Molecular dynamics simulations are used to probe the response of a self-assembled monolayer of hexadecanethiol chemisorbed on a gold(111) surface to compression with a flat surface and a surface with atomic-scale roughness in the form of surface asperities. The film responds elastically with only a uniform change in tilt angle upon compression to low loads with the flat surface. In contrast, our results show that the asperities are able to get closer to the substrate relative to the flat surface at comparable loads without completely disrupting the film and that at higher loads the monolayer film disorders by creating a large number of gauche defects in order to accommodate the asperity. The findings presented in this work support conclusions based on recent scanning tunneling microscopy studies on self-assembled monolayers that imaging occurs at surface asperities which penetrate the film.