Screening currents have long been known to impact the stress state in tape-wound superconducting coils. In recent years the advent of REBCO tape has led to the development of tape-wound coils by a number of organizations. While several groups have been computing screening currents and ac losses in REBCO tape in a variety of applications for several years, little has been published about the stress due to the screening currents. This problem is challenging due to the need to analyze thousands of REBCO turns which are not bonded together. The T-A formulation of Maxwell's equations employing a homogenization technique enables efficient estimation of the current distribution while structural calculations employing contact elements allow conservative estimation of stresses. Computational results are compared with observed degradation in a test coil. Future coil designs that include the effects of screening current strains are proposed.