Civil Engineering Reference
In-Depth Information
The symbolic integration limits
L
and
L
exp
indicate integration over the entire structure or
over the wind exposed part of the structure. The modal matrices
K
on the
left hand side of Eq. 4.9 are all diagonal due to the orthogonal properties of the eigen-
modes. However, we shall later see that motion induced parts of the load will be moved
to the left hand side of the modal equilibrium equation, thus rendering non-diagonal
mass, damping and stiffness matrices for the combined flow and structural system. For
educational reasons the development of the necessary theory is divided into three parts,
depending on the complexity of the problem. The first part of the presentation is dealing
with the situation that the relevant eigen-frequencies are well separated and each mode
only contains one component. The next is dealing with the same situation but now with
each mode containing all three components. The final presentation is considering the
situation that a full multi-mode investigation is required.
M
,
C
and
0
4.2 Single mode single component response calculations
In this first section it is assumed that the eigen-frequencies are well spaced out on the
frequency axis. Furthermore, the cross sectional shear centre is assumed to coincide (or
nearly coincide) with the centroid and there are no other significant sources of mechani-
cal or flow induced coupling between the three displacement components. These as-
sumptions imply that coupling between modes may be ignored, and that each mode
shape only contains one component, i.e. any of the
mo
N
mode shapes is purely horizon-
tal, vertical or torsion. The response covariance between modes will then be zero, and
thus, the variance of the total dynamic horizontal, vertical or torsion displacement re-
sponse can be obtained as the sum of contributions from each mode, i.e. the variance of a
displacement component is the sum of all variance contributions from excited modes
containing displacement components exclusively in the
y
,
z
or
θ
direction (see Eq. 2.27).
2
y
E.g.
is the sum of all variances associated with the relevant number of modes con-
taining only horizontal displacements, and so on. Thus,
σ
⎡
⎤
∑
2
σ
⎢
⎥
i
y
2
⎡⎤
⎢
σ
⎥
i
y
y
⎢⎥
⎢
⎥
2
∑
2
⎢⎥
σ
=
σ
(4.15)
⎢
⎥
z
i
z
⎢⎥
⎢
⎥
i
z
2
σ
⎢⎥
⎢
⎥
θ
⎣⎦
⎢
∑
2
σ
⎥
i
θ
⎢
⎥
⎣
i
⎦
θ
()
i
x
Given an arbitrary horizontal, vertical or torsion mode shape
φ
with eigen-
frequency
ω
and damping ratio
ζ
, the time domain displacement response contribu-
i
i
tion of this mode is simply
