You are reading the online version of the LimitState:GEO manual, which is also available as a pdf.

Modelling Guide

Model Definition and Solver

Model Definition

Problem geometries are built up using Geometry objects (see Problem Geometry Terminology). The two key geometry objects relevant to model definition are:

Solid

This is a 2D polygon defining a body of soil or other material. Its extent is defined by the surrounding Boundary objects.

Boundary

boundary This is a straight line that defines the edge or boundary of a Solid, or an interface between two Solids.

Generally the problem will be defined in terms of Solid objects. Boundary objects are automatically generated around Solid objects. Single Solid objects should be used for bodies of one material type. A problem such as a simple slope stability problem may thus consist of one solid, while a simple bearing capacity problem might consist of two solids, the footing and the underlying soil.

Boundary objects are used to define interface properties and set boundary conditions. In the example of the footing and the underlying soil, the soil/footing interface properties may be independently defined within the Boundary object that is the interface between the Solid objects representing the soil and the footing.

Solver Specification

The specification of the LimitState:GEO solver is as follows:

1. The software is designed to generate the optimal layout of slip-lines that make up the critical or failure translational sliding block mechanism for a specified plane strain problem.
2. The slip-lines are restricted to those that connect any two nodes within a predefined grid.
3. Slip-lines are restricted to those that connect nodes within a single Solid object, or between a node within a Solid object, and a node lying on an adjacent Boundary object.
4. The solution is given in the form of an adequacy factor. This is the factor by which specified load or material self weight must be multiplied by to cause collapse.
5. In LimitState:GEO problems involving rotational kinematics are modelled by permitting Solid objects to rotate as a single body. If such bodies rotate into a deformable body, then the rotational kinematics are modelled as equivalent translational kinematics at the interface between these bodies.

Solutions are generated using the upper bound theory of plasticity. Plasticity theory is a very common technique utilized in Geotechnical Engineering. It is assumed that the user is fully familiar with the advantages and limitations of plasticity theory. Discussion of some of the advantages and limitations may be found here.

Adequacy Factor and Factors of Safety

Introduction

Many different definitions of factors of safety (FoS) are used in geotechnical engineering. Three in common usage are listed below:

1. Factor on load.
2. Factor on material strength.
3. Factor defined as ratio of resisting forces (or moments) to disturbing forces (or moments).

The calculation process used to determine each of these factors for any given problem will in general result in a different failure mechanism, and a different numerical factor. Each FoS must therefore be interpreted according to its definition.

In general any given design is inherently stable and will be well away from its ultimate limit state. Therefore, in order to undertake a ULS analysis it is necessary to drive the system to collapse by some means. This can be done implicitly or explicitly. In many conventional analyses the process is typically implicit. In a general numerical analysis it must be done explicitly.

There are three general ways to drive a system to ULS corresponding to the three FoS definitions previously mentioned:

1. Increasing an existing load in the system.
2. Reducing the soil strength
3. Imposing an additional load in the system

LimitState:GEO solves problems using Method 1 by means of the Adequacy Factor. However it can be straightforwardly used to find any of the other two types of Factors of Safety.

Note that partial factor based design codes such as Eurocode 7 do not explicitly compute a factor of safety, but pre-apply factors to problem parameters. Application of this approach in LimitState:GEO is described in more detail in Use of Partial Factors.

Method 1

Consider the problem depicted below. The question that is posed by Method 1 is as follows: how much bigger does the load need to be to cause collapse, or, by what factor a does the load need to be increased to cause collapse: This factor a is the same as the Adequacy factor as used in LimitState:GEO.

---

---

Method 2

Consider the problem depicted below. The question that is posed by Method 2 is as follows: how much weaker does the soil need to be under the design load to cause collapse, or, by what factor F does the soil strength need need to be reduced to cause collapse. This factor F is the factor of safety on strength.

---

---

If it is required to determine the Factor of Safety on the soil strength in LimitState:GEO, then the recommended approach is to set up a series of Scenarios (see Scenario Manager) with partial factors on material properties across a suitable range according to problem type. The solution that produces a Adequacy Factor of 1.0 is the Factor of Safety on soil strength. It may be necessary to interpolate results to determine the Factor of Safety.

