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A rectangular section beam subjected to a bending moment M varying along its length is required to develop the same maximum bending stress at any cross-section. If the depth of the section is constant, then its width will vary as
- a)M
- b)√M
- c)M2
- d)1/M
Correct answer is option 'A'. Can you explain this answer?
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Understanding Bending Stress in Beams
When a rectangular beam is subjected to a varying bending moment (M), the stress at any section can be calculated using the formula:
- Bending Stress (σ) = M * c / I
Where:
- M = Bending moment
- c = Distance from the neutral axis to the outermost fiber (half the depth for rectangular sections)
- I = Moment of inertia of the beam's cross-section
Beam Section Characteristics
For a rectangular beam with a constant depth (d), the moment of inertia (I) is given by:
- I = (b * d^3) / 12
Where:
- b = Width of the beam
- d = Constant depth of the beam
Condition for Constant Bending Stress
To maintain a constant maximum bending stress (σ) across any section, we need to ensure that the product M * c remains proportionate to the moment of inertia (I).
Since depth is constant, the only variable is the width (b).
Deriving Width Variation
Rearranging the bending stress formula gives:
- b = (12 * M) / (σ * d^3)
This indicates that as the bending moment M increases, the width b must also change to keep the bending stress constant. Simplifying the relationship shows that:
- b varies directly with M
Therefore, if we consider the options provided:
- a) M
- b) √M
- c) M^2
- d) 1/M
The correct choice is option 'a' (b varies linearly with M).
Conclusion
In summary, for a rectangular beam with a constant depth subjected to a varying bending moment, the width must increase linearly with the bending moment to maintain a constant maximum bending stress at any cross-section.
When a rectangular beam is subjected to a varying bending moment (M), the stress at any section can be calculated using the formula:
- Bending Stress (σ) = M * c / I
Where:
- M = Bending moment
- c = Distance from the neutral axis to the outermost fiber (half the depth for rectangular sections)
- I = Moment of inertia of the beam's cross-section
Beam Section Characteristics
For a rectangular beam with a constant depth (d), the moment of inertia (I) is given by:
- I = (b * d^3) / 12
Where:
- b = Width of the beam
- d = Constant depth of the beam
Condition for Constant Bending Stress
To maintain a constant maximum bending stress (σ) across any section, we need to ensure that the product M * c remains proportionate to the moment of inertia (I).
Since depth is constant, the only variable is the width (b).
Deriving Width Variation
Rearranging the bending stress formula gives:
- b = (12 * M) / (σ * d^3)
This indicates that as the bending moment M increases, the width b must also change to keep the bending stress constant. Simplifying the relationship shows that:
- b varies directly with M
Therefore, if we consider the options provided:
- a) M
- b) √M
- c) M^2
- d) 1/M
The correct choice is option 'a' (b varies linearly with M).
Conclusion
In summary, for a rectangular beam with a constant depth subjected to a varying bending moment, the width must increase linearly with the bending moment to maintain a constant maximum bending stress at any cross-section.
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A rectangular section beam subjected to a bending moment M varying along its length is required to develop the same maximum bending stress at any cross-section. If the depth of the section is constant, then its width will vary asa) Mb) √Mc) M2d) 1/MCorrect answer is option 'A'. Can you explain this answer?
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