In this Letter, we investigate the nanomechanical properties of self-assembled 1,3,5-benzenetrisamide whiskers with atomic force microscopy (AFM) bending experiments. We use force mapping to acquire spatially resolved force measurements over the full length of a whisker segment spanning a channel of a structured glass substrate. This allows validation of the experimental boundary conditions directly from the AFM data and a reliable determination of Young's modulus. The presented technique can be generalized for the mechanical characterization of other one-dimensional materials.