Why does the Principle of Virtual Work work without higher math?

[observer1]

The last chapter of most introductory textbooks on STATICS introduces VIRTUAL WORK.

It is rarely taught (I studied the syllabi of colleagues).

I understand the Principle of Virtual Work, having researched and studied the Calculus of Variations, Hamilton's Principle, the Lagrangian and related items.

But I am trying to give myself a short, concise justification for why VIRTUAL WORK... works, but WITHOUT the higher math justifications.

For example, the opening chapter of one text on Statics states:

"The principle of virtual work was pioneered by Bernoulli. It provides an alternative methof for solving equilibrium problems... The PVW states that if a system exists in equilibrium, then the sum of all the work done by virtual displacments is 0"

Well (and please forgive me): whoppie-do, so it does. Like magic, it works.

Well, can someone justify why it works WITHOUT recourse to the higher mathematics I mentioned above?

I am hoping for a concise justification on why it should work... without the math.

 

[Dr.D]

This does not answer the question of the OP directly, but it is relevant to the topic of VW.

The practical application of VW requires a mechanism (methodology) for dealing with kinematics in a wide variety of situations. Without a formal study of kinematics, particularly the use of vector loop equations and their derivatives, this methodology is usually missing. This makes the application of VW difficult for all but the simplest problems, and consequently leads to its neglect.

 

[zwierz]

I think that in statics this principle is useless indeed while its dynamical version the Lagrange–d'Alembert principle is very important

 

[gleem]

I think that in statics this principle is useless..."

How so?

 

[zwierz]

Statics problems are the problems about solving systems of linear algebraic equations. You can obtain these systems by means of ordinary equations of statics and solve them by standard means. Why make it more complicated than it is

 

[Dr.D]

I think that in statics this principle is useless indeed while its dynamical version the Lagrange–d'Alembert principle is very important

I have to disagree here. Broadly speaking, there are two classes of statics problems:
(1) The structural problem, where the geometry is essentially unchanging and the problem is to find internal loads;
(2) The machine problem, where the geometry is variable while the external loads are known, with the problem being to find the equilibrium position.

For the second class (as listed above), the Principle of Virtual Work is invaluable and I use it frequently. I also made it a point to teach it to my students when I was teaching because it is so very important. The equations to be solved are rarely ever linear, but just obtaining the proper equations can be quite a challenge if you do not apply Virtual Work.

 

[zwierz]

Yes to find an equilibrium is in general nonlinear problem indeed. I'm used to that the topic "statics" is only about your first item. Perhaps the tradition varies from country to country. ok Actually in our courses statics is frequently dropped. Engineers study statics, theorists do not

Nevertheless:

obtaining the proper equations can be quite a challenge if you do not apply Virtual Work.

would you bring an example please

 

[Dr.D]

would you bring an example please

I have a good example, but unfortunately, I am unable to upload the figure. The system is a ship's hatch cover with built-in opening mechanism. All weights and part dimensions are known. The problem is to find the equilibrium position (partially opened) for a specified hydraulic cylinder pressure.