# In the Principle of Virtual Work, what's the meaning of virtual?

I'll talk here of the principal of virtual work in the context of virtual forces.

The idea of virtual work may seem strange at first. But it's one of the most fascinating I've encountered.

The problem I had at first with the idea of a set of virtual forces was the "definition" given in most textbooks. Usually, it goes something like this: "In this statement, the term virtual means the forces are not related to the deformations of the body."

But, as far as I know, the forces applied to a body are always related to the deformations. This relationship may be more or less complicated, but it's a relationship that exists. This made me think that virtual forces are forces that are virtual or imaginary in a litteral sense. We imagine them to be there, but they're actually not there. So, whenever you find a "definition" such as the one you find above, all you'll be reading is something like: "a virtual force is a force that is virtual".

It's a childish idea (in the best sense of the word childish), but many applications show it to be a very powerfull one: amazingly, it works.

Would you agree with meaning of virtual force such as I wrote it?

## 3 Answers

The principle of virtual work is an energy method of structural analysis. While there is technically a principle of real work its utility is low because it's limited to determining displacement at the point of a single concentrated real load in the direction of that applied load.

The principle of virtual work gives us far more power and flexibility in analyzing structures because we can use a virtual force or moment applied at the location and orientation of our choosing and obtain whatever displacement is of interest to us. It's called a virtual load because this load does not actually exist on the real-world structure. You're correct that it's simply an imaginary construct created for the purposes of our calculations.

So, we apply a virtual load $$Q$$ to the structure. (A magnitude of 1 is usually chosen for convenience though in principle you could choose any value.) This load creates virtual internal forces, $$u$$, in the structure. Next, we add our real loads to the structure and our virtual load goes along for the ride, doing both external and internal virtual work in the process. By the conservation of energy, we can then say...

$$Q \bullet \delta_P = \sum F_Q \bullet dL$$

$$Q =$$ virtual external load

$$\delta_p =$$ real displacement (usually we're solving for this - a displacement or rotation at a node)

$$F_Q =$$ virtual internal forces produced by the virtual external load

$$dL =$$ real internal deformations. This will generally require determining the internal forces due to the real loads.

Essentially, we're mixing and matching loads from the virtual system with displacements from the real system. It's a very powerful method, and one that is leveraged to develop the flexibility method. Tip of the hat to Mr. Bernoulli.

Fundamentals of Structural Analysis by Leet et al. includes a nice visual proof of the principle of virtual work.

Imagine a simply supported beam loaded at say L/3 by P. So at any cross-section, actually any point it has stresses and related strains.

Now we apply a virtual load of 100N at mid-span. This virtual load will deform the beam in a way that the sum of virtual work done by the contributory stress times deformation in all infinitesimally small particles of the beam is minimum.

It is closely related to the concept of least work.

And the virtual deformation and virtual stresses and strains have to be added to the existing real forces.

It is exact within small-angle deformations which allow superposition.

Virtual force here implies that while the force does not act on the structure that one is analyzing, it does "act" on the paper as we analyze. The net work done as a result of the virtual force, also called virtual work has to be zero because physically there is no work being done.