A bronze bushing is to be installed into a steel sleeve as indicated in Figure 13–6. The bushing has an inside diameter of 2.000 in and a nominal outside diameter of 2.500 in. The steel sleeve has a nominal inside diameter of 2.500 in and an outside diameter of 3.500 in.

Question

1. Specify the limits of size for the outside diameter of the bushing and the inside diameter of the sleeve in order to obtain a heavy drive fit, FN3. Determine the limits of interference that would result.
2. For the maximum interference from 1, compute the pressure that would be developed between the bushing and the sleeve, the stress in the bushing and the sleeve, and the deformation of the bushing and the sleeve. Use [latex]E = 30 × 10^6[/latex] psi for the steel and [latex]E = 17 × 10^6[/latex] for the bronze.
Use [latex] u = 0.27[/latex] for both materials.

Accepted Answer

For 1, from Table 13–4, for a part size of 2.50 in at the mating surface, the tolerance limits on the hole in the outer member are +1.2 and -0. Applying these limits to the basic size gives the dimension limits for the hole in the steel sleeve:
2.5012 in
2.5000 in
For the bronze insert, the tolerance limits are +3.2 and +2.5. Then the size limits for the outside diameter of the bushing are
2.5032 in
2.5025 in

The limits of interference would be 0.0013 to 0.0032 in.
For 2, the maximum pressure would be produced by the maximum interference, 0.0032 in. Then, using [latex]a=1.00 \mathrm{in}, b=1.25 \mathrm{in}, c=1.75 \mathrm{in}, E_{0}=30 imes 10^{6} \mathrm{psi}, E_{i}=17 imes 10^{6} \mathrm{psi}[/latex], and [latex] u_{o}= u_{i}=0.27[/latex] from Equation (13-3),

[latex]\begin{aligned}p=& \frac{\delta}{2 b\left[\frac{1}{E_{o}}\left(\frac{c^{2}+b^{2}}{c^{2}-b^{2}}+ u_{o} ight)+\frac{1}{E_{i}}\left(\frac{b^{2}+a^{2}}{b^{2}-a^{2}}- u_{i} ight) ight]} \p=& \frac{0.0032}{(2)(1.25)\left[\frac{1}{30 imes 10^{6}}\left(\frac{1.75^{2}+1.25^{2}}{1.75^{2}-1.25^{2}}+0.27 ight)+\frac{1}{17 imes 10^{6}}\left(\frac{1.25^{2}+1.00^{2}}{1.25^{2}-1.00^{2}}-0.27 ight) ight]} \p=& 3518\mathrm{psi}\end{aligned}[/latex]

The tensile stress in the steel sleeve is
[latex]\sigma_{o}=p\left(\frac{c^{2}+b^{2}}{c^{2}-b^{2}} ight)=3518\left(\frac{1.75^{2}+1.25^{2}}{1.75^{2}-1.25^{2}} ight)=10846 \mathrm{psi}[/latex]

The compressive stress in the bronze bushing is
[latex]\sigma_{i}=-p\left(\frac{b^{2}+a^{2}}{b^{2}-a^{2}} ight)=-3518\left(\frac{1.25^{2}+1.00^{2}}{1.25^{2}-1.00^{2}} ight)=-16025 \mathrm{psi}[/latex]

The increase in the diameter of the sleeve is
[latex]\begin{aligned}&\delta_{o}=\frac{2 b p}{E_{o}}\left[\frac{c^{2}+b^{2}}{c^{2}-b^{2}}+ u_{o} ight] \&\delta_{o}=\frac{2(1.25)(3518)}{30 imes 10^{6}}\left[\frac{1.75^{2}+1.25^{2}}{1.75^{2}-1.25^{2}}+0.27 ight]=0.00098 ext { in }\end{aligned}[/latex]

The decrease in the diameter of the bushing is

[latex]\begin{aligned}&\delta_{i}=\frac{2 b p}{E_{i}}\left[\frac{b^{2}+a^{2}}{b^{2}-a^{2}}- u_{i} ight] \&\delta_{i}=\frac{2(1.25)(3518)}{17 imes 10^{6}}\left[\frac{1.25^{2}+1.00^{2}}{1.25^{2}-1.00^{2}}-0.27 ight]=0.00222 ext { in }\end{aligned}[/latex]

Note that the sum of [latex]\delta_{0}[/latex] and [latex]\delta_{i}[/latex] equals 0.0032, the total interference, [latex]\delta[/latex].

