Energy Method
Often it is very difficult to determine the exact buckling load in complex structures using the Euler formula, due to the difficulty in deciding the constant K. Therefore, maximum buckling load often is approximated using energy conservation. This way of deciding maximum buckling load is often referred to as the energy method in structural analysis.
The first step in this method is to suggest a displacement function. This function must satisfy the most important boundary conditions, such as displacement and rotation. The more accurate the displacement function, the more accurate the result.
In this method, there are two equations used (for small deformations) to approximate the "inner" energy (the potential energy stored in elastic deformation of the structure) and "outer" energy (the work done on the system by external forces).
where is the displacement function and the subscripts and refer to the first and second derivatives of the displacement. Energy conservation yields:
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