Derivation of sagging expression

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The sagging expression for a centrally loaded bar is derived as δ = W l^3/(4bd^3Y), where Y is the Young's modulus. The discussion emphasizes the use of calculus for this derivation, referencing the Euler–Bernoulli beam theory, particularly the section on three-point bending. It notes that shear force remains constant at half the central load and changes sign at the midpoint. The bending moment is described as varying linearly, reaching its maximum at the center of the beam. Understanding these principles is crucial for accurately deriving the sagging expression.
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A bar of length l, breadth b, and depth d when loaded at the centre by a load W sags by
an amount given by
δ = W l^3/(4bd^3Y). Here Y is the young's modulus of the wire. My textbook says that we have to use little calculus to derive this expression. Can you please derive this?
 
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It gets pretty complicated. But take a look at the Wikipedia article "Euler–Bernoulli beam theory", especially the section "Three-point bending" and see if that is any help.

"The shear is constant in absolute value: it is half the central load, P/2. It changes sign in the middle of the beam. The bending moment varies linearly from one end, where it is 0, and the center where its absolute value is PL/4. The deformation of the beam is described by a polynomial of third degree over a half beam (the other half being symmetrical)."
 
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