Flexible Beam Tensile Force 1

[YoungTurk]

Hello All,

Here's a simple beam problem that just doesn't sit right with me. I have a thin, flexible beam, a strap really, fixed in the center and loaded on each end. The ends are restrained such that the end to end distance can be idealized as fixed.

Analyzing the beam as "perfectly flexible" per Roark table 8.10 case 5 gives tremendous tension in my "cable" relative to the transverse load. Since in reality my strap is riveted at the ends, this results in very high analytical shear stress in my rivets (33X tranverse load in this case).

On the other hand, pure beam flexure theory gives extraordinary deflection and doesn't allow one to derive axial loads (and hence rivet shear) at the beam ends.

My mentor has implemented the flexible beam theory, but the rivets required just seem excessive. What, if anything, am I missing? Is there a generally accepted solution which captures both the tensile and bending behaviors?

I have discussed this one a bit around work and looked through a few handy references to no avail.

 

[rb1957]

beam theory doesn't answer your problem (as you've found out) 'cause it doesn't allow the beam to generate endload.

membrane theory allows the beam to react the applied transverse load by tension in the beam (like a pressure vessel reacts the applied transverse pressure with in-plnae tension).

looking at the picture in Roark, and reading your post, you've flipped the beam around and are appling P/2 at each end and reacting that in the middle (which is ok). it means that the ends of your beam are in guides, allowing them to deflect in one direction and react load in the other. The vector sum of these two forces (the applied load and the transverse reaction) define the angle of your beam (its a three force body, so all the forces intersect at a point, in this case the middle support). the magnitude of this transverse force can be determined by energy principles (least work).

good luck

 

[MikeHalloran]

Do you not believe the force developed is real?

I have on a couple of occasions pulled a car out of a ditch by myself using a similar setup. Okay, the whole car wasn't in the ditch; just the two wheels at one end. It's what happens when you try to do a K-turn on a narrow road with ditched margins, and you're not as good as you ... never were.

Lucky for me, on both occasions there was a sturdy tree on the opposite side of the road, and I had enough rope to reach from the car to the tree.

The technique required is simplicity itself:
1 - Tie the rope to the car.
2 - Carry the other end of the rope to the tree, wrap it around the tree, retighten it a couple times, then tie it off. The objective is to get the rope as tight as possible.
3 - Then walk to the middle of the road, and lift the rope midway between the tree and the car.

It may be necessary to repeat 2 and 3 a few times; the mechanical advantage available is tremendous, but falls off fast, and doesn't move the car very far. A rock can be positioned to fall in behind a tire and chock the car as you inch it out of the ditch.

It's not a beam problem, it's a trig problem. Figure out how much axial force is required to suspend a mass at the center of a cable with some arbitrary large deflection. Then figure the axial force required to suspend the same mass at half the deflection, and repeat, until the cable approaches perfect straightness.
















Mike Halloran
Pembroke Pines, FL, USA

 

[SWComposites]

The ends of the riveted strap are not close to fixed; all mechanically fastened connections have a fair amount of flexibility. To accurately model the strap, the rivets should be idealized as springs connected to rigid ground. See the various posts on this forum for fastener flexibility equations. Further, significant strap displacement probably puts you into the realm of large displacements, and thus simply (small displacemnet) beam theory is not correct.

 

[rb1957]

i don't think it'd matter much how fixed the ends are (not disputing your observation SW) ... it's just that the beam is so flexible that i don't think it'll support any significant moment, and will deform like a cable.

actually, maybe the end condition might be the critical section, perhaps depending on how sharp a bend the support makes the beam adopt (that english sounds backwards, but i'm not a writer) ... consider the ends of the beam being supported by a sharp cornered block or by a block with a radius corner

 

[YoungTurk]

Thank you all for your insights.

Mike and rb both show that based on my initial assumptions the cable analysis provides the correct value, though I'm still curious if its possible to couple the beam and cable solutions...

SW, however, points out a key flaw in my assumptions. The Roark equations I quoted use the EA term for axial stiffness. Using Huth to calculate the shear stiffness of the riveted ends reveals that they are significantly less stiff than the axial stiffness of the strap.

Assuming a series of springs model to compute a substitute for the axial stiffness term in Roark, my axial load drops by a factor of about 3. Iterated to determine an appropriate number of rivets, and the number of rivets required drops to a much more realistic figure.

More specifically, I substitute the following equation for the (EA) term in Roark:

k=(1/EA+1/"rivet shear stiffness")^(-1)

Where my rivet shear stiffness is from Huth, "Influence of Fastener Flexibility on the Prediction of Fastener and Fatigue Life for Multiple-Row Joints", ASTM STP 927, pg. 235

What do you think?

 

[rb1957]

?

the problem is statically determinate, therefore the solution of the forces isn't stiffness based

though the solution of deflection would be,
but then the deflection of the cable would depend on it's stiffness + the deflection of the supports;
i'd calculate the two effects separately, just so i could see them individually and also if there was a problem with the answer i'd have a better chance of improving the solution (i'd know which term is dominant)

you can combine the beam and cable results ... it's just that the beam absorbs very little energy bending (compared with the cable stretching) that it's probably just a 2nd order effect.

 

[YoungTurk]

"the problem is statically determinate, therefore the solution of the forces isn't stiffness based"

I'd have to disagree with you here. Since the flexible beam/cable solution I'm using isn't based on rigid body assumptions, the geometry is determined by the deflection, which is determined by the axial stiffness. A rubber band would obviously deflect much further and hence the horizontal reactions would be much lower.

I am looking at the supports, however, as rigid; so the deflection there is zero. My new solution just looks at the flexibility of the fasteners, not the material the supports are through. I consider this conservative with respect to the load in the strap since any support deflection will reduce the tensile load in the strap.

 

[YoungTurk]

Woops... the EA isn't a traditional stiffness term (that would be L/EA), so my parallel springs solution doesn't apply. I think I'm still on the right track but without seeing a derivation of Roark's formula, its not as easily adapted to include the rivet shear stiffnes as I'd hoped. Has anybody got an idea of where I'd find a derivation of said formula? Perhaps Mike has given me a good hint, but I'm a bit rusty on limits...

I'll answer my other question while I'm at it. The coupled tensile/bending solution is given by case one in the table I listed initially. Its nonlinear, so it is most easily solved iteratively (still love my TI-85). That solution only reduces the tensile load by about 10%, though.