Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

600
PRESSuRE VESSELS
Since the ends of the cylinder are open, the longitudinal stress σ long = σ x = 0. Therefore,
a condition of plane stress exists in the cylinder wall. From the generalized Hooke’s
law, the relationship between the longitudinal strain ε long and the radial and circumferential
stresses can be expressed as that for an open-ended cylinder, namely,
ε long
ν =−
E ( σ r + σ θ ) = constant
Poisson’s ratio ν and the modulus of elasticity E are constants for the material constituting
the cylinder. Hence, the sum of the radial and circumferential stress must equal a constant
value. For convenience, we will denote this constant as 2α:
σ + σ = constant = 2α
The circumferential stress can be expressed as
r
θ
θ = 2 − r
(a)
σ α σ
The expression for σ θ from Equation (a) can be substituted into Equation (14.12) to give
Equation (b) may be written in the form
Integration of Equation (c) yields
dσr
σr
− α
+ 2 = 0
(b)
dr r
dσ
r dr
=− 2
(c)
σ − α r
r
β
ln( σr − α) = − 2lnr
+ lnβ
= ln
2
r
where β is a constant of integration. After exponentiation, we can express the radial stress
as
β
σr = α + (d)
2
r
The constants α and β are evaluated in terms of the pressures at the inner and outer
surfaces, respectively, of the cylinder. On the inner surface at r = a, the radial normal stress
is equal to the magnitude of the internal pressure; that is, σ r = –p i . The minus sign is used
to convert a positive pressure to a compressive normal stress. On the outer surface at r = b,
the radial normal stress is equal to the magnitude of the external pressure; that is, σ r = –p o .
From these known boundary conditions, the values of α and β can be determined as
2 2
a pi
− b po
α =
2 2
b − a
2 2
ab( pi
− po)
β =−
2 2
b − a
(e)