XFEM for reinforced concrete

Aalborg University Structural and Civil Engineering, 10th Semester School of Engineering and Science Studyboard of Civil Engineering Sohngårdsholmsvej 57 www.bsn.aau.dk

Synopsis:

Title:

Numerical analysis of a Reinforced Concrete Beam in Abaqus 6.10

Project period: Project group:

B10K, spring 2011 B122b

Baldvin Johannsson

Poul Reitzel

Supervisors: Christian Frier Print runs: 10 Number of pages: Appendix: Completed:

This report deals with the modeling of cracks in a three-dimensional reinforced concrete beam subjected to three point bending. The commercial nite element program Abaqus 6.10 is used to model the beam. In Abaqus 6.10 the eXtended Finite Element Method, abbreviated XFEM, is used to model the cracking in cooperation with the Concrete Damaged Plasticity material model, abbreviated CDP, to model the tensile and compressive behavior. Prior to modeling the three-dimensional beam, benchmark tests are performed to validate the quality of the implementation of the XFEM and the CDP material model. Load-deection curves are analyzed for two numerical models; with and without the XFEM, and compared to analytical load-deection relations from EN 1992-1-1 [2004]. The cross-sectional stress distribution is plotted for various stages on the loaddeection curves for the numerical models and compared to a theoretical cross-sectional stress distribution at the corresponding stages. Crack widths, spacing and patterns are analyzed and compared to expressions given in EN 1992-1-1 [2004]. This report has shown that it is possible to combine the CDP material model with the XFEM in order to visualize cracks in reinforced concrete beams. In general, good agreement is found between analytical and numerical results. However, it is concluded that improvements are necessary to the implementation of the XFEM in Abaqus 6.10.

The report's content is freely available, but the publication (with source indications) may only happen by agreement with the authors.

Preface This report is a product of Poul Reitzel's and Baldvin Johannsson's project work at the 4th semester of the master degree in Structural and Civil Engineering at Aalborg University. The project has been completed within the period of 1st of February, 2011, to the 10th of June, 2011, under the supervision of Christian Frier. The report is prepared and made in accordance and compliance with the current curriculum of the 4th semester of the master programme in Civil Engineering at Aalborg University, Denmark. The project is based on the theme "Numerical analysis of a Reinforced Concrete Beam in Abaqus 6.10". The project aims at the increase of knowledge to apply an advanced computational method for the evaluation of crack formation and propagation in reinforced concrete. The project report consists of two parts, the main project and the appendix. A reference to the appendix can be: Appendix B. The main project examines the use of the extended nite element method, abbreviated XFEM, in combination with the Concrete Damaged Plasticity material model, abbreviated CDP, in Abaqus 6.10 to model a three-dimensional reinforced concrete beam loaded to failure. The project report uses the Harvard method of bibliography with the name of the source author and year of publication inserted in brackets after the text, for example: Irwin [1958]. The list of all references is found in the bibliography at the end of the report. The les used in Matlab and Abaqus 6.10 can be found on the attached CD. An introduction to the les and how they work can be found in Appendix B. The authors expect the reader to have a basic knowledge of the standard nite element method. Experience with computational engineering within the nite element method framework will make the interpretation of certain technical terms easier.

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Table of contents Chapter 1 Introduction

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Chapter 2 Fracture mechanics for concrete

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2.1 2.2 2.3

The fracture process in concrete . . . . . . . . . . . . . . . . . . . . . . . . . 8 Linear elastic fracture mechanics . . . . . . . . . . . . . . . . . . . . . . . . 10 Non-linear fracture mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Chapter 3 Smeared vs. discrete crack modelling

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Chapter 4 The eXtended Finite Element Method

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3.1 3.2 4.1 4.2

4.3 4.4 4.5

4.6 4.7 4.8 4.9

Smeared crack concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Discrete crack concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Discontinuities and high gradients . . . Level-set method . . . . . . . . . . . . . 4.2.1 Description of closed interfaces . 4.2.2 Description of open interfaces . . General formulation of the XFEM . . . Choice of enriched nodes . . . . . . . . . Global enrichment functions . . . . . . . 4.5.1 Weak discontinuities . . . . . . . 4.5.2 Strong discontinuities . . . . . . 4.5.3 Singularities . . . . . . . . . . . . Cohesive segments method . . . . . . . . Phantom-node method . . . . . . . . . . Numerical integration of the weak form Governing equations . . . . . . . . . . .

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Chapter 5 Discontinuous modeling in Abaqus

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Chapter 6 Verication of the XFEM and Abaqus

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6.1 6.2 6.3 6.4 6.5

Crack-hole interaction . . . . . . . . . . . . . . . . . . . . . . Verication of the CDP material model . . . . . . . . . . . . Crack propagation in a concrete beam in three point bending Crack formation analysis of a reinforced concrete plate . . . . Concluding remarks . . . . . . . . . . . . . . . . . . . . . . .

Chapter 7 Results and discussion of 3d beam analysis 7.1

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Model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 v

Group B122b - Spring 2011

7.2 7.3

7.1.1 Boundary conditions . . . . . . . 7.1.2 Input for the material model . . 7.1.3 Reinforcement Model . . . . . . List of studies . . . . . . . . . . . . . . . Results and discussion . . . . . . . . . . 7.3.1 Load-deection curves . . . . . . 7.3.2 Cross-sectional stress distribution 7.3.3 Crack widths and spacing . . . . 7.3.4 Crack pattern . . . . . . . . . . .

TABLE OF CONTENTS . . . . . . . . .

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69 70 72 74 74 74 79 84 87

Chapter 8 Conclusion

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Chapter 9 Suggestions for future work

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Appendix A Concrete Damaged Plasticity material model

A.1 Concrete Damaged Plasticity . . . . . . . . . . . . . . . . . A.1.1 Stress-strain relations . . . . . . . . . . . . . . . . . A.2 Yield function . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.1 Damage and stiness degradation for unixial loading A.2.2 Flow rule . . . . . . . . . . . . . . . . . . . . . . . .

Appendix B Guide to Appendix CD B.1 B.2 B.3 B.4 B.5 B.6 B.7 B.8 B.9 B.10 B.11 B.12 B.13 B.14 B.15 B.16 B.17 B.18 B.19 B.20 B.21 B.22 B.23 B.24 B.25

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DVD 1 . . . . . . . . . . . . . . . . . . . . . . . . 3D-Beam XFEM.cae . . . . . . . . . . . . . . . . Maximum and minimum reinforcement ratio.pdf Crack widths and spacing.pdf . . . . . . . . . . . Shear deformation.pdf . . . . . . . . . . . . . . . Ultimate load and corresponding deection.pdf . Seed45XFEM.odb . . . . . . . . . . . . . . . . . Seed50XFEM.odb . . . . . . . . . . . . . . . . . Seed60XFEM.odb . . . . . . . . . . . . . . . . . Seed70XFEM.odb . . . . . . . . . . . . . . . . . Seed80XFEM.odb . . . . . . . . . . . . . . . . . Seed150XFEM.odb . . . . . . . . . . . . . . . . . DVD 2 . . . . . . . . . . . . . . . . . . . . . . . . 3D-Beam CDP.cae . . . . . . . . . . . . . . . . . Seed45CDPFull.odb . . . . . . . . . . . . . . . . Seed50CDPFull.odb . . . . . . . . . . . . . . . . Seed60CDPFull.odb . . . . . . . . . . . . . . . . Seed70CDPFull.odb . . . . . . . . . . . . . . . . Seed80CDPFull.odb . . . . . . . . . . . . . . . . Seed150CDPFull.odb . . . . . . . . . . . . . . . . Seed50CDPRed.odb . . . . . . . . . . . . . . . . Seed60CDPRed.odb . . . . . . . . . . . . . . . . Seed70CDPRed.odb . . . . . . . . . . . . . . . . Seed80CDPRed.odb . . . . . . . . . . . . . . . . Seed150CDPRed.odb . . . . . . . . . . . . . . . .

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TABLE OF CONTENTS

Bibliography

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Resumé På nuværende tidspunkt er der ingen tilgængelige resultater for en fuldstændig analyse af revnedannelse i armerede betonkonstruktioner i Abaqus 6.10, der ved hjælp af den såkaldte "eXtended Finite Element Method", forkortet XFEM, er i stand til at forudsige revnemønstre og revnevidder. Samtidig tilbyder Abaqus 6.10 muligheden for, at modellere udviklingen i stivheden for konstruktioner hele vejen frem til brud ved hjælp af materialemodellen Concrete Damaged Plasticity, forkortet CDP. Modeller, der kan forudsige revnevækst, revnemønstre og revnevidder i armerede betonkonstruktioner, er påkrævede i henhold til at overholde krav til revnevidder og revneafstande fra gældende normer. En mere præcis analyse af revnevidder og revneafstande er motivationen bag den forestående rapport. Analyse af konstruktioner med komplekse geometrier vil ofte kræve anvendelse af numeriske værktøjer som fx. nite element metoden. Denne afhandling omhandler kohæsiv revnevækst i beton inden for rammerne af XFEM. Hovedparten af arbejdet er relateret til modellering af revnevækst i en tre-dimensionel armeret betonbjælke ud fra et modelleringsteknisk perspektiv. XFEM er baseret på berigelse af ytningsfeltet i elementer, der er gennemskåret af en diskontinuitet. Konceptet for berigelsen er baseret på lokale enhedsytningsfelter (partition of unity). Den anvendte berigelse er elementlokal, det vil sige kun elementer der er gennemskåret af diskontinuiteten berøres af berigelsen. Modelleringsarbejdet i rapporten tager udgangspunkt i en tre-dimensionel armeret betonbjælke udsat for trepunkts bøjning og anvendelsen af materialemodellen CDP. Det undersøges hvorvidt aktiveringen af XFEM forbedrer resultaterne med hensyn til last-ytningskurver, spændingsfordelinger i et revnet tværsnit samt revnedannelse. Resultaterne sammenlignes med retningslinjer og formler fra gældende normer i EN 1992-1-1 [2004]. Gode resultater er opnået med hensyn til såvel forudsigelse af den fulde last-ytningsrespons og spændingsfordelinger i tværsnittet. Last-ytningskurverne viste sig, at være uafhængige af aktiveringen af XFEM, hvorimod aktiveringen af XFEM havde positiv indydelse på spændingerne i tværsnittet. Resultaterne for revnevidder, revneafstand og revnemønster er kun opnåelig ved aktiveringen af XFEM. Resultaterne viste, at en forbedring af den nuværende implementering af XFEM i Abaqus 6.10 er nødvendig på grund af antagelser vedrørende propagering og initiering af revner.

1

Introduction

1

In this chapter the motivation for this project is described followed by a presentation of the problem to be handled. This leads to the problem formulation of the project, covered in the report. Finally, a list of objectives is stated to answer the problem formulation. Concrete is a stonelike material obtained by permitting a mixture of cement, sand and gravel or other aggregate and water, to harden in forms of the desired shape of the structure. Additives like silica fume and y ash can be added to the concrete mixture for increased strength and workability. Concrete has become a popular material in civil engineering for several reasons, such as the low cost of the aggregate, the accessibility of the needed materials and its high compressive strength, which makes it suitable for members primarily in compression such as columns. For example the pillars of the wellknown Storebælt bridge is constructed of reinforced concrete, see gure 1.1.

Figure 1.1.

The Storebælt bridge. Zeelandsite [2011]

On the other hand concrete is a relatively brittle material with low tensile strength compared to the compressive strength. For this reason, concrete elements are reinforced 3


1. Introduction

with steel bars in areas of the cross-section which is subjected to tension to provide some tensile resistance. In this way, reinforced concrete acts a bi-material utilizing the high compressive strength of concrete and the tensile strength of steel. A.H. Nilson, D. Darwin and Charles W. Dolan [2004] Methods of determining the ultimate strength of concrete based on linear elasticity or plasticity is widely developed. However, the possibilities for analysis in the serviceability limit state, e.g. deection and crack width estimations, are lacking and empirical in EN 1992-1-1 [2004]. Moreover, the proposed deection formula in EN 1992-1-1 [2004] does not take reinforcement arrangement or geometry into account. The last statement is also true for the estimation of crack width and spacing in EN 1992-1-1 [2004]. For structures with a complex geometry the Finite Element Method, abbreviated FEM, must be used, which should be able to model the cracking of the concrete. Within the Finite Element framework it is desirable to represent the actual stress distribution and the development hereof, e.g. by employing an advanced material model, to precisely model the section stresses at every load stage. A principle sketch of the stress development in a cross-section for increasing bending moment is shown in gure 1.2.

Figure 1.2.

Principle sketch of the stress development for increasing bending moment. Søren Madsen [2009]

Cracking of the concrete is unavoidable due to its low tensile strength and low extensibility. The downside of cracking is twofold. Cracking exposes the reinforcing steel to the surrounding environment, which can cause corrosion of the steel. Secondly, a cracked construction loses its aesthetical qualities and appears unsafe to reside in. Studies performed by Arya and Ofori-Darko [1996] reveal that crack spacing is a governing factor in the rate of corrosion. They found, that several small cracks are more severe than one big crack. Other studies performed by Schieÿl and Raupach [1997] and Mohammed et al. [2001] show, that the crack width inuences the time to initiation of the corrosion. To avoid this, the engineer must ensure that the crack widths and spacing are within the allowable limits put forth by the governing code. In addition, cracking of a member will cause reduction in bending stiness, which inuences the deection of the member. It is important to accurately assess the inuence of cracks on the deection of concrete members in the serviceability limit state. Piyasena [2002] The standard FEM is well established as a robust and reliable numerical technique for studying the behavior of a wide range of engineering and physical problems. All physical phenomena encountered in engineering mechanics are modeled by dierential equations, usually too complicated to be solved by analytical methods. For problems 4

Master Thesis where the solution variables behave in a continuous manner, the FEM is a highly suited method for approximating the solution to the dierential equation governing the addressed problem. However, a large number of models in continuum mechanics involve solutions that are discontinuous in local parts of the solution domain. For example, displacements change discontinuously across cracks. In the FEM special care must be taken in the construction of an appropriate mesh, as element topology must align with the geometry of the discontinuity, and this is not desirable in applications, where the crack location is unknown a-priori. In applications, such as crack analysis, where the FEM encounters problems, other, more appropriate numerical techniques exist. One class of techniques is the so-called enriched methods, which are advantageous for problems having non-smooth and non-regular solution characteristics. A popular enriched method is the so-called extended nite element method, abbreviated XFEM. The XFEM was implemented by Dassault Systémes Simulia Corp. [2010] in their latest version of Abaqus (6.10), which puts the engineer in a position of being able to qualitatively estimate crack patterns, spacing and widths for arbitrary geometries. Abaqus 6.10 also oers advanced material models to accurately model the behavior of concrete shown in gure 1.2. Thomas-Peter Fries and Andreas Zilian. [2010] This report approaches the problem of modeling a three-dimensional reinforced concrete beam loaded to failure. The report aims at identifying the location of crack initialization and propagation independent of the mesh, and estimating the crack widths and spacing. Moreover, it is desired to reect the damage process of crushing and cracking in the cross-sectional stress distribution. The above introduction leads to the following specic problem formulation for this project:

Application of the extended nite element method and the Concrete Damaged Plasticity material model for cracking simulation in a three-dimensional reinforced concrete beam using the commercial nite element program Abaqus. In order to handle the described problem, the following objectives for the project are set:

Review of linear and non-linear fracture mechanics for concrete. Review of available material models for modeling of cracks in concrete. Introduction to the XFEM with focus on crack modeling. Description of the implementation of the XFEM in Abaqus 6.10. Examination of the quality of the implementation of the Concrete Damaged Plasticity material model and the XFEM in Abaqus 6.10, by performing four benchmark tests. Review of existing guidelines with respect to crack width and spacing in EN 1992-1-1 [2004]. Analysis of load-deection curves and stress distributions using the Concrete Damaged Plasticity material model, with and without the XFEM. Analysis of crack width, crack spacing and crack pattern using the Concrete Damaged Plasticity material model and the XFEM. 5


1. Introduction

The boundary conditions, dimensions and reinforcement arrangement in the threedimensional reinforced concrete beam are shown in gure 1.3. This specic beam is analyzed, because the dimensions and material properties are typical for construction beams.

Figure 1.3.

Boundary conditions, dimensions and reinforcement arrangement in the examined reinforced concrete beam.

Note, that shear reinforcement is not incorporated into the model in order to simplify the modeling work load in Abaqus 6.10.

