User:EAS4200c.f08.Blue.E/nov12th

$$r=(y^2+z^2)^{1/2}$$

$$T=2\int \phi dA=2c(\frac{J}{a^2}-\pi a^2)$$

$$J=\int_A r^2 dA=\frac{1}{2}\pi a^4$$

$$T= GJ\theta$$

$$\sigma_{yz}=\frac{\delta \phi}{\delta z}=-G \theta_z$$

$$\sigma_{zy}=\frac{\delta \phi}{\delta y}= -G\theta_y$$

$$\tau=\frac{T r}{J}$$

$$u_x(x,y)=0$$

i,e no warping

since

$$\gamma_{yx}=\frac{\delta_{yx}}{G}=\frac{\delta_u}{\delta_y}-\theta_z$$

$$\gamma_{zx}=\frac{\delta_{zx}}{G}=\frac{\delta_u}{\delta_z}-\theta_y$$

$$\sigma_{yx}=-G\theta_z$$

$$\sigma_{zx}=G\theta_y$$

therefore

$$\delta u=(G+1)\theta_z\ \delta y=(G+1)\theta_y \delta z$$

since there is no warping

$$\delta y= /delta z = 0$$

therefore

$$u=o$$