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Three-dimensional Non-linear FE-analysis of Reinforced Concrete Bending Tests of Driven Prefabricated Reinforced Concrete Piles

Master’s Thesis in the International Master’s Programme Structural Engineering

MATTHEW BATMAN NILS RAMFJELL Department of Civil and Environmental Engineering Division of Structural Engineering Concrete Structures CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden 2005 Master’s Thesis 2005:34

MASTER’S THESIS 2005:34

Three-dimensional Non-linear FE-analysis of Reinforced Concrete

Bending Tests of Driven Prefabricated Reinforced Concrete Piles

Master’s Thesis in the International Master’s Programme Structural Engineering

MATTHEW BATMAN

NILS RAMFJELL

Department of Civil and Environmental Engineering Division of Structural Engineering

Concrete Structures CHALMERS UNIVERSITY OF TECHNOLOGY

Göteborg, Sweden 2005

Three-dimensional Non-linear FE-analysis of Reinforced Concrete Bending Tests of Driven Prefabricated Reinforced Concrete Piles Master’s Thesis in the International Master’s Programme Structural Engineering MATTHEW BATMAN

NILS RAMFJELL

© MATTHEW BATMAN & NILS RAMFJELL, 2005

Master’s Thesis 2005:34 Department of Civil and Environmental Engineering Division of Structural Engineering Concrete Structures Chalmers University of Technology SE-412 96 Göteborg Sweden Telephone: + 46 (0)31-772 1000 Cover: Results from the 3D non-linear analyses of the pile model in the from of cracking and strains, plus the load-deformation curves from the empirical tests and the FE-analysis, Chapter 6. Reproservice / Department of Civil and Environmental Engineering Göteborg, Sweden 2005

I

Three-dimensional Non-linear FE-analysis of Reinforced Concrete Bending Tests of Driven Prefabricated Reinforced Concrete Piles Master’s Thesis in the International Master’s Programme Structural Engineering MATTHEW BATMAN NILS RAMFJELL Department of Civil and Environmental Engineering Division of Structural Engineering Concrete Structures Chalmers University of Technology

ABSTRACT

Driven prefabricated reinforced concrete piles are used quite commonly and effectively as foundations, however, there is still little known about the behaviours of these driven pile elements. To acquire a better understanding of a pile’s behaviour this research project aims to establish 3D numerical models of the spliced area of driven prefabricated reinforced concrete piles, which are verified with existing full-scale load tests. The experimental data used in the verification process originates from empirical tests performed by the Swedish National Testing and Research Institute, SP, in cooperation with Skanska Sverige AB, in 1991. The experiments consisted of pure bending tests on driven, spliced concrete piles. During these tests, the global behaviour, i.e. load-deflection response, was recorded. These tests do not, however, address problems corresponding to the nature of driven concrete piles.

Bonding is often a crucial parameter when dealing with reinforced concrete structures. Therefore, initially, the bond model was carefully established and studied with FE-analyses that isolated the behaviour of the bond-mechanism. A numerical model of the pile was then created, significant characteristics of the structure were gradually introduced, and numerous results were analysed. The results from these different analyses were closely studied until the final numerical model could be concluded as appropriately simulating the pile tested by SP.

The initial responses of the final numerical model adequately simulate the tested pile, however, the complete failure of the structure was never identified in this thesis. Nevertheless, it is not the “most correct” numerical model that is the most important result of this thesis. That, which is most interesting, regarding the understanding of the behaviour of a pile, is how different properties and mechanisms affect the pile’s response. The resulting effects of these tests have been identified and documented.

Although it is known that handling, transporting and driving of a pile has considerable influence on the behaviour of the pile in reality, that which is unknown are the exact consequences of these physical loads, e.g. how bond-slip is affected. It is a complicated process trying to simulate these so-called consequences with a model, but it can be concluded, that the effects from handling, transporting and driving of a pile have the most influence on creating a good simulation and numerical model of the driven spliced prefabricated reinforced concrete pile.

Key words: Bond, bending test, driven pile, non-linear finite element analysis, reinforced concrete, splice, three-dimensional modelling

II

Tredimensionell ickelinjär FE-analys av armerad betong Böjprovningar av slagna förtillverkade armerad betong pålar Examensarbete inom civilingenjörsprogrammet Väg och vattenbyggnad MATTHEW BATMAN NILS RAMFJELL Institutionen för bygg- och miljöteknik Avdelningen för konstruktionsteknik Betongbyggnad Chalmers tekniska högskola

SAMMANFATTNING

Förtillverkade slagna armerade betong pålar är idag en vanligt förekommande grundläggningssmetod. Det råder i dagsläget viss osäkerhet kring dessa pålars verkningssätt. Målet med detta examensarbete är att etablera en verifierad numerisk modell över skarvområdet på en påle och genom denna modell öka förståelsen för dessa pålars verkningssätt. Modellen verifieras mot befintliga experimentella provförsök vilka utfördes 1991 under ett samarbete mellan Statens Provnings- och Forskningsinstitut och Skanska Sverige AB. Testerna utgjordes av böjprovningar av ett antal skarvade slagna betongpålar. Under dessa tester registrerades pålarnas verkningssätt i form av dess lastnedböjnings respons. Syftet med dessa tester var att undersöka styvhets- och hållfasthetsegenskaper i skarven och området närmast skarven, således studerades inte responsen på detaljnivå närmare.

Vidhäftning är ofta en betydelsefull parameter när armerade betong konstruktioner skall analyseras. Med hänsyn till detta inleddes arbetet med att noggrant studera vidhäftningsmodellen i analyser där vidhäftningsegenskaperna kunde undersökas i en isolerad miljö. Efter detta skapades den numeriska modellen av den laboratorietestade pålen. Denna modell byggdes upp successivt, där betydelsefulla parametrar introducerades gradvis på ett sådant sätt att dess effekter kunde identifieras. Resultaten från dessa simuleringar analyserades noggrant och utifrån dessa resultat kunde den slutgiltiga modellen skapas.

Den etablerade modellen av pålstrukturen fångar den initiala responsen på ett överensstämmande sätt. Dock har inte brottet för modellen identifierats, detta på grund av mjuk- och hårdvaruproblem. Hursomhelst har många viktiga resultat erhållits beträffande enskilda parametrars påverkan av pålens verkningssätt. Dessa resultat har analyserats och sammanställts.

Det är sedan tidigare känt att transportering, hantering och slagning har en påtaglig effekt på pålars beteende. Vad som det däremot råder viss osäkerhet kring är hur och i vilken utsträckning dessa laster påverkar pålstrukturens egenskaper, till exempel vidhäftningen. Det är en komplicerad process med många osäkerheter när dessa effekter skall introduceras i en FE-modell. Dock kan det konstateras att dessa effekter från hantering, transport och slagning har en betydande effekt på den simulerade pålens verkningssätt.

Nyckelord: Vidhäftning, böjprovning, slagna pålar, icke-linjär finit element analys, armerad betong, skarv, tredimensionell modellering

CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2005:34 I

Contents ABSTRACT I

SAMMANFATTNING II

CONTENTS I

PREFACE V

NOTATIONS VI

1 INTRODUCTION 1

1.1 Background 1

1.2 Purpose 2

1.3 Goal and Objectives 3

1.4 Scope and Limitations 3

1.5 Outline 4

2 METHOD 5

2.1 Study of Piling Procedures and Techniques 5

2.2 Discussions 5

2.3 Literature and Software Studies 6 2.3.1 Literature Study 6 2.3.2 LUSAS study 6

2.4 Software 7

2.5 Strategy 7

3 PILING TECHNIQUES, THEORY AND PROCEDURES 9

3.1 Piles 9

3.2 Production of Pile Elements 12

3.3 The Piling Process 16

3.4 Load Effect 19

3.5 The Problem at Hand With Piles 21

3.6 Earlier Test Cases and Results 25

4 MATERIALS AND MATERIAL MODELLING 28

4.1 Materials 28 4.1.1 Concrete 28 4.1.2 Steel 29 4.1.3 Reinforced Concrete 29 4.1.4 Bond Mechanism 30

4.2 Constitutive modelling 31

CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2005:34 II

4.2.1 Elasticity 31 4.2.2 Plasticity 31 4.2.3 Fracture Mechanics 34

4.3 Material Models 35 4.3.1 von Mises Steel Model 35 4.3.2 Multi-cracking Concrete Model 36 4.3.3 Mohr-Coulomb Bond-mechanism Model 39

5 FINITE ELEMENT MODELLING 41

5.1 The Finite Element Method 41 5.1.1 Non-linear FEM 41 5.1.2 FEM and Reinforced Concrete 44

5.2 Finite Element Modelling with LUSAS 45 5.2.1 General Description of the FE-program 45 5.2.2 Model Attributes 46 5.2.3 Non-linear Solution Procedures 48

6 THE DEVELOPMENT OF THE NUMERICAL MODEL 50

6.1 The Numerical Model 50

6.2 Alternative Approaches 56

7 VERIFICATION AND VALIDATION OF THE NUMERICAL MODEL 59

7.1 Verification of Individual Components 59

7.2 Validation of Reinforced Concrete Models 60 7.2.1 Bond Action Between Concrete and Reinforcement 61 7.2.2 Bond Action Between Spliced Reinforcement Bars 68

7.3 Validation of the Pile Model 75 7.3.1 Validation of the Established Model 75 7.3.2 Salient Factors Critical to the Response of the Model 80

8 CONCLUSIONS 85

8.1 Outcomes 85

8.2 Future Applications and Improvements 87

9 REFERENCES 90

CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2005:34 III

APPENDICES

Appendix A1 The Laboratory Tested Splice

Appendix A2 The Laboratory Tested Pile Element

Appendix B1 SP’s Experimental Test Reports

Appendix C1 Verification of the LUSAS Multi-cracking Concrete

Material Model

Appendix C2 Verification of the LUSAS Non-associative Mohr

Coulomb Material Model

Appendix C3 Verification of the LUSAS von Mises Material Model

Appendix D1 The Confined Pullout Tested FE-model

Appendix D2 The Unconfined Pullout Tested FE-model

Appendix D3 The Force Transfer/Bond-slip Model

Appendix E1 Detailed Description of the Established FE-model

Appendix E2 Additional Results of the FE-analysis

CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2005:34 IV

CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2005:34 V

Preface This Masters Thesis, which focuses on three-dimensional non-linear FE-analyses of reinforced concrete models, has been conducted at Skanska Teknik’s Gothenburg office between October 2004 and March 2005. In cooperation with Skanska Teknik, this research has been carried out with the Department of Civil and Environmental Engineering, Chalmers University of Technology, Sweden. This thesis concludes the authors’ Masters of Science Degree in Civil Engineering, specialising in Structural Engineering, at Chalmers University of Technology.

The initiators, as well as our supervisors, of this thesis research, M.Sc.C.E. Techn. Lic. Gunnar Holmberg and M.Sc.C.E. Techn. Lic. P.O. Svahn, have been a fountain of information regarding the thesis research and theory. They have constantly supported us throughout the project’s duration, even when it seemed like we had run into a brick wall. We would firstly like to thank them for the opportunity to carry out the study at Skanska Teknik’s Gothenburg office, in which we were in direct contact with experienced engineers, within and outside the field of research. They shall also be thanked for always being available and helping us throughout the duration of the project.

Skanska Teknik’s personal shall also be thanked for their help and support during the project, and their eagerness in accepting us as fellow engineers. Special thanks shall go to M.Sc.C.E. Marcus Davidson, structural designer at Skanska Teknik, for his support and expert guidance regarding LUSAS, as well as occasionally acting as our sounding board.

We would like to thank all the “LUSAS engineers” at Skanska Teknik in Göteborg, Stockholm and Malmö for their extreme patience with us regarding the availability of the LUSAS software. We would also like to thank our opponents, Marica Eriksson and Linda Kjellholm, for their help and views during the project.

Göteborg March 2005

Matthew Batman & Nils Ramfjell

CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2005:34 VI

Notations Roman Upper Case Letters

C Velocity of stress-wave E Young’s modulus E Elastic stiffness tensor

sF Steel force extF External force vector intF Internal force vector fG Fracture energy I Identity tensor K Spring stiffness K Stiffness matrix L Length M Moment P Applied load

kP Elastic buckling load of pile R Residual vector

1S , 2S , 3S Slip parameters which defines bond-slip curve according to CEB-FIP V Shear force Z Impedance

Roman Lower Case Letters

c Cohesion cd Mid-deflection

ckf Concrete characteristic compressive strength

sf Equivalent fracture stress

tf Concrete tensile strength

yf Steel yield strength

ch Cohesion hardening parameter n Normal vector cr Roughness-cohesion parameter s Local stress vector lt Longitudinal component of traction force

nt Normal component of traction force q Side resistance of pile w Crack width w Displacement vector

cw Crack bandwidth

uw Ultimate crack width y Extra deformation of pile

CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2005:34 VII

Greek Upper Case Letters Φ Yield function

*Φ Plastic potential

Greek Lower Case Letters

δ Initial deflection of pile ε Strain tensor

eε Elastic strain tensor

fε Limiting plastic strain

pε Plastic strain tensor

xε , yε , zε Strains in X-, Y- respective Z-direction

0ε End of softening curve φ Friction angle

iφ Initial friction angle

fφ Final friction angle γ Plastic multiplier

ψγ Residual force norm

dγ Displacement norm

dtγ Incremental displacement norm κ Hardening parameter λ Plastic multiplier µ Friction factor

lccµ Reduction factor due to driving load influence

pν Particle velocity

hν Hammer velocity ρ Density σ Stress σ Stress tensor

nσ Normal stress

sσ Soil stress

pσ Pile stress , ,1 2 3σ σ σ Principal stresses

τ Shear stress maxτ Maximum bond stress

υ Poisson’s ratio ψ Dilatation

CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2005:34 1

1 Introduction Driven prefabricated reinforced concrete piles are often used within the building industry, that is, in the foundations of bridges, buildings, harbours and tunnels. The function of a pile is to help stabilise and transfer loads from a super structure, through otherwise unstable soil material, to a preferred and more stable foundation. Although piles are used quite commonly and effectively in modern times, there is still a lot unknown about the behaviour of driven pile elements when loaded from a super structure and/or the surrounding soil.

Numerous laboratory tests have been made over the years on driven reinforced concrete piles, but it is both costly and time consuming trying to replicate the exact environments and loads that are produced in reality. An example of these earlier established tests include static load tests conducted on piles in-situ. Prefabricated concrete piles are statically loaded in compression, after installation, until the geo-technical bearing capacity is reached. Although the capacity load is recorded for the pile, the exact failure type and internal mechanisms contributing to the failure remain unknown due to the surrounding soil environment.

To reduce the costs and time frames corresponding to laboratory tests, and increase awareness concerning “hidden” failure types and mechanisms, it is of great interest to simulate the testing with so-called “finite element” models. Simulation with these models can hopefully decrease costs and increase efficiency, plus generate responses that have otherwise been assumed.

Within today’s engineering analyses, the Finite Element Method, FEM, is widely used and its role as a calculation tool is becoming increasingly more accepted within the building industry. In today’s world, concrete structures are being repeatedly and increasingly analysed with the use of non-linear finite element programs. These types of analyses include many contributing factors, such as the choice of material models, solution methods, finite elements, etcetera. These factors are defining for the outcome of the analysis and how well the analysis reflects the response of the real structure.

1.1 Background

Piles can neither be inspected nor repaired once a construction has been erected on top of the piles. The cost of constructing a super structure generally corresponds to an amount that far exceeds the cost of building the pile foundation. Therefore, it is of great importance that care is taken and quality used when building a foundation of piles. Since the failure of piles cannot be accurately analysed, due to the enclosed environment of soil, it is not exactly known what sort of failure the piles are exposed to. Every pile construction is exposed to varying loads and boundary conditions, and consequently, different forces and moments.

Major uncertainties pertaining to the behaviour of driven concrete piles arise from the dynamic driving load that inserts the piles into the ground. Considerable damage, i.e. cracking, is caused in concrete piles before loading, mostly due to this installation process, but also to the handling and transporting of the piles. The effects of these


influences on an installed pile’s behaviour are hard to estimate and simulate in laboratory tests, and therefore, even harder to simulate with a FE-model.

In addition to varying loads and load combinations, a pile’s behaviour is also affected by critical internal mechanisms. These mechanisms arise partly due to that pile elements are “spliced,” that is, special splice elements join two pile elements together. It is due to this spliced area that many imperfections and discontinuities occur.

Two of the most critical internal mechanisms are the effects due to the details of the splice and the bond-slip between the pile’s reinforcement and the splice’s anchorage reinforcement. These critical mechanisms occur in and around the splice connection, and these mechanisms are exaggerated by the influence of the driving of the pile into the ground. Much is unknown about these mechanisms and their influence on the behaviour of the pile element.

In reality, these problem areas cannot be analysed with known, established, calculation methods. For example, the Commission of Pile Research’s report 96:1 (1998) recommends using a reduction factor that relates the dynamic driving load’s effect to the pile. However, even this factor is based on established calculations and tests, which have not fully taken respect to these internal mechanisms in combination with driving loads.

Creating numerical models of spliced pile elements, according to the finite element method, is one step towards broadening the understanding of these specific problem areas. On the global scale, this increased understanding can even lead to improved piling and measuring techniques. In addition, the development of new numerical models helps to create new piling methods and broaden the technical and economic availability for pile foundations. Real load tests are costly and numerical simulation of real load tests can increase the efficiency of the development of new piles.

1.2 Purpose

The purpose with this Masters Thesis is to acquire a better understanding of the effects of the splice, the interaction between concrete and reinforcement, and the driving load on the behaviour of a driven prefabricated reinforced concrete pile, in the ultimate limit state. This increased understanding will lead to an increase in the development, and improvement, in the dimensioning and designing of spliced constructions for driven prefabricated reinforced concrete piles.


1.3 Goal and Objectives

The main goal of this thesis is to establish a sound three-dimensional numerical model of the spliced area for driven prefabricated concrete piles, which is verified with existing real load tests.

The project can be divided into four major parts:

1. Evaluation and application of appropriate and applicable constitutive models that reflect the expected behaviours of each material. Establishment of a method for generation of the numerical model of the immediate area surrounding the splice and the adjacent concrete.

2. Establishment of a model where the non-linear behaviour of reinforced concrete and the bond mechanism between concrete and the reinforcement is appropriately reflected, and apply the model in FE-analyses of simple load tests using the commercial FE-program, LUSAS.

3. Establishment of numerical models of the immediate area surrounding a driven prefabricated reinforced concrete pile’s splice connection.

