Has anyone ever seen this formula? 2

[Lion06]

I am looking at some calcs submitted that are showing the deflection of a column with an eccentric axial load (equal end moments). The equation being used is Pe(L^2)/(16EI). I could certainly go through the problem and use Castigliano's method or something similar, but it seems like so much work to verify this. I have looked in teh AISC manual and my mechanics of materials book and I can't find this (either with eccentric axial load or just end moments - assuming this would be virtuallly identical since the axial load contributes little to deflection).
Has anyone ever come across an equation like this or know where I might be able to find this or something similar?

 

[minorchord2000]

You can find it in the PCI Design Handbook. I have the 4th edition where it shows up on page 11-12 under Design Information for camber (deflection) and rotation coefficients for prestress force and loads, case 2, where the load is eccentric to the compression member, where the Moment = Pe and the upward camber is Ml^2/16EI.

 

[csd72]

It looks correct to me.

That is the deflection at mid height.

It matches the formula that I have for an end moment on a beam where M=Pe

csd

 

[Lion06]

Thank you both.
csd- what reference do you have that formula?

 

[prex]

Can't agree with the above replies.
That formula represents indeed the midspan deflection of a simply supported beam with a moment Pe at one end.
However:
1)the maximum deflection is not at midspan in that case, it is at 0.577L and the max.deflection is f=PeL²/15.57EI (of course this doesn't make a big difference, but to be correct...): to check this see Roark 5th ed. Table 3 Case 3e, or go to the first site below under Beams -> Simply supported -> Conc.moment
2)StructuralEIT is calling for a beam (or column) with a constant eccentricity of the load all over the length (equal end moments): in this case the bending moment is constant in the beam, and the maximum deflection is exactly twice, or f=PeL²/8EI; this comes from the superposition of two moments, one at each end, and of course the max.deflection is now at midspan and is two times the deflection with one end moment.

prex

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[COEngineeer]

I agree with prex. simply supported with end moment is :
ML^2/9*(sqrt 3 EI) which is exactly what prex mentioned.

I am a little loss with the second point he made though.

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[Lion06]

Does anyone know if ACI requires a minimum e at the top and bottom of a column? It clearly states that the minimum e for M2 is 0.6+0.03h, but I can't find anything for M1. Can you assume min. e at the top only and no e at the bottom? This doesn't seem right to me, but I don't see anything in ACI regarding this.