1. Variational formulation of the FEM Principle of Stationary Potential Energy: Among all admissible displacement functions u, the actual ones are those which render the total potential energy  stationary  W  + P,where is strain energy function, external forces potential. Meaning of the variables: - displacement -strain -stress -external body forces -external surface forces

2. Illustrative Example: Using the principle of stationary potential energy, evaluate the displacement u 0 of the end point of spring on the Fig.1 Given: spring stiffness k, loading force F Accumulated strain energy in the spring: Potential of external force: Total potential energy: Its stationary value renders the trivial result Fig.1 Loaded spring Fig.2 shows clearly, that the equilibrium displacement u 0 corresponds to minimum potential energy: Fig.2 Displacement vs. energy

3. Discretization of continuum – basic idea State and characteristics of continuum are described by continuous functions – e.g. displacements u(x,y,z), v(x,y,z), w(x,y,z). To solve any problem on digital computer, it must be first discretized – continuous functions must be expressed by finite number of scalar parameters. In the most popular displacement version of FEM, unknown functions of displacement are approximated with the help of apriori selected, known simple functions - so called shape functions, which are defined on each element: Shape functions are multiplied by unknown coefficients a i, b j, c k, or deformation parameters, which must be evaluated. Inserting the approximation into the functional  (u,v,w), we obtain  as a function of a finite number of parameters  (a 1,a 2,a 3,...). Principle of stationary value of  then leads to a system of equations for unknown values of deformation parameters: