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The state of stress at a point can be described by σx = 40 MPa , σy = 64 MPa , and τxy = 18 MPa . A second coordinate system is rotated by θ = 25 ∘ A) What is the normal stress in the direction of the x′-axis? B) What is the shear stress in the y′-direction for the faces with a normal in the x′-direction? C) What is the normal stress in the direction of the y′-axis?

The state of stress at a point can be described by σx = 40 MPa , σy = 64 MPa , and τxy = 18 MPa . A second coordinate system is rotated by θ = 25 ∘ A) What is the normal stress in the direction of the x′-axis? B) What is the shear stress in the y′-direction for the faces with a normal in the x′-direction? C) What is the normal stress in the direction of the y′-axis?

Elements Of Electromagnetics
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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The state of stress at a point can be described by σx = 40 MPa , σy = 64 MPa , and τxy = 18 MPa . A second coordinate system is rotated by θ = 25 ∘

A) What is the normal stress in the direction of the x′-axis?

B) What is the shear stress in the y′-direction for the faces with a normal in the x′-direction?

C) What is the normal stress in the direction of the y′-axis?

The image depicts a three-dimensional stress element, which is rotated at an angle θ to illustrate the transformation of stress components between two coordinate systems: (x, y) and (x', y'). ### Description and Components: 1. **Coordinate Axes:** - The original coordinate system is labeled as \(x\) and \(y\). - The rotated coordinate system is labeled as \(x'\) and \(y'\), showing how stresses transform at an angle \(θ\). 2. **Stress Components:** - \(\sigma_x\) and \(\sigma_{x'}\) are normal stresses acting perpendicular to the original and rotated planes, respectively. - \(\sigma_y\) and \(\sigma_{y'}\) are normal stresses acting perpendicular to the original and rotated planes, respectively. - \(\tau_{xy}\) and \(\tau_{x'y'}\) are shear stresses acting parallel to the original and rotated planes, respectively. 3. **Forces and Directions:** - Arrows indicate the direction of these stresses, which act on the faces of the stress element. - Normal stresses (\(\sigma\)) act perpendicular to the surface, while shear stresses (\(\tau\)) act parallel. 4. **Rotation:** - The element is rotated by an angle \(θ\) to show how the stress components transform from \((x, y)\) to \((x', y')\). This visualization is crucial for understanding stress transformation in mechanics, particularly in materials engineering and structural analysis, where analyzing stresses in different orientations helps in assessing material strength and failure conditions.
Transcribed Image Text:The image depicts a three-dimensional stress element, which is rotated at an angle θ to illustrate the transformation of stress components between two coordinate systems: (x, y) and (x', y'). ### Description and Components: 1. **Coordinate Axes:** - The original coordinate system is labeled as \(x\) and \(y\). - The rotated coordinate system is labeled as \(x'\) and \(y'\), showing how stresses transform at an angle \(θ\). 2. **Stress Components:** - \(\sigma_x\) and \(\sigma_{x'}\) are normal stresses acting perpendicular to the original and rotated planes, respectively. - \(\sigma_y\) and \(\sigma_{y'}\) are normal stresses acting perpendicular to the original and rotated planes, respectively. - \(\tau_{xy}\) and \(\tau_{x'y'}\) are shear stresses acting parallel to the original and rotated planes, respectively. 3. **Forces and Directions:** - Arrows indicate the direction of these stresses, which act on the faces of the stress element. - Normal stresses (\(\sigma\)) act perpendicular to the surface, while shear stresses (\(\tau\)) act parallel. 4. **Rotation:** - The element is rotated by an angle \(θ\) to show how the stress components transform from \((x, y)\) to \((x', y')\). This visualization is crucial for understanding stress transformation in mechanics, particularly in materials engineering and structural analysis, where analyzing stresses in different orientations helps in assessing material strength and failure conditions.
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