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taking MT 2000 lbs. from col. 5 of Table 66, Rule (324) gives W = 112 × 11 × 2000 ÷ (3 × 112) = 20 cwt. as the “limit of Elasticity," up to which point, by col. 4 of Table 70, the deflections are nearly as calculated for perfect Elasticity; but with heavier strains the Ratio progressively rises, becoming finally as much as 13 13 with the breaking-down load.

"Steel."-The transverse elasticity of Steel is more perfect than even that of wrought iron, as shown by Table 107, the deflections being simply proportional to the load, and therefore the Modulus of Elasticity constant up to the "limit of Elasticity," or ths of the Breaking-down load.

"Timber."-Timber beams have very imperfect elasticities, as shown by cols. 4, 8, in Table 112, the value of the Modulus of Elasticity falling off regularly as the load is increased. The constants for the deflection of Timber in col. 4 of Table 105 were for the most part obtained with 3rd to th of the breaking weights, and the Rules (658), &c., will be correct enough for practice within those limits.

DEFLECTION WITH SAFE LOAD.

(693.) It is usual in practice to make the Working or Safe load on beams a certain Standard fraction of the breaking weight by the use of a "Factor of Safety" (880): in that case, the ordinary Rules (658), &c., admit of certain modifications by which calculations of the deflection may be simplified very considerably.

By Rule (659) it is shown that the deflection of a rectangular W x L3 beam of any material is proportional to : now if with d3 x b

the same beam we take lengths in the ratio 1, 2, 3, &c., obviously the transverse strengths, or the load W, would vary in inverse ratio to those lengths, becoming 1,,, &c., and with those weights the deflections (being = to W x L3) become 1 x 13 = 1·0; × 23 4; 1 × 33 = 9, &c., which are in the direct ratio of the square of the lengths.

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Then, with depths d in the ratio 1, 2, 3, &c., W would be in the ratio of the square of d, and become 12, 22, 32, or 1, 4, 9 ;

1

the deflections being proportional to become =

W d

1.0;

13

4

9

= 11;

=

23

33

, &c.; or inversely as the depths simply.

Then for breadths in the ratio 1, 2, 3, &c., W would vary in the ratio 1, 2, 3 also, and the deflections being proportional to

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cases, showing that the deflection with safe load is independent of the breadth.

(694.) From all this we get the general law that with similar beams, all loaded in proportion to their strength, the deflections are proportional to the square of the length, divided by the depth, and are independent of the breadth. This is true for all sections, whether circular, square, girder-sections, &c., so long as the beams compared are similar and are loaded to the same extent in proportion to their strength. We then have the Rules:(695.)

8 = L2 x C÷d.

d = L2 x C÷8.

L = √ox d÷C.

In which L = length of beam between supports in feet: d= the depth in inches: = deflection in inches: and C = a constant for the material, mode of fixing, loading, &c.

These Rules may be applied correctly to beams or girders of all sections, whether parallel from end to end, or with bellied flanges, also with any mode of fixing, any method of distributing the load, and any material, so long only as the Constant C is adapted to all the circumstances of the case. The effect of defect of Elasticity (688) is also covered by these rules, because the value of C is supposed to have been found by experiment from beams loaded to a certain degree in proportion to their ultimate strength, and will therefore apply without correction to all beams similarly loaded, &c., &c.

(696.) The most useful values of C are Cs for the Safe Load, and C, for the Breaking Weight, giving &, and d or the deflections with Safe and Breaking loads respectively. Table 64 gives these values for 54 varieties of British Cast iron, the mean for C, being 01971, or say 02, by col. 11; and for C1 = ⚫0785

by col. 8. Table 67 gives, in cols. 2 and 4, the values of Cs and CB for many materials, these being in fact the deflections of a bar 1 inch square and 1 foot long with the Safe and Breaking Loads respectively: in that particular case, C, and C are identical with 8 and B.

It will be observed that the values of Cs and C, for the two Standard cases, are not simply proportional to the load, or "Factor of Safety." In Table 67, col. 7 gives the Factor of Safety, and col. 8 the ratios of the deflections with Safe and Breaking loads respectively: thus with Cast iron the ratio of the loads is 3 to 1 by col. 7, but the ratio of the corresponding deflections is 4 to 1 by col. 8: again with Ash, the ratio of the Loads = 5 to 1, but the ratio of the deflections 10.7 to 1·0. Thus, a beam of Ash, say 15 feet long, 7 inches deep, 3 inches wide, would give for the breaking weight by Rule (324), the Value of MT for safe load being 136 lbs. by col. 3 of Table 67, W = 72 × 3 × 136 ÷ 15 1333 lbs., the deflection with which 153 x 1333 × ⚫00026 by rule (659) is 8 =

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73 × 3

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1.13 inch de

Now applying Rule (695) and taking 0354, we obtain dg

flection with safe load.
Cg from col. 4 of Table 67 at
71.13 inch deflection,
weight 8B

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152 x .0354

as before: but with the breaking

152 × ·375 ÷ 7 = 12 inches deflection, &c.

