In this lecture, we'll cover:

• What the strain–displacement matrix $B$ is and why it matters,
• How local element displacements are expressed using shape functions,
• How the membrane, bending and shear parts of the $B$ matrix are formed,
• How the derived $B$ matrix provides a key building block for the element stiffness matrix.

In this lecture, we work through the derivation of the local strain–displacement matrix $B$ for an element. We begin by defining $B$ as the matrix that links generalised strains at a point in the element, specified by the local coordinates $r$ and $s$, to the element’s nodal displacement vector. To get there, we first express the displacement field inside the element as a weighted sum of nodal displacements using the shape functions, which act as functions of the local coordinates and determine how much each node contributes at a given point.

We then substitute this displacement representation into the strain expressions and differentiate with respect to the local coordinate directions. Doing this shows that the entries of $B$ are built from derivatives of the shape functions, and that the matrix can be separated into membrane, bending and shear components for each node. By the end of the lecture, we will have fully derived the local generalised strain–displacement matrix.

Next up

In the next lecture, we will revisit and review the element stiffness matrix formulation.