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Derivation of sagging expression

by sharan swarup
Tags: youngs modulus
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sharan swarup
#1
Mar10-12, 02:30 AM
P: 98
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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Bill_K
#2
Mar10-12, 05:13 AM
Sci Advisor
Thanks
Bill_K's Avatar
P: 4,160
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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