---
title: "Plastic moment capacity of frame, 60 kN"
subject: "Theory of Structures II"
chapter: 5
key: "semester_5/theory_of_structures_ii:plastic-moment-capacity-frame-60kn-40kn"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/plastic-moment-capacity-frame-60kn-40kn
site: BCT Engineering
---

# Plastic moment capacity of frame, 60 kN

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

- Asked in: 2070 Chaitra, 10 marks
- Video: [Plastic Analysis Problem No -  23 ( Finding Plastic Moment Capacity in a Frame )](https://www.youtube.com/watch?v=HsnNrS1vR3M) — 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/plastic-moment-capacity-frame-60kn-40kn
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-5

## Question

Find the plastic moment capacity of the frame shown in figure below during collapse. [Figure: portal frame; A (left base) hinged, F (right base) fixed; left column AB with 3Mp, 20 kN/m UDL horizontally; beam B-C-D 2Mp with 60 kN vertical at C (4 m from B, total 6 m); right column D-E-F with 40 kN horizontal at E, Mp; heights 2 m and 4 m marked.]

## Short answer

Three mechanisms analyzed: beam mechanism gives M_p = 24 kN·m, sway mechanism gives M_p = 60 kN·m, combined gives M_p = 48 kN·m. The sway mechanism governs, requiring M_p = 60 kN·m.

## Answer

#### Assumptions
- Portal frame: A (left base) hinged, B (0,4), C (4,4), D (6,4), F (right base, 6,0) fixed, E at 2 m above F. Columns AB and DF are 4 m high; beam BCD is 6 m with the 60 kN at C (4 m from B).
- Plastic moments: AB $3M_p$, beam BCD $2M_p$, DEF $M_p$. $M_p$ is the value to be found.
- Loads taken as ultimate: 20 kN/m on AB (horizontal, to the right), 60 kN vertical at C and 40 kN horizontal at E (to the right).
- Degree of indeterminacy $= 5 - 3 = 2$; critical sections B, C, D, E, F (A is hinged): 5 - 2 = 3 independent mechanisms. The required $M_p$ is the largest from all mechanisms.

#### Mechanisms

**Mechanism 1: beam mechanism**

No sway; hinges at B (beam end, $2M_p$, weaker than the column $3M_p$), at C (4 m from B, $2M_p$) and at D, where the hinge forms in the weaker column DF ($M_p$). The left part of the beam (4 m, B to C) rotates $\theta/3$ and the right part (2 m, C to D) rotates $2\theta/3$, so the hinge rotations are $\theta/3$ at B, $\theta$ at C and $2\theta/3$ at D.

| Hinge | Rotation | Plastic moment | Internal work |
|---|---|---|---|
| C (under 60 kN) | $1\theta$ | $2M_p$ | $2\,M_p\theta$ |
| D (top of column DF) | $\frac{2}{3}\theta$ | $M_p$ | $\frac{2}{3}\,M_p\theta$ |
| B (beam end) | $\frac{1}{3}\theta$ | $2M_p$ | $\frac{2}{3}\,M_p\theta$ |

Internal work $= \frac{10}{3}\,M_p\,\theta$.

| Load | Displacement along the load | External work |
|---|---|---|
| 60 kN at C | $\frac{4}{3}\theta$ | $80\,\text{kN·m}\,\theta$ |
| 40 kN at E | $0\theta$ | $0\,\text{kN·m}\,\theta$ |
| 20 kN/m on AB | - | $0\,\text{kN·m}\,\theta$ |

**Equation:** $M_p\times \frac{10}{3}\,\theta = 80\,\theta$ (kN·m), so $M_p = 24$ kN·m.

**Mechanism 2: sway mechanism**

The columns (4 m high) rotate $\theta$ about A and F, and the beam translates $4\theta$. Hinges at B (beam end, $2M_p$ is smaller than the column $3M_p$), D (in the right column, $M_p$) and F.

| Hinge | Rotation | Plastic moment | Internal work |
|---|---|---|---|
| B (beam end) | $1\theta$ | $2M_p$ | $2\,M_p\theta$ |
| D (column) | $1\theta$ | $M_p$ | $1\,M_p\theta$ |
| F (fixed base) | $1\theta$ | $M_p$ | $1\,M_p\theta$ |

Internal work $= 4\,M_p\,\theta$.