4. Verifying the models by comparison of the empirical load test results and the results from the FE-analyses.

1.4 Scope and Limitations

This thesis work concentrates on the analysis of driven prefabricated reinforced concrete piles. There are many types of piles available, but this pile is the most common pile used today by Skanska Sverige AB and within Swedish industry, and is therefore studied in this analysis. However, standard pile dimensions are not used in this study due to the fact that earlier empirical tests use piles with non-standard dimensions. The dimensions of the studied pile are 400 mm in width and breadth.

The ABB-splice, known as “Axiell Belastad Betong” splice in Swedish, or Axially Loaded Concrete splice in English, which connects these prefabricated driven concrete piles, is the only splice analysed in this study for the same reasons as mentioned above for the piles. Since the immediate area surrounding the splice is assumed to be the area of weakness, i.e. failure, it is concentrated on in this analysis.

Due to the fact that in the empirical tests of the piles, where the structures are only loaded with a bending moment, the axial forces that occur in real loading situations are not taken into consideration. Therefore, only a bending moment is assumed as a load case in this study. Hopefully, in the future, this model can be further tested with an axial force. It should also be pointed out that only static load cases are analysed.

The analyses are carried out in the FEM-program LUSAS, where the implemented features, such as the material models for concrete, steel and the bond mechanism, are used. Limitations in this program may affect the accuracy of the results of the


analyses. The improvement of material models is, however, outside the scope of this thesis.

This thesis has also been limited by two factors outside the control of the researchers. The capacities of the computers, on which the analyses have been processed at Skanska Teknik, control the speed of the analyses. As well, the number of available LUSAS licenses at Skanska Teknik is limited, which limits the number of processed analyses, and therefore affects the scope of this thesis.

1.5 Outline

The different methods used during the course of this Masters Thesis, such as, the way in which information was gathered, knowledge obtained and results processed, are explained in Chapter 2. Chapter 3 describes the production methods of piles and the methods used today in the piling process. This chapter also explains the function of piles in foundations, problems encountered with piles and earlier test cases and results from studies made on piles.

The underlying theory describing the characteristics and behaviour of the individual materials, which prefabricated concrete piles consist of, is found in the beginning of Chapter 4. Constitutive modelling, which explains the theories of elasticity, plasticity and fracture mechanics, is described next in Chapter 4. Finally, this chapter discusses the theory behind the material models in LUSAS that are used in this analysis.

Once the necessary theory concerning the different materials and material modelling has been described, the Finite Element Method, as well as the relationships between FEM and non-linearities, and FEM and reinforced concrete, is described in Chapter 5. The commercial FE-program, LUSAS, is also described in this chapter, together with the non-linear solution procedures found in LUSAS.

The developed FE-model used to model the spliced pile is described in detail in Chapter 6, whilst also explaining the decisions and assumptions that were made concerning the model. Alternative models, which also were established, for example, in order to lower the computational time, are then presented. Chapter 7 focuses on the analyses and verifications of the different parameters, methods and models applied in this study. Finally, in Chapter 8, the established model is discussed and proposals for future applications and improvements are made.


2 Method Once the goal and scope of the study have been determined, the next major question asked is “How is this goal reached within the defined scope?” The answer to this question is “with a well defined, methodical, but somewhat flexible, strategy.” In creating a well-structured and thorough report, it is important, among other things, to have a structured method, which is discussed in this chapter.

Firstly, the relevance of field trips conducted at two factories, where prefabricated concrete piles and their components are manufactured, is briefly explained. The manner in which discussions and consultations with the supervisors, and other experts within the relevant fields, is taken up next. In-depth literature studies, concerning finite element theory, piling and LUSAS, are also illustrated in this chapter. Next, the programs used to calculate and present the study, are exemplified. Finally, the strategy involved with building a numerical model is discussed and illustrated with a flow chart.

2.1 Study of Piling Procedures and Techniques

Understanding the fundamentals regarding prefabricated piles begins with the manufacturing process. In early October 2004, a field trip was made to Skanska Sverige AB’s pile factory in Fjärås, and splice factory in Kungsbacka. The aim of this trip was to acquire information and knowledge concerning the composition, dimensions and production process of prefabricated driven reinforced concrete piles and ABB-splices. The factories’ respective managers conducted tours of each factory. Skanska Teknik’s pile expert, Gunnar Holmberg, was also on hand to expand on the technical aspects involved with piling. Further discussions with Gunnar Holmberg were also conducted during the course of the study.

2.2 Discussions

During the study, ongoing discussions have been made with numerous experts within the fields of piles, reinforced concrete, FEM and LUSAS. As mentioned earlier, Gunnar Holmberg has helped with any questions concerning the subject of piling. Marcus Davidson, LUSAS expert at Gothenburg’s department of Skanska Teknik, has been available in assisting with questions concerning the FE-program, LUSAS. Information concerning certain specific details in LUSAS has also been obtained by correspondence with LUSAS’ support team in the United Kingdom. Queries concerning in-depth reinforced concrete and FEM-theory have been discussed thoroughly with Skanska Teknik’s reinforced concrete and FEM specialist P.O. Svahn, during the course of the thesis.


2.3 Literature and Software Studies

Studying such topics as FEM and non-linear analyses of reinforced concrete structures requires specific and detailed information. How this information is gathered is explained in Section 2.3.1. Implementing this information in a numerical model cannot be done without first becoming familiar with the FE-program. The process of learning LUSAS is illustrated in Section 2.3.2.

2.3.1 Literature Study

An extensive literature study, from information regarding the basics about pile production, to standards and codes, to detailed and specific theory concerning non-linear bond-mechanism models, has been carried out. The sources of the information necessary to conduct this study are, for example, books, codes, Masters Thesis and PhD. Thesis, which have been from borrowed from personal collections, Skanska Teknik’s library, the Department of Structural Engineering Mechanics’ library, as well as Chalmers’ own library. The LUSAS manuals available online and other Internet sources are also used as sources to find suitable information and literature.

Literature that was required in order to perform this study included material concerning the theories of FEM, non-linear numerical solution methods, material modelling, reinforced concrete and piling.

One source that is studied thoroughly is the lecture notes handbook, Numerical Modelling of Deformation and Failure of Materials (Jirásek 2000). In combination with the LUSAS manuals, this handbook is a source of good and relevant information concerning numerical modelling. In addition to these sources, Karin Lundgren’s PhD. Thesis (Lundgren 1999), has also been studied and analysed in-depth.

2.3.2 LUSAS study

Since the researchers had no prior knowledge or experience of the finite element program, LUSAS, before this study began, it was a necessity to become familiar with LUSAS from day one. Simple examples concerning model building were continually carried out during the first weeks of the study period, whilst at the same time literature studies were made. LUSAS’ online help manuals (LUSAS 2004), which consist of, for example, theory and solver manuals, were constant partners in the LUSAS learning process.

The next step, after achieving a basic understanding for LUSAS, was to study the different types of elements and material models available in the program, and subsequently, narrowing down the available material models to a few specific models that best suit this particular study. Again, the LUSAS manuals were of great help regarding the explanation of the theory of the program. Element types and material models were chosen for concrete, reinforcing steel and the bond-mechanism model, and these models were accordingly tested separately. The tests were conducted to distinguish whether the material models responded as expected.


LUSAS’ theory was also compared with pure finite element theory, as well as theory relating to different types of material models and numerical solution methods.

2.4 Software

There are many different types of software used in this study. LUSAS 13.6-2 is principally used to conduct the finite element calculations. Microsoft Word and Microsoft PowerPoint are used to illustrate the study in both paper and presentation form. MathType, a program for expressing mathematical equations, is a supplement program to Microsoft Word. Another Microsoft Office attribute, Microsoft Excel, is used to calculate and illustrate results.

2.5 Strategy

Considering the restricted scope of a Master Thesis, the task is approached with regards to LUSAS’ capabilities, i.e. the models are established according to the program’s limitations. A more scientific approach would be to identify the features of the task and then investigate which FE-program is most suitable for the problem. However, with regards to the extent of the study, and the limited access to calculation programs, the chosen approach is evaluated as the most reasonable.

The numerical models are established in an iterative, stepwise manner, where all the components of the models are analysed and tested separately, and successively assembled into a global model, as illustrated in Figure 2.1. The main reason for adopting this procedure is due to the numerous numbers of different parameters that influence the analysis. Each parameter needs to be thoroughly investigated until an estimate for the parameter can be identified. The “most correct” parameters are then tested in isolated cases, until a more exact value can be confirmed. This is a basic but important condition for the study to succeed. The parameters are then verified with confirmed data, for example laboratory tests, codes and other analyses.


Figure 2.1 Flow chart illustrating the process involved in building a numerical model

The confirmation of the parameters is an ongoing process, which begins at the individual material models’ level and continues into the analysis of the numerical pile model. Other examples of basic features, which are analysed and verified in the first stage, include element type, incrementation method and material models, as seen in Figure 2.1.

The results of the first stage of verifications are applied in, for example, the individual reinforcement, concrete and bond-mechanism models. The model attributes, material models and solution methods are tested on a more global scale and also verified. An example of the verification process at this level is the confirmation of a reinforced concrete model’s behaviour, under a tensile load, in comparison with earlier tests made by Jonas Magnusson (Magnusson 2000).

Finally, the tested and verified factors are gathered and applied in the global numerical model of the pile element. This model is compared with earlier tests results conducted by, SP, Swedish National Testing and Research Institute.


3 Piling Techniques, Theory and Procedures Pile is a little word that greatly influences, and positively improves, the standard of living of the human race. Without piles, building possibilities would be extremely restricted. There are a large variety of piles available today, but one standard pile is commonly used in Sweden, as explained in the beginning of this chapter. As well, a general description of the function of a pile is explained in this section, together with the different load capacities associated with piles.

The production details of creating an ABB-splice and reinforced concrete piles are outlined next. Subsequently, the methods and intricacies involved with driving the piles into the ground are discussed. The technical responses created from a static load, so-called load effects, and the predicted load capacity, in terms of an axial load versus bending moment relation, are also considered in this chapter. After that, the known problem areas and mechanisms of pile elements, and also possible areas/mechanisms that might influence the behaviour of piles, which play a significant role in this study, are explained. Finally, the conditions, tests and results ensuing from laboratory studies of piles loaded with bending moments are discussed.

3.1 Piles

Geological and geo-technical conditions, construction types, climate, tradition, culture, plus building standards, contribute to the fact that different pile systems are used in different countries. Some examples of the most common types of piles used in Sweden are prefabricated driven concrete piles, driven wooden piles, and combination and steel piles. A few of these examples are illustrated in Figure 3.1.

Figure 3.1 Examples of concrete, timber and steel piles (Broms 2005)


Today, within the Swedish building industry, prefabricated driven concrete piles are referred to, quite simply, as concrete piles. Concrete piles represent approximately 60-70 % of piles installed in Sweden, as illustrated in the diagram in Figure 3.2.

Figure 3.2 Statistics for driven piles in Sweden (Pile Statistics for Sweden 2003)

A prefabricated driven concrete pile is a slender construction element that is inserted somewhat vertically inclined into the ground. A piled foundation is a system consisting of piles, where pile caps, slabs and beams are also normally included in this system.

The main function of a pile is to transfer and/or spread vertical and horizontal loads, from a superstructure, through weak or unstable soil layers, to more stable soil or bedrock. Piles can be loaded with many different types of loads and many different types of load combinations, as in Figure 3.3. A pile’s function is to increase the load carrying capacity, and/or reduce settling of a superstructure, by improving the foundation conditions. One distinguishing factor of piles is that they can bear both tension and compression forces.


Figure 3.3 Examples of different load conditions for piles (Broms 2005)

Piles are not only used in bad soil conditions, but even for economical and technical motivations. Piles are even suitable as foundation elements for constructions in water, such as, bridge, quay or pier supports.

Piles are divided into two families, according to the way the load is transferred into the ground. One category of piles are so-called end-bearing piles, as seen in Figure 3.4 (a), in which the load is transferred through the pile down to the bedrock, whilst the other category, friction piles, Figure 3.4 (b), transfer the load between the skin of the pile and the surrounding soil as shear forces. All piles are, however, not always categorised as end-bearing or friction piles, but as something in-between. For example, the load can be partly transferred to the bedrock by the pile’s end and partly to the surrounding soil by the pile’s skin.

Figure 3.4 Load transfer function of (a) an end-bearing pile and (b) a friction pile (Holm and Olsson 1993)

When dimensioning piles, it is quite important to be aware of the distinction between the terms geo-technical bearing capacity and structural capacity. The term bearing


capacity, Figure 3.5, defines the maximum load that the pile can carry and/or transfer before failure occurs in either the pile or in the soil/bedrock. Whilst load capacity, Figure 3.5, defines the actual strength of the pile-structure, including features such as splices and rock shoes, before, during and after installation. The term structural capacity is discussed further in Section 3.4.1.

Figure 3.5 Definitions of structural capacity and geo-technical bearing capacity of a pile (Holmberg 2001)

3.2 Production of Pile Elements

In Sweden, concrete piles are manufactured indoors in factories. Prefabricated concrete piles, depending on the region where production occurs, can differ slightly in composition. A standard pile in the clay-ridden city of Gothenburg can be slightly different to the standard pile manufactured in the bedrock rich city of Oslo. A pile’s dimension and composition are often based on the environment it shall be placed in, as well as the customer’s preference.

A standard concrete pile element’s length can vary between 3 - 13 metres, where specially manufactured piles can be as long as 20 metres. Concrete piles consist of several elements, assembled by special splice devices. These steel splices have, at the very least, the same strength as the concrete cross-section. The splices connect two pile elements so as they work together as one. Today, this is the most established method known and used in emulating a pile continuum.

Design Load

The structural capacityincludes the effects of theinitial deflection of the pileand the buckling of the pilein the surrounding soil.

The geo-technical bearingcapacity of the pile is theload that the pile cantransfer to the sub-ground.


Piles manufactured in Sweden contain non-tensioned reinforcement, whilst, for example, in the USA, pre-stressed reinforcement is used. Piles are cast with high strength concrete, such as C45/55 quality concrete or higher. The cross-section is normally four-sided, but can also be cast in other shapes, Figure 3.6. The standard dimensions are a square shaped cross-section with 235 and 270 (275) mm side lengths. However, specially manufactured piles can have side-lengths up to 500 mm long. Piles that are driven down to bedrock have so-called rock shoes mounted on the ends that come into contact with bedrock, Figure 3.6.

Figure 3.6 A prefabricated concrete pile with examples of different cross-sections (Broms 2005)

The first stage in the manufacturing of prefabricated concrete piles is the fabrication of the splice mechanism. The most common splice produced and applied these days is the ABB-splice. A standard ABB-splice consists of four reinforcement bars, two locking devices and two male connections, as seen in Figures 3.7 (a) and (b).

Figure 3.7 (a) The ABB-splice’s components and (b) the connection joints of ABB-splices


Figure 3.8 The locking device and lock pin used to lock the splices together, with the male and female connectors

The production phases of an ABB-splice is as follows:

1. The bottom plates of the splice are produced and machine-stamped with four holes. Sincethe plates are not completely flat after they have been stamped, they are “flattened out.” Itis extremely important to have a flat and straight bottom plate when splices are joinedtogether.

2. The “splice walls ” are made from “cut” rolled steel and then carefully bent and weldedinto a four-edged form.

3. The locking devices are produced of rolled steel, machine-drilled, and then carefullyinspected by hand.

4. The lock pins are produced by a lathe-machine with rolled, high strength steel. The lockpins are prestressed in the locking devices, passing through the male connection, asillustrated in Figure 3.8.

5. The locking devices and its corresponding reinforcement, plus the male connection and itscorresponding reinforcement, are welded together with a special method known asfriction welding. The friction welds can be seen in Figure 3.7 (a).

6. A robot then welds together all of the above mentioned parts, consequently producing theABB-splice.


Figure 3.9 (a) Concrete piles’ reinforcement cage before casting in concrete and (b) the casting of the piles

Figure 3.10 (a) The drying/hardening of the piles and (b) examples of rock shoes

The concrete pile elements are then manufactured in the following order:

1. The reinforcement is manufactured with stirrups in-situ with B500 steel, as seen in Figure 3.9(a).

2. The reinforcement, ABB-splices and rock shoes are lifted into the casts, and the concrete is castwith a concrete cover that is normally around 30 mm, Figure 3.9 (b). The splice’s fourreinforcement anchorage bars are bent inwards so as they fit inside the pile’s four reinforcementcorner bars. This creates an area of the pile that contains two reinforcement bars over the lengthof the anchorage reinforcement, Figure 3.11.

3. Once the concrete is cast, it is then transferred to a hardening area, Figure 3.10 (a). Hardening ofconcrete piles is carried out with help of either steam or warm water so as to reach the desiredconcrete strength needed to lift the piles out of the casts.

4. The piles are lifted out after one day of drying/hardening, when they have the equivalent strengthof approximately 20-30 MPa, Figure 3.10 (b).

5. After seven days, and strength of approximately 48-51 MPa, the piles can be loaded for transportand are subsequently transported to the building site.


Figure 3.11 A simple figure illustrating how the splice’s reinforcement, i.e. anchorage reinforcement, after bending, is placed inside the main reinforcement

3.3 The Piling Process

Once the piles have been manufactured, the next step is transporting the pile from the factory to the building site. How the pile is handled, i.e. lifted, moved, etc, and transported, has a critical influence on the response of the pile, before, during and after hammering. For example, a pile is influenced during the handling process by a dynamic load.

When the piles are delivered to the building site, the piles are driven into the ground with a piling machine. A detailed picture of a piling machine is illustrated in Figure 3.12 and an example of a piling machine on site is illustrated in Figure 3.13. The pile machine’s hammer strikes a “drive cap,” which is placed between the hammer and pile. A mast, known as a guider, controls the inclination of a pile’s insertion into the ground. The maximum inclination is 4:1, both forwards and backwards.


Figure 3.12 An example of a pile driving machine and its components (Holm and Olsson 1993)

The main difference in piling machines available today are the different types of driving hammers that can be used to drive the pile into the ground. There are numerous types of hammers, for example, wire and hydraulic driven hammers, and vibration, air and diesel hammers, where each has its own advantages and disadvantages. The most common hammer used in Sweden is the hydraulic driven drop hammer, and there are also different varieties of this, with the difference being the height where the hammer is released and the weight of the hammer.

The hydraulic driven drop hammer works as a free falling weight. The “hammer weight” is lifted with a hydraulic cylinder to a prescribed height. The cylinder is then drawn in faster than the hammer falls, i.e. the hammer falls freely. The hammer can weigh between 3 to 5 tons. It should be noted, that different types of hammers have different methods of drawing in the cylinder.

The drive cap acts like a spring between the pile and hammer, whilst also helping to centre the strike on the pile. This spring like action helps to avoid fractures in the top of a pile. A drive cap normally consists of 50 -75mm of plywood.


