GATE 2022 ME (ME2) – Question 58
A cylindrical billet of 100 mm diameter and 100 mm length is extruded by a direct extrusion process to produce a bar of L-section. The cross sectional dimensions of this L-section bar are shown in the figure. The total extrusion pressure (p) in MPa for the above process is related to extrusion ratio (r) as $p=K_s\sigma_m[0.8+1.5\ln(r)+2l/d_0]$, where $\sigma_m$ is the mean flow strength of the billet material in MPa, l is the portion of the billet length remaining to be extruded in mm, $d_0$ is the initial diameter of the billet in mm, and $K_s$ is the die shape factor. If the mean flow strength of the billet material is 50 MPa and the die shape factor is 1.05, then the maximum force required at the start of extrusion is ______ kN (round off to one decimal place).
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Correct answer: 2417.15 to 2441.45
Explanation
**Step 1: area of the L-section.** Overall height 60 mm and width 50 mm, both legs 10 mm thick. Add the two legs and subtract the corner square that is counted twice:
$$A_f=60(10)+50(10)-10\times10=1000\text{ mm}^2 .$$
**Step 2: extrusion ratio.** The billet area is
$$A_0=\frac\pi4(100)^2=7853.98\text{ mm}^2,\qquad r=\frac{A_0}{A_f}=7.854 .$$
**Step 3: pressure at the start of the stroke.** At the start, the whole billet remains, so $l=100$ mm and $\dfrac{2l}{d_0}=2$:
$$p=K_s\sigma_m\big[0.8+1.5\ln r+2\big]=1.05(50)\big[0.8+1.5(2.0612)+2\big]=52.5\times5.892=309.3\text{ MPa}.$$
**Step 4: force.**
$$F=p\,A_0=309.3\times7853.98=2.429\times10^6\text{ N}=\mathbf{2429.3\ kN}.$$