Civil Engineering Reference
In-Depth Information
frequency domain. This requires the establishment of a frequency domain description of
the wind field as well as the structural properties, and it involves the establishment of
frequency domain transfer functions, one from the wind field velocity pressure distribution
to the corresponding load, and one from load to structural response. We shall see that this
implies the perception of wind as a stochastic process, and a structural response calculation
based on its modal frequency-response-properties. The important input parameters to this
solution strategy are the statistical properties of the wind field in time and space, and the
eigen-modes and corresponding eigen-frequencies of the structural system in question. The
outcome is the statistical characteristics of the structural response.
Thus, apart from the geometry and mass properties of the structural system, it is
necessary to know its eigen
−
modes and corresponding eigen
−
frequencies. These are the
results of eigen
value calculations. The theory of such calculations may be found in
many classical text books, see e.g. Timoshenko, Young & Weaver [1], Clough &
Penzien [2] and Meirovitch [3]. It has been considered unnecessary to include any of
such theory in this topic, except for a simple example shown in chapter 4.1. I.e., it will
be taken for granted that sufficient information regarding the eigen
−
value solution has
already been provided. Most often, such information has been obtained from a finite
element calculation of a discretised structural system, and thus, the eigen
−
−
modes are
given as more or less ample vectors representing eigen
mode displacements along the
span. In the following it is tacitly assumed that such an eigen
−
−
value analysis has been
performed in vacuum or in still air.
It should be acknowledged that in the mathematical development of the basic theory
in this topic it is for convenience assumed that eigen
modes are continuous functions.
This simplifies and helps on the comprehension of the various steps behind the theory.
After the final expressions of response are developed, the vector-matrix operations
involved in a purely numerical format of the solution strategy are presented wherever it
is considered necessary.
In structural dynamics where a modal solution procedure is adopted it is also
necessary to quantify modal eigen
−
damping properties. This is another subject that will
not be treated in this topic. It is taken for granted that the modal damping ratio is known
from elsewhere (e.g. standards or handbooks).
−
1.2 Random variables and stochastic processes
A physical process is called a stochastic process if its numerical outcome at any time or
position in space is random and can only be predicted with a certain probability. A data
set of observations of a stochastic process can only be regarded as one particular set of
realisations of the process, none of which can with certainty be repeated even if the
conditions are seemingly the same. In fact, the observed numerical outcome of all
physical processes is more or less random. The outcome of a process is only
deterministic in so far as it represents a mathematical simulation whose input parameters
has all been predetermined and remains unchanged.
