---
title: "Collapse load of portal frame, P = 2ql"
subject: "Theory of Structures II"
chapter: 5
key: "semester_5/theory_of_structures_ii:collapse-load-portal-p-2ql"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/collapse-load-portal-p-2ql
site: BCT Engineering
---

# Collapse load of portal frame, P = 2ql

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

- Asked in: 2076 Asoj, 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-p-2ql
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-5

## Question

Evaluate the collapse load for the given portal frame. Assume $P = 2ql$. [Figure: portal frame ABDE, A hinged, E fixed, width l and height l; uniform load q on column AB; beam BD with 2MP and a vertical load P at C; columns AB and DE with MP.]

## Short answer

Virtual work for beam, sway, and combined mechanisms yields the lowest collapse load from the combined mechanism: q = 14Mp/(3l^2) and P = 28Mp/(3l).

## Answer

#### Assumptions
- Portal frame A-B-D-E with width $l$ and height $l$: A hinged, E fixed. Columns AB and DE have $M_p$, beam BD has $2M_p$; C is the mid-point of BD.
- Loads: UDL $q$ per unit length on AB acting horizontally (towards the right), and $P = 2ql$ vertical at C. Both increase together; find the collapse value of $q$ (and $P$) in terms of $M_p$ and $l$.
- Take $l = 1$ for the working (all lengths in units of $l$); restore $l$ at the end. Reactions: A hinged (2), E fixed (3), so degree of indeterminacy = 2; critical sections B, C, D, E (and the hinge A) give 4 - 2 = 2 independent mechanisms (beam, sway) and their combination.

#### Mechanisms

**Mechanism 1: beam mechanism**

No sway. Hinges at B (in the column, the weaker member), under P (at mid-span C) and at D (in the column DE). Joint rotations: $\theta/2$ at B and D, $\theta$ at C.

| Hinge | Rotation | Plastic moment | Internal work |
|---|---|---|---|
| C (under P) | $1\theta$ | $2M_p$ | $2\,M_p\theta$ |
| 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 |
|---|---|---|
| P = 2ql at C (per unit $q$) | $\frac{1}{4}\theta$ | $\frac{1}{2}q l\,\theta$ |
| q on AB | - | $0q l\,\theta$ |

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

**Mechanism 2: sway mechanism**

The beam translates by $l\theta$; both columns rotate $\theta$ (A is hinged). Hinges at B, D (in the columns) 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 |
|---|---|---|
| P = 2ql at C (per unit $q$) | $0\theta$ | $0q l\,\theta$ |
| q on AB | - | $\frac{1}{2}q l\,\theta$ |

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

**Mechanism 3: combined mechanism**

The beam and sway mechanisms are combined so that the hinge at B cancels. Hinges at C, D and E.

| Hinge | Rotation | Plastic moment | Internal work |
|---|---|---|---|
| C (under P) | $1\theta$ | $2M_p$ | $2\,M_p\theta$ |
| 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 |
|---|---|---|
| P = 2ql at C (per unit $q$) | $\frac{1}{4}\theta$ | $\frac{1}{2}q l\,\theta$ |
| q on AB | - | $\frac{1}{4}q l\,\theta$ |

**Equation:** $q\times \frac{3}{4}\,\theta = \frac{7}{2}\,M_p\theta$, so $q = \frac{14}{3}\,M_p = 4.6667\,M_p$ per $l^2$.

A mechanism with the sway hinge inside the loaded column (at 0.618 of the height, the section of maximum moment) gives 5.236, which is higher.

#### Collapse load
The lowest value governs (combined mechanism):
$$
q_c = \frac{14}{3}\frac{M_p}{l^2} = 4.667\,\frac{M_p}{l^2},\qquad P_c = 2q_cl = \frac{28}{3}\frac{M_p}{l} = 9.333\,\frac{M_p}{l}
$$

**Answer: $q_c = 14M_p/(3l^2)$ and $P_c = 28M_p/(3l)$.**

## Check yourself

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

1. In the beam mechanism, where do hinges form?
   A. At B, A and E
   B. At B (column), C under P and D (column)
   C. At C only
   D. At A, B and D
   Answer: B. Hinges at B and D in the weaker columns and at C under P. (Key point: Beam mechanism)

2. Why combine the beam and sway mechanisms?
   A. To cancel the hinge at B and get lower q
   B. To add the loads
   C. To make the frame stiffer
   D. To increase M_p
   Answer: A. Combining eliminates the hinge at B and gives lower q. (Key point: Combined mechanism)

3. Beam and sway each give q = 6Mp/l^2, combined gives 14Mp/(3l^2). Which governs?
   A. 6 Mp/l^2
   B. The average
   C. 14Mp/(3l^2), because the lowest value governs
   D. The highest value
   Answer: C. Lowest upper bound is the collapse load: 4.667 Mp/l^2. (Key point: Governing value)

4. If q_c = 14Mp/(3l^2) and P = 2ql, what is P_c?
   A. 28Mp/(3l)
   B. 14Mp/(3l)
   C. 7Mp/(3l)
   D. 14Mp/(3l^2)
   Answer: A. P_c = 2 q_c l = 28Mp/(3l). (Key point: Point load)

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