Beam Visualizations

Euler–Bernoulli beam theory is a method for calculating the behavior of beams under load. It gives us the differential equation relating beam deflection \(v\) to loads \(M\). \[M = -EI \frac{d^2v}{dx^2}\] Where

• \(M\) is the internal moment distribution
• \(v\) is the beam deflection
• \(x\) is the location on the beam. x=0 is usually on the left.

We can be lazy with physical dimensions since the visualizations have no basis in physical reality. Don't do this in any serious engineering application.
\(E\) and \(I\), relate to the beam cross section geometry and material properties. We can just set them to 1, making our equation \[M = -\frac{d^2v}{dx^2}\] This is a second order ordinary differential equation. \(M\) is a function of just \(x\), therefore it can be solved using direct integration: \[v(x) = \int \int M dx dx\] For example, the cantilever beam in the first visualization has an internal moment distribution \[M = F(x-L)\] Integrating twice gives \[v = F \left(\frac{x^3}{6} - \frac{Lx^2}{2} + C_1x+C_2\right)\] Applying our two boundary conditions for the cantilever beam

1. when \(x=0, v=0\)
2. when \(x=0, v'=0\)

gives \(C_1=C_2=0\).
Our final equation for \(v\) is then \[v = \frac{Fx^2}{6}(x-3L)\] which is what is used in the visualization for the cantilever beam.