Handbook of Railroad Construction

Chapter VIII Wooden Bridges

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.

Tension

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:—

7.07 to 10 as 1,000 to 1,414 lbs.

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:—

WL²
80E = bd³,

and the value of W is

bd³ × 80E
L²,

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

d = ∛(WL²
80Eb),

and

b = WL²
80Ed³,

Where W represents the weight in pounds,
      L represents the length in feet,
      E represents a constant,
      d represents the depth in inches,
      b represents the breadth in inches.

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.

Material.	Resistance to extension.	Resistance to compression.	Ratio.	Distance of axis from top, in fractions of the depth.
Wrought iron,	90	66	90 66	90 156	or 0.58
Cast-iron,	20	111	20 111	20 131	or 0.15
Wood,	2	1	2 1	⅔	or 0.66

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

S = 4bd²
e,

Where S represents the ultimate strength in lbs.
      b represents the breadth in inches,
      d represents the depth in inches,
      e represents the length in inches,

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.

	lbs. per square inch.
Mean of 17 experiments by Barlow (p. 270)	62,720
Weisbach’s Mechanics (Vol. ii., p. 71)	60,500
Overman’s Mechanics, (p. 408, 409)	61,333
Brown, Rennie, and Telford, (mean)	65,251
	———
The mean	62,451
	———
Reducing by 4 for safety	15,613

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.

We have by Weisbach	56,000
We have by Rondelet	70,000
We have by Hodgkinson	65,000
	———
The mean	63,667
	———
Reducing by 4	15,917
	———
In round numbers	16,000	lbs. per square inch.

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:—

By Weisbach	20,000	pounds.
By Barlow	18,233	pounds.
By Overman	20,000	pounds.
By Rennie	18,000	pounds.
By Hodgkinson	16,577	pounds.
By the British Iron Commission	15,711	pounds.
	———
The mean	18,087	pounds.
	———
Reducing by 4	4,522	pounds.
	———
In round numbers	4,500	pounds.

154. Compression.

By Weisbach	109,800	pounds.
By Hodgkinson	107,520	pounds.
By Iron Commission	100,000	pounds.
Stirling’s toughened	130,000	pounds.
	———
Mean of Common	105,773	pounds.
	———
Mean of Stirling’s	130,000	pounds.
	———
Reducing by 4 for safety (Common)	26,443	pounds.
	———
Reducing by 4 for safety (Stirling’s)	32,500	pounds.
	———
In round numbers (Common)	25,000	pounds.
	———
In round numbers (Stirling’s)	30,000	pounds.

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

Wrought Iron.	Cast-Iron.
15,000	4,500	Tensile strength,
11,000	25,000	Compressive strength.

For additional remarks on iron, see chap. IX.

156. Nature and Strength of American Woods.

Name of the wood.	Weight per cubic foot.	Resistance to Extension.	Resistance to Compression.	Value of S.	Elasticity.
White Pine	26	12,000	6,000	1,229	———
Yellow Pine	31	12,000	6,000	1,185	———
Pitch Pine	46	12,000	6,000	1,727	4,900
Red Pine	35	12,000	6,000	1,527	7,359
Virginia Pine	37	12,000	6,000	1,456	———
Spruce	48	12,000	6,000	1,036	———
Larch	33	12,000	6,000	907	2,465
Tamarack	26	12,000	6,000	907	———
White Cedar	22	8,000	4,000	766	———
Canada Balsam	34	12,000	6,000	1,123	———
White Oak	48	15,000	7,500	1,743	8,595
Red Oak	41	15,000	7,600	1,687	———
Live Oak	72	15,000	7,200	1,862	———
White Beech	44	18,000	9,100	1,380	5,417
Red Beech	48	18,000	9,000	1,739	———
Birch	44	15,000	7,000	1,928	———
Black Birch	41	15,000	7,200	2,061	———
Yellow Birch	36	15,000	7,200	1,335	———
Ash	38	16,000	8,100	1,795	6,581
Black Ash	35	16,000	8,000	861	———
Swamp Ash	57	16,000	8,000	1,165	———
Hickory	51	15,000	7,200	2,129	———
Butternut	54	15,000	7,600	1,465	———
Ironwood	54	16,000	8,100	1,800	———
Rock Elm	45	16,000	8,011	1,970	2,799

The mean tensile strength of wood is	14,080	lbs.
Reducing by 4 for safety	3,520	lbs.
Reducing for want of seasoning	2,000	lbs.
The reduced mean compressive strength	1,000	lbs.
Reduced resistance to detrusion	150	lbs.
Ratio of tensile to compressive strength	2 to 1.
Mean value of S in formula (WL = 4Sbd2) for the woods most used in practice	1,250.

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.

Material.	Tension.	Compression.	Cross Strain.	Sum.	Weight per cubic ft.	Sum divided by weight per cub. ft.
Cast-Iron	300	1,666	31.68	1,997.68	450	4.4
Wrought Iron	1,000	733	55.40	1,788.40	480	3.7
Wood	133	66	5.60	204.60	30	6.8

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

T = Sa,

Where T represents the whole strength,
      S represents the strength per square inch,
      a represents the area of section in inches.

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

Wrought Iron

a = W
15,000,

Cast-Iron

a = W
4,500,

Wood

a = W
2,000.

Compression

161. Wrought Iron

a = W
12,000,

Cast-Iron

a = W
25,000,

Wood

a = W
1,000.

Cross Strain

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

W = 4sbd²
L,

Where W represents the breaking weight in pounds,
      s represents the constant in the table of woods,
      b represents the breadth in inches,
      d represents the depth in inches,
  and L represents the length in inches,

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

W = 4sbd²
4L,

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

W = 5000bd²
4L.

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

d = √(W × 4L
5000b) = the depth,

b = W × 4L
5000d² = the breadth.

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

Let the span, or length, be 20 feet,
The breadth 12 inches, and depth 18,

required the load.

The formula

W = 5000bd²
4L

becomes

W = 5000 × 12 × 18²
4 × 240 = 20,250 lbs.

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

d = √(15000 × 1440
5000 × 16) = √270 = 16 inches,

also,

b = 15000 × 1440
5000 × 256 = 21600000
1280000 = 16 inches.

Cast-Iron

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

850bd² = WL, from which we have

b = LW
850d² = the breadth,

and d = √(L × W
850b) = the depth.

Wrought Iron

164.

952bd² = WL, whence

b = WL
700d² = the breadth,

and d = √(LW
700b) the depth.

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:—

Length	16
Height	1
Area of top flange	1.0
Area of lower flange	6.1

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.

W = 26ad
L,

Where W is the breaking weight in tons,
      a the area of the bottom flange,
      d the depth of the girder in inches,
      L the length in inches.

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:—

W = 26ad
6L,

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

a = 6WL
26d

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.

Span	30 feet,	Whence—
Length	34 feet,	Length	34 feet,
Load	10 tons at centre.	Span	30 feet,
		Depth	25½ inches,
		Lower flange	32.58 square inches,
		Upper flange	5.34 square inches,

a = 6 × 10 × 12 × 30
26 × 34 × 12
16 = 32.58

and 32.58
6.1 = 5.34.

and the area of the top flange will be

36
6 = 6,

whence the following dimensions:—

Length	30	feet,
Depth	23	inches,
Lower flange	36	square inches,
Upper flange	6	square inches,

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.

W = 2240bd³
L2,

Where W represents the weight in lbs., which may be safely borne,
      b represents the breadth in inches,
      d represents the depth in inches,
  and L represents the length in feet.

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:—

A	4,000,000
B	181,100
C	370,000
D	940,000
E	307,242
F	300,000

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

A	8,007,500
B	204,500
C	1,442,500
D	3,640,000
E	1,170,000
F	600,000

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.

