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II. For short pillars, below 30 times their diameter in height with flat ends, or 15 times their diameter in pillars with rounded ends, the above formulæ do not apply. III. For solid columns, when the length is less than 30d, the formula is

49Wa

W +37a

= W1. Here W crushing weight of short column in tons; a sectional area of solid part of columns in inches. In hollow columns the thickness of metal should not be less d WC than 12 W+C = W tons. Here W as found above for long columns; C crushing force of material x sectional area of column; and W' crushing weight of short columns.

1631e. The formula for a rectangular pillar of oak, fixed at

7,200a

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the case may be. (Rankine, from Gordon, and Hodgkinson.)

For

16319. In order to give lateral stiffness to a flat-ended pillar, its ends should be spread so as to form a capital and base, whose abutting surfaces should be “ faced " in the lathe, or planed, to make them exactly plane and perpendicular to the axis of the pillar. the same reason, when a cast iron pillar consists of two or more lengths, the ends of those lengths should be made truly plane and perpendicular to the axis of the pillar by the same process, so that they may abut firmly and equally against each other; and they should be fastened together by at least four bolts passing through projecting flanges. (Rankine.)

1631h, Hollow Columns.-With an equal quantity of metal, a round column cast hollow is far stronger than one cast solid. The best form for cast iron columns is to make the inner diameter five-eighths of the size of the exterior diameter. The ring thus formed of the section of the column increases in strength according to the thinness, but the size of it must be kept within practical limits. If, in casting a hollow column, the core is driven to one side, the column of course cannot be loaded to its full resistance; it will not carry more than the thinnest part of it is strong enough to bear. Hollow columns, therefore, require particular care in casting them. Hodgkinson noticed in hollow pillars above 30 times as long as their diameter, that although the pillars were generally thicker on one side than the other, yet in bending, the compressed was always the thinner side; and as cast iron resists compression with above six times the force with which it sustains tension, no danger resulted from this almost unavoidable difference of thickness.

I. The formula given by him for a hollow cylinder not less than 15 times its diameter 130 tons d3-76-d13-76 in height, with rounded ends, is W tons.

21-7

=

II. For the same when not less than 30 times its height, with flat ends and fixed by discs,

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III. When the length is equal to 20 diameters, the value of W is 77,817 lbs. or 34.7

tons.

Gregory has adopted an average of d36 in both of these formulæ.

IV. For short pillars, below those lengths, the formulæ do not apply, as the strength of the column becomes modified in consequence of its being then partially crushed as well as bent.

Stancheons and Struts.

1631. Gordon's formula for the ultimate strength of wrought iron struts of a solid rectangular section fixed at the ends, as deduced from Hodgkinson's experiments, is

34,000

1+3000/2 W or load

62

a or sect. area

(b least thickness). For other forms of cross section. approximate rules have been given. But it may be, in many cases, more satisfactory to take into Account the least "radius of gyration" of the cross section; and for that purpose the formula may be put in the following shape :

36,000
1+

12

+36,000 72 *

W
a

Here r2 is the mean of the

squares of the distances of the particles of the cross section from a neutral axis traversing its centre of gravity in that direction which makes 2 least. For hinged ends, take 4, or 9,000 is to be substituted for 36,000. The value of r2 for a solid rectangle, least dimenFor a thin rectangular cell,

b2
12'

12

sion=b, then For a thin square cell, side=b, then 6.

breadth=b, depth=h, the

12 h +b h2h+3b

For a solid cylinder, diameter = b, then For a thin

62 16.

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cylindrical cell, diameter = b, then

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62

For an angle iron of equal ribs, breadth of each = b, then For an angle iron of unequal ribs, greaterb, lesser=h, then For a 24་

=

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62
24'

12 A+B (Rankine, who follows out the subject further.)

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cross of equal arms For Hiron, breadth of flanges=b, their joint area = B, area of web A, then 1631k. Stancheons of cast iron are recommended to be used in lieu of cast iron columns. The form shown in fig. 613x. is generally considered as the best for use; the flanges which divide the length into three equal parts, are found to add considerably to the strength of the casting in resisting the tendency of its load to produce deflection from the vertical position. Hodgkinson's experiments show that while cast iron is the better material for a pillar whose length does not exceed 26 times the diameter, wrought iron is the better material when the length exceeds that limit. For pillars with hinged ends, about 13 times is the limit, but these results are roughly approximate only. In order to stiffen wrought iron struts, they are made of various forms in cross section, such as angle iron, T iron, double T iron, channel iron, &c. The cross is a very convenient form as in cast iron; it is generally built by riveting bars of simple forms together. Thus it may be made up of T irons riveted back to back, or four angle irons riveted back to back; or by one flat bar, two narrower flat bars, and four angle irons, all riveted together, as fig. 613y., and as used for the strut diagonals of the Warren girders in the Crumlin viaduct. The stiffest form for a wrought iron strut is that of a cell or built tube, fig. 614a., which may be cylindrical, rectangular, or triangular, as fig. 614b. When a wrought iron strut is considered as hinged at the end, that is generally effected by its abutting at each end against a cylindrical pin, by which it is connected with some other piece of the frame-work, in the manner already described for tie-bars. To fix its ends in direction, as it seldom has large abutting faces, it is in general necessary to fasten it to the adjoining pieces of the structure by several bolts or rivets.

