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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.
M = -Px
Slope and elastic curve
35. Using the boundary conditions
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.