Multilevel Multiplicative Interval Uncertainty in Linear Structural Analysis

Interval treatments of uncertainty are plagued by dependency issues in their implementation. Ignoring the dependency among interval values can lead to a significant overestimation of the bounds on calculated uncertain responses. However, the clever implementation of operation order and other dependency effects in interval computations can result in models for the uncertainty that provide sharp bounds for multidomain uncertainty dependency in physical systems. In this paper, we explore an interval analysis of structural systems—specifically, systems where one or more groups of structural elements are subject to a global uncertainty in a value of a property and, concomitantly, each element has an additional intragroup uncertainty. Consider a typical scenario where the steel for structural elements is from a single batch (has unknown but identical modulus and strength) while each element has uncertainty in its stiffness from both modulus and geometric variations. To address the relationship between overall group uncertainty and local uncertainty, a product of interval variables is employed. The dependency for element level uncertainty (i.e., area) is addressed by the element-by-element methods of Muhanna and Mullen, while the group dependency is addressed by a new group-by-group method where a single interval value multiplies the entire substructure matrix. Example calculations are presented to compare the results of the new multilevel fixed-point method with conventional interval finite element calculations and with all combinations of endpoints, sensitivity-based methods, and two optimization methods. In general, only the fixed-point methods provide a guaranteed enclosure. Both solution accuracy, as well as computational requirements of the methods, will be assessed.