Vytautas Tamulėnas

VILNIAUS GEDIMINO TECHNIKOS UNIVERSITETAS STATYBOS FAKULTETAS TILTŲ IR SPECIALIŲJŲ STATINIŲ KATEDRA

Vytautas Tamulėnas

ARMUOTO BETONO ELEMENTŲ, VEIKIAMŲ TRUMPALAIKE IR CIKLINE APKROVOMIS, ĮTEMPIŲ IR DEFORMACIJŲ BŪVIO EKSPERIMENTINIAI IR TEORINIAI TYRIMAI

EXPERIMENTAL AND THEORETICAL INVESTIGATION OF STRESS- STRAIN BEHAVIOR OF REINFORCED CONCRETE MEMBERS SUBJECTED TO SHORT-TERM AND CYCLIC LOADING

Baigiamasis magistro darbas

Statinių konstrukcijų studijų programa, valstybinis kodas 621H21001 Tiltų ir viadukų specializacija Statybų inžinerijos studijų kryptis

Vilnius, 2014

Vilniaus Gedimino technikos universitetas ISBN ISSN Statybos fakultetas Egz. sk...... Tiltų ir specialiųjų statinių katedra Data ...... -.....-.....

Antrosios pakopos studijų Statinių konstrukcijų programos magistro baigiamasis darbas Armuoto betono elementų, veikiamų trumpalaike ir cikline Pavadinimas apkrovomis, įtempių ir deformacijų būvio eksperimentiniai ir teoriniai tyrimai Autorius Vytautas Tamulėnas Vadovas prof. habil. dr. Gintaris Kaklauskas

Kalba: anglų

Anotacija Dauguma gelžbetoninių statinių bei konstrukcinių elementų, pvz., tiltai, keliai, geležinkelio pabėgiai, vėjo jėgainių pamatai ir kt., yra nuolat veikiami pasikartojančių apkrovų. Ciklinis konstrukcijų apkrovimo pobūdis sukelia betono pleišėjimą, spartų deformacijų augimą, mažina jų laikomąją galią. Tokių statinių eksploatacija reikalauja didelių finansinių sąnaudų. Minėtos problemos negali būti išspręstos nagrinėjant vien tik pavienius jų sprendimo atvejus, todėl, nagrinėjant skirtingus apkrovimo atvejus, šiame darbe buvo nuspręsta atlikti konceptualius armuoto betono elementų tyrimus. Magistro baigiamasis darbas susideda iš dviejų tarpusavyje susijusių temų: 1) universalios konstrukcinių elementų nuovargio vertinimo metodikos kūrimo ir 2) galimybių naudoti inovatyvias medžiagas statinių ilgaamžiškumui užtikrinti vertinimo. Buvo atlikti eksperimentiniai betoninių prizmių, veikiamų cikline gniuždymo apkrova, deformacijų elgsenos tyrimai. Sukurtas gelžbetoninių elementų nuovargio prognozavimo metodas. Skaitinei analizei atlikti buvo parinktos gelžbetoninės kolonos. Skaičiavimo rezultatai palyginti su plačiausiai taikomais nuovargio vertinimo metodais. Parodyta, kad normų metodai yra neadekvačiai konservatyvūs. Tiriant inovatyvių medžiagų taikymo galimybes ir atsižvelgiant į Lietuvos statybinių konstrukcijų gamybos poreikį, baigiamojo darbo antrojoje dalyje dėmesys skiriamas pluoštu armuotų polimerų konstrukciniams kompozitams ir biologiniam savaime gyjančiam betonui. Sprendžiant konstrukcijų pleišėjimo bei korozijos problemas, aptartos svarbiausios šių medžiagų savybės, susijusios su patikimu gelžbetoninių konstrukcijų projektavimu. Atlikti eksperimentiniai bazalto lakštais sustiprintų tempiamųjų gelžbetoninių elementų tyrimai. Gauti rezultatai panaudoti formuluojant gaires ir perspektyvas tolesniems moksliniams tyrimams. Darbo apimtis: 82 puslapiai teksto, 81 formulė, 34 paveikslai, 6 lentelės ir 105 bibliografiniai šaltiniai.

Prasminiai žodžiai: biologinis betonas; ciklinė apkrova; nuovargis; pluoštu armuotų polimerų armatūra; laipsniškas silpnėjimas; savaime gyjantis; įrąžų persiskirstymas. Vilnius Gediminas Technical University ISBN ISSN Faculty of Civil Engineering Copies No...... Department of Bridges and Special Structures Date ...... -.....-.....

Master Degree Studies Structural Engineering study programme Master Graduation Thesis Experimental and Theoretical Investigation of Stress-Strain Behavior of Title Reinforced Concrete Members Subjected to Short-Term and Cyclic Loading Author Vytautas Tamulėnas Academic Prof Dr Habil Gintaris Kaklauskas supervisor

Thesis

language: English

Annotation A variety of concrete structures, i.e. bridge elements, roads, railway sleepers, foundations of wind turbines are constantly exposed to cyclic loads leading to the cracking of concrete, strength loss, rapid growth of deformations, and huge maintenance investments. The above-mentioned problems can not be resolved by examining their individual cases. Therefore, it has been decided to carry out a conceptual research of reinforced concrete composites subjected to different loading conditions. Present study consists of two interrelated topics: 1) the development of comprehensive fatigue assessment method, and 2) investigation of the versatility and relevance of innovative materials for improvement of structural durability. An experimental investigation on deformation behavior of concrete prisms subjected to cyclic compression has been performed. A new method to predict fatigue behavior of RC elements has been proposed. The results of numerical analysis on columns subjected to repeated loading has been compared to the most extensive fatigue assessment methods reported in the literature. It was obtained that the code techniques present inadequately high safety margin. Investigating the application of innovative materials and considering demands of Lithuanian civil engineering, the second part of the study focused on FRP reinforcement and biological self-healing concrete as the most promising alternatives for avoiding cracking of concrete and corrosion of steel reinforcement. Problematic issues (particularly, material properties of innovative materials) related to the design of RC structures have been discussed. In addition, the experimental investigation of RC ties strengthened with basalt FRP sheets has been performed. The obtained results has been used to formulate the guidelines and prospects for further research. The total scope of the master’s thesis – 82 pages, 81 formulae, 34 figures, 6 tables and 105 references.

Keywords: biological concrete; cyclic loading; fatigue; FRP reinforcement; progressive degradation; self-healing; stress redistribution. Contents

INTRODUCTION ...... 14

Reasons for Investigation ...... 15

Research Object ...... 16

Main Objective and Tasks ...... 16

Research Methods ...... 17

Reporting Results at Scientific Conferences ...... 17

Structure of the Master’s Thesis ...... 17

Acknowledgements ...... 18

1. FATIGUE ASSESSMENT METHODS FOR REINFORCED CONCRETE ELEMENTS SUBJECTED TO REPEATED LOADING ...... 19

1.1. Literature Survey on S–N Approach Models ...... 20

1.1.1. Wöhler Curves ...... 20

1.1.2. Refined Wöhler Curves ...... 22

1.1.3. Goodman Diagrams ...... 25

1.2. Damage Accumulation Models...... 26

1.3. Methodology of Fatigue Assessment for Concrete in Compression ...... 27

1.3.1. Time-Dependent Material Model ...... 27

1.3.2. Determination of Resistant Number of Cycles ...... 31

1.3.3. Evolution Law for Strain ...... 32

1.3.4. Algorithm of Concrete Capacity for Redistribution Processes ...... 34

1.4. Numerical Analysis of Reinforced Concrete Columns ...... 46

1.4.1. Numerical Disclosure of Presented Algorithm ...... 46

1.4.2. Graphical Illustration of Analysis Results ...... 46

1.4.3. Comparative Analysis ...... 48

1.5. Experimental Investigation of Concrete Prisms Subjected to Cyclic Loading ...... 49

1.6. Concluding Remarks of Chapter 1 ...... 50

2. INNOVATIVE MATERIALS FOR REINFORCED CONCRETE STRUCTURES ...... 52

2.1. FRP Reinforcement for Concrete Structures ...... 52

2.1.1. FRP Materials, Properties and Types of Manufacturing ...... 53

2.1.2. Experimental Investigation of RC Structures Strengthened with FRP ...... 57

2.1.3. FRP Applications ...... 59

2.1.4. Peculiarities of Structural Application and Design ...... 62

2.2. Biological Self-healing Concrete ...... 65

2.2.1. Mechanism of Self-healing using Bacteria ...... 66

2.2.2. Effect of healing agent additions on concrete strength ...... 68

2.2.3. Principal Obstacles Limiting the Application of Self-healing Concrete ...... 69

2.2.4. Self-healing Concrete Applications ...... 69

2.3. Concluding Remarks of Chapter 2 ...... 70

GENERAL CONCLUSIONS AND RECOMMENDATIONS ...... 71

REFERENCES ...... 72

LIST OF PUBLICATIONS BY THE AUTHOR ON THE TOPIC OF THE MASTER’S THESIS 81

List of Figures

Fig. 1.1. Types of fatigue regarding to applied load cycles (according to Hsu 1981) ...... 19 Fig. 1.2. Wöhler (S-N) curve (Elfgren and Gylltoft 1997)...... 20 Fig. 1.3. Fatigue strength of steel reinforcement (Hordijk 1991)...... 21 Fig. 1.4. Different types of loading that can be used in fatigue tests (Thun 2006) ...... 21 Fig. 1.5. Graphical representation of equation (1.2) (Tepfers and Kutti 1979)...... 23 Fig. 1.6. S-N diagram for fatigue of concrete in compression (Stemland et al. 1990) ...... 25 Fig. 1.7. Fatigue strength of concrete by a Goodman diagram for compressive stress (Schläfli et al. 1998) ...... 25 Fig. 1.8. Static stress-strain relationship for concrete in compression (CEB-FIP 2012) ...... 27 Fig. 1.9. Scheme of theoretical model (Zanuy et al. 2009) ...... 29 Fig. 1.10. (a) Load history with two different stress level steps; and (b) strain increment employing equivalent number of cycles (Zanuy et al. 2009) ...... 30 Fig. 1.11. S–N relations based on different stress ratios R ...... 31 Fig. 1.12. S–N relations based on different rate of loading f ...... 32 Fig. 1.13. Schematic variation of concrete variables with cycle ratio ...... 33 Fig. 1.14. Initial stress-strain relationship for concrete under compressive fatigue loading ...... 37 Fig. 1.15. A flow-chart of proposed algorithm ...... 45 Fig. 1.16. Cross-sections of the studied columns ...... 46 Fig. 1.17. Evolution of longitudinal strain with number of cycles regarding to reinforcement ratio 47 Fig. 1.18. Evolution of maximum stress with number of cycles regarding to reinforcement ratio .... 47 Fig. 1.19. Evolution of maximum stress level with number of cycles regarding to reinforcement ratio ...... 48 Fig. 1.20. Fatigue testing machine and fatigue failure of concrete prism ...... 49 Fig. 1.21. The rate of change of displacement regarding the number of load cycles ...... 50 Fig. 2.1. Structure and coating of FRP bars ...... 56 Fig. 2.2. Bond action of Schöck ComBAR reinforcement ...... 57 Fig. 2.3. Carbon FRP sheet for structural reinforcement and containment ...... 57 Fig. 2.4. Test setup (Kesminas and Tamulenas 2014)...... 58 Fig. 2.5. Crack pattern of all the specimens (Gribniak et al. 2014) ...... 59 Fig. 2.6. Fiberglass grid form for bridge decks ...... 60

Fig. 2.7. Externally-bonded carbon FRP sheets for shear strengthening of a reinforced concrete bridge girder ...... 60 Fig. 2.8. Comparison of characteristic and design tensile strength and elasticity modulus of different types of reinforcement (Timinskas et al. 2013) ...... 63 Fig. 2.9. A bacteria of self-healing concrete (Arnold 2011) ...... 66 Fig. 2.10. Autonomous self-healing mechanism of biological concrete ...... 66 Fig. 2.11. Clay pallets filled with healing agent (50% of concrete volume) (Arnold 2011) ...... 67 Fig. 2.12. Schematic representation ofself-healing process in biological concrete (Jonkers 2011) ... 67 Fig. 2.13. Compressive strength development of cement stone specimen containing: a) 5.8×108 cm-3 bacteria; b) 1% peptone of cement weight; c) 1% calcium glutamate of cement weight; d) 1% calcium lactate of cement weight (Jonkers and Schlangen 2009) ...... 68

List of Tables

Table 1.1. Main characteristics of analyzed columns ...... 46 Table 1.2. Comparison of results of RC columns A, B, C and D under different models ...... 48 Table 2.1. Physical and mechanical properties of different FRPs ...... 54 Table 2.2. Physical and mechanical properties of polyester, epoxy, and vinyl-ester resins ...... 55 Table 2.3. Main characteristics of experimental materials ...... 58 Table 2.4. Application of internal and external FRP reinforcements ...... 61

Symbols

퐸 − concrete modulus of deformation;

퐸1−2 − relative modulus of deformation at 푁⁄푁푓 = 0,1;

퐸2 − reduction rate of concrete stiffness in second fatigue stage;

퐸푐 − concrete initial tangent modulus of deformation;

퐸푓푎푖푙 − concrete modulus of deformation at the failure instant;

퐸푟푒푓 − modulus of deformation taken as reference;

퐸푠 − steel modulus of elasticity; 푓 − frequency;

푓푐 − concrete compressive strength;

푓푢 − steel yielding stress; 푁 − number of load cycles;

푁푒푞 − equivalent number of cycles;

푁푓 − number of load cycles to failure; 푅 − ratio minimum stress / maximum stress;

푆푚푎푥 − normalized maximum stress;

푆푚푎푥푙푖푚 − normalized threshold stress;

푆푚푖푛 − normalized minimum stress; 휀 − strain;

휀0 − maximum concrete strain at 푁 = 1;

휀1−2 − relative strain at 푁⁄푁푓 = 0,1;

휀2 − strain rate in second fatigue stage;

휀푐 − concrete strain at 푓푐;

휀푓푎푖푙 − failure strain;

휀푙푖푚 − concrete strain at 0,5푓푐 in softening curve;

휀푚푎푥 − maximum strain;

휀푚푖푛 − minimum strain; 휌 − steel reinforcement ratio; 휎 − stress;

휎푚푎푥 − maximum stress;

휎푚푖푛 − minimum stress.

Abbreviations

AFRP – aramid fiber-reinforced polymer; AR – alkali-resistant; BFRP – basalt fiber-reinforced polymer; CFRP – carbon fiber-reinforced polymer; EU – European Union; FRP – fiber-reinforced polymer; GFRP – glass fiber-reinforced polymer; HCF – high-cycle fatigue; LCF – low-cycle fatigue; NSM – near surface mounted; RC – reinforced concrete; SHCF – super-high cycle fatigue; UV – ultraviolet; VARTM – vacuum assisted resin transfer molding; VGTU Vilnius Gediminas Technical University.

