-Coupling Reliability methods with Finite Elements-

Structural reliability analysis of a dike with a sheet pile wall

-Coupling Reliability methods with Finite Elements- A Rippi

Structural reliability analysis of a dike

with a sheet pile wall

Coupling Reliability methods with Finite Elements

by

A RIPPI

in partial fulfilment of the requirements for the degree of

Master of Science

in Civil Engineering

at the Delft University of Technology

to be defended publicly on Wednesday November 25 2015 at 1100

Graduate Aikaterini Rippi Student ID 4325583 E k-rippihotmailcom

Thesis committee Prof dr ir S N Jonkman TU Delft

Dr ir R B J Brinkgreve TU Delft and Plaxis bv

Dr ir T Schweckendiek TU Delft and Deltares

Dr A Teixeira Deltares

An electronic version of this thesis is available at httprepositorytudelftnl

MSc Thesis A Rippi i

MSc Thesis A Rippi ii

Preface

This thesis is the final challenge in the master Hydraulic Engineering at Delft University of

Technology The report ldquoStructural reliability analysis of a dike with a sheet pile wall Coupling

Reliability methods with Finite Elementsrdquo was completed at Deltares as a part of a larger

research project namely TO2 in collaboration with Toegepast Natuurwetenschappelijk

Onderzoek (TNO) I chose that subject first of all because it combines two things that I enjoyed

a lot at TU Delft probabilities and flood defences systems Secondly it was an opportunity for

me to get acquainted with FEM and geotechnics that I was always interested in

Different people have contributed to the successful completion of this thesis First of all I would

like to express my appreciation to my graduation committee for their guidance and particularly

my daily supervisor Timo Schweckendiek Together we had many fruitful and interesting

discussions on the subject that triggered and motivated me for keep searching Especially I

want to express my gratitude and my thanks to Ana Teixeira and Jonathan Nuttall employees of

Deltares as they were also some of the main contributors to this research They stood by me not

only as colleagues and thesis mentors but also as friends I would like also to thank the rest of

Deltares employees whom ensured a friendly and easy going working environment Last but not

least I want to thank my friends and especially Panagiotis Apostolidis and my family for their

love support and advice throughout all my studies

Katerina Rippi

Delft November 2015

MSc Thesis A Rippi iii

MSc Thesis A Rippi iv

Abstract

Some dike sections in the Netherlands failed to comply with the safety standards and one of the

most promising countermeasures is the construction of retaining walls inside the dike The

Dutch design codes for dikes with retaining walls rely on Finite Element Analysis (FEM) in

combination with partial safety factors However this can lead to conservative designs For this

reason in this research a reliability analysis is carried out with FEM calculations aiming to

demonstrate the feasibility of reliability analysis for such a soil-structure interaction problem

The case study concerns a (river) dike with an anchored sheet pile wall modelled in PLAXIS The

sensitivity and reliability analyses were enabled by coupling the uncertainty software package

OpenTURNS and PLAXIS through a Python interface The most relevant (ultimate) limit states

concern the anchor the sheet pile wall and global instability (soil body failure) The case was

used to investigate the applicability of the First Order Reliability Method (FORM) and

Directional Sampling (DS) to analysing these limit states Finally also the system reliability was

evaluated using sampling-based methods (DS)

Due to the considerable number of random variables before starting the reliability analysis a

sensitivity analysis was conducted for each limit state This indicated the most important soil

layers to be accounted for and the variables to be considered as stochastic The sensitivity

analysis and later on the reliability analysis were based on analytical formulations of the limit

state functions The anchor and the sheet pile limit states were formulated in terms of their

yield stress for global instability loss of equilibrium in the FEM analysis was used to define

failure Moreover the systemrsquos reliability was evaluated by taking into account all the three limit

states that were mentioned previously

The goal is to implement the coupling between FEM and reliability methods in order to analyse

the components of such a system (ie anchor sheet pile wall and dikersquos soil body) estimate the

probability of failure and identify the most important soil properties that affect the behaviour of

each component and the system as a whole The results of this research can be used to assess

and optimize the current design procedure for dikes with retaining walls

MSc Thesis A Rippi v

MSc Thesis A Rippi vi

Table of Contents

Preface ii

Abstract iv

List of abbreviations x

1 Introduction 1

11 Project objective and main research questions 2

12 Research approach and outline 3

2 System description and current design concept 7

21 System description and forces configuration 7

22 Current design concept 12

23 Safety standards 17

3 Literature study 21

31 Background 21

311 Finite Element Modeling 21

312 Uncertainties and Sensitivity analysis 24

32 Previous Studies 29

33 Overview 34

4 Structural Reliability Analysis 35

41 Basics of Reliability Analysis 35

42 Overview of Reliability Analysis Methods 36

421 Level III Methods 36

422 Level II Methods 40

423 Level I Methods (semi-probabilistic) 42

424 Response Surface Techniques (RS) 44

43 Coupling Reliability Analysis with FEM 46

431 The functionality and possibilities of OT 46

432 Coupling OpenTURNS-Plaxis 47

44 Overview 49

5 Failure Mechanisms and Limit State Functions 53

51 Introduction to the system analysis and the limit states 53

52 Limit State Functions 55

521 Serviceability Limit State 55

522 ULS for Structural Members 57

MSc Thesis A Rippi vii

523 ULS for Soil Failure 60

53 Overview 66

6 Case Study-Dike with an anchored sheet pile wall 69

61 Case Description 69

62 Soil Parameters 70

63 Finite Element Model 73

64 Deterministic Analysis 74

641 Calculation Scheme and Design Values 75

642 Construction Stages 77

643 Determination of the structural elementsrsquo characteristics 78

65 Overview 81

7 Reliability analysis results with stochastic soil properties 85

71 Method description 85

72 Mean values calculations 88

73 Sensitivity Analysis Results 91

74 Soil Shear Failure 98

75 Anchor Failure 102

76 Sheet pile wall failure 105

8 Conclusions and Recommendations 121

81 Conclusions 121

82 Recommendations 123

References 125

Appendix A 129

OpenTURNS features 129

A1 Fourier Amplitude Sensitivity Test (FAST) 129

A2 Optimization Algorithms in FORM 132

Principles of optimization algorithms 132

Convergence criteria 134

Evaluation of the algorithms performance 135

A3 Distribution Types 143

Uniform Distribution 143

Normal Distribution 144

Lognormal Distribution 144

Truncated Normal Distribution 145

Appendix B 147

MSc Thesis A Rippi viii

Plaxis 2D (2015) features 147

B1 Mohr Coulomb failure criterion 147

B2 φ-c Reduction Technique 150

B3 Initial Stress Generation 151

B4 Interface Strength 151

Appendix C NEN 6740 - Table 1 153

Appendix D 155

Input Files for the Reliability Analysis 155

Appendix E 159

Characteristic and mean values 159

Appendix F 161

Reliability methods 161

F1 Generation of random samples in Monte Carlo 161

F2 Other Sampling Methods 162

F3 First Order Second Moment (FOSM) Method 165

MSc Thesis A Rippi ix

MSc Thesis A Rippi x

List of abbreviations

CDF Cumulative Distribution Function

CoV Coefficient of Variation

CUR Civieltechnisch Centrum Uitvoering Research en Regelgeving

DS Directional Sampling

FAST Fourier Amplitude Sensitivity Analysis

FEA Finite Element Analysis

FEM Finite Element Model

FERM Finite Element Reliability Method

FORM First Order Reliability Method

FOSM First Order Second Moment

LEM Limit Equilibrium Method

LRFD Load and Resistance Factor Design

LSF Limit State Function

MC Monte Carlo

OT OpenTURNS

PDF Probability Distribution Function

RFEM Random Finite Element Method

RS Response Surface

SA Sensitivity Analysis

SLS Serviceability Limit State

SORM Second Order Reliability Method

ULS Ultimate Limit State

VNK Veiligheid Nederland in Kaart

MSc Thesis A Rippi xi

The roots of education are bitter but the fruit is sweet

Aristotle

MSc Thesis A Rippi 1

1 Introduction

In the Netherlands according to the Flood Protection Program (Hoogwater

Beschermingsprogramma1) and the Room for the River program (Ruimte voor de Rivier2)

alternative structural techniques for the reinforcement of existing dikes or for future dike

constructions additional to the conventional ones have been introduced and tend to be

attractive The heightening of the dike crest and the construction of a stability berm are some of

the most common current actions for dike strengthening Additionally filter layers geotextiles

and drainage systems can be applied in advance in order to prevent several failure mechanisms

Recently cantilever or anchored sheet pile walls and diaphragm walls tend to be used as an

alternative procedure for embankment reinforcing (see Figure 11) Such an alternative can be

chosen for an improvement of the macro stability of the dikersquos inner slope and additionally for

saving space in the land area that needs to be habited

Figure 11 Design options for dike reinforcement (source Flood Defences 2015)

For that purpose engineers need a concrete and unequivocal design methodology for such

combined structures In the meantime Deltares in cooperation with the Water Board of

Rivierenland is working on a design guideline which will be applicable to design the so-called

Type II stability fences3 A draft of such a guideline is elaborated in the report of Larsen et al

(2013) In this report suggestions and recommendations are outlined for the design of such

structures with Finite Element Modeling (FEM) and partial factors Remarks are also made for

the advantages and the limitations of such models and how they should be handled so that

reasonable and trustworthy results can be retrieved from the analysis

1 httpwwwhoogwaterbeschermingsprogrammanldefaultaspx 2 httpalbertawatercomhow-is-water-governedwhat-is-room-for-the-river 3 Type II structures are structures which in combination with a soil construction fulfil water retaining functionalities

Introduction


The concept behind the recommended design criteria (Larsen et al 2013) is the definition and

the evaluation of several partial factors with the view to determine overall safety factors These

factors are further discussed in section 22 However validation of these factors shall be carried

out before being used in any case otherwise either the safety of the structure is jeopardized or

the structure will be overdesigned and cost inefficient

For improving the design criteria researchers for a long time focused on enhancing structural

models (beams shells etc) and constitutive laws (elasticity plasticity damage theories etc)

With the development of computer science a great amount of work has been devoted to

numerically evaluate approximated solutions of the boundary value problems describing the

mechanical system FEM is probably nowadays the most applied approach for the solution of

these problems

However the increasing level of detail of the constitutive models and the constant enhancement

of the computational tools do not solve the problem of identification of the model parameters

and the inherent physical and modelling uncertainties Moreover in most civil engineering

applications the intrinsic randomness of materials (soil rock concrete etc) or loads (water

elevation wind earthquake motion etc) is such that deterministic models are using average or

later on characteristic values of the properties at best lead to rough representations of the

reality

As a counteraction a semi-probabilistic methodology has been developed that was based on the

application of characteristic and design values by using partial factors Current design codes

such as Eurocode provide target reliabilities for different types of structures and structural

elements according to the potential consequences of failure However these partial factors are

not always equally suitable and efficient for all types of structural applications since they have

been calibrated under specific conditions Besides the consequences of failure of flood defences

such as dikes can be comparable to the investments in increasing the reliability of such systems

are For that reason it would be advisable for these systems to define target reliabilities based

on a risk assessment (ie tailor-made solution) rather than using the standard partial factors

coming from general geotechnical design codes which may be either too low or too high for a

given flood defence system

One step of such a risk assessment is accounting for randomness and spatial variability of the

mechanical properties of materials is one of the tasks of stochastic or probabilistic mechanics

which has developed fast in the last decade In this master thesis project the uncertainty of soil

properties is going to be treated in terms of its contribution to failure For that purpose

probabilistic methods are going to be implemented for a dike with an anchored sheet pile wall

(see Figure 12) simulated in a FEM software that is specialized in soil mechanics The successful

implementation and in future research the verification of such methods can be considered as

the most preferable and cost efficient way to design structures with high safety requirements

and not only for the validation of the partial factors Of course such a procedure tends to be

time consuming However the gradual improvement of the current probabilistic methods in

combination with the state-of-the-art computer capabilities as well as the scientific knowledge

gained in terms of different systems behaviour and failure modes can introduce a more

optimized way of designing structures with considerable investments

Introduction


Figure 12 Reinforced dike section with an anchored sheet pile wall

11 Project objective and main research questions At the beginning of this research it has been observed that the design procedure being followed

for dikes with retaining walls modeled in FEM can lead to an overestimation of the critical loads

and thus to a potential cost inefficient final structure An example of the magnitude of the design

values in such a case study is given in the next chapter where also a more detailed explanation

of the current design concept is described

The objective of this study is to implement a full probabilistic analysis for evaluating the

reliability of a dike reinforced with an anchored sheet pile wall Such a research analysis can

subsequently come up with valuable recommendations for the improvement of the present

design approach Soilrsquos and structural elementsrsquo behavior should be taken into account both

separately and as a system Some of the reliability methods can deal with system reliability

problems such as Monte Carlo (MC) and Directional Sampling (DS) whereas for others like

FORM and SORM additional methods should be applied that use reliability information for each

individual limit state function to obtain the systemrsquos reliability

In principle the probability of failure of different limit states is to be computed individually

while in the sequence the system reliability is going to be estimated As far as the soil failure

mechanisms are concerned this research will focus mainly on the global stability of the dike

slope while for the sheet pile wall and the anchorrsquos failure the exceedance of the steel yield

stress is going to be considered

The scope of this thesis is not only to test different reliability models robustnessrsquo in conjunction

with FEM simulations but also to get a better insight into the specific system behavior (ie of a

dike with a retaining wall) analyze each component separately and investigate its response

under certain load conditions Furthermore the minimization of the computational effort and

time could also be carried out meaningfully under the constraint of sufficient accuracy The

accuracy should be such that the probability of failure is acceptable for the ultimate limit state

(ULS) in a normal design process and according to the current safety standards Finally the

Sheet pile

wall

Anchor

Dike section

Introduction


robustness of the coupling between the reliability model and the FEM will be tested via their

capability of adapting to new input parameters without encountering convergence errors

during execution

The main research question of this master thesis project is thus formed as follows

How can the probability of failure of a dike with a sheet pile wall due to global instability modeled

by a Finite Element Model be analyzed

Essential questions regarding the soil models and failure criteria as well as the reliability

methods and the systems behavior are generated which are listed below

Subquestion 1 Which reliability methods are computationally tractable in

combination with FEM

Subquestion 2 How robust (convergence) are the tractable methods

Subquestion 3 Which limit states are relevant for the envisaged application of

retaining walls in dikes and how can they be formulated using FEM

analysis outcomes

Subquestion 4 What is the contribution of different uncertainties in the failure

mechanisms of the system

Subquestion 5 Can response surface techniques help to increase the efficiency and

robustness of the reliability model

Subquestion 6 How can the current design approach for dikes with sheet piles be

improved

In the next chapters the above research question and the related subquestions are going to be

answered by following the methodology that is described in the next section

12 Research approach and outline In this section an overview is given of how the aforementioned objective and sub-questions are

approached The thesis is mainly divided into 8 chapters and 6 appendices In Figure 13 the

thesis outline is illustrated and an indication of which subquestions (Sq1-6) are treated to

which chapter is given In Chapter 1 an introduction into this research content is made and a

first illustration of the system under investigation is presented together with the main research

questions

Chapter 2 has been devoted to the description of the system (ie a dike with an anchored sheet

pile wall) and to the case study that has been adopted for being analyzed in terms of its

reliability The forces configuration is also illustrated and the results of the case study according

to the current design concept are presented and evaluated Last but not least the new

recommended safety standards are included and the current required reliability of the specific

dike section is presented

Introduction


In Chapter 3 a literature study related to FEM uncertainties and sensitivity analysis is

presented Moreover previous studies that are associated with the objective of this research are

mentioned whereas some of their results were also taken into account for proceeding with this

research

The main scope of this thesis is the implementation of different reliability methods on a specific

case study with the view to investigate and analyze its behavior Some of these methods are

continuously mentioned through the test and thus in Chapter 4 an overview is given of the

most well-known reliability methods Eventually an evaluation of these methods is made based

on literature and preliminary testing with simple case studies and the procedure of their

coupling with FEM is discussed The evaluation helped to qualitatively answer subquestion 1

while also an introduction of how special reliability methods such Response Surfaces were used

in this thesis that partially covers subquestion 5 In this chapter an introduction into the Limit

State Functions (LSF) concept is made and how they are considered for the reliability analysis of

a system An LSF actually represents a failure mode that can be detected in a structural or soil

element and it is expressed as a function of several variables In Chapter 5 the failure

mechanisms and the corresponding LSF that are related to the specific case study are identified

and formulated which answers subquestion 3

In Chapter 6 the case study whose reliability is to be evaluated is presented as it was modeled

in FEM The boundary conditions are specified and the soil and structural properties are

indicated and illustrated Moreover a deterministic design is also taking place in order to

roughly estimate the structural elementsrsquo properties This will help to make a qualitative

comparison between the current design procedure and the design according to a fully

probabilistic approach by referring thus to subquestion 6

Eventually the results of the aforementioned analysis are presented in Chapter 7 The results

mainly include the estimated probability of failure of each of the system components as well as

of the system as a whole by considering the soil properties as stochastic Moreover an

interpretation of the failure points is made and an assessment of the level of impact of the

random variables on the systemrsquos behavior is carried out At that chapter subquestions 2 4 and

5 are mainly treated

Last but not least in Chapter 8 the general conclusions are presented together with some

valuable recommendations for future consideration and research Furthermore a reflection on

the methodology and how the different research questions were approached is made

Introduction


Figure 13 Thesis outline

Introduction



2 System description and current design concept

In this section the system to be analyzed is described more in detail and the individual

components are identified Moreover the forcesrsquo configuration is explained and illustrated as

they would have been calculated with the conventional way in parallel with a qualitative

evaluation of the potential expected deformations Finally the current design concept and the

safety standards that are referred to such a system are introduced

21 System description and forces configuration The system to be investigated in this thesis consists of a dike section reinforced with a one-layer

anchored sheet pile wall The system has been simulated by FEM and later on it was coupled

with a reliability package for carrying out a reliability analysis In the figure below an

illustration of the system is depicted and the several elements of an anchored sheet pile wall are

showed

Figure 21 System layout and different components of the anchored sheet pile wall

In particular an anchored sheet pile wall consists of the sheet piles which are anchored in the

soil via a tie rod The rod is grouted most of the times in a sandy layer with a cemented grout

body and it both alleviates the sheet piles from the axial forces exerted by the upper structure

and keeps the wall stable in case of excessive developed moments due to the lateral earth

pressure Last but not least a waling system is applied mainly downstream of the sheet piles in

order to transfer the loads from the piles to the anchors in such a manner so as to avoid

excessive local stresses on the intersection between the sheet piles and the tie rod

Cross-Section

Top View

System description and current design concept


The main load configuration acting on a dike section as well as a sheet pile wall is depicted in

Figures 22-24 together with the possible displacements As far as the soil body stability is

concerned most of the slope stability analysis computer programs are based on the limit

equilibrium concept according which a soil mass tends to slide down under the influence of

gravity The transitional or rotational movement is considered on an assumed or known slip

surface below the soil There an equilibrium should be achieved among the driving and the

resisting forces In that case the driving moments consist of the soil weight the water pressures

and the loads around the center of the slip surface such as a possible vertical load on the top of

the dike crest (ie traffic load) The magnitude of the water pressures is controlled by the water

elevation on the river side

In Figure 22(a) the external and internal forces acting on a slice of the slip surface are

illustrated The driving forces are the soil weight W the water pressure U and any additional

load that can contribute to the rotation of the slip surface which in turn activate the lateral

active earth pressure The resisting forces consist of the lateral passive earth pressure

(Rankine 1857) and the shear strength (Terzaghi 1943) Their expressions are written as

follows

[kN] (21)

[kN] (22)

[kPa] (23)

where is the active lateral earth pressure coefficient and the passive lateral earth

pressure [kNm3] is the unit weight of the corresponding soil layer and H [m] is the thickness

of the soil layer that the lateral earth pressure comes from (in case of multiple soil layers there

should be a distinction among the different forces) and [kPa] is the effective cohesion of the

soil layer The safety factor that is written in the Figure 22(a) as FS is defined by the ratio of the

shear strength (excluding the pore water pressure) divided by the weight of the earth body

(including the pore water pressure)

The loss of equilibrium between the driving and the resisting moments lead to the rotation and

instability of a slip surface as it is illustrated in Figure 22(b) A slide surface can take various

shapes according to the Limit Equilibrium Method (LEM) that is followed According to Bishop

method this surface tends to be circular in Uplift-Van method a horizontally compressed zone

can be also considered whereas in Spencer method the shape of the slide body can be arbitrary

The slip surface can be located either in the landside or in the river side of the dike depending

on the load the soil characteristics and the design of the dike (ie inner or outer berm

reinforcing revetment on the outer slope etc) Most of the times the landside area of the dike is

jeopardized due to different failure mechanisms that are described in section 523 The

instability of the outer slope is mainly relevant in case of sea dikes and a potential damage can

be usually reconstructed until the next flooding



(a)

(b) Figure 22 (a) External and internal forces acting on a slip surface of a dike section and (b) possible deformation pattern

As a counteraction for the slope instability a sheet pile wall can be installed inside the dike The

special thing about these structures is that they make possible a greater freedom in form and

functionality than a traditional dike design This structure derives its strength from the

materials used such as steel which are able to withstand higher pressures than clay for instance

The general stability is due to friction and wedging in the bottom

After the reinforcement of the inner side of the dike with a retaining wall the strength capacity

of the wall is also important for the global stability of the system The forces to be taken into

account for the sheet pile wall stability are the active and the passive earth pressures (effective

pressures) and the water pressure as they are illustrated in Figure 23(a) In that figure the



forces were simplified in a singular triangular shape as in a homogeneous soil body in order to

explain and depict the overall picture of the acting forces However the stresses distribution

over depth can be more complex depending on the variety of the soil layers that are present In

Figure 23(a) it can be noticed that a sheet pile wall which is installed close to the inner berm

might not have significant instability issues as there is the passive side that contributes to

resisting forces However in the passive side the soil can differ and be weaker than this of the

active side Therefore the passive force in that case might not be very supportive and thus the

wall shall be designed cautiously

Additionally the anchor resistance shall be reassured for the sake of the stability of both the

sheet pile wall and the overlying soil body In Figure 23(a) the forces acting on the anchorage

are displayed The distributed load qs comes from the earth pressure of the upper soil layer (it

might not be uniformly distributed as it is depicted in the figure but more in a trapezoidally type

of load) whereas the tension force Ft4 stems both from the earth pressure and the displacement

of the sheet pile wall The bending and the tension capacity of the anchor are determinant for its

stability In Figure 23(b) a possible displacement pattern of the retaining wall and the

anchorage is illustrated

Furthermore corrosion is an additional weakening impact on the wall that depends on the

water level and the pore water pressures near the structure The thickness of the wall and the

material properties are playing a key role to the resistance towards corrosion This will not be a

subject of this thesis however it should be taken into account in case of design purposes

(a)

4 where O is the average circumference of the pile P [kPa] is the shaft friction at depth z and L is the length of the pile



(b) Figure 23 (a) Mobilizing forces acting on the sheet pile wall and the anchorage and (b) possible deformation pattern

It should be mentioned that as far as the pore pressures are concerned in Figure 23(a) only the

hydrostatic pore pressures are depicted Generally the pore pressures (or active pore pressures)

are the sum of the steady state pore pressures and the excess pore pressures

[kNm2] (24)

Steady state pore pressures can be hydrostatic (based on a horizontal phreatic line) or non-

hydrostatic (based on steady state groundwater flow) Excess pore pressures are based on

loading of undrained soils In FEM these various pore pressures are taken into account

automatically In Figure 24 an example of a potential distribution of the active pressures

nearby the sheet pile wall is depicted as it was deduced from Plaxis Therefore as it can be

noticed from the figure the distribution can indeed sometimes approximated as triangular



Figure 24 Example distribution of the active pore pressures acting nearby the sheet pile wall

The knowledge of the type of forces that are exerted on both the structural components and the

soil body as well as of the most expected form of displacements is valuable in order for

someone to be able to evaluate the results deduced from FEM and detect potential modelling

errors Therefore this validation was necessary before starting with the reliability analysis

In the next sections the current design approach of the system described above is elaborated

Additionally the Dutch recommended safety standards are presented for this type of structures

which show the need of carrying out a reliability analysis For that purpose a case study was

adopted that it is presented first as it was designed according to the current regulations while

finally a comparison is made between the original and the new case study as it was found to be

according to the reliability analysis

22 Current design concept In the Netherlands dike reinforcement is becoming a major issue concerning the design of flood

defenses Special reinforcing structures such as cofferdams anchored sheet piles and diaphragm

walls are used for strengthening the dike Especially in the context of the Flood Protection

Program and the Room for the River program many dike reinforcement projects have been

suggested

In the report of Larsen et al (2013) the design procedure of a dike reinforced with sheet pile

wall is prescribed using FEM The safety philosophy that is followed in this technical report

actually composes the current procedure for designing dikes with sheet pile walls and it is

linked with the usage of partial safety factors Particularly the required overall safety factor

FEM that should be compared with the one from FEM calculations is determined as follows

SFEM

b d m n

(25)

where



Partial safety factor indicating the uncertainties of the soil composition and

the water pressures (also called schematization factor)

Partial safety factor which is related to the calculation model and the way the

calculations have been carried out (also called model factor)

Partial safety factor which is related to the material parameters (also called

material factor)

Partial safety factor associated with damage caused during the soil tests (also

called loss factor)

Safety factor of load

Each of these partial factors shall be at first calibrated by conducting a full probabilistic analysis

of the system under consideration In most of the cases these partial factors have been already

calibrated on previous similar projects and then they are reused for any similar case In

Schweckendiek et al (2013) a new approach of using partial factors for flood defences is

proposed whose application is still under consideration At the moment the aforementioned

partial factors for dike reinforced with sheet pile walls modelled in FEM are estimated

according to mostly engineering judgement and political negotiations

In sequence the FEM design procedure that is applied for sheet pile walls inside soil structures

is based on a so-called ldquosafety calculationrdquo (for information for this type of calculation see

Appendix B2) In this calculation an artificial reduction to the soil strength parameters (friction

angle and cohesion) is applied in order to evaluate the overall safety factor and confirm if this is

lower or higher than the required one that was estimated according to Eq 25 Furthermore the

resulting moments and forces developed on the structural elements during the specific

calculation are used for their design

This is a quite helpful and indicative calculation type available in Plaxis that can give a sense of

the structurersquos safety factor and the possible failure mechanisms that can occur under the

predefined load configuration However such a method of calculating the developed stresses

can be misleading in terms of the moments and forces acting on the structural elements and the

total deformations of the system

In this thesis the reliability of a dike with an anchored sheet pile wall is going to be evaluated

and in particular the case study of Breedeveld (2011) is to be used for that purpose In Figure

26 the location of the existing dike section is presented while in Figure 25 the structure as it

was modelled in FEM is illustrated It is essential to mention that the retaining wall does not

exist in reality but it was placed so as to implement and demonstrate the current design

regulations in the report of Breedeveld (2011) The anchorrsquos angle has been taken equal to 30deg

from the vertical which is generally a steep anchor inclination That is expected to reduce the

part of the horizontal force that is undertaken by the anchor and thus the most of it exerts on

the sheet pile wall The angle is usually taken 30deg-45deg from the horizontal axis (CUR 2005) but

this is of course depended on the load conditions and the construction requirements for each



case The influence of the anchor orientation and the construction methods that are generally

followed for an anchored sheet pile wall are not considered in this thesis The case study was

taken as it was modelled in the aforementioned project and the reader should be aware that this

serves the proceedings of a reliability analysis and not of a perfectly constructed retaining wall

Figure 25 Original dike section in Varik-Heesselt as it is presented in Breedeveld (2011)

In particular based on this case study and trying to understand the design criteria a

comparison was carried out between the ldquosafety calculationrdquo and a normal ldquoplastic calculationrdquo

(ie elastoplastic drained or undrained analysis) In both cases the boundary conditions the

external loads and the input parameters are the same The difference between them is the

calculation procedure followed in order to come up with final stresses which in the case of the

ldquoplastic calculationrdquo does not include a reduction of the soil strength properties In Figures 27

and 28 the results from both calculations are presented In Figure 27 the total deformations

are exhibited while in Figure 28 the moments and the axial forces on the sheet pile are shown

Figure 26 Location of the Varik-Heeselt dike section (section number TG-078) in the Dike ring 43

BetuweTieler-en Culemborgerwaarden west (source Breedeveld 2011)



As it is realised from this figure the discrepancy between the two results is quite high while as

far as the deformations are concerned no realistic conclusions can be drawn from a ldquosafety

calculationrdquo Moreover taking as design moments the moments that are deduced from the

ldquosafety calculationrdquo will lead to a much heavier and more expensive sheet pile cross section than

the ldquoplastic calculationrdquo

It is also essential to mention that the required safety factor was estimated up to FEM =18

according to the partial-factor procedure that is defined in Eq 25 Such a safety factor as

prerequisite for coming up with the design values of the moments and forces of the sheet pile

and the procedure of the ldquosafety calculationrdquo in FEM can lead to a conservative and financially

inaccessible design

(a) ldquoPlastic calculationrdquo Deformed mesh and maximum deformations (umax= 008 m)

(b) ldquoSafety calculationrdquo Deformed mesh and maximum deformations (umax= 9041103 m)

Figure 27 Deformed mesh and maximum deformations in the end of (a) plastic calculation and

(b) safety calculation

30deg



(a) ldquoSafety

calculationrdquo

Mmax = 9392 kNmm Nmax= -4402 kNm

(b) ldquoPlastic

calculationrdquo


Figure 28 Bending moments and axial forces exerted on the sheet pile during the (a) safety

calculation and (b) plastic calculation

On the other hand the aforementioned design procedure guaranties a strong structure capable

of probably undertaking more than the expected loads and thus ensuring the safety against

flooding of the landside However the knowledge on fully probabilistic methods that has been

obtained as well as the development of the technology can introduce a reliability analysis of the

system as a preferable way for the future design concept Such a procedure should be

accompanied also by field test that would be able to validate FEM results as well as in-situ

measurements and inspections of the soil properties for calibrating their statistical

characteristics

Due to the conservative results that the aforementioned procedure came up with and because of

the simplifications that this case study was later subjected to in order to be used in this thesis a

new design was carried out in Chapter 6 and the properties of the structural elements were

redefined

The inclination towards probabilistic methods has not only been created due to the incomplete

design regulations for the system under consideration but also the current proposed safety

standards which are discussed in the section below recommends a risk based safety assessment

of the primary flood defenses which in principle implies the evaluation of the failure probability

In the next section an overview of the new recommended safety standards related to primary

flood defenses is given and the required reliability for the stability of the aforementioned case

study is elaborated



23 Safety standards An important factor for the evaluation of a structurersquos reliability is the safety standards that

have been set and according which the acceptance or not of the failure probability is made After

the safety assessment5 of 2011 it has been declared that 33 (1225 km) of the primary flood

defenses (3767 km) in the Netherlands does not comply with the safety standards For this

purpose a new session of investigations has commenced in order to assess the reliability of the

existing dikes and the potential amendment of the safety standards for future constructions

Taking into account this latest information and within the framework of the Flood Risk in the

Netherlands 2 project6 (VNK2) a new Delta program was founded whose one of the main

decisions was the ldquoDelta Safety Decisionrdquo This decision contains a proposal for new safety

standards of the primary dikes as they are depicted in Figure 29 which are stricter than the

previous ones and formulated in terms of failure probability

According to this project a new policy regarding the safety assessment of the flood defenses has

been settled and proposed for the future design criteria which is based on the evaluation of the

acceptable flood risk (probability of failure (x) consequences) of each dike ring rather than the

probability of exceedance in order to achieve a level of protection that is in balance with the

societal value (Cost Benefit Analysis and life loss)

VNK has been already using reliability evaluation techniques in order to calibrate partial safety

factors Especially in geotechnical engineering the high inherent uncertainty of the soil

properties renders the evaluation of the structural reliability essential for the safety assessment

of the structure Subsequently this implies the investigation of the response of the different

reliability methods applied in a real case study of a dike The techniques that are used to

evaluate the probability of failure are discussed thoroughly in Chapter 4 and in Appendix F In

case of a complicated structure such as a dike with a retaining wall both the reliability of the

dike body and the reliability of the wall shall be evaluated and eventually a common standard

for the reliability of the system shall be defined

5 A safety assessment is an evaluation of the function of the flood protection structure and it is being conducted every approximately 6 years after the reassessment of the existing hydraulic boundary conditions 6 in Dutch Veiligheid Nederland in Kaart 2 or VNK2 It has been initiated by the Ministry of Infrastructure and the Environment (IampM) the Association of Regional Water Authorities (UvW) and the Association of Provincial Authorities (IPO)



Figure 29 Maximum admissible flooding probability for primary defences according to Delta

programme 2015 (Deltaprogramma 2014)

As far as the retaining walls are concerned CUR (2005) design procedure distinguishes the

following three safety classes for retaining walls with corresponding reliability indexes

Table 21 Safety classes and corresponding reliability indices (CUR 2005)

Class I Relatively simple constructions no personal safety risks and relatively minor

damage in the case of overall failure ϐ=25

Class II Considerable damage in the case of overall failure minor personal safety

risks ϐ=34

Class III Major damage in the case of overall failure andor considerable personal

safety risks ϐ=42



Additionally as far as a dike section is concerned the required overall reliability index shall be

concluded according to the new norms that are shown in Figure 29 However the reliability

index of the different failure mechanisms of a dike shall be estimated separately

According to Vergouwe et al (2014) that is part of the VNK2 report the major failure

mechanisms in dike ring 43 where the case study under investigation is located (see Figure 26)

is uplift and piping and macro-instability of the inner slope Especially in Figure 210 the

estimated percentage of the contribution of each failure mechanism to the overall probability of

flooding7 is indicated In Vergouwe et al (2014) the overall flooding probability of the specific

dike section that the case study concerns was evaluated to 17800 Therefore the macro-

instability on which this thesis is concentrated with a contribution percentage of 226 has a

probability of about 310-5 (

) and thus a target reliability index β of

approximately 4 This reliability level can be considered as a benchmark for evaluating the one

that will be entailed in the end of this thesis

Figure 210 Percentage contribution of the different failure mechanisms to the probability of flooding (source Vergouwe et al 2014)

It should be mentioned though that the aforementioned safety standards concerning the dike

ring 43 were defined according to the probability of exceedance of a certain water level that the

dike has to retain Therefore this complies with the old norms and not with those appeared in

Figure 29 However for large engineered systems such as flood defence systems with large

potential consequences and substantial investments it is worthwhile to assign target reliability

levels based on a risk assessment of the area surrounded by the specific dike ring There are

three widely used types of criteria for evaluating the risks related to floods and major industrial

hazards (Vrijling et al 2011)

Economic criteria

Individual Risk Criteria

Societal Risk criteria

7 The dependencies among the failure mechanisms were not taken into account Therefore this probability stated in Vergouwe et al (2014) is not exactly equal to the probability of flooding



