Solving Beam Deflection Problem with Variable Moment of Inertia

In summary, the conversation discusses the use of a deflected beam as a model for a work problem. The deflection equation for this model is provided, assuming the constant value of I throughout the beam. However, the conversation then delves into how to obtain delta_max if I is different on either side of the force. Multiple equations are suggested, including one that calculates the maximum deflection at a specific x-coordinate, which may not necessarily be at midspan. The conversation ends with a request for a reference or derivation of these equations, and a final equation for calculating deflection at midspan.
  • #1
baron.cecil
8
0
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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  • #2
No suggestions?

Are there any superposition principles I can use for this problem?
 
  • #3
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
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
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.
 
  • #6
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
Deflection at midspan is delta = F*(L^3)(I1 + I2)/[24*E*(I1^2 + I2^2 + 14*I1*I2)].
 

1. What is a beam deflection problem?

A beam deflection problem is a type of engineering problem that involves analyzing and determining the amount of deflection (or bending) that occurs in a beam when a load is applied to it. This is important in structural design to ensure that the beam can safely support the applied load without breaking or failing.

2. What factors affect the deflection of a beam?

The deflection of a beam can be affected by several factors, including the type of material the beam is made of, its cross-sectional shape, the magnitude and location of the applied load, and the beam's length and support conditions. In general, stiffer and stronger materials will experience less deflection, and longer beams will experience more deflection.

3. How is beam deflection calculated?

There are several methods for calculating beam deflection, including analytical formulas, numerical methods, and experimental testing. The most common approach is to use the Euler-Bernoulli beam theory, which involves solving a differential equation to determine the deflection at any point along the beam's length.

4. What are the types of supports for a beam?

There are several types of supports that can be used for a beam, including simply supported, fixed, cantilever, and continuous supports. A simply supported beam is supported at both ends and is free to rotate, while a fixed support prevents both rotation and translation. A cantilever beam is fixed at one end and free at the other, and a continuous support is a combination of different support types along the length of the beam.

5. How can beam deflection be minimized?

To minimize beam deflection, engineers can use several techniques, including increasing the beam's stiffness by using stronger or thicker materials, changing the shape of the beam's cross-section, and adding additional supports or braces. Additionally, distributing the load evenly along the beam's length can also help reduce deflection.

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