Abstract: According the principles of the inequality variations,the existence and the uniqueness of the solution have been verified for elastic contact problems,which contain friction forces between the contact surfaces.Thus the contact problems are made to be equivalent to the quardratic programming problems.By using the basic idea of supplement,the objects of a contact system is discreted and then lumped to the contact boundaries,then an resultant stiffness matrix is formed and a linear supplementary equation of non negative variables,υ adn λ,is also derived in the light of the non penetration condition of the normal displacements of the contact surfaces and the Conlumb friction law,etc.By the direct use of the Lemke algorithm of quardratic programming method,the scope of solution is limited to the contact boundaries.The whole field information is therefore obtained from further solution of the subelements.Such an algorithm enables to eliminate effectively the convergence instability which used to appear in the iteration methods in solving the problems.The results of calculations of the lug tooths contact stresses are fitted with the experimental data quite well.