A tapered (structural steel) column with a circular cross-section varies linearl

Question

A tapered (structural steel) column with a circular cross-section varies linearly from a radius of 25 mm to 15 mm over a length of 250 mm. If the column is hanging from its broad end, determine the maximum stress and the total change in length due to the weight. If there is an additional axial force of P applied at the bottom end, determine the maximum stress. When is the stress due to P of the same order of magnitude as that due to the weight of the bar? When can you ignore the stress induced in the bar due to its own weight? Please be though in working of equations, written response preferred. It's a multi tier question so I'm upping the points offered!!

Explanation / Answer

EXTENSION OF A SOLID CONICAL BAR DUE TO SELF WEIGHT IS GIVEN BY

F = EXTERNAL FORCE + FORCE DUE TO SELF WEIGHT

= P + Y* 1/3 * PI * D^2 * 1/4 * X

WHERE, Y = WEIGHT / VOLUME

EXTENSION = F . dX / A*E

INTEGRATING THE ABOVE TERM FROM

0 TO L(lenght of the bar)

NOW FOR THE SECOND PART OF THE QUESTION

EXTENSION OF A BAR DUE TO EXTERNAL FORCE = p .dx / A*E

extension of a bar due to self weight = Y * L^2 /A*E

y = W / A X L

SO , EXTENSION CAN BE IGNORED IF IT HAS LARGE CROSS SECTIONAL AREA OR LARGE LENGTH COMPARED TO ITS WIEGHT

HOPE THIS IS HELPFUL TO YOU

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