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

p14.20 The cylindrical pressure vessel shown in Figure
P14.20/21 has an inside diameter of 610 mm and a wall thickness
of 3 mm. The cylinder is made of an aluminum alloy that has an
elastic modulus of E = 70 GPa and a shear modulus of G = 26.3 GPa.
Two strain gages are mounted on the exterior surface of the cylinder
at right angles to each other; however, the angle θ is not known.
If the strains measured by the two gages are ε a = 360 µε and ε b =
975 µε, respectively, what is the pressure in the vessel? Notice that,
when two orthogonal strains are measured, the angle θ is not needed
to determine the normal stresses.
p14.21 The cylindrical pressure vessel shown in Figure
P14.20/21 has an inside diameter of 900 mm and a wall thickness
of 12 mm. The cylinder is made of an aluminum alloy that has an
elastic modulus of E = 70 GPa and a shear modulus of G = 26.3 GPa.
Two strain gages are mounted on the exterior surface of the cylinder
at right angles to each other. The angle θ is 25°. If the pressure
in the vessel is 1.75 MPa, determine
FIGURE p14.20/21
90° θ
x
(a) the strains that act in the x and y directions.
(b) the strains expected in gages a and b.
(c) the normal stresses σ n and σ t .
(d) the shear stress τ nt .
t
b
y
a
n
14.5 Stresses in Thick-Walled cylinders
Equilibrium
The determination of stresses in a thick-walled cylinder was first accomplished by the French
mathematician Gabriel Lamé in 1833. His solution models a thick cylinder as a series of thin,
adjacent cylinders that each exert stress on the adjacent cylinders. Figure 14.9a represents the
cross section of a relatively long, open-ended, thick-walled cylinder that has an inside radius
a and an outside radius b. The cylinder is subjected to an inside pressure p i and an outside
pressure p o . The circumferential stress σ θ and the radial stress σ r are to be determined at any
radial distance r from the central axis O of the cylinder. Because of the symmetry of the cylinder
about the central axis O, there will be no shear stress on the planes on which σ θ and σ r
act. Consequently, the circumferential and radial stresses are principal stresses.
A free-body diagram that exposes both the circumferential and radial normal stresses
is required. To expose the radial stresses, we will consider a thin cylinder of inner radius r,
thickness dr, and arbitrary longitudinal length ∆x. The circumferential stresses are exposed
by cutting this cylinder in half with a vertical diametric plane. The free-body diagram
shown in Figure 14.9b shows the forces acting on the annular element expressed in terms
of σ θ and σ r . The condition of symmetry dictates that σ θ and σ r are independent of θ. Both
σ θ and σ r are assumed to be tensile normal stresses.
In the lateral direction (i.e., the z direction), the resultant force due to the radial stress
σ r acting on the inner surface of the annular element is to be determined. Consider a differential
element located at θ that has dimensions of dr in the radial direction, r dθ in the
circumferential direction, and ∆z in the longitudinal direction. The product of the radial
stress σ r and the area of the differential element r dθ ∆z gives the radial force on the inner
surface. The z component of this radial force is found by multiplying by sin θ. Thus, the
resultant force in the z direction created by σ r on the inner surface is found from
π
∫ ( r∆ x) σrsinθdθ = σr2r∆x
0
Similarly, the radial force acting on the outer surface of the annular element is found
from the radial stress σ r + dσ r acting on the area of the differential element (r + dr) dθ ∆z.
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