DNV-RP-0419
Edition September 2016
Amended October 2021
Analysis of grouted connections using the
finite element method
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RECOMMENDED PRACTICE
DNV recommended practices contain sound engineering practice and guidance.
©
DNV AS September 2016
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FOREWORD
The numbering and/or title of items containing changes is highlighted in red.
Amendments October 2021
Topic
Rebranding to DNV
Reference
Description
All
This document has been revised due to the rebranding of DNV
GL to DNV. The following have been updated: the company
name, material and certificate designations, and references to
other documents in the DNV portfolio. Some of the documents
referred to may not yet have been rebranded. If so, please see
the relevant DNV GL document. No technical content has been
changed.
Editorial corrections
In addition to the above stated changes, editorial corrections may have been made.
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Changes - current
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CHANGES – CURRENT
Section 1 Introduction............................................................................................ 6
1.1 Objectives......................................................................................... 6
1.2 Validity..............................................................................................6
1.3 Definitions.........................................................................................6
1.4 Acronyms, abbreviations and symbols.............................................. 6
1.5 References........................................................................................ 9
1.6 Grouted connection concepts............................................................9
Section 2 Basic considerations.............................................................................. 12
2.1 Limit state safety format................................................................ 12
2.2 Empirical basis for the resistance................................................... 14
2.3 Characteristic resistance.................................................................15
2.4 Types of failure...............................................................................15
2.5 Analysis of grouted connections..................................................... 15
Section 3 Materials................................................................................................17
3.1 Material models for steel................................................................ 17
3.2 Material models for grout............................................................... 17
3.3 Selection of material model and properties.................................... 25
Section 4 Contact interactions.............................................................................. 27
4.1 Contact modelling........................................................................... 27
Section 5 Finite element method modeling........................................................... 30
5.1 Solution schemes............................................................................ 30
5.2 Modeling schemes...........................................................................32
5.3 Geometric extent............................................................................ 33
5.4 Element selection............................................................................39
Section 6 Boundary conditions and load application............................................. 42
6.1 Boundary conditions....................................................................... 42
6.2 Load application..............................................................................43
Section 7 Limit state analyses.............................................................................. 48
7.1 Ultimate limit state.........................................................................48
7.2 Buckling.......................................................................................... 49
7.3 Fatigue limit state.......................................................................... 49
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Contents
Changes – current.................................................................................................. 3
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CONTENTS
Appendix A Constitutive formulations for grout.................................................... 52
A.1 Stress modeling basics................................................................... 52
A.2 Yield surfaces................................................................................. 58
A.3 References...................................................................................... 80
Appendix B Contact modeling methodologies....................................................... 81
B.1 Penetration: Pressure interaction in the normal direction...............81
B.2 Sliding: Shear interaction on the surface........................................83
Changes – historic................................................................................................ 85
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Contents
7.5 Accidental limit state...................................................................... 51
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7.4 Serviceability limit state................................................................. 51
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. DNV-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. DNV-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 Terms
This recommended practice uses terms as defined in DNV-ST-0126 /9/. Additional terms used are:
Term
Definition
dynamic
a load or load effect that is dependent on time and inertia effects
static
a load or load effect that is independent of time
quasi-static
a load or load effect that is dependent on time but with negligible inertia effects
1.4 Acronyms, abbreviations and symbols
1.4.1 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
1D, 2D, 3D
Description
1-, 2-, and 3-dimensional
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SECTION 1 INTRODUCTION
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Abbreviation
Description
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
1.4.2 Symbols
1.4.2.1 Latin characters
A
area
c
speed of dilatation
d
cohesion
D
diameter
E
modulus of elasticity (Young’s module)
F
load
GF
fracture energy
h
height
K
shape parameter
l
length
L
length
p
pressure
q
equivalent stress
R
resistance
S
load effect
u
displacement
t
time or thickness or deviatoric stress
Δt
time step or increment
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w
width
1.4.2.2 Greek characters
β
material friction angle
δ
small length, imperfection magnitude
e
engineering strain
ε
true strain (logarithmic strain)
ϵ
flow potential eccentricity
γf
load partial safety factor
γm
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
1.4.2.3 Scripts
c
compressive
d
design value or dynamic
e
elastic
g
grout
k
characteristic value
p
plastic or pressure
s
steel
t
tensile
y
yield
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Table 1-2 DNV 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/
DNV-OS-C401
Fabrication and testing of offshore structures
/6/
DNV-RP-C203
Fatigue design of offshore steel structures
/7/
DNV-RP-C208
Determination of structural capacity by non-linear finite element analysis methods
/8/
DNV-SE-0190
Project certification of wind power plants
/9/
DNV-ST-0126
Design of wind turbine support structures
/10/
DNV-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.
