Slender Concrete Columns Rigidity

[Gus14]

Concrete column buckling load (Pc) decreases significantly as the rigidity (EI) increases. The rigidity (EI) increases as the section stiffness (Ig) increases and as the reinforcement moment of inertia about the centroidal cross section (Ise) increases.

My question is, how to calculate the (Ise). In the reinforced concrete mechanics book by James MacGregor. I found some equations to calculate (Ise) depending on:
1. The column width to length ratio.
2. Number of bars per face.

Any idea how he derived these equations?

 

[Retrograde]

You can calculate the cracked bending stiffness using the method of transformed sections.

There is an example here: http://www.ce.memphis.edu/4135/PDF/Notes/Chap_5_2_flex-trans.pdf

 

[Gus14]

Thank you Retrograde for replying.

Yeah I did it that way. His equations don't address the case where the column length is more than twice the width ( like slender concrete walls ). In that case I can get significant increase in Ise, as the software results agree.

It's just that the increase felt too big and remaining within the 1.4 moment magnification ratio really worries me, so I will try my best to increase the width and not be critical.