The document discusses static and kinematic indeterminacy of structures. It defines different types of supports for 2D and 3D structures including fixed support, hinged/pinned support, roller support, and their properties. It also discusses internal joints like internal hinge, internal roller, and internal link. The document explains concepts of static indeterminacy, kinematic indeterminacy, and degree of freedom for different types of structures.
2. Structure
⢠A structure refers to a system of connected parts used to support a load.
⢠A structure defined as an assembly of different members connected to each other which
transfers load from space to ground.
⢠Mainly of two types :
1. Load Bearing Structure
2. Framed Structure
3. Support System
⢠Supports are used in structures to provide it stability and strength.
⢠Main types of support :
1. Fixed Support
2. Hinged or Pinned Support
3. Roller Support
4.Vertical Guided Roller Support
5. Horizontal Guided Roller Support
5. 2-D Support
Fixed Support :
ďNo. of Reaction - 3 (RCX ,RCY ,MCZ)
⢠RCX -Reaction at joint â C â in x-direction
⢠RCY -Reaction at joint â C â in y-direction
⢠MCZ -Moment at joint â C â about z-direction
ďDisplacement in x-direction at joint â C â is zero ( i.e yCX = O )
ďDisplacement in y-direction at joint â C â is zero ( i.e yCY = O )
ďRotation about z-direction at joint â C â is zero ( i.e θCZ = O )
6. 2-D Support
Hinged or Pinned Support :
ďNo. of Reaction - 2 (RAX ,RAY )
⢠RAX -Reaction at joint âA â in x-direction
⢠RAY -Reaction at joint â A â in y-direction
ďDisplacement in x-direction at joint â A â is zero ( i.e yAX = O )
ďDisplacement in y-direction at joint â A â is zero ( i.e yAY = O )
ďRotation about z-direction at joint â A â is not zero ( i.e θAZ â O )
7. 2-D Support
Roller Support :
ďNo. of Reaction - 1 (RBY )
⢠RBY -Reaction at joint â B â in y-direction
ďDisplacement in x-direction at joint â B â is not zero ( i.e yBXâ O )
ďDisplacement in y-direction at joint â B â is zero ( i.e yBY = O )
ďRotation about z-direction at joint â B â is not zero ( i.e θBZ â O )
8. 2-D Support
Vertical Guided Roller Support :
ďNo. of Reaction - 2 (RAX , MAZ)
⢠RAX -Reaction at joint â A â in x-direction
⢠MAZ -Moment at joint â A â about z-direction
ďDisplacement in x-direction at joint â A â is zero ( i.e yAX = O )
ďDisplacement in y-direction at joint â A â is not zero ( i.e yAYâ O )
ďRotation about z-direction at joint â A â is zero ( i.e θAZ =O )
9. 2-D Support
Horizontal Guided Roller Support :
ďNo. of Reaction - 2 (RAY , MAZ)
⢠RAY -Reaction at joint â A â in y-direction
⢠MAZ -Moment at joint â A â about z-direction
ďDisplacement in y-direction at joint â A â is zero ( i.e yAY = O )
ďDisplacement in x-direction at joint â A â is not zero ( i.e yAX â O )
ďRotation about z-direction at joint â A â is zero ( i.e θAZ =O )
10. 2-D Support
Spring Support :
ďNo. of Reaction - 1 (RAY )
⢠RAY -Reaction at joint â A â in y-direction
ďDisplacement in y-direction at joint â A â is zero ( i.e yAY = O )
12. 2-D INTERNAL JOINTS
Internal Hinge or Pin :
ďCharacteristics :
⢠Moment at â C â is zero (i.e. M@C = O)
ďDisplacement in y-direction at joint â C â is not zero ( i.e yCY â O )
ďDisplacement in x-direction at joint â C â is not zero ( i.e yCX â O )
ďRotation about z-direction at joint â C â are may be different at either side ( i.e θC1 â θC2 )
13. 2-D INTERNAL JOINTS
Internal Roller :
ďCharacteristics :
⢠Canât transfer horizontal reaction (axial thrust)(i.e. FBX = O)
ďDisplacement in y-direction at joint â B â may not be zero ( i.e yBY â O )
ďDisplacement in x-direction at joint â B â is not zero ( i.e yBX â O )
14. 2-D INTERNAL JOINTS
Internal Link :
⢠Portion â BC â is known as internal link.
