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Two dimensional stress tensor at a point is given by a matrix
[σxx τxy τyx σyy] = [100 30 30 20 ]MPa
The maximum shear stress is _______ MPa.
- a)15
- b)30
- c)42
- d)50
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Two dimensional stress tensor at a point is given by a matrix[σxx τx...
= 60 ± 50 σmax = 60 + 50 = 110
σmin = 60 - 50 = 10
Most Upvoted Answer
Two dimensional stress tensor at a point is given by a matrix[σxx τx...
To find the maximum shear stress, we need to calculate the principal stresses and then subtract the smaller principal stress from the larger one.
Given stress tensor matrix:
[σxx τxy]
[τyx σyy] = [100 30]
[30 20] MPa
1. Calculate the eigenvalues of the stress tensor matrix. The eigenvalues represent the principal stresses.
The characteristic equation for a 2x2 matrix is given by:
|σxx - λ τxy |
|τyx σyy - λ | = 0
Using the given stress tensor matrix, we have:
(100 - λ)(20 - λ) - (30)(30) = 0
λ^2 - 120λ + 1600 - 900 = 0
λ^2 - 120λ + 700 = 0
Solving this quadratic equation, we find the eigenvalues λ1 and λ2:
λ1 ≈ 113.416
λ2 ≈ 6.584
2. The maximum shear stress (τmax) is given by the difference between the two principal stresses:
τmax = |λ1 - λ2|
≈ |113.416 - 6.584|
≈ 106.832 MPa
Therefore, the maximum shear stress is approximately 106.832 MPa.
The correct answer is not listed among the options provided. However, if we round the value to the nearest whole number, the closest option is 107 MPa, which is not listed. Therefore, it seems that there may be an error in the options or the provided correct answer.
Given stress tensor matrix:
[σxx τxy]
[τyx σyy] = [100 30]
[30 20] MPa
1. Calculate the eigenvalues of the stress tensor matrix. The eigenvalues represent the principal stresses.
The characteristic equation for a 2x2 matrix is given by:
|σxx - λ τxy |
|τyx σyy - λ | = 0
Using the given stress tensor matrix, we have:
(100 - λ)(20 - λ) - (30)(30) = 0
λ^2 - 120λ + 1600 - 900 = 0
λ^2 - 120λ + 700 = 0
Solving this quadratic equation, we find the eigenvalues λ1 and λ2:
λ1 ≈ 113.416
λ2 ≈ 6.584
2. The maximum shear stress (τmax) is given by the difference between the two principal stresses:
τmax = |λ1 - λ2|
≈ |113.416 - 6.584|
≈ 106.832 MPa
Therefore, the maximum shear stress is approximately 106.832 MPa.
The correct answer is not listed among the options provided. However, if we round the value to the nearest whole number, the closest option is 107 MPa, which is not listed. Therefore, it seems that there may be an error in the options or the provided correct answer.
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Two dimensional stress tensor at a point is given by a matrix[σxx τxy τyx σyy] = [100 30 30 20 ]MPaThe maximum shear stress is _______ MPa.a)15b)30c)42d)50Correct answer is option 'D'. Can you explain this answer?
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