In this lecture, we'll cover:

• How to define stress resultants in a local reference frame,
• How to group stress resultants into membrane forces, bending moments, torsion, and transverse shear forces,
• How to visualise these resultants on an infinitesimal plate element,
• How to derive the stress resultant vector by integrating stresses through the thickness,
• How to form a generalised local constitutive matrix and identify its membrane, bending, shear, and coupling parts.

The lecture begins by shifting from stresses to stress resultants. We separate the resultants into three membrane forces, three bending/torsional moments, and two transverse shear forces, and we relate each group to the physical action it represents on a plate element. We then show how the resultants arising from normal stresses and in-plane shear stresses act on the element, including the interpretation of torsion and in-plane shear distortion.

We then move on to the derivation, where we integrate the local stress vector through the thickness to obtain the stress resultants. Using the same transformation matrix previously introduced for strains, we replace the local stresses with the constitutive matrix times the generalised strain vector. This leads us to a generalised local constitutive matrix, which we begin to decompose into membrane, bending, shear, and membrane-bending coupling contributions. By the end of the lecture, you will understand how the constitutive matrix ties stresses and strains together so that these resultants can be computed.

Next up

In the next lecture, we will evaluate the integrals for each of these constitutive sub-matrices and introduce the shear correction factor needed for the Reissner–Mindlin formulation.