When a shaft of diameter D is subjected to a twisting moment T and a bending moment M, then the maximum normal stress is given by

Explanation:

When shaft subjected to pure bending develops normal stress which is given by:

\({σ _b} = \frac{M}{I}{y_{max}} \Rightarrow \frac{M}{{\frac{{\pi {d^4}}}{{64}}}} \times \frac{d}{2} = \frac{{32M}}{{\pi {d^3}}}\)

When shaft subjected to pure twisting moment develops shear stress which is given by:

\({τ _t} = \frac{T}{J}{r_{max}} \Rightarrow \frac{T}{{\frac{{\pi {d^4}}}{{32}}}} \times \frac{d}{2} = \frac{{16T}}{{\pi {d^3}}}\)

The combined effect of bending and torsion produces principal stress which is given by:

\({σ _{1,2}} = \frac{{{σ _x} + {σ _y}}}{2} \pm \sqrt {{{\left\{ {\frac{{{σ _x} - {σ _y}}}{2}} \right\}}^2} + {τ ^2}} \)

\({σ _{1,2}} = \left\{ {\frac{{\frac{{32M}}{{\pi {d^3}}}}}{2}} \right\} \pm \sqrt {{{\left\{ {\frac{{\frac{{32M}}{{\pi {d^3}}}}}{2}} \right\}}^2} + {{\left\{ {\frac{{16T}}{{\pi {d^3}}}} \right\}}^2}} \)

\({σ _1} = \frac{{16}}{{\pi {d^3}}}\left\{ {M + \sqrt {{M^2} + {T^2}} } \right\}\)

\({σ _2} = \frac{{16}}{{\pi {d^3}}}\left\{ {M - \sqrt {{M^2} + {T^2}} } \right\}\)

from the above:

σ_Max = \({σ _1} = \frac{{16}}{{\pi {d^3}}}\left\{ {M + \sqrt {{M^2} + {T^2}} } \right\}\)