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.
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1.5 References
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.
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.
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In offshore applications, structural members with cylindrical cross sections dominate as these in general are
favorable when exposed hydro-dynamic and static loading.
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.
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
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Common for all jackets with vertical piles is the need for shear keys to provide axial capacity of the grouted
connection.
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.
Figure 2-1 Illustration of the limit state safety format
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SECTION 2 BASIC CONSIDERATIONS
design load Sd does not exceed the design resistance
, why the design requirement that the
Rd can be written as
(2.1)
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
th
th
the 98 percentile of the load and the 5 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
(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 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
(2.3)
A design resistance
ϕ
Rd is obtained by multiplying the characteristic resistance Rk by a given resistance factor
(2.4)
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The limit state is then formally expressed by the equation:
ϕ relates to the material factor γm as
(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 Sd does not exceed the design resistance
said load effect i.e.
Rd for
(2.6)
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.
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The resistance factor
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 DNVST-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.
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 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.
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2.3 Characteristic resistance
It is generally recommended to compare results attained from FEA with analytical expression from available
standards on grouted connections, e.g. DNV-ST-0126 /9/, NORSOK N-004 /17/, and ISO 19902 /16/ as a
plausibility check of the FEA response.
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2.5.1 Plausibility check
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 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.
3.1.2 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 DNV-RP-C208 /7/ or
EN 1993-1-5 /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.
3.2.1 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.
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SECTION 3 MATERIALS
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 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
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
dε = dεe + dεp.
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 f(σ) = 0 where the function
dependent not only on the stress σ but any number of material constants.
f may be
Post-yielding the hardening rule describes the change in the yield surface, why formally f(σ,εp) = 0, and
the plastic strain increments are determined by the flow rule dεp = dλ∂g(σ)/∂σ 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 f(σ,εp) = 0,
whereby the magnitude of the plastic strain increment is found.
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
The plastic potential
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 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
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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.
Following the notation that positive stresses and strains are tensile, the compressive elastic limit is thus
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 DNVOS-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
defined by the stress point
E = Ecdk.
Hereafter the material yields and hardens to the plastic limit taken to occur at the stress point
response in both tension and compression, i.e. that
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
(3.1)
in which the shape parameter
of elasticity, i.e.
m = E/Ecsk.
m is this adaptation becomes the ratio between the initial and static modulus
The resulting compressive hardening is sketched in Figure 3-1.
Figure 3-1 Tensile strain softening and compressive strain hardening
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followed by a power-law hardening up till the total compressive strength. Hereafter the behavior is taken to
be ideal-plastic.
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.
ut = 2GF/σt is experienced by the grout, why
utmay be described as the crack displacement. Obviously then, the corresponding crack strain εtp will depend
p
on the element length l as εt = ut/l, why as mentioned a mesh size dependency is introduced explaining
The tensile strength σt is thus depleted when an elongation of
why the grout elements should then ideally be equal sized cubes, unless the strain softening is specified
based on the fracture energy.
3.2.3 Drucker-Prager
The original linear Drucker-Prager yield criterion is a straight line in the
p-q plane given as
(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
(3.3)
where ψ is the dilation angle. A geometric interpretation of ψ is included in the p-q diagram shown in Figure
pl
3-2. In the original Drucker-Prager model the plastic strain increment dε 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
pl
increment dε 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.
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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.
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
t: Lode angle
(3.4)
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
(3.5)
The term
resembles the Lode angle
θ in the deviatoric plane in that
per definition.
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
Hence the variation of
envelope in the deviatoric plane (see Figure 3-3).
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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
p-q plane and the
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-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].
