determinate and indeterminate structures

This topic I am uploading here contains some basic topics in structural analysis which includes types of supports, reactions for different support conditions, determinate and indeterminate structures, static and kinematic indeterminacy,external and internal static indeterminacy, kinematic indeterminacy for beams, frames, trusses.
need of finding indeterminacy, different methods available to formulate equations to solve unknowns.

1. STRUCTURAL ANALYSIS-II v shiva Assistant Professor, Dept. of Civil Engineering, Anurag Engineering College, Kodada Mobile: [phone]-2020 vempati shiva

2. UNIT – I (TOPIC - I) What is Structure? Structural analysis? Types of connections(supports) Equations of equilibrium Determinate and indeterminate structures Degree of static indeterminacy Internal static indeterminacy External static indeterminacy Degree of static indeterminacy for: Beams Frames(2-D and 3-D) Trusses(2-D and 3-D) Kinematic indeterminacy 03-08-2020 vempati shiva

3. By the end of the session: Student will know the difference between determinate and indeterminate structure. student will be able to calculate static and kinematic indeterminacy in a structure. student will understand the requirement of compatibility equations that are to be formulated in order to solve unknowns. 03-08-2020 vempati shiva

4. STRUCTURE A structure consists of a series of connected parts used to support a load. Connected parts are called structural elements which include beams, columns and tie rods. The combination of structural elements and the materials from which they are composed is referred to as structural system. 03-08-2020 vempati shiva

5. STRUCTURAL ANALYSIS Structural analysis may be defined as the prediction of performance of a given structure under stipulated loads or other external effects, such as support movements and temperature changes. Performance characteristics include axial forces, shear forces, and bending moments, deflections, and support reactions. These unknown quantities may be obtained by formulating a suitable number of independent equations. These equations can be obtained from the following relations which govern the behaviour of structure: Equilibrium Stress – Strain relation kinematics 03-08-2020 vempati shiva

6. CONNECTIONS Structural members are jointed together by rigid(fixed) or/and flexible(hinged) connections. A rigid connection or joint prevents relative translations and rotations of the member ends connected to it (Original angles between members intersecting at rigid joint are maintained after the deformation also). Rigid joints capable of transmitting forces as well as moments also. A hinged joint prevents only relative translations of member ends. These are capable of transmitting only forces. 03-08-2020 vempati shiva

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8. EQUATIONS OF EQUILIBRIUM A structure initially at rest and remains at rest when subjected to a system of forces and couples is said to be in a state of static equilibrium. The conditions of zero resultant force and zero resultant couple can be expressed as: For planar structure: Σ Fx = 0 Σ Fy = 0 Σ M z = 0 When a planar structure subjected to concurrent coplanar force system, the above requirements for equilibrium reduces to Σ Fx = 0 Σ Fy = 0 03-08-2020 vempati shiva

9. For space structure: Σ Fx = 0 Σ Fy = 0 Σ Fz = 0 Σ Mx = 0 Σ M y = 0 Σ Mz = 0 03-08-2020 vempati shiva

10. INDETERMINATE STRUCTURE If the unknown forces in a structure are more than the equilibrium equations then such structure is referred to as indeterminate structure. Indeterminacy can be of static indeterminacy or kinematic indeterminacy. 03-08-2020 vempati shiva

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12. External static indeterminacy( Dse ): When an all the external reactions cannot be evaluated from the static equilibrium equations alone, then such structure is referred to as externally indeterminate. Internal static indeterminacy( Dsi ): When an all the internal forces cannot be evaluated from the static equilibrium equations alone, then such structure is referred to as internally indeterminate. Degree of Static indeterminacy(Ds) = Dse + Dsi 03-08-2020 vempati shiva

13. Need of degree of indeterminacy: Degree of indeterminacy = Total no. of unknown forces – Total no. of equilibrium equations Degree of indeterminacy gives us the additional equations(compatibility equations) required to find unknown forces. Thus the Degree of indeterminacy is equal to the number of additional equations(other than static equilibrium equations) required to solve the unknown forces in that structure. 03-08-2020 vempati shiva