Method 3

Consider the problem depicted below. The question that is posed by Method 3 is as follows: If the soil is failing around the structure, what is the ratio R of resisting forces to disturbing forces. In this example, the factor R = A∕(P + S) is a factor of safety.

---

---

Note that if R > 1:

• the passive earth pressure and base friction significantly exceed the active earth pressure.
• The system is therefore completely out of equilibrium.
• The assumed earth pressures are not possible without some external disturbing agent.

In a numerical analysis, equilibrium is required at all times. Therefore in order to undertake a Method 3 type analysis, it is necessary to apply a ‘hypothetical’ external force H in the direction of assumed failure (as depicted below) and increase this force until failure occurs. It is then possible to determine the ratio of other resisting to disturbing forces as before (but H itself is ignored in this calculation). For this method it is therefore necessary to pre-determine the mode of failure.

---

---

Further information on the Adequacy Factor

As has been previously mentioned, LimitState:GEO provides solutions in terms of Adequacy Factor. An Adequacy Factor may be applied to any load or to the self weight of any body of material. The Adequacy Factor that is returned by LimitState:GEO when it has completed solving is the factor by which all the specified loads/self weights must be multiplied by to cause collapse. The Adequacy Factor is similar to a Factor of Safety on load.

It is important to note that if there are several actions driving collapse, yet an Adequacy Factor is applied only to one of them, then the Adequacy Factor may seem to have a misleadingly high sensitivity to parameter changes. For example in the Gravity Wall problem shown below, both the surface load and weight of soil behind the wall are driving it to collapse. If the Adequacy Factor is applied only to the surface load, but the load on the wall is dominated by the soil self weight, then large changes in Adequacy Factor will be required to cause any change in collapse state.

---

---

Use of Partial Factors

Introduction

LimitState:GEO is designed to work closely with the Eurocode 7 approach to Ultimate Limit State design. It has therefore adopted the Eurocode 7 definitions of actions and partial factors, which may be used if required in any analysis. These are sufficiently broad based enough to cover the needs of most other design codes.

In Eurocode 7 Design Approach 1 (as adopted in the UK), partial factors are pre-applied to loads and/or material properties prior to analysis. Assessment of safety is then undertaken by testing whether in the subsequent analysis, the available resistance to collapse exceeds the actions causing collapse. In LimitState:GEO this is equivalent to checking whether the Adequacy Factor (applied to any unfavourable load or self weight), is greater than 1.0.

The setting of Partial Factor values is carried out using the Scenario Manager. The available factors that may be set are shown below. Further details about the Scenario Manager may be found in Scenario manager.

---

---

The general principles implemented in LimitState:GEO are described below. However with respect to Eurocode 7, the following is not to be taken as a definitive guide. The engineer is expected to apply their own understanding of Eurocode 7, especially with regard to some of the subtleties that can arise in certain situations. If there are any inconsistencies between what is described below and the documented Eurocode, then the documented Eurocode should be followed.

Factoring of Actions (Loads)

Eurocode 7 specifies three different types of actions (loads). These are all available within LimitState:GEO:

1. Permanent
2. Variable
3. Accidental

The relevance of each action is the nature of the partial factor to be applied to it, with the corresponding values taken from the Scenario Manager. A Variable action will typically have a higher partial factor applied to it in comparison to a Permanent action.

Actions may be loads applied to external boundaries or may arise from the self weight of a block of material. The above settings can thus be applied to both Boundary loads and to Solids. Self weights are regarded as Permanent actions in LimitState:GEO.

Eurocode 7 also requires that each action is assessed as to its effect on the overall stability calculation. If it contributes to stability then it is Favourable, if it contributes to collapse then it is Unfavourable. Its Loading Type affects the value of partial factor to be applied to it. The following Loading Types may be applied to any Solid or Boundary:

Favourable:

Apply the favourable partial factors to any loads on a boundary or to the self weight of the materials within a solid.

Unfavourable:

Apply the unfavourable partial factors to any loads on a boundary or to the self weight of the materials within a solid.

