ON THE STRENGTH OF PILLARS. 
405 
be required to crush the material, one-half only of the strength may be considered as 
available for resistance to flexure, whilst the other half is employed to resist crushing; 
and when, through the shortness of the pillar, the breaking pressure is so great as to 
be nearly equal to the crushing force, we may consider that no part of the strength 
of the pillar is applied to resist flexure. 
We may, therefore, separate these effects by taking, in imagination, from the pillar 
(by reducing its breadth, since the strength is as the breadth) as much as would 
support the pressure, and consider the remainder as resisting flexure to the degrees 
indicated by the previous rules (Arts. 36 and 38.). 
42. Suppose then c to be the force which would crush the pillar without flexure ; cl 
the utmost pressure the pillar, as flexible, would bear to break it without being weak- 
ened by crushing (Art. 6 .) ; b the breaking weight, as calculated by Articles 36 and 
38. ; y the real breaking weight. 
Then, supposing a portion of the pillar, equal to what would just be crushed by 
the pressure d , to be taken away, we have c — d = crushing strength of the remaining 
part, and y — d the weight actually laid upon it 
Whence _ f = the part of this 
remaining portion of the pillar which has to resist crushing, .*. 1 — J ^ = c ' —,i — 
the part to sustain flexure. 
But the strength of the pillar, if rectangular, may be supposed to be reduced by 
reducing either its breadth, or the calculated strength of the whole, to the degree in- 
dicated by the fraction last obtained. In the circle this mode is not strictly appli- 
cable ; but we obtain a near approximation to the breaking weight, y , by reducing 
the calculated value of b in that proportion. We have, therefore, 
c — d : c — y : ; b ; y, 
.'. cy — dy — be— by, 
b c 
y b + c — d 
whence 
43. It has been shown (Art. 7-) that about one-fourth of the crushing weight was 
the utmost that a flexible pillar could be broken with, without the material being 
crushed, and the breaking weight reduced in consequence. 
In the Table following, I will give the values of y, as deduced from all the pillars in 
c be 
Article 39, taking d = — ; and, therefore, y = b + 3 c - Before concluding this sub- 
ject, I may, however, observe here, that there is a falling off in the strength of pillars, 
in consequence of the pressure, through all stages, from the smallest force necessary 
to break them to the largest, though this diminution does not become so obvious as to 
need correcting for, till the breaking weight is about one-fourth of the crushing weight. 
This reduction in strength will be shown from a comparison of the strengths of the 
thicker with the more slender pillars, the length being the same, the falling off always 
increasing with the diameter. This will be rendered evident from the decrease in the 
