Analysis of grouted connections using the finite element method
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1 RECOMMENDED PRACTICE DNVGL-RP-0419 Edition September 2016 Analysis of grouted connections using the finite element method The electronic pdf version of this document found through is the officially binding version. The documents are available free of charge in PDF format.
2 FOREWORD DNV GL recommended practices contain sound engineering practice and guidance. September 2016 Any comments may be sent by to 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 this document 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 of this document.
3 CHANGES CURRENT General This is a new document. Changes current Recommended practice, DNVGL-RP-0419 Edition September 2016 Page 3
4 CONTENTS CHANGES CURRENT... 3 Sec.1 Introduction Objectives Validity Definitions Terms Acronyms, abbreviations and symbols Acronyms and abbreviations Symbols References Grouted connection concepts...8 Sec.2 Basic considerations Limit state safety format Load and resistance factor design principle Design by load and resistance factor design and the finite element method Empirical basis for the resistance Characteristic resistance Types of failure Analysis of grouted connections Plausibility check...14 Sec.3 Materials Material models for steel Linear Nonlinear Material models for grout Linear and nonlinear models Plasticity Drucker-Prager Willam-Warnke Lubliner-Lee-Fenves Selection of material model and properties Specifying nonlinear properties...22 Sec.4 Contact interactions Contact modeling Normal contact Sliding contact Calibration...24 Sec.5 Finite element method modeling Solution schemes Explicit and implicit solution method Choosing a solution scheme Modeling schemes Submodeling Geometric extent Overall geometry...29 Contents Recommended practice, DNVGL-RP-0419 Edition September 2016 Page 4
5 5.3.2 Features Tolerances and imperfections Symmetry utilization Element selection Element shape, order, and integration scheme Element density...35 Sec.6 Boundary conditions and load application Boundary conditions Load application Global and local loads Ultimate limit state load cases Finite limit state load cases Serviceability limit state load cases Accidental limit state load cases...42 Sec.7 Limit state analyses Ultimate limit state Response assessment Buckling Fatigue limit state Serviceability limit state Accidental limit state...46 App. A Constitutive formulations for grout App. B Contact modeling methodologies Contents Recommended practice, DNVGL-RP-0419 Edition September 2016 Page 5
6 SECTION 1 INTRODUCTION 1.1 Objectives This 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 Validity The 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 /16/. It is further assumed that the fabrication of the structure comply with the standard requirement associated with the governing standard. 1.3 Definitions Terms This recommended practice uses terms as defined in DNVGL-ST-0126 /9/. Additional terms used are: Term dynamic static quasi-static Definition a load or load effect that is dependent on time and inertia effects a load or load effect that is independent of time a load or load effect that is dependent on time but with negligible inertia effects 1.4 Acronyms, abbreviations and symbols Acronyms and abbreviations Acronyms and abbreviations as shown in Table 1-1 are used in this recommended practice. Table 1-1 Acronyms and abbreviations Abbreviation Description 1D, 2D, 3D 1-, 2-, and 3-dimensional ALS accidental limit state FEA finite element analysis FEM finite element method FLS fatigue limit state LRFD load and resistance factor design rebar reinforcement bar SCF stress concentration factor SLS serviceability limit state ULS ultimate limit state Recommended practice, DNVGL-RP-0419 Edition September 2016 Page 6
7 1.4.2 Symbols Latin characters A c d D E F G F h K l L p q R S u t Δt w area speed of dilatation cohesion diameter modulus of elasticity (Young s module) load fracture energy height shape parameter length length pressure equivalent stress resistance load effect displacement time or thickness or deviatoric stress time step or increment width Greek characters β material friction angle δ small length, imperfection magnitude e engineering strain ε true strain (logarithmic strain) flow potential eccentricity γ f load partial safety factor γ m material safety factor ν Poisson ratio φ resistance factor θ Lode angle ρ specific density Ψ( ) load effect function ψ material dilation angle s engineering stress (nominal stress) σ true stress Scripts c d e g k p s t y compressive design value or dynamic elastic grout characteristic value plastic or pressure steel tensile yield Recommended practice, DNVGL-RP-0419 Edition September 2016 Page 7
