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## Q. 10.4

A semi-circular groove of radius 3 mm is machined in a 50 mm diameter shaft which is then subjected to the following combined loading system:

(a) a direct tensile load of 50 kN,

(b) a bending moment of 150 Nm,

(c) a torque of 320 Nm.

Determine the maximum value of the stress produced by each loading separately and hence estimate the likely maximum stress value under the combined loading

## Verified Solution

For the shaft dimensions given, D/d = 50/(50 – 6) = 1.14 and r/d = 3/44 = 0.068

Nominal stress  $σ_{nom}=\frac{P}{A} =\frac{50\times 10^{3}}{\pi \times (22\times 10^{-3})^{2}} =32.9 MN/m^{2}$

From Fig. 10.23         $K_{t}=2.51$

Maximum stress  $= 2.51 × 32.9 = 82.6 MN/m^{2}$

(b) For bending

Nominal stress $σ_{nom}=\frac{32M}{\pi d^{3}} =\frac{32\times 150}{\pi \times (44\times 10^{-3})^{3}} =18 MN/m^{2}$

and from Fig. 10.24   $K_{t}=2.24$

Maximum stress Maximum stress  $= 2.24 × 18 = 40.3 MN/m^{2}$

(c) For torsion

Nominal stress $τ_{nom }=\frac{16 T}{\pi d^{3}} =\frac{16\times 320}{\pi \times (44\times 10^{-3})^{3}} =19.1 MN/m^{2}$

and from Fig. 10.25    $Kt_{s}=1.65$

Maximum stress $= 1.65 × 19.1 = 31.5 MN/m^{ 2}$

(d) For the combined loading the direct stresses due to bending and tension add to give a total maximum direct stress of $82.6 + 40.3 = 122.9 MN/m^{2}$ which will then act in conjunction with the shear stress of $3 1.5 MN/m^{2}$ as shown on the element of Fig. 10.49.

Then either by Mohrs circle or the use of eqn.$(13.11)^{†}$the maximum principal stress will be
$σ_{1} = 130.5 MN/m^{2}$.
With a maximum shear stress of $τ_{max}= 69 MN/m^{2}$.