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experiment, the effect of other conditions may be most readily found by the use of Constants: we then have the ratios

Beam supported at ends and weight in

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In all these cases, the weight is supposed to be constant. There is considerable uncertainty in the deflection of beams fixed at one end, arising from irregularities in fixing. This is shown by Mr. Fincham's experiments, who found the ratio to vary from 18.6 to 44 5, the mean of 14 experiments being 28, whereas, the theoretical ratio, as we have shown, is 32.

(668.) "Ratio of Round and Square Sections."-Theoretically a round bar deflects more than a square one in the ratio of 1.7 to 1.0, the weight, &c., being the same in both cases, and this ratio should be the same for all materials. It is probable that this ratio is correct for light strains, but when the breaking weight is approached the conditions are changed, and the ratio of stiffness seems to change also: from the inadequate experiments we have the experimental ratio is for wrought iron 1.6 to 1.0, and for cast iron 1.5 to 1.0.

"Cast-iron

Sections."—When the top and bottom flanges of a girder are equal to one another the theoretical Rule for deflection is

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In which D = the total depth, B = breadth of flanges, and d = the depth between top and bottom flanges, b = breadth of flange

minus the thickness of the vertical web, all in inches; L = the length of the beam between supports in feet; W = central load in tons, lbs., &c., dependent on C, the value of which is given by col. 4 of Table 105: col. 6 of Table 64, &c.

"Old Rule."-The Rule commonly used, although not so correct in principle, will give results which agree better with experiment: this rule becomes

(670.) 8 = L3 × W × C ÷ { D3 × B) − (ď3 × b }.

"Unequal-flanged Sections."-In ordinary cases, the flanges of cast-iron girders are unequal, as in Fig. 79, which is the section of large girders experimented upon by Mr. Owen, whose results are given by Table 68. For such sections we have the Rule:(671.)

B

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14, d = 14

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12

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d = L3× W × C÷ {D3× B) − (ď3 × b] + [d3 × b}. In which D = total depth, d = total depth minus that of the bottom flange, do the depth between top and bottom flanges: breadth of bottom flange, b = the breadth of bottom flange minus that of the top one, b breadth of top flange, minus the thickness of the web. Thus in Fig. 79, D 121, do 111, B = 12, b = 12 31 81, b = 31 − 1 = 21: then taking C from col. 6 of Table 64 = ⚫00002886, and W = say 7 tons, or 15,680 lbs., and L 15680 ×·00002886÷{143 × 12) – (1213 × 81] + [1133 × 21 =1349 inch deflection of a parallel beam, but in our case the flanges were bellied, as in Fig. 131, when the deflections are greater in the ratio 1·44 to 1·0 (701), hence ⚫1349 × 1.44

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16 feet, we obtain 8

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1942 inch deflection. The experimental deflections were very variable, as shown by Table 68, ranging in 11 specimens from 14 inch to 42 inch, or in the ratio 1 to 3.

"Cast Iron and T Sections."-We found in (344) that the transverse strength of these sections depends on their position, being greater in than in T in the Ratio 3.08 to 1.0 in that particular case. But the stiffness of such beams is the same in either position, as shown by Mr. Hodgkinson's experiments: thus in Fig. 72, G and H were practically identical, and with a length of 6 feet the results were

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With 14 lbs., the deflections were 032 inch, and ⚫025 inch.

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Showing almost perfect equality up to 112 lbs., which is about 3rd of the breaking weight in position T (364 lbs.), but th only in position (1120 lbs.). The old Rule (670) will give the same deflection in either position: thus with Fig. 72, and W say 112 lbs. we obtain 8 = 61 × 112 × ⚫00002886 ÷ {1·553 × 5) — (1·253 × 4·64} = ·1853 inch deflection: experiment gave 273 inch.

