In this lecture, we'll cover:

• Why the strain–displacement matrix $B$ must be evaluated using derivatives in different coordinate systems,
• How the Jacobian matrix maps derivatives from natural coordinates $(r,s)$ to element coordinates $(x',y')$,
• How to assemble the full element $B$ matrix by stacking the nodal contributions,
• How Gauss numerical integration is used to evaluate the stiffness matrix,
• How sampling points, weighting factors, and the Jacobian determinant are combined in the integration scheme.

In this lecture, we focus on how to calculate the element stiffness matrix. We start by looking at the $B$ matrix and the problem of differentiating shape functions, which are naturally expressed in the local $(r,s)$ coordinate system, while the stiffness formulation needs derivatives with respect to $x'$ and $y'$. To bridge that gap, we use the Jacobian matrix and its inverse, which lets us transform the readily calculated $r$- and $s$-derivatives into the derivatives needed for $B$. We also note that the full element $B$ matrix is built by placing the nodal $B$ matrices side by side.

We then turn to the integration itself and show how Gauss numerical integration replaces the area integral with a weighted sum over a small number of sampling points. For a two-by-two scheme, we evaluate the integrand at four specific points in the natural coordinate space, use the corresponding weights, and include the determinant of the Jacobian to account for the area transformation from $\mathrm{d}A$ to $\mathrm{d}r\,\mathrm{d}s$. The key idea is that the stiffness matrix is obtained by sampling $B^T D B$ at each Gauss point, multiplying by the weights and Jacobian determinant, and summing the results.

Next up

In the next section, we will begin coding up the shell element stiffness matrix step by step.