STR

STE

468

distance x from the vertex ; now, the sum of all the-, in

a

the altitude d of the pyramid, is equal to the whole pyramid ;but every pyramid is one-third of a prism, having the samebase and altitude, or it is equal to one-third of the base mul-tiplied by its altitude, namely, X a; that is, d times

o 3

b x^

are equal to the sum of as many times —; consequently,

the sum of all thefb*

fb a?d ’

or the whole momentum of resist-

ance, is

as above; therefore, p l —

fbd>

= b rf s , or

p =

fb d* b<P

31

l

3 '

b d* f fCorollary 1.—Because p == — X and —

is a given

quantity, P is as

b d>

l

ratio

; that is, the power is as the breadthof the depth directly, and length

Corollary 2.—Because fI = b d 2 x

-, and is a given

and duplicatereciprocally.

f and I

quantity, p l will be as b d? ; that is, the power, or weight,multiplied by the length, is as the breadth multiplied by thesquare of the depth.

Corollary 3.—In square timber, the power multiplied bythe length is as the cube of the breadth or depth.

Corollary 4.—Because p = X

f f

j, and are given,

the force, or power, required to break similar prisms is as thesquare of their corresponding dimensions.

Proposition XIII. Figure 11.— To determine the relativestrength of a prismatic rectangular beam, projecting horizon -tally from a wall, supposing it to turn round some interme-diate axis, in the section of fracture.

Let a b c d be a longitudinal section of the beam, parallelto two vertical sides ; now it has been shown, by Proposi-tion IY. Corollary 1, that here is a certain line passingthrough some point, E, on which the beam turns in the actof bending, or breaking : that all the fibres above this lineare in a state of tension, while those below are in a state ofcompression ; and that the compression or extension of anyfibre is as its distance from that line; it also appears, thatp.CB = AE./-J- kb .f; where f is the whole force ofextension or compression. Let m equal the extension of afibre in the upper part, at the instant of breaking ; n equalthe compression of a fibre in the lower surface at the sameinstant; and let b equal the breadth of the beam ; a = a e,e = EB,( = BC, x — any indeterminate part of a e, or a,from e towards A: then, since, by Proposition I. the force ofa fibre is as the distance to which it is extended or compressed,

a : m : : x : ——, the force of a fibreat the distance x; buta

the energy, or resistance, of this force, is as the length of the

lever, x, upon which it acts; consequently, as •sura of all the resistances in the breadth, i

and the

m b x*

the

whole resistance of the extended fibres in

of

mbx 2 x m b a*

and when x = a, then

3 a ’ " ' 3 a

In the same manner, it may be shown, that the su®

n b e 1 i thet®”

the resistances of the compressed fibres is —-—, aD ’

fore, the whole sum of all the extended and co®P

. m b a 2 + n b e* . 7 _

fibres is - - - = 4E./+EB./ = (r '

- to e ^

Now, if the ratio of m to n be known, that ot a

also be known; for fa =

t _ n ^ e

J — q >

therefore

n b e

-,/ =m b a

b a

njjf. ;and y®- 5 3

cons 6 '

or ncs 1 ** 1

quently, n ; m : : a : e,

3 3

Hence, it will be seen,

that tb«ature

distance of the axis of fracture depends on the » aCL ““ ,dtimber. Now, let d ~ a + e, the depth of the bea®, alet a and e be equal to each other, then will m and n be a .

equal to each other ; and m h a * + ” h e - ~ ^

3

mb d

"fT

b d? zz p /, or p =

b d?

X

m

l ” 6~"

It will readily appear, that all the Corollaries in the F 6ceding Propositions will equally flow from this; 1 c j)nevertheless, be observed, that the relative strength olbeams as are equal rectangular prisms of homogeneous ■ ,

ture, upon supposition, is only one-half of what is exp rin the former. r y

The celebrated Galileo, who first undertook the ^‘? c A u ;-e

ot the law of resistance, supposes that at the place of l r - stthe body breaks at once, so that the whole of the 6bres_with their ultimate force: a supposition that would yrect, were the matter of bodies of a homogen eoUunelastic texture. he® 8 '

Mariotte, an eminent French philosopher and g j 9 t-

tician, who first corrected the Galilean notion of equ* tat e,ances made by the fibres of a beam at the section of c0 „.havmg observed that all bodies bend before they brei1 *’ x er*

,1,0 fihrea ns sn Trirmv he.nt snrinffS, which neve ^

sidered the fibres as so many bent springs, . V tan ce>their utmost force till stretched to a certain ^ nnnV e,%

the con

side

tio^Si-

the axis of fracture was still considered as in

of the beam. Now, as it has been shown, that ^ e x

state of bending, or breaking, is both compresse gU rf“-

ded, it follows that this axis cannot be in the COI !A v veen 1

but must have some indeterminate situation ^ ^ g upP° sl g

.

uuc muse nave some mueierminube -- . q

upper and lower superficies. To compare the t teS iat»-tions with each other : in the act of breaking, V rC e >’ u ‘of a fibre, according to Galileo, is a constan ^ a fib jsection of fracture ; therefore, let f denote the fx v

then its momentum will be f x, and the sum of a y/ j n tl®

be the fluent oifxx an d the sum of all tb jbjf

jeptb, 2

breadth will befbd*

X -: and when xr=. d, tb e

for the relative strength. NoW)

it has

been

sbe^

1 a fd 8 '

a - . tba t tl> e A d‘

in Proposition XII., which is that of Mariotte, ^

tive strength and by Proposition ’

3 f b <* at 6

fbd'fbjj and^’

~ 3

by substituting/"for m ; but*