GATE Papers >> Mechanical >> 2018 >> Question No 157

Question No. 157 Mechanical | GATE 2018

A steel wire is drawn from an initial diameter $ \left(d_i\right) $ of 10 mm to a final diameter $ \left(d_f\right) $ of 7.5 mm. The half cone angle $ (\alpha) $ of the die is $ 5^\circ $ and the coefficient of friction $ \left(\mu\right) $ between the die and the wire is 0.1. The average of the initial and final yield stress $ \left[{\left(\sigma_Y\right)}_{avg}\right] $ is 350 MPa. The equation for drawing stress $ \sigma_f $ , (in MPa) is given as:

                              $ \sigma_f={\left(\sigma_Y\right)}_{avg}\left\{1+\frac1{\mu\;cot\alpha}\right\}\left[1-\left(\frac{d_f}{d_i}\right)^{2\mu\;cot\alpha}\right] $

The drawing stress (in MPa) required to carry out this operation is _________ (correct to two decimal places).


Answer : 315.00 to 317.00


Solution of Question No 157 of GATE 2018 Mechanical Paper

Given, di = 10 mm 

df = 7.5 mm                    $ \cot\alpha=\cot5^\circ=11.43 $

$ \begin{array}{l}\mathrm\alpha=5^\circ\\\mathrm\mu=0.1\\{\left({\mathrm\sigma}_\mathrm Y\right)}_\mathrm{avg}=350\;\mathrm{MPa}\\{\mathrm\sigma}_\mathrm f={\left({\mathrm\sigma}_\mathrm Y\right)}_\mathrm{avg}\left\{1+\frac1{\mathrm{μcotα}}\right\}\left[1-\left(\frac{\mathrm{df}}{\mathrm{di}}\right)^{2\mathrm{μcotα}}\right]\\{\mathrm\sigma}_\mathrm f=350\left\{1+\frac1{0.1\times11.43}\right\}\left[1-\left(\frac{7.5}{10}\right)^{2\times0.1\times11.43}\right]\\{\mathrm\sigma}_\mathrm f=350\times1.8749\times\left[1-\left(0.75\right)^{2.286}\right]\\{\mathrm\sigma}_\mathrm f=350\times1.8749\times0.482=316.248\;\mathrm{MPa}\end{array} $

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