(835.) Table 131 gives the results of experiments by Captain James, on Cast-iron bars of various sizes, all 9 feet long, the effect of velocity being here shown not as affecting the deflection, but as governing the breaking weight. In these experiments, the breaking statical weight was first found for each size of bar and the corresponding ultimate deflection; then lighter loads were caused to pass over similar bars at a certain fixed velocity, the load being increased continuously by increments of 56 lbs. until the bar broke, &c. Theoretically, as we have seen (775), the rolling load with which the beam breaks should be half the equivalent dead weight, the maximum effect being attained by a certain velocity such that the rolling body can fall by gravity the height due to the ultimate deflection in the same time as it takes to traverse the half-length of the beam. TABLE 131. Of the STRENGTH of BEAMS of CAST IRON, 9 feet long, to bear Loads rolling over them at different Velocities. Velocity d=14 inch, b=4 inches. d 2 inches, b = 1 inch. d 3 inches, b = 1 inch. (836.) We can calculate with approximate accuracy the velocities with which the different bars should break:-thus, the observed ultimate deflection of the 3 x 1-inch bars was 2.25 inches; the load would fall that height by gravity by the Rule (832) in (2.25 193) = 0.107 second, and as the rolling load has to travel half the length of a 9-foot bar, or 4 feet in that time, its horizontal velocity must be 4.5÷ 0.107 42 feet per second. By experiment, the bar broke with a velocity of 43 feet per second with 515, or nearly half the dead breaking load, as in col. 7 of Table 131. = Again, with the bar 2 × 1 inch, the observed ultimate deflection was 3.2 inches: the load would fall that height in (3.2193) 0.1288 second, hence the horizontal velocity would be 4.50∙1288 35 feet per second: the experimental velocity was 36 feet, with 5155, or nearly half the dead breaking load, as in col. 5. : Again with the bars 4 × 1 inch, the observed ultimate deflection was 4.45 inches; the load would fall that height in (4·45 ÷ 193) √/ = 0·1516 second, hence the horizontal velocity will be 4.50.1516 30 feet per second: the experimental velocity, however, was 43 feet per second, with 431 the dead breaking load, as in col. 3, &c. = (837.) "Effect of the Inertia of the Beam."-We have so far regarded the beam as having no weight, therefore yielding no resistance to impact by its vis-inertia, and for small bars, such as we have considered, the effect of inertia on the result would be so small that it might be neglected without serious error. For instance, the bar 2 inches deep, 1 inch wide, and 9 feet long, should break by the ordinary rules (324) with 22 x 1 x 20639917 lbs. in the centre, with an ultimate deflection of 0785 × 9o ÷ 2 = 3.18 inches (695). Now, that deflection being produced by a statical or dead load of 917 lbs., would equally be produced by a dynamic or falling load of half that amount, or 458 lbs., as shown in (775), neglecting for the moment the inertia of the beam. By (781) it is shown that in resisting impact, the power of a heavy beam is to that of a light one, as the inertia of the beam plus the falling weight, is to the I + w falling weight alone, or The inertia of a beam being taken as equal to half its weight between bearings, and the total weight in our case being 56 lbs., the inertia would be 28 lbs. ; hence the effect of the inertia would be to increase the falling 458 + 28 weight from 458 lbs., to 458 × 458 = 486 lbs., an increase of 28 lbs. only, or 5 per cent., which is so small that it may be neglected with impunity. (838.) But where the weight of the beam is great in proportion to the falling load, the case is very different. Say we have a beam weighing 2 tons between bearings, its inertia being therefore 1 ton, and that a rolling load was calculated to produce a deflection of 2 inches when the inertia was neglected, then the effect of the inertia would be to reduce the deflection 1 to 2 x = 1 inch, or to half. In that case the rolling 1+1 load might be increased to 2 tons, and the dynamic deflection would then be 2 inches, or the same as that due to a statical or dead load of 2 tons, the inertia doubling the power of this particular beam in resisting a falling or rolling load. (839.) In most practical cases of Railway bridges the highest attainable velocity is very much below that which we have shown (832) to be necessary, in order to obtain the maximum dynamic deflection. This fact, together with the resistance from the inertia of the bridge itself, causes