Vízügyi Közlemények, 1935 (17. évfolyam)

Kivonatok, mellékletek - Kivonat a 3. számhoz

25 treats of a special question, when, in the sliding of an earth mass, the back surface of the retaining wall itself is a surface of sliding. Prof. Jáky (1934), starting from Boussinesq'

s suppositions, sets up all the equations of equilibrium to be satisfied mathematically — including Kötter'

s equation — and has been the first to state the differential equation determining the form of the sliding slope. b) The author describes his own method as follows : According to Boussinesq, it can be assumed that the stresses n

x, t

1 and o

h acting in the side of the OA В differential sector are in linear proportion to the h distance (fig. 9. The figure in the Hungarian text is placed incorrectly ; it should be turned so that the common side of the a and « angles is horizontal), and generally we may write: t=h fj (ß) ...etc. (see equation 10). The conditions of equilibrium are expressed in a polar coordinate system by Cauchy's partial differential equations (No. 11), where substituting those under No. 10., the equation system No. 12 is obtained, in which only total differential quotients are present. If we put next to the AB lamina an ABC differential triangle, the hypothenuse AB of which is a differential element of the curved sliding slope, then, in consequence of an equilibrium, the relations expressed in equations No. 13 come into being between the stresses t and n acting in the sliding slope, and the stresses 0

h, t, and «, mentioned above. As the fractional term — attains a maximum in the n sliding slope, and this equals tgrp, the equation No. 14 can easily be derived ; then the components of the stress can solely be expressed with the stress t and the angle A (see equation 15). Making use of equationis 16 and 17, after substitution, the equation 18 and d t 19 are valid for t and -73dß These equations show that the sliding stress t is in direct proportion to the pole distance h, and besides, it is a function of a, the inclination of the sliding slope and ß, the pole angle. Substituting the equations 18 and 19 into the formula of -j

7, — 21 tg cp -j-j j- we get the equation No. 20, which is nothing but Kötter'

s equation, only in another form. This proves that Boussinesq'

s suppositions are correct, because Kötter in his derivation omits these suppositions, and arrives at his result without any others. c

) Now we can easily come to the equation of the curved sliding slope. Eliminating the stress t from the equation 18 and 19 (and at the same time h also falls out), we get the second differential equation (No. 21) : which is the equation of the sliding slope that has been sought for. The А, В, С terms in the equation No. 21 are functions of («,/?). The integral of this differential equation cannot be written in a closed form, but it is certain dot that = 0 is one singular solution of the equation, i. e. the sliding slope can dß also be a plane. In this, I is defined by the equation 23, and the sliding slope (« <£[)