Geo5FEM.pdf

7/28/2019 Geo5FEM.pdf

1/113

GEO 4FEM

Users guide version 3.0

FINE 2004


2/113


3/113

Users guide

CONTENT

INTRODUCTION.......................................................................................................................................4

1. APPLICATION OF THE PROGRAM GEO FEM .............................................................................5

1.1BASIC INFORMATION ABOUT MODELING..............................................................................................51.2SOILSCONSTITUTIVE MODELS AND THEIR DATA ..............................................................................8

1.2.1LINEAR MODEL (LM)............................................................................................................................91.2.2MODIFIED LINEAR MODEL (MLM) .......................................................................................................91.2.3MOHR-COULOMB MODEL (MC) .........................................................................................................111.2.4DRUCKER-PRAGER MODEL (DP) ........................................................................................................121.2.5MODIFIED MOHR-COULOMB (MCM).................................................................................................121.2.6MODIFIED CAM-CLAY MODEL (MCC)................................................................................................141.2.7K0PROCEDURE, DETERMINATION OF INITIAL (GEOSTATIC) STRESSES................................................191.3RIGID BODIES .......................................................................................................................................201.4INPUT OF GEOMETRY ...........................................................................................................................201.4.1INPUT OF INTERFACESMARGINS OF THE TOPOLOGICAL MODEL......................................................201.4.2ASSIGNING SOILS ................................................................................................................................211.4.3FREE NODES ........................................................................................................................................221.4.4FREE LINES .........................................................................................................................................231.4.5CORRECTOR OF SPECIFIED GEOMETRY ...............................................................................................241.5MESH GENERATOR...............................................................................................................................251.6BOUNDARY CONDITIONS......................................................................................................................281.7BEAMS...................................................................................................................................................291.8HEREDITY OF FEATURES (BOUNDARY CONDITIONS, BEAMS, SOILS) ................................................321.9CONTACTS (INTERFACES)....................................................................................................................321.10ANCHORS ............................................................................................................................................361.11PROPS..................................................................................................................................................391.12GEO-REINFORCEMENTS.....................................................................................................................401.13SURCHARGE........................................................................................................................................42

1.14WATER................................................................................................................................................441.15REMOVING SOIL (ACTIVITY) AND ASSIGNING SOILS TO REGIONS (REGIONS) ...............................441.16ANALYSIS ............................................................................................................................................461.17ANALYSIS SETTING.............................................................................................................................491.17.1SOLUTION METHOD ..........................................................................................................................491.17.2UPDATE OF THE STIFFNESS MATRIX .................................................................................................501.17.3INITIAL LOAD STEP ...........................................................................................................................511.17.4MAXIMUM NUMBER OF ITERATIONS PER A SINGLE LOAD STEP ........................................................511.17.5CONVERGENCE CRITERION ...............................................................................................................511.17.6SETTING THENEWTON-RAPHSON METHOD......................................................................................521.17.7SETTING THE ARC-LENGTH METHOD................................................................................................531.17.8SETTING A VERSION OF THE ARC-LENGTH METHOD ........................................................................53

1.17.9SETTING ARC-LENGTH ......................................................................................................................541.17.10AUTOMATIC ARC LENGTH CONTROL ..............................................................................................541.17.11LINE SEARCH METHOD....................................................................................................................551.17.12PLASTICITY.....................................................................................................................................561.17.13SETTING THE BASIC PARAMETERS DRIVING THE SAFETY FACTOR ANALYSIS.................................571.17.14SETTING DRIVING PARAMETERS FOR RELAXATION OF THE REDUCTION FACTOR...........................581.17.15COURSE OF ANALYSIS.....................................................................................................................591.18RESULTS..............................................................................................................................................631.19STABILITY...........................................................................................................................................69

2. EXAMPLE 1 TERRAIN SETTLEMENT FILE EXAMPLE1.GMK.........................................71

3. EXAMPLE 2 ANALYSIS OF COLLECTOR LINING FILE EXAMPLE2.GMK..................77

4. EXAMPLE NO. 3 SHEETING STRUCTURE FILE EXAMPLE3.GMK................................915. EXAMPLE 4 STABILITY ANALYSIS OF SLOPES FILE EXAMPLE4.GMK...................106

___________________________________________________________________________________________________________________________________________________________

GEO 4 FINE 2004 3


4/113

Geo 4 - FEM

Introduction

The user manual consists of two chapters.

Chapter 1 this chapter provides the basic information about the program GEO FEM. The user

is strongly recommended to read this chapter carefully.

Chapter 2 tutorial part, contains several examples of individual types of problems and

provides comparison of the results found from the finite element method and analytical

solutions:

- Example 1 terrain settlement

- Example 2 analysis of collector lining

- Example 3 retaining wall analysis

- Example 4 slope stability analysis.

Individual solutions are accompanied by the results provided by other programs for the

verification purposes.

The theoretical grounds of the finite element method including the description of the specific

approaches for the solution of various geotechnical problems are summarized in the Theoretical

guide.

___________________________________________________________________________________________________________________________________________________________

4 GEO 4 FINE 2004


5/113

Users guide

1.Application of the program GEO FEM

The program GEO FEM is intended for the modeling of various geotechnical tasks such as:

Terrain settlement Retaining wall structures

Slope stability

Beams on elastic foundation

Analysis of underground excavations including linings, etc.

In number of cases the results can be also derived using analytical solutions available from a

group of programs GEO. However, the program GEO FEM allows the user to obtain a more

complex view on a given task and compare the results. Moreover, in case of more complex tasks,

for which the analytical solutions are not available, the numerical analysis using the finiteelement method (FEM) is usually the only option.

On the other hand, care must be taken when using FEM for the analysis of complex engineering

tasks. To obtain reliable results with FEM a certain experience of the user is generally required.

The results may be strongly affected not only by the selection of input parameters, but also by

the selected finite element mesh, boundary conditions, solution technique, etc. All in all

however, the correct selection of input parameters is the basis of the success of computation. The

most difficult part is that most constitutive models require parameters that can be obtain only

from laboratory measurements of soils. To ease this task, the manual also offers approximate

values of input parameters. However, it is the user responsibility to select such parameters that

most closely correspond to the actual soil or rock material. At all events this task deserves aspecial attention.

1.1Basic information about modeling

There are two basic steps to follow when using the program

- Topology of the structure input and mesh generation

- Step by step definition of individual construction stages including analysis andsubsequent post-processing of results

The first step is to create a structure in the topology regime (input information about soils,

specify interfaces between soils, beams, linings, boundary conditions, etc.) followed by

generation of the finite element mesh, Fig. 1.

___________________________________________________________________________________________________________________________________________________________

GEO 4 FINE 2004 5


6/113

Geo 4 - FEM

mode To olo

Fig. 1Topology input

Individual calculation stages (stages of construction) are defined subsequently. The stages of

construction serve to activate or deactivate parts of a structure, to replace the soil material in a

selected region, to add and remove anchors, to either define or modify loads applied to thestructure, to change parameters of the selected beams, etc. The actual calculation including the

stability analysis and post-processing of results for a given stage is also carried out within this

step.

___________________________________________________________________________________________________________________________________________________________

6 GEO 4 FINE 2004


7/113

Users guide

add new stagemodeStage

remove last stagemodeTopology

view current sta e

Fig. 2Working with calculation stages

The calculation stage does not allow for the option to change predefined mesh, to add boundary

conditions, contact and beam elements these objects must be all defined already in the

topology regime objects not required in the current calculation stage can be then deactivated.To that end, the build in automatic corrector of topology can substantially simplify the input of

more stages at the same time (e.g., step by step tunnel excavation). See also Section 1.4.

The first calculation stage always serves to determine initial (geostatic) stress state. The actual

stages of the analyzed engineering task are specified next. It is necessary to realize that the

modeling process proceeds from one stage to the other results from the previous stage form the

basis for calculations carried out in the subsequent stage.

