264 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



[1844. 



HYDRAULICS. 



General Sketch of a Theory of the Conlracl.ioji of Feins in Water 

 discharged from Orifices in thin Plane JValls. By M. B.bver. 



In comparing the different esperinienls upon the flow of water from 

 vertical openings in thin plane walls, we are .struck by the great varia- 

 tion undergone by the coefficient by which, in each case the theoretic 

 formula for the discharge must be multiplied. On examining the facts 

 more strictly, we find two sorts of distinct variations, of which one de- 

 pends solely upon the amount of the charge, and the other upon the 

 form of the orifice. The existence of the first is a certain sign that 

 the formula used does not accord with the experiments. The second 

 proves that the water of the reservoir is subject to a law of motion, 

 the effect of which is modified by the form of the opening. While 

 reflecting upon tliese difficulties, I was struck by a very simple idea, 

 and one which deserved a rigorous investigation. This investigation 

 is the subject of my memoir. To present the results in a proper 

 light, I will here give, in a concise manner, the sketch of my work. 

 I assume hypothetically, that the molecules of water in the reservoir, 

 move towards the centre of the orifice, with velocities which are in- 

 versely proportional to the square of their distances from that cen- 

 tre. Hence it follows, that molecules equi-distant from that centre 

 will have the same velocity, and are situated upon the circumference 

 of a hemisphere described from that centre with a radius equal to, or 

 greater, than that of the orifice. As soon as the molecules have ar- 

 rived at the hemisphere described with the radius of the orifice itself, 

 their velocity is decomposed into two others, of which one is parallel 

 to the axis of the orifice, and the other perpendicular to this axis. 

 The first gives the velocity perpendicular to the plane of the orifice, 

 and the other represents the velocity of contraction. But in order to 

 determine, in conformity with the hypothesis adopted, both of these 

 velocities, it is required to find the mean distance from the plane of the 

 orifice, of the particles in the section of a hemisphere passing through 

 its axis ; that is, the mean distance of the molecules upon the peri- 

 phery of a semicircle of the same diameter. We arrive, by this 

 means, for circular orifices, at results conformable to those of the ex- 

 periments of Bossnt, Poleni, Eytehvein, &c. By this examination we 

 find, for the orifices in question, the variation in the .lischarga de- 

 pendent upon the form of the orifice ; nothing more is wanting than to 

 seek for that which depends upon the charge, or to determine the true 

 velocity of discharge, which is done by the known methods. Thus, 

 in determining the coefficient of contraction {k) of vertical rectangular 

 orifices, we arrive at the general formula, 







V'«+l 



('- 



) 



Where /, is the base of the rectangle, b, its height, -n, the ratio of the 

 circumference of a circle to its diameter ; m is determined by means 



of the equation »t— ^ ~°~ ^\ where H is the charge above the 



lower edge of the rectangular orifice, a, is the height of a column of 

 water equivalent to the difference of the atmospheric pressure upon 

 the surface of the water in the reservoir, and upon the centre of tlie 

 orifice. This value taken for the following table is equal to O-0U2O 



metre. 



In order to show the correspondence of the formula with observa- 

 tions, I have compared it with the admirable experiments made by 

 M. Poncelet, et Lesbros, at Metz, in 1828; calculating only the co- 

 efficients of the first experiments in the six tables, trom No. 4 to 

 No. 9, 1 get 



These difi'erences do not exceed those which the results of experi- 

 ment several times repeated, show. The calculated values uf A, are 

 found a little too large, because all the other small corrections have 

 been neglected, such as the friction ou the edges of the orifice, the 

 temperature, the resistance of the air, &c. 



In the memoir I have employed an approximate formula, which 

 ditt'ers very little from the exact value, and which is formed by sup- 

 posing that the velocities, in rectangular orifices, are as the square 

 roots of the charges above their centres. 



All these formula, however, suppose that the level in the reservoir 

 remains constant, which is not the case in practice, except when the 

 charge is ten, or twelve, times greater than the radius of the orifice. 

 In small charges there is a depression of the level above the orifice, 

 for whicli allowance must be made, in order to obtain exact results ; 

 for this reason it is necessary to multiply all the formula by a factor 

 which depends upon the depression ; by this means I obtain equations 

 which are applicable, at the same time, both to great and small 

 charges, and even to overfalls. 



Finally, the different forms of the veins of water are determined by 

 means of the theoium mentioned above, that the force of contraction 

 is proportional to the radius of the orifice. Hence it follows that the 

 contraction in the diagonal sections of a square orifice is greater than 

 that in the sections passing through the centres of parallel sides, and 

 as the contraction may be regarded as a force acting perpendicularly 

 upon the axis of the vein, it follows that the particles of water in the 

 larger sections approach the axis, whilst the particles, in the smaller 

 sections, are farther from it, which explains the forms found by expe- 

 riment. — Cumptts Rendus, translated for the American Journal oj' the 

 Franklin Institute. 



ON THE FLEXURE OF BEAMS. 



Report upon a Note relative to the Flexure of Beams Loaded in a 

 f'erlical Position; presented June 20///, 1842. By M. E. Lamarle. 

 Committee, Ponctkt and Lionville. 



In this note M. L marie has chiefly proposed to establish the fol- 

 lowing principles: — 



1. The loads, which beams, loaded vertically, can support without 

 permanent alteration, are independent of their lengths, and siniply 

 proportional to their sections, so long as the ratio of their lengths to 

 the least dimensions to their transverse section does not reach a cer- 

 tain limit. 



2. Beyond that limit, and in all cases of practical application, the 

 maximum load may reach, but can never exceed, the pressure corres- 

 ponding to the initial flexure. 



M. Lamarle also shows that (the pieces being supposed prismatic,) 

 it is sufficient to know the greatest change in length compatible with 

 the preservation of elasticity, in order to determine numerically the 

 limit alluded to. He remarks besides that the results furnished, by 

 calculation, accord with the facts generally observed, and that they 

 imply the consequences announced by M. Duleau, in the following 

 terms: " A rectangular bar pressed vertically, resists until the corn- 

 er 'c 

 pressingweight attains the value, Q = 2-^. This vpeight causes the 



piece to assume a curvature in the direction of its smallest dimension, 

 and it at once folds together." The deductions of the author rest 

 essentially upon the analysis given by M. Lagrange, for the problem 

 of the flexure of pieces loaded vertically, but by imposing the condi- 

 tion of not surpassing the force capable of producing a permanent 

 alteration, and by expressing this condition numerically, M. Lamarle 

 has introduced into the question an element of which advantage had 

 not yet been taken to solve it practically. The introduction of this 

 element fixes the degree of convergence of the series which are ob- 

 tained, and permits the deduction from the general solution, of rules 

 valuable to the builder. 



We know, and Lagrange has proved, that the flexure of pieces 

 pressed vertically, becomes possible only when the pressure has ob- 

 tained a certain minimum value. If the pieces are prismatic, the load 

 corresponding to the initial flexure, increases in the inverse ratio of 

 the squares of their lengths. The contractions which it produces, in- 

 dependent of all flexure, are, therefore, more considerable in reference 

 to the unit of length, in proportion as the pieces are stouter, and we 

 may conceive that for a given cross section there exists always a length 

 below which there is already an alteration of elasticity, even when the 

 load is too small to cause a commencement of flexure. Hence the first 

 principle anDOuuced by M. Lamarle. 



