multiplas usermat manual 5.0.679

MULTIPLAS

GENERAL MULTISURFACE PLASTICITY

Release

Dynamic Software and Engineering GmbH

MULTIPLAS – ELASTOPLASTIC MATERIAL MODELS FOR ANSYS


USER’S MANUAL

February 2013

Release 5.0.679 for ANSYS 14.0 and ANSYS


Weimar, Steubenstraße 25, Germany

ELASTOPLASTIC MATERIAL


14.5


License agreement Copyright of this product and its attachments by DYNARDO GmbH. All unauthorized copying, multiplica-tion and reengineering of this product and its documentation is strictly prohibited. Guarantee reference dynardo GmbH takes greatest care in developing software and tests all products thoroughly. Despite, the user is fully responsible for the application of this software and is obliged to check correctness of the re-sults obtained. dynardo GmbH takes no liability for any damage caused by the use of this software, by incorrectness of results, or by misinterpretation. Registered trademark All trademarks and product name specified in this documentation are registered trademarks of dynardo GmbH

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USER’S MANUAL - multiPlas 5.0.679, May, 2013

CONTENT1 INTRODUCTION................................................................................................................. 4

2 INSTALLATION INSTRUCTIONS ....................................................................................... 5

2.1 Dll-version ................................................................................................................................... 5

2.1.1 How to use ANSYS with multiPlasDll .......................................................................................... 5

2.2 Custom executable-version ......................................................................................................... 5

2.2.1 How to start ANSYS Mechanical APDL with multiPlas ............................................................... 5

2.2.2 How to use ANSYS Workbench with multiPlas ........................................................................... 6

2.3 License ........................................................................................................................................ 6

3 THEORY OF THE MULTIPLAS MATERIAL MODELS IN ANSYS ...................................... 7

3.1 Basics of elasto-plasticity in multiPlas ......................................................................................... 7

3.2 Multisurface plasticity .................................................................................................................. 8

3.3 Computed yield surfaces ............................................................................................................. 9

3.3.1 Introduction yield surfaces of basic material models .................................................................. 9

3.3.2 MOHR-COULOMB isotropic yield criterion ............................................................................... 10

3.3.3 MOHR-COULOMB anisotropic yield criterion ........................................................................... 12

3.3.4 Yield criterion according to DRUCKER-PRAGER .................................................................... 14

3.3.5 Combination of flow condition according to MOHR-COULOMB and DRUCKER-PRAGER or TRESCA and von MISES ......................................................................................................................... 15

3.3.6 Concrete modeling using modified DRUCKER-PRAGER model.............................................. 16

3.4 Dilatancy .................................................................................................................................... 22

4 COMMANDS ..................................................................................................................... 23

4.1 Material Models ......................................................................................................................... 23

4.2 TBDATA-Declaration ................................................................................................................. 24

4.2.1 LAW = 0 – linear elastic model.................................................................................................. 24

4.2.2 LAW = 1 – jointed rock material model ..................................................................................... 25

4.2.3 LAW = 2 – Modified Drucker-Prager model with joints ............................................................. 27

4.2.4 LAW = 5 – Modified Drucker-Prager, temperature dependent ................................................. 29

4.2.5 LAW = 9 – Concrete .................................................................................................................. 31

4.2.6 LAW = 10 – jointed rock material model (linear elastic base material) ..................................... 39

4.2.7 LAW = 11 – Fixed crack model ................................................................................................. 41

4.2.8 LAW = 14 – Menetrey-Willam model for concrete .................................................................... 42

4.3 Numerical control variables ....................................................................................................... 46

4.3.1 Choice of the numerical control variables ................................................................................. 46

4.3.2 Remarks for choosing the material parameters ........................................................................ 46

4.3.3 Remarks and tips for using multiPlas in nonlinear structural analysis ...................................... 46

4.4 Remarks for Postprocessing ..................................................................................................... 48

5 VERIFICATION EXAMPLES ............................................................................................. 49

5.1 Example 1 – Earth pressure at rest ........................................................................................... 49

5.2 Examples 2 to 4 - Earth pressure at rest and active earth pressure ......................................... 51

5.3 Examples 5 to 8 - Kienberger Experiment G6 [6-13] ................................................................ 53

5.4 Example 9 - MOHR-COULOMB anisotropic ............................................................................. 55

5.5 Example 10 – Concrete-model DRUCKER-PRAGER singular (LAW=9) ................................. 56

5.6 Example 11 – Concrete-model DRUCKER-PRAGER singular (LAW=9) ................................. 58

5.7 Example 22 – Single Joint Shear-Test (LAW=1, 10) ................................................................ 60

5.8 Example 23 – Single Joint Tensile-Test (LAW=1, 10) .............................................................. 61

6 REFERENCES ................................................................................................................. 62

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1 INTRODUCTION

This manual describes the use of Dynardo’s software product multiPlas for ANSYS. multiPlas is a library of elasto-plastic material models for ANSYS.

The elasto-plastic material models in multiPlas, enable the user to simulate elasto-plastic effects of artifi-cial materials, e.g. steel or concrete, and natural born materials, e.g. soil or rock, in geotechnics, civil en-gineering - as well as - mechanical engineering.

In the context of finite element calculations with ANSYS, multiPlas provides an efficient and robust algo-rithm for the handling of single and multi-surface plasticity. The material models are based on elasto-plastic flow functions with associated and non-associated flow rules. One special feature of the multiPlas material models is the combination of isotropic and anisotropic yield conditions.

The multiPlas material models are available for structural volume elements, e.g. SOLID 185 or SOL-ID 186, structural shell elements, e.g. SHELL 181 or SHELL 281, and structural plane elements assuming plane stress, plane strain or axisymmetric conditions, e.g. PLANE 182 or PLANE 183.

The following material models and features are provided:

Model Application Flow Rule Stress-Strain

Response Temperature Dependency

isotropic Material Models:

Tresca Steel, ... associative bilinear, ideal elastic-plastic

Mohr-Coulomb Soil, Rock, Stone, Masonry, ... non-associative

bilinear, residual strength yes

von Mises Steel, ... associative bilinear, ideal elastic-plastic

Drucker-Prager Soil, Stone, ... associative bilinear, ideal elastic-plastic

modified Drucker-Prager

Stone, Cement, Concrete, ... associative

bilinear, ideal elastic-plastic yes

Concrete Concrete, Cement, Stone, Brick, ... non-associative

nonlinear hardening and softening yes

Tension cut off rotated cracking associative residual strength anisotropic Material Models:

Mohr-Coulomb

Joints, jointed Rock, Cohesive Zones, ... non-associative

bilinear, residual strength

Masonry_Ganz Masonry, ... non-associative nonlinear hardening and softening yes

Tsai / Wu Wood, ... associative bilinear, ideal elastic-plastic

boxed value Wood, ... associative multilinear hardening and softening

Tension cut off fixed cracking, Cohesive Zones associative

residual strength / exponential softening

Additionally, all Mohr-Coulomb Models are coupled with a tension cut-off yield surface.

In simulations of joint materials (e.g. jointed rock), it is possible to arrange the joint sets arbitrarily. Isotropic and anisotropic Mohr-Coulomb yield surfaces can be combined in manifold ways. Up to 4 joint sets can be associated with an isotropic strength definition.

MultiPlas has been successfully applied in nonlinear simulations of and concrete as well as in stability analysis of soil or jointed rock.

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2 INSTALLATION INSTRUCTIONS multiPlas is provided either as customized executable (ANSYS.EXE) for ansys or as a dll (usermatLib.dll).

2.1 Dll-version The custom executable multiPlas package is delivered as a single zip-file, e.g. multiP-lasDll_5_0_679_ansys145_64bit.zip. Please extract this file into an arbitrary directory, e.g. C:\Program Files\ANSYS Inc\v145. A new sub-directory, multiPlasDll_5.0.679 is created. Please notice the full path to your multiPlas installation. There is no further installation required for multiPlas.

2.1.1 How to use ANSYS with multiPlasDll In order to use the multiPlas dll in your simulations (ANSYS Mechanical APDL or ANSYS workbench) the environment variable ANS_USER_PATH must be set before starting ANSYS. The variable can be set globally for the user in the Windows Control Panel (Systemsteuerung):

• WindowsXP: 1. “Systemsteuerung” öffnen 2. “System” öffnen 3. auf Tab “Erweitert” wechseln 4. “Umgebungsvariablen” auswählen 5. “Neu” auswählen 6. unter “Name der Variablen” ANS_USER_PATH eintragen 7. unter “Wert der Variablen” den absoluten Pfad zur multiPlas dll eintragen 8. Mit „Ok“ die Änderungen bestätigen

• Windows 7: 1. „Systemsteuerung“ öffnen 2. eventuell „Benutzerkonten und Jugendschutz“ öffnen 3. „Benutzerkonten“ öffnen 4. „Eigene Umgebungsvariablen ändern“ auswählen 5. „Neu“ auswählen 6. unter „Name der Variablen“ ANS_USER_PATH eintragen 7. unter „Wert der Variablen“ absoluten Pfad zur multiPlas dll eintragen 8. Mit „Ok“ die Änderungen bestätigen

After setting this variable ANSYS can be started in the standard way. Alternatively, the variable can be set on the command line:

set ANS_USER_PATH=<multiPlasDllDir> where <multiPlasDllDir> must be replaced by the full (absolute) path to your multiPlas dll installation. AN-SYS (mechanical APDL or workbench) must be started from the same command line. Otherwise this va-riable is not known to ANSYS. If this variable was successfully set the following message can be found in the ANSYS output: User Link path (ANS_USER_PATH): <multiPlasDllDir> where <multiPlasDllDir> should be the full (absolute) path to the multiPlas dll.

2.2 Custom executable-version The custom executable multiPlas package is delivered as a single zip-file, e.g. multiP-las_5_0_679_ansys145_64bit.zip. Please extract this file into an arbitrary directory, e.g. C:\Program Files\ANSYS Inc\v145. A new sub-directory, multiPlas_5.0.679 is created. Please notice the full path to your multiPlas installation. There is no further installation required for multiPlas.