An important factor with the driving of piles is that the hammering is stopped when the pile element has the required capacity to carry the load. This is completely dependent on the surrounding conditions, weight of hammer, etcetera.

Figure 3.13 A pile machine driving experimental tests (Holmberg 2001)

Once a pile element has been driven into the ground a new pile element is mounted to the driven pile. The piles are connected by the ABB-splices, which are placed on the end of the concrete piles. The splices are connected by the insertion of so-called lock pins into the locking devices, according to Figure 3.8.

Another phenomenon also occurs in the spliced zone, due to the prestressing created from the lock pins, as illustrated in Figure 3.14. When the pin is driven into the dub and locking device, the hole into which the pin is inserted becomes marginally larger. When the pile is driven, this increase in the diameter of the hole allows for a small deformation in the hole in the load’s direction, back and forth, in relation to the driving load. This back and forth motion contributes to a slight opening in the locking device, Figure 3.14, contributing to an initial slip when the pile is loaded with a moment. For more information about how this influences the model, see Chapter 6.

Figure 3.14 The locking device with small gaps induced due to the force of insertion of the locking pin


3.4 Load Effect

A pile that is axially loaded with a compressive force has a load effect that consists of a combination of a compressive normal force, a bending moment and a shear force, as illustrated in Figure 3.15. This occurs due to the initial curvature and deflection of the pile. This initial curvature and deflection causes even an extra bending of the pile when under the earth’s surface. This extra bending creates a compressive side resistance in the soil on the pile due to the subsoil reaction, which is illustrated as q in Figure 3.15.

Figure 3.15 A pile loaded with an axial compression force (Commission of Pile Research 1998)

The bending moment in the pile increases non-linearly with an increasing axial load. The pile’s “sideways” deformation should be observed carefully. When the deformation exceeds the threshold value that causes the soil to become plastic, the bending moment of the pile increases rapidly.

The relationship between the load effects, i.e. normal force and bending moment, in a pile can be presented graphically with so-called load effect curves, as seen in Figure 3.16. These curves depend upon the piles material, cross-section, stiffness and curvature, plus the soils “side resistance.”

The curves in Figure 3.16 illustrate the load effect curves for concrete piles installed in differing soil qualities with different “side resistances.” Curve L1 represents soft soil whilst curve L2 corresponds to compact soil. Curve a) and b) are the boundaries for when the soil begins to perform plastically.


Figure 3.16 The load effect curve for concrete piles in the ultimate limit state, based on the M vs. N curve from the Commission of Pile Research (1998)

A pile’s load capacity, with respect to the compressive normal force, decreases when a bending moment is acting in combination on a pile, and therefore, the capacity must be calculated with the influence of this phenomenon, according to the Commission of Pile Research (1998). This leads to the fact that an axially loaded pile’s load capacity for normal forces, acting together with the influence of bending moments, must be associated with the values of these bending moments.

This can be illustrated with a cross-sectional control, which ensures that the most strained cross-section of a pile can handle the load effects without exceeding the pile’s cross-section capacity. The piles cross-section capacity for normal forces and bending moments, for piles manufactured of different materials, is illustrated graphically in Figure 3.17 with so-called cross-sectional capacity curves.


Figure 3.17 The cross-section capacity curves in the ultimate limit state, based on the M vs. N curve from the Commission of Pile Research (1998)

3.5 The Problem at Hand With Piles

Slender concrete piles are primarily designed to carry axial compressions, i.e. the stresses that appear due to the axial driving forces needed in the installation of the piles. Piles are also designed for all the load effects that arise due to the handling, transportation and driving of piles. Due to many varying factors, e.g. the bending stiffness of the pile, unintended imperfections arise before, during and after the piling process. These so-called unintended imperfections can result in curvature of the pile elements and/or angular deviation in the joint/s, which lead to bending moments in the pile, as in Figure 3.15.

A pile can be considered as a column, after the installation process, when the pile is surrounded by an elastic medium, e.g. soil. Consequently, the behaviour of a pile, when exposed to an axial force, is not only dependent on it’s own mechanical properties, i.e. bending stiffness and compressive strength, but on other factors, such as, the initial deflection of the pile and the modular of sub-grade reaction of the soil.

The splice in a spliced pile plays also an important and more complicated role in the behaviour of the pile structure. The splice influences both the initial deflection and the bending stiffness of the pile. The angular deviation of the joint influences the initial


deflection of the pile, whilst both the joint’s rigidity against rotation and bending stiffness of the anchorage zone influences the bending stiffness of the pile. In determining the structural capacity of a spliced pile, subjected to an axial force, as suggested by Holmberg (2004), it is necessary to take respect to:

• Axial forces and bending moments, before, during and after installation.

• Bending stiffness of the pile.

• Modulus of sub-grade reaction of the soil.

• Angular deviation in the joint after the installation of the pile.

• Bending stiffness of the jointed area after the installation of the pile.

When dimensioning concrete piles and splices, and thus the moment capacity for these structures, it is vital that the designer is aware of the applied axial loads. The resulting bending moment increases, due to imperfections, with an increasing applied axial force, i.e. the moment increases with increasing axial force. The capacity of the relationship between the normal force and bending moment can be described with a curve, as earlier illustrated in Figure 3.17. The curve represents a capacity boundary for the combination of normal forces and bending moments. If the normal force-bending moment relationship falls outside of this curve, the structure fails. It should be noted that the estimated bending stiffness of the spliced area of a pile is different for different load combinations. (Holmberg 2004)

However, a spliced driven prefabricated reinforced concrete pile element exposed to axial forces and bending moments, does not, unfortunately, behave as predictably as, for example, a simply supported beam. The imperfections that come hand in hand with the inclusion of the spliced connection, and, for example, the influence of surrounding soil, do not give a response that is as pure as a simply supported beam’s response. Therefore, it would be interesting to see if this N-M relationship, which is based on calculation tests, can be confirmed or denied.

It is known that the driving load applied to a pile influences the pile’s capacity in a negative way, for example, cracking in the concrete occurs in the spliced area as well as tensile cracks predominantly occur in the rest of the pile. Fatigue’s effects do not however influence the calculations for the axial load and bending moment capacities of critical sections of a pile and a joint. Nevertheless, it can be rather complicated trying to estimate the effects of the impact load during installation on a joint’s behaviour, capacity and structure, e.g. prestressing, gaps and welded parts in the joint, and stress concentrations in the concrete caused by the joint. (Holmberg 2004)

The repeated and large load effects which a pile is exposed to during driving can result in a certain amount of fatigue in the pile. The effect of the driving load shall be taken into account in pile design and one way that the effect of the driving load is considered is with a reduction factor.

The reduction factor, lccµ , takes respect to the driving load’s influence, and is determined with respect to the number of driving hits, the hammer’s drop height, arising load effects during driving and the pile material’s strength. This type of


reduction factor is determined by patterns and is used in combination with concrete’s compressive strength for concrete piles (Commission of Piles 1998). Since the actual influence of the installation process on a pile’s material characteristics is hard to determine, and still relative unidentified, this reduction factor can still be improved with increased knowledge of the effects from the installation process.

According to the stress wave theory, the effects from the driving load are substantial. The impact of the hammer on the top of the pile creates a stress wave that propagates in the pile with an acoustic speed. When the wave reaches the pile tip, it will be reflected upwards again. The character of the reflected wave depends on the resistance of the soil/rock material surrounding the tip: high resistance gives compressive reflection, whilst low resistance gives tensile reflection.

In the extreme case when the pile tip is free, as illustrated in Figure 3.19, the compressive pulse, when reaching the pile tip, is reflected as a mirror image tensile pulse, thus creating tensile cracks in the pile. When this tensile pulse reaches the pile top against which the hammer is still resting, it is reflected as a compressive pulse travelling downwards.

When the pile tip rests against a rigid bottom, such as bedrock, as illustrated in Figure 3.18, the compressive pulse is reflected as an identical compressive pulse travelling upwards in the pile. When this pulse reaches the hammer, it pushes the hammer upwards and hammer rebound is obtained, whilst part of the pulse is reflected downwards. During the first reflection at the bottom, the initial compressive force and the reflected compressive force are superimposed, thus leading to a maximum compressive force against the rigid base. This maximum compressive force is twice the maximum force in the travelling pulse, which leads to a doubling of the compressive force at the tip. The concrete in these ends have therefore very high stress concentrations and can be exposed to severe cracking. (Commission of Pile Research 1984)

It should be noted that, only in Figures 3.18 and 3.19, the plus sign resembles compression and minus tension.

Figure3.18 Stress-wave analysis, rigid base (Broms 2005)


Figure3.19 Stress-wave analysis, soft base (Broms 2005)

Apart from the damages caused by the driving load, piles are also negatively affected when being handled and transported, by both a moment and a dynamic load. The bi-effects are predominantly bending cracks in the concrete but they can also be hard to estimate. According to Holm and Olsson (1993), the dimensioning criteria concerning bending cracks, which arise due to the handling and transporting of piles, should not be greater than 0.4 mm in breadth.

The cracking that results from pre-loading factors, i.e. handling, transportation and driving of a pile, significantly influences both the concrete’s behaviour and the bond-slip relationship between concrete and reinforcement. These effects can be assumed to affect the concrete’s parameters such as elastic module, tensile strength and fracture energy with 10-30%, as well as considerably affecting the bond-mechanism’s cohesion strength.

An acknowledged internal mechanism that is produced when a spliced pile is loaded with a bending moment is the separation of the splice components, as illustrated in Figures 3.20 and 3.21. Even before the pile is loaded, a small gap can already exist between the two splice elements, due to the driving load. This gap is accentuated with the application of a bending moment. The consequences of this phenomenon are known to be negative, and assumptions exist concerning what occurs, but what really happens is still unknown.

Figure3.20 A simple model of the slippage in the splice area due to driving


Figure3.21 An exaggerated but simple model of the slippage in the splice area due to a bending moment after hammering

Another big grey area is the bond-slip mechanism that occurs between the splice’s anchorage reinforcement and the pile’s principal reinforcement, also known as splicing. In this part of the pile, forces are transferred from the anchorage reinforcement to the principal reinforcement, which create zones with large stress concentrations. The immediate area surrounding the locking device is another area in which not much is known. Due to the locking device’s sharp edges and the effects from the driving load and the resulting bending moments, it is assumed that the accumulation of stresses in the compressive area of the pile result in crushing. It is possible that this mechanism affects the global response of the pile but unfortunately this mechanism is outside the scope of this study.

3.6 Earlier Test Cases and Results

When predicting moment capacities for a spliced pile it is important to validate the calculations with full-scale tests. It is recommended that bending tests of spliced piles be carried out on piles previously exposed to driving tests. The tests should include measurements of the bending stiffness of the jointed section of the pile and the rigidity against rotation within the joint. There are three known methods to dimension spliced piles based on tests. These methods are:

• Tests according to recommendations from SBN Godkännanderegler 1975:8, which is further explained in this chapter.

• Tests with spliced piles in column press with an eccentricity, from CEN (European Committee for Standardisation).

• Tests with horizontally lying spliced piles, loaded with a bending moment combined with an axial normal force. This type of test was developed at Skanska Teknik and Skanska Grundläggning and was conducted by SP. (Holmberg 2004)


SP, employed by Pålgruppen AB, carried out two documented test cases of loaded piles in December 1991. The tests were carried out according to SBN Godkännanderegler 1975:8. The aim of these tests was to try and simulate the bending moments that the piles are loaded with once they have been hammered into the ground, so as to determine the capacity and the response of the structures.

The first round of tests involved six different types of piles, which consisted of different pile dimensions, reinforcement patterns and ABB-splices. The second round of tests, a complement to the first test case, tested three of the earlier tested pile types again.

The piling tests included two 3 m long piles, connected by a splice system, with the total length being 6 m. The piles were vertically driven down to bedrock, which was at a depth of 0.5 m under the earth’s surface, before being further driven 3000 times, thus simulating the procedure used in reality. The two pile elements were shortened to a length of 1.5 meters and then placed horizontally on supports. The splice was centrically placed between the supports, simulating a simply supported system, and loaded with a bending moment, as seen in Figure 3.22. The eventual deflection is placed faced down. For more information concerning these tests see Appendix A1, A2 and B1.

Figure 3.22 The performance of the tests according to SBN 1975:8

The tests were carried out with a cubic strength of 60 MPa, 8Ø20 Ks600s for the main reinforcing steel and 4Ø20 Ks400s for the anchoring steel. In the first test, 1991-12-05, the concrete strength is 60.5 MPa. In the second test, 1991-12-18, the cubic strength is 63.7 MPa. The static load was continually increased until failure was reached. The load and bending deformations were registered during the tests. On the basis of these recorded results, the maximum moment and maximum bending stiffness for the corresponding moment was calculated.

Unfortunately, the types of failure and crack patterns were not recorded when the testing was conducted. It can be assumed that the piles were handled and transported, and consequently bending cracks are assumed to have been present. When the piles were driven 3000 times, they were resting on bedrock, and consequently tensile cracks are also assumed to have been present before the bending moment was applied.


Although tests were carried out on various reinforcement patterns, this study concerns only the reinforcement pattern, i.e. pile type, and the splice seen in Figures 3.23 and 3.24. Therefore, only information and results concerning this pile and splice type and reinforcement pattern referred to in this study.

Figure 3.23 Reinforcement arrangement of the tested pile, with a concrete cover of 50 mm

Figure 3.24 Sketch of the tested splice, where two of the four locking devices are illustrated


4 Materials and Material Modelling The composite structure, reinforced concrete, consists of several different materials, that is, concrete and reinforcing steel, plus a bond mechanism. Each material has its own characteristics and behaviour, as illustrated in the beginning of this chapter. These different responses are then described in detail with proven theories, that is, elastic and plastic theory, as well as fracture mechanics. Defined material models for concrete, reinforcing steel and the bond mechanism, which apply these proven theories, are available in the FE-program, LUSAS. The description of the underlying theory of the three applied material models round off this chapter.

4.1 Materials

Since concrete piles consist of both concrete and reinforcing steel, it is important to consider these materials’ individual constitutions, characteristics and behaviours before further studying the combined structure of concrete and reinforced steel. Furthermore, modelling of a reinforced concrete structure is performed with a conglomerate of different material models, thus solidifying the need to explain each material in detail, as outlined in the following sections.

4.1.1 Concrete

Hardened concrete is a heterogeneous material consisting of cement paste, aggregates and water. However, a macroscopic analysis of concrete, that is assuming that concrete is homogeneous, is best presumed when examining concrete structures. Thus homogeneity, as well as isotropy, is assumed until cracking begins.

Since concrete has two main material response phases, that is, before and after cracking, different material models are used to describe concrete’s behaviour. Linear elastic models are generally used to illustrate the concrete’s behaviour before cracking. Whereas, crack models based on fracture mechanics and plasticity models are frequently used to model the non-linear behaviour of concrete after cracking.

Concrete’s failure modes are quite dissimilar depending upon the stress states, i.e. compression and tension, as can be seen in Figure 4.1.


Figure 4.1 A typical uniaxial stress-strain response for concrete (Plos 2000)

Although the failure modes are unalike, failure is induced by micro-cracks in both stress states. During compression, the micro-cracks remain well distributed until failure, whereas during tension, micro-cracks localise into narrow zones and eventually connect to real cracks. There are numerous theories used to model the behaviour of concrete, but generally, plastic theory is used to model compressed concrete whilst concrete in tension uses fracture mechanics.

4.1.2 Steel

Steel has, characteristically, a linear elastic behaviour between a zero stress state and the yield stress state, and thereafter deforms plastically, whether it is in compression or tension, as is visualised in Figure 4.2. Therefore, a combination of linear elastic and plastic material models are normally used to model steel’s behaviour.

Figure 4.2 A typical uniaxial stress-strain response for steel (Plos 2000)

4.1.3 Reinforced Concrete

Both steel and concrete, materials with different material behaviours, as mentioned earlier, combine to create the composite reinforced concrete. Ordinary reinforcing steel has a compressive strength approximately 15 times greater than common structural concrete, and a tensile strength greater than 100 times concretes tensile strength. Although steel has much better strength properties than concrete, arguments


against using steel alone, such as the high cost of steel, property changes under high temperatures, corrosion, etcetera, lead to the hypothesis that a good solution is the combination of both materials, i.e. reinforced concrete. The combination of these materials produces a material that has good qualities to resist such forces as compressive, tensile and tensile-bending stresses.

A reinforced concrete beam in bending can be used to exemplify reinforced concrete’s behaviour. There are three distinct stages in the load versus deflection response, that is, the uncracked linear elastic, the cracking and the plastic stages. The cracking of the concrete stage and the yielding of the steel and crushing of the concrete stage cause the non-linear response in the composite material’s behaviour. Other non-linearities can also arise due to, for example, bond-slip between the surrounding concrete and reinforcing steel.

4.1.4 Bond Mechanism

Within reinforced concrete, the bond mechanism is defined as the interaction between reinforcement and concrete. This interaction allows for a transfer of stresses between the two materials, thus contributing to the possibility that both the materials’ strength capacities, i.e. concrete-compression and steel-tension, are utilised. This bond mechanism has therefore a strong influence on the fundamental behaviour of a structure, for example, on crack development, spacing and width, plus ductility of the structure (Lundgren 1999).

In a purely physical and somewhat simplified perspective, the ribs on the reinforcement bar constitute the bond mechanism. The ribs transfer the main part of the stresses between the reinforcement and the surrounding concrete. As can be seen in Figure 4.3, the shape of the ribbing leads to a resulting force, known as traction, which can be divided into a shear and a normal component.

Figure 4.3 Contact stresses on a deformed bar embedded in concrete (Plos 2000)


4.2 Constitutive modelling

Different materials have different material models, which consist of, for example, elastic or plastic behaviours and linear or non-linear responses, and furthermore, there are different theories, which describe these different behaviours/responses, as outlined in the following sections.

4.2.1 Elasticity

Almost all engineering materials exhibit a certain amount of elasticity. Elasticity is defined with the assumption that, if the external forces that are producing deformations are removed, the deformations will disappear. Linear elasticity is the simplest constitutive theory and Hooke’s law is, in tensor notation, obtained as

σ = E : ε (4.1)

4.2.2 Plasticity

Plasticity material models allow a material to deform plastically, that is, when Hooke’s elastic law is no longer valid. A constitutive plasticity model consists of three conditions:

1. Yield condition

2. Flow Rule

3. Hardening Rule

The material model is initially assumed to have a linear elastic response. A region in the stress space, which is bounded by a yield surface, defines the linear elastic stress space. This so-called yield surface also defines when the deformations become plastic. Once the stress state reaches the yield surface, plastic deformation starts, i.e. the stress state lies on the yield surface, and the change of strain is divided into an elastic part,

eε , and plastic part, pε .

e p= +ε ε ε (4.2)

If the stress state remains inside the elastic domain, i.e. within the yield surface, the deformation process is purely elastic and plastic strain does not occur. Plastic flow starts when the stress state reaches the yield surface. During plastic flow the stress state remains on the yield surface.

