FIG. 3.

b

inch, as the bursting pressure of the boiler. Again, as the forces in this direction are not as the squares, but simply as the diameters, it is clear that at 40 lbs. on the square inch, we have in a hoop an inch in depth, or that portion of a cylinder whose diameter is 6 feet, exactly double the force applied to the points bb, which was acting on the points a a, in the diameter of 3 feet. Now assuming the plates to be a quarter of an inch thick, as in the 3-feet boiler, it follows, if the forces at the same pressure be doubled in the large cylinder, that the thickness of the plates must also be doubled in order to sustain the same pressure with equal security; or what is the same thing, the 6-feet boiler must be worked at half the pressure at which the 3-feet boiler is worked in order to increase the same degree of safety. From these facts we may conclude, that boilers having increased dimensions, should also have increased strength in the ratio of their diameters; or, in other words, the plates of a 6-feet boiler should be double the thickness of the plates of a 3-feet boiler, and so on in proportion as the diameter is increased.*

b

*It will render these discussions clearer to bear in mind the following formulæ, which will be seen to follow at once from the equation d IP=2lc T established in the preceding note.

(1.) Given the pressure at which a cylindrical boiler is to work and its diameter, to find the requisite thickness of plates.

Or for iron plates, T being taken at 34,000 lbs. as above, the required thickness in inches is equal to the diameter in inches multiplied by the pressure per square inch in lbs. and divided by 68,000.

The relative power of force applied to cylinders of different diameters becomes more strikingly apparent when we reduce them to their equivalents of strain per square inch, as applied to the ends and circumference of the boiler respectively. In the 3-feet boiler, working at 40 lbs. pressure, we have a force equal to 720 lbs. upon an inch width of plate, and one quarter of an inch thick, or 720 × 4 = 2880 lbs., the force per square inch of sectional area at every point of the circumference of the boiler.

Let us now compare this with the actual strength of the riveted plates themselves, which taken as before at 34,000 lbs. on the square inch, we arrive at the ratio of the working strain to the strength of the circumference as 2880 to 34,000, or nearly as 1 to 12, and 40 × 12 = 480 lbs. per square inch nearly, as the ultimate bursting pressure of the boiler.

These deductions appear to be true in every case as regards the resisting powers of cylindrical boilers, to a force radiating in every direction from the axis towards the circumference; but the same law is, however, not maintained when applied to the ends, or, to speak technically,

(2.) Given the thickness and diameter of a cylindrical boiler, to find its bursting pressure.

Or for wrought iron plates the bursting pressure in lbs. per square inch is equal to the thickness in inches multiplied by 68,000, and divided by the diameter in inches.

Example. Required the thickness of the plates of a boiler, 20 feet long and 5 feet in diameter, in order to stand the pressure of steam at 204 lbs. per square inch, the tenacity of the metal being 34,000 lbs. per square inch.

Here by formula (3) we have.

d = 5 x 12=60, P=204, and T=34,000,

..C=

60 × 204 2 × 34,000

=18 inch.

to the angle-iron and riveting where the ends are attached to the circumference. Now, to prove this, let us take the 3-feet boiler, where we have about 113 inches in the circumference, and upon this circular line of connexion we have, at 40 lbs. to the square inch, to sustain a pressure of 18 tons, which is equal to a strain of 360 lbs. acting longitudinally upon every inch of the circumference. Apply the same force to a 6-feet boiler, with a circumference or line of connexion equal to 226 inches, and we shall find it exposed to exactly four times the force, or 72 tons; but in this case it must be borne in mind that the circumference is doubled, and consequently the strain, instead of being in the quadruple ratio, is only doubled, or a force equal to 720 lbs. acting longitudinally as before upon every inch of the circumference of the boiler.

Again, if we refer to the comparative merits of the plates composing cylindrical vessels subjected to internal pressure, they will be found in this anomalous condition, that their strength in their longitudinal direction is twice that in the curvilinear direction. This appears by a comparison of the two forces, wherein we have shown that the ends of the 3-feet boiler, at 40 lbs. internal pressure, sustain 360 lbs. of longitudinal strain upon each inch of a plate a quarter of an inch thick; whereas plates of the same thickness have to bear in the curvilinear direction a strain of 720 lbs. This difference of strain is a difficulty not easily overcome, and all that we can accomplish in this case will be to exercise a sound judgment in crossing the joints, in the quality of the workmanship, and the distribution of the material. For the attainment of these objects, the following Table, which exhibits the proportionate strength of cylindrical boilers from 3 to 8 feet in diameter, may be useful.*

* In this discussion I have not taken into account the effect that boilers constructed with central flues will have upon the flat ends in diminishing

TABLE of equal strengths in the external shell of Cylindrical Boilers from 3 to 8 feet diameter, showing the thickness of metal in each respectively, for a bursting pressure of 450 lbs. to the square inch.

Boilers of the simple form, and without internal flues, are subjected only to one species of strain; but those constructed with internal flues are exposed to the same tensile force which pervades the simple form; and further, to the force of compression, which tends to collapse or crush the material of the internal flues. In the cylindrical boiler with round flues, the forces are diverging from the

the strain upon the anterior or the exterior circumference where the ends are united to the cylinder. On the contrary, I have purposely omitted that element of strength in the calculation, and for the sake of illustration I have assumed that the boiler has no internal flues, and that the circumference is subjected to the whole strain in the inverse ratio as before noticed- of the diameters. In cases where boilers are constructed with internal flues, there will be this advantage as regards strength; namely, that the outer shell will be relieved from a longitudinal strain to the extent of their respective areas or circumferences.

central axis as regards the outer shell, and converging as applied to every separate flue which the boiler contains.

These two forces in a steam-boiler are in constant operation; the tendency of the one being to tear up the external plates and force out the ends, and the other to destroy the form and to force the material into the central area of the flues. These two forces operate in a widely different manner upon the resisting powers of the boiler, which, taken in the direction of its exterior envelope, has to resist a tensile strain operating in every direction from within, and the internal flues, acting as an arch, offer a powerful resistance to compression from without.

In previous editions of this work I had at this point alluded to our ignorance of the laws which determine the rupture of cylindrical vessels subjected to an external pressure, and had said that these could only be learnt by experiments conducted on so extensive a scale, and with so great a degree of accuracy, that for the present the hope of them must be abandoned. At the same time I had shown that the resistance of internal flues would be in the inverse ratio of their diameters, which was at that time. the most that was known upon the subject of collapse.

We need no longer, however, continue to construct our boiler flues as heretofore in ignorance of the laws which regulate their powers of resistance. Under the sanction of the Royal Society I have made a very extensive series of experiments upon the subject, this results of which will be found in full in the volumes of the Transactions of that body for 1858 and 1859. These experiments have fully established the formulæ of strength, and have, I believe, a very important bearing on the future construction of these vessels. I, in common with other engineers, had acted under the impression that perfectly cylindrical tubes, when subjected to a uniform pressure converging upon their centre, were equal in their powers of resistance,