Within continuum dislocation theory, one-dimensional energy functional of a bent beam, made of a single crystal, is derived. By relaxing the continuously differentiable minimizer of this energy functional, we construct a sequence of piecewise smooth deflections and piecewise constant plastic distortions reducing the energy and exhibiting polygonization. The number of polygons can be estimated by comparing the surface energy of small angle tilt boundaries with the contribution of the gradient terms from the weak minimizer in the bulk energy.