DNVGL-RP-0419 Analysis of grouted connections using the ...

RECOMMENDED PRACTICE

DNVGL-RP-0419 Edition September 2016

Analysis of grouted connections using the finite element method

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The electronic pdf version of this document found through http://www.dnvgl.com is the officially binding version. The documents are available free of charge in PDF format.

FOREWORD
DNV GL recommended practices contain sound engineering practice and guidance.
© DNV GL AS September 2016

Any comments may be sent by e-mail to [email protected]

This service document has been prepared based on available knowledge, technology and/or information at the time of issuance of this document. The use of thisdocument by others than DNV GL is at the user's sole risk. DNV GL does not accept any liability or responsibility for loss or damages resulting from any use ofthis document.

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Recommended practice, DNVGL-RP-0419 – Edition September 2016 Page 3

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CONTENTS
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Sec.1 Introduction.................................................................................................. 61.1 Objectives ..............................................................................................61.2 Validity...................................................................................................61.3 Definitions..............................................................................................6

1.3.1 Terms ..........................................................................................6

1.4 Acronyms, abbreviations and symbols ...................................................61.4.1 Acronyms and abbreviations............................................................61.4.2 Symbols .......................................................................................7

1.5 References .............................................................................................81.6 Grouted connection concepts .................................................................8

Sec.2 Basic considerations ................................................................................... 112.1 Limit state safety format ......................................................................11

2.1.1 Load and resistance factor design principle ......................................122.1.2 Design by load and resistance factor design and the finite element

method ......................................................................................13

2.2 Empirical basis for the resistance.........................................................132.3 Characteristic resistance ......................................................................132.4 Types of failure ....................................................................................132.5 Analysis of grouted connections...........................................................14

2.5.1 Plausibility check .........................................................................14

Sec.3 Materials ..................................................................................................... 153.1 Material models for steel ......................................................................15

3.1.1 Linear ........................................................................................153.1.2 Nonlinear....................................................................................15

3.2 Material models for grout .....................................................................153.2.1 Linear and nonlinear models..........................................................153.2.2 Plasticity.....................................................................................163.2.3 Drucker-Prager............................................................................183.2.4 Willam-Warnke ............................................................................193.2.5 Lubliner-Lee-Fenves .....................................................................20

3.3 Selection of material model and properties ..........................................213.3.1 Specifying nonlinear properties ......................................................22

Sec.4 Contact interactions .................................................................................... 234.1 Contact modeling .................................................................................23

4.1.1 Normal contact ............................................................................234.1.2 Sliding contact.............................................................................244.1.3 Calibration ..................................................................................24

Sec.5 Finite element method modeling ................................................................. 265.1 Solution schemes .................................................................................26

5.1.1 Explicit and implicit solution method ...............................................265.1.2 Choosing a solution scheme ..........................................................28

5.2 Modeling schemes ................................................................................285.2.1 Submodeling ...............................................................................28

5.3 Geometric extent..................................................................................285.3.1 Overall geometry .........................................................................29


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5.3.2 Features .....................................................................................29
5.3.3 Tolerances and imperfections ........................................................315.3.4 Symmetry utilization ....................................................................34

5.4 Element selection .................................................................................345.4.1 Element shape, order, and integration scheme.................................345.4.2 Element density...........................................................................35

Sec.6 Boundary conditions and load application ................................................... 376.1 Boundary conditions.............................................................................376.2 Load application ...................................................................................38

6.2.1 Global and local loads...................................................................386.2.2 Ultimate limit state load cases .......................................................396.2.3 Finite limit state load cases ...........................................................406.2.4 Serviceability limit state load cases ................................................426.2.5 Accidental limit state load cases.....................................................42

Sec.7 Limit state analyses .................................................................................... 437.1 Ultimate limit state ..............................................................................43

7.1.1 Response assessment...................................................................43

7.2 Buckling ...............................................................................................447.3 Fatigue limit state ................................................................................447.4 Serviceability limit state.......................................................................467.5 Accidental limit state............................................................................46

App. A Constitutive formulations for grout ............................................................. 47App. B Contact modeling methodologies ................................................................ 68


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SECTION 1 INTRODUCTION
1.1 ObjectivesThis recommended practice provides principles and guidance for structural analysis of grouted connections by the finite element method. That is, how to calculate load effects using the finite element method.

It is in general generic in its recommendations why it may be used for any structure. The focus is however on offshore structures with special attention to wind energy applications.

It is not intended to replace formulas for resistance in codes and standards for the cases where they are applicable and accurate, but to present methods that allows for using nonlinear FE-methods to determine resistance for cases that is not covered by codes and standards or where accurate recommendations are lacking.

It is not the purpose of the recommended practice to provide specific design requirements for grouted connections, but rather to provide recommendations on how to build, load, and solve finite element models of grouted connections. There hereby obtained structural response is then assumed qualified based on valid design requirements from an applicable offshore standard, e.g. DNVGL-ST-0126 /9/.

The present recommended practice is thus not a standalone document on the design of grouted connections, but rather a supporting document for any valid design standard. See further [1.2].

1.2 ValidityThe recommended practice assumes design by the load and resistance factor method taken to be qualified through the use of applicable offshore standards, e.g. DNVGL-ST-0126 /9/, Norsok N-004 /17/, and ISO 19902 /16/. It is further assumed that the fabrication of the structure comply with the standard requirement associated with the governing standard.

1.3 Definitions

1.3.1 TermsThis recommended practice uses terms as defined in DNVGL-ST-0126 /9/. Additional terms used are:

1.4 Acronyms, abbreviations and symbols

1.4.1 Acronyms and abbreviationsAcronyms and abbreviations as shown in Table 1-1 are used in this recommended practice.

Term Definitiondynamic a load or load effect that is dependent on time and inertia effectsstatic a load or load effect that is independent of timequasi-static a load or load effect that is dependent on time but with negligible inertia effects

Table 1-1 Acronyms and abbreviations

Abbreviation Description1D, 2D, 3D 1-, 2-, and 3-dimensionalALS accidental limit stateFEA finite element analysisFEM finite element methodFLS fatigue limit stateLRFD load and resistance factor designrebar reinforcement barSCF stress concentration factorSLS serviceability limit stateULS ultimate limit state


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1.4.2 Symbols
1.4.2.1 Latin characters

1.4.2.2 Greek characters

1.4.2.3 Scripts

A areac speed of dilatationd cohesionD diameterE modulus of elasticity (Young’s module)F loadGF fracture energyh heightK shape parameterl lengthL lengthp pressureq equivalent stressR resistanceS load effectu displacementt time or thickness or deviatoric stressΔt time step or incrementw width

β material friction angleδ small length, imperfection magnitudee engineering strainε true strain (logarithmic strain)∈ flow potential eccentricityγf load partial safety factorγm material safety factorν Poisson ratioφ resistance factorθ Lode angleρ specific densityΨ(⋅) load effect functionψ material dilation angles engineering stress (nominal stress)σ true stress

c compressived design value or dynamice elasticg groutk characteristic valuep plastic or pressures steelt tensiley yield


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1.5 References
1.6 Grouted connection conceptsGrouted connections are in the present context taken to be a structural connection between two overlapping steel components one being larger than the other where a grout is cast in the void between the two to form a load transferring snug fit body between said steel components.

Other types of grouted connections can be envisaged but are not considered in the present context.

In offshore applications, structural members with cylindrical cross sections dominate as these in general are favorable when exposed hydro-dynamic and static loading.

These connections can therefore be described in terms of an axial extent with either a constant (cylindrical) or varying (conical) diameter.

In general the functional requirement to a grouted connection is that it can transfer any individual or combined axial, shear, and bending loading from one steel component to the other.

As the interface between the steel and cast grout only provides marginal passive shear capacity this resistance cannot be relied on in the design, why providing the axial capacity of the connection splits these into two principal classes of connections as already indicated namely:

— vertical cylindrical connections with shear keys, and— inclined or conical connection without shear keys.

Table 1-2 DNV GL service documents

/1/ DNV-OS-C502 Offshore Concrete Structures

/2/ DNV-RP-C204 Design against Accidental Loads

/3/ DNV-RP-C207 Statistical Representation of Soil Data

/4/ DNV Technical Report No. 2011-1415

Capacity of Cylindrical Shaped Grouted Connections with Shear Keys – Background Report, Joint Industry Project

/5/ DNVGL-OS-C401 Fabrication and testing of offshore structures

/6/ DNVGL-RP-C203 Fatigue design of offshore steel structures

/7/ DNVGL-RP-C208 Determination of structural capacity by non-linear finite element analysis methods

/8/ DNVGL-SE-0190 Project certification of wind power plants

/9/ DNVGL-ST-0126 Design of wind turbine support structures

/10/ DNVGL-ST-0145 Offshore substations

Table 1-3 Other documents

/11/ EN 1990 Eurocode: Basis of Structural Design, 2002.

/12/ EN 1993-1-5 Eurocode 3: Design of Steel Structures, Part 1-5: Plated Structural Elements, 2006.

/13/ EN 1993-1-6 Eurocode 3: Design of Steel Structures, Part 1-6: Strength and Stability of Shell Structures, 2007.

/14/ IIW Document XIII-1823-07/XIII-2151r4-07/XV-1254r4-07

Recommendations for Fatigue Design of Welded Joints and Components, International Institute of Welding (IIW/IIS), Edited by A. Hobbacher, 2008.

/15/ ISO 2394 General Principles on Reliability for Structures, Second Edition, 1998.

/16/ ISO 19902 Fixed Steel Offshore Structures – Petroleum and Natural Gas Industries, 2007.

/17/ Norsok N-004 Design of Steel Structures, Norsk Standard, Rev. 3, 2013.

/18/ NRL MP 82039 U The Analysis of Load-Time Histories by means of Counting Methods, J.B. de Jonge, National Aerospace Laboratory (NRL) – The Netherlands, 1982.

/19/ Grouted Connections for Offshore Wind Turbine Structures – Part 2: Structural Modelling and Design of Grouted Connections, Fehling, E.; Leutbecher, T.; Schmidt, M.; Ismail, M., Steel Construction 6 (2013), Issue 6, pp. 216-228, Ernst & Sohn Verlag.


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Shear keys are protuberances on the steel surface on the interface that when subsequently cast into the
grout provides a mechanical resistance to relative sliding between the two bodies being the steel and grout hereby creating a passive shear capacity of the interface.

Traditionally, inclined piles have been used for offshore oil and gas jacket platforms. In these designs the piles are either driven through the jacket legs or through external sleeves. With sufficient long overlap these grouted connection types can due to their inclination provide sufficient axial capacities.

Common for both is that the jacket is put in place first, and the piles are then driven essentially using the jacket as a template for the piling. This is referred to as post-piling.

Alternatively, the piles can be driven vertically. Vertical piles can be used with sleeves for jackets where, as for the inclined piles, the jacket is used as a template for the post-piling.

Figure 1-1 Illustration of typical grouted connection for jacket foundations

For jackets, the piles may alternatively be driven using a removable template prior to the installation of the jacket. This is referred to as pre-piling and entails a design where the jacket legs are then stabbed into the annuli of the pre-driven piles, thus entailing that the jacket leg ends with a protruding vertical section. As this approach requires the stabbing (or lowering) of the jacket into the piles, these types of connections typically need relatively thick grout annuli to ensure sufficient play during installation.

Common for all jackets with vertical piles is the need for shear keys to provide axial capacity of the grouted connection.

Illustrations of post- and pre-piling foundation concepts for jackets are shown in Figure 1-1.

In the special case of monopile foundations, overturning loads are carried as bending in contrast to the jacket foundations where overturning ideally is carried by an axial pull-push force couple in the piles.

In this case then, the grouted connection does not need to carry an upward pull but instead only a downward push combined with significant bending.

Two designs of grouted connections for monopiles are:

— cylindrical connections with shear keys, and — conical connection without shear keys

as illustrated in Figure 1-2.

Inclined pile in leg

Inclined pile in sleeve

Vertical pile in sleeve

Leg in pile


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Because overturning loads preferably should be transferred via compression though the thickness of the
grout at the top and bottom of the connection it is generally recommended that in the case of shear keys these are placed near the mid height of the connection. In the case of a conical design it is for the same reasons recommended to keep the cone angle small, say 1° to 3° relative to vertical.

Figure 1-2 Illustration of the grouted connection for monopile foundations. Left cylindrical with shear keys. Right conical without shear keys

Transition Piece

Grout

Monopile

Shear Keys


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SECTION 2 BASIC CONSIDERATIONS
2.1 Limit state safety formatA limit state can be defined as “A state beyond which the structure no longer satisfies the design performance requirements”. See e.g. /15/.

Limit states can be divided into the following groups:

— Ultimate limit states (ULS) corresponding to the ultimate resistance for carrying loads.

— Fatigue limit states (FLS) related to the possibility of failure due to the effect of cyclic loading.

— Accidental limit states (ALS) representing failure due to an accidental event or operational failure.

— Serviceability limit states (SLS) corresponding to the criteria applicable to normal use or durability.

The safety format that is used in limit state standards is schematically illustrated in Figure 2-1 showing the probability density distribution of the load (formally the load effect) in blue and the resistance in red.

The distinction between load and action effect is important in cases where the relationship between the load and the load effect (response) in nonlinear. For the purpose of illustration a simple linear relation is assumed as this facilitates the introduction of the load and resistance factor design principle.

Figure 2-1 Illustration of the limit state safety format

The limit state is then formally expressed by the equation: , why the design requirement that the design load does not exceed the design resistance can be written as

(2.1)

Mean oad

Mean esistance

Characteristic esistance

Design esistance ( )

Resistance actor

Requirement

Load and esistance

Prob

abili

tyen

sity

Design oad ( )

d = d d d d ≤ d


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2.1.1 Load and resistance factor design principle
As illustrated by Figure 2-1 the load and resistance are both recognized as fundamentally stochastic quantities, i.e. each described by their own probability density function. For the purpose of illustration the normal distribution has been assumed for both quantities in Figure 2-1.

Implicitly then, the value of each quantity varies about a mean value, why the first step is the introduction of the characteristic value.

The characteristic value is the first level of safety embedded in the load and resistance factor design (LRFD) principle. It is normally defined as a specific percentile of the load and resistance respectively. Typically the 98th percentile of the load and the 5th percentile of the resistance are applied. It should be noted that the choice of these percentiles inherently forms part of the overall safety, why these may vary based on application in low-, normal-, or high safety class.

Assuming then that a large load combined with a small resistance represent the worst condition, the second level of safety embedded in the LRFD principle is expressed through the application of safety factors on both load and resistance. As illustrated in Figure 2-1 these factors then increases the load to the design load magnitude and reduces the resistance to the design strength. The limit state requirement is then judged based on these design quantities.

As pointed out initially, it is important to distinguish between loads and load effects – in particular for nonlinear systems. Moreover, as a structure is in general exposed to more than a single load, a generalization of the LRFD principle is described in the following.

A design load Fd is obtained by multiplying the characteristic load Fk by a given load factor γf The magnitude of the load factor is dependent on the load type. It is applied to account for:

— possible unfavorable deviations of the loads from the characteristic values— the reduced probability that various loads acting together will act simultaneously at their characteristic

value— uncertainties in the model and analysis used for determination of load effects.

A design load effect Sd is the most unfavorable combined load effect. Taking the load effect to be a single quantity derivable by the load effect function Ψ(⋅), the design load effect from say n design loads Fd,i may be expressed as

A design resistance Rd is obtained by multiplying the characteristic resistance Rk by a given resistance factor φ

The resistance factor φ relates to the material factor γm as

The magnitude of the material factor is dependent on the strength type. It is applied to account for:

— possible unfavorable deviations in the resistance of materials from the characteristic values— possible reduced resistance of the materials in the structure, as a whole, as compared with the

characteristic values deduced from test specimens.

The design requirement is then that the design load effect Sd does not exceed the design resistance Rd for said load effect i.e.

(2.2)

(2.3)

(2.4)

(2.5)

(2.6)

d = f k

d = Ψ( d,1, d,2, ⋯ , d, ) d = k

= 1m

d ≤ d
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2.1.2 Design by load and resistance factor design and the finite element
methodThe finite element method (FEM) is a generalized numerical technique for finding approximate solutions to boundary value problems for partial differential equations. The application of the method is commonly referred to as finite element analysis (FEA).

The practical application of FEA in structural design typically consist of building a model of structure using appropriate elements that through constitutive relations (material formulations) all together constitutes a set of partial differential equations representing the resistance (stiffness) of the structure. Applying loads and other boundary conditions to this model then sets up the boundary value problem the solution to which is the response of the structure quantified e.g. as stresses, strains, and displacements.

In relation to the load and resistance factor design (LRFD) principle, FEA may then be seen as the load effect function Ψ(⋅) in Eq. (2.3).

In static or quasi-static analyses the design load effect can therefore be determined by applying design loads to a FEM model that represents the characteristic resistance of the structure.

The FEM model should aim to represent the resistance as the characteristic values according to the governing standard. In general that means 5% fractile in case a low resistance is unfavorable and 95% fractile in case a high resistance is unfavorable. Fractile magnitudes should be used in accordance with the governing LFRD standard.

2.2 Empirical basis for the resistanceAll engineering methods, regardless of level of sophistication, need to be calibrated against an empirical basis in the form of laboratory tests or full scale experience. This is the case for all design formulas in standards. In reality the form of the empirical basis vary for the various failure cases that are covered by the standards, from determined as a statistical evaluation from a large number of full scale representative tests to cases where the design formulas are validated based on extrapolations from known cases by means of analysis and engineering judgments.

It is of paramount importance that capacities determined by nonlinear FEA methods build on knowledge that is empirically based. That can be achieved by calibration of the analysis methods to

— experimental data,— established practice as found in design standards, or— full scale experience.

2.3 Characteristic resistanceThe characteristic resistance should represent a value which will imply that there is less than 5% probability that the resistance is less than this value. Often lack of experimental data prevents an adequate statistical evaluation so the 5% shall be seen as a goal for the engineering judgments that in such cases are needed.

The characteristic resistance given in design standards is determined also on the basis of consideration of other aspects than the maximum load carrying resistance. Aspects like post-ultimate behavior, sensitivity to construction methods, statistical variation of governing parameters etc. are also taken into account. In certain cases these considerations are also reflected in the choice of the material factor that will be used to obtain the design resistance. It is necessary that all such factors are considered when the resistance is determined by nonlinear FEA.

2.4 Types of failureIt is important to recognize that for grouted connections the definition of failure as given by the design standards is for the entire connection. That is, the design provisions set forth in standards such as DNVGL-ST-0126 /9/, Norsok N-004 /17/, and ISO 19902 /16/ are all addressing the entire grout body as one structural component that subsequently is either safe or failed.

Apart from buckling of the steel, overall failure of grouted connection is difficult to assess using FEA as it entails progression of local cracking and crushing of the grout.


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For the grout body it is therefore necessary to rely on engineering judgment if the design strength of the
grout is predicted to be exceeded by the FEA. Here possible adverse effects of load redistribution should be carefully considered. This could either be assessed by re-analysis using a material model that exhibits only design strengths, or by continuing the analysis based on the characteristic material strength until the loading is scaled also by the material safety factor, i.e. by γf γm.

2.5 Analysis of grouted connectionsGrouted connections are in principle made up of three discrete continuum bodies that interact with each other through frictional contact. In itself this contact interaction makes the response of a grouted connection nonlinear. Add to this, the inherent nonlinear behavior of a brittle material like grout, and the combined effect is a complex general nonlinear response.

The effect of this nonlinear response will be felt not only by the grouted connection itself but also by the surrounding steel structure. The influence is however normally dissipated at a distance of 1.5 times the diameter of the connection, and significant only in, say one- to half a diameter distance above and below the grouted connection overlap.

Analysis of grouted connection may also depending on the type of connection be encumbered by a complex loading environment necessitating assessment of a many rather than a few load cases for the different limit states.

Analysing grouted connections is because of these nonlinearities normally a time consuming and somewhat delicate affair requiring both skills and insight from the analyst.

Lacking previous experience with the analysis of grouted connections, or faced with a complex load environment or a novel design, valuable insight may be gained from models with linear materials and comparatively coarse meshes.

Irrespectively of experience, it is generally recommended to always conduct a thorough examination of the loading environment as a first step. Especially in cases where the analyst conducting the local detail FEA of the grouted connection is relying on loads derived from global simulations executed by a different analyst.

2.5.1 Plausibility checkIt is generally recommended to compare results attained from FEA with analytical expression from available standards on grouted connections, e.g. DNVGL-ST-0126 /9/, Norsok N-004 /17/, and ISO 19902 /16/ as a plausibility check of the FEA response.


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SECTION 3 MATERIALS
The two principal materials used in grouted connections are steel and grout. Both materials exhibit nonlinear behavior when strained, however the nonlinear response is most pronounced for the grout.

