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Strength reduction

In strength the reduction method the strength characteristics is typically used to assess the stability of geotechnical structures such as slopes. The idea is to reduce the strength characteristics of the materials, i.e. the cohesion \(c\) and friction angle \(\varphi\), by a factor \(FoS\) until a loss of stability or failure of the system occurs.

To determine the stability, first self weight and additional loads are applied on the structure using standard calculation steps. From this equilibrium state, the strength reduction is started (step type *Reduction).

Performing the strength reduction (or \(\phi-c\) reduction) within the general workflow of a nonlinear analyses required some changes to the method detailed in Sections Newton Raphson and Automatic time stepping:

  • The strength reduction is carried out in an incremental non-linear analysis. Instead of incrementing over (pseudo) time, the progress of the calculation is controlled by increments of the factor of safety \(\Delta FoS\).
  • At the beginning of the analysis, the factor of safety is set to \(FoS_0\), where \(FoS_0\) corresponds to the initial value, which can be specified by the user. Typically, it is \(FoS_0 = 1.0\).
  • Subsequently, the \(FoS\) is incrementally changed. The first increment is \(\Delta FoS = \Delta FoS_{max}\). \(\Delta FoS_{max}\) is the maximum allowed increment size and can be specified by the user. Based on experience, \(\Delta FoS_{max} = 0.2\) is a good starting value. The factor of safety in the following increment \(n+1\) is thus \({^{n+1}}FoS = {^n}FoS + {^n+1}\Delta FoS\).

  • The size of \({^{n+1}}\Delta FoS\) depends on whether the previous increment achieved convergence or not.

    • In the case of convergence, \({^{n+1}}\Delta FoS\) is increased adaptively based on the number of iterations \(N_{\text{iter}}\) required for the previous increment to converge, relative to the maximum allowed number of iterations \(N_{\text{iter,max}}\). Let \(\rho = N_{\text{iter}} / N_{\text{iter,max}}\):

      \[ {^{n+1}}\Delta FoS = \begin{cases} 1.75 \cdot {^n}\Delta FoS & \rho < 0.25 \\ 1.50 \cdot {^n}\Delta FoS & 0.25 \leq \rho < 0.75 \\ 1.25 \cdot {^n}\Delta FoS & \rho \geq 0.75 \end{cases} \]

      The increment is clamped to \({^{n+1}}\Delta FoS \leq \Delta FoS_{max}\). This adaptive scheme accelerates the analysis when convergence is easy and remains conservative when convergence is already marginal, thereby reducing the risk of a subsequent cutback.

    • If no convergence is achieved, the increment is reduced by a factor of three, \({^{n+1}}\Delta FoS = {^n}\Delta FoS/3\), and the analysis continues from the last converged state.

  • The incremental analysis is carried out until \({^{n+1}}\Delta FoS\) falls below a user-defined minimum value \(\Delta FoS_{min}\). \(\Delta FoS_{min}\) can be specified by the user; we recommend \(\Delta FoS_{min}=10^{-3}\). The \(FoS\) at the termination of the analysis corresponds to the final factor of safety.

An exemplary schematic progression of the \(FoS\) in such a calculation is shown in Figure 1.


Figure 1: Solution strategy to determine the Factor of Safety FoS in a strength reduction analysis

The friction angle \(\varphi^i\) and cohesion \(c^i\) at each increment \(i\) of the strength reduction analysis are calculated according to:

\[ \varphi^{i} = \tan^{-1}\left(\dfrac{\tan \varphi}{FoS_i}\right) ~~~\text{and}~~~ c^{i} = \dfrac{c}{FoS_i} \]

Per default, numgeo also performs a simultaneous reduction of the dilatancy angle \(\psi^i\) during a strength reduction analysis, i.e.

\[ \psi^{i}=\tan^{-1}\left(\dfrac{\tan \psi}{FoS_i}\right) \]

This option can be turned off by the user, see the corresponding section in the Reference Manual.

