What is the sway mechanism result?

W_c = 5/4 Mp (internal 5 Mp theta, external 4W theta)

At joints B and C, in which member does the hinge form?

The weaker member of those meeting at the joint

What is the aim in combining the beam and sway mechanisms?

To eliminate a hinge (at B) and lower the load factor

What is the collapse load?

1.1 Mp, with hinges at A, under first W, and C

2078 Bhadra · 10 marks

Collapse load of portal frame, 8 m span

Short answer

Virtual work evaluates beam, sway, and combined mechanisms; the combined mechanism gives the minimum collapse load, Wc = 11Mp/10 = 1.1Mp.

Find the collapse load of the following frame. [Figure: portal frame ABCD, A fixed, D hinged; column AB (2Mp) of height 4 m with horizontal W at B, beam BC (3Mp) of span 8 m with two vertical W loads each 2 m from the ends, column CD (Mp).]

Assumptions

Portal frame: A (0,0) fixed, B (0,4), C (8,4), D (8,0) hinged. Column AB $2M_p$ , beam BC $3M_p$ (span 8 m), column CD $M_p$ ; both columns are 4 m high.
Loads: W horizontal (to the right) at B, and two vertical loads W at 2 m and 6 m from B. Find $W_c$ in terms of $M_p$ (kN·m, lengths in m).
Hinges may form at A, B, under the loads and at C (the weaker of the members meeting at a joint). The fixed base gives 3 and the hinge 2 reactions, so the degree of indeterminacy is 5 - 3 = 2. There are 5 critical sections (A, B, two load points, C), hence 5 - 2 = 3 independent mechanisms (two beam mechanisms, one for each load, and one sway); their combinations are also examined.

Mechanisms

Mechanism 1: beam mechanism

Joints do not rotate or sway. Hinges at B (in the weaker column AB), under the load 2 m from C and at C (in the weaker column CD). The rotations of the beam parts are $\theta/4$ (B to hinge, 6 m) and $3\theta/4$ (hinge to C, 2 m): the hinge under the load has $\theta$ .

Hinge	Rotation	Plastic moment	Internal work
under W at 6 m from B (2 m from C)	$1\theta$	$3M_p$	$3\,M_p\theta$
C (in column CD)	$\frac{3}{4}\theta$	$M_p$	$\frac{3}{4}\,M_p\theta$
B (in column AB)	$\frac{1}{4}\theta$	$2M_p$	$\frac{1}{2}\,M_p\theta$

Internal work $= \frac{17}{4}\,M_p\,\theta$ .

Load	Displacement along the load	External work
W horizontal at B	$0\theta$	$0W\,\theta$
W at 2 m	$\frac{1}{2}\theta$	$\frac{1}{2}W\,\theta$
W at 6 m	$\frac{3}{2}\theta$	$\frac{3}{2}W\,\theta$

Equation: $W_c\times 2\,\theta = \frac{17}{4}\,M_p\theta$ , so $W_c = \frac{17}{8}\,M_p = 2.1250\,M_p$ .

Mechanism 2: sway mechanism

The columns rotate $\theta$ about the bases and the beam moves horizontally by $4\theta$ . Hinges at A, B (in AB) and C (in CD).

Hinge	Rotation	Plastic moment	Internal work
A (fixed base)	$1\theta$	$2M_p$	$2\,M_p\theta$
B (in column AB)	$1\theta$	$2M_p$	$2\,M_p\theta$
C (in column CD)	$1\theta$	$M_p$	$1\,M_p\theta$

Internal work $= 5\,M_p\,\theta$ .

Load	Displacement along the load	External work
W horizontal at B	$4\theta$	$4W\,\theta$
W at 2 m	$0\theta$	$0W\,\theta$
W at 6 m	$0\theta$	$0W\,\theta$

Equation: $W_c\times 4\,\theta = 5\,M_p\theta$ , so $W_c = \frac{5}{4}\,M_p = 1.2500\,M_p$ .

Mechanism 3: combined mechanism

Combining a beam mechanism (hinge under the load 2 m from B) with the sway mechanism removes the hinge at B. Hinges at A, under the first W, and at C (in CD).

Hinge	Rotation	Plastic moment	Internal work
under W at 2 m from B	$1\theta$	$3M_p$	$3\,M_p\theta$
C (in column CD)	$1\theta$	$M_p$	$1\,M_p\theta$
A (fixed base)	$\frac{3}{4}\theta$	$2M_p$	$\frac{3}{2}\,M_p\theta$

Internal work $= \frac{11}{2}\,M_p\,\theta$ .

Load	Displacement along the load	External work
W horizontal at B	$3\theta$	$3W\,\theta$
W at 2 m	$\frac{3}{2}\theta$	$\frac{3}{2}W\,\theta$
W at 6 m	$\frac{1}{2}\theta$	$\frac{1}{2}W\,\theta$

Equation: $W_c\times 5\,\theta = \frac{11}{2}\,M_p\theta$ , so $W_c = \frac{11}{10}\,M_p = 1.1000\,M_p$ .

A search of all other hinge combinations (including the hinge under the load at 6 m with the sway mechanism, 1.5 $M_p$ , and sway with hinge in the beam, 1.5 $M_p$ ) gave higher values.

Collapse load

The lowest value is from the combined mechanism:

Answer: $W_c = 1.1\,M_p$ (hinges at A, under the first W and at C; $M_p$ in kN·m, $W_c$ in kN with lengths in m).

✓ Answer checked by Claude Opus 5.5 (high effort)

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Collapse load of portal frame, 8 m span

Assumptions

Mechanisms

Collapse load

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