It thus becomes clear that each topological change within a calculation stage calls for a new

finite element mesh, which inevitably results in a loss of all calculations performed so far the

geometrical data already specified for individual stages, however, will remain unchanged. It is

therefore necessary only to reanalyze all calculation stages.

___________________________________________________________________________________________________________________________________________________________

GEO 4 FINE 2004 7


8/113

Geo 4 - FEM

1.2Soils constitutive models and their data

The current version of the program GEO FEM assumes fully drained (steady state) conditions

after redistribution of pore pressure at the end of consolidation process (long term conditions

the loads were applied for sufficiently long time for the soil to become fully consolidated, the

pressure and displacements reached their constant values and the loads are taken by the skeleton

grains only). The determination of the pore pressure distribution can be therefore performedindependently of the analysis of the skeleton deformation. The program further assumes

incompressibility of grains and works exclusively with the effective soil parameters ef, cef. The

skeleton deformation thus solely depends on the predefined distribution of pore pressure, applied

loads and constitutive (material) models of the soils.

The material model attempts to describe the soil (or rock) behavior as close to reality as possible

clearly, it is necessary to specify for each soil the most reliable material model.

Fig. 3Input of basic material parameters of soilsDetail information about actual input of soils (for a new user) is described in the tutorial manual

GEO4, Chapter 2.2.3 Soils and rocks. Here we present only the most relevant information

regarding individual material models implemented in GEO FEM. Material models in program

GEO FEM can be subdivided into two main groups: linear and nonlinear models. Linear models

give relatively fast, but not very accurate estimate of the true material response. These models

can be used in cases, where only the stress or deformation states of a soil mass are of interest.

They provide no information about locations and possible mechanisms of failure. They can be

used to model soil behavior in regions, where only the local failure with no effect on the

evolution of global failure occurs, but which may cause premature loss of convergence.

Providing the main interest is in a reliable description of the soil behavior it necessary to employ

nonlinear models. Individual models will be now described in more details.

___________________________________________________________________________________________________________________________________________________________

8 GEO 4 FINE 2004


9/113

Users guide

1.2.1Linear model (LM)

The linear model is the basic material model that assumes linear relationship between stress and

strain given by Hooks law. The following data are required:

- Self weight of soil [kN/m3]

- Poissons ratio [-]

- Elastic modulus E [MPa]

In a one dimensional problem Hooks law describes the linear dependence of stress on strain

via the Young modulus E (modulus of elasticity), Fig. 4. Within this framework the linear

model provides a linear variation of displacements as a function of applied loads.

Fig.4 Stress-strain diagram for LM

1.2.2Modified linear model (MLM)

It is clear that for soils the linear behavior is acceptable only for relatively low magnitudes of

applied loads. This becomes evident upon unloading that usually shows a rather small amount of

elastic deformation compare to the overall deformation. The modified linear model attempts at

least to some extent to take this into account by considering different modulus for loading and

unloading as plotted in Fig. 5. A drop in the material stiffness along a given loading path

attributed to the plastic yielding is reflected through a certain secant modulus Esec. An elastic

response is assumed upon unloading. To increase clarity of model formulation the elastic

modulus for the unloading branch is replaced by the unloading-reloading modulus Eur that

governsthe response of a soil upon unloading and subsequent reloading up to the level of stress

found in the material point prior to the unloading.

With reference to Fig. 5 these modules are given by:

Secant modulusEsec [MPa]: == /tansecE

Unloading/reloading modulusEur[MPa]:el

urE == /tan

During primary loading the response of a soil is therefore governed by the secant modulus whileupon unloading it follows the path set by the unloading-reloading modulus. An approximate

value of this modulus is 3*secant modulus. In any case, both parameters should be provided by

reliable experimental measurements.

___________________________________________________________________________________________________________________________________________________________

GEO 4 FINE 2004 9


10/113

Geo 4 - FEM

(a) (b)

Obr.5 (a) True stress-strain curve, (b) Stress-strain diagram for LMM

Nonlinear models

Two groups of nonlinear models can be again specified. The first class of problems originates

from a classical Mohr-Coulomb failure criterion. In particular, the Drucker-Prager, Mohr-Coulomb and Modified Mohr-Coulomb models fall within this category. A common feature of

these models is a possibility of evolution of unbounded elastic strains along hydrostatic axis.

This is evident from Fig. 6 that shows projections of the yield surface into deviatoric and

meridian planes, respectively

(a) (b)

Fig.6 Projection of yield surfaces into (a) deviatoric, (b) meridian plane

Employing nonlinear models allows us to capture the typical nonlinear response of soils, Figs.

5a and 7, in much better way compare to the MLM model. These models describe evolution of

permanent (plastic) deformation of a soil material. The onset of plastic deformation is controlled

by so-called yield surface. The yield surface can be either constant (elastic-rigid plastic material),

___________________________________________________________________________________________________________________________________________________________

10 GEO 4 FINE 2004


11/113

Users guide

or depend on the current state of stress (material with hardening/softening). The current versions

of models assume constant yield surface with no material hardening/softening.

Fig.7 Stress-strain diagram for nonlinear models

The input parameters include all parameters of the linear model. In addition the models require

the following parameters:

- Angle of internal friction []

- Cohesion of soil c [kPa]

- Angle of dilation []

Angle of internal friction determines the onset of plastic deformation. The user should obtain

these values from reliable experimental measurements. The angle of dilation controls an amount

of plastic volume strain developed during plastic shearing and is assumed constant during plastic

yielding. The value = 0 corresponds to the volume preserving deformation while in shear.

Clays, regardless of overconsolidated layers, are characterized by a very low amount of dilation

( 0). As for sands, the angle of dilation depends on the angle of internal friction. A value of

orientation for non-cohesive soils (sand, gravel) with the angle of internal friction >30 is given

by =-30. A negative value of dilation angle is acceptable only for rather loose sands. The

default value of = 0 is assumed. Unlike the MLM the nonlinear models require to specify only

the elastic modulus E. A drop in material stiffness is a result of evolution of plastic strains and

corresponding redistribution of stresses. This consequently yields an instantaneous tangentmaterial stiffness as a function of the current state of stress represented in Fig. 7 by an

instantaneous tangent modulusET.

1.2.3Mohr-Coulomb model (MC)

The Mohr-Coulomb yield surface can be defined in terms of three limit functions that plot as anon-uniform hexagonal cone in the principal stress space. Projections of the MC yield surface

into deviatioric and meridian planes appear in Fig. 6. As evident from Fig. 6a the MC yieldfunction has corners, which may cause certain complications in the implementation of this model

into the finite element method. The advantage on the other hand is the fact that the traditional

___________________________________________________________________________________________________________________________________________________________

GEO 4 FINE 2004 11


12/113

Geo 4 - FEM

soil mechanics and partially also the rock mechanics are based on this model. Details can be

found in the Theoretical guide.

1.2.4Drucker-Prager model (DP)

The Drucker-Prager model (sometimes also known as the extended von Mises model) modifies

the Mohr-Coulomb yield function to avoid singularities associated with corners. Unlike theMohr-Coulomb model the Drucker-Prager yield surface is smooth and plots as a cylindrical conein the principal stress space. Similarly to the MC model the DP yield surface depends on the

effective mean stress m as shown in Fig. 6a. The current version of the DP model implemented

in GEO MKP builds upon the assumption of triaxial extension. In other words, the yield surfaceprojection into deviatoric plane touches the inner corners of the Mohr-Coulomb hexagon

( = 300), where is the Lode angle. As displayed in Fig. 8 the DP yield function unlike theMC model does not depend on this parameter. Details can be found in the Theoretical guide.

