where P is the horizontal pressure of the earth against the wall
exerted at one-third its height, w the weight of unit volume of
the material, x the height of the wall, and the angle of repose of
the material.
That the pressure so given exceeds the maximum
possible pressure we do not doubt; and, conversely, if we put
we may have equal confidence that P′ will be less than the maximum pressure which, if exerted by the wall against the earth, will be borne without disturbance. But like every pure theory the principles of conjugate pressures in earth may lead to danger if not applied with due consideration for the angle of repose of the material, the modifications brought about by the limited width of artificial embankments, the possible contraction away from the masonry, of clayey materials during dry weather for some feet in depth and the tendency of surface waters to produce scour between the wall and the embankment. Both the Neuadd and the Fisher Tarn dams are largely dependent upon the support of earthen embankments with much economy and with perfectly satisfactory results.
Fig. 16.—Showing Stresses at base of model dam determined experimentally.
In the construction of the Vyrnwy masonry dam Portland cement concrete was used in the joints. When more than six months old, 9 in. cubes of this material never failed under compression below 111 tons per sq. ft. with an average of 167 tons; and the mean resistance of all the blocks tested between two and three years after moulding exceeded 215 tons per sq. ft., while blocks cut from the concrete of the dam gave from 181 to 329 tons per sq. ft. It has been shown that the best hydraulic lime, or volcanic puzzuolana and lime, if properly ground while slaking, and otherwise treated in the best-known manner, as well as some of the so-called natural (calcareous) cements, will yield results certainly not inferior to those obtained from Portland cement. The only objection that can in any case be urged against most of the natural products is that a longer time is required for induration; but in the case of masonry dams sufficient time necessarily passes before any load, beyond that of the very gradually increasing masonry, is brought upon the structure. The result of using properly treated natural limes is not to be judged from the careless manner in which such limes have often been used in the past. Any stone of which it is desirable to build a masonry dam would certainly possess an average strength at least as great as the above figures for concrete; the clay slate of the Lower Silurian formation, used in the case of the Vyrnwy dam, had an ultimate crushing strength of from 700 to 1000 tons per sq. ft. If, therefore, with such materials the work is well done, and is not subsequently liable to be wasted or disintegrated by expansion or contraction or other actions which in the process of time affect all exposed surfaces, it is clear that 15 to 20 tons per sq. ft. must be a perfectly safe load. There are many structures at present in existence bearing considerably greater loads than this, and the granite ashlar masonry of at least one, the Bear Valley dam in California, is subject to compressive stresses, reaching, when the reservoir is full, at least 40 to 50 tons per sq. ft., while certain brickwork linings in mining shafts are subject to very high circumferential stresses, due to known water-pressures. In one case which has been investigated this circumferential pressure exceeds 26 tons per sq. ft., and the brickwork, which is 18 in. thick and 20 ft. internal diameter, is perfectly sound and water-tight. In portions of the structure liable to important changes of pressure from the rise and fall of the water and subject to the additional stresses which expansion and contraction by changes of temperature and of moisture induce, and in view of the great difficulty of securing that the average modulus of elasticity in all parts of the structure shall be approximately the same, it is probably desirable to limit the calculated load upon any external work, even of the best kind, to 15 or 20 tons per sq. ft. It is clear that the material upon which any high masonry dam is founded must also have a large factor of safety against crushing under the greatest load that the dam can impose upon it, and this consideration unfits any site for the construction of a masonry dam where sound rock, or at least a material equal in strength to the strongest shale, cannot be had; even in the case of such a material as shale the foundation must be well below the ground.
