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experimental data for Timber pillars, and are compelled to resort to theory, in order to obtain the values of Mp for other kinds of Timber this we have done in (303), and we have thus obtained most of the numbers in Table 34. This method is of course not so satisfactory as direct experiment, but it is shown in (139) that as applied to Timber, the theoretical and experimental results practically agree with one another.

(292.) Table 58 gives the strength of square pillars of Red Deal calculated by the rule (282): we have selected the case of flat at one end and round at the other as approximating to ordinary conditions more nearly than any other. In most cases timber pillars are nominally flat at both ends, but this supposes that the surfaces between which the pillar is strained are perfectly parallel and unyielding, conditions which are seldom realised in practice: for example, when a soft-wood Bressummer is supported by a pillar, the effect of flexure in the latter is to compress the fibres of the former unequally, the soft wood yielding, so that the result is little if any better than it would have been with a round end. If we suppose that the foot of the pillar is well bedded on a large stone or cast-iron plate, and the upper end loaded by the Bressummer in the usual way, we should have in effect a pillar flat at one end and round at the other, being the conditions assumed in Table 58.

(293.) Say we take the case of a pillar 7 inches square; then by col. 5 of Table 34, the value of Mp for a pillar of Riga Fir, with one flat and one round end = 7.14 tons, and with a length of say 12 feet, rule (282) becomes 7·14 × 7122, or 7·14 × 2401 ÷ 144 = 119 tons, the breaking weight by flexure. This requires correction for incipient crushing, being greater than 4th Cp given by col. 4 of Table 58 (169). By col. 4 of Table 32, the specific resistance of Red Deal to crushing, or C = 2.75 tons per square inch, and as we have 72 49 square inches area, Cp becomes 2.75 × 49 = 135 tons, as in col. 2: 3 Cp = 101 tons, col. 3: and Cp 33.8, col. 4. the rule (164) becomes Po tons, as in Table 58. By the use of col. 4, we can easily determine when the correction for incipient crushing is necessary: thus, for a pillar 3 inches square 10, 12, and 14 feet long, the

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TABLE 58.-Of the STRENGTH of SQUARE PILLARS of

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breaking weights by flexure 5.8, 4.0, and 2.9 tons respectively, which are all less than 6.18 tons or Cp by col. 4; the correction is therefore not required, as shown by the Table, but for the shorter lengths, 5, 6, 7, 8, and 9 feet, that correction is necessary.

(294.) Table 58 may be adopted for conditions of fixing other than that of flat at one end and round at the other, as in that Table. By (149) it is shown that the breaking weights by flexure are in the ratio 1, 2, 3, for the three cases both ends pointed,—one flat, one pointed,—and both ends flat respectively. Thus a pillar 6 inches square, 16 feet long = 36 tons by flexure from Table 58: then with both ends pointed we have 36 ÷ 2 18 tons, which being less than 24.8 tons given by col. 4, correction for incipient crushing is not required (163). The same pillar with both ends flat 36 × 3 ÷ 2 = = 54 tons breaking weight by flexure, which being greater than 24-8 or Cp by col. 4, correction for crushing will be necessary. Taking

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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 × 99 ÷ (54 +74·2) = 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 × 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 × 71⁄2 would give a breaking weight 9.6 × 7÷ 3 = 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. calculated from Table 58.

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TABLE 59.-Of the STRENGTH of RECTANGULAR PILLARS of RED DEAL: one end flat and the other rounded.

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CHAPTER VIII.

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÷L2; the experimental law (147) for cast iron being d36 ÷ L', and for wrought iron and steel d÷L3. Timber pillars, however, follow the theoretical law precisely.

TABLE 60.-CAST-IRON PILLARS: comparison of Theory with

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(297.) "Cast Iron."-Taking first, cast iron; Table 67 shows that a bar 12 inches long and 1 inch square deflects 0.0785

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