Discontinuity Layout Optimization

LimitState:GEO

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Discontinuity Layout Optimization

Introduction

At the heart of LimitState:GEO is a solution engine which uses the Discontinuity Layout Optimization (DLO) numerical analysis procedure to find a solution. The procedure was developed at the University of Sheffield and was first described in a paper published in the Proceedings of the Royal Society (Smith & Gilbert 2007a). In essence DLO can be used to identify critical translational sliding block failure mechanisms, output in a form which will be familiar to most geotechnical engineers (for example see Retaining wall for a Coulomb wedge analysis of a retaining wall, or for a method of slices solution for a slope stability analysis problem see Slope stability). However while traditional methods can typically only work with mechanisms involving a few sliding blocks, DLO has no such limitations. It can identify the critical translational failure mechanism for any geotechnical stability problem, to a user specified geometrical resolution. This can be illustrated by examining the critical solution to the vertical anchor uplift problem in cohesive soil problem, as shown in Anchor.

Figure 22:

Example of Coulomb wedge analysis of a gravity retaining wall

Figure 23:

Example of slope stability by method of slices

Figure 24:

Example of the DLO solution of vertical anchor uplift in a cohesive soil

Limit Analysis and Limit Equilibrium

Determination of the ultimate limit state (ULS) or ‘collapse state’ of any geotechnical problem has traditionally been carried out using a range of approaches. The most commonly used approaches are Limit Equilibrium and Limit Analysis.

Limit Analysis procedures are rigorously based upon the theorems of plasticity while Limit Equilibrium typically involves a similar form of calculation but relaxes one or more of the conditions of plasticity theory to make the problem easier to solve. For example the method of slices used to analyse slope stability problems is a Limit Equilibrium method. The sliding mass is divided up into a number of independent blocks. Equilibrium of each block is not fully determined; instead an assumption is made concerning inter-slice forces. This may result in a solution that does not fully satisfy equilibrium, or the yield condition for all blocks, but is typically found to give reasonable results.

Limit Analysis is used to determine many of the bearing capacity and retaining wall formulae typically used by geotechnical engineers.

DLO is a limit analysis method that effectively allows free choice of slip-line orientation, and the critical solution identified may involve the failing soil mass being divided into a large number of sliding blocks. Accuracy can be assessed by determining the influence of nodal refinement. DLO also readily handles variation of soil parameters, and heterogeneous bodies of soil.

DLO: How Does it Work?

Basic principles

Discontinuity Layout Optimization (DLO), as its name suggests, involves the use of rigorous mathematical optimization techniques to identify a critical layout of lines of discontinuity which form at failure. These lines of discontinuity are typically ‘slip-lines’ in planar geotechnical stability problems and define the boundaries between the moving rigid blocks of material which makeup the mechanism of collapse. Associated with this mechanism is a collapse load factor, which will be an upper bound on the ‘exact’ load factor according to formal plasticity theory. Thus in essence the procedure replicates and automates the traditional upper bound hand limit analysis procedure which has been used by geotechnical engineers for many years. DLO is general, and can be applied to a wide range of geotechnical stability problems involving cohesive and/or frictional soils.

In order to allow a wide range of different failure mechanisms to be identified, a large number of potential lines of discontinuity must be considered. In order to achieve this, closely spaced nodes are distributed across the problem domain and potential lines of discontinuity are created to connect each node to every other node, thereby providing a very large search space. In numerical terms, if there are $n$ nodes, there are approximately $n(n - 1)∕2$ possible slip-lines and approximately $2n(n- 1)∕2$ possible slip-line mechanism topologies. Thus for example 500 nodes give rise to $≈$ 125,000 possible slip-lines and of the order of 10 $38000$ possible mechanism topologies.

A simple example involving the bearing capacity of a footing is given in this figure. The thin black lines indicate the set of potential discontinuities to be considered and these connect an initial set of nodes (for sake of clarity very coarse nodal refinement has been used, and only near-neighbour nodes have been inter-connected). Whilst there is no restriction on the pattern of nodal distribution utilized, square or triangular grids are generally most convenient. In LimitState:GEO uniform square grids aligned to the global $x$ and $y$ axes are utilized. The solution will clearly be restricted to sets of discontinuities that connect these nodes, and is thus the solution will be influenced by the starting positions of the nodes. However when fine nodal refinement is used, the exact positions of individual nodes will have relatively little influence on the solution generated.

