In this lecture, we'll cover:

• The distinction between the finite element equations, the finite element method, and the direct stiffness method
• How finite element equations are used to derive element stiffness matrices
• How to directly derive the stiffness matrix for an axially loaded bar element
• How the stiffness matrix relates nodal forces to nodal displacements
• How element stiffness matrices form the foundation of the direct stiffness method for analysing whole structures

In this lecture, we set the scene for the section by clarifying the key terminology and the overall roadmap. We distinguish between the finite element method as a broad analytical framework, the direct stiffness method as a specific implementation, and the finite element equations as the tools used to derive element stiffness matrices. We explain that while we can directly derive the stiffness matrix for a simple axial bar, finite element equations become essential for more complex elements such as beams and higher-dimensional elements.

We then focus on understanding the element stiffness matrix in a concrete way, beginning with a simple spring and Hooke’s law. By extending the force–displacement relationship from a single spring to a two-node axial bar element, we show how the constant of proportionality generalises into a matrix. We derive the local bar stiffness matrix and interpret its physical meaning: it relates nodal forces to nodal displacements, ensures equilibrium, and captures how differences in displacement lead to axial force. Finally, we position this stiffness matrix at the centre of the wider process, highlighting that once individual element stiffness matrices are known, the direct stiffness method allows us to assemble them into a structure stiffness matrix and ultimately analyse complete structures.

Next up:

In the next lecture, we begin deriving the finite element equations formally using the Principle of Minimum Potential Energy applied to a general 3D element.