139. Wooden bridging, owing to its cheapness and fitness for universal application, has been and is being adopted in all parts of the country. Almost any variety of form may be seen upon our railroads, and though less durable than stone or iron, it may with proper precaution be made to last a long time.

OF THE FORCES AT WORK IN BRIDGES.

140. There are four distinct strains to which a piece of timber or a bar of metal may be exposed, each of which tends to destroy the piece in a different manner. The amount and character of these strains, depend upon the position of the bar or beam, and upon the direction of the force.

A beam may be pulled apart by stretching,—Tension.

It may be destroyed by crushing,—Compression.

It may be broken transversely,—Cross strain.

It may be crushed across the grain,—Detrusion.

141. If one thousand pounds were hung from the end of a suspended timber, so that the direction of the weight coincides with the axis of the timber, then will the tension upon the beam be one thousand pounds.

If the direction of the force is vertical, and the beam is inclined, then the strain is increased by as much as the diagonal of inclination exceeds the vertical; for example, let one thousand pounds be suspended from the lower end of a beam ten feet long, inclined at an angle of 45°. The diagonal being ten, the vertical will be 7.07 feet, and the strain is increased as follows:—

As the angle of inclination, from the horizontal, increases, the strain from a given load decreases, until the beam is vertical, when a weight acts with its least power.

COMPRESSION.

142. If a vertical post is loaded with one thousand pounds, the compressive strain upon that post will also be one thousand pounds. If a post is inclined, the amount of strain is increased, as noticed in the case of tension, and to the same amount, that is, depending upon the inclination.

A piece of wood or metal acting as a post, or pillar, must not only be able to resist crushing, but also bending or bulging laterally.

143. A cylinder of which the length is only seven or eight times the diameter, will not bulge by any force that can be applied to it longitudinally, but will split. When the length exceeds this, it will be destroyed by a similar movement to that produced by a cross strain. When the length of a cast-iron pillar is thirty diameters, the fracture is produced by bending alone; when less, partly by bending and partly by fracture. When the column is cast hollow, and enlarged towards the middle, the strength is increased in a very great ratio.

144. The formula for finding the weight which any beam acting as a post, will support before bending, is, according to Barlow, who considers the weight as varying inversely as the length, as follows:—

and the value of W is

and the weight being given, and the sectional dimensions assumed, we have

and

CROSS STRAIN.

145. The amount of strain caused by any weight applied in a transverse direction, to a beam supported at both ends, is as the breadth, as the length inversely, and as the square of the depth. Whatever depression takes place, tends to shorten the upper, and to extend the under-side; whence the fibres of the top part suffer compression, and those of the bottom extension. The amounts of compression and extension must of course be equal, and therefore if any material resists these two strains in a different degree, the number of fibres opposing each will also be different.

The top being compressed, while the bottom is extended, of course at some point within the beam there exists a line which suffers neither compression nor extension. The position of this line (the neutral axis) depends upon the relative power of the material to oppose the strains, upon its form and upon its position. Thus if wood resists two thousand pounds per square inch of extension, and one thousand pounds of compression, the axis will be twice as far from the top as from the bottom.

In some materials the neutral axis changes its place while the bar is at work; thus wrought iron, after being a little compressed, will bear a great deal more compression than when in its original state; also the lower fibres, after being extended, will resist less than at first; the effect of which two actions is to move the neutral axis up.

146. The following table shows the relative resisting powers of wood, wrought and cast-iron; with the corresponding positions of the axis, with sufficient accuracy for practice.

Thus in beams subjected to a cross strain, as well as to a direct extensile or compressive one, the resistance is effected by the incompressibility and inextensibility of the material.

147. The formula for dimensioning any beam to support a given weight transversely is

DETRUSION.

148. Detrusion, or crushing across a fixed point, is such as occurs wherever a brace abuts against a chord, or where a bridge bears upon a bolster or wall plate; also the shearing of bolts, pins, and rivets.

GENERAL RESISTANCE OF MATERIALS.

149. The resistance to extension, to compression, (as regards simple crushing,) and to detrusion, is as the area of cross section; i. e., if we double the area, we double the strength. The resistance to a cross strain is as the breadth, as the length inversely, and as the square of the depth; i. e. if we double the breadth we double the strength; if we double the length, we divide the strength by two; and if we double the depth, we multiply the strength by four.

ACTUAL STRENGTH OF MATERIALS.

150. Any material will bear a much larger load for a short time than for a long one. The weight that does not so injure materials as to render them unsafe, is from one third to one fourth only of the ultimate strength. Throughout the present work one fourth will be the most that will in any case be used.

WROUGHT IRON.

151. Extension.

Or in round numbers 15,000 lbs. per square inch, is the resistance of wrought iron to extension, to be used in practice.

152. Compression.—Great discrepancies appear among writers on the strength of materials, as to the compressive strength of wrought iron. Though all estimate the resistance to compression, as great as to extension, yet no one in summing up the general result of experiment, places the former at more than from 50 to 75 per cent. of the latter. William Fairbairn gives, as the relative resistances to extension and compression in bars applied as girders, 2 to 1.

