Specific applications

LimitState:GEO

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Specific applications

Introduction

In this chapter, specific guidance on modelling specific problem types with LimitState:GEO is given. Example problems made available with the software are briefly described. More specific guidance on particular modelling issues are then given in later chapters.

Example problems

The LimitState:GEO software comes supplied with a range of example files covering a broad range of applications. These present recommended approaches to modelling these problem types. If guidance is being sought on modelling a specific type of problem then following the method used in these files or modifying these files to represent the required problem can be an effective approach.

Additional information on particular modelling issues associated with each example file is given in the Project Details box (accessible under the Tools menu item).

The example files may be accessed 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.

Examples types include problems involving:

Slope stability
Embankment stability
Reinforced earth
Retaining walls
Propped excavation
Quay wall
Tunnel
Pipeline
Reinforced earth walls
Foundations on heterogeneous soils

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 static slope problem with no externally applied loads, it is necessary to apply the adequacy factor to either

the self weight of the soil itself, or
an artificially imposed pseudo static horizontal acceleration,

which raises some interesting theoretical issues.

Slopes in cohesionless soils (adequacy on self weight)

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 $tan ϕ′$ . 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 $tan ϕ′$ is sufficiently high. An example of the mechanism generated by this process is shown below.

Figure 43:

Visualising the failure mechanism for a cohesionless slope by applying a small value of $′ c$ . Note that the actual failure mechanism would be shallow parallel slope failure of infinitesimal depth.

An example file (cohesionless_slope_multi_PF.geo) illustrating the above process is included in the LimitState:GEO ‘example files’ directory. See this Appendix for further information on accessing 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 $′ ϕ$ angle of 30 degrees, the required factor on $′ tan ϕ$ is 1.5011 to produce failure (i.e. to reduce $′ ϕ$ 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/m $2$ .

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 cohesionless soils (adequacy on horizontal acceleration)

An alternative to using an adequacy factor on self weight to drive a cohesionless slope problem to failure is to use an adequacy factor applied to a horizontal pseudo static acceleration (see Modelling seismic loads). In this case a similar procedure to that described in the preceding section is required, however this approach has the advantage that instead of generating *locked* results for statically stable slopes, a collapse mechanism will be found with a corresponding adequacy factor. This adequacy factor will reduce towards zero as the switch over point is approached. After this point *unstable* will be returned as before. It is thus unnecessary to impose a small cohesion into the problem to visualise the collapse mechanism at switch over.

An example file (cohesionless_slope_multi_PF_kh.geo) illustrating the above process is included in the LimitState:GEO ‘example files’ directory. See this Appendix for further information on accessing example files. It will be seen that the solution changes to *unstable* between an applied factor of 1.5 and 1.55 as was found in the abovesection. The collapse mechanism for the lowest value of Adequacy factor found is depicted below

Figure 44:

Visualising the failure mechanism for a cohesionless slope by applying an artificial horizontal acceleration. The failure mechanism modelled is one involving a very shallow almost parallel slope failure as expected.

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.

Figure 45:

Slope failure mechanism in a cohesive soil, touching boundaries at the left and right edges.

Modelling soil reinforcement

Introduction

Soil reinforcement is typically modelled using the Engineered Element material in LimitState:GEO. This is a material that can be applied to Boundaries to provide essentially one dimensional (in section) objects such as soil nails, geotextiles, and sheet pile walls. The special properties of an Engineered Element allow soil to flow through or past it if required as would be expected of e.g. a soil nail. This would otherwise not be possible in a plane strain program such as LimitState:GEO.

In other modelling approaches, soil reinforcement is often only modelled by its effect such as an imposed point force on a slip surface. This makes it challenging to model complex failure mechanisms and requires the user to pre-judge to some extent the effect of the reinforcement. In LimitState:GEO the software is given full freedom to determine the critical failure mechanism involving soil reinforcement. The parameters required for the soil reinforcement may thus not be familiar to users familiar with other approaches. However these parameters are typically required when estimating e.g. an imposed point force on a slip surface.