2.2.1 How to start ANSYS Mechanical APDL with multiPlas The ANSYS Mechanical APDL Product Launcher can be used to start ANSYS Mechanical APDL with multiPlas. After starting the launcher, choose the “Customization/Preferences” tab. In the field “Custom ANSYS executable” browse to the ANSYS.EXE in your multiPlas installation directory. This procedure is summarized in Fig. 2-1.

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Fig. 2-1: start multiPlas in ANSYS Mechanical APDL via launcher

Another possibility is to start ANSYS Mechanical APDL from command line using the option “–custom <multiPlasDir>\ANSYS.EXE”, where <multiPlasDir> must be replaced by the full (absolute) path to your multiPlas installation.

The following command line starts ANSYS Mechanical APDL with multiPlas in graphical mode: "C:\Programme\ANSYS Inc\v145\ansys\bin\winx64\ansys.exe" –g –custom “<multiPlasDir>\ANSYS.EXE”

The corresponding command line for batch mode is: "C:\Programme\ANSYS Inc\v145\ansys\bin\winx64\ansys.exe" –b –i <InputFile> -o <OutputFile> –custom “<multiPlasDir>\ANSYS.EXE”

An example of the windows-batch script is included in the shipment.

Note: The path to the customized multiPlas executable must be enclosed in quotation marks.

For any further command line options please take a look at the ANSYS operations guide: Operations Guide, chapter 3, Running the ANSYS Program, 3.1. Starting an ANSYS Session from the Command Level

2.2.2 How to use ANSYS Workbench with multiPlas In order to enable multiPlas in ANSYS Workbench the solver settings in Mechanical must be customized. In ANSYS Mechanical:

• Select “Solve Process Settings…“ from menu “Tools“ • Choose the solver settings to be modified and select “Advanced…” • Add the following option to the field “Additional Command Line Arguments”: –custom “<mul-

tiPlasDir>\ANSYS.EXE”

2.3 License In addition the multiPlas license file, e.g. dynardo_client.lic, must be copied into one of the following direc-tories:

the application installation directory the "%Program Files%/Dynardo/Common files" directory (Unix: "~/.config/Dynardo/Common") the users home directory (Unix: $HOME, Windows: %HOMEPATH%) the current working directory

For any further questions of licensing, please contact your system administrator or write an E-mail to [email protected]

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3 THEORY OF THE MULTIPLAS MATERIAL MODELS IN ANSYS

3.1 Basics of elasto-plasticity in multiPlas The material models in multiPlas uses a rate-independent plasticity. The material models are characte-rized by the irreversible strain that occurs once yield criteria are violated. It is assumed that the total strain vector can be divided into an elastic part and a plastic part.

pleltot εεε += (3-1) where: εel – elastic strain vector (EPEL) εpl – plastic strain vector (EPPL) The plastic strains are assumed to develop instantaneously, that is, independent of time. The elastic stress domain is limited by the yield criterion

0),( ≤κσF (3-2) where: σ - stress vector κ - hardening parameter If the computed (trial) stress, using the elastic deformation matrix, exceeds the yield criteria (F>0), then plastic strain occurs. The evolution of plastic strains is computed using the flow rule

σ

λε∂∂= Q

d pl (3-3)

where: λ - plastic multiplier (which determines the amount of plastic strains) Q - plastic potential (which determines the direction of plastic strains)

The plastic strains reduce the stress state so that it satisfies the yield criterion (F=0). By using associated flow rules, the plastic potential is equal the yield criterion and the direction of plastic strains is perpendicu-larly to the yield surface. FQ = (3-4) By using non-associated flow rules FQ ≠ (3-5) effects that are known from experiments, like dilatancy, can be reproduced more realistically. The hardening / softening function Ω(κ) describes the expansion and/or the reduction of the initial yield surface as well as the translation of the yield criterion in the stress domain. For the strain driven harden-ing/softening equations in multiPlas the scalar value κ serves as a weighting factor for plastic strain.

pleq

pl ddd εεκκ == )( (3-6)

The introduction of separate softening functions for each strength parameter allows the formulation of an orthotropic softening model which depends on the failure mode. Existing relations, for example shear-tension interaction (mixed mode), can be described by the model. The numerical implementation of the plasticity models is carried out using the return-mapping method [6-17], [6-18], [6-22]. The return mapping procedure is applied at integration point level for the local itera-tive stress relaxation. It consists of two steps: 1. elastic predictor step:

tot1i

*i

trial1i dD ++ ε+σ=σ (3-7)

2. plastic corrector step (local iterative procedure):

σ∂∂−=

λσ Q

Dd

d

(3-8)

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3.2 Multisurface plasticity The consideration of different failure modes rsp. failure mechanisms of a material is possible by a yield surface built up from several yield criteria. In the stress domain then a non-smooth multisurface yield cri-terion figure develops. The elastic plastic algorithm has to deal with singularities at intersections from different yield criteria (e.g. F1 to F2 as represented in Fig. 3-1).

pl1dε

pl2dε

F1 F2

Fig. 3-1 Intersection between the two flow criteria F1 and F2 The consistent numerical treatment of the resulting multi-surface plasticity must deal with the possibility that many yield criteria are active simultaneously. This leads to a system of n=j equations:

∑=

∂∂

∂∂

−∂∂

∂∂

=

∂∂ YCactiveofSet

jj

j

n

n

njT

n

T

n dFQ

DF

dDF

1

λλκ

κσσε

σ (3-9)

The solution of this system of equations generates the stress return to flow criteria or within the intersection of flow criterias. Contrary to single surface plasticity exceeding the flow criterion is no longer a sufficient criterion for activity of the plastic multiplier for each active yield criterion. An activity criterion needs to be checked.

0d j ≥λ (3-10) This secures that the stress return within the intersection is reasonable from a physical point of view.

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3.3 Computed yield surfaces

3.3.1 Introduction yield surfaces of basic material model s

Fig. 3-2 Cut in the deviator plane of different flo w figures Miscellaneous yield criteriona of soil or rock mechanics generally describe flow figures which lie in be-tween the flow figure of Mohr-Coulomb and of Drucker-Prager. . The difference in the area, surrounded by the yield surface (elastic stress domain) in the deviator-cut-plane, is 15% at its maximum. In the standard literature of soil mechanics, the general usage of Mohr-Coulomb material models is sug-gested. Yield graphs according to Drucker and Prager do in fact generally overestimate the bearing strength. For brittle materials (concrete/rock) composite flow conditions on the basis of Mohr-Coulomb as well as composite flow conditions on the basis of Drucker-Prager are used in the standard literature.

σ3

MOHR-COULOMB (Sonderfall: TRESCA) ARGYRIS DRUCKER/PRAGER (Sonderfall: V. MISES)

σ2 σ1

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3.3.2 MOHR-COULOMB isotropic yield criterion

Fig. 3-3 MOHR-COULOMB isotropic yield criterion The yield criterion is:

ϕ−

ϕΘ−Θσ+ϕσ= cosc3

sinsincossinF Sm (3-11)

where:

3zyx

m

σ+σ+σ=σ (3-12)

2S I=σ (3-13)

( )23

2

3

II

233

3sin −=Θ (3-14)

σm hydrostatic stress I2 second invariant of the deviatoric main stresses I3 third invariant of the deviatoric main stresses Θ Lode-angle

-σ1

σ1 = σ2 = σ3

-σ2

σF

-σ3

-σ3

τ

Θ = - 30° Θ = 30° compressive meridian (Θ = 30°) ϕ tensile meridian (Θ = - 30°)

C

σM

-σ2 -σ1

These yield criteria depend only on the two ma-terial parameters: cohesion c and inner friction angle ϕ

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Necessary material parameters in the ANSYS material model MOHR-COULOMB isotropic:

ϕ – inner friction angle (phi) C – cohesion ft – tensile strength (in case of tension cut off) ψ – dilatancy angle (psi)

The residual strength can be defined. The residual strength is initiated after the yield strength has been exceeded. The uniaxial compressive strength corresponds with the friction angle and cohesion as shown below:

+=−

=2

45tan2sin1

cos2 ϕϕϕ

ccfc (3-15)

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3.3.3 MOHR-COULOMB anisotropic yield criterion For the definition of joints, separation planes or strength anisotropies the position of the yield surface de-pends on the position of the two joint-angles: Fig. 3-4 Angle definition of the joint The yield criterion is:

0tanRe =−⋅− Cns ϕστ (3-16)

Fig. 3-5 MOHR-COULOMB anisotropic yield criterion

where:

τRes – shear stress in the joint

σn – normal stress perpendicular to the joint Necessary material parameters in the ANSYS material model MOHR-COULOMB anisotropic:

α,β – position angle of the family or separation planes ϕ – friction angle C – cohesion ft – tensile strength (in case of tension cut off) ψ – dilatancy angle

The two angles „First Angle“ (α) and „Second Angle“ (β) describe the position of the joint /

separation plane.

y y y x

z x z α β

1. α - rotation against positive rotational direction about the z-axis

2. β - rotation in positive rotational direction about the y-axis

x z

|τRes|

ϕ

C

σn

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Residual strength can be defined. The residual strength is initiated after the yield strength has been ex-ceeded.

x

y z

xJ

yJ

zJ

α=0°, β=90°

x

y z

xJ

yJ

zJ

α=90°, β=90°

x

y z

xJ

yJ

zJ

α=90°, β=0° Fig. 3-6 Examples for the angle definition of joints

x, y, z – Element coordinate system xJ, yJ, zJ – joint coordinate system

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3.3.4 Yield criterion according to DRUCKER-PRAGER

Fig. 3-7 Flow conditions according to DRUCKER-PRAGE R The Drucker-Prager yield criterion is:

ytmS~F σ−σβ+σ= (3-17)

The plasticity potential is:

mSQ σδβ+σ= (3-18)

where:

σm hydrostatic stress s. (3.12) I2 second invariant of the deviatoric main stresses s. (3.13) δ dilatancy factor

The Drucker-Prager yield criterion can approximate the Mohr-Coulomb failure condition as circumlocutory cone or as inserted to a cone (see Fig. 3-8). Using the material library multiPlas calculation of arbitrary

-σ1

σ1= σ2= σ3

-σ2

σF

-σ3 √3 c cotϕ

-σ3 τ

σM

-σ2 -σ1

interim values or blending with Mohrparameters in the ANSYS multiPlas

β and y

~

σ . Both parameters are connected to cohesion and angle of friction by the following formula:

(ϕβ

sin33

sin6

+⋅=

(σ

sin33

cos6~+⋅⋅= c

y

Fig. 3-8 Drucker- Prager yield criterion

3.3.5 Combination of flow condition according to MOHRand DRUCKER

As shown in Fig. 3-8 the MOHRtic stress domain. The difference of the surrounded area in the dev For some problem formulations it can be necessary the MOHR-COLOUMB yield criterionvergence or even divergence, it can be reasonable to use a combination of the yield criteria to stabilize the numerical computation. It has to be kept in mind that this combination is reasonable only for numerical stabilization. It leads ievitably to differences in the COULOMB. The return-mapping of the stress is not commutaded exactly for both criteria COULOMB and DRUCKER-PRAGER. Therefore, the permissibility of these result has to be checked idividually!

