---
title: "Define shape factor"
subject: "Theory of Structures II"
chapter: 5
key: "semester_5/theory_of_structures_ii:define-shape-factor"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/define-shape-factor
site: BCT Engineering
---

# Define shape factor

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

- Asked in: 2072 Chaitra, 1.5 marks; 2070 Chaitra (old course), 2 marks; 2069 Asar, 1 marks (asked 3 times)
- Video: [How to calculate plastic, elastic section modulus and Shape Factor of a cross-section?](https://www.youtube.com/watch?v=T2XMORhJ49Q) — Bold Learning
- 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/define-shape-factor
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-5

## Question

Define shape factor.

## Short answer

Shape factor is the ratio of plastic moment capacity M_p to yield moment M_y, equal to plastic section modulus Z_p divided by elastic modulus Z_e. It indicates reserve strength beyond first yield; typical values: rectangle 1.5, I-section~1.15.

## Answer

**Shape factor** is the ratio of the plastic moment capacity $M_p$ of a cross-section to its yield moment $M_y$ (the moment at which the extreme fibre first reaches the yield stress):
$$
S = \frac{M_p}{M_y} = \frac{\sigma_y Z_p}{\sigma_y Z_e} = \frac{Z_p}{Z_e}
$$
where $Z_p$ is the plastic section modulus (the sum of the first moments of the areas above and below the equal-area axis, taken about that axis) and $Z_e$ is the elastic section modulus. It depends only on the shape of the section and shows the reserve of strength after first yield.

Typical values: rectangle 1.5, solid circle 1.70, I-section about 1.10 to 1.15, thin-walled circular tube about 1.27, triangle 2.34, T-section about 1.7 to 2.0.

## Check yourself

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

1. The shape factor of a section is the ratio of:
   A. yield moment to plastic moment
   B. plastic moment to yield moment
   C. plastic modulus to the area
   D. collapse load to working load
   Answer: B. S = M_p / M_y, plastic moment capacity over the yield moment. (Key point: Definition)

2. The shape factor equals:
   A. Z_e / Z_p
   B. sigma_y Z_p
   C. Z_p / Z_e
   D. Z_p x Z_e
   Answer: C. Since M_p = sigma_y Z_p and M_y = sigma_y Z_e, S = Z_p/Z_e. (Key point: In terms of moduli)

3. What does a rectangular section's shape factor of 1.5 show?
   A. The section has 50 percent reserve strength after first yield.
   B. The section yields at 1.5 times the load.
   C. The section has 1.5 times the area.
   D. It is independent of the section shape.
   Answer: A. The shape factor depends only on shape and indicates the reserve of strength beyond first yield. (Key point: Meaning)

## Related in Theory of Structures II

- [Shape factor, plastic hinge properties, T-beam](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Ashape-factor-t-beam-plastic-hinge) — Theory of Structures II, chapter 5

## Related in other subjects

- [Shape factor algebra and concentric spheres](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Ashape-factor-algebra-concentric-spheres) — Heat Transfer, chapter 4
- [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