Problem find the shear stress how do you find the y bar

Question

Problem: Find the Shear Stress. How do you find the y bar value of the hollow shaft? How did they arrive at that formula?

D - 30mm

d - 20mm

A? - area of the cross-section isolated by the horizontal cut; i.e., above the location of the shear stress being determined (i.e., above y)

y? - vertical distance between the centroidal axis and the CG of A?

Since: - dM_xz/dx = F_xy

∴ (Dfyx/dx = Fxy/Iz )A? y

This term is known as the 'shear force per unit length' or 'shear flow'

To calculate the maximum shear stress:

Shear force,

F_xy = 30 * 9.81N = 294.9N

Shear stress,

Τ = FxyA?y? /lb

We have: I = π/64 (6.5 * 10^-7) m⁴ = 3.19 * 10⁴mm⁴

b = 10 mm

A? = ½ (πD² - πd²/4)

= π/8 (D² - d²)

= π/8 (30² - 20²) = 196.3mm²

y? = D + d/2π = 30 + 20/ 2π

= 7.96mm