For the cylindrical shell given in Example 11.15, determine the stresses due to internal pressure and applied external loading by the method in WRC Bulletin 107 .

Question

Accepted Answer

1 . For the WRC method, the following shell-nozzle parameters are required:

[latex]\beta=0.875 \frac{r_{o}}{R_{m}}=0.875 imes \frac{5}{42.5}=0.103[/latex]

[latex]\gamma=\frac{R_{m}}{T}=\frac{42.5}{1}=42.5[/latex]

2 . The constants below are determined from various figures in WRC and the stresses determined as follows:

From Fig. 4C [latex]N_{\phi} /\left(P / R_{m} ight)=7.3 imes \frac{12,000}{42.5 imes 1}=2060 [/latex] psi

From Fig. 2 C-1 [latex]M_{\phi} / P=0.088 imes \frac{6 imes 12,000}{(1)^{2}}=6340[/latex] psi

From Fig. 3B [latex]\begin{aligned} N_{\phi} /\left(M_{L} / R_{m}^{2} \beta ight) &=4.5 imes \frac{150,000}{(42.5)^{2}(0.103)(1)} \ &=3630 ext{ psi} \end{aligned}[/latex]

From Fig. 1B [latex]\begin{aligned} \frac{M_{\phi}}{\left(M_{L} / R_{m} \beta ight)} &=0.045 imes \frac{6 imes 150,000}{(42.5)(0.103)(1)^{2}} \ &=9250 ext{ psi} \end{aligned}[/latex]

From Fig. 3C [latex]\frac{N_{x}}{\left(P / R_{m} ight)}=6.0 imes \frac{12,000}{42.5 imes 1}=1690[/latex] psi

From Fig. 1C-1 [latex]\frac{M_{x}}{P}=0.125 imes \frac{6 imes 12,000}{(1)^{2}}=9000[/latex] psi

From Fig. 4B [latex]\begin{aligned} \frac{N_{x}}{\left(M_{L} / R_{m}^{2} \beta ight)} &=1.3 imes \frac{150,000}{(42.5)^{2}(0.103)(1)} \&=1050 ext{ psi}\end{aligned}[/latex]

From Fig. 2B [latex]M_{x} /\left(M_{L} / R_{m} \beta ight)=0.072 imes \frac{6 imes 150,000}{(42.5)(0.103)(1)^{2}}=14,800[/latex] psi

3 . Using the internal pressure stresses determined for Example 11.15 and combining them with these stresses gives:

	[latex]\sigma_{\phi}[/latex]
	[latex]A_U[/latex]	[latex]A_L[/latex]	[latex]B_U[/latex]	[latex]B_L[/latex]
Pressure membrane	+23,800	+23,800	+23,800	+23,800
Pressure bending	-3,400	+3,400	-3,400	+3,400
P membrane	-1,690	-1,690	-1,690	-1,690
P bending	-9,000	+9,000	-9,000	+9,000
[latex]M_L[/latex] membrane	-3,630	-3,630	+3,630	+3,630
[latex]M_L[/latex] bending	-9,250	+9,250	+9,250	-9,250
Total	-880	+37,100	+24,880	+25,860

	[latex]\sigma_{x}[/latex]
	[latex]A_U[/latex]	[latex]A_L[/latex]	[latex]B_U[/latex]	[latex]B_L[/latex]
Pressure membrane	+13,600	+13,600	+13,600	+13,600
Pressure bending	+1,700	-1,700	+1,700	-1,700
P membrane	-2,060	-2,060	-2,060	-2,060
P bending	-6,340	+6,340	-6,340	+6,340
[latex]M_L[/latex] membrane	-1,050	-1,050	+1,050	+1,050
[latex]M_L[/latex] bending	-14,800	+14,800	+14,800	-14,800
Total	-11,240	+32,960	+20,460	+5,460

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