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At a point in a loaded member, a state of plane stress exists and the strains are εx = 270 x 10-6, εy = -90 x 10-6 and εxy = 360 x 10-6. If E = 200 GPa, μ = 0.25 and G = 80 GPa, then the value of normal stress (major) (in MPa) is (Answer up to one decimal place)
Correct answer is '52.8'. Can you explain this answer?
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At a point in a loaded member, a state of plane stress exists and the...
Given information:
- Strains: εx = 270 x 10^-6, εy = -90 x 10^-6, εxy = 360 x 10^-6
- Young's modulus: E = 200 GPa
- Poisson's ratio: μ = 0.25
- Shear modulus: G = 80 GPa
- We need to find the value of normal stress (major).
Solution:
Step 1: Calculate the normal strains:
Normal strains are given by the formula:
εx = (σx - μσy) / E
εy = (σy - μσx) / E
Plugging in the given values, we can rearrange the equations to solve for σx and σy:
σx = Eεx + μEεy
σy = Eεy + μEεx
Substituting the given values:
σx = (200 x 10^9 Pa)(270 x 10^-6) + (0.25)(200 x 10^9 Pa)(-90 x 10^-6)
σy = (200 x 10^9 Pa)(-90 x 10^-6) + (0.25)(200 x 10^9 Pa)(270 x 10^-6)
Calculating these values:
σx = 54 MPa
σy = -81 MPa
Step 2: Calculate the shear stress:
Shear stress is given by the formula:
τxy = Gεxy
Plugging in the given values:
τxy = (80 x 10^9 Pa)(360 x 10^-6)
Calculating the shear stress:
τxy = 28.8 MPa
Step 3: Calculate the normal stresses:
The normal stresses can be calculated using the formula:
σ1 = (σx + σy) / 2 + sqrt(((σx - σy) / 2)^2 + τxy^2)
σ2 = (σx + σy) / 2 - sqrt(((σx - σy) / 2)^2 + τxy^2)
Substituting the calculated values:
σ1 = (54 MPa - 81 MPa) / 2 + sqrt(((54 MPa - (-81 MPa)) / 2)^2 + (28.8 MPa)^2)
σ2 = (54 MPa - 81 MPa) / 2 - sqrt(((54 MPa - (-81 MPa)) / 2)^2 + (28.8 MPa)^2)
Calculating these values:
σ1 = 52.8 MPa
σ2 = -79.8 MPa
Step 4: Determine the major normal stress:
The major normal stress is the higher value between σ1 and σ2. In this case, the major normal stress is σ1 = 52.8 MPa.
Therefore, the value of the normal stress (major) is 52.8 MPa.
- Strains: εx = 270 x 10^-6, εy = -90 x 10^-6, εxy = 360 x 10^-6
- Young's modulus: E = 200 GPa
- Poisson's ratio: μ = 0.25
- Shear modulus: G = 80 GPa
- We need to find the value of normal stress (major).
Solution:
Step 1: Calculate the normal strains:
Normal strains are given by the formula:
εx = (σx - μσy) / E
εy = (σy - μσx) / E
Plugging in the given values, we can rearrange the equations to solve for σx and σy:
σx = Eεx + μEεy
σy = Eεy + μEεx
Substituting the given values:
σx = (200 x 10^9 Pa)(270 x 10^-6) + (0.25)(200 x 10^9 Pa)(-90 x 10^-6)
σy = (200 x 10^9 Pa)(-90 x 10^-6) + (0.25)(200 x 10^9 Pa)(270 x 10^-6)
Calculating these values:
σx = 54 MPa
σy = -81 MPa
Step 2: Calculate the shear stress:
Shear stress is given by the formula:
τxy = Gεxy
Plugging in the given values:
τxy = (80 x 10^9 Pa)(360 x 10^-6)
Calculating the shear stress:
τxy = 28.8 MPa
Step 3: Calculate the normal stresses:
The normal stresses can be calculated using the formula:
σ1 = (σx + σy) / 2 + sqrt(((σx - σy) / 2)^2 + τxy^2)
σ2 = (σx + σy) / 2 - sqrt(((σx - σy) / 2)^2 + τxy^2)
Substituting the calculated values:
σ1 = (54 MPa - 81 MPa) / 2 + sqrt(((54 MPa - (-81 MPa)) / 2)^2 + (28.8 MPa)^2)
σ2 = (54 MPa - 81 MPa) / 2 - sqrt(((54 MPa - (-81 MPa)) / 2)^2 + (28.8 MPa)^2)
Calculating these values:
σ1 = 52.8 MPa
σ2 = -79.8 MPa
Step 4: Determine the major normal stress:
The major normal stress is the higher value between σ1 and σ2. In this case, the major normal stress is σ1 = 52.8 MPa.
Therefore, the value of the normal stress (major) is 52.8 MPa.
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At a point in a loaded member, a state of plane stress exists and the...
Larger will be the major normal stress.
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At a point in a loaded member, a state of plane stress exists and the strains are εx = 270 x 10-6, εy = -90 x 10-6 and εxy = 360 x 10-6. If E = 200 GPa, μ = 0.25 and G = 80 GPa, then the value of normal stress (major) (in MPa) is (Answer up to one decimal place)Correct answer is '52.8'. Can you explain this answer?
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