In this lecture, we'll cover:

• A consolidation of the steps used to derive the finite element stiffness matrix for a bar element
• The role of the strain–displacement matrix $[\mathbf{B}]$ and its relationship to the interpolation matrix $[\mathbf{H}]$
• A step-by-step review of how to construct the displacement interpolation matrix using a trial polynomial
• The process of forming the element stiffness matrix by integrating $[\mathbf{B}^T] [\mathbf{C}] [\mathbf{B}]$ over the element volume
• The significance of the trial displacement function and when the resulting stiffness matrix is exact or approximate
• A brief introduction to the need for local and global coordinate transformations in upcoming lectures

In this lecture, we consolidate the full procedure for deriving the element stiffness matrix using the finite element equation, focusing on the bar element as our working example. We revisit the key ingredients of the formulation: the material constant $[\mathbf{C}]$ (Young’s modulus in the case of a bar) and, most importantly, the strain–displacement matrix $[\mathbf{B}]$. We remind ourselves that strain is obtained from the derivative of displacement, and that displacement within an element is interpolated from nodal values using the interpolation matrix $[\mathbf{H}]$. By differentiating $[\mathbf{H}]$, we obtain $[\mathbf{B}]$, and from there we construct the stiffness matrix by integrating $[\mathbf{B}^T] [\mathbf{C}] [\mathbf{B}]$ over the element volume. When broken into clear steps — defining $[\mathbf{H}]$, differentiating to find $[\mathbf{B}]$, combining with $[\mathbf{C}]$, and integrating — the overall process becomes systematic and manageable.

We also reflect on how we construct the displacement interpolation matrix in the first place. We begin with a trial polynomial whose number of unknown coefficients matches the number of degrees of freedom, then use boundary conditions to express those coefficients in terms of nodal displacements. This allows us to write displacement anywhere within the element as an interpolation of nodal values. Importantly, we recognise that for a simple axially loaded bar, the chosen polynomial exactly represents the displacement field, which is why the finite element formulation produces the exact stiffness matrix. For more complex elements, however, the trial function may only approximate the true displacement field, meaning the resulting stiffness matrix is itself approximate. Finally, we preview the next step in the course: introducing local and global coordinate systems and the transformation matrices needed to analyse bars within larger, arbitrarily oriented structures.

Next up:

In the next lecture, we address coordinate transformations and how to convert element stiffness matrices from local to global coordinates so they can be assembled into a structure-level system.