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Need help with this  Mechanics of Materials review  How to draw the element with the numbers (c)  Use Mohr’s circle to find the principal stresses and the angle through which you have to rotate the original element to get the principal stresses. In the figure provided, draw the element that has the principal stresses and show the angle of rotation as well as the stresses.   (d). Use Mohr’s circle to find the maximum in-plane shear stress and the angle through which you have to rotate the original element. In the figure provided, draw the element and show the angle of rotation as well as the stresses.

Need help with this  Mechanics of Materials review  How to draw the element with the numbers (c)  Use Mohr’s circle to find the principal stresses and the angle through which you have to rotate the original element to get the principal stresses. In the figure provided, draw the element that has the principal stresses and show the angle of rotation as well as the stresses.   (d). Use Mohr’s circle to find the maximum in-plane shear stress and the angle through which you have to rotate the original element. In the figure provided, draw the element and show the angle of rotation as well as the stresses.

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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Need help with this  Mechanics of Materials review 

How to draw the element with the numbers

(c)  Use Mohr’s circle to find the principal stresses and the angle through which you have to rotate the original element to get the principal stresses. In the figure provided, draw the element that has the principal stresses and show the angle of rotation as well as the stresses.

 

(d). Use Mohr’s circle to find the maximum in-plane shear stress and the angle through which you have to rotate the original element. In the figure provided, draw the element and show the angle of rotation as well as the stresses.

 

