Handbook of Railroad Construction; For the use of American engineers. / Containing the necessary rules, tables, and formulæ for the location, construction, equipment, and management of railroads, as built in the United States.

by George L. Vose

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.

Fig. 60.

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.