STATIONARY ENERGY PRINCIPLE

It is a fact for linear systems that the internal strain energy must equal the external work performed on the structure. For a single degree-of-freedom system, we can use this principle to solve for the displacement. However, for a multi degree-of-freedom system, a different approach is required. The energy plots, shown in Figure, illustrate that a new energy function  can be defined.

 

It is apparent that the solution at the point of minimum potential energy is where the internal energy equals the external energy. Therefore, the major advantage of the use of the potential energy function is that the solution must satisfy the following equation for all displacement degrees-of-freedom  :

The energy function written in matrix form is:

The resultant load vector R associated with the four types of loading is:

Therefore, the node equilibrium equation for all types of structural systems can be written as the following matrix equation:

 

The only approximation involved in the development of this equation is the assumption of the displacement patterns within each element. If the same displacement approximation is used to calculate the kinetic energy, the resulting mass matrix is termed a consistent mass matrix.

Another important fact concerning compatible displacement-based finite elements is that they converge from below, to the exact solution, as the mesh is refined. Therefore, the displacements and stresses tend to be lower than the exact values. From a practical structural engineering viewpoint, this can produce very dangerous results. To minimize this problem, the structural engineer must check statics and conduct parameter studies using different meshes.

Reference

Three-Dimensional Static and Dynamic Analysis of Structures A Physical Approach With Emphasis on Earthquake Engineering Edward L. Wilson