K1.2 refers to local flange bending, it's a check to see if you need to provide stiffeners on a column due to a gusset pulling at the flange. The stiffeners are required according to k1.2 if the flange is thinner than
0.4*(Pbf/Fyc)^1/2
Where Fyc is the column yield stress in ksi, and Pbf is the computed force to the flange times 5/3.
K1.3 is for local web yielding, if you are away from the end of the member by more than the depth of the member, the following applies:
R/(tw*(N+5k) < 0.66*Fy
R is the reaction
tw is the web thickness
N is bearing length
and k is distance from outer face of flange to web toe of fillet.
This is from AISC ASD 9th edition. You can get the latest spec from AISC's website for free. It's updated for strength level design rather than allowable stress methods.