To determine the ratio of the maximum torque for a square tube versus a round tube, we need to consider the torsional formulas for each shape.
For a square tube:
The maximum torque (T_square) for a square tube is given by the formula:
T_square = (τa / τmax_square) * J_square * G * L / b
Where:
τa is the allowable shear stress
τmax_square is the maximum shear stress for the square tube
J_square is the torsional constant for a square tube (J_square = (b^4 - (b-2t)^4) / 6)
G is the shear modulus of the material
L is the length of the tube
b is the side length of the square tube
t is the wall thickness of the square tube
For a round tube:
The maximum torque (T_round) for a round tube is given by the formula:
T_round = (τa / τmax_round) * J_round * G * L / (2r)
Where:
τmax_round is the maximum shear stress for the round tube
J_round is the torsional constant for a round tube (J_round = π * (r^4) / 2)
r is the radius of the round tube (r = b/2)
To find the ratio of maximum torque for the square tube versus the round tube, we can divide the equation for the square tube by the equation for the round tube:
Ratio = T_square / T_round
Ratio = [(τa / τmax_square) * J_square * G * L / b] / [(τa / τmax_round) * J_round * G * L / (2r)]
Ratio = [(b^4 - (b-2t)^4) / 6] / [(π * (r^4) / 2)]
For the given values b = 2 in and t = 0.03 in, we can substitute these values into the equation and calculate the ratio.
b = 2 in
t = 0.03 in
r = b/2 = 2/2 = 1 in
Ratio = [(2^4 - (2-2*0.03)^4) / 6] / [(π * (1^4) / 2)]
Ratio = [(16 - (1.94)^4) / 6] / [(π * 1^4) / 2]
Using a calculator, we can compute this ratio:
Ratio ≈ 0.949
Therefore, the ratio of the maximum torque for a square tube versus a round tube, with b = 2 in and t = 0.03 in, is approximately 0.949.