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

# Degrees of indeterminacy of frame with hinges

Theory of Structures II · Chapter 1: Introduction

- Asked in: 2078 Kartik, 3+2 marks
- Video: [Static Determinacy, Indeterminacy and Stability of a Plane Frame - Solved Examples](https://www.youtube.com/watch?v=gcPP8DCa2sw) — AF Math & Engineering
- 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-frame-hinges-1
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-1

## Question

Determine the external and internal degree of static indeterminacy and also the kinematic indeterminacy of the structure given below. [Figure: two-bay frame with internal hinges, supports as drawn (two hinged/roller bases and two fixed bases).]

## Short answer

External static indeterminacy = 5, internal = 0, total = 5. Kinematic indeterminacy = 16, or 9 if members are considered axially inextensible.

## Answer

The figure is read as a single-storey frame on four columns (two hinged bases, two fixed bases) joined by a continuous girder having two internal hinges: $j = 8,\ m = 7,\ r = 2(2) + 2(3) = 10,\ c = 2$.

#### Static indeterminacy
- External: $D_{se} = r - 3 - c = 10 - 3 - 2 = 5$ (the two hinges give two extra equations of condition for the reactions).
- Internal: the frame has no closed loop, so $D_{si} = 0$.
- Total: $D_s = 3m + r - 3j - c = 21 + 10 - 24 - 2 = 5$.

#### Kinematic indeterminacy
Each internal hinge adds one independent rotation ($c' = 2$).

- With axial deformation: $D_k = 3j - r + c' = 24 - 10 + 2 = 16$
- Neglecting axial deformation: $D_k = 16 - m = 16 - 7 = 9$

**Answer: external $D_{se} = 5$, internal $D_{si} = 0$ (total 5); kinematic $D_k = 16$ (9 if members are inextensible).**

## Check yourself

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

1. For this frame, what is the reaction count r with two hinged/roller bases and two fixed bases?
   A. 10
   B. 8
   C. 12
   D. 9
   Answer: A. r = 2(2) + 2(3) = 10. (Key point: Data of the frame)

2. External Dse = r - 3 - c with r = 10 and c = 2 gives:
   A. 7
   B. 3
   C. 5
   D. 8
   Answer: C. 10 - 3 - 2 = 5; the two hinges give two extra equations. (Key point: External indeterminacy)

3. Why is the internal static indeterminacy zero?
   A. Hinges remove internal redundants
   B. The frame has no closed loop
   C. The supports are fixed
   D. Members are inextensible
   Answer: B. Internal indeterminacy comes from closed loops; with none, Dsi = 0. (Key point: Internal indeterminacy)

4. Using Dk = 3j - r + c' with j = 8, r = 10, c' = 2, Dk is:
   A. 12
   B. 16
   C. 14
   D. 9
   Answer: B. 24 - 10 + 2 = 16; subtracting m = 7 for inextensible members gives 9. (Key point: Kinematic indeterminacy)

## 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
- [Indeterminacy of frame with internal hinges](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Adegrees-indeterminacy-trapezoid-frame) — 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
- [Explain indeterminacies; degrees of frame](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aexplain-indeterminacies-degrees-two-bay-frame) — Theory of Structures II, chapter 1