Diameter Change Press and Shrink Fit Formulae and Calculator

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Diameter Change Press and Shrink Fit Formulae and Calculator

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Diameter Change for Cylinder Press and Shrink Fits Analysis Equations and Calculator

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Figure 1 Press Fit Cylinders Contact and Interference

Change in Diameter
Eq. 1
Δd = d ε_θ

The change in diameter of the inner member when subjected to contact pressure p_c
Eq. 2
Δd_i = - ( p_c d_c ) / E [ ( d_c² + d_i² ) / ( d_c² - d_i² ) -v ]

The change in diameter of the outer member when subjected to contact pressure p_c
Eq. 3
Δd_o = ( p_c d_c ) / E [ ( d_o² + d_c² ) / ( d_o² - d_c² ) +v ]

The original difference in diameters of the two cylinders when the material of the members is the same
Eq. 4
δ = Δd_o + Δd_i

Eq. 4a
Δd_o = p_c d_c / E [ ( d_o² + d_c² ) / ( d_o² - d_c² ) +v ]

Eq. 4b
Δd_i = p_c d_c / E [ ( d_c² + d_i² ) / ( d_c² - d_i² ) -v ]

The total change in the diameters of hub and hollow shaft due to contact pressure at their contact surface when the material of the members is the same
Eq. 5
δ = Δd_s + Δd_h = d_s - d_h

Eq. 5a
Δd_s = p_c d_s / E_s [ ( d_s² + d_i² ) / ( d_s² - d_i² ) - v_s ]

Eq. 5b
Δd_h = p_c d_h / E_h [ ( d_o² + d_h² ) / ( d_o² - d_s² ) - v_h ] exactly

Eq. 5c
δ = p_c d_c / [ ( d_c² + d_i² ) / ( E_s ( d_c² - d_i² ) ) -v ] + p_c² d_c / E_h [ ( d_o² + d_h² ) / ( d_o² - d_s² ) - v ]

Eq. 5d - (approx.)
p_c d_c = [ ( d_c² + d_i² ) / ( E_s ( d_c² - d_i² ) ) + ( d_o² + d_c² ) / ( E_h ( d_o² - d_c² ) ) - v_s / E_s + v_h / E_h )

Figure 2 Tangential stress, σ_θ

Figure 3, radial stress σ_r

The shrinkage stress in the band (Fig. 2 & 3)
Eq. 6
σ_θ = E δ / d_c

The contact pressure between cylinders at the surface of contact when the material of both the cylinders is same (Fig. 2)
Eq. 7
p_c = ( E δ ( d_c² - d_i² ) ( d_o² - d_c² ) ) / ( 2 d_c³ ( d_o² - d_i² ) )

The tangential stress at any radius r of outer cylinder (Fig. 2)
Eq. 8
σ_θ-o = p_c d_c² / ( d_o² - d_c² ) [ 1 + d_o² / ( 4 r_o² ) ]

The tangential stress at any radius r of inner cylinder (Fig. 2)
Eq. 9
σ_θ-i = - p_c d_c² / ( d_o² - d_c² ) [ 1 + d_i² / ( 4 r_i² ) ]

The radial stress at any radius r of outer cylinder (Fig. 2)
Eq. 10
σ_θ-o = - p_c d_c² / ( d_o² - d_c² ) [ d_o² / ( 4 r_o² ) -1 ]

The radial stress at any radius r of inner cylinder (Fig. 2)
Eq. 11
σ_θ-i = p_c d_c² / ( d_c² - d_i² ) / 1 - d_i² / ( 4 r_i² ) ]

The tangential stress at outside diameter of outer cylinder (Fig. 2 & 3)
Eq. 12
σ_θ-oo = 2 p_c d_c² / ( d_o² - d_c² )

The tangential stress at inside diameter of outer cylinder (Fig. 2 & 3)
Eq. 13
σ_θ-oi = p_c [ ( d_o² + d_c² ) / ( d_o² - d_c² ) ]

The tangential stress at outside diameter of inner cylinder (Fig. 2 & 3)
Eq. 14
σ_θ-io = - p_c [ ( d_c² + d_i² ) / ( d_c² - d_i² ) ]

The tangential stress at inside diameter of inner cylinder (Fig. 2 & 3)
Eq. 15
σ_θ-ii = - 2 p_c d_c² / ( d_c² - d_i² )

The radial stress at outside diameter of outer cylinder (Fig. 2 & 3)
Eq. 16
σ_r-oo = 0

The radial stress at inside diameter of outer cylinder (Fig. 2 & 3)
Eq. 17
σ_r-oi = - p_c

The radial stress at outside diameter of inner cylinder (Fig. 2 & 3)
Eq. 18
σ_r-io = - p_c

The radial stress at inside diameter of inner cylinder (Fig. 2 & 3)
Eq. 19
σ_r-ii = 0

The semiempirical formula for tangential stress for cast-iron hub on steel shaft
Eq. 20
σ_θ = E_o δ / ( d_c + 0.14 d_o )

Timoshenko equation for contact pressure in case of steel shaft on cast-iron hub
Eq. 21
p_c = E_o δ / d_c [ ( 1 - ( d_c / d_o )² ) / ( 1.53 + 0.47 ( d_c / d_o )² ) ]
for E_s / E_c = 3

The allowable stress for brittle materials
Eq. 22
σ_all = σ_su / n = [ ( E_c δ ( 1 + ( d_c / d_o )² ] / ( d_c [ 1.53 + 0.47 ( d_c / d_o )² ]

where

d_s = shaft outer diameter, m (in)
d_i = shaft inner diameter, m (in)
d_o = Outer hub diameter, m (in)
d_h = Outer hub inner diameter, m (in)
r_o = radius within outer cylinder material, m (in)
r_i = radius within inner cylinder material, m (in)
d_c = contact surface diameter, compressive, m (in)
p_c = contact pressure MPa (psi)
σ = stress, MPa (psi)
δ = total change in diameter (interference), m (in)
E = modulus of elasticity, GPa (Mpsi)
E_h = modulus of elasticity of cast iron, GPa (Mpsi)
E_s = modulus of elasticity of steel, GPa (Mpsi)
v = Poisson’s ratio
n = factor of safety

Source:

Machine Design Handbook 2nd Edition, 2001
Prof. Dr. K. Lingaiah
Former Principal, Professor and Head, Department of Mechanical Engineering
University Visvesvaraya College of Engineering, Bangalore University, Bangalore