Cement-Based Overlay for Orthotropic Steel Bridge Decks

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Department of Civil Engineering

Rasmus Walter

Cement-Based Overlayfor Orthotropic Steel Bridge Decks

A Multi-Scale Modeling Approach

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Cement-Based Overlay for OrthotropicSteel Bridge Decks

A Multi-Scale Modeling Approach

Rasmus Walter

Ph.D. Thesis

Department of Civil Engineering

Technical University of Denmark

2005



Cement-Based Overlay for Orthotropic Steel Bridge DecksA Multi-Scale Modeling ApproachCopyright (c), Rasmus Walter, 2005Printed by DTU-TrykDepartment of Civil EngineeringTechnical University of DenmarkISBN number: 87-7877-181-1



Preface

This thesis is submitted as a partial fulfilment of the requirements for the Danish Ph.d.degree. The thesis is divided into two parts. The first part introduces the motivation andhighlights the major conclusions and findings. The second part is a collection of sevenpapers, presenting the research in greater details.

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Acknowledgements

I gratefully acknowledge the support of my main supervisor, Assoc. Prof. Henrik Stang,Technical University of Denmark, as well as my co-supervisors Assoc. Prof. John ForbesOlesen and Professor Niels-Jørgen Gimsing both at the Technical University of Denmark,and Tina Vejrum, COWI A/S Consultant Engineers.

Furthermore, I would like to thank Professor V. Li for my stay at his group at TheUniversity of Michigan, which has been very fruitful for my work. The assistance, in thelaboratory, of M. Lepech and S. Wang both at The Advanced Civil Engineering MaterialResearch Laboratory, University of Michigan, are acknowledged. Financial support fromThe Knud Højgaard Foundation, for my stay at The University of Michigan, is gratefullyacknowledged.

Finally, former master students, S. Siggurdson, B. Jansen, M. Østergaard, and M. Langeare greatly acknowledged for performing experiments useful to my work, during theirmaster thesis work.

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Abstract

The success of the traditional orthotropic steel bridge deck may be due to its high strengthto weight ratio. However, fatigue damage has been experienced within heavily traffickedroutes due to the low stiffness of this deck in combination with increasing traffic intensityand wheel pressure. This thesis investigates a system to stiffen an orthotropic steel bridgedeck, using a cement-based overlay. The investigation is based on nonlinear fracturemechanics and aims to determine the performance of the bridge deck in terms of crackingbehavior. The main goal of applying a cement-based overlay to an orthotropic steel bridgedeck, is to increase the deck stiffness and thereby reduce the stresses in fatigue sensitivesteel parts. Cracking of the cement-based overlay will have considerable influence onthe composite action and durability of the system. The system has to be economicallybeneficial and show a good performance with regards to cracking behavior. Since crackingplays a major role on the performance, cracking behavior of the overlay is the main focusof the present thesis.

The strategy utilized in the present thesis is based on multi scale modeling, which spansfrom modeling and experiments on the steel-concrete interface scale, to modeling of areal size structure. The multi scale concept is utilized by identifying mechanical behav-ior on the steel-concrete interface scale and later applying the mechanical behavior onthe structural scale. On the steel-concrete interface scale, normal cracking (Mode I) andcombined normal and shear cracking (mixed mode cracking) are analyzed through exper-iments and modeling. The aim and outcome of the study on the interface scale, is a setof constitutive parameters which later are applied on the structural scale. The compositeaction between an overlay and steel plate is analyzed experimentally through small beamand plate elements with spans in the range of 0.8 to 1.0 meter. Through these tests, the

numerical tools applied are verified by comparing experimental results to numerical re-sults. Cracking between the overlay and steel plate (debonding) is also analyzed throughsmall scale experiments and further investigated numerically. The investigation showsthat debonding is initiated from a defect in the overlay, e.g. an overlay crack. Debondinginitiation is observed for a certain crack width of the overlay. Significant findings on thecomposite elements, such as the overlay crack width which initiates debonding, are alsoobserved when modeling a full size structure.

A set of theoretical tools have been established to analyze the performance of an or-thotropic steel bridge deck stiffened with a cement-based overlay, with respect to crackingbehavior. For a given design situation, it might be possible to give an estimate on the

crack pattern and maximum crack width when applying a given cement-based overlay.

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This thesis demonstrates a nonlinear investigation of a real size structure, with emphasison the performance of the overlay system, using different cement-based materials. Ef-fects, such as traffic load, early age shrinkage, and temperature gradients are taken into

account, and it is showed that all these effects might have a significant influence on thecracking behavior. The overlay performance, e.g. the relation between the magnitudeof axle load and maximum crack width, is dependent on the constitutive parameters of the overlay material. Temperature gradients and early age shrinkage, have a considerableinfluence on the relationship between axle load and maximum crack width. The analysisshows that cracking of the overlay, for the given structure and design regulations, might beunavoidable. Therefore, the challenge in certain design situations, might be to minimizethe maximum crack width of the overlay.



Resume

De traditionelle orthotrope stalbrodæks succes kan tilskrives deres høje styrke-til-vægtforhold. Der er dog, indenfor de senere ar, observeret udmattelsesskader pa de tradi-tionelle orthotrope stalbrodæk. Udmattelsesskaderne er især observeret pa orthotropestalbrodæk der er placeret pa højt trafikerede strækninger og skyldes den lave stivhed af et orthotropt stalbrodæk i kombination med stigende trafikintensitet og højere hjultrykDenne afhandling undersøger et system til at forstærke orthotrope stalbrodæk ved anven-delsen af et cementbaseret dæklag. Undersøgelsen baserer sig pa ikke-lineær brudmekanikmed det mal, at bestemme stalbrodækkets mekaniske opførelse med fokus pa revnedan-nelse. Hovedformalet med at anvende et cementbaseret dæklag er at forøge stivheden af stalpladen i det orthotrope stalbrodæk. Ved at forøge stivheden, sænkes spændingerne i dekritiske stalsamlinger og levetiden af brodækket forøges. Holdbarhed og samvirke mellemdet cementbaseret dæklag og stalpladen, er begge forhold der er styret af revnedannelse.Anvendelsen af systemet er afhængigt af om det er økonomisk fordelagtigt, samtidig medat det udviser en stor modstandsdygtighed overfor revnedannelse. Idet revnedannelsen har

en markant indflydelse pa levetiden af systemet, er revnedannelse centralt i nærværendeafhandling.

Strategien i afhandlingen baserer sig pa multi-skala modellering, der spænder fra mo-dellering og eksperimenter af en skilleflade mellem beton og stal, til modelleringen af et brodæk pa konstruktionsniveau. Den betydningsfulde mekaniske opførelse pa skille-fladeniveau er identificeret, der senere hen er medtaget i analysen af et brodæk pa kon-struktionsniveau. Pa skillefladeniveau er revnedannelse i normalretningen (Mode I) ogkombineret revnedannelse i normal- og forskydningsretningen, analyseret gennem eksper-imenter og modellering. Malet og resultatet af analysen pa skillefladeniveau, er et sæt

af konstitutive parametre til senere anvendelse pa konstruktionsniveau. Samvirke mellemstal og det cementbaseret dæklag er analyseret gennem en række eksperimenter bestaendeaf et antal plade- og bjælkeforsøg, med spænd i størrelsesordenen 0.8 til 1.0 m. Vedat modellere plade- og bjælkeforsøgene kan de, i afhandlingen anvendte, numeriske Fi-nite Element (FE) metoder verificeres ved at sammenligne numeriske og eksperimentelleresultater. Revnedannelse mellem det cementbaseret dæklag og stalpladen er ligeledesanalyseret gennem forsøg og FE. Undersøgelsen viser at delaminering mellem dæklagetog stalpladen initieres pga. en defekt i dæklaget, til eksempel revnedannelse i dæklaget.Delamineringen initieres oftest, for en bestemt revnevidde i dæklaget, hvilket ogsa erobserveret ved FE modellering af et brodæk pa konstruktionsniveau.

Igennem Ph.D. arbejdet er der etableret et sæt teoretiske værktøjer til at analysere et or-

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thotropt stalbrodæk med vægt pa revnedannelse. I en given design situation, er det evt.muligt at give et bud pa revnemønster og maksimum revnevidde. Afhandlingen demon-strer en ikke-lineær FE analyse af et orthotropt stalbrodæk forstærket med forskellige ce-

mentbaseret materialer. Forhold som trafiklast, svind i tidlig alder, og temperaturgradien-ter er medtaget i analysen, der viser at alle disse forhold har betydning for revnedannelse.Relationen mellem aksellast og maksimum revnevidde afhænger af dæklagets konstitu-tive parametre. Temperaturgradienter og svind i tidlig alder pavirker ligeledes relationenmellem aksellast og maksimum revnevidde. Analysen viser, for den givne konstruktion ognorm, at revnedannelse i overlaget er uundgaeligt. Udfordringen vil, i et givent tilfælde,være at minimere den maksimale revnevidde af det cementbaseret dæklag.



Table of Contents

I Introduction and Summary 1

1 Introduction 31.1 Orthotropic Steel Bridge Decks . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Fatigue in Orthotropic Steel Bridge Decks . . . . . . . . . . . . . . . . . . 41.3 Stiffening of Orthotropic Steel Bridge Decks . . . . . . . . . . . . . . . . . 4

1.3.1 Conventional Surfacing . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.2 Steel Plate Reinforcements and Concrete Filled Ribs . . . . . . . . 51.3.3 Synthetic Overlays . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.4 Cement-Based Overlays . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Proposed Cement-Based Overlay System . . . . . . . . . . . . . . . . . . . 61.5 Overview of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5.1 Aim and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5.2 Strategy and Method . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5.3 Scope and Original Features . . . . . . . . . . . . . . . . . . . . . . 8

2 Interface Characterization 132.1 Studies on Interface Fracture in Mode I . . . . . . . . . . . . . . . . . . . . 142.2 Studies on Mixed Mode Interface Fracture . . . . . . . . . . . . . . . . . . 16

2.2.1 Mixed Mode Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.2 Mixed Mode Experiments . . . . . . . . . . . . . . . . . . . . . . . 20

3 Composite Elements 253.1 Numerical Studies on Composite Beams . . . . . . . . . . . . . . . . . . . 25

3.2 Test Methods to Obtain Constitutive Parameters of Overlay . . . . . . . . 273.3 Verification of Numerical Modeling Tools . . . . . . . . . . . . . . . . . . . 283.4 Testing of Composite Plates . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 Structural Behavior 374.1 Linear Elastic Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2 Nonlinear Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5 Conclusions 455.1 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . . 46

Bibliography 49

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II Appended Papers 53

Paper I Cohesive Mixed Mode Fracture Modelling and Experiments ,submitted to: Journal of Engineering Fracture Mechanics

Paper II Wedge Splitting Test for a Steel-Concrete Interface , Journal of Engineering Fracture Mechanics 72(17), pp. 2565-2583, 2005.

Paper III Cement-Based Overlay in Negative Bending - Experimental and FEM

Studies , submitted for publication

Paper IV Experimental Investigation of Fatigue in a Steel-Concrete Interface

in: 5th International Conference on Fracture Mechanics of Concrete andConcrete Structures, Vail Colorado, USA, pp. 839-845, 2004.

Paper V Debonding of FRC Composite Bridge Deck Overlay , in the proceedings of:7th International Symposium on brittle matrix composites - BMC 7,Warsaw, Poland, pp. 191-200, 2003.

Paper VI Method for Determination of Tensile Properties of ECC , in the

proceedings of: ConMat’05, Vancouver, Canada, 2005.

Paper VII Analysis of Steel Bridge Deck Stiffened with Cement-Based

Overlay , submitted to: ASCE - Journal of Bridge Engineering



Additional reading (not included in thesis)

[1] Walter R., Olesen J. F. & Stang H.:Interface Mixed Mode Model ,

in: 11th International Conference on Fracture - ICF11, Turin, Italy, 2005.

[2] Walter R., Stang H., Gimsing N.J. & Olesen J.F.:High Performance Composite Bridge Decks Using SCSFRC , in: High Performance Fiber Reinforced CementComposites - HPFRCC 4, RILEM Workshop, June, Ann Arbor, Michigan, USA,pp. 495-504, 2003.

[3] Walter R., Li V.C. & Stang H.:Comparison of FRC and ECC in a Composite ,Bridge Deck in: 5th International PhD Symposium in Civil Engineering,June 16-19, Delft, The Netherlands, pp. 477-484, 2004.

[4] Walter R., Gimsing N.J. & Stang H.:Composite steel-concrete orthotropic bridge ,deck in: 10th Nordic Steel Construction Conference, June 7-9, Copenhagen,Denmark, pp. 519-530, 2004.



Part I

Introduction and Summary

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Chapter 1

Introduction

1.1 Orthotropic Steel Bridge DecksThe development of orthotropic steel bridge decks can be traced back to the 1930s and1940s, when German and American engineers used the principles of ship decks in bridgeengineering. After World War II, rebuilding of the long span bridges in Germany, beingshort on steel supply, lead German engineers to the closed rib stiffener design as we knowthe decks today, (Dowling 1968), (Wolchuk 1963). The typical orthotropic steel bridgedeck consists of a top steel plate with a number of closed rib stiffeners welded to thebottom face. Typically, each 4 meters in the longitudinal direction, a transverse beam isinstalled to distribute load to other parts of the bridge, cf. Figure 1.1.

Figure 1.1 Orthotropic steel bridge deck (AISC 1962).

The primary task of a highway bridge is to provide a flat surface that is capable of carryinglarge numbers of heavy concentrated wheel loads. Lightness is very important since itaffects the cost-effectiveness of a long span structure. Whole life cost is also importantsince bridges are designed for a long service life. Orthotropic steel bridge decks providelightness, but their record of durability on routes with heavy traffic is not satisfactory.

Repairs on highway bridges have been necessary on important bridges within 20 years

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Introduction 1.2 Fatigue in Orthotropic Steel Bridge Decks

or less, cf. Smith & Bright (2003). There are three primary reasons for the durabilityproblems on orthotropic steel bridge decks. These are increasing traffic volume, increasingaxle loads, and tire designers’ development of higher pressure tires with more concentrated

loads (Jong et al. 2004).

1.2 Fatigue in Orthotropic Steel Bridge Decks

Fatigue is a critical factor in the design of orthotropic steel bridge decks. Compared to aconcrete deck, the thin plate weld is relatively flexible, and high stresses are induced due tohighly concentrated wheel loads. Passing of a truck causes a transverse bending momentover the stiffening web, and fatigue damage due to the transverse bending moment inthe asphalt pavement and steel plate, have been observed in many bridges, cf. (Flint &Smith 1992). The transverse moment induces high stresses in the joint, where the top

steel plate is welded to the bottom rib. This joint has in many cases been characterized asthe most common fatigue sensitive detail in orthotropic steel bridge decks, see e.g. (Jonget al. 2004). Furthermore, the splice joint between the ribs, and the joint between therib and transverse beam, have also experienced fatigue damage in orthotropic steel bridgedecks. The problem of fatigue in orthotropic steel bridge decks has achieved internationalattention and numerous research projects are carried out to find a solution to stiffeningthe orthotropic steel bridge deck.

1.3 Stiffening of Orthotropic Steel Bridge Decks

A well-known fatigue example is the bascule part of the Van Brienenoord bridge in Rot-terdam, (Kolstein & Wardenier 1998),(Kolstein & Wardenier 1999). Since the discoveryof fatigue cracks in this bridge, research in the area of retrofitting and alternative systemsto stiffening orthotropic steel bridge decks has been investigated.

1.3.1 Conventional Surfacing

Studies by Kolstein & Wardenier (1997) and later by Wolchuk (2002), show the stiff-ness contribution of surfacing due to its composite behavior with the steel deck plate.Assuming a rigid bond between the surfacing and underlying steel plate, surfacing may

well contribute to the strain and stress distribution using the elementary bending theory.When using the surfacing as an integral part of the deck system, a considerable amountof stress reduction is observed. However, surfacing materials are generally visco-elasticto plastic and behaves elastically at low temperatures only. The elastic moduli of a gen-eral surfacing material depends highly on the temperature. Experimental recordings byKolstein & Wardenier (1997) and Smith & Cullimore (1987), show, when using a thickpolyurethane surfacing layer, that a stress reduction factor of 6 may be expected at -200Cand about 4 at +300C. However, polyurethane materials are distinctly different than con-ventional asphalt materials. It is relatively soft throughout a wide temperature range. Forthe mastic asphalt pavements used on European and Japanese orthotropic steel bridgedecks, the elastic moduli ranges from close to that of Portland cement concrete at -200C

to 2GPa at a temperature of +300C. A new approach has been proposed by Smith &

4 Department of Civil Engineering - Technical University of Denmark



1.3 Stiffening of Orthotropic Steel Bridge Decks Introduction

Bright (2003), combining lightweight asphalt, conventional asphalt and a layer of glassfiber mesh embedded just beneath the chip-sealed surface, cf. Figure 1.2. Applying thissystem was found to increase the durability by a factor 10.

Figure 1.2 Lightweight layered surfacing system, after Smith & Bright (2003).

1.3.2 Steel Plate Reinforcements and Concrete Filled Ribs

Several kinds of retrofit applications to improve the fatigue performance of orthotropicsteel bridge decks have been studied by Machida et al. (2004). Two methods are sum-marized here. One approach is to bolt an overlaid steel plate with a thickness of 12 mmto the underlying steel plate, cf. Figure 1.3(a). By this, they achieve a stress reductionof about 40 %. A second approach is to fill the ribs with a lightweight self-compactingconcrete, cf. Figure 1.3(b). This method performs rather poor compared to the steelreinforcement case and reduces the critical stresses with 10 % to 20 %. They recommendcombining this method with other alternative stiffening solutions to achieve a satisfactoryresult, e.g. to combine a cement-based overlay with concrete filled ribs.

(a) (b)

Figure 1.3 Retrofit approaches by Machida et al. (2004), (a) steel plate reinforcement,(b) concrete filled ribs.

1.3.3 Synthetic Overlays

The use of synthetic layers to reduce the stress range in orthotropic steel bridge deckshas been investigated by De Backer et al. (2004). They have carried out investigationson orthotropic railroad bridge decks to reduce the stresses at the sensitive fatigue details.Their approach is to install a synthetic layer on top of the steel plate to improve the load

dispersion. This system has been investigated for different synthetic layers both as an

Department of Civil Engineering - Technical University of Denmark 5



Introduction 1.4 Proposed Cement-Based Overlay System

independent layer, a glued on layer, and as a sandwich structure with a steel plate on top.The synthetic layers investigated were of different types, such as neopren mats, rubbermats, and polyurethane panels. The investigations showed a general trend, a glued on

layer with high elastic moduli achieved a remarkable stress reduction in the steel plate. Inone example, a 40 mm thick neoprene layer simply placed on top of the steel plate allowsonly for a 20 % reduction in stresses, whereas in the case of a rigid connection between theneoprene layer and steel plate a stress reduction by 50 % is achieved. One of the possibleproblems, using rubber like dispersion layers with a considerable thickness, is the highPoisson ratio of the material. A local vertical compression will result in a considerabletransversal compression and consequently deformation of the synthetic layer.

1.3.4 Cement-Based Overlays

An alternative, less temperature dependent, and with relatively high elastic moduli, is touse a cement-based overlay to reduce the stress range in the fatigue sensitive details. Thisidea has been investigated by several authors, see e.g. (Battista & Pfeil 2000), (Braamet al. 2003) and (Jong & Kolstein 2004).

The application of a cement-based overlay for retrofitting an orthotropic steel bridge deckhas already been carried out, as a pilot test, in practice. An area of the bascule part of theVan Brienenoord bridge in Rotterdam, The Netherlands has been chosen as a test area andreinforced by a 50 mm cement-based overlay using a fiber reinforced concrete with a fiberconcentration of 5 kg/m2 (Buitelaar 2002). The test area has the size of 60 m2 and theoverlay was placed in October 2000. A traditional steel reinforcement was also applied,using 24 kg/m2. The steel reinforcement consisted of a special welded mesh of threelayers with a spacing of 50x50 mm with a steel bar diameter of 8 mm. Initial conclusionsshow that the stresses in fatigue sensitive details, compared to a traditional orthotropicsteel bridge deck are reduced from approximately 128 MPa to 28 MPa. The economicalinvestment for placing the overlay is equal to the cost of a traditional bituminous wearingcourse of melted asphalt.

1.4 Proposed Cement-Based Overlay System

The main subject of this thesis is an investigation of cement-based overlays for orthotropicsteel bridge decks. In the proposed system, a typical deck consist of a 40-60 mm thickcement-based overlay bonded to the steel plate. It is suggested, in this system, to achievecomposite action through adhesion between the overlay and underlying steel deck. Thecurrent system leaves out mechanical shear connectors as used in traditional concrete-steelstructures. Adhesion between the cement-based overlay and steel plate is ensured by sandblasting of the steel plate prior to casting of the overlay. The proposed overlay system isshown in Figure 1.4 along with a traditional steel deck.

The motivation for leaving out mechanical shear connectors is mainly based on two rea-sons: (i) using shear connectors creates undesirable stress concentrations, and (ii) a system

with small shear connectors in large numbers will be costly with regards to labor. Since




1.5 Overview of the Thesis Introduction

Figure 1.4 (a) Typical steel system, with a center span of 300 mm between the supporting ribs and a 12 mm steel plate. (b) Cement-based overlay system with a 50 mmthick cement-based overlay.

adhesion between the overlay and steel deck plays a central role on the composite action,interface fracture is of main focus in the present thesis.

The thesis concerns numerical and experimental work on different length scales. In ex-ample, when considering the interface between the cement-based overlay and steel deck,crack openings are several orders of magnitudes smaller than an actual piece of bridgedeck. The results presented in this thesis are based on numerical and experimental studies

on the length scale of a steel-concrete interface, which later are implemented on the struc-tural scale. The method illustrates how to link the gap between the important featureson the small length scale and modeling on structural scale.

1.5 Overview of the Thesis

The thesis can be divided into two main parts. The first part introduces the motivationand background of this study and highlights the major conclusions and findings. Thesecond part is a collection of seven papers, explaining the research in greater details.

This first part, introduces the motivation and the problem at hand. Additionally, strategyand scope are included, along with an overview of the papers appended. The next threechapters highlight the major conclusions and findings, on three different length scales,from micro mechanical studies to numerical simulations on the structural scale. Finallyin Chapter 5, an overall conclusion is drawn together with recommendations for furtherwork.

The order, in which the papers are presented, does not follow a chronical time line in thethesis work as a whole, but in an order making it easy to relate the individual resultsfrom each paper to each other. Moreover, the order in which the papers are presented is

also discussed in relation to the strategy employed in the PhD study.




Introduction 1.5 Overview of the Thesis

1.5.1 Aim and Motivation

The present study investigates the application of casting a cement-based overlay to stiffen

the steel plate in an orthotropic steel bridge deck. The motivation for doing so is thewell-known fatigue problems associated with orthotropic steel bridge decks as describedin the present chapter. The goal is to integrate detailed materials modeling with thestructural modeling. The overall aim is, with emphasis on cracking behavior, to modelthe structural response of an orthotropic steel bridge deck reinforced with a cement-basedoverlay.

1.5.2 Strategy and Method

The strategy and the method of modeling a bridge deck using an integrated material-structural approach is explained. The strategy used in the present study can be charac-terized as a multi scale approach. This study takes into account various lengths scalesand shows how these can be linked together. In order to get an overview of the study,all the work carried out can roughly be dived into three length scales, from: (i) inter-face characterization, to (ii) material-interface interaction, to (iii) structural design. Thelength scales are shown in Figure 1.5.

The smallest length scale, fundamental interface behavior, includes fracture mechanicalstudies of a steel-concrete interface. Here, the aim is to model steel-concrete interfacefracture utilizing finite elements. Constitutive parameters are obtained through testing,and the results on the small length scale are then later used in analysis at larger length

scales. The intermediate length scale, material-interface interaction, concerns studieson the composite behavior between a concrete overlay and a steel plate. Studies onthis length scale are based on numerical simulations and experiments. One of the maingoals on this length scale is to analyze the effects of debonding (delamination betweenthe overlay and steel plate). The significance of the overlay material and steel-concreteinterface in relation to debonding behavior is analyzed. A sound connection between theoverlay and steel plate is of vital importance in regards to the composite action betweenthe two materials. The third and final length scale, the structural scale, is analyzed usingnumerical simulations only. The aim on this length scale is to study the performance of the orthotropic steel bridge deck on a large length scale. This is carried out by applying

experience from from smaller length scales.

1.5.3 Scope and Original Features

The study can be regarded as a collection of seven papers from I to VII . The aim of each individual part is given in each individual paper, and will not be repeated in greatdetails here. The paper order from I to VII is organized according to the length scalesas sketched in Figure 1.5. So that Paper I is characterized by the smallest length scaleto Paper VII dealing with the largest length scale.

Paper I concerns numerical mixed mode modeling of a steel-concrete interface. It de-

scribes and presents a nonlinear fracture mechanical model, which can be applied to





Figure 1.5 The length scales in the study can be organized as 1. steel-concrete interface behavior, 2. material-interface behaviour, and 3. structural design.

model the connection between steel and concrete. Additionally, it introduces an exper-imental set-up to measure the fracture properties of a steel-concrete interface in orderto feed the model with constitutive parameters. As this paper is placed on the smallestlength scale, the model is the backbone of the study, since this model is used throughoutmost of the papers at larger length scales.

Paper II is also placed at the smallest length scale and deals with experimental testing of a steel-concrete interface. The paper introduces a modification of the well-known WedgeSplitting Test (WST) for concrete to test the fracture mechanical Mode I behavior of asteel-concrete interface. Additionally, an inverse analysis, in order to obtain the Mode Ifracture parameters, are introduced and applied on composite steel-concrete specimens.

Paper III is based on the intermediate length scale and deals with a composite deckelement. The aim of the study is to analyze the debonding behavior between a cement-based overlay and a steel plate. It especially investigates the composite beam subjectedto negative bending (the overlay is subject to tension). The paper investigates a macrocrack propagating through the overlay, which subsequently causes debonding. By using

a fairly simple deck element (can be viewed as a part of the orthotropic steel deck), both





experimental and numerical studies are carried out. Furthermore, the correlation betweennumerical and experimental results is studied, and can be viewed as the first link betweentwo length scales.

Paper IV is also placed in the intermediate length scale. This paper deals with anexperimental set-up to determine the fatigue properties of a steel-concrete interface. Thespecimen is also a composite deck element, which has been analyzed numerically andexperimentally. A four point bending configuration is applied, with and without a pre-notch, exposed to cyclic loading. Different kinds of steel plates were applied in order toinvestigate the influence of surface roughness.

Paper V is a pure numerical paper dealing with debonding between the cement-basedoverlay and steel plate. The paper discusses modeling of the overlay and discrete interfa-

cial cracking. It solely concerns the situation when a crack propagates through the overlayand how this affects debonding between the overlay and steel plate. Through a parametricinvestigation, the parameters which affect the debonding process are analyzed. The paperaims to analyze the parameters which might be important with regards to debonding, e.g.the fracture energy of the steel-concrete interface, the fracture energy of the overlay, etc.

Paper VI proposes a test and inverse analysis to determine the fracture properties of strain hardening materials, in this case Engineered Cementitious Composites (ECC). Inorder to analyze the bridge deck based on fracture mechanics, constitutive parametersof the overlay material are of great importance. Fibre reinforced cement-based materialscan be categorized into two main groups, either tension softening or strain hardening

materials. A large amount of research has been carried out in the field of experimentaldetermination of fracture properties for tension softening materials. In the case of strainhardening materials as ECC, little research on testing methods to determine fractureproperties has been carried out. This paper contributes to research in standard fracturemechanical test methods for strain hardening materials.

Paper VII can be classified on the largest length scale dealing with structural design.This paper looks at a part of an orthotropic steel bridge deck reinforced with a cement-based overlay. The study investigates the performance of the reinforced bridge deck, andtakes into account temperature loads, traffic load, and shrinkage at early age. The study

is based on numerical analysis and takes into account results from papers on smallerlength scales. In particular Paper I , which is very important since it describes how theinterface between the cement-based overlay and orthotropic steel bridge deck is modeled.Furthermore, some of the phenomena described in Paper III and Paper V are importantand can directly be related to what is observed on the structural length scale.

Additional work has been carried out which is not included in the thesis. Paper (Walteret al. 2005) deals with interfacial mixed mode modeling. Most of this paper is reportedin appended Paper I , however some FE implementation and mixed mode debonding isdiscussed. Both papers (Walter et al. 2003) and (Walter, Gimsing & Stang 2004) includeresults, which are not presented in the appended papers. Both papers include experi-

mental studies on three dimensional composite plates. However, the results observed do





not differ significantly from what can be observed in experiments on the two dimensionalcomposite beams which have been analyzed experimentally in Paper III and numericallyin Paper V . Another published paper (Walter, Li & Stang 2004) has been extended and

published in the form of Paper III .




Interface Characterization 2.1 Studies on Interface Fracture in Mo de I

2.1 Studies on Interface Fracture in Mode I

The difference between interfacial steel-concrete fracture and concrete fracture can be

explained by the so-called wall effect , cf. Figure 2.1. As a crack propagates along a steel-concrete interface, the crack path is dominated by the presence of cement paste and lackof large aggregates which makes an interface fracture different from cracking observed inconcrete.

Figure 2.1 Schematic illustration of the wall-effect. A crack path (path 1) close to a steel wall might be dominated by matrix crack propagation where as in the case of crack propagation in the concrete material (path 2) crack propagation takes place in combination of aggregates and matrix.

The difference in concrete and steel-concrete interface fracture has been analyzed in Paper II . The test set-up applied is similar to the well-known Wedge Splitting Test (WST),originally proposed by Linsbauer & Tschegg (1986). The size and shape of the wedgesplitting specimen is the same as described by Bruhwiler & Wittmann (1990). The ideais to replace half of the specimen with a steel block, cf. Figure 2.2.

The specimen is placed on a linear support and two loading devices equipped with rollerbearings are placed on top of the specimen. A steel profile shaped as a wedge is placedbetween the bearings. Moving the actuator of the testing machine results in a splittingforce between the two bearings. The experimental results obtained in the test are load

versus the splitting force. As the goal of the study is to achieve a stress-crack openingrelationship, an inverse analysis is needed. This has been carried out utilizing the crackedhinge model originally proposed by Ulfkjær et al. (1995) and further developed by Olesen(2001). The advantage of this model is that it yields closed form analytical solutions,which can be implemented in an inverse analysis program. This was achieved by modifyingan already established inverse analysis program by Østergaard (2003). The model andinverse analysis has been calibrated and verified using finite elements. It can be concludedthat the inverse analysis to obtain the stress-crack opening relationship for a steel-concreteinterface using the wedge splitting test, is acceptable. The optimization strategy employedis always able to find the global minimum.

The proposed test and inverse analysis has been employed to investigate the fracture




2.1 Studies on Interface Fracture in Mo de I Interface Characterization

(a) (b)

Figure 2.2 (a) Geometry of the tested bimaterial WST specimen. The hatched part rep-resents a steel block. (b) Load configuration.

properties of a steel-concrete interface. For comparison with concrete fracture, full con-crete specimens were also tested and analyzed. Two batches were cast, in each batch,three composite and three full concrete specimens were cast. The steel surface was sand-blasted prior to casting. A self-compacting concrete was used, since it is believed that

a self-compacting feature enhances the steel-concrete bond. Vibration of the concretemight cause water to separate from the mix and create a weak interface. Comparisonbetween steel-concrete and concrete fracture can be viewed by comparing bilinear stress-crack opening relationships. The bilinear stress-crack opening relationships have beenobtained utilizing the cracked hinge model and are displayed in Figure 2.3.

0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1

w [mm]

σ ( w ) / f t [ M P a / M P a ]

0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1

w [mm]

σ ( w ) / f t [ M P a / M P a ]

InterfaceConcrete

InterfaceConcreteBatch 1 Batch 2

Figure 2.3 Stress-crack opening relationship for (a) batch no. 1 and (b) batch no. 2 determined using the inverse analysis.

It is observed from Figure 2.3 that a higher drop in the beginning of the σ − w curve is

present for the interface tests compared to the full concrete specimens. This is due to the




Interface Characterization 2.2 Studies on Mixed Mode Interface Fracture

wall effect , since the shape of the σ − w curve is influenced by the fracture strength of aggregates and matrix. Crack propagation in the composite specimen takes place in thematrix, whereas in the full concrete specimens cracking takes place in a combination of

aggregates/matrix. The higher stress drop in the beginning of the stress-crack openingrelationship might be due to the fact that the matrix is more brittle compared to theaggregates.

2.2 Studies on Mixed Mode Interface Fracture

Cracking between the overlay and underlying steel bridge deck would in many cases becharacterized by fracture in a combination of normal and shear stresses. This type of fracture is offend referred to as mixed mode fracture. Hence, pure Mode I cracking asstudied in the previous section might not be sufficient to describe the behavior. Together

with a brief summary of a mixed mode model applied in Paper I , experimental work onmixed mode testing for steel-concrete interfaces is presented. This has not, as far as theauthor is informed, been carried out on steel-concrete interfaces in the past.

Pure Mode I cracking can be modeled using the FCM. In order to compensate for theinfluence of shear on interface fracture, a mixed mode model based on the FCM, has beenstudied and applied. Previous modeling on discrete mixed mode fracture in cement-basedmaterials has been carried out by Loureno & Rots (1997) and later by Cervenka et al.(1998). These models have a limitation on the shape of the stress-crack opening relation-ship, and require the pure Mode II fracture energy as input, which is complex to measure

experimentally. The aim of the mixed mode studies is to have a model, which can be im-plemented into a commercial finite element program. Another demand is the possibilityto suply the model with constitutive parameters obtained in experiments. Therefore, themodel used in this thesis, based on a model first presented by Wernersson (1994), is easyto correlate with data obtained in experiments. The finite element implementation of themodel is described in (Walter et al. 2005), whereas Paper I gives a review of the modelalong with experimental studies. The implementation of the model in a commercial FEcode DIANA (2003) has been carried out, and is aimed at fracture mechanical studies onthe structural level.

2.2.1 Mixed Mode Model

When discussing mixed mode cracking in Linear Elastic Fracture Mechanics (LEFM), acommon parameter is the so-called phase angle ψk, as a function of Mode I and II stressintensity factors K I and K II

ψk = arctan

K II

K I

(2.1)

which is directly related to the stress state at the vicinity of the crack tip. Materials havebeen classified by He et al. (1990) using the phase angle. He related the phase angle to thetotal critical energy release rate, Gc, which shows two typical behaviors, which can define

the difference between ductile and brittle materials. It is stated that for brittle materials




2.2 Studies on Mixed Mode Interface Fracture Interface Characterization

the energy release rate increases significantly for increasing phase angle ψk, which meansthat it is weaker in Mode I than II. Concrete may be considered as a quasi-brittle materialcompared to e.g. steel. The ratio between the critical energy release rates in Mode I and

II, GII c/GIc , has for concrete been measured larger than 1 and in some cases around 5,see e.g. Carpenteri & Swartz (1991).

A stress intensity based failure criterion, was suggested by Wu (1967) in the form:

K I

K Ic

m

+

K II

K II c

n

≤ 1.0 (2.2)

where the exponents m and n are material constants. By using the present failure criterionit is possible to define a failure criterion based on the state of stress, e.g. the combination

of normal and shear stress. Carpenteri & Swartz (1991) utilized studies using the presentfailure criterion on concrete. However, a theory based on LEFM is not applicable in thecase of concrete due to its large fracture process zone, cf. Peterson (1981).

The aim of the mixed mode studies in the present thesis is to obtain experimental dataand implement them into a numerical model. Fracture behavior of a steel-concrete in-terface can be characterized as a discrete process, and the following model is targeted ata standard FE interface element. Configuration of a standard two-dimensional interfaceelement is shown in Figure 2.4.

Figure 2.4 A three node interface element and node stresses and displacements.

where σ and τ , are the normal and shear stresses, respectively, acting across the inter-face. The crack opening in normal and tangential direction are expressed by δ n and δ t,respectively. The 2-D relationship between the gradients of stress and crack opening in

the normal and tangential direction is given by:στ

=

D11 D12

D21 D22

δ nδ t

(2.3)

The Dij components describe the relation between gradients of stress and crack displace-ments. In pure elastic mode, no coupling is assumed between normal and shear modeand the off-diagonal terms are set to zero, D12 = D21 = 0. Furthermore, the diagonalelements D11 and D22, are assigned large values in the elastic state to model continuousgeometry. After peak stress, it is important to couple the two crack modes, Mode I andII. The situation where the off-diagonal terms are set to zero is equal to a situation with

two independent springs.





The coupling, after peak stress, is based on two given curves in pure Mode I and II, cf.Figure 2.5.

Figure 2.5 Curves in pure Mode I and II. Until peak stress linear elasticity is assumed.

The coupling between the pure Mode I and II curves, the current stresses (σ, τ ), andcrack deformations (δ n, δ t), can be found assuming the following criteria:

δ kn

δ max kn

m

+

δ kt

δ max kt

n

= 1.0 (2.4)

σk

= δ k

n

σmax k

δ max kn (2.5)

τ k = δ ktτ max k

δ max kt

(2.6)

where n and m are material constants. The kink points on the pure Mode I and IIcurves (e.g. (σmax1, δ max1

n ) with (τ max1, δ max1t )) are linked using Equations (2.4)-(2.6).

Thus, for any given mixed mode combination, the current stresses can be found usinglinear interpolation. Equation (2.4) can be plotted in a δ n − δ t space, cf. Figure 2.6(a).An example on how the coupling affects the σ-δ n curve, is shown in Figure 2.6(b). As

observed, two crack paths are considered, which can be defined by their mixed mode angleψ. The crack path with a mixed mode angle ψ1, is Mode I dominated compared to thecrack path defined by mixed mode angle ψ2. As a result, the σ-δ n curve diminish forincreased mixed mode angle, whereas the τ -δ t curve (not shown) expands for increasingmixed mode angle. From Figure 2.6, it is furthermore observed, that a mixed mode angleof 900 results in a situation where the Mode I contribution is equal to zero and a theMode II contribution is equal to the pure Mode II curve. Decreasing the mixed modeangle results in a situation where the Mode II contribution diminishes and the Mode Icontribution increases. When the mixed mode angle reaches 00 the Mode II contribution iszero and the Mode I contribution is equal to the pure Mode I curve. From the constructedsurfaces σ(δ n, δ t) and τ (δ n, δ t), it is now possible, via differentiation, to determine the Dij

elements in Equation 2.3.





Dilation effects are not considered in the present mixed mode formulation. In the presentapplication it is assumed that the effect of dilation is negligible, since interface cracks arecharacterized by low surface roughness and Mode I dominated crack growth.

(a) (b)

Figure 2.6 (a) Consider two crack paths in the δ t-δ n space, their (b) corresponding bilinear curve, e.g. the σ− δ n curve, changes as a function of the mixed mode angle ψ.

0

0.5

1

0

0.5

1

0

0.5

1

δn/ δ

n

max3[mm/mm]

δt/ δ

t

max3[mm/mm]

σ /

σ m a x 1

[ M

P a / M P a ]

0

0.5

1

0

0.5

1

0

0.5

1

δn/ δ

n

max3[mm/mm]

δt/ δ

t

max3[mm/mm]

τ / τ m a x 1

[ M

P a / M P a ]

Figure 2.7 Model visualization, for the case m=n=2 using bilinear softening relationships in pure Mode I and II. (a) Variation of the normal stress σ with respect to the normal and tangential displacements across the crack. (b) Variation of the

shear stress τ with respect to the normal and tangential displacements across the crack.

The interface response can also be visualized in a 3D surface. The visualization is carriedout in terms of a stress surface in a displacement space e.g. σ versus the normal andtangential crack openings, cf. Figure 2.7(a). Two curves, in pure Mode I and II, aregiven as input along with a value of the material constants, m = n = 2. The surfaceplot shows the amount of normalized normal stress σ, the z-axis, for various values of normal crack-opening δ n and tangential crack opening δ t. As observed in the figure, in

pure Mode I deformation, when δ t = 0, a full bilinear response σ − δ n is observed, which





is equivalent to the pure Mode I curve given as input. As the tangential crack openingis increased, the response on the normal stress diminishes to a minimum. Along withthe normal stress response, a diagram in Figure 2.7(b) displays the response of the shear

stress τ . It is clearly seen that the shear stress response is affected oppositely of the caseof normal stress.

2.2.2 Mixed Mode Experiments

In order to obtain experimental data for the mixed mode model, a set-up has been de-veloped. Full stress displacement curves from different mixed mode angles have to becollected and combined in accordance with the criterion in Equation (2.4). A simple testset-up, developed for a uniaxial testing machine, is presented here. The idea is to have asteel block with a plane rotated with a certain angle α, cf. Figure 2.8.

Figure 2.8 Schematic representation of the set-up to test an interface exposed to mixed mode loading. Only Mode I loading and displacement are recorded during the experiment.

As observed in the figure, by varying the inclination angle α, it is possible to test thesteel-concrete interface exposed to different mixed mode combinations of normal andshear stress. The experiment is carried out by gluing the concrete part to the top part of the loading device using a fast curing polymer. An important parameter, in order to carryout a stable test, is the stiffness of the system. Insufficient stiffness causes the two parts,the steel and concrete part, to rotate with respect to each other, producing non-physicalexperimental data. This phenomenon was first described by Hillerborg (1989). As therotational stiffness of the set-up is of crucial importance the proposed set-up is not capableof testing steel-concrete interfaces for large mixed mode angles. As the inclination angle

α of the specimen increases, the specimen becomes long and slender and consequently





very flexible, and hence unsuitable for testing. In the present study, the stiffness has beenmeasured for the specific set-up and the maximum testing angle α was found to be 30degrees.

A test program to obtain constitutive parameters for the mixed mode model was per-formed. A total number of nine specimens, three for each of the mixed mode angles 00,150, and 300, were tested. A two step inverse analysis to couple the experiments in theδ n − δ t space, was established. First step is to translate each experiment into a bilin-ear stress-crack opening relationship (for Mode I) and a bilinear stress-crack tangentialopening (for Mode II). This is shown for an experiment on a test with α=300, in Figure2.9.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Crack deformation δn,δ

t[mm]

S t r e s s σ , τ

[ M P a ]

Experimental results Mode II

Bilinear approximation Mode IIExperimental results Mode IBilinear approximation Mode I

Mixed Mode Angle ψ =300

Figure 2.9 Example on experimental data from a test with α = 300 and the approximation

of two bilinear curves in Mode I and II.

As displayed in Figure 2.9, an experiment on a specimen with an inclination angle of 30 0

produces a larger Mode I response compared to the Mode II response. The translation of the experimental data into a bilinear shape makes later numerical interpretation simple.The final step in the inverse analysis is to couple the bilinear curves obtained in the δ n−δ t

and σ−τ space. The kink points in the bilinear curves are coupled according to Equation

(2.4). When applying a bilinear shape, a total number of 4 kink points need to be coupled.Two stress kink points need to be coupled in the σ− τ space, and two crack deformationkink points need to be coupled in the δ n − δ t space. Coupling of the stress kink pointsare shown in Figure 2.10(a)-(b), and coupling of the deformation kink points are shownin Figure 2.11(a)-(b).

As observed in the figures, coupling of the experimental data is possible and acceptable.However, a large amount of scatter is present for the kink points in the deformationmeasurements, Figure 2.11(a), though the scale of the deformations has to be consideredwhen comparing to the deformations in Figure 2.11(b).

It should be noted that only reliable results are obtained from experiments on low mixed





mode angles, and via extrapolation of these, it is possible to derive the pure Mode IIcurve. The final results, which can be given as input in the constitutive model are: (i)a pure Mode I and II curve, and (ii) exponents m and n for use in the failure criterion

(Equation (2.4)). The values of the pure Mode I and II curves are shown in Table 2.1.

Mode I σmax1 [MPa] σmax2 [MPa] δ max2n [mm] δ max3

n [mm] Gf [n/mm]n-direction 3.0 0.4 0.02 0.5 0.12Mode II τ max1 [MPa] τ max2 [MPa] δ max2

t [mm] δ max3t [mm] Gf [n/mm]

t-direction 3.5 0.5 0.02 0.77 0.23

Table 2.1 Pure Mode I and II parameters (Figure 2.5), obtained in the inverse analysis.

The results in Table 2.1 are found for the exponents m = n = 2. Optimization of theexponents can not be justified on the small amount of data available and has not beenutilized in the present study. The two curves are found as two bilinear curves where a1and a2 are the slopes of the two line segments and b2 is the cross point of the second linesegment and the normalized stress axis (y-axis).

Since data has solely been collected for low mixed mode angles, extrapolated data forhigh mixed mode angles is less reliable. The ideal case, and a major improvement of thedata collected, is to carry out tests using a biaxial testing machine capable of changingthe ratio of shear and normal deformation. Then by testing different mixed mode angles

a full set of stress deformation curves in a mixed mode angle range from 00

to 900

couldbe collected.

0 1 2 3 40

1

2

3

4

σ [MPa]

τ [ M

P a ]

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

σ [MPa]

τ [ M

P a ]

Figure 2.10 (a) Plot of coupling stress kink points, ( σmax1, τ max1) (b) Plot of coupling stress kink points, ( σmax2, τ max2).





0 0.01 0.02 0.03 0.04 0.050

0.01

0.02

0.03

0.04

0.05

δn

[mm]

δ t

[ m m ]

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

δn

[mm]

δ t

[ m m ]

Figure 2.11 (a) Plot of coupling crack deformation kink points, ( δ max2n , δ max2

t ) (b) Plot of coupling crack deformation kink points, ( δ max3

n , δ max3t ).




Chapter 3

Composite Elements

An important issue in the cement-based overlay system for stiffening orthotropic steelbridge decks, is cracking of the overlay. The composite strength of the system is closelyrelated to cracking of the overlay, and subsequently cracking of the steel-concrete interface(debonding). Penetration of a vertical crack in the overlay, might for a certain crack width,initiate large stresses at the interface. Increased interfacial stresses will consequently leadto debonding. Using numerical tools can help identifying important parameters, whichinfluence the performance, with regards to cracking of a steel deck stiffened with a cement-based overlay. Through testing and numerical simulations it might be possible to identifythe significance of overlay and interface cracking in relation to the composite behavior.The major outcome of Paper V is a number of numerical parametric studies on differentconstitutive parameters of the overlay and interface with regards to the composite behaviorbetween a cement-based overlay and steel plate. Both Paper III and Paper IV containsnumerical and experimental studies of overlay fracture. In the present chapter, a shortoverview is given on numerical and experimental studies on small composite elements.The reader is referred to the appended papers for a review in greater details.

3.1 Numerical Studies on Composite Beams

The composite behavior between the overlay and a the steel plate can be analyzed using asimple three point bending test. Consider a composite beam exposed to negative bending

as shown in Figure 3.1. The composite beam is viewed as the very top part of the bridgedeck, turned up-side down for convenience. Cracking and debonding can be analyzed asdiscrete processes, vertical cracking in the overlay and horizontal cracking at the interface,respectively.

Loading of the composite beam will at some point cause cracking of the overlay as theoverlay reaches its tensile strength. As the vertical crack propagates through the overlay,its crack front will at some stage be opposed by the steel plate. The opposition of the steelplate will lead to an increase of normal stress in the plane perpendicular to the verticalcrack tip, i.e. in the plane of the steel concrete interface. The increase in horizontalstresses is likely to introduce cracking of the steel-concrete interface. This situation can

be analyzed using finite elements. Consider a close up look of the part where the overlay

25



Composite Elements 3.1 Numerical Studies on Composite Beams

Figure 3.1 Three point bending set-up: simulating a negative bending moment in a bridge deck.

crack initiates as sketched in Figure 3.2(a). Two situations are analyzed: (i) an overlay

Crack Mouth Opening Displacement (CMOD) of 0 mm, and (ii) an overlay crack openingof 0.03 mm. The shear and normal stress can be plotted along the interface to analyzethe problem, cf. Figure 3.2(b).

(a)

0 0.5 1 1.5 2−2

0

2

X−coordinate/hc

[mm/mm]

S t r e s s [ M P

a ]

σ, CMOD=0.03mm

σ, CMOD=0.00mm

τ, CMOD=0.00mm

τ, CMOD=0.03mm

(b)

Figure 3.2 Stress distribution along the interface for a CMOD value of zero and 0.03 mm. (a) Interfacial forces and configuration. (b) Stress distribution along the interface versus the x-coordinate normalized with the concrete height hc.Dashed lines represent shear stress τ and solid line represent the normal stress σ.

The first situation corresponds to a sound overlay with no cracking, whereas the secondsituation corresponds to the initiation of a small overlay crack. The interfacial stresses(normal, σ, and shear, τ ) can be plotted in a stress vs. x-coordinate diagram (x = 0is the location of the vertical overlay crack). It is observed from Figure 3.2(b), that theinterfacial stresses change dramatically for an increase of crack opening from 0 to 0.03

mm. The normal stresses change from compression to tension, which in many cases are




3.2 Test Methods to Obtain Constitutive Parameters of Overlay Composite Elements

critical for a concrete-steel interface.The debonding behavior can also be studied in terms of the overlay fracture energy. In-creasing the toughness of the overlay does not necessarily eliminate the effects of debond-

ing. This is shown in a FE study on a composite beam with three different overlaymaterials (denoted A1, A2 and A3). Each of the materials considered posses a differentfracture energy, Gf . The global behavior in terms of load versus crack mouth opening, isdisplayed, cf. Figure 3.3(a).

