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For an inclined plane in a rectangular block subjected to two mutually perpendicular normal stresses 1000 MPa and 400 MPa and shear stress 400 MPa, the maximum normal stress will be
- a)1200 MPa
- b)700 MPa
- c)600 MPa
- d)200 MPa
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
For an inclined plane in a rectangular block subjected to two mutuall...
σ1 = 
= 700 + √9 × 104 + 16 × 104
= 700 + 500 = 1200MPa
Most Upvoted Answer
For an inclined plane in a rectangular block subjected to two mutuall...
To determine the maximum normal stress on an inclined plane in a rectangular block subjected to two mutually perpendicular normal stresses and a shear stress, we need to consider the stress transformation equations. The stress transformation equations allow us to relate the normal and shear stresses on one plane to the normal and shear stresses on another plane.
Given:
Normal stress σ1 = 1000 MPa
Normal stress σ2 = 400 MPa
Shear stress τ = 400 MPa
1. Determining the principal stresses:
The principal stresses are the maximum and minimum normal stresses experienced by the inclined plane. They can be determined using the stress transformation equations:
σ1' = (σ1 + σ2)/2 + √((σ1 - σ2)/2)^2 + τ^2
σ2' = (σ1 + σ2)/2 - √((σ1 - σ2)/2)^2 + τ^2
Substituting the given values, we get:
σ1' = (1000 + 400)/2 + √((1000 - 400)/2)^2 + 400^2
= 700 + √(600/2)^2 + 400^2
= 700 + √90000 + 160000
= 700 + 300 + 400
= 1400 MPa
σ2' = (1000 + 400)/2 - √((1000 - 400)/2)^2 + 400^2
= 700 - √(600/2)^2 + 400^2
= 700 - √90000 + 160000
= 700 - 300 + 400
= 800 MPa
2. Determining the maximum normal stress:
The maximum normal stress is the larger of the two principal stresses. In this case, it is σ1' = 1400 MPa.
Therefore, the correct answer is option 'A' - 1200 MPa.
In summary:
- The principal stresses are determined using the stress transformation equations.
- The maximum normal stress is the larger of the two principal stresses.
- In this case, the maximum normal stress is σ1' = 1400 MPa, which corresponds to option 'A' - 1200 MPa.
Given:
Normal stress σ1 = 1000 MPa
Normal stress σ2 = 400 MPa
Shear stress τ = 400 MPa
1. Determining the principal stresses:
The principal stresses are the maximum and minimum normal stresses experienced by the inclined plane. They can be determined using the stress transformation equations:
σ1' = (σ1 + σ2)/2 + √((σ1 - σ2)/2)^2 + τ^2
σ2' = (σ1 + σ2)/2 - √((σ1 - σ2)/2)^2 + τ^2
Substituting the given values, we get:
σ1' = (1000 + 400)/2 + √((1000 - 400)/2)^2 + 400^2
= 700 + √(600/2)^2 + 400^2
= 700 + √90000 + 160000
= 700 + 300 + 400
= 1400 MPa
σ2' = (1000 + 400)/2 - √((1000 - 400)/2)^2 + 400^2
= 700 - √(600/2)^2 + 400^2
= 700 - √90000 + 160000
= 700 - 300 + 400
= 800 MPa
2. Determining the maximum normal stress:
The maximum normal stress is the larger of the two principal stresses. In this case, it is σ1' = 1400 MPa.
Therefore, the correct answer is option 'A' - 1200 MPa.
In summary:
- The principal stresses are determined using the stress transformation equations.
- The maximum normal stress is the larger of the two principal stresses.
- In this case, the maximum normal stress is σ1' = 1400 MPa, which corresponds to option 'A' - 1200 MPa.
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For an inclined plane in a rectangular block subjected to two mutually perpendicular normal stresses 1000 MPa and 400 MPa and shear stress 400 MPa, the maximum normal stress will bea) 1200 MPab) 700 MPac) 600 MPad) 200 MPaCorrect answer is option 'A'. Can you explain this answer?
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