Normal construction steel is in general required to be very ductile and will exhibit strain hardening up until 20% elongation as a typical design requirement. From an engineering strength perspective moreover, steel excel by being equally capable in tension and compression.

Contrary, grouts are typically very brittle materials exhibiting vastly different compressive and tensile strength. Moreover, the elastic behavior of grouts is typically strain rate dependent causing typically a stiffer response to a rapid loading.

Nonlinear material models should therefore in general be applied in FEA of grouted connections if the true behavior is to be captured. However, in the design of grouted connection a linear elastic simplification may – dependent on the type of analysis – be either sufficient or indeed an assumption inherent to the structural design checks. Thus, guidance on both linear and nonlinear material models will be given in the following sections.

3.1 Material models for steel

3.1.1 LinearFor most purposes the isotropic linear elastic material model will be sufficient for analysis of grouted connections in the fatigue and serviceability limit states.

In offshore wind energy applications the foundation designs are typically driven by fatigue, why also in the ultimate limit state, a linear elastic material assumption in general will be sufficient.

3.1.2 NonlinearExceptions from the above are geometric and materially nonlinear buckling assessment (push over analyses) and designs where the ultimate limit state is governing.

In these cases where yielding is to be captured by the analysis a nonlinear material model with isotropic kinematic hardening should be used for the steel.

Guidance on relevant material models and strain hardening behavior can be found in DNVGL-RP-C208 /7/ or EN 1993-1-5 /12/.

3.2 Material models for groutThe structural behavior of grout is very complex if all of the materials characteristics are to be captured. Not only is it a very brittle material, it is also exhibits a very nonsymmetrical strength that is pressure dependent with in general very low tensile strength and comparatively very high compression strength.

3.2.1 Linear and nonlinear modelsThe isotropic linear material model representing Hooke’s Law is the most basic material model that can be envisaged. As the model ignores all nonlinear effects such as cracking and crushing, it obviously falls short of a full description of the true grout behavior. It does however by virtue of its simplicity offer high computational efficiency in terms of both minimal speed and minimal memory use.

It is recommended to gain initial insight into the structural response of the structure by conducting an extreme limit state assessment first based on an assumed linear elastic material behavior of the grout using the mean dynamic modulus of elasticity.

For a more accurate description of the grout behavior a nonlinear model is needed. A multitude of such models exist particularly in research papers, however in commercial general FEM software a more limited subset of these will be available to describe the nonlinear behavior of the grout.

The simplest nonlinear model applicable for the grout is the pressure dependent linear Drucker-Prager model which is generally available in all nonlinear FEM software. The Drucker-Prager model stems from an effort to deal with plastic deformation of soils but has later been applied to e.g. rock and concrete.


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A number of extensions to the Drucker-Prager model exist and are typically available in commercial FEM
software, making the Drucker-Prager family of material models very versatile. Typical extensions are noncircular yield surface in the deviatoric plane to match different yield values in triaxial tension and compression for the linear model, hyperbolic or power-law pressure dependence rather that linear, and a capping of the compressive hydrostatic pressure.

More advanced models available in commercial general FEM software are e.g. the Willam-Warnke model available in Ansys or the Lubliner-Lee-Fenves model available in Abaqus. These models are however only applicable under low to medium compressive confinement. If crushing under high confinement pressures is to be captured the capped Drucker-Prager model is typically the only model available directly in commercial general FEM software.

The yield criterions for the suggested three nonlinear models are described in detail in App.A and are, together with advice on the selection of parameters, summarized in the following sections. First however, a quick recap of general concepts of plasticity is presented to facilitate the description of the material models.

3.2.2 PlasticityThe basic assumption in plasticity is that the total deformation can be divided into an elastic and plastic part. Historically this is done using additive decomposition, i.e. that the total strain ε is the sum of an elastic strain εe that is fully recoverable and an inelastic (plastic) strain εp that cannot be recovered.

In terms of strain increments the plasticity model can be directly formulated as .

Any nonlinear material model has three principal components:

— a yield criterion defining when the material initially deviates from the linear elastic behavior,— a hardening rule which prescribes the hardening of the material and the change in yield condition with

the progression of plastic deformation, and— a plastic flow rule that defines how the material deform in its plastic condition by relating increments of

plastic deformation to the stress components.

Choosing any nonlinear material model for an FEA therefore implies selecting a model for not only yielding – but also the plastic behavior in terms of hardening and plastic flow.

In 3D stress space, the yield criterion is a surface formally described by where the function f may be dependent not only on the stress σ but any number of material constants.

Post-yielding the hardening rule describes the change in the yield surface, why formally , and the plastic strain increments are determined by the flow rule in which the function g is the plastic potential that defines the direction of the plastic strain increment and λ is the plastic multiplier to be determined such that the stress state lies on the yield surface, i.e. by the hardening rule , whereby the magnitude of the plastic strain increment is found.

The plastic potential g can be any scalar function which, when differentiated with respect to the stress gives the plastic strains. If the plastic potential is taken to be the yield function, i.e. g = f the material is said to have associated flow, as opposed to the general case of non-associated flow.

Associated flow is typically used with classic Mises yield plasticity for steel. For grout materials however, these in general exhibits a rater low dilation angle of say 10° to 20° implying non-associated flow.

3.2.2.1 Strain HardeningThe strain hardening and/or softening of the hardening rule can be either an analytical formulation such as e.g. Johnson-Cook plasticity for steel, or given a tabular form of yield stress versus plastic strain which is the most basic and common form typically available in commercial FEM software.

It is recommended to model the compressive hardening of the grout based on the DNV-OS-C502 /1/ recommendations. This approach entails a linear-elastic behavior up till 60% of the compressive strength followed by a power-law hardening up till the total compressive strength. Hereafter the behavior is taken to be ideal-plastic.

Following the notation that positive stresses and strains are tensile, the compressive elastic limit is thus

= e + p

( ) = 0

, p = 0 p = ( )/ , p = 0


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defined by the stress point where E is the elastic modulus, σc the compressive
strength, and α is the portion of the compressive response that behaves linear-elastic set to 60% in DNV-OS-C502 standard which is recommended in lieu of actual material data. It is further recommended that the characteristic compressive dynamic modulus of elasticity Ecdk is used to describe the linear elastic part of the response in both tension and compression, i.e. that E = Ecdk.Hereafter the material yields and hardens to the plastic limit taken to occur at the stress point

in which Ecsk is the characteristic compressive static modulus of elasticity after with the behavior is ideal plastic.

The compressive stress-strain relationship is then expressed as

in which the shape parameter m is this adaptation becomes the ratio between the initial and static modulus of elasticity, i.e. .

The resulting compressive hardening is sketched in Figure 3-1.

Figure 3-1 Tensile strain softening and compressive strain hardening

For the tensile response a linear strain softening is recommended. In case the FEM software has the ability to model this based on a specified fracture energy GF, this method is preferable to the classic tabular definition of stress and strain.

If the tensile strain softening is give directly via stress-strain data, a mesh size dependency is introduced why in this case the element size should preferably be very homogeneous.

Assuming the softening to be linear as recommended the strain softening is as sketched in Figure 3-1.

The tensile strength σt is thus depleted when an elongation of is experienced by the grout, why ut may be described as the crack displacement. Obviously then, the corresponding crack strain will depend on the element length l as , why as mentioned a mesh size dependency is introduced explaining why the grout elements should then ideally be equal sized cubes, unless the strain softening is specified based on the fracture energy.

(3.1)

( ce, ce) = − (1/ , 1) c

c0, c0 = −(1/ csk, 1) c

c( ) = for 0 > ≥ ce+ ( − 1) c + c( − ) c−

for ce > ≥ c0c0 for c0 >

= / csk

( )

( )

t = 2 F/ t tp tp = t/


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3.2.3 Drucker-Prager
The original linear Drucker-Prager yield criterion is a straight line in the p-q plane given as

where q is the equivalent von Mises stress, p is the hydrostatic pressure (positive in compression), and d,β are the material constants, commonly denoted cohesion and friction angle, reflecting the intercept and slope of the yield surface in the meridional p-q plane as illustrated in Figure 3-2 which also shows the cone shape of the Drucker-Prager yield surface in principal stress space.

For the linear Drucker-Prager model the plastic flow potential g takes the form

where ψ is the dilation angle. A geometric interpretation of ψ is included in the p-q diagram shown in Figure 3-2. In the original Drucker-Prager model the plastic strain increment is assumed to be normal to the yield surface. This correspond to it being normal to the circular yield envelope in the deviatoric plane, and to the yield trace in the meridional p-q plane. This condition is attained for ψ = β, i.e. by assuming the dilation angle equal to the friction angle whereby associated flow is attained.

Choosing any other dilation angle ψ < β will result in non-associated flow. In this condition the plastic strain increment is still assumed normal to the yield envelope in the deviatoric plane, but at an angle ψ to the q-axis in the p-q plane as shown in Figure 3-2.

Choosing ψ = 0 will cause the inelastic deformation to be incompressible, whereas the material dilates for ψ ≤ β, hence the reference to ψ as the dilation angle.

For grout materials this dilation angle is typically small, say 10° to 20° implying non-associated flow should be used in the modeling.

Figure 3-2 Drucker-Prager yield surface illustrated in principal stress space and in the meridional p-q plane. Also, in the p-q plane the geometric interpretation of the dilation angle ψ is shown for hardening

As shown in Figure 3-2 the standard linear Drucker-Prager model assumes a circular yield envelope in the deviatoric plane akin to the Mises yield assumption and a linear variation with the hydrostatic pressure. Grout materials in general do however not conform well to this, in that these exhibit a both a nonlinear variation with the hydrostatic pressure, and a non-circular yield envelope in the deviatoric plane, i.e. a Lode angle dependency.

The hydrostatic pressure dependency can be addressed by e.g. a hyperbolic or a power-law extension of the linear variation in the meridional p-q plane described in [A.2.1.4].

The Lode angle dependency is typically addressed by the introduction of a Lode angle dependent alternative deviatoric stress measure expressed as

(3.2)

(3.3)

(3.4)

= + tan

= − tan

pl

pl

Hardening

= 2 1 + 1 − 1 − 1 3


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where q is the equivalent von Mises stress and K is a shape parameter for the failure envelope in the
deviatoric plane that – to ensure convexity of the yield surface is confined to .

The Drucker-Prager yield criterion of Eq. (3.2) is thus simply reformulated using the deviatoric stress measure t instead of the equivalent stress q, and it becomes simply

The term resembles the Lode angle θ in the deviatoric plane in that per definition. Hence the variation of t is linear with (see [A.1.6]).

Using a shape parameter K = 1 implies t = q and thus recovers the original circular trace of the yield envelope in the deviatoric plane (see Figure 3-3).

Figure 3-3 Drucker-Prager yield surface extension illustrated in the meridional p-q plane and the deviatoric plane

It is recommended that if the Drucker-Prager material model is selected for the analysis of grouted connections, that the extended formulation using the deviatoric stress measure t is used, and that the shape parameter K of the yield envelope in the deviatoric plane is calibrated based on strength data for the compressive and tensile meridians of the grout.

Examples of stress states on the tensile meridian are uniaxial tension and biaxial compression. On the compressive meridian stress states such as uniaxial compression and biaxial tension resides.

It is recommended to fit the yield surface using the uniaxial tension and compression strengths together with the biaxial compression strength. In lieu of detailed stress data the biaxial compression strength is normally between 1.10 to 1.20 times the uniaxial compressive strength.

It should be noted that even in this extended version of the Drucker-Prager material model it will still be difficult to get a tight match of the tensile region without compromising the compressive strength of the model. Hence, if a tight matching of tension is needed, it is recommended to use either the 5-parameter Willam-Warnke material model or the Lubliner-Lee-Fenves material model.

3.2.4 Willam-WarnkeThe yield surface of the Willam-Warnke material model is described in detail in [A.2.3].

The basic 3-parameter model is akin to the previously described extended Drucker-Prager material model, why it is not commonly available in commercial FEM software. The extended 5-parameter model is however adopted in the Ansys FEM software.

The specific implementation of the criterion in the Ansys FEM software is also described in [A.2.3] and will in the present context be taken as the practical implementation that can be used if Ansys is the chosen analysis software and is hereafter referred to as the Ansys concrete model.

It is important to recognize that the Ansys concrete model is a failure criterion based on the 5-parameter Willam-Warnke yield criterion rather than an implementation of said criterion as a yield surface with plastic flow rule. Moreover, it should be noted that the Ansys implementation is a discrete approach that does not ensure continuity of the entire failure surface as illustrated in Figure 3-4 and explained in detail in [A.2.3.2].

(3.5)

0.778 ≤ ≤ 1.0

= + tan ( / )3 cos(3 ) = ( / )3 cos(3 )

Deviatoric plane Drucker-Prager (Mises)

=0.8

Curve

Tensile meridian

Compressive meridian


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As a failure criterion rather than an actual yield criterion, the typical application of the Ansys concrete model
will be to assume a linear elastic behavior up until failure where after all stiffness is lost. The loss of stiffness is abrupt in the Ansys implementation, why the model is prone to convergence issues.

The shape of the failure surface in triaxial compression is governed by the choice of two additional compressive strengths specified at a high level of hydrostatic pressure. These are by Ansys suggested to be taken as the uni- and bi-axial compressive strength at a hydrostatic pressure equal to the uniaxial compressive strength, and in lieu of actual data are suggested taken as 1.45 and 1.725 times the uniaxial compressive strength respectively.

The shape of the failure criterion shown in Figure 3-4 is based on these recommendations.

Calibration of the model is discussed in further detail in [A.2.3].

Figure 3-4 Yield surface of the original 5-parmeter Willam-Warnke model shown left and the Ansys concrete model shown right. Both are shown in principal stress space with the intercept of the plane stress condition indicated in dashed line, together with the meridians in outline border line. The discontinuous gap zone of the Ansys concrete model between the tension-compression-compression region and the tension-tension-tension region is shown in thin red

3.2.5 Lubliner-Lee-FenvesThe yield surface of the Lubliner-Lee-Fenves material model is described in detail in [A.2.2]. The implementation of this criterion is closely tied to the Abaqus FEM software where it is referred to as the concrete damage plasticity model.

Unlike the Ansys concrete model, the damage plasticity model is a full material model with a yield criterion and subsequent plasticity. It further has the capability to model progressive stiffness degradation making it ideal for cyclic assessments.

The yield surface of the damage plasticity model is unlike the Ansys concrete model insured continuous in the entire stress space. It is in shape akin to the Ansys concrete model except for the triaxial compressive region. Here a linear variation along the meridians is assumed the slop of which is given by the parameter Kc that can assume any value between 1/2 and 1.

The implication of the Kc parameter on the yield surface is shown in Figure 3-5. The recommended default value for the Kc parameter is 2/3 for which a very close resemblance with the failure surface of the default Ansys concrete model previously shown in Figure 3-4 is achieved.

Both the damage plasticity model and the Ansys concrete model exhibit a near perfect match to a Rankin tension cutoff assumption, why both these models are superior in their representation of tensile cracking.

If cracking of the grout is the primary focus of the FEA, it is therefore recommended to use one of these two material models.


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Apart from the uniaxial strengths, the damage plasticity model yield surface is defined by the biaxial
compression strength and the previously described Kc parameter.

In lieu of detailed stress data the biaxial compression strength is normally between 1.10 to 1.20 times the uniaxial compressive strength, and the Kc parameter may be taken as 2/3 for grout that is moderately confined. If a high degree of confinement exists in the structure a lower Kc value may be needed.

Figure 3-5 Damage plasticity model yield surface illustrated in principal stress space for the lower and upper limit of the shape parameter Kc together with the default value of 2/3. The relative scale between the two yield surfaces is accurate. Moreover, the intercept of the plane stress condition is shown in dashed line, together with the meridians in outline border line

The plastic flow potential g used in the concrete damage plasticity model is a hyperbolic extension of the Drucker-Prager flow potential given as

where q is the equivalent von Mises stress, p is the hydrostatic pressure (positive in compression), and σt is the tensile yield stress. Specific to the flow potential is then is the dilation angle ψ measured in the p-q plane at high confining pressure and ∈ commonly referred to as the eccentricity, that defines the rate at which the function approaches the asymptote (the flow potential tends to a straight line as the eccentricity tends to zero).

The flow potential eccentricity ∈ is in practice a small positive number that defines the rate at which the hyperbolic flow potential approaches its asymptote defined by the dilation angle ψ.

The default flow potential eccentricity is ∈ = 0.1, which implies that the material has almost the same dilation angle over a wide range of confining pressure stress values. Increasing the value of the eccentricity ∈ provides more curvature to the flow potential, implying that the effective dilation angle increases more rapidly as the confining pressure decreases.

3.3 Selection of material model and propertiesThe Lubliner-Lee-Fenves and Willam-Warnke material models are considered superior in detail why either is recommended for general application. Second to these is the extended Drucker-Prager material model with Lode angle dependency.

The use of the classic linear Drucker-Prager material model is discouraged.

Pertaining to material properties, it is generally recommended to use characteristic values. In general that means 5% fractile in case a low resistance is unfavorable and 95% fractile in case a high resistance is unfavorable. Fractile magnitudes should be used in accordance with the governing LFRD standard see further [2.1].

(3.6)

= ( t tan )2 + 2 − tan


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It is advised that the upper fractile of material properties may be most onerous, in that e.g.
— low tensile strength may through excessive cracking lead to favorable redistribution of stresses— low modulus of elasticity can have a beneficial impact on contact pressure— low crushing strength may lead to favorable redistribution of stress in steel.

It is therefore not generally possible to analyze all aspects of a grouted connection based on just one characteristic material model.

Depending on the type of connection and the response investigated, the impact of a weak or strong grout model may be more or less pronounced.

It is therefore recommended that the response based on a weak model is establish first and scrutinized for any possible un-conservatisms due to favorable stress redistribution. If based on this, such behavior cannot be ruled out it is recommended to assess the same limit state condition based on a strong material model.

Guidance on statistical determination of characteristic values based on laboratory testing is given in e.g. DNV-RP-C207 /3/ or EN 1990 /11/.

3.3.1 Specifying nonlinear propertiesGeneral engineering practice is to specify material properties as engineering – or nominal – quantities. That is, as capacities relative to a constant reference specimen. Hence, we have strengths in terms of engineering stress i.e. force F per original undeformed reference area A0, and engineering strain as , i.e. elongation being the current length l minus the original undeformed reference length l0 relative to again the original undeformed reference length l0.

In reality when a material is loaded it will strain incrementally with the application of the load and thus deform. The true strain accounts for the fact that the reference length is continually changing by defining a strain increment based on each small change in length dl relative to the current length l, and the defining the total strain as accumulation of these strain increments i.e.

The corresponding true stress is defined as or simply the force F per current deformed area A.

Assuming that the material volume remains constant, i.e. the true stress σ and strain ε can be related to the engineering stress s and strain e as1

For very small deformations, within the elastic range say, the cross-sectional area of the material undergoes negligible change and both definitions of stress are more or less equivalent. However, for large deformations the effect of accounting for the deformed body is significant.

Hence, as it is precisely the true stress and strain definition that is used at least internally in any finite element method, it is important that nonlinear material strength data is input as true stress and strain quantities, i.e. transformed from nominal or engineering measures to true do using Eq. (3.8).

1

(3.7)

(3.8)

= / 0 = ( − 0)/ 0

= /

= 0 = ln ( 0) = /

0 0 =

= (1 + ) and = ln (1 + )

= = 00 = 0 0 = 00 0 = ⇒ 0 = 0 = − 0+ 00 = − 00 + 1 = 1 + ⇒ = (1 + ) and = ln 0 = ln − 0+ 00 = ln 1 + − 00 = ln(1 + )


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SECTION 4 CONTACT INTERACTIONS
In finite element analysis the inclusion of contact interaction requires the handling to two principal problems, namely penetration and sliding. The methodology for this is typically either an exact Lagrange method or an approximate penalty method as explained in App.B.

4.1 Contact modelingDepending on the software used for the analysis various specialized implementation of the basic contact interaction methodologies, i.e. Lagrangian or penalty, will typically both be available for use in the analysis if an implicit formulation is used. Explicit formulations will be limited to the penalty method (see further [5.1]).

It is recommended to use the penalty method due to its robustness and computational efficiency.

A surface-to-surface approach is further recommended over the basic node-to-surface approach.

As the penalty method is not exact, it is important that it be calibrated to the accuracy needed. This is important as high accuracy will be at the cost of additional computational effort. As the size of the areas with contact interaction is substantial it is important to get the penalty stiffness scaled appropriately for the type of grouted connection being analyzed.

