Question 5.7: Derive N𝜃 for the case of applied bending moment M0 at edge ...
Derive N_{\theta} for the case of applied bending moment M_{0} at edge x=0 for a short cylinder of length l.
The four boundary conditions are as follows:
At x=0,
-D\left(\frac{ d ^{2} w}{ d x^{2}}\right)=M_{0}-D\left(\frac{ d ^{3} w}{ d x^{3}}\right)=0
At x=l,
-D\left(\frac{ d ^{2} w}{ d x^{3}}\right)=0-D\left(\frac{ d ^{3} w}{ d x^{3}}\right)=0
From Eq. (5.26), the second derivative is given by
w=A_{1} \sin \beta x \sinh \beta x+A_{2} \sin \beta x \cosh \beta x +A_{1} \cos \beta x \sinh \beta x+A_{4} \cos \beta x \cosh \beta x (5.26)
\frac{ d ^{2} w}{ d x^{2}}=2 \beta^{2}(A_{1} \cos \beta x \cosh \beta x+A_{2} \cos \beta x \sinh \beta x
-A_{3} \sin \beta x \cosh \beta x-A_{4} \sin \beta x \sinh \beta x) (1)
whereas the third derivative is expressed as
\frac{ d ^{3} w}{ d x^{3}}=2 \beta^{2} [A_{1}(\cos \beta x \sinh \beta x-\sin \beta x \cosh \beta x)+A_{2}(\cos \beta x \cosh \beta x-\sin \beta x \sinh \beta x)
-A_{3}(\sin \beta x \sinh \beta x+\cos \beta x \cosh \beta x)
-A_{4}(\sin \beta x \cosh \beta x+\cos \beta x \sinh \beta x)] (2)
Substituting Eq. (1) into the first boundary condition gives
A_{1}=\frac{-M_{0}}{2 D \beta^{2}}Substituting Eq. (2) into the second boundary condition gives
A_{2}=A_{3} ,
and from the third and fourth boundary conditions, the relationships
A_{3}=\frac{M_{0}}{2 D \beta^{2}}\left(\frac{\sin \beta l \cos \beta l+\sinh \beta l \cosh \beta l}{\sinh ^{2} \beta l-\sin ^{2} \beta l}\right)And
A_{4}=\frac{-M_{0}}{2 D \beta^{2}}\left(\frac{\sin ^{2} \beta l+\sinh ^{2} \beta l}{\sinh ^{2} \beta l-\sin ^{2} \beta l}\right)are obtained.
From Eq. (5.19),
N_{\theta}=\frac{-E t w}{r} (5.19)
N_{\theta}=\frac{E t w}{r}= \frac{E t}{r} (A_{1} \sin \beta x \sinh \beta x+A_{2} \sin \beta x \cosh \beta x + A_{3} \cos \beta x \sinh \beta x+A_{4} \cos \beta x \cosh \beta x) (3)
Using the values of A_{1}, A_{2}, A_{3}, A_{4} thus obtained and the terminology of Table 5.3, Eq. (3) reduces to
N_{\theta}=\frac{E t}{r} \frac{M_{0}}{2 D \beta^{2}}[-\sin \beta x \sinh \beta x+\frac{C_{3}}{C_{1}}(\sin \beta x \cosh \beta x
+\cos \beta x \sinh \beta x)-\frac{C_{2}}{C_{1}} \cos \beta x \cosh \beta x]
or
N_{\theta}=2 r M_{0} \beta^{2}\left(-V_{8}+\frac{C_{3}}{C_{1}} V_{2}-\frac{C_{2}}{C_{1}} V_{7}\right)
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 | |||