INTRODUCTION

The effect of cyclic loading on reinforced concrete (RC) structures has become relevant in the last decades from the point of view of the adequate structural design and safety assessment. It is supposed that well designed and properly built RC structures should serve for centuries. Unfortunately, this is not the case. The present tendency of optimization in structural design makes the materials to work at higher stresses and in case of RC members this becomes clear by employing more slender elements. It also explains a larger variation of stresses between the maximum and the minimum loads, which are tend to cause fatigue problems. Insufficient cracking resistance causes excessive cracking in the structures followed by the corrosion of reinforcement and, consequently, a complete deterioration of the buildings. That is the case of approach slabs, road pavements or bridge deck slabs, where the dead load is much smaller than the live load. An appropriate design of durable RC structures has to consider all the possible deterioration mechanisms, including both time- and cycle-dependent effects. Cyclic loading is one of these effects and might be a determinant factor for a good structural performance. Therefore, a fatigue performance becomes an important limit state that must be considered by all the designers who are related to the structures (e.g. roads, wind power plants, offshore structures, different types of machinery foundations, etc.) that are often exposed to repeated loading. Next to the development of comprehensive and universal fatigue assessment methods, it should not be forgotten an application of innovative structural materials improving physical and mechanical properties of RC structures. Repair and strengthening of existing structures with externally bonded laminates or fiber-reinforced polymer (FRP) sheets might be an effective solution for the cracking- induced problems. Fiber-reinforced polymers are considered to be a promising alternative to steel reinforcement, especially in concrete structures subjected to aggressive environment or to the effects of electromagnetic fields. Another alternative to conventional materials for RC structures is the use of biological concrete. A novel technique based on the application of bacteria causing a self-healing mechanism in concrete is becoming a revolutionary up-to-date approach for long-term structural maintenance. It helps structures to prevent from excessive cracking and other structural defects. Therefore, considering the adequate design of RC structures, civil engineers have to combine both the comprehensive methods for long-term structural assessment and advanced materials.

Reasons for Investigation

A variety of concrete structures, i.e. bridge elements, roads, railway sleepers, foundations of wind turbines are constantly exposed to cyclic loads leading to the cracking of concrete, strength loss and rapid growth of deformations. Such a fatigue effect significantly reduces the service life of RC structures. Therefore, it is essential to identify the coherent solution of evaluating deformation response of RC elements subjected to cyclic loading. In general, principal theoretical models of fatigue effect on reinforced concrete members split into two major groups. The first group of models, which is also the most common one, is based on S-N (stress versus number of cycles) relationships and the static stress state. These models only provide a resistant number of load cycles under constant stress levels. However, the idealization of constant stress level is not adequate as acting cyclic loads cause the redistribution of stresses in concrete and reinforcement. Thus, the concrete stresses can be reduced up to a few times. Neglecting such a phenomena, the structural load-carrying capacity is being overestimated and design becomes economically inefficient. The second group includes damage models, which do consider the fatigue deterioration of concrete. However, hard applicability to structural concerns and large computational efforts make them generally inadequate for practical applications. Moreover, current design codes provide the assessment of fatigue effect only for newly designed structures. They do not cover any issues, e.g., fatigue-carrying capacity when the structure will be subjected to higher loads in the future, on already existing structures. It is important to mention that almost a half of the budget of construction industry in European Union (EU) is spent on the maintenance and repair of existing buildings. In order to reduce these costs, the huge efforts of scientists and engineers concentrate on improving the properties of concrete and development of innovative construction materials. However, due to the lack of legislative framework, insufficient professional qualification, and conservative attitude to unusual ways of dealing with a problem the wider application of these materials are still limited. Therefore, it is necessary to investigate applicability of innovative materials highlighting problematic issues important for the design of RC structures following by formulation of targets for the further research. The above-mentioned problems can not be resolved by examining their individual cases. Therefore, it has been decided to carry out a conceptual research of reinforced concrete composites subjected to different loading conditions. Present study consists of two interrelated topics: 1) the development of comprehensive fatigue assessment method, 2) investigation of the versatility and relevance of innovative materials for improvement of structural durability. Considering the demand

15 and opportunities of Lithuanian civil engineering, it has been decided to investigate the application of FRP reinforcement and biological self-healing concrete as the most promising alternatives for avoiding cracking of concrete and corrosion of steel reinforcement. The results of performed research will allow to formulate the guidelines and prospects for further research.

Research Object

Having a conceptual nature, the study investigates reinforced concrete composites subjected to different loading conditions. Therefore, it has two interrelated objects. The first object is a fatigue behavior of compressive reinforced concrete. The second – innovative concrete materials with increased cracking-resistance.

Main Objective and Tasks

The study has two objectives: 1. To propose a fatigue assessment method considering a time-dependent stress and strain behavior of reinforced concrete columns subjected to cyclic compressive loading; 2. To investigate applicability of innovative materials ensuring durability of reinforced concrete members.

In order to achieve these objectives, the following problems have to be solved: 1. To review a literature of commonly used material models and calculation techniques of fatigue analysis. 2. To perform an experimental investigation on cracking and deformation of reinforced concrete elements subjected to short-term and cyclic loading. 3. To develop a model for time-dependent stress-strain analysis of RC columns subjected to fatigue loading taking into account degradation of concrete. 4. Using the proposed model to predict deformation response of a reinforced concrete columns subjected to high-cycle loading. 5. To perform a parametric analysis of stress-strain behavior of the columns subjected to repeated loading. 6. To evaluate applicability of innovative materials emphasizing their advantages over conventional structural materials and durability issues. 7. To formulate the topics and give recommendations for the further research.

Research Methods

Seeking to achieve the aim of the work the research methods such as theoretical study, numerical research, experimental and comparative analysis were used.

Reporting Results at Scientific Conferences

The author has made 4 presentations at 3 scientific conferences:  The Eighteenth International Conference Mechanics of Composite Materials, Riga (Jūrmala), Latvia, 2014.  The Seventeenth Lithuanian Conference of Young Scientists Science – Future of Lithuania, Vilnius, Lithuania, 2014.  The Conference of Project “Promotion of Students’ Scientific Activities” Studentų moksliniai tyrimai 2012/2013, Vilnius, Lietuva, 2013.

Structure of the Master’s Thesis

The first part, a Chapter 1, reviews generally prevailing fatigue assessment methods for reinforced concrete elements subjected to repeated loading. Following the literature review, a new method to predict fatigue behavior of RC elements under cyclic actions has been proposed. A time-dependent model including the progressive degradation of concrete in compression is applied. According to the proposed algorithm determining the concrete capacity for redistribution processes, the numerical analysis on four columns having different reinforcement ratios has been performed. The obtained results have been compared to the most extensive fatigue assessment methods reported in the literature. Additionally, an experimental investigation on deformation behavior of concrete prisms subjected to cyclic compression has been performed. The second part, a Chapter 2, is dedicated to revise properties, application area, manufacturing and design peculiarities of internal and external FRP reinforcement and biological self-healing concrete. The main problems restricting application of these materials in building industry have been highlighted. Furthermore, the problematic issues (related to the material properties of innovative materials) important for the design of RC structures have been discussed. In order to evaluate the influence of various parameters on deformation behavior, the experimental investigation of RC ties strengthened with basalt FRP sheets has been performed. In addition, formulation of targets for the further research has been proposed.

General conclusions as well as recommendations for further research summarize the present study. It is followed by an extensive list of references and a list of 10 publications by the author on the topic of the master’s thesis. The total scope of master’s thesis – 82 pages, 81 formulae, 34 figures, 6 tables and 105 references.

Acknowledgements

The author expresses sincere gratitude and acknowledgement to his supervisor, Professor Gintaris Kaklauskas, Head of the Department of Bridges and Special Structures of Vilnius Gediminas Technical University, for providing a support and guidance along the way. Special gratitude and thanks are due to Dr. Viktor Gribniak, Associate Professor and Researcher at Vilnius Gediminas Technical University, for his help and friendship throughout this research. The financial support provided by the Research Council of Lithuania is gratefully acknowledged. Last but not the least, the author wishes to express his sincere gratitude to his family, friends and colleagues at the Department of Bridges and Special Structures for all-round support and friendly atmosphere created throughout this research.

1. FATIGUE ASSESSMENT METHODS FOR REINFORCED CONCRETE ELEMENTS SUBJECTED TO REPEATED LOADING

Fatigue failure could be defined as a failure that takes place below the stress limit of a material when it has been subjected to cyclic loading. Fatigue tests are often very expensive and time- consuming to perform due to the many factors that influence the fatigue capacity. A few of these factors are: maximum load level, load frequency, stress ratio etc. Regarding to the applied load cycles to a specimen fatigue tests are classified into three main groups. If a fatigue test takes only a few load cycles it is called a low-cycle fatigue (LCF). The limit that is used is approximate up to 103 load cycles. If the test keeps on longer than this limit it is named a high-cycle fatigue (HCF). There is also a third limit for a fatigue test, approximately 107 load cycles, called a super-high cycle fatigue (SHCF). The limits that have been mentioned here are not categorical ones; there are other ones to be found in the literature. In Fig. 1.1 Hsu (1981) presents several examples of some structures and to which fatigue category they do belong.

SUPER HIGH- LOW-CYCLE FATIGUE HIGH-CYCLE FATIGUE CYCLE FATIGUE

HIGHWAY AND AIRPORT RAILWAY BRIDGES, STRUCTURES SUBJECTED TO PAVEMENTS AND HIGHWAY EARTHQUAKES BRIDGES PAVEMENTS,CONCR ETE RAILROAD TIES

STRUCTURES

SEA STRUCTURES SEA

MASS RAPID TRANSIT TRANSIT MASS RAPID

0 101 102 103 104 105 106 107 5×107 108 5×108 Fig. 1.1. Types of fatigue regarding to applied load cycles (according to Hsu 1981)

According to Hsu (1981), it is necessary to look at the fatigue problem with a wider prospect and consider that rules and equations derived in research projects regarding high-cycle fatigue cannot be used in a study regarding low-cycle fatigue since the two ranges are different. Two reasons for this difference are the rate of loading and the effect of time in a fatigue test. Some researchers have claimed that the rate of loading is trivial in a fatigue test but others have shown that it is of great importance regarding low-cycle fatigue. The same could be said concerning the effect of time in a fatigue test.

1.1. Literature Survey on S–N Approach Models

The first approach to fatigue assessment to be developed, which is also the most common one, is based on stress versus number of load cycles curves (S–N curves) and the static stress state. These models only provide a resistant number of load cycles without considering any redistribution of stresses or the strain development of concrete.

1.1.1. Wöhler Curves

Steel Fatigue Fatigue capacity is normally described by Wöhler curves. They are named after the German engineer August Wöhler who carried out studies of the fatigue capacity of railway axles in the late nineteenth century (Wöhler 1858). The curve is also called an S-N curve (Stress-Number-curve). The

Fig. 1.2 shows the number of load cycles for different applied amplitudes of stresses (σA) and σB corresponds to the static failure load. If the number of load cycles at failure, N, is presented on a logarithmic axle the curve becomes linear. As can be seen in Fig. 1.2, the fatigue life decreases with increasing number of cycles. There is also a fatigue limit, which implies that no fatigue failure will occur if the loading does not exceed this limit. Steel is such a material but for concrete no such limit has been detected. The reason for this difference is that steel is a strain-hardening material (the strength increases at large strains) and concrete is a strain-softening material (the strength decreases at large strains).

Fig. 1.2. Wöhler (S-N) curve (Elfgren and Gylltoft 1997)

Fatigue resistance is determined by testing, and fatigue reliability of a structural element over its design service is verified if the fatigue resistance Rfat is larger than the effect of fatigue loading Sfat. The fatigue safety values of steel reinforcement and concrete are determined separately. For steel reinforcement the relevant parameters are: (1) the stress ranges; (2) the number of load cycles; and (3) discontinuities both in the cross section and the layout of the steel reinforcement 20 resulting in stress concentration at possible fatigue damage locations. Test results are plotted on a double-logarithmic scale (see Fig. 1.3) and, usually, a 5% fractile-criteria is used to determine the slope of the S–N curve and the detail category defined as the fatigue strength at 2×106 load cycles.

Fig. 1.3. Fatigue strength of steel reinforcement (Hordijk 1991).

The Wöhler curves are used for simplifications when the loading of the material has a constant mean value and constant amplitude. Then the fatigue life can be estimated directly from the Wöhler diagram of the material. On the other hand, the majority of structures are subjected to variable amplitude loading and such an application of the Wöhler diagram becomes problematic. Therefore, further described methods for estimating the fatigue life of the structures should be taken into account.

Concrete Fatigue There are several factors that influence a fatigue process. Some of them are: the maximum and minimum stress levels, the stress ratio, loading frequency (lower frequency gives lower number of cycles to failure) etc., see e.g. Holmen (1979), Hsu (1981) or Zhang et al. (1996).

Fig. 1.4. Different types of loading that can be used in fatigue tests (Thun 2006) 21

The load can also be varied in many ways in a fatigue test (see Fig. 1.4). One of the loading type is pulsating sinusoidal load (which implies that the mean stress, σm, is equal to zero and the amplitude is σa, see Fig. 1.4 a)) representing stress reversals under which the member is subjected to alternate between tension and compression. Such a stress reversals are observed to occur in the deck slabs due to the passage of a moving load (Schläfli and Brühwiler, 1998). Also the load could be the same as in

Fig. 1.4 a) but with a mean stress equal to the amplitude and σmin = 0. A more general case

(representing a pure compression or a pure tension of the member) is shown in Fig. 1.4 c) where σm ≠ 0 and σmin ≠ 0. There is also a type of loading called irregular loading which is the most realistic one a structure is exposed to (see Fig. 1.4 d)). In this case it is a bit more difficult to decide loading cycle, mean value or the amplitude. There are several methods that could be used to determine these parameters, for instance, the peak count method, range pair count or the rain flow count method etc. A description of the methods could be found in the document EuroLightCon (BE96-3942/R34 2000).

1.1.2. Refined Wöhler Curves

Aas-Jakobsen’s fatigue equation In order to improve the Wöhler curve Aas-Jakobsen (1970) examined the influence of the minimum stress, σmin, on the fatigue strength. He showed that the relationship between σc,max / fc and

σc,min / fc (also known as maximum and minimum stress levels, Smax, Smin, respectively) is linear for fatigue failure at N = 2·106 load cycles. If R is defined as the ratio between the minimum and maximum stresses, σc,min / σc,max, then the relationship between σc,max / fc (fc equal to the static strength) and R should also be linear. Combining these linear relationships he derived the following expression:

휎푐,푚푎푥 = 1 − 훽(1 − 푅) log 푁, (1.1) 푓푐 where 훽 − the slope of the S-N curve when 푅 = 0, 훽 = 0,064;

푅 − ratio between maximum and minimum stresses (휎푐,푚푖푛⁄휎푐,푚푎푥); 푁 − number of loading cycles to fatigue failure.

