Question 5.8: Determine the maximum stress at point A of the thin cylinder...

A free-body diagram of junction A is shown in Figure 5.14b. The deflection at point A in the thick cylinder due to P is obtained from Table 5.3 by letting \beta x equal to \beta l . Hence,

For \beta_{1}=0.7421, \quad D_{1}=0.01145 E, , and \beta x=\beta l, the following values are obtained:

Thus, the expression for w_{ p } due to P is given by

The deflection compatibility equation at point A is

or from Tables 5.2 and 5.3 with x=0,

= \frac{Q_{0}}{2 \beta_{2}^{3} D_{2}}-\frac{M_{0}}{2 \beta_{2}^{2} D_{2}}

and with \beta_{2} =1.0495 and D_{2} =0.00145E, the equation becomes

= 298.30 Q_{0}-313.07 M_{0}

Or

5.206 M_{0}-4.077 Q_{0}=-P                               (1)

The rotation at point A due to P is obtained from Table 5.3 as

Or

The rotation compatibility equation at point A is

Hence,

Or

= -657.13 M_{0}+313.07 Q_{0}

which reduces to

12.58 M_{0}+Q_{0}=-3.48 P                              (2)

Solving Eqs. (1) and (2) yields

Hence, the maximum axial stress is

The circumferential bending moment is given by

The circumferential force N_{\theta} is given by Eq. (5.19) as.

= 17.80P.

Table 5.2 Various discontinuity functions.

\text { Functions }^{\text {) }}
Edge functions
w	\frac{-M_{0}}{2 \beta^{2} D}	\frac{Q_{0}}{2 \beta^{3} D}	\Delta_{0}	0
\theta	\frac{M_{0}}{\beta D}	\frac{-Q_{0}}{2 \beta^{2} D}	0	\theta_{0}
M_{x}	M_{0}	0	2 \beta^{2} D \Delta_{0}	2 \beta D \theta_{0}
N_{\theta}	-2 M_{0} \beta^{2} r	2 \beta r Q_{0}	\frac{E t \Delta_{0}}{r}	0
Q_{0}	0	Q_{0}	4 \beta^{3} D \Delta_{0}	2 \beta^{2} D \theta_{0}
General functions
w	\frac{-M_{0}}{2 \beta^{2} D} B_{\beta x}	\frac{Q_{0}}{2 \beta^{3} D} C_{\beta x}	\Delta_{0}\left(2 C_{\beta x}-B_{\beta x}\right)	\frac{\theta_{0}}{\beta}\left(C_{\beta x}-B_{\beta x}\right)
\theta	\frac{M_{0}}{\beta D} C_{\beta x}	\frac{-Q_{0}}{2 \beta^{2} D} A_{\beta x}	2 \beta \Delta_{0}\left(A_{\beta x}-C_{\beta x}\right)	\theta_{0}\left(A_{\beta x}-2 C_{\beta x}\right)
M_{x}	M_{0} A_{\beta x}	\frac{Q_{0}}{\beta} D_{\beta x}	2 \beta^{2} D \Delta_{0}\left(A_{\beta x}-C_{\beta x}\right)	2 \beta D \theta_{0}\left(D_{\beta x}-A_{\beta x}\right)
N_{\theta}	-2 M_{0} \beta^{2} r B_{\beta x}	2 \beta r Q_{0} C_{\beta x}	\frac{E t}{r} \Delta_{0}\left(2 C_{\beta x}-B_{\beta x}\right)	\frac{E t \theta_{0}}{r \beta}\left(C_{\beta x}-B_{\beta x}\right)
Q_{x}	-2 \beta M_{0} D_{\beta x}	Q_{0} B_{\beta x}	4 \beta^{3} D \Delta_{0}\left(B_{\beta x}-D_{\beta x}\right)	2 \beta^{2} D \theta_{0}\left(2 D_{\beta x}+B_{\beta x}\right) B11:F16
a) Clockwise moments and rotation are positive at point 0. Outward forces and deflections are positive at point 0. M_{0}=\mu M_{x}.

Table 5.3 Various functions of short cylinders.

