The tensile strength of carbon/epoxy composite rods used in tension-leg platforms, prestressed concrete, and post-tensioned concrete, for example, is highly sensitive to the method of load introduction to the ends of the rods by grips or anchors. The present paper analyzes the stresses imparted by the grips to a round-sectioned carbon/epoxy rod by various distributions of shear traction applied to the sides of the rod. The stresses were obtained using the linear theory of elasticity of anisotropic body and a series solution. It was found that severe concentrations of longitudinal and hoop stresses can exist in the rod near the loaded edges of the grips. Due to the singular nature of stresses at the grip edge, it is suggested to use an average stress criterion to estimate the tensile strength of the gripped rod. By applying the average stress criterion to a gripped intermediate modulus carbon/epoxy rod, a clear inverse relationship is shown between the tensile strength and the diameter of the rod. This relationship is independent of any size-dependent variation in inherent material strength. By changing the distribution of the applied shear tractions, the high stress concentrations can be mitigated, resulting in higher strength composite rods.