A circular shaft is made of gray cast iron, which may be...

A circular shaft is made of gray cast iron, which may be considered to be linearly elastic up to its ultimate strength $\sigma_u=$ $145 \mathrm{MPa}$. The shaft is subjected to a moment $M=5.50 \mathrm{kN} \cdot \mathrm{m}$ and torque $T=5.00 \mathrm{kN} \cdot \mathrm{m}$. Determine the diameter $d$ of the shaft if the factor of safety against brittle fracture is $S F=4.00$.

Key Concepts

Combined Loading

When a shaft is subjected to more than one type of load, such as bending moments and torsional torques, the resulting stress state is a combination of stresses induced by each load condition. Understanding how to analyze combined loading is essential because it allows for the superposition of bending stress, which is usually calculated using the section modulus, and torsional stress, calculated based on the polar moment of inertia. This combined analysis ensures that the design is safe under the actual service conditions, accounting for the synergistic effects of multiple stress types.

Geometric Properties of Circular Sections

The geometry of a circular cross-section directly influences its resistance to bending and torsion through its moment of inertia and polar moment of inertia, respectively. The moment of inertia quantifies the distribution of area about a given axis and is vital for calculating bending moments, while the polar moment of inertia is used in computing torsional stresses. Recognizing these properties helps in designing shafts by correlating physical dimensions, such as diameter, with mechanical performance criteria.

Failure Criteria for Brittle Materials

Brittle materials, unlike ductile ones, tend to fail by sudden fracture without significant plastic deformation. For such materials, the design must ensure that the maximum principal stress does not exceed the material's ultimate strength divided by the required factor of safety. This approach is critical because it takes into account the sensitivity of brittle materials to tensile stresses, making failure predictions more conservative.

Factor of Safety

The factor of safety is a design parameter used to provide a margin against uncertainties in loading, material properties, and potential flaws. In engineering design, it is implemented by ensuring that the calculated stresses are significantly below the material's ultimate strength when the factor is applied. This conservative approach enhances reliability and reduces the risk of unexpected failure, particularly in critical applications where brittle fracture can have catastrophic consequences.

Material Properties in the Linear Elastic Regime

Materials that exhibit linear elastic behavior have a proportional relationship between stress and strain up to a certain limit. In design analyses for such materials, it is assumed that the material will deform in a predictable manner under load without any permanent deformation until the ultimate strength is reached. This assumption simplifies analysis and is crucial for determining safe dimensions and load capacities in engineering components.

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