# 18 - 20 Rectangular beam in maxima and minima problems

**Problem 18**

The strength of a rectangular beam is proportional to the breadth and the square of the depth. Find the shape of the largest beam that can be cut from a log of given size.

**Solution:**

$b^2 + d^2 = D^2$

$2b\dfrac{db}{dd} + 2d = 0$

$\dfrac{db}{dd} = -\dfrac{d}{b}$

Strength:

$S = bd^2$

$\dfrac{dS}{dd} = b (2d) + d^2 \dfrac{db}{dd} = 0$

$2bd + d^2 \left( -\dfrac{d}{b} \right) = 0$

$2bd = \dfrac{d^3}{b}$

$2b^2 = d^2$

$d = \sqrt{2} \, b$

$\text{depth } \, = \sqrt{2} \, \times \, \text{ breadth}$ *answer*

**Problem 19**

The stiffness of a rectangular beam is proportional to the breadth and the cube of the depth. Find the shape of the stiffest beam that can be cut from a log of given size.

**Solution:**

$b^2 + d^2 = D^2$

$2b\dfrac{db}{dd} + 2d = 0$

$\dfrac{db}{dd} = -\dfrac{d}{b}$

Stiffness:

$k = bd^3$

$\dfrac{dk}{dd} = b(3d^2) + d^3 \dfrac{db}{dd} = 0$

$3bd^2 + d^3 \left( -\dfrac{d}{b} \right) = 0$

$3bd^2 = \dfrac{d^4}{b}$

$3b^2 = d^2$

$d = \sqrt{3} \, b$

$\text{depth } \, = \sqrt{3} \, \times \, \text{ breadth}$ *answer*

**Problem 20**

Compare for strength and stiffness both edgewise and sidewise thrust, two beams of equal length, 2 inches by 8 inches and the other 4 inches by 6 inches (See Problem 18 and Problem 19 above). Which shape is more often used for floor joist? Why?

**Solution:**

^{2}

Stiffness, k = bd

^{3}

For 2" × 8":

Oriented such that the breadth is 2"

S = 8(2^{2}) = 32 in^{3}

k = 8(2^{3}) = 64 in^{4}

Oriented such that the breadth is 8"

S = 2(8^{2}) = 128 in^{3}

k = 2(8^{3}) = 1024 in^{4}

For 4" × 6":

Oriented such that the breadth is 6"

S = 6(4^{2}) = 96 in^{3}

k = 6(4^{3}) = 384 in^{4}

Oriented such that the breadth is 4"

S = 4(6^{2}) = 144 in^{3}

k = 4(6^{3}) = 864 in^{4}

2" x 8" is stiffer than 4" x 6" and it is the commonly used size for floor joists. In fact, some local codes required a minimum depth of 8".