In which W = weight or load in lbs., tons, &c., dependent on

the value of C.

breadth, in inches.

length between bearings, in Feet.

deflection, in inches.

Constant derived from experiment, in lbs., tons, &c.

(665.) The mean value of C for most ordinary Materials is given by col. 4, &c., of Table 105. Table 64 gives in col. 6 the mean value for 54 different kinds of Cast iron = .00002886. Table 106 gives the result of experiments on the Deflection of wrought-iron bars by Mr. Hodgkinson; cols. 3, 3, 3, 3 having been calculated by Rule (659), the value of C being taken at 00001565 from col. 4 of Table 105: thus for the bar in which L = 13.5 feet, d = 1.515; b = 5.523, and say 13.5 x 112 × ⚫00001565 1.5159 x 5.523

W = 112 lbs., we obtain 8 = = 0.2246 inch deflection, as in col. 3 of Table 106. Col. 3 of Table 70 has also been calculated by this rule: col. 4 shows very clearly the effect of defect of Elasticity, the Ratio rising from 1.0 with light loads to 13 13 with Breaking-down load.

Table 107 gives the result of Mr. Fairbairn's experiments on the deflection of Steel; cols. 3, 3, 3, 3 have been calculated by Rule (659), the value of C being taken at 00001433 from col. 4

=

=

1000 lbs., we obtain 8

of Table 105; thus for the bar in which L = 4.5 feet; d 1.054; b 1·054, and say W 4.5 x 1000 × ⚫00001433 1.054 x 1.054

of Table 107.

=

=

= 1.058 inch deflection, as in col. 3

(666.) Table 108 gives the result of special experiments on Blaenavon Cast iron; it will be found that with small loads, say up to 3rd of the breaking weight, the deflections will be given by Rule (659) with moderate accuracy, as shown by col. 7, but as the load increases the experimental deflections exceed the calculated ones more and more. This fact is due to defect of elasticity, leading to the necessity for special Rules for Cast iron under heavy strains: this matter is fully considered in (688). Col. 7 has been calculated by Rule (659), taking the value of C for Blaenavon iron, at 00003133 from col. 6 of Table 64. Thus for the bar in Table 108, in which L d = 1.522; b = 3.066; and say W = 112 lbs., we obtain 8 = 13.53 x 112 × 00003133

1.522 x 3.066

in col. 7 of Table 108.

=

13.5;

= 0.7987, or say 0.8 inch, &c., as

Table 112 gives in cols. 2, 6 the deflections of two large beams of American Pine from the experiments of Mr. Edwin Clark; cols. 3 and 7 give the calculated deflections by Rule (659). Taking the value of C by Tredgold's experiments in col. 4 of Table 105, at 0002661 inch, we obtain for say 3653 lbs. 153 x 3653 × ⚫0002661 6 = = 0.1582 inch deflection, as in

123 × 12

col. 3: experiment gave 0.15 in col. 2. It will be observed that up to the safe load, say th of the breaking weight (888), the deflections as calculated agree fairly with experiment, but as the load is increased, the actual deflections are more and more in excess of those given by the rule, this being due to defect of Elasticity (692).

(667.) "Effect of Modes of Fixing and Loading."-When the deflection for the Standard case of a beam, having the load in centre and supported at the ends, is known by calculation 'or

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

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

b

(669.) 8 = L3× W × C÷ Wx

D

D' × B) = (d' × ₺ }.

{D'

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) − (ď × b}·

66

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.)

=

8 = 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: B 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 = 14, d · 14 - 13 121, do 111, B = 12, b = 12 31 81, b = 311 = 21: then taking C from col. 6 of Table 64

=

=

=

=

00002886, and W =

=

say 7 tons, or 15,680 lbs., and L = 16 feet, we obtain & 163 X 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 = 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