ON THE STRENGTH OF PILLARS. 
395 
Pillars enlarged in the Middle. 
20. It has been stated above, that long pillars of uniform thickness, with both ends 
alike, always break first, in or near to the middle: this was the case even when they had 
discs upon the ends, to give the utmost firmness to their fixings. It seemed evident, 
then, that the pillar was always too weak there. I felt, therefore, desirous to ascer- 
tain the effect of strengthening the middle of pillars ; but as the ends could not be 
reduced at pleasure, since they would be crushed without bending, in the manner of 
our shortest pillars in Table II., I increased the diameter in the middle, leaving that 
of the ends the same ; and conceiving it best to make the experiments upon pillars 
whose form was as simple as possible, models were made of the form of two frustums 
of cones, the bases of which met in the middle of the pillar, the end diameter being 
1 inch, and the middle 1^, 1|, If, 2 inches. The sides, therefore, were straight, and 
regularly tapering from the middle to the two ends. Tables VI. and VII. contain the re- 
sults of the experiments, the pillars being all of the same length. In the first of these 
Tables, the extreme ends of the pillars were rounded, that the pressure might be through 
the axis ; and in the second, the ends had large discs upon them, turned flat. 
Table VI. shows, that in all the pillars with rounded ends, those with increased 
middles were stronger than uniform pillars of the same weight, the increase being 
about one-seventh of the weight borne by the former. 
In the pillars with discs, Table VII., those with the middle but little increased, had 
no advantage, with regard to strength, over the uniform ones. But the pillars with 
the middle diameter half as great again as the end ones, bore from one-eighth to one- 
ninth more than uniform pillars of the same weight with discs upon the ends. 
Strength of long Pillars as dependent upon their Dimensions. 
21. We shall now investigate the relative strengths of long pillars, as influenced by 
variations in their diameter and length. From the theory of Euler, it would appear 
that the power of a pillar to resist incipient flexure is directly as the fourth power of 
the diameter, and inversely as the square of the length. The inquiry in this paper is 
with respect to the resistance of pillars to fracture ; and, as has been mentioned be- 
fore, I have not been able to find the point to which Euler’s computations refer. His 
measures of the strength, however, seem not very widely different from those which 
apply to the breaking point. 
Strength as dependent upon the Diameter. 
22. I shall first ascertain, from the first and second Tables, the power n of the dia- 
meter to which the strength is proportional, in pillars of the same length. Com- 
paring the resistances of the pillars whose diameters were ‘50 and 1765 inches, and 
the length 60^ inches, in Table I., we find that the former was broken with 143 lbs., 
and the latter with 15560. 
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