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of cast-iron one inch square would bear a pressure of 93,000 pounds, and would sustain a pressure of 15,300 pounds without permanent alteration. The proper load for square or round iron pillars is, therefore, about 15,000 pounds to the square inch, when the pillar is not more than one foot in length, and the ends flat; for if the ends are round, the pillar will not sustain more than as great a pressure. A vertical cylindrical pillar of cast-iron 34 feet in length and 74 inches in diameter, will sustain permanently, without bending, a crushing weight of about 160,000 pounds; and a vertical column of fir 10 inches in diameter and 16 feet long, will sustain with safety a crushing weight of about 161,000 pounds.
To find the greatest transverse strength of a rectangular beam of New England fir, when fixed at one end and loaded at the other:
Multiply the breadth by the square of the depth, both in inches, and this product by 1100, and divide the last product by the length in inches, and the quotient will be the weight in pounds.
For a permanent load, only of the result obtained by the rule should be taken for fir and the softer kinds of wood, though most of the heavier kinds of wood, as beech, birch, and ́ oak, will sustain with safety, as a permanent load, as great a pressure as that given by the rule; whilst cast-iron will sustain 7 times, and malleable iron 8 times as great a pressure as a permanent load. The strength of a rectangular beam supported at both ends and loaded in the middle, is 4 times as great as that given by the above rule.
STRENGTH OF ROPES, ETC.
Ropes are stronger in proportion to the fineness of the strands of which they are composed. Damp hempen ropes are stronger than when dry; and such as are tarred are stronger than untarred, twisted than spun, and unbleached than bleached. Silk ropes are nearly three times as strong as hempen or flaxen ropes of the same size.
To find the strength of a hempen rope :
Multiply the square of the circumference in inches by 200, and the product will be the strength in pounds.
A cylinder of rolls of paper glued together, and presenting at the end an area of, 1 square inch, will support a weight of 30,000 pounds; whilst a similar cylinder composed of small hempen strings glued together lengthwise, will sustain 92,000 pounds, its strength exceeding that of the best iron.
A square beam hewn from a round log or cylinder is only 100 as strong as was the cylinder before it was squared. If it be required to hew a beam from a tree that shall possess the greatest absolute strength, its length and breadth must be to the diameter of the tree in the ratio of .82 and .58 to 1.
The lateral strength of square timber is to that of the tree whence it is hewn, as 10 to 17, nearly. A piece of the best bar-iron 1 inch square will suspend a weight of 77,370 pounds; but thin iron wires, arranged parallel to each other, and presenting a surface at their extremities of 1 square inch, will carry a weight of 126,340 pounds.
1. What weight will break a spar of New England fir, its breadth being 2 inches, depth 3 inches, and length 5 feet? Ans. 330 lbs.
2. What weight will break a spar of New England fir 10 feet long and 6 inches square, the weight being uniformly distributed throughout the length? Ans. 3,976 lbs.
3. Find the proper load for a bar of long, 2 inches thick, and 3 inches deep.
malleable iron 10 feet Ans. 1,320 lbs.
4. What weight will a bar of cast-iron 6 feet long, 1 inch thick, and 3 inches deep, sustain permanently?
Ans. 770 lbs.
5. A joist of New England fir is 25 feet long, 3 inches thick, and 7 inches deep; what weight uniformly distributed over it, will break it when supported at both ends?
Ans. 4,512 lbs.
6. A birch plank is 10 feet long, 5 inches deep, and 2.24 inches thick; what weight will it permanently sustain at its centre, when both ends are supported? Ans. 2,053 lbs.
To find the greatest possible deflection of ash, beech, birch, elm, New England fir, American oak, and teak, before rup
For ash, Multiply the depth of the beam, in inches, by 400; for beech, by 615; for birch, by 600; for elm, by 500; for fir, by 757; for oak, by 724; for teak, by 818; and divide the square of the length also in inches, by the product, and the quotient will be the greatest deflection, when the beam is supported at both ends; but if the beam is fixed only at one end, the deflection will be eight times greater.