14. BEAMS Find the degree of static indeterminacy for the following beams: Number of reactions(re) = 3 Number of internal forces( i ) = Total number of unknown forces(re + i ) = 3 + 0 = 3 Number of parts(n) = 1 Total number of equations of equilibrium available = 3n = 3 X 1 = 3 Degree of static indeterminacy = (re + i ) – 3n = (3 + 0) – 3 = 0 (Determinate) 03-08-2020 vempati shiva

15. Number of reactions(re) = 4 Number of internal forces( i ) = Total number of unknown forces(re + i ) = 4 + 0 = 3 Number of parts(n) = 1 Total number of equations of equilibrium available = 3n = 3 X 1 = 3 Degree of static indeterminacy = (r + i ) – 3n = (4 + 0) – 3 = 1 (Indeterminate) 03-08-2020 vempati shiva

16. Number of reactions(re) = 6 Number of internal forces( i ) = Total number of unknown forces(re + i ) = 6 + 0 = 6 Number of parts(n) = 1 Total number of equations of equilibrium available = 3n = 3 X 1 = 3 Degree of static indeterminacy = (r + i ) – 3n = (6 + 0) – 3 = 3 (Indeterminate) 03-08-2020 vempati shiva

17. Number of reactions(re) = 4 Number of internal forces( i ) = 2 Total number of unknown forces(re + i ) = 4 + 2 = 6 Number of parts(n) = 2 Total number of equations of equilibrium available = 3n = 3 X 2 = 6 Degree of static indeterminacy = (r + i ) – 3n = (4 + 2) – 6 = (determinate) 03-08-2020 vempati shiva

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19. Number of reactions(re) = 4 Number of internal forces( i ) = 1 Total number of unknown forces(re + i ) = 4 + 1 = 5 Number of parts(n) = 2 Total number of equations of equilibrium available = 3n = 3 X 2 = 6 Degree of static indeterminacy = (r + i ) – 3n = (4 + 1) – 6 = -1 (Determinate but UNSTABLE) 03-08-2020 vempati shiva

20. STABILITY Structural stability is the major concern of the structural designer. To ensure the stability, a structure must have enough support reaction along with proper arrangement of members. The overall stability of the structure can be divided into: External stability Internal stability 03-08-2020 vempati shiva

21. External stability For stability of structures there should be no rigid body movement of structure due to loading. So it should have proper supports to restrain translation and rotation motion. There should be min. 3 no. of externally independent support reactions. All reactions should not be parallel All reactions should not be linearly concurrent otherwise rotational unstability will setup. 03-08-2020 vempati shiva

22. Internal stability For internal stability no part of the structure can move rigidly relative to the other part so that geometry of the structure is preserved, however small elastic deformations are permitted. To preserve geometry enough number of members and their adequate arrangement is required. For geometric stability there should not be any condition of mechanism. Mechanism is formed when there are three collinear hinges. Hinge 03-08-2020 vempati shiva

23. FRAMES 03-08-2020 vempati shiva

24. FRAMES Frames always forms closed loops. Each time when you cut the loop it releases three unknown forces(axial force, shear force and moment). If there are ‘c’ closed loops then 3c is the internal static indeterminacy. For 2-D Frames: External static indeterminacy ( Dse ) = re – 3 Internal static indeterminacy ( Dsi ) = 3c – r r re = number of external reactions c = number of closed loops r r = number of reactions released.(in case of any internal hinges) r r = Σ (m’ – 1) m’ = number of members connecting the hinge. 03-08-2020 vempati shiva

25. For 3-D Frames: External static indeterminacy ( Dse ) = re – 6 Internal static indeterminacy ( Dsi ) = 6c – r r re = number of external reactions c = number of closed loops r r = number of reactions released.(in case of any internal hinges) r r = Σ 3 (m’ – 1) m’ = number of members connecting the hinge. 03-08-2020 vempati shiva

