A stepped steel shaft, shown below, is used in a spur gear reducer. The shaft is subjected to a constant axial stress (A=40 MPa), a constant bending moment that results in a variable stress (M=60 MPa) due to loads by bearings and gears, and a steady torque (T=80 MPa). However, these stress values do not take into account stress concentration factors. The shaft has a D=45 mm, d=30 mm, and r=6 mm, resulting in ki,bend=1.34, Kraxial 1.48, and k₁,tor 1.2. The shaft is made of steel with S,=1006 MPa, S,=648 MPa, and H₂=229. The size of the shaft results in a gradient factor of CG=0.9. The shaft has a fine-ground finish. A 90% reliability is required. Verify the stress concentration factors are correct for this shaft geometry. Calculate the safety factor relative to infinite life (106 cycles). If the safety factor is less than one (i.e., shaft fails), find the number of cycles to failure.

A stepped steel shaft, shown below, is used in a spur gear reducer. The shaft is subjected to a constant axial stress (A=40 MPa), a constant bending moment that results in a variable stress (M=60 MPa) due to loads by bearings and gears, and a steady torque (T=80 MPa). However, these stress values do not take into account stress concentration factors. The shaft has a D=45 mm, d=30 mm, and r=6 mm, resulting in ki,bend=1.34, Kraxial 1.48, and k₁,tor 1.2. The shaft is made of steel with S,=1006 MPa, S,=648 MPa, and H₂=229. The size of the shaft results in a gradient factor of CG=0.9. The shaft has a fine-ground finish. A 90% reliability is required. Verify the stress concentration factors are correct for this shaft geometry. Calculate the safety factor relative to infinite life (106 cycles). If the safety factor is less than one (i.e., shaft fails), find the number of cycles to failure.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Concept explainers

Combined Loading

Any functioning component, such as machine parts, structural members, pressure vessels, and so on, are all under the influence of more than one force. When anybody is under the influence of more than one force such as shear forces, bending moments, axial forces, torsion, and so on, then such body is said to be under combined loading. A very practical example of combined loading is the crankshaft of an internal combustion engine (IC engine). The crankshaft of the IC engine is under high rpm, which is caused due to the torsional forces. Due to the presence of piston, counterbalances, and internal imbalances, the crankshaft experiences lateral forces and due to the rise in temperature, the crankshaft undergoes thermal expansion, which pulls the crankshaft along the axial direction and lateral direction. The presence of unbalanced loads along the periphery of the crankshaft also induces a bending moment.

Question

1) A stepped steel shaft, shown below, is used in a spur gear reducer. The shaft is subjected to a constant axial stress (A=40 MPa), a constant bending moment that results in a variable stress (M=60 MPa) due to loads by bearings and gears, and a steady torque (T=80 MPa). However, these stress values do not take into account stress concentration factors. The shaft has a D=45 mm, d=30 mm, and r=6 mm, resulting in \(k_{t,bend}=1.34\), \(k_{t,axial}=1.48\), and \(k_{t,torsion}=1.2\). The shaft is made of steel with \(S_u=1006\) MPa, \(S_y=648\) MPa, and \(H_B=229\). The size of the shaft results in a gradient factor of \(C_g=0.9\). The shaft has a fine-ground finish. A 90% reliability is required. Verify the stress concentration factors are correct for this shaft geometry. Calculate the safety factor relative to infinite life (\(10^6\) cycles). If the safety factor is less than one (i.e., shaft fails), find the number of cycles to failure.

**Diagram Explanation:**
- The diagram shows a stepped shaft with two different diameters, \(D\) and \(d\).
- The shaft is subjected to a bending moment \(M\) at both ends and a torque \(T\).
- There are forces \(A\) applied axially at the center of the larger diameter section.
- The radius \(r\) of the fillet between the two diameters is indicated.
- The visual aids support understanding of the various forces and geometries impacting the shaft's stress analysis.

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