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
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Figure 3-3 Drucker-Prager yield surface extension illustrated in the meridional
deviatoric plane
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-compressioncompression region and the tension-tension-tension region is shown in thin red
3.2.5 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
that can assume any value between 1/2 and 1.
Kc
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.
The implication of the
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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compressive strength, and in lieu of actual data are suggested taken as 1.45 and 1.725 times the uniaxial
compressive strength respectively.
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
(3.6)
q is the equivalent von Mises stress, p is the hydrostatic pressure (positive in compression), and σt is
ψ measured in the p-q plane
at high confining pressure and ϵ commonly referred to as the eccentricity, that defines the rate at which the
where
the tensile yield stress. Specific to the flow potential is then is the dilation angle
function approaches the asymptote (the flow potential tends to a straight line as the eccentricity tends to
zero).
ϵ 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 ϵ = 0.1, which implies that the material has almost the same
The flow potential eccentricity
dilation angle over a wide range of confining pressure stress values. Increasing the value of the eccentricity
ϵ provides more curvature to the flow potential, implying that the effective dilation angle increases more
rapidly as the confining pressure decreases.
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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.
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].
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 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 s = F/A0 i.e. force F per original undeformed reference area A0, and engineering strain as e = (l –
l0)/l0, 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 dε = dl/l 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.
(3.7)
σ = F/A or simply the force F per current deformed area A.
Assuming that the material volume remains constant, i.e. A0l0 = Al the true stress σ and strain ε can be
1
related to the engineering stress s and strain e as
The corresponding true stress is defined as
(3.8)
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3.3 Selection of material model and properties
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
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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.
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 modelling
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.
4.1.1 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 [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
(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
can be estimated based on the radial stiffness of the combined cross section
as
(4.2)
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SECTION 4 CONTACT INTERACTIONS
or
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 DNV-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
gives rise to by friction, are the only forces acting on the grout, i.e.
principal stresses
,
σp together with the shear it
, and
, then the
can by application of Mohr’s circle for plane stress be derived as:
(4.3)
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 considerations
If 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 Calibration
It 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.
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R is the radius of the tubular in question, being then either Ri for the inner tubular with thickness ti
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.
where
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.
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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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.
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 schemes
In FEM the general equation of motion for a continuum can at any time
t be formulated as
(5.1)
where
is mass matrix,
is the damping matrix,
the stiffness matrix,
is the vector of the external
forces, and u 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
(5.2)
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 timedependent, then these can be solved using either the explicit or implicit method.
5.1.1 Explicit and implicit solution method
The 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.
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SECTION 5 FINITE ELEMENT METHOD MODELING
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
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 problems
When 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 considerations
While 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.
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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.
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.
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 scheme
Historically, 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 schemes
It 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 DNV-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 pilesleeve 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 DNV-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 Submodeling
A 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
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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.
5.3 Geometric extent
Any 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
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 geometry
The 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.
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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.
5.3.2 Features
It 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.
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 spaces
Spacers 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 keys
Shear 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 Rebar
Rebars 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 Seals
Seals 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.
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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].
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 supports
Inherent to the grouted connections is the need for temporary support during offshore installation. The
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 Externals
External 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.
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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.
Tolerances 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.
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.
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5.3.3 Tolerances and imperfections
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.
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 Imperfections
Imperfections 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 DNV-OS-C401 /5/ are recommended and conform
to the requirements of e.g. DNV-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 outof-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 R/2,
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.
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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).
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
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 utilization
Being 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.
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Figure 5-2 Illustration of initial imperfections in the monopile for a shear key connection
It 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 scheme
It 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
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. DNV-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.
st
Consider e.g. a solid brick element. A linear 1 order, fully integrated version has 8 nodes and 8 internal
nd
integration points assuming standard [2×2×2] Gauss integration. A 2 order, fully integrated version
has 20 nodes and 27 internal integration points assuming standard [3×3×3] Gauss integration. Hence,
nd
it computationally significantly more expensive to establish the element stiffness matrix for the 2 order
st
element is than 1 order counterpart. A reduced integration would mean a [1×1×1] and [2×2×2] Gauss
st
nd
st
integration for the 1 and 2 order formulation respectively, i.e. only 1 integration point for the 1 order
nd
element equivalent to constant stress and 8 integration points for the 2 order element equivalent to linear
stress.
