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Required information Problem 03.061 - Consider the maximum torque and angle of twist for a square tube versus a round tube. Two steel thin-wall tubes in torsion of equal length are to be compared. The first is of square cross section, side length b, and wall thickness t. The second is a round of diameter b and wall thickness t. The largest allowable shear stress is τa/l​ and is to be the same in both cases. Problem 03.061.a - Determine the ratio of the maximum torque for a square tube versus a round tube. Determine the ratio of maximum torque for the square tube versus the round tube for b=2 in and t=0.03in. The ratio of maximum torque for the square tube versus the round tube is

User Seer
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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.

User Nazia Jan
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