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## Q. 12.8

A steel shaft supported in bearings A and B at its ends carries a pulley (diameter = 600 mm) at point C as shown in Figure 12.18. The shaft is transmitting power in such a way that power is input by a torque T at A and is taken off through a belt over-running the pulley where the tensions in the belt are 1250 N in tight side and 250 N in the slack side. Calculate the required shaft diameter if the working stresses are $\sigma_n$ = 85 MPa and τ = 42.5 MPa.

## Verified Solution

Clearly from Figure 12.18 given in the problem, portion AC of the shaft is subjected to the torque:

T = (1250 − 250)(300)Nmm = 300 ×10³ Nmm

Shear stress developed on any element placed on the shaft periphery is

$\tau=\frac{16 T}{\pi d^3}=\frac{(16)(300)\left(10^3\right)}{\pi d^3} N mm =\frac{4800\left(10^3\right)}{\pi d^3} N mm$

Again, the shaft AB is subjected to downward bending load at point C of magnitude = 1500 N.
The reaction forces at the bearings are

$R_{ A }=\frac{1500(1.5)}{2.5} N =900 N \text { and } R_{ B }=(1500-900) N =600 N$

Bending moment at C is

M = (900)(1.0)N m = 900 N m = 900 × 10³ N mm

Normal stress on the shaft element at AC is

$\sigma_{x x}=\frac{32 M}{\pi d^3}=\frac{(32)(900)\left(10^3\right)}{\pi d^3}$

Principal normal stress on the element is

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

$=\frac{(16)(900)\left(10^3\right)}{\pi d^3}+\sqrt{\left[\frac{(16)(900)\left(10^3\right)}{\pi d^3}\right]^2+\left[\frac{4800\left(10^3\right)}{\pi d^3}\right]^2}$

$=\frac{(16)(1848.68)\left(10^3\right)}{\pi d^3} \frac{ N }{ mm ^2}$

Therefore,

$\frac{(16)(1848.68)\left(10^3\right)}{\pi d^3}=85$

or                  d = 48.03mm

and maximum shear stress on the element is

$\tau_{\max }=\sqrt{\left[\frac{(16)(900)\left(10^3\right)}{\pi d^3}\right]^2+\left[\frac{4800\left(10^3\right)}{\pi d^3}\right]^2}=\frac{(16)(948.68)\left(10^3\right)}{\pi d^3}$

Therefore,

$\frac{(16)(948.68)\left(10^3\right)}{\pi d^3}=42.5$

or                d = 48.44mm

We select the shaft diameter as d = 48.44 mm.