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.

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:

\[ \nu^i = \min\left\{ 0.5\left(1 - \dfrac{\sin \varphi^i}{\beta}\right) ; 0.499 \right\} ~~~\text{with:}~~~ \beta = \dfrac{\sin \varphi}{1-2\nu} \]

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

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 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`	\(1 \cdot 10^{-2}\)	\(1 \cdot 10^{-2}\)	\(1 \cdot 10^{-2}\)	\(1 \cdot 10^{-5}\)	\(1 \cdot 10^{-5}\)	1

Additional Post-Processing Methods for Post-Analysis FoS Evaluation

After the strength reduction analysis completes, numgeo provides six “post-processing” methods to identify a “critical” FoS from the computed data points \(\bigl\{\,(FoS_i,\,\max(\text{Disp}_i))\bigr\}\). The set of points arises from each converged increment of the reduction procedure, as described in the previous sections. By default, these points are sorted such that \(FoS_1 < FoS_2 < \dots < FoS_n\). The displacement is also filtered to ensure monotonicity (\(\max(\text{Disp}_{i+1}) \ge \max(\text{Disp}_i)\)), thus ensuring a meaningful curve.

The six post-processing approaches are:

Raw (Linear) – Curvature
Raw (Linear) – Knee
Raw (Log) – Curvature
Uniform (Linear) – Curvature
Uniform (Linear) – Knee
Uniform (Log) – Curvature

These methods rely on incremental or “piecewise” finite difference estimates of derivatives, or geometric distances (for “knee detection”). In the following, we denote:

\[ \{\,(\,FoS_i,\;d_i\,)\}_{i=1}^n \]

where \(d_i = \max(\text{Disp}_i)\). The index \(n\) is the total number of data points after the filtering step.

1. Raw (Linear) – Curvature

In the raw data (linear scale), numgeo computes discrete first and second derivatives:

\[ d1_i = \frac{d_{i+1} - d_{i-1}}{\,FoS_{i+1} - FoS_{i-1}\,} \quad\text{(central difference)}, \]

with forward/backward differences at the endpoints (\(i=1\) or \(i=n\)). Then it computes

\[ d2_i = 2\,\Bigl(\frac{\,d_{i+1}- d_i\,}{\,FoS_{i+1}- FoS_i\,} \;-\; \frac{\,d_i - d_{i-1}\,}{\,FoS_i - FoS_{i-1}\,}\Bigr)\,/\,\bigl(FoS_{i+1}- FoS_{i-1}\bigr), \]

and curvature at index \(i\) by

\[ \kappa_i = \frac{\,\bigl\lvert d2_i\bigr\rvert\,}{\,\bigl(1 + d1_i^{2}\bigr)^{3/2}}. \]

The program identifies the maximum \(\kappa_i\) across \(i=1,\dots,n\) as the index for the critical FoS in raw space.

2. Raw (Linear) – Knee

For the knee detection, numgeo connects the first point \((FoS_1,d_1)\) and the last point \((FoS_n,d_n)\) with a chord (straight line). For each intermediate point \(i=2,\dots,n-1\), it computes the perpendicular distance from that chord:

\[ d^\perp_i = \dfrac{\bigl\lvert (\,d_n - d_1\,)\,FoS_i \;-\;(\,FoS_n - FoS_1\,)\,d_i \;+\; FoS_n\,d_1 \;-\; d_n\,FoS_1 \bigr\rvert}{\,\sqrt{(\,d_n - d_1\,)^2 + (\,FoS_n - FoS_1\,)^2}\,}. \]

The point \(i\) of maximum \(d^\perp_i\) is taken as the critical FoS in the raw (linear) sense.