0 0.5 1 1.5 20

100

200

300

400

500

600

•

•

•

CMOD − Vertical crack [mm]

M o m e n

t [ k N m m ]

0 0.5 1 1.5 20

0.5

1

1.5

2


c r a c k l e n g t h

/ h

c [ m m / m m ] A2

Mat. A2, Gf=0.3 N/mm



Debondingstarts

A3

A1

Figure 3.3 Graphical representation of three hypothetical cases: —A1, -.-. A2, - - - A3.(a) The bending moment M = PL/4 versus the crack opening of the vertical crack - CMOD. (b) The interface crack length normalized with respect to the

overlay height hc versus CMOD.

Debonding starts approximately for the same vertical crack opening (CMOD) for eachof the materials A1-A3. However, for the material with the largest amount of fractureenergy A3, debonding initiates for a higher load level. Additionally, the debonding cracklength can be plotted as a function of the crack opening of the vertical crack (CMOD).This is illustrated in Figure 3.3(b). This clearly illustrates that the fracture energy of theoverlay hardly influences the relation between CMOD and the length of the interfacialcrack.

3.2 Test Methods to Obtain Constitutive Parameters of Overlay

An important issue in the investigation of the cement-based overlay as a stiffening systemto steel bridge decks are constitutive parameters. Standard test methods can be used,if available, to characterize materials used in experimental investigations on compositeelements.

The well-known advantage of fiber reinforced concrete is its ability to sustain larger de-formation after the first crack is formed. The fibers will typically stay unbroken aftercrack initiation and the fibers that cross a crack will resist further opening. Depending on

the so-called crack bridging effect , fiber reinforced composites can show different failure




Composite Elements 3.3 Verification of Numerical Modeling Tools

modes. If the fiber bridging effect is increasing during crack initiation and propagation,multiple cracks can form. This behavior is also known as strain hardening behavior. Onthe other hand, if the fibers cannot carry more load after the formation of the first crack,

then further deformation is characterized by opening of a single crack. This behavior is interms called tension softening. Experimental tests to obtain constitutive parameters forboth strain hardening and tension softening materials is an important issue with regardsto fracture mechanical modeling. To model cracking in the overlay, constitutive para-meters are important, and should reflect the material behavior observed in experiments.In this study, tension softening and strain hardening materials are analyzed as overlaymaterial. In the case of tension softening materials, well established test methods are athand, see e.g. (RILEM 2000) or (RILEM 2001). Furthermore, Østergaard (2003) madea full review and has compared standard test methods for tension softening materials.In the case of strain hardening materials, little or no work has been performed so far

to establish a standard test method. Paper VI contributes to the work in the field of atest method to achieve the constitutive strain hardening parameters, using a four-pointbending set-up.

3.3 Verification of Numerical Modeling Tools

Experimental tests on composite beams using both tension softening and strain hardeningmaterials have been performed. Numerical tools applied in this thesis are verified exper-imentally in Paper III , by comparing the modeling work to experimental data. Previousexperimental studies have been carried out on beams reinforced with concrete overlays.

Early studies by Silfwerbrand (1984), show results from tests on concrete beams withconcrete overlays. More recently, studies by Granju (1996), investigate the influence of fibers in cement-based overlays. Paper III contributes to these studies, and the majoroutcome of this paper is the treatment of the significant influence of overlay defects andtheir relationship to debonding.

Three different fiber reinforced composites have been tested as overlay material. Twotension softening materials: Fibre Reinforced Concrete (FRC), and Fibre ReinforcedDensit(FRD), as well as one strain hardening material, known as Engineered Cemen-titious Composites (ECC). The significant findings in this study are the experimental

and numerical investigations of macro crack formation in the overlay and its influenceon debonding. Furthermore, the verification of the numerical modeling work is carriedout by comparing FE results with experimental data. Two experimental set-ups, one forFRC and FRD beams, cf. Figure 3.4, and one for ECC beams, cf. Figure 3.5, have beenproposed. The experimental set-ups applied, have been utilized as the primary set-up totest tension softening and strain hardening materials in composite with steel.

The general experimental results of a test program is displayed in Figure 3.6. In the fig-ure, tests on FRC, FRD, and ECC composite beams are shown in a load vs. displacementdiagram. As observed, the composite beam with the ECC material exhibits a larger loadbearing capacity compared to FRC and FRD. Additionally, a second y-axis shows the

measured debonding signal. This shows clearly the significance of macro cracks and their




3.3 Verification of Numerical Modeling Tools Composite Elements

Figure 3.4 Experimental set-up for FRD and FRC composite beams. Since a single crack is formed at midspan a clip gauge is placed to measure the crack opening displacement (COD). The test set-up simulates a part of a stiffening overlay cast on a steel bridge deck loaded in negative bending.

Figure 3.5 Experimental set-up for ECC composite beams, the crack width of the overlay cracks are monitored and debonding is measured. The test set-up simulates a part of a stiffening overlay cast on a steel bridge deck loaded in negative bending.

Figure 3.6 Results from three representative tests of composite beams with FRD, FRC

and ECC overlay materials. Result are plotted in a load versus displacementdiagram with a second y-axis showing the corresponding debonding.




Composite Elements 3.3 Verification of Numerical Modeling Tools

initiation of debonding. The two tension softening materials, FRC and FRD, form onesingle crack and for deflection values of 2-3 mm, debonding is observed. The FRD mate-rial, which has a much stronger matrix than FRC, initiates debonding for approximately

for the same deflection value as for the FRC material. This concludes, that the governingmechanism for debonding is the magnitude of the crack width, also evident by numericalstudies as showed in the previous section. Furthermore, observing the experimental dataof the ECC composite beam, debonding initiates for a large beam deflection value com-pared to FRC and FRD. This is due to the fact that the ECC composite beam behavioris characterized by multiple cracking. The largest crack width, before localization in theECC beam, was measured using a micro camera in the range of 100 micron. In compar-ison, crack openings of the discrete crack in the FRD and FRC tests, had a value in therange of 0.1 to 0.2 mm at debond initiation. It should be noted that when comparingthese crack openings to the numerical crack opening in the previous section, debonding

was measured 50 mm from the point where the vertical crack propagates. For a morecomplete and detailed review of the experimental set-up, and how debonding is measured,the reader is referred to Paper III .

The numerical tools applied using the finite element method has been verified comparingnumerical results with experimental results. The comparisons are shown in Figures 3.7(a)-(b) and 3.8. As observed in the figures, the numerical results correlate well with theexperimental data. It should be noted that for the FRC and FRD beams, the comparisonsare shown in a load vs. Crack Opening Displacement (COD). When using numerical tools,it is possible to extract more information on the composite beam behavior than mightbe possible in the experiments. In the present case, it is possible via the numericalcalculation to extract information on the deformation state in the steel plate. As verylarge deformations are achieved, yielding of the steel plate starts (marked in Figure 3.8),before a macro crack localizes in the ECC composite beam.




3.3 Verification of Numerical Modeling Tools Composite Elements

0 0.2 0.4 0.6 0.8 1

0

1

2

3

COD [mm](a)

L o a d

[ k N ]

0 0.2 0.4 0.6 0.8 1

0

1

2

3

COD [mm](b)

L o a d

[ k N ]

FRC composite beam FRD composite beam

FE modelFE model

Figure 3.7 Comparison between experimental and numerical results. (a), and (b) show comparison between numerical and experimental results for a FRC, and FRD composite beam test, respectively.

0 10 20 300

1

2

3

4

5

6

7

Deflection [mm]

P [ k

N ]

Start yielding of steel plate

Crack localization

FE Model

ECC composite beam

Figure 3.8 Comparison between experimental and numerical results for an ECC composite beam test.




Composite Elements 3.4 Testing of Composite Plates

3.4 Testing of Composite Plates

In addition to the two-dimensional composite beam tests in negative bending, testing

of full three-dimensional composite plates have been performed. The primary set-up isa double-cantilever plate subjected to negative bending. The plate had the dimensions1 m x 2 m and an overlay thickness of 50 mm and a steel plate thickness of 8 mm.To ensure an interfacial bond between the overlay and steel plate, the steel surface wassand blasted prior to casting of the overlay. A test program of three composite plateswere carried out, each cast in separate batches. A self-compacting steel fiber reinforcedconcrete comprised of 1 % by volume hooked-end steel fibers was used. Fibres wereplaced in random orientation along with standard matrix components, cement, fly ash,micro silica, water and aggregates with a maximum diameter of 16 mm. To improvethe fresh properties of the mixture super-plasticizer was added. Mechanical properties

of the material were determined using the RILEM standard three-point bending test(RILEM 2000). The average ultimate tensile strength was determined to be 4.6 MPa.Furthermore, the ultimate average compressive strength, determined using cylindricalspecimens, was 47 MPa.

To test the composite plate in negative bending, a set-up is used where the middle partof the plate is placed on a fixed line support allowing rotation. Figure 3.9 shows the ideaof having a fixed and a free edge and a single load point P .

In order to test for different negative moment distribution, the set-up allows changingthe eccentricity of the load, denoted ξ . Furthermore, the set-up also allows testing for

different combinations between interfacial shear stresses and a negative bending moment.The actual set-up is depicted in Figure 3.9(b). The middle part of the specimen with thedimensions 1 m x 2 m is placed on a line support. A stiff crossbeam was used to applya force P on each side of the line support. To ensure a stable set-up, the crossbeam issupported using a spring on each side. To check the applicability of equal load on eachside of the composite plate, the elongation of the springs were monitored throughout eachexperiment.

The experiment is conducted with a 200 kN hydraulic testing actuator placed on thecrossbeam. The crossbeam distributes the total load 2P from the hydraulic actuator into

two load points P , acting on approximately 0.1 m x 0.1 m area on the concrete sideof the plate. To obtain an even reaction force on the line support, a soft material waslaid between the specimen and line support. The test was manually force-controlled, i.e.open-loop controlled. During the experiment, the deflection was recorded at the free edgein the line of the point load, using LVDTs.

Applying a point load P creates a bending moment along the clamped edge of the com-posite plate. The magnitude and distribution of the negative bending moment can becalculated in an elastic manner, using an analytical solution by Gere & Timoshenko(1999). The distribution of the elastic bending moment along the clamped edge is illus-trated in Figure 3.10. The magnitude of the bending moment is given as a ratio between

the moment and the point load. The maximum bending moment along the clamped edge




3.4 Testing of Composite Plates Composite Elements

(a) (b)

Figure 3.9 Experimental set-up. (a) Concept of experimental set-up, half composite plate

is subjected to a load point P with an eccentricity of ξ . (b) Experimental set-up in details, the composite plate is loaded using a double cantilever plate,the cross beam applies the two load points on each half of the composite plate.

(i.e. y/a=0) can be obtained for different ratios between the load eccentricity ξ and lengtha.

The three composite plates used in the test program, each had a different eccentricity assummarized in Table 3.1. For each test the maximum elastic bending moment has beencalculated as a function of the point load P according to the concept in Figure 3.10.

The results are presented graphically in two ways, cf. Figure 3.11. The magnitude of thepoint load P is plotted versus the maximum deflection, measured at the free edge on theload line, Figure 3.11(a). Furthermore, each test is also presented in a maximum elasticbending moment versus deflection diagram, cf. Figure 3.11(b).

All the tests showed a good and acceptable load bearing capacity and good ductility inthe nonlinear deformation range. For the eccentricities tested, the composite plate didnot have any problems sustaining the interfacial shear stresses between the overlay andsteel plate. Visible cracks, along the clamped support, were observed during testing.Cracks formed, in all test series, for a load magnitude close to the threshold part of

the load deflection diagram. Total failure of the composite plates are characterized by





Figure 3.10 The distribution of elastic bending moments along the clamped edge for var-ious ratios of the plate width a and load eccentricity ξ , after Timoshenko (1999).

Test No. Eccentricity ξ [mm] ξ/a [mm/mm] M [Nmm/mm]1 300 0.300 0.33P 2 250 0.250 0.32P 3 200 0.200 0.30P

Table 3.1 The eccentricity and maximum elastic bending moment M for each test.

the formation of a major crack along the clamped support, which eventually leads todebonding between the fiber reinforced overlay and underlying steel plate. Formation of a macro crack along the clamped support initiates normal tensile stresses at the interfacebetween the steel plate and overlay. Tensile stresses are in many cases severe for interfacialbond, but in the elastic state, the normal stresses at the interface are compressive.

Numerical simulation of the set-up and comparison to tests results are found in (Sigurdsson2003). The conclusion upon the numerical investigations is a good agreement betweenexperimental and numerical results. Similar modeling concepts were applied as describedin appended papers and in previous section for composite beam elements.




3.4 Testing of Composite Plates Composite Elements

Figure 3.11 Experimental results from each of the three tests. (a) Graphical representa-

tion of the three tests in a load-deflection diagram. (b) The same results plot-ted as maximum elastic bending moment vs. deflection. The maximum elastic bending moments is calculated using the concepts by Timoshenko (1999).




Chapter 4

Structural Behavior

The main purpose of the cement-based overlay is to stiffen the underlying orthotropicsteel bridge deck. The durability of the cement-based overlay is influenced by cracking of the overlay and cracking between the overlay and steel deck (debonding). Both crackingof the overlay and debonding can reduce the composite action between the steel deck andoverlay. In a design situation numerous effects, which can cause overlay cracking, haveto be taken into account. Apart from mechanical loading such as traffic, environmentalloading can have considerable influence on the overlay and its composite behavior withthe steel deck. Especially, shrinkage and temperature gradients have to be taken intoaccount. Paper VII is a purely numerical paper looking at a part of an orthotropicsteel bridge deck, using results from papers I-IV. The overlay system is investigated, forrealistic purposes, on an existing bridge. The Farø Bridges, located in southern Denmarklinking Copenhagen to the European Continent, are analyzed. The Farø Bridges werebuild between 1980 and 1985, and their orthotropic steel bridge decks are a part of a steelbox girder, which span 80 m, cf. Figure 4.1 for a cross-sectional view of the girder.

Figure 4.1 Cross-sectional view of steel girder on the Farø Bridges.

The bridge deck is designed with a 12 mm deck plate with trapezoidal ribs with a thicknessof 6 mm. The ribs are 300 mm high and placed with a center distance of 620 mm. Themain goal of the study is to analyze the performance of the overlay system in a real

size structure. In the steel box girder design, a number of bulkheads are placed every 4

37



Structural Behavior 4.1 Linear Elastic Studies

meters, cf. Figure 4.2. The bridge deck section close to the transverse bulkheads is of special concern since the overlay is subject to tension due to a negative bending moment.

Figure 4.2 Box girder, structural principle. Bulkheads every 4 m.

Studies on the performance of the overlay system are based on nonlinear fracture me-chanics utilized using finite elements. Global effects, such as traffic and dead load, aretaken into account. Emphasis is put on the situation where the cement-based overlay is

exposed to maximum negative bending, which is considered to be the most critical situ-ation. A local model of 3.9 times 8 meters, has been modeled in three dimensions usingthe software package DIANA (2003). The applied mesh is shown in Figure 4.3.

As illustrated, in addition to the orthotropic steel deck plate, a part of the bulkhead ismodeled as well. The steel and overlay parts have been modeled using standard 20-nodesolid elements. The connection between the overlay and underlying steel deck has beenmodeled using an 8-node interface element. The constitutive formulation of the interfaceis based on the mixed mode model as described in Chapter 2. The model presented inChapter 2 was presented in a two-dimensional configuration, however in the present case

a three-dimensional formulation is needed. The applied three dimensional formulation ispresented in details in Paper VII . Furthermore, global bending moments and shear forcesare found by using simple two-dimensional beam models. The exterior effects from trafficand dead load is then applied as boundary conditions. The magnitude of traffic load hasbeen chosen according to ENV 1991-3 (1991).

4.1 Linear Elastic Studies

Lack of transverse bending stiffness is one of the main problems causing fatigue in tra-ditional orthotropic steel bridge decks. Concentrated wheel loads induce considerabletransverse bending moments, which are critical to the connection between the steel plate

and rib. Consider a load system of a double-axle tandem with an axle load of 260 kN




4.1 Linear Elastic Studies Structural Behavior

Figure 4.3 Applied finite element mesh of the deck part. To model steel and overlay a solid 20-node element is used. The connection between the overlay and steel plate is modeled using an 8-node interface element.

together with global traffic and dead load. Figure 4.4, shows a deformation plot of a steelbridge deck (no overlay) exposed to the given loading system.

(a) (b)

Figure 4.4 Result from linear elastic FE calculations. (a) Wheel load causing transverse bending. (b) Three dimensional view of deformation caused by wheel load.

The figure illustrates the transverse deformations induced by a concentrated wheel load,which gives rise to high stresses in the intersection between the steel plate and rib. Thebenefits of the overlay system contra upgrading the steel deck by increasing the steel platethickness, can be shown through a parametric study. The parametric study investigatesthe influence of: (i) different steel plate heights, and (ii) different overlay height for a

steel deck thickness of 12 mm. In the study, full composite action is considered, thus no




Structural Behavior 4.1 Linear Elastic Studies

cracking of the overlay is modeled. The focus in the study is the maximum von Misesstress in the intersection between the steel plate and rib. The results from the linearelastic study is presented in Figure 4.5.

40 60 80 1000

100

200

300

Overlay thickness [mm]

0 10 20

0

100

200

300

tee p ate t c ness mm

v o n M i s e s s t r e s s [ M P a ]

Steel plate thicknessvariation (no overlay)

Overlay thicknessvariation, steel plate

thickness 12 mm

Figure 4.5 Result of linear elastic studies for traffic and an axle load of 260 kN. The leftmost line represents the von Mises stress of different steel plate thicknesses without an overlay. The other line represents results from a steel plate of 12 mm and different overlay thicknesses.

The graph shows the maximum von Mises stress in the intersection between the steel plateand rib for different orthotropic deck geometries. The first parametric study investigatesa steel deck without any overlay. In this study, the geometry of the Farø Bridge is applied,and the only modification to the original design is changing of the steel plate thickness. Asobserved in Figure 4.5, the von Mises stress has been calculated for steel plate thicknessesof 8, 12, 14, 18, and 20 mm. In the second parametric study, von Mises stresses have beencalculated in the similar connection, where the only modification to the original design isdifferent overlay thicknesses. The results show a reduction of the von Mises stresses for

increasing steel plate thickness. For an 8 mm plate the von Mises stress is around 200MPa, while for a 20 mm plate, it is reduced to around 25 MPa. However, comparing thestress level to that of the overlay system, it is observed that even applying a thin overlaycause a significant reduction in the von Mises stresses.

Stress reductions in the intersection of the rib and steel plate through linear elastic studiesand experiments have also been investigated by other authors. Numerical simulations onan orthotropic steel bridge deck have been reported by Buitelaar et al. (2004). This studyshows a stress reduction factor of 21 in the steel plate near the rib. A paper by Jong &Kolstein (2004), reports strain measurements on the Caland Bridge in Rotterdam, TheNetherlands, prior to and after upgrading the deck with a cement-based overlay. Four

weeks prior to the repair work, strain gauges at various locations collected data. After




Structural Behavior 4.2 Nonlinear Studies

4.1. The development of cracks for various stages of axle loads is illustrated in Figure4.6(a)-(d).

Figure 4.6 Crack patterns of the overlay (concrete) for traffic and an axle load of: (a) 50 kN, (b) 180 kN, (c) 260 kN, and (d) 290 kN.

As observed in the figure, the first cracks develop in the area on top of the bulkhead,denoted localized crack area no. 1, cf. Figure 4.6(a). Even for a small axle load of approximately 50 kN, the concrete overlay reaches a tensile value of 2 MPa. When furtherloading is applied, the maximum crack width is increased and more cracks develop. Atsome point, in this case for an axle load of approximately 290 kN, cracks develop in the

direction of the bridge axis. Cracking of the overlay in the direction of the bridge axis isdenoted crack area no. 2, cf. Figure 4.6(d).

The maximum crack width can for fixed traffic load, and variable tandem load, be viewedin a load vs. crack width diagram, cf. Figure 4.7.

The figure displays the axle load as a function of the maximum crack width for the fourmaterials considered: Concrete, FRC, FRD, and ECC. In the case of ECC, the resultsare presented as load vs. strain. In the load range considered, the maximum crack widthis always located in crack area no. 1 as defined in Figure 4.6(a). In addition to the crackwidth, initiation of cracking in crack area no. 2, as defined in Figure 4.6(d), is marked.

In the load range considered, only concrete and FRC initiates cracking in crack area no.




4.2 Nonlinear Studies Structural Behavior

0 0.02 0.04 0.06 0.08 0.10

100

200

300

400

500

600

700

•

•

•⊕

⊕

Crack width [mm] or Strain (ECC) in percent

A x l e

l o a d [ k N ]

• − Debonding initiates

⊕ − Crack area no.2 initiates

ConcreteFRC

FRD

ECC

Figure 4.7 Numerical results of three overlay materials for fixed traffic load and variable axle loading. The results are presented in a load vs. crack width diagram for concrete, FRC, and FRD. In the case of ECC material the result are presented as load vs. strain in percent.

2. In the case of tension softening materials, increasing crack width will at some pointlead to debonding. Debonding is initiated for a certain crack width and is also markedin Figure 4.7. It is noted that debonding is initiated for approximately the same crackwidth for: Concrete, FRC, and FRD.

Apart from traffic loading and dead load, effects such as shrinkage and temperature gra-dients might also have a significant influence on the overlay performance. These effectshave also been analyzed utilizing finite elements, and are included in Paper IV . These two

effects have a considerable influence on the cracking behavior and the stress state of theoverlay material.

In the case of temperature loading, cooling of the overlay (bottom warmer than top) isof special concern. Since the overlay is restrained from moving due to its bond withthe steel deck, cooling of the overlay results in a situation with tensile stresses. This, incombination with traffic load will consequently lower the load level where cracking anddebonding are initiated.

When using high performance concretes as overlay material, early age shrinkage can haveconsiderable influence on the cracking behavior. After casting of the overlay, the cement

starts to hydrate and the material will experience a macroscopic volume change, hence




Structural Behavior 4.2 Nonlinear Studies

internal stresses will develop in the overlay. This can, in worst case, lead to cracking of the overlay and needs to be controlled. Another aspect is the level of internal stresses,due to shrinkage of the overlay material, when the bridge is opened for traffic. Paper IV

shows modeling of shrinkage and creep in the overlay which aims to give a stress historyat early age.




Chapter 5

Conclusions

Fatigue damage of orthotropic steel bridge decks has achieved international attention inthe past years. The work within this thesis contributes to the ongoing research. Varioussystems to stiffening an orthotropic steel bridge deck have been proposed by several au-thors. A promising system is to use a cement-based overlay as investigated in this thesis.The strategy of the present study is to investigate the cement-based overlay system usingmulti scale modeling based on fracture mechanics. The study as a whole, can be dividedinto three length scales: (i) interface behavior, (ii) material interface interaction, and (iii)structural design.

On the steel-concrete interface length scale, a robust test method and inverse analysis toobtain Mode I fracture parameters for a steel-concrete interface has been established. A

modification of the well known Wedge Splitting Test (WST) has proven to be a good andreliable test method to investigate a steel-concrete interface. However, with regards tothe cement-based overlay system, cracking might not always occur in pure Mode I. Mixedmode cracking has been investigated theoretically as well as experimentally. A mixedmode model by Wernersson (1994), which is easy to implement into finite elements codes,showed good correlation between experiments and theory. However, the set-up developedin the present study is limited to tests on low mixed mode angles. Results on interfacefracture under high mixed mode angles have been extrapolated.

Studies on composite beams show that when a crack propagates through the overlay, the

interface stresses, at some point, change dramatically from compression to tension. Sincetension is critical to the interface, the change in interface stresses might at some pointlead to debonding. In the case of a composite beam with an overlay material such asECC subjected to negative bending, debonding is prevented as long as the material isin its hardening state. This is due to the fact that a material like ECC forms multiplecracks, and one single crack never evolves to a stage where the risk of debonding becomescritical. Experimental tests on small composite elements also served a good purpose inverifying the numerical tools applied in the thesis.

Linear elastic studies on a real size structure, utilizing the overlay system, show a con-siderable stress reduction in fatigue sensitive steel parts. Upgrading an orthotropic steel

bridge deck by adding an overlay might be more cost-effective than increasing the steel

45



Conclusions 5.1 Recommendations for Future Work

plate thickness. A nonlinear method to give an estimate on the performance, with regardsto cracking behavior, of an orthotropic steel bridge deck stiffened with a cement-basedoverlay, is demonstrated on a real size structure. The numerical investigation has been

carried out on the orthotropic steel bridge deck of the Farø Bridges located in Denmark.Effects such as traffic load, early age shrinkage and temperature gradients, have beentaken into account. Cooling of the bridge deck (bottom warmer than top), has a con-siderable influence on the cracking behavior. Since the overlay is restrained from movingdue to its bond with the steel plate, the cooling situation initiates internal stresses in theoverlay.

The current system with an adhesive connection between the overlay and steel deck de-pends highly on the interface and debonding would be unacceptable. However, the non-linear analysis shows, that cracking of the overlay might be unavoidable for the axle loads

found in codes. However, the solution might be durable by minimizing the maximumcrack width, and thereby avoiding debonding. Minimizing the crack width might be acentral issue when applying a cement-based overlay that exhibits localized cracking be-havior (a tension softening material). Since, a certain crack width, will at some point,lead to debonding. A more promising solution is to use a cement-based overlay that ex-hibits multiple cracking behavior (a strain hardening material). Since the study shows,that the overlay, in the load range considered, never reaches a crack width that initiatesdebonding.

5.1 Recommendations for Future Work

The outcome of the present study has contributed to the work within stiffening of anorthotropic steel bridge deck using a cement-based overlay. Looking at the thesis as awhole, the author feels that a theoretical background, to model a steel deck reinforcedwith a cement-based overlay has been established. However, one important aspect is fullscale testing. With all the important parameters pointed out in this study, it would beof importance to see whether it is possible to reproduce the full scale behavior using thenumerical tools and concepts described in this thesis. However, contributions are stillneeded for every part of this study. Numerical testing of interfacial mixed mode fracture,

presented in Paper I could well be improved. In the present study, a uniaxial testingmachine has been applied. It is not possible to test high mixed mode angle using theuniaxial set-up presented in this thesis. An improvement would be to use a biaxial testingmachine to test a steel-concrete interface exposed to different mixed mode angles. Byusing a biaxial testing machine, it would be possible to vary the amount of normal andshear stress, and thereby test the interface for higher mixed mode angles than possible inthe current uniaxial test set-up.

As shown in Paper IIV , temperature gradients and shrinkage can have major influenceon the composite behavior on the structural scale. In this thesis, no experimental workhas been carried out on shrinkage and temperature gradients and its influence on the

composite behavior.




5.1 Recommendations for Future Work Conclusions

Another important issue, which has not been analyzed in greater details in the presentthesis is fatigue. In order to give a full recommendation on the cement-based overlaysystem the effect of cyclic loading and its influence on the overlay crack width has to

be considered. Somehow, the influence of cyclic loading should be implemented in theconstitutive models to analyze the effect of fatigue. Cyclic loading will consequentlyincrease the maximum overlay crack width and thereby increase the risk of debonding.




Conclusions 5.1 Recommendations for Future Work




Bibliography

Dowling, P. J. (1968), The Behaviour of Stiffened Plate Bridge Deck under Wheel Loading,PhD thesis, Imperial College London.

ENV 1991-3 (1991), Eurocode 1 - Basis of Design and Actions on Structures, Part 3:Traffic Loads on Bridges , European Commitee for Standardisation.

Flint, A. R. & Smith, B. W. (1992), Strengthening and refubishment of servern crossingpart5: Other background research and development, in ‘Proc. Inst. Civil Engr.’,Vol. 94, pp. 51–60.

Gere, J. & Timoshenko, S. (1999), Mechanics of Materials , Stanley Thomes, Fourth SIedition.

Granju, J. L. (1996), ‘Thin bonded overlays: About the role of fiber reinforcement on the

limitation of their debonding’, Adv. Cement Based Mat 4(1), 21–27.

He, M., Cao, H. & Evans, A. (1990), ‘Mixed-mode fracture: The four point shear speci-men’, Acta Metal. Mater. 38, 839–846.

Hillerborg, A. (1989), ‘Stability problems in fracture mechanics testing’, Fracture of con-crete and rock: recent developments pp. 369–378.

Hillerborg, A., Moder, M. & Petersson, P. (1976), ‘Analysis of crack formation and crackgrowth in concrete by means of fracture mechanics and finite elements’, Cem. Concr.Res. 6(6), 773–782.

Jong, F. B. P. & Kolstein, M. H. (2004), Strenghening a bridge deck with high performanceconcrete, in ‘Orthotropic Bridge Conference, Sacremento, USA’, ASCE, pp. 328–347.

Jong, F. B. P., Kolstein, M. H. & Bijlaard, F. S. K. (2004), Strain measurement tests atorthotropic steel bridge decks with a heavy vehicle simulator, in ‘Prooceddings of the10th Nordic Steel Construction Conference, Copenhagen, Denmark’, pp. 401–412.

Kolstein, M. H. & Wardenier, J. (1997), ‘Stress reduction due to surfacing on orthotropicsteel decks’, IABSE Workshop, IABSE Reports, Vol. 76, Zurich, Lausanne .

Kolstein, M. H. & Wardenier, J. (1998), A new type of fatigue failures in steel or-thotropic bridge decks, in ‘Proceedings of the fifth Pacific Structural Conference,Korea’, pp. 483–488.

Kolstein, M. H. & Wardenier, J. (1999), Laboratory tests of the deckplate weld at theintersection of the through and the crossbeam of steel orthotropic bridge decks, in ‘Proceedings of the Eurosteel’, pp. 411–414.

Linsbauer, H. N. & Tschegg, E. K. (1986), ‘Fracture energy determination of concretewith cube shaped specimens (in german)’, Zement und Beton 31, 38–40.

Loureno, P. B. & Rots, J. G. (1997), ‘Multisurface interface model for analysis of masonry

structures’, Journal of Engineering Mechanics 123(7), 660–668.




Bibliography

Machida, F., Wakabayashi, N., Shimozato, T., Masui, T., Ono, S. & Miki, C. (2004),‘Orthotropic steel bridge decks study on fatigue improvement of weld connection totrough stiffeners in orthotropic steel bridge decks.’, International Institute of Weld-

ing, Document XII-2024-04 .

Olesen, J. F. (2001), ‘Fictitious crack propagation in fiber-reinforced concrete beams’,Journal of Engineering Mechanics 127(3), 272–280.

Østergaard, L. (2003), Early-Age Fracture Mechanics and Cracking of Concrete. Ex-periments and Modelling, PhD thesis, Department of Civil Engineering, TechnicalUniversity of Denmark, Lyngby, Denmark.

Peterson, P. (1981), Crack growth and development of fracture zones in plain concreteand similar materials, Technical report, Report TVBM-1006, Division of Building

Materials, Lund Institute of Technology.

RILEM (2000), ‘Test and design methods for steel fiber reinforced concrete. recommen-dations for the three point bending test’, Materials and Structures 33, 3–5. Preparedby RILEM-Committee-TDF-162, Chairlady L. Vandewalle.

RILEM (2001), ‘Test and design methods for steel fiber reinforced concrete. recommen-dations for uniaxial tension test’, Materials and Structures 34(3–6). Prepared byRILEM-Committee-TDF-162, Chairlady L. Vandewalle.

RILEM TC-108 (1996), Interfacial Transition Zone in Concrete , RILEM Report 11,

Chapman and Hall.

Sigurdsson, S. (2003), Composite bridge decks of sfrc/steel, Master’s thesis, Departmentof Civil Engineering, Technical University of Denmark, Lyngby.

Silfwerbrand, J. (1984), ‘Composite action between partially chipped concrete bridge deckand overlay’, Bulletin No. 142, Dept. of Structural Mechanics and Engineering, Royal Institute of Technology, Stockholm, Sweden (in swedish) p. 149.

Smith, J. W. & Bright, S. (2003), Upgrading orthotropic bridge decks with fiber reinforcedcomposites, in ‘High Performance Materials in Bridges’, pp. 463–472.

Smith, J. W. & Cullimore, M. S. G. (1987), Stress reduction due to surfacing on a steelbridge deck, in ‘Int. Conf. Steel and Aluminimum Structures, Cardiff’, pp. 806–816.

Ulfkjær, J., Krenk, S. & Brincker, R. (1995), ‘Analytical model for fictitious crack prop-agation in concrete beams’, Journal of Engineering Mechanics 121(1), 7–15.

Walter, R., Gimsing, N. & Stang, H. (2004), ‘Composite steel-concrete orthotropic bridgedeck’, 10th Nordic Steel Construction Conference, Copenhagen, Denmark, pp.519-530 .

Walter, R., Li, V. C. & Stang, H. (2004), ‘Comparison of frc and ecc in a composite bridge

deck’, 5th International PhD Symposium, Delft, The Netherlands, pp. 477-484 .




Bibliography

Walter, R., Olesen, J. F. & Stang, H. (2005), ‘Interfacial mixed mode model’, In: The 11th International Conference on Fracture, Turin, Italy .

Walter, R., Stang, H., Gimsing, N. J. & Olesen, J. F. (2003), ‘High performance compositebridge decks using scsfrc’, The Fourth International Workshop on High Performance Fiber Reinforced Cement Composites, Ann Arbor, Michigan pp. 495–504.

Wang, J. & Maji, A. K. (1995), Experimental studies and modeling of the concrete/rockinterface, in B. O. & W. M., eds, ‘Interface Fracture and Bond’, ACI SP-156, pp. 45–68.

Wernersson, H. (1994), ‘Fracture characterization of wood adhesive joints’, Report TVSM-1006, Lund University, Division of Structural Mechanics .

Wolchuk, R. (1963), Design Manual for Ortotropic Steel Plate Deck Bridges , AmericanInstitue of Steel Construction.

Wolchuk, R. (2002), ‘Structural behaviour of surfacing on steel orthotropic decks andconsiderations for practical design’, Structural Engineering International 2, 124–129.

Wu, E. (1967), ‘Application of fracture mechanics to anisotropic plates’, ASME Journal of Applied Mechanics 34, 967–974.




Part II

Appended Papers

53



Paper I —

Cohesive Mixed Mode Fracture Modelling and Experiments

Paper submitted to: Journal of Engineering Fracture Mechanics



C o h e s i v e M i x e d M o d e F r a c t u r e M o d e l l i n g

a n d E x p e r i m e n t s

R a s m u s W a l t e r * & J o h n F . O l e s e n

D e p a r t m e n t o f C i v i l E n g i n e e r i n g T e c h n i c a l U n i v e r s i t y o f D e n m a r k D K - 2 8 0 0 K g s .

L y n g b y , D e n m a r k , * e - m a i l : r w @ b y g . d t u . d k

P a p e r s u b m i t t e d t o J o u r n a l o f E n g i n e e r i n g F r a c t u r e M e c h a n i c s

A b s t r a c t

A n o n l i n e a r m i x e d m o d e m o d e l o r i g i n a l l y d e v e l o p e d b y W e r n e r s s o n ( 1 9 9 4 ) , b a s e d o n

n o n l i n e a r f r a c t u r e m e c h a n i c s , i s d i s c u s s e d a n d a p p l i e d t o m o d e l i n t e r f a c i a l c r a c k i n g i n

a s t e e l - c o n c r e t e i n t e r f a c e . T h e m o d e l i s b a s e d o n t h e p r i n c i p l e s o f H i l l e r b o r g s c t i t i o u s

c r a c k m o d e l , h o w e v e r , t h e M o d e I s o f t e n i n g d e s c r i p t i o n i s m o d i e d t a k i n g i n t o a c c o u n t

t h e i n u e n c e o f s h e a r . T h e m o d e l c o u p l e s n o r m a l a n d s h e a r s t r e s s e s f o r a g i v e n c o m b i n a -

t i o n o f M o d e I a n d I I f r a c t u r e . A n e x p e r i m e n t a l s e t - u p f o r t h e a s s e s s m e n t o f m i x e d m o d e

i n t e r f a c i a l f r a c t u r e p r o p e r t i e s i s p r e s e n t e d , a p p l y i n g a b i - m a t e r i a l s p e c i m e n , h a l f s t e e l

a n d h a l f c o n c r e t e , w i t h a n i n c l i n e d i n t e r f a c e a n d u n d e r u n i a x i a l l o a d . L o a d i n g t h e i n -

c l i n e d s t e e l - c o n c r e t e i n t e r f a c e u n d e r d i e r e n t a n g l e s p r o d u c e s l o a d - c r a c k o p e n i n g c u r v e s ,

w h i c h m a y b e i n t e r p r e t e d u s i n g t h e n o n l i n e a r m i x e d m o d e m o d e l . T h e i n t e r p r e t a t i o n o f

t e s t r e s u l t s i s c a r r i e d o u t i n a t w o s t e p i n v e r s e a n a l y s i s a p p l y i n g n u m e r i c a l o p t i m i z a t i o n

t o o l s . I t i s d e m o n s t r a t e d h o w t o p e r f o r m t h e i n v e r s e a n a l y s i s , w h i c h c o u p l e s t h e a s s u m e d

i n d i v i d u a l e x p e r i m e n t a l l o a d - c r a c k o p e n i n g c u r v e s . T h e i n d i v i d u a l l o a d - c r a c k o p e n i n g

c u r v e s a r e o b t a i n e d u n d e r d i e r e n t c o m b i n a t i o n s o f n o r m a l a n d s h e a r s t r e s s e s . R e l i a b l e

r e s u l t s a r e o b t a i n e d i n p u r e M o d e I , w h e r e a s e x p e r i m e n t a l d a t a f o r s m a l l m i x e d m o d e

a n g l e s a r e u s e d t o e x t r a p o l a t e t h e p u r e M o d e I I c u r v e .

K e y w o r d s N o n l i n e a r f r a c t u r e m e c h a n i c s , m i x e d m o d e f r a c t u r e , s t e e l - c o n c r e t e i n t e r f a c e .

1 I n t r o d u c t i o n

S i n c e H i l l e r b o r g e t a l . ( 1 9 7 6 ) i n t r o d u c e d t h e c t i t i o u s c r a c k m o d e l , d i s c r e t e M o d e I c r a c k -

i n g i n c o n c r e t e h a s b e e n t h e s u b j e c t o f i n t e n s i e d r e s e a r c h , w h i c h h a s d e m o n s t r a t e d t h e

u s e f u l n e s s o f t h e c o n c e p t s o f t h i s c o h e s i v e c r a c k m o d e l . T h e f o c u s o f t h i s s t u d y i s t h e

i n u e n c e o f s h e a r o n t h e p r o c e s s o f d i s c r e t e c r a c k i n g , w h i c h i n t h e t e r m i n o l o g y o f f r a c t u r e

m e c h a n i c s i s c a l l e d m i x e d m o d e c r a c k i n g . M i x e d m o d e c r a c k i n g w i l l b e t r e a t e d b o t h t h e -

o r e t i c a l l y a n d e x p e r i m e n t a l l y . I n t h e p r e s e n t s t u d y t h e m i x e d m o d e i n t e r f a c i a l c r a c k i n g o f

a s t e e l - c o n c r e t e i n t e r f a c e i s i n f o c u s , w h e r e t h e i n t e r f a c e i s d e n e d a s a r e g i o n o f c o n c r e t e

m o r t a r n e a r t h e b o u n d a r y b e t w e e n t h e t w o m a t e r i a l s . E x p e r i m e n t a l e x p e r i e n c e s h o w s

t h a t i n t e r f a c i a l c r a c k i n g o f a s t e e l - c o n c r e t e i n t e r f a c e u s u a l l y o c c u r s a t a c e r t a i n d i s t a n c e

1



f r o m t h e p h y s i c a l b o u n d a r y b e t w e e n t h e t w o m a t e r i a l s , c f . ( R I L E M T C - 1 0 8 1 9 9 6 ) . P h y s -

i c a l l y , t h e i n t e r f a c i a l t r a n s i t i o n z o n e b e t w e e n c o n c r e t e a n d s t e e l h a s a n i t e t h i c k n e s s o n

t h e m i c r o s c a l e , w h i c h i s r e l a t e d t o t h e p e n e t r a t i o n o f t h e c e m e n t p a s t e i n t o t h e r o u g h

s t e e l s u r f a c e . I n t h e p r e s e n t s t u d y i n t e r f a c i a l c r a c k i n g i s d e n e d a s t a k i n g p l a c e c l o s e t o

o r i n s i d e t h e i n t e r f a c i a l t r a n s i t i o n z o n e .

R e c o r d i n g s o f M o d e I b e h a v i o r o f s t e e l - c o n c r e t e i n t e r f a c e s h a v e a l r e a d y b e e n m a d e , s e e

e . g . ( W a l t e r e t a l . 2 0 0 5 ) . L e s s e x p e r i m e n t a l r e s e a r c h h a s b e e n c a r r i e d o u t o n m i x e d m o d e

c r a c k i n g o f c e m e n t - b a s e d i n t e r f a c e s , h o w e v e r , s e r v a l n u m e r i c a l m o d e l s t o d e s c r i b e i n t e r f a -

c i a l m i x e d m o d e c r a c k i n g b a s e d o n n o n l i n e a r f r a c t u r e m e c h a n i c s h a v e b e e n p r o p o s e d , s e e

e . g . L o u r e n ç o & R o t s ( 1 9 9 7 ) o r C e r v e n k a e t a l . ( 1 9 9 8 ) . I n t h e p r e s e n t s t u d y m i x e d m o d e

m o d e l l i n g i s b a s e d o n a m o d e l o r i g i n a l l y p r e s e n t e d b y W e r n e r s s o n ( 1 9 9 4 ) . T o i n t r o d u c e

t h e c o n c e p t s a n d i d e a s o f t h i s m i x e d m o d e m o d e l , a b r i e f s u m m a r y o f t h e c o n c e p t s o f

c o u p l i n g s t r e s s i n t e n s i t y f a c t o r s i n l i n e a r e l a s t i c f r a c t u r e m e c h a n i c s ( L E F M ) f o r m i x e d

m o d e l o a d i n g i s g i v e n . T h e s e c o n c e p t s a r e w e l l u n d e r s t o o d a n d t h e s u m m a r y e x p l a i n s

h o w s o m e o f t h e s e c o n c e p t s c a n b e a d o p t e d t o n o n l i n e a r c o h e s i v e c r a c k m o d e l l i n g . W h e n

d e s c r i b i n g m i x e d m o d e c r a c k i n g i n l i n e a r e l a s t i c f r a c t u r e m e c h a n i c s t h e p h a s e a n g l e ψk i s

i n t r o d u c e d , w h i c h i s a f u n c t i o n o f t h e M o d e I a n d I I s t r e s s i n t e n s i t y f a c t o r s K I a n d K II .

ψk = arctan

K II

K I

( 1 )

T h u s , t h e p h a s e a n g l e i s d i r e c t l y r e l a t e d t o t h e s t r e s s s t a t e i n t h e v i c i n i t y o f t h e c r a c k t i p .

R e l a t i n g t h e p h a s e a n g l e t o t h e t o t a l c r i t i c a l e n e r g y r e l e a s e r a t e s h o w s t w o t y p i c a l b e h a v -

i o u r s , w h i c h c a n e x p l a i n t h e d i e r e n c e b e t w e e n d u c t i l e a n d b r i t t l e m a t e r i a l s . M a t e r i a l s

h a v e b e e n c l a s s i e d b y H e e t a l . ( 1 9 9 0 ) u s i n g t h e p h a s e a n g l e . T h e r e i t i s s t a t e d , t h a t f o r

b r i t t l e m a t e r i a l s t h e e n e r g y r e l e a s e r a t e i n c r e a s e s s i g n i c a n t l y w h e n i n c r e a s i n g t h e p h a s e

a n g l e ψk , t h u s t h e m a t e r i a l i s w e a k e r i n M o d e I t h a n i n M o d e I I . T h e c r i t i c a l e n e r g y

r e l e a s e r a t e r a t i o b e t w e e n M o d e I I a n d I , GII c/GIc , f o r c o n c r e t e i s l a r g e r t h a n 1 a n d i n

s o m e c a s e s i t h a s b e e n m e a s u r e d i n t h e r a n g e o f 5 , s e e , e . g . C a r p e n t e r i & S w a r t z ( 1 9 9 1 ) .

S o i n t h i s r e s p e c t c o n c r e t e m a y b e c o n s i d e r e d a s a b r i t t l e m a t e r i a l , a n d c o n s e q u e n t l y a l s o

t h e s t e e l - c o n c r e t e i n t e r f a c e , s i n c e i n t e r f a c i a l f r a c t u r e i s r e l a t e d t o t h e b e h a v i o u r o f t h e

c o n c r e t e .

A s t r e s s i n t e n s i t y b a s e d f a i l u r e c r i t e r i o n , h a s b e e n s u g g e s t e d b y W u ( 1 9 6 7 ) i n t h e f o r m

K I

K Ic

m

+ K II

K II c

n

≤ 1.0 ( 2 )

w h e r e m, n a r e m a t e r i a l e x p o n e n t s . U s i n g t h i s c r i t e r i o n i t i s p o s s i b l e t o d e n e a f a i l u r e

c r i t e r i o n b a s e d o n t h e s t a t e o f s t r e s s , i . e . t h e c o m b i n a t i o n o f n o r m a l a n d s h e a r s t r e s s , s e e

f o r i n s t a n c e ( C a r p e n t e r i & S w a r t z 1 9 9 1 ) . H o w e v e r , L E F M i s n o t a p p l i c a b l e i n t h e c a s e o f

c o n c r e t e f r a c t u r e d u e t o i t s l a r g e f r a c t u r e p r o c e s s z o n e , c f . P e t e r s o n ( 1 9 8 1 ) . T h e r e f o r e ,

t h e p r e s e n t s t u d y a d o p t s s o m e o f t h e c o n c e p t s b y a p p l y i n g t h e s a m e f o r m o f i n t e r a c t i o n a s

g i v e n i n E q u a t i o n ( 2 ) . T h e p r e s e n t a n a l y s i s i s c a r r i e d o u t u s i n g c o h e s i v e c r a c k m o d e l l i n g

a n d i s n o n l i n e a r i n t h e s e n s e t h a t s o f t e n i n g a f t e r c r a c k i n i t i a t i o n i s i n c l u d e d . A s t h e

2



i n t e r f a c e h a s r e a c h e d i t s p e a k l o a d , e . g . i n u n i a x i a l t e n s i o n , t h e i n t e r f a c e d e g r a d a t i o n i s

n o t n e c e s s a r i l y a b r u p t a s i t m a y i n v o l v e a n a m o u n t o f e n e r g y d i s s i p a t i o n b e f o r e c o m p l e t e

s e p a r a t i o n .

I n t h e g e n e r a l t h e o r y o f e l a s t i c i t y , m a t e r i a l s c a n b e c l a s s i e d i n t w o m a i n g r o u p s : h y p e r -

a n d C a u c h y e l a s t i c m a t e r i a l s . T h e s t r a i n e n e r g y f u n c t i o n o f a h y p e r - e l a s t i c m a t e r i a l ,

ws =

σijdij , i s e q u a l t o a p o t e n t i a l . H e n c e t h e s t r a i n e n e r g y f u n c t i o n i s n o t d e p e n d e n t

o n t h e s t r a i n p a t h . C o n t r a r y t o t h i s e v e r y s t a t e o f s t r e s s f o r a C a u c h y m a t e r i a l i s u n i q u e

a n d d e n e d b y t h e s t r a i n p a t h . C r a c k i n g i n b o t h h y p e r - a n d C a u c h y m a t e r i a l s h a v e b e e n

m o d e l l e d b y s e r v a l a u t h o r s u s i n g n o n l i n e a r f r a c t u r e m e c h a n i c s . N e e d l e m a n ( 1 9 8 7 ) f o r m u -

l a t e d a c o h e s i v e c r a c k m o d e l t o s t u d y i n t e r f a c i a l d e b o n d i n g w i t h h y p e r - e l a s t i c m a t e r i a l

p r o p e r t i e s . H e d e r i v e d n o r m a l a n d s h e a r s t r e s s e s f r o m a n e l a s t i c p o t e n t i a l w h i c h o n l y

d e p e n d e d o n t h e n o r m a l a n d t a n g e n t i a l d i s p l a c e m e n t s .

A n o n l i n e a r i n t e r f a c i a l m i x e d m o d e m o d e l w i t h p a t h d e p e n d e n c y h a s b e e n d e n e d b y

( W e r n e r s s o n 1 9 9 4 ) , b a s e d o n c o h e s i v e c r a c k m o d e l l i n g . I n t h i s c a s e , c o n t r a r y t o t h e c a s e

o f a p o t e n t i a l , t h e t o t a l f r a c t u r e e n e r g y ,

Gf , i s d e n e d b y

Gf =

Γ

(σdδ n + τ dδ t) ( 3 )

w h e r e Γ i s t h e d e f o r m a t i o n p a t h t h a t r e s u l t s i n c o m p l e t e f a i l u r e o f t h e c o n s i d e r e d c r a c k .