Neutral:

Do not apply any factors to the loads on this boundary or to the self weight of the materials within a solid. (NB the type of action, permanent, variable or accidental has no relevance in this case.)

By default all Boundaries and Solids are set to Neutral when first created. It is up to the user to explicitly set them to Favourable or Unfavourable if required.

The purpose of the Neutral setting is to:

1. ensure that settings for any new problems that do not require analysis with partial factors, remain unambiguous and unaffected by partial factors.
2. ensure that for any problem that is to be analysed using partial factors (such as when using Eurocode 7), that the user must make explicit decisions about the nature of the actions i.e. change the setting to either Favourable or Unfavourable.
3. to facilitate modelling of problems where partial factors are to be applied to effects of actions rather than to the source actions themselves. In these cases Neutral might be used for the self weights of certain bodies.

Automatic factoring of source actions only is implemented in the current version of LimitState:GEO. To factor effects of actions a procedure similar to that described in Factor of Safety - Method 3 may be followed manually.

Note that in the Wizards, external loads are preset to Favourable or Unfavourable as appropriate. The self weight of structural elements such as footings may also be set to Unfavourable where they are unambiguous unfavourable actions.

For certain problems it can be a matter of debate as to whether the self weight of a soil body acts favourably, unfavourably or both. Thus in the LimitState:GEO Wizards, soil body self weights are always set to Neutral and should be amended by the user as appropriate.

In Eurocode 7, Neutral is equivalent to Favourable in Design Approach 1, Design Combination 1. In Design Combination 2, the factors on permanent actions are the same for both Favourable and Unfavourable effects, and therefore the setting is irrelevant.

For certain problems it can be unclear at the start whether a particular load is Favourable or Unfavourable. LimitState:GEO provides additional assistance in these cases. Following determination of the collapse load LimitState:GEO performs a check on all external actions to determine whether they acted favourably or unfavourably. If these are inconsistent with the original specifications, then the user is alerted to this (see Favourable / Unfavourable Settings) and may alter the specification and re-solve.

In a very small number of cases it is possible that the amended Favourable / Unfavourable settings may result in a different collapse mechanism and another set of inconsistent Favourable / Unfavourable settings. This is not a inherent problem with LimitState:GEO but simply a consequence of the Partial factor values. As always in these cases it is up to the engineer to apply their own judgement consistent with the principles underpinning the design code.

Factoring of Material properties

Partial factors may also be applied to material properties. Different factors are applied to the key parameters controlling collapse: the drained cohesion intercept (c′), the tangent of the angle of shearing resistance (tan′) and the undrained cohesion (c_u). In general self weight (regarded as a material property rather than as contributing to an action) is not factored.

Modelling slope stability problems

Introduction

In conventional slope analysis software a global factor of safety is applied to the soil shear strength parameters, and the analysis of many trial slip surfaces is conducted. In LimitState:GEO (in common with most generic numerical analysis software) a different approach must be taken.

To solve a slope problem with no externally applied loads, it is necessary to apply the adequacy factor to the self weight of the soil itself, which raises some interesting theoretical issues.

Slopes in cohesionless soils

In a purely cohesionless soil, geotechnical theory tells us that the slope is:

• stable for any friction angle greater than, or equal to, the slope angle, and
• unstable for any friction angle less than the slope angle.

Collapse is therefore entirely independent of the self weight of the soil (crudely speaking, as the self weight of the material goes up, the normal stresses go up, but frictional stresses also go up in exact proportion). For the above reasons, in a slope that is shallower than the angle of friction, factoring up the self weight on a frictional soil will not cause collapse and LimitState:GEO will return a *locked* result. Conversely, if the slope is steeper than the angle of friction then it will be found to be *unstable* under its own self weight.

To find the point of transition between these two states, a series of scenarios should be set up (see Scenario Manager) each with a progressively increased partial factor on tan′. Solving will indicate at what point the problem changes from being *locked* to being *unstable*.

To visualize the failure mechanism, simply repeat the process with a very small cohesion (say 0.01 kN/m2) so that the mechanism becomes visible when the partial factor on tan′ is sufficiently high. An example of the mechanism generated by this process is shown below.