ďCharacteristics :
⢠Two internal pins at B & C
⢠Portion BC contains only axial load because moment at B and C is zero.(i.e. M@B &
M@C = O)
ďDisplacement in y-direction at joint â B & C â is not zero ( i.e YBY , yCY â O )
ďDisplacement in x-direction at joint â B & C â is not zero ( i.e YBx ,YCX â O )
15. 2-D INTERNAL JOINTS
Torsional Spring Support :
ďCharacteristics :
⢠θZ -Rotational resistance at joint in z-direction
17. 3-D Support
Fixed Support :
ďNo. of Reaction - 6 (RCX ,RCY ,RCZ ,MCX ,MCY ,MCZ )
⢠RCX ,RCY ,RCZ - Reaction at joint â C â in x,y,z-direction
⢠MCX ,MCY ,MCZ -Moment at joint â C â about x,y,z-direction
ďDisplacement in x,y,z-direction at joint â C â is zero ( i.e yCX , yCY ,yCZ = O )
ďRotation about x,y,z-direction at joint â C â is zero ( i.e θCX , θCY , θCZ = O )
18. 3-D Support
Hinged or Pinned Support :
ďNo. of Reaction - 3 (RCX ,RCY ,RCZ)
⢠RCX ,RCY ,RCZ - Reaction at joint â C â in x,y,z-direction
ďDisplacement in x,y,z-direction at joint â C â is zero ( i.e yCX , yCY ,yCZ = O )
ďRotation about x,y,z-direction at joint â C â is not zero ( i.e θCX , θCY , θCZ â O )
19. 3-D Support
Roller Support :
ďNo. of Reaction - 1 (RCY)
⢠RCY - Reaction at joint â C â in y-direction
ďDisplacement in y-direction at joint â C â is zero ( i.e yCY = O )
ďDisplacement in x,z-direction at joint â C â is not zero ( i.e yCX , yCZ â O )
ďRotation about x,y,z-direction at joint â C â is not zero ( i.e θCX , θCY , θCZ â O )
20. Equilibrium Equation
⢠When a body is in static equilibrium, no translation or rotation occurs in any
direction.
⢠Since there is no translation, the sum of the forces acting on the body must
be zero.
⢠Since there is no rotation, the sum of the moments about any point must be
zero.
22. Equilibrium Equation
PIN JOINT SPACE FRAME (3-DTruss)
No. of Equilibrium Equation : 3
⢠â Fx = O
⢠â Fy = O
⢠â Fz = O
23. Equilibrium Equation
RIGID JOINT PLANE FRAME (2-D Frame)
No. of Equilibrium Equation : 3
⢠â Fx = O
⢠â Fy = O
⢠â Mz = O
24. Equilibrium Equation
RIGID JOINT SPACE FRAME (3-D Frame)
No. of Equilibrium Equation : 6
⢠â Fx = O
⢠â Fy = O
⢠â Fz = O
⢠â Mx = O
⢠â My = O
⢠â Mz = O
NOTE :Above equilibrium equations are used to find members forces and moments , To find out support reaction
equilibrium equation for any type of structure always remains 3(i.e. â Fx = O â Fy = O â Mz = O )for 2-D and 6 for 3-D
structure.
â˘
25. Static Indeterminacy
Statical Determinant Structure :
⢠If condition of static equilibrium are sufficient to analyse the structure , it is called
Statical Determinant Structure.
⢠Bending moment and Shear force are independent of material properties and cross
section.
⢠Stresses are not induced due to temp. changes and support settlement.
