91Ó°ÊÓ

Torque, often represented as \( T \), is a measure of the rotational force applied to an object. When we apply torque to a shaft, it causes twisting, and the resultant force is vital for checking the material's strength. For engineering calculations, torque is typically measured in Newton-meters (N·m). In this scenario, a torque of \( T = 1000 \, \text{N} \cdot \text{m} \) is applied. The goal is to determine how this torque affects the shearing stresses within the shaft. To better understand the role of torque in calculations, it's crucial to consider:

• Material Properties: Knowing how a material responds to torque can help in selecting the right material for the job. Common properties to consider include elasticity and yield strength.
• Geometry of the Shaft: The shape and dimensions of the shaft are key factors in calculating how torque will affect it.
• Torque Application Points: Calculating moments and torques accurately depends significantly on comprehension of where and how they are applied across the shaft.

Given these elements, torque calculation is essential to assure the structural integrity of circular shafts under rotational loads.

The polar moment of inertia, \( J \), is a measure of a shaft's ability to resist torsion. It directly influences how the shaft will deform under a torque and is derived from the shaft's geometry.For solid circular shafts, the polar moment of inertia is calculated using the formula:\[J = \frac{\pi d^4}{32}\]where \( d \) is the shaft diameter. For instance, calculating \( J \) for shaft AB using the diameter 56 mm, involves converting it to meters, which gives \( 0.056 \text{ m} \).By inserting this value into the formula, you can find the polar moment of inertia:\[J_{AB} = \frac{\pi (0.056)^4}{32} \approx 6.97 \times 10^{-6} \text{ m}^4\]Similarly, for shaft CD with a diameter of 42 mm:\[J_{CD} = \frac{\pi (0.042)^4}{32} \approx 1.30 \times 10^{-6} \text{ m}^4\]This measure is critical in ensuring the shaft can handle the applied torque without exceeding maximum stress limits.

Analyzing stress in a circular shaft helps determine how much load the shaft can safely carry.The primary focus is on the maximum shearing stress, represented as \( \tau_{max} \), which occurs when the torque is applied to the shaft.The formula to calculate maximum shearing stress in these situations is:\[\tau_{max} = \frac{T \cdot c}{J}\]where \( c \) represents the outer radius of the shaft, and \( J \) is its polar moment of inertia. For shaft AB:

• The outer radius \( c = \frac{56}{2} = 0.028 \text{ m} \)
• Substitute into the stress formula: \( \tau_{max, AB} = \frac{1000 \cdot 0.028}{6.97 \times 10^{-6}} \approx 40.17 \, \text{MPa} \)

For shaft CD:

• The outer radius \( c = \frac{42}{2} = 0.021 \text{ m} \)
• Substitute into the stress formula: \( \tau_{max, CD} = \frac{1000 \cdot 0.021}{1.30 \times 10^{-6}} \approx 16.15 \, \text{MPa} \)

Thus, performing a thorough circular shaft stress analysis allows engineers to ensure the structural safety and material efficiency of shafts under torsion.