Moment of Inertia: Non Uniform Rod

[Andrew VanFossen]

Homework Statement

Hello,

I am looking to determine the moment of inertia for the cantilever beam pictured below. I want I to be a function of L1 and T1.

Known variables: L, D, P, t
Dependent variables: T2, L2
Design variables: T1, L1

Homework Equations

L2 = L - L1
T2 = T1 - t

Standard Moment of Inertia: I = π/4 * (Ro^4 - Ri^4)
I am looking to modify the above equation to be include 2 sections of variable thickness

The Attempt at a Solution

I1 = π/64 * (D^4 - (D - T1)^4)
I = I1 * (1 - L1/L)^4

The above solution is valid for a linearly tapered beam, not a beam with 2 discrete sections. I have not been able to find any derived moments of inertia for the above shape. Any help would be greatly appreciated!

 

[haruspex]

I have not been able to find any derived moments of inertia for the above shape.

Moments of Inertia are additive. Just break it up into pieces of convenient shape, find the MoI of each (about the same axis) and add them up.