Civil Engineering Reference
In-Depth Information
Appendix A
TIME DOMAIN SIMULATIONS
A.1 Introduction
It is in the following taken for granted that the stochastic space and time domain
simulation of a process
x
implies the extraction of single point or simultaneous
multiple point time series from known frequency domain cross spectral information
about the process. The process may contain coherent or non-coherent properties in space
and time. Thus, a multiple point representation is associated with the spatial occurrence
of the process. For a non-coherent process there is no statistical connection between the
simulated time series that occur at various positions in space, and thus, the simulation
may be treated as a representation of independent single point time series. This type of
simulation is shown in chapter A.2. For a coherent process there is a prescribed
statistical connection between each of the spatial representatives within a set of
M
simulated time series. E.g., if the simulated time series represent the space and time
distribution of a wind field, there will be a certain statistical connection between the
instantaneous values
()
that matches the spatial properties of the
wind field. Such a simulation is shown in chapter A.3. The simulation procedure
presented below is taken from Shinozuka [23] and Deodatis [24].
Simulating time series from spectra is particularly useful for two reasons. First, there
are some response calculations that render results which are more or less narrow banded
(or contain beating effects), and thus, they do not necessarily comply with the
assumptions behind the peak factor given in Eq. 2.45. These cases may require separate
time domain simulations to establish an appropriate peak factor for the calculation of
maximum response. This application will usually only require single point simulations.
Secondly, if the relevant cross spectra of the wind field properties in frequency domain
are known, there is always the possibility of a time domain simulation of the entire wind
field, or those of the flow components that are deemed necessary. Together with the
buffeting load theory in chapter 5.1 this is a tempting option, as time domain step-wise
load effect integration may be performed, and thus, the response calculation may be
carried out in time domain instead of the frequency domain approach that is shown in
chapter 6. The mathematical procedure for such an approach may be found in many text
books, see e.g. Hughes [25]. The main advantage is that such an approach may contain
many of the non-linear effects that had to be simplified or discarded in the linear theory
that was required for a frequency domain solution. The disadvantage is that motion
induced load effects can only be fully included if a new set of indicial functions are
introduced (see e.g. Scanlan [26]). These may not be readily available.
x tm M
,
=
1,2,....,
