---
title: "Indeterminacy of frame with internal hinges"
subject: "Theory of Structures II"
chapter: 1
key: "semester_5/theory_of_structures_ii:degrees-indeterminacy-trapezoid-frame"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/degrees-indeterminacy-trapezoid-frame
site: BCT Engineering
---

# Indeterminacy of frame with internal hinges

Theory of Structures II · Chapter 1: Introduction

- Asked in: 2068 Chaitra, 3 marks
- Video: [Internal Hinges | Structural Analysis | Part-5 | Civil engineering](https://www.youtube.com/watch?v=AQSfEYNoD3s) — Civil Engineering Mastery
- 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/degrees-indeterminacy-trapezoid-frame
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-1

## Question

Determine the static indeterminacy (external/internal) and kinematic indeterminacy for the structure as shown. [Figure: trapezoidal two-storey frame with two internal hinges (one on each of two levels), left base hinged, right base fixed.]

## Short answer

External static indeterminacy is 2, internal 1 (total 3). Kinematic indeterminacy is 15 considering axial deformation, or 9 if axial deformation is neglected.

## Answer

Reading of the figure: trapezoidal two-storey frame with two inclined columns on each side, a floor beam and a top beam forming one closed loop, one internal hinge at each of two levels, left base hinged, right base fixed.

$j = 6,\ m = 6,\ r = 2 + 3 = 5,\ c = 2$

#### Static indeterminacy
- External: $r - 3 = 2$
- Internal: $3 \times 1\ \text{loop} - 2 = 1$
- Total: $D_s = 3m + r - 3j - c = 18 + 5 - 18 - 2 = 3$

#### Kinematic indeterminacy
- Axial deformation considered: $D_k = 3j - r + c' = 18 - 5 + 2 = 15$
- Axial deformation neglected: $D_k = 15 - 6 = 9$

**Answer: external 2, internal 1 (total 3); $D_k = 15$ (9 if inextensible).**

## Check yourself

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

1. External Ds with r = 5 is:
   A. 5
   B. 3
   C. 2
   D. 0
   Answer: C. r - 3 = 2. (Key point: External part)

2. Internal Ds = 3 x 1 loop - 2 hinges =
   A. 1
   B. 3
   C. 2
   D. 5
   Answer: A. 3 - 2 = 1. (Key point: Internal part)

3. Dk with axial deformation (3j - r + c' = 18 - 5 + 2) is:
   A. 13
   B. 15
   C. 9
   D. 20
   Answer: B. 15; inextensible: 15 - 6 = 9. (Key point: Dk)

## Related in Theory of Structures II

- [External/internal static and kinematic indeterminacy of frame](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Adegrees-indeterminacy-frame-asar-2070) — Theory of Structures II, chapter 1
- [Static and kinematic indeterminacy of frame](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Adegrees-indeterminacy-frame-old-2070) — Theory of Structures II, chapter 1
- [Degrees of indeterminacy of frame with hinges](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Adegrees-indeterminacy-frame-hinges-1) — Theory of Structures II, chapter 1
- [Static and kinematic indeterminacy of frame](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Adegrees-indeterminacy-frame-bhadra) — Theory of Structures II, chapter 1
- [Indeterminacy and stability of frame and truss](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aindeterminacy-stability-frame-and-truss-baishakh) — Theory of Structures II, chapter 1
- [Indeterminacy and stability of truss and frame](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aindeterminacy-stability-truss-and-frame-bhadra) — Theory of Structures II, chapter 1