Potential Energy - in Structural FEA

The structural potential energy of a loaded structure is measured relative to some datum - usually the unloaded configuration. It is considered to comprise two components which could be "released" if the shape of the structure is physically changed from the loaded to the unloaded geometry; the components are the stored potential energy associated with the stress field and the potential energy of the external forces.

N.B. It is essential that this is understood - it is not the energy released by unloading, which for a conservative system is zero.

The total potential energy of an elastic body is defined as the sum of the total strain energy (U) which is stored potential energy

and the work potential (WP) of the external forces (the negative signs indicates thast work is done on the structure to pull each force back to the unloaded position):

Then the Principle of Minimum Potential Energy (which can be deduced from the Principle of Virtual Work) states that in a stable equilibrium position the total potential energy has a minimum value.

Or formally: "Of all the possible geometric configurations that a body may assume the true one for stable equilibruium is identified by a minimum value for the total potential energy".

Giving N equations representing the N equations of equilibrium.

Using the Principle of Minimum Potential Energy to find the Approximate Solution.
The importance of this principle in FEA is that it is a very useful method of determining approximate solutions. The method, which is a generalisation of the Raleigh-Ritz method, assumes a deflected shape consistent with the boundary conditions which defines the displacements. The N equations derived from:

are then a set of equilibrium equations in Pi and hence represent an approximate set of equations for the stiffness of the structure. Obviously the better the assumed deflected shape approximates to the true shape the more accurate the stiffness equations.

This is the method used to determine the stiffness matrix for most structural finite elements.

David J Grieve, 17th November 2012.