In this lecture, we'll cover the following:

• Deriving the strain–displacement matrix $B$ from generalised strain definitions.
• Expressing rotations and displacements using interpolation (shape) functions.
• Rewriting generalised strains in terms of nodal displacements.
• Identifying the structure, dimensions, and partitioning of the $B$ matrix.
• Highlighting the need for the Jacobian to compute shape function derivatives.

In this lecture, we focus on constructing the strain–displacement matrix $B$ by expressing the generalised strains in terms of nodal displacements. We begin with the known strain definitions for bending and transverse shear, and systematically substitute interpolation (shape) functions for the displacement and rotation fields. This allows us to rewrite all strain components as functions of nodal values, which naturally leads to the identification of the $B$ matrix as the operator linking nodal displacements to generalised strains.

We then examine the structure of $B$, noting its dimensions and how it is assembled per node before forming the full matrix for the element. We distinguish between bending and shear contributions within the matrix and show how it integrates into the broader formulation, including its role in computing stress resultants. Finally, we recognise a key computational challenge: evaluating derivatives of shape functions with respect to physical coordinates, which motivates the introduction of the Jacobian matrix in the next lecture.

Next up

In the next lecture, we will see how the Jacobian matrix enables computation of the shape function derivatives needed to assemble the $B$ matrix.