A s r e p o r t e d b y s e r v a l a u t h o r s , s e e e . g . ( C e r v e n k a e t a l . 1 9 9 8 ) , a c e m e n t i t i o u s i n t e r f a c e

c a n b e d e s c r i b e d a s a p a t h d e p e n d e n t m e d i a , a n d d i e r e n t a m o u n t s o f f r a c t u r e e n e r g y

a r e c o n s u m e d i n t h e c a s e s o f p u r e M o d e I a n d I I f a i l u r e s . T h e s e o b s e r v a t i o n s s u p p o r t a

m o d e l t a k i n g i n t o a c c o u n t p a t h d e p e n d e n c y a n d c o h e s i v e c r a c k m o d e l l i n g w i t h s o f t e n i n g .

2 M i x e d M o d e M o d e l

T h e m i x e d m o d e m o d e l b y W e r n e r s s o n ( 1 9 9 4 ) i s b r i e y p r e s e n t e d h e r e . F o r a f u l l r e v i e w o n

t h e m i x e d m o d e m o d e l t h e r e a d e r i s r e f e r r e d t o t h e o r i g i n a l w o r k b y W e r n e r s s o n ( 1 9 9 4 ) .

T h e r e a s o n a n d m o t i v a t i o n f o r u s i n g t h e p r e s e n t m i x e d m o d e m o d e l i s t h e p o s s i b i l i t y

o f i n c l u d i n g t h e f r a c t u r e b e h a v i o u r w h i c h i s e x p e c t e d w h e n m o d e l l i n g a s t e e l - c o n c r e t e

i n t e r f a c e . T h e m a i n f e a t u r e s g o v e r n i n g t h e m i x e d m o d e b e h a v i o u r o f a s t e e l - c o n c r e t e

i n t e r f a c e a r e a s l i s t e d :

• D i s c r e t e c r a c k i n g ( c r a c k i n g a l o n g a n i n t e r f a c e )

• S t r e s s s o f t e n i n g ( i n c l u d e t e n s i o n a n d s h e a r s o f t e n i n g )

• M i x e d m o d e c r a c k i n g ( c o u p l i n g o f n o r m a l a n d t a n g e n t i a l c r a c k o p e n i n g )

• P a t h d e p e n d e n c y ( t h e a m o u n t o f f r a c t u r e e n e r g y c o n s u m e d d e p e n d s o n t h e f r a c t u r e

m o d e a n d p a t h , i . e . h o w M o d e I a n d I I a r e c o m b i n e d d u r i n g c r a c k i n g )

A l i m i t a t i o n t o t h e p r e s e n t s t u d y i s t h a t o n l y m o n o t o n i c l o a d i n g i s c o n s i d e r e d , t h u s n o

e e c t s f r o m c y c l i c l o a d i n g a r e t a k e n i n t o a c c o u n t . I n t h e c a s e o f u n l o a d i n g , t h e m o d e l

3



w i l l f o l l o w t h e s a m e p a t h a s f o l l o w e d d u r i n g l o a d i n g , h o w e v e r , t h i s i s a v i o l a t i o n o f t h e

e x p e c t e d m a t e r i a l b e h a v i o r , b u t a c c e p t e d a s a l i m i t a t i o n i n t h e p r e s e n t s t u d y s i n c e o n l y

m o n o t o n i c c r a c k g r o w t h i s c o n s i d e r e d . F i n a l l y , i t i s a s s u m e d t h a t n o c o m p r e s s i o n f a i l u r e

o c c u r s , t h e m i x e d m o d e m o d e l o n l y a p p l i e s t o t e n s i l e l o a d u n d e r t h e i n u e n c e o f s h e a r .

2 . 1 G e n e r a l I n t e r f a c e D e s c r i p t i o n

T h e n a l g o a l f o r t h e m i x e d m o d e m o d e l l i n g i s t h e i m p l e m e n t a t i o n o f a c o n s t i t u t i v e m o d e l

f o r d i s c r e t e c r a c k s i n a n i t e e l e m e n t c o d e . F o r i n s t a n c e t h e m i x e d m o d e m o d e l m a y b e

a p p l i e d i n a n i n t e r f a c e e l e m e n t . U s u a l l y , t o m o d e l a c o n t i n u o u s g e o m e t r y , a F E i n t e r f a c e

i s m o d e l l e d w i t h a t h i c k n e s s o f z e r o , a n d e v e r y n o d e i s a s s o c i a t e d w i t h a n o r m a l a n d a

t a n g e n t i a l d i s p l a c e m e n t a n d a s s o c i a t e d n o r m a l a n d s h e a r s t r e s s e s , c o r r e s p o n d i n g t o t h e

c r a c k o p e n i n g . T h i s i s i l l u s t r a t e d i n F i g u r e 1 s h o w i n g a t h r e e n o d e i n t e r f a c e e l e m e n t .

F i g u r e 1 : A t h r e e n o d e i n t e r f a c e e l e m e n t w i t h n o d e s t r e s s e s a n d d i s p l a c e m e n t s i n d i c a t e d

T h e r e l a t i o n s h i p b e t w e e n s t r e s s e s a n d c r a c k o p e n i n g i n p l a n e s t r a i n i s g i v e n b y t h e f o l -

l o w i n g e x p r e s s i o n

στ

=

D11 D12D21 D22

δ nδ t

( 4 )

w h e r e σ a n d τ a r e t h e n o r m a l a n d s h e a r s t r e s s , r e s p e c t i v e l y . T h e d i s p l a c e m e n t s δ n a n d δ sa r e t h e n o r m a l a n d t a n g e n t i a l d i s p l a c e m e n t , r e s p e c t i v e l y . T h e Dij c o m p o n e n t s r e l a t e t h e

s t r e s s e s t o t h e n o r m a l a n d t a n g e n t i a l d i s p l a c e m e n t , δ n a n d δ t , r e s p e c t i v e l y . I n p u r e e l a s t i c

m o d e n o c o u p l i n g i s a s s u m e d b e t w e e n n o r m a l a n d s h e a r m o d e a n d a s a c o n s e q u e n c e , t h e

o d i a g o n a l t e r m s a r e s e t t o z e r o , D12 = D21 = 0 . A f t e r p e a k s t r e s s i t i s i m p o r t a n t t o

c o u p l e t h e t w o c r a c k m o d e s , M o d e I a n d I I . A s i t u a t i o n w i t h t h e o d i a g o n a l t e r m s s e t t o

z e r o i s e q u a l t o h a v i n g t w o i n d e p e n d e n t s p r i n g s . T h e m o d e l d e r i v e d h e r e a l l o w s f o r t h e

i m p l e m e n t a t i o n o f a c o n s t i t u t i v e r e l a t i o n s h i p t o r e p r e s e n t a g e n e r a l i n t e r f a c e b e h a v i o u r .

2 . 2 C o u p l i n g o f M o d e I a n d I I

C o n s i d e r t w o s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p s , o n e f o r t h e n o r m a l o p e n i n g a n d a n o t h e r

f o r t h e t a n g e n t i a l d i s p l a c e m e n t o f t h e c r a c k : σ(δ n) a n d τ (δ t), v i s u a l i z e d i n F i g u r e 2 . T h e

c u r v e s a r e d e s c r i b e d i n t w o p a r t s , a n e l a s t i c a n d a n o n l i n e a r p a r t . T h e e l a s t i c p a r t i s

d e s c r i b e d a s t h e i n i t i a l a s c e n d i n g p a r t f r o m z e r o s t r e s s t o p e a k s t r e s s a n d i s c h a r a c t e r i z e d

b y a v e r y l a r g e s t i n e s s , D n a n d D t , t o m o d e l i n i t i a l c o n t i n u o u s g e o m e t r y o f t h e i n t e r f a c i a l

z o n e . T h i s p a r t h a s n o p h y s i c a l m e a n i n g b u t i s p u r e l y m o d e l l e d h a v i n g a s l o p e i n o r d e r f o r

4



t h e n u m e r i c a l s o l v e r a p p l i e d t o b e a b l e t o c o n v e r g e . I n t h i s , n o n - p h y s i c a l e l a s t i c s t a g e , t h e

t w o m o d e s a r e u n c o u p l e d . T h e p o s t p e a k b e h a v i o r i s d e s c r i b e d b y a d e s c e n d i n g , s o f t e n i n g

p a r t , w h i c h r e l a t e s t h e n o r m a l a n d s h e a r s t r e s s e s a c t i n g a c r o s s t h e c r a c k t o t h e n o r m a l

( δ n ) o r t a n g e n t i a l ( δ t ) o p e n i n g s , r e s p e c t i v e l y .

F i g u r e 2 : U n i a x i a l s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p s i n ( a ) p u r e M o d e I a n d ( b ) p u r e

M o d e I I . T h e c u r v e s a r e d e s c r i b e d b y a s t i , l i n e a r a s c e n d i n g p a r t u n t i l p e a k s t r e s s a n d

a m u l t i l i n e a r p o s t p e a k s o f t e n i n g p a r t .

E a c h l i n e a r s e g m e n t o f t h e c o m p l e t e s t r e s s - c r a c k d e f o r m a t i o n r e l a t i o n s h i p i s t r e a t e d i n -

d i v i d u a l l y . T h e i n d i c e s ( k ) a n d ( k + 1 ) r e f e r t o t w o s u c c e s s i v e k i n k p o i n t s , w h i c h d e n e a

l i n e a r s e g m e n t o n t h e u n i a x i a l s o f t e n i n g c u r v e . T h e s u b s c r i p t ' m a x ' i s a p p l i e d a s n o t a t i o n

w h e n r e f e r r i n g t o t h e p u r e M o d e I a n d I I u n i a x i a l c u r v e s . M a x i m u m i s u s e d , s i n c e i t i s

a s s u m e d t h a t t h e m a x i m u m M o d e I a n d I I f r a c t u r e e n e r g y c o n s u m p t i o n o n l y c a n b e o b -

t a i n u n d e r p u r e M o d e I a n d I I u n i a x i a l c o n d i t i o n s , r e s p e c t i v e l y . T h e c o u p l i n g o f t h e t w o

M o d e I a n d I I u n i a x i a l s o f t e n i n g c u r v e s i s c a r r i e d o u t b y e x p a n d i n g t h e u n i a x i a l c u r v e s

i n t h e ( δ n − δ t ) - p l a n e . T h i s r e s u l t s i n a s i t u a t i o n w h e r e t h e n o r m a l a n d s h e a r s t r e s s e s

d e p e n d , n o t o n l y o n t h e i r c o r r e s p o n d i n g d i s p l a c e m e n t , b u t a l s o o n b o t h s h e a r a n d n o r m a l

o p e n i n g , a c c o r d i n g t o t h e f o l l o w i n g e q u a t i o n s :

σ(δ n, δ t) ( 5 )

τ (δ n, δ t) ( 6 )

I n o r d e r t o d e s c r i b e h o w t h e s t r e s s e s d e p e n d o n t h e r e l a t i v e d i s p l a c e m e n t δ n a n d δ t , i t

i s c o n v e n i e n t t o i n t r o d u c e a p o l a r c o o r d i n a t e s y s t e m . T h e m i x e d m o d e a n g l e

ψa n d

d i s p l a c e m e n t δ a r e g i v e n b y

ψ = arctan

δ tδ n

( 7 )

δ =

δ 2n + δ 2t ( 8 )

5



F r o m E q u a t i o n s ( 7 ) - ( 8 ) δ n a n d δ t m a y b e e x p r e s s e d i n t e r m s o f t h e m i x e d m o d e d i s -

p l a c e m e n t a n d a n g l e , δ a n d ψ . T h i s a l l o w s ( 5 ) a n d ( 6 ) t o b e w r i t t e n i n t e r m s o f δ a n d

ψ :

σ(δ, ψ) ( 9 )

τ (δ, ψ)( 1 0 )

A f a i l u r e i n p u r e M o d e I c o r r e s p o n d s t o δ t = 0 ⇒ ψ = 00a n d p u r e M o d e I I c o r r e s p o n d s

t o δ n = 0 ⇒ ψ = 900. T h e n o r m a l s t r e s s - c r a c k o p e n i n g a n d s h e a r s t r e s s - c r a c k t a n g e n t i a l

d i s p l a c e m e n t c u r v e s m a i n t a i n t h e i r s t e p w i s e l i n e a r s h a p e f o r c o n s t a n t m i x e d m o d e a n g l e

ψ . T h e c u r v e s v a r y s m o o t h l y a s a f u n c t i o n o f ψ , a c c o r d i n g l y , f o r i n c r e a s i n g v a l u e o f ψ ,

t h e n o r m a l s t r e s s - c r a c k o p e n i n g c u r v e w i l l d i m i n i s h f r o m a m a x i m u m a t ψ = 00, w h i l e

t h e s h e a r s t r e s s - c r a c k s l i d i n g c u r v e w i l l e x p a n d t o w a r d s a m a x i m u m f o r ψ = 900.

L e t

δ maxn

a n d

δ maxt

b e t w o s p e c i c v a l u e s c o r r e s p o n d i n g t o p u r e M o d e I a n d I I , a n d d e n o t e d

b y ' m a x ' s i n c e i t i s a s s u m e d t h a t t h e y a t t a i n m a x i m u m v a l u e i n u n i a x i a l r e s p o n s e . I n o r d e r

t o n d t h e c r a c k d i s p l a c e m e n t s δ n a n d δ t f o r a n y g i v e n m i x e d m o d e a n g l e t h e f o l l o w i n g

c r i t e r i o n i s a s s u m e d : δ n

δ maxn

m

+

δ t

δ maxt

n

= 1.0 ( 1 1 )

w h e r e m a n d n a r e m a t e r i a l p r o p e r t i e s . T h e f u n c t i o n i n e q u a t i o n ( 1 1 ) c a n b e p l o t t e d o v e r

t h e δ n− δ t a r e a . T h i s i s c a r r i e d o u t f o r t h e v a l u e s o f t h e c o e c i e n t s m = n = 2.0 a n d f o r

t w o u n i a x i a l k i n k p o i n t s d e n o t e d k a n d k + 1 , c f . F i g u r e 3

F i g u r e 3 : F o r a g i v e n m i x e d m o d e s t a t e t h e c r a c k o p e n i n g s δ n a n d δ t c a n b e f o u n d a s a

f u n c t i o n o f t h e m i x e d m o d e a n g l e ψ

6



A s s e e n f r o m F i g u r e 3 t h e c r a c k d i s p l a c e m e n t s δ n a n d δ t c a n b e r e l a t e d t o t h e d i s p l a c e m e n t

δ , a s d e n e d i n E q u a t i o n ( 8 ) b y δ n = δcos(ψ) a n d δ t = δsin(ψ) .

I n o r d e r t o d e n e a p r o p e r f u n c t i o n f o r δ , t h e f u n c t i o n δ (ψ)i s i n t r o d u c e d f o r t h e c a s e

w h e r e n = m:

δ (ψ) =

cosψ

δ maxn

m

+

sinψ

δ maxt

m− 1

m

( 1 2 )

A n e x p r e s s i o n f o r δ (ψ) i s e x t r a c t e d f o r t w o c a s e s , m e q u a l t o 1 a n d 2 .

δ (ψ) =δ maxt

ccosψ + sinψf or m = n = 1 ( 1 3 )

δ (ψ) =δ maxt

c2cos2ψ + sin2ψf or m = n = 2 ( 1 4 )

w h e r e c = δ t / δ n . T h e s t r e s s e s σ a n d τ , f o r a g i v e n m i x e d m o d e a n g l e , a r e r e l a t e d t o t h e

u n i a x i a l s t r e s s b y t h e r e l a t i o n

σ = σmaxδ (ψ)cos(ψ)

δ maxn

( 1 5 )

τ = τ maxk

δ (ψ)sin(ψ)

δ maxt

( 1 6 )

2 . 3 B i l i n e a r M o d e I a n d I I C u r v e s

A s s h o w n i n s e r v a l c a s e s , e . g . ( O l e s e n 2 0 0 1 ) , u s i n g t h e c t i t i o u s c r a c k m o d e l a p p l y -

i n g a b i l i n e a r s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p g i v e s a g o o d c o r r e l a t i o n b e t w e e n n u m e r i -

c a l / a n a l y t i c a l a n d e x p e r i m e n t a l r e s u l t s . A s t h i s m o d e l r e q u i r e s p u r e M o d e I a n d I I s o f t -

e n i n g r e l a t i o n s a c o n v e n t i o n a s s h o w n i n F i g u r e 4 i s u s e d . T h e s t r e s s - c r a c k d i s p l a c e m e n t

c u r v e s f o r p u r e M o d e I a n d I I a r e r e p r e s e n t e d i n a b i l i n e a r f o r m .

2 . 3 . 1 M o d e l V i s u a l i z a t i o n

A p p l y i n g t h e p r o p o s e d c o n v e n t i o n , t h e i n t e r f a c e r e s p o n s e c a n b e v i s u a l i z e d a s a 3 D s u r f a c e .

A c a s e s t u d y i s s h o w n u s i n g t h e f o l l o w i n g p a r a m e t e r s i n p u r e M o d e I a n d I I :

σf , τ f [ M P a ] an1, at1 [ m m

−1

] an2, at2 [ m m

−1

] bn2, bt2 GI f , GII

f [ n / m m ]

M o d e I 3 . 0 2 5 0 . 3 0 . 1 0 . 1 0

M o d e I I 4 . 0 2 5 0 . 1 0 . 1 0 . 2 7

T a b l e 1 : P a r a m e t e r s f o r m o d e l l i n g p u r e M o d e I a n d I I s o f t e n i n g c u r v e s u s e d i n c a s e s t u d y

T h e v i s u a l i z a t i o n i s c a r r i e d o u t i n t e r m s o f a s t r e s s s u r f a c e o v e r a d i s p l a c e m e n t a r e a , i . e .

σ v e r s u s t h e n o r m a l a n d t a n g e n t i a l c r a c k d i s p l a c e m e n t , c f . F i g u r e 5 ( a ) . T h e s u r f a c e p l o t

7



F i g u r e 4 : C o n v e n t i o n f o r b i l i n e a r d e s c r i p t i o n o f M o d e I a n d I I s o f t e n i n g r e l a t i o n s h i p s ( a )

s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p a n d ( b ) s t r e s s - c r a c k t a n g e n t i a l o p e n i n g r e l a t i o n s h i p

s h o w s t h e a m o u n t o f n o r m a l s t r e s s σ o n t h e z - a x i s f o r v a r i o u s v a l u e s o f n o r m a l c r a c k -

o p e n i n g δ n a n d t a n g e n t i a l c r a c k s l i d i n g δ t . A s o b s e r v e d i n t h e g u r e , i n p u r e M o d e I

d e f o r m a t i o n , w h e n δ t = 0, a f u l l b i l i n e a r r e s p o n s e σ − δ n i s r e t r i e v e d . A s t h e a m o u n t o f

t a n g e n t i a l c r a c k d i s p l a c e m e n t i s i n c r e a s e d , t h e n o r m a l s t r e s s d i m i n i s h e s t o a m i n i m u m .

A l o n g w i t h t h e n o r m a l s t r e s s r e s p o n s e , a d i a g r a m i n F i g u r e 5 ( b ) d i s p l a y s t h e r e s p o n s e

o f t h e s h e a r s t r e s s τ . I t i s s e e n t h a t t h e s h e a r s t r e s s r e s p o n s e i s a e c t e d i n a n o p p o s i t e

m a n n e r o f t h e n o r m a l s t r e s s , i . e . t h e s h e a r s t r e s s d i m i n i s h e s a s t h e n o r m a l c r a c k o p e n i n g

i n c r e a s e s .

F i g u r e 5 : M o d e l v i s u a l i z a t i o n , f o r t h e c a s e m = n = 2 u s i n g a u n i a x i a l b i l i n e a r s o f t e n i n g

r e l a t i o n s i n p u r e M o d e I a n d I I g i v e n i n T a b l e 1 . ( a ) V a r i a t i o n o f t h e n o r m a l s t r e s s σ w i t h

r e s p e c t t o t h e n o r m a l a n d t a n g e n t i a l d i s p l a c e m e n t s a c r o s s t h e c r a c k . ( b ) V a r i a t i o n o f t h e

s h e a r s t r e s s τ w i t h r e s p e c t t o t h e n o r m a l a n d t a n g e n t i a l d i s p l a c e m e n t s a c r o s s t h e c r a c k .

8



2 . 3 . 2 P a t h D e p e n d e n c y

A s a l r e a d y m e n t i o n e d i n t h e i n t r o d u c t i o n t h e m o d e l t a k e s i n t o a c c o u n t p a t h d e p e n d e n c y

a c c o r d i n g t o E q u a t i o n ( 3 ) . T h e a m o u n t o f f r a c t u r e e n e r g y r e q u i r e d t o s e p a r a t e t h e i n -

t e r f a c e t o g e t a s t r e s s f r e e c r a c k s u r f a c e i s d e p e n d e n t o n t h e p a t h , i . e . t h e a m o u n t o f

n o r m a l c r a c k o p e n i n g a n d t a n g e n t i a l c r a c k d i s p l a c e m e n t e x p e r i e n c e d a l o n g t h e p a t h t o a

s t r e s s f r e e c r a c k s u r f a c e . S t a r t i n g f r o m a f u l l y b o n d e d i n t e r f a c e e n d i n g u p w i t h a f u l l y

s e p a r a t e d i n t e r f a c e , c a n b e a c h i e v e d i n v a r i o u s w a y s . T o s e p a r a t e t h e f r a c t u r e s u r f a c e

i n p u r e M o d e I c r a c k i n g m a y c o n s u m e l e s s f r a c t u r e e n e r g y t h a n s e p a r a t i o n t h r o u g h p u r e

M o d e I I c r a c k i n g . F o r a c e m e n t i t i o u s i n t e r f a c e a p a t h d e p e n d e n c y m a y e x i s t . I n o r d e r t o

i l l u s t r a t e p a t h d e p e n d e n c y , p u r e M o d e I a n d I I p a r a m e t e r s a r e c h o s e n a c c o r d i n g t o T a b l e

1 . T h e p u r e M o d e I f r a c t u r e e n e r g y i s t a k e n t o b e s m a l l e r t h a n p u r e M o d e I I f r a c t u r e

e n e r g y .

I n o r d e r t o v i s u a l i z e t h e p a t h d e p e n d e n c y a n a c a d e m i c e x a m p l e o f t h r e e d i e r e n t p a t h s

e n d i n g a t t h e s a m e p o i n t , w i t h a f u l l y s e p a r a t e d , s t r e s s f r e e i n t e r f a c e , a r e c o m p a r e d i n

t e r m s o f f r a c t u r e e n e r g y c o n s u m e d . T h e t h r e e p a t h s a r e d e n o t e d A , B a n d C , c f . F i g u r e

( 6 ) . P a t h A i s a r a d i a l p a t h , w h i c h o p e n s i n a m i x e d m o d e , a n d i s c h a r a c t e r i z e d b y a

c o n s t a n t m i x e d m o d e a n g l e ψ . F r o m t h e s t a r t i n g p o i n t t o t h e e n d p o i n t , p a t h B f o l l o w s

a p a t h w i t h a m i x e d m o d e a n g l e o f 900

( p u r e M o d e I I ) u n t i l i t r e a c h e s t h e a m o u n t o f

t a n g e n t i a l c r a c k d i s p l a c e m e n t o f t h e e n d p o i n t . T h e n i t c h a n g e s i t s o p e n i n g m o d e t o p u r e

M o d e I u n t i l i t e n d s u p w i t h t h e s a m e a m o u n t o f n o r m a l a n d t a n g e n t i a l c r a c k o p e n i n g a s

p a t h A . T h e n a l p a t h C , l e a d s t o i n t e r f a c i a l s e p a r a t i o n i n a s e q u e n c e o p p o s i t e t h a t o f

p a t h B , s t a r t i n g i n p u r e M o d e I a n d e n d i n g w i t h p r o p a g a t i o n i n p u r e M o d e I I . F o r e a c h

p a t h A , B a n d C , t h e a m o u n t o f f r a c t u r e e n e r g y c o n s u m e d i s c a l c u l a t e d . T h i s i s c a r r i e d

o u t f o r v a r i o u s e n d p o i n t s , c h a n g i n g t h e m i x e d m o d e a n g l e ψ , s u c h t h a t ψ ∈ [0; π2

].

I n c a s e A , f o r e a c h m i x e d m o d e a n g l e c a l c u l a t e d , t h e t o t a l a m o u n t o f f r a c t u r e e n e r g y

c o n s u m e d c o n s i s t s o f a M o d e I a n d M o d e I I c o n t r i b u t i o n . I n F i g u r e 7 ( a ) t h e a m o u n t o f

M o d e I a n d M o d e I I e n e r g y c o n s u m e d , GI f a n d GII

f , r e s p e c t i v e l y , i s s h o w n a l o n g w i t h t h e

t o t a l f r a c t u r e e n e r g y a s a f u n c t i o n o f t h e m i x e d m o d e a n g l e ψ . A s e x p e c t e d , f o r a l o w

m i x e d m o d e a n g l e t h e t o t a l f r a c t u r e e n e r g y c o n s u m e d i s d o m i n a t e d b y M o d e I e n e r g y . T h e

o p p o s i t e i s t h e c a s e f o r a l a r g e m i x e d m o d e a n g l e , w h e n t h e f r a c t u r e e n e r g y i s d o m i n a t e d

b y M o d e I I f r a c t u r e e n e r g y . F u r t h e r m o r e , a s t h e m i x e d m o d e a n g l e i n c r e a s e s t h e t o t a l

a m o u n t o f e n e r g y c o n s u m e d i n c r e a s e s , w h i c h i s a c o n s e q u e n c e o f t h e p a r a m e t e r s g i v e n i n

T a b l e 1 , w h e n t h e p u r e M o d e I e n e r g y i s s m a l l e r t h a n t h e p u r e M o d e I I f r a c t u r e e n e r g y .

T h e t h r e e d i e r e n t p a t h s A , B a n d C , a n d t h e i r t o t a l a m o u n t o f f r a c t u r e e n e r g y c o n s u m e d

a r e c o m p a r e d a s a f u n c t i o n o f t h e m i x e d m o d e a n g l e , c f . F i g u r e 7 ( b ) . I n g e n e r a l p a t h B

c o n s u m e s t h e l e a s t a m o u n t o f e n e r g y a n d p a t h C t h e l a r g e s t , w h e r e a s p a t h A i s i n t e r m e -

d i a t e . T h i s c a n a g a i n b e e x p l a i n e d b y t h e p a r a m e t e r s g i v e n i n T a b l e 1 , s i n c e p a t h C i s

d o m i n a t e d b y M o d e I I c r a c k p r o p a g a t i o n i t c o n s u m e s m o r e f r a c t u r e e n e r g y t h a n t h e t w o

o t h e r p a t h s .

9



F i g u r e 6 : T h r e e p a t h s A , B a n d C t o s t u d y t h e p a t h d e p e n d e n c y o f t h e m o d e l . P a t h A i s

t h e r a d i a l p a r t l e a d i n g t o f u l l s e p a r a t i o n i n a m i x e d m o d e o p e n i n g , w h e r e a s t h e B a n d C

p a t h s o p e n s i n p u r e M o d e I a n d I I , a l l e n d i n g u p i n t h e s a m e e n d p o i n t . T h i s i s c a r r i e d

o u t f o r e a c h e n d i n g p o i n t , i . e . f o r ψ ∈ [0; π2

]

3 E x p e r i m e n t a l S e t - u p

A t e s t s e t - u p i s d e v e l o p e d w i t h t h e a i m o f o b t a i n i n g t h e c o n s t i t u t i v e p a r a m e t e r s f o r t h e

i n t e r f a c i a l m i x e d m o d e m o d e l a s d e s c r i b e d i n t h e p r e v i o u s s e c t i o n s . T h e s e t - u p w a s

d e s i g n e d f o r a s t e e l - c o n c r e t e i n t e r f a c e . I n o r d e r t o h a v e a g e n e r a l m o d e l c o v e r i n g a r a n g e

o f m o d e m i x i t y i t i s n e c e s s a r y t o m e a s u r e t h e s t r e s s - s e p a r a t i o n c u r v e f o r a n i n t e r f a c e

l o a d e d u n d e r v a r i o u s m i x e d m o d e a n g l e s . T h e e x p e r i m e n t a l l y m e a s u r e d c u r v e s m a y t h e n

b e i n t e r p r e t e d a c c o r d i n g t o t h e n o n l i n e a r m i x e d m o d e m o d e l .

M u c h w o r k h a s b e e n c a r r i e d o u t o n p u r e M o d e I c r a c k s i n f u l l c o n c r e t e s p e c i m e n s r e c o r d i n g

t h e c o m p l e t e s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p u s i n g d i e r e n t s e t - u p s , s e e e . g . ( Ø s t e r g a a r d

2 0 0 3 ) f o r a c o m p a r i s o n o f s t a n d a r d t e s t m e t h o d s . T h e a i m i n t h e p r e s e n t s t u d y i s t o

d e v e l o p a s i m p l e u n i a x i a l t e s t m e t h o d t o d e t e r m i n e t h e m i x e d m o d e b e h a v i o r o f a s t e e l -

c o n c r e t e i n t e r f a c e . T o c a p t u r e t h e w h o l e s t r e s s - d e f o r m a t i o n c u r v e o f b r i t t l e m a t e r i a l s ,

t h e s t i n e s s o f t h e m a t e r i a l t h a t s u r r o u n d s t h e f r a c t u r e z o n e i s o f c r u c i a l i m p o r t a n c e .

P e t e r s o n ( 1 9 8 1 ) s u g g e s t t h e f o l l o w i n g c r i t e r i o n t o b e m e t i n o r d e r t o p r o d u c e a s t a b l e

t e s t .

K s > −As

dσ(w)

dw( 1 7 )

w h e r e

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i s t h e s l o p e o f t h e σ − w c u r v e , w d e n o t e s M o d e I c r a c k o p e n i n g , a n d As i s

t h e c r o s s - s e c t i o n a l a r e a o f t h e f r a c t u r e z o n e . T h e s t i n e s s K s h o l d s a c o n t r i b u t i o n f r o m

t h e s p e c i m e n o u t s i d e t h e f r a c t u r e z o n e a n d t h e s t i n e s s o f t h e t e s t i n g m a c h i n e i t s e l f . A s

1 0



0 30 60 900

0.1

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0.3

0.4

0.5

Mixed Mode Angle ψ [Deg]

F r a c t u r e E n e r g

y [ N / m m ]

0 30 60 900.1

0.2

0.3

0.4

0.5

0.6

Mixed Mode Angle ψ [Deg]

F r a c t u r e E n e r g

y [ N / m m ]

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F i g u r e 7 : ( a ) F r a c t u r e e n e r g y o f p a t h A c o n s u m e d f o r d i e r e n t m i x e d m o d e a n g l e s ( b )

T o t a l f r a c t u r e e n e r g y c o n s u m e d c o m p a r e d f o r t h r e e d i e r e n t p a t h s A , B a n d C a s a

f u n c t i o n o f t h e m i x e d m o d e a n g l e .

t h e c r i t e r i o n s u g g e s t s a s m a l l s p e c i m e n , e . g . l o w h e i g h t a n d s m a l l a r e a , t e s t e d i n a v e r y

s t i m a c h i n e i s n e e d e d t o p e r f o r m a s t a b l e t e s t o n b r i t t l e m a t e r i a l s . H o w e v e r , t o o b t a i n

r e p r e s e n t a t i v e r e s u l t s f o r t h e s p e c i c m a t e r i a l a l o w e r b o u n d o n t h e s p e c i m e n s i z e e x i s t ,

s e e e . g . G u s t a f s s o n ( 1 9 8 5 ) f o r f u r t h e r d i s c u s s i o n s o n s p e c i m e n s i z e .

F u r t h e r m o r e , w h e n p e r f o r m i n g a u n i a x i a l t e s t , t h e r o t a t i o n a l s t i n e s s o f t h e m a c h i n e a n d

s p e c i m e n m u s t m e e t a m i n i m u m r e q u i r e m e n t . I n s u c i e n t r o t a t i o n a l s t i n e s s o f t h e s y s t e m

c a u s e s t h e t w o p a r t s o f t h e s p e c i m e n t o r o t a t e w i t h r e s p e c t t o e a c h o t h e r . T h i s b e h a v i o r

r e s u l t s i n a n u n e v e n s t r e s s d i s t r i b u t i o n o f t h e f r a c t u r e p r o c e s s z o n e . T h i s p h e n o m e n o n h a s

b e e n a n a l y z e d b y H i l l e r b o r g ( 1 9 8 9 ) a n d h e c o n c l u d e s t h a t i n s u c i e n t r o t a t i o n a l s t i n e s s

c a u s e s a n o n - p h y s i c a l p l a t e a u o n t h e s t r e s s - d e f o r m a t i o n c u r v e . T h i s p l a t e a u i s c a u s e d

b y i n c r e a s i n g e c c e n t r i c i t y o f t h e s y s t e m a s t h e s p e c i m e n s t a r t s t o b e n d . S i m i l a r s t u d i e s

s u p p o r t e d b y F E c a l c u l a t i o n s h a v e b e e n c a r r i e d o u t b y Ø s t e r g a a r d ( 2 0 0 3 ) . A c c o r d i n g

t o t h e s t u d i e s b y H i l l e r b o r g ( 1 9 8 9 ) r o t a t i o n a l i n s t a b i l i t y c a n b y a v o i d e d i f t h e s t i n e s s

o u t s i d e t h e f r a c t u r e z o n e K s f u l l l s t h e c r i t e r i o n i n e q u a t i o n ( 1 8 ) .

K s > I dσ(w)

dw( 1 8 )

H e r e I i s t h e m o m e n t o f i n e r t i a o f t h e c r o s s - s e c t i o n o f t h e f r a c t u r e p r o c e s s z o n e . B a s e d

o n t h e b e f o r e m e n t i o n e d c o n s i d e r a t i o n s a s e t - u p t o d e t e r m i n e t h e c o m p l e t e s t r e s s - c r a c k

n o r m a l / t a n g n t i a l o p e n i n g r e l a t i o n s h i p f o r a n i n t e r f a c e h a s b e e n d e v e l o p e d . T h e s e t - u p i s

p r e s e n t e d i n F i g u r e 8 , w h i c h s h o w s t h e i n t e r f a c e s p e c i m e n , a n d h o w i t i s m o u n t e d i n t h e

t e s t i n g m a c h i n e .

T h e i n t e r f a c e s p e c i m e n i s c h a r a c t e r i z e d b y t h e i n c l i n a t i o n a n g l e α. B y v a r y i n g t h e a n g l e

α i t i s p o s s i b l e t o t e s t t h e i n t e r f a c e e x p o s e d t o d i e r e n t m i x e d m o d e c o m b i n a t i o n s o f

n o r m a l a n d s h e a r s t r e s s . T h e e x p e r i m e n t i s c a r r i e d o u t b y g l u i n g t h e c o n c r e t e p a r t o f

t h e s p e c i m e n t o t h e t o p p a r t o f t h e l o a d i n g d e v i c e u s i n g a f a s t c u r i n g p o l y m e r i c a d h e s i v e .

1 1



F i g u r e 8 : S c h e m a t i c r e p r e s e n t a t i o n o f t h e s e t - u p t o t e s t a n i n t e r f a c e e x p o s e d t o m i x e d

m o d e l o a d i n g . O n l y M o d e I l o a d i n g a n d d i s p l a c e m e n t a r e r e c o r d e d d u r i n g t h e e x p e r i m e n t .

T h e l o a d i n g d e v i c e c o n s i s t o f t w o i n t e r c h a n g e a b l e s t e e l b l o c k s r i g i d l y c o n n e c t e d t o t h e

m a c h i n e .

T h e s p e c i m e n s i z e i s c h o s e n i n a c c o r d a n c e w i t h t h e c r i t e r i o n s d i s c u s s e d a b o v e . T h e

s p e c i m e n , i n c l u d i n g s t e e l a n d c o n c r e t e p a r t s , i s c u b e d a n d m e a s u r e s 7 5 x 7 5 m m

2w i t h a

n o t c h d e p t h o f 1 2 . 5 m m , l e a v i n g a f r a c t u r e a r e a o f 5 0 x 5 0 m m

2. T h e n o t c h a r e e s t a b l i s h e d

b y p r e v e n t i n g t h e c o n c r e t e f r o m b o n d i n g t o t h e s t e e l p a r t . T h i s i s e n s u r e d b y a s t r i p o f

p l a s t i c t a p e p r i o r t o c a s t i n g . T h e m a t e r i a l t e s t e d i s a s e l f - c o m p a c t i n g c o n c r e t e w i t h a

m a x i m u m a g g r e g a t e s i z e o f 1 6 m m . A c c o r d i n g t o ( A S T M C 1 9 2 - 9 6 1 9 9 6 ) t h e a r e a t e s t e d

i s g r e a t e r t h a n t h r e e t i m e t h e a g g r e g a t e s i z e a n d i s t h e r e b y a r e p r e s e n t a t i v e a r e a . T h e

p r e p a r a t i o n o f t h e s p e c i m e n s a r e c a r r i e d o u t b y r s t s a n d b l a s t i n g t h e s t e e l p a r t , t h u s

m i n i m i z i n g t h e r i s k o f d e f e c t s , a n d t h e n p o u r i n g t h e s e l f - c o m p a c t i n g c o n c r e t e .

T h e t e s t s a r e c o n d u c t e d i n a 1 0 0 k N e l e c t r o - m e c h a n i c a l 6 0 2 5 I n s t r o n t e s t i n g m a c h i n e

e q u i p p e d f o r c l o s e d - l o o p c o n t r o l . T h e d i s p l a c e m e n t a c r o s s t h e f r a c t u r e p r o c e s s z o n e i s

m e a s u r e d u s i n g t w o s t a n d a r d I n s t r o n c l i p g a u g e s w i t h a m a x i m u m t r a v e l l e n g t h o f 4 m m .

T h e s t i n e s s o f t h e m a c h i n e i n c l u d i n g t h e s e t - u p a n d t h e s p e c i m e n o u t s i d e t h e f r a c t u r e

z o n e h a s b e e n m e a s u r e d t o 0 . 6 4 k N / m . I n o r d e r t o e n s u r e n e c e s s a r y s t i n e s s t h r o u g h -

o u t t h e w h o l e e x p e r i m e n t s p e c i a l a t t e n t i o n h a s t o b e p a i d t o t h e f a s t e n i n g o f b o l t s i n

t h e s e t - u p . T h e b o l t s h a v e t o b e p r e s t r e s s e d , a s a m i n i m u m , w i t h a v a l u e h i g h e r t h a n

c o r r e s p o n d i n g t o t h e p e a k l o a d i n t h e s p e c i c e x p e r i m e n t . T o e n s u r e s t a b l e t e s t i n g w i t h

s u c i e n t r o t a t i o n a l s t i n e s s , a s s p e c i e d i n e q u a t i o n s ( 1 7 ) , ( 1 8 ) a n o t c h d e p t h o f 1 2 . 5

m m h a s p r o v e n s u c i e n t .

1 2



4 T e s t P r o g r a m

A t e s t p r o g r a m h a s b e e n c a r r i e d o u t c o n s i s t i n g o f s p e c i m e n s w i t h d i e r e n t i n c l i n a t i o n s o f

t h e i n t e r f a c e . T h e a n g l e o f i n c l i n a t i o n i s m e a s u r e d w i t h r e s p e c t t o a p l a n e n o r m a l t o t h e

a x i s o f l o a d i n g , a n d t h r e e d i e r e n t v a l u e s o f t h e i n c l i n a t i o n a n g l e w e r e t e s t e d : α = 00 , 150 ,

a n d 300

. F o r e a c h i n c l i n a t i o n o r m i x e d m o d e a n g l e , α, a t o t a l n u m b e r o f t h r e e s p e c i m e n s

w e r e t e s t e d . A l l s p e c i m e n s w e r e c a s t a c c o r d i n g t o t h e m i x d e s i g n s h o w n i n T a b l e 2 .

T a b l e 2 : M i x d e s i g n

M i x k g / m

3

C e m e n t ( P o r t l a n d , C E M I 5 2 . 5 ) 2 4 5

F l y a s h 9 4 . 5

S i l i c a f u m e 1 0 . 5

W a t e r 1 4 2 . 9

A i r e n t r a i n i n g a g e n t 0 . 4

P l a s t i c i z e r 4 . 2

S a n d , 0 0 - 0 4 m m 7 5 2 . 6

A g g r e g a t e s , 0 4 - 0 8 m m 4 5 0 . 6

A g g r e g a t e s , 0 8 - 1 6 m m 5 9 4 . 0

T h e s p e c i m e n s w e r e d e - m o l d e d a f t e r 2 4 H o u r s , c u r i n g o f t h e s p e c i m e n s w a s c o m p l e t e d i n

1 0 0 % h u m i d i t y a n d t e s t i n g w a s c a r r i e d o u t a f t e r 2 8 d a y s o f c u r i n g .

5 T e s t R e s u l t s a n d I n v e r s e A n a l y s i s

T h e r e c o r d i n g s d u r i n g a n e x p e r i m e n t c o n s i s t o f t h e d e f o r m a t i o n s i g n a l δ measured a n d t h e

l o a d s i g n a l P . T h e d e f o r m a t i o n m e a s u r e m e n t c a n b e t r a n s f o r m e d i n t o t h e c r a c k - o p e n i n g

v a l u e δ a n d f u r t h e r i n t o M o d e I a n d I I c o m p o n e n t s a s s k e t c h e d i n F i g u r e 9 ( a ) .

T h e m e a s u r e d d e f o r m a t i o n s i g n a l δ measured c o n s i s t o f t w o c o n t r i b u t i o n s , o n e d u e t o e l a s t i c

d e f o r m a t i o n s a n d o n e d u e t o t h e c r a c k o p e n i n g . O n l y r e c o r d i n g s i n t h e c r a c k e d s t a t e

o f t h e e x p e r i m e n t a r e o f i n t e r e s t . T h e c r a c k - o p e n i n g v a l u e δ a r e f o u n d b y s u b t r a c t i n g

t h e e l a s t i c r e s p o n s e f o u n d f r o m t h e o v e r a l l r e s p o n s e m e a s u r e d . T h i s m a y , a c c o r d i n g t o

Ø s t e r g a a r d ( 2 0 0 3 ) , b e d o n e i n t h e f o l l o w i n g w a y .

δ = δ measured(σ)− δ measured(σmax) +σmax − σ

K i( 1 9 )

H e r e K i i s t h e i n i t i a l e x i b i l i t y o f t h e s p e c i m e n , i n s i d e t h e m e a s u r e d l e n g t h , δ measured i s

t h e m e a s u r e d d e f o r m a t i o n a t t h e s t r e s s σ a n d δ measured(σmax) i s t h e m e a s u r e d d e f o r m a t i o n

a t p e a k s t r e s s σmax , c f . F i g u r e 9 ( b ) . T h e c r a c k - o p e n i n g v a l u e δ m a y t h e n b e s p l i t i n t o a

M o d e I a n d I I c o n t r i b u t i o n .

T h e r e m a y e x i s t n u m e r o u s s o l u t i o n s t o t h e p r e s e n t c o n s t i t u t i v e m o d e l f o r a s i n g l e s t r e s s -

c r a c k n o r m a l / t a n g e n t i a l o p e n i n g c u r v e . A n i n v e r s e a n a l y s i s i s n e e d e d t o o b t a i n r e l i a b l e

1 3



( a ) ( b )

F i g u r e 9 : ( a ) T r a n s f o r m i n g δ measured t o M o d e I a n d I I c o m p o n e n t s ( b ) T r a n s f o r m a t i o n o f

t h e d e f o r m a t i o n δ measured m e a s u r e d i n t h e e x p e r i m e n t t o t h e c r a c k - o p e n i n g m e a s u r e m e n t

δ .

r e s u l t s . I d e a l l y t h e m o d e l a s s u m e s t h e e x i s t e n c e o f a p u r e M o d e I a n d M o d e I I s t r e s s -

c r a c k n o r m a l / t a n g e n t i a l o p e n i n g r e l a t i o n s h i p a l o n g w i t h a f a i l u r e c r i t e r i o n i n t h e s h e a r

n o r m a l s t r e s s p l a n e . S i n c e i t s v e r y d i c u l t t o p e r f o r m a p u r e M o d e I I e x p e r i m e n t i t i s

d e m o n s t r a t e d h o w d a t a f r o m d i e r e n t m i x e d m o d e a n g l e s m a y b e u s e d t o e x t r a p o l a t e t h e

p u r e M o d e I I c u r v e .

T h e s t r a t e g y f o r o b t a i n i n g t h e n e c e s s a r y p a r a m e t e r s f o r t h e c o n s t i t u t i v e m i x e d m o d e

m o d e l c a n b e s u b d i v i d e d i n t o t w o s t e p s . A n a m o u n t o f e x p e r i m e n t s p e r f o r m e d u n d e r

s p e c i c m i x e d m o d e a n g l e s a r e o n l y a f r a c t i o n o f t h e w h o l e f a i l u r e s u r f a c e f o r ψ ∈ [0; π2

].

F i r s t s t e p i s t o a p p r o x i m a t e a b i l i n e a r s t r e s s - c r a c k d e f o r m a t i o n c u r v e f r o m e a c h e x p e r i -

m e n t . F i n a l l y , a l l t h e a p p r o x i m a t e d b i l i n e a r c u r v e s a r e c o m b i n e d , i n t h e δ n − δ t - s p a c e ,

a c c o r d i n g t o t h e f a i l u r e c r i t e r i o n g i v e n i n E q u a t i o n ( 1 1 ) . T h e i n v e r s e a n a l y s i s p r o c e s s i s

s h o w n s c h e m a t i c a l l y i n F i g u r e 1 0 .

F o r e a c h s t e p a n i n v e r s e a n a l y s i s i s r u n , c o r r e l a t i n g t h e e x p e r i m e n t a l d a t a t o t h e m o d e l

i n o r d e r t o o p t i m i z e t h e s o u g h t p a r a m e t e r s . A s s h o w n i n t h e g u r e , r s t s t e p i s t o

a p p r o x i m a t e e a c h e x p e r i m e n t w i t h t w o b i l i n e a r s t r e s s - c r a c k o p e n i n g c u r v e s , o n e a s n o r m a l

s t r e s s v s . n o r m a l o p e n i n g a n d a s e c o n d a s s h e a r s t r e s s v s . t a n g e n t i a l o p e n i n g . U s i n g t h i s

i n f o r m a t i o n , k i n k p o i n t s f o r e a c h c u r v e a r e u s e d i n s t e p 2 i n o r d e r t o d e n e a f a i l u r e

c r i t e r i o n i n t h e δ n − δ t s p a c e . B o t h s t e p s a r e e x p l a i n e d i n t h e f o l l o w i n g .

1 4



F i g u r e 1 0 : S c h e m a t i c r e p r e s e n t a t i o n o f t h e t w o s t e p i n v e r s e a n a l y s i s p r o c e d u r e

5 . 1 I n v e r s e A n a l y s i s S t e p 1

F i r s t s t e p i n t h e i n v e r s e a n a l y s i s i s t o a p p r o x i m a t e e a c h i n d i v i d u a l e x p e r i m e n t t o t w o

b i l i n e a r s t r e s s - c r a c k o p e n i n g c u r v e s . I t i s a s s u m e d t h a t a f t e r p e a k l o a d t h e m i x e d m o d e

a n g l e i n t h e f r a c t u r e p r o c e s s z o n e ψ i s c o n s t a n t a l o n g t h e c r a c k i n g p l a n e a n d e q u a l t o t h e

a n g l e α a s s h o w n i n F i g u r e 8 . I n t h i s c a s e t h r e e e x p e r i m e n t s h a v e b e e n p e r f o r m e d f o r

e a c h m i x e d m o d e a n g l e e . g . α ∈ [00, 150, 300] . A t o t a l n u m b e r o f 1 5 b i l i n e a r c u r v e s a r e

t h e n f o u n d , s i n c e a n e x p e r i m e n t w i t h α = 00o n l y p r o d u c e s o n e b i l i n e a r c u r v e .

T h e b i l i n e a r a p p r o x i m a t i o n s i n M o d e I a n d I I a r e o b t a i n e d i n t h r e e s t e p s , r s t δ , a s

e x p l a i n e d e a r l i e r , i s d e c o m p o s e d i n t o δ n a n d δ t a c c o r d i n g t o t h e a n g l e α. T h e m a x i m u m

s t r e s s a n d m a x i m u m c r a c k d e f o r m a t i o n m e a s u r e m e n t δ n o r δ t i s t h e n x e d i n t h e f u r t h e r

a n a l y s i s . T h e y r e p r e s e n t t h e s t a r t i n g a n d e n d i n g p o i n t s o f t h e t w o l i n e s e g m e n t s s o u g h t .

T h e t w o u n k n o w n s i n t h e t t i n g p r o c e d u r e s a r e h e n c e t h e s l o p e s o f t h e t w o l i n e s e g m e n t s ,

an1; at1 a n d an2; at2 , a s s h o w n i n F i g u r e 4 . T h e s e s l o p e s a r e c a l c u l a t e d b y m i n i m i z i n g

t h e d i e r e n c e b e t w e e n e x p e r i m e n t a l a n d m o d e l d a t a . T h e t w o s l o p e s i n t h e b i l i n e a r

r e l a t i o n s h i p c a n b e e s t i m a t e d b y o p t i m i z i n g e a c h o f t h e f o l l o w i n g e x p r e s s i o n s :

min(an1,an2)

Σ(Qn − Qn)2 ( 2 0 )

min(at1,at2)

Σ(Qt − Qt)2

( 2 1 )

1 5



α T y p e f t [MPa] a n1 [mm−1] a n2 [mm−1] b n2 [−] G

I f [N/mm]

0

0A v g . 3 . 0 2 4 8 . 6 0 . 2 6 0 . 1 2 0 . 1 2 1

0

0S t d e v . 0 . 1 3 5 . 1 0 . 1 3 0 . 0 2 0 . 0 1 7

1 5

0A v g . 2 . 9 3 6 0 . 7 0 . 1 3 0 . 1 0 0 . 1 0 2

1 5

0S t d e v . 0 . 0 6 8 . 0 0 . 0 5 0 . 0 5 0 . 0 4 6

3 0

0A v g .

