APPENDIX B.

FLUID CATENARY.

It has already been shown, in Art. 26, that when the effective building material of the arch consists of a fluid under uniform pressure, the entire load being carried by the fluid arch, and no part of it by the surrounding tubular arch of metal, the internal sectional area of the tube must at all points be proportional to sec 0. Hence the weight of the fluid per unit of arc-length will also be proportional to sec 6, while its weight per unit of horizontal length (2) will be proportional to sec2 0, or to 1 + tan2 0.

The same considerations imply that the diameter d of the tube will at all points be proportional to sec ; and if the tube is so built that the working stress in hoop-tension shall have any determined value ƒ per square inch of metal section, the thickness of the tube will at all points be proportional to the diameter or to sec 0. If, then, we assume that the thickness of the tube will be small in comparison with its diameter, its weight per unit of arc-length will be very nearly proportional to d multiplied by the thickness, and therefore proportional to sec 0; so that this part of the load, like the weight of the fluid itself, will have an intensity per unit of horizontal length, proportional to sec2 0.

Therefore, in the case which we are supposing, wherein the arch is equilibrated under its own weight, including the fluid and its containing envelope, the entire load per unit of x is proportional to sec2 0, and the form of the arch can be determined accordingly.

If the tube were formed in a series of short, straight lengths, each length being cylindrical in form, and if AB and BC in Fig. 556 are taken to represent the centre lines of two such contiguous lengths, having the inclinations 1 and 02, the vertical force BG, in the parallelogram ABCG, may be taken as consisting of one-half of the weight of the tube AB and one-half of BC; while in the same force diagram, AB, or BC, will represent the horizontal thrust at the crown, and AB the inclined thrust in the first length.

If each side of the polygon subtends the same horizontal length

Ax

=

AB1 = BC1, it will follow that the weight of each pipe must be proportional to Ax sec2 0, and at each joint of the polygon the vertical BG will be proportional to the mean of the squares of AB and BC.

The curve will be the same as that of a suspension chain, when the load consists only in the weight of the chain, and when the links are made of uniform strength-i.e. not of uniform section, but with a section proportional to the stress.

For in the arch constructed as we have supposed, the weight w per foot of arc-length is always proportional to the direct thrust S, so that S = mw, m being a constant multiplier.

When the tube is charged with hydraulic pressure, the weight of the water will probably form a large part of the total weight w, and the multiplier m will vary in each case according to the FIG.55

diameter d, the pressure p, and the working strength f. If, however, the fluid is compressed air, or any elastic gas, the fluid weight will be very small in itself, and will form a constant fraction of the weight of the pipe, so that m will be theoretically independent of p or d (within certain limits), and will depend only on the working strength f.

The curve will then have one definite figure expressed by the function

y = m hyp. log. (sec)

in which x and y are the co-ordinates shown in Fig. 55; and m will be the radius of curvature at the crown.

Thus, if we assume for a steel tube a uniform working stress of

6 tons per square inch, its thickness in inches will be

as the weight of each square inch of steel section is about 3.36 lbs. per foot lineal, the weight of the tube per lineal foot would be 3.36παρ

pd2π which is equal to 20th of the actual stress S, or 12 × 2240' 4 For a tube of drawn steel, therefore, the curve of the equilibrated arch would have a radius of about 2000 feet at the crown.

We should, however, add to this weight of tube about 7 per cent. for the weight of the contained air under any given pressure p; and if, for the sake of example, we further add 53 per cent. for the estimated supplementary weight of connections, etc. (including loss of strength at rivet-holes), the multiplier m will be reduced to 1250.

With this estimated weight of the built tube, we should have the following values for the co-ordinates x and y in Fig. 55. For m = 1250.

Thus a tubular arch of 1200 feet span, with a rise of one-eighth of the span, would suffer no compressive stress from its own weight, so long as the tube was charged to a pneumatic pressure p sufficient to produce a hoop-tension of 6 tons per square inch.

So far as the weight of the tubes is concerned, the multiplier m would be nearly the same for different diameters and different thicknesses of pipe (within moderate limits); and the calculated co-ordinates would have the same value, whether the arch consisted of one large tube or of several smaller tubes, grouped together as in Fig. 55a, except that the weight of bracing would then have to be taken into account.

APPENDIX C.

EXPERIMENTS ON THE TENSILE STRENGTH OF

SOCKET JOINTS.

THE object of these tests was to find, by actual experiment, the resistance which a socket joint offers to a direct longitudinal pull inflicted by the pipe itself, just as it would be at the joints of a bend, for example, under the action of the displacing force.

The pull was, therefore, applied to the pipe axially, and was measured by means of a Wicksteed testing machine, as the measurement obtained in this way was more accurate than any that could be made by means of hydraulic pressure.

It is possible that the resistance of the lead joint would be affected, to some slight extent, by the presence or absence of internal hydraulic pressure-either diminished by water lubrication or increased by lateral pressure upon the lead—while the egress of the lead would, no doubt, be aided by the fluid pressure upon the annular end; but it does not seem probable that any of these forces or resistances would very much help or hinder the action of the great axial stress that is sometimes transmitted through the pipe itself. At any rate, the tests were made without the presence of any internal fluid pressure, and for the purpose that has been mentioned.

The socket joints that were thus tested included several different forms, such as are commonly employed in water-pipes of large diameter; but as the power of the testing machine was limited to 50 tons, the sockets were necessarily limited to the small diameter of 3.5 inches in the bore, although in other respects the sections were such as would be used in the largest pipes; and it seems reasonable to suppose that, with any given section, the resistance would be nearly proportional to the circumference measured at the line of shear.

The pipes and sockets were of cast-iron without any bituminous or other coating, and in general the surfaces were left as they came from the mould; but in one case the socket was smoothly bored, as specified in the description of joint No. 2c.

L

In all cases the lead was well driven up by a caulking tool at the mouth of the socket. Whenever gaskin was employed, a considerable movement of the pipe took place, during its compression, without starting the lead; but in all cases the load here recorded is the load under which the lead began to move, and from this point the resistance generally decreased as the joint slowly drew out.

To ascertain the ultimate resistance, it was necessary to apply the load with extreme slowness.

SOCKET JOINT No. 1.

Length of socket, 4.0 inches, entirely filled with lead. Diameter of socket, 3.5 inches, plain cylindrical form. Diameter of pipe, 2.92 inches, plain cylindrical form.

The same socket as in No. 1-no gaskin used. Pipe 2.92 inches outside diameter, with a shoulder 3.25 inches in diameter, and inch thick.

The socket of No. 2c was bored to a smooth surface.