Civil Engineering Reference
In-Depth Information
7.2 The mean value
For the calculation of the mean value of cross sectional forces all quantities are time
invariants and thus, Eq. 6.3 still holds, implying that the global displacements are given
by
1
−
rKR
=⋅
(7.11)
Similarly, the mean values of local forces and displacements (see Eqs. 7.8 and 7.9) are
defined by
Fkd
dAr
=⋅
⎫
⎪
⎬
mmm
(7.12)
=⋅
⎪
⎭
mm
and thus, the mean value of cross sectional forces is given by
(
)
(
)
m m
−
⎡
1
⎤
FkAr kAKR
=⋅
⋅
=⋅
⋅
⋅
(7.13)
mm m
⎣
⎦
Eqs. 7.8 - 7.13 are identical to that which one will usually encounter in an ordinary finite
element formulation. The establishment of
k
and
A
as well as the ensuing strategy
for the calculation of global displacements and element force vectors may be found in
many text books, see e.g. Hughes [25] or Cook et.al. [29]. Nonetheless, the brief
summary presented above has been i
n
cluded for the sake of completeness. The only part
that is special is the development of
R
, which has previously been shown in chapter 6.2.
7.3 The background quasi-static part
For the determination of the quasi-static part of the cross sectional response forces the
mean part of the load as well as any motion induced contributions are obsolete.
According to Eq. 5.8 the fluctuating part of the load on a line-like structure is given by
(
)
⎡
qxt
,
⎤
y
⎢
⎥
VB
ρ
ˆ
(
)
(
)
q
xt
,
=
q xt
qxt
θ
,
=
B v
⋅
=
⋅
B v
(7.14)
⋅
⎢
⎥
z
q
q
2
⎢
⎥
(
)
,
⎣
⎦
where
v
and
B
are defined in Eqs. 5.9 and 5.12, and recalling that this was developed
for a horizontal type of structure. As mentioned above the quasi-static part may be
determined by a formal finite element formulation, or alternatively, by the use of static
influence functions based on a direct establishment of the equilibrium conditions.
