For finding the required plastic moment, which mechanism value is chosen?

The largest M_p from all mechanisms

How are the plastic moments of members related?

In proportion to I: AB = 3Mp, BC = 1.5Mp, CD = 2Mp

2076 Chaitra · 10 marks

Plastic moment capacity of frame

Short answer

Applying virtual work to beam, sway, and combined mechanisms, the largest required plastic moment capacity is from the sway mechanism: Mp = 120 kN·m.

Find the plastic moment capacity of the frame shown in figure below. [Figure: portal frame ABCD, A and D hinged; column AB (3I) with 20 kN/m UDL horizontally; beam BC (1.5I) with 80 kN vertical at 2 m from B, span 6 m (2 m + 4 m); column CD (2I) with 100 kN horizontal at 2 m above D, heights 2 m + 2 m marked.]

Assumptions

Portal frame: A (0,0) and D (6,0) hinged, B (0,4), C (6,4); both columns are 4 m high, the beam is 6 m with the 80 kN at 2 m from B.
The plastic moments are in proportion to $I$ : $M_{AB} = 3M_p$ , $M_{BC} = 1.5M_p$ , $M_{CD} = 2M_p$ , where $M_p$ is the plastic moment capacity to be found. The loads are taken as ultimate (collapse) loads: 20 kN/m on AB and 100 kN on CD (2 m above D), both horizontal towards the right, and 80 kN vertical on the beam.
Method: virtual work for each mechanism, $M_p\sum(\text{factor}\times\theta) = \sum P\delta$ ; the required $M_p$ is the largest value obtained from all mechanisms.
Two hinged bases give 4 reactions, so the degree of indeterminacy is 1 and a mechanism needs 2 hinges (sway) or 3 hinges (beam).

Mechanisms

Mechanism 1: beam mechanism

No sway. Hinges at B, under the 80 kN load (2 m from B) and at C, all in the beam ( $1.5M_p$ is smaller than the column moments $3M_p$ and $2M_p$ ). The beam parts rotate $2\theta/3$ (hinge at B) and $\theta/3$ (hinge at C), with $\theta$ under the load.

Hinge	Rotation	Plastic moment	Internal work
under the 80 kN load	$1\theta$	$\frac{3}{2}M_p$	$\frac{3}{2}\,M_p\theta$
B (beam end)	$\frac{2}{3}\theta$	$\frac{3}{2}M_p$	$1\,M_p\theta$
C (beam end)	$\frac{1}{3}\theta$	$\frac{3}{2}M_p$	$\frac{1}{2}\,M_p\theta$

Internal work $= 3\,M_p\,\theta$ .

Load	Displacement along the load	External work
80 kN	$\frac{4}{3}\theta$	$\frac{320}{3}\,\text{kN·m}\,\theta$
100 kN	$0\theta$	$0\,\text{kN·m}\,\theta$
20 kN/m on AB	-	$0\,\text{kN·m}\,\theta$

Equation: $M_p\times 3\,\theta = \frac{320}{3}\,\theta$ (kN·m), so $M_p = \frac{320}{9} = 35.56$ kN·m.

Mechanism 2: sway mechanism

Both columns rotate $\theta$ about their hinged bases, the beam translates by $4\theta$ to the right. Hinges form at B and C in the beam, each rotating $\theta$ .

Hinge	Rotation	Plastic moment	Internal work
B (beam end)	$1\theta$	$\frac{3}{2}M_p$	$\frac{3}{2}\,M_p\theta$
C (beam end)	$1\theta$	$\frac{3}{2}M_p$	$\frac{3}{2}\,M_p\theta$

Internal work $= 3\,M_p\,\theta$ .

Load	Displacement along the load	External work
80 kN	$0\theta$	$0\,\text{kN·m}\,\theta$
100 kN	$2\theta$	$200\,\text{kN·m}\,\theta$
20 kN/m on AB	-	$160\,\text{kN·m}\,\theta$

Equation: $M_p\times 3\,\theta = 360\,\theta$ (kN·m), so $M_p = 120$ kN·m.

Mechanism 3: combined mechanism

Beam and sway mechanisms combined so that the hinge at B cancels; hinges under the 80 kN load and at C.

Hinge	Rotation	Plastic moment	Internal work
under the 80 kN load	$1\theta$	$\frac{3}{2}M_p$	$\frac{3}{2}\,M_p\theta$
C (beam end)	$1\theta$	$\frac{3}{2}M_p$	$\frac{3}{2}\,M_p\theta$

Internal work $= 3\,M_p\,\theta$ .

Load	Displacement along the load	External work
80 kN	$\frac{4}{3}\theta$	$\frac{320}{3}\,\text{kN·m}\,\theta$
100 kN	$\frac{4}{3}\theta$	$\frac{400}{3}\,\text{kN·m}\,\theta$
20 kN/m on AB	-	$\frac{320}{3}\,\text{kN·m}\,\theta$

Equation: $M_p\times 3\,\theta = \frac{1040}{3}\,\theta$ (kN·m), so $M_p = \frac{1040}{9} = 115.56$ kN·m.

Required plastic moment

The largest required $M_p$ comes from the sway mechanism (the others give 35.56 and 115.56 kN·m); a frame with $M_p$ smaller than this would collapse in sway, while a larger value is safe against all mechanisms.

Answer: $M_p = 120$ kN·m (so $M_{AB} = 360$ , $M_{BC} = 180$ , $M_{CD} = 240$ kN·m).

✓ Answer checked by Claude Opus 5.5 (high effort)

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