USER’S MANUAL -

values or blending with Mohr-Coulomb failure conditions are possible as well. Necessary material multiPlas material model DRUCKER/PRAGER are

connected to cohesion and angle of friction by the following formula:

)ϕϕ

sin

)ϕϕ

sin

cos

β =

σ~ =y

Prager yield criterion as circumlocutory cone (left) or inserted to a cone (right).

Combination of flow condition according to MOHRand DRUCKER -PRAGER or TRESCA and von MISES

the MOHR-COULOMB and the DRUCKER-PRAGER yield criterion. The difference of the surrounded area in the deviator cut plane is

For some problem formulations it can be necessary to limit the elastic stress domainCOLOUMB yield criterion. In cases if MOHR-COULOMB or TRESCA alone lead to poor co

divergence, it can be reasonable to use a combination of the yield criteria to stabilize

It has to be kept in mind that this combination is reasonable only for numerical stabilization. It leads ievitably to differences in the results contratry to the sole usage of the yield criterion by MOHR

mapping of the stress is not commutaded exactly for both criteria PRAGER. Therefore, the permissibility of these result has to be checked i

15

- multiPlas 5.0.679, May, 2013

mb failure conditions are possible as well. Necessary material

connected to cohesion and angle of friction by the following formula:

( )ϕϕ

sin33

sin6

−⋅

( )ϕϕ

sin33

cos6

−⋅⋅c

as circumlocutory cone (left) or inserted to a cone (right).

Combination of flow condition according to MOHR -COULOMB or TRESCA and von MISES

PRAGER yield criterion differ in the elas-tor cut plane is 15% at the maximum.

stress domain to the area given by COULOMB or TRESCA alone lead to poor con-

divergence, it can be reasonable to use a combination of the yield criteria to stabilize

It has to be kept in mind that this combination is reasonable only for numerical stabilization. It leads in-results contratry to the sole usage of the yield criterion by MOHR-

mapping of the stress is not commutaded exactly for both criteria – MOHR-PRAGER. Therefore, the permissibility of these result has to be checked in-

16


3.3.6 Concrete modeling using modified DRUCKER-PRAGER mod el The yield condition consists of two yield criteria (equations (3-19), (3-20)), whereby the concrete strength can be described closed to the reality as well in the compressive as in the tensile domain.

1ytmtS1~F Ωσ−σβ+σ= (3-19)

( )zd

zdt RR

RR

+−= 3β )(3

2~

zd

zdyt

RR

RR

+=σ

2ycmcS2~F Ωσ−σβ+σ= (3-20)

( )du

duc RR

RR

−−=

2

3β )2(3~

du

duyc

RR

RR

−=σ

where: σm hydrostatic stress

I2 second invariant of the deviatoric main stresses Rz uniaxial tensile strength

Rd uniaxial compression strength Ru biaxial compression strength

Ω hardening and softening function (in the pressure domain Ω1 = Ω2 = Ωc, in the tensile domain Ω1 = Ωt).

The plasticity potentials are:

mttS1Q σβδ+σ=

(3-21)

mccS2Q σβδ+σ=

The yield condition is shown in Fig. 3-9 and Fig. 3-10 in different coordinate systems. The comparison with the concrete model made by Ottosen [6-15] is shown in Fig. 3-9 and illustrates the advantages of the Drucker-Prager model consisting of two yield criteria. While there is a very good correspondence in the compressive domain, the chosen Drucker-Prager model can be well adjusted to realistic tensile strength. In opposite to that, the Ottosen model overestimates these areas significantly! A further advantage lies within the description of the yield condition using the three easily estimable and generally known parameters Rz, Rd and Ru.

Fig. 3-9 Singular Drucker-Prager flow conditions – Illustrated in the octaeder system

where: δt, δc are dilatancy factors

17


-σm

σy

σx

F1

F2

a) b) Fig. 3-10 Singular Drucker-Prager flow condition: a) yield surface in the main stress domain; b) Illu stration in the σσσσx-σσσσy-ττττxy-space

3.3.6.1 Nonlinear deformation behaviour in case of pressure load In general the uniaxial stress-strain relationship of concrete is characterized by three domains: A linear elastic domain which generally reaches up to about a third of the compressive strength. This is followed by an increasingly bent run until the compressive strength is reached. The

nonlinear relation between stress and strain is caused by an initially small number of micro-cracks which merge with higher stress levels.

The achievement of the compressive strength is associated with the forming of fracture surfaces and cracks which are aligned parallel to the largest main stress.

The softening area is characterized by a decreasing strength. Finally, it leads to a low residual strain level. The slope of the decreasing branch is a measure for the brittleness of the material.

Fig. 3-11 shows the typical nonlinear stress-strain relation of normal concrete in uniaxial compressive tests [6-9].

Fig. 3-11 Nonlinear stress-strain relation (uniaxia l compression test) of normal concrete used in codes (DIN 1045-1 [6-9] and EC2 [6-10) In Fig. 3-12 the stress-strain relation which is available in multiPlas is shown. Thereby linear softening (mlaw = 0, 2) or parabolic-exponential softening (mlaw = 1) can be chosen. Up to reaching the strain εu the parabola equation (as seen in Fig. 3-11) is used.

18


Fig. 3-12 stress-strain relation in multiPlas (mlaw =0,2; mlaw = 1)

3.3.6.2 Nonlinear deformation behavior during tensi le load Concrete tends to soften relatively brittle with local appearances of cracks. For including this into the context of a continuum model, a homogenized crack and softening model is needed. The crack itself does not appear in the topology description of the structure - but is described by its impact on stress and deformation state [6-16],[6-21]. The softening process is formulated respectively to the energy dissipation caused by the occurance of cracks. For the complete cracking, the fracture energy concerning the crack surface has to be Gf -dissipated. The used model has its origin within the crack band theory of Bažant / Oh [6-6]. It states that cracks develop within a local process zone. Its width hPR (crack band width) is a material specific constant. To avoid a mesh dependency of the softening and to assess the fracture energy correctly, a modification of the work equation is necessary. For a given width of the crack band and a given fracture energy, the volume fracture energy can be computed via:

PR

ff h

Gg = (3-22)

where: gf volume fracture energy Gf fracture energy hPR crack band width For meshing of the structure with elements which are larger than the expected width of the crack band the stress-strain relationship has to be modified in such a way that the volume fracture energy reaches the following value:

hG

gh

hg f

fPR

INT,f == (3-23)

where: gf,INT volume fracture energy at the integration point h equivalent length

σd

Rd

Rd/3

σr

εml εu ε

σu

εr

19


This model guaranties a consistent dissipation of fracture energy during the softening process for different sizes of elements. The stress-strain lines available in multiPlas are shown in Fig. 3-13. Thereby, a linear elastic behaviour is assumed until the tensile strength is reached. After that, one of the following is assumed as consequence of tensile fracturing: ** linear softening until the strain limit is reached εtr (mlaw = 0, 2) or ** exponential softening (mlaw = 1)

Fig. 3-13 Stress-strain relation in multiPlas ( mlaw=0, 2 ; mlaw = 1) For the exponential softening model (mlaw = 1) one should assume

2t

f

f

EGh ≤ (3-24)

for the length h in order to avoid the numerically unstable „snap-back“ phenomena. This is preferably achieved by choosing a proper mesh size. In multiPlas, the equivalent length will calculated automatically.

f t

ε

σ

hPR

hPR h

h

(a)

(b)

(a)

(b)

"Snap-Back", wenn h groß

Über h gemittelte Spannungs-Dehnungslinie

ft/E εtr

20


3.3.6.3 Temperature dependency Information on the temperature dependencies of the material behaviour are included in DIN EN 1992-1-2 [6-11]. As an example the temperature dependencies from pressure level are shown in Fig. 3-14, Fig. 3-15 and Tab. 3-1.

Fig. 3-14 Temperature dependency of concrete pressu re resistance from [6-11]

Fig. 3-15 Temperature dependency of concrete compre ssion strain from [6-11]

Betondruckfestigkeitsentwicklung

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 100 200 300 400 500 600 700 800 900 1000 1100 1200

Temperatur (°C)

bez.

Dru

ckfe

stig

keit

quarzh. Zuschlägekalksteinh. Zuschläge

temperaturabhängige Stauchungen

0

0,005

0,01

0,015

0,02

0,025

0,03

0,035

0,04

0,045

0,05

0 100 200 300 400 500 600 700 800 900 1000 1100 1200

Temperatur (°C)

Sta

uchu

ng

εmlεr

rel.

com

pres

sion

str

engt

h

temperature (°C)

temperature (°C)

com

pres

sion

str

ain

21


Tab. 3-1 Temperature dependency of concrete materia l values from [6-11] In multiPlas up to 11 temperatures-pressure points and respective strains εm can be predefined. The associated limit strains are assumed according to Tab. 3-1. Interim values are linearly interpolated. The temperature dependency of the concrete tensile strength is implemented in multiPlas using the Data from [6-11] (s. Fig. 3-16).

Fig. 3-16 Temperature dependency of the tensile str ength of concrete from [6-11] For the temperature dependency of steel reinforcement we refer to [6-11]. It can be taken into account by the standard parameters of ANSYS (tb,bkin oder tb,mkin).

