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.
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.
To determine 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 hole punched through the steel plate:
Area = π * (diameter/2)^2
Given that the diameter of the punch is 20 mm, we get:
Area = π * (20/2)^2 = π * 10^2 = 100π mm^2
Converting square millimeters to square meters, we have:
Area = 100π * (1/1000)^2 = 0.0001π m^2
Now, we can calculate the maximum shearing stress:
Shear stress = 250 kN / Area
Converting kilonewtons to newtons:
Shear stress = 250,000 N / (0.0001π m^2)
Simplifying:
Shear stress = 250,000 / (0.0001π)
Calculating the value:
Shear stress ≈ 795,774.72 kPa
Therefore, the maximum shearing stress developed in the material is approximately 795,774.72 kPa.
To determine the maximum shearing stress developed in the material, we can use the formula for shear stress:
Shear stress (τ) = Force (F) / Area (A)
First, let's calculate the area of the circular punch:
Area (A) = π * radius²
The radius of the punch is half its diameter, so:
Radius = 20 mm / 2 = 10 mm = 10 * 10⁻³ m (converted to meters)
Now, we can calculate the area:
A = π * (10 * 10⁻³ m)²
Next, let's convert the thickness of the steel plate into meters:
Thickness = 10 mm = 10 * 10⁻³ m
Now, we can calculate the force required to drive the punch through the plate:
Force (F) = 250 kN = 250 * 10³ N (converted to Newtons)
Finally, we can substitute the values into the shear stress formula:
Shear stress (τ) = (250 * 10³ N) / [π * (10 * 10⁻³ m)² * (10 * 10⁻³ m)]
Simplifying the equation:
Shear stress (τ) = (250 * 10³ N) / [π * (10 * 10⁻³ m)³]
Calculating the solution:
Shear stress (τ) ≈ 79.577 N/mm²
Therefore, the maximum shearing stress developed in the material is approximately 79.577 N/mm².