6

Fracture mechanics for concrete

2

This chapter presents an introduction to cracking in concrete with focus on the fracture process in compression and tension. This is followed by a description of the fundamental concepts of linear elastic fracture mechanics, LEFM. Within the scope of LEFM the theory in Grith [1921]/Irwin [1958] is presented along with the cohesive crack model proposed by Dugdale [1960] and Barrenblatt [1959]. Following the description of LEFM, the work done by A. Hillerborg and Peterson [1976], within the eld of nonlinear fracture mechanics, is presented. Concrete is a heterogeneous anisotropic non-linear inelastic composite material, which is full of aws that may initiate crack growth when the concrete is subjected to stress. Failure of concrete typically involves growth of large cracking zones and the formation of large cracks before the maximum load is reached. This fact, and several properties of concrete, points toward the use of fracture mechanics. Furthermore, the tensile strength of concrete is neglected in most serviceability and limit state calculations. Neglecting the tensile strength of concrete makes it dicult to interpret the eect of cracking in concrete. This may be accounted for by applying a fracture mechanics approach. Five arguments stated by the ACI Committee [1992] suggests why fracture mechanics should be adopted into certain aspects of design of concrete structures: 1. It is not sucient to specify how cracking is initiated, e.g. by a stress criterion, but also how it will propagate. The growth of a crack requires the consumption of a certain amount of energy, called the fracture energy. Therefore, crack propagation can only be studied through an energy criterion. 2. The calculations must be objective, i.e. mesh renement, choice of coordinates etc. must not aect the results. This entails that the energy dissipated through cracking is constant, which is done by specifying the energy dissipated per unit length of the crack. 3. Two basic types of structural failure may be stated: brittle and plastic. Plastic failure occurs in materials with a long yield plateau and the structure develops plastic hinges. For materials with a lack of yield plateau, the fracture is brittle, 7


2. Fracture mechanics for concrete

which implies the existence of softening. During softening the failure zone propagates throughout the structure, so the failure is propagating. 4. The area under the load-displacement curve determines the amount of energy consumed during failure process. This energy determines the ductility of the structure, and a limit state analysis cannot give an indication of this, because the post-peak response is not taken into account. 5. Fracture mechanics may opposite to strength criterions predict the inuence of the structural size on the failure load and ductility. The ve arguments stated above motivate towards using fracture mechanics in the modeling of concrete when cracking is of interest. Thus fracture mechanics may lead to a physical explanation of cracking in concrete, that the current codes, e.g. EN 1992-1-1 [2004], do not by their present empirical formulas.

2.1 The fracture process in concrete In 1983 Wittmann [1983] suggested to dierentiate between three dierent levels of cracking in concrete. The levels are categorized as follows: Micro cracks that can only be observed by an electron microscope. Meso cracks that can be observed using a conventional microscope. Macro cracks that visible to the naked eye. Micro cracks occur on the level of the hydrated cement, where cracks form in the cement paste. Meso cracks form in the bond between aggregates and the cement paste. Finally, macro cracks form in the mortar between the aggregates.

The fracture process in compression The compressive stress-strain curve for concrete can be divided into four regions, see gure 2.1. The gure describes four dierent states of compressive cracking.

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2.1. The fracture process in concrete

Figure 2.1.

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The compressive stress-strain curve for concrete. The curve is divided into four regions for dierent states of cracking. J.P. Ulfkjær [1992]

Initial cracks on the micro-level, caused by shrinkage, swelling and bleeding, are observed in the cement paste prior to loading. For loads of approximately 0 − 30 % of the ultimate load the stress-stain curve is approximately linear and no growth of the initial cracks is observed. Between approximately 30 − 50 % of the ultimate load a growth in bonding cracks between the cement paste and aggregates is observed. The cement paste and the aggregates have dierent elastic modulii, which increases the non-linearity of the stressstrain curve. Beyond 50 % of the ultimate load macro-cracks start to slowly form in the mortar, running between the aggregates parallel with the load direction. At app. 75 % of the ultimate load a more complex crack formation is established, where the bonding cracks and the cracks in the mortar coalesce until nally failure occurs. J.P. Ulfkjær [1992]

The fracture process in tension The tensile strength of concrete is, much like the compressive strength, dependent on the strength of each link in the cracking process, i.e micro-cracks in the cement paste, meso-cracks in the bond and macro-cracks in the mortar. Consider a concrete rod under pure tensile loading, see gure 2.2. The fracture process initiates with crack growth of existing micro cracks at approximately 80 % of the ultimate tensile load. This is followed by formation of new cracks and a halt in formation of others due to stress redistribution and the presence of aggregates in the crack path. These cracks are uniformly distributed throughout the concrete specimen. When the ultimate tensile load is reached, a localized fracture zone will form in which a macro-crack, that splits the specimen in two, will form. The fracture zone develops in the weakest part of the specimen. J.P. Ulfkjær [1992]

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Figure 2.2.


A concrete rod subjected to pure tensile loading. Outside the fracture zone, the cracks are uniformly distributed. Inside the fracture zone a macro-crack forms which splits the rod in two.

In the following section the basis of linear elastic fracture mechanics is presented.

2.2 Linear elastic fracture mechanics Grith [1921] was the rst to develop a method of analysis for the description of fracture in brittle materials. Grith found that, due to small aws and cracks, stress concentrations arise under loading, which explains why the theoretical strength is higher than the observed strength of brittle materials. Grith studied the inuence of a sharp crack on an arbitrary body with the thickness t loaded remotely from the crack-tip with an arbitrary load F , see gure 2.3.

Figure 2.3.

Arbitrary body with an internal crack of length a subjected to an arbitrary force, F.

By superposition, the potential energy of the body is given by 2.1.

Π = Π e + ΠF + ΠK + Πc 10

(2.1)

2.2. Linear elastic fracture mechanics Πe ΠF ΠK Πc

The The The The

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elastic energy content in the body. potential of the external forces. total kinetic energy in the system. fracture potential.

The fracture potential, Πc , is the energy that dissipates during crack growth. By assuming that crack growth is only dependent on the crack length, a, the equilibrium equation can be stated, by requiring that the potential energy of the system equals zero, see 2.2.

∂Π =0 t · ∂a

(2.2)

Grith [1921] introduced a parameter, the energy release rate, G, and dened a fracture criteria, see equation 2.3.

G=−

∂Πc =R t · ∂a

(2.3)

where R is the fracture resistance of the material, which is assumed to be constant in LEFM. The total potential energy of a system increases when a crack is formed because a new surface is created, thus increasing the fracture potential. However, the formation of a crack consumes an amount of energy, G, in the form of surface energy and frictional energy. If the energy release rate is larger than the energy required to form a crack, see 2.4, crack growth is unstable.

∂R ∂G > =0 ∂a ∂a

(2.4)

The method proposed by Grith [1921] was based on energy considerations, but is not adequate in design situations. For this reason, Irwin [1958] developed the stress intensity factor, abbreviated SIF, concept. The SIF can be understood as a measure of the strength of a singularity, understood in the sense that the SIF amplies the magnitude of the stresses around the singularity. The literature distinguishes between the three dierent fracture modes shown in gure 2.4.

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Figure 2.4.


Top: Crack mode I. Middle: Crack mode II. Bottom: Crack mode III. NDT [2011]

Mode I fracture is the condition in which the crack plane is perpendicular to the direction of the applied load and mode II fracture is the condition in which the crack plane is parallel to the direction of the applied load. Mode III fracture corresponds to a tearing mode and is only relevant in three dimensions. Mode I and mode II fracture is also referred to as an opening and in-plane shear mode, respectively. Irwin [1958] showed that the stress variation near a crack tip in a linear elastic material is dependent on the distance to the crack tip, called r. More precisely, the stress is singular at the crack-tip with a square-root singularity in r, see equation 2.5.

K σij = √ · fij (θ) + higher order terms 2πr σij K θ, r fij

(2.5)

The stress tensor. The stress intensity factor. The polar coordinates at the crack-tip. A trigonometric function.

From equation 2.5 it can be seen, that a linear relationship exists between the stress and the SIF, which reects the linear nature of the theory of elasticity. In practical calculations, only the rst order term of equation 2.5 is included. This is because, that for r → 0, the rst order term approaches innity while the higher order terms are constant or zero. Because the stress tends towards innity when r → 0 a stress criterion as a failure criterion is not appropriate. For this reason, Irwin derived a relationship between the SIF and the energy release rate, G, see 2.6. √ (2.6) K = G·E 12

2.3. Non-linear fracture mechanics

Master Thesis

The fracture criterion can thereby be written as

K = Kc

(2.7)

It should be noted that the global energy balance criteria by citebib:grith is equivalent to the local stress criteria by Irwin [1958]. Moreover, Kc is also referred to as the fracture toughness of the material, and is regarded as a constant in LEFM. The Grith [1921]/Irwin [1958] theory assumes that the stresses in the vicinity of the crack-tip tend to innity. This contravenes the principle of linear elasticity, relating small strains to stresses through Hooke's law. In the fracture process zone, abbreviated FPZ, ahead of the crack-tip, plastic deformation of the material occurs. Specically for concrete debonding of aggregate from the cement matrix and microcracking occurs. Moreover, cracks coalesce, branch and deect in the FPZ. To describe this highly non-linear phenomenon, non-linear fracture mechanics, abbreviated NLFM, must be adopted.

2.3 Non-linear fracture mechanics The rst attempt to analyze plasticity at the crack-tip was done by Dugdale [1960] and Barrenblatt [1959]. Dugdale [1960] and Barrenblatt [1959] independently proposed two models, in which closing forces were included at the crack-tip. The closing forces are also referred to as cohesive forces, and the small zone, over which they act, is termed the cohesive zone. The stress singularity that arises at the crack-tip using the Grith [1921]/Irwin [1958] theory vanishes when the approaches suggested by Dugdale [1960] or Barrenblatt [1959] are used. In the model suggested by Dugdale [1960], the cohesive closure stress is the yield strength of the considered material. In the model suggested by Barrenblatt [1959], the cohesive closure stress is a characteristic material molecular force of cohesion and has a generally unknown variation along the FPZ. The principle of the cohesive zone model by Barrenblatt [1959] is shown in gure 2.5.

Figure 2.5.

The cohesive zone model by Barrenblatt [1959]. A body with a crack of length 2a subjected to tension, σ . Cohesive stresses, q(x), act along a cohesive zones of length c at each crack-tip. 13



Inspired by the concept of Dugdale [1960] and Barrenblatt [1959], A. Hillerborg and Peterson [1976] redened the FPZ by introducing a so-called ctitious crack in front of the real crack-tip. The purpose of introducing a ctitious crack was to improve the description of the tractions acting in FPZ. The closure stress in the FPZ has a maximum value of ft , i.e. the tensile strength of the material, at the boundary of the FPZ and is zero at the tip of the real crack. The variation in-between is given by a softening law, relating stresses to the crack opening displacement, w. Similar to elastic materials with a constitutive law described by e.g. Hooke's law, the tension softening law is the constitutive law in the FPZ. Thus the tension softening law describes the transition between the continuous state and the discontinuous state of the material behavior. Figure 2.6 illustrates a typical tensile load-displacement response of concrete and the related ctitious crack ahead of the real crack. Note that the FPZ extends only over the length of the tension softening region BCD, see gure 2.6. Tension softening is the relationship between the cohesive stress and the crack opening displacement in the FPZ. Note that the relation between the closure stress and the crack opening displacement is non-linear, and that a degradation of the Young's modulus occurs gradually inside the FPZ. J.L. Asferg [2006]

Figure 2.6.

14

(a) Typical tensile load-displacement curve of concrete with letters indicating the crack-state: A: Uncracked, linear-elastic behavior, B: The tensile strength has been reached and microcracking and tension softening occur, C: Stress bridging D: The crack becomes traction-free. (b) The FPZ related to the load-displacement curve. The variation of the cohesive stresses is indicated and the crack is divided into a traction-free zone, a microcracking/bridging zone and a microcracking zone.

2.3. Non-linear fracture mechanics

Master Thesis

As previously mentioned energy is absorbed during crack growth in order to form the new crack surfaces. The amount of absorbed energy is the fracture energy, Gf , which is equal to the area under the tension-softening curve, see gure 2.7 and equation 2.8.

Figure 2.7.

Z

wc

σ(w) dw

Gf =

Tension-softening.

(2.8)

0

As a nal remark, the ctitious crack model proposed by A. Hillerborg and Peterson [1976] assumes, that the FPZ has negligible width. For this reason, the model belongs to the class of discrete crack models. The ctitious crack model by A. Hillerborg and Peterson [1976] is the basis for the cohesive segments method used in Abaqus 6.10. The cohesive segments method is described in chapter 4.

15

Smeared vs. discrete crack modelling

3

This chapter presents the concepts of smeared and discrete crack models for concrete. Popular techniques, e.g. the XFEM, available for discrete crack modeling are discussed. Advantages and drawbacks are identied and pointed out for a discrete vs. smeared approach to crack modeling. The purpose of this chapter is to illustrate the motivation for working with the XFEM in this report. In the late 1960's D. Ngo and A.C. Scordelis [1967] performed a numerical simulation of discrete cracks in concrete. At the same time, Rashid [1968] successfully applied a smeared crack model for concrete. Discrete crack simulation aims at the initiation and propagation of dominating cracks, whereas the smeared crack model is based on the observation, that the heterogeneity of concrete leads to the formation of many, small cracks which, only in a later stage, nucleates to form one larger, dominant crack. The smeared crack model captures the deterioration process by smearing the eect of microcracks, that is, a reduction in stiness, over a given volume. With respect to the problem formulation stated in chapter 1, the objective of the numerical model is to

Identify the location of crack initialization. Predict arbitrary crack propagation paths. Handle multiple cracks and the coalescence of cracks. Estimate crack width and spacing. Simulate the damage process up to failure.

Since the pioneering work by D. Ngo and A.C. Scordelis [1967] and Rashid [1968] work has been done to improve the initially presented crack concepts. With respect to the ve criteria presented above for a feasible numerical model, popular techniques available for numerical modeling of cracks within the smeared and discrete crack concepts are presented in the following sections.

17


3. Smeared vs. discrete crack modelling

3.1 Smeared crack concept In the smeared crack model a cracked body is represented as a continuum. This is done by smearing the eect of one or more cracks attributed to a representative volume surrounding an integration point. Smeared crack models are based on the concept of a crack band model, where the cracking strain is set equal to the crack opening, w, divided by the length of the fracture process zone, lp , see equation 3.1, Z.P. Bazant and B. Oh [1983]. The length of the fracture process zone is also referred to as the localization band. de Borst et al. [2004]

cr =

w lp

(3.1)

The eect of cracks is translated into a stiness-deterioration in the integration point. When a combination of stresses satises a specied failure criterion, e.g. the maximum principal stress reaching the tensile strength of the concrete, cracking is initiated. Until the initiation of cracks, the concrete is modeled as an isotropic material. At the onset of cracking, the initial isotropic stress-strain law is replaced by an orthotropic law. This is done in order to represent the gradual reduction of stiness normal to the crack direction, called tension-stiening. The introduction of tension-stiening is motivated by the fact, that in reinforced concrete, the volume attributed to an integration point contains a number of cracks, and due to the bond between the concrete and reinforcement, the intact concrete between the cracks adds stiness to the model. de Borst et al. [2004] Moreover, shear stiness is added as a representation of some eects of aggregate interlock and friction within the crack. The orthotropic constitutive matrix relating stresses to strains in a two-dimensional, cracked setting is given by 3.2.

  µE 0 0   Ds =  0 E 0  0 0 βG

(3.2)

The tension-stiening eect is represented by the parameter µ, which gradually decreases to zero as a function of the normal strain, nn , µ = µ(nn ), n referring to the direction normal to the crack direction. E and G are the Young's modulus and shear modulus, respectively, and β is the so-called shear retention factor representing aggregate interlock and friction within the crack. de Borst et al. [2004] The smeared crack concept suers from three major drawbacks outlined in the following. Firstly, the model is based on the concept of a crack band in which the exact location of the crack inside an element is unknown. In a case where crack width or spacing is of interest a discrete approach should be preferred over a smeared approach. Secondly, the smeared crack concept has convergence problems as a mesh-renement will aect the width of the localization band. Lastly, the strain imposed by a crack inside an element implies adjacent elements to be strained as well. This is illustrated in gure 3.1 and is referred to as stress-locking. Rots and Blaauwendraad [1989] 18

3.2. Discrete crack concept

Figure 3.1.

Master Thesis

Stress-locking is a consequence of displacement compatibility in smeared cracking. Strain of inclined crack at element 2 induced locked-in stress at element 1.

Stress-locking refers to the situation, where tensile stresses in adjacent elements is still in the elastic regime and refuses to decrease. This results in locked-in stresses at locations, such as the sides of a crack, where the stress should be zero. This drawback is a consequence of approximating a strong discontinuity using the assumption of displacement continuity. For the above mentioned reasons, a smeared crack model does not satisfy the ve specied criteria for a feasible numerical model in this report. In the following section the discrete crack concept will be introduced. Rots and Blaauwendraad [1989]

3.2 Discrete crack concept The discrete crack approach is the counterpart to the smeared crack approach. A crack is modeled as a geometrical discontinuity, that is, a discontinuity in displacement across the crack. In the work done by D. Ngo and A.C. Scordelis [1967], a discrete crack was initiated when the nodal force exceeded the tensile strength of the concrete. This approach suered from two drawbacks, namely: forcing the crack to propagate along element boundaries and implying a continuous change in nodal connectivity. The latter drawback refers to remeshing and the possibility that element edges do not conform to and recreate the intended crack geometry. This is especially the case for curved cracks. A more sophisticated approach to modeling of discrete cracks is the interface-technique, where interface elements are inserted along element boundaries. An example of an interface element can be seen in gure 3.2. As indicated, the thickness is almost zero. Moreover, a large dummy stiness is assigned to the interface element in order not to aect the stiness of the structure being investigated. Upon cracking the large dummy stiness is set to zero. Rots and Blaauwendraad [1989]

Figure 3.2.

Standard three-noded two-dimensional interface element. J.L. Asferg [2006]

19


3. Smeared vs. discrete crack modelling

Interface elements are based on a traction-separation description for modeling of cohesive cracks. The relation between the stress and crack opening gradients for a two-dimensional conguration are given by equation 3.3. J.L. Asferg [2006]

" # " #" # σ˙ D11 D12 δ˙n = τ˙ D21 D22 δ˙t

σ˙ τ˙ D δ˙n δ˙t

(3.3)

Gradient the of normal stresses acting along the interface. Gradient the of shear stresses acting along the interface. Constitutive matrix. Index 1 and 2 refer to the normal and tangential direction, respectively. Gradient the of crack opening in the normal direction. Gradient the of crack opening in the tangential direction.