26. Static Indeterminacy
Statical Indeterminant Structure :
⢠If condition of static equilibrium are not sufficient to analyse the structure , it is called
Statical Indeterminant Structure.
⢠Bending moment and Shear force are dependent on material properties and cross
section.
⢠Stresses are induced due to temp. changes and support settlement.
27. Static Indeterminacy
Static Indeterminacy = External Indeterminacy + Internal Indeterminacy
Ds = Dse + Dsi
External Indeterminacy : If no. of reactions are more than equilibrium equation is
known as Externally Indeterminant Structure.
No of Reactions = 4 Equilibrium Equations=3 for 2-D and 6 for 3-D structure.
Beams is externally indeterminate to the first degree
28. Static Indeterminacy
Internal Indeterminacy : If no. of Internal forces or stresses canât evaluated
based on equilibrium equation is known as Internally
Indeterminant Structure.
⢠Member forces ofTruss can not be determined based on statics alone, forces in the
members can be calculated based on equations of equilibrium.Thus, structures is
internally indeterminate to first degree.
29. Static Indeterminacy
(A) Rigid Jointed Plane Frame :
⢠External Indeterminacy,Dse : R-E
⢠Internal Indeterminacy,Dsi : 3C-râ
OR R = No. of external unknown reaction
Ds = 3m+R-3j-râ E = No. of Equilibrium Equation = 3
m = No. of members , j = joints
C = No. of close loop
râ = Total no. of internal released or
= No. of members
connected -1
with internal hinge
=(mâ-1)
30. Static Indeterminacy
⢠Some of the example for the râ :
⢠râ = 2 ( Moment and Horizontal Reaction Released
⢠at joint â B â )
⢠râ = 1 (Only Vertical Reaction Released at joint
⢠â B â)
râ = 2-1 =1 (i.e. member connected to hinges = 2)
â˘
⢠râ = 3-1 =2 (i.e. member connected to hinges = 3)
B
B
31. Static Indeterminacy
(B) Rigid Jointed Space Frame :
⢠External Indeterminacy,Dse : R-E
⢠Internal Indeterminacy,Dsi : 6C-râ
OR R = No. of external unknown reaction
Ds = 6m+R-6j-râ E = No. of Equilibrium Equation = 6
m = No. of members , j = joints
C = No. of close loop
râ = Total no. of internal released or
= No. of members
3 * connected -1
with internal hinge
= 3(mâ-1)
32. Static Indeterminacy
(C) Pinned Jointed Plane Frame :
⢠External Indeterminacy,Dse : R-E
⢠Internal Indeterminacy,Dsi : m+E-2j
OR R = No. of external unknown reaction
Ds = m+R-2j E = No. of Equilibrium Equation = 3
m = No. of members
j = joints
33. Static Indeterminacy
(D) Pinned Jointed Space Frame :
⢠External Indeterminacy,Dse : R-E
⢠Internal Indeterminacy,Dsi : m+E-3j
OR R = No. of external unknown reaction
Ds = m+R-3j E = No. of Equilibrium Equation = 6
m = No. of members
j = joints
35. Kinematic Indeterminacy
⢠Kinematic Indeterminacy = Degree of Freedom
⢠If the displacement component of joint canât be determined by
Compatibility Equation , it is called Kinematic Indeterminant Structure.
Degree of Kinematic Indeterminacy(Dk) :
⢠It is defined as total number of unrestrained displacement (translation
and rotation) component at joint.
