418: On the Resistance of Solids. 
14. When the beam is rectangular, and its depth = d; 
d i " Qfyarte  4fbare 
a=} and making y= 0, we have 7 Sen ea Fea the 
: - 4fb 2x3 de bd? 
fluent of which is “ — and when x = goat becomes az = 
W = the transverse strength of the beam. 
15. When a beam is supported at both ends, its strength to 
bear a weight in the middle of its length, is to that of a beam 
of the same length, supported at one end, as 4 is to 1. Hence 
2fld 
- — =W2= the strength of a beam supported at both ends. 
16, When the beam is square, and the force acts in the di- 
rection of its diagonal BC; EF will be 
the neutral line, BD=a, and EF=2a. 
Also, by sim. trians. d:@a—a::2a: 
2(a—x)=y. 
Therefore fal i dasa js ac 
la la 
‘ a3 ‘ 
and, when x=a, the fluent is Bas! or, 
; 3 
because 2a =d = the diagonal; ds = 
W= the strength of the beam supported at one end. 
And when a beam of the same length is supported at both ends, 
{* -W= the strength of the beam. 
17. When the beam is a cylinder, and its radius ="; a=7, 
and, by the property of the circle, y=2 / (1? a"). 
: Qfy 2a 4 fx2x af (12—22 
Therefore — = = 
xz=r, and p=3'141=5, &c. is 
' 
and its fluent when 
fpr 
AL 
y . 
cylinder supported at one end; or, f m =W= thestrength sup- 
= W= the strength ef a 
ported at both ends. 
18. From the preceding examples it appears, that when the 
Jength and the areas of the sections are the same, the transverse 
strengths of the beams will be nearly in the following propor- 
tion ; viz. : 
The strength of a square beam being unity, or .. 1000 
The strength of the same beam when the force is t 0-7071 
in the direction of the diagonal, willbe... .. 
And that of a cylinder .. ee a ee 0846 
It is not, however, to be expected that these proportions will 
perfectly agree with experiments, unless the material be homo- 
geneous, For this reason, timber is very unfit for the purpose 
of making experiments to establish a pure mechanical nee 
. , en 
