In this lecture, we'll cover the following:

• How to compute the substitute transverse shear strain-displacement matrix in practice.
• Identification and dimensions of each contributing matrix term.
• Derivation of the natural shear interpolation matrix from assumed shear strain field.

In this lecture, we work through how to actually compute the substitute transverse shear strain-displacement matrix, focusing especially on the central term $(A P^{-1} T)$. We begin by reviewing the dimensions and roles of each matrix in the formulation, identifying which components can be precomputed (such as the Jacobian-based $C$ matrix and the stacked $B_s$ matrices at mid-side sampling points) and which must be evaluated within the standard Gauss point loop. This helps us isolate the main challenge: determining the combined $(A P^{-1} T)$ term efficiently.

We then reinterpret $(A P^{-1} T)$ as an interpolation matrix for transverse shear strains in natural coordinates. By revisiting the assumed shear strain field and sampling it at appropriate points, we derive explicit expressions for the interpolated shear strains. From this, we construct the natural shear interpolation matrix directly in a compact $2 \times 8$ form, avoiding the need to compute $A$, $P$, and $T$ separately. By the end of the lecture, we have all components required to assemble the substitute transverse shear matrix and are ready to implement the formulation in code, with only minor modifications needed to the shear stiffness calculation.

Next up

In the next lecture, we will implement these derivations in code, modifying the shear stiffness calculation and validating the results against an OpenSeesPy benchmark.