This paper addresses the issue of analytically predicting the strength of cross plied composite laminates on the basis of effective, rather than measured, lamina properties. In particular, the typically nonlinear transverse properties are replaced by linear elastic properties with secant moduli selected to predict the correct strength when the transverse strains match the longitudinal strain of the fibers at failure. The basic lamina-to-laminate theory is the classical linear model for predicting Young's moduli and Poisson's ratios. However, the usual one-phase homogeneous model of orthotropic laminae for strength predictions is rejected as inappropriate for fiber-reinforced resin matrices. Failure occurs in one constituent or the other, and not in accordance with any homogenized mathematical theory. The theory developed here is a pseudo-single-phase model in which the fiber dominates under some stress states and the resin matrix may dominate under others. A complete two-phase theory is needed to account properly for matrix-dominated failures and residual thermal stresses. The theory is illustrated with extensive examples of carbon-epoxy cross-plied laminates but, with appropriate different choices of material properties, it can be applied to almost any fiber-reinforced composite.