Civil Engineering Reference
In-Depth Information
Such a typical stochastic process is illustrated in Fig. 1.2. It may for instance be a
short term representation of the fluctuating along wind velocity, or the fluctuating
structural displacement response at a certain point along its span. As can be seen, it is
taken for granted that the process may be split into a constant mean and a stationary
fluctuating part. There are two levels of randomness in this process. Firstly, it is random
with respect to the instantaneous value within the short term period between 0 and
T
.
I.e., regarding it as a set of successive individual events rather than a continuous
function, the process observations are stored by two vectors, one containing time
coordinates and another containing the instantaneous recorded values of the process. The
stochastic properties of the process may then be revealed by performing statistical
investigations to the sample vector of recorded values. For the fluctuating part, it is a
general assumption herein that the sample vector of a stochastic process will render a
Gaussian probability distribution as illustrated to the right in the figure. This type of
investigation is in the following labelled time domain statistics.
The second level of randomness pertains to the simple fact that the sample set of
observations shown in Fig. 1.2 is only one particular realisation of the process. I.e. there
is an infinite number of other possible representatives of the process. Each of these may
look similar and have nearly the same statistical properties, but they are random in the
sense that they are never precisely equal to the one singled out in Fig. 1.2. From each of
a particular set of different realisations we may for instance only be interested in the
mean value and the maximum value. Collecting a large number of different realisations
will render a sample set of these values, and thus, statistics may also be performed on the
mean value and the maximum value of the process. This is in the following labelled
ensemble statistics.
In wind engineering
()
X
xxt
=+ may be a representative of the wind velocity
fluctuations in the main flow direction. The time invariant part
x
is then the commonly
known mean wind velocity, given at a certain reference height (e.g. at 10 m) and
increasing with increasing height above the ground, but at this height assumed constant
within a certain area covered by the weather system. The fluctuating part
k
k
k
()
k
x t
represents the turbulence component in the along wind direction. The mean wind
velocity is a typical stochastic variable for which long term ensemble statistics are
applicable, while the turbulence component is a stochastic variable whose statistical
properties are primarily interesting only within a short term time domain window.
Likewise, the relevant structural response quantities, such as displacements and cross
sectional stress resultants, may be regarded as stochastic processes. In the following, it is
to be taken for granted that the calculation of structural response, dynamic or non-
dynamic, are performed within a time window where the load effects are stationary [i.e.
the static (mean) load effects are constant and the dynamic (fluctuating) load effects are
Gaussian with a constant standard deviation].
