Plane Stress in Solid Mechanics: Reducing 3D Problems to 2D

Overview

Plane stress is a simplification used in solid mechanics for thin components where all stresses act in a single plane. In this video we learn that stresses in the thickness direction are negligible (sigma Z, tau XZ, tau YZ are zero), reducing a 3D problem to a 2D problem described by sigma X, sigma Y and tau XY. The discussion covers why this is useful, how the 2D stress state is represented, and the necessary engineering judgment since real Z-direction stresses are not exactly zero.

Key takeaway

When the plate is thin and loaded in the XY plane, plane stress can greatly simplify analysis by collapsing the stress tensor from a 3x3 to a 2x2 form, enabling easier solutions for common components such as plates, vessels, and gears.

Plane Stress in Solid Mechanics

The video introduces plane stress as a common simplification in solid mechanics, enabling the reduction of a three-dimensional stress state to a two-dimensional problem for certain geometries and loading conditions. A component is in plane stress when all stresses lie in the same plane, which is often a valid approximation for thin structures. In a typical stress analysis at a point, six components define the stress state: three normal stresses and three shear stresses. Under plane stress, the components in the thickness direction vanish: sigma Z, tau XZ, and tau YZ are zero. This leaves only sigma X, sigma Y, and tau XY as nonzero components, turning a three-dimensional problem into a two-dimensional one and reducing the 3x3 stress tensor to a simpler 2x2 form. The video emphasizes that this is an approximation and that engineering judgment is required to assess its applicability in real problems.

Three practical illustrations are discussed to motivate when plane stress can be used: perforated plates, pressure vessels with thin walls, and spur gear teeth. In the perforated plate example, loads are applied in the XY plane, but the thickness direction still must be considered to determine whether Z stresses are negligible. The top and bottom faces of a thin plate have zero normal and shear stresses on free surfaces, which supports the plane stress assumption since stress variations through thickness are small if the plate is thin. For pressure vessels, a thin wall relative to the diameter causes radial stresses to be near zero, reinforcing plane stress viability. Gear teeth are another potential candidate when the gear width is narrow enough. The video concludes with a summary: plane stress is a useful simplification that turns a 3D solid mechanics problem into a 2D one by assuming zero stresses in one direction, typically applicable to thin in-plane loaded structures.