In this lecture, we'll cover:

• How to evaluate the integrals for each of the generalised constitutive matrices,
• Why membrane–bending coupling vanishes at the element level for a homogeneous, symmetric middle plane,
• How the Reissner–Mindlin shear correction factor of $(5/6)$ modifies the shear constitutive matrix,
• How these results fit into the larger goal of building the element stiffness matrix, $K$.

In this lecture, we work through the generalised constitutive matrices one by one and perform the required integrations. For bending, we find that the matrix becomes $t^3/12$ times the usual plane stress constitutive matrix; for membrane action, it becomes $t$ times the same plane stress matrix; and for shear, it becomes $t$ times the shear constitutive matrix, with an important correction factor. We also show that the membrane–bending coupling term integrates to zero when the middle plane is a neutral plane and the material is homogeneous, which means there is no membrane–bending coupling at the element level under these assumptions.

We then introduce the $(5/6)$ shear correction factor to account for the Reissner–Mindlin assumption of constant transverse shear stress through the thickness. This factor adjusts the shear contribution so that the total work done by shear is correct, even though the local distribution is simplified compared with the true parabolic stress profile. By the end of the lecture, we have the generalised constitutive matrices needed for element stiffness calculations, and are ready to move on to the strain–displacement matrix in the next lecture.

Next up

In the next lecture, we see how to build the strain–displacement matrix, $B$.