# Area Moment of Inertia

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1. Jan 20, 2017

### mhajinaw

1. The problem statement, all variables and given/known data
I'm reading a paper and I'm trying to understand how does the author arrived from equation (1) to the following buckling load equation (2). I know that the author substitutes equation (1) with the dimensions of the geometry but I still could not understand how he comes to last term 6e/t in equation (2).

2. Relevant equations
P=Aσmax / [1+ec/r2] (1)

P=Btσy / [1+6e/t] (2)

where;
A=cross-sectional area of column
e=eccentricity
B,t=geometry of specimen (refer to attached figure)
c= distance from neutral axis to the outer fiber where σmax occurs
r, radius of gyration, r2=I/A where I is the moment of inertia computed about the bending axis

3. The attempt at a solution
I tried to play around by substituting r2 with I/A. But I don't know how can I expand the results afterwards. Maybe there are some simplifications that I missed.

2. Jan 24, 2017

### PhanthomJay

The first equation represents loading on a relatively short eccentric/ compression loaded column; the second appears to be that same formula when the max stress reaches yield on a eccentrically loaded rectangular solid column of width B and depth t (found by replacing c with t/2, the distance from section centroid to outer fiber, and by replacing r^2 with I/A, where I is Bt^3/12, and A is Bt). That's the only sense I can make of the 2nd equation.

Last edited: Jan 24, 2017