Elasticity/Rotating rectangular beam

Example : Rotating Rectangular Beam

File:Elastic rotating beam.png A rotating rectangular beam

The body force potential is given by

$\text{(52)}V=-\frac{\rho {\dot{\theta }}^2}{2}(x_1^2+x_2^2)$

Hence,

$\text{(53)}{\mathrm{\nabla }}^{2}V=V_{,11}+V_{,22}=-\frac{\rho {\dot{\theta }}^2}{2}(2+2)$

or,

$\text{(54)}{\mathrm{\nabla }}^{2}V=-2\rho {\dot{\theta }}^2$

The compatibility condition (in terms of stress) is

$\text{(55)}{\mathrm{\nabla }}^4\phi +(2-\frac{1}{\alpha }){\mathrm{\nabla }}^{2}V=0$

Plug $V$ in to get

$\text{(56)}{\mathrm{\nabla }}^4\phi -(2-\frac{1}{\alpha })2\rho {\dot{\theta }}^2=0$

Since $V$ is even in $x_1$ and $x_2$ and BCs are homogeneous, assume

$\text{(57)}\phi =A{x}_1^4+B{x}_1^2{x}_2^2+C{x}_2^4+D{x}_1^2+E{x}_2^2$

Hence,

$\begin{array}{cc}\text{(58)}{\sigma }_{11}& ={\phi }_{,22}+V=2B{x}_1^2+12C{x}_2^2+2E-\frac{\rho {\dot{\theta }}^2}{2}(x_1^2+x_2^2)\\ \text{(59)}{\sigma }_{22}& ={\phi }_{,11}+V=12A{x}_1^2+2B{x}_2^2+2D-\frac{\rho {\dot{\theta }}^2}{2}(x_1^2+x_2^2)\\ \text{(60)}{\sigma }_{12}& =-{\phi }_{,12}=-4B{x}_1{x}_2\end{array}$

The traction BCs are

$\begin{array}{ccc}\text{(61)}\text{at}{x}_1=\pm a& & t_1=t_2=0\Rightarrow {\sigma }_{11}={\sigma }_{12}=0\\ \text{(62)}\text{at}{x}_2=\pm b& & t_1=t_2=0\Rightarrow {\sigma }_{12}={\sigma }_{22}=0\end{array}$

Apply BCs at $x_2=\pm b$.

$\begin{array}{cc}\text{(63}){\sigma }_{22}=0& =12A{x}_1^2+2B{b}^2+2D-\frac{\rho {\dot{\theta }}^2}{2}(x_1^2+b^2)\\ \text{(64}){\sigma }_{12}=0& =-4Bb{x}_1\end{array}$

Therefore,

$\begin{array}{cc}\text{(65)}B& =0\\ \text{(66)}A& =\frac{\rho {\dot{\theta }}^2}{24}\\ \text{(67)}D& =\frac{\rho {\dot{\theta }}^2{b}^2}{4}\end{array}$

We then have,

$\text{(68)}\phi =\frac{\rho {\dot{\theta }}^2}{24}{x}_1^4+C{x}_2^4+\frac{\rho {\dot{\theta }}^2{b}^2}{4}{x}_1^2+E{x}_2^2$

Plug into compatibility equation

$\text{(69)}{\phi }_{,1111}+2{\phi }_{,1122}+{\phi }_{,2222}-(2-\frac{1}{\alpha })2\rho {\dot{\theta }}^2=0$

to get

$\text{(70)}24(\frac{\rho {\dot{\theta }}^2}{24}+C)-(2-\frac{1}{\alpha })2\rho {\dot{\theta }}^2=0$

or,

$\begin{array}{cc}C& =(2-\frac{1}{\alpha })\frac{\rho {\dot{\theta }}^2}{12}-\frac{\rho {\dot{\theta }}^2}{24}\\ & =(3-\frac{2}{\alpha })\frac{\rho {\dot{\theta }}^2}{24}\text{(71)}\end{array}$

Apply BCs at $x_1=\pm a$.

$\begin{array}{cc}\text{(72)}{\sigma }_{11}=0& =2B{a}^2+12C{x}_2^2+2E-\frac{\rho {\dot{\theta }}^2}{2}(a^2+x_2^2)\\ & =(3-\frac{2}{\alpha })\frac{\rho {\dot{\theta }}^2}{2}{x}_2^2+2E-\frac{\rho {\dot{\theta }}^2}{2}(a^2+x_2^2)\text{(73)}\end{array}$

Strong BCs imply that

$\text{(74)}(3-\frac{2}{\alpha })\frac{\rho {\dot{\theta }}^2}{2}-\frac{\rho {\dot{\theta }}^2}{2}=0$

which cannot be true. So weak BCs on ${\sigma }_{11}$ need to be applied at $x_1=\pm a$.

$\begin{array}{ccc}\text{(75)}\text{at}{x}_1=\pm a& & \underset{-b}{\overset{b}{\int }}{\sigma }_{11}d{x}_2=0\end{array}$

Hence,

$\text{(76)}(3-\frac{2}{\alpha })\frac{\rho {\dot{\theta }}^2}{2}\frac{2b^3}{3}+4Eb-\frac{\rho {\dot{\theta }}^2}{2}(2a^{2}b+\frac{2b^3}{3})=0$

or,

$\text{(77)}(2-\frac{2}{\alpha })\frac{\rho {\dot{\theta }}^2{b}^2}{3}+4E-\rho {\dot{\theta }}^2{a}^2=0$

Hence,

$\text{(78)}E=\frac{\rho {\dot{\theta }}^2}{4}[a^2-(2-\frac{2}{\alpha })\frac{b^2}{3}]$

The stress field is, therefore,

$\begin{array}{cc}\text{(79}){\sigma }_{11}& =(3-\frac{2}{\alpha })\frac{\rho {\dot{\theta }}^2}{2}{x}_2^2+\frac{\rho {\dot{\theta }}^2}{2}[a^2-(2-\frac{2}{\alpha })\frac{b^2}{3}]-\frac{\rho {\dot{\theta }}^2}{2}(x_1^2+x_2^2)\\ \text{(80}){\sigma }_{22}& =\frac{\rho {\dot{\theta }}^2}{2}{x}_1^2+\frac{\rho {\dot{\theta }}^2{b}^2}{2}-\frac{\rho {\dot{\theta }}^2}{2}(x_1^2+x_2^2)\\ {\sigma }_{12}& =0\end{array}$

or,

$\begin{array}{cc}\text{(81)}{\sigma }_{11}& =\frac{\rho {\dot{\theta }}^2}{2}[(a^2-x_1^2)+2(1-\frac{1}{\alpha })(x_2^2-\frac{b^2}{3})]\\ \text{(82)}{\sigma }_{22}& =\frac{\rho {\dot{\theta }}^2}{2}(b^2-x_2^2)\\ {\sigma }_{12}& =0\end{array}$

The displacements can be found in the standard manner.

File:Elastic rotating beam sigxx.png Stresses (${\sigma }_{11}$) in a rotating rectangular beam

File:Elastic rotating beam sigxx.png Stresses (${\sigma }_{22}$) in a rotating rectangular beam