TABLE SHOWING THE LOAD IN POUNDS SAFELY BORNE BY CAST-IRON COLUMNS.
HOLLOW CYLINDERS.											H AND + SECTIONS.
Diameter in inches.	Length or height in feet.									Metal thickness.		Length or height in feet.
	6	8	10	12	15	18	20	22	24			6	8	10	12	15	18	20	22	24
2	6000	5000	4000	3000	2500	1800	1500	1300	1100	¼		4000	3000	2400	1800	1400	1100	1000	900	800
3	16000	14000	13000	11000	9000	7000	6000	5000	5000	⅜		12000	11000	10000	9000	8000	7000	5000	4000	3000
4	30000	29000	26000	24000	22000	18000	16000	14000	13000	½		25000	23000	21000	18000	16000	13000	12000	9000	6000
5	50000	37000	45000	42000	39000	37000	31000	28000	26000	⅝		36000	34000	31000	28000	25000	23000	21000	20000	18000
6	59000	57000	55000	52000	49000	44000	41000	38000	36000	¾		40000	38000	37000	36000	35000	34000	32000	30000	28000
7	101000	99000	96000	92000	88000	81000	76000	72000	68000	13 16		60000	59000	58000	57000	56000	54000	53000	51000	49000
8	131000	129000	126000	122000	118000	109000	105000	100000	96000	⅞		100000	98000	96000	94000	91000	88000	83000	78000	70000
9	169000	167000	164000	160000	156000	146000	141000	136000	131000	1		140000	130000	126000	120000	114000	110000	106000	100000	90000
10	210000	200000	200000	200000	190000	180000	180000	170000	170000	1⅛		190000	180000	170000	160000	150000	140000	130000	125000	120000
11	250000	250000	240000	240000	240000	230000	220000	220000	210000	1¼		230000	220000	210000	200000	190000	180000	170000	160000	150000
12	300000	300000	290000	290000	290000	270000	270000	260000	260000	1½		280000	260000	250000	240000	230000	220000	200000	190000	180000
14	450000	430000	410000	380000	370000	350000	330000	320000	300000	1¾		360000	320000	310000	300000	290000	280000	270000	260000	240000
16	520000	500000	480000	460000	440000	420000	400000	370000	350000	2		460000	430000	400000	370000	350000	330000	310000	300000	280000
18	650000	630000	610000	590000	560000	520000	470000	430000	400000	2½		560000	530000	510000	480000	440000	410000	380000	350000	330000
20	800000	760000	740000	690000	650000	590000	540000	490000	450000	3		600000	580000	550000	520000	500000	460000	430000	400000	380000
Diameter in inches.	6	8	10	12	15	18	20	22	24	Metal thickness.		6	8	10	12	15	18	20	22	24
	Length or height in feet.											Length or height in feet.

Of the Truss

171. The most simple bridge that could be built, consists of a single piece of timber placed across the opening to be spanned. This form is applicable to spans under twenty feet. The proper dimensions are found by the formula:—

d = √(4wL
5000b) = the depth.

Example.—The depth of a beam of twenty feet span, and twelve inches wide, to support a load of twenty thousand two hundred and fifty lbs. is

d = √(4 × 20,250 × 20 × 12
5000 × 12) = 18 inches.

A beam 12 × 18, and of 20 feet span, will therefore bear safely a load of 20,250 lbs., applied at the centre.

In this manner is formed the following table, giving the scantling of sticks for railroad stringer bridges, of twenty feet span and under.

Span.	Breadth.	Depth.
5	12	12
10	12	13
12	12	15
15	12	18
18	12	20
20	12	21	inches.

The first scantlings exceed the requirement of the rule, but are none too large to resist the shocks to which such sticks are exposed.

Cross-ties of plank, 2 or 3 by 6 or 8 inches, and plank braces underneath, (as shown in the fig. at the end of chapter VIII.,) should be bolted to the main timbers; the same bolt passing through the tie beam and plank. The longitudinal pieces should be firmly notched and bolted to the wall-plates, and these latter either built in or scribed on to the masonry.

172. For a span of from 20 to 50 feet, we may use the combination shown in fig. 61. The piece A B, must be so strong as not to yield between A and D, or D and B. The pieces C E must be stiff enough to resist the load coming upon them which is as follows. A locomotive engine of the heaviest class will not exceed fifty tons weight, each pair of driving wheels will support ten tons, and on each side five tons, 2240 × 5 = 11,200 lbs.; or to allow for shocks and extra strains, 15,000 lbs. Each brace, then, must support seven thousand five hundred pounds, which for compression simply would require only seven and one half square inches of sectional area; but the brace being inclined, the strain is increased as follows:—

A E to E C as 7,500 to X.

And A E being ten feet, and A D fifteen feet, E C becomes eighteen feet, whence

10 to 18 as 7,500 to 13,500 lbs.; which would require only thirteen and one half inches for compression, or a piece 4 × 3½. But is this enough for flexure?

On page 124 the load which may be safely borne, by a rectangular post of wood, is shown by the formula

W = 2240bd³
L².

Substituting for b and d, the dimensions 4 × 4, we have

W = 2240 × 4 × 4³
18², or 573,440
324, = 1,770,

which is evidently too small.

Placing 6 × 7, for b × d, we have

W = 2240 × 6 × 7³
18² = 14,227;

exceeding by a small amount the requirement.

173. It is evidently immaterial whether we support the point D upon C, or suspend it as in fig. 62, provided we prevent any motion in the feet of the inclines A c B c. Abutting them against A B, throws a tension against A B, found as follows:—

Representing by c D, the applied weight, draw D E parallel to c B; also E f parallel to A B; E f is the tension. The graphic construction gives results near enough for practice. Rigorously we have

A c D, similar to E c f; also,

A c, to E c, as A D, to E f; and

Ef = Ec × AD
Ac.

When a d and c d are differently inclined, proceed as follows. See fig. 102, p. 200 inverted.

Let d b represent the weight; e h shows the tension. The triangles a c d, and a b e, are similar; as also e b h and d b c; whence

b e = a b, c d
a c, and e h = c b, b e
d c = tension.

In practice place w for b d; i. e. the actual weight.

In this plan, if the chord is able to resist the cross strain between A and D, it will also resist the tension. This cross strain is found by the formula already given and illustrated.

174. From what precedes, we have the following dimensions for bridges such as are shown in figs. 61 and 62. The details of 62, at f and c, and at E, 61, are shown in figs. 62 A, 62 B, and 61 C.

Span.	Rise.	A B.	C E.		Rod b.
20	8	12 × 12	(5 ×	8)—2	1¼ inches.
25	10	12 × 15	(5 ×	9)—2	1⅜ inches.
30	12	12 × 18	(5 ×	10)—2	1½ inches.
35	13	12 × 20	(5 ×	10)—2	1⅝ inches.
40	14	14 × 21	(5 ×	12)—2	1⅝ inches.
45	15	14 × 22	(6 ×	12)—2	1¾ inches.
50	16	14 × 24	(6 ×	12)—2	1¾ inches.

The braces, (column 4,) being in pairs and blocked together. In spans exceeding twenty-five feet, the braces d f, and the rods f g, should never be omitted. The size of the rod g f, is found by considering A, d, f, as a small bridge.

175. In all light bridges, like the one under consideration, all parts should be fastened by bolts, to prevent springing by reaction. A bridge with but little inertia, or dead weight, tends to jump up when the engine has passed over it. Fastening takes the place of weight in a large span.

As soon as the rise admits, the points C, on each side of the bridge, should be connected to resist lateral motion. When the height is not enough for this, the same points may be joined to a floor beam extended out beyond the truss.

Though the dimensions are given for this plan up to fifty feet span, it is very seldom advisable to go beyond twenty-five or thirty feet; as frames consisting of a few long timbers are not so rigid, and free from vibration, as those made of a greater number of short pieces.

176. In extending this system one hundred or two hundred feet, we see at once that the pieces A c, B c, would become very long and would need to be made large and heavy. We should always so proportion any beam in a bridge that it is at once able to resist all of the several strains to which it may be exposed, without being unnecessarily large.

As to compression, the above system might be extended to almost any amount; but the braces would yield by flexure.

Instead of producing the braces A c, A′ c′, fig. 64, to their intersection, we stop at c and c′, insert c c′; to prevent the approach of these points, suspend the points B and B′ from c and c′, and commence again with the braces B D, B′ D; and so on as far as necessary.

To prevent the backward motion of the points B, and B′, either the chord A A′, or the counter-braces m, m, are necessary.

