Fuzzy sets application in the problems of structural mechanics and optimal design

Fuzzy sets approaches to the consideration of optimal design of the elastic hinge-rod system under conditions of uncertainty was proposed. The restrictions on stability and strength were applied, as well as restrictions on the natural frequency. The considered optimal design model in mathematical programming belongs to the class of so-called "resource distribution" problems, to which the dynamic programming method was adapted. A computational algorithm of step by step approach, based on the use of the dynamic programming method, is proposed and its convergence is shown for the truss optimization problem with deterministic data. For the case of uncertainty in setting information about the load and natural frequency, the fuzzy formulation of the optimal design problem was performed. Numerical illustration of the search for the optimal truss volume includes such steps of fuzzy modeling as: operation of fuzzification of initial fuzzy values; carrying out calculations related to the solution of the problem of optimization with deterministic data of the level of trust. Constructing a fuzzy result set; defuzzification of the obtained fuzzy results (expected value of volume). The use of fuzzy random uncertainty approaches for the hinge-rod system was demonstrated. This involved modeling situations where loads were randomly attached and their magnitudes were represented as fuzzy values with triangular membership functions. Additionally, the application of fuzzy analysis approaches was demonstrated using a simple beam as an example.