6. Illustrations and validations

Typical soil mechanics problems are analyzed next to illustrate and validate the program.

6.2. Bearing capacity of strip and circular footings

The bearing capacity of strip and circular footings is analyzed, and the performance of two material models is investigated : the Drucker-Prager and a smooth Mohr-Coulomb. The role of the yield condition is studied for standard footing problems and the results are compared to those obtained with analytical methods proposed by different authors. A systematic analysis of the influence of several parameters is performed. The problems analyzed next include : axisymmetric superficial foundations, strip superficial foundations, including the influence of a rigid layer at finite distance and the influence of a cohesion gradient. The bearing capacity is captured using a static approach. The load applied on the footing is progressively increased until localization of strain into a mechanism occurs. The capacity of the model to capture such a mechanism is clearly demonstrated (Fig. 6.2a and 6.2b).

Axisymmetric superficial foundation

Figure 6.3 shows the geometry. Material data are E = 3000 kN/m², Poisson's ratio = 0.38, cohesion C = 1 kN/m² and dilatancy = 0° (incompressible plastic flow). The value of the friction angle is first varied between 20° and 45° and the corresponding bearing capacities are illustrated in figure 6.4 for different yield surfaces : smooth Mohr-Coulomb, Drucker-Prager with internal and external adjustments to Mohr-Coulomb. These numerical predictions are compared with the analytical results given by three different methods. Two of these methods are based on limit analysis, that is Terzaghi’s method adjusted by Vesic [VES,1975] for circular footings and the method developed by Salençon and Matar [SAL, 1982]. The third analytical method is based on the slip-line method and was developed by Cox [COX, 1961]. All the results are presented in figure 6.4 for comparizon purposes. It can be seen that all the theoretical and numerical methods predict the same increase of the ultimate bearing stress with increase of the friction angle. However, this increase varies depending on the method considered and that variation is not only observed for the numerical methods but also for the analytical ones illustrating the sensitivity of the problem. From figure 6.4 it can be seen that the ultimate bearing stress is bounded by the values obtained with the Drucker-Prager material calibrated to the two extreme values, external and internal Mohr-Coulomb apices. For a friction angle greater than 36.8° no clear failure could be obtained with the external adjustment of Drucker-Prager criterion as illustrated by the vertical asymptote. It can also be observed that the bearing stress predicted with the smooth Mohr-Coulomb condition and the one obtained with the method developed by Salençon and Matar are in a very close agreement. Furthermore, the agreement is improved for increasing friction angle. This is partly due to the fact that the smooth Mohr-Coulomb condition approximates the original Mohr-Coulomb one (used by Salençon and Matar) more closely for higher values of the friction angle.

The analysis of the above results suggests that the calibration of the Drucker-Prager surface is best when using an intermediate adjustment value between the tensile and compression Mohr-Coulomb edges. This calibration will lead to results which are closer to the ones obtained with the Mohr-Coulomb criterion, especially for axisymmetric computations.

Infinitely long superficial foundation

A systematic analysis of a superficial foundation under plane strain conditions is performed.

The geometry is the same as for the axisymmetric problem. The depth to a rigid layer is considered infinite. The friction angles are varied between 0° and 45° and results compared with results published by Chen [CHE, 1975]. The size adjustment of the Drucker-Prager criterion used for these analyses corresponds to enforcing a failure load identical to the Mohr-Coulomb solution under plane strain conditions and deviatoric flow; the resulting Drucker-Prager constants are given by equation (3.3.a). This adjustment was initially proposed in [CHE, 1982] for associated flow and it is extended in [ZSOIL] for non-associated flow.

It is observed that the predictive capability of the model is quite satisfactory with a tendency to undershoot the ultimate loads at high friction angles (and correspondingly high ultimate loads).

The influence of a rigid layer at a finite depth and of a cohesion gradient are analyzed next.. Results are compared with the ones obtained by the standard superposition method and the ones resulting from Matar and Salençon's improved formula [MAT, 1979]. Again, coherent results are obtained which confirm the conservative prediction obtained by superposition. These results are summarized next in table 6.1.