Question

Find the width of the strongest beam that can be cut of cylindrical log of wood whose diameter is ‘d'

Answer 1

Given that, the diameter of log is ‘d’

 

Let the strongest section which can be cut out of cylindrical log be ABCD as shown in figure.

 

Let the width of strongest section be ‘b’ and depth be ‘d’.

 

Now, moment of inertia of ABCD about axis X-X is


I = b.h3/2

ymax = h/2

 

∴ Z = 1/ymax = bh3/12 / h/2 = bh2/6 …(1)

 

Since, ‘b’ and ‘h’ are dependent variables, we can write

 

b2 + h2 = d2 (from triangle ABC)

 

h2 = d2 – b2 …(2)

Substituting the value of h2 from equation (2) in equation (1)

 

Z = b(d2-b2)/6 …(3)

 

In equation (3)b is the only variable, since d is constant.

 

Now, for a beam to be strongest, its section modulus ‘Z’ should be maximum

 

⇒ dZ/db = 0

 

⇒ d/db (db2 – b2 / 6) = 0

 

Or d2 – 3b2 = 0

⇒  d2 = 3b2 …(4)

 

Putting in equation (2)

 

 h2 = 3b2 – b2 = 2b2

 

⇒ h = √2b<!--

 

<!--∴ h/b = √2, hence proved.