This document provides information on structural analysis including loads and response, determinacy and indeterminacy, and calculating the degree of static and kinematic indeterminacy for beams, frames, and trusses. It defines key terms like deflection, reaction, moment, and displacement. It explains how to determine if a structure is statically determinate or indeterminate by comparing the number of equilibrium equations to the number of unknown reactions. It provides formulas for calculating the degree of static indeterminacy for beams, frames, and trusses. It also discusses kinematic indeterminacy and provides a formula for calculating the degree of 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