Interval Algebra to Deal with Pattern Loading and Structural Uncertainties

Structural and loading uncertainties, bounded from above and below, are considered within a finite-element formulation to determine conservative bounds for the displacement and force response quantities. Discretization of a continuum with material uncertainties is illustrated using a linear elastic beam. This yields the elements of the stiffness matrix with uncertainties and the components of the force vector with uncertainties, to be defined in bounded intervals. Then, the response quantities become uncertain, yet bounded, in a multidimensional rectangular prism. The discretized linear static interval equation is solved using the triangle inequality and linear programming to determine the conservative bounds for the response quantities. For the case when only loading uncertainties are considered, the problem reduces to the pattern loading problem of structural design. The proposed formulation is applied to the structural analysis of frames with material uncertainty under static loads with uncertainties.