⎥
2
0
2
0
⎣
⎦
⎣
⎦
n
n
(9.66)
where
a
k
are given in Eqs. 9.58 and 9.59.
Thus, all wind forces and motion induced forces at element level have been
established. What remains before any response calculations can be performed is to
establish equilibrium conditions expressed in global structural degrees of freedom.
and
c
ae
0
0
9.4 The global analysis
As previously mentioned (see Chapter 9.2) it is taken for granted that displacements as
well as forces at element level comprise a time-invar
ia
nt mean value (the
s
tatic part) and
a stationary fluctuating (dynamic) part, i.e.
()
()
t
t
d
=+
d
d
,
FFF
and
=+
n
n
n
n
n
n
tot
tot
