NEW CONFIRMATION OF THE FORMULA FOR THE CALCULATION OF ROCK FILL DIKES

In the paper entitled "Una formula para el caloulo de los diques de escollera" (A Formula for the Calculation of Rock Fill Dikes) published July 1938* there was presented the expression, 
P=NA^3d/((cosa- sina)^3(d-1)^3) 
where 
P = weight of the individual stones or blocks in kilograms 
N = 15 for dikes of natural rook fill 
N = 19 for dikes of artificial block fill 
A = 2h = total height of the wave that breaks on the dike,measured in meters 
d = specific weight of material of the stones in tons (metric) per oubic meter 
a = angle of the dike's side slope with the horizontal

In the paper entitled "Una formula para el oaloulo de los diques de esoollera" (A Formula for the Calculation of Rook Fill Dikes) published July 1938* there was presented the expression, p _ NA. 3 d (oos«*-sin«<) 3 (d-1) 5 where P = weight of the individual stones or blocks in kilograms N = 15 for dikes of natural rook fill N • 19 for dikes of artificial block fill A -2h • total height of the wave that breaks on the dike, measured in meters d • specific weight of material of the stones in tons (metric)  per oubic meter a.* angle of the dike's side slope with the horizontal Before the preliminary determination of those tentative values of N, each based only on a single observed case -the natural rook fill dike of Orio and the artificial rook fill dike of San Juan de Luz, -the publication mentioned above stated thatj "It only remains now to determine the coefficient N and verify if it is sensibly constant, as seems inferred from the material submitted, or varies with the other elements of the formula."In the worst case, a coefficient similar to the classic and variable coefficient C of the formula of uniform flow, V • CVRS, will be considered." In spite of the fourteen years intervening, during whioh the reasoning followed for the deduction of the formula has been refined, in a manner that might have been advantageously taken into aocount by the translators of the paper, the coefficients 15 and 19 still stand, due to the satisfactory results always obtained in numerous cases of practical application.
•Translated and published in the January 1949 Bulletin of the Beach Erosion Board, Office Chief of Engineers, Department of the Army, Wash.D.C.The coefficient K of that publication is denoted by N in this paper to avoid confusion with K » coth-7r£, and similarly, the angle The formula is actually derived for the upper slope of the dikes, and a generalization for all the depths of the structure was indicated only tentatively at the end of the paper.In this connection substantially the following was stateds "This generalization of the formula assumes a certain margin of security, but it is not logical to apply on the sea the strict results obtained from theoretical formulas when on land it is usual to multiply them by ample safety factors."I should be most grateful to my colleagues who, acquainted in detail with concrete practical cases, would kindly furnish me information for refining the coefficients." As a result of the ideas presented in the first publication of 1938 it is possible to refine somewhat the coefficients in the artiole entitled "Generalizaoion de la formula para el olloulo de los diques de esoollera y comprobaoiSn de sus ooefioientes", (Generalization of the Formula for the Calculation of Rock-fill Dikes and Determination of its Coefficients) published in May 1950* in the Revista de Obras Publioas.On oomparing them with results with the established experience obtained from the Argel dikes, the degree of approximation sufficient for practical application was definitely confirmed.
Besides the practical confirmation of the coefficients mentioned in the article of the Revista de Obras Publicas, the composition of the formula was verified.In this connection it was substantially stated: As was a matter of record, in the XVTI International Congress of Navigation held in Lisbon, it is satisfying that the formula deduced in the Amerioan report of Epstein and Tyrrel for reflecting rook fil| dikes, starting from our expression for pressures of reflection P > " , is similar to that which was obtained in 1938 for breakwaters.
In effect, the American formula is s s H3 *t = R t (s-l)3 (^A-rP and ours: Expressing the American formula in our notation, it becomes: whioh is ours, except only that it includes the factor oos 3 «t in the coefficient.
It is a matter of record that on establishing our formula it was indicated that the coefficient should vary with the data of the problem.Practically the angle <*• , whioh varies most for the upper part of the dike, will not vary much, for from cotan«<i, -3, corresponding to present rook fill dikes, it cannot get much steeper than ootan •c"fi^2, even in the reflecting dikes, because of the enormous weight of the stones this requires.Between those maximum limits the relation isj cos 3 at 2 which would represent only a small difference in the weight of the stones, and even less in their size, whose relation would be y 1.2 = 1.06.Only direct observation can determine properly N or Rt, including oases of very steep slopes.
Another interesting confirmation of our formula is given now in the article entitled "Notes on Determination of Stable Underwater Breakwater Slopes" published by Kaplan in the Bulletin of the Beach Erosion Board for July of the present year.*In this article, starting from Blanohet's formula for the destruction of stone piles by a current Ki/g^yr \fcn(« 0 -*) in whioh v a velocity limit for the disintegration of the stone pile K^ « a dimensional coefficient whioh should be constant for stones of a given shape •Kenneth Kaplan, "Notes on Determination of Stable Underwater Breakwater Slopes", Bulletin of the Beach Erosion Board, Vol. 6, No. 3, p.where the coefficient K, also duly corrected, becomes vl/2,.1/6S ( w l in which K f is dimensional.
As was deduced from our publication of 1938, the maximum velocity of the wave breaking over steep slopes, being theoretically equal to its velocity, the following is obtained C a v' •ffi -*P <•>' Prom the expressions (4) ( 5) and ( 6), in connection with our definitions Yf • Pj w^ s dj V s d-L and the angle of the natural slope being approximately *< 0 2 ^T. the following expression is readily obtained 4

