Michell trusses in the presence of microscale material randomness: limitation of optimality

The classical problem of a Michell (optimal) truss concerns a minimum-weigh t design of a planar truss that transmits a given load to a given rigid fou ndation with the requirement that the axial stresses in the bars of the tru ss stay within an allowable range sigma (0) less than or equal to sigma les s than or equal to sigma (0). The present study considers this problem when the truss is made of a material with random microstructure, that is, when co is a random field. The trusses tending to the optimal state can be deter mined through a net of characteristics generalized to a stochastic setting. While in the classical case of a homogeneous material this net gives the m inimum weight as its spacing tends to zero, the presence of a random micros tructure prevents the attainment of this state. Basically, the finer the ne t, the stronger the scatter of characteristics, which forces one to use mor e structural material to compensate for these fluctuations. In effect, ther e is a limitation to the attainment of the optimality of the Michell truss made of a hypothetical perfectly homogeneous material.