practicable dimensions, the ratio to the large tube being as 1.0 to 5.34, and this was doubtless the most prudent course to adopt. But as a matter of fact, and a wonderful illustration of the correctness of the law R', we may show that the strength of the great tube at Conway, 400 feet span and weighing 1080 tons, may be calculated with approximate accuracy from the experimental strength of a little model 3 feet span, weighing only 4 lbs. In this case the ratio of the dimensions is 400 3.75 = 106.7 to 1.0: the log. of 106.7 or 2.028164 × 1.9 = 3.8535, the natural number due to which, or 7137 !! is the ratio of the breaking weights. By col. 2 in Table 78 the breaking load of the little model No. 8 was 0.3009 ton, hence the strength of the great tube would be 0·3009 × 7137 = 2147 tons, differing very little from 2145 tons as found in (473) from the strength of the 75-foot model. Then following out the ratio we may obtain the general dimensions of the great tube from those of the little one; thus, the depth of the little tube being 3 inches or 25 feet, that of the great one comes out 25 x 106.7 = 26 7 feet the actual depth was 25.5 feet. The breadth would be 106.7 × 1·9 ÷ 12 = 16.8 feet, the actual breadth = 14.7 feet. The thickness of plate comes out ⚫03 × 106.7 3.2 inches, and the width of the top plate being 16.8 feet or 202 inches, the area at the top would be 202 × 3·2 = 646 square inches: the actual area was 565 square inches, &c., which for practical reasons was arranged as cells of thin plateiron inch thick.

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If, in calculating the strength of the Conway tube from the 75-foot model, we had taken the theoretical ratio R2, we should obtain 89.15 × 400° 75° 2536 tons in the centre breaking weight, or 2536 540 1996 tons net, exclusive of the weight of the tube itself.

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(476.) It is an inconvenience of this 6th law (461) that all the dimensions should be in an exact given ratio, a condition that is sometimes difficult to fulfil in practice. The third law (455) is more elastic, for though the value of M is adapted strictly only to beams that are mathematically "similar," still small departures from that primary condition are unimportant, and variations in the dimensions are met by the rule.

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Applying this 3rd law to the 75-foot model tube and the Conway Bridge as before, we have the areas of the whole crosssection 55 47, and 1530 square inches respectively. Then, the model tube being 4.5 feet deep, we get the value of M by the rule M = (W × L) ÷ (a x D), which, taking L and D both in feet, becomes in our case (89.15 x 75) ÷ (55.47 × 4.5) = 26.8 the value of M. In the Conway tube L 400, D = 25.5, and a = 1530 square inches; hence the rule W = a x D x M÷L, becomes 1530 × 25.5 × 26·8 ÷ 400 = 2614 tons gross, or 2614 540 = 2074 tons net central breaking weight. This differs but little from 1996 tons as found by the 6th law in (475), where the theoretical ratio R2 is taken as a basis, but is much in excess of 1605 tons as found in (473) with the corrected experimental ratio R1, which is unquestionably the most correct.

(477.) The 1st law in (453), that the breaking weights of similar beams are simply proportional to their respective crosssectional areas, may be illustrated from the same examples. Thus, the areas being 55 47, and 1530 square inches, and the gross breaking weight of the small tube being 89.15 tons, that of the large one will be 89.15 × 1530 ÷ 55.47 = 2459 tons gross, or 2459

540 = 1919 tons net.

(478.) The 9th law (464) gives us valuable practical rules for cast-iron girders more particularly:-thus for the proportions recommended by Mr. Hodgkinson, where the flanges have areas in the ratio of 6 to 1, he gives, from his own experiments, the value of M at 26 for the breaking weight in tons. Fig. 79 gives the section of large girders experimented upon by Mr. Owen (see Table 68): the mean breaking weight by thirteen experiments was 38.3 tons with a length of 16 feet between bearings. The area of the flanges was 6 to 1, and that of the bottom one 1.75 × 12 = 21 square inches; then the rule W = ax dx M÷ 1, becomes 21 × 14 × 26 ÷ 192 = 39.81 tons. The multiplier 26 is strictly applicable only to girders with flanges in the proportions of 6 to 1. (See 351.)

(479.) This law may also be applied to plate-iron tubular beams; but some caution is necessary here, because the rule, taking the bottom flange alone as the index of the strength, pre

supposes that the other parts of the beam, such as the top plate, &c., are not weaker than the bottom flange or plate, &c. In cast-iron girders this condition is generally realised, because the tensile strength of cast iron is only th of its compressive strength, and for that reason the bottom flange is usually the weak part. But with wrought iron, although the tensile and crushing strains are nearly equal (377), the tendency of thin plates to wrinkle or corrugate with less than the crushing strain, causes the top flange or plate to be frequently the weak member, and therefore to govern the case.

