A flat plate specimen of a soft metal alloy is to be tested by pulling on round pins inserted through the ends of the plate, as illustrated below

(a) Calculate the required diameter of the pins, given that the plate is 8.0 mm thick, the maximum tensile load is 4.0 kN and the maximum allowable average bearing stress is 50 MPa.
(b) What stresses are apparent in the pins?

To calculate the required diameter of the pins (or the bearing area), we need to consider the maximum allowable average bearing stress on the plate.

(a) Calculation of the required diameter of the pins:

The formula for bearing stress is:

σ = F / A

where:
σ = average bearing stress
F = force
A = bearing area

In this case, the maximum allowable average bearing stress (σ) is given as 50 MPa, and the maximum tensile load (F) is given as 4.0 kN.

First, let's convert the load from kN to N:
4.0 kN = 4,000 N

Next, let's calculate the required bearing area (A):
σ = F / A
50 MPa = 4,000 N / A
A = 4,000 N / 50 MPa

Now, let's convert the units of stress and force:
1 MPa = 1 N/mm²

A = 4,000 N / 50 N/mm²
A = 80 mm²

Since we have two pins, the total required bearing area (A_total) is:
A_total = 2 * A
A_total = 2 * 80 mm²
A_total = 160 mm²

The formula for the area of a circle is:
A = π * r²

where:
A = area
π ≈ 3.14 (a constant)
r = radius

Let's calculate the required radius (r):
A_total = π * r²
160 mm² = 3.14 * r²

Solving for r:
r² = 160 mm² / 3.14
r² ≈ 50.8 mm²

Taking the square root on both sides:
r ≈ √50.8 mm²
r ≈ 7.13 mm

Finally, the required diameter (d) is twice the radius (r):
d ≈ 2 * r
d ≈ 2 * 7.13 mm
d ≈ 14.26 mm

Therefore, the required diameter of the pins would be approximately 14.26 mm.

(b) Calculation of the stresses in the pins:

To calculate the stresses in the pins, we need to consider the bearing stress and the area of the pins.

The bearing stress (σ_bearing) can be calculated using the equation:

σ_bearing = F / A_pin

where:
σ_bearing = bearing stress
F = force (4.0 kN or 4,000 N)
A_pin = area of one pin (π * r², where r is half the diameter of the pin)

Let's calculate the bearing stress in the pins:
A_pin = π * (d_pin / 2)²
A_pin = π * (14.26 mm / 2)²
A_pin ≈ 160 mm²

σ_bearing = 4,000 N / 160 mm²
σ_bearing ≈ 25 N/mm²

Therefore, the apparent stress in the pins would be approximately 25 N/mm².