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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_{c}^{2} + d_{i}^{2} ) / ( d_{c}^{2} - d_{i}^{2} ) -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_{o}^{2} + d_{c}^{2} ) / ( d_{o}^{2} - d_{c}^{2} ) +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_{o}^{2}+ d_{c}^{2}) / ( d_{o}^{2}- d_{c}^{2}) +v ]*Eq. 4b
Δd _{i}* =

*p*

_{c}d_{c}/ E [ ( d_{c}^{2}+ d_{i}^{2}) / ( d_{c}^{2}- d_{i}^{2}) -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_{s}^{2}+ d_{i}^{2}) / ( d_{s}^{2}- d_{i}^{2}) - v_{s}]Eq. 5b

*Δd _{h}* =

*p*

_{c}d_{h}/ E_{h}[ ( d_{o}^{2}+ d_{h}^{2}) / ( d_{o}^{2}- d_{s}^{2}) - v_{h}] exactlyEq. 5c

*δ = p _{c} d_{c} / [ ( d_{c}^{2} + d_{i}^{2} ) / ( E_{s} ( d_{c}^{2} - d_{i}^{2} ) ) -v ] + p_{c}^{2} d_{c} / E_{h} [ ( d_{o}^{2} + d_{h}^{2} ) / ( d_{o}^{2} - d_{s}^{2} ) - v ] *

Eq. 5d - (approx.)

*p _{c} d_{c} = [ ( d_{c}^{2} + d_{i}^{2} ) / ( E_{s} ( d_{c}^{2} - d_{i}^{2} ) ) + ( d_{o}^{2} + d_{c}^{2} ) / ( E_{h} ( d_{o}^{2} - d_{c}^{2} ) ) - 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_{c}^{2} - d_{i}^{2} ) ( d_{o}^{2} - d_{c}^{2} ) ) / ( 2 d_{c}^{3} ( d_{o}^{2} - d_{i}^{2} ) ) *

The tangential stress at any radius r of outer cylinder *(Fig. 2*)

Eq. 8

*σ _{θ-o} = p_{c} d_{c}^{2} / ( d_{o}^{2} - d_{c}^{2} ) [ 1 + d_{o}^{2} / ( 4 r_{o}^{2} ) ] *

The tangential stress at any radius r of inner cylinder *(Fig. 2*)

Eq. 9

*σ _{θ-i} = - p_{c} d_{c}^{2} / ( d_{o}^{2} - d_{c}^{2} ) [ 1 + d_{i}^{2} / ( 4 r_{i}^{2} ) ] *

The radial stress at any radius r of outer cylinder *(Fig. 2*)

Eq. 10

*σ _{θ-o} = - p_{c} d_{c}^{2} / ( d_{o}^{2} - d_{c}^{2} ) [ d_{o}^{2} / ( 4 r_{o}^{2} ) -1 ] *

The radial stress at any radius r of inner cylinder *(Fig. 2*)

Eq. 11

*σ _{θ-i} = p_{c} d_{c}^{2} / ( d_{c}^{2} - d_{i}^{2} )* /

*1 - d*

_{i}^{2}/ ( 4 r_{i}^{2}) ]The tangential stress at outside diameter of outer cylinder* (Fig. 2* & 3)

*Eq. 12
σ _{θ-oo} = 2 p_{c} d_{c}^{2} / ( d_{o}^{2} - d_{c}^{2} )*

The tangential stress at inside diameter of outer cylinder* (Fig. 2* & 3)

Eq. 13

*σ _{θ-oi} = p_{c} [ ( d_{o}^{2} + d_{c}^{2} )* /

*( d*]

_{o}^{2}- d_{c}^{2})The tangential stress at outside diameter of inner cylinder *(Fig. 2* & 3)

Eq. 14

*σ _{θ-io} = - p_{c}*

*[ ( d*/

_{c}^{2}+ d_{i}^{2})*( d*]

_{c}^{2}- d_{i}^{2})The tangential stress at inside diameter of inner cylinder *(Fig. 2* & 3)

Eq. 15

σ_{θ-ii} = - 2 *p _{c}*

*d*/

_{c}^{2}*( d*

_{c}^{2}- d_{i}^{2})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}*) / ( 1.53 + 0.47 (*

^{2}*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*[ 1.53 + 0.47 (

_{c}*d*/

_{c}*d*)

_{o}*]*

^{2}where

*d _{s}* = shaft outer diameter, m (in)

*d*= shaft inner diameter, m (in)

_{i}*d*= Outer hub diameter, m (in)

_{o}*d*= Outer hub inner diameter, m (in)

_{h}r

_{o}= radius within outer cylinder material, m (in)

r

_{i}= radius within inner cylinder material, m (in)

*d*= contact surface diameter, compressive, m (in)

_{c}*p*= contact pressure MPa (psi)

_{c}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

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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