Overview

In this lecture, we will cover the following:

• Why 3D problems are often approximated as 2D problems
• The concept of a plane stress condition
• The assumptions required for a structure to be in plane stress
• How plane stress affects the 3D stress–strain relationship
• The derivation and interpretation of the reduced $[\mathbf{D}]$ and $[\mathbf{C}]$ matrices for plane stress
• The distinction between plane stress and plane strain

In this lecture, we explore the idea of plane stress as a simplified 2D representation of a 3D stress state. We begin by recognising that full 3D problems are computationally more complex, and that in engineering practice we frequently apply simplifying assumptions to reduce this complexity. Plane stress applies when one dimension of a structure (such as the thickness of a plate) is very small compared with the other two, and when loading is applied only in-plane. Under these conditions, stresses acting on the faces normal to the thin direction are negligibly small and can reasonably be taken as zero.

We then examine the implications of this assumption for the 3D stress–strain equations. By setting the out-of-plane normal and shear stresses to zero, we simplify the constitutive relationship and obtain a reduced matrix form appropriate for plane stress. Importantly, we see that although out-of-plane stresses vanish, the out-of-plane strain does not necessarily equal zero due to Poisson’s ratio effects. This highlights the key distinction between plane stress and plane strain. Finally, we derive the reduced stiffness $[\mathbf{C}]$ matrix for a plane stress element and place it in the broader context of material modelling, showing how it emerges as a simplification of the full 3D formulation.

Next up:

In the next lecture, we examine the complementary simplification — the plane strain condition — which applies to long structures where strain in one direction is negligible.