In order to check then if a certain system abides by the target reliability that is required the

failure probability of each failure mechanism shall be calculated based on a LSF The

aggregation of the failure probabilities of all the modes gives the failure probability of the

system which is supposed to be compared with the required in order to reassure the reliability

of the structure Therefore this thesis is dealing with the calculation of this failure probability of

a certain system modelled in FEM analysis and understanding of the systemrsquos behaviour and

finally the investigation of the robustness of coupling FEM with reliability methods for

evaluating the reliability of a structure In the next chapter an overview over the research has

been done until now concerning coupling and reliability methods applications is presented and

a discussion over the most important is made


3 Literature study

In this chapter an overview of some principles related to FEM and the uncertainties in

geotechnical engineering is presented More precisely an introduction into FEM concept and a

discussion over the different types of FEM is made Moreover the uncertainties in geotechnical

engineering are stressed and the general framework according to which they are handled is

described Finally previous studies that are associated with the application of FEM on soil

structures and the reliability analysis are discussed

31 Background

311 Finite Element Modeling

FEM is a numerical method whose essence is to convert a problem described by partial

differential equations over space and time into one by dividing the space-time continuum into a

set of discrete elements and assuming that the unknowns vary over each element FEM solution

process is as follows

1 Divide structure into pieces (elements with nodes) (discretizationmeshing)

2 Connect the elements at the nodes to form an approximate system of equations for the

whole structure (forming element matrices)

3 Solve the system of equations involving unknown quantities at the nodes (eg

displacements)

4 Calculate desired quantities (ie strains and stresses) at selected elements

The properties of each element are set the same as the material properties that have been

defined by the user Then the Deterministic Finite Element Method (the finite element method

that was formulated with deterministic variables) can be used in conjunction with means and

standard deviations of the input variables to obtain reliability estimates

In the simple coupling of FEM with reliability analysis each parameter that is considered as

stochastic is given a particular probability density function estimated either by field tests or by

engineering judgement This approach is also called ldquosingle random variablerdquo and it assumes

that the spatial correlation length is infinite In other words the soil is considered to be

homogeneous and the stochastic property assigned to the soil is taken at random from a certain

probability distribution

Nevertheless a more realistic model should take into account the spatial correlation within

smaller regions where the stochastic property is allowed to vary For that purpose the Random

Finite Element Method (RFEM) was introduced by Fenton and Vanmarcke (1990) in which the

random variables are correlated to one another using auto-correlation functions

The most sophisticated method of FEM is the Stochastic Finite Element Method introduced by

Ghanem and Spanos (1991) which accounts for the uncertainties in the input parameters

Literature study


implicitly within the finite element calculation This aims at representing the complete response

probability distribution function (PDF) in an intrinsic way Two steps are basically applied for

that purpose

Discretization of the problem and definition of the random vector with an unknown

joint PDF and

Expansion of the response in a particular basis of random vectors with a finite variance

called the polynomial chaos

In this thesis the Deterministic Finite Element Method is to be used coupled with reliability

analysis According to Waarts (2000) coupling FEM with reliability methods (FERM) would lead

to the following advantages

In comparison to standard finite element analysis it gives direct insight into the

structural reliability and decisive parameters

Structures designed using FERM will either be safer orand more economically built in

comparison to structures designed using safety factors and classical constitutive models

FERM is expected to be useful for the determination of LSF that cannot be explicitly formulated

and that differ in each case such as soil limit state in different structural schematizations

Moreover it is likely to be valuable in areas where little knowledge exists on the systems

reliability of structures with multiple components (ie a dike with a sheet pile wall)

FEM in practice

The conventional method for stability analysis in a soil body is represented by LEM although

FEM is increasingly used by designersresearchers The latter has been proved to be quite

realistic for the progressive behaviour (ie stress-strain development in different construction

phases) of a soil system under the effect of stress redistribution in comparison with classical

models Especially in their master thesis Johansson amp Sandeman (2014) compared the

deformations and the forces measured at a deep excavation supported by anchored sheet pile

wall in a railway tunnel located in Gothenburg in Sweden with 1D finite element software the

2D finite element software (Plaxis) and hand calculations They proved that Plaxis produces

reliable results for horizontal deformations in the sheet pile wall and anchor forces when

compared to in-situ measurements

Moreover in Gonzaacutelez et al (2013) it was proved that the simplification of reality done by the

classical methods such as Blumrsquos Engelrsquos Kreyrsquos methods etc (for further information for these

methods a reference is made to Gonzaacutelez et al (2013)) allows us to generally understand the

behaviour of the system wall-soil Nonetheless the results that came out of this analysis were

found to be quite conservative whereas FEM managed to give a more realistic interpretation of

the wallrsquos movement

Seed et al (2006) investigated the performance of the New Orleans Flood Protection Systems

during hurricane Katrina in August 2005 by simulating the different levee sections with FEM As

an example in Figure 31 pictures of the failure of the Lower Ninth ward IHNC East Bank in

New Orleans after the hurricane Katrina are presented FEM was proved to be an efficient tool

to realistically interpret the shape and the triggers of the failure mechanism

Literature study


Figure 31 Failure of the Lower Ninth ward IHNC East Bank in New Orleans modelled in FEM with Plaxis (source Seed et al 2006) At the bottom right the shear strain contours are depicted that are developed through the embankment and at the bottom left the deformed mesh is presented illustrating the failure mode of the levee

In recent implementation of design concepts and technical recommendations such as CUR 166

ndash Damwandconstructies (2005) which refers to the design of retaining walls it can be observed

that guidelines for Finite Element Analysis (FEA) have been set parallel to the existing analytical

or empirical calculation methods Therefore FEA tends to become more and more accepted as

an alternative for Serviceability Limit State (SLS) as well as for ULS design

According to Wyllie and Mah (2004) LEM give an estimation of the factor of safety with no

information on deformation of the slope However in numerical analysis the failure surface can

evolve during the calculation in a way that is representative of the natural evolution of the

physical failure plane in the slope In that way a better insight into the evolution of failure

mechanisms can be gained

Cundall (2002) compared the characteristics of numerical solutions and LEM in solving the

factor of safety of slopes and concluded that continuum mechanics-based numerical methods ie

FEM have the following advantages

No pre-defined slip surface is needed

The slip surface can be of any shape (in contrast with Bishop and Kranz stability)

Multiple failure surfaces are possible

No static assumptions are needed

Literature study


Structures (such as footings embankments etc) and structural elements (such as

beams cables sheet piles etc) and interface can be included without concern about

compatibility

It is also important to recognize that LEM only identifies the onset of failure whereas FEM

includes the effect of stress redistribution and progressive failure after its initiation Numerical

models can also be used to determine the factor of safety of a slope in which a number of failure

mechanisms can exist simultaneously or where the mechanism of failure may change as

progressive failure occurs (Hoek et al 2000)

On the other hand the accuracy of FEM is dependent on usersrsquo settings such as the refinement

of the mesh generation the loading steps etc Moreover even if FEM can be functional and easy

to use it is essential that the user has a deep knowledge of the subject under investigation (soil

and structural mechanics) in order to be sceptical and critical with the FEM results able to

interpret the output behaviour of the structure and changecorrect everything that seems

peculiar Moreover the usage of FEM implies also some basic knowledge on numerical methods

and the general background of the FEM so as to solve possible numerical errors

An ideal way of modelling in FEM would be a prior calibration of the program according to data

related with the structurersquos properties and performance (ie stress generation) However this

means that enough field measurements should be carried out and under specific conditions in

order to be considered as a base for FEM calibration Unfortunately such field tests are not

always available and if so their reliability is on doubt For example measurements related with

the developed moments or displacements of a retaining wall inside a dike section are rare to be

found but even if there are some available they cannot represent the situation near the failure

domain where most of the engineers are worried about

312 Uncertainties and Sensitivity analysis

Uncertainty in geotechnical engineering can be categorized into aleatoric epistemic and

decision model uncertainty as they are depicted in Figure 32 (Baecher amp Christian (2003))

Aleatoric uncertainty is associated with the natural variability which is defined as the intrinsic

randomness of natural processes Such variability can be expressed by changes of the soil

properties over time at a certain location (temporal variability) or over space at a single time

(spatial variability) Epistemic uncertainty can be divided into the site characterization model

and parameters uncertainty and it is attributed to lack of information about events or lack of

understanding the physical laws that prohibits the ability of creating a realistic model Finally

the decision uncertainty describes the difficulty of being aware of social objectives defining

social values such as discount rates and predicting the planning horizon

Literature study


Figure 32 Categories of uncertainty in geotechnical engineering (modified from Baecher amp

Christian (2003))

Below the general steps of an uncertainty study are described and an introduction into the

sensitivity analysis concept is made

Global methodology of an uncertainty study

A first step of an uncertainty study8 can be described as ldquothe definition of the problemrdquo Initially

the variables of interest (or else the output variables) of which the uncertainty is to be

quantified shall be specified In sequence given several input variables for which the user may

have data andor expertengineering judgment a model denoted usually by a mathematical

function should be introduced that enables the computation of the set variable of interest

After the general context has been staged we should choose the criteria with which the

uncertainty can be evaluated The most complete measure of uncertainty when dealing with a

random vector is the probability distribution In order to assess the value of this distribution

function the following criteria can be followed

Probability of exceeding a threshold the aim is to assess the probability that the variable

of interest exceeds a threshold important for the goals at stake

Quantiles the aim is to assess the threshold that a variable of interest may exceed with a

probability equal to a given value

Central dispersion the aim is to obtain an ldquoorder of magnituderdquo of the uncertainty by

specifying the average value and the variance of a variable of interest

8 In the engineering world this is also called reliability analysis However in this section the general uncertainty study is described In particular reliability describes the ability of a system or component to function under stated conditions for a specified time period

Literature study


The next step is to define a model to represent and quantify the uncertainties of the input

variables One shall investigate each variable as a singularity and come up with the most

suitable probability density function (mostly depending on available data) Besides it is

essential to assess also the potential correlations among the variables that can be included in a

joint probability density function which is discussed later in this section

Eventually the uncertainties of the input variables shall be translated in terms of uncertainty on

the variables of interest This procedure is called uncertainty propagation and can be carried

out via several reliability methods (approximation methods or sampling methods) that are

extensively elaborated in Chapter 4 In Figure 33 a schematization of the different steps during

an uncertainty study is given

Figure 33 Methodology of uncertainty study

Last but not least a better understanding of uncertaintiesrsquo influence can be achieved by

analyzing the contribution of the different uncertainty sources to the uncertainty of the

variables of interest via a sensitivity analysis Such an analysis aims at identifying important

parameters for the system response besides it gives a better insight into the model used to

quantify the uncertainties Before conducting an uncertainty analysis it is advisable to filter out

parameters of less significance in order to reduce the modeling computational effort Below

different methods for sensitivity analysis are described

Sensitivity analysis

Sensitivity analysis (SA) is the study of how the variation in the output of a model can be

apportioned quantitatively and qualitatively to the variation in the model parameters (Saltelli

et al 2004) Saltelli (2004) proposes one possible way of grouping these methods into three

classes (i) screening methods (ii) global SA methods and (iii) local SA methods

i Screening methods

Screening is a particular instance of a sampling-based method The objective here is to identify

which input variables are contributing significantly to the output uncertainty in high-

dimensionality models rather than exactly quantifying sensitivity (ie in terms of variance)

Screening tends to have a relatively low computational cost when compared to other

Literature study


approaches and can be used in a preliminary analysis to weed out less influential variables

before applying a more informative analysis to the remaining set One of the most commonly

used screening methods is the elementary effect method

ii Global SA methods

Global SA techniques incorporate the whole range of variation and the probability density

function of the input parameters to calculate their influence on the output Many global

sensitivity analysis techniques are now available such as

Variance-based methods

o Sobolrsquos method9

o Fourier Amplitude Sensitivity Test (FAST)10

o Analysis of Covariance (ANCOVA)

Regression-based methods11

Both FAST and Sobolrsquos method rely on the assumption of parameter independence while

ANCOVA can also handle correlated input parameters The main principles of FAST method are

presented in Appendix A1 For more information about the other methods the reader can

betake himself to the related references

iii Local SA methods

Local SA methods provide the slope of the calculated model output in the parameter space at a

given set of values (Turanyi and Rabitz 2004) This basically means that local methods involve

taking the partial derivative of the output Y with respect to an input factor Xi |

|

where the

subscript indicates that the derivative is taken at some point in the space of the input

Examples for these are (Schweckendiek 2006)

the α-values in a FORM calculation (further discussion in section 422)

Local Probabilistic Sensitivity Measure

In reliability analysis the local sensitivities will be more important than the global ones in the

sense that the influence of all variables in specific points such as the design point cannot be

identified by the global methods However local SA can only inspect one point at a time and the

sensitivity index of a specific parameter is dependent on the central values of the other

parameters

In this thesis the influence of the different parameters on the response of the system is

evaluated according to local sensitivity indices However before the main part of the reliability

analysis starts it was necessary to filter out the less influencing variables in order to reduce the

number of the major variables and make the analysis more efficient and computationally

affordable For this purpose the global sensitivity method FAST was used whose results are

presented in Chapter 7

9 created by Sobolrsquo (1993) 10 introduced by Cukier et al (1973) 11 introduced by Helton et al (2005)

Literature study


Probabilities as a measure of uncertainties

Most engineers today use the concept of probabilities as the standard way to talk about

uncertainty One way to quantify a probability of a random variable is the calculation of the

cumulative probability function (CDF) of non-exceedance which can be obtained by

the probability distribution function For a random vector this reads

( ) (11)

From this the joint probability density function of this random vector can be determined as

( )

( )

(12)

This function is depicted in Figure 34 by means of contour levels The joint probability density

function has the shape of a hill as it is showed in the 3D figure and the inclination of the ellipses

reveals a correlation between the two variables and

Figure 34 Joint probability density function in 2D and 3D

The functions that join or couple multivariate distribution functions to their one-dimensional

marginal distribution functions are called copulas These are going to be used in order to define

the joint density probability distribution from which the random variables are taking their

values However no detailed explanation is made in this report regarding copulas and its

characteristics as it is automatically implemented in OT whereas the user has to define only the

type of copula that heshe considers the most appropriate (a reference is made to Nelson (1999)

for further information) According to the type of copula that is selected the order of the values

of the variables can be chosen within an iterative reliability method In this thesis an

independent copula was used considering that the input variables are independent the ones

from the others due to the time limit

Literature study


32 Previous Studies In this section a summary of the most relevant researches on the topic of FEM applied in soil

structures and coupling with reliability methods is given

To begin with an attempt to estimate the reliability of a structure modeled in FEM has been

made by Waarts (2000) by introducing an optimized reliability method in terms of

computational effort and efficiency

In particular in Waarts (2000) two adopted reliability methods are introduced both making

use of a response surface (a detailed explanation of the reliability methods and the response

surface techniques is made in Chapter 4 and Appendix F) These adaptive response surfaces are

used in combination with FORM and DS respectively The accuracy and the effectiveness of

these methods are investigated on the basis of artificial LSFs and a comparison is made with the

existing standard reliability methods The most efficient combinations of response surface

techniques and reliability methods were with FORM (FORM-ARS) and DS (DARS) Comparing

these two methods DARS predominated over FORM-ARS as it can cope with a much wider

range of limit state functions

In a later stage the above best performing reliability method (ie DARS) is further investigated

in terms of its efficiency on the basis of complex structures reliability In Figure 35 a couple of

case studies used to verify the performance of DARS are given

Figure 35 Left Two-story two-bay frame structure Right overview of a sheet pile installation

(source Waarts 2000)

The conclusions of this research showed that DARS serves its purpose and that the benefit from

using it increases with increasing number of random variables In Figure 36 the Limit State

Function Evaluations (LSFE) that are carried out as a function of the number of variables are

presented

Literature study


Figure 36 Number of LSFE as a function of the number of variables n The upper line depicts the

required number of LSFE N=160 (it is supposed that on average 3 LSFE per sample are required

and consequently the number of LSFE equals 3x160n=480n) The lower line shows the

performance of DARS (source Waarts 2000)

More recently a methodology for assessing the reliability of geotechnical structures and in

particular deep excavations has been developed by Schweckendiek (2006) More precisely this

study deals with the application of different reliability methods in combination with FEM which

carries out the LSFE The reliability methods are applied initially on simple examples in order to

be calibrated and eventually the most efficient methods are used for the reliability assessment

of a deep excavation with a sheet pile wall The several failure mechanisms are treated in detail

concerning the structural elements of the excavation as well as the soil medium

Finally the failure probability of the system is estimated according to a fault tree

schematization of a deep excavation with one layer anchored sheet pile wall Uncertainties in

the soil properties the phreatic levels and also the corrosion of the sheet pile wall were taken

into account In Figure 37 a picture of the case study is shown together with a FEA illustration

and the importance factors of different soil properties in terms of the probability of failure after

a FORM analysis

Literature study


Figure 37 Finite Element simulation and contribution of several structural and soil properties in

the failure mechanism of an anchored retaining wall in layered soil (source Schweckendiek 2006)

For a similar case study as in Schweckendiek (2006) a cantilever and an anchored sheet pile

wall the safety of the structure based on LEM is evaluated in a research conducted by Gonzaacutelez

et al (2013) Moreover in order to compare the efficiency of the classical methods with FEM

the finite element software Plaxis has been used This research shows the limitation of the

classical models and the efficiency of FEM as far as an anchored sheet pile wall in concerned

More precisely LEM results found to be conservative as they do not take into account the

confinement around the free length of the anchor rod that increases the passive pressure After

comparing numerical results with the classical methods in anchored walls the failure

mechanism given by the FEM suggests an intermediate failure state where the sheet pile wall

describes a translation movement on the deep zone and at the same time a rotation movement

around the anchor point that is closer to the reality

Furthermore Moumlllmann and Vermeer (2008) conducted a reliability analysis of a dike failure A

case study at river Elbe in Saxony Germany has been used and the failure probabilities of

different dike cross sections involving different failure modes were compared More precisely

overflowwave overtopping upliftpiping slope stability and damage of the revetment on the

waterside were taken into account For each failure mode the particular failure probability was

determined using the software package PC-Ring (Vrouwenvelder et al 1999) while the

reliability method used for the assessment of the failure probability was FORM That was

coupled with a slope stability software where Bishoprsquos approach was performed

However this method is limited to circular slip surfaces and prescribed pore pressure

distributions within the dike In order to overcome this limitation FEM was used for further

analysis In combination with FEM an adaptive response surface technique was used in order to

represent the limit state function of each failure mode In Figure 38 the simulated structure

Literature study


together with the response surface of the slope instability failure mode is showed Note that this

structure is approached with a single soil layer (same soil within and under the dike) and that

no structural element is implemented inside the dike

Figure 38 Dike model for the probabilistic FEA of dike stability and the corresponding response

surface of the limit state function for 3 stochastic input parameters (source Moumlllmann and

Vermeer 2008)

The failure probability of each of the aforementioned failure mechanisms were finally calculated

and compared with those recommended in each case while the performance of the coupling

between FEM and response surface method was assessed

Additionally a more recent investigation in terms of dikes with sheet piles employing FEA was

carried out by Breedeveld (2011) The main objective of this study was to display the

implementation of partial factors for design purposes using FEM and especially the software

Plaxis in simulating dikes with a sheet pile wall The dike was firstly simulated as a single

structure without reinforcement while in sequence the occurring stresses and pore pressures

were calculated with an existing sheet pile wall The results of the distribution of the effective

stresses within and below the dike are illustrated in Figure 39 His case study has been

introduced in Chapter 2 while in Chapter 6 a more detailed explanation of how it was modelled

in FEM is given This case study will be used as an example case in this thesis

(a)

Literature study


(b)

Figure 39 Effective stress distribution before (a) and after (b) the installation of the sheet pile wall (source Breedeveld 2011)

Last but not least Bach (2014) has been occupied with the reliability analysis of bored piles and

the case study that he examined is presented in Figure 310 The major objectives of this

research were (1) Propose models to calibrate resistance factors for the Load and Resistance

Factor Design (LRFD) (see section 423 for further explanation of this method) approach and

find a suitable model aiming to directly determine reliability of a bored pile considering some

types of defect that may occur in the bored pile (2) Select a quality control method and evaluate

its reliability when applied to bored piles

For that purpose he made a coupling calculation between the finite element software Plaxis

(version 90) and the numerical probabilistic toolbox Prob2B in order to design bored pile

foundations in light of the reliability-based design Two failure modes the geotechnical failure

mode and the structural failure mode were proposed in order to comprehensively assess the

reliability of an axially loaded pile The parameter uncertainty was considered through the use

of statistical parameters and probability distributions for material parameters in soil models

The soil parameters were treated as random variables The geometry parameters of pile were

used as deterministic quantities because a change in pile shape in the calculation process

requires establishing a new mesh which is now impossible with regard to the Plaxis software

Figure 310 Nine calculation cases considered with different pile toe levels (source Bach 2014)

Literature study


For the reliability of the pile the ULS of the pile was investigated and finally a displacement

criterion of a percentage of the pile diameter used depending on the soil type and

characteristics that the structure is founded on

33 Overview In this chapter concepts that are related to FEM probabilities uncertainty and sensitivity

analysis were discussed The introduction of FEM applications and the principles of how to

handle uncertainties will help the reader to better understand and follow the process of this

thesis

Furthermore preliminary researches related to several aspects of this thesis are presented that

mainly concern

Reliability methods and their applications [Waarts 2000 Schweckendiek 2006 Bach

2014]

Coupling FEM with reliability methods [Waarts 2000 Schweckendiek 2006 Bach

2014]

(Reliability) Analysis of geotechnical structures and especially dikes reinforced with

retaining walls [Waarts 2000 Schweckendiek 2006 Moumlllmann and Vermeer 2008

Breedeveld 2011 Gonzaacutelez et al 2013 Bach 2014]

As someone might have noticed from the overview of the existing research on the topic both the

performance of different reliability methods and the simulation of a dike with a sheet pile wall

in FEM have been carried out These references were quite helpful in order to get an idea of

coupling FEM with reliability methods as well as FEM and reliability methods individually

Moreover a first dive into the system under investigation and the variables to be handled was

made However noone has analysed the reliability of such a complicated system that is more

and more often seemed to be applied in the Netherlands Thus it would be essential to evaluate

the implementation and the results of the reliability analysis of a dike with sheet piles simulated

in FEM The conclusions of such a research are going to contribute to design and optimization

concepts and hopefully to a better understanding of the system behaviour


4 Structural Reliability Analysis

The aim of this chapter is to introduce the basic theory of a reliability analysis and to summarize

the principles of different reliability methods that are widely used for the uncertainty

evaluation of a system Particularly three of them (MC DS and FORM)12 were used in this thesis

and are described in this chapter Finally a brief evaluation of the selected reliability methods

and the way that they are applied on this thesis by coupling them with FEM are discussed

41 Basics of Reliability Analysis In the consideration of the reliability of an element the determination of the probability of

failure is the central issue The limit between failure and non-failure is defined as a limit state

and the reliability is the probability that this limit state is not exceeded The limit states are

interpreted through the so-called limit state functions (LSF) whose general form is

(41)

in which R is the strength or more general the resistance to failure and S is the load or that

which is conducive to failure (ldquosolicitationrdquo) The basic principle of structural design is that the

resistance needs to be higher than the load or in other words that the LSF is larger than zero

(Zgt0) The main objective of the design is to ensure that this performance criterion is valid

throughout the lifetime of a structure Nevertheless the majority of the quantities that both

resistance and load consists of are uncertain Therefore a probability of satisfying the preceding

criterion is estimated and expresses the reliability of the structure P(Zgt0) The probability of

failure is

(42)

Hence the probability of failure is complement to the reliability to the sense that

(43)

The LSF is plotted in Figure 41 indicating the failure space and the limit state function by Z=0

12 Besides RS techniques were used for proceeding with the analysis but most for solving numerical problems rather than evaluating the failure probability The RS techniques are still under investigation and need to be tested before being applied in real cases However it is highly possible that this would be a potential way of evaluating the failure probability in the future

Structural Reliability Analysis


Figure 41 Limit Sate function

In this case the design concept is based on the evaluation of the so-called design point which is

the point in the failure space with the greatest probability density Generally this point is

located on the border between the safe and the unsafe areas

In the structural domain the Joint committee on structural safety (1981) proposed a level-

classification of the calculation methods This classification includes the following three levels

Level III these methods calculate the probability of failure by considering the probability

density functions of all strength and load variables The reliability of an element is linked

directly to the probability of failure

Level II this level comprises a number of methods for determining the probability of

failure and thus the reliability It entails a linearization of the LSF in a carefully selected

point These methods approximate the probability distribution of each variable by a

standard normal distribution

Level I at this level no failure probabilities are calculated The level I calculation is a

design method according to the standards which consider an element sufficiently reliable

if a certain margin is present between the representative values of the strength and the

loads This margin is created by taking the so-called partial safety factors into account in

the design

In the next section the different reliability methods to be used in this thesis are further

elaborated

42 Overview of Reliability Analysis Methods

421 Level III Methods

The foundation of the Level III failure probability calculation is the mathematical formulation of

the subset of the probability space which involves failure (Zlt0) Level III reliability methods



(also known as fully probabilistic methods) compute the probability of failure based on the

exact probability density function and the exact limit state functions Therefore such methods

are considered to be the most accurate Well-known sampling methods are (Quasi-) MC

simulation Importance Simulation DS and the Latin Hybercube Simulation Below the former is

discussed as it represents the fundamental theory on which the sampling methods are based on

while DS is also further explained as it is continuously used in this thesis For the rest a brief

explanation is provided in Appendix F There are also other sampling methods such as the

Numerical and the Directional integration which will not be described in this study However

for further information about them a reference though is made to Waarts (2000) In Level III

methods errors can only occur by a too low number of simulations

Crude Monte Carlo Simulation

The MC method uses the possibility of drawing random numbers from a uniform probability

density function between zero and one If and are the

vectors for the resistance and the load respectively that consists of n variables then by taking

multiple realizations of the uniform probability distribution a value can be determined for

every and

By repeating this procedure a large number of times the failure probability can be estimated

(44)

where is the total number of simulations (nm draws from the uniform distribution in which

m is the number of base variables) and is the number of simulations for which Zlt0

The simulationrsquos relative error is

(45)

If the expected value of the relative error is zero the standard deviation is

radic

(46)

Based on the central limit theorem the error is normally distributed provided is sufficiently

large The probability that the relative error is smaller than the given value E is then

(47)

Thus for the reliability of the relative error is smaller than For the wanted k and

E the required number of simulations can be determined with



(48)

If for instance a reliability of 95 is required for a maximum relative error E=01 the required

number of simulations amounts to

(49)

The number of simulations is therefore still dependent on the probability of failure

MC simulation is applicable whatever the complexity of the deterministic model is However the

number of samples required to get an accurate estimation of may be dissuasive especially

when the value of is small (10-k) Thus crude MC is not applicable when small values of are

sought andor when the CPU cost of each run of the model is non-negligible

In Appendix F and especially in sections F1 and F2 a further elaboration of the MC simulation

is presented Moreover the methods of Quasi-MC Importance sampling and Latin Hybercube

are described

Directional Sampling

The directional simulation method is an accelerated sampling method It implies a preliminary

iso-probabilistic transformation as for FORM and SORM methods however it remains based on

sampling and is thus not an approximation method Below the method as it is described in Open

TURNS 15 Reference Guide (2015 pp190-193) is presented

Each simulation of the directional simulation algorithm is made of three steps Using the

probability distribution of a random vector we seek to evaluate the following probability

( ) (410)

Here is a deterministic vector and ( ) the limit state function which enables the

definition of the event For the iteration these steps are the

following

Let A point is drawn randomly on S according to a uniform

distribution

In the direction starting from the origin and the passing through solutions of the

equation ( ) (ie limits of ) The set of values of that belong to is

deduced for these solutions it is a subset of

Then one calculates the probability By property of independent

standard variable

is a random variable distributed according to chi-square

distribution which makes the computation effortless



Finally the estimate of the probability after N simulations is the following

sum

(411)

The main idea is that for each direction we go along the direction by step of a specified length

from the origin to the maximum distant point and we check if there is a sign change on each

segment so formed (see Figure 42) We go until the maximum distant point Then for all the

segments where a sign change is detected we research the root λ in the segment (there are

also other strategies available in OT however all of them follow the same principle of

searching) The following figure illustrates the principle of an iteration in dimension 2

Figure 42 DS of a 2-dimensional problem

Similar to MC method the outcome of the estimated probability of failure is a random variable

and the error in the estimate can be made as small as possible by taking a sufficient number of

samples For DS the standard deviation σ of the estimated failure probability can be quantified

as follows (Melchers 2002)

radic

sum

(412)

From this equation relative errors and the confidence intervals can be estimated As can be

seen the error in the estimated probability of failure will decrease with increasing number of

sampled directions Equation (412) can be used to determine the number of sampled directions

that is required for a reliable estimate of the failure probability The convergence criterion in

this method is usually the coefficient of variation (CoV) which is defined as



422 Level II Methods

Level II methods (known also as fully probabilistic methods with approximations) can take all

the probabilistic properties of the random variables into account but they include

approximations of the limit state function and therefore their use and outcomes should be

inspected and evaluated in order to be considered as reliable However experience in that

methods has shown that the computational effort is profoundly decreasing in comparison with

Level III methods and that the application of them can provide important parameters such as

the reliability index (β) and the influence factors (α) Some of the most known Level II methods

are First Order Second Moment (FOSM) Method First and Second Order Reliability Methods

(FORM and SORM) and the Point Estimate method (PEM) In the next paragraphs FORM is

elaborated as it will be mainly used in this thesis while in Appendix F3 the principles of FOSM

are discussed

First Order Reliability Method (FORM)

The principles of FORM are based on the First-Order Second-Moment (FOSM) approach which

first introduced the reliability index concept This method is presented in Appendix F3 in detail

However the reliability index estimated using the FOSM approach is not ldquoinvariantrdquo which

means that the index value depends on the limit state function format (Farrokh 2007) Hasofer

and Lind (1974) proposed an invariant definition of the reliability index that leads to a

geometric interpretation known as first-order reliability method (FORM)13 The first step of this

method is the transformation of the random variables to equivalent standard normally

distributed variables and the whole procedure is carried out in u-space (or else standard space)

For variables that are normally distributed this step is as follows

(413)

For other types of distributions there are procedures available for carrying out this

transformation such as Generalised Nataf and Rosenblatt transformations In this study these

transformations are not discussed however for further explanation a reference is made to the

Reference Guide of OpenTURNS 15 (2015)

In the sequence the limit state function Z is expressed in terms of

The second step is the approximation of the function with the first two terms of the Taylor-

polynomial The approximation reads (CUR 1997)

( ) ( ) sum

(414)

where is the point where the linearization takes place This approximation of Z is linear and

according to the central limit theorem it is normally distributed The expected value of the LSF

can be approximated by the expected value of the linearized function

13 VNK2 uses FORM in view of the calculation time required In case of inaccuracies VNK2 switches to more time-consuming techniques such as DS whereby a large number of draws are made from the probability density functions of the various stochastic variables (uncertain quantities)



( ) sum

(415)

While the standard deviation (sum of contributions of each variable to the variance of Z) is

defined as

radicsum

( )

(416)

Using the definition of FOSM for the reliability index [ ]

where [ ] is the mean value of

the limit state function and substituting Eq 415 and Eq 416 in it the reliability index can be

approximated However linearization in different points can lead to different values of the

reliability index Hasofer and Lind definition of the reliability index overcomes this limitation

and renders it equal to the minimum distance from the origin to the design point

(417)

where radic

Looking for the design point is basically an optimization problem Many analytical and

numerical approaches can be used for that purpose A relatively straightforward method to do

this is by firstly assuming that the design point is the mean value (the starting point can also be

another point according to an engineering judgement in order to accelerate the optimization

procedure) The obtained β-value is used to determine a new point in which the LSF is

linearized In this case the importance factors αi are calculated as

( )

radicsum (

( ) )

(418)

where is the design point and is the number of variables expresses the

contribution of the variance of each variable to the total variance of Z in the design point Figure

43 illustrates the linear approximation of the limit state function and the aforementioned

parameters for a two dimensional problem The probability that Zlt0 can be determined using

the standard normal CDF

(

) (419)



The new calculation point is determined by

(420)

After some iterations the chosen optimization algorithm finds the final design point and the

new reliability index In Appendix A2 an overview of the different optimization algorithms

available in OT is presented together with an evaluation of their performance in the specific

case study

Figure 43 Two-dimensional illustration of u-space LSF and design point

423 Level I Methods (semi-probabilistic)

At the beginning of the probabilistic concepts incorporation in the field of structural

engineering the most notable development was the implementation of LFRD At the past single

factors of safety on the ratio of total resistance to total load were used in order to ensure that

the stresses developed from an applied load on a structure were lower than the allowable ones

LRFD replaces those factors by introducing a set of partial safety factors on the individual

components of load and resistance These partial factors can be selected such as they account

for uncertainties associated with the resistance and the load respectively

The current design philosophy in CUR 166 and Eurocode is characterized by the use of

characteristic values of the parameters (in Appendix E a further explanation of the

characteristic values is given together with the way that they are converted into the mean

values for being used in the reliability analysis later on) The values of the partial safety factors

are additional factors to the characteristic values Therefore the characteristic values are

multiplied with the aforementioned safety factors ( ) and the following criterion has

to be satisfied for a reliable structural performance (see Figure 44)

(421)



The main disadvantage of LRFD is the fact that there is a set of load and resistance factors that

need to be calibrated in order to cover the specific cases that are likely to occur This is because

of the lack of sufficient statistical data that are necessary for this calibration procedure

Consequently it cannot reassure that all the designs to which the set of factors is applied result

in the reliability level that was aimed for As a result in majority of the cases a conservative

calibration that gives the required reliability is chosen which can lead to ldquoover-designrdquo

structures

Figure 44 Design with partial factors for the load and resistance parameters (source

Schweckendiek 2006)

The evaluation of an elementrsquos reliability starts in principle with calculating the probability of

failure and subsequently the reliability for the given strength and load In practise the problem

is often that the strength is unknown but it has to be determined for a given reliability The

determination of the required reliability can be estimated with the help of Level II and III

methods by iteratively adjusting the strength in the calculation until a sufficiently small

probability of failure is found

In the design domain regulations and guidelines follow the standard that the characteristic

value14 of the strength is divided by a factor and that the characteristic value of the load is

multiplied by a factor as it is described in Eq 421

The link between Level I and the estimation of the failure probability has been achieved through

Level II methods The design point that results from a Level II method calculation is the point

with the greatest joint probability density of the strength and the load and it is therefore

possible that when failure occurs the strength and load values will be close to that point The

design values can be specified as

(422)

(423)

14 The characteristic value of a parameter is its main representative value and its value can be specified as a mean value an upper or lower value or a nominal value according to EN 1990 definition (2002)



where are the coefficient of variation for the resistance and the solicitation respectively

15 and are the importance factor and β is the reliability factor These can be also expressed

in terms of characteristic values as follows

(424)

