Bounds of structural response for all possible loading combinations

In this paper, a new treatment of load uncertainties in structural problems based on fuzzy set theory is introduced. A fuzzy finite-element method (FFEM) for treating uncertain loads in static structural problems is developed. First, the problem is discretized, which results in a system of fuzzy algebraic equations. An efficient algorithm for calculating guaranteed inclusions for the solution of such fuzzy systems is implemented. In the case of uncertain loading, the resulting system of equations is linear, and only the right-hand side vector contains fuzzy values. Solutions due to all load combinations, as a special case of load uncertainties, can be obtained by a single computation, leading to a set of possible displacement values corresponding to all loading cases. In conventional analysis, due to the combinatorial nature of the problem, calculating a possible structural response for all possible loading combinations becomes computationally intractable for typical structures. Examples show that the possible interval values calculated using FFEM provide a sharp bound on possible nodal displacements and element forces calculated by combinatorial calculation of all loading conditions. Thus the extreme values of element forces for all loading combinations can be calculated using FFEM. Comparison of the extreme values of element forces with typical design loading schemes show that the current design procedures can lead to nonconservative predictions of element forces. The method therefore is applicable to static analysis of any structural system that can be analyzed using finite-element methods.