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is to the breadth as v? is to I, or nearly as 1:4142136 to 1; or as 7 to 5. Its two sides must be to the diameter of the tree as the vį and vs to l. The required proportions are obtained by dividing a diameter ab, fig. 613h., into three equal parts ac, cd, db, and drawing the lines ce and df at right-angles to ab; the points atbf being joined, the figure is that of the strongest beam that can be cut out of a cylinder. The strength of a square beam, fig. 615g., cut from the same cylinder, is to the strength of the strongest beam Dearly as 101 to 110, although the square beam would contain more timber nearly in the ratio of 5 to 4.714. The stiffest beam is to the strongest as 0.97877 to 1, as regards power of bearing a load ; but as I •04382 to 1 as regards amount of deflection, in equal lengths between the supports.

1628w. Buffon, during his extensive series of experiments on oak timber, from 20 to 28 feet in length, and from 4 inches to 8 inches square in section, found that the heart-wood which was densest was also strongest, and the side on which the beam was laid also affected the strength; for when the annual layers were horizontal, and the strength 7, the layers laid vertically gave a strength of 8. He also found that beams which had each supported, without breaking, a load of 9,000 lbs. during one day, broke at the end of five or six months with a weight of 6,000 lbs., that is to say, they were unable to carry for six months two-thirds of the weight they bore for one day.

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Fig. 613i.

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1628r. The transverse section of a cast iron girder previous to Hodgkinson's experiments

was that of Tredgold, consisting of equal flanges at top and bottom, as A,
fig. 613i; and that of Lillie and Fairbairn, in 1825, with a single flange, as
B; Hodgkinson deduced a section of greatest strength having areas of flanges
as 6 to 1, as C. Taking this form as unity, the ratios will stand :-
For Hodgkinson and Fairbairn, as

For Hodgkinson and Tredgold, as

For Fairbairn and Tredgold, as -

1: 820
(Fairbairn, Application, &c. p. 25: Tredgold, Cast Iron, 1824, p. 55, describes
the advantages of his own form of section.)

1628y. Hodgkinson's complete section for a cast iron girder is shown in fig.
613). Its chief principle is, that the bottom flange must contain six times the
area of the top flange. The several dimensions are taken thus :-1. For the
depth, the total dimension D. II. For the bottom flange, the width B, and
for the two thicknesses, one is taken at the centre bb; the other b at the end.

III. For the top flange, the width T, and for the two thicknesses, one is taken at the centre tt, the other t at the end. In this manner the dimensions of the flanges are

quite independent of the thickness of the rib. V. For the rib
the two dimensions r and rr are measured as continued to the
extreme top and bottom surfaces of the girder, with the same
view of making these dimensions independent of those of the
fanges, and promoting exaetness in defining the entire section.
Hodgkinson's complex rule for obtaining the weight a girder
will carry, is så } bd-(6-0,") d,'=W. Here W=tons, or
b'eaking weight; 1 feet, or length between supports ; d whole
depth ; d, depth to bottom flange ; b breadth of bottom flange ;
and by thickness of vertical rib. The simp'er rule usually em-

ployed, as = W tons, or the breaking weight which should
not be less than four times the permanent load distributed; and
it gives a result less by 7 per cent. than the complex rule above
described, therefore an excess of strength is obtained.

1628z. The proportions of the rib are undetermined, but it -B

is evident that they should bear some ratio to those of the

flanges. It must be sufficiently rigid to prevent läteral werkFig. 613;.

Moreover the very theory which maintains the principle of the neutral aris (par. 1630c) also recognises the increase of the forces of compression and extension upward and downward from the neutral aris, and would therefore lead to the adoption of a rib tapered in both directions. In practice it is found desirable to taper the rib so as to meet each of the flanges with a thickness corresponding to that of the flange, for if any very great disproportion exists, the operation of casting the beam cannot be so perfectly performed, from unequal shrinkage of the metal, and an imperfect casting or one hasing flaws in it, renders futile all calculations of strength.

