Pressure Vessels and Shrink Fits

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Pressure Vesselsand Shrink Fits
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Thin-Walled Pressure Vessels

• If the wall thickness is lt 1/10 the inner
  radius, the vessel may be considered thin-walled.
• In thin walled pressure vessels, the inner and
  outer radii are set equal to r, and the thickness
  is t.

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Thin-Walled Pressure Vessels

• The stress is assumed to be uniform throughout
  the thickness. Usually only two stress
  components are significant, axial and tangential.

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Thin-Walled Stresses (Review)

• Axial, or longitudinal stress, is
• Circumferential, or tangential stress, is

(also for a sphere)
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Thin-Walled Stresses (Review)

• Stress in the radial direction, ?r, varies
  from -p (p pressure) at the interior of the
  vessel to zero at the exterior (for internally
  pressurized vessels). ?r is usually ltlt than
  either ?? or ?a, and is usually neglected.

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Thick-Walled Pressure Vessels

• Thick-walled vessels often operate under much
  higher pressures than thin-walled, and the radial
  stress component cannot usually be ignored.
• Stresses vary through the wall thickness, unlike
  the assumption for thin-walled vessels.

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Thick-Walled Nomenclature
a inside radius b outside radius r
arbitrary radius u radial displacement pi
inside pressure po outside pressure ?r radial
stress ?? tangential stress ?z axial stress
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Thick-Walled, Internal Pressure (po 0)
?? is gt ?r. Note variation with r.
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Thick-Walled, Internal Pressure
Notes ?r is always compressive max. at r a.
(max at i.d.) ?? is always tensile, also max. at
r a.
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Thick-Walled, External Pressure (pi 0)
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Thick-Walled, External Pressure
Notes ?r is always compressive max. at r b.
(max. at o.d.) ?? is also compressive, max. at r
a.
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Thick-Walled, Axial (or Longitudinal) Stress
?z (pia2 pob2)/(b2 a2)
(For either internal or external pressure)
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Uses for Thick-Walled Equations

• Use is for high pressures in a thick walled
  vessel. Straightforward application of formulas.
• A second use is in the calculation of shrink
  fits, either for assembly or to create very
  strong composite structures. In these cases, the
  contact pressure p between the parts, is treated
  as po or pi in the preceding equations.

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Shrink Fit Nomenclature
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Determining Contact Pressure, p

• In many cases, the designer may choose a
  maximum tangential stress ??. Then, p may be
  solved for (as pi for the hub) using

Watch your as, bs, and cs!
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Determining Contact Pressure, p

• Otherwise, the designer may choose to specify
  an interference ?, or the contact pressure p
  itself and then solve for the necessary ? to
  achieve that p.

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Contact Pressure
If the hub shaft are of the same material, this
condenses to
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Shrink Fits

• Once p is determined, assume a friction factor
  f usually 0.15 lt f lt .20. The assembly force F
  to assemble a shrink-fit assembly is given by
  Eq.
• F 2?bpfL, where L is the length of the fit.
• Holding Torque T is given by
• T Fb 2 ?b2fpL

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Press Fit, Example Steel Shaft Press Fit Onto
Cast Fe Disc
a 25mm b 50 mm c 125 mm L 100 mm Es 210
Gpa ?s .3 Eh 70 Gpa ?h .25
Assumptions max. tangential stress NTX 30 MPa,
contact pressure is uniform, and f 0.15. FIND
radial interference ?, assembly force, torque
capacity.
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Flywheels
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Flywheels

• A flywheel is a typically a disc which rotates
  on a shaft. They are used to smooth out small
  oscillations and to store energy (kinetic energy
  of rotation). Examples cars, hybrids, punch
  press

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Flywheels

• Design and analysis is similar to what we have
  covered. An added component is consideration of
  the radial forces developed by the rotation a
  term of ??2 is introduced.