0

0,2

0,4

0,6

0,8

1

1,2

0 100 200 300 400 500 600 700 800

Temperatur (°C)

bezo

gene

Bet

onzu

gfes

tigke

it

TtA

TtE

rel.

tens

ile s

tren

gth

temperature (°C)

22


3.4 Dilatancy The strict adherence of the stability postulations of Drucker usually requires an associated yield constitu-tive law (e.g. dilatancy angle = friction angle). But in reality, for some materials, the calculated volume strains are significantly larger than those, determined by experiments. In that cases, a deformation beha-viour closer to reality can be described by using non-associated flow rules. The dilatancy angle describes the ratio of normal and shear translation in the Mohr-Coulomb shear criterion. It has two limits: Dilatancy angle = friction angle => maximum plastic normal strain at shear strain, associated plasticity Dilatancy angle = 0.0 => no plastic normal strain at shear (not recommended limit case because of result-ing numerical problems) By replacing the friction angle ϕ by the dilatancy angle ψ within the yield condition (rsp. plastic potential function), a non-associated flow rule is obtained. In addition to that it has to be considered, that the dila-tancy angle is only physically reasonable for dilatancy angle ψ ≤ friction angle ϕ. A dilatancy angle larger than the friction angle leads to generation of energy within the system. For the Drucker-Prager yield condition, the plastic deformation behaviour can be controlled via a non-associated flow rule by manipulating the dilatancy factor δ. Then the plastic potential is modified accord-

ing to the equations ( mSQ σδβ+σ= (3-18) and ( (3-21). Thereby: dilatancy factor δ = 1 => associated plasticity dilatancy factor δ ≤ 1 => non-associated plasticity For the compressive domain 0 ≤ δc ≤ 1 is true. In the tensile domain of the Drucker-Prager, a yield condi-tion of δt = 0,1 ... 0,25 is recommended. In case of prevalent shear strain, the dilatancy factor δ can be calculated via the ratio of the beta values from Fig. 3-8 with the help of the friction angle ϕ and the dilatancy angle ψ. From equation (4.1), this could calculated by:

( )( )ϕ+

ϕ⋅=ϕβsin33

sin6 ( )

( )ψ+ψ⋅=ψβ

sin33

sin6

( )( )ψβϕβδ = (3-25)

Please note that a non-associated flow rule however lead to asymmetric deformation matrices and may result in pure convergence behaviour.

23


4 COMMANDS

4.1 Material Models LAW Material Model 0 Linear elastic (orthotropic) material model 1 Mohr-Coulomb / Tresca (isotropic), tension cut off (isotropic)

Mohr-Coulomb (anisotropic), tension cut off (anisotropic) for up to 4 joint sets 2 Drucker-Prager / modified Drucker-Prager (ideal elastic-plastic) 5 Drucker-Prager / modified Drucker-Prager, temperature dependent 9 Concrete (nonlinear hardening / softening, Temperature Dependency) 10 Mohr-Coulomb (anisotropic), tension cut off (anisotropic) for up to 4 joints 11 fixed crack model (x-direction) 14 Menetrey-Willam model for concrete The material library multiPlas was implemented within the ANSYS-user-interface “usermat”. MultiPlas is activated and the multiPlas material model is defined by following APDL commands:

TB,USER,mat,,80 ! mat – material number TBDATA,1,LAW, , , ! allocation of the data field with the selected material model (LAW) and the

! material parameters (see the following sections 4.2) The nonlinear material parameters of the multiPlas material model must be provided in the TBDATA-field.

The linear elastic material properties (Young’s modulus, Poisson’s ratio, shear modulus) must be pro-vided in the standard ANSYS way. In ANSYS mechanical APDL the linear elastic properties can be de-fined by following commands (assuming isotropic behavior):

EX,mat,youngs_modulus NUXY,mat,poisons_ratio

Please refer to the ANSYS manual for the full definition of an orthotropic material. In ANSYS workbench the linear elastic material properties must be defined in “material data” or “engineering data” in your work-bench project.

In addition the number of state variables must be defined for the multiPlas material model using the fol-lowing APDL command:

TB,STATE,mat,,num_state_vars ! mat – material number ! num_state_vars – number of state variables

The number of state variables depends on the material model and is defined the following sections.

24


4.2 TBDATA-Declaration In this section the tbdata-field definition for the input parameters of the individual material models is de-scribed. Furthermore, the state variables and the relationship between activity numbers and yield surfac-es are explained. In this section the following unit system is assumed: N, m, kg, s, ° (degrees), °C. This unit system can be converted into any other consistent unit system. Note for workbench users: Please choose the right u nit system in workbench. The workbench unit system and the unit system of the multiPlas paramet ers must be identical. Otherwise the simula-tion will most likely fail.

4.2.1 LAW = 0 – linear elastic model 1 2 3 4 5 6 7 8 9 10

1-10

LAW

11-20

21-30

31-40

41-50

51-60

wr

61-70 Elem Intpt

71-80

This material law has no state variables.

25


4.2.2 LAW = 1 – jointed rock material model Failure of jointed rock is represented by a combination of Mohr-Coulomb and Rankine (tension cut-off) yield surfaces. The material model differentiates between anisotropic failure of one of the joint sets and isotropic failure of intact rock (material between joints). Up to 4 independent joint sets can be defined.

1 2 3 4 5 6 7 8 9 10

1-10 rock

LAW phiR cR psiR phiR* cR* ft,R ft,R* number of joint sets

11-20 1. joint

phiJ1 cJ1 psiJ1 phiJ1* cJ1* ft,J1 alphaJ1 betaJ1 ft,J1*

21-30 2. joint


31-40 3. joint


41-50 4. joint

phiJ4 cJ4 psiJ4 phiJ4* cJ4* ft,J4 alphaJ4 betaJ4 ft,J4* tempd

51-60

T1 T2 βc2 T3 βc3 T4 βc4 T5 βc5 wr

61-70

Elem Intpt eps geps maxit cutmax

71-80

Intact rock (base material between the joint sets) parameters:

• parameters of the isotropic Mohr-Coulomb yield surface parameter description unit range

phiR initial friction angle [°] 0° ≤ phiR < 90° cR initial cohesion [N/m²] cR ≥ 0

psiR dilatancy angle [°] 0° ≤ psiR ≤ phiR phiR* residual friction angle [°] 0° ≤ phiR* ≤ phiR cR* residual cohesion [N/m²] 0 ≤ cR* ≤ cR

• parameters of the isotropic Rankine (tension cut-off) yield surface parameter description unit range

ft,R initial uni-axial tensile strength [N/m²] ft,R > 0 ft,R* residual uni-axial tensile strength [N/m²] 0 < ft,R* ≤ ft,R

Note: The Rankine yield surface is only active if phiR=0° or ft,R < cR/tan(phiR)

• temperature dependency rsp. moisture dependency1 of initial cohesion parameter description unit range

tempd switch for activating temperature dependency [-] 0 – disabled 1 – enabled

T1 reference temperature (100% cohesion cR) [°C] T2 … T5 additional temperature points defined in ascend-

ing order (if the temperature is smaller than the previous one, than this temperature points and

all following points are ignored)

[°C] Ti > Ti-1 i=2…5

βc2 … βc5 relative cohesion at individual temperature points cR(Ti) = βci cR

[-] βci ≥ 0 i=2…5

1 Volume-referred moisture as temperature equivalent interpreted

26


Joint material parameters (up to 4 joint sets can b e defined)

• parameters of the anisotropic Mohr-Coulomb yield surface parameter description unit range

phiJi initial friction angle of joint i [°] 0° ≤ phiJi < 90° cJi initial cohesion of joint i [N/m²] cJi ≥ 0

psiJi dilatancy angle of joint i [°] 0° ≤ psiJi ≤ phiJi phiJi* residual friction angle of joint i [°] 0° ≤ phiJi* ≤ phiJi cJi* residual cohesion of joint i [N/m²] 0 ≤ cJi* ≤ cJi

• parameters of the anisotropic tension cut-off yield surface parameter description unit range

ft,Ji initial uni-axial tensile strength of joint i [N/m²] ft,Ji > 0 ft,Ji* residual uni-axial tensile strength of joint i [N/m²] 0 < ft,Ji* ≤ ft,Ji

Note: The tension cut-off yield surface is only active if phiJi=0° or ft,Ji < cJi/tan(phiJi)

• direction of anisotropic joint system: As shown in Fig. 3-4 and Fig. 3-6, the transformation from the element coordinate system into the joint coordinate system is defined by two angles (alpha, beta). parameter description unit range

alphaJi negative rotation about element Z-axis (joint i) [°] betaJi positive rotation about new Y-axis (joint i) [°]

State variables

In this model 15 state variables are defined: state variable description

1 number representing the activity of individual yield surfaces (failure modes) 2 accumulated plastic multiplier of the isotropic Mohr-Coulomb yield surface 3 accumulated plastic multiplier of the isotropic Rankine yield surface 4 accumulated tangential plastic strains due to Mohr-Coulomb failure of joint 1 5 accumulated normal plastic strains due to Mohr-Coulomb failure of joint 1 6 accumulated normal plastic strains due to tensile failure of joint 1 7 accumulated tangential plastic strains due to Mohr-Coulomb failure of joint 2 8 accumulated normal plastic strains due to Mohr-Coulomb failure of joint 2 9 accumulated normal plastic strains due to tensile failure of joint 2

10 accumulated tangential plastic strains due to Mohr-Coulomb failure of joint 3 11 accumulated normal plastic strains due to Mohr-Coulomb failure of joint 3 12 accumulated normal plastic strains due to tensile failure of joint 3 13 accumulated tangential plastic strains due to Mohr-Coulomb failure of joint 4 14 accumulated normal plastic strains due to Mohr-Coulomb failure of joint 4 15 accumulated normal plastic strains due to tensile failure of joint 4

Activity encoding

The activity of individual yield surface is encoded into a single number. The following table shows the encoding if a single yield surface fails. If multiple yield surfaces fail the corresponding num-bers are added, e.g. 11 represents shear failure of intact rock and joint 1.

number activity 1 shear failure (Mohr-Coulomb yield surface) of intact rock

10 shear failure (Mohr-Coulomb yield surface) of joint 1 100 shear failure (Mohr-Coulomb yield surface) of joint 2


100000 tensile failure (Rankine yield surface) of intact rock 1000000 tensile failure (tension cut-off yield surface) of joint 1

10000000 tensile failure (tension cut-off yield surface) of joint 2 100000000 tensile failure (tension cut-off yield surface) of joint 3

1000000000 tensile failure (tension cut-off yield surface) of joint 4

27


4.2.3 LAW = 2 – Modified Drucker-Prager model with joints This material model describes failure of a jointed material such as jointed rock. Failure of the base ma-terial (the material between the joints) is represented by two isotropic Drucker-Prager cones assuming ideal plastic behavior and associated flow . Joint failure is described by a combination of anisotropic Mohr-Coulomb yield surface and anisotropic tension cut-off yield surface . Up to 4 independent joint sets can be defined. The joint behavior is ideal plastic but with residual strength and non-associated flow can be applied.