(480.) Taking again the 75-foot model tube: after the top plate had been duly strengthened until the beam failed by the bottom plate giving way under the tensile strain, we had an area of 22 45 square inches at the bottom, with a depth of 4.5 feet, and a length of 75 feet. In the Conway tube the bottom area was 500 square inches, the depth 25·5 feet, and the length 400 feet hence the rule M (W x L) (a x D), taking L and D in feet, becomes (89.15 × 75) ÷ (22·45 × 4·5) = 66. Then for the Conway tube, the rule W = a × D × M ÷ L, becomes 500 x 25.5 × 66 ÷ 400 2104 tons gross, or 2104 540 1564 tons net.

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(481.) We have thus obtained by different rules various values for the strength of the Conway tube; the variations are not very great, and any of the results are sufficiently correct for the requirements of practice, a small error being in any case amply covered by the Factor of Safety, which for a Railway Bridge would not be less than 6. Collecting these results :By Rule 6, with R1 we found in (473) 2145 tons, gross.

(482.) The difference between R and R in the 6th law is of course most considerable when R, or the difference in the sizes of the model and the beam, is very great, and in such cases

more particularly the 6th Rule (461) will no doubt with R19 give more correct results than any of the others. For instance, if we calculate the strength of the Conway tube from that of the little model as in (475) by the 6th law as before, but with R2 instead of R19, we should have a difference or error of 59 per cent. The ratio R is in this case 106.7 to 1, hence we obtain 0.3092 × 106·7* = 3520 tons gross, instead of 2207 tons as in (475); a difference of 3520 ÷ 2207 1.59, or 59 per cent.; this being due to the difference between 106.72 11385, and 106.719 = 7137.

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(483.) The 7th law in (462) is a very useful one, enabling us to reason direct from a girder of any form whose strength is known by experiment, to another similar girder for any span and load whose sizes we require. For instance, Fig. 156 is the section of a cast-iron girder 4 feet between bearings, whose breaking weight was found by Mr. Hodgkinson to be 6.456 tons in the centre. Now, say that we require to find from this the sizes for a girder of similar section, to bear safely 20 tons in the centre, with a length of 25 feet. With Factor 3, we have 20 × 3 = 60 tons breaking weight:-then by Rule 7, putting w and for the breaking weight and length of the experimental girder, and W and L for the breaking weight and length of the girder required, we have:-

(484.)

MG = W x L ÷ / w × 1,

3!

which in our case becomes 60 × 25 ÷ 6·456 × 4·5 =3·71, the value of Mc, a multiplier, which, applied to all the crosssectional dimensions of Fig. 156, will give the corresponding dimensions of Fig. 157. Thus, for the depth we have 5.125 x 3.71 = 19 inches; for the breadth of the bottom flange 4.16 x 3.71 15.43, or say 15 inches, &c., &c., and we thus obtain all the sizes in Fig. 157 from direct experiment on a Similar girder.

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The most useful application of this method is in cases where for particular reasons the section required is of unusual form, such as to be not easily calculated by the ordinary rules. In such a case the most satisfactory course is to make a model girder, as large as conveniently possible, which, being experi

mentally broken, will supply data for calculating the sizes of the girder required (901).

(485.) "Unit Girders."-The 7th law in (462) will also enable us to find, from direct experiment or otherwise, the sizes of "Unit" girders 1 foot long with a breaking weight of 1 ton, from which we may easily obtain the sizes necessary for similar girders with other loads and lengths by the Rule:

In which W = the load on the centre of the girder in lbs., tons, &c., dependent on M; L the length in feet, and Mr = = a constant. Thus, taking again the girder, Fig. 156, the rule in our case becomes M1 = √6·456 × 4·5 = 3.074:-now dividing all the cross-sectional dimensions of Fig. 156 by 3.074, we obtain the corresponding sizes of the "UNIT" girder, Fig. 160: thus for the depth we obtain 5·125 ÷ 3·074 = 1.667: for the bottom flange 4.163.074 1.353 inch, &c., &c.

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(487.) Then, to apply this to other cases, say for a girder as in (483), 60 tons breaking weight and 25 feet long, the rule W x L = Mu becomes 60 × 25 = 11.4, and by multiplying all the cross-sectional sizes of the "Unit” girder, Fig. 160, by 11.4 we obtain the corresponding sizes for the girder required. Thus, the depth will be 1.667 x 11·4 19 inches; the width of bottom flange, 1·353 × 11.4 15.4, say 15 inches, &c., &c., as in Fig. 157.

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We have thus obtained the series of "Unit" girders, Figs. 158 to 163, any one of which may be used in the way we have illustrated all of these Unit sections have been calculated from girders experimented upon by Mr. Hodgkinson. In making a selection we should be guided by the special requirements of practice for instance, where great stiffness is required, say for carrying a water-tank, where considerable deflection would strain and endanger the joints, we should select a deep one, such as Fig. 159; in other cases great depth might be inadmissible, and say Fig. 162 would be selected; in other cases we may require a wide top flange for a Bressummer to carry a wall, and Fig. 158 would be the most suitable, &c. See (618) and Fig. 208.