Welcome to the Engineering Video Series.
This tutorial discusses convergence in SLOPE/W analyses,
including a demonstration of how to assess convergence,
typical reasons for convergence issues in limit equilibrium analyses,
and methods for improving convergence.
SLOPE/W uses an iterative procedure to determine the factor of safety of each slip surface.
This procedure runs until the computed factor of safety reaches a converged solution,
or in other words, when the factor of safety from the current iteration
is sufficiently similar to the factor of safety computed by the previous iteration.
The solver judges convergence based on the tolerable difference specified in the Advanced tab of the KeyIn Analyses window.
In addition to reaching a converged factor of safety, the Morgenstern and Price, and Spencer methods search for
a lambda value that produces a factor of safety satisfying both force and moment equilibrium.
The limit equilibrium method of slices assumes that the factor of safety is the same along the entire slip surface.
This means that the strength of each slice is reduced by the same factor to bring the system into a state of limiting equilibrium.
Therefore, large variations in strength along a slip surface will cause convergence issues.
In the scenarios, there is no lambda value causing the moment and force factors of safety to be equivalent.
SLOPE/W may generate one of the following error messages when convergence issues arise:
Error 981: A factor of safety could not be calculated at the zero lambda value or at the specified required lambda value.
And Error 994: Cannot find intersection of factor of safety and lambda.
Large variations in strength along a slip surface are caused by angularity in the slip surface,
high variability in the inputted strength properties, and concentrated loads resulting from
reinforcement, surface loads, point loads, or seismic loads.
SLOPE/W includes two methods for finding the lambda value that satisfies both moment and force equilibrium.
The first is the linear search method.
This method steps through lambda at regular intervals, calculating the factor of safety for
force equilibrium and factor of safety for moment equilibrium at each lambda value.
When the relationships between factor of safety and lambda are very steep or nonlinear,
as illustrated here, the linear search technique may terminate before finding the intersection
between the moment and force factor of safety plots.
These relationships are often steep when reinforcement is present in an analysis.
In these instances, the second lambda search method called the root finder method, may improve convergence.
The root finder method calculates the factor of safety for force equilibrium and
factor of safety for moment equilibrium at three lambda values near zero.
The results are used to estimate the lambda value at the intersection point.
The factors of safety for force and moment equilibrium are then calculated at this lambda value.
This forecasting technique is used until the difference between the two factors of safety is within a specified tolerance.
In this slope stability analysis, the slip surface is fully specified.
An error message appears in the Slip Surfaces window when I attempt to solve the analysis.
When I go to Draw | Graph to look at the factor of safety versus lambda plot, the graph window contains the message:
No valid values found.
Therefore, SLOPE/W could not compute one value for the factor of safety for this slip surface
likely due to the sharp change in the slip surface at this point.
When the Block Specified method is used, as in this second analysis,
there are many more slip surfaces generated.
A factor of safety was computed for many of the slip surfaces.
However, when I filter my results to show only the invalid slip surfaces, I see that there are still a
substantial number of slip surfaces with an undefined factor of safety.
If I select a slip surface with Error 994
and open the Draw Graph window,
I can see that the factor of safety versus lambda plot for moment equilibrium does not intersect the force equilibrium plot.
It appears that the lambda search method terminated before reaching the intersection point.
I will go back to the Define View, open KeyIn Analyses,
and select the Advanced tab.
Here I can change the lambda search method to Root Finder.
After resolving the analysis, I see that the list of invalid slip surfaces has decreased.
Thus, the lambda root finder method improved convergence.
The Grid and Radius method used in the third analysis, creates much smoother slip surfaces.
Consequently, there are no convergence issues in this third analysis
and SLOPE/W was able to compute a factor of safety for each slip surface.
Here is a SLOPE/W analysis with a similar geometry; however, the materials were modified
such that the top of the domain is very weak relative to the bottom.
The slip surfaces were defined with the Grid and Radius method to limit angularity
and the Root Finder lambda search method was selected.
In the Results View, you can see that a third of the slip surfaces are invalid due to convergence
issues and the computed factor of safety for the critical slip surface is quite high.
In the second analysis, I assumed that the slip surfaces will not penetrate this higher strength material
and changed the bottom portion of the domain to impenetrable bedrock.
The results of this analysis show that all of the slip surfaces are valid.
You will also see that the computed values for factor of safety are much lower than the first analysis.
This analysis includes two anchor reinforcement loads. By going to KeyIn | Reinforcement Loads,
I can see that a concentrated force distribution was selected for both of the anchors.
After solving this analysis, a factor of safety was computed for only a small portion of the trial slip surfaces.
In the Draw Graph window, I have plotted shear resistance versus slice.
This graph shows the large change in shear resistance at the anchor locations.
As previously discussed, variability in strength properties along the slip surface leads to convergence issues.
I will go back to Define View, open the KeyIn Reinforcement Loads window, and change the force distribution of both anchors to distributed.
I will then resolve the analysis.
The modified analysis was able to compute a factor of safety for many more of the trial
slip surfaces when the reinforcement load was distributed.
If I open the Draw Graph window again, I can see that the shear resistance changes more continuously with the slice number.
In conclusion, convergence issues in SLOPE/W occur when one factor of safety cannot be computed for a given slip surface.
These issues are generally caused by strength variation over a slip surface.
Convergence issues may be limited by:
creating smoother slip surfaces, for example by using the Grid and Radius method;
ensuring that the applied material properties are representative of the field conditions and by been conscious of material strength variability;
by distributing the applied loads;
and by selecting the Root Finder lambda search method.
Finally, the factor of safety versus lambda plot is an important tool for assessing convergence.
For more information on GeoStudio and its features, please take a look at the other tutorial videos on the GEO-SLOPE website.
Thank you for watching.