1629. Hodgkinson gradually varied the form of section of girder in his experiments until the widths and depths of the flanges were as follows:- Top flange 2:33 inches wide, 0:31

1 feet



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inch deep; bottom flange 6.67 inches wide, 0-66 inch deep: the areas being 720 and 4.4 inches. The rib was 266 inch thick, and the total depth 54 inches. The constant or C was found to be 514 for cwts., or 26 for tons. (Warr.)

16299. It will scarcely be within our province to describe all the forms of sections, and the results of the experiments made by Fairbairn in obtaining his box beam or plate girder in wrought iron, but it is to be noted that all the cylindrical tubes broke by extension at the rivets before the tube could fail by compression. Fairbairn in his Application of Cast and Wrought Iron to Building Purposes, edit. 1857–8, p. 80, notices that although the plate girder be inferior in strength to the box beam, it has nevertheless other valuable properties to recommend it. On comparing the strength of these separate beams, weight for weight, it will be found that the box beam is as 100: 93. The plate beam is in some respects superior to the box beam; it is of more simple construction, less expensive, and more durable, from the circumstance that the vertical plate is thicker than the side plates of the box beam. It is also easier of access to all its parts for the purposes of cleaning, &c.

1629b. Fairbairn has formed a comparison between a wrought iron and a cast iron g rder for a span of 30 feet. The plate girder, fig. 613k, would be 31 feet 6 inches in length, and would be composed of plates 22 inches deep and faths thick; with angle iron 6.74 Iths thick, riveted on both sides at the bottom of the plate, and angle iron ; inch thick at the top, the width over the top being 7 inches, and the bottom 5, inches. The breaking weight of this

6 x 22 x 75 beam, taking the constant at 75, would be adq=W; or =27.5 tons in the middle, or 55 tons distributed equally over the surface. In the edition of 1857–8, the angle irons are described as 3 inches by 3 inches, 1 inch thick for the bottom, and 4 inches by 4 inches, 1 inch thick at the top; it would, therefore, be 84 inches et over at the top, and about 6 inches at the bottom. Now a cast 54. iron girder of the best form and strongest section and calculated Fig. 613. to support the same load, would weigh about 2 tons, the plate beam about 18 cwts., or less than one half. To secure uniformity of strength in a rectangular box beam, the top is required to be about twice the sectional area of the bottom; hence resulted the use of cells in that portion.

1629c. Fig. 6131. is a plate beam having a single plate for the vertical web, while each of the flanges consists of a flat bar and a pair of angle irons riveted to each other and to the vertical web. Fig. 613m. is a bor beam, in which there are two vertical webs. Fig. 613n. is a plite girder of greater dimensions than fig. 6131. ; the Hanges contain more than one layer of flat bars, and the web, which consists of plates with their largest dimensions vertical, is

db stillened by vertical T iron ribs at thie joints of those plates, as shown in the plan or horizontal section luttered A. The pieces should abut closely and truly against each other, having end surfaces made exactly perpendicular to the axis of the beam. The thickness of the web is seldom made less than this inch, and except for the largest beams, this is in general more than sufficient to resist the shearing stress. Above each of the points of support, the vertical ribs must be placed either closer or made larger, so that they may be jointly capable of safely bearing as pillars the entire share of the load which rests on that point of support. A pair of vertical T iron ribs riveted back to back through tbe web plates (as A, fig. 613n.) may be held to act as a pillar of cross-shaped section.

1629d. The rib or web of a plate beam, as fig. 6131, having little or nothing to do with the pressure directly, has been replaced in some cases by simple upright struts or diagonal braces between the flanges, which in cast iron girders are in one casting, but experience has proved this not altogether politic, particularly in cast iron. Hodgkinson remarked that such beams were weaker than those with a solid rib. Rankine observes that transverse ribs or feathers on cast iron beams are to be avoided, as forming lodgments for air bubbles, and as tending to cause cracks in cooling. Open work in the vertical web is also to be avoided, partly for the same reasons, and partly because it too much diminishes the resistance to distortion by the shearing action of the load.

1629e. “Where the span renders it impracticable to roll a beam in one piece," Fairbairn, page 91, notices that “convenient weighis might be rolled into sections of the proper forin

and being united by properly proportioned covering plates at top and bottom, and te

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