# Principle of Virtual Work to the FE method

• Trying2Learn
In summary, the principle of virtual work is a reformulation of Hamilton's principle to account for non-conservative forces. The development of the FE equations give us the term "internal virtual work" which is equivalent to stress and variation of strain. When the internal virtual work is equal to the external virtual work, we recover the equilibrium equation and the boundary condition.
Trying2Learn
TL;DR Summary
PVW, FE, Equilibrium Equations
(I do understand there are many ways to formulate the FE method, but I wish to understand this one.)

1. I understand that Hamilton's Principle is just that: a principle ( a law the recapitulates the equations of motion)
2. I also understand that the Principle of Virtual Work (PVW) is a reformulation of Hamilton's Principle to account for non-conservative forces (in bodies that are rigid)

HOWEVER

If my focus was ONLY the PVW alone, I can understand this and where it came from: both are forces acting through virtual displacements)

But the development of the FE equations give (when we apply Gauss theorem, definition of strain, etc.)

Now, I can anticipate that this new term on the right is the INTERNAL virtual work: stress and variation of strain, internal

However, it seems so ad hoc, that books get away with this.

It seems to me that this statement below is CRITICAL:

When the Internal virtual work is equal to the External Virtual work, we recover the equilibrium equation and the boundary condition.

It seems to me that this BOLD statement above is the beating heart of the FE method (from the mechanical engineering perspective), but all books give it lip service and appear to simply "tweak" the PVW and "add this term."

I am sorry to say I am not entirely sure of where my confusion lies, but it seems to be that the BOLD BLUE statement above, must rise above the development and must become a restatement of the PVW.

I do not know what I am trying to ask, but could someone comment on this post?

Maybe I am beating this, senseless.

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I don't get the math, unfortunately, but your BOLD statement seems to be the equivalent of saying the obvious. If A is equal to B then we have equilibrium and the boundary is the equal sign between them. Of course I may be oversimplifying it. But you may also be overcomplicating it.
Just a thought.

GramInvents51 said:
I don't get the math, unfortunately, but your BOLD statement seems to be the equivalent of saying the obvious. If A is equal to B then we have equilibrium and the boundary is the equal sign between them. Of course I may be oversimplifying it. But you may also be overcomplicating it.
Just a thought.

I now think I was overcomplicating it.

Thank you!

## 1. What is the principle of virtual work?

The principle of virtual work states that the work done by the internal forces of a system in equilibrium is equal to the work done by the external forces acting on the system. This principle is used in the finite element method to solve for the unknown displacements and stresses in a structure.

## 2. How is the principle of virtual work used in the finite element method?

In the finite element method, the principle of virtual work is used to create a system of equations that can be solved to determine the displacements and stresses in a structure. This is done by considering the virtual work done by the external forces and the virtual work done by the internal forces, and setting them equal to each other.

## 3. Can the principle of virtual work be applied to any type of structure?

Yes, the principle of virtual work can be applied to any type of structure, as long as it is in equilibrium. This includes structures made of different materials, with complex geometries, and subjected to various loading conditions.

## 4. Are there any limitations to using the principle of virtual work in the finite element method?

One limitation of using the principle of virtual work in the finite element method is that it assumes linear behavior of the materials and small displacements. This means that it may not accurately predict the behavior of structures under large deformations or non-linear material properties.

## 5. What are the benefits of using the principle of virtual work in the finite element method?

The principle of virtual work allows for a systematic and efficient approach to solving complex structural problems. It also provides a way to validate the results obtained from the finite element method, as the principle is based on the fundamental laws of mechanics. Additionally, it allows for the analysis of structures with different boundary conditions and loading scenarios.

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