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5.4 Element selection
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.
nd
For grouted connections it is generally recommended to use 2 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 density
When 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
2 times the overlap height times the circumference, i.e. ~235 m 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
nd
of 2 order elements.
st
If 1 order element are used e.g. in an explicit solution scheme a substantially larger number of elements
nd
should be used. Typically then, 3 to 4 elements should be used to discretize any single edge of a 2 order
3
3
elements, meaning in general 3 to 4 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.
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 2
though the thickness.
nd
For the grout body a minimum of three 2
recommended.
nd
order, reduced integration element
order, reduced integration elements through the thickness is
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.
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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.
Figure 5-3 Suggested geometry idealization and minimum meshing of weld bead shear keys
nd
A suggested minimum mesh is shown in Figure 5-3 based in 2 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 t/2 and 3t/2 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 DNV-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.
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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.
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 preprocessing 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 conditions
Provided 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,
inclusion of the surrounding soil should be considered if the stiffness of the soil is significant as discussed
previously in [5.3.2.7].
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SECTION 6 BOUNDARY CONDITIONS AND LOAD APPLICATION
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 application
For 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 loads
As 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
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.
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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.
6.2.1.1 Local loads
For 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 cases
Because 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.
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
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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.
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 outof-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 assessment
It 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. DNV-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 cases
As 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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just one overturning load case of maximum magnitude axial load combined with maximum bending load
magnitude.
—
—
—
—
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 load
Fatigue 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 correlation
Once 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 Directionality
The 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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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:
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 cases
The 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 cases
The 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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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 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.
DNV-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 state
Because 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 assessment
The 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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SECTION 7 LIMIT STATE ANALYSES
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 Buckling
Because 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 DNV-RP-C208 /7/ or
EN 1993-1-6 /13/.
7.3 Fatigue limit state
The 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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If the linear elastic capacity of the grout is exceeded the extent of cracking and crushing should be assessed.
(See also [2.4]).
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. DNV-ST-0126 /9/ for grout and DNV-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.
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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.
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 state
The 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 state
Guidance on the analysis of the accidental limit state may be found in DNV-RP-C204 /2/.
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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.
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 basics
At a given point in a continuum the stress state is described by the Cauchy stress tensor:
(A.1)
When a body is in equilibrium the components of the stress tensor in every point of the body satisfy the
general equilibrium equation
(A.2)
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
(A.3)
1
where
is the Kronecker delta ,
expressed in elastic moduli
K is the bulk modulus and G is the shear modulus; or alternatively
(A.4)
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.
(A.5)
1
Shorthand in tensor indices notation
.
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APPENDIX A CONSTITUTIVE FORMULATIONS FOR GROUT
The 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.6)
where
I1, I2, I3, are the first, second, and third stress invariants given as
(A.7)
The characteristic equation has three roots (eigenvalues)
stress tensor. These are when ordered as
which all are real due to the symmetry of the
,
, and
principal stresses which are unique for a given stress tensor, hence the invariant property of
, the
I1, I2 and I3.
A.1.2 Principal stress space
For 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
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A.1.1 Principal stresses and stress invariants
As mentioned, the principal stresses
stress invariants
are unique for a given stress tensor and the invariant property of the
. The principal stresses can therefore be combined to form the stress invariants as
(A.9)
A.1.3 Hydrostatic and deviatoric stress
The Cauchy stress tensor
can be expressed as the sum of a deviatoric stress tensor
s 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
(A.10)
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
(A.11)
and the deviatoric stress is thus
(A.12)
The deviatoric stress tensor is also known simply as the stress deviator.
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(A.8)
As the deviatoric stress tensor of second order, it itself also has a set of invariants
be shown that the principal directions of deviatoric stress tensor
. Moreover, it can
s are identical to those of the Cauchy
stress tensor . Therefore, following the same procedure used to find the invariants
Eq. (A.6), the characteristic equations is
of the stress tensor,
(A.13)
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
the deviatoric eigenvalues
stresses
, or alternatively, as a function of Cauchy stress components
or
or the principal
as
(A.14)
A.1.5 Derived state quantities in principal stress space
Because 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 stresses
Observing 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
(A.15)
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A.1.4 Deviatoric stress invariants
In this plane of maximum shear stress the normal stress is non-zero and equal to
(A.16)
A.1.5.2 Octahedral stresses
Alternatively, 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.
