A long cylindrical shell is subjected to end moment M0. Plot the value of Mx from βx = 0 to βx = 4.0. Also, determine the distance x at which the moment is about 7% of the original applied moment M0.

Question

A long cylindrical shell is subjected to end moment [latex]M_0[/latex]. Plot the value of [latex]M_x[/latex] from [latex]β_x[/latex] = 0 to [latex]β_x[/latex] = 4.0. Also, determine the distance x at which the moment is about 7% of the original applied moment [latex]M_0[/latex].

Accepted Answer

From Eq. 5.25, [latex]M_{x}=M_{0} A_{\beta_{x}}[/latex]

[latex]M_{x}=-D\left(\frac{d^{2} w}{d x^{2}}\right)=\frac{1}{2 \beta}\left(2 \beta M_{0} A_{\beta x}+2 Q_{0} D_{\beta x}\right)[/latex]

[latex]Q_{x}=\left(2 \beta M_{0} D_{\beta x}-Q_{0} B_{\beta x}\right)[/latex] (5.25)

The values of [latex]A_{\beta x}[/latex] are obtained from Table 5.1 and a plot of [latex]M_{x}[/latex] is shown in Fig. 5.11. From Table 5.1 the values of βx at which [latex]M_{x}[/latex] is equal to 7% of [latex]M_{0}[/latex] are about 2.00.

Table 5.1 Values of Functions [latex]A_{\beta x}, B_{\beta x}, C_{\beta x}, D_{\beta x}[/latex]
[latex]\beta _{x}[/latex]	[latex]A_{\beta x}[/latex]	[latex]B_{\beta x}[/latex]	[latex]C_{\beta x}[/latex]	[latex]D_{\beta x}[/latex]
0	1.0000	1.0000	1.0000	1.0000
0.05	0.9976	0.9025	0.9500	0.0475
0.10	0.9907	0.8100	0.9003	0.0903
0.15	0.9797	0.7224	0.8510	0.1286
0.20	0.9651	0.6398	0.8024	0.1627
0.30	0.9267	0.4888	0.7077	0.2189
0.40	0.8784	0.3564	0.6174	0.2610
0.50	0.8231	0.2415	0.5323	0.2908
0.55	0.7934	0.1903	0.4919	0.3016
0.60	0.7628	0.1431	0.4530	0.3099
0.80	0.6354	-0.0093	0.3131	0.3223
1.00	0.5083	-0.1108	0.1988	0.3096
1.20	0.3899	-0.1716	0.1091	0.2807
1.40	0.2849	-0.2011	0.0419	0.2430
1.60	0.1959	-0.2077	-0.0059	0.2018
1.80	0.1234	-0.1985	-0.0376	0.1610
2.00	0.0667	-0.1794	-0.0563	0.1231
2.50	-0.0166	-0.1149	-0.0658	0.0491
3.00	-0.0423	-0.0563	-0.0493	0.0070
3.5	-0.0389	-0.0177	-0.0283	-0.0106
4.0	-0.0258	0.0019	-0.0120	-0.0139
5.0	-0.0045	0.0084	0.0019	-0.0065
6.0	0.0017	0.0031	0.0024	-0.0007
7.0	0.0013	0.0001	0.0007	0.0006

β x = 2.00

or

[latex]x=\frac{2}{\beta}[/latex]

and

[latex]x=1.56 \sqrt{r t}[/latex] (1)

The significance of the quantity [latex]1.56 \sqrt{r \cdot t}[/latex] is apparent from Fig. 5.11. It shows that a moment applied at the end dissipates very rapidly and reduces as much as 94% of the original moment at [latex]x=1.56 \sqrt{r t}[/latex]. This indicates that any other force applied at that distance [latex]x [/latex] can be analyzed without regard to the applied moment [latex]M_{0}[/latex].

Question 5.3: A long cylindrical shell is subjected to end moment M0. Plot...

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