of jointed pipe is subjected to any bending forces, the longitudinal stresses or fibre stresses due to the bending moment at any joint must be transmitted through the joint itself.

Thus, if we take first the case of a line of flanged pipes united by ordinary flange bolts, and neglect in the first instance any question of initial stress, it appears certainly that while the compressive stresses on one side of the neutral axis are transmitted by the pressure of the flanges, the tensile stresses on the other side must be transmitted through the bolts, and that the stress in each bolt will be nearly proportional to its distance from the neutral axis.

The bolts will, in most cases, be spaced at uniform intervals around the "pitch-circle," whose diameter we may denote by D1, and, on the assumption above mentioned, we may take it that the transverse strength of the bolts will be nearly the same as though they were replaced by a continuous circle of metal, or thin pipe, having the diameter D, and a sectional area equal to the combined areas of all the bolts in the circle. The maximum stress per square inch in the outermost bolt would then be given by the same formula f1 = in which A now

4M

AD,'

represents the combined sectional area of all the bolts, and D1 the diameter of the pitch-circle.

EXAMPLE. Suppose that the steam-pipe already illustrated in Fig. 22 is united to the valve-chest at the base of the cantilever by a flange joint, as sketched in Fig. 226, the flange bolts, 12 in number, being uniformly spaced around the pitch-circle, which is 21 inches diameter. Then, if each bolt has a diameter of 1 inch inside the thread, its net sectional area will be about 1-23 square inches, and the combined section of the 12 bolts will have an area of 14.76 square inches.

The bending moment of 398.7 inch-tons, above calculated, would then produce in the outermost bolt a tensile stress which may be approximately estimated at

As the flange joint will also have to resist the direct stress due to the force V of 7.95 tons, the additional stress of 7.95

=0·50 ton per square inch would bring up the total 14.76 stress to 5.64 tons per square inch in the outermost bolt, without making any allowance for the initial stress that may be imposed upon the bolts. In the practical process of "making" the

joint, the flange bolts will no doubt be so firmly screwed up as to throw them all into considerable tension, and to bring the flanges together with a certain compressive force. Whatever compressive force may be requisite for this purpose in a joint otherwise unstrained, will probably also be necessary when the bolts are utilized to transmit such structural stresses as those above considered; and it would therefore appear that the initial stress (required for tightness) should be added to the structural stresses.

If we go on to consider next the case of a riveted pipe of steel plate in which the lengths are united, as in ordinary boiler work, by a lap joint, single or double riveted, it will be seen that in all such cases the longitudinal stresses due to any bending moment, or to any direct longitudinal forces, will have to be resisted by the shearing strength of the rivets. And we may again assume that the shearing stress borne by the individual rivets around the circumferential joint will be proportional to the direct stress, at each point in the circle, taking effect in the metal of the pipe itself the stress being zero at the neutral axis and reaching a maximum at the two edges of the circle farthest from the neutral axis.

Thus, for any bending moment M we may find the maximum shearing stress by the formula already employed-i.e. if A denotes the whole rivet section around the circumferential joint, and D, the diameter at the joint, the maximum shearing stress will be

It is unnecessary to remark that in such riveted joints the net plate section would have to be proportioned to the longitudinal stress as well as the rivet section.

CHAPTER IV.

THE BUCKLING TENDENCY IN STRAIGHT PIPES UNDER

UNIFORM FLUID PRESSURE.

Art. 19. Axial Stress in the Straight Fluid Column. In the case of a long straight pipe supported only at the ends, or supported between those points in such a way as to permit of free lateral flexure, it will sometimes be necessary to consider the pipe as a fluid column liable to buckling under the axial fluid stress.

The conditions under which it would be possible for a pressure-pipe to be actually buckled in this way are possibly of somewhat rare occurrence, but the tendency will certainly be present under conditions that are sometimes met with in actual engineering practice, as, for example, in the straight steam-pipe AB illustrated in Art. 15.

