thus found will have to be borne in direct tension by the row of bolts uniting the joint BC.

The tangential stress in the metal of the cylinder itself is determined in the same way by taking any diametral section through the metal of the cylinder; and it is obvious that the sum of the tensile stresses in the metal cut through by the section is equal and opposite to the internal stress of the fluid acting across the plane of the same section.

In this example, as well as in the previous one, it is assumed that the cylinder BC and the conical junction-ring are standing above the water-level. If they were submerged, the tensile stress in the cylinder and the lifting force due to the internal pneumatic pressure would both be reduced by the contrary action of the external fluid pressure; but this point may be left for discussion at a later period.

EXAMPLE 3.-The vessel shown in section in Fig. 11 may be taken to represent either the barrel of a plunger-pump or the cylinder of a hydraulic engine or ram; and it may be required to find, in either case, the vertical forces and stresses acting upon the vessel and upon the framing to which it is attached. The circle drawn in Fig. 11a to the diameter d2, which is the internal diameter of the cylinder, may be taken as a ground plan or horizontal projection of the vessel's bottom, which is visible from above; or if we regard the figure as an inverted plan, the inner circle will represent the base of the ram, or the circular opening in the stuffing box; so that the annular space between the two circles will be a horizontal projection of the internal shoulder which is visible from below.

The vertical force exerted by the fluid pressure upon the vessel as a whole, and borne by the framing which supports the outer flanges, will be the algebraical sum of all the vertical components, or the difference between the lifting pressure at the shoulder and the downward pressure upon the bottom of the cylinder. Therefore, subtracting the negative area of the annular space in Fig. 11a from the positive area of the larger circle, we have remaining the area of the inner circle, and this area multiplied by p will give us the downward pressure upon the engine-framing. This force is obviously equal to the load P1 upon the ram, and is independent of the diameter d. At the same time it will be at once seen that the vertical tensile stress in the sides of the cylinder below the framing will have the greater value

dg

FIG. //A

с

EXAMPLE 4.-Suppose a cast-iron pipe to be formed as a bend curving through 180° as sketched in Fig. 12, and to be connected at each end to the straight vertical pipes by an ordinary flange-joint. It is required to find the stress in the flangebolts when the whole line of pipe is charged with a hydraulic pressure p.

The vertical lifting force which must be resisted by these bolts will evidently be the algebraical sum of all the vertical components of the radial pressures acting upon all the elements of area of the internal surface. In the straight vertical pipes all these vertical components will be zero, because normals drawn to all the elements of internal surface will be horizontal; but in the bend these normals will be inclined to the horizontal, some in one direction and some in the opposite

FIG. 12

direction. The dotted centre-line ACB in Fig. 12, if it were traced upon the sides of the pipe itself, would lie in vertical planes forming tangents to the pipe at all cross-sections of the bend; and we may conceive the centre-line as dividing the pipe into two troughs of semicircular section. The upper trough will contain all those elements of surface which are exposed to pressure having a vertical component in the upward (positive) direction; while the lower trough will contain all those parts of the surface on which the pressure has a downward vertical component. The horizontal projections of these surfaces are easily traced in the sectional plan of Fig. 12a. The figure HF ABFB11E is the projection of the positive area, while the figure ABHB1ÁG is the projection of the negative area; and subtracting the latter from the former, we have remaining the areas of the two circles which represent the open ends of the bend.

FIG. 12A

A

E

A1

B

B1

4'

The lifting force which has to be resisted at each joint is therefore pd2, where p is the fluid pressure and d the internal diameter of the pipe.

The same result might have been obtained still more simply by taking an imaginary section across the bend at the plane AB, and regarding the bend with its contained water, as forming a body whose equilibrium is to be considered. For the stress or pressure of fluid upon fluid, across this plane, must be balanced by the tensile resistance of the bolts.

CHAPTER III.

CYLINDERS, PIPES, AND BENDS UNDER UNIFORM PRESSURE.

Art. 7. Tangential Stress or Hoop-tension. In determining the strength of any steam boiler, hydraulic cylinder, or cylindrical pressure-pipe, the engineer will naturally begin by calculating the "hoop-tension" as it is called, or, in other words, the tensile stress which is transmitted round the circumference of the cylindrical shell, and which acts everywhere in a direction tangential to the circle. It will presently be seen that this is not, by any means, to be regarded as the only stress that we have to deal with; but it is the simplest part of the problem, and may conveniently be taken first.

