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Press Fit Cylinders Contact and Interference
Figure 1 Press Fit Cylinders Contact and Interference

Change in Diameter
Eq. 1
Δd = d εθ

The change in diameter of the inner member when subjected to contact pressure pc
Eq. 2
Δdi = - pc dc [ ( dc + di2 ) / ( dc - di2 ) -v ]

The change in diameter of the outer member when subjected to contact pressure pc
Eq. 3
Δdo = - pc dc / [ ( do2 + dc2 ) / ( do2 - dc2 ) +v ]

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

Eq. 4a
Δdo = pc dc / E [ ( do2 + dc2 ) / ( do2 - dc2 ) +v ]

Eq. 4b
Δdi
= pc dc / E [ ( dc + di2 ) / ( dc - di2 ) -v ]

Eq. 4c
δ = pc dc / E [ ( do2 + dc2 ) / ( do2 - dc2 ) +v ] + pc dc / E [ ( dc + di2 ) / ( dc - di2 ) -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 = Δds + Δdh = ds + dh

Eq. 5a
Δds = pc dc / Es [ ( ds + di2 ) / ( ds - di2 ) -v ]

Eq. 5b
Δdh = pc dc / Eh [ ( do + dh2 ) / ( do - ds2 ) - v ]

Eq. 5c - exactly
δ = pc dc / Es [ ( ds + di2 ) / ( ds - di2 ) -v ] + pc dc / Eh [ ( do + dh2 ) / ( do - ds2 ) - v ]

Eq. 5d - (approx.)
pc dc = [ ( dc2 + di2 ) / ( E ( dc2 - di2 ) ) + ( do2 + dc2 ) / ( E ( do2 - dc2 ) ) - vs / Es + vh / Eh )

Tangential stress
Figure 2 Tangential stress, σθ

radial stress
Figure 3, radial stress σr

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

The contact pressure between cylinders at the surface of contact when the material of both the cylinders is same (Fig. 2)
Eq. 7
pc = ( E δ ( dc2 - di2 ) ( do2 - dc2 ) ) / ( 2 dc3 ( do2 - di2 ) )

The tangential stress at any radius r of outer cylinder (Fig. 2)
Eq. 8
σθ-o = pc dc2 / ( do2 - dc2 ) [ 1 + do2 / ( 4 ro2 ) ]

The tangential stress at any radius r of inner cylinder (Fig. 2)
Eq. 9
σθ-i = - pc dc2 / ( do2 - dc2 ) [ 1 + di2 / ( 4 ri2 ) ]

The radial stress at any radius r of outer cylinder (Fig. 2)
Eq. 10
σθ-o = - pc dc2 / ( do2 - dc2 ) [ do2 / ( 4 ro2 ) -1 ]

The radial stress at any radius r of inner cylinder (Fig. 2)
Eq. 11
σθ-i = pc dc2 / ( do2 - di2 ) / 1 - di2 / ( 4 ri2 ) ]

The tangential stress at outside diameter of outer cylinder (Fig. 2 & 3)
Eq. 12
σθ-oo = 2 pc dc2 / ( do2 - dc2 )

The tangential stress at inside diameter of outer cylinder (Fig. 2 & 3)
Eq. 13
σθ-oi = pc [ ( do2 + dc2 ) / ( do2 - dc2 ) ]

The tangential stress at outside diameter of inner cylinder (Fig. 2 & 3)
Eq. 14
σθ-io = - pc [ ( dc2 + di2 ) / ( dc2 - di2 ) ]

The tangential stress at inside diameter of inner cylinder (Fig. 2 & 3)
Eq. 15
σθ-ii = - 2 pc dc2 / ( dc2 - di2 )

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

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

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

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

The semiempirical formula for tangential stress for cast-iron hub on steel shaft
Eq. 20
σθ = Eo δ / ( dc + 0.14 do )

Timoshenko equation for contact pressure in case of steel shaft on cast-iron hub
Eq. 21
pc = Eo δ / dc [ ( 1 - ( dc / do )2 ) / ( 1.53 + 0.47 ( dc / do )2 ) ]
for Es / Ec = 3

The allowable stress for brittle materials
Eq. 22
σall = σsu / n [ ( Ec δ ( 1 + ( dc / do )2 ] / ( dc [ 1.53 + 0.47 ( dc / do )2 ]

where

ds = shaft outer diameter, m (in)
di = shaft inner diameter, m (in)
do = Outer hub diameter, m (in)
dh = Outer hub inner diameter, m (in)
ro = radius within outer cylinder material, m (in)
ri = radius within inner cylinder material, m (in)
dc = contact surface diameter, compressive, m (in)
pc = contact pressure MPa (psi)
σ = stress, MPa (psi)
δ = total change in diameter (interference), m (in)
E = modulus of elasticity, GPa (Mpsi)
Eh = modulus of elasticity of cast iron, GPa (Mpsi)
Es = modulus of elasticity of steel, GPa (Mpsi)
v = Poisson’s ratio
n = factor of safety

Related:

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