The pieces A c, A′c′ must support all of the load, including the weight of the bridge, lying within the rectangle B c, B′ c′. The next set of braces must sustain that part of the load only which comes over the centre of the bridge. Thus the braces should decrease in size as the centre is approached. The rods c B, c′ B′, must resist a tension equal in amount to the pressure on the braces, only being vertical they do not need the increase given to the braces on account of their inclination.

177. There is another method of stiffening a beam, as shown in figs. 65 and 66, by trussing rods, and a post. The dimensions being the same, the forces in both cases will be equal. The second, fig. 66, leaves the passage beneath the bridge clear.

The tension on the rods A c, B c, fig. 66, tends to draw the points A and B together, an effort which is resisted by the top chord A B.

In extending this system, as in art. 176, the rods become either very long, or very large, from the small angle of inclination; evils which remedied as before, by supporting the post c B, fig. 64, from the foot of the first rod, fig. 64, and commencing again from c.

To prevent the motion of the triangle c B G, fig. 64, about the angle B, we must introduce either the upper chord c c′, or the counter rod c A. If the lower chord is omitted the rod D B must be of the same size as E B. In this truss, either the top or the lower chord simply may theoretically be omitted, due allowance being made in the size of the rods. In practice it is never advisable to omit either, as both are required for lateral bracing, and for support of the road-way.

Having said thus much of the general ideas that apply to all bridges, let us now look at some of the plans most in use; and to become familiar with the subject, work out the dimensions of an example of each kind.

178. As rods, nuts, and washers are used in all bridges, the following table may not be out of place:—

Column 1 gives the diameter of rod.
Column 2 strength at 15,000 lbs. per square inch.
Column 3 the weight per lineal foot.
Column 4 side of the square nut.
Column 5 the thickness of the same.
Column 6 the dimensions of washers.
Column 7 the thickness of washers.
Column 8 breadth (side to side) six-sided nut.
Column 9 breadth (across angles) six-sided nut.
Column 10 thickness of six-sided nut.
Column 11 number of screw threads per inch.
Column 12 gives the diameter of rod.

1	2	3	4	5	6	7	8	9	10	11	12
Diameter.	Strength of Rod.	Weight per Foot.	Square Nut.	Thickness.	Square of Washer.	Thickness of Washer.	Six-Sided Nut.	Six-Sided Nut.	Six-Sided Nut.	Screw.	Diameter.
½	2,940	0.66	1¼	¾	2½	¼	1⅜	1½	9 16	12	½
¾	6,630	1.49	1½	⅞	3	¼	1¾	2	⅞	10	¾
1	11,775	2.65	2	1	4	⅜	1⅞	2¼	1⅛	8	1
1⅛	14,910	3.36	2	1⅛	4½	⅜	2⅛	27 16	1¼	7	1⅛
1¼	18,405	4.17	2¼	1¼	5	½	2¼	211 16	17 12	7	1¼
1⅜	22,260	5.02	2½	1⅜	5½	½	2½	2⅞	17 16	6	1⅜
1½	26,505	5.97	2¾	1½	6	½	2⅝	3⅛	111 16	6	1½
1⅝	31,095	7.01	2⅞	1⅝	6½	⅝	2⅞	35 16	113 16	5	1⅝
1¾	36,075	8.13	3	1¾	7	⅝	3	3½	2	5	1¾
1⅞	41,415	9.33	3¼	1⅞	7½	⅝	3¼	3¾	2⅛	4½	1⅞
2	47,130	10.62	3½	1⅞	8	⅝	3½	4	2¼	4½	2
2⅛	53,190	12.00	3¾	2	8½	¾	3⅝	43 16	2⅜	4	2⅛
2¼	59,640	13.40	4	2⅛	9	¾	3¾	46 16	2½	4	2¼
2⅜	66,450	15.00	4⅛	2¼	9½	¾	4	4⅝	2⅝	4	2⅜
2½	73,620	16.70	4¼	2½	10	¾	4¼	4⅞	2¾	3½	2½
1	2	3	4	5	6	7	8	9	10	11	12

179. Let us now assume the following data:—

Span	200	feet,
Rise (centre to centre of chords)	25	feet,
Width	20	feet,
Length of panel	15	feet,
Weight (bridge and load) per lineal ft.	4,000	lbs.

Howe’s Bridge

Lower Chords.—The tension, at the centre of the lower chord, is found by dividing the product of the weight of the whole bridge and load by the span, by eight times the height, or

T = W × S
8h,

which becomes, with the above data,

T = 800000 × 200
200 = 800,000 lbs.

Here the tension and the total weight are equal, a result which can occur only when the rise is one eighth of the span. This is the best ration between these dimensions, as then the horizontal and vertical forces are equal.

As to the proportion of the panel, (or the rectangle inclosed by the chords and any two adjacent posts,) the ratio of base to height should be such as to make the inclination of diagonal about 50° from the horizontal; if much less, the timbers become large and heavy; and if more, the number of pieces is unnecessarily increased.

The braces at the end of a long span, may be nearer to the vertical than those near the centre, as they have more work to do. If the end panel be made twice as high as long, and the centre panel square, the intermediates varying as their distance from the end, a good architectural effect is produced.

To determine the size of the lower chords, to resist the above 800,000 pounds of tension, proceed as follows: Each side truss will support one half of the whole load, or 400,000 pounds; which, at 2,000 pounds per inch, will require 200 square inches of section. Four sticks of 8 × 12 inches, give an area of 384 square inches, which must be reduced as follows: Deduct 72 square inches for the area cut out by the splicing blocks, 40 inches for the bolts connecting the pieces, 28 inches for inserting the foot blocks, and 10 inches for inserting the washer, and we have remaining 234 square inches; which exceeds by a little the exact demand. This excess (about one seventh) is a necessary allowance for accidental strain, to which all bridges are subjected.

The splices used in bridge framing are shown in fig. 67 A and fig. 67 B. For the first, the depth of insertion and length of the block depend upon the tension upon the chord. The following dimensions have been much used and are perfectly reliable:—

Span of Bridge.	A C	B C	C D
Feet.	Feet.	Inches.	Feet.
50	1.00	1½	1.50
100	1.25	2	2.00
150	1.75	2½	2.25
200	2.00	3	2.75

There is no need of cutting more than one notch, as in the figure; the resistance of the triangles is thereby lessened, and the work increased.

In fig. 67 B, the rods must of course be able to resist the tension upon the one piece which is cut.

Upper Chord.—The upper chords of a bridge suffer compression, to the same amount numerically, as the tension on the lower chord; as whatever tension is thrown by any brace upon the lower chord, reacts as just so much compression upon the upper. In the case at hand, 800,000 lbs. in all, or 400,000 on each chord.

The resistance to compression being one thousand pounds per inch, renders necessary four hundred inches of section to each chord; four pieces 8 × 12 give in all three hundred and eighty-four inches of section, which requires no reduction, as the whole chord pressing together and being properly framed is not weakened by splicing. The splicing blocks in the upper are merely plain pieces, inserted one half inch, the only duty being to keep the sticks at the proper horizontal distance.

The spaces between the pieces should be large enough to allow the rods to pass without cutting the chords; (two inches answers every purpose). The bolts for splicing, have no very great strain to bear. In small spans from ½ to ⅝, and in large bridges from ⅝ to an inch is enough.

The object in framing a built beam for a bridge chord, is to make a stick which shall be uniformly strong. This is done by cutting the pieces in the centre of the panel, and by having no two joints in either chord in one panel; though in long spans this cannot always be done. Figs. 67 D and 67 E (page 153)

Braces

The whole load being	800,000 pounds,
Each truss supports	400,000 pounds,
Each set of braces	200,000 pounds,
Each brace (there being 4)	50,000 pounds,

which must be increased for inclination as follows: The length of diagonal is twenty-nine feet, (the height being twenty-five and length 15,) whence

25 to 29 as 50,000 to 58,000 lbs.; which would need fifty-eight square inches, or 7 × 8 for compression; which, however, is quite too small for flexure. 12 × 12 placed in the formula gives

W = 2240 × bd³
L²,

or

W = 2240 × 12 × 1728
841 = 55,296 lbs.

