Vízügyi Közlemények, 1935 (17. évfolyam)

Kivonatok, mellékletek - Kivonat a 3. számhoz

26 by the equation 24. This latter is the same as the equation No. 7 of Ran/cine'

s theory, which proves that Rankine's planes really belong to the mathematically perfect solution of the problem, and it seems to be unnecessary to introduce the terms őr до

к ^Г — a

s the characteristics of a continuum of infinite extent. But these planes of Rankine's can only extend as far as the PB

2 straight line that starts from the top-point В of the retaining wall, because it can be proved that at point P

2, where the curved and the plane portions of the sliding slope meet, the stresses can only be the same (equation 18 = equation 23),if cos (<*„ -f- ß

0

—cp)'—0, or a

Q-\-ß

0= 90°-j-cp

, consequently BP

2 is the other sliding Rankine plane, because they intersect at the angle of 90°+<f. The coordinates of the point P

2 (« = a

0, ß =ß

0) determine also the one constant of the differential equation of the sliding slope, while the other constant can be stated from the presumed inclination angle S of the earth pressure acting upon the back (ßj of the retaining wall, by the aid of the equation No. 27 ; for this equation gives the relation existing between the inclination (a

,) of the sliding slope and the angle S (see fig. 10). In this way the solution of the differential equation goes through the point P'

2 (a

0, ß

0) and A' («

1

; ß as is shown in figure 11. The inclination angle b of the earth pressure varies between the limits of+<p and — cp (see figure 12). d) By a detailed analysis of the differential equation, it can be proved that the tangent at the starting point A of the curve cc = f (ß) is horizontal, which means that all solutions of the differential equation start from the point A, and have there a common horizontal tangent (fig. 13). Similarly it can be stated that the point В («j = cp, ß

1 = 180°—

cp), which is characteristic of the natural slope (

ßj = 180°—cp), is always a solutiom of the differential equation, i. e. all curves also pass through the point В (fig. 13). Finally it can also be proved (see equations 30—36) that all solutions of the differential equation are concave regarding the ß <X axis, i. e. this bundle of rays having two poles is pressed into rather a limited space. From the boundary curves H

1 and H

2, which limit this space, H

1 is the ^4 7? straight line gained by connecting the points A and B, because this is the last, still possible concave curve ; and the other curve is H

2, the end-tangent of which (da\ 2 \<Za/.B

= —^appears as a maximum (equation 38). For any curve lying between these two boundary curves a good approximation is given by an w-rate parabola (equation 39), the n exponent of which can be determined from the end-tangent in point B. Having a knowledge of this parabola, the function « = / (ß) is perfectly known, and to any ß

1 inclination of the retaining wall the angle «

1

( and as a final result, the angle b of the direction of the earth pressure can be determined. After all, the components of the pressure (N

ï and T

s) are defined by the equations 41 and 42. e) According to the formula No. 41, the value of the earth pressure acting on a retaining wall that is inclined at the angle ß

v is defined by the varying magnitude of the angle cc

v From the point of view of computing the dimensions of a wall, the extreme values, especially the maximum of the earth pressure, are of interest. A graphical solution is shown in figure 15, where the variation of the к hydrostatic coefficient (equation 44) with the angle a

2 is depicted. There are rela-