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When same principal tensile stresses P1 and P2 in two mutually perpendicular direction X and Y is subjected in a rectangular element. The maximum shear would occur along the
- a)Plane normal to X axis
- b)Plane normal to Y axis
- c)Plane at 450 to axis
- d)Plane at 450 and 1350 to te Y direction
Correct answer is option 'D'. Can you explain this answer?
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When same principal tensile stresses P1 and P2 in two mutually perpen...
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The shear stress, τ = - (σ1-σ2) sin 2θ/2
Where θ is the inclination of the angle of the plane of the major principal stress.
For shear stress to be maximum
Sin (2θ) = 1
So, θ = 450 and 1350 with y-direction.
Most Upvoted Answer
When same principal tensile stresses P1 and P2 in two mutually perpen...
Explanation:
To understand why the maximum shear occurs along a plane at 45° and 135° to the Y direction, let's first review the concept of principal stresses and shear stresses.
Principal Stresses:
Principal stresses are the normal stresses acting on a plane through a point in a material. They represent the maximum and minimum stresses experienced by the material. In this case, the principal stresses are P1 and P2 acting in the X and Y directions, respectively.
Shear Stresses:
Shear stresses are the stresses that act parallel to a given plane and are responsible for the deformation of the material. The maximum shear stress occurs when the shear stress is at its maximum value.
Analysis:
When the principal stresses P1 and P2 act on a rectangular element, the maximum shear stress occurs along a plane at 45° and 135° to the Y direction. This can be explained by considering the Mohr's circle representation of stress.
Mohr's Circle:
Mohr's circle is a graphical method used to determine the principal stresses and maximum shear stress for a given stress state. In this case, we can construct a Mohr's circle with P1 and P2 as the two principal stresses.
Construction of Mohr's Circle:
1. Plot the principal stresses P1 and P2 on the x-axis and y-axis of the circle, respectively.
2. Draw a circle with a radius equal to the difference between P1 and P2.
3. The center of the circle represents the average normal stress, which is the average of P1 and P2.
4. The points on the circle represent the states of stress at different planes.
5. The angle between the x-axis and the line connecting the center of the circle to a point on the circle represents the orientation of the plane.
Maximum Shear Stress:
The maximum shear stress occurs when the radius of the Mohr's circle is perpendicular to the line representing the plane. In this case, the line representing the plane at 45° and 135° to the Y direction is perpendicular to the radius of the circle.
Therefore, the maximum shear stress occurs along the plane at 45° and 135° to the Y direction, which is option D.
To understand why the maximum shear occurs along a plane at 45° and 135° to the Y direction, let's first review the concept of principal stresses and shear stresses.
Principal Stresses:
Principal stresses are the normal stresses acting on a plane through a point in a material. They represent the maximum and minimum stresses experienced by the material. In this case, the principal stresses are P1 and P2 acting in the X and Y directions, respectively.
Shear Stresses:
Shear stresses are the stresses that act parallel to a given plane and are responsible for the deformation of the material. The maximum shear stress occurs when the shear stress is at its maximum value.
Analysis:
When the principal stresses P1 and P2 act on a rectangular element, the maximum shear stress occurs along a plane at 45° and 135° to the Y direction. This can be explained by considering the Mohr's circle representation of stress.
Mohr's Circle:
Mohr's circle is a graphical method used to determine the principal stresses and maximum shear stress for a given stress state. In this case, we can construct a Mohr's circle with P1 and P2 as the two principal stresses.
Construction of Mohr's Circle:
1. Plot the principal stresses P1 and P2 on the x-axis and y-axis of the circle, respectively.
2. Draw a circle with a radius equal to the difference between P1 and P2.
3. The center of the circle represents the average normal stress, which is the average of P1 and P2.
4. The points on the circle represent the states of stress at different planes.
5. The angle between the x-axis and the line connecting the center of the circle to a point on the circle represents the orientation of the plane.
Maximum Shear Stress:
The maximum shear stress occurs when the radius of the Mohr's circle is perpendicular to the line representing the plane. In this case, the line representing the plane at 45° and 135° to the Y direction is perpendicular to the radius of the circle.
Therefore, the maximum shear stress occurs along the plane at 45° and 135° to the Y direction, which is option D.
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When same principal tensile stresses P1 and P2 in two mutually perpendicular direction X and Y is subjected in a rectangular element. The maximum shear would occur along thea)Plane normal to X axisb)Plane normal to Y axisc)Plane at 450 to axisd)Plane at 450 and 1350 to te Y directionCorrect answer is option 'D'. Can you explain this answer?
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