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A point in a strained material is subjected to stress shown is Figure. Using Mohr's circle method, determine the normal and tangential stresses across the oblique place. Also find Principal Stresses and their directions and Maximum Shear Stress. Check the answers analytically?
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A point in a strained material is subjected to stress shown is Figure....
Analysis of Stresses using Mohr's Circle Method
Given Data:
- Normal stress on the plane, σx = 50 MPa
- Shear stress on the plane, τxy = 30 MPa
Step 1: Constructing Mohr's Circle
- Plot the normal stress (σx) on the x-axis and shear stress (τxy) on the y-axis.
- Mark point A at (σx, τxy) = (50, 30).
- Draw a circle with diameter AB passing through point A.
- Determine the center O and radius R of the circle.
Step 2: Finding Stresses
- The center of the Mohr's circle represents the average normal stress, σavg = (σx1 + σx2) / 2 = (50 + 0) / 2 = 25 MPa.
- The radius of the Mohr's circle represents the maximum shear stress, τmax = R = (σx1 - σx2) / 2 = (50 - 0) / 2 = 25 MPa.
- The principal stresses are given by σ1 = σavg + τmax = 25 + 25 = 50 MPa and σ2 = σavg - τmax = 25 - 25 = 0 MPa.
- The direction of principal stresses can be found using the formula tan(2θ) = 2τxy / (σx1 - σx2) = 2*30 / (50 - 0) = 1.2, which gives θ = 30 degrees.
Step 3: Analytical Verification
- The normal stress on the oblique plane can be calculated using the formula σ = (σx + σy) / 2 + [(σx - σy) / 2] * cos(2θ) + τxy * sin(2θ).
- Substitute the values to get σ = (50 + 0) / 2 + [(50 - 0) / 2] * cos(60) + 30 * sin(60) = 25 + 25 * 0.5 + 30 * 0.87 = 40 MPa (approx).
- The tangential stress on the oblique plane can be calculated using the formula τ = -[(σx - σy) / 2] * sin(2θ) + τxy * cos(2θ).
- Substitute the values to get τ = -[(50 - 0) / 2] * sin(60) + 30 * cos(60) = -25 * 0.87 + 30 * 0.5 = 5 MPa (approx).
Therefore, the normal stress on the oblique plane is approximately 40 MPa, the tangential stress is approximately 5 MPa, the principal stresses are 50 MPa and 0 MPa, and the maximum shear stress is 25 MPa, with a direction of 30 degrees.
Given Data:
- Normal stress on the plane, σx = 50 MPa
- Shear stress on the plane, τxy = 30 MPa
Step 1: Constructing Mohr's Circle
- Plot the normal stress (σx) on the x-axis and shear stress (τxy) on the y-axis.
- Mark point A at (σx, τxy) = (50, 30).
- Draw a circle with diameter AB passing through point A.
- Determine the center O and radius R of the circle.
Step 2: Finding Stresses
- The center of the Mohr's circle represents the average normal stress, σavg = (σx1 + σx2) / 2 = (50 + 0) / 2 = 25 MPa.
- The radius of the Mohr's circle represents the maximum shear stress, τmax = R = (σx1 - σx2) / 2 = (50 - 0) / 2 = 25 MPa.
- The principal stresses are given by σ1 = σavg + τmax = 25 + 25 = 50 MPa and σ2 = σavg - τmax = 25 - 25 = 0 MPa.
- The direction of principal stresses can be found using the formula tan(2θ) = 2τxy / (σx1 - σx2) = 2*30 / (50 - 0) = 1.2, which gives θ = 30 degrees.
Step 3: Analytical Verification
- The normal stress on the oblique plane can be calculated using the formula σ = (σx + σy) / 2 + [(σx - σy) / 2] * cos(2θ) + τxy * sin(2θ).
- Substitute the values to get σ = (50 + 0) / 2 + [(50 - 0) / 2] * cos(60) + 30 * sin(60) = 25 + 25 * 0.5 + 30 * 0.87 = 40 MPa (approx).
- The tangential stress on the oblique plane can be calculated using the formula τ = -[(σx - σy) / 2] * sin(2θ) + τxy * cos(2θ).
- Substitute the values to get τ = -[(50 - 0) / 2] * sin(60) + 30 * cos(60) = -25 * 0.87 + 30 * 0.5 = 5 MPa (approx).
Therefore, the normal stress on the oblique plane is approximately 40 MPa, the tangential stress is approximately 5 MPa, the principal stresses are 50 MPa and 0 MPa, and the maximum shear stress is 25 MPa, with a direction of 30 degrees.
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A point in a strained material is subjected to stress shown is Figure. Using Mohr's circle method, determine the normal and tangential stresses across the oblique place. Also find Principal Stresses and their directions and Maximum Shear Stress. Check the answers analytically?
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A point in a strained material is subjected to stress shown is Figure. Using Mohr's circle method, determine the normal and tangential stresses across the oblique place. Also find Principal Stresses and their directions and Maximum Shear Stress. Check the answers analytically? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A point in a strained material is subjected to stress shown is Figure. Using Mohr's circle method, determine the normal and tangential stresses across the oblique place. Also find Principal Stresses and their directions and Maximum Shear Stress. Check the answers analytically? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A point in a strained material is subjected to stress shown is Figure. Using Mohr's circle method, determine the normal and tangential stresses across the oblique place. Also find Principal Stresses and their directions and Maximum Shear Stress. Check the answers analytically?.
A point in a strained material is subjected to stress shown is Figure. Using Mohr's circle method, determine the normal and tangential stresses across the oblique place. Also find Principal Stresses and their directions and Maximum Shear Stress. Check the answers analytically? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A point in a strained material is subjected to stress shown is Figure. Using Mohr's circle method, determine the normal and tangential stresses across the oblique place. Also find Principal Stresses and their directions and Maximum Shear Stress. Check the answers analytically? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A point in a strained material is subjected to stress shown is Figure. Using Mohr's circle method, determine the normal and tangential stresses across the oblique place. Also find Principal Stresses and their directions and Maximum Shear Stress. Check the answers analytically?.
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