Structural Mechanics
The total potential energy, is the sum of the elastic strain energy, U, stored in the deformed body and the potential energy, V, of the applied forces:
This energy is at a stationary position when an infinitesimal variation from such position involves no change in energy:
The principle of minimum total potential energy may be derived as a special case of the virtual work principle for elastic systems subject to conservative forces.
The equality between external and internal virtual work (due to virtual displacements) is:
where
- = vector of displacements
- = vector of distributed forces acting on the part of the surface
- = vector of body forces
In the special case of elastic bodies, the right-hand-side of (3) can be taken to be the change, of elastic strain energy U due to infinitesimal variations of real displacements. In addition, when the external forces are conservative forces, the left-hand-side of (3) can be seen as the change in the potential energy function V of the forces. The function V is defined as:
where the minus sign implies a loss of potential energy as the force is displaced in its direction. With these two subsidiary conditions, (3) becomes:
This leads to (2) as desired. The variational form of (2) is often used as the basis for developing the finite element method in structural mechanics.
Read more about this topic: Minimum Total Potential Energy Principle
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