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PLAXIS v. 8.2FEM computations in practice

ProgrammeMesh generationSoil–structure couplingInitial stressesGround water flowTypes of analysisIntegration algorithms

ProgrammeMesh generation

Element types in PLAXIS

Mesh refinement

Soil–structure couplingInitial stressesGround water flowTypes of analysisIntegration algorithms

Mesh generation: Element types in PLAXISTwo types of triangular elements (isoparametric)

Plane strain Axial symmetry

Nodes

Gauss points

6-noded element 15-noded element

Mesh generation: Element types in PLAXISShape/weight functions for 6-noded triangle

Quadratic interpolation in two (three) directions

Mesh generation: Element types in PLAXISShape/weight functions for 15-noded triangle

4th order interpolation in two (three) directions

Mesh generation: Element types in PLAXISAdvantages of higher order elements

Good for the description of continuous strain and stress variations, e.g. failure in a zone

Good description of a continuous displacement field with relatively few elements

Disadvantages of higher order elements Failure loads may be dependent on the mesh Poor description of discontinuous stress and strain,

e.g. failure along a line

Note: According to the manual, the 15-noded element is superior. However, you should test whether two grids provide the same result.

Mesh generation: Element types in PLAXISFailure along a line modelled with CST-elements

Mesh generation: Element types in PLAXISFailure along a line modelled with higher-order

elements

Mesh generation: Mesh RefinementPLAXIS: automatic unstructured mesh generation

No possibility of making a so-called structured mesh The mesh size cannot be set explicitly The mesh is generated based on random seeds

The mesh size may be changed implicitly Globally by means of global coarseness Locally by means of local coarseness

Mesh generation: Mesh RefinementGlobal coarseness

Average “element length” in the entire model

Very coarse: nc = 25 approx. 50 elements

Coarse: nc = 50 approx. 100 elements

Medium: nc = 100 approx. 250 elements

Fine: nc = 200 approx. 500 elements

Very fine: nc = 400 approx. 1000 elements

Note: The number of elements is independent of type

Coarseness

Mesh generation: Mesh RefinementDefinition of Local coarseness

At a point Along a line

Mesh generation: Mesh RefinementDefinition of Local coarseness

At a point Along a line

Note: A line = two points

ProgramMesh generationSoil–structure couplingInitial stressesGround water flow Types of analysisIntegration algorithms

Soil–Structure CouplingInterface strength defined by Mohr-Coulomb

Elastic behaviour at an interface:

Plastic behaviour at an interface:

Reduced parameters

Adhesion:

Friction:

Dilatation: for else

Rigid interface (Rinter = 1.0)

Applied at interfaces between soil and soil

Soil–Structure Coupling

In manual settings for interfaces, overlap and slip between soil and structure can be allowed

Id the slip/overlap becomes too big, the system of equations becomes ill-conditioned (inaccurate)

The fictive thickness ti of the interface can be set, so that the problem is avoided


Permeability Interfaces in PLAXIS are always fully impermeable Inactive interfaces are fully permeable Interfaces can be turned off in the flow phase

Note: Plates are fully permeable!

Real thickness of interface, δinter

Only used in the Hardening Soil model Typical thickness: δinter = a few grain diameters

Used in the computation of changes of the void ratio related to dilatation cut-off

Significant impact on the capacity of tension piles


Real thickness of interface, δinter

Significant impact on the capacity of tension piles


ProgrammeMesh generationSoil–structure couplingInitial stresses

Effective stresses Pore pressure

Ground water flowTypes of analysisIntegration algorithms

Initial stresses: Effective StressesOver-Consolidation Ratio (OCR)Pre-Overburden Pressure (POP)

Initial stresses: Effective StressesPre-consolidation pressure

Jaky’s equation: (used in HS)

is applied in order to determine which provides the position of the cap in the Hardening Soil model

Initial stresses: Effective StressesRatio between horizontal and vertical stress:

Initial stresses: Effective StressesProcedures for determination of initial stresses

The K0-procedure for (nearly) horizontally layered soil

Gravity loading for all other situations

Note: No strength of cohesion-less soil w/o load

Initial stresses: Effective StressesThe K0-procedure for horizontally layered soil

Initial stresses defined in “Initial Conditions”

For full gravity, in addition to K0 one must define

To avoid points with plastic behaviour, the at-rest earth pressure coefficient must lie within a given interval, here given for the case of cohesion-less soil

Initial stresses: Effective StressesThe K0-procedure for horizontally layered soil

If the state of stress found by the K0-procedure does not provide static equilibrium, a plastic nil-step is included, i.e. a step with no additional load

If the K0-procedure provides completely misleading results, the solution of the plastic nil-step may diverge

At the end of the plastic nil-step, it is recommended to “Reset displacements to zero” (check this option)

Initial stresses: Effective StressesGravity loading

Initial stresses are equal to zero (in the computation) The weight of soil is introduced as a body force Often plastic behaviour is observed at a number of

integration points in the finite element model This happens in cohesion-less soil, unless

A number of plastic points can be accepted Poisson’s ratio should be set so that a realistic value

of the at-rest earth pressure is obtained All displacements are reset after the gravitation step