37. 2-D Support
Fixed Support :
ďDegree of Freedom - O
ďDisplacement in x-direction at joint â C â is zero ( i.e yCX = O )
ďDisplacement in y-direction at joint â C â is zero ( i.e yCY = O )
ďRotation about z-direction at joint â C â is zero ( i.e θCZ = O )
38. 2-D Support
Hinged or Pinned Support :
ďDegree of Freedom - 1 (θAZ )
⢠θAZ -Rotation about z-direction at joint â A â
ďDisplacement in x-direction at joint â A â is zero ( i.e yAX = O )
ďDisplacement in y-direction at joint â A â is zero ( i.e yAY = O )
ďRotation about z-direction at joint â A â is not zero ( i.e θAZ â O )
39. 2-D Support
Roller Support :
ďDegree of Freedom - 2(θBZ ,yBX )
⢠yBX -Displacement in x-direction at joint â B â
⢠θBZ -Rotation about z-direction at joint â B â
ďDisplacement in x-direction at joint â B â is not zero ( i.e yBXâ O )
ďDisplacement in y-direction at joint â B â is zero ( i.e yBY = O )
ďRotation about z-direction at joint â B â is not zero ( i.e θBZ â O )
40. 2-D Support
Vertical Guided Roller Support :
ďDegree of Freedom - 1(yAY)
⢠yAY -Displacement in y-direction at joint â A â
ďDisplacement in x-direction at joint âA â is zero ( i.e yAX = O )
ďDisplacement in y-direction at joint â A â is not zero ( i.e yAYâ O )
ďRotation about z-direction at joint â A â is zero ( i.e θAZ =O )
41. 2-D Support
Horizontal Guided Roller Support :
ďDegree of Freedom - 1(yAX)
⢠yAX -Displacement in x-direction at joint â A â
ďDisplacement in y-direction at joint âA â is zero ( i.e yAY = O )
ďDisplacement in x-direction at joint â A â is not zero ( i.e yAX â O )
ďRotation about z-direction at joint â A â is zero ( i.e θAZ =O )
43. 3-D Support
Fixed Support :
ďDegree of Freedom - O
ďDisplacement in x,y,z-direction at joint â C â is zero ( i.e yCX , yCY ,yCZ = O )
ďRotation about x,y,z-direction at joint â C â is zero ( i.e θCX , θCY , θCZ = O )
44. 3-D Support
Hinged or Pinned Support :
ďDegree of Freedom - 3 (θCX , θCY , θCZ )
⢠θCX , θCY , θCZ -Rotation about x,y,z-direction at joint â C â
ďDisplacement in x,y,z-direction at joint â C â is zero ( i.e yCX , yCY ,yCZ = O )
ďRotation about x,y,z-direction at joint â C â is not zero ( i.e θCX , θCY , θCZ â O )
45. 3-D Support
Roller Support :
ďDegree of Freedom - 5 (yCX , yCZ , θ CX , θCY , θCZ )
⢠yCX , yCZ -Displacement in x,z-direction at joint â C â
⢠θCX , θCY , θCZ - Rotation about x,y,z-direction at joint â C â
ďDisplacement in y-direction at joint â C â is zero ( i.e yCY = O )
ďDisplacement in x,z-direction at joint â C â is not zero ( i.e yCX , yCZ â O )
ďRotation about x,y,z-direction at joint â C â is not zero ( i.e θCX , θCY , θCZ â O )
47. 2-D INTERNAL JOINTS
Internal Hinge or Pin :
ďDegree of Freedom â 4(yCX ,yCY ,θC1 ,θC2 )
ďDisplacement in y-direction at joint â C â is not zero ( i.e yCY â O )
ďDisplacement in x-direction at joint â C â is not zero ( i.e yCX â O )
ďRotation about z-direction at joint â C â are may be different at either side and not zero (
i.e θC1 â θC2 )
48. 2-D INTERNAL JOINTS
Free End :
ďDegree of Freedom â 3(yBX , yBY , θ BZ )
ďDisplacement in x-direction at joint â B â is not zero ( i.e yBX â O )
ďDisplacement in y-direction at joint â B â is not zero ( i.e yBY â O )
ďRotation about z-direction at joint â B â is not zero ( i.e θBZ â O )
B
