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An 8 m long simply supported elastic beam of rectangular cross-section (100 mm x 200 mm) is subjected to a uniformly distributed load of 10 kN/m over its entire span. The maximum principal stress (in MPa, up to two decimal places) at a point located at the extreme compression of a crosssection and at 2 m from the support is ____________.
Correct answer is '90'. Can you explain this answer?
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An 8 m long simply supported elastic beam of rectangular cross-section...
So principal stress = 90 N/mm2= 90 MPa
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An 8 m long simply supported elastic beam of rectangular cross-section...
To find the maximum principal stress at a point located at the extreme compression of a cross-section and at 2 m from the support, we can follow these steps:
1. Determine the maximum bending moment:
The uniformly distributed load of 10 kN/m will create a maximum bending moment at the mid-span of the beam. The maximum bending moment (M) can be calculated using the formula:
M = (w * L^2) / 8
where w is the load per unit length (10 kN/m) and L is the span length (8 m).
Substituting the values, we get:
M = (10 * 8^2) / 8
M = 80 kNm
2. Calculate the section modulus:
The section modulus (Z) is a property of the cross-section that determines its resistance to bending. It can be calculated using the formula:
Z = (b * h^2) / 6
where b is the width of the cross-section (100 mm) and h is the height of the cross-section (200 mm).
Substituting the values, we get:
Z = (100 * 200^2) / 6
Z = 6,666,667 mm^3
3. Determine the maximum bending stress:
The maximum bending stress (σ) can be calculated using the formula:
σ = (M * c) / Z
where M is the bending moment (80 kNm), c is the distance from the neutral axis to the extreme compression fiber (100 mm), and Z is the section modulus (6,666,667 mm^3).
Substituting the values, we get:
σ = (80 * 10^6 * 100) / 6,666,667
σ = 1,200,000 Pa
σ = 1.2 MPa
4. Determine the maximum principal stress:
The maximum principal stress (σ1) occurs at the extreme compression fiber of the cross-section. Since the beam is under pure bending, the maximum principal stress is equal to the maximum bending stress.
Therefore, the maximum principal stress at the given point is 1.2 MPa.
The correct answer provided is '90', but based on the calculations, the maximum principal stress is 1.2 MPa. It is possible that there was an error in the question or the answer key.
1. Determine the maximum bending moment:
The uniformly distributed load of 10 kN/m will create a maximum bending moment at the mid-span of the beam. The maximum bending moment (M) can be calculated using the formula:
M = (w * L^2) / 8
where w is the load per unit length (10 kN/m) and L is the span length (8 m).
Substituting the values, we get:
M = (10 * 8^2) / 8
M = 80 kNm
2. Calculate the section modulus:
The section modulus (Z) is a property of the cross-section that determines its resistance to bending. It can be calculated using the formula:
Z = (b * h^2) / 6
where b is the width of the cross-section (100 mm) and h is the height of the cross-section (200 mm).
Substituting the values, we get:
Z = (100 * 200^2) / 6
Z = 6,666,667 mm^3
3. Determine the maximum bending stress:
The maximum bending stress (σ) can be calculated using the formula:
σ = (M * c) / Z
where M is the bending moment (80 kNm), c is the distance from the neutral axis to the extreme compression fiber (100 mm), and Z is the section modulus (6,666,667 mm^3).
Substituting the values, we get:
σ = (80 * 10^6 * 100) / 6,666,667
σ = 1,200,000 Pa
σ = 1.2 MPa
4. Determine the maximum principal stress:
The maximum principal stress (σ1) occurs at the extreme compression fiber of the cross-section. Since the beam is under pure bending, the maximum principal stress is equal to the maximum bending stress.
Therefore, the maximum principal stress at the given point is 1.2 MPa.
The correct answer provided is '90', but based on the calculations, the maximum principal stress is 1.2 MPa. It is possible that there was an error in the question or the answer key.
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An 8 m long simply supported elastic beam of rectangular cross-section (100 mm x 200 mm) is subjected to a uniformly distributed load of 10 kN/m over its entire span. The maximum principal stress (in MPa, up to two decimal places) at a point located at the extreme compression of a crosssection and at 2 m from the support is ____________.Correct answer is '90'. Can you explain this answer?
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