MECH 360 - Question about strain energy problem with beams

[theBEAST]

Homework Statement

I need to find the total strain energy in the beam with a moment applied:

The Attempt at a Solution

For my attempt I decided to split the integral into two parts since the moment is different depending on whether or not you are on the left side of the moment or on the right side.

I know there is an easier method; that the loading has a kind of skew-symmetry, so you might be able to take a short cut by evaluating the left side integral with a coordinate system x starting from A and going to the right, and the right side integral with a coordinate system X starting from B and going to the left.

However, for my attempt I decided to do the integral I circled with red below. When I run through the algebra I do not get the correct answer which is:

So I would like to know... Is my method correct or did I just do my algebra wrong. I feel like what I did should work... I spent a few hours on this and kept getting the wrong answer (might be just because I was tired and slept very little :zzz:)

 

[paisiello2]

Maybe if you drew a bending moment diagram you would see your mistake.

 

[theBEAST]

Maybe if you drew a bending moment diagram you would see your mistake.

Hmmm, so I drew it out and I get this:

I guess it makes sense since there is a moment in the beam so we should expect a discontinuity so I am not sure where my mistake is still :(.

 

[SteamKing]

Are you sure you have the correct BM diagram?

 

[theBEAST]

Are you sure you have the correct BM diagram?

Hmmm, I went over my moment balance equation several times and got the same equations, then I plugged in x = a and got the values... I feel like I am doing something really dumb and I can't seem to figure out what.

 

[theBEAST]

So I did it with Macaulay Equations and got the same answer:

 

[paisiello2]

Well, it looks like to me that either your BMD is wrong or you are misreading it because the formula you are entering appears wrong.

 

[SteamKing]

Hmmm, I went over my moment balance equation several times and got the same equations, then I plugged in x = a and got the values... I feel like I am doing something really dumb and I can't seem to figure out what.

You expression for M(x) for a <= x < L is suspect. When x = a, M(a) = Mo * (b/L),
and when x = L, M(L) = 0. You should check your definition of M(x) and see if it
returns these values for M(a) and M(L).

 

[theBEAST]

Well, it looks like to me that either your BMD is wrong or you are misreading it because the formula you are entering appears wrong.

But is the integral I circled in red correct, is my method correct?

 

[paisiello2]

Yes, the integral and your approach look right. Just the execution is off.