The solid spindle A B has a diameter ds=1.5 in. and is made...

The solid spindle $A B$ has a diameter $d_{s}=1.5$ in. and is made of a steel with an allowable shearing stress of 12 ksi, while sleeve $C D$ is made of a brass with an allowable shearing stress of 7 ksi. Determine the largest torque $\mathbf{T}$ that can be applied at $A$. The torques shown are exerted on pulleys $A, B,$ and $C .$ The diameter of shaft $A B$ is 1.3 in. and that of $B C$ is 1.8 in. Knowing that both shafts are solid, determine the maximum shearing stress in (a) shaft (b) shaft $B C$ $A B$

The solid spindle $A B$ has a diameter $d_{s}=1.5$ in. and is made of a steel with an allowable shearing stress of 12 ksi, while sleeve $C D$ is made of a brass with an allowable shearing stress of 7 ksi. Determine the largest torque $\mathbf{T}$ that can be applied at $A$.
The torques shown are exerted on pulleys $A, B,$ and $C .$ The diameter of shaft $A B$ is 1.3 in. and that of $B C$ is 1.8 in. Knowing that both shafts are solid, determine the maximum shearing stress in (a) shaft (b) shaft $B C$ $A B$

Key Concepts

Torsion in Circular Shafts

This concept involves understanding how applied torques produce shear stresses in circular shafts. The fundamental relationship is given by the torsion formula, where the shear stress (?) is proportional to the applied torque (T) and the distance from the center (r), and inversely proportional to the polar moment of inertia (J). This relationship is essential for determining the allowable torque that can be applied to a shaft without exceeding material limits.

Polar Moment of Inertia

The polar moment of inertia (J) is a geometric property of a cross-section that quantifies its resistance to twisting under a torque. For a solid circular shaft, it is calculated using the formula J = (? * d^4) / 32, where d is the diameter of the shaft. This property is key to linking the applied torque to the resulting shear stresses, as it appears in the denominator of the torsional stress equation.

Allowable Shearing Stress

Allowable shearing stress is the maximum stress that a material can safely withstand in shear without failure. In design problems, each material’s allowable shearing stress is used to ensure that the induced stresses from applied loads do not exceed these limits. This concept is crucial when comparing the performance of materials with different mechanical properties, like steel and brass, to ensure safe operation under torsional loads.

Composite Shaft Analysis

In problems involving multiple materials or sections, such as shafts and sleeves made of different materials, composite shaft analysis is used. This involves assessing each component individually based on its geometry and material properties while considering the overall load distribution. Ensuring that each section operates below its allowable stress helps in determining the maximum overall applied torque and in predicting where the highest stresses occur.

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