2 . 7 9 4 4 . 7 0 . 4 0 0 . 0 8 0 . 0 7 7

3 0

0S t d e v . 0 . 1 5 1 7 . 4 0 . 2 4 0 . 0 2 0 . 0 2 0

T a b l e 3 : M o d e 1 b i l i n e a r p a r a m e t e r s a l o n g w i t h t h e i r s t a n d a r d d e v i a t i o n o b t a i n e d i n t h e

i n v e r s e a n a l y s i s a c c o r d i n g t o s t e p 1 . T h e a v e r a g e e s t i m a t e i s b a s e d o n t h r e e e x p e r i m e n t s

f o r e a c h m i x e d m o d e a n g l e .

w h e r e Qn a n d Qt a r e v e c t o r s c o n t a i n i n g e x p e r i m e n t a l v a l u e s f r o m M o d e I a n d M o d e I I

c o n t r i b u t i o n s . T h e v a l u e s Qn a n d Qt d e n o t e v e c t o r s w i t h v a l u e s b a s e d o n t h e e s t i m a t e d

p a r a m e t e r s . A n e x a m p l e o f a n a p p r o x i m a t e d b i l i n e a r s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p

r e l a t i o n s h i p f o r a n e x p e r i m e n t p e r f o r m e d w i t h a m i x e d m o d e a n g l e α = 300 a n d i t s

c o r r e l a t i o n t o t h e e x p e r i m e n t i n M o d e I a n d I I a r e s h o w n i n F i g u r e 1 1 .

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Crack deformation δn,δ

t[mm]

S t r e s s σ , τ

[ M P a ]

Experimental results Mode IIBilinear approximation Mode IIExperimental results Mode IBilinear approximation Mode I

Mixed Mode Angle ψ =300

F i g u r e 1 1 : E x a m p l e o n e x p e r i m e n t a l d a t a f r o m a t e s t w i t h α = 300 a n d t h e a p p r o x i m a t i o n

o f t w o b i l i n e a r c u r v e s i n M o d e I a n d I I .

T h e p r e l i m i n a r y r e s u l t s f r o m t h e i n v e r s e a n a l y s i s S t e p 1 a r e s u m m a r i z e d i n T a b l e s 3

a n d 4 . A s m e n t i o n e d e a r l i e r , f o r e a c h a n g l e α a t o t a l n u m b e r o f t h r e e e x p e r i m e n t s h a v e

b e e n c a r r i e d o u t . I n T a b l e 3 t h e a v e r a g e r e s u l t s f r o m t h e i n v e r s e a n a l y s i s a l o n g w i t h t h e

s t a n d a r d d e v i a t i o n s a r e g i v e n .

5 . 2 I n v e r s e A n a l y s i s S t e p 2

A f t e r r e t r i e v i n g a t o t a l n u m b e r o f 1 5 b i l i n e a r r e l a t i o n s h i p s t h e n a l s t e p i s t o c o u p l e t h e

c u r v e s i n t h e σ − τ a n d δ n − δ t s p a c e s . T h i s i s a l s o c a r r i e d o u t u s i n g a n i n v e r s e a n a l y s i s

1 6



α T y p e τ t [MP a] a t1 [mm−1] a t2 [mm−1] b t2 [−] G

II f [N/mm]

1 5

0A v g . 0 . 9 5 2 3 5 0 . 8 6 0 . 1 1 0 . 0 1 7

1 5

0S t d e v . 0 . 0 9 3 0 0 . 7 2 0 . 0 2 0 . 0 2 0

3 0

0A v g . 1 . 4 1 9 0 0 . 7 9 0 . 1 2 0 . 1 1 5

3 0

0S t d e v . 0 . 0 6 3 5 0 . 5 0 0 . 0 3 0 . 0 0 7

T a b l e 4 : M o d e 2 b i l i n e a r p a r a m e t e r s a l o n g w i t h t h e i r s t a n d a r d d e v i a t i o n o b t a i n e d i n t h e

i n v e r s e a n a l y s i s a c c o r d i n g t o s t e p 1 . T h e a v e r a g e e s t i m a t e i s b a s e d o n t h r e e e x p e r i m e n t s

f o r e a c h m i x e d m o d e a n g l e .

p r o c e d u r e b a s e d o n a n u m e r i c a l o p t i m i z a t i o n r o u t i n e . S i n c e t h e e x p e r i m e n t a l d a t a c o n s i s t

o f b i l i n e a r c u r v e s , t h e k i n k p o i n t s o n t h e c u r v e s h a v e t o b e c o u p l e d . A t o t a l n u m b e r o f

f o u r v a l u e s h a v e t o b e d e t e r m i n e d , t w o s t r e s s a n d c r a c k o p e n i n g k i n k p o i n t s , a c c o r d i n g

t o t h e e x p r e s s i o n s s h o w n i n ( 2 2 ) t o ( 2 5 ) .

min(σmax

1,τ max1

)Σ(Qs1 − Qs1)2 ( 2 2 )

min(σmax

2,τ max2

)Σ(Qs2 − Qs2)2 ( 2 3 )

min(δmaxn1

,δmaxt1

)Σ(Qw1 − Qw1)2 ( 2 4 )

min(δmaxn2

,δmaxt2

)Σ(Qw2 − Qw2)2 ( 2 5 )

H e r e

Qs1i s a v e c t o r c o n t a i n i n g k i n k p o i n t s a c c o r d i n g t o t h e b i l i n e a r r e l a t i o n s h i p s f o u n d

i n t h e p r e v i o u s i n v e r s e a n a l y s i s s t e p . E a c h o p t i m i z i n g s t e p i s t t e d u s i n g t h e c r i t e r i o n

i n E q u a t i o n ( 1 1 ) . T h e e x p o n e n t s h a v e b e e n x e d s u c h t h a t m = n = 2, w h i c h i s a n

a s s u m p t i o n . I t m i g h t b e p o s s i b l e t o o b t a i n d i e r e n t t s , u s i n g d i e r e n t e x p o n e n t s , h o w -

e v e r , t h e l i m i t e d e x p e r i m e n t a l d a t a d o e s n o t j u s t i f y a n o p t i m i z a t i o n o f t h e s e e x p o n e n t s .

A f u r t h e r a s s u m p t i o n i s t h e b i l i n e a r s h a p e o f t h e p u r e M o d e I a n d I I u n i a x i a l c u r v e s ,

w h e r e a s a d i e r e n t s h a p e m i g h t b e a b l e t o i m p r o v e t h e t . T h e o n l y u n k n o w n p a r a m e t e r s

w h i c h h a v e t o b e e s t i m a t e d a r e t h e k i n k p o i n t s o f t h e p u r e M o d e I a n d I I s t r e s s - c r a c k n o r -

m a l / t a n g e n t i a l o p e n i n g c u r v e s . A f u l l s e t o f c o n s t i t u t i v e p a r a m e t e r s c o n s i s t o f 8 v a r i a b l e s ,

w h e r e t w o a r e f o u n d i n e a c h o f t h e f o u r s u b s t e p s , u s i n g t h e e x p r e s s i o n s 2 2 t o 2 5

C o u p l i n g o f t h e s t r e s s k i n k p o i n t s c a n b e v i e w e d i n F i g u r e s 1 2 ( a ) a n d ( b ) , u s i n g t h e i n v e r s e

a n a l y s i s p r o c e d u r e f r o m e x p r e s s i o n s ( 2 2 ) - ( 2 5 ) . C o u p l i n g o f t h e t w o c r a c k o p e n i n g k i n k

p o i n t s a r e s h o w n i n F i g u r e s 1 3 ( a ) a n d ( b ) , w h i c h a r e f o u n d u s i n g o p t i m i z i n g e x p r e s s i o n s

( 2 4 ) - ( 2 5 ) .

A s s e e n f r o m t h e g u r e s , t h e c o u p l i n g r e l a t i o n s h i p i s c a p a b l e o f l i n k i n g t h e d i e r e n t k i n k

p o i n t s o b t a i n e d f r o m t h e b i l i n e a r c u r v e s . H o w e v e r , a l a r g e a m o u n t o f s c a t t e r i s o b s e r v e d

i n t h e c o u p l i n g o f t h e r s t c r a c k o p e n i n g k i n k p o i n t , c f . F i g u r e 1 3 ( a ) . T h e s c a l e b e t w e e n

t h e d i e r e n t g r a p h s , h a s t o b e c o n s i d e r e d w h e n c o m p a r i n g t h e c u r v e s .

1 7



F i g u r e 1 2 : ( a ) P l o t o f c o u p l i n g p e a k s t r e s s , (

σmax1 , τ max

1 ) ( b ) P l o t o f c o u p l i n g k i n k p o i n t

s t r e s s , ( σmax2 , τ max

2 ) .

T h e n a l r e s u l t s a n d o u t c o m e o f S t e p 2 i n t h e i n v e r s e a n a l y s i s i s t h e p u r e M o d e I a n d I I

u n i a x i a l s t r e s s - c r a c k o p e n i n g c u r v e s . T h e s e a r e o b t a i n e d s t r a i g h t f o r w a r d b y c o n v e r t i n g

t h e k i n k p o i n t s i n t o t h e c o n v e n t i o n s p e c i e d i n F i g u r e 4 . T h e r e s u l t s f o r t h e b i l i n e a r

p a r a m e t e r s a r e s h o w n i n T a b l e 5 .

σf , τ f [ M P a ] an1, at1 [ m m

−1] an2, at2 [ m m

−1] bn2, bt2 GI

f , GII f [ n / m m ]

M o d e I 3 . 0 1 5 9 0 . 2 8 0 . 1 3 0 . 1 2

M o d e I I 3 . 4 8 4 4 0 . 2 0 0 . 1 5 0 . 2 4

T a b l e 5 : P u r e M o d e I a n d I I u n i a x i a l p a r a m e t e r s o b t a i n e d i n t h e i n v e r s e a n a l y s i s , a c c o r d -

i n g t o F i g u r e 4 .

A s s e e n f r o m T a b l e 5 t h e f r a c t u r e e n e r g y o f t h e p u r e M o d e I I r e l a t i o n s h i p i s a p p r o x i m a t e l y

t w i c e a s l a r g e a s t h e p u r e M o d e I c u r v e . H o w e v e r , t h e r e s u l t s f r o m u s i n g t h i s e x p e r i m e n t a l

a p p r o a c h h a v e t o b e c o n s i d e r e d c a r e f u l l y . I t s h o u l d b e n o t e d t h a t n o e x p e r i m e n t a l d a t a

e x i s t f o r h i g h m i x e d m o d e a n g l e a n d a l l d a t a h e r e a r e e x t r a p o l a t e d f r o m e x p e r i m e n t s

w i t h s m a l l m i x e d m o d e a n g l e s . T h i s i s o n e o f t h e d r a w b a c k s u s i n g a u n i a x i a l s e t - u p a s

s h o w n i n t h i s p a p e r . T h e p r o b l e m i s t h a t i n o r d e r t o t e s t l a r g e m i x e d m o d e a n g l e s a h i g h

i n c l i n a t i o n a n g l e i s n e e d e d h e n c e a l o n g s p e c i m e n . A s a l r e a d y d i s c u s s e d e a r l i e r , t o p e r f o r m

s t a b l e t e s t o n b r i t t l e m a t e r i a l s t h e l e n g t h o f t h e s p e c i m e n i s a n i m p o r t a n t p a r a m e t e r . T o

t e s t a v e r y s l e n d e r s p e c i m e n m i g h t n o t b e p o s s i b l e d u e t o t h e l o w s t i n e s s t h a t i t p o s s e s .

M o r e i d e a l l y , a b i a x i a l t e s t i n g m a c h i n e w o u l d b e p r e f e r a b l e f o r o b t a i n i n g d a t a f o r l a r g e

m i x e d m o d e a n g l e s . T h i s i s r e c o m m e n d e d f o r f u r t h e r r e s e a r c h . T h e p r e s e n t r e s u l t s a n d

m e t h o d s h o u l d t o g e t h e r r e e c t t h e m e t h o d o f i n v e r s e a n a l y s i s a n d t h e p r o c e d u r e t o o b t a i n

c o n s t i t u t i v e p a r a m e t e r s f o r t h e n o n l i n e a r m i x e d m o d e m o d e l .

1 8



F i g u r e 1 3 : ( a ) P l o t o f c o u p l i n g c r a c k o p e n i n g s k i n k p o i n t 1 , ( δ max

n1 , δ maxt1 ) ( b ) P l o t o f

c o u p l i n g c r a c k o p e n i n g s k i n k p o i n t 2 , (

δ

max

n2 , δ

max

t2) .

6 C o n c l u s i o n s

A n o n l i n e a r m i x e d m o d e m o d e l r s t p r e s e n t e d b y W e r n e r s s o n ( 1 9 9 4 ) m i g h t b e a p p l i c a b l e

f o r t h e m o d e l l i n g a s t e e l - c o n c r e t e i n t e r f a c e e x p o s e d t o m i x e d m o d e l o a d i n g . I t m i g h t b e

i m p o r t a n t t o c a p t u r e t h e p h y s i c a l p a t h d e p e n d e n c y b e h a v i o r o b s e r v e d i n e x p e r i m e n t s o n

c e m e n t i t i o u s i n t e r f a c e s , i n n u m e r i c a l m o d e l s o f s t e e l - c o n c r e t e i n t e r f a c e s . P a t h d e p e n d e n c y

i s t a k e n i n t o a c c o u n t f o r t h e m i x e d m o d e m o d e l p r e s e n t e d , a n d i s i l l u s t r a t e d v i a a c a s e

s t u d y . A n e x p e r i m e n t a l s e t - u p w i t h t h e a i m t o c h a r a c t e r i z e t h e c o n s t i t u t i v e p a r a m e t e r s o f

a s t e e l - c o n c r e t e i n t e r f a c e e x p o s e d t o m i x e d m o d e l o a d i n g h a s b e e n p r e s e n t e d . T h e s e t - u p

i s s i m p l e a n d o n l y g i v e s a n i n i t i a l i d e a o f t h e c o n s t i t u t i v e p a r a m e t e r s o f a s t e e l - c o n c r e t e

i n t e r f a c e b u t m o r e s o p h i s t i c a t e d m e t h o d s a r e n e e d e d t o o b t a i n e d i m p r o v e d r e s u l t s . T h e

p r o b l e m i s t o e s t a b l i s h h i g h m i x e d m o d e a n g l e s i n t h i s s i m p l e u n i a x i a l c o n g u r a t i o n , w h i c h

r e q u i r e l o n g s p e c i m e n s t h a t w o u l d h a v e v e r y l o w s t i n e s s c a u s i n g u n s t a b l e t e s t i n g . I d e a l l y

m i x e d m o d e t e s t s c o u l d b e p e r f o r m e d i n a b i a x i a l t e s t i n g m a c h i n e r e c o r d i n g n o r m a l a n d

s h e a r l o a d i n g u n d e r d i e r e n t m i x e d m o d e a n g l e s . T e s t d a t a h a s o n l y b e e n o b t a i n e d f o r

s m a l l m i x e d m o d e a n g l e s , h o w e v e r , t h e i n v e r s e a n a l y s i s c a n e x t r a p o l a t e d a t a f o r M o d e I I

d o m i n a t e d f r a c t u r e b e h a v i o u r . T h e e x t r a p o l a t i o n i s b a s e d o n a t w o s t e p i n v e r s e a n a l y s i s ,

w h i c h i s b a s e d o n e x p e r i m e n t a l d a t a f r o m e x p e r i m e n t s u n d e r d i e r e n t m i x e d m o d e a n g l e s .

R e f e r e n c e s

A S T M C 1 9 2 - 9 6 ( 1 9 9 6 ) , ` S t a n d a r d p r a c t i c e f o r m a k i n g a n d c u r i n g c o n c r e t e t e s t s p e c -

i m e n s i n t h e l a b o r a t o r y ' , A m e r i c a n S o c i e t y o f T e s t i n g S t a n d a r d s ( A S T M ) , W e s t

C o n s h o h o c k e n , P e n n s y l v a n i a .

1 9



C a r p e n t e r i , A . & S w a r t z , S . ( 1 9 9 1 ) , F r a c t u r e M e c h a n i c s T e s t M e t h o d s f o r C o n c r e t e , C h a p -

m a n n a n d H a l l , C h a p m a n n a n d H a l l , c h a p t e r 3 , p p . 1 2 9 1 9 7 .

C e r v e n k a , J . , K i s h e n , J . M . C . & S a o u m a , A . E . ( 1 9 9 8 ) , ` M i x e d m o d e f r a c t u r e o f c e m e t i -

t o u s b i m a t e r i a l i n t e r f a c e s ; p a r t i i : N u m e r i c a l s i m u l a t i o n ' , E n g i n e e r i n g F r a c t u r e M e -

c h a n i c s 6 0

( 1 ) , 9 5 1 0 7 .

G u s t a f s s o n , P . ( 1 9 8 5 ) , F r a c t u r e m e c h a n i c s s t u d i e s o f n o n y i e l d i n g m a t e r i a l s l i k e c o n c r e t e ,

T e c h n i c a l r e p o r t , R e p o r t T V B M - 1 0 0 7 , D i v i s i o n o f B u i l d i n g M a t e r i a l s , L u n d I n s t i t u t e

o f T e c h n o l o g y .

H e , M . , C a o , H . & E v a n s , A . ( 1 9 9 0 ) , ` M i x e d - m o d e f r a c t u r e : T h e f o u r p o i n t s h e a r s p e c i -

m e n ' , A c t a M e t a l . M a t e r . 3 8 , 8 3 9 8 4 6 .

H i l l e r b o r g , A . ( 1 9 8 9 ) , ` S t a b i l i t y p r o b l e m s i n f r a c t u r e m e c h a n i c s t e s t i n g ' , F r a c t u r e o f c o n -

c r e t e a n d r o c k : r e c e n t d e v e l o p m e n t s p p . 3 6 9 3 7 8 .

H i l l e r b o r g , A . , M o d é e r , M . & P e t e r s s o n , P . ( 1 9 7 6 ) , ` A n a l y s i s o f c r a c k f o r m a t i o n a n d c r a c k

g r o w t h i n c o n c r e t e b y m e a n s o f f r a c t u r e m e c h a n i c s a n d n i t e e l e m e n t s ' , C e m . C o n c r .

R e s . 6 ( 6 ) , 7 7 3 7 8 2 .

L o u r e n ç o , P . B . & R o t s , J . G . ( 1 9 9 7 ) , ` M u l t i s u r f a c e i n t e r f a c e m o d e l f o r a n a l y s i s o f m a s o n r y

s t r u c t u r e s ' , J o u r n a l o f E n g i n e e r i n g M e c h a n i c s 1 2 3 ( 7 ) , 6 6 0 6 6 8 .

N e e d l e m a n , A . ( 1 9 8 7 ) , ` A c o n t i n u u m m o d e l f o r v o i d n u c l e a t i o n b y i n c l u s i o n d e b o n d i n g ' ,

J o u r n a l o f A p p l i e d M e c h a n i c s 2 9 , 5 2 5 5 3 1 .

O l e s e n , J . F . ( 2 0 0 1 ) , ` F i c t i t i o u s c r a c k p r o p a g a t i o n i n b e r - r e i n f o r c e d c o n c r e t e b e a m s ' ,

J o u r n a l o f E n g i n e e r i n g M e c h a n i c s 1 2 7 ( 3 ) , 2 7 2 2 8 0 .

Ø s t e r g a a r d , L . ( 2 0 0 3 ) , E a r l y - A g e F r a c t u r e M e c h a n i c s a n d C r a c k i n g o f C o n c r e t e . E x -

p e r i m e n t s a n d M o d e l l i n g , P h D t h e s i s , D e p a r t m e n t o f C i v i l E n g i n e e r i n g , T e c h n i c a l

U n i v e r s i t y o f D e n m a r k , L y n g b y , D e n m a r k .

P e t e r s o n , P . ( 1 9 8 1 ) , C r a c k g r o w t h a n d d e v e l o p m e n t o f f r a c t u r e z o n e s i n p l a i n c o n c r e t e

a n d s i m i l a r m a t e r i a l s , T e c h n i c a l r e p o r t , R e p o r t T V B M - 1 0 0 6 , D i v i s i o n o f B u i l d i n g

M a t e r i a l s , L u n d I n s t i t u t e o f T e c h n o l o g y .

R I L E M T C - 1 0 8 ( 1 9 9 6 ) , I n t e r f a c i a l T r a n s i t i o n Z o n e i n C o n c r e t e , R I L E M R e p o r t 1 1 ,

C h a p m a n a n d H a l l .

W a l t e r , R . , Ø s t e r g a a r d , L . , O l e s e n , J . F . & S t a n g , H . ( 2 0 0 5 ) , ` W e d g e s p l i t t i n g t e s t f o r a

s t e e l - c o n c r e t e i n t e r f a c e ' , J o u r n a l o f E n g i n e e r i n g F r a c t u r e M e c h a n i c s 7 2 ( 1 7 ) , 2 5 6 5

2 5 8 3 .

W e r n e r s s o n , H . ( 1 9 9 4 ) , ` F r a c t u r e c h a r a c t e r i z a t i o n o f w o o d a d h e s i v e j o i n t s ' , R e p o r t T V S M -

1 0 0 6 , L u n d U n i v e r s i t y , D i v i s i o n o f S t r u c t u r a l M e c h a n i c s .

2 0



W u , E . ( 1 9 6 7 ) , ` A p p l i c a t i o n o f f r a c t u r e m e c h a n i c s t o a n i s o t r o p i c p l a t e s ' , A S M E J o u r n a l

o f A p p l i e d M e c h a n i c s 3 4 , 9 6 7 9 7 4 .

N o t a t i o n

αi n t e r f a c e a n g l e o n m i x e d m o d e s p e c i m e n

δ m i x e d m o d e d i s p l a c e m e n t

δ n n o r m a l c r a c k o p e n i n g

δ t t a n g e n t i a l c r a c k o p e n i n g

δ measured m e a s u r e d d e f o r m a t i o n

δ maxnk k i n k p o i n t o n t h e p u r e M o d e I I u n i a x i a l s t r e s s - c r a c k n o r m a l o p e n i n g c u r v e

δ maxtk k i n k p o i n t o n t h e p u r e M o d e I I u n i a x i a l s t r e s s - c r a c k t a n g e n t i a l o p e n i n g c u r v e

Qi v e c t o r c o n t a i n i n g e x p e r i m e n t a l d a t a

ψm i x e d m o d e a n g l e

ψk p h a s e a n g l e o f s t r e s s i n t e n s i t y f a c t o r s

σn o r m a l s t r e s s

σf p e a k n o r m a l s t r e s s o n p u r e M o d e I u n i a x i a l s t r e s s - c r a c k n o r m a l o p e n i n g c u r v e

σmaxk k i n k p o i n t o n t h e p u r e M o d e I u n i a x i a l s t r e s s - c r a c k n o r m a l o p e n i n g c u r v e

τ s h e a r s t r e s s

τ f p e a k s h e a r s t r e s s o n p u r e M o d e I I u n i a x i a l s t r e s s - c r a c k t a n g e n t i a l o p e n i n g c u r v e

τ maxk k i n k p o i n t o n t h e p u r e M o d e I I u n i a x i a l s t r e s s - c r a c k t a n g e n t i a l o p e n i n g c u r v e

an1 s l o p e o f p u r e M o d e I u n i a x i a l s t r e s s - c r a c k n o r m a l o p e n i n g c u r v e

an2 s l o p e o f p u r e M o d e I u n i a x i a l s t r e s s - c r a c k n o r m a l o p e n i n g c u r v e

As a r e a o f f r a c t u r e z o n e

at1 s l o p e o f p u r e M o d e I I u n i a x i a l s t r e s s - c r a c k t a n g e n t i a l o p e n i n g c u r v e

at2 s l o p e o f p u r e M o d e I I u n i a x i a l s t r e s s - c r a c k t a n g e n t i a l o p e n i n g c u r v e

bn2 p a r a m e t e r o f p u r e M o d e I u n i a x i a l s t r e s s - c r a c k t a n g e n t i a l o p e n i n g c u r v e

bt2 p a r a m e t e r o f p u r e M o d e I I u n i a x i a l s t r e s s - c r a c k t a n g e n t i a l o p e n i n g c u r v e

Dij s t i n e s s c o m p o n e n t s f o r i n t e r f a c e d e s c r i p t i o n

Gf f r a c t u r e e n e r g y

GII f p u r e M o d e I I f r a c t u r e e n e r g y

GI f p u r e M o d e I f r a c t u r e e n e r g y

2 1



I m o m e n t o f i n e r t i a o f f r a c t u r e z o n e

K II M o d e I I s t r e s s i n t e n s i t y f a c t o r

K I M o d e I s t r e s s i n t e n s i t y f a c t o r

K i i n i t i a l e x i b i l i t y o f s p e c i m e n

K s s t i n e s s o f s e t - u p o u t s i d e t h e f r a c t u r e p r o c e s s z o n e

me x p o n e n t t o c o u p l e M o d e I a n d I I

n e x p o n e n t t o c o u p l e M o d e I a n d I I

Qi v e c t o r c o n t a i n i n g e s t i m a t e d d a t a

wM o d e I c r a c k o p e n i n g

2 2



Paper II —

Wedge Splitting Test for a Steel-Concrete Interface

Journal of Fracture Mechanics, 72(17), pp. 2565-2583, 2005.



W e d g e S p l i t t i n g T e s t f o r a S t e e l - C o n c r e t e I n t e r f a c e

R a s m u s W a l t e r * , L e n n a r t Ø s t e r g a a r d , J o h n F . O l e s e n & H e n r i k S t a n g


L y n g b y , D e n m a r k , e - m a i l : r w @ b y g . d t u . d k

J o u r n a l o f E n g i n e e r i n g F r a c t u r e M e c h a n i c s 7 2 ( 1 7 ) , p p . 2 5 6 5 - 2 5 8 3 .

A b s t r a c t

T h i s p a p e r p r e s e n t s a t e s t m e t h o d d e s i g n a t e d f o r t h e d e t e r m i n a t i o n o f t h e s t r e s s - c r a c k

o p e n i n g r e l a t i o n s h i p o f a s t e e l - c o n c r e t e i n t e r f a c e . T h e m e t h o d i s b a s e d o n t h e w e l l

k n o w n W e d g e S p l i t t i n g T e s t ( W S T ) , a n d i t i s i l l u s t r a t e d h o w t o o b t a i n t h e s t r e s s - c r a c k

o p e n i n g r e l a t i o n s h i p t h r o u g h a n i n v e r s e a n a l y s i s . T h i s i n v e r s i o n m e t h o d u t i l i z e s t h e

c r a c k e d h i n g e m o d e l , m o d i e d s u c h t h a t i t d e s c r i b e s t h e p r o b l e m a t h a n d . I n t h i s

p a p e r , p u r e c o n c r e t e a n d s t e e l - c o n c r e t e c o m p o s i t e s p e c i m e n s a r e t e s t e d a n d c o m p a r e d .

I t t u r n s o u t t h a t i n t e r f a c i a l c r a c k i n g o f a b i m a t e r i a l s p e c i m e n u s u a l l y b e h a v e s a s o n e

o f t h e p a r e n t m a t e r i a l s , i n t h i s c a s e c o n c r e t e . T h e s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p o f

b o t h t h e c o n c r e t e a n d b i m a t e r i a l s p e c i m e n s a r e o b t a i n e d t h r o u g h t h e p r o p o s e d i n v e r s e

a n a l y s i s . T h e r e s u l t s s h o w , t h a t i n t e r f a c i a l c r a c k i n g i s d o m i n a t e d b y t h e s o - c a l l e d w a l l -

e e c t a n d i t s b e h a v i o r c a n b e d e s c r i b e d a s q u a s i b r i t t l e . H o w e v e r , d u e t o t h e w a l l - e e c t ,

i n t e r f a c i a l c r a c k i n g i s m o r e b r i t t l e t h a n f o r t h e p u r e c o n c r e t e .

K e y w o r d s N o n l i n e a r f r a c t u r e m e c h a n i c s , s t e e l - c o n c r e t e i n t e r f a c e , i n v e r s e a n a l y s i s , b i -

m a t e r i a l W S T s p e c i m e n .


I n c o m p o s i t e s t r u c t u r e s a n d m a t e r i a l s , t h e w e a k e s t p a r t i s o f t e n t h e i n t e r f a c e b e t w e e n

d i e r e n t m a t e r i a l s . S o , n a t u r a l l y , a l o t o f a t t e n t i o n h a s b e e n p a i d t o t h e u n d e r s t a n d i n g o f

t h e c h a r a c t e r i s t i c s o f b i m a t e r i a l i n t e r f a c e s . I n t h e p r e s e n t s t u d y a s t e e l - c o n c r e t e i n t e r f a c e i s

t h e f o c u s a n d i s d e n e d a s t h e r e g i o n o f t h e c o n c r e t e m o r t a r n e a r t h e b o u n d a r y b e t w e e n t h e

t w o m a t e r i a l s . E x p e r i m e n t a l e x p e r i e n c e s h o w s t h a t i n t e r f a c i a l c r a c k i n g o f a s t e e l - c o n c r e t e

i n t e r f a c e u s u a l l y o c c u r s a t a c e r t a i n d i s t a n c e f r o m t h e p h y s i c a l b o u n d a r y . P h y s i c a l l y , t h e

i n t e r f a c i a l t r a n s i t i o n z o n e b e t w e e n c o n c r e t e a n d s t e e l h a s a n i t e t h i c k n e s s o n t h e m i c r o

s c a l e , w h i c h i s r e l a t e d t o t h e p e n e t r a t i o n o f t h e c e m e n t p a s t e i n t o t h e r o u g h s t e e l s u r f a c e ,

c f . ( R I L E M T C - 1 0 8 1 9 9 6 ) . I n t h e p r e s e n t s t u d y i n t e r f a c i a l c r a c k i n g i s d e n e d a s t a k i n g

p l a c e c l o s e t o o r i n s i d e t h e i n t e r f a c i a l t r a n s i t i o n z o n e . T h i s z o n e i s d o m i n a t e d b y t h e

p r e s e n c e o f c e m e n t p a s t e a n d l a c k o f a g g r e g a t e s c a u s e d b y t h e w a l l - e e c t .

T h e r s t m o d e l d e s c r i b i n g a n i n t e r f a c i a l c r a c k b e t w e e n t w o d i s s i m i l a r m a t e r i a l s w a s d e -

v e l o p e d b y W i l l i a m s ( 1 9 5 9 ) . L a t e r , t e s t s e t - u p s t o m e a s u r e t h e f r a c t u r e r e s i s t a n c e o f

1



b i m a t e r i a l i n t e r f a c e s u s i n g l i n e a r e l a s t i c f r a c t u r e m e c h a n i c s c o n c e p t s h a v e b e e n d e v e l -

o p e d , i . e . t h e n o t c h e d f o u r - p o i n t b e n d i n g b e a m s p e c i m e n p r e s e n t e d b y C h a r a l a m b i d e s

e t a l . ( 1 9 8 9 ) . S o a r e s & T a n g ( 1 9 9 8 ) p r o p o s e d a s e t - u p t o c h a r a c t e r i z e t h e l i n e a r e l a s t i c

f r a c t u r e m e c h a n i c a l p r o p e r t i e s o f a r o c k / m o r t a r i n t e r f a c e u s i n g a t e s t w h i c h t h e y c a l l : T h e

B i m a t e r i a l B r a z i l i a n S p e c i m e n . S i n c e t h e i n t r o d u c t i o n o f t h e n o n l i n e a r f r a c t u r e m e c h a n -

i c s b a s e d c t i t i o u s c r a c k m o d e l b y H i l l e r b o r g e t a l . ( 1 9 7 6 ) , p a p e r s h a v e b e e n p u b l i s h e d

u t i l i z i n g t h i s t h e o r y . T e s t s p e c i m e n s h a v e b e e n p r o p o s e d t o m e a s u r e t h e s t r e s s - c r a c k

o p e n i n g r e l a t i o n s h i p o f a c e m e n t - b a s e d i n t e r f a c e . W a n g & M a j i ( 1 9 9 5 ) i n v e s t i g a t e d a

r o c k / m o r t a r i n t e r f a c e a n d p r o p o s e d a h i n g e d u n i a x i a l t e n s i o n s e t - u p t o d e t e r m i n e t h e

s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p . T h e s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p w a s d e s c r i b e d

b y a s i n g l e l i n e a r r e l a t i o n s h i p . V e r i c a t i o n w a s c a r r i e d o u t c o m p a r i n g e x p e r i m e n t s o n

b i m a t e r i a l i n t e r f a c e s u s i n g t h e s o - c a l l e d C o m p a c t T e n s i o n s p e c i m e n w i t h F E s i m u l a t i o n s

u s i n g c o n s t i t u t i v e i n p u t o b t a i n e d f r o m t h e u n i a x i a l t e n s i o n t e s t s . R e c e n t l y , ? a n d C h a n -

d r a K i s h e n & S a o u m a ( 2 0 0 4 ) u s e d t h e W S T t o e v a l u a t e t h e m o d e I f r a c t u r e e n e r g y o f

l i m e s t o n e - c o n c r e t e i n t e r f a c e s . B o t h s t u d i e s i n c l u d e d c o m p a r i s o n b e t w e e n f u l l c o n c r e t e

a n d l i m e s t o n e s p e c i m e n s , a n d a s e x p e c t e d , i n b o t h c a s e s , a m o r e b r i t t l e b e h a v i o r w a s

o b s e r v e d i n t h e b i m a t e r i a l t e s t s .

T h i s p a p e r p r o p o s e s a t e s t s e t - u p t o d e t e r m i n e t h e f r a c t u r e p r o p e r t i e s o f a s t e e l - c o n c r e t e

i n t e r f a c e u s i n g a s p e c i m e n w h e r e o n e h a l f i s s t e e l w h i l e t h e o t h e r h a l f i s c o n c r e t e . T h e

a p p l i e d s e t - u p i s s i m i l a r t o t h e w e l l - k n o w n W S T o r i g i n a l l y p r o p o s e d b y L i n s b a u e r &

T s c h e g g ( 1 9 8 6 ) . T h i s n e w t e s t m e t h o d , i s d e s i g n a t e d f o r t h e c h a r a c t e r i z a t i o n o f t h e m o d e

I c r a c k p r o p a g a t i o n t a k i n g p l a c e n e a r t h e s t e e l - c o n c r e t e i n t e r f a c e . I t i s h y p o t h e s i z e d

t h a t t h i s p r o p e r t y m a y b e d e s c r i b e d u s i n g t h e s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p o r i g i n a l l y

d e n e d b y H i l l e r b o r g e t a l . ( 1 9 7 6 ) . T h e m o s t s t r a i g h t f o r w a r d t e s t t o d e t e r m i n e t h e

s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p w o u l d b e t h e u n i a x i a l t e s t s i n c e n o i n v e r s e a n a l y s i s i s

n e c e s s a r y . H o w e v e r t h i s t e s t i s r a t h e r d e m a n d i n g w i t h r e s p e c t t o l a b o r a t o r y e q u i p m e n t

a n d s t a e x p e r i e n c e . T h i s g i v e s r e a s o n s f o r u s i n g t h e W S T w h i c h s t o r e s l i t t l e e l a s t i c

e n e r g y d u r i n g t e s t i n g , t h u s r e s u l t i n g i n a s t a b l e e x p e r i m e n t , a n d i s w e l l s u i t e d f o r i n v e r s e

a n a l y s i s , s e e e . g . Ø s t e r g a a r d ( 2 0 0 3 ) .

T h e n e e d f o r t h i s i n t e r f a c i a l m a t e r i a l d e s c r i p t i o n a r i s e s f r o m a n o n g o i n g r e s e a r c h p r o j e c t

w i t h f o c u s o n t h e a p p l i c a t i o n o f s t r e n g t h e n i n g s t e e l b r i d g e d e c k s u s i n g a c e m e n t - b a s e d

o v e r l a y . T h e g o a l i s t o c a s t a c e m e n t - b a s e d o v e r l a y o n t o p o f t h e s t e e l b r i d g e d e c k i n o r d e r

t o f o r m a t o p c o m p o s i t e s l a b . H e n c e t h e f r a c t u r e p r o p e r t i e s o f a s t e e l - c o n c r e t e i n t e r f a c e

i s o f g r e a t i n t e r e s t , s e e , e . g . , W a l t e r e t a l . ( 2 0 0 4 ) .

2 T h e B i m a t e r i a l W S T S p e c i m e n

T h e p r o p o s e d t e s t m e t h o d u s e s t h e s h a p e o f t h e w e l l - k n o w n w e d g e s p l i t t i n g s p e c i m e n a s

d e s c r i b e d b y e . g . B r ü h w i l e r & W i t t m a n n ( 1 9 9 0 ) . T h i s s p e c i m e n i s c h a r a c t e r i z e d b y a

g r o o v e a n d a s t a r t e r n o t c h , a n d t h e i d e a i s t o r e p l a c e h a l f o f t h e s p e c i m e n w i t h a s t e e l

b l o c k , c f . F i g u r e 1 ( a ) . T h e s p e c i m e n i s p l a c e d o n a l i n e a r s u p p o r t . T w o s t e e l l o a d i n g

d e v i c e s e q u i p p e d w i t h r o l l e r b e a r i n g s a r e p l a c e d o n t o p o f t h e s p e c i m e n . A w e d g e s h a p e d

2



s t e e l p r o l e i s p l a c e d b e t w e e n t h e b e a r i n g s . M o v i n g t h e a c t u a t o r o f t h e t e s t i n g m a c h i n e

r e s u l t s i n a s p l i t t i n g f o r c e P sp a c t i n g b e t w e e n t h e t w o p a r t s o f t h e s p e c i m e n , c f . F i g u r e

1 ( b ) .

( a ) ( b )

F i g u r e 1 : ( a ) G e o m e t r y o f t h e t e s t e d b i m a t e r i a l W S T s p e c i m e n . T h e h a t c h e d p a r t r e p -

r e s e n t s a s t e e l b l o c k ( b ) L o a d c o n g u r a t i o n

D u r i n g t h e t e s t , t h e l o a d i n t h e v e r t i c a l d i r e c t i o n P v a n d t h e C r a c k M o u t h O p e n i n g

D i s p l a c e m e n t ( C M O D ) a r e r e c o r d e d . F o r s t a b l e e x p e r i m e n t s c l o s e d l o o p c o n t r o l w i t h

C M O D a s t h e c o n t r o l p a r a m e t e r i s a p p l i e d . F r i c t i o n a l f o r c e s o f t h e b e a r i n g a r e n o t t a k e n

i n t o c o n s i d e r a t i o n s i n c e t h e c o n t r i b u t i o n w i l l b e l e s s t h a n 2 % R o s s i e t a l . ( 1 9 9 1 ) . T h e

r e l a t i o n b e t w e e n t h e h o r i z o n t a l f o r c e a n d t h e s p l i t t i n g f o r c e i s c a l c u l a t e d u s i n g :

P sp =P v

2tan(αw)( 1 )

I n t h e c u r r e n t p a p e r , a n a n g l e o f αw = 150i s a p p l i e d .

I t i s f u r t h e r m o r e a s s u m e d , t h a t a l l c r a c k i n g i s p u r e m o d e I . H o w e v e r , s i n c e t h e b i m a t e r i a l

W S T s p e c i m e n c o n s i s t o f t w o m a t e r i a l s w i t h d i e r e n t e l a s t i c m o d u l i t h e i n t e r f a c e i s i n

f a c t n o t a p l a n e o f s y m m e t r y . C h a n d r a K i s h e n & S a o u m a ( 2 0 0 4 ) a n a l y z e d t h e a s s u m p t i o n

o f p u r e m o d e I c r a c k i n g i n a b i m a t e r i a l W S T s p e c i m e n . T h e i r a p p r o a c h w a s t o m e a s u r e

t h e s t r e s s i n t e n s i t y f a c t o r s u s e d i n L i n e a r E l a s t i c F r a c t u r e M e c h a n i c s ( L E F M ) t h e o r y . I n

t h e i r i n v e s t i g a t i o n t h e f r a c t u r e t o u g h n e s s o f a l i m e s t o n e c o n c r e t e i n t e r f a c e w a s m e a s u r e d

t o K 1c = 0.28MP a√

m a n d K 2c = 0.01MP a√

m . T h e y c o n c l u d e t h a t t h e m o d e I I f r a c t u r e

t o u g h n e s s i s r e l a t i v e l y l o w .

I n t h e p r e s e n t s t u d y , t h e a s s u m p t i o n o f p u r e m o d e I c r a c k i n g i s a n a l y z e d u s i n g t h e c o m -

m e r c i a l n i t e e l e m e n t p a c k a g e ( D I A N A 2 0 0 3 ) . T h e c o n n e c t i o n b e t w e e n s t e e l a n d c o n c r e t e

i s m o d e l l e d u s i n g s t a n d a r d i n t e r f a c e e l e m e n t s w i t h a t h i c k n e s s o f z e r o . S t e e l a n d c o n c r e t e

a r e b o t h m o d e l l e d a s s u m i n g p u r e e l a s t i c b e h a v i o r w i t h a n e l a s t i c m o d u l u s o f 2 1 0 G P a a n d

3



3 0 G P a , r e s p e c t i v e l y . T h e a p p l i e d m e s h i s s h o w n i n F i g u r e 2 ( a ) a n d i s f o u n d b e t o p r e c i s e

t h r o u g h a c o n v e r g e n c e a n a l y s i s .

( a )

−10 −5 0 5

0

10

20

30

40

50

Stress [MPa]

C r a c k l i n e c o o r d i n a t e

y

[ m m ]

σ τ

( b )

F i g u r e 2 : ( a ) A p p l i e d m e s h i n t h e F E a n a l y s i s t o i n v e s t i g a t e t h e a m o u n t o f p u r e m o d e I

e n e r g y c o n s u m e d u s i n g a b i m a t e r i a l W S T s p e c i m e n . ( b ) S t r e s s d i s t r i b u t i o n a t p e a k l o a d

f o r n o r m a l s t r e s s , σ , a n d s h e a r s t r e s s , τ .

A l l n o n l i n e a r b e h a v i o r i s a s s u m e d t o t a k e p l a c e a t t h e i n t e r f a c e . T h e c o n s t i t u t i v e l a w o f

t h e i n t e r f a c e e m p l o y s a m i x e d m o d e m o d e l i m p l e m e n t e d i n t h e c o m m e r c i a l n i t e e l e m e n t

p a c k a g e u s i n g u s e r - s u p p l i e d s u b r o u t i n e s . T h e m o d e l i s b a s e d o n a n c o u p l i n g o f t h e u n i a x i a l

σ−

δ n r e l a t i o n s h i p a n d τ −

δ s r e l a t i o n s h i p . C o n s i d e r a t w o d i m e n s i o n a l c o n g u r a t i o n ,

w h e r e t h e i n t e r f a c e e l e m e n t r e l a t e s t h e s t r e s s e s a c t i n g o n t h e i n t e r f a c e t o t h e r e l a t i v e

d i s p l a c e m e n t s , i . e . c r a c k o p e n i n g a n d c r a c k s l i d i n g . T h e c r a c k d e f o r m a t i o n i n o p e n i n g

m o d e i s d e n o t e d δ n , w h i l e s l i d i n g m o d e i s d e n o t e d b y δ s . T h e i r r e l a t i o n t o t h e n o r m a l , σ ,

a n d s h e a r , τ , s t r e s s i s g i v e n b y : στ

=

D11 D12

D21 D22

δ nδ s

( 2 )

I n t h e c r a c k e d s t a t e a p e r m a n e n t c o u p l i n g i s p r e s e n t b e t w e e n t h e n o r m a l a n d s h e a r d e f o r -

m a t i o n d e s c r i b e d t h r o u g h t h e s t i n e s s c o m p o n e n t s Dij . T w o b i l i n e a r c u r v e s i n p u r e m o d e

I a n d m o d e I I d e f o r m a t i o n a r e g i v e n a s i n p u t . T h e m o d e l u t i l i z e s t h e f o l l o w i n g c r i t e r i o n

t o d e s c r i b e t h e s t a t e o f d e f o r m a t i o n f o r e v e r y p o i n t o n t h e c u r v e s : δ n

δ maxn

2

+

δ s

δ maxs

2

= 1.0 ( 3 )

w h e r e δ maxn a n d δ max

s i s t h e m a x i m u m c r a c k d e f o r m a t i o n i n n o r m a l a n d s l i d i n g m o d e g i v e n

i n t h e t w o b i l i n e a r c u r v e s , r e s p e c t i v e l y . F o r f u r t h e r d e t a i l s o n t h e m o d e l s e e W a l t e r e t a l .

( 2 0 0 5 ) o r W e r n e r s s o n ( 1 9 9 4 ) .

4



T h e a m o u n t o f f r a c t u r e e n e r g y c o n s u m e d i n m o d e I , GI f , a n d m o d e I I , GII

f , a l o n g t h e s t e e l

c o n c r e t e i n t e r f a c e f o r e v e r y n o d e c a n b e c a l c u l a t e d u s i n g t h e f o l l o w i n g f o r m u l a s

GI

f

= Γ

σ(δ n)dδ n GII

f

= Γ

τ (δ s)dδ s ( 4 )

T o i n v e s t i g a t e t h e a m o u n t o f m o d e I a n d I I e n e r g y w h i c h i s c o n s u m e d , t h r e e n u m e r i c a l

s i m u l a t i o n s a r e i n v e s t i g a t e d . I n e a c h c a s e t h e m o d e I f r a c t u r e e n e r g y o f t h e b i l i n e a r

c u r v e i s s e t t o 0 . 1 N / m m , w h e r e a s t h e b i l i n e a r c u r v e i n m o d e I I i s v a r i e d a c c o r d i n g t o :

GII f ∈ [0.05;0.1; 0.2]N / m m . T h e m a x i m u m n o r m a l a n d s h e a r s t r e s s i s s e t t o 2 M P a i n

a l l t h r e e c a s e s . F i g u r e 2 ( b ) s h o w s t h e n o r m a l a n d s h e a r s t r e s s d i s t r i b u t i o n a l o n g t h e

i n t e r f a c e o f t h e W S T s p e c i m e n a t p e a k l o a d . F o r e a c h n u m e r i c a l s i m u l a t i o n t h e a m o u n t

o f m o d e I a n d I I e n e r g y c o n s u m e d a r e c a l c u l a t e d u s i n g e q u a t i o n 4 . I n e v e r y c a s e t h e m o d e

I e n e r g y c o n s u m e d a c c o u n t s f o r 9 9 . 5 % o f t h e t o t a l a m o u n t o f e n e r g y c o n s u m e d . T h u s ,

w h e n E s/E c = 7 t h e p u r e m o d e I c r a c k i n g a s s u m p t i o n w i l l p r o d u c e a m a x i m u m e r r o r o f

0 . 5 % .

100

101

102

103

104

105

106

0

0.2

0.4

0.6

0.8

Es /E

c

G f

I I / G f

I [ % ]

F i g u r e 3 : T h e r a t i o o f c o n s u m e d e n e r g y GII f /GI

f i n p e r c e n t , a c c o r d i n g t o f o r m u l a ( 4 ) ,

v e r s u s t h e e l a s t i c m o d u l i r a t i o E s/E c .

F u r t h e r , a n a d d i t i o n a l i n v e s t i g a t i o n , v a r y i n g t h e r a t i o o f t h e t w o e l a s t i c m o d u l i a r e c a r r i e d

o u t t o d e t e r m i n e t h e a m o u n t o f m o d e I I e n e r g y c o n s u m e d f o r d i e r e n t E s/E c r a t i o s . T h e

e l a s t i c m o d u l u s E c i s k e p t c o n s t a n t a t 3 0 G P a , w h e r e a s E s i s v a r i e d a c c o r d i n g t o t h e

f o l l o w i n g r a t i o s : E s/E c ∈ [1; 7; 70; 700; 700000]. T h e r e s u l t s a r e s h o w n i n F i g u r e 3 a s

t h e r a t i o o f c o n s u m e d e n e r g y GII f /GI

f i n p e r c e n t v e r s u s t h e e l a s t i c m o d u l i r a t i o E s/E co n a l o g a r i t h m i c s c a l e . A s o b s e r v e d t h e a m o u n t o f m o d e I I e n e r g y c o n s u m e d g r o w s f o r

i n c r e a s i n g e l a s t i c m o d u l i r a t i o . H o w e v e r , a t s o m e p o i n t a t h r e s h o l d i s o b s e r v e d .

