Test: Kinematic Indeterminacy - Civil Engineering (CE) MCQ

Concept: 
kinematic indeterminacy for Rigid jointed plane 2D framed, if axial deformation is neglected 

D_k = 3j - R
where j = Number of joints
R = Number of unknown support reactions
where j = 4 , R = 6 , m = 3
D_k = 3×4-6 = 6
If axial deformation is neglected,
Number of inextensible members (For which axial deformation is neglected) = 3
kinematic Indeterminacy = Dk - no of in extensible members = 6 - 3 = 3
Therefore kinematic indeterminacy of a single-bay, single-storeyed portal has its column ends fixed and if axial deformation is neglected is 3.

Concept:
The degree of kinematic indeterminacy for the building frame is
D_K = 3J – R_e – m
Where,
m = number of the axially rigid member
R_e = Support Reaction
J = Number of Joints

Calculation:
Given, 
m = 2, j = 3, R_e = 2

D_K = 3J – R_e – m = 3 × 3 – 4 – 2
D_K = 3

Concepts:
The kinematic indeterminacy is given as
D_k­ = 3j-r-m
j = no of joints
r = no of support reactions
m = no of axially rigid members 

Calculation:
No of Joints, j = 3
No of support reactions, r = 5 (3 at support A, 1 at each support B and C)
No of axially rigid members, m = 0 (assumed as not given in Question )
D_k­ = 3j-r-m
D_k = 3 × 3 – 5 – 0
∴ D_k = 4

Concept:
Degree of Kinematic Indeterminacy:
Degree of kinematic indeterminacy (D_k) refers to the total number of independent available degree of freedom of all joints. The degree of kinematic indeterminacy may be defined as the total number of unrestrained displacement component of all joints.
Degree of kinematic indeterminacy for rigid jointed plane frame and beam is given by,
D_k = 3j - r_e
Where, j = Number of joint, r_e = Number of external recation

Calculation:
Degree of kinematic indeterminacy (D_k) = 3j - r_e
Here, j = 3 and r_e = 4
D_k = 3 × 3 - 4 = 5

Concept:
Static Indeterminacy: If the equilibrium equations are enough to analyze for unknown reactions, the structure is said to be statically indeterminate.
D_s = D_se + D_si
D_se = r – S
D_si = 3 × Number of closed loop (For portal frame)
D_si = m- 2j+ 3 (For truss structure)
Kinematic Indeterminacy: It is the total number of possible degree of freedom of all the joints.
D_k = 3J - r (For portal frame)
D_k = 2J-r (For truss structure)
where, D_se = External Indeterminacy, D_si = Internal Indeterminacy, D_k = Kinematic Indeterminacy, r = No. of unknown reactions, S = No. of equilibrium equation, m = No. of members & J = No. of joints

Given,
J = 9, r = 7, m = 10(inextensible)
∴ D_k = 3j - r = 27 - 7 = 20
For members are extensible, D_k = 20 - 10 = 10

Concept:
If the number of unknown displacement components are greater than the number of compatibility equations, for those structures additional equations based on equilibrium must be written in order to obtain sufficient number of equations for the determination of all the unknown displacement components. The number of these additional equations necessary is known as degree of kinematic indeterminacy or degree of freedom of the structure.

Where
D_k = Kinematic Indeterminacy
J = number of joints
J’ = Number of hybrid joints
r_e = Number of external reactions
r_r = Number of released reactions

Concept:
Static Indeterminacy: If the equilibrium equations are enough to analyze for unknown reactions, the structure is said to be statically indeterminate.
D_s = D_se + D_si
D_se = r – S
D_si = 3 × Number of closed loop (For portal frame)
D_si = m- 2j+ 3 (For truss structure)
Kinematic Indeterminacy: It is the total number of possible degree of freedom of all the joints.
D_k = 3J - r (For portal frame)
D_k = 2J-r (For truss structure)
where, D_se = External Indeterminacy, D_si = Internal Indeterminacy, D_k = Kinematic Indeterminacy, r = No. of unknown reactions, S = No. of equilibrium equation, m = No. of mombers & J = No. of joints
∴ D_k = 3j - r = 12 - 6 = 6
For axial deformation is neglected, Dk = 6 - 3 = 3

Concept:
Static Indeterminacy: If the equilibrium equations are enough to analyze for unknown reactions, the structure is said to be statically indeterminate.
D_s = D_se + D_si
D_se = r – S
D_si = 3 × Number of closed loop (For portal frame)
D_si = m- 2j+ 3 (For truss structure)
Kinematic Indeterminacy: It is the total number of possible degree of freedom of all the joints.
D_k = 3J - r (For beam & portal frame)
D_k = 2J - r (For truss structure)
where, D_se = External Indeterminacy, D_si = Internal Indeterminacy, D_k = Kinematic Indeterminacy, r = No. of unknown reactions, S = No. of equilibrium equation, m = No. of mombers & J = No. of joints
D_k = 3J - r + H = 3 × 4 – 5 + 1 = 8
D_k = 8 - 3 = 5

Kinematic Indeterminacy (D_K): 

• Also known as the degree of freedom (DOF)
• It refers to the number of independent components of joint displacement with the specified set of axes.
• If the plane frame is extensible, the degree of kinematic indeterminacy = (3j - r)
• If plane frame is inextensible & unbraced, Degree of kinematic indeterminacy = 3j - (r + m)
• Here j = no. of Joints, r = No. of Unknown Equations, m = no. of Plastic hinges equation (m'-1), m' = No. of member meeting at hinges 

Calculation:
As per the above diagram:
No. of joints (j) = 7
No. of unknown Equation (r) = 3(Fixed end) +1 (Roller Support ) +2 (Hinged Support ) +1 (Spring as prop) = 7
No. of member meeting at hinges = 5
m = 5 - 1 = 4
Putting in formula D_k = 3j - (r + m)
D_k = 3 × 7 - (7 + 4)
D_k = 21 - 11
D_k = 10
So the D_k of the following beam is 10.

Concept:
If the number of unknown reaction are more than the number of equilibrium equation available then structure is called indeterminate structure and degree of indeterminacy is given by static indeterminacy.
Some important cases are:

• Pin jointed plane frame, D_s = m + r - 2j
• Pin joined space frame, D_s = m + r - 3j
• Rigid jointed plane frame, D_s = 3m + r - 3j
• Rigid jointed space frame, D_s = 6m + r - 6j

Where,
m = number of members
r = number of external reactions
j = number of joints.
Data Given by the figure,
m = 7
r = 6
J = 5
We know that,
For pin jointed plane frame: Ds = m + r - 2J
D_s = 7 + 6 - 2 x 5
D_s = 3