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Northrop, E. B.; Chittenden, H. A., Jr. (ed.) / The Wisconsin lumberman, devoted to the lumbering interests of the northwest
(August, 1874)

Steam boilers. Strength of cylinder, sphere and flat surfaces,   pp. 513-515 PDF (1.1 MB)

Page 513

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The Wisconsin Lumbernman.                  513
STEAM BOILERS.
Strength of Cylinder, Sphere and Flat
Surfaces.
In analyzing the various forms of
boiler shells they are found to resolve
themselves into the cylinder, oval,
sphere, chambered and flat surfaces.
THE CYLINDER.
According to the well known law
of hydrostatics, the pressure of steam
in a close vessel is exerted equally in
all directions. In acting against the
circumference of a cylinder, the
pressure must therefore be regarded
as radiating from the axis, and ex-
erting a uniform tensional strain
throughout the enclosing material.
Its .tendency to cause longitudinal
rupture, or to rend the cylinder in
lines parallel to its axis, may be con-
sidered as a force acting and re-act-
ing in opposite directions to divide
the cylinder in two. As it must be
exerted on equal areas in order that
the action and reaction may be equal,
this divellant force may be consider-
ed as the pressure exerted on the
semi-circumference, and tending to
rupture the cylinder in a plane drawn
through the diameter. It follows,
however, from the pressure acting
equally in all directions, that the
whole amount exerted on the semi-
circumference is not equally effective
in producing a strain perpendicular
to the diameter through which the
cylinder may be assumed to rend.
If we examine the force tending
to cause rupture through the hori-
zontal diameter, we shall find the
pressure is exerted directly upwards
and downwards only along the ver-
tical diameter. As we recede right
and left from this line, the pressure
is exerted diagonally with diminish-
ing vertical effect, to produce tension
at the extremities of the horizontal
diameter, and, unit vanishes alto-
gether when we reach these points.
The radial pressure at any point,
may be resolved into two forces, the
one vertical and the other horizontaL
It is evident the latter has no ten-
sional eftect at there extremities. By
taking the component vertical forces
at an infinite number of points in the
semi-circumference it can be proved
that their sum  is equal to the full
pressure exerted on a line equal in
length to the diameter.
We may consider the cylinder as
composed of a number of rings of a
unit's length, placed side by side,
each of which resists the pressure in-
dependently of the rest.
Hence the force, tending to rup-
ture the cylinder longitudinally, is
represented by multiplying the diam-
eter by the pressure on each unite of
surface. As this applies only to a
cylinder of a unit's length, it is evi-
dent that the total amount of force,
tending to divide thescylinder in lines
parallel to its axis, is found by multi-
plying the above product by the
length of the cylinder. The practical
truth of this has been proved by ex-
periment.
The retaining force opposed to
this pressure, is evidently the resist-
ance of the material at the two op-
posite sides which bear the strain.
The manner in which the strain is
borne by the material depends great-
ly on its thickness. WVLen this is
considerable, compared with the
diameter, as in hydraulic presses and
cannon, the inner layers of the ma-
terial are more severely taxed than
those on the outside. This differ-
ence may be so great that the latter
render no material assistance to the
former. If we take two straight bars,
of the same material and section but
of different lengths, and submit them
to the same tensile strain, they will
be stretched, within certain limits, in
proportion to their length. Suppose
a bar 1 foot long is stretched 1-10
inch by a given weight, then a similar
bar 10 feet long would be elongated
1 inch by a similar weight, the exten-
sion being simply a factor of the
length. In a cylnder, say of 3 inches
diameter, and 21-2 inches thick, we

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