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LimitState:GEO

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Model components

Material Models

Introduction

LimitState:GEO is an ultimate limit state analysis program. Thus only the parameters defining the yield surface are required for the material models used by LimitState:GEO. In LimitState:GEO the following models are available:

Mohr-Coulomb
Tension and/or compression cut off
Rigid

In addition, material models may be combined to generate more complex yield surfaces. Note that while Engineered Elements (e.g. soil nails) are defined by setting a material property, they are a special case. Engineered Elements are discussed further here.

Mohr-Coulomb Material

The yield surface is defined as depicted in Mohr-Coulomb yield surface in terms of parameters $c$ and $ϕ$ where $c$ is the intercept on the shear stress ( $τ$ ) axis and $ϕ$ the angle of shearing resistance. For drained analysis the values $c′$ and $ϕ′$ are used, while for undrained analysis the parameter $c u$ is used. A linear variation of $cu$ with depth may be specified. While it is possible to derive a value of $ϕu$ from a shear box or triaxial test, interpretation and use of this parameter requires care . The parameter $ϕu$ is typically taken as zero, and this is assumed in LimitState:GEO. If an analysis using a non-zero value of $ϕu$ is required then it is recommended that the user defines the problem in LimitState:GEO as if it was a drained problem but used the undrained parameters $cu$ and $ϕu$ .

Figure 31:

Mohr-Coulomb yield surface

Cutoff Material (Tension and/or Compression)

The yield surface is defined as depicted in here in terms of a limiting tensile stress $σT$ and a limiting compressive stress $σ C$ . The limiting tensile stress is typically used to model tension cutoff so that tension cracks may be modelled by LimitState:GEO or to indirectly represent tensile yield in adjacent structural elements without fully modelling the structural element. The limiting compressive stress is typically used to represent crushing of materials, or to indirectly represent compressive yield in adjacent structural elements (e.g. a retaining wall prop) without fully modelling the structural element.

No distinction is made in these values for drained or undrained analyses. This material model may be utilized on its own, or in conjunction with a another material e.g. Mohr-Coulomb (see Combined materials). Additionally no partial factors are defined for these parameters.

Note that if the yield condition lies on this surface, then a shear stress may still be maintained. It is only the normal stress that is limited. To avoid this, a more rounded yield surface may be defined using multiple materials see Combined materials.

Figure 32:

Tension cutoff and crushing yield surface

Rigid

This simple material type is fully rigid. No slip-line can pass through it at any load. Use of this material type for solids for which it is known that they will not deform, should enhance the solver efficiency, since it reduces the number of nodes and slip-lines that need to be considered.

Engineered Element

In LimitState:GEO, Engineered Elements, that are essentially one dimensional (in section) such as soil nails, geotextiles, and sheet pile walls may be modelled as a special material that may be assigned to a Boundary object. (In versions of LimitState:GEO previous to 2.0, Engineered Elements were termed ‘Soil Nails’ and had a more limited capability.)

Engineered Elements are modelled as objects superimposed ‘on top’ of the existing soil model. They may translate and rotate independently of the surrounding soil (such as a soil nail), or may be constrained to move with the soil (such as a sheet pile). They may be considered to be divided into a series of individual segments between the assigned nodes that run along the element.
Within the model formulation, the relative slip displacement ( $si$ ) and normal displacement ( $n i$ ) between the soil and each segment $i$ of the Engineered Element is computed. These values may vary along its length due to variations in soil displacement and or element displacement.
The Engineered Element itself is treated as a rigid object between each Vertex along the element, but is allowed to rotate (bend) at each vertex. The rotation ( $θj$ ) at each vertex $j$ is computed.
The additional work done within the system, due to the presence of an Engineered Element is calculated from $∑g ∑v i=1 (T si + N ni)+ j=1 Mp θj$ , where $T$ is the pullout resistance per unit length per unit width, $N$ is the resistance per unit length per unit width to lateral displacement, $g$ is the number of segments, $v$ is the number of vertices and $Mp$ is the plastic moment resistance of the element per unit width.
Where the relative slip $s$ is different on either side of an element segment, then the average of the two values is used in the energy equation. Where an element lies on a soil surface, then the relative slip on the ‘surface’ side is zero.
Where the relative normal displacement $n$ is different on either side of an element segment, then the average of the two values is used in the energy equation. This circumstance arises only when the nail lies on the surface of a soil.
The optimizer will determine the absolute displacement of the soil nail that minimizes the energy dissipated within the problem.

