for a submerged cylinder with a vertical axis, we should have

Art. 54. Effect of Water-ballast in Compartments.-The pontoons employed in the erection of girder bridges have sometimes been built in compartments, with the intention of admitting a certain amount of water-ballast, so as to bring the pontoon to a deep draught or to raise it again through a certain height by pumping out the whole or a part of the water.

When the compartments are partly filled with water, the stability of the pontoon will be somewhat diminished, and the b2 metacentric height BM, will be less than the quantity ,which 3D'

was found in Art. 48 for a pontoon of the same width and displacement. But the effect of the water-ballast in reducing the metacentric height will be great or small according to the width of the individual tanks, or the number of compartments contained in the whole width of the pontoon.

To illustrate the question, we will suppose the pontoon to be divided, transversely, into a certain number N of equal compartments, the tanks extending throughout the whole length of the pontoon, and all having the same width between bulkheads, as sketched in the cross-section, Fig. 45, in which N = 4.

If all the tanks are filled to the same level, and if D denotes now the vertical height of the exterior water-line above the

water in the tanks, the net displacement of the pontoon (apart from water-load) will be equal to the length multiplied by 26D.

When the vessel rolls so as to bring the exterior water-mark to the inclined position SL, the water surface in each tank will

be parallel to SL1; and the figure of the area of displacement will be changed by the transference of a parallelogram from No. 1 tank to No. 4; and of another parallelogram from No. 2 to No. 3.

Taking in succession the computed area of each parallelogram multiplied by the distance through which it is transferred (horizontally and vertically), and dividing the sum by the displacement area 26D, we have the co-ordinates of the curve of buoyancy,

So long as the water plays up and down upon vertical walls and bulkheads without reaching the exterior corners of the pontoon or the interior corners of the tanks, the curve of buoyancy will be parabolic; and for any value of 8 within these limits the corresponding height Bm will be—

For the upright position, where y = 0, we have the meta

It is then easy to see the way in which the metacentric height decreases, as the number of compartments is decreased, thus

b2

The quantity

3D

is, of course, the metacentric height for the

same pontoon, carrying the same external load, and having the same net displacement, but with the compartments pumped dry.

b2

The negative quantity (12)

3D

negative metacentric height for each individual tank whose

b,2

half-breath is b, = and might be written in which case the

N'

3D

If all the tanks are partly filled, but some higher than others, the net displacement for the same external load will still remain the same, and we may still use the above expressions if D is understood to denote the mean depth of the contained water below the external water-level.

In this article we have considered only the stability of the pontoon in the transverse plane, and have assumed the tanks to extend throughout the whole length. But the subdivision of the pontoon longitudinally by transverse bulkheads, and the stability in the longitudinal plane, may be treated in the same manner.

CHAPTER IX.

BENDING STRESSES IN FLOATING PONTOONS.

Art. 55. Distribution of the Vertical Forces.-The bending stress that takes effect in any floating vessel will, of course, depend wholly upon the distribution of the load and the distribution of the supporting buoyant pressure. In shipbuilding it is mainly the longitudinal stresses that have to be considered; and the principal bending moments are the hogging moment due to the inadequacy of the buoyant support at the ends of the vessel, and the contrary sagging moment which sometimes takes place when the centre of the ship's length lies in the trough of the sea.

When the ship floats in calm water, it is easy to see that, if the weight of ship and cargo per lineal foot were everywhere proportional to the area of the immersed section, there would be no longitudinal bending stress whatever. This equable distribution of the opposing forces is not quite attained in practice, and in calm water the ship is generally subjected to a hogging stress owing to the fineness of the water-lines at the ends. But in pontoons that are built for the support of heavy concentrated loads, the distribution of the opposing forces is often such as to produce a much greater bending stress (in proportion to the total load), and the bending may take effect not only in the longitudinal but in the transverse plane.

Sometimes the problem would be very unmanageable were it not for the fact that, when the water is calm, we know, almost exactly, how the supporting buoyant pressure must be distributed.

For, whatever may be the form of a pontoon, we have already seen that the vertical buoyant pressure per square inch of horizontal section is everywhere measured by the vertical depth of the floor below the water-line, so that a section of the vessel, with the water-line drawn across it as a datum, gives us at once a diagram of the distributed buoyant force.

We need only consider the case of a flat-bottomed rectangular pontoon, and it is obvious that if the load is symmetrically trimmed so as to bring the vessel upright, i.e. if the floor is a horizontal plane, the buoyant pressure will be a uniformly distributed force as in Fig. 36; and if the centre of gravity does not coincide with the centre-line of the pontoon, the diagram will have the very simple form sketched in Fig. 36a, and the actual distribution can easily be found.

The calculation will, therefore, be very simple when we know the disposition of the load; but it is here that a difficulty sometimes arises, because the actual pressure of the external load at any given point is often affected by the elastic deflection of the pontoon itself under the bending moment. Another complexity arises when the deflection of the pontoon is great enough (as it sometimes may be) to sensibly affect the figure of the area of displacement, and therefore the distribution of the buoyant

pressure.

We may assume that the weight of the pontoon itself will generally have a nearly constant value per square foot of floor; and as it will be borne by a uniformly distributed water pressure, these opposite forces may be eliminated from both sides of the account, as they will produce no bending moment under any circumstances. What we have to consider, therefore, is the external load, opposed by a buoyant force of equal magnitude, and the moments produced by these opposite forces.

Art. 56. General Expressions for the Bending Moment and the Vertical Shearing Force.-The question we have in hand involves only a somewhat special application of well-known principles. We may consider the pontoon as a girder of uniform depth, subjected to the action of certain vertical forces which we may treat as positive or negative according to the upward or downward direction in which they act, taking the buoyant pressures as positive and the loads as negative. Then, at any vertical section taken through the girder, the shearing force S will be simply the algebraical sum of all those forces which act anywhere to the right of the section, while the bending moment M will be the algebraical sum of the moments of those forces.

Thus if P denotes any vertical pressure, upward or downward, whose line of action or centre of action lies to the right of the section by a horizontal distance X, its moment at the section