Related Resources: calculators

### 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 (δc) radially, and the inside cylinder, or shaft, decreases an amount (δs) radially. Figure 1
Geometry of press fit cylinders

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

Eq. 1
δc = ( ( p R ) / Ec ) [ ( r2o + R2 ) / ( r2o - R2 ) + vc ]

decrease in the inside cylinder, or shaft, radially (δs ) is given by:

Eq. 2
δs = ( ( p R ) / Es ) [ ( R2 + r2i ) / ( R2 - r2i ) + vs ]

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

Eq. 3
δ = δc + | δs | = ( ( p R ) / Ec ) [ ( r2o + R2 ) / ( r2o - R2 ) + vc ] + ( ( p R ) / Es ) [ ( R2 + r2i ) / ( R2 - r2i ) + vs ]

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 [ ( ( r2o + R2 ) ( R2 - r2i ) ) / ( 2 R2 ( r2o + r2i ) )]

If the inner shaft is solid, meaning the inside radius (ri) 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 / ro )2 ]

Where:

Ec = Modulus of elasticity collar (psi, MPa),
Es = Modulus of elasticity shaft (psi, MPa),
vc = Poisson's ratio collar,
vs = Poisson ratio shaft,
δ = radial interference (in, m),
R = Interface radius (in, m),
ri = inside radius (in, m),
ro = external radius (in, m)

Related:

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