---
title: "Collapse load of portal frame with 1.5W"
subject: "Theory of Structures II"
chapter: 5
key: "semester_5/theory_of_structures_ii:collapse-load-portal-1-5w-1"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/collapse-load-portal-1-5w-1
site: BCT Engineering
---

# Collapse load of portal frame with 1.5W

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

- Asked in: 2075 Chaitra, 10 marks
- 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-1-5w-1
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-5

## Question

Determine collapse load in the portal frame shown in figure below. [Figure: portal frame; horizontal 1.5W at the top-left joint; vertical W at mid-span of the beam (1.5Mp, spans 3 m + 3 m); left column 2Mp, 3 m + 2 m high, fixed base; right column Mp hinged at its base.]

## Short answer

Using virtual work, the beam, sway and combined mechanisms give 1.833Mp, 1.111Mp and Mp; the lowest is the combined mechanism with hinges at A, C and D. Therefore Wc = Mp, with Mp in kN·m and lengths in m.

## Answer

#### Assumptions
- Portal frame: A (0,0) fixed, B (0,3), C (3,3) at mid-span, D (6,3), E hinged at the base of the right column. The figure gives "3 m + 2 m": the left column AB is taken as 3 m high and the right column DE as 2 m high (so E is 1 m above the level of A). Beam BD is 6 m ($1.5M_p$), AB is $2M_p$ and DE is $M_p$.
- Loads: 1.5W horizontal (to the right) at B and W vertical at C. $W_c$ is found in terms of $M_p$ (kN·m, lengths in m).
- Method: virtual work, $\sum M_p\theta = \sum P\delta$. Reactions: 3 at A and 2 at E, degree of indeterminacy $= 5 - 3 = 2$; critical sections A, B, C, D give 4 - 2 = 2 independent mechanisms (beam and sway), plus the combination.

#### Mechanisms

**Mechanism 1: beam mechanism**

No sway. Hinges at B (beam end, $1.5M_p$ is smaller than the column $2M_p$), under W (mid-span) and at D (in the weaker column DE, $M_p$). The rotations are $\theta/2$ at B and D and $\theta$ at C.

| Hinge | Rotation | Plastic moment | Internal work |
|---|---|---|---|
| C (under W) | $1\theta$ | $\frac{3}{2}M_p$ | $\frac{3}{2}\,M_p\theta$ |
| B (beam end) | $\frac{1}{2}\theta$ | $\frac{3}{2}M_p$ | $\frac{3}{4}\,M_p\theta$ |
| D (column DE) | $\frac{1}{2}\theta$ | $M_p$ | $\frac{1}{2}\,M_p\theta$ |

Internal work $= \frac{11}{4}\,M_p\,\theta$.

| Load | Displacement along the load | External work |
|---|---|---|
| 1.5W horizontal at B | $0\theta$ | $0W\,\theta$ |
| W at C | $\frac{3}{2}\theta$ | $\frac{3}{2}W\,\theta$ |

**Equation:** $W_c\times \frac{3}{2}\,\theta = \frac{11}{4}\,M_p\theta$, so $W_c = \frac{11}{6}\,M_p = 1.8333\,M_p$.

**Mechanism 2: sway mechanism**

Left column AB (3 m) rotates $\theta$ about A; the beam moves $3\theta$ to the right; the right column DE (2 m high) rotates $3\theta/2$ about E. Hinges at A, B (beam end) and D (in DE).

| Hinge | Rotation | Plastic moment | Internal work |
|---|---|---|---|
| D (column DE) | $\frac{3}{2}\theta$ | $M_p$ | $\frac{3}{2}\,M_p\theta$ |
| A (fixed base) | $1\theta$ | $2M_p$ | $2\,M_p\theta$ |
| B (beam end) | $1\theta$ | $\frac{3}{2}M_p$ | $\frac{3}{2}\,M_p\theta$ |

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

| Load | Displacement along the load | External work |
|---|---|---|
| 1.5W horizontal at B | $3\theta$ | $\frac{9}{2}W\,\theta$ |
| W at C | $0\theta$ | $0W\,\theta$ |

**Equation:** $W_c\times \frac{9}{2}\,\theta = 5\,M_p\theta$, so $W_c = \frac{10}{9}\,M_p = 1.1111\,M_p$.

**Mechanism 3: combined mechanism**

Mechanisms 1 and 2 are combined so that the hinge at B cancels. Hinges at A, C and D.

| Hinge | Rotation | Plastic moment | Internal work |
|---|---|---|---|
| D (column DE) | $\frac{5}{4}\theta$ | $M_p$ | $\frac{5}{4}\,M_p\theta$ |
| C (under W) | $1\theta$ | $\frac{3}{2}M_p$ | $\frac{3}{2}\,M_p\theta$ |
| A (fixed base) | $\frac{1}{2}\theta$ | $2M_p$ | $1\,M_p\theta$ |

Internal work $= \frac{15}{4}\,M_p\,\theta$.

| Load | Displacement along the load | External work |
|---|---|---|
| 1.5W horizontal at B | $\frac{3}{2}\theta$ | $\frac{9}{4}W\,\theta$ |
| W at C | $\frac{3}{2}\theta$ | $\frac{3}{2}W\,\theta$ |

**Equation:** $W_c\times \frac{15}{4}\,\theta = \frac{15}{4}\,M_p\theta$, so $W_c = 1\,M_p$.

#### Collapse load
The combined mechanism is the lowest (the sway mechanism gives $1.111M_p$ and the beam mechanism $1.833M_p$), and a search over all hinge positions confirmed it:

**Answer: $W_c = M_p$** (hinges at A, C and D; the loads at collapse are 1.5$M_p$ horizontal and $M_p$ vertical, with $M_p$ in kN·m and lengths in m).

## Check yourself

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

1. The beam mechanism for this frame gives:
   A. W_c = 11/6 Mp = 1.833 Mp
   B. W_c = 10/9 Mp
   C. W_c = Mp
   D. W_c = 5/3 Mp
   Answer: A. Internal 11/4 Mp theta = W x 3/2 theta, so W_c = 11/6 Mp. (Key point: Beam mechanism)

2. In sway, the right column is 2 m high; its rotation is:
   A. theta
   B. 3 theta/2
   C. 2 theta
   D. theta/2
   Answer: B. Beam moves 3 theta; the 2 m column rotates 3 theta/2. (Key point: Sway geometry)

3. In sway, what is the external work of 1.5W?
   A. 1.5W x 3 theta = 4.5 W theta
   B. 1.5W x 2 theta
   C. zero
   D. W x 3 theta
   Answer: A. Gives sway W_c = 5/4.5 = 10/9 Mp. (Key point: Sway load work)

4. Which mechanism governs and what is W_c?
   A. Beam, 1.833 Mp
   B. Sway, 1.111 Mp
   C. Combined, W_c = Mp with hinges at A, C and D
   D. Combined, W_c = 2 Mp
   Answer: C. Combined mechanism gives 15/4 Mp theta = W x 15/4 theta, so W_c = Mp. (Key point: Result)

## Close variants set in other exams

- [Collapse load of portal frame with W/2](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Acollapse-load-portal-w2-l) — Theory of Structures II, chapter 5, asked 3×
- [Collapse load of portal frame, height 2L](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Acollapse-load-portal-2w-3w-2l) — Theory of Structures II, chapter 5, asked 3×

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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
- [Collapse load of portal frame, 8 m span](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Acollapse-load-frame-w-2mp-8m) — Theory of Structures II, chapter 5
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