---
title: "Collapse load of portal frame with W/2"
subject: "Theory of Structures II"
chapter: 5
key: "semester_5/theory_of_structures_ii:collapse-load-portal-w2-l"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/collapse-load-portal-w2-l
site: BCT Engineering
---

# Collapse load of portal frame with W/2

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

- Asked in: 2073 Shrawan, 10 marks
- Video: [structure analysis 2 :- find the value of plastic moment (Mp)  for the portal frame..](https://www.youtube.com/watch?v=w43VPVlSPvc) — UNIQUE HOME DESIGNER'S
- 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-w2-l
- 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; W/2 horizontal at the top of the left column (height L, Mp); beam 3Mp with W vertical at mid-span (L/2 + L/2); right column Mp of height 2L/3; left base hinged, right base fixed.]

## Short answer

Sway mechanism governs; plastic analysis gives collapse load Wc = 8Mp/L with hinges at B, D, and E.

## Answer

#### Assumptions
- Portal frame: A (0,0) hinged, B (0,L), C (L/2, L), D (L, L), E fixed at the base of the right column, which is $2L/3$ high (so E is at height $L/3$ above A). Column AB $M_p$, beam BD $3M_p$ (span L), column DE $M_p$.
- Loads: $W/2$ horizontal (to the right) at B and $W$ vertical at mid-span C. $W_c$ in terms of $M_p$ and $L$.
- Degree of indeterminacy $= 5 - 3 = 2$; critical sections B, C, D, E: $4 - 2 = 2$ independent mechanisms (beam and sway) and their combination.

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

**Mechanism 1: beam mechanism**

Lengths in units of $L$ (take $L = 1$ in the tables). No sway; hinges at B (in the weaker column AB), at C and at D (in the column DE, $M_p$). Rotations $\theta/2$ at B and D, $\theta$ at C.

| Hinge | Rotation | Plastic moment | Internal work |
|---|---|---|---|
| C (under W) | $1\theta$ | $3M_p$ | $3\,M_p\theta$ |
| 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 $= 4\,M_p\,\theta$.

| Load | Displacement along the load | External work |
|---|---|---|
| W/2 horizontal at B | $0\theta$ | $0W\,\theta$ |
| W at C | $\frac{1}{4}\theta$ | $\frac{1}{4}W\,\theta$ |

**Equation:** $W_c\times \frac{1}{4}\,\theta = 4\,M_p\theta$, so $W_c = 16\,M_p$ per $L$.

**Mechanism 2: sway mechanism**

The right column DE (height $2L/3$) rotates $\theta$ about the fixed base E, so the beam moves $2L\theta/3$; the left column AB (height $L$, hinged base) rotates $2\theta/3$. Hinges at B (in AB), D (in DE) and E.

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

Internal work $= \frac{8}{3}\,M_p\,\theta$.

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

**Equation:** $W_c\times \frac{1}{3}\,\theta = \frac{8}{3}\,M_p\theta$, so $W_c = 8\,M_p$ 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 |
|---|---|---|---|
| D (column DE) | $\frac{5}{4}\theta$ | $M_p$ | $\frac{5}{4}\,M_p\theta$ |
| C (under W) | $1\theta$ | $3M_p$ | $3\,M_p\theta$ |
| E (fixed base) | $\frac{3}{4}\theta$ | $M_p$ | $\frac{3}{4}\,M_p\theta$ |

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

| Load | Displacement along the load | External work |
|---|---|---|
| W/2 horizontal at B | $\frac{1}{2}\theta$ | $\frac{1}{4}W\,\theta$ |
| W at C | $\frac{1}{4}\theta$ | $\frac{1}{4}W\,\theta$ |

**Equation:** $W_c\times \frac{1}{2}\,\theta = 5\,M_p\theta$, so $W_c = 10\,M_p$ per $L$.

#### Collapse load
The sway mechanism governs (beam 16 and combined 10 are higher):

**Answer: $W_c = 8M_p/L$** (hinges at B in column AB, D in column DE and E).

## Check yourself

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

1. With critical sections B, C, D, E (4) and degree of indeterminacy 2, how many independent mechanisms are there?
   A. 4
   B. 1
   C. 6
   D. 2
   Answer: D. 4 - 2 = 2 (beam and sway); the combined mechanism is the third to check. (Key point: Independent mechanisms)

2. In the beam mechanism the hinge at C has internal work 3 Mp theta. Why?
   A. Rotation there is 3 theta
   B. The beam has plastic moment 3Mp and rotates theta at C
   C. Two hinges coincide at C
   D. The load is 3W
   Answer: B. The beam is 3Mp and the hinge rotation under the load is theta, so work is 3Mp x theta. (Key point: Beam mechanism hinge moments)

3. In the sway mechanism, which hinges form and what is the internal work?
   A. B, C and D, work 5 Mp theta
   B. B, D and E, work (8/3) Mp theta
   C. C, D and E, work 5 Mp theta
   D. A, B and E, work 4 Mp theta
   Answer: B. Hinges form at D (1), E (1) and B (2/3): internal work = 8/3 Mp theta. (Key point: Sway mechanism)

4. Beam, sway and combined give 16, 8 and 10 Mp/L. Which governs and what is Wc?
   A. Beam, 16 Mp/L
   B. Combined, 10 Mp/L
   C. Sway, 8 Mp/L
   D. Sway, 16 Mp/L
   Answer: C. The smallest value governs, so Wc = 8 Mp/L from the sway mechanism. (Key point: Governing mechanism)

## Close variants set in other exams

- [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 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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