CHAPTER V.

FLUID ARCHES.

Art. 24. Pressure Pipes used as Tubular Arches.-In extensive works of water supply, gravitation mains are often carried across rivers or valleys by bridges constructed for that purpose; and in several cases it has been proposed to construct such bridges in the form of a tubular arch, in which the pipes themselves constitute the hollow arched ribs. This principle has been followed in a few structures, designed for the conveyance of either water or gas, and may perhaps be found useful in other situations.

Such an arch must evidently be regarded as an upright bend, like the segmental bend of Art. 10, in which the “displacing force" acts as a distributed pressure in a radial (upward) direction, and in opposition to the load or weight of the structure; and the load being thus in some measure supported by the fluid stress, the compressive stress in the tubular arch would be correspondingly reduced. If the displacing force were great enough, it might evidently relieve the metal arch of all its usual compressive stress, so that the load would be carried wholly by the arch of water, and not at all by the arch of steel; and if the displacing force were still greater, it would tend to lift the whole superstructure, or if the tubes were anchored down to the abutments would throw the metal arch into tension, like the chain of a suspension bridge inverted.

The structure considered simply as an arch, having a given figure and carrying a given load, will, of course, be subject to the well-known principles by which the axial thrust at any point may be calculated, and by which the bending stresses due to deformation under the load, or due to change of temperature, may also be approximately determined in arched ribs of known section; but, on the other hand, we have to bring into the calculation, in addition to all these, the displacing forces which are due

to fluid reaction, and which will constitute the special feature of such structures.

It will hardly be necessary to pursue this question into all its details, and it will be sufficient to take a few simple cases which will illustrate the effects of the fluid stress in tubular arches charged with hydraulic or pneumatic pressure.

Art. 25. Direct Longitudinal Stress in Charged Tubular Arches. Let us suppose the bridge sketched in Fig. 25 to consist of two (or more) arched ribs of hollow cylindrical section, placed side by side and united by suitable cross-bracing; and let each of these arched ribs form part of a continuous line of pressure main, the ribs being also abutted upon the skew-backs at B and D by exterior flanges, which take a bearing upon suitably formed annular bedplates.

For the sake of simplicity, we shall suppose the load to consist wholly of dead load, including, of course, the weight of

the arched pipes and their fluid contents, and that the figure of the arch is designedly adapted to the known distribution of the load, so that the line of pressure shall coincide with the neutral axis of the rib or the curved centre-line of the arched pipe. And we may perhaps also assume that the fluid pressure is uniform throughout the pipe, as it would be practically if the fluid were gas or compressed air, and as it would be very nearly if the pipe were charged under a great hydrostatic head, such as would be likely to occur at the valley crossings of a gravitation main.

Then, if the internal diameter d of the arched pipe is first assumed to be uniform throughout the arch, the axial fluid stress will at all points have the same value, P = pd2; while the

4'

arch thrust will, of course, differ slightly at different points in the curve.

Thus, if we take first the case of a uniform load-i.e. a load of uniform weight w per foot of span, supported by a parabolic arch-we may denote the total load by W, the span (BD) by L, and the rise of the arch (CF) by D. The horizontal thrust at the crown of the arch will then be given by the familiar formula—

and the direct axial thrust at any point in the rib will be—

CH sec.

where is the varying inclination of the rib.

φ

(1)

(2)

The horizontal distance CG to the intersection of the tangent BG will, in this example, be equal to one-fourth of the span, and the point & will, of course, coincide with the centre of gravity of the uniform load upon the half-span.

But, in most cases, the centre of gravity will be somewhat nearer to the abutment, and consequently the curve of pressure will be more nearly segmental than parabolic. In the equilibrated segmental arch the centre of gravity of the half-load must coincide with the intersection G of the two equal tangents, and the true value of the horizontal thrust would then be

in which is the angle subtended by the whole arc BCD, and is the angular inclination of the tangent BG to the horizontal. In either form of rib, however, we may express the direct thrust by CH sec ; and we have already seen that this thrust must represent the algebraical sum of the compressive axial stresses in the fluid and in the solid portions of the arch. The direct longitudinal stress in the metal of the tube will therefore be

If Q is positive the tube will be in compression, but if it is a negative quantity it will mean that the tube is in tension like the chain of a suspension bridge.

EXAMPLE 1.-Let us suppose the bridge sketched in Fig. 25 to consist of a pair of tubular arched ribs of riveted steel plate, the internal diameter of each tube being 36 inches throughout the span, and the same as the internal diameter of the horizontal water pipes AB and DE of the gravitation main, in which the arched tube forms a part of the continuous line of pipe. Let the span (BD) be 120 feet, and the rise 12 feet, or one-tenth of the span, and suppose, first, that the arch is of parabolic form, and the load uniformly distributed.