Fig. 8 -DP and MC yield surfaces in the deviatoric plane

1.2.5Modified Mohr-Coulomb (MCM)

Similarly to the DP model the Modified Mohr-Coulomb model smoothes out the corners of theMC yield surface. As suggested in Fig. 9 the projection of the MCM yield surface into deviatoric

plane passes through all corners of the Mohr-Coulomb hexagon and as the MC yield function

the MCM yield function depends on the effective mean stress m and the Lode angle . Withreference to Figs. 6 and 9 one can expect a slightly stiffer response of the material when using

the MCM plasticity model compare to the MC and DP models, see the Theoretical guide.

Fig. 9 -MCM and MC yield surfaces in the deviatoric plane

___________________________________________________________________________________________________________________________________________________________

12 GEO 4 FINE 2004


13/113

Users guide

To illustrate an effect of the used model on structural response we present an example of shallow

foundation loaded by distributed loading q, Fig. 10a. A certain simplification of this task is an

assumption of an infinitely stiff foundation loaded by prescribed displacements, Fig. 10b. Thegeometrical model and finite element mesh for individual tasks appear in Fig. 11. Influence of

soil and foundation self-weight on the response is neglected. The results are plotted in Fig. 12.

Fig. 10 Task assignment: foundation strip

Fig. 11 Geometrical model and finite element mesh

Fig. 12 Analysis results

___________________________________________________________________________________________________________________________________________________________

GEO 4 FINE 2004 13


14/113

Geo 4 - FEM

Fig. 12 shows a significantly stiffer response of the soil to external loading when using the MCM

model compare to the DP and MC models, which in the present example show similar behavior.

Owing to the fact that one of the objectives of this task was to determine a limit load it wasnecessary to employ the Arc-length method in the analysis. In similar problems with load control

conditions the use of the Newton-Raphson iteration scheme may lead to a loss of convergence

and premature termination of the analysis. This phenomenon is demonstrated by the green curvein Fig. 12. In order to increase stability of the solution process it is possible to switch to

displacement control conditions as shown in Figs. 10 and 11. The resulting response isrepresented by the black curve in Fig. 12. In this case we plot the variation of prescribed

displacements and the resultant of corresponding reaction forces developed in supports withprescribed displacements.

1.2.6Modified Cam-clay model (MCC)

The MCC model was originally developed for triaxial loading conditions. Experimental

measurements on soft clays provided the background for the development of the constitutivemodel plotted in Fig. 13a that shows the variation of void ratio e as a function of the logarithm

of effective mean stress meff. The graph consists of a normal consolidation line (NCL) and a setof swelling lines. On first loading the virgin soil moves down along the NCL. Next, suppose that

the soil was consolidated to a certain level of stress, which is termed the preconsolidation

pressure, and subsequently unloaded up the current swelling line. Then, upon reloading the soilinitially moves down along the swelling line until reaching the stress state given by the

parameterpc, which existed prior to the unloading. At this point the soil begins to move againdown the normal consolidation line (primary loading compression line).

The MCC model further introduces distinction between plastic yielding and ultimate collapse

using the concept of critical state line, which assumes that the soil is found at the critical state

when during continuous loading there is no change in void ratio (zero increment of volumetricplastic strain) and effective mean stress. This state is represented by point 3 in Fig. 13b.

(a) (b)

Fig. 13 (a) constitutive model (b) projection of the yield surface into the meridian plane

Clearly, see Fig. 13b, the yield surface is smooth without the possibility of evolution of tensilestresses. The MCC model allows unlike the first group of models a direct modeling of strain

___________________________________________________________________________________________________________________________________________________________

14 GEO 4 FINE 2004


15/113

Users guide

hardening or softening for normally consolidated or overconsolidated soils, nonlinear

dependence of volumetric strain on effective mean stress and limit conditions of ideal plasticity.

When using the MCC model the soil loaded in shear can be plastically deformed withoutcollapse (points 1,2 for hardening, point 2 for softening) until reaching the critical state (points 3

and 2 for hardening and softening, respectively). The soil deforms further in shear under the

assumption of ideal plasticity without change ofe and meff

.

The MCC model requires, apart from the self-weight and Poisson ratio, an input of the following

parameters:

- Bulk modulus K [MPa]

- Slope of swelling line [-]

- Slope of NCL [-]

- Initial void ratio e [-]

- Slope of the critical state line M [-]

- Preconsolidation pressure pc [kPa]- Coefficient OCR (overconsolidation ratio) OCR [-]

Fig. 14 Determining parameter pc

From the above parameters the main attention deserves the determination of the initial value of

preconsolidation pressures pc, which controls the onset of plasticity under hydrostatic loading

conditions, Fig. 13. The program offers two options. The first approach exploits the K0procedure, Section 1.2.7, assuming that the soil is normally consolidated. In such a case the

value ofpc follows from, Fig. 14

( ) cc Kp 00

213

1+= ,

( )hOCR zc == ,

sin10 =K .

This approach ensures that the material point is found at or below the yield surface prior to

subsequent loading. If the value ofpc0 is specified by the user the program takes the larger of the

two for the analysis. The instantaneous value of bulk modulusKat a certain time is given by

( )( )

sin11

+

=

teff

m

t

t

eK .

___________________________________________________________________________________________________________________________________________________________

GEO 4 FINE 2004 15


16/113

Geo 4 - FEM

The instantaneous shear modulus also required in the analysis is provided by

( )( )

tt KG

=

12

213.

Therefore, when using the K0 procedure, the value ofKspecified on input is not relevant for the

actual analysis. On the other hand, when employing the standard procedure this material

parameter must be input. It can be derived from the elastic shear modulus G0 as

( )( )

00

213

12GK

+= ,

where is the Poisson ration. Parameters and can be estimated from the expressions

3.2

cC= ,( )( ) s

C

+

=

1

13.1 ,

where Cc is the coefficient of one-dimensional compression and Cs is the corresponding

coefficient for one-dimensional swelling. These parameters follow from a simple oedometric

test. It thus remains to specify an approximate expression for parameter M. Provided that the

triaxial extension experiment is carried out (assumption consistent with the implemented version

of the DP model) the value of M is provided by

cv

cvM

sin3

sin6

+= ,

and under the assumption of triaxial compression we get

cv

cvM

sin3

sin6

= ,

where cv is the value of frictional angle at constant volume (the soil is found at critical state).

This variable is given by

sinsin1

sinsinsin

=cv ,

It is clear that for clayey soils, for which the dilation angle equals approximately to zero, the

cv equals approximately to frictional angle .

The following study illustrates the model behavior depending on the selection of parameterpc.

We consider a very simple problem of a settlement of flexible foundation. This problem is

described in details in the Example 1 of this tutorial. In all cases the parameter OCR = 1 was

considered. The distribution of the equivalent plastic deformation displayed in Fig. 15 wasderived with the help of K0 procedure and setting pc=1kPa in the input dialog window

(minimum allowable value). The maximum equivalent plastic strain was found around 7%.

However, when setting pc=100kPa we arrive at quite different distribution of equivalent plastic

strain, see Fig. 16. In this case we simply assume that up to a certain depth (m


17/113

Users guide

geostatic stress. Results are shown in Fig. 16. Thus, if it is necessary to run the standard

procedure in the first stage it is reasonable to specify several layers for inputtingpc as suggested

in Fig. 18. The corresponding results are plotted in Fig. 19. In this example we set in the first

layer (up to depth of 2m)pc=25kPa (=20kN/m3). The maximum equivalent plastic strain did not

exceed 2.7%.

