Fuzzy Behavior of Beams on Winkler Foundation

Abstract

This paper addresses and solves the classical beam‐on Winkler‐foundation problem with fuzzy definitions for the Winkler‐spring stiffness, the flexural rigidity of the beam, and the concentrated load. The paper demonstrates through an example the extension of the classical solutions to include fuzzy quantities. Fuzzy equations for deflection, slope, moment, and shear are derived. These are then solved numerically using the vertex method, and the results are shown as fishnet plots. Classical solutions with nonfuzzy values of the parameters are also plotted. These plots show that the curves obtained using extreme values for non‐fuzzy parameters are coincident with the curves formed by the intersection of the horizontal plane at membership equal to zero with the fuzzy behavioral fishnet surface. An index of fuzziness is defined. This is useful in quantitatively describing the propagation of uncertainties in the behavior. The index of fuzziness can be used by the designer to provide adequate margins of safety in regions of maximum fuzziness.