We analyze analytically and numerically the stress distribution along the plug-rock interface and within an axially loaded plug emplaced in a borehole in rock. Experiments show that the interface strength increases with decreasing plug radius and with increasing plug length. Axial strength decreases as a power law of plug radius. An increase in the modulus ratio (ratio of plug modulus to rock modulus) increases the interface strength, until it levels off at a ratio of about 5.0. For short plugs, the tensile stresses may reach a magnitude significant to be of concern for long-term stability of the plug and of the host rock. Our studies suggest designing friction plugs with a length to radius ratio of at least 8.0. In practice, especially for short-term performance, e.g., emergency flood control or temporary diversions, a ratio of 8.0 may not be necessary or justified. For permanent abandonment plugs, it is essential to reduce the tensile stresses in the plug and in the host rock to a level that will minimize the risk of long-term deterioration. We recommend in situ experiments on larger diameter plugs to assess the validity of the proposed size effect extrapolation obtained in this study.