In addition, numgeo offers the possibility to modify the Poisson's ratio \(\nu\) during a strength reduction analysis such that the inequality \(\sin \varphi \leq 1 - 2\nu\) is guaranteed. If this option is activated, \(\nu^i\) is updated following the work of Zheng et al.1:

\[ \nu^i = \min\left\{ 0.5\left(1 - \dfrac{\sin \varphi^i}{\beta}\right) ; 0.485 \right\} ~~~\text{with:}~~~ \beta = \dfrac{\sin \varphi}{1-2\nu} \]

This option is turned off by default. To enable it, refer to the Reference Manual.

Two material implementations of the Mohr-Coulomb model are available in numgeo and can be used in strength reduction analyses:

  • Mohr-Coulomb-2 returns an elastic Jacobian. The stress update itself is exact (closed-form return mapping in principal stress space), but the Jacobian used to assemble the global stiffness matrix is the elastic one. This implementation is very robust and remains stable in complex simulations, in particular those involving contact.

  • Mohr-Coulomb-3 returns the consistent elasto-plastic Jacobian. The resulting global stiffness matrix reflects the softening of the plastic zone, which yields faster convergence per increment but may fail in more complex simulations.

Default: Mohr-Coulomb-2 with elastic Jacobian

By default, numgeo assumes that Mohr-Coulomb-2 in used. Because the Jacobian is only approximate, the Newton-Raphson iteration reduces to a fixed-point iteration and convergence is at best linear. Tight tolerances on the solution increment or the energy are therefore impractical; the relevant convergence measure is the residual of the force balance.

Extrapolation and Convergence criteria (default)

*Extrapolation, fos-linear: For the strength reduction, numgeo uses by default a linear extrapolation of the solution increment with the factor of safety, as detailed in Section Initial solution estimate. See *Extrapolation for more information.

*Controls: Per default, numgeo uses tolerances for the strength reduction that deviate the default values given in Section Convergence criteria:

dof \(\epsilon^{d}\) \(\epsilon^e\) \(\epsilon^{r}\) \(\epsilon^{\Delta d}\) \(\epsilon^{\Delta e}\) \(r^\epsilon\)
u \(1 \cdot 10^{3}\) \(1 \cdot 10^{-12}\) \(1 \cdot 10^{-3}\) \(1 \cdot 10^{-6}\) \(1 \cdot 10^{-6}\) 1

With these values the increment and energy tolerances are effectively deactivated, so that convergence is decided by the residual force criterion alone. The default maximum number of iterations per increment is \(N_{\text{iter,max}} = 100\).

Solution accelerators

Because the fixed-point iteration of the default setup converges only linearly, the number of iterations per increment can become large in the vicinity of the critical factor of safety. We recommend the use of solution accelerators to reduce the computational cost of the analysis. Experience shows that Anderson acceleration of depth 1 or 2 works very well for strength reduction analyses and noticeably reduces the overall computation time. See Accelerators for details.

Alternative: Mohr-Coulomb-3 with elasto-plastic Jacobian

If Mohr-Coulomb-3 is used instead of Mohr-Coulomb-2, the consistent elasto-plastic Jacobian is available and the Newton-Raphson iteration recovers (locally) quadratic convergence. In this case tighter tolerances are appropriate and fewer iterations per increment are required.

Convergence criteria for Mohr-Coulomb-3

When using Mohr-Coulomb-3 with the elasto-plastic Jacobian, the default strength-reduction controls must be overwritten by the user with the following values:

dof \(\epsilon^{d}\) \(\epsilon^e\) \(\epsilon^{r}\) \(\epsilon^{\Delta d}\) \(\epsilon^{\Delta e}\) \(r^\epsilon\)
u \(1 \cdot 10^{-2}\) \(1 \cdot 10^{-2}\) \(1 \cdot 10^{-2}\) \(1 \cdot 10^{-5}\) \(1 \cdot 10^{-5}\) 1

The maximum number of iterations per increment should be set to \(N_{\text{iter,max}} = 16\).

  1. H. Zheng, D. F. Liu, and C. G. Li. Slope stability analysis based on elasto-plastic finite element method. International Journal for Numerical Methods in Engineering, 64(14):1871–1888, 2005. doi:10.1002/nme.1406. ↩