the deflection & being proportional to the load n, so long as the stress is kept within the elastic limit. But, putting this relationship in another form, let us say that when the beam bends to a certain deflection 8, it will carry a certain distributed load-no more and no less-and evidently the load that it will carry with that particular deflection is simply—

Then taking this quantity n, as representing, for our present purposes, what may be called the distributed reactionary force of the elastic beam, we have only to compare its magnitude with the magnitude of the displacing force n, against which it is opposed, and which is also a uniformly distributed force. This comparison will show us at once whether the pipe will buckle or whether it will straighten itself.

Putting the two formula for n and for n, side by side, we see that the displacing force n will be greater than the opposing

Apparently, then, the question at issue does not depend upon the value that may be arbitrarily selected for the deflection & in making this comparison. As the deflection increases the elastic reaction of the pipe increases uniformly, but so also does the displacing force, and the question depends only on the relative rates at which these forces increase. If, then, we find that with a deflection of one inch the displacing force would overpower the elastic resistance, we may conclude that the same thing would be true if the deflection were only a quarter of an inch or only the thousandth part of an inch, and we need not go any further to discover a reason why the pipe should begin to curve from a condition of initial straightness. The critical value of the axial fluid stress P, that will determine the question of buckling or not buckling, may be denoted by P., and we may write EI If P is greater than P., the pipe will buckle; but

P =

с

48

5 12.

if P is less than Pe, the pipe will not buckle; and if we forcibly drag it out of the straight, it will spring back again as soon as we let it

go.

Art. 22. Exact Determination of the Critical or Buckling Pressure. The reasoning put forward in the last article is not quite correct, for we assumed that the unknown curve of the beam would be a curve of segmental form. We had no right to assume, as we have done, that the radius of curvature would be uniform throughout, and if it is not uniform the displacing force will not be uniformly distributed.

Without making any tentative guesses as to the true form of the curve, we may set in order the known relationships that are concerned in this question, and its solution will then become manifest.

1. With any given distribution of load, represented, let us say, by the ordinates of a load diagram, we can construct the corresponding curve of moments, and it is certain that this curve can have only one form for the given distribution of load.

2. From this diagram of moments, we can, by a similar process and with equal certainty, construct the deflection curve for a homogeneous beam of uniform section.

3. In the deflection curve thus constructed we can find the slope at any given point, and also the radius of curvature R, or the quantity which is often called the curvature" of the

1 R'

beam; and it is also known that this quantity points proportional to the bending moment.

1

4. It follows, of course, that if this varying quantity is Ꭱ

represented by the ordinates of a diagram, that diagram must be bounded by a curve similar to the curve of moments; and it also follows that the curve must be similar to the curve of the load diagram from which the whole series of diagrams has to be constructed, because the load or displacing force is proportional to Р and is in fact n =

1

R'

R

There is nothing to fix the form of any one of the curves à priori, but this last condition furnishes the required solution, and fixes the form of them all. The actual distribution of the displacing force depends upon the curve of the pipe, while it governs the form of the diagram of moments and the form of the deflection curve. But the deflection curve is the curve of the pipe.

in Fig. 24 represent by its ordinates the unknown Le load or displacing force n; while B shall repre-ing forces, and C the bending moments resulting tribution of the load. Then, to follow out the ations consequent upon these bending stresses, let its ordinates the slope of the bent beam, and E deflection. It is known that B can always be A by integration, or from C by differentiation; en shown elsewhere1 by the author that just as

usually constructed from A, so D and E can be om C. Indeed, it will easily be seen that the varying values are shown in diagram D, is simply l of the deflection ordinate y in diagram E; and

1

tity will be proportional to the differential

R

eflection is small; while a well-known theorem at the same quantity is proportional also to the ent M represented by the ordinates of diagram C. relationships are generally to be traced in any

on

66

Continuous Girders," in the Transactions of the Inst. C.E., 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 to the curve C, because n =

Р

R®

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

1 R'

either the deflection ordinate y, the curvature the bending

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, in which d1 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.

с

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