A hollow circular shaft has an external diameter of 160mm and an internal diameter of 120mm. Calculagte the polar second moment of areaof the shaft and the maximum and minimum shear stress produced when the applied moment is 40kN m.
D=150 mm =0.15 m d= 120 mm = 0.12 m
T=40 kN•m = 40000 N•m.
J=π(D²-d²)/32 =...
T/J=Gθ/L=τ(min)/r,
τ(min) = Tr/J=Td/2J=...
T/J=Gθ/L=τ(max)/R,
τ(max) = TR/J=TD/2J=...
To calculate the polar second moment of area (also called the polar moment of inertia), you need to use the formula:
J = (π/2) * (D^4 - d^4)
where J is the polar second moment of area, π is a mathematical constant approximately equal to 3.14159, D is the external diameter of the shaft, and d is the internal diameter of the shaft.
In this case, the external diameter (D) is given as 160mm and the internal diameter (d) is given as 120mm. Let's begin by converting these measurements to meters:
D = 160mm = 0.16m
d = 120mm = 0.12m
Next, substitute these values into the formula to calculate J:
J = (π/2) * (0.16^4 - 0.12^4)
Now, evaluate the expression inside the brackets using a calculator:
J = (π/2) * (0.01024 - 0.0020736)
J ≈ 0.004977 * π
Finally, multiply the result by π to get the polar second moment of area:
J ≈ 0.01565 m^4
Now, to calculate the maximum and minimum shear stress (τmax and τmin), you need to use the following formulas:
τmax = M * rmax / J
τmin = M * rmin / J
where τmax is the maximum shear stress, τmin is the minimum shear stress, M is the applied moment, rmax is the maximum distance from the shaft's center to its outer edge, rmin is the minimum distance from the shaft's center to its inner edge, and J is the polar second moment of area.
In this case, the applied moment (M) is given as 40kN m. Let's convert it to Newton meters (Nm):
M = 40kN m = 40,000 N m
The maximum and minimum distances from the shaft's center to its outer and inner edges, respectively, can be calculated as half of the external and internal diameters:
rmax = D/2 = 0.16m/2 = 0.08m
rmin = d/2 = 0.12m/2 = 0.06m
Now substitute these values into the formulas to calculate the shear stresses:
τmax = (40,000 N m) * 0.08m / (0.01565 m^4)
τmin = (40,000 N m) * 0.06m / (0.01565 m^4)
Evaluate the expressions to get the shear stresses:
τmax ≈ 2033 N/m^2
τmin ≈ 1525 N/m^2
Therefore, the maximum shear stress produced when the applied moment is 40kN m is approximately 2033 N/m^2, and the minimum shear stress is approximately 1525 N/m^2.