3. Raw (Log) – Curvature

Here, numgeo applies a logarithmic transform to the displacement data:

\[ X_i = \ln\!\bigl(d_i\bigr) \quad\text{if}\; d_i>0,\quad \text{else}\; X_i = -10^8\;(\text{fallback}). \]

Then we treat \(\text{FoS}_i\) as \(Y_i\). Numerically, we compute discrete first and second derivatives of \(Y_i\) w.r.t. \(X_i\):

\[ \begin{align} d1Log_i &= \dfrac{\,Y_{i+1}- Y_{i-1}\,}{\,X_{i+1}- X_{i-1}\,}, \\ d2Log_i &= 2\,\Bigl(\frac{Y_{i+1}- Y_i}{X_{i+1}-X_i} - \frac{Y_i - Y_{i-1}}{X_i - X_{i-1}}\Bigr)\bigl/\bigl(X_{i+1}- X_{i-1}\bigr). \end{align} \]

We then define a log‐space curvature,

\[ \kappa^\text{log}_i = \dfrac{\bigl\lvert d2Log_i\bigr\rvert}{\Bigl(1 + (d1Log_i)^{2}\Bigr)^{3/2}}. \]

The index \(i\) that maximizes \(\kappa^\text{log}_i\) is reported as the critical FoS in raw log‐space.

4. Uniform (Linear) – Curvature

numgeo also resamples the data onto a uniform FoS grid. Let

\[ FoS_{\text{min}} = FoS_1 ~~~;~~~FoS_{\text{max}} = FoS_n \]

and let

\[ dx_{\text{candidate}} = \min\Bigl(\frac{FoS_{\text{max}}}{1000}, \frac{d_n}{10^{-5}}\Bigr). \]

We define

\[ n_{\text{sub}} = \bigl\lfloor \frac{FoS_{\text{max}} - FoS_{\text{min}}}{dx_{\text{candidate}}} + 0.5 \bigr\rfloor, \]

where \(\lfloor\cdot\rfloor\) is “floor” for half‐away‐from‐zero rounding. Then

\[ \begin{align} \Delta x^{new} &= \frac{\,FoS_{\text{max}} - FoS_{\text{min}}\,}{\,n_{\text{sub}}\,},\\ N_{\text{uniform}} &= n_{\text{sub}} + 1. \end{align} \]

Hence, the uniform FoS array is:

\[ FoS^u_i = FoS_{\text{min}} + (i-1)\,\Delta x^{new}, \quad i=1,\dots,N_{\text{uniform}}, \]

with forced endpoints \(FoS^u_1= FoS_{\text{min}}, \, FoS^u_{N_{\text{uniform}}}= FoS_{\text{max}}\). The displacements are linearly interpolated onto these uniform FoS points. On that uniform dataset \(\{(FoS^u_i,\, d^u_i)\}\), we compute the same linear derivatives and curvature as in (1), picking the maximum as the critical FoS in uniform (linear) space.

5. Uniform (Linear) – Knee

On that same uniform data, the chord is drawn from \((FoS^u_1, d^u_1)\) to \((FoS^u_{N_{\text{uniform}}}, d^u_{N_{\text{uniform}}})\). Each intermediate point’s perpendicular distance is computed with the same formula from method (2). The maximum distance identifies the uniform‐space knee.

6. Uniform (Log) – Curvature

Finally, we log‐transform the displacements \(d^u_i\):

\[ \begin{align} X^u_i &= \ln(d^u_i)\quad(\text{or fallback for near-zero}),\\ Y^u_i &= FoS^u_i \end{align} \]

then compute the finite‐difference derivatives in \(X^u\) just like method (3). The point of maximum log‐space curvature is the critical FoS in “Uniform (Log) – Curvature.”

Notes on increments and scaling:

Monotonic Increments: The raw data points \(\{(FoS_i,d_i)\}\) come from each converged increment in the step type *Reduction. Because “non-converged” increments are halved, the final set of points can be quite fine near the critical region. This ensures the “knee” or strong curvature is more accurately located.
Uniform Grid Rounding: The half‐away‐from‐zero rounding \(\lfloor x + 0.5 \rfloor\) is used to emulate consistent behavior across compilers and matching Python reference scripts. Without this, minor differences can arise in the final \(\Delta x^{new}\).
Log‐Space Fallback: If any displacement is effectively zero or negative, we assign a fallback \(\log\) of \(-10^8\). In practice, these points rarely dominate the maximum curvature but ensure the code remains robust for small displacements.

All six of these post-processing approaches are automatically reported at the end of the strength reduction analysis, giving you multiple data-driven indicators for a “critical” factor of safety. Select whichever suits best your engineering judgment or project requirements.

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. ↩