Basics of structure analysis ppt

Basics of structure analysis ppt

1. 5/25/2021 1 “BASICS OF STRUCTURAL ANALYSIS” Dr. Shrishail B. Anadinni Associate Dean(Core Branches) School of Engineering Presidency University, Bengaluru

2. Structural Systems Structure: Assemblage of Elements which is capable of resisting loads without substantial deformation (Stable)

3. Equilibrium: A structure is in equilibrium under a system of applied loads when the resultant force in any direction and the resultant moment about any point are zero. For a system of coplanar forces this may be expressed by the three equations of static equilibrium: ΣH = 0/ ΣF X =0 ΣV = 0/ ΣF Y =0 ΣM = 0/ ΣM Z =0 Equilibrium and Conditions of Equilibrium

4. Types of Supports                                         1. Knife Edge Support or Simple Support: 2. Roller Support: 3. Hinged Support 4. Fixed Support

5. The DEGREE OF STATIC INDETERMINACY ( DoI ) is the number of redundant forces in the structure. Therefore, the degree of static indeterminacy ( DoI ) can be calculate by taking difference between the number of unknown reaction components and the number of available conditions of equilibrium. External Degree of Indeterminacy (Beams and Frames)(De) Internal Degree of Indeterminacy (Trusses)(Di) Total Degree of Indeterminacy(Dt = De + Di De ( DoI ) = No. of Reaction Components – No. of available Equilibrium conditions Di = m – (2j -3) Degree of Static Indeterminacy

6. Degree of Indeterminacy will decide the Stability of the Structure: Unstable Structure (If DoI = – v e ) Just stable (If DoI = 0) Over stable(If DoI = + ve )

7. Degree of Static Indeterminacy No. of Reaction components = 03 No. of available equilibrium conditions = 03 DoI = No. of Reaction – No. of equilibrium Components conditions DoI = 03 – 03 \ DoI = 0 Hence, it is a statically determinate structures also called just stable structure. Example 1: Cantilever Beam

8. Degree of Static Indeterminacy No. of Reaction components = 03 No. of available equilibrium conditions = 03 DoI = No. of Reaction – No. of equilibrium Components conditions DoI = 03 – 03 \ DoI = 0 Hence, it is a statically determinate structure also called just stable structure. Example 2: Simply Supported Beam

9. Degree of Static Indeterminacy No. of Reaction components = 02 No. of available equilibrium conditions = 03 DoI = No. of Reaction – No. of equilibrium Components conditions DoI = 02 – 03 \ DoI = – 01 Hence, it is an unstable structure. Example 3: Simply Supported Beam

10. Degree of Static Indeterminacy No. of Reaction components = 04 No. of available equilibrium conditions = 03 DoI = No. of Reaction components – No. of equilibrium conditions DoI = 04 – 03 DoI = 01; Hence, it is over stable structure. Also, called as Indeterminate structure. Example 4: Propped Cantilever Beam

11. Degree of Static Indeterminacy No. of Reaction components = 04 No. of available equilibrium conditions = 02 DoI = No. of Reaction components – No. of available equilibrium conditions DoI = 04 – 02 DoI = 02; Over stable structure. Example 5: Fixed Beam

12. Degree of Static Indeterminacy No. of Reaction components = 04 No. of available equilibrium conditions = 03 DoI = No. of Reaction components – No. of available equilibrium conditions DoI = 04 – 03 DoI = 01; Over stable structure. Example 6: Continuous Beam

13. Degree of Static Indeterminacy No. of Reaction components = 05 No. of available equilibrium conditions = 02 DoI = No. of Reaction components – No. of available equilibrium conditions DoI = 05 – 02 DoI = 03; Over stable structure. Example 7: Continuous Beam

14. Degree of Static Indeterminacy No. of Reaction components = 06 No. of available equilibrium conditions = 03 DoI = No. of Reaction – No. of equilibrium Components conditions DoI = 06 – 03 \ DoI = 03; Over stable or Indeterminate structure. Alternate Approach 3 x No. of sections = 03 x 1 = 03 Example 8: Portal Frame

15. Degree of Static Indeterminacy No. of Reaction components = 05 No. of available equilibrium conditions = 03 DoI = No. of Reaction – No. of equilibrium Components conditions DoI = 05 – 03 \ DoI = 02; Over stable or Indeterminate structure. Example 9: Portal Frame

16. Degree of Static Indeterminacy 1 2 3 4 5 6 7 1 2 3 4 5 No. of Reaction components = 03 No. of available equilibrium condition = 03 DoI ( De ) = No. of Reaction – No. of equilibrium Components conditions De = 03 – 03 = 0 No. of Members, m = 07 No. of joints, j = 05 D i = m – (2j – 3) = 07 – ([2*5] – 3) =0 D t = D e + D i = 0 + D t = 0 Example 10: Trusses

17. Degree of Static Indeterminacy No. of Reaction components = 03 No. of available equilibrium condition = 03 DoI ( De ) = No. of Reaction – No. of equilibrium Components conditions De = 03 – 03 = 0 No. of Members, m = 06 No. of joints, j = 04 D i = m – (2j – 3) = 06 – ([2*4] – 3)=01 D t = D e + D i = + 01 D t = 01 1 4 2 3 4 5 6 3 1 2 Example 11: Trusses

18. Degree of Static Indeterminacy No. of Members, m = 15 No. of joints, j = 08 D i = m – (2j – 3) D i = 15 – ([2*8] – 3) D i = 02 No. of Reaction components = 04 No. of available equilibrium condition = 03 DoI ( De ) = No. of Reaction – No. of equilibrium Components conditions De = 04 – 03 = 01 D t = D e + D i D t = 01 + 02 D t = 03 Example 12: Trusses

