Page:The American Cyclopædia (1879) Volume IX.djvu/127

HYDROMECHANICS 119 Ibs., or a little less than one ton. In the con- struction of walls for resisting only the hydro- static pressure of water, as that pressure is in proportion to the depth, the strength of the wall should be in the same proportion. If strength were not given to the lower layers by superincumbent pressure, the inclination of the slope should he 45 ; but in consequence of this pressure it may be less, varying with the mate- rials and their manner of being put together. In the construction of dams or barrages the varying circumstances of cases allow of the dis- play of a good deal of engineering skill. A barrage suitable for restraining a body of water which is never strongly moved in a lateral di- rection against it, as at the outlet of a canal or a reservoir fed by an insignificant stream, would not be adapted to a mountain torrent, where the surface of the reservoir can scarcely ever be large enough to prevent, by the inertia offered by a large mass of water, the walls from being subjected to a strong lateral force from the action of the current. Under such circum- stances it is usual to give a curved surface to the facings, in a vertical as well as in a hori- zontal direction ; the curves in both directions being calculated from the following elements : 1, the ascertained hydrostatic pressure; 2, the nature of the materials, such as the weight of stone and tenacity of the hydraulic cement used ; and 3, an estimate of the maximum force of flowing water which may at any time be brought against the structure during a freshet. This force, it will readily be seen, will have a different direction and a differ- ent point of application in different cases, depending upon the depth and extent of the reservoir. The top of the dam is therefore given a greater horizontal section than would be called for if hydrostatic pressure alone had to be opposed. The hydrostatic pressure at any point against the surface of a contain- ing vessel is the resultant of all the forces collected at that point, and is therefore at right angles to that surface. In a cylindrical or spherical vessel these resultants are in the direction of the radii, and in the sphere vary in direction at every point. Centre of Pres- sure. The centre of pressure is that point in a surface about which all the resultant pres- sures are balanced. The cases are innumer- able, and often require elaborate mathemati- r,G.s.-Centre of Pressure, cal investigation. The simplest case and its general application only will be considered here, viz., that of the centre of pressure against a side of a rectangular vessel. Let any base in the triangle ABC, fig. 8, rep- resent the pressure at B ; then will D E rep- resent the pressure at E, and all lines paral- lel to it will represent the pressures at corre- sponding heights. The finding of the centre of pressure now consists in finding the centre of gravity of the triangle ABC, which will be at H, the intersection of the bisecting lines E and D B, and at one third the height of the side A B ; consequently the centre of hy- drostatic pressure against the rectangular side A B is at G, one third the distance from the bottom to the surface of the liquid. The ave- rage intensity of pres- sure against A B being atE, one half the depth FIG. 9. Principle of Archimedes. of A B, therefore the total pressure on the rectangular side A B will be the same as if it formed the bottom of the vessel and was pressed upon by a column of water of half the depth of A B. In general, the total pressure on any surface, plain or curved, is equal to the weight of a liquid col- umn whose base is equal to that surface, and whose height is the distance of the centre of gravity of the surface from the surface of the liquid. Principle of Archimedes. A solid im- mersed in liquid loses an amount of weight equal to that of the liquid it displaces. This is called the principle of Archimedes, and is demonstrated as follows : Let a J, fig. 9, be a solid immersed in a liquid. The vertical sec- tion c d will be pressed downward by a force equal to the weight of the column of water e c, and it will be pressed upward by a force equal to that exerted by a column of water equal to e d ; therefore the upward or buoyant pressure exceeds the downward pressure by the weight of a column of water equal to the section c d. Now, this section also exerts a downward pressure ; and if the body is denser than the liquid, the downward pressure will be greater than the excess of the upward pres- sure of the liquid, and the body will sink if not supported ; but if the body is less dense than the liquid, the downward pressure of the col- umn e d will be less than the upward pressure exerted against it, and the body will float. This principle may be experimen- tally demonstrated by the hydrostatic balance, fig. 10. From a balance, 5, is sus- pended a cylindri- cal vessel, a, from which again is sus- FIG. 10. Experimental Vcrifl- ponded a solid cylin- cation of the principle of der, c,which isof such Archimedes. bulk and dimensions as just to fill the vessel a when introduced. The whole system is first balanced by weights at the other end of the beam, and then e is immersed in water. The equilibrium will be destroyed, and that the body c loses a portion