Plane Stress and Plane Strain

1. Introduction

In solid mechanics, simplifying a three-dimensional stress or strain state to a two-dimensional approximation is often valid and practical, especially in aerospace structural analysis. Two such common idealizations are:

• Plane Stress – Suitable for thin flat elements like aircraft skins.
• Plane Strain – Applicable to long structural members where deformation in one direction is restricted.

2. Plane Stress Condition

2.1 Assumptions

Plane stress assumes negligible stress in the thickness direction (say, zz-axis):

Non-zero stress components:

2.2 Strain Relations

The 3D strain–stress relations reduce to:

Where:

• = Young’s modulus
• = Poisson’s ratio
• = Shear modulus

3. Plane Strain Condition

3.1 Assumptions

Plane strain assumes zero strain in the zz-direction:

Non-zero strain components:

Typical in long bodies where the length dimension is constrained (e.g., long fuselage sections or nozzles).

3.2 Stress–Strain Relations

From Hooke’s law in 3D:

Solving for :

So, out-of-plane stress is non-zero, even though .

4. Effective Elastic Moduli (Plane Strain)

In many numerical methods (like FEM), plane strain uses modified elastic constants:

• Effective Young’s Modulus:

Effective Poisson’s Ratio:

5. Comparison Table

Feature	Plane Stress	Plane Strain
Common Scenario	Thin plates, aircraft skins	Long, constrained structures
Governing Equations	2D Hooke’s Law	Modified 3D Hooke’s Law

6. Relevance in Aerospace Applications

• Plane Stress:
  • Used for thin sheets under in-plane loads (e.g., fuselage skins, wing panels).
  • Critical in analyzing riveted and composite skin structures.
• Plane Strain:
  • Found in long aerospace structures where end constraints dominate (e.g., nozzle flanges, spars under axial compression).
  • Also relevant in structural dynamic simulations where transverse motion is minimal.