Let d be the internal diameter of the cylinder whose crosssection is drawn in Fig. 13a, and through the cylinder take any diametral section such as that shown by the line AA, dividing the vessel into two semi-cylindrical troughs. Also in Fig. 13, which is a longitudinal section on the same plane, let AB represent a unit of length, say 1 inch lineal, of the cylindrical vessel, so that the rectangle AA,BB, is the projection of the internal surface of one of the troughs on a length of 1 inch. We have already seen that when the vessel is subjected to internal pressure, the force which tends to separate the two troughs from each other is equal to the pressure p multiplied by the area of the rectangle, and if the length AB is 1 inch, the force will be simply pd.

If, then, we may assume that, on each lineal inch of the cylinder, this force is resisted by the direct tension in the same lineal inch of the metal shell, whose thickness is denoted by t, we should have for the mean tensile stress per square inch of metal the well-known expression for the hoop tension, or

We can make this assumption without any doubt in the case of a long pipe of uniform diameter and uniform thickness if the metal is fairly homogeneous. In the case of a long boiler or cylinder with closed ends, we could make the same assumption with safety, and it would be nearly true for a lineal inch of shell taken at the central portion of the length. But near the ends of the boiler the force would be resisted in part by the tensile strength of the cover, and the hoop-tension in the shell would be somewhat diminished. If, however, we went on to calculate the exact distribution of the stress between the ends and the adjacent shell, by reference to their relative extensibilities, the calculation might be seriously interfered with by change of temperature, initial stress, and unequal elasticity. In most cases, therefore, engineers will prefer to leave out of account any assistance that could be derived from the strength of the endcovers, and will generally design the shell of sufficient strength throughout to resist the hoop-tension as found by the usual formula.

Apart from this matter, however, there are two other points in connection with the hoop-tension which may sometimes have to be considered. If the cylinder is surrounded by another fluid under the pressure p1, its effect on each lineal inch would be to produce a compressive force pid1, in which d1 is the external diameter of the cylinder; so that the compressive stress in a

boiler-flue, for example, would be mean f Pid And if the

=

2t

cylinder is subjected at the same time to an internal and an external pressure, the hoop-tension would have for its mean

These expressions for the mean value of the hoop-tension per square inch of metal section will be equally true whether

the shell is very thin or very thick; and in the case of a thin cylinder, we may assume that the stress will be distributed almost uniformly over all parts of the metal section, so that we may take the formula as giving practically the maximum tensile stress in the metal.

But when the cylinder is made of great strength to resist very high pressures, as in the case of many hydraulic cylinders, the thickness may become so great, relatively to the diameter, that the stress is very unequally distributed, and the maximum stress may then be considerably greater than the mean value as above given. If we imagine the cross-section to be divided into a number of concentric rings, as shown in Fig. 13a, the stress will be most severe upon the innermost ring, and least upon the outermost ring.

The formula that has generally been proposed by mathematicians for calculating the maximum stress is a little too complex for ready use, owing to the fact that it takes into account the external pressure P1; but in most cases this will represent only the atmospheric pressure which, for practical purposes, may be neglected, and we then have the following values for the maximum stress in terms of the mean stress:

Max. f
Mean f

1.010 1.013 1.017 1.020 1.026 1.034 1.052 1.109 1.139 1.190 1.233 1.300 1.417 1.667 2.500

These figures, however, are calculated on the assumption that the cylinder has been manufactured in such a manner as to throw no initial stress upon the concentric rings. It is well known that if the outer rings are subjected to initial tension, throwing the inner rings into initial compression, the maximum tensile stress in the latter, due to internal fluid pressure, will be correspondingly reduced; and by this means the hooptension in a thick cylinder may be more evenly distributed over the whole section. Perhaps, however, this question is of more interest to the gun-maker than to the hydraulic engineer, and need not here be considered in detail.

Art. 8. Axial Fluid Stress. If pressure is to be maintained in any vessel, the fluid stress or pressure must be resisted in all