Beam deflection by superposition

[Jim Newt]

Homework Statement

In this problem, I'm trying to solve for the Reaction, Rb

Homework Equations

I believe for the distributed load, the deflection equation would be:

v = (-qx/24EI) * (L^3 - 6Lx^2 + x^3)

For the support, Rb, the deflection equation could be:

v = (-Pbx/6LEI) * (L^2 - b^2 - x^2)

where L = 3L

total deflection at Rb = db1 - db2 = 0

I could solve for Rb with the proper equations.

The Attempt at a Solution

For db1:

When I plug in x=L and L=3L, I end up with db1 = (11qL^4) / 12EI.
This answer matches the solution manual.

I can't figure out db2...I think I might be using the wrong deflection equation, but I'm not sure. The correct answer for db2 is db2 = (4RbL^3) / 9EI

How do I go about solving for db2?

 

[Jim Newt]

Ok, maybe my whole method is bunk. Given the standard beam deflection equations, how would you solve for Rb?

 

[PhanthomJay]

Ok, maybe my whole method is bunk. Given the standard beam deflection equations, how would you solve for Rb?

You set up the equations perfectly! You may have made a math error in your db2 equation, perhaps forgetting to convert l to 3L. Note that in using the beam tables, you could have used db2 = (R_ba²b²)/3EIl, the deflection at the point of the applied reaction load. The result is the same.

 

[Jim Newt]

Hey thanks buddy! I went through it again and ended up with the correct answer. Its always the tidbit math that gets me...

 

[DeeNos]

Hello, it seems like you are on the right track in using the superposition method to solve for the reaction Rb. To solve for db2, you will need to consider the deflection equation for the support, which you have correctly identified as v = (-Pbx/6LEI) * (L^2 - b^2 - x^2). However, you will need to substitute in the values of L=3L and x=L to get the deflection at Rb. This will give you:

db2 = (-PbL^2/6LEI) * (L^2 - b^2 - L^2)

Simplifying this equation further, you will get:

db2 = (-PbL^4/6LEI)

Now, to solve for Rb, you can use the total deflection equation db1 - db2 = 0 as you have mentioned. Substituting in the values of db1 and db2, you will get:

(11qL^4) / 12EI - (-PbL^4/6LEI) = 0

Solving for Rb, you should get:

Rb = (11qL^4) / (12L^3) = (11qL) / 12

I hope this helps in solving your problem. Keep up the good work!