Related Variational and Energy Principles - Mechanics of Structures: Variational and Computational Methods - page 151

FIGURE 3.8
Curved bar with in-plane loading.
EXAMPLE 3.11 Flexibility and Stiffness of a Beam
In Example 3.2, it was shown that a stiffness coefficient k ij can be considered to be the force
developed at i due to a unit displacement at j with other end displacements set to zero.
This definition lends itself to implementation using Castigliano's second theorem. Equation
(3.12) provides a displacement in terms of the applied loading. This type of relationship
is referred to as a flexibility equation. In contrast to flexibility, stiffness is an expression for
a force in terms of the displacements. Castigliano's theorem, part II, leads to flexibility
relationships, which can be converted to stiffness equations.
To illustrate the use of Castigliano's theorem, part II in deriving a stiffness matrix, consider
a beam. It would appear that there are two independent displacements ∗ —one deflection
∗ Chapter 4 contains a discussion on the subject of the independence of these two displacements.