Evaluate elongation because of self weight of tapering rod, Mechanical Engineering

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Determine elongation because of self weight of tapering rod:

Sol.: Take a tapering rod hung vertically and firmly fixed at top position. The rod is of length L tapers uniformly from diameter dl  to d2, Let sides AC and BD meet at the point E when produced. Extension L in the length L of the rod ABDE because of self weight is

δL = extension of conical rod AEB because of self weight - extension because of conical segment CEB because of self weight - extension in rod length L because of weight of segment CED
692_Evaluate elongation because of self weight of tapering rod.png 
where P is weight of segment CED and w is specific weight of bar material
2156_Evaluate elongation because of self weight of tapering rod1.png 
Substituting the value of tensile load P in above expression, we get
1996_Evaluate elongation because of self weight of tapering rod2.png 
From geometrical configuration
1222_Evaluate elongation because of self weight of tapering rod6.png 
This gives :
940_Evaluate elongation because of self weight of tapering rod4.png 
If rod is conical,  that is, d2 = 0
1817_Evaluate elongation because of self weight of tapering rod5.png 
which is same expression as derived previously. 


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