Pertaining to contact calibration it is important to consider that contact accuracy affects not only the relative movement but also the straining and stressing of the individual bodies. Hence, while a distance of say 0.1 mm may in general be considered sufficiently accurate for displacements given the general size of typical grouted connections, as an ‘error’ on the contact enforcement it would in terms of stresses and strains general be unactable.

4.1.1 Normal contactFor the basic contact interaction in the direction normal to the surfaces, it is the amount of overclosure or penetration that is to be limited to attain an acceptable level of accuracy.

Areas of particular interest are the top and bottom most regions of the grout body where bending and shear loads acting on the connections have their peak effect and of course at the shear keys if present.

If in these areas, a too weak penalty stiffness is used the resulting overclosure will potentially mask any real peak or concentration in contact pressure, and subsequently in the derived stressing of the grout.

Particularly for shear keys, it is important that a sufficiently strict enforcement of normal contact is attained in the model. See also [5.3.2.3].

The part of a grouted connection that is affected by bending and shear may be estimated using the beam on an elastic foundation, or Winkler foundation, as an analog. Doing so, the elastic length may be taken as a measure for how large a region that is affected by bending and shear loads. Typically, half this length will be considered to be significantly affected by bending.

For a grouted connection the elastic length le for any of the two steel tubulars can be taken as

with E being the elastic modulus of steel and I the moment of inertial for the tubular in question. The foundation spring stiffness can be estimated based on the radial stiffness of the combined cross section as

where R is the radius of the tubular in question, being then either Ri for the inner tubular with thickness ti or Ro for the outer tubular with thickness to. The grout has the thickness tg and elastic modulus Eg, and the coefficient of friction between the steel and grout is μ.

(4.1)

(4.2)

e = 4 rD4

rD

rD = (2 + 3 )i2i + o2o + gg


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4.1.2 Sliding contact
It is acknowledged that grout does exhibit an ability to bond to a steel surface and that some surface imperfections and general irregularities will always exist in real structures which will give rise to some – at least initial – passive shear capacity of the grout-steel interface. However, cyclic loads integral to offshore structures are considered very likely to erode any initial passive shear capacity, why design standards such as DNVGL-ST-0126 /9/ explicitly excludes relying on any passive shear capacity in the design of grouted connections. It is therefore in general recommended not to include any passive shear capacity associated with the plain steel-grout interaction.

Although the loading of an offshore structure is in general dynamic, it is typically occurring at a frequency low enough to make a quasi-static assumption valid for grouted connections. Hence, it is generally recommended to use a static Coulomb friction model for the sliding interface.

The coefficient of friction should be based on test data for the specific type of grout being used in the design and should account for potential water lubrication effects if the connection is submerged.

Moreover, due consideration for variations in the steel surface conditions for the actual connections, and the possibility of polishing wear over time due to the inherent dynamic nature of the loading, should be exercised and reflected in the analysis assumptions for the connection.

This is of particular importance for connections exposed to bending loads, e.g. connections in monopile foundations or pre-piled jackets with significant pile stick-up.

Friction coefficients should be chosen carefully as they impact the predicted stressing of the grout body.

Assuming plane stress for the grout, i.e. , and that the contact pressure σp together with the shear it gives rise to by friction, are the only forces acting on the grout, i.e. , , and , then the principal stresses can by application of Mohr’s circle for plane stress be derived as:

with σI representing the maximum compressive stress in the grout, and σII the maximum tension.

It is difficult to generally quantify the impact of a lower and upper bound friction as the equilibrium between the external loading and the resulting contact pressure and shear for a given coefficient of friction varies.

It is therefore recommended to assess the stressing of the grout body assuming both a lower bound friction coefficient typically a 5% fractile and an upper bound friction coefficient, typically a 95% fractile, i.e. to conduct two separate analyses for the same loading to assess it safely.

4.1.2.1 Special considerationsIf the FEM software used for the analyses support smoothing of contact surfaces based on underlying geometry, e.g. cylindrical smoothing, it is generally recommended to make use of this in the analysis as it will improve the accuracy of the interface sliding behavior.

4.1.3 CalibrationIt is generally recommended that, unless a pure Lagrangian contact is used, the stiffness of the penalty contact is initially calibrated. The calibration should be conducted prior to any FEA assessment of the connection and is recommended included as part of the documentation.

In general it is the stressing of the grout and steel that is the principal quantity sought by FEA, why it recommended using convergence of the predicted maximum compressive and tensile stress, i.e. the minimum and maximum principal stress, in the grout as a tool for calibrating the penalty stiffness.

For all connections exposed to bending and/or shear loading it is recommended to conduct this convergence study on a plain cylindrical grouted connection representing typical dimensions equal to those of the actual connection being investigated both in geometry and maximum loading.

It is recommended to use a fictitious 2 times the diameter of the connection as the grout overlap to ensure a reasonable representation of bending interaction at top and bottom of the connection.

(4.3)

= 0 = p = 0 = p I, II I, II = +2 ± −2 2 + 2 = 12 p 1 ± 1 + 4 2


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The model is recommended to be meshed using the same element types and sizes as planned for the use
in the modeling of the actual connection being investigated. Recommendations on element types and mesh densities are given in [5.4].

As explained the magnitude of the assumed coefficient of friction has an implication on the split between compressive and tensile stressing in the grout. It is therefore recommended that the convergence is studied firstly with a frictionless contact formulation where the normal contact is calibrated based on the maximum compressive stress predicted in the grout, followed by a high friction model where the sliding contact is calibrated based on the maximum tensile stress predicted in the grout.

For connections with shear keys it is further recommended that a simplified geometry representing at least 3 sets of shear keys be analyzed for convergence of the predicted maximum compressive and tensile stress in the grout when exposed to axial loading alone.

As bending and shear loads are probable for any practical application of grouted connections, it is the general recommendation that the proposed plain bending calibration is conducted always as the first step and that the hereby calibrated contact formulation is then subsequently checked for the axial load on the shear key model.

Pertaining to this, it should be noted that although an axially loaded shear key connection in principle can be simplified to a 2D rotational symmetric model, it is not considered reasonable to use this simplification if the final model is to be analyzed in full 3D. This primarily motivated by differences in the implementation of contact in 2D and 3D.

Finally, if an explicit FEM formulation is chosen for the analysis, additional care should be exercised when selecting the contact penalty stiffness, as a high stiffness in this formulation make the contact interface prone to contact chatter where a slave node when approaching contact constantly changes from overclosed, then severely corrected by a high penalty, rendering it open, which is also wrong why it is then moved back to being overclosed, and on and on, i.e. chattering.

This is to be avoided as it not only introduces significant noise in the solution; it potentially causes overestimation of stresses in the grout and is further likely to hurt the stable time increment causing unnecessary running times for any analyses. See further [5.1.1].


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SECTION 5 FINITE ELEMENT METHOD MODELING
Performing FEA of grouted connections requires choices additional to just the material and contact formulations already discussed.

Suitable software has to be chosen for the analysis. The software should be tested and documented suited for the nonlinear analysis. For general analysis of grouted connections the minimum requirements to the capabilities of the software would be:

— nonlinear material behavior, i.e. yielding and plasticity— nonlinear geometry behavior, i.e. stress stiffening and second order load effects— frictional contact interaction.

With a software chosen, the task is then to select a solution- and modeling scheme for the analysis based on an idealized representation of the actual geometry. This idealization requires choices pertaining to the geometric extent and element formulation.

5.1 Solution schemesIn FEM the general equation of motion for a continuum can at any time t be formulated as

where is mass matrix, is the damping matrix, the stiffness matrix, is the vector of the external forces, and is the displacement vector. Hence, expressing that the sum of inertia, damping, and internal forces due to the motion are to equated the external forces for any given time.

In FEA solutions of this general equation of motion is traditionally referred to as dynamic solutions as opposed to static solutions where there is no dependency on time, why the equation of motion reduce to simply an equilibrium between internal and external forces or

Recall that FEM is a generalized numerical technique for finding approximate solutions to boundary value problems for partial differential equations. If these like the general equation of motion in Eq. (5.1) are time-dependent, then these can be solved using either the explicit or implicit method.

5.1.1 Explicit and implicit solution methodThe explicit method directly calculates the state of a system at a later time from its current state, while the implicit method solves an equation involving both the current and later state of the system.

Let Y denote the system, then expressed mathematically, Y(t) is the current state and is the state at a later time. For the explicit method the solution is then simply whereas for the implicit method the equation is solved to find .

In FEA more computations are required to solve the equation of the implicit method than doing the updating associated with the explicit method. In dynamic analyses, explicit solvers are therefore attractive for large equation systems, as the solution scheme does not require matrix inversion or iterations, and thus, is much more computational effective for solving the same time step than solvers based on the implicit scheme.

However, the stiffer the system is, the smaller the time step of the explicit method needs to be for the system to remain stable and the error on the result bounded, whereas the implicit method does not suffer from this in that it is unconditionally stable for large time steps.

In fact, it is not just the stiffness that influences the maximum stable time step of the explicit solution scheme. The issue of a stable time step may be thought of in the following way.

In a continuum, the effect of a load (stresses and strains) travels with the speed of dilatation c in the material (the speed of sound). Because the future state is directly calculated from the present one, for the

(5.1)

(5.2)

( ) + ( ) + ( ) = ( )

=

( + Δ ) ( + Δ ) = ( ( )) ( ), ( + Δ ) = 0 ( + Δ ) (∙) = 0 (∙) Δ

Δ


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effect of any change to be captured in any and all nodes of the model, the time step need to be smaller
than or equal to the time it takes to travel the shortest distance lmin between two nodes in the continuum.

The speed of dilatation in a continuum is with E being the modulus of elasticity and ρ the specific density. Hence, the shortest distance lmin between two nodes in the continuum will be covered in time of lmin/c, why the stable time step needs to be .

In general therefore, the inherent requirement of a very small time step makes the explicit solution scheme well suited for analysis of short duration transient loadings as seen in e.g. impact scenarios. For transients of longer durations the number of time increments will however be much larger than that for an implicit solution scheme.

5.1.1.1 Nonlinear problemsWhen expanding the FEM to include nonlinear behavior in any form, the balance between the two solution schemes is altered. In principle, nonlinearities will have no effect on the stable time step of the explicit solution scheme, provided that it does not entail excessive changes in stiffness from either material behavior or contact interaction.

The implicit scheme on the other hand needs to be expanded by typically the Newton-Raphson algorithm to handle the nonlinear behavior. This adds additional iteration solutions to each time increment and thus to the computational cost of the implicit scheme.

For moderately nonlinear problems, an implicit Newton-Raphson solution scheme will however likely still require less computational effort than the explicit solution for analyses not involving rapid transients.

Excessively nonlinear problems may however defeat the implicit Newton-Raphson solution scheme rendering the explicit solution scheme the only viable option for analyzing the case irrespectively of the computational cost.

5.1.1.2 Practical considerationsWhile inertia loads are important generally for the global response of offshore structures and in particular for wind energy applications, for the grouted connection inertia effect is normally ignored locally and included solely through the application of load effects from a global dynamic analysis of the foundation structure (see further [6.2]).

It is therefore in general sufficient to use static or quasi-static approaches to analyze the detailed local behavior of a grouted connection.

A truly static approach will necessitate the use of the implicit solution methodology, whereas quasi-static and for that matter fully dynamic analyses can be conducted using both implicit and explicit solution schemes as previously described.

If the explicit solution methodology is used in a quasi-static analysis, a number of options exist to speed up the analysis. Firstly, as time is introduced artificially to an essentially static condition, the duration of the event is essentially arbitrary. The time needed to complete an explicit analysis is directly proportional to the duration of the event being analyzed hence the quicker the event the faster the analysis will be to complete. Alternatively, the maximum time step may be tweaked either by limiting the shortest distance between two nodes or by reducing the speed of dilatation via mass scaling.

The first approach is typically impractical as element sizes in general are dictated by geometry and requirements to result accuracy in areas of interest. Scaling the mass is therefore generally the only practical method for accelerating the analysis. Scaling the mass will increase the inertia effects, but as the analysis is assumed quasi-static, these are per definition insignificant, why it is an acceptable tweak.

Caution should however be exercised when selecting a mass scale factor, as too large a scaling will inevitably, even in a quasi-static framework, introduce inertia that will impede the flow of forces to an extent where the results may become erroneous.

If a quasi-static analysis is to be conducted using the explicit solution methodology, it is recommended to choose the duration of the event close to the actually expected duration, say half the natural period of the structure. Mass scaling can then be used to speed up the analysis if needed.

If mass scaling is used, it is recommended to continue the analysis beyond the time where the desired loading is applied keeping the loading constant. This will allow for checking that a static equilibrium has been achieved by the quasi-static solution.

Δ

= / Δ ≤ min /


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Further, it should always be checked that the kinetic energy is small compared to the deformation energy
say less than 1% to have confidence in an explicit quasi-static solution.

5.1.2 Choosing a solution schemeHistorically, implicit and explicit FEM solvers have evolved separately and with different focus areas for their primary application. When having to choose between the two solution schemes, the choice is often impaired by a sometime significant difference in available features, e.g. material models and element types.

Apart from the issue of available features, there is the also the more practical issue of the computational effort needed to complete the analysis. Explicit solvers typically require far less memory and scale significantly better over multiple CPUs than implicit solvers. On the other hand, explicit solvers typically require far more time steps than an implicit solver.

It is generally recommended to use a static implicit solution scheme if possible. Only if truly transient load cases such as ship impact are to be assessed is a dynamic solution recommended be it using the implicit or explicit solution scheme.

5.2 Modeling schemesIt is generally recommended to use a full 3D modeling approach based on 3D continuum elements and surface contact interaction.

Idealized modeling approaches based on e.g. compression struts are possible as a simplified description of the steel-grout-steel interaction as described in e.g. /4/ presenting the underlying assumptions for the analytical design formulas of DNVGL-ST-0126 /9/ and in /19/ presenting and comparing a compression strut method with a full 3D nonlinear modeling.

These modeling schemes are however only applicable for grouted connections with shear keys uniformly distributed over the entire height of the grouted connection, why they cannot be generally recommended.

However, for the subset of grouted connections with shear keys over the entire height – typically jacket pile-sleeve applications or pre-piled jackets – the compression strut methodology do pose a simplified alternative to a full 3D modeling.

Because of the close tie with the analytical design formulas of DNVGL-ST-0126 /9/ using a compression strut idealization of the grouted connection is likely to yield results comparable to said analytical approach.

5.2.1 SubmodelingA submodel is in the present context taken to be a detailed FEM model of a reduced portion of a larger previously analyzed FEM model which response is used to drive the submodel.

In a sense then, all analyses of grouted connections for offshore application will likely be a submodel as they are typically designed for loads derived as cross sectional forces from a global model of the entire offshore structure (see further [6.2]).

For large structures with small critical features the submodeling scheme is generally attractive. It is however ill suited for cases where contact interaction exist across the cut face between the submodel and the global model. This primarily due to potential sliding in the contact interface which may cause a body to ‘fall of the edge’ of the surface it is in contact with, within the confines of the reduced geometry of the submodel.

It is therefore not recommended to attempt submodeling within the grouted connection itself. It may however serve as an excellent method for gaining detailed insight to e.g. fatigue prone attachment to the steel where a submodel can be defined within the confines of one continuum, or where the entire contact condition can be contained within the submodel, e.g. contact between temporary support brackets of say a jacket leg and the rim of the pile it is stabbed into for a pre-pied jacket.

5.3 Geometric extentAny grouted connection is considered to have three principal components: Outer tubular, grout, and inner tubular. Additional to this are a number of possible internal features such as spacers and shear keys.

Externally, the grouted connection may also have a number of features attached to it, e.g. boat landing


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fenders and ladder for monopile foundations. And there may be additional structural components attached
to one of the tubulars in close proximity to the grouted connection, e.g. lattice bracing in a jacket foundation.

The implication of all of these features in terms of possible effect on the grout and steel in the grouted connection should be carefully assessed and if significant included in the model.

5.3.1 Overall geometryThe full extent of the grout body and the surrounding steel obviously shall be included in the model, as should the steel above and below the grout body to an extent where local disturbances from both the grouted connection and the assumed boundary conditions are insignificant for the predicted response of the grouted connection.

It is the general recommendation for steel tubulars that steel up to a minimum distance of 3 times the diameter D between submodeling cut faces and the location of interest to the analysis is included in the modeling. For a simple grouted connection geometry, e.g. a monopile foundation, the cut face should then be placed at a distance 3 times the outer diameter of the transition piece above the grout body and equally 3 times the distance of the monopile below the grout body.

For grouted connections surrounded by more complex steel work as e.g. vertical pile-sleeve connections in jackets, or pre-piled jackets, it is recommended to include any primary braces, brackets, etc. if attracted to grouted connection tubular within a distance of 3 times the diameter of said tubular, and then included a minimum of 3 times the diameter of each individually attached member.

Within the grouted connection it is generally recommended to include the full extent of grout body consider to be effective in the connection.

Design standards for grouted connections typically consider a grouted connection to have an effective length shorter than the actual cast length to allow to potential weak grout at each end of the cast.

The prevailing approach is to reduce the cast length of the grouted connection by whichever is the largest of 2 times the nominal grout thickness or one shear key spacing if these are present.

While the concern of weak grout is valid and a reduced effective length is a reasonable approach within the context of design formulas, for FEA some considerations are needed before simply transferring these reductions onto the modeled extent of the grout body.

Firstly, it is evident that if the sole purpose of the FEA is to assess the performance of only the grout body, then any reduction in the size of said grout body will be a conservative approach. However, if the FEA is used to also assess the performance of the steel in the vicinity and surrounding the grout body, then fictitiously removing grout from the connection will change the stress distribution in the steel, perhaps in a way that will cause elevated stressing at fatigue prone welds or attachments near the grout. The same concern is relevant for e.g. buckling assessments.

In general is should be noted that variations in not only the extent of the grout body but also in the assumed stiffness of the grout, will have influence on the deformations and stresses predicted by the FEA, why it is generally recommended that a design envelope on the extent and stiffness is analyzed.

It is recommended that for FEA assessment of the grout body, the extent of said body is taken to vary with 2 times the nominal grout thickness tg taken equal at the top and bottom, i.e. that the grout is assumed either at its nominal length or with the nominal thickness of the grout tg removed at top and bottom. Reduction of the grout body extent based on shear key spacing is not recommended in FEA.

For FEA assessment of the surrounding steel, it is recommended that additional to the variations in grout extent described above, the stiffness and strength of the grout is also varied. See further [3.3].

5.3.2 FeaturesIt is generally recommenced to carefully consider and include any feature or welded attachment to the steel in the immediate vicinity, say 1 times the diameter of the grouted connection, to facilitate fatigue assessment based on accurate detailed stress predictions.


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5.3.2.1 Thickness tapers and weld reinforcements
Thickness tapers should from a design point of view always be made such at the change in thinness occur on the side of the steel not interacting with the grout, thus leaving the interface flat and unaffected by the taper. Likewise weld should – irrespectively of taper – be ground flat to avoid unintended shear key like effects in the grouted connection.

If weld reinforcements are left un-grinded these should be included in the model geometry and are recommended model to the same level of detail as an actual shear key.

Tapers should always be included in the model, and if these are placed such that they affect the interface it is recommended that they are also modeled to the same level of detail as an actual shear key.

5.3.2.2 Internal spacesSpacers are typically used to ensure a minimum thickness of the grout and to guide the installation. Spacer may also be present to provide protection of grout seals and possible rebars. These are generally of no significance for the overall response of the grouted connection, why they normally may be ignored in the FEM model.

Special consideration should however be paid to spacers locally if they are in direct steel-to-steel contact in a connection carrying significant bending load. Here, lacking the comparatively larger flexibility of the grout, contact forces will be focused on the spacer in turn leading to stress concentrations that may prove of significance for the steel in terms of fatigue behavior and buckling resistance.

5.3.2.3 Shear keysShear key are typically constructed using a number of layered weld beads shown idealized as a half-circle in illustrations and on design drawings.

Shear keys should be included at their nominal spacing and minimum height. When modeling shear keys it is important that an angle reasonable smaller than 90° is introduced between the wall and the shear key to safeguard against numerical over-constraints of the contact interaction.

For weld bead shear keys it is recommended that the geometry is based more on the actual combined profile of the weld beds rather than the idealized half circle shape used e.g. design drawings. See further Figure 5-3, in [5.4.2].

5.3.2.4 RebarRebars may be used in grouted connections to increase the tensile capacity of the grout body. This is common in e.g. vertical pile-sleeve connections for jacket structures if a short overlap is desired.

If rebars are present in the design it is recommended that these be included in the modeling as discrete bars using e.g. beam elements. It is recommended to embed the rebar elements in the grout rather than attempt modeling contact interaction between rebars and grout.

