# Rigorous statement of virtual work principle?

• abhishekkgp
In summary, the principle of virtual work states that the virtual work of the total force acting on a particle( or system of particles) in equilibrium is zero for any arbitrary(not necessarily infinitesimal) virtual displacement.
abhishekkgp
rigorous statement of virtual work principle??

in the texts on mechanics the virtual work principle is always stated in 'infitesimal form'.
is there a "proper" way to write the principle of virtual work in which we don't leave it in terms of INFINITESIMALS.

Sure. You can replace infinitesimals by their definitions. It is like trying to explain what does it mean that the derivative of a certain function at a certain point is zero without using the concept of the derivative. In one dimension you can try to appeal to the intuitive concept of the "direction of the tangent line". But in more dimensions?

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You don't need calculus at all to 'do' virtual work.

Look at a book on structural engineering.

Sure. You can replace infinitesimals by their definitions. It is like trying to explain what does it mean that the derivative of a certain function at a certain point is zero without using the concept of the derivative. In one dimension you can try to appeal to the intuitive concept of the "direction of the tangent line". But in more dimensions?

can you please state the the principal of virtual work without using infinitesimals..
(i don't know how to type mathematical symbols here... like how to type greek symbols and partial derivative symbols. how do i do that?)

abhishekkgp said:
(i don't know how to type mathematical symbols here... like how to type greek symbols and partial derivative symbols. how do i do that?)

and follow the links there to learn some LaTex code if you are new to it.
If you can explain why do not want infinitesimals - for which reason, I will try to help you.

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i started studying calculus in a non rigorous way. in physics the non rigorous calculus works perfectly well. then i started reading rigorous calculus( from apostol) and since then any non rigorous analsis always leaves me thinking whether the statement is true in strict mathematical sense.

for example... consider a function f(x,y). chose a point (x1,y1) on the x-y plane.
assume that the function f(x,y) is differentiable along EVERY line passing through (x1, y1). one may think that under this assumption it can be stated that the function f(x,y) is continuous at (x1,y1). but this conclusion is wrong and you are probably aware of this.

for this reason i wanted a rigorous definition of the principle of virtual work.

moreover, the principle of virtual work states that the virtual work of the total force acting on a particle( or system of particles) in equilibrium is zero for any arbitrary(not necessarily infinitesimal) virtual displacement. now the total force on i-th particle can be written as the sum of constraint forces and applied forces. but the fact that the FORCES OF CONSTRAINTS do NOT perform zero virtual work for finite displacements comes in our way. the forces of constraints perform zero virtual work( rather in second order of infinitesimals) only under infinitesimal displacements. this is why we can say that the virtual work of the applied forces is also zero for infinetesimal displacements( and not for finite displacements).

THANKS FOR YOU REPLIES.

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the principle of virtual work states that the virtual work of the total forces acting on a particle( or system of particles) in equilibrium is zero for any arbitrary(not necessarily infinitesimal) virtual displacement.

This is the difficult approach to Virtual Work.

Did you read my post#3? You haven't commented.

can you name a common book on structural engineering so that i can download it.

also, what other approach is there to virtual work. why the one i stated is "difficult". is it wrong ??

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For instance in "Structural analysis" by Kassimali infinitesimals are not being avoided. And in the applications you really need to know how to calculate a derivative of a function and what does it mean.

i saw a text on "structural analysis". in the index i didnt find virtual work but 'virtual crack extension method' . i saw it but it didnt look like virtual work method.

even in the analytical mechanics text while doing the lagrangian formulation infinitesimals are extensively used in both of the two books i am reading.
but it just doesn't feel right to equate an infinetesimal quantity to zero!
some friend of mine told me that even the methods of 'non standard( or non rigorous)' calculus
are equivalent to rigorous calculus and that some mathematician( i forgot the name he told me) has proved this. is this right?

even if that friend of mine is right i still like to understand calculus in rigorous manner( although it becomes very tough and non intuitive! )

Virtual Work has been discussed before at Physics Forums, have you done a search?

eg

VW is sometimes called by other names eg the 'unit load method.'

Books?

Structural Analysis uing Virtual Work
by
F Thompson & GG Haywood

a recently revived classic is

Mechanics of Internal Work
by
Irving Porter Church

When you do VW calculations it is vital to distinguish between the the real and the imaginary.

virtual work has been discussed many times in physics forums... i saw many threads but my query was not discussed there. i ll see the books you have stated( if i can download them) and then reply. the link to thread you have given... i had seen it before but there my question is not discussed.

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Your original question was how to present the statement of VW in non infinitesimal terms.

Structural Engineers use such a form all the time, which was why I recommended it to you.

Your reply indicates you haven't fully appreciated that either the forces, or the displacements ( or both but that is not useful) are imaginary.

The work is not zero, but calculable.
But it is not real work.