In practice, smaller braces than 12 × 12 would answer, because the four braces in a set may be fastened together, making a post of four pieces 8 × 12, or in all a built post of 44 × 12 inches; twelve being the depth, whence

W = 2240 × 44 × 1728
841 = 202,511 lbs.;

the forty-four inches being made by blocking the braces four inches apart. The second set of braces are to be treated in the same manner, the weight to be supported being only the rectangle included by those braces; i. e. the whole bridge and load less the two end panels.

As the centre of the span is approached, the pressure on the braces becomes very small; and the scantling of the braces will be reduced to about 6 × 7 inches.

Rods

The weight upon the first set of rods is the same as that upon the end sets of braces; in the present case 800000 ÷ 2 = 400000 on each side truss, and 400000 ÷ 2 = 200000 on each end; and if there are five rods in each set, each rod bears 40,000 lbs. Referring to the table on p. 146, opposite to 41,415 lbs., is the diameter 1⅞ inches; whence the first set must contain five rods, of 1⅞ inches diameter. The second set decrease in size as the weight is lessened by the two end panels. The nut and washer for the rod are also found in the same table.

Counterbracing

180. When a load is placed on the point C′, fig. 64, the truss tends to sink at that point, and a corresponding rise takes place at C. This motion changes the figure A B C E, from a rectangle to an oblique angled figure; the diagonal E B being shortened, and A C lengthened. This motion is easily checked by the introduction of the counter brace E B.

The action which this timber is called upon to resist, being caused by the moving or variable load on one panel, the brace must resist the load coming thereon, (say fifteen feet,) and is thus the same size as the brace at the centre of the span.

The counter braces may be so confined between the braces, at the intersection, as not to move laterally or vertically, but must not be fastened to the braces; because the action of the separate timbers is thus trammelled.

The manner of adjusting the braces and counter braces to the chord is shown in fig. 67 C. It was formerly the custom to abut the braces against a block on one side of the chord, and to screw the rod against a block on the opposite side; the whole strain acting to crush the chord crosswise. This has been remedied by the arrangement shown in the figure, the two blocks being cast in one piece and connected by a small hollow cylinder passing between the chord sticks.

This system is known as Howe’s bridge, and may be seen in almost any section of the country; and though in many cases badly proportioned, and of bad material, if properly made answers a very good end.

The following table has been formed for the use of engineers and builders, giving, together with the table of nuts and washers, all dimensions required.

Span.	Rise.	Panel.	Chords.	End braces.	Centre braces.	End rod.	Centre rod.
50	10	7	2—8 × 10	72	5 × 5	2—1⅛	2—1
75	12	9	2—8 × 10	82	5 × 5	2—1½	2—1
100	15	11	3—8 × 10	92	6 × 6	2—1¾	2—1
150	20	13	4—8 × 12	102	6 × 7	3—2	3—1
200	25	15	4—8 × 16	122	7 × 7	5—2	5—1

Pratt’s Bridge

181. Assume the following data for an example:—

Span	100	feet.
Rise	12	feet.
Panel	10	feet.
Weight per lineal ft.	2,500	lbs.

The tension on the lower, or the compression on the upper chord, will be

250000 × 100
96 = 260,417 lbs.

The manner of dimensioning the chord, and of splicing, is the same as already described for Howe’s.

Suspension Rods

The first sets of rods, A B, A′ B′, must sustain the whole weight of the bridge and load; which is 250,000 lbs. Each side 125,000 lbs.; and each end set of rods 62,500 lbs.; and if each set has four rods, each rod must support 15,625 lbs.

The rod being inclined, this amount is increased by the following proportion:—

12 (height) to 15.8 (diagonal) as 15,625 to 20,573 lbs.

This is half-way between the tabular numbers for rods of 1¼ and 1⅜ inches in diameter; 1⅜ will therefore answer. The next set of rods must be considered as supporting the whole load, less the two end panels, and so on as already explained for Howe’s bridge. The manner of applying the rods to the chords is shown in fig. 68 A. The bevel block should be connected with the block at the foot of the post, so as to prevent crushing the chord.

Counter Rods

As both top and bottom chords are always used in this bridge, the counter rods have only the variable load on one panel to resist. The action is, in amount, the same as that on the counter braces in Howe’s bridge; but acts in a different direction, and in the other diagonal.

The weight of a passing load cannot be more than two thousand pounds per lineal foot. The panel being ten feet long, the whole weight coming on two sets of counter rods, (one set in each side truss,) is twenty thousand pounds; or ten thousand pounds on each set; and if there are put three rods in each set, we have 3,333 pounds per rod, which increase for inclination as follows:—

12: 15.8:: 3333: 4389 lbs., requiring a rod of three fourths inch diameter.

The posts in this structure, correspond to the braces in the Howe bridge; only being vertical, they need not be so large.

182. The following table gives all the dimensions necessary for proportioning this truss.

Span.	Rise.	Chords.	End post.	C post.	End rod.	C rod.	Counter rod.
50	10	2—8 × 10	5 × 5	42	2—1⅜	2—1	1—1½
75	12	2—8 × 10	6 × 6	52	2—1⅝	2—1	1—1½
100	15	3—8 × 10	7 × 7	62	2—1¾	2—1	2—1⅛
125	18	3—8 × 10	8 × 8	62	3—1⅞	3—1	2—1⅜
150	21	4—8 × 12	9 × 9	62	3—2⅛	3—1	3—1⅛
200	24	4—8 × 16	10 × 10	62	5—1⅞	5—1	3—1⅛

And the following, the sizes of counter rods, for different panels.

Length of panel.	Height of panel.	Approximate diagonal of panel.	Diameter of the rod.
			One in a set.	Two per set.	Three per set.
10	12	16	1⅝	1⅛	⅞
11	13	17	1⅝	1¼	1⅛
12	14	18	1¾	1¼	1⅛
13	15	20	1¾	1¼	1⅛
14	16	21	1⅞	1⅜	1⅛
15	18	23	1⅞	1⅜	1⅛
16	21	26	2	1⅜	1⅛
18	25	27	2	1⅜	1⅛

The advantage possessed by this bridge, over Howe’s plan, is that the panel diagonals may be adjusted by the screws; by which control is had over the form of the truss, and of the duty done by the several parts. Change of form cannot be had by working upon verticals. Howe’s bridge must be adjusted by wedging the braces and the counter braces.

183. The manner of drawing the bevel block in this bridge, is shown in fig. 68 b. The proportions of the block depend upon the proportions of the panel; and the dimensions, upon the size of washer used.

Let C C be the centre line of the post, and A B the chord. Let o m, and o n, be the panel diagonals, and H and y, the length of the washers.

The depth of insertion of the block into the chord, depends upon the horizontal strain upon it. In a span of one hundred and fifty feet, with the rods at an angle of 50°, two inches have been found ample at the end of the truss, and one half inch at centre.

From D, perpendicular to m m, lay off D E; equal to H, also from E, at right angles to n n, make E E′ = y. From E′ draw the vertical E′ L.

The strain upon the rod o m, being represented by o m; and that upon o n, by o n, the resultant is shown, both in direction and amount, by o V. It is not necessary that this should pass through the centre of the post, as the excess of tension on o m, over that on o n, is absorbed by the lower chord.

Note.—Screwing up truss bridges, is a more scientific operation than is generally supposed. Many builders commence at each end, and lift the bridge from the scaffolding. By this method the greater part of the load is often borne by a few of the end sets of rods. The better method is to begin at the centre and work both ways towards the ends, being sure that each set of rods does its duty before the next is touched. The lift to be made by each set of rods, should first be calculated, and tested while screwing up, with the level.

Lattice Bridges

184. Town’s lattice, consists of a simple lattice-work of plank, 3 × 12 inches, treenailed together at an angle of forty, forty-five, or fifty degrees. It possesses great stiffness, without by any means having the material disposed in the best manner. Such bridges might well be made by the mile, and cut off to order according to the span.

The improved lattice, by Hermann Haupt, Esq., C. E., avoids all of the evils attendant upon the common lattice, and gives a very cheap, strong, and rigid bridge. In this plan the braces are placed in pairs, with vertical tie planks between them; by which the twisting seen in the common lattice, is removed. The braces are also brought to the vertical, as the point of support is reached, by which a good bearing is given to the end sets of timbers.