1 2 3 4 5 6 7 8 9 10

1-10 isotrop

LAW Rd Rz Ru number of joint sets

11-20 1. joint


21-30 2. joint


31-40 3. joint


41-50 4. joint


51-60

wr

61-70 Elem Intpt

eps geps maxit

cutmax

71-80

Base material (material between the joint sets) par ameters:

• parameters defining the isotropic Drucker-Prager cones parameter description unit range

Rd uni-axial compressive strength [N/m²] Rd > Rz Rz uni-axial tensile strength [N/m²] Rz > 0 Ru bi-axial compressive strength [N/m²] Ru > Rd










28


State variables


1 number representing the activity of individual yield surfaces (failure modes) 2 accumulated tangential plastic strains due to Mohr-Coulomb failure of joint 1 3 accumulated normal plastic strains due to Mohr-Coulomb failure of joint 1 4 accumulated normal plastic strains due to tensile failure of joint 1 5 accumulated tangential plastic strains due to Mohr-Coulomb failure of joint 2 6 accumulated normal plastic strains due to Mohr-Coulomb failure of joint 2 7 accumulated normal plastic strains due to tensile failure of joint 2 8 accumulated tangential plastic strains due to Mohr-Coulomb failure of joint 3 9 accumulated normal plastic strains due to Mohr-Coulomb failure of joint 3

10 accumulated normal plastic strains due to tensile failure of joint 3 11 accumulated tangential plastic strains due to Mohr-Coulomb failure of joint 4 12 accumulated normal plastic strains due to Mohr-Coulomb failure of joint 4 13 accumulated normal plastic strains due to tensile failure of joint 4

Activity encoding

The activity of individual yield surface is encoded into a single number. The following table shows the encoding if a single yield surface fails. If multiple yield surfaces fail the corresponding num-bers are added, e.g. 100001 represents failure of both Drucker-Prager yield surfaces.

number activity 1 compressive-compressive Drucker-Prager yield surface failure (base material)



100000 tensile-compressive Drucker-Prager yield surface failure (base material) 1000000 tensile failure (tension cut-off yield surface) of joint 1



29


4.2.4 LAW = 5 – Modified Drucker-Prager, temperature depe ndent This material model represents failure of the material by two Drucker-Prager cones . The first Drucker Prager cone (tension-compression) is defined by the uni-axial tensile strength and the uni-axial compres-sive strength. The parameters of the second Drucker-Prager cone (compression-compression) are ob-tained from the uni-axial compressive strength and the bi-axial compressive strength. Furthermore, ideal plastic behavior and an associated flow rule are assumed. A special feature of this model is the tem-perature dependency of the strength parameters.

1 2 3 4 5 6 7 8 9 10

1-10

LAW Rd Rz Ru

11-20

T1

21-30

T2 T3 T4 T5 T6 T7 T8 T9 T10 T11

31-40

βc2 βc3 βc4 βc5 βc6 βc7 βc8 βc9 βc10 βc11

41-50

51-60

utz TtS TtE βtE wr

61-70 Elem Intpt

eps geps maxit

cutmax

71-80

Base material parameters at reference temperature:



Temperature dependency:

• temperature dependent uni-axial tensile strength parameter description unit range

utz reduction function switch 0 - DIN EN 1992-1-2

1 - user defined linear function

[-] 0, 1

TtS reference temperature linear reduction of tensile strength starts at this

temperature (Fig. 3-16)

[°C]

TtE linear reduction of tensile strength ends at this temperature (Fig. 3-16)

[°C] TtE > TtS

βtE tensile strength scaling factor at TtE Rz(TtE)= βtE Rz

[-] βtE > 0

Note: The parameters TtS, TtE, βtE must be provided if utz=1 (user-defined). If utz=0 (DIN EN 1992-1-2) these parameters are automatically set by multiPlas to following values: TtS=100 °C, TtE=600 °C, βtE=1E-12.

30


• temperature dependent uni-axial compressive strength and bi-axial compressive strength parameter description unit range

T1 reference temperature (100% strength Rd, Ru) [°C] T2 … T11 additional temperature points defined in ascend-



[°C] Ti > Ti-1 i=2…11

βc2 … βc11 corresponding compressive strength scaling fac-tors at individual temperature points

Rd(Ti) = βci Rd and Ru(Ti) = βci Ru

[-] βci ≥ 0 i=2…11

Note: The scaling factors between temperature points are linearly interpolated.

State variables

In this model 1 state variables is defined state variable description

1 number representing the activity of individual yield surfaces (failure modes)

Activity encoding



10 tensile-compressive Drucker-Prager yield surface failure (base material)

31


4.2.5 LAW = 9 – Concrete Failure of concrete is described by two Drucker-Prager yield surfaces. The first Drucker-Prager cone (tension-compression) is defined by the uni-axial tensile strength and the uni-axial compressive strength. The second Drucker-Prager cone (compression-compression) is based on the uni-axial compressive strength and the bi-axial compressive strength. Various types of hardening/softening functions differen-tiating between tensile and compressive failure can be used in this model. Furthermore, non-associated flow rules are applied.

In the following table only those parameters are summarized, which are available in all models. The TBDATA-declarations of the individual models can be found in the following subsections.

1 2 3 4 5 6 7 8 9 10 1-10 LAW Rd Rz Ru δψt δψc 11-20 21-30 31-40

41-50

51-60 mlaw wr 61-70 Elem Intpt eps geps maxit cutmax 71-80

Base material parameters:



• dilatancy parameters parameter description unit range

δψt dilatancy factor in tension (recommendation δψt = 0,25)

[-] 0 < δψt ≤ 1

δψc dilatancy factor in compression (recommendation δψc = 1,00)

[-] 0 < δψc ≤ 1

• mlaw - softening/hardening/dilatancy function switch

mlaw hardening/softening functions dilatancy function

tension compression tension compression 0 linear softening function power hardening function

and linear softening func-tion

constant constant

1 fracture energy based exponential softening

function

power hardening function and combination of power

softening function and exponential softening

function

constant constant

5 multi-linear softening function

power hardening function and linear softening func-

tion

step-wise con-stant

constant

6 fracture energy based exponential softening

function

power hardening function and fracture energy based rational softening function

constant constant

10 linear softening function power hardening function and linear softening func-

tion

constant constant

32


State variables


1 number representing the activity of individual yield surfaces (failure modes) 2 hardening variable for tensile failure (work hardening) 3 hardening variable for compressive failure (work hardening) 4 failure switch for tensile-compressive Drucker-Prager yield surface

(<0 compressive failure, otherwise tensile failure)

Activity encoding



10 tensile-compressive Drucker-Prager yield surface failure (base material)

4.2.5.1 Law=9, mlaw=0 – linear softening Law=9, mlaw=0 describes the stress-strain relationship in compression by a combination of nonlinear hardening function and linear softening function. According to DIN EN 1992-1-2 the ultimate plastic strain in compression is set to 0.02. In tension the post-peak behavior is described by a linear softening func-tion. Fig. 4-1 shows the hardening/softening functions and the corresponding material parameters.

Fig. 4-1: hardening and softening functions of law 9 mlaw 0 (left - compression, right - tension)

Additionally, this material law supports temperature dependent strength and softening parameters. The dependency of individual parameters on the temperature according to DIN EN 1992-1-2 is illustrated in Fig. 3-14, Fig. 3-15 and Fig. 3-16.

The material parameters of the law=9, mlaw=0 material model are summarized in the following table.

1 2 3 4 5 6 7 8 9 10 1-10 LAW Rd Rz Ru δψt δψc 11-20 κcm1 Ωci Ωcr T1 21-30 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 31-40 βc2 βc3 βc4 βc5 βc6 βc7 βc8 βc9 βc10 βc11

41-50 κcm2 κcm3 κcm4 κcm5 κcm6 κcm7 κcm8 κcm9 κcm10 κcm11

51-60 Ωtr κtr utz mlaw TtS TtE βtE wr 61-70 Elem Intpt eps geps maxit cutmax 71-80

33


Base material parameters

• hardening/softening parameters in compression parameter description unit range

κcm1 plastic strain at uni-axial compressive strength Rd (κcm1 = εml – Rd/E)

[-] 0 < κcm1 < 0.02

Ωci relative stress level at start of nonlinear harden-ing in compression (recommendation Ωci=0.33)

[-] 0 ≤ Ωci ≤ 1

Ωcr residual relative stress level in compression (recommendation Ωcr = 0.2)

[-] 0 < Ωcr ≤ 1

• hardening/softening parameters in tension parameter description unit range

κtr plastic limit strain in tension [-] κtr ≥ 0 Ωtr residual relative stress level in tension

(recommendation Ωcr = 0.2) [-] 0 < Ωtr ≤ 1


• temperature dependent uni-axial tensile strength parameter description unit range

utz reduction function switch 0 - DIN EN 1992-1-2

1 - user defined linear function

[-] 0, 1



[°C]


[°C] TtE > TtS


[-] βtE > 0






[°C] Ti > Ti-1 i=2…11



[-] βci ≥ 0 i=2…11

κcm2…κcm11 plastic strain at uni-axial compressive strength Rd at temperature Ti

[-] 0 < κcm1 < κcr(T)

Note: The scaling factors between temperature points are linearly interpolated. The dependency of plastic limit strain κcr on temperature T is given in Fig. 3-15.