Interface-elements have been used in a number of applications, e.g. C. M. López [2008], where non-linear fracture is modeled for uniaxial tension loading of plain concrete. More recently, an analysis of ber reinforced polymer, FRP, strengthened reinforced concrete members has been performed, N. Khomwan, S.J. Foster and S.T. Smith [2010]. The stress transfer between the FRP and the reinforced concrete is modeled using 2-dimensional interface-elements. A bond-stress slip law was used for the description of the stress transfer at the interface. The use of interface-elements, however, puts a constraint on the crack propagation path. This is so, because the crack is constrained to propagate along the inserted interfaceelements. This constrain renders the modeling of a-priori unknown crack paths dicult. A remedy is re-meshing at every simulation stage, however the mesh must conform to the crack geometry, which in the case of curved or intersecting cracks is dicult to obtain. Several attempts have been made to construct eective remeshing algorithms, e.g. by A.R. Ingraea [1985], that reduced the mesh bias. However, such algorithms are computationally expensive. The missing possibilities of identifying the location of crack initialization and prediction of arbitrary crack propagation paths render the method unpreferable according to the ve specied criteria for a feasible numerical model in this report. The mesh-dependence was to a large extent alleviated by the advent of so-called enriched methods. An example of such enriched methods is the XFEM described in chapter 4. General for all enriched methods is to enrich the polynomial approximation space such that non-smooth solutions can be modeled independent of the mesh. This enables the class of methods to model discontinuities at arbitrary locations inside element interiors, such as a displacement discontinuity imposed by a crack. Moreover, enriched methods put no restriction on the number of cracks in the model, or whether the cracks are predened or initiated by fullling a material fracture criterion. For j cracks in the model, the polynomial approximation space is expanded to a sum over the j nodal sets describing the j cracks. Finally, by belonging to the class of discrete modeling, the XFEM is able to estimate crack width and spacing. According to the ve specied criteria for a feasible 20

3.2. Discrete crack concept

Master Thesis

numerical model in this report, the XFEM seems a valid candidate. With respect to the last point referring to the simulation of complete failure, this is a question of the numerical solver implemented in Abaqus 6.10. Fries and Belytschko. [2000] A drawback of the discrete crack concept is that it is intended for the representation of dominating cracks, thus neglecting the eect of microcracks known to occur in heterogeneous materials like concrete. However, the XFEM has attractive properties with respect to crack modeling and will be used further on in this project.

21

The eXtended Finite Element Method

4

This chapter presents the general formulation of the XFEM. Initially, the background of the XFEM is presented and an introduction to the concept of discontinuities is given. This is followed by a description of a formulation of the approximation space in the XFEM. A convenient method for choosing the set of so-called enriched nodes is presented and the concept of blending elements is introduced. Various enrichment types are discussed, followed by a description of two methods for numerical integration of the weak form. Finally, the XFEM approximation to the displacement is derived using the variational principle. Unless stated otherwise the sources used in this chapter are Fries and Belytschko. [2000] and Thomas-Peter Fries and Andreas Zilian. [2010] . Discontinuities and singularities in eld quantities are observed in many areas of civil engineering, e.g. singular stresses and strains in the vicinity of a crack-tip, or a jump in displacement across a crack. For the numerical approximation of these non-smooth variables two fundamentally dierent approaches exist. The rst method relies on polynomial approximation, based on nite element shape functions, and requires the mesh to conform to the discontinuities. Moreover, a rened mesh is required in areas where eld quantities exhibit high gradients, and remeshing is required in order to model the evolution of interfaces, e.g. cracks, boundary layers and phase transition. However, for complex geometries an eective remeshing algorithm can be dicult to construct, as the elements must conform to the geometry of the discontinuity or projection errors are introduced. Moreover, this is computationally expensive and not suited for evolving interfaces. The second, fundamentally dierent, method is based on enriching the polynomial approximation space with discontinuous functions, such that non-smooth solutions can be modeled independent of the mesh. This is a basic principle of the XFEM, and was developed by Belytschko and Black [1999] and N.Moës and Belytschko [1999], based on the partition of unity concept pioneered by Melenk and Babuska [1996]. In the following chapter, concepts behind the XFEM for treating discontinuities will be described. In order to have a common terminology for the description of discontinuities, the following section is dedicated to this cause.

23


4. The eXtended Finite Element Method

4.1 Discontinuities and high gradients Discontinuities are observed in the real world where eld quantities change rapidly over a length scale that is small compared to the observed domain. For a reinforced concrete beam, a jump in stress occurs across the material interface separating the concrete and the reinforcement. At the onset of cracking, stresses and strains change discontinuously across a crack and become singular at the crack-tip. Moreover, the displacement eld is discontinuous across the crack. An understanding of these phenomena is important in numerical modeling of reinforced concrete, and knowledge of the formation and propagation of cracks is vital for reliability and damage analysis of structures. For the remainder of this report, the following denition will be adopted for a discontinuity:

A discontinuity is a rapid change of a eld quantity over a length negligible in comparison to the dimensions of the observed domain. In reality a eld quantity may never change over a length of zero. However, this is justied when compared to the length scale of the observed domain. In the case where the length scale is small but has to be accounted for, the term "high gradient" will be used. The location in space, over which eld quantities or their gradients change discontinuously, will be termed interface. A mathematical description of interface is: Consider a d-dimensional domain Ω ∈ Rd , then a manifold Γ ∈ Rd−1 is called interface. In this way, an interface is a surface in three dimensions, a line in two dimensions and a point in one dimension. Examples of two types of interfaces are given in gure 4.1.

Figure 4.1.

Example of (a) open interface and (b) closed interface.

The interfaces in gure 4.1 dier by having and not having a free end in the domain. Figure 4.1a is a so-called open interface, because the interface ends inside the domain. An example of an open interface is a crack. Figure 4.1b is a so-called closed interface, because the interface does not have any free ends inside the domain. An example of a closed interface is a material interface. The topological dierence between an open and a closed interface is described by the level-set method, described in section 4.2. A distinction is made between moving and xed interfaces. A xed interface is treated by a Lagrangian description, meaning the relative position of the interface is unchanged during deformation of the body. A moving interface is treated by an Eulerian description, 24

4.2. Level-set method

Master Thesis

meaning the interface moves through the domain. The initial position of the interface is given, and the future position is then part of the solution. In this report, the cracks appearing in the 3-dimensional concrete beam are xed interfaces. However, the crack propagates and one would assume the interface to be moving. This is not the case, since the crack propagation speed is unknown and not part of the solution in the displacement variational principle. For this reason, the propagation of the crack is treated as a quasistatic process and thus, as a xed interface. A nal distinction is made between so-called strong and weak discontinuities. Strong discontinuities refer to a jump in a eld quantity across an interface, whereas a weak discontinuity refers to a jump in the gradient of the eld quantity across an interface. A discontinuity in the gradient is also referred to as a kink in the eld variable. An example of a strong and weak discontinuities is given in gure 4.2.

Figure 4.2.

Example of (a) strong discontinuity and (b) weak discontinuity in a eld quantity represented as a surface. The interface is represented as a bold line.

With the denition of strong and weak discontinuities, the displacement exhibits a strong discontinuity across a crack and a weak discontinuity across a material interface. An accurate description of the location of the interfaces, e.g. cracks, is necessary in order to enrich the solution appropriately. This issue is addressed in the following section.

4.2 Level-set method The level-set method is a technique for locating interfaces and is useful in combination with the XFEM, because it facilitates the construction of the enrichment, as will be shown later. However, the method is not a part of the XFEM but is widely used in combination hereof. The level-set method denes interfaces implicitly by the zero-level of a scalar function. The method is restricted by the requirements, that the scalar function must be a continuous function and change sign across the interface. The signed distance function is a particularly useful function in this regard, because it fulls the requirements to the scalar

25



function and is easy to implement into a code. Examples of eligible level-set functions for a one-dimensional bar with a discontinuity located at x = 0 are shown in gure 4.3.

Figure 4.3.

Eligible level-set functions. The red line is the signed distance function and the black line is an arbitrary level-set function. The interface is located at the red circle on a one-dimensional bar discretized with nodes indicated as blue stars.

The signed distance function is given by equation 4.1. As the name suggests, the signed distance function computes the distance from the discontinuity to a given point and assigns a sign to the distance.

φ(x) = ± min kx − x∗ k , ∗ x ∈ Γ

x∗ Γ k.k Ω

The The The The

∀x ∈ Ω

(4.1)

coordinates of a node on the interface. set of all nodes x∗ on the p interface. Euclidean norm kzk = z12 + z22 + ... + zn2 . considered domain.

The sign in equation 4.1 is determined by the sign-equation sign(n · (x − x∗ )), where n ∇Φ is the normal vector to Φ given by n = k∇Φk , where ∇ is the dierential operator and 26

4.2. Level-set method

Master Thesis

k∇Φk = 1 holds for the signed distance function. By convention, n points from the Φnegative subdomain into the Φ-positive subdomain, which is the reason for the existence of the sign-equation. Because exact functional representation of discontinuities is often inconvenient, the discontinuities are stored in a discrete way. This is done by evaluating the signed distance function at the nodes, and using standard nite element shape functions to interpolate in-between, see equation 4.2. Note that an error is introduced in this approximation, which decreases with mesh renement.

Φh =

X

Ni (x) · Φ(xi )

(4.2)

i∈I

Φh and Φ x and xi i and I Ni

The approximated level-set function value and exact level-set function value. Nodal coordinate and the coordinates of node i. Node i in the set of all nodes I . Standard Finite Element shape function belonging to node i.

As previously mentioned, the topological dierence between an open and a closed interface is described by the number of level-set functions needed to describe the discontinuity. In the following two subsections, open and closed interfaces are described using the level-set method.

4.2.1 Description of closed interfaces Consider a domain Ω ∈ Rd containing an interface. Ω can be decomposed into two subdomains, Ω1 and Ω2 , such that Ω = Ω1 ∪ Ω2 and the interface Γ12 = Ω1 ∩ Ω2 . Ω1 and Ω2 may consist of disconnected regions. The interface is then given by the set

Γ12 = {x : φ(x) = 0} This situation is depicted in gure 4.4, where the signed-distance function has been used as the level-set function.

27


Figure 4.4.


(a) The domain Ω is decomposed into two subdomains Ω1 and Ω2 . Γ12 describes the interface. The normal vector, n, points from the φ-negative subdomain into the φ-positive subdomain. (b) Contour values of the signed-distance function.

For more than 2 subdomains, 1 level-set function is no longer sucient. In general, for closed interfaces, n level-set functions can separate 2n subdomains.

4.2.2 Description of open interfaces Consider a domain Ω ∈ Rd partially cut by an interface. Where one level-set function is able to describe a closed interface, the description of an open interface requires a second level-set function, γ , to describe where the interface ends. The interface is then given by the set

Γ12 = {x : φ(x) = 0 and γ(x) ≤ 0} γ has to fulll the same requirements as those put on φ. However, in computational implementations, γ is often chosen a straight line orthogonal to the tip of the interface and φ is the signed-distance function, which is zero across the interface and extended tangentially from the crack tip. In other words, γ is not necessarily a signed-distance function, but used to dene the end of the open interface, and φ describes a closed interface. In Abaqus 6.10, γ and φ are both signed-distance functions. A crack is a typical open interface, and an example is depicted in gure 4.5.

28

4.3. General formulation of the XFEM

Figure 4.5.

Master Thesis

(a) The domain Ω partially cut by a crack. (b) The signed-distance function φ discribing the crack path. (c) The level-set function γ dening the crack tips.

In the following section an XFEM approximation of the displacement is presented.

4.3 General formulation of the XFEM The XFEM is a numerical method that enables the local enrichment of approximation spaces by including known solution properties into the approximation space. Consider an n -dimensional domain Ω ∈ Rn , which is discretized by nel elements, numbered from 1 to nel . I is the set of all nodes. The general formulation of the standard XFEM for the approximation of the unknown displacement u(x) is of the form shown in equation 4.3.

uh (x) =

X

Ni (x) · ui +

i∈I | {z

}

Strd. F EM approx.

X i∈ |

Mi1 (x) · a1i + . . . +

I1∗ {z

Enrichment 1

}

X i∈ |

Mim (x) · am i

(4.3)

∗ Im

{z

Enrichment m

}

29

Group B122b - Spring 2011 uh (x) Ni (x) ui I Mim (x) am i ∗ Im


The approximation of the displacement. Standard FEM shape function of node i. The degree of freedom of the standard FEM part at node i. The set of all nodes. The local enrichment function of node i belonging to the m 'th enrichment. The degree of freedom of the m 'th enrichment of node i. ∗ ⊂ I. The nodal subset of the m 'th enrichment, Im

Note that the approximation in 4.3 consists of a standard nite element part plus additional m enrichment terms. This form of enrichment is called extrinsic enrichment, because special enrichment terms are added to the polynomial approximation space, resulting in more functions and nodal unknowns to be evaluated. The alternative to extrinsic enrichment is intrinsic enrichment, where the standard nite element shape functions are replaced by special shape functions, which are able to capture the nonsmooth solution. This is done by expanding the function basis for the elements cut by a discontinuity, and results in no additional unknowns. However, the amount of computational work needed to establish the special shape functions makes intrinsic enrichment unappealing in comparison to extrinsic enrichment. In Abaqus 6.10 extrinsic enrichment is used. Each enrichment consists of a local enrichment function, Mim (x), and additional nodal unknowns am if the m'th i describing the character of the m'th discontinuity, e.g. discontinuity is a crack, the local enrichment function reects the inuence of the crack by introducing a jump in the displacement eld. The local enrichment function is given by equation 4.4.

Mim (x) = Ni∗ (x) · ψ m (x) Ni∗ (x) ψ m (x)

(4.4)

Partition of unity function of node i. Global enrichment function of the m 'th enrichment.

The global enrichment function, ψ m (x), incorporates the special knowledge of the solution properties into the approximation space. The partition of unity functions, Ni∗ (x), only build a partition of unity in a local part of the domain, that is, in elements whose nodes are all in the nodal subset I ∗ . In elements that are fully enriched the property of 4.5 holds, that is, Ni∗ (x) build a partition of unity.

X

Ni∗ (x) = 1,

∀x ∈ Ω∗j , ∀j = 1, . . . , m.

(4.5)

i∈Ij∗

The partition of unity concept is a well-known property of the standard nite element shape functions. For this reason Ni∗ (x) = Ni (x) is an option, but not a necessity. For 30

4.4. Choice of enriched nodes

Master Thesis

example, Ni∗ (x) can be chosen quadratic while using linear standard nite element shape functions, Ni (x). In Abaqus 6.10 Ni∗ (x) = Ni (x). The approximation on the form shown in 4.3 does not have the Kronecker-δ property. This is seen by evaluating the approximation at x = xk

uh (xk ) =

X

X

Ni (xk ) · uk +

i∈I

i∈

Mi1 (xk ) · a1k + . . . +

I1∗

X i∈

Mim (xk ) · am k

(4.6)

∗ Im

which for i = k yields

uh (xk ) = uk +

X i∈

Mi1 (xk ) · a1k + . . . +

I1∗

X i∈

Mim (xk ) · am k

(4.7)

∗ Im

Consequently, uh (xk ) 6= uk , which renders the imposition of essential boundary conditions dicult and the computed unknowns are no longer the sought functions values. It is therefore desirable to recover the Kronecker-δ property, which is achieved by making the enrichment terms vanish at the nodes. This is done by shifting the approximation. The shifted approximation of the displacement is shown in 4.8. Note, that the expression for the local enrichment function 4.4, has been inserted. For brevity in the expression, only one enrichment term has been included.

uh (x) =

X i∈I

Ni (x) · ui +

X i∈

Ni∗ (x) · [ψ(x) − ψ(xi )] · ai

(4.8)

I1∗

By evaluating 4.8 in x = xk it can be shown, that the Kronecker-δ property is recovered. Abaqus 6.10 is using the shifted form of the approximation. The following section will describe a method for choosing the nodal subset I ∗ .

4.4 Choice of enriched nodes For computational eciency, partition of unity enrichments are preferably localized to the sub-domains where they are needed. In other words, only a nodal subset needs enrichment. By enrichment the authors allude to including enrichment terms in the approximation of the displacement. The nodal subset I ∗ is built from the nodes of the elements cut by a discontinuity. A convenient method for choosing the enriched nodes is the level-set method described in section 4.2. The level-set function determines whether or not an element is cut by a discontinuity. The signed distance function is used for determining whether an element is cut or not:

31



Cut element: Uncut element:

min (sign(Φ(xi ))) · max (sign(Φ(xi ))) < 0

i∈I el

i∈I el

min (sign(Φ(xi ))) · max (sign(Φ(xi ))) > 0

i∈I el

i∈I el

I el is the set of element nodes. In other words, an element is cut if the level-set function changes sign in the element. For identifying an element containing a crack-tip the following two criteria of the signed distance function must be met simultaneously:

min (sign(Φ(xi ))) · max (sign(Φ(xi ))) < 0

i∈I el

i∈I el

min (sign(γ(xi ))) · max (sign(γ(xi ))) < 0

i∈I el

i∈I el

The reason for identifying the crack tip element is to properly enrich it so that stress and strain singularities are accounted for. However, if only one element is enriched sub-optimal convergence rates are obtained. This is due to the fact, that mesh renement reduces the area over which the singularity is accounted for. An alternative method that accounts for this problem is the branch enrichment approach, where nodes within a certain radius from the crack tip are enriched. In this way the enriched area is kept constant during mesh renement.

