Kinetic & strain energy formulas can be used as a basis for Rayleigh-Ritz or finite element analysis of vibrating structures.
Equations for common structures and loading conditions are given in strain_energy.pdf
The cases include:
Uniform Bar under a Constant Axial Load
Bar under a General Axial Load
Beam in Bending
Circular Rod in Torsion
Beam or Rod in Shear
Rectangular Plate in Bending
Rectangular Plate in Bending & Shear, Stress-Strain Formulation
Circular Plate in Bending
Rectangular Plate, In-plane Displacement
Axial Spring
Ring, In-Plane, Extensional Displacement
See also: Continuum Mechanics
– Tom Irvine