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A rectangular section beam subjected to a bending moment M varying along its length is required to develop 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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Solution:
Given, a rectangular section beam subjected to a bending moment M varying along its length is required to develop same maximum bending stress at any cross-section.
Let us consider a rectangular section beam with constant depth and width=b.
Assume that the bending moment M varies linearly along the length of the beam. i.e., M varies from M1 at one end to M2 at the other end.
To develop the same maximum bending stress at any cross-section, the bending stress σb should be same at any cross-section of the beam.
σb = (M*y)/I
where y is the distance of the neutral axis from the extreme fibers and I is the moment of inertia of the cross-section.
Let us assume that the neutral axis is at a distance of d/2 from the top fiber.
Then, the moment of inertia of the cross-section is given by
I = (b*d^3)/12
Maximum bending stress occurs at the top and bottom fibers of the beam. At these fibers, y=d/2.
σmax = (M*d)/I
= (M*d)/(b*d^3/12)
= 12M/(bd^2)
From the above equation, we can infer that the maximum bending stress is directly proportional to the bending moment M and inversely proportional to the square of the depth d.
To develop the same maximum bending stress at any cross-section, the width of the beam should vary linearly along the length of the beam.
Let the width of the beam vary from b1 at one end to b2 at the other end.
Then, the maximum bending stress at any cross-section of the beam is given by
σmax = 12M/(bd^2)
= 12M/[(b1+(b2-b1)x)*d^2/4]
= 48M/[d^2*(b1+(b2-b1)x)]
From the above equation, we can infer that the maximum bending stress is directly proportional to the bending moment M and inversely proportional to the square of the depth d and the width of the beam.
To develop the same maximum bending stress at any cross-section, the width of the beam should vary linearly along the length of the beam.
Therefore, the correct option is A.
Given, a rectangular section beam subjected to a bending moment M varying along its length is required to develop same maximum bending stress at any cross-section.
Let us consider a rectangular section beam with constant depth and width=b.
Assume that the bending moment M varies linearly along the length of the beam. i.e., M varies from M1 at one end to M2 at the other end.
To develop the same maximum bending stress at any cross-section, the bending stress σb should be same at any cross-section of the beam.
σb = (M*y)/I
where y is the distance of the neutral axis from the extreme fibers and I is the moment of inertia of the cross-section.
Let us assume that the neutral axis is at a distance of d/2 from the top fiber.
Then, the moment of inertia of the cross-section is given by
I = (b*d^3)/12
Maximum bending stress occurs at the top and bottom fibers of the beam. At these fibers, y=d/2.
σmax = (M*d)/I
= (M*d)/(b*d^3/12)
= 12M/(bd^2)
From the above equation, we can infer that the maximum bending stress is directly proportional to the bending moment M and inversely proportional to the square of the depth d.
To develop the same maximum bending stress at any cross-section, the width of the beam should vary linearly along the length of the beam.
Let the width of the beam vary from b1 at one end to b2 at the other end.
Then, the maximum bending stress at any cross-section of the beam is given by
σmax = 12M/(bd^2)
= 12M/[(b1+(b2-b1)x)*d^2/4]
= 48M/[d^2*(b1+(b2-b1)x)]
From the above equation, we can infer that the maximum bending stress is directly proportional to the bending moment M and inversely proportional to the square of the depth d and the width of the beam.
To develop the same maximum bending stress at any cross-section, the width of the beam should vary linearly along the length of the beam.
Therefore, the correct option is A.
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A rectangular section beam subjected to a bending moment M varying along its length is required to develop 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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