Equation (1.1) is valid for 0 ≤ 푅 ≤ 1 it means only for the stresses, which do not alternate between compression and tension.

Tepfers’ and Kutti’s fatigue equation Tepfers and Kutti (1979) made an extensive study of equation (1.1) for ordinary and lightweight concrete with the intention of proposing a fatigue relationship common for both types of concrete. They proposed the following equation (also see Fig. 1.5):

휎푐,푚푎푥 = 1 − 0,0685(1 − 푅) log 푁, (1.2) 푓푐 where 푁 − number of load cycles to fatigue failure;

푅 − ratio between maximum and minimum stresses (휎푐,푚푖푛⁄휎푐,푚푎푥);

휎푐,푚푎푥 − maximum compressive stress under pulsating load;

휎푐,푚푖푛 − minimum compressive stress under pulsating load;

푓푐 − static compression strength of concrete.

Fig. 1.5. Graphical representation of equation (1.2) (Tepfers and Kutti 1979)

Hsu fatigue equation Hsu (1981) studied the work done by Aas-Jakobsen (1970) and Tepfers and Kutti (1979) and even though he considered equation (1.2) being a big step forward in the development of the S-N curve, it had, in his opinion, two essential weaknesses. The first one is when 푅 = 1, equation (1.2) becomes

휎푐,푚푎푥⁄푓푐 = 1 and 휎푐,푚푎푥 equals to a constant. He points out that this is theoretically incorrect because when 푅 approaches unity a repeated load becomes a sustained load. It has by other researchers been established that sustained strength of concrete is time-dependent. Therefore time must be included in

23 the relationship. The second weakness is that it does not include the rate of loading as a variable and this must be considered at least in low-cycle fatigue. After some assumptions, following equations were established: (1) for high-cycle fatigue:

휎푐,푚푎푥 = 1 − 0,0662(1 − 0,556푅) 푙표푔 푁 + 0,0294 푙표푔 푓 ; (1.3) 푓푐 (2) for low-cycle fatigue:

휎푐,푚푎푥 = 1,20 − 0,2푅 − 0,133(1 − 0,779푅) 푙표푔 푁 + 0,053(1 − 0,445푅) 푙표푔 푓 ; (1.4) 푓푐

where 푁 − number of load cycles to fatigue failure;

푅 − ratio between maximum and minimum stresses (휎푐,푚푖푛⁄휎푐,푚푎푥);

휎푐,푚푎푥 − maximum stress in repetitive loading;

휎푐,푚푖푛 − minimum stress in repetitive loading;

푓푐 − static compression strength of concrete; 푓 − frequency of the repetitive loads expressed in cycle per sec.

Stemland’s et al. fatigue equation Another model of fatigue assessment proposed by Stemland et al. (1990) was developed to predict the relation between maximum stress level 푆 푚푎푥, minimum stress level 푆 푚푖푛 and number of load cycles to fatigue failure 푁. The main intention with the constant amplitude tests performed on nonreinforced concrete by Stemland et al. (1990) was to evaluate the effect of the change in minimum stress level on the fatigue life. Their design proposal for fatigue in compression has the following equation: 2 푙표푔 푁 = (12 + 16푆푚푖푛 + 8푆푚푖푛)(1 − 푆푚푎푥); (1.5) As can be seen in Fig. 1.6 the slope of the curves changes at log 푁 = 6. They found that this is approximately where the experimental results started to deflect towards longer lives than indicated by the equation. Therefore, they proposed that if log 푁 is greater than log 푁 = 6, there should be a correction factor, X, established for calculating the longer fatigue life of the member. Then the value of log 푁 should be multiplied by the factor X as follows: 2 log 푁1 = 푋 ∙ [(12 + 16Smin + 8Smin)(1 − Smax)], (1.6) where 푋 = 1 − 0,2 ∙ (log 푁 − 6). (1.7)

Fig. 1.6. S-N diagram for fatigue of concrete in compression (Stemland et al. 1990)

Still one problem with the model is that the time effects are omitted. Nevertheless, this model has some positive features as the distinction between the inclination in the low- and high-cycle region and the simplicity in practical use (Sørensen 1993).

1.1.3. Goodman Diagrams

One of the key limitations to the S-N curves was the inability to predict life at stress ratios different from those under which the curves were developed. In predicting the life of a component, a more useful presentation of fatigue life test data is the modified Goodman diagram (see Fig. 1.7).

Fig. 1.7. Fatigue strength of concrete by a Goodman diagram for compressive stress (Schläfli et al. 1998)

Fatigue of concrete is in principle defined by a pair of stresses, i.e. the maximum and minimum stress values as the most important fatigue relevant parameters for normal stress. A Goodman diagram best represents the effect of this pair of stresses on the fatigue strength as a function of the number of load cycles. These diagrams, while still limited by specimen geometry, surface condition, and material characteristics, afford the user to predict life at any stress ratio.

1.2. Damage Accumulation Models

Damage accumulation models enable to determine the degree of damage due to varying stresses. The most common hypothesis is the linear damage accumulation by Palmgren-Miner. The rule was first proposed by Palmgren (1924) and independently by Miner (1945). It describes the sum of damage due to each of the individual cycles and is convenient to use as an approximation for high cycle fatigue. Palmgren-Miner rule suggests that failure occurs when the following condition is fulfilled: 퐼 푛 ∑ 푖 = 1. (1.8) 푁푖 푖=1

Here 푛푖 is the number of load cycles at some stress condition and 푁푖 is the number of load cycles required to cause failure at that condition. According to Holmen (1982), the Palmgren-Miner hypothesis underestimates damage for variable amplitude loading – the value of minimum stress and small amplitudes also have an influence on damage. Cornelissen et al. (1984) report that the hypothesis is conservative with large scatter when based on S–N and therefore must be used with care. Palmgren–Miner hypothesis is a useful approximation, rather easy to apply, that might be accurate enough to use in design. But damage accumulation in fatigue is usually a complicated mixture of several different mechanisms, and the assumption of linear damage accumulation inherent in the hypothesis should be viewed skeptically. If parts of the material's microstructure become unable to bear load as fatigue progresses, the stress must be carried by the surviving microstructural elements. The rate of damage accumulation in these elements then increases, so that the material suffers damage much more rapidly in the last portions of its fatigue lifetime. If on the other hand cyclic loads induce strengthening mechanisms such as molecular orientation or crack blunting, the rate of damage accumulation could drop during some part of the material's lifetime. Palmgren–Miner hypothesis ignores such effects, and often fails to capture the essential physics of the fatigue process.

1.3. Methodology of Fatigue Assessment for Concrete in Compression

Most code provisions do not consider the concrete capacity for redistribution because they are not based on time-dependent material behavior. A realistic description should allow reproduction of the material changes ant the stress-strain evolution. The model presented herein accounts for these effects, keeping in mind the applicability to practical problems.

1.3.1. Time-Dependent Material Model

Time-dependent material model was first introduced by Zanuy et al. (2009) and aimed at the uniaxial behavior of concrete under cyclic loading. As already stated before, the fatigue response of concrete in compression is highly dependent on the stress level. Therefore, a reliable stress-strain relationship for the short-term (static) behavior of concrete is necessary. For that reason, the generally accepted nonlinear σ-ε curve for normal-strength concrete in compression of the Model Code 2010 (CEB-FIP 2012) is taken to provide the initial stress-strain state of the fatigue process (푁 = 1) and is also used as a failure envelope for the cyclic process (see Fig. 1.8). The expression is as follows:

Fig. 1.8. Static stress-strain relationship for concrete in compression (CEB-FIP 2012)

 for ascending branch of the σ-ε curve: 휀 휀 2 퐸푐 − ( ) 푓푐푚 휀푐 휎 = 푓푐푚; (1.1) 휀푐 휀 1 + (퐸푐 − 2) 푓푐푚 휀푐

 for descending branch of the σ-ε curve: 1 2 휀 2 4 휀 −1 (1.2) 휎 = [( 휉 − 2) ( ) + ( − 휉) ( )] 푓푐푚; 휀푙푖푚⁄휀푐 (휀푙푖푚⁄휀푐) 휀푐 휀푙푖푚⁄휀푐 휀푐 with 2 휀푙푖푚 휀푐 휀푙푖푚 휀푐 4 [( ) (퐸푐 − 2) + 2 − 퐸푐 ] 휀푐 푓푐 휀푐 푓푐 휉 = 2 ; (1.3) 휀푐 휀푙푖푚 [(퐸푐 − 2) ( ) + 1] 푓푐 휀푐 where 휎 − concrete stress;

퐸푐 − concrete initial tangent modulus of deformation; 휀 − concrete strain;

푓푐푚 − mean value of concrete compressive strength;

휀푐 − concrete strain that corresponds to the peak stress 푓푐;

휀푙푖푚 − concrete strain at 0,5푓푐 in the softening part of the curve.

To consider the fatigue effect of concrete, a time-dependent material model based on maximum concrete strain 휀푚푎푥 is proposed. Selected variable is able to define the material state at a determined instant of the fatigue life (푁⁄푁푓) (see Fig. 1.9). According to Zanuy et al. (2009), the model assumes that the reproduction of the fatigue process cycle by cycle is numerically ineffective, so analytical expression for the maximum strain with respect to the number of cycles is required. The equation of 휀푚푎푥 is expressed as a function of the number of cycles and the loading conditions:

푁 휎푚푎푥 휎푚푖푛 휀푚푎푥 = 푓 ( , , ). (1.4) 푁푓 푓푐 푓푐

The failure strain is calculated by means of the envelope concept. It considers that the static law σ-ε also defines the failure boundary of the cyclic problem. This seems to be a simple tool for calculating failure strain and provides reliable results.

Fig. 1.9. Scheme of theoretical model (Zanuy et al. 2009)

In order to identify the relative lifetime instant, the resistant number of cycles (푁푓) is necessary.

To obtain 푁푓, the S-N curves proposed by Hsu (refer to equation (1.6)) are selected due to the most extensive approach on the influence of several parameters. As it has already mentioned before, cyclic loading causes a redistribution of stresses in concrete components. This means that a process of variable stress limits is developed in each material fiber. Therefore, the direct application of equation (1.4) is not possible and the process is estimated by a number of shorter processes for which constant stress limits can be presumed. To relate them to each other, an accumulation criterion is required, thus the concept of the equivalent number of cycles (푁푒푞) is now introduced as a new accumulation rule

(Zanuy et al. 2009). The parameter 푁푒푞 is the number of load cycles that is necessary to be applied in

29 a fatigue process with constant limits (휎푚푎푥,휎푚푖푛) until a total strain of 휀푚푎푥 is reached. It may be simply calculated using the complete analytical expression (refer to equations (1.7) to (1.16)). The new concept can be easily explained by imagining a concrete specimen, which is first subjected to 푁푛 load cycles with constant limits (휎푚푎푥,푛, 휎푚푖푛,푛) and then to 푁(푛+1) additional cycles varying between (휎푚푎푥,(푛+1), 휎푚푖푛,(푛+1)) (see Fig. 1.10 (a)). The concept of the equivalent number of cycles allows to obtain the strain increase due to the last 푁(푛+1) excursions by means of the following expression:

푁푒푞,푛 + 푁(푛+1) 휎푚푎푥,(푛+1) 휎푚푖푛(푛+1) ∆휀 = 휀푚푎푥,(푛+1) − 휀푚푎푥,푛 = 푓 ( , , ) − 휀푚푎푥,푛. (1.5) 푁푓 푓푐 푓푐

Fig. 1.10. (a) Load history with two different stress level steps; and (b) strain increment employing equivalent number of cycles (Zanuy et al. 2009)

The calculation of the strain increment is explained in Fig. 1.10 (b). There, 푁푒푞,푛 is the number of load cycles needed to develop a total strain 휀푚푎푥,푛 in the fatigue process under stress levels (휎푚푎푥,(푛+1),

휎푚푖푛,(푛+1)). The additional number of cycles 푁(푛+1) must be introduced from point B, leading to the strain increase EF. It is apparent from Fig. 1.10 (b) that this value is smaller than the one obtained without introducing the equivalent number of cycles (segment CD), which is based on generally accepted damage accumulation model presented by Palmgren-Miner (Miner 1945).

1.3.2. Determination of Resistant Number of Cycles

S-N curves for calculating resistant number of cycles (푁푓) suggested by Hsu are selected due to more extensive approach on the influence of several parameters that other authors neglect and do not evaluate. Equation (1.6) for high-cycle fatigue is established to determine the resistant number of cycles (푁푓) as follows: 1 + 0,0294 log 푓 − 푆 log 푁 = 푚푎푥. (1.6) 푓 0,0662(1 − 0,556푅)

Fig. 1.11. S–N relations based on different stress ratios R

Fig. 1.12. S–N relations based on different rate of loading f

As can be seen from the Fig. 1.11 and Fig. 1.12, the resistant number of cycles (푁푓) increases with decreasing maximum stress level. Low stress ratio (푅 → 0, usually meaning a large stress range) and high frequency loading entail more fatigue like condition, i.e. the number of cycles determines the degree of degradation. On the other hand, a high stress ratio (푅 → 1, meaning a small stress range) and low frequency loading imply sustained loading condition, i.e. the degree of degradation depends on how long the load has been applied. As in static loading, high stress rate results in high static strength; therefore, increased loading rate may lead to higher fatigue strength (Byggtjänst 2000).

1.3.3. Evolution Law for Strain

According to Zanuy et al. (2009), the evolution law for the total strain is determined from the data available in the literature. The proposed curve of strain evolution (see Fig. 1.13) reproduce three typical stages of the fatigue process of concrete. Second-order parabolic equations (equations (1.7) and (1.9)) are used for the first and third stages, whereas a linear expression (Eq. (1.7) and (1.9)) is employed for the second stage, so that the experimental S-shaped evolution (see Fig. 1.13) is attained. According to Holmen (1982), the transition points between stages 1-2 and 2-3 are supposed to occur at 10% and 80% of the fatigue live, respectively.

