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Plane stress versus plane strain

Short answer

Plane stress applies to thin plates loaded in their plane: sigma_z=0 while epsilon_z is nonzero, example a gusset plate. Plane strain applies to long bodies with constant transverse load: epsilon_z=0 while sigma_z is nonzero, examples a dam or tunnel. The constitutive matrix uses different E and nu relations.

Differentiate between plane stress and plane strain problems with suitable examples, conditions and necessary figures.

Definitions

Plane stress: a thin plate loaded only by forces in its own plane, so the stresses through the thickness are zero: σz=τxz=τyz=0\sigma_z=\tau_{xz}=\tau_{yz}=0. The stress state is two-dimensional, {σ}={σx,σy,τxy}T\{\sigma\}=\{\sigma_x,\sigma_y,\tau_{xy}\}^T, although the strain εz=−νE(σx+σy)≠0\varepsilon_z=-\dfrac{\nu}{E}(\sigma_x+\sigma_y)\neq0.

Plane strain: a long prism (length much larger than cross-section) with uniform cross-section and load that does not vary along its length and is perpendicular to it. Movement along the axis is prevented, so εz=γxz=γyz=0\varepsilon_z=\gamma_{xz}=\gamma_{yz}=0, but σz=ν(σx+σy)≠0\sigma_z=\nu(\sigma_x+\sigma_y)\neq0.

Comparison

PointPlane stressPlane strain
GeometryThin plate, thickness ≪\ll other dimensionsLong body, length ≫\gg cross-section
LoadingIn the plane of the plate, uniform over thicknessPerpendicular to the length, constant along it
Zero componentsσz=τxz=τyz=0\sigma_z=\tau_{xz}=\tau_{yz}=0εz=γxz=γyz=0\varepsilon_z=\gamma_{xz}=\gamma_{yz}=0
Non-zero out-of-planeεz=−ν(σx+σy)/E\varepsilon_z=-\nu(\sigma_x+\sigma_y)/Eσz=ν(σx+σy)\sigma_z=\nu(\sigma_x+\sigma_y)
[D][D] matrixE1−ν2[1ν0ν10001−ν2]\dfrac{E}{1-\nu^2}\begin{bmatrix}1&\nu&0\\\nu&1&0\\0&0&\frac{1-\nu}{2}\end{bmatrix}E(1+ν)(1−2ν)[1−νν0ν1−ν0001−2ν2]\dfrac{E}{(1+\nu)(1-2\nu)}\begin{bmatrix}1-\nu&\nu&0\\\nu&1-\nu&0\\0&0&\frac{1-2\nu}{2}\end{bmatrix}
ExamplesDeep beam, gusset plate, plate with a hole, shear wallGravity dam, retaining wall, tunnel, long embankment, strip footing

The plane strain matrix is obtained from the plane stress one by replacing E→E/(1−ν2)E\to E/(1-\nu^2) and ν→ν/(1−ν)\nu\to\nu/(1-\nu).

Figures

 Plane stress           Plane strain
   thin plate           long dam / tunnel
  ->+------+<-          ||||||||||||| load
  ->|      |<-          +-----------+   z (long)
  ->+------+<-         ==============>  no strain
  t small, sz=0         ez = 0

In plane stress the plate is free to change thickness; in plane strain each slice of unit thickness of the long body is prevented from changing length by neighbouring slices.

✓ Answer checked by Claude Opus 5.5 (high effort)

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