The parameters defining the behaviour of an Engineered element are the Pullout Factors $Tc$ and $Tq$ , the Lateral factors $Nc$ and $Nq$ and the plastic moment $Mp$ . The theoretical background to these parameters is given in Theory of Engineered Elements. Guidance on defining Engineered element geometry using LimitState:GEO is given in Engineered element geometry.

Simple reinforcement pullout test

The simple pullout test modelled in the figure below illustrates how the parameter $Tc$ works. The reinforcement (coloured green in the diagram) is embedded in both a weightless rigid block (height 1m) on the left and a body of soil of length 2m on the right. It is assumed that it has a pullout resistance of 0.1 kN/m $2$ in the soil. The aim of the test is to determine what force is required to displace the rigid block to the left, thus pulling the reinforcement out of the soil. To ensure the reinforcement remains embedded in the rigid block but can pullout of the soil it is necessary to define two Engineered Element materials. The first is given a high value of $T c$ (e.g. 1E+30 kN/m $2$ ) and is assigned to the portion of the reinforcement in the rigid block (dark green in the diagram). The second is given a value of $Tc$ of 0.1 kN/m $2$ and is assigned to the portion of the reinforcement in the soil block (light green in the diagram). A unit stress with Adequacy factor (red arrows) is applied to the left hand edge of the rigid block. On solving the block displaces to the left and the light green portion of the reinforcement pulls through the soil. An Adequacy factor of 0.2 is returned, indicating that a force of 0.2 kN/m was required to pull out the 2m of reinforcement, with $Tc$ = 0.1 kN/m $2$ .

Figure 46:

Simple reinforcement pullout test.

Simple reinforcement lateral displacement test

The simple lateral displacement test modelled in the figure below illustrates how the parameter $Nc$ works. The reinforcement, for example a soil nail, (coloured green in the diagram) is embedded in a body of soil of width 3m and attached to two weightless rigid blocks (width 0.5m) either side of the soil body. It is assumed that it has a lateral displacement resistance of 0.1 kN/m $2$ in the soil. The aim of the test is to determine what force is required to displace the rigid blocks upwards, thus displacing the reinforcement through the soil. To ensure that the reinforcement remains fixed to the rigid blocks but can displace through the soil it is necessary to define two Engineered Element materials. The first is given a high value of $Nc$ (e.g. 1E+30 kN/m $2$ ) and is assigned to the portion of the reinforcement adjacent to the rigid blocks (dark green in the diagram). The second is given a value of $Nc$ of 0.1 kN/m $2$ and is assigned to the portion of the reinforcement in the soil block (light green in the diagram). A unit stress with Adequacy factor (red arrows) is applied to the base of each rigid block (i.e. over a total width of 1m). On solving the blocks displace upwards and the light green portion of the reinforcement pulls through the soil. An Adequacy factor of 0.3 is returned, indicating that a force of 0.3 kN/m was required to displace the 3m of reinforcement upwards, with $Nc$ = 0.1 kN/m $2$ .

Figure 47:

Simple reinforcement lateral displacement test.

Soil reinforcement predefined types

Introduction

LimitState:GEO provides five predefined Engineered Element parameter sets, representing five different common soil reinforcement types. These are described as follows. Each parameter set contains fixed values and suggested values which may be modified.

Soil nail (rigid)

A rigid soil nail has its ‘Plastic Moment’ set to 1e+30 kNm/m (effectively infinite) to ensure that the element behaves as a rigid body and cannot form any plastic hinges. The properties $Tc$ , $Tq$ , $Nc$ , and $N q$ are ideally determined by field tests. However estimates using theory may be made. For example if a nail of diameter $B$ is embedded in a cohesive soil of undrained shear strength $cu$ then plasticity theory could be used to estimate the ‘Pullout Factor’ $Tc$ as $πBcu$ and the ‘Lateral Factor’ $Nc$ as $~ 9cuB$ based on laterally loaded pile theory for widely spaced piles. Both values might then be factored. For an undrained cohesive soil the pullout and lateral factors $Tq$ and $Nq$ are independent of depth and therefore would be set to zero.