There is, however, no distinctive relationship between a stress state and its corresponding strain state due to the non-linear elastic-plastic stress-strain material response. The correlation between a stress state and its strain state depends on both the loading history and the history of the stress-strain state in the current material


point. This non-linear material response is generally illustrated with an incremental approach so as to capture this non-linear behaviour. The strain shown in equation 4.3 is now incremented, together with its elastic and plastic components.

e p= +ε ε ε (4.3)

The plastic part of the incremental strain, equation 4.4, although also a function of the incremental stress, is a function of the stress-strain history of a particular material point, which includes the plastic strain and the current stress state. The development of the plastic strain is described with the flow rule.

( , , )p pf κ=ε ε ε (4.4)

The hardening parameter, κ, is determined by the history of the evolution of the plastic strain. With an increasing plastic strain, the yield surface changes, and this phenomenon is defined with the hardening rule, i.e. it takes the strain hardening and strain softening effects into consideration.

Yield Conditions

For a 3D stress state, the yield condition stipulates that when the stress state reaches the yield surface in the principal stress space, yielding occurs. There are many different types of yield surfaces, depending on the material wishing to be modelled. For example, typical yield criteria for reinforced concrete is the von Mises yield surface for steel models and the Mohr-Coulomb yield surface for concrete models.

The yield surface is described by a yield function, Φ , which defines an elastic domain in stress space, bounded by the yield surface. The yield function depends on the current stress state and a hardening parameter,κ , which is determined by the history of the evolution of plastic strain.

( ),=Φ Φ κσ (4.5)

If the stress satisfies the yield function, i.e. 0Φ = , yielding occurs. If 0Φ > , the current stresses result in an elastic stress state. It is impossible to get a stress state outside these yield surfaces, i.e. Φ cannot be greater than zero. This is summarised in

the Kuhn-Tucker loading conditions 0Φ ≤ , 0λ ≥i

and 0λΦ =i

, where λi

is the plastic multiplier. Hence we obtain the three different states:

• Elastic loading 0Φ < , 0λ =i

• Plastic loading 0Φ = , 0λ >i

• Neutral loading 0Φ = , 0λ =i


Flow Rules

The flow rule describes the magnitude and the direction of the plastic strain, pε , in any point on the yield surface. The flow rule can either be associated or non-associated. An associated flow rule assumes that the plastic strain develops in a direction that is normal to the current yield surface, for the current stress state, as can be seen in Figure 4.4. In equation 4.6, an associated flow rule is defined, where λ represents the scalar multiplier, which controls the magnitude of the plastic strain, whilst the gradient of the yield function, Φ , with respect to the stress, σ , determines the direction. It can therefore be stated that this flow rule is associated with the yield function.

pΦλ ∂=

∂ε

σ (4.6)

However, all materials do not exhibit such behaviours that an associated flow rule provides. Therefore, for these types of cases, a non-associated flow rule has to be adopted. A non-associated flow rule assumes that the direction of the plastic strain is “non-normal” to the current yield surface, in the current stress state, as can be seen in Figure 4.4. In the associated flow rule, the direction of the plastic flow depends exclusively on the yield function and the stress state, but in the non-associated rule this relation is no longer valid. Therefore, a new function has to be introduced, a so-called plastic potential, ( )Φ σ . Analogously, the gradient of ( )Φ σ provides the direction of the plastic flow, and by using this extra function, it is possible to simulate the sought after behaviour. (Jirásek 2000)

*

pΦλ ∂=∂

εσ

(4.7)

Figure 4.4 Associated and non-associated plastic flow visualised in a Mohr-coulomb yield surface

Hardening rule

Hardening, or strain hardening, illustrates the stress behaviour of a material, even after yielding has occurred. Hardening describes the phenomenon of stresses continuing to increase even after non-linear strains appear. Consequently, the decrease of stresses, after non-linear strains appear, is termed softening.


When modelling hardening behaviour, during plastic response, the initial yield surface is altered with increasing plastic strains. Put simply, the hardening rule determines how the yield surface changes with plastic loading. Theoretically, the hardening rule expresses the yield surface as a function of the current stress state, the plastic strain and a hardening parameter,κ .

Isotropic hardening and kinematic hardening are two of the most common types of hardening theories used. As can be presumed for isotropic hardening, the yield surface expands uniformly. This theory is quite simple to use but does not account for the so-called Bauschinger effect, i.e. that plastic deformation in one direction reduces the plastic strength in the opposite direction. However, a theory that does take this effect into account is kinematic hardening. This theory assumes that the yield surface translates as a rigid body, with no change in size, shape or orientation. (Plos 2000)

4.2.3 Fracture Mechanics

Plasticity theory works quite well when modelling compressive failure of concrete, but is unfortunately not so suitable modelling tensile behaviour and failure. To model the effect of cracking concrete, there are a number of methods available, which are based on fracture mechanics. The easiest way to describe why this type of approach is convenient in simulating the cracking behaviour of concrete is by illustrating a uniaxial tensile test of a concrete specimen. The behaviour of the specimen can be seen in Figure 4.5 as it is loaded until failure.

Figure 4.5 Response of concrete specimen, which is loaded until failure (Plos 2000)

The load-displacement relation cannot directly be translated into a stress-strain relation since different lengths of the specimen would give different stress-strain relations. Instead, the material properties must be subdivided into two parts, one that represents the elastic behaviour outside the fracture zone, and another that represents the behaviour inside the cracking zone, i.e. the crack propagation. The area under the “stress-crack deformation” curve represents the energy that is dissipated during the fracture process and is called the fracture energy, fG . (Plos 2000)


As previously mentioned, there are a number of different approaches, with different pros and cons, to model cracking concrete. However, in LUSAS, the micro-plane model is implemented, which is a variant of the so-called smeared crack approach. This means that the cracking behaviour is smeared out over an integrated volume element and no special elements or predefined cracks are required. This model enables regions with distributed cracking and crack orientations to be identified. However, this approach tends often to result in an overly ductile behaviour when simulating structures as no localisations in the cracking process occur. In order to correct this incongruity, it is sometimes convenient to weaken certain areas where crack propagation is expected. The concrete material model in LUSAS is further treated in Section 4.3.

4.3 Material Models

A material model is an important ingredient in a numerical model’s constitution, along with the parameters associated with the model. For a general overview of how material models influence the creation of a numerical model, see Figures 2.1 and 5.5.

4.3.1 von Mises Steel Model

The von Mises yield criterion states that yielding depends on the maximum and minimum principal stresses, plus the intermediate principal stress. Yielding occurs in von Mises when the deviatoric part of the stress reaches a critical value in the principal stress space. The von Mises yield criterion is commonly expressed as

( ) ( ) ( )2 2 2 21 3 2 3 2 1 y

1 f2

Φ σ σ σ σ σ σ = − + − + − − (4.8)

where 1σ , 2σ and 3σ are the principal stresses and yf is the yield strength. Note that hardening is not included in this yield function. Since the von Mises yield criterion is independent of both the angle in the deviatoric plane and of the hydrostatic stress, the yield surface forms a circular cylindrical prism around the hydrostatic axis in the principal stress space, as can be seen in Figure 4.6. Yielding of metals can normally be assumed to be independent of the hydrostatic stress and can thus be modelled with von Mises.


Figure 4.6 Von Mises yield surface in the principal stress space (Plos 2000)

4.3.2 Multi-cracking Concrete Model

The multi-cracking concrete model implemented in LUSAS is a so-called micro plane model. This model assumes that, at any single material point, there can be a number of permissible cracking directions. In theory, these cracking directions can be either pre-defined or developed during the analysis. In this LUSAS model, the directions are pre-defined and, when modelling in 3D, the directions are defined with Bazant and Oh’s 21 integration spherical integration rule. Each direction defines a plane of possible cracking, and for each one of these planes there is a separate yield surface and set of yield parameters. The local coordinate system for a cracking plane is illustrated in Figure 4.7.

Figure 4.7 The local coordinate system for a crack-plane (LUSAS 2004)

Within this model, a hyperbolic yield surface, which is asymptotic to a Coulomb friction surface, is used, as seen in Figure 4.8. The value of the yield function, Φ , depends upon the local stresses ( , , )T

r s ts s s=s , the equivalent fracture stress, sf , the friction factor, µ, and the roughness-cohesion parameter, cr . The yield surface changes


shape as cracking progresses and, in the limit of full fracture, takes the shape of a friction plane.

Figure 4.8 Yield surface in local stress space

The yield criterion is proposed as

( ) ( )2

2 2 2 2 2 2rc r s t s2

c c

s 11 r s 4r s s f2 r 2r

µΦ µ = + + − + + −

(4.9)

where cr is the cohesion divided by the tensile strength and µ is the friction factor. The equivalent fracture stress is a function of the accumulated fracture strain parameter, according to equation 4.10.

The coupling between the directions arises from the static constraint, which enforces the condition that the local stresses on each crack plane are transformations of the global stress tensor.

The model assumes that the material can soften and eventually lose all strength in tensile loading, in any one of the predefined cracking directions. An exponential softening curve is assumed and, for direct tension loading in one direction, this gives the normal stress-strain relationship shown in Figure 4.9.


Figure 4.9 The stress-strain behaviour normal to a crack-plane

The shape of the softening curve, or the crack normal stress, rs , is governed by equation 4.10, where pε is the equivalent plastic strain parameter and 0ε defines the end of the curve.

( / )p 05s tf f e ε ε−= (4.10)

The end of the softening curve, 0ε , is received from the prescribed fracture energy, as can be seen in equation 4.10, where fG denotes the fracture energy, cw denotes the element gauss point characteristic length and tf denotes the tensile strength or crack initiation value.

f0

c t

5Gw f

ε = (4.11)

By reviewing equation 4.11, it can be stated that the end of the softening curve, 0ε , automatically adjusts itself with regards to the current element size with the variable cw . The variable cw is determined according to equation 4.12, and in LUSAS, is defined as both the element gauss point characteristic length and the crack bandwidth.

( )13

cw V∆= (4.12)

where V∆ is the integrated volume. The reason for this is to avoid the model becoming mesh dependent, according to the LUSAS Manual (2004). However, studying equations 4.10 - 4.12, it can be noted that, due to the definition of cw , the shape of the softening curve is actually dependent of the meshing. Different element sizes will result in different cracking behaviour. It is important to be aware of this characteristic of the material model.


4.3.3 Mohr-Coulomb Bond-mechanism Model

In this specific case, the interaction between the concrete and the reinforcing steel is modelled in LUSAS with a non-associated Mohr-Coulomb frictional material model. This model is applicable in this analysis since it has the ability to describe the behaviour of dilatant frictional materials, which exhibit increasing shear strength with increasing confining stress. This 3D elasto-plastic material model works like a Mohr-Coulomb material model. As for any other elastic plastic model it is impossible to have a stress state outside the yield surface. When the stress state reaches the yield surface, plastic flow starts to occur. As seen in Figure 4.10, the Mohr-Coulomb surface is visualised in the principal stress space, where the surface becomes an irregular hexagonal pyramid, centred around the hydrostatic stress axis.

Figure 4.10 The Mohr-Coulomb yield surface in the principal stress space (LUSAS 2004)

The yield criterion, where c is the material cohesion and Φ is the angle of internal friction, can be derived as

( ) cos [ ( ) ( )sin ] tan1 3 1 3 1 31 1 1c2 2 2

Φ σ σ φ σ σ σ σ φ φ= − = − + + − (4.13)

where 1 2 3σ σ σ> > .

However, the application of the Mohr-Coulomb yield criteria, when modelling the interaction between the steel and the reinforcement, is more easily visualised in the shear-normal stress plane, see Figure 4.11. The normal stress, nσ , and the shear stress, τ , on a fracture plane is defined as

nσ = ⋅ ⋅n σ n (4.14)

( )2τ = ⋅ ⋅ − ⊗ ⋅ ⋅n σ I n n σ n (4.15)

where n is the normal to the fracture plane, I is the second order identity tensor and σ is the stress tensor.


Figure 4.11 The Mohr-Coulomb yield criteria visualised in the shear-normal stress plane

The Mohr-Coulomb yield criterion is defined by straight lines in the shear-stress plane and is a function of the maximum and minimum principal stresses. The material cohesion, c , and the angle of internal friction, Φ , determine the yield function.

( tan )nc 0Φ τ σ φ= − − = (4.16)

During loading, the bond-mechanism between concrete and reinforced steel, as mentioned earlier, produces a combination of shear and normal stresses due to the ribbed reinforced steel bars. The inclined shape of the ribs produces a so-called increase in volume during plastic flow. This effect is called dilatation. Manipulating the dilatation makes it possible to control the direction in which the plastic flow propagates. The dilatancy is defined by an angle, Ψ , which describes how much the plastic flow deviates from the associated case, i.e. from the normal to the yield surface, according to Figure 4.12. The result of controlling the development of the plastic strain is that the development of the normal and shear stresses are controlled. Since dilatation occurs in this interaction, the non-associated flow rule is needed to describe the plastic strain. Otherwise, by using an associated flow rule, there is risk that the normal stresses will increase too quickly. By using a non-associated flow rule, the normal stresses can be partly controlled with a dilatation angle, together with the cohesion and friction angle.

Figure 4.12 The definition of the dilatation angle, Ψ


5 Finite Element Modelling The fundamentals regarding the finite element method are briefly outlined in this chapter. Due to the non-linear characteristics of reinforced concrete structures, the theories concerning non-linear FEM are discussed in more detail. As well, the underlying theory describing the relationship of reinforced concrete and FEM is also considered. The FE-program LUSAS and its features, which combine to produce the numerical models, are also described.

5.1 The Finite Element Method

FEM is considered a powerful tool in today’s engineering world and can be used to model arbitrary structural geometries, whilst predicting a pattern for multi-directional stress states. Many physical phenomena encountered in engineering mechanics are modelled by partial differential equations, and usually, the problem addressed is too complicated to be solved by classical analytical methods. The finite element method is a numerical approach by which general partial differential equations can be solved in an approximate manner. The differential equation, which describes the problem considered, is assumed to be valid for a certain region. It is a characteristic feature of the finite element method, which instead of seeking approximations that are valid for the entire region, the region is divided into smaller parts, i.e. finite elements, and the approximation is then carried out for each element.

5.1.1 Non-linear FEM

Some materials or structures have a non-linear behaviour. In order to catch the non-linear response, the solution is generally carried out in a step-wise manner, i.e. the load is applied as increments. To obtain reliable results in non-linear problems it is often necessary to adopt an iterative method by means of iterations within each increment to achieve equilibrium. When the solution is carried out in an iterative manner the method is said to be implicit, and on the other hand, when the solution is carried out instantly, it is dubbed explicit.

The response of concrete structures is often strongly non-linear. Analyses of reinforced concrete often exhibit local maxima when the concrete cracks, with snap-through or snap-back behaviour in the load-displacement response, see Figure 5.1.


Figure 5.1 Possible behaviours of reinforced concrete structures

In general, when solving these types of finite element problems, an implicit incremental solution procedure is adopted. The incrementation can be carried out in three different ways: displacement control, load control or with more advanced methods, i.e. arc-length method. When using displacement control, the prescribed displacement is successively increased in the incrementation process, and respectively the load, if the load control method is used. When implementing these two methods it is not possible to catch eventual snap-back behaviour. On the other hand, the arc-length method is useful to trace snap-back behaviour. This method uses the arc-length method as a control parameter for the incrementation, instead of the load or the displacement. This means that the next equilibrium point is sought at a certain distance along the equilibrium path for the load-displacement response, as seen in Figure 5.2.

Figure 5.2 The arc-length method

5.1.1.1 Newton-Raphson Iteration

As previously mentioned, an iterative solution method is often needed when modelling cracking concrete. A commonly used method is the Newton-Raphson method, by which the stiffness matrix is updated after each iteration. However, this is


a very expensive method in terms of the number of calculations. Therefore, a family of modified Newton-Raphson methods has been developed, where the stiffness matrix is updated only once in each increment. However, these methods often lead to a higher number of iterations, but require less work due to the fact that the stiffness matrix is updated less frequently. In Figure 5.3, the Newton-Raphson method is illustrated and the method is further explained in the following text.

Figure 5.3 The Newton-Raphson method

The FE-set is obtained from the variational formulation and the residual vector R is defined as

( )int ext= −R F w F (5.1)

where intF is the internal force vector and extF is the prescribed external force vector. The displacement vector, w , is computed in an iterative procedure. By using the Taylor expansion of the residual vector we obtain

k 1 k δ+ ∂≈ + =∂RR R w 0w

(5.2)

where the residual is set to zero.

1k 1 kδ

−−∂ = − ⋅ = − ⋅ ∂

Rw R K Rw

(5.3)

k 1 k+ = + ∂w w w (5.4)

It can be seen in equation 5.3, that the stiffness parameter is updated in each increment. It is sometimes suitable to modify this method, as mentioned earlier, in order to lower the number of calculations. Modified Newton-Raphson methods usually start the first iteration within the increment with an updated stiffness matrix, a so called predictor, and then processes the following iterations without updating the stiffness matrix, see Figure 5.4.


Figure 5.4 The modified Newton-Raphson method

5.1.2 FEM and Reinforced Concrete

Due to the strongly non-linear behaviour of reinforced concrete structures, and the large time-cost attributed with non-linear analyses, it is of importance to create an effective finite element model. This means taking measures, such as minimising the number of elements and making appropriate simplifications, whilst the model still represents the salient features of the structure. It is also of importance to find an appropriate solution method by means of incrementation procedures, iteration methods and convergence criteria.

When establishing the numerical model, it is suitable and logical to build the model in a stepwise manner, whilst gradually introducing the non-linear characteristics of the problem. The building process is an iterative procedure where important parameters are gradually identified and obtained.

A numerical model is a graphical representation consisting of geometric features, i.e. points, lines, surfaces and volumes, and assigned attributes i.e. materials, loads, boundary conditions and meshes. Since this analysis is three-dimensional and non-linear, and examines the stress distribution in a structure that consists of three different material models, the numerical model is quite complicated. Apart from the fact that the model consists of three different material models, there are many parameters and assumptions that need to be determined, as well as there are many different steps involved in the solution method. In Figure 5.5, a flow chart illustrates, simply, the features involved in creating the numerical model analysed in this finite element study. A more detailed overview of the steps involved in creating this numerical model is illustrated in Figure 2.1.


Figure 5.5 Flow chart illustrating the factors involved in creating a numerical model

5.2 Finite Element Modelling with LUSAS

The finite element program LUSAS is one of hundreds of FE-programs available today. Each program has its own predefined features, e.g. material models, and consequently corresponding strengths and weaknesses. LUSAS is implemented in this research’s analyses and some of the significant implemented features and procedures are described in the following sections.

