Uniformly varying stress

Finally, the complete solution of and will be of the form,

(3.35)

Notice that the terms containing represent a displacement which would be possible in a rigid body.

If the cylinder is not displaced by rotation we can omit .

If there is no lateral displacement, .

Since, the plane through does not get displaced, then . The displacements are then given as:

(3.36)

Any cross-section of the cylinder distorts into a paraboloid of revolution about the vertical axis.

Plane strain

In a state of plane strain the displacements are functions of only and displacement vanishes. All components of strain and stress are independent of , so that ;

(3.37)

The situation of plane strain occurs in bodies of long cylindrical form.

Plane stress

The state of plane stress occurs in a planer body.

In general, the stress components are equal to zero but the displacements and are not independent of .

(3.38)