ε εfail

εmax (N/Nf)

ε0

1 2 3

0,0 0,1 0,8 1,0 N/Nf Fig. 1.13. Schematic variation of concrete variables with cycle ratio

The strain evolution curve is defined as follows: 휀 푁 푁 푁 2 푁 푚푎푥 ( ) = 1 + 퐴 + 퐵 ( ) , 0,0 ≤ < 0,1; (1.7) 휀0 푁푓 푁푓 푁푓 푁푓

휀푚푎푥 푁 푁 푁 ( ) = 휀1−2 + 휀2 ( − 0,1) , 0,1 ≤ < 0,8; (1.8) 휀0 푁푓 푁푓 푁푓 2 휀푚푎푥 푁 푁 푁 푁 ( ) = 휀1−2 + 휀2 ( − 0,1) + 퐶 ( − 0,8) , 0,8 ≤ < 1; (1.9) 휀0 푁푓 푁푓 푁푓 푁푓 where

퐴 = 20(휀1−2 − 1) − 휀2; (1.10)

퐵 = 100(1 − 휀1−2) + 10휀2; (1.11)

휀푓푎푖푙 퐶 = 25 ( − 휀1−2 − 0,9휀2) ; (1.12) 휀0 where

휀푚푎푥 − maximum strain;

휀0 − maximum concrete strain at 푁 = 1;

휀1−2 − relative strain at 푁⁄푁푓 = 0,1;

휀2 − strain rate in second fatigue stage;

휀푓푎푖푙 − failure strain;

푁 − number of load cycles; 푁푓 − number of load cycles to failure.

Equations (1.7) to (1.9) provide the maximum strain with respect to its initial value 휀0 at the first load cycle (푁 = 1). The ratio 휀1−2 defines the relative strain at 0,1푁푓, which is the transition between stages 1 and 2, and 휀2 gives the constant strain rate of the second domain. Both are defined according to Holmen’s approach (Holmen 1982): 1,184 휀1−2 = ; (1.13) 푆푚푎푥 0,74037 휀2 = ; (1.14) 푆푚푎푥

1 휀푓푎푖푙 휀2 ≤ ( − 휀1−2) ; (1.15) 0,9 휀0 where

푆푚푎푥 = 휎푚푎푥⁄푓푐 (1.16) where

푆푚푎푥 − normalized maximum stress;

휎푚푎푥 − maximum stress;

푓푐 − concrete compressive strength.

1.3.4. Algorithm of Concrete Capacity for Redistribution Processes

According to the experimental tests found in the literature (Johansson 2004), even under high initial compressive stresses where fatigue of concrete could be expected, reinforced members fail systematically due to the brittle fatigue fracture of the steel reinforcing bars. This fact is explained by the large concrete capacity for redistribution of stresses. A simple algorithm based on the time-dependent material model suggested by Zanuy et al. (2009) is presented here in thesis. The algorithm displays a redistribution of stresses of reinforced concrete columns subjected to high-cycle fatigue loading. A step-by-step procedure is being introduced in order to get a total evolution of the degradation mechanism of concrete until it reaches failure instant. The limit state of failure of RC columns is governed either by the yielding of reinforcement (e.g. 400 MPa, referring to S400) or by concrete reaching the maximum number of load cycles (푁푓).

Determination of sectional parameters of reinforced concrete member

In the beginning, the dimensions of the cross-section are selected as follows:

퐴푐 = ℎ ∙ 푏, (1.17) where 2 퐴푐 − cross sectional area of concrete, m ; ℎ − height of a column, m; 푏 − width of a column.

휋 ∙ 푑2 퐴 = 푛 ∙ , (1.18) 푠 4 where 2 퐴푠 − cross sectional area of reinforcement, m ; 푛 − number of reinforcement bars; 푑 −diameter of reinforcement bars.

A reinforcement ratio ρ is counted as follows:

휌 = (퐴푠⁄퐴푐) ∙ 100%. (1.19)

Determination of material properties of reinforced concrete member

According to Model Code 2010, material properties are selected as follows:

푓푐푚 = 푓푐푘 + 8, (1.20) where

푓푐푚 − mean value of concrete compressive strength, MPa;

푓푐푘 − characteristic compressive strength of concrete, MPa;

(1⁄3) 퐸푐 = 21500[(푓푐푚)/10] , (1.21)

where

퐸푐 − modulus of elasticity for normal weight concrete as well as

concrete initial tangent modulus of deformation, MPa; 퐸푠 − steel modulus of elasticity, MPa;

2 2 2 √(퐸푐 ∙ 휀푐 ∙ 0,25 + 퐸푐 ∙ 휀푐 ∙ 푓푐푚 − 푓푐푚 ) 2 휀푐 ∙ 퐸푐 휀푐 (1.22) 휀푙푖푚 = 휀푐 + + , 2푓푐푚 4푓푐푚 2 where

휀푙푖푚 − concrete strain at 0,5푓푐푚 in softening curve;

휀푐 − concrete strain at the peak stress 푓푐푚.

Determination of loading process

Due to the cyclic loading of a column, a constant degradation of the stresses appears in concrete.

The loading process (starting from the first load cycle and continuing to the 푁푓 number of load cycles causing the failure state) is divided into a smaller blocks ∆푁 = 푥 ∙ 푁푓, consisting of a certain number of load cycles for which constant stress limits can be assumed. Each loading block ∆푁 accounts as one iteration. The number of load cycles forming the block ∆푁 is taken to maintain a total amount of iterations (needed to reach the final failure state) as small as possible, still having in mind that the accuracy of the calculation would be assured. According to experimental studies by Schläfli and Brühwiler (1998), a great importance of evaluating the fatigue process in the final failure of structural elements under large amplitude external load (when the concrete stress overcomes 60% of static strength) is needed. Therefore, a value of maximum loading force 푃푚푎푥(kPa) has to be taken such a that the maximum loading stress in concrete

휎푐,푚푎푥 would be in a range over the 0,6푓푐푚. Then the value of minimum loading force 푃푚푖푛(kPa) is selected to be 50% of the 푃푚푎푥. 푃푚푎푥 and 푃푚푖푛 are calculated as follows: 3 푃푚푎푥 = 푘푚푎푥 ∙ 푓푐푚 ∙ 10 ∙ 퐴푐, (1.23) 3 푃푚푖푛 = 푘푚푖푛 ∙ 푓푐푚 ∙ 10 ∙ 퐴푐, (1.24) where

푘푚푎푥 − coefficient of maximum loading force 푃푚푎푥;

푘푚푖푛 − coefficient of minimum loading force 푃푚푖푛.

Calculation of starting values of initial maximum and minimum concrete strains at the first load cycle

In order to get starting values of maximum 휎푐,푚푎푥,0 and minimum 휎푐,푚푖푛,0 loading stresses in concrete, starting values of initial maximum concrete strain 휀0,푚푎푥,0 and initial minimum concrete strain 휀0,푚푖푛,0 at the first load cycle (corresponding to the 휎푐,푚푎푥,0 and 휎푐,푚푖푛,0, respectively; see Fig. 1.14) have to be calculated as follows:

Fig. 1.14. Initial stress-strain relationship for concrete under compressive fatigue loading

−3 푃푚a푥 ∙ 10 휀0,푚푎푥,0 = ; (1.25) 퐸푐 ∙ 퐴푐 + 퐸푠 ∙ 퐴푠 −3 푃푚푖푛 ∙ 10 휀0,푚푖푛,0 = . (1.26) 퐸푐 ∙ 퐴푐 + 퐸푠 ∙ 퐴푠

Calculation of starting values of maximum and minimum concrete stresses

Starting values of maximum 휎푐,푚푎푥,0 and minimum 휎푐,푚푖푛,0 loading stresses in concrete are calculated as follows (see Fig. 1.14):

휎푐,푚푎푥,0 = 휀0,푚푎푥,0 ∙ 퐸푐; (1.27)

휎푐,푚푖푛,0 = 휀0,푚푖푛,0 ∙ 퐸푐. (1.28)

Calculation of starting value of failure strain

A failure strain 휀푓푎푖푙,0 is derived from the equations (1.1) and (1.2) substituting 휎 and 휀 with

휎푐,푚푎푥,0 and 휀푓푎푖푙,0, respectively, and is calculated as follows (see Fig. 1.14):

 when 휎푐,푚푎푥,0 < 0,5푓푐푚:

2 휀푙푖푚√휎푐,푚푎푥,0 ∙ (4푓푐푚 ∙ 휀푐(휀푙푖푚 ∙ 휉 − 2휀푐) + 휎푐,푚푎푥,0 ∙ (휀푙푖푚 ∙ 휉 − 4휀푐) ) 휀 = 푓푎푖푙,0 2휎 ∙ (2휀 − 휀 ∙ 휉) 푐,푚푎푥,0 푐 푙푖푚 (1.29)

휀푙푖푚 (휎푐,푚푎푥,0 ∙ (휀푙푖푚 ∙ 휉 − 4휀푐)) − , 2휎푐,푚푎푥,0 ∙ (2휀푐 − 휀푙푖푚 ∙ 휉)

37 where 2 휀푙푖푚 휀푐 휀푙푖푚 휀푐 4 [( ) (퐸푐 − 2) + 2 − 퐸푐 ] 휀푐 푓푐 휀푐 푓푐 휉 = 2 ; (1.30) 휀푐 휀푙푖푚 [(퐸푐 − 2) ( ) + 1] 푓푐 휀푐

 when 휎푐,푚푎푥,0 ≥ 0,5푓푐푚:

2 2 2 √(푓푐푚 − 휎푐,푚푎푥,0)(퐸푐 ∙ 휀푐 ∙ (푓푐푚 − 휎푐,푚푎푥,0) + 4퐸푐 ∙ 푓푐푚 ∙ 휀푐 ∙ 휎푐,푚푎푥,0 − 4푓푐푚 ∙ 휎푐,푚푎푥,0) 휀푓푎푖푙,0 = 2 × 2푓푐푚 (1.31) 휀 (퐸 ∙ 휀 ∙ (푓 − 휎 )) 푐 푐 푐 푐푚 푐,푚푎푥,0 휀푐(2푓푐푚 ∙ 휎푐,푚푎푥,0) × 휀푐 + 2 + 2 . 2푓푐푚 2푓푐푚

Calculation of starting value of number of load cycles to failure

According to Hsu (1981), the number of load cycles to failure, 푁푓, is a function of several different parameters. These parameters are the stress ratio, 푅, normalized maximum stress, 푆푚푎푥, and load frequency, 푓. The case of columns subjected to high-cycle fatigue is analyzed herein. Then the starting value of 푁푓,0 is calculated as follows:

1 + 0,0294 log 푓 − 푆푚푎푥,0 log 푁푓,0 = , 푁푓,0 ≥ 1000, (1.32) 0,0662(1 − 0,556푅0) where

푅0 = 휎푐,푚푖푛,0⁄휎푐,푚푎푥,0, (1.33)

푆푚푎푥,0 = 휎푐,푚푎푥,0⁄푓푐푚. (1.34)

Calculation of starting value of maximum concrete strain

As it has already been mentioned, maximum concrete strain is expressed in terms of the relative number of cycles 푁/푁푓 (see equations (1.7) to (1.9)). In order to get a starting value of maximum concrete strain 휀푚푎푥,0 (see Fig. 1.14), a starting number of load cycles 푁0 has to be selected and further calculations made as follows:

푁0 = ∆푁 = 푥 ∙ 푁푓,0; (1.35) 2 푁0 푁0 푁0 휀푚푎푥,0 = 휀0,푚푎푥,0 (1 + 퐴0 + 퐵0 ( ) ) , 0,0 ≤ < 0,1, (1.36) 푁푓,0 푁푓,0 푁푓,0 where

퐴0 = 20(휀1−2,0 − 1) − 휀2,0, (1.37)

퐵0 = 100(1 − 휀1−2,0) + 10휀2,0, (1.38) 1,184 0,74037 휀1−2,0 = , (1.39) 휀2,0 = , (1.40) 푆푚푎푥,0 푆푚푎푥,0 where

휀0,푚푎푥,0 − starting value of the maximum concrete strain at N = 1;

휀1−2,0 − starting value of the relative strain at 푁0⁄푁푓,0 = 0,1;

휀2,0 − starting value of the strain rate in second fatigue stage;

휀푓푎푖푙,0 − starting value of the failure strain;

푁0 − starting value of the number of load cycles;

푁푓,0 − starting value of the number of load cycles to failure;

푆푚푎푥,0 − starting value of the normalized maximum stress according to equation (1.34).

Calculation of starting value of secant modulus of deformation

Secant modulus of deformation 퐸푠푒푐,0 is calculated as follows (see Fig. 1.14):

휎푐,푚푎푥,0 퐸푠푒푐,0 = . (1.41) 휀푚푎푥,0

Calculation of altered maximum and minimum concrete strains

After calculation of the starting values of maximum concrete strain 휀푚푎푥,0 and secant modulus of ∗ ∗ deformation 퐸푠푒푐,0, altered maximum 휀푚푎푥,0 and minimum 휀푚푖푛,0 concrete strains are found as follows: −3 ∗ 푃푚푎푥 ∙ 10 휀푚푎푥,0 = , (1.42) 퐸푠푒푐,0 ∙ 퐴푐 + 퐸푠 ∙ 퐴푠 −3 ∗ 푃푚푖푛 ∙ 10 휀푚푖푛,0 = . (1.43) 퐸푠푒푐,0 ∙ 퐴푐 + 퐸푠 ∙ 퐴푠

Calculation of starting value of equivalent number of load cycles needed to develop a total strain

∗ The maximum concrete strain 휀푚푎푥,0 was obtained after 푁0 number of load cycles under stress levels 휎푐,푚푎푥,0 and 휎푐,푚푖푛,0. 푁푒푞,0 is the number of load cycles needed to develop the same total strain ∗ 휀푚푎푥,0 except under stress levels 휎푐,푚푎푥,1, 휎푐,푚푖푛,1. It is derived from the equation (1.7) substituting ∗ 휀푚푎푥,0, 휀0,푚푎푥,0, 퐴0, 퐵0 and 푁푓,0 with 휀푚푎푥,0, 휀0,푚푎푥,1, 퐴1, 퐵1 and 푁푓,1, respectively, and calculated as follows:

퐴1 − ( ) + √퐷0 푁푓,1 푁0 푁푒푞,0 = , 0,0 ≤ < 0,1, (1.44) 퐵1 푁푓,0 2 ( 2 ) 푁푓,1 where

2 ∗ 퐴1 퐵1 1 − 휀푚푎푥,0 퐷0 = ( ) − 4 ( 2 ) ( ), (1.45) 푁푓,1 푁푓,1 휀0,푚푎푥,1

퐴1 = 20(휀1−2,1 − 1) − 휀2,1, (1.46)

퐵1 = 100(1 − 휀1−2,1) + 10휀2,1, (1.47) 1,184 0,74037 휀1−2,1 = , (1.48) 휀2,1 = , (1.49) 푆푚푎푥,1 푆푚푎푥,1 where

휀0,푚푎푥,1 − altered value of the maximum concrete strain at N = 1 according to eq. (1.55);

휀1−2,1 − altered value of the relative strain at 푁0⁄푁푓,0 = 0,1;

휀2,1 − altered value of the strain rate in second fatigue stage;

휀푓푎푖푙,1 − altered value of the failure strain according to equation (1.52);

푁0 − starting value of the number of load cycles according to equation (1.35);

푁푓,0 − starting value of the number of load cycles to failure according to eq. (1.32);

푁푓,1 − altered value of the number of load cycles to failure according to eq. (1.56);

푆푚푎푥,1 − altered value of the normalized maximum stress according to eq. (1.58).