Rebars are typically welded to the sleeve. In this case it is recommended that only the part of the rebar not welded to the sleeve is embedded in the grout, and that the welded part is attached to sleeve either by match meshing or by kinematic node-to-node constrains. This way the overall contact interaction between sleeve and grout may coexist with the rebar modeling without causing over-constraints that would impede the solution.

5.3.2.5 SealsSeals are generally required to keep the grout in place while it is being casted offshore. The exception being pre-piled jacket foundations where the internal soil plug normally serves as the lower barrier for the grout.

Seals can be either inflatable or more commonly steel reinforced rubber lips. Irrespectively, the seals are typically more flexible than the grout, why they are generally safe to ignore.

Provided that the seal have significantly lower compressive stiffness than the grout, it is recommended that the seals are ignored in the modeling and that the nominal lower extent of the grout be take equal to the expected upper elevation of the seal with due consideration of potential settlement resulting from the seal carrying the full weight of the uncured grout.

5.3.2.6 Temporary supportsInherent to the grouted connections is the need for temporary support during offshore installation. The


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strength of a grouted connection may be compromised by early age movement, why until the grout has
cured and attained sufficient strength, it is imperative that the weight and loading on e.g. the transition piece of a monopile foundation is carried by temporary supports to an extent sufficient to keep it from moving.

It is innate to the grouted connection that it will exhibit some relative movement between the two steel tubulars when loaded, why it is important that after the grout has cured sufficient play is left between the temporary support and the tubular it rested on during installation. This is typically achieved using shims that are removed after then grout has cured.

If this cannot be achieved, the FEA should account for the load path through the temporary supports.

Irrespectively, the attachment of the temporary supports to the primary is likely to be close to the grout, why although they will not affect the performance of the grout body, the local stressing at the support will be affected by the grouted connection. It is therefore recommended to include the supports in the FEM model to allow of an accurate assessment of the fatigue performance of the attachment welds.

5.3.2.7 ExternalsExternal features of the primary steel should be included to the extent where it either impacts the stressing of the grout body or the response of the grouted connection affects the stressing of the steel.

Examples of primary steel features that should be included are:

— joke plates and shear plates or bracing attached to vertical pile-sleeve connections for jackets

— bracing attached to the led for traditional pile in leg jackets

— bracing attached to the leg in pre-piled jackets.

For grouted connection placed in the immediate vicinity of the seabed, the effect of the soil interaction should also be considered. Albeit typically considered a boundary condition, for pile-sleeve connections and in particular for pre-piled jackets connections, the local resistance of soil may be significant for the stressing of the grout body. It is therefore recommended to include the lateral soil support in the grouted connection model either directly as a continuum model having contact interaction with the pile, or as an elastic support in cases where the connection placed in the immediate vicinity of sea bed in stiff soil and/or it is exposed to significant bending.

Depending on the type and location of the grouted there may also be a number of secondary structures attached externally to foundation in the region of the grouted connection.

Provided that these are either sufficiently flexible or local, these will in general not affect the performance of the grout body within the grouted connection.

However, as with the temporary supports, the effect of the grouted connection in terms of stresses at these welded attachments may affect the fatigue performance of said welds.

It is therefore generally recommended to include any stiff secondary structure in the FEM model to facilitate detailed and accurate fatigue predictions of all welds onto the primary steel of the grouted connection. The zone in which the grouted connection has a notable influence on the stress in the steel may be taken as 1 times the outer diameter of the respective primary steel tubulars above and below the grout body.

An example of secondary structures to carefully consider are e.g. boat landings and possibly external J-tubes for monopiles placed near the mean water level.

5.3.3 Tolerances and imperfectionsTolerances falls in two classes: Fabrication and installation. Assuming fabrication is performed according to the requirements of the governing standard (see [1.2]) the model may be built using the nominal geometry. In the following tolerances are therefor take to refer solely to installation tolerance.

Tolerances should be included in all limit states. Only exception is the SLS assessment of expected long term settlement particular to conical grouted connections in monopile foundations (see further [7.4]).

Tolerances to include are horizontal, vertical, and inclination eccentricities as illustrated in Figure 5-1.


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Figure 5-1 Horizontal, vertical, and inclination tolerance for grouted connections

Horizontal tolerances are to be taken as the largest possible deviation for the ideal concentricity of the two steel tubulars. For grouted connections in monopiles this is typically well controlled as the transition piece can be positioned with relatively high accuracy. For traditional connections with piles driven either internally in the jacket leg or through a pile sleeve the concentricity is typically less accurate. Pre-piled jacket connections do however typically need to allow for a significant horizontal tolerance primary due to limitations associated with the driving of the piles.

Vertical tolerances are to be taken as the largest possible deviation in the relative axial positioning between the two steel tubulars. This impacts the height of the grout body for all types of grouted connections. For conical connections if further impacts the thickness of the grout. This is not the case for cylindrical connections but here it changes the relative positioning of the shear keys. The vertical tolerance is chiefly related to uncertainties in the driving of the piles down to the desired penetration depth.

Inclination tolerances are to be taken as the largest possible relative inclination between the two steel tubulars. The principal impact of this tolerance is on the grout thickness but it also affects the relative positioning of the shear keys if present. The latter being more pronounced for large diameter connections.

For traditional connections with piles driven either internally in the jacket leg or through a pile sleeve the relative inclination is typically small due to the use of internal guides in the leg or sleeve. For pre-piled connections be it jackets with legs stabbed into the piles or monopiles with transition pieces positioned after the driving of the pile, the inclination tolerances are typically bigger.

Together these three tolerances form an envelope on the possible final size and shape of the cast grout body within the grouted connection which implications should be assessed in the FEA.

The principle implications of the three tolerances are:

— additional local bending— thickness variation of the grout— angle variation in the compressive struts between shear keys if present.

It is important to recognize that each and any of these configurations are to be equally acceptable if occurring in the actual structure, why it is necessary that the design assessment encompasses all combinations of these tolerances.

In general however, it is normally the extreme grout thickness configuration that together with axial offset of shear keys – if present – forms the worst case(s).

It is recommended that as a minimum, the configuration leading to the thinnest grout be assessed for loading applied such as to produce maximum compression and maximum tension in the grout. This will normally coincide with maximum bending bearing down on the thinnest grout location.

For axially dominated connections with shear keys it is further recommended that the maximum axial offset without any other tolerance be assessed for peak downward load followed by peak load reversal, i.e. maximum push down onto the pile followed by maximum pull up from the pile.

- +

-

+

- +


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Finally, it should be added that tolerances affect not only the grout body itself, but also the steel in that e.g.
girth welds between cans may end up closer to the grout due to vertical tolerances and thus may be exposed to additional local bending. This should be considered in the fatigue assessment of the steel.

5.3.3.1 ImperfectionsImperfections need only be considered in relation to buckling assessment. The size and extent of the imperfections should conform to the requirements of the chosen governing standard.

The fabrication tolerances given in e.g. Eurocode 3 /12/ or DNVGL-OS-C401 /5/ are recommended and conform to the requirements of e.g. DNVGL-ST-0126 /9/, Norsok N-004 /17/, and ISO 19902 /16/.

For tubulars three types of imperfections are normally considered: Out-of-roundness, eccentricity, and out-of-straightness. According to Eurocode 3 /12/ the three differently types may be treated independently and no interaction need normally be considered.

Eccentricity is covered by the installation tolerances already described.

It is recommended to focus on the dimple associated with the out-of straightness tolerance when assessing the buckling capacity of the grouted connection. The out-of-roundness tolerance can be important for connections exposed to external pressure why if this is the design case it should be investigated. However, due to the shear bulk of the grouted connection, it will in this case likely be relevant only for the plain tubulars above and below the grout body.

For the grouted connection it is recommended to assume and initial imperfection dimple that extends from a maximum value , i.e. 1‰ of the diameter decreasing linearly to zero over a distance of , i.e. half the radius, to each side around the circumference and over a length in the axial direction as illustrated for a monopile connection with shear keys in Figure 5-2.

Figure 5-2 Illustration of initial imperfections in the monopile for a shear key connection

For all types of grouted connections it is recommended to assess buckling using the proposed dimple imperfection shape at the top and bottom of the grout. For connections without shear keys, the cases with the imperfection at the center of the shear keys are obviously obsolete and may be ignored.

For grouted connections dominated by axial loading and thus likely to have shear keys spread over a larger portion of the overlap, it is recommended to also investigate a case with the imperfection peak placed

= /200 /2 4√

Transition Piece – Outward

Peak at grout top

Peak at shear key center

Monopile – Inward

Peak at grout bottom

Peak at shear key center

inward


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centered on the worst loaded shear key, likely to be more towards the top of the connection if a substantial
number of shear keys are part of design.

For assessment of the buckling capacity of the steel outside the grout overlap, it is recommended to perform a linear eigenvalue buckling analysis of the model pre-stressed with the governing ULS load case and then subsequently use the first mode shape to control an inertial imperfection of magnitude identical to that used for the prescribed dimple imperfection, i.e. .

5.3.4 Symmetry utilizationBeing that most grouted connections are either cylindrical or conical they will in general exhibit geometric symmetry. Provided that the loading is also symmetric, this may be utilized to half the size of the model.

It is however an important prerequisite that the loading is symmetric. This is e.g. the case for

— bending, shear, and axial loading of a monopile foundation, or— pure axial loading of pile in a jacket foundation.

In other loading conditions, e.g. torsional, the symmetry condition cannot be utilized.

It is in general recommended to utilize symmetry only when it is evident that the loading of the connection is symmetric and when an omnidirectional load generalization in adopted.

5.4 Element selectionIt is generally recommended to use continuum solid elements for all structural components of the grouted connection. Shell elements may be considered if the connection is without shear keys or any other geometric features. However, the computational gain of shell element versus the versatility of the solid element is generally not considered sufficient to recommend shell elements in general for these types of connections in the region with the grouted connection.

Structural elements such as shells, beam, and spring elements may however be used to lighten the computational work associated with the inclusion of external features.

As a rule of thumb, the use of structural elements should however only be contemplated for features that are a distance of 1 time the descriptive diameter away from the grout.

Most commercial FEM software includes automated methods to e.g. tie a shell mesh to a solid mesh and coupling contestants can be used to tie a beam or spring to e.g. a shell edge or solid surface.

These methods are all acceptable, but all require diligence and response verification based on engineering judgment and experience. For the novice user it is therefore recommended not to use structural elements, but rather stick to 3D continuum elements e.g. with transition to a coarse mesh outside the focus region at the grout overlap. Here again, a distance of 1 time the descriptive diameter away from the grout is recommended before the mesh is made coarse.

5.4.1 Element shape, order, and integration schemeIt is recommended using brick shaped 3D continuum elements to the largest extent possible within the confines of the actual geometry. The use of tetrahedron shaped elements is generally discouraged as these elements are prone to be too stiff and sensitive to small internal angles. If transitional elements are needed due to the geometry, a triangular prism shaped element may likely serve for this purpose as the grouted connections in general exhibit rotationally symmetric geometry. They should however be avoided in regions of particular interest.

Most elements exist in various formulations each describing a set of assumptions on how the element deforms and how associated stresses are integrated within the element.

The deformation is given by the assumed shape function of the element and may be constant, linear, or any higher order function. The associated strains and stresses are calculated by numerical integration of the partial derivatives of the shape function. This may be done in many ways, but is typically done using Gauss integration in either a full-, reduced-, or hybrid integration formulation.

In general higher order elements are preferred for accurate stress estimates; elements with simple shape

= /200


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functions (constant or linear) will require more elements to give the same stress accuracy as higher order
elements. Constant stress elements are not recommended used in the area of interest.

The required stress accuracy should however be seen in relation to the purpose of the analysis or rather the use of the stress and strain results in the following code checks.

Consider e.g. fatigue assessment. If fatigue in e.g. the steel is assessed using an SN curve based on a detail classification using e.g. DNVGL-RP-C203 /6/, then a linear variation of the stress through the thickness of the steel is a reasonable accuracy. If on the other hand a notch stress or fracture mechanics approach is used then a linearization would not be acceptable without a much larger element density.

The availability of elements and the computational cost associated with the chosen element needs to be weighed against the attained accuracy.

Consider e.g. a solid brick element. A linear 1st order, fully integrated version has 8 nodes and 8 internal integration points assuming standard [2×2×2] Gauss integration. A 2nd order, fully integrated version has 20 nodes and 27 internal integration points assuming standard [3×3×3] Gauss integration. Hence, it computationally significantly more expensive to establish the element stiffness matrix for the 2nd order element is than 1st order counterpart. A reduced integration would mean a [1×1×1] and [2×2×2] Gauss integration for the 1st and 2nd order formulation respectively, i.e. only 1 integration point for the 1st order element equivalent to constant stress and 8 integration points for the 2nd order element equivalent to linear stress.

All of the previously listed formulations will typically be available if an implicit solution scheme is chosen, but typically, the higher order elements with full integration will be unavailable in an explicit solver.

One reason for this is that explicit solver are typically developed for the purpose of analyzing rapid transient events with large deformations and large rotations and for such cases simple element formulations give a more robust numerical model and analysis than higher order elements.

Irrespectively of the solver chosen, care should be taken when selecting element formulations and integration rules. Formulations with (selective) reduced integration rules are less prone to locking effects than full integrated simple elements; however the reduced integration elements may produce zero energy modes (“hourglassing”) and may require hourglass control. When hourglass control is used, the hourglass energy should be monitored and shown to be small compared to the internal energy of the system (typically less than 5%).

Hourglass and locking effects are however not normally an issue until full plastic crushing of grout and/or steel is developed. Hence, it is typically only heavily loaded shear key designs that may give rise to these conditions in extreme events.

For grouted connections it is generally recommended to use 2nd order elements with reduced integration for the ultimate and fatigue limit states for both steel and grout. However dependent on the behavior of the grout, fully integrated elements may be used/needed to combat hourglassing.

5.4.2 Element densityWhen considering a grouted connection it is often difficult in practice to fulfil the ideal of global mesh convergence simply due to the sheer size of the structure and practical limitations on computer resources.

Consider e.g. a typical cylindrical monopile foundation with a pile diameter of say 5 m and a grout overlap of 1.5 times the diameter i.e. 7.5 m. For such a connection the contact interface area will be roughly some 2 times the overlap height times the circumference, i.e. ~235 m2 which is massive compared to a typical descriptive element length in terms of thickness of say 50 to 100 mm.

The following recommendation on minimum mesh requirements is therefore given based on the assumed use of 2nd order elements.

If 1st order element are used e.g. in an explicit solution scheme a substantially larger number of elements should be used. Typically then, 3 to 4 elements should be used to discretize any single edge of a 2nd order elements, meaning in general 33 to 43 or a 27 to 64 fold increase in the number of elements required. It should however be noticed that the increase in computational cost of additional elements is near constant in the explicit scheme why this is feasible if desired or needed.


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The aspect ratio of the solid elements should be kept below 1.5 in the regions of interest for the analysis.
For grouted connections this would be e.g. at the top and bottom of grout body, at shear keys if present, and at fatigue prone attachments. Outside these regions the element aspect ratio may be increased but should be kept less than 3.

If the grouted connection is exposed to significant bending, the ±60° section of the circumference centered at the bending neutral axis may be considered as a candidate for a coarser mesh.

It is not recommended to use mesh ties in the region of the grouted connection why element transition should be done gradually. As grouted connections in general have rotational symmetric geometry, and as solid brick elements are preferred, this leads typically to the use of swept meshes. Hence, it is generally only along the circumference that the mesh may be made coarser by increasing the element aspect ratio.

If attachments are to be modeled, it is recommended to use a meshing approach of sweeping a free mesh based on quadrilateral elements through the thickness to facilitate a smooth transition between the perhaps irregular shape of the attachment and the overall ordered swept mesh of the steel.

It is generally recommended that the plated steel is meshed with one 2nd order, reduced integration element though the thickness.

For the grout body a minimum of three 2nd order, reduced integration elements through the thickness is recommended.

In the case of shear keys, a minimum of 6 element faces are recommended for a shear key both on the steel and grout side of the interface.

It is further recommended that the meshed geometry of the shear key is taken to reflect the true shape of the shear key rather than the idealized shape typically used on design drawings.

Figure 5-3 Suggested geometry idealization and minimum meshing of weld bead shear keys

A suggested minimum mesh is shown in Figure 5-3 based in 2nd order elements together with a suggested geometry simplification based on a shear key built by 3 or 4 layered weld beads.

The suggested mesh further places nodes strategically at distances and away from the weld toe on the primary steel plate to facilitate easy post-processing of hot-spot stress extrapolation using in this case the recommendations DNVGL-RP-C203 /6/. Alternatively, the mesh can be made to suit hot-spot extrapolation according to e.g. IIW Fatigue Recommendations /14/.

Apart from these specific recommendations on minimum mesh sizes the analyst should make sure that the element mesh is adequate for representing all relevant failure modes and hot spot stresses.

Real 3 weld beads Real 4 weld beadsIdealized

Minimum detail mesh

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SECTION 6 BOUNDARY CONDITIONS AND LOAD APPLICATION
As stated in the introduction, grouted connections are in principle made up of three discrete continuum bodies that interact with each other through frictional contact.

In general therefore, when analyzing a grouted connection only one of the three components will be fully constrained. This will typically be one of the steel components, e.g. the foundation pile, while the grout and other steel component will be if not free then under-constrained allowing them to perform rigid body motions at least initially until contact and hereby restraint is established.

Irrespectively of the solution scheme being used it is important that this condition is appreciated and addressed in the first step of the analysis of any grouted connection.

If not, the implications are for an implicit solution scheme that the model will like not be solvable as the first increment of the first step is unlikely to converge. For the explicit scheme, the solution will work unhindered by this, but a lot of noise is like to manifest itself in then predicted response due to the inherent impact loading generated as the rigid bodies move into contact.

Various methods can be used to address this initial contact condition. Most commercial FEM software will offer methods for strain-free moving of nodes on the slave surface to the master surface as part of the pre-processing of the model input. It may also offer the possibility of directly prescribing a contact opening and associated state that then overrides the actual condition existing in the model definition.

It is recommended to use one of these methods to resolve any initial gaps or overclosures in the model definition, unless the grout material is influenced by shrinkage and gaps are intended in the model.

If gaps are present in the model by intent, it is recommended to always perform initial solution steps dedicated to resolving this initial contact condition. Again this is not an uncommon scenario why most commercial FEM software will offer methods to facilitate the solution of this by offering stabilization of the contact solution typically though some sort of fictitious viscous damping. Alternatively, stabilization may be introduced directly by adding weak springs and viscous dampers (dashpods) elements.

In this case it is recommended to solve the initial contact condition by easing the bodies into contact one by one using a small amount of gravitational acceleration on the modeled mass. It is recommended to do this in two solution steps. In the first step the rigid body moment of both steel components is to be fully restraint so that only the grout body can move. In the second step, the restraint of one of the steel parts it then ramped down to full release whereby the last steel body and/or the grout depending of which of the steel bodies it rests on after the first step, is eased into contact.

Once contact has been established between all three bodies any automatic stabilization or manually added springs and dampers should be removed (deactivated) for the model and the full gravitational acceleration applied.

If the FEM software allows for changes to be made to the coefficient of friction between analysis steps, it is generally recommended to solve the first two steps without friction as this will likely speed up the convergence of the contact solution. The desired friction coefficient may then be applied in the last step together with the full gravity.

6.1 Boundary conditionsProvided the submodeling cut face is at a distance 3 times the diameter of the tubular away from the part of the model in particular interest, a rigid plane assumption can be applied to the cut face. Said rigid plane can then be driven by a single master node at the center of the tubular cut face.

Automated modeling of this condition is typically provided by commercial FEM software either by kinematic constraint equations or by the use of rigid elements. It is recommended to use either of these methods, as this safeguard against an ill conditioned system of equations that potentially may result from e.g. enforcing a near-rigid condition using excessively stiff normal structural elements.

It is recommend using rigid planes with a master node at all cut faces as it allows for easy application of global loads and easy reaction force extraction at the supports.

If the geometric extent of the model stretches into the soil, say e.g. in the case of a pre-piled jacket,


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inclusion of the surrounding soil should be considered if the stiffness of the soil is significant as discussed
previously in [5.3.2.7].

It will normally be acceptable to assume a clamp boundary condition at the end of one of the steel members – typically the foundation pile. However it is recommended that the validity of this assumption is checked by review of the deformation response of the global model.

Alternatively, and beam model extending either down to the apparent fixation depth of the pile or representing the entire pile supported by soil springs, may be attached to the submodel at the master node of the pile cut face. This approach adds insignificantly to the overall size of the model in terms of computational effort, why it is recommended for grouted connection located in the vicinity of the seabed, i.e. for jacket structures.

