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RED DEAL: one end flat, and the other rounded.
from cols. 2 and 3 of Table 58, the values of Cp and Cp, or 99 and 74.2 respectively, the Rule (164) becomes 54 x 99 = (54 + 742) = 41.8 tons.
Thus the reduced strengths under the 3 different forms at the ends are 18, 32, and 41:8 tons, those due by flexure only, being 18, 36, and 54 tons respectively.
(295.) “ Rectangular Pillars." — Table 58 may easily be applied to rectangular pillars: these always fail by bending in the direction of the least dimension, so that a rectangular pillar may be regarded as a number of square ones; thus 2 x 6 is equivalent to three 2-inch pillars, the breaking weight of which, say 8 feet long, would be = 1.8 X 3 = 5.4 tons. Again, a pillar 7 feet long, and say 3 x 7} would give a breaking weight = 9.6 x 73 = 24 tons, &c.: from this it will be seen that the strength of rectangular pillars of all kinds is simply proportional to their larger dimension. Table 59 has been thus calculated from Table 58.
ON THE CONNECTION BETWEEN THE STRENGTH OF PILLARS, AND
THE TRANSVERSE STRENGTH AND DEFLECTION OF A BEAM OF THE SAME MATERIAL,
(296.) By the theory of the strength of pillars in (136), &c., it is shown that the transverse strain on a beam multiplied by the distance between supports in inches, and divided by four times the deflection produced by that strain, will give the equivalent longitudinal strain which tends to break the same beam as a pillar (137). It is also shown (147) that the theoretical strength of pillars is directly proportional to d', and inversely as L.
We now propose to test the accuracy of the theoretical laws which connect the transverse and longitudinal strains by comparing calculation with experiment. This comparison is rendered very difficult by the fact that the strengths of cast iron, wrought iron, and steel pillars do not follow precisely the theoretical law d'L; the experimental law (147) for cast iron being d3 : L', and for wrought iron and steel d$* • L'. Timber pillars, however, follow the theoretical law precisely. TABLE 60.-CAST-IRON PILLARS : comparison of Theory with
(297.) “Cast Iron.”—Taking first, cast iron ; Table 67 shows that a bar 12 inches long and 1 inch square deflects 0·0785 inch, with 2063 lbs. in the centre; hence the rule in (138), namely W = w x1=(8 x 4), becomes 2063 x 12 = (.0785 x 4) =78840 lbs., or 35•2 tons breaking weight as a pillar 1 inch square, the strain being in the centre, or the pillar having pointed ends. By (361) the ratio of the strength of square to round is by experiment 1.5 to 1.0; hence we have 35.2:15 = 23.5 tons for a round pillar 1 foot long: calculating in this way, we obtain col. 2 of Table 60, which gives the strength with various lengths.
Mr. Hodgkinson found by his experiments that the strength of cast-iron pillars is governed by L"?, instead of Las by theory :—the effect of this divergence is very great, for instance, with a length of 10 feet, L' = 100, but L"? = 50 only, giving thus double strength to that due by theory for that particular length. By col. 1 of Table 34, the experimental strength of a pillar 1 inch diameter and 1 foot long, with both ends pointed, is 14.73, or say 15 tons, and admitting the strength to be inversely as L''?, we obtain col. 3 of Table 60, which shows that the theoretical and experimental strengths agree when the length is about 4£ feet, or 54 times the diameter.
The length with which theory and experiment agree will not, however, be the same for all diameters, because according to theory, the strength varies directly as D`, whereas by experiment it is as D3•. Thus, if the strength of a pillar l-inch diameter 1.0, then another of the same length, but 6 inches diameter, would by theory have a strength of 69 = 1296, whereas by the experimental ratio it would be 636 = 633 only, or about half. The effect of this divergence of the laws is shown by Table 60 to be that with a pillar 2 inches diameter, the results coincide with a length of about 114 feet, or 70 times the diameter: with a 3-inch pillar they coincide with a length of 19} feet, or 78 times the diameter.
(298.) “ Wrought Iron."-By Table 67 a bar of wrought iron 1 inch square and 1 foot between supports, loaded transversely with 2000 lbs., or •893 ton in the centre, deflects • 0313 inch; hence the equivalent load as a pillar will be •893 x 12 = (.0313 x 4) = 85.6 tons, breaking weight of a pillar 1 inch square and 1 foot long, with both ends pointed. From this we obtain by the ratio given in (225) 85.6 - 1.7 = 50•4 tons for a cylindrical pillar 1 inch diameter and 1 foot long. By col. 1 of Table 34, the experimental strength is 42.79 tons :—this ratio, 50.4 to 42.79 will prevail for all lengths, for, as we have seen in (296), the strength varies inversely as L’, both theoretically and experimentally. But it will not be the same for all diameters, theory giving D*, and experiment D*'s, the effect of which is shown by Table 61, where col. 2 is the theoretical, and col. 3 the experimental strength for different diameters. It will be observed that the two rules agree in their results with a diameter of :67 inch :—with larger diameters the theoretical results are in excess, and with smaller diameters in defect.
TABLE 61.-PILLARS of WROUGHT IRON and STEEL: comparison of
Theory and Experiment.
(299.) “ Steel Pillars.”—By Table 67, a bar 1 inch square and 1 foot long between bearings, loaded transversely with 5600 lbs. in the centre, deflects .0802 inch: hence by Rule (138) W = 5600 x 12 = (-0802 x 4) = 209500 lbs. is the equivalent strain as a pillar 1 inch square, or 209500 = 1.7 = 123300 lbs., or 55 tons, for a cylindrical pillar 1 inch diameter with both ends pointed. The experimental strength by col. 1 of Table 34 = 48.44 tons. With Steel, as we found with wrought iron, a comparison of the theoretical with the experi