Certain current alternative analysis methods do not take account of the lateral resistance of soil nails and so $Nc$ and $Nq$ could be set to zero when comparing with these analyses. However in this situation, occasionally LimitState:GEO may identify a failure mode where the nails simply ‘float’ out of the ground, because they have no lateral resistance. It is therefore recommended that a small value of $Nc$ is set of the order of 5 kN/m $2$ .

Soil nail (can yield at vertices)

This type of soil nail behaves as the rigid soil nail except that a finite value of plastic moment is set so that the nail may form plastic hinges at Vertices along the nail. The plastic moment (in kNm) of each nail can be estimated from a knowledge of the nail cross section geometry and the yield strength of the nail material. To convert this to the value required by LimitState:GEO for plane strain modelling it is necessary to multiply by the number of nails present per metre width (into the diagram).

Soil nail (flexible at vertices)

This type of soil nail behaves as the rigid soil nail except that a zero value of plastic moment is set so that the nail may freely hinge at Vertices along the nail if required.

Sheet pile wall (rigid)

To represent a rigid sheet pile wall the values of ‘Pullout Factor’ $Tc$ , ‘Lateral Factor’ $Nc$ and ‘Plastic moment’ $Mp$ are all set to 1e+30 i.e. effectively infinity.

Setting the ‘Pullout Factor’ $T c$ to infinity prevents the wall pulling out by shearing along the wall/soil interface. In effect this would prevent relative shear displacement between a wall and the surrounding soil. However in reality shear may still occur just adjacent to the wall within the soil only. LimitState:GEO always models shear along a boundary such as this as potentially occurring using either the wall/soil interface strength (if specified) or the soil shear strength only. In practice the lowest strength will always be used. The effect is that the wall/soil interface strength is thus modelled as the same as the soil itself.

If it is desired to model a lower interface strength then this could be set as the value of ‘Pullout Factor’ $Tc$ . In practice this will be the same on both sides of the wall and will only represent a cohesion. If a frictional interface strength is required that differs from the surrounding soil then it would be necessary to model the sheet pile wall as a Solid zone of narrow width rather than using an Engineered Element, or to model narrow soil zones on either side of the pile with modified shear parameters.

Setting the ‘Lateral Factor’ $Nc$ to infinity prevents any soil flowing through or past the element as would be expected for a sheet pile wall.

Setting the ‘Plastic moment’ $Mp$ to infinity ensures that the element behaves as a rigid body and cannot form any plastic hinges.

Sheet pile wall (can yield at vertices)

This type of sheet pile wall behaves as the rigid sheet pile wall except that a finite value of plastic moment is set so that the wall may form plastic hinges at Vertices along the nail. The plastic moment per metre width (into the diagram) can be estimated from a knowledge of the wall thickness and the yield strength of the wall material.

Other

This ‘Application’ type is used for user defined reinforcement properties which do not fit the predefined values of the other types. All parameters are freely editable.

Modelling seismic loads

Modelling of seismic loading using DLO

In LimitState:GEO horizontal and/or vertical pseudo static acceleration may be applied to any problem. These may be applied with or without adequacy applied to them. Often it may be required to check a design against seismic loading and the question may be posed as follows: ’What value of horizontal acceleration is required to cause collapse’. This may be straightforwardly done by applying adequacy to the horizontal acceleration (details of how to enter seismic parameters via the user interface in LimitState:GEO are given in Setting seismic parameters).

