3. Macro command description#

According to the recommendations, the justification for the stability of dams and embankment dikes, the verification of the stability of the slopes are essential as well as during the design and operation of such structures.

The objective of this analysis is to prevent sudden slope failure by evaluating quantitatively the risk of rupture, i.e., the safety factor. The stability of slopes depends on both geometry and mechanical properties embankment, and mechanical and hydraulic loads such as gravity, the hydrostatic pressure and the interstitial pressure generated by the infiltration of water into the structure.

The macro command CALC_STAB_PENTE allows you to calculate the safety factor via two approaches:

  1. Method SRM: the method of gradually reducing parameters related to

    the shear strength of the material (called the SRM method). This method was proposed by Griffiths [1] in 1999. The implemented SRM method was validated only on slopes modelled in 2D. Use on 3D models is also possible but the relevance of the result must be verified by the user.

  2. Method LEM: a series of methods based on limit equilibrium theory (named method LEM),

    including simplified Bishop procedures, Fellenius, Spencer, and Morgenstern-Price [2]. These methods divide the slope into several slices that are considered rigid bodies. Given a potential fracture surface, the safety factor is calculated by resolution of the balance of forces and moments of all the slices. The safety factor is defined by the relationship between the strength of resistance and the driving force along the fracture surface. CALC_STAB_PENTE is also equipped with optimization algorithms to locate fracture surfaces, minimizing the safety factor.

The SRM method takes advantage of the relevance and fidelity of the finite element method to analyze the stability of slopes without the need to assume in advance the sliding circles.

Indeed, using the Mohr-Coulomb failure criterion, method SRM defines the safety factor as being the relationship between the initial material parameters and those reduced leading to the discrepancy of the nonlinear calculation under the loads applied:

\(\mathit{FS}=\dfrac{c}{{c}_{i}}=\dfrac{\mathrm{tan}(\varphi )}{\mathrm{tan}({\varphi }_{i})}=\dfrac{\mathrm{tan}(\psi )}{\mathrm{tan}({\psi }_{i})}\)

where

  • \(\psi\) is the angle of expansion of the material,

  • \(c\) and \(\varphi\) represent the strength of cohesion respectively and the angle of friction of the embankment.

CALC_STAB_PENTE allows you to define the SRM zone by an independent mesh group.

The benefits of the SRM method are as follows:

  • Possibility to apply various types of mechanical loads, such as

    hydrostatic pressure and the force of inertia under seismic conditions (pseudo-static).

  • Constraint field calculation true to reality without the need to presuppose

    the fracture surfaces, which leads to the result of FS that is more relevant than that of method LEM.

  • Natural visualization of the fracture surface characterized by high plastic deformation.

Avertissement

If the user is not experienced enough, method SRM will take generally more calculation time and the relevance of the result is not guaranteed because of the misconfigured FEM calculation.

Unlike method SRM, the LEM method ** is independent of the calculation FEM. However, method LEM requires pre-assuming potential fracture surfaces and makes an assumption on the interaction forces between the slices. Given the simplicity and efficiency of calculation, method LEM has become the most popular in the hydraulic industry and is recommended by the regulations for the design and verification of embankments.

At the output of the macro-command, we get a table to the left of the equality sign, whose content depends on the analysis method chosen:

  • In case of SRM: the table contains the safety factors tested during the iterations,

    as well as the values for the maximum total displacement in zone SRM. This table makes it easy to both the recovery of the stability result and the verification of its relevance. In fact, the safety factor is in the last row of the table. The convergence of the algorithm is verified by a progressively « flattened » plot of the safety factor based on maximum displacement.

  • In case of LEM: the table contains critical surface safety factors

    during the refinement of the mesh, as well as the geometric parameters of the fracture surface.

    • For Bishop and Fellenius procedures, the fracture surface is described

      by the x-axes of the entry and exit points, the radius, and the coordinates of the center of the circle.

    • For the Spencer and Morgenstern-Price procedures,

      the fracture surface is described by the abscissa of the entry and exit points, and the ordinates of the intermediate points describing the multi-linear surface.

The macro command provides the option to visualize the sliding surface. If requested, a concept such as evol_noli will be produced for this purpose.