Page:Cassells' Carpentry and Joinery.djvu/53

Rh should be cut as shown at Fig. 147. By arranging boards as in Fig. 148 a better joint is made than that shown at Fig. 149. When mouldings are prepared from wood which has been cut so that the annual rings are nearly parallel to the breadth (see Fig. 150), there is almost sure to be more or less shrinkage, which will, of course, take place in the breadth and thus produce an

Fig. 150.—Shrinkage of Moulding.

open mitre as shown, although the workmanship may be first rate. Fig. 151 shows the best arrangement, the annual rings being at right angles to the breadth.

By mathematical investigation Fig. 152 shows the graphic method of finding the strongest rectangular beam that can be cut

Fig. 151.—Best Arrangement of Grain in Mouldings.

out of a round log of timber. The diameter is divided into three equal parts, and perpendiculars are raised on opposite sides on the inner ends of the outer divisions. The four points in which the circumference is touched are then joined to give the beam $$AEBF$$. The proportion is $$ \frac{FB}{AF} = \frac{1}{\surd 2'}$$ because by Euclid II. 14, $$ \sqrt{AD \times DB} = FD$$, and by Euclid I. 47, $$FB = \sqrt{(FD)^2 + (DB)^2}$$ also $$ AF = \sqrt{(AB)^2 - (FB)^2}$$. Let $$AB = 1$$, then $$FD = \sqrt{ \tfrac{2}{3} \times \tfrac {1}{3} } = \frac{\surd 2}{3}$$, $$FB = \sqrt{ \left ( \frac{\surd 2}{3} \right )^2 + \left ( \tfrac{1}{3} \right )^2 } = \sqrt{ \tfrac{2}{9} + \tfrac{1}{9} } = \sqrt{ \tfrac{1}{3} } = \frac{1}{\surd 3}$$, $$AF = \sqrt{ 1^2 - \tfrac{1}{3} } = \sqrt{ \tfrac{2}{3} } = \frac{\surd 2}{\surd 3}$$ and $$\frac{FB}{AF} = \frac{1}{\surd 3} \div \frac{\surd 2}{\surd 3} = \frac{1}{\surd 2}$$

Fig. 152.—Strongest Beam from a Round Log. When the diameter of log $$AB = d$$, the depth $$AF = \frac{\surd 2}{\surd 3} = \frac{1.414}{1.732} = 0.816d$$ and the bredth $$FB = \frac{1}{\surd 3} = \frac{D}{1.732} = 0.577d.$$ It must be fully understood, of course, that the above shows only the mathematical calculation corresponding to the graphic

Fig. 153.—Stiffest Beam from a Round Log.

diagram, and does not in any way prove the statement that this beam will be the strongest that can be cut out of a round log. The calculations necessary to prove that statement would probably be a laborious matter. But given such a beam, its strength could be calculated by ordinary formula, and then another beam slightly narrower, and a beam slightly broader, both inscribed in the circle, could be tested by the same formula.