In the DLO procedure the problem is formulated in entirely in terms of the relative displacements along discontinuities, e.g. each potential line of discontinuity can be assigned a variable that defines the relative slip displacement along that discontinuity. When relative displacements are used, compatibility can be straightforwardly checked at each node by a simple linear equation involving these variables. Finally an objective function may be defined in terms of the total energy dissipated in sliding along all discontinuities. This is a linear function of the slip displacement variables. A linear optimization problem is thus defined, the solution of which identifies the optimal subset of discontinuities that produce a compatible mechanism with the lowest energy dissipation (thick lines in the below figure). The accuracy of the solution obtained depends on the prescribed nodal spacing. As is evident from here, a key benefit of the procedure, compared with comparable ‘element based’ procedures, is that singularities can be identified without difficulty (potential fan zones centred on a given node can be identifies if critical by linking multiple lines of discontinuity to that node). Finally, while the fact that discontinuities are clearly free to ‘cross over’ one another might appear problematic, it can be shown that compatibility is implicitly enforced at ‘cross over’ points, and their presence is very beneficial as the search space is dramatically increased. Further discussion of this issue and a detailed description of the DLO procedure is given in Smith & Gilbert (2007a).

Figure 25:

DLO analysis of the undrained stability of a footing

Adaptive Solution Procedure

While LimitState:GEO utilizes the core methodology described above, it also makes use of an adaptive solution procedure described by Gilbert & Tyas (2003) in order to significantly reduce memory requirements and also reduce the time needed to obtain a solution. The procedure makes use of the fact that the solution identified by the linear programming solver can be used to determine the forces on any slip-line connecting any pair of nodes, even if the solver did not explicitly consider that slip-line in determining its solution. In essence the procedure operates as follows:

Set up an initial ‘ground state’ in which nodes are connected by slip-lines only to nearest neighbour nodes vertically, horizontally and diagonally.
Solve this problem utilising the DLO procedure. This will give a valid upper bound solution, but of relatively low accuracy, utilising only these short slip-lines.
Utilizing this solution, search through all potential slip-lines connecting every node to every other node and determine whether yield is violated on any potential slip-line.
If yield is violated on a potential slip-line, add this slip-line to the linear optimization problem set and re-solve. Repeat from Stage 3.
If yield is not violated on any slip-line then the correct solution has been arrived at, as if all possible slip-lines had been included in the original linear programming problem.

Rotational failure mechanisms

Implementation

The current implementation of DLO in LimitState:GEO generates solutions based on translational mechanisms. In order to model mechanisms involving rotation of structural elements such as cantilever retaining walls or eccentrically loaded footings, it is necessary to set the Model Rotations value in the Project level Property Editor to Along Edges. (If this value is reset to False then a purely translational solution will be found.)

With Along Edges set, LimitState:GEO allows rotations to be modelled along Boundaries. This allows Solids to rotate as rigid bodies and to transmit these rotations into translational deforming bodies by modelling localized rotational deformations along the boundary. This is an approximation to modelling rotational and translational failure everywhere, but is generally accurate enough for most problems while keeping the size of the numerical problem smaller. The approach is depicted below, where a series of small rotational elements are modelled along the length of the boundary. This maintains the upper bound status of the solution. However in some cases modification to the boundary properties are required to ensure collapse loads are not overestimated (see following subsection).

Figure 26:

Modelling of rotational elements along a rotating boundary

Figure 27:

Geometry of log-spiral

In the following analysis $c$ and $ϕ$ may represent either $c′$ and $ϕ ′$ for drained parameters or $cu$ (and $ϕu = 0$ ) for undrained parameters. The geometry of each rotational element is depicted in the figure above. In general the shape of the slip surface in this element will be a log-spiral. From the geometry of the log-spiral, the following expression can be obtained for $r 0$ :

r = -----l------ 0 (1+ eπ tanϕ)

(4)

where $l$ is the length of segment along the boundary. If the boundary rotates by an angle $ω$ , then the resultant effect is an equivalent rotation about the segment midpoint $M$ by $ω$ , accompanied by an additional dilation:

nω = ulω

(5)

where

1 u = 0.5 - ------πtanϕ-- (1 + e )

(6)

Outside the small rotational element, movement is purely translational and equal to the translation of the segment midpoint. If there is also a cohesion $c$ present, then the work done $W$ on the arc $AB$ overcoming the cohesion $c$ for a relative body rotation of $ω$ is given by the following expression:

2 W = cωul-- tan ϕ

(7)

The derivation of this equation may be found in the Appendix.

