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Fig shows a continuous beam subjected to a collapse loud system. If the plastic moments of AB,BC,CD and DE are 2MP,1.5MP,1.5MP and MP respectively,Find the value of MP

enter image description here

Subject: Structure Analysis 2

Topic: Determination of Collaps Load for Beams

Difficulty: Hard

1 Answer
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1) Collapse of the Span AP :

With plastic hinges at A,B and at the centre of the Span.

enter image description here

Using Virtual work principle

EWD = IWD

MP=4006=66.67KN.m

MP=66.67Kn.m

3) Collapse of the span CD:

with plastic hinges at C,D and at the centre of the Span.

enter image description here

Using virtual work done:

FWD = IWD

100γ=1.5MPθ+1.5MPθ+1.5MPθ+MPθ

θ=γ3γ=3θ

1003θ=MPθ(1.5+1.5+1.5+1)

300θ=5.5MPθ

MP=3005.5=54.55KN.m

MP=54.55KN.m

(50γ)+(80γ)+(50γ)=2MPθ+2MPθ+2MPθ+1.5MPθ

MP=5207.5=69.33

MP=69.33Kn,m

2) Collapse of the span BC

with the plastic hinges at B,C and at the centre of the span.

enter image description here

using virtual work done

EWD = IWD

25128γ=1.5MPθ+1.5MPθ+1.5MPθ+1.5MPθ

θ=γ4γ=4θ

4) Collapse of the span CD

with the plastic hinges at D and understand the load at c.

enter image description here

EWD = IWD

(45γ)+(45γ)=MPθ+MPθ+MPα

270θ=MPθ(1+1+2)

MP=2704=65.7Kn.m

MP=65.7KN.m

From all the above possible consideration, the collapse Mechanism for the span AB requires the greatest value of MP

MP=69.33KN.m

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