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

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Generic principles

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, material self weight or body acceleration 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.

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Figure 35: Method 1 Factor of Safety

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.

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Figure 36: Method 2 Factor of Safety

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.

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Figure 37: Method 3 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.

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Figure 38: Method 3 usage of hypothetical external force to determine Factor of Safety

Application of the Adequacy Factor

The Adequacy factor may be applied to one or more of the following parameters that result in a force within a problem:

  1. an applied load (see Specifiying Loads)
  2. a material self weight (see Specifiying Self Weight Loads)
  3. a body acceleration (see Specifiying Seismic Accelerations)

The reference in brackets refers to the part of the manual that details how the Adequacy factor is set on the given parameter.

In many problems, the Adequacy factor will be applied to a load. For problems such a slope stability where there is no externally applied load, then the Adequacy factor may be applied to a material self weight or a body acceleration. This is discussed further in Modelling Slope Stability Problems.

For seismic problems, it is often most convenient to apply the adequacy factor to a body acceleration. This is discussed further in Modelling Seismic Problems.

Where the Adequacy factor is applied to more than one parameter, then it is applied equally to each.

Note that when the Adequacy factor is applied to a material self weight and this material lies below the water table, then adequacy is also applied to the water pressure. In effect this means that for static groundwater conditions the Adequacy factor is applied to the buoyant weight of the soil.

Adequacy Factor Sensitivity

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/seismic accelerations 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.

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Figure 39: Gravity Wall Problem

Adequacy Factor Direction

Occasionally, LimitState:GEO may generate a failure mechanism that is unexpected by the user. This can sometimes relate to how the Adequacy factor is used.

When an Adequacy factor is applied to a load or self weight or acceleration, then associated with the factor is an Adequacy Direction (AD) . This direction is defined as follows:

Load
The AD is in the direction of application of the load and relates to the area of application of the load only.
Material Self weight
The AD is directed vertically downwards and relates to the entire zone to which the Adequacy factor is applied.
Acceleration
The AD is in the direction of the seismic acceleration (horizontal or vertical) taking into account the sign. This AD relates to the entire problem geometry.

The mathematical formulation of DLO utilised in LimitState:GEO requires that the identified critical failure mechanism must result in net positive work being done by the parameter to which the Adequacy factor is applied. In simple terms it means that the failure mechanism must result in collapse that involves net movement in the AD.

For example the movement of a load on a rigid foundation at collapse must be in the direction of the AD. Note that the component of movement of the load perpendicular to the AD is unrestricted.

For investigating the collapse of a slope in a single zoned body of homogeneous soil, the Adequacy factor is often applied to the soil self weight. In this case the identified collapse mechanism must involve more of the soil moving in the direction of the AD than away, i.e. more must move downwards than upwards.

This principle will normally give rise to intuitively expected solutions and does not normally require further consideration. However consider the issues involved in determining the active and passive collapse loads for the gravity wall depicted below. Active collapse of the wall would result in a required force on the wall less than that for passive collapse. However even though LimitState:GEO finds the smallest Adequacy factor that will result in collapse, it will find the larger passive value because only this mechanism results in the applied load moving in the direction of its application (the AD) as shown below . Intuitively this is what would normally be expected.

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Figure 40: Investigating passive failure of a gravity wall

The challenge does however remain as to how to determine the active force required for collapse. It is not possible to simply reverse the direction of the force to which the Adequacy factor is applied. In this case LimitState:GEO will return ‘Unstable’ since there is no positive value of the force that would bring the problem just to the point of collapse (only positive values of Adequacy factor can be found).

In order to find the active force, it is necessary to additionally apply a passive dead load DH against the wall (where D is the applied stress in kN/m2 , and H is the height of the wall in m) that exceeds the expected active force (any value will do) as depicted shown below. In this figure the green arrows represent the dead load (D ) while the red arrows represent the load to which Adequacy factor is applied. Let this be a unit stress (1 kN/m2 ). The solver will thus find the maximum value of Adequacy factor (AF) to cause collapse in the AD (i.e. active direction) as shown shown below. The required active load may be computed as a force of (D - AF × 1)× H .

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Figure 41: Investigating active failure of a gravity wall

A further example illustrating the effect of AD in problems involving seismic actions is given in Adequacy Factor Direction.

Finally it should be noted that where an Adequacy factor is applied to more than one parameter, then the identified collapse mechanism must involve movement in at least one of the specified Adequacy directions.

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.

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Figure 42: Available Partial Factors in the 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 ϕ ) and the undrained cohesion (cu ). In general self weight (regarded as a material property rather than as contributing to an action) is not factored.

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