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mental strength will not be affected by the length, both materials being governed by L’; but the diameter affects the comparison considerably, as shown by Table 61, where col. 5 follows the theoretical law Dʻ, and col. 6, the experimental law D, the two laws coinciding nearly in their results with the diameter of 0.74 inch.
(300.) The superiority of steel over wrought iron as a pillar is shown to be remarkably small: with 1 inch diameter and 1 foot long, theory gives 50.4 to 55.0, or 55.0 = 50.4 = 1.09, an increase of 9 per cent. only. The experimental strengths are 42.79 and 48.44, or 48•44 = 42.79 = 1.13, an increase of 13 per cent. This applies to long pillars only; with short pillars requiring correction for incipient crushing the superior crushing strength of steel will give it much greater advantage as a pillar. We have shown (133) that the value of C for wrought iron in pillars is 19 tons per square inch, whereas for Steel (268) it is as much as 52 tons; a very short steel pillar, where the strength depends almost exclusively on the resistance to crushing, will have 52 = 19 = 2.74 times the strength of a similar one of wrought iron.
(301.) From the preceding investigation it will be evident that in pillars of cast iron, wrought iron, and steel, the divergence of the theoretical from the practical laws governing the strength, renders the former unreliable for those materials.
Fortunately, under these circumstances, we have practical rules whose general accuracy has been experimentally proved as shown by our various Tables, and more particularly by (959) and Table 150.
(302.) “ Timber Pillars.”—The experiments on Timber pillars in Table 57 show that they follow precisely the theoretical law D4 - L', which simplifies comparisons very considerably. Unfortunately the experimental information available is very scanty, this however will only enhance the value of the theoretical investigation, as we shall obtain thereby a knowledge of the strength of pillars for many kinds of Timber of which nothing is known experimentally. As the few experiments we have agree well with the theoretical results, as shown for Dantzic Oak by (139), we may have the more confidence in the theory as applied to other cases. Thus the mean error of all the calculated strengths in Table 57 is shown in (291) to be less than } per cent.
(303.) We may now find the value of the constant Mp for Timber pillars from the transverse load, and corresponding deflection by the theoretical law (138), namely W = wxl; (8 x 4). Taking the values of w and 8 from cols. 3 and 4 of Table 67, we obtain for pillars 12 inches long, 1 inch square, both ends pointed, the values of Mp:
We obtain from this col. 4 in Table 34 for pillars with both ends pointed; then adopting Mr. Hodgkinson's Ratios 1, 2, 3 for, both ends pointed, 1 pointed and i flat, and both ends flat respectively (149), we have obtained cols. 5 and 6.
The theoretical ratio of the strength of square and round pillars (519) is 1.7 to 1.0, but the experimental ratio (361) is 1.5 to 1.0: adopting the latter we obtain cols. 1, 2, 3, in Table 34. For Dantzic Oak and Red Deal we have taken the experimental values of Mp which agree the best with Table 57.
The strength of Timber Pillars may be found from the Modulus of Elasticity: thus in Table 34, the pillars are arranged in the order of their strength, and in col. 7 of Table 105 we have the Modulus of Elasticity. Taking as examples Teak, Pitch-pine, and Cedar, we have strong, medium, and weak pillars: Teak gives for & cylindrical pillar with both ends pointed 2413410 x .0046 = 11100 lbs., col. 1 of Table 34 gives 11150 lbs.: Pitch-pine = 1224840 x .0046 = 5634 lbs., Table 34 gives 5600 lbs.: Cedar 448237 x .0046 = 2246 lbs., Table 34 gives 2247 lbs., &c.
ON THE WRINKLING STRAIN.
(304.) When a rectangular pillar is made of thin wroughtiron plates, and the sizes are such as to preclude yielding by flexure, it is necessary not only to have sufficient area to resist crushing, but also considerable thickness to prevent failure by Wrinkling or Corrugation.
