In the sway mechanisms, why does the 2W vertical load do no external work?

Because the beam translates horizontally, so there is no vertical displacement at J

Because it is smaller than the 3W load

Because the plastic moment cancels it

Mechanism values are 0.5833 Mp, 0.3111 Mp and 0.2444 Mp. Which is the collapse load and why?

0.2444 Mp, the smallest, since collapse occurs at the lowest load

0.5833 Mp, the largest, since the frame must carry it

2075 Asoj · 8 marks

Collapse load of rectangular portal frame

Short answer

By virtual work, the beam, panel and sway-with-load mechanisms give 0.5833Mp, 0.3111Mp and 0.2444Mp; the lowest is the sway mechanism with a hinge under the 3W load. Thus Wc = 11/45 Mp = 0.2444Mp.

Determine the collapse load $W_c$ for the rectangular portal frame shown in figure below. [Figure: portal frame, fixed bases; left column 3 m + 5 m (total 8 m) and right column 6 m; beam span 4 m + 3 m with 2W vertical at the junction point and 3W horizontal at 5 m above the base of the left column.]

Assumptions

Portal frame with fixed bases A and D. Left column AB is 8 m (3 m + 5 m), right column CD 6 m, so the base D is 2 m higher than A; beam BC has span 7 m (4 m + 3 m). All members have the same plastic moment $M_p$ .
Loads: 2W vertical at J (4 m from B), 3W horizontal (to the right) on the left column at H, 5 m above A. $W_c$ is found in terms of $M_p$ (kN·m, lengths in m).
Method: virtual work. The degree of indeterminacy is 6 - 3 = 3 and there are 6 critical sections (A, H, B, J, C, D), so there are 6 - 3 = 3 independent mechanisms: the beam mechanism (B, J, C), the panel (sway) mechanism (A, B, C, D) and the beam mechanism of the left column under 3W (A, H, B). They are combined below.

Mechanisms

Mechanism 1: beam mechanism

Hinges at B, under 2W (4 m from B, 3 m from C) and C; no sway. The beam parts rotate $3\theta/7$ (left, 4 m) and $4\theta/7$ (right, 3 m).

Hinge	Rotation	Plastic moment	Internal work
J (under 2W)	$1\theta$	$M_p$	$1\,M_p\theta$
C (beam end)	$\frac{4}{7}\theta$	$M_p$	$\frac{4}{7}\,M_p\theta$
B (beam end)	$\frac{3}{7}\theta$	$M_p$	$\frac{3}{7}\,M_p\theta$

Internal work $= 2\,M_p\,\theta$ .

Load	Displacement along the load	External work
2W vertical at J	$\frac{12}{7}\theta$	$\frac{24}{7}W\,\theta$
3W horizontal at H	$0\theta$	$0W\,\theta$

Equation: $W_c\times \frac{24}{7}\,\theta = 2\,M_p\theta$ , so $W_c = \frac{7}{12}\,M_p = 0.5833\,M_p$ .

Mechanism 2: panel (sway) mechanism

Hinges at A, B, C and D. The left column (8 m) rotates $\theta$ and the beam moves $8\theta$ , so the right column (6 m) rotates $4\theta/3$ .

Hinge	Rotation	Plastic moment	Internal work
C	$\frac{4}{3}\theta$	$M_p$	$\frac{4}{3}\,M_p\theta$
D (fixed base)	$\frac{4}{3}\theta$	$M_p$	$\frac{4}{3}\,M_p\theta$
A (fixed base)	$1\theta$	$M_p$	$1\,M_p\theta$
B	$1\theta$	$M_p$	$1\,M_p\theta$

Internal work $= \frac{14}{3}\,M_p\,\theta$ .

Load	Displacement along the load	External work
2W vertical at J	$0\theta$	$0W\,\theta$
3W horizontal at H	$5\theta$	$15W\,\theta$

Equation: $W_c\times 15\,\theta = \frac{14}{3}\,M_p\theta$ , so $W_c = \frac{14}{45}\,M_p = 0.3111\,M_p$ .

Mechanism 3: sway with the hinge at the horizontal load

The hinge in the left column forms under the 3W load (5 m above A), where the moment is largest. The lower part of the column (5 m) rotates $\theta$ ; the upper part (3 m) and the beam translate by $5\theta$ ; the right column rotates $5\theta/6$ , and hinges form at A, H, C and D.

Hinge	Rotation	Plastic moment	Internal work
A (fixed base)	$1\theta$	$M_p$	$1\,M_p\theta$
H (under 3W, 5 m above A)	$1\theta$	$M_p$	$1\,M_p\theta$
C	$\frac{5}{6}\theta$	$M_p$	$\frac{5}{6}\,M_p\theta$
D (fixed base)	$\frac{5}{6}\theta$	$M_p$	$\frac{5}{6}\,M_p\theta$

Internal work $= \frac{11}{3}\,M_p\,\theta$ .

Load	Displacement along the load	External work
2W vertical at J	$0\theta$	$0W\,\theta$
3W horizontal at H	$5\theta$	$15W\,\theta$

Equation: $W_c\times 15\,\theta = \frac{11}{3}\,M_p\theta$ , so $W_c = \frac{11}{45}\,M_p = 0.2444\,M_p$ .

Other mechanisms give larger values: for example, the combined mechanism with hinges at A, J, C and D gives $\frac{22}{69}M_p = 0.319M_p$ , and the panel mechanism (Mechanism 2) gives $0.311M_p$ .

Collapse load

$W_c$ is the smallest value found:

Answer: $W_c = \dfrac{11}{45}M_p \approx 0.244\,M_p$ (hinges at A, H, C and D; $M_p$ in kN·m, $W_c$ in kN, lengths in m).

✓ Answer checked by Claude Opus 5.5 (high effort)

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