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

# Indeterminacy: frame with hinge and truss

Theory of Structures II · Chapter 1: Introduction

- Asked in: 2071 Shrawan, 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-and-truss
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-1

## Question

Obtain the degree of static and kinematic indeterminacies for the given structures. [Figure: (i) a two-bay, two-storey frame with an internal hinge, supports hinged/fixed/roller as drawn; (ii) a multi-panel truss with double diagonals, hinged at one end and roller at the other.]

## Short answer

(i) Frame: static indeterminacy = 8, kinematic = 22 (12 if members inextensible). (ii) Truss: static indeterminacy = 4, kinematic = 17.

## Answer

#### (i) Two-bay, two-storey frame with an internal hinge
Reading of the figure: three column lines over two storeys with floor beams at two levels. Supports: one fixed, one hinged, one roller; one internal hinge.

$j = 9,\ m = 10\ (6\ \text{columns} + 4\ \text{beams}),\ r = 3 + 2 + 1 = 6,\ c = 1$

- Static: $D_s = 3(10) + 6 - 3(9) - 1 = 8$ (external $6-3 = 3$; internal $3 \times 2\ \text{closed loops} - 1 = 5$).
- Kinematic, axial deformation allowed: $D_k = 3(9) - 6 + 1 = 22$.
- Kinematic, inextensible members: $D_k = 22 - 10 = 12$.

#### (ii) Truss with double diagonals
Reading of the figure: four panels with top and bottom chords, five verticals and two diagonals in each panel (X-bracing). Hinge at one end, roller at the other.

$j = 10,\ m = 4 + 4 + 5 + 8 = 21,\ r = 3$

- Static: $D_s = m + r - 2j = 21 + 3 - 20 = 4$ (external 0, internal 4, one for each X-panel).
- Kinematic: $D_k = 2j - r = 20 - 3 = 17$.

**Answer: (i) $D_s = 8$, $D_k = 22$ (12 if inextensible); (ii) $D_s = 4$, $D_k = 17$.**

## Check yourself

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

1. For the two-storey frame (m=10, r=6, j=9, c=1), Ds is:
   A. 9
   B. 8
   C. 7
   D. 6
   Answer: B. 30 + 6 - 27 - 1 = 8. (Key point: Frame Ds)

2. Internal static indeterminacy of the frame is found as:
   A. 3 x loops - c = 6 - 1 = 5
   B. r - 3
   C. 2j - r
   D. m - 2j
   Answer: A. Each closed loop gives 3 redundants and each hinge reduces by one; with 2 loops, 6 - 1 = 5. (Key point: Internal indeterminacy of frame)

3. For the X-braced truss (m=21, r=3, j=10), Ds = m + r - 2j equals:
   A. 0
   B. 4
   C. 3
   D. 24
   Answer: B. 21 + 3 - 20 = 4, one per X-panel. (Key point: Truss Ds)

4. Dk for this truss is:
   A. 20
   B. 3
   C. 17
   D. 23
   Answer: C. Dk = 2j - r = 20 - 3 = 17. (Key point: Truss Dk)

## Related in Theory of Structures II

- [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
- [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
- [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
- [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
- [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
- [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

## Related in other subjects

- [Frame: roller displacement and hinge rotation](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aframe-roller-hinge-rotation-2071b) — Theory of Structures I, chapter 3
- [AFD, SFD, BMD for frame with internal hinge](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Aframe-afd-sfd-internal-hinge-5kn-per-m) — Applied Mechanics, chapter 6
- [Truss indeterminacy; forces in BC, BG, HG, GD](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Atruss-indeterminacy-bc-bg-hg-gd) — Applied Mechanics, chapter 7
- [Static determinacy and stability of a truss](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Astatic-determinacy-stability-truss) — Strength of Materials, chapter 1