Indeterminacy and Stability of a Structure

By Mukul Yadav|Updated : September 6th, 2022

Indeterminacy and Stability of a Structure is a topic of Structural engineering which is the branch of civil engineering in which discussions about structures have been made. Indeterminacy and stability of structure depend on the type of support used in the structure. A structure can be classified as stable or unstable based on the determinacy of the system of its members. An unstable structure is also known as a mechanism and its determinacy will be less than zero.

Indeterminacy is the number of restraints offered by the structure in excess of the number of equilibrium equations for that particular structure. Equilibrium equations involve horizontal equilibrium, vertical equilibrium, and rotational stability. Indeterminacy and stability of a structure are key parameters for governing the strength of the whole members.

This article contains fundamental notes on the "Indeterminacy and Stability of a Structure" topic of the "Structural Analysis" subject.

Table of Content

What is Indeterminacy and Stability of a Structure

The indeterminacy of a structure is the number of redundant reactions present in the structure. and stability of structure implies that all members of the structure is stable against its movement in rotation as well as in translation.

Statically Determinate Structures

Statically determinate structures are structures that have zero degrees of indeterminacy. These structures have just a sufficient number of reactions equal to the number of equilibrium equations. In such type of structure, equilibrium conditions are sufficient to analyze the structure. Such a type of structure has no redundant reaction, so we can say that such a structure is just a stable structure, or we can say that by removing any of the reactions, the structure will convert into a mechanism.

Statically Indeterminate Structures

Statically indeterminate structures are structures that have more than zero degrees of indeterminacy. These structures have more support reactions to the number of equilibrium equations. In such type of structure, along with conditions of equilibrium, some compatibility equations are required to analyze the structure. Such a type of structure has some redundant reactions, so we can say that such a structure is stable.

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Difference Between Determinate and Indeterminate Structure

Indeterminacy and stability of a structure will depend upon whether the structure is determinate or indeterminate so, it will important to understand the difference between them. A statically determinate structure is stable and can be determined from equilibrium equations alone. However, a statically indeterminate structure has more unknown forces than available equilibrium equations.

Only equilibrium equations are required to analyze a determinate structure, and it is sufficient to find unknowns of the member. But compatibility conditions are also required for the analysis of indeterminate structures, along with the equilibrium equations for finding the unknowns of the structure. 

To define the indeterminacy and stability of a structure, it is required to know the terms which are used. These terms are described below in detail.

Stable/Unstable structure: A stable structure will not collapse when disturbed. Stability may also be defined as "The power to recover equilibrium." In general, there are many ways that a structure may become unstable, including the buckling of compression members and yielding/rupture of members; however, for linear structural analysis, the main concern is instability caused by insufficient reaction points or poor layout of structural members.

Internally Stable: An internally stable structure would maintain its shape if all the support reactions were removed. An internally unstable structure may remain stable with sufficient external support reactions. An example is shown below in Figure.

byjusexamprepExternal Determinacy: The ability to calculate all of the external reaction component forces by using only static equilibrium. A structure that satisfies this requirement is the externally statically determinate. Conversely, a structure for which the external reactions component forces cannot be calculated using only equilibrium is an externally statically indeterminate structure.

Internal Determinacy: The ability to calculate all of the external reaction component forces and internal forces by using only static equilibrium. A structure that satisfies this requirement is the internally statically determinate. A structure for which the internal forces cannot be calculated using only equilibrium is an internally statically indeterminate structure. Typically, if one talks about 'determinacy', internal determinacy is meant.

Redundant: Indeterminate structures effectively have more unknowns than can be solved using the three equilibrium equations (or six equilibrium equations in 3D). These extra unknowns are called redundant reactions.

Degree of Indeterminacy: The degree of indeterminacy is equal to the number of redundant. An indeterminate structure with 2 redundant may be said to be statically indeterminate to the second degree.

Static Indeterminacy

Static indeterminacy is one of the important terms which helps to define the both indeterminacy and stability of a structure. It can be related as, If a structure cannot be analyzed for external and internal reactions using static equilibrium conditions alone, then such a structure is called an indeterminate structure.

DS = DSe + DSi

Where,

DS = degree of static-indeterminacy

DSe = External static-indeterminacy

DSi = Internal static-indeterminacy

External static indeterminacy:

It is related to the structure's support system, and it is equal to the number of external reaction components in addition to the number of static equilibrium equations.

DSe = re - 3 For 2D

Because in 2D, there are 3 number of equilibrium equations only

DSe = re – 6 For 3D

Because in 3D, there are 6 equilibrium equations.

Where, re = total external reactions

Internal static indeterminacy:

It refers to the geometric stability of the structure. If, after knowing the external reactions, it is impossible to determine all internal forces/internal reactions using static equilibrium equations alone, then the structure is said to be indeterminate.

For geometric stability, a sufficient number of members is required to preserve the shape of a rigid body without excessive deformation.

DSi = 3C - rr …… For 2D

DSi = 6C - rr …… For 3D

where C = number of closed loops.

and

rr = released reaction

rr = ∑(mj - 1) …… For 2D

rr = 3∑(mj - 1) ……. For 3D

where mj = number of members connecting with J number of joints.

and J = number of the hybrid joint.

