Question

Derive the formula of elongation of tapered bar with rectangular cross section under it's own weight?

Answer 1

Explanation:

Elongation of Prismatic bar under its self weight And comparison between ... Load acting on any cross-section of the member by method of section ... Introduction of Pure Bending and Bending equation.

Answer 2

Answer:

$\Delta =\dfrac{WL}{2AE}$

Explanation:

Lets take

Length of the bar = L

Cross sectional area = A

Young modulus = E

Total weight of the bar = W

Weight per unit volume

$\rho=\dfrac{W}{AL}$

The wight of the lower part

w=ρ A dx

The elongation of elemental part

$d\Delta =\dfrac{\rho\ A\ x}{AE}dx\\\Delta =\int_{0}^{L}\dfrac{\rho\ A\ x}{AE}dx\\\Delta =\int_{0}^{L}\dfrac{\rho\ x}{E}dx\\\Delta =\left[\dfrac{\rho\ x^2}{2E}\right]_0^L\\\Delta =\dfrac{\rho\ L^2}{2E}\\\rho=\dfrac{W}{AL}\\\Delta =\dfrac{WL}{2AE}$

Therefore the elongation due to self weight of the bar will be

$\Delta =\dfrac{WL}{2AE}$