The Locomotive - Lighthouse Survival Blog

1901.] THE LOCOMOTIVE. 83
(It may happen that the actual resultant force on the inner shell acts inward instead
of outward; hut the assumption that it acts the other way can do no harm because if it-
is wrong, we shall merely come out, in the end, with a negative value for >S„, and this
would show that S 2 is a compressive strain, instead of a tension, which would be pre-
cisely the result that would be reached if we assumed, at the outset, that the resultant
force acted inwardly.)
As before, we may now assume that the stress on the inner shell, due to an outwardly-
directed force of
rr PAR& i
la — ?_ pounds
*i
42?
acting on each unit containing _^? square inches of area, is the same as would be pro-
duced by a uniform steam pressure, acting outwardly with an intensity just sufficient to
is, by a steam pressure of
PAR A P
give a total pressure of 1\ Ta— 2 pounds on every —i_2 scpiare inches of area; that
AR Ta PAR 2 \ .
Ri j '
_TaR — 2 PAR x 2
#! " AL\
pounds per square inch. Finding, by the same method as was used for the outer shell
the tensile stress, S 2 , that such a pressure would produce in a shell of radius R 2 and
thickness t 2 ,
we have
q _TaR —PAR x
"
2 " " t 2A
2
which is the formula given for S 2 in the original article.
It will be noted that we have made no allowance for the fact that on each unit of
each plate there is an area of a square inches taken up by the stay-bolt, and therefore not
exposed to steam pressure. This could be readily taken into account; but the formula?
would be considerably more complicated, and the gain in accuracy would be too small
to be of practical importance. Hence we have neglected it.
To apply the foregoing formula?, we need to know T, the stress on the stay-bolt per
square inch of its cross-section. To obtain this, we have to consider how much the
various parts of the boiler stretch when a given steam pressure is applied. Let us sup-
pose that the letters i?, and R 2 represent the radii of the two shells before the pressure
is applied, and that when the full working pressure is reached, the radius of the outer
shell becomes R x + A i? n and the radius of the inner one becomes i? 2 + A 2? 2 . Now
nothing can make the shells stretch except the tensile stress that comes on them; and
while the stresses denoted by S t and 8 2 are not the total stresses on the shells, yet they
are, by hypothesis, the extra stresses that are due to the fact that the plates are circular
instead of flat; and hence we shall take S t and S 2 to be the significant stresses for our
present purposes, and shall consider the extension that is due to these forces alone.
The circumference of the outer shell, after the stretch, is 3.1416 (R 1 + A R-,), as
against 3.1416 i?,, its circumference before the stretch. Hence the total stretch of the
outer shell is 3.1416 (R l + A -K,)— 3.14167^ = 3.1416A7?!. Now we know that the
stretch due to a given force is equal to the length of the piece stretched, multiplied by
the stress per square inch of sectional area, and multiplied again by a constant which
depends upon the nature of the material, and which we will denote by l\ Hence in the