To vary the size of the braces, as the strain upon them decreases, would be both inconvenient and expensive; but the same effect may be produced by varying the distance between them, making it greater as the centre is approached.

S. W. Hall’s Wooden Truss and Arch Bridge

Inverting Mr. Haupt’s design for a lattice of improved construction, (which consists of vertical ties and inclined braces,) we have the base of the above-named bridge; where the inclined timbers are used to resist tension, as below.

This being a very good plan, and the arrangement for building being such as to secure the thorough execution of the work in its most minute detail; it is thought best to extract at some length from a letter from the inventor, dated July 31st, 1856, not however being confined to the matter therein.

The first claim, is for a new form of truss, formed of posts vertical, or nearly so, and tension pieces, inclining downwards toward the centre; thus differing from nearly all other plans. Timber resists double as much extension as compression; and when large enough to resist the simple tension, does not have to be increased as in resisting compression for flexure; but requires a larger allowance for joints, as tension tends to pull the joints apart, while compression forces them together.

The following result was obtained, showing the superior strength of timber work in resisting by tension. Two models, containing the same amount of timber, were tested. The one built with vertical ties and braces, broke by crippling the brace, under 2,400 lbs.; while that constructed with verticals and suspenders, inclining towards the centre, sustained 4,200 lbs. with no visible change of form.

The second claim is for more efficient bearings and connections than common, and this with less cutting away of timber. The arch and arch braces have a full, fair bearing at top and bottom. The first sets of tension braces, (those extending from the top of the arch braces towards the centre,) are sustained by two pins at each joint; which gives six pin bearings, or twelve for one set of braces, of six inches each, (the pin being two inches in diameter, and plank three inches thick,) equal in all, to seventy-two inches of bearing surface at least, for each five feet lineal of bridge, or one hundred and forty-four inches for ten feet.

The third claim, is that the bearings, at joints, are central, and that the shrinkage of the timber is towards and not from them as in many plans.

The pin holes are bored by machinery smooth and true; the treenails when of wood are of seasoned oak or locust, turned to a perfect fit, and when of iron are made hollow.

These bridges, after three years, stand within an inch of their shape as framed without exception. One indeed supporting an aqueduct, which throws upon the truss a constant load of 2¾ tons per foot, not including the weight of the bridge, without any apparent settling.

The connections being fast, prevent reaction and vibration from variable loads, the strains in this case are reversed, the bridge tending to spring up instead of settling.

The fourth claim, is for the small brace connecting the lower with the intermediate chord; by which additional connections are obtained, and smaller timbers rendered available.

The fifth claim, is the formation of a stronger chord than by the plan of using a few large sticks. The chord being made of a great number of small pieces, the strength is of course less affected at any one point, by a joint, than when only a few pieces are used.

In the bridges built by the above engineer, are to be seen some of the most perfect built beams in the country. The following conditions being observed, the most uniform, and highest average strength possible is obtained.

First. To cut but one stick in any one panel.

Second. To cut no stick at the centre of the bridge.

Third. To place every joint in the middle of the collateral piece.

The chords are cut by two rows of pins, two inches each; and if the chord be fifteen inches, the cutting at centre, where there is no joint, is but four fifteenths of the whole section. To resist the parting of any two sticks, there is the resistance to shearing of ninety-six pins; and the section of each being three square inches, the whole resistance is two hundred and eighty-eight inches of area. If the intermediate chord has the same number, the whole area to resist shearing, in the lower chord, is five hundred and seventy-six square inches. The bearing surface of each pin in the chord stick is 2 × 3 inches, or six inches; and 96 × 6 = 576: and in both chords 1152 inches.

	Sq. in.
The whole chord timber, (both chords,) is (6 × 3 × 14),	252
In the intermediate chord, (6 × 3 × 12),	216
	———
Whole timber section,	468
Deduct 4 pins for both chords, 468 – (4 × 2 × 3 × 6) or 468 – 144 =	324
Deduct for joints 3 × 10 + 3 × 8 or 324 – 54, Square inches of available area.	270

Comparing the amount thus cut away with that cut away in other plans, we have the following figures;—

A. Hall’s Bridge, (actual bridge,)	30 100
B. Howe’s bridge, (actual bridge,)	35 100
C. Page 163, (Handbook R. R. Construction,)	39½ 100
D. McCallum’s, (Susquehanna bridge,)	58 100

The sixth claim, is the peculiarly convenient form for applying an arch,—the superiority consisting in convenience for attachment; in the connections being less affected by shrinkage than when posts are locked into arches; in the timbers not being weakened by cutting. The arch is loaded with the tension timbers, inwardly, and acts as a general arch brace, transferring at once all of the several tensions to the abutment, thus really combining the arch with the truss.

The liability of this plan to decay, certainly appears to be less than that of most plans of wooden bridges now in use; as will be plainly seen by observing the position of the joints; falling rain finds a much easier access to almost any other joint than the pin hole. The timber work being made of plank, all the timbers are small, and are thus much more likely to be sound.

The bridges built upon this plan upon the Alleghany Valley, and upon the Williamsport and Elmira roads, illustrate plainly the design.

185. Applying arch braces to lattice bridges, has suggested The Arch-brace truss bridge, in which the whole strength lies in a series of differently inclined braces, extending from the abutment to the head of each post; a very light lattice being used to prevent reaction, or as a counter-brace or stiffener. See fig. 69.

In trusses consisting of a series of triangles, when the span is large, (150 to 200 feet,) the immense weight coming at the feet of the second and third sets of braces, causes settling or depressing at twenty or thirty feet off from the abutment, which can hardly be removed. The remedy for such settling, is to transfer the load at once to the abutment; which is completely done in the above-named bridge. Each brace does its duty directly and well. Before the lattice-work is fastened, the bridge should be loaded with a maximum load. Then by fastening the diagonals, the recoil is prevented; and the effect of a passing load is to ease the counterbracing lattice, without otherwise affecting the truss.

Note.—A model of this bridge, made by the writer, of the following dimensions:—

Length,	7 feet.
Height,	1 foot.
Width,	1 inch.
Chords,	¼ × ½ inch.
Braces,	¼ × ⅓ inch.
Lattice,	¼ × 1 16 inch.

Supported 2,500 lbs. at centre, besides a variable load of 150 lbs. applied as a rolling weight in the most disadvantageous manner. It represented a span of one hundred and fifty feet, and according to Weisbach’s formula for testing a model, proved the actual structure, (as far as can be proved by a model,) both strong and rigid to any desired amount. The longest bridge ever built upon this principle, was that of Schaffhausen, over the Rhine, which had a single span of three hundred and ninety feet. This bridge was not stiff, having no lattice, but was very strong. B. H. Latrobe, Esq. has adopted this form upon the Baltimore and Ohio Railroad.

The calculations for the parts of this bridge are as follows:—

The Span being	150	feet,
The Rise	20	feet,
The Panel	15	feet,
Weight per foot of bridge and load	3,000	lbs.

The half number of panels is five; the diagonals of which, neglecting fractions, are

√20² + 15² = 25 feet,
√20² + 30² = 37 feet,
√20² + 45² = 49 feet,
√20² + 60² = 64 feet,
√20² + 75² = 78 feet.

The weight upon each of these sets of braces, is the weight of the length of one panel; which, in the present case, is 3,000 × 15 = 45,000 lbs. As there is a brace under each chord stick, and assuming four sticks in each chord, we divide by eight, and have, in round numbers, 6,000 lbs. per brace; and correcting for inclination, as follows, we have the numbers below.

20: 25:: 6000: 10000
20: 37:: 6000: 15000
20: 49:: 6000: 20000
20: 64:: 6000: 25000
20: 78:: 6000: 30000.

The last column has the several weights coming upon the different braces at their several inclinations; to resist which, the scantling might be very small, for compression, but flexure requires larger dimensions.

These braces should be confined laterally and vertically, as they pass each post, but not connected therewith; as this would not permit a free action of the brace, without straining transversely the post.