By substituting function 423 with 421 and 422 the partial safety factors are defined as

(425)

(426)

where 16 and are the values for load or resistance respectively to which a probability of

(non-)exceedance of 95 corresponds (non-exceedance for the load and exceedance for the

resistance) (for a standard normal distributed parameter this is 1645)

424 Response Surface Techniques (RS)

In case that the models presented above tend to be time consuming for the limit state function

evaluation or convergence problems of the optimization algorithms occur a better methodology

is recommended known as response surface The RS is mainly used when (unknown) response is

only available from experiments or complex FEM computations (for example large highly-non-

linear FEM models) Indeed once a RS has been built up the various standard methods may be

applied at a negligible cost An analytical limit state function replaces the real response function

The main idea is that the response consisting of a complex function of input variables is

approximated by a simple function of the input variables

A list of possible response surfaces techniques is given below

Linear and quadratic Taylor expansions

Polynomial RS based on least squares

Kriging method

Polynomial chaos expansion

A detailed explanation of the aforementioned methods can be found in OpenTURNS 15 (2015a)

The first method is associated with the approximation of the model response around a specific

set of input parameters while the rest seek a global approximation of the model over its whole

domain of definition The most sophisticated one is the polynomial chaos expansion but one the

15 α values are negative for solicitation (ie RV for which increasing their values lead to adversely effects on the reliability) and positive for resistance (ie RV for which increasing their values implies a beneficial effect on the reliability) 16 is negative and can be negative or positive



other hand its complexity does not make it attractive In this research a polynomial RS is

applied based on least squares

In combination with FEA the standard procedure is as follows

1 Select the most important random variables on the basis of engineering basis

2 A SA is carried out in combination with FEM

3 Reduce the stochastic variables if needed according to the SA

4 A RS is constructed through the response data

5 A reliability calculation is carried out using the RS instead of the real response

A polynomial RS is generally constructed by fitting a quadratic function to the sampling points

whose general expression is

sum

sum

sum sum

(427)

The type of the reliability method to be used is of little importance since the time consuming

LSFE (using FEM) are replaced by analytical expressions However of main importance is now

the accuracy of the RS compared to the real response There are several parameters in order to

evaluate the goodness of fit of the RS to the real model among which is the well-known R-

squared

In Figure 45 an example of a quadratic response surface is showed that is fitted on the data

response indicated with the black circles This was a 6-dimensional problem that means 6

different variables were included in the system (non-visualized dimensions are kept constant at

their mean value) In this thesis the response of the system is to be acquired after the coupling

of FEM with a reliability method and that would give the ldquoblackrdquo circles that are shown in Figure

45 The response surface technique applies a curve fitting on the real response of the system

(ldquoblackrdquo circles) whose analytical formula can later be used for the reliability analysis of similar

type of systems with negligible computation time



Figure 45 Quadratic response surface with 28 terms (ie ) for 6 dimensions ( ie x

=[ x1 hellip x6])

43 Coupling Reliability Analysis with FEM The limit state function evaluations (LSFE) will be carried out with the software Plaxis 2D 2015

which is a special two-dimensional finite element software used to perform deformations and

stability analysis for various types of geotechnical applications Moreover considering the case

study under investigation in this thesis Plaxis in contrast with other FEM such as Abacus

Comsol DIANA etc offers several techniques to realistically simulate structural elements such

as sheet pile walls and anchors and their interaction with soil while the variety of the

constitutive models for the soil body that are available and the ability to include the history of

the construction phases can lead to a better analysis of the systemrsquos behavior in terms of the

stress level and the deformations It is essential at that point to mention that using FEM for this

purpose means that the limit state formulation is implicit and can only be solved numerically

The reliability analysis is carried out through an uncertainty package In this section firstly a

description of the reliability package and its possibilities are given Finally an explanation of the

coupling procedure between the reliability tool and FEM is given together with the calculation

process that was followed for the parameters manipulation

431 The functionality and possibilities of OT

The reliability analysis tool to be utilized in this research is OpenTURNS (OT) which is a

scientific library usable as a Python module dedicated to the treatment of uncertainties and it is

still under development during the work of this thesis Several reliability packages are available

such as OpenEarth (applied in Matlab) Prob2b (applied in Probox software of TNO) and

Probabilistic Toolkit (software of Deltares) Plaxis 2015 features Python connection possibility

and thus coupling was decided to be carried out with an uncertainty package in Python and OT

is the most developed one Moreover until now only OT is an open source package and can be



used easily from anyone as there is a wide community of experienced people supporting with

their knowledge while many related manual reports are already available for starting learning

A list of the current available reliability methods in OT is given in Figure 46 In this figure apart

from the standard methods the alternative method of the Adaptive Response surfaces is

introduced that is mainly used when (unknown) response is only available from complex FE

computations

Figure 46 Reliability methods available in OpenTURNS

In this thesis the methods to be basically applied are the FORM and the DS analysis while MC is

also used mainly for confirming the application of FORM Additionally the RS technique has

been mainly implemented for enhancing the performance of FORM and DS analysis

OT handles 47 types of distributions amongst which the ones used in this research are Normal

Lognormal Uniform and Truncated Normal distribution Regarding the joint distributions that

are available to be selected for random vectors 11 types of copulas are existing in OT amongst

which the most known ones are the Independent the Gumbel and the Normal copula In this

thesis the independent copula has been used during the reliability analysis which means that

the variables were assumed to be independent among each other However it is strongly

advised a further elaboration and research considering correlation matrices for specific soil

parameters

432 Coupling OpenTURNS-Plaxis

The finite element simulation will be carried out in Plaxis 2015 Plaxis is a finite element

software for plane strain and axi-symmetric modelling of soil and rock behaviour Moreover it

supports a fully automatic mesh generation allowing for a virtually infinite number of 6-node

and 15-node elements



The coupling of reliability analysis and FEA requires an interface for the communication

between each other When a reliability tool is coupled with another software program the

reliability program carries out the whole reliability analysis and it uses the other program only

for the evaluation of the limit state function More precisely OT should be able to read and

amend Plaxis output values for important variables such as material parameters pore pressures

generation and stresses development and corresponding deformations inside the dike

Respectively Plaxis has to be also capable of obtaining the new values that has been set by OT

for the variables that are treated as stochastic during an iterative process according to the

reliability assessment In Figure 47 an illustration of the coupling methodology and its function

is shown

Figure 47 Coupling scheme OpenTURNS-Plaxis

In principle an input file is firstly required where the user set the preferable reliability method

to be used the stochastic input parameters and their probability distributions the joint

probability distribution and the corresponding correlation matrix and finally the limit state

function is formed depending on the situation In Appendix D an example of the input files that

were used for soil sheet pile wall and anchor analysis is shown However such input files

should be interpreted so as to be readable by both Plaxis and OT Therefore an input interpreter

was created which is actually a python script that helps OT to start up the reliability analysis

according to the assigned method variables distributions and LSF As it was mentioned before

the evaluation of the limit state function is conducted by Plaxis For that purpose the input

interpreter should be also able to send the next set of input parameters to Plaxis However an

additional means of connecting the interpreter with Plaxis is also needed This is can be



achieved via a Plaxis interface which actually call Plaxis to make the calculations while it also

transfers the required value of the limit state function to input interpreter and this in turn to OT

Eventually the probability of failure is obtained as a model result However it is also essential

that Plaxis simulation procedure converges to the desirable criteria and under the physical

boundary conditions that have been determined Likewise the convergence criteria of the

reliability methods shall be manipulated so as the optimization algorithms to able to converge

efficiently (see Appendix A2 for further explanation)

44 Overview In section 42 a summary of the principles of the main reliability methods was made From

Level III methods MC and DS were presented whereas from Level II the basics of a FORM

analysis were introduced Besides the concept of the Level I method was explained Even

though this method was not used in this research for the evaluation of the probability of failure

it was applied in section 643 for the deterministic analysis of the system under investigation

and the preliminary calculations of the dimensions In Appendix F more reliability methods are

introduced and they are further described In the sequence their implementation in conjunction

with Plaxis was discussed through the coupling of FEM with OT For more information

concerning OT special features (ie SA optimization algorithms and probability distributions) a

reference is made to Appendices A and E

Before any of these reliability methods was applied on the case study of this thesis their

performance and their compatibility with Plaxis were tested with simple examples At first the

probability of failure was evaluated with FORM SORM Importance Sampling (IS) crude MC and

DS for simple artificial linear and non-linear limit state functions

and

respectively where R B and F are the random variables distributed normally

or lognormally The crude MC technique has been applied in order to validate the failure

probability in case the various methods give different results Secondly a flexible circular

footing over a one-layer soil material was used from Plaxis tutorial in order to test the coupling

between Plaxis and the reliability method and reassure the functionality of the interface In this

pilot example the objective was to determine the probability that the settlements of the footing

exceed a certain threshold of settlements and thus The respective

components that were taken into account for the failure mechanism was the cohesion (c) the

friction angle (φ) and the specific weight (γ) of the soil

From a qualitative evaluation of the reliability methods that has been done and according to the

experience gained from the aforementioned applications of some methods valuable conclusions

were drawn in terms of their efficiency More precisely regarding the sampling methods it does

not required previous knowledge of the failure domain apart from the IS method in which a

starting point inside the failure domain shall be provided Moreover the calculation effort

depends on the magnitude of the failure probability and the required accuracy whereas for the

IS it depends also on the selection of the sampling region In Waarts (2000) it has been proved

that DS tends to be more efficient than MC for low dimensional problems as it is indicated in

Figure 48 (number of random variables nlt100)



Figure 48 Required number of samples for MC and DS as a function of the random variables


As far as the approximation reliability methods (FORM and SORM) are concerned it is not

necessary to be aware of the failure region in advance Furthermore it has been observed that

the required iterations and the calculation time is quite lower in comparison with the sampling

methods However the accuracy of the method is highly dependent on the shape of the LSF

In Table 41 a summary of the most applied reliability methods evaluation is presented

According to this evaluation it was decided that the methods of FORM and DS as well as the

combination of them with RS techniques are going to be tested and evaluated in terms of their

efficiency and robustness These methods were chosen from both Level II and Level III methods

due to their expected reduced computational time the non-requirement of previous knowledge

about the LSF and their satisfactory performance in similar case studies In Chapter 5 the

different failure mechanisms and the relevant LSF that are going to be utilized in this thesis are

presented whereas in Chapter 7 the results deduced from the reliability analysis are elaborated

In Chapter 7 there is also an indication of which reliability method was used for each systemrsquos

component the selection of which was based on the aforementioned evaluation

Table 41Evaluation of reliability methods

Method Previous knowledge Accuracy Calculation Effort

FORM not required full accuracy for

Gaussian variables and linear LSF

depends on LSF linearity and the number of random

variables

SORM not required exact up to 2nd order

LSF error dependent on the shape of LSF


variables



Crude MC not required can be controlled by convergence criteria

depends on the magnitude of failure probability and the

required accuracy

IS required can be controlled by convergence criteria

depends on the choices made for the sampling region

DS not required can be controlled by convergence criteria

for low dimensional problems (nlt100) DS is more efficient

than MC

Especially RS were created based on a SA sampling in order basically to increase the efficiency

of FROM and DS analysis in case of FEM numerical errors Precisely an analytical formula or a

response surface was needed that can approximately represent the performance of the structure

under investigation and that is called in case of Plaxis errors This formula was essential in

order to keep the reliability analysis running by providing a response value for the LSF when

Plaxis calculation is unsuccessful and thus incapable of returning a result It should be

mentioned that in case a RS is implemented for the reliability analysis Figure 47 can be

reduced to a graph consisting of only the input file OT and the input interpreter as no Plaxis

calculations would be necessary any more The LSFE will then be performed through the

response surface that is provided by the user Due to this limited use of the RS method it will

not be further treated in this thesis




5 Failure Mechanisms and Limit State Functions

The reliability analysis of a structure requires the definition of the different failure modes that

are relevant to the corresponding structural elements In this chapter the possible failure

mechanisms of the systemrsquos elements expressed in LSF are identified and their role in the

system reliability is explained Based on these LSF the reliability of the system components was

evaluated These LSFs focus on the ULS whose exceedance implies failure of the corresponding

component

51 Introduction to the system analysis and the limit states For system reliability analysis all the possible failure mechanisms have to be identified and

summarized in a fault tree In Figures 51 and 52 the fault trees of a dike section without

structural elements and a retaining structure with sheet piles are depicted respectively In this

project the combination of these two separate structures ie a reinforced dike section with an

anchored sheet pile wall is to be simulated and studied in terms of its failure modes

As it is described later in this chapter failure can be expressed in different ways depending on

the structure and stakeholders demands and safety standards This research is mainly focused

on the ULS of failure and Mohr Coulomb was chosen as the soil failure criterion due to its

efficiency on detecting failure Such a failure criterion is actually defining the stress strain

relationship and the gradual weakening of the soil (ie the gradual reduction of the strength

parameters such as the friction angle and the cohesion due to the stresses development until

soil collapses) under a stress condition In Plaxis new version there is the ability of gradually

reducing structural properties such the strength of steel elements However the results

retrieved of such a safety analysis have not been yet investigated and for that reason such

calculations are not included in this project

As far as the dike section is concerned the failure mechanism to be investigated in this project is

the macro instability of the slope rather than the other mechanisms depicted in Figure 51 On

the other hand because of Plaxis limitations to distinguish among the different structural

elements failure mechanisms each element was considered as a singular case for investigating

its failure mode In the next section a distinction is also made between the ULS and the SLS

For a reinforced dike with retaining walls there are basically four classes of structural elements

(see Figures 12 and 21)

∙ Retaining wall (ie sheet piles)

∙ Anchors

∙ Walings

∙ Soil structure (ie dike section)

Failure Mechanisms and Limit State Functions


For the reliability analysis of each of the elements the probability of failure is to be determined

In the next sections of this chapter the respective LSF that are considered for each class are

further elaborated according to the ULS criterion while in the end the general combined fault

tree is to be schematized

Figure 51 Fault tree of a dike section

Figure 52 Fault tree for retaining structures using anchored sheet piles (source Schweckendiek 2006)

The system failure in this research is considered as a serial system of the anchor sheet pile

wall and soil body LSFs The fault tree of the system failure is presented in Figure 53 together

with the LSF of each component as they are formulated according to sections 522 and 523



Figure 53 Fault tree of the system failure

It should be mentioned at that point that the probability of failure of the system differs from the

overall probability of flooding which takes into account any potential failure mechanism of the

system combined with the uncertainties of the water elevation More precisely as far as the dike

safety is concerned a probability of failure (where ) under a certain

water level is estimated from the fault tree of Figure 51 which in sequence is multiplied

with the occurrence probability of the corresponding water level in order to estimate the

overall probability of failure for the specific water elevation Then that product is accumulated

over a required range of water levels that can jeopardise the overall stability of the structure in

order to calculate the overall probability of failure or else the probability of flooding This

probability is then compared to the one established from the safety standards so as to reassure

the safety or not of the structure can be estimated as follows

int

sum (51)

In this thesis the probability is to be estimated where is a specific water level

and especially the design water level as it is considered to be the most challenging part of the

procedure described above gives the probability of occurrence of the water level (ie it is

the PDF of ) After setting up the steps and implementing them successfully for the estimation

of the calculation of the overall probability of failure is just a repetition of the same

procedure for more water levels It should be mentioned that in this thesis failure consists only

of the macro-instability failure mode whereas the rest are excluded for the time being A

simplified alternative for the overcoming the integral of Eq51 is to repeat the calculations for

certain water levels and then sum their products instead of integrating the full range of the

them

52 Limit State Functions

521 Serviceability Limit State

The SLS is evaluated in design calculations in order to avoid excessive deformations of a

structure that could lead to the loss of its service requirements and its functionality In some

cases the SLS criteria are used to prevent damage to adjacent structures in the surroundings



For example concerning the design of a dike section attention should be paid to deformations

caused to buildings that are located in the inland part of the dike in case of an extreme event

In Figure 54 an example of a deformed dike is depicted In this figure some of the potential

locations that excessive deformations might be experienced are illustrated such as

∙ the vertical settlement of top of dike (arrow 1)

∙ the horizontal displacement of the top of the sheet pile (it might also be the middle part

of the sheet pile that ends up with the largest deformation according to the loading

conditions and the specific soil structure different deformations might occur) (arrow 2)

∙ Uplift upstream of the dike (arrow 3) special attention shall be paid to this type of

deformation not only due to the direct effect on inland structures but also as a sign of

developing piping mechanism

(a)

(b)

Figure 54 Figure (a) shows the initial undeformed structure while (b) illustrates the deformed mesh (not to scale) (the plotted length of the anchor does not have any physical meaning) In (b) the possible locations and directions (black arrows) of potentially excessive deformations

According to the current Dutch regulations (Larsen et al 2013) the SLS criteria for designing

reinforced dikes with sheet pile walls require that

∙ the settlement on the top of dike does not exceed 10 cm and

∙ the horizontal displacements of the sheet pile wall does not overcome 10 cm

In this research the SLS criteria and especially the dike settlements were used roughly for the

preliminary determination of the structures characteristics that are discussed in section 643

Dike Anchor

Sheet pile wall

1

2 3



More precisely displacements at the top of the dike were also taken into account as

complementary to the safety factor in order to determine the required length of the sheet pile

wall As far as the reliability analysis in concerned the LSF were formulated based on the ULS of

the different systemrsquos components that are presented in the next section

522 ULS for Structural Members

In the design process one is most interested in the ULS of a failure mechanism This state

describes the situation wherein the acting extreme loads are just balanced by the strength of the

construction If that limit state is exceeded the construction will lose its functionality and thus

collapse or fail In general most attention is paid to the behaviour of the structure after

completion However during construction there are also periods in which the construction may

fail The different phases of construction are listed in section 642 In the present section the

analytical LSF of the structural elements are given as they are going to be used in the reliability

analysis

Sheet pile wall

The most relevant failure mode for the sheet pile wall is the exceedance of the yield strength

which corresponds to the ultimate steel strength The response of the structure is mainly due to

bending moments and the axial forces (shear forces are considered to be negligible) Where an

axial force is present allowance should be made for its effect on the moment resistance

Figure 55 The water and earth pressures as well as the anchor forces lead to bending moments and axial forces on the sheet pile wall

In Figure 55 an example of the axial forces and the bending moments that can be developed in

the sheet pile wall with one anchor layer are illustrated after an extreme water level loading

Accordingly the maximum stresses on the sheet pile wall are composed of a bending moment

and a normal force component17

[

] (52)

where [kNm] and [kN] are the bending moment and the axial normal force

respectively that depend on the depth level where they are calculated over the sheet pile length

[m3] is the elastic section modulus and [m2] the cross-sectional area of the sheet pile

wall

17 the vertical anchor force component is reducing by its interaction with the soil over depth



Bending moment and axial force can be variable over the depth and that is why they are

expressed as a function of z-depth FEM has the advantage to take into account second order

effects ie a stiffer structure will experience higher bending moments than a more flexible one

Taking into account the above the limit state function can be formed as the difference between

the maximum developed stress and the yield stress

[

] (53)

where and can be characterized as the load variables while and can be considered

as the resistance variables and are assumed to be constant over depth

Concerning the permissible displacements for the ULS an upper limit for the maximum

horizontal displacements of the top of the sheet pile has been set to 150L where L is the

vertical length of the sheet pile wall (Larsen et al 2013) This requirement was mainly used in

section 643 where the structural properties were determined after the deterministic analysis

Anchors

Anchors are loaded by their reaction to the horizontal loads on the retaining walls The failure of

the anchor element is actually represented by the failure of the steel members of the anchor

(tubes bars cables etc) that are loaded by traction forces

As it is illustrated in Figure 56 the anchor experiences an axial tensile N that is nearly constant

over its length

Figure 56 Axial loading of anchor inside a dike

The elastic behaviour of an anchor involves only a relationship between axial force N and

displacement (elongation) u of the form

[ ] (54)

where EA [kN] is the anchor stiffness consisting of the steel Youngrsquos modulus E [kNm2] and the

anchor cross section A [m2] and L [m] in the length of the anchor



Similarly to the sheet pile wall the limit state function of the anchor involves the certain yield or

ultimate strength of the steel members and the maximum stress that the anchor experiences

during its loading Consequently the LSF is as following

(55)

where [kN] is the calculated anchor force and [m2] is the cross sectional area of the

anchor (both of them considered to be constant over the depth) It is essential to mention that

the anchor is also subjected to bending moments due to soil settlements (that are implicitly

illustrated via the uniformly distributed load q over the tie rod) that should be taken into

account in order to investigate the displacements of the tie rod itself However in this thesis

only the axial forces on the anchor are considered without taking into account the individual

deformations and its reaction with the surrounding soil

Walings

The waling is the element that transfers the loads from the retaining wall to the anchors (see

Figure 21) The loading of the walings can be schematized as a continuous beam on several

supports as it is depicted in Figure 57

Figure 57 Loading of walings

If Nα [kNm] is the calculated anchor force per m then the developed bending in the support Ms

and in the opening Mo can be approximated as follows

[ ] (56)

(57)

where La is the mutual anchor distance Considering as the design moment of the waling to be

the limit state function can be formulated as follows

Waling Sheet pile Anchor



(58)

For a conservative design of the waling the limit state function will give the same or lower

failure probability than the anchor itself Therefore it will actually not be necessary to carry out

this analysis The anchor failure will be the determinant mechanism (Schweckendiek 2006)

523 ULS for Soil Failure

In this thesis emphasis is given on the dike global instability which actually consists of several

failure modes Figure 58 indicates the possible failure mechanisms of flood defences The

combination if these can lead to the overall instability of the dike and thus to soil body failure

However Plaxis is not capable of accounting for all of them More precisely Plaxis assumes the

soil to be a continuous body and thus it can model movements in the scale of soil bodies (using a

relative fine mesh in relation to the moving soil body) but not on a soil particle level Plaxis can

simulate the groundwater flow in a soil body and from this someone can deduce input for a

piping analysis (see failure mechanism G in Figure 58) (using other softwaremethods) but as

it has been mentioned above it cannot determine movement of soil particles due to

groundwater flow Moreover Plaxis cannot deal with the flow and waves occurring in ldquoopen

waterrdquo ie water outside the soil in a canal a lake or sea for instance

Figure 58 Possible failure mechanisms of flood defences (source Flood Defences 2015)

Therefore Plaxis can handle the failure modes (C) (D) (E) (J) (K) and (L) of those illustrated in

Figure 57 In this thesis loads due to extreme temperature conditions or ships are not

considered while the settlements are taken into account implicitly though the mechanisms (C)

(D) and (E) These mechanisms are thus investigated in this thesis and they mainly refer to the

macro-instability of the inner and outer slope of the dike or the overall dike body (ie horizontal

sliding (mechanism D))



For these failure modes soil failure is defined as shear failure In Figure 59 the most relevant

patterns of the macro-instability regarding the shear strength of the soil are illustrated

Applying FEA the most critical failure mode is determined automatically However it is not

always straightforward what the trigger mechanism of the failure was and that is why more

investigation and FEA tests are needed in order to obtain a clear view

(a) Outer slope failure (b) ldquoActive siderdquo failure

(c) ldquoPassive siderdquo failure (d) Overall failure

(e) ldquoKranzrdquo stability

Figure 59 Macro-instability failure modes in a dike with an anchored sheet pile wall

In Figure 59 failure modes (a)-(d) follows Mohr-Coulomb failure criterion In principle the

total stress state inside a dike section consists of the effective stresses and the pore pressures

according to Terzaghi principle

[

] (59)

Mohr-Coulomb failure criterion basically considers as limit for the failure to occur the

maximum shear stress that the soil can withstand This is determined from the friction

angle the cohesion of the soil and the current stress state The drained shear strength

(similarly in for the undrained shear strength) in the soil according to Mohr-Coulomb is defined

as follows



[

] (510)

where is the effective friction angle In Appendix B1 a detailed explanation of this failure

criterion is attached

Taking into account the above mentioned failure mechanism (a) is mainly triggered by the

increase of the pore pressures in the outer slope which subsequently causes a decrease in the

effective stresses ( ) which at the same time leads to a reduction of the shear strength

(considering Mohr-Coulomb failure criterion) When the shear stresses in the soil exceed the

shear strength a slip plane forms and a soil wedge collapses

Failure modes (b) and (c) are primarily determined from the active and passive effective

stresses respectively As far as the ldquoactive siderdquo failure is concerned displacements of the sheet

pile wall towards the downstream side of the dike lead to development of active stresses

where is the active pressure coefficient During the ldquoactiverdquo failure the

retaining soil is allowed to relax which leads to a decrease of horizontal stresses and

simultaneously to a decrease in shear strength Similarly to failure mode (a) a soil wedge

collapses The result is an increased earth pressure on the wall for which it is not designed

However this type of failure in dikes is not such determinant for the soil and the sheet pile wall

failure because there is always the resistance of the passive side of the dike that keeps the active

side stable enough Such a mechanism is thus more relevant for deep excavations with retaining

walls

In mechanism (c) the passive soil resistance is exceeded by the horizontal loads In this case the

wall moves inland and a wedge of soil compresses The shear strength is larger due to the

deformation of the wall and the horizontal stresses increase stresses where is

the passive pressure coefficient ( ) This failure mechanism usually occurs due to an

underestimation of the sheet pile length or due to the presence of a weak soil layer in the

passive side

Failure mode (d) is actually a combination of (b) and (c) possibly accompanied also with the

development of excess pore pressures as it was discussed in failure mechanism (a)

Moreover too short anchors could cause a soil block to collapse as it is showed in Figure 59(e)

(ldquoKranzrdquo stability)

As it has been already mentioned for mechanism (d) failure modes can be correlated and

combined with each other in order to lead to a final failure state Therefore the common failure

probability would be smaller than the sum of the singular probabilities In this thesis the total

probability of failure of the soil body due to global instability is to be determined This is due to

the fact that the application of FEM is capable of simulating the combination of the failure

modes depicted in Figure 58 and thus the overall failure (related to (C) (D) and (E) modes of

Figure 58) can be evaluated However with the classical engineering approach for the stresses

calculation each one of the failure modes depicted in Figure 59 should be separately evaluated



Below a description of the available methods to formulate the LSF of the soil failure is given

After the evaluation of these alternatives a selection was made regarding the most suitable LSF

for the current case study

Limit State functions for soil failure

Soil mechanical failure can be analysed in several ways (Schweckendiek 2006)

1 Excessive Deformations

2 φ-c Reduction

3 Relative Shear Resistance

4 Plaxis definition of soil collapse

The possibilities and limitations of the aforementioned methods are briefly discussed below


Similarly to the SLS a limit state function can be formulated by deformations that are

unacceptable As it is illustrated in Figure 54(b) there are some points (the top of the dike the

sheet pile top and the inland soil level behind the dike) the displacements of which shall be

limited to the minimum possible during the structurersquos lifetime in order for the structure to

meet the required service standards Therefore the calculated deformations at locations 1 2

and 3 could give a clear indication of failure given the maximum acceptable deformation

In case that there is a limit value for each location then the limit state function can be formed as

follows

[ ] (511)

with this criterion designer can control the developed deformation on the system and improve

the structure so as to limit the displacements to the required level and subsequently increase

the reliability

On the other hand such an approach might exhibit several problems during the reliability

analysis that are listed below

Many failure mechanisms suddenly happen and as a result no significant displacements

are observed before moving very close to failure In sequence this may cause problems

for the iterative procedures of some reliability methods such as FORM and DS

The determination of suitable maximum admissible deformations is not

straightforward They have to be large enough to serve as failure criterion and

simultaneously they shall not be larger than the values that can be calculated within

the limits of equilibrium in FEM-calculations This requires previous knowledge on the

analysed system as well as on the feasibility of FEM-calculations on the specific subject

Last but not least the location of the undesirable displacements in a dike section is also

under investigation as there can be multiple vulnerable spots on the dike that should

be kept under a certain range of deformations in order not to cause a collapse and that

they are not always known in advance



2 φ-c Reduction

An available type of calculation in Plaxis is the so-called ldquoSafety Calculationrdquo with which global

safety factors can be computed In Appendix B2 a more detailed elaboration of this method is

exhibited This feature can also be used in reliability analysis by formulating the LSF as follows

(512)

where is the Multiplier Safety Factor obtained by the φ-c reduction The main concept is

that if the safety factor is smaller than 1 then it is considered as failure This method can provide

us with the probability of failure for a general soil body collapse However even if this method

seems to be quite simple there are some issues during its implementation in a reliability

analysis

For complex limit state functions there can be convergence problems for some

reliability methods with iterative procedures (ie FORM) This can be explained from the

fact that in a φ-c reduction soil strength properties follows a certain path as it is

described in Appendix B2 independently of the values for the random variables that

have been set from the reliability method

The safety factor is a general safety factor regarding the failure of the system

Therefore sometimes it is not straightforward what the ldquotriggerrdquo factors are that led to a

certain type of failure

A safety calculation in Plaxis is time-consuming and the outcome can be unstable (Plaxis

calculationsrsquo convergence depends on the number of calculation steps)

Plaxis cannot handle safety factors below 1 Therefore a new limit of the safety factor

shall be introduced in this limit state function


In this method the basic idea is to define soil failure according to the failure criterion that Plaxis

is set to use for the analysis In this research the Mohr Coulomb failure criterion is applied in

combination with a linear ndashelastic perfectly-plastic stress-strain relationship (that is the so-

called Mohr-Coulomb model) For this model the initiation of plasticity is considered as failure

Therefore the maximum shear resistance is defined just before plastic yielding occur for any

given stress state Of course the occurrence of plasticity does not directly indicate the failure of

the soil structure however this model can give a first estimate for the stress state and the

deformations

According to this method the relative shear resistance is defined as the ratio between the

mobilized shear stress and the maximum shear resistance In Figure 510 the Mohr-Coulomb

model is illustrated in a Cartesian stress-plane In the graph the principle stresses σ1 and σ3 are

indicated More precisely the principle stresses are defined as follows

radic

(513)

(514)



radic

(515)

Figure 510 Mohrs stress circle and Mohr-Coulomb failure criterion

Hence the mobilized shear stress is

radic

(516)

And the maximum shear resistance which is the distance of the mean stress point to the yield

surface is defined as

(517)

Then the relative shear resistance is a measure for load-resistance ratio that can be determined

in any integration point in the soil continuum

(518)

However this criterion requires prior knowledge of the possible relevant failure mechanisms

and their potential locations on the soil structure This is necessary in this method in order to

choose a suitable cluster of integration points where the average value of the relative shear

strength is to be determined Therefore the limit state function to be considered is not

straightforward and it needs problem investigation in advance


In Plaxis the construction stages are analysed by performing a Load advancement ultimate level

procedure which is controlled by a total multiplier

(519)

where is the load that is applied by Plaxis is the load at the beginning of the

calculation phase (ie the load that has been reached at the end of the previous calculation



phase) and is the defined load configuration (ie water level soil weight vertical load

on the top of the dike etc) Thus the main idea is that Plaxis applies an incremental load until it

reaches the defined one When has reached an ultimate level (which by default is 1)

the current phase is finished However if a staged construction calculation has not properly

finished the multiplier fails to converge to the desired ultimate level at the end of the

phase and a warning of ldquoSoil collapsesrdquo appears in the Log info box

In other words a collapse load has been reached In this case the total specified load has not

been applied In physical terms this means that the current value of the stiffness parameter CSP

is less than 0015 (Plaxis 2D 2015b) It is a measure for the amount of plasticity that occurs

during the calculation When the solution is fully elastic CSP is equal to unity whereas at failure

it approaches zero Therefore soil reaches its upper limit of plasticity and it collapses which can

be visualized as a settlement of the dike body Such a warning in the analysis is thus assumed to

be a possible failure situation

53 Overview In this chapter an overview of the relevant possible LSF of the structural elements and the dike

body respectively has been given The different LSFs are formulated and discussed whereas the

failure of each component is going to be evaluated in the ULS of each component

The system failure in this research is considered as the combination of the anchor sheet pile

wall and soil body LSF Particularly for the anchor and the sheet pile failure their yield stress

was chosen as a limitation for their failure by adopting the next LSFs

[(

)] sheet pile wall LSF (520)

anchor LSF (521)

As far as the soil body is concerned in this research method 4 is to be used in order to identify

soil failure The limit state function is then formed as a single value in case of a successful and an

unsuccessful computation Therefore the limit state function was chosen to be equal to 1

(actually the value of is retrieved from Plaxis that in case of a successful calculation is

1) in case of a successful Plaxis analysis and -1 in case that soil collapses in any phase (error in

Plaxis Log file ldquoSoil Collapsesrdquo) This formula can be written as follows

(522)

This accounts for the global instability of the dike body in contrast with the other methods that

are related to a certain dike area (methods 1 and 3) or they are cannot handle overall failure

due to numerical issues (method 2) More precisely methods 1 and 3 were excluded due to the

prior knowledge that is required in method 3 while in method 1 a definition of the maximum

admissible deformations shall be first introduced However attention shall be paid to the type

of the warning that Plaxis gives because it might also be that the maximum number of load steps

was not sufficient In that case the phase must be recalculated using a larger number of steps



Someone could argue that such a failure can be considered as a system failure rather than a soil

failure alone Partially this is true as the incapability of the sheet pile wall or the anchor to keep

the upstream part of the dike stable can lead to non-convergence and thus according to Plaxis

definition to failure This incapability can be translated into various scenarios of failures that

stem from either the structural elements or the soil body weakness and that are illustrated in

Figure 511 This picture shows the different scenarios that can take place due to macro

instability that was discussed in 523 and the failure of the retaining wallrsquos elements

Figure 511 Combined failure scenarios (initial and developed mechanisms) (source Larsen et al 2013)

However this does not mean that the sheet pile wall or the anchor fail because of exceeding

their yield stress but rather due to possible unacceptable deformations This is also an

advantage of the inherent residual strength of the structural elements that allows a large

deformation before they fail However the residual strength of the elements was not taken into

account in this thesis as they were considered as elastic Moreover the message for the ldquosoil

collapsesrdquo can appear in a phase prior to the installation of the wall which means that this type

of failure is exclusively attributed to the deficiency of the soil strength Therefore it is not

always straightforward what is failing first and lead to the system failure Consequently by



considering such an analysis representative of a system failure it is likely to underestimate the

overall probability of failure whereas if it is considered as a soil failure only it might

overestimate the total probability of failure since it is later combined with the rest LSF of the

anchor and the sheet pile wall In this research it will be considered only for the soil failure that

brings the results to the safe side even if it does not lead to the best optimization In this thesis a

serial system of the anchor the sheet pile and the soil body LSF was considered the fault tree of

which is depicted in Figure 53

Another way to evaluate the reliability of the different elements could be the reach of a

threshold displacement according to the SLS criteria Such a threshold could be also considered

for the system as a whole rather than the individual elements However in that case someone

should be cautious regarding the choice of the value for the maximum deformations as well as

the location of their occurrence

Concluding in this section the possible failure mechanisms of a flood defence (Figure 58) were

shown Then it was clarified which of these mechanisms can be modelled in Plaxis and they

were further explained and schematized in Figure 58 with the presence of the retaining wall as

they would have to be handled in case of hand calculations Plaxis can incorporate these

mechanisms automatically and thus no distinguish among the different patterns of these

specific modes is necessary Last but not least in Figure 511 the possible failure modes for the

system of the dike and the retaining wall that can be simulated in Plaxis are depicted In this

thesis the plasticity of the structural elements is not considered and thus possible plastic hinges

in the anchor and the sheet pile are not taken into account Therefore scenario 3 is excluded

