This thesis takes an interdisciplinary approach to the problem of the aleatory uncertainty manifest in the design of engineering structures that are subject to random loading, with specific application to continuous gust loading on aircraft and wave loading on offshore structures. The main focus is on aircraft gust loading because this is the area in which more significant progress is made. A review of the literature on gust loading is carried out to evaluate the sufficiency of existing methods and the possibility of a unified certification model is discussed. In order to obtain reliable probabilistic design loads using conventional stochastic simulation techniques, a large number of simulations are required to derive probability distributions that have adequately low sampling variability in the area of interest. A novel method, called the Efficient Threshold Upcrossing method, is developed that reduces the required number of simulations by at least 2 orders of magnitude. The method is initially developed for the efficient derivation of short-term offshore structural response statistics and is subsequently applied to the modelling of aircraft response to continuous turbulence. The ETU method was successfully extended to take into account long-term statistics of nonlinear aircraft response and it was shown that reliable design exceedance curves can be obtained by as little as 4% of the computational cost of the conventional method. The current methods for the computation of design loads for nonlinear aircraft are limited to discrete, `1~-~cosine' gust encounters as the continuous turbulence models are only applicable to linear aircraft response. However, the most significant outcome of this thesis is that this is no longer the case, because the ETU method provides a way to calculate nonlinear response statistics in the time domain at a significantly lower computational cost. Mathematical models of a simple offshore structure, and both linear and nonlinear aircraft, are developed and a more robust technique is introduced for simulating patches of continuous turbulence. These models, which have the ability to generate random inputs, are used to derive response probability distributions for each of the test structures. The results obtained by applying the new approaches to these data sets show that they offer a marked improvement in performance.
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