5



3 T h e B i m a t e r i a l C r a c k e d H i n g e M o d e l

T h e c r a c k e d h i n g e m o d e l m a y b e e m p l o y e d t o o b t a i n t h e s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p

u s i n g e x p e r i m e n t a l W S T r e s u l t s . H o w e v e r , s i n c e t h e o u t c o m e o f t h e W S T i s a g l o b a l l o a d -

d e e c t i o n c u r v e r a t h e r t h a n t h e s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p d i r e c t l y , t h e l a t t e r h a s

t o b e d e t e r m i n e d b y a n i n v e r s e a n a l y s i s . H e r e , t h e a d v a n t a g e o f t h e c r a c k e d h i n g e m o d e l

i s t h a t i t y i e l d s c l o s e d - f o r m a n a l y t i c a l s o l u t i o n s , w h i c h c a n b e i m p l e m e n t e d i n a s i m p l e

p r o g r a m w r i t t e n f o r t h e p u r p o s e o f i n v e r s e a n a l y s i s . T h e i d e a o f a c r a c k e d h i n g e w a s

o r i g i n a l l y d e v e l o p e d b y U l f k j æ r e t a l . ( 1 9 9 5 ) a n d f u r t h e r d e v e l o p e d b y P e d e r s e n ( 1 9 9 6 ) ,

S t a n g & O l e s e n ( 1 9 9 8 ) , S t a n g & O l e s e n ( 2 0 0 0 ) , a n d O l e s e n ( 2 0 0 1 ) . F u r t h e r m o r e , t h e

p r o p o s e d m e t h o d h a s p r o v e n t o b e r o b u s t a n d a c c u r a t e f o r n o r m a l c o n c r e t e , s e e Ø s t e r g a a r d

( 2 0 0 3 ) .

T h e b a s i c i d e a o f t h e c r a c k e d h i n g e i s t o m o d e l t h e n o n l i n e a r b e h a v i o r , i n t h i s c a s e

i n t e r f a c i a l c r a c k i n g , b e t w e e n t w o r i g i d b o u n d a r i e s . I n s i d e t h e b o u n d a r i e s , i n d e p e n d e n t

h o r i z o n t a l s p r i n g e l e m e n t s a r e g i v e n a f u l l e l a s t i c a n d n o n l i n e a r d e s c r i p t i o n , w h i l e t h e

m o d e l l i n g o u t s i d e t h e h i n g e i s c o n d u c t e d u s i n g c l a s s i c a l e l a s t i c t h e o r y . T h e s p e c i c h i n g e

w i d t h o f t h e s t e e l a n d c o n c r e t e p a r t s a r e g i v e n a s ss/2 a n d sc/2, r e s p e c t i v e l y , w h i l e t h e

s u m i s e q u a l t h e t o t a l b a n d w i d t h s, c f . F i g u r e 4 . T h e c o n s t i t u t i v e b e h a v i o r o f t h e s t e e l

a n d c o n c r e t e p a r t s a r e m o d e l l e d u s i n g e l a s t i c m o d u l i o f E s a n d E c , r e s p e c t i v e l y . I n t h e

c r a c k e d s t a t e t h e n o n l i n e a r b e h a v i o r o f t h e i n t e r f a c e i s d e s c r i b e d t h r o u g h a s t r e s s - c r a c k

o p e n i n g r e l a t i o n s h i p :

σ =

E s S t e e l p a r t

E c c o n c r e t e p a r t

σw(w) = g(w)f t I n t e r f a c e i n c r a c k e d s t a t e

( 5 )

T h e s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p i s a p p r o x i m a t e d b y a b i l i n e a r s o f t e n i n g c u r v e , s e e

E q u a t i o n ( 6 ) a n d F i g u r e 5 .

g(w) = bi − aiw =

b1 − a1w 0 ≤ w < w1

b2 − a2w w1 ≤ w ≤ w2( 6 )

w h e r e b1 = 1; w h i l e t h e l i m i t w1 i s g i v e n b y t h e i n t e r s e c t i o n o f t h e t w o l i n e s e g m e n t s . T h e

i n t e r s e c t i o n o f t h e s e c o n d l i n e s e g m e n t a n d t h e a b s c i s s a i s d e n o t e d w2 :

w1 =1− b2

a1

−a2

, w2 =b2a2

( 7 )

T h e d e f o r m a t i o n o f t h e h i n g e i s d e s c r i b e d t h r o u g h t h e a n g u l a r d e f o r m a t i o n ϕ = ϕs + ϕc ,

c f . F i g u r e 4 . T h e m e a n v a l u e o f t h e c u r v a t u r e f o r e a c h p a r t o f t h e h i n g e c a n b e e x p r e s s e d

a s

κ∗s = 2ϕs

ss; κ∗c = 2

ϕc

sc( 8 )

T h e d i s t r i b u t i o n o f t h e l o n g i t u d i n a l s t r a i n s a t t h e d e p t h y c a n b e e x p r e s s e d b y t h e m e a n

c u r v a t u r e

6



F i g u r e 4 : G e o m e t r y , l o a d i n g a n d d e f o r m a t i o n o f t h e b i m a t e r i a l h i n g e e l e m e n t

F i g u r e 5 : D e n i t i o n o f p a r a m e t e r s o f t h e b i l i n e a r s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p

s = (y − y0)κ∗s; c = (y − y0)κ∗c ( 9 )

I n t h e c a s e w h e r e t h e s t r i p h a s c r a c k e d , t h e d e f o r m a t i o n u(y), c a n b e o b t a i n e d a s t h e s u m

o f t h e e l a s t i c d e f o r m a t i o n a n d t h e c r a c k o p e n i n g

w.

u(y) =ss2

σw

E s+

sc2

σw

E c+ w ( 1 0 )

B y c o m b i n i n g e q u a t i o n s ( 9 ) a n d ( 1 0 ) , w e m a y w r i t e

σw =

(y − y0)(ϕs + ϕc)− w

2

ss

E s+

sc

E c

−1

( 1 1 )

7



T h e m o m e n t M a n d t h e n o r m a l f o r c e N a c t i n g a t t h e c r a c k i n g p l a n e c a n b e c a l c u l a t e d

u s i n g e q u i l i b r i u m . F r o m F i g u r e 6 ( c ) i t i s s e e n t h a t t h e m o m e n t a n d n o r m a l f o r c e i s g i v e n

b y

N = P sp ( 1 4 )

M = P sp

d2 −

h

2

+ P spd1tan(αw) ( 1 5 )

T h e C M O D , d e p e n d s o n t h r e e d i e r e n t c o n t r i b u t i o n s . T h e e l a s t i c d e f o r m a t i o n o f t h e

s p e c i m e n c a u s e s a n o p e n i n g o f C M O D , d e n o t e d δ e . S e c o n d , a c o n t r i b u t i o n δ COD d e n o t e s

t h e c r a c k o p e n i n g o f t h e s p e c i m e n . F i n a l l y , a c o n t r i b u t i o n δ g h a s t o b e t a k e n i n t o a c c o u n t

s i n c e t h e l i n e w h e r e C M O D i s m e a s u r e d i n t h e e x p e r i m e n t i s l o c a t e d d i e r e n t l y t h a n t h e

a c t u a l c r a c k m o u t h . T h u s , C M O D i s g i v e n b y :

CMOD = δ e + δ COD + δ g ( 1 6 )

H a v i n g a w e d g e s p l i t t i n g s p e c i m e n c o n s i s t i n g o f o n e m a t e r i a l , δ e c a n b e d e t e r m i n e d u s i n g

h a n d b o o k s o n s t r e s s a n a l y s i s , e . g . T a d a e t a l . ( 1 9 8 5 ) . H o w e v e r , s i n c e t h e s p e c i m e n d i e r s

i n g e o m e t r y a n d c o n s i s t o f t w o m a t e r i a l s h a v i n g d i e r e n t e l a s t i c m o d u l i , t h e a p p r o x i m a t i o n

f o r m u l a e s b y T a d a e t a l . ( 1 9 8 5 ) a r e n o t a p p l i c a b l e . U s i n g t h e s e f o r m u l a s w i l l p r o d u c e a n

e r r o r o n t h e d e t e r m i n a t i o n o f E c o f a p p r o x i m a t e l y 2 0 % . F o r h i g h e r p r e c i s i o n δ e m a y s i m p l y

b e d e t e r m i n e d u s i n g a F E m o d e l o f t h e a d o p t e d W S T - g e o m e t r y . T h i s a p p r o a c h h a s b e e n

u s e d h e r e . T h u s , t h e e l a s t i c c o n t r i b u t i o n t o t h e C M O D , δ e , i s c a l c u l a t e d a c c o r d i n g t o t h e

f o l l o w i n g f o r m u l a

δ e =1

2

P sp

tν 2

1

E s+

1

E c

( 1 7 )

w h e r e t i s t h e t h i c k n e s s o f t h e s p e c i m e n , a n d t h e t e r m v2 i s d e t e r m i n e d f r o m a l i n e a r

e l a s t i c F E a n a l y s i s .

O p e n i n g o f t h e c r a c k m o u t h c a u s e d b y t h e p r e s e n c e o f t h e c r a c k h a s b e e n d e r i v e d b y

O l e s e n ( 2 0 0 1 ) :

δ COD =sf t

E ∗1− bi + 2αθ

1− βi, β i =

f tais

E ∗i ∈ [1, 2] ( 1 8 )

w h e r e t h e t e r m s

β ia n d

bim u s t b e c h o s e n a c c o r d i n g t o o n e o f t h e t h r e e c r a c k i n g p h a s e s .

T h e t h i r d a n d n a l t e r m δ g , w h i c h i s d u e t o a g e o m e t r i c a l a m p l i c a t i o n s i n c e t h e r e i s a

c e r t a i n d i s t a n c e , b − h, f r o m t h e c r a c k m o u t h , l o c a t e d a t h, t o t h e l i n e w h e r e C M O D i s

m e a s u r e d , l o c a t e d a t b . T h e c o n t r i b u t i o n c a n b e f o u n d f r o m

δ g = 2(b− h)

δ COD2αh

− sf thE ∗

θ0−I 1− β 1

( 1 9 )

w h e r e θ0−I r e p r e s e n t s t h e n o r m a l i z e d a n g u l a r d e f o r m a t i o n o f t h e h i n g e a t i n i t i a t i o n o f t h e

c r a c k . F o r f u r t h e r e x p l a n a t i o n s e e Ø s t e r g a a r d ( 2 0 0 3 ) .

9



4 I n v e r s e A n a l y s i s

M e t h o d s f o r t h e e x t r a c t i o n o f t h e c o n s t i t u t i v e p r o p e r t i e s o f t h e c r a c k e d r e g i o n o f q u a s i -

b r i t t l e m a t e r i a l s h a s b e e n i n v e s t i g a t e d s i n c e H i l l e r b o r g e t a l . ( 1 9 7 6 ) p r o p o s e d t h e c t i t i o u s

c r a c k m o d e l i n 1 9 7 6 . T h e m e t h o d s h a v e b e e n a n a l y t i c a l l y a s w e l l a s n u m e r i c a l l y b a s e d

a n d u s u a l l y a p p l i e d t o p u r e c o n c r e t e , e m p l o y i n g t h e t h r e e p o i n t b e n d i n g s p e c i m e n . A s u c -

c e s s f u l a p p r o a c h w a s r e p o r t e d b y W i t t m a n n e t a l . ( 1 9 8 7 ) i n w h i c h a s i m p l e n i t e e l e m e n t

b a s e d m e t h o d p r o v e d a p p l i c a b l e f o r t h e t h r e e p o i n t b e n d i n g s p e c i m e n . T h e o u t p u t w a s

a b i l i n e a r s t r e s s - c r a c k o p e n i n g c u r v e . A n o t h e r s t r a t e g y p r e s e n t e d b y K i t s u t a k a ( 1 9 9 7 )

y i e l d s a m u l t i - l i n e a r c u r v e b a s e d o n a n i n c r e m e n t a l a p p r o a c h l i n k i n g s p e c i c p a r t s o f t h e

l o a d - d i s p l a c e m e n t c u r v e t o s p e c i c p a r t s o f t h e s t r e s s - c r a c k o p e n i n g d i s p l a c e m e n t c u r v e .

L a t e l y , t h e p r o p o s e d m e t h o d s o f i n v e r s e a n a l y s i s t e n d t o b e m o r e a n d m o r e c o m p l e x .

S t r a t e g i e s b a s e d o n e v o l u t i o n a r y a l g o r i t h m s h a v e b e e n p r o p o s e d V i l l m a n n e t a l . ( 2 0 0 4 ) ,

w h i l e a l s o t h e E x t e n d e d K a l m a n F i l t e r p r o c e d u r e ( E K F ) h a s b e e n a p p l i e d b y B o l z o n e t a l .

( 2 0 0 2 ) . W h i l e c o m p l e x m e t h o d s m a y b e j u s t i e d f o r c o m p l e x p r o b l e m s , i n v e r s e a n a l y s i s

o f p u r e m o d e I c r a c k i n g o f c o n c r e t e , a s s u m i n g v a l i d i t y o f t h e c t i t i o u s c r a c k m o d e l , i s i n

f a c t a s i m p l e p r o b l e m a n d s h o u l d b e t r e a t e d a s s u c h .

A s i m p l e a p p r o a c h t o t h e i n v e r s e a n a l y s i s o f t h e c o n c r e t e - s t e e l i n t e r f a c e i s a p p l i c a t i o n

o f t h e s e m i - a n a l y t i c a l h i n g e m o d e l p r o p o s e d b y O l e s e n ( 2 0 0 1 ) , u t i l i z i n g e . g . t h e f a m i l i a r

s i m p l e x o p t i m i z a t i o n a l g o r i t h m . A m e t h o d b a s e d o n t h e a b o v e t e c h n i q u e s a p p l i e d t o p u r e

c o n c r e t e e m p l o y i n g t h e W S T w a s p r o v e n t o b e f a s t a n d r e l i a b l e i n ( Ø s t e r g a a r d 2 0 0 3 ) . A s

w i l l b e s h o w n i n t h e p r e s e n t p a p e r , t h i s i s a l s o t h e c a s e f o r t h e c o n c r e t e - s t e e l i n t e r f a c i a l

p r o b l e m . T h e m e t h o d p r o p o s e d g e n e r a l l y y i e l d s t h e s o l u t i o n t o t h e i n v e r s e p r o b l e m i n a

m a t t e r o f m i n u t e s r a t h e r t h a n h o u r s o r d a y s f o r t h e m o r e c o m p l e x m e t h o d s o r F E - m e t h o d s .

E q u a t i o n s ( 1 ) - ( 1 8 ) f o r m t h e b a s i s f o r i n t e r p r e t a t i o n a n d i n v e r s e a n a l y s i s o f t h e W S T .

T h e i n t e r p r e t a t i o n p r o c e e d s b y b a l a n c i n g t h e i n t e r n a l m o m e n t a n d n o r m a l f o r c e w i t h t h e

e x t e r n a l o n e s . T h e i d e a i n t h e i n v e r s e a n a l y s i s i s t o u s e a s t e p w i s e a l g o r i t h m , w h e r e t h e

o p t i m i z a t i o n p r o b l e m i s s o l v e d i n s t e p s c o r r e s p o n d i n g t o t h e d i e r e n t p h a s e s o f c r a c k

p r o p a g a t i o n . F i r s t , t h e o p t i m i z a t i o n i s c o n d u c t e d i n t h e e l a s t i c p h a s e w i t h t h e m o d u l u s

o f e l a s t i c i t y o f t h e c o n c r e t e p a r t a s t h e o n l y f r e e p a r a m e t e r , a n d o n l y c o n s i d e r i n g t h e

o b s e r v a t i o n s b e l o n g i n g t o t h e e l a s t i c p h a s e . T h i s r s t p a r t , n a m e d S t e p I , s e e E q u a -

t i o n 2 0 , w i l l r e s u l t i n a f a s t a n d r e l i a b l e d e t e r m i n a t i o n o f t h e m o d u l u s o f e l a s t i c i t y . I t

i s i m p o r t a n t t o r e a l i z e t h a t t h e i n i t i a l g u e s s o n t h e t e n s i l e s t r e n g t h w i l l d e t e r m i n e h o w

m a n y o b s e r v a t i o n s t o i n c l u d e i n t h e o p t i m i z a t i o n . B u t w i t h r e a s o n a b l e i n i t i a l g u e s s e s

a n d b y g l o b a l l y r e - r u n n i n g t h e o p t i m i z a t i o n p r o c e s s ( i n c l u d i n g a l l s t e p s ) t h i s i s a m i n o r

p r o b l e m , s i n c e t h e g l o b a l i t e r a t i o n s w i l l c o n v e r g e a t t h e t r u e p h a s e c h a n g e p o i n t . H a v i n g

d e t e r m i n e d a n e s t i m a t e f o r t h e m o d u l u s o f e l a s t i c i t y , t h e n e x t s t e p i s t o f o r m u l a t e a n

o p t i m i z a t i o n s t r a t e g y f o r t h e c r a c k e d p h a s e s . I t t u r n s o u t t h a t t h e b e s t s t r a t e g y i s t o

s e p a r a t e t h e p r o b l e m i n t o t w o , s u c h t h a t f t a n d a1 a r e d e t e r m i n e d r s t ( S t e p I I ) , w h i l e a2

a n d b2 a r e d e t e r m i n e d s u b s e q u e n t l y ( S t e p I I I ) . T h i s i s d u e t o r e s u l t s s h o w i n g t h a t l o c a l

m i n i m a m a y b e a v o i d e d u s i n g t h i s a p p r o a c h . N o t e i n t h i s c o n t e x t t h a t m a n y m e t h o d s

1 0



f o r i n v e r s e a n a l y s i s a r e p r o n e t o n d i n g l o c a l m i n i m a , s e e e . g . U l f k j æ r & B r i n c k e r ( 1 9 9 3 ) ,

V i l l m a n n e t a l . ( 2 0 0 4 ) a n d B o l z o n e t a l . ( 2 0 0 2 ) . W i t h t h e p r o p o s e d m e t h o d , l o c a l m i n -

i m a a r e e n t i r e l y a v o i d e d . I n c o n t r a s t t o S t e p I , a l l o b s e r v a t i o n s m u s t b e i n c l u d e d i n t h e

o p t i m i z a t i o n f o r S t e p I I a n d I I I . I f o n l y o b s e r v a t i o n s b e l o n g i n g t o t h e a c t u a l p h a s e ( e . g .

p h a s e I ) w e r e c o n s i d e r e d , a s p u r i o u s s o l u t i o n m a y b e f o u n d . T h i s s o l u t i o n r e p r e s e n t s t h e

m i n i m u m w h e r e t h e c o n s t i t u t i v e p a r a m e t e r s h a v e b e e n s e l e c t e d s u c h t h a t n o o b s e r v a t i o n

b e l o n g s t o t h e c o n s i d e r e d p h a s e .

N o t e t h a t i t i s n o t p o s s i b l e i n t h e c r a c k e d s t a g e s t o d e t e r m i n e b o t h t h e b i m a t e r i a l m o d u l u s

o f e l a s t i c i t y , E ∗ a n d t h e b i m a t e r i a l h i n g e w i d t h s f r o m E q u a t i o n 1 2 . H o w e v e r , t h i s i s n o

p r o b l e m s i n c e o n l y t h e r a t i o E ∗/s e n t e r s t h e c a l c u l a t i o n s , s e e O l e s e n ( 2 0 0 1 ) . T h u s , a s s u m -

i n g t h a t t h e h i n g e w i d t h p a r a m e t e r s sc a n d ss a r e k n o w n f r o m a n i n i t i a l F E M - c a l i b r a t i o n ,

E c i s k n o w n f r o m t h e i n i t i a l o p t i m i z a t i o n o f t h e r s t s l o p e o f t h e l o a d - C M O D c u r v e ,

a n d t h a t t h e s t e e l m o d u l u s o f e l a s t i c i t y i s k n o w n a p r i o r i , t h e E ∗/s- r a t i o m a y b e c a l c u -

l a t e d . T h i s r e s u l t i s u s e d f o r t h e d e t e r m i n a t i o n o f t h e c o n s t i t u t i v e p a r a m e t e r s o f t h e c r a c k .

U t i l i z i n g a l l o b s e r v a t i o n s i n t h e c r a c k e d p h a s e s , a n d u s i n g t h e m e a n s q u a r e o f d i e r e n c e s

b e t w e e n o b s e r v a t i o n s a n d p r e d i c t i o n s a s a n e r r o r n o r m , t h e o p t i m i z a t i o n p r o b l e m r e a d s :

S t e p I - D e t e r m i n a t i o n o f E c

minE

1

N 0

max

N 0max

0P sp

−P sp

2

( 2 0 )

s u b j e c t t o E > 0

S t e p I I - D e t e r m i n a t i o n o f f t a n d a1

min(f t,a1)

1

N max

N max

0 P sp − P sp

2

( 2 1 )

s u b j e c t t o f t > 0

S t e p I I I - D e t e r m i n a t i o n o f a2 a n d b2

min(a2,b2)

1

N max

N max0

P sp − P sp

2( 2 2 )

1 1



w h e r e N 0max a n d N max r e p r e s e n t t h e l a s t o b s e r v a t i o n b e l o n g i n g t o p h a s e 0 a n d t h e t o t a l

n u m b e r o f o b s e r v a t i o n s , r e s p e c t i v e l y . T h e o p t i m i z a t i o n i s r e s t r i c t e d s u c h t h a t o n l y p h y s -

i c a l m e a n i n g f u l s o l u t i o n s a r e f o u n d ( E c > 0 , f t > 0 e t c . ) . M o r e d e t a i l s c o n c e r n i n g t h e

i n v e r s e a n a l y s i s a n d t h e v a l i d a t i o n w i t h r e g a r d t o n i t e e l e m e n t a n a l y s i s m a y b e f o u n d i n

Ø s t e r g a a r d ( 2 0 0 3 ) . T h e e x i s t e n c e o f l o c a l m i n i m a h a s b e e n i n v e s t i g a t e d b y a l a r g e n u m -

b e r o f s t a r t g u e s s e s o f t h e c o n s t i t u t i v e p a r a m e t e r s a n d t h e r e s u l t s a r e c l e a r : A l l s t a r t i n g

g u e s s e s w i l l e v e n t u a l l y r e s u l t i n a c o n v e r g e n c e a t t h e g l o b a l m i n i m u m .

T a b l e 1 : R e s u l t s f r o m i n v e r s e a n a l y s i s w i t h 1 2 d i e r e n t m a t e r i a l s t o g e t h e r w i t h t h e o p t i m a l

c h o i c e s o f sc a n d ss . T h e n u m b e r g i v e n o n e a c h p l a c e i n t h e m i d d l e p a r t o f t h e t a b l e r e f e r s

t o t h e i n p u t v a l u e o f t h e r e l e v a n t c o n s t i t u t i v e p a r a m e t e r , w h i l e t h e n u m b e r i n p a r e n t h e s i s

i s t h e e r r o r o n t h e d e t e r m i n a t i o n o f t h i s v a l u e u s i n g t h e m e t h o d o f i n v e r s e a n a l y s i s u t i l i z i n g

t h e s- p a r a m e t e r s g i v e n t o t h e r i g h t .

M a t e r i a l f t [ M P a ] a1 [ m m

−1] a2 [ m m

−1] b2 [ - ] E c [ G P a ] E s [ G P a ]

sc

h

ss

h1 2 ( 8 % ) 1 0 ( 3 % ) 0 . 2 ( - 5 % ) 0 . 1 ( - 7 % ) 3 0 ( - 2 % ) 2 1 0 1 . 0 4 0 . 0 8

2 2 ( 2 ) 2 0 ( - 3 ) 0 . 2 ( - 7 ) 0 . 1 ( - 4 ) 3 0 ( - 2 ) 2 1 0 0 . 7 4 0 . 2

3 2 ( 4 ) 3 0 ( - 2 ) 0 . 2 ( 0 ) 0 . 1 ( - 1 ) 3 0 ( - 2 ) 2 1 0 0 . 8 6 0 . 2 5

4 2 ( 5 ) 4 0 ( 0 ) 0 . 2 ( 5 ) 0 . 1 ( 1 ) 3 0 ( - 3 ) 2 1 0 0 . 9 8 0 . 1 5

5 2 ( 3 ) 2 0 ( - 2 ) 0 . 6 ( 0 ) 0 . 1 ( - 4 ) 3 0 ( - 3 ) 2 1 0 0 . 8 0 . 2

6 2 ( 3 ) 2 0 ( - 2 ) 1 . 2 ( 2 ) 0 . 1 ( - 6 ) 3 0 ( - 3 ) 2 1 0 0 . 7 4 0 . 2 1

7 2 ( 4 ) 2 0 ( - 3 ) 0 . 2 ( - 1 ) 0 . 2 5 ( - 1 ) 3 0 ( - 2 ) 2 1 0 0 . 8 4 0 . 2

8 2 ( 4 ) 2 0 ( - 4 ) 0 . 2 ( 0 ) 0 . 4 ( - 1 ) 3 0 ( - 2 ) 2 1 0 0 . 9 0 . 1 8

9 2 ( 2 ) 2 0 ( - 2 2 ) 0 . 2 ( 1 ) 0 . 7 ( 1 ) 3 0 ( - 3 ) 2 1 0 0 . 8 3 0 . 1 8

1 0 2 ( 2 ) 2 0 ( - 3 ) 0 . 2 ( - 9 ) 0 . 1 ( - 6 ) 4 0 ( - 3 ) 2 1 0 0 . 7 7 0 . 1 9

1 1 2 ( 2 ) 2 0 ( - 3 ) 0 . 2 ( - 2 ) 0 . 1 ( - 2 ) 3 0 ( - 5 ) 6 0 1 . 2 2 0 . 1 1

1 2 2 ( 3 ) 2 0 ( - 3 ) 0 . 2 ( - 5 ) 0 . 1 ( - 4 ) 3 0 ( - 3 ) 1 2 0 0 . 9 4 0 . 1 7

T h e c r a c k e d h i n g e m o d e l i s d e p e n d e n t o n t h e w i d t h o f t h e h i n g e i n b o t h t h e s t e e l a n d t h e

c o n c r e t e p a r t s . T h u s , i n i t i a l l y , a c a l i b r a t i o n i s c o n d u c t e d i n o r d e r t o i d e n t i f y t h e m o s t

s u i t a b l e v a l u e s o f t h e s e w i d t h p a r a m e t e r s . T h i s i s d o n e u s i n g a n i t e e l e m e n t m o d e l o f

t h e W S T f r o m w h i c h l o a d - C M O D c u r v e s a r e g e n e r a t e d f o r a n u m b e r o f d i e r e n t s t r e s s -

c r a c k o p e n i n g c u r v e s . T h e s e l e c t e d v a l u e s o f t h e s t r e s s - c r a c k o p e n i n g p r o l e a n d t h e e r r o r

o n t h e d e t e r m i n a t i o n o f t h e s e p a r a m e t e r s t h r o u g h a n i n v e r s e a n a l y s i s u s i n g t h e o p t i m a l

w i d t h p a r a m e t e r s a r e g i v e n i n T a b l e 1 . T h e o p t i m a l s - p a r a m e t e r s h a v e b e e n d e t e r m i n e d

b y a n o p t i m i z a t i o n i n w h i c h t h e t h e s q u a r e d s u m o f d i e r e n c e s b e t w e e n t h e p r e d i c t e d

p a r a m e t e r s a n d t h e i n p u t p a r a m e t e r s h a s b e e n m i n i m i z e d :

min(sc,ss)

f t − f t

f t

2

+

a1 − a1

a1

2

+ ... +

E c −E c

E c

2

( 2 3 )

1 2



I t i s i m p o r t a n t t o r e a l i z e t h a t t h e r e s u l t s o b t a i n e d b y t h e m e t h o d o f i n v e r s e a n a l y s i s a r e

b a s e d o n t h e e n t i r e l o a d - C M O D c u r v e s , i . e . f r o m o n s e t o f l o a d i n g u n t i l t h e s p e c i m e n

h a s b e e n f u l l y s e p a r a t e d . I f t h e t a i l s o f t h e l o a d - C M O D c u r v e s a r e o m i t t e d , t h e e r r o r s

o n t h e d e t e r m i n a t i o n o f t h e c o n s t i t u t i v e p a r a m e t e r s w i l l i n c r e a s e s i g n i c a n t l y . T h i s i s

d e m o n s t r a t e d i n F i g u r e 7 , w h e r e t h e F E M g e n e r a t e d P - C M O D c u r v e h a s b e e n t r u n c a t e d

a t d i e r e n t v a l u e s o f C M O D - a n d t h e i n v e r s e a n a l y s i s s u b s e q u e n t l y h a s b e e n c o n d u c t e d

o n t h e r e m a i n i n g c u r v e . T h e p a r a m e t e r b2 h a s b e e n s e l e c t e d s i n c e i t o n l y e n t e r s t h e e q u a -

t i o n s a f t e r t h e c r a c k p r o l e h a s t u r n e d i n t o a b i l i n e a r f u n c t i o n - t h u s a d e t e r m i n a t i o n

o f t h i s p a r a m e t e r ( l i k e w i s e a2 ) r e q u i r e s m o r e P - C M O D c u r v e t o b e k n o w n t h a n e . g . t h e

t e n s i l e s t r e n g t h , f t , o r i n p a r t i c u l a r m o d u l u s o f e l a s t i c i t y , E c , d o e s . A s F i g u r e 7 s h o w s ,

t h e d e t e r m i n a t i o n o f b2 b y t h e m e t h o d o f i n v e r s e a n a l y s i s i s s p u r i o u s a s l o n g a s t h e P -

C M O D c u r v e i s t r u n c a t e d b e f o r e t h e s e c o n d p o i n t o f t r a n s i t i o n . H o w e v e r , o n l y a f e w

d a t a p o i n t s a f t e r t h i s t r a n s i t i o n p o i n t a r e n e c e s s a r y t o m a k e t h e a l g o r i t h m c o m e u p w i t h

a r e a s o n a b l e r e s u l t , b u t t h e q u a l i t y o f t h e r e s u l t c o n t i n u e s t o i m p r o v e e a c h t i m e a n e w

p o r t i o n o f t h e P - C M O D c u r v e i s i n c l u d e d . I t i s t h e r e f o r e r e c o m m e n d e d t o o b t a i n a s m u c h

o f t h e P - C M O D c u r v e a s p o s s i b l e i n o r d e r t o i n c r e a s e t h e a c c u r a c y o f t h e r e s u l t s . N o t e

a l s o t h a t w h e r e a t o t a l C M O D o f 4 m m s e e m s s u c i e n t f o r m a t e r i a l n o . 8 , t h i s w i l l n o t

b e t h e c a s e f o r e . g . m a t e r i a l n o . 9 w h e r e t h e h i g h v a l u e o f b2 r e s u l t s i n a f a r m o r e d u c t i l e

r e s p o n s e o f t h e m a t e r i a l . I t i s m o r e r e l e v a n t t o r e q u i r e t h a t f o r e x a m p l e t h e e x p e r i m e n t

i s c o n t i n u e d u n t i l t h e l o a d i s l e s s t h a n 0 . 5 % o f t h e p e a k l o a d .

1 4



!"#$ %&&&&' &

F i g u r e 7 : L o a d - C M O D c u r v e f o r m a t e r i a l n o . 8 a s g e n e r a t e d b y F E M . T h e h o l l o w c i r c l e s

m a r k t h e t r a n s i t i o n p o i n t s w h e r e , r e s p e c t i v e l y , t h e c r a c k i n i t i a t e s , c h a n g e s f r o m l i n e a r t o

b i l i n e a r s t r e s s d i s t r i b u t i o n , a n d c h a n g e s f r o m b i l i n e a r t o b i l i n e a r s t r e s s d i s t r i b u t i o n w i t h

a s t r e s s f r e e t a i l . T h e s o l i d s q u a r e s m a r k t h e e r r o r o n t h e d e t e r m i n a t i o n o f b2 , g i v e n t h a t

t h e F E M - c u r v e i s t r u n c a t e d c o r r e s p o n d i n g t o t h e x - c o o r d i n a t e o f t h e s q u a r e .

5 E x p e r i m e n t a l R e s u l t s

T h e u s e o f t h e W S T t o d e t e r m i n e t h e f r a c t u r e p r o p e r t i e s o f a s t e e l - c o n c r e t e i n t e r f a c e

u s i n g t h e i n v e r s e a n a l y s i s d e s c r i b e d i n t h e p r e v i o u s s e c t i o n i s i l l u s t r a t e d . F u l l c o n c r e t e

a n d b i m a t e r i a l W S T s p e c i m e n s a r e t e s t e d a n d t h e i r f r a c t u r e p r o p e r t i e s a r e c o m p a r e d .

T h e c o n c r e t e u s e d i n t h i s s t u d y w a s c o m p o s e d o f c e m e n t , c o a r s e a g g r e g a t e s 1 6 m m m a x -

i m u m g r a i n s i z e , n e a g g r e g a t e s , w a t e r a n d v a r i o u s a d m i x t u r e s , s u c h a s , y a s h , s i l i c a

f u m e a n d p l a s t i c i z e r t o i m p r o v e t h e f r e s h p r o p e r t i e s o f t h e m i x t u r e . T h e m i x d e s i g n i s

g i v e n i n T a b l e 3 .

T h e m i x i s o p t i m i z e d f o r s e l f - c o m p a c t i n g p r o p e r t i e s a n d d o e s t h e r e f o r e n o t n e e d m e c h a n -

i c a l v i b r a t i o n a f t e r c a s t i n g . T h i s i s b e l i e v e d t o o p t i m i z e t h e s t e e l - c o n c r e t e b o n d , s e e e . g .

( S c h i e s s l & Z i l c h 2 0 0 1 ) . I n t h e c a s e o f t h e b i m a t e r i a l W S T s p e c i m e n s , t h e s t e e l s u r f a c e

w a s s a n d b l a s t e d p r i o r t o c a s t i n g t o i m p r o v e t h e b o n d a n d m i n i m i z e t h e r i s k o f d e f e c t s .

T h e t e s t p r o g r a m c o n s i s t s o f 1 2 t e s t s p e c i m e n s i n t o t a l , s i x c o n c r e t e a n d s i x c o n c r e t e - s t e e l

s p e c i m e n s . S p e c i m e n s w h e r e c a s t i n t w o b a t c h e s w i t h 3 o f e a c h t y p e p e r b a t c h .

S a m p l e r e s u l t s o f t h e W S T e x p e r i m e n t s a r e g i v e n i n F i g u r e 8 , w h e r e t h e s p l i t t i n g f o r c e

P sp v e r s u s C M O D i s p l o t t e d . T h e F i g u r e a l s o c o m p a r e s r e s u l t s o n b o t h t h e f u l l c o n c r e t e

a n d b i m a t e r i a l W S T s p e c i m e n s .

I n t h e c a s e o f b i m a t e r i a l W S T s p e c i m e n s , c r a c k i n g t o o k p l a c e c l o s e t o t h e p h y s i c a l b o u n d -

1 5



a r y b e t w e e n t h e s t e e l a n d c o n c r e t e b l o c k . T h i s b e h a v i o r i n d i c a t e s t h a t t h e a c t u a l b o u n d -

a r y i s s t r o n g e r t h a n t h e c e m e n t p a s t e i t s e l f . I n e a c h b i m a t e r i a l t e s t , p a t c h e s o f c e m e n t

p a s t e w a s l e f t o n t h e s t e e l b l o c k . F i g u r e 9 s h o w s a n a c t u a l p i c t u r e o f a b i m a t e r i a l s p e c i -

m e n a f t e r t e s t i n g . T h e l e f t p a r t o f t h e W S T s p e c i m e n i s t h e s t e e l b l o c k . I t i s c l e a r l y s e e n

t h a t s o m e c e m e n t p a s t e i s s t i l l b o n d e d t o t h e s t e e l b l o c k .


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1 6



T a b l e 5 : R e s u l t s f r o m i n v e r s e a n a l y s i s o n f u l l c o n c r e t e W S T s p e c i m e n s . * N o t e f a i l u r e o f

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r e s u l t s i n T a b l e s 4 a n d 5 , i t s h o u l d b e n o t e d t h a t t h e r e e x i s t s o m e s c a t t e r b e t w e e n t h e

t w o b a t c h e s . N o d i r e c t e v i d e n c e e x i s t i n o r d e r t o f u l l y u n d e r s t a n d t h e d i e r e n c e b u t s o m e

i n d i c a t i o n s r e g a r d i n g t h e b e h a v i o r o f t h e m i x d e s i g n m i g h t e x p l a i n t h e d i e r e n c e . B y

c o m p a r i n g t h e r e s u l t s o f t h e t e s t s o n t h e c o n c r e t e s p e c i m e n s i t i s n o t e d t h a t t h e t e n s i l e

s t r e n g t h f t i s h i g h e r i n b a t c h n o . 2 t h a n i n b a t c h n o . 1 . T h e o p p o s i t e i s t h e c a s e w i t h t h e

b i m a t e r i a l t e s t s . A p o o r b o n d , d u e t o p o s s i b l e s h r i n k a g e , a t t h e e d g e o f t h e b i m a t e r i a l

s p e c i m e n s m i g h t b e t h e e x p l a n a t i o n . T h i s i s f u r t h e r s u p p o r t e d b y t h e v i s u a l i n s p e c t i o n

o f t h e b i m a t e r i a l s p e c i m e n s a f t e r t e s t i n g , w h i c h r e v e a l e d c l e a r s i g n s o f s h r i n k a g e o n t h e

b i m a t e r i a l s p e c i m e n s i n b a t c h n o . 2 .

5 . 2 D i s c u s s i o n

T h e r e s u l t s f r o m t h e b i m a t e r i a l a n d c o n c r e t e s p e c i m e n s c a n b e v i e w e d i n a n o r m a l i z e d

s t r e s s v e r s u s c r a c k o p e n i n g d i a g r a m . F i g u r e 1 0 s h o w s , f o r e a c h s p e c i m e n t y p e , t h e a v e r a g e

s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p f o r b a t c h n o . 1 a n d b a t c h n o . 2 , r e s p e c t i v e l y .

I t i s s e e n t h a t f o r e a c h b a t c h a c o n s i s t e n t p a t t e r n i s o b s e r v e d . T h e b i m a t e r i a l t e s t s e x h i b i t s

a h i g h e r s l o p e i n t h e r s t p a r t o f t h e s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p , d e n o t e d a1 , c f .

F i g u r e 5 . I n g e n e r a l , t h e s h a p e o f t h e σ−w c u r v e i s i n u e n c e d b y t h e s i z e o f a g g r e g a t e s a n d

t h e m a t r i x / a g g r a g a t e b o n d . W h e n a c r a c k p r o p a g a t e s t h r o u g h t h e c o n c r e t e t h e f r a c t u r e

s t r e n g t h o f t h e i n d i v i d u a l p a r t s , m a t r i x / a g g r e g a t e s a n d t h e i r b o n d , d e t e r m i n e s i f t h e c r a c k

p r o p a g a t e s t h r o u g h t h e a g g r e g a t e , t h e m a t r i x o r t h r o u g h a c o m b i n a t i o n o f b o t h . I n t h e

c a s e o f a p o o r c e m e n t p a s t e o r a p o o r m a t r i x / a g g r e g a t e b o n d , a g g r e g a t e p u l l o u t i s o f t e n

o b s e r v e d . T h i s b e h a v i o r l e a d s t o a l o n g t a i l o f t h e σ − w c u r v e . T h e p r e s e n t m i x d e s i g n ,

T a b l e 3 , c o n s i s t o f a m i x w i t h o p t i m i z e d b o n d i n g p r o p e r t i e s a n d a s a r e s u l t t h e c r a c k i n g

h a s l e a d t o a g g r e g a t e b r e a k a g e . A s o b s e r v e d i n F i g u r e 1 0 t h e b i m a t e r i a l s p e c i m e n s s h o w

h i g h e r s t r e s s d r o p i n t h e b e g i n n i n g o f t h e σ − w c u r v e . T h i s i s d u e t o t h e w a l l - e e c t ,

w h i c h i s p r e s e n t i n t h e r e g i o n c l o s e t o t h e s t e e l - c o n c r e t e b o u n d a r y . D u e t o t h e w a l l -

e e c t , c r a c k p r o p a g a t i o n i n t h e b i m a t e r i a l s p e c i m e n s t a k e s p l a c e i n t h e m a t r i x w h e r e a s i n

t h e f u l l c o n c r e t e s p e c i m e n s c r a c k i n g t a k e s p l a c e i n a c o m b i n a t i o n o f a g g r a g a t e s / m a t r i x .

1 8



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F i g u r e 1 0 : S t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p f o r ( a ) b a t c h n o . 1 a n d ( b ) b a t c h n o . 2

d e t e r m i n e d u s i n g t h e i n v e r s e a n a l y s i s

T h e h i g h e r s t r e s s d r o p i n t h e b e g i n n i n g o f t h e s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p m i g h t b e

d u e t o t h e f a c t t h a t t h e m a t r i x m i g h t b e m o r e b r i t t l e c o m p a r e d t o t h a t o f a g g r e g a t e s .

T h e w a l l - e e c t a n d t h e t w o d i e r e n t c r a c k p a t h s f o r c o n c r e t e a n d b i m a t e r i a l s p e c i m e n s

a r e i l l u s t r a t e d i n F i g u r e 1 1 . A s i l l u s t r a t e d , c l o s e t o t h e s t e e l b l o c k a c r a c k w i l l m a i n l y

p r o p a g a t e i n t h e m a t r i x . O n t h e c o n t r a r y , p a t h 2 , c o n s i s t o f a c o m b i n a t i o n o f m a t r i x a n d

a g g r e g a t e s a s o b s e r v e d i n t h e f u l l c o n c r e t e W S T s p e c i m e n s .

F i g u r e 1 1 : S c h e m a t i c i l l u s t r a t i o n o f t h e w a l l - e e c t . A c r a c k p a t h ( P a t h 1 ) c l o s e t o a s t e e l

w a l l m i g h t b e d o m i n a t e d b y m a t r i x c r a c k p r o p a g a t i o n w h e r e a s i n t h e c a s e o f c r a c k p r o p -

a g a t i o n i n t h e c o n c r e t e m a t e r i a l ( p a t h 2 ) c r a c k p r o p a g a t i o n t a k e s p l a c e i n c o m b i n a t i o n o f

a g g r e g a t e s a n d m a t r i x .

1 9





R e s . 6 ( 6 ) , 7 7 3 7 8 2 .

K i t s u t a k a , Y . ( 1 9 9 7 ) , ` F r a c t u r e p a r a m e t e r s b y p o l y l i n e a r t e n s i o n - s o f t e n i n g a n a l y s i s ' , J o u r -

n a l o f E n g i n e e r i n g M e c h a n i c s 1 2 3 ( 5 ) , 4 4 4 4 5 0 .

L i n s b a u e r , H . N . & T s c h e g g , E . K . ( 1 9 8 6 ) , ` F r a c t u r e e n e r g y d e t e r m i n a t i o n o f c o n c r e t e

w i t h c u b e s h a p e d s p e c i m e n s ( i n g e r m a n ) ' , Z e m e n t u n d B e t o n 3 1 , 3 8 4 0 .






P e d e r s e n , C . ( 1 9 9 6 ) , N e w p r o d u c t i o n p r o c e s s e s , m a t e r i a l s , a n d c a l c u l a t i o n t e c h n i q u e s f o r

b e r r e i n f o r c e d c o n c r e t e p i p e s , P h D t h e s i s , D e p a r t m e n t o f S t r u c t u r a l E n g i n e e r i n g

a n d M a t e r i a l s , T e c h n i c a l U n i v e r s i t y o f D e n m a r k , L y n g b y , D e n m a r k .

R I L E M T C - 1 0 8 ( 1 9 9 6 ) , I n t e r f a c i a l T r a n s i t i o n Z o n e i n C o n c r e t e , R I L E M R e p o r t 1 1 ,

C h a p m a n a n d H a l l .

R o s s i , P . , B r ü h w i l e r , E . , C h h u y , S . , J e n q , Y . - S . & S h a h , S . P . ( 1 9 9 1 ) , F r a c t u r e p r o p e r t i e s

o f c o n c r e t e a s d e t e r m i n e d b y m e a n s o f w e d g e s p l i t t i n g t e s t s a n d t a p e r e d d o u b l e

c a n t i l e v e r b e a m t e s t s , i n S . S h a h & A . C a r p i n t e r i , e d s , ` F r a c t u r e M e c h a n i c s T e s t

M e t h o d s f o r C o n c r e t e ' , C h a p m a n n & H a l l , c h a p t e r 2 , p p . 8 7 1 2 8 .

S c h i e s s l , A . & Z i l c h , K . ( 2 0 0 1 ) , ` T h e e e c t o f t h e m o d i e d c o m p o s i t i o n o f S C C o n s h e a r

a n d b o n d b e h a v i o r ' , P r o c e e d i n g s o f t h e S e c o n d I n t e r n a t i o n a l S y m p o s i u m o n s e l f -

C o m p a c t i n g C o n c r e t e , T o k y o p p . 5 0 1 5 0 6 .

S o a r e s , B . J . & T a n g , T . ( 1 9 9 8 ) , ` B i m a t e r i a l b r a z i l i a n s p e c i m e n f o r d e t e r m i n i n g i n t e r f a c i a l

f r a c t u r e t o u g h n e s s ' , J o u r n a l o f E n g i n e e r i n g F r a c t u r e M e c h a n i c s 5 9 ( 1 ) , 5 1 7 1 .

S t a n g , H . & O l e s e n , J . F . ( 1 9 9 8 ) , O n t h e i n t e r p r e t a t i o n o f b e n d i n g t e s t s o n f r c m a t e r i a l s ,

i n ` F R A M C O S - 3 ' , V o l . I , A e d i c a t i o P u b l i s h e r s , F r e i b u r g , G e r m a n y , p p . 5 1 1 5 2 0 .

S t a n g , H . & O l e s e n , J . F . ( 2 0 0 0 ) , A f r a c t u r e m e c h a n i c s - b a s e d d e s i g n a p p r o a c h t o f r c , i n

` 5 t h R I L E M S y m p . o n F i b e r - R e i n f o r c e d C o n c r e t e ' , B E F I B .

T a d a , H . , P a r i s , P . & I r w i n , G . ( 1 9 8 5 ) , T h e s t r e s s a n a l y s i s o f c r a c k s h a n d b o o k , P a r i s

P r o d u c t i o n s I n c o r p o r a t e d , 2 2 6 W o o d b o u r n e D R . , S t . L o u i s , M i s s o u r i , U S A .

U l f k j æ r , J . P . & B r i n c k e r , R . ( 1 9 9 3 ) , I n d i r e c t d e t e r m i n a t i o n o f t h e σ - w r e l a t i o n o f H S C

t h r o u g h t h r e e - p o i n t b e n d i n g , i n ` F r a c t u r e a n d D a m a g e o f C o n c r e t e a n d R o c k -

F C D R - 2 ' , E & F N S p o n , p p . 1 3 5 1 4 4 .

2 1



U l f k j æ r , J . , K r e n k , S . & B r i n c k e r , R . ( 1 9 9 5 ) , ` A n a l y t i c a l m o d e l f o r c t i t i o u s c r a c k p r o p -

a g a t i o n i n c o n c r e t e b e a m s ' , J o u r n a l o f E n g i n e e r i n g M e c h a n i c s 1 2 1 ( 1 ) , 7 1 5 .

V i l l m a n n , B . , V i l l m a n n , T . & S l o w i k , V . ( 2 0 0 4 ) , D e t e r m i n a t i o n o f s o f t e n i n g c u r v e s b y

b a c k w a r d a n a l y s e s o f e x p e r i m e n t s a n d o p t i m i z a t i o n u s i n g a n e v o l u t i o n a r y a l g o r i t h m ,

i n V . C . L i , C . K . Y . L e u n g , K . J . W i l l a m & S . L . B i l l i n g t o n , e d s , ` P r o c e e d i n g s

o f t h e F i f t h I n t e r n a t i o n a l C o n f e r e n c e o n F r a c t u r e M e c h a n i c s o f C o n c r e t e a n d C o n -

c r e t e S t r u c t u r e s / V a i l , C o l o r a d o / U S A / 1 2 - 1 6 A p r i l ' , V o l . 1 , I a - F r a M C o S , I a - F r a M C o S ,

p p . 4 3 9 4 4 6 .

W a l t e r , R . , G i m s i n g , N . & S t a n g , H . ( 2 0 0 4 ) , ` C o m p o s i t e s t e e l - c o n c r e t e o r t h o t r o p i c b r i d g e

d e c k ' , 1 0 t h N o r d i c S t e e l C o n s t r u c t i o n C o n f e r e n c e , C o p e n h a g e n , D e n m a r k , p p . 5 1 9 -

5 3 0 .

W a l t e r , R . , O l e s e n , J . F . & S t a n g , H . ( 2 0 0 5 ) , ` I n t e r f a c i a l m i x e d m o d e m o d e l ' , I n : T h e

1 1 t h I n t e r n a t i o n a l C o n f e r e n c e o n F r a c t u r e , T u r i n , I t a l y .

W a n g , J . & M a j i , A . K . ( 1 9 9 5 ) , E x p e r i m e n t a l s t u d i e s a n d m o d e l i n g o f t h e c o n c r e t e / r o c k

i n t e r f a c e , i n B . O . & W . M . , e d s , ` I n t e r f a c e F r a c t u r e a n d B o n d ' , A C I S P - 1 5 6 , p p . 4 5

6 8 .



W i l l i a m s , M . L . ( 1 9 5 9 ) , ` T h e s t r e s s a r o u n d a f a u l t o r c r a c k i n d i s s i m i l a r m e d i a ' , B u l l e t i n

o f t h e S e i s m o l o g i c a l S o c i e t y o f A m e r i c a 4 9 ( 2 ) , 1 9 9 2 0 4 .