Fig 613

Fig. 613y.

Fig. 614a.

Fig. 6146.

16317. Cast iron, from its great resistance to crushing, is peculiarly well suited for struts, especially those of moderate length. The best form containing a given quantity of metal is that of a hollow cylinder (fig. 614c.); the thickness of metal is seldom less than of the diameter. I. The formula for the cylinder has already been given; II. for a cast iron strut 80,000

of a cross shape (fig. 614d.) the whole width being d, then 1+

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of sectional area. III. For a hollow square (fig. 614e.), d = diagonal, 1+

322 -
800 d2
-W, as before.

W, as before.

These

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IV. For a hollow cylinder (fig. 614f.), d= diameter 12 formulæ refer to struts fixed at both ends. V. When they are hinged at the ends, the second term of each division is to be made four times as great. 1631m. With the ends fixed; I. the formulæ for a hollow tube (fig. 614c.) a = 3000, then 16s (or sectional area inches)

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II. The formula for a cross with equal arms (fig. 614d) a=1000.
III. The formulæ for an angle with equal sides (fig. 614e) a = 1000.
IV. When hinged at the ends, take 4.

1631n. Detrusion, or shearing, denominates that kind of fracture, which would occur in use of shears if their edges were blunt; or when the punch of a punching machine

makes a hole in a plate. Fairbairn has deduced the following laws from his experiments : 1. That the ultimate resistance to shearing in any bolt or rivet, is proportional to the sectional area of the bar torn asunder. II. That the ultimate resistance of any bar to a shearing strain is nearly the same as the ultimate resistance of the same bar to a direct longitudinal tensile strain.

Table of the RESISTANCE OF MATERIALS TO SHEARING AND DISTORTION,

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4h

=

16310. To find the length between the end of a beam and the foot of a strut or of a rafter, necessary to resist the thrust of the latter, so as to prevent the detrusion of the beam, the formula to be used is ' the length in inches. Here b the breadth of the beam in inches; h horizontal thrust in pounds; and S, the cohesive strength in pounds of a square inch of the material. Tredgold states that 4 is a sufficient value for a factor of safety in this case; S=600 lbs. per square inch for fir, and 2,300 lbs. for oak, The strap or bolt usually employed to bind the rafter and beam together, should be at as acute an angle as possible, and holds the rafter in its place should the end of the beam give way.

1631p. Iron fastenings to joints. In forming eyes by welding, at the end of iron bars for chain links and other purposes, the bar is found to be weaker than in its plain form. In iron plate work, the joints are made by riveting on which the whole efficacy of the built-up plate work depends. Taking the strength of the plain plate as 100, a double

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riveted plate will be 70; and a single-riveted, 56. Again, with single plate jointings having a top and bottom covering plate over the joint, and with half-inch rivets, as A, fig. 6131, the plates were torn asunder through the rivet-holes, with 24:41 tons in the square inch. With a double plate, having a single covering plate on the side of the joint, as B, the plates broke asunder by shearing off the rivets close to the plate, with 18.73 tons per square inch; but the rivets having been made larger, a similar strength to the previous Experiment was realised. A plain plate broke with 22.78 tons mean value.

1631q. Fairbairn recommends the flanges or double plates to be used as long as possible, and the joints to be carefully united by covering plates, chain-riveted, as C, with three or more rows of rivets according to the widths of the plates. Eight rivets are required in each of the lines, four on each side of the joints, to give sufficient strength, and the area of the rivets collectively should be equal to the area of the jointed plates taken transversely through one line of the rivets, the area of the parts punched out in that line being deducted These proportions give the required security to the joint, and afford nearly the same strength to a tensile strain as the solid plate; that is, if the covering plates be as much thicker as will give the same area of section through the rivet-holes as the unperforated double plate. (par. 1629e.)

1631r. Rivets are made of the most tough and ductile iron. It is essentially necessary that the rivet should tightly fit its hole; and the longitudinal compression to which rivets are subjected during the formation of its head, whether by hand or machinery, tends to produce that result. The diameter of a rivet for plates less than half an inch thick, is

about double the thickness of the plate. For plates of half an inch thick and upwards, about once and a half the thickness of the plate. The length of the rivet before clenching (which is effected whilst the rivet is red-hot). measuring from the head, equals the sum of the thickness of the plates to be connected, added to 24 inches multiplied by the diameter of the rivet. A good rivet may be bent double whilst cold without showing any signs of fracture; and the head when hot should stand being hammered down to less than in. in thickness without cracking at the edge. They should also stand having a punch of nearly their own diameter driven right through the shank of the rivet when hot, without cracking the iron round the hole. (C. G. Smith.)