In general it is recommended that Adequacy be applied to horizontal acceleration only or to both horizontal and vertical acceleration. Use of Adequacy on vertical acceleration only should be done with caution as it may result in correct but unexpected results as described in Seismic Adequacy Direction example)

Example 1 - modelling a simple mechanics problem

Consider the sliding block problem in the figure below. The rigid block is of dimensions 1m $×$ 1m and of unit weight 1 kN/m $3$ . It rests on a rigid platform separated by a frictional interface with $tan ϕ = 0.5$ . A fixed unit stress (green arrows) of 1 kN/m $3$ is applied to the left hand face of the block. The question is ‘what horizontal acceleration $kh$ applied right to left will cause failure’.

To determine this using LimitState:GEO the geometry and parameters are entered as described. An example file of this problem (simple_sliding_block_seismic.geo) is included in the LimitState:GEO ‘example files’ directory (see Appendix: Accessing Example Files for further information on accessing example files).

Figure 48:

Simple sliding block example under horizontal seismic loading.

To find the required horizontal acceleration to cause failure, set Horizontal Accel. k $h$ (g) to 1.0, Adequacy (on k $h$ ) to True and solve. LimitState:GEO will return an Adequacy factor of 1.5 and show a sliding failure. This means that the horizontal acceleration must be 1.5g to cause failure. (Note that the Horizontal Accel. k $h$ (g) could have been set to e.g. 0.3, in which case an Adequacy factor of 5 would have been returned, still giving a required horizontal acceleration of 1.5g to cause failure). This can be checked by simple mechanics:

Resolving horizontally,

khW = P + W tanϕ

(15)

where $W$ is the weight of the block and $P$ is the applied fixed force. For the given units this gives:

kh × 1.0 = 1.0+ 1.0× tan ϕ = 1.5

(16)

Example 2 - checking the Mononobe-Okabe equation

Consider the simple retaining wall problem in the figure below. The wall is modelled as a weightless rigid block of height 1m resting on a smooth base. It retains soil of angle of shearing resistance 30 $o$ , and the soil/wall interface angle of shearing resistance is also set to 30 $o$ . The seismic active earth pressure acting on the wall due to a horizontal acceleration of 0.1g is required.

To determine this using LimitState:GEO the geometry is entered as described. An example file of this problem (simple_retaining_wall_seismic_1.geo) is included in the LimitState:GEO ‘example files’ directory (see Appendix: Accessing Example Files for further information on accessing example files).

To find the required active pressure, set Horizontal Accel. k $h$ (g) to 0.1 and the Adequacy (on k $h$ ) to False. To determine an active earth pressure it is necessary to set one of the normal loads (e.g. the permanent load) on the left hand face of the wall to 1.0 kN/m $2$ without Adequacy and another (e.g. the variable load) on the left hand face of the wall to -1.0 kN/m $2$ with Adequacy (see Adequacy Factor Direction for an explanation of why this is required to obtain active earth pressures). On solving with a medium resolution, LimitState:GEO will return an Adequacy factor of 0.8377 and show a compound wedge failure mechanism.

This means that the active earth pressure is the sum of the applied loads i.e. 1.0 - 0.8377 = 0.1623 kN/m. This is slightly higher than the value of 0.1611 kN/m given by Mononobe-Okabe equation for this problem. The reason for the discrepancy is that the Mononobe-Okabe equation is based on the assumption of a single wedge failure mechanism whereas in fact for a rough wall the true mechanism differs slightly from this.

Figure 49:

Simple retaining wall example under horizontal seismic loading (1).

It is possible to pose this problem in an alternative form asking ‘what horizontal acceleration is required to give a resultant active earth pressure on the wall of 0.1611 kN/m’. This is modelled by the example file simple_retaining_wall_seismic_2.geo as depicted in the figure below. In this case the Horizontal Accel. k $h$ (g) is set to 1.0 and the Adequacy (on k $h$ ) to True. One of the normal loads (e.g. the permanent load) on the left hand face of the wall is set to 0.1611 kN/m $2$ without Adequacy. On solving with a medium resolution, LimitState:GEO will return an Adequacy factor of 0.09665 and again show a compound wedge failure mechanism. This implies that the wall will fail at a horizontal acceleration of 0.09665. This is close to the expected horizontal acceleration of 0.1, but again LimitState:GEO has found a slightly more optimal failure mechanism.

