Buckling Load Equation for Eccentrically Loaded Rectangular Solid Column

[mhajinaw]

Homework Statement

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).

Homework Equations

P=Aσ_max / [1+ec/r²] (1)

P=Btσ_y / [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 σ_max occurs
r, radius of gyration, r²=I/A where I is the moment of inertia computed about the bending axis

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.

 

[PhanthomJay]

Homework Statement

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).
View attachment 111853

Homework Equations

P=Aσ_max / [1+ec/r²] (1)

P=Btσ_y / [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 σ_max occurs
r, radius of gyration, r²=I/A where I is the moment of inertia computed about the bending axis

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.

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.