Flexible matrix composites are a class of fiber-reinforced polymers characterized by a low modulus of elasticity and high ultimate strain of the matrix material. Such composites are attractive for power transmission shafts, which are commonly made via processes that cause undulation (waviness) along the path of the reinforcement fibers, such as filament winding and braiding. Fiber undulations can be expected to reduce the in situ modulus and strength of the composite material in the fiber direction. The reported investigation proposes and evaluates a method for determining the effective in situ properties of the plies in filament wound tubes so that classical lamination theory (CLT) can be used to calculate effective ply-level stresses and to predict the overall modulus and strength of tubes loaded in axial compression. An experimental method is proposed to back-calculate the undulation-influenced ply properties from representative filament wound tubes using CLT together with other required ply properties determined via simpler conventional tests. This approach, along with an interactive failure criterion proposed to predict fiber microbuckling in the presence of combined compression and shear on the fibers, is able to accurately predict the axial compressive modulus and strength of a variety of tubes made with different winding angles and matrix moduli. In general, the fiber-direction compressive strength of the composites increased with increasing matrix modulus and decreased in the presence of undulation. The reduction in strength due to undulation was more apparent with increasing matrix modulus. The fiber-direction modulus of elasticity was not very sensitive to matrix modulus in undulated composites. Undulation reduced the fiber-direction modulus significantly relative to unidirectional composites, although the percent reduction could not be correlated with matrix modulus.