the deflection from a rolling load to be in most cases very slightly in excess of the statical deflection from a dead load. Thus, in the case of the Ewell bridge experimented upon by H.M. Commissioners, the length between bearings was 48 feet, the weight of the bridge 30 tons, hence its inertia 15 tons, and the central statical deflection produced by an Engine and Tender weighing 39 tons, was 215 inch. With velocities of = inches respectively, which are irregular, and increase very slightly with increase of velocity; this is what might have been expected, as we shall see. (840.) We have shown (831) that at a certain horizontal velocity the dynamic deflection would be double the statical or •215 × 2 = 43 inch in our case: but to produce that deflection, the load must traverse the half-length of the beam in the time necessary for a body to fall that height by gravity, which in our case by Rule (832) would be (43 ÷ 193) √ = ·047 second. The horizontal velocity must therefore be 24 ÷·047 = 511 feet per second, or 348 miles per hour! whereas the highest velocity attained was 51 miles per hour only, or about 4th of that necessary to produce the maximum deflection :-it would therefore (833) have little more effect than a dead load, which was the fact, as shown by the experiments. Moreover, even with 348 miles per hour, if that had been attainable, we should not have had the full dynamic deflection, because the inertia of the bridge would reduce it from 43 to 43 x 39 =.311 inch. CHAPTER XXI. COLLAPSE OF TUBES. (841.) "Experimental Results."-The laws governing the strength of cylindrical tubes in resisting collapse under external pressure are so obscure that it seems hopeless to attempt to discover them by any theoretical investigation, and we are compelled to obtain Rules from experiment. Nothing was experimentally known until Mr. Fairbairn investigated the matter, and Engineers had to work in the dark. The laws obtained by Mr. Fairbairn are very remarkable, and differ entirely from the theoretical ones; for example, with a perfectly cylindrical tube the strain generated by external pressure would be simply a crushing one, and the strength in that case should be directly proportional to the thickness, and inversely as the diameter, the length having no effect on the result. But Mr. Fairbairn found with tubes made of thin wrought-iron plates, that the strength was directly proportional to the 2.19 power of the thickness, and inversely as the length as well as the diameter. (842.) Table 132 gives the general results of Mr. Fairbairn's experiments; they were for the most part on tubes without cross-joints, and with one longitudinal joint only, a fact which it is necessary to observe, as the strength is affected by it considerably (848). Direct evidence of the effect of length is given by many of these experiments; for example, Nos. 4, 5, and 9 were all of the same dimensions except the lengths, which were in the ratio 1, 2, 3; col. 4 shows that the strengths were almost precisely in inverse ratio, or 3, 2, 1, being in fact 140, 93, and 47 lbs. respectively. (843.) In five cases the ends of the tubes were free, in all the rest they were fixed as with an ordinary boiler flue, but either way the result was about the same. We should expect the pressure to be less with free ends than with fixed ones, but this result was realised in two cases only, Nos. 1 and 16, the difference by col. 6, being 11 6 and 14 per cent. only. In the other three cases the free-ended tubes were stronger than the average, Nos. 5, 9, and 19 giving 11.8, 12.7, and 4.5 per cent. respectively it would appear from this, that fixing the ends of a tube has no effect on the strength; which again is an unexpected result. No. 28 was not collapsed by 450 lbs. per square inch, nor was it likely to fail with that pressure, the calculated collapsing strain being 1396 lbs. by col. 5. : (844.) The tube No. 33 was of Steel, with which we should have expected greater strength than with iron, but a comparison of cols. 4 and 5 shows that the calculated pressure by the Rules for iron tubes = 298 lbs., but experiment gave with steel 220 lbs. only, showing that for some unknown reason, the strength of the steel tube was 26 per cent. less than that of an iron one, which is an unsatisfactory result requiring further experimental investigation: a similar anomaly, however, was observed with steel chain (102).. From these experiments Mr. Fairbairn obtains for cylindrical wrought-iron tubes the Rules:: (845.) P = 33.6 x (100 t)2-19 (L x d). = (846.) p5.6 x (100 t)2:19 (L x d). |
