9

account of its increase in width, and

1

times as great on account

n3n

or n2

n2

of its increase of length. It will thus be altogether or n2 times as strong."

15. We thus see how erroneous a conclusion would be drawn from thin models, as to the buckling of the top, in larger and similar structures, as was the case with the preliminary experiments on thin circular and elliptical tubes, which were condemned on this account; for the top of these tubes was merely a pillar failing from flexure, and had similar tubes been constructed of larger dimensions, then the strength of the top to resist buckling or flexure would have been as the square of the lineal dimensions of increase, until the pillars or thickness became of sufficient dimensions to resist buckling altogether, and to fail by crushing. The curvature in the top of these tubes was, in fact, favourable to the resistance of flexure; a flat plate would have buckled with less strain.

16. "The limit of 16 tons per inch cannot, however, be exceeded practically, whatever the form. This limit, it will be observed, is nearly approached in the inch square bars in experiment No. 16, which were 15 inches long; in the next experiment the bar was practically crushed with about 16 tons, although it carried considerably more; and we shall hereafter see that with the large models, with cells, the top was destroyed by compression with about 14.8 tons per square inch.

17. However valuable, in numberless practical cases, these interesting experimental confirmations of the theory of pillars may prove, they are evidently not directly applicable to the structure which led to them. The top of the Britannia Bridge, though closely analogous, is not a simple pillar or plate subjected to compression. When failure takes place in the top of an ordinary tube by actual compression, it will be found to have sustained little more than 16 tons per square inch of its transverse section, and no alteration of its manner of construction would be of any avail. But should it fail by buckling or flexure, its breaking strain would fall short of this quantity, and any contrivance or method of construction which would prevent flexure, would approximate it to this limit. The importance of some

such contrivance with thin tubes is thus evident. The necessity for preventing flexure in a column has already been alluded to, and (as in a beam broken transversely) the comparative depth, or leverage with which the material resists flexure, has been shown to be the most important consideration. We may double the width of a plank, and when we stand on the centre, it bends only half as much as before; but if we double its thickness, we have only one-eighth part of its original deflection. It, indeed, assumes another name, and we call it a beam, though in both cases the quantity of material is precisely the same.

18. A plate employed as a pillar similarly resists flexure in a much higher ratio than in the simple proportion of its thickness, such stiffness or strength being analogous to the transverse stiffness of a beam. Hence, as in the beam, it will also be highly advantageous to distribute any given material in a pillar in such a manner as to insure the greatest possible depth in the direction in which it is liable to bend. If the pillars are short as compared with their diameter, such precautions are useless, the cubic inch of wrought iron cannot be put in better form; but if it were rolled into a very long and very thin plate one inch broad, and placed on edge, the smallest force would bend it. If we shorten this thin plate by increasing its thickness, but maintaining the same height of one inch, we shall increase its resistance to flexure in proportion directly to the cube of the thickness, and in proportion inversely to the length, since the length will diminish in the same proportion as the thickness increases, therefore the strength will increase directly as the square of the increasing thickness, or inversely as the square of the decreasing length, until the plate arrives at such a thickness that it will fail partly by crushing. The law will now begin to vary as we go on increasing the thickness at the expense of the length; and ultimately, as we approach to the cube itself again, the strength, instead of varying as the square of the increasing thickness, will cease to vary at all with the thickness, its strength will, therefore, have varied during these changes, as every power of the thickness between 0 and the square of the thickness, while the resistance itself would be represented progressively by every quantity between 16 tons and 0, the quantity of material or section and height, having remained constant, so that n square

inches of sectional area on the top of a tube may resist any compression between 0 and n x 16 tons, according to the form in which it is applied.

19. We should thus use the thickest plates we can get for the top of a tube, until their thickness is such that any variation in the thickness causes no corresponding variation in their resistance to compression-beyond this we get no further advantage. If, however, we are compelled to use thin plates, we should arrange them so as to insure depth to resist buckling. If one cubic inch, when rolled out into a long strip so as as to fail by flexure, were, for instance, formed into corrugations, as in fig. 21, it would in this form support considerably more than in the form

Fig. 21.

of a straight plate, for instead of being a mere line in section with no depth, it would now possess a depth equal to the versed sine of the corrugations, or equal to the distance between each convexity; and in this corrugated form we should attain the maximum resistance to pressure, viz. 16 tons, with our plates much thinner than when used straight. The depth would be still further increased if we folded our corrugated plate round upon itself, so as to complete a series of tubes, fig. 22, taking care to unite carefully the points of contact. There are numberless familiar Fig. 22.

examples of stiffness obtained by such method of construction. An ordinary paper fan, and many household articles in tin, though constructed of thin and pliable material, are extremely strong and rigid from the depth acquired by the bending of the material. The domestic tea-board and dust-shovel are striking examples. It thus becomes a question, with a given section of material of given thickness, how to construct the strongest form of pillar to resist crushing, and how near we can with this form approach to the limit of 16 tons per square inch.

20. Since a flat plate, for the reasons explained, will bend

i

sooner than a curved plate, it would be readily inferred that a round tube, of moderate dimensions and of given thickness and section, would be a stronger form than the same plate in a rectangular form, in which the resistance to crumpling must depend solely on the four angles; and since the rigidity afforded by the angles is extended throughout the four sides of a rectangular tube, in some manner proportionate to the distance from the angles, it would be concluded that a square tube would be stronger than a rectangular tube constructed with the same plate, inasmuch as the central portions of the longer sides of the rectangle will be less maintained in form on account of their greater distance from the angles; similarly increased strength might be expected from this form:. These assumptions were all submitted to experiment and confirmed.

For this purpose a number of tubes or cells of wrought iron were constructed, all 10 feet long and either 4 or 8 inches square, or of rectangular form about 4 x 8 inches; their ends were perfectly flat, and they were compressed by the intervention of a lever between two parallel disks of steel, with arrangements for maintaining the pressure perfectly vertical, the cells being supported laterally. The direct object was to ascertain the value of each particular form of cell, and to ascertain the resistance per square inch of section in each case. The lateral dimensions of these cells were so large, that with a length of 10 feet the pillars were not destroyed by flexure, as in a long pillar, but by absolute buckling or crushing; the strongest possible form should, therefore, give about 16 tons per square inch of section.

21. Similar experiments were then made with circular cells under precisely similar circumstances for comparison. The cylinders varying from 14 to 6 inches in diameter, the diameter being small, in some cases, as compared with the length; some of these pillars failed by flexure, and followed the laws of long pillars, the resistance increasing nearly inversely as the square of the length; but where the diameter was 6 inches, the length being 10 feet, flexure could not take place, and the cells failed by buckling or crushing, as in all the rectangular pillars, and in such pillars the strength is independent of the length.'

22. The annexed drawings (figs. 23, 24, 25, 26) show how the failure in the rectangular and cylindrical tubes took place.

From the foregoing researches it will be observed, that in order to determine the maximum powers of resistance to compression in the use of iron plates, the square box with thin plates next to the plate itself is the weakest experimented

upon; the next in the order of strength is the rectangular form with a division across the centre as at a, fig. 24, but the best distribution of the material is in the cylindrical form. This latter cannot, however, be accomplished conveniently in ship building, but the rectangular or cellular construction is applicable in all cases where resistance to compression as well as tension is required in the hull and upper decks of vessels.