W i t t m a n n , F . H . , R o e l f s t r a , P . E . , M i h a s h i , H . , H u a n g , Y . - Y . & Z h a n g , X . - H . ( 1 9 8 7 ) ,

` I n u e n c e o f a g e o f l o a d i n g , w a t e r - c e m e n t r a t i o a n d r a t e o f l o a d i n g o n f r a c t u r e e n e r g y

o f c o n c r e t e ' , M a t e r i a l s a n d S t r u c t u r e s 2 0 , 1 0 3 1 1 0 .

7 A p p e n d i x

C o n s i d e r i n g t h e s o l u t i o n o f t h e c r a c k e d h i n g e m o d e l , t h i s a p p e n d i x g i v e s a s h o r t d e s c r i p -

t i o n o f t h e o r i g i n a l s o l u t i o n b y O l e s e n ( 2 0 0 1 ) . A n a l y s i s o f t h e h i n g e e l e m e n t m a k e s i t

p o s s i b l e t o d e t e r m i n e t h e e x t e r n a l n o r m a l f o r c e N a n d b e n d i n g m o m e n t M f o r a n y g i v e n

v a l u e o f t h e a n g u l a r h i n g e d e f o r m a t i o n ϕ, a s s h o w n f o r t h e b i m a t e r i a l s p e c i m e n i n F i g u r e

4 . T h e p r o b l e m i s s o l v e d i n f o u r p h a s e s , o n e f o r e a c h s t a t e o f c r a c k p r o p a g a t i o n . P h a s e 0

r e p r e s e n t s t h e e l a s t i c s t a t e , w h e r e n o c r a c k h a s f o r m e d , w h i l e p h a s e s I , I I a n d I I I r e p r e s e n t

d i e r e n t s t a g e s o f p r o p a g a t i o n ( l i n e a r , b i l i n e a r a n d b i l i n e a r w i t h s t r e s s - f r e e t a i l ) . T h e s o -

l u t i o n i s p r e s e n t e d i n t e r m s o f n o r m a l i z e d p r o p e r t i e s , w h e r e t h e f o l l o w i n g n o r m a l i z a t i o n s

a r e u s e d :

β 1 =f ta1s

E , β 2 =

f ta2s

E , c =

(1 − b2)(1 − β 1)

β 2 − β 1( 2 4 )

2 2



µ =6M

f th2t, ρ =

N

f tht, θ =

hEϕ

sf t, α =

d

h( 2 5 )

T h e h i n g e s o l u t i o n i s e x p r e s s e d i n t e r m s o f t h e n o r m a l i z e d c r a c k d e p t h α a n d t h e n o r m a l -

i z e d m o m e n t µ a s f u n c t i o n s o f t h e n o r m a l i z e d h i n g e d e f o r m a t i o n θ a n d t h e n o r m a l i z e d

n o r m a l f o r c e ρ :

E l a s t i c p h a s e : 0 ≤ θ ≤ 1− ρ

α = 0 ( 2 6 )

µ = θ ( 2 7 )

P h a s e I : 1− ρ < θ ≤ θI - I I

:

α = 1− β 1 −

(1 − β 1)

1− ρ

θ− β 1

( 2 8 )

µ = 4

1− 3α + 3α2 − α3

1− β 1

θ + (6α − 3)(1 − ρ) ( 2 9 )

P h a s e I I : θI - I I < θ ≤ θI I - I I I

:

α = 1− β 2 −1− b2

2θ−

(1 − β 2)

(1− b2)2

4θ2(β 1 − β 2)− β 2 +

b2 − ρ

θ

( 3 0 )

µ = 4

1− 3α + 3α2 − α3

1− β 2

θ + (6α − 3)(1 − ρ)

−(1 − b2)

3α2 −

c

2θ

21− β 2

( 3 1 )

P h a s e I I I : θI I - I I I

< θ :

α = 1 − 1

2θ

1 +

(1 − b2)2

β 1 − β 2+

b22β 2− 4ρθ

( 3 2 )

µ = 4

1− 3α + 3α2 − α3

θ + (6α − 3)(1 − ρ)− 3α2

+1

4θ2

1− b2

β 2

1 − b2

β 2+ c

1 +

β 1c

1 − β 1

+ c

2θ

2( 3 3 )

2 3



δ s c r a c k s l i d i n g d e f o r m a t i o n

s t r a i n

c, s s t r a i n o f c o n c r e t e a n d s t e e l p a r t

κc u r v a t u r e

κ∗c , κ∗

s m e a n c u r v a t u r e o f c o n c r e t e a n d s t e e l p a r t

µ n o r m a l i z e d c r o s s s e c t i o n a l m o m e n t

ρ n o r m a l i z e d c r o s s s e c t i o n a l n o r m a l f o r c e

σ,σw n o r m a l s t r e s s


θn o r m a l i z e d h i n g e r o t a t i o n

θ0−I

p o i n t o f t r a n s i t i o n b e t w e e n p h a s e 0 a n d I

θI −II p o i n t o f t r a n s i t i o n b e t w e e n p h a s e I a n d I I

θII −II I p o i n t o f t r a n s i t i o n b e t w e e n p h a s e I I a n d I I I

ϕa n g u l a r d e f o r m a t i o n

ϕc, ϕs a n g u l a r d e f o r m a t i o n o f c o n c r e t e a n d s t e e l p a r t

a0, am c o o r d i n a t e s f o r t h e n o t c h

a1, a2, b1, b2 p a r a m e t e r s i n t h e b i l i n e a r s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p

b d i s t a n c e t o l i n e o f C M O D - m e a s u r e m e n t

bm w i d t h o f t o p s p e c i m e n

cc o n s t a n t

dd e p t h o f c r a c k

d1, d2 c o o r d i n a t e s f o r t h e l o a d p o i n t

Dij s t i n e s s c o m p o n e n t s f o r i n t e r f a c e d e s c r i p t i o n

E ∗ b i m a t e r i a l e l a s t i c m o d u l u s

E c e l a s t i c m o d u l u s o f c o n c r e t e

E s e l a s t i c m o d u l u s o f s t e e l

f t t e n s i l e s t r e n g t h

g(w) n o r m a l i z e d f u n c t i o n d e s c r i b i n g s h a p e o f s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p

Gf f r a c t u r e e n e r g y

GI f p u r e m o d e I e n e r g y

GII f p u r e m o d e I I e n e r g y

hh e i g h t o f h i n g e a n d l i g a m e n t

2 5



K 1c m o d e I s t r e s s i n t e n s i t y f a c t o r

K 2c m o d e I I s t r e s s i n t e n s i t y f a c t o r

Ls i d e l e n g t h o f W S T - c u b e

M c r o s s s e c t i o n a l m o m e n t

N c r o s s s e c t i o n a l n o r m a l f o r c e

N max n u m b e r o f e x p e r i m e n t a l o b s e r v a t i o n s

P v v e r t i c a l l o a d

P sp s p l i t l o a d

st o t a l h i n g e w i d t h

sc, ss h i n g e w i d t h - c o n c r e t e a n d s t e e l p a r t

tt h i c k n e s s o f s p e c i m e n

uh o r i z o n t a l d i s p l a c e m e n t o f l o a d p o i n t

w c r a c k o p e n i n g

w1, w2 c r a c k o p e n i n g l i m i t s i n t h e b i l i n e a r s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p

yc o o r d i n a t e a l o n g c r a c k d i r e c t i o n

y0 d e p t h o f n e u t r a l a x i s

2 6



C e m e n t - B a s e d O v e r l a y i n N e g a t i v e B e n d i n g - E x p e r i m e n t a l a n d

F E M S t u d i e s

R a s m u s W a l t e r * , J o h n F . O l e s e n * , V i c t o r L i * * & H e n r i k S t a n g *

* D e p a r t m e n t o f C i v i l E n g i n e e r i n g , T e c h n i c a l U n i v e r s i t y o f D e n m a r k , D K - 2 8 0 0 K g s .


* * D e p a r t m e n t o f C i v i l a n d E n v i r o n m e n t a l E n g i n e e r i n g , T h e U n i v e r s i t y o f M i c h i g a n ,

A n n A r b o r , M I , 4 8 1 0 9 - 2 1 2 5 , U S A

P a p e r s u b m i t t e d f o r p u b l i c a t i o n

A b s t r a c t

A n e w t y p e o f c o m p o s i t e b r i d g e d e c k i s c u r r e n t l y u n d e r r e s e a r c h . T h e c o n c e p t i s t o

a c h i e v e c o m p o s i t e a c t i o n b y t h e a d h e s i o n b e t w e e n a t h i n l a y e r o f c e m e n t - b a s e d m a t e r i a l

c a s t o n a s t e e l p l a t e . O f s p e c i a l c o n c e r n i s t h e p e r f o r m a n c e o f t h e c o m p o s i t e d e c k i n

n e g a t i v e b e n d i n g . B a s e d o n e x p e r i m e n t s a n d a n o n l i n e a r 2 D n i t e e l e m e n t m o d e l a

s t u d y h a s b e e n c o n d u c t e d i n r e l a t i o n t o v e r t i c a l c r a c k i n g o f t h e o v e r l a y a n d i n t e r f a c i a l

c r a c k i n g ( d e b o n d i n g ) b e t w e e n t h e o v e r l a y a n d s t e e l p l a t e . T h r e e t y p e s o f c e m e n t - b a s e d

c o m p o s i t e s , F i b e r R e i n f o r c e d C o n c r e t e ( F R C ) , F i b e r R e i n f o r c e d D e n s i t

R ( F R D ) a n d

E n g i n e e r e d C e m e n t i t i o u s C o m p o s i t e ( E C C ) h a v e b e e n t e s t e d a s o v e r l a y m a t e r i a l a n d

c o m p a r e d . F i n a l l y , t h e p a p e r s h o w s m o d e l i n g o f t h e t e s t s e t - u p a p p l y i n g t h e d i e r e n t

m a t e r i a l s . T h e m o d e l i n g i s c a r r i e d o u t u s i n g n o n l i n e a r f r a c t u r e m e c h a n i c s a n d t h e F i n i t e

E l e m e n t M e t h o d ( F E M ) . A g o o d c o r r e l a t i o n b e t w e e n e x p e r i m e n t a l a n d n u m e r i c a l r e s u l t s

i s o b t a i n e d . T h e s t u d y s h o w s , b o t h n u m e r i c a l l y a n d e x p e r i m e n t a l l y , t h a t d e b o n d i n g i s

r e l a t e d t o m a c r o c r a c k i n g o f t h e o v e r l a y .

K e y w o r d s C o m p o s i t e b e a m , c r a c k i n g , d e b o n d i n g , o v e r l a y , F E M , F i b r e r e i n f o r c e d c o n -

c r e t e .


A l a r g e n u m b e r o f s t e e l b r i d g e d e c k s , i n p a r t i c u l a r s t e e l o r t h o t r o p i c b r i d g e d e c k s , s u e r

s i g n i c a n t l y f r o m i n c r e a s e d t r a c i n t e n s i t y a n d h i g h e r w h e e l l o a d s . S t r e s s c o n c e n t r a t i o n s

a n d l a r g e a m p l i t u d e s o f s t r e s s e s c a u s e d b y t r a c g i v e r i s e t o f a t i g u e p r o b l e m s i n w e l d e d

j o i n t s a n d s u r f a c i n g , s e e , e . g . , K o l s t e i n & W a r d e n i e r ( 1 9 9 8 ) , ( 1 9 9 9 ) . W h e n i m p r o v i n g t h e

f a t i g u e b e h a v i o r o f a b r i d g e d e c k , o n e s t r a t e g y i s t o r e d u c e t h e s t r e s s r a n g e i n t h e f a t i g u e

s e n s i t i v e d e t a i l s . T r a d i t i o n a l r o a d p a v e m e n t m a y c o n t r i b u t e t o d i s p e r s i o n o f c o n c e n t r a t e d

1



w h e e l l o a d s , h o w e v e r , t r a d i t i o n a l a s p h a l t i c l a y e r s a r e c h a r a c t e r i z e d b y v i s c o e l a s t i c b e h a v -

i o r a n d a r e s t r o n g l y t e m p e r a t u r e d e p e n d e n t , t h u s t h e i r c o n t r i b u t i o n t o d i s t r i b u t i o n o f

c o n c e n t r a t e d l o a d s m i g h t b e i n s u c i e n t . A p r o m i s i n g s o l u t i o n m i g h t b e t o s t i e n t h e

s t r u c t u r e b y c a s t i n g a c e m e n t - b a s e d o v e r l a y t o f o r m a t o p c o m p o s i t e p l a t e . A t y p i c a l

s t i e n i n g s o l u t i o n c o n s i s t s o f a 4 0 - 6 0 m m l a y e r o f b e r r e i n f o r c e d c e m e n t - b a s e d m a t e r i a l

b o n d e d t o t h e s t e e l p l a t e s e e , e . g . , W a l t e r e t a l . ( 2 0 0 4 ) . W h e r e a s c o n v e n t i o n a l c o m p o s i t e

c o n s t r u c t i o n s a c h i e v e c o m p o s i t e a c t i o n b y m e c h a n i c a l f a s t e n e r s , t h e i d e a h e r e i s t o a c h i e v e

c o m p o s i t e a c t i o n o n l y t h r o u g h a d h e s i o n . T h e a d h e s i o n b e t w e e n t h e c e m e n t - b a s e d o v e r l a y

a n d s t e e l i s e n h a n c e d b y s a n d b l a s t i n g o f t h e s t e e l p l a t e , c r e a t i n g a c l e a n s u r f a c e a n d

m i n i m i z i n g t h e r i s k o f d e f e c t s . F o r t h i s p a r t i c u l a r s y s t e m , d e b o n d i n g o f t h e o v e r l a y i s o f

s p e c i a l i n t e r e s t s i n c e t h e c o m p o s i t e s t r e n g t h i s c l o s e l y r e l a t e d t o a f u l l y b o n d e d s t r u c t u r e .

T h e d e b o n d i n g m e c h a n i s m i s s i g n i c a n t l y i n u e n c e d b y v e r t i c a l c r a c k i n g o f t h e o v e r l a y .

D u r i n g e x u r a l c r a c k i n g o f t h e o v e r l a y , t h e d i s t r i b u t i o n o f s h e a r a n d n o r m a l s t r e s s e s a l o n g

t h e s t e e l - c o n c r e t e i n t e r f a c e c h a n g e s d r a m a t i c a l l y f r o m t h a t o f t h e e l a s t i c p h a s e . I n t h e

f r a c t u r e p r o c e s s z o n e o f a n i n t e r f a c i a l c r a c k , h i g h s t r e s s c o n c e n t r a t i o n s d e v e l o p i n a p l a n e

p e r p e n d i c u l a r t o t h e c r a c k a n d c o n s e q u e n t l y l e a d t o d e b o n d i n g .

D e b o n d i n g i s a w e l l - k n o w n p r o b l e m f r o m r e p a i r o f e x i s t i n g s t r u c t u r e s b y t h i n c e m e n t -

b a s e d o v e r l a y s , s e e , e . g . , G r a n j u ( 2 0 0 1 ) , ( 1 9 9 6 ) . I t i s s h o w n t h a t c r a c k i n g o f t h e o v e r l a y

i n d u c e s h i g h i n t e r f a c i a l t e n s i l e s t r e s s e s l e a d i n g t o d e b o n d i n g b e t w e e n t h e t w o m a t e r i a l s .

O n c e i n i t i a t e d , t h e i n t e r f a c i a l c r a c k w i l l p r o p a g a t e i n a m i x e d m o d e c h a r a c t e r i z e d b y

t e n s i l e a n d s h e a r s t r e s s e s a l o n g t h e i n t e r f a c e . T h e u s e o f b e r s i n t h e o v e r l a y m i g h t b e

b e n e c i a l i n o r d e r t o i n c r e a s e t h e v e r t i c a l c r a c k r e s i s t a n c e a n d c o n s e q u e n t l y r e d u c e t h e

r i s k o f d e b o n d i n g . T h i s s t u d y a i m s a t i n v e s t i g a t i n g t h e h y p o t h e s i s t h a t d e b o n d i n g i s

c l o s e l y r e l a t e d t o c r a c k i n g o f t h e o v e r l a y .

F o c u s o f t h i s s t u d y i s o n a s m a l l s t r u c t u r a l e l e m e n t , a c o m p o s i t e b e a m , e x p o s e d t o t h r e e

p o i n t b e n d i n g . T h e s t r u c t u r a l p e r f o r m a n c e g o v e r n e d b y t h e c o m p o s i t e a c t i o n i s i n v e s -

t i g a t e d . I n a d e s i g n s i t u a t i o n a p p l y i n g a n o v e r l a y t o s t i e n a s t e e l d e c k , o n e w o u l d b e

c o n c e r n e d a b o u t s e v e r a l f a c t o r s . D i e r e n t k i n d s o f l o a d i n g m i g h t h a v e a n i m p a c t o n t h e

s t r e n g t h a n d d u r a b i l i t y o f a s t i e n i n g o v e r l a y . I n p a r t i c u l a r , m e c h a n i c a l / e n v i r o n m e n t a l

l o a d i n g a n d s h r i n k a g e o f t h e o v e r l a y w o u l d n e e d t o b e t a k e n i n t o a c c o u n t w h e n d e s i g n i n g

a s t i e n i n g o v e r l a y . E n v i r o n m e n t a l l o a d i n g s u c h a s t e m p e r a t u r e g r a d i e n t s a n d s h r i n k a g e

o f t h e o v e r l a y i s n o t c o n s i d e r e d i n t h i s s t u d y . U n t i l n o w , e x p e r i m e n t a l t e s t s o n c o m p o s i t e

b e a m s w i t h o v e r l a y m a t e r i a l s h a v e b e e n p e r f o r m e d , a l t h o u g h m o s t l y w i t h t h e o v e r l a y o n

t h e c o m p r e s s i v e s i d e o f t h e b e a m s a n d w i t h f o c u s o n c o n c r e t e - c o n c r e t e b o n d s . S i l f w e r -

b r a n d ( 1 9 8 4 ) i n v e s t i g a t e d t e s t s p e c i m e n s ( c o n c r e t e o v e r l a y o n c o n c r e t e b e a m s ) e x p o s e d

t o t h r e e p o i n t b e n d i n g w i t h f o c u s o n i n t e r f a c e b o n d i n g a g e n t s a n d f a i l u r e m o d e s . T h e

s i g n i c a n c e o f t h e p r e s e n t s t u d y i s t o t e s t t h e s t r u c t u r a l p e r f o r m a n c e o f d i e r e n t b e r

r e i n f o r c e d c o m p o s i t e s b o n d e d t o s t e e l .

T o a n a l y z e t h e e x p e r i m e n t s , n u m e r i c a l m o d e l i n g o f t h e t e s t s e t - u p i s c a r r i e d o u t . F u r -

t h e r m o r e , t h e e e c t o f m a t r i x t o u g h n e s s a n d t h e b e r b r i d g i n g e e c t o n d e b o n d i n g w i l l

b e a n a l y z e d . T h r e e b e r r e i n f o r c e d c e m e n t - b a s e d o v e r l a y s a r e t e s t e d a n d m o d e l e d n u -

m e r i c a l l y . O r d i n a r y F i b e r R e i n f o r c e d C o n c r e t e ( F R C ) , F i b e r R e i n f o r c e d D e n s i t R ( F R D )

a n d a n E n g i n e e r e d C e m e n t i t i o u s C o m p o s i t e m a t e r i a l ( E C C ) a r e a l l t e s t e d a n d m o d e l e d .

2



T h e m o d e l i n g w o r k , c a r r i e d o u t u s i n g F E M , a d o p t s t h e s o - c a l l e d F i c t i t i o u s C r a c k M o d e l

( F C M ) f o r t h e t e n s i o n s o f t e n i n g b e h a v i o r o f t h e o v e r l a y m a t e r i a l s d e v e l o p e d b y H i l l e r b o r g

e t a l . ( 1 9 7 6 ) . T h e a d v a n t a g e o f t h e m o d e l i s t h e s i m p l i c i t y a n d t h e g o o d c o r r e l a t i o n

b e t w e e n t h e o r y a n d e x p e r i m e n t a l t e s t s . T h e t w o t y p e s o f c r a c k i n g - v e r t i c a l l y t h r o u g h

t h e o v e r l a y a n d i n t e r f a c e d e b o n d i n g b e t w e e n s t e e l a n d o v e r l a y - w i l l b e d e s c r i b e d b y

t h e F C M . F o r s m a l l c r a c k o p e n i n g s , a n i n t e r f a c i a l f r a c t u r e b e t w e e n s t e e l a n d c o n c r e t e

s h o w s t h e a b i l i t y t o t r a n s f e r s t r e s s e s a c r o s s t h e c r a c k . T h e e x p e r i m e n t s a r e m o d e l e d i n

a t w o - d i m e n s i o n a l n o n l i n e a r n i t e e l e m e n t m o d e l t o s i m u l a t e t h e o v e r l a y c r a c k i n g a n d

d e b o n d i n g . I n p u t f o r t h e m o d e l i n g w o r k i s d e t e r m i n e d i n d e p e n d e n t l y u s i n g u n i a x i a l

t e n s i o n t e s t s .

2 R e s e a r c h S i g n i c a n c e

T h i s s t u d y c o n t r i b u t e s t o o n g o i n g r e s e a r c h p r o j e c t s c o n c e r n i n g t h e i m p r o v e m e n t o f f a t i g u e

l i f e o f t r a d i t i o n a l o r t h o t r o p i c s t e e l b r i d g e d e c k s . T h e p r e s e n t w o r k i s c o n c e r n e d w i t h t h e

p e r f o r m a n c e o f t h r e e d i e r e n t t y p e s o f b e r r e i n f o r c e d c e m e n t i t i o u s c o m p o s i t e s a p p l i e d

a s a s t i e n i n g o v e r l a y t o s t e e l b r i d g e d e c k s . T h e w o r k i s b a s e d o n e x p e r i m e n t a l a s w e l l

a s n u m e r i c a l s t u d i e s , a n d i t c o n t r i b u t e s t o t h e u n d e r s t a n d i n g o f t h e c o m p o s i t e a c t i o n

o f d i e r e n t t y p e s o f b e r b a s e d c e m e n t i t i o u s o v e r l a y s o n s t e e l p l a t e s , w i t h e m p h a s i s o n

f r a c t u r e p r o c e s s e s i n t h e o v e r l a y .

3 E x p e r i m e n t a l P r o c e d u r e s

3 . 1 M a t e r i a l s a n d S p e c i m e n P r e p a r a t i o n

T h e w e l l - k n o w n a d v a n t a g e o f b e r r e i n f o r c e d c o n c r e t e i s i t s a b i l i t y t o s u s t a i n m a j o r d e -

f o r m a t i o n a f t e r t h e r s t c r a c k i s f o r m e d . T h e b e r s w i l l t y p i c a l l y r e m a i n u n b r o k e n a f t e r

c r a c k i n i t i a t i o n a n d t h e b e r s t h a t c r o s s a c r a c k w i l l r e s i s t f u r t h e r o p e n i n g . D e p e n d i n g

o n t h e c r a c k b r i d g i n g e e c t , b e r r e i n f o r c e d c o m p o s i t e s c a n s h o w d i e r e n t f a i l u r e m o d e s .

I f t h e a v e r a g e b e r b r i d g i n g e e c t i s i n c r e a s i n g d u r i n g c r a c k i n i t i a t i o n a n d p r o p a g a t i o n ,

m u l t i p l e c r a c k s c a n f o r m . T h i s b e h a v i o r i s a l s o k n o w n a s s t r a i n h a r d e n i n g . O n t h e o t h e r

h a n d , i f t h e b e r s c a n n o t c a r r y m o r e l o a d a f t e r t h e f o r m a t i o n o f t h e r s t c r a c k , t h e n f u r -

t h e r d e f o r m a t i o n i s g o v e r n e d b y o p e n i n g o f a s i n g l e c r a c k . T h i s b e h a v i o r i s c a l l e d t e n s i o n

s o f t e n i n g . I n t h e f o l l o w i n g , c o m p o s i t e b e a m s c o m p r i s i n g m a t e r i a l s w i t h t e n s i o n s o f t e n i n g

a n d s t r a i n h a r d e n i n g b e h a v i o r a r e e x a m i n e d . T w o m a t e r i a l s , c l a s s i e d a s t e n s i o n s o f t e n i n g

m a t e r i a l s , a r e t e s t e d : F i b e r R e i n f o r c e d C o n c r e t e ( F R C ) a n d F i b e r R e i n f o r c e d D e n s i t R

( F R D ) . A m a t e r i a l k n o w n a s E n g i n e e r e d C e m e n t i t i o u s C o m p o s i t e ( E C C ) i s a l s o t e s t e d

a n d c a n b e c h a r a c t e r i z e d a s a s t r a i n h a r d e n i n g m a t e r i a l , s e e , e . g . , L i ( 2 0 0 2 ) .

T h e E C C m a t e r i a l c o n t a i n s 2 % b y v o l u m e p o l y - v i n y l - a l c o h o l ( P V A ) b e r s , w i t h a l e n g t h

o f 8 m m , i n r a n d o m o r i e n t a t i o n a l o n g w i t h s t a n d a r d m o r t a r m a t r i x c o m p o n e n t s , c e m e n t ,

n e a g g r e g a t e ( 0 . 3 m m m a x i m u m g r a i n s i z e ) , w a t e r , a n d v a r i o u s a d m i x t u r e s t o i m p r o v e

t h e f r e s h p r o p e r t i e s o f t h e m i x t u r e , c f . T a b l e 1 . T h e E C C m a t e r i a l e x h i b i t s s t r a i n -

3



C e m e n t W a t e r S u p e r P l a s t i c i z e r F l y a s h S a n d P V A b e r s

0 . 1 - 0 . 3 m m lf = 8 m m

[k g / m

3] [k g / m

3] [k g / m

3] [k g / m

3] [k g / m

3] [k g / m

3]5 8 3 2 9 8 1 7 . 5 7 0 0 4 6 7 2 6

T a b l e 1 : M i x d e s i g n f o r E C C

C e m e n t W a t e r F l y a s h S i l i c a S u p e r A g g r . A g g r . A g g r . S t e e l b e r s

F u m e P l a s t i c i z e r 0 - 4 m m 4 - 8 m m 8 - 1 6 m m lf = 3 0 m m

[k g / m

3] [k g / m

3] [k g / m

3] [k g / m

3] [k g / m

3] [k g / m

3] [k g / m

3] [k g / m

3] [k g / m

3]2 4 5 1 4 0 9 4 . 5 1 0 . 5 2 . 8 7 5 3 . 9 4 5 2 . 9 5 7 0 . 3 7 8

T a b l e 2 : M i x d e s i g n f o r F R C

h a r d e n i n g b e h a v i o r i n t e n s i o n . T h i s b e h a v i o r i s a c h i e v e d t h r o u g h a m i c r o - m e c h a n i c d e s i g n

a p p r o a c h , ( L i 2 0 0 2 ) . T h e F R C m a t e r i a l u s e d i n t h i s s t u d y w a s c o m p o s e d o f c e m e n t , c o a r s e

a g g r e g a t e s ( 1 6 m m m a x i m u m g r a i n s i z e ) , n e a g g r e g a t e s , a d m i x t u r e s , a n d w a t e r . A b e r

v o l u m e o f 1 % h o o k e d - e n d s t e e l b e r s , w i t h a l e n g t h o f 3 0 m m , w a s a p p l i e d , c f . T a b l e 2 f o r

m i x d e s i g n . T h e F R D m a t e r i a l c o n t a i n s a m a x i m u m g r a i n s i z e o f 2 m m , 2 % b y v o l u m e

s t e e l b e r , w i t h a l e n g t h o f 8 m m , a n d D e n s i t

R b i n d e r , t y p e J o i n t - C a s t , c f . T a b l e 3 f o r

m i x d e s i g n . A l l t h e m a t e r i a l s u s e d i n t h i s s t u d y e x h i b i t s e l f - c o m p a c t i n g p r o p e r t i e s a n d

n o m e c h a n i c a l v i b r a t i o n i s n e e d e d d u r i n g c a s t i n g . I t i s b e l i e v e d t h a t t h i s f e a t u r e i s o f

g r e a t i m p o r t a n c e t o a c h i e v e a g o o d b o n d b e t w e e n t h e o v e r l a y a n d u n d e r l y i n g s t e e l p l a t e .

V i b r a t i o n m i g h t c a u s e w a t e r t o s e p a r a t e f r o m t h e m i x a n d c r e a t e a w e a k i n t e r f a c e b e t w e e n

t h e f r e s h m i x a n d s t e e l p l a t e . A l l t h e s t e e l p l a t e s w e r e p r e p a r e d b y a n i n i t i a l s a n d b l a s t i n g

f o l l o w e d b y c a s t i n g o f t h e c e m e n t - b a s e d l a y e r w i t h i n 2 4 h o u r s . S a n d b l a s t i n g t e c h n i q u e i s

a d o p t e d t o e n s u r e a c l e a n s u r f a c e w i t h a m i n i m u m r i s k o f a d h e s i o n d e f e c t s .

D e n s i t B i n d e r W a t e r A g g r e g a t e s S t e e l b e r s

T y p e : J o i n t c a s t 0 - 2 m m lf = 8 m m

[k g / m

3] [k g / m

3] [k g / m

3] [k g / m

3]1 0 5 7 2 1 0 1 3 0 0 1 5 7

T a b l e 3 : M i x d e s i g n f o r F R D

3 . 2 E x p e r i m e n t a l S e t - u p

T h e p r e s e n t s t u d y i s l i m i t e d t o m o n o t o n i c m e c h a n i c a l l o a d i n g . A p p l y i n g m e c h a n i c a l l o a d

t o t h e c o m p o s i t e s y s t e m m i g h t i n t r o d u c e c r a c k i n g o f t h e o v e r l a y a n d w o u l d b e o f i n t e r e s t

i n a d e s i g n s i t u a t i o n . C r a c k f o r m a t i o n i n t h e o v e r l a y a n d a s s o c i a t e d d e b o n d i n g u n d e r

n e g a t i v e b e n d i n g i s o f p a r t i c u l a r c o n c e r n . H e n c e , a t e s t s e t - u p i s u s e d , s i m u l a t i n g a

p a r t o f a s t i e n i n g o v e r l a y c a s t o n a s t e e l b r i d g e d e c k , l o a d e d i n n e g a t i v e b e n d i n g . A

c o m p o s i t e b e a m , t u r n e d u p - s i d e d o w n , s u b j e c t e d t o t h r e e - p o i n t b e n d i n g i s u s e d a s t h e

p r i m a r y t e s t s e t - u p . T h e c o m p o s i t e b e a m c o n s i s t s o f a s t e e l p l a t e a n d a t h i n c e m e n t -

b a s e d l a y e r . A d d i t i o n a l l y , i n a d e s i g n s i t u a t i o n c o n c e r n i n g d u r a b i l i t y , c r a c k s p a c i n g a n d

4



c r a c k w i d t h i s o f s i g n i c a n c e . T h u s , w h e n t e s t i n g t h e E C C c o m p o s i t e b e a m s , e m p h a s i s

i s a l s o p u t o n o b s e r v a t i o n o f c r a c k s p a c i n g , c r a c k w i d t h , a n d d e b o n d i n g b e t w e e n t h e s t e e l

p l a t e a n d o v e r l a y . C r a c k w i d t h d u r i n g l o a d i n g i s m o n i t o r e d u s i n g a m i c r o s c o p e ( O l y m p u s

O V M 1 0 0 0 N ) w i t h a m a g n i c a t i o n o f 2 0 0 t i m e s . A s i n g l e c r a c k i s s e l e c t e d a n d m e a s u r e d

t h r o u g h o u t t h e e x p e r i m e n t . T e s t i n g o f t h e c o m p o s i t e b e a m i s p e r f o r m e d i n c l o s e d - l o o p

c o n t r o l , u s i n g t h e s t r o k e o f t h e c r o s s h e a d a s t h e c o n t r o l l i n g p a r a m e t e r . T h e s e t - u p i s

d e p i c t e d i n F i g u r e 1 .

F i g u r e 1 : E x p e r i m e n t a l s e t - u p f o r E C C c o m p o s i t e b e a m s , t h e c r a c k w i d t h o f o v e r l a y

c r a c k s a r e m o n i t o r e d a n d d e b o n d i n g i s m e a s u r e d . T h e t e s t s e t - u p s i m u l a t e s a p a r t o f a

s t i e n i n g o v e r l a y c a s t o n a s t e e l b r i d g e d e c k l o a d e d i n n e g a t i v e b e n d i n g .

D e b o n d i n g b e t w e e n t h e s t e e l p l a t e a n d E C C i s m e a s u r e d u s i n g t w o L V D T s , t e r m e d

O m e g a - g a g e s , m o u n t e d o n e i t h e r s i d e , 5 0 m m f r o m t h e l o a d p o i n t . T h e O m e g a - g a g e s

a r e a t t a c h e d w i t h o n e e n d t o t h e s t e e l p l a t e a n d t h e o t h e r t o t h e E C C l a y e r . T h e y m e a -

s u r e a n y e l a s t i c d e f o r m a t i o n b e t w e e n t h e t w o e n d p o i n t s a n d a d d i t i o n a l l y c a p t u r e a n y

s t e e l - E C C i n t e r f a c i a l c r a c k i n g .

( a ) ( b )

F i g u r e 2 : ( a ) P l a c i n g o f O m e g a - g a g e s , b e a m v i e w e d f r o m e n d . ( b ) A c t u a l p i c t u r e o f

O m e g a - g a g e s o n b e a m

T h e s t e e l - F R C a n d s t e e l - F R D c o m p o s i t e b e a m s f o r m e d o n e d i s c r e t e c r a c k i n t h e o v e r l a y

a t m i d s p a n , w h e r e t h e n e g a t i v e b e n d i n g m o m e n t a t t a i n s i t s m a x i m u m . S i n c e t h e c r a c k

h a r d l y f o r m s e x a c t l y a t m i d s p a n , a c k n o w l e d g e d t h r o u g h a s e r i e s o f p i l o t t e s t s , a n o t c h i s

a p p l i e d i n o r d e r t o r e c o r d t h e c r a c k o p e n i n g d u r i n g l o a d i n g . T h e n o t c h w i t h a d e p t h o f

5



1 / 8 o f t h e b e a m h e i g h t i s s a w n p r i o r t o t e s t i n g , w h i c h c o n s e q u e n t l y w i l l l o w e r t h e p e a k

l o a d b u t i s o f n o s i g n i c a n c e i n t h e p r e s e n t s t u d y . A c l i p g a g e i s m o u n t e d o n t h e b o t t o m

f a c e o f t h e b e a m , a c r o s s t h e n o t c h , m e a s u r i n g t h e c r a c k o p e n i n g d i s p l a c e m e n t ( C O D ) o f

t h e v e r t i c a l o v e r l a y c r a c k . T o e n s u r e s t a b l e c r a c k g r o w t h t h r o u g h o u t t h e e x p e r i m e n t , t h i s

s i g n a l i s u s e d a s t h e c o n t r o l l i n g p a r a m e t e r i n t h e c l o s e d - l o o p c o n t r o l o f t h e e x p e r i m e n t .

F u r t h e r m o r e , a t t e n t i o n i s p u t o n i n t e r f a c i a l d e b o n d i n g a n d t h e o p e n i n g o f t h e o v e r l a y c r a c k

( C O D ) . A g a i n , t o d e t e c t a n d m e a s u r e p o s s i b l e d e b o n d i n g , t w o O m e g a - g a g e s a r e m o u n t e d

o n e i t h e r s i d e , 5 0 m m f r o m t h e l o a d p o i n t t o r e c o r d i n t e r f a c i a l c r a c k i n g , a s d e p i c t e d i n

F i g u r e 2 . T h e s e t - u p f o r s t e e l - F R C a n d s t e e l - F R D b e a m s i s d e p i c t e d i n F i g u r e 3 .

F i g u r e 3 : E x p e r i m e n t a l s e t - u p f o r F R D a n d F R C c o m p o s i t e b e a m s . S i n c e a s i n g l e c r a c k

i s f o r m e d a t m i d s p a n a c l i p g a u g e i s p l a c e d t o m e a s u r e t h e c r a c k o p e n i n g d i s p l a c e m e n t

( C O D ) . T h e t e s t s e t - u p s i m u l a t e s a p a r t o f a s t i e n i n g o v e r l a y c a s t o n a s t e e l b r i d g e d e c k

l o a d e d i n n e g a t i v e b e n d i n g .

3 . 3 U n i a x i a l T e s t i n g o f M a t e r i a l s i n T e n s i o n

T o d e t e r m i n e t h e t e n s i o n p r o p e r t i e s o f t h e m a t e r i a l s t e s t e d , u n i a x i a l t e n s i o n t e s t s h a v e

b e e n p e r f o r m e d . T w o d i e r e n t t e s t s e t - u p s w e r e u s e d . T h e p r i n c i p l e o f t e s t i n g a t e n s i o n -

s o f t e n i n g m a t e r i a l c o m p a r e d t o a s t r a i n h a r d e n i n g m a t e r i a l d i e r s s l i g h t l y . F o r t h e t e n s i o n

s o f t e n i n g m a t e r i a l s F R C a n d F R D , a c y l i n d r i c a l s p e c i m e n w i t h a h e i g h t o f 3 0 0 m m a n d

d i a m e t e r o f 1 5 0 m m , w a s u s e d , a c c o r d i n g t o R I L E M ( 2 0 0 1 ) . T o c a p t u r e t h e s t r e s s - c r a c k

o p e n i n g r e l a t i o n s h i p a n o t c h i s s a w n i n t h e m i d d l e . F u r t h e r d e t a i l s o n t h e s e t - u p c a n b e

f o u n d i n Ø s t e r g a a r d ( 2 0 0 3 ) .

I n t h e e x p e r i m e n t a d e f o r m a t i o n u i s m e a s u r e d a c r o s s t h e s p e c i m e n , i n t h i s s p e c i c c a s e

a g a g e l e n g t h o f 4 0 m m i s u s e d . I n o r d e r t o t r a n s f o r m t h e d e f o r m a t i o n m e a s u r e m e n t u t o

t h e c r a c k o p e n i n g w a m e t h o d i s u s e d a c c o r d i n g t o f o r m u l a ( 1 ) .

w = uavg(σ)− uavg(σmax) +σmax − σ

K i( 1 )

w h e r e K i i s t h e i n i t i a l s l o p e o f t h e c u r v e a n d uavg i s t h e a v e r a g e s i g n a l r e c o r d e d a c r o s s t h e

g a g e l e n g t h . F i g u r e 4 s h o w s t h e p r i n c i p l e o f t r a n s f o r m i n g t h e d e f o r m a t i o n s i g n a l r e c o r d e d

t o a c r a c k o p e n i n g .

6



F i g u r e 4 : T r a n s f o r m a t i o n o f t h e d e f o r m a t i o n u m e a s u r e d i n t h e e x p e r i m e n t t o t h e c r a c k -

o p e n i n g m e a s u r e m e n t w , c f . F o r m u l a 1

I n t h e c a s e o f a s t r a i n h a r d e n i n g m a t e r i a l , m u l t i p l e c r a c k s a r e e x p e c t e d , a n d t h e e l o n g a t i o n

o f t h e m a t e r i a l c a n b e i n t e r p r e t e d a s s t r a i n . T h e p r i n c i p l e o f t h e t e s t s e t - u p c a n b e f o u n d

i n L i e t a l . ( 2 0 0 2 ) . I n t h i s s t u d y , o n l y t h e s t r a i n h a r d e n i n g p h a s e i s o f i n t e r e s t a n d w h e n

a m a c r o c r a c k d e v e l o p s t h e u n i a x i a l t e s t i s s t o p p e d .

F o r t h e s t e e l u s e d i n t h e t e s t s e t - u p s , t h e e l a s t i c m o d u l u s i s a p p r o x i m a t e l y 2 1 0 G P a

w i t h a y i e l d s t r e s s o f 3 1 0 M P a ( m a n u f a c t u r e r ' s d a t a ) . T h e s e v a l u e s a r e i m p o r t a n t i n t h e

E C C c o m p o s i t e b e a m t e s t s a s t h e s t e e l w i l l r e a c h i t s y i e l d s t r e s s , a s w i l l b e s h o w n i n t h e

f o l l o w i n g .


4 . 1 R e s u l t s f r o m U n i a x i a l T e s t s

T o e x p l a i n t h e r e s u l t s i n t h e c o m p o s i t e b e a m t e s t s , t h e t h r e e m a t e r i a l s a n d t h e i r p r o p e r t i e s

i n t e n s i o n h a v e b e e n c h a r a c t e r i z e d u s i n g u n i a x i a l t e n s i o n t e s t s . R e s u l t s f r o m t h e t e n s i o n

s o f t e n i n g m a t e r i a l s , F R C a n d F R D a r e g i v e n i n F i g u r e 5 ( a ) a n d E C C i s s h o w n i n F i g u r e

5 ( b ) .

N o t e t h a t i n t h e c a s e o f t e n s i o n s o f t e n i n g m a t e r i a l s , t h e r e s u l t s a r e g i v e n a s l o a d v e r s u s

c r a c k o p e n i n g a n d i n t h e E C C c a s e , t h e r e s u l t i s g i v e n a s l o a d v e r s u s s t r a i n . A t o t a l o f

t h r e e s p e c i m e n s f o r e a c h i n d i v i d u a l m a t e r i a l h a v e b e e n t e s t e d . T h e s t a n d a r d d e v i a t i o n

i s g i v e n i n t h e F i g u r e s y m b o l i z e d w i t h e r r o r b a r s . C o m p a r i n g t h e t w o t e n s i o n s o f t e n i n g

m a t e r i a l s , i t c a n b e s e e n t h a t F R D e x h i b i t s a h i g h e r r s t c r a c k i n g s t r e n g t h t h a n t h a t

o f F R C . A f t e r p e a k s t r e s s , F R D d r o p s s i g n i c a n t l y m o r e t h a n F R C , m a i n l y d u e t o i t s

m o r e b r i t t l e m a t r i x . T h e d i e r e n c e i n m a t r i x s t r e n g t h a n d b r i t t l e n e s s i s m a i n l y d u e t o

d i e r e n t w a t e r c e m e n t r a t i o s , F R D h a s a w / c - r a t i o a r o u n d 0 . 2 w h e r e a s t h e w / c - r a t i o f o r

F R C l i e s a r o u n d 0 . 5 . A f t e r c r a c k d e v e l o p m e n t , t h e e e c t o f b e r s t r a n s f e r r i n g s t r e s s e s

a c r o s s t h e c r a c k i s r e v e a l e d a s a p l a t e a u i n t h e s t r e s s - C M O D c u r v e a f t e r a c r a c k o p e n i n g

o f a p p r o x i m a t e l y 0 . 0 2 5 m m . T h e u l t i m a t e c r a c k o p e n i n g f o r F R D a n d F R C a r e 4 a n d 2

m m , r e s p e c t i v e l y . T h e E C C m a t e r i a l s h o w s a s t r a i n h a r d e n i n g b e h a v i o r a n d a s s e e n f r o m

t h e F i g u r e , i t e x h i b i t s a n u l t i m a t e s t r a i n c a p a c i t y o f a r o u n d 2 - 3 % . A t t h i s u l t i m a t e s t a g e ,

7



0 0.05 0.1 0.15 0.20

1

2

3

4

5

6

7

S t r e s s [ M P

a ]

CMOD [mm]

(a)

0 1 2 30

1

2

3

4

5

6

7

S t r e s s [ M P

a ]

Strain [%]

(b)

FRD

FRC

ECC

F i g u r e 5 : R e s u l t s f r o m u n i a x i a l t e n s i o n t e s t i n g . ( a ) M a t e r i a l s F R C a n d F R D s h o w n a s

s t r e s s v e r s u s c r a c k m o u t h o p e n i n g r e l a t i o n s h i p ( C M O D ) ( b ) S t r a i n h a r d e n i n g m a t e r i a l

E C C s h o w n a s s t r e s s v s . s t r a i n .

a c r a c k w i l l l o c a l i z e a n d t h e b e h a v i o r c a n t h e n b e d e s c r i b e d i n a s t r e s s - c r a c k o p e n i n g

r e l a t i o n s h i p .

4 . 2 R e s u l t s f r o m C o m p o s i t e B e a m T e s t s

T h e f a i l u r e m o d e o f t h e E C C c o m p o s i t e b e a m i s c h a r a c t e r i z e d b y f o r m a t i o n o f m u l t i p l e

c r a c k s a l o n g t h e b e a m , w h e r e a s i n t h e c a s e o f F R D a n d F R C c o m p o s i t e b e a m s , a s i n g l e

c r a c k f o r m s a t m i d s p a n . D u r i n g t h e e x p e r i m e n t , a t t e n t i o n h a s b e e n p a i d t o t h e e x u r a l

c r a c k g r o w t h . I n t h e E C C c a s e , t h e c r a c k w i d t h w a s o b s e r v e d d u r i n g t h e e x p e r i m e n t

a n d f o r t h e F R D a n d F R C b e a m s , t h e c r a c k o p e n i n g d i s p l a c e m e n t ( C O D ) f o r t h e s i n g l e

c r a c k a t m i d s p a n w a s m e a s u r e d . I n F i g u r e 6 ( a ) , l o a d v e r s u s C O D i s p l o t t e d f o r F R D

a n d F R C b e a m s a l o n g w i t h t h e s t a n d a r d d e v i a t i o n r e p r e s e n t e d b y e r r o r b a r s . F i g u r e 6 ( b )

s h o w s t h e l o a d - d e e c t i o n d i a g r a m o f t h e E C C c o m p o s i t e b e a m a l o n g w i t h t h e c r a c k w i d t h

r e p r e s e n t e d b y a s e c o n d y - a x i s .

T h e d i e r e n c e i n s c a l i n g o f t h e t w o y - a x e s r e p r e s e n t i n g t h e l o a d i n F i g u r e s 6 ( a ) a n d ( b ) ,

s h o u l d b e n o t e d w h e n c o m p a r i n g t h e t w o r e s u l t s . I t i s s e e n f r o m F i g u r e s 6 ( a ) a n d ( b ) , t h a t

c r a c k i n g o f E C C a n d F R D i n i t i a t e s a r o u n d t h e s a m e l o a d , a p p r o x i m a t e l y 2 0 0 0 N , w h e r e a s

i n t h e c a s e o f F R C , a l o w e r c r a c k i n g s t r e n g t h i s o b s e r v e d . C o m p a r i n g F R D a n d F R C , i t

i s o b s e r v e d t h a t i n b o t h c a s e s , a p l a t e a u i s p r e s e n t f o r a c e r t a i n l o a d v a l u e . F R D h a s a

h i g h e r f r a c t u r e e n e r g y , h e n c e a h i g h e r p l a t e a u . O n t h e c o n t r a r y , f o r t h e E C C c o m p o s i t e

b e a m , a n i n c r e a s e o f t h e l o a d i s o b s e r v e d f o r i n c r e a s e d d e e c t i o n u n t i l s o m e p o i n t w h e r e

f a i l u r e o c c u r s . F o r t h e E C C b e a m , t w o f a i l u r e m o d e s a r e p o s s i b l e : ( i ) l o c a l i z a t i o n o f a

c r a c k a n d ( i i ) r e a c h i n g o f t h e y i e l d s t r a i n i n t h e s t e e l p l a t e . N u m e r i c a l s t u d i e s , s e e l a t e r

i n t h i s p a p e r , s h o w t h a t e v e n f o r s m a l l s t r a i n c a p a c i t i e s o f t h e E C C , l e s s t h a n 1 % , i n t h i s

8



0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

COD [mm]

(a)

L o a d [ k N

]

0 10 20 300

1

2

3

4

5

6

Deflection [mm]

(b)

L o a d [ k N

]

0 10 20 300

5

10

15

20

25

30

C r a c k W i d t h

[ µ m ]

FRD

FRC

ECC

Crack

F i g u r e 6 : E x p e r i m e n t a l r e s u l t s f r o m c o m p o s i t e b e a m t e s t s ( a ) f o r l a y e r u s i n g t e n s i o n

s o f t e n i n g m a t e r i a l s F R C a n d F R D p r e s e n t e d i n a l o a d v e r s u s C O D d i a g r a m . ( b ) f o r

l a y e r u s i n g a s t r a i n h a r d e n i n g m a t e r i a l E C C i n a l o a d v e r s u s d e e c t i o n d i a g r a m w i t h a

s e c o n d y - a x i s s h o w i n g a s s o c i a t e d c r a c k w i d t h . N o t e t h e d i e r e n t s c a l i n g o n t h e y - a x i s

r e p r e s e n t i n g t h e l o a d .

s p e c i c t e s t s e t - u p , y i e l d i n g o f t h e s t e e l p l a t e i s t h e m a i n c a u s e o f f a i l u r e . Y i e l d i n g o f t h e

s t e e l p l a t e w i l l s u b s e q u e n t l y c a u s e c r a c k l o c a l i z a t i o n i n t h e E C C . I n t h i s s t u d y , y i e l d i n g

o f t h e s t e e l p l a t e i s t h e g o v e r n i n g f a i l u r e m o d e a n d i s t a k e n i n t o a c c o u n t .

F i g u r e 6 ( b ) s h o w s t h e c r a c k w i d t h o f t h e r s t c r a c k o f t h e E C C p l o t t e d v e r s u s t h e d e e c -

t i o n . T h e r s t f e w c r a c k s f o r m a t a l o w l o a d l e v e l a n d a s t h e b e a m i s m o n i t o r e d d u r i n g

t h e w h o l e e x p e r i m e n t , o n e c r a c k i s s e l e c t e d a n d m o n i t o r e d u n t i l c o m p l e t e f a i l u r e o f t h e

b e a m . I t i s s e e n t h a t t h e c r a c k w i d t h r e m a i n s i n t h e r a n g e o f 1 0 - 3 0 m i c r o n .

