9/3/2018 Determinacy, Indeterminacy and Stability - EngineeringWiki

Determinacy, Indeterminacy and Stability

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Contents
1 Introduction
2 Important Terms
3 External Determinacy
3.1 Reaction Components
3.2 Equations of Condition
4 Internal Determinacy
4.1 Members and Joints
5 Stability
6 Example Problem
7 References

Introduction
Before beginning to analyse a structure, it is important to know what kind of structure it is. Different
types of structures may need to be analysed using different methods. For example, structures that are
determinate may be completely analysed using only static equilibrium, whereas indeterminate
structures require the use of both static equilibrium and compatibility relationships to find the internal
forces. In addition, real structures must be stable. This means that the structure can recover static
equilibrium after a disturbance. There is no point analyzing a structure that is not stable.

This section will explain the concepts of determinacy, indeterminacy and stability and show how to
identify determinate, indeterminate and stable structures.

Important Terms
Stable/Unstable
A stable structure is one that will not collapse when disturbed. Stability may also be defined as
"The power to recover equilibrium."[1] In general, there are may ways that a structure may
become unstable, including buckling of compression members, yielding/rupture of members, or
nonlinear geometric effects like P-Delta; however, for linear structural analysis, the main concern
is instability caused by insufficient reaction points or poor layout of structural members.
Internally Stable[2]
In internally stable structure is one that would maintain its shape if all the reactions supports were
removed. A structure that is internally unstable may still be stable if it has sufficient external
support reactions. An example is shown in Figure 1.
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External Determinacy
The ability to
calculate all of the
external reaction
component forces
using only static
equilibrium.[2] A
structure that Figure 1: Internal Stability
satisfies this
requirement is
externally statically determinate. A structure for which the external reactions component forces
cannot be calculated using only equilibrium is externally statically indeterminate.

Internal Determinacy
The ability to calculate all of the external reaction component forces and internal forces using
only static equilibrium.[2] A structure that satisfies this requirement is internally statically
determinate. A structure for which the internal forces cannot be calculated using only equilibrium
is internally statically indeterminate. Typically if one talks about 'determinacy', it is internal
determinacy that is meant.
Redundant
Indeterminate structures effectively have more unknowns than can be solved using the three
equilibrium equations (or six equilibrium equations in 3D). The extra unknowns are called
redundants.[2]
Degree of Indeterminacy
The degree of indeterminacy is equal to the number of redundants.[2] An indeterminate structure
with 2 redundants may be said to be statically indeterminate to the second degree or "2º S.I."

External Determinacy
If a structure is externally determinate, then all of the reactions may be calculated using equilibrium
alone. To calculate external determinacy, the following equations are used:[2]

Double subscripts: use braces to clarify Statically unstable externally ~Eq. 1

r = 3 + ec Statically determinate externally ~Eq. 2
r > 3 + ec Statically indeterminate externally ~Eq. 3

where r is the number of reaction components, and ec is the number of equations of condition.
Both of these are described in detail below.

The degree of indeterminacy is given by the following equation:[2]

ie = r − (3 + ec ) ~Eq. 4

Reaction Components
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In the equations above, r is equal to the total number of reaction components as follows:

Support Type Image Reactions r

Roller r = 1

Pin r = 2

Fixed r = 3

For multiple reaction points, r is the sum of all the components for all the reaction points in the
structure.

Equations of Condition

Additionally, ec is the "number of equations of

condition." These are release conditions within
the structure that provide extra equilibrium
equations beyond the three for global
equilibrium.[3]

For example, if an internal hinge is added to the

structure, as shown in Figure 2, then there is one
equation of condition. If there was no internal
hinge in this example, then the structure would
be indeterminate and it would not be possible to
find the reaction forces or the internal forces
(since is has four reaction components). The
addition of the hinge provides an additional
equilibrium condition which forces the internal
moment to be equal to 0 at point B (
∑ MB = 0 ). This may be seen if the
Figure 2: Structure with an Internal Hinge
structure is split into two free body diagrams as
shown in the lower part of Figure 2. At point B,
there are three internal force components that exist in equal and opposite action/reaction pairs on either
side of point B:

1. Axial force - BAB

x
and BBC
x

2. Shear force - BAB

y
and BBC
y

3. Moment - MBAB and MBBC

So, MBAB
BC
= M
B
= 0 , because they are action reaction pairs.
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Therefore, only one extra equilibrium equation is possible due to the introduction of the hinge: either
AB BC
∑ MB = 0 or ∑ M = 0 but not both because the two equations are not independent. So, for
B

each internal hinge in a structure, there is a single equation of condition: ec = 1 .

For a structure with an internal roller, such as that shown in

Figure 3, both the force transfer in the direction of the roller
and the moment are equal to zero at the location of the roller.
This provides two extra equilibrium equations, and therefore
two equations of condition. For the structure shown in Figure
3, the extra equations are: Figure 3: Structure with an Internal
Roller
AB BC
∑ MB = 0 ( or ∑ MB = 0 )

and
AB BC
∑ Bx = 0 ( or ∑ Bx = 0 )

So, for each internal roller, there are two equations of condition: ec = 2 .

