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Chapter 9

Chapter 9. Deflection of Beams. 9.1 Introduction . -- Concerning about the “deflection” of a beam -- Special interest: the maximum deflection -- Design: to meet design criteria. 9.1 Introduction . (4.21). M = bending moment E = modulus I = moment of inertia.

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Chapter 9

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  1. Chapter 9 Deflection of Beams

  2. 9.1 Introduction -- Concerning about the “deflection” of a beam -- Special interest: the maximum deflection -- Design: to meet design criteria

  3. 9.1 Introduction (4.21) M = bending moment E = modulus I = moment of inertia If M is not a constant, i.e. M=M(x) (9.1) or -- will be explained in Sec. 9.3

  4. y = y(x)

  5. 9.2 Deformation of a Beam under Transverse Loading (9.1) Since M(x) = -Px (9.2)

  6. Example:a beam subjected to transverse loads Moment Diagram and the Deformed Configuration: Mmax occurs at C

  7. In addition to M(x) and 1/, we need further information on: 1. Slope at various locations 2. Max deflection of a beam 3. Elastic curve: y = y(x)

  8. 9.3 Equation of the Elastic Curve (9.4) (9.5)

  9. 9.3 Equation of the Elastic Curve The curvature of a plane curve at Point Q(x,y) is (9.2) For small slope dy/dx  0, hence Therefore, (9.3) (9.4) Fially,

  10. -- the governing diff. equation of a beam -- the governing diff. equation of the “elastic curve” EI = flexural rigidity

  11. (9.4) Integrating Eq. (9.4) once (9.5) Since, It follows, (9.5’)

  12. (9.5) Integrating Eq. (9.5) once again, (9.6) or

  13. C1 & C2 are determined from the B.C.s Boundary conditions: -- the conditions imposed on the beam by its supports Examples:

  14. Three types of Statically Determinate Beams: 1. Simply supported beams: 2. Overhanging beams: 3. Cantilever beams:

  15. 9.4 Direct Determination of the Elastic Curve from the Load Distribution (9.4) (9.31) (9.32)

  16. (9.33)

  17. 9.5 Statically Determinate Beams Applying equations of equilibrium: (9.37) Conclusion: -- This is a statically indeterminate problem. -- Because the problem cannot be solved by means of equations of equilibrium

  18. 9.5 Statically Determinate Beams By adding (1) deflection y = y(x) and  = (x), the problem can be solved. i.e. five unknowns with six equations

  19. Statically indeterminate to the 1st degree: -- one redundant support Statically indeterminate to the 2nd degree: -- two redundant supports

  20. 9.6 Using Singular functions to Determine the Slope and Deflection of a Beam (9.44) (9.45)

  21. (9.46) (9.47)

  22. 9.7 Method of Superposition

  23. 9.8 Application of Superposition to Statically Indeterminate Beams

  24. 9.9 Moment-Area Theorems (9.54) (9.55)

  25. (9.57) (9.56) (9.59) (9.60)

  26. 9.10 Application To Cantilever Beams And Beams With Symmetric Loads

  27. 9.11 Bending-Moment Diagrams By Parts

  28. 9.12 Application Of Moment-area Theorems To Beams With Unsymmetric Loadings (9.61) (9.62) (9.63)

  29. 9.13 Maximum Deflection

  30. 9.14 Use Of Moment-Area Theorems With Statically Indeterminate Beams

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