Stress and Strain - Structural and Stress Analysis - page 179

7.11 P LANE S TRESS

In some situations the behaviour of a structure, or part of it, can be regarded as
two-dimensional. For example, the stresses produced in a flat plate which is sub-
jected to loads solely in its own plane would form a two-dimensional stress system;
in other words, a plane stress system. These stresses would, however, produce strains
perpendicular to the surfaces of the plate due to the Poisson effect (Section 7.8).

An example of a plane stress system is that produced in the walls of a thin cylindrical
shell by internal pressure. Figure 7.22 shows a long, thin-walled cylindrical shell sub-
jected to an internal pressure p . This internal pressure has a dual effect; it acts on the
sealed ends of the shell thereby producing a longitudinal direct stress in cross sections
of the shell and it also tends to separate one-half of the shell from the other along
a diametral plane causing circumferential or hoop stresses. These two situations are
shown in Figs. 7.23 and 7.24, respectively.

Suppose that d is the internal diameter of the shell and t the thickness of its walls. In
Fig. 7.23 the axial load on each end of the shell due to the pressure p is

π d 2
4
This load is equilibrated by an internal force corresponding to the longitudinal direct
stress, σ L , so that

p ×

p π d 2
4

σ L π dt

=

which gives

pd
4 t

σ L = (7.62)

p
p

p

L

F IGURE 7.22 Thin cylindrical shell under internal
pressure

p

F IGURE 7.23 Longitudinal
stresses due to internal pressure

s L