8 1.5 References 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 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, /12/ EN Eurocode 3: Design of Steel Structures, Part 1-5: Plated Structural Elements, /13/ EN Eurocode 3: Design of Steel Structures, Part 1-6: Strength and Stability of Shell Structures, /14/ IIW Document XIII /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, /15/ ISO 2394 General Principles on Reliability for Structures, Second Edition, /16/ ISO Fixed Steel Offshore Structures Petroleum and Natural Gas Industries, /17/ Norsok N-004 Design of Steel Structures, Norsk Standard, Rev. 3, /18/ NRL MP U The Analysis of Load-Time Histories by means of Counting Methods, J.B. de Jonge, National Aerospace Laboratory (NRL) The Netherlands, /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 , Ernst & Sohn Verlag. 1.6 Grouted connection concepts Grouted 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. Recommended practice, DNVGL-RP-0419 Edition September 2016 Page 8
9 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. Inclined pile in leg Inclined pile in sleeve Vertical pile in sleeve Leg in pile 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. Recommended practice, DNVGL-RP-0419 Edition September 2016 Page 9
10 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. Transition Piece Grout Shear Keys Monopile Figure 1-2 Illustration of the grouted connection for monopile foundations. Left cylindrical with shear keys. Right conical without shear keys Recommended practice, DNVGL-RP-0419 Edition September 2016 Page 10
11 SECTION 2 BASIC CONSIDERATIONS 2.1 Limit state safety format A 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. Requirement Probabilityensity Design oad ( ) Mean oad Designesistance ( ) Resistanceactor Characteristic esistance Mean esistance Load and esistance Figure 2-1 Illustration of the limit state safety format The limit state is then formally expressed by the equation: d = d, why the design requirement that the design load does not exceed the design resistance can be written as d d d d (2.1) Recommended practice, DNVGL-RP-0419 Edition September 2016 Page 11
12 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 98 th percentile of the load and the 5 th 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 F d is obtained by multiplying the characteristic load F k by a given load factor γ f d = f k (2.2) 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 S d 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 F d,i may be expressed as d = Ψ( d,1, d,2,, d, ) (2.3) A design resistance R d is obtained by multiplying the characteristic resistance R k by a given resistance factor φ d = k (2.4) The resistance factor φ relates to the material factor γ m as = 1 m (2.5) 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 S d does not exceed the design resistance R d for said load effect i.e. d d (2.6) Recommended practice, DNVGL-RP-0419 Edition September 2016 Page 12
13 2.1.2 Design by load and resistance factor design and the finite element method The 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 resistance All 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 resistance The 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 failure It 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 /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. Recommended practice, DNVGL-RP-0419 Edition September 2016 Page 13
14 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 γ f γ m. 2.5 Analysis of grouted connections Grouted 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 Plausibility check It 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 /16/ as a plausibility check of the FEA response. Recommended practice, DNVGL-RP-0419 Edition September 2016 Page 14
15 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 Linear For 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 Nonlinear Exceptions 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 /12/. 3.2 Material models for grout The 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 Linear and nonlinear models The 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. Recommended practice, DNVGL-RP-0419 Edition September 2016 Page 15
16 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 Plasticity The 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 ε e that is fully recoverable and an inelastic (plastic) strain ε p that cannot be recovered. In terms of strain increments the plasticity model can be directly formulated as = e + p. 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 () = 0 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, p = 0, and the plastic strain increments are determined by the flow rule p = ()/ 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, p = 0, 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 Strain Hardening The 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 Recommended practice, DNVGL-RP-0419 Edition September 2016 Page 16