∗ Itip = {i : kxi − xi ∗ k < r}

where kxi − xi ∗ k is the Euclidean distance between a point, xi , and the crack-tip, xi ∗ . Abaqus 6.10 uses this approach, for static cracks, with an enrichment radius of three times the characteristic element length. For evolving cracks the crack tip enrichment has not been implemented in Abaqus 6.10. The mathematical description of the enrichment functions used in Abaqus 6.10 are described in section 4.5. Since only a subset of the nodes are enriched three types of elements can be dened: 1. A standard element if none of its nodes are enriched. 2. A fully enriched element if all of its nodes are enriched. 3. A partly enriched element if some of its nodes are enriched. Figure 4.6 illustrates a one-dimensional and a two-dimensional example of the three element types and Ni∗ (x), here chosen as a linear function.

32

4.5. Global enrichment functions

Figure 4.6.

Master Thesis

Discretized domains in one- and two dimensions with nodal subset I ∗ . (a) and (c) show the reproducing-, blending- and standard elements. (b) and (d) show that the function Ni∗ (x), here linear, only builds a partition of unity in reproducing elements and varies linearly from one to zero over the blending element.

The fully enriched elements are called reproducing elements, because the approximation shown in 4.3 is able to reproduce any enrichment function exactly in Ω. The partly enriched elements are also referred to as blending elements because the enrichment is blended over the element. This is because the partition of unity functions do not build a partition of unity, which introduces parasitic terms into the approximation if linear or higher order global enrichment functions are chosen. In Abaqus 6.10 blending elements do not exist, because the meshing algorithm is constructed such that element boundaries conform to interfaces. Moreover, only discontinuous enrichment functions, with constant variation, are used, which eectively eliminates potential problems caused by blending elements.

4.5 Global enrichment functions As previously mentioned distinction is made between weak and strong discontinuities. Moreover, singularities in the stress and strain eld near a crack tip must be reected in the solution. In the following the global enrichment functions, used to reect these phenomena in the solution, are presented.

33



4.5.1 Weak discontinuities A weak discontinuity refers to a kink in the solution, that is, a jump in the gradient of the solution. A well-known example of a weak discontinuity is at a material interface, where stresses or strains change discontinuously across the material interface. For weak discontinuities one choice for the global enrichment function is the abs-function, which is shown in 4.9. Note that the abs-function, also referred to as abs-enrichment, uses the levelset function. As previously mentioned, the level-set function is convenient in corporation with the XFEM, because it nds use in the construction of the global enrichment function.

ψ(x) = abs(φ(x)) = |φ(x)|

(4.9)

The gradient of the abs-function is given in equation 4.10.

∇ψ(x) = sign(φ(x)) · ∇φ(x)

(4.10)

However, this type of enrichment function leads to trouble in blending elements. This is because the function has a linear variation in the domain Ω. As previously mentioned, troubles arise because the partition of unity function, N ∗ (x), does not build a partition of unity in the blending elements. When ψ(x) is multiplied with N ∗ (x), a parasitic term is introduced into the approximation. However, this is only the case if the order of N (x) and N ∗ (x) is the same. For example, if both shape functions are chosen linear, the enrichment term becomes parabolic because of the linear variation of ψ(x) and thus, a linear term is summed with a parabolic term in the blending elements. In other words, the standard FEM part cannot compensate for the error introduced by the parasitic term. A remedy for this problem is to choose N (x) one order higher than N ∗ (x), in which case the standard FEM part will be able to compensate for the error introduced in the approximation by the parasitic term. An improvement to the abs-function was introduced by N.Moës and Belytschko [1999]. The improved function is the so-called modied abs-enrichment, which has the property of being non-zero only in the fully enriched elements. By being zero in the blending elements, no parasitic terms are introduced into the approximation and thus optimal convergence rates can be obtained. The modied abs-enrichemt is shown in 4.11.

ψ(x) =

X i∈I

X φi Ni (x) |φi | Ni (x) −

(4.11)

i∈I

Referring to the problem at hand in this project, a reinforced concrete beam, the reinforcement and the beam are meshed as independent parts in Abaqus 6.10. This entails that no elements contain more than one material property and thus, no weak discontinuities are present in the model because the material interface is coincident with the element boundaries. In the following section two global enrichment functions for strong discontinuities will be presented. 34

4.5. Global enrichment functions

Master Thesis

4.5.2 Strong discontinuities A strong discontinuity refers to a jump in the solution. Two popular choices for a global enrichment function are the Heaviside function and the sign-function. Both functions utilize the level-set function, again proving its usefulness in the context of the XFEM. Although dierent in structure, the two functions yield identical results as they span the same approximation space. The Heaviside function and the sign-function are shown in equations 4.12 and 4.13, respectively.

ψ(x) = H(φ(x)) =

( 0

: φ(x) ≤ 0

1

: φ(x) > 0

   −1 ψ(x) = sign(φ(x)) = 0   1

(4.12)

: φ(x) < 0 : φ(x) = 0

(4.13)

: φ(x) > 0

The gradient of these enrichment functions is zero. Note, that the functions do not cause trouble in blending elements, because they are constant in Ω. In Abaqus 6.10 the jumpfunction in 4.14 is used. ( −1 : φ(x) < 0 ψ(x) = H(φ(x)) = (4.14) 1 : φ(x) ≥ 0

4.5.3 Singularities At the crack tip a global enrichment function with a singular derivative is needed. Moreover, the function must be discontinuous along the crack. In practice the four global enrichment functions in equations 4.15 to 4.18 are often used.

√

θ 2 √ θ 2 ψ (x) = r sin sin θ 2 √ θ ψ 3 (x) = r cos 2 √ θ 4 ψ (x) = r cos sin θ 2 ψ 1 (x) =

r sin

(4.15) (4.16) (4.17) (4.18)

The functions depend on a local polar coordinate system at the crack-tip, see gure 4.7, where θ = 0 is tangent at the crack-tip.

35


Figure 4.7.


The polar coordinate system around the crack-tip. xtip and ytip are the Cartesian coordinates of the crack-tip. (r, θ) are the radial and angular coordinates, respectively, from the pole to a point, (x, y).

The four global enrichment functions in equations 4.15 to 4.18 are a result of linear elastic fracture mechanics, LEFM. They span the linear asymptotic crack-tip function of elasto-statics, and 4.15 takes the discontinuity in displacement into account. These are important in crack modeling, because if only sign-enrichment was used, the crack would be virtually extended to the boundary of the element in which the crack-tip is present. Using crack-tip enrichment functions ensures that the crack ends exactly at the location of the crack-tip. Moreover, ψ 1 (x) to ψ 4 (x) are an analytical result from LEFM to the near tip behavior, that is, the accuracy of the approximation is increased by including analytical results in the approximation. Abaqus 6.10 takes advantage of these four functions in representing the singular stress and strain eld near singularities. However, this is only the case for stationary cracks, because accurate modeling of the crack-tip singularity requires constantly keeping track of the crack location, and the degree of the singularity depends on the location in non-isotropic material, e.g. concrete. Moving cracks are modeled with the so-called cohesive segments method and phantom nodes. This matter is addressed in the following.

4.6 Cohesive segments method The XFEM-based cohesive segments method can be used to simulate crack initiation and propagation along an arbitrary, solution-dependent path. The cohesive segments method is based on the insertion of a cohesive segment through an element once a decohesion criteria is met, e.g. a damage criteria. The segments are not, as previously described, restricted to being located along element boundaries, but can be located at arbitrary locations and in arbitrary directions, allowing for the resolution of complex crack patterns. For this reason, a cohesive segment is not to be mistaken for an interface element, described in chapter 3. The segment is taken to extend through the element to the boundary in which it is inserted. The method is based on the partition of unity property of nite element shape functions and enriching the approximation space with discontinuous functions, as 36

4.7. Phantom-node method

Master Thesis

described in the previous sections. The displacement jump across a crack is described using the Phantom node method in Abaqus 6.10.

4.7 Phantom-node method This section is based on the sources M.J McNary [2009], T.Rabczuk et al. [2008], Song et al. [2006] Equivalent to the XFEM it is possible to model discontinuities at arbitrary locations in the mesh using the so-called Phantom-node method. The Phantom-node method is based on adding an extra element on top of an existing, cracked element. In contrast to the XFEM the crack kinematics is obtained by overlapping elements instead of introducing additional degrees of freedom. Abaqus 6.10 has a limitation of 20 degrees of freedom available per node. This limit is exceeded in special cases when using the XFEM, which has prompted Dassault Systémes Simulia Corp. [2010] to use the Phantom node method in its implementation, which is an alternative approach within the framework of the XFEM. Figure 4.8 shows an example of a cracked domain Ω0 , supported along the boundary Γu and subjected to a traction, t, applied along the boundary Γt .

Figure 4.8.

The principle of the Phantom-node method. On top of the nodes in the cracked elements phantom nodes are added and the integration is performed over the hatched area. Solid circles represent real nodes and hollow circles represent phantom nodes. f (X) is the signed-distance level-set function evaluated at the Cartesian coordinate X .

37



In the cracked elements phantom nodes are added on top of the real nodes and thus, leading to an additional element on top of the cracked element. Each element consists of + − − a real subdomain and a phantom subdomain, e.g. Ω+ 0 and Ωp , where Ω0 = Ω0 ∪ Ωp with reference to gure 4.8. Then the displacement eld in the real domain can be interpolated using the degrees of freedom in the phantom domain. Initially the real node and the phantom node are tied together. When cracking occurs, e.g. by fullling a damage criterion, each phantom node and its corresponding real node are no longer tied together and can move apart. The magnitude of the separation, i.e. the crack opening, is governed by a cohesive law until the cohesive strength of the element is zero. This relation is governed by the cohesive segments method. In Abaqus 6.10 the relation between crack opening and traction is linear and given as input to the model by prescribing a fracture energy. This is described in detail in chapter 5. While the XFEM is suitable for modeling the singular stresses and strains around a cracktip, the Phantom-node method is only applicable to cohesive crack modeling. In this fashion, the crack is extended to the element boundary and the singular eld is replaced by a cohesive traction. However, the Phantom-node method has been extended to model the crack-tip inside an element by T. Rabczuk and Wall [2008]. Abaqus 6.10 uses the former approach, that is, the crack-tip always ends at an element boundary. This is a simplied approach to crack modeling, as mesh-sensitivity is introduced for crude mesh densities. For this reason the possibility of a precise evaluation of the crack-tip location and propagation is limited. Figure 4.9 illustrates the dierence between the XFEM and the Phantom-node method for a one-dimensional bar with an inter-element discontinuity.

Figure 4.9.

38

The interpolation basis of the XFEM vs. the Phantom-node method for a onedimensional element. T.Rabczuk et al. [2008]

4.8. Numerical integration of the weak form

Master Thesis

Note that the displacement jump, [[u]], in gure 4.9 is identical for the two methods. In the Phantom-node method the two elements representing the cracked element do not share nodes and therefore have independent displacement elds. Both elements are only partially active, which is represented numerically in the denition of the displacement eld by introducing the jump function, see equation 4.14, which is active based on the signed-distance level-set function. The displacement jump over a crack is then dened as the dierence between the displacement elds of the two elements. The approximation of the displacement, using the Phantom-node method, is given in equation 4.19.

uh (X) =

X

NI (x) · uI · H(f (X)) +

I ∈ {w0+ , wp− }

X

NJ (x) · uJ · H(−f (X))

J ∈ {w0− , wp+ } (4.19)

− + − where w0+ , w0− , wp+ and wp− are the nodes belonging to Ω+ 0 , Ω0 , Ωp and Ωp , respectively. It has been shown that the Phantom-node method is equivalent to the XFEM. An example is the similarity between enriched degrees of freedom in the XFEM and the phantom degrees of freedom in the Phantom-node method. From a computational point of view, the Phantom-node method is superior to the XFEM, however more simple for previously mentioned reasons, e.g. crack-tip position. Note, that the Phantom-node method is based on the XFEM, in the sense that the standard FEM approximation space is enriched. Moreover, both methods reect the jump in the displacement eld by piecewise integration. Furthermore, both methods use the level-set method in the topological description of the discontinuity and in the evaluation of the global enrichment functions. The numerical integration method used in both methods is described in the following section.

4.8 Numerical integration of the weak form Strong discontinuities will be present in the reinforced concrete beam treated in this report, that is, a jump in the displacement eld across the cracked elements. This complicates the numerical integration in the cracked elements, because standard Gauss quadrature requires the integrand to be polynomial. Within the nite element framework, this corresponds to the integration of element-wise smooth, continuous shape functions. If the numerical quadrature of the weak form is not performed correctly, the advantage of including local enrichment functions is lost. Two remedies for the numerical quadrature of the weak form exist: 1. Element decomposition. 2. Integrand transformation. Element decomposition is a widely used technique, because it is computationally less demanding than the integrand transformation. Only element decomposition is described in the following, because Abaqus 6.10 utilizes this technique. 39



Element decomposition refers to a sub-division into smaller elements in the element containing the discontinuity. The sub-division is carried out in the reference element, that is, the element described in the isoparametric space. Isoparametric mapping is used to map the reference element from the isoparametric space to the physical space. Figure 4.10 shows a triangular element cut by a crack. The triangle is decomposed into two dierent elements: A smaller triangle and a quadrilateral, both aligning with the geometry of the crack.

Figure 4.10.

Element decomposition of a triangle into two sub-elements: A triangle and a quadrilateral. Four Gauss points are placed in each sub-element.

Standard Gauss quadrature is then performed over the sub-elements, in which the integrand is now smooth. Note that the sub-elements each have been assigned a number of Gauss points, in this case four. The number of Gauss points and the type of sub-elements used in Abaqus 6.10 are unknown to the authors. The crack is then an internal boundary of the domain of integration. A mathematical description of the integration is given in equation 4.20.

Z

f (x)non-smooth dx =

Ωe

K Z X k=1

Ωke

f (x)smooth dx

(4.20)

The sub-division is done by non-overlapping elements that must conform to the same requirement as the continuous problem. Note that element sub-division is not equivalent to remeshing, as no additional degrees of freedom are introduced. The sub-elements are only introduced for the purpose of integration. Moreover, the basis function is associated with the nodes of the parent domain, which implies that no restriction is imposed on the shape of the sub-elements. Recall, that the discontinuity is dened as the zero-level of the interpolated level-set function, see equation 4.2.

Φh =

X i∈I

40

Ni (x) · Φ(xi ) = 0

4.9. Governing equations

Master Thesis

Dierent shape functions can be used for the interpolation than for the approximation of the displacement. Linear shape functions are particularly useful, as the discontinuity is a straight line in the reference element, and also when mapped in the real element. This simplies the integration signicantly, as an element sub-division algorithm is more easily constructed, than for a curved discontinuity. However, linear shape functions only allows for a piecewise linear representation of discontinuities. Bi-linear shape functions produce curved discontinuities, and an ecient element sub-division algorithm is dicult to construct. For this reason, if bi-linear or higher order shape functions are used, linearization of the discontinuity is often used. The discontinuity is linearized by drawing a straight line between the points on the element edges that are cut by the discontinuity, see gure 4.11.

Figure 4.11.

(a) A curved interface in a bi-linear element can be (b) linearized by neglecting the curvature of the interface or (c) the element is decomposed into two triangles and linear interpolation is assumed.

The nodal values of the level set function are interpolated using the same shape functions as in the approximation of the displacement. However, it is unknown whether a linearization of the interface is performed. In either case, when the sub-division has been performed, standard Gauss quadrature is adopted. The element decomposition approach is favorable in computational implementation, because existing nite element integration schemes do not need any modication. In the following section the weak form of the equilibrium equations are presented. Systèmes [2010]

4.9 Governing equations The weak form of the equilibrium equations, the displacement u ¯ being the primary variable, can be stated by the principle of virtual work, see equation 4.21.

Z

T

Z

δ σ dΩ = Ω

T

Z

δu b dΩ + Ω

δuT tˆ dΓt ,

∀δu

(4.21)

Γt

41

Group B122b - Spring 2011 Ω Γt σ b tˆ


The domain of integration. The traction surface. Strain tensor in vector form. Stress tensor in vector form. Prescribed body force vector. Prescribed traction vector.

where superscript T refers to the transpose. Prescribed displacements u are imposed on Γu , while tractions tˆ are imposed on Γt . The internal boundary of the crack, Γc , is assumed to be traction-free. The domain Ω is bounded by Γ = Γt ∪ Γc ∪ Γu , as shown in gure 4.12.

Figure 4.12.

Domain Ω supported on Γu and loaded on Γt . An internal crack is dened along the boundary Γc .

σ · n = tˆ on Γt σ · n = 0 on Γc u=u ˆ on Γu The two-dimensional nite element discretized form of the weak form is stated in 4.22.

    fx tx Z   Z Z   fy   ty  B T · Dep · B dΩ · u =   dΩ +   dΓ ⇔ K · u = f + ft   0 Γ0 Ω Ω 0 0

(4.22)

where fx = Nstd · bx and fy = Nstd · by . bx and by are the body forces in the horizontal and vertical direction, respectively. ft contains the tractions tx and ty in the horizontal 42

4.9. Governing equations

Master Thesis

and vertical direction, respectively. Dep is the elasto-plastic constitutive matrix chosen according to the adopted material model of the concrete, see chapter 5. The strain distribution matrix B is given by equation 4.23.

