This module calculates the contact pressure between two isotropic elastic cylindrical parts subject to an interference fit (e.g., press fit, shrink fit).  The modules also calculates the change in radii of the members, the resulting tangential stresses, and maximum torque that can be transmitted by the interference.

An "interference fit" is used when two cylindrical parts are assembled by shrink-fitting or press-fitting one part upon another (a common means of coupling a hub to a shaft). A press fit is obtained by machining the hole in the hub (the outer member) to a slightly smaller diameter than that of the shaft (the inner member, note that the shaft does not have to be solid). The two parts are then forced together slowly using a press (normally with oil applied at the intersection to act as a lubricant). The subsequent elastic deformation of both the shaft and the hub act to create large normal and frictional forces between the parts. The frictional force transmits the shaft torque to the hub and also resists axial motion. 

Only relatively small parts can be press-fitted without exceeding the capacity of typical presses. Therefore, shrink-fitting is used for larger parts. A shrink fit is made by heating the hub to temporarily expand its inside diameter and/or an expansion fit can be made by cooling the shaft to temporarily reduce its outside diameter. The heated and cooled parts can then be slipped together with minimal axial force, and when the assembly returns to room temperature, the dimensional changes in the parts creates the required interference for frictional contact. 

Note that the amount of interference needed to create a tight fit varies with the diameter of the shaft and the application. The radial interference, d, is the difference between the change in radii of the hub and shaft. A common standard is to assume approximately 0.001 to 0.002 units of diametral interference (note that the radial interference is one half the diametral interference) per unit of shaft diameter, with the low end of the range being applied to large diameter shafts. 

There are two standards for fits and tolerances in the United States, one based on inch units (ANSI Standard B4.1-1967, R87) and the other on metric units (ANSI B4.2-1978, R94) which is more recent. These standards contain detailed recommendations on fits and tolerances and serve as valuable guides for determining the required fit for a given application.

Regardless of the fitting method, an interference fit creates a contact pressure (also termed radial pressure, interference pressure, etc.), p, between the two parts at the transition radius (also termed the common radius), r. This contact pressure causes radial stresses equal to -p in each member at the contacting surfaces. An interference fit creates the same stress state in the shaft as would a uniform external pressure on its outer surface. the hub experiences the same stresses as a thick-walled cylinder subjected to an internal pressure. Note that the stresses resulting from the contact pressure, in either part, at any radial position, can be calculated using the Thick Walled Cylinder Module. 

The maximum calculated radial and tangential stresses of the members (shown in the output for this module) must be kept below the yield strengths of the materials to maintain the interference fit. If the materials yield the hub will become loose on the shaft. For the best fit, the American Gear Manufactures Association (AGMA) recommends a surface finish of at least 32 min rms on the contacting diameters of both members.

Assumptions:

In the equations used in this module (based on the deformations of the two members caused by the interference), both the shaft and the hub are considered to be the same length (contact length, L). In the case of a hub that has been press-fitted to a shaft, this assumption is seldom true as the shaft is generally longer than the hub. This would result in an increased pressure at each end of the hub. It is customary to allow for this condition by the employment of a stress concentration factor. The value of the concentration factor depends upon the contact pressure and the design of the hub, but its theoretical value is seldom greater than 2.0. Please see the references for the calculation of the stress concentration factor.

Figure 1. Module input form

• Contact pressure (or radial interference depending on the analysis option that was selected)
• Change in radii of the members
• Tangential stress at r of the members
• Torque transmitted by the interference

The results are in the same units used to enter the inputs and the material properties.

Format

The numeric solutions displayed in the results section can be formatted using the results format option from the module pull down menu. The number of digits to the right of the decimal point and the type of display can be selected. The available selections include a short (6 decimal places) or a long (12 decimal places) display in either a decimal  or scientific format (E used for the exponent symbol). Once the format has been changed, the results window will update upon the next calculation. To see the current results under the new format, simply select the calculate button again without changing  any of the inputs.

Errors

The module will verify that proper material properties and geometrical data were entered.

Figure 2. Module tabulated results.

Variables:

E1 = Elastic Modulus for the inner member

E2 = Elastic Modulus for the outer member

n1 = Poisson's Ratio for the inner member

n2 = Poisson's Ratio for the outer member

m = Coefficient of friction at the interface of the members. Note that the American Gear Manufactures Association (AGMA) recommends a value of between 0.12-0.15 for hydraulically expanded hubs and 0.15-0.20 for shrink or press fit hubs.

L = Contact length

r = Nominal radius at the interference, common radius

r_o = Outer radius of the outer member

r_i = Inner radius of the inner member (note r_i = 0 for a solid member)

p = Contact pressure, Interference Pressure

d = Radial Interference

d_i = Change of the outer radius of the inner member

d_o = Change of the inner radius of the outer member

s_i = Tangential stress at the outer surface of the inner member

s_o = Tangential stress at the inner surface of the outer member

T = Torque that the pressed joint will transmit