Euler-buckling

A vertical beam submitted to a vertical force can exhibit an instability. The critical load for instability depends on the end conditions of the beam. The picture displays fours different end conditions. The Euler buckling problem is formulated as a constrained optimization. The total energy, potential + deformation, is minimum for stable equilibrium. The constraints are the end conditions and the geometric equations of the beam.

In the four demo Jupyter notebook, this is done using the Pyomo Optimization Modeling in Python package.
Four cases are considered, as in the picture from left to:

• PP: pinned below, pinned above
• CP: clamped below, pinned above
• CC: clamped below, clamped above
• CF: clamped below, free above

Dependencies

Pyomo and Ipopt, for optimization, are installed automatically from the notebooks if needed.
Bokeh and Holoviews, for graphics, need to be installed beforehands.