In this lecture, we'll cover:

• The concept of plane strain as a simplification of full three-dimensional behaviour
• Why plane strain is appropriate for long structures with negligible strain in one direction
• Practical examples such as tunnels and retaining walls
• The strain assumptions in plane strain (which components are zero)
• How these assumptions reduce the stress–strain (constitutive) matrix
• The distinction between plane strain and plane stress, particularly the role of $\sigma_{zz}$

In this lecture, we focus on the concept of plane strain as a simplification of full three-dimensional elastic behaviour. We begin by revisiting the basic definition of strain and use it to explain why negligible strain arises in structures that are very long in one direction. From this, we establish that plane strain is typically assumed for long structures whose geometry does not change along their longitudinal direction, such as tunnels and retaining walls. We use a retaining wall example to visualise how the significant deformations and strains occur in the transverse (x–y) plane, while strain in the longitudinal (z) direction is negligible.

We then translate this physical understanding into mathematical assumptions. In a state of plane strain, we set the normal strain in the z-direction and the shear strains involving the z-direction to zero. This allows us to reduce the full three-dimensional constitutive $[\mathbf{C}]$ matrix to a simplified form appropriate for plane strain. Importantly, we clarify that plane strain does not imply plane stress: the normal stress $\sigma_{zz}$ is not necessarily zero. However, if we choose to neglect stresses in the z-direction for practical purposes, the stress–strain relationships can be simplified further. This completes our treatment of the two common material behaviour models, plane stress and plane strain, that we will use in subsequent analyses of elastic behaviour.

Next up:

In the next lecture, we consolidate all of the material models developed so far into a concise summary.