3D Elastic Solutions for Laterally Loaded Discs: Generalised Brazilian and Point Load Tests

This paper investigates the application of a double Fourier series technique to the construction of an elastic stress field in a cylindrical bar subject to lateral boundary loads. The lateral loads, including the constant load boundary conditions, are represented by two Fourier series: one on the perimeter of the circular section (r ₀, θ) and the other on the longitudinal curved surface parallel to the bar axis (z). The technique invokes acceptable potential functions of the Papkovich–Neuber displacement field, satisfying the governing partial differential equations, to assign appropriate odd and even trigonometric Fourier terms in cylindrical coordinates (r, θ, z). The generic solution decomposes the problem of interest to a state of stress caused by two independent boundary conditions along the z axis and θ-polar angle, both superimposed on a solution for which these potentials are the product of the trigonometric terms of the independent variables (θ, z). Constants appearing in the resultant second-order partial differential equations are determined from the generally mixed (tractions and/or displacements) boundary conditions. While the solutions are satisfied exactly at the ends of an infinite bar, they are satisfied weakly on average, in the light of Saint Venant’s approximation at the two ends of a finite bar. The application of the proposed analysis is verified against available elastic solutions for axisymmetric and non-axisymmetric engineering problems such as the indirect Brazilian Tensile Strength and Point Load Strength tests.