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

τ
pr
τ abs max = +
2t
p
2
591
STRAINS IN THIN-wALLEd
PRESSuRE VESSELS
Out-of-plane
Mohr’s circles
R = pr p
2t
+ 2
In-plane
Mohr’s circle
σ radial = −p
pr
σ long = 2t
σ hoop =
σ
pr
t
FIGURE 14.8
Mohr’s circle for inner surface of a cylinder.
The maximum out-of-plane shear stresses occur on planes that are rotated at 45° with respect
to the radial direction. The magnitude of these stresses is
1
τ = σ − σ =
⎛ pr
−
⎞
⎝ t ⎠ = pr
2 ( ) 1
0
2
2t
absmax hoop radial (14.7)
Stresses on the Inner Surface
The stresses σ long and σ hoop act on the inner surface of the cylindrical pressure vessel, and
these stresses are the same as those on the outer surface because the vessel is assumed to
be thin walled (Figure 14.6). Pressure inside the vessel pushes on the cylinder wall, creating
a normal stress in the radial direction equal in magnitude to the internal pressure. Consequently,
the inner surface is in a state of triaxial stress and the third principal stress is
equal to σ p3 = σ radial = −p.
Mohr’s circle for the inner surface of a cylindrical pressure vessel (subjected to an
internal gage pressure) is shown in Figure 14.8. The maximum in-plane shear stresses on
the inner surface are the same as those on the outer surface. However, the maximum outof-plane
shear stresses on the inner surface are increased because of the radial stress caused
by the pressure:
1
τ = σ − σ =
⎡ pr
− − p
⎤
⎣⎢ t ⎦⎥ = pr
t
+ p
2 ( ) 1
( )
2
2 2
absmax hoop radial (14.8)
14.4 Strains in Thin-Walled pressure Vessels
Since pressure vessels are subjected to either biaxial stress (on outer surfaces) or triaxial
stress (on inner surfaces), the generalized Hooke’s law (Section 13.8) must be used to relate
stress and strain. For the outer surface of a spherical pressure vessel, Equations (13.24) can
be rewritten in terms of the tangential stress σ t :
1 1 pr pr pr
εt = ( σt − υσ t ) =
⎛
−υ ⎞
(1 υ)
E
E⎝
2t
2t
⎠ = 2tE
− (14.9)