Forcing a curved beam flat against a rigid surface 1

[ggarnier]

Problem: Pre-curved beam, radius of elastic curve constant (r &gt;&gt; thickness), like one leaf of a truck spring.<br><br>Rest concave side up on rigid surface (that is, resting at the center of its span, tangent, on the rigid surface.<br><br>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.)<br><br>What is the downward force, and what is the distribution of reaction force on the underside of the beam as it rolls out flat?<br><br>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 <b>very grateful</b> to have it faxed to 815-371-0649.<br><br>Thanks,<br><br>Gary Garnier<br>Raytek Corp.<br><br><br>

 

[Imagineer]

It would appear to be a simple beam.&nbsp;&nbsp;You know the deflection you want to achieve, i.e., the distance from the beam to the plate.<br><br>For a point the load the formula is:<br>&nbsp;&nbsp;&nbsp;deflection = PL^3/(48EI)<br><br>Solve for load given your deflection.&nbsp;&nbsp;This may or may not get a permament deflection.&nbsp;&nbsp;To achieve permanent deflection you must exceed the yield strength of the steel, probably 36ksi or 50ksi.<br><br>If you trying to permanently flatten the beam.&nbsp;&nbsp;I'd suggest that you send it to a bending company and have them reverse bend it through their rollers.<br><br>End reactions will depend on the kind of support. <p> Imagineer<br><a href=mailto: > </a><br><a href= > </a><br>

 

[ggarnier]

My intent is to use this part as a clamp, not exceeding the elastic limit in use (imagine the &quot;rigid surface&quot; 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 &quot;beam&quot;.<br><br>While your formula (Pl^3/48EI) for deflection applies to a simply supported beam, center-loaded, simply supported, my case is quite different.<br><br>Thanks for taking a look at it.<br><br>Gary Garnier<br><br>

 

[Imagineer]

Now that I've looked at the question again.&nbsp;&nbsp;Maybe you should look at it as a cantilever.&nbsp;&nbsp;Follow me... If you hold the one end down, then the other end will move up the corresponding distance.&nbsp;&nbsp;If you look at it in section, it now looks a lot like a cantilever with a fixed end.<br><br>Forgive the crude drawing but...&nbsp;&nbsp;we start something like this:<br><br>A&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;B<br>&nbsp;&nbsp;&nbsp;(&nbsp;&nbsp;&nbsp;&nbsp;&nbsp<br>&nbsp;&nbsp;&nbsp;&nbsp;(&nbsp;&nbsp;&nbsp<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;( )<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;~<br>--------------- Flat Surface<br><br>Push down end A, we now have:<br><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;~~&nbsp;&nbsp;&nbsp;(sorry no flat curving symbols in ascii)<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;~~&nbsp;&nbsp;&nbsp;<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;~~<br>&nbsp;&nbsp;&nbsp;&nbsp;~~<br>---------------<br><br>&nbsp;&nbsp;Again, we are back to a basic deflection equation, i.e., <br>deflection = PL^3/(3EI)<br><br>Hope this helps,&nbsp;&nbsp;If not, could you restate? <p> Imagineer<br><a href=mailto: > </a><br><a href= > </a><br>

 

[ggarnier]

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?<br><br>Thanks,<br><br>Gary Garnier<br>

 

[Imagineer]

Basically the load will be transmitted out from the point load at about 45 degree angles (depending on the material).&nbsp;&nbsp;The beam will spread transfer the load points to the thickness of the beam.<br><br>/tt<br>Without Beam&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;With Beam<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;¦&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;¦<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;/ \&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;---------------<br>&nbsp;&nbsp;&nbsp;&nbsp;/&nbsp;&nbsp;&nbsp;\&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;¦&nbsp;&nbsp;d&nbsp;&nbsp;¦&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br>&nbsp;&nbsp;&nbsp;/&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;---------------<br>&nbsp;&nbsp;/&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;/&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\<br>&nbsp;/&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;/&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\<br>/&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;/&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\<br><br>d is the thickness of beam (approximately)<br><br> <p> Imagineer<br><a href=mailto: > </a><br><a href= > </a><br>

 

[ggarnier]

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.<br><br>Thanks again.

 

[Imagineer]

No problem.&nbsp;&nbsp;Just curious, what does Roark's say the solution to this is?<br> <p> Imagineer<br><a href=mailto: > </a><br><a href= > </a><br>

 

[ggarnier]

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.

 

[dkengland]

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).&nbsp;&nbsp;&nbsp;

 

[who]

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&quot; 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 &quot;beef&quot; 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.