  T T ∂x Nstd ∂x Nenr  T T  B= ∂y Nstd ∂y Nenr  T T T T ∂y Nstd ∂x Nstd ∂y Nenr ∂x Nenr

(4.23)

T are standard FEM shape functions, and N T are the local enrichment functions where Nstd enr given by equation 4.4. The strain distribution matrix is of dimension 3 × 2 · (nel + n∗el ), and nel and n∗el are the numbers of element nodes and enriched element nodes, respectively.

43

Discontinuous modeling in Abaqus

5

This chapter presents a summary of the methods described in chapters 2-4, used in Abaqus 6.10 in relation to discontinuous modeling. With regard to the methods of discontinuous modeling, preliminary choices made in Abaqus 6.10 for this report regarding the element type used for the numerical discretization and the material properties of the examined concrete and steel are presented. The choices are common for the benchmark tests considering concrete and the reinforced concrete beam considered in the problem formulation of this project. In Abaqus 6.10 cracking in concrete and steel is modeled in a discrete fashion, that is, a strong discontinuity is introduced in the displacement eld by the Phantom-node method, in which the displacement jump is reproduced by introducing the jump function. For the remainder of this report the Phantom-node method will be referred to as the XFEM because the methods are equivalent. The crack is represented as an open interface, but since the crack is always extended to the boundary of the element, in which it is present, only one level-set function is used for the topological description of a crack. The levelset function is chosen as the signed-distance function and is interpolated using the same interpolation functions as the approximation of the displacement. The crack is modeled by inserting a cohesive segment in the cracked element. The cohesive segments method is based on the ctitious crack model by A. Hillerborg and Peterson [1976], that is, nonlinear fracture mechanics is used. The adopted crack initiation criterion is the maximum principal stress criterion, in which a crack is initiated if the maximum principal stress reaches the tensile strength of the concrete. The crack propagates perpendicular to the direction of the maximum principal stress. The evolution of the crack is governed by the fracture energy, which represents the tension-softening behavior of the concrete during cracking. The relationship between the crack opening displacement and the closing stresses acting on the crack is linear, in which case the critical crack opening displacement, wc , is

1/2 · ft · wmax = Gf ⇔ wmax = 2 · Gf /ft

(5.1)

which is found by simple geometrical considerations, see gure 5.1. 45


Figure 5.1.

5. Discontinuous modeling in Abaqus

Linear relationship between the closing stress and crack opening displacement upon cracking.

The modeling of cracked concrete is illustrated in gure 5.2.

Figure 5.2.

The modeling concept used in Abaqus 6.10 to model cracks in concrete.

The constitutive behavior of concrete is modeled using the so-called Concrete Damaged Plasticity material model, abbreviated CDP. In tension the CDP material model is used in coordination with the XFEM and the cohesive segments method, that is, CDP is used until tensile crack initiation is detected, at which point a cohesive segment is inserted and the XFEM is activated. For compression the CDP material model is used without the XFEM. The CDP material model is a continuum-damage based constitutive relation. The stress-strain relation is given by equation 5.2.

σ = (1 − d)D0el : ( − pl ) = Del : ( − pl ) 46

(5.2)

Master Thesis D0el Del d

Initial, undamaged elastic stiness matrix of concrete. Degraded elastic stiness matrix of concrete. Scalar degradation variable.

The scalar degradation variable has an initial value of zero for intact material and increases towards one for complete loss of material stiness. Based on the observation that concrete is anisotropic, the scalar degradation variable is dierent in tension and compression. Since the scalar degradation variable is not used for tension, it will not be further discussed in this report. The compressive scalar degradation variable is determined according to equation 5.3.

(5.3)

dc = 1 − σc /fc

Unless stated otherwise the concrete examined in this report has the following material properties:

fc = 30 M P a fcm = 38 M P a ft = 1.95 M P a Ecm = 33 GP a ν = 0.2 ψ = 38◦ G = 0.08 N/mm

Characteristic compressive strength. Mean compressive strength. Mean tensile strength. The secant Young's Modulus between σ = 0 M P a and σ = 0.4fcd . Poisson's ratio. Angle of dilatancy. Fracture energy.

The ultimate tensile strength is determined by ft =

√

fcm · 0.1.

Abaqus 6.10 requires a specication of the uniaxial relationship between the inelastic strain and the stress after yielding has occurred. The uniaxial behavior is then extended to multiaxial directions. The uniaxial tensile and compressive response of the examined concrete is shown in gures 5.3 and 5.4, respectively.

Figure 5.3.

Response of concrete due to uniaxial tension. The blue circles represent the values used in Abaqus 6.10. 47



The blue circles on the uniaxial tensile response in gure 5.4 are used as input in Abaqus 6.10 to describe the tensile behavior of the concrete. However, Abaqus 6.10 requires the cracking strain versus stress. At the onset of cracking, the cracking strain is equal to zero and the total strain is equal to ft /Ecm = 5.24 · 10−5 . The exact values given as input in Abaqus 6.10 are given in table 5.1. More details regarding the stress-strain relations of the CDP material model are shown in appendix A. Stress [MPa] 1.732 Table 5.1.

Cracking strain [-] 0

The relationship between tenstile stress and direct cracking strain used as input for tensile behavior in the CDP model.

Figure 5.4.

Reponse of concrete due to uniaxial compression. The blue circles represent the values used in Abaqus 6.10.

The non-linear compressive reponse shown in gure 5.4 is based on an empirical stressstrain relation given in EN 1992-1-1 [2004]. The blue circles on the uniaxial compressive response in gure 5.4 are used as input in Abaqus 6.10 to describe the compressive behavior of the concrete. Note that the points shown in gure 5.4 are not directly used, as Abaqus 6.10 requires the specication of the inelastic strains. For this reason the elastic strain at each point has been subtracted from the total strain. The exact values are given in table 5.2. The non-linear relationship is shown in equation 5.4. 48

Master Thesis Stress [MPa] 6.61 9.68 12.58 15.34 17.93 20.36 22.64 24.76 26.72 28.52 30.17 31.66 32.99 34.16 35.18 36.04 36.74 37.29 37.68 37.92 38.00 37.92 37.68 37.29 36.75 36.05 35.19 34.18 33.01 31.69 30.21 28.58 26.79 24.85 Table 5.2.

σc =

η k

Inelastic strain [-] 0 0.0000066 0.000018 0.000035 0.000056 0.000082 0.00011 0.00014 0.00019 0.00023 0.00028 0.00034 0.00040 0.00046 0.00053 0.00060 0.00068 0.00076 0.000857 0.00095 0.0010 0.0011 0.0012 0.0013 0.0014 0.0016 0.0017 0.0018 0.0019 0.0021 0.0022 0.0024 0.0025 0.0027

Stress-strain values used as input for the compressive behavior in the CDP model.

kη − η 2 · fcm 1 + (k − 2)η

(5.4)

c /c1 1.05 · E |fc1c |

c1 = 0.22% is the strain corresponding to the ultimate compressive strength for the examined concrete in this report according to EN 1992-1-1 [2004]. As previously mentioned, the uniaxial behavior of concrete is extended to multiaxial directions in 49



order to capture its three-dimensional behavior. A main dierence in the description of the constitutive behavior of concrete between one-dimensional and three-dimensional modeling, is the introduction of the yield surface. The yield surface is described in a three-dimensional principal stress space, and is used to describe the stress evolution upon yielding. However, the shape of the yield surface is controlled by so-called hardening variables, also referred to as the equivalent plastic strains, ˜pl . The stress-strain relations for the general three-dimensional behavior of concrete is given by equation 5.5. Note, that equation 5.5 is the eective stress contrary to the Cauchy stress in 5.2, and is obtained by dividing equation 5.2 by (1 − d). Further information regarding the hardening variables is given in A.

σ ¯ = D0el : ( − ˜pl )

(5.5)

Tensile cracking is initiated for a principal tensile strain corresponding to a principal tensile stress equal to ft . Similarly, compressive elastic degradation is initiated for a principal compressive stress of fc . At the stage of the loading process where a tensile crack forms a discontinuity in the displacement eld of the cracked element is formed. Upon crack initiation a cohesive segment is inserted and the XFEM is activated in the cracked domain. In the uncracked domain solely CDP is used. Unless stated otherwise, the steel examined in this report has the following material properties:

fy = 400 M P a E = 210 GP a ν = 0.3 G = 5 N/mm

Characteristic yield strength. Modulus of elasticity. Poisson's ratio. Fracture energy.

The element type used for the numerical discretization in all benchmark tests and in the three-dimensional reinforced concrete beam, investigated in this report, is the C3D8 element from the Abaqus 6.10 library. The C3D8-element is a linear, 8-node, isoparametric three-dimensional hexahedron with full integration. The elements are integrated using eight integration points. The C3D8-element is shown in gure 5.5.

50

Master Thesis

Figure 5.5.

2 × 2 × 2 integration point scheme in a C3D8-element. The node numbering and

integration point numbering follows the convention in Abaqus 6.10. The integration points are shown in bold dots. MIT [2011]

The implementation of the XFEM in Abaqus 6.10 will be veried in the following chapter based on the modeling choices made in this chapter.

51

Verification of the XFEM and Abaqus

6

This chapter presents four dierent benchmark tests used to evaluate dierent aspects of the eectiveness of the XFEM tool and the CDP material model available in Abaqus 6.10. A description of the individual benchmark tests are given with respect to model setup and the available basis for a comparison, such as existing analytical, experimental or numerical results for the given problem. Finally the quality of the obtained results is used to assess the quality of the XFEM implementation and the CDP material model in Abaqus 6.10. In this chapter three benchmark tests are evaluated using the XFEM implementation in Abaqus 6.10, and one benchmark tests is carried out to verify the compressive behavior of the examined concrete using the CDP material model. Benchmark tests are critical to the understanding of the capabilities of the XFEM implementation and the CDP material model in Abaqus 6.10, because they serve as a source for comparison. Moreover, all examined benchmark tests are simple in geometry and loading. This makes the numerical model computationally light, which is convenient in order to verify the XFEM and the CDP material model eciently. Finally the authors believe, that since a learning process is involved, it is advantageous to start with a simple, well-dened benchmark problem before resorting to advanced, three-dimensional modeling of a reinforced concrete beam with more uncertainty in the outcome of the analysis. The following list of benchmark tests are performed in this chapter. 1. Crack-hole interaction for studying the inuence of a hole on the crack propagation path in a steel plate. 2. Verication of the compressive behavior of concrete using the CDP material model. 3. Crack propagation in a concrete beam in three point bending. 4. Crack formation analysis of a reinforced concrete plate. The benchmark test introduced in point 1 above deal with analyzing a two-dimensional steel plate with isotropic and linear elastic material behavior. The reason for analyzing steel is to start out with a simple material model before analyzing more complex materials, 53


6. Verication of the XFEM and Abaqus

e.g. concrete, that requires a more sophisticated material model. Moreover, a non-linear material behavior entails that the weak form of the XFEM is described in an incremental form, putting higher demands on the quality of the implementation of the numerical solver in Abaqus 6.10. The material behavior of steel is described using a linear elastic perfectly plastic material model. The material properties of the steel in test 1 and 2 are shown in table 5. This is followed by the benchmark tests introduced in point 2, 3 and 4, where concrete material properties are implemented. The material properties of the concrete in test 2 and 3 are dierent from the properties of the concrete in test 4, and are thus described in the introduction to the respective benchmarks. The CDP model described in appendix A is used in the context of describing the non-linear behavior of concrete. All benchmark tests are discretized using the C3D8 element described in section 5. The C3D8 element is a three-dimensional element and is used for four reasons: rstly, Abaqus 6.10 does not oer the XFEM tool for analyzing non-linear material models with planar elements. Secondly, only linear continuum elements are allowed when the XFEM is activated. Thirdly, in order to use the same element in all benchmark tests, thus eliminating a potential variable between the benchmarks on steel and concrete. Finally, the quality of the C3D8 element is tested as a candidate for modeling of the three-dimensional reinforced concrete beam. Although three-dimensional elements are used, the thickness of the specimens in benchmark test 1, 2 and 3 is irrelevant for the results. In all numerical simulations a poor element discretization, also referred to as element meshing in most commercial FEM software, can have a signicant eect on the reliability of the results. Therefore the benchmark problems will be modeled with carefully considered element discretization, i.e. structured and symmetrical meshes. In section 6.3 dierent mesh structures will be tested in order to verify the importance hereof.

6.1 Crack-hole interaction The following benchmark test concern crack-hole interaction for studying the inuence of a hole on the crack propagation path in a steel plate loaded in tension. The purpose of the analysis is to show, that the XFEM in Abaqus 6.10 can simulate crack growth without the need of remeshing while accounting for a complex geometry. Dierent starting locations of the crack were chosen to see how the proximity of the hole aects the stress eld, and thus the crack propagation path. The dimensions of the steel specimen are 7 mm × 21 mm × 0.1 mm, see gure 6.2. The numerically obtained results on the crack propagation path are compared with numerically obtained results by G. Ventura and Belytschko [2003]. Three models are created with three dierent distances between the crack and the hole. The three models have a distance from the center of the hole to the crack of A: 75 mm, B: 150 mm and C: 225 mm. Figure 6.2 shows setup A with applied boundary conditions and load. The load in the models is increased until Abaqus 6.10 fails to achieve equilibrium in the model. An initial crack of 2 mm is placed in each model.

54

6.1. Crack-hole interaction

Figure 6.1.

Master Thesis

Load and boundary conditions applied to the steel plate model.

Figure 6.2 shows the numerical results obtained by G. Ventura and Belytschko [2003]. The crack is clearly aected by the presence of the hole. It propagates towards the hole and ends on the periphery of the hole. The numerical results are shown in gure 6.3, and are seen to be similar with the dierence being the smoothness of the crack paths. It is observed, that the crack path becomes less aected by the hole the further the initial crack is placed from the hole.

Figure 6.2.

Numerical results of crack paths in a steel plate obtained by G. Ventura and Belytschko [2003].

55


Figure 6.3.


Crack propagation paths for three dierent initial crack locations: A: 75 mm, B: 150 mm and C: 225 mm are the distances from the center of the hole to the crack.

The results show that Abaqus 6.10 is capable of modeling crack propagation with respect to a change in geometry, such as the hole in the benchmark problem, without the need for remeshing.

6.2 Verication of the CDP material model In the following benchmark test a concrete cylinder, with the properties shown in table 5 and the dimensions shown in gure 6.4, is examined using the CDP material model available in Abaqus 6.10. Figure 5.4 shows the stress-strain relation given as input to the model. The benchmark test is carried out to verify, that the stress-strain response in compression is identical to the empirical stress-strain relation given as input, based on EN 1992-1-1 [2004], see equation 5.4.

Figure 6.4.

Dimensions in mm and boundary conditions for a section of the examined cylinder.

Note that, due to symmetry about the z-axis, only a section of the cylinder is modeled. The section is supported from moving in the longitudinal and horizontal direction in the way shown in gure 6.4. The loading is displacement based in order to get the post-peak response of the stress-strain relation as output. The stress-strain relation is requested in an arbitrary integration point. The location of the integration point is irrelevant because 56

6.3. Crack propagation in a concrete beam in three point bending

Master Thesis

the stress-state is uniform throughout the cylinder. Figure 6.5 shows the stress-strain relation given as output from Abaqus 6.10. The gure also compares the relationship to the expected relationship from EN 1992-1-1 [2004].

Figure 6.5.

Stress-strain relation for a cylinder loaded in compression calculated in Abaqus 6.10 and according to EN 1992-1-1 [2004].

The compressive behavior is as expected, and for this reason the implementation of the CDP material model in Abaqus 6.10 is acceptable. The tensile behavior is examined in 7

6.3 Crack propagation in a concrete beam in three point bending At this point two benchmark problems concerning the compressive response in concrete and crack propagation in steel have been successfully used to verify the CDP material model and the XFEM tool in Abaqus 6.10. In this section a concrete beam in three point bending with a 20 mm notch in the midspan will be used to further verify the XFEM tool. This is because the main purpose of the report is to model a three-dimensional concrete beam, and since concrete is a nonlinear, anisotropic material, this puts higher demands on the numerical solver. All dimensions are shown in gure 6.6. The applied load, P , reaches a value of 9 kN , at which stage Abaqus 6.10 fails to obtain equilibrium in the beam. The material properties of the concrete are shown in table 5. Experimental results obtained by J.Davies [1996], see gure 6.7, are used for comparison of the observed crack pattern.

57


Figure 6.6.

Figure 6.7.


The setup of a notched beam in three point bending. Numerical and experimental results of the crack propagation path are compared.

Experimental results obtained by J.Davies [1996] for a notched beam in three point bending.

The intention is to obtain numerical results similar to the experimental work performed by J.Davies [1996]. The crack propagation path observed in the experiment is the focus of the analysis. The microcracks developing in branches from the main crack are not possible to model in Abaqus 6.10, because cracks are not allowed to branch in the current implementation of the XFEM. Furthermore the inhomogeneous distribution of the aggregates in the concrete cause crack formation such as illustrated in gure 6.8. This is not to be expected in the numerical analysis.

58


Figure 6.8.

Master Thesis

Illustration of a possible crack path in concrete.