As far as practice is any guide, from 8,000 to 12,000 pounds per inch is the most to be used. The ratio of 90 to 66, seems to express very nearly the action as in the most reliable structures; which will, therefore, be adopted, or 11,000 pounds per square inch nearly. The resistance to compression is very much greater after wrought iron has been somewhat compressed.

CAST-IRON.

153. Extension.—This material is seldom used to resist a tensile force. That the tables may be complete, however, the following is given:—

154. Compression.

155. Following are given the condensed results of the preceding figures, which may be relied upon as giving perfectly safe dimensions in practice.

For additional remarks on iron, see chap. IX.

156. Nature and Strength of American Woods.

157. The lateral adhesion of fir was found, by Barlow, to be six hundred pounds per square inch. (Lateral adhesion is the resistance which the fibres offer to sliding past each other in the direction of the grain; as, in pulling off the top of a post where it is halved on to the chord.)

158. As regards the nature of timber, seasoning, time of cutting, etc., although these are important items, still, generally, commercial considerations outbalance all else. The most complete treatise on the nature of woods, is “Du Hamel, L′exploitation des bois;” from which it appears that the best oaks, elms, and other large trees, are the product of good lands, rather dry than moist. They have a fine, clear bark, the sap is thinner in proportion to the diameter of the trunk, the layers are less thick, but more adherent the one to another; and more uniform than those of trees growing on moist places. The grain of the latter may look very fine and compact, but microscopic examination shows the pores to be full of gluten.

The density of the same species of timber, in the same climate, but on different soils, will vary as 7 to 5; and the strength, both before and after seasoning, as 5 to 4.

In trees not beyond their prime, the density of the butt is to that of the top, as 4 to 3; and of centre to circumference, as 7 to 5. After maturity, the reverse occurs in both cases.

Oak, in seasoning, loses from ¼ to ⅓ of its weight; but its strength is increased from 30 to 40 per cent.

GENERAL TABLE OF THE NATURE OF MATERIALS.

159. The tensile strength of wrought iron assumed as 1,000.

The advantage possessed by iron over wood, is in durability only. The above figures show how much more of the strength of the material is consumed by its own weight in iron than in wood. In actual practice, however, the method of making joints and other details often render iron the lightest material.

RULES FOR PRACTICE.

TENSION.

160. The tensile strength of any material, is expressed by the formula

whence the necessary area of section of any material to resist a tensile strain, is found by the following rules:—

Wrought Iron

Cast-Iron

Wood

COMPRESSION.

161. Wrought Iron

Cast-Iron

Wood

CROSS STRAIN.

162. The power of any material to resist a cross strain, is shown by the formula

and to reduce the load to one fourth of the breaking weight

and finally, by substituting for 4s, 4 × 1,250, (1,250 of the table of woods,) we have

Also, knowing the weight to be supported, and requiring the dimensions, we take out the values of d and b, and have

As an example of the use of the formula, take the following:—

required the load.

The formula

becomes

Again, the weight to be supported being 15,000 lbs., length 30 feet, breadth 16 inches, the formula for the depth becomes

also,

163. The formula, expressive of the strength of a cast-iron beam, is

from which we have

WROUGHT IRON.

164.

whence

165. Mr. Hodgekinson found, that by arranging the material in a cast-iron beam, as in fig. 60, that the resistance per unit of section was increased over that of a simple rectangular beam, in the ratio of 40 to 23. He makes the general proportion of such girders as follows:—

In this consummate disposition of material, the areas of top and bottom flanges are made inversely proportional to the power of cast-iron to resist compression and extension.

166. Mr. Fairbairn found, that in wrought iron flanged girders, (under which come the various rails, chap. XIII.,) the top web should contain double the area of the lower one. This agrees with the conclusion adopted on page 129, as wrought iron resists more extension than compression.

167. In cast-iron girders, on no account should there be introduced webs, or openings of any kind, either from economic or ornamental motives; as the uniformity of cooling is thereby very much opposed.

168. Mr. Hodgekinson gives, as the result of his experiments, the following formula for dimensioning the cast-iron girder above referred to.

As it is not considered safe to load a cast-iron beam with more than one sixth of the breaking load, the formula may be expressed as follows:—

for the weight in tons which may be safely borne, and transforming

for the area of the lower flange.

Example.—Required the dimensions of a cast-iron beam, of Mr. Hodgekinson’s form, for a span of thirty feet, to support a load of ten tons at the centre.

and 32.58
6.1 = 5.34.

and the area of the top flange will be

whence the following dimensions:—

OF POSTS.

169. A post may be very well able to resist the compressive strain thrown upon it by any load, but may bulge, or bend, laterally.

The formula by which beams are dimensioned for this requirement, changes with the material, and with the form of section. For rectangular posts of wood, we have the formula below.

170. The value of the formula for the strength of cast-iron posts, seems to depend more upon the authority consulted than upon the nature of iron. For example, assume the length of a post as twenty feet, and the diameter as ten inches; the load which may be safely borne is, according to six different authorities, as follows:—

and assuming the length as ten feet, and diameter as ten inches, we have

showing not only a great difference in the unit resistance taken, but also in the effect of the ratio between the length and diameter.

Such being the discrepancy, there will be given no formula; but in place of such, the table following, which is calculated from the rules least opposed to experimental evidence.