Estimation of the resource of a structure with the tensile crack on the basis of fuzzy simulation

Abstract

Statement and solution of the problem. The problems of optimal design of structure, which is operated under the constraints of strength, stability and rigidity are considered. As an additional restriction, the possibility of the crack presence in a stretched element is considered. With a small, compared with the yield strength, destructive stress, the construction element is in critical condition. So that, the calculation of strength can be carried out according to the well-known Irwin criterion of a linear destructive mechanics. An algorithm for solving the optimal design problem using dynamic programming is proposed. A numerical-analytical solution is obtained in the form of recurrent formulas. In addition to the deterministic optimal design solution there is a situation where a tensile crack in the rod of the truss is considered. The size of the crack is not clearly defined. The influence of the fuzzy problem of the initial crack length on the number of load cycles during non-destruction for separate elements of the structure is estimated. The truss consists of four elements in a deterministic formulation. The effect of a predetermined accuracy of calculation on the convergence of the process, when the optimal design problem is solved with using dynamic programming, is estimated. For the case of sufficient predetermined accuracy for the truss, a search was made for the influence of the length of the crack on the cross-sectional area of the rods and the volume of the structure. Numerical experiments of the influence of the fuzzy problem of the crack length for different values of the modal value spread on the volume of the structure are carried out. Conclusion. Convergence of the solution starts when relative error is approximately equal 0.003. Influence of crack length on the volume of rod structure shows its growing after the value of 15 cm in deterministic problem. Investigation shows the principle possibility of the theory of fuzzy sets using in problems of optimal design of hinged-rod systems. As an example, the fuzzy numbers with triangular form of membership function were used in the procedure. The stages such as fuzzification, optimal design and defuzzification let us estimate the result of fuzzy problem. So, the result of the possible value of volume for situation, when modal value of crack length is 25 mm and deviation has different domain, is shown in the article.