49. 2-D INTERNAL JOINTS
AxialThrust Release:
ďDegree of Freedom â 4(yCX1 , yCX2 ,yCY , θ CZ )
ďDisplacement in x-direction at joint â C â is not zero ( i.e yCX1 , yCX2 â O )
ďDisplacement in y-direction at joint â C â is not zero ( i.e yCY â O )
ďRotation about z-direction at joint â C â is not zero ( i.e θCZ â O )
50. 2-D INTERNAL JOINTS
Shear Release:
ďDegree of Freedom â 4(yCY1 , yCY2 ,yCX , θ CZ )
ďDisplacement in x-direction at joint â C â is not zero ( i.e yCX â O )
ďDisplacement in y-direction at joint â C â is not zero ( i.e yCY1 , yCY2â O )
ďRotation about z-direction at joint â C â is not zero ( i.e θCZ â O )
51. 2-D INTERNAL JOINTS
Frame Joint:
ďDegree of Freedom â 5(yOY ,yOX , θ OAZ , θ OBZ , θ OCZ )
ďDisplacement in x-direction at joint â O â is not zero ( i.e yOX â O )
ďDisplacement in y-direction at joint â O â is not zero ( i.e yOY â O )
ďRotation about z-direction at joint â O â is not zero ( i.e θ OAZ , θ OBZ , θ OCZ â O )
52. 2-D INTERNAL JOINTS
Internal Roller:
ďDegree of Freedom â 5(yCX1 , yCX2 , θ CZ1 , θ CZ2 , yCY)
ďDisplacement in x-direction at joint â C â is not zero ( i.e yCX1 , yCX2â O )
ďDisplacement in y-direction at joint â C â is not zero ( i.e yCY â O )
ďRotation about z-direction at joint â C â is not zero ( i.e θ CZ1 , θ CZ2 â O )
53. Kinematic Indeterminacy
(A) Rigid Jointed Plane Frame :
⢠Dk : 3j-R+râ
⢠Dk(NAD) : Dk-mâ
R = No. of external unknown reaction
NAD=Neglecting Axial Deformations mâ = No. of axially rigid members
(Beams are azially rigid or stiffness is râ = Total no. of internal released or
infinite) = No. of members
connected -1
with internal hinge
=(m-1)
54. Kinematic Indeterminacy
(B) Rigid Jointed Space Frame :
⢠Dk : 6j-R+râ
⢠Dk(NAD) : Dk-mâ
R = No. of external unknown reaction
NAD=Neglecting Axial Deformations mâ = No. of axially rigid members
râ = Total no. of internal released or
= No. of members
3* connected -1
with internal hinge
=3(m-1)
56. Kinematic Indeterminacy
(D) Pinned Jointed Space Frame :
⢠Dk : 3j-R
⢠Dk(NAD) : 0
R = No. of external unknown reaction
NAD=Neglecting Axial Deformations j = No. of joints
57. Stability of Structure
External stability :
⢠For any 2-D structure 3 no. of reactions and foe 3-D structure 6 no. of
reactions are required to keep structure in stable condition.
⢠All reactions should not be Parallel.
⢠All reactions should not be Concurrent (line of action meets at one point).
Unstable Structure because all
reactions are parallel.
Unstable Structure because all
reactions are concurrent.
58. Stability of Structure
Internal stability :
⢠No part of structure can move rigidly releative to other part.
⢠For geometric stability there should not be any condition of mechanism.
⢠Static Indeterminacy should not be less than zero.(i.e. Ds >=0)(But it is not
mandatory, sometimes structure is not stable though this conditions satisfied)
⢠For internal stability following conditions should be satisfied :
(1) Pinned Jointed Plane Truss : m>=2j-3 m = No. of members
(2) Pinned Jointed Space Truss : m>=3j-6 j =No. of joints
(3) Rigid Jointed Plane Frame : 3m>=3j-3
(4) Rigid Jointed Space Frame : 6m>=6j-6
60. Stability of Structure
Example of unstable structure :
Unstable because of local member failure.
Geometric unstable because of no
diagonal member.