Initial stresses: Pore PressureTotal pore pressure in PLAXIS

pactive = psteady + pexcess

Excess pore pressure can only be determined in undrained conditions: “Cluster is Undrained”

Two different approaches Phreatic level ~ simple definition Ground water flow ~ (anisotropic) Laplace equation

Initial stresses: Pore PressureGround water table (phreatic levels)

Generally In a cluster

Note: Phreatic level is defined at geometry lines

Inaccurate Accurate

Initial stresses: Pore PressureGround water table (phreatic levels)

Generally In a cluster

Initial stresses: Pore PressureGround water flow

Steady state• Solution of Laplace (or Poisson) equation

Boundary conditions• Dirichlet: potential = position of ground water

table• Neumann: flux = flow through a boundary• Seepage: mixture of Dirichlet and Neumann

conditions

Initial stresses: Pore PressureComputation

Phreatic level• Approximate solution• All clusters turned on, whether they are active or not• This is avoided by the setting “Cluster is dry”

Ground water flow• Better, but more “expensive”• Only active clusters are included

Note: Time dependent flow can be analysed with the PLAXIS Groundwater Flow module

ProgrammeMesh generationSoil–structure couplingInitial stressesGround water flow

Steady state solution Consolidation

Types of analysisIntegration algorithms

Ground water flow: Steady State SolutionDarcy’s law

,

Flow in the direction of decreasing potential

Potential (groundwater head [m])

Seepage velocity [m/s]

Permeability [m/s]

Vertical position [m]

Pore pressure [Pa]

Ground water flow: Steady State SolutionEquation of continuity for incompressible fluid

Note: kx = ky → Laplaceequation in the potential

Ground water flow: Steady State SolutionPermeability in partially saturated soil

Reduction factor

VS

Ground water flow: Steady State SolutionDiscretization

,

Local: ,

Global: , p = w ( y - )

Gradient of shape functions

Potential in the element nodes

Inflow at nodes [m3/s]

Ground water flow: Steady State SolutionExample: Flow under a dam

NB: Interface instead of a plate

Ground water flow: Steady State SolutionExample: Flow under a dam

Note: Interface instead of a plate

Ground water flow: ConsolidationEffective stresses and pore pressure

Effective stress rate and strain rate

,

Interpolation

, ,

Ground water flow: ConsolidationFE formulation of the equations of equilibrium

Note: The residual leads to a self-correcting solution

Internal forces (weight)

Surface traction

Residual

Ground water flow: ConsolidationFE formulation of the equations of equilibrium

Incremental equilibrium:

Stiffness of grain skeleton:

Coupling matrix:

External forces:

Constitutive matrix

Ground water flow: ConsolidationEquation of continuity for an incompressible fluid

Steady state part of the solution

Porosity (pore volume/grain volume)

Bulk modulus for pore fluid

Ground water flow: ConsolidationFE formulation of the equations of continuity

Note: q = 0 in PLAXIS 8.2

Flux through the boundary

Ground water flow: ConsolidationCombined system of equations

Incremental form (elastic material)

,

ProgramMesh generationSoil–structure couplingInitial stressesGround water flowTypes of analysis

Plastic calculation c-φ-reduction

Integration algorithms

Types of analysis: Plastic CalculationPrimary loading to ultimate failureMaterial behaviour

Elastic Plastic Pore pressure

Geometry Normal computation: linear (small strain) Updated Mesh: Updated Lagrange formulation

• Strengthening of anchors during deformation• Large deformations ~ soft soil deposits• May follow a “normal” (i.e. geometrically linear) computation

by use of the option “Reset displacements to zero”• A “normal” computation cannot follow an UM computation

Types of analysis: c-φ-reductionDetermination of factors of safety

Sf =

Same reduction of cohesion and friction terms

Initial value: ΣMsf = 0.1

Subsequent values are found automatically

ProgrammeMesh generationSoil–structure couplingInitial stressesGround water flowTypes of analysisIntegration algorithms

Tolerated error Over relaxation Maximum iterations Desired minimum and maximum Arc-length control

Integration Algorithms

Integration Algorithms: Tolerated ErrorSet by the parameter tolerated errorThe default is 0.03 ( total external load)

Integration Algorithms: Over RelaxationApplied in order to speed up the processUpper limit

Theoretical: 2.0 In practice: 1.5

Integration Algorithms: Maximum IterationsPermitted number of iterations in a load step

The default value of Maximum iterations is 50 Values of 1 to 100 are possible If more iterations are needed, a warning is given in

the Log info box on the General page

Integration Algorithms: Desired Min. & Max.Desired minimum and maximum number of

iterations in a single load incrementCan be set to values in the interval 1 to 100Default settings

Desired minimum = 4 ; Desired maximum = 10

Soil with a low angle of friction Desired minimum = 3 ; Desired maximum = 7

Soil with a high angle of friction (and HS model) Desired minimum = 7 ; Desired maximum = 15 Maximum iterations = 75

Integration Algorithms: Arc-length ControlIncreased stability of numerical solution

Default for c-φ-reduction Computation of collapse load

Note: Arc-length control may lead to unintended unloading → restart without arc-length control

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Documents

soil slide

noded element mesh generation

line slide

mesh size

mesh refinementplaxis

soilstructure coupling

practice slide

local coarseness slide