Short Shaft Buckling Calc for Side Load and Compression

[lancer360]

I'm working on a problem dealing with buckling on a short shaft with combined compression and side loading. The slenderness ratio is only 10. The side loading can be significant (as much as 1:1), but as the compressive load goes up the side load decreases. I haven't been able to find a formula that directly address this issue.

I've been studying up in Roark's and have been playing around with a version of the secant formula (12.4-2, 7th edition) for an eccentrically loaded column. I have been taking the moment generated by the side load and using that to calculate the eccentric distance of the compressive force to generate the same moment. My length is 35 inches so for a particular case of 100 kips of compression and 100 kips of side load I end up with an eccentricity of 35 inches. I have a feeling this is a bit outside the original intent of the equation.

I also have tried the same formula, but using the side force to calculate the deflection in the shaft and then using that deflection as the eccentric distance. I then reduce the maximum allowable compressive stress by the stress generated by the side load. I didn't figure I would get the same answer, but they aren't even close.

Is anyone familiar with the secant method for buckling that could give me a little guidance, or tell me if I going the wrong direction. Any other methods I should look at instead?

 

[rb1957]

i think the key thing about the side load is that it creates a moment in the shaft, presumably the shaft has a cantilevered end. this sounds more like a beam column than a true euler column.

 

[lancer360]

This is a short cylinder shaft approx. 3' long and 14" dia. It is being used to lift a heavy load up. At the end of the shaft is a footing with a roller and a small cylinder is pushing sideways on the foot. Subtracting the roller friction from the sideways cylinder force gives the resultant side force that is transmitted through the vertical lifting cylinder. It is no where close to a Euler column. The critical slenderness ratio is 107 and this cylinder has a ratio of only 10. One of the reasons I was questioning the use of the Secant formula is I couldn't seem to find a reference to what slenderness ratio's it was valid for? Any ideas?

 

[rb1957]

have you drawn a free body diagram of the shaft ?

i thought the secant equation applied to columns with an off-set (axial) load (creating a moment) ... doesn't sound like that's what you have ??

i don't think the secant equation is limited to a slenderness range.

it sounds to me that you're using slenderness ratio as the pass/fail criteria; ie for your load you calculate the critical slenderness ratio given your geometry, you then calc your actual slenderness ratio and show it good. this is the same as calculating the critical load for your column and comparing this to the applied load (which is what i'm more used to seeing). The caution i'd suggest is that it's not particularly clear to me (at least) when plasticity comes into play, and whne you need to consider Johnson-Eular (or some other similar method) dto deal with plasticity.

 

[lancer360]

I am only using the critical slenderness ratio to determine that I am in the short column range not intermediate or long column where Euler would apply. I was 99.9% sure it was a short column, but it never hurts to document it.

Yes, the Secant formula is for a column with an eccentric axial load. This eccentric load generates a moment. What I was trying to do was calculate the moment generated by the side load. Then use that moment to calculate the eccentric distance for the axial load that would generate the same moment in the shaft. It seems like that should work, but I was hoping to get some input from this site from people more familiar with the Secant formula and its application than I am. In essence, I am transposing the transverse load at the end of the shaft to a moment at the end of the rod and a transverse load at the base of the shaft. Then use the secant formula. This does neglect the shear stress at the base of the rod, but it is only around 700psi so it should have negligible affect.

I have the formula set up to calculate P critical with the maximum stress set equal to the material yield strength. Then I will compare the P critical to the actual compressive force.

Thoughts?

 

[rb1957]

i think you have a beam column (a cantilevered beam with a transverse load and a compressive axial load). i think you should analyze as such; but applying the secant equation with e = L*d/P (L=lateral load, d=length of beam, P=compressive load, e=eccentricity) should be ok.

 

[lancer360]

I was crunching the numbers using the cantilevered beam based on Roark's Table 8.8-1a as well as the Secant method and came up with the following:

Load Case 1
Axial Load = 111 kips
Transverse Load = 108 kips
Secant Method = 20.6 ksi
Beam Method = 12.4 ksi

Load Case 2
Axial Load = 615 kips
Transverse Load = 82 kips
Secant Method = 19.3 ksi
Beam Method = 12.9 ksi

I'm inclined to use the beam method numbers, not because they are lower, but the formulas exactly match my load case and with a slenderness ratio that is 1/10 of the critical slenderness ratio I think it is a safe assumption that this shaft would fail by plastic deformation not elastically.

Thoughts?

 

[GregLocock]

I haven't tried it but I'd have thought that if you set up the actual derivation for Euler buckling you could check. The solution may be ugly with your sideforce, but that's what puters are for.

Cheers

Greg Locock

SIGlease see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[lancer360]

Good news! I found a typo in my Mathcad sheet and now the Secant method and the beam method agree within 20 psi! That gives me a fair bit of confidence that I've got the right answer now. Thanks for the help!

 

[rb1957]

IMHO that's just coincidence, particularly as for the eccentrically loaded column (secant method) the critical section is at the mid-span whilst for the beam column the critical section is somewhere else, probably closer to the cantilevered end.

personally i'd go with the beam column, 'cause that's what you've got.