Fig. 15 K0 procedure, pc = 1kPa, 2nd

stage

Fig. 16 K0 procedure, pc = 100kPa, 2nd

stage

___________________________________________________________________________________________________________________________________________________________

GEO 4 FINE 2004 17


18/113

Geo 4 - FEM

Fig. 17 Standard analysis, pc = 100kPa, 1st

stage

Fig. 18 Standard analysis in the 1st

stage, subdivision of soil mass into layers to input pc

Fig. 19 Standard analysis in the 1st

stage, pc = 25-305kPa (different for each layer), 2nd

stage

___________________________________________________________________________________________________________________________________________________________

18 GEO 4 FINE 2004


19/113

Users guide

1.2.7K0 procedure, determination of initial (geostatic) stresses

Apart from other soil parameters describing the material model the program makes possible to

enter the coefficient of lateral earth pressure at rest K0. This coefficient can be used to generate

the initial stress state in the 1st calculation stage when selecting the option K0-procedure Fig.

20. In such a case the stress x is found from the expression K0.z, where K0 is specified by the

user. If K0 is not specified it is determined from the expression K0=/(1-). The K0-procedureshould be used only in problems with horizontal surfaces without surcharge, water or anchors.

When using the standard analysis it is also possible in the 1st stage of construction to use the

Poisson number different from the one used in other stages of construction. If the option K0 for

computation of geostatic stress (Fig. 20) in the material data set is checked, then the Poisson

ratio for the 1st calculation stage is given by the expression = K0 /(1+ K0). This enables us todetermine an arbitrary value of the lateral earth pressure in the 1st stage of construction.

Fig. 20 Input of the coefficient of lateral earth pressure at restK0

___________________________________________________________________________________________________________________________________________________________

GEO 4 FINE 2004 19


20/113

Geo 4 - FEM

Fig. 21 1st

stage of construction selection of the procedure for the determination of initial stresses

1.3Rigid bodies

The program makes possible to specify so called rigid bodies within the geometrical model

characterized only by their self-weight. The material is assumed to be infinitely stiff. Such

bodies serve to model massive concrete structures and walls in both standard and stability

analysis.

.

Fig. 22 Input of rigid bodies

1.4Input of geometry

1.4.1Input of interfaces margins of the topological model

One of the first and most important input data is the selection of the margins of the geometrical

model.

___________________________________________________________________________________________________________________________________________________________

20 GEO 4 FINE 2004


21/113

Users guide

Fig. 23 Input of margins of the geometrical modelThe width of the geometrical model can be usually estimated without much of a trouble (care

must be take in the stability analysis to provide for sufficiently large space surrounding the

critical region). The depth of a mesh however is quite important. The lowest point of a mesh can

be imagined as incompressible subsoil. If there is no such layer of the soil or rock material in the

geological profile it is possible to assume that at a certain depth from the ground the internal

forces will vanish so that there will be no deformation. This will be the lowest point of the

geometrical model. If you are not certain about the margins of the geometrical model it is useful

to proceed as follows:

- First enter larger margins, use coarser mesh and compute changes in the stress

distributions within a soil body.- In the next step modify the initial margins (regions with no apparent deformation or

changes in stresses can be cut off), generate new and finer mesh and carry out a new and

more accurate analysis.

After selecting the margins the model creation proceeds by defining geological profile via

interfaces. The procedure is described in details in the Tutorial guide GEO4, Chapter 3.2.1

Input of interfaces. Interfaces can be also imported from other modules of the software

package GEO using clipboard.

1.4.2Assigning soils

The procedure of assigning soils into geological profile is described in details in the manualGEO4 Tutorial guide, Chapter 2.2.4 Assign. The only difference compared to other

programs appears in calculation stages. Here, the soils are not assigned to individual interfaces

___________________________________________________________________________________________________________________________________________________________

GEO 4 FINE 2004 21


22/113

Geo 4 - FEM

but rather to regions that are automatically created after geometry correction is completed, see

Chapter 1.4.5. When a soil is assigned in the topology regime, this action automatically carries

over to all regions that are found in a given geological layer.

1.4.3Free nodes

Program enables an input of arbitrary number of free nodes at any point inside or outside thestructure, Fig. 24.

Fig. 24 Free nodes

Free nodes have several main functions

- Nodes to define structure (tunnel opening, lining, sheeting, beams)

- Auxiliary points for the mesh refinement

- Points to define a boundary condition, to input forces, etc.

If a free node is found inside or on the boundary of the structure, it becomes automatically a part

of the finite element mesh. This option allows an adjustment of the finite element mesh or makes

possible to create own mesh.

___________________________________________________________________________________________________________________________________________________________

22 GEO 4 FINE 2004


23/113

Users guide

Fig. 25 Mesh with no free nodes

Fig. 26 Mesh with free nodes

1.4.4Free lines

An important tool for creating a geometrical model is definition of free lines. The lines are

defined between individual points (segments, arcs, circles) or about individual points (circles).

The lines may intersect each other and may have an arbitrary number of contact points.

___________________________________________________________________________________________________________________________________________________________

GEO 4 FINE 2004 23


24/113

Geo 4 - FEM

Fig. 27 Various types of free lines

1.4.5Corrector of specified geometry

The program contains a build in automatic corrector of the specified geometry. This means that

prior to the mesh generation the program automatically locates all points of intersection of lines,

locates all closed regions and creates a corresponding model.

Such regions can be then deactivated or they can be assigned a new soil. The main advantage of

this system becomes evident when creating geometrical model for tunnels (step by step

excavation) or for sheeted structures. Creating even a very complicated model thus becomes

rather simple and can be performed very efficiently.

Fig. 28 Regions after running the geometry corrector

___________________________________________________________________________________________________________________________________________________________

24 GEO 4 FINE 2004


25/113

Users guide

1.5Mesh generator

Correctly generated finite element mesh is the major step in achieving accurate and reliable

results. The program GEO FEM has an automatic mesh generator that may substantially simplify

this task. Nevertheless, certain rules should be followed when creating a finite element mesh:

- The basic mesh density can be specified in the dialog window Mesh generator. The finerthe mesh the better the results computation as well as post-processing, however, mayslow down substantially. The goal thus becomes to find an optimum mesh density this

mainly depends on the user experiences. Meshes generated in example problems may

serve as a certain initial hint.

- The finite element mesh should be sufficiently fine in locations in which large stressgradients are expected to occur (point supports, corners, openings, etc.). To that end, it is

possible to specify the mesh refinement in the neighborhood of these locations. The mesh

refinement can be specified around individual points or lines. The spread of refinement

should be at least 3-5 times the desired refinement in the center of the refinement. Also,

both values (density and spread of refinement) should be reasonable in view of the

prescribed mesh density that applies to the surrounding region. This assures a smoothtransition between regions with different mesh densities. Singular lines should be tackled

in the same way. For more complicated problems it is useful to first carry out the

analysis with a rather coarse mesh and then after examining the results to refine the mesh

accordingly

Fig. 29 Mesh without local refinement

___________________________________________________________________________________________________________________________________________________________

GEO 4 FINE 2004 25


26/113

Geo 4 - FEM

, ,

Fig. 30 Input of local refinement

,

Fig. 31 New mesh after refining a region around a point placed in the center of an opening

As a default setting for the mesh generation the program assumes 6-node triangular elements

with mesh smoothing. The accuracy of the results more or less corresponds to twice as fine meshcomposed of 3-node triangular elements. For research and testing purposes the program allows

for the option to use constant strain (3-node) triangular elements (to select this option select the

Extended option). The stability analysis, however, can be performed with 6-node triangular

elements only. In case of nonlinear analysis, these elements should be used exclusively.

In the Warning structure critical locations dialog window the user is prompted for

possible locations on the structure that may cause problems during automatic mesh generation.

When positioning the cursor on individual warnings the corresponding critical region on a

structure is highlighted with a red color. The following items are checked:

- Whether the distance of the two points is greater than one tenth of the required elementedge length

___________________________________________________________________________________________________________________________________________________________

26 GEO 4 FINE 2004


27/113

Users guide

- Whether the distance of a point from a line is greater than one tenth of the element edgelength

- Whether the area of a region is greater than twice the element edge length

- Whether points and/or lines are found within a structure (in the soil).

