PRINCIPLE  OF  VIRTUAL WORK

Consider a body in equillibrium under the action of a set of loads and constraints.  The action of the loads produces a dispalcement u, a function of the coordinates i.e.u= u(x,y,z).   Under the action of this load the work done = Load x displacement = Work, W.  This work must equal the strain energy V developed in the body due  to the deformation.  This strain energy may be computed from the stresses and strains developed in the body energy = stress x strain  integrated over the whole body.

    V = W
Now imagine an additional imaginary (virtual) displacement  as shown below. This virtual displacement is arbitrary except where there is a constraint. At locations  with a constraint the imaginary displacement is 0. This requirement is often stated as " the virtual displacements are compatible". The virtual displacement will also cause virtual strains.
Clearly the applied load will now do additional work  Total Work = External Load x (actual displaceemnt + virtual displacement) = Actual Work + External Virtual Work. = W + EVW

Similarly the Total Strain energy = stress x (actual strain + virtual strain) = Actual strain energy +  internal virtual strain energy = V + IVSE.
Equating   W + EVW = V + IVSE
But W  = V, thus

EVW = IVSE

This is the principle of virtual work !  External virtual work equals the internal virtual strain energy.
DERIVATION OF ELEMENT STIFFNESS USING PVW
We now illustrate the use of the above principle in deriving element stiffness matrices.

Step 1 Assume displacement (deformation) say linear. Thus u looks like


 
 


Ni, Nj are called shape functions.

Now we can construct a  variation (assumed displacement).
 

Step 2 Strain   and Stress  We can rewrite
 
  Similarly


 
 

 Step 3 PVW
Internal Virtual  Strain Energy = External Virtual Work

Internal Virtual Strain Energy is


 

External Virtual Work is


 

Equating  the internal  virtual strain energy and external virtual work and cancelling the virtual displacements on both sides we have:
 

 

We note here that this method of deriving the element stiffness matrix is very general and as long as expressions for work and energy are available. The same method can be used for all types of problems irrespective of dimnesion (2D/3D) complexity of the problem (thermal effects, magnetic effects, coupled flow and structure etc.).