In this lecture, we'll cover the following:

• How stress resultants (moments and shear forces) are derived from stress distributions through the plate thickness
• The formulation of the stress resultant vector and its components (bending moments, torsion, and transverse shear forces)
• How to construct the generalised constitutive matrix by integrating through the thickness
• The separation of bending and shear contributions within the constitutive formulation
• The introduction and role of the shear correction factor $(5/6)$ in improving transverse shear predictions
• How these developments complete the mechanical description of a plate element in preparation for stiffness matrix formulation

In this lecture, we establish the crucial link between stresses and stress resultants by integrating stress components across the plate thickness. We show how bending moments, torsional moments, and transverse shear forces emerge naturally from these integrations, and we reformulate the relationship in terms of generalised strains and a generalised constitutive matrix. By exploiting the properties of the transformation matrix and separating bending and shear behaviour, we obtain compact expressions for the constitutive matrices, including the $(t^3/12)$ scaling for bending and a thickness-proportional term for shear.

We also address the limitation of assuming constant transverse shear stress through the thickness and introduce the shear correction factor $(5/6)$ to account for the true parabolic distribution. With this correction, we ensure accurate prediction of shear forces. This lecture completes the mechanical foundation of the plate element: we now have displacement fields, strain–displacement relationships, and stress–strain (and stress resultant) formulations fully defined, setting the stage for developing the element stiffness matrix.

Next up

Next, we will explore shape functions, which provide the interpolation framework needed to express displacements and rotations within an element.