---
title: "Collapse mechanisms: continuous beam ABCD"
subject: "Theory of Structures II"
chapter: 5
key: "semester_5/theory_of_structures_ii:collapse-continuous-beam-abcd-80kn-5knm"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/collapse-continuous-beam-abcd-80kn-5knm
site: BCT Engineering
---

# Collapse mechanisms: continuous beam ABCD

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

- Asked in: 2072 Kartik, 10 marks
- Video: [Plastic Analysis Problem No - 7 ( Finding Plastic Moment Capacity in a Three span continuous Beam )](https://www.youtube.com/watch?v=Y9o_rUrkV1M) — 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-continuous-beam-abcd-80kn-5knm
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-5

## Question

A prismatic continuous beam ABCD is fixed at A and simply supported at B, C and D. It is subjected to factored loads as shown. Find the collapse mechanisms and draw BMD. [Figure: beam with 80 kN, 5 kN/m UDL, 60 kN and 40 kN loads; dimensions 4 m, 4 m, 5 m, 3 m, 2 m and 1 m as marked; plastic moment $M_p$ in each span.]

## Short answer

Collapse occurs in span AB with hinges at A, mid-span E, and B; required plastic moment Mp = 80 kN·m.

## Answer

#### Assumptions (the figure is not fully clear)
- Continuous beam A-B-C-D, prismatic, with the same plastic moment $M_p$ in all spans: A fixed, B, C and D simple supports. AB = 8 m with 80 kN at E, 4 m from A; BC = 5 m with 5 kN/m over the whole span and 60 kN at F, 3 m from B; CD = 3 m with 40 kN at G, 2 m from C. The given loads are the factored (collapse) loads, so $M_p$ is the value required.
- Degree of indeterminacy $= 6 - 3 = 3$ (A fixed 3, and B, C, D 1 each). Each span can fail on its own, and the largest $M_p$ among the spans governs.

#### Mechanism 1: span AB (hinges at A, E and B)
AB is fixed at A and continuous at B, so it acts as a fixed-ended beam. With rotation $\theta$ at A and at B and $2\theta$ at E, the deflection under the load is $4\theta$.
$$
M_p(\theta + 2\theta + \theta) = 80\times 4\theta \;\Rightarrow\; M_p = 80\ \text{kN·m}
$$
This is the usual $PL/8 = 80\times 8/8$ for a fixed beam with a central load.

#### Mechanism 2: span BC (hinges at B, F and C)
BC is also treated as a fixed-ended span. Part BF (3 m) rotates $2\phi$ and part FC (2 m) rotates $3\phi$ (since $3\times2\phi = 2\times3\phi$), so the deflection under the point load is $6\phi$. Hinge rotations: B $2\phi$, F $5\phi$, C $3\phi$.
- Internal work $= M_p(2\phi + 5\phi + 3\phi) = 10M_p\phi$.
- External work $= 60\times6\phi + 5\times\tfrac12\times5\times6\phi = 435\,\phi$.
$$
M_p = \frac{435}{10} = 43.5\ \text{kN·m}
$$

#### Mechanism 3: span CD (hinges at C and G)
CD has a simple end at D, so two hinges are enough. Part CG (2 m) rotates $\theta$ about C and the deflection under the load is $2\theta$; part GD (1 m) rotates $2\theta$ about D.
- Internal work $= M_p\left[\theta + (\theta + 2\theta)\right] = 4\,M_p\theta$.
- External work $= 40\times2\theta$.
$$
M_p = \frac{80}{4} = 20\ \text{kN·m}
$$

#### Collapse mechanism and required $M_p$
| Span | Hinges | Required $M_p$ (kN·m) |
|---|---|---|
| AB | A, E, B | 80 |
| BC | B, F, C | 43.5 |
| CD | C, G | 20 |

The largest requirement is for span AB, so the collapse mechanism is the failure of AB with hinges at A, E and B (a beam mechanism), and **$M_p = 80$ kN·m**. Spans BC and CD stay elastic and carry the moments below.

#### Bending moment diagram at collapse
The hinges give $M_A = -80$, $M_E = +80$ and $M_B = -80$ kN·m. The hinge at B holds the moment $-M_p$ while BC and CD remain elastic as a two-span beam, so the three-moment equation at C (with $M_B = -80$, $M_D = 0$) gives the hogging support moment
$$
M_C = -27.43\ \text{kN·m}
$$
The sagging moments are then $M_F = +38.54$ kN·m under the 60 kN load in BC and $M_G = +17.52$ kN·m under the 40 kN load in CD. All are below $M_p$, so no other hinge forms and the diagram is statically admissible.

| Section | A | E | B | F | C | G | D |
|---|---|---|---|---|---|---|---|
| BM (kN·m) | -80 | +80 | -80 | +38.54 | -27.43 | +17.52 | 0 |

Shape (sagging above the line): in AB, a triangle with peak $+80$ at E and $-80$ at A and B; in BC, a parabola starting at $-80$ at B, rising to about $+38.54$ near F and falling to $-27.43$ at C; in CD, straight lines from $-27.43$ at C through $+17.52$ at G to 0 at D.

**Answer: $M_p = 80$ kN·m; collapse mechanism of span AB with plastic hinges at A, at E (under the 80 kN load) and at B.**

## Check yourself

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

1. Beam ABCD with A fixed and B, C, D simple supports: what is the degree of indeterminacy?
   A. 2
   B. 4
   C. 3
   D. 6
   Answer: C. Reactions 3+1+1+1 = 6, minus 3 equilibrium equations = 3. (Key point: Degree of indeterminacy)

2. In span AB (hinges at A, E, B) with 80 kN at the centre of 8 m, what Mp is required?
   A. 80 kN.m
   B. 160 kN.m
   C. 40 kN.m
   D. 43.5 kN.m
   Answer: A. Mp(4 theta) = 80 x 4 theta gives Mp = 80 kN.m (= PL/8). (Key point: Span AB mechanism)

3. Spans need 80, 43.5 and 20 kN.m. Which governs and why?
   A. CD, the smallest
   B. BC, the middle value
   C. AB, the largest requirement; the others stay elastic
   D. The sum of all three
   Answer: C. A single Mp must satisfy all spans, so the largest requirement (AB) governs. (Key point: Governing span)

4. Why is the moment diagram checked after finding Mp = 80 kN.m?
   A. To find the shear force
   B. To confirm no moment exceeds Mp so no other hinge forms
   C. To reduce Mp
   D. To locate the supports
   Answer: B. Moments in BC and CD (e.g. -27.43, +38.54, +17.52) are all below Mp, so the diagram is statically admissible. (Key point: BMD check)

## Related in Theory of Structures II

- [Collapse mechanism: continuous beam ABCD, BMD](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Acollapse-continuous-beam-abcd-100kn-6knm) — Theory of Structures II, chapter 5
- [Plastic moment capacity of frame, 60 kN](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aplastic-moment-capacity-frame-60kn-40kn) — Theory of Structures II, chapter 5
- [Moment distribution: frame with 80 kN, 60 kN](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Amoment-dist-frame-80kn-10knm-60kn) — Theory of Structures II, chapter 3
- [Collapse load of fixed beam with UDL](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Acollapse-load-fixed-beam-udl) — Theory of Structures II, chapter 5
- [Collapse load of two-span beam](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Acollapse-load-two-span-beam-20kn) — Theory of Structures II, chapter 5
- [Three moment theorem: SFD and BMD of beam](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Athree-moment-beam-100kn-60knm-20kn) — Theory of Structures II, chapter 2

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