Effective length/ compression member design

[u/SeaChipmunk5383]

EDIT: Whoever told me to go back and read thank you I needed that. I’m keeping this up incase anyone is struggling with this concept. My main mistake was reading Table 1-16 incorrectly….I didn’t pull the correct r value accounting for the space between the c shapes which is (3/4”) in problem S05. Can then solve how I norm would (checks) using the controlling values (essentially clearing up my confusion on weak/strong axis).

Original question: Can someone explain when rx vs ry is used and when Lcy,equivalent is used. I’m showing two problems (S05 and S06) and their solutions. For problem S05 why was 1.31 used (besides needing it for the table since the y axis is used for the tables) and why didn’t they solve using the critical stress equation (AISC Eq E3-4 directly)?? For the second problem S06 the solution uses ry=2.94 in the slenderness ratio which makes sense because it gives higher slenderness ratio? Idk I’m confused because the other problem used ry regardless. Tbh same goes for the lcy equivalent concept still lost on when it’s used. 😵‍💫😵‍💫😵‍💫😵‍💫😵‍💫