Bending a beam from straight to an arc with axial force

[Oseania]

Hi,

If I have a straight beam and I start to push its ends towards each other with force F, the beam will obviously bend into an arc. What is the height/amplitude at the center of the arc when the beam length is l and the beam ends have been displaced by d. How much force (P) the center part can withstand before the beam starts to become straight again.

 

[Mech_Engineer]

What research have you done so far? This sounds a bit like homework, and we need to know more about your background before we can answer appropriately.

 

[Oseania]

Well, definitely not a homework. Haven’t done those in a decade. I stumbled into this problem in one design where a tall coil spring was used in a very space limited application. I started to wonder if the coil spring could be replaced with a thin metal sheet which would be bent into an arc by an external force. Mechanical engineering is not exactly my field and it has been some quite some time since I last studied strenght of materials etc. At the moment I don’t have my old books from which to check so any help into the right direction would be really helpful.

 

[Mech_Engineer]

Generally speaking what you're describing sounds like a beam bending calculation, albeit with the added complication of both axial and transverse loads.

How familiar are you with beam bending calculations/theory? See here: https://en.wikipedia.org/wiki/Bending

Edit: You can also look here: MIT Open Courseware Solid Mechanics, Bending

 

[Dr.D]

Sounds like the classical problem of the elastica. Are you up to speed on elliptic integrals?

 

[jrmichler]

Are you trying to permanently bend the beam, or do you want to use it as a spring? Either way, we need some dimensions and/or forces. And a diagram is always helpful.

 

[Oseania]

Hi,

Here is a diagram of the problem. The steel plate could be made from some spring steel grade, eg. SAE 1080. The idea is that the plate functions as a spring so we have a repetitive cycle from straight to arc and back again. Of course in real life the plate could be pre-bent just a bit, so that the plate would always bend into the correct direction. From mathematical perspective it is probably easier to analyze the movement from straight-to-arc.

 

[jrmichler]

You have an Euler column. Search the term to find the equation. Euler columns can work very well in applications where you want a rigid support that deflects at a particular load. Some things to watch for:

1) If the load is applied suddenly, the inertia of the column will (for a few milliseconds) act to stiffen the column.
2) You need to check that you do not overstress the column at peak deflection. Just run a test where you deflect it twice too far, and see if it comes back to straight.
3) A slight bevel on the ends will cause it to buckle in the desired direction. Do not prebend it.

The equation really does work. I once used some carbon fiber rods as a combination linear pushrod and Euler column overload device. They deflected at exactly the calculated load.