  The buckling of spherical shells is plagued by a strong sensitivity to
imperfections. Traditionally, imperfect shells tend to be characterized
empirically by the knockdown factor, the ratio between the measured buckling
strength and the corresponding classic prediction for a perfect shell.
Recently, it has been demonstrated that the knockdown factor of a shell
containing a single imperfection can be predicted when there is detailed a
priori knowledge of the defect geometry. Still, addressing the analogous
problem for a shell containing many defects remains an open question. Here, we
use finite element simulations, which we validate against precision
experiments, to investigate hemispherical shells containing a well-defined
distribution of imperfections. Our goal is to characterize the resulting
knockdown factor statistics. First, we study the buckling of shells containing
only two defects, uncovering nontrivial regimes of interactions that echo
existing findings for cylindrical shells. Then, we construct statistical
ensembles of imperfect shells, whose defect amplitudes are sampled from a
lognormal distribution. We find that a 3-parameter Weibull distribution is an
excellent description for the measured statistics of knockdown factors,
suggesting that shell buckling can be regarded as an extreme-value statistics
phenomenon.
