In this lecture, we'll cover the following:

• How to express the interpolated shear strain within an element in terms of shear strains at selected sample points
• How to transform shear strains between natural and Cartesian coordinate systems using the Jacobian
• How to construct a shape function matrix that interpolates shear strains across the element
• How to relate shear strains at sample points to nodal degrees of freedom using standard shear strain–displacement matrices
• How to assemble the substitute transverse shear strain matrix that avoids shear locking

In this lecture, we work through the full derivation needed to express the transverse shear strain at any point in an element as a function of shear strains evaluated at carefully chosen sample points. We begin by combining earlier results to interpolate shear strains in natural coordinates, then use the Jacobian to map these into Cartesian coordinates. By systematically assembling these relationships, we show that the shear strain field can be written using a shape function matrix that interpolates values from the four edge midpoints of the element.

We then connect these sampled shear strains to the nodal degrees of freedom by stacking standard transverse shear strain–displacement matrices evaluated at each sample point. Bringing all components together, we derive the substitute transverse shear strain matrix. Crucially, we emphasise that this formulation eliminates shear locking due to the strategic placement of sample points, allowing us to use standard $2\times 2$ Gauss quadrature to eliminate zero-energy modes. This provides a stable and accurate framework for evaluating the shear stiffness matrix, setting up the next step of actually computing it in practice.

Next up

Next, we will focus on the practical computation of the substitute transverse shear strain matrix, deriving it in a compact form that avoids the need to compute each sub-matrix separately.