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

σ θ dr ∆x
dθ
599
STRESSES IN THICk-wALLEd
CyLINdERS
a
b
p
O i
r
dr
z
y
O
θ
σ r·2r ∆x
r
(σ r +dσ r )·2(r +dr)∆x
σ θ
σ r
dr
σ r
σ θ
σ r
p o
FIGURE 14.9a Cross section of a
thick-walled cylinder.
σ θ dr ∆x
FIGURE 14.9b Free-body diagram exposing
circumferential and radial normal stress.
The resultant force in the z direction created by σ r + dσ r on the outer surface of the annular
element can be expressed as
π
∫ ( r + dr) ∆ x( σr + dσr)sin θdθ = ( σr + dσr)2( r + dr)
∆x
0
Note that the thickness dr of the annular element and the incremental radial stress dσ r are
independent of θ and are thus constants in this integration.
The circumferential normal stress σ θ acts on the exposed z surfaces (i.e., on the y axis)
at the top and bottom of the annular element. The area exposed at each of these sections is
dr ∆x. Since the thickness dr may be made infinitesimally small, σ θ may be considered
constant over the thickness of the element. Thus, the resultant force that is created by σ θ
acting in the z direction on each exposed surface is σ θ dr ∆x.
The sum of all forces in the z direction can be expressed as
∑ F = 2σ dr∆ x + σ ⋅2 r∆x − ( σ + dσ
)2( r + dr) ∆ x = 0
z θ
r r r
Since 2∆x appears in each term, it can be cancelled out. Further, the second-order term
dσ r dr is negligibly small. Therefore, the relationship between the radial and circumferential
stresses can be expressed as a function of the radius r as
Lamé Solution
dσr
σr
− σ θ
+ = 0
(14.12)
dr r
It is reasonable to assume that strains in the longitudinal direction (i.e., the x direction in
Figure 14.9b) of the thick-walled cylinder are equal. This means that plane transverse sections
which are flat and parallel before pressures p i and p o are applied remain flat and parallel
after the pressures are applied. This assumption will be true for at least a cylinder with
open ends, and it will also be nearly true for a closed cylinder at sections well removed
from the ends of the cylinder 1 .
1
Seely, Fred B., and James O. Smith. Advanced Mechanics of Materials, 2nd ed. (New York: John Wiley &
Sons, Inc., 1952.)