5. Nonlinear analysis

Both the single phase and the coupled two phase problem are generally nonlinear and must be solved iteratively. Nonlinearity results from plasticity in the solid and from the free surface evolution in the fluid, unknown a priori, which leads to saturation ratios, storage coefficients and permeabilities which vary in time.

The numerical solution is obtained using finite elements with a modified Newton-Raphson iteration scheme. The types of analyses addressed herein raise a number of questions related to incompressibility, localisation, numerical stability, etc., which are beyond the scope of this article, theses aspects are discussed in detail in [ZSOIL].

The following types of analyses, plus all meaningful combinations, are possible: initial state, stability, ultimate load, consolidation, creep, excavation/construction, and flow. The initial state, stability and ultimate load analysis require only the standard material data, Young’s modulus E, Poisson’s ratio n, cohesion C, friction angle , specific weight .

· Initial state analysis : gravity and preexisting loads are applied progressively to simulate the time-history of the site. The analysis produces a ²zero-displacement non-zero stress² state by superposition of gravity with the stress state produced by gravity.

· Stability analysis using a C - reduction algorithm : This approach to soil stability problems defines the safety factor as the ratio of the sum over the slip surface of available shear strength on the sum of mobilized shear strength. That is :

where C is the cohesion, ' the effective stress normal to the slip surface, the angle of friction, SF the safety factor, , the yield stress according to Mohr-Coulomb criterion. The C - reduction algorithm proceeds in steps, as described in Table 5.1:

· Ultimate load analysis : The load is increased progressively until instability is detected in the form of evidence of a failure mechanism, numerical divergence or nonconvergence of the iterative scheme.

· Primary consolidation : This analysis requires hydraulic characteristics like k_x, k_y, the Darcy coefficient, w the water specific weight, _F the water compressibility, e₀ the initial void ratio and the compression index from the oedometric test, if nonlinear consolidation occurs. The driver is also time-dependent and requires specification of a time increment t subject to numerical stability conditions.

· Creep and secondary consolidation : Creep is a time-dependent analysis which requires the specification of two creep parameters, plus t the time increment. Creep is split, for convenience, into volumetric and deviatoric components. Secondary consolidation can be modelled as volumetric creep.

· Excavation and construction stages : Excavation and construction stages can be simulated in association with any driver except initial state and stability drivers, which are fixed in time, while all other include a real or a fictitious time evolution. Excavation/construction stages require that the user specifics time dependent existence functions for the elements which may be removed or added at some specific time. The unloading of the medium, after excavation can be controlled by the user, using a load-time function, in order to simulate three-dimensional effects.

· Flow : Steady state and transient flow with time-dependent free surface can be simulated. The same parameters as for primary consolidation are sufficient to characterize the flow, except for special situations which may requires additional informations.

Any meaningful combination of drivers is supported.