These warnings suggest locations, in which the mesh generator experience problems. The

following possibilities may occur:

- The mesh is not generated => this calls for a new input of a geometrical data

- The mesh is generated => in this case it is up to the user to decide whether the mesh isreasonable; in any case, the warning can be further ignored and the analysis can be

carried out.

-

,

Fig. 32 Warning suggesting critical regions in the FE mesh

___________________________________________________________________________________________________________________________________________________________

GEO 4 FINE 2004 27


28/113

Geo 4 - FEM

Fig.33 Zoom into the critical region the two points are to close to each other

1.6Boundary conditions

The program automatically generates standard boundary conditions. Therefore, in routine

problems the user does not have to enter the step of specifying supports. The standard boundary

conditions are:

- Smooth pin in nodes on the bottom edge of the geometrical model

- Sliding pin along the vertical edges

,

Fig. 34 Standard boundary conditions

Furthermore, the user has an option to specify support conditions in free nodes of the mesh

(smooth or sliding pin, spring support, prescribed displacements). These supports become active

from the 2nd calculation stage they are not accounted for in the 1st stage when generating initialstresses.

___________________________________________________________________________________________________________________________________________________________

28 GEO 4 FINE 2004


29/113

Users guide

,

Fig. 35 Assigning boundary conditions along a line

Fig. 36 Line boundary conditions active in the analysis

1.7Beams

The beam elements serve to model beams, linings, sheeting walls, etc. The internal forces such

as moment, normal and shear forces developed along a beam can be displayed. The beam

elements must be defined when creating the geometrical model in the topology regime. They can

be assigned to already defined lines. The corresponding line then represents the beam axis. In the

1st calculation stage the beam elements are excluded by default from the analysis to be

included in the subsequent stages they must be first activated.

The beam elements are formulated on the basis of Mindlin theory with three degrees of freedom

in each node. The theory assumes that the plane cross-section normal to the beam axis before

deformation remains plane after deformation but not necessarily normal to the deformed beamaxis. At present, the internal forces are evaluated at the element nodes.

___________________________________________________________________________________________________________________________________________________________

GEO 4 FINE 2004 29


30/113

Geo 4 - FEM

Fig. 37 Input of beam parameters

Contact (interface) elements can be assigned to both sides of a beam. A correct definition of

contacts is quite important especially when modeling sheeting walls more details are provided

in Section 1.8 Contacts. The program automatically includes the beam self-weight into the

analysis. This feature, however, can be turned off.

Fig.38 Plotting beam elements

___________________________________________________________________________________________________________________________________________________________

30 GEO 4 FINE 2004


31/113

Users guide

In the 1st calculation stage the beam elements are excluded from the analysis by default the

user, however, has the option to activate them. The beam activity can be set in the Edit beam

element dialog window. This window also serves to make changes of the selected beam

parameters (e.g., change of lining thickness). Activity or information about changes made to the

beam parameters in the current calculation stage are displayed in the table.

The inputted beam can be split into several beam elements as a result of geometry correction insuch cases it is useful to take advantage of edit mode that allows editing of more then one beam

element at the same time (group edit mode) - to that end, select the required beams and edit all

selected beams the same way at the same time by clicking the Edit selected button.

Fig. 39 Editing beam elements in stages

Fig. 40 Activating beam elements

___________________________________________________________________________________________________________________________________________________________

GEO 4 FINE 2004 31


32/113

Geo 4 - FEM

1.8Heredity of features (boundary conditions, beams, soils)

Typical engineering tasks involve analysis of more construction stages. Providing you make a

change in topology of a structure after defining several stages e.g., change of beam parameters,

inserting a new support, assigning a new soil in place of the existing one the program, after

accounting for the required changes in a given stage (or in topology regime), automatically

carries over those changes to all subsequent stages of construction.

If one of the successive calculation stages already experienced the change of parameters

(function Change parameters), the program asks for a next step the new parameters can be

assigned to either all-subsequent stages or only to those that have not experienced the change of

parameters yet.

Fig. 41 Dialog window to select a way of assigning changed parameters to subsequent stages

1.9Contacts (interfaces)

The contact elements are used in applications that require studying an interaction of a structure

and a soil. They can be further used to model joints or interfaces of two distinct materials (soil

rock interface). The typical example of using contact elements is the modeling of sheetingstructures, retaining walls or tunnel lining. In such applications the contact elements are used to

model a relatively thin layer of a soil or rock loaded primarily in shear.

Fig. 42 Location of contact elemenets when modeling retaining walls

Formulation and numerical implementation of the contact element are described in details in the

Theoretical guide of this manual. Here we just recall that the formulation assums a contact

element with zero thickness (Fig. 43) that relates the contact stresses and jumps in the

displacement field developed along the interface.

___________________________________________________________________________________________________________________________________________________________

32 GEO 4 FINE 2004


33/113

Users guide

Fig. 43 Construction of a sheeting wall represented by beam an contact elements

Parameters of the contact element adjacent to a beam element are specified directly in the Edit

beam element dialog window option Contact left or Contact right, respectively.

Fig. 44 Input of contact element parameters

___________________________________________________________________________________________________________________________________________________________

GEO 4 FINE 2004 33


34/113

Geo 4 - FEM

Specification of the contact element location (left or right) depends on the beam element

orientation (element numbering) the most simple check of the correct input is to select one

option of the two options and view the graphical representation the contact is represented by a

dashed line displayed next to the beam.

Obr. 45 Contact visualization

Contacts (interfaces) can be further specified independantely along soil interfaces or on free lines

using the New contact dialog window.

.

Fig. 46 Input of contact element parameters

Two options of the contact element material model are available. In particular, one may select

either the elastic model with the possibility of plotting contact stresses while assuming the elastic

behavior along the interface or the plastic model. The plastic model is based on the classical

Mohr-Coulomb model extended by including the tesion cut-off. This model is therefore well

suited when modeling tensile separation. When modeling, e.g., sheeting structures. this model

should be used exclusively.

The basic model parameters are the cohesion c, coefficient of friction , and angle of dilation .The parameters c and can be specified also indirectly by reduction of parameters c and tan()

___________________________________________________________________________________________________________________________________________________________

34 GEO 4 FINE 2004


35/113

Users guide

of a soil along the contact. If the contact is assumed between two soils then the one used for the

reduction is the one having smaller values of c and of the two.

The contact parameters are then defined as

,soilc

cc =

.)tan( soil= If no better information regarding the reduction parameters is available one may use the

following values. For steel structures in sandy soils one may set the reduction parameters equal

to 2/3 while for clays use the value of 1/3. These parameters usualy attain higher values when

concrete structures are used. In general, the reduction parameters should be less than 1. The

dilation angle plays the same role as in case of standard soil models. Just recall that by setting

= 0 we apriory assume elastic behavior in tension/compression. The plastic deformation is thus

limitted to shear.

Additional parameters of the contact material model are the elastic stiffnesses in the normal and

tangential directions kn and ks, respectively. They can be imagined as spring stiffnesses along agiven inteface, see Fig. 47. A reliable selection of the values of these parameters is not an easy

task and is usually problem dependent. To shed a light on this subject one may relate these

stiffnesses to the material paramaters of a soil along the contact. The following relations then

apply

t

Ekn = ,

t

Gks = ,

where tis the assumed (fictitious) thickness of the interface layer

G is the shear modulus of elasticity

Eis the elastic modulus

In case of distinct materials (E1, E2, G1, G2)we take the lower value ofks a kn.

