Ex. 49. It is observed that two opposite walls of an ancient building are each 3o out of the vertical, the inclination being outward; to bring them into the perpendicular, the following means are employed; at certain intervals iron bars are placed across the building, their ends passing through the walls and projecting on the outside, on these ends strong plates or washers are screwed; the rods are then heated and expand, in this state the washers are screwed tightly against the outside of the walls and the rods allowed to cool, when they contract and draw the walls together; the process being continued until the walls become vertical.* If we suppose the rods to be 50 ft. long and 3 square inches in section, and to be fastened 15 ft. above the joint of the masonry, round which walls will be made to turn; and if the range of temperature is from 60° F. to 240° F.; determine the number of times the bars must be heated before the operation is complete, and the pressure which would tend to draw the walls together if they were entirely immovable. Ans. (1) 27 times. (2) 100,572 lbs. 7. Resistance to rupture by tearing or tenacity. When a strain which elongates a bar attains a certain magnitude, the bar will break. If we determine by experiment this force in lbs. per square inch, we obtain the tenacity of the substance. It is manifest that the strain which will tear a bar whose section is n square inches will be n times the tenacity. Ex. 50. How great a strain will a cylindrical bar of wrought iron bear which is of an inch in diameter? and by what fraction of its length would it lengthen under this strain if the elasticity continued perfect? Ans. (1) 3298 lbs. (2) 0.0023. Ex. 51. How many iron wires of an inch in diameter must be put together to sustain a strain of 3 tons? Ans. 13. * The walls of Armagh Cathedral were restored to a vertical position by this process. Daniell's Chemistry, p. 103. Ex. 52. What is the length of a bar of wrought iron which being suspended vertically would break by its own weight? Ans. 19,880 ft. Ex. 53. What strain will a bar of oak 11⁄2 in. square sustain ? Ans. 38,925 lbs. Ex. 54. What strain will a cylindrical bar of larch 14 in. in diameter sustain? Ans. 17,671 lbs. Ex. 55. If a rope be made of wires whose diameter is d, show that the number of wires in each square inch of the section of the rope is very of an inch in diameter must be put together Ans. 115. to form a rope a square inch in section? Ex. 57. If the number of wires of an inch in diameter which must be put together to form a rope one square inch in section be determined by each of the formulæ in Ex. 55, what is the difference between the results? Ans. 4.8. Ex. 58.-Show that the number of lbs. weight in a foot length of a rope made of iron wire is given by the formula (circ. in inches)2 x 0.244 very nearly; the specific gravity of iron wire being assumed to be the same as that of wrought iron. Ex. 59. Show that if a rope of hemp has the same strength as a rope of iron wire, the circumference of the latter is about of the circumference, and its weight about of the weight of the former. 8. Resistance to rupture by compression. There are as many as five forms which the results of crushing assume in different bodies. They are enumerated as follows by Mr. Rankine :* (1) Crushing by splitting, when the substance divides in a direction nearly parallel to the direction of the presThis occurs in the case of hard homogeneous substances of a glassy texture. sure. (2) Crushing by shearing, when the substance divides along a plane inclined at a certain angle to the direction of the force, the upper part of the substance sliding upon the lower. This fact was ascertained, and its conditions investigated, by Mr. Hodgkinson. It takes place in the case of substances of a granular texture, such as cast iron, * Applied Mechanics, p. 303. See also Mr. Moseley's Mechanics of Engineering, pp. 549, 579, and most kinds of stone and brick. To exhibit its effects the height of the block to be crushed must be at the least one and a half times its thickness. In the above cases the resistance to crushing is considerably greater than the tenacity. In the case of cast iron the resistance is more than six times the tenacity. (3) Crushing by bulging, when the material spreads like compressed dough. This takes place with ductile substances, such as wrought iron in short blocks. In this case the resistance is somewhat less than the tenacity, being in wrought iron about of