A constitutive equation for nonlinear viscoelasticity is used to model the mechanical response of solid polymers such as polycarbonate. The nonlinearity arises from a reduced time which causes stress relaxation to accelerate with increasing strain. The constitutive equation can account for the occurrence of yield in a homogeneous uniaxial constant strain rate test. The constitutive equation is used in a study of the pure bending of beams. It is assumed that the classical assumption of beam theory is valid, i.e., plane sections remains plane. At each fixed time, the strains vary linearly through the depth of the beam. At a fixed material element the strain varies in time with the curvature. This spatial variation of the strains combined with the nonlinear dependence of the reduced time on strain leads to a significantly different response from that given by traditional beam theory. The implications of this for the bending moment history, stress distributions, and other factors that relate to beam design are discussed.