6.2 Load applicationFor grouted connections in offshore structures, loads are generally derived from global simulations of the entire structure typically simplified to a framework of beams. Within the context of a global beam model of the structure the grouted connection is typically represented with sufficient accuracy as either three overlaid beams or one equivalent single beam.

As addressed previously in [5.2.1] this implies that the detailed FEA of the grouted connection is in principle a submodel to the global model. Submodels are in general FEA driven by the displacements of the global model. However, for grouted connections it is recommended to instead use the cross sectional response forces of the global beam model to drive the submodel, as this approach is better suited to extract design load cases from the global simulations of the entire structure which may then be used for assessment of the individual limit states.

Moreover as described in [5.3] the nature of the grouted connection necessitates the inclusion of a substantial part of the steel surrounding the grouted connection. The cut faces between the submodel of the grouted connection and the global model will thus be at some distance from the connection itself.

This is relevant to consider as it is only at these cut faces that global loads may be transferred easily to the submodel. Hence, it should be considered if significant loads are acting on the geometric extent of the submodel in global model and if so, how to account for these in the submodel.

In the following loads acting on the boundary cut faces of the submodel will be referred to as global loads, as opposed to local loads acting on the submodel itself.

6.2.1 Global and local loadsAs described the recommended approach is to apply cross sectional response forces from the global model as loads on the submodel. It is further generally recommended to apply the deadweight of the modeled structure via a downward gravitational acceleration.

In general, global loads should be applied at the submodel cut face(s) above the grouted connection and boundary constraints should be applied only below the grouted connection.

This is easily achieved for grouted connections in monopile foundations where there is a single cut face above and below the grouted connection.

For jacket structures the situation is however typically more complicated as part of the jacket structure needs to be included in the submodel. In these cases it may seem attractive to simply flip ones view of the connection and consider the cross sectional response forces from the global model at the cut face in the pile to be the driving load while imposing a clamped boundary condition at all remaining cut faces in the jacket structure. This is strongly discouraged because such an assumption fundamentally changes the local flexibility of the jacket-to-pile structure and likely makes it unreasonably stiff whereby the validity of the predicted response is compromised.

If no significant loads are acting on the geometric extent of the submodel, which is likely the case for near seabed grouted connections in jacket structures, the recommended approach of applying loads at the cut faces above the grouted connection can readily be applied.

If on the other hand the local loads are significant due to the extent and/or location of the grouted


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connection, it is recommended to scale the loading at the cut faces above the connection such that the
predicted cross sectional response just below the grouted connection is match in the submodel.

This scaling is recommended to be done with due consideration of the underlying nature of the load and what is the dominating component, say axial, in-plane bending, or out-of-plane bending; and that the scaled load is ensured to be conservative for the grouted connection.

6.2.1.1 Local loadsFor offshore structures the following loads may act locally on the grouted connection:

— hydrodynamic loads from waves and current— hydrostatic loads if the connection is not flooded— ice loads if the connection is at sea level— impact loads from ship collision again if the connection is at sea level.

The direct application of hydrodynamic loads locally on the submodel is in general very difficult and normally not necessary as the sheer bulk of a grouted connection is such that in general it is reasonable to represent the load by a constant shear and associated linear bending load over the height of the connection, as implicitly assumed in the previous discussion on global load application.

For the same reason, hydrostatic pressure is normally only of significance when it causes an external overpressure on members that are not flooded. If this is the case, it is recommended to include the external pressure in the ULS buckling assessment. In all other limit state assessments it is recommended to use exercise engineering judgment on the significance of any overpressure.

For monopile foundations where the grouted connections are typically placed near the sea level local loads from sea ice and ship impacts are a possibility. Local loading is of particular importance if they act close to the bottom of the connection near the seal.

It is generally recommended that local loads from sea ice and ship impacts are applied as such, i.e. locally on the submodel of the grouted connection, irrespectively of the overall bulk of the connection.

6.2.2 Ultimate limit state load casesBecause of the nonlinear behavior of the grouted connection it is not possible to assess any ultimate limit state (ULS) condition by linear superposition of individual loads. Instead each combined loading has to be analyzed individually as the response of the grout is load path dependent.

The loading recommended applied to the structure in a minimum of two load steps:

— Step 1: Static deadload— Step 2: Environmental load.

The base case of the static deadload may be reused as a pre-stressed start condition for a number of subsequent ULS load cases for that particular geometry.

From the global simulation of the structure a number of time series for the response above and below the grouted connection normally forms the basis for the design loads to be applied to the local submodel. Each set of time series represent a simulation of the combined action of environmental loads from wind and waves under a set of different conditions, e.g. direction of the wave, the direction of wind, the operational state of the turbine if present, etc. All of these thus represent a realization for a worst case assumption.

It is recommended to analyze all these time series and extract simultaneously occurring sets of sectional response force repressing the following the conditions:

— peak axial loading— peak bending loading— peak torsion loading.

For each case both the minimum and maximum load magnitude should be extracted, i.e. a total of six cases at each end of the grouted connection.

In general it is recommended that all of these are analyzed. However, depending on the type of connection and the symmetry of not only the geometry but also the load, a subset of these may prove sufficient.


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For a monopile foundation the axial load normally does not vary significantly. Moreover, the monopile
foundations normally exhibit geometric rotational symmetry, why it normally would be acceptable to consider just one overturning load case of maximum magnitude axial load combined with maximum bending load magnitude.

For torsion loads on monopiles, bending will via friction provide additional shear capacity in the grouted connection, why torsion should be considered not only at its overall peak magnitude but also at its maximum magnitude occurring at zero-crossing bending. For this case it is recommended to combine it with the minimum axial force.

For jacket structures the loading normally do not exhibit rotational symmetry relative to the individual pile. Although grouted connections in jacket structures are generally dominated by axial loads, bending should be assessed carefully. It should be considered that the flexibility of the jacket joint above the grouted connection will have an effect on bending load transferred to the pile. For a standard 4-legged jacket in-plane and out-of-plane bending relative to the jacket joint above the connection normally will be the governing bending cases. For 3-legged jackets other directions relative to the joint geometry may prove governing.

It is recommended to pay special attention to not only the magnitude of the bending loading but also its orientation. Hence, a set of load cases for bending and corresponding axial load is recommended for the ULS assessment of jackets. It should further be noticed that because of the nonlinear behavior of the grouted connection, one should be cautious in pre-judging relative severity between load cases having various combination of axial, moment, and shear loading.

As touched upon previously, fixation of the submodel is recommended done at the bottom of the modeled pile, why in particular for jackets a set of cross sectional loads likely need to be applied at various cut faces in the leg and braces of the jacket. Attention should therefore paid to ensure that the combined set of loads do match the expected load distribution over the submodel – in particular over the height of the grout but also generally.

6.2.2.1 Buckling assessmentIt is recommended to assess buckling of the grouted connection via a push-over analysis. That is, by increasing the environmental load proportional to a reference load case until the structure buckles.

FEA methodologies for this using are explained in e.g. DNVGL-RP-C208 /7/.

It is recommended to select base load cases for buckling push-over analyses from the set of ULS load cases analyzed from steel yielding and grout capacity.

In principle only the environmental load should be increased until buckling occurs in the push-over analysis. Thus, the deadweight of the structure should be kept constant at its actual magnitude. However, as additional deadweight in general will be onerous for the buckling capacity, the entire ULS load case may conservatively be proportionally ramped up during the push-over analysis if judged more convenient from a practical analysis point of view.

For connections without shear keys (conical monopiles), buckling is generally driven by the overturning load on the connection.

For connections with shear keys, buckling may be caused not only overturning loads but also by axial loads. In the case of a monopile connection with shear keys, the axial load due to environmental loads is normally small, why buckling of grouted connections in monopiles generally is governed by the overturning load irrespectively of shear keys or not.

For grouted connections in jackets on the other hand, the axial load is generally the principal loading on the connection, why for these types of connections buckling should be assessed for load combinations describing both worst bending and worst axial load conditions of the grouted connection.

6.2.3 Finite limit state load casesAs it would in general be impractical to analyze the submodel for all the response time series obtained from the global simulations of the structure, it is recommended to use transfer functions relating stress to loading derived from the submodel response for a number of representative load cases.


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It is recommended that the selection of these representative load cases is done based on a detailed
statistical analysis of the global load. This analysis should address and quantify the following characteristics of the simulated response to be applied as global loads on the submodel:

— governing load component— causality and correlation between load components— directionality of governing load— markov data for governing load.

Based on the characteristic of the global load a manageable subset of load cases may be determined.

6.2.3.1 Governing loadFatigue in offshore structures is normally driven by environmental loads, from wind and waves. Typically waves are driving the fatigue. In the case of offshore wind turbine foundations however, wind loading is typically either dominating or equal in contribution to the fatigue loading. In regions with sea ice, fatigue due to ice loading may additionally contribute to the accumulated fatigue damage.

Irrespectively of the contribution ratio between wind and wave loading, the common denominator for the two is that they cause an overturning and shearing load on the structure. This is also the case for any eventual seasonal loading from sea ice.

For a grouted connection in a monopile foundation the governing load for will thus normally be bending, whereas the overturning predominantly will be carried as axial loads in the grouted connections between jacket and piles.

6.2.3.2 Causality and correlationOnce the governing fatigue load component for the grouted connection has been established, the causality and correlation between it and all other global load component acting on the submodel of the grouted connection can be determined from statistical analysis of all available time series from the global load simulations.

When doing so, the underlying nature of the environmental loading should be appreciated. For a typical offshore structure there will be a strong correlation between wind and wave loading as wind speeds and wave height are by nature strongly correlated. However, if the structure carries a wind turbine, the nature of the wind loading on the structure changes dramatically as the turbine is actively controlled to produce as close to a rated power as possible. Hence, the wind loading on the foundation will exhibit a relatively constant load magnitude over a large range of wind speeds when the wind turbine is producing power, thus wreaking the correlation between wind and wave loading. Only when the wind turbine is not actively controlled is the normal strong correlation restored.

Likewise, in the case of sea ice loading then this will normally be strongly correlated to sea current only. Obviously, there are no waves when there is sea ice, but there is wind. However, wind and current are normally not strongly correlated.

In general therefore, is recommended that the correlation between the governing load component and all other global load component acting on the submodel is assessed for subsets of fatigue load conditions if the structure is exposed to sea ice loading and/or wind turbine loading.

Additional subdividing may be necessary to obtain subsets with reasonable correlation between the individual load components. However, in general this is normally not necessary as the nature of the loading within the describe subsets tends to correlate well.

6.2.3.3 DirectionalityThe environmental loads from wind, waves, current, and sea ice will in general not be aligned relatively to each other. Moreover, the direction of environmental loads is likely to vary over time.

It should be carefully considered if the load directionally can be simplified to an omni-directional load, or if individual load directions need to be assessed individually. Pertaining to this, it is important to consider not only the nature of the load environment, but also the type of structure being assessed.

Monopile foundations will in general exhibit a symmetric response in the grouted connection, whereas jackets will not. Hence, special attention should be paid to loading directionally for jackets.


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6.2.3.4 Markov data
The fatigue accumulation in grout is highly dependent on both the stress range and the mean stress. Rain flow counting of ranges alone is thus not sufficient. Instead Markov data should be established for the governing load component. Methods for Markov matrices based on rain flow counting is given in e.g. /18/.

The Markov data should be collected individually for each subset of fatigue load conditions identified as needed to establish reasonable correlation between the individual load components.

Within each of these subsets of fatigue load simulations a strong correlation is likely present and when determined it may be used to select one fatigue load case to describe the entire subset.

From the Markov data for each subset of fatigue loading, the envelope of the governing load can be extracted. Using this load range of the governing load together with the established correlations between it and all other load components, a load case can be defined for each identified subset of the fatigue loading.

6.2.4 Serviceability limit state load casesThe serviceability limit state is of particular relevance for the conical grouted connections as these will settle when carrying axial load. As the conical grouted connection only offers axial resistance in one direction only its use is exclusive to the monopile type of foundation.

Conical connections will settle in two tempi. First the connection will settle as the deadweight of the structure above the connection is transferred to the connection by the removal of the temporary supports. Secondly, the connection will wedge down as a result of bending moving the transition piece from moved side to side.

For wind turbine application this cyclic bending is significant not only in number of occurrences but also in magnitude. Hence, it is recommended that the SLS be analyzed for series of bending load reversals with the aim of tracking not only the settlement but also the cracking of the grout.

Cracking of the grout will have a significant effect on the settlement of a conical grouted connection if vertical cracks develop over the entire height of the grouted connection.

The settlement per bending load range is highly nonlinear as it is load path dependent. It is therefore recommended that a minimum of 50 load reversals of a representative bending moment is analyzed.

The hereby predicted settlement may then be extrapolated to the full lifetime of the structure typically by assuming a logarithmic trend.

It is recommended that the load range cycled for the settlement assessment be taken as the maximum load range occurring on average hourly or less frequent over the entire design life of the structure.

The magnitude of the range may be derived by binning the peak moment of each moment range present in the Markov data and assuming a range of twice this magnitude with a zero mean.

6.2.5 Accidental limit state load casesThe accidental load cases for grouted connections in offshore structures in normally exclusively impact loads from ship collisions. Guidance on load assessment for these conditions may be found in DNV-RP-C204 /2/.


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SECTION 7 LIMIT STATE ANALYSES
The general design of grouted connections is in a LRFD approach conducted by assessment of the structural performance in various limit states. Typically the governing limit states for a grouted connection are yielding and buckling in the ULS and fatigue of steel and grout in the FLS.

Additionally, the SLS settlement of conical connections in monopile foundations and the ALS condition typically associated with connection that may experience accidental boat impacts needs to be assessed.

The design requirements for all limit states are to be taken as defined by the chosen governing standard, e.g. DNVGL-ST-0126 /9/ (see further [1.1] and [1.2]).

Before any of these limit states can be assessed, it is necessary first to address the initial condition of the structure. Grouted connections in offshore foundations are typically used to create a structural connection between various large steel components during the installation process, why the grout is normally cast offshore and typically either fully or partly submerged.

To establish the initial condition of the grouted connection prior to any limit state assessment, the temporary condition(s) during the installation needs to be assessed. Here, items such as heat development during curing, potential subsequent shrinkage, and early age movement should be considered and their possible impact on the initial condition of the grout body established and accounted for in the setup of any limit state analysis.

7.1 Ultimate limit stateBecause tolerance exists in the local submodel – but not in the global model used for load simulation, it is generally important that the imperfection is placed at its most disadvantageous position for the load case in question. In general this will mean orienting the imperfection such that the associated minimum grout thickness coincide with the location in the pile exposed to the severest bending stress, i.e. perpendicular to the bending neutral axis of the load case.

If multiple bending cases need to be assessed this implies that different imperfect submodel geometries need to be built as FEM models. However, as the geometry of the grouted connection itself normally exhibit rotational symmetry, this is in practice easily attained either at a model assembly level or by an initial rotation/translation of the grout and pile model mesh. Note however, that if as recommended the deadweight of the submodel is applied using a gravitational acceleration; the direction of this acceleration may need to be changed depending on how the imperfection is conceived to reflect the real geometry.

It is generally recommended to analyze the connection assuming both the initially perfect geometry and a relevant set of installation tolerances (see further [5.3.3]).

Offshore structures are typically designed with a certain corrosion allowance, i.e. a specified thickness that the steel is assumed to diminish by due to corrosion. Normally, the full extent of the corrosion allowance is deducted from the initial wall thickness of the structure when assessing the ULS, as for a steel structure this is in general the most onerous condition.

A weakening of the steel stiffness is however not universally conservative for a grouted connection. If the wall thickness of the steel is reduced by a corrosion allowance at the top and bottom extremities of the grouted connection, this will likely result in a contact pressure between steel and grout that is lower than that corresponding to the non-corroded condition when the connection is exposed to bending.

The contact pressure is key to assess not only the possibility of wear in the steel-grout interface but also tensile stresses caused by shearing of the grout due to the friction of the interface.

Hence, it is generally recommended to analyze the connection assuming both the initial non-corroded geometry and the fully corroded condition, if the design encompasses corrosion allowance with the region from half a diameter above to half a diameter below the grout body.

7.1.1 Response assessmentThe utilization of the grout should be judged based on the tensile and compressive stress magnitudes, i.e. by inspection of the maximum and minimum principal stress respectively.


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If the linear elastic capacity of the grout is exceeded the extent of cracking and crushing should be assessed.
(See also [2.4]).

The magnitude of cracking and crushing of the grout may be asserted by inspection of the maximum and minimum principal plastic strains respectively. Vector plots of these will show not only the magnitude but also the normal direction to any cracking or crushing plane.

When assessing stresses and strains derived from FEA, it is important to recognize that these quantities are determined in the element integration points whereas the displacements are determined at the nodes.

In FEA post-processing, results are typically examined as contour plots on the element faces based on nodal averages of the quantity in question, e.g. stress and strain. Nodal values of stresses and strains are obtained by extrapolation of the integration point values using the element shape functions. Hence, if the solution exhibit a large gradient over the element, non-physical errors like tensile stress in exceedance of the yield limit specified in the material description may present itself in these contour plots.

It is therefore recommended to use vector plots of the stress and strains at the integration point in conjunction with the typical nodal average contour plots when assessing the response of the grout body.

It is further recommended that if large stress gradients manifest themselves in the grout response, then in particular if associated with crushing, it be considered if a finer mesh more capable of capturing this gradient is needed to accurately judge the response of the grout.

Irrespectively, it is recommended always to compare the stress-strain response obtained in the integration points with the yielding assumptions of the material model being used, e.g. via a tensor plot of the integration point stresses and strains.

7.2 BucklingBecause of the shear bulk of the grouted connection the assessment of buckling will be of importance only for the steel components. Depending on the type of grouted connection different buckling conditions should be investigated.

For conical connections in monopile foundations, buckling need only be assessed for the steel at the extremities of the grout body.

It is recommended to assess the buckling capacity by means of a push-over analysis.

It is further generally recommended to use a linear elastic stiff grout model based on the characteristic dynamic modulus of elasticity of the grout. Contrary for the steel and nonlinear material model should be used. See further [3.1.2].

If a nonlinear material model is used for the grout, it is recommended to calibrate it to upper fractile strengths as a weak grout based on lower fractiles in general will be favorable for the steel.

The full completion of the push-over analysis may be forfeited if the onset of buckling can be observed not to occur at a load proportionally factor of 2. That is, in case the structure exhibit additional load capacity beyond double the ULS load case this in general will be considered sufficient proof of adequate buckling capacity, why determination of the actual maximum capacity is not necessary.

As for the general ULS cases, installation tolerance are to be included in modeling together with local imperfections to the geometry as described in [5.3.3].

Additional guidance on the general buckling assessment of the steel may be found in DNVGL-RP-C208 /7/ or EN 1993-1-6 /13/.

7.3 Fatigue limit stateThe assessment of fatigue is normally based on the assumption of linear damage accumulation also known as the Palmgren-Miner rule. This approach originates from fatigue in steel where fatigue is typically accumulated in the linear elastic range of its response.

The typical approach for varying amplitude fatigue loading is to use rain flow counting to identify stress ranges and number of occurrences. This approach is valid for plated steel structures where the fatigue damage accumulation is typically assumed to be independent of mean stress.


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This assumption is however not valid for a material like grout, why the rain flow counting needs to be
expanded to also identify mean stress of each range and count the occurrence of stress ranges at various levels of mean stress, i.e. establish the full Markov matrix.

Because grouted connections are typically assessed as submodels driven by a the response of a simplified global model, the Markov data will typically be derived by performing rain flow counting on the response of the global model rather than directly on the stresses in the submodel, as the computational resources needed for a direct re-simulation of the submodel in general are prohibitively large.

In these cases it is recommended to establish the response of the grouted connection for various representative load cases based on representative correlations between the forces driving the submodel as described in [6.2.3] and then use these to map stresses at individual locations in the submodel to load magnitudes in the Markov data.

Dependent on the type of grouted connection and the nature of the loading environment one or more sets of Markov matrices and associated load case will need to be assessed individually and summed up to get the final total fatigue damage accumulation in the grout and steel.

For each set, the load case should be analyzed as an up loading to a load magnitude that will encompass any and all load magnitudes in the associated Markov matrix. The uploading should be done gradually, and the response determined in reasonably closed space intervals, say 100 or more for the entire peak to peak load range.

From such a response, the stress at any location may be established as a function of load. Using these functions the Markov load data may be mapped into stress data and using an appropriate SN curve for the steel and grout respectively, the fatigue damage may be calculated.

As offshore grouted connections typically are submerge, it is important that the SN curve used in the fatigue assessment accounts for this condition. In particular for the grout this is of paramount importance as grout exhibits significantly lower fatigue resistance in the wet condition. SN curves suitable for the wet condition may be taken from e.g. DNVGL-ST-0126 /9/ for grout and DNVGL-RP-C203 /6/ for steel.

