[Ths]2012 defl-01

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Lecture 02
DEflection 01
Introduction - Double integration Method

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  • 10/15/11
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  • [Ths]2012 defl-01

    1. 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
    2. 2. 10/15/11 DEFLECTIONS of STRUCTURES Tharwat Sakr ■ Definition Deflection is defined as the displacement of various points of the structure from their original positions including <ul><li>linear deformations of points. </li></ul><ul><li>rotational deformations of lines (slopes) from their original position. </li></ul>
    3. 3. 10/15/11 DEFLECTIONS of STRUCTURES Tharwat Sakr ■ Importance of Deflections <ul><li>Bad appearance and uncomforting of occupants </li></ul><ul><li>Cracking of plaster. </li></ul><ul><li>Drainage problems. </li></ul><ul><li>Damage of walls and non structural Elements. </li></ul>■ All Codes and Standards specify limits of deflection as excessive deflections lead to problems as:
    4. 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
    5. 5. 10/15/11 DEFLECTIONS of STRUCTURES Tharwat Sakr ■ Sources of Deflections <ul><li>The external loads. </li></ul><ul><li>Temperature variation </li></ul><ul><li>Differential settlements between supports cause various deformations. </li></ul>
    6. 6. 10/15/11 DEFLECTIONS of STRUCTURES Tharwat Sakr ■ Factors Affecting Deflections <ul><li>Span and Configuration </li></ul><ul><li>The Applied Loads </li></ul><ul><li>Material of the structure </li></ul><ul><li>Cross section of the Structure </li></ul>
    7. 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.
    8. 8. 10/15/11 <ul><li>Plane section before deformation remains plane after deformations (Bernoulli's law ) </li></ul>THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Main Assumptions
    9. 9. 10/15/11 <ul><li>Stress is Proportional to Strain (Elastic Material) (Hook’s Low) </li></ul>THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Main Assumptions
    10. 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. 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. 12. 10/15/11 THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE
    13. 13. 10/15/11 THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Curvature of Elastic Line (Property of Curves) Curvature of elastic line
    14. 14. THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Curvature of Elastic Line (Property of Curves)
    15. 15. 10/15/11 THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Curvature of Elastic Line (Property of Curves) To be in x, y
    16. 16. 10/15/11 As  is so small THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Curvature of Elastic Line (Property of Curves)
    17. 17. 10/15/11 As Then THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Curvature of Elastic Line (Property of Curves)
    18. 18. 10/15/11 Lead to THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Curvature of Elastic Line (Property of Curves)
    19. 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. 20. 10/15/11 THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Moment-Curvature Relationship Moment - Curvature Relationship
    21. 21. 10/15/11 THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Moment-Curvature Relationship
    22. 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. 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. 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. 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. 26. 10/15/11 Is the Relation between Deflection and Moment THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE Sign Convention
    27. 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. 28. 10/15/11 THE DOUBLE INTEGRATION METHOD
    29. 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. 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. 31. THE DOUBLE INTEGRATION METHOD Boundary Conditions At Roller support At Hinged support At Fixed support
    32. 32. THE DOUBLE INTEGRATION METHOD Continuity Conditions At Any intermediate point
    33. 33. 10/15/11 Standard Cases of Beam Deflection Tharwat Sakr
    34. 34. 10/15/11 Standard Cases of Beam Deflection Tharwat Sakr

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