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### Thick Wall Cylinder Press or Shrink Fits Equations and Calculator

Pressure Vessel Design Formula and Calculators Resources

**Thick Wall Cylinder Press or Shrink Fits Interference and Pressure Equations and Calculator**

If two thick-walled cylinders are assembled by either a hot/cold shrinking or a mechanical press-fit, a pressure is developed at the interface between the two cylinders. At the interface between the two cylinders, at a radius (R), the outside cylinder, or collar, increases an

amount (δ_{c}) radially, and the inside cylinder, or shaft, decreases an amount (δ_{s}) radially.

Figure 1

Geometry of press fit cylinders

Preview: Thick Walled Cylinder Press Fit Interference Calculator

Using the geometry of Figure 1, the increase in the outside cylinder, or collar, radially (δ_{c}) is given by:

Eq. 1

δ_{c} = ( ( p R ) / E_{c} ) [ ( r^{2}_{o} + R^{2} ) / ( r^{2}_{o} - R^{2} ) + *v*_{c} ]

decrease in the inside cylinder, or shaft, radially (δ_{s} ) is given by:

Eq. 2

δ_{s} = ( ( p R ) / E_{s} ) [ ( R^{2 } + r^{2}_{i} ) / ( R^{2 } - r^{2}_{i} ) + *v*_{s} ]

The difference between the radial increase (δ_{c}) of the collar, a positive number, and the radial decrease (δ_{s} ) of the shaft, a negative number, is called the radial interference (δ) at the interface (R) and is given by:

Eq. 3

δ = δ_{c} + | δ_{s} | = ( ( p R ) / E_{c} ) [ ( r^{2}_{o} + R^{2} ) / ( r^{2}_{o} - R^{2} ) + *v*_{c} ] + ( ( p R ) / E_{s} ) [ ( R^{2 } + r^{2}_{i} ) / ( R^{2 } - r^{2}_{i} ) + *v*_{s} ]

When the radial interference (δ) is determined from a particular fit specification, Eq. 3

If the collar and shaft are made of the same material, then the modulus of elasticity’s and Poisson ratio’s are equal and so Eq. (3) can be rearranged to give an expression for the interface pressure (p) given in Eq. (4).

Eq. 4

p = ( E δ ) / R [ ( ( r^{2}_{o} + R^{2} ) ( R^{2 } - r^{2}_{i} ) ) / ( 2 R^{2} ( r^{2}_{o} + r^{2}_{i} ) )]

If the inner shaft is solid, meaning the inside radius (r_{i}) is zero, then Eq. (4) for the interface pressure (p) simplifies to the expression in Eq. (5).

Eq. 5

p = ( E δ ) / ( 2 R ) [ 1 - ( R / r_{o} )^{2} ]

Where:

E_{c} = Modulus of elasticity collar (psi, MPa),

E_{s} = Modulus of elasticity shaft (psi, MPa),

*v*_{c} = Poisson's ratio collar,

*v*_{s} = Poisson ratio shaft,

δ = radial interference (in, m),

R = Interface radius (in, m),

r_{i} = inside radius (in, m),

*r _{o}* = external radius (in, m)

Related:

- Thick Walled Cylinder Stress Pressure Vessel Equations and Calculator
- Pressure Vessel External Pressure Calculations
- Pressure Vessel, Thin Wall Longitudinal Stress Calculator
- Pressure Vessel, Thin Wall Hoop and Longitudinal Stresses Equation and Calculator
- Pressure Vessel Design Calculations Handbook
- Identifying and Reducing Stresses in Pressure Vessels

Source:

Marks' Calculations for Machine Design,

Thomas H. Brown, Jr. Ph.D., PE

Faculty Associate

Institute for Transportation Research and Education

NC State University

Raleigh, North Carolina

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