The buckling of a slender column of solid elastic material, under the action of an end-thrust, is a familiar fact, depending upon known mechanical principles, and the same principles must have an application to the cases which we now have to consider. It might, indeed, be sufficient to treat the question by direct reference to the longitudinal compressive forces or stresses; but in this problem, as in previous ones, it will probably be better to trace the buckling action of the fluid stress to its distinct proximate cause, viz. the unbalanced pressure of the fluid against the internal surface of the pipe.

In some cases the metal pipe may itself be subjected to a direct compressive stress Q due to the fluid pressure upon the annular end of the pipe, as in previous examples, and it is easy to see that this end-thrust would be accompanied by a certain buckling tendency, just as in any other hollow cylindrical column; but it is not quite so plainly apparent that the fluid stress P will act in a similar manner, and that the pipe may be buckled without undergoing any compressive longitudinal stress, and solely by the action of internal pressure.

Art. 20. Displacing Forces due to Elastic Deflection. To illustrate the manner in which a pipe, initially straight, may be subjected to the action of displacing forces, let us suppose the pipe ABC in Fig. 23 to be carried at each end in expansion joints, which are themselves firmly fixed to a substructure, while between them the pipe is free to deflect in any transverse direction, the ends being regarded as merely supported by the glands, and not rigidly fixed in direction.

We may further suppose that the weight of the charged pipe is counterbalanced, and then it must be admitted that the internal fluid pressures in the straight cylindrical pipe will exactly balance each other, and can produce no displacing force whatever so long as the pipe is straight. But a precisely similar statement might be made in regard to the analogous case of the solid column, in which the load can produce no bending moment so long as the column is straight; and when the fact is acknowledged, in either case, the real question at issue is not disposed of. In both cases there may be no recognisable force by which we can account for the beginnings of deflection; but in both cases the requisite force is forthcoming as soon as deflection commences. In the solid column, the deflection gives rise at once to a certain bending moment; and in the pipe the deflection gives rise at once to a certain displacing force consequent upon the curvature of the pipe.

To ascertain the stability of the pipe, therefore, we shall have to pursue the same inquiry as in the case of a slender column— i.e. assuming hypothetically a certain slight curvature, we must calculate the displacing force that would arise simultaneously with that curvature, and see whether it is greater or less than the resisting forces that would be called into play at the same instant of time.

FIG. 23

B1
B

Assuming, then, that the pipe ABC is, or may be, bent elastically out of the straight line to a slight certain curvature AB,C, or slight deflection d=BB1, it has already been found in Art. 9 that if R is the radius of curvature, the displacing force will be a

distributed radial pressure whose value per foot lineal is n=

Р

R

and as will be taken to be very small in comparison with the span l, the pressure may be called a lateral or transverse pressure, acting upon the pipe just as a distributed load acts upon a beam that is supported at each end. This brings the problem at once within sight of an easy approximate solution.

Art. 21. The Conditions of Equilibrium broadly considered.Without stopping to inquire how or by what means the straight pipe could ever become bent by internal pressure, let us assume that it has acquired a slight curvature (within the elastic limit), and let us ask the more important question whether the deflection will now increase, or whether the pipe will straighten itself by its own elastic reaction or resilience. Evidently the question will depend upon the relative magnitude of the forces which are here opposed to each other-the displacing force, tending to push on the deflection, and the elastic resilience of the pipe, tending to resist any further deflection, and to restore the pipe, if it can, to the straight form.

We may take first the displacing force, and to make this calculation as simple as possible, let us assume, in the first instance, that the curve ABC in Fig. 23 is a very flat segment of a circle. Then the radius of that circle will be very 72 nearly R= and the displacing force will be equivalent to a 88' uniformly distributed load (or lateral pressure) whose value per foot lineal will be n=

Р 88
=P nearly.

Ꭱ 12

With this uniform load upon a beam supported at each end, the bending moment at the centre of the span can at once be calculated by the well-known rule, and we have for that bending moment MB

=

nl2

8

=PS.

We may next turn to the elastic reaction or resilience of the pipe, which is the only force concerned in resisting the displacing force. The elastic pipe will be a beam of uniform section, and if I denotes the moment of inertia of its section, we know very well that it will take a certain deflection-no more and no less -under a uniformly distributed load of any given intensity n1; and by the well-known rule that deflection will be