Civil Engineering Reference
In-Depth Information
shear centre of the cross section) may be split into the sum of a time invariant mean part
and a fluctuating part
(
)
() ( )
r
x t
,
=
r
x
+
r
x t
,
⎫
⎪
⎬
tot
(4.1)
(
)
()
( )
q
x t
,
=
q
x
+
q
x t
,
⎪
⎭
tot
each containing three components (horizontal, vertical and torsion), i.e.:
T
T
(
)
()
xr
⎡
r
r
θ
⎤
xt
,
⎡
r
r
θ
⎤
r
=
⎣
r
=
⎣
(4.2)
⎦
⎦
yz
yz
T
T
()
(
)
xqq
θ
⎡
⎤
xt
,
⎡
q
q
q
θ
⎤
q
=
q
=
(4.3)
⎣
y
z
⎦
⎣
y
z
⎦
In the following the mean values of the response are considered trivial. The entire focus
is on the calculation of the variances of the fluctuating displacement components. The
solution will be based on a modal frequency domain approach. Thus, it is assumed that a
sufficiently accurate eigen-value solution is available, and that it contains the necessary
number of eigen-frequencies and corresponding eigen-modes. That they are orthogonal
goes without saying. Scaling of mode shapes is optional, but consistency is required such
that the relative difference between cross sectional displacement and rotation compo-
nents is maintained. It is taken for granted that the eigen-value solution has been ob-
tained in vacuum or in still air conditions. Such a solution has usually been obtained
from some finite element formulation (see Chapter 9), and for line-like beam type of
elements the eigen-modes will then occur as vectors usually containing six components
in each element node, three displacements and three rotations. In the development of the
theory below the number of eigen-value components is reduced, focusing on the degrees
of freedom associated with
r
,
r
and
r
θ
. Thus, the mode shape components associated
with an arbitrary mode is the displacements
that has been extracted from a finite element type of solution. It should be noted that the
mode components are formally treated as continuous functions, and therefore the two
other rotation components may be retrieved from the first derivatives of
φ
,
φ
and the cross sectional rotation
φ
y
z
. It is
then only the x-axis displacement (i.e. the component in the direction of the main span)
that is entirely discarded, but this is not considered important since it is not directly asso-
ciated with any flow induced load.
φ
and
φ
y
z
Example 4.1
Given a simply supported beam with a single symmetric channel type of cross section as shown in
Fig. 4.2. The system contains three displacement components:
y
(
x,t
),
z
(
x,t
) and θ(
x,t
), all referred
to the shear centre, which in this case does not coincide with the centroid. Disregarding any exter-
nal loading and damping contributions, the differential equilibrium conditions are given by (see
Timoshenco, Young & Weaver [1], chapter 5.21):
