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Title: Thick-Walled Cylinders (Notes,3.14)


1
Thick-Walled Cylinders(Notes,3.14)
  • MAE 316 Strength of Mechanical Components
  • NC State University Department of Mechanical and
    Aerospace Engineering

2
Cylinders (3.14)
  • Applications
  • pi 0
  • Submarine
  • Vacuum chamber
  • Shrink fit
  • Buried pipe
  • po 0
  • Gun barrel
  • Liquid- or gas-carrying pipe
  • Hydraulic cylinder
  • Gas storage tank

3
Thin-Walled Pressure Vessels (Review)
  • For a thin-walled pressure vessel, ri/t gt 10,so
    hoop stress (st) variation in the
    radialdirection is minimal
  • Radial stress (sr) is equal to -p on the inner
    surface, zero on the outer surface, and varies in
    between.
  • sr is negligible compared to st.

(hoop stress)
(longitudinal stress)
t
sr
4
Thick-Walled Cylinders (3.14)
  • For thick-walled pressure vessels
  • Maximum shear stress
  • If the ends of the cylinder are capped, must
    include longitudinal stress.

t
5
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

6
Thick-Walled Cylinders (3.14)
  • Special case Internal pressure only (po 0)

st/pi
7
Thick-Walled Cylinders
  • Compare previous result with thin-walled pressure
    vessel case (po 0)

8
Thick-Walled Cylinders
  • Continued

9
Thick-Walled Cylinders
  • Special case External pressure only (pi 0)

10
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.

11
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.

12
Press and Shrink Fits(3.16)
  • MAE 316 Strength of Mechanical Components
  • NC State University Department of Mechanical and
    Aerospace Engineering

13
Press and Shrink Fits (3.16)
Press together or shrink inner
Inner member (external pressure only)
Outer member (internal pressure only)
  • Assume inner member has slightly larger outer
    radius than inner radius of outer member.
  • R is the shared radius between the two pieces
    before they are pressed together.
  • Interference pressure will develop upon assembly.

14
Press and Shrink Fits (3.16)
Press together or shrink inner
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.

14
15
Press and Shrink Fits (3.16)
  • Once d is known we can calculate p, or vice
    versa.
  • Typically, d is very small, approximately 0.001
    in. or less.

16
Press and Shrink Fits (3.16)
  • If the materials are the same
  • E Ei Eo
  • ? ?i ?o
  • If the inner member is not hollow, ri 0.

17
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 30x106
    psi, ? 0.3).
  • Solid shaft
  • ri 0 in, R 0.5 in. (nominal)
  • Tolerances 2.3x10-3/1.8x10-3 in.
  • Gear hub
  • R 0.5 in. (nominal), ro 1 in
  • Tolerances 0.8x10-3/0 in.

18
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.
  • Eb 105 Gpa
  • Es 210 Gpa
  • ? 0.5
  • radial interference d 0.025 mm
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