case, and such as is founded upon unerring data derived from experimental research. On this head I am fortunate in having before me the calculations of Professor W. R. Johnson, of the Franklin Institute of America, whose inquiries into the strength of cylindrical boilers are of great value, and of which the following is a short and useful abstract:

1st. To know the force which tends to burst a cylindrical vessel in the longitudinal direction, or, in other words, to separate the head from the curved sides, we have only to consider the actual area of the head, and to multiply the units of surface by the number of units of force applied to each superficial unit. This will give the total divellent force in that direction.

To counteract this, we have, or may be conceived to have, the tenacity of as many longitudinal bars as there are lineal units in the circumference of the cylinder. The united strength of these bars constitutes the total retaining or quiescent force, and at the moment when rupture is about to take place, the divellent and the quiescent forces must obviously be equal.

6 2nd. To ascertain the amount of force which tends to rupture the cylinder along the curved side, or rather along the opposite sides, we may regard the pressure as applied through the whole breadth of the cylinder upon each lineal unit of the diameter. Hence the total amount of force which would tend to divide the cylinder in halves, by separating it along two lines, on opposite sides, would be represented by multiplying the diameter by the force exerted on each unit of surface, and this product by the length of the cylinder. But even without regarding the length, we may consider the force requisite to rupture a single band in the direction now supposed, and of one lineal unit in breadth; since it obviously makes no difference whether the cylinder be long or short, in respect to

the ease or difficulty of separating the sides. The divellent force in this direction is therefore truly represented by the diameter multiplied by the pressure per unit of surface. The retaining or quiescent force, in the same direction, is only the strength or tenacity of the two opposite sides of the supposed bond. Here also, at the moment when a rupture is about to occur, the divellent force must exactly equal the quiescent force.'

In the subsequent portion of the paper, Mr. Johnson appears to reason on the supposition that there are no joints in the plates, and that the tenacity of the iron is equal to 60,000 lbs.—rather more than 26 tons to the square inch. Now we have shown by the results of the experiments already adduced, that ordinary boiler plates will not bear more than 23 tons to the square inch, and as nearly one-third of the material is punched out for the reception of the rivets, we must still further reduce the strength, and take 15 tons, or about 34,000 lbs.* on the square inch, as the tenacity of the material, or the pressure at which a boiler would burst.

This I should consider in practice as the maximum power of resistance of boiler plates in their riveted state, and I will now trouble you to follow me in a very concise and I trust not uninteresting investigation, as to the bearing powers of boilers, and the pressure at which they can be worked with safety.

It follows from the general principles which have been stated, that the divellent force tending to rupture the boiler plates in longitudinal lines parallel to the axis of the boiler is in the direct ratio of the diameter of the

* By experiment it is found that the strength of the riveted joints of boilers is only about one-half the strength of the plate itself; but taking into consideration the crossing of the joints, 34,000 lbs. may reasonably be taken as the tenacity of the riveted plates, or the bursting pressure of a cylindrical boiler.

boiler; whereas the divellent force tending to rupture the plates in transverse lines, formed by section taken perpendicular to the axis of the boiler, is in the ratio of the squares of the diameters. THE THICKNESS OF THE

PLATES OF CYLINDRICAL BOILERS SHOULD BE IN PRO

PORTION TO THEIR DIAMETERS; for, as the force tending to burst a boiler of a given length and subject to a given pressure varies simply as the diameter of the boiler, and as the resistance of the plates varies as their thickness, it follows that the thickness of the plates, in order to resist a given pressure of steam per square inch, must be in proportion to the diameter of the boiler. It will be afterwards shown that the weakest portions of the plates are in the longitudinal lines drawn parallel to the axis of the boiler.* For example, let us take two boilers, one

* Let A B C D represent a cylindrical boiler; A B QE a longitudinal section passing through the axis of the boiler; and D C a transverse section perpendicular to the axis; put = A B the length of the boiler, 2r=d=ae the internal diameter, c=. =Aa Ee the thickness of the boiler

E

A

FIG. 1.
с

plates, T=the tenacity in lbs. of a square inch of the material, and P= the pressure of the steam in lbs, for every square inch; then the pressure of the steam tending to produce a longitudinal rupture in the section ABQE will act upwards and downwards upon the internal section, having its area equal to d x l;

.. Pressure of the steam to produce longitudinal rupture = area internal

section x P=dlP

(1)

3 feet in diameter and the other 6 feet, and suppose each to be subject to a pressure of 40 lbs. on the square inch. In this condition, it is evident that the area of the end of a 3-feet boiler is to the area of the 6-feet boiler as 1 to 4;

This result shows that the divellent force tending to produce longitudinal rupture, that is, to separate the plates along the lines A B and E Q, is in the direct ratio of the diameter of the boiler.

Again, we have

Resistance of the section ABQ E to rupture = area section × T=2lcT. (2) Now when rupture is about to take place, formula (1) becomes equal to (2);

.. dlP=2lcT,

which gives the thickness of the boiler plates when rupture is upon the point of taking place, under the pressure, P, of the steam. Hence it appears that the thickness of the plates of cylindrical boilers should be in proportion to their diameters.

The section A BQ E, passing through the axis of the cylinder, is obviously the weakest longitudinal section.

We shall now consider the conditions of rupture through the transverse section CD.

Pressure of steam tending to produce rupture in the section CD=area

This result shows that the divellent force tending to produce transverse rupture varies as the square of the internal diameter of the boiler. Area of the material in the section DC=πc (c+d);

.. Resistance section DC to rupture=nc (c + d)T

(5)

But when rupture is about to take place, formula (4) must be equal to formula (5);

παρ

.. Tс (c+ d)T:

+1) T d

= ;

neglecting as being very small, we get

C=

dP 4T

(6)

Comparing this expression with (3), we find that the transverse section has double the strength of the longitudinal section. From this proposition: it follows that a boiler constructed with plates of uniform thickness is most liable to undergo rupture in its longitudinal sections.

calculate

and by a common process of arithmetic we may the divellent force in the direction of the axis, and shall find that the edges of the plates forming the cylindrical part of the 3-feet boiler are subject (at 40 lbs. on the square inch) to a pressure of 40,712 lbs.-upwards of 18 tons; whereas the plates of the 6-feet boiler have to sustain a pressure of 162,848 lbs., or 72 tons, which is quadruple the force to which the boiler only one-half the diameter is exposed; and the circumferences being only as 2 to 1, there is necessarily double the strain upon the cylindrical plates of the large boiler.

Let us, for the sake of illustration, suppose the two cylindrical boilers, such as we have described, to be divided into a series of hoops of 1 inch in width; and, taking one of these hoops in the 3-feet boiler, we shall find it exposed, at a pressure of 40 lbs. on the square inch, to a force of 1440 lbs. acting on each side of a line drawn through the axis of a cylinder 36 inches diameter and 1 inch in width, and which line forms the diameter of the circle. Now this force causes a strain upon each of the points a a in the direction of the arrows in the annexed

FIG. 2.

a

α

diagram of the 3-feet circle of =720 lbs., and as

1440
2

suming the pressure to be increased till the force becomes equal to the tenacity or retaining powers of the iron at a a, it is evident, in this state of the equilibrium of the two forces, that the least preponderance on the side of the internal pressure would produce fracture. And suppose we take the plates of which the boiler is composed at one quarter of an inch thick, and the ultimate strength of the material at 34,000 lbs. on the

square inch, we shall have

34,000

=472 lbs. per square

36 × 2