GATE CE 2016 Set 2 — Question 48
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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 concentrated load () as shown in the figure. The beam has two different segments having different plastic moment capacities () as shown.
The minimum value of load () at which the beam would collapse (ultimate load) is
Correct answer
(A) 7.5 Mₚ / L
Solution
We evaluate possible collapse mechanisms using the principle of virtual work ().Mechanism 1: Hinges at A (left support), C (under load), and B (right support).
Let be the displacement under the load .
Rotation at A:
Rotation at B:
Rotation at C:
Internal Work:
External Work:
Mechanism 2: Hinges at A (left support), D (junction of and ), and B (right support).
Let be the displacement at the junction (at distance from left).
Rotation at A:
Rotation at B:
Rotation at D:
Displacement under load (at ):
Internal Work:
External Work:
Comparing mechanisms, the collapse load is the minimum of the upper bounds: .
Let be the displacement under the load .
Rotation at A:
Rotation at B:
Rotation at C:
Internal Work:
External Work:
Mechanism 2: Hinges at A (left support), D (junction of and ), and B (right support).
Let be the displacement at the junction (at distance from left).
Rotation at A:
Rotation at B:
Rotation at D:
Displacement under load (at ):
Internal Work:
External Work:
Comparing mechanisms, the collapse load is the minimum of the upper bounds: .
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