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Again, with a bar of Cast iron 3 inches deep, 4 inches wide, and 11 feet long, taking M, for safe load = 688 lbs. from col. 10 of Table 64, Rule (324) gives W = 32 × 4 × 688 11 2251 lbs. Safe load, with which Rule (659) gives 113 × 2251 × 00002886

8

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=8 inch deflection.

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By Rule (695) we obtain much more easily ds 112 x 023 8 inch also. We have here taken the deflection per lb. = .00002886 from col. 6 of Table 64, or col. 4 of Table 105. The deflection of the same bar with the Breaking weight becomes B 112 x 0785÷3 3.17 inches.

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But these Rules need not be restricted to ds and 8B, but will apply equally well to any standard fraction of the breaking weight, so long as the great principle is maintained, that the beams compared shall be loaded always in some given and constant proportion to their strength. Thus for Wrought iron

and Steel, we found (374) (376) it convenient to take the "limit of Elasticity" and the "Working Safe Load" as data: then putting dg and ds for the deflections with those strains, we have for Wrought iron, the Rules:

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Thus, with a wrought-iron bar 2 inches deep, 4 inches wide, and 12 feet long, taking from col. 5 of Table 66, Mr = 1500 lbs. for the safe working load, we have by Rule (324) W = 22 x 4 x 1500 ÷ 12 = 2000 lbs., with which the deflection by Rule (659) 123 × 2000 × ⚫00001565

becomes &=

(697) we have d

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= 1.69 inch. By Rule

= 122 ×·0235 ÷ 2 = 1·69 inch also.

the bar will be strained to the

Again: with a Steel bar 1 inch deep, 5 inches wide, and 10 feet long, the load by which "limit of elasticity " will be W =

112 × 5 × 5600 ÷ 10 =

6300 lbs., with which the deflection by Rule (659) becomes 109 × 6300 × ⚫00001433

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we obtain dg

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5.35 inches. By Rule (698)

102 x 08021·55·35 inches also. (699.) When the load is not in the centre, and when the beam is not supported at both ends as is assumed in the ordinary rules, the case is complicated, but by combining the data given in (431) and (667) the matter may be simplified.

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Thus, for example, if the beam of ash in (696) had been built into walls at both ends, the safe central load would have been 1.50, or 50 per cent. greater, but the deflection with that increased load would have been 1·0, or precisely as before.

Again: if the bar of cast iron in (696) had been fixed at one end, and the safe load had been equally distributed, that load would be 0.5, or half only of its former amount, but the deflection with that reduced load would be 6 times greater than before, and becomes 8 x 6 = 4.8 inches, &c.

(700.) Again: say we require an Oak Bressummer to carry the front of a house, the estimated distributed load being 19 tons and the span 12 feet, the ends being built into the walls in the usual way. Say we try 12 inches square, then by col. 3 of Table 67, MT = 102 lbs. for Safe load: then Rule (324) becomes W = 12 x 12 × 102 ÷ 12 = 14688 lbs., or 6.56 tons for the ordinary case of a beam merely supported at ends and loaded in the centre; but by (431) in our case, 6·56 × 3 19.68 tons safe distributed load, or very nearly the actual load.

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Then, for the deflection, Rule (695) gives 8 = 122 ×·04÷ 12 0.48 inches for the deflection with safe load in centre, which by (699) becomes 48 x 1·25 = 0·6 inch deflection, when the load is distributed and the ends built in, as in our case. We have taken the value of Cs 04 from col. 4 of Table 67.

(701.) Although, as stated in (695), the value of C、 should strictly be adapted to the special section of the beam, we may with moderate accuracy apply the value for simple rectangular beams to ordinary girders, at least where the beam is parallel. But when the flanges are bellied, as in Fig. 131, the deflections are about 40 per cent. greater than with parallel beams, as shown by Mr. Hodgkinson's experiments. For ordinary parallel girders of cast iron loaded to of the Breaking Weight, we have the Rule:

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For cast-iron girders reduced in section progressively towards the ends, or having bellied flanges, as in Fig. 131, the Rule becomes :

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