Answer:
Pmax = 38251.73 N
Explanation:
Given info
L = 1.47 m
D = 112 mm ⇒ R = D/2 = 112/2 mm = 56 mm
d = 101 mm ⇒ r = D/2 = 101/2 mm = 50.5 mm
a) We can apply the following equation in order to get Q (First Moment of Area):
Q = 2*(A₁*y₁-A₂*y₂)
where
A₁ = π*R² = π*(56 mm)² = 3136 π mm²
y₁ = 4*R/(3*π) = 4*56/(3*π) mm = 224/(3*π) mm
A₂ = π*r² = π*(50.5 mm)² = 2550.25 π mm²
y₂ = 4*r/(3*π) = 4*50.5/(3*π) mm = 202/(3*π) mm
then
Q = 2*(3136 π mm²*224/(3*π) mm-2550.25 π mm²*202/(3*π) mm)
⇒ Q = 62437.833 mm³
b) If τallow = 83 MPa = 83 N/mm²
P = ?
We can use the equation
τ = V*Q / (t*I) ⇒ V = τ*t*I / Q
where
t = D - d = 112 mm - 101 mm = 11 mm
I = (π/64)*(D⁴-d⁴) = (π/64)*((112 mm)⁴-(101 mm)⁴) = 2615942.11 mm⁴
Q = 62437.833 mm³
we could also use this equation in order to get Q:
Q = (4/3)*(R³-r³)
⇒ Q = (4/3)*((56 mm)³-(50.5 mm)³) = 62437.833 mm³
then we have
V = (83 N/mm²)*(11 mm)*(2615942.11 mm⁴) / (62437.833 mm³)
⇒ V = 2942.255 N
Finally Pmax = V = 38251.73 N
