Contains questions on finding degree of static and kinematic indeterminacy

1. Structural Analysis
LOADS RESPONSE?
 Deflection
 Reaction
 Moment
 Rotation
 Displacement

2. Pin / Hinge Roller Fixed
Fx
Fy Fy
Fx
Fy
Mz
Supports
Reactions

3. Determinacy and Indeterminacy
P
L/2 L/2
A B
Ra
Hx
Rb
Ra = ?
Rb = ?
Hx = ?
Ra = P/2
Rb = P/2
Hx = 0
Equilibrium
equations
⅀ Fx = 0
⅀ Fy = 0
⅀ Mz = 0
P

4. P
L/2 L/2
A B
Ra
Hx
Rb
⅀Fx = 0 → Hx = 0
⅀Fy = 0 → Ra + Rb = P
⅀Ma = 0 → P(L/2) - Rb(L) = 0
P(L/2) = Rb(L)
Rb = P/2
Ra = P - Rb
P – P/2
Ra = P/2
3 equations, 3 unknowns → Statically Determinate beam
Unknowns can be solved using available equations
P

5. P
A B C
P
Ra
Hx
Rb Rc
 3 Equations, 4
Unknowns
 Statically
Indeterminate Beam
 Cannot solve for the
unknowns
 Reactions – Equations = 0 → Determinate
 Reactions – Equations ˃ 0 → Indeterminate
 Reactions – Equations < 0 → Unstable

6. Indeterminacy
Static Indeterminacy
(associated with force)
Kinematic Indeterminacy
(associated with displacement)
Degree of freedom (DOF)
Externally
Indeterminate
Internally
Indeterminate
Degree of Static Indeterminacy (DSI) = External + Internal

7. 1. Find DSI ?
No. of unknown reactions (r) =
No. of equilibrium equations (e) =
DSI = r-e =

8. 2. Find DSI ?
No. of unknown reactions (r) =
No. of equilibrium equations (e) =
DSI = r-e =

9. 3.
Find DSI ?
2 2
3 3
3 3
3 R – 3n
R = no. of unknown reactions
n = no. of members
Hinge
DSI= 10 – 3x3 = 1

10. c
4. Find DSI ?
Sliding support
2
3
2
3
3
DSI= 7-(3x2)= 1

11. 5.
c
Find DSI ?
DSI= 10 – (3x2) = 4
3 3
3
2 3
2

12. 6. Find DSI ?
c c
c

13. Frames
Open Cut Concept
1. Supports should be fixed
2. No. of supports = no. of open
structures
DSI = 3C-R (2D)
6C-R (3D)
C = no. of cuts
R = no. of reactions introduced
R=2 DSI= (3x2) – 2 = 4
7.

14. 8.
1
2 2
Find DSI ?
DSI= (3x3) – 5 = 4

15. 9.
Find DSI ?
11 1 1
DSI= (3x6) – 4 = 14

16. 10.
Find DSI ?
Hinge → R = m-1 2D
3(m-1) 3D
m = no. of members joining
Here, R= m-1
2-1=1
DSI = (3x1) – 1 = 2

17. 11.
Find DSI ?

18. 12.
c
c
3-1=2
3-1=2
Find DSI ?

19. 13.
c
c
3(m-1)
6C - R
Find DSI ?

20. Trusses
Trusses – all the joints are pin (hinge) joints
all members are tie members (take axial load only)
 m members will have m loads
 r support reaction
 Each joint (j) will have 2
equilibrium equations
DSI = (m+r) -2j
m

21. 14.
c
Find DSI ? Method 1→ c as joint
m = 8
r = 3
j = 5
DSI = m+r – 2j
8+3 – (2x5) = 1
Method 2→ c not as joint
m = 6
r = 3
j = 4
DSI = 6+3 – (2x4) = 1

22. 15. Find DSI ?
m =
r =
j =
DSI = m+r – (2xj)

23. Find DSI ?16.
c c
Gate 2008
DSI = 3C-R

24. Find DSI ?17.
Rigid Wall
Beams clamped
3D
Fx, Fy, Fz, Mx, My, Mz
DSI = Reactions – Equations
12 – 6 = 6

25. Find DSI ?18.
c
Gate 2014
45º

26. Find DSI ?19.
P
c
Internal hinge → gives 1 extra equation

27. Find Internal and External indeterminacy ?20.
Ext Indeterminacy = Rns – Eqns
(2+2) – 3 = 1
Total DSI = m+r – 2j
15+4 – (2x8) = 3
Int Indeterminacy = Total – Ext
3-1 = 2

28. Kinematic Indeterminacy
Degrees of freedom or independent coordinated required
to locate the structure in displaced configuration,
relative to the original configuration
DOF = 0 DOF = 3
DOF = 1 DOF = 2
DKI = 3j-r + additional DOF due to hinge
c
c
DOF = 4 linear
DOF = m-1 angular
Δx
Δy
θ1
θ2

29. 21. Find DKI ?

30. 22. Find DKI ?

31. 23. Find DKI ?
All members are inextensible
Inextensible members : Rigid members with
no axial deformation
DK’ = DKI – no. of members
DK’ = 3j-r- m
(3x4)-6 – 3 = 3

32. 24.
All members are inextensible
Find DKI ?

33. 25.
c
All members are inextensible
Find DKI ?

34. 26.
All members are inextensible
Find DSI, DKI ? Gate 2001

35. 27.
All members are inextensible
Find DKI ? Gate 2004

36. 28.
Axial deformation neglected
Find DKI ?

37. 29.
Truss
DKI = 2j-r (2D)
3j-r (3D)
Gate 2016