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Obviously, for the above problem, the maximum shear stress is

(\tau)_{\max }=\sqrt{\left\lgroup \frac{\sigma_n}{2} \right\rgroup^2+\tau^2}

or (\tau)_{\max }=\frac{16}{\pi d^3} \sqrt{M^2+T^2}=\frac{16 T_{ e }}{\pi d^3}

where

T_{ e }=\sqrt{M^2+T^2}