Remark: switch to LAW = 5 (with same input like LAW=9) – for temperature dependency without me-chanical softening

4.2.5.2 Law=9, mlaw=1 – exponential softening By setting parameter mlaw=1, the softening behavior of concrete in tension is described an exponential function. The shape of the exponential function is controlled by the mode I area specific fracture energy. The softening model is regularized with respect to the element size. In compression the pre-peak harden-ing behavior is described by a power hardening law. The softening behavior in compression is represented by a combination of a power softening law and an exponential softening function. Fig. 4-2 shows the hardening/softening functions and the corresponding material parameters.

34


Fig. 4-2: hardening and softening functions of law= 9, mlaw=1 (left - compression, right - tension)


1 2 3 4 5 6 7 8 9 10 1-10 LAW Rd Rz Ru δψt δψc 11-20 κcm κcu Ωci Ωcu Ωcr 21-30 31-40

41-50

51-60 Gft Ωtr mlaw wr 61-70 Elem Intpt eps geps maxit cutmax 71-80



κcm plastic strain at uni-axial compressive strength Rd (κcm = εml – Rd/E)

[-] 0 < κcm < κcu

κcu plastic strain defining the end of power softening law and start of exponential softening law

(κcu = εu – Ωcu * Rd/E)

[-] κcu > κcm


[-] 0 ≤ Ωci ≤ 1

Ωcu relative stress level at κcu Ωcr < Ωcu ≤ 1 Ωcr residual relative stress level in compression

(recommendation Ωcr = 0.2) [-] 0 < Ωcr < Ωcu


Gft area specific mode-I fracture energy [Nm/m²] Gft > 0 Ωtr residual relative stress level in tension


Note: The fracture energy is regularized with respect to the element size.

4.2.5.3 Law=9, mlaw=5 – steel-fiber concrete In law=9, mlaw=5 the stress-strain relationship in compression is similar to law 9, mlaw=0. It is described by a combination of nonlinear hardening function and linear softening function. According to DIN EN 1992-1-2 the ultimate plastic strain in compression is set to 0.02. In tension the influence of steel fibres is taken into account. The post-peak behavior in tension is described by a multi-linear softening function. Fig. 4-3 shows the hardening/softening functions and the corresponding material parameters. Further-more, the dilatancy factor is not constant in tension. As shown in Fig. 4-4, the dilatancy factor is described in terms of the tension hardening variable as step-wise constant function.

35



Fig. 4-4: dilatancy functions of law=9, mlaw=5 (lef t - compression, right - tension)


1 2 3 4 5 6 7 8 9 10 1-10 LAW Rd Rz Ru δψc 11-20 κcm Ωci Ωcr 21-30 31-40

41-50

51-60 mlaw wr 61-70 Elem Intpt eps geps maxit cutmax 71-80 κt1 κt2 κt3 Ωt1 Ωt2 Ωt3 δψt1 δψt2 δψt3 δψt4




[-] 0 < κcm < 0.02


[-] 0 ≤ Ωci ≤ 1


[-] 0 < Ωcr ≤ 1

• hardening/softening parameters in tension parameter description unit range κt1, κt2, κt3 plastic strain points in tension, cf. Fig. 4-3 [-] 0 < κt1< κt2< κt3

Ωt1, Ωt2, Ωt3 residual relative stress level points in tension, cf. Fig. 4-3

[-] 0 < Ωti ≤ 1 i=1…3

δψt1 dilatancy factor in range 0 ≤ κ ≤ κt1 [-] 0 < δψt1 ≤ 1

36


parameter description unit range δψt2 dilatancy factor in range κt1 < κ ≤ κt2 [-] 0 < δψt2 ≤ 1 δψt3 dilatancy factor in range κt2 < κ ≤ κt3 [-] 0 < δψt3 ≤ 1 δψt4 dilatancy factor in range κ > κt3 [-] 0 < δψt4 ≤ 1

4.2.5.4 Law=9, mlaw=6 – fracture energy based softe ning By setting parameter mlaw=6, fracture energy (mode-I) based softening functions are applied in tension and in compression. In order to regularize the model (remove mesh dependency), both fracture energy values are scaled by an equivalent element size. Additionally, a power hardening law is used to describe the nonlinear pre-peak behavior in compression. Fig. 4-5 shows the hardening/softening functions and the corresponding material parameters.



1 2 3 4 5 6 7 8 9 10 1-10 LAW Rd Rz Ru δψt δψc 11-20 κcm Ωci Ωcr Gfc 21-30 31-40

41-50





[-] 0 < κcm < κcu


[-] 0 ≤ Ωci ≤ 1


[-] 0 < Ωcr < Ωcu

Gfc area specific mode-I fracture energy in com-pression

[Nm/m²] Gfc > 0


Gft area specific mode-I fracture energy in tension [Nm/m²] Gft > 0 Ωtr residual relative stress level in tension


Note: The fracture energies are regularized with respect to the element size.

37


4.2.5.5 Law=9, mlaw=10 – linear softening Law=9, mlaw=10 is quite similar to law=9, mlaw=0. It describes the stress-strain relationship in compres-sion by a combination of nonlinear hardening function and linear softening function. In contrast to law=9, mlaw=0 the ultimate plastic strain in compression is not a constant and must be defined by the user as additional input. In tension the post-peak behavior is described by a linear softening function. Fig. 4-6 shows the hardening/softening functions and the corresponding material parameters.

Fig. 4-6: hardening and softening functions of law 9 mlaw 10 (left - compression, right - tension)

Additionally, this material law supports temperature dependent strength and softening parameters. The dependency of individual parameters on the temperature according to DIN EN 1992-1-2 is illustrated in Fig. 3-14, Fig. 3-15 and Fig. 3-16. In contrast to law=9, mlaw=0 the temperature dependency of the ulti-mate plastic strain in compression is not based on DIN EN 1992-1-2, but a multi-linear function describes the temperature dependency of the ultimate plastic strain Up to four temperature points and the corres-ponding ultimate plastic strains can be defined for this function.


1 2 3 4 5 6 7 8 9 10 1-10 LAW Rd Rz Ru δψt δψc 11-20 κcm1 Ωci Ωcr T1 21-30 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 31-40 βc2 βc3 βc4 βc5 βc6 βc7 βc8 βc9 βc10 βc11

41-50 κcm2 κcm3 κcm4 κcm5 κcm6 κcm7 κcm8 κcm9 κcm10 κcm11

51-60 Ωtr κtr utz mlaw TtS TtE βtE wr 61-70 Elem Intpt eps geps maxit cutmax 71-80 κcr1 κcr2 κcr3 κcr4



κcm1 plastic strain at uni-axial compressive strength Rd (κcm1 = εml – Rd/E)

[-] 0 < κcm1 < κcr1

κcr1 ultimate plastic strain in compression [-] κcr1 > κcm1 Ωci relative stress level at start of nonlinear harden-

ing in compression (recommendation Ωci=0.33) [-] 0 ≤ Ωci ≤ 1


[-] 0 < Ωcr ≤ 1





• temperature dependent uni-axial tensile strength

38


parameter description unit range utz reduction function switch

0 - DIN EN 1992-1-2 1 - user defined linear function

[-] 0, 1



[°C]


[°C] TtE > TtS


[-] βtE > 0






[°C] Ti > Ti-1 i=2…11



[-] βci ≥ 0 i=2…11

κcm2…κcm11 plastic strain at uni-axial compressive strength Rd at temperature Ti

[-] 0 < κcmi < κcri

κcr2…κcr4 ultimate plastic strain at temperature Ti [-] κcri > κcmi Note: The scaling factors between temperature points are linearly interpolated.

39


4.2.6 LAW = 10 – jointed rock material model (linear elas tic base ma-terial)

Failure of jointed rock is represented by a combination of Mohr-Coulomb and Rankine (tension cut-off) yield surfaces. In contrast to law 1, the material model assumes linear elastic material behavior of the base material (rock between joints). Up to 4 independent joint sets can be defined.

1 2 3 4 5 6 7 8 9 10

1-10 rock

LAW number of joint sets

11-20 1. joint


21-30 2. joint


31-40 3. joint


41-50 4. joint


51-60

wr

61-70

Elem Intpt eps geps maxit cutmax

71-80










State variables


1 number representing the activity of individual yield surfaces (failure modes) 2 accumulated tangential plastic strains due to Mohr-Coulomb failure of joint 1 3 accumulated normal plastic strains due to Mohr-Coulomb failure of joint 1 4 accumulated normal plastic strains due to tensile failure of joint 1 5 accumulated tangential plastic strains due to Mohr-Coulomb failure of joint 2 6 accumulated normal plastic strains due to Mohr-Coulomb failure of joint 2 7 accumulated normal plastic strains due to tensile failure of joint 2 8 accumulated tangential plastic strains due to Mohr-Coulomb failure of joint 3 9 accumulated normal plastic strains due to Mohr-Coulomb failure of joint 3

10 accumulated normal plastic strains due to tensile failure of joint 3 11 accumulated tangential plastic strains due to Mohr-Coulomb failure of joint 4

40


state variable description 12 accumulated normal plastic strains due to Mohr-Coulomb failure of joint 4 13 accumulated normal plastic strains due to tensile failure of joint 4

Activity encoding

The activity of individual yield surface is encoded into a single number. The following table shows the encoding if a single yield surface fails. If multiple yield surfaces fail the corresponding num-bers are added, e.g. 110 represents shear failure of joint 1 and joint 2.

number activity 1 not used



100000 not used 1000000 tensile failure (tension cut-off yield surface) of joint 1



41


4.2.7 LAW = 11 – Fixed crack model Law 11 implements a fixed crack model for tensile failure. The crack normal vector is aligned with the x-direction of the element coordinate system. In this model the post-peak behavior is represented by a frac-ture energy based exponential softening law. shows the uni-axial stress strain curve in tension.

Fig. 4-7: uni-axial stress-strain curve of law=11 i n tension

The material parameters of the fixed crack model (law=11) are summarized in the following table.

1 2 3 4 5 6 7 8 9 10 1-10 LAW ft 11-20 21-30 31-40

41-50

51-60 Gft ftr wr 61-70 Elem Intpt eps geps maxit cutmax 71-80

Base material parameters: parameter description unit range

ft tensile strength (in x-direction) [N/m²] ft > 0 ftr residual tensile strength [N/m²] ftr ≤ ft Gft area specific mode I fracture energy in tension [Nm/m²] Gft > 0


State variables


1 number representing the activity of individual yield surfaces (failure modes) 2 hardening variable for tensile failure (work hardening)

Activity encoding

The activity of individual yield surface is encoded into a single number. The following table shows the encoding if a single yield surface fails.

number activity 1 tensile failure in x-direction

42


4.2.8 LAW = 14 – Menetrey-Willam model for concrete Failure of concrete is described by the Menetrey-Willam yield surface. Various types of harden-ing/softening functions differentiating between tensile and compressive failure can be used in this model. Furthermore, non-associated flow rules are applied.