The model created in Abaqus 6.10 is a three-dimensional continuum model. A planar shell model is a more eective and simple model to use; however, as previously mentioned, Abaqus 6.10 does not oer the XFEM tool for analyzing nonlinear material models with planar elements. The Concrete Damaged Plasticity material model is used. The load is applied as a pressure, see gure 6.9. This is preferable to the 'Concentrated Force' option in Abaqus 6.10, in order to ensure that stress concentrations at the load surface do not occur. To prevent crushing at the supports, the beam is supported by a xed and a rolling 15 mm wide steel block, see gure 6.9.

Figure 6.9.

The modied loading and boundary conditions for the three point bend beam model in Abaqus 6.10.

The procedure is to ensure that the applied pressure and the resulting pressure at the supports do not exceed the prescribed compressive yield stress of the concrete. The global load-displacement curve has been plotted in gure 6.10 to illustrate, where the numerical 59



solver fails to obtain equilibrium. The load-displacement curve looks reasonable, as failure occurs abruptly upon reaching the maximum load carrying capacity of the plain concrete.

Figure 6.10.

Load-displacement curve for the three point bend beam.

Unlike the previous models an initial crack is not inserted prior to the loading, since the purpose is to determine whether Abaqus 6.10 can correctly initiate crack formation. By correct initiation the authors allude to crack formation at the location of the notch, as observed in the experiment by J.Davies [1996]. In the following the beam will be modeled with dierent mesh structures with the purpose of identifying a quality mesh from a poor mesh. Figure 6.11 shows a poorly constructed mesh, in the sense that the mesh is unstructured and unsymmetrical around the notch. The crack propagates from one side of the centerline of the notch to the other. Compared to the experimental results in gure 6.7 the propagation path is not similar. It can be observed that the initial crack propagation is skewed to the right. The experimental results indicate that the crack should propagate straight up from the notch. However, due to the fact that an intersection of two elements lies in that path, Abaqus 6.10 cannot simulate a straight crack path from the notch, hence the skewness to the right.

60


Figure 6.11.

Master Thesis

A poorly constructed mesh of a notched beam in three point bending. The gure shows the area around the notch where a crack is formed. The crack propagation path is inconsistent with experimental results.

In gure 6.12 the mesh has been made more structured around the notch. The crack path is straighter but the initial crack path is incorrect due to the asymmetric element distribution.

Figure 6.12.

An average constructed mesh of a notched beam in three point bending. The gure shows the area around the notch where a crack is formed. The crack propagation path is not correct despite being improved.

Figure 6.13 shows a structured and symmetrical mesh. The crack path is very similar to the experimental path in gure 6.7. However the small branches of cracks in gure 6.7 cannot be modeled. Moreover, the crack path does not deviate from a straight line as seen in the experiment by J.Davies [1996]. This is due to the inhomogeneous distribution of aggregates in concrete, which cannot be implemented into the numerical model. This issue was illustrated in gure 6.8.

61


Figure 6.13.


A well constructed mesh of a notched beam in three point bending. The gure shows the area around the notch where a crack is formed. The crack propagation path is similar to the results of the experiment conducted by J.Davies [1996].

In conclusion, the importance of the mesh structure depends on the problem being investigated. If a straight crack path is desired, e.g. in the analysis of pure mode I fracture, a symmetric, structured mesh is suggested. However, when analyzing crack propagation in heterogeneous materials like concrete, where cracking is arbitrary, a structured mesh is not an advantage with respect to realistic crack propagation paths. In other words, the randomness of the aggregate distribution can be accounted for by using a random distribution of elements. Furthermore it is apparent that the XFEM tool in Abaqus 6.10 can model crack propagation of geometrically simple concrete specimens in three dimensions with the desired precision.

6.4 Crack formation analysis of a reinforced concrete plate The nal benchmark test performed to verify the XFEM tool in Abaqus 6.10 considers a three-dimensional reinforced concrete plate. Boundary conditions, dimensions and applied loads can be seen in gure 6.14. A tensile stress of 1.75 M P a is applied at each end of the plate. There is no initial crack in the model. The boundary conditions along the right boundary include in-plane and out-of-plane restrainment. The left side boundary is restrained out of plane in order to ensure that Poisson deformation can occur.

62

6.4. Crack formation analysis of a reinforced concrete plate

Figure 6.14.

Master Thesis

The boundary conditions and applied loads to the reinforced concrete plate model in Abaqus 6.10.

The concrete and reinforcement used in the analysis have characteristic values shown in table 6.1. The reinforcement is embedded into the concrete, which is an option in Abaqus 6.10 that implies innite bond strength at the interface between the concrete and the reinforcement. Despite the fact that these conditions are not optimal, the authors were not able to successfully model the bond properties at the interface in Abaqus 6.10 any dierent. The characteristic compressive strength is obtained from S.B Bhide and M.P Collins [1989]. Information regarding the modulus of elasticity and Poisson's ratio are not described in S.B Bhide and M.P Collins [1989]. As a result these material properties have been assigned the values shown in table 6.1. The CDP material model is used for this benchmark problem. The compressive behavior of the concrete is described in 15 points along the curve in gure 6.15. The curve is constructed based on EN 1992-1-1 [2004]. The √ ultimate tensile strength is determined based on ft = fc · 0.1. The stress-strain behavior in tension is modeled as linear between 0 M P a − 1.52 M P a. The maximum principal stress damage criterion is used with a value of ft as the maximum principal stress at cracking.

63

Group B122b - Spring 2011 fc = 23.4 M P a ft = 1.52 M P a E = 19 GP a ν = 0.2 ψ = 38◦ G = 0.04 N/mm fy = 414 M P a d = 6.6 mm sx = 89 mm Table 6.1.

Figure 6.15.


Characteristic compressive strength. Characteristic tensile strength. Modulus of elasticity. Poisson's ratio. Angle of dilatancy. Fracture energy. Yield strength of reinforcing steel Diameter of reinforcement bars. Spacing of reinforcement bars.

Material properties of the examined concrete and steel in benchmark 4.

The stress-strain relation used to describe the compressive behavior of the concrete for this benchmark problem.

The purpose of the benchmark is to compare the crack formation in the concrete plate at failure to experimental results obtained by S.B Bhide and M.P Collins [1989]. The formation of cracks perpendicular to the direction of the applied load are expected. These cracks are so-called dilatational cracks, which occur because of volumetric expansion of the concrete. Figure 6.16 compares the crack formation in the concrete plate obtained from Abaqus 6.10 with the experimental results. The crack formation is similar, however more structured in the Abaqus 6.10 model. This is because Abaqus 6.10 cannot account for the inhomogeneous distribution of the aggregates in the concrete.

64

6.5. Concluding remarks

Figure 6.16.

Master Thesis

A comparison of crack formation at failure for a 890 mm × 890 mm × 70 mm plate. Right: results obtained in Abaqus 6.10. Left: results obtained from an experiment by S.B Bhide and M.P Collins [1989].

It is observed from gure 6.16, that Abaqus 6.10 is able to represent the dilatational response.

6.5 Concluding remarks With respect to the forward work of modeling a 3-dimensional reinforced concrete beam, the benchmarks tests have been valuable in the sense, that Abaqus 6.10 is capable of accounting for an advanced geometry. With the XFEM implementation, interelement discontinuities are possible and cracks can be allowed to propagate, as indicated in the crack-hole analysis. Abaqus 6.10 reects the intended stress-strain relation in concrete when loaded in compression and using the CDP material model. Abaqus 6.10 is capable of using the XFEM in conjunction with analysis of crack propagation in non-linear materials. Crack initialization and propagation according to a failure criterion are working options in Abaqus 6.10. A structured mesh is not necessarily an advantage when dealing with concrete due to the inhomogeneous aggregate distribution, as indicated in the three point bend test. Abaqus 6.10 is capable of modeling composite materials, e.g. reinforced concrete, by assuming innite bond strength between the reinforcement and concrete.

65

Results and discussion of 3d beam analysis

7

This chapter presents the results obtained from analyzing a three-dimensional beam loaded in three point bending. Dimensions, boundary conditions and material properties are specied for the concrete and the reinforcing steel used for the beam. The adopted material model and the modeling procedure in Abaqus 6.10 are described. This is followed by an introduction to, and the motivation for, the studies performed on the beam. Finally, the results are presented and discussed. This project investigates a three-dimensional reinforced concrete beam. The boundary conditions, dimensions and reinforcement arrangement in the beam are shown in gure 7.1.

Figure 7.1.

Boundary conditions, dimensions and reinforcement arrangement in a reinforced concrete beam.

67


7. Results and discussion of 3d beam analysis

The analytical models used for the deection comparison is based on Bernoulli-Euler theory, and therefore the dimensions of the beam are chosen such that the beam primarily deects due to bending. The material properties of the concrete and the reinforcement are shown in tables in chapter 5, and are repeated below for convenience. Material data for concrete C30:

fc = 30 M P a fcm = 38 M P a ft = 1.95 M P a Ecm = 33 GP a ν = 0.2 ψ = 38◦ G = 0.08 N/mm

Characteristic compressive strength. Mean compressive strength. Mean tensile strength. The secant Young's Modulus between σ = 0 M P a and σ = 0.4fcd , according to EN 1992-1-1 [2004]. Poisson's ratio. Angle of dilatancy. Fracture energy.

Material data for reinforcement steel:

fy = 400 M P a E = 210 GP a ν = 0.3

Characteristic yield strength. Modulus of elasticity. Poisson's ratio.

Note, that the mean values of the concrete strength parameters are used, because the focus of the report is not to design the beam, but to obtain a realistic estimate of the crack widths, crack spacing and crack pattern. Moreover, a realistic estimation of the deection, crack widths, crack spacing etc. is of interest, rather than an estimation which is inuenced by a safety factor. The total area of the provided reinforcement falls within the limits of the minimum and maximum reinforcement areas according to EN 1992-1-1 [2004]. For this reason the reinforcement ratio is normal, and the assumption, in the ultimate limit state, of simultaneous tensile yielding in the reinforcement and compressive failure in the concrete is not violated. This is desired when comparing the cross-sectional stress distribution determined in Abaqus 6.10 to the expected analytical distribution, which assumes normal reinforcement ratio in the beam. The calculation of the reinforcement ratio and the minimum and maximum limits can be found in appendix B.3. In the following section the numerical model of the beam in Abaqus 6.10 is described with respect to boundary conditions, input for the material model of the concrete and the steel and modeling of the reinforcement.

68

7.1. Model setup

Master Thesis

7.1 Model setup In this section the following topics will be discussed in the stated order: boundary conditions, material model and reinforcement model. The input les used in this chapter can be found in appendix B.2 and B.14, for a model with and without the activation of the XFEM, respectively. The input le for appendices B.2 and B.14 is found on appendices B.1 and B.13, respectively.

7.1.1 Boundary conditions The beam is simply supported along two lines parallel with the x-axis, with a mutual distance of 5000 mm, see gure 7.1. It was found in Abaqus 6.10 that the support conditions do not lead to noticeable stress concentrations. The beam has been extended by 100 mm from the support in order to increase the area over which the reaction stresses can be distributed. Note that the total length of the beam is 5200 mm, and the length between the supports is 5000 mm. Figure 7.2 shows the condition at one of the two supports.

Figure 7.2.

The condition at one of the supports on the beam. The beam is supported along a line parallel with the x-axis, and is extended by 100 mm from the support.

The boundary conditions of the beam model are as follows: U1 = X U2 = Y U3 = Z

BC 1 (Fixed support) 0 0 0 Table 7.1.

BC 2 (Roller support) 0 0

BC 3 (Load) 35 mm

The boundary conditions of the beam model.

The X, Y and Z direction in table 7.1 are shown in gure 7.1. The load is deection controlled and applied at the top midpoint of the beam over an area of 250 mm×250 mm; the top midpoint being the center of deection. In this way the simulation can continue after the ultimate load has been reached. The load could not be applied along a line, as the supports, because localized crushing of the concrete was observed causing the analysis to abort prematurely.

69



7.1.2 Input for the material model In chapter 5 a decision was made to use the Concrete Damaged Plasticity model to describe the constitutive behavior of the concrete. The stress-strain relations used as input for the compressive and tensile behavior in the CDP model was described in chapter 5, and are shown in table 5.2 and 5.1, respectively. As described in chapter 5, the tensile behavior in the model combining the CDP model with the XFEM is described without any tension stiening because the Phantom-node Method is activated at the onset of cracking. For the model without the XFEM, that is, solely CDP, the tension stiening eect could not be modeled. The authors have attempted to model the tension stiening eect but without success. Since it is not within the scope of this report to model tension stiening using the CDP model the issue is not pursued further. The maximum principal stress criterion, MaxPS, for traction separation laws is used in the model, with the sub option Damage Evolution activated, in Abaqus 6.10. The used values are given in table 7.2 MaxPS [MPa] 1.732 Table 7.2.

Fracture Energy [N/mm] 0.08

The values of the maximum principal stress, MaxPS and the fracture energy, in the sub option Damage Evolution, in Abaqus 6.10.

In the CDP model four parameters control the evolution and the shape of the yield surface and the ow potential. These are outlined in the following. The dilation angle, ψ , is included in the ow potential and can be visualized in the p−q plane, under high conning pressures, as the angle between the direction of the plastic strain increment and vertical, see gure 7.3. p and q are the hydrostatic and the deviatoric stress tensors, respectively. A value of ψ = 38◦ is chosen inspired by Jankowiak and Lodygowski [2005]. Secondly, the so-called eccentricity, , denes the rate at which the Drucker-Prager hyperbolic ow function approaches the asymptote. The ow potential tends to a straight line as the eccentricity tends to zero. The asymptote corresponds to a linear DruckerPrager model, see gure 7.3. The default value is 0.1, which is accepted in this report. Note that the friction angle is not given as input to the CDP model. The determination of the friction angle, β , is further described in appendix A.

70

7.1. Model setup

Figure 7.3.

Master Thesis

The Drucker-Prager hyperbolic function in the p − q plane. The Drucker-Prager linear function is shown with dashed lines. β is the friction angle and ψ is the dilation angle.

The third parameter of inuence is the ratio of biaxial compressive yield stress, fcb , to the uniaxial compressive yield strength, fc0 , that is fcb /fc0 . The ratio is important for the shape of the yield surface in the principal stress space, as it determines the location of the point of biaxial compression, see gure A.1 in appendix A. The default value is 1.16, which is accepted in this report, and corresponds to a ultimate biaxial compressive yield strength of 38 M P a · 1.16 = 44.08 M P a. The last parameter of inuence on the yield surface is K , which describes the ratio of the second invariant on the tensile meridian, to that on the compressive meridian. The default value is K = 2/3, which is accepted in this report. The inuence of K on the yield surface in the deviatoric plane is illustrated in gure 7.4, where an additional yield surface for K = 1 is plotted for comparison.

71



Figure 7.4.

Yield surfaces in the deviatoric plane. (T.M.) refers to tensile meridian and (C.M.) refers to compressive meridian. Si is the i'th principal stress, i = 1, 2, 3.

The four nal values of the parameters used in Abaqus 6.10 are summarized in table 7.3.

fcb /fc0 1.16 Table 7.3.

ψ 38◦

0.1

K 2/3

The values of the four nal values of the parameters used in Abaqus 6.10.

7.1.3 Reinforcement Model The reinforcement is modeled with the wire-option in Abaqus 6.10. The wire is meshed with truss elements that have only uniaxial degrees of freedom. The wire is embedded into the solid beam model using the Constraints-option, which entails innite bond between the concrete and the reinforcement. Kinematically the translational degrees of freedom of the embedded nodes are constrained to the interpolated values of the degrees of freedom of the host elements, that is, the solid elements constituting the beam. The thickness of the wire is not geometrically modeled; however a cross-sectional area is specied as input. The stress-strain relation, corresponding to the input shown in table 7.4, is shown in gure 7.5. No limit on the value of the total strain is specied.

72

7.1. Model setup

Master Thesis Yield stress [MPa] 400

Table 7.4.

Plastic strain strain [-] 0

The relationship between yield stress and plastic strain used as input for tensile behavior of the reinforcing steel.

Figure 7.5.

The employed stress-strain relationship for the reinforcing steel.

The wires only add stiness to the host elements in which they are located. Figure 7.6 shows an embedded wire superposed on a host element.

Figure 7.6.

Illustration of a wire embedded in a host element.

73



7.2 List of studies The list of studies performed on the beam is: A study of the load-deection curve of the beam using the CDP material model with and without the XFEM. This is done in order to investigate the inuence of the activation of the XFEM. An analytical expression from EN 1992-1-1 [2004] for the load-deection curve serve as a guideline for the expected result. A study of the stress distribution along the beam height in a section located at the midspan of the beam. The stress-distribution is plotted at various load-stages. The stress-distribution is plotted for two models; with and without the XFEM. The stress-distributions are compared to the theoretical distribution and cross-reference is made to the yield surface. A study of crack width, spacing and formation using the CDP material model and the XFEM in the serviceability limit state. Crack widths, spacing and formation are examined for a deection of 1/250'th of the beam span, equal to 20 mm, and compared to guidelines by EN 1992-1-1 [2004]. A deection of 1/250'th of the beam span is the maximum allowable deection in the serviceability limit state suggested by EN 1992-1-1 [2004]. Note, that the ultimate limit state is not of the same importance as the serviceability limit state when crack width, spacing and formation are analyzed. Crack widths are of importance for the durability of the beam and therefore the cracks caused by the service load should be analyzed. The main purpose of the two rst analyses in the list of studies is to verify the the CDP material model and the XFEM for a three-dimensional beam. The nal analysis in the list of studies is performed to answer the problem formulation of the report. In the following section the results from the above mentioned list of studies will be presented and discussed in the mentioned order.

