## Approximate Methods: Virtual Work Method

The statement of the equilibrium equations applied to a set is as follows. Assuming that at equilibrium is the symmetric Cauchy stress distribution on and that is the displacement vector distribution and knowing the relationship , then the equilibrium equation seeks to find such that the associated satisfies the equation:

where is the body forces vector distribution on , is the mass density, and is the space of all possible displacement functions applied to , i.e., . The term “Kinematically admissible” in indicates that the space of possible solutions must satisfy the boundary conditions imposed on (as stated below) and any differentiability constraints.

The boundary conditions for the equations of equilibrium are usually given on two parts of the boundary of denoted . On the first part, , the external traction vectors are known so we have the boundary conditions for since ( is the normal vector to the boundary). On the second part, , the displacement is given.

Alternatively, the statement of the principle of virtual work states that at equilibrium, where is the space of at least once differentiable vector functions on , i.e., :

where

As shown in the principle of virtual work section, one of its major applications is to find approximate solutions for the equations of equilibrium. This can be obtained by assuming that the solution u has a particular form with a finite number of unknowns, i.e., by looking for in a subset that is finite dimensional but still able to approximate functions in . The example in the principle of virtual work section utilizes polynomial approximations to find an approximate solution for the displacement field. The finite element method in the next sections utilizes piecewise affine functions to approximate the displacements. In this case the set is expanded to allow for possible displacement functions that are not differentiable across element boundaries: .

### Problems

Solve the problems in the Rayleigh Ritz method section using the Virtual Work Method.