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A freely supported beam at its ends carries a central concentrated load, and maximum bending moment is M. If the same load be uniformly distributed over the beam length, then what is the maximum bending moment?
- a)M
- b)M/2
- c)M/3
- d)2M
Correct answer is option 'B'. Can you explain this answer?
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Given:
- A freely supported beam at its ends carries a central concentrated load, and maximum bending moment is M.
To find:
- If the same load be uniformly distributed over the beam length, then what is the maximum bending moment?
Solution:
- Let the length of the beam be L.
- Let the central concentrated load be W.
- Let the uniformly distributed load be w.
Maximum bending moment for concentrated load:
- The maximum bending moment occurs at the center of the beam when it is loaded with a central concentrated load.
- Maximum bending moment, M = WL/4
Maximum bending moment for uniformly distributed load:
- The maximum bending moment occurs at the center of the beam when it is loaded with a uniformly distributed load.
- Maximum bending moment, M' = wL²/8
Comparing M and M':
- To find the relationship between M and M', we need to equate the total load in both cases.
- Total load in case of central concentrated load, W = Total load in case of uniformly distributed load, wL
- Equating the total loads, we get w = W/L
Substituting w in the expression for M', we get:
M' = (W/L)L²/8
M' = W*L/8
Comparing M and M':
- M = WL/4
- M' = W*L/8
- Dividing M by M', we get:
M/M' = (WL/4) / (W*L/8)
M/M' = 2
- Therefore, M' = M/2
Answer:
- If the same load be uniformly distributed over the beam length, then the maximum bending moment is M/2.
- A freely supported beam at its ends carries a central concentrated load, and maximum bending moment is M.
To find:
- If the same load be uniformly distributed over the beam length, then what is the maximum bending moment?
Solution:
- Let the length of the beam be L.
- Let the central concentrated load be W.
- Let the uniformly distributed load be w.
Maximum bending moment for concentrated load:
- The maximum bending moment occurs at the center of the beam when it is loaded with a central concentrated load.
- Maximum bending moment, M = WL/4
Maximum bending moment for uniformly distributed load:
- The maximum bending moment occurs at the center of the beam when it is loaded with a uniformly distributed load.
- Maximum bending moment, M' = wL²/8
Comparing M and M':
- To find the relationship between M and M', we need to equate the total load in both cases.
- Total load in case of central concentrated load, W = Total load in case of uniformly distributed load, wL
- Equating the total loads, we get w = W/L
Substituting w in the expression for M', we get:
M' = (W/L)L²/8
M' = W*L/8
Comparing M and M':
- M = WL/4
- M' = W*L/8
- Dividing M by M', we get:
M/M' = (WL/4) / (W*L/8)
M/M' = 2
- Therefore, M' = M/2
Answer:
- If the same load be uniformly distributed over the beam length, then the maximum bending moment is M/2.
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