19. Degree of Static Indeterminacy Total Degree of Indeterminacy can be calculated by the formula given below: Pin-jointed Plane truss D t = (m + r) – 2j where m = No. of members r = No. of unknown reactions j = No. of joints Pin-jointed Space frame D t = (m + r ) – 3j where m = No. of members r = No. of unknown reactions j = No. of joints Rigid frame D t = (3m+r ) – 3j where m = No. of members r = No. of unknown reactions j = No. of joints

20. Degree of Static Indeterminacy No. of Members, m = 15 No. of joints, j = 08 No. of unknown reactions, r = 04 D t = (m + r) – 2j D t = (15 + 4) – 2(8) = 19 – 16 \ D t = 03

21. Degree of Static Indeterminacy No. of Members, m = 45 No. of joints, j = 30 No. of unknown reactions, r = 10 D t = (3m + r) – 3j D t = ([3*45] + 10) – 3(30) = 145 – 90 \ D t = 55 Alternate Approach No . of unknown reactions = 10 No. of equilibrium equations = 3 D e = 10 – 03 = 07 D i = 3 x No. of sections = 3 x 16 = 48 D t = D e + D i = 07 + 48 D t = 55 Rigid Frame

22. Degree of Freedom or Kinematic Indeterminacy of Structure K inematic indeterminacy or Degree of Freedom (DK): T he number of independent displacement components in all the Joints of a structure is called degree of freedom or Kinematic Indeterminacy .

23. Degree of Freedom or Kinematic Indeterminacy of Structure D A X = 0 D A Y = 0 q A = 0 D B X D B Y q B BEAMS: Cantilever Beam (Axial Extensible Member) Degree of Kinematic Indeterminacy = 3

24. Degree of Freedom or Kinematic Indeterminacy of Structure D A X = 0 D A Y = 0 q A = 0 D B Y q B BEAMS: Cantilever Beam (Axial Rigid or Inextensible Member) DK (When Members are inextensible) = DK (Extensible members) – (No. of axially rigid members) Degree of Kinematic Indeterminacy (DK) = 02

25. Degree of Freedom or Kinematic Indeterminacy of Structure D A X = 0 D A Y = 0 q A = 0 D B X = 0 D B Y = 0 q B = Degree of Kinematic Indeterminacy = 0 BEAMS: Fixed Beam Note: Degree of Static Indeterminacy = 3

26. Degree of Freedom or Kinematic Indeterminacy of Structure BEAMS: Continuous Beam If beam is Extensible, DK = 09 D A X = 0 A Y = q A = 0 D B X D B Y = 0 q B D C X D C Y = 0 q C D D X D D Y = 0 q D D E X D E Y q E If beam is inextensible or Rigid (No Axial deformation considered) DK (When Members are inextensible) = DK (Extensible beam) – (No. of axially rigid members) DK (When Members are inextensible) = 09 – 04 DK = 05

27. Degree of Freedom or Kinematic Indeterminacy of Structure Alternate Approach No. of joints, j = 03 No. of unknown reactions, r = 03 DK = 2j – r DK = (2 x 3) – 3 DK = 03 Pin Jointed Truss or Frame

28. Degree of Freedom or Kinematic Indeterminacy of Structure Pin Jointed Truss Alternate Approach No. of joints, j = 05 No. of unknown reactions, r = 05 DK = 2j – r DK = (2 x 5) – 5 DK = 05

29. Degree of Freedom or Kinematic Indeterminacy of Structure Alternate Approach No. of joints, j = 04 No. of unknown reactions, r = 03 DK = 2j – r DK = (2 x 4) – 3 DK = 05 Pin Jointed Truss

30. Degree of Freedom or Kinematic Indeterminacy of Structure Rigid Frame (Extensible Members) Alternate Approach No. of joints, j = 04 No. of unknown reactions, r = 06 DK = 3j – r DK = (3 x 4) – 6 DK = 06

31. Degree of Freedom or Kinematic Indeterminacy of Structure Alternate Approach No. of joints, j = 04 No. of unknown reactions, r = 06 DK = (3j – r) – No. of axially rigid Members DK = ([3x4] – 6) – 03 DK = 03 Rigid Frame (Rigid or Inextensible Members)

32. Degree of Freedom or Kinematic Indeterminacy of Structure

33. Compatibility equations Compatibility equations  are those additional equations which are required to analyze the statically indeterminate structures. Number of Compatibility equations depend upon the static degree of indeterminacy of a structure. Propped Cantilever beam Fixed Beam Continuous beam Portal Frames .

34. Linear Analysis A linear static analysis is an analysis where a linear relation holds between applied forces and displacements It is applicable to the structural problems, where stress is Proportional to Strain (Elastic range of stress strain curve) Deformation in the structural members are small i.e. within the elastic limit In a linear static analysis the model’s stiffness matrix is constant, and the solving process is relatively short compared to a nonlinear analysis on the same model . Since stiffness matrix is constant, therefore solution for any structural problem is simple and easy. Non Linear Analysis A nonlinear analysis is an analysis where a nonlinear relation holds between forces and displacements. It is applicable to the structural problems, where s tress is not proportional to strain ( Elasto - plastic range stress strain curve) Deformation in the structural members considerably large i.e. In the plastic range In Non-linear analysis, stiffness matrix which is not constant during the load application. This is opposed to the linear static analysis, where the stiffness matrix remained constant. As a result, a different solving strategy is required for the nonlinear analysis. Linear and Non Linear Analysis

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