If there are additional members that frame into a single internal hinge, then there is an additional
equation of condition for each additional member.[2] For example, for three members connected at a
hinge, then there are two extra independent equilibrium equations that are added to the system (because
in an equal and opposite action/reaction pair, there can only be two sides). So, for a hinge connection
with multiple elements, ec = n − 1 where n is equal to the number of members connected to the
hinge.[2]

Similarly, for a roller connection with multiple members, each additional member adds two equations
of condition, so ec = 2 ∗ (n − 1) .[2]

In summary:

Internal Equations
Support Type of Condition
Hinge ec = n − 1 ~Eq. 5
Roller ec = 2 ∗ (n − 1) ~Eq. 6

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

WARNING: This method of determining external determinacy is not valid for indeterminate structures
which contain closed loops.

Internal Determinacy

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If a structure is internally determinate, then all of the reactions and internal forces may be calculated
using equilibrium alone. Internal determinacy is generally much more important than external
determinacy in structural analysis. To calculate internal determinacy, the following equations are used:
[2]

3m + r < 3j + ec Statically unstable ~Eq. 7

3m + r = 3j + ec Statically determinate ~Eq. 8
3m + r > 3j + ec Statically indeterminate ~Eq. 9

where m is the total number of members in the structure, r is the number of reaction components,
j is the total number of joints in the structure, and ec is the number of equations of condition. The
meaning of r and ec are the same as for #External Determinacy above. The definition of members
and joints will be discussed below.

The degree of indeterminacy is given by the following equation:[2]

ie = 3m + r − (3j + ec ) ~Eq. 10

Members and Joints

There is no specific
way that a structure
must be split into
members and joints for
the purposes of the
determinacy analysis.
Any division of the
structure is okay as
long as the members
and joints are
consistent with each
other; however, joints
should be placed at
least at the following
locations:

1. Free ends Figure 4: Determination of the Number of Members and Joints

2. Reactions
3. Intersections of three or more elements

For an example of how to calculate the numbers of members and joints, see Figure 4.

Stability

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An unstable structure generally cannot be analysed. Therefore, it is useful to know if a structure is

stable or unstable before a structural analysis is conducted. There are four main ways that a structure
may be geometrically unstable. These apply only to linear geometric stability and not to instability
caused by buckling, member yielding or nonlinear geometry.

1. There are not enough

reactions[3]: This will
generally be clear from an
application of the determinacy
equations (Eq. 7-9).
2. The reactions are parallel[4]:
All of the reaction components
point in the same direction. An Figure 5: Instability due to Parallel Reactions
example of such a situation is
shown in Figure 5. In this
example, the horizontal
equilibrium ∑ Fx = 0
cannot be solved and there will
be a net horizontal force on the
system with no resistance.
3. The reactions are
concurrent[4]: All of the
reaction components meet at a
point. An example of such a
situation is shown in Figure 6.
Effectively, the system is free
to rotate as a rigid body around Figure 6: Instability due to Concurrent Reactions
the point that the reaction
components meet at.
4. There is an internal collapse
mechanism[3]: This is any
situation in which there is an
internal mechanism in the
system that will cause it to
deform between the supports.
In some such situations, this
will be clear from the use of
the determinacy equations, but
in others, it may not. In all Figure 7: Instability due to an Internal Collapse Mechanism
such cases, though, the
instability will become clear during the structural analysis because it will be impossible to solve
for all of the internal forces. An example internal collapse mechanism is shown in Figure 7.

Example Problem

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Determine whether the following structures shown in Figure 8 are externally determinate, internally
determinate, externally indeterminate, internally indeterminate or unstable. If a structure is
indeterminacy, determine how many degrees of indeterminacy it has.

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 .

Then, is this structure statically determinate? No, 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 = 10 , 3j + ec = 10 , and ie = 0 .

Then, is this structure statically determinate? No, it is unstable due to the same reason above.

b) External Determinacy:

r = 3, ec = 0 .

Therefore,

ie = 0 .

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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 (or "3º S.I.").

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,

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3m + r = 20 and 3j + ec = 20 ,

so this structure is internally statically determinate (or "S.D.").

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 design of the structure, 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.

However, if the right hand pin were a fixed-end support this case would be considered a
stable, statically determinate structure.

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References
1. http://www.colorado.edu/engineering/CAS/courses.d/Structures.d/IAST.Lect23.d/IAST.Lect23.pdf
2. Kassimali, A. (2011). Structural Analysis: SI Edition (4th ed.). Stamford, CT: Cengage Learning.
3. Leet, K.M., Uang, C-M., Gilbert, A.M. (2011). Fundamentals of Structural Analysis (4th ed.).
New York, NY: McGraw Hill.
4. Hibbeler, R.C. (2012). Structural Analysis (8th ed.). Upper Saddle River, NJ: Pearson Prentice
Hall.

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