Calculation of alternating maximum and minimum concrete stresses

After the first loading iteration, the degradation of stresses in concrete has appeared and new decreased values of maximum 휎푐,푚푎푥 and minimum 휎푐,푚푖푛 concrete stresses have to be found. The same has to be done after each of the following loading iterations. The alternating maximum 휎푐,푚푎푥,푛 and minimum 휎푐,푚푖푛,푛 concrete stresses are calculated as follows:

∗ 휎푐,푚푎푥,푛 = 휀푚푎푥,(푛−1) ∙ 퐸푠푒푐,(푛−1); (1.50)

∗ 휎푐,푚푖푛,푛 = 휀푚푖푛,(푛−1) ∙ 퐸푠푒푐,(푛−1). (1.51)

Calculation of alternating failure strain

An alternating failure strain 휀푓푎푖푙,푛 (it is changing after the each of the loading iterations) is derived from the equations (1.1) and (1.2) substituting 휎 and 휀 with 휎푐,푚푎푥,푛 and 휀푓푎푖푙,푛, respectively,and calculated as follows:

 when 휎푐,푚푎푥,푛 < 0,5푓푐푚:

2 휀푙푖푚√휎푐,푚푎푥,푛 ∙ (4푓푐푚 ∙ 휀푐(휀푙푖푚 ∙ 휉 − 2휀푐) + 휎푐,푚푎푥,푛 ∙ (휀푙푖푚 ∙ 휉 − 4휀푐) ) 휀 = 푓푎푖푙,푛 2휎 ∙ (2휀 − 휀 ∙ 휉) 푐,푚푎푥,푛 푐 푙푖푚 (1.52)

휀푙푖푚 (휎푐,푚푎푥,푛 ∙ (휀푙푖푚 ∙ 휉 − 4휀푐)) − ; 2휎푐,푚푎푥,푛 ∙ (2휀푐 − 휀푙푖푚 ∙ 휉)

where 2 휀푙푖푚 휀푐 휀푙푖푚 휀푐 4 [( ) (퐸푐 − 2) + 2 − 퐸푐 ] 휀푐 푓푐 휀푐 푓푐 휉 = 2 ; (1.53) 휀푐 휀푙푖푚 [(퐸푐 − 2) ( ) + 1] 푓푐 휀푐

 when 휎푐,푚푎푥,0 ≥ 0,5푓푐푚:

2 2 2 √(푓푐푚 − 휎푐,푚푎푥,푛)(퐸푐 ∙ 휀푐 ∙ (푓푐푚 − 휎푐,푚푎푥,푛) + 4퐸푐 ∙ 푓푐푚 ∙ 휀푐 ∙ 휎푐,푚푎푥,푛 − 4푓푐푚 ∙ 휎푐,푚푎푥,푛) 휀푓푎푖푙,푛 = 2 × 2푓푐푚 (1.54) 휀 (퐸 ∙ 휀 ∙ (푓 − 휎 )) 푐 푐 푐 푐푚 푐,푚푎푥,푛 휀푐(2푓푐푚 ∙ 휎푐,푚푎푥,푛) × 휀푐 + 2 + 2 . 2푓푐푚 2푓푐푚

Calculation of alternating initial maximum concrete strain at the first load cycle

In order to get a value of the maximum concrete strain 휀푚푎푥,푛, the initial maximum concrete strain

휀0,푚푎푥,푛 at 푁 = 1 (corresponding to the 휎푐,푚푎푥,푛) has to be calculated as follows:

휎푐,푚푎푥,푛 휀0,푚푎푥,푛 = . (1.55) 퐸푐

Calculation of alternating number of load cycles to failure

Because of the constant degradation of the stresses in concrete (휎푐,푚푎푥 is always decreasing), normalized maximum stress 푆푚푎푥 is changing as well. Therefore, the alternating number of load cycles to failure 푁푓,푛 has to be calculated after each of the following iterations. Alternating number of load cycles to failure 푁푓,푛 is calculated as follows:

1+0,0294 log 푓−푆푚푎푥,푛 0,0662(1−0,556푅 ) (1.56) 푁푓,푛 = 10 푛 , 푁푓,푛 ≥ 1000; where

푅푛 = 휎푐,푚푖푛,푛⁄휎푐,푚푎푥,푛 ; (1.57)

푆푚푎푥,푛 = 휎푐,푚푎푥,푛⁄푓푐푚. (1.58)

Calculation of alternating maximum concrete strain

After 푁푛 number of load cycles, alternating maximum concrete strain 휀푚푎푥,푛 is calculated as follows:

푁푛 = 푁푒푞,(푛−1) + ∆푁 = 푁푒푞,(푛−1) + 푥 ∙ 푁푓,푛, (1.59) 2 푁푛 푁푛 푁푛 휀푚푎푥,푛 = 휀0,푚푎푥,푛 (1 + 퐴푛 + 퐵푛 ( ) ) , 0,0 ≤ < 0,1; (1.60) 푁푓,푛 푁푓,푛 푁푓,푛

푁푛 푁푛 휀푚푎푥,푛 = 휀0,푚푎푥,푛 (휀1−2,푛 + 휀2,푛 ( − 0,1)) , 0,1 ≤ < 0,8, (1.61) 푁푓,푛 푁푓,푛

2 푁푛 푁푛 푁푛 휀푚푎푥,푛 = 휀0,푚푎푥,푛 (휀1−2,푛 + 휀2,푛 ( − 0,1) + 퐶 ( − 0,8) ) , 0,8 ≤ < 1,0, (1.62) 푁푓,푛 푁푓,푛 푁푓,푛 where

퐴푛 = 20(휀1−2,푛 − 1) − 휀2,푛, (1.63)

퐵푛 = 100(1 − 휀1−2,푛) + 10휀2,푛, (1.64)

휀푓푎푖푙,푛 퐶푛 = 25 ( − 휀1−2,푛 − 0,9휀2,푛), (1.65) 휀0,푚푎푥,푛

1,184 0,74037 1 휀푓푎푖푙,푛 휀1−2,푛 = , (2.66) 휀2,푛 = , (2.67) 휀2,푛 ≤ ( − 휀1−2,푛), (1.68) 푆푚푎푥,푛 푆푚푎푥,푛 0,9 휀0,푚푎푥,푛 where

휀0,푚푎푥,푛 − alternating maximum concrete strain at N = 1;

휀1−2,푛 − alternating relative strain at 푁푛⁄푁푓,푛 = 0,1;

휀2,푛 − alternating strain rate in second fatigue stage;

휀푓푎푖푙,푛 − alternating failure strain;

푁푛 − alternating number of load cycles;

푁푓,푛 − alternating number of load cycles to failure;

푆푚푎푥,푛 − alternating normalized maximum stress.

Calculation of alternating secant modulus of deformation

Alternating secant modulus of deformation 퐸푠푒푐,푛 is calculated as follows:

휎푐,푚푎푥,푛 퐸푠푒푐,푛 = . (1.69) 휀푚푎푥,푛

Calculation of alternating maximum concrete strain

After calculation of the alternating maximum concrete strain 휀푚푎푥,푛 and alternating secant ∗ ∗ modulus of deformation 퐸푠푒푐,푛, the alternating maximum 휀푚푎푥,푛 and minimum 휀푚푖푛,0 concrete strains are found as follows: −3 ∗ 푃푚푎푥 ∙ 10 휀푚푎푥,푛 = ; (1.70) 퐸푠푒푐,푛 ∙ 퐴푐 + 퐸푠 ∙ 퐴푠 −3 ∗ 푃푚푖푛 ∙ 10 휀푚푖푛,푛 = . (1.71) 퐸푠푒푐,푛 ∙ 퐴푐 + 퐸푠 ∙ 퐴푠

Calculation of alternating equivalent number of load cycles needed to develop the alternating total strain

∗ After 푛 loading iterations, the alternating maximum concrete strain 휀푚푎푥,푛 is obtained at 푁푛 =

푁푒푞,(푛−1) + ∆푁 number of load cycles under stress levels 휎푐,푚푎푥,푛 and 휎푐,푚푖푛,푛. The number of load ∗ cycles 푁푒푞,푛 needed to develop the same total strain 휀푚푎푥,푛 except under stress levels 휎푐,푚푎푥,(푛+1),

휎푐,푚푖푛,(푛+1). It is derived from the equations (1.60) to (1.62) substituting 휀푚푎푥,푛, 휀0,푚푎푥,푛, 퐴푛, 퐵푛, 퐶푛 ∗ and 푁푓,푛 with 휀푚푎푥,푛, 휀0,푚푎푥,(푛+1), 퐴(푛+1), 퐵(푛+1), 퐶(푛+1) and 푁푓,(푛+1), respectively, and calculated as follows:

퐴(푛+1) − ( ) + √퐷푛 푁푓,(푛+1) 푁푛 푁푒푞,푛 = , 0,0 ≤ < 0,1, (1.72) 퐵(푛+1) 푁푓,푛 2 ( 2 ) 푁푓,(푛+1)

∗ 휀푚푎푥,푛 − 휀1−2,(푛+1) 휀0,푚푎푥,(푛+1) + 0,1 (1.73) 휀2,(푛+1) 푁푛 푁푒푞,푛 = , 0,1 ≤ < 0,8, 푁푓,(푛+1) 푁푓,푛

푁푒푞,푛 = 푁푓,(푛+1) ∙ √5 ×

휀∗ √((−20휀 − 14휀 + 20 푚푎푥,푛 ) ∙ 퐶 + 5휀2 ) 1−2,(푛+1) 2,(푛+1) 휀 (푛+1) 2,(푛+1) 0,푚푎푥,(푛+1) (1.74) × + 10퐶(푛+1)

−5휀2,(푛+1) + 8퐶(푛+1) 푁푛 +푁푓,(푛+1) ∙ , 0,8 ≤ < 1,0, 10퐶(푛+1) 푁푓,푛 where

퐴(푛+1) = 20(휀1−2,(푛+1) − 1) − 휀2,(푛+1), (1.75)

퐵(푛+1) = 100(1 − 휀1−2,(푛+1)) + 10휀2,(푛+1), (1.76)

2 ∗ 퐴(푛+1) 퐵(푛+1) 1 − 휀푚푎푥,푛 퐷푛 = ( ) − 4 ( 2 ) ( ), (1.77) 푁푓,(푛+1) 푁푓,(푛+1) 휀0,푚푎푥,(푛+1)

휀푓푎푖푙,(푛+1) 퐶(푛+1) = 25 ( − 휀1−2,(푛+1) − 0,9휀2,(푛+1)), (1.78) 휀0,푚푎푥,(푛+1) 1,184 휀1−2,(푛+1) = , (1.79) 푆푚푎푥,(푛+1)

0,74037 1 휀푓푎푖푙,(푛+1) 휀2,(푛+1) = , (1.80) 휀2,(푛+1) ≤ ( − 휀1−2,(푛+1)). (1.81) 푆푚푎푥,(푛+1) 0,9 휀0,푚푎푥,(푛+1) Then starting again from the step 12 (see Fig. 1.15) until the equivalent number of load cycles

푁푒푞,푛 reaches the number of load cycles to failure 푁푓,(푛+1) or, in other words, ratio between the equivalent number of load cycles and the number of load cycles to failure 푁푒푞,푛⁄푁푓,(푛+1) reaches unity: 푁푒푞,푛⁄푁푓,(푛+1) → 1.

Ac , As ρ I III Pmax , Pmin Ec , Es , fcm εc , εlim II

ε0,max,0 , ε0,min,0 IV

σc,max,0 , σc,min,0 V

εfail,0 VI

R0 , Smax,0 Nf,0 VII

ε1-2,0 , ε2,0 A0 , B0 + Nf,0 N0 εmax,0 VIII

Esec,0 IX

ε*max,0 , ε*min,0 X

Neq,0 XI

σc,max,n , σc,min,n XII

εfail,n XIII

ε0,max,n XIV

Rn , Smax,n Nf,n XV

ε1-2,n , ε2,n An , Bn ,Cn + Neq,(n-1) , N f,n Nn εmax,n XVI

Esec,n XVII

ε*max,n , ε*min,n XVIII

Neq,n XIX

Fig. 1.15. A flow-chart of proposed algorithm

1.4. Numerical Analysis of Reinforced Concrete Columns

1.4.1. Numerical Disclosure of Presented Algorithm

A numerical research is carried out by applying an algorithm presented above. Four columns subjected to cyclic loading have been analyzed (see Fig. 1.16 and Table 1.1). Results of numerical analysis are summarized by graphical illustrations in Fig. 1.17 to Fig. 1.19, and in Table 1.2.

Table 1.1. Main characteristics of analyzed columns

ρ, f , E , E , f, No. cm c s S S % MPa GPa GPa Hz 0,max 0,min A 0,503 B 1,131 38 33,5 205 5 0,80 0,40 C 1,689 D 2,182

Fig. 1.16. Cross-sections of the studied columns

1.4.2. Graphical Illustration of Analysis Results

As can be seen from the Fig. 1.17, total strain evolution has three distinct stages. In the first stage, as well as in the second, the reinforcement ratio has no significant influence on the total strain. Meanwhile, the third stage, particularly at the failure instant, exhibits rather essential dependency of

46 total strain on reinforcement ratio. This leads to more rapid total strain growth with increasing reinforcement ratio.

Fig. 1.17. Evolution of longitudinal strain with number of cycles regarding to reinforcement ratio

Fig. 1.18. Evolution of maximum stress with number of cycles regarding to reinforcement ratio

Fig. 1.18 and Fig. 1.19 demonstrate evolution of maximum stresses with number of load cycles. The biggest stress drop in comparison with its initial value can be observed in a member with the highest reinforcement ratio. Therefore, a phenomenon of redistribution process of stress in concrete becomes more obvious with increasing reinforcement ratio.