The length of beam, therefore, in which flexure is to be checked, is the distance between posts in any panel.

In panel No. 1, it will be 25 feet.
In panel No. 2, it will be 18 feet.
In panel No. 3, it will be 17 feet.
In panel No. 4, it will be 16 feet.
In panel No. 5, it will be 16 feet.

and applying the formula

2240bd³
L² = W

we get, in round numbers, the following dimensions, the braces being bolted and blocked together:—

For the 1st panel, 25 feet long, 8 × 10
For the 2d panel, 37 feet long, 8 × 10
For the 3d panel, 49 feet long, 8 × 10
For the 4th panel, 64 feet long, 8 × 10
For the 5th panel, 78 feet long, 8 × 10.

For the lattice-work, a double course on each side of each truss, in long spans, (150 to 200 feet); and a single course in shorter spans, of 3 × 6 plank, treenailed at intersections, is ample.

GENERAL TABLE OF DIMENSIONS FOR ARCH BRACE TRUSS.
Span.	Rise.	Chords.	Ties.	Braces.	Lattice.
50	10	2–8 × 10	1–8 × 10	2–6 × 6	2 × 9 or 3 × 6
75	12	2–8 × 10	1–8 × 10	2–6 × 6	2 × 9 or 3 × 6
100	15	3–8 × 10	2–8 × 10	3–6 × 6	2 × 9 or 3 × 6
150	20	4–8 × 12	3–8 × 10	4–6 × 8	2 × 9 or 3 × 6
200	25	4–8 × 16	3–8 × 10	4–6 × 9	2 × 9 or 3 × 6

Fig. 69 A, shows the method of bringing the arch braces to the chord. To find the dimensions of the cast-iron block, make a complete drawing of all of the braces, at their proper angles, and then draw in the block around the feet, as shown in fig. 69 A.

Note.—The centre of pressure of the braces in fig. 69 A, is not, as might seem, at C; because the vertical components of the forces, coming down the brace, are much less in the braces at small angles than in those at the end of the span. The load applied to each brace being the same, and the inclines being found, we find the centre of pressure, or the centre of bridge seat as follows:—

The length of the brace is to the vertical height, as the applied load to the vertical pressure. In fig. 69 A, we have the following lengths of braces: a, 25; b, 37; c, 49; d, 64; e, 78; f, 92; and g, 106; and the weights corresponding thus,

a, 25: 20:: 6000: 4800.
b, 37: 20:: 6000: 3243.
c, 49: 20:: 6000: 2450.
d, 64: 20:: 6000: 1870.
e, 78: 20:: 6000: 1540.
f, 92: 20:: 6000: 1304.
g, 106: 20:: 6000: 1132.

In fig. 69 A, assume the foot of the fifth brace (B) as the centre of pressure, and adding the moments, (or products of vertical components on the braces by their distance from B,) and we have the sum on the land side 18,928, and on the water side 16,930; showing that the centre is taken too far from the land side. In the same manner A will be found too far from the land side. A third trial will give the place.

Fig. 69 B, shows the manner of splicing the arch braces: being subjected to compression, they are spliced in the same manner as the upper chords. Fig. C, shows the lower chord spliced. Figs. D and E, the connection of the posts, chord, and lattice. Figs. F, G, and H, the casting for applying the upper end of the arch brace to the chord. Fig. 69 K, the method of supporting the tracks at the end of the span, where the arch braces will not allow the floor beams to bear upon the lower chord.

McCallum’s Patent Railroad Bridge

186. This bridge represents a class of structures in which the upper chord is curved upwards (7½ feet in 200 in the Susquehanna bridge, New York and Erie Railroad), which curved chord has the effect of distributing an applied load at once to all of the braces directly, by means of the chord, as well as indirectly, by means of the braces, as in the common trusses. To this bridge is applied the arch braces A B, A B, fig. 70, which serves to aid the 2d, 3d, and 4th pair of diagonal braces in bearing their load.

The great distributing power of the curved chord, is shown by the fact that a bridge of 125 feet span, actually supported a railroad train before the diagonal bracing was introduced. The whole strain was thrown through the curved chord and arch braces to the abutments. The bridge is counterbraced by the pieces d d and d d, adjustable by screws at the ends.

The following test was applied to a span of 190 feet of this plan of bridge. Placing the load as near as possible to the centre, the following deflections were produced.

Load.	Deflection.
41.40 tons,	0.013 feet,
95.35 tons,	0.038 feet,
140.70 tons,	0.061 feet,
187.20 tons,	0.061 feet.

Upon removing the load, the bridge entirely recovered its form.

187. As the span increases, the benefit derived from the curved chord also augments; and though in the latter part of the present chapter its application to small spans is shown, it may not be worth while to adopt it.

Bridges transferring the load directly, from each panel to the abutment, would not be aided, to an amount worth the increased expense, by adopting the curved top chord.

In case of any settling at the centre of the span, the reverse effect is seen from that produced in a truss with horizontal chords; i. e., when the ends of the upper chords in the latter draw in, those of the former push out; and when in such bridges, arch braces are not used, the top chords of adjoining spans must be wedged apart, in place of tying together as in common plans, over the centre of the piers.

The Arch

188. The arch has been applied to long spans for a great while, and when care has been taken to prevent flexure, answers very well. The repair of such bridges, if any of the arch timbers decay, is difficult; but is effected, in the largest arches.

The most correct ideas on wooden arch bridge building, are to be found in Weibeking’s Traite d’une parte essential de construire les grandes pents en charpente. This engineer, (General Director of Roads and Bridges in Bavaria,) has built a great number of wooden arches of the best description, which show him to be master of both the science and the art.

The general plan of his bridges is shown in fig. 71. They consist of curved ribs formed of long pieces scarfed and bolted together, from which the road-way is supported by posts.

The bridges of Neucettringin, Freysingin, Bamberg, Scharding, Wertach, Vilshoven, and Altenmarkt, all testify to the good judgment of this man. The spans vary from one hundred to two hundred feet; and the width from twenty-five to thirty-two feet. The proportions which he gives for the ratio of rise to span, are valuable; as they are the result of his own experience. He states, generally, that one tenth of the span is the best rise; but that for convenience, it is better to keep it lower. The following table shows the dimensions he has adopted in practice.

189.

Name.	Span.	Rise.	Width.	Rad. of Arch.	Scantling of Arch.
Bamberg,	208	16.9	32	422	13½ × 15½
Scharding,	194	18.8	25	258	12½ × 15½
Vilshoven,	179	11.1	27	378	13½ × 15½
Freysingin,	153	11.6	25	246	12½ × 14½
Ettringin,	139	8.0	25	305	12½ × 15½
Ersingin,	126	7.0	25	285	11½ × 14½
Augsberg,	114	10.6	25½	158	12½ × 14½
Neucettringin,	103	6.8	25	200	13½ × 15½

The last column shows the scantling of the arch timbers; these being placed three deep, in spans of less than 150 feet; and in larger spans, 3 deep at centre, and 5 deep at ends. Mr. Weibeking’s formula for determining the scantling of ribs, is as follows:—

W × (S/2)²
Rn.0011 = Scantling in sq. ft.

Where R is the rise of the arch;
      n, the number of ribs;
      W, width of bridge;
  and S, span of bridge.

Example.—Required the scantling of the ribs of a bridge of 300 feet span, 20 feet wide, and 30 feet high. The formula becomes,—

20 × 22500
30 ×.0011 = 16½ sq. feet of

section, of all of the arches; or two parallel arches, 2½ feet wide, by 3¼ feet deep each.

From 100 to 150 feet span, he makes the rise 1
20 span,
From 150 to 200 feet span, he makes the rise 1
18 span,
From 200 to 300 feet span, he makes the rise 1
15 span,
From 300 to 400 feet span, he makes the rise 1
14 span,
From 400 to 500 feet span, he makes the rise 1
13 span.

190. The bridge built by Mr. Burr across the Delaware at Trenton, New Jersey, is a good specimen of an arch. It is composed of white pine planks, from thirty-five to fifty feet long, and of a scantling 4 × 12. These planks are laid close together, breaking joint, having an entire depth of three feet. The arches are stiffened by horizontal tie beams, supporting the road-way, and by diagonal bracing. The spans are 160, 180, and 200 feet, and the rise twenty-seven feet.

