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43 - L' in the case of pillars with both ends pointed, and to

L" in those with both ends flat :- for practical purposes a mean between these extremes may be taken for all modes of bearing at the ends, and we obtain the law 236 = L'7. For wrought-iron and steel pillars, the experimental law is 136 : L’.

(148.) The effect of this divergence of the experimental from the theoretical law is very considerable :thus, if the strength of a pillar 1 inch diameter 1:0, then another, of the same length, &c., but 6 inches diameter, would, by theory, have a strength of 64 = 1296, whereas by the experimental ratio it would be 63-6 633 only, or about half.

Again, as to the length :say we have a pillar 10 feet long whose strength = 1.0, then the same pillar with a length of 1 foot would, by theory, have a strength of 10% = 100, but by the experimental ratio it would be 1017 = 50 only.

It will be seen from this that it is impossible to give general " ratios” for the strengths of pillars of different materials, which will be correct for all diameters and lengths. Mr. Hodgkir son has given a series of numbers as the ratios of strength for castiron, wrought-iron, steel, and timber pillars, but these are simply misleading, for if they are correct for a particular diameter and length, they must, of necessity, be incorrect for all other dimensions.

(149.) Effect of Form at the Ends."-One of the remarkable results of Mr. Hodgkinson's experimental researches was to show that the strength of pillars is potentially governed by the character of the bearings at their ends : it was found that a long pillar of any material with both ends flat and well bedded, being pressed between two perfectly parallel planes, had a strength 3 times that of a similar pillar with both ends pointed or rounded, so that the strain was exactly in the axis. It was also found that with one end flat and the other pointed, the strength was an arithmetical mean between the other two.

We have, therefore, the ratios 1, 2, 3 for the strength of three similar pillars-with both ends pointed—one end pointed and one flat-and both ends flat respectively. That these ratios are practically correct may be shown by the tables in this chapter.

Thus, in Table 44, pillars of wrought iron 7.56 feet long 1.02 inch diameter failed with 1825, 3355, and 5280 lbs. respectively, the ratio being pretty nearly 1, 2, 3. Others 5.04 feet long and about 1.02 inch diameter, failed with 3938, 8137, and 12,990 lbs. respectively, which is rather in excess of the ratio 1, 2, 3.

With steel, Nos. 28 to 30 in the same table, we have in col. 6 numbers nearly in the ratio 1, 2, 3, which would have followed the law almost exactly but for the fact that Nos. 29 and 30 required correction for incipient crushing (163).

For Dantzic oak, in col. 7 of Table 57, we have 3197, 6109, and 9625 lbs., which are nearly in the ratio 1, 2, 3.

With cast-iron pillars Table 38 shows similar results: thus comparing Nos. 1 and 13, we have in col. 6, 143 and 487 lbs., where the ratio should be 1 to 3 :- Again, in Nos. 2 and 14, we have 1902 and 6238 lbs. where the same ratio should have prevailed. In these cases, however, the diameters of the pillars compared with one another are not precisely identical, which may account in part for the divergence of the experiments from the standard ratio 1 to 3.

(150.) It will be evident from all this, that it is highly expedient, wherever possible, to secure flat ends for pillars :with cast iron this is easily done by casting sole-plates at both ends, but even then great care should be taken that they are well bedded and guarded from the effects of unequal settlement of foundations, &c.

Connecting-rods, with jointed ends as usual, must be regarded as pillars with both ends pointed (204).

The piston-rod of a steam-engine may be taken as a pillar, flat at one end, where it is connected to the piston, and further steadied by the gland: the upper end being jointed at the cross-head, is assimilated to a pillar with pointed end.

(151.) “ Cast-iron Pillars.”—Mr. Hodgkinson's experiments have supplied very full information on the strength of pillars of cast iron; for solid cylindrical pillars we have the following general rules :(152.)

F = Mp X D3 : L'7. (153.)

D = SY (F XL"? Mp). (154.)

L = "V (Mp X 136 = F). (155.)

Mp = F X L'7- D36.

In which F = the breaking weight of the pillar by flexure

in lbs., tons, &c., dependent on Mp. D = the diameter of the pillar at the centre, in

inches. L = the length of the pillar in feet. Mp = constant multiplier, the value of which is given

in Table 34.

TABLE 34.-Of the Value of Mp, being the THEORETICAL BREAKING

WEIGHT OF PILLARS, 1 Foot long.

