MECH 360 - Question about strain energy problem with beams

In summary, the conversation was about finding the total strain energy in a beam with a moment applied. The person asking for help had attempted to solve the problem by splitting the integral into two parts, but was getting the wrong answer. They also considered using a bending moment diagram to find their mistake, but were unsure of where they went wrong. Another person suggested that their expression for M(x) may be incorrect. Ultimately, it was determined that the method and integral used were correct, but the execution was off and the person needed to double check their calculations and definitions.
  • #1
theBEAST
364
0

Homework Statement


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

rCICdQ0.png


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.

BamS31S.png


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:

fbHCyIe.png


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:)
 
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  • #2
Maybe if you drew a bending moment diagram you would see your mistake.
 
  • #3
paisiello2 said:
Maybe if you drew a bending moment diagram you would see your mistake.

Hmmm, so I drew it out and I get this:
WC6AnVV.jpg


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 :(.
 
  • #4
Are you sure you have the correct BM diagram?
 
  • #5
SteamKing said:
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.
 
  • #6
So I did it with Macaulay Equations and got the same answer:
P5KEAJb.jpg
 
  • #7
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.
 
  • #8
theBEAST said:
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).
 
  • #9
paisiello2 said:
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?
 
  • #10
Yes, the integral and your approach look right. Just the execution is off.
 

Related to MECH 360 - Question about strain energy problem with beams

What is the purpose of solving strain energy problems in beams?

The purpose of solving strain energy problems in beams is to determine the internal stresses and deformations in a beam under a given load. This is important in designing and analyzing structures, as it helps engineers ensure that the beam can withstand the expected loads without failing.

What is strain energy in relation to beams?

Strain energy is the potential energy stored within a material when it is deformed. In the case of beams, strain energy is the energy stored in the beam due to bending or shear stresses. It is directly related to the internal stresses and deformations in the beam.

What is the difference between strain energy and strain?

Strain energy and strain are two different concepts. Strain energy is the potential energy stored within a material due to deformation, while strain is the measure of the amount of deformation that a material experiences. Strain energy is dependent on strain, but they are not interchangeable terms.

How do you calculate strain energy in a beam?

The equation for calculating strain energy in a beam is U = (1/2) * ∫(M^2/EI)dx, where M is the bending moment, E is the modulus of elasticity, I is the moment of inertia, and dx is the differential length of the beam. This equation is derived from the strain energy density formula, which takes into account the bending stress and deformation of the beam.

What are the assumptions made in solving strain energy problems in beams?

There are several assumptions made when solving strain energy problems in beams, including: the beam is made of a homogeneous and isotropic material, the beam is initially straight and has a constant cross-section, the beam is subjected to small deformations, and the material follows Hooke's Law. These assumptions allow for simplified calculations and provide accurate results for most engineering applications.

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