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Plane State of Strain

Some common engineering problems such as a dam subjected to water loading, a tunnel under external pressure, a pipe under internal pressure, and a cylindrical roller bearing compressed by force in a diametral plane, have significant strain only in a plane; that is, the strain in one direction is much less than the strain in the two other orthogonal directions. If small enough, the smallest strain can be ignored and the part is said to experience plane strain.

Assume that the negligible strain is oriented in the z-direction. To reduce the 3D strain matrix to the 2D plane stress matrix, remove all components with z subscripts to get,

where exy = eyx by definition.

The sign convention here is consistent with the sign convention used in plane stress analysis.

Coordinate Transformation
The transformation of strains with respect to the {x,y,z} coordinates to the strains with respect to {x',y',z'} is performed via the equations,

The rotation between the two coordinate sets is shown here,

where q is defined positive in the counterclockwise direction.