An operational method is used to obtain known formulas  for the expected values of the mean squares and the variances of estimates of variance components obtained from the analysis of variance of an $r$-way crossed classification. The results are independent of normality assumptions.
"An Operational Approach to the $r$-Way Crossed Classification." Ann. Math. Statist. 31 (1) 16 - 22, March, 1960. https://doi.org/10.1214/aoms/1177705983