So,

Ds = m + r­e – 2j ….. For 2D truss

DSe = re - 3 & DSi = m – (2j – 3)

DS = m + re – 3j ….. For 3D truss

DSe = re – 6 & DSi = m – (3j - 6)

DS = 3m + re – 3j - rr ….. 2D Rigid frame

Ds = 6m + r­e – 6j - rr ….. 3D rigid frame

DS = (re – 6) + (6C – rr) ….. 3D rigid frame

Kinematic Indeterminacy

Kinematic indeterminacy is another important term that helps to define the both indeterminacy and stability of a structure. This term is associated with the degree of freedom of the structure. It can be described as if the number of unknown displacement components is greater than the number of compatibility equations; for these structures, additional equations based on equilibrium must be written to obtain a sufficient number of equations for the determination of all the unknown displacement components. The number of these additional equations necessary is known as the degree of kinematic indeterminacy or degree of freedom of the structure. Kinematic indeterminacy can find out with the help of the degree of freedom of the joints.

(i) Each joint of the plane pin jointed frame has 2 degrees of freedom.

(ii) Each joint of the space pin-jointed frame has 3 degrees of freedom.

(iii) Each joint of a plane rigid jointed frame has 3 degrees of freedom.

(iv) Each joint of space rigid jointed frame has 6 degrees of freedom.

The degree of kinematic indeterminacy is given by:

  1. Dk = 3j - re ………. For 2D Rigid frame when all members are axially extensible.
  2. Dk = 3j - re - m ………. For a 2D Rigid frame, if 'm' members are axially rigid/inextensible.
  3. Dk = 3(j + j’) - re – m + rr …… For 2D Rigid frame when J' = Number of Hybrid joints available.
  4. Dk = 6(j + j’) - re – m + rr ….. For 3D Rigid frame
  5. Dk = 2(j + j’) - re – m + rr ….. For 2D Pin jointed truss.
  6. Dk = 3(j + j’) - re – m + rr …… For 3D Pin jointed truss.

Examples of Static Indeterminacy

Here are some examples described in detail which will help to understand the degree of indeterminacy and stability of a structure.

Notations used in examples

ie is the degree of Indeterminacy

ec is the number of equations of condition,

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where n is the number of members connected to the hinge or roller.

1. Determination of the Number of Members and Joints

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2. Instability due to Parallel Reactions

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3. Instability due to Concurrent Reactions

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4. Instability due to an Internal Collapse Mechanism

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5. Mixed upbyjusexamprep

a) External Determinacy:

ie=r(3+ec)  
r=4,ec=1 (The hinge on the left at the pin does not provide any additional equations of condition).
Therefore,
ie=0.
But this structure is not a statically determinate structure. It is unstable because if we take a free-body diagram of the left side of the beam and take a sum of moments about the center hinge, the sum of moments will be non-zero due to the vertical reaction at the left pin (but we know that it has to be zero due to the existence of the pin).
Internal Determinacy:
ie=(3m+r)(3j+ec)
m=2,r=4,j=3,ec=1 (Again, the hinge on the left at the pin does not provide any additional equations of condition).
Therefore,
3m+r=103j+ec=10, and ie=0.
Hence, It is a determinate but unstable structure, as described earlier.

b) External Determinacy:

r=3,ec=0.
Therefore,
ie=0.

Then is this structure statically determinate? No, because the reactions are concurrent through the pin on the right.

Internal Determinacy:
m=2,r=3,j=3,ec=0.
Therefore,
3m+r=9 and 3j+ec=9,
so the structure appears internally determinate, but it is still unstable due to the concurrent reactions.

c) External Determinacy:

r=3,ec=0.

Therefore,

ie=0.
Since there are no sources of instability, this structure is externally statically determinate.
Internal Determinacy:
m=6,r=3,j=6,ec=0.
Therefore,
3m+r=21 and 3j+ec=18,
so this structure is internally statically indeterminate to three degrees.

d) External Determinacy:

r=5,ec=2.

Therefore,

ie=0.
Since there are no sources of instability, this structure is externally statically determinate.
Internal Determinacy:
m=5,r=5,j=6,ec=2.
Therefore,
3m+r=20 and 3j+ec=20,
so this structure is internally statically determinate

e) External Determinacy:

r=7,ec=2. (Due to the three members connected to the internal hinge)
Therefore,
ie=2.
This structure can be described as 2 degrees externally statically indeterminate.
Internal Determinacy:
m=3,r=7,j=4,ec=2.
Solving,
3m+r=16 and 3j+ec=14,

Again, this structure is found to be 2 degrees internally statically indeterminate.

f) External Determinacy:

r=4,ec=2.
Therefore,
ie=1.

Due to the structure's design, the internal roller cannot be supported, and the structure is classified as unstable.

Internal Determinacy:
m=2,r=4,j=3,ec=2.
Solving,
3m+r=10 and 3j+ec=11,

We can safely say that this structure is unstable, both by the equations of determinacy and by understanding how the structure will bend under loading.

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Indeterminacy and Stability of a Structure FAQs

  • Stability of a structure means that structure is stable against its rotational motion as well as its translation motion

  • Determinacy simply means that number of extra unknowns in the structure in excess to the number of equilibrium equations

  • Indeterminacy of structure means the number of redundant forces present in the structure. Redundant forces makes the structure more stable than the determinate one.

  • When we provide more support and/or member to structure than required for static stability, it makes structure indeterminate. The excess member increases rigidity and stability of the structure.

  • The displacement method is used to solve statically indeterminate structures. It is used when unknowns are rotations and displacement in the structure.

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