Page 300 - Introduction to Continuum Mechanics

284 Thick-walled Circular Cylinder under Internal and External Pressure

5.18 Thick-walled Circular Cylinder under internal and External Pressure
Consider a circular cylinder subjected to the action of an internal pressure p/ and an external
pressure p 0. The boundary conditions for the plane strain problem are:

These boundary conditions can be easily shown to be satisfied by the following stress field

These components of stress are taken from Eq. (5.17.10) with B = 0 and represent therefore,
a possible state of stress for the plane strain problem, where T^ = v (T^ + TQQ). We note that
4Br6 2
if B is not taken to be zero, then ua = —=— (1 -v ) which is not acceptable because if we start
by
from a point at 6 =0, trace a circuit around the origin and return to the same point, 0 becomes
2jt and the displacement at the point takes on a different value. Now applying the boundary
conditions given in Eqs. (5.18.1), we find that

We note that if only the internal pressure />/ is acting, T n is always a compressive stress and
TQQ is always a tensile stress.

The above stress components together with T zz = v (7^ + TQQ) constitute the exact plane
strain solution for the cylinder whose axial end faces are fixed.
As discussed in the last section, the state of stress given by Eqs. (5.18.3) above and with
=
^ b® regarded as an approximation to the problem of a cylinder which is very
TZZ 0» can a so
thin in the axial direction, under the action of internal and external pressure with traction-free
end faces. However, the strain field is not given by Eq. (5.17.11), which is for the plane strain
case. For the plane stress case,