STR

474

STR

produce fracture ; while the same, or a similar rod, fixed ina wall, and acted upon at the distance of a foot, is brokenwitli a weight of 932 pounds.’’

He farther remarks, that “ a beam of fir, 6 feet long and2 inches square, supported at each end, broke with 7441b.;and the mean of several experiments of similar pieces of thesame dimension, fixed at each end, required 1,1051b.; whilethe fragments of the same, 3 feet long, broke with one end ina wall, required at a medium 4001b.”

Upon the experiments of Buffon and Belidor , this writerobserves, that “ Another discrepancy between theory andexperiment is, where the strength ought to be inversely asthe length ; it shows itself in the above experiments, butis very remarkable in those of M. De Buffon ; and thoughour preceding remarks will explain very satisfactorily thisdeviation, we are almost afraid to offer it as an illustration,after seeing it treated as an inexplicable paradox by somewriters of the .first eminence. We have seen that if w bethe computed weight, independent of the deflection, the abso-lute weight will be w cos. d, d being the angle of deflection;and as this deflection, both from theory and practice, is foundto increase as the square of the length ; it follows, that whenthe length is quadrupled, the depth of deflection will be six-teen times greater ; that is, the sine of the angle of deflectionwill be sixteen times more in one case than in the other,while the radius will be only four times longer ; and there-fore the angle in the one case about four times what it is inthe other, (supposing, in a rough way, the angle to vary asthe sine). Consequently, if w X cos. d is the weight whichbreaks the shorter beam, JwX cos. 4 D ought to be thatwhich breaks the longer one, and this, we presume, willnearly, if not entirely, account for the decrease of the strengthin Buffon’s experiments. We cannot perceive but that thisreasoning is perfectly legitimate, yet we are astonished thatit should not have occurred to so keen a mathematician asthe one to whom we have alluded, or to some one of thewriters on this subject ; and on this account we offer it withsome hesitation.”

This excellent writer concludes thus: “ Our limits willnot admit of reporting here the nature of the experiments,nor the calculations founded upon them, which led to thisdetermination ; but we hope soon to see them laid before thepublic in another form. We can only give here the result,which, as far as it is at present ascertained, is as follows :

“ The centre of tension and centre of compression arenearly, or exactly, coincident with the centre of gravity; andthe neutral line, whatever may be the figure of the section,is so posited, that the rectangle of the area of tension intothe distance of its centre of gravity from the said line, is tothe rectangle of the area of compression into the distance ofits centre of gravity, as 1 to 3.

“ From which theorem, the neutral line for any formedbeam may be determined, and the absolute strength maythen be found as follows, viz. Let d denote the distance ofthe centre of tension, and l the length of the beam, all ininches ; D the angle of deflection, and f the strength of directcohesion on a square inch; then, without considering theincreased length of lever,

“ 1. When a beam is fixed at one end,2 fad

w =

l COS. D '

“ 2. When the beam is supported at both ends,8 f a d

w

l

X COS. D.

“ 3. When the beam is fixed at both ends,12/a d

W =3

l

X COS. D.

“ And when the beam is fixed atone end at any

ang

le,

for*

the

mula 1 will still apply ; only increasing or decrease '.'"leangle of deflection by the quantity of the or

inclination, according as that inclination is dovvnwupwards. , e „ds,

“ And when the beam is supported, or fixed, at bo ^[queand either resting obliquely, or acted upon by anforce, the two latter formulae become,

“4. For the beam supported at each end,

8 fad

T~

w =

COS. D

X-.

COS. I

1 5 For the beam fixed at eaoli end,

W :

12 fa d cos. D

J x -

fores

l " COS. I

where x denotes the angle, which the direction of themakes with the direction of the beam. . e pre*

“ Note 1.—It should have been observed, that .^.^nyceding theorem for determining the neutral line, ,s P 1 at .j 0 tl>* ndrawn from experiments on fir beams. A different < gell t1 :3 may be necessary in other kinds of wood, but a ithat ratio has not been found. bsei' ve< ^’

“Note 2.—The deflection, d, as we have beforeis not a necessary datum in estimating the strength o j ntr o-for any practical purposes of building, &c., it is ITresult ofduced in order to reconcile the theory with tl tren gtbiexperiments made upon the absolute and ultimate ^ gnin which cases, particularly in long beams, it a \\ c& seSimportant quality, and must not be omitted ; an 1 pri 01 ’where it is required, it must be drawn from so ^experiment on the same kind of wood, by mefollowing theorem, viz. , p eC tion, 0

“Let l, d, d represent the length, depth, and e . r peanbany beam; l', d', the length and depth of any 0

td_o

whose deflection, d', is required ; then D = pf

“ Example 1.— The iinch of fir being 13,000/5.,

a

uar e

strength of direct cohesion esS atfJ ‘.

O lb., required the weigh * deep} <>n

break a rectanqular bar, 30 inches long , 2 W c ,, a jio- - - - ’ - a tea' 1 '}

1 inch in breadth, when fixed at one end in a ifrolft °^‘weight acting at the other; the deflection, compu e

othe f

experiments, having been found to be 5 inches. geC tion“ First, find the neutral line : here, since t1 ^ eaC li o

rectangle, the centre of tension and compressio ^ th e "T‘ c ij

.the centres of their respective areas ; therefore ^ _^ v vln

of tension x, the depth of compression tve theref^also denote these areas ; and we must have,

e»c

e .dep’

,th

X

T

(2 - «)’

1 : 3

or 3 = 4 — 4 x + a? or x* + 2 x = 2.

Whence x = — 1 + f 3 = .732 = a;.732

also

= .366 = d;

_ 5 _ 1 _ j 666666 6 '

tan. of deflection = — - — ,,

30 6

When the angle d = 9° 34', and i* 3therefore, by formula 1,

2 fa d 2 x 13,000 X -732 X

w —

Z. cos. p

30 x -9 86