Figure 50:

Simple retaining wall example under horizontal seismic loading (2).

Example 3 - correct use of Adequacy Direction

Consider again the sliding block problem in the figure below with the same parameters as used in Example 1. In this case the question posed is ‘what value of horizontal acceleration $kh$ applied right to left in combination with a vertical acceleration $k = 0.5k v h$ applied downwards will cause failure. To find the required solution, set Horizontal Accel. k $h$ (g) to 1.0, Adequacy (on k $h$ ) to True , Vertical Accel. k $v$ (g) to 0.5, Adequacy (on k $v$ ) to True and solve. LimitState:GEO will return an Adequacy factor of 2.0 and show a sliding failure similar to Example 1.

This can again be checked by simple mechanics:

khW = P + (1 + kv)W tan ϕ

(17)

where $W$ is the weight of the block and $P$ is the applied fixed force. For the given units this gives:

kh × 1.0 = 1.0 + (1 + 0.5kh)× 1.0× tan ϕ

(18)

and

0.75kh = 1.5

(19)

or $kh = 2$ .

To facilitate simple comparison of several solutions, it is helpful to model the scenario such that the Adequacy factor returned is a value of 1.0. To do this set Horizontal Accel. k $h$ (g) to 2.0, and Vertical Accel. k $v$ (g) to 1.0.

If now the acceleration on which Adequacy is applied is changed then the results in the table below are obtained. The anomalous result in the final row relates to the issue of Adequacy Direction (see Adequacy Factor Direction). In the first two cases the failure mechanism involves sliding in the direction of the applied horizontal acceleration and, due to dilation, a small upward movement of the block, against the direction of the applied vertical acceleration. In the third case, the solver is restricted to finding a failure mode in the specified direction of the applied vertical acceleration only. The sliding mode is thus not possible and since both blocks are rigid, a bearing type failure cannot be found and a result of *locked* is returned.


Adequacy on $kh$	Adequacy on $kv$	Result (Adequacy factor)

Yes	No	1.0
Yes	Yes	1.0
No	Yes	locked

Table 1:

Result of applying Adequacy to different parameters

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, $ϕ$ =30 degrees, $γ$ = 20 kN/m $3$ , 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 $tanϕ$ are decreased by 1% to give $c′ = 4.95kP a$ and $ϕ = 29.75o$ 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 $ϕ$ 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, $o ϕ = 29.63$ . For the coarse nodal density the error is 2.5% : $′ c$ =4.875 kPa, $o ϕ = 29.39$ .

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 m $2$ . If the total area is less than 0.00025 m $2$ , 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 $ϕ$ . 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 $ϕ$ less than $o ~ 30$ .

$q∕γB$




	Rough		Smooth

$ϕ′$ (degrees)	plane strain	axisymmetric	plane strain	axisymmetric

10	0.217	0.161	0.140	0.103

15	0.591	0.466	0.350	0.267

20	1.42	1.21	0.79	0.64

25	3.25	3.04	1.73	1.49

30	7.38	7.76	3.83	3.56

35	17.2	21.0	8.79	9.02

40	42.8	61.9	21.6	25.1

45	117	209	58.8	79.9

50	371	855	186	309

Table 2:

Comparison of plane strain and axisymmetric collapse pressures ( $q$ ) for a footing of width or diameter B, founded on soil of unit weight $γ$ .

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:

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.
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

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 $cu$ or $′ c$ as appropriate. If a solution is then found, try gradually reducing $cu$ or $′ c$ until no solution is found. The cause of the problem should then be much clearer.
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 $ϕ′$ back to its original value.
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.

In some circumstances the problem may relate to the issue of adequacy direction.

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