Effect of strength-modulus correlation on reliability of randomly heterogeneous beams

The strength reliability of linearly elastic, brittle, stochastically heterogeneous beams, is studied on the basis of the weakest link approach. The analysis is formulated by a functional perturbation method, resulting in an analytical solution of the failure probability of the beam. Heterogeneity (material morphology) is random and confined to longitudinal direction only, under Bernoulli assumptions. The problem is statically indeterminate and external loads are not random. The stress field is random and functionally dependent on morphology. In particular, local strength is also a function of modulus. Therefore, the strength reliability of the beam is morphology dependent, both through static indeterminacy and local strength-modulus correlation. The above is also coupled with the probabilistic nature of strength, associated with surface defects and irregularities. The case of single indeterminacy (clamped––simply supported beam) is investigated, for simplicity. It is shown that the strength of the beam is significantly affected by material morphology and that the effect can be either positive (increased strength) or negative, depending on the strength-moduli correlation. For example, for an effective grain size of L/10, and a compliance statistical variance of 1/12, the morphology effect on the allowable design load, was found to be in the order of 10%. Calculation of size effect, corresponding to strength, showed a complex, non-classical grain size dependency.