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, \({^{n+1}}\Delta FoS = {^n}\Delta FoS\). If no convergence is achieved, the increment is halved, \({^{n+1}}\Delta FoS = {^n}\Delta FoS/2\).
- 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.
Extrapolation and Convergence criteria
*Extrapolation, none: For the strength reduction, numgeo does not use an extrapolation of the solution in the first iteration as detailed in Section Initial solution estimate per default. See *Extrapolation for more information.
*Controls: Per default, numgeo uses adaptive tolerances for the strength reduction. Deviating from the default values given in Section Convergence criteria numgeo uses the following tolerances:
| dof | \(\epsilon^{d}\) | \(\epsilon^e\) | \(\epsilon^{r}\) | \(\epsilon^{\Delta d}\) | \(\epsilon^{\Delta e}\) | \(r^\epsilon\) |
|---|---|---|---|---|---|---|
u |
\(1 \cdot 10^{-8}\) | \(1 \cdot 10^{-2}\) | \(1 \cdot 10^{-2}\) | \(1 \cdot 10^{-5}\) | \(1 \cdot 10^{-5}\) | 1 |
Large changes in the factor of safety (\(\Delta \text{FoS}\)) can significantly impact the investigated system, resulting in large initial energy increments \(\Delta {^t}W^{(i=0),\alpha}_{int}\) (see Unbalanced energy). Since \(\Delta {^t}W^{(i=0),\alpha}_{int}\) is used as a reference for determining convergence, the accuracy of solution control is sensitive to \(\Delta \text{FoS}\). Large \(\Delta \text{FoS}\) values increase \(\Delta {^t}W^{(i=0),\alpha}_{int}\), thus softening the convergence criterion and potentially leading to an overestimation of the final FoS.
To address this, we scale \(\epsilon^e\) and \(\epsilon^r\) by a scaling factor \(s^{\text{FoS}}\) that depends on \(\Delta \text{FoS}\):
This scaling mitigates the influence of large \(\Delta \text{FoS}\) values, thereby maintaining the robustness of the convergence criterion. Deviation from Section Convergence Criteria, in strength reduction analysis the energy and force convergence criteria read:
and