Stresses due to frictional sliding contact between a rigid cylinder and laminated glass fiber composites are calculated in this study. A novel analytical formulation based on Cholesky decomposition, Fourier transforms, and singular integral equation is presented to solve the stress and displacement fields both at the contact patch and sub-surface, which can take into account transition of laminae with different fiber angle. An augmented finite element algorithm having an adaptive mesh refinement algorithm is also developed to verify the results of the analytical formulation. A perfect match between these results reveals the success of the new analytical formulation. Then, the formulation is implemented for various loading types and composite-related parameters to reveal the effects of those on the surface and sub-surface stresses. The results and discussions presented can be beneficial for the structural design of laminated composites under severe contact conditions.