Solving Beam Deflection Problem with Variable Moment of Inertia

[baron.cecil]

Hello,

I am doing a problem for work where I use a deflected beam as a model. Basically, I am using a beam with two fixed ends and a force directly in the middle. The deflection equation for this model is:

delta_max=FL^3/(192EI)

This assumes I is constant through the entire beam. However, how would I obtain delta_max if I is different to the left and right of the force (I_1 and I_2)?

Thank you!

P.S. Please see attached images for visuals.

 

[baron.cecil]

No suggestions?

Are there any superposition principles I can use for this problem?

 

[nvn]

Always use I1 ≤ I2, and place the x-axis origin at the beam end having area moment of inertia I1. The maximum deflection occurs at x = L*(I1 + 3*I2)/(I1 + 7*I2), and is delta_max = {F*(L^3)/[12*E*(I1^2 + I2^2 + 14*I1*I2)]}*[(I1 + 3*I2)^3]/[(I1 + 7*I2)^2].

 

[baron.cecil]

So the maximum deflection (delta_max = {F*(L^3)/[12*E*(I1^2 + I2^2 + 14*I1*I2)]}*[(I1 + 3*I2)^3]/[(I1 + 7*I2)^2]) occurs at the center of the beam as well, or not neccessariy?

And do you have a reference for this equation or a derivation?

 

[nvn]

The maximum deflection occurs at the x coordinate given in post 3, which is not necessarily at midspan. I don't have a reference. If you want to study derivation of beam problems, study your favorite mechanics of materials, strength of materials, or structural analysis textbooks.

 

[baron.cecil]

I guess I'm not so much interested in the derivation, just a source of where you got the equations from, unless you pulled them off the top of your head.

Do you know the equation for delta_max as a function of x? I mostly need to the know the deflection at the midpoint...I should've stated that earlier.

 

[nvn]

Deflection at midspan is delta = F*(L^3)(I1 + I2)/[24*E*(I1^2 + I2^2 + 14*I1*I2)].