In Chapter 7 the results after the reliability analysis of the aforementioned LSFs are given It

was essential though before proceeding with the reliability analysis to get a better

understanding into the system under investigation For this purpose in the next chapter a

description of the case study as it has been simulated in Plaxis is given while also the different

structural properties are defined for the following reliability analysis The soil variables to be

considered as stochastic are presented and a first insight into of the systemrsquos behaviour is

obtained through the mean values calculation


6 Case Study-Dike with an anchored sheet pile wall

In this chapter a description of the case study is presented together with the relevant soil and

structural parameters In sequence a deterministic analysis follows in order to specify the basic

structural characteristics and obtain a first sense of the stresses magnitude developed on the

structure This was carried out based on the recommendations given in the CUR 166 (2005)

The aim of this deterministic analysis is to redefine the structural properties in order to avoid

having an overdesigned structure understand the current design procedure and see if there are

any possibilities of improvement by applying the proposed reliability analysis

61 Case Description The case study concerns a dike section with an anchored sheet pile wall which was initially used

to showcase the current methodology of designing dikes with sheet piles using partial factors

For more information a reference is made to Breedeveld (2011)

In this research this case study has been modified and simplified for the needs of the thesis and

it is illustrated in Figure 61 while in Table 61 the soil materials are identified Details for the

properties of the soil layers are given in section 62 We assume the structure to be

homogeneous to the third dimension and therefore a plane-strain model is applied The dike

soil materials (layers 3 and 4 in Figure 61) were adopted from the case study used in

Breedeveld (2011) that was presented in Chapter 2 Especially in Figure 25 the original case

study as it was modeled in Plaxis is shown

Table 61 Soil materials that corresponds to Figure 61

1 Aquifer Sand layer 2 Aquitard Clay layer 3 Dike new material Clay layer 4 Dike old material Clay layer

The geometrical properties the groundwater level and the design water level were taken

identical to the original case study as well as the soil properties of the dike materials The soil

layers under the dike were modified for the need of the thesis and were modeled using random

average quantities for which the statistics were chosen arbitrarily but in a realistic range

according to the NEN6740- Table 1 (see Appendix C) included in the Dutch code for

geotechnical structures Moreover the vertical load was removed in order to examine the

impact of the water elevation on the dike stability

Case Study ndash Dike with an Anchored Sheet Pile Wall


Figure 61 Case study Dike with anchored sheet pile wall Dimensions and Soil materials

62 Soil Parameters The soil parameters for the dike material were obtained from the report of Breedeveld (2011)

where for some parameters the characteristic values were available whereas for some others

the mean values were given For the soil layers below the dike Table 1 of NEN6740 was used as

it was mentioned before which gives characteristic values For those parameters whose value

was a characteristic 18 one a transformation was made according to their probability

distributions in order to acquire the mean values that are required for the reliability analysis

Only the properties required for the use of the Mohr-Coulomb model are presented

The parameters of each soil layer are presented in Tables 63(a) and (b) and the front number of

each layer indicates the corresponding soil layer as showed in Figure 61

The distribution types and the coefficients of variation for each parameter were chosen

according to the knowledge that has been obtained until now about the physically possible

ranges of such parameters and the recommendations that have been given in several researches

up to now (Baecheramp Christian 2003 Schweckendiek 2006 GriffithsampFenton 2007 Phoon

2008 Wolters 2012 Bach 2014) The distributions and the CoV of all parameters are displayed

in Table 62 They were considered to be the same for all soil layers Particularly for the

saturated and unsaturated volumetric weight γsat and γunsat respectively a relationship was

established in order to derive the one from the other In general γunsat varies between the real

dry weight and γsat depending each time on the degree of saturation An estimated maximum

difference between γunsat and γsat is approximately 4 kNm3 (Brinkgreve 2015) In this thesis γsat

was expressed as the summation of γunsat and a variable with uniform distribution in the range

of [0 2]19 Initially a deterministic relationship was defined between the two variables but later

on it was proved the variable was important for the soil failure and thus it was decided to

consider it as random

18 for the definition of the characteristic value see Appendix E 19 The range of [0 4] was not chosen in order to avoid extreme conditions



Table 62 Soil parameter distributions

Soil parameter Symbol DistributionRelation COV Unit

Unsaturated

Volumetric weight γunsat Normal (microσ) 5 [kNm3]

Saturated Volumetric

weight γsat γsat= γunsat+U(02) 5 [kNm3]

Cohesion c Lognormal (microσ0) 20 [kPa]

Friction angle φ Truncated normal (microσ045) 10 [ ˚]

Youngrsquos Modulus E Lognormal (microσ0) 25 [kPa]

Poissonrsquos ratio ν Truncated normal (microσ005) 10 [-]

Interface strength Rinter Truncated normal (microσ0099) 20 [-]

Table 63 (a) Soil statistical properties under the dike and (b) soil statistical properties in the dike

(a) 1 SAND (very silty)

Soil property Symbol Characteristic

value

Mean

value STD Unit

Saturated

Volumetric

weight

γsat 20 22 11 [kNm3]

Unsaturated

Volumetric

weight

γunsat 19 21 103 [kNm3]

Cohesion c 0 0 0 [kPa]

Friction angle φ 30 36 36 [ ˚]

Dilatancy angle ψ 0 0 0 [ ˚]

Youngrsquos Modulus E 20000 30769 7692 [kPa]

Poissonrsquos ratio ν 025 03 003 [-]

Interface strength Rinter 044 066 013 [-]



2 CLAY (clean medium)


value

Mean

value STD Unit

Saturated

Volumetric

weight

γsat 17 185 093 [kNm3]

Unsaturated

Volumetric

weight

γunsat 17 185 093 [kNm3]







(b) 3 DIKE NEW (very sandy clay)


value

Mean

value STD Unit

Saturated

Volumetric

weight

γsat 17 185 093 [kNm3]

Unsaturated

Volumetric

weight

γunsat 17 185 093 [kNm3]









4 DIKE OLD (little sandy medium clay)


value

Mean

value STD Unit

Saturated

Volumetric

weight

γsat 195 212 106 [kNm3]

Unsaturated

Volumetric

weight

γunsat 19 207 103 [kNm3]







63 Finite Element Model The structure has been modelled with the finite element software Plaxis 2D 2015 In Figure 62

the model as it was created and the automatically generated mesh are presented The ldquoboxrdquo

around the dike section was used to refine the mesh in this area The mesh in the interfaces

between the sheet pile and the soil was also refined for a better representation of the potential

high stresses that can develop in that area

Figure 62 FEM model and generated mesh for case study



The sheet pile has been modelled with elastic20 plate elements and the free anchor length with

an elastic fixed-end-anchor element The grout body of the anchor was not explicitly modelled

but it is implied in the end of the anchor as its final point is considered to be fixed in the soil A

fixed-end-anchor was chosen because even if it is a simplification of an anchorage system it can

still give reliable results in terms of the anchor axial force which is actually included in the limit

state function However the interaction of the soil and the grout body in that case cannot be

simulated The interaction and the modelling of a grouted body in the soil are still under

investigation due to several difficulties in modelling the real conditions

For all the steel members a Youngrsquos modulus of E=210 GPa was used while the specific

structural parameters are to be specified after the deterministic analysis that is described in the

next section

64 Deterministic Analysis The design concept of dikes with sheet piles modelled in FEM is based on the combination of

several partial factors in order to evaluate the overall required safety factor Additionally the

design recommendations of CUR 166 (2005) are considered for using the design or the

characteristic values of the structural parameters and the loads The design procedure (see

section 22) that has been initially followed in this structure is reported in Breedeveld (2011)

and the following structural elements have been defined accordingly

Cross sectional area of the sheet pile (per meter)

Length of the sheet pile wall

Steel quality of the piles

Cross sectional area of the anchor (per meter)

Free length of the anchor (without the grounded body)

Steel quality of the anchorrsquos reinforcement

Profile of waling

The required overall safety factor was calculated up to 18 according to the current design

procedure described in section 22 according which the above mentioned structural parameters

were defined Moreover apart from the overall safety factor in Breedeveld (2011) additional

partial factors are applied on the deduced forces and bending moments of the anchor and the

sheet pile wall as they are defined in CUR 166 (2005) In particular in Table 64 the different

partial factors that are applied on the several structural developed actions are given

Table 64 Partial factors for the different structural actions according to CUR 166 (2005)

Structural action Partial factor γ

Sheet pile normal force Fn 115

Sheet pile bending moment M 115

Anchor normal force FA 125

Waling bending moment Mw 110

20 The elements were considered as elastic in order to avoid numerical problems in FORM analysis For a better explanation a reference is made to the evaluation of the optimization algorithms in Appendix A2



However as it is mentioned in section 62 alterations have been made in the soil layers below

the dike while also the vertical load has been removed On the top of that in section 22 it was

shown that the large applied partial factors led to an overestimation of the design values for the

moments of the sheet pile Therefore a redesign of the sheet pile and the anchor is carried out

in this section in order to determine the new required characteristics of the structural elements

according to the new loading situation with less strict partial factors In particular in this thesis

the required overall safety factor was considered 12 according to CUR 166 (2005) calculation

scheme that is described in the next section The reason for using a lower safety factor was to

avoid an overestimation of the design forces and moments of the sheet pile wall that would

render a reliability analysis meaningless as the structure would be quite safe Such an analysis is

also important in order to show the discrepancy between the two different design procedures

and indicate the optimal one

However the reliability analysis was carried out for the mean values of the soil parameters and

for a safety factor of 10 in order to account only for soil properties uncertainties A safety factor

of 10 was not directly used as the design procedure followed in section 643 accounted only for

the macro-instability of the system and thus other failure mechanisms such as piping

overtopping infiltration etc were not taken into account In that way we prevent coming up

with a retaining wall that would be volatile under other crucial failure mechanisms

In section 641 the calculation scheme to be used in this thesis for the deterministic analysis is

further described In the sequence in section 642 the construction phases that were followed

in Plaxis simulation are listed and illustrated and finally in section 643 the results of the

preliminary design are displayed

641 Calculation Scheme and Design Values

In this research only the design philosophy suggested in CUR 166 is applied and the desired

safety factor of the structure is considered to be 12 for structures classified in ldquosafety class IIIrdquo

in order to avoid an overestimation of the structurersquos dimensions as it happened in the original

case study At this point it should be also clarified that the redesign was conducted considering

only the overall stability of the structure and no other failure mechanisms such as piping

internal erosion or overtopping

The prescribed calculation schemes in CUR 166 (2005) for designing structures with retaining

walls are basically the following

Calculation scheme A Calculations with design values The calculations are executed

using the design values for the soil parameters retaining height water levels and

stiffness of the structure It uses two different soil stiffnesses When using a high soil

stiffness the anchor force is generally relatively high The advantage of this scheme is

that it requires relative little effort The disadvantage is that the deformations may be

overestimated due to the use of design values in every stage However a designer is not

interested in deformations when performing a ULS calculation

Calculation scheme B Calculations with characteristic values The calculations are

executed using characteristic values of the soil strength (crsquo and φrsquo) Design values are



used for the retaining height water levels external loads and stiffness of the soil Here

again two different soil stiffnesses are used In the end of the governing stage a phi-c

reduction should be carried out As the characteristic values for the soil parameters are

used the deformations are most probably smaller than in case of calculation scheme A

The safety margin lies in the fact that phi-c reduction is imposed to obtain a safety factor

of 115 (class II) or 12 (class III)

For the soil stiffness parameters two options are given in CUR 166 a low and a high value

(61)

where is the (low and the high) design value the (low and the high) characteristic value

and is the mean value of the stiffness parameter In Woltersrsquos master thesis (2012) both

have been applied and the appeared to be governing for the bending moment in the wall

and the anchor force

In this research the Calculation scheme B was used and the as the soil stiffness

parameter It should be mentioned that regarding the soil parameters the characteristic values

were available with an exemption for the stiffness for which the mean values were given The

groundwater level was manually schematized according to TAW (2004) because of lack of data

while the maximum water level in the river side was taken equal to the design water level

according the Dutch regulations for the specific dike region

In Tables 65 and 66 the partial factors according to CUR 166 and the corresponding design

values for the different soil properties are presented As someone can notice in these tables

only the strength parameters (phi and c) and the stiffness parameter (E) are exhibited For the

rest parameters the design value is identical to the characteristic value (see Tables 63(a) and

(b)) as a partial factor of 1 is used

Table 65 Partial factors and design values for the soil layers under the dike

1 SAND (very silty)

Soil property Symbol Partial factor γk Design value Unit

Cohesion c 11 0 [kPa]

Friction angle φ 12 2569 [ ˚]

Youngrsquos Modulus E 13 1538462 [kPa]

2 CLAY (medium)







Table 66 Partial factors and design values for the soil materials in the dike

3 DIKE NEW (medium clay)





4 DIKE OLD (stiff clay)





642 Construction Stages

The construction and the gradual loading of the dike ware modelled as follows (see Figure 63)

1 K0-procedure for the generation of the initial stresses under horizontal groundwater

level

2 Dike self-weight under horizontal groundwater level

3 Rise groundwater level to the phreatic water elevation

4 Interpolation21 of the clay layer under the dike and activation of the sheet pile wall and

the anchor

5 Apply extreme water level conditions on the dike structure

6 (φ-c reduction for the determination of the safety factor)

21 Due to different permeability between the clay and the sand layer an approximation of the developed pore pressures is carried out through interpolation between the two layers



1

2

3

4

5

Figure 63 Construction and loading stages

For the soil modelling the Mohr-Coulomb model was used with a non-associative flow rule

(φneψ) For phases 1-4 the calculations were conducted under drained conditions whereas in

phases 5 and 6 the undrained behaviour of clay layers was considered

643 Determination of the structural elementsrsquo characteristics

According to the calculation scheme that was described above the design characteristics of the

structural elements were obtained from phase 6 In particular the target overall safety factor is

12 with which initially the length of the sheet pile wall was calculated

+370 NAP

+700 NAP

+1050 NAP



Firstly a length of 10 m and a profile of AZ 12 with a steel quality of 240 kNm2 yield stress

were used while for the anchor a cross-section of 628 mm2 and S355 were chosen for the first

calculation However even if the safety factor reached 12 it seemed to be quite unstable during

the calculation steps while the developed anchor stress was above the ultimate yield stress

What is more the deformations resulted in that case were more than 30 cm Therefore in order

to stabilize the safety factor and to decrease the displacements the length of the sheet pile was

increased 1 m (so 11 m in total) while also a better steel quality for the sheet pile was chosen

S355 Furthermore the anchor cross section was increased to 933 mm2 with a steel quality of

MW450 Last but not least the waling was designed based on the anchor force

In the following paragraphs the design procedure of each element is described providing also

the results of moments and forces that were retrieved from Plaxis

Sheet Pile wall

As it was partially mentioned in the previous paragraph the length of the sheet piles is

considered to be sufficient when the safety factor of the system22 reaches at least 12 For the

reasons discussed above the sheet pile length was decided to be 11 m and with a profile of AZ

12 S355

With this configuration the safety factor reached is 13 as it is depicted at the right graph of

Figure 64 Someone can notice that the safety factor from 117 in the situation without a sheet

pile structure increased to 13 with the structure

Figure 64 Safety factor deduced from the phi-c reduction calculation without the anchored sheet pile wall at the left (safety factor=117) and with the wall at the right (safety factor=13) The steps on the x-axis represent the calculation steps have been carried out in order to reach an equilibrium after incrementally reducing the strength parameters ie phi and c

The design moment is to be taken for a safety factor of 12 that is reached in step 34 where it

was found to be 1519 kNmm and the corresponding axial force 9133 kNm Therefore

considering Eq 52 for checking the exceedance of the ultimate yield stress and applying the

prescribed partial factor of 115 for both the moment and the axial force we get

22 Plaxis always gives the safety factor of the system which is under investigation



In Figure 65 the maximum bending moments with a safety factor 10 and 12 are respectively

showed From the values being shown in that figure someone can easily distinguish the

difference between the two moments It must be also mentioned in that point that the required

safety factor defined in Chapter 2 was estimated at 18 (and not 12 that is considered here)

which gives excessive design moments (almost 7 times larger) For that reason in this research

a lower safety factor was assumed avoiding thus the design of a quite conservative structure

Figure 65 Bending moments on the sheet pile for a safety factor safety factor=10 at the left and

safety factor=12 at the right

The explanation behind this discrepancy is that in case of the phi-c reduction procedure the

artificially decreasing strength of the soil and on the other hand the non-reduced strength and

stiffness23 parameters of the structural elements tends to lead in an increased arching

phenomenon that makes the structure attract the most of the load

In this research the reliability of the structure will be evaluated for a safety factor of 1 or else

during phase 5 and by considering the mean values of the soil parameters As someone can

notice from Figure 65 the magnitude of the maximum bending moment for a safety factor =10

is quite low which implies a possible minor contribution to the overall failure

At this point it is essential to mention that Mohr-Coulomb model is capable of simulating the

elastic behaviour and the state at failure Therefore in case that plasticity of the soil or the

structural elements plays an important role a different constitutive model is recommended to

be applied

Anchor

For a safety factor 12 the anchor force was found to be 10387 kNm Considering a mutual

anchor distance of 3 m the total axial force that is undertaken by the rod multiplied also with

the prescribed design partial factor is

23 ldquoStrengthrdquo and ldquostiffnessrdquo are usually confused terms Strength is a measure of the maximum load that can be placed on a material before it permanently deforms or breaks and it is represented by the yield stress σy in metals and by φ and c in soil Stiffness is a measure of the amount of deflection that a load causes in a material and it is represented by the Youngrsquos modulus E and the Poisson ratio ν both in soil and metals



Therefore choosing an anchor with a cross sectional area of A = 933 mm2 and a steel quality of

MW450 ( ) the design stress level is

In comparison with the sheet piles the anchor does develop large stresses as it was concluded

from the results in the deterministic calculations This is probably attributed to the high

stiffness of the anchor that apparently attracts the load and thus the anchor develops large axial

force

Waling

The type of the waling system is defined according to the anchor force as it was described in

section 522 Therefore by choosing a waling type of 2UPE 200 the developed bending

moment can be estimated as follows

where is the design bending moment of the walling system according to the chosen profile

In Table 67 the characteristics of each structural element that was described above are

displayed in detail

65 Overview In this chapter an introduction of the case study as it has been used in this research is made The

soil materials and their properties are summarized and their mean and design values are

defined The probability distributions of the soil parameters to be considered as stochastic were

proposed according to the literature study and the engineering judgement Moreover the mesh

generation and the modelling features are discussed

Last but not least due the observed overestimation of the retaining wallrsquos design moment and

axial force according to the current design regulation a new deterministic analysis was carried

out in order to redefine the structural properties and come up with a less conservative structure

This makes the reliability analysis of the system meaningful in a sense that an extremely safe

structure would give a failure probability of zero Besides the scope of a reliability analysis is

also the optimization of the construction that corresponds to a financially attractive and

simultaneously safe structure This can be made by conducting a risk analysis whose first step

would be the analysis that is carried out in this research Therefore the aim is to start with the

marginal required structural properties according to the safety standards In a risk analysis that

should conducted for several dimensions of the structure until the most optimized one is

identified However this is out of the scope of this thesis



In Table 67 the structural properties of the sheet pile wall the anchor and the waling are

summarized as they were found in the deterministic analysis

Table 67 Design parameters of structural elements

SHEET PILES

Property Symbol Value Unit

Profile - AZ 12 [-]

Steel quality - S355 [-]

Length L 11 [m]

Moment of inertia I 21430 [cm4m]

Elastic section modulus Wel 1245 [cm3m]

Mass w 943 [kgm]

Sheet pile thickness d 85 [mm]

Cross sectional area A 1201 [cm2m]

ANCHOR


Steel quality - MW450 [-]

Free length Lafree 104 [m]

angle φ 30 [deg]

Cross sectional area A 933 [mm2]

Mutual anchor distance s 3 [m]

WALING


Profile - 2UPE200 [-]



In the original case study the sheet pile wall is 162 m long and it has a profile of AZ 28 (S240)

while the anchor is an MW450 with a cross-section of 1134 mm2 As it is noticed both sheet

pilesrsquo and anchorrsquos properties have been reduced after the new deterministic analysis However



it should be mentioned that in the original case there was a vertical load on the top of the dike

that was displaced while the clay layers under the dike was merged into one in the new case

study These changes might lead to less heavy structure However the difference among the

design properties between the new and the original case study is mainly due to the required

safety factor that was assumed In the new case study the safety factor was set to 12 whereas in

the original cases study a safety factor of 18 was considered as it was calculated according to

the partial safety factors

In the next chapter the results after the reliability analysis of the just designed case study are

presented




7 Reliability analysis results with stochastic soil properties

In this chapter the probability of failure of the different structural elements that were discussed

in Chapter 4 will be estimated by varying the soil properties of the several soil layers under and

inside the dike At the beginning a better insight into the system behavior is gained by

inspecting the deterministic Plaxis calculations for the mean values of the soil variables After

that a global SA is carried out in order to evaluate the most important soil parameters and

inspect their impact on the different structural elements in combination with each other and

individually Finally the results regarding the probability of failure of the anchor the sheet pile

wall and the soil are presented as they deduced by the reliability analysis (concerning the ULS of

the elements) Finally the systemrsquos reliability is evaluated The methodology that was followed

during the probabilistic analysis is described below and it is mainly dedicated to the DS

rationale that was used in order to obtain the influence factors and to evaluate the results of

Plaxis calculations

71 Method description For the calculation of the failure probability of the systemrsquos different components FORM and DS

were used for the retaining wall and the soil body respectively From these methods the

probability of failure and the important factors are going to be deduced for each component (ie

anchor sheet pile wall and soil body) However before starting with the reliability analysis it is

essential to carry out a sensitivity analysis in order to obtain a first impression of what are the

most influencing variables and thus determine the input random parameters for the reliability

analysis In the figure below the steps for the reliability analysis as they were followed in this

thesis are presented

Figure 71 Steps of the reliability analysis

Below the methodology of both the sensitivity and the reliability analysis are discussed and

explained in detail


In this thesis the FAST method was utilized for conducting the sensitivity analysis which is

further elaborated in Appendix A1 Here an explanation is given for the quantitative measure of

sensitivity which is represented by Sensitivity Indices The first-order sensitivity index of an

Reliability analysis results with stochastic soil properties


input variable pi is the measure of the main (direct) effect of pi on the output variance

(where i ne j) the second-order sensitivity indices measures the interaction effect of pi and pj on

the output variance Other higher-order indices are defined in the same manner The total

sensitivity index is the sum of all sensitivity indices involving factor pi (Homma amp Saltelli

1996) For example the total sensitivity index of factor 1 for a model with 3 input factors is

given as

The first-order (Si) and total ( ) indices can be interpreted as following (Pandya et al 2008)

high pi is an influent parameter

and both small pi is not an influent parameter (neither alone nor in interaction

with other parameters)

and nearly the same no interaction of pi with the other parameters

and very different high interaction of pi with the other parameters

Reliability analysis

As far as the FORM analysis is concerned section 422 gives sufficient information about the

methodology that is followed in order to obtain the failure probability and the influence factors

(ie α-values) The main concept and principles of DS are briefly explained in section 421

whereas here a more precise description of the implementation of DS and the how the α2-

values can be deduced is given

In particular DS is a method in which ldquodirectionsrdquo are sampled The concept of this method is

schematically depicted in Figure 72(a) For each sampled direction it is evaluated whereas

along this line the LSF Z equals to zero or not This procedure is repeated for a number of

directions and this method is applied in the standard normal space (u-space see also section

422) using equations to transform the standard normal variables u1 hellip un to their

corresponding ldquorealrdquo variables (in the x-space) X1 hellipXn and vice versa

(a) (b)

Figure 72 (a) Schematic view of the DS method in the u-space The numbers 0 1 2 3 4 show successive LSF evaluations along one direction (b) Schematic view of the line search algorithm β is the distance from the origin in the u-space along one direction The numbers 0 1 2 3 4 correspond to the ones shown in figure (a) (source Hydra-Ring 12 2015)



The search procedure for the location on the line where Z=0 is often referred to as the ldquoline

search methodrdquo and it is depicted in Figure 72 (a) and (b) In both figures the numbers 0-4 refer

to successive Z-function evaluations along one direction Figure 72(a) shows 3 evaluated

directions in the u-space whereas Figure 72(b) shows the (fictional) value of the Z-function

along one direction (where ||u|| is the distance from the origin along the line) The maximum

length of the direction line as well as the step size along the direction (for example the distance

between 0 and 1 along one line) can be steered in order to increase the efficiency of the method

and presumably decrease the computational time

In the sequence a method was developed in this thesis in order to estimate the α2-values of the

random variables out of a DS analysis as OT does not have an available method so far For that

purpose a transformation of the output samples to u-space was firstly carried out Then the

distance to the origin of all samples (in u-space) that are located on a direction where failure

(Z=0) was detected is calculated as follows

radicsum

i=1hellip (71)

where n is the number of random variables and uij is the ith ldquofailurerdquo sample of the jth random

variable is equivalent with the reliability index of this direction βi The α-values for each

random variable can then be calculated as (for more information about these values see

sections 422- Eq 417 and 423)

(72)

After that three different methods were created in order to evaluate the influence of the

random variables on the response of the LSF ie the α2-values (square of Eq 72)

Shortest distance ldquoβminrdquo

Average 10

Average all

In the Shortest distance method the sample with the smallest distance to the origin is

considered to be the design point and thus the corresponding influence factors are estimated

according to that point The smallest distance is also equivalent to the reliability index βmin This

method is well-known for estimating the design point out of MC techniques (Vrijling amp Van

Gelder 2002) However for a soil failure modeled in FEM this might not be always indicative of

the real influence that the soil properties can have on the system failure Therefore the last two

techniques were used as additional in order to get a better insight into the dominant soil

properties In Figure 73 a general scheme of the methodology that was followed for obtaining

the α2-values is depicted where also the different techniques are presented



Figure 73 DS methodology for calculating α2-values

The second method Average 10 takes into account the α2-values of the samples in the failure

domain whose distance to the origin lies within the 10 higher than βmin and averages them

Therefore it is firstly required the calculation of βmin as it was described for the Shortest distance

method Such a technique helps to identify the other directions that are close to the one which

gives the shortest distance to the origin and thus inspect if the important variables that were

indicated according to the first method are influential for other failure mechanisms as well

Last but not least the third method Average all averages the α2-values of all the samples

located in the failure domain This shows the overall contribution of the random variables to the

response of the LSF under investigation Concerning the soil failure LSF that is described in

section 73 such a method can be more suitable for distinguishing the soil properties that are

involved in the majority of the failure mechanisms rather than only in the one that the first

method indicates

72 Mean values calculations The results of the deterministic analysis that are presented in the previous chapter included a

degree of conservatism due to the load and material factors (partial safety factors) In order to

overcome this limitation structural reliability calculations were carried out and are discussed in

the next sections by considering relevant soil parameters as random However before starting

the probabilistic calculations by coupling Plaxis with OT that cannot be fully interpreted and

verified to a certain extent it is essential to first carry out some deterministic calculation Such

calculations can help to get a better insight into the behavior of the system and show

qualitatively what kind of response to wait later in the analyses These calculations were taken

place based on the mean values of the soil properties

Deformations

The analysis of this case study focuses on the ULS of the structure For the stresses and

deformation calculations the Mohr-Coulomb model is used that is generally accepted for the



simulation of the soil behavior until failure and for detecting the failure modes (a more detailed

explanation of this model is provided in Appendix B1) However this model is not so suitable

for analyzing the deformations of soft soils or the settlements around the wall

The deformation pattern that would more likely occur due to primarily the soil body failure can

be illustrated by conducting a ϕ-c reduction calculation In Figure 74 (a)-(c) the displacement

pattern after the ϕ-c reduction of the mean values of the friction angle and the cohesion

respectively As it can be seen a slip surface in the inner side of the dike has been created that

rotates towards the inland

(a) deformed mesh

(b) shadings

(c) arrows

Figure 74 Deformations after the ϕ-c reduction (mean values)

Stresses on the structural elements and the soil body

The anchor and the sheet pile wall are installed in phase 4 In phase 5 the anchor normal force

increases and especially for the mean values it amounts to Na = 7983kNm

As far as the sheet pile wall is concerned the bending moments and the normal forces that are

developed on the wall are indicated in Figure 75 as they were deduced from phase 5 Similarly

to the anchor the maximum stresses occur in phase 5 and are mainly located in the area of the

wall which is surrounded by the clay layer (brownish soil layer in Figure 74(a)) As someone

can observe the bending moments are not significant taking into account that the maximum one

observed reaches 2132 kNm



Figure 75 Bending moments and normal forces for the mean values calculation

For the interpretation of the stress field of the soil body the effective principle stresses and the

relative shear stress τrel (see Eq 517) were illustrated in Figure 76 (a) and (b) respectively In

Figure 76 (a) the crosses indicate the direction and the magnitude of the principle stresses σ1rsquo

and σ3rsquo and from which an active soil behavior (

) can be noticed on the river side of

the retaining wall whereas a passive behavior (

) is dominant in the inland side

(a) Effective principle stresses

(b) Relative

shear stress τrel

Figure 76 Effective principle stresses and relative shear stress in mean values calculation

In Figure 76 (b) the distribution of the relative shear strength shows the potential developing

plastic area in case of failure that expands from the river side of the wall and around it It can

also betoken the possible shape of formation of a slip surface that starts from the outer side of

the dike and goes around the wall



73 Sensitivity Analysis Results In section 312 different types of SA are presented and that can be applied through OT In this

research due to the time limit and the amount of parameters FAST method was used for

carrying out the SA as it allows the calculation of the first order sensitivity indices as well as the

total order indices and the computational time required is less than the other available methods

In this section the results from the conducted SA are presented separately for the anchor the

sheet pile and the soil performance Regarding the sheet pile wall and the anchor the sensitivity

of the total developed stresses towards the soil properties was evaluated according to Eq 53

and 55 respectively The influence of the soil parameters on the soil behaviour was evaluated

based on the calculation of the overall safety factor after a safety calculation (ϕ-c reduction

method) The total order indices are considered for distinguishing the most important

parameters while their difference with the first order indices Si indicates the level of interaction

effect amongst the parameters on the output variance

As far as the anchor and the sheet pile wall are concerned the SA was carried out in two steps

(1) SA with all the soil parameters (listed in Table 62) and (2) SA with the most important

parameters found in step 1 In Figure 77(a) and (b) the sensitivity indices from step (1) are

presented as they were found for the anchor and the sheet pile respectively Step 1 mainly

provides an overview of the contribution of the variablesrsquo variance to the total variance of the

limit state response and it helps to identify the most influencing parameters Step 2 was

basically carried out in order to build the response surface based on the SA output that was

later utilized in the FORM analysis as a complementary function in case of Plaxis errors At that

point it should be mentioned that the Dikenew material was not considered neither for the SA

nor for the reliability analysis of the structural elements as its influence was assumed to be

insignificant relative to the other soil layers However Dikenew was considered in case of the

soil body investigation

From Figure 77(a) it can be concluded that as far as the anchor is concerned the shear stiffness

G of the clay layer seems to be the most crucial soil property In particular the anchorrsquos stress

level seems also to be sensitive towards the Poisson ratio of the clay layer ν the friction angle φ

of the sand layer and the strength parameters φ c of the Dikeold material Moreover the

interface strength Rinter between the Dikeold material and the sheet pile wall seems to display

an additional notable influence as well The unit soil weight γ does not appear to affect the

anchor performance significantly

As far as the sheet pile wall is concerned the shear stiffness of the clay layer and the Dikeold

material seem to be dominant according to Figure 77(b) From these two soil layers it appears

that also the strength properties (φ c and Rinter) as well as the unit soil weight γ have a relative

impact on the stress level of the wall Furthermore the sand layer contributes via especially its

friction angle and shear stiffness



(a)

(b)

Figure 77 Total and First order sensitivity indices for the (a) anchor and the (b) sheet pile in step 1

As it was mentioned before step 2 was used for building the response surfaces for the anchor

and the sheet pile limit states The input random variables in this step are the most influencing

ones that were found in step 1 In Table 71 the soil properties that were considered as random

during step 2 are presented for both anchor and sheet pile It is essential to mention that the RS

was constructed by fitting a quadratic polynomial function to the sampling points of the SA

whose general expression is given in Eq 425 In Figure 78(a) and (b) an illustration of these

response surfaces is made However the figures depict the response of the anchor and the sheet

pile stress level over the fluctuation of only two variables amongst the 14 that are presented in

Table 71



Table 71 Soil properties considered for the RS construction of the anchor and the sheet pile respectively

Soil layer Anchor Sheet pile

Clay Sand

Dikeold

The linear regression of all the parameters was based on least squares and the R2 parameter

was found 0989 for the sheet pile and 0996 for the anchor Therefore the regression was quite

representative for the domain of the samples over which it was adjusted However there were

many residuals (ie difference between the observed and the predicted value) in the order of

103 and 104 which means that a possible expansion of this RS to an extended domain of the

input variables may lead to an underestimation of the limit state response

(a)

(b)

Figure 78 Polynomial response surface adjusted for the (a) anchor and (b) sheet pile stress level For the anchor the friction angle of the sand layer and the shear stiffness of the clay layer were accounted for this illustration whereas for the sheet pile wall the shear stiffness of the sand and the cohesion of the clay layer were used

The SA for the soil performance was also conducted in two steps However in that case there

was no need of a response surface construction since the LSF for the reliability analysis (Eq

522) takes the value of -1 in case of Plaxis error due to soil collapse Eq 522 though takes only



two different values -1 and 1 for soil collapse or not respectively which cannot be used for

carrying a SA For that reason the φ-c reduction method was used and the sensitivity of the

value of the safety factor (Msf) was investigated instead The second step of the SA was carried

out in order to divide the soil properties into two different groups of variables and thus

investigate the sensitivity of the soil in a more efficient way rather than including all the

parameters at one SA It should be mentioned that in the soil analysis soil properties of the

Dikenew material were added In Figure 79(a) and (b) the total and the first order indices are

depicted for step 1 and 2 respectively Step 1 includes all the stiffness and strength parameters

of all the soil layers whereas in step 2 the most important variables that were found in step 1

together with the unit soil weight γ are included Subsequently step 1 detects the most

influencing stiffness and strength properties and in step 2 the impact of γ is investigated

