EN224: Linear Elasticity

Division of Engineering

6.2 Plane Stress Approximation

We may find a similar approximate solution to thin plates stretched in their own plane.

Consider the cylindrical solid shown above. Assume that the height of the cylinder is much smaller than any relevant cross-sectional dimension.

Find

With

To derive the plane stress field equations, we make two approximations:

1. Assume

The justification for this approximation is that on to satisfy the boundary conditions. In addition, the equilibrium equation

shows that on , since on . Thus, we expect

2. Find field equations for the through thickness averages of the stress and displacement components.

Assumption (1) allows us to determine the out of plane strain component

Thus, the field equations reduce to

Now take a thickness average of these, and note that

So that

Where

The boundary conditions reduce to

The field equations for plane stress are almost identical to those for plane strain, except for the term involving Poisson’s ratio in the constitutive law.

Note that, while the plane strain solution is an exact solution to a three dimensional boundary value problem, the plane stress solution is an approximate solution, and is exact only in the limit of vanishing plate thickness. The through thickness averages of field quantities may or may not approximate the actual fields (for example, if the plate is loaded so that it bends, then there would be a significant variation in stress and strain through the thickness).