Reaction forces at any point

[georgacus]

Hi,

I'm trying to calculate the reaction forces on a vertical continuous beam (seawater caissons on offshore platforms if anyone's interested). I'm doing this in Excel. So far I've been able to calculate everything up until deflections if the beam was a cantilever but I'm not sure how to calculate the deflections when they're fixed at the supports annotated in the diagram (red arrows). Any ideas on how to go about doing this in Excel?

 

[georgacus]

Sorry I mean how to calculate the reactions, not deflections.

 

[BAretired]

One way is as follows:
Consider the beam simply supported at the upper and lower supports. Calculate deflection Δ at middle support due to the applied load. Calculate end reactions.

Under the action of a unit load, calculate deflection Δ[sub]u[/sub] at middle support due to unit load. Calculate reactions due to unit load.

Calculate reaction at middle support, R[sub]2[/sub] = Δ/Δ[sub]u[/sub].

Adjust end reactions for R[sub]2[/sub].

BA

 

[georgacus]

Thanks for your help. I used your idea but I split my beam (3 spans) into 3 individual beams, fixed at both ends. Then I was easily able to find the reaction forces at each end. Here's the output from STAAD.Pro and Excel. The scales are exaggerated but the actual forces turned out to be roughly the same.

 

[georgacus]

STAAD output

 

[rb1957]

assuming fixed at the middle support in probably unconservative, and fixed at the top ... sounds odd.

you could analyze as a beam, cantilevered at the base with two supports (doubly propped cantilever). This is doubly redundant, but the singly propped cantilever is a well known book solution, so then it's "only" a matter of using displacements of the singly propped cantilever to determine the reaction at the second prop (unit load method). you can solve your structure two ways to check your solution. This solution allows the beam to develop it's "correct" internal moment, as a continuous beam, probably less than your fixed assumption.

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[BAretired]

I suspect that none of your support points is fixed against rotation. The assumption that they are fixed against translation may be wrong too. You have to determine the boundary conditions before the problem can be solved.

BA

 

[georgacus]

The caissons I'm analyzing are fixed at multiple locations along its length using guides. These guides are assumed to fix the caisson in place and I'm modelling it assuming that no rotation is allowed at the guide locations. So according to that, the spreadsheet outputs a similar solution to the software I use for comparison.

 

[BAretired]

Well, if all support points are fixed, you can treat each beam as a fixed end beam and solve it easily. The continuity of the beam is not a consideration.

BA

 

[rb1957]

how does the picture, showing reactions at specific points, line up with the thread title "reactions at any point" ... maybe just an ESL thing ?

I think BA's point is that nothing is truly fixed, but then it seems you're doing a hand calc to verify a FEA ?

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[georgacus]

Yes I'm doing a hand calc to verify the FEA.

I should've been a bit more clear, those three points were arbitrary points and depending on the situation, there can be more or less of them. I just put them there as an example.

 

[rb1957]

that sounds confusing to me, but then i'm not seeing the bigger picture ...

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