Find dmension of strongest rectangular beam, Mechanical Engineering

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Q. Find the dimension of strongest rectangular beam that can be cut out of log of 25 mm diameter.

Sol.: b2 +  d2 = 252

d2 = 252 -  b2

As M/I = σ/y; M = σ(I/y) = σ.Z

M is maximum when Z is maximum
Z =  I/y = (bd3/12)/(d/2) =  bd2/6 =  b.(252 -  b2)/6

The value of Z maximum at  dZ/db = 0;

That is, d/db[252b/6 -  b3/6] = 0

252/6 - 3b2/6 = 0b2 = 625/3;

b = 14.43 mm .......ANS

d = 20.41 mm .......ANS
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