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Prediction of response variability of structures due to uncertain parameters such as material properties and loads is a significant concern in practical engineering. As a non-probabilistic uncertainty analysis approach, the interval methods aim to find the upper and lower response bounds of structures where the uncertainties are described based on intervals, including convex models for correlated interval variables, interval processes and interval fields for dynamic and spatial uncertainties. In many cases, the structural response is a variable regarding physical parameters such as location, node, time, or other degrees of freedom. Different kinds of interval methods have been proposed and developed for response bounds analysis, mainly including interval arithmetic-based methods, optimization methods, perturbation methods, etc. Besides, the simulation methods are also frequently performed to verify the results of the other interval methods. But different from the situation in probabilistic problems, the current simulation strategy for interval uncertainty analysis is still far from a practical and credible approach. When the upper and lower bounds rather than the statistical moments or failure probability are concerned, the direct simulation strategy is much more inefficient because most of the input samples contribute nothing to the resulting structural response bounds. Therefore, to improve the efficiency of the simulation method, this work proposes a sequential simulation method for response bounds analysis of structures with interval uncertainties, through which we aim to find the upper and lower bounds by implementing the simulation procedure sequentially. The proposed sequential simulation strategy suggests an initial sampling and simulation, from which those contributive samples that result in the current upper or lower response bound are retained. In the subsequent simulation sequence, local samples within neighborhoods of the present contributive samples as well as a set of global samples are generated for the input interval uncertainties. With such a simulation procedure, the response bounds are expected to expand outwards sequentially until convergence. By illustration and comparison, the proposed sequential simulation method proves to be much more efficient than the direct simulation method, because it omits a large amount of unnecessary computation that makes little or no contribution to the output response bounds.