STRUCTURE ANALYSIS: STATICALLY DETERMINATE AND INDETERMINATE STRUCTURES

SIGNIFICANCE OF STRUCTURE ANALYSIS 

Structures have always been admired for their beauty of architecture and their aesthetic structural design. They possess a trait—almost indefinable—that embodies design ingenuity, connection to place, and, above all, imagination. People have been analysing these structures and making us aware of the stunning architectural facts. 

From ancient age to the modern era, we have seen enormous examples such as Taj Mahal, Burj Khalifa, Gardens of the Bay, Eiffel Tower and many more, who steal our hearts at every glance. Every structure is designed from a different concept. In the modern era, high raised buildings with irregular shapes are in trend. These fascinating constructional pieces demand colossal effort to make its structural design effectively. Hence, the importance of structural analysis and design cannot be denied. For designing of any structure, their foundation support and stability is the most crucial factor to consider. Hence, to understand the concept and application of static determinacy and indeterminacy is essential. 

APPLICATIONS OF STATIC INDETERMINANCY AND DETERMINANCY: 

1. Before designing any structure, you can know the structure is stable or not.

2. The degree of indeterminacy of structure can be determined by making an unstable structure to stable structure to change the support or a member.

3. This will help you to prevent the settlement of foundation support.

4. You can analyse the structure behaviour at different conditions and how to make them stable through knowing these behaviours.

So, let’s understand the method for calculating stability which has been introduced in structural engineering for analysis purpose.

STATICALLY DETERMINATE STRUCTURES: The flexural rigidity and cross-sectional area of the member does not depend upon the bending moment and shear force. Because of temperature disparity and lack of fit or differential settlement (i.e. settlement of foundation support), there is no stress.

Or

If total no of unknown reactions is equal to the total no of equilibrium equation in structure, then the structure is called a static determinate structure. Equilibrium conditions are adequate for analysis of structure.

STATICALLY INDETERMINATE STRUCTURE: The flexural rigidity and cross-sectional area of the member depends upon the bending moment and shear force. Stresses are generated due to temperature variation and lack of fit or differential settlement (i.e. settlement of foundation support).

Or

If total no of unknown reactions is more than total no of equilibrium equation in structure, then the structure is called a static indeterminate structure.

STATIC INDETERMINANCY (REDUNDANT STRUCTURE): If equilibrium conditions are not enough to analyse the internal and external reactions, then a structure is called indeterminate structure.

D s = External static indeterminacy + Internal static indeterminacy

where, 

Ds = degree of static indeterminacy

REACTIONS OF DIFFERENT SUPPORTS:

  1. Fixed support – 3

2. Hinge support – 2

3. Roller support – 1

EXTERNAL REDUNDANCY OR EXTERNAL STATIC INDETERMINANCY (E):  It depends on the unknown support reactions and no equilibrium equations with the new equation for hinge or pin support.

Equilibrium equations for 2D structure:

∑ Fx = 0

∑ Fy = 0

∑ Mxy = 0

Equilibrium equations for 3D structure:

 ∑ Fx = 0 ∑ Fz = 0

∑ Fy = 0 ∑ Myz = 0

∑ Mxy = 0 ∑ Mzx = 0

E = R – re + extra equation for pin or link

where, 

E = External static indeterminacy

R =total no of reactions

re = no of equilibrium equations 

FORMULA FOR EXTERNAL INDETERMINANCY: 

For 2D frame and truss: Dse = R – 3 

For 3D frame and truss: Dse = R – 6

INTERNAL REDUNDANCY OR INTERNAL STATIC INDETERMINACY (I): It depends on the internal reactions like shear force, bending moment and axial force.

For 2D structure:

Shear force = Fy

Axial force = Fx

Bending moment = Mxy

For 3D structure:

Shear force = FZ, Fy

Axial force = Fx

Bending moment = Mxy, Myz, Mzx

2D STRUCTURE AND FRAME: 

For continuous beam

I = 0

Ds = R – re

Example-

Continuous beam

Here,
R =9, re = 4
EXTERNAL REDUNDANCY
E = 9-(3+1)
E = 5
Externally static indeterminate beam

INTERNAL REDUNDANCY
I = 0
Internally static determinate beam
Then Ds = 0 + 5 = 5
Therefore, this is a statically indeterminate beam to 5 degree

RIGID FRAME:

I = 3a

Ds = (3m+R) – (3j+α)

If we will not consider axial force in member than

Ds = (2m+R) – (2j+α)

α = ∑ (m’ – 1)

Where,

a = no. of completely closed loops area of the frame

α = no of the equation for hinge or link

m’ = no. of connecting members to hinge or link

Example-

Rigid frame

Here, R = 9, re = 3, a = 0


EXTERNAL REDUNDANCY
E = 9 – 3
E = 6
Externally indeterminate frame

INTERNAL REDUNDANCY
I = 3a
There is no closed loop
Therefore a = 0
I = 3 x 0 = 0
Internally determinate frame
Ds = 6 + 0 = 6
This frame is statically indeterminate and the degree of indeterminacy of frame is 6.