| Load | Displacement along the load | External work |
|---|---|---|
| 60 kN at C | $0\theta$ | $0\,\text{kN·m}\,\theta$ |
| 40 kN at E | $2\theta$ | $80\,\text{kN·m}\,\theta$ |
| 20 kN/m on AB | - | $160\,\text{kN·m}\,\theta$ |

**Equation:** $M_p\times 4\,\theta = 240\,\theta$ (kN·m), so $M_p = 60$ kN·m.

**Mechanism 3: combined mechanism**

Combine 1 and 2 so that the hinge at B cancels; hinges under the 60 kN load, at D and at F.

| Hinge | Rotation | Plastic moment | Internal work |
|---|---|---|---|
| C (under 60 kN) | $1\theta$ | $2M_p$ | $2\,M_p\theta$ |
| D (column) | $1\theta$ | $M_p$ | $1\,M_p\theta$ |
| F (fixed base) | $\frac{1}{3}\theta$ | $M_p$ | $\frac{1}{3}\,M_p\theta$ |

Internal work $= \frac{10}{3}\,M_p\,\theta$.

| Load | Displacement along the load | External work |
|---|---|---|
| 60 kN at C | $\frac{4}{3}\theta$ | $80\,\text{kN·m}\,\theta$ |
| 40 kN at E | $\frac{2}{3}\theta$ | $\frac{80}{3}\,\text{kN·m}\,\theta$ |
| 20 kN/m on AB | - | $\frac{160}{3}\,\text{kN·m}\,\theta$ |

**Equation:** $M_p\times \frac{10}{3}\,\theta = 160\,\theta$ (kN·m), so $M_p = 48$ kN·m.

A sway mechanism with hinges at B, under the 40 kN load at E, and at F (value 53.3 kN·m) and other mechanisms were also checked; they all require less.

#### Required plastic moment
The largest requirement comes from the sway mechanism:

**Answer: $M_p = 60$ kN·m** (so AB $= 180$, beam $= 120$ and column DF $= 60$ kN·m).

## 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 4 Mp theta and external work is 80 theta. What is Mp?
   A. 20 kN.m
   B. 60 kN.m
   C. 48 kN.m
   D. 80 kN.m
   Answer: A. Mp = 80/4 = 20 kN.m. (Key point: Beam mechanism)

2. In the sway mechanism, what loads do work?
   A. Only the 60 kN vertical
   B. The 40 kN at E (80) and the UDL on AB (160)
   C. Only the 40 kN
   D. All three equally
   Answer: B. The horizontal loads move with the sway (80 + 160 = 240 theta); the vertical 60 kN does no work. (Key point: Sway external work)

3. Mechanisms give Mp = 20, 60 and 48 kN.m. Which is taken and why?
   A. 20, the smallest
   B. 48, the combined one
   C. 60, the largest, since the section must resist every mechanism
   D. The average, 42.7
   Answer: C. For a required Mp (design), the largest value over all mechanisms governs, unlike a collapse load where the smallest governs. (Key point: Required Mp rule)

4. With Mp = 60 kN.m, what are the capacities of AB, beam BCD and DEF?
   A. 60, 60, 60
   B. 180, 120, 60
   C. 120, 180, 60
   D. 60, 120, 180
   Answer: B. AB = 3Mp = 180, beam = 2Mp = 120, DEF = Mp = 60 kN.m. (Key point: Resulting member capacities)

## Related in Theory of Structures II

- [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
- [Plastic moment capacity of frame](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aplastic-moment-capacity-frame-80kn-20knm) — Theory of Structures II, chapter 5
- [Collapse load of frame with UDL on column](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Acollapse-load-frame-20kn-10knm) — 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 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×

## Related in other subjects

- [AFD, SFD, BMD of frame with 60 kN/m UDL](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Aframe-afd-sfd-bmd-60knm-20knm-12kn) — Applied Mechanics, chapter 6
- [Parabolic arch 100 m: UDL left half and 80 kN](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aarch-100m-80kn-udl-d-20m) — Theory of Structures I, chapter 6, asked 4×
- [AFD, SFD, BMD of frame with 60 kN/m UDL](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Aframe-afd-sfd-bmd-60knm-hinge) — Applied Mechanics, chapter 6
- [Frame AFD, SFD, BMD (30 and 20 kN/m, hinge)](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aframe-afd-sfd-bmd-30-20) — Strength of Materials, chapter 1