TABLE 13–4 Force and Shrink Fits (FN)
Nominal size range (in)	Class FN1			Class FN2			Class FN3			Class FN4			Class FN5
Nominal size range (in)	Limits of interference	Standard limits		Limits of interference	Standard limits		Limits of interference	Standard limits		Limits of interference	Standard limits		Limits of interference	Standard limits
Over To	Limits of interference	Hole	Shaft	Limits of interference	Hole	Shaft	Limits of interference	Hole	Shaft	Limits of interference	Hole	Shaft	Limits of interference	Hole	Shaft
0–0.12	0.05	+0.25	+0.5	0.2	+0.4	+0.9				0.3	+0.4	+0.95	0.3	+0.6	+1.3
	0.5	-0	+0.3	0.85	-0	+0.6				0.95	-0	+0.7	1.3	-0	+0.9
0.12–0.24	0.1	+0.3	+0.6	0.2	+0.5	+1.0				0.4	+0.5	+1.2	0.5	+0.7	+1.7
	0.6	-0	+0.4	1.0	-0	+0.7				1.2	-0	+0.9	1.7	-0	+1.2
0.24–0.40	0.1	+0.4	+0.75	0.4	+0.6	+1.4				0.6	_0.6	+1.6	0.5	+0.9	+2.0
	0.75	-0	+0.5	1.4	-0	+1.0				1.6	-0	+1.2	2.0	-0	+1.4
0.40–0.56	0.1	+0.4	+0.8	0.5	+0.7	+1.6				0.7	+0.7	+1.8	0.6	+1.0	+2.3
	0.8	-0	+0.5	1.6	-0	+1.2				1.8	-0	+1.4	2.3	-0	+1.6
0.56–0.71	0.2	+0.4	+0.9	0.5	+0.7	+1.6				0.7	+0.7	+1.8	0.8	+1.0	+2.5
	0.9	-0	+0.6	1.6	-0	+1.2				1.8	-0	+1.4	2.5	-0	+1.8
0.71–0.95	0.2	+0.5	+1.1	0.6	+0.8	+1.9				0.8	+0.8	+2.1	1.0	+1.2	+3.0
	1.1	-0	+0.7	1.9	-0	+1.4				2.1	-0	+1.6	3.0	-0	+2.2
0.95–1.19	0.3	+0.5	+1.2	0.6	+0.8	+1.9	0.8	+0.8	+2.1	1.0	+0.8	+2.3	1.3	+1.2	+3.3
	1.2	-0	+0.8	1.9	-0	+1.4	2.1	-0	+1.6	2.3	-0	+1.8	3.3	-0	+2.5
1.19–1.58	0.3	+0.6	+1.3	0.8	+1.0	+2.4	1.0	+1.0	+2.6	1.5	+1.0	+3.1	1.4	+1.6	+4.0
	1.3	-0	+0.9	2.4	-0	+1.8	2.6	-0	+2.0	3.1	-0	+2.5	4.0	-0	+3.0
1.58–1.97	0.4	+0.6	+1.4	0.8	+1.0	+2.4	1.2	+1.0	+2.8	1.8	+1.0	+3.4	2.4	+1.6	+5.0
	1.4	-0	+1.0	2.4	-0	+1.8	2.8	-0	+2.2	3.4	-0	+2.8	5.0	-0	+4.0
1.97–2.56	0.6	+0.7	+1.8	0.8	+1.2	+2.7	1.3	+1.2	+3.2	2.3	+1.2	+4.2	3.2	+1.8	+6.2
	1.8	-0	+1.3	2.7	-0	+2.0	3.2	-0	+2.5	4.2	-0	+3.5	6.2	-0	+5.0
2.56–3.15	0.7	+0.7	+1.9	1.0	+1.2	+2.9	1.8	+1.2	+3.7	2.8	+1.2	+4.7	4.2	+1.8	+7.2
	1.9	-0	+1.4	2.9	-0	+2.2	3.7	-0	+3.0	4.7	-0	+4.0	7.2	-0	+6.0
3.15–3.94	0.9	+0.9	+2.4	1.4	+1.4	+3.7	2.1	+1.4	+4.4	3.6	+1.4	+5.9	4.8	+2.2	+8.4
	2.4	-0	+1.8	3.7	-0	+2.8	4.4	-0	+3.5	5.9	-0	+5.0	8.4	-0	+7.0
3.94–4.73	1.1	+0.9	+2.6	1.6	+1.4	+3.9	2.6	+1.4	+4.9	4.6	+1.4	+6.9	5.8	+2.2	+9.4
	2.6	-0	+2.0	3.9	-0	+3.0	4.9	-0	+4.0	6.9	-0	+6.0	9.4	-0	+8.0
4.73–5.52	1.2	+1.0	+2.9	1.9	+1.6	+4.5	3.4	+1.6	+6.0	5.4	+1.6	+8.0	7.5	+2.5	+11.6
	2.9	-0	+2.2	4.5	-0	+3.5	6.0	-0	+5.0	8.0	-0	+7.0	11.6	-0	+10.0
5.52–6.30	1.5	+1.0	+3.2	2.4	1.6	+5.0	3.4	+1.6	+6.0	5.4	+1.6	+8.0	9.5	+2.5	+13.6
	3.2	-0	+2.5	5.0	0	+4.0	6.0	-0	+5.0	8.0	-0	+7.0	13.6	-0	+12.0

Question 13.2: A bronze bushing is to be installed into a steel sleeve as i...

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