Civil Engineering Reference
In-Depth Information
6.2 The mean value of the response
The mean value of the response is the load effects of the mean flow induced load as
defined in Eq. 5.11. It may readily be calculated according to standard static equilibrium
type of procedures in structural mechanics. Such procedures are in general
mathematically formulated within a finite element type of description where the solution
strategy is based on the displacement method, i.e. f
or
a chosen discrete model containing
N
number of nodes the mean displacement vector
r
is obtained from
Kr R
(6.3)
⋅
=
where
K
is the static stiffness matrix and
R
is the mean load vector. A line like
structure will in general be modelled by beam or beam-column type of elements, in
which case there will us
ua
lly be six degrees of freedom in each node (as illustrated in
Fig. 6.4.a). Thus,
r
and
R
are
6
⋅
by one vectors and
K
is a
6
⋅
by
6
⋅
matrix.
Herein, the establishment of
K
and the ensuing str
at
egy for the calculation of
r
will not
be further pursued. However, the establishment of
R
is presented below.
Let us consider a typical finite element type of modelling with six load components in
each node. According to Eq. 5.11 the mean value of the evenly distributed load on an
element is given by
q
⎡
⎤
⎡⎤
⎢⎥
DC
BC
BC
y
D
2
V
⎢
⎥
ρ
()
x
q
q
θ
q
=
=
⋅
b
where
b
=
(6.4)
⎢⎥
⎢
⎣⎦
⎢
⎥
z
q
q
L
2
⎢
⎥
2
⎣
⎦
M
At an arbitrary node
p
the load contribution from an adjoining element
m
(see
Fig. 6.4.b and c) is then
⎡⎤
⎢⎥
Q
y
2
⎛ ⎞
L
V
L
ρ
()
m
m
Q
x
Q
=
=
q
⋅
=
⋅
⋅
b
(6.5)
⎢⎥
⎜ ⎟
p
z
m
⎜ ⎟
q
m
2
2
2
m
⎢⎥
⎝ ⎠
Q
θ
p
⎢
⎣⎦
where
L
is the element length,
q
b
is the
b
vector that contains the properties
associated with element
m
,
and where it has for simplicity been assumed that the nodal
discretisation is such that
q
may be taken constant within the length of the element
(otherwise,
may be replaced by the result of a simple span-wise integration).
L
2
