We investigate the morphology and mechanics of a naturally curved elastic
arch loaded at its center and frictionally supported at both ends on a flat,
rigid substrate. Through systematic numerical simulations, we classify the
observed behaviors of the arch into three distinct types of configurations in
terms of the arch geometry and the coefficient of static friction with the
substrate. A linear theory is developed based on a planar elastica model
combined with Amontons-Coulomb's frictional law, which quantitatively explains
the numerically constructed phase diagram. The snapping transition of a loaded
arch in a sufficiently large indentation regime, which involves a discontinuous
force jump, is numerically observed. The proposed model problem allows a fully
analytical investigation and demonstrates a rich variety of mechanical
behaviors owing to the interplay between elasticity, geometry, and friction.
This study provides a basis for understanding more common but complex systems,
such as a cylindrical shell subjected to a concentrated load and simultaneously
supported by frictional contact with surrounding objects