Let Fig. 58 be a square pillar so short in proportion to the side of the square as to avoid the probability of failure by bending. We have seen in (201) that the absolute crusbing strength of wrought iron in the form of pillars is 19 tons per square inch; but if the plates are very thin, and the breadth unsupported, or the distance, say from A to B, very considerable, the plate would fail by wrinkling or corrugation near the centre-line C, with a strain much less than that required to crush the material. At, and near the corners, wrinkling would be prevented by the support which those corners afford, but the centre is only imperfectly supported by them, and the more imperfectly as the breadth of the plate, or the distance from a corner is greater.
(305.) Fig. 59 is half the tubular pillar Fig. 58, and we may admit as self-evident, that the edges D and E being supported at one side only, or from F and G, will fail by wrinkling with a strain much less than the plate A, B, whicb was supported at both sides, although the distance from a sapport is the same in both cases. In Fig. 60, we have a pillar where the distance H, J, or the distance from a support on one side is įth of the width of the plate A, B, which was supported on both sides :in the absence of experimental information we may assume that this would probably give equality of strength, the edge J failing by wrinkling with the same strain per square inch as the centre C of the wide plate A, B. In Fig. 61, we have the application of the same principles to a pillar of I section, the analogy of which with Fig. 60 is obvious :—this will be useful when we come to consider the crushing strain on the top flange of a plateiron (395) or lattice girder.
(306.) A rectangular pillar of thin plate-iron may fail in one of three ways. 1st, by Flexure; 2nd, by Crushing; 3rd, by Wrinkling; each being governed by laws peculiar to itself and differing from the other two. Of course it will actually fail from that particular strain to which its power of resistance is the least.
Taking No. 15, in Table 55, as an example; col. 9 shows that by flexure it would fail with 2,628,000 lbs. or 1173 tons, and the area by col. 4 being 1.532 square inch, this is equal to 1173 : 1.532 = 766 tons per square inch. But we have seen (201) that the absolute crushing strength of wrought iron in pillars is only 19 tons per square inch, or oth of the theoretical breaking weight by flexure in this case. Col. 12 shows that even this reduced strain was not borne by the pillar, which really failed by wrinkling with 7.108 tons per square inch, or little more than įrd of the crushing strain, and Togth of the strength due by flexure: thus by Wrinkling Crushing
Flexure the breaking strains per square inch, were 7.108 19.0
766 tons the Ratios of which are :1:0
108 Here evidently the pillar failed by Wrinkling, the actual breaking load being togth only of the bending strength and aşth of the crushing strength.
By increasing the length of the pillar, the resistance to flexure might be reduced until it became even less than the wrinkling strain : thus, for the sake of illustration, with 10 times the length the resistance to flexure (being proportional to LP) would become 766 = 100 = 7.66, or nearly the wrinkling strain.
With intermediate lengths we should obtain a mixed result, part of the strength being employed in resisting flexure, and part in resisting “incipient wrinkling” (249), (253). searching for the laws governing Wrinkling, it will therefore be necessary to take the experiments on short pillars, where the strength is dominated almost exclusively by that strain.
(307.) “ Laws of Wrinkling.”—The laws governing the wrinkling strain may be obtained from Mr. Hodgkinson's experiments on square pillars in Table 55, and on tubular beams of thin plate-iron in (406), and Table 77. They may be expressed by the rules :
(308.) Ww = Ntv ; bw x M. (309.)
bw = (Viw Mw = W
(311.) Mv = Ww X wbw two In which Ww = the compressive strain in tons per square inch with which the plate will wrinkle; t = the thickness of the plate in inches; bw = the breadth in inches of a plate supported at both edges, as in Fig. 58; where the corners are connected by angle-irons in the usual way, the breadth must be measured between their edges, as at C in Fig. 62. When the plate is supported at one edge only, as in Fig. 59, four times the distance projecting beyond the angle-iron must be taken for the value of bw, as explained in (305). My = the Multiplier found from experiment, the mean value of which in rectangular pillars - 80, and in Beams 104, as shown by Table 62.
Table 63 gives the Wrinkling strain for plates of different thickness and breadth calculated by the rule.
(312.) The value of Mw for pillars may be found from the experiments in Table 55; selecting the short pillars for reasons given in (306) we obtain Table 62, the mean being 79.45, say 80. Thus taking No. 15 as an example which failed with 7.108 tons per square inch by col. 12; then rule (311) becomes 7.108