STR
467
STR
‘ten,
by the last Proposition, the strain at C from the
A B + iBC.CD.T
^'ght w is
2ab+ bc.cd.w
AD 2 A D
^ + A C .) C D . W
5 -; and, by Proposition VII., the strain at
« AB
°>8unr,„„- . , . AC.CD.T
AD
2 A D
: :ab + ac:2ac.
i C(J ^position X. Figure 8 .—The strain of a prismatic«)! % supported at the two ends, A and b, by its own weight atint Sect ‘ on > C, is equal to the rectangle of the two segments0 tie weight of the beam, divided by twice its length ; thatit, i^BC2 A B
l^ceive the parts ac and cb of the beam to be accumu-in “heir centres of gravity, p and G. Let w equal the*<i t°f t ‘ ie w hole beam ; the weight of the part a c will be
B C • W
.4 a > and the weight of the part b c will be- : —. By
Pr 0 n • . ab j
Position VIII., the strain at c, from the weight of the
AF.CB.AC.T
‘th
AB
and the
"' n at c, from the weight of the part b c, is
•V
AB
X
GB.AC.CB.TAB 1
A B‘
AB"
AB
0vv > A f + g b = —^; therefore, the whole strain by the
A
'“ole
height, is
AC.BC.T2 AB
h^ r °ttary 1 .—Hence, the strain of a beam at any point, c,tli e a Weight suspended there, is double what it would be athji atJ1 e point, C, from the same weight being uniformly dis-te< l over it; because, by Proposition VII. the stress is
^<5® • "W ,, _ .. T AB.BC.T
, and by Proposition X. it is
.C«
'Vej
2 A B
2.—Because, in the same beam, w and ab being■ “be strain at any point, c, is as rectangle of the twoetlts A c and C b.
THE SAME BY FLUXIONS.
"C, *= W, equal to the whole weight of a = A c, b =
i V,
: A B, and x any indeterminate part, a p. Then the
of ap = and the fluxion of ^ ^ equal
to
* ei ght
((f o-~ acting at P, Then, by Proposition VIII., the
at n f . ,. . . bx wx bw xx
a c > trom a weight acting at P, is — X —— = —-—;
ll l
U re > by taking the fluent, the strain at c, from the: „ . iwa? „ ,
ei ght of a p is So the strain at c, from the
^ *
°f b c, will be found to be —therefore, the
H l
° e 8 tn;„ 4 . , . ba^w + aFw
dln tending to break the beam at c, is--
a + bxabw , . a + b X abw
but a + b = l; therefore,
2 P
2P
a b t
A c X c b X W
21
2 A B
Proposition XI. Figure 10. — If a prismatic beam, a b,be fixed in a wall at A; the strain at any section, c D ,from itsown weight will be as the square of the distance, B c , of thatsection from the extremity.
The strain which the weight of the part B C occasions atthe section c d, is the same as if the whole weight of BO werecollected in its centre of gravity, f, and the length of thelever, c f.
Let w represent the whole weight of abj then, becausethe beam is prismatic, the weight of the part B C will be
B ° W ; therefore, by Proposition IV., Corollary 2, the strain
ab
at c is C f x
BC.T
A B
B C B C— X-
. W BC’.WB — 2ab ’
but in the
same beam, w and 2 a b are given quantities, consequently,the strain at the section c d is as b c a .
Proposition XII. Figure 11. — To determine the relativestrength of a rectangular prismatic beam, abcd, projectinghorizontally from a wall, supposing the beam to be incompres-sible at the fulcrum.
Let a b c d be considered as a longitudinal section, and letA b be the section of fracture; let a b c be considered as abent lever, whose fulcrum is at b; a b, b c, the two arms;and p the power employed to overcome all the resistances inthe section A b. Now, in the act of breaking, all the fibresin A b will be stretched, and this the more in the same ratioas they are more remote from the axis of fracture passingthrough b; and those at a will break first. Let /be a forcejust sufficient to destroy the cohesion of a fibre: also, let lequal the length, d equal the depth, b equal the breadth orhorizontal dimension, and x equal any indeterminate distancefrom B; then, because the tension of any fibre is as its exten-sion (by Proposition I.) and the extension as the distance of
f
the fibre from the axis of fracture, d :x ::f r—, the force or
tension of a fibre; therefore, the force of all the fibres in thef b x
breadth is, * / -^-; but the momentum of any force applied to alever, is that force multiplied by its distance from the fulcrum,
fbx
Jb x-
consequently,*^-^- X x =■ ^ -, the momentum of the fibres
d
in the breadth ; and the sum of all these momenta upon the■p b x f* b ^
lever x is the fluent of -—^— = ~ g ^ -; and when x = d,
fbx > fb d\
3d 3 ’
, , , f bd ‘ / bd ’ f
therefore, p 1= - = bd i x or p = -r- X-^-.
O o I Q
now, the momentum of the power is p l,
THE SAME 'WITHOUT FLUXIONS.
The momentum of the fibres in the breadth is
bx?
fbx !*
as
above ; but — is the section of a pyramid, having b and dfor the dimensions of its base, and for its altitude a, at the