The principle is that

$$\Sigma$$ Internal Work = $$\Sigma$$ External Work

If one of these is transferred to the other side the equation can be written equal to zero, but this practise can is sometimes misleading.

The calculation of the sums can involve differential or integral equations. All differential equations can also be presented in integral form, perhaps this is a way for you to avoid differentials?

abhishekkgp said:
but it just doesn't feel right to equate an infinetesimal quantity to zero!

Look, suppose you have f'(x)=df(x)/dx = 0. It makes sense, right? But you can also write it as

df(x)=f'(x)dx=0

The meaning is the same.

$$d$$ or $$\delta$$ means: "take the linear part of the change of whatever is after the symbol".

For instance in "Structural analysis" by Kassimali infinitesimals are not being avoided..

here you have written that infinitesimals are not being avoided. you perhaps wanted to say that infinitesimals are not being used.

Look, suppose you have f'(x)=df(x)/dx = 0. It makes sense, right? But you can also write it as

df(x)=f'(x)dx=0

The meaning is the same.

yes.. you are right.. but can you do it for the virtual work principal for EXTERNAL( or applied forces).
have you read the lagrangian formalism( probably in your analytical mechanics course) ?

Both integrals and infinitesimals are being used. For instance Eq. (7.26) reads:

$$dW=M(d\theta)$$

They are not needed only when you deal with linear functions. Then they can be replaced by finite increments $$\Delta$$.

Studiot said:
Your original question was how to present the statement of VW in non infinitesimal terms.

Structural Engineers use such a form all the time, which was why I recommended it to you.

Your reply indicates you haven't fully appreciated that either the forces, or the displacements ( or both but that is not useful) are imaginary.

The work is not zero, but calculable.
But it is not real work.

The principle is that

$$\Sigma$$ Internal Work = $$\Sigma$$ External Work

If one of these is transferred to the other side the equation can be written equal to zero, but this practise can is sometimes misleading.

The calculation of the sums can involve differential or integral equations. All differential equations can also be presented in integral form, perhaps this is a way for you to avoid differentials?

i agree. in my post#6 i have said the same thing. internal work( work done by constraints) = external work( work done by applied or external forces). this is true for arbitrary displacements(not necessarily infinitesimals).

but the problem is that in the lagrangian formalism we write that VW of the EXTERNAL (in case of dynamic equilibrium d alembert's principle is used and reversed effective forces are also added to it)forces on every particle is zero for infinitesimal virtual displacements. we cannot say that VW of external forces is zero for finite virtual displacements.

i understand that virtual displacements are imaginary.. they don't actually take place... we "freeze" time and then perform virtual displacements... this also ensures that VW of constraint forces is zero for infinitesimal virtual displacements because then the displacements are perpendicular to the constraint force.

I am still not sure whether you are talking about VW as applied to dynamics or statics?

Here is a pretty general statement of the principle.

Suppose there is defined in some region R a stress field which satisfies the equation

$$\frac{{\partial {T_{ij}}}}{{\partial {x_i}}} + \rho {b_j} = 0$$

and also in R there is a velocity field with components $${{v_i}}$$ then
if

$${D_{ij}} = \frac{1}{2}\left( {\frac{{\partial {v_i}}}{{\partial {x_i}}} + \frac{{\partial {v_j}}}{{\partial {x_i}}}} \right)$$

are the components of the deformation rate tensor then the virtual work expended by combining the two fields is found by forming the product
$${{T_{ij}}}$$$${D_{ij}}$$

and volume integrating it over the region R.

$$VW = \int {\int {\int {{T_{ij}}} } } {D_{ij}}dV$$

http://arxiv.org/abs/physics/0510204"

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## 1. What is the virtual work principle?

The virtual work principle is a fundamental concept in mechanics that states that the work done by the internal and external forces on a system in equilibrium is equal to zero. It is used to analyze the behavior and stability of structures and mechanical systems.

## 2. What is the significance of the virtual work principle?

The virtual work principle allows engineers and scientists to model and predict the behavior of complex mechanical systems without having to solve complicated equations of motion. It also helps in the design and optimization of structures and mechanical systems.

## 3. How is the virtual work principle applied?

The virtual work principle is applied by considering the equilibrium of a system and writing the equations of virtual work for each force acting on the system. These equations are then solved to determine the unknown forces and displacements within the system.

## 4. Can the virtual work principle be applied to any system?

Yes, the virtual work principle can be applied to any system as long as it is in equilibrium. It is commonly used in structural engineering, mechanical engineering, and other fields of mechanics.

## 5. How does the virtual work principle relate to energy conservation?

The virtual work principle is closely related to energy conservation. Since the work done by internal and external forces on a system in equilibrium is equal to zero, it can be concluded that the total energy of the system remains constant. This is known as the principle of conservation of energy.

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