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INPUT   DATA

♥ EXAMPLE Of Input/Output

Title

Outside diameter of outer member, do		m
Contact surface diameter, dc		m
Inside diameter of inner member, di		m
Modulus of elasticity of outer member, Eo		109N/m2
Poisson's ratio of outer member, υo;
Modulus of elasticity of inner member, Ei		109N/m2
Poisson's ratio of inner member, υi;
Total interference, δ		m

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OUTPUT   VARIABLES   &   GRAPHS

PRESSURE	Values	Units
♦ Pressure at contact surface, pc		N/m2

STRESSES	Lamé's equation	Birnie's equation	Units
♦ Stress on surface at do, σo			N/m2
♦ Stress on surface at dc for outer member, σco			N/m2
♦ Stress on surface at dc for inner member, σci			N/m2
♦ Stress on surface at di, σi			N/m2

Interface Pressure Due to Press or Shrink Fits

If two thick-walled cylinders (eg. outer collar and inner shaft) are assembled together by either a hold/cold shrinking or mechanical press-fit, a pressure is developed at the interface between the two cylinders. This pressure p_c is given by the first equation below. Once p_c is determined, then the tangential stresses at the various surfaces can be estimated via two models. These models are: Lamé's equation for use in conjunction with the maximum shear theory of failure, and Birnie's equation for use with the maximum strain theory of failure.

where
     p_c = pressure at contact surface
     δ = total interference
     d_i = inside diameter of inner member, may be zero for solid shaft
     d_c = diameter of contact surface
     d_o = outside diameter of outer member
     E_i = modulus of elasticity of inner member
     υ_i = Poisson's ratio of inner member
     E_o = modulus of elasticity of outer member
     υ_o = Poisson's ratio of outer member
     σ_o = tangential stress of surface at d_o, Lamé's equation
     σ_co = tangential stress at d_c of outer member, Lamé's equation
     σ_ci = tangential stress at d_c of inner member, Lamé's equation
     σ_i = tangential stress of surface at d_i, Lamé's equation
     σ^/_o = tangential stress of surface at d_o, Birnie's equation
     σ^/_co = tangential stress at d_c of outer member, Birnie's equation
     σ^/_ci = tangential stress at d_c of inner member, Birnie's equation
     σ^/_i = tangential stress of surface at d_i, Birnie's equation

Tips

    ◊ Use link EXAMPLE Of Input/Output  to demo data entry expectations and results; you may edit & use it as starting point

BIBLIOGRAPHY

• Hicks TG; Mechanical Engineering Formulas, Pocket Guide; McGraw Hill, New York, 2003, Section 3.
• Brown TH; Marks' Calculations for Machine Design; McGraw Hill, New York, 2005, Chapter 3.