# Beam Deflection Problem

1. Jun 21, 2009

### 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.

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• ###### def I_1 & I_2.jpg
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Last edited: Jun 21, 2009
2. Jun 21, 2009

### baron.cecil

No suggestions?

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

3. Jun 22, 2009

### 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].

4. Jun 22, 2009

### 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?

5. Jun 22, 2009

### 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 text books.

6. Jun 22, 2009

### 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.

7. Jun 22, 2009

### nvn

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