D e b o n d i n g o f t h e o v e r l a y , m e a s u r e d u s i n g O m e g a - g a g e s , i s s h o w n f o r t h e F R C , F R D a n d

E C C c o m p o s i t e b e a m s i n F i g u r e 7 . T h e s i g n i c a n t r e l a t i o n b e t w e e n a n o v e r l a y m a c r o -

c r a c k a n d d e b o n d i n g i s c l e a r l y i l l u s t r a t e d i n F i g u r e 7 a n d s u p p o r t s t h e h y p o t h e s i s , s e e ,

e . g . , G r a n j u ( 2 0 0 1 ) , ( 1 9 9 6 ) , t h a t d e b o n d i n g i s c l o s e l y r e l a t e d t o d e f e c t s i n t h e o v e r l a y .

T h e F R C a n d F R D c o m p o s i t e b e a m s d e v e l o p a m a c r o - c r a c k a t a n e a r l y s t a g e c o m p a r e d

t o t h a t o f E C C a n d d e b o n d i n g i s o b s e r v e d f o r a d e e c t i o n a r o u n d 2 m m . I n c o n t r a s t ,

t h e E C C - s t e e l i n t e r f a c e r e m a i n s i n t a c t e v e n f o r l a r g e d e f o r m a t i o n s . I t i s c l e a r l y o b s e r v e d

t h a t t h e E C C c o m p o s i t e b e a m e x h i b i t s a m o r e d u c t i l e b e h a v i o r w h e n e x p o s e d t o n e g a t i v e

b e n d i n g .

A c t u a l p i c t u r e s o f t h e c o m p o s i t e b e a m s p e c i m e n s a r e s h o w n i n F i g u r e 8 ( a ) - ( b ) . I n t h e

r s t p i c t u r e a F R D s p e c i m e n i s s h o w n . A s i n g l e m a c r o c r a c k i s o b s e r v e d a l o n g w i t h

f o r m a t i o n o f i n t e r f a c i a l d e b o n d i n g . I n t h e o t h e r c a s e , h a v i n g a n E C C c o m p o s i t e b e a m ,

m u l t i p l e c r a c k s a r e f o r m e d a n d d e b o n d i n g h a p p e n s w h e n a c r a c k l o c a l i z e s . F o r m a t i o n o f

m u l t i p l e c r a c k s i n t h e c o m p o s i t e b e a m i s m a r k e d a n d c l e a r l y s e e n i n t h e p i c t u r e .

9



0 5 10 15 200

1

2

3

4

5

6

L o a d [ k N

]

Deflection [mm]0 5 10 15 20

0

2

4

6

8

10

12

0 5 10 15 200

2

4

6

8

10

12

0 5 10 15 200

2

4

6

8

10

12

D e b o n d i n g [ m m ]

Load/defECCFRDFRC

DebondingECCFRDFRC

F i g u r e 7 : R e s u l t s f r o m t h r e e r e p r e s e n t a t i v e t e s t s o f c o m p o s i t e b e a m s w i t h F R D , F R C

a n d E C C o v e r l a y m a t e r i a l s . R e s u l t s a r e p l o t t e d i n a l o a d v e r s u s d i s p l a c e m e n t d i a g r a m

w i t h a s e c o n d y - a x i s s h o w i n g t h e c o r r e s p o n d i n g d e b o n d i n g .

5 N u m e r i c a l M o d e l l i n g

T h e e x p e r i m e n t s d e s c r i b e d i n t h e p r e v i o u s s e c t i o n s a r e a n a l y z e d u s i n g F E M . T h e a i m

o f t h e n i t e e l e m e n t a n a l y s i s i s t o e x t e n d t h e i n t e r p r e t a t i o n o f t h e r e s u l t s a n d o b s e r v a -

t i o n s o b t a i n e d i n t h e t e s t s t o g a i n a b e t t e r u n d e r s t a n d i n g o f t h e b e h a v i o r o f t h e p r e s e n t

c o m p o s i t e b e a m s . T h e F E m o d e l s h o u l d b e a b l e t o s i m u l a t e t h e c o m p o s i t e b e a m s i n a r e -

a l i s t i c w a y . P h e n o m e n a s u c h a s c r a c k i n g o f t h e o v e r l a y a n d b o n d b e t w e e n t h e o v e r l a y a n d

s t e e l p l a t e h a v e t o b e t a k e n i n t o a c c o u n t . F o r t h e E C C c o m p o s i t e b e a m , y i e l d i n g o f t h e

s t e e l p l a t e a l s o n e e d s t o b e t a k e n i n t o c o n s i d e r a t i o n . T h e p r e s e n t n o n l i n e a r n i t e - e l e m e n t

a n a l y s i s w a s c a r r i e d o u t w i t h D I A N A ( 2 0 0 3 ) .

5 . 1 F i n i t e E l e m e n t M o d e l

T h e m o d e l c o n s i s t s o f a s t e e l p l a t e a n d a n o v e r l a y e i t h e r o f F R C , F R D o r E C C . T h e

c o m p o s i t e b e a m i s m o d e l e d a s a p l a n e s t r u c t u r e a n d o n l y t w o d i m e n s i o n s a r e c o n s i d e r e d .

A s t h e b e a m i s l o a d e d , c r a c k i n g c a n o c c u r i n t h e o v e r l a y a n d a t t h e i n t e r f a c e b e t w e e n t h e

o v e r l a y a n d s t e e l p l a t e . T h e c o n s t i t u t i v e m o d e l i n g o f t h e o v e r l a y d e p e n d s o n w h e t h e r t h e

o v e r l a y m a t e r i a l i s t e n s i o n s o f t e n i n g o r s t r a i n h a r d e n i n g . I n t h e c a s e o f t e n s i o n - s o f t e n i n g

m a t e r i a l s , F R C a n d F R D , c r a c k i n g i s t r e a t e d a s a d i s c r e t e p r o c e s s a n d c r a c k i n g o f t h e

o v e r l a y , l o c a t e d a t t h e n o t c h , t a k e s p l a c e a t m i d s p a n . M i d s p a n c r a c k i n g i s m o d e l e d u s i n g

s t a n d a r d i n t e r f a c e e l e m e n t s a v a i l a b l e i n t h e s o f t w a r e p a c k a g e D I A N A ( 2 0 0 3 ) . I n t h e

c a s e o f a s t r a i n - h a r d e n i n g m a t e r i a l , m u l t i p l e c r a c k i n g o f t h e o v e r l a y o c c u r s a n d s t a n d a r d

s h e l l e l e m e n t s a r e u s e d , a p p l y i n g c r a c k i n g i n a s m e a r e d m a n n e r . F i n a l l y , t h e i n t e r f a c i a l

c r a c k i n g b e t w e e n t h e o v e r l a y a n d s t e e l p l a t e i s i n a l l c a s e s t r e a t e d a s a d i s c r e t e p r o c e s s

u s i n g s t a n d a r d i n t e r f a c e e l e m e n t s . T h e m o d e l i n g c o n c e p t a n d t h e a p p l i e d m e s h a r e g i v e n

1 0



5 . 3 M o d e l l i n g o f I n t e r f a c e

T o s i m u l a t e t h e i n t e r f a c e b e t w e e n t h e o v e r l a y a n d u n d e r l y i n g s t e e l p l a t e , a m o d e l b a s e d o n

f r a c t u r e m e c h a n i c s i s u s e d . T h e s t e e l / o v e r l a y d e b o n d i n g p h e n o m e n o n i s o f t e n c o n s i d e r e d

a d i s c r e t e p r o c e s s a n d m o d e l e d a s t a k i n g p l a c e a t t h e b i m a t e r i a l i n t e r f a c e . A s d e b o n d i n g

t a k e s p l a c e i n a m i x e d m o d e , r e p r e s e n t e d b y n o r m a l a n d s h e a r s t r e s s e s , a m i x e d m o d e

i n t e r f a c e m o d e l i s a p p l i e d . T h e m i x e d m o d e m o d e l t a k e s i n t o a c c o u n t g r a d u a l s o f t e n i n g

o f s h e a r a n d n o r m a l s t r e s s e s f o r n o r m a l a n d s l i d i n g c r a c k p r o p a g a t i o n . T h e n o r m a l s t r e s s

i n M o d e I i s d e n o t e d σ a n d c r a c k o p e n i n g n o r m a l t o t h e i n t e r f a c e i s d e n o t e d δ n . S i m i l a r l y ,

M o d e I I f r a c t u r e i s d e s c r i b e d b y t h e s h e a r s t r e s s , τ , a n d t h e c r a c k s l i d i n g d e f o r m a t i o n , δ t .

T h e c o n s t i t u t i v e l a w o f t h e m i x e d m o d e m o d e l i s b a s e d o n a c o u p l i n g o f t h e u n i a x i a l σ−δ nr e l a t i o n s h i p a n d τ −δ t r e l a t i o n s h i p , c o r r e s p o n d i n g t o M o d e I a n d I I f r a c t u r e , r e s p e c t i v e l y .

C o n s i d e r a t w o d i m e n s i o n a l c o n g u r a t i o n , w h e r e t h e s t r e s s e s a c t i n g o n t h e i n t e r f a c e a r e

r e l a t e d t o t h e r e l a t i v e d i s p l a c e m e n t s , i . e . , c r a c k o p e n i n g a n d c r a c k s l i d i n g . T h e r e l a t i o n

i s g i v e n b y :

σ

τ

=

D11 D12

D21 D22

δ nδ t

( 2 )

I n t h e c r a c k e d s t a t e t h e c o u p l i n g b e t w e e n t h e n o r m a l a n d s h e a r d e f o r m a t i o n i s d e s c r i b e d

t h r o u g h t h e s t i n e s s c o m p o n e n t s Dij . T w o b i l i n e a r c u r v e s i n p u r e M o d e I a n d I I d e f o r -

m a t i o n i s g i v e n a s i n p u t . T h e m o d e l u t i l i z e s t h e f o l l o w i n g c r i t e r i o n t o d e s c r i b e t h e s t a t e

o f d e f o r m a t i o n f o r e v e r y p o i n t o n t h e b i l i n e a r c u r v e s :

σ

σmax

m+ τ

τ max

n= 1.0 ( 3 )

w h e r e σmax a n d τ max a r e t h e m a x i m u m c r a c k i n g s t r e s s e s , a c c o r d i n g t o t h e t w o b i l i n e a r

u n i a x i a l c u r v e s d e n e d i n p u r e M o d e I a n d I I . F o r f u r t h e r d e t a i l s o n t h e m o d e l , s e e ( W a l t e r ,

O l e s e n & S t a n g 2 0 0 5 ) , ( W a l t e r & O l e s e n 2 0 0 5 ) . T h e m o d e l i s a p p l i e d a s a u s e r - s u p p l i e d

s u b r o u t i n e a n d i m p l e m e n t e d i n t h e c o m m e r c i a l F E s o f t w a r e p a c k a g e . T h e p a r a m e t e r s

u s e d i n t h i s s t u d y a r e n o t o b t a i n e d d i r e c t l y b u t a r e b a s e d o n e x p e r i m e n t a l w o r k ( W a l t e r

& O l e s e n 2 0 0 5 ) . A s a c o n s e r v a t i v e v a l u e , a f r a c t u r e e n e r g y o f 0 . 1 N / m m w a s u s e d a s p u r e

M o d e I a n d I I f r a c t u r e e n e r g y a n d w a s a p p l i e d i n a l l n u m e r i c a l s i m u l a t i o n s . T h e r o l e o f

t h e i n t e r f a c i a l f r a c t u r e p a r a m e t e r s h a s b e e n s t u d i e d ( W a l t e r e t a l . 2 0 0 3 ) . I t s h o w s t h a t

f o r t h e s p e c i c s e t - u p u s e d , i n t e r f a c i a l c r a c k p r o p a g a t i o n i s h i g h l y d o m i n a t e d b y M o d e

I c r a c k i n g . F u r t h e r m o r e , p r e v i o u s s t u d i e s h a v e b e e n p e r f o r m e d i n o r d e r t o a n a l y z e t h e

i n t e r f a c i a l b o n d . A n i n t e r f a c i a l c r a c k p r o p a g a t e s , f o r t h e m a t e r i a l s u s e d i n t h i s s t u d y , i n

t h e o v e r l a y c l o s e t o t h e s t e e l p l a t e . H e n c e , t h e c o n s t i t u t i v e b e h a v i o r c a n b e t r e a t e d a s

t h e o v e r l a y m a t r i x w i t h o u t b e r s . W h e n u s i n g t h i s a p p r o a c h , t h e w a l l e e c t h a s t o b e

c o n s i d e r e d , a s b i g a g g r e g a t e s a r e r a r e l y p r e s e n t c l o s e t o t h e s t e e l p l a t e , s e e , e . g . , ( W a l t e r ,

Ø s t e r g a a r d , O l e s e n & S t a n g 2 0 0 5 ) , f o r e x p e r i m e n t a l s t u d i e s o n t h e i n t e r f a c i a l b o n d .

1 2



5 . 4 C o n s t i t u t i v e M o d e l l i n g o f E C C

M o d e l i n g o f t h e E C C o v e r l a y i s c a r r i e d o u t u s i n g a c r a c k b a n d m o d e l . T h i s i s a m o d i c a -

t i o n o f t h e s o - c a l l e d c t i t i o u s c r a c k m o d e l b y H i l l e r b o r g e t a l . ( 1 9 7 6 ) , f u r t h e r d e v e l o p e d

b y B a º a n t & O h ( 1 9 8 3 ) . T h e m a t e r i a l i s m o d e l e d i n t h r e e p h a s e s , c f . F i g u r e 1 0 . F o r s m a l l

d e f o r m a t i o n s - t h e r s t p h a s e - t h e m a t e r i a l i s a s s u m e d t o b e h a v e l i n e a r e l a s t i c a l l y , w i t h

a n e l a s t i c m o d u l u s d e n o t e d b y E . S e c o n d p h a s e i s t h e s t r a i n - h a r d e n i n g p h a s e , d e s c r i b e d

b y a s t r e s s - s t r a i n r e l a t i o n s h i p . I n t h i s p h a s e , t h e t o t a l s t r a i n o f t h e m a t e r i a l c o n s i s t s o f

t w o s t r a i n p a r t s , a n e l a s t i c s t r a i n e a n d a c r a c k i n g s t r a i n cr , a c c o r d i n g t o t h e f o l l o w i n g

f o r m u l a

tot = e + cr ( 4 )

A t s o m e p o i n t , t h e t e n s i l e c a p a c i t y i s e x h a u s t e d , d u e t o t h e f a c t t h a t b e r b r i d g i n g o f a

c e r t a i n c r a c k h a s r e a c h e d i t s l o a d c a r r y i n g c a p a c i t y , a n d s o f t e n i n g t a k e s p l a c e . A t t h i s

p o i n t , t h e m a t e r i a l m o v e s f r o m a s t r e s s - s t r a i n s t a g e i n t o a s t r e s s - c r a c k o p e n i n g s t a g e .

S i m i l a r c o n s t i t u t i v e m o d e l i n g o f E C C h a s b e e n c a r r i e d o u t b y K a b e l e ( 2 0 0 1 ) .

F i g u r e 1 0 : C o n s t i t u t i v e m o d e l i n g o f E C C . T h e m a t e r i a l i s d e s c r i b e d i n t h r e e p h a s e s : a

l i n e a r e l a s t i c r a n g e , a s t r a i n h a r d e n i n g r a n g e a n d a l i n e a r s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p .

5 . 5 C o n s t i t u t i v e M o d e l l i n g o f F R C a n d F R D

M o d e l i n g o f t h e F R C c o m p o s i t e b e a m i s c a r r i e d o u t c o n s i d e r i n g a s i n g l e c r a c k i n t h e

o v e r l a y , m o d e l e d u s i n g i n t e r f a c e e l e m e n t s . A l a r g e a m o u n t o f w o r k h a s b e e n c a r r i e d o u t

m o d e l i n g F R C a n d i t t u r n s o u t t h a t a b i l i n e a r s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p p r o v i d e s

r e a s o n a b l e r e s u l t s , s e e , e . g . , O l e s e n ( 2 0 0 1 ) . T h e c o n s t i t u t i v e d e s c r i p t i o n o f t h e F R C

m a t e r i a l c o n s i s t s o f t w o p a r t s : a l i n e a r e l a s t i c a n d a b i l i n e a r t e n s i o n s o f t e n i n g p a r t , c f .

F i g u r e 1 1 .

T h e t w o p a r t s o f t h e b i l i n e a r s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p a r e c o n c e p t u a l l y a s s o c i a t e d

w i t h t h e m a t r i x c r a c k i n g a n d t h e b e r b r i d g i n g , r e s p e c t i v e l y . T h e t w o l i n e s e g m e n t s h a v e

a s l o p e o f a1 a n d a2 , r e s p e c t i v e l y a n d c a n b e e x p r e s s e d b y t h e f o l l o w i n g f o r m u l a :

1 3



F i g u r e 1 1 : D e n i t i o n o f p a r a m e t e r s o f a b i l i n e a r s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p , u s e d

i n t h e c o n s t i t u t i v e m o d e l i n g o f F R C a n d F R D .

σ(w)f t

= bi − aiw =

b1 − a1w, 0 ≤ w < w1

b2 − a2w, w1 ≤ w ≤ w2

( 5 )

w h e r e b1 = 1 ; a n d w h e r e t h e l i m i t s w1 a n d w2 a r e g i v e n b y t h e t w o l i n e s e g m e n t s , a n d t h e

i n t e r s e c t i o n o f t h e s e c o n d l i n e s e g m e n t a n d t h e a b s c i s s a , r e s p e c t i v e l y .

5 . 6 P a r a m e t e r s f r o m U n i a x i a l T e s t s

U s i n g t h e s i m p l e i n t e r p r e t a t i o n d e s c r i b e d i n t h e p r e v i o u s s e c t i o n s , t h e u n i a x i a l t e n s i o n

t e s t y i e l d s d i r e c t l y t h e s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p ( E q . 5 ) f o r t h e s p e c i c m a t e r i a l .

A p p r o x i m a t i o n o f t h e e x p e r i m e n t a l r e s u l t t o a b i l i n e a r c u r v e i s o p t i m i z e d u s i n g n u m e r i c a l

t o o l s l i k e M a t l a b

R . T h e b e s t t i s o b t a i n e d b y r s t l o c k i n g t h e t e n s i l e s t r e n g t h f t a n d

t h e u l t i m a t e c r a c k o p e n i n g w2 . N e x t s t e p i s t h e n t o c a l c u l a t e t h e t w o s l o p e s a1 a n d a2

u s i n g a n o p t i m i z i n g r o u t i n e .

A n e x a m p l e , c o m p a r i s o n o f t h e b i l i n e a r a p p r o x i m a t i o n t o t h e e x p e r i m e n t a l r e c o r d e d , f o r

a r e p r e s e n t a t i v e F R C s p e c i m e n , i s s h o w n i n F i g u r e 1 2 .

F r o m T a b l e 4 , t h e p a r a m e t e r s f o r F R C a n d F R D a r e g i v e n a c c o r d i n g t o t h e b i l i n e a r s t r e s s -

c r a c k o p e n i n g r e l a t i o n s h i p a s s h o w n i n F i g u r e 1 1 . T h e r e s u l t s a r e g i v e n a s a n a v e r a g e a n d

t h e s t a n d a r d d e v i a t i o n f r o m t h r e e t e s t s o f e a c h m a t e r i a l .

F R C F R C F R D F R D

A v e r a g e S T D A v e r a g e S T D

f t [ M P a ] 2 . 3 0 0 . 1 6 4 . 8 0 . 3 5

a1 [ m m

−1] 2 6 . 7 3 . 2 0 6 4 2 4 . 1

a2 [ m m

−1] 0 . 4 7 0 . 1 3 0 . 3 7 0 . 2 4

b2 [ - ] 0 . 3 5 0 . 0 5 0 . 4 6 0 . 1 6

T a b l e 4 : P a r a m e t e r s f o r t h e b i l i n e a r s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p o b t a i n e d f r o m t h e

u n i a x i a l t e n s i o n t e s t f o r m a t e r i a l s F R C a n d F R D .

1 4



0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

w [mm]

σ [ M

P a ]

Uniaxial testBilinear approximation

F i g u r e 1 2 : C o m p a r i s o n o f t h e u n i a x i a l e x p e r i m e n t a l m e a s u r e m e n t s , f o r a F R C s p e c i m e n ,

a n d t h e b i l i n e a r a p p r o x i m a t i o n .

F r o m t h e u n i a x i a l t e s t o f t h e E C C s p e c i m e n s , t h e i n t e r p r e t a t i o n f o r c o n s t i t u t i v e m o d e l

p a r a m e t e r s i s s t r a i g h t f o r w a r d . T h e m e a s u r e m e n t s f r o m t h e t e s t g i v e t h e s t r e s s - s t r a i n

c u r v e d i r e c t l y a n d t h e s t r e s s - c r a c k o p e n i n g p h a s e i s n o t c o n s i d e r e d . T a b l e 5 s h o w s t h e

a v e r a g e t e n s i l e s t r e n g t h f t , u l t i m a t e s t r a i n u , a n d u l t i m a t e s t r e s s f u . T h e r e s u l t s a r e

g i v e n a s a n a v e r a g e b a s e d o n t h r e e e x p e r i m e n t s a l o n g w i t h t h e i r s t a n d a r d d e v i a t i o n .

E C C E C C

A v e r a g e S T D

f t [ M P a ] 4 . 2 0 . 3 7

u [ % ] 1 . 9 0 . 7 1

f u [ M P a ] 5 . 3 0 . 9

T a b l e 5 : P a r a m e t e r s o b t a i n e d f r o m t h e u n i a x i a l t e n s i o n t e s t o f E C C , a c c o r d i n g t o F i g u r e

1 0 .

6 M o d e l V e r i c a t i o n

T o v e r i f y t h e F E m o d e l i n g , t h e n u m e r i c a l r e s u l t s a r e c o m p a r e d w i t h e x p e r i m e n t a l r e s u l t s .

F i t t i n g t h e r e s u l t s c a n b e a l o n g a n d o f t e n s l o w p r o c e s s d e p e n d i n g o n t h e a c c e s s i b l e c o m -

p u t a t i o n a l p o w e r . T h e u l t i m a t e t e s t i s w h e n t h e c o n s t i t u t i v e p a r a m e t e r s f r o m t h e u n i a x i a l

t e s t s p r o d u c e a g o o d c o r r e l a t i o n b e t w e e n n u m e r i c a l a n d e x p e r i m e n t a l r e s u l t s . H o w e v e r ,

a s a l r e a d y s h o w n s o m e s c a t t e r i s p r e s e n t f o r t h e r e s u l t s o b t a i n e d i n t h e u n i a x i a l t e n s i o n

t e s t s a n d m i g h t n o t p r o d u c e a p e r f e c t c o r r e l a t i o n . I n o r d e r t o i l l u s t r a t e t h i s p h e n o m e n o n

a n d t o v e r i f y t h e n u m e r i c a l m o d e l i n g w o r k , t w o n u m e r i c a l c u r v e s a r e c o m p a r e d t o t h e

e x p e r i m e n t a l r e s u l t s . O n e c u r v e s h o w s t h e n u m e r i c a l r e s u l t u s i n g t h e a v e r a g e p a r a m e t e r s

1 5



o b t a i n e d i n t h e u n i a x i a l t e n s i o n t e s t s c o m p a r e d t o t h e c o m p o s i t e b e a m t e s t s f o r F R C a n d

F R D , c f . F i g u r e 1 3 ( a ) a n d ( b ) , r e s p e c t i v e l y .

0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2

2.5

3

COD [mm](a)

L o a d

[ k N ]

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

COD [mm](c)

L o a d

[ k N ]

0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2

2.5

3

COD [mm](b)

L o a d

[ k N ]

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

COD [mm](d)

L o a d

[ k N ]

Model using uniaxial parameters

Model using uniaxial parameters

Model fit

Model fit

F i g u r e 1 3 : C o m p a r i s o n b e t w e e n e x p e r i m e n t a l a n d n u m e r i c a l r e s u l t s . ( a ) a n d ( b ) s h o w

n u m e r i c a l r e s u l t u s i n g a v e r a g e s t r e s s - c r a c k o p e n i n g p a r a m e t e r s f r o m u n i a x i a l t e n s i o n t e s t s

c o m p a r e d t o F R C a n d F R D c o m p o s i t e b e a m t e s t s , r e s p e c t i v e l y . ( c ) a n d ( d ) s h o w b e s t

t u s i n g t r i a l a n d e r r o r a n d e x p e r i m e n t a l r e s u l t s , f r o m a r e p r e s e n t a t i v e F R C a n d F R D

c o m p o s i t e b e a m t e s t , r e s p e c t i v e l y .

T h e t h i c k s o l i d l i n e i n F i g u r e 1 3 ( a ) a n d ( b ) a r e o b t a i n e d b y u s i n g t h e a v e r a g e c o n s t i t u t i v e

p a r a m e t e r s f r o m t h e u n i a x i a l t e n s i o n t e s t s a s t h e m o d e l i n p u t . S o m e d i e r e n c e s a r e

o b s e r v e d w h e n c o m p a r i n g t h e c o m p o s i t e b e a m e x p e r i m e n t t o n u m e r i c a l r e s u l t s u s i n g

u n i a x i a l t e s t p a r a m e t e r s . I n t h i s c a s e , f o r t h e t w o e x a m p l e s i n F i g u r e 1 3 ( a ) a n d ( b ) , u s i n g

u n i a x i a l p a r a m e t e r s p r o d u c e s a r e s u l t c h a r a c t e r i z e d b y h i g h e r f r a c t u r e e n e r g y , o b s e r v e d

b y a h i g h e r t h r e s h o l d . T h e o n l y e x p l a n a t i o n f o r t h i s p h e n o m e n o n i s t h e g e n e r a l s c a t t e r

1 6



o b s e r v e d i n t h e u n i a x i a l t e s t s , h o w e v e r , t h e r e s u l t s a n d t h e v e r i c a t i o n o f t h e m o d e l a r e

a c c e p t a b l e . F u r t h e r m o r e , F i g u r e s 1 3 ( c ) a n d ( d ) , s h o w a n e x a m p l e o f c u r v e t t i n g . C u r v e

t t i n g h a s b e e n c a r r i e d o u t u s i n g t r i a l a n d e r r o r . O n l y o n e , r e p r e s e n t a t i v e , e x p e r i m e n t i s

s t u d i e d , w h i c h i s a s t r o n g l i m i t a t i o n o f t h i s s t u d y . I t h a s n o t b e e n t h e p u r p o s e t o t a l l

t h e c u r v e s b u t t o s h o w t h e p r i n c i p l e o f c o m p a r i n g n u m e r i c a l a n d e x p e r i m e n t a l r e s u l t s .

F u r t h e r m o r e , t h e f o l l o w i n g c o n c l u s i o n s c a n b e d r a w n f r o m t h e n u m e r i c a l s o l u t i o n s . I t

i s c o n r m e d t h a t t h e n o r m a l s t r e s s e s p e r p e n d i c u l a r t o t h e i n t e r f a c e , f o r a n u n - c r a c k e d

s p e c i m e n i n t h e s e t - u p c o n s i d e r e d , a r e c o m p r e s s i v e a n d n o t c r i t i c a l t o t h e b o n d . W h e n a

v e r t i c a l m a c r o - c r a c k f o r m s a n d p r o p a g a t e s i n t h e o v e r l a y , i t w i l l c a u s e a c r i t i c a l c h a n g e

i n t h e i n t e r f a c i a l n o r m a l s t r e s s e s , c h a n g i n g t h e m f r o m c o m p r e s s i o n t o t e n s i o n . F o r a n y

f u r t h e r d e f o r m a t i o n o f t h e b e a m , d e b o n d i n g o f t h e o v e r l a y t a k e s p l a c e i n a c o m b i n a t i o n o f

s h e a r a n d n o r m a l t e n s i l e s t r e s s e s . A p a r a m e t r i c s t u d y u s i n g t h i s m o d e l h a s b e e n c a r r i e d

o u t , s e e W a l t e r e t a l . ( 2 0 0 3 ) . T h i s s t u d y s h o w s t h a t t h e f r a c t u r e e n e r g y o f t h e i n t e r f a c e

h a s l i t t l e i n u e n c e o n t h e g l o b a l c o m p o s i t e b e a m r e s p o n s e . M o r e o v e r , a s o b s e r v e d i n t h e

e x p e r i m e n t s , d e b o n d i n g i s c l o s e l y r e l a t e d t o t h e f o r m a t i o n o f a m a c r o c r a c k i n t h e o v e r l a y .

I n t h e c a s e o f a d i s c r e t e c r a c k i n g p r o c e s s , d e b o n d i n g w i l l a l w a y s o c c u r w h e n a n o v e r l a y

m a c r o c r a c k h a s r e a c h e d a n o p e n i n g i n t h e r a n g e o f 0 . 1 - 0 . 2 m m .

I n T a b l e 6 , t h e b i l i n e a r s t r e s s - c r a c k o p e n i n g p a r a m e t e r s u s e d i n t h e c o m p a r i s o n o f n u -

m e r i c a l r e s u l t s a n d e x p e r i m e n t s a r e g i v e n . F o r e a c h t e n s i o n s o f t e n i n g m a t e r i a l , F R C a n d

F R D , t h e a v e r a g e p a r a m e t e r a n d t h e i r s t a n d a r d d e v i a t i o n f r o m t h e u n i a x i a l t e s t s a l o n g

w i t h t h e p a r a m e t e r s f r o m t h e t t e d c u r v e a r e g i v e n .

F R C F R C F R C F R D F R D F R D

A v e r a g e S T D B E A M F I T A v e r a g e S T D B E A M F I T

f t [ M P a ] 2 . 3 0 . 1 6 1 . 5 4 . 8 0 . 3 5 3 . 2

a1 [ m m

−1

] 2 6 . 7 3 . 2 1 5 . 0 6 4 . 0 2 4 . 1 8 4 . 0

a2 [ m m

−1] 0 . 4 7 0 . 1 3 0 . 5 7 0 . 3 7 0 . 2 4 0 . 5 0

b2 [ - ] 0 . 3 5 0 . 0 5 0 . 4 1 0 . 4 6 0 . 1 6 0 . 4 2

T a b l e 6 : C o m p a r i s o n b e t w e e n a v e r a g e s t r e s s - c r a c k o p e n i n g p a r a m e t e r s o b t a i n e d f r o m t h e

u n i a x i a l t e n s i o n t e s t o f F R C , F R D a l o n g w i t h t h e s t a n d a r d d e v i a t i o n ( S T D ) a n d t h e o n e s

o b t a i n e d t t i n g a c u r v e f r o m t h e c o m p o s i t e b e a m t e s t

V e r i c a t i o n o f t h e n u m e r i c a l m o d e l u s i n g a s t r a i n h a r d e n i n g m a t e r i a l , E C C , i s c a r r i e d o u t

i n t h e s a m e m a n n e r a s f o r F R C a n d F R D . A c o m p a r i s o n i s m a d e b e t w e e n t h e a v e r a g e

p a r a m e t e r s o b t a i n e d i n t h e u n i a x i a l t e n s i o n t e s t s , a r e p r e s e n t a t i v e c o m p o s i t e b e a m e x -

p e r i m e n t , a n d a t t e d c u r v e . A g a i n , t h e t t e d c u r v e i s o b t a i n e d u s i n g t h e t r a i l a n d e r r o r

m e t h o d . T h e c u r v e s a r e c o m p a r e d a n d p l o t t e d , c f . F i g u r e 1 4 .

T h e p o i n t w h e r e t h e s t e e l p l a t e s t a r t s t o y i e l d , w a s n o t m e a s u r e d i n t h e e x p e r i m e n t . T h e

y i e l d i n g p o i n t i s e x t r a c t e d f r o m t h e n u m e r i c a l r e s u l t s a n d s h o w n a l o n g w i t h t h e p o i n t

w h e r e a c r a c k l o c a l i z e s i n t h e o v e r l a y . I n p r e s e n t c a s e , t h e u n i a x i a l p a r a m e t e r s p r o d u c e

a r e s u l t o f a ' w e a k e r ' c o m p o s i t e b e a m . B y t h e t e r m ' w e a k ' i s m e a n t t h a t t h e s t e e l p l a t e

s t a r t s t o y i e l d a t a l o w e r l o a d l e v e l a n d a c r a c k l o c a l i z e s a t a l o w e r l o a d l e v e l t h a n i n t h e

c a s e o f t h e t t e d r e s u l t . T h e v a l u e s f r o m t h e t t e d c u r v e a r e c o m p a r e d w i t h t h e o n e s

1 7



0 5 10 15 20 25 30 350

1

2

3

4

5

6

7

Deflection [mm](a)

P [ k

N ]

0 5 10 15 20 25 30 350

1

2

3

4

5

6

7

Deflection [mm](b)

P [ k

N ]


Crack localization


Crack localization

Model using uniaxialparameters

Model fit

F i g u r e 1 4 : C o m p a r i s o n b e t w e e n e x p e r i m e n t a l a n d n u m e r i c a l r e s u l t s . ( a ) N u m e r i c a l r e s u l t

u s i n g a v e r a g e p a r a m e t e r s f r o m u n i a x i a l t e n s i o n t e s t . ( b ) B e s t t u s i n g t r i a l a n d e r r o r o n

a n e x p e r i m e n t a l r e s u l t f r o m a r e p r e s e n t a t i v e E C C c o m p o s i t e b e a m t e s t .

o b t a i n e d i n t h e u n i a x i a l t e n s i o n t e s t o f E C C , c f . T a b l e 7 . A g a i n , t a k i n g t h e s c a t t e r i n t o

a c c o u n t , v e r i c a t i o n o f t h e m o d e l i s a c c e p t a b l e .

E C C E C C E C C

A v e r a g e S T D B E A M F I T

f t [ M P a ] 4 . 2 0 . 3 7 4 . 3

u[ % ] 1 . 9 0 . 7 1 2 . 3

f u [ M P a ] 5 . 3 0 . 9 6 . 2

T a b l e 7 : C o m p a r i s o n b e t w e e n a v e r a g e u n i a x i a l p a r a m e t e r s , c f . F i g u r e 1 0 , o b t a i n e d f r o m

t h e u n i a x i a l t e n s i o n t e s t o f E C C a l o n g w i t h t h e s t a n d a r d d e v i a t i o n ( S T D ) a n d t h e o n e s

o b t a i n e d t t i n g c u r v e s f r o m c o m p o s i t e b e a m t e s t s


A p r o p o s e d s y s t e m t o s t i e n a s t e e l p l a t e w i t h a b e r r e i n f o r c e d c e m e n t - b a s e d o v e r l a y

h a s b e e n i n v e s t i g a t e d w i t h r e s p e c t t o c o m p o s i t e b e a m e l e m e n t s . T o s t u d y t h e c o m p o s i t e

b e h a v i o r o f t h e c e m e n t - b a s e d o v e r l a y a n d u n d e r l y i n g s t e e l p l a t e , a s p e c i c t h r e e p o i n t

b e n d i n g s e t - u p h a s b e e n u s e d t o s i m u l a t e t h e v e r y t o p p a r t o f a b r i d g e d e c k e x p o s e d t o

n e g a t i v e b e n d i n g . O n l y m o n o t o n i c m e c h a n i c a l l o a d i n g h a s b e e n c o n s i d e r e d . T w o c l a s s e s

o f m a t e r i a l s h a v e b e e n i n v e s t i g a t e d : t e n s i o n s o f t e n i n g a n d s t r a i n h a r d e n i n g m a t e r i a l s .

F i n a l l y , n u m e r i c a l m o d e l i n g o f t h e t e s t s e t - u p h a s b e e n p e r f o r m e d u s i n g t h e F i n i t e E l e m e n t

M e t h o d ( F E M ) . B a s e d o n t h i s w o r k , t h e f o l l o w i n g c o n c l u s i o n s c a n b e d r a w n .

1 8



• O v e r a l l , t h i s s t u d y s u p p o r t s t h e h y p o t h e s i s t h a t d e b o n d i n g o f a n o v e r l a y i s r e l a t e d

t o d e f e c t s i n t h e o v e r l a y . O f m a j o r c o n c e r n i n t h e p r e s e n t s t i e n i n g s y s t e m w o u l d

b e t h e i n i t i a t i o n o f a b e n d i n g c r a c k , i . e . , i n n e g a t i v e b e n d i n g .

•T h e t w o t e n s i o n - s o f t e n i n g m a t e r i a l s t e s t e d , F i b e r R e i n f o r c e d C o n c r e t e ( F R C ) a n d

F i b e r R e i n f o r c e d D e n s i t R ( F R D ) e x h i b i t a n o v e r a l l g o o d s t r u c t u r a l p e r f o r m a n c e i n

t h e t e s t s e t - u p c o n s i d e r e d . A r e a s o n a b l e s c a t t e r i n t h e e x p e r i m e n t s w a s o b s e r v e d .

L e s s s c a t t e r i s p r e s e n t i n t h e c a s e o f F R C c o m p a r e d t o F R D . A s e x p e c t e d a l o w e r

l o a d c a p a c i t y w a s o b s e r v e d i n t h e c a s e o f F R C .

• F o r t h e o v e r l a y w i t h s t r a i n h a r d e n i n g m a t e r i a l , E C C , m u l t i p l e c r a c k i n g i n t h e o v e r l a y

p r e v e n t e d s t e e l / E C C d e b o n d i n g i n t h e u l t i m a t e l i m i t s t a t e . F u r t h e r m o r e , t i g h t c r a c k

w i d t h b e l o w 3 0 m i c r o n f a v o r a b l e f r o m a d u r a b i l i t y v i e w p o i n t w a s o b s e r v e d .

• A g o o d c o r r e l a t i o n b e t w e e n e x p e r i m e n t a l a n d n u m e r i c a l r e s u l t s w a s o b t a i n e d . H o w -

e v e r , t o g e t a p e r f e c t t u n i a x i a l p a r a m e t e r s h a v e t o b e m o d i e d i n a t t i n g p r o c e -

d u r e . I n t h i s s t u d y , t t i n g w a s p e r f o r m e d u s i n g t r i a l a n d e r r o r p r o d u c i n g , i n m o s t

c a s e s , a s l i g h t d i e r e n c e b e t w e e n m a t e r i a l p a r a m e t e r s f r o m t h e u n i a x i a l t e n s i o n t e s t s

a n d t h e o n e s f r o m c u r v e t t i n g .

T h e c o n c l u s i o n s c o n c e r n i n g s t r u c t u r a l p e r f o r m a n c e , ( l i m i t e d t o m o n o t o n i c m e c h a n i c a l

l o a d i n g u s i n g s m a l l b e a m e l e m e n t s ) , c a n b e d r a w n f o r t h r e e d i e r e n t b e r r e i n f o r c e d

m a t e r i a l s i n a n o v e r l a y s y s t e m w i t h t h e a i m t o s t i e n a s t e e l d e c k . T o g i v e a f u l l r e c o m -

m e n d a t i o n f o r t h e s e m a t e r i a l s , b r o a d e r c o n s i d e r a t i o n s r e g a r d i n g t o p i c s s u c h a s : s h r i n k a g e ,

e n v i r o n m e n t a l l o a d i n g a n d f a t i g u e h a v e t o b e i n c l u d e d i n f u r t h e r r e s e a r c h .

8 A c k n o w l e d g e m e n t s

F i n a n c i a l s u p p o r t f r o m T h e K n u d H ø j g a a r d F o u n d a t i o n , D e n m a r k , i s g r a t e f u l l y a c k n o w l -

e d g e d . T h e a s s i s t a n c e i n t h e l a b o r a t o r y w o r k , w i t h E C C , f r o m M . L e p e c h a n d S . W a n g

b o t h a t T h e A d v a n c e d C i v i l E n g i n e e r i n g M a t e r i a l R e s e a r c h L a b o r a t o r y , U n i v e r s i t y o f

M i c h i g a n , U S A , i s a c k n o w l e d g e d . D e n s i t A / S i s a c k n o w l e d g e d f o r s u p p l y i n g m a t e r i a l s t o

t h e F R D t e s t s . F i n a l l y f o r m e r s t u d e n t M . L a n g e i s g r e a t l y a c k n o w l e d g e d f o r p e r f o r m i n g

c o m p o s i t e b e a m e x p e r i m e n t s o f F R C a n d F R D d u r i n g h i s M a s t e r T h e s i s w o r k .

R e f e r e n c e s

B a º a n t , Z . P . & O h , B . H . ( 1 9 8 3 ) , ` C r a c k b a n d t h e o r y f o r f r a c t u r e o f c o n c r e t e ' , M a t e r i a l s

a n d S t r u c t u r e s 1 6 , 1 5 5 1 5 7 .

D I A N A ( 2 0 0 3 ) , D I A N A U s e r ' s M a n u a l R e l e a s e 8 . 1 , j u n e e d n , T N O B u i l d i n g a n d C o n -

s t r u c t i o n R e s e a r c h , P . O . B o x 4 9 , 2 6 0 0 A A D e l f t , T h e N e t h e r l a n d s .

1 9



W a l t e r , R . , O l e s e n , J . F . & S t a n g , H . ( 2 0 0 5 ) , ` I n t e r f a c i a l m i x e d m o d e m o d e l ' , I n : T h e

1 1 t h I n t e r n a t i o n a l C o n f e r e n c e o n F r a c t u r e , T u r i n , I t a l y .


s t e e l - c o n c r e t e i n t e r f a c e ' , J o u r n a l o f E n g i n e e r i n g F r a c t u r e M e c h a n i c s 7 2 ( 1 7 ) , 2 5 6 5

2 5 8 3 .

W a l t e r , R . , S t a n g , H . , O l e s e n , J . F . & G i m s i n g , N . J . ( 2 0 0 3 ) , ` D e b o n d i n g o f F R C c o m -

p o s i t e b r i d g e d e c k o v e r l a y ' , B r i t t l e M a t r i x C o m p o s i t e s B M C 7 , W a r s a w , P o l a n d , p p .

1 9 1 - 2 0 0 .

2 1



Paper IV —

Experimental Investigation of Fatigue in a Steel-Concrete Interface

Paper in: 5th International Conference on Fracture Mechanics of Concrete and Concrete Structures,

Vail, Colorado, USA, pp. 839-845, 2004.



E x p e r i m e n t a l I n v e s t i g a t i o n o f F a t i g u e i n a S t e e l - C o n c r e t e

I n t e r f a c e

R . W a l t e r , B . H . J a n s e n , M . R . Ø s t e r g a a r d & J . F . O l e s e n



P a p e r i n t h e p r o c e e d i n g s o f : 5 t h I n t e r n a t i o n a l C o n f e r e n c e o n F r a c t u r e o f C o n c r e t e a n d C o n c r e t e

S t r u c t u r e s - F r a m C o S - 5 , V a i l , C o l o r a d o , U S A , p p . 8 3 9 - 8 4 5 , 2 0 0 4 .

A b s t r a c t


a c h i e v e c o m p o s i t e a c t i o n b y t h e a d h e s i o n b e t w e e n a t h i n l a y e r o f b e r r e i n f o r c e d c o n c r e t e

a n d a s t e e l p l a t e . O f s p e c i a l c o n c e r n i s t h e f a t i g u e s t r e n g t h o f t h e b o n d b e t w e e n t h e

c o n c r e t e a n d s t e e l p l a t e . I n t h i s p a p e r , e m p h a s i s i s p u t o n t h e s t e e l - c o n c r e t e i n t e r f a c e a n d

h o w i t p e r f o r m s u n d e r c y c l i c l o a d i n g . A s e r i e s o f t e s t s h a v e b e e n c a r r i e d o u t o n c o m p o s i t e

b e a m s m a d e b y c a s t i n g a l a y e r o f s e l f - c o m p a c t i n g c o n c r e t e d i r e c t l y o n t o t h e s a n d b l a s t e d

s u r f a c e o f a s t e e l p l a t e . A f o u r p o i n t b e n d i n g s e t - u p w a s u s e d , s u b j e c t e d t o s t a t i c a s

w e l l a s c y c l i c l o a d i n g . A g o o d p e r f o r m a n c e o f t h e s t e e l - c o n c r e t e i n t e r f a c e u n d e r c y c l i c

l o a d i n g w a s o b s e r v e d . F u r t h e r m o r e , i n o r d e r t o a n a l y z e t h e s t r e s s / d e f o r m a t i o n s i t u a t i o n

i n t h e i n t e r f a c e , n u m e r i c a l m o d e l i n g o f t h e t e s t s e t - u p u s i n g t h e n i t e e l e m e n t m e t h o d

i s c a r r i e d o u t . T h i s a n a l y s i s m a i n l y s h o w t h a t t h e f a t i g u e t h r e s h o l d l o a d c o r r e s p o n d s t o

t h e s t a t i c l o a d l e v e l a t w h i c h i n t e r f a c i a l c r a c k s a r e i n i t i a t e d i n t h e n u m e r i c a l m o d e l .

K e y w o r d s F R C , f a t i g u e , f r a c t u r e m e c h a n i c s , s t e e l - c o n c r e t e i n t e r f a c e .


A t y p i c a l o r t h o t r o p i c s t e e l b r i d g e d e c k i n E u r o p e c o n s i s t s o f a 1 2 m m s t e e l p l a t e w i t h

s u p p o r t i n g r i b s w e l d e d t o t h e b o t t o m f a c e . T h i s t y p e o f b r i d g e d e c k s u e r s s i g n i c a n t l y

f r o m i n c r e a s i n g t r a c l o a d s , p a r t i c u l a r l y f r o m h e a v y t r u c k s . T h e i n c r e a s e d t r a c i n t e n s i t y

a n d h i g h e r w h e e l l o a d s r e s u l t s i n f a t i g u e c r a c k s i n b o t h w e l d e d s t r u c t u r e s a n d s u r f a c i n g .

T h e r e f o r e t h e d e v e l o p m e n t o f a n e n t i r e l y n e w c o n c e p t o f d e c k s y s t e m s i s o f i n t e r e s t .

I n c r e a s i n g t h e l o c a l s t i n e s s o f a t y p i c a l o r t h o t r o p i c s t e e l d e c k , u s i n g a c e m e n t - b a s e d

o v e r l a y t o f o r m a c o m p o s i t e p l a t e , i s t h e s u b j e c t o f a n o n g o i n g r e s e a r c h p r o j e c t . E v e n

c a s t i n g a t h i n l a y e r i n c r e a s e s t h e l o c a l s t i n e s s s i g n i c a n t l y a n d m i g h t b e b e n e c i a l i n a

f a t i g u e s i t u a t i o n . I t i s h o p e d , t h a t t h e c o n c e p t u n d e r d e v e l o p m e n t c a n e i t h e r b e a p p l i e d

t o r e t r o t t i n g o r i n t h e d e s i g n o f n e w b r i d g e d e c k s .

T o a c h i e v e h i g h l o c a l s t i n e s s , t h e s t e e l d e c k m u s t a c t c o m p o s i t e l y w i t h t h e c e m e n t - b a s e d

o v e r l a y . S i n c e t h e o v e r l a y i s t r e a t e d a s a n i n t e g r a l p a r t o f t h e s t r u c t u r a l d e c k s y s t e m , t h e

1



s t r e n g t h o f t h e s t e e l - c o n c r e t e i n t e r f a c e i s o f m a j o r i m p o r t a n c e . E m p h a s i s i n t h i s c a s e i s

p u t o n t h e i n t e r f a c e , e s p e c i a l l y , h o w i t p e r f o r m s u n d e r c y c l i c l o a d i n g .

A t y p i c a l d e c k a c c o r d i n g t o t h e p r o p o s e d s y s t e m c o n s i s t s o f a 4 0 - 6 0 m m l a y e r o f S e l f -

C o m p a c t i n g S t e e l F i b e r R e i n f o r c e d C o n c r e t e , ( S C S F R C ) , b o n d e d t o a n 8 - 1 4 m m s t e e l

p l a t e ( W a l t e r , S t a n g , G i m s i n g & O l e s e n 2 0 0 3 ) . W h e r e a s c o n v e n t i o n a l c o m p o s i t e c o n -

s t r u c t i o n s a c h i e v e c o m p o s i t e a c t i o n b y m e c h a n i c a l f a s t e n e r s , t h e i d e a h e r e i s t o a c h i e v e

c o m p o s i t e a c t i o n o n l y t h r o u g h a d h e s i o n . T h e a d h e s i o n b e t w e e n c o n c r e t e a n d s t e e l i s e n -

h a n c e d b y s a n d b l a s t i n g o f t h e s t e e l p l a t e . T h e m a j o r c o n c e r n i s h o w t h e i n t e r f a c e b e t w e e n

t h e c o n c r e t e o v e r l a y a n d s t e e l p l a t e p e r f o r m s u n d e r c y c l i c l o a d i n g .

T o i l l u s t r a t e s t i n e s s e e c t s u n d e r t h e i n u e n c e o f a t h i n o v e r l a y o n a s t e e l p l a t e , a n

e x a m p l e t a k e n f r o m W o l c h u k ( 2 0 0 2 ) i s u s e d . A c o m p o s i t e s e c t i o n c o n s i s t i n g o f a n o v e r l a y

f u l l y b o n d e d t o a s t e e l p a r t i s e x a m i n e d t h r o u g h a l i n e a r s t a t i c a n a l y s i s , c f . F i g u r e 1 ( a ) .