Although in case of fully plastic behavior the selection of parameters kn and ks is not essential,

the values assigned to these parameters are decisive for the success of the solution of a given

nonlinear problem. Providing these values are two large (above 100000 kN/m3) the iteration

process may ocillate. On the other hand, setting the values ofks and kn too low (below 10000

kN/m3) may result in an unrealistic distribution of displacements.

The default setting in the program is 10000 kN/m3.

Fig.47 Visualization of elastic stiffnesses

___________________________________________________________________________________________________________________________________________________________

GEO 4 FINE 2004 35


36/113

Geo 4 - FEM

1.10Anchors

Anchors as stabilizing or reinforcing elements are represented by elastic tensile-compressive bar

element with constant normal stiffness. The maximum allowable tensile force the element can

sustain controls tensile failure of an anchor element. The bar element is anchored into the soil

only at its staring and end points. No mutual interaction between the soil and anchor along the

anchor length is considered.

Anchor elements are defined by their starting and end points and by their stiffness. The anchor

can be positioned anywhere within a structure, not necessarily connected to any existing

geometry lines or finite element mesh nodes. The program automatically links the anchor

degrees of freedom to the actual degrees of freedom of the predefined finite element mesh.

To simplify its input, the anchor starting point can be either fixed on (projected to) the ground,

individual interfaces, or beam elements, or it can be inputted numerically. The anchor stiffness is

specified in terms of the elastic modulus and its area. The program makes also possible to enter

the anchor diameter the area is then determined automatically. Other important parameters are

the pre-stress force and tensile strength (the anchor breaks when the tensile strength is exceeded).

For elements with no pre-stress the pre-stress force is set equal to zero. Sufficiently large valueof the anchor tensile strength may be specified to avoid anchor failure. By default the anchor

does not support a compressive force anchor elements loaded in compression during a certain

stage of calculation are temporarily disabled. If tension occurs in subsequent analysis run, e.g.,

due to change in loading, geometry or material parameters of soil, the program automatically

introduces these elements back into the analysis. The program makes also possible to include

compressive response of an anchor. However, for elements loaded primarily in compression we

recommend to specify these elements as props see Section 1.11 Props.

Fig. 48 Input of anchor

The anchor deforms during analysis. Such deformation together with deformation of a

surrounding soil may cause the reduction of the specified pre-stress force in the anchor.Providing we wish to achieve a specific pres-stress force in the anchor, it is necessary to either

___________________________________________________________________________________________________________________________________________________________

36 GEO 4 FINE 2004


37/113

Users guide

post-stress the anchor to a given value in the next calculation stage or to use a sufficiently large

magnitude of the pre-stress force right from the beginning to compensate for a possible drop (the

resulting anchor force after completion of the calculation step is displayed at the anchor head

below the prescribed pres-stress force).

In subsequent stages the program allows only for the anchor post-stressing (change of the initialpre-stress force).

Fig. 49 Post-stressing of anchor

Introducing pre-stressed anchors into the soil may lead to plastic deformation of the soil in the

vicinity of the anchor head or root, respectively the program then often fails to converge.

___________________________________________________________________________________________________________________________________________________________

GEO 4 FINE 2004 37


38/113

Geo 4 - FEM

Fig. 50 Plastic region in the vicinity of anchor head and root

In such a case we recommend to make the following changes in the input:

- To place a beam element under the anchor head (this results into a better transition ofload into the soil).

- To place the anchor root into a sufficiently stiff soil (use the elastic or modified elasticmaterial model for the soil layer around an anchor.

When performing the stability analysis the actual pre-stressed anchor is automatically replacedby loading due to a point forces acting at the anchor head.

,

Fig. 51 Modeling anchor during stability analysis

___________________________________________________________________________________________________________________________________________________________

38 GEO 4 FINE 2004


39/113

Users guide

1.11Props

Props are supporting elements represented by elastic bar element with constant normal stiffness.

The props can sustain only compressive loading. When found in tension they are removed from

the analysis. Similar to anchors, the props are linked to the finite element mesh in two points. If

the prop is positioned into the soil then there is no interaction between the soil and prop along

the prop length considered.

Prop elements are defined by their starting and end points and by their stiffness. The program

automatically links the prop degrees of freedom to the actual degrees of freedom of the

predefined finite element mesh.

To simplify its input, the prop starting point can be either fixed on (projected to) the ground,

individual interfaces, or beam elements, or it can be inputted numerically. The prop stiffness is

specified in terms of the elastic modulus and its area. The program makes also possible to enter

the diameter of a circular prop the area is then determined automatically.

In subsequent stages the prop cannot be edited it can be either removed or inputted again.

,

Fig. 52 Input of prop

___________________________________________________________________________________________________________________________________________________________

GEO 4 FINE 2004 39


40/113

Geo 4 - FEM

Fig. 53 Example of using props

1.12Geo-reinforcements

Geo-reinforcements are tensile reinforcing elements (geotextiles, geogrids) represented again by

elastic bar elements and specified by their end points and axial stiffness. Unlike anchors or

props, the geo-reinforcement is linked to an underlying finite element mesh along its entire

length. However, similar to anchors the program introduces the geo-reinforcement end points

into the finite element mesh automatically so the geo-reinforcement can be specified anywhere

within the mesh. Owing to its geometrical characteristics, the geo-reinforcement calls for the

input of the cross-sectional stiffness taken with respect to 1m of its width, Fig. 54. The user

should contact the manufacturer for this information.

Fig. 54 Input of geo-reinforcement

___________________________________________________________________________________________________________________________________________________________

40 GEO 4 FINE 2004


41/113

Users guide

The program allows us to consider the geo-reinforcement also in compression by default

however, the part of geo-reinforcement found in compression is disabled for the analysis. This

state is simulated in Fig. 55 showing distribution of normal tensile forces over active parts of

individual geo-reinforcements. The compressive part of the geo-reinforcement is excluded from

the analysis. Similar to anchors, however, it can be automatically activated once loaded again in

tension.

Fig. 55 Evolution of tensile forces in geo-reinforcements

When introducing the geo-reinforcement into a soil body it is necessary to keep in mind a

sufficient anchorage of the reinforcement since the program does not check the geo-

reinforcement against shear failure. A sudden increase of the normal force as shown in Fig. 56suggests singularity in contact stresses and probable shear failure of the geo-reinforcement. From

that point of view the results displayed in Fig. 56 is misleading and essentially unrealistic. In

such a case, the reinforcement should be either removed from the analysis or ensure its sufficient

anchorage as plotted in Fig. 57.

___________________________________________________________________________________________________________________________________________________________

GEO 4 FINE 2004 41


42/113

Geo 4 - FEM

,

Fig. 56 Insufficiently anchored geo-reinforcement

,

Fig. 56 Sufficiently anchored geo-reinforcement

1.13Surcharge

An arbitrary number of loads can be specified in individual stages. The surcharge may act either

on the existing interface (including ground surface) or can be applied anywhere within a soil

body.

___________________________________________________________________________________________________________________________________________________________

42 GEO 4 FINE 2004


43/113

Users guide

t

Fig. 58 Input of surcharge

In subsequent calculation stages the existing surcharge can be either removed or it is possible to

change its magnitude.

Fig. 59 Change of surcharge magnitude

Note that applying a surcharge directly on the ground surface may lead to excessive plastic

deformations in the vicinity of surcharge and the analysis may fail to converge. In such a case,

one may either place a beam element under the applied surcharge, or choose an elastic ormodified elastic material model for the soil under the surcharge.

___________________________________________________________________________________________________________________________________________________________

GEO 4 FINE 2004 43


44/113

Geo 4 - FEM

1.14Water

There are three options in the program to introduce ground water:

- The ground water table can be specified as a continuous interface below and above theground surface.

- The pore pressure values are inputted via isoline. The first isoline always coincides withthe ground surface. The remaining isolines are introduced the same way as interfacesbetween individual soil layers. The pore pressure values are inserted into the table List

of interfaces in the left bottom part of screen. The values between isolines follow from

linear interpolation.