Note that if an Engineered Element is used to represent a truly one dimensional object such as a soil nail that has a pullout resistance $T1$ per unit length, a resistance $N1$ per unit length to lateral displacement and a plastic moment of $Mp1$ , then if there are $m$ nails present per unit width the following values should be used. $T = mT1$ is the pullout resistance per unit length per unit width, and $N = mN1$ is the resistance per unit length per unit width to lateral displacement of the nails and $Mp = mMp1$ .

2D objects such as sheet piles may be modelled by setting the resistance $N$ per unit length per unit width to lateral displacement to infinity (in LimitState:GEO any number $> 1 × 1030$ is regarded as infinity). This prevents any flow through or past the Engineered Element as required for a sheet pile.

Suitable values for $T$ and $N$ will normally be selected based on experimental evidence, or on appropriate theory for the situation. In certain circumstances, the resistances $T$ and $N$ will be a function of the normal stress and pore pressure around the element. In common with many approaches in the literature, LimitState:GEO allows $T$ and $N$ to be computed as linear functions of the nominal vertical effective stress at the element as follows:

T = Tc + Tqσ′v

(10)

N = Nc + Nqσv′

(11)

where $Tc$ , $Tq$ , $Nc$ , and $Nq$ are constants defined in the Engineered Element material. The vertical effective stress $σ′ v$ is computed by LimitState:GEO prior to solving based on the weight of overburden per unit area above the element midpoint minus the pore pressure at the element midpoint. This is computed separately for each straight line portion of element (i.e. section between vertices).

Note that if a water table is present that intersects the element, then the solver automatically adds a vertex at the intersection and will therefore compute separate stresses for the parts of the element above and below the water table.

Also note that it is very likely that the vertical effective stress along the element is modified by the overall failure mechanism. While the vertical effective stress and pore pressure may be derived from the final solution determined by LimitState:GEO, they cannot be used in the solution procedure due to the indirect representation of the element in the numerical model. If required the element parameters could be refined iteratively using the results of a previous solution.

Finally if element rotation occurs, it is necessary to employ are relatively high nodal density along the element to ensure accuracy.

Combined Materials

Any number of materials may be combined by assigning them simultaneously to soil layers (Solid objects) or interfaces (Boundary objects).

It is therefore effectively possible to build up any convex yield surface as a series of individual linear yield surfaces. Typical example uses of this facility are:

To model an undrained material with zero tension cutoff. Specify the undrained shear strength $cu$ in a Mohr-Coulomb model and combine with a Tension cutoff material with $σ = T$ 0.0 and a limiting compressive stress $σ = C$ 1.0E30. (It is not possible to specify infinity in LimitState:GEO so a very large value is specified instead.)
To model a material with a non-linear but convex yield surface. Such a yield surface may be approximated by two or more linear yield surfaces. LimitState:GEO will always work with the yield surface that produces the lowest adequacy factor. An example of such an application is in modelling the non-linear Hoek-Brown yield surface for rock masses.
To model uncertainty in soil parameters. In many circumstances the properties of a given soil layer will be uncertain. Normally the most conservative shear strength values would be selected. However for soils with both $c$ and $ϕ$ values, the most conservative shear strength mobilized will depend on the prevalent normal stress. To be sure of a conservative result, the lowest values of $c$ and of $ϕ$ might separately be selected, which may result in an unnecessary degree of conservatism. With LimitState:GEO this is unnecessary and multiple soil models may be assigned to a body of soil. LimitState:GEO will automatically utilize the weakest material consistent with the prevalent normal stresses.
Note that when combined materials are used which have different specified unit weight, it will be necessary to indicate which values are to be used (minimum, average or maximum).

Representation of Water Pressures

Modelling of Water Pressures using DLO

The effect of water pressures is readily incorporated into the DLO computations. As shown by Smith & Gilbert (2007b), the presence of water requires an additional work term in the Linear Programming formulation equal to the average water pressure multiplied by the dilation occurring on any slip-line.

Water Table with Hydrostatic Water Pressures

A water table (or phreatic surface) combined with an assumption of hydrostatic water pressures below this provides a commonly used representation of the water pressure conditions within a body of soil, without resorting to full seepage computations. With this idealization a water table is defined for the problem and the water pressure $u$ at any point is given by the hydrostatic pressure as follows:

u = γwd

(12)

where $γw$ is the unit weight of water and $d$ is the depth of the point vertically below the water table. The water pressure is taken as zero above the water table. This method of determining water pressures provides a reasonable approximation for engineering purposes, especially for water tables of shallow gradient.

