Strength reliability of statically indeterminate heterogeneous beams

The strength reliability of linearly elastic (up to failure) beams, made from random heterogeneous microstructures is studied, based on the weakest link approach. Heterogeneity is confined to the longitudinal direction. The problem is statically indeterminate, and the local stress at each point in any cross section is a function of the stiffness morphology of the whole beam. External loading is not random, but reaction forces are, due to their statistical correlation with the beam morphology. The case of one degree of indeterminacy is studied here, for simplicity. The strength and reliability of the beam, being a stochastic function of local stresses, is therefore morphology dependent, in addition to (coupled with) the classical inherent probabilistic nature, associated with surface defects and irregularities. This dependence is found analytically as a function of external loading shape. A simple design formula for the bound of these effects on the beam strength has been found, covering any possible external loading. For example, for a beam of 10 grains (compliance correlation length of 0.1L) and a 10% compliance variance, the bound of the heterogeneity effect on strength is about 8%.