Cantilevered tapered rod 1

[TeejT]

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!

 

[Ltwine]

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.

 

[GregLocock]

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

 

[Ltwine]

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.

 

[Ltwine]

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