The horizontal perforated sheet metal plates are commonly used in the process industries as trays in distillation columns, important internal parts for fractionating the input liquid mixture. Normally, the operating performance of such trays is satisfactory. However, cases have been reported of abnormally high levels of tray vibration during operation at particular conditions. The trays then experienced fatigue cracking accompanied by the loosening of bolts and fixings, which led to expensive failures. The excitation of structural resonance was suspected as a component in flow-induced vibration. Using linear stress superposition, a simple but robust analytical method is developed to provide high-quality predictions for the stress and strain distributions for in-plane loaded thin perforated plates with periodic hole arrangements. This approach is built on the classical solution for the elastic stress field around a single circular hole in a large plate. The perforated plates with square penetration patterns are investigated in this article, although the same approach is applicable to any regular penetration pattern. Stress concentration factors as well as the effective elastic constants, which can be used to describe the bending properties of the perforated plates, are then verified against both the established theoretical solutions and the results from finite element simulations. Excellent agreement to both previously published physical experiments and complex modelling is observed in all cases, with small-to-medium (up to 40%) hole-area fraction. The proposed analytical method is much simpler and computationally efficient than finite element analysis. The computed effective elastic constants are used in a finite element modal analysis to estimate the free vibration frequencies of a stiffened distillation column tray example; the first 30 vibration modes are found to be almost uniformly distributed between 25 and 70 Hz, which matches the vibration frequency range reported from plant operations.