5.2.1 General Description of the FE-program

The commercial FE-program LUSAS is used to carry out the non-linear analysis of the spliced area of the pile. LUSAS is a MS Windows based programme and has a graphical user interface in which the model is developed. The program is installed on a regular personal computer and utilises no extra computational power in terms of processing servers. LUSAS has quite a large library of predefined features, for example, different material-models, solution methods and element types. It is also possible to program your own features. However, creating plug-ins is not a reasonable feature in the scope of this Master Thesis, as it requires a very deep knowledge of LUSAS, as well as within the art of programming.


5.2.2 Model Attributes

Model attributes include the geometry, mesh, loading and supports involved in creating a numerical model. The mesh attribute describes the element type and discretisation of the geometry. Geometric attributes specify any relevant geometrical information or properties involved in the model. The behaviour of the element material is defined by material attributes, support attributes specify how the structure is restrained and loading attributes specify how the structure is loaded. In the following text, the theory behind the model attributes is discussed.

Some of the model attributes and the corresponding theories assigned to the numerical models are quite fundamental. However, some of these attributes and theories are not so clear and straightforward, whilst at the same time critical with respect to the models’ behaviours. In this chapter, these not so straightforward and critical model attributes and theories are described.

So as to capture the behaviour of the pile, for example, a 3D stress field, modelling is made in 3D. In partly fulfilling this 3D modelling criteria, the numerical models incorporate the hexahedral volume feature, as illustrated in Figure 5.6.

Figure 5.6 A hexahedral volume body (LUSAS 2004)

The HX8 3D solid continuum element is one of many available elements in LUSAS and is used in the numerical model when modelling the concrete, bond-mechanism and reinforcement in 3D. This element type is numerically integrated and has three nodal degrees of freedom, i.e. translations in X, Y, and Z directions. It should be noted that the local and global coordinate systems are defined in the same directions. The HX8 element consists of eight Gauss points, i.e. 2 2 2× × , and its interpolation order, i.e. shape of the basis functions, is linear.

As well, element discretisation is an important factor when meshing the model. Firstly, the element length ratio size has a big influence on the accuracy of the results. For the most accurate results, an ideal ratio is 1:1, but more than acceptable results are received with ratios up to 1:4. However, ratios up to 1:10 are assumed as reasonable according to LUSAS manuals (2004). Secondly, the intensity of the mesh also has an influence on the accuracy of the results. If the mesh is too sparse, the results can be unreliable. Of course, it is best to create a very tight, intense mesh so as the results are as accurate as possible. However, this causes the calculation time to increase dramatically. The most appropriate way to find “correct” mesh intensities is to test the model and confirm that the results converge towards reasonable values.


Another element type available in LUSAS, that is, a 3D joint element with no rotational stiffness, is identified as JNT4. It is used to model the “induced” deformation in the locking device, and is suitable for use in interaction between 3D solid elements, i.e. HX8 elements. The element shape is defined as a line, which connects the corresponding nodes between two arbitrary bodies.

The joint elements may be used to release degrees of freedom between elements, e.g. a hinged shell, or to provide non-linear support conditions, e.g. friction-gap condition. This element connects two nodes by three springs in the local X, Y and Z-directions, as illustrated in Figure 5.7. The element has four nodes and the 3rd and 4th nodes are used to define the local X-axis and local XY-plane. The interpolation order for this element type is quadratic.

Figure 5.7 A JNT4 3D-joint element (LUSAS 2004)

Joint material models are used in conjunction with joint elements to define the material properties for linear and non-linear joint models. The non-linear joint material model used in the numerical models is identified as a frictional contact model and its feature predefines an initial gap, Figure 5.8. If an initial gap is used in a spring, then the positive local X-axis for this spring must go from node 1 to 2. If nodes 1 and 2 are coincident, the relative displacement of the nodes in the local X-direction must be negative to close an initial gap.

Figure 5.8 Frictional contact material model for joints (LUSAS 2004)


Support conditions describe the way in which the model is supported or restrained and applied to each degree of freedom. Three valid support conditions used in this analysis are free, fixed and spring stiffness degree of freedoms. The degrees of freedom depend on the chosen element type used in the model. Since this analysis is three-dimensional, the degrees of freedom for this structural problem are translational freedoms in the X, Y and Z-directions, which restrict movement along each axis direction.

Loads describe the external influences to which the model is subjected. The loads are assigned to the model geometry and are effective over the whole of the feature to which they are assigned. The feature-based loads used in this analysis are structural loads and prescribed loads.

The theory behind constitutive material models is discussed in Section 4.2, and in Section 4.3, the theory behind the material models used in the numerical models is considered in detail.

5.2.3 Non-linear Solution Procedures

In LUSAS, the incremental-iterative solution is based on Newton-Raphson iterations. The program handles all of the three earlier mentioned incremental procedures; load, displacement and arc-length controlled methods. It is also possible to combine the arc-length method with the load and displacement controlled methods, i.e. switch the iteration method at a certain predefined point. The incrementation procedure can be specified in three different ways, manually, automatic or through predefined load curves. During the actual analyses, the automatic alternative was adopted. Automatic incrementation means that LUSAS automatically adjusts the increment size based on the history of convergence. LUSAS allows, however, the user to control the incrementation through constraints such as maximum change in the increment size and the specified starting increment size. If convergence is not achieved within the increment after a specified number of iterations, the program will automatically reduce the step length by a predefined factor.

In LUSAS, a modified Newton-Raphson method, combined with a line searchers feature, is implemented. The purpose of the line searchers feature is to accelerate the iteration process.

In order to monitor the convergence, there are several convergence criteria implemented in the program. The selection of appropriate convergence criteria is of utmost importance. An excessively tight tolerance may result in unnecessary iterations and consequent waste of computer resources, whilst a slack tolerance may provide incorrect answers. In this type of problem, which is characterised by the non-linear material behaviour, there are three suitable convergence criteria implemented in LUSAS:

• Euclidian residual norm

• Euclidian displacement norm


• Euclidian incremental displacement norm

The Euclidian residual norm, equation 5.5, is defined by the norm of the residuals as a percentage of the external forces. In this norm, as well as in the following two norms, only the translational degrees of freedom are considered. The application of only translational and non-rotational degrees of freedom is due to the inconsistency of the units.

2ext

2

100ψγ = ×RF

(5.5)

The Euclidian displacement norm, equation 5.6, is defined by the norm of the iterative displacements as a percentage of the norm of the total displacements.

2d

2

100γ∂

= ×ww

(5.6)

The Euclidian incremental displacement norm, equation 5.7, is defined by the norm of the iterative displacements as a percentage of the norm of the displacement for the actual increment. (LUSAS 2004)

2dt

2

100γ∂

= ×∆ww

(5.7)


6 The Development of the Numerical Model The main goal of this thesis is to establish a sound three-dimensional numerical model of the spliced area for prefabricated driven concrete piles, which is verified with existing full-scale load tests. The established numerical model should be able to appropriately capture the non-linear behaviour of reinforced concrete, which is strongly dependent on correct modelling of the bond mechanism’s behaviour.

The art of modelling the bond-mechanism between concrete and reinforcement is still relative new in the world of the finite element method. This is due to the fact that in creating material models of the bond mechanism, it is not easy capturing the true behaviour of the particular bond-slip mechanism.

In this chapter, the numerical model, that best models the behaviour of the pile element, is described in detail, including the specific parameters and information applied in LUSAS. The alternative approaches, models, solutions, etcetera, that have been tested, but deemed less suitable, are also discussed in this chapter.

6.1 The Numerical Model

As discussed earlier, a numerical model consists of many features, in which many analyses, with different numerical models, are conducted in search of the “most correct” model. The combination of the following parameters and modelling features is the key to the numerical model that most correctly simulates the behaviour of the laboratory tested pile elements.

Geometry and Meshing

The pile is modelled with half the length, and half the width of the cross-section of the empirical tested pile, as illustrated in Figures 6.1 and 6.2. The reasons behind this are described in the following text.

Figure 6.1 Geometry of the FE-model


Figure 6.2 The cross-section of the FE-model

The main design of the pile element is the same as in real life, that is, it is modelled with rectangular features. One of the biggest changes in the pile’s geometry design is the geometry of the reinforcement bars. The reinforcement bars have a circular cross-section in reality, whilst in the model they are modelled with a square cross-section, Figure 6.2. The square form of the reinforcement is much easier to model, i.e. design and mesh, in LUSAS, as well as contributing to minimising the calculation time for the FE-analyses. It should be noted that the area of the reinforcement’s square cross-section is equivalent to the area of the circular reinforcement.

Another assumption made regarding the design of the pile concerns the amount of reinforcement modelled relative to the amount of reinforcement in the pile. The stirrups and stirrup cage are not modelled due to the miniscule effect they have on the studied behaviour of the pile, i.e. bending due to moment. As well, the reinforcement placed in the compression zone is assumed to have little influence on the response of the pile, since the pile’s failure is due to tensile stresses. Therefore, the principal and anchorage reinforcement bars, located in the compression zone, are modelled as one continuum bar, Figures 6.1 and 6.2.

As mentioned earlier in Section 3.5, tensile cracks occur due to the driving load applied when installing a pile that rests on a rigid base. The piles studied in the empirical tests were also exposed to driving loads whilst resting on bedrock, and therefore, it was assumed that tensile cracks already existed before the bending moment was applied. These tensile cracks are modelled as crack bands in the numerical model, Figure 6.1, which are initiated directly when the load is applied. In these bands, the concrete is prominently “weakened” by a lowering of the tensile strength and the fracture energy. With the inclusion of these crack bands ductile failure is avoided.


The width of the cracked zones applied in the numerical model is 20 mm, which is considered larger than the true crack zone. However, the fact that the crack zones were chosen as 20 mm in width was no coincidence. Numerical problems occur when the crack bands are modelled too thinly.

The concrete, reinforcement and bond-mechanism are modelled with hexahedral volume features and HX8 three-dimensional continuum solid elements, so as to simulate the 3D stress field in the pile, as illustrated in Figures 6.1 and 6.2. The element ratio is approximately 1:3 throughout the whole model, except for in the bond-mechanism model, where the ratio is approximately 1:8. The accuracy of the bond-mechanism’s behaviour is somewhat sacrificed so as to keep the calculation time as low as possible. If the bond-mechanism’s element ratio were changed to 1:4, then the calculation time would be approximately twice as large as the present calculation time.

Boundary Conditions

Since the studied pile element is simply supported, and its geometry is symmetrical in both the longitudinal and transversal directions, certain simplifications can be assumed. The pile’s supports are simulated with a fixed area in the Y-translation under the left end of the model, Figure 6.1.

To minimise the calculation time of the analysis, symmetry planes are created in both the longitudinal direction in the middle of the cross-section, as well as in the transversal direction in the splice, Figures 6.1 and 6.2. The symmetry plane in the cross-section is simulated with fixed translation boundary conditions, on the XY-surface along the pile, in the Z-direction. The symmetry plane created in the splice is simulated with fixed translations in the X-direction in the anchorage reinforcement and the compressed zone of the concrete.

Normally, in a simply supported continuous beam, symmetry is simulated with fixed translations in the whole cross-section surface where the symmetry plane is situated. However, the spliced area does not behave in the same fashion as a continuous beam. Since the splice is situated in the middle of the simply supported system, the spliced area is exposed to the largest deformation when a bending moment is applied. In the tensile zone of the spliced area, the two splices separate from one another, unlike a composite beam element, whilst the splices are still in contact with each other in the compressed zone. The symmetry plane simulates the contact in the compressed zone, plus the continuation of the reinforcement anchorage bar, with fixed X-translations combined with a slip device, which simulates existing plastic deformations in the locking device, Figure 6.1.

The initial slip phenomenon that occurs in the locking device, as explained in Section 3.3, is modelled with a joint element that behaves as a manipulated spring, Figure 6.3. Initially, when the model is loaded, the anchorage bar is unsupported, but when the deformation in the longitudinal direction reaches 0.3 mm, the spring is activated. The spring has been assigned very stiff material properties so it can be considered as fixed. The slip-support device is visualised in Figure 6.3. To avoid numerical problems, weak springs are added in the Y and Z directions to the fixed X-boundary.


Figure 6.3 The slip-support device, which represents the existing deformations in the locking device due to the driving process

Loading

The load applied to the model is an incremental displacement load, which initially has the incremental length of 0.1 mm in the negative Y-direction and acts over an area of 10 000 mm2 on the top surface of the pile, as illustrated in Figure 6.1. This simulates the load applied to the pile in the laboratory tests. The initial increment length load is chosen as 0.1 mm since it makes the numerical solution process easier, as discussed later in this chapter.

Material Models

As explained in Section 4.3, three material models are used in this analysis. The von Mises stress-potential material model is used to model the pile’s reinforcement steel. The pile’s principal and anchorage reinforcement have the same characteristics except for the tensile strengths. The steel quality for the pile’s reinforcement and the anchorage reinforcement are Ks600 respective Ks400. The von Mises material model’s properties are illustrated in Table 6.1.

Table 6.1 Material properties of reinforcement steel

Reinforcement Steel

Elastic Properties

E = 200 GPa

υ = 0.3

Plastic Properties

yf = 600 MPa principal reinforcement

yf = 400 MPa anchorage reinforcement


The LUSAS multi-cracking concrete material model is used to model the concrete in this study. These properties are based on the properties of the concrete used in the empirical tests. Important adjustments, however, have been made to the elasticity module, E , the tensile strength, tf , and the fracture energy, fG . In the laboratory tests, the pile had been hammered with 3000 hits before it was loaded with a bending moment. Due to the effects on the concrete from the driving load, the concrete’s characteristics are somewhat affected, and subsequently, the concrete’s E , tf and fG can be assumed to be approximately 10-30% lower than the characteristic values. The concrete’s characteristic E , and fG values are reduced with 10% in this FE-model. Driven concrete’s tensile strength shall not be estimated higher than 3.2 MPa for the tested piles, and is thus defined accordingly. These values are illustrated in Table 6.2. For the concrete’s characteristic values used in the empirical tests see Appendix B1.

Furthermore, before the fracture energy is decreased with 10-30% due to the driving load’s effects, the fracture energy is also decreased with 50% due to the behaviour of the concrete model, as described in Appendix C1. This approximation of fG is based on the element ratio used in the numerical model which is often 1:3, and no greater than 1:4, with an exception being made for several bond-mechanism elements.

As explained earlier in this chapter, crack bands are applied in the model to simulate tensile cracks caused by the driving load, Figure 6.1. These crack bands are of course modelled with the multi-cracking concrete model, but with necessary modifications to ensure that cracking in these zones is initiated instantaneously, i.e. in the first increment of loading. The tensile strength and fracture energy are decreased dramatically. It should be noted that in reality, once these tensile cracks occur, there is no bond action between the reinforcement and the concrete in these crack zones. However, it is quite difficult to model this phenomenon and bond-slip is therefore assumed to occur in these modelled crack bands. The influence of this assumption is not significant in the global response of the pile model.

Table 6.2 Material properties of concrete and of “weakened” concrete in crack bands

Concrete Concrete in Crack Bands

Elastic Properties Elastic Properties

E = 33.6 GPa E = 33.6 GPa

υ = 0.2 υ = 0.2

Plastic Properties Plastic Properties

tf = 3 MPa tf = 0.1 MPa

fG = 80 Nm/m2 fG = 1 Nm/m2

0ε = 0 0ε = 0


Certain material models’, so-called bond-mechanism models, primary function is to model the interaction between two materials. However, none of the models implemented in LUSAS have the necessary characteristics needed to model appropriate bond-slip behaviour of the interaction between concrete and reinforcement. Therefore, a non-associative Mohr-Coulomb model is used, which, as mentioned in Section 4.3, appropriately models the traction stress, whilst taking into account the dilatant behaviour of the bond-mechanism. This model is not theoretically defined by LUSAS as a bond-mechanism model, but as a material model instead.

Different varieties of the Mohr-Coulomb model are needed to fully capture the true response of two different bond-mechanism phenomena that exist in the reinforced concrete pile. The interaction between concrete and reinforcement, as explained in Chapter 7, has a different behaviour to the interaction between the two reinforcement bars, and thus the model properties are applied accordingly. The biggest differences in the model properties are the values associated with the friction and dilatancy angles, as is illustrated in Table 6.3.

Table 6.3 Material properties of the bond-mechanism between concrete and reinforcement and two reinforcement bars

Concrete-reinforcement Reinforcement-reinforcement


E = 33.6 GPa E = 33.6 GPa

υ = 0 υ = 0


c = 1 MPa c = 3.75 MPa

iφ = 20° iφ = 1°

fφ = 20° fφ = 1°

ψ = 7° ψ = 1°

ch = 0 ch = 0

fε = 1 fε = 1

As mentioned in Chapter 3, the driving load creates cracking in the concrete in the installation stage, which significantly influences the bond-slip relationship. Due to this effect on the bond-mechanism, the parameters defining this model must also be changed accordingly. The model applied between concrete and reinforcement still behaves as a true friction model, with infinitely increasing shear and normal forces.


However, due to a decrease in the bond-slip effect, the size of the cohesion is notably reduced to 1 MPa.

The interaction between the reinforcement bars is also affected by the driving load in the same manner, and is consequently modelled with low friction and dilatancy angles, which prevents the shear forces from becoming too large, and allowing the force distribution between the two bars to behave as predicted.

Solution Methods

As mentioned in Section 5.2, the Newton-Raphson incrementation-iterative method is applied in the analysis of this numerical model. The automatic incrementation feature is chosen to control the method of incrementation. This means that the incrementation is displacement controlled, and if the analysis has trouble converging, the incrementation method can switch to the arc-length method. The incrementation process starts with an initial incremental length of 0.1 mm and then adjusts the length automatically according to the convergence history. As explained in Section 5.2, the non-linear convergence criteria specifies which stage the iterative corrections can be assumed to have restored the structure to equilibrium. The three most significant criteria regarding this analysis, that is, the residual force norm, the incremental displacement norm and the displacement norm, are illustrated in Table 6.4.

Table 6.4 Convergence criteria

ψγ (residual force norm) 7%

dγ (displacement norm) 3%

dtγ (incremental displacement norm) 3%

6.2 Alternative Approaches

A few alternative numerical models were designed and tested during the course of this study. The main aim of these alternative models was to reduce the calculation time by means of simplifications and assumptions. However, for one reason or another, these alternative models did not produce results as accurate as the model described in Section 6.1. The alternative models are described in general, together with the pros and cons of the models.