---

---

An example file (cohesionless_slope_multi_PF.geo) illustrating the above process is included in the LimitState:GEO ‘example files’ directory. To open the file, locate it in:

[ROOT]/example files/

where [ROOT] is the installation path of LimitState:GEO, usually:

C:/Program Files/LimitState/GEO2.0/

Alternately, open the Example Files html page from the LimitState:GEO program menu and open the file from there:

Start > All Programs > LimitState:GEO 2.0 > Example Files

The file demonstrates more clearly the use of multiple partial factors to obtain the factor on strength. In this example the slope angle is 21.04 degrees. With a soil ′ angle of 30 degrees, the required factor on tan′ is 1.5011 to produce failure (i.e. to reduce ′ to 21.04 degrees). If the example file is solved, a series of scenarios is examined with partial factors on soil strength ranging from 1.0 to 2.0 in 0.05 intervals. Each scenario returns either a *locked* or *unstable* result. It will be seen that the solution changes from *locked* to *unstable* between an applied factor of 1.5 and 1.55 as expected.

Note that this changeover is sudden in theory and this is mirrored in LimitState:GEO. For example imagine a brick lying on a plank. If one edge of the plank is lifted and the plank’s angle to the horizontal is gradually increased, the brick will remain stable until the plank angle reaches the angle of friction between the brick and the plank. At this point the brick slides. Beyond this point the situation is inherently unstable. The situation is either stable or unstable, and is entirely unaffected by the weight of the brick. (If however there is cohesion and friction between the brick and the plank, then weight will have an effect, but only when the angle of the plank ≥ the friction angle. By analogy, in this type of problem LimitState:GEO will give solutions of *locked* for the type of case where the slope angle is less than the friction angle, but will give an adequacy factor for problems with slope angle greater than the friction angle).

Note that in general LimitState:GEO will predict a slightly higher factor than given by theory for pure cohesionless slopes since it must find a finite shallow failure mechanism, rather than an infinitesimally thick parallel slope failure mechanism. The predicted factor will also naturally depend on the chosen nodal density.

Once the switch over point from *locked* to *unstable* has been identified, this determines the solution to the stability problem. The purpose of introducing the small value of c′ is simply to aid visualization of the failure mechanism (it has nothing to do with finding the solution to the stability problem. The adequacy factor generated at this stage should be ignored.) The example file ‘cohesionless_slope_visualisation.geo’ (also included with the ‘example files’) illustrates this stage of the process for the original problem in ‘cohesionless_slope_multi_PF.geo’. In this case the partial factor value of 1.55 was taken as representative of the solution and the value of c′ for the soil set to 0.01kN/m².

If it is required to reuse the partial factor set in the ‘cohesionless_slope_multi_PF.geo’ example file, it can be exported to a .csv file and reused in other problems by importing it. See Scenario Manager Import/Export for further information on the import and export functions available in the Scenario manager.

Slopes in cohesive soils

In a purely cohesive soil, for certain slope geometries, the failure surface will always touch the edge of the domain (see for example the figure below). While counter-intuitive it is entirely in accordance with slope theory which predicts a failure mechanism of semi-infinite extent for e.g. finite shallow slopes in a semi-infinite domain of purely cohesive soil .

Thus changing the distance to the outer boundary of the problem (whether in the wizard or by dragging a boundary in the viewer) will not succeed in bringing the mechanism fully within the boundaries. However it will be found that the adequacy factor will converge to a fixed value.

---

---

Solution Accuracy

Introduction

This section discusses solution accuracy within the context of the DLO numerical method itself. For discussion of the accuracy of limit analysis in general, refer to Advantages and Limitations of Limit Analysis.

As with any numerical method, solution accuracy is dependent on the resolution of the underlying model. With DLO this relates to the distribution of nodes within Solid and Boundary objects. The method will provide the most critical sliding block mechanism that can be generated using slip-lines connecting any of the nodes. In many cases a sufficiently accurate solution will be generated for a coarse distribution of nodes. To assess solution accuracy, it is recommended that the nodal resolution be progressively refined, thereby allowing an assessment of the convergence characteristics to be made (towards the ‘exact’ solution).

