Elasticity/Warping of rectangular cylinder

In both these cases, we get trivial values of ${\displaystyle C_{1}=C_{2}=0\,}$.

Let

${\displaystyle \eta =-k^{2}~~~;~~k>0}$

Then,

${\displaystyle {\begin{aligned}f^{''}(x_{1})+k^{2}f(x_{1})=0~~\Rightarrow &~~f(x_{1})=C_{1}\cos(kx_{1})+C_{2}\sin(kx_{1})\\g^{''}(x_{2})-k^{2}g(x_{2})=0~~\Rightarrow &~~g(x_{2})=C_{3}\cosh(kx_{2})+C_{4}\sinh(kx_{2})\end{aligned}}}$

Therefore,

${\displaystyle {\bar {\psi }}(x_{1},x_{2})=\left[C_{1}\cos(kx_{1})+C_{2}\sin(kx_{1})\right]\left[C_{3}\cosh(kx_{2})+C_{4}\sinh(kx_{2})\right]}$

Apply the BCs at ${\displaystyle x_{2}=\pm b\,}$ ~~ (${\displaystyle {\bar {\psi }}_{,2}=2x_{1}}$), to get

${\displaystyle {\begin{aligned}\left[C_{1}\cos(kx_{1})+C_{2}\sin(kx_{1})\right]\left[C_{3}\sinh(kb)+C_{4}\cosh(kb)\right]&=2x_{1}\\\left[C_{1}\cos(kx_{1})+C_{2}\sin(kx_{1})\right]\left[-C_{3}\sinh(kb)+C_{4}\cosh(kb)\right]&=2x_{1}\end{aligned}}}$

or,

${\displaystyle F(x_{1})G^{'}(b)=2x_{1}~~;~~~F(x_{1})G^{'}(-b)=2x_{1}\,}$

The RHS of both equations are odd. Therefore, ${\displaystyle F(x_{1})}$ is odd. Since, ${\displaystyle \cos(kx_{1})\,}$ is an even function, we must have ${\displaystyle C_{1}=0\,}$.

Also,

${\displaystyle F(x_{1})\left[G^{'}(b)-G^{'}(-b)\right]=0}$

Hence, ${\displaystyle G'(b)\,}$ is even. Since ${\displaystyle \sinh(kb)\,}$ is an odd function, we must have ${\displaystyle C_{3}=0\,}$.

Therefore,

${\displaystyle {\bar {\psi }}(x_{1},x_{2})=C_{2}C_{4}\sin(kx_{1})\sinh(kx_{2})=A\sin(kx_{1})\sinh(kx_{2})}$

Apply BCs at ${\displaystyle x_{1}=\pm a\,}$ (${\displaystyle {\bar {\psi }}_{,1}=0}$), to get

${\displaystyle Ak\cos(ka)\sinh(kx_{2})=0\,}$

The only nontrivial solution is obtained when ${\displaystyle \cos(ka)=0}$, which means that

${\displaystyle k_{n}={\frac {(2n+1)\pi }{2a}}~~,~~~n=0,1,2,...}$

The BCs at ${\displaystyle x_{1}=\pm a\,}$ are satisfied by every terms of the series

${\displaystyle {\bar {\psi }}(x_{1},x_{2})=\sum _{n=0}^{\infty }A_{n}\sin(k_{n}x_{1})\sinh(k_{n}x_{2})}$

Applying the BCs at ${\displaystyle x_{1}=\pm b\,}$ again, we get

${\displaystyle \sum _{n=0}^{\infty }A_{n}k_{n}\sin(k_{n}x_{1})\cosh(k_{n}b)=2x_{1}~~\Rightarrow ~~~\sum _{n=0}^{\infty }B_{n}\sin(k_{n}x_{1})=2x_{1}}$

Using the orthogonality of terms of the sine series,

${\displaystyle \int _{-a}^{a}\sin(k_{n}x_{1})\sin(k_{m}x_{1})dx_{1}={\begin{cases}0&{\rm {if}}~m\neq n\\a&{\rm {if}}~m=n\end{cases}}}$

we have

${\displaystyle \int _{-a}^{a}\left[\sum _{n=0}^{\infty }B_{n}\sin(k_{n}x_{1})\right]\sin(k_{m}x_{1})dx_{1}=\int _{-a}^{a}\left[2x_{1}\right]\sin(k_{m}x_{1})dx_{1}}$

or,

${\displaystyle B_{m}a={\frac {4}{ak_{m}^{2}}}\sin(k_{m}a)}$

Now,

${\displaystyle \sin(k_{m}a)=\sin \left({\frac {(2m+1)\pi }{2}}\right)=(-1)^{m}}$

Therefore,

${\displaystyle A_{m}={\frac {B_{m}}{k_{m}\cosh(k_{m}b)}}={\frac {(-1)^{m}32a^{2}}{(2m+1)^{3}\pi ^{3}\cosh(k_{m}b)}}}$

The warping function is

${\displaystyle \psi =x_{1}x_{2}-{\frac {32a^{2}}{\pi ^{3}}}\sum _{n=0}^{\infty }{\frac {(-1)^{n}\sin(k_{n}x_{1})\sinh(k_{n}x_{2})}{(2n+1)^{3}\cosh(k_{n}b)}}}$

The torsion constant and the stresses can be calculated from ${\displaystyle \psi }$.