# Deflection of hanging beam in two parts

Can anyone quickly explain the analytical method for finding the deflection at the base of a beam shaped like this? Note the top beam is a truncated cone and the bottom is a uniform cylinder.

First work out the tip deflection due to F as if it was applied at the end of L1, and an additional moment F*L2 and the angular deflection (slope) of the tapered beam. https://www.researchgate.net/profile/Viral-Mevcha/post/How-can-one-find-the-deflection-of-a-non-prismatic-cantilever-beam/attachment/59d6282e79197b80779868b1/AS%3A329079610986497%401455470101705/download/2.pdf
Call that x1 and a1

Then work out the deflection for the straight bit as if was a cantilever by itself. x2=FL^3/(3EI)

Then tip deflection of the system as shown is x1+x2+a1*L2

• Excellent paper. How to get the slope of the tapered beam?
– r13
Commented May 4, 2023 at 0:26
• Hmm. You asked for it! pdfs.semanticscholar.org/72f3/… Commented May 4, 2023 at 0:50
• Thanks for the PDF. Shouldn't X1 be the result of the tip load (F) and moment (F*L2)?
– r13
Commented May 4, 2023 at 10:51

I am assuming that you need to find the horizontal deflection here. Since this is a non prismatic beam, you can use the strain energy method to find the deflection. You can look up some problems on strain energy method on Youtube for introduction and try to solve this one by this method.

• All I need is the deflection at the bottom of the part. I assumed there would be a good method using beam deflection equations but I am finding it quite tricky. Commented May 3, 2023 at 15:19

We need to do this in two parts. first the top part.

Let's call the stress on top and bottom $$\delta_t\quad \delta_ b$$

$$\delta_b= \delta_t*k \quad k= \frac{A_t}{A_b}$$ We have strain as stress divided by Young's modulus. $$\epsilon =\frac{\delta}{200GPa}$$

A total defection on the top part is the integral of strain.

$$\epsilon_{total top}=\frac{L_{top}\delta_t k/2}{200GPaint}$$

then we add the deflection of the bottom part $$\epsilon_B =\frac{L_B\delta_b}{200GPa}$$ The total deflection is the sum of these two.