As it is observed in Figure 79(a) and (b) the safety factor appeared to be sensitive towards the

sandrsquos friction angle while also the strength properties (c and φ) and the shear stiffness (G) of

the clay layer play an important role Moreover the cohesion of the Dikeold material and the

unit soil weight of the clay sand and Dikenew layer seem to be influencing as well As it has

been mentioned above for this SA the LSF was different than the one considered in the

reliability analysis Therefore even if the SA at that point helps to evaluate in general the most

dominant soil properties the engineering judgement should also be included for the selection of

random variables that are going to be utilized in the reliability analysis

(a)



(b)

Figure 79 Sensitivity indices for the soil in step 1 and 2

It is also advisable that the difference between the total and the first order sensitivity indices be

investigated in order to have an insight into the most interactive variables The magnitude of the

interaction effect of a variable is crucial for the system behaviour because a variable might not

be important as a singularity but it is possible its combination with another variable to have a

considerable effect on the limit state under investigation Therefore this can be considered as

an additional criterion for choosing the set of the random variables to be used in the reliability

analysis In Figure 710(a) and (b) the difference between the two indices is presented for the

anchor and the sheet pile in (a) and the soil in (b) as it was deduced from the first step of the SA

Regarding the sheet pile wall the most interactive variables are the shear stiffness and the unit

weight of the clay layer the friction angle of the sand and the cohesion shear stiffness interface

strength and unit weight of the Dikeold material Concerning the anchorrsquos limit state the

cohesion of the Dikeold material seems to have the highest interaction with the rest of the

variables Last but not least regarding the soil performance the stiffness of all the soil layers

and the friction angle of the clay and the sand layer have a higher interaction effect amongst the

other parameters The importance of the aforementioned variables can be enhanced

considering both total indices and the difference that they display between their first and total

indices



(a)

(b)

Figure 710 Interaction effect amongst the parameters in (a) anchor and sheet pile and in (b) soil This effect is expressed as the difference between the total order indices T and the first order indices S as they were deduced from the first step of the SA for both anchorsheet pile and soil respectively

Considering the discussion above some preliminary conclusions can be drawn regarding the

influence of the several variables to the different components of the system

1 both anchor and sheet pile wallrsquos limit states are sensitive mostly towards the shear

stiffness of the soil

2 sheet pile wall stress level is also affected by the unit soil weight more than the anchor

does

3 soil body is mostly influenced by the strength properties and the unit soil weight

4 the interaction effect among the soil properties is more profound for the soil and the

sheet pile limit states rather than the anchorrsquos

It must be mentioned that this SA is a global one that gives an overview of the impact of the soil

parameters in terms of the system behaviour However near the design point a local sensitivity



can possibly give a better estimation of the variablesrsquo impact Near the design point the

structural elements as well as the soil body usually reaches plasticity that implies a non-linear

behaviour Such a behaviour is unlikely to be initiated during a SA that usually samples around

the mean values This can be counteracted by conducting more iterations so as to have a larger

amount of available samplings but this can make the SA to be quite time consuming and thus to

lose its benefit to give a fast preliminary estimation of the most influencing soil layers and

properties

Besides the purpose of a preliminary global SA is to reduce the number of the stochastic

parameters to a reasonable amount so as the initial reliability analysis to be more efficient and

computationally accessible The parameters that were eventually considered as stochastic in

each LSF are summarized in the table below and they were chosen according to both the SA and

the engineering judgement In the last column the stochastic variables used also for the

systemrsquos analysis are presented

Table 72 Stochastic variables for each LSF

Soil

Failure Anchor Failure

Sheet Pile failure

System failure

Sa

nd

γunsat radic

c

φ radic radic radic radic

G radic radic radic radic

v

radic

Rinter radic

Cla

y

γunsat radic

radic radic

c radic radic radic radic



v

radic

Rinter radic radic radic

Dik

e_n

ew

γunsat radic

c radic

φ radic

G

v

Rinter

Dik

e_o

ld

γunsat

radic radic



G

radic radic radic

v

radic




74 Soil Shear Failure As far as the soil failure is concerned the limit state function that was implemented is given

below (see also sections 52 and 53) It was formulated as a single value depending on the

success of computation in Plaxis

(71)

The reliability method that was chosen in that case is DS due to the formulation of the limit state

function and the convergence issues of FORM There were used 12 input stochastic soil

parameters as it is indicated in Table 72 (see also input file of Soil Failure in Appendix D) The

variables were chosen based on the SA that was presented in the previous section For the soil

failure Poisson ratio was initially included in the analysis but it was noticed that values close to

04 during Plaxis calculations were leading to unrealistic behaviour of the dike More precisely

it has been observed that Poisson ratio equal to 04 can lead to swelling effects on the dike

which consequently lead to a raise of the dike crest instead of collapsing as it would be expected

in a soil failure analysis Therefore Poisson ratio was excluded from the reliability analysis of

the soil failure

The results of the reliability analysis according to DS are presented below In particular 300

iterations were carried out and 3 different failure directions were found Specifically in Table

73 the probability of failure the reliability index the number of iterations and the duration of

the analysis are firstly shown In the sequence the points from each failing direction with the

lowest reliability index are shown

Table 73 Reliability results of DS for the soil failure

DS Pf 1310-8 β 55 Number of LSF calls 1840 Number of iterations (or directions)

300

Elapsed time (hr) 34

Failure points (in real space) Parameter 1st direction 2nd direction 3rd direction Dikenew_phi [deg] 2277 2543 3021 Dikenew_c [kPa] 529 557 444 Dikenew_gammaunsat [kNm3]

1748 1848 1973

Dikenew_gammasat [kNm3]

1948 1984 1997

Sand_phi [deg] 2782 2039 3899 Sand_G [kPa] 808437 715473 715785 Sand_gammaunsat [kNm3]

2183 2122 2097

Sand_gammasat [kNm3] 2352 2145 2283 Clay_G [kPa] 56934 90063 80134 Clay_c [kPa] 704 1593 794



Clay_phi [deg] 1039 1629 1077 Clay_gammaunsat [kNm3]

1772 1413 1612

Clay_gammasat [kNm3] 1833 1433 1719 Dikeold_c [kPa] 1621 2978 915 Dikeold_phi [deg] 2474 3149 3281

According to the Shortest distance method the failure point from the 2nd direction turned out to

be the design point the importance factors of which are depicted in Figure 714 (first pie chart

at the left) However before concluding for the final design point an assessment of the above

failing directions was made

In particular the combination of the variables from each direction was later implemented in

Plaxis in order to illustrate the failure This is also necessary for verifying and assessing the

failure that Plaxis is giving and inspecting the error in the logfile in order to check if it is a ldquorealrdquo

(ie error=Soil collapses) or a ldquofakerdquo failure (ie error=Load advancement failed or Not enough

load steps) In Figure 711 the ldquorealrdquo failure is defined

Moreover a demonstration of the displacements is also essential in order to certify the failure

shape that normally has the dike crest settled down A different output of the displacements

formation (ie dike crest goes up) would probably imply a ldquofakerdquo failure However it should be

mentioned that a ldquofakerdquo failure in Mohr Coulomb model can be a ldquorealrdquo failure in another model

that can more realistically simulate the deformation patterns of the structure In general even

though Mohr Coulomb can be successfully applied for detecting failure it is not very suitable for

the determination of the displacements and especially for soft soils such that the dike under

investigation is constructed by It should be also mentioned at that point that in this case study

the system is subjected to unloading conditions24 (ie decrease of effective stresses) in soft soil

layers after phase 4 which cannot be modelled successfully by the Mohr-Coulomb model In that

case other constitutive models such as the Soft Soil Model and the Hardening Soil model are

presumably more reliable for simulating such situations For further information for this model

a reference is made to Plaxis (2015) Therefore someone should be always critical and verify

the failure points based on hisher engineering judgement

Figure 711 Prerequisites for real failure in Plaxis

24 From phase 4 to 5 there is an increase in water pressures and thus due to constant total stresses there is a decrease of the effective stresses This can be characterised as an unloading situation



In order to make it more understandable the different failure points were illustrated in Plaxis

and their deformed mesh and total displacements are presented in Figure 712 (a)-(c) for each

failure direction As one can observe in the first two directions the dike crest settles down

whereas in the 3rd one the dike body seems to swell and move to the right (the blue lines that

surrounds the dike soil layers indicate the initial dike geometry) However even if the 2nd

direction gave a normal pattern of displacements the error message was ldquoNot enough load

stepsrdquo The combination of the variables of the 3rd direction was then tested again with the

Hardening Soil model (HS) in order to illustrate the deformations with a more suitable model

and it was proved that the deformations seem to be realistic as it is depicted in Figure 713 With

Mohr Coulomb only the 1st direction seems to comply with the diagram in Figure 711 and that

would properly be considered as the real design point

(a) Deformed mesh

(b) Total displacements |u| in m (shadings)

(c) Total displacements direction (arrows)

1st direction 2nd direction 3rd direction

Figure 712 Illustration of the design point in soil body failure

(a) Deformed mesh (b) Total displacements |u| in m (shadings)


Figure 713 Displacements pattern of the 3rd failure direction according to HS



However taking into account that the other directions showed characteristics of failing behavior

they should also be accounted for the determination of the importance factors Excluding them

from the analysis would not be wise as the error message or the deformation schematization can

be a limitation of FEM or Mohr Coulomb model respectively In any case the points that lead to

failure shall be treated critically in order to decide if they should be included or not In this case

the points from the three directions seem to be realistic as both strength and stiffness

parameters are low enough in order to lead to a potential failure Therefore they were finally

considered for the analysis and the importance factors were averaged over all the points as it is

depicted in Figure 714 (last graph at the right)

Figure 714 Importance factors α2 for soil failure

Another option is also to average over the 10 range of close to the lowest reliability index that

was calculated with the shortest distance method (ie Average 10 method) in order to come

up with the α2 values However such methodology can lead to averaging over points of the same

direction that are not very different with each other Therefore the rest of the failure points are

automatically omitted from the consideration This can be also verified from Figure 714 where

the pie chart of the Shortest Distance and the Average 10 are quite similar whereas the Average

all chart introduces the importance of other soil variables as well

From a general perspective the soil properties of the clay layer seems to be determinant

whereas also sand and Dikeold material play an important role to the soil failure More precisely

according to the Average all method the unit weight and the friction angle of the clay layer

turned out to be the most influencing while also the cohesion of the Dikeold material and

friction angle of the sand layer contribute to the failure domain From a qualitative point of view

the weakening and consequently the settlement of the Dikeold material pushes the subsoil

creating an additional surcharge for the underlying clay and sand layer which act like the

foundation soil the incapability then of the clay layer to withstand the overlying load due to the

low unit weight in combination with the low friction angle of the clay as well as the low stiffness

and strength of the sand layer can lead to the creation of an inner slip surface and thus failure

Since the clay layer can be considered as a foundation soil the importance of the friction angle

and the unit weight can be also explained from the Terzaghirsquos bearing capacity theory (Terzaghi



1943) where these two factors are crucial for the determination of the ultimate bearing capacity

of the soil However these can be better verified by conducting large scale experiments in dikes

and inspecting the sensitivity of the dikersquos stability towards the soil properties

75 Anchor Failure The reliability of the support system is estimated based on the anchor and the waling probability

of failure However the load on the waling is proportional to the anchor force as it appeared in Eq

57 Therefore after the estimation of the anchor reliability by calculating the probability of

exceedance of a certain admissible anchor force the waling can be designed deterministically

Besides the failure probability of the waling must be lower than the anchorrsquos (given that no

uncertainties of the strength and the structural properties are considered) For the anchor failure

the LSF was formulated as follows

(72)

where [kN] is the anchor force is the yield stress and the cross

section area The reliability method to be utilized in this analysis is FORM and the selection of the

random variables to be used was made according to the sensitivity analysis results and the

engineering judgment and they are listed in Table 72 In particular the variables from the second

step of the sensitivity analysis were considered which will also help at a later stage to compare

FORM sensitivity method and FAST In Table 74 the results of the reliability analysis with FORM

are listed Additionally the design point together with the importance factors of each variable is

presented It should be mentioned at that point that the sensitivity analysis helped to define a

starting point closer to the design point and thus reduce the computational time required for the

analysis More precisely the analysis lasted approximately 16 hours whereas by starting from

mean values from which by default FORM starts the iterations it would take almost a day

Table 74 Reliability results for the anchor failure with FORM

FORM Pf 28410-6 β 45 Number of LSF calls 119 Maximum number of iterations

100


Design point Parameter value α-value Clay_phi [deg] 1814 -0059 Clay_nu [-] 033 0086 Clay_c [kPa] 1487 -0089 Clay_G [kPa] 41739 0911 Clay_Rinter [-] 051 -0021 Sand_Rinter [-] 069 -0051 Sand_nu [-] 030 0007 Sand_phi [deg] 3259 -0174 Sand_G [kPa] 1130770 0048 Dikeold_phi [deg] 2842 -0020 Dikeold_nu [-] 036 -0045



Dikeold_c [kPa] 1873 -0045 Dikeold_G [kPa] 243183 -0326 Dikeold_Rinter [-] 053 -0072

Figure 715 Importance factors α2 in anchor failure from FORM analysis

The influence coefficients in Figure 715 indicate that this limit state is governed by the shear

stiffness of the clay layer G Furthermore the shear stiffness of the Dikeold material seems to

contribute significantly to the anchor yield stress exceedance It can be concluded that the

problem is still in the elastic domain as far as the soil behaviour is concerned In case of

predominately plastic behaviour the strength properties of the soil (φ and c) become more

important In Figure 716 a demonstration of the design point as it was acquired by the FORM

analysis is presented The figures indicate that presumably due to the clay layer withdrawal in the

passive side as it is depicted in Figure 716(b) with the red shadings sheet pile moves to the right

and subsequently anchor is tensioned enough to reach its yield stress25



Figure 716 Design point illustration from the anchor failure

25 A better representation of the soil-anchor interaction could be possibly obtained if the anchor was modelled as a node-to-node element and the anchorage body as an embedded element



The calculations were repeated with the 6 most influencing parameters both with FORM and DS

in order to investigate the effect of reducing the number of random variables and in order to

validate the results of FORM analysis In Table 75 the reliability results of respectively FORM and

DS are presented while in Figure 717 the importance factors of both methods are juxtaposed The

design point and the importance factors of DS were estimated based on the Average 10 method

(see section 71)

Table 75 Reliability results of FORM and DS with reduced variables in anchor failure

FORM Pf 8510-6 DS Pf 1510-5 β 43 β 42 Number of LSF calls

28 Number of LSF calls

776

Maximum number of iterations

100 Number of iterations

100

Elapsed time (hr) 034 (starting point close to the design point)

Elapsed time (hr)

115

Design point Design point Parameter value α-value value α-value Clay_nu [-] 033 0119 033 0242 Clay_c [kPa] 1404 -0026 1535 -0212 Clay_G [kPa] 43236 0928 44548 0807 Sand_phi [deg] 3163 -0110 2954 0203 Dikeold_G [kPa] 239465 -0330 251723 -0366 Dikeold_Rinter [-] 053 -0064 060 -0263

Figure 717 Importance factors α2 with FORM and DS analysis respectively



As it can be noticed comparing FORM results from Table 74 and 75 the reliability index changes

from β=45 with 14 parameters to β=43 with the 6 most influencing parameters This verifies the

importance of the most influencing parameters that it was found in the analysis with the 14

parameters The small discrepancy though between the two reliability indexes might be

attributed to the interaction effect of the soil variables on the output performance

The estimation of failure probability with DS is almost the same with FORM As far as the

importance factors are concerned DS also came up with the conclusion that the shear stiffness of

the clay and the Dikeold material are the most crucial soil properties for the anchor stress level

However according to DS the stiffness of clay ν and the interface strength Rinter between the

Dikeold material and the sheet pile wall seem to also contribute to the limit state of the anchor

The reason why the strength of the clay layer under the dike and the Dikeold material (the part of

the dike that comes in contact with the anchor and the sheet pile wall) are the most important

variables can be explained by the principle of soil arching According to this principle the soil

columns on both sides of the rigid sheet pile wall are more compressive than the soil columns on

the top of the wall (ie overlying dike body) because of the higher stiffness of the wall when

compared with soils As such soil columns on both sides tend to settle more than the soils on top

of the rigid wall and this differential settlement causes a downward shear force acting along the

sides of soil columns on top of the wall As such the vertical load on the wall becomes larger than

the sole weight of soil columns on its top and the anchor that actually receives the most of this

vertical load reaches eventually its yielding stress

Last but not least it should be mentioned that the response surface which was constructed based

on the SA worked efficiently in most of the Plaxis errors cases in a sense that it gave reasonable

results of the anchor stress level

76 Sheet pile wall failure For the reliability analysis of the sheet pile wall the strength property ie the yield stress was

considered constant and equal to 355 kPa Therefore the LSF that was applied for this case was

formulated as follows

[

] (73)

where [kNm] and [kN] are the maximum bending moment and the axial force

that are exerted on the wall respectively is the elastic section modulus ( =1245 cm3m)

and is the cross sectional area ( =1201 cm2m for a AZ12 profile)

In sections 643 and 72 the maximum bending moments of the sheet pile wall were calculated

with the design values of the soil properties and the mean values respectively It has been

observed that the bending moments were not so significant and determinant in comparison with

the anchor force and the safety factor of the system that were also considered (in section 643) in

order to determine the structural properties

In a later stage a SA was carried out for the sheet pile wall that has been further discussed in

section 73 Figure 718 shows the stress level of the wall that was developed after 3150 iterations



by considering as random variables the ones that are defined in Table 72 As someone can

observe the stress level is generally quite low and even an order lower than the yield stress (ie

355105 Pa)

That small magnitude of bending moments can be probably attributed to the counteraction of the

active earth pressures from the passive ones that due to the homogeneity of the soil in both sides

of the wall are quite similar

Figure 718 Stress level [in Pa] of the sheet pile wall after 3150 iterations of the SA

The reliability analysis has been carried out with FORM method and considering as random

variables those listed in Table 72 However due to the low level of stresses that are developed

and the simultaneously high yield stress the method converged after almost 35 days of

calculations As it can be noticed in Table 76 the analysis came up with an extremely low

probability of failure It should be mentioned at that point that the convergence of the analysis

was successful thanks to the RS that was fitted This means that a Plaxis error of soil collapse

preceded and it activated the RS which implies that the soil failure is more likely to occur before

the sheet pile failure



Table 76 Reliability results for the sheet pile wall failure with FORM


100

Elapsed time (hr) 84 (35 days)

Design point Parameter value α-value Clay_phi [deg] 1164 0261 Clay_c [kPa] 586 0336 Clay_G [kPa] 14275 0663 Clay_Rinter [-] 052 -0016 Clay_gammaunsat [kNm3]

1848 0000

Clay_gammasat [kNm3] 1948 0000 Sand_phi [deg] 3020 0000 Sand_G [kPa] 1193650 0000 Dikeold_phi [deg] 1360 0406 Dikeold_c [kPa] 592 0438 Dikeold_G [kPa] 168798 0000 Dikeold_Rinter [-] 053 -0024 Dikeold_gammaunsat [kNm3]

2275 -0150

Dikeold_gammasat [kNm3]

2375 0000

This can be also verified from the importance factors that are depicted in Figure 719 where apart

from the shear stiffness of the clay layer which seems to be the predominant soil property for the

sheet pile LSF the strength parameters (φ and c) of the clay and the Dikeold materials also play

an important role This implies that the soil behaviour has already passed into its plastic domain

where the governing properties are the strength parameters before the sheet pile exceeds its

yield stress



Figure 719 Importance factors α2 of the sheet pile stress level in the plastic domain of soil

Nevertheless in order to investigate the potential most influencing soil properties for the sheet

pile LSF in the elastic domain as well the available output data of the FORM analysis have been

assessed The highest observed value of the sheet pile stress before its failure was around

235000 kPa After tracing the results it seems that when FORM tries to reduce further the

stiffness and strength properties of the soil layers soil collapses before the sheet pile failure The

vectors of the random variable that led to a stress level on the sheet pile wall in the range of

230000-235000 kPa were chosen in order to evaluate the importance factors The method that

was followed in that case is the Shortest Distance and the results are shown in Figure 720

Figure 720 Importance factors α2 of the sheet pile stress level in the elastic domain of soil

As it can be noticed from Figure 720 the results in the elastic domain regarding the influence of

the stochastic parameters are comparable with anchorrsquos analysis The stiffness parameter of the

clay layer (69) is the most dominant variable for the system behaviour whereas the stiffness of

the sand (15) and the Dikeold (4) layers seems also to have a significant influence This is not



surprising as the nature of the loads that generate high stresses on the anchor as well as the sheet

pile is the same namely the horizontal earth and water forces

Concluding in this case study the sheet pile wall is not expected to show a decisive behaviour to

the systemrsquos reliability according to the reliability analysis and the deterministic analysis (see

section 643) It should be stressed the fact that the fitted RS in a lot of calculations seemed not to

perform suitably for the sheet pile and this can be attributed to the fact that the it was created

based on the SA which was far from the plastic domain In that case a better fit of a RS is

recommended However such an action is still under investigation that for future considerations

in the design concept will be contemplated as a valuable technique for complex structures

modelled in FEM whose reliability analysis with conventional methods is time consuming and the

coupling may lead to numerical problems In the next section the analysis of the systemrsquos

reliability is presented

77 System Failure The estimation of the systemrsquos reliability was based on the systemrsquos fault tree of Figure 53

according to which the general LSF was formulated as the minimum of the three LSF that were

presented above for the soil the sheet pile and the anchor and it is given below

[ ]

[(

)]

(74)

According to this LSF the failure that occurs first is counted as a system failure The analysis was

carried out with DS reliability method and the variables that were taken as stochastic are listed in

Table 72 As it was discussed in section 51 the systemrsquos probability of failure differs from the

probability of flooding for which a combination between the systems reliability and the water

level uncertainty shall be made

In Table 77 the reliability analysis results are presented as they were deduced from DS Similarly

to the soil failure in Figure 721 the importance factors according to the Shortest Distance the 10

Average and the Average all methods are illustrated

Table 77 Reliability analysis results from the system failure


300




Figure 721 Importance factors α2 of the system failure

The analysis took almost one and a half days and it came with indicative valuable though results

about the system behaviour According to the averaged importance factors in Figure 721

(deduced by the Average all method) we can notice features from both soil and anchor failure

For example the unit weight (14) the friction angle (7) and the cohesion (8) of the clay

layer as well as the friction angle (8) and the stiffness (7) of the sand layer that played an

essential role in the soil failure analysis Moreover the stiffness of the clay (11) and the Dikeold

(9) layers that appeared to be important for the anchor and sheet pile LSF They seem to be also

essential in the system analysis

However the other two methods (Shortest Distance and Average 10) came up with the cohesion

and the interface strength of the Dikeold material to be the most influencing soil properties for

the system For that reason an investigation was conducted regarding the output results of the

system analysis in order to obtain an insight into the different reasons of the system failure More

precisely the failure points were divided into those that come from Plaxis Error and those that

come from the yield stress exceedance of the structural elements It was then observed that many

failure points originate from Plaxis errors that are related to ldquoNot enough load stepsrdquo (Error codes

102 and 112) rather than ldquoSoil seems to collapserdquo (Error code 101)

Therefore in order to estimate the actual important soil properties the vectors of the random

variables that led to errors of ldquoSoil seems to collapserdquo and to anchorsheet pile (SP) failure were

studied separately The importance factors are depicted in Figure 722 which actually verifies the



importance of the variables that were also deduced from the Averaged all method in Figure 721

(which are basically the unit weight and the stiffness of the clay layer and the friction angle of the

sand layer) It should be mentioned that the failure of the structural elements contributes 65

whereas the soil failure contributes 35 on the total probability of failure The latter would be

possibly lower than 2410-3 that was initially found as several failure points are not attributed to

real failure but to Plaxis numerical errors

Figure 722 Influence factors α2 for the structural elements (left) and the soil LSF (right) respectively

The system analysis can lead to valuable conclusions for the system behaviour as a whole and

individually for the different elements given though that an inspection of the results is followed

Moreover such an analysis is taking into account correlations among the three sub-failure

mechanisms implicitly For example the different LSFs may be affected by the same soil variable

such as the shear stiffness of the clay layer for the anchor and the sheet pile LSF In that case the

most dominant failure mechanism will occur first and the rest are excluded26 However during

the system analysis someone is not aware of what failed or not and therefore heshe is not

capable of distinguishing the weaknesses of the system that need further improvement Besides

with this technique is not also easy to inspect the failure mechanisms that occur in order to get

some knowledge about the systemrsquos behaviour Therefore the reliability analysis of the

components is recommended to be carried out separately if time is available

78 Comparison between Global Sensitivity Analysis and Local Sensitivity

Analysis (FORM)

The aim of this section is a comparison between the importance factors that were deduced in

section 75 according to a FORM analysis and the results of the global SA that were presented in

section 73 for the anchor failure

The output of the global SA is actually the total indices that show the influence of the random

variables The importance factors of FORM and the total indices of global SA cannot be directly

26 This method is not capable though of accounting for the simultaneous occurrence of different failure mechanisms as the system was considered a serial system



compared as they are different values More precisely the total index of a random variable is the

ratio between the total variance and the portion of the total variance that stems from the

uncertainty of the specific variable and its interactions with the rest (see Eq A10 in Appendix A)

On the other hand importance factors of FORM express the ratio between the normalized value of

the variable in the design point and the reliability index (see Eq 72) Therefore the comparison

between these values is carried out qualitatively in terms of the variables that are distinguished

as the most influencing

In particular according to global SA the shear stiffness G of the clay layer and the cohesion c of

the Dikeold material seem to be the most determinant soil properties Moreover the strength

reduction parameters Rinter of the Dikeold and the clay layers and the friction angle φ of the

Dikeold and the sand layer appear to have a relatively notable contribution to the anchors limit

state

On the other hand FORM analysis has concluded to the same results as the global SA with the

difference that Dikeold material affects the anchorrsquos stress level more with its shear stiffness

rather than its cohesion In Table 78 the results of both FORM and global SA are presented

separately for each soil layer and property (in blue for the global SA and in black for the FORM

analysis)

Table 78 Important soil properties according to global SA and FORM analysis for the anchor failure (radicradic=very important radic=less important but still considerable)

FORM global SA Sand Clay Dikeold

ν

radicradic

ϕ radicradic

radic

c radic radic radicradic

G radicradicradicradic radicradic

Rinter

radic radicradic

The validation of the most important parameters was carried out with the repetition of the

analysis with DS as it was showed in section 75 The small difference of the probability of failure

as well as the same tendency of the important factors of both DS and FORM verified the reliability

of FORM analysis Therefore the similarity of the results deduced according to global SA with the

ones given by FORM indicates the fact that in this case study the global SA should be considered

as a trustable way of SA for obtaining a first overview of the dominant variables that affect the

limit state under investigation Furthermore it is capable of giving a general insight into the

outputrsquos (in this case anchor stress level) behavior and the soil layers that can potentially stand as

predominant during the reliability analysis For example in order to obtain a first sense of the

reliability and the magnitude of the expected probability of failure someone can inspect the

output samples of the LSF deduced from the global SA In Figure 723 the samples of the anchor

stress level are presented as they were deduced after 3150 iterations during the global SA

(computation time =75 hr = 3 days) The fact that after 3150 iterations and variables

combinations the anchor did not reach its ultimate stress level (ie 450 kPa) implies that the

probability of failure might be small This can also help for qualitative decision making concerning

a systemrsquos behavior without conducting a reliability analysis Moreover global SA proved the high



interaction (large difference between first and total order indices) among the parameters that can

affect the reliability analysis and can help to decide the amount of the random variables

Figure 723 Anchor stress level during global SA (step 1)

However as someone can observe from Table 78 there are also some discrepancies concerning

the dominance of some variables This can be attributed to the fact that the influence of all

parameters in specific points such as the design point cannot be identified by the global SA

Moreover it is possible that the samples used in the SA could not reach the field where the

system has a plastic behavior and where presumably other variables may also affect the limit

state

As general conclusions for global SA some possibilities and limitations are listed below

Possibilities

bull Indicates the level of interaction effect on the output variance

bull Indicate the amount of the random variables that should be considered in the reliability

analysis

bull Gives a general insight into the output behavior

bull Valuable for future use in the construction of response surfaces

Limitations

bull Cannot rely on global SA for the evaluation of the most important parameters near the

design point

bull Difficult to reach plastic zone (large amount of samples are probably needed)

79 Discussion

In this chapter the reliability analysis results were presented for the soil the anchor the sheet

pile and eventually the system failure taking soil properties of the several soil layers as stochastic



Initially a SA was carried out for each system component in order to identify the major soil

properties and reduce the number of the stochastic parameters The reliability analysis was

performed either with FORM or DS and in both cases the design point and the importance factors

were estimated Eventually the list of subquestions that is proposed in section 11 was answered

The way they were treated and the corresponding conclusions are elaborated below

Which reliability methods are computationally tractable in combination with FEM

Before starting analyzing the reliability of the system it was essential to evaluate and

assess the performance of the available reliability methods in order to be aware of their

performance in terms of computational time and flexibility in different kind of LSFs After

a literature review and the implementation of some of them in simple LSF problems a

general overview has been obtained for the most applied reliability methods concerning

the previous knowledge required for the system behavior in order to provide the right

input parameters its accuracy and the computational effort Eventually FORM and DS

were chosen to be employed for this case study considering the no preliminary knowledge

of the LSF is required their reliable accuracy and the limited computational effort that

they have in comparison with other methods

The anchorrsquos and sheet pilersquos reliability has been evaluated with FORM method whereas

soil and systemrsquos reliability have been assessed by DS Carrying out a FORM analysis and

steering the algorithmrsquos parameters in order to converge proved to be more difficult than

it was expected However an efficient performance was finally achieved and the failure

probability of the anchor failure was successfully estimated together with the importance

factors (or else the α values) The analysis was repeated with the most important variables

and they were validated with DS The FORM analysis took approximately one and half

hour to converge (with 14 random variables) after providing a starting point that it was

estimated to be close to the design point whereas for the validation it took almost half an

hour On the other hand in case of the sheet pile wall FORM analysis took almost 4 days

The computational time also depends on the LSF to be approximated the number of

variable and the complexity of the problem that each time is analyzed As far as the soil

analysis is concerned DS took approximately one and a half day to converge for 300

iterations and 14 random variables

Which limit states are relevant for the envisaged application of retaining walls in dikes and

how can they be formulated using FEM analysis outcomes

The relevant LSFs of a dike with an anchored sheet pile wall were investigated For each

element ie the sheet pile the anchor and the soil body a separate LSF has been set up

The focus of this research was on the ULS that in case of the structural elements it was

represented by the ultimate yield stress of the anchor and the sheet pile respectively as

they were expressed in Eq 72 and 73 The determination of the soil failure has been also

elaborated and the available alternatives that can be considered in that case were

elaborated After a short evaluation the Plaxis definition of soil failure was selected that it

was formulated as it is shown in Eq 71 Finally the systemrsquos LSF was formed as a

combination of the aforementioned limit states More precisely the system was



considered as a serial and thus system failure occurs when one of the elements fails first

(see Eq 74)

How robust (convergence) are the tractable methods

The robustness of the different reliability methods can be controlled by the convergence

criteria that are defined by the user and depend on the required accuracy In case of FORM

the convergence criteria consist of a set of errors that has mostly to do with the

approximation of the reliability index and the limit state threshold whereas in DS the

convergence is controlled through the coefficient of variation (CoV)

In order to reassure the robustness of FORM the method has been validated with DS The

validation showed that the FORM method complies with the result of DS with a small

difference though in the design point and the importance factors FORM analysis seemed

to converge sufficiently and faster than DS However the choice of the optimization

algorithm plays a key role to the methodrsquos efficiency In this research the convergence

errors were manipulated for increasing the efficiency of the method More precisely it has

been observed that by relaxing the relative error and the constraint error (the explanation

of these errors and an evaluation of the different optimization algorithms are available in

Appendix A2) FORM convergence can be accelerated

The reliability analysis of the soil body and the system was performed with DS Similarly

to FORM analysis in order to improve the performance of the searching algorithm some

of the methodrsquos parameters should be stipulated for improving its efficiency and

reliability Therefore the maximum step size of the algorithms was tripled and it was

proved that for the same time period double iterations were feasible to be carried out

This does not mean that the larger the step size the better it is The choice of the step size

must be made wisely depending on the reliability index that it is expected and

respectively it is allowed to change for achieving a better performance In this case study

it was proved that after 300 iterations the CoV of DS tends to be high and it can therefore

still be improved by carrying out more iterations In Figure 724 an example of a DS

convergence graph concerning the system analysis is shown The middle red line shows

the probability estimation whereas the green lines show the boundaries At that example

the CoV was 040

DS seems to perform successfully both in soil failure with one LSF and in system failure

where 3 different LSF were included In such cases performing an approximation method

such as FORM would not be wise as the output from each iteration does not provide

information for going to the next that a FORM analysis normally needs In contrast it just

gives an indication of failure or not This means that only a random sampling method can

handle it and converge successfully as it was proved



Figure 724 Directional Sampling convergence graph (at confidence level 095) of the system LSF

What is the contribution of different uncertainties in the failure mechanisms of the system

According to the global SA and the reliability analysis of FORM and DS the friction angle

the cohesion and the shear stiffness properties appeared to have an influence on the

different LSFs This is not surprising as they all have large CoV of 10 20 and 25

respectively Thus their uncertainty generally showed an impact on the systemrsquos

behaviour It is also essential to mention that the global SA has shown that the uncertainty

of the interface strength Rinter (CoV=20) has also a notable contribution to the structural

elements LSFs However the FORM analysis which is more representative close to the

design point came up with a less important impact of that variable

The reliability analysis showed that as far as the structural elements (anchor and sheet

pile) are concerned the shear stiffness of the clay and the Dikeold material is determinant

and especially for the sheet pile the strength parameters of the soft soils seem also to play

a crucial role In contrast with the anchor where the soil body seems still to behave in an

elastic manner during the sheet pile failure soil appears to reach plasticity in more

locations where the strength properties (ϕ and c) are also influential The most dominant

failure mode appeared to be the anchor failure whereas the soil failure is always preceded

the sheet pile failure The dominance of the anchor failure can be explained from the fact

that the most influent parameter in this LSF is the stiffness that is also the variable with

the highest uncertainty (CoV=25) Therefore the plastic limit of the anchor is more

likely to be reached

Regarding the soil body reliability the analysis detected 3 different failure directions in

which the clay and the sand layer below the dike seem to be the most influent More

specifically the friction angle φ the cohesion c the unit weight γ of the clay layer the

friction angle of the sand layer and the cohesion of the Dikeold material turned out to play

an important role The decrease of the clay layerrsquos unit weight lead to the incapability of

the layer to withstand the overlying load of the dike body and at the same time the

weakening of the sand layer contribute to the creation of an inner sliding surface Even if



the uncertainty of the unit weight is small (CoV=5) it is still important for the stability

of the system as the maximum deformations appear in that layer (see Figure 712) Last

but not least the illustration of the several failure points in Plaxis allows someone to

inspect the failure mode and realize if it is a real failure of the structure or if it is a

weakness of the modelling

Finally during the system analysis multiple failure directions were detected whereas the

pie charts of the importance factors include soil properties that affect both soil and

structural elements as it was expected However due to Plaxis numerical errors several

failure points were found to be not realistic and they should be excluded in order to come

up with a more reliable failure probability which will be presumably lower The most

suitable method for estimating the importance factors though seems to be the Average all

method which is not distorted from the Plaxis numerical errors However it is

recommended that a further research be done on how realistic are the failure points

coming from Plaxis warnings and to verify if they comply with Figure 711 failure

definition

In Table 79 a synopsis of the most important soil properties for the different LSFs is

presented It is also apparent the common contribution that some of them have in

different LSFs In the last two columns of the table the results of the influencing soil

variables for the system LSF are shown with the Average 10 and Average all method

respectively For the reasons that were explained in the previous paragraph the Average

all method is considered to be the most suitable in that case and as it can be noticed from

the table it gives the most representative picture of the governing soil variables of each

