Putting the several curves one below the other, in the order of cause and effect as we usually conceive that relationship, let the diagram A in Fig. 24 represent by its ordinates the unknown. intensity of the load or displacing force n; while B shall represent the shearing forces, and C the bending moments resulting from this distribution of the load. Then, to follow out the elastic deformations consequent upon these bending stresses, let D represent by its ordinates the slope of the bent beam, and E the curve of deflection. It is known that B can always be obtained from A by integration, or from C by differentiation; and it has been shown elsewhere1 by the author that just as

B and C are usually constructed from 4, so D and E can be constructed from C. Indeed, it will easily be seen that the slope s, whose varying values are shown in diagram D, is simply the differential of the deflection ordinate y in diagram E; and that the quantity Ꭱ will be proportional to the differential

1

of s, if the deflection is small; while a well-known theorem informs us that the same quantity is proportional also to the bending moment M represented by the ordinates of diagram C.

All these relationships are generally to be traced in any

1 Vide Paper on "Continuous Girders," in the Transactions of the Inst. C.E., vol. lxxiv.; also “Bridge Construction," chap. viii. p. 113.

E

elastic beam of uniform stiffness; but, in the case we are now considering, the 'special condition which determines the form of all the curves, is that the curve A must of necessity be similar

Hence, it follows that, in the deflection curve E, in which y is some unknown function of x, that function must be such that its second and its fourth differential shall have the same form, and this condition is satisfied by the curve of sines.

Thus, in diagram E, let & represent a small deflection of the pipe at the centre of its length, and for all values of x between zero and π (or 180°) let the corresponding ordinates be y 8. sin x; then the curve so constructed may be taken to represent either of the curves A, C, or E, so that, by using a suitable vertical scale in each case, it will measure for us

=

either the deflection ordinate y, the curvature the bending

1 R'

moment M, or the intensity of the displacing force n. In the same way, it will be seen that the curve of cosines will give us the vertical shearing force in B, or the slope of the pipe in D.

It is now obvious that the distribution of the displacing force is not uniform, as we assumed it to be in the last article ; and that the curve of the bent pipe is not exactly a flat segment. When the deflection is small there is but little difference between the two curves; but taking now the curve of sines, which satisfies all the mechanical conditions, we shall have to modify slightly the results that were obtained by using the other; and we find for the critical value of the axial fluid stress

This expression will be recognized at once1 as being identical with a well-known formula for the "breaking load" or "buckling load" of a slender column of solid material; and, so far as the buckling tendency is concerned, it is evident that the axial fluid stress P in a straight pipe is equivalent to so much endthrust, although the fluid pressure takes effect, not upon the end of the pipe, but upon its internal surface.

1 Vide Paper on the "Practical Strength of Columns," Trans. Inst. C.E., vol. lxxxvi.; also "Bridge Construction," chap. x. p. 153.

The critical fluid pressure per square inch may be denoted by P, and if the metal of the pipe is absolutely free from longitudinal compressive stress, its value will be—

Art. 23. Effective Length for Flexure and Effective Axial Thrust.—The length l, in the formulæ above given, is, of course, the length of the span AC between supports, or the length of pipe that is free to deflect in any one direction, just as in the column formula it represents the length of a column with rounded or hinged ends. When the pipe is carried between two expansion joints, as in Fig. 23, we can hardly attribute to those joints any rigid fixity of direction; and the length would, therefore, be measured from joint to joint.

But in other cases, as perhaps in the steam-pipe illustrated in Fig. 22, the pipe may be rigidly fixed at one end A, while the other is carried in an expansion joint, which is itself fixed in position, although affording no fixity of direction; and in such a case the effective length l for flexure might probably be estimated at two-thirds to three-quarters of the total length.

In this, as in so many other respects, the pipe may be considered simply as a column subjected to a certain load, or exterior end-thrust.

The effective value of this axial thrust must next be considered, and a little reflection will show that, in all cases, it must be taken to be the algebraical sum of the longitudinal stresses P and Q in the fluid and in the solid portions of the column, as already indicated in Arts. 11 and 12.

The axial fluid stress P is equivalent to so much end-load, and, if the metal of the pipe is also subjected to the end-thrust Q, due to fluid pressure upon its annular end, the total axial thrust will be P + Q = pd,27, in which d, is the diameter of the bore of the gland.

To avoid buckling in this case, the effective thrust P+Q must, of course, be less than the critical value P. as determined in Art. 22.

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The external load being thus determined by the bore d1, that load will, of course, be transmitted through the whole length ABC, and will represent the sum of the longitudinal stresses

4

P and Q at every cross-section between A and C in Fig. 23. If we make the intervening length of pipe with an internal diameter d equal to the bore of the gland, the fluid axial stress at any intervening cross-section will be equal to the total external thrust pd; and, therefore, the metal pipe will be relieved of all longitudinal compressive stress throughout the free length intervening between the expansion joints. But notwithstanding its entire freedom from any longitudinal compression, the pipe will be buckled by fluid pressure alone if the effective axial stress P is greater than the critical value P

It would, of course, be possible to go a step further, and by making the intervening length of pipe with an internal diameter greater than the bore d1, the pipe would be thrown into longitudinal tension throughout its free length; but would, nevertheless, be buckled if the algebraical sum P+Q, or the endthrust pd (which is the same thing), were greater than the quantity P.

4

In short, it is evident that the buckling of such a pipe would not depend upon the transmission of compressive stress through the metal of the pipe, but upon the compressive stress transmitted through the column as a whole, including the fluid and the solid portions of its cross-section; and obviously, if the two ends of a straight pipe are closed by covers, so that the pressure P upon each cover is transmitted in direct tension Q-P throughout the pipe, the algebraical sum P + Q will be zero, and there will be no buckling tendency, however flexible the pipe may be.

Here, however, it may be as well to remark that if such a tubular column with closed ends were subjected to an external load W, that load would at once become the measure of the buckling tendency, and also of the effective axial thrust P + Q. In such a case the longitudinal stress Q in the metal of the pipe might be either positive or negative or zero, but the buckling tendency would in either case be that due to the load W.

Whatever the external conditions may be, the fluid stress P will be in operation in all cases, but the longitudinal tensile stress Q, in the metal of the pipe will depend upon those conditions very largely. As already mentioned, it will often be difficult to ascertain the longitudinal stress Q, and it may then be

difficult to say whether there is any buckling tendency; but in other cases the conditions are more clearly defined, and when an expansion joint is introduced into the pipe the conditions are quite definite, and there can be no doubt that the buckling tendency will then have the values above found.

It would hardly be justifiable, however, to employ a "column formula" of the usual kind in determining the requisite strength of such pipes, for the pipe differs from the column in one important respect. The column whose breaking weight is given by the "column formula," or by recorded experiments, is subjected to direct compressive stress as well as to the liability of buckling, while the pipe suffers no direct compressive stress, or, at all events, none worth speaking of.

If, then, the section of the pipe is such that the quantity EIT is greater than the axial stress P+Q by a reasonable margin of safety, we may reasonably conclude that the pipe will not begin to bend; and if it does not begin to bend, the buckling tendency will exist only as a purely inoperative tendency, producing no stress and no actual effect of any kind upon the pipe.