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Principle of virtual work

(1) Principle of virtual work statement can be said as:

"Virtual work = Work done by real forces in moving through virtual displacements
                        = (Real Force) * virtual displacement
Therefore, the equation of virtual work that you get is:
F * du = (K*u) * du (where, du is the virtual displacement and u is real displacement, and Ku is the real internal force and F is the real external force)
(2) However, I was going through the lecture notes of Duke university, we have at the following URL:

"In other words, the external virtual work of a virtual force ( F ) moving through a real displacement (D ) equals the integral of the virtual stresses associated with F times the real strains associated with D , over the volume of the solid.

That is, in the proof provided he considers "virtual forces" and "real displacement"?
Am I ineterprating wrong anywhere?

(3) I shall be extremely grateful if you can lead me where I am going wrong.Please help - whether it should be "Real Foces x virtual displacement or Virtual forces x Real displacement? whether (1) or (2) is correct?

with warm regards,

please someone help me with the doubt above

an anxious/eager learner

Arash_Yavari's picture

I'm not sure if I see the source of confusion here, but let me comment on "principle of virtual work".

Principle of virtual work in elasticity states that the work done by all the real "forces" (inertial, body forces, and tractions) in any virtual deformation of the body is zero. This is nothing but a weak form of balance of linear momentum and is widely used, for example, in finite element formulations.

In structural analysis something similar is referred to as the method of virtual work (or unit load method). Suppose one is interested in finding a (real) "displacement" component in a structure. In this method one puts a unit force (or moment) at the point of interest and in the desired direction. The unit (virtual) force induces some virtual reactions and internal forces. Work of external virtual forces (the applied unit force and the reactions if there are any support settlements) acting on real displacements is equal to the virtual internal energy (energy of virtual internal forces acting on real deformations) stored in the structure. This will give the unknown displacement.

I hope this helps.


Thanks Arash for your response.

However, if you check this link (also mentioned above) in my first post:

it defines and derives virtual work as product of VIRTUAL FORCESand REAL DISPLACEMENTS.As I've seen in FE formulations the definition used for virtual work is product of REAL FORCES AND VIRTUAL DISPLACEMENTS (as also you've stated).

My question is then, how, in this link this definition holds good?

please help


Arash_Yavari's picture

Dear Kajal:

As Ajit also pointed out, there is no unique way of defining "virtual work". What you see in those notes is fine and helps to find displacements and rotations in determinate structures.

Dear Kajal,

1. As the Wikipedia article on the Principle of Virtual Work (PVW) makes it clear:

(i) Virtual work could arise either through real forces acting through virtual displacements or virtual forces acting through real displacements.

(ii) If you insist on virtual displacements, then that's a special case, and it has a special name: Principle of Virtual Displacements (PVD). Similarly for the other special case---the Principle of Virtual Forces (PVF).

2. There is a reason why most books prefer to present PVD while discussing PVW:

In analysis, if you have a displacement-primary formulation, then compatibility of displacements is, of course, always ensured. That is to say, the strain-field derived from an arbitrary displacement field (arbitrary, within reason) is always compatible; you don't have to worry imposing the fourth order strain compatibility equations as an additional constraint on your system. But the converse is not true. If you start with an arbitrary stress (i.e. strain) field, then the displacement field that it implies does not necessarily satisfy compatibility requirement.

That's why, even in analytical theory itself (let alone in FEM), the displacement-primary formulation enjoys a somewhat favored status. Especially so, if you are dealing with continua (say the plane-stress/strain situation or stress analysis for 3D solids), and not the discrete or "lumped" systems (such as trusses and frames).

The force/stress-primary formulation (i.e. PVF if it's the weak form you are talking about) has its own advantages, but for the reason mentioned above, practically speaking, the application of such a formulation remains limited only to the simpler civil structural analysis (discrete systems like trusses and frames)---not for continuum analysis in general.

And, of course, today everybody uses FEM, and most FE formulations are displacement-primary too, for the same reason. This makes PVD even more relevant when it comes to PVW.

Hope this helps. I would like to know if I am missing any other point too.

Thanks for the response-Arash and Ajit.

1) From your respective responses, I presume that what is given in the notes in my link is the principle of virtual forces and what is used in FE formulations is principle virtual displacements.Right?

2) The one discussed in the notes corresponding to my link is used in determining displacements in determinate structures as pointed out by Arash.

3)Now if I want to derive,

virtual work = Real force x Virtual displacement using the same example- that is the same procedure as given in the notes, can you tell me how to go about that?

please help


Dear Kajal,

Before replying further, I would appreciate knowing the following from you:

- Whether you are a student; if yes, the usual particulars about your institution and program; if not, those about your occupation and background(s); your location... Best if you update your iMechanica profile section itself.

- What use---whether in theory or in application---you have in mind in raising your second query above.

And, oh... Do you have a typo in saying "determinate"?


Arash_Yavari's picture

Dear Kajal:

I should first emphasize that your structure does not have to be determinate to use the result of your "notes". In the case of determinate structures you can directly calculate the displacements.

Instead of putting a virtual force on your structure, you could impose a virtual displacement and following a similar line of arguments find a very similar result: Work of real forces acting on virtual displacements is equal to the corresponding stored energy. In the proof you would start with your real forces and corresponding real displacements and then impose the (compatible) virtual displacements. Balance of energy then will give you everything. I don't know where you would use this but  perhaps it can help in calculating stiffness matrices.

In short, what you see in those notes as "virtual work theorem" or any variants of it are all consequences of balance of energy.


Arash, you said where would we use this?-in FE formulations what is used is principle of virtual displacements,right?

Thank you Ajit for your response.

1)About my background- I've completed my masters in Structural Engineering and am planning my Ph.D in US/likewise.I've graduated from Chennai

Currently am preparing for the challenge of Ph.D reading as much on my specialised topic related to Finite Element Engineering-my topic of resaerch not yst decided.the question pertaining to derivation of principle of virtual work as Real force x Virtual dislacement is for a pure academic interest to understand facts and things

No, "Determinate" was not a typo-in fact its was what even Arash had pointed in his post above.

Awaiting your valuable response.

with warm regards,



I am genuinely surprised that anyone would want to confirm things of so straight-forward a nature after finishing his master's in structural engineering from India. Or, seek detailed work-outs.

Still, having said that, I ran the following query in Google and a majority of the first 10 hits it returned seems relevant: "Principle of Virtual Work in Solid Mechanics." 

Out of those hits, Alan Bower's book is well-known to iMechanicians and seems to carry a detailed section on what you seek; IIT Madras' prescribed syllabus for the PhD aspirants in Applied Mechanics Department, in a way, goes to confirm that the surprise that I express above was right; from what I have browsed of J. N. Reddy's book on Energy Principles, it seems very well suited to what you seek; and while I have not yet had an opportunity to consult Holzapfel's book, I remember it being recommended by Zhigang (Suo) in the recent past at iMechanica.

All in all, I guess it would be a good (and pleasurable) exercise to work out what you seek through self-studies alone. (I am certain you could do it.)


If a system is statically determinate, why would one at all seek more complicated procedures like PVW or MWR (Method of Weighted Residuals)? (This question is rhetorical in nature.) ... OK, therefore to make it all a little bit interesting: It would be interesting to see how PVW, Principle of Stationary Total Potential Energy (PSTPE), and MWR work out in systems that are (a) overdetermined and (b) underdetermined. "...exercise left for the reader..."


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