PlaneStress

Plane stress is a two-dimensional state of stress in which the stresses perpendicular to a given plane are neglected. It is commonly used for thin plates and shell structures where the thickness is small compared to in-plane dimensions and bending-induced through-thickness stresses are limited. In this state, the normal stress in the thickness direction, σzz, and the shear stresses on planes normal to that direction, τxz and τyz, are taken as zero. The in-plane stresses σxx, σyy and the in-plane shear τxy (or γxy) can be nonzero.

Constitutive relations for plane stress follow from three-dimensional elasticity with σzz = τxz = τyz = 0. For isotropic materials, the in-plane strain–stress relations are εxx = (1/E)(σxx − νσyy), εyy = (1/E)(σyy − νσxx), and γxy = (1/G)σxy, where G = E/[2(1+ν)]. The out-of-plane normal strain εzz is not constrained to zero and will adjust so that compatibility and constitutive requirements are satisfied.

Governing equations of equilibrium reduce to two in-plane balance equations: ∂σxx/∂x + ∂τxy/∂y + fx = 0 and ∂τxy/∂x + ∂σyy/∂y + fy = 0, where fx and fy are body forces per unit volume. In numerical analysis, plane-stress elements are used in finite element methods to model thin structures.

Applications include automotive panels, aircraft skins, and electronic enclosures. Limitations arise for thick bodies or regions where through-thickness stresses are significant, or near free edges where σzz and τxz, τyz cannot be neglected.