The document discusses deflections in structures. It defines deflection as the displacement of points in a structure from their original positions. Deflections are important because excessive deflections can cause problems like cracking and damage. The document outlines several methods for calculating deflections, including the double integration method. It also discusses factors that affect deflections like loads, span, and material. The derivation of the differential equation of the elastic line and its application in the double integration method are presented.

1. 10/15/11 DEFLECTIONS of STRUCTURES Tharwat Sakr Definition, Importance and Causes of Deflections Derivation of Moment – Curvature Relation The Double Integration Method 1 3 2

4. 10/15/11 DEFLECTIONS of STRUCTURES Tharwat Sakr ■ Importance of Deflections ■ In Addition the analysis of indeterminate structures requires the calculation of deflections as in advance step

7. 10/15/11 DEFLECTIONS of STRUCTURES Tharwat Sakr ■ Methods to calculate Deflections 1. The Double Integration Method. 5. The Real Work Method. 2. The Moment Area Method. 3. The Elastic Load Method. 4. The Conjugate Beam Method. 6. The Virtual Work Method.

10. 10/15/11 3. The Depth – Span Ration is very small ( Slender Members) THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Main Assumptions Slender Beam Deep Beam

11. 10/15/11 3. The Deflection is very – small compared to Span (Small deflection) THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Main Assumptions

12. 10/15/11 THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE

13. 10/15/11 THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Curvature of Elastic Line (Property of Curves) Curvature of elastic line

14. THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Curvature of Elastic Line (Property of Curves)

15. 10/15/11 THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Curvature of Elastic Line (Property of Curves) To be in x, y

16. 10/15/11 As  is so small THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Curvature of Elastic Line (Property of Curves)

17. 10/15/11 As Then THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Curvature of Elastic Line (Property of Curves)

18. 10/15/11 Lead to THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Curvature of Elastic Line (Property of Curves)

19. 10/15/11 For Very small values of deflection (y) Is the equation of radius of curvature of any curve and its simplification for very small values of y THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Curvature of Elastic Line (Property of Curves)

20. 10/15/11 THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Moment-Curvature Relationship Moment - Curvature Relationship

21. 10/15/11 THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Moment-Curvature Relationship

22. 10/15/11 THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Moment-Curvature Relationship Elongation at any fiber Strain can be defined as From Hook’s Law

23. 10/15/11 THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Moment-Curvature Relationship From the stress formula By Substitution From Is the differential equation of the elastic line subjected to moment

24. 10/15/11 THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Sign Convention Deflection Slope Curvature Is the rate of change of the displacement Is the rate of change of the slope Represents the displacement of the structure Investigating the moment – deflection sign

25. 10/15/11 THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Sign Convention Positive Moment Slope decreased with x Negative Moment Slope Increased with x

26. 10/15/11 Is the Relation between Deflection and Moment THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Sign Convention

27. THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Concepts Deflection at any point Slope angel at any point Curvature at any point Moment at any point

28. 10/15/11 THE DOUBLE INTEGRATION METHOD

29. 10/15/11 The method is based on the direct application of the differential equation of elastic line THE DOUBLE INTEGRATION METHOD Theoretical Bases

30. THE DOUBLE INTEGRATION METHOD Basic Procedure Derive an Equation of the Moment M as function of position variable x (M(x)) Apply the differential equation of elastic line and integrate twice Apply the Boundary and Continuity conditions to obtain the integration constants Substitute with the integration constants into the deflection and slope equations

31. THE DOUBLE INTEGRATION METHOD Boundary Conditions At Roller support At Hinged support At Fixed support

32. THE DOUBLE INTEGRATION METHOD Continuity Conditions At Any intermediate point

33. 10/15/11 Standard Cases of Beam Deflection Tharwat Sakr

34. 10/15/11 Standard Cases of Beam Deflection Tharwat Sakr