reasoning as to the cause. Considering the pillar as having two functions, one to support the direct crushing weight, and the other to resist flexure; when the pressure necessary to break the pillar is very small because of its great length in proportion to its diameter, then the whole strength of the material may be considered as employed in resisting flexure. When the breaking weight is half of that required to crush the material, one half only of the strength may be considered as available for resistance to flexure, the other half being employed in resisting crushing. When, through the shortness of the pillar, the breaking weight is nearly equal to the crushing strain, we may consider that no part of the strength of the pillar is applied to resist flexure, &c. It was found by experiment, that when the load on a pillar was only of the crushing strain, there was a sensible falling off in the strength as calculated by the rules in (151), due therefore to "Incipient" rather than absolute Crushing.

As the combined result of reasoning and experiment, Mr. Hodgkinson gives the rule :

(164.) In which F

the breaking weight by flexure as due by the rules in (151) (156), &c.

Cp the crushing strain due to the area of the section and the specific strength of the material.

=

Po the reduced actual breaking weight; all in the same terms.

This rule requires some caution in its application; where F is less than Cp, the effect of it would be to make the calculated strength greater than F. Now, the strength of a pillar can never be greater than is due to flexure, hence there is a limit beyond which the rule must not be applied :—when F is exactlyCp, the effect of the rule is nil: when F is greater than Cp, the rule is necessary, and will reduce the calculated strength of the pillar as due by flexure:-when F is less than

Cp the rule will give the erroneous result of making Po greater than F.

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=

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Thus, say Cp 80 and F = 20, or exactly Cp; then, 80 × 75 60, and the rule (164) gives Po 20 x 80 (20+60) = 20, or the same value as F, the effect of the rule being nil. Again, say Cp 80, F = 10, then P。 = 10 × 80 ÷ (10+60) 11.43, which is greater than 10, or the value of F, and is impossible, showing that the rule has been applied in a case where it was not admissible. Again, say F

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=

=

=

80

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30, and Cp as before:-then the rule gives Po = 30 × 80 ÷ (30+60) 26.67 tons, which is less than 30, the value of F, and is a correct result.

(165.) We may now search for the lengths of pillars, with which the correction given by this rule becomes nil, which will happen when the length is such that F or the breaking weight by flexure is Cp.

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The mean crushing strain of cast iron is 43 tons per square inch as given in (132); a pillar 1 inch diameter will be crushed with 43 × 7854 = 33.73 tons, and the required length of pillar will be that which breaks by flexure with 33·77 ÷ 4 8.44 tons. For pillars with both ends pointed, the value of Mp as given by Table 34, is 14.73 tons, and the rule (154), namely, L = 1 (Mp × D3·6 ÷ W), becomes in our case (14·73 × 1÷ 8.44)17 1.387 feet, or 16.64 inches. The length with which the correction becomes nil is therefore in this case 16.64 times the diameter. Similarly, with one end flat, and the other pointed, the value of Mp being in that case 29 46 tons by Table 34; we have (29.46 × 1 ÷ 8·44) 1·7 2.086 feet, or 25 inches, the length being thus 25 times the diameter. With both ends flat Mp 44.19 tons, and the length comes out (44.19 x 18.44) 2.648 feet, or 31.78 inches, the length being thus 31 78 times the diameter.

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(166.) Mr. Hodgkinson, adopting 49 tons per square inch as the crushing strain of the particular iron used in his experiments, gives the length at 15 times the diameter for pillars with both ends pointed, and 30 times the diameter in those with both ends flat.

These ratios are, however, not constant for all diameters, as is

shown by Table 39, which has been calculated for cast-iron pillars by the rule:—

(167.) LD (Mp × D3·o ÷ ÷ C1) × 12 ÷ D.

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=

In which LD the length in terms of the diameter with which correction for incipient crushing is nil; Cp = the crushing strain due to the area of section and the specific crushing strength of cast iron, or 43 tons; Mp the multiplier for pillars, given by Table 34; and D

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1=

the diameter in inches. This table shows that the ratio of length to the diameter is reduced as the diameter is increased, in the case of cast iron considerably, and still more so with wrought-iron pillars (202).

TABLE 39.-Of the LENGTH OF CYLINDRICAL PILLARS, in terms of the Diameter with which correction for "Incipient Crushing' becomes nil.

(168.) As an example of the application of the rule (164) we may take the pillar 6 inches diameter, inch thick, and 14 feet long, which we found in (160) to break by flexure with 151 tons. By a table of areas, 6 inches 28.3, and 5 inch the area of the annulus

=

= 28.3

19.6 =

=

19.6, hence 8.7 square inches,

and the mean crushing strength of cast iron being 43 tons per square inch (132), we obtain 8.7 x 43 = 374 tons for the value of Cp, and 374 × = 280 for Cp. Then the rule (164) becomes Pc 151 × 374 ÷ (151 + 280) 131 tons, the reduced breaking weight.

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(169.) The fact (164) that the correction for incipient crushing is necessary for those cases only where F is greater than Cp, supplies an easy method of finding beforehand where it is required, so as to save the labour of going through the whole calculation. Thus, taking No. 34 in Table 38, col. 10 gives 148,000 lbs. for the value of Cp, then Сp becomes 148000 ÷ 4 = 37,000; by col. 9, F = 19750 lbs., which is less than C, therefore the correction for incipient crushing is not necessary, and the breaking load of the pillar is simply that due by flexure. Again, in No. 40, col. 10 gives Cp = 38690 lbs., | Cp becomes 38690 ÷ 4 = 9672 lbs., but F by col. 9 is 141,060 lbs., which being greater than Cp, requires correction by the rule in (164) by which we obtain 32,100 lbs. for the reduced breaking load, as in col. 7.

The effect of the application of this rule is in some cases remarkably great: for instance in Table 55, No. 9, col. 9 gives 10,750,000 lbs. for the value of F, but Po 79380 lbs. only by

col. 7, orth of F.

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The rule for incipient crushing applies not only to cast-iron pillars, but equally to all other materials. It is used for wrought iron, steel, and timber in Tables 44, 57, and its correctness is proved by the general agreement of the calculations with the experimental results as shown by (959) and Table 150, that agreement being to a great extent due to the use of the rule.

(170.) " Square Pillars of Cast Iron."-It is shown in (359) that the theoretical ratio of the strengths of square to round bars, either as pillars or beams, is 1.7 to 1.0, but the experimental