- Values of the coefficientRu are specified for individual isolines. The first isoline alwayscoincides with the ground surface. The remaining isolines are introduced the same way as

interfaces between individual soil layers. The values are inserted into the table List of

interfaces in the left bottom part of screen. The values between isolines follow from

linear interpolation.

The simplest way to check the input of water is to plot the distribution of pore pressure as shown

in Fig. 60.

Fig. 60 Plot of pore pressure distribution

1.15Removing soil (Activity) and assigning soils to regions (Regions)

The program allows for removing soils from individual regions. As an example we consider an

embankment analysis. In such a case, it must be accounted for already in the topology regime

when creating the overall geometrical model. In the 1st calculation stage, however, it can be

deactivated.

___________________________________________________________________________________________________________________________________________________________

44 GEO 4 FINE 2004


45/113

Users guide

Fig. 61 Modeling embankment 1st

calculation stage

In the next stage it can be again reactivated.

Fig. 62 Modeling embankment activity of embankment body

Similar approach applies also to underground or open excavations (tunnels, sheeting structures).

With regard to this option we should mention the case, in which the soil to be removed is found

below the GWT. There are two cases to be considered. In the first case we assume that the soil

subjected to excavation is completely enclosed by active beam elements. The beam is thenconsidered to be impermeable and both the soil and water are removed (removing total stresses

___________________________________________________________________________________________________________________________________________________________

GEO 4 FINE 2004 45


46/113

Geo 4 - FEM

inactive region is free of water). Owing to impermeability of the beam elements, the pore

pressure distribution remains unchanged, see Fig. 63.

Fig. 63 Pore pressure distribution after removing soil from region enclosed by active beams

In all other cases we assume that water in the excavated region is still active. This state is evident

from the pore pressure distribution shown in Fig. 64. Its effect can be removed by changing the

ground water table, Fig. 65.

Fig. 64 Pore pressure distribution after removing soil

Fig. 65 Pore pressure distribution after chaniging the ground water table

In the regime Regions it is further possible to replace existing soils in a given region with a

new one. This option can be also used to change the previously selected material model, e.g.,

from the elastic into plastic material model. This option, however, should be used with caution.

1.16AnalysisThe actual analysis is carried out after pressing the Analysis button.

___________________________________________________________________________________________________________________________________________________________

46 GEO 4 FINE 2004


47/113

Users guide

Fig. 66 Structure analysis

During analysis the program attempts to arrive at such a solution that satisfies for given loading

and boundary conditions the global equilibrium. In most cases this step results into an iterative

process. The process of iteration and convergence of the solution is displayed on the screen. The

analysis can be stopped any time by pressing the STOP button. The results are then available

for the last converged load increment.

Fig. 67 Process of iteration

When the analysis is completed the program displays the final results and provides information

about the course of analysis.

___________________________________________________________________________________________________________________________________________________________

GEO 4 FINE 2004 47


48/113

Geo 4 - FEM

Fig. 68 Screen after completion of analysis

The correct results are obtained when 100% of the applied load is reached. Due to convergence

failure the program may stop before reaching the desired load level (only a fraction of the total

applied load is reached).

Fig. 69 Analysis failed to converge plot of equivalent plastic strain

___________________________________________________________________________________________________________________________________________________________

48 GEO 4 FINE 2004


49/113

Users guide

Such a result calls for certain modifications of the model the following steps can be followed:

- Increase the stiffness of the structure.

- Decrease the applied loads, split the soil excavation in more steps.

- Improve material parameters of existing soils.- Add reinforcing members (beams, anchors).

- Change parameters settings affecting the iteration process (increase number of iterations).

An explanation of the failure of analysis can be provided by plots of plastic strains, Fig. 69.

1.17Analysis setting

The default setting of parameters that drive the solution analysis is optimized to ensure sufficient

accuracy and efficiency of the analysis. Nevertheless, an experienced user may require to change

the default setting, or to examine the influence of parameters on the accuracy and course of the

analysis. The parameters setting can be adjusted in the Analysis setting dialog window.

The change of standard setting deserves, however, a word of caution. Prior to making anychanges, the user should be well aware of possible consequences. In particular, improper setting

may substantially slow down the computation process, may cause divergence and eventually lead

to incorrect results. Details, regarding individual parameters and their optimal setting, can be

found in the Theoretical guide.

The default setting can be always recovered by pressing the Standard button.

1.17.1Solution method

The program GEO FEM serves to analyze geotechnical problems characterized by nonlinear

response of a soil or rock body. A successful analysis of most of such problems calls for aniterative solution of a given boundary value problem. Applying the finite element method (FEM)

then leads to an incremental form of the equilibrium conditions written as

fuKT = , (1)

where KT is the instantaneous (tangent) stiffness matrix of a structure, u is the vector of nodal

displacement increments and f corresponds to the vector of out-of-balance force increments.

Eq. (1) can be solved only approximately using a suitable numerical method. The goal of the

method is to arrive, during the process of iteration, at such a state of stress and strain that

satisfies the condition f= 0, see also the theoretical guide of the program. The program GEO

FEM offers two basic methods, Fig. 70:

1. TheNewton-Raphson method NRM,2. TheArc-length method ALM.

___________________________________________________________________________________________________________________________________________________________

GEO 4 FINE 2004 49


50/113

Geo 4 - FEM

Fig. 70 Analysis setting method selection

1.17.2Update of the stiffness matrix

Both methods are described in details in the theoretical guide of the program. Recall that the

full Newton-Raphson method assumes that an instantaneous tangent stiffness matrix is formed

each new iteration. Forming a new tangent stiffness matrix only at the beginning of a new load

increment leads to so-called modified Newton-Raphson method. If the stiffness matrix is formed

only once at the beginning of the solution analysis we obtain so called initial stress method.

Individual methods can be selected from the Analysis settings dialog window section

Stiffness update, Fig. 71. Individual options correspond to:

1. Keep elastic initial stress method,2. Each iteration full Newton-Raphson method,3. Each load step modified Newton-Raphson method.

Fig. 71 Newton-Raphson method stiffness update option

___________________________________________________________________________________________________________________________________________________________

50 GEO 4 FINE 2004


51/113

Users guide

The default setting assumes the full Newton-Raphson algorithm (stiffness update after each

iteration). Recall that in GEO FEM formulation of the stiffness matrix is consistent with the

stress update algorithm; see the theoretical guide for more details. Such a formulation then

ensures quadratic convergence of the full Newton-Raphson (NRM) unlike the modified NRM or

the initial stress method that, in comparison with the full NRM, require considerably moreinteractions to attain equilibrium. On the other hand, it is fair to mention that the computational

cost per iteration is mainly determined by the calculation and factorization of the tangent

stiffness matrix. Assuming elastic response of a structure it is clearly meaningless to set up the

structural stiffness matrix more then once (stiffness update keep elastic). On the contrary,

increasing the degree of nonlinearity suggests more frequent stiffness reformations (stiffness

update Each iteration). Details can be found in the theoretical guide.

1.17.3Initial load step

As already intimated in the introductory part of this chapter the actual analysis is carried out

incrementally in several load steps until the overall prescribed load is reached. Note that theprogram requires setting the initial load step only. This can be done in the Analysis settings

dialog window, Fig. 70. This parameter represents the ratio between the load applied in a given

load step to overall prescribed load. Depending on the course of iteration this parameter is

adaptively adjusted. The default setting assumes 25% of the total prescribed load. Similarly to

what we have already mentioned it holds that increasing the solution complexity from the

nonlinear response point of view requires reduction of this parameter. In case of elastic response,

however, this parameter can be set to 1, which corresponds to the solution of a given problem in

one load step.

