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maximum deflection and the rest as before: see Fig. 175. Thus in the case of the beam in (654) we obtain m = (5 + 2) 5 x 2 x 2 411

= 1.5 foot from the weight, therefore 3

3 2 1.5 = 0·5 foot, or 6 inches from the centre.

As the distance of the weight from the centre increases, so does m increase, and it becomes a maximum when the weight is at the support, when z = (4L), and it then becomes m = (5 + 5) 5 x 5 x 2 25

= 4.227 feet from the weight (and 3

3 the support), or 5 4.227 = 0.773 foot from the centre, which is equal to 773 - 5 = • 1546 of the half-length of the beam. From this it appears that wherever a weight is placed, the point of maximum deflection can never be more than 0.1546 of the half-length distant from the centre.

(657.) Comparing the curves A and B in the Diagram, Fig. 213, we observe this remarkable fact; that the deflection at any and every point of a beam with a central load is greater than would be produced by that same weight at any other point: for example, the curve B = the deflection at every point

: throughout the length by a given central weight, and A = the deflection at the same point by the self-same weight placed there. This of course is due to the fact that the beam is less strained by a weight out of the centre than by the same weight in the centre. When strained at every point to the same extent, or in proportion to the strength, the deflections are given by the line C.

(658.) “ Laws of Deflection.”—The Deflection of beams varies very much with the methods of fixing and loading (667): to simplify the matter we may take as a Standard case, that of a horizontal beam, supported at each end and loaded in the centre: other conditions may be considered afterwards. We then have the Rules

L' X W XC (659.)

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TABLE 105.-Of the DEFLECTION of Beams 1 foot long between Supports, 1 inch square, with 1 lb. in the

Centre, being the Mean per lb. up to about One-third the Breaking Weight.

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00003744

00001866

221

00001514

00001336

4 1

Hodgkinson

Fairbairn
Hodgkinson

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99

• 0000145

0000258

1 7 1 3 2 5

.0002695 .0002832

.000257 000227

Box
Ebbels
Barlow
Tredgold

Mean
Barlow
Ebbels

Beech

.00002886

00001565
.00001433
·00001386
00003893
·00004123
.000036+
·0003975
.0002625
·0002551
*0002600
.0003200

0003282
.0003536
.0008889
•0003460
.0004051
•0003800
•0004670
.0006288
.0003478
.0003804
.0002602
0002223

.0003343

000311

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14,968,800
27,603,900
30, 146,600
31,169,000
11,096,800
10,477,900
11,868,100
1,086,800
1,645,720
1,693,450
1,661,390
1,350,000
1,316,270
1,221,720

488,237
1,248,550
1,066,410
1,136,850

925,053

687,022
1,242,090
1,135,650
1,660,270
1,943, 330

.00014940
.00008106
.00007423
.00007179
.00020166
·0002133

0001885
.002059
.001360
.001321
·001347
.001658
.001700
.001832
.004604
.001792
002098
• 001968
.002419
.003257
.001802
.001970
. 001348
. 001152

Birch ..
Cedar of Lebanon
Cherry-tree
Chestnut, Horse

Spanish

1

Tredgold

Ebbels
Tredgold
Ebbels
Barlow
Ebbels
Barlow
Tredgold

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..

Scotch
Christiania, Spruce..
Quebec
New England

Norway
Larch

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0003017

.0002681

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·0007463 •0007078

·0003704
.0003168

Barlow
Tredgold

Barlow
Tredgold

Mean
Tredgold

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16
1
1

Lime-tree
Mahogany, Honduras

Spanish ..
Maple, common
Oak, English

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0005287
0006068

·0002939
0002520

Barlow
Tredgold
Ebbels

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•0003596 ·0002392 •0003172 .0002895 .0002395 · 0001944 ·0001562 .0004873 .0003810 .0002712 .0003411 ·0001923 .0003966 .000 1050 ·0003330 •0003942 .0002239 • 0001447 •0003708 .0005378 ·0003527 .0002318 • 0002661 ·0003216 •0005663 0001211 0001790 .0005161 •0007627

