Stress and Strain - Structural and Stress Analysis - page 180

t

s C

p

s C

d

F IGURE 7.24 Circumferential stress due to internal
pressure

s C

s C

s L

s L s L

s L

s C

F IGURE 7.25
Two-dimensional stress
system

s C

(a) (b)

Now consider a unit length of the half shell formed by a diametral plane (Fig. 7.24).
The force on the shell, produced by p , in the opposite direction to the circumferential
stress, σ C , is given by

p × projected area of the shell in the direction of σ C

Therefore for equilibrium of the unit length of shell

2 σ C × (1 × t ) = p × (1 × d )

which gives

pd
2 t

σ C = (7.63)

We can now represent the state of stress at any point in the wall of the shell by con-
sidering the stress acting on the edges of a very small element of the shell wall as
shown in Fig. 7.25(a). The stresses comprise the longitudinal stress, σ L , (Eq. (7.62))
and the circumferential stress, σ C , (Eq. (7.63)). Since the element is very small, the
effect of the curvature of the shell wall can be neglected so that the state of stress may
be represented as a two-dimensional or plane stress system acting on a plane element
of thickness, t (Fig. 7.25(b)).

In addition to stresses, the internal pressure produces corresponding strains in thewalls
of the shell which lead to a change in volume. Consider the element of Fig. 7.25(b).
The longitudinal strain, ε L , is, from Eq. (7.13)

σ L
E −

ν σ C
E

ε L =