LSF

Table 79 Important soil properties for the different LSFs (radicradic=very important radic=less important but still considerable the most important parameters for each LSF is painted in orange)

Soil material

Soil property

Anchor Sheet pile

Soil body

System Average

10

System Average

all

Clay

G radicradic radicradic

radic radicradic

φ

radic radicradic

c

radic radic radic

Rinter γ

radicradic

radicradic

Sand

G

radic radic

φ

radicradic radicradic

c

Rinter γ

Dikeold

G radicradic

radic

φ

radic

c

radicradic radic radicradic radic

Rinter radic

radicradic radic



γ

radic radic

Rinter was taken from the validation of FORM with DS Figure 717 (right graph)

taken from the sensitivity of the sheet pile in the soil elastic domain Figure 720

Can response surface techniques help to increase the efficiency and robustness of the

reliability model

In this research RS techniques were used only as a source for providing a reasonable

value to the FORM analysis in order to continue running in case of Plaxis errors These

response surfaces were constructed based on the SA results for both the anchor and the

sheet pile wall Apparently the domain of the SA is limited and most of the times it is

incapable of approaching the design point However RS worked efficiently in many cases

where Plaxis failed to give an output result such as in case of the sheet pile LSF Moreover

the RS could be used individually for the estimation of the failure probability (instead of

Plaxis calculations) but a better fit is advised to be firstly achieved that would be

representative in a wider plastic domain of the structural elements Therefore the RS

technique can be used to enhance the efficiency of the reliability analysis of a complex

system in the sense that it can be considered as solution for FEM errors What is more RS

techniques are a quite promising method for the reliability analysis of a system as it can

also be used alone without coupling with FEM and eliminate the computational effort

However this requires more investigation of such a technique

How can the current design approach be improved

In section 22 the current design concept of the case study that is under investigation in

this research is described At that section the design values of the moments and the forces

on the structural elements are presented as well as the required total safety factor FEM

=18 is given as it was initially calculated by Breedeveld (2011) It has been observed that

a significant overestimation is made in terms of the design values that might later lead to

heavy and cost inefficient structures For that purpose alterations was made on the

original case study in terms of the structural elements and some soil properties and a new

deterministic design was made in order to come up with a less heavy structure This

would also render it possible to investigate the reliability of the system rather than taking

zero probabilities or non-convergence Moreover it would finally help to see if there are

any potentials for optimizing the design by applying a full probabilistic analysis

According to the soil reliability analysis the probability of failure was found 1310-8 and

the reliability index β=55 As far as the anchor failure is concerned the probability of

failure found to be 28410-6 with a reliability index β=45 according to the FORM analysis

The sheet pile failure seem not be significant as it will more likely occur after the soil

failure Eventually the system analysis was performed the probability of failure was

found to be 2410-3 and the reliability index β=30 which is high enough to consider the

system safe

The systemrsquos reliability index that came out of this analysis cannot be directly compared

with the required one (ie β =40) because they stem from different probabilities The



former refers to the probability of failure due to the global instability given a design water

level whereas the latter refers to the probability of flooding that incorporates also the

water level uncertainty It must be mentioned that in reality there is no sheet pile wall

inside the dike section in the dike ring 43 but this section is under investigation for

reinforcement purposes as it does not comply with the safety standards after a recent

inspection that has been carried out Moreover the original case study taken from

Breedeveld (2011) was subjected to multiple alterations for the needs of the current

research Therefore a comparison of the reliability index with the required one from the

regulations would not be meaningful to make However what it can be compared in that

case is the order of magnitude of the reliability index After the analysis a β of 30 was

estimated for the probability of failure whereas for the overall probability of flooding a β

of 40 is required whose magnitude is the comparable with the former This can firstly be

considered a sort of the probability ldquovalidationrdquo in the sense that the analysis gave

realistic results and secondly it can imply that the structure that has been investigated in

this research is likely to be safe towards macro-instability as its coupling with the water

level uncertainty would certainly give a reliability index above 40 Furthermore a

reliability index of 30 indicates that there is room for optimizing the design concept of

such a system if someone considers also the fact that the soil strength of the subsoil was

further reduced for the needs of the current study

Due to the limitations of the modeling and the alterations of the original case study these

reliability indices should not be taken into account as an indicative number for the safety

of this dike section but rather as an order of magnitude in order to realize what the

potentials for improving the design procedure are This research can also contribute to

the way of designing not only the system but also each structural element as the reliability

of each can be calculated in advance According to the methodology developed in this

thesis a comparison can later on be made for a real case situation between the results of a

full reliability analysis and the current design procedure In that sense a tangible

improvement can be suggested for the design concepts and even to introduce a new way

for the future design of dikes with retaining walls

It should be stressed that the conclusions are based on the results of this case study with its

geometry and set of material parameters and the variation coefficients Changes in the geometry

the material parameters or the statistical properties could lead to different results The

generalization of these conclusions should thus be treated carefully




8 Conclusions and Recommendations

81 Conclusions The advances in structural reliability analysis have made a large variety of methods available The

applicability and efficiency of these methods depends on the problem that is analyzed and on the

models that are used One the other hand the necessity for optimizing the design procedure of

several structures is becoming more and more intense A solution to this demand can be the

reliability analysis of a system for which rare information about its application on real-world

problems is available This work can contribute to making probabilistic analysis more accessible

for dikes reinforced with a sheet pile wall and at the same time to analyzing such a complex

system in terms of the factors that mainly influence the system behavior In particular the main

research question of this research was

How can the probability of failure due to global instability of a dike with a sheet pile wall modeled


For that purpose in this research a reliability analysis was conducted by making use of a soil-

structure model that was simulated in FEM The output of the FEM calculations was utilized as an

input in the probabilistic model that was used in this thesis in order to assess the reliability of a

dike with an anchored sheet pile wall In that case the load was basically the earth and water

pressure whose impact was investigated on the anchor (LSF 1) the sheet pile (LSF 2) and the soil

body (LSF 3) respectively as well as on the system as a whole

It was demonstrated that the calculation effort of this kind of analysis does not necessarily have to

be extremely high Especially when previous knowledge about the system behavior and the

reliability models used is available This can be achieved by carrying out a sensitivity analysis In

this thesis a global SA was conducted that was proved capable of giving a representative

estimation of the most influencing variables and give a general overview of the systemrsquos

performance However close to the design point local SA such as FORM becomes more suitable

As far as the reliability of the structural elements is concerned their limit state was evaluated in

terms of the exceedance of their yielding stress The reliability assessment was carried out with

FORM from which the probability of failure and the influence factors were concluded More

precisely the anchor failure mode seemed to be the most determinant one with the shear

stiffness of the soft soils to be the most important soil properties The sheet pile failure is quite

rare and it is unlikely to occur before the soil failure since the soil body has already entered its

plastic domain

The soil and the system failure were challenging to assess as there are multiple criteria that

someone can choose In this thesis regarding the soil failure the Plaxis definition of soil collapse

was used and it was evaluated with DS In that case the unit weight and the strength parameters

of the soft soil layers seemed to be influencing The detected failure directions were investigated

Conclusions amp Recommendations


in terms of their validity More precisely the failure points were illustrated in Plaxis in order to

visualize the failure and see if it is real failure or not (real if dike crest settles) That procedure

revealed the possibilities and limitations of FEM modeling The advantage is that a potential

failure can be detected and its visualization is possible On the other hand one should be aware of

the limitations related with the constitutive model that is used In this study the Mohr Coulomb

method was applied that although it can efficiently identify possible failure mechanisms the

representation of the deformations is not so realistic

The failure of the system was considered as a serial system of LSF 1 2 and 3 and it was also

assessed with DS The importance factors stemmed from the reliability analysis indicated soil

features that were important in all the LSFs However it should be stressed that the frequency of

the Plaxis numerical errors was such that quite a few failure points seem to be unrealistic In this

thesis the identification of those errors and their partially exclusion from the analysis was

achieved The latter was attained by the construction of the so-called Response Surfaces (RS)

based on a preliminary sensitivity analysis Such a technique found to be valuable in order to

overcome such a problem however a study for fitting more suitable RS in each problem is

recommended Nevertheless the further elimination of those numerical errors is strongly

recommended to be investigated in future research given the results from the current study

Retrieving the importance factors from a DS analysis was also a challenging part In this study

additional methods were developed in order to evaluate the importance factors and define the

design point after a DS analysis It revealed that alternative possible approaches of the

importance factors (Average 10 and Average all methods) can presumably provide a better

understanding of the systemrsquos behaviour Such methods seemed to be valuable for excluding the

ldquonoiserdquo of the possible fake failure points that originated from FEM numerical errors However

due to the inherent randomness of DS in terms of the selection of the directions the design point

shall be inspected and compared with the other failure points that were found Then someone can

contemplate if it is necessary to proceed with an averaging of the importance factors in order to

get a clearer view the variables influence Of course such a procedure takes time that is not always

available but according to the requirements of each research the most preferable assumptions

should be made

Last but not least the reliability analysis came up with relatively low probabilities of failure which

implies that there is still room for optimizing the design concept of a dike with retaining wall

More precisely the original case study of Breedeveld (2011) has been altered by reducing the

strength properties of the subsoil and installing a less heavy sheet pile wall after following a less

strict deterministic design It was then observed that even then the reliability analysis came up

with a respectively high reliability index It should be stressed that the failure probability of the

system should be also coupled with the uncertainty of the water level in order to obtain an overall

picture of the total probability of failure Nevertheless this research proves the potentials to

optimize the design of such a system which presumably will lead to a less expensive construction

However additional failure mechanisms such as piping internal erosion etc should be also

investigated and the design properties of the system (such as the length and cross section area of

the sheet pile wall) shall be based on them



82 Recommendations The following topics are recommended for further research

Additional research is required on FEM modeling both for the structural elements and the

soil body More in particular in this research it has been proved that Mohr Coulomb

model cannot realistically estimate the deformations in a soft soil In that case there are

other more advanced models such Hardening Soil model and Soft Soil model that are

capable of simulating the soil displacements and the stress-strain relationship closer to

the actual ones However these methods require more parameters to be specified and

their coupling with reliability packages might not be so robust and functional

As far as the structural elements modeling is concerned there are two major topics that

future research should focus on Firstly the anchor rod can be simulated in Plaxis as a

node-to-node element and at the bottom an embedded beam can be installed in order to

represent the anchorage body This would provide a better reaction with the surrounding

soil body and it could probably simulate the anchor-soil interaction more realistically

Another alternative would be to simulate it as a plate element in order to be able to test

the moments that are also developed on the anchor separately Secondly it is

recommended that the elements be investigated close to their plastic area and a reliability

analysis to be carried out with plastic elements instead In this research an attempt was

made for such an analysis but it appeared not to be functional in cooperation with FORM

analysis

Another issue that is recommended to be investigated is the inspection and the validation

of the failure that Plaxis is giving During the soil failure analysis different types of failure

were observed from which some might not be realistic but due to the modeling limitations

Therefore an attempt should be made in order to further eliminate the effect of Plaxis

numerical errors on the reliability analysis that can lead to fake failure modes A check of

the failure points can be also conducted by using a more suitable constitutive model for

the deformations rather than Mohr Coulomb However this is not always feasible for each

individual case Therefore the illustration of the failure mechanisms in large scale

experiments can give a better insight into the systemrsquos behavior and valuable knowledge

can be acquired of the expected failure modes In this way also 3D effects can be

investigated see what is their effect to the dike instability and at what extend they shall be

considered

Apart from the soil body the stresses and the displacements of the structural elements

shall be also verified It is thus recommended that field tests and measurements take place

in embedded elements like sheet piles and anchor in order to validate Plaxis results and

learn what is the most suitable manner for the their simulation (fixed-end-anchor node-

to-node element plate etc)

Concerning the reliability analysis of the specific case study a research on the potential

relevant LSFs is recommended for the different elements Specifically in this thesis the

LSF were chosen in order to represent the ULS of the element There is the possibility that

the SLS is used and certain acceptable deformations can be settled as a threshold for the



limit states However the choice of this displacement limit is not straightforward and a

qualitative research shall also be conducted in order to identify the impacts of different

values Besides it is not only the value that should be selected but also the location on the

structure that is going to be checked

It is also advised that the potential correlations among the soil parameters to be included

In Wolters (2012) and Bach (2014) correlation matrices are introduced for specific soil

properties that can be included in a future research and reveal what is their effect on the

reliability analysis The influence of correlations depends on what kind of variables are

correlated and what is the influence of those variables on the system behavior However a

preliminary analysis with independent variables is advisable to be carried out at the

beginning so as for someone to be able to analyze the effect of the variables individually

and then evaluate the correlation effect more wisely

It would be also advisable a comparison to be made between the design of a real case

according to a full probabilistic analysis and partial safety factors In that way a direct

insight into the advantages and disadvantages of each method can be gained and

fundamental improvements of the design concept can be made for dikes with retaining

walls

At this moment the most reliable way of conducting a reliability analysis is with sampling

methods like MC or DS However such methods are usually unattractive due to their large

computational time Thus the tolerance of sampling methods accuracy is lowered in order

also to reduce the number of iterations or approximating methods are used like FORM

analysis However even if approximating methods are used they are usually preferred to

be validated with sampling methods in order for their results to be acceptable A real

validation of the failure probability would mean that a structure should be experimentally

tested in different positions and under the same conditions in order to see if the

probability of failure that is calculated in each case is the same Therefore a real

validation of the failure probability cannot yet be initiated but the computational effort of

the sampling methods can be eliminated with the development of technology and the

improvement of the computer science


References

Abdo T amp Rackwitz R (1990) A new beta-point algorithm for large time-invariant and time

variant reliability problems Proceedings of the 3rd IFIP WG 75 Conference Berkeley pp1-12

California USA

Ang AHS and Tang W (1984) Probability Concepts in Engineering Planning and Design Volume

IBasic Principles John Wiley and Sons New York USA

Baecher GB and Christian JT (2003) Reliability and Statistics in Geotechnical Engineering

Chichester West Sussex John Wiley amp Sons Ltd

Behaviour of Soils and Rocks GEO-ENGINEERING - CIE4361 (Vol 2015) Retrieved from

httpsblackboardtudelftnlbbcswebdavpid-2386735-dt-content-rid-

7926659_2courses32281-14150221-possibilities26limitationspdf

Breedeveld J (2011) Technisch Rapport Analyse Macrostabiliteit Dijken met de Eindige Elementen

Methode Bijlage Voorbeeldcase Deltares Delft the Netherlands

Brinkgreve RBJ (2015) Personal discussion TU Delft the Netherlands

Brinkgreve RBJ and Bakker HL (1991) Non-linear finite element analysis of safety factors In

G Beer JR Booker and J P Carter (Eds) Proceedings of the Seventh International Conference on

computer methods and advances in Geomechanics Rotterdam A A Balkema Brookfield

Cukier RI Fortuin M Shuler KE Petschek AG and Schaibly JH (1973) A study of the

sensitivity of coupled reaction systems to uncertainties in rate coefficients I Theory California

University USA

Cundall PA (2002) A discontinuous future for numerical modeling in soil and rock Published in

Proceedings of the third international conference Discrete Element Methods Numerical Modeling

of Discontinua Edited by Cook and Jensen Santa Fe pp 3-4

CUR (2005) Handbook Quay Walls Gouda the Netherlands

CUR-publication 190 (1997) Probabilities in civil engineering Part 1 Probabilistic design in theory

Stichting CUR Gouda

CUR 166 ndash Damwandconstructies (2005) Technical Recommendation 4e herziene uitgave 2005

The Netherlands

Deltaprogramma (2014) Synthesedocument Veiligheid Achtergronddocument B1 Ministrie van

Infrastructuur en Milieu

EN 1990 2002 Basis of Structural Design European Committee for Standardization

References


Farrokh N (2007) Tools and Strategies for Dealing with Uncertainty in Geotechnics In DV

Griffiths and GA Fenton (Ed) Probabilistic methods in Geotechnical Engineering (pp 71-95) New

York NY Springer Vienna

Fenton GA and Vanmarcke EH (1990) Simulation of random fields via local average

subdivision J Eng Mech ASCE 116(8)1733ndash1749

Flood Defences HYDRAULIC ENGINEERING ndash CIE5314 (Vol 2015) Retrieved from


7621002_2courses29759-

131404CIE5314_FloodDefences_05_failure_mechanisms_stability_Schweckendiekpdf

Gemeentewerken-Rotterdam (2003) Alle triaxiaal proeven rotterdam Technical report Gemeentewerken Rotterdam Rotterdam the Netherlands

Ghanem R and Spanos P (1991) Stochastic Finite Elements ndashA Spectral Approach

Springer New York

Gonzaacutelez A Cuadrado A Josa A and Olivella S (2013) Safety assessment of limit equilibrium

methods for the design of cantilever and anchored sheet pile walls Escola de Camins Barcelona

Griffiths DV and Fenton GA (Eds) (2007) Probabilistic Methods in Geotechnical Engineering Lecture notes monographs edited works and proceedings in the field of Mechanics Engineering Computer Science and Applied Mathematics Italy Hasofer AM and Lind NC (1974) An exact and invariant first-order reliability format Journal of

Engineering Mechanics Division ASCE 100(1) 111-121

Hoek E Read J Karzulovic A and Chen ZY (2000) Rock slopes in civil and mining engineering

Published in Proceedings of the International Conference on Geotechnical and Geological

Engineering GeoEng2000 19-24 November Melbourne

Homma T and Saltelli A (1996) Importance measures in in global sensitivity analysis of

nonlinear models Reliab Eng Syst Saf 52(1) pp1-17

Hydra-Ring 12 (2015) Probabilistic toolbox for the WTI2017 Technical Reference Manual Version

12 Deltares Delft

Jaky J (1944) ldquoThe coefficient of earth pressure at rest In Hungarian (A nyugalmi nyomas tenyezoje)rdquo J Soc Hung Eng Arch (Magyar Mernok es Epitesz-Egylet Kozlonye) 355ndash358

Johansson E amp Sandeman E (2014) Modelling of Deep Excavation in Soft Clay A comparison of different calculation methods to in-situ measurements Chalmers University of Technology Gothenburg Sweden

Joint Committee on Structural Safety (1981) General principles on reliability for structural design

International Association for Bridge and Structural Engineering

References


Larsen H Lubking P and Breedeveld J (2013) Ontwerp stabiliteitsschermen (type II) in primaire

waterkeringen (groene versie) Deltares Delft

Melchers R E (2002) Safety and risk in structural engineering Prog Struct Engng Mater

4 193ndash202

Moumlllman A amp Vermeer P (2008) Reliability analysis of a dike failure case study Elbe river

University of Stuttgart Germany

Nelson RB (1999) Lecture Notes in Statistics An Introduction to Copulas Springer New York

OpenTURNS 15 (2015a) Reference Guide OpenTURNSrsquo methods for Step C uncertainty

propagation pp155-161

OpenTURNS 15 (2015b) Use Cases Guide for the Textual User Interface pp190-192

Pandya N Marsdea E Floquet P and Gabas N (2008) Sensitivity analysis of a model for

atmospheric dispersion of toxic gases In B Braunschweig and X Joulia (Ed) 18th European

Symposium on Computer Aided Process Engineering-ESCAPE18 (pp1143-1148) Elsevier BV

Phoon K (Ed) (2008) Reliability-based Design in Geotechnical Engineering Computations and Applications Taylor amp Francis Britain

Plaxis 2D (2015a) Tutorial manual Delft the Netherlands

Plaxis 2D (2015b) Reference Manual The Netherlands

Plaxis (2015) Material Models Manual The Netherlands

Powell MJD (1994) A direct search optimization method that models the objective and

constraint functions by linear interpolation In S Gomez amp JP Hennart (Ed) Advances in

Optimization and Numerical Analysis (pp51-67) Dordrecht Kluwer Academic

Rankine W (1857) On the stability of loose earth Philosophical Transactions of the Royal Society

of London Vol 147

Saltelli A (2004) What is Sensitivity Analysis In A Saltelli K Chan and EM Scott (Ed)

Sensitivity Analysis (pp 3-13) John Wiley amp Sons Publication

Saltelli A S Tarantola F Campolongo and M Ratto (2004) Sensitivity Analysis in Practice

A Guide to Assessing Scientific Models John Wiley amp Sons Ltd

Schittkowski K (1985) NLPQL A Fortran subroutine solving constrained nonlinear

programming problems Annals of Operations Research 5 485-500

Schweckendiek T (2006) Structural Reliability applied to Deep Excavations Coupling Reliability

Methods with Finite Elements TU Delft Delft

References


Schweckendiek T Vrouwenvelder ACWM Calle EOF Jongejan RB and Kanning W (2013)

Partial Factors for Flood Defenses in the Netherlands Modern Geotechnical Codes of Practice ndash

Development and Calibration Fenton GA et al (eds) Special Geotechnical Publication Taylor amp

Francis

Seed RB Bea R G Abdelmalak R I Athanasopoulos A G Boutwell G P Bray J D Briaud J-L Cheung C Cobos-Roa D Cohen-Waeber J Collins B D Ehrensing L Farber D Hanemann M Harder L F Inkabi K S Kammerer A M Karadeniz D Kayen RE Moss R E S Nicks J Nimmala S Pestana J M Porter J Rhee K Riemer M F Roberts K Rogers J D Storesund R Govindasamy A V Vera-Grunauer X Wartman J E Watkins C M Wenk Jr E and Yim SC (2006) Investigation of the Performance of the New Orleans Flood Protection Systems in Hurricane Katrina on August 29 2005 Volume I Main Text and Executive Summary New Orleans USA

Steenbergen RDJM (2013) Lecture notes Probabilistic Design Delft the Netherlands TU Delft

TAW (Technische Adviescommissie voor de Waterkeringen) (2004)Technisch Rapport Waterspanningen bij dijken The Netherlands

Terzaghi K (1942) Theoretical Soil Mechanics New York Wiley ISBN 978-0-471-85305-3 Teunissen JAM (2009) SBW Analyse macrostabiliteit van dijken met Eindige Elementen

Modellen Achtergronden bij Activiteit EEM 04a Opstellen stappenplan Deltares the Netherlands

Terzaghi K (1943) Theoretical Soil Mechanics John Wiley and Sons New York

Turaacutengi T amp Rabitz H (2004) Local Methods In A Saltelli K Chan and EM Scott (Ed)


Vergouwe R Huting RJM and van der Scheer P (2014) Veiligheid Nederland in Kaart 2 Overstromingsrisico dijkring 43 Betuwe Tieler- en Culemborgerwaarden The Netherlands

Vrijling JK Schweckendiek T and Kanning W (2011) Safety Standards of Flood Defenses Keynote at ISGSR 2011 (International Symposium on Geotechnical Safety and Risk) Munich Vogt Schuppener Straub amp Braumlu (eds)

Vrijling JK and Gelder PHAJM van (2002) Probabilistic Design in Hydraulic Engineering Lecture notes TUDelft Faculty of Civil Engineering and Geosciences Delft Vrouwenvelder ACWM Steenbergen HMGM and Slijkhuis KAH (1999) Theoriehandleiding PC-Ring Delft the Netherlands TNO

Waarts PH (2000) Structural reliability using Finite Element Analysis An appraisal of DARS

(Directional Adaptive Response surface Sampling) Delft University of Technology Delft the

Netherlands

Wyllie DC amp May CW (2004) Rock Slope Engineering (4th edition) Spon Press London


Appendix A OpenTURNS features

In this Appendix an overview of the most relevant of OT is given Firstly in section 112 an

introduction of the applicable methods in SA is made while in section 52 the results from such an

analysis are presented In Appendix A1 a more detailed elaboration is made of how the FAST

sensitivity method that was applied in this project works

In Appendix A2 an explanation of the different optimization algorithms available for FORM

analysis is given The theory behind these algorithms and their convergence criteria are discussed

while finally an evaluation of their performance after applying them in case of the anchor failure

is presented This evaluation has been conducted in order to decide about the most relatively

reliable algorithm that should be adopted for the reliability analysis

Finally in Appendix A3 the types of probability distributions that have been used in this study

for the random variables and which are available in OT are displayed

A1 Fourier Amplitude Sensitivity Test (FAST) In this section the extended FAST sensitivity method is discussed as it is also described in the

Reference Guide of OpenTURNS 15 (2015) FAST is based on the Fourier decomposition of the

variance of the model response ( ) the latter being represented by its Fourier expansion

is an input random vector of independent components Its key idea is to

recast this representation as a function of a scalar parameter by defining

exploring the support of the input random vector

For each input the same procedure is realized in three steps

1) Sampling

Deterministic space-filling paths with random starting points are defined ie each input Xi is

transformed as follows

( ( )) (A1)

Appendix A ndash OpenTURNS features


Figure A1 Search curves in the input space as they are defined from the transformation (the blue

dot indicates the direction of the current search path)

where is the number of input variables N is the length of the discretization of the s-space with

s varying in (-π π) by step 2πN is a random phase-shift chosen uniformly in [0 2π]

is a set of integer frequencies assigned to each input Xi The frequency

associated with the input of interest is set to the maximum admissible frequency satisfying the

Nyquist criterion (which ensures to avoid aliasing effects)

(A2)

with M the interference factor usually equal to 4 or higher It corresponds to the truncation level

of the Fourier series ie the number of harmonics that are retained in the decomposition realised

in the third step of the procedure And the maximum frequency of the complementary set of

frequencies is

(A3)

with the index ldquo-irdquo which meaning ldquoall but irdquo

2) Simulations

Output is computed such as

Then is expanded onto a Fourier series

sum [ ] (A4)

where and are Fourier coefficients defined as follows



int

(A5)

int

(A6)

These coefficients are estimated thanks to the following discrete formulation

sum

(A7)

sum

(A8)

3) Estimations by frequency analysis

The quantitative measure of sensitivity is represented by Sensitivity Indices The first-order

sensitivity index of an input pi is the measure of the main (direct) effect of pi on the output

variance (where i ne j) the second-order sensitivity indices measures the interaction effect of

pi and pj on the output variance Other higher-order indices are defined in the same manner The

total sensitivity index is the sum of all sensitivity indices involving factor pi (Homma amp Saltelli


given as



and both small pi is not an influent parameter (neither alone nor in interaction with

other parameters)



Total indices are especially suited to apportion the model output variation to the input factors in a

comprehensive manner The FAST method calculates the first-order and the total sensitivity

indices whereas Sobolrsquo method in addition to these also provides all higher-order sensitivity

indices to determine quantitatively the interaction between parameters However the

computational cost and calculation time of Sobolrsquos method tends to be higher than that of the

FAST method

Combining equations A1-A8 the first-order indices are estimated as follows

sum (

)

sum (

)

(A9)

where is the total variance the portion of D arising from the uncertainty of the ith input and N

the size of the sample using to compute the Fourier series

Subsequently the total-order indices are estimated as follows



sum (

)

sum (

)

(A10)

where is the part of the variance due to all inputs except the ith input

A2 Optimization Algorithms in FORM

Principles of optimization algorithms

The possible optimization algorithms in OT are

Abdo-Rackwitz (ARF)

Cobyla

Sequential Quadratic Programming (SQP)

Below a description of the main principles of the optimization methods mentioned above is

quoted ARF and SQP methods are elaborated according to Abdo and Rackwitz (1990) while

Cobylarsquos principles according to Powell (1994) Finally an evaluation of the performance of the

different methods follows after applying them in case of the anchor failure

The general optimization problem (objective function) can be written in the classical form

subject to inequality constraints (constraint function) (A11)

The Lagrangian function of the general problem is defined by

sum (A12)

where are the so-called Lagrangian multipliers The SQP method (developed by Schittkowski

(1985)) replaces the original problem by a sequence of quadratic programming problems which

are exactly solvable and which approximate the original one This is done by approximating the

Lagrangian function by its second order Taylor expansion in an initial point u0

(A13)

where

sum



sum

sum

in which is the gradient operator and represents the Hessian matrix27 of the function f in

the point u0 The optimality conditions for any iteration point k of the sequence of quadratic

expansions are

sum [ sum

] (A14)

(A15)

The exact calculation of the Hessian matrix is generally too expensive and cannot be efficiently

implemented for a general case Therefore the gradient information obtained in each point

during iteration is used to build up an approximation of this matrix using one of the known

update formulas

The new iteration point is defined by

(A16)

where is the step length and is a direction in which a line search is performed The process

stops when the optimality conditions of the original problem are satisfied

The most time consuming part in this algorithm is the updating of the Hessian matrix and the

solution of the system of equations A fair approximation of the Hessian of non-quadratic

functions is also obtained with about n updates of the matrix This means that the approximation

used in the few (say ten) iterations to reach convergence cannot be very good when the problem

has large number of variables The rounding errors during the updating process in large problems

can make the approximate Hessian to become singular Close to singularity the search direction

can be significantly distorted In this case the algorithm has to restart the iteration with a unit

Hessian matrix in the point where singularity occurred

The posterior algorithm ARF developed by Abdo and Rackwitz (1990) overcomes this problem

by obtaining a constant approximation of the true Hessian matrix Only the contribution of the

objective function to the Hessian is considered The numerical decomposition of the matrix

contains the scalar product of the gradients of the constraints as elements in each iteration

27 Hessian is a square matrix of second-order partial derivatives of a scalar-valued function or scalar field It describes the local curvature of a function of many variables



Cobyla algorithm has been developed by Powell (1994) and it constructs linear polynomial

approximation to the objective and constraint functions by interpolation at the vertices of

simplices (a simplex in n dimensions is the convex hull of n+1 points n being the number of

variables) It generates the next vector of variables from function values at the vertices

of a nondegenerate simplex in In this case there are unique linear

functions and say that interpolate f and at the vertices

and the optimization problem A15 by the linear programming problem

( ) (A18)

( )

The iterative use of expression A18 puts this method in the class of ldquosequential linear

programming algorithmsrdquo It is also essential to notice that in this method the gradients are

derived by interpolation at the vertices of simplices and not analytically as it happens in SQP and

ARF

Additionally Cobyla algorithm makes use of a trust region bound ρ so that the trust region

condition on the new vector of variables is

(A19)

Such a region gives the user some control over the steps that are taken automatically and which

respond satisfactorily to the fact that there may be no finite solution to the linear programming

problem The algorithm also employs a merit function of the form

( ) ( ) [ ( ) ] (A20)

in order to compare the goodness of two different vectors of variables Here is a parameter that

is adjusted automatically depending on the how close to optimizing the objective function the

analysis is and the subscript ldquo+rdquo means that the expression in square brackets is replaced by zero

if and only if its value is negative so we have ( ) ( ) whenever is feasible Parameters

and are changing ( only reduces) automatically according to the improvement of the

optimization problem that sequential vectors might cause Therefore if the change in the merit

function does not happen to improve the optimization then these values are changing Especially

regarding the parameter user gives the initial and the final values of it namely and (it

is recommended that be a reasonable change to make the variables for a coarse exploration

of the calculation while should be approximately the required distance from the final vector

of variables to the solution of the optimization problem)

Convergence criteria

The convergence of the aforementioned algorithms is controlled by the evaluation of the

following errors expressed in the standard space (for the relationship between the real space (x-

space) and the standard space (u-space) see section 422) (OpenTURNS 15 (b) 2015 pp 190-

192)



The absolute error which is the distance between two successive iterates

(A21)

The constraint error which is the absolute value of the limit state function minus the

threshold

(A22)

The relative error which is the relative distance between two successive iterates (with

regards to the second iterate)

(A23)

The residual error which is the orthogonality error (lack of orthogonality between the

vector linking the center and the iterate and the limit state function)

( ) (A24)

The algorithm converges if all the final error values are less than the maximum value specified by

the user The convergence can be also controlled by altering the maximum number of iterations

which should be higher in case of stricter required errors What is more there is the possibility to

manipulate the starting point that the algorithm uses for its first iteration Usually the default

starting point in a FORM analysis consists of the mean values of the random variables However if

there is already some knowledge about the potential design point it can be set as the starting

point of the algorithm in order to accelerate the convergence and save a lot of time

The aforementioned errors the maximum number of iterations and the starting point were

manipulated during the analysis in order to understand the functionality of the different

algorithms and to improve their efficiency

Evaluation of the algorithms performance

An evaluation of the above optimization algorithms has been made in order to make a decision of

the most suitable for the current project In principle a literature overview regarding the

performance of the algorithms is presented Then the 3 algorithms were tested considering the

anchor failure limit state function (see Eq 55) with a stress limit of 450000 Pa and their results

are discussed below

According to Abdo and Rackwitz (1990) the SQP method seems to have better convergence

behavior for highly curved constraint functions which is especially true when very expensive

structural state functions are involved such as finite element structural analyses However the

mentioned algorithm fails to reach convergence at a problem dimension (amount of random

variables) of around n=50 Therefore for increased dimensionality SQP is less efficient in terms

of storage and CPU time compared with other methods due to singularity of the updated Hessians

matrices In OT a warning is introduced since an analysis has started that a default

implementation of Hessian is being used and that the computation can be severely wrong



Abdo and Rackwitz (1990) also proved that ARF algorithm can handle problems of up to 2000

variables while the storage requirements and the CPU time are much smaller than with SQP

Therefore optimization problems of high dimensionality can be evaluated

In OT a warning appears at the beginning of a FORM calculation saying that a default

implementation of the gradient is being used and that the computation can be wrong A similar

message also appears for the SQP algorithm saying that a default implementation of the Hessian

matrix is used and that the computationrsquos results might be unreliable Moreover in both

algorithms it is advisable to check the values of the random variables that were used during the

analysis in order to ensure that they fluctuate within acceptable boundaries (Figure A4 is an

example of a variable that surpassed the minimum boundary at some stage)

Firstly the Cobyla algorithm was applied on the case study in combination with a FORM analysis

Cobyla does not require a gradient evaluation of the limit state function and does not use a default

implementation of the Hessians matrix as ARF and SQP does while the computational time is

quite small comparable to the aforementioned methods

The Cobyla algorithm was tested with (i) 3 (ii) 10 and (iii) 19 parameters respectively in order to

check its efficiency28 As it has been mentioned in the previous section the convergence criteria to

be steered are the errors (absolute relative residual and constraint) the maximum number of

iterations and the starting point At first the default errors were used with a value of 10-3 but it

was realised that such an order of errors lead to non-convergence of the algorithm Therefore the

algorithm was then checked after relaxing the maximum errors to values of 01 05 10 50 and

