---
title: "Define plastic moment capacity of beam"
subject: "Strength of Materials"
chapter: 7
key: "semester_3/strength_of_materials:plastic-moment-capacity-definition"
source: https://www.bctengineer.com/q/semester_3/strength_of_materials/plastic-moment-capacity-definition
site: BCT Engineering
---

# Define plastic moment capacity of beam

Strength of Materials · Chapter 7: Theory of Flexure

- Asked in: 2066 Jestha (old course), 4 marks
- Video: [Basics of Bending Stress part 4 - Plastic section modulus (Z) and plastic moment capacity](https://www.youtube.com/watch?v=jkJuaClWyZU) — Prometheus Engineering Group
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/strength_of_materials/plastic-moment-capacity-definition
- Whole chapter: https://www.bctengineer.com/md/semester_3/strength_of_materials/chapter-7

## Question

Define the plastic moment capacity of beam.

## Short answer

Plastic moment capacity is the moment at which the entire cross-section yields, with tension and compression zones reaching yield stress. It equals σ_y times the plastic section modulus, and a plastic hinge forms.

## Answer

The **plastic moment capacity** ($M_p$) of a beam is the maximum bending moment that the cross-section can carry when the whole section has yielded, that is, when every fibre on the tension side and on the compression side has reached the yield stress $\sigma_y$.

- Up to the yield moment $M_y = \sigma_y Z$, stress varies linearly and only the extreme fibres yield.
- As the moment increases, yielding spreads inward from the outer fibres.
- At $M_p$ the section is fully plastic. The neutral axis then divides the area into two equal halves (equal-area axis), and the tension and compression forces form a couple:

$$
M_p = \sigma_y\,S
$$

where $S = A_t\bar y_t + A_c\bar y_c$ is the **plastic section modulus**, the sum of the first moments of the tension and compression areas about the equal-area axis.

At $M_p$ a plastic hinge forms and the section rotates at constant moment. The ratio $M_p/M_y = S/Z$ is the **shape factor**: 1.5 for a rectangle, about 1.7 for a circle, and about 1.12 to 1.15 for I-sections.

## Check yourself

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

1. What is plastic moment capacity M_p?
   A. Moment when the extreme fibres first yield
   B. Moment at the ultimate tensile strength
   C. Moment at which the beam first cracks
   D. Maximum moment the section can carry when the whole section has yielded
   Answer: D. M_p is reached when every fibre has reached sigma_y. (Key point: Definition)

2. Where does the neutral axis lie at M_p?
   A. At the equal-area axis dividing the section into two equal halves
   B. At the centroid always
   C. At the extreme fibre
   D. At the base
   Answer: A. For equal tension and compression forces, areas above and below the axis are equal. (Key point: Neutral axis at M_p)

3. How is M_p expressed?
   A. M_p = sigma_y Z, Z the elastic modulus
   B. M_p = sigma_y S, S the plastic section modulus
   C. M_p = sigma_y/S
   D. M_p = sigma_y A
   Answer: B. S is the sum of first moments of the tension and compression areas about the equal-area axis. (Key point: Formula)

4. What is the shape factor and a typical value for a rectangle?
   A. M_y/M_p, 0.67
   B. S x Z, 1.5
   C. M_p/M_y = S/Z, 1.5
   D. S/Z, 1.12
   Answer: C. Shape factor is M_p/M_y; rectangle gives 1.5, I-sections about 1.12 to 1.15. (Key point: Shape factor)

## Related in Strength of Materials

- [Limiting plastic moment is 1.5 times elastic moment](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aplastic-moment-rectangular-1-5) — Strength of Materials, chapter 7
- [Beam thrust and bending moment diagrams with max moments](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Abeam-thrust-bmd-2kn-5kn-1knm) — Strength of Materials, chapter 1
- [I-section beam: moment of resistance and UDL](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Ai-section-moment-resistance-udl-10m) — Strength of Materials, chapter 7
- [Deflection at A: beam with load and moment](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Abeam-deflection-point-a-moment-and) — Strength of Materials, chapter 7
- [Symmetrical I-beam: moment, flange and web share](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Asymmetrical-i-beam-max-moment-flange) — Strength of Materials, chapter 7
- [Define shear force and bending moment](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Adefine-shear-force-bending-moment) — Strength of Materials, chapter 1, asked 2×

## Related in other subjects

- [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
- [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
- [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 and shape factor of a beam](https://www.bctengineer.com/question.md?key=semester_5%2Fmechanics_of_solids%3Ashape-factor-plastic-moment-rectangle) — Mechanics of Solids, chapter 12