In the following table only those parameters are summarized, which are available in all models. The TBDATA-declarations of the individual models can be found in the following subsections.

1 2 3 4 5 6 7 8 9 10 1-10 LAW Rd Rz Ru ψ 11-20 21-30 31-40

41-50

51-60 mlaw wr 61-70 Elem Intpt eps geps maxit cutmax 71-80

Base material parameters:



• dilatancy parameters parameter description unit range

ψ dilatancy angle [°] d

z

R

R

2tan

2

1 << ψ

• mlaw - softening/hardening/dilatancy function switch

mlaw hardening/softening functions

tension compression 0 linear softening function power hardening function and linear softening function 1 fracture energy based exponen-

tial softening function power hardening function and combination of power softening function and exponential softening function

6 fracture energy based exponen-tial softening function

power hardening function and fracture energy based rational softening function

State variables


1 number representing the activity of individual yield surfaces (failure modes) 2 hardening variable (work hardening) 3 switch defining failure type (tensile or compressive)

Activity encoding

The activity of individual yield surface is encoded into a single number. The following table shows the encoding if a single yield surface fails.

number activity 1 failure of the Menetrey-Willam yield surface

4.2.8.1 Law=14, mlaw=0 – linear softening Law=14, mlaw=0 describes the stress-strain relationship in compression by a combination of nonlinear hardening function and linear softening function. In tension the post-peak behavior is represented by a linear softening function. Fig. 4-8 shows the hardening/softening functions and the corresponding material parameters.

43




1 2 3 4 5 6 7 8 9 10 1-10 LAW Rd Rz Ru ψ 11-20 κcm κcr Ωci Ωcr 21-30 31-40

41-50

51-60 Ωtr κtr mlaw wr 61-70 Elem Intpt eps geps maxit cutmax 71-80




[-] 0 < κcm < κcr

κcr plastic limit strain in compression [-] κcr > κcm Ωci relative stress level at start of nonlinear harden-

ing in compression (recommendation Ωci=0.33) [-] 0 ≤ Ωci ≤ 1


[-] 0 < Ωcr ≤ 1




4.2.8.2 Law=14, mlaw=1 – exponential softening By setting parameter mlaw=1, the softening behavior of concrete in tension is described an exponential function. The shape of the exponential function is controlled by the mode I area specific fracture energy. The softening model is regularized with respect to the element size. In compression the pre-peak harden-ing behavior is described by a power hardening law. The softening behavior in compression is represented by a combination of a power softening law and an exponential softening function. Fig. 4-9 shows the hardening/softening functions and the corresponding material parameters.

44




1 2 3 4 5 6 7 8 9 10 1-10 LAW Rd Rz Ru ψ 11-20 κcm κcu Ωci Ωcu Ωcr 21-30 31-40

41-50





[-] 0 < κcm < κcu

κcu plastic strain defining the end of power softening law and start of exponential softening law

(κcu = εu – Ωcu * Rd/E)

[-] κcu > κcm


[-] 0 ≤ Ωci ≤ 1

Ωcu relative stress level at κcu Ωcr < Ωcu ≤ 1 Ωcr residual relative stress level in compression

(recommendation Ωcr = 0.2) [-] 0 < Ωcr < Ωcu


Gft area specific mode-I fracture energy [Nm/m²] Gft > 0 Ωtr residual relative stress level in tension



4.2.8.3 Law=14, mlaw=6 – fracture energy based soft ening By setting parameter mlaw=6, fracture energy (mode-I) based softening functions are applied in tension and in compression. In order to regularize the model (remove mesh dependency), both fracture energy values are scaled by an equivalent element size. Additionally, a power hardening law is used to describe the nonlinear pre-peak behavior in compression. Fig. 4-10 shows the hardening/softening functions and the corresponding material parameters.

45


Fig. 4-10: hardening and softening functions of law =14, mlaw=6 (left - compression, right - ten-sion)


1 2 3 4 5 6 7 8 9 10 1-10 LAW Rd Rz Ru δψt δψc 11-20 κcm Ωci Ωcr Gfc 21-30 31-40

41-50





[-] 0 < κcm < κcu


[-] 0 ≤ Ωci ≤ 1


[-] 0 < Ωcr < Ωcu

Gfc area specific mode-I fracture energy in com-pression

[Nm/m²] Gfc > 0


Gft area specific mode-I fracture energy in tension [Nm/m²] Gft > 0 Ωtr residual relative stress level in tension


Note: The fracture energies are regularized with respect to the element size.

46


4.3 Numerical control variables eps local convergence criteria Return Mapping geps criteria for singular systems of equations for multi-area activity maxit Maximum amount of local iterations of the return mapping process cutmax amount of local bisections prior the activation of a global bisection activation of the output control (debug- resp. control-modus for developers or users when necessary) wr output key Elem element number Intp number of the integration point

4.3.1 Choice of the numerical control variables eps: 10-6, shall not be chosen to small (10-4 at KN and m); maxit: 10 local iteration steps shall be enough (at least as many as active yield surfaces); but not to be chosen too small geps: 10-20 for double precision; cutmax 4

4.3.2 Remarks for choosing the material parameters No material parameter should be ever set to 0.0. Even for residual strengths values above eps*100 should be chosen. Dilatancy angles close to zero imply ideally smooth friction surfaces in a physical sense and can lead to extreme convergence difficulties. This results from tension component which can not be removed in case of shear failure. The dilatancy angle therefore should always be set at least to 1°. The tension strength is limited to the intersection point of the Mohr-Coulomb-line (-plane) and the normal friction axis (C /tan(phi)).

4.3.3 Remarks and tips for using multiPlas in nonlinear s tructural analysis

If an oscillation can be seen for a certain imbalance value, the convergence for the load step can be achieved by increasing the convergence criteria in ANSYS slightly above the oscillation value. In the fol-lowing load case the convergence criteria can be set to the smaller value again. In case of frequent error messages (**local return mapping failed**) the local number of iterations should be increased and the yield areas should be checked. This output only occurs if wr ≥ 1. In case where problems occur from processing the polyhedral yield figure (in case of unfortunate physi-cally problematic choice of parameters) the calculations should be performed by using isotropic yield cri-teria with the whole load at first. Then a following calculation with activation of anisotropic yield criteria (this is especially the case for primary stress conditions) can be done. Do never chose dilatancy or friction angle as 0.0 because this can lead to unbalanced forces which can not be relocated! A cohesion c = 0 (e.g. sand) implies that the material does not have any uniaxial compressive or tensile strength. First, the material therefore has to be iterated into a stable position. This leads very often to convergence difficulties, so it is advised to use an adequately small value instead of zero for the cohesion while using the MOHR-COULOMB (LAW = 1) yield conditions. The automatic time stepping is called directly from the routine (it can be switched on via: autots,on). The global load step bisection can be disabled by choosing a large value for cutmax. In the case of a lo-cal bisection, no hints are written out.

47


Be careful not to use too large values for maxinc and simultaneous suppression of the global bisection. This may lead to a large computational effort in a Newton-Raphson-Equilibrium iteration! In this case, re-quest the cause by use the global bisection! Multi surface plasticity fundamentally is a physical path dependent phenomenon. Therefore a global in-crementation in order to represent the relocation of force correctly is of utmost importance. In the multi surface routines, softening (residual strength) is only introduced in at the equilibrium states (so only after reaching the global Newton-Raphson equilibrium). Therefore, a global incrementing is im-portant in the case of softening. The value dtmin in the tb-data-fiel has to be identical to the value of dtmin that is used by ANSYS in the solution-phase (deltim,dtanfang,dtmin,dtmax,...). If no convergent solution could be found: -decrease incrementation (dtmin,...) -increase the global convergence criteria (cnvtol,f,...) Newton-Raphson, full (usage of consistent elasto-plastic tangent) or Newton-Raphson, init (starting stiff-ness) is supported. For the practical problems Newton-Raphson, init is recommended. Especially when considering geometric nonlinearities or when working with EKILL / EALIVE the full Newton-Raphson me-thod is necessary.

48


4.4 Remarks for Postprocessing Plastic effective strain EPPLEQV: The plastic effective strain shows the quantitative activity and is used for pointing out

the areas in which local load shifting or material failure / crack forming take place.

)](21

[32 2

zx,pl2

yz,pl2

xy,pl2

z,pl2

y,pl2

x,pleqv,pl ε+ε+ε+ε+ε+ε=ε

Plastic activity value The plastic activity shows which yield surfaces are active in the current equilibrium state. A yield surface is marked active if this yield surface has a positive plastic multiplier. Consequently the yield surface was violated during the return-mapping procedure and the stress state at the end of the return-mapping process is on this yield surface. This allows the identification of failure mechanism type and of the cause of load shifting. The plastic activity is path depended. A yield surface can be activated and deactivated more than once during a load case. The plastic activity value is stored in the first state variable of nonlinear multiPlas material models. The meaning of the activity value depends on the individual material models. The encoding of the activity val-ue is shown in the subsections of section 4.2. In order to plot the activity value the state variables must be stored in the result file. Please execute the following command before solving the global system of equations:

OUTRES,SVAR,<Freq> (<Freq> can be replaced by an integer, LAST or ALL. Please refer to the ANSYS APDL manual for a comprehensive description of the OUTRES command)

In the post-processing (/post7) the activities can be plotted with one of the following commands: PLESOL,SVAR,1 PLNSOL,SVAR,1

49


5 VERIFICATION EXAMPLES

5.1 Example 1 – Earth pressure at rest

Material assumptions (material 1 to 3), sand: Angle of inner friction ϕ = 30° Cohesion c = 0 Constrained modulus ES = 40000 kN/m² Coefficient of earth pressure at rest k0 = 0,5 Density ρ = 1,8 t/m³ Therefore:

