Equation of Virtual Work VS Potential Energy

[rdbateman]

For linear elastic materials (3D General)...

The virtual work equation looks exactly like the equation for potential energy except that in the potential energy equation, the internal work term has a (1/2) coefficient. Why are these different?

 

[Studiot]

Do you know why the potential energy has a 1/2 coefficient?

It has to do with the method of application of the loads, which is different for VW.

 

[rdbateman]

That doesn't answer the question.

 

[Studiot]

Actually I did provide an answer, albeit very short.

But I still don't know if you understand where the 1/2 comes from.

 

[PJC01]

The virtual work equation and the potential energy equation are both fundamental equations in the study of mechanics and are closely related. However, they have different origins and serve different purposes.

The virtual work equation is derived from the principle of virtual work, which states that the virtual work done by external forces on a system in equilibrium is equal to the virtual work done by internal forces within the system. This equation is used to analyze the equilibrium of a system and determine the unknown displacements or forces.

On the other hand, the potential energy equation is derived from the principle of conservation of energy, which states that the total energy of a system remains constant. This equation is used to analyze the energy state of a system and determine the potential energy stored in the system.

In the case of linear elastic materials in 3D, the virtual work equation and the potential energy equation are mathematically equivalent. However, the (1/2) coefficient in the potential energy equation is a result of the integration of the internal work term, which is the work done by internal forces over a finite displacement. This integration is not present in the virtual work equation, which considers infinitesimal virtual displacements.

Therefore, the (1/2) coefficient in the potential energy equation is a result of the energy being stored in the system over a finite displacement, whereas the virtual work equation considers the energy stored over an infinitesimal displacement. In summary, the differences between the two equations stem from their different origins and their intended use in analyzing different aspects of a system's behavior.