Hence, the octahedral normal stress can be expressed as
(A.17)
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.18)
A.1.5.3 Other common stress invariants – von Mises and Tresca
Evaluating 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
(A.19)
associated with the von Mises yield criterion stating that material yields when the second deviatoric stress
invariant J2 reaches a critical value, i.e. when
and
, where
is the yield stress in pure shear
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 J2 it is also known as J2-plasticity. Finally, it may be
added that as the von Mises yield criterion is independent of first stress invariant I2 it is independent of the
hydrostatic pressure.
Another common stress invariant is the Tresca stress
(A.20)
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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.
.
A.1.6 Haigh-Westergaard coordinates, meridians, and lode angles
In the principal stress space it is often convenient to use the Haigh-Westergaard coordinates
as
defined
(A.21)
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
The first coordinate
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
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
(A.22)
If
then the principal stresses are ordered as
.
A.1.6.1 Meridians
The 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.
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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.2 Yield surfaces
Returning 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-Prager
The Drucker-Prager yield criterion is in its original form (see (A.1)) a pressure dependent von Mises
1
2
criterion , why it can be seen as a smooth version of the Mohr-Coulomb yield criterion .
The Drucker-Prager yield criterion has the form
(A.23)
where
I1 is the first invariant of the Cauchy stress σ and J2 is second invariant of the deviatoric stress s. The
A and B are determined from experiments.
constants
In Haigh-Westergaard coordinates, see Eq. (A.22), this can be expressed as
(A.24)
1
The von Mises yield criterion is
with σy being the yield stress in unidirectional tension. See
further the discussion of common stress invariants [A.1.5.3] and below.
2
The Mohr Coulomb yield criterion assumes that yielding is controlled by the maximum shear stress and
that this yielding shear stress depends on the normal stress which may be expressed as
where τ is the shear stress, σ is the normal stress (negative in compression),
is the angle of internal friction.
c is the cohesion, and ϕ
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 below.
ϕ = 0° it reduces to the Tresca (or
ϕ = 90° the Rankine (or tension-cutoff) criterion is recovered as
The criterion is a generalization of other known criteria in that for
maximal shear) criterion, whereas for
also shown in the sketch below.
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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 and B are as stated initially material parameters that needs to be determined by
A and B directly, rewriting the yield criterion in terms of the
principal stress σi using Eqs. (A.12) and (A.14) the following is obtained
The constants
experiments. However, instead of testing for
(A.25)
Denoting the uniaxial tensile and compressive strength
σt and σc respectively the criterion implies
(A.26)
(A.27)
Which solved simultaneously gives
(A.28)
i.e. the constants
A and B expressed by the uniaxial tensile and compressive strength σt and σc.
A.2.1.2 Parameter fitting using cohesion and friction angle
As 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 ϕ that describes the Mohr-Coulomb yield
criterion.
The Mohr-Coulomb yield criterion in Haigh-Westergaard space is
(A.29)
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A.2.1.1 Parameter fitting using uniaxial strengths
that the two surfaces coincides for a Lode angle
. At this point the Mohr-Coulomb yield criterion in
Eq. (A.29) reduces to
(A.30)
or rearranged to
(A.31)
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
(A.32)
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.33)
A.2.1.3 Alternative p‐q formulation
Recalling Eqs. (A.11) and (A.19), the Drucker-Prager criterion of Eq. (A.23) may be expressed as
(A.34)
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 MohrCoulomb 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.
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Assuming that the Drucker-Prager yield surface circumscribes the Mohr-Coulomb yield surface implies
A.2.1.4 Extended formulations
As 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
(von Mises) is recovered.
for all Lode angles whereby the original circular trance of Drucker-Prager
The implications of both the hyperbolic and power-law extensions are illustrated in the meridional
in Figure A-2 together with effect of the alternative deviatoric stress measure
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p‐q plane
t in the deviatoric plane.
