Nouvelle méthode d'application de quelques fonctions de la variable complexe aux calculs des sous-pressions agissant sous les ouvrages de retenue

A new method for the application of few complex variable functions to the calculation of uplift affecting dam structures

Professeur à l'Université Catholique d'Amérique à Washington, D.C.

Résumé

The integral of Schwarz and Christoffer is used to find the complex variable function which transforms the positive half plane into a polygon with one point located in the infinity. The polygon represents the field of flow under a dam with one non-symmetrical cutoff built on an infinitely deep layer of pervious soil. The half plane represents a field of flow in similar conditions but without cutoff. By regrouping the members of the transforming function, the polygon can be transformed into a half plane with the dam located non-symmetrically about the origin of coordinates. A second transformation brings it to a symmetrical position. Since the solution for a symmetrical dam without cutoff is known, this solution can be used for the case of flow under a dam with cutoff by calculating conjugated points by use of the transforming formula. This method is due to Pavlovsldi and is exposed in his treaty of 1922 (5, 11). The two transformations, the first, from the cutoff case to a non- symmetrical dam and the second, to a similar symmetrical dam can easily be made for the bottom line of the dam by calculation as well as by a simple graphical procedure. The new application of Pavlovskii method, which is due to Firchakoy, consists of the use of the transfoming function, derived for a simple cutler case, for the analysis of multi-cutoff dams and of dams with the base located Mellow the ground line, i.e., of dams with any arbitrary shape of the underground profile inc1uding an impervious layer located at a given depth. The underground profile is first divided into several sections, each containing one cutler and a common point with adjacent sections. The conform transformation of each section is made separately. Special monographs may he used. However, the writer found it more practical to use the commonly available tables of hyperbolic functions and that of elliptical integrals, which he illustrates by numerical examples. The accuracy of the new method in calculating the uplift acting in particular points of the dam foundation profile is in general greater than 2 to 3 per cent; only in single points, for example, in re-entrant angles it decreases to 5 per cent, hut the accuracy in calculating the total up lift must he estimated as insuring an accuracy of a least 3 per cent. The numerical values of the uplift calculated in various points of the dam can serve as departing points for the construction of the flow net. However, the flow net is of relatively minor interest when the uplift is already calculated. The gradient of escaping velocities can also he determined from the calculations provided by the method.