A cylindrical shell that is 84 -in. ID by 1.0-in. nominal thickness contains a nozzle 8 -in. ID by 1.0-in. nominal thickness. The design pressure is 400 psi and the allowable stress of the material is 17.5 ksi. The nozzle is subjected to an inward radial loading of 12,000 lb and an applied moment

Question

A cylindrical shell that is 84 -in. ID by 1.0-in. nominal thickness contains a nozzle 8 -in. ID by 1.0-in. nominal thickness. The design pressure is 400 psi and the allowable stress of the material is 17.5 ksi. The nozzle is subjected to an inward radial loading of 12,000 lb and an applied moment in the longitudinal direction of 150,000 in.-1b. What are the combined stresses on the longitudinal axis due to these two external loadings using the Mershon method and the curves in Appendix K ? The vessel is not subjected to cyclic loading, and therefore no stress concentration factors need be considered.

Accepted Answer

1 . The shell parameter is

[latex]\lambda=\frac{d_{o}}{\sqrt{D_{m} T}}=\frac{10}{\sqrt{(85)(1)}}=1.08[/latex]

2 . Using this parameter, the constants from the radial loading on the longitudinal axis are:

From Fig. K.5 [latex]M_{x} / P=0.127[/latex]

From Fig. K.6 [latex]M_{\phi} / P=0.086[/latex]

From Fig. K.8 [latex]N_{x} T / P=0.160[/latex]

From Fig. K.8 [latex]N_{\phi} T / P=0.176[/latex]

3 . Using these constants, the stresses due to the radial loading are determined as follows:

[latex]M_{x} ext { from } P ext { (bending) }=0.127\left[\frac{(6)(12,000)}{(1)^{2}} ight]=9140 \ psi[/latex]

[latex]M_{\phi} ext { from } P ext { (bending) }=0.086\left[\frac{(6)(12,000)}{2} ight]=6190 \ psi[/latex]

[latex]N_{x} ext { from } P ext { (membrane) }=0.160\left[\frac{(12,000)}{(1)^{2}} ight]=1920 \ psi[/latex]

[latex]N_{\phi} ext { from } P( ext { membrane })=0.176\left[\frac{(12,000)}{(1)^{2}} ight]=2110 \ psi[/latex]

4 . Using the parameter in item (1), the constants from the applied longitudinal moment on the longitudinal axis are determined as follows:

From Fig. K.3 [latex]M_{x} \frac{d_{o}}{M_{L}}=0.170[/latex]

From Fig. K.3 [latex]M_{\phi} \frac{d_{o}}{M_{L}}=0.104[/latex]

From Fig. K.4 [latex]N_{x} \frac{d_{o} T}{M_{L}}=0.076[/latex]

From Fig. K.4 [latex]N_{\phi} \frac{d_{0} T}{M_{L}}=0.260[/latex]

5 . Using these constants, the stresses due to the longitudinal moment are determined as follows:

[latex]M_{x} ext { from } M_{L} ext { (bending) }=0.170\left[\frac{(6)(150,000)}{(10)(1)^{2}} ight]=15,300 ext { psi }[/latex]

[latex]M_{\phi} ext { from } M_{L} ext { (bending) }=0.104\left[\frac{(6)(150,000)}{(10)(1)^{2}} ight]=9360 \ psi[/latex]

[latex]N_{x} ext { from } M_{L} ext { (membrane) }=0.076\left[\frac{(150,000)}{(10)(1)^{2}} ight]=1140 \ psi[/latex]

[latex]N_{\phi} ext { from } M_{L} ext { (membrane) }=0.260\left[\frac{(150,000)}{(10)(1)^{2}} ight]=3900 \ psi[/latex]

6 . Summaries of membrane stresses, bending stresses, and combined stresses at various locations for external loadings are given in Tables 11.6, 11.7, and 11.8.

Table 11.6 Summary of Membrane Stresses at Various Locations
	[latex]\sigma_{\phi}[/latex]				[latex]\sigma_{x}[/latex]
	[latex]A_{U}[/latex]	[latex]A_{L}[/latex]	[latex]B_{U}[/latex]	[latex]B_{L}[/latex]	[latex]A_{U}[/latex]	[latex]A_{L}[/latex]	[latex]B_{U}[/latex]	[latex]B_{L}[/latex]
Due to P	-2,110	-2,110	-2,110	-2,110	-1,920	-1,920	-1,920	-1,920
Due to [latex]M_{L}[/latex]	-3,900	-3,900	+3,900	+3,900	-1,140	-1,140	+1,140	+1,140
Total	-6,010	-6,010	+1,790	+1,790	-3,060	-3,060	-780	-780

