Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN: 9780190698614
Author: Sadiku, Matthew N. O.
Publisher: Oxford University Press
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Stress transformation
The term stress indicates the measure of internal forces that develop due to an external force's application on an object. When a specific external force is applied to an object, internal stress generates in the material. The value of stress can obtain b…
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### Problem Description The cantilever bar illustrated is composed of a brittle material. Various forces are applied at specific points along the bar: - At point \( D \): - \( F_x = 300 \) lb - \( F_y = 250 \) lb - \( F_z = 100 \) lb The task is to determine the precise location of the critical stress element at the cross-section at point \( A \). The following needs to be determined: 1. The magnitude of each stress component considering the stress concentration at \( A \). 2. The principal stresses and the maximum shear stress at this location. ### Diagram Explanation The diagram shows a cantilever bar fixed at the wall (left side) extending horizontally to the right. It comprises two cylindrical segments: 1. **Segment OA**: - Diameter: \( \frac{1}{2} \) inch - Radius: \( \frac{1}{8} \) inch - Length: 9 inches from the fixed point \( O \) to point \( A \). 2. **Segment AB (extending to point C)**: - Diameter at B: 1 inch - Diameter at C: \( \frac{1}{2} \) inch - Diameter at connection (cross-section): \( \frac{3}{4} \) inch - Length from \( A \) to \( C \) (total of segments): 14 inches **Points of Interest**: - **Point O**: Fixed end of the bar on the wall. - **Point A**: Critical cross-section location. - **Point B and C**: Provide dimensional context for intermediate sections. - **Point D**: Location where forces are applied (12 inches from \( C \)). **Forces**: - **\( F_x \)**: Acts horizontally along the x-axis. - **\( F_y \)**: Acts vertically, downward initially. - **\( F_z \)**: Acts perpendicularly outward from the page. **Additional Dimensions**: - Overall horizontal length of the cantilever bar is crucial for stress calculations. - Various diameter transitions along the bar signify potential concentration points for stress. For further detail-oriented study, focus on evaluating stress distribution using the provided measurements and applied forces to calculate stress elements accurately at the designated sections. This includes analyzing normal and shear stresses, utilizing appropriate equations for stress
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Transcribed Image Text:### Problem Description The cantilever bar illustrated is composed of a brittle material. Various forces are applied at specific points along the bar: - At point \( D \): - \( F_x = 300 \) lb - \( F_y = 250 \) lb - \( F_z = 100 \) lb The task is to determine the precise location of the critical stress element at the cross-section at point \( A \). The following needs to be determined: 1. The magnitude of each stress component considering the stress concentration at \( A \). 2. The principal stresses and the maximum shear stress at this location. ### Diagram Explanation The diagram shows a cantilever bar fixed at the wall (left side) extending horizontally to the right. It comprises two cylindrical segments: 1. **Segment OA**: - Diameter: \( \frac{1}{2} \) inch - Radius: \( \frac{1}{8} \) inch - Length: 9 inches from the fixed point \( O \) to point \( A \). 2. **Segment AB (extending to point C)**: - Diameter at B: 1 inch - Diameter at C: \( \frac{1}{2} \) inch - Diameter at connection (cross-section): \( \frac{3}{4} \) inch - Length from \( A \) to \( C \) (total of segments): 14 inches **Points of Interest**: - **Point O**: Fixed end of the bar on the wall. - **Point A**: Critical cross-section location. - **Point B and C**: Provide dimensional context for intermediate sections. - **Point D**: Location where forces are applied (12 inches from \( C \)). **Forces**: - **\( F_x \)**: Acts horizontally along the x-axis. - **\( F_y \)**: Acts vertically, downward initially. - **\( F_z \)**: Acts perpendicularly outward from the page. **Additional Dimensions**: - Overall horizontal length of the cantilever bar is crucial for stress calculations. - Various diameter transitions along the bar signify potential concentration points for stress. For further detail-oriented study, focus on evaluating stress distribution using the provided measurements and applied forces to calculate stress elements accurately at the designated sections. This includes analyzing normal and shear stresses, utilizing appropriate equations for stress
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Step 1: Given data
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Step 2: Solution (a) The precise location of critical stress element at the cross-section at A
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Step 3: Solution(b) The magnitude for each of the stress components cosidering the stress concentraion at A
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Step 4: Solution (c) Principal stresses and Maximum shear stress
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