# Principle of virtual work invention

## Main Question or Discussion Point

i want to throughly understand the principle of virtual work. why this principle was invented?

"sum of work done by forces is zero". such a definition is very vague for me. can anyone explain?

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I will try explain you, but I sorry for my bad english.

You know that Work is Force * "desplazamiento" W=F*d
In Static problem the displaicement d are 0, then you can imagine virtual Force to solve some problems knowing that finally the virtual work will be zero.

presbyope
Gold Member
The example I've seen is a ball rolling down a frictionless surface. Since there is no friction, the work done by the surface is zero. That simplifies the math if you're doing Langrangian mechanics.

If there is no friction, what force or couple acts to cause the ball to rotate?

In any event this is an example of real work, not virtual work.

It would be very helpful if sr chandran would tell us in what context he is studying VW. The answer can be very mathematical or more practical depending upon the field of study.

I has used VW in Static because displaicement are little. I teach this method for solve structures like trusses ("cerchas" in spanish). You suppose that has solve (forces and elongations of bars) a very complex structure (Structure 0) and you need to know what is the displaicement of one nude of the structure. You can imagine a virtual force in the nude and calculate all the structure only with this virtual force (Structure 1 virtual). After this, you can obtain the displaicement of the node with Structure 0 and Structure 1 using VW.
Whit VW and lineal equations of compatibility of movements you obtain lineal equations of equilibrium. Then VW conected static lineal equations of compatibilty of displaicement with equilibirum.

presbyope
Gold Member
Yes, the ball will slide on the surface and not rotate. The force of constraint is normal to the surface. There is no friction to overcome. The virtual work done by the forces of constraint is zero. The constraints are always defined to ensure this.

In statics, since the total force is zero, the virtual work done by the total force must also be zero. This allows solutions like Rafa has described. Clearly the displacements must be virtual since nothing really moves in statics.

I can't reconcile my picture of virtual work with your description.

When you do a virtual work calculation you introduce either imaginary displacements or imaginary forces and calculate the 'virtual work' carried out by either the real forces acting over these imaginary displacements or the virtual work carried out by the imaginary forces over the real displacements in the system.

The virtual work equation does not say the virtual work is zero it says

The sum of the virtual external work = the sum of the internal virtual work

Which is a very different thing.

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Studiot, your picture of virtual work is the consequence of my explication of VW.
In every nude of a roof truss, the equilibrium tell us that the sum of forces will be 0. You can multiply this equilibrium for a vector, in particular for a displacement vector (in this case you have a VW=0)
You can move every nude of the truss virtualy withou work. Then you have a sum over all nudes of the truss and other sumatory over the equilibirum forces in every nude, if you operate you can separate in two sumatories of works. And the results are:
The sum of the virtual external work = the sum of the internal virtual work.

If I have a blackboard may be I can explain better.
Don`t hesitate in correcting my english. I try to improve my wiriting english.

Hello Rafa, I am not disagreeing with anything you have said, my comments were directed at presbyope's posts.

I just think it is important to point out that virtual work is not another energy theorem.
The 'work' calculated is neither real work nor complementary work so the true energy theorems of Castigliano the reciprocal theorem and so on cannot be used in conjunction with VW.

Some authors call VW the unit load or unit force method, but it can be much more comprehensive than simply applying a unit force in the direction of interest to obtain the deflection of a structure.

In fact VW can also be applied to dynamical systems as well as static ones.

Studiot,
Thanks
I never used the VW applied to dynamical systems, but I agree that virtual work is not another energy theorem.
Srchandram...did you understand?