Engineering Transactions, 3, 1, pp. 35-55, 1955

### O Pewnych Zagadnieniach Mechaniki Gruntów

W. Pogány

Two problems of soil mechanics are discussed: (1) the stress distribution under a foundation together with the resulting problem of foundation setting, and (2) the problem of soil pressure against a wall. A short historical account is given.
(1) Stress Distribution under a Foundation. In classical physics (Newton, Leibnitz, Lagrange, Ciapeyron an expression for stresses is obtained, according to which the stresses vanish only at infinite depth. The first mathematical theory was created by Boussinesg in 1885. It yielded results which were closer to reality and constituted a basis for the practical methods of Steinbrenner and Haefeli. The notion of «stress bulb» also came from Boussinesq.
In 1934 the concentration ceofficient was introduced in Boussinesq's equations by Fröhlich.
Prandtl and his school (Caquot, Résal, Maar) applied the theory of plasticity to the considered problem, and obtained equations for unit load in the cases of loose and cohesive soils. On the other hand, Terzaghi, Bendel and others introduced approximate formulae based on laboratory tests and taking into consideration the composition of the soil. They obtained a differential equation for foundation setting, in which time is represented. This is therefore a rheologic equation. A practical method of solution of this equation has been given by Ghersevanov and was simplified by Pietkowski.
(2) Soil Pressure. This problem was first treated by Coulomb (1773), who gave a formula constituting a basis for a series of graphical methods. (Culmann, Poncelet, Rebhann, Winkler, Szily). In the special case considered by Coulomb the three forces appearing in the problem (weight of the prism of soil between the wall and the slip plane, soil pressure and reaction force) have always a common point of intersection, which is not true in general. An effort has been therefore made (Fellenius, Krey, Terzaghi) to avoid this error by assuming a curved surface of slip. The author gives his own experimental method (1951), which in practice has proved to be useful although theoretically is not without reproach. The theory of elasticity has been used in 1857 by Rankine to solve the problem in question. This method, from the theoretical point of view, is better than that of Coul om b, the results, however, are rather inferior. Kötter (1903), Reissner (1924) and Kármán tried to improve and to generalize the theory of Rankine.
The rheological aspect of this problems was treated in Saint-Venant's. papers (1871), but systematic investigations have been undertaken only in recent times (H. Umstätter, 1948). These investigations are based on Maxwell's equations (1868), which take into account the elastic and plastic state. In this manner the angle of internal friction of the soil can be eliminated and a function obtained in which viscosity is represented, this being a variable paramater. In this manner the author transforms the Kármán-Nádai equations for active and passive soil pressure.

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