191. The bridge over the Susquehanna, at Columbia, built in the same manner, consists of twenty-nine arches, each two hundred feet clear span, supported on two abutments and twenty-eight stone piers. The clear water-way of this bridge is 5,800 feet; and the entire length, including piers and abutments, one and one fourth miles. There are three sets of arches, which allow of two carriage roads and one railroad, the whole width being thirty feet.

192. An arch to support a passing railroad train must be very rigid. It is customary to connect them with a light truss, which effectually counter braces the arch, and prevents that change of form which would otherwise take place; depending entirely upon the arch for strength.

Wherever the load is applied, the arch tends to sink, and a corresponding rise takes place at the opposite point. A load placed at E, fig. 71, settles the arch at that point and causes it to rise at C. A load placed at the curve of the arch depresses the centre, and elevates the haunches. To counteract these movements a light, stiffening frame may be used, its strength being able to resist the variable load passing over the bridge. The strain thrown by the arch upon the truss, advances from the opposite end to meet the train, passes it at the centre, and finally goes off from the bridge behind the load.

When the arch is the truss, or when a truss is made with curved chords, the counteracting effect of the truss is not completely obtained. We should not depend upon the curved chord as an arch, but only as a member of the truss.

193. Many combinations of arch and truss have been built in America for railroad bridges. The principle of connecting the two systems is by some thought bad, as they can hardly be made to bear equal parts of the load; whence each must have more than half the necessary strength of the whole. Others maintain that by a proper arrangement of parts a perfect adjustment may be made, by which the load may be placed more or less on either. There seems to be no very good reason why the two systems should be combined, as either may be made strong enough to bear the largest loads.

Both arches and arch braces, however, are very usefully applied to bridges which have been made too light.

194. The manner of applying arches is well shown in the bridges of the Pennsylvania Central Railroad, built by Hermann Haupt, Esq.

These bridges are on Howe’s plan, to which have been added strong wooden arches. The systems are connected by adjusting the counter braces against the arch by set screws. The arrangement is simple and effectual. The name of the builder is sufficient to warrant the stability of the bridge.

195. However nicely we may form an arch, it will settle more or less when the scaffolding is removed, according to its flatness; which depression increases with time. Mr. Weibeking expresses it in inches as follows:—

0.806R
S

Where R, shows the rise,
  and S, shows the span.

To allow for this settling, the curve when laid down on the platform for building the arch, should be made a little more convex than the completed arch is required to be; the amount of excess being that shown by the formula.

As a bridge composed of a curved rib when the span is large yields at D, C, and E, fig. 71, when the load is applied in the middle, the strength must of course be increased by increasing the depth of the rib; and not to make this too heavy, a framed or built beam should be used as in fig. 72. Here it must be remembered that the two ribs must be so framed as to resist both tension and compression; for when a load is placed at D, the lower rib will be extended at d, and compressed at c′, and e; while the upper one will be compressed at D, and extended at C and E.

Of the Road-Way

196. The flooring of any system is about the same; consisting of transverse floor beams, placed either on the top or bottom chords, (according as the road-way is more or less elevated above the water-way,) which support longitudinal timbers, upon which are placed cross-ties. In some cases, two curves of diagonal plank have been placed across the floor beams, spiked at right angles to each other, by which the bridge is considerably stiffened laterally.

General dimensions for the floor may be thus:—

Transverse timbers, 3 feet from centre to centre,	8 ×	14
Track strings, notched 2 inches to floor beams,	12 ×	14
Cross-ties placed one foot apart, (clear,)	3 ×	6

Lateral Bracing

197. To prevent vibration in a horizontal direction, a system of diagonal bracing is necessary. The chief pressure upon these braces is caused by wind; and may be found by considering the bridge as turned over upon the side, and loaded with a weight equal to the maximum pressure of the wind, which may be taken as forty pounds per square foot.

It is unnecessary to vary the size of these braces, except in very long spans, when they should increase from the centre to the ends. For short spans, (less than one hundred feet,) a brace 5 × 5 is large enough. For larger spans 7 × 7 is sufficient.

198. Diagonal bracing, when it can be introduced, is a very desirable part of a bridge. When the road is on the lower chord this cannot have place in full, but may be applied as in fig. 74.

By increasing the height of truss in any bridge, the tension and compression on the chords is lessened; but the length of posts and rods is increased. As a general thing, one eighth of the span gives the best results.

199. In framing a large bridge, it is customary to cut the top chord sticks a little longer than to dimension; to allow for compression in settling.

200. Bridges in exposed situations have been sometimes blown off from the masonry. If a bridge slides off from the masonry, the whole force of the wind must be fifteen twenty-fourths of the whole weight of the bridge; but if, as is generally the case, the masonry is rough, (and not hammered,) no amount of wind will cause the bridge to slide.

The bridge will upset, turning about its lower edge, when the whole pressure of the wind, multiplied by half the height of truss, overbalances the whole weight, multiplied by the half width. In very exposed places the rod A D, fig. 74, answers a very good end; when the road is upon the upper chord, and a rod from B to the masonry, when upon the lower.

Oblique Bridges

201. The effect of running a train over a skew bridge, is to depress one side before the other; as the load comes upon the centre of one truss before it does upon the opposite one. This produces a side rocking in the engine, dangerous alike to the bridge and to itself.

The floor timbers transferring the load to the chords should not be at right angles to the axis of the road, but parallel to the abutment. Thus in fig. 75, a wheel at B, throws one third of its weight upon the abutment at E; and two thirds upon the chord at C; while in fig. 75 A, the wheel at B, throws two thirds of the load upon C, but one third also upon D.

202. In a very long, oblique span, the floor timbers may be arranged as in fig. 76, that is, inclined at the entrance and exit of the bridge, but at right angles at the middle of the span.

203. The preservation of timber in wooden bridges may be accomplished by covering with boards, whitewashing, painting, and by Kyanizing. Covering and whitewashing are the best, if care is taken to prevent dry rot by giving a good circulation of air about the timbers. The oil in paints prevents the escape of moisture from within as well as the entrance of that from without; and should not be used unless the wood is well seasoned. The best plan is to thoroughly whitewash and cover the frame of the bridge, and to paint the outside of the covering.

204. In framing two or more continuous spans, the chords should always be connected over the piers; as there is thus given something for the upper chords to pull against, and a counter thrust for the lower.

205. Bridges should never, when it can be avoided, be placed either upon a curve or upon a grade; particularly upon the former, as the effect of a load is thereby very much increased, in the first case causing a lateral, and in the second a vertical shock.

Pile Bridging

206. In shallow water, in marshes, and in similar situations, where an embankment would be expensive, pile bridging is very useful. Indeed, whenever we are at liberty to obstruct the passage beneath the road, it is well to adopt this system, unless over twenty feet high. It is cheaper than any other, easier to repair, the parts are quite independent of each other, and such bridges last full as long as other wooden structures.

Different plans for pile bridging are given in figs. 77 to 82. Figs. 77 to 81 show plans for temporary pile work, to be used during construction. Nothing lighter than fig. 82, ought to be permanently used. A pile bridge upon a curve may need stronger lateral bracing upon the convex, than upon the concave side of the curve; and also in running water; in which cases, such a form as fig. 81 may do good service.

(For pile-driving, and for proper dimensions, see chap. XII.)

Trestling

207. Trestling is a system of vertical posts, and of caps and braces, used both for temporary and for permanent works; temporarily to pass a road over low ground where embankments are to be made, and permanently over deep, dry gorges, where the amount of earthwork or masonry would be too great.

American railroads show all sizes and arrangements of trestling, from twenty to two hundred feet high. Figs. 83 and 84 show temporary works, and fig. 85 permanent.

The main part of the design in trestles is to connect the several posts and caps by well-formed triangles; the equilateral being the best.

