The axial compressive strength and failure time of unidirectional, viscoelastic composites are investigated. Effects of nonlinear shear behavior and fiber misalignment are emphasized because they are important strength-limiting factors in those strongly anisotropic composites which fail by local buckling in the shear mode of deformation. We first describe the basic buckling model and then, neglecting hereditary effects, predict the compressive strengths of an untoughened and a rubber-toughened carbon/epoxy composite. Next, using a nonlinear viscoelastic constitutive equation for shear behavior, failure time for constant load and compressive strength for increasing load history are predicted by a numerical method. Additionally, approximate analytical formulas are developed which enable one to easily estimate buckling response as a function of initial fiber misalignment angle as well as loading and material parameters.