A large class of tribological contacts involve two components, each carrying finite roughness, sliding over one another for many repeated cycles of operation. Although the initial contact may be plastic the steady state situation is one of elastic stress fields in each consistent with the presence of a low shear strength boundary film on one or both of the surfaces. In order to relate the overall or macroscopic coefficient of friction to the properties of the film it is necessary, in principle, to have information about the topography of the surfaces and the rheology of the boundary layer. For some specific surface topographies and/or idealised film rheologies this relation may be particularly simple but, in the case of commercial lubricants, the situation is more complex: such lubricants invariably contain a chemically complex additive package aimed at producing protective boundary layers which are physically robust enough to survive prolonged service conditions but slippery enough to maintain low coefficients of friction. In the present study we look quantitatively at the influence of the pressure dependence of the shear strength of a surface layer on the global friction coefficient of a contact which is made up of an array of asperities whose heights vary in a statistical manner representative of real surfaces. The analysis results in plots of coefficient of friction versus a non-dimensional service or load parameter which also includes information on surface topography and the hardness of the softer surface.