100 Finally the combination of the errors that led to convergence was 01 05 01 02 for the

maximum absolute relative residual and constraint error respectively After the analysis the

number of LSF calls in case (i) (ii) and (iii) was 25 even if the maximum one was set to 100 and

the probability of failure was around 035 The probability of failure was quite high and that is

why an inspection of the output file was carried out In Figures A2 and A3 the values of the

anchor limit state function are presented in the right graph during the iterations It was then

observed that the threshold of 450 kPa was never surpassed during the iterations

28 All of the algorithms were first checked with very simple LSF (linear and quadratic) and their results were verified by MC and DS



Figure A2 Friction angle and cohesion progress during a FORM analysis with Cobyla and with 3 random variables The figure at the right illustrates the limit state function evolution



Figure A3 Friction angle and cohesion progress during a FORM analysis with Cobyla and with 19 input parameters The last figure at the right illustrates the limit state function evolution

An additional action was then taken by relaxing the maximum number of iterations from 100 to

200 but no improvement was noticed (the LSF calls were again 25) It was also given extreme

values to the errors and especially to the absolute constraint and relative error in order to check

the response of the algorithm but that did not lead to better results Furthermore the

parameter (see theory part for explanation) was manipulated but even then no further

improvement was noticed in terms of the results In Figures A2 and A3 the performance of the

algorithm during the analysis can be seen through the values of the variables that were assigned

during the iterations

As someone can notice from the graphs above it is obvious that Cobyla does not perform very

well in high dimensionality problems as the variables do not seem to vary a lot More specifically

in Figure A3 with the 19 parameters it can be noticed that the random variable does not change

at all apart from two peaks at the beginning and at the end of the analysis respectively The

incapability of the algorithm to change the parameters in order to reach the limit of the anchor

stress ie 450 kPa (while here a maximum of approximately 267 kPa is reached) leads to a poor

approximation of both the design point and the probability of failure In Figure A2 with only 3



parameters the variability of the parameter is higher during the analysis which leads to a better

approach of the limit (around 370 kPa which is larger than with the 19 parmaters) but even then

a failure probability of 035 was not a realistic value On the other hand Cobyla is easy to use only

for small number of variables (ie less than 9) with linear LSF otherwise the linear

approximations can be highly inefficient Apparently the limit state function of the anchor stress

level as a function of the soil parameters is much more complex in order to be approximated by

this algorithm

Therefore the other available optimization algorithms ie ARF and SQP were examined as well

Because of the similarity of the two algorithms only the ARF performance is presented below and

a link with SQP is made when it is needed

The ARF algorithm was tested initially with 19 parameters The settings for the errors that were

used were 01 for the absolute relative residual and constraint error respectively and the

maximum number of iterations was set to 100

Figure A4 Friction angle and cohesion progress during a FORM analysis with ARF algorithm and with 19 input parameters The figure at the right illustrates the limit state function evolution while the red spots indicate the points where the 450 kPa threshold was surpassed



The algorithm worked efficiently which means that the algorithm managed to converge towards

the threshold of 450 kPa of the limit state function as it is depicted in Figure A4 (see red dots) It

was also noticed that the random variables fluctuated a lot during the iteration steps even if the

amount was significant (19 parameters) The capability of the algorithm to vary all the 19

variables evenly and in a wide range makes it possible to find the real design point and thus to

come up with a more reliable failure probability The resulted probability of failure was 3810-9

and the reliability index was ϐ=58

However the analysis took almost 4 days to finish which is a long duration and the number of LSF

calls counted at 4151 Moreover even if the results were promising after tracing all the 19

parameters in order to inspect the algorithmrsquos performance it was found out that some variables

took unacceptable values29 during the analysis As an example in Figure A5 Poissonrsquos ratio

received negative values (see red circle) at the beginning of the iterations There were also other

such examples in the same analysis such as the friction angle which took sky-high values and the

interface strength which took negative values as well as values above 1 which is unacceptable It

should be mentioned that the design point of the algorithm was completely reasonable but even

then such a performance can prevent the algorithm from converging to the right design point

quickly and increase the computational effort

Figure A5 Poisson ratio progress during the FORM analysis with ARF algorithm

Taking into account the above implications there are two main concerns at that stage (a) the

high computational time and (b) the unreasonable values that the variables took Firstly in order

to reduce the computational time the maximum errors were relaxed More precisely it was

noticed that after relaxing the maximum relative error a lot which is more related to the

29 In some cases a value of ldquoinfrdquo was given to the variables that led to termination of the analysis without results



relaxation factor (in this case an extreme value of 1000030 was tried) and increasing the absolute

error to 05 the number of LSF calls was reduced to 724 and the analysis was finished after

almost 10 hours In Figure A6 the values that the cohesion and the friction angle took during the

iterative procedure while the red spots in the right figure indicate the points that are included in

the failure space Figure A4 shows the progress of friction angle after relaxing the errors

Figure A6 Progress of the friction angle during the FORM analysis with ARF algorithm with errors

05 10000 01 01 for the absolute relative residual and constraint error respectively

As a general conclusion after the test of different combination of maximum errors by relaxing the

constraint absolute and relative error too much successive iterations of a certain variablersquos value

seem to have a wide difference which can detain the algorithm from converging Therefore the

wisest in this case study is to keep the maximum errors below or around 10 except for the

relative and the constraint error that can relax more than 1 (depending on the case) in order to

reduce the computational time to the minimum possible

To prevent having unacceptable values of the parameters and in order to increase the efficiency of

the algorithm the random variables were reduced to 11 after a SA and some of them were

truncated in a certain range according to observations and experience until now However the

algorithm then started to give an error due to zero gradient of the objective function at a specific

point (similarly SQP gave an error due to zero hessian matrix at a specific point) while the upper

limit of the LSFs could not be reached As a countermeasure the errors were relaxed (05 10 05

10 for the absolute relative residual and constraint error respectively) in order for the algorithm

not to be trapped in a specific point while also the step of the algorithm was manipulated It

was finally proved that this kind of error stemmed from the fact that the structural elements were

considered as elastoplastic and thus a limit of the stress level was considered that misled the

algorithm The elements were eventually considered as elastic and the numerical problem was

solved

30 Practically such a value has the meaning that there is no limitation of the relative error during the optimization In the standard space (where these errors are expressed) a value also greater than 10 can be also quite high However here extreme values were given due to the time limit to test multiple values



Before figuring out that the elastoplastic behaviour was giving the problem Cobyla as a gradient

free algorithm was tested again as a potential solution but with changing the starting point close

to the design one that was found with ARF It is worthwhile to take a look into the performance of

the algorithm with 19 random variables in Figure A7 In that way we boost Cobyla to converge

and it eventually performed quite differently than in Figure A3 and without giving convergence

problems The computational time was estimated to be 9 hours and the probability of failure was

very small in the order of 10-120 However the validation of the failure probability with DS

showed that the result of Cobyla was far from correct and thus ARF was preferred for continuing

with the rest of the analysis For educational reasons a FORM analysis with ARF was again

performed after changing the starting point and it was found that the computational time was

reduced to 4 hours Therefore it is still worthy to be aware of the fact that by changing the

starting point the performance of the algorithm can improve a lot However that requires a prior

knowledge of the potential design point that usually does not exist

Figure A 7 Cobyla performance after changing the starting point closer to the failure point



It must be also mentioned that by reducing the number of the variables to 11 as it was mentioned

before the algorithms could not converge to the errors that has been set and the computational

time increased compared with the case of 19 variables Taking into account the high interaction

among the variables that was proved in section 73 through the SA it can be possible that the

combination of the soil variables could lead to failure rather than the variables individually

Therefore an analysis was tried again with the complete set of the parameters and it was realised

that the algorithm did converge to the limit value

A3 Distribution Types

Uniform Distribution

If the probability of a result is between a lower limit a and an upper limit b and if every result

between these limits is equally probable a uniform distribution should be used This distribution

is defined by the two limit parameters and the probability density function is written

(A25)

and the CDF is

(A26)

Figure A8 Probability density function of a uniform distribution with a=-2 and b=2

The mean and the variance of the distribution are also defined by the limit parameters

(A27)

(A28)



Normal Distribution

One of the most commonly used distribution types is the normal distribution or Gauss-

distribution and it is defined by its first two central moments the mean micro and the standard

deviation σ The PDF is given by

radic

(A29)

and its CDF is

int

radic

(A30)

Figure A9 Probability density function of a normal distribution with different mean and standard

deviation values (notice that the curve with micro=0 and σ2=1 belongs to the standard normal distribution)

The normal distribution with mean and standard deviation is called the standard

normal distribution In the most reliability methods such as FORM SORM DS etc the analysis is

carried out in the standard normal space which actually means that the stochastic parameters are

transformed into standard normal variables

Lognormal Distribution

If a variable X has a lognormal distribution the variable Y=ln(X) has a normal distribution The

probability density function is defined as

radic (

) (A31)

where and the mean and the standard deviation of the normally distributed random

variable Y Sometimes it is useful to work with the Yrsquos parameters rather than Xrsquos These

parameters can be expressed as follows



radic

(A32)

(A33)

where and are the mean and the standard deviation of Y In Figure A10 an example is given

of a log-normally distributed variable with different values of mean and standard deviation

Figure A10 Probability density function of a log-normal distribution with different mean and standard deviation values

Truncated Normal Distribution

The truncated normal distribution is the probability distribution of a normally distributed

random variable whose value is either bounded below or above or both

Suppose has a normal distribution and lies within the interval

Then conditional on has a truncated normal distribution whose probability

density function is

(

) (

)

(A34)

and otherwise and is the probability density function and the CDF of the standard

normal distribution similar to Eq (1) and (2) If

and

then the CDF is

defined as

(A35)

Then the mean value and the variance of the conditional will be respectively



(A36)

[

(

) ] (A37)

In the next figure the PDF histograms of a variable with a mean of 0 and a standard deviation 1

were plotted for two cases (a) variable is normally distributed (b) variable is truncated-normally

distributed with =-2 and b=2

Figure A 11 Truncated normal distribution in comparison with normal distribution


Appendix B Plaxis 2D (2015) features

In this Appendix the basic features of Plaxis 2D that were treated in this thesis are discussed First

of all an introduction over the Mohr Coulomb model is made as the soil failure criteria stem from

that model Then the initial stress generation in Plaxis is discussed and how that was applied in

the current master thesis while also the φ-c reduction technique is described which was used in

the SA of the soil failure Last but not least a more detailed explanation of the interface strength

Rinter is given which represents the modeling of the soil-structure interaction in Plaxis and it has

been considered as a random variable in the current project

B1 Mohr Coulomb failure criterion Soil tends to behave in a highly non-linear way under load This non-linear stress-strain behaviour

can be modelled at several levels of sophistication There are different constitutive models

available for simulating the behaviour of the soil whose parameters can change depending on the

level of sophistication that is required in each case

As someone can notice in Table B1 different models can be applied in different cases of soil

behaviour Hardening Soil model is becoming more and more interesting in soil structures as it

has been proved to approach the plastic behaviour of the soil and the non-linear stress-strain

relationship in a reliable and promising level Moreover UBCSAND31 and the Hypoplastic model

are more sophisticated models capable of representing several aspects of soil behaviour

however the amount of the model parameters is large and for that reason they are recommended

only in case that enough soil data are available for determining these parameters

Table B 1 Features of material models (source Behaviour of Soils and Rocks 2015)

ModelFeature Elasticity Failure Hardening Softening Small strain stiffness

Cyclic loading

Liquefaction Creep

Mohr-Coulomb x x

Duncan-Chang x x

Hardening Soil x x x

HSsmall x x x

x

Modified Cam-Clay

x x x x

Soft Soil x x x

Soft Soil Creep x x x

x

UBCSAND x x x

x x

Hypoplastic (x) x x x

x (x)

31 UBC are the initials for University of British Columbia in Vancouver in Canada

Appendix B ndash Plaxis 2D (2015) features


In this master thesis the Mohr-Coulomb model is going to be applied which is simple but an

efficient model to describe the soilrsquos failures condition Below a description of that model is made

The Mohr-Coulomb model is a linear elastic perfectly plastic model (see Figure B1) The linear

elastic part is based on Hookersquos law while the perfectly plastic part is based on the Mohr-Coulomb

failure criterion formulated in a non-associated plasticity framework

The main principle of elastoplasticity is that total strains and strain rates consist of elastic and

plastic strain components

(B1)

(B2)

in which the elastic strain rate component abides by the Hookersquos law and thus it can be expressed

as follows

(B3)

where is the elastic stress-strain matrix and the effective stress rate The plastic strain

component is given by

(B4)

where is a scalar defining the magnitude of the plastic strains called plastic multiplier and

is

a vector describing their direction with to be the so-called plastic potential function

Figure B1 Basic principle of elastoplasticity

The occurrence of (elasto)plasticity can be indicated by the yield function which is

illustrated in Figure B2 in principle stress space and it is derived according to the Mohr-Coulomb

failure criterion expressed in the next formula

(B5)



Considering the sign of this yield function the different soil states can be recognised as follows

if flt0 (inside yield contour) pure elastic behaviour

if f=0 and dflt0 unloading from aplastic state (=elastic behaviour)

if f=0 and df=0 elastoplastic behaviour

if fgt0 (outside yield contour) impossible stress state

Figure B2 Mohr-Coulomb yield surface in principle stress space

Finally in Mohr-Coulomb model the calculation of the plastic strains is achieved by the so-called

non-associated flow rule where with the dilatancy angle

This makes a difference between the friction and the dilatancy angle which prevents the model

from overestimating potential uplift

Concluding this model uses 5 input parameters

Stiffness parameters E (Youngrsquos modulus) and v (Poissonrsquos ratio) as they are included in

the elastic stress-strain matrix according to Hookersquos law

Strength parameters c (cohesion) and (friction angle)

Dilative behaviour ψ (dilatancy angle)

The author is aware of the possibilities and limitations of this model In particular the model is

capable of

a first order approach of the soil behaviour in general

describing quite well the (drained) failure behaviour

It should be mentioned at that point that the Youngrsquos modulus E cannot be steered via the

Python interface (as it appears to be a readonly parameter) and thus the shear stiffness G was

considered instead that is defined as follows

[kPa]

f=0 (yield contour)

flt0

fgt0



On the other hand the model has also limitations that are referring to

the linear elastic behaviour until failure (no strain- or stress path-dependent stiffness

behaviour)

overestimation of the shear strength in case of an undrained behaviour

However this model is considered reliable for detecting failure and giving realistic stresses

magnitudes in the soil and structural elements

Moreover in this project a non-associative behaviour was assumed by considering a zero

dilatancy angle However it has been proved by Teunissen (2009) that in case of deep failure

mechanisms in a dike structure with soil materials of a friction angle above 30deg problems arise in

the numerical analysis in finite element calculations Particularly the strength in the sliding

surface decreases due to rotation and the ground stresses are relieving around it by releasing

energy which might lead to calculation implications These phenomena are not treated in this

thesis but it is advisable to check and review the results after such an analysis

B2 φ-c Reduction Technique In this section the ldquoSafety Calculationrdquo is applied as it is described in Plaxis (Plaxis 2D (a) 2015)

In the Safety Calculation the φ-c reduction technique (Brinkgreve amp Bakker 1991) is adopted

where the shear strength parameters tanφ and c of the soil as well as the tensile strength are

successively reduced until failure of the structure occurs The dilatancy angle ψ is in principle not

affected by the phic reduction procedure However the dilatancy angle can never be larger than

the friction angle When the friction angle φ has reduced so much that it becomes equal to the

(given) dilatancy angle any further reduction of the friction angle will lead to the same reduction

of the dilatancy angle The strength of interfaces if used is reduced in the same way

The total multiplier is used to define the value of the soil strength parameters at a given

stage in the analysis

(B6)

where the strength parameters with the subscript ldquoinputrdquo refer to the properties entered in the

material sets and the parameters with the subscript ldquoreducedrdquo refer to the reduced values used in

the analysis is set to 10 at the beginning of a calculation to set all material strengths to

their input values

The incremental multiplier is used to specify the increment of the strength reduction of the

first calculation step (this increment is by default 01 but user can change it) The strength

parameters are successively reduced automatically until all required steps have been performed

(by default these steps are set to 100 but this can be subjected to changes depending on the

situation) It must always be checked whether the final step has resulted in a fully developed

failure mechanism In other words if has reached an almost constant value while the



deformation is continuing that indicates an equilibrium state In that case the factor of safety is

given by

(B7)

B3 Initial Stress Generation The first construction phase in Plaxis was the initial stress generation before the dike has been

constructed In that phase for horizontal soil surfaces with any soil layers parallel to this surface

and horizontal phreatic level32 a K0-procedure can be applied where K0 is the ratio between

horizontal and vertical stresses

(B8)

In practice the value of for a normally consolidated soil is often assumed to be related to the

friction angle by Jakyrsquos empirical expression (Jaky 1944)

Using very low or very high K0-values in the K0 procedure may lead to stresses that violate the

Mohr-Coulomb failure criterion In this case Plaxis automatically reduces the lateral stresses such

that the failure condition is obeyed Hence these stresses points are in a plastic state and are thus

indicated as plastic points Although the corrected stress state obeys the failure condition it may

result in a stress field which is not in equilibrium

B4 Interface Strength In FEM for a proper modelling of soil-structure interaction interface elements can be added to

plates in order to simulate the thin zone of intensely shearing material at the contact between the

plate and the soil

The strength properties of the interface are linked to the strength properties of the adjacent soil

layer and they are reduced by a factor of as follows

(B9)

(B10)

(B11)

where index ldquoirdquo refers to the interfacersquos parameters whereas index ldquosoilrdquo refers to the adjacent soil

parameters The behaviour of the interfaces is defined according to the Coulomb criterion which

distinguishes between elastic behaviour where small displacements can occur within the

interface and plastic interface behaviour when permanent slip may occur

32 For non-horizontal surfaces a Gravity loading is recommended



When the interface is elastic then both slipping (relative movement parallel to the interface) and

gapping (relative displacements perpendicular to the interface) can occur The magnitudes of the

interface displacements are

(B12)

(B13)

where is the shear modulus of the interface is the 1-D compression modulus of the

interface is the virtual thickness of the interface (generated automatically from Plaxis) is

the elastic interface normal stiffness and is the elastic interface shear stiffness

The shear and compression moduli are related as follows

(B14)

(B15)

Taking into account the above relations the displacements can be highly influenced by the elastic

parameters and thus the strength reduction factor can have a considerable impact on the

systemrsquos behaviour In the Reference manual of Plaxis 2D (2015) a value of 23 is generally

recommended while a value of 1 would not normally be used It should be stated that in the

framework of this thesis there are no available measurements in order to define the sensitivity to

this parameter and hence it is considered uncertain variable following a truncated normal

distribution

Appendix C ndash NEN 6740 ndash Table 1


Appendix C NEN 6740 - Table 1



Appendix D Input Files for the Reliability Analysis

Below an example of the input files that were used for the anchor the sheet pile the soil body and

the system failure is presented in Figures D1 D2 D3 and D4 respectively In particular

regarding the input file of the anchor failure the meaning and the scope of each line is briefly

explained in Table D1

Figure D 1 Anchor analysis input file (with FORM method)

APPENIDX D ndash Input Files for the Reliability Analysis


Table D1 Explanation of input components

A Reliability method and its parameters eg FORM with a maximum number of iterations 100 a maximum absolute error of 05 a relative error of 1000 a residual error of 05 and a constraint error of 05

B Number of random variables

C Soil material as it is specified in Plaxis eg AKleiVenigOnder is the clay layer under the dike

D Soil parameter eg Rinter is the interface strength of the clay layer

E Probability distribution and its parameters eg Truncated normal distribution with mean 066 standard deviation 013 and bounds from 00 to 10

F Copula type

G I Limit state function I is the threshold of the function

H Response surface function that is activated in case of Plaxis errors

Figure D 2 Sheet pile wall analysis input file (with FORM method)



Figure D 3 Soil analysis input file (with DS method)



Figure D 4 System analysis input file (with DS method)


Appendix E Characteristic and mean values

The characteristic value of a parameter implies that there is 5 probability that the real value is

higher (load) or lower (resistance) than the characteristic In Figure 22 the probability density

function of a standard normal distributed parameter (mean value equals to zero and standard

deviation 1) is depicted In case of such a distribution the characteristic value is 164σ higher than

the mean value

Figure E1 Characteristic value illustration for a normal distribution (source Wolters 2012)

In this study full probabilistic calculations are taking place in order to estimate the corresponding

structural reliability According to the probabilistic design concept the mean values and the

standard deviations are used instead of characteristic values In that case characteristic values

and shall be converted to their mean values and as follows

where

(E1)

(E2)

and are the values for load or resistance respectively to which a probability of (non-

)exceedance of 95 corresponds (non-exceedance for the load and exceedance for the resistance)

(for a standard normal distributed parameter this is 1645) The definition of the rest parameters

is given in section 423



Appendix F Reliability methods

In this appendix an overview of different reliability methods is given More precisely in F1 an

explanation is given concerning the generation of random samples that is applied in MC as well as

in other sampling methods In Appendix F2 more advanced sampling methods than crude MC are

presented that can be used as accelerating reliability methods in the sense that the computational

time can be reduced Finally in Appendix F3 some basic principles of the FOSM are elaborated

that can help to better understand the concept of FORM as well

F1 Generation of random samples in Monte Carlo

The non-exceedance probability of an arbitrary random variable is uniformly distributed between

zero and one regardless of the distribution of the variable In formula (CUR ndashpublication 190

1997)

(F1)

is the uniformly distributed variable between zero and one and is the non-exceedance

probability Thus for the variable X

(F2)

in which is the inverse of the PDF of X Using this formula a random number X can be

generated from an arbitrary distribution by drawing a number of from the uniform

distribution between zero and one

More or less the same way base variables of a statistical vector can be drawn from a known joint

probability distribution function However the joint PDF must then be formulated as the product

of the conditional probability distributions of the base variables of the vector In formula this is

( )

By taking m realizations of the uniform probability distribution between zero and one a value can

be determined for every

(F3)

APPENDIX F ndash Reliability methods


If the variables are statistically independent this can be simplified to

(F4)

By inserting the values for the LSF one can check whether the obtained vector ( ) is

located in the safe area

F2 Other Sampling Methods

Quasi-Monte Carlo Simulation

As an alternative to pseudo-random number generation (used in Crude MC) the quasi-random

number sequences are introduced from where the method Quasi-MC took its name Quasi-MC is a

technique which approximates the probability of failure using low discrepancy sequences33

x1hellipxN instead of randomly generated sequences

If the domain of failure is the goal is to estimate the following

probability

int ( )

(F5)

The main principle of this method is the integral of the function on [ ] can be

approximated by using some low discrepancy sequence x1hellipxN as follows

int ( )

sum

(F6)

The low discrepancy sequence is generated in Δ according to Lebesgue measure and then may be

transformed to any measure micro thanks to the inverse CDF technique in order to approximate the

integral

int ( )

sum

(F7)

In Figure 4 a comparison is made between quasi-random sample and pseudo-random sample in

order to illustrate the difference in sampling for the two methods

33 The discrepancy of a sequence is low if the number of points in the sequence falling into an arbitrary set B is close to proportional to the measure of B as would happen on average (but not for particular samples) in the case of a uniform distribution



Figure F1 Example of Quasi random Sample (left) and Pseudo Random Sample (right) (source

Waarts 2000)

This method a priori provides an asymptotically higher convergence rate than traditional MC but

no general rule can guarantee a better efficiency of the Quasi-MC sampling than the classical MC

sampling The advantages of Quasi-MC sampling tend to disappear with the increase of the

number of dimensions of the input variables It is recommended to use the Quasi-MC technique

with high sampling sizes or with very low dimensionality (in OT this method is valid only if the

input random variables are independent) (OpenTURNS 15 2015)

Importance Simulation

The main idea of the Importance Sampling method is to replace the initial probability distribution

of the input variables by a more efficient one Efficient means that more events will be counted

in the failure domain and thus reduce the variance of the estimator of the probability of

exceeding a threshold In other words importance sampling starts with the observation that if we

are going to sample randomly we should distribute the points to put the most of them in the

region that contains information (ie the failure region) and to waste as few as possible

In this method a sampling density is used instead of the actual probability density function

For a MC simulation the probability of failure is calculated by

sum ( ( )) ( )

( )

(F8)

where ( ( )) ( )

( ( )) ( )

The problem is that prior knowledge is needed of the failure area which in most cases is not

available In Figure 5 examples of the different sampling methods are shown



(a) (b)

(c)

Figure F2 Comparison of Importance Sampling (a) MC (b) and Numerical integration (c) (source

Latin Hypercube Simulation

Latin Hypercube Sampling is a sampling method enabling to better cover the domain of variations

of the input variables thanks to a stratified sampling strategy (in OT this method is applicable in

the case of independent input variables) Stratified sampling tries to place the sampling points so

that more of them will be found in regions where the variance of the limit state function g(x) is

largest In Figure F3 a comparison is made between the LHS and MC in terms of their sampling

technique The sampling procedure according to Baecher and Christian (2003) is based on the

next steps

1 Divide the range of each variable into several intervals of equal probability

2 If there are D variables to be sampled each in k regions then for each variable the k

sampling points are placed in independent random order

3 K samplings are made by choosing the first randomized value for each variable the second

randomized value for each variable and so on until k sets of randomized variables have

been chosen



Figure F 3 Comparison between Latin Hypercube sampling and MC (source OpenTurns 15 2015a)

F3 First Order Second Moment (FOSM) Method The First-Order Second-Moment (FOSM) approach (Ang amp tang 1984) provides analytical

approximation for the mean and the standard deviation of a parameter of interest as a function of

the mean and standard deviations of the various input factors and their correlations

Consider Z to be a function of random variables x1 x2 hellip xn that is

(F9)

In general case are correlated with covariance matrix [C] ie [C]=[σ][R][σ] where [σ]

is a diagonal matrix of standard deviations and [R] is the (positive-definite and symmetric)

correlation matrix with diagonal elements Rii=1 and non-diagonal elements Rij=ρij (ρij is the

correlation coefficient between variables i and j) In scalar notation Cij= σiσjRij (Farrokh 2007)

The first order approximation to the mean variance and standard deviation of the Z function is

based on the first terms of a Taylor series expansion of Z The following first-order estimates of

mean and variance are obtained

[ ]

(F10)

[ ] (F11)

Where the vector b denotes evaluated at the mean values of ie

(F12)

If there is no correlation among the variables Eq 223 can be written as



sum

(F13)

Then the reliability index can be calculated in 6 steps (Baecher amp Christian 2003)

1 Identify all variables that affect the mechanism that is researched

2 Determine the best estimate (usually the mean value) of each variable E[ ] and use these

to calculate the best estimate of the function E[Z]

3 Estimate the uncertainty in each variable and in particular its variance Var[ ]

4 Perform sensitivity analyses by calculating the partial derivatives of Z with respect to each

of the uncertain variables or by approximating each derivative by the divided difference

5 Use the equation of Var[Z] to obtain the variance of the function Z

6 Calculate the reliability index

[ ]

(F14)


Structural reliability analysis of a dike

with a sheet pile wall

Coupling Reliability methods with Finite Elements

by

A RIPPI

in partial fulfilment of the requirements for the degree of

Master of Science

in Civil Engineering

at the Delft University of Technology

to be defended publicly on Wednesday November 25 2015 at 1100

Graduate Aikaterini Rippi Student ID 4325583 E k-rippihotmailcom

Thesis committee Prof dr ir S N Jonkman TU Delft

Dr ir R B J Brinkgreve TU Delft and Plaxis bv

Dr ir T Schweckendiek TU Delft and Deltares

Dr A Teixeira Deltares

An electronic version of this thesis is available at httprepositorytudelftnl



Preface


















Katerina Rippi

Delft November 2015



Abstract





























Table of Contents

Preface ii

Abstract iv


1 Introduction 1








31 Background 21




33 Overview 34











44 Overview 49








53 Overview 66









65 Overview 81









81 Conclusions 121


References 125

Appendix A 129












Appendix B 147








Appendix D 155


Appendix E 159


Appendix F 161





















MC Monte Carlo

OT OpenTURNS



RS Response Surface








Aristotle


1 Introduction






















Introduction












these problems







reality

























Introduction




























Sheet pile

wall

Anchor

Dike section

Introduction




during execution











analysis outcomes






improved








questions







Introduction





research





























Introduction



Introduction













showed









Cross-Section

Top View


















follows

[kN] (21)

[kN] (22)

[kPa] (23)





















(a)

































(a)








[kNm2] (24)
























suggested






SFEM

b d m n

(25)

where








material factor)


called loss factor)



























































inaccessible design





30deg



(a) ldquoSafety

calculationrdquo


(b) ldquoPlastic

calculationrdquo











characteristics




redefined







study is elaborated







































risks ϐ=34


safety risks ϐ=42


























Economic criteria

















3 Literature study







31 Background










displacements)


















Literature study




that purpose


joint PDF and














FEM in practice






















Literature study




















Literature study




compatibility
































Literature study



Christian (2003))




















Literature study

























i Screening methods





Literature study






















|

where the









parameters








Literature study







( ) (11)


( )

( )

(12)















Literature study
























presented

Literature study


















a FORM analysis

Literature study



























Literature study







Vermeer 2008)













(a)

Literature study


(b)



















Literature study








thesis


mainly concern


2014]


2014]

























(41)








failure is

(42)


(43)






















the design


elaborated






















every and


(44)




(45)


radic

(46)



(47)





(48)



(49)








are described








( ) (410)



following


distribution





standard variable






sum

(411)












radic

sum

(412)



















are discussed











(413)








( ) ( ) sum

(414)







( ) sum

(415)


defined as

radicsum

( )

(416)







(417)

where radic







( )

radicsum (

( ) )

(418)






(

) (419)




(420)




case study

















(421)









structures


Schweckendiek 2006)















(422)

(423)







(424)


(425)

(426)
















Kriging method



















sum

sum

sum sum

(427)





squared












=[ x1 hellip x6])






























computations












parameters

















is shown




































and










































variables




variables





required accuracy





than MC





















component
























∙ Anchors

∙ Walings



























int

sum (51)










them



















(a)

(b)








Dike Anchor

Sheet pile wall

1

2 3















analysis

Sheet pile wall










[

] (52)




wall









[

] (53)







Anchors





over its length




[ ] (54)








(55)








Walings







[ ] (56)

(57)






(58)





































[

] (59)





as follows



[

] (510)


















walls






passive side





















2 φ-c Reduction












follows

[ ] (511)



the reliability

















2 φ-c Reduction




(512)





analysis





















deformations





radic

(513)

(514)



radic

(515)



radic

(516)



(517)



(518)









(519)
























[(


anchor LSF (521)







(522)



















































































































Unsaturated












value

Mean

value STD Unit

Saturated

Volumetric

weight

γsat 20 22 11 [kNm3]

Unsaturated

Volumetric

weight

γunsat 19 21 103 [kNm3]











value

Mean

value STD Unit

Saturated

Volumetric

weight

γsat 17 185 093 [kNm3]

Unsaturated

Volumetric

weight

γunsat 17 185 093 [kNm3]









value

Mean

value STD Unit

Saturated

Volumetric

weight

γsat 17 185 093 [kNm3]

Unsaturated

Volumetric

weight

γunsat 17 185 093 [kNm3]











value

Mean

value STD Unit

Saturated

Volumetric

weight

γsat 195 212 106 [kNm3]

Unsaturated

Volumetric

weight

γunsat 19 207 103 [kNm3]

























next section













Profile of waling

































































(61)

















1 SAND (very silty)





2 CLAY (medium)





















level




the anchor






1

2

3

4

5









+370 NAP

+700 NAP

+1050 NAP














Sheet Pile wall




12 S355































be applied

Anchor












force

Waling






displayed in detail






















SHEET PILES


Profile - AZ 12 [-]


Length L 11 [m]



Mass w 943 [kgm]



ANCHOR




angle φ 30 [deg]



WALING


















presented
















Plaxis calculations











explained in detail












given as



















(a) (b)

















radicsum

i=1hellip (71)





(72)




Average 10

Average all
























method indicates










Deformations













(a) deformed mesh

(b) shadings

(c) arrows






















(b) Relative

shear stress τrel














































(a)

(b)










Table 71





Clay Sand

Dikeold







(a)

(b)


























(a)



(b)


















indices



(a)

(b)







does














properties








Soil


Sheet Pile failure

System failure

Sa

nd

γunsat radic

c



v

radic

Rinter radic

Cla

y

γunsat radic

radic radic




v

radic


Dik

e_n

ew

γunsat radic

c radic

φ radic

G

v

Rinter

Dik

e_o

ld

γunsat

radic radic



G

radic radic radic

v

radic







(71)










the soil failure








300



1748 1848 1973


1948 1984 1997


2183 2122 2097





1772 1413 1612








































(a) Deformed mesh



















































(72)














100



























(see section 71)




776



100


Elapsed time (hr)

115































[

] (73)















355105 Pa)

















100



1848 0000


2275 -0150


2375 0000






yield stress


































[ ]

[(

)]

(74)











300













































Analysis (FORM)




















state





analysis)



ν

radicradic

ϕ radicradic

radic



Rinter

radic radicradic



























state


Possibilities



analysis



Limitations


design point


79 Discussion

















































(see Eq 74)




























the CoV was 040





















































definition








LSF


Soil material

Soil property

Anchor Sheet pile

Soil body

System Average

10

System Average

all

Clay


radic radicradic

φ

radic radicradic

c

radic radic radic

Rinter γ

radicradic

radicradic

Sand

G

radic radic

φ


c

Rinter γ

Dikeold

G radicradic

radic

φ

radic

c


Rinter radic

radicradic radic



γ

radic radic




reliability model

































system safe







































































plastic domain



































should be made

































analysis












considered




























walls














References



California USA
















University USA






Stichting CUR Gouda


The Netherlands




References









7621002_2courses29759-




Springer New York











12 Deltares Delft





References





4 193ndash202


















of London Vol 147









References





Francis














Netherlands






















1) Sampling



( ( )) (A1)










(A2)




frequencies is

(A3)


2) Simulations



sum [ ] (A4)




int

(A5)

int

(A6)


sum

(A7)

sum

(A8)








given as




other parameters)








FAST method


sum (

)

sum (

)

(A9)






sum (

)

sum (

)

(A10)





Abdo-Rackwitz (ARF)

Cobyla









sum (A12)





(A13)

where

sum



sum

sum



expansions are

sum [ sum

] (A14)

(A15)




update formulas


(A16)

























( ) (A18)

( )




ARF



(A19)




( ) ( ) [ ( ) ] (A20)
















192)




(A21)


threshold

(A22)



(A23)



( ) (A24)
















are discussed below




































































this algorithm



























































solved
































(A25)

and the CDF is

(A26)



(A27)

(A28)



