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

# Collapse load P of portal frame

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

- Asked in: 2071 Shrawan, 10 marks
- Video: [Plastic Analysis Problem No -  26 ( Finding Collapse Load in a Frame )](https://www.youtube.com/watch?v=-amh6vptYwc) — 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-p-p-p
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-5

## Question

For the given portal frame with same plastic moment capacity Mp for all members calculate the value of P at collapse. [Figure: portal frame ABCDEF, A and F hinged, height L; span L divided into three parts of L/3; horizontal P at the top-left corner B and vertical loads P at C and D on the beam; all members $M_p$.]

## Short answer

The portal frame with hinged bases has three independent mechanisms. Sway and combined mechanisms both yield P = 2M_p/L, while beam mechanism gives P = 6M_p/L. The minimum collapse load is P_c = 2M_p/L.

## Answer

#### Assumptions
- Portal frame ABCDEF: A and F hinged, columns AB and EF of height $L$, beam BE of span $L$ with C and D at the third points. All members have plastic moment $M_p$.
- Loads: $P$ horizontal (to the right) at B, and $P$ vertical at C and at D.
- Both bases are hinged, so the degree of indeterminacy is $4 - 3 = 1$; critical sections B, C, D, E: $4 - 1 = 3$ independent mechanisms (two beam mechanisms, one sway). The hinge position at a joint is in the beam end (same $M_p$).

#### Mechanisms (tables with $L = 1$)

**Mechanism 1: beam mechanism**

Taking $L = 1$. No sway; hinges at B, C and E. The beam part BC (length $L/3$) rotates $2\theta/3$ and CE ($2L/3$) rotates $\theta/3$ the other way, so C has $\theta$ and C deflects $2\theta/9$.

| Hinge | Rotation | Plastic moment | Internal work |
|---|---|---|---|
| C | $1\theta$ | $M_p$ | $1\,M_p\theta$ |
| B | $\frac{2}{3}\theta$ | $M_p$ | $\frac{2}{3}\,M_p\theta$ |
| E | $\frac{1}{3}\theta$ | $M_p$ | $\frac{1}{3}\,M_p\theta$ |

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

| Load | Displacement along the load | External work |
|---|---|---|
| P horizontal at B | $0\theta$ | $0P\,\theta$ |
| P at C | $\frac{2}{9}\theta$ | $\frac{2}{9}P\,\theta$ |
| P at D | $\frac{1}{9}\theta$ | $\frac{1}{9}P\,\theta$ |

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

**Mechanism 2: sway mechanism**

Both columns rotate $\theta$ about the hinged bases; the beam translates by $L\theta$. Hinges at B and E only (the beam stays straight).

| Hinge | Rotation | Plastic moment | Internal work |
|---|---|---|---|
| B | $1\theta$ | $M_p$ | $1\,M_p\theta$ |
| E | $1\theta$ | $M_p$ | $1\,M_p\theta$ |

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

| Load | Displacement along the load | External work |
|---|---|---|
| P horizontal at B | $1\theta$ | $1P\,\theta$ |
| P at C | $0\theta$ | $0P\,\theta$ |
| P at D | $0\theta$ | $0P\,\theta$ |

**Equation:** $P\times 1\,\theta = 2\,M_p\theta$, so $P = 2\,M_p$ per $L$.

**Mechanism 3: combined mechanism**

Combine 1 and 2 so that the hinge at B cancels; hinges at C and E.

| Hinge | Rotation | Plastic moment | Internal work |
|---|---|---|---|
| C | $1\theta$ | $M_p$ | $1\,M_p\theta$ |
| E | $1\theta$ | $M_p$ | $1\,M_p\theta$ |

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

| Load | Displacement along the load | External work |
|---|---|---|
| P horizontal at B | $\frac{2}{3}\theta$ | $\frac{2}{3}P\,\theta$ |
| P at C | $\frac{2}{9}\theta$ | $\frac{2}{9}P\,\theta$ |
| P at D | $\frac{1}{9}\theta$ | $\frac{1}{9}P\,\theta$ |

**Equation:** $P\times 1\,\theta = 2\,M_p\theta$, so $P = 2\,M_p$ per $L$.

The beam mechanism with the hinge under the load at D, and the combined mechanism with the hinge under D (value 3), are higher.

#### Collapse load
The sway and the combined mechanism give the same minimum, and the beam mechanism is much higher, so

**Answer: $P_c = \dfrac{2M_p}{L}$.**

## Check yourself

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

1. Both bases are hinged, so the degree of indeterminacy is 1. With 4 critical sections, how many independent mechanisms?
   A. 4
   B. 1
   C. 5
   D. 3
   Answer: D. 4 - 1 = 3 (two beam mechanisms and one sway). (Key point: Number of independent mechanisms)

2. In the sway mechanism, hinges at B and E rotate theta each and P moves L theta. What is P?
   A. Mp/L
   B. 2 Mp/L
   C. 6 Mp/L
   D. Mp/(2L)
   Answer: B. P L theta = 2 Mp theta gives P = 2 Mp/L. (Key point: Sway mechanism)

3. Why is the beam mechanism not critical (P = 6 Mp/L)?
   A. It gives a higher load than sway
   B. It has fewer hinges
   C. It lacks a hinge at C
   D. It gives the lowest load
   Answer: A. 6 Mp/L is much larger than 2 Mp/L, and collapse occurs at the lowest load. (Key point: Beam mechanism)

4. In the combined mechanism the hinge at B cancels. Which hinges remain?
   A. B and E
   B. C and E
   C. A and F
   D. C and D
   Answer: B. Adding beam and sway mechanisms cancels B, leaving hinges at C and E, giving P = 2 Mp/L. (Key point: Combined mechanism)

## Related in Theory of Structures II

- [Collapse value of P: portal frame with horizontal P/2](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Acollapse-p-portal-p-half-ml) — Theory of Structures II, chapter 5
- [Collapse load P: portal frame with UDL 2P/l](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Acollapse-load-portal-p-p-udl) — Theory of Structures II, chapter 5
- [Collapse moment of portal frame, load factor 1.75](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Acollapse-moment-portal-load-factor-1) — Theory of Structures II, chapter 5
- [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×
- [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
- [Plastic moment capacity Mp of frame](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aplastic-moment-capacity-frame-3mp-2mp) — Theory of Structures II, chapter 5

## Related in other subjects

- [Define plastic moment capacity of beam](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aplastic-moment-capacity-definition) — Strength of Materials, chapter 7
- [Frame vertical deflection at D](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aframe-vw-d-vertical-2071o) — Theory of Structures I, chapter 3