In this lecture, we'll cover the following:

• How to calculate the element stiffness matrix in code.
• How to compute the Jacobian matrix from shape function derivatives and nodal coordinates.
• How to transform derivatives from local to global coordinates using the Jacobian.
• How to construct the strain–displacement $(B)$ matrix node by node.
• How to apply Gauss quadrature to numerically integrate and assemble the stiffness matrix.
• How the number of sampling points affects accuracy and convergence of the stiffness matrix.

In this lecture, we walk through the full implementation of the element stiffness matrix for a four-node plate element. We translate the governing equation into code by iterating over Gauss quadrature sampling points, computing shape functions and their derivatives at each point, and assembling contributions to the stiffness matrix. A key step is forming the Jacobian matrix from nodal coordinates and shape function derivatives, which allows us to transform derivatives from local to global coordinates and compute its determinant and inverse for use in later steps.

We then construct the strain–displacement matrix, $(B)$, by combining derivatives of shape functions with respect to global coordinates and assembling contributions node by node into a full $5\times 12$ matrix. With the $B$ matrix, Jacobian determinant, and constitutive matrix in place, we evaluate and accumulate the stiffness contributions at each sampling point. Finally, we explore how increasing the number of Gauss sampling points affects the result, observing clear convergence and noting that two sampling points are sufficient despite higher-order schemes offering marginal gains at increased computational cost.

Next up

Next, we will refine this implementation by separating the stiffness matrix into distinct bending and shear contributions, providing greater flexibility for later use.