A further complication to the fatigue assessment of grout is that the SN curves available in various standards – irrespectively of wet or dry condition – typically are based on uniaxial loading of unconfined test specimens.

As the loading environment of grouted connections typically exhibit significant multi-dimensional stressing, it is recommended that fatigue is assessed based on the stressing of the grout in the particular direction exhibiting the largest compressive stress for the individual stress cycle.

A scheme to facilitate this approach would be to establish load-to-stress mappings as described previously for all six stress components at the location be analyzed, i.e. a mapping for the entire Cauchy stress tensor σ. Based on such a mapping, a simple post processing algorithm may be followed to assess the damage of say e.g. a moment range ΔM with mean moment of .

The first step is to find the largest compressive stress in the moment range to . Focusing on only this portion of the response, solving the eigenvalue problem at each known response within this range will give the eigenvalues from which the largest compressive at each of these points is and the response point where this compressive stress attains it largest magnitude can be determined. This know, it is then a small matter of using the eigenvalue solution ν at this specific response point to transform the full Cauchy stress tensor σ for all response points into the direction of the maximum compressive stress as from which the stress range may be established.

However, the issues of mean load dependency and multi-dimensional stressing describe so far are not the only effects that need to be accounted for in the fatigue assessment of grouted connections.

It is an implicit assumption of the linear damage accumulation method, that a specific stress range produces the exact same damage when occurring at the same mean stress irrespectively of when it occurs in the life time of the structure.

Hence, because of the load-path dependency inherent to grouted connections it is difficult to argue fatigue based on nonlinear grout models. On the other hand, cracking of the grout may lead to significant redistribution of the stress in the grouted connection.

− ½Δ + ½Δ = = [ 1, 2, 3] 3 = min

= T


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It is therefore recommended that the fatigue life of the connection is assessed assuming two different
material behaviors of the grout: Linear elastic and nonlinear.

It is further recommended that the fatigue damage is calculated from the response of a model geometry both with and without installation tolerances.

In the case of fatigue assessment based on a nonlinear material model for the grout, it is important that the cracked response used for the assessment of stresses represents a steady state response as closely as possible. That is, that additional load cycles will cause only marginal crack propagation.

It is therefore recommended to cycle the loading back and forth within the peak magnitudes of the individual load case until a steady state can be declared from inspection of the developing crack pattern.

The final load sequence of this cycling may then be used to establish the load-to-stress mapping.

Finally, it should be noted that the stiffness of the grout has an impact not only on the stressing of the grout but also for the steel.

7.4 Serviceability limit stateThe serviceability limit state is principally of interest only for conical monopile exhibiting settlements over time. This settlement will lead to an increase in the passive compression of the grout. It will also cause an increase in the hoop stresses of the grout body why it has the potential to cause progressive cracking. Finally, it will cause a reduction in the play left after installation between the temporary supports of the transition piece and the monopile rim.

It is recommended to assess the settlement of the transition piece base on the response of a model with any installation tolerance. The material model for the grout should be nonlinear.

It is further recommended to cycle the SLS load back and forth at full magnitude with a zero mean value at least 50 times to obtain a reasonable trend of the settlement from which a long term settlement may be estimated.

When fitting a trend to the settlement it is generally advisable to ignore the first few cycles.

The predicted maximum settlement during the lifetime of the structure should be considered in relation to the ULS and FLS.

As described, the effect of the settlement is in general an increase of the passive compressive stress in the grout, why adverse implication on in particular fatigue being highly sensitive to the mean stress may arise from excessive settlement.

Finally, it should be considered that the settlement response due to cyclic bending of conical connections is highly sensitive to the assumed friction coefficient of the steel-grout interface.

It is thus of particular importance that the sliding behavior of the contact modeling is accurate.

7.5 Accidental limit stateGuidance on the analysis of the accidental limit state may be found in DNV-RP-C204 /2/.


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APPENDIX A CONSTITUTIVE FORMULATIONS FOR GROUT
For grout and concrete numerous constitutive relations of varying complexity exist in the literature. However, only a handful of these have matured into being readily available in commercial finite element codes. These range from basic Drucker-Prager, to highly specialized models such as the Willam-Warnke model available in Ansys or the Lubliner-Lee-Fenves model available in Abaqus.

Before addressing these material models, a short recap of the basics of stress modeling is presented in the next section to facilitate the following description of the various models.

A.1 Stress modeling basicsAt a given point in a continuum the stress state is described by the Cauchy stress tensor:

When a body is in equilibrium the components of the stress tensor in every point of the body satisfy the general equilibrium equation

Using e.g. the general constitutive relation for a linear elastic material with the additional assumptions of the material being homogeneous and isotropic this reduces to the well-known Hook’s law

where is the Kronecker delta1, K is the bulk modulus and G is the shear modulus; or alternatively expressed in elastic moduli

where E is Young’s modulus (also known as the modulus of elasticity), and ν is Poisson’s ratio.

At the same time, equilibrium requires that the summation of moments with respect to an arbitrary point is zero, which leads to the conclusion that the stress tensor is symmetric, i.e.

1Shorthand in tensor indices notation .

A.1.1 Principal stresses and stress invariantsThe components of the stress tensor depend on the orientation of the coordinate system at the point under consideration. However, the stress tensor itself is a physical quantity and as such, it is independent of the coordinate system chosen to represent it. Hence, there are certain invariants associated with the stress tensor which are also independent of the coordinate system. Two sets of such invariants are commonly known as the principal stresses and the stress invariants .

The principal stresses are merely the eigenvalues of the stress tensor, thus they are the roots to

(A.1)

(A.2)

(A.3)

(A.4)

(A.5)

(A.6)

= = 11 12 1321 22 2331 32 33

, + = 0

= = 3 (13 ) + 2 ( − 13 )

= 1 (1 + ) −

=

= 1 if =0 if ≠

| − | = − = 11 − 12 1321 22 − 2331 32 33 − = − 3 + 1 2 − 2 + 3 = 0characteristic equation


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where I1, I2, I3, are the first, second, and third stress invariants given as
The characteristic equation has three roots (eigenvalues) which all are real due to the symmetry of the stress tensor. These are when ordered as , , and , the principal stresses which are unique for a given stress tensor, hence the invariant property of I1, I2 and I3.

A.1.2 Principal stress spaceFor each eigenvalue there is a non-trivial solution in the equation .

These solutions are the eigenvectors defining the planes where the principal stresses act, why they are also known as the principal directions and the planes they define as principal planes.

The principal stresses and principal directions characterize the stress at a given point independent of the orientation. Hence, in a coordinate system with axes oriented to the principal directions, known as the principal stress space, the principal stresses will be the normal stresses and there will be no normal shear stresses, why the stress tensor reduces to a diagonal matrix as

As mentioned, the principal stresses are unique for a given stress tensor and the invariant property of the stress invariants . The principal stresses can therefore be combined to form the stress invariants as

A.1.3 Hydrostatic and deviatoric stressThe Cauchy stress tensor can be expressed as the sum of a deviatoric stress tensor and a hydrostatic (or mean) stress p. In solid mechanics it is common practice to sign the hydrostatic pressure such that it is positive in compression. Following this notation the relation becomes

i.e. a mean part which tends to change the volume of the stressed body, and a deviatoric part which tend to distort said body analogous to the bulk- and shear modulus parts of Hook’s law, Eq. (A.3).

With pressure positive in compression it become

and the deviatoric stress is thus

(A.7)

(A.8)

(A.9)

(A.10)

(A.11)

(A.12)

1 = 11 + 22 + 33 = 2 = 22 2332 33 + 11 1331 33 + 11 1221 22 = 11 22 + 22 33 + 11 33 − 122 − 232 − 132 = 12 − 3 = 11 22 33 + 2 12 23 31 − 122 33 − 232 11 − 132 22 = det ( ) 1 = max( 1, 2, 3) 3 = min( 1, 2, 3) 2 = 1 − 1 − 3

( − ) = 0

= = 1 0 00 2 00 0 3

1 = 1 + 2 + 3 2 = 1 2 + 2 3 + 1 3 3 = 1 2 3

= − = = −

= −13 = −13( 1 + 2 + 3) = −13 1 = + = + 11 12 1321 22 2331 32 33 = 0 00 00 0 + 11 12 1321 22 2331 32 33 = 11 + 12 1321 22 + 2331 32 33 +


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The deviatoric stress tensor is also known simply as the stress deviator.
A.1.4 Deviatoric stress invariantsAs the deviatoric stress tensor of second order, it itself also has a set of invariants . Moreover, it can be shown that the principal directions of deviatoric stress tensor are identical to those of the Cauchy stress tensor . Therefore, following the same procedure used to find the invariants of the stress tensor, Eq. (A.6), the characteristic equations is

where , , and are the first, second, and third deviatoric stress invariants, respectively. Their values are the same (invariant) regardless of the orientation of the coordinate system chosen.

These deviatoric stress invariants can be expressed as a function of the deviatoric stress components or the deviatoric eigenvalues , or alternatively, as a function of Cauchy stress components or the principal stresses as

A.1.5 Derived state quantities in principal stress spaceBecause of its simplicity and inherent invariant characteristics the principal stress space is useful when considering the state of the elastic medium at a particular point.

A.1.5.1 Peak tension, compression, and shear stressesObserving e.g. the stress tensor in principal stress space, Eq. (A.8), it is seen to be a diagonal matrix, i.e. free of shear stresses. Hence, with the principal stresses ordered as it is immediately evident that the maximum tensile stress at the specific stress point is σ1 whereas the maximum compressive stress is σ3. Moreover, it can be shown that the maximum shear stress is

i.e. is equal to one-half the difference between the largest and smallest principal stress, and acts on the plane that bisects the angle between the directions of the largest and smallest principal stresses, why the plane of the maximum shear stress is oriented 45° from the principal stress planes.

In this plane of maximum shear stress the normal stress is non-zero and equal to

A.1.5.2 Octahedral stressesAlternatively, taking outset in the mean stress, i.e. the hydrostatic pressure p, this stress will be the normal stress in a plane whose normal vector makes equal angles with each of the principal axes (i.e. a plane having direction cosines equal to ). There will be a total of eight such plane why they are known as octahedral planes, and the normal and shear components of the stress tensor on these planes are called octahedral normal stress and octahedral shear stress , respectively.

(A.13)

(A.14)

(A.15)

(A.16)

| − | = − = 11 − 12 1321 22 − 2331 32 33 − = − 3 + 1 2 − 2 + 3 = 0characteristic equation

1 2 3

1 = = 0 2 = 12 = − 1 2 − 2 3 − 3 1 = 16[( 11 − 22)2 + ( 22 − 33)2 + ( 33 − 11)2] + 122 + 232 + 312 = 16[( 1 − 2)2 + ( 2 − 3)2 + ( 3 − 1)2] = 13 12 − 2 3 = det = 13 = 1 2 3 = 227 13 − 13 1 2 + 3

1 ≥ 2 ≥ 3

max = 12| 1 − 3|

n = 12| 1 + 3|

1/√3 oct oct


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Hence, the octahedral normal stress can be expressed as
i.e. then mean normal stress or hydrostatic pressure, whose value is the same in all eight octahedral planes as is the shear stress on the octahedral plane which is expressed as

A.1.5.3 Other common stress invariants – von Mises and TrescaEvaluating the state of a material is integral in the assessing the capacity of said material, why stress invariants pertinent to various yield criterions are common. These are typically named after the father of the yield criterion why among the most commonly used we find e.g. the equivalent von Mises stress

associated with the von Mises yield criterion stating that material yields when the second deviatoric stress invariant reaches a critical value, i.e. when , where is the yield stress in pure shear and is the yield stress in unidirectional tension, hence the yield criterion may also be expresses as

which is the common form. Because the equivalent von Mises stress is always a positive value, i.e. it is sometime also referred to as the equivalent tensile stress. Likewise, because of its sole dependency on the second deviatoric stress invariant it is also known as -plasticity. Finally, it may be added that as the von Mises yield criterion is independent of first stress invariant it is independent of the hydrostatic pressure.

Another common stress invariant is the Tresca stress

associated with the Tresca, or maximal shear stress, yield criterion which compared to the von Mises criterion predicts the same yield point for uniaxial and biaxial loading, but for other conditions will be a more conservative criterion predicting yielding prior to that of the von Mises criterion. Additionally, referring to Eq. (A.15), the Tresca stress is seen to be exactly twice the magnitude of the maximum shear stress, i.e. .

A.1.6 Haigh-Westergaard coordinates, meridians, and lode anglesIn the principal stress space it is often convenient to use the Haigh-Westergaard coordinates defined as

The first coordinate ξ is proportional to the hydrostatic pressure p, and the scaling factor is chosen such that ξ equals the distance to the hydrostatic projection of the stress point from the origin. The second coordinate ρ is proportional to the equivalent von Mises stress q, and it equals the distance of the stress point from the hydrostatic axis. Finally, the third coordinate θ is the polar angle in the deviatoric projection of the stress point, i.e. the Lode angle.

The Haigh-Westergaard coordinates thus represents cylindrical coordinates in the principal stress space, with the coordinate ξ measured along the hydrostatic axis (the equisectrix), coordinate ρ as the radius, and

(A.17)

(A.18)

(A.19)

(A.20)

(A.21)

oct = 13( 1 + 2 + 3) = 13 1 = −

oct = 13 [( 1 − 2)2 + ( 2 − 3)2 + ( 3 − 1)2] = 13 2 12 − 6 2 = 23 2

= 3 2 = 12[( 1 − 2)2 + ( 2 − 3)2 + ( 3 − 1)2] = 3√2 oct

2 2 = 2 = y/√3 y y ≤ 3 2 = ≥ 0 2 2 1

tresca = 1 − 3 tresca ≤ y

max = 12| tresca| ( , , )

= 1√3 1 = √3 = 2 2 = 23 cos(3 ) = 3 = 3√32 323/2 ; = 3 12 3 1/3


DNV GL AS

coordinate θ as the polar angle measured from the projection of the first principal axis onto the deviatoric
plane. Consequently, the principal stresses can be expressed as

If then the principal stresses are ordered as .

A.1.6.1 MeridiansThe meridians are simply the trace of the yield surface in the meridional planes, being any plane that contains the hydrostatic axis, i.e. a plane uniquely determined by a Lode angle. These are of interest in them self as their variation with the hydrostatic pressure for most materials is not linear, and as the deviatoric section of the strength envelope of most materials is not circular, why the meridians in halfplanes characterized by different values of the Lode angle are in general different.

Two extreme cases are represented by the meridians with Lode angles and . The former corresponds to stress states with two equal principal stresses smaller than the third one, i.e. the locus of stress states satisfying the condition , and is called the tensile meridian. The latter corresponds to stress states with two equal stresses larger than the third one, i.e. the locus of stress states satisfying the condition , and is called the compressive meridian.

Examples of stress states on the tensile meridian are uniaxial tension and biaxial compression. On the compressive meridian stress states such as uniaxial compression and biaxial tension resides.

A.2 Yield surfacesReturning to the actual focus of the yield surfaces, having set the basis for stress modeling in the previous section, these can now easily be formulated and visualized in principal stress space and associated deviatoric and meridional planes.

A.2.1 Drucker-PragerThe Drucker-Prager yield criterion is in its original form (see (A.1)) a pressure dependent von Mises criterion1, why it can be seen as a smooth version of the Mohr-Coulomb yield criterion2.

The Drucker-Prager yield criterion has the form

where I1 is the first invariant of the Cauchy stress σ and is second invariant of the deviatoric stress s. The constants A and B are determined from experiments.

In Haigh-Westergaard coordinates, see Eq. (A.22), this can be expressed as

1 The von Mises yield criterion is with σy being the yield stress in unidirectional tension. See further the discussion of commonstress invariants [A.1.5.3] and below.

2 The Mohr Coulomb yield criterion assumes that yielding is controlled by themaximum shear stress and that this yielding shear stress depends on thenormal stress which may be expressed as

where τ is the shear stress, σ is the normal stress (negative in compression), c is the cohesion, and is the angle of internal friction.

The yield surface is a cone with a hexagonal cross section in the deviatoric stress space, and assuming that this failure envelope inscribes that of the Drucker-Prager (von Mises) criterion in the deviatoric plane it looks as sketched to the right.

The criterion is a generalization of other known criteria in that for = 0° it reduces to the Tresca (or maximal shear) criterion, whereas for = 90° the Rankine (or tension-cutoff) criterion is recovered as also shown in the sketch to the right.

(A.22)

(A.23)

(A.24)

123 = 1√3 + 23 coscos( − 2 /3)cos( + 2 /3)

0 ≤ ≤ /3 1 ≥ 2 ≥ 3

= 0 = /3

1 > 2 = 3 1 = 2 > 3

2 = + 1 2

1√2 − √3 =

y ≤ 3 2 Mohr-Coulomb( = )

Tresca( = )

Rankine( = )

=

Deviatoric plane

= =

=

Drucker-Prager(von Mises)

= − tan


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A.2.1.1 Parameter fitting using uniaxial strengths
The constants A and B are as stated initially material parameters that needs to be determined by experiments. However, instead of testing for A and B directly, rewriting the yield criterion in terms of the principal stress using Eqs. (A.12) and (A.14) the following is obtained

Denoting the uniaxial tensile and compressive strength and respectively the criterion implies

Which solved simultaneously gives

i.e. the constants A and B expressed by the uniaxial tensile and compressive strength and .

A.2.1.2 Parameter fitting using cohesion and friction angleAs the Drucker-Prager yield surface is a smooth version of the Mohr-Coulomb yield surface, it is often expressed in terms of the cohesion c and angle of internal friction that describes the Mohr-Coulomb yield criterion.

The Mohr-Coulomb yield criterion in Haigh-Westergaard space is

Assuming that the Drucker-Prager yield surface circumscribes the Mohr-Coulomb yield surface implies that the two surfaces coincides for a Lode angle . At this point the Mohr-Coulomb yield criterion in Eq. (A.29) reduces to

or rearranged to

which compared to the Drucker-Prager yield criterion in Haigh-Westergaard space, Eq. (A.24), makes it evident that constants A and B then becomes

If on the other hand, it is assumed that the Drucker-Prager surface inscribes the Mohr-Coulomb surface, then the two surfaces as to coincide at a Lode angle θ = 0, which gives

A.2.1.3 Alternative p-q formulationRecalling Eqs. (A.11) and (A.19), the Drucker-Prager criterion of Eq. (A.23) may be expressed as

(A.25)

(A.26)

(A.27)

(A.28)

(A.29)

(A.30)

(A.31)

(A.32)

(A.33)

(A.34)

16[( 1 − 2)2 + ( 2 − 3)2 + ( 3 − 1)2] = + ( 1 + 2 + 3) t c

1√3 t = + t

1√3 c = − c

= 2√3 c t

c + t ; = 1√3 t − c

c + t

t c

√3 sin + 3 − sin cos + 3 − √2 sin( ) = √6 cos = 3

√3 sin 23 − sin cos 23 − √2 sin( ) = √6 cos 1√2 − 2 sin3 + sin = √12 cos3 + sin

= √12 cos3 + sin = 6 cos√3(3 + sin ) ; = 2 sin√3(3 + sin )

= 6 cos√3(3 − sin ) ; = 2 sin√3(3 − sin ) = + tan( )


DNV GL AS

where q is the equivalent von Mises stress, p is the hydrostatic pressure (positive in compression), and d,
β are the material constants, commonly denoted cohesion and friction angle due to their kinship with the Mohr-Coulomb parameters of cohesion c and angle of internal friction .

They are however not of the same magnitudes, but reflect the intercept and slope of the yield surface in the meridional p-q plane as illustrated in Figure A-1 which also shows the cone shape of the Drucker-Prager yield surface in principal stress space.

Figure A-1 Drucker-Prager yield surface illustrated in principal stress space and in the meridional p-q plane

A.2.1.4 Extended formulationsAs observable from Figure A-1, the original Drucker-Prager yield criterion is characterized by having a circular yield envelope in the deviatoric plane and a linear variation in the meridional plane, why it is also commonly referred to at the Linear Drucker-Prager failure criterion.

As mentioned previously in the discussion of Lode angles and meridians (see [A.1.6]); in reality actual materials do not exhibit neither a linear variation with the hydrostatic pressure, nor a circular yield envelope in the deviatoric plane. Hence, the linear Drucker-Prager criterion is an approximation to the real behavior of the material.

Because of this, various extensions have subsequently been added to original Drucker-Prager yield criterion addressing both these issues.

