Question

derive the expression for elongation of a conical bar due to its self_weight​

Answer 1

Answer:

δL=ρL²/2E

Explanation:

Consider a bar of circular cross section and uniform diameter throughout. Consider it to be suspended from a rigid support and its top end, such that it is in a hanging in a vertical position as shown in the figure.

Let,

A = Uniform cross sectional area of the bar

E = Young’s modulus for the bar

L = Length of the bar

ρ = Weight of the bar, per unit length, for the material of the bar

Consider an element of length ‘dy’ at a distance of ‘y’ from the bottom of the bar being elongated due to the force ‘P’, at section x-x, as shown in the figure.

Weight of the portion below x-x = P = ρ × A × y

Change in the length of the element ‘dy’ = Pl/AE=ρ×(A×y)×dy/AE

=ρ×y×dyE

For total change in the length of the bar, we need to integrate along the length

Total change in length = ∫L0 ρy.dy/E

On integrating, we get,

δL=ρL²/2E

This is the expression for the elongation of a uniform bar under self weight

Hope my answer being helpful for you if it's then mark it as brainliest ;)

Thnk you :-)

Note:

I am not sure that my answer is right so you can check it or ask ur supervisor