Note that in the software it is possible to draw a water table with a vertical edge. This requires careful interpretation. With a water table of any gradient less than vertical, the water pressure on the inclined surface would be taken as zero. The software maintains this assumption at the limit where the water table has a vertical edge. Thus the water pressure is zero on the vertical edge, but a small distance to the side it is computed as normal according to the depth of water vertically above. Exactly below the vertical edge the water pressure is computed according to the depth below the base of the edge. These definitions are shown below. If it is required to model a water pressure on for example a concrete dam face, but not continue the same water pressures beyond the dam, then the vertical (or steeply inclined) face of the water table should be drawn inside the dam itself as shown below. The software will give a warning on Solve if a near vertical water table has been drawn.

Figure 33:

Computation of water pressures at a vertical edge

Figure 34:

Example of how to define water table to act with hydrostatic pressure on the outer face of a dam.

$ru$ values

An alternative approach to representing water pressures is to utilize the average pore pressure ratio $ru$ . This may be is defined as:

ru = -u- γh

(13)

and thus the water pressure $u$ at any point may be given by

u = ruγh

(14)

where $γh$ is the overburden pressure which may be represented by the defining $h$ as the vertical depth of the point in question below the ground surface.

Seismic loading

Modelling of seismic loading using DLO

In common with most Limit Analysis procedures, the DLO method can be extended to handle pseudo-static accelerations to enable analysis of problems subject to seismic or earthquake loading. If the seismic accelerations are as follows:

$kh$ = horizontal acceleration / $g$

$k v$ = vertical acceleration / $g$

where $g$ is the acceleration due to gravity (9.81 m/s $2$ ) , then additional inertial body forces of magnitude $khW$ and $kvW$ are imposed on a body of weight $W$ .

The adopted sign convention is that positive $kh$ acts in the negative $x$ -direction (i.e. right to left) and positive $kv$ acts in the negative $y$ -direction (i.e. downwards).

Modelling of water pressures during seismic loading

The effects of horizontal and vertical accelerations on free and pore water pressures in pseudo static calculations often require consideration of other factors in addition to $k h$ and $k v$ , and it is up to the engineer to decide how to incorporate these appropriately into a limit analysis approach as taken by LimitState:GEO.

Within LimitState:GEO the effect of seismic accelerations are only assumed to modify the unit weight of water used in the water pressure calculations to $(1+ kv)γw$ . The horizontal acceleration $kh$ is assumed not to affect water pressures. This has the effect of correctly modelling lateral earth pressures on a retaining wall according to the method of Matsuzawa et al. (1985), for low permeable backfill soils. To address highly permeable soils by the method of Matsuzawa et al. (1985) it would be necessary to modify $kh$ by the ratio of $γdry$ to $γsat$ of the soil being modelled. If more than one soil is modelled then a representative or average $kh$ factor would need to be utilised. In addition an additional horizontal dynamic water pressure cannot be modelled directly. Note also that the ‘static’ water pressure would be computed as $u = (1+ kv)γwz$ where $z$ is the depth below the water table.

If required, water pressures can be additionally modified using the $ru$ factor described here.

Soil Reinforcement

In LimitState:GEO soil reinforcement may be modelled directly as e.g. a long thin Solid or using the special material type Engineered Element. The latter is recommended in LimitState:GEO. The advantage of the latter over the former for a plane strain analysis program such as LimitState:GEO is that the latter is implemented such that it can allow soil to flow around or ‘through’ the reinforcement as would be expected for e.g. a soil nail. This can however be suppressed for sheet reinforcement such as a geotextile fabric. Modelling of bending failure is also much more straightforward for an Engineered Element. The theory behind Engineered Elements is discussed here. The practical use of Engineered Elements in LimitState:GEO is discussed here.

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More

LimitState:GEO

Try / Buy

Model components

Material Models

Introduction

Mohr-Coulomb Material

Cutoff Material (Tension and/or Compression)

Rigid

Engineered Element

Combined Materials

Representation of Water Pressures

Modelling of Water Pressures using DLO

Water Table with Hydrostatic Water Pressures

values

Seismic loading

Modelling of seismic loading using DLO

Modelling of water pressures during seismic loading

Soil Reinforcement

$ru$ values