the tenacity. (4) Crushing by crippling, which is characteristic of fibrous substances, and takes place when the thrust acts along the fibres in timbers and in bars of wrought iron that are too long to yield by bulging. It consists in a lateral yielding, and sometimes separation of the fibres. In the case of dry timber the resistance is about of the tenacity, in the case of moist timber about th of the tenacity; consequently moist timber is only half as strong as dry when subjected to a crushing force. (5) Crushing by crossbreaking, which is the mode of fracture in columns and struts where the length greatly exceeds the diameter. Under the breaking load they yield sideways, and are broken across like beams under a transverse pressure. Limestone (granular) 4,000 Ex. 60. What must be the height of a column of cast iron producing that pressure per square inch which would crush a short column of the same material? Ans. 35,805 ft. Ex. 61. Compare the heights of columns of cast iron, wrought iron, cast brass, and larch fir, which would produce the pressure per square inch requisite for crushing short columns of their respective materials? Ans. 1·475: 0439:0-116: 1. 9. Ultimate and proof strength and working stress. It must be borne in mind that no material is in practice subjected to the strain or thrust which it is capable of supporting. This will appear very clearly from the following definitions :* (1) The ultimate strength of a solid is the stress required to produce fracture in some specified way. (2) The proof strength is the stress required to produce the greatest strain in some specified way consistent with safety. A stress exceeding the proof strength, though it does not produce immediate fracture, will produce it by long application or frequent repetition. (3) The working stress is always made less than the proof strength in a certain ratio determined by experience. In the cases of wrought-iron boilers, timber, brick, and stone, the ultimate strength is from 2 to 3 times the proof strength, and from 8 to 10 times the working stress. In the following examples the working stress is assumed to be th of the ultimate strength : Ex. 62. A wall of brickwork 3 ft. thick, is supported at intervals of 10 ft. by sandstone columns 9 in. in diameter; to what height can the wall be carried? Ans. 7.6 ft. Ex. 63. If in the last example the columns had been of brickwork 2 ft. thick, to what height would the work then be carried? Ans. 10.8 ft. Ex. 64. To what height could the wall in Ex. 44 be carried with safety so far as the strength of the columns is concerned ? Ans. 34-26 ft. Ex. 65. Make the same determination with regard to Ex. 45. Ans. 45.8 ft. * Rankine, Applied Mechanics, p. 273. Ex. 66. What would have been the heights in each of the last examples if the columns had been of brickwork? What if of limestone? What if of granite ? Ans. Brickwork, 2.9 ft. 3.9 ft. 19.3 ft. 38.6 ft. Ex. 67. A wall of brickwork, 50 ft. high and 3 ft. thick is to be carried by columns of brick 20 ft. apart, from centre to centre; determine the least diameter consistent with safety. Make the same determination if the columns were of granite. Ans. 73 in. brickwork. 23 in. granite. 10. Strength of cast-iron columns. The columns in the preceding examples are supposed to follow the law of the crushing of short columns. It may be instructive to add the following particulars, which have reference to the crushing of cast-iron columns exceeding that length. The greatest part of our knowledge of this subject is due to experiments conducted by Mr. Hodgkinson, who thus states his conclusions with regard to the form of the ends of iron columns :-'1st. A long circular pillar, with its ends flat, is about three times as strong as a pillar of the same length and diameter with its ends rounded in such a manner that the pressure would pass through the axis. 2nd. If a pillar of the same length and diameter as the preceding has one end rounded and one flat, the strength will be twice as great as that of one with both ends rounded. 3rd. If, therefore, three pillars be taken, differing only in the forms of their ends, the first having both ends rounded, the second having one end rounded and one flat, and the third both ends flat, the strength of these pillars will be as 1-2-3 nearly.' Mr. Hodgkinson further considers that the breaking weight w of a hollow column is given in tons by the formula, W=MX D3.5-3.5 .... and that of a solid column by the formula D3-5 w=m × 11.63 |