In this lecture, we'll cover the following:

• How to compute the equivalent nodal force vector for distributed actions on plate elements.
• Application of Gauss quadrature to numerically integrate distributed loads.
• Construction of the shape function matrix and its role in load distribution.
• Implementation of the equivalent force vector calculation in Python.
• Validation using regular and irregular (quadrilateral) elements.

In this lecture, we walk through how to calculate the equivalent nodal force vector when distributed actions, such as surface loads, are applied to plate elements. We revisit the integral formulation and show how it is evaluated numerically using Gauss quadrature, incorporating weighting factors, shape functions, and the determinant of the Jacobian. We demonstrate how to construct the shape function matrix and use it to map distributed loads onto nodal forces.

We then implement this procedure in Python, first validating it on a simple rectangular element where the load distribution is uniform and predictable, and then applying it to a more general quadrilateral element where the distribution is non-uniform. Finally, we encapsulate the logic into a reusable function and integrate it into our growing utilities file. With this, we've assembled the key computational tools needed to move forward and build a full plate analysis solver in the next section.

Next up

Having developed the key element-level tools, the next section focuses on expanding these into a full multi-element plate solver.