---
title: "Static and kinematic indeterminacy with examples"
subject: "Theory of Structures II"
chapter: 1
key: "semester_5/theory_of_structures_ii:indeterminacy-define-examples"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/indeterminacy-define-examples
site: BCT Engineering
---

# Static and kinematic indeterminacy with examples

Theory of Structures II · Chapter 1: Introduction

- Asked in: 2075 Chaitra, 6 marks; 2074 Chaitra, 6 marks; 2074 Asoj, 5 marks; 2070 Chaitra, 5 marks; 2070 Asar, 4 marks (asked 5 times)
- Video: [Degree of Static and Kinematic Indeterminacy - Structural Analysis 2](https://www.youtube.com/watch?v=8shs6UjnFcw) — Ekeeda
- 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/indeterminacy-define-examples
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-1

## Question

Define degree of static and kinematic indeterminacies with suitable expressions and give suitable examples (pin jointed, rigid jointed and hybrid structures) to explain the concept.

## Short answer

Static indeterminacy (Ds) is the number of extra unknown forces beyond equilibrium equations; kinematic indeterminacy (Dk) is the number of independent joint displacements. Plane truss: Ds=(m+r)-2j, Dk=2j-r; rigid frame: Ds=3m+r-3j-c, Dk=3j-r. Example: square truss with both diagonals (j=4,m=6,r=3) has Ds=1, Dk=5.

## Answer

**Static indeterminacy (DSI)** is the number of unknown forces (reactions and internal forces) in excess of the available equilibrium equations. It is the number of redundants that must be removed to make the structure statically determinate. **Kinematic indeterminacy (DKI)** is the number of independent joint displacements (translations and rotations) that must be known to fix the deformed shape of the structure; it is the number of unknowns in the displacement method.

#### Expressions
- Plane truss: $D_s = (m + r) - 2j$ and $D_k = 2j - r$
- Plane rigid frame / beam: $D_s = (3m + r) - (3j + c)$
- Frame, axial deformation allowed: $D_k = 3j - r + c'$
- Frame, members inextensible: $D_k = 3j - r - m + c'$

Here $m$ = members, $j$ = joints (including supports and free ends), $r$ = reaction components, $c$ = equations of condition from internal hinges (or hinged bar ends), $c'$ = extra independent rotations created by internal hinges.

#### Example 1: Pin-jointed truss
Square panel ABCD with both diagonals, hinge at A and roller at B.

$j = 4,\ m = 6,\ r = 3$

$D_s = 6 + 3 - 2(4) = 1$ (internal, one diagonal is redundant); $D_k = 2(4) - 3 = 5$.

#### Example 2: Rigid-jointed frame
Portal frame with both bases fixed: $j = 4,\ m = 3,\ r = 6$.

$D_s = 3(3) + 6 - 3(4) = 3$

$D_k = 3(4) - 6 = 6$ with axial deformation, and $6 - 3 = 3$ if axial deformation is neglected (rotations of the two top joints and one side sway).

```text
   B ________ C
   |          |
   |          |
  ///        ///
  (fixed)  (fixed)
```

#### Example 3: Hybrid (rigid joints plus pin-ended bar)
The same portal frame with a pin-ended diagonal brace BD: $j=4,\ m=4,\ r=6$, and the brace has two hinged ends ($c = 2$).

$D_s = 3(4) + 6 - 3(4) - 2 = 4$

$D_k = 12 - 6 = 6$ (extensible); with inextensible members $D_k = 6 - 4 = 2$, because the brace stops sway and only the rotations of B and C remain.

A hinge or a pin-ended bar lowers $D_s$, but a hinge raises $D_k$. So hinges make the force method easier (fewer redundants) and the displacement method harder (more unknown displacements).

## Check yourself

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

1. What does the degree of static indeterminacy represent?
   A. The number of joint displacements to be found
   B. The number of redundants to remove to make the structure statically determinate
   C. The number of members in the structure
   D. The number of reactions at the supports
   Answer: B. DSI is the excess of unknown forces over available equilibrium equations, i.e. the redundants that must be removed to get a determinate structure. (Key point: Meaning of DSI)

2. The degree of kinematic indeterminacy is best described as:
   A. Unknown forces beyond equilibrium equations
   B. Number of hinges in the frame
   C. Number of independent joint displacements needed to fix the deformed shape
   D. Number of closed loops
   Answer: C. DKI counts independent joint translations and rotations; they are the unknowns of the displacement method. (Key point: Meaning of DKI)

3. For a plane truss, which pair of expressions is correct?
   A. Ds = (m + r) - 2j and Dk = 2j - r
   B. Ds = 2j - r and Dk = (m + r) - 2j
   C. Ds = 3m + r - 3j and Dk = 3j - r
   D. Ds = m - 2j and Dk = j - r
   Answer: A. For a plane truss Ds = m + r - 2j and Dk = 2j - r. (Key point: Truss formulas)

4. A portal frame with both bases fixed (j = 4, m = 3, r = 6) has Ds equal to:
   A. 6
   B. 2
   C. 4
   D. 3
   Answer: D. Ds = 3m + r - 3j = 9 + 6 - 12 = 3. (Key point: Fixed portal frame)

5. What does a hinge (or pin-ended bar) do to Ds and Dk?
   A. Lowers Ds and raises Dk
   B. Raises Ds and lowers Dk
   C. Lowers both
   D. Raises both
   Answer: A. A hinge adds an equation of condition, lowering Ds, but creates an extra independent rotation, raising Dk. (Key point: Effect of hinges)

## 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-bhadra) — 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
- [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
- [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
- [Define kinematic indeterminacy](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Adefine-kinematic-indeterminacy) — Theory of Structures II, chapter 1
- [Indeterminacy: frame with hinge and truss](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Adegrees-indeterminacy-frame-and-truss) — Theory of Structures II, chapter 1

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