weight produces a greater deflection than the one preceding, and that the departures from uniformity increase nearly as the squares of the weight applied. This is shown clearly by Table 111, where the load is divided into 20 parts, and the Ratio of the deflections of Cast-iron beams is given by col. 2, while those of Timber are given in col. 6. These Ratios were obtained by a Diagram (Fig. 216), in which the experimental deflections were plotted, and the irregularities equalized by a curve. In cols. 3 and 7 the deflections are assumed as supposed to be due with perfect elasticity, and in cols. 4 and 8 the defect of elasticity is given on the hypothesis that it varies as W2. We have thus obtained cols. 5 and 9, comparing which with cols. 2 and 6 they will be found to agree very well up to about half the breaking load, beyond which they become irregular. This, however, is unimportant, as in practice beams of Cast iron are seldom loaded above and Timber ones or th of the breaking weight.

The effect of defect of elasticity is shown by Table 108 also; with perfect elasticity the deflections would have been simply proportional to the load, as in col. 7, but col. 4 shows that they increase more rapidly than the weights throughout. But, with loads not exceeding 3rd of the breaking weight, the departures from uniformity are not great, and within that limit, the ordinary rules are correct enough for practical purposes; where, however, great exactness is necessary, they require correction. For a bar 1 inch square and 1 foot long we have the Rule:

(689.) 8 = (00002397 × W) + (·000000006827 × W2). In which W = the weight in centre in lbs., and 8 = deflection in inches.

Thus, the mean strength of British Cast iron by col. 7 of Table 64 is 2063 lbs. breaking weight, and by col. 8 the mean deflection 0785 inch. With of that weight, or 687 6 lbs., the deflection by col. 11 = .01971 inch.

8

=

By rule (689) we get with 2063 lbs., = (00002397 × 2063) +(000000006827 × 20632) = .0785 inch, and with 687.6 lbs.,

=

(00002397 × 687·6) + (·000000006827 × 687·62)

=

TABLE 111.-Of the RATIOS of DEFLECTION in BEAMS of CAST IRON and TIMBER.

02480247 + ·0004 = ·0251 05020494·0018·0512 07660741 + ·0040 0781 10380988 + ·0071 = ·1059 13201235 + ·0111 = ·1346 16131482 + 0160 = .1642 19181729 + ·0218·1947 2236 1976 + ·0285 = ·2261 25682223 + 0360 = ·2583 2915 2470 +0445 = 2915 32782717 +0538 = 3255 36582964 + ·0641 = ·3605 40533211 +0752 = ·3963 4470 3458+0872 = ·4330 49073705 + ·1001 = ·4706 53653952 + ·1140 = ⚫5092 5845 4199 + ·1286 = ·5485 64504446 + ·1442 = •5888 73604693 + ·1606 = ·6299 4940 +1780 = ⚫6720 (8) (9)

.6

.65

.7

⚫75

4320+ ·2048 = ·6368 7421 4590 + ·2312 = ·6902 81634860 + 2592 ·7452 90005130+ ·2888 = ·8018 1.00 1.00005400+ ·3200 = 8600 1·000 (4)

(5)

(6)

(7)

01971 inch, or precisely the same as the experimental results. Thus, while the loads are in the ratio 1 to 3, the deflections are in the ratio 1 to 078501971 3.983, or nearly 1 to 4. With any other dimensions for rectangular bars we have the Rule:

(690.)

=

In which L = length between supports, in feet; d= depth in inches; b = breadth in inches; and z = the constant for the thickness of metal, as in (934) and col. 7 of Table 18.

By this rule col. 5 of Table 108 has been calculated. Taking the weight W in tons, and the rest as before, the Rule be

(692.) "Wrought Iron."-The elasticity of wrought iron is very much more perfect than that of cast iron, as we found by its behaviour under tensile (620) and compressive strains (626). Table 106 shows the same result under transverse strains, the deflection in cols. 2, 2, 2, 2 being nearly in the simple ratio of the weights up to the "limit of Elasticity," or half the breakingdown load (see Table 67). We may therefore admit that certainly within that limit the deflections are given accurately by

the Rules.

TABLE 112.-Of the DEFLECTION, &c., of Two BEAMS of AMERICAN RED PINE, 12 inches square, 15 feet long.

But with strains beyond that limit defect of elasticity manifests itself very clearly, as shown by col. 4 of Table 70. Thus,

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 × 13 1.0; × 23 4; 3 × 33 = 9, &c., which are in the direct ratio of the square of the lengths.

=

=

=

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

4

9

23

;

=

33

13

, &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

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.) 8L2x C÷d.

=

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 8 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 C 0785

=