The shaft has an outer diameter of 100 mm and an inner diameter of 80 mm. If it is subjected to

step 1 So let's solve this question. We have j equals to 5 by 32, D note 4, minus D .I, for substitutin

The shaft has an outer diameter of 100 mm and an inner...

The shaft has an outer diameter of $100 \mathrm{mm}$ and an inner diameter of $80 \mathrm{mm}$. If it is subjected to the three torques, determine the absolute maximum shear stress in the shaft. The smooth bearings $A$ and $B$ do not resist torque.

Key Concepts

Torsion in Circular Shafts

Torsion in circular shafts involves the application of a torque that causes the shaft to twist about its longitudinal axis. The relationship between the applied torque, the resulting shear stress, and the geometry of the shaft is fundamental in mechanical design, and it is described by the formula ? = T?/J, where ? is the shear stress, T is the torque, ? is the distance from the center to the point of interest, and J is the polar moment of inertia.

Polar Moment of Inertia for Hollow Sections

The polar moment of inertia is a geometrical property that measures a shaft's resistance to torsion. For hollow circular shafts, it is calculated as J = (?/32)(D_o^4 - D_i^4), where D_o is the outer diameter and D_i is the inner diameter. This value is crucial because it directly influences the magnitude of the shear stress experienced by the material under a given torque.

Shear Stress Distribution in Torsion

The shear stress in a circular shaft subjected to torsion is not uniformly distributed; it increases linearly from zero at the center to a maximum value at the outer surface. This concept is critical in engineering design because it allows the determination of the maximum stress, which is used to ensure that the material's strength is not exceeded, hence avoiding failure.

Superposition of Multiple Torque Loads

When a shaft is subjected to multiple torques, the stresses induced by each torque can be superimposed to determine the overall effect. This principle of superposition assumes that the effects of individual torques can be linearly added, taking into account their directions and magnitudes, to identify the absolute maximum shear stress in a structure.

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