Function
w	\frac{M_{0}}{2 \beta^{2} D}\left[\frac{-C_{2}}{C_{1}} V_{7}+\frac{C_{3}}{C_{1}} V_{2}-V_{8}\right]	\frac{Q_{0}}{2 \beta^{3} D}\left[\frac{C_{4}}{C_{1}} V_{7}-\frac{C_{5}}{C_{1}} V_{5}-\frac{C_{6}}{C_{1}} V_{6}\right]	\frac{\theta_{0}}{\beta}\left[\frac{C_{6}}{C_{1}} V_{5}-\frac{C_{5}}{C_{1}} V_{6}-\frac{C_{4}}{C_{1}} V_{8}\right] \quad \Delta_{0}\left[V_{7}-\frac{C_{3}}{C_{1}} V_{1}-\frac{C_{2}}{C_{1}} V_{8}\right]
\theta	\frac{M_{0}}{2 \beta D}\left[\frac{C_{2}}{C_{1}} V_{1}+\frac{2 C_{3}}{C_{1}} V_{7}-V_{2}\right]	\frac{-Q_{0}}{2 \beta^{2} D}\left[\frac{C_{4}}{C_{1}} V_{1}+\frac{C_{5}}{C_{1}} V_{4}+\frac{C_{6}}{C_{1}} V_{3}\right]	\theta_{0}\left[\frac{C_{6}}{C_{1}} V_{4}-\frac{C_{5}}{C_{1}} V_{3}-\frac{C_{4}}{C_{1}} V_{2}\right] \quad \beta \Delta_{0}\left[-V_{1}+2 \frac{C_{3}}{C_{1}} V_{8}-\frac{C_{2}}{C_{1}} V_{2}\right]
M_{x}	M_{0}\left[\frac{C_{2}}{C_{1}} V_{8}-\frac{C_{3}}{C_{1}} V_{1}-V_{7}\right]	\frac{Q_{0}}{\beta}\left[-\frac{C_{4}}{C_{1}} V_{8}-\frac{C_{5}}{C_{1}} V_{6}+\frac{C_{6}}{C_{1}} V_{5}\right]	2 \beta D \theta_{0}\left[\frac{C_{6}}{C_{1}} V_{6}+\frac{C_{5}}{C_{1}} V_{5}-\frac{C_{4}}{C_{1}} V_{7}\right] 2 \beta^{2} D \Delta_{0}\left[-V_{8}+\frac{C_{3}}{C_{1}} V_{2}-\frac{C_{2}}{C_{1}} V_{7}\right]
N_{\theta}	2 B^{2} r M_{0}\left[-\frac{C_{2}}{C_{1}} V_{7}+\frac{C_{3}}{C_{1}} V_{2}-V_{8}\right]	2 \beta r Q_{0}\left[\frac{C_{4}}{C_{1}} V_{7}-\frac{C_{5}}{C_{1}} V_{5}-\frac{C_{6}}{C_{1}} V_{6}\right]	\frac{E t \theta_{0}}{\beta}\left[\frac{C_{6}}{C_{1}} V_{6}+\frac{C_{5}}{C_{1}} V_{5}-\frac{C_{4}}{C_{1}} V_{7}\right] \frac{E t \Delta_{0}}{r}\left[V_{7}+\frac{C_{3}}{C_{1}} V_{1}-\frac{C_{2}}{C_{1}} V_{8}\right]
Q_{x}	-\beta M_{0}\left[\frac{C_{2}}{C_{1}} V_{2}-\left(\frac{2 C_{3}}{C_{1}}+1\right) V_{8}\right]	Q_{0}\left[\frac{C_{4}}{C_{1}} V_{2}+\frac{C_{5}}{C_{1}} V_{3}-\frac{C_{6}}{C_{1}} V_{4}\right]	2 \beta^{2} D \theta_{0}\left[\frac{C_{6}}{C_{1}} V_{3}+\frac{C_{5}}{C_{1}} V_{4}+\frac{C_{4}}{C_{1}} V_{1}\right]-2 \beta^{3} D \Delta_{0}\left[-V_{2}+2 \frac{C_{3}}{C_{1}} V_{7}+\frac{C_{2}}{C_{1}} V_{1}\right]
	Constants		Variables
	C_{1}=\sinh ^{2} \beta l-\sin ^{2} \beta l		V_{1}=\cosh \beta x \sin \beta x-\sinh \beta x \cos \beta x
	C_{2}=\sinh ^{2} \beta l+\sin ^{2} \beta l		V_{2}=\cosh \beta x \sin \beta x+\sinh \beta x \cos \beta x
	C_{3}=\sinh \beta l \cosh \beta l+\sin \beta l \cos \beta l		V_{3}=\cosh \beta x \cos \beta x-\sinh \beta x \sin \beta x
	C_{4}=\sinh \beta l \cosh \beta l-\sin \beta l \cos \beta l		V_{4}=\cosh \beta x \cos \beta x+\sinh \beta x \sin \beta x
	C_{5}=\sin ^{2} \beta l		V_{5}=\cosh \beta x \sin \beta x
	C_{6}=\sinh ^{2} \beta l		V_{6}=\sinh \beta x \cos \beta x
			V_{7}=\cosh \beta x \cos \beta x
			V_{8}=\sinh \beta x \sin \beta x