---
title: "Collapse load of rectangular portal frame"
subject: "Theory of Structures II"
chapter: 5
key: "semester_5/theory_of_structures_ii:collapse-load-portal-2w-3w-rect"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/collapse-load-portal-2w-3w-rect
site: BCT Engineering
---

# Collapse load of rectangular portal frame

Theory of Structures II · Chapter 5: Introduction to plastic analysis

- Asked in: 2075 Asoj, 8 marks
- Video: [Plastic Analysis Problem No -  24 ( Finding Collapse Load in a Frame )](https://www.youtube.com/watch?v=ume4u1BAVqI) — Stan Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/collapse-load-portal-2w-3w-rect
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-5

## Question

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.]

## 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.

## Answer

#### 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).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. For this frame there are 6 critical sections and the degree of indeterminacy is 3. How many independent mechanisms are there?
   A. 6
   B. 3
   C. 9
   D. 2
   Answer: B. Independent mechanisms = critical sections minus degree of indeterminacy = 6 - 3 = 3 (beam, panel/sway, joint-type). (Key point: Number of independent mechanisms)

2. In the beam mechanism, the hinge rotations sum to give internal work 2 Mp*theta, and 2W moves 12/7 theta. What is Wc from this mechanism?
   A. 0.2444 Mp
   B. 0.3111 Mp
   C. 1.714 Mp
   D. 0.5833 Mp
   Answer: D. Wc x (24/7)theta = 2 Mp theta gives Wc = 7/12 Mp = 0.5833 Mp. (Key point: Beam mechanism work equation)

3. In the sway mechanisms, why does the 2W vertical load do no external work?
   A. Because it is smaller than the 3W load
   B. Because the beam translates horizontally, so there is no vertical displacement at J
   C. Because the plastic moment cancels it
   D. Because it acts at a hinge
   Answer: B. In pure sway the beam only moves sideways, so the vertical load has zero displacement along its line and does no work. (Key point: Work of the horizontal load in sway)

4. Mechanism values are 0.5833 Mp, 0.3111 Mp and 0.2444 Mp. Which is the collapse load and why?
   A. 0.5833 Mp, the largest, since the frame must carry it
   B. 0.2444 Mp, the smallest, since collapse occurs at the lowest load
   C. 0.3111 Mp, the middle one
   D. The average of the three
   Answer: B. The true collapse load is the lowest value from all mechanisms: Wc = 11/45 Mp = 0.244 Mp, with hinges at A, H, C and D. (Key point: Governing collapse load)

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- [Collapse load of rectangular portal frame](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Acollapse-load-rect-portal-2p-3p) — Theory of Structures II, chapter 5
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