A2
.. .wdi whioh is definitely our formula; since this expression is obtained likewise readily and without introducing previous simplifications, for any one of the processes followed in our report of 1938, in whioh K" is likewise dimensional.
If, in order to simplify the applications, we define the speoifio weights in tons per cubic meter, with whioh they are reduced to the relative densities, and that of the water, salt water included, being practically d^ a 1 the formula is reduced to which is our plain, convenient practical formula onoe more confirmed in an interesting wayj whose degree of approximation is more than sufficient in such complex problems, where only a slight variation in the height of the wave might give rise to greater variations than those resulting from this degree of approximations whioh might also be increased by simply refining the coefficient in the manner already demonstrated in the first report of 1938.
*Actually this result does not apply to calculations in a contrary sense for the generalization of the formula for submerged slopes.
NA 3 d (cos*.-sin<) s (d-1) 5 *This artiole was translated and published in the January 1951 issue of The Bulletin of The Beach Erosion Board, Office Chief of Engineers, Depar ment of the U.S. Army, and by the Waterways Experiment Station; Translati No. 51-2.**See the publication "Caloulo de diques vertioales" (Calculation of Vertical Dikes) published 1938, translated and published in French and in English in the "Bulletin de l'Assooiation Internationale Permanente des Congres de Navigation", No. 23, July 1939.NEW CONFIRMATION OF THE FORMULA FOR THE CALCULATION OF ROCK FILL DIKES in which Pt • P • weight of the individual stones H • A s 2h a wave height s = d s relative density of the material /A. a natural batter of the rook fill -1 r s tan •*• = slope of the rook fill Rfc and N are coefficients.
20. **Ch.Blanchet, "Formation et destruction par un oourant d'eau des massifs en pierres", in "La Houille Blanche", March 1946.D * lineal dimension of the stones w m specific weight of the water wi a specific weight of the stones ed • angle of slope with the horizontal o*o = angle with the horizontal of the stones' natural slopeCalling W the weight limit of the individual stones, "a" the major axis of the corresponding orbitary ellipse and T the wave period, of the coefficient K is a function of the acceleration of gravity "g" and the specific weights of the water and of the stones* Actually the case whioh is of most interest is that concerning submerged stones, or stones entirely enveloped by water* In this case it is neoessary to take into account, besides the hydrodynamic thrust, the vertical hydrostatic foroe.It is desirable to substitute the speoifio weight w]_ of the stones, without deviating from Blanohet's formula, by wi -w corresponding to these submerged stones.Therefore the formula becomessupposed to admit also, with a oertain degree of practical approximation for suoh complicated problems, the application of the troohoidal theory, even in the oase of steep slopes* It is important to remember that in our first report of 1938, besides deducing the formula for the direot action of the wave breaking upon the breakwater, the same formula was also deduced, thus confirming its generalization, upon the basis of the water's descent on the slope at a reduced velooity due mainly to the roughness and permeability of the rook fill* Hence, following our own previous deduction it may be stated thatNEW CONFIRMATION OF THE FORMULA FOR TFIE CALCULATION OF ROCK FILL DIKESThus from the equations (l), (2), and (3) the formula at onoe is obtained, this being duly corrected, in whioh•\fsin ( mc 0 -«) X-sin<*)3(d-l) 3