Alternative: Tied Mesh

Figure 6.4 The tied mesh approach, which consists of two bodies connected with the tied mesh feature

It is assumed that the most critical area of the pile is the double reinforced zone, i.e. the area in which bond-slip occurs between the principal reinforcement bar and the anchorage bar. It would therefore be favourable to model this part of the pile with desired mesh intensity and the remaining part of the pile with simplified mesh intensity. In LUSAS, there is a “tied mesh” feature available, which can be used in order to connect different mesh intensities in a model.

The tied mesh function makes it possible to connect volumes or surfaces with different mesh intensity, i.e. the number of nodes and position of the nodes between bonded volumes do not need to coincide. In this model, the tied mesh feature connects two bodies with different designs and mesh intensity. The body to the right has the same design and mesh intensity as the model described in Section 6.1, however, the body to the left has a much more simplified structure, so as to minimise the processing time, Figure 6.4.

The body to the left is modelled as one volume body with a linear material model, with the properties of an equivalent cracked concrete cross-section, i.e. no reinforcement, and is meshed with a somewhat “non-tight” mesh. These simplifications were made with the intention of significantly reducing the processing time. However, it is suggested in the LUSAS manuals (2004) that the tied mesh features should be used with care since they may cause numerical problems and may provide incorrect solutions. This was certainly true for this model.

Simple tests were initially conducted concerning the tied mesh feature and it seemed to perform well. However, when the tied mesh feature was applied to the numerical pile model the results were unreliable. For example, large stress concentrations appeared at the nodes in the connected surfaces, which subsequently affected the results in a negative manner.


Alternative: Beam Elements

Figure 6.5 The beam element approach, which consists of a beam element connected with solid elements by a rigid link device

As mentioned in the tied mesh alternative, it would be ideal if the “less interesting” part of the pile, i.e. the left side, could be simplified. Another way to simplify the model is to replace the left side of the pile with a beam element, as illustrated in Figure 6.5. This beam element is also modelled with the characteristics of an equivalent concrete cross-section. The presence of the beam element dramatically reduces the number of elements in the model.

There are two varieties of applying the beam element to the solid element, with a rigid link constraint or with an overlapping modelling feature. In the first variety, a rigid link connection is tested. A rigid link creates a rigid fixity between two features. The nodes in the assigned features may be constrained to be rigid, the group of nodes may translate and/or rotate but their positions relative to one another remain constant. Translation displacements and rotational freedoms can be constrained using this type of constraint.

A complication however occurred when this constraint feature was included in the model. Within LUSAS, warnings occur when a geometric feature has been “over constrained.” This means that a certain geometric feature, e.g. a line, is defined with more than one constraint. In this model, the surface connecting the beam and the solid elements is defined with the rigid link constraint. However, since the support conditions defining the symmetry line of the cross-section are also defined to a surface connected by a coinciding line, the LUSAS solver rejects the model.

One way to try and avoid this problem is use an overlapping method. This method overlaps the line feature used to define the beam element onto another existing line in the solid body. This type of constraint creates, however, an immediate zone in which the model is much stiffer than originated. This model did not result in reasonable results either.


7 Verification and Validation of the Numerical Model

The verification and validation process occurs on both local and global levels, as illustrated in Figure 2.1. As explained in this chapter, and in more detail in the Appendices, the individual components of the numerical model are first tested and the results, responses, etcetera, are verified with proven and established theories and methods. The second stage, the validation stage, occurs next, for example, testing and comparing the results of a simple reinforced concrete block with the CEB-FIP Model Code 1990 and other established studies. Finally, the pile models are tested, analysed and validated with the laboratory tests conducted by SP.

7.1 Verification of Individual Components

As mentioned in earlier chapters, there are many components that contribute to a working numerical model and all of these components need to be controlled at some stage during the study. Some of the more simple controls of assumptions and methods used, such as verifying the intensity of the mesh, the response due to symmetry lines, element types, increment step length and convergence criteria, are not presented due to the fact that they are considered elementary in the FE-model building process.

Controls of, for example, the material models are essential in contributing to creating a suitable, accurate and acceptable final numerical model. These verifications and controls are summarised in the following text and explained in more detail in Appendices C1, C2 and C3.

von Mises Material Model

The von Mises material model is used to model the reinforcing steel used in analyses throughout this study. A simple tensile load was applied to a steel bar and analysed in LUSAS. As expected, the model responds according to the ideally elastic-plastic theory, as presented in Appendix C3.

Multi-cracking Concrete Material Model

A non-linear control has been conducted on the LUSAS multi-cracking concrete material model. The control consists of tensile tests, whereas the non-linear model is loaded with a prescribed displacement controlled load, as can be seen in Appendix C1.

The elastic response of the model is verified according to elasticity theory whilst the plastic response is somewhat more complicated to calibrate. It was unfortunately noticed quite late in the study that the fracture energy applied to a concrete model is not equivalent to the fracture energy dissipated from the analysis. Modifications to the applied fracture energy used throughout this study are made according to the results from theses simple tests. For more details regarding this problem see Appendix C1. Figure 7.1 illustrates the cracked cube and the stress versus displacement response for the cube.


Figure 7.1 (a) Cracking in the Gauss points for test case 1 and (b) stress versus displacement curve for test case 1

Non-associative Mohr-Coulomb Material Model

This material model is verified according to Appendix C2, where an analysis of a cube, with the non-associative material properties, is exposed to a shearing load, Figure 7.2 (a). When the friction angle is set to 45°, the corresponding normal and shear forces at the yield surface are, as expected, equal, as illustrated in Figure 7.2 (b).

Figure 7.2 (a) The sheared cube and (b) the response from test 1 due to a friction angle of 45°

7.2 Validation of Reinforced Concrete Models

In Section 7.1, some of the individual components of the numerical model are verified. In this chapter, structures modelling the interaction of steel and concrete, i.e. bond-slip, are analysed. As earlier mentioned, the interaction between the concrete and the steel is a very important feature when modelling reinforced concrete structures. Another significant aspect is the bonding between spliced reinforced bars, i.e. the transfer of a load between two reinforcing bars. The bond model is validated


with the help of these two factors and is discussed in more detail in the Sections 7.2.1 and 7.2.2.

7.2.1 Bond Action Between Concrete and Reinforcement

In order to validate the bond-slip model, simple pullout tests are conducted with a reinforced concrete FE-model, which are explained in more detail in Appendices D1 and D2. The FE-model’s parameters are calibrated, partly according to data from the CEB-FIP Model Code (1990), and partly according to data from experimental tests. The empirical data used for validation is taken from Magnusson (2000), where the anchorage of ribbed bars is analysed. The model is validated according to these two different data sets due to the difficulty in finding additional reliable, suitable and established theories. After studying the available “relevant” theories and experimental tests, it can be stated that not only is it difficult locating results pertaining to simple bond-slip pullout tests, but also that bond behaviour is a tremendously complex mechanism.

The complexity of the bond-slip modelling dilemma can partly be explained by the large number of parameters which influence the behaviour of the bond-mechanism, for example the concrete strength, support conditions, presence of stirrups, concrete cover, etcetera. On account of this complexity, a more “overall” behaviour, e.g. parameters such as the magnitude of bond stresses, and slip and crack propagation, of the modelled bond-mechanism is studied and validated.

Additional important factors that must be considered are the handling, transportation and driving of a pile, in which all the components of a pile structure are affected. It is known that the bonding strength decreases due to these physical loads, but not to which extent. It can thereby be stated that not only is modelling the bond action quite difficult, but it is an extremely essential feature contributing to the behaviour of a reinforced concrete structure. With these aspects in mind, the behaviour of the modelled bond mechanism is further studied and subsequently presented in this section.

Figure 7.3 The principal bond stress slip relation according to CEB-FIP Model code (1990)


In the CEB-FIP Model Code (1990), a bond-slip theory, according to Figure 7.3, is presented. The different values of the slip and the maximum bond stress are presented in Table 7.1.

Table 7.1 Parameters for defining the bond stress-slip relationship according to CEB-FIP Model Code (1990)

Unconfined Concrete Confined Concrete

1S 0.6 mm 1.0 mm

2S 0.6 mm 3.0 mm

3S 1 - 2.5 mm Clear rib spacing

maxτ 1 - 2 * ckf 1.25 - 2.5 * ckf

As discussed in Chapter 6, the bond-mechanism is modelled with a friction Mohr-Coulomb material model that is applied to solid elements. Initially, in order to study the influence of the different parameters, a confined reinforced structure was analysed. The studied geometry is visualised in Figure 7.4.

Figure 7.4 The confined geometry on which pullout analyses have been carried out. Two symmetry planes are used and the concrete block’s degrees of freedom are fixed in the top surface in the Y-direction. The reinforcement bar is loaded with a displacement load in the Y-direction.


The parameters of interest pertaining to the model are the friction angle, φ , the dilatancy angle, ψ , and the cohesion strength, c . Parameter studies have been conducted on these variables regarding their influence on the global behaviour of a reinforced concrete model. The main effects that these variables have on a pullout test of a reinforced concrete model are presented in this section, and further details, such as material properties and solution procedures, are presented in Appendix D1.

In Figures 7.5 - 7.7, the results of the parameter studies are presented. Note that the empirical data from Haga and Olausson (1998) are also presented as reference values in the diagrams in Figures 7.5 - 7.7.

Figure 7.5 Parameter study of the bond model regarding the effects of the friction angle

By studying the diagrams in Figures 7.6 and 7.7, it can be stated that both the cohesion and the dilatancy angle have a large impact on the magnitude respective growth of the bond stresses. The value of the cohesion seems to be especially important in achieving a reasonable and rational size of the bond stress. However, the size of the friction angle has less impact on the bond behaviour, Figure 7.5, and is therefore accordingly chosen as 20°.


Figure 7.6 Parameter study of the bond model regarding the effects of the dilatancy angle

Figure 7.7 Parameter study of the bond model regarding the effects of cohesion

Despite the pile’s moderately thick concrete cover of 50 mm, the pile’s geometry is considered unconfined rather than confined. The failure mode, splitting of the concrete, Figure 7.9, is more likely to occur in an unconfined case than a pullout failure, i.e. shearing of the concrete between the ribs, which occurs in confined cases.


With this in mind, the confined bond model is further analysed as an unconfined model, Figure 7.8. As earlier mentioned, the angle of friction is chosen as 20° and in the unconfined analyses the influence of the dilatancy angle and the cohesion is further studied.

The geometry of the unconfined model analysed is presented in Figure 7.8 and the cracking of the model is illustrated in Figure 7.9. As can be seen in Figure 7.8, the concrete cover of the model is quite small, only 16 mm. The reason for using this concrete cover thickness depends on the fact that the data in the CEB-FIP Model Code (1990) is based on this condition. More detailed information about the model can be found in Appendix D2.

Figure 7.8 The unconfined geometry of the pullout analyses. Two symmetry planes are used and the concrete block’s degrees of freedom in the Y-direction are fixed in the top surface. The reinforcement bar is loaded with a displacement load in the Y-direction.


Figure7.9 The different phases of the cracking process illustrated in the concrete from the unconfined pullout test. Note that the steel and bond-mechanism models have been removed to better illustrate the cracking.

Studying the results from the FE-analyses of the unconfined pullout model, as illustrated in Figure 7.10, it can be seen that the bond-mechanism model possesses the ability to provide behaviours similar to the recommended theory located in the CEB-FIP Model Code (1990), which are also presented in the Figure 7.10 and in Table 7.1. For more information see Appendix D2.

Upper and lower boundaries are plotted in Figure 7.10, according to values recommended by CEB-FIP Model Code (1990). These curves are used as reference values when refining the choices of φ , ψ and c . Different combinations of these parameters are tested and the relevant results illustrated in Figure 7.10. It should be noted that with a low cohesion, e.g. 2 MPa, a softening response is not achieved with this model. At this stage it is assumed that the analysis was not run long enough to allow for the response to soften. This analysis should be examined in future studies.


Figure 7.10 Results of the unconfined pullout tests

It should be noted that there is no presence of transverse reinforcement in the FE-model, which the CEB-FIP Model Code (1990) prescribes, and therefore, the absence of transverse reinforcement can be assumed to contribute to uncertainties in the validation process. However, the presence of stirrups results in large bond-stresses and a ductile behaviour, whilst the absence of stirrups results in lower bond-stresses, a more brittle failure and less slip, and consideration to this must be taken.

In Magnusson’s experimental studies of anchorage regions (Magnusson 2000), the absence of stirrups almost halved the maximum bond stress, as well as decreasing the magnitude of the slip. In the tests without stirrups, the failure is of a brittle splitting type, whilst the tests with stirrups result in a more ductile failure. It is of great importance to be aware of this effect when validating the results of the FE-analyses.

Considering these facts, it is assumed reasonable to use a combination of parameters for the bond-slip model that result in lower maximum bond stresses than the values presented in the CEB-FIP Model Code (1990). A bond-slip model, which provides a smaller slip and more brittle behaviour, is also preferable, since there is no transverse reinforcement used in the FE-model. With these preferences in mind, it can be assumed that the combination of a friction angle of 20°, dilatancy of 7° and cohesion of 5 MPa results in suitably simulating the behaviour of the unconfined pullout test model.

As can be seen in Figure 7.10, the initial response is very stiff. When implementing this bond model in the load transfer and global pile models, Section 7.2.2 respective 7.3.2, the initial response is also too stiff. The parameters in this bond model, and the load transfer model, are analysed for “non-driven” concrete, whilst in the global pile model respect is taken to the influences of driven concrete, which is explained in more detail in Section 7.3.


7.2.2 Bond Action Between Spliced Reinforcement Bars

An important characteristic of the spliced pile area is the splicing between the main reinforcement bars and the anchored splice reinforcement bars. It is of quite important to accurately model the load transfer between the bars to correctly capture the structural behaviour of the pile. A bond action between the bars that is too large will result in a cross sectional response that is too stiff, whilst on the other hand, a bond action that is too low will result in a response that is too “soft”.

As mentioned in Section 7.2.1, the bond action between the bars is quite a complex phenomenon with no definite predictability, especially after the structure has been handled and driven. In order to investigate and verify the bonding between the bars a number of analyses are carried out, for example, where the bond phenomenon has been isolated in simpler models, as well as in analyses of the complete pile structure.

As mentioned in Chapter 6, the bonding between the anchorage bar and the main reinforcement bar consists of a Mohr-Coulomb friction model. In order to investigate whether the force can be correctly transferred between two bars with the Mohr-Coulomb model, simplified tests are performed on unconfined models. The geometry, load and supports of one of the test cases, i.e. model 1, are visualised in Figures 7.11, 7.12 and 7.13.

Figure 7.11 The geometry of model 1, in which the load transfer test has been performed. The degrees of freedom in the Y-direction are fixed in the left bar’s bottom surface, whilst the other bar is loaded with a displacement load also in the Y-direction.


Figure 7.12 A side-view of the geometry of the FE-model

Figure 7.13 A view from the top of the geometry of the FE-model

The FE-models, in which the force transfer tests were performed, consist of two main bond-mechanism models, i.e. between reinforcement and concrete, and reinforcement and reinforcement. To study the response of the bond-mechanism, several different analyses are conducted, in which both the properties of the bond-mechanism and the anchorage length are varied. Six tests were conducted and the corresponding important bond-mechanism properties and anchorage lengths of the six models are illustrated in Table 7.2, whilst the remaining properties are presented in Appendix D3. The properties of the concrete and steel materials are uniform in all of the tests, and these properties are also illustrated in Appendix D3.

With an increasing anchorage length, and the same load and material properties, it is expected that the models can tolerate greater loads for the same amount of normalised strain. Two different anchorage lengths are implemented in the six tests. Models 1, 2 and 3 are modelled with 0.08 m anchorage lengths and models 4, 5 and 6 are modelled with 0.2 m anchorage lengths. The results of these analyses are presented in Figures 7.14 and 7.15, in terms of the load-slip respective load-normalised strain behaviours.


Figure 7.14 Load-slip responses of the force transfer tests

Figure 7.15 Load-strain responses of the force transfer tests


Studying Figures 7.14 and 7.15, it can be seen that a slight softening behaviour is achieved in models 1, 2 and 3. The effects of the force-transferring phenomenon are analysed in the models with the same material properties but with differing anchorage lengths. For example, the combinations of models 1 and 4, models 2 and 5 or models 3 and 6 can be compared. The relationship regarding the changes in anchorage lengths is common for all three combinations. All the combinations result in tolerating higher loads for longer anchorage lengths. It should be noted that these analyses were not run until failure occurred due to time considerations.

Three major changes to the bond-mechanisms are introduced in the models with the intention of studying the global response of the structure. These changes affect the bonding strengths between concrete and reinforcement, and reinforcement and reinforcement. The properties that are varied are the friction angle, dilatancy and cohesion strength, as illustrated in Table 7.2.

Table 7.2 Anchorage lengths and significant bond-mechanism model properties

In models 1 and 4, the bond-mechanism models are assigned the same material properties that were estimated from the unconfined pullout tests, i.e. φ = 20°, ψ = 7° and c = 5 MPa. The results of these applied properties are illustrated in Figures 7.14 and 7.15. Models 2 and 5 were then assigned different bond-properties of φ = 1°, ψ = 1° and c = 3.75 MPa in the bond model between the two steel reinforcement bars. The modification of these parameters limits the bond-slip behaviour by allowing for slight increases in the shear stresses with increasing normal stresses. However, these modifications did not result in large changes in the global responses, as illustrated in Figures 7.14 and 7.15.

Finally, to induce a global behaviour that is “less stiff”, the bond models between both the concrete and reinforcement, and the two reinforcement bars, are modified in models 3 and 6, compared with models 1 and 4. The bond model’s properties between the steel bars are the same as in models 2 and 5, however, the cohesion strength in the bond model between concrete and reinforcement is drastically reduced to 1 MPa. These modifications result not only in decreasing the global response’s stiffness, but also reducing the global response’s maximum load, as illustrated in Figures 7.14 and 7.15.

In Figures 7.16 and 7.17, the bond stresses produced in the layer between the reinforcement bars, from models 1 and 2, are presented. As can be seen in Figure


7.16, the shear stresses in the bonding layer are quite large. This is due to the non-associative model, which allows for increasingly high shear and normal stresses.

In Figure 7.17, the shear stresses are much lower. This is due to the properties of the bond model, which allow for very slight increases in shear forces in combination with increasing normal forces.

The shear stresses behave as expected, as illustrated in Figures 7.16 and 7.17. The shears stresses are largest in the ends of the bond layer when the load is initially applied. Thereafter, the stresses grow towards the middle of the layer whilst they decrease in the ends.

