Natural Frequencies of the Moderately Thick Plate Models with Uniformly Distrubuted Mass

A four-node finite element is developed for modeling plates according to the Mindlin plate theory and it is constructed with the assumed shear strain approach. The element is previously verified in a static analysis on the benchmark problems of moderately thick and extremely thin plate models and compared to the other elements known from the literature. As starting interpolations, a complete cubic polynomial for the transverse displacement field and quadratic polynomials for the two rotation fields are used, and they are problem dependent at the same time. Some unfavorable terms are excluded from the derived shear strain expression to avoid locking phenomena in the thin geometry conditions. In this paper, the proposed element is tested for the dynamic analysis calculating the natural frequencies of plate vibrations with the uniformly distributed mass. The influence of the element consistent mass matrix is analyzed on the first 12 vibration modes. The results are verified on the circular plate model and compared to the existing analytical solutions as well as the results of other four-node elements from the literature. The goal of this paper is to demonstrate the efficiency of the proposed assumed strain element also in the dynamic analysis of plane structures.