The actual weight of such pipes, made of half-inch steel plate with double butt joints, but without any stiffeners, may be estimated at 2 cwt. per foot lineal, while the contained water in each pipe would weigh 4 cwt. per foot, so that the total weight of the two pipes would be 12 cwt. per foot lineal; but we will here suppose that the total dead weight of the structure as sketched, including roadway girders, spandrils, and bracing, amounts to 20 cwt. per foot of span, or W = 120 tons, and we will examine the equilibrium of the bridge under this uniform load of 1 ton per foot when the pipes are charged under a hydrostatic head of 320 feet.

Taking a cross-section through the arch at the crown, we have the horizontal thrust due to the load W, or

But the axial fluid stress in each pipe under a head of 320 feet will be 63 tons (as already calculated in Art. 9), so that a fluid stress of 126 tons will act in direct opposition to the arch thrust at the crown; and it follows that the compressive stress in the tubes themselves can only amount to—

QCP 150 — 126 = 24 tons

or 12 tons in each tube.

=

At the springing of the parabolic rib, we should have—

and here the arch thrust would be increased to C = 150 x 1.077 161.5 tons; and the compressive stress in the metal tube, or the thrust delivered upon the annular bedplate, would be about

To be a little more accurate, we should perhaps say that if the head is exactly 320 feet at C it will be 332 feet at B, and the axial fluid stress being thus increased to 130-7 tons, the thrust of the tubes would be—

or 15.4 tons in each tube.

QB

=

161.5 130.7 = 30.8 tons

EXAMPLE 2.—If, in the same bridge, we suppose the pipes to be charged under a hydrostatic head of 381 feet (measured at the crown), it is easy to calculate that the fluid stress at the crown would be P = 150 tons, very nearly ; and the tubes would then be relieved of all longitudinal stress at the crown; while at the springing we should have a hydrostatic head of 393 feet, and as P would then amount to 154-8 tons, there would be at the springing a slight compressive stress

Q = 161.5 154.8 67 tons

Or, again, if we suppose the hydrostatic head at B to be 402 feet, the axial fluid stress at that point would be nearly 161.5 tons. In this case there would be no compressive stress in the tubes at B, and they would exert no thrust upon the abutment. The thrust of the fluid arch would, of course, be delivered upon the fluid in the buried main AB, the horizontal component of that thrust being exactly met and balanced by the fluid axial stress in the horizontal main, while its W vertical component would be just and would be carried as a vertical force or 2 load upon the abutment wall. But, at the crown of the arch, we should now find a slight tensile stress; for although the head would be reduced at that point to 390 feet, yet the fluid axial stress would amount to 156.7 tons; and as the algebraical sum of the stresses in tube and in fluid must be 150 tons, the tubes must be under a tensile stress of 6.7 tons.

Lastly, if the hydrostatic head were greater than 402 feet, it is obvious that the tubes would suffer a longitudinal tensile stress throughout the length BCD, the stress being slightly greater at the crown than at the springing.

EXAMPLE 3.-It may perhaps be worth while to work out the calculation for a segmental rib, which would be the more usual, and generally the better form for the arch. Thus, without altering the dimensions given in the last example, we may easily find (by similar triangles) that with a span of 120 feet, and a rise of 12 feet, the segmental rib would have a radius of 156 feet, that 156-12 12 0 13 0 5 and that sin 2 5' 2 13

cot 2

=

60

5'

that cosec

=

=

We may again assume a total load W of 120 tons, but if the line of pressure is to be a segmental curve passing through B, C, and D, the load upon the half span must have its centre of gravity in the same vertical line with the point G, which is the intersection of the tangents BG and CG.

With this distribution of load the horizontal thrust at the crown will be

H = 120 × 12 = 144 tons

while at the springing the inclined thrust will be—

When the pipe is charged under any given hydrostatic head, the axial fluid stress is easily calculated, and the compressive or tensile stress in the arched tube may readily be found by simple subtraction as before.

It may be noted, however, that under a head of about 396 feet (measured at B) the axial fluid stress would be 156 tons, and the tube would exert no thrust at the springing. This corresponds with the calculation of the displacing force given in Art. 10, where it was found that the total lifting force, or the vertical resultant of all the unbalanced fluid pressures distributed round the arc BCD, would be expressed by 2P sin 2; for in this case P 156 tons, and the lifting force would therefore be

=

We have not, hitherto, taken into account any bending stresses which may be due either to change of temperature, or to the initial adjustment of the rib