### Shaft Design with Semicircular Groove#### Problem Statement:A shaft is to have a semicircular groove machined around its circumference at a specified location. The following parameters are given:- **Shaft Diameter (D):** 0.055 m- **Reduced Cross-sectional Diameter (d):** 0.05 m- **Torsion (T):** 2500 Nm- **Yield Strength (Sy):** 500 MPa- **Safety Factor (η):** 2Using these parameters and the stress concentration figures referenced in the textbook (Fig. A-15-15, p. 1046) along with Eq. 4.21, determine the minimum radius of the groove (r). This analysis will employ two different failure criteria:1. Maximum Distortional Energy Failure Theory2. Maximum Shear Stress Failure Theory#### Data:- **d:** 0.05 m- **D:** 0.055 m- **T:** 2500 Nm- **Sy:** 500 MPa- **η:** 2#### Failure Criteria to Consider:(a) Maximum Distortional Energy Failure Theory(b) Maximum Shear Stress Failure Theory### Diagram ExplanationBelow the problem statement, a diagram is presented illustrating the shaft with the semicircular groove. Relevant dimensions include:- **D (Shaft Diameter):** Labeled on the thicker middle segment of the shaft.- **d (Reduced Diameter):** Labeled on the narrower cross-sectional area of the shaft.- **r (Groove Radius):** The radius to be calculated, shown as transitioning between the shaft diameter (D) and the reduced cross-sectional diameter (d).### Calculation DetailsTo solve this problem, reference the specific equation (Eq. 4.21) from the provided textbook and use the stress concentration figures. This will involve:1. Calculating the stress concentration factor.2. Applying the given failure criteria to determine the allowable stresses.3. Using these stresses to compute the minimum radius \( r \) ensuring the shaft remains safe under the specified torsional load.These computations will ensure that the shaft design adheres to safety requirements, thereby preventing failure under operational stress conditions.

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A shaft is to have a semicircular grove machined around the circumference at a specified location. The shaft diameter and reduced cross-sectional diameter are known as well as the torsion that the shaft will experience and the material's yield strength. Using the stress concentration figures given the textbook as Fig. A-15-15 (p. 1046) and the equations given as Eq. 4.21, determine minimum radius of the groove (r). Using the both of the following failure criteria: (use a safety factor of n = 2) d = 0.05 m, D = 0.055 m, T = 2500 Nm, Sy = 500 MPa, n = 2 (a) Using the maximum distortional energy failure theory (b) Using the maximum shear stress failure theory

A shaft is to have a semicircular grove machined around the circumference at a specified location. The shaft diameter and reduced cross-sectional diameter are known as well as the torsion that the shaft will experience and the material's yield strength. Using the stress concentration figures given the textbook as Fig. A-15-15 (p. 1046) and the equations given as Eq. 4.21, determine minimum radius of the groove (r). Using the both of the following failure criteria: (use a safety factor of n = 2) d = 0.05 m, D = 0.055 m, T = 2500 Nm, Sy = 500 MPa, n = 2 (a) Using the maximum distortional energy failure theory (b) Using the maximum shear stress failure theory

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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### Shaft Design with Semicircular Groove

#### Problem Statement:
A shaft is to have a semicircular groove machined around its circumference at a specified location. The following parameters are given:
- **Shaft Diameter (D):** 0.055 m
- **Reduced Cross-sectional Diameter (d):** 0.05 m
- **Torsion (T):** 2500 Nm
- **Yield Strength (Sy):** 500 MPa
- **Safety Factor (η):** 2

Using these parameters and the stress concentration figures referenced in the textbook (Fig. A-15-15, p. 1046) along with Eq. 4.21, determine the minimum radius of the groove (r). This analysis will employ two different failure criteria:
1. Maximum Distortional Energy Failure Theory
2. Maximum Shear Stress Failure Theory

#### Data:
- **d:** 0.05 m
- **D:** 0.055 m
- **T:** 2500 Nm
- **Sy:** 500 MPa
- **η:** 2

#### Failure Criteria to Consider:
(a) Maximum Distortional Energy Failure Theory
(b) Maximum Shear Stress Failure Theory

### Diagram Explanation
Below the problem statement, a diagram is presented illustrating the shaft with the semicircular groove. Relevant dimensions include:
- **D (Shaft Diameter):** Labeled on the thicker middle segment of the shaft.
- **d (Reduced Diameter):** Labeled on the narrower cross-sectional area of the shaft.
- **r (Groove Radius):** The radius to be calculated, shown as transitioning between the shaft diameter (D) and the reduced cross-sectional diameter (d).

### Calculation Details
To solve this problem, reference the specific equation (Eq. 4.21) from the provided textbook and use the stress concentration figures. This will involve:
1. Calculating the stress concentration factor.
2. Applying the given failure criteria to determine the allowable stresses.
3. Using these stresses to compute the minimum radius \( r \) ensuring the shaft remains safe under the specified torsional load.

These computations will ensure that the shaft design adheres to safety requirements, thereby preventing failure under operational stress conditions.

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