Common extensions are for the meridional plane to introduce either a hyperbolic or a power-law variation. Likewise the circular yield envelope in the deviatoric plane is commonly extended by the introduction of an alternative deviatoric stress measure formulated as function of the Lode angle θ and a shape parameter K such that for all Lode angles whereby the original circular trance of Drucker-Prager (von Mises) is recovered.

The implications of both the hyperbolic and power-law extensions are illustrated in the meridional p-q plane in Figure A-2 together with effect of the alternative deviatoric stress measure t in the deviatoric plane.

Figure A-2 Hyperbolic and power-law extension to the Drucker-Prager yield surface illustrated in the meridional p-q plane together with the deviatoric stress measure t shown in the deviatoric plane

= =

( , , ) = | =1

Deviatoric plane Drucker-Prager (Mises)

=0.8

Curve

Power-Law: t Hyperbolic:


DNV GL AS

Common for all these variations on the Drucker-Prager yield criterion is that none of them imposes any limit
on pure hydrostatic compression. Hence, their applicability for capturing compressive failure at high confining pressure is inherently impaired. If this type of crushing of the material is to be captured, a capping of the yield surface at high confining pressures is needed. Methodologies for obtaining such a capping of the yield surface exist, but are considered outside the present scope.

A.2.1.5 Abaqus implementation notesThe implementation of the Drucker-Prager yield criterion in Abaqus (see /A.5/) takes its outset in the alternative p-q formulation presented previously. Abaqus offers the full suite of variations on the Drucker-Prager yield criterion being linear, hyperbolic, and power-law formulations as well as the possibility of a non-circular failure envelope in the deviatoric plane – although this only for the linear model.

Focusing on the linear model the Drucker-Prager yield criterion is in Abaqus formulated as

in which the cohesion d and the fiction angle β are the governing material parameters, p is the hydrostatic pressure (positive in compression) and t is the deviatoric stress defined in Eq. (A.36).

where q is the equivalent von Mises stress and K is a shape parameter for the failure envelope in the deviatoric plane that – to ensure convexity of the yield surface is confined to .

The term resembles the Lode angle θ in the deviatoric plane in that per definition. Hence the variation of t is linear with .

Using a shape parameter K = 1 implies t = q and thus recovers the original circular trace of the yield envelope in the deviatoric plane (see Figure A-2).

Denoting the uniaxial tensile and compressive strength σt and σc respectively and adopting the original Drucker-Prager formulation by selecting K = 1 the cohesion d and the fiction angle β can be determined as

A.2.1.6 Ansys implementation notesThe implementation of the Drucker-Prager yield criterion in Ansys (see /A.6/) offers the same full suite of variations on the Drucker-Prager yield criterion being linear, hyperbolic, and power-law formulations as well as the possibility of a non-circular failure envelope in the deviatoric plane – although this only for the capped Drucker-Prager model.

Focusing again on linear model the Drucker-Prager yield criterion the Ansys implementation strictly follows the original formulation, although with an implicit tie to the Mohr-Coulomb criterion in that the material parameters for the model are chosen to be the cohesion c and the internal angle of friction β associated with the Mohr-Coulomb yield criterion.

Hence, in Ansys the linear Drucker-Prager yield criterion is formulated as

in which p is the hydrostatic pressure (negative in compression), J2 is the second invariant of the deviatoric stress, and β, σy are material parameters defined as

(A.35)

(A.36)

(A.37)

(A.38)

(A.39)

= − tan − = 0

= 2 1 + 1 − 1 − 1 3

0.778 ≤ ≤ 1.0 ( / )3 cos(3 ) = ( / )3 cos(3 )

= 1 + 13 tan t ; tan = 3 c − t

c + t

= 3 + 2 − y = 0

= 2 sin√3(3 − sin )
DNV GL AS

with φ being the internal angle of friction, and
where c is the cohesion.

Implicitly then, the Ansys implementation forces the circular trace of the Drucker-Prager yield surface in the deviatoric plane to inscribe the corresponding Mohr-Coulomb yield surface which must be considered a conservative approximation to the Mohr-Coulomb yield criterion.

Ansys also offers and alternative linear formulation expressed as

where q is the equivalent von Mises stress, p is the hydrostatic pressure (negative in compression), and α, σy are the governing material parameters.

Comparing this expression with that for the Abaqus implementation, Eq. (A.35), and noting the difference in the sign on the hydrostatic pressure, it is observed that they are identical for K = 1.

Hence, this model may be fitted to the uniaxial tensile and compressive strength σt, σc by Eq. (A.37) with α = tanβ and σy = d, i.e. by

A.2.2 Lubliner-Lee-Fenves (Abaqus concrete damage plasticity model)The Lubilner-Lee-Fenves model is a combination of the original Lubliner model /A.2/ with the modifications for cyclic loading proposed by Lee-Fenves /A.3/.

Together, this forms the basis for the Abaqus concrete damage plasticity model /A.5/ giving it the ability to not only describe first yield of the material, but also the stiffness degradation associated with cyclic loading causing transgression of the yield surface i.e. cracking and crushing.

In the present context however, only the yield criterion associated with first yield is addressed. Thus, the implications of incorporating stiffness degradation by means of a damage concept will be ignored. Hence, the model as described presently will resemble a plasticity model only.

Striped of damage, the yield criterion for the Abaqus concrete damage plasticity model is

where q is the equivalent von Mises stress, p is the hydrostatic pressure (positive in compression), is the maximum principal stress, in relation to which the Macauley bracket is defined as , and α, β, γ are material constants defined as

i.e. dependent on the ratio σb/σc between the biaxial- and uniaxial compressive strengths σb and σc,in which σt is the uniaxial tensile strength, and

where Kc is a shape parameter controlling the yield envelope in the deviatoric plane for , i.e. for purely compressive stress states.

(A.40)

(A.41)

(A.42)

(A.43)

(A.44)

(A.45)

(A.46)

y = 6 cos√3(3 − sin )

= + − y = 0

y = 1 + 13 t ; = 3 c − t

c + t

= 11 − ( − 3 + ⟨ max⟩ − ⟨− max⟩) − c = 0

max max = max( 1, 2, 3) ⟨∙⟩ ⟨ ⟩ = 12(| | + ) = b − c2 b − c

= b c⁄ − 12 b c⁄ − 1

= c

t(1 − ) − (1 + ) = 3(1 − c)2 c − 1

max < 0


DNV GL AS

For , i.e. for stress states exhibiting tension, the corresponding shape parameter is
i.e. directly dependent on the material constant β.

The model introduces the concept of a biaxial compressive strength σb such that for biaxial compression in plane stress, i.e. , the yield criterion reduces to the classic Drucker-Prager yield criterion described previously in [A.2.1] by choosing the material parameter α as defined in Eq. (A.44).

In terms of strengths, the material model is thus defined by uniaxial tensile and compressive strength σt and σc together with the biaxial compressive strength σb and shape parameter Kc.The biaxial compressive strength is typically proportional to the uniaxial compressive strength, why it is commonly expressed via the ratio portion σb/σc. For concrete, experimental values of this this ratio are typically in range 1.10 to 1.16 according to Lubliner et al. /A.2/. Willam and Warnke /A.4/ however suggest it to be commonly equal to 1.20 for concrete.

Regarding the shape parameter Kc controlling the yield envelope in the deviatoric plane for purely compressive stress states, then the model allows for this to be in the range .

The effect the shape parameter Kc has on the yield envelope in the deviatoric plane is illustrated in Figure A-3 together with a schematic of the yield envelope in plane stress. From the figure it is observed that choosing a Kc value approaching results in a ‘Rankine-like’ compression cutoff for purely compressive stress states, whereas choosing Kc equal to 1 leads to a Drucker-Prager like yield criterion for the purely compressive locus of the stress states.

In terms of the full yield surface, the effect of the shape parameter Kc is illustrated in principal stress space in Figure A-4 for the same lower bound ( ), upper bound (Kc = 1), and default value ( ).

The latter is as indicated the default value of Kc based on experimental observations by Lubliner et al. /A.2/ indicating this to be an appropriate shape of the deviatoric variation of the yield criterion for concrete in the purely compressive locus of the stress states.

Figure A-3 Concrete damage plasticity model yield surface illustrated for plane stress in the σ1-σ2 plane and in the deviatoric plane for pure compression shown the effect of the shape parameter Kc

(A.47)

max > 0

t = + 32 + 3

max = 0

½ < c ≤ 1

½

c → 12 c = 23

Deviatoric plane - Pure ompression

Drucker-Prager(Mises)

'Rankine-like' cutoff

Uniaxial compression

Uniaxial tension

Plane tress

Biaxial compression

Biaxial tension


DNV GL AS

Figure A-4 Concrete damage plasticity model yield surface illustrated in principal stress space for the lower and upper limit of the shape parameter Kc together with the default value of 2/3. The relative scale between the yield surfaces is accurate. Moreover, the intercept of the plane stress condition is shown in dashed line, together with the meridians in outline border line. Note that outside the purely compressive locus of the stress states, the yield surface is the same for all choices of the shape parameter Kc, as also evident from the three surfaces shown

A.2.3 Willam-Warnke (Ansys concrete model)The Willam-Warnke model /A.4/ is a three-parameter smooth version of the Mohr-Coulomb yield criterion. The three-parameter model assumes – like the Mohr-Coulomb model and for that matter the linear Drucker-Prager model – that the meridians are straight, i.e. linear functions of the hydrostatic pressure.

The three-parameter Willam-Warnke yield criterion is expressed in terms of the ‘average’ stress scalars σα and τα defined in terms of principal stresses as

such that the yield criterion is

in which σc is the uniaxial compressive strength, z is a material factor, and r(θ) is the assumed elliptical trace of the yield surface in the deviatoric plane given as

in which θ is the Lode angle and r1, r2 are the position vectors that describe the meridians at θ = 0 and , i.e. at the tensile and compressive meridians respectively.

The ‘average’ stress scalars σα and τα are related to the octahedral stresses and stress invariants as

(A.48)

(A.49)

(A.50)

(A.51)

= 13 ( 1 + 2 + 3) = 1√15 ( 1 − 2)2 + ( 2 − 3)2 + ( 3 − 1)2

= 1c

+ 1( ) c− 1 = 0

( ) = 2 2( 22 − 12) cos + 2(2 1 − 2) 4( 22 − 12) cos2 + 5 12 − 4 1 24( 22 − 12) cos2 + ( 2 − 2 1)2

= 3 = oct = 13 1 = = 35 oct = 25 2 = 215 12 − 3 2


DNV GL AS

hence the scalar σα represents the hydrostatic pressure p (signed negative in compression). Another
interpretation by Willam-Warnke /A.4/ is that the stress scalars σα and τα represents the mean distribution of normal and shear stresses on an infinitesimal spherical surface at the stress point.

The free parameters of the model are thus z, r1, and r2, hence the three-parameter name.

These three parameters may be fitted to the uniaxial tensile and compressive strength σt and σc together with the biaxial compressive strength σb.Introducing the two strength ratios αz = σt/σc and αu = σb/σc the uniaxial and biaxial conditions are characterized by

Substituting these strength into the yield criterion of Eq. (A.49), the three free parameters becomes in terms of the strength ratios αz = σt/σc and αu = σb/σcOr expressed directly in terms of the uni- and bi-axial strengths σt, σc and σb as

The resulting yield surface is a cone with an ellipsoidal trace in the deviatoric plane as illustrated in Figure A-5 showing a schematic of the yield surface in the meridional σα-τα plane normalized by the compressive strength σc, and the elliptic trace of the yield surface in the deviatoric plane. The full yield surface is illustrated in Figure A-6 together with plane stress yield envelope.

Convexity of the yield surface requires that the r1, r2 position vectors describing the tensile and compressive meridians respectively are restricted to .

Smoothness and continuity at the compressive meridian ( ) additionally necessitates .

Both of the above requirements will be satisfied if for all position vectors r.

Within these constrains, the apex of the conical yield surface lies on the hydrostatic axis at z giving a triaxial tensile strength of zσc, and the opening angle ϕ of the cone varies between on the tensile meridian and on the compressive meridian.

Figure A-5 Willam-Warnke 3-parameter model yield surface illustrated as a hydrostatic section in the normalized meridional σα-τα plane for θ = 0 and in the deviatoric plane for the validity range

Test Stress state σα /σc τα /σc θ r(θ)

Uniaxial tension 0 r1

Biaxial compression 0 r1

Uniaxial compression r2

(A.52)

(A.53)

1 = t; 2 = 3 = 0 13 z 215 z 1 = 0; 2 = 3 = b −23 u 215 u 1 = 2 = 0; 3 = c −13 215 3

= u z

u − z ; 1 = 65 u z2 u + z

; 2 = 65 u z3 u z + u − z

= b t

c( b − t) ; 1 = 65 b t

c(2 b + t) ; 2 = 65 b t3 b t + c( b − t)

12 > 12 = 3 12 ≤ 1 12 2 < ≤ 2 tan 1 = − 1⁄ tan 2 = − 2⁄

Hydrostatic section for Deviatoric plane


1'Rankine-like' cutoff

12 < 12 ≤ 1


DNV GL AS

Figure A-6 Willam-Warnke 3-parameter model yield envelope for plane stress in the σ1-σ2 plane shown together with the yield surface illustrated in principal stress space with the intercept of the plane stress condition is shown in dashed line, together with the meridians in outline border line

The linear Willam-Warnke model will thus degenerate to the Drucker-Prager model of a circular cone if r1 = r2 = r0 corresponding to producing a circular yield envelope in the deviatoric plan as shown in Figure A-5 in which case the two free parameters are z and r0 and the yield criterion reduces to

The Drucker-Prager model is also recovered if

in which case the resulting Drucker-Prager yield surface effectively is fitted to the uni- and bi-axial compressive strengths σc and σb at the cost of matching the uniaxial tensile strength σt.A.2.3.1 Extended five-parameter modelWhile the linear 3-parameter model offers a good match to the experimentally observed behavior of concrete under low to moderate confinement, in the high pressure regime the correspondence especially on the compressive meridian suffers from the implicitly assumed linear meridians.

To address this, Willam-Warnke /A.4/ proposes an extended 5-parameter model in which the linear meridians are replaced by second order parabolas. Thus, the position vectors r1, r2 that describe the tensile and compressive meridians respectively becomes dependent of the hydrostatic pressure or average stress σα as

and the yield criterion of Eq. (A.49) reduces to

(A.54)

(A.55)

(A.56)

(A.57)

Uniaxial tension t

Uniaxial compression c

Biaxial compression

12 = 1

= 1c

+ 10 c− 1 = 0

z = u3 u −

1( ) = 0 + 1 c + 2 c 2 at = 0, the tensile meridian

2( ) = 0 + 1c

+ 2c

2at = 3, the compressive meridian

= 1( , ) c− 1 = 0


DNV GL AS

where the elliptical trace of the yield surface in the deviatoric plane remains unaltered as
however with the position vectors r1, r2 now functions of the pressure σα as defined in Eq. (A.56).

Yielding thus implies

The yield surface can then be made to match test data by fitting the six degrees of freedom a0, a1, a2 and b0, b1, b2 to the uniaxial tension and compression strengths σt and σc, and the biaxial compression strength σb, together with two strength values in the high compression regime.

Willam-Warnke suggest these additional two strengths be selected at a hydrostatic pressure ξ as

i.e. one on the tensile meridian and one on the compressive meridian.

This with the additional constraint following from triaxial tension state where r1 = r2 must be fulfilled, implies that the two parabolas must pass through a common apex at the equisectrix ( ), which imposes the constraint

which in conjunction with the above gives a full set of constraints as listed in the table below for the position vectors r1, r2, again using the two strength ratios αz = σt/σc and αu = σb/σc, that are to be solved simultaneously.

The extended model with meridians given by second order parabolas thus ends up having a total of five free parameters.

Solving for r1 on the tensile meridian gives

(A.58)

(A.59)

(A.60)

(A.61)

Test Stress state σα /σc τα /σc θ r(σα ,θ)

Uniaxial tension 0 r1(σα)

Biaxial compression 0 r1(σα)

High compression regime Tensile meridian −ξ ρ1 0 r1(σα)

Uniaxial compression r2(σα)

High compression regime Compressive meridian −ξ ρ2 r2(σα)

Triaxial tension ξ0 0 r2(σα)

(A.62)

( , ) ( , ) = 2 2( 22 − 12) cos + 2(2 1 − 2) 4( 22 − 12) cos2 + 5 12 − 4 1 24( 22 − 12) cos2 + ( 2 − 2 1)2

= 0 ⇒c

= ( , )

for = −c

1 =c

at = 02 =

cat = 3

1 = 2 = 3 for 0 =

c 1( 0) = 2( 0) = 0

1 = t; 2 = 3 = 0 13 z 215 z

1 = 0; 2 = 3 = b −23 u 215 u

1 = 2 = 0; 3 = c −13 215 3 3

1 = 2 = 3 3

0 = 23 u 1 − 49 u2 z + 215 u 1 = 13(2 u − z) 2 + 65 z − u2 u + z 2 = 65 ( z − u) − 65 z u + 1(2 u + z)(2 u + z) 2 − 23 u + 13 z − 29 z u


DNV GL AS

in addition to which r1 = r2 = 0 for triaxial tension gives
and then subsequently solving for r2 on the compressive meridian gives

The yield surface will be smooth and convex if the following constraints are fulfilled

The 5-paramter yield surface is conical with ellipsoidal traces in the deviatoric plane of varying shapes following the ratio as illustrated in Figure A-7 showing a schematic of the yield surface in the meridional σα-τα plane normalized by the compressive strength σc, and the elliptic trace of the yield surface in the deviatoric plane. Compared to the original 3-parameter model, the parabolic shape of the position vectors r1, r2 that for the 5-parameter model depend on the pressure σα causes the elliptic yield envelope in the deviatoric plane to vary with the pressure with the validity range , such that as the pressure increase, so does the ratio.

Hereby – as intended – a better description of the compressive meridian is achieved as seen from the full yield surface illustrated in Figure A-8 together with plane stress yield envelope.

This is however at the cost of the model now exhibiting a convex shape on the tensile meridian which effectively means additional tensile capacity compared to that of the 3-parameter model.

Figure A-7 Willam-Warnke 5-parameter model yield surface illustrated as a hydrostatic section in the normalized meridional σα-τα plane for θ = 0 and in the deviatoric plane for the validity range

(A.63)

(A.64)

(A.65)

1( , 0) = 0 ⇒ 0 + 1 0 + 2 02 = 0 ⇒ 0 = − 1 − 12 − 4 0 22 2

0 = − 0 1 − 02 2 1 = + 13 2 + 65 − 3 23 − 1 2 = 2 0 + 13 − 215( 0 + )( + 0) − 13 0 + 13

0 > 0 ; 1 ≤ 0 ; 2 ≤ 00 > 0 ; 1 ≤ 0 ; 2 ≤ 0 and 1( )2( ) > 12 1( ) 2( )⁄

12 < 12 ≤ 1 12

Hydrostatic section for

Deviatoric plane


1'Rankine-like' cutoff

< ≤


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Figure A-8 Willam-Warnke 5-parameter model yield envelope for plane stress in the σ1-σ2 plane shown together with the yield surface illustrated in principal stress space with the intercept of the plane stress condition is shown in dashed line, together with the meridians in outline border line

This can be observed from the convex shape of the plane stress yield envelope between uniaxial tension and uniaxial compression shown in Figure A-8, and from the meridians of the full yield surface shown in the same figure.

The differences are however marginal compared to the gained realism in high pressure domain.

A.2.3.2 Ansys ImplementationThe Ansys implementation takes its outset in the extended 5-parameter Willam-Warnke model. However, only in the purely compressive region of the response is the model used directly. In the remaining regions, a revised or alternate failure criterion is incorporated into the Ansys concrete model as described in the following (see further /A.6/).

Firstly, the failure surface is divided into four distinct domains of the principal stress spaces being:

1) compression-compression-compression, i.e. pure compression

2) tension-compression-compression

3) tension-tension-compression

4) tension-tension-tension, i.e. pure tension

Of these, Domain 1, i.e. pure compression, is as already stated taken to fail as described by the 5-parameter Willam-Warnke model. In the opposite direction, i.e. the pure tension of Domain 4, failure is taken to follow the Rankine maximum tension cutoff criterion. For the remaining two domains that then need to bridge the gap between the Willam-Warnke and Rankine model, then in the Domain 3 (tension-tension-compression) the Rankine criterion is used with a linear ramping down to zero tensile strength at the uniaxial compression interface with the Willam-Warnke model. In the remaining Domain 2 (tension-compression-compression) a revised version of the Willam-Warnke model is used in which the contribution of the tensile stress to the hydrostatic pressure and Lode angle is ignored in combination with a linear ramping down of the revised failure limit to zero at the interface with the Rankine model.

