Utilizing high-resolution transmission electron microscopy, in tandem with molecular dynamics simulations, we conduct a fundamental study of the mechanical force-structure relationship of carbon nanotubes. We study the response of tubes to asymmetrical radial compressive forces, which are directly related to observed structure distortions in HRTEM (high-resolution transmission electron microscopy). We show that the magnitude of such forces can be readily estimated and is related to the shape of the distortion, characterized by percent compression and the radius of the tube, as well as the number of layers for very narrow tubes. The elasticity and resilience of the walls depend on the tube radius and the number of layers. We further suggest that even very large forces produce reversible, elastic distortions, indicating that radial mechanical forces should not be a feasible method to cut a nanotube without causing severe structural damage.