7.3 Results and discussion 7.3.1 Load-deection curves Figure 7.8 shows the load-deection curves for ve dierent methods of analysis. These are: 1. A Bernoulli-Euler based analytical solution to the deection assuming a homogenous uncracked section, see equation 7.1. 2. A Bernoulli-Euler based analytical solution to the deection assuming a fully cracked section, i.e. concrete carries no tension, see equation 7.2. 3. An analytical solution to the deection given in EN 1992-1-1 [2004], see equation 7.3. 4. A numerical solution from Abaqus 6.10 where the Concrete Damaged Plasticity model is used. 74

7.3. Results and discussion

Master Thesis

5. A numerical solution from Abaqus 6.10 where the Concrete Damaged Plasticity model is used in combination with the XFEM. The load-deection equation for point 1 above is given by

uuncr =

uuncr P Ig

P · l3 48 · Ecm · Ig

(7.1)

The deection assuming a homogenous uncracked section. The applied load. Gross moment of inertia, Ig = b · h3 /12.

The load-deection equation for point 2 above is given by

ucr =

P · l3 48 · Ecm · Icr

(7.2)

where the moment of inertia for a fully cracked section, Icr , is given by Icr = b · x3 /12 + n · As · (d − x)2 . The equation for determining Icr assumes a fully cracked section and that the concrete in the compression zone behaves linear elastic until the steel is yielding. The equation for Icr is established based on gure 7.7.

Figure 7.7.

ucr x n d As

A fully cracked section for determining Icr .

The deection assuming a fully cracked section. 5·As ·fy The neutral axis depth given by x = 4·b·f . c Modular ratio of steel to concrete given by n = Es /Ecm . Eective depth of tension reinforcement. Total area of reinforcement. 75



The load-deection equation for point 3 above, suggested in EN 1992-1-1 [2004], is given by

u = ucr ·

1−

Mcr Ma

2 !

+ uuncr ·

Mcr Ma

2

where the cracking moment, Mcr , is given by Mcr =

u Ma fr fctm h

(7.3)

fr ·Ig h−x .

The deection of the beam according to EN 1992-1-1 [2004]. The midspan moment, equal to Ma = P4·l . Modulus of rupture, given by fr = max(fctm , (1.6 − h/1000) · fctm ). Mean exural tensile strength of concrete. Beam height.

Figure 7.8.

A comparison of the load-deection curves for two analytical solutions, the beam assumed uncracked and fully cracked, a solution recommended in EN 1992-1-1 [2004] and the two numerical solutions from the XFEM and CDP models to the displacement of the beam as a function of the load.

The analytical displacement curves end when the ultimate load is reached. The ultimate load is calculated by determining the moment of resistance, Mr , see equation 7.4, and ·l isolating Pult in Mr = Pult 4 . The ultimate load is found to be Pult = 262.8 kN . Equation 7.4 is established based on equilibrium between the tensile and the equivalent rectangular compressive stress distribution in a fully cracked cross-section, see gure 7.9. 76


Master Thesis

The rectangular compressive stress distribution is assumed equivalent with the compressive stress distribution shown in gure 7.11.

Figure 7.9.

Equivalent stress distribution at ultimate load in a fully cracked cross-section.

Mr = d −

4 x 10

4 b xfc 5

(7.4)

Note that equation 7.3 assumes, that the beam will crack but not fully crack, and will thus behave in a manner intermediate between the uncracked and fully cracked conditions. The displacement based on equation 7.3 shows a kink at a load of Pcr = 14.78 kN , which is the analytical value for the load causing crack initiation. The load has been determined by evaluating the cracking moment and isolating P in the equation for the midspan moment. The displacement kink at the cracking load is better visualized in gure 7.10. In the numerical results the cracking load was found to be Pcr = 16.90 kN , which is in close agreement with the analytical value.

77


Figure 7.10.


Close-up of the load-deection curves from gure 7.8 illustrating the displacement kink represented by equation 7.3 at a load of 14.78 kN . A displacement kink is present at a load of 16.90 kN in the numerical models.

Note, that the load-displacement curves with and without the XFEM are nearly coincident up to a level of the ultimate load. This illustrates, that the activation of the XFEM does not inuence the load-deection behavior. With this observation it can be concluded that, if a deection or ultimate load capacity estimation is desired, the XFEM does not need to be activated. Recall, that the XFEM introduces additional degrees of freedom to the system, which increases the computational eort. As seen from gure 7.8, the load-displacement curves from Abaqus 6.10, with and without the XFEM, falls within the analytical limits for deections smaller than approximately 25 mm. This does not imply that the numerical model is inaccurate in the ultimate limit state. However, the purpose of this report is to investigate crack widths, patterns and spacing in the serviceability limit state and compare the results to guidelines from the EN 1992-1-1 [2004]. For the sake of this purpose, it is convenient to have agreement between the load-deection curves using the XFEM in Abaqus 6.10 and equation 7.3. The load-displacement curves shown in gure 7.8 are for converged results. The convergence test is based on a uniform h -renement, that is, the element size is varied in the same proportion in the entire mesh. Convergence tests on the ultimate load and the corresponding displacement are performed in the following section.

78


Master Thesis

7.3.2 Cross-sectional stress distribution Figure 7.11 shows a principle sketch of the actual stress development at a cross-section for increasing bending moment.

Figure 7.11.

Principle sketch of the stress development at increasing bending moment.

Note that the distribution of the compressive stresses is non-linear close to failure. The purpose of investigating the cross-sectional stress distribution in the beam is to determine the validity of the CDP material model and the XFEM. The success criterion in this analysis is to observe a stress development in a cracked section of the beam in Abaqus 6.10 similar to the theoretical development in the distribution shown in gure 7.11. The numerically obtained cross-sectional stress-distributions, with and without the XFEM in Abaqus 6.10, are shown in gure 7.12.

79


Figure 7.12.


A comparison of the stress distribution in a cracked section of the beam. The black line represents the stress-distribution without the XFEM and the blue line represents the stress-distribution with the XFEM model. The arrow represents the direction of the tensile force in the 4 steel reinforcement bars. The load stages are shown in gure 7.13.

Figure 7.13 shows the load stages of gure 7.12 on a load-deection curve.

80


Figure 7.13.

Master Thesis

Stages 1-6 of gure 7.12 illustrated with points on the load-deection curve from models with and without the XFEM.

Note, that for all load-stages the concrete carries tension if the XFEM is not activated. Based on the results shown in gure 7.12 it can be concluded, that Stage 1: Initially, the stress-distribution is linear. The tensile stresses in the extreme tension bers are close to the tensile strength of the concrete (1.95 M P a). Stages 2-4: The compressive stress distribution in the concrete is linear and the tensile stresses are constant and equal to the tensile strength of the concrete without the XFEM, and zero with the XFEM because the section is cracked. Stage 5: The stresses in the midspan section have reached the uniaxial compressive stress limit, causing crushing in the concrete, aecting the stresses in the neighboring, cracked section. The concrete exhibits compression softening. Stage 6: Crushing has occurred in the midspan section, resulting in a stress reduction in the adjacent, cracked cross-section. The eective load carrying area is reduced due to crushing according to the CDP material model. Crushing has not yet occurred in the part of the section, where the largest compressive stresses are present. The stress-distributions presented above are for converged results. The convergence test is based on a uniform h -renement and was performed to study the convergence of the ultimate load and the corresponding deection. The results are shown in gures 7.14 and 7.15, for the ultimate load and the corresponding deection, respectively. The stressdistributions are plotted for the nest mesh using reduced integration.

81


Figure 7.14.

Figure 7.15.


Convergence study on the ultimate load for the numerical models. The bold line represents the analytically determined ultimate load.

Convergence study on the displacement at the ultimate load level for the numerical models. The bold line represents the analytically determined deection at a level of the ultimate load.

Recall, that the analytical ultimate load is determined using equation 7.4, and that the analytical ultimate deection is determined using equation 7.3. 82


Master Thesis

The convergence tests show, that the results, with and without the XFEM, converge using approximately 2800 elements. Note that reduced integration when the XFEM is used is not possible, because the results do not converge. Generally the stiness of the beam in Abaqus 6.10 is underestimated when compared to the analytical estimations of the deection. A factor contributing to the larger displacements observed using Abaqus 6.10 is that the displacement contribution from shearing strains is neglected in the analytical equations, which is based on Bernoulli-Euler beam theory. This contribution has been determined to constitute 1, 5% of the total deection, see appendix B.5. However, the main contributing factor to the dierence in displacement is the cross-sectional stress distribution. As previously mentioned, the analytical equation for the displacement assumes the crosssectional stress distribution in gure 7.7, which is an elastic distribution. Moreover, the analytical equation for the ultimate load assumes the cross-sectional stress distribution in gure 7.9, which is a plastic distribution. This could explain why the ultimate load is more accurately approximated than the deection corresponding to the ultimate load, because the observed stress distribution at the ultimate load level, see stage 5 in gure 7.12, is plastic and matches the plastic stress distribution shown in gure 7.9. An additional convergence test has been carried out for a load corresponding to an analytical deection of 20 mm, see gure 7.16. This load has been determined to be 202.23 kN . The convergence test is based on a uniform h -renement and was performed to study the convergence of the deection in the serviceability limit state.

Figure 7.16.

Convergence study on the displacement at the serviceability limit state for the numerical models. The bold line represents a deection of 20 mm.

The ultimate load converges to the analytically determined resistance of the beam. The overestimation in the deection of the beam is approximately 10% in the ultimate limit state. The deection is underestimated by approximately 7% in the serviceability limit 83



state. This corresponds to the observed load-deection in comparison to the analytical load-deection curve from EN 1992-1-1 [2004], see gure 7.14. The results used in the convergence tests for the ultimate limit state and the serviceability limit state can be found in appendix B.1 and B.13. In conclusion the results are satisfying. The XFEM has been successfully applied to the CDP material model. The load-deection relation is unaected by the activation of the XFEM. The same conclusion applies for the estimation of the ultimate load capacity and the corresponding deection. It is dicult to state a conclusion on the cross-sectional stress-distribution without the activation of the XFEM, because the tension-stiening eect was not successfully implemented in the CDP material model. The compressive stress distribution, with and without the XFEM, are similar to the expected, analytical distribution.

7.3.3 Crack widths and spacing Cracks formed in reinforced concrete can be classied into two main categories, namely those occurring by externally applied loads, and those occurring independently of the loads. Cracks caused by shrinkage and temperature change fall into the second category. This report considers cracks caused by externally applied loads, and shrinkage and temperature induced cracking will not be considered in this section. Only cracks formed in the tensile zone of the beam will be considered in the following. Piyasena [2002] The maximum crack width that will not endanger the corrosion of the steel reinforcement depends on the environment in which the structure is placed. The EN 1992-1-1 [2004] puts a limit to the maximum allowable crack width of 0.3 − 0.4 mm depending on the exposure class, e.g. the environment, and whether the reinforcing bars are bonded or not. The limit to the maximum allowable crack width put by the EN 1992-1-1 [2004] is regardless of the nature of the cracking. For exposure classes X0 and XC1, that is, for concrete inside buildings with very low humidity or permanently submerged in water, respectively, the maximum allowable crack width is set to guarantee acceptable appearance of the concrete, as corrosion is not imminent. For the other exposure classes the limit to the maximum allowable crack width is put to avoid corrosion of the reinforcement. In this report, the amount of reinforcement in the beam falls within the requirements to the minimum and maximum reinforcement areas put by the EN 1992-1-1 [2004]. A limit to the minimum reinforcement area has been put by the EN 1992-1-1 [2004], because a minimum amount of bonded reinforcement is required to control cracking. This section will investigate the possibilities within the XFEM in Abaqus 6.10 to successfully model crack widths and spacing in the serviceability limit state, that is, for a vertical displacement of 20 mm measured at the midspan of the beam. The equation for the crack spacing, sr,max , is given in 7.5.

sr,max = k3 c + k1 k2 k4 φ/ρp,e

84

(7.5)

7.3. Results and discussion φ c k1 k2 k3 and k4 ρp,e Ac,e

Master Thesis

The bar diameter. Cover of the longitudinal reinforcement. A coecient which takes into account the bonding properties of the bonded reinforcement, = 0.8 for high bond bars. A coecient which takes into account the distribution of the longitudinal strains, = 0.5 for bending. Recommended values are 3.4 and 0.425 found in the National Annex. = As /Ac,e . The eective tension area.

h The eective tension area is given by Ac,e = b·hc,e , where hc,e = min 2.5 · (h − d), h−x 3 , 2 , where x is the neutral axis depth, see gure 7.17.

Figure 7.17.

The eective tension area Ac,e of a beam.

Equation 7.5 assumes that the bonded reinforcement is xed at closed centres within the tension zone, which is fullled by satisfying the inequality: spacing ≤ 5(c + φ/2). The equation for the crack width, wk , is given in 7.6.

σs − kt ρct,e (1 + αe ρp,e ) p,e f

wk = sr,max

σs αe kt

Es

≥ 0.6

σs Es

(7.6)

The stress in the tension reinforcement. The modular ratio, = Es /Ecm . A coecient, = 0.6 for short term loading.

The values of the coecients are chosen to t to the current model and all inequalities above are fullled. The calculations are shown in appendix B.4. The analytical crack width and spacing for a load corresponding to a displacement of 20 mm is calculated in appendix B.4 and is based on a cracked cross-sectional analysis, see gure 7.7. The numerically estimated crack width and spacing is a part of the output from the model in Abaqus 6.10. Figure 7.18 shows an example crack for a mesh consisting of 348 elements. The crack width is measured as the dierence in the position of the original and the deformed coordinates, marked with red dots in gure 7.18. The crack spacing is determined in the 85



same way. Since numerous cracks appear, the largest crack is chosen to represent the numerically estimated crack width. This is done for all mesh densities.

Figure 7.18.

The crack width and spacing is measured as the dierence in the position of the original and the deformed coordinates, marked with red dots

Table 7.5 shows the relation between the stress in the tension reinforcement for various mesh densities, the analytically and the numerically determined crack width and spacing.

nel 6708 4600 2796 2060 1227 348

σs (numerical), [MPa] 300.8 299.3 292.3 302.9 296.2 307.0

Table 7.5.

wk /sr,max (analytical), [mm] 0.159/166.924 0.159/166.924 0.159/166.924 0.159/166.924 0.159/166.924 0.159/166.924

wk /sr,max (numerical), [mm] 0.35/208.3 0.39/200.0 0.27/546.9 0.40/187.8 0.40/487.5 0.46/−

The analytically determined crack width and spacing according to equation 7.6 and 7.5, respectively, and the numerically obtained crack width and spacing in Abaqus 6.10 for a various number of elements, nel .

In average the analytical determined crack width is 58% smaller than the numerically determined crack width. The dierence between the analytical and the numerically obtained crack spacing is arbitrary, because the mesh density greatly inuences the spacing. However, a tendency towards a spacing of approximately 200 mm exist. The mesh consisting of 348 consists of a single crack, i.e. no crack spacing can be calculated. Either the semi-empirical equation proposed by EN 1992-1-1 [2004] is inaccurate, or the assumption behind innite bond, due to the embedment of the reinforcement in the Abaqus 6.10 model, violates the results to an unsatisfactory degree. No conclusion can be made with regard to the accuracy of the equations proposed by EN 1992-1-1 [2004], as no experimental work regarding crack width measurement or crack spacing on the exact beam in this report is available. With regard to the numerical results it is known, that crack width is greatly inuenced by the bond force acting on the interface between the reinforcing steel and the concrete Piyasena [2002]. Since the embedding of the reinforcing steel in the concrete implies that no slip can occur, the deformation of the concrete must conform to that of the reinforcing bars, which further widens the cracks. In other words, cracking results in relaxation in the concrete tensile stresses upon cracking due to slip, 86


Master Thesis

and this phenomenon cannot be modeled when the reinforcement is embedded. However, the employed method of determining the crack widths does not consider the contribution from elastic or plastic deformation of the cracked elements, which must be nonzero.

7.3.4 Crack pattern In this section crack patterns are discussed for six dierent mesh densities. The crack patterns are recorded at a midspan deection of 20 mm, corresponding to 1/250'th of the beam span, that is, in the serviceability limit state, see gure 7.19

Figure 7.19.

The crack patterns at U 2 = 20 mm for dierent mesh renements.

It is observed that the crack spacing depends on the mesh density to some extent. The crack height is similar, except for the mesh consisting of 348 elements. The reason must be found in the previously described cohesive segments method used in Abaqus 6.10, which assumes that a crack ends at an element edge, thus propagating through the entire 87



element. The crack in the crudest mesh does not propagate further than the rst element, because equilibrium would not be obtainable. Generally, the appearance of the exural cracks is as expected, with the maximum crack width at the tension face and zero width near the neutral axis. No shear cracks were observed. It is dicult to conclude on the accuracy of the crack pattern, because no experimental results are at hand for an identical case. However, experimental results for a beam loaded in four-point bending is performed by R.I. Gilbert and S. Nejadi [2011] and shown in gure 7.20.

Figure 7.20.

