A pressurized cylindrical ring might have been discussed since the field equations can be integrated directly.

**LamÃ©â€™s Solution
for Rings under Pressure **

A pressurized
cylindrical ring might have been discussed in previews pages since the field
equations can be integrated directly. However, since it and derivative
solutions are so elegant, practical, and intriguing because of their optimal
properties, this simple case has been reserved as an introduction to this
chapter devoted to this general class of structures.

Consider a thick ring
as shown in Figure 7.1a of outer radius, *b*, and inner radius, *a*,
subjected to uniform internal and external pressures *p _{a}* and

Thus the equilibrium equations reduce to
a single requirement:

and from the conjugate relationship
between the first invariant and elastic rotation [Equation (5.13)]

This stress
distribution for internal and external pressure considered sepa-rately is
plotted in Figure 7.1c.

The solution by the stress function
approach is equally straightforward. Since nothing can be a function of Î¸,
the stress function must be of type I.

We have seen, when considering the pure
bending of a ring, that the *B* term contributes to the displacements by
introducing [Equation (6.28)] an elastic rotation * *Ï‰_{z}= 4BÎ¸/E which is not single-valued and therefore not
possible for complete rings. Therefore, taking B as zero, the stresses are:

giving the same solution for the stress field [Equation (7.3)].

Since there are no shears or rotation,
all points move radially outward and

Clearly this LamÃ©
solution* leads to many practical applications. For thin rings *b* - *a* = *t* << *r*, the strength-of-materials
approximation: Ïƒ* _{r}* =0 (assumed) ÏƒÎ¸ = (

**Example 7.1**

A thick, oval, cylindrical pressure
vessel has the cross-sectional dimensions shown below. Assume plane stress. By
superpoisition of elasticity solutions, determine the maximum stress and the
yield pressure.

Note: Slight discrepancy at *B*
where the neutral axis of the beam is at the centroid and of the ring at - e
= - .057*R.*
However, this section is not critical and resulting error at *C* is, by
St. Venantâ€™s principle, negligible.

Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail

Civil : Principles of Solid Mechanics : Rings, Holes, and Inverse Problems : Lameâ€™s Solution for Rings under Pressure |

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