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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.