Equating the moment of resistance (2) to the overturning moment (1), we have
| and | ||
| (3) |
That, is to say, for such a monolith to be on the point of overturning under the horizontal pressure due to the full depth of water, its base must be equal to that depth divided by the square root of twice the density of the monolith. For a density of 2⋅5 the base would therefore be 44⋅7% of the height.
We have now to consider what are the necessary factors of safety, and the modes of their application. In the first place, it is out of the question to allow the water to rise to the vertex a of such a masonry triangle. A minimum thickness must be adopted to give substance to the upper part; and where Factors of safety. the dam is not used as a weir it must necessarily rise several feet above the water, and may in either event have to carry a roadway. Moreover, considerable mass is required to reduce the internal strains caused by changes of temperature. In the next place, it is necessary to confine the pressure, at every point of the masonry, to an intensity which will give a sufficient factor of safety against crushing. The upper part of the dam having been designed in the light of these conditions, the whole process of completing the design is simple enough when certain hypotheses have been adopted, though somewhat laborious in its more obvious form. It is clear that the greatest crushing pressure must occur, either, with the reservoir empty, near the lower part of the water face ab, or with the reservoir full, near the lower part of the outer face ac. The principles hitherto adopted in designing masonry dams, in which the moment of resistance depends upon the figure and weight of the masonry, involve certain assumptions, which, although not quite true, have proved useful and harmless, and are so convenient that they may be continued with due regard to the modifications which recent investigations have suggested. One such assumption is that, if the dam is well built, the intensity of vertical pressure will (neglecting local irregularities) vary nearly uniformly from face to face along any horizontal plane. Thus, to take the simplest case, if abce (fig. 13) represents a rectangular mass already designed for the superstructure of the dam, and g its centre of gravity, the centre of pressure upon the base will be vertically under g, that is, at the centre of the base, and the load will be properly represented by the rectangle bfgc, of which the area represents the total load and the uniform depth of its uniform intensity.
Fig. 13.—Factor of Safety Diagram.
At this high part of the structure the intensity of pressure will of course be much less than its permissible intensity. If now we assume the water to have a depth d above the base, the total water pressure represented by the triangle kbh will have its centre at d/3 from the base, and by the parallelogram of forces, assuming the density of the masonry to be 2⋅5, we find that the centre of pressure upon the base bc is shifted from the centre of the base to a point i nearer to the outer toe c, and adopting our assumption of uniformly varying intensity of stress, the rectangular diagram of pressures will thus be distorted from the figure bfgc to the figure of equal area bjlc, having its centre o vertically under the point at which the resultant of all the forces cuts the base bc. For any lower level the same treatment may, step by step, be adopted, until the maximum intensity of pressure cl exceeds the assumed permissible maximum, or the centre of pressure reaches an assigned distance from the outer toe c, when the base must be widened until the maximum intensity of pressure or the centre of pressure, as the case may be, is brought within the prescribed limit. The resultant profile is of the kind shown in fig. 14.
Having thus determined the outer profile under the conditions
hitherto assumed, it must be similarly ascertained that the water
Fig. 14.—Diagram showing lines of pressure in Masonry Dam.
face is everywhere capable of resisting the vertical pressure of the
masonry when the reservoir is empty, and the base of each compartment
must be widened if necessary in that direction also. Hence in dams above 100 ft. in
height, further adjustment of the outer profile may be required by reason of the deviation
of the inner profile from the vertical. The effect of this process is to give
a series of points in the horizontal planes at which the resultants of all forces above those
planes respectively cut the planes. Curved
lines, as dotted in fig. 14,
drawn through these points give the centre of pressure, for the
reservoir full and empty respectively, at any horizontal plane.
These general principles were recognized by Messrs Graeff and
Delocre of the Fonts et Chaussées, and about the year 1866 were put
into practice in the Furens dam near St Etienne. In 1871 the late
Professor Rankine, F.R.S., whose remarkable perception of the
practical fitness or unfitness of purely theoretical deductions gives
his writings exceptional value, received from Major Tulloch, R.E.,
on behalf of the municipality of Bombay, a request to consider the
subject generally, and with special reference to very high dams, such
as have since been constructed in India. Rankine pointed out that
before the vertical pressure reached the maximum pressure permissible,
the pressure tangential to the slope might do so. Thus
conditions of stress are conceivable in which the maximum would be
tangential to the slope or nearly so, and would therefore increase the
vertical stress in proportion to the cosecant squared of the slope.
It is very doubtful whether this pressure is ever reached, but such a
lim.it rather than that of the vertical stress must be considered when
the height of a dam demands it. Next, Rankine pointed out that, in
a structure exposed to the overturning action of forces which fluctuate
in amount and direction, there should be no appreciable tension
at any point of the masonry. But there is a still more important
reason why this condition should be strictly adhered to as regards
the inner face. We have hitherto considered only the horizontal
overturning pressure of the water; but if from originally defective
construction, or from the absence of vertical pressure due to weight of
masonry towards the water edge of any horizontal bed, as at ab in
fig. 14, water intrudes beneath that part of the masonry more readily
than it can obtain egress along bc, or in any other direction towards
the outer face, we shall have the uplifting and overturning pressure
due to the full depth of water in the reservoir over the width ab added
to the horizontal pressure, in which case all our previous calculations
would be futile. The condition, therefore, that there shall be no
tension is important as an element of design; but when we come to
construction, we must be careful also that no part of the wall shall
be less permeable than the water face. In fig. 13 we have seen that
the varying depth of the area bjlc approximately represents the
varying distribution of the vertical stress. If, therefore, the centre
of that became so far removed to the right as to make j coincident
with b, the diagram of stresses would become the triangle j ′l ′c′, and
the vertical pressure at the inner face would be nil. This will
evidently happen when the centre of pressure i′ is two-thirds from the
inner toe b′ and one-third from the outer toe c′; and if we displace the
centre of pressure still further to the right, the condition that the
centre of figure of the diagram shall be vertically under that centre of
pressure can only be fulfilled by allowing the point j ′ to cross the
base to j ′′ thus giving a negative pressure or tension at the inner toe.
Hence it follows that on the assumption of uniformly varying stress
the line of pressures, when the reservoir is full, should not at any
horizontal plane fall outside the middle third of the width of that
plane.
Rankine in his report adopted the prudent course of taking as the safe limits certain pressures to which, at that time, such structures were known to be subject. Thus for the inner face he took, as the limiting vertical pressure, 320 ft. of water, or nearly 9 tons per sq. ft., and for the outer face 250 ft. of water, or about 7 tons per sq. ft.