Normal Distribution




radic

(A29)

and its CDF is

int

radic

(A30)










radic (

) (A31)






radic

(A32)

(A33)









density function is

(

) (

)

(A34)



and

then the CDF is

defined as

(A35)




(A36)

[

(

) ] (A37)



























Cyclic loading

Liquefaction Creep

Mohr-Coulomb x x

Duncan-Chang x x


HSsmall x x x

x

Modified Cam-Clay

x x x x

Soft Soil x x x


x

UBCSAND x x x

x x


x (x)











(B1)

(B2)


as follows

(B3)



(B4)


is






(B5)



















capable of






[kPa]

f=0 (yield contour)

flt0

fgt0





behaviour)





















(B6)




their input values










given by

(B7)





(B8)










plate and the soil



(B9)

(B10)

(B11)











(B12)

(B13)





(B14)

(B15)







distribution




















F Copula type
















the mean value






where

(E1)

(E2)

















1997)

(F1)



(F2)







( )



(F3)




(F4)










probability

int ( )

(F5)



int ( )

sum

(F6)



integral

int ( )

sum

(F7)







Waarts 2000)
















sum ( ( )) ( )

( )

(F8)

where ( ( )) ( )

( ( )) ( )





(a) (b)

(c)









next steps






been chosen








(F9)








[ ]

(F10)

[ ] (F11)


(F12)




sum

(F13)










[ ]

(F14)




Preface


















Katerina Rippi

Delft November 2015



Abstract





























Table of Contents

Preface ii

Abstract iv


1 Introduction 1








31 Background 21




33 Overview 34











44 Overview 49








53 Overview 66









65 Overview 81









81 Conclusions 121


References 125

Appendix A 129












Appendix B 147








Appendix D 155


Appendix E 159


Appendix F 161





















MC Monte Carlo

OT OpenTURNS



RS Response Surface








Aristotle


1 Introduction






















Introduction












these problems







reality

























Introduction




























Sheet pile

wall

Anchor

Dike section

Introduction




during execution











analysis outcomes






improved








questions







Introduction





research





























Introduction



Introduction













showed









Cross-Section

Top View


















follows

[kN] (21)

[kN] (22)

[kPa] (23)





















(a)

































(a)








[kNm2] (24)
























suggested






SFEM

b d m n

(25)

where








material factor)


called loss factor)



























































inaccessible design





30deg



(a) ldquoSafety

calculationrdquo


(b) ldquoPlastic

calculationrdquo











characteristics




redefined







study is elaborated







































risks ϐ=34


safety risks ϐ=42


























Economic criteria

















3 Literature study







31 Background










displacements)


















Literature study




that purpose


joint PDF and














FEM in practice






















Literature study




















Literature study




compatibility
































Literature study



Christian (2003))




















Literature study

























i Screening methods





Literature study






















|

where the









parameters








Literature study







( ) (11)


( )

( )

(12)















Literature study
























presented

Literature study


















a FORM analysis

Literature study



























Literature study







Vermeer 2008)













(a)

Literature study


(b)



















Literature study








thesis


mainly concern


2014]


2014]

























(41)








failure is

(42)


(43)






















the design


elaborated






















every and


(44)




(45)


radic

(46)



(47)





(48)



(49)








are described








( ) (410)



following


distribution





standard variable






sum

(411)












radic

sum

(412)



















are discussed











(413)








( ) ( ) sum

(414)







( ) sum

(415)


defined as

radicsum

( )

(416)







(417)

where radic







( )

radicsum (

( ) )

(418)






(

) (419)




(420)




case study

















(421)









structures


Schweckendiek 2006)















(422)

(423)







(424)


(425)

(426)
















Kriging method



















sum

sum

sum sum

(427)





squared












=[ x1 hellip x6])






























computations












parameters

















is shown




































and










































variables




variables





required accuracy





than MC





















component
























∙ Anchors

∙ Walings



























int

sum (51)










them



















(a)

(b)








Dike Anchor

Sheet pile wall

1

2 3















analysis

Sheet pile wall










[

] (52)




wall









[

] (53)







Anchors





over its length




[ ] (54)








(55)








Walings







[ ] (56)

(57)






(58)





































[

] (59)





as follows



[

] (510)


















walls






passive side





















2 φ-c Reduction












follows

[ ] (511)



the reliability

















2 φ-c Reduction




(512)





analysis





















deformations





radic

(513)

(514)



radic

(515)



radic

(516)



(517)



(518)









(519)
























[(


anchor LSF (521)







(522)



















































































































Unsaturated












value

Mean

value STD Unit

Saturated

Volumetric

weight

γsat 20 22 11 [kNm3]

Unsaturated

Volumetric

weight

γunsat 19 21 103 [kNm3]











value

Mean

value STD Unit

Saturated

Volumetric

weight

γsat 17 185 093 [kNm3]

Unsaturated

Volumetric

weight

γunsat 17 185 093 [kNm3]









value

Mean

value STD Unit

Saturated

Volumetric

weight

γsat 17 185 093 [kNm3]

Unsaturated

Volumetric

weight

γunsat 17 185 093 [kNm3]











value

Mean

value STD Unit

Saturated

Volumetric

weight

γsat 195 212 106 [kNm3]

Unsaturated

Volumetric

weight

γunsat 19 207 103 [kNm3]

























next section













Profile of waling

































































(61)

















1 SAND (very silty)





2 CLAY (medium)





















level




the anchor






1

2

3

4

5









+370 NAP

+700 NAP

+1050 NAP














Sheet Pile wall




12 S355































be applied

Anchor












force

Waling






displayed in detail






















SHEET PILES


Profile - AZ 12 [-]


Length L 11 [m]



Mass w 943 [kgm]



ANCHOR




angle φ 30 [deg]



WALING


















presented
















Plaxis calculations











explained in detail












given as



















(a) (b)

















radicsum

i=1hellip (71)





(72)




Average 10

Average all
























method indicates










Deformations













(a) deformed mesh

(b) shadings

(c) arrows






















(b) Relative

shear stress τrel














































(a)

(b)










Table 71





Clay Sand

Dikeold







(a)

(b)


























(a)



(b)


















indices



(a)

(b)







does














properties








Soil


Sheet Pile failure

System failure

Sa

nd

γunsat radic

c



v

radic

Rinter radic

Cla

y

γunsat radic

radic radic




v

radic


Dik

e_n

ew

γunsat radic

c radic

φ radic

G

v

Rinter

Dik

e_o

ld

γunsat

radic radic



G

radic radic radic

v

radic







(71)










the soil failure








300



1748 1848 1973


1948 1984 1997


2183 2122 2097





1772 1413 1612








































(a) Deformed mesh



















































(72)














100



























(see section 71)




776



100


Elapsed time (hr)

115































[

] (73)















355105 Pa)

















100



1848 0000


2275 -0150


2375 0000






yield stress


































[ ]

[(

)]

(74)











300













































Analysis (FORM)




















state





analysis)



ν

radicradic

ϕ radicradic

radic



Rinter

radic radicradic



























state


Possibilities



analysis



Limitations


design point


79 Discussion

















































(see Eq 74)




























the CoV was 040





















































definition








LSF


Soil material

Soil property

Anchor Sheet pile

Soil body

System Average

10

System Average

all

Clay


radic radicradic

φ

radic radicradic

c

radic radic radic

Rinter γ

radicradic

radicradic

Sand

G

radic radic

φ


c

Rinter γ

Dikeold

G radicradic

radic

φ

radic

c


Rinter radic

radicradic radic



γ

radic radic




reliability model

































system safe







































































plastic domain



































should be made

































analysis












considered




























walls














References



California USA
















University USA






Stichting CUR Gouda


The Netherlands




References









7621002_2courses29759-




Springer New York











12 Deltares Delft





References





4 193ndash202


















of London Vol 147









References





Francis














Netherlands






















1) Sampling



( ( )) (A1)










(A2)




frequencies is

(A3)


2) Simulations



sum [ ] (A4)




int

(A5)

int

(A6)


sum

(A7)

sum

(A8)








given as




other parameters)








FAST method


sum (

)

sum (

)

(A9)






sum (

)

sum (

)

(A10)





Abdo-Rackwitz (ARF)

Cobyla









sum (A12)





(A13)

where

sum



sum

sum



expansions are

sum [ sum

] (A14)

(A15)




update formulas


(A16)

























( ) (A18)

( )




ARF



(A19)




( ) ( ) [ ( ) ] (A20)
















192)




(A21)


threshold

(A22)



(A23)



( ) (A24)
















are discussed below




































































this algorithm



























































solved
































(A25)

and the CDF is

(A26)



(A27)

(A28)



Normal Distribution




radic

(A29)

and its CDF is

int

radic

(A30)










radic (

) (A31)






radic

(A32)

(A33)









density function is

(

) (

)

(A34)



and

then the CDF is

defined as

(A35)




(A36)

[

(

) ] (A37)



























Cyclic loading

Liquefaction Creep

Mohr-Coulomb x x

Duncan-Chang x x


HSsmall x x x

x

Modified Cam-Clay

x x x x

Soft Soil x x x


x

UBCSAND x x x

x x


x (x)











(B1)

(B2)


as follows

(B3)



(B4)


is






(B5)



















capable of






[kPa]

f=0 (yield contour)

flt0

fgt0





behaviour)





















(B6)




their input values










given by

(B7)





(B8)










plate and the soil



(B9)

(B10)

(B11)











(B12)

(B13)





(B14)

(B15)







distribution




















F Copula type
















the mean value






where

(E1)

(E2)

















1997)

(F1)



(F2)







( )



(F3)




(F4)










probability

int ( )

(F5)



int ( )

sum

(F6)



integral

int ( )

sum

(F7)







Waarts 2000)
















sum ( ( )) ( )

( )

(F8)

where ( ( )) ( )

( ( )) ( )





(a) (b)

(c)









next steps






been chosen








(F9)








[ ]

(F10)

[ ] (F11)


(F12)




sum

(F13)










[ ]

(F14)



Preface


















Katerina Rippi

Delft November 2015



Abstract





























Table of Contents

Preface ii

Abstract iv


1 Introduction 1








31 Background 21




33 Overview 34











44 Overview 49








53 Overview 66









65 Overview 81









81 Conclusions 121


References 125

Appendix A 129












Appendix B 147








Appendix D 155


Appendix E 159


Appendix F 161





















MC Monte Carlo

OT OpenTURNS



RS Response Surface








Aristotle


1 Introduction






















Introduction












these problems







reality

























Introduction




























Sheet pile

wall

Anchor

Dike section

Introduction




during execution











analysis outcomes






improved








questions







Introduction





research





























Introduction



Introduction













showed









Cross-Section

Top View


















follows

[kN] (21)

[kN] (22)

[kPa] (23)





















(a)

































(a)








[kNm2] (24)
























suggested






SFEM

b d m n

(25)

where








material factor)


called loss factor)



























































inaccessible design





30deg



(a) ldquoSafety

calculationrdquo


(b) ldquoPlastic

calculationrdquo











characteristics




redefined







study is elaborated







































risks ϐ=34


safety risks ϐ=42


























Economic criteria

















3 Literature study







31 Background










displacements)


















Literature study




that purpose


joint PDF and














FEM in practice






















Literature study




















Literature study




compatibility
































Literature study



Christian (2003))




















Literature study

























i Screening methods





Literature study






















|

where the









parameters








Literature study







( ) (11)


( )

( )

(12)















Literature study
























presented

Literature study


















a FORM analysis

Literature study



























Literature study







Vermeer 2008)













(a)

Literature study


(b)



















Literature study








thesis


mainly concern


2014]


2014]

























(41)








failure is

(42)


(43)






















the design


elaborated






















every and


(44)




(45)


radic

(46)



(47)





(48)



(49)








are described








( ) (410)



following


distribution





standard variable






sum

(411)












radic

sum

(412)



















are discussed











(413)








( ) ( ) sum

(414)







( ) sum

(415)


defined as

radicsum

( )

(416)







(417)

where radic







( )

radicsum (

( ) )

(418)






(

) (419)




(420)




case study

















(421)









structures


Schweckendiek 2006)















(422)

(423)







(424)


(425)

(426)
















Kriging method



















sum

sum

sum sum

(427)





squared












=[ x1 hellip x6])






























computations












parameters

















is shown




































and










































variables




variables





required accuracy





than MC





















component
























∙ Anchors

∙ Walings



























int

sum (51)










them



















(a)

(b)








Dike Anchor

Sheet pile wall

1

2 3















analysis

Sheet pile wall










[

] (52)




wall









[

] (53)







Anchors





over its length




[ ] (54)








(55)








Walings







[ ] (56)

(57)






(58)





































[

] (59)





as follows



[

] (510)


















walls






passive side





















2 φ-c Reduction












follows

[ ] (511)



the reliability

















2 φ-c Reduction




(512)





analysis





















deformations





radic

(513)

(514)



radic

(515)



radic

(516)



(517)



(518)









(519)
























[(


anchor LSF (521)







(522)



















































































































Unsaturated












value

Mean

value STD Unit

Saturated

Volumetric

weight

γsat 20 22 11 [kNm3]

Unsaturated

Volumetric

weight

γunsat 19 21 103 [kNm3]











value

Mean

value STD Unit

Saturated

Volumetric

weight

γsat 17 185 093 [kNm3]

Unsaturated

Volumetric

weight

γunsat 17 185 093 [kNm3]









value

Mean

value STD Unit

Saturated

Volumetric

weight

γsat 17 185 093 [kNm3]

Unsaturated

Volumetric

weight

γunsat 17 185 093 [kNm3]











value

Mean

value STD Unit

Saturated

Volumetric

weight

γsat 195 212 106 [kNm3]

Unsaturated

Volumetric

weight

γunsat 19 207 103 [kNm3]

























next section













Profile of waling

































































(61)

















1 SAND (very silty)





2 CLAY (medium)





















level




the anchor






1

2

3

4

5









+370 NAP

+700 NAP

+1050 NAP














Sheet Pile wall




12 S355































be applied

Anchor












force

Waling






displayed in detail






















SHEET PILES


Profile - AZ 12 [-]


Length L 11 [m]



Mass w 943 [kgm]



ANCHOR




angle φ 30 [deg]



WALING


















presented
















Plaxis calculations











explained in detail












given as



















(a) (b)

















radicsum

i=1hellip (71)





(72)




Average 10

Average all
























method indicates










Deformations













(a) deformed mesh

(b) shadings

(c) arrows






















(b) Relative

shear stress τrel














































(a)

(b)










Table 71





Clay Sand

Dikeold







(a)

(b)


























(a)



(b)


















indices



(a)

(b)







does














properties








Soil


Sheet Pile failure

System failure

Sa

nd

γunsat radic

c



v

radic

Rinter radic

Cla

y

γunsat radic

radic radic




v

radic


Dik

e_n

ew

γunsat radic

c radic

φ radic

G

v

Rinter

Dik

e_o

ld

γunsat

radic radic



G

radic radic radic

v

radic







(71)










the soil failure








300



1748 1848 1973


1948 1984 1997


2183 2122 2097





1772 1413 1612








































(a) Deformed mesh



















































(72)














100



























(see section 71)




776



100


Elapsed time (hr)

115































[

] (73)















355105 Pa)

















100



1848 0000


2275 -0150


2375 0000






yield stress


































[ ]

[(

)]

(74)











300













































Analysis (FORM)




















state





analysis)



ν

radicradic

ϕ radicradic

radic



Rinter

radic radicradic



























state


Possibilities



analysis



Limitations


design point


79 Discussion

















































(see Eq 74)




























the CoV was 040





















































definition








LSF


Soil material

Soil property

Anchor Sheet pile

Soil body

System Average

10

System Average

all

Clay


radic radicradic

φ

radic radicradic

c

radic radic radic

Rinter γ

radicradic

radicradic

Sand

G

radic radic

φ


c

Rinter γ

Dikeold

G radicradic

radic

φ

radic

c


Rinter radic

radicradic radic



γ

radic radic




reliability model

































system safe







































































plastic domain



































should be made

































analysis












considered




























walls














References



California USA
















University USA






Stichting CUR Gouda


The Netherlands




References









7621002_2courses29759-




Springer New York











12 Deltares Delft





References





4 193ndash202


















of London Vol 147









References





Francis














Netherlands






















1) Sampling



( ( )) (A1)










(A2)




frequencies is

(A3)


2) Simulations



sum [ ] (A4)




int

(A5)

int

(A6)


sum

(A7)

sum

(A8)








given as




other parameters)








FAST method


sum (

)

sum (

)

(A9)






sum (

)

sum (

)

(A10)





Abdo-Rackwitz (ARF)

Cobyla









sum (A12)





(A13)

where

sum



sum

sum



expansions are

sum [ sum

] (A14)

(A15)




update formulas


(A16)

























( ) (A18)

( )




ARF



(A19)




( ) ( ) [ ( ) ] (A20)
















192)




(A21)


threshold

(A22)



(A23)



( ) (A24)
















are discussed below




































































this algorithm



























































solved
































(A25)

and the CDF is

(A26)



(A27)

(A28)



Normal Distribution




radic

(A29)

and its CDF is

int

radic

(A30)










radic (

) (A31)






radic

(A32)

(A33)









density function is

(

) (

)

(A34)



and

then the CDF is

defined as

(A35)




(A36)

[

(

) ] (A37)



























Cyclic loading

Liquefaction Creep

Mohr-Coulomb x x

Duncan-Chang x x


HSsmall x x x

x

Modified Cam-Clay

x x x x

Soft Soil x x x


x

UBCSAND x x x

x x


x (x)











(B1)

(B2)


as follows

(B3)



(B4)


is






(B5)



















capable of






[kPa]

f=0 (yield contour)

flt0

fgt0





behaviour)





















(B6)




their input values










given by

(B7)





(B8)










plate and the soil



(B9)

(B10)

(B11)











(B12)

(B13)





(B14)

(B15)







distribution




















F Copula type
















the mean value






where

(E1)

(E2)

















1997)

(F1)



(F2)







( )



(F3)




(F4)










probability

int ( )

(F5)



int ( )

sum

(F6)



integral

int ( )

sum

(F7)







Waarts 2000)
















sum ( ( )) ( )

( )

(F8)

where ( ( )) ( )

( ( )) ( )





(a) (b)

(c)









next steps






been chosen








(F9)








[ ]

(F10)

[ ] (F11)


(F12)




sum

(F13)










[ ]

(F14)




Abstract





























Table of Contents

Preface ii

Abstract iv


1 Introduction 1








31 Background 21




33 Overview 34











44 Overview 49








53 Overview 66









65 Overview 81









81 Conclusions 121


References 125

Appendix A 129












Appendix B 147








Appendix D 155


Appendix E 159


Appendix F 161





















MC Monte Carlo

OT OpenTURNS



RS Response Surface








Aristotle


1 Introduction






















Introduction












these problems







reality

























Introduction




























Sheet pile

wall

Anchor

Dike section

Introduction




during execution











analysis outcomes






improved








questions







Introduction





research





























Introduction



Introduction













showed









Cross-Section

Top View


















follows

[kN] (21)

[kN] (22)

[kPa] (23)





















(a)

































(a)








[kNm2] (24)
























suggested






SFEM

b d m n

(25)

where








material factor)


called loss factor)



























































inaccessible design





30deg



(a) ldquoSafety

calculationrdquo


(b) ldquoPlastic

calculationrdquo











characteristics




redefined







study is elaborated







































risks ϐ=34


safety risks ϐ=42


























Economic criteria

















3 Literature study







31 Background










displacements)


















Literature study




that purpose


joint PDF and














FEM in practice






















Literature study




















Literature study




compatibility
































Literature study



Christian (2003))




















Literature study

























i Screening methods





Literature study






















|

where the









parameters








Literature study







( ) (11)


( )

( )

(12)















Literature study
























presented

Literature study


















a FORM analysis

Literature study



























Literature study







Vermeer 2008)













(a)

Literature study


(b)



















Literature study








thesis


mainly concern


2014]


2014]

























(41)








failure is

(42)


(43)






















the design


elaborated






















every and


(44)




(45)


radic

(46)



(47)





(48)



(49)








are described








( ) (410)



following


distribution





standard variable






sum

(411)












radic

sum

(412)



















are discussed











(413)








( ) ( ) sum

(414)







( ) sum

(415)


defined as

radicsum

( )

(416)







(417)

where radic







( )

radicsum (

( ) )

(418)






(

) (419)




(420)




case study

















(421)









structures


Schweckendiek 2006)















(422)

(423)







(424)


(425)

(426)
















Kriging method



















sum

sum

sum sum

(427)





squared












=[ x1 hellip x6])






























computations












parameters

















is shown




































and










































variables




variables





required accuracy





than MC





















component
























∙ Anchors

∙ Walings



























int

sum (51)










them



















(a)

(b)








Dike Anchor

Sheet pile wall

1

2 3















analysis

Sheet pile wall










[

] (52)




wall









[

] (53)







Anchors





over its length




[ ] (54)








(55)








Walings







[ ] (56)

(57)






(58)





































[

] (59)





as follows



[

] (510)


















walls






passive side





















2 φ-c Reduction












follows

[ ] (511)



the reliability

















2 φ-c Reduction




(512)





analysis





















deformations





radic

(513)

(514)



radic

(515)



radic

(516)



(517)



(518)









(519)
























[(


anchor LSF (521)







(522)



















































































































Unsaturated












value

Mean

value STD Unit

Saturated

Volumetric

weight

γsat 20 22 11 [kNm3]

Unsaturated

Volumetric

weight

γunsat 19 21 103 [kNm3]











value

Mean

value STD Unit

Saturated

Volumetric

weight

γsat 17 185 093 [kNm3]

Unsaturated

Volumetric

weight

γunsat 17 185 093 [kNm3]









value

Mean

value STD Unit

Saturated

Volumetric

weight

γsat 17 185 093 [kNm3]

Unsaturated

Volumetric

weight

γunsat 17 185 093 [kNm3]











value

Mean

value STD Unit

Saturated

Volumetric

weight

γsat 195 212 106 [kNm3]

Unsaturated

Volumetric

weight

γunsat 19 207 103 [kNm3]

























next section













Profile of waling

































































(61)

















1 SAND (very silty)





2 CLAY (medium)





















level




the anchor






1

2

3

4

5









+370 NAP

+700 NAP

+1050 NAP














Sheet Pile wall




12 S355































be applied

Anchor












force

Waling






displayed in detail






















SHEET PILES


Profile - AZ 12 [-]


Length L 11 [m]



Mass w 943 [kgm]



ANCHOR




angle φ 30 [deg]



WALING


















presented
















Plaxis calculations











explained in detail












given as



















(a) (b)

















radicsum

i=1hellip (71)





(72)




Average 10

Average all
























method indicates










Deformations













(a) deformed mesh

(b) shadings

(c) arrows






















(b) Relative

shear stress τrel














































(a)

(b)










Table 71





Clay Sand

Dikeold







(a)

(b)


























(a)



(b)


















indices



(a)

(b)







does














properties








Soil


Sheet Pile failure

System failure

Sa

nd

γunsat radic

c



v

radic

Rinter radic

Cla

y

γunsat radic

radic radic




v

radic


Dik

e_n

ew

γunsat radic

c radic

φ radic

G

v

Rinter

Dik

e_o

ld

γunsat

radic radic



G

radic radic radic

v

radic







(71)










the soil failure








300



1748 1848 1973


1948 1984 1997


2183 2122 2097





1772 1413 1612








































(a) Deformed mesh



















































(72)














100



























(see section 71)




776



100


Elapsed time (hr)

115































[

] (73)















355105 Pa)

















100



1848 0000


2275 -0150


2375 0000






yield stress


































[ ]

[(

)]

(74)











300













































Analysis (FORM)




















state





analysis)



ν

radicradic

ϕ radicradic

radic



Rinter

radic radicradic



























state


Possibilities



analysis



Limitations


design point


79 Discussion

















































(see Eq 74)




























the CoV was 040





















































definition








LSF


Soil material

Soil property

Anchor Sheet pile

Soil body

System Average

10

System Average

all

Clay


radic radicradic

φ

radic radicradic

c

radic radic radic

Rinter γ

radicradic

radicradic

Sand

G

radic radic

φ


c

Rinter γ

Dikeold

G radicradic

radic

φ

radic

c


Rinter radic

radicradic radic



γ

radic radic




reliability model

































system safe







































































plastic domain



































should be made

































analysis












considered




























walls














References



California USA
















University USA






Stichting CUR Gouda


The Netherlands




References









7621002_2courses29759-




Springer New York











12 Deltares Delft





References





4 193ndash202


















of London Vol 147









References





Francis














Netherlands






















1) Sampling



( ( )) (A1)










(A2)




frequencies is

(A3)


2) Simulations



sum [ ] (A4)




int

(A5)

int

(A6)


sum

(A7)

sum

(A8)








given as




other parameters)








FAST method


sum (

)

sum (

)

(A9)






sum (

)

sum (

)

(A10)





Abdo-Rackwitz (ARF)

Cobyla









sum (A12)





(A13)

where

sum



sum

sum



expansions are

sum [ sum

] (A14)

(A15)




update formulas


(A16)

























( ) (A18)

( )




ARF



(A19)




( ) ( ) [ ( ) ] (A20)
















192)




(A21)


threshold

(A22)



(A23)



( ) (A24)
















are discussed below




































































this algorithm



























































solved
































(A25)

and the CDF is

(A26)



(A27)

(A28)



Normal Distribution




radic

(A29)

and its CDF is

int

radic

(A30)










radic (

) (A31)






radic

(A32)

(A33)









density function is

(

) (

)

(A34)



and

then the CDF is

defined as

(A35)




(A36)

[

(

) ] (A37)



























Cyclic loading

Liquefaction Creep

Mohr-Coulomb x x

Duncan-Chang x x


HSsmall x x x

x

Modified Cam-Clay

x x x x

Soft Soil x x x


x

UBCSAND x x x

x x


x (x)











(B1)

(B2)


as follows

(B3)



(B4)


is






(B5)



















capable of






[kPa]

f=0 (yield contour)

flt0

fgt0





behaviour)





















(B6)




their input values










given by

(B7)





(B8)










plate and the soil



(B9)

(B10)

(B11)











(B12)

(B13)





(B14)

(B15)







distribution




















F Copula type
















the mean value






where

(E1)

(E2)

















1997)

(F1)



(F2)







( )



(F3)




(F4)










probability

int ( )

(F5)



int ( )

sum

(F6)



integral

int ( )

sum

(F7)







Waarts 2000)
















sum ( ( )) ( )

( )

(F8)

where ( ( )) ( )

( ( )) ( )





(a) (b)

(c)









next steps






been chosen








(F9)








[ ]

(F10)

[ ] (F11)


(F12)




sum

(F13)










[ ]

(F14)



Abstract





























Table of Contents

Preface ii

Abstract iv


1 Introduction 1








31 Background 21




33 Overview 34











44 Overview 49








53 Overview 66









65 Overview 81









81 Conclusions 121


References 125

Appendix A 129












Appendix B 147








Appendix D 155


Appendix E 159


Appendix F 161





















MC Monte Carlo

OT OpenTURNS



RS Response Surface








Aristotle


1 Introduction






















Introduction












these problems







reality

























Introduction




























Sheet pile

wall

Anchor

Dike section

Introduction




during execution











analysis outcomes






improved








questions







Introduction





research





























Introduction



Introduction













showed









Cross-Section

Top View


















follows

[kN] (21)

[kN] (22)

[kPa] (23)





















(a)

































(a)








[kNm2] (24)
























suggested






SFEM

b d m n

(25)

where








material factor)


called loss factor)



























































inaccessible design





30deg



(a) ldquoSafety

calculationrdquo


(b) ldquoPlastic

calculationrdquo











characteristics




redefined







study is elaborated







































risks ϐ=34


safety risks ϐ=42


























Economic criteria

















3 Literature study







31 Background










displacements)


















Literature study




that purpose


joint PDF and














FEM in practice






















Literature study




















Literature study




compatibility
































Literature study



Christian (2003))




















Literature study

























i Screening methods





Literature study






















|

where the









parameters








Literature study







( ) (11)


( )

( )

(12)















Literature study
























presented

Literature study


















a FORM analysis

Literature study



























Literature study







Vermeer 2008)













(a)

Literature study


(b)



















Literature study








thesis


mainly concern


2014]


2014]

























(41)








failure is

(42)


(43)






















the design


elaborated






















every and


(44)




(45)


radic

(46)



(47)





(48)



(49)








are described








( ) (410)



following


distribution





standard variable






sum

(411)












radic

sum

(412)



















are discussed











(413)








( ) ( ) sum

(414)







( ) sum

(415)


defined as

radicsum

( )

(416)







(417)

where radic







( )

radicsum (

( ) )

(418)






(

) (419)




(420)




case study

















(421)









structures


Schweckendiek 2006)















(422)

(423)







(424)


(425)

(426)
















Kriging method



















sum

sum

sum sum

(427)





squared












=[ x1 hellip x6])






























computations












parameters

















is shown




































and










































variables




variables





required accuracy





than MC





















component
























∙ Anchors

∙ Walings



























int

sum (51)










them



















(a)

(b)








Dike Anchor

Sheet pile wall

1

2 3















analysis

Sheet pile wall










[

] (52)




wall









[

] (53)







Anchors





over its length




[ ] (54)








(55)








Walings







[ ] (56)

(57)






(58)





































[

] (59)





as follows



[

] (510)


















walls






passive side





















2 φ-c Reduction












follows

[ ] (511)



the reliability

















2 φ-c Reduction




(512)





analysis





















deformations





radic

(513)

(514)



radic

(515)



radic

(516)



(517)



(518)









(519)
























[(


anchor LSF (521)







(522)



















































































































Unsaturated












value

Mean

value STD Unit

Saturated

Volumetric

weight

γsat 20 22 11 [kNm3]

Unsaturated

Volumetric

weight

γunsat 19 21 103 [kNm3]











value

Mean

value STD Unit

Saturated

Volumetric

weight

γsat 17 185 093 [kNm3]

Unsaturated

Volumetric

weight

γunsat 17 185 093 [kNm3]









value

Mean

value STD Unit

Saturated

Volumetric

weight

γsat 17 185 093 [kNm3]

Unsaturated

Volumetric

weight

γunsat 17 185 093 [kNm3]











value

Mean

value STD Unit

Saturated

Volumetric

weight

γsat 195 212 106 [kNm3]

Unsaturated

Volumetric

weight

γunsat 19 207 103 [kNm3]

























next section













Profile of waling

































































(61)

















1 SAND (very silty)





2 CLAY (medium)





















level




the anchor






1

2

3

4

5









+370 NAP

+700 NAP

+1050 NAP














Sheet Pile wall




12 S355































be applied

Anchor












force

Waling






displayed in detail






















SHEET PILES


Profile - AZ 12 [-]


Length L 11 [m]



Mass w 943 [kgm]



ANCHOR




angle φ 30 [deg]



WALING


















presented
















Plaxis calculations











explained in detail












given as



















(a) (b)

















radicsum

i=1hellip (71)





(72)




Average 10

Average all
























method indicates










Deformations













(a) deformed mesh

(b) shadings

(c) arrows






















(b) Relative

shear stress τrel














































(a)

(b)










Table 71





Clay Sand

Dikeold







(a)

(b)


























(a)



(b)


















indices



(a)

(b)







does














properties








Soil


Sheet Pile failure

System failure

Sa

nd

γunsat radic

c



v

radic

Rinter radic

Cla

y

γunsat radic

radic radic




v

radic


Dik

e_n

ew

γunsat radic

c radic

φ radic

G

v

Rinter

Dik

e_o

ld

γunsat

radic radic



G

radic radic radic

v

radic







(71)










the soil failure








300



1748 1848 1973


1948 1984 1997


2183 2122 2097





1772 1413 1612








































(a) Deformed mesh



















































(72)














100



























(see section 71)




776



100


Elapsed time (hr)

115































[

] (73)















355105 Pa)

















100



1848 0000


2275 -0150


2375 0000






yield stress


































[ ]

[(

)]

(74)











300













































Analysis (FORM)




















state





analysis)



ν

radicradic

ϕ radicradic

radic



Rinter

radic radicradic



























state


Possibilities



analysis



Limitations


design point


79 Discussion

















































(see Eq 74)




























the CoV was 040





















































definition








LSF


Soil material

Soil property

Anchor Sheet pile

Soil body

System Average

10

System Average

all

Clay


radic radicradic

φ

radic radicradic

c

radic radic radic

Rinter γ

radicradic

radicradic

Sand

G

radic radic

φ


c

Rinter γ

Dikeold

G radicradic

radic

φ

radic

c


Rinter radic

radicradic radic



γ

radic radic




reliability model

































system safe







































































plastic domain



































should be made

































analysis












considered




























walls














References



California USA
















University USA






Stichting CUR Gouda


The Netherlands




References









7621002_2courses29759-




Springer New York











12 Deltares Delft





References





4 193ndash202


















of London Vol 147









References





Francis














Netherlands






















1) Sampling



( ( )) (A1)










(A2)




frequencies is

(A3)


2) Simulations



sum [ ] (A4)




int

(A5)

int

(A6)


sum

(A7)

sum

(A8)








given as




other parameters)








FAST method


sum (

)

sum (

)

(A9)






sum (

)

sum (

)

(A10)





Abdo-Rackwitz (ARF)

Cobyla









sum (A12)





(A13)

where

sum



sum

sum



expansions are

sum [ sum

] (A14)

(A15)




update formulas


(A16)

























( ) (A18)

( )




ARF



(A19)




( ) ( ) [ ( ) ] (A20)
















192)




(A21)


threshold

(A22)



(A23)



( ) (A24)
















are discussed below




































































this algorithm



























































solved
































(A25)

and the CDF is

(A26)



(A27)

(A28)



Normal Distribution




radic

(A29)

and its CDF is

int

radic

(A30)










radic (

) (A31)






radic

(A32)

(A33)









density function is

(

) (

)

(A34)



and

then the CDF is

defined as

(A35)




(A36)

[

(

) ] (A37)



























Cyclic loading

Liquefaction Creep

Mohr-Coulomb x x

Duncan-Chang x x


HSsmall x x x

x

Modified Cam-Clay

x x x x

Soft Soil x x x


x

UBCSAND x x x

x x


x (x)











(B1)

(B2)


as follows

(B3)



(B4)


is






(B5)



















capable of






[kPa]

f=0 (yield contour)

flt0

fgt0





behaviour)





















(B6)




their input values










given by

(B7)





(B8)










plate and the soil



(B9)

(B10)

(B11)











(B12)

(B13)





(B14)

(B15)







distribution




















F Copula type
















the mean value






where

(E1)

(E2)

















1997)

(F1)



(F2)







( )



(F3)




(F4)










probability

int ( )

(F5)



int ( )

sum

(F6)



integral

int ( )

sum

(F7)







Waarts 2000)
















sum ( ( )) ( )

( )

(F8)

where ( ( )) ( )

( ( )) ( )





(a) (b)

(c)









next steps






been chosen








(F9)








[ ]

(F10)

[ ] (F11)


(F12)




sum

(F13)










[ ]

(F14)


Documents

-Coupling Reliability methods with Finite Elements-