EN224: Linear Elasticity
Division of Engineering
6. 2D Boundary Value Problems III: Plane problems
Anti-plane shear problems are nice and simple, but we rarely load a solid so as to cause anti-plane shear deformation. The assumption of in-plane deformation is more useful.
Plane stress and plane strain solutions to the governing equations of linear elasticity approximate the following three dimensional boundary value problem.
Consider the cylindrical solid shown above.
The boundary conditions on will be left unspecified for the time being.
6.1 Plane Strain Approximation
Since the loading appears to cause the solid to deform transverse to the axis of the cylinder, it is natural to attempt to approximate the solution by assuming in-plane deformations.
The state of strain follows as
while the stress state (for an isotropic solid) is
The equilibrium equations reduce to
and the boundary conditions on B become
We can now examine the boundary conditions on . Evidently
Thus, the plane strain solution is the exact solution to a three dimensional problem involving a cylinder whose ends are constrained by rigid frictionless walls.
Generalized plane strain
The plane strain solution may be corrected to provide approximate solutions to cylindrical solids with other end constraints. As an example, suppose we wish to find an approximate solution for a three dimensional solid with traction free.
Let represent the exact solution to the three dimensional boundary value problem.
Let represent the plane strain solution.
Suppose that where is a correction that must be added to the plane strain solution in order to satisfy the boundary conditions on . Consider the boundary conditions associated with :
We cannot easily find the exact solution to , but we may easily find a Saint-Venant solution, which will be accurate away from the ends of the cylinder, by enforcing the boundary conditions in weak form.
Evidently, the weak form of corresponds to either case I or case II in our list of Saint-Venant problems (i.e. a cylinder loaded by axial force and/or bending moments at its ends). Thus, the approximate solution to is
and are the principle moments of area about the centroid, and the coordinate system is chosen so that are measured from the centroid and are aligned with the principal axes of inertia.
The main conclusion is that the correction only affects the axial stress component.