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Cast Iron

(Ibs. 33,000 66,000 99,000 56,100 112,200 168.300 tons 14.73 | 29.46 | 44:19 25 50

lbs. Wrought Iron

95,848 197,700 299,620 162,900 336,000 498,500

Itons 42:79 88.26 133.8 72.74 150 223 Steel

lbs. 108,500 217,000 325,500 184,400 368,900 553,300

tons 48.44 96.88 145.3 82.35 164:7 247 Dantzic Oak..

(lbs.

6,000 12,000 18,000 9,000 18,000 27,000 Itons 2.68 5.36 8.04 4.02 8.04 12.06

lbs. Red Deal

5,333 10,666 16,000 8,000 16,000 24,000

2.38 4:76 7.14 3.57 7:14 10.71 Teak

lbs. 11, 150 22,300 33,450 16,730 33,460 50, 190 Red Pine

8,520 17,040 25,560 12,780 25,560 38,340 Canadian Oak 8,360 16,720 25,080 12,540 25,080 37,620 Deal

7,933 15,866 23,800 11,900 23,800 35,700 Ash

7,773 15,516 23,319 11,660 23,320 | 34,980 Beech

6,222 12,444 18,666 9,333 18,666 28,000 Pitch Pine

5,600 11,200 16,800 8,403 16,806 25,209 English Oak 5,440 10,880 16,320 8,160 16,320 24,480 Riga Fir

5,200 10,400 15,600 7,800 15,600 23,400 Larch..

4,108 8,216 12,324 6,162 12,324 18,486 Memel Deal

4,087 8,174 12,261 6,130 12,260 18,390 Elm

3,154 6,308 9,462 4,731 9,462 14,193 Willow

2,600 5,200 7,800 3,902 7,804 11,706 Cedar

2,247 4,494 6,741 3,370 6,740 10,110 (1) (2) (3)

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(156.) “ Hollow Cast-iron Pillars.”—For hollow pillars, instead of D3-6, we have D3-6 – dze, in which D = the external,

TABLE 35.–Of the POWERS of NUMBERS for PILLARS, &c.

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12.3 32.3 5:01

13:8 38.2 5.42

15.6 41.8 5.80 17.4 52.2 6.20 19:4 60.5 6.60

21.4 69:6 7.00 23.6 79.8 7.42 26.0 90.9 7.84 28.4 103:0 8.29 31:1 116 8.73 | 33.8 9:19 36.7 147 9.64 39.7 164 10:1 42.9 183 10.6 46:4 203 11:1 49.8 225 11.6 53.6 248 12:1 57.6 273 12.6 61.5 300 13:1 65.7 328 13.6 69.7 359 11.2 74.5

391 14.7 79:1 426 15.3 83.9 463

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150 157 165 172 181 188 196 205 214 223 232 241 251 260

1175 1251 1330 1413 1500 1590 1685 1783 1882 1992 2102 2218

270 2338 281 2462 292 2591 302 2724 313

2863 325 3007 337 3158 348 3310 360 3470 373 3631 385 3805 398

3981 423 4351 452 4746 480 5165 510 5611 541 6083 573 6584 605 7115 640 7675 11 2 21 23

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and d = the internal diameter, and the rest being as before, we have the rules :

(157.) F = Mp X (D36 – 086) - L17.
(158.) L = ") {Mp (D*** – 43-6) =- F).
(159.) Mp = F x L1 = (D96 - ).

It should be clearly understood, that these rules give the breaking load of long flexible pillars, or those whose length is so great in proportion to their diameter, that they will fail by bending simply. Short pillars require correction for “Incipient Crushing," as explained and illustrated more fully in (163).

(160.) Tables 35, 36 give the 3.6 and 1.7 powers of numbers to facilitate calculations of the strength of solid and hollow pillars of cast iron, wrought iron, and steel :-Thus, say we

TABLE 36.-Of the 1.7 POWER of NUMBERS for CALCULATING the

STRENGTH of PILLARS.

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• 129 .211 .308 • 420 •545 .684 .836 1.00 1.46 1.99 2.59 3.25 3.97 4.75 5.58 6.47 7.42 8:41 9.46

10.56 11.70 12.90 14:14 15.43 18.14 21:03 24:10 27.33 30.73 31.30 38.02 41.90 45.94 50.10 58.94 68.33 78.29 88.80

61 7 73 8 87 9 94 10

99.8 111.4 123.5 136:1 149.2 162.8 176.9 191.5 206.5 222 238 254 286 324 362 401 442 485 529

24 25 26 28 30 32 34 36 38 40

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