Crack pattern for a beam loaded in four-point bending. R.I. Gilbert and S. Nejadi [2011]

The observed crack pattern is similar in two ways to the numerically obtained results in this report. Firstly, the beam does not crack directly at the midspan of the beam. Secondly, the observed crack heights are approximately 60% of the beam height, which corresponds to the observed height of the cracks in the Abaqus 6.10 model. Note, that the dimensions, loading and reinforcement arrangement dier from the beam analyzed in this report, and for this reason a comparison should be taken lightly. Only two cracks are observed, which is less than the expected number. The number of cracks showed to be strongly dependent on the rate of loading. A low load rate was chosen for symmetry in cracking and equilibrium reasons. A high load rate generally caused errors, such as convergence problems and level-set function errors. Moreover, Abaqus 6.10 requires that cracks initiate in the center of the elements, which is also observed in gure 7.19. Finally, the adopted equations in EN 1992-1-1 [2004] does not take loading and geometry into account, naturally putting a limit to the quality of the comparison between the semiempirical results from EN 1992-1-1 [2004] and the numerically obtained results regarding crack width and spacing.

88

Conclusion

8

The problem statement of the project reads

Application of the extended nite element method and the Concrete Damaged Plasticity material model for cracking simulation in a three-dimensional reinforced concrete beam using the commercial nite element program Abaqus. Problem In this report a three-dimensional reinforced concrete beam has been analyzed with respect to load-deection behavior, cross sectional stress distribution, crack width, crack spacing and crack pattern. The dimensions of the beam and the reinforcement conguration was presented in gure 7.1.

Analysis In order to answer the problem statement a non-linear fracture mechanics approach was used. Abaqus 6.10 oers the cohesive segments method, which is based on the ctitious crack model proposed by A. Hillerborg and Peterson [1976]. This model was adopted to model the cracking behavior in the tensile region of the beam given a fracture energy of Gf = 0.08 N/mm. The maximum principal stress criterion, MaxPS, was adopted as the crack initiation criterion with a value of 1.95 M P a, equal to the tensile strength of the examined concrete. The Concrete Damaged Plasticity material model, abbreviated CDP, was adopted to model the inelastic stress-strain relation in the compressive region. The compressive behavior was successfully modeled in chapter 6. The CDP material model was chosen due to its manner of degrading stiness discretely in the Gauss points. The quality of the implementation of the XFEM was tested for three benchmarks in chapter 6. The rst benchmark considers crack hole interaction in a steel specimen. The second benchmark considers crack initiation and propagation in a concrete beam, without reinforcement, loaded in three-point bending. The nal benchmark considered crack formation in a reinforced concrete specimen. The benchmark results were considered satisfactory, although mesh-sensitivity was observed.

89


8. Conclusion

Results The purpose of the analysis of the three-dimensional beam was twofold. Firstly, to verify the modeling technique adopted in this project for a three-dimensional reinforced concrete beam, and secondly, to analyze the crack widths, spacing and pattern and compare the results to the results obtained by use of current empirical formulas given in EN 1992-1-1 [2004]. Load-deection curves were obtained in Abaqus 6.10 for numerical models with and without the XFEM. These were plotted versus analytical load-deection relations for uncracked, partly cracked and fully cracked sections. The analytical equation assuming a partly cracked section is provided by EN 1992-1-1 [2004] and was the main basis of comparison. The load-deection curves were similar with and without the XFEM. Within the serviceability limit state the numerical results showed a stier response than the load-deection relation given by EN 1992-1-1 [2004]. In the ultimate limit state the load-deection relation given by EN 1992-1-1 [2004] showed stier response than the numerical results. This was concluded to be due to the elastic stress distribution assumed by the expression given in EN 1992-1-1 [2004]. Moreover, the analytical expressions do not account for the deection contribution due to shearing strain. However, this was determined to constitute only 1.5 % of the total deection. It was concluded that the inclusion of the XFEM does not improve the estimation of the load-deection considerably. Furthermore the modeling technique adopted in this report proved to have no negative impact on the load-deection curve. The cross sectional stress distribution in a cracked cross section of the beam was analyzed. The distributions from the numerical models, with and without the XFEM, were compared. The purpose of the analysis was to verify the validity of the CDP material model, with and without the XFEM. Results showed similarity in the compressive stress distribution in every load stage when compared to the expected, analytical distribution. It was not possible to model the tension stiening behavior in the concrete in the CDP material model, and for this reason the tensile stresses did not degrade to zero. However, the stresses degraded to zero when the XFEM was used. Convergence tests showed, that the numerically estimated ultimate load converged to the analytically determined ultimate load for approximately 2800 elements. The numerically determined deection at the ultimate load was approximately 10% higher, with and without the XFEM, which was explained by the fact, that the analytical equation proposed in EN 1992-1-1 [2004] assumes a linear stress distribution, whereas the numerically obtained stress distribution was plastic. In the serviceability limit state analysis, the numerically determined deection was approximately 7% smaller, with and without the XFEM, than the analytically determined deection. The numerically determined deection in the serviceability limit state showed a better approximation to the analytically determined deection, because the cross sectional stress distributions were similar. This behavior also corresponded to the observed loaddeection behavior. Cracking in the beam was analyzed with respect to crack width, crack spacing and crack pattern. The analysis was made using the XFEM in order to visualize the cracks, which is not possible without the XFEM. Comparison was made to the EN 1992-1-1 [2004] with respect to crack width and crack spacing. The numerical model cracked in two 90

Master Thesis locations, which was below the expected number of cracks. This was explained to be strongly dependent on the rate of loading. The crack width was overestimated in the numerical model in comparison to the suggested equation in EN 1992-1-1 [2004]. This was explained by the assumption in the model, that the reinforcing steel was embedded in the concrete beam, thus allowing no bond slip. The crack spacing was found to be mesh dependent in the sense, that cracks must initiate in the center of an element. This is a limitation to the implementation of the XFEM in Abaqus 6.10. For three out of ve mesh densities the results on the crack spacing showed agreement with the analytically estimated spacing in EN 1992-1-1 [2004]. The observed crack patterns for various mesh densities showed similarities with an experimental study on a concrete beam subjected to four-point bending in two ways. Firstly, the crack heights were similar and equal to approximately 60% of the beam height and secondly, no cracks were observed in the midspan of the beam. It was noted, that geometry, loading and reinforcement arrangement diered between the experimental study and the study in this report. For that reason, the comparison was taken lightly. This report has shown that the XFEM can be successfully combined with the CDP material model in Abaqus 6.10 for accurate load-deection estimations and cross-sectional stress-distributions. Regarding cracking, the current implementation of the XFEM in Abaqus 6.10 contains few limitations, which inuences the quality of the estimation of crack width, crack spacing and crack pattern. It was found, that the XFEM is not necessary in order to accurately estimate the load-deection behavior of a reinforced concrete beam loaded to failure. The cross-sectional compressive stress distribution can be successfully modeled with and without the XFEM. However, the tensile stress distribution requires the implementation of the tension-stiening eect in the CDP material model in order to yield accurate results. With respect to the estimation of crack widths, crack spacing and crack patterns the implementation of the XFEM in Abaqus 6.10 needs improvements, generally by improving the cohesive segments method, so that cracks does not have to end on elements boundaries, by allowing cracks to initiate in arbitrary locations and not only on the center of element boundaries and lastly, allowing the coalescence of cracks.

91

Suggestions for future work

9

The implementation of the XFEM in Abaqus 6.10 has presented itself as an area of future research because of the aforementioned limitations. The cohesive segments method contains the limitation that a crack must end on an element boundary. This limitation could be alleviated, if crack-tip related quantities, such as the stress intensity factor, were used to calculate the size of the crack propagation increment and the direction of the crack. If the concrete is modeled as non-linear, this cannot be done using the J-integral, which assumes linear-elastic behavior of the material. However, since the aim of this report was to mainly consider cracking the serviceability limit state, a linear material model could be adopted and the J-integral could be used. This is because the concrete behaves approximately linear in the serviceability limit state. This has not been done in this report, because the J-integral is not implemented for evolving cracks in Abaqus 6.10. Finally, crack-tip enrichment can be incorporated in the XFEM to allow cracks to end inside elements and to improve the estimation of crack-tip related quantities, such as the stress intensity factor. The bond properties used in this report can be improved. A physically realistic description of the stress transfer at the interface between the steel and the concrete is necessary, as well as modeling the bond-slip that occurs at the interface. Shear reinforcement should also be included for a more realistic estimation of cracking. Moreover, a sensitivity study on the parameters used in this report should be performed. This report has lacked experimental data for a comparison of the list of studies in this project. Experimental studies on the inuence of geometry, reinforcement arrangement and loading conditions on would assist in the evaluation of the quality of the obtained results and lead to potential improvements to the used equations from EN 1992-1-1 [2004] in this study.

93

Appendix

1

Concrete Damaged Plasticity material model

A

This appendix presents the Concrete Damaged Plasticity material model, abbreviated CDP, used in this report. Unless stated otherwise the sources used in this chapter are Systèmes [2010], Jankowiak and Lodygowski [2005].

A.1 Concrete Damaged Plasticity The Concrete Damaged Plasticity model incorporated in Abaqus 6.10 is primarily intended for providing a tool of analysis in the case of cyclic loading. However, the model is also suitable for monotonic loading by being the only model capable of modeling the eects of irreversible damage. The model is intended for fairly low conning pressures, that is, less than four to ve times the ultimate compressive stress of concrete in uniaxial compression. For low conning pressures the concrete behaves in the brittle manner, with cracking in tension and crushing in compression. The constitutive theory in the Concrete Damaged Plasticity is aiming at the description of the irreversible damage with those two failure mechanisms, and thus, low conning pressure is assumed. In the following the stress-strain relations, yield function, ow rule and damage and stiness degradation are described.

A.1.1 Stress-strain relations The total strain is decomposed into an elastic part, el , and a plastic part, pl , see equation A.1.

= el + pl

(A.1)

The stress-strain relations are governed by scalar damaged elasticity, see equation A.2.

3


A. Concrete Damaged Plasticity material model

σ = (1 − d)D0el : ( − pl ) = Del : ( − pl ) D0el Del d

(A.2)

Initial, undamaged elastic stiness of concrete. Degraded elastic stiness of concrete. Scalar degradation variable.

The scalar degradation variable, d, can attain values between 0 and 1. In a material point with d = 0 the concrete is undamaged. d = 1 corresponds to a fully damaged, while intermediate values represent the irrecoverable, plastic deformation endured by the concrete for pre-peak loading conditions. Failure of concrete, that is, crushing or cracking, is therefore associated with a degradation of the elastic stiness. Within the context of scalar-damage theory the stiness degradation is isotropic and a single parameter, d, is used for the description. The eective stress at a material point is dened in equation A.3.

σ ¯ = D0el : ( − pl )

(A.3)

The Cauchy stress is related to the eective stress through the scalar degradation variable, see equation A.4

σ = (1 − d)¯ σ

(A.4)

The factor (1−d) represents the eective area of the load-carrying part of the cross-section. In the absence of damage the eective stress in equation A.3 is equal to the Cauchy stress. However, for d 6= 0 the eective stress is more representative than the Cauchy stress, as the eective, that is, the uncracked or uncrushed area, is carrying the external forces. For this reason the eective stress is used for plasticity analysis. The evolution of the degradation variable is controlled by the eective stress and a hardening parameter, e pl . The eigenvalues of the eective stress tensor, that is, the principal eective stresses, are used in the evolution equations for the hardening variables. Note, that two single hardening variables are used; one for compression and one for tension, see equations (A.5A.6).

ˆpl max ˆpl min ˆ¯i σ ˆ¯ ) r(σ 4

ˆ ˜pl ¯ ) · ˆpl c = −(1 − r(σ min

(A.5)

˜pl t

(A.6)

ˆ = r(σ ¯ ) · ˆpl max Maximum eigenvalue of the plastic strain tensor pl . Minimum eigenvalue of the plastic strain tensor pl . Eigenvalues, or principal stresses, of the eective stress tensor σ ˆ. Stress weight factor, see equation A.7.

A.2. Yield function

Master Thesis

ˆi Σ3i=1 σ ¯ ˆ r(σ ¯) = 3 ˆi ¯ Σi=1 σ

(A.7)

where h.i is the Macauley bracket and |.| refers to the absolute value. The stress weight factor attains values between 0 and 1 and is used to assign a weight to the principal strains in relation to their corresponding principal stress. The plastic strain tensor contains strains in directions parallel to the directions of the global Cartesian coordinate system. The values are determined using the specied uniaxial behavior of the concrete.

A.2 Yield function The yield function, F (¯ σ, e pl ) describes a surface in eective stress space, which determines the states of failure or damage. The yield surface used in Abaqus 6.10 are given by equation A.8 and shown in gure A.1.

F (¯ σ, e pl ) = α, γ p¯ q¯ ˆ¯max σ α and β

1 ˆ¯max − γ −σ ˆ¯max − σ ¯c (e pl q¯ − 3α¯ p + β(e pl ) σ c )≤0 1−α

(A.8)

Dimensionless material constants. Eective hydrostatic pressure. The Mises equivalent eective stress. The algebraically maximum eigenvalue of σ ¯ Dimensionless coecients.

5


Figure A.1.


Yield surface in plane stress used in the Concrete Damaged Plasticity in Abaqus 6.10. σc0 and σb0 refer to the unixial and equibiaxial compressive yield stress, respectively.

The denition of the Mises equivalent eective stress, q , and the eective hydrostatic pressure, p, follows from equations A.9 and A.10, respectively. r 3 q= S:S (A.9) 2 where S is the deviatoric part of the stress tensor given by S = σ ¯ + p · I . I is the identity matrix.

1 p = − · trace(¯ σ) 3

(A.10)

Microcracking and crushing are represented by increasing the value of the hardening parameter e pl , which also inuences the value of the degradation parameter, d, as previously mentioned.

6

A.2. Yield function

Master Thesis

A.2.1 Damage and stiness degradation for unixial loading Figure A.2 and A.3 shows the uniaxial tensile and compressive behavior of concrete, respectively, used in the Concrete Damaged Plasticity model in Abaqus 6.10. If the concrete is unloaded at any point on the softening branch, the elastic stiness is reduced. The eect of the damage is dierent in tension and compression, and the degraded response of concrete is taken into account by introducing two independent scalar damage variables, dt and dc , for tension and compression respectively.

Figure A.2.

Reponse of concrete due to uniaxial tension. Jankowiak and Lodygowski [2005]

Figure A.3.

Reponse of concrete due to uniaxial compression. Jankowiak and Lodygowski [2005]

7



The stress-strain relations for uniaxial tension and compression are given by equations A.11 and A.12, respectively.

σt0 = (1 − dt )E0 ( − ˜pl t )

(A.11)

σc0 = (1 − dc )E0 ( − ˜pl c )

(A.12)

where E0 is the initial stiness of concrete, and ˜pl ˜pl c are the plastic strains in t and tension and compression, respectively. The eective uniaxial cohesion stresses in tension and compression are given by equations A.13 and A.14, respectively.

σ ¯t =

σt = E0 ( − ˜pl t ) 1 − dt

(A.13)

σ ¯c =

σc = E0 ( − ˜pl c ) 1 − dc

(A.14)

Note, that eective uniaxial cohesion stresses act on the available load-carrying area. For this reason the nucleation and propagation of cracks increases eective stresses by increasing the scalar damage parameter.

A.2.2 Flow rule The CDP material model in Abaqus 6.10 uses a nonassociated potential ow given by equation A.15.

∂G(¯ σ) ˙pl = λ˙ ∂σ ¯

(A.15)

where the ow potential, G, is chosen as the Drucker-Prager hyberbolic function, see equation A.16.

G=

σt0 ψ

8

p (σt0 tan(ψ))2 + q¯2 − p¯ tan ψ

Parameter dening the rate at which G approaches the asymptote. Uniaxial tensile stress of concrete at failure. Angle of dilatancy measured in the p − q plane at high conning pressure.

(A.16)

Guide to Appendix CD

B

B.1 DVD 1 Appendix DVD number one containing all .pdf les and output databases from Abaqus 6.10 with the XFEM activated.

B.2 3D-Beam XFEM.cae Input le to Abaqus 6.10 with the XFEM activated.

B.3 Maximum and minimum reinforcement ratio.pdf Determination of the maximum and minimum reinforcement ratios according to EN 19921-1 [2004].

B.4 Crack widths and spacing.pdf Determination of the analytical crack width and spacing according to EN 1992-1-1 [2004] for a deection of 20 mm.

B.5 Shear deformation.pdf Determination of the shearing strain contribution to the deection according to EN 19921-1 [2004].

9


B. Guide to Appendix CD

B.6 Ultimate load and corresponding deection.pdf Determination of the ultimate load and the corresponding deection according to EN 19921-1 [2004].

B.7 Seed45XFEM.odb Output database for the three-dimensional reinforced concrete beam in Abaqus 6.10. The mesh consists of 6708 elements and the XFEM is activated.






10

B.13. DVD 2

Master Thesis

B.13 DVD 2 Appendix DVD number two containing all output databases from Abaqus 6.10 without XFEM activated.

B.14 3D-Beam CDP.cae Input le to Abaqus 6.10 without the XFEM activated.

B.15 Seed45CDPFull.odb Output database for the three-dimensional reinforced concrete beam in Abaqus 6.10. The mesh consists of 6708 elements and the XFEM is not activated. The model uses full integration.




B.19 Seed80CDPFull.odb Output database for the three-dimensional reinforced concrete beam in Abaqus 6.10. The mesh consists of 1227 elements and the XFEM is not activated. The model uses full integration. 11


B. Guide to Appendix CD


B.21 Seed50CDPRed.odb Output database for the three-dimensional reinforced concrete beam in Abaqus 6.10. The mesh consists of 4600 elements and the XFEM is not activated. The model uses reduced integration.





12

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