Benchmarking results

When comparing LimitState:GEO results with known analytical solutions, it is important to interpret them with regard to the sensitivity of the result to parameter variation.

For example, the bearing capacity problem is notoriously sensitive to small changes in the angle of friction. The predicted bearing capacity of a surface footing on a cohesionless soil of high strength can double with an increase of friction angle of  3 degrees.

In these circumstances it is logical to think about the problem in terms of the input values rather than the output values which is the philosophy taken, for example, by Eurocode 7 Design Approach 1.

To illustrate this, consider the example, of the simple footing problem, with 1m wide footing, soil parameters: c′=5 kPa, =30 degrees, γ = 20 kN/m³ , smooth base, and soil domain 0.75m high by 2.25m wide. The exact benchmark solution for the collapse load is 268 kN. The results obtained from LimitState:GEO using a symmetrical half space model are as follows:

• Coarse 293 kN (9% variation from 268kN)
• Medium 282 kN (5% variation from 268kN)
• Fine 277 kN (3% variation from 268kN)

However if c′ and tan are decreased by 1% to give c′ = 4.95kPa and = 29.75o then the solution at a fine nodal resolution is the same as the exact benchmark solution. Thus the error in input parameters is  1%. It would be hard to predict c′ and to an accuracy of 1% from site investigation/lab test data.

For the medium nodal density the error is  1.5% : c′=4.925 kPa, = 29.63o. For the coarse nodal density the error is  2.5% : c′=4.875 kPa, = 29.39o.

LimitState:GEO is regularly benchmarked against a known set of limit analysis solutions from the literature. These tests are described in more detail here and may be accessed via the internet. Reference to these results can provide useful guidance as to the expected accuracy of the software over a range of problem types.

Other factors that can affect accuracy are described in the following sections.

Interfaces

To maximize computational efficiency, the solver does not model slip-lines that cross the interface (Boundary object) between one solid and another. The smoothness of the solution in the vicinity of the boundary is thus dependent on the nodal density on that boundary. LimitState:GEO automatically assigns a higher nodal density on boundaries and the net effect on the solution is usually very small. However for coarse nodal resolutions, the effect can be noticeable. Thus if a Solid object is split in half by a new Boundary object, then the value of the solution may increase slightly. The user may individually set the nodal density on boundaries (see Setting nodal density).

When using Engineered Elements it is also necessary to ensure a high nodal density is present on the relevant boundaries to ensure accuracy.

Small solid areas

As with all numerical software, numerical tolerance issues can cause the generation of unexpected results. This can occur when the software attempts to compute solutions based on numerically very small problem sizes. To assist users, LimitState:GEO undertakes a pre-solve check and issues a warning when the total area of the problem is less than 0.25 m². If the total area is less than 0.00025 m², then an error is issued.

Singularities

The DLO procedure is particularly suited to identifying singularities, i.e. fan zones. However it must be noted that for frictional soils, stress levels can increase exponentially around the fan. This means that solutions with singularities may be particularly sensitive to the number of slip-lines and thus nodal resolution in these areas. To improve the accuracy of a solution, increasing the number of nodes within Solid objects which contain dense patterns of slip-lines is likely to be beneficial.

Model Extent

In many cases, the problem domain is not of finite size, but is typically of semi-infinite extent, e.g. a footing on horizontal soil. In this case the user must specify a suitable domain that is large enough not to influence the solution. Since LimitState:GEO generates mechanism based solutions, it is straightforward to determine if the boundaries do or do not influence a solution; if the mechanism intersects or touches any of the boundaries, then it is likely that they have an influence, and may lead to a significant increase in load capacity. If the intention is to model an infinite or semi-infinite body of soil, then the boundaries should be moved outwards until the mechanism does not intersect or touch the given boundary.

Failure mechanisms dominated by rotations

In problems where significant rotational displacements of the soil occur (e.g. if modelling a vane test), the Edge Rotations model employed by LimitState:GEO can produce a mechanism that dissipates too much energy. Refer to the section on Rotational failure mechanisms for further discussion of this issue and approaches that may be employed to minimize its effect.

