# Analytical bending of a deformed beam

1. Feb 21, 2017

### saybrook1

Hi guys; I have an analytical solution for the deformation of a beam due to a couple with moments C_1 and C_2 with boundary conditions y=0 and x=±(L/2) where L≡length of the beam. The derivation from the Bernoulli-Euler equation is below:

\begin{align*}
y''=\frac{\epsilon}{2}(C_1+C_2)-\frac{\epsilon}{L}(C_1-C_2)x
\end{align*}
Where ε = 1/E*I, I=Moment of inertia, E=Young's Modulus and L= beam length.
\begin{align}
y'=\frac{\epsilon}{2}(C_1+C_2)x-\frac{\epsilon}{2L}(C_1-C_2)x^2+C_3\\
y=\frac{\epsilon}{2}(C_1+C_2)\frac{x^2}{2}-\frac{\epsilon}{2L}(C_1-C_2)\frac{x^3}{3}+C_3x+C_4
\end{align}
For y=0 at x=-L/2;

\begin{align}
0=\frac{\epsilon}{4}(C_1+C_2)(\frac{-L}{2})^2-\frac{\epsilon}{6L}(C_1-C_2)(\frac{-L}{2})^3+C_3(\frac{-L}{2})+C_4\\
C_3=-\frac{\epsilon}{4}(C_1+C_2)(\frac{-L}{2})+\frac{\epsilon}{6L}(C_1-C_2)(\frac{-L}{2})^2+\frac{2C_4}{L}
\end{align}
For y=0 at x=L/2;

\begin{align}
0=\frac{\epsilon}{4}(C_1+C_2)(\frac{L}{2})^2-\frac{\epsilon}{6L}(C_1-C_2)(\frac{L}{2})^3+C_3(\frac{L}{2})+C_4
\end{align}
Plugging in equation (4) yields:

\begin{align}
C_4=-\frac{\epsilon L^2}{16}(C_1+C_2)
\end{align}
And plugging (6) back in to (4) yields:

\begin{align}
C_3=\frac{\epsilon L}{24}(C_1-C_2)
\end{align}
With both integration constants determined, we arrive at our expression for vertical displacement:

\begin{align}
y=\frac{\epsilon}{4}(C_1+C_2)(x^2-\frac{L^2}{4})+\frac{\epsilon}{6}(C_1-C_2)(\frac{Lx}{4}-\frac{x^3}{L})
\end{align}

This expression lets me come up with the displacement for an undeformed beam given certain values of C_1 and C_2. I was hoping that someone may be able to help me figure out a way in which I could take the displacement data(set of y-values) of a deformed beam and apply certain values of C_1 and C_2 in order to create a new displacement profile(new y-values). Any help or a point in the right direction would be greatly appreciated. Thanks and please let me know if you need any more information.

2. Feb 21, 2017

### Nidum

Why don't you just tell us in simple words and pictures what you are trying to do overall in this project ?

Last edited: Feb 21, 2017
3. Feb 21, 2017

### saybrook1

Okay, I'll try my best - I've got an FEA model where I apply a heat load to a beam and simulate the thermal expansion. After that, I apply a couple and bend the beam back in another step of the simulation. Often times I will plot the displacement of the top center line of the beam after heat deformation and after subsequent bending.

Right now, I can get the model to agree pretty well with an analytical solution - Ideally, I would like to be able to take this top center line of displacement data after heat deformation - and then use that data to analytically apply the couple(with moments C_1 and C_2) and then match this precisely with my FEA models.

Right now - I add the heat displacement data to the "bending" displacement values that I get from the above equation. This matches decently with the FEA but not perfectly because the above equation is bending an undeformed beam whereas the model takes the deformed beam and then bends it.

Please let me know if you have any more questions and thanks for the follow-up.

4. Feb 21, 2017

### Mapes

In an old paper, I used the virtual work/flexibility method to calculate the deformation and forces (Eqs. (15)-(32)) in statically indeterminate offset beams undergoing thermal expansion under a constraint. I haven't used the method since then, but it seemed powerful and might be of interest. It seems like the main challenge would be in adapting to the unique curved profile of your deformed beam.

5. Feb 21, 2017

### saybrook1

That would absolutely be the main challenge - I have been scanning for papers like this; Thank you very much!! - I'll let you know what happens; as it stands, the method I'm using now is functional but I would definitely like to add this extra degree of accuracy.

6. Feb 22, 2017

### Nidum

Ok . That is a lot clearer now .

This plot from your earlier post is representative of the magnitude of deflections that you are getting ? If so the centre point deflection is only about 0.3% of the beam length ?