17 defined by the stress point ( e c, e c ) = (1/, 1) c where E is the elastic modulus, σ c 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 E cdk is used to describe the linear elastic part of the response in both tension and compression, i.e. that E = E cdk. Hereafter the material yields and hardens to the plastic limit taken to occur at the stress point 0 c, 0 c = (1/ csk, 1) c in which E csk is the characteristic compressive static modulus of elasticity after with the behavior is ideal plastic. The compressive stress-strain relationship is then expressed as for 0> c e c () = + ( 1) c + c for ( e 0 ) c > c c 0 c for 0 c > (3.1) in which the shape parameter m is this adaptation becomes the ratio between the initial and static modulus of elasticity, i.e. = / csk. 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 G F, 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 σ t is thus depleted when an elongation of t = 2 F / t is experienced by the grout, why p u t may be described as the crack displacement. Obviously then, the corresponding crack strain t will depend on the element length l as p t = t /, 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. Recommended practice, DNVGL-RP-0419 Edition September 2016 Page 17
18 3.2.3 Drucker-Prager The original linear Drucker-Prager yield criterion is a straight line in the p-q plane given as = + tan (3.2) 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 = tan (3.3) 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 pl 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 pl 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. Hardening 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 = (3.4) Recommended practice, DNVGL-RP-0419 Edition September 2016 Page 18
19 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 = + tan (3.5) The term (/) 3 resembles the Lode angle θ in the deviatoric plane in that cos(3) = (/) 3 per definition. Hence the variation of t is linear with cos(3) (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). Compressive meridian Tensile meridian =0.8 Curve Deviatoric plane Drucker-Prager (Mises) 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 Willam-Warnke The 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]. Recommended practice, DNVGL-RP-0419 Edition September 2016 Page 19
20 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 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-tensiontension region is shown in thin red Lubliner-Lee-Fenves The 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 K c that can assume any value between 1/2 and 1. The implication of the K c parameter on the yield surface is shown in Figure 3-5. The recommended default value for the K c 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. Recommended practice, DNVGL-RP-0419 Edition September 2016 Page 20
21 Apart from the uniaxial strengths, the damage plasticity model yield surface is defined by the biaxial compression strength and the previously described K c 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 K c 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 K c 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 K c 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 = ( t tan ) tan (3.6) where q is the equivalent von Mises stress, p is the hydrostatic pressure (positive in compression), and σ t 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 properties The 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]. Recommended practice, DNVGL-RP-0419 Edition September 2016 Page 21
22 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/ Specifying nonlinear properties General 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 = / 0 i.e. force F per original undeformed reference area A 0, and engineering strain as = ( 0 )/ 0, i.e. elongation being the current length l minus the original undeformed reference length l 0 relative to again the original undeformed reference length l 0. 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. = 0 = ln ( 0 ) (3.7) 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. 0 0 = the true stress σ and strain ε can be related to the engineering stress s and strain e as 1 = (1 + ) and = ln (1 + ) (3.8) 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 = = = = = 0 = = 0+ 0 = =1+ =(1+) and =ln =ln 0+ 0 =ln1 + 0 =ln(1+) Recommended practice, DNVGL-RP-0419 Edition September 2016 Page 22
23 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 modeling Depending 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 Normal contact For 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 [ ]. 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 l e for any of the two steel tubulars can be taken as 4 e = 4 rd (4.1) with E being the elastic modulus of steel and I the moment of inertial for the tubular in question. The foundation spring stiffness rd can be estimated based on the radial stiffness of the combined cross section as rd = (2 + 3) 2 i + o 2 + g i o g (4.2) where R is the radius of the tubular in question, being then either R i for the inner tubular with thickness t i or R o for the outer tubular with thickness t o. The grout has the thickness t g and elastic modulus E g, and the coefficient of friction between the steel and grout is μ. Recommended practice, DNVGL-RP-0419 Edition September 2016 Page 23
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