Table 11.7 Summary of Bending Stresses at Various Locations
	[latex]\sigma_{\phi}[/latex]				[latex]\sigma_{x}[/latex]
	[latex]A_{U}[/latex]	[latex]A_{L}[/latex]	[latex]B_{U}[/latex]	[latex]B_{L}[/latex]	[latex]A_{U}[/latex]	[latex]A_{L}[/latex]	[latex]B_{U}[/latex]	[latex]B_{L}[/latex]
Due to P	-6,190	+6,190	-6,190	+6,190	-9,140	+9,140	-9,140	+9,140
Due to [latex]M_{L}[/latex]	-9,360	+9,360	+9,360	-9,360	-15,300	+15,300	+15,300	-15,300
Total	-15,550	+15,550	+3,170	-3,170	-24,440	+24,440	+6,160	-6,160

Table 11.8 Summary of Membrane-Plus-Bending at Various Locations
	[latex]\sigma_{\phi}[/latex]				[latex]\sigma_{x}[/latex]
	[latex]A_{U}[/latex]	[latex]A_{L}[/latex]	[latex]B_{U}[/latex]	[latex]B_{L}[/latex]	[latex]A_{U}[/latex]	[latex]A_{L}[/latex]	[latex]B_{U}[/latex]	[latex]B_{L}[/latex]
Membrane	-6,010	-6,010	+1,790	+1,790	-3,060	-3,060	-780	-780
Bending	-15,550	+15,550	+3,170	-3,170	-24,440	+24,440	+6,160	-6,160
Total	-21,560	+9,540	+4,960	-1,380	-27,500	+21,380	+5,380	-6,940

7 . In addition to the stresses from the external loadings, the stresses from internal pressure must be combined. These stresses may be determined as shown in Table 11.5. However, exact values may be used if available. For this geometry, assume the following values were determined:

Table 11.5 Stress Index I
Stress	Spherical Shells and Formed Heads		Cylindrical Shells
	Spherical Shells and Formed Heads		Longitudinal Plane		Transversc Planc
	Inside	Outside	Inside	Outside	Inside	Outside
[latex]\sigma_{n}[/latex]	2.0 - (r/R)	2.0 - (r/R)	3.1	1.2	1.0	2.1
[latex]\sigma_{t}[/latex]	-0.2	2.0 - (r/R)	-0.2	1.0	-0.2	2.6
[latex]\sigma_{r}[/latex]	[latex]-4 T_{s} / D_{m}[/latex]	0	[latex]-2 T_{s} / D_{m}[/latex]	0	[latex]-2 T_{s} / D_{m}[/latex]	0

	Longitudinal Axis			Membrane	Bending
	Membrane	Bending		Membrane	Bending
[latex]\sigma_{n}\left(=\sigma_{\phi} ight)[/latex]	1.40	[latex]\mp[/latex]0.20	[latex]\sigma_{n}\left(=\sigma_{x} ight)[/latex]	0.70	[latex]\pm[/latex]0.35
[latex]\sigma_{t}\left(=\sigma_{x} ight)[/latex]	0.80	[latex]\pm[/latex]0.10	[latex]\sigma_{t}\left(=\sigma_{\phi} ight)[/latex]	0.85	[latex]\pm[/latex]0.55

[latex]S=\frac{P D_{m}}{2 T}=\frac{400 imes 85}{2 imes 1}=17,000 \ psi[/latex]

8 . The total combined stresses from internal pressure and external loadings are given in Table 11.9.

Table 11.9 Summary of Stresses due to Internal Pressure and External Loadings
	[latex]\sigma_{\phi}[/latex]				[latex]\sigma_{x}[/latex]
	[latex]A_{U}[/latex]	[latex]A_{L}[/latex]	[latex]B_{U}[/latex]	[latex]B_{L}[/latex]	[latex]A_{U}[/latex]	[latex]A_{L}[/latex]	[latex]B_{U}[/latex]	[latex]B_{L}[/latex]
Membrane	+23,800	+23,800	+23,800	+23,800	+13,600	+13,600	+13,600	+13,600
Bending	-3,400	+3,400	-3,400	+3,400	+1,700	-1,700	+1,700	-1,700
[latex]P+M_L[/latex]	-21,560	+9,540	+4,960	-1,380	-27,500	+21,380	+5,380	-6,940
Total	-1,160	+36,740	+25,360	+25,820	-12,200	+33,280	+20,680	+4,960

Question 11.15: A cylindrical shell that is 84 -in. ID by 1.0-in. nominal th...

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