Max Force Loads on Aluminum Pipe Calculations

One of the parameters of interest would be the maximum allowable bending moment of the aluminum pipe through a radial force on the pipe supported on two ends.

Axial bending load geometry parameter set up across Pipe

Cross Section of Pipe showing two areas of integration against stress distribution

$ M = 4 \sigma_y \frac{E}{\sigma_y R} \int_                         0                     ^              \frac{ \sigma_y R }{E}               \!

y ^2 ( \sqrt( r_0 ^2 - y^2 )              -                       \sqrt(r_i ^2 - y^2) \,

\mathrm{d}y + \int_ \frac{ \sigma_y * R }{E})       ^             ( r_i    )                                        \! $

$ y * \sqrt( (r_0)^2 - y^2 ) - \sqrt( (r_i)^2 - y^2 )  \mathrm{d}y$ + $ \int_  {r_i }     ^    {r_0 }\!          y * \sqrt{( (r_0) ^2) - (y^2 )} \mathrm{d}y $