As a result of the above, a smooth transition is obtained between Domains 1, 3, and 4, and between Domains 1, 2 and 4. However, the combined yield surface will be disjoint at the interface between Domains 2 and 3. This fact will be illustrated and commented on further after the Ansys model is presented in detail in the following.

Being that the Ansys concrete model takes its outset in the Willam-Warnke model which basically is


Uniaxial tension t

Biaxial compression

0 ≥ 1 ≥ 2 ≥ 3 ⇔ 1 ≥ 0 ≥ 2 ≥ 3 ⇔ 1 ≥ 2 ≥ 0 ≥ 3 ⇔ 1 ≥ 2 ≥ 3 ≥ 0 ⇔


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formulated in a stress space that is normalized by the uniaxial compressive strength σc, but also needs to
be extended to four discrete stress domains; a generalized failure criterion is adopted. This is expressed as

where the two stress measures F and S then takes on various forms for each of the four domains.

Domain 1 – Compression-Compression-Compression

In the pure compression domain, the five-parameter Willam-Warnke model is used. The failure criterion is thus as defined in Eq. (A.57) which expressed in the generalized form of Eq. (A.66) gives F1 = τα and

, each defined by Eqs (A.48) and (A.58) respectively. Thus, by insertion

and

in which r1, r2 are the pressure dependent position vectors that describe the tensile and compressive meridians respectively given in Eq. (A.56) and θ is the Lode angle defined by the principal stresses as

Introducing the normalized hydrostatic pressure with the hydrostatic pressure defined as i.e. negative in compression, the second order parabolas of Eq. (A.56) describing the

position vectors r1, r2 reduces to

In the Ansys implementation the six degrees of freedom a0, a1, a2 and b0, b1, b2 are fitted to the uniaxial tension strength σt, the uniaxial compression strength σc, and the biaxial compression strength σb, together with two strength values in the high compression regime.

As a variation to the original Willam-Warnke proposal, these two strengths are taken to represent the ultimate compressive strengths of biaxial and uniaxial compression superimposed on an ambient hydrostatic stress state . Hence, they describe a stress state residing on the tensile and compressive meridian respectively but not necessarily in the same deviatoric plane as assumed by Willam-Warnke.

Of these in total six stresses needed to fix the six degrees of freedom associated with the position vectors r1, r2, the biaxial compressive strength σcb and the two ultimate compressive strengths and together with the associated ambient hydrostatic stress , may in the Ansys model all default to depend on the uniaxial compressive strength σc as

albeit with the limitation on the hydrostatic pressure.

As the elliptical trace r of the failure envelope in the deviatoric plane remains identical to that of both the 3- and 5-paramter Willam-Warnke models, its properties on the tensile and compressive meridians remains, i.e. that r = r1 for θ = 0 and r = r2 for respectively. Hence, the same set of equations can be established to fix the six degrees of freedom a0, a1, a2 and b0, b1, b2 for the tensile and compressive meridians. Using the two strength ratios and for uniaxial tension and biaxial

(A.66)

(A.67)

(A.68)

(A.69)

(A.70)

c− = 0

1 = ( , ) -1 = 1√15 ( 1 − 2)2 + ( 2 − 3)2 + ( 3 − 1)2

1 = 2 2( 22 − 12) cos + 2(2 1 − 2) 4( 22 − 12) cos2 + 5 12 − 4 1 24( 22 − 12) cos2 + ( 2 − 2 1)2

cos = 2 1 − 2 − 3√2 ( 1 − 2)2 + ( 2 − 3)2 + ( 3 − 1)2 = / c = ⅓( 1 + 2 + 3)

1 = 0 + 1 + 2 22 = 0 + 1 + 2 2

bu cu ha cbu cu ha

b = 1.2 c bu = 1.45 c cu = 1.725 c ha = cu ≥ −√2 c = 3

z = t c⁄ u = b c⁄


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compressions, together with the three strength ratios for the high compression regime being for
the ambient hydrostatic pressure, and and for the ultimate uni- and bi-axial compression, the system of equations becomes

In the Ansys implementation this set of equations are solved numerically in the following order:

1/ Fit the tensile meridian r1 to the uniaxial tension strength and biaxial compression strengths as

2/ Determine the resulting hydrostatic pressure ξ0 at triaxial tension as defined by r1, i.e. the positive root of the equation

3/ Fit the compressive meridian r2 to the hydrostatic pressure ξ0, uniaxial compression strengths as

Hereby the complete 5-paramter Willam-Warnke model is obtained and is then used unaltered to describe the pure compression region of the Ansys concrete model.

Domain 2 – Tension-Compression-Compression

In the tension-compression-compression domain, Ansys applies a modification to the 5-parameter Willam-Warnke model, in which it is assumed that the ‘average’ stress scalars σα and τα are independent of the magnitude of the tensile stress σ1. The variation of the yield criterion with the tensile stress is instead taken to be a linear variation between uniaxial tension and the plane stress (σ1 = 0) failure envelope for biaxial compression being the interface between Domain 1 and 2.

Hence, the stress measures entering the generalized form of Eq. (A.66) are expressed as

Test Stress state ξ = σα/σc τα/σc θ r(σα,θ)

Uniaxial tension 0 r1(σα)

Biaxial compression 0 r1(σα)

Ultimate biaxial compression 0 r1(σα)

Uniaxial compression r2(σα)

Ultimate uniaxial compression r2(σα)

Triaxial tension 0 r2(σα)

(A.71)

(A.72)

(A.73)

(A.74)

h = ha/ c c = cu/ c b = bu/ c

1 = t; 2 = 3 = 0 13 z 215 z 1 = 0; 2 = 3 = b −23 u 215 u 1 = ha; 2 = 3 = ha + bu − h − 23 b 215 b 1 = 2 = 0; 3 = c −13 215 3 1 = 2 = ha; 3 = ha + cu − h − 13 c 215 c 3 1 = 2 = 3 0 3

1 13 z13 z

21 −23 u −23 u

21 − h − 23 b − h − 23 b

21

2

3

= 215 z

u

b

1( 0) = 0 + 1 0 + 2 02 = 0 ⇒ 0 = − 1 − 12 − 4 0 22 2

1 − 13 191 − h − 13 c − h − 13 c21 0 0

21

2

3

= 215 1c0

-2 = 1√15 ( 2 − 3)2 + 22 + 32


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which is the same as Eq. (A.67) only with σ1 = 0, and
in which p1, p2 are the pressure dependent position vectors that describe the tensile and compressive meridians respectively and ϑ is the revised ‘Lode’ angle. By comparison with Eq. (A.68), this is seen to be the same expression only with the linear variation (1-σ1/σt) added. However, as the Willam-Warnke part is assumed independent of the tensile stress magnitude by setting σ1 = 0, a revised set of position vectors p1, p2 are introduced together with the revised ‘Lode’ angle ϑ.

For the position vectors, the coefficients a0, a1, a2 and b0, b1, b2 are maintained, however the with σ1 = 0, the hydrostatic pressure reduces to , why with the normalized hydrostatic pressure denoted

the revised position vectors following Eq. (A.70) becomes

and the revised ‘Lode’ angle following Eq. (A.69) reduces to

Obviously then, continuity is ensured between Domain 1 and 2, as for their interface defined by σ1 = 0, the linear variation (1 - σ1/σt) = 1, and , as well as . Likewise for uniaxial tension σ1 = σt and σ2 = σ3 = 0, i.e. the interface between Domain 2 and 4, F2 = 0 and the linear variation gives S2 = 0 ensuring continuity with the Rankine criterion of Domain 4.

The interface between Domain 2 and 3 is however discontinuous as already mentioned.

Domain 3 – Tension-Tension-Compression

In the tension-tension-compression domain a linear ramping of the Rankine maximum tension cutoff criterion is adopted. The linear variation is taken to be (1 + σ3/σc) why the pure Rankine criterion is maintained at the interface between Domain 3 and 4 where σ3 = 0. Likewise for uniaxial compression σ3 = –σc being the interface between Domain 3 and 1, continuity is also ensured by this linear variation. Thus, using the Rankine criterion (see Domain 4) the stress measures entering the generalized form of Eq. (A.66) becomes

and

Domain 4 – Tension-Tension-Tension

In the pure tension domain, the Rankine maximum tension cutoff criterion is adopted. Hence, the stress measures entering the generalized form of Eq. (A.66) becomes

and

(A.75)

(A.76)

(A.77)

(A.78)

(A.79)

(A.80)

(A.81)

2 = 1- 1t

2 2( 22 − 12) cos + 2(2 1 − 2) 4( 22 − 12) cos2 + 5 12 − 4 1 24( 22 − 12) cos2 + ( 2 − 2 1)2

= 13( 2 + 3) = / c 1 = 0 + 1 + 2 22 = 0 + 1 + 2 2

cos = − 2 − 3√2 ( 2 − 3)2 + 22 + 32 = | =1,2 cos = cos

3 = 1

3 = t

c1 + 3

c

4 = 1

4 = t/ c


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A.2.3.3 Combined yield surface
With the yield surface defined for each of the four domains, the combined yield surface of the Ansys concrete model thus follows the generalized form of Eq. (A.66) and becomes

The combined yield surface will be continuous between all domains except for the interface between Domain 2 and 3 i.e. it is discontinuous between the tension-compression-compression domain and the tension-tension-compression domain.

The resulting yield envelope in plane stress is illustrated overleaf in Figure A-9. This figure also shows the full yield surface in principal stress space. The afore mentioned discontinuity between Domain 2 and 3 is visible in both illustrations, but due to scale most dominantly in the plane stress yield envelope, why it has been highlighted in red line for the yield surface in principal stress space.

Compared to the original 5-paramter Willam-Warnke model shown previously in Figure A-8 it is observed that the introduction of the Rankine criterion adopted in pure tension (Domain 4) adds a small amount of additional tensile capacity in this region. Contrary, a small amount of capacity is removed from the tension-tension-compression region (Domain 3) by the adopted linear scaling of the Rankine criterion. The tension-compression-compression region (Domain 2) does however, apart from not matching the plane stress yield envelope of Domain 3, also introduce a significant reduction in tensile capacity for this region (Domain 2) as the linear scaling of the Willam-Warnke yield criterion cause a very steep drop towards no tensile capacity at low hydrostatic pressures.

Figure A-9 Ansys concrete model yield envelope for plane stress in the σ1-σ2 plane shown together with the yield surface illustrated in principal stress space with the intercept of the plane stress condition is shown in dashed line, together with the meridians in outline border line. The discontinuous gap zone between Domain 2 and 3, i.e. between the tension-compression-compression region and the tension-tension-tension region is for the full surface in principal stress space shown in thin red

(A.82)c

− = 0 ; = 1 for 0 ≥ 1 ≥ 2 ≥ 3 (Domain 1)2 for 1 ≥ 0 ≥ 2 ≥ 3 (Domain 2)3 for 1 ≥ 2 ≥ 0 ≥ 3 (Domain 3)4 for 1 ≥ 2 ≥ 3 ≥ 0 (Domain 4)


Uniaxial tension t

Biaxial compression

Domain 3

Domain 2


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A.3 References
/A.1/ Soil Mechanics and Plastic Analysis or Limit Design,Drucker, D. C., and W. Prager, Quarterly of Applied Mathematics, Vol. 10, pp. 157165, 1952

/A.2/ A Plastic-Damage Model for Concrete,Lubliner, J., J. Oliver, S. Oller, and E. Oñate, International Journal of Solids and Structures, Vol. 25, No. 3, pp. 229326, 1989

/A.3/ Plastic-Damage Model for Cyclic Loading of Concrete Structures,Lee, J., and G. L. Fenves, Journal of Engineering Mechanics, vol. 124, no.8, pp. 892–900, 1998

/A.4/ Constitutive Model for the Triaxial Behavior of Concrete,Willam, K. J., and E. D. Warnke, Proceedings, International Association for Bridge and Structural Engineering, Vol. 19, ISMES, Bergamo, Italy, p. 174, 1975

/A.5/ Abaqus Theory Manual,Simulia, Dassault Systèmes Simulia Corp., Providence, RI, USA, Ver. 610, 2010

/A.6/ Theory Reference for ANSYS and ANSYS Workbench,ANSYS Inc. Canonsburg, PA, USA, Rel. 14.0, 2011


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APPENDIX B CONTACT MODELING METHODOLOGIES
In finite element analysis the inclusion of contact interaction requires the handling to two principal problems, namely penetration and sliding.

While contact interaction can be between any number of bodies down to self-contact of one body, for the purpose of this general introduction contact between two deformable bodies is chosen as a simple system to illustrate the basics.

Figure B-1 Basic master – slave contact definition between two deformable bodies

The basic 2-body contact problem is illustrated in Figure B-1 where the two meshed bodies and their contact surfaces are illustrated. The figure also introduces the concept of a master and slave surface, which is the fundamental approach used in finite element analysis.

B.1 Penetration: Pressure interaction in the normal directionThe basic problem in contact is to detect and resolve penetrations between the contacting bodies. Therefore, to handle contact in finite element analysis the first step is to have a tracking of the slave surface nodes position relative to the master surface as part of the solution scheme.

Considering again the basic 2-body contact problem, the tracking solution can result in thee possible configurations namely open, closed, or overclosed as illustrated in Figure B-2.

Figure B-2 Simplified schematic contact interaction in the normal direction

For these three configurations, commonly denoted contact states, the open state obviously needs no further handling. Likewise, the closed state does not warrant any further handling in that it either resembles the 2 bodies coming into initial contact by just touching, or the solved condition of a previous overclosure. Thus, the closed state is per definition in equilibrium being it either without any interface forces for the initial just touching scenario, or with a set of interface forces derived from a previously resolved overclosure.

Consequently, only if the tracking yields any overclosures is there a need for introducing additional contact constraints that subsequently must be solved to resolve the overclosed condition whereby the system is returned to being in a closed state equilibrium.

Slave urface

Master urface

Open Closed Overclosed

Slave urface

Master urface


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Figure B-3 Illustration of the contact interaction solution scheme by either the Lagrangian or Penalty method

Finite element analysis basically solves the linear system of equations where is the nodal forces vector, is the global stiffness matrix, and is the nodal displacements vector.

Within this regime of solving linear equations two different methods of introducing the contact interaction are commonly used namely the Lagrangian method and the penalty method. For the contact penetration interaction the problem and solution methods are illustrated in Figure B-3.

In the Lagrangian approach, the overclosure triggers a set of constraints between the penetrating slave node and the master surface that enforces the gap distance to be equal to zero, i.e. back to the closed state. Hence, the Lagrangian method represents an exact solution to the contact interaction.

The penalty method on the other hand, enforces the contact condition via ‘springs’ inserted between the penetrating slave node and the master surface as sketched in Figure B-4. The stiffness of these ‘springs’ is termed the penalty stiffness κn. Hence, if the penalty stiffness is chosen sufficiently large, the penalty method yields a stiff approximation to the Lagrangian method, albeit allowing for a finite amount of penetration between the two contact surfaces.

Figure B-4 Schematic diagram of the linear penalty method

Irrespectively of the method applied, obtaining a converged solution where there is no penetration requires and iterative procedure through which the two bodies are ‘eased’ into equilibrium with the contact forces acting between them while the penetration is resolved.

Ideally then, the exact solution yielded by the Lagrangian method should be the preferred choice for enforcing contact in finite element analyses. However, in practice doing so comes at a price in terms of additional computational effort.

As mentioned, the Lagrangian method enforces contact by means of constraints whereby additional degrees of freedom are effectively added which in turn will not only increase the number of equations but also require a reformulation of the global stiffness matrix. The penalty method on the other hand needs only to update the nodal force vector with a balanced pair of forces equal to the ‘spring’ reaction to the present overclosure, and thus do not introduce additional degrees of freedom to the system.

Moreover, for contact conditions prone to chattering, i.e. multiple nodes moving in and out of contact during the iterative solution, the Lagrangian method notoriously suffers significantly in performance due to its exact and in essence infinitely rigid interface constraining, leading to a slower convergence not only in the individual contact iterations to resolve overclosures, but also in the general load incrementation associated with the generally nonlinear solution method.

Contrary, the penalty method excels at providing computational performance in these conditions given its more forgiving ‘soften’ spring approximation to the contact penetration. This while at the same time providing a reasonable accuracy provided that an adequate penalty stiffness is used.

Solved – Closed

Slave urface

Master urface

Overclosed

Lagrangian ethod

Penalty ethod

=

Slave urface

Master urface

Solved

Contact pressure

Overclosure

(Linear penalty stiffness)


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In relation to this it should be noted that although the penalty method in theory converges towards the
Lagrangian approach as the penalty stiffness escapes towards infinity, in practice choosing too large penalty stiffness will cause the global stiffness matrix to become so ill-conditioned that the general solution accuracy is lost.

Nevertheless, weighing the pros and cons of the two methods in relation of the present task of solving the contact interaction between the steel and the grout in a grouted connection, the computational cost of using the Lagrangian approach combined with its susceptibleness to contact chatter, makes it in light of the large areas of the two interfaces an impractical choice.

Thus, the penalty method with due calibration of the penalty stiffness is recommended for modeling the contact interaction of grouted connections.

B.2 Sliding: Shear interaction on the surfaceWhen the two contacting surfaces have frictional characteristics, solving the contact interaction needs to be extended from the previously discussed method for resolving penetrations to include a handing of the stick/slip characteristics of friction.

In principle, this can like the contact penetration be done using either a Lagrangian approach of introducing displacement constraints and solving for these simultaneously with those arising from the penetration, or it can be done using a penalty method with a new set of ‘springs’ acting in the surface tangent plane.

Having previously argued the selection of the penalty method for enforcing penetration contact based on computational efficiency and robustness, the same considerations applied to the shear interaction only further enforces the previous choice. Obviously, the Lagrangian method will produce also for the shear interaction an exact solution. However, let it suffice to state that that exactness, in relation to the highly nonlinear transgression from sticking to slipping friction, adds even more computational effort not only in terms of additional degrees of freedom, but also in the amount of iterations required and size of stable load increments.

Hence, in the following the discussion will be limited to only implementation of friction by means of the penalty method. Moreover, to reduce the complexness, the discussion is limited to that of classical static Coulomb friction, i.e. a linear relation between contact pressure and frictional shear without any upper limit. The proportionality factor is thus the classic static Coulomb friction coefficient denoted μ whereby the friction model becomes as illustrated in Figure B-5.

Figure B-5 Schematic of the linear Coulomb friction model and the penalty implementation of it

The first thing to note is that the shear interaction by nature is a limited capacity unlike the penetration interaction which in principle offers unlimited capacity. That is, while the contact pressure in principle is without any upper limit, the frictional shear capacity is limited by whatever magnitude of contact pressure that exists and thus has a finite magnitude.

Phenomenologically, the resulting behavior is that when in contact two surfaces will stick for any magnitude

Contact pressure

(constant friction coefficient)

Sticking region

Shear stress

Total slip

(shear stiffness)

Sticking region

Shear stress

Sticking friction Slipping friction

Critical shear stress

Elastic slip e


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of external shear force up until the frictional shear capacity proportional to the presently prevailing contact
pressure is reached. External shear loading beyond this finite shear capacity will result in the two surfaces sliding relative to each other.

The two conditions are denoted static and kinematic friction respectively.

In reality, the transition from static to kinematic friction will be accompanied with a reduction in the friction coefficient where the friction coefficient in the kinematic region normally will exhibit a dependency on the slip rate (velocity) typically described by a logarithmic decline.

In classical static Coulomb friction, the change in friction coefficient in the slipping condition is ignored why the kinematic behavior becomes that of a constant shear capacity termed the critical shear stress .

This simplification does however not remove the stick-slip behavior of the shear interaction. It merely simplifies the slipping condition to be a constant capacity.

The behavior as it manifests itself in a penalty formulation is illustrated in Figure B-5.

Akin to the penalty implementation for penetration interaction, the penalty shear interaction is a stiff approximation to the Lagrangian method, where the frictional shear capacity is modeled via ‘springs’ between the slave nodes in contact and the master surface. The stiffness of these ‘springs’ is denoted the shear penalty stiffness κs and if chosen appropriately will result in a good approximation to the exact Lagrangian approach.

As a result of the finite shear penalty stiffness of the ‘springs’ it will require a finite relative displacement between two contacting points to build up before the full frictional shear capacity, the critical shear stress, is achieved in the interface. This displacement is denoted the elastic slip γe and will for the penalty method be a constant contribution to the total slip γ in the slipping state as illustrated in Figure B-5.

Consequently, in the sticking state there will in fact be a total slip γ ranging between zero and the elastic slip γe as required to attain equilibrium with the prevailing magnitude of external shear load on the interface.

In terms of specifying an appropriate shear penalty stiffness then as it is common practice to do so indirectly through the elastic slip γe as it represent an actual physical length measure that can be related to the physical model.

crit = c

s = crit/ e = c / e


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