Civil Engineering Reference
In-Depth Information
5.3.7
Along-wind response of a structure with distributed mass—
separation of background and resonant components
In the case of wind loading, the method described in the previous section is not an
efficient one. For the vast majority of structures, the natural frequencies are at the high
end of the range of forcing frequencies from wind loading. Thus, the resonant
components become very small as
j
increases in Equation (5.16). However, the
contributions to the mean and background fluctuating components for
j
greater than 1 in
Equation (5.16)
may not be small
. Thus, it is necessary to include higher modes (
j
>1) in
Equation (5.16) not for their resonant contributions, but to accurately determine the mean
and background contributions. For example, Vickery (1995) found that over 20 m odes
were required to determine the mean value of a response and over 10 values were need to
compute the variance. Also for the background response, cross-coupling of modes cannot
be neglected, i.e. Equation (5.23) is not valid.
A much more efficient approach is to separately compute the mean and background
components as for a quasi-static structure. Thus, the total peak response, can be taken
to be:
(5.25)
where is the peak background response equal to
g
B
σ
B
; and is the peak resonant
response computed for the
j
th mode equal to
g
j
σ
R, j
. This approach is illustrated in Figure
5.1.
g
B
and
gj
are peak factors which can be determined from Equation (5.15); in the case
of the resonant response, the cycling rate,
υ,
in Equation (5.15), can be taken as the
natural frequency,
n
j
.
The mean-square value of the quasi-static fluctuating (background) value of any
reponse,
r,
is:
(5.26)
where
I
r
(z)
is the influence line for
r,
i.e. the value of
r
when a unit load is applied at
z
.
The resonant component of the response in mode
j
can be written to a good
approximation as:
(5.27)
