A circular punch 20 mm in diameter is used to punch a hole through a steel plate 10 mm thick. If the force necessary to drive the punch through the metal is 250 kN, determine the maximum shearing stress developed in the material.

Answer 1

To find the maximum shearing stress developed in the material, we can use the formula for shear stress:

Shear stress = Force / Area

First, let's find the area of the circular punch.

The area of a circle is given by the formula:
Area = π * (diameter/2)^2

Given that the diameter of the punch is 20 mm, we can calculate the area:
Area = π * (20/2)^2 = π * 10^2 = 100π mm^2

Next, let's convert the area from mm^2 to m^2:
Area = 100π * (1 m / 1000 mm)^2 = 100π * (1/1000)^2 = 100π * 1/1000^2 = 100π/10^6 m^2

Now let's substitute the values into the formula for shear stress:
Shear stress = Force / Area
Shear stress = 250 kN / (100π/10^6 m^2) = (250/100π) * 10^6 N/m^2 = 2500/π * 10^6 N/m^2 ≈ 795.77 MPa

Therefore, the maximum shearing stress developed in the material is approximately 795.77 MPa.