ABSTRACT
Let us denote by x= (x1,x2, . . . ,xn) the structural location (e.g., distance from one support) at which we measure the value of a structural parameter (e.g., the value of the beam cross section A(x)). If we perform such a measurement for different beams (i.e., different realizations of the beam) we will observe that the value of the cross section A at location x will randomly vary from one measurement to the next. In other words, the cross section A(x) at x is a random variable. If we additionally measure the beam cross section A(y) at a second location y= (y1, y2, . . . , yn) we will observe a similar situation, i.e., the measurements will again vary randomly from one realization to the next. The cross section A(y) at y is also a random variable. If we measure the values of the cross sections at both locations x and y we observe, in general, that their values will be different and that this difference will also vary randomly from one realization of the beam to the next. However, we can also observe that values from adjacent loacations do not differ as much as values that are measured at locations
further apart. Such a behavior is an example of a covariance structure, many different types of which may be modeled by random fields. Now let us define more precisely what we mean by random fields. A random field
H(x) is a real-valued random variable whose statistics (mean value, standard deviation, etc.) may be different for each value of x (Matthies, Brenner, Bucher, and Soares 1997; Matthies and Bucher 1999), i.e.,
H ∈ R; x = [x1,x2, . . .xn]T ∈ D ⊂ Rn (5.1) The mean value function is defined as
H¯(x) = E[H(x)] (5.2) whereby the expectation operator E is to be taken at a fixed location x across the ensemble, i.e., over all possible realizations H(x,ω) of the random field (see Fig. 5.2). The spatial correlation, i.e. the fact that we observe a specific dependency structure
of random field values H(x) and H(y) taken at different locations x and y is described by the auto-covariance function
CHH(x, y) = E[{H(x)− H¯(x)}{H(y)− H¯(y)}] (5.3) With respect to the form of the auto-covariance function we can classify the random fields. A random field H(x) is called weakly homogeneous if
H¯(x) = const. ∀x ∈ D; CHH(x,x + ξ) = CHH(ξ) ∀x ∈ D (5.4) This property is equivalent to the stationarity of a random process. If the covariance function depends on the distance only (not on the direction), i.e.
