Deflection of beams and shafts

Deflection of beams and shafts PowerPoint PPT Presentation


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Deflection of beams and shafts

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1. Deflection of beams and shafts

9. where ? = the radius of curvature at a specific point on the elastic curve (1/? is refer to as the curvature)

10. M = the internal moment in the beam at the point where ? is to be determined

11. E = the material’s modulus of elasticity I = the beam’s moment of inertia computed about the neutral axis

12. Using the flexure formula,

13. Slope and displacement by integration

25. Boundary and continuity conditions

31. Example 1 The cantilevered beam shown in fig. is subjected to a vertical load P at its end. Determine the equation of the elastic curve. EI is constant.

33. Solution I Elastic curve: as shown in fig. Moment function M = -Px Slope and elastic curve

35. Using the boundary conditions

37. Thus

38. we get

39. Maximum slope and displacement occur at A(x=0), for which

41. so we get,

43. Solution 2

45. at x = 0; VA = -P thus C1’ = -P, so,

47. at x = 0; M = 0, so C2’ = 0 and the solution proceed as before.

48. Slope and displacement by the moment area method

49. Theorem 1 The angle between the tangents at any two points on the elastic curve equals the area under the M/EI diagram between these two points.

50. Theorem 2 The vertical deviation of the tangent at a point (A) on the elastic curve with respect to the tangent extended from another point (B) equals the moment of the area under the M/EI diagram between these two points (A and B).

51. This moment is computed about point (A) where the vertical deviation (tA/B) is to be determined.

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