18_Principle of virtual work

The book(Darly L.Logan - A First Course in the Finite Element Method) explains the principle of virtual work as follows.

 

If a deformable body in equilibrium is subjected to arbitrary virtual (imaginary) displacements associated with a compatible deformation of the body, the virtual work of external forces on the body is equal to the virtual strain energy of the internal stresses.

 

In other words, the change in internal energy is equal to the amount of work done externally.

 

 

Where $\delta {U^{(e)}}$ is the virtual strain energy due to internal stresses and $\delta {W^{(e)}}$ is the virtual work of external forces on the element. 

In matrix form.

 

 

where $\delta \left\{ d \right\}$ is the vector of virtual nodal displacements, $\delta \left\{ \psi  \right\}$ is the vector of virtual displacement functions $\delta u$, $\delta v$, and $\delta w$, $\delta \left\{ {{\psi _s}} \right\}$ is the vector of virtual displacement functions acting over the surface where surface tractions occur, $\left\{ P \right\}$ is the nodal force matrix, $\left\{ {{T_s}} \right\}$ is the surface force per unit area matrix, and $\left\{ X \right\}$ is the body force per unit volume matrix.

equation can be summarized as follows.

 

 

Shape functions are used to relate displacement to nodal displacements.

 

 

where $\left[ {{N_s}} \right]$ is the shape function matrix evaluated on the surface where traction $\left[ {{T_s}} \right]$ occure.

Strains and stresses are related to nodal displacements and strains as follows.

 

 

Therefore equation can be summarized

 

 

In static problems, $\left\{ {\ddot d} \right\} = 0$

 

 

 

Therefore, the equation $KU = F{\text{ }}$ can be derived.