Fig. 1.19. Evolution of maximum stress level with number of cycles regarding to reinforcement ratio

1.4.3. Comparative Analysis

In order to evaluate the adequacy of proposed technique, the obtained results have been compared with two generally prevailing fatigue assessment methods. The first one is taken from Model Code 2010 and is based on Stemland et al. The second model proposed by Hsu is applied due to the most extensive approach on the influence of several parameters. To assess the stress redistribution of concrete, the Palmgren-Miner damage accumulation hypothesis have been adopted for the models of Stemland et al. and Hsu, whereas the concept of the equivalent number of cycles (푁푒푞) has been applied for the model of Zanuy et al. The comparison of results is given in Table 2. According to Stemland et al. and Hsu, the maximum number of load cycles to failure (푁푓) considering the stress redistribution of concrete is 1.8 to 31.9 and 2.1 to 93.3 times, respectively, greater than that under constant stress limits.

Table 1.2. Comparison of results of RC columns A, B, C and D under different models

Stemland et Stemland et al. Hsu Hsu Zanuy et Reinforcement al. (constant (stress (constant (stress al. ratio No. stress limits) redistribution) stress limits) redistribution) S400 휌, % 푁푓 푁푓 푁푓 푁푓 푁푓 A 0,503 8629 15621 41149 88391 127616 B 1,131 8629 45035 41149 336943 691906 C 1,689 8629 117687 41149 1197729 3108498 D 2,182 8629 275388 41149 3841468 11683885

Although Stemland et al., Hsu and Zanuy et al. models do take into account stress redistribution processes, the estimated maximum number of load cycles differs significantly. The higher reinforcement ratio in RC columns makes the Palmgren-Miner damage accumulation rule to be inaccurate while giving the smaller resistant number of cycles, thus overestimating failure state of concrete. Therefore, it could be stated that the code techniques (generally based on Palmgren-Miner hypothesis), present inadequately high safety margin.

1.5. Experimental Investigation of Concrete Prisms Subjected to Cyclic Loading

In order to verify the adequacy of the proposed fatigue assessment technique and to determine the influence of reinforcement on fatigue behavior, an experimental investigation of concrete prisms subjected to cyclic compression has been performed (Fig. 1.20). A test program consisted of eight concrete prisms of 400 mm in length with 100×100 mm cross section. There were seven specimens made of normal concrete and one made of fiber-reinforced concrete (fiber percentage was 0.5% by volume). Mean compressive strength and deformation modulus of concrete were 45.1 MPa and

35.8 GPa, respectively. The frequency of loading rate was 5 Hz. The normalized maximum (푆푚푎푥) and minimum (푆푚푖푛) stresses were 0.75 and 0.1, respectively.

Fig. 1.20. Fatigue testing machine and fatigue failure of concrete prism

The rate of change of displacement regarding the number of load cycles were measured during the test (Fig. 1.21). The dashed line with squares depicts the fatigue behavior of steel fiber-reinforced concrete specimen (No.3 F3). A fiber reinforcement considerably improves serviceability properties

49 of concrete elements. The foremost advantages of fiber reinforcement are high fatigue and impact resistance as well as plasticity Having steel fibers inside, it possesses larger stress redistribution capacity, thus reach 1.63 and 4.52 times higher number of load cycles till failure than that of specimens made of normal concrete (No.6 P7 and No.7 P8, respectively). However, even the results of identical specimens (No.6 P7 and No.7 P8) are characterized by a large scatter.

Fig. 1.21. The rate of change of displacement regarding the number of load cycles

This shows that in order to ensure the adequacy of results, the quantitative analysis and additional experimental research have to be performed. In addition, the concrete strain measurement has to be performed during the tests.

1.6. Concluding Remarks of Chapter 1

The first chapter aims at contributing to a better understanding of fatigue influence on stress and strain behavior of reinforced concrete columns subjected to high-cycle loading. Having conducted thorough literature review, investigated methodology of fatigue assessment and performed numerical and comparative analysis as well as experimental tests, the following remarks can be stated:

1. A time-dependent material model for concrete under fatigue is necessary to adequately predict the fatigue behavior of reinforced concrete under cyclic compression.

2. Using the time-dependent material model proposed in the literature, a simplified and transparent iterative analysis technique of reinforced concrete columns subjected to high-cycle fatigue loading has been proposed.

3. The proposed technique allows evaluating the progressive time-dependent degradation and stress redistribution of concrete.

4. A numerical analysis of reinforced concrete columns subjected to high-cycle fatigue loading has shown that the resistant number of cycles, next to well established parameters such as maximum stress level, stress ratio and loading frequency, is greatly affected by reinforcement ratio. The members with higher reinforcement ratio have larger redistribution capability and reach higher number of load cycles till failure.

5. The performed comparative analysis of prevailing fatigue assessment methods has shown that the higher reinforcement ratio in RC columns makes a generally accepted Palmgren-Miner damage accumulation rule to be inaccurate while giving the smaller resistant number of cycles, thus overestimating the failure state of concrete.

6. The experimental investigation on fatigue behavior of concrete prisms subjected to cyclic compression verified that internal reinforcement significantly improves fatigue-resistance resulting in increased number of load cycles till failure.

2. INNOVATIVE MATERIALS FOR REINFORCED CONCRETE STRUCTURES

Investigating the application of innovative materials and considering demands of Lithuanian civil engineering, the second part of the study focusses on FRP reinforcement and biological self-healing concrete as the most promising alternatives for avoiding cracking of concrete and corrosion of steel reinforcement. Problematic issues (particularly, material properties of these materials) related to the design of RC structures are being discussed.

2.1. FRP Reinforcement for Concrete Structures

Steel and concrete are the principal materials for the industry of up-to-date construction. Nevertheless, there are applications which require an alternative material to be used. Fiber reinforced polymers are considered to be a promising alternative to steel reinforcement, especially in concrete structures subjected to aggressive environment or to the effects of electromagnetic fields (Alsayed et al. 2000). Numerous concrete structures such as bridges, dams and off-shore structures are exposed to de-icing salts, combinations of temperature, moisture, and chlorides which reduce the alkalinity of the concrete and result in the corrosion of steel reinforcement. At present almost a half of the budget of the construction industry is spent on the repair and reconstruction of existing buildings (Cigna et al. 2003). In order to cope with corrosion problems, engineers have turned to alternative metallic reinforcement, such as epoxy-coated steel bars, cathodic protection, and increased thickness of concrete cover. While adequate in certain situations, such methods may still be unable to entirely eliminate the problems of steel corrosion (ACI 440 2006). Therefore, due to the non-corrosive nature, higher strength and lower unit weight of FRPs relative to conventional steel reinforcement, the use of FRP materials in adverse environment is gaining recognition. Different kinds of materials are used in the production of the FRP reinforcement. Carbon fiber reinforced polymers (CFRP) have the best mechanical properties (amongst other FRP composites), but materials for its production are hardly accessible. In terms of mechanical properties and production complexity, basalt (BFRP) and aramid (AFRP) bars are somewhere in the middle, but they are seldom used in practice. The glass fiber reinforced polymer (GFRP) bars are the most popular among other FRP types due to the combination of relatively low-cost with environmental resistance of structural fibers. With high durability, GFRP bars have a tensile strength up to 5–6 times higher than structural steel. However, the low elastic modulus of the polymer composites (in respect to the steel) generally leads to increased deformations of GFRP reinforced elements. Thus, the serviceability limit state often

52 becomes the governing criterion in the design of such elements. A number of techniques have been proposed for predicting deformational response of FRP reinforced concrete (RC) elements (Faza and GangaRao 1992; Bischoff 2007), though the lack of experimental data is still evident. Moreover, various types of surface coatings are used in the production of the composite bars. The surface treatment determines the quality of bond between the bars and the concrete matrix. Complex, uneven and rough shape of bars ensures good bond properties; however, such surface treatment may result in more complex manufacturing processes. There is still no global consensus on the most effective shape of FRP bars. Standardization of the shape would allow a more extensive use of FRP reinforcement in the construction. Over the past decades, external bonding of FRP plates or sheets has been widely used for the strengthening of RC structures. Due to the high tensile strength and low weight (comparing to conventional steel) FRPs have become an ideal material for use in the construction industry. Another advantage of FRP over the steel as external reinforcement is its easy handling, hence minimal time and labor are required to implement them. However, an engineer should be aware of reliability of the applied strengthening technique. Often concrete cover separation or plate end interfacial debonding become the failure modes of FRP strengthening (Smith and Teng 2002; Oehlers et al. 2003). The behavior of the interface between the FRP and the concrete is the key factor controlling debonding failures in FRP-strengthened RC structures (Lu et al. 2005). Inadequacy of existing design techniques in capturing highly complex behavior of externally bonded RC composite is one of the main reasons of huge investments into reconstruction.

2.1.1. FRP Materials, Properties and Types of Manufacturing

FRP composites are made from fiber, resin, interface, fillers, and additives. Fibers of higher deformation modulus contribute to the mechanical strength of the FRP, whereas the resin helps to transfer or distribute the stress from one fiber to another and to protect the fiber against environmental and mechanical damage. The interface between the fiber and the matrix is known to significantly affect the performance of FRP composites. In addition to these three basic components, the fillers serve to reduce cost and shrinkage. The additives help to improve the mechanical and physical properties of the composites as well as the workability. The main part of FRP reinforcement is fibers. Table 2.1 presents physical and mechanical properties of various kinds of fibers.

Table 2.1. Physical and mechanical properties of different FRPs

Tensile Deformation Coefficient of Density Elongation Poisson's Type of FRP strength modulus thermal expansion ratio kg/m3 MPa GPa % 10–6/°C E – glass 2500 3450 72,4 2,4 5,0 0,22 S – glass 2500 4580 85,5 3,3 2,9 0,22 AR – glass 2270 1800–3500 70–76 2,0–3,0 – – Carbon 1700 3700 250 1,2 -0,6 up to -0,2 0,20 Carbon 1950 2500–4000 350–800 0,5 -1,2 up to -0,1 0,20 (high-modulus) Carbon 1750 4800 240 1,1 -0,6 up to -0,2 0,20 (high-strength) Aramid -2,0 longitudinal 1440 2760 62 4,4 0,35 (Kevlar 29) 59 radial Aramid -2,0 longitudinal 1440 3620 124 2,2 0,35 (Kevlar 49) 59 radial Aramid -2,0 longitudinal 1440 3450 175 1,4 0,35 (Kevlar 149) 59 radial Aramid -2,0 longitudinal 1390 3000 70 4,4 0,35 (Technora H) 59 radial Aramid 1430 3800–4200 130 3,5 – – (SVM) Bazalt 2800 4840 89 3,1 8,0 – (Albarrie)

There are four main materials used for production of fibers dominating in civil engineering industry: glass, carbon, aramid, and basalt:  Glass fiber reinforced polymers (GFRP). Relatively low cost comparing to other kinds of FRPs makes glass fibers the most commonly used in the construction industry. However, a relatively low deformation modulus, the low humidity and alkaline resistance as well as low long-term strength due to stress rupture are the main disadvantages of GFRP. In case of demand of better alkaline resistance so-called AR glass FRP is being used.  Carbon fiber reinforced polymers (CFRP). These fibers have high deformation modulus and fatigue strength, do not absorb water. Though, a comparatively high energy requirement for the production of carbon fibers leading to the high costs is one of the major disadvantages. Furthermore, their disadvantages include anisotropy (reduced radial strength) as well as potential galvanic corrosion in direct contact with steel (Carolin 2003).  Aramid fiber reinforced polymers (AFRP). These fibers have high static, and impact strengths. Nevertheless, their use is limited by the reduced long-term strength (stress rupture) as well as

sensitivity to UV radiation. Another drawback of aramid fibers is that they are difficult for cutting and processing (Tuakta 2005).  Basalt fiber reinforced polymers (BFRP). Such fibers have excellent resistance to high temperatures, high tensile strength, and good durability. The other advantages are high resistance to acids, superior electro-magnetic properties, resistance to corrosion, resistance to radiation and ultraviolet (UV) light, and good resistance to vibration (Banibayat 2011).

Depending on the type of FRP, fibers combined with matrix consisting of resins, fillers and additive are used for production of bars or sheets. The resins are the basic components of a matrix. There are two major types of resins: thermoplastic and thermosetting polymers. The latter are most popular for production of FRP elements. Unlike thermoplastic polymers, once thermosetting polymers are scured, they cannot be reheated or reformed. Thermosets are usually brittle in nature, but offer high rigidity, thermal and dimensional stability, higher electrical, chemical, and solvent resistance. Table 2.2 gives physical and mechanical properties of the most widely used thermosets.

Table 2.2. Physical and mechanical properties of polyester, epoxy, and vinyl-ester resins

Thermosetting resins Properties Polyesters Epoxy Vinyl-ester Density, kg/m3 1200–1400 1200–1400 1150–1350 Tensile strength, MPa 34,5–104 55–130 73–81 Deformation modulus, GPa 2,1–3,45 2,75–4,10 3,0–3,5 Poisson's ratio 0,35–0,39 0,38–0,40 0,36–0,39 Coefficient of thermal expansion, 10-6/°C 55–100 45–65 50–75 Saturation, % 0,15–0,6 0,08–0,15 0,14–1,30

The processes used to produce FRP structural elements are the following (Hoffard and Malvar 2005; Banibayat 2011):  Pultrusion involves pulling rolls of material through a series of tooling devices, resin baths, and heated dies to merge, shape, and cure the resulting composite into a solid part. Pultrusion produces continuous lengths of structural shapes with constant cross-sections.  Hand Lay-up/Contact Molding is used to fabricate face skins over a panel core. Resin is manually applied to the core. Assembly of pre-cured face sheets are then placed on top of a wet corrugated sheet of core to produce a sandwich panel. The hand lay-up method lends itself to composite fabrication and repair in the field.

 Vacuum Assisted Resin Transfer Molding (VARTM) uses a vacuum to infuse resin into reinforcement fibers or fabrics that are placed in an evacuated mold. The mixture is allowed to cure under vacuum. The main advantage of VARTM over pultrusion is the unlimited size and geometry possibilities of the components.  Automated wet lay-up manufacturing process essentially consists of laying up the fibers that are impregnated with polymeric resin such that it yields usable composite bars when cured. FRP bars are made using a programmable arm with controlled movement in the three orthogonal directions to manufacture the desired lengths to the required shape. Cost of production of FRP bars by this method is believed to be reduced because the production method is simple and designed to reduce human involvement.

In addition to the already mentioned properties of FRP, it is essential to secure the sufficient bond strength between the reinforcement and structural elements. Depending on the application, FRP reinforcement can be categorized into two main groups: internal and external. Internal reinforcement is usually manufactured in the shape of bars. As shown in Fig. 2.1, the core of bars is made of fibers and resin and the surface might be deformed or sand-coated. The technology of surface coating is very important for the bond properties. Sand-coated surfaces often do not assure sufficient bond quality that leads to the slip of reinforcement in concrete. Therefore, deformed surface is more prominent for structural application. Recent investigations by Gribniak et al. (2012) and Timinskas et al. (2013) have revealed that grooved surface reinforcement (for instance, Schöck ComBAR) is characterized by high quality (strength) of the bond. It could be explained by well-balanced shape of the surface of the bar: the shear strength of concrete in the grooves is proportional to bar surface-to-core connection strength (see Fig. 2.2).

