GATE CE 2015 Set 2 — Question 46
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Structural Engineering → Connections & Plastic Analysis → Plastic Analysis (Beams)
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Question
A fixed end beam is subjected to a load, at 1/3rd span from the left support as shown in the figure. The collapse load of the beam is:
Correct answer
(C) 15.0 Mₚ/L
Solution
To find the collapse load, we consider possible collapse mechanisms and apply the principle of virtual work. Mechanism 1: Plastic hinges at X, under the load W, and at Z.
Let the virtual displacement under the load be .
Rotation at :
Rotation at :
Internal Work ():
Since and the load are in the region and is in the region:
External Work ():
Equating Mechanism 2: Plastic hinges at X, at the junction Y, and at Z.
Let the virtual displacement at be .
Rotation at :
Rotation at :
Internal Work ():
At , the plastic moment is the minimum of the two segments, i.e., .
External Work (): The displacement under the load (at ) is .
Equating The collapse load is the minimum of the loads calculated from all possible mechanisms. Thus, .
Let the virtual displacement under the load be .
Rotation at :
Rotation at :
Internal Work ():
Since and the load are in the region and is in the region:
External Work ():
Equating Mechanism 2: Plastic hinges at X, at the junction Y, and at Z.
Let the virtual displacement at be .
Rotation at :
Rotation at :
Internal Work ():
At , the plastic moment is the minimum of the two segments, i.e., .
External Work (): The displacement under the load (at ) is .
Equating The collapse load is the minimum of the loads calculated from all possible mechanisms. Thus, .
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