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FIGURE 10.15
Percentage period elongations (PE) and amplitude decays (AD). See Fig. 10.10 for definitions.
stable, the selection of
t should be based on the analysis of the accuracy of each particular
method.
It is known that large integration steps may cause period elongation (PE) and amplitude
decay (AD) (Fig. 10.10). This can be demonstrated by the integration of
2
q i
+ ω
i q i
=
0
(10.128)
q i
(
0
) =
1
.
0 ,
q i
˙
(
0
) =
0
(10.129)
which is one of the equations in Eq. (10.127) with
0. The result is shown
in Fig. 10.15. The curves in Fig. 10.15 indicate that, in general, when
ζ i =
0 and P i =
T is smaller than
0.01, where T is the period corresponding to Eq. (10.128), the integration is accurate using
any of the integration methods. But when
t
/
T is larger than 0.01, the different integration
schemes exhibit different characteristics. For a given
t
/
t
/
T , the Wilson
θ
method with
θ =
4 introduces less amplitude decay and period elongation than the Houbolt method,
and the Newmark method only introduces period elongation, but no amplitude decay. See
Bathe (1982) for further discussions of accuracy.
1
.
10.6
Dynamic Analysis Based on Ritz Vectors
If the modal superposition method of Section 10.4.1 with the usual free vibration mode
shapes is applied to structures under earthquakes forces, it is found in the references by
Wilson that the results are not as accurate as those obtained with basis vectors that take into
account the spatial distribution of the dynamic loading. These basis vectors, called Ritz vec-
tors , have been used in wave propagation and foundation analyses [Bayo and Wilson, 1985].
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