1. Find the ultimate deflection of an ash plank 2 inches thick, and 40 feet long. Ans. 288 inches.
2. A spar of ash 4 inches deep and 6 feet long is fixed at one end, and is broken by a weight applied at the free end; what was the greatest deflection before rupture?
Ans. 26 inches, nearly.
3. A plank of teak (the least flexible of the various woods) is 180.9 inches in length, and 2 inches in thickness; what will be its greatest deflection before rupture, when supported at both ends? Ans. 20 inches.
4. A beam of fir is 50 feet long and 5 inches deep; required its greatest deflection when supported at both ends. Ans. 7 feet, 11.11 inches.
5. Required the strength of a hempen rope 4 inches in cirAns. 3,200 lbs. 6. Required the strength of a silk cord 2 inches in circumference. Ans. 2,400 lbs.
T81. MISCELLANEOUS EXERCISES IN MENSURATION
1. How many square feet are there in a floor 16 feet 9 inches square? Ans. 280.5625 square feet.
2. What is the area of a board 12 broad?
feet long and 9 inches Ans. 9 square feet.
3. What is the difference between the superficial contents of a floor 28 feet long and 20 broad, and that of two others, of only half its dimensions? Ans. 280 feet.
4. It is required to cut off a piece containing a square yard and a half from a plank 26 inches broad; what must be the length of the piece? Ans. 6.23 feet.
5. What is the area of a rhombus, whose length is 6.2 chains and perpendicular breadth 5.45 chains?
Ans. 3.36 acres.
6. How many square yards of painting in a rhomboid, whose length is 37 feet, and breadth 5 feet 3 inches?
Ans. 21 yards.
7. What is the area of a trapezoid, whose parallel sides are 14 and 24 feet, and the perpendicular distance between them 8 feet 3 inches? Ans. 161.9 feet.
8. What is the length of a rectangle, whose area is 784 feet and breadth 12? Ans. 65 feet.
9. What is the length of a board that contains 18 square feet, its parallel ends being 15 and 9 inches? Ans. 18 feet.
10. What is the area of a triangle, whose base is 36 and its altitude 12 rods? Ans. 1.35 acres. 11. What is the area of an equilateral triangle whose side is 40 rods? Ans. 4.33 acres, nearly.
12. What is the length of a perpendicular falling from either angle of an equilateral triangle on the opposite side, the side of the triangle being 12 chains? Ans. 10.3923 chains.
13. The area of an equilateral triangle is 1 acre, 2 roods, and 15 rods; required the length of its side.
Ans. 24.2687 rods.
14. The area of an equilateral triangle is 24 acres; required the length of its side. Ans. 23.542 chains. 15. The base of an isosceles triangle is 25 chains, and each of the other sides 30 chains; how many acres does it contain? Ans. 34.087 acres.
16. The sides of a scalene triangle are 16, 18, and 24 chains; how many acres does it contain? Ans. 14.4 acres.
17. The distance between the feet of two rafters is 20 feet; and one of the rafters is 14 and the other 18 feet in length; required the height of the ridge above the plates on which the rafters stand. Ans. 12.2376 feet.
18. What is the area of a regular pentagon, whose side is 25 feet, and its apothegm 17.2 feet? Ans. 1,075 feet.
19. How many square rods are contained in an octagon, whose side is 12 feet? Ans. 2.77 rods. 20. The area of a pentagon is 4 acres; required its perimeAns. 96.435 rods.
21. The wheel of a carriage turns once and a half in a rod; required its diameter. Ans. 3.5 feet.
22. Required the value of a circular garden at 8 cents per square foot, its diameter being 6 rods. Ans. $615.8164.
23. If a horse be tied to a stake by a cord, what must be its length in order that he may feed on an acre of ground? Ans. 235.504 feet.
24. What is the radius of a circle that contains 2 acres? Ans. 2.5231 chains.
25. The diameter of a circle is 36 feet 3 inches; required the length of an arc of 30 degrees. Ans. 9.49 feet.
26. The chord of an arc of a circle is 12. feet, and its height, or versed sine, 2 feet; required the diameter of the circle. Ans. 20 feet.