(4)

6
5
1
12
3
3

1,201,340
1,806,000
1, 244, 240
1,492,230
1,803, 760

873,800
946,950

886,500
1,133,850
1,592,920
1,255,450

877,514
1,089, 260
1,066,670
1,297,300
1,095,900
1,929,440

971,440
1,165,050

803,273
1,224,840
1,838,600
1,623,450
1,313,280

762,850
1,025,880
2,413,410

837,050
566,409
(7)

.001863
·001239
• 001798
.001500
.001241
.002561
·002363
·002524

001973
·001405
•001782
·002550
.002070

002098
. 001725
·002042
.001160
. 002304
·001921
·002786
•001827
.001216
·001378
.001660
·002933
002182
.000927
·002673
.003951

(8)

Mean

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Barlow

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.0002384
·0005319
.0001327

.0002092
•0003327
.0002937

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1

Ebbels
Barlow

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Canadian
Adriatic

Dantzio
Pear-tree
Pine, Pitch

Red..

American
Plane-tree
Poplar, Lombardy
Sycamore
Teak
Walnut
Willow

(1)

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Tredgold
Ebbels

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(2)

(3)

(5) TABLE 106.-Of the DEFLECTION of

Lengths between supports, 13.5 ft
Depth

1.027 inch. Breadth ..

... 5.510 inches.

Length between supports, 13.5 ft.
Depth
1.515 inch,

Per-
Weight in Breadth ::

5.523 inches.

manent Centre,

Set.
Deflection. Modulus of

Elasticity in
By Ex- Calcu- Lbs. per Square
periment. lated. Inch.

Deflection. Modulus of

Elasticity in By Ex Cal- Lbs. per Square periment. culated. Inch.

lbs. 28

56

..

112 168 224 280 336 392 448 504 560 616

672 728 781

inches.
• 051 0561 31,092,700
• 112 •1123 27,672,000

-3585 •3613 27,817,800
• 232
• 2246 26,717,000

• 7200 •7226 27,701,900 • 344 •3368 27,028,500 .001 |1.0700 1.084 27,960, 700 • 458 •4491 27,067,900 .002 1.3940 1:445 28,616,000 •571 •5614 27,139,000.003 1.7410 1.804 27,477,800 .684 .6737 27,186,600 .003 2.1590 2:168 27,778,500 .800 •7860 27,118,600 .004 2.5065 2.529 27,851,000

.916 .8982 27,067,900 006 2.8015 2.890 28,478,100 1.005 1.010 27,754,700 .007 3.1635 3.252 28,371,800 1.124 1.223 27,573,600| +008 3.6095 3.613 27,629,000 1.222 1.235 27,895,000 .010 3.9595 3.974 27,705,300 1.332 1.374 27,921,400 .011 4.3370 4.336 27,593,300 1.434 1.459 28,096,500 .017 1.547 1.572 28,047,700 .019

..

..
1.693 1.684 27,459, 600 .019
1.824 1.797 27,264,200 .019
1.933 1.909 27,256,900 .020
2.044 2.021 27,293,000 .021
2.165 2.133 27,199, 100 .022
2.256 2.246 27,470,000
2.461 2:470 27,705,800
2.546 2.582 27,998,100
2.659 2:695 27,978,100
3:057 3:144 28,387,100 .024
3.567 3.592 27,803,300
4:160 4.042 26,820,600 .268

840 896 952

1008 1064 1120

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1232 1288 1344

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1568 1792 2016

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: : :

“Limit of Elasticity" = 788 lbs. (2) (3)

.9040

.9032

27,579,300

1.084

1.093
1.299
1.536

1.264
1.445

27,384,900
26,870,000
25,970,500

“Limit of Elasticity” = 1722 lbs.
(3)

(4)

(2)

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