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Cantilevered tapered rod

Cantilevered tapered rod

Cantilevered tapered rod

(OP)
I want to calculate the deflection of a cantilevered tapered rod with an applied point load P acting perpendicular to the length of the rod at the free end. Let's say at the supported end r = R and at the free end r = r.

I'm not so familiar with this stuff... I have this for a cylindrical rod.

v (deflection) = - (P*L^3) / (3*E*I)

I = 1/4*(pi)*r^4

P = const.
L = const.
E = const.
I = variable from r = r to r = R

Help please!

RE: Cantilevered tapered rod

There is not a closed form analytical solution for tapered sections, in general. I solve them using a piecewise linear approach based on M/I=E/R, for small slices all the way along the beam. this has the advantage that I can vary in any way you like.

FEA is an alternative of course.


 

Cheers

Greg Locock

I rarely exceed 1.79 x 10^12 furlongs per fortnight

RE: Cantilevered tapered rod

You can solve this.  Look at Castigliano's Theorem.  I'll post more when I get into the office.

RE: Cantilevered tapered rod

Setup deflection formula, perform intergration from free end towards support.

RE: Cantilevered tapered rod

i disagree with greg,

I = i(x) ... what that means is that moment of inertia of the beam is a function of x.  you've got M = m(x) (the moment in the beam as a function of x, slope = int(M/EI) dx = 1/E*int(m(x)/i(x)) dx; and displacement = 1/E int(int(m(x)/i(x)dx)dx) ... clear as mud.

if you just looking for the maximum value, that's probably in reference books, like Roark (maybe wiki, maybe xcalc.com).

if you're after the solution, this gets messy Very quickly, so i often resort to greg's solution (piecewise linear approximations)

if you're after the solution for a student problem, you shouldn't be posting here ...

RE: Cantilevered tapered rod

I is function of r, and M & r are function of x (in direction of length). Messy intergration. "Strength of Material" text will show you how to setup.

RE: Cantilevered tapered rod

Ok, I have a little more time to write now.  

I disagree with Greg that there isn't a closed-form solution to the problem.  As long as you have a I as a function of x, then it's no problem.

I prefer to use Castigliano's Theorem.  It relates the internal strain energy to the external work done.  

1. Set x=0 at the free end
2. Write the loading as a function of x
3. Write the MOI as a function of x (this is where the messy integration is going to come in)
4. Place a fake point load at the tip and label it P1
5. Develop an expression for the moment as a function of x (in terms of the distributed load from step 2, AND the fake point load from step 4)
6. Determine the partial derivative of the moment function (from step 5) with respect to P1
7. Set up the integral (from x=L to x=0) of (M/EI)*(partial M/partialP1)dx
8. E will be constant, so pull that out and integrate what is left with respect to x
9. Substitute 0 for P1 everywhere it appears
10. You now have the deflection of the tip of the cantilever!!

RE: Cantilevered tapered rod

There is a closed form solution for r = constant, a special case.

RE: Cantilevered tapered rod

r doesn't have to be constant to get a closed form solution.
I just posted a procedure above.

RE: Cantilevered tapered rod

My understanding is that closed form is any solution where you don't need to use a numerical approximation to get an answer.  If you can get x=0.3", then you're solution is closed form.  If you have to use infinte series or some other numerical approximation, then you don't have a closed form solution.

RE: Cantilevered tapered rod

for this case, x=0, 0.001, 0.01, 0.1, 0.1001.....1.0, an infinate series.

RE: Cantilevered tapered rod

LTwine-

Your links confirm my idea that any solution which can be "solved" is closed form, it doesn't matter if it is integration with variables.  By definition, integration contains variables.

Not having a closed form solution is when you use a numerical approximation, it says so right in the links you provided, and this is what I've always learned.  Read the links again.

RE: Cantilevered tapered rod

What are you talking about?  Not trying to be an ass here, seriously, but brush up on your math.  Integration is NOT an infinite series.  By your definition, all integration is an infinite series.   

RE: Cantilevered tapered rod

(OP)
StructuralEIT -
Develop an expression for the moment as a function of x (in terms of the distributed load from step 2, AND the fake point load from step 4)

I don't understand why I'm developing this expression when the one I have doesn't even include M.
I have:
v (deflection) = - (P*L^3) / (3*E*I)

Thanks for the input.

RE: Cantilevered tapered rod

I missed a line - both r & I vary with x.
As pointed out by the linked material, there are at least two camps on what presents a "closed form solution". Basically I want to point out that there wasn't anything wrong with Greglocock's statement, though you may not agree with it.

Teejt:

v = summation of (P*L[x]^3)/(3*E*I[x]), from x = 0 to 1.
Hope I didn't make mistake this time.

RE: Cantilevered tapered rod

StructuralEIT, when I was young and going to the technical college one day a week, I tried to use methods I learned at school, but I wasn't permitted to use calculus. In those days, almost everyone was convinced that calculus was haarrrd so they shied away from it, my boss said nobody was confident enough to check it, in any case, there were means and methods that could be used with the desired accuracy.

In this case, we would divide the length into segments and calculate the M/EI for the junctions and then use those; the smaller the segments the closer we got to your answer. Perhaps six segments would be close enough for design purposes.

 

Michael.
Timing has a lot to do with the outcome of a rain dance.

RE: Cantilevered tapered rod

Deflection at free end is 4 P L^3/(3 pi E r r0^3) where

r=radius at free end and
r0= radius at fixed end....

obtained by integration of y''=M(x)/(E I(x))

Ed.R.
 

RE: Cantilevered tapered rod

Nice job Ed.

That looks like a closed-form solution to me.  

paddington-
I don't disagree, I was just pointing out that there is a way to do it with the varying I that didn't involve stepping it.

RE: Cantilevered tapered rod

This is to correct myself:

The intergration over length by assuming a solid tapering bar does not present correct solution, since the mass is much less, thus the rigidity, along the spring coil.

RE: Cantilevered tapered rod

Not wishing to drag this on, yes EdR's solution is closed form for one subset of tapered beams. It is not general purpose , although the same idea can be applied to (I think) all beams where I(x)=I(0)*(1-a*x/L)^n, where 1>a>=0 and n>=0. In this case n=4, and dx/I(x) is an integral that can be solved. But there are infinitely many other functions for I(x) that give a tapered beam, but for which an analytical solution of the integral is not possible.

The universe is full of integrals that cannot be solved analytically, even when modelling relatively straightforward physical systems.

Cheers

Greg Locock

I rarely exceed 1.79 x 10^12 furlongs per fortnight

RE: Cantilevered tapered rod

I have revisit my rusty math memory bank, and have to admit I was wrong on concept over "closed form solution". EDR has found it, and StructralEIT was correct to disagree on the claim - "there is no closed form solution...".

My appology to both.

RE: Cantilevered tapered rod

Actually I messed up cylinderical tapering rod with tapering spring coil. What a miss, only 1000 miles apart.  

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