Energy Methods

Derivation of stiffness as defined from an energy approach.

From the direct approach, we saw that:

employing hat functions (linear piece wise functions)

where u is the displacement, L is the element length, t is time, is the hat function. The potential energy inner product determines K, without proof the potential energy (V) is defined as:

where , substituting results in

evaluating this integral yields , and applying the limits of integration

this can be placed in matrix form, and dropping the t for convenience, yields.

or