In this lecture, we'll cover the following:

• How to compute stress resultants using the relation between the constitutive matrix, strain–displacement matrix, and nodal displacements
• How local coordinates (r, s) determine where stress resultants are evaluated within an element
• How to extract element-level nodal displacements from global displacement vectors
• How to construct and reuse the strain–displacement matrix for each evaluation point
• How to handle duplicate stress resultant values at shared grid points by averaging

In this lecture, we walk through the full process of evaluating stress resultants across an element mesh using previously developed finite element components. We apply the relationship between the generalised constitutive matrix, the strain–displacement matrix, and the nodal displacement vector to compute bending moments and shear forces at specific points. A key idea is that the strain–displacement matrix depends on local coordinates (r, s), which allows us to evaluate stress resultants at any location within an element by varying these coordinates across an evaluation grid.

We then implement this numerically by looping over all evaluation points, extracting the relevant element displacements, constructing the strain–displacement matrix, and assembling the stress resultants. A practical issue arises because multiple elements contribute values at shared global coordinates, leading to duplicate but slightly different results. To address this, we identify duplicates using coordinate-based masking and compute averaged values, producing a clean, unique dataset of stress resultants across the grid. This prepares us for visualisation in the next stage.

Next up

Next, we will visualise these bending moments using heat maps and explore the Wood and Armer method for computing practical design moments.