On occasion, ASTM standard test specimens fail in one of the transition regions between the gauge and grip sections due to the stress concentration present there. Then one can seek stress relief by increasing the fillet radius in the transition regions. The intent here, therefore, is to quantify the stress relief afforded by such means. We use finite elements to determine the stress concentration factors for the various dimensions admitted by ASTM E8/E8M-15a. The finite element results are verified by current verification techniques recommended by ASME, and are accurate to three figures.