26. Find the degree of static indeterminacy for the frames shown: No of external reactions = 3; No. of static equilibrium equations = 3 External static indeterminacy( Dse ) = 3 – 3 = 0; No. of closed loops(c) = 0; No. of unknowns in a closed loop = 3c . Internal static indeterminacy( Dsi ) = 3c – r r . = 3X0 – 0 = 0; Ds = Dse + Dsi; ∴ degree of static indeterminacy(Ds) = 0 + 0 = 0 = 0 (Determinate structure) Alternatively: To make the given frame into a closed loop, I need to add one rotational restraint at support A and one horizontal restraint and one rotational restraint at support D. so total 3 restraints I need to add to make it into a closed loop. As we know every closed loop has 3 unknown forces, now the above frame has 3 unknown forces. But as we have added 3 restraints to make it into closed loop , subtract those 3 restraints from 3 unknown forces which comes out to be 0 and it is a determinate structure. 03-08-2020 vempati shiva

27. Find the degree of static indeterminacy for the frames shown: No of external reactions = 6; No. of static equilibrium equations = 3 External static indeterminacy( Dse ) = 6 – 3 = 3; No. of closed loops(c) = 0; No. of internal releases( r r ) = 0; No. of unknowns in a closed loop = 3c . Internal static indeterminacy( Dsi ) = 3c – r r . = 3X0 – 0 = 0; Ds = Dse + Dsi; ∴ degree of static indeterminacy(Ds) = 3 + 0 = 3 = 3 (Indeterminate structure) 03-08-2020 vempati shiva

28. Find the degree of static indeterminacy for the frames shown: No of external reactions = 9; No. of static equilibrium equations = 3 External static indeterminacy( Dse ) = 9 – 3 = 6; No. of closed loops(c) = 4; No. of internal releases( r r ) = 0; No. of unknowns in a closed loop = 3c . Internal static indeterminacy( Dsi ) = 3c – r r . = 3X4 – 0 = 12; Ds = Dse + Dsi; ∴ degree of static indeterminacy(Ds) = 6 + 12 = 18 = 18 (Indeterminate structure) 1 2 3 4 03-08-2020 vempati shiva

29. Find the degree of static indeterminacy for the frames shown: No of external reactions = 12; No. of static equilibrium equations = 3 External static indeterminacy( Dse ) = 12 – 3 = 9; No. of closed loops(c) = 1; No. of internal releases( r r ) = Σ (m’ – 1) = Σ (2’ – 1) = 1 No. of unknowns in a closed loop = 3c . Internal static indeterminacy( Dsi ) = 3c – r r . = 3X1 – 1 = 2; Ds = Dse + Dsi; ∴ degree of static indeterminacy(Ds) = 9 + 2 = 11 = 11 (Indeterminate structure) 1 03-08-2020 vempati shiva

30. Find the degree of static indeterminacy for the Plane frames shown: No of external reactions = 12; No. of static equilibrium equations = 3 External static indeterminacy( Dse ) = 12 – 3 = 9; No. of closed loops(c) = 3; No. of internal releases( r r ) = Σ (m’ – 1) ; = (3 – 1) + (3 – 1) = 4 ; No. of unknowns in a closed loop = 3c . Internal static indeterminacy( Dsi ) = 3c – r r . = 3X3 – 4 = 5; Ds = Dse + Dsi; ∴ degree of static indeterminacy(Ds) = 9 + 5 = 14 = 14 (Indeterminate structure) 1 2 3 03-08-2020 vempati shiva

31. Find the degree of degree of indeterminacy for the space FRAMES shown: No of external reactions = 4X6 = 24; No. of static equilibrium equations = 6 External static indeterminacy( Dse ) = 24 – 6 = 18; No. of closed loops(c) = 1; No. of internal releases( r r ) = Σ 3 (m’ – 1) = 0 ; No. of unknowns in a closed loop = 6c . Internal static indeterminacy( Dsi ) = 6c – r r . = 6X1 – 0 = 6; Ds = Dse + Dsi ; ∴ degree of static indeterminacy(Ds) = 18 + 6 = 24 = 24 (Indeterminate structure) 03-08-2020 vempati shiva

