Page 100 - Dynamic Loading and Design of Structures

Page 77

leading to a corresponding normalized value, HN=He/L,n (Wyatt, 1981). The solution of eqn
(3.15) for uniform weighting (i.e.) in conjunction with the simple exponential form
for R , which is
u

(3.17)

2
(plotted as Figure 3.2b), then constitutes an excellent approximation to J for any weighting in
which γz) is of uniform sign throughout the structure. This covers the fundamental mode of
(
most cantilever structures (towers, chimneys and translational modes of buildings) and other
single-span structures; it is not restricted to HN>>1 . The less sophisticated approximation
2
J =2/H remains applicable for any weighting at large H , with error of order 1/HN- For the
N
N
example given, with the first cantilever mode approximated by y(z)=(z/H)1.5, H 0.64H=64
e=
2
m, so that with Ln=6 m, the normalized effective size HN=10.7, and J =20.17.
For white noise excitation (i.e. Sp invariant with frequency) there is a closed form solution
for the response variance, i.e. for displacement Yj in mode j,

(3.18)

in which nj is the natural frequency and δs the damping expressed as logarithmic decrement.
i
As the peak spectral response ordinate comprises dynamic magnifier , this implies an
effective bandwidth . For practical values of , the magnifier is so large, and the
response bandwidth so small compared to the bandwidth of the input spectrum, that this result
gives an excellent approximation to the area under the resonance peak, identified as on
Figure 3.2(d), i.e.

(3.19)

in which Yj=Pj/Kj is the mean value of modal displacement. Clearly the r.m.s. value
follows

(3.20)

The corresponding value of modal narrowband (quasi-resonant) contribution to any structural
load effect F (e.g. stresses or stress resultants such as bending moments) can be obtained by
multiplying the displacement by the respective modal influence coefficient β Fj (say), i.e.
. Values for the modal influence coefficients can now generally be
obtained from the computer modal solution output, but if in doubt concerning accuracy of
modelling, or when using hand computation, they should be evaluated as the static effect of
the