117. The strength of a beam, or the weight that it would carry without fracture, is determined by the relation which exists between the moment of rupture and the moment of resistance to rupture, termed the equality of moments as explained in Art. 44.

118. The moment of resistance is the sum of the forces due to the resistance of the fibres of the beam at the place of rupture, to tearing and crushing, multiplied by their respective distances from the neutral axis of the cross section.*

119. As the position of the neutral axis varies for each description of wood owing to the different degrees of resistance to tension and compression, it will be sufficient for the carpenter's purpose to know, as shown by writers on the Strength of Materials, that in solid beams of rectangular cross section, the moment of resistance is proportional to the area of the section at the point of rupture, multiplied by the depth of the beam and by a constant number, which has to be found by experiment for each description of wood.† . Therefore, if k be taken to represent the constant, B to represent the breadth of the beam, D its depth, and L its length between the supports - all in the same terms - and W the weight as before, we have

Moment of resistance MR =BxDxDxк=BxD2xк.

This being taken equal to the moment of rupture M (Sect. 1) we have for the strength of a beam supported at both ends (Art. 47)

BxD2xк = WxL/4; or 4xкxB,xD2/L = W, [14]

* This is usually represented by the moment of inertia of the section multiplied by the intensity of stress on the extreme fibres per unit of area divided by the distance from the neutral axis.

† Tate ' On the Strength of Materials.' a formula which gives the weight that would fracturo a beam when loaded in the middle.

In practice it is more convenient to take the length of the beam in feet, the other dimensions being in inches, which requires the formula to be modified thus: -

BxD2/Lx.к/3 = W. [15]

The constant k must be ascertained by experiment, and as the divisor 3 in the formula will also be constant, the formula may be still further simplified by taking c = к/3 as

B x D2/L xc = W; [16] where c is the constant number in the Table, and can be determined by the equation

Lxw/BxD2 = c. [17]

120. In this manner the strength of any beam with a rectangular cross section may be obtained, no matter how the load may be placed upon it.

121. When a square beam is strained in the direction of its diagonal the strength is decreased in the proportion of 0.7071 to 1.*

122. The strength of a solid cylinder is as the cube of its diameter (= D),† therefore D3xc/Lx1.7 = W. [18]

The ratio of the strength of square beams to cylinders being the same as their stiffness.

A hollow cylinder is both stronger and stiffer than a solid one containing the same quantity of material; therefore, where it is desirable to combine strength and lightness, cylinders may be made hollow. In timber this is rather too expensive an operation to be often employed; but there are cases where it is useful. The strength of a tube, or hollow cylinder, is to the strength of a solid one as the difference between the fourth powers of the exterior and interior diameters of the tube divided by the exterior diameter, is to the cube of the diameter of a solid cylinder: the quantity of material in each being the same.

* ' Philosophical Magazine,' vol. 1., p. 418, and Stoney's 'Theory of Strains,' p, 51.

† Emerson's ' Mechanics,' sect. viii.; Gregory's ' Mechanics,' vol. i.

123. A beam with a triangular cross section, supported at the ends, is about one-tenth stronger when the base is upwards than when it is downwards, but in the former case the sharp angle has to be cut partly away to give a bearing on the supports.

124. The strongest beam that can be cut out of a round tree is that of which the depth is to the breadth as the square root of 2 is to 1; * or nearly as 7 is to 5. And the strength of a square beam cut from the same cylinder, or round tree, is to the strongest beam nearly as 101 is to 110; but the square beam would contain more timber nearly in the ratio of 5 to 4.714.