Poisson’s ratio: 0

0

k1

k

+=ν = 0,333

Shear modulus: ( ) SE12

21G

ν−ν−= =10000 kN/m²

Young’s modulus: ( ) G12E ν+= =26670 kN/m² Boundary conditions: lower boundary y = 0: ux = uy = uz = 0 side boundary x = 0 bzw. 10: ux = 0 side boundary z = 0 bzw. 1: uz = 0

10 m

10 m

1 m

y

x z

FE-model

theory see [6-19]

50


Elements: Solid185 Load history: 1st Load step: installation 1st layer (material 1) 2nd Load step: installation 2nd layer (material 2) 3rd Load step: installation 3rd layer (material 3) Reference solution: (bsp1.dat) path plots along the boundary at x = 0

Stresses in load step 3

Coefficient of earth pressure sx / sy in load step 3

51


5.2 Examples 2 to 4 - Earth pressure at rest and ac tive earth pressure

Material assumption (material 1): Angle of inner friction ϕ = 30°; residual strength ϕr = 30° Angle of dilatancy ψ = 30° Cohesion c = 0 Elastic modulus ES = 40000 kN/m² Coefficient of earth pressure at rest k0 = 0,5 Density ρ = 1,8 t/m³ Therefore: Poisson’s ration: ν = 0,333 Shear modulus: G =10000 kN/m² Young’s modulus: E =26670 kN/m² Elements: Solid45 Load history: 1st Load step: Earth pressure in a result of the gravity Boundary conditions: lower boundary y = 0: ux = uy = uz = 0 side boundary x = 0 bzw. 10: ux = 0 side boundary z = 0 bzw. 1: uz = 0 2nd load step: Activation of the active earth pressure by rotation of the side boundary x = 0 about the base point, horizontal top point displacement: 3cm

10 m

10 m

1 m

FE-model

52


Reference solution: Example 2 - Calculation with LAW = 1 MOHR-COULOMB (bsp2.dat) Number of substeps / iterations: 4 / 18 cpu-time: (1x 4-M CPU 1,70 GHz) 21,9 sec

Stresses in load step 2 as path plots along the boundary x = 0

Coefficient of earth pressure sx / sy in load step 2 as a path plot along the border x = 0 U,sum = 0,047487 m EPPL,EQV = 0,003927

53


5.3 Examples 5 to 8 - Kienberger Experiment G6 [6-1 3]

**Calculation:

- material and geometric nonlinear - Convergence bound at 1% to 2% of the L2-norm of the residual forces. In ANSYS, the convergence criterion is defined as default value of 0.1 % of the L2-norm of the residual forces. That means that all residual forces have to be transferred to the load vector ex-cept 0.1 % of the Root Mean Square. In the following calculations, a convergence criterion be-tween 1% and 2% has been used. According to experience, convergence criteria between 1% and 2% are precisely enough, to verify equilibrium conditions. (Cohesion = 0 means no tensile- resp. compressive strength of the material. Hence result con-vergence problems, so that the convergence bound has to be increased compared with the de-fault value to achieve a solution.)

**Element types: Solid185 und Shell181 **Load history Load step 1: self-weight Sand Load step 2: load

54


Reference solution: Example 5 - Calculation with LAW = 1 MOHR-COULOMB (bsp5.dat)

Vertical displacement uy (mm) of the steel tube - arc

55


5.4 Example 9 - MOHR-COULOMB anisotropic (bsp9.dat)

FE-model

Rock with 2 joint blades 1st joint α = 85°, β = 0° 2nd joint α = -5°, β = 90° Elements: SOLID 185

3000 kN/m²

1850 kN/m²

Equivalent plastic strain EPPL,EQV max eppl = 0,281 E-03

Total deformation usum (m) max usum = 0,019961 m

56


5.5 Example 10 – Concrete-model DRUCKER-PRAGER sing ular (LAW=9)

(eld.dat) Uniaxial compressive tests:

Stress-strain diagram 20°C, mlaw = 0


57



58


5.6 Example 11 – Concrete-model DRUCKER-PRAGER sing ular (LAW=9)

(elz.dat) Uniaxial tensile tests:



59




60


5.7 Example 22 – Single Joint Shear-Test (LAW=1, 10 ) (bsp22.dat)

61


5.8 Example 23 – Single Joint Tensile-Test (LAW=1, 10) (elz.dat)

62


6 REFERENCES

[6-1] ANSYS Users Manual for ANSYS Rev. 11.0, Analy sis Guides, ANSYS Inc., Houston, Ca-nonsburg

[6-2] ANSYS Users Manual for ANSYS Rev. 11.0, Comma nds, ANSYS Inc., Houston, Canonsburg

[6-3] ANSYS Users Manual for ANSYS Rev. 11.0, Elem ents, ANSYS Inc., Houston, Canonsburg

[6-4] ANSYS Users Manual for ANSYS Rev. 11.0, Theo ry, ANSYS Inc., Houston, Canonsburg

[6-5] ANSYS Users Manual for ANSYS Rev. 11.0, Verif ication, ANSYS Inc., Houston, Canonsburg

[6-6] Bažant, Z.P.; Oh, B.H.: Crack band theory fo r fracture of concrete. Materials and Struc-tures, RILEM, 93 (16), S. 155-177

[6-7] Chen, W.F.: Constitutive Equations for Engine ering Materials. Vol. 2 Plasticity and Model-ing. Elsevier Amsderdam - London - New York - Tokyo , (1994)

[6-8] Deutscher Ausschuss für Stahlbeton, Heft 525 - Erläuterungen zu DIN 1045-1, Ausgabe 2003, Beut-Verlag

[6-9] DIN 1045-1 Tragwerke aus Beton, Stahlbeton un d Spannbeton, Teil1: Bemessung und Konstuktion, Beuth-Verlag, Ausgabe Juli 2001

[6-10] DIN EN 1992-1-1 Eurocode 2: Bemessung und Ko nstruktion von Stahlbeton- und Spannbe-tontragwerken. Teil 1-1: Allgemeine Bemessungsregel n und Regeln für den Hochbau, Ausgabe Oktober 2006

[6-11] DIN EN 1992-1-2 Eurocode 2: Bemessung und Ko nstruktion von Stahlbeton- und Spannbe-tontragwerken. Teil 1-2: Allgemeine Regeln – Tragwe rksbemessung für den Brandfall, Ausgabe Nov 2005

[6-12] Hintze, D.: Zur Beschreibung des physikalisc h nichtlinearen Betonverhaltens bei mehr-achsigem Spannungszustand mit Hilfe differentieller Stoffgesetze unter Anwendung der Methode der finiten Elemente. Hochschule für Archit ektur und Bauwesen Weimar, Disser-tation (1986)

[6-13] Kienberger, H.: Über das Verformungsverhalte n von biegeweichen, im Boden eingebette-ten Wellrohren mit geringer Überschüttung. Rep. Öst erreich, Bundesministerium f. Bauten u. Technik Straßenforschung, Heft 45, 1975

[6-14] Krätzig, W.; Mancevski, D.; Pölling, R.: Mod ellierungsprinzipien von Beton. In: Baustatik-Baupraxis 7. Hrsg. Meskouris Konstantin (RWTH Aache n). Balkema Verlag, Rotterdam 1999. S. 295-304

[6-15] Ottosen, N.S.: A Failure Criterion for Concr ete. Journal of the Eng. Mech. Div. ASCE. 103, EM4, S. 527-535 (1977)

[6-16] Pölling, R.: Eine praxisnahe, schädigungsori entierte Materialbeschreibung von Stahlbeton für Strukturanalysen. Ruhr-Universität Bochum, Diss ertation (2000)

[6-17] Schlegel, R.: Numerische Berechnung von Maue rwerkstrukturen in homogenen und dis-kreten Modellierungsstrategien. Dissertation, Bauha us-Universität Weimar, Univer-sitätsverlag (2004), ISBN 3-86068-243-1

[6-18] Simo, J.C.; Kennedy, J.G.; Govindjee, S.: No n-smooth multisurface plasticity and viscop-lasticity. Loading / unloading conditions and numer ical algorithms. Int. Journal for numer-ical methods in engineering. Vol. 26, 2161-2185 (19 88)

[6-19] Vermeer, P.A.: Materialmodelle in der Geotec hnik und ihre Anwendung. Proceedings Finite Elemente in der Baupraxis 1995

[6-20] Vonk, R.A.: Softening of concrete loaded in compression. Dissertation, Delft University of Technology (1992)

[6-21] Weihe, S.: Modelle der fiktiven Rissbildung zur Berechnung der Initiierung und Ausbrei-tung von Rissen. Ein Ansatz zur Klassifizierung. In stitut für Statik und Dynamik der Luft- und Raumfahrtkonstruktionen, Universität Stuttgart, Dissertation (1995)

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[6-22] Will, J.: Beitrag zur Standsicherheitsberech nung im geklüfteten Fels in der Kontinuums- und Diskontinuumsmechanik unter Verwendung implizit er und expliziter Berechnungsstra-tegien: Bauhaus Universität Weimar, Dissertation 19 99, Berichte Institut für Strukturme-chanik 2/99

[6-23] Ganz, H.R.: Mauerwerkscheiben unter Normalkr aft und Schub. ETH Zürich, Institut für Baustatik und Konstruktion. Dissertation. Birkhäuse r Verlag Basel (1985)

[6-24] Mann,W.; Müller,H.: Schubtragfähigkeit von g emauerten Wänden und Voraussetzungen für das Entfallen des Windnachweises. Berlin: Ernst u. Sohn. In: Mauerwerk-Kalender (1985)

[6-25] Berndt, E.: Zur Druck- und Schubfestigkeit v on Mauerwerk – experimentell nachgewiesen an Strukturen aus Elbsandstein. Bautechnik 73, S. 2 22-234 Ernst & Sohn, Berlin (1996)

[6-26] Grosse, M.: Zur numerischen Simulation des p hysikalisch nichtlinearen Kurzzeittragver-haltens von Nadelholz am Example von Holz-Beton-Ver bundkonstruktionen. Dissertation, Bauhaus-Universität Weimar (2005)

[6-27] Pluijm, R. van der: Shear behaviour of bed j oints. Proc. 6th Noth American Masonry Confe-rence, S. 125-136 (1993)

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