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A cantilever of span L subjected to uniformly distributed gravity loading w/m is subjected to an upward force P at a distance L/2 from the free end. If the deflection at the free end of the beam is zero, (wL/P) is
- a)1.50
- b)2.00
- c)0.67
- d)0.83
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
A cantilever of span L subjected to uniformly distributed gravity loa...
Deflection at free end due to vertical load
δ1 = 
Deflection at free end due to support P
δ2 = 
= 
∴ Deflection at free end is zero
= 0.83
Most Upvoted Answer
A cantilever of span L subjected to uniformly distributed gravity loa...
Given:
- Span of cantilever beam, L
- Uniformly distributed gravity loading, w/m
- Upward force, P at a distance L/2 from the free end
- Deflection at the free end of the beam is zero
To find: (wL/P)
Solution:
1. Determine the reaction forces:
- At the fixed end, there is only a vertical reaction force, R1, which balances the sum of the forces in the vertical direction: R1 = wL/2 + P.
- At the free end, there is a horizontal reaction force, R2, and a vertical reaction force, R3, which balance the sum of the forces in the horizontal and vertical directions, respectively: R2 = 0 and R3 = wL/2 + P.
2. Calculate the moment equation:
- Taking moments about the fixed end, we have: M = P(L/2) + (wL/2)(L/2) - R1L = 0.
- Solving for R1, we get: R1 = wL/4 + P/2.
3. Calculate the deflection equation:
- Using the moment equation, we can derive the deflection equation: y = (w/(24EI))(x^2)(x^2 - 4Lx + 6L^2) - (P/(6EI))(L/2 - x)^3.
- Substituting x = L, y = 0, we get: 0 = (wL^4)/(16EI) - (PL^3)/(48EI).
- Solving for P, we get: P = (wL^2)/8.
4. Calculate (wL/P):
- Substituting the value of P in (wL/P), we get: (wL/P) = 8.
Therefore, the correct answer is option 'D' (0.83).
- Span of cantilever beam, L
- Uniformly distributed gravity loading, w/m
- Upward force, P at a distance L/2 from the free end
- Deflection at the free end of the beam is zero
To find: (wL/P)
Solution:
1. Determine the reaction forces:
- At the fixed end, there is only a vertical reaction force, R1, which balances the sum of the forces in the vertical direction: R1 = wL/2 + P.
- At the free end, there is a horizontal reaction force, R2, and a vertical reaction force, R3, which balance the sum of the forces in the horizontal and vertical directions, respectively: R2 = 0 and R3 = wL/2 + P.
2. Calculate the moment equation:
- Taking moments about the fixed end, we have: M = P(L/2) + (wL/2)(L/2) - R1L = 0.
- Solving for R1, we get: R1 = wL/4 + P/2.
3. Calculate the deflection equation:
- Using the moment equation, we can derive the deflection equation: y = (w/(24EI))(x^2)(x^2 - 4Lx + 6L^2) - (P/(6EI))(L/2 - x)^3.
- Substituting x = L, y = 0, we get: 0 = (wL^4)/(16EI) - (PL^3)/(48EI).
- Solving for P, we get: P = (wL^2)/8.
4. Calculate (wL/P):
- Substituting the value of P in (wL/P), we get: (wL/P) = 8.
Therefore, the correct answer is option 'D' (0.83).
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A cantilever of span L subjected to uniformly distributed gravity loading w/m is subjected to an upward force P at a distance L/2 from the free end. If the deflection at the free end of the beam is zero, (wL/P) isa) 1.50b) 2.00c) 0.67d) 0.83Correct answer is option 'D'. Can you explain this answer?
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