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Forcing a curved beam flat against a rigid surface

Forcing a curved beam flat against a rigid surface

Forcing a curved beam flat against a rigid surface

(OP)
Problem: Pre-curved beam, radius of elastic curve constant (r >> thickness), like one leaf of a truck spring.

Rest concave side up on rigid surface (that is, resting at the center of its span, tangent, on the rigid surface.

Apply enough vertical force at the ends of the beam to just bring the beam flat against the rigid surface. (Ends unrestrained in direction of beam's length.)

What is the downward force, and what is the distribution of reaction force on the underside of the beam as it rolls out flat?

I'm told that Roark does not cover this case, but I don't have a copy, so I can't say first hand. If Roark or any reference or text has the problem worked out, I'd be very grateful to have it faxed to 815-371-0649.

Thanks,

Gary Garnier
Raytek Corp.


RE: Forcing a curved beam flat against a rigid surface

It would appear to be a simple beam.  You know the deflection you want to achieve, i.e., the distance from the beam to the plate.

For a point the load the formula is:
   deflection = PL^3/(48EI)

Solve for load given your deflection.  This may or may not get a permament deflection.  To achieve permanent deflection you must exceed the yield strength of the steel, probably 36ksi or 50ksi.

If you trying to permanently flatten the beam.  I'd suggest that you send it to a bending company and have them reverse bend it through their rollers.

End reactions will depend on the kind of support.

Imagineer


RE: Forcing a curved beam flat against a rigid surface

(OP)
My intent is to use this part as a clamp, not exceeding the elastic limit in use (imagine the "rigid surface" as the part being clamped). The preformed radius is intended to concentrate more of the reaction force in the center of the span of this "beam".

While your formula (Pl^3/48EI) for deflection applies to a simply supported beam, center-loaded, simply supported, my case is quite different.

Thanks for taking a look at it.

Gary Garnier

RE: Forcing a curved beam flat against a rigid surface

Now that I've looked at the question again.  Maybe you should look at it as a cantilever.  Follow me... If you hold the one end down, then the other end will move up the corresponding distance.  If you look at it in section, it now looks a lot like a cantilever with a fixed end.

Forgive the crude drawing but...  we start something like this:

A          B
   (     )
    (   )
     ( )
      ~
--------------- Flat Surface

Push down end A, we now have:

          ~~   (sorry no flat curving symbols in ascii)
        ~~   
      ~~
    ~~
---------------

  Again, we are back to a basic deflection equation, i.e.,
deflection = PL^3/(3EI)

Hope this helps,  If not, could you restate?

Imagineer


RE: Forcing a curved beam flat against a rigid surface

(OP)
Now you have it. No need to restate, just to add that I need to know the distribution of forces along the underside of the beam as well as the two forces at A & B. Intuition tells me that it will not be uniform, but if not, then what?

Thanks,

Gary Garnier

RE: Forcing a curved beam flat against a rigid surface

Basically the load will be transmitted out from the point load at about 45 degree angles (depending on the material).  The beam will spread transfer the load points to the thickness of the beam.

/tt
Without Beam                         With Beam
      ¦                                  ¦
     / \                           ---------------
    /   \                             ¦  d  ¦     
   /     \                         ---------------
  /       \                         /        \
 /         \                       /          \
/           \                     /            \

d is the thickness of beam (approximately)

Imagineer


RE: Forcing a curved beam flat against a rigid surface

(OP)
Thanks for the replies on this. Most helpful was a fax (from Australia!) with pages from Roark - while it is true that the exact case is not solved in Roark, there are equations I can use, and most importantly, I decided I should get my own copy.

Thanks again.

RE: Forcing a curved beam flat against a rigid surface

No problem.  Just curious, what does Roark's say the solution to this is?

Imagineer


RE: Forcing a curved beam flat against a rigid surface

(OP)
I haven't worked through the equations for this (just got the fax today. Roark doesn't address this directly, but it does deal with a curved beam having a non-uniform load (2nd order) along its length - by playing with coefficients, I hope I can come up with a loaded condition close to what my design calls for.

RE: Forcing a curved beam flat against a rigid surface

You have probably solved this problem by now, but use Roark with caution. I have'nt looked at Roark yet, but many beam equations are based on the assumption of small deflections, i.e., The angle between sucessive tangents on the deflection curve, a = dy/dx. For slender beams in which deflections may be large, one may have to resort to equations based on the exact expression, a = atan (dy/dx).   

RE: Forcing a curved beam flat against a rigid surface

I am working on a similar problem and hoping you would be able to help.

I am also considering using a pre-bowed bar to push something against a relatively flat surface.  It is a ~12" length bar mounted on both ends.  I calculated the deflection for a STRAIGHT bar under an uniformly distributed load, and I know the deflection will be too much.  Since I have no room to "beef" up the bar, I am thinking of using a pre-bowed bar.  I would like to know the maximum deflection of the pre-bowed bar at the center (the weakest point) under the same load.  Ideally the deflection will be equal to or less than the pre-bowed amount.

Any suggestions as to how to approach this problem?

Thanks.

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