Deflection of tapered beam cantilever

[SJB]

Hi and thank you for taking a look at my problem..

I'm very familiar with doing deflection calculations for beams of constant cross-section but I find myself needing to now do calculations on a beam of varying cross-section and I'm completely lost.

It's a relatively simple case in that the beam is a round/tubular, constant/symmetrical taper. It's supported in cantilever at the large end and the load is a single point load at the tip. I've attached a simple sketch if it helps.

As a starting point, is it even possible to derive an equation for the deflection of the beam or can this only be calculated by FEA? (excuse my ignorance!)

If anyone can offer any help/pointers/equations I would be very grateful.

Thanks for your time.

Simon

 

[piygar]

This is certainly possible to derive eq'n of delection.

You will need to derive the deflection equation for you case, from equation:
E*I*d2y/dx2 = -M,where I is varying with position of section, and will be a function of x.
Therfore

E*d2y/dx2 = -M/Ix, and integrate this equation twice.

you will need to do this carefully and in the end you will get a big,scary expression of deflection.

You can definitely varify your result with FEA.

Take a look at Roark also for some empirical formula,depending upon ratio of both end area of inertias.

 

[SJB]

Hello Piygar,

Thank you for your reply, I'll give this a go and if successful will post back the equation in case it is useful to others.

Thanks

Simon

 

[nvn]

SJB: The cantilever tip deflection would be as follows, with x = 0 at the fixed support.

$$y_{\,\mathrm{max}}=\frac{1}{E}\int_{0}^{L} \int_{0}^{x}\frac{M(x)}{I(x)}\,dx\ dx$$