# Principal Virtual Work vs Principal of Minimum Potential Energy

## Main Question or Discussion Point

Hello

Just had a quick question for a structures class. When attempting a problem, whats the main difference between applying PVW and PMPE? What would classify a problem in using one over the other?

Thanks

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OK PMPE is the calculation of the actual energies involved when we vary a parameter for a system.
The system need not be in equilibrium.
There need be no external forces acting.

For example the potential energy of a soap bubble depends on its surface area, which in turn depends on its shape, and PMPE says it will take a shape which minimises this.
Looking at shapes we find that as we vary the shape the one with the minimal surface area is a sphere, which is the shape of a soap bubble.

Virtual Work is more difficult conceptually and is not a minimum theorem. In fact the energies (work) calculated have no reality, hence the term.

The system must be in equilibrium.
There must be external forces acting.

Structural VW stems from the fact that in responding to loads any system must satisfy two independent conditions

Static equilibrium of forces (internal and external) acting (set1)
&
Geometrical compatibility of strains or displacements (set2)

Since work is the product of force and distance any such multiplication has the dimensions of work.

So multiplying set 1 by set 2 has the dimensions of work. This is known as the virtual work since either set 1 or 2 may be different from the actual ones applied to the system.

VW is particularly if we wish to calculate a single displacement of a complicated set of loads and replace the set with a single load in the direction of the desired displacement.

Thanks for the detailed response. that definitely clears up a lot of things.