Fig. 2.1. Structure and coating of FRP bars

Unlike internal reinforcement, which is used in the production of new structures, where the bond quality is mostly depended on the surface production technology, external reinforcement is generally

used for strengthening of the existing structural elements. Gluing FRP sheets or strips (see Fig. 2.3) onto damaged elements results in inadequate bond strength.

Fig. 2.2. Bond action of Schöck ComBAR Fig. 2.3. Carbon FRP sheet for structural reinforcement reinforcement and containment

Applicability and efficiency of strengthening with FRP composites depend mainly on the material and the type of member to be strengthened. In general, applications where the accessibility conditions allow wrapping of the member with FRP composites, such as FRP wrapping of RC columns, has no problem regarding debonding issue (Buyukozturk et al. 2004). Considering the use of FRPs for external strengthening and external reinforcement of other types of concrete elements, its effectiveness is often related to the connection mechanism that allows the stress transfer between FRP sheets and concrete surface. Xiong et al. (2007), Kim et al. (2008a; 2008b), Skuturna et al. (2008), Diab et al. (2009), and Daugevicius et al. (2012) found that an additional anchoring is essential in order to assure FRP-to-concrete bond strength.

2.1.2. Experimental Investigation of RC Structures Strengthened with FRP

As already mentioned before, the effectiveness of external strengthening is related to the interface properties between FRP sheets and concrete surface. Recent investigations by Gribniak et al. (2014) revealed that the mechanical anchoring is unnecessary if the bond strength is designed proportionally to the overall stiffness of the composite element. This finding was proven by the experimental investigation of RC ties strengthened with basalt FRP sheets. A brief description of experimental trial as well as obtained results are presented herein.

A test program consisted of six RC ties of 1000 mm in length with 80×80 mm cross section: two reference specimens (T1-ref and T2-ref) and four externally strengthened with BFRP sheets. Two specimens strengthened with one (T5-1LS and T6-1LS), one with two (T4-2LS), and one with three (T3-3LS) layers of BFRP sheets were tested (Fig. 2.4). The layers of basalt fibers were applied on two opposite sides along the entire length of every specimen. All specimens were made of normal strength concrete. The ordinary Portland cement (CEM II/A-LL 42.5 N) and crushed granite aggregate (4/8 and 8/11 mm nominal size) were used. Water/cement, aggregate (4/8 mm)/cement, aggregate (8/11 mm)/cement, and sand/cement ratio by weight were taken as 0.45, 1.22, 1.83, and 2.61, respectively. Main material parameters are presented in Table 2.3.

Fig. 2.4. Test setup (Kesminas and Tamulenas 2014)

Table 2.3. Main characteristics of experimental materials

Concrete Steel reinforcement BFRP reinforcement Pre-load, Element f , f , E , Ø, f , Axial stiffness, Number Axial stiffness, cm ctm cm su kN MPa MPa GPa mm MPa As Es, MN of layers Af Ef, MN T1-ref – – – T2-ref T3-3LS 2×3 3.1 45.31 3.35 33.1 12 1120 21.5 T4-2LS 2×2 2.0 30 T5-1LS 2×1 0.55 T6-1LS 2×1 0.55

Ties were tested using a displacement-controlled machine with 600 kN capacity. The loading rate was 0.1 mm/min. The tests were carried out in two load cycles. The first one, a preload cycle, after which strengthening with BFRP was done. The second loading cycle was performed up to 70% of the ultimate strength of internal reinforcement. Linear variable displacement transducers (LVDT) (Fig. 2.4) recorded the strains on concrete surface and reinforcement. During the both load cycles, a crack development was observed and final crack pattern indicated (Fig. 2.5).

T1-ref T2-ref T3-3LS T4-2LS T5-1LS T6-1LS 100 Side I II III IV 100 Side I II III IV 100 Side I II III IV 100 Side I II III IV 100 Side I II III IV 100 Side I II III IV FRP FRP FRP FRP FRP FRP FRP FRP

90 90 6 90 1 90 90 90 6 4 7 7 4 5 4 2 4 8 8 1 80 80 3 80 80 80 80 8 3 2 2 7 1 3 70 70 70 70 5 70 70 1 2 6 3 6 1 7 3 60 60 4 60 60 60 9 60 3 9 5 5 5 1 6 50 50 1 50 50 8 50 50 2 1 6 4 6 10 8 5 40 40 40 40 1 40 40 2 6 4 2 4 2 1 30 30 30 30 3 30 30 2 7 7 10 8 5 5 20 20 20 20 3 20 3 5 5 4 3 11 2 10 10 10 10 4 10 7 * 10 9 0 0 0 0 0 0 preload crack preload crack preload crack preload crack preload crack preload crack 2nd load crack 2nd load crack 2nd load crack 2nd load crack * shrinkage crack Fig. 2.5. Crack pattern of all the specimens (Gribniak et al. 2014)

The performed experimental research revealed that the FRP bond strength has to be designed proportionally to the overall stiffness of the composite element – including cracked concrete and internal and external reinforcement as well. While using stiff FRP sheets, the reduction of bond strength might result in a dangerous increment of bond stresses related to a sudden failure of FRP bond.

2.1.3. FRP Applications

There is a huge variety of applications in which FRPs can be effectively used in structural engineering. FRP composites are used both for new construction and in strengthening or repair of existing buildings. Generally, two main categories of FRP application can be defined: FRP bars, rods and tendons as internal reinforcement as well as FRP sheets, wraps and laminates as external reinforcement. This section outlines some of the most common uses of FRPs in the civil infrastructure. FRP reinforcing bars comprising FRP grids have been extensively used as internal reinforcement for a number of concrete structures including bridges, tunnels underground precast chambers and

59 highway pavements. FRP grids are often used as lightweight reinforcement in the curtain walls, where lower requirements for concrete cover results in thinner and lighter facade panels. Due to their excellent resistance for corrosion, internal FRP reinforcement has been widely used in marine structures and in systems for slope protections and stability. Moreover, FRP composites are characterized by having inertness for electric-magnetic inductivity, thus, they are used for production of maglev rails. Another high promising potential use of FRP materials is to fabricate specific structural components entirely out of FRP, such as bridge decks, girders, or to use prefabricated FRP stay-in- place reinforcement panels for construction of concrete bridges decks (see Fig. 2.6). The replacement of conventional concrete bridge decks with FRP composite bridge decks oﬀers a viable solution for the rehabilitation of existing bridges. The beneﬁts of FRP replacement decks are their low weight (increasing the live load capacity of the bridge structure), increased durability (highly resistant to corrosion and fatigue), lower or competitive life-cycle cost, and rapid bridge construction using large prefabricated FRP reinforcements.

Fig. 2.7. Externally-bonded carbon FRP sheets for

shear strengthening of a reinforced concrete bridge Fig. 2.6. Fiberglass grid form for bridge decks girder

In order to increase strength and ductility of RC columns, FRP sheets (wraps) can be used as a confining reinforcement applied around them. Another FRP application as an external reinforcement is concrete filled FRP tubes. The FRP outer shell protects the concrete core from exposure to harsh environmental conditions and provides confinement for the concrete thereby increasing the strength and ductility of the pile. Summarizing the major FRP application options, the utilization of FRP is classified in Table 2.4.

Table 2.4. Application of internal and external FRP reinforcements

Internal External

Reference Reinforcement Application Reference Reinforcement Application Precast piles, Teng et al. 2007; concrete- Bridge deck FRP tubes Fam et al. 2003a; filled fender One- slabs (outer shells) Fam et al. 2003b piles and Rizkalla et al. dimensional columns 1998 (longitudinal) Flexural bars Shahawy et al. Sheets and strengthening Barrier walls 1996a; Shahawy et pre-cured of slabs and al. 1996b laminates beams Cheng and Hutchinson et al. Shear Karbhari 2006; Strands and Prestressed 2003; FRP sheets strengthening Rizkalla and rods girders Teng et al. 2003 of girders Tadros 1994; Benmokrane et al. 2006 El-Hacha and Near surface Repair of Bridge deck Benmokrane et Rizkalla 2004; mounted concrete slabs and al. 2000; Two- De Lorenzis and (NSM) rods bridge deck systems Lopez-Anido, dimensional Nanni 2002 and bars girders 1997 grids Underground Ilki et al. 2008; Seismic Benmokrane, structures, Demers et al. retrofit of 1999 Wrapping chambers 2004; columns and sheets, jackets Rizkalla et al. Prestressed Neale, 2000; special Shear stirrups 2006 girders Seible et al. 1997 structures Highway Eddie et al. 2001; Dowels Niroomandi et al. Joint pavements 2010; strengthening Keller, 2003; Bridge FRP sheets Prestressing Engindeniz et al. of RC Tezuka, 1994; cables, cable tendons 2005 structures Mufti et al. 1991 reinforcement

In spite of extensive attempts to apply FRP reinforcement in civil engineering, there are some aspects limiting this process. The two main reasons are as follows:  There are already a design guides for FRP reinforced concrete elements in USA, Canada, Japan, and Italy, though, no design codes for FRP reinforcement are developed. Due to the absence of design codes in most cases the responsibility of structural safety and serviceability fully lies on the designer.  There are not enough reliable experimental data on long-term degradation of mechanical properties of FRP materials. Therefore, common design practice is based on increased values of safety factors that leads to higher costs of elements with FRP reinforcement and makes such constructions economically inefficient.

2.1.4. Peculiarities of Structural Application and Design

Structural elements reinforced with FRPs may deteriorate due to environmental, physical, or chemical conditions, leading to loss of strength and stiffness. The degree of damage and deterioration depends on a variety of factors such as the type and volume of fibers, and resin matrix, the exposed environment, and the manufacturing process (Malvar 1998). Most composites exhibit a long-term static strength that is significantly lower than the short-term strength. For Polyester E-glass tendons, the long-term static strength at 10,000 hours (about 1 year) has been reported to be 70% of the short-term static strength (Wolff and Miesser 1989; Taerwe 1993). Sultan et al. (1995) report that remaining strengths for hand laid-up fiberglass after 10 to 15 years becomes 40% the short-term static strength. Slattery (1994) reports that long-term tests on fiberglass composites with epoxy resin showed failure of about half of the samples tested at a sustained stress of only 50% of ultimate, after about 7 years. Some of the samples ruptured at levels as low as 33% of ultimate. According to Hawkins et al. (1996), an E- glass composite wraps applied as confinement for circular highway columns failed after 3 years under sustained stresses around 32% of the manufacturer’s reported strength. For Kevlar fibers, the 100-year sustained strength is around 60% of the short-term strength (Taerwe 1993; Horn et al. 1977). Test data on carbon fibers shows very few failures after several years and a sustained stress of 80% of the short-term ultimate value (Slattery 1994). Tests on aramid bar showed sustained to short-term strength ratios of 75%, 70%, 60% and 50% for exposures to 20°C air and 20°C, 40°C and 60°C alkaline environments, respectively (at 10,000 hours) (Scheibe and Rostasy 1995). The estimated 100-year sustained strength of an aramid rod decreased from 60% in air to 50% of the short-term strength in an alkaline environment (Horn et al. 1977; Gerritse 1992; Gerritse and Den Uijl 1995). Dolan et al. (1997) found long-term strength (at 5500 hours and for GFRP tendons embedded in concrete) of about 55% of the short-term value. Attention should be paid to protect FRP materials, particularly glass and aramid bars, from alkali environment in concrete. In case of CFRP, the decrease in strength and stiffness might reach about 20% (Takewaka and Khin 1996). While the type of glass fibers, resin and manufacturing process may lower the tensile capacity even in the range of 25–100% (Rostasy 1997). In addition to that, according to Nkurunziza et al. (2005), the reduction of strength due to alkali can be influenced by high temperature and stress level. Another report states that a reduction in the tensile strength of 41% was observed after alkali exposure for 42 days at a temperature of 60 oC (Micelli and Nanni 2004).

Regarding AFRP, the tensile strength and stiﬀness of AFRP rods in elevated temperature alkaline solutions either with or without tensile stress have been reported to decrease between 10–50% and 0–20% of initial values, respectively (Takewaka and Khin 1996; Rostasy 1997; Sen et al. 1998). A protection from alkali environment may be assured at the manufacturing stage of FRP by using proper coating materials. However, the coated surface may be damaged under construction. Moreover, fibers are deteriorated due to chemical attack through uncovered (by cutting) endings of the bars. Thus, construction of elements with GFRP or AFRP bars requires increased accuracy. A relatively low modulus of elasticity (comparing to steel) is characteristic for the most of FRP bars (see Fig. 2.8). This leads to a smaller structural rigidity provided by these bars in respect to RC elements. Moreover, deformation modulus might significantly decrease in time. According to Arockiasamy et al. (2000), the increase in deflections over the instantaneous values for a period of 470 and 610 days is up to 115% and 125%, respectively. Structural elements with FRP reinforcement with low modulus of elasticity may not meet serviceability (a limitation of strain and deflection) requirements. In order to solve this problem, design of FRP reinforced elements often is based on condition of relative stiffness nf × ρ nfρnfρ(FRP bar-to-concrete deformation modulus ratio multiplied by longitudinal reinforcement ratio) equivalent to conventionally (with steel bars) reinforced elements (Baena et al. 2012). As can be observed from Fig. 2.8, such a design depending on the type of FRP reinforcement may lead to 2–3 times increased cross-area of FRP bars. Consequently, this increases the cost of the structural element.