Figure 7.16 The shear stresses produced in the bond-mechanism between the two reinforcement bars in model 1 where the deformation is (a) 0.05 mm, (b) 0.9 mm, (c) 0.25 mm, and (d) 0.87 mm


Figure 7.17 The shear stresses produced in the bond-mechanism between the two reinforcement bars in model 2 where the deformation is (a) 0.05 mm, (b) 0.9 mm, (c) 0.24 mm, and (d) 0.73 mm

The crack development in model 1 is illustrated in Figure 7.18. It is assumed that the crack patterns are unrealistic, that is the cracking process is “more ductile” than it should be. This seems to be due to the consumption of too much energy in the cracking process. The initial response is satisfactory, but it can be ascertained that it is not easy capturing the correct failure mode, i.e. splitting failure, since the concrete cracks in all the gauss points. The number of cracks increase with increasing load instead of the cracking localising where the initial cracking begins. Since there is little known information concerning this phenomenon, validating these results is not possible. Nonetheless, that which can be presented from these results is the influence of certain bond-parameters on the response of the structure.


Figure 7.18 The crack development in model 1, at deformations of 0.18 mm, 0.26 mm, 0.40 mm respective 0.87 mm, as viewed from the side and top

When implementing this bond model in the global pile model, the initial response is too stiff. The parameters in this bond model are analysed for “non-driven” concrete, whilst in the global pile model respect is taken to the influences of driven concrete, which is explained in more detail in Section 7.3.


7.3 Validation of the Pile Model

The numerical model that produces a response that best agrees with the behaviour of the experimental tested pile is analysed in detail in Section 7.3.1. Furthermore, the influence of individual parameters and mechanisms is closely analysed in the following chapter, Section 7.3.2.

7.3.1 Validation of the Established Model

The primary verification data used for the established numerical model are the load-displacement curves, which were documented during the experimental tests performed by SP. The deflection of the pile was recorded in three points during the testing, one in the middle of the test element and two at a distance of 700 mm from the centre, according to Figure 3.17. This is the only data available from the experimental tests, and unfortunately, information such as crack pattern and failure modes were not recorded. Further information about the experimental tests can be found in Appendix B1. The results from the FE-analyses, for example, crack propagation, stress distributions and strains, were also analysed in order to verify the numerical model.

In Figures 7.19 and 7.20, the load-displacement responses resulting from the analyses of the FE-model and the experimental tests are illustrated. Studying the graphs in Figure 7.19 and 7.20, it can be stated that there is good agreement of the numerical model and the experimental tests’ responses.


Figure 7.19 Comparison between the load-deflection responses of the established FE-model and the experimental tests, in the outer measurement points

Figure 7.20 Comparison between the load-deflection responses of the established FE-model and the experimental tests, in the middle measurement points


Figure 7.21 Crack propagation in the FE-model, where the mid-deflection is (a) 0.5 mm, (b) 3.0 mm and(c) 13.5 mm

Studying Figure 7.21, it can be seen that cracking starts immediately in the crack bands and that bending cracks are the dominating crack type in these bands. Splitting cracks starts to propagate later in the analysis. It can also be seen that shear cracking is initiated in the area next to the end of the anchorage bar.


Figure 7.22 Concrete strains, xε , in the FE-model at a late stage, when the mid-deflection, cd , is 13.5 mm

In Figure 7.22, the strains in the longitudinal direction are presented. It can be seen then that large strains, i.e. deformations due to bending cracks, are concentrated in the crack bands and in the immediate area to the left of the end of the anchorage bar.

The parameters that were initially used in the bond-mechanism are a friction angle of 20°, a dilatancy angle of 7° and cohesion of 5 MPa. However, when analysing the results of these early analyses, the response of the structure was too stiff compared to the experimental tests made by SP. It was also noticed that the bond stresses are too large. As earlier mentioned, there is quite a large difference between the responses of driven concrete and “non-driven” concrete. Therefore, the parameters of the bond-mechanism have also been adjusted in order to match the overall behaviour of the driven pile structure. After testing the different concrete-reinforcement bond-slip parameters, the combination consisting of a friction angle of 20°, dilatancy angle of 7° and cohesion of 1 MPa provided satisfying results. The resulting bond stresses of this analysis can be seen in Figure 7.23. These modifications seem reasonable with regard to the impact from the driving process.

Figure 7.23 The bond stresses between the concrete and the principal reinforcement bar at (a) an early stage, cd = 5 mm and (b) a later stage, cd = 13.5 mm, where the surrounding concrete is considerably cracked.


When implementing the bond model, which was established in the pullout tests, into the global pile model, the initial response is too stiff and large bonds stresses occur between the bars. It is reasonable to assume that the stress transfer between the bars should be drastically reduced due to the driving process’ effect on the bond. Therefore, another bond-mechanism model between the bars was used: φ = 1°, ψ = 1° and c =3.75 MPa.

In Figure 7.24, the bond stresses in the layer between the reinforcement bars are presented. It can be seen in the different figures that the highest shear stresses are around 4.5 MPa, which are reasonable with regard to the impact of the driving process.

Figure 7.24 Bond stresses between the anchorage reinforcement bar and the principal reinforcement bar at (a) an early increment when cd = 5 mm and (b) a later increment when cd = 13.5 mm

Figure 7.25 Tensile stresses in the steel at (a) an early increment when cd = 5 mm and (b) a later increment when cd = 13.5 mm


7.3.2 Salient Factors Critical to the Response of the Model

Building an extensive FE-model is often made in an iterative, step-wise manner, where the effects of the different components are acknowledged one after another. In the establishment of the FE-model of the pile, changes were implemented successively so that the corresponding effects could be isolated and identified. In this section, the influences from distinctive features, such as crack bands, boundary conditions and the bond-mechanism model, are identified and analysed. With regards to the numerical model’s long computational time and the variety of influencing factors, focus in this study concentrates on the initial response, i.e. from when loading begins until the response begins to level out. However, some analyses were run longer and these results are also considered.

Figure 7.26 The load versus “mid-deflection” responses considering the different modifications made to the model


Figure 7.27 The load-deflection responses at the outer measurement point considering the different modifications made to the model

Model 1

In model 1, the entire cross-sectional surface of the splice is fixed in the longitudinal direction, except for the principal reinforcement bar in the tensile zone. As illustrated in Figures 7.26 and 7.27, the initial response of model 1 is very stiff. Studying the strains, xε , in Figure 7.28, it can be noted that there is a localisation of the crack propagation in the region immediately next to the symmetry plane.

Figure 7.28 Strains, xε , at load 65 kN and a large mid-deflection of 23 mm


Model 2

In model 2, the boundary conditions of the cross-sectional surface of the splice are modified. In this model, only the concrete in the compression zone, the top reinforcement and the anchorage reinforcement bar are fixed. Studying Figures 7.26 and 7.27, it can be noted that the initial response is somewhat “less stiffer” in comparison with model 1 when studying the mid-deflection. It can also be seen that the curve tends towards a slightly lower value of the “yielding load” when compared with model 1.

Figure 7.29 Strains, xε , at load 45 kN


Studying the strains in Figures 7.29 and 7.30, it can be seen that strains, xε , are equally large in the “bending crack zone” during the initial response of the structure. This indicates that no crack localisation initially occurs. However, in a later stage, a significant localisation occurs in the immediate area to the left of the end of the anchorage bar. This behaviour of the bending cracks is not reasonable.


Model 3

The initial slip that occurs in the locking device, due to deformations induced during the driving process, is simulated in this model. The initial slip of 0.3 mm is applied to the anchorage bar, which means that the anchorage bar responds freely until the actual slip is reached. Once the deformation corresponding to the initial slip occurs, the bar behaves with fixed boundary conditions in the X-direction. Otherwise, as in model 2, only the top reinforcement bar and the concrete in the compression zone is fixed, whilst the rest of the concrete cross-section is free in this model.

Studying Figures 7.26 and 7.27, it can be seen that this modification only effects the stiffness of the response at a very early stage, i.e. the curve “snaps” to the right, aligning itself with the empirical responses. Compared with the previous models, it can be noted that models 1, 2 and 3’s curves are parallel with one another, which clarifies the fact that this modification only influences the initial response. A similar effect is achieved in both the mid-deflection results as well as the outer deflection results.

Model 4

Implementing crack bands, with a drastically lowered tensile strength and fracture energy, results in largely impacting the structure’s initial behaviour. As in model 2, only the top reinforcement bar, the concrete in the compression zone and the anchorage bar’s X-translations are fixed, and the rest of the cross-section is free. The initial stiffness of the response is noticeably “less stiffer” in comparison with the other models. This change in the stiffness behaviour is recognised in both the mid-deflection and outer deflection responses, even though the changes were slightly less in the outer deflection response.

Studying Figure 7.31, where the strains, xε , are presented, it can be noticed that the crack propagation is localised in all three of the crack bands, as well as in the immediate area to the left of the end of the anchorage bar.



Model 5

In this analysis, relative ideally plastic properties are applied to the interface layer between the anchorage bar and the principal reinforcement bar. The combination of a friction angle of 1°, dilatancy of 1° and cohesion of 7.5 MPa are applied in the layer between the bars. The initial stiffness, concerning the mid-deflection response, is slightly “lowered,” whilst the response in the outer measurement point seems to be unaffected. Even though this analysis has not run especially long, the same tendency of the response occurs, as noticed in model 2, where the “yielding load” is higher compared to the experimental curves.

Figure 7.32 Strains, xε , at a load of 40 kN

Studying Figure 7.32, where the strains, xε , are illustrated, it can be noticed that the crack propagation has localised naturally in all three bands, without the help of predefined crack bands.

Figure 7.33 The development of bond stresses in the layer between the anchorage and the principal reinforcement bar. The upper end is where the locking device is located. The applied load, P , is equal to (a) 22 kN, (b) 35 kN and (c) 42 kN.


8 Conclusions The suitability, applicability and appropriateness of the established driven prefabricated concrete pile numerical model are discussed in this chapter. Critical details that have been discovered and analysed are also discussed. Subsequently, future applications of this model, and improvements in this analysis and in the dimensioning of piles, are also discussed.

8.1 Outcomes

The load-deflection response of the spliced area of the pile’s resulting numerical model agrees exceptionally well with the empirical results. The model also provides reasonable results regarding bond-stresses and global crack-patterns. This agreement, though, has occurred partly by using actual and tested characteristics and parameters, and partly by choosing characteristics and parameters that produce the response resembling the empirical results. For example, it is easy to define such parameters as the steel’s elasticity module and tensile strength. On the other hand, it is hard to estimate the influences of such pre-loading factors as handling, transportation and driving of the concrete, e.g. which cohesion value shall be used for the bond-mechanism between two reinforcement bars. This may not be the most scientific method to conduct an analysis but since this field of research within finite element modelling is still relatively unexplored, and the scope of this study limited, this method can be deemed acceptable.

Simulating the exact response of the pile is one important result, but another equally important result is how the different parameters and mechanisms influence the response of the numerical pile model. As explained in Section 7.3.2, slight changes in the model result in considerable changes in the behaviour of the model. The most notable factors that influence the pile’s response are the bond-mechanism models and the effects of handled/driven concrete, i.e. the predefined crack bands and the “initial slip” mechanism. All of these factors are included in the model but not all of these factors are based on verified facts. Many realistic assumptions have been concluded, e.g. the effect of driven concrete on the fracture energy, and even though the results are more than acceptable, further analysis of these factors is recommended.

The combination of these factors results in an acceptable initial stiffness that is very similar to the response of the empirical tests, Figures 7.19 and 7.20. However, each of these factors individually influences the initial stiffness of the response. The inclusion of the “initial slip” mechanism in the model almost directly produces a “snap” to the right in the response curve. This snap aligns the model’s curve almost perfectly with the empirical test’s curves. The greatest influence on the model’s response is the addition of the crack bands. Cracking in the bands occurs directly after loading has begun and they result in significantly decreasing the initial stiffness of the model. The combination of the crack bands and correctly modelling the boundary conditions in the splice produce an even “less stiffer” response. The modifications of the bond-mechanisms due to the weakening effects from the handling, transporting and driving of the pile also somewhat decrease the stiffness of the response.


Another important factor, which influences the behaviour of the pile, is the splice itself. The splice’s details have been partly modelled in this study, and the adaptation of certain parameters in order to match the empirical test’s response can therefore be misleading, since it is not known what the influence of all of the splice’s details has on the global response. However, it can be assumed that the most critical factors that contribute to the global response of the pile model have been included.

Be that as it may, it is important to be aware of the fact that the model is only validated according to one type of testing procedure. There is still a possibility that the behaviour of the model will not coincide to the same extent if other empirical testing procedures are compared. On the other hand, according to the agreement of the results, and with a few small adjustments, this model is acceptable in simulating the behaviour of the experimental pile, which the FE-model has been based on.

According to the results from numerous tests of different models regarding meshing and geometry, e.g. symmetry lines, it can also be concluded that the numerical model is the most time and computational effective model that simulates the global response of the pile.

Numerical modelling of reinforced concrete structures has often a complicated nature with many critical details. In the building of the numerical model of the spliced concrete pile, the bond-slip mechanism is a very essential feature for the overall behaviour of the structure. The most difficult part in the building process was establishing a suitable bond-mechanism. The concrete-reinforcement bond’s characteristics are based on empirical values, theoretical data and advice from respective experts, whilst, in order to achieve proper force transference between the spliced bars, a special, “weaker” model was used in the interfacing layer. According to the results presented in Section 7.3, the bond-mechanisms have a significant role in the initial load-deflection response of the global model.

In order to catch the true response of the structure, the model is built with solid elements. Analyses with certain simplifications, such as the usage of beam elements and tied mesh were performed, but unfortunately the accuracy of the results was not satisfactory. However, by modelling with the symmetry assumption, the number of degrees of freedom is drastically reduced. Nonetheless, the computational time is still quite large, as it takes about ten days to run a complete analysis with a personal computer possessing a CPU of 1.4 GHz.

The Multi-cracking Concrete model implemented in LUSAS has shown to be some what mesh dependent regarding the modelling of the fracture energy. In some situations, the dissipated fracture energy in the analysis can be many times higher than the value used for input. This is a characteristic, which is very important to be aware of when using this material model since it has an evident affect on the results. The cracking patterns produced in the analyses seem reasonable, and can be assumed as those that occur in reality. However, consideration must be taken to the fact that the concrete model is not entirely reliable and other crack patterns may occur.

It should also be noted that critical factors that contribute to producing accurate results are the element ratio and mesh intensity, plus the combination of increment length and value of the norms.


It can therefore be concluded that with the establishment of a sound method, appropriate and applicable constitutive models, models that reflect the non-linear behaviour of reinforced concrete and the bond-mechanism, that a sound three-dimensional numerical model of the spliced area for driven prefabricated concrete piles has been constructed and validated with existing full-scale load tests.

8.2 Future Applications and Improvements

Of course, improvements can always be made to a FE-model, and in the established numerical model analysed in this study, this is no exception. Although the current numerical model produces very agreeable and acceptable results there are areas that can be improved. One suggested improvement regards the predefined crack bands that are applied to the model, which arise in the preloading stage. All the concrete in the crack bands consist of “weakened” concrete, thus resulting in through-cracks, which subsequently free the reinforcement bars in these zones. The bi-effects of predefining crack bands are that the cracking due to bending is strictly localised to these zones. In reality, the preloading tensile cracks that occur are not through cracks. Therefore, it would be appropriate to apply the weakened concrete to a depth that corresponds with the depth that the cracks reach in reality.

The splice connection consists of many features, as explained in Chapter 3. However, due to time considerations and complications in modelling the splice’s details, certain splice details were ignored in this study. However, it is seen as a natural progression to model all of the details of the splice in order to receive an even truer response of the pile. One of the details not modelled is the locking device, which creates high stress concentrations in the compression zone of the concrete. This detail seems like an important factor to include in the model since it would contribute to a better simulation of a pile’s behaviour. It should be noted that problems could arise concerning symmetry when modelling the locking device, which would result in a less simplified model and longer calculation times.

The established numerical model has been validated with results that are produced from one type of experimental test on one type of driven pile. In determining whether or not this numerical model is acceptable, the model should be applied and validated with other pile and test types. Furthermore, the development of a FE-model that can be applied to all types of driven prefabricated concrete piles could confirm, or lead to changes, of the reduction factor used to take respect to the influences of the driving load, as mentioned in Section 3.5.

As well as comparing the model to other pile types, a simple change to the model can be made to the cross-section. Since there are other experimental tests made by SP, with other cross-section types, a change in the model’s cross-section, and thereafter a comparison with the empirical tests, can further confirm the suitability of the numerical model.

During the development of the FE-model it has been concluded that the driving process has a crucial impact on the pile’s behaviour. It would be of great interest to validate this statement with experimental tests of non-driven piles, which are conducted in the same manner as the tests conducted by SP. With these tests, the


numerical pile model’s responses can then be compared with these tests. This would help clarify uncertainties surrounding the affected parameters, such as the bond-mechanism and the driving induced tensile cracks.

The established model can hopefully be used in further analyses, where the load conditions are modified. A natural progression is the application of an axial force, in addition to the bending moment. This load combination simulates the real load combination that occurs after the installation of a pile. Experimental tests loaded both axially and with a moment already exist with which this type of analysis can be verified. The results could then be compared and discussed with the normal force versus bending moment capacity curves, as illustrated in Figure 3.14.

The established model can also be implemented and analysed in a more global model, where geo-technical properties and influences are considered, such as the modulus of sub-grade reaction of the soil and initial deflection of the piles, including angular deviation in the joint.

Another simple addition to the global model can be the inclusion of all of the principal reinforcement bars and the shear reinforcement in the model. This would naturally result in increasing the stiffness of the pile, thus requiring the model’s parameters to be further modified. The inclusion of two reinforcement bars in the compressed zone, instead of the current reinforcement bar, can also be added to the model. It should be noted that the increase in computational time might not be worth the slight improvement in the pile’s response.

LUSAS 13.6-2 contains two concrete models, that is, the currently applied Multi-cracking Concrete Model and the Multi-cracking Concrete with Crushing Model. It may be necessary to apply the crushing model to a numerical model that contains the locking device.

It is also possible that different combinations of the parameters used to define the material models, such as the friction and dilatancy angles and the cohesion strength, can be applied to the numerical model and similar results produced. The application of different combinations can be realistic after some of the suggested changes are made to the model. It is possible that the currently applied combinations compensate for the lack of certain details that have not been modelled, for example, the locking device, and if these details are added to the model the parameters shall be modified accordingly.

Testing of different combinations should, however, be initially conducted at a more local level, for example with such reinforced concrete models used in the pullout and load transfer tests. This would not only drastically reduce the calculation times but also help in isolating the studied parameters and mechanisms.

Before testing these simpler models and the global pile model, it is recommended that the concrete model is tested and analysed with the next available version of the FE-program, that is, LUSAS 13.7. The LUSAS support team suggested that in version 13.7, a new concrete cracking-crushing model would be released, which provides better convergence. Even if this is the case, a more definite understanding of the fracture energy is recommended, and therefore, it is suggested that further tests be conducted on this material model.