32. Find the degree of degree of indeterminacy for the space FRAMES shown: No of external reactions = 6X6 = 36; No. of static equilibrium equations = 6 External static indeterminacy( Dse ) = 36 – 6 = 30; No. of closed loops(c) = 16; No. of internal releases( r r ) = Σ 3 (m’ – 1) = 0 ; No. of unknowns in a closed loop = 6c . Internal static indeterminacy( Dsi ) = 6c – r r . = 6X16 – 0 = 96; Ds = Dse + Dsi ; ∴ degree of static indeterminacy(Ds) = 30 + 96 = 126 = 126 (Indeterminate structure) 03-08-2020 vempati shiva

33. Find the degree of degree of indeterminacy for the space FRAMES shown: No of external reactions = 2X6 + 2X3 = 18; No. of static equilibrium equations = 6 External static indeterminacy( Dse ) = 18 – 6 = 12; No. of closed loops(c) = 1; No. of internal releases( r r ) = Σ 3 (m’ – 1) = 3(2-1)=3 ; No. of unknowns in a closed loop = 6c . Internal static indeterminacy( Dsi ) = 6c – r r . = 6X1 – 3 = 3; Ds = Dse + Dsi ; ∴ degree of static indeterminacy(Ds) = 12 + 3 = 15 = 15 (Indeterminate structure) 03-08-2020 vempati shiva

34. TRUSSES Trusses are most common type of structure used in constructing building roofs, bridges and towers etc. A truss can be constructed by straight slender members joined together at their end by bolting, riveting or welding. Classification of trusses: Plane trusses(2-D) Space trusses(3-D) 03-08-2020 vempati shiva

35. External static indeterminacy( Dse ) = re – 3 Internal static indeterminacy( Dsi ) = m – (2j – 3) re = no. of external reactions; m = no. of members; j = no. of joints; Degree of static indeterminacy(Ds) = Dse + Dsi Example-1: re = 3; Dse = 3 – 3 = 0; ∴ external static indeterminacy = 0 m = 3; j = 3; Dsi = m – (2j - 3); Dsi = 3 – (2X3 – 3); Dsi = 0; ∴ internal static indeterminacy = 0; Ds = Dse + Dsi Ds = 0; ∴Degree of static indeterminacy = 0 Alternatively: Ds = Dsi + Dse; Ds = re – 3 + m – (2j - 3); Ds = re – 3 + m – 2j + 3; Ds = m + re -2j; Ds = 3 + 3 – 2X3; Ds = 6 – 6; Ds = 0; 03-08-2020 vempati shiva

36. Example-2: re = 6; Dse = 6 – 3 = 3; ∴external static indeterminacy = 3 m = 3; j = 4; Dsi = m – (2j - 3); Dsi = 3 – (2X4 – 3); Dsi = -2; ∴internal static indeterminacy = -2; Ds = Dse + Dsi Ds = 3 – 2; ∴Degree of static indeterminacy = 1 Alternatively: Ds = Dsi + Dse; Ds = re – 3 + m – (2j - 3); Ds = re – 3 + m – 2j + 3; Ds = m + re -2j; Ds = 3 + 3 – 2X3; Ds = 6 – 6; Ds = 0; 03-08-2020 vempati shiva

37. Example-2: re = 4; Dse = 4 – 3 = 1; ∴external static indeterminacy = 1 m = 23; j = 13; Dsi = m – (2j - 3); Dsi = 23 – (2X13 – 3); Dsi = 0; ∴internal static indeterminacy = 0; Ds = Dse + Dsi Ds = 1 – 0; ∴Degree of static indeterminacy = 1 03-08-2020 vempati shiva

38. Fig: 1 Ans : Ds = 1 Fig: 2 Ans : Ds = 0 Find the degree of static indeterminacy for the following trusses 03-08-2020 vempati shiva

