---
title: "Shape factor, plastic hinge properties, T-beam"
subject: "Theory of Structures II"
chapter: 5
key: "semester_5/theory_of_structures_ii:shape-factor-t-beam-plastic-hinge"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/shape-factor-t-beam-plastic-hinge
site: BCT Engineering
---

# Shape factor, plastic hinge properties, T-beam

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

- Asked in: 2068 Baishakh, 10 marks
- Video: [Steel structure- T section shape factor (In Hindi)](https://www.youtube.com/watch?v=SRYhOAVbDws) — Yogendra Ram Meena
- 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/shape-factor-t-beam-plastic-hinge
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-5

## Question

Define shape factor and write properties of plastic hinge. Find shape factor of the given T-beam section. [Figure: T-section; flange width 2000 mm and thickness 150 mm; web depth 950 mm and web width 175 mm (overall depth measured as printed).]

## Short answer

Shape factor S = Zp/Ze = Mp/My. A plastic hinge forms at full yield, rotates at constant Mp, occurs at points of maximum moment, and reduces indeterminacy by one. For the T-section, Zp = 9.92×10^7 mm³, Ze = 5.48×10^7 mm³; S ≈ 1.81.

## Answer

#### Shape factor
The shape factor is $S = \dfrac{M_p}{M_y} = \dfrac{Z_p}{Z_e}$, the ratio of the plastic moment of the section to the yield moment.

#### Properties of a plastic hinge
1. It forms at a section where the moment reaches $M_p$ and the whole depth of the section has yielded.
2. It rotates at a constant moment $M_p$ (like a hinge that carries a moment) and offers no extra resistance to rotation.
3. It forms at the points of maximum moment: fixed supports, joints, under concentrated loads, and at the section of zero shear under a UDL.
4. Rotation takes place only in the direction of the applied moment; on unloading or reversal the hinge locks and behaves elastically.
5. It extends over a finite length of the member (the plastic hinge length), but is treated as a point in analysis.
6. Each hinge reduces the degree of indeterminacy by one; when enough hinges form (degree + 1), the structure becomes a mechanism and collapses.

#### Shape factor of the T-section
**Assumed dimensions:** flange 2000 mm × 150 mm; web 175 mm wide, 950 mm deep below the flange; total depth $D = 150 + 950 = 1100$ mm. (If 950 mm is the overall depth, the web is 800 mm and the shape factor becomes 1.84.)

*Area:* $A = 2000\times150 + 950\times175 = 300000 + 166250 = 466250$ mm².

**1. Elastic neutral axis (distance from the top):**
$$
\bar y = \frac{300000\times75 + 166250\times(150+475)}{466250} = 271.11\ \text{mm}
$$
**2. Moment of inertia about the elastic neutral axis:**
$$
I = \frac{2000\times150^3}{12} + 300000(271.11-75)^2 + \frac{175\times950^3}{12} + 166250(625-271.11)^2 = 4.542\times10^{10}\ \text{mm}^4
$$
**3. Elastic section modulus** (extreme fibre in the web at $1100 - 271.11 = 828.89$ mm):
$$
Z_e = \frac{4.542\times10^{10}}{828.89} = 5.480\times10^{7}\ \text{mm}^3
$$
**4. Plastic neutral axis** divides the area into two equal halves, $A/2 = 233125$ mm². The flange area is 300000 mm² $> 233125$, so the axis lies in the flange, at depth
$$
y_p = \frac{233125}{2000} = 116.56\ \text{mm below the top}
$$
**5. Plastic section modulus** (moment of each half-area about the plastic axis):
$$
Z_p = 2000\times116.56\times\frac{116.56}{2} + 2000\times33.44\times\frac{33.44}{2} + 166250\times\left(33.44 + 475\right) = 9.923\times10^{7}\ \text{mm}^3
$$
**6. Shape factor:**
$$
S = \frac{Z_p}{Z_e} = \frac{9.923\times10^{7}}{5.480\times10^{7}} = 1.81
$$

**Answer: $S \approx 1.81$** (elastic $Z_e = 5.48\times10^7$ mm³, plastic $Z_p = 9.92\times10^7$ mm³, plastic neutral axis in the flange 116.6 mm from the top).

## Check yourself

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

1. What is the shape factor of a section?
   A. Ratio of yield moment to plastic moment
   B. Ratio of plastic moment to yield moment, Zp/Ze
   C. Ratio of depth to width
   D. Ratio of Ze to the area
   Answer: B. S = Mp/My = Zp/Ze, always at least 1. (Key point: Definition of shape factor)

2. Which describes a plastic hinge correctly?
   A. It rotates freely with zero moment
   B. It rotates at constant moment Mp and only in the direction of the applied moment
   C. It forms where the moment is zero
   D. It can rotate both ways at will after unloading
   Answer: B. A plastic hinge carries Mp while rotating, rotates only in the direction of the moment and locks on reversal. (Key point: Plastic hinge behaviour)

3. How is the plastic neutral axis located, and where does it lie for the T-section?
   A. Through the centroid, 271 mm from the top
   B. It divides the area into two equal halves; here it lies in the flange 116.6 mm from the top
   C. At the flange-web junction, 150 mm
   D. At mid-depth of the web
   Answer: B. Equal-area axis: A/2 = 233125 mm2 is less than the flange area 300000, so it is in the flange at 233125/2000 = 116.56 mm. (Key point: Plastic neutral axis)

4. With Ze = 5.48x10^7 mm3 and Zp = 9.92x10^7 mm3, what is the shape factor?
   A. 0.55
   B. 1.5
   C. 1.81
   D. 5.4
   Answer: C. S = Zp/Ze = 9.92/5.48 = 1.81. (Key point: Shape factor value)

## Related in Theory of Structures II

- [Define plastic hinge](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Adefine-plastic-hinge) — Theory of Structures II, chapter 5, asked 2×
- [Plastic hinge versus elastic hinge](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Adefine-plastic-hinge-compare-elastic) — Theory of Structures II, chapter 5
- [Plastic hinge and its length](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Adefine-plastic-hinge-length) — Theory of Structures II, chapter 5
- [Elastic vs plastic analysis; plastic hinge](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aelastic-vs-plastic-analysis-plastic-hinge) — Theory of Structures II, chapter 5
- [Define shape factor](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Adefine-shape-factor) — Theory of Structures II, chapter 5, asked 3×
- [Slope deflection: continuous beam, B sinks 19 mm](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aslope-deflection-beam-sinks-19mm) — Theory of Structures II, chapter 3

## Related in other subjects

- [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
- [Ultimate moment of T-beam (2250 mm flange)](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Aultimate-moment-of-t-beam-2250) — Design of RCC Structure, chapter 4
- [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
- [Principal stresses at web-flange junction](https://www.bctengineer.com/question.md?key=semester_4%2Fstrength_of_materials_mechanical%3Aprincipal-stresses-web-flange-junction) — Strength of Materials, chapter 9