The finest example of this system of building is the Genesee high bridge, over Genesee River near Portageville, on the Buffalo and New York Railroad; built by H. C. Seymour, Esq. It is eight hundred feet long, and two hundred and thirty feet above the river. It has eight stone piers, thirty feet high, upon which are placed trestles one hundred and ninety feet high, seventy-five feet wide at base, and twenty-five at top. Upon the top of all is placed a bridge fourteen feet high. To build this viaduct was used 1,500,000 feet, board measure of timber, which covered, when standing, two hundred and fifty acres; also, sixty tons of bolts. The whole time occupied in building was but eighteen months, the whole cost being $140,000.

Drawbridges

208. In crossing rivers or bays open to navigation, it is required from any companies building a bridge, to leave a free passage for shipping. This is done by making that part of the bridge over the channel movable; (a draw).

Draws may lift up, (being counterbalanced,) may slide back upon the fixed part of the bridge, or may turn on a pivot. Fig. 86 shows a draw much used at present, and answering every purpose. Each half of the movable part must be calculated as a small bridge. The rods c c c support the overhanging part of the draw while open. The whole revolves upon a centre pin and a set of rollers.

Centres

209. Centres are temporary wooden frames, used in the construction of stone arches. Their duty is to hold the masonry, while it is unable to support itself.

For arches from five to fifteen feet span, a centre made of boards or planks, fig. 87, is all that is necessary. For longer spans, when the ground beneath the arch can be used, the form, fig. 88, answers well. When there is no support but the abutments or piers, something similar to fig. 89 must be adopted. This is the plan adopted by George Rennie, chief engineer at the Waterloo bridge over the Thames at London.

Centres are strained in a different manner as the arch progresses; first at the haunches, and last at the crown. Excess of weight at any point causes a settling at such, and a rise takes place at some other place. By loading the arch temporarily, such motions are checked.

These frames are placed vertically upon the pieces F F, which being connected with the braces D D by the folding wedges c c, admit of adjustment of the height of the centre. The distance between the ribbed frames depends upon the form of the arch, and the span, or upon the weight to be supported; varying from one to four feet. The centres are covered with a course of narrow plank, placed parallel with the axis of the arch, upon which the voussoirs rest.

210. The method of putting a bridge upon the masonry is shown in figs. 90, and 91; the former when the road-way is upon the upper, and the latter when upon the lower chord.

211. In figs. 92 to 100, are given several plans for spans from five to seventy-five feet. Fig. 92, shows the simple beam braced beneath with diagonal plank; the bolts passing through the ties, stringers, and braces. The stringers are bolted to the wall plates, and when the bridge is upon a curve notched also, by cutting the bolster. Fig. 92 A shows the plan. This form answers for openings from five to twenty feet. From fifteen to thirty feet, we may use figs. 94, 95, 96, and 97. From twenty-five, to fifty and sixty feet, figs. 93, 97, and 98. And from fifty to seventy-five feet, figs. 99 and 100.

The following tables give reliable dimensions for bridges upon the above plans.

Span.	Bolsters.	Stringers.	Ties.	Braces.	Bolts.
5	12 × 12	12 × 12	6 × 10	2 × 10	1 inch
10	12 × 12	12 × 13	6 × 10	2 × 10	1 inch
15	14 × 14	12 × 18	6 × 10	2 × 10	1 inch
20	14 × 14	12 × 24	6 × 10	2 × 10	1 inch

The ties being notched three inches on to the stringers, without cutting the latter.

Span.	Rise.	Bolster.	Stringer.	Braces.	Rod.
15	6	12 × 12	12 × 12	2—5 × 6	1 inch
20	7	14 × 14	12 × 12	2—5 × 8	1¼ inch
25	8	14 × 14	12 × 15	2—5 × 9	1⅜ inch
30	10	15 × 15	12 × 18	2—5 × 10	1½ inch

Span.	Rise.	Stringer.	Post.	Rod.
15	5	12 × 12	8 × 8	1⅛
20	6	12 × 13	9 × 9	1¼
25	7	12 × 15	10 × 10	1½
30	8	12 × 18	10 × 12	1⅝

Span.	Rise.	Stringer.	Braces.	Rods.	Lattice.
15	5	2—8 × 8	2—5 × 5	1	2 × 6
20	6	2—8 × 9	2—5 × 6	1¼	2 × 8
25	7	2—8 × 10	2—5 × 8	1⅜	2 × 9
30	9	2—8 × 12	2—5 × 9	1½	2 × 10

Span.	Rise.	Stringer.	Post.	Rods.	Braces.
15	5	2—8 × 8	8 × 8	1	4 × 5
20	6	2—8 × 9	9 × 9	1⅛	4 × 6
25	7	2—8 × 10	10 × 10	1¼	5 × 6
30	9	2—8 × 12	10 × 12	1½	6 × 6

Span.	Rise.	Chords.	Braces.	Posts.	Rods.
25	8	2—6 × 8	6 × 6	6 × 8	1
40	10	2—7 × 9	6 × 7	8 × 8	1⅛
50	10	2—8 × 10	6 × 8	8 × 10	1¼

Span.	Rise.	Chords.	Posts.	Braces.	Rods.
25	8	2—6 × 8	8 × 8	5 × 5	1⅛ or 2—⅞
40	10	2—7 × 9	9 × 9	5 × 8	1⅜ or 2—1⅛
50	10	2—8 × 10	10 × 10	6 × 8	1¾ or 2—1¼

Span.	Rise.	Chords.	Posts.	Braces.	Rods.
25	8	2—6 × 8	6 × 8	5 × 5	2—1 or 1—1⅜
40	10	2—7 × 9	6 × 9	5 × 8	2—1⅛ or 1—1⅝
50	10	2—8 × 10	6 × 12	5 × 10	2—1¼ or 1—1¾

Span.	Rise.	Chord.	Posts.	Braces.	Lattice.
50	8	2—8 × 10	1—8 × 10	2—6 × 7	2 × 6
60	9	2—8 × 10	1—8 × 10	2—6 × 7	2 × 6
75	10	3—8 × 10	2—8 × 10	3—6 × 8	2 × 6

Span.	Rise.	Chords.	Posts.	Braces.	Centre braces.	Rods.
						No. 1.	No. 2.
50	8	2—8 × 10	1—8 × 10	2—6 × 7	5 × 5	1½	1
60	9	2—8 × 10	1—8 × 10	2—6 × 7	5 × 5	1⅝	1⅛
75	10	3—8 × 10	2—8 × 10	2—6 × 8	5 × 6	1¾	1¼

212. In dimensioning small bridges, like the above, in estimating the maximum load, more regard must be given to the weight of momentary loads than (as in large bridges) the weight per lineal foot, as the weight of the bridge itself, when under fifty or sixty feet span, is inconsiderable. The greatest load that can come upon a single post or rod, is that from the driving wheels of a locomotive. If the whole engine weighs forty tons, there will be ten tons on each pair of drivers, or five tons or 11,200 lbs. on each wheel; which, being applied over a length of ten feet only, may be considered as at a single point, and all parts must be able to bear such load. In large spans, where the weight is great, if the truss is strong enough to support the bridge and load, it will safely resist the effects of a sudden application of passing trains.

Note.—On the Static and Dynamic deflection of Bridges. Considerable variance of opinion exists as to the relative deflection of bridges, produced by stationary and by moving loads. Neither experiment nor theory has exactly settled the point.

Experiments upon the Elwell bridge, (Epsom and Croydon Railway, England).

Velocity in feet per second.	Deflection in inches.
0	0.215
25	0.215
31	0.230
32	0.225
54	0.245
75	0.235

The bridge being a cast-iron girder of forty-eight feet span, load thirty-nine tons.

Experiments on the Godstone bridge, (S. E. R. R. England).

Velocity in feet per second.	Deflection in inches.
0	0.19
22	0.23
40	0.22
73	0.25

Cast-iron girder, thirty feet span, load thirty-three tons.

Mr. W. H. Barlow, (Eng.) observed, “that in case of a timber viaduct, a freight train, at a low speed, produced a certain deflection; but an extra train, with a much lighter engine, seemed to push the bridge like a wave before it.”

The Britannia tubular bridge was depressed three fourths of an inch by two locomotives and a train of two hundred and eighty tons standing still; but at seventy miles per hour, the deflection was sensibly less.