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Figure A-1 Drucker-Prager yield surface illustrated in principal stress space and in the meridional
p-q plane
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 notes
The 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 DruckerPrager yield criterion being linear, hyperbolic, and power-law formulations as well as the possibility of a noncircular 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
(A.35)
in which the cohesion
d and the fiction angle β are the governing material parameters, p is the hydrostatic
t is the deviatoric stress defined in Eq. (A.36).
pressure (positive in compression) and
(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
.
3
The term (r/q) resembles the Lode angle θ in the deviatoric plane in that
per definition.
t is linear with
.
Using a shape parameter K = 1 implies t = q and thus recovers the original circular trace of the yield
Hence the variation of
envelope in the deviatoric plane (see Figure A-2).
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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
Drucker-Prager formulation by selecting
(A.37)
A.2.1.6 Ansys implementation notes
The 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
(A.38)
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
in which
(A.39)
with
ɸ being the internal angle of friction, and
(A.40)
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
(A.41)
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.
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σt and σc respectively and adopting the original
K = 1 the cohesion d and the fiction angle β can be determined as
Denoting the uniaxial tensile and compressive strength
and
σy = d, i.e. by
σt, σc by Eq. (A.37) with
(A.42)
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
(A.43)
where
q is the equivalent von Mises stress, p is the hydrostatic pressure (positive in compression),
the maximum principal stress,
as
, and
in relation to which the Macauley bracket
is
is defined
α, β, γ are material constants defined as
(A.44)
i.e. dependent on the ratio
σb/σc between the biaxial- and uniaxial compressive strengths σb and σc,
(A.45)
in which
σt is the uniaxial tensile strength, and
(A.46)
where Kc is a shape parameter controlling the yield envelope in the deviatoric plane for
purely compressive stress states.
For
, i.e. for
, i.e. for stress states exhibiting tension, the corresponding shape parameter is
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Hence, this model may be fitted to the uniaxial tensile and compressive strength
i.e. directly dependent on the material constant
β.
σb such that for biaxial compression in
, the yield criterion reduces to the classic Drucker-Prager yield criterion described
The model introduces the concept of a biaxial compressive strength
plane stress, i.e.
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
σc together with the biaxial compressive strength σb and shape parameter Kc.
σt and
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
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
choosing a
compressive locus of the stress states.
In terms of the full yield surface, the effect of the shape parameter
in Figure A-4 for the same lower bound (
Kc is illustrated in principal stress space
), 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
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(A.47)
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 DruckerPrager 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
and τα defined in terms of principal stresses as
σα
(A.48)
such that the yield criterion is
(A.49)
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
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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
in which
θ is the Lode angle and r1, r2 are the position vectors that describe the meridians at
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.51)
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.
hence the scalar
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
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(A.50)
σα /σc
r(θ)
Uniaxial tension
0
r1
Biaxial compression
0
r1
Stress state
τα /σc
r2
Uniaxial compression
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/σc
(A.52)
Or expressed directly in terms of the uni- and bi-axial strengths
σt, σc and σb as
(A.53)
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
meridians respectively are restricted to
r1, r2 position vectors describing the tensile and compressive
.
Smoothness and continuity at the compressive meridian (
) additionally necessitates
for all position vectors r.
Both of the above requirements will be satisfied if
Within these constrains, the apex of the conical yield surface lies on the hydrostatic axis at
tensile strength of
meridian and
.
zσc, and the opening angle φ of the cone varies between
z giving a triaxial
on the tensile
on the compressive meridian.
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θ
Test
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
r2 = r0 corresponding to
r1 =
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
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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
The Drucker-Prager model is also recovered if
(A.55)
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 model
While 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
(A.56)
and the yield criterion of Eq. (A.49) reduces to
(A.57)
where the elliptical trace of the yield surface in the deviatoric plane
remains unaltered as
(A.58)
however with the position vectors
r1, r2 now functions of the pressure σα as defined in Eq. (A.56).
Yielding thus implies
(A.59)
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(A.54)
(A.60)
i.e. one on the tensile meridian and one on the compressive meridian.
This with the additional constraint following from triaxial tension state where
that the two parabolas must pass through a common apex
imposes the constraint
r1 = r2 must be fulfilled, implies
at the equisectrix (
), which
(A.61)
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.