61. Examples
* Calculate static indeterminacy and comment on stability of structure :
1) Dse =R-E =3-3=0 Ds=0
Dsi =3C-râ= 0 (no close loop) Stable and Determinate Structure
2) Dse =R-E =7-3=4 Ds=3 (internal hinge is part of internal
indeterminacy)
Dsi =3C-râ=3*0-(2-1)=-1 (no close loop)
Stable and Indeterminate Structure
3) Dse=R-E=6-3=3 Ds=1
Dsi=3C-râ=3*0-(2*(2-1))=-2 (no close loop)
Stable and Indeterminate Structure
62. Examples
* Calculate static indeterminacy and comment on stability of structure :
4) Dse =R-E =7-3=4 Ds=2
Dsi =3C-râ=3*0-2*(2-1)=-2 (no close loop,
two hinges & not link,link is only vertical)
Stable and Indeterminate Structure
5) Dse =R-E =6-3=3 Ds=2 (Guided roller)
Dsi =3C-râ=3*0-(2-1)=-1 (no close loop and one releases)
Stable and Indeterminate Structure
6) Dse=R-E=6-3=3 Ds=1
Dsi=3C-râ=3*0-2=-2 (no close loop and two releases)
Stable and Indeterminate Structure
63. Examples
* Calculate static indeterminacy and comment on stability of structure :
7) Dse =R-E =5-2=3(no axial load) Ds=3
Dsi =3C-râ=0 (no close loop)
Stable and Indeterminate Structure
8) Dse =R-E =10-3=7 Ds=6
Dsi =3C-râ=3*0-(2-1)=-1 (no close loop)
Stable and Indeterminate Structure
9) Dse=R-E=4-3=1 Ds=12
Dsi=3C-râ=3*4-(2-1)=11 (4 close loop and one hinge)
Stable and Indeterminate Structure
64. Examples
* Calculate static indeterminacy and comment on stability of structure :
10) Dse =R-E =3-3=3 Ds=1
Dsi =3C-râ=-2 (no close loop and two releases)
Stable and Indeterminate Structure
11) Dse =R-E =5-3=2 Ds=2
Dsi =3C-râ=3*0-0=0 (no close loop)
Stable and Indeterminate Structure
12) Dse=R-E=8-3=5 Ds=3
Dsi=3C-râ=3*0-2=-2 (0 close loop and two releases)
Stable and Indeterminate Structure
65. Examples
* Calculate static indeterminacy and comment on stability of structure :
13) Dse =R-E =8-3=5 Ds=14
Dsi =3C-râ=3*3-0=9 ( 3close loop and no releases because guided roller
is support not internal joint)
Stable and Indeterminate Structure
14) Dse =R-E =11-3=8 Ds=7
Dsi =3C-râ=3*2-7=-1 (two close loop and 7 releases)
Stable and Indeterminate Structure
15) Dse=R-E=8-3=5 Ds=16
Dsi=3C-râ=3*6-7=11 (6 close loop and 7 releases)
Stable and Indeterminate Structure
66. Examples
* Calculate static indeterminacy and comment on stability of structure :
16) Dse =R-E =12-6=6 Ds=12
Dsi =6C-râ=6*1-0=6 ( one close loop and zero releases)
Stable and Indeterminate Structure
17) Dse =R-E =16-6=10 Ds=13
Dsi =6C-râ=6*1-3(2-1)=3 (one close loop and 3 releases)
Stable and Indeterminate Structure
18) Dse=R-E=3-3=0 Ds=2
Dsi=m+E-2j=13+3-2*7=2 (13 members and 7 joints)
Stable and Indeterminate Structure
67. Examples
* Calculate static indeterminacy and comment on stability of structure :
19) Dse =R-E =4-3=1 Ds=3
Dsi =m+E-2j=17+3-2*9=2 (17 members and 9 joints)
Stable and Indeterminate Structure
20) Dse =R-E =6-3=3 Ds=0
Dsi =m+E-2j=12+3-2*9=-3 (12 members and 9 joints)
Stable and Determinate Structure
21) Dse=R-E=3-3=0 Ds=2
Dsi=m+E-2j=19+3-2*10=2 (19 members and 10 joints)
Stable and Indeterminate Structure