How Is Downward Displacement Calculated in a Hanging Prismatic Bar?

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The discussion focuses on deriving the formula for the downward displacement of point C in a hanging prismatic bar under its own weight. The key formula presented is d(c) = W(L^2 - l^2) / (2EA), with participants exploring how to arrive at this result. They discuss the relationship between the weight distribution along the bar and the incremental displacement changes, emphasizing the need for integration. The conversation highlights the importance of considering how the load varies along the length of the bar. Overall, the derivation involves integrating the weight distribution to find the displacement at a specific point.
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A prismatic bar AB of length L, cross-sectional area A ,modulus of elasticity E ,and weight W hangs vertically under its own weight
a>derive a formula for the downward displacement d(c) of point C which is located at a distance l from the lower end of the bar

The answer is W(L^2-l^2)/(2EA)

how do we get it?

Thanks for any hints or answers!
 
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Well, so delta = PL/EASo P varies along the y direction...P = W*y/L, right?

and if we're doing incremental displacement changes, the L is your dy

d delta = integral W*y/LEA * dx

so you integrate from your l to L. Shouldn't your answer also have an L term in the denominator?
 

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