In this lecture, we'll cover:

• How to calculate the local element stiffness matrix for the membrane contribution,
• How to use Gauss numerical integration with a two-by-two sampling scheme,
• How to define the shape functions and their derivatives for a four-node quadrilateral element,
• How to construct the Jacobian matrix and its inverse to map between natural and physical coordinates,
• How to build the $B$ matrix and assemble the membrane part of the stiffness matrix in code,

In this lecture, we work through how to turn the theoretical expression for the membrane stiffness into code. We start by setting up a four-node quadrilateral element, defining its material properties, thickness, nodal coordinates, and the Gauss integration points and weights. We then focus on the membrane strain-displacement matrix and the plane stress constitutive matrix, and we use these to prepare the terms needed for the stiffness calculation.

We also work carefully through the role of the Jacobian, showing how it connects derivatives with respect to the natural coordinates to derivatives with respect to the physical element coordinates. From there, we build the membrane $B$ matrix by placing the appropriate derivative terms into the correct positions for each node, then evaluate the stiffness contribution at each Gauss point and accumulate the result. The key idea is that we are sampling $B^T D B$ over the element and summing those contributions to form the membrane stiffness matrix.

Next up

In the next lecture, we will build the bending and shear stiffness matrices by following the same approach.