During the disassembly phase of the refurbishment of the National Institute of Standards and Technology 4.45 MN Deadweight Machine, workers confirmed previously suspected damage in certain conical contact joints within the apparatus. A finite element analysis of the contact joints is performed with the aim of understanding some of the potential origins of failure in order to prevent future damage in the machine. Parametric studies are performed using a finite element model while varying geometry, material constitutive relations, and frictional forces at the interface of the convex and concave cones. The numerical solutions treat nonlinearities originating from elastic-plastic material relations, frictional forces, and contact between the surfaces. These results provide insights into the mechanisms of stress generation in such joints, and how these stresses are influenced by different structural parameters.