Euler Buckling
Index
Materials loaded in compression may exhibit two types of elastic response. At low stress levels the sample behaves elastically, obeying  Hooke's law: s = E e.  Samples with a high length to area ratio show another deformation mode at high stresses. The material buckles above a stress threshold and the sample will show a lateral displacement. This is the Euler buckling. If strain rate is controlled, reversing the strain after buckling permits the sample to regain its initial configuration, indicating that the buckling response is elastic. 
                                                                  Leonhard Euler, Switzerland, (1707 to 1783) 
The diagram shows the behavior of a slender sample with ends constrained to lie on the compression axis but free to rotate. The threshold force at which buckling occurs for this case is given for a rod of circular crosssection by: 
F = ( n2 p2 E I / L2 ) = (n2 p2 E A2 / 4 L2 ),  where n is a mode number, I the geometric moment of inertia, A area, L length, and E Young's modulus. The lowest energy mode occurs first and for this n = 1.