1.17.4Maximum number of iterations per a single load step

This parameter represents the maximum number of iterations allowed for a single load step to

reach the state of equilibrium. It can be set in the Analysis settings dialog window, Fig. 70.

Exceeding this value prompts the program to automatically reduce the current value of the

assumed load step and restarts the solution from the last load level that complies with the state of

equilibrium. Similar action is taken when oscillation or divergence of the program is imminent.

1.17.5Convergence criterion

For the incremental solution strategy based on an iterative method to be effective, it is necessary

to select suitable criteria (preset tolerances for reaching equilibrium) for the termination of the

iteration process. Recall that loose convergence criteria may result in inaccurate results while too

tight convergence tolerances may lead to unjustified increase of computational cost spent to

arrive at results of superfluous accuracy. In GEO FEM the convergence is checked against the

change of nodal displacement increments, the change of out-of-balanced forces and also the

change of internal energy. The last criterion gives a certain idea about how both displacements

and forces approach their equilibrium values. The above parameters, suitable for convergence

test, are described in details in the theoretical guide. Convergence tolerances for individual

parameters can be set in the Analysis settings dialog window, Fig. 70 The corresponding

settings are:

1. Displacement error tolerance tolerance for the change of displacement incrementnorm.

2. Out-of-balanced forces tolerance tolerance for the change of out-of-balance forcenorm.

___________________________________________________________________________________________________________________________________________________________

GEO 4 FINE 2004 51


52/113

Geo 4 - FEM

3. Energy error tolerance - tolerance of the change of internal energy.

The default setting is 0.01 for all convergence tolerances.

1.17.6Setting the Newton-Raphson method

In Section 1.15.4 we already pointed out the possibility of restarting the program in case ofdivergence. This state manifests itself either by very slow convergence, divergence or oscillation

of the process of iteration. With the Newton-Raphson method the course of iteration can be

driven by setting the parameters in theAnalysis settings section Newton Raphson, Fig. 72.

The corresponding parameters are

1) Relaxation factor it represents the value of reduction of the current load step for therestart providing the solution fails to converge. A new value of the assumed load step

is found from the expression:

new load step = old load step / relaxation factor.

2) Max. No. of relaxations for a single load step this parameter determines how manytimes it is possible to invoke the above action during the entire analysis. Exceeding

this value prompts the program to terminate the analysis. The results are then

available for the last successfully converged load level.

3) Min. No. of iterations for a single load step this parameter allows for possible

acceleration of the analysis. In particular, providing the number of iterations to

converge in the last load step is less than the minimum one set, the load step for a new

load increment is increased as follows:

new load step = old load step * relaxation factor.

The default setting of the above parameters corresponds to values displayed in Fig. 72.

Fig. 72 Parameters driving the iteration process

___________________________________________________________________________________________________________________________________________________________

52 GEO 4 FINE 2004


53/113

Users guide

1.17.7Setting the Arc-length method

The Arc-length method is relatively robust method particularly suitable for the solution of

problems that require the search for the collapse load of a structure. Stability analysis of earth

structures (slopes, embankments) is an example of such a task. The principles of the method are

summarized in the theoretical guide. Here we just remind the essential difference between the

NRM and the Arc-length method (ALM). Unlike the NRM where the solution is driven purelyby prescribing load increments, the ALM introduces an additional parameter representing a

certain constraint on the value of load increment in a given load step. The value of load step thus

depends on the course of iteration and is directly related to the selected arc length; see the

Theoretical guide. The basis of the method is an assumption that the prescribed load varies

proportionally during calculation. This means that a respective level of the applied load can be

expressed as:

FF = , (2)

where F represents the current fraction of the total applied load, is the coefficient ofproportionality and F corresponds to the overall prescribed load. Note that with ALM the loadvectorF represents only a certain reference load that is kept constant during the whole response

calculation. The actual value of the load at the end of calculation is equal to the multiple ofF;

< 1 represents the state when the actual bearing capacity of a structure is less than prescribed

reference load; if at the end of response calculation exceeds 1, the program automatically

adjusts the arc length in order for the solution to converge to value = 1 within a selected

tolerance equal to 0.01 (1% the maximum applied load). This value cannot be changed.

1.17.8Setting a version of the Arc-length method

The literature offers a number of ALM formulations. The program GEO FEM supports themethod suggested by Crisfield and consistently linearized method proposed by Ramm. The latter

one is considerably simple, at least from the formulation point of view, than the Crisfield

method. On the other hand it is reportedly less robust. The default setting is the Crisfield method.

The method can be set in theAnalysis settings section Arc-length, Fig. 73.

Fig. 73 - Arc-length setting a version of the method

___________________________________________________________________________________________________________________________________________________________

GEO 4 FINE 2004 53


54/113

Geo 4 - FEM

1.17.9Setting arc-length

The arc length is the basic parameter affecting the response calculation. An indicator for the

selection of arc length can be a course of iteration in the previous solution stage. In any case, the

program GEO FEM enables the following setting, Fig. 74:

1)Determine from load step the arc length is determined automatically from theinitial load step. This parameter is discussed in details in Section 1.15.3.

2)Assign from the previous stage the value of arc length at the end of previouscalculation stage is used as a starting value for a new stage. Clearly, this option

becomes active in the second stage of construction.

3) Input the value of arc length can be directly prescribed.

Providing the structure response cannot be determined a priory we recommend to use the first

option. Depending on the course of calculation it is possible to adjust the value of arc length and

repeat the calculation. At no event, however, it is possible to ensure convergence for an arbitrary

value of arc length selected. Similarly to NRM, if the convergence problems occur the program

allows for the reduction of the current arc length and restarts the calculation. Parameters

Relaxation factorandMaximum No. of iteration affect the process of iterations the same way as

the parameters described already in Section 1.15.6. The next parameter driving the iteration

process is theMaximum No. of load steps, Fig. 74. The program always carries on the prescribed

number of load steps providing:

parameter exceeds 1,

or the maximum number of relaxations of arc length is exceeded.

Providing the analysis is terminated due to exceeding the maximum number of prescribed loadsteps and parameter is less than 1, it is necessary to increase the number of steps and restart the

analysis.

1.17.10Automatic arc length control

Automatic arc length control strategy constitutes very important part of implementation of any

numerical method. The program GEO FEM makes possible to adaptively adjust the current arc

length for a new load step depending on the course of iteration in the previous step by activating

option Optimize in theAnalysis settings section Arc-length. The program will then attempt

to select a value of arc length that keeps the desired number of iterations in each load step

needed for convergence option Optim. No. of iter. in a single load step, Fig. 75. As for defaultsetting this option is disabled. Further details can be found in the theoretical guide.

The next parameter driving the process of iteration is Ratio load/displacement. This parameter

represents a scalar factor, which adjusts the scales of load given by parameter and

displacement vectoru. providing this parameter is sufficiently large the analysis is essentiallydriven by load increment. Setting this parameter equal to 0 (default setting) we obtain so-called

cylindrical ALM and the analysis will be driven by displacement increment. This approach is

more stable and recommended by the authors. Nevertheless, the program GEO FEM allows for

optimization of this parameter by activating the option Optimize, Fig. 75. In such a case the

current value of this parameter is set equal to the Bergan current stiffness parameter that provides

a scalar measure of the degree of nonlinearity. With increasing the degree of nonlinearity this

parameter is decreasing. In the vicinity of collapse load the value of this parameter approacheszero and the solution is driven by displacement increment. This strategy thus supports the use of

___________________________________________________________________________________________________________________________________________________________

54 GEO 4 FINE 2004


55/113

Users guide

cylindrical method having this parameter equal to zero. As for the default setting this option is

turned off. Further details can be found again in the theoretical guide.

Fig. 74 - Arc-length arc length setting

Fig. 75 Arc-length automatic arc length control

1.1

Documents

Geo5FEM.pdf