Probabilistic buckling of spherical shells: from single to multiple defects

Thin shell structures are pervasive across length scales, from pollen grains to storage tanks. Their high slenderness ratios create pronounced sensitivity to imperfections, traditionally studied through single-defect analyses characterized by knockdown factors κ—the ratio between measured buckling strength and classical predictions for perfect shells. However, realistic shells contain multiple defects whose interactions remain poorly understood. We employ finite element simulations validated by high-precision desktop experiments with tomographic imaging to investigate knockdown factor statistics of elastic hemispherical shells with random defect distributions. For shells with two interacting defects, we uncover an interaction regime governed by the classical buckling wavelength. Extending to multiple defects distributed lognormally, we find that resulting knockdown factors follow a three-parameter Weibull distribution, establishing shell buckling as an extreme value statistics problem. By systematically removing imperfections, we reveal that the most severe defect largely dictates buckling onset—a weakest-link mechanism that enables simplified analysis of complex shells through single-defect descriptions. This probabilistic framework provides fundamental understanding of imperfect shell buckling and opens pathways for designing functional mechanisms exploiting defect interactions, with implications spanning biological structures to aerospace applications.