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

# Collapse load of portal frame, height 2L

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

- Asked in: 2071 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-2w-3w-2l
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-5

## Question

Find the collapse load for the portal frame shown in figure below. [Figure: portal frame of span 2L (L + L) and height 2L; 3W horizontal at the top-left joint; 2W vertical at mid-span; beam 2Mp; columns Mp; left base hinged, right base fixed.]

## Short answer

Sway mechanism governs; collapse load Wc = 0.5Mp/L with hinges at B, D, and E.

## Answer

#### Assumptions
- Portal frame: span $2L$ ($L + L$), height $2L$. A hinged, E fixed (right base). Columns $M_p$, beam $2M_p$.
- Loads: $3W$ horizontal (to the right) at the top-left joint B and $2W$ vertical at mid-span C. Take $L = 1$ in the tables.
- Reactions $2 + 3 = 5$, degree of indeterminacy $= 2$; critical sections B, C, D, E: two independent mechanisms (beam, sway) plus the combined mechanism.

#### Mechanisms

**Mechanism 1: beam mechanism**

Taking $L = 1$. No sway; hinges at B (in column AB, $M_p$), at mid-span C ($2M_p$) and at D (in column DE, $M_p$). Rotations $\theta/2$ at B and D, $\theta$ at C.

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

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

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

**Equation:** $W_c\times 1\,\theta = 3\,M_p\theta$, so $W_c = 3\,M_p$ per $L$.

**Mechanism 2: sway mechanism**

Both columns are $2L$ high. They rotate $\theta$ (A is hinged), the beam moves $2L\theta$ to the right. Hinges at B (in AB), D (in DE) and E.

| Hinge | Rotation | Plastic moment | Internal work |
|---|---|---|---|
| B (column AB) | $1\theta$ | $M_p$ | $1\,M_p\theta$ |
| D (column DE) | $1\theta$ | $M_p$ | $1\,M_p\theta$ |
| E (fixed base) | $1\theta$ | $M_p$ | $1\,M_p\theta$ |

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

| Load | Displacement along the load | External work |
|---|---|---|
| 3W horizontal at B | $2\theta$ | $6W\,\theta$ |
| 2W at C | $0\theta$ | $0W\,\theta$ |

**Equation:** $W_c\times 6\,\theta = 3\,M_p\theta$, so $W_c = \frac{1}{2}\,M_p = 0.5000\,M_p$ per $L$.

**Mechanism 3: combined mechanism**

Combine 1 and 2 so that the hinge at B cancels. Hinges at C, D and E.

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

Internal work $= \frac{7}{2}\,M_p\,\theta$.

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

**Equation:** $W_c\times 4\,\theta = \frac{7}{2}\,M_p\theta$, so $W_c = \frac{7}{8}\,M_p = 0.8750\,M_p$ per $L$.

#### Collapse load
The sway mechanism governs:

**Answer: $W_c = \dfrac{M_p}{2L} = 0.5\,\dfrac{M_p}{L}$.**

## Check yourself

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

1. In the beam mechanism, internal work is 3 Mp theta and external work is W theta (L = 1). What is Wc?
   A. 3 Mp
   B. 0.5 Mp
   C. 1.5 Mp
   D. 6 Mp
   Answer: A. Wc x 1 = 3 Mp gives Wc = 3 Mp per L. (Key point: Beam mechanism result)

2. With A hinged and E fixed, in the sway mechanism hinges form at:
   A. A, B and E
   B. B, D and E
   C. C, D and E
   D. B, C and D
   Answer: B. The hinged base A cannot take moment, so hinges form at B (in AB), D (in DE) and E. (Key point: Sway mechanism hinges)

3. In sway, beam moves 2L theta. What is the external work?
   A. 2W theta
   B. 3W theta
   C. 6W theta
   D. 8W theta
   Answer: C. 3W x 2 theta = 6W theta; the vertical 2W does no work. (Key point: Sway external work)

4. Beam, sway and combined give 3, 0.5 and 0.875 Mp/L. What is Wc?
   A. 0.875 Mp/L
   B. 0.5 Mp/L
   C. 3 Mp/L
   D. 7/8 Mp/L from combined
   Answer: B. The lowest value, sway, governs: Wc = 0.5 Mp/L. (Key point: Governing mechanism)

## 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 with 1.5W](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Acollapse-load-portal-1-5w-1) — 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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