39. DEGREE OF KINEMATIC INDETERMINACY or DEGREE OF FREEDOM( Dk ) Degree of kinematic indeterminacy refers to the total number of independent available degree of freedom at all joints. The degree of kinematic indeterminacy may be defined as the total number of unrestraint displacement components at all joints. S.NO TYPE OF JOINT POSSIBLE DEGREE OF FREEDOM 1. 2 -D Truss joint Two degree of freedoms are available 1. ∆x 2. ∆y 2. 3 –D Truss joint Three degree of freedoms are available 1. ∆x 2. ∆y 3. ∆z 3. 2 –D Rigid joint Three degree of freedoms are available 1. ∆x 2. ∆y 3. θ z 4. 3 –D Rigid joint six degree of freedoms are available ∆x 2. ∆y 3. ∆z 4. θ x 5. θ y 6. θ z 03-08-2020 vempati shiva

40. Plane truss (2–D truss) Dk = 2j – re space truss (3–D truss ) Dk = 3j – re Rigid jointed Plane frame (2–D frame) Dk = 3j - re – m” Rigid jointed space frame (3–D frame) Dk = 6j - re – m” Dk = degree of freedom; re = no. of external reactions; J = no. of joints; M” = no. of axially rigid members; 03-08-2020 vempati shiva dy dx dy dy dx dx

41. Find the degree of kinematic indeterminacy for the following BEAMS: If the beam is axially flexible: No. of external reactions = 3; No. of joints = 2; No. of axially rigid members = 0; Degree of freedom ( Dk ) = 3j – re – m”; Dk = 3X2 – 3 – 0 ; Dk = 3 If the beam is axially rigid: No. of external reactions = 3; No. of joints = 2; No. of axially rigid members = 1; Degree of freedom ( Dk ) = 2j – re – m” ; Dk = 2X3 – 3 – 1 ; Dk = 2 03-08-2020 vempati shiva

42. Find the degree of kinematic indeterminacy for the following BEAMS: If the beam is axially flexible: No. of external reactions = 4; No. of joints = 2; No. of axially rigid members = 0; Degree of freedom ( Dk ) = 3j – re – m”; Dk = 3X2 – 4 – 0 ; Dk = 2 If the beam is axially rigid: As the beam is already axially restrained by reactions. Dk will be same as the previous case. ∴ Dk = 2 03-08-2020 vempati shiva

43. Find the degree of kinematic indeterminacy for the following FRAMES : No. of external reactions = 5; No. of joints = 4; No. of axially rigid members = 0; No. of released reactions rr = Σ (m’ – 1); rr = 2 – 1 = 1 Degree of freedom ( Dk ) = 3j – re – m” + rr ; Dk = 3X4 – 5 – 0 + 1 ; Dk = 8 03-08-2020 vempati shiva

44. Find the degree of kinematic indeterminacy for the FRAME shown, consider all beams are axially rigid: No. of external reactions = 7; No. of joints = 9; No. of axially rigid members = 4; No. of released reactions r r = Σ (m’ – 1) = 0 ; Degree of freedom ( Dk ) = 3j – re – m” + rr ; Dk = 3X9 – 7 – 4 + 0 ; Dk = 16 03-08-2020 vempati shiva

45. Find the degree of kinematic indeterminacy for the following TRUSSES : No. of external reactions = 3; No. of joints = 3; Degree of freedom ( Dk ) = 2j – re Dk = 2X3 – 3 ; Dk = 3 03-08-2020 vempati shiva

46. Find the degree of kinematic indeterminacy for the following TRUSSES : No. of external reactions, re = 3; No. of joints, j = 4; Degree of freedom ( Dk ) = 2j – re Dk = 2X4 – 3 ; Dk = 5 03-08-2020 vempati shiva

47. Find the degree of Static indeterminacy and kinematic indeterminacy for the following plane structures : 03-08-2020 vempati shiva

48. E nd of first topic from Unit-I 03-08-2020 vempati shiva