On the contrary, it has already been shown, in Art. 11, that the axial fluid stress P will be generally augmented in such cases by a positive or compressive stress Q due to the hydraulic pressure upon the annular end of the pipe; and the displacing force at any adjacent bend will be proportional to the augmented axial stress P+Q, as in the example worked out in that article.

Remembering that the magnitude of the displacing forces is always governed by the effective axial stress, which is the algebraical sum of the stresses P and Q, it will be worth while to notice here that, although the stress Q is always a positive quantity at the end of the sliding neck, yet it is quite possible to construct an expansion joint in such a form that the compressive stress shall be confined to a short length of neck, while the remainder of the pipe shall be free from any longitudinal stress, or shall be subjected to longitudinal tension. Thus, for example, if we make the joint in the form which is diagrammatically sketched in Fig. 20, so that the external diameter d of the sliding neck is just equal to the internal diameter d of the pipes AB and CD, it will easily be seen that neither of these pipes will suffer any longitudinal stress, because the opposite hydraulic pressures upon the two annular ends of the length BC will exactly balance each other.

-A

B

FIG. 20

-D-· E

And if beyond D we make the internal diameter of the pipe DE greater than d1, the hydraulic pressure upon the annular surface at D must be borne by the pipe DE in tension.

In short, the expansion joint fixes, in the most definite manner, the value of the gross effective axial stress which must be transmitted through the pipes in its immediate neighbourhood. For if we regard the pipe BCDE, together with its fluid contents, as constituting a body whose circular end at B is subjected to the hydraulic pressure, we shall see at once that this external force must be transmitted through the pipes CD

and DE, and in both cases must determine the value of the effective stress P+Q.

If d is the diameter of the sliding neck, or the bore of the

gland, the external end pressure will be P1 = Pd ̧27; and if d

1

denotes the diameter (as great or as small as we please) of any length, BC, CD, or DE, we shall in all cases have —

and as P = pd2, the stress Q is determined by the equation—

4'

and may be either positive or negative, according as d is less than or greater than d. This equation will apply to the pipe right and left of the expansion joint, so far as the pipe is subjected to no other external forces, acting in the longitudinal direction, and would apply to the whole length of the bend AB in Fig. 19. In such a vertical bend, therefore, or in any analogous case, the lifting force, or displacing force, will really depend upon the diameter d, of the sliding neck; and as the force cannot be resisted by longitudinal tension in the pipe, we have now to consider whether it can be resisted in any other way, or whether it is absolutely necessary to load the pipe with a weight of nearly 4 tons per lineal foot, as calculated in Art. 11.

Art. 14. The Pipe-bend acting as a Girder.-If we suppose the segmental bend sketched in Fig. 19 to be fitted at each end with a cylindrical neck which slides without friction in the gland of the stuffing-box, and that the stuffing-boxes are themselves so fixed to a firm substructure as to resist any displacement, the bend may be regarded as a girder supported at each end by the normal reaction of its bearing in the gland. Between these supports the pipe will be subjected to a uniformly distributed pressure acting normally to the axis, and due to the unbalanced fluid pressure. The girder may be taken as supported, and not fixed at the ends, because the joint could hardly possess fixity of direction, and on this assumption the bending moments can be readily determined.

Finding first (by the usual principles) the external normal

forces or reactions at A and B, we have for the value of each

in which (P +Q) represents the gross axial stress. The maximum bending moment will naturally take effect at the middle of the span, where its value will be expressed by

To construct a complete diagram of moments for this case, let ADB, in Fig. 21, be the neutral axis or centre line of the segmental bend, subtending the angle 0, and let O be its centre

0

2 1),

of curvature. At right angles to OA and OB, draw the tangents AE and BE, and join OE by a straight line cutting the arc at the crown D. Of course, DE will be the length R(sec and the axial stress P+Q, multiplied by DE, will give us the bending moment at D. Then, through the three points, A, E, and B, draw the circular arc AEB, the circle passing also through 0, and between the two arcs draw any number of polar ordinates, such as gk normal to the arc ADB. Then at any point, g, the bending moment, will be proportional to gk, and equal to that length multiplied by P+Q.

The diagram may be drawn in another form as follows:

Considering the forces P+Q and S as external forces, their resultant H will have the direction of the chord-line AB, and

0
2

the magnitude H = (P+Q) sec It follows that if we simply

draw the chord AB, the moment at any point g in the arc will be proportional to the perpendicular gm, and will be equal to

0

that ordinate multiplied by H, or by (P +Q) sec 2'

EXAMPLE. In the particular case illustrated in Art. 11, the angle e subtended by the bend ADB was 45°, and sec would therefore be equal to

2

1-08248; so that with the given radius of 20 feet, the quantity R(sec-1) would be

20 × 0.08248 = 1.6476 feet

Then, if the external diameter d1 of the sliding neck is 39 inches, as assumed in that example, the gross axial stress P + Q, under a head of 320 feet, will be 73.94 tons, and the bending moment at the centre of the arc will be—

MD 73.94 x 1·6476 = 125.82 foot-tons

=

Art. 15. The Pipe-bend acting as a Cantilever.-Let the horizontal pipe AB, in Fig. 22, be connected with the vertical pipe DE by a bend BCD turning through an arc of 90°, and let the pipe be fitted with an expansion joint at B. This case, or something very much like it, is of frequent occurrence in the steam pipes of large marine engines, the pipe DE being very often attached to the side of the valve chest by a flanged joint at E.

We may regard the bent pipe BCDE with its fluid contents as forming together one body, which as a whole is subjected to the external forces H and V. The former of these is of course the gross axial stress P+Q, or pd2, taking effect at the

4'

sliding joint, whose bore is d, while the vertical force V will really be determined by the area of the orifice in the side of the valve chest. This will generally be a circular orifice, whose diameter d is the same as that of the pipe, and then, of course, V = P = pd2TM

4

The force H must evidently be resisted by the bent pipe acting as a cantilever, fixed at the base E; and at the same time the flange joint at E will in all probability have to resist

the force V in direct tension, because the straight pipe AB will hardly afford any material resistance at B to the action of this force.

At the base EE, of the cantilever we shall have, therefore : (1) a bending moment M. = HX, in which X. denotes the

height measured from the base E to the horizontal line of action. of the force H, or to the centre of the expansion joint; (2) a shearing force parallel to H and having the same magnitude; (3) a direct tensile stress due to the vertical force V, which may be taken as uniformly distributed over the horizontal crosssection of the pipe.

The bending moment M, will be resisted by vertical longitudinal stresses, which at E, and E2 will have to be algebraically added to the vertical stress due to the force V.

Instead of referring to the axial fluid stress, or pressure of fluid upon fluid, we might arrive at the same results by considering the pressure of the fluid on the internal surfaces of the bend. This unbalanced pressure would produce, of course, a displacing force acting normally to the curve of the bend, and uniformly distributed around the arc; but the horizontal and the vertical components of the unbalanced pressures are easily measured by the method already described, and the projections