Adapting plane strain results to 3D

It is not possible to give specific guidance on adapting plane strain results to three-dimensional cases, since the situation will vary for any given geometry.

Shape factors are generally available for converting plane strain bearing capacity results to 3D geometries. However these are generally only valid for surface footings on homogenous soil bodies. It is not necessarily valid to utilize these factors for other soil geometries e.g. layered soils.

However it can generally be said that in most cases any 3D collapse mechanism will be more unfavourable than an equivalent plane strain mechanism (though defining the equivalence of mechanisms requires careful consideration). Thus a plane strain analysis such as is provided by LimitState:GEO will in general be conservative.

However it should be noted that there are exceptions to this principle. This is clearly illustrated by comparing the plane strain and axisymmetric collapse pressures for a footing founded on the surface of a cohesionless soil with angle of shearing resistance . Example results (from limit analysis) for rough and smooth footings are given in the below table. It can be seen that the axisymmetric collapse pressure is lower than the plane strain collapse pressure for values of less than ~ 30o.

Further discussion of the relationship between plane strain and 3D collapse pressures for footings may be found in Salgado et al. (2004) and Lyamin et al. (2007).

Troubleshooting

Insoluble Problems

Normally when a model is set up, it is preferable to carry out initial modelling utilising a coarse nodal distribution in order to check that the material models, geometry and boundary conditions have been correctly set up. Any error in these will become apparent in the nature of the solution.

Certain circumstances will lead to no solution being found. These include the following:

1. Parts of the problem are unstable, and the computed adequacy factor can be reduced down to zero (or beyond). Examples of such situations include:
   1. Slopes in cohesionless soils steeper than the angle of shearing resistance of the soil.
   2. Zero or insufficient fixed boundaries to hold the soil in place.
   3. Zero strength soil.
   4. Related to (b), free floating or or partially supported blocks of soil within the problem. This can sometimes not be immediately apparent if there is a narrow gap between blocks. Zooming in on suspect blocks may assist in identifying such areas. Alternatively select the suspect solid and drag it with the mouse. If it does not drag the adjacent solid with it, then it is not fully connected and there is a gap.
2. No solution will be found if the problem is locked. In this case no value of the adequacy factor will induce failure. Examples of such situations include:
   1. Slopes in frictional or cohesive-frictional soils shallower than the angle of shearing resistance of the soil.
   2. Problems where the adequacy factor is applied to both disturbing and restoring forces. For certain problems, failure cannot be induced if each force is increased simultaneously.
   3. Problems where boundary conditions are set such that deformation of a body of soil is fully constrained, for example attempting to indent a soil contained within fully fixed boundaries.
   4. Problems where frictional soils are modelled with boundary conditions that are highly restricted (but not necessarily fully fixed everywhere). This is due to geometrical locking. In plasticity models of frictional soils, shear deformation is accompanied by dilation. i.e. if there is no room available for the dilation to take place, then no solution will be found.

If an attempt is made to solve such a problem LimitState:GEO will output a suitable error message.

Troubleshooting Insoluble Problems

1. If not immediately obvious, it is recommended that the soil properties in all or suspected zones are changed to increase the influence of cohesion by increasing c_u or c′ as appropriate. If a solution is then found, try gradually reducing c_u or c′ until no solution is found. The cause of the problem should then be much clearer.
2. If the above does not work for a drained material, try also setting the friction to zero when first increasing the value of c′ and then as c′ is gradually reduced, simultaneously increase ′ back to its original value.
3. An alternative to the above is to reduce the soil self weight, or set it to zero. This should help to identify cases where the soil is collapsing under its own self weight.

Problems giving solutions that appear incorrect

On occasion LimitState:GEO can find a solution, with an accompanying adequacy factor and/or collapse mechanism that does not lie within the anticipated values. This can occur if the adequacy factor is inadvertently applied to parts of the problem that were not initially intended. Commonly this occurs with the adequacy factor applied to Solids. It is worth checking that the adequacy factor is applied as required by interrogating each solid and checking the setting in the PE. The settings for all solids can be checked by selecting them all using the rectangle select function, and then selecting Solids in the PE.