A modified extreme-value-based methodology is discussed for computing statistical bounds associated with the magnitude of the frequency response of a specified number of structures with high levels of random parameter uncertainty. The methodology, intended for small numbers of uncertain parameters, is capable of constructing accurate statistical bounds in terms of quantiles associated with the extreme value distribution. Quantiles can be constructed for an ensemble of structural responses across the entire frequency range without using Monte Carlo Simulation. To test the methodology, statistical bounds for the energy of an L-shaped structure with low and high levels of uniformly-distributed length and thickness variability are obtained: i) via direct integration using an ANSYS Finite Element model, and ii) via Statistical Energy Analysis (SEA). Comparisons are shown with bounds obtained using Monte Carlo simulation. The merit of direct integration for computing bounds associated with the responses of an ensemble of structures with high levels of random parameter uncertainty is demonstrated by its simplicity, high accuracy, and absence of statistical scatter.

The ability to predict the effect of dimension and thickness variability on the dynamic
response of realistically uncertain engineering structures is examined in this thesis.
Initially, available methods for predicting key response statistics and probabilities, for
both low and high frequencies are examined to establish their strengths and limitations
for specified levels of random dimension variability. For low frequency applications,
the ability of Direct Integration (DI) and the First-Order Reliability Method (FORM) to
predict exceedance probability is examined. For high frequency applications, the ability
of the methods of Statistical Energy Analysis (SEA) and DI to predict the mean and
standard deviation of the energy response is examined.

The use of Extreme Value (EV) theory is included as a way to bound responses using
simulated or measured responses. The strengths and limitations of Monte Carlo
simulation methods are explored for both low and high frequency responses of
randomly uncertain structures using both simple mode superposition plate-structure
solutions and (commercially available) finite element solutions for coupled plate
structures.

To address, without the need to undertake expensive Monte Carlo simulation, the
problem of predicting response bounds for structures with varying levels of uncertainty,
a novel DI-EV method is developed and examined. It is tested first on a simple plate
structure, then on a coupled plate structure, with low-level and high-level random
dimension and thickness uncertainty. In addition, the method is compared with the
SEA-EV method.

The thesis shows that the results from the existing SEA-EV bounding approach gives
good bounds only at particular frequencies and mainly for low levels of dimension
variability. In contrast, the proposed DI-EV bounding approach compare extremely well
with Monte Carlo simulations, which is not only at all frequencies but also with both
low-level and high-level uncertainties, for simple and coupled plate structures with
dimension and thickness variation.

A method is presented for predicting the amplitude bounds associated with the dynamic response of engineering structures with varying levels of parameter uncertainty. In particular, a novel SEA Extreme-Value-based bounding method is discussed and tested across the frequency range on a
non-simple built-up structure with high and low-level random uncertainties. This method does not use expensive Monte-Carlo simulation and is therefore computationally very efficient. The paper shows that for a built-up structure, with both low and high-level uncertainty in a single parameter, the proposed SEA-EV-based prediction method compares extremely well at high frequency with
FE-based Monte-Carlo simulation.