This paper attempts to provide a simple analytical solution for the calculation of stress tensors induced in a linear elastic ring of a brittle material subjected to parabolic loading over two symmetric strips of its outer perimeter. Using the Michell- Fourier series technique and Airy stress functions, stress components are investigated in a tensile positive fashion. Results are obtained in series form to facilitate parametric studies on the influence of the loading angle and geometric characteristics of the ring on the maximum tensile stress. Results indicate that for a ring with a small hole, the type of the imposed stress is crucial in determining the distributions of the stress components inside the ring domain, although it becomes almost negligible for a ring with a relatively large hole. In contrast, a large inner to outer ratio influences significantly the stress values at critical points of the ring experiencing the maximum tensile stress.
Stress and displacement analysis of cylindrical annular bodies under arbitrary tractions is one of the classic topics of Elasticity, with its major applications in Machine Design Theory and strength measurement of materials . The ring test has been widely accepted as an indirect laboratory technique for measuring the tensile strength of a material in the shape of a circular solid cylinder (disc) containing concentric holes subjected to lateral compression until a tensile failure occurs. The growing application of the ring test in geotechnical engineering, particularly in rock mechanics, is directly tied to the ease of sample preparation and its unique breakage mechanism in a pure tensile mode. There is a vast body of literature devoted to calculating the induced stresses in the ring test aimed at producing relationships for estimating the tensile strength of a test material as functions of the ring’s geometrical aspect ratio and the imposed contact conditions. Timoshenko , Filon , Ripperger and Davis , Bortz and Lund , Hobbs , Jaeger and Hoskins , Chianese and Erdlac , and Kourkoulis and Markides , among others, have proposed analytical solutions for the ring problem. However, they are often limited to simplified stress assumptions at the contact, e.g. line-forces or uniform radial stresses, or Muskhelishvili's complex potentials  to account for the complicated boundary conditions, e.g. parabolic contact loadings. Despite the generality of complex-variable schemes; however, such approaches require excessive and complex mathematical rigour, limiting their application to broader disciplines. In contrast, focusing on the analytical-methodological aspect, this study attempts to construct a simpler and more convenient easy-to-use treatment for the study of stresses when the ring is subjected to parabolic radial compressive stresses over two finite arcs of its lateral surface (Figures 1 and 2).