### Understanding Mohr's Circle #### Problem Statement (b) The Mohr’s circle for the above centre and radius is given below, #### Mohr's Circle Explanation Mohr's Circle is a graphical representation of the state of stress at a point. The diagram provided shows Mohr's Circle for a given center and radius. Below is a detailed breakdown of each component: 1. **Axes**: - The horizontal axis represents the normal stress, denoted as \(\sigma\). - The vertical axis represents the shear stress, denoted as \(\tau_{xy}\). 2. **Center and Radius**: - The center \(C\) of the circle is located at \(C(20,0)\). - The point \(A\) is located at \(A(60, -30)\). - The radius \(R\) of the circle is the distance from \(C\) to \(A\). 3. **Principal Stresses (\(\sigma_1\) and \(\sigma_2\))**: - \(\sigma_1\) is the maximum principal stress, which lies on the extreme right edge of the circle: \(\sigma_1\). - \(\sigma_2\) is the minimum principal stress, which lies on the extreme left edge of the circle: \(\sigma_2\). 4. **Shear Stresses (\(\tau_{xy}\), \(\tau_{yx}\))**: - The positive shear stress occurs both on the top and bottom of the vertical axis at the maximum distance from the center of the circle. 5. **Angle \(2\theta\)**: - The angle \(2\theta\) between the center \(C\) and point \(A\) is marked. This angle is significant for finding the orientation of principal planes. #### Diagram Interpretation - Point **(-20,30)**: Represents a stress state on the left side of the circle along the shear and normal stress axes. - Point **(60, -30)**: Represents another stress state on the right side of the circle. - Central Point \(C(20,0)\): Represents the average normal stress. - \(\sigma_x\) and \(\sigma_y\): The extensions along the normal stress axis. By using Mohr's Circle, one can find the principal stresses and the maximum shear stresses and their orientations from complex stress
Transcribed Image Text:### Understanding Mohr's Circle #### Problem Statement (b) The Mohr’s circle for the above centre and radius is given below, #### Mohr's Circle Explanation Mohr's Circle is a graphical representation of the state of stress at a point. The diagram provided shows Mohr's Circle for a given center and radius. Below is a detailed breakdown of each component: 1. **Axes**: - The horizontal axis represents the normal stress, denoted as \(\sigma\). - The vertical axis represents the shear stress, denoted as \(\tau_{xy}\). 2. **Center and Radius**: - The center \(C\) of the circle is located at \(C(20,0)\). - The point \(A\) is located at \(A(60, -30)\). - The radius \(R\) of the circle is the distance from \(C\) to \(A\). 3. **Principal Stresses (\(\sigma_1\) and \(\sigma_2\))**: - \(\sigma_1\) is the maximum principal stress, which lies on the extreme right edge of the circle: \(\sigma_1\). - \(\sigma_2\) is the minimum principal stress, which lies on the extreme left edge of the circle: \(\sigma_2\). 4. **Shear Stresses (\(\tau_{xy}\), \(\tau_{yx}\))**: - The positive shear stress occurs both on the top and bottom of the vertical axis at the maximum distance from the center of the circle. 5. **Angle \(2\theta\)**: - The angle \(2\theta\) between the center \(C\) and point \(A\) is marked. This angle is significant for finding the orientation of principal planes. #### Diagram Interpretation - Point **(-20,30)**: Represents a stress state on the left side of the circle along the shear and normal stress axes. - Point **(60, -30)**: Represents another stress state on the right side of the circle. - Central Point \(C(20,0)\): Represents the average normal stress. - \(\sigma_x\) and \(\sigma_y\): The extensions along the normal stress axis. By using Mohr's Circle, one can find the principal stresses and the maximum shear stresses and their orientations from complex stress
--- ### Problem Statement: Finding Normal and Shear Stresses #### Task e: **Objective:** Determine the normal and shear stresses in the plane represented by the dashed line. **Instructions:** To find the appropriate values for the normal and shear stresses, identify the angle \( \theta \) to use for the stress transformation equations: \[ \sigma_{x'} = \frac{\sigma_x + \sigma_y}{2} + \frac{\sigma_x - \sigma_y}{2}\cos{2\theta} + \tau_{xy}\sin{2\theta} \] \[ \tau_{x'y'} = \left( \frac{\sigma_x - \sigma_y}{2} \right)\sin{2\theta} + \tau_{xy} \cos{2\theta} \] **Given Data:** - Horizontal stress (\(\sigma_x\)): 20 MPa - Vertical stress (\(\sigma_y\)): 40 MPa - Shear stress (\(\tau_{xy}\)): 30 MPa - Angle (\(\theta\)): **15°** (specified by the dashed line in the figure) #### Mohr’s Circle Construction: **Tasks:** 1. **Indicate Point Q:** - Add a point in the Mohr's circle from part b that represents the given state of stress. - Label this point as Q. - Clearly show necessary angles for clarity. 2. **Finding Perpendicular Plane Stress:** - Identify the state of stress in the plane perpendicular to the dashed line. - Mark this in the Mohr's circle on the first page of the problem and label it as point S. - Calculate normal and shear stresses for this plane. - Show detailed calculations. **Note:** The necessary value for \(\theta\) to be used in the stress transformation equations is directly provided as **15°**. #### Diagram Analysis: - **Figure Description:** - A square element subject to different levels of stress is shown: - 20 MPa applied horizontally. - 40 MPa applied vertically. - 30 MPa applied as shear stress. - A dashed line indicates a plane at **15°** to the horizontal axis, along which stresses need to be calculated. #### Conclusion: Proceed with the given information and use the specified formulas to compute the normal and shear stresses for the designated plane and the perpendicular
Transcribed Image Text:--- ### Problem Statement: Finding Normal and Shear Stresses #### Task e: **Objective:** Determine the normal and shear stresses in the plane represented by the dashed line. **Instructions:** To find the appropriate values for the normal and shear stresses, identify the angle \( \theta \) to use for the stress transformation equations: \[ \sigma_{x'} = \frac{\sigma_x + \sigma_y}{2} + \frac{\sigma_x - \sigma_y}{2}\cos{2\theta} + \tau_{xy}\sin{2\theta} \] \[ \tau_{x'y'} = \left( \frac{\sigma_x - \sigma_y}{2} \right)\sin{2\theta} + \tau_{xy} \cos{2\theta} \] **Given Data:** - Horizontal stress (\(\sigma_x\)): 20 MPa - Vertical stress (\(\sigma_y\)): 40 MPa - Shear stress (\(\tau_{xy}\)): 30 MPa - Angle (\(\theta\)): **15°** (specified by the dashed line in the figure) #### Mohr’s Circle Construction: **Tasks:** 1. **Indicate Point Q:** - Add a point in the Mohr's circle from part b that represents the given state of stress. - Label this point as Q. - Clearly show necessary angles for clarity. 2. **Finding Perpendicular Plane Stress:** - Identify the state of stress in the plane perpendicular to the dashed line. - Mark this in the Mohr's circle on the first page of the problem and label it as point S. - Calculate normal and shear stresses for this plane. - Show detailed calculations. **Note:** The necessary value for \(\theta\) to be used in the stress transformation equations is directly provided as **15°**. #### Diagram Analysis: - **Figure Description:** - A square element subject to different levels of stress is shown: - 20 MPa applied horizontally. - 40 MPa applied vertically. - 30 MPa applied as shear stress. - A dashed line indicates a plane at **15°** to the horizontal axis, along which stresses need to be calculated. #### Conclusion: Proceed with the given information and use the specified formulas to compute the normal and shear stresses for the designated plane and the perpendicular
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