A new method of predicting the strength of cross-plied fibrous composite laminates is based on expressing the classical maximum-shear-stress failure criterion for ductile metals in terms of strains. Starting with such a formulation for classical isotropic materials, the derivation is extended to orthotropic materials having a longitudinal axis of symmetry to represent the fibers in a unidirectional composite lamina. The only modification needed to represent those same fibers with properties normalized to the lamina rather than fiber is a change in axial modulus. A mirror image is added to the strain-based “lamina” failure criterion for fiber-dominated failures to reflect the cutoffs caused by the presence of orthogonal fibers. It is found that the combined failure envelope is now almost identical with the well-known maximum-strain failure model in the tension-tension and compression-compression quadrants, but is truncated in the shear quadrants. The new predictions refer only to fiber-dominated failures of efficient, well-interspersed, laminate patterns. This restriction permits the laminate strengths to be defined by only half as many material properties as were used in the past. Premature matrix failures are excluded from consideration for conventional fiber/polymer composites. The new failure envelope is compared with published biaxial test data.