User:Eas4200c.f08.aeris.guan/hw6/1110

Bending and Torsional Analysis Continued
To continue torsional analysis, let us consider two cases of different cross sections. Case 1 is a thin-walled cross section that is closed. An example of this kind of cross section would be a NACA airfoil. It is two lateral surfaces denoted by S with corresponding subscripts as shown in the figure below. Case 2 is a solid cross section with only one lateral surface. Both cross sections are uniform and non circular.  In case 2, solid sections with a single contour boundary have phi as an arbitrary constant and thus can be chosen to be zero. This is due to a traction free boundary condition meaning that the traction force t_z is equal to zero. In case 1, using the Prandtl stress function phi, the stress-free boundary condition says that the derivative of phi with respect to the contour length is zero on lateral surfaces o and i. This means that phi on S_o and S_i are different constants. Differing from case 2, these two constants cannot be set to equal zero. Now consider a uniform bar with circular, solid cross section. The Prandtl stress function of this geometry is as follows.  $$ \phi(y,z)=c\left(\frac{y^2}{a^2}+\frac{z^2}{b^2}-1\right) $$ where a=b=radius and thus constituting a circle. If a did not equal b, the shape of the cross section would be an ellipse. Applying equation 3.19 in the book, it is possible to solve for the factor C. $$
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\nabla^2{\phi}=\frac{\partial^2{\phi}}{\partial{y}^2}+\frac{\partial^2{\phi}}{\partial{z}^2}=-2G\theta$$ $$\nabla^2{\left[C(\frac{y^2}{a^2}+\frac{z^2}{a^2}-1)\right]}={C}\nabla^2{\left[(\frac{y^2}{a^2}+\frac{z^2}{a^2}-1)\right]}$$ $$=C\nabla{(\frac{2y}{a^2}+\frac{2z}{a^2})}=\frac{2C}{a^2}\nabla{(y+z)}=\frac{4C}{a^2}$$ Applying the compatibility equation yields the following $$C=-\frac{a^2}{2}G\theta$$