The B31.1/B31.3 Codes do not necessarily calculate and evaluate stress as you have learned from principles. Except for the use of SL in calculating SA, these Codes evaluate yield and fatigue loads independent of one another.
Max shear is the radius of Mohr's Circle. Stress Intensity is the diameter of Mohr's Circle. The diameter is easier to come by (S1-S3), so use that number and compare to double the allowable (using Sy rather than Sy/2 - another easy number - Sy). Simplification (for a slide rule).