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(672.) "Wrought-iron I Sections."-The Rules we have given for cast-iron L, T, and I sections will apply equally to wroughtiron ones, with the proper value of C, which for lbs. = ⚫00001565 by col. 4 of Table 105. Thus, with Fig. 154, by Rule (669) D1 = 10000, d' or 83* = 5862, b = 42 - 1 = 41, &c., L3 or 183 = 5832, and with W = say 304 cwt. or 33,880 lbs. we obtain ♪ = 5832 × 33880×·00001565÷ J10000 x 43)-(5862 × 411

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1369 inch deflection: experiment gave 16 inch. Table 73 gives in col. 8 the experimental deflection of a series of rolled beams of ordinary equal-flanged sections: col. 9 has been calculated by Rule (669) and shows an error of 14 per cent. The old Rule (670), although not so correct in principle, will give results which agree better with experiment; col. 10 has been calculated by that rule; the mean error of the whole is +5 per cent. only.

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Unequal Sections."-When the top and bottom flanges are unequal, as in Fig. 80, the most correct method of calculation will be to estimate from the bottom or the line N. A., as we found to be necessary in calculating the strength in (378): we then have the rule ::

(673.)

♪ = L × W ×·00001565 ÷ (D3 d'] x B)

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TABLE 109.-Of the DEFLECTION of ROLLED IRON I BEAMS.

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The values of D, d, d,, B, b, and C, are given by Fig. 80, and the rest as in (664). Figs. 89, 90 are sections of beams experimented upon by Mr. Fairbairn, the deflections being given in cols. 2, 5 of Table 109. Thus in Fig. 89, with 885 lbs., the Rule gives

♪ = 11a × 885 × ·00001565 ÷ { 73 — 63] × 21)

+(63 - ·383] × ·325)+(38 x 4} = .04751 inch

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deflection experiment gave 04 inch, &c. Cols. 3, 6 of Table 109 have been calculated by this rule, and show a fair agreement with experiment: the calculated deflections with Fig. 89 show an error of +11.9 per cent., and of Fig. 90, +2.3 per cent.: the mean of the whole being + 6.4 per cent.

It is remarkable that this rule applied to sections with equal flanges does not give satisfactory results: col. 11 of Table 73 has been calculated by it and shows a mean error of +42 per cent.; while Rule (669) gave - 14, and Rule (670), + 5 per cent. In order to render Table 73 directly available for practical purposes, we have given in col. 13 the experimental

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deflection for each section by 1 cwt. in the centre of a beam 1 foot between supports, and as the deflections are simply proportional to the cube of the length and the weight, we have the Rule:

(674.)

D

8 = TD x L3 x W.

the Tabular number in col. 13; L = length in = deflection in inches.

In which To feet; W = weight in cwts.; and 8 Thus with No. 6, say L = 20 feet; W = 35 cwt. then we obtain 00000239 × 8000 × 35 = 0·6692 inch deflection: experiment gave inch, col. 8.

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(675.) "Wrought-iron T Sections."-This form of section in wrought iron should always be loaded with the top flange uppermost in the ordinary case of a beam supported at both ends, for reasons given in (377), we must then calculate the deflections by measuring the depths from the bottom or from the line N. A. in Fig. 132, and we have the Rule:—

(676.)

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ď] × B) + (ď3 × b}.

d = L3 × W ×·00001565 ÷ { D3 In which the values of D, d, B, b, are given by Fig. 132 and the rest as in (664). Thus, Fig. 87 is the section of a bar, which with a length of 10 feet, deflected inch with 2 cwt. in the centre: then the Rule gives 8 = 103 × 224 × ⚫00001565÷

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(21 - 213] × 21) + (21 x ) = 0.2359 inch deflection: experiment gave 0.25 inch, &c. Table 71 gives in col. 11 the experimental deflections of a series of T iron bars: col. 12 has been calculated by the Rule.

When the depth is equal to the breadth and the thickness is the same all over, the rule becomes

(677.) 8 = L3 × W × 00001565 ÷ (D1 — d1).

In which the section A is regarded as composed of two bars B, C, as in Fig. 70, which is not strictly correct, as explained in (337), a more correct rule would be:

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