A set of conditions had not been formulated on the boundary of an elastic continuum since the time of Saint-Venant. This limitation prevented the formulation of a direct stress calculation method in elasticity for a continuum with a displacement boundary condition. The missed condition, referred to as the boundary compatibility condition, is now formulated in polar coordinates. The augmentation of the new condition completes the Beltrami-Michell formulation in polar coordinates. The completed formulation that includes equilibrium equations and a compatibility condition in the field as well as the traction and boundary compatibility condition is derived from the stationary condition of the variational functional of the integrated force method. The new method is illustrated by solving an example of a mixed boundary value problem for mechanical as well as thermal loads. Patnaik, Surya N. and Hopkins, Dale A. Glenn Research Center; NASA Headquarters NASA/TM-2005-213634, E-15117 WBS-22-708-24-05 MICHELL THEOREM; POLAR COORDINATES; PARAMETERIZATION; BOUNDARY CONDITIONS; BOUNDARY VALUE PROBLEMS; EQUILIBRIUM EQUATIONS; LOADS (FORCES); STRESS ANALYSIS; ELASTIC PROPERTIES