New Formulas Improve E-Modulus Calculations: Disc Shaped Bodies and Rectangular or Square Bars

Authors: A. Decneut

Abstract

Development work on the GRINDO-SONIC test equipment has produced a simple and accurate means for measuring the fundamental frequency of grinding wheels and honing sticks. This has long been recognised suitable for expressing the relative hardness of similarly shaped bodies. Absolute grading, however, requires that Young’s modulus of the bulk material be calculated. Timoshenko has set forth a basic relation between the natural frequency and the E-modulus of circular plates. The validity of his formula is limited to very thin discs due to the fact that shear forces were not taken into account. McMaster corrected this formula for perforated discs with relatively small central holes. Five years of practical experience with the GRINDO-SONIC revealed the need for a wider field of application. To cover the gap, experiments have been performed on a large number of bodies with widely differing shapes. The results show the unquestionable evidence of cross-influence between hole size and disc thickness, which was neglected by McMaster. A mathematical treatment of the experimental data applying a multiple regression technique allowed the computation of a shape factor P(δ,β) for hole-to-diameter ratios ranging from 0.0 to 0.8 and for thickness-to-diameter ratios up to 30%. For practical purposes, form-factor tables have been edited, together with formulas, greatly simplifying the calculation of the modulus of elasticity.