Shafts are generally manufactured from ductile materials, therefore, principle shear stress theory can be used for shaft design which can sustain various combination of loads. Using Mohr’s circle, shear strength of a ductile material is related to yield strength \(\sigma_y\) as
\[ \tau_y = \frac{\sigma_y}{2} \]5.6.1.1 Simple bending
Suppose a hollow shaft is to be designed for a bending moment \(M\). Internal and external diameters of the shaft are \(d_i\) and \(d_o\). Using flexure formula, the maximum bending strength of the shaft is determined as
\[ \frac{\sigma_y}{N} = \frac{32 M}{\pi d_o^3 (1 - k^4)} \]where \(k = d_i / d_o\) and \(N\) is the factor of safety. For solid shafts (\(k = 0\)):
\[ \frac{\sigma_y}{N} = \frac{32 M}{\pi d_o^3} \]5.6.1.2 Simple torsion
Suppose a hollow shaft is to be designed for a torsional moment \(T\). Internal and external diameters of the shaft are \(d_i\) and \(d_o\). Using torsion formula, the maximum shear stress in the shaft is given by
\[ \frac{\tau_y}{N} = \frac{16 T}{\pi d_o^3 (1 - k^4)} \]where \(k = d_i / d_o\) and \(N\) is the factor of safety. For solid shafts (\(k = 0\)):
\[ \frac{\tau_y}{N} = \frac{16 T}{\pi d_o^3} \]5.6.1.3 Combined bending and torsion
Most transmission shaft supporting gears and pulleys are subjected to a combined load of bending and torsional moments. Suppose a hollow shaft is to be designed for a combined loads of bending moment \(M\) and torsional moment \(T\). Internal and external diameters of the shaft are \(d_i\) and \(d_o\). Using Mohr’s circle for combined stresses, the combined loads can be replaced by an equivalent bending moment \((M_e)\) or equivalent torsional moment \((T_e)\), given by
\[ M_e = \frac{1}{2}\left( M + \sqrt{M^2 + T^2} \right) \] \[ T_e = \sqrt{M^2 + T^2} \]Using flexure formula and torsion formula, the maximum bending and shear stresses in the shaft are determined as
\[ \frac{\sigma_y}{N} = \frac{32 M}{\pi d_o^3 (1 - k^4)} \] \[ \frac{\tau_y}{N} = \frac{16 T_e}{\pi d_o^3 (1 - k^4)} \]where \(k = d_i / d_o\). For solid shafts, \(k = 0\).
The standard procedure for design of shafts against combined bending and torsion is specified in ASME code which takes the permissible shear stress \(\tau_y\) as the minimum of \(0.3 \times \sigma_{yt}\) and \(0.18 \times \sigma_{ut}\). Key-ways are assumed to reduce these values by 25%. The code is based maximum shear stress theory and takes into account the shock and fatigue in operating conditions. Accordingly, the maximum shear stress in a shaft is given by
\[ \frac{\tau_y}{N} = \frac{16}{\pi d^3}\,\sqrt{(k_b M)^2 + (k_t T)^2} \]where \(k_b\) and \(k_t\) are combined shock and fatigue factors applied to bending and torque, respectively.