Lateral contact of slender prismatic bodies

Many nominally two-dimensional contact problems, such as the contact of parallel rollers or spur gears, exhibit slow axial variation in contact conditions due to manufacturing errors, misalignment or generalized beam deflections of the contacting bodies. In this paper, an asymptotic method is used to develop a solution to such problems, the inner problem being the two-dimensional Hertzian contact theory and the outer problem the bending of the contacting bodies as beams separated by a nonlinear elastic foundation. The method is illustrated in the case of two cylindrical rollers that are slightly misaligned and pressed together by forces applied at their ends. The classical Hertzian elliptical contact area is obtained at low loads, but with increasing load the contact area evolves first into a dog-bone shape and then into two separated contact areas surrounding the point forces that would be predicted by an elementary beam theory solution of the problem.