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!
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
FEA is an alternative of course.
Cheers
Greg Locock
I rarely exceed 1.79 x 10^12 furlongs per fortnight
RE: Cantilevered tapered rod
RE: Cantilevered tapered rod
RE: Cantilevered tapered rod
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
RE: Cantilevered tapered rod
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
RE: Cantilevered tapered rod
I just posted a procedure above.
RE: Cantilevered tapered rod
Check out from links below.
h
http://
RE: Cantilevered tapered rod
RE: Cantilevered tapered rod
RE: Cantilevered tapered rod
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
RE: Cantilevered tapered rod
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
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
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
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
RE: Cantilevered tapered rod
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
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
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
My appology to both.
RE: Cantilevered tapered rod