θ
r(σα ,θ)
Uniaxial tension
0
r1(σα)
Biaxial compression
0
r1(σα)
0
r1(σα)
Test
High compression regime
σα /σc
Stress state
Tensile meridian
−ξ
τα /σc
ρ1
r2(σα)
Uniaxial compression
High compression regime
Compressive meridian
Triaxial tension
−ξ
ρ2
r2(σα)
ξ0
0
r2(σα)
The extended model with meridians given by second order parabolas thus ends up having a total of five free
parameters.
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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
The yield surface can then be made to match test data by fitting the six degrees of freedom
r1 on the tensile meridian gives
(A.62)
in addition to which
r1 = r2 = 0 for triaxial tension gives
(A.63)
and then subsequently solving for
r2 on the compressive meridian gives
(A.64)
The yield surface will be smooth and convex if the following constraints are fulfilled
(A.65)
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
increase, so does the
, such that as the pressure
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.
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Solving for
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
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.
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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.
Firstly, the failure surface is divided into four distinct domains of the principal stress spaces being:
1)
2)
3)
4)
compression-compression-compression, i.e. pure compression
tension-compression-compression
tension-tension-compression
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
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
(A.66)
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
(A.67)
and
(A.68)
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A.2.3.2 Ansys Implementation
The 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/).
(A.69)
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
(A.70)
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
hydrostatic stress state
and uniaxial
compression superimposed on an ambient
. 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
r2, the biaxial compressive strength σcb and the two ultimate compressive strengths
the associated ambient hydrostatic stress
compressive strength
and
r1,
together with
, may in the Ansys model all default to depend on the uniaxial
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 3and 5-paramter Willam-Warnke models, its properties on the tensile and compressive meridians remains, i.e.
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
that
the two strength ratios
and
for uniaxial tension and biaxial compressions, together
with the three strength ratios for the high compression regime being
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for the ambient hydrostatic
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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
and
for the ultimate uni- and bi-axial compression, the system of
equations becomes
θ
r(σα,θ)
Uniaxial tension
0
r1(σα)
Biaxial compression
0
r1(σα)
Ultimate biaxial compression
0
r1(σα)
Test
ξ = σα/σc
Stress state
τα/σc
Uniaxial compression
r2(σα)
Ultimate uniaxial compression
r2(σα)
Triaxial tension
r2(σα)
0
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
(A.71)
2/ Determine the resulting hydrostatic pressure ξ0 at triaxial tension as defined by
the equation
r1, i.e. the positive root of
(A.72)
3/ Fit the compressive meridian
r2 to the hydrostatic pressure ξ0, uniaxial compression strengths as
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pressure, and
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 WillamWarnke model, in which it is assumed that the ‘average’ stress scalars σα and τα are independent of the
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
magnitude of the tensile stress
compression being the interface between Domain 1 and 2.
Hence, the stress measures entering the generalized form of Eq. (A.66) are expressed as
(A.74)
which is the same as Eq. (A.67) only with
σ1 = 0, and
(A.75)
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,
in which
the hydrostatic pressure reduces to
, why with the normalized hydrostatic pressure denoted
the revised position vectors following Eq. (A.70) becomes
(A.76)
and the revised ‘Lode’ angle following Eq. (A.69) reduces to
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(A.73)
σ1 = 0,
. Likewise for uniaxial tension
Obviously then, continuity is ensured between Domain 1 and 2, as for their interface defined by
the linear variation (1 -
σ1/σt) = 1, and
, as well as
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
(A.78)
and
(A.79)
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
(A.80)
and
(A.81)
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
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(A.77)
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 tensiontension-compression region (Domain 3) by the adopted linear scaling of the Rankine criterion. The tensioncompression-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-compressioncompression region and the tension-tension-tension region is for the full surface in principal
stress space shown in thin red
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(A.82)
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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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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 direction
The 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,
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APPENDIX B CONTACT MODELING METHODOLOGIES
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.
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
vector,
is the global stiffness matrix, and
where
u is the nodal displacements vector.
F is the nodal forces
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
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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.
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.
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 surface
When 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.
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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.
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
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.
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Figure B-5 Schematic of the linear Coulomb friction model and the penalty implementation of it
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