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Chapter 4: Problem 28

Determine how the bending stress at the surface of a shaft of diameter \(d\) changes with values for the bending moment \(M\).

Short Answer

Expert verified
The bending stress at the surface of a shaft directly varies with the bending moment and inversely with the cube of the diameter of the shaft. Therefore, as the bending moment increases or diameter decreases, the bending stress increases.

Step by step solution

01

Apply the Bending Stress Formula

The formula for bending stress in terms of shaft diameter and bending moment is \(\sigma = \frac{32M}{\pi d^3}\). This is the equation that relates the bending stress, shaft diameter, and bending moment.
02

Discuss the Relation

This formula tells us that the bending stress on the surface of the shaft is directly proportional to the bending moment. This implies that as \(M\) increases, \(\sigma\) increases, and as \(M\) decreases, \(\sigma\) decreases. On the contrary, \(\sigma\) is inversely proportional to the cube of the diameter \(d\). This means that as \(d\) increases, \(\sigma\) decreases, and vice versa.
03

Conclusion

From the discussed relations, it is concluded that when subjected to the same bending moment, a shaft with a smaller diameter will experience a greater bending stress compared to a shaft with a larger diameter. Similarly, for a given diameter, bending stress increases with the increase in bending moment.

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Most popular questions from this chapter

A cylinder is internally pressurized to a pressure of \(100 \mathrm{MPa}\). This causes tangential and axial stresses in the outer surface of 400 and \(200 \mathrm{MPa}\), respectively. Make a Mohr circle representation of the stresses in the outer surface. What maximum shear stress is experienced by the outer surface?

A 12-in. length of aluminum tubing (with properties of \(E=10.4 \times 10^{6}\) psi and \(\alpha=12 \times 10^{-6}\) per degree Fahrenheit) having a cross- sectional area of \(1.5\) in. \({ }^{2}\) is installed with "fixed" ends so that it is stress-free at \(60^{\circ} \mathrm{F}\). In operation, the tube is heated throughout to a uniform \(260^{\circ} \mathrm{F}\). Careful measurements indicate that the fixed ends separate by \(.008\) in. What loads are exerted on the ends of the tube, and what are the resultant stresses?

The inner surface of a hollow cylinder internally pressurized to \(100 \mathrm{MPa}\) experiences tangential and axial stresses of 600 and \(200 \mathrm{MPa}\), respectively. Make a Mohr circle representation of the stresses in the inner surface. What maximum shear stress exists at the inner surface?

A 30 -mm-diameter shaft transmits \(700 \mathrm{~kW}\) at \(1500 \mathrm{rpm}\). Bending and axial loads are negligible. (a) What is the nominal shear stress at the surface? (b) If a hollow shaft of inside diameter \(0.8\) times outside diameter is used, what outside diameter would be required to give the same outer surface stress? (c) How do weights of the solid and hollow shafts compare?

Is using force flow to study stress in a component an art or a science? Can the concept of force flow be used to study problems where numerical values for loads are indeterminate?
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