Transcript Normal Stress (1.1-1.5)
1
Thick-Walled Cylinders
(Notes,3.14) MAE 316 – Strength of Mechanical Components NC State University Department of Mechanical and Aerospace Engineering Thick-Wall Cylinders
Cylinders
(3.14)
r
Applications p i = 0 Submarine Vacuum chamber Shrink fit Buried pipe 2
σ r σ l z
p o p i r i r o σ r
σ t
p o = 0 Gun barrel Liquid- or gas-carrying pipe Hydraulic cylinder Gas storage tank Thick-Wall Cylinders
Thin-Walled Pressure Vessels
(Review)
t
pr i t
(hoop stress)
l
pr i
2
t
(longitudinal stress) p t r i r o For a thin-walled pressure vessel, r i /t > 10, so “hoop” stress (
σ t
) variation in the radial direction is minimal
σ r σ t
Radial stress (
σ r
) is equal to -p on the inner surface, zero on the outer surface, and varies in between.
σ r
is negligible compared to
σ t
.
3 Thick-Wall Cylinders
Thick-Walled Cylinders
(3.14) For thick-walled pressure vessels
r
t
i
2
i
2 2
r o i
2
r r o
2
r i
2 2 (
p o
p i
) /
r
2 2
r o i
2
r r o
2
r i
2 2 (
p o
p i
) /
r
2 p o Maximum shear stress max 1 2 (
t
r
) p i If the ends of the cylinder are capped, must include longitudinal stress.
l
r o i
2 2
r i
2 2 4 Thick-Wall Cylinders
σ r σ t
t r i r o
Thick-Walled Cylinders
Examples of closed cylinders include pressure vessels and submarines.
Examples of open cylinders include gun barrels and shrink fits.
Radial displacement of a thick-walled cylinder
u r
1
E
(
i r o i
2
o r i
2 E Young's modulus
o
) Poisson's ratio 1
E
(
p i
(
r o
2
o
)
i i
)
o
2 u r 5 Thick-Wall Cylinders
Thick-Walled Cylinders
(3.14) Special case: Internal pressure only (p o = 0)
r
r r o
2
i
2
p i r i
2 1
r o
2
r
2 &
t
r r o
2
i
2
p i r i
2 1
r o
2
r
2 (
t
(
r
) max ) max
p i
p i
@ (
r i
2 (
r o
2
r o
2
r i
2 ) )
r
r i
@
r
r i u r
E
(
p i r i r o
2 2
r r i
2 ) ( 1 ) ( 1 )
r o
2
r
2 (
u r
)
r
r i
p i r i E
r o
2
r o
2
r i
2
r i
2 (
u r
)
r
r o
E
2 (
r o p i r i
2 2
r o r i
2 ) 6 Thick-Wall Cylinders p i r i r o
σ r σ t σ t /p i
Thick-Walled Cylinders
Compare previous result with thin-walled pressure vessel case (p o = 0)
t
p i
(
r i
2 (
r o
2
r o
2
r i
2 ) ) @
r
r i r o t
t
t r i p i
(
r o
(
r i
2
i
for t
r o
2 )
o
r i
)
p i r i
2 (
r i
walled) (
r i
t
) 2
i
t
p i
t
r i
2
r i
2
p i r t i p r i i t
(inside) 2
r i
2
r t i
p i
σ r
r i
σ t
r o 7 Thick-Wall Cylinders
Thick-Walled Cylinders
Continued…
t
(
r o
2 2
p r i i
r i
2 2 ) @
t r o
i
o r i t r
r o
t
(
r o
for t 2
p r i i
2
i o
i
r i
r (thin ) (
r i
2
p r i i
walled) 2
i
t
2
p r i i
2
r t i
t
p r i i t
(outside)
t
same on inside and outside 8 Thick-Wall Cylinders p i r i r o
σ r σ t
Thick-Walled Cylinders
Special case: External pressure only (p i = 0)
r
r o r
2
o
2
p o r i
2
r i
2
r
2 1 &
t
r o r
2
o
2
p o r i
2 1
r i
2
r
2 (
r
) max
p o
@ (
t
) max (
r
2
o
2
p r o o
r i
2 2 )
r
r o
@
r
r i u r
E
(
p o r o
2
r o
2
r r i
2 ) ( 1 ) ( 1 )
r i
2
r
2 (
u r
)
r
r i
(
u r
)
r
r o
E
2
p o
(
r o
2
r o
2
r i
2
r i
2 )
p o r o E
(
r o
2 (
r o
2
r i
2 )
r i
2 ) p o 9 Thick-Wall Cylinders r i r o
σ r σ t
Example
Find the tangential, radial, and longitudinal stress for a pipe with an outer diameter of 5 inches, wall thickness of 0.5 inches, and internal pressure of 4000 psi.
10 Thick-Wall Cylinders
Example
Find the maximum allowable internal pressure for a pipe with outer radius of 3 inches and wall thickness of 0.25 inches if the maximum allowable shear stress is 10000 psi.
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Press and Shrink Fits
(3.16) MAE 316 – Strength of Mechanical Components NC State University Department of Mechanical and Aerospace Engineering Press and Shrink Fits
Press and Shrink Fits
(3.16) Press together or shrink inner p u ri r i r f + δ u ro p r f r o Inner member (external pressure only) Outer member (internal pressure only) Assume inner member has slightly larger outer radius than inner radius of outer member.
Interference pressure will develop upon assembly.
Press and Shrink Fits
Press and Shrink Fits
(3.16)
u ri u ro
pR E i
R
2
R
2
r i
2
r i
2
pR E o
r o r o
2 2
R
2
R
2
i
o
(inner) (outer) For compatibility
u ro
u ri
pR
1
E o
r o
2
r o
2
R
2
R
2
o
1
E i
R
2
R
2
r i
2
r i
2
i
Once
δ
is known we can calculate p, or vice versa.
Typically,
δ
is very small, approximately 0.001 in. or less.
15 Press and Shrink Fits
Press and Shrink Fits
(3.16) If the materials are the same: E = E i
υ = υ i
= E
= υ o
o 2
pR
3
E
(
r o
2 (
r o R
2 2 )(
r i
2 )
R
2
r i
2 ) If the inner member is not hollow, r i = 0.
2
pR E
(
r o
2
r o
2
R
2 ) 16 Press and Shrink Fits
Example
A solid shaft is to be press fit into a gear hub. Find the maximum stresses in the shaft and the hub. Both are made of carbon steel (E = 30x10 6 psi, ν = 0.3).
Solid shaft r i = 0 in, R = 0.5 in. (nominal) Tolerances: +2.3x10
-3 /+1.8x10
-3 in.
Gear hub R = 0.5 in. (nominal), r o = 1 in Tolerances: +0.8x10
-3 /0 in.
17 Press and Shrink Fits
Example
A bronze bushing 50 mm in outer diameter and 30 mm in inner diameter is to be pressed into a hollow steel cylinder of 100 mm outer diameter. Determine the tangential stresses for the steel and bronze at the boundary between the two parts.
E b E s ν = 105 Gpa = 210 Gpa = 0.5
radial interference
δ
= 0.025 mm 18 Press and Shrink Fits