The thin wall shaft is fixed at and loaded by concentrated torques at sections and of direction and magnitude indicated in the figure. The shaft is composed of three segments, and of equal lengths m, and diameters indicated in the figure. The three segments are hollow tubes made of steel with shear modulus GPa and have equal wall thickness mm.
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To find the maximum shear stress in the shaft, we need to calculate the torque applied at each section and then determine the maximum shear stress using the shear formula.
1. First, let's calculate the torques at sections A and B. The torque at each section can be determined using the formula:
Torque = Force × Distance
In this case, the torque at section A is -100 N·m (negative sign indicates counterclockwise direction) and at section B is 150 N·m (clockwise direction).
2. Next, we need to determine the polar moments of inertia for each segment of the shaft. The polar moment of inertia (J) is given by the formula:
J = π/32 * (D^4 - d^4)
Where D is the outer diameter and d is the inner diameter of the hollow tube segment.
Since all three segments have equal diameters and wall thicknesses, we can calculate the polar moment of inertia for one segment and multiply it by three to get the total polar moment of inertia.
3. Now, we can determine the maximum shear stress using the shear formula:
Shear stress = (Torque * r) / J
Where r is the radius of the shaft.
We need to calculate the shear stress at each section A and B. Since the outer diameter varies for each segment, we need to calculate the radius for each section using the respective outer diameter.
4. Finally, we compare the shear stress at each section to find the maximum shear stress in the shaft.
Remember to check the units throughout the calculations and use consistent SI units for all quantities.