---
title: "Plastic hinge and its length"
subject: "Theory of Structures II"
chapter: 5
key: "semester_5/theory_of_structures_ii:define-plastic-hinge-length"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/define-plastic-hinge-length
site: BCT Engineering
---

# Plastic hinge and its length

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

- Asked in: 2072 Chaitra, 4 marks
- Video: [Plastic Hinge Length - Plastic Analysis of Beams - Steel Structures - GATE](https://www.youtube.com/watch?v=PgtMBXEy8EA) — Ekeeda GATE & ESE
- 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-plastic-hinge-length
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-5

## Question

Define plastic hinge and explain how its length is determined.

## Short answer

A plastic hinge is a yielded section; its length is where moment exceeds yield moment. For a point-loaded simply supported beam, Lp = L(1 minus shape factor reciprocal).

## Answer

#### Plastic hinge
A **plastic hinge** is a section of a beam where the plastic moment $M_p$ has been reached: the whole depth has yielded, so the section rotates freely at a constant moment $M_p$, like a hinge that still carries $M_p$. In reality yielding spreads outwards from the section of maximum moment over a finite length, called the **plastic hinge length** $L_p$.

#### How the length is determined
The hinge length is the length of the member over which the bending moment exceeds the yield moment $M_y$, so part of the depth is yielded. It follows from the BMD at collapse by finding the points where $M = M_y = M_p/S$ ($S$ = shape factor).

**Simply supported beam, central point load $P$:** $M = \dfrac{P}{2}x$ for $x \le L/2$ and $M_{max} = \dfrac{PL}{4} = M_p$. The yield moment is reached at $x_y = \dfrac{L}{2S}$ from each support, so
$$
L_p = L - 2x_y = L\left(1 - \frac1S\right)
$$
For a rectangular section ($S = 1.5$): $L_p = L/3$.

**Simply supported beam, UDL $w$:** $M = \dfrac{wLx}{2} - \dfrac{wx^2}{2}$ and $M_{max} = \dfrac{wL^2}{8} = M_p$. Setting $M = M_p/S$ gives $x_y = \dfrac L2\left(1 - \sqrt{1 - 1/S}\right)$, so
$$
L_p = L\sqrt{1 - \frac1S}
$$
For a rectangular section: $L_p = 0.577L$.

So the hinge length depends on the shape factor and on the loading and span. (The larger $S$, the longer the hinge; in the idealised analysis, $L_p \to 0$.)

## Check yourself

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

1. The plastic hinge length is the length of the member over which:
   A. The shear exceeds yield
   B. The moment exceeds the yield moment My
   C. The deflection is zero
   D. The moment equals zero
   Answer: B. It is the zone where M exceeds My, so part of the depth has yielded. (Key point: Plastic hinge length meaning)

2. How is the edge of the hinge zone found from the BMD at collapse?
   A. Where M = Mp/S (S = shape factor)
   B. Where M = S Mp
   C. Where shear is zero
   D. Where M = 0
   Answer: A. My = Mp/S, so the zone ends where the moment drops to Mp/S. (Key point: Yield point in BMD)

3. For a simply supported beam with central point load, what is Lp?
   A. L/S
   B. L sqrt(1 - 1/S)
   C. L(1 - 1/S)
   D. L/2
   Answer: C. x_y = L/(2S) from each support, so Lp = L - 2x_y = L(1 - 1/S), i.e. L/3 for rectangular. (Key point: Central point load result)

4. For a rectangular section (S = 1.5) under UDL, Lp equals:
   A. L/3
   B. 0.577 L
   C. 0.5 L
   D. 0.75 L
   Answer: B. Lp = L sqrt(1 - 1/1.5) = 0.577 L. (Key point: UDL result)

## 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
- [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
- [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

## Related in other subjects

- [Frame SFD, BMD (100 kN, 20 kN/m, hinge)](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aframe-sfd-bmd-100kn-hinge-60deg) — Strength of Materials, chapter 1