Shaft A B has a 30 -mm diameter and is made of a steel with...

Shaft $A B$ has a 30 -mm diameter and is made of a steel with an allowable shearing stress of 90 MPa; shaft $B C$ has a 50 -mm diameter and is made of an aluminum alloy with an allowable shearing stress of 60 MPa. Neglecting the effect of stress concentrations, determine the largest torque $\mathbf{T}$ that can be applied at $A$

Key Concepts

Weakest Link Approach

This design principle emphasizes that in a system comprising multiple components—each with different dimensions or material properties—the overall load capacity is determined by the part most susceptible to failure. The component with the lowest threshold for stress (the weakest link) essentially limits the maximum applied load that can be safely transmitted through the entire system.

Material Properties and Allowable Stress

Materials exhibit different properties, notably their allowable shear stress, which is the maximum stress the material can endure before yielding or failure. When shafts are made of different materials, each with their own allowable stresses, it becomes important to consider these limits in design to ensure that the component with the lowest allowable stress governs the overall design criteria.

Polar Moment of Inertia

The polar moment of inertia is a geometric property that quantifies an object's resistance to torsional deformation. For circular shafts, it is a function of the diameter raised to the fourth power, meaning small changes in diameter can lead to significant changes in torsional rigidity. This concept is critical in designing shafts to withstand potential torsional loads without exceeding material stress limits.

Shear Stress

Shear stress in the context of torsion is calculated using the formula that relates the applied torque, the shaft's radius, and its polar moment of inertia. This relationship helps determine whether the material’s allowable shear stress is exceeded under a given loading, making it a fundamental concept in assessing the structural integrity of the shaft.

Torsion in Circular Shafts

This concept deals with the analysis of how shafts twist under an applied torque. It involves understanding that when a circular shaft is subjected to a torque, shear stress develops across its cross-section, and that stress is highest at the outer surface. The torsion equation quantifies this relationship and is essential for predicting the behavior of shafts under loading.

Transcript

Please give Ace some feedback