16318. Steel rivets, fully larger in diameter than those used in riveting iron plates of the same thickness, being found to be greatly too small for riveting steel plates, the probability is suggested that the proper proportion for iron rivets is not, as generally assumed, a diameter equal to the thickness of the two plates to be joined. The shearing strain of steel rivets is found to be about a fourth less than the tensile strain. (Kirkaldy). 1631t. In the bridge over the Thames for the Charing Cross Railway, the holes were drilled and not punched. This is a point upon which engineers differ considerably; but most firms punch the holes. At Fairbairn's works at Manchester, drilling holes was considered to be more expensive without adding to the strength. Mr. Parkes thinks that the punching injured the iron considerably, and thought Fairbairn's experiments went to show it. (Society of Engineers, Transactions, 1865).

163ļu. Pins, keys, and wedges are exposed, like rivets, to a shearing stress. The formula for finding their proper sectional area is the same. They must be held tightly in their aeats; and in order that a wedge or key may not slip out of its seat, its angle of obliquity ought not to exceed the angle of repose of iron upon iron, which, to provide for the contingency of the surfaces being greasy, may be taken at about 4°. (Rankine).

1631v. If a bolt or screw has to withstand a shearing stress, its diameter is to be determined like that of a cylindrical pin. If it has to withstand tension, its diameter is to be determined by having regard to its tenacity. In either case the effective diameter of the bolt is its least diameter; that is, if it has a screw upon it, the diameter of the spindle inside the thread. The projection of the thread is usually one-half of the pitch; and the pitch should not in general be greater than one-fifth of the effective diameter, and may be considerably less. In order that the resistance of a screw or screw-bolt to rupture, by stripping the thread, may be at least equal to its resistance to direct tearing asunder, the length of the nut should be at least one half of the effective diameter of the screw; and it is often in practice considerably greater; for example, once and a half that diameter. The head of a bolt is usually about twice the diameter of the spindle and of a thickness which is usually greater than gths of that diameter. (Rankine).

For

1631w. Wushers are flat plates of iron, placed at the sides of timbers to secure them against the crushing action of the head and nut of a bolt whilst being screwed up. fir, the diameter of the washer is made about 3 times that of the bolt; and for oak. about 2 times. When a bolt is placed oblique to the direction of the beam which i traverses, a notch should be cut in the timber perpendicular to the bolt, to receive the pressure of the washer equally, or notched to receive a bevelled washer of cast iron, on side of which fits the wood, and the other fits the axis of the bolt.

TORSION.

1631x. Torsion, or the resistance of bodies to being twisted, is found: I. When a bod is fastened at one end and a force is applied at the other. II. When the force at one en is greater than at the other end. III. When the forces at the ends are in opposite direction: and are so applied as to twist the body. As this fact chiefly, if not entirely, concerns machiner in motion, we refer the student for more specific details to Warr, Dynamics, p. 269, who give a table of "modulus of torsion" of various timbers and metals, derived from experiment made by Bevan, in Phil. Trans. 1829, p. 128. Approximate formulæ are given by Hurst :I. When the shaft is circular, Wd. And CW. II. When the shaft is squar 3/4W 8. Here d diameter inches; W weight pounds permanently sustained by the shaf I length of lever in feet, at the end of which W acts; s side in inches; and C, cast steel 590 wrought iron 335; cast iron 330; gun metal 170; brass 150; copper 135; lead 34.

5C

:

1631y. In the Artizan for 1857 and 1858 is an instructive Enquiry into the Strength Beams and Girders, by S. Hughes, deserving attention. The chief authorities for the da contained in that article, and also in this section, are quoted herein.

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a, Stress parallel to the fibres; b, ditto perpendicular to the fibres.

The above values of the safe load may be taken for structures subject to travelling loads. When subject to dead loads, these values may, in the case of iron and steel, be multiplied by. G. S. Clarke, Graphic Strains, 4to, 1880, p. 138.

1632a.

TABLE OF STRENGTH OF VARIOUS TIMBERS.

The primitive horizontal or transverse strength of oak is taken at 1000; its supporting or primitive vertical strength at 807; and its cohesive or absolute strength at 1821; being deduced from pieces 19.188 lines English square. The relative strengths of other woods are given :

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1633. Steel is now largely superseding wrought iron in all uses to which the latter material was usually applied. Nearly every section of L, T, and E. as well as rolled joists I, are now made in steel to specification. Railway metals or rails have been made of steel for some years. Plates, sheets, and bars for every purpose of bridge girder, roof, and boiler making, are now commonly in use, as also for cylindrical and octan ular columns which have to carry great weights; also for ship armour and gun mounts. Steel is most useful when bulk and weight is a consideration; the constructional cost, as a rule, can be brought down almost to that of iron; the price per ton is more, but less weight is required. The kind mostly used is called mild steel, containing about 0.18 per cent. of carbon, bearing a tensile stress 30 to 35 tons per square inch with the fibre, and 28 to 30 across the fibre. Much higher results can be obtained for special purposes, but the manufacture for ordinary structural purposes cannot be fully relied upon beyond 30 tons tensile. The Committee of the British Association advised a maximum of

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