In this section, we'll cover:

• Deriving an expression for the stiffness matrix of an axially loaded bar
• Clarifying the meaning of the finite element equations and the direct stiffness method
• Using the principle of minimum potential energy to derive the finite element equations
• Constructing the element stiffness matrix from the finite element equations
• Transforming the stiffness matrix from local to global coordinates to assemble the primary stiffness matrix

In this section, we focus on determining the stiffness matrix for an axially loaded bar, which relates the forces applied at the ends of the bar to the resulting displacements. We begin by clarifying what we mean by the finite element equations and how they relate to the direct stiffness method, setting the conceptual foundation for what follows. We then outline the derivation of the finite element equations using the principle of minimum potential energy, an approach that not only underpins bar elements but extends to more complex elements such as beams and 2D or 3D elements.

Once we have established the finite element equations, we use them to construct the stiffness matrix for an axially loaded bar. Finally, we address the essential step of transforming the stiffness matrix from its local coordinate system to a global reference frame so that multiple element matrices can be assembled into the primary stiffness matrix representing the entire structure. By the end of this section, we understand both the origin of the finite element equations and how they are used in practice to model structural behaviour.

Next up:

In the next lecture, we clarify the distinction between the finite element equations, the finite element method, and the direct stiffness method, and derive the element stiffness matrix for a simple bar.