The modelling of the bond action is a critically important factor within the art of simulating the response of reinforced concrete. As it has been pointed out throughout this thesis, knowledge about modelling the bond action with the finite element method is very limited. Therefore, it is highly recommended that this mechanism is studied in more detail with simple tests such as the pullout and load transfer tests analysed in this study. It is even recommended to apply the modified parameters that result from handled and driven concrete in these simple models. Isolating problem areas and complicated mechanisms are necessary in understanding their influence on the global scale.

One more characteristic of the model that justifies improvement is the computational time. In the present model, the calculation time is quite large and reducing this would facilitate the use of the model.


9 References Boverket (2004): Boverkets handbok om betongkonstruktioner BBK 04, (Boverket´s

Handbook on Concrete Structures BBK 04), Boverket, Byggavdelningen, Karlskrona, Sweden

Broms (2005), Prof. Bengt B.: Foundation Engineering, http://www.geoforum.com/knowledge/texts/index.asp?Lang=Eng [2005-01-31]

CEB Bulletin d’Information 195 (1990): CEB-FIP Model Code 1990, Comité Euro-International du Béton, Paris, France, 1990

Commission of Pile Research (1984): Second International Conference on the Application of Stress-wave Theory on Piles, Roland Offset AB, Linköping, Sweden

Commission of Pile Research (1995): Beräkning av dimensionerade lastkapacitet för slagna pålar med hänsyn till pålmaterial och omgivande jord, Rapport 84a, (Dimensioning Load Capacities of Driven Piles with respect to the Pile’s Material and the Surrounding Soil, Report 84a), Roland Offset AB, Linköping, Sweden

Commission of Pile Research (1998): Dimensioneringsprinciper för pålar, Rapport 96:1, (Dimensioning Principles of Piles, Report 96:1), Roland Offset AB, Linköping, Sweden

Concrete Structures (2003): Concrete Structures - Advanced Course, Compendium, Lecture Notes and Extracts, Department of Structural Engineering, Chalmers University of Technology, ARB NR:13, Göteborg, Sweden, 2003

Crisfield M.A. (1991): Non-linear Finite Element Analysis of Solids and Structures, Volume 1, John Wiley & Sons Ltd., West Sussex, England, 1991

Crisfield M.A. (1997): Non-linear Finite Element Analysis of Solids and Structures, Volume 2, Advanced Topics, John Wiley & Sons Ltd., West Sussex, England, 1997

Geoforum.com (1998-2005): http://www.geoforum.com/ [10-2004 to 03-2005]

Haga K. and Olausson K. (1998): Olinjär finit elementanalysis av utdragsförsök (Non-linear Finite Element Analysis of a Pullout Test), Masters Thesis. Department of Structural Engineering, Chalmers University of Technology, Publication no. 98:7, Göteborg, Sweden, 1998

Holm G., Olsson C. (1993): Pål Grundläggning, (Foundation Piles), AB Svensk Byggtjänst, Solna, Sweden

Holmberg G. (2001): Fatigue of Concrete Piles of High Strength Concrete Exposed to Impact Load, Licentiate Thesis. Department of Structural Engineering, Chalmers University of Technology, Publication no. 01:3, Göteborg, Sweden, 2001

Holmberg G. (2004): Provningsmetoder för skarvar och bergskor, (Testing Methods of Splices and Rock Shoes), PM, Skanska Teknik AB, Reference no. 4201.001, 2004


Jefferson A. D. (1999): A Multi-crack Model for the Finite Element Analysis of Concrete, Proc BCA Concrete Conf, 1999

Jirásek M. (2000): Numerical Modelling of Deformation and Failure of Materials, Lecture Notes, Department of Structural Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague, Czech, 2000

Kröplin B., Weihe S. (1997): Constitutive and Geometrical Aspects of Fracture Induced Anisotropy, Institute for Statics and Dynamics of Aerospace Structures, University of Stuttgart, Stuttgart, Germany, 1997

Lundgren K. (1999): Three-Dimensional Modelling of Bond in Reinforced Concrete, Ph.D. Thesis. Department of Structural Engineering, Chalmers University of Technology, Publication no. 99:1, Göteborg, Sweden, 1999

LUSAS (2004): LUSAS Manual 13.6, Lusas Ltd, Surrey, United Kingdom, 2004

Magnusson J. (2000): Bond and Anchorage of Ribbed Bars in High-Strength Concrete, Ph.D. Thesis. Department of Structural Engineering, Chalmers University of Technology, Publication no. 00:1, Göteborg, Sweden, 2000

Ottosen N., Petersson H. (1992): Introduction to the Finite Element Method, Prentice Hall, Essex, England, 1992

Plos M. (2000): Finite Element Analysis of Reinforced Concrete Structures, Department of Structural Engineering, Chalmers University of Technology, Compendium 96:14, Göteborg, Sweden, 2000

Pålstatistik för Sverige 2003 (Pile Statistics for Sweden 2003), information 2004:1 www.palkommissionen.org [2005-02-25]

Thuresson P.A. (1991): Provning av betongpålar, (Testing of Concrete Piles), Report, Byggnadsteknik, Swedish National Testing and Research Institute, Borås, Sweden, Reference no. 91B1,3019, 1991

Thuresson P.A. (1991): Provning av betongpålar, (Testing of Concrete Piles), Report, Byggnadsteknik, Swedish National Testing and Research Institute, Borås, Sweden, Reference no. 91B1,3019 A, 1991


Appendix A1 - The Laboratory Tested Splice


Appendix A2 - The Laboratory Tested Pile Element


Appendix B1 - SP’s Experimental Test Reports


Appendix C1 - Verification of the LUSAS Multi-cracking Concrete Material Model The non-linear multi-crack concrete material model is used to represent the non-linear material effects associated with the cracking of concrete. The model contains HX8 elements and is loaded with a 1 mm prescribed displacement, as in Figure C1. The concrete analysed is C50/60 and the material model’s properties are illustrated in Table C1.

Table C1 The material model’s properties

Concrete’s Material Properties

E = 37 GPa

υ = 0.15

ρ = 2.3 kg/m3

tf = 2.5 MPa

fG = 100 Nm/m2

Note: The convergence tolerances need to be slacker for concrete than for other non-softening non-linear models. Typical values for problems that are dominated by fracture behaviour are 2% for the force residual norm, ψγ , and 1% for the displacement norm, dγ .

This multi-cracking concrete FE-model is modelled as both a cube and as a rectangular block, as seen in Figure C1, in which both models are meshed with one HX8 element. The load applied to the first two test cases is in the Y-direction, whilst the load applied in the third test case is applied in the X-direction.


Figure C1 a) test case 1, b) test case 3 and c) test case 6

When the length of a structure is less than a so-called critical length, LCR, the concrete material’s behaviour is able to “soften,” i.e. can continue past the yield point. All these tests were conducted with respect taken to the critical length. The softening behaviour occurs, as illustrated in the stress versus displacement Figure C2, for all the different tests.

Figure C2 The stress versus displacement relationship for the different test cases


The relationship illustrated in the above figure contains elastic and plastic states. The elastic state is verified with elasticity theory, as shown in Table C2.

Table C2 Elastic deformations

Elastic Deformation: Elastic Theory Elastic Deformation: LUSAS

0.025 mm 0.025 mm

The non-linear behaviour of the concrete model, as explained earlier, should soften. This softening behaviour is due to the fracture energy dissipated when the first crack appears. One way to verify the non-linear behaviour of concrete is to confirm that the area under the stress versus displacement curve, i.e. the fracture energy, is equal to the applied fracture energy.

This concrete model’s behaviour, regarding fracture energy, is deemed hard to control with the multi-cracking concrete model. Since the element characteristic length cannot be predefined, that is, it is already defined as part of the model, the fracture energy dissipated in this concrete model is not that which is expected. For example, the area under the curve is calculated for the cube model, test 1, and the resulting fracture energy is approximately double the applied fracture energy, Table C3. The areas under the curves for all the models, together with the applied fracture energies, are presented in Table C3. The relationship between the crack building, load direction and the element ratio length is simply illustrated in Figure C3.

Table C3 The differences between the applied and the integrated fracture energies

Load Case

Geometry Load Direction

Element Ratio

Applied fG [Nm/m2]

Integrated fG

[Nm/m2]

Test 1 Cube Y 1:1 100 200

Test 2 Rectangle Y 1:1.5 100 261

Test 3 Rectangle Y 1:2 100 316

Test 4 Rectangle Y 1:4 100 500

Test 5 Rectangle X 1:1.5 100 174

Test 6 Rectangle X 1:2 100 159

Test 7 Rectangle X 1:4 100 125


Figure C3 Tests of the concrete model with different load directions, geometry and crack directions, plus the resulting calculated dissipated fracture energies

It is stated in LUSAS theory that the concrete material model is not mesh and element dependent (LUSAS 2004), but it can be clearly seen that this is not the case. Since the element characteristic length cannot be defined, whereby the fracture energy cannot be controlled, an approximation of the predefined fracture energy in the concrete model has been made.

This approximation is based on the element ratio recommended by LUSAS, that is, 1:4. According to the results from tests 4 and 7, upper and lower boundaries can be determined for the fracture energy since these ratios are also 1:4. The applied fracture energy’s boundaries can be set to 20 - 80% of the characteristic fracture energy. Taking respect to both crack building directions, a 50% reduction of the fracture energy is thus recommended regarding this phenomenon.

It is very hard to approximate the relationship of the applied fracture energy and the dissipated fracture energy in the analyses. This model is a limitation in the analysis,


and ideally, it would be much more accurate if the element characteristic length could also be defined pre-analysis.

The crack direction of the two different types of models is illustrated in Figure C4.

Figure C4 Cracking in the Gauss points for a few certain test cases: (a) test case 3 and (b) test case 6

The model is verified in the elastic state with correct displacements and a linear stress versus strain response. The plastic state is somewhat more complicated, and regarding the applied fracture energy, the element ratios and direction of crack building are taken respect to in further application of this model.


Appendix C2 - Verification of the LUSAS Non-associative Mohr-Coulomb Material Model An analysis of a cube, with the non-associative material properties, is exposed to a shearing load and analysed in LUSAS. The non-associative Mohr-Coulomb material model is analysed and verified according to the defined parameters and results presented below.

Table C4 Test parameters for the two non-associative Mohr-Coulomb models

Test 1 Test 2


E = 35 GPa E = 35 GPa

υ = 0 υ = 0


c = 1 MPa c = 1 MPa

iφ = 45° iφ = 0.1°

fφ = 45° fφ = 0.1°

ψ = 45° ψ = 0.1°

ch = 0 ch = 0

fε = 1 fε = 1


Figure C5 Results from test 2

Note: The results from test 1 and an illustration of the deformed cube are illustrated in Section 7.1.

As explained in Section 7.1, test 1 illustrates the analysis when the friction and dilation angle are defined as 45°, and consequently the shear and normal stresses should be equal. In test 2, since the angles are defined as almost zero, the resulting angle in Figure C5, as expected, is also very close to zero. An angle of zero results in only shear stresses and no normal stresses.


Appendix C3 - Verification of the LUSAS von Mises Material Model A simple tensile load was applied to a 100 100 500× × mm steel bar, with the characteristics of k500 steel, and analysed in LUSAS. The model responds according to elastic-plastic theory and is verified according to the results presented in Tables C5 and C6, and Figure C6.

Table C5 Material properties of k500 steel

Steel’s Material Properties

Elastic Properties

E = 200 GPa

υ = 0.2

ρ = 7.8 kg/m3

Plastic Properties

yf = 500 MPa

Table C6 Elastic deformations

Elastic Strain: Elastic Theory Elastic Strain: LUSAS

0.0025 approx. 0.0025

Figure C6 a) The deformed steel model and b) the simplified stress-strain curve for the steel model


Appendix D1 - The Confined Pullout Tested FE-model Geometry, Boundary Conditions and Meshing

The confined pullout analyses that were performed in order to investigate the parameters of the bond slip model were carried out on the FE-model presented in Figure D1. In this model, the element type HX8 was exclusively used in all volumes.

Figure D1 Geometry, boundary conditions and material assignments of the FE-model


Material Properties

Table D1 The material properties of concrete, steel and the bond-mechanism for the test cases

Concrete Steel Bond-mechanism

Elastic Properties Elastic Properties Elastic Properties

E = 30 GPa E = 192 GPa E = 30 GPa

υ = 0.15 υ = 0.3 υ = 0

ρ = 2.4 kg/m3 ρ = 7.8 kg/m3 ρ = 0 kg/m3

Plastic Properties Plastic Properties Plastic Properties

tf = 2.2 MPa yf = 569 MPa c = varying

fG = 90 Nm/m2 iφ = varying

fφ = varying

ψ = varying

ch = 0

fε = 1

Solution Procedure

The analysis was carried out in a displacement-controlled manner with the initial increment length of 0.00625 mm and the restricting parameter, “maximum change of load factor,” is set to the same value. The convergence criteria are set according to Table D2.

Table D2 Convergence criteria





Appendix D2 - The Unconfined Pullout Tested FE-model Geometry, Boundary Conditions and Meshing

The unconfined pullout analyses, that were performed in order to calibrate the parameters of the bond slip model, were carried out on the FE-model presented in Figure D2. In this model, the element type HX8 was exclusively used in all volumes.

Figure D2 Geometry, boundary conditions and material assignments of the FE-model


Material Properties

Table D3 The material properties of concrete, steel and the bond-mechanism for the unconfined test cases

Concrete Steel Bond-mechanism

Elastic Properties Elastic Properties Elastic Properties

E = 30 GPa E = 192 GPa E = 30 GPa

υ = 0.15 υ = 0.3 υ = 0

ρ = 2.4 kg/m3 ρ = 7.8 kg/m3 ρ = 0 kg/m3

Plastic Properties Plastic Properties Plastic Properties

tf = 2.9 MPa yf = 569 MPa c = varying

fG = 50 Nm/m2 iφ = varying

fφ = varying

ψ = varying

ch = 0

fε = 1

Solution Procedure

The analyses were carried out with a displacement-controlled load, with the initial increment length of 0.00625 mm and the restricting parameter, “maximum change of load factor,” set to the same value. The convergence criteria are set according to Table D4. With regard to the LUSAS limitation of allowed increments processed in the plot file, i.e. result file, the parameter “incplt” has to be increased from the default value of 1 if more than 1245 increments need to be run. If the value of “incplt” is set to 2, the results of every second increment will be stored in the plot file.


Table D4 Convergence criteria





Appendix D3 - The Force Transfer/Bond-slip Model Modelling the bond-slip between concrete and steel with the finite element method has still not been perfected in the world of finite elements, and consequently, there is an absence of established tests and methods to validate results. In order to gain a better understanding of this mechanism, a simple reinforced concrete model is tested with a tensile load, as illustrated in Figures 7.11, 7.12 and 7.13, simulating the bond mechanisms between concrete and reinforcement. The concrete and steel material properties of the model are presented in Table D5.

It should be noted that it is assumed that the amount of concrete found between two adjacent reinforcement bars is minimal and concrete is therefore not modelled between these two bars. Only the bond-mechanism model is located between the bars.

Table D5 The material properties of concrete, steel and the bond-mechanism for all the test cases

Concrete Steel


E = 30 GPa E = 192 GPa

υ = 0.15 υ = 0.3

ρ = 2.4 kg/m3 ρ = 7.8 kg/m3


tf = 2.2 MPa yf = 569 MPa

fG = 90 Nm/m2

The simulation of the load transfer between two reinforcement bars is tested with different test cases, i.e. different anchorage lengths, Table D6 - D8. In combination with gradually increasing anchorage lengths, and otherwise keeping the same material characteristics, the model responds with relative increasing yield loads, as expected, Figure 7.12. As the anchorage length increases, the yield loads should converge towards the yield stress of the steel.


Table D6 The bond-mechanism model’s material properties for Models 1 and 4

Concrete/Reinforcement Reinforcement /Reinforcement


E = 30 GPa E = 30 GPa

υ = 0 υ = 0

ρ = 0 kg/m3 ρ = 0 kg/m3


c = 5 MPa c = 5 MPa

iφ = 20° iφ = 20°

fφ = 20° fφ = 20°

ψ = 7° ψ = 7°

ch = 0 ch = 0

fε = 1 fε = 1





E = 30 GPa E = 30 GPa

υ = 0 υ = 0

ρ = 0 kg/m3 ρ = 0 kg/m3


c = 5 MPa c = 3.75 MPa

iφ = 20° iφ = 1°

fφ = 20° fφ = 1°

ψ = 7° ψ = 1°

ch = 0 ch = 0

fε = 1 fε = 1





E = 30 GPa E = 30 GPa

υ = 0 υ = 0

ρ = 0 kg/m3 ρ = 0 kg/m3


c = 1 MPa c = 3.75 MPa

iφ = 20° iφ = 1°

fφ = 20° fφ = 1°

ψ = 7° ψ = 1°

ch = 0 ch = 0

fε = 1 fε = 1


Appendix E1 - Detailed description of the Established FE-model Geometry, Boundary Conditions and Meshing

The geometry and boundary conditions of the developed FE-model can be seen in Figure E1. All the features of the model are meshed with the LUSAS element type HX8, except for the joint at the “slip support,” which has been assigned JNT4 joint elements.

Figure E1 Geometry, boundary conditions and material assignments of the FE-model


Material Models

The characteristics of the material models used are presented in Section 6.1.

Solution Method

The parameters used for the non-linear analysis are presented in Section 6.1

Special Notes

• The system parameter LFRADD is set to the value 4000. If the default value of 1000 had been used, the scratch file would probably have grown too large and subsequently increased the computational time.

• The system parameter DECAYL, is set to the value 106 instead of the default 104. The LUSAS Solver monitors the condition of the stiffness matrix and DECAYL is the tolerance threshold. Any values above this threshold result in warnings. If three warnings are achieved, the analysis will be terminated. However, according to the LUSAS Manual, values up to 107 are reasonable.


Appendix E2 - Additional results of the FE-analysis In this appendix additional results of the FE-analysis are presented.

Figure E2 Concrete strains, xε , when mid-deflection = 2 mm


Figure E5 Concrete strains, 1ε , when mid-deflection = 6 mm

Figure E6 Concrete strains, 1ε , when mid-deflection = 13,5 mm


Figure E7 Concrete strains, 1ε , when mid-deflection = 20 mm

Figure E8 Steel stresses, xσ , when mid-deflection = 6 mm


Figure E9 Steel stresses, xσ , when mid-deflection = 20 mm

Figure E10 Concrete stresses, xσ , when mid-deflection = 2 mm


Figure E13 Concrete stresses, xσ , when mid-deflection = 13,5 mm


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