We study the augmented method and augmented-mixed (with reduced symmetry) method for the elastic contact problem under the Signorini and Tresca-friction boundary conditions. For both two methods, we establish the well-posedness for the variational inequalities and the Lagrange multiplier problems in the continuous and discrete senses. We adopt the RT0-element for stress tensor and P1-element for displacement in the finite element discretization, and establish the convergence analysis.