Stress field equations for a disk subjected to self-equilibrated arbitrary loads: revisited

Closed-form expressions for stress distribution in a disk subjected to self-equilibrated loads having arbitrary magnitudes and directions have been derived following the approach of Timoshenko. The stress field equations are simplified to a readily usable form by intelligently employing the solution to the problem of a concentrated load acting on a semi-infinite plate (Flamant’s solution) and the solution of Lame’s problem. The proposed equations eliminate a series of involved and avoidable steps to be followed in the existing methods for stress determination, which makes these new equations computationally more efficient. The isochromatic fringes plotted using the proposed equations for disks subjected to various numbers of loads revealed a subtle aspect that the fringe order at the free boundary of a disk is non-zero if the loads are acting in the non-radial direction whereas it is zero for radial loads. This heuristic information can further simplify the determination of contact forces from isochromatic fringes, which is the current focus of many of the researchers working in the field of granular materials. Graphical abstract