Example-

Rigid frame

Here, R = 6, re = 3, a = 3
EXTERNAL REDUNDANCY
E = R – re
E= 6 – 3
E = 3
Externally indeterminate frame

INTERNAL REDUNDANCY
I = 3a
I = 3×3 = 6
Internally indeterminate frame
Then,
Ds = 6 + 3 =9
This frame is statically indeterminate, and the degree of indeterminacy of the frame is 9.
Or
You can also find static indeterminacy from this formula:
Ds = (3m + R) – (3j + α)

α = ∑ (m’ – 1)

where,
m = no. of members
j = no. of joints
R = no. of reactions
a = no. of completely closed loops area of the frame
α = no of the equation for hinge or link
m’ = no. of connecting members to hinge or link

For checking static determinacy, indeterminacy and stability of truss
if 3m + R = 3j + α then, frame is stable
if 3m + R > 3j + α then, frame is indeterminate
if 3m + R < 3j + α then, frame is deficient frame or unstable

Example-

For hybrid frame structure

2D frame structure with internal hinge

Here, m =12, R = 6, j =10
Static indeterminacy
Ds = (3m + R) – (3j + α)
Ds = (3 x 12 + 6) – (3 x 10 + α)
α = ∑ (m’ – 1)
α = (4 – 1) = 3
Ds = (36 + 6) – (30 + 3)
Ds = 42 – 33
Ds = 9
This frame is statically indeterminate and the degree of indeterminacy of the frame is 9.

PIN JOINTED TRUSS:
I = (m + re)-2j
Ds = (m + R)-2j

where,
m = no. of members
re = no. of equilibrium equations
R = no. of reactions
j = no. of joints

Example-

Pin jointed truss

Here,

R = 3, re = 3, m = 13, j = 8

EXTERNAL REDUNDANCY

E = 3 – 3

E = 0

Externally determinate truss

INTERNAL REDUNDANCY

I = (13+3) – (2 x 8)

I = 16 – 16 

I = 0

Internally determinate truss

Ds = 0 + 0 = 0

This truss is statically determinate 

For checking static determinacy, indeterminacy and stability of truss

if m + R = 2j then, truss is stable

if m + R > 2j then, truss is indeterminate

if m + R < 2j then, truss is deficient truss or unstable

FOR 3D STRUCTURE AND FRAME:

3D truss,

Ds = (m + R) – 3j

3D frame structure,

Ds = (6m+R) – (6j+α)

α = ∑ (m’ – 1) 

STABILITY CHECK FOR STRUCTURE:

If static indeterminacy of structure is negative, it signifies that the structure is unstable, and even though the degree of indeterminacy is zero or positive, the stability of the structure is not confirmed. Therefore, we need to examine or verify some cases- 

1. If reactions are parallel so that structure will be unstable.

2. If reactions are concurrent.

Note: sometimes structures found are the statically determinate structure, but they are not stable.

Example-

Here, R = 2, re = 2, a = 2 (two closed loop)
EXTERNAL REDUNDANCY
E = R- re
E= 2 -2 = 0
Externally determinate

INTERNAL REDUNDANCY
I = 3 x 2 =6
Internally indeterminate
Ds = 0 + 6 = 6
This frame is statically indeterminate to 6 degree but structure is unstable because all reactions are parallel.

Example-

Here, R = 3, re = 3, m = 8, j = 6

EXTERNAL REDUNDANCY 

E = 3 – 3

E = 0

Externally determinate truss 

INTERNAL DETERMINACY

I = (8+3) – 6×2

I = 11 – 12

I = -1

Internally indeterminate truss

This truss is statically indeterminate and unstable due to negative indeterminacy. 

Example-

Here, R = 3, re = 3

EXTERNAL REDUNDANCY
E = 3-3 = 0
Externally static determinate

INTERNAL REDUNDANCY
I = 0
Also, internally determinate
Then, Ds = 0

Therefore, the structure is statically determinate but unstable because of all reactions are concurrent means they are meeting at the same point.

Hope this will help you in preparation and study of structure analysis If you have any doubts in solving problems of structure analysis stability and determinacy then feel free to write to us! Eye on structures will try to bring you concepts and real-life structure issues. So get ready to build up your strong structure knowledge with us.

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