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:(adsbygoogle = window.adsbygoogle || []).push({});

\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.

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# Analytical bending of a deformed beam

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