Design and Stress Analysis of Extraterrestrial ... - The Black Vault

Let us examine the equilibrium of an element taken from a shell
(see Fig. 2.31) with central angle d,, for which within this element
we separate 'he inner shell from the outer shell. Gas pressure p,
pressure in the inner shell space pf 1 , and contact pressure P,, which
is an averaged pressure of the forces of the connection between the
shells, act on the internal shell.
PressurespM and p act on the external shell; we disregard the
pressure on the external surface (if it is substantial, it must be
taken into account). If we applied the vectors of the peripheral
stresses a' and a", we write the condition for shell element
equilibrium, indicated in Fig. 2.31, according to formula (2.33).
We obtain
M ,P-xrf hU" = P, + j ,
where p is
the current value of the gas or liquid pressure.
Eliminating (p,., + p R) from these equations we obtain
S(2.39)
In this equation there are two unknowns a' and al; we supplement
equation (2.39) with the strain compatibility equation.
A peculiarity of strains in an axisymmetric (in this case,
cylindrical) element is
the inter-relationship (usually called
compatibility) of the strains in
radial and peripheral directions.
Let us examine the elongation of element ab (Fig. 2.33) up to the
dimensions a 1 bl; in view of the axial symmetry, points a and a,, b and
bI remain on the radii of this element.
The full relative strain of the element in
C is easily determined:
a peripheral direction
S~ n
-S I , I -