Fig. 2.8. Comparison of characteristic and design tensile strength and elasticity modulus of different types of reinforcement (Timinskas et al. 2013)

Effect of UV radiation is another aspect that is of vital importance for application of FRP composites as an external reinforcement (Bank et al. 1995; Odagiri et al. 1997). Aramids are most

63 vulnerable to UV attack. A thin Kevlar 29 fabric exposed to Florida sun for 5 weeks lost 49% of its strength (DuPont, 1992). Tests of FRP materials exposed to UV rays carried out by Kato et al. (1997) and Tomosawa et al. (1998) have shown for AFRP rods around 13% reduction in tensile strength after 2500 h exposure, 8% reduction for GFRP rods after 500 h (no reduction thereafter). Glass and particularly carbon FRPs are less sensitive to effects of UV radiation, though majority of resins will be affected by UV. To prevent effect of UV radiation, the structural measures or material modifications (extra matrix additives, pigmented gel coatings, painting) are used. Design recommendations for FRP reinforced concrete elements exist in USA (ACI 440 2006), Canada (CSA 2010; CSA 2012), Japan (JSCE 1997) and Italy (CNR 2007); the International Federation for Structural Concrete had developed technical report considering application of FRP reinforcement in RC structures (fib 2007), though, there are no design codes for such a type of reinforcement. Current European (CEN 2004), American (ACI 318 2011) and Russian (NIIZhB 2006) design codes of structural concrete are adapted to the elements reinforced with steel bars, but, may be inadequate for design of structures with composite bars. As noted before, the lack of reliable experimental data results in unreasonably increased values of safety factors. According to the ACI 440 recommendations, in order to ensure the serviceability limit state of existing structures, a characteristic value of tensile strength of GFRP, AFRP, CFRP has to be reduced by 80%, 70%, 45%, respectively. Following report (Schöck, 2006), despite the high (well above 1000 MPa) short-term tensile strength of the ComBAR bars, reduction of the strength value is recommended to be 435 MPa for design purposes. For deformation analysis of FRP RC elements, the Italian design guide (CNR 2007) applies empirical expressions from the Eurocode 2 (CEN 2004) with a multiplier that simply increases deformations of a cracked element twofold. However, recent investigations by Gribniak et al. (2013) have revealed that such methodology is too rough as deformations are mainly depended on bond properties of FRP bars embedded in concrete. Similar results were obtained by Miàs et al. (2013b) who investigated long-term deflections of FRP RC elements. However, it is important to note that long-term deflection increment was depended on longitudinal reinforcement ratio increasing with increased cross-section of FRP reinforcement (Miàs et al. 2013a). In fact, a design of concrete elements reinforced with FRP bars should be based on the experimental results of the structural stiffness and the bond properties between FRP bars and concrete. This problem can be solved developing standard shape of FRP bars and anchoring measures for the external reinforcement. Standardization of the shape would allow a more extensive use of FRP reinforcement in the construction industry.

2.2. Biological Self-healing Concrete

A durability, relatively low cost, and unlimited design options are the governing factors why more than 70% of people across the world are living in reinforced concrete structures. Concrete is highly resistant to compression but considerably weaker for acting tensile forces resulting in a crack opening. Consequently, the aggressive environmental effects, acting through the cracks, affect the conventional steel reinforcement, initiate corrosion process, and further erode the concrete composition, thus greatly reducing the lifespan of a structure. As already mentioned before, the half of the budget of construction industry is spent on the maintenance and repair of existing buildings. In order to reduce these costs, the huge efforts of scientists and engineers concentrate on improving the properties of concrete and development of advanced construction materials. There are already several alternatives for normal concrete such as engineered cementitious composites (also known as bendable concrete), steel fiber reinforced concrete, and concrete reinforced with composite FRP reinforcement, which in one or another way do significantly improve the performance of structures. Due to rather simple manufacture, autonomous functioning and effective use in hard-to-reach locations one of the most adequate ways to solve the cracking problems of concrete structures is the use of biological self-healing concrete. About a decade ago, Dutch researcher Dr. H. M. Jonkers from the Technical University of Delft has developed a novel technique based on the application of bacteria causing a self-healing mechanism in concrete. The essential characteristic of such a concrete is the ability to reproduce an autonomous self-healing process which prevents structures from further cracking and other structural defects. Traditional concrete shows some self-healing capacity as well which is due to excess of non-hydrated cement particles present in the material matrix. These particles can experience secondary hydration by crack seeping water resulting in precipitation of fresh hydration products (i.e. calcium carbonate) which can seal or heal smaller cracks. However, the addition of excess cement in concrete is undesirable from both an economical and environmental point of view. Cement is expensive and, what is more, its production contributes significantly to global atmospheric CO2 emissions (Jonkers and Schlangen 2009).

2.2.1. Mechanism of Self-healing using Bacteria

Unlike the normal, biological concrete contains two additional self-healing agents causing biochemical reactions in concrete. The first of these components – a bacteria capable of surviving in an extreme alkaline environment (Fig. 2.9). The second one – calcium-based nutrients (i.e. calcium lactate), which are vital for bacterial feeding and consistent support of healing processes.

Fig. 2.9. A bacteria of self-healing concrete (Arnold 2011)

During the self-healing process of concrete, the soluble nutrients are converted into insoluble calcium carbonate (CaCO3). The CaCO3 solidifies in the opened cracks, thereby sealing them up (Fig. 2.10). Acting on such a principle, the bacteria is able to heal up to 0.5 mm wide cracks and can lie dormant and intact up to 200 years that is much longer than the standard lifetime of any structure. Bacteria

Ca(C3H5O2)2 + 7O2 CaCO3 + 5CO2 + 5H2O

Crack Sealed crack

Fig. 2.10. Autonomous self-healing mechanism of biological concrete

Fig. 2.11. Clay pallets filled with healing agent (50% of concrete volume) (Arnold 2011)

In order to avoid the self-healing mechanism in a concrete mixing stage, it is important to ensure that bacteria and nutrients are not in direct contact. In this case, the nutrients are introduced to the concrete within separate expanded clay pellets 2-4 mm in diameter, which do prevent the beginning of a spontaneous microbiological activity of bacteries (Fig. 2.11).

Fig. 2.12. Schematic representation ofself-healing process in biological concrete (Jonkers 2011)

When concrete structures suffers from the developing cracks (Fig. 2.12), the protective layer of clay pellets breaks down. Then incoming water, which is rich in dissolved nutrients, and oxygen induce microbiological activity. In this way, the cracks are sealed, a protection from harmful environmental effects to steel reinforcement is ensured, and a decreased intensity of steel corrosion by the oxygen consumption during the biochemical reactions is provided.

2.2.2. Effect of healing agent additions on concrete strength

As incorporation of healing agents to concrete mixture may have negative effects on material properties, the development of compressive strength of control specimen without additions as well as specimen with healing agent additions was investigated (Jonkers and Schlangen 2009). Incorporation of a high number of bacteria (5.8 × 108 cm-3 cement stone) appeared to have a slightly negative effect on compressive strength development as specimen with bacteria appeared almost 10% weaker than that of control specimen (Fig. 2.13 a).

Compressive strength (MPa) Compressive strength (MPa)

Control Peptone

KontrolinisControl BakterinisBacteria

Curing time (days) Curing time (days) a) b)

Compressive strength (MPa) Compressive strength (MPa)

Control Control Ca Glutamate Ca Lactate

Curing time (days) Curing time (days) c) d) Fig. 2.13. Compressive strength development of cement stone specimen containing: a) 5.8×108 cm-3 bacteria; b) 1% peptone of cement weight; c) 1% calcium glutamate of cement weight; d) 1% calcium lactate of cement weight (Jonkers and Schlangen 2009)

Effect of nutrient incorporation on development of strength appeared strongly dependent on its identity. Addition of peptone resulted in a catastrophic development as initial strength (after 3 and 7 days curing) was already less than 50% of the control. Further curing resulted in an even lower compressive strength (Fig. 2.13 b). Strength development of the specimen containing calcium glutamate appeared, similar to the bacterial specimen, parallel to control specimen, though at a 25%

68 lower level (Fig. 2.13 c). It was obtained that the additions of calcium lactate do not significantly affect strength development as values were only a little lower or higher than that of control specimen (Fig. 2.13 d).

2.2.3. Principal Obstacles Limiting the Application of Self-healing Concrete

However, the practical application of self-healing biological concrete in construction industry is limited by several factors. One of the most important is that a part of the conventional aggregate (i.e. gravel) is being replaced by the clay pellets (granules) filled with nutrients; thus reducing the strength of concrete. In order to avoid this problem, the further research and development of stronger granular shell materials and optimal content of granular nutrients in concrete are carried on (Arnold 2011). A price is another key factor limiting the wider use of biological concrete. At present, the price is still more than double than that of normal concrete. It is likely that the massive application of biological concrete will reduce its production costs. It is also important to point out that the concrete price itself often accounts for only 1-2% of the total value of the building. Therefore, a twofold increase of a concrete cost during construction essentially is paying off considering an expanded entire operational period of the structure and a significant reduce of maintenance expenses. What is more, the lack of design codes governing the composition of biological concrete makes it dificult to select the amount of bacteria and nutrients required for effective self-healing mechanism.

2.2.4. Self-healing Concrete Applications

The development of self-healing concrete for sectors such as structural basement walls, tunnel- lining, concrete floors, highway bridges and marine structures is recently considered. The autonomous recovery and self-repair of damaged elements are of importance for the structures located in hard-to-reach places; i.e. underground building elements, tunnels, marine structures are severely damaged or even collapse due to the corrosion of exposed reinforcement. In all the structures, where the limited width of the crack opening is governed (i.e. tower abutments, bridge decks, railway sleeper), the use of biological concrete along with steel or polymer fibers is very rational and efficient. The fibers constrain the width of opening crack, which leads to a more rapid and effective self-healing mechanism of reinforced concrete structures.

2.3. Concluding Remarks of Chapter 2

1. On the base of performed extensive analysis of literature sources it can be concluded that in design of FRP reinforced concrete elements the main attention should be pointed out on:  long-term degradation of mechanical properties;  proper selection of FRP material in a severe environmental conditions;  bond properties as the governing criteria for deformational analysis.

2. The study on applicability of FRP materials has revealed that further research should aim at:  experimental investigation on long-term mechanical processes in concrete elements with FRP reinforcement;  development of standard shape of internal FRP bars and anchoring measures for external reinforcement;  development of design procedures for the application of the unified internal and external reinforcements.

3. The review of experimental investigations on effect of healing agent additions on concrete strength revealed that no significant loss of concrete strength has been observed.

4. Due to the autonomous recovery and self-repair of damaged elements, the application of biological concrete is of importance for the structures located in hard-to-reach places.

5. In case the limited crack width is governed, the use of biological concrete along with the steel or polymer fibers is recommended.

GENERAL CONCLUSIONS AND RECOMMENDATIONS

Having conducted thorough literature review on the methodology of fatigue assessment and applicability of innovative materials, completed experimental investigations, and performed numerical and comparative analysis, the following conclusions can be drawn:

1. Considering fatigue behavior, the resistant number of cycles is greatly affected by reinforcement ratio. Concrete members with higher reinforcement ratio possess larger stress redistribution capacity and reach higher number of load cycles till failure, i.e. have higher fatigue-resistance.

2. The performed comparative analysis of prevailing fatigue assessment methods has shown that the higher reinforcement ratio in RC columns makes a generally accepted Palmgren-Miner damage accumulation rule to be inaccurate while giving the smaller resistant number of cycles, thus overestimating the failure state of concrete. As a result, the code techniques present inadequately high safety margin.

3. There are two major options of improving the durability of reinforced concrete structures: 1) it is necessary to improve the design methods; 2) to apply innovative composite materials with increased durability-resistance.

4. In the context of Lithuanian civil engineering demands, it is recommended to develop a production of biological concrete (for reduction of building maintenance and repair costs). Due to the autonomous recovery and self-repair of damaged elements, the application of biological concrete is of importance for the structures located in hard-to-reach places.

5. Application of FRP reinforcement can be useful for avoiding corrosion-induced problems. Regarding the design of FRP reinforced concrete elements, attention should be pointed out on long- term degradation of mechanical properties, proper selection of FRP material in a severe environmental conditions, and bond properties as the governing criteria for deformational analysis.

The study has revealed that further research should aim at:  experimental investigation on long-term mechanical processes in concrete elements with FRP reinforcement and incorporated bacteria;  development of anchoring measures for external reinforcement;  development of design procedures and reglamentation of the use of innovative materials for the construction, maintenance and reconstruction of RC structures.

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LIST OF PUBLICATIONS BY THE AUTHOR ON THE TOPIC OF THE MASTER’S THESIS

Papers in the Reviewed Scientific Journals

Gudonis, E.; Timinskas, E.; Gribniak, V.; Kaklauskas, G.; Arnautov, A. K.; Tamulėnas, V. 2013. FRP reinforcement for concrete structures: state-of-the-art review of application and design. Engineering Structures and Technologies 5(4), 147–158.

Timinskas, E.; Jakštaitė, R.; Gribniak, V.; Tamulėnas, V.; Kaklauskas, G. 2013. Accuracy analysis of design methods for concrete beams reinforced with fiber reinforced polymer bars. Engineering Structures and Technologies 5(3), 123–133.

Bacinskas, D.; Kamaitis, Z.; Jatulis, D.; Kilikevicius, A.; Gudonis, E.; Danielius, G.; Tamulenas, V.; Rumsys, D. 2014. Field load testing and structural evaluation of steel truss footbridge. In 9th International Conference “Environmental Engineering”. May 22–23, 2014, Vilnius, Lithuania. Vilnius: Technika, 1–6.

Tamulenas V.; Gelažius, V. 2014. Stress-strain analysis of reinforced concrete members subjected to cyclyc loading. Proceedings of the 17th Conference for Junior Researchers „Science – Future of Lithuania“, Civil Engineering, Vilnius, Lithuania, April 30, 2014. Vilnius: Technika.

Kesminas, D.; Tamulenas V. 2014. Investigation of deformation and cracking behavior of RC ties strengthened with FRP sheet. Proceedings of the 17th Conference for Junior Researchers „Science – Future of Lithuania“, Civil Engineering, Vilnius, Lithuania, April 30, 2014. Vilnius: Technika.

Rimkus, A.; Meškėnas, A.; Tamulėnas V. 2014. Sumaniųjų tiltų taikymo perspektyvos Lietuvoje. Proceedings of the 17th Conference for Junior Researchers „Science – Future of Lithuania“, Civil Engineering, Vilnius, Lithuania, April 30, 2014. Vilnius: Technika.

Garškaitė, A.; Tamulėnas V. 2014. Sumaniųjų medžiagų, konstrukcinių sprendimų ir statinių priežiūros priemonių taikymo perspektyvos Lietuvoje. Proceedings of the 17th Conference for Junior Researchers „Science – Future of Lithuania“, Civil Engineering, Vilnius, Lithuania, April 30, 2014. Vilnius: Technika.

Jakštaitė, R.; Timinskas, E.; Tamulėnas, V.; Gribniak, V. 2013. Kompozitais armuotų sijų tempiamojo betono įtempių ir deformacijų tyrimai. Proceedings of the 16th Conference for Junior Researchers “Science – Future of Lithuania”, Civil Engineering, Vilnius, Lithuania, March 20-22, 2013. Vilnius: Technika. ISSN 2029-7149. ISBN 9786094575365.

Papers in Popular Scientific Journals

Kaklauskas, G.; Gribniak, V.; Meškėnas, A.; Rimkus, A.; Tamulėnas, V. 2014. Betonas, bakterijomis gydantis plyšius konstrukcijose. Statyba ir architektūra 4: 32-33. ISSN 0131-9183.