Examination of equation ??eqn:rotationWork:: and equation ??eqn:rotationDilation:: indicates that $l$ gets smaller, both the dilation and work done associated with the rotational elements tends to zero. A reduction in $l$ corresponds to an increase in nodal density on the rotating boundary. However any sliding on the boundary is not coupled to the rotations, thus a relative slip $s$ will give rise to an additional dilation component of $n = stan ϕ$ , and the overall dilation or normal displacement is given by the following equation:

n = s tan ϕ + ulω

(8)

Only the material properties in the solid are used for parameters in the (edge) rotational equations. If a separate material is defined on the boundary then LimitState:GEO utilizes the weakest of the solid and boundary material to determine dilation and energy dissipation due to sliding along this boundary.

It should be noted that LimitState:GEO assumes that the boundary segments and thus rotational elements are small. LimitState:GEO makes no checks to see whether the modelled rotational elements remain within the adjacent solid and do not extend into adjacent solids which may have different properties. Hence this mode should only be used with high nodal densities. It should also be noted that in circumstances where the actual true failure mechanism happens to be similar to the modelled rotational mechanism for a few nodes, then increasing the nodal density may cause a increase in collapse load rather than decrease as would normally be expected.

Additionally any work done against gravity by the soil within the rotational elements is neglected. As the size of the elements reduces (as nodal density increases), this work term tends to zero.

Accuracy of edge rotations model

The edge rotations model is an approximation to modelling rotational and translational failure everywhere, but is generally accurate enough for most problems while keeping the size of the numerical problem smaller. It works well for problems that are dominated by translational displacements in the soil (even if rigid elements are undergoing significant rotation, see for example the case in the below figure).

Figure 28:

Mechanism involving rigid body rotation, but translation (shear) deformation of soil

However for problems where strong rotational soil displacements occur such as in eccentrically loaded footings, the method tends to overestimate collapse loads because in the process of modelling the collapse mechanism beneath the footing, it models additional sliding between the soil and the boundary interface. This leads to extra work either against cohesion or against gravity (due to additional dilation). Examples of measures that may be used to remove this effect in the Edge Rotations model are described below:

The Boundary strength parameters may be reduced by a suitable scale factor.
For eccentrically loaded footings placed on the horizontal surface of a cohesionless soil. The following modification has been found to give satisfactory results:
$tan ϕrb = tan ϕb × max (1- tan ϕs, 0)$ (9)

where $tanϕrb$ is the reduced value of $tanϕb$ to be used on the Boundary, and $ϕs$ is the value of angle of shearing resistance in the adjacent Solid.
For undrained eccentrically loaded footings it is recommended that the cohesion on the Boundary is set to zero.
The eccentric footing problem is likely to be a class of problem most significantly affected by this issue, and the above reductions will be conservative in most cases. It should be noted that if failure by sliding or rotation is possible, then the above strength reduction method may result in an enhanced likelihood of sliding failure. While this may not be the intended effect, a collapse state has still been found.

For problems with internal corners on the rotating solid, the solver can be assisted by linking the adjacent vertices either side of the corner as shown in the below figure of a footing with attached sheet pile. This allows all the soil ‘internal’ to the corner to fully rotate. A common problem where this should be undertaken is that of a stem wall analysis as shown in the below figure of a stem wall. The LimitState:GEO Stem Wall Wizard automatically generates these lines.

Figure 29:

Modelling rotation of a footing with rigidly connected sheet pile below. Note connection across internal corners by internal Boundaries in order to allow soil adjacent to the footing/pile to rotate freely with the footing/pile when using the edge rotations model.

Figure 30:

Modelling rotation of a stem wall. Note connection across internal corners by internal Boundaries, in order to allow soil adjacent to the wall to rotate freely with the wall when using the edge rotations model.

In all cases it is recommended that calibration tests are carried out where strong rotational displacements are dominant.

The verification tests available on the LimitState website may assist in assessing this issue for a range of problem types.

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LimitState:GEO

Try / Buy

Discontinuity Layout Optimization

Introduction

Limit Analysis and Limit Equilibrium

DLO: How Does it Work?

Basic principles

Adaptive Solution Procedure

Rotational failure mechanisms

Implementation

Accuracy of edge rotations model