Stochastic analysis of structures with uncertain-but-bounded parameters via improved interval analysis

The stochastic analysis of linear structures, with slight variations of the structural parameters, subjected to
zero-mean Gaussian random excitations is addressed. To this aim, the fluctuating properties, represented
as uncertain-but-bounded parameters, are modeled via interval analysis. In the paper, a novel procedure
for estimating the lower and upper bounds of the second-order statistics of the response is proposed.
The key idea of the method is to adopt a first-order approximation of the random response derived
by properly improving the ordinary interval analysis, based on the philosophy of the so-called affine
arithmetic. Specifically, the random response is split as sum of two aliquots: the midpoint or nominal
solution and a deviation. The latter is approximated by superimposing the responses obtained considering
one uncertain-but-bounded parameter at a time. After some algebra, the sets of first-order ordinary
differential equations ruling the midpoint covariance vector and the deviations due to the uncertain
parameters separately taken are obtained. Once such equations are solved, the region of the response
covariance vector is determined by handy formulas.
To validate the procedure, two structures with uncertain stiffness properties under uniformly
modulated white noise excitation are analyzed.