A b e n d i n g m o m e n t w i t h a m a g n i t u d e o f 4 0 0 0 N m m / m m i s a p p l i e d t o t h e c o m p o s i t e

s e c t i o n a n d i l l u s t r a t e s t h e b e n d i n g m o m e n t , w h i c h m i g h t b e c r i t i c a l t o t h e w e l d i n g o f

t h e l o n g i t u d i n a l r i b s i n a n o r t h o t r o p i c s t e e l d e c k . T h e g e o m e t r y i s d e n e d b y a n o v e r l a y

h e i g h t h c a n d a s t e e l h e i g h t hs . T h e m a x i m u m s t r e s s f o r a g i v e n b e n d i n g m o m e n t i s

o b s e r v e d f o r v a r y i n g t h i c k n e s s o f t h e o v e r l a y , c f . F i g u r e 1 ( b ) , w h e r e σc a n d σs a r e t h e

m a x i m u m s t r e s s e s a t t h e e d g e s o f t h e o v e r l a y a n d s t e e l p l a t e , r e s p e c t i v e l y .

( a )

0 10 20 30 40 500

20

40

60

80

100

120

140

hc[mm]

σ m a x

n=32

n=8

n=2

n=2

n=8

n=32

n=Es /E

c

hs=14mm

σs

σc

( b )

F i g u r e 1 : F a i l u r e m o d e s . ( a ) : N o t c h e d c o m p o s i t e b e a m , b e s i d e t h e i n t e r f a c i a l c r a c k i n g a

n u m b e r o f c r a c k s i n t h e b u l k m a t e r i a l i s o b s e r v e d . ( b ) : S t r e s s r e d u c t i o n : ( σs , · · · σc ) ,

d u e t o o v e r l a y c o n t r i b u t i o n t o t h e d e c k p l a t e r i g i d i t y f o r d i e r e n t h e i g h t s hc , i n t h e c a s e

o f t h r e e d i e r e n t e l a s t i c r a t i o s d e n o t e d n.

S t r e s s r e d u c t i o n i n t h e s t e e l d e c k i s d e s i r a b l e b e c a u s e i t s u b s t a n t i a l l y e n h a n c e s f a t i g u e

s a f e t y . H o w e v e r , a f a t i g u e p r o b l e m m a y a r i s e a t t h e s t e e l - c o n c r e t e i n t e r f a c e . T h i s w i l l b e

d i s c u s s e d i n t h e p r e s e n t p a p e r a s i t i s t h e m a i n t o p i c o f t h e p r o j e c t b e h i n d i t ( J a n s e n &

Ø s t e r g a a r d 2 0 0 3 ) .

2



M i x k g / m

C e m e n t ( R A P I D ) 2 4 5

F l y a s h 9 4 . 5

S i l i c a f u m e 1 0 . 5

W a t e r 1 4 2 . 9



S a n d , 0 0 - 0 4 m m 7 5 2 . 6

A g g r e g a t e s , 0 4 - 0 8 m m 4 5 0 . 6

A g g r e g a t e s , 0 8 - 1 6 m m 5 9 4 . 0

F i b e r s , l = 3 0 m m , D = 0 . 5 m m 7 8


O n e p e r c e n t o f h o o k e d e n d s t e e l b e r s w e r e u s e d w i t h a d i a m e t e r o f 0 . 5 m m a n d a l e n g t h o f

3 0 m m . W o r k a b i l i t y a n d c o m p r e s s i v e s t r e n g t h o f a l l b a t c h e s w e r e s i m i l a r . F o r e a c h b a t c h ,

t h r e e c y l i n d e r s w e r e c a s t . T h e a v e r a g e u l t i m a t e s t r e n g t h f o r a l l b a t c h e s w a s f c = 4 4 . 6 M P a

w i t h a s t a n d a r d d e v i a t i o n o f 2 . 7 M P a .


B o t h s t a t i c a n d c y c l i c t e s t s o n t h e c o m p o s i t e b e a m s h a v e b e e n c a r r i e d o u t . I n t h e s t a t i c

t e s t s , t h e m a x i m u m i n t e r f a c i a l s h e a r s t r e s s a n d t h e f r a c t u r e b e h a v i o r a r e o f i n t e r e s t .

D u r i n g t h e c y c l i c t e s t s e m p h a s i s i s p u t o n t h e m a x i m u m n u m b e r o f c y c l e s p r i o r t o f a i l u r e .

4 . 1 S t a t i c t e s t s

T o m e a s u r e t h e s l i p o f t h e s t e e l - c o n c r e t e i n t e r f a c e d u r i n g l o a d i n g o f t h e c o m p o s i t e b e a m , a

s p e c i a l e n d - s l i p m e a s u r e m e n t a r r a n g e m e n t i s a p p l i e d . A c l i p g a u g e w a s m o u n t e d b e t w e e n

t w o s m a l l e x t e n d i n g r o d s , w h i c h h a d b e e n g l u e d o n t o t h e s p e c i m e n , o n e o n t h e s t e e l p l a t e

a n d a n o t h e r o n t h e c o n c r e t e , c f . F i g u r e 3 .

F i g u r e 3 : M o u n t i n g o f c l i p g a u g e t o m e a s u r e e n d - s l i p

4



B a t c h N o . S t e e l p l a t e h c [ m m ] h s [ m m ] τ c [ M P a ]

1 F l a t 6 2 1 0 3 . 0 9 ±0 . 2 4

1 C h e c k e r 6 2 8 3 . 1 9 ±0 . 1 5

2 F l a t 6 2 1 0 3 . 3 0 ±0 . 1 3

2 C h e c k e r 6 2 8 3 . 2 9 ±0 . 1 0

3 + 4 F l a t 7 0 8 3 . 0 4 ±0 . 0 2

3 + 4 C h e c k e r 7 0 8 3 . 0 7 ±0 . 1 3

T a b l e 2 : R e s u l t s o f s t a t i c t e s t s f o r e a c h b a t c h u s i n g a t a n d c h e c k e r s t e e l p l a t e s , a l l b e a m s

u n - n o t c h e d .

T o i l l u s t r a t e t h e e e c t o f a n i n t e r f a c i a l n o t c h , t w o s t a t i c e x a m p l e s a r e a n a l y z e d u s i n g t h e

e n d - s l i p m e a s u r e m e n t . T h e l o a d v s . e n d - s l i p d i a g r a m s h o w e d i n F i g u r e 4 r e v e a l s t h a t a

n o t c h e d b e a m g i v e s m o r e i n f o r m a t i o n o n t h e c r a c k p r o p a g a t i o n d u r i n g t h e t e s t t h a t o f a n

u n - n o t c h e d b e a m . T h i s i n f o r m a t i o n i s e s s e n t i a l t o t h e c o m p a r i s o n a n d v a l i d a t i o n o f t h e

n u m e r i c a l m o d e l a n d i s d i s c u s s e d f u r t h e r , i n t h e l a t t e r . A n u m b e r o f s t a t i c e x p e r i m e n t s

u s i n g t h e f o u r p o i n t b e n d i n g s e t - u p w e r e c a r r i e d o u t t o d e t e r m i n e t h e m a x i m u m l o a d o f

e a c h b a t c h . T h i s h a s b e e n r e p e a t e d f o r e a c h t y p e o f s t e e l p l a t e a n d t h e r e s u l t s a r e g i v e n

i n T a b l e ( 2 ) .

0 0.01 0.02 0.030

5

10

15

20

25

30

35

P [ k

N ]

end−slip [mm]

Un−notched

Notched

Failure

F i g u r e 4 : L o a d e n d - s l i p c u r v e s o f a t y p i c a l c o m p o s i t e b e a m i n s t a t i c l o a d u s i n g a n u n -

n o t c h e d a n d n o t c h e d c o n g u r a t i o n .

G l o b a l f a i l u r e i s a s s o c i a t e d w i t h d e b o n d i n g o f t h e i n t e r f a c e . S i n c e t h e c o m p o s i t e b e a m

g e o m e t r y i s n o t t h e s a m e f o r e a c h e x p e r i m e n t , t h e m a x i m u m l i n e a r s t a t i c s h e a r s t r e s s a c t -

i n g a t t h e s t e e l - c o n c r e t e i n t e r f a c e i s c a l c u l a t e d u s i n g t h e m a x i m u m l o a d a t g l o b a l f a i l u r e .

T h e m a x i m u m s h e a r s t r e s s w a s c a l c u l a t e d u s i n g t h e c l a s s i c s h e a r f o r m u l a f o r b e a m s , s e e

e . g . G e r e & T i m o s h e n k o ( 1 9 9 9 ) , a s s u m i n g a f u l l y b o n d e d s t e e l - c o n c r e t e i n t e r f a c e . T h i s

m a x i m u m s h e a r s t r e s s a t f a i l u r e i s d e n o t e d τ f a n d i s s h o w n f o r e a c h b a t c h i n T a b l e 2 .

5



4 . 2 F a t i g u e T e s t s

T h e e x p e r i m e n t a l r e s u l t s r e l a t e d t o c y c l i c l o a d i n g i n v o l v e 1 4 s p e c i m e n s , 6 w i t h a a t s t e e l

p l a t e a n d 8 w i t h a c h e c k e r s t e e l p l a t e . D u r i n g t h e f a t i g u e t e s t s , a c y c l i c l o a d w a s a p p l i e d

b e t w e e n t w o v a l u e s

P maxa n d

P mini n t h e s h a p e o f a s i n u s o i d a l w a v e a n d a t a f r e q u e n c y o f

5 H z . R e s u l t s f r o m t h e c y c l i c t e s t s a r e g i v e n f o r e a c h b e a m i n T a b l e 3 . F o r e a c h t y p e o f

p l a t e , a n a d d i t i o n a l t w o t e s t s w e r e c a r r i e d o u t a p p l y i n g a n i n t e r f a c e n o t c h w i t h a l e n g t h

o f a0 = 4 0 m m . T h e r e s u l t s f o r e a c h s e r i e s o f a t a n d c h e c k e r s t e e l p l a t e s a r e p r e s e n t e d

g r a p h i c a l l y i n F i g u r e s 5 ( a ) a n d ( b ) , w h e r e t h e r a t i o τ max/τ f i s p l o t t e d a g a i n s t l o g ( N f ) ,

τ max b e i n g t h e m a x i m u m s h e a r s t r e s s a t t h e i n t e r f a c e a n d N f t h e n u m b e r o f c y c l e s a t

f a i l u r e . A f a t i g u e t h r e s h o l d v a l u e f o r τ max/τ f o f a p p r o x i m a t e l y 6 5 % i s o b s e r v e d f o r t h e

u n - n o t c h e d b e a m s . T h e n o t c h e d b e a m s e x h i b i t a l o w e r t h r e s h o l d v a l u e .

B e a m h c [ M P a ] h s [ M P a ] τ max/τ f L o g ( n f )

F U 1 6 2 1 0 0 . 6 4 ± 0 . 0 4 * 6 . 9 1

F U 2 7 0 8 0 . 6 5 ± 0 . 0 1 6 . 8 9

F U 3 6 2 1 0 0 . 6 9 ± 0 . 0 5 6 . 0 5

F U 4 7 0 8 0 . 7 0 ± 0 . 0 1 5 . 0 6

F N 1 6 2 1 0 0 . 6 5 ± 0 . 0 3 6 . 4 7

F N 2 6 2 1 0 0 . 7 0 ± 0 . 0 3 3 . 2 3

C U 1 6 2 8 0 . 5 9 ± 0 . 0 3 * 7 . 0 0

C U 2 7 0 8 0 . 6 8 ± 0 . 0 1 * 6 . 8 8

C U 3 7 0 8 0 . 6 9 ± 0 . 0 3 * 6 . 1 8

C U 4 6 2 8 0 . 6 9 ± 0 . 0 3 5 . 2 6

C U 5 7 0 8 0 . 7 5 ± 0 . 0 3 3 . 9 0

C U 6 6 2 8 0 . 7 8

±0 . 0 4 3 . 4 1

C N 1 6 2 8 0 . 6 6 ± 0 . 0 2 6 . 5 6

C N 2 6 2 8 0 . 6 0 ± 0 . 0 2 6 . 4 4

* T e s t s t o p p e d b e f o r e f a i l u r e

T a b l e 3 : R e s u l t s o f c y c l i c t e s t s . F U : ( F l a t s t e e l p l a t e u n - n o t c h e d ) ; F N : ( a t s t e e l p l a t e

n o t c h e d ) ; C U : ( C h e c k e r s t e e l p l a t e u n - n o t c h e d ) ; C N : ( C h e c k e r s t e e l p l a t e n o t c h e d ) .

5 N u m e r i c a l M o d e l i n g

T o s t u d y t h e b e h a v i o r o f t h e t e s t s e t - u p a n u m e r i c a l m o d e l i s u s e d . A t w o d i m e n s i o n a l

c o n g u r a t i o n o f t h e c o m p o s i t e b e a m i s c o n s i d e r e d . T h e s e t - u p i s m o d e l e d a s a h a l f b e a m ,

u s i n g a c o m m e r c i a l n i t e e l e m e n t p r o g r a m p a c k a g e ( D I A N A 2 0 0 3 ) . T h e h a l f b e a m i s

m o d e l e d a s s u m i n g a n o n l i n e a r b e h a v i o r i n t h e o v e r l a y a n d t h e i n t e r f a c e . T h e s t e e l p l a t e

i s m o d e l e d a s a l i n e a r e l a s t i c m a t e r i a l . T h e s t e e l - c o n c r e t e i n t e r f a c e i s m o d e l e d u s i n g s o -

c a l l e d i n t e r f a c e e l e m e n t s . T h e i n t e r f a c e e l e m e n t s h a v e a t h i c k n e s s o f z e r o a n d a r e p l a c e d

b e t w e e n t h e s t e e l a n d c o n c r e t e a n d r e l a t e t h e f o r c e s a c t i n g o n t h e i n t e r f a c e t o t h e r e l a t i v e

6



t h e c o h e s i o n a n d t e n s i l e s t r e n g t h . F o r f u r t h e r d e t a i l s s e e e i t h e r L o u r e n ç o & R o t s ( 1 9 9 7 )

o r W a l t e r , S t a n g , O l e s e n & G i m s i n g ( 2 0 0 3 ) .

T h e s t e e l i s m o d e l e d a s a l i n e a r e l a s t i c m a t e r i a l w i t h t h e f o l l o w i n g e n g i n e e r i n g c o n s t a n t s :

E s = 2 1 0 G P a , ν = 0 . 3 . T h e e l e m e n t s m o d e l i n g t h e o v e r l a y a n d s t e e l p l a t e a r e 8 - n o d e

i s o p a r a m e t r i c p l a n e s t r e s s e l e m e n t s .

5 . 2 C r a c k P r o p a g a t i o n

T o i l l u s t r a t e t h e i n t e r f a c i a l c r a c k b e h a v i o r , w i t h a n d w i t h o u t a n o t c h , a n u m e r i c a l e x a m p l e

i s s t u d i e d . F i g u r e 6 s h o w s h o w t h e i n t e r f a c i a l c r a c k p r o p a g a t e s . F o r a g i v e n l o a d P t h e

g u r e i l l u s t r a t e s w h e r e c r a c k i n g o c c u r s a l o n g t h e b e a m a x i s a s d e n e d i n F i g u r e 7 . I n

t h e c a s e o f a n i n t e r f a c i a l n o t c h , c r a c k i n g s t a r t s a t t h e t i p o f t h e n o t c h a t a r e l a t i v e l y l o w

l o a d v a l u e a n d p r o p a g a t e t o w a r d s t h e l o a d p o i n t . I n t h e c a s e o f a n u n - n o t c h e d c o m p o s i t e

b e a m , i n t e r f a c i a l c r a c k i n g i n i t i a t e s s o m e w h e r e i n b e t w e e n t h e s u p p o r t a n d l o a d , a n d t h e

c r a c k p r o p a g a t e s t o w a r d s t h e s u p p o r t a n d l o a d p o i n t , r e s u l t i n g i n a m o r e d r a m a t i c g l o b a l

f a i l u r e .

0 100 200 300 4000

10

20

30

40

P [ k

N ]

Beam x−axis [mm]

notchun−notched

Crack initiation

Failure

F i g u r e 6 : C r a c k p r o p a g a t i o n a l o n g t h e i n t e r f a c e , c a l c u l a t e d u s i n g F E M . C r a c k p r o p a g a t i o n

i s p l o t t e d f o r a g i v e n p o s i t i o n o n t h e x - a x i s a c c o r d i n g t o F i g u r e ( 7 ) v e r s u s t h e l o a d f o r a

n o t c h e d a n d u n - n o t c h e d b e a m .

T h e s i g n i c a n t d i e r e n c e s b e t w e e n t h e t w o c a s e s a r e : w h e r e t h e i n t e r f a c e c r a c k i n i t i a t e s

a n d h o w i t p r o p a g a t e s . R u n n i n g a n e x p e r i m e n t w i t h o u t a n o t c h i s s o m e w h a t l e s s u s e f u l f o r

t h e s t u d y o f t h e i n t e r f a c i a l c r a c k p r o p a g a t i o n . H o w e v e r , w h e n a p p l y i n g a n o t c h , t h e c r a c k

i n i t i a t e s a t t h e n o t c h t i p c l o s e t o t h e s u p p o r t a n d p r o p a g a t e s t o w a r d s t h e l o a d p o i n t . T h i s

i s s u i t a b l e f o r c l o s e d - l o o p t e s t i n g u s i n g t h e e n d - s l i p a s t h e c o n t r o l l i n g p a r a m e t e r . P h o t o s

o f s a m p l e b e a m s , w i t h a n d w i t h o u t a n o t c h , a r e s h o w n a f t e r f a i l u r e i n F i g u r e 8 . I t r e v e a l s

t h a t t h e n o t c h e d b e a m h a s c r a c k s r u n n i n g f r o m t h e i n t e r f a c e i n t o t h e o v e r l a y , w h e r e a s

t h e u n - n o t c h e d b e a m h a s o n l y o n e d o m i n a n t i n t e r f a c e c r a c k .

8



0 0.01 0.02 0.03 0.040

5

10

15

20

25

30

35

End−slip [mm]

P [ k N

]

0 0.01 0.02 0.03 0.040

5

10

15

20

25

30

35

End−slip [mm]

P [ k N

]

F i g u r e 9 : ( a ) : F i t t i n g o f F E - c a l c u l a t i o n s t o e x p e r i m e n t s f o r a c h e c k e r s t e e l p l a t e . ( b ) :

F i t t i n g o f F E - c a l c u l a t i o n s t o e x p e r i m e n t s f o r a a t s t e e l p l a t e .


A t e s t s e t - u p u s i n g a c o m p o s i t e b e a m c o n s i s t i n g o f a c e m e n t - b a s e d o v e r l a y c a s t u p o n a

s a n d b l a s t e d s t e e l p l a t e h a s b e e n i n v e s t i g a t e d e x p e r i m e n t a l l y u n d e r s t a t i c a s w e l l a s c y c l i c

l o a d i n g . A c h e c k e r a n d a a t s t e e l p l a t e w e r e u s e d i n t h e e x p e r i m e n t s . T h e d i e r e n t s t e e l

p l a t e s d i d n o t r e v e a l a n y s i g n i c a n t d i e r e n c e s i n s t a t i c o r f a t i g u e s t r e n g t h . A m o d i e d

v e r s i o n o f t h e t e s t s e t - u p w a s u s e d , i n t r o d u c i n g a n i n t e r f a c i a l n o t c h b e t w e e n t h e c o n c r e t e

a n d t h e s t e e l p l a t e n e a r t h e s u p p o r t . T h i s s e t - u p h a s p r o v e n s u i t a b l e f o r s t u d y i n g t h e

i n t e r f a c i a l c r a c k p r o p a g a t i o n d u r i n g l o a d i n g . A n u m b e r o f d i e r e n t s p e c i m e n s w e r e u s e d

i n t h e s t u d y o f a c o m p o s i t e b e a m e x p o s e d t o c y c l i c l o a d i n g . A p r i m a r y r e s u l t i s t h a t t h e

s t e e l - c o n c r e t e i n t e r f a c e s h o w s a g o o d f a t i g u e r e s i s t a n c e , a s a f a t i g u e t h r e s h o l d l e v e l w a s

o b s e r v e d a r o u n d 6 5 % o f t h e s t a t i c f a i l u r e l o a d f o r t h e u n - n o t c h e d b e a m s .

N u m e r i c a l m o d e l i n g w a s c a r r i e d o u t u s i n g F E M . T h e m o d e l h a s b e e n v a l i d a t e d , u s i n g t h e

n o t c h e d b e a m s e t - u p . T h e m o d e l i n g h e l p e d r e v e a l t h e d i e r e n t f r a c t u r i n g b e h a v i o r o f t h e

n o t c h e d a n d u n - n o t c h e d c o m p o s i t e b e a m s . F i n a l l y , t h e n u m e r i c a l m o d e l i n g c o n r m s t h a t

c r a c k i n g a t t h e i n t e r f a c e o c c u r s a t a l o a d l e v e l o f a p p r o x i m a t e l y 6 5 % o f t h e s t a t i c f a i l u r e

l o a d f o r a n u n - n o t c h e d b e a m , w h i c h c o r r e s p o n d s w e l l t o t h e o b s e r v e d f a t i g u e t h r e s h o l d

l o a d . A s e x p e c t e d , t h e n o t c h e d b e a m s h a v e a l o w e r t h r e s h o l d i n f a t i g u e , s i n c e i n t e r f a c i a l

c r a c k i n g i s i n i t i a t e d a t a l o w e r l o a d .

R e f e r e n c e s



1 0



G e r e , J . & T i m o s h e n k o , S . ( 1 9 9 9 ) , M e c h a n i c s o f M a t e r i a l s , S t a n l e y T h o m e s , F o u r t h S I

e d i t i o n .

J a n s e n , B . J . & Ø s t e r g a a r d , R . M . ( 2 0 0 3 ) , C r a c k i n g a n d f a t i g u e o f t h e i n t e r f a c e i n c o m -

p o s i t e b r i d g e d e c k s , M a s t e r ' s t h e s i s , D e p a r t m e n t o f C i v i l E n g i n e e r i n g , T e c h n i c a l

U n i v e r s i t y o f D e n m a r k , L y n g b y D e n m a r k .



S i g u r d s s o n , S . ( 2 0 0 3 ) , C o m p o s i t e b r i d g e d e c k s o f s f r c / s t e e l , M a s t e r ' s t h e s i s , D e p a r t m e n t

o f C i v i l E n g i n e e r i n g , T e c h n i c a l U n i v e r s i t y o f D e n m a r k , L y n g b y .

W a l t e r , R . , S t a n g , H . , G i m s i n g , N . J . & O l e s e n , J . F . ( 2 0 0 3 ) , ` H i g h p e r f o r m a n c e c o m p o s i t e

b r i d g e d e c k s u s i n g s c s f r c ' , T h e F o u r t h I n t e r n a t i o n a l W o r k s h o p o n H i g h P e r f o r m a n c e

F i b e r R e i n f o r c e d C e m e n t C o m p o s i t e s , A n n A r b o r , M i c h i g a n p p . 4 9 5 5 0 4 .



1 9 1 - 2 0 0 .

W o l c h u k , R . ( 2 0 0 2 ) , ` S t r u c t u r a l b e h a v i o u r o f s u r f a c i n g o n s t e e l o r t h o t r o p i c d e c k s a n d

c o n s i d e r a t i o n s f o r p r a c t i c a l d e s i g n ' , S t r u c t u r a l E n g i n e e r i n g I n t e r n a t i o n a l 2 , 1 2 4 1 2 9 .

1 1



Paper V —

Debonding of FRC Composite Bridge Deck Overlay

Paper in the proceedings of: 7th International Symposium on brittle matrix composites - BMC 7

Staszic Palace, Warsaw, Poland, pp. 191-200, 2003.



D e b o n d i n g o f F R C C o m p o s i t e B r i d g e D e c k O v e r l a y

R a s m u s W a l t e r , H e n r i k S t a n g , J o h n F . O l e s e n , N i e l s J . G i m s i n g



P a p e r i n t h e p r o c e e d i n g s o f : 7 t h I n t e r n a t i o n a l S y m p o s i u m o n b r i t t l e m a t r i x c o m p o s i t e s - B M C 7

S t a s z i c P a l a c e , W a r s a w , P o l a n d , p p . 1 9 1 - 2 0 0 , 2 0 0 3 .

A b s t r a c t


a c h i e v e c o m p o s i t e a c t i o n b y t h e a d h e s i o n b e t w e e n a t h i n l a y e r o f b e r r e i n f o r c e d c o n c r e t e

c a s t o n a s t e e l p l a t e . O f s p e c i a l c o n c e r n i s t h e s t r e n g t h o f t h e b o n d b e t w e e n t h e c o n c r e t e

a n d s t e e l p l a t e i n t h e c a s e o f v e r t i c a l c r a c k i n g o f t h e o v e r l a y . B a s e d o n a n o n l i n e a r

2 D n i t e e l e m e n t m o d e l a t h e o r e t i c a l p a r a m e t e r s t u d y h a s b e e n c o n d u c t e d i n r e l a t i o n

t o d e b o n d i n g o f a c o n c r e t e o v e r l a y . D i s c r e t e c r a c k t h e o r y w a s u s e d t o m o d e l v e r t i c a l

c r a c k i n g o f t h e o v e r l a y a n d i n t e r f a c i a l c r a c k i n g ( d e b o n d i n g ) b e t w e e n t h e s t e e l p l a t e a n d

c o n c r e t e . T w o p a r a m e t e r s t u d i e s w e r e m a d e . O n e c o n c e r n s t h e f r a c t u r e e n e r g y o f t h e

o v e r l a y a n d s t e e l - c o n c r e t e i n t e r f a c e . T h e s e r e s u l t s s h o w t h a t t h e c o m p o s i t e p e r f o r m a n c e

i s d e p e n d e n t p r i m a r i l y o n f r a c t u r e e n e r g y o f t h e c o n c r e t e o v e r l a y , a n d l e s s o n t h e f r a c t u r e

e n e r g y o f t h e s t e e l - c o n c r e t e i n t e r f a c e . I n t h e o t h e r , t h e m i x e d m o d e b e h a v i o u r o f t h e

s t e e l - c o n c r e t e i n t e r f a c e i s s t u d i e d i n r e l a t i o n t o t h e i n t e r f a c i a l f a i l u r e c r i t e r i o n .

K e y w o r d s D e b o n d i n g , F R C , f r a c t u r e m e c h a n i c s , s t e e l - c o n c r e t e i n t e r f a c e .


A l a r g e n u m b e r o f s t e e l b r i d g e d e c k s s u e r s i g n i c a n t l y f r o m i n c r e a s e d t r a c i n t e n s i t y

a n d h i g h e r w h e e l l o a d s . T h i s r e s u l t s i n f a t i g u e c r a c k s i n b o t h w e l d e d s t r u c t u r e s a n d

s u r f a c i n g . D e v e l o p m e n t o f a n e n t i r e l y n e w c o n c e p t o f d e c k s y s t e m s i s o f i n t e r e s t . A

t y p i c a l d e c k a c c o r d i n g t o t h i s s y s t e m c o n s i s t s o f a 4 0 - 6 0 m m l a y e r o f S e l f - C o m p a c t i n g

S t e e l F i b e r R e i n f o r c e d C o n c r e t e , ( S C S F R C ) , b o n d e d t o a n 8 m m s t e e l p l a t e ( W a l t e r

e t a l . 2 0 0 3 ) . W h e r e a s c o n v e n t i o n a l c o m p o s i t e c o n s t r u c t i o n s a c h i e v e c o m p o s i t e a c t i o n b y

m e c h a n i c a l f a s t e n e r s , t h e i d e a h e r e i s t o a c h i e v e c o m p o s i t e a c t i o n o n l y t h r o u g h a d h e s i o n .

T h e a d h e s i o n b e t w e e n c o n c r e t e a n d s t e e l i s e n h a n c e d b y s a n d b l a s t i n g o f t h e s t e e l p l a t e .

S i n c e a c o n t r o l l i n g f a c t o r o f t h e c o m p o s i t e s t r e n g t h i s r e l a t e d t o t h e v e r t i c a l c r a c k i n g o f

t h e o v e r l a y , t h e c o m p o s i t e p l a t e s u b j e c t e d t o a n e g a t i v e b e n d i n g m o m e n t i s o f i n t e r e s t .

D u r i n g e x u r a l c r a c k i n g o f t h e o v e r l a y , t h e d i s t r i b u t i o n o f s h e a r a n d n o r m a l s t r e s s e s a l o n g

t h e s t e e l - c o n c r e t e i n t e r f a c e c h a n g e s d r a m a t i c a l l y f r o m t h a t o f t h e e l a s t i c p h a s e . I n t h e

f r a c t u r e p r o c e s s z o n e o f a d i s c r e t e c r a c k , h i g h s t r e s s c o n c e n t r a t i o n s d e v e l o p i n a p l a n e

p e r p e n d i c u l a r t o t h e c r a c k d i r e c t i o n d u e t o t h e p r e s e n c e o f t h e s t e e l p l a t e t h a t o p p o s e s

1



t h e o p e n i n g o f t h e e x u r a l c r a c k . T h e d e b o n d i n g p r o b l e m i s k n o w n f r o m t h e r e p a i r i n g

o f e x i s t i n g s t r u c t u r e s b y t h i n c e m e n t - b a s e d o v e r l a y s , s e e e . g . G r a n j u ( 1 9 9 6 ) o r G r a n j u

( 2 0 0 1 ) . T h e y s h o w t h a t c r a c k i n g o f t h e o v e r l a y i n d u c e s h i g h i n t e r f a c i a l t e n s i l e s t r e s s e s

l e a d i n g t o d e b o n d i n g b e t w e e n t h e t w o m a t e r i a l s . O n c e i n i t i a t e d , t h e i n t e r f a c i a l c r a c k w i l l

p r o p a g a t e i n a m i x e d m o d e c h a r a c t e r i z e d b y t e n s i l e a n d s h e a r s t r e s s e s a l o n g t h e i n t e r f a c e .

T h e a i m o f t h i s p a p e r i s t o s t u d y t h e b e h a v i o u r o f t h e i n t e r f a c e b e t w e e n c o n c r e t e a n d s t e e l

w i t h a f r a c t u r e m e c h a n i c a l a p p r o a c h . I n t h i s s t u d y t h e s o - c a l l e d c t i t i o u s c r a c k m o d e l

( F C M ) d e v e l o p e d b y H i l l e r b o r g e t a l . ( 1 9 7 6 ) w i l l b e a p p l i e d . T h e a d v a n t a g e o f t h e m o d e l

i s t h e s i m p l i c i t y a n d t h e g o o d c o r r e l a t i o n b e t w e e n t h e o r y a n d e x p e r i m e n t a l t e s t s . T h e

b a s i c i d e a o f t h e F C M i s a r e l a t i o n s h i p b e t w e e n t h e s t r e s s a n d t h e c r a c k o p e n i n g . T h i s

r e l a t i o n s h i p d e s c r i b e s t h e s t r e s s e s t r a n s f e r r e d a c r o s s t h e c r a c k . T h i s m e t h o d h a s a l s o b e e n

a p p l i e d t o b e r r e i n f o r c e d c o n c r e t e , s e e e . g . O l e s e n ( 2 0 0 1 ) . T h e t w o t y p e s o f c r a c k i n g -

v e r t i c a l l y t h r o u g h t h e o v e r l a y a n d i n t e r f a c e d e b o n d i n g b e t w e e n s t e e l a n d c o n c r e t e - w i l l

b e d e s c r i b e d b y t h e F C M . F o r s m a l l c r a c k o p e n i n g s , a n i n t e r f a c i a l f r a c t u r e b e t w e e n s t e e l

a n d c o n c r e t e s h o w s t h e a b i l i t y t o t r a n s f e r s t r e s s a c r o s s t h e c r a c k . T h i s p h e n o m e n o n i s

c a l l e d ' i n t e r l o c k i n g ' a n d i t s s i g n i c a n c e i n a d e b o n d i n g p r o c e s s i s d e s c r i b e d i n G r a n j u

( 2 0 0 1 ) . T h e ' i n t e r l o c k i n g ' e e c t i s s y n o n y m o u s w i t h t h e F C M . T h e m a i n f o c u s o f t h e

p r e s e n t p a p e r i s t h e p r e s e n t a t i o n o f a t w o - d i m e n s i o n a l n o n l i n e a r n i t e e l e m e n t m o d e l t o

s i m u l a t e t h e d e b o n d i n g p r o c e s s d i s c u s s e d a b o v e . F u r t h e r m o r e , t h e m o d e l i s u s e d t o c a r r y

o u t t w o t h e o r e t i c a l p a r a m e t e r s t u d i e s i n o r d e r t o i l l u s t r a t e t h e i n u e n c e o f d e b o n d i n g o n

t h e o v e r a l l b e h a v i o u r .

2 N u m e r i c a l M o d e l l i n g

T h e p r o b l e m o f a v e r t i c a l c r a c k p e n e t r a t i n g a c o n c r e t e l a y e r b o n d e d t o a s t e e l p l a t e a n d

t h e s u b s e q u e n t d e b o n d i n g i s s t u d i e d i n a t h r e e p o i n t b e n d i n g s e t - u p a c c o r d i n g t o F i g u r e

1 . T h e m o d e l l i n g i s c a r r i e d o u t a s a t w o d i m e n s i o n a l c o m p o s i t e b e a m m o d e l , c o n s i s t i n g o f

a c o n c r e t e l a y e r b o n d e d t o a t h i n s t e e l p l a t e . T h e v e r t i c a l c r a c k i n g z o n e i n t h e c o m p o s i t e

b e a m u n d e r t h e a p p l i e d l o a d P c o r r e s p o n d s t o t h e z o n e n e a r a m i d s p a n s u p p o r t i n a

b r i d g e s t r u c t u r e . N o t e t h a t t h e c o m p o s i t e b e a m i s t u r n e d u p s i d e d o w n f o r c o n v e n i e n c e .

F i g u r e 1 : E x p e r i m e n t a l s e t - u p : S i m u l a t i n g a n e g a t i v e b e n d i n g m o m e n t i n a b r i d g e d e c k

T h e v e r t i c a l c o n c r e t e c r a c k i n g i s m o d e l l e d u s i n g s t a n d a r d i n t e r f a c e e l e m e n t s a s i m p l e -

2



m e n t e d i n t h e a p p l i e d s o f t w a r e P a c k a g e D I A N A ( 2 0 0 3 ) . T h e i n t e r f a c i a l m i x e d m o d e

f r a c t u r e i s m o d e l l e d u s i n g a c o m p o s i t e i n t e r f a c e m o d e l o r i g i n a l l y d e v e l o p e d b y L o u r e n ç o

& R o t s ( 1 9 9 7 ) , a l s o i m p l e m e n t e d i n D I A N A . T h e m i x e d m o d e b e h a v i o u r a n d t h e s i g n i -

c a n c e o f c o n s t i t u t i v e m a t e r i a l p a r a m e t e r s a r e i n v e s t i g a t e d t h r o u g h t w o p a r a m e t e r s t u d i e s .

F i r s t l y , t h e i n u e n c e o f a t o u g h c o n c r e t e o v e r l a y a n d d i e r e n t f r a c t u r e e n e r g i e s o f t h e s t e e l -

c o n c r e t e i n t e r f a c e a r e i n v e s t i g a t e d . T h e n , a p a r a m e t e r s t u d y i s c a r r i e d o u t c o n c e r n i n g t h e

f a i l u r e c r i t e r i o n f o r t h e s t e e l - c o n c r e t e i n t e r f a c e .

2 . 1 M a t e r i a l C h a r a c t e r i s a t i o n

C o n s i d e r i n g t h e c o n c r e t e - s t e e l i n t e r f a c e , t h e s t a t e o f s t r e s s i s d e s c r i b e d t h r o u g h n o r m a l

a n d s h e a r s t r e s s e s . I n m a n y c a s e s t h e f a i l u r e s u r f a c e o f t h e s t r e s s s t a t e i n a s t e e l - c o n c r e t e

i n t e r f a c e i s w e l l d e s c r i b e d t h r o u g h a M o h r - C o u l o m b c r i t e r i o n . T h e f a i l u r e c r i t e r i o n i n t h e

a p p l i e d c o n s t i t u t i v e m o d e l , c o n s i s t o f t w o s u r f a c e s , d e n o t e d f 1 a n d f 2 , r e p r e s e n t i n g s h e a r

a n d t e n s i o n f a i l u r e r e s p e c t i v e l y , c f . F i g u r e 2 .

F i g u r e 2 : F a i l u r e s u r f a c e s f 1 a n d f 2 , d e n e d b y t h e c o h e s i o n c, s l o p e t a n ( ϕ) , a n d t h e

t e n s i l e s t r e n g t h f t .

N o n l i n e a r s o f t e n i n g o f t h e i n t e r f a c e i s c h a r a c t e r i z e d b y d e g r a d a t i o n o f t h e t e n s i l e s t r e s s f ta n d t h e c o h e s i o n c. T h e n o n l i n e a r s o f t e n i n g o f t h e t e n s i l e s t r e s s a n d c o h e s i o n i s a s s u m e d

t o b e h a v e a c c o r d i n g t o t h e f o l l o w i n g l a w

c(s) = c0exp

−

c0

GII f

s

; f t(w) = f toexp

−

f t0

GI f

w

( 1 )

w h e r e c0 a n d f t0 a r e i n i t i a l v a l u e s o f t h e c o h e s i o n a n d t e n s i l e s t r e n g t h , a n d r e p r e s e n t t h e

f r a c t u r e e n e r g y o f M O D E I a n d I I r e s p e c t i v e l y , s a n d w r e p r e s e n t t h e s l i p a n d o p e n i n g

a f t e r c r a c k i n i t i a t i o n . T h e n o n l i n e a r e x p o n e n t i a l b e h a v i o u r g i v e n i n E q u a t i o n s 1 ( a - b ) ,

o r i g i n a t e s f r o m e x p e r i m e n t s o n m o r t a r / b r i c k i n t e r f a c e s b y d e r P l u i j m ( 1 9 9 2 ) .

A t t h e i n t e r s e c t i o n b e t w e e n t h e f a i l u r e s u r f a c e s f 1 a n d f 2 , c f . F i g u r e 2 , a p e r m a n e n t

c o u p l i n g b e t w e e n t e n s i o n a n d s h e a r f a i l u r e i s p r e s e n t . T h e s o f t e n i n g o f t h e t w o s u r f a c e s f 1a n d f 2 a r e c o u p l e d a s s u m i n g i s o t r o p i c s o f t e n i n g , m e a n i n g t h a t t h e p e r c e n t a g e o f s o f t e n i n g

o n t h e c o h e s i o n i s a s s u m e d t o b e t h e s a m e o n t h e t e n s i l e s t r e n g t h . T h i s i s e x p l a i n e d

3



i n f u r t h e r d e t a i l s i n L o u r e n ç o e t a l . ( 1 9 9 8 ) . F u r t h e r m o r e , f o r s i m p l i c i t y , a l l c a l c u l a t i o n s

a r e c a r r i e d o u t a s s u m i n g a s s o c i a t e d p l a s t i c i t y : t a n ( ϕ) = t a n ( ψ ) , w h e r e ψ i s t h e d i l a t a t i o n

a n g l e .

C r a c k i n g o f t h e o v e r l a y i s a s s u m e d t o p r o p a g a t e a t m i d s p a n 0 . 5 L, u s i n g d i s c r e t e c r a c k

t h e o r y . N o n l i n e a r e e c t s o f t h e b u l k m a t e r i a l a r e n o t c o n s i d e r e d i n t h i s c a s e . T h e s t r e s s -

c r a c k o p e n i n g r e l a t i o n s h i p o f c o n c r e t e c a n i n m a n y c a s e s b e m o d e l l e d u s i n g a b i l i n e a r c u r v e

( O l e s e n 2 0 0 1 ) . I n t h e p r e s e n t s t u d y a b i l i n e a r c u r v e i s a p p l i e d a c c o r d i n g t o e q u a t i o n ( 2 ) .

σ(w)

f t= bi − aiw =

b1 − a1w, 0 ≤ w < wb2 − a2w, w1 ≤ w ≤ w1

( 2 )

w h e r e b1 = 1 ; a n d t h e l i m i t w1 a r e g i v e n b y t h e i n t e r s e c t i o n o f t h e t w o l i n e s e g m e n t s a n d

w2 c o r r e s p o n d t o t h e i n t e r s e c t i o n b y t h e s e c o n d l i n e a n d t h e a b s c i s s a .

T h e s t e e l a n d c o n c r e t e o u t s i d e t h e c r a c k p a t h s i s m o d e l l e d a s a l i n e a r e l a s t i c m a t e r i a l s

w i t h t h e e n g i n e e r i n g c o n s t a n t s E s = 2 1 0 G P a , ν s = 0 . 3 , E c = 3 0 G P a a n d ν c = 0 . 1 5 . T h e

e l e m e n t s r e p r e s e n t i n g t h e b u l k m a t e r i a l a r e 8 - n o d e i s o p a r a m e t r i c p l a n e s t r e s s e l e m e n t s o r

t r i a n g u l a r f o r m e s h r e n i n g , c f . F i g u r e 3 f o r t h e a p p l i e d m e s h .

T h e t w o c r a c k p a t h s - t h e v e r t i c a l a n d i n t e r f a c e c r a c k - a r e m o d e l l e d u s i n g s o - c a l l e d i n t e r -

f a c e e l e m e n t s , a v a i l a b l e i n t h e a p p l i e d F E M - p a c k a g e ( D I A N A 2 0 0 3 ) . A t w o d i m e n s i o n a l

c o n g u r a t i o n i s c o n s i d e r e d , w h e r e t h e i n t e r f a c e e l e m e n t r e l a t e s t h e f o r c e s a c t i n g o n t h e

i n t e r f a c e t o t h e r e l a t i v e d i s p l a c e m e n t o f t h e t w o s i d e s o f t h e i n t e r f a c e . I n t h e p r e s e n t

c a s e , t h e i n t e r f a c e e l e m e n t i s d e s c r i b e d b y a t r a c t i o n v e c t o r a n d a v e c t o r w h i c h r e p r e s e n t s

t h e r e l a t i v e d i s p l a c e m e n t s . I n t h e e l a s t i c r e g i m e t h e r e l a t i o n s h i p b e t w e e n t r a c t i o n a n d

r e l a t i v e d i s p l a c e m e n t i s g i v e n b y :

twts

=

kw 00 ks

uwus

( 3 )

w i t h kw a n d ks a s s i g n e d l a r g e p e n a l t y v a l u e s t o m o d e l i n i t i a l c o n t i n u o u s g e o m e t r y . T h e

i n t e r f a c e e l e m e n t s h a v e a t h i c k n e s s o f z e r o a n d a r e p l a c e d b e t w e e n t h e s t e e l a n d c o n c r e t e

a n d a t m i d s p a n t o r e p r e s e n t a p o s s i b l e v e r t i c a l c r a c k , c f . F i g u r e 3 .

F i g u r e 3 : H a l f b e a m m e s h u s e d i n F E M c a l c u l a t i o n s , t h e t w o t h i c k l i n e s r e p r e s e n t i n t e r f a c e

e l e m e n t s .

4



2 . 2 S t u d y o f D e b o n d M e c h a n i s m

C o n s i d e r a l o a d P a c t i n g o n a c o m p o s i t e b e a m a s s h o w n i n F i g u r e 1 . T h e l o a d c a u s e s a

' n e g a t i v e b e n d i n g m o m e n t ' , c h a r a c t e r i s e d b y t e n s i l e s t r e s s e s i n t h e c o n c r e t e l a y e r . T h i s

w i l l e v e n t u a l l y , a t s o m e s t a g e , l e a d t o c r a c k i n g o f t h e c o n c r e t e . T h e v e r t i c a l c r a c k p r o p -

a g a t e s t h r o u g h t h e c o n c r e t e o v e r l a y , b u t a t s o m e s t a g e i t i s o p p o s e d b y t h e s t e e l p l a t e .

T h e o p p o s i t i o n o f t h e s t e e l p l a t e w i l l l e a d t o a n i n c r e a s e o f n o r m a l s t r e s s i n a p l a n e p e r -

p e n d i c u l a r t o t h e v e r t i c a l c r a c k t i p , i . e . i n t h e p l a n e o f t h e s t e e l c o n c r e t e i n t e r f a c e . T h e

h o r i z o n t a l s t r e s s i n t e n s i t y i s l i k e l y t o i n t r o d u c e c r a c k i n g , a l o n g t h e ' w e a k e s t l i n k ' - w h i c h

i n t h i s c a s e i s a s s u m e d t o b e t h e i n t e r f a c e b e t w e e n s t e e l a n d c o n c r e t e .

T h e i n i t i a t i o n o f a n i n t e r f a c i a l c r a c k i s i l l u s t r a t e d t h r o u g h a c a s e s t u d y . A c o m p o s i t e

b e a m i s s t u d i e d g i v e n t h e g e o m e t r y : L= 8 0 0 m m , hc = 5 0 m m , hs = 8 m m , b = 1 0 0 m m a n d a

v e r t i c a l c r a c k d e s c r i b e d t h r o u g h E q u a t i o n ( 2 ) g i v e n t h e v a l u e s : f t = 3 M P a , a1 = 1 0 m m

−1,

a2 = 0 . 1 m m

−1, b2 = 0 . 5 .

T h e d i s t r i b u t i o n o f i n t e r f a c e s t r e s s e s i s i l l u s t r a t e d f o r d i e r e n t o p e n i n g s o f t h e v e r t i c a l

c r a c k . T h e o p e n i n g o f t h e v e r t i c a l c r a c k , c a l l e d t h e c r a c k m o u t h o p e n i n g d i s p l a c e m e n t

( C M O D ) , i s c a l c u l a t e d a t m i d s p a n , c f . F i g u r e 4 - a . F o r a n o p e n i n g o f 0 a n d 0 . 0 3 m m t h e

s t r e s s d i s t r i b u t i o n i s s h o w n a s a f u n c t i o n o f t h e x - c o o r d i n a t e n o r m a l i s e d w i t h t h e c o n c r e t e

h e i g h t hc , c f . F i g u r e 4 - b . T h e x - c o o r d i n a t e i s m e a s u r e d f r o m m i d s p a n .

( a )

0 0.5 1 1.5 2−2

0

2

X−coordinate/hc

[mm/mm]

S t r e s s [ M P a ]

σ, CMOD=0.03mm

σ, CMOD=0.00mm

τ, CMOD=0.00mm

τ, CMOD=0.03mm

( b )

F i g u r e 4 : S t r e s s d i s t r i b u t i o n a l o n g t h e i n t e r f a c e f o r a C M O D v a l u e o f z e r o a n d 0 . 0 3 m m

( a ) I n t e r f a c i a l f o r c e s a n d c o n g u r a t i o n ( b ) S t r e s s d i s t r i b u t i o n a l o n g t h e i n t e r f a c e v e r s u s

t h e x - c o o r d i n a t e n o r m a l i s e d w i t h t h e c o n c r e t e h e i g h t hc . D a s h e d l i n e s r e p r e s e n t s h e a r

s t r e s s τ a n d s o l i d l i n e r e p r e s e n t t h e n o r m a l s t r e s s σ .

I t i s o b s e r v e d t h a t f o r a n u n c r a c k e d c o n c r e t e o v e r l a y ( C M O D = 0 m m ) , t h e n o r m a l i n t e r f a c e

s t r e s s e s a r e g o v e r n e d b y c o m p r e s s i o n . A s t h e c r a c k i s i n i t i a t e d a n d f u r t h e r o p e n e d , t h e

5



n o r m a l s t r e s s e s c h a n g e s f r o m c o m p r e s s i o n t o t e n s i o n , i n m a n y c a s e s c r i t i c a l t o t h e b o n d

b e t w e e n s t e e l a n d c o n c r e t e . F u r t h e r t h e s h e a r s t r e s s e s a r e s i g n i c a n t l y i n c r e a s e d .

3 P a r a m e t r i c S t u d y I - F r a c t u r e E n e r g y

T h e p u r p o s e o f t h i s p a r a m e t r i c s t u d y i s t o s i m u l a t e t h e g l o b a l b e h a v i o u r o f t h e s e t - u p

g i v e n i n F i g u r e 1 f o r d i e r e n t f r a c t u r e e n e r g i e s o f t h e o v e r l a y a n d s t e e l - c o n c r e t e i n t e r f a c e .

T h e d i e r e n t s i m u l a t i o n s p e r f o r m e d a r e g r o u p e d i n t o 3 d i e r e n t g r o u p s d e n o t e d A - C .

S i m u l a t i o n A 1 i s t h e r e f e r e n c e c a s e , a n d e a c h o f t h e o t h e r g r o u p s i n v o l v e s s i m u l a t i o n s

w h e r e p a r a m e t e r s h a v e b e e n c h a n g e d i n r e l a t i o n t o t h i s c a s e . T h e d i e r e n t g r o u p s o f

s i m u l a t i o n s a r e :

A . V a r i a t i o n o f t h e b i l i n e a r σ − w r e l a t i o n o f t h e v e r t i c a l c r a c k a s d e n e d i n E q u a t i o n

( 2 )

B . V a r i a t i o n o f M o d e I e n e r g y f o r t h e s t e e l - c o n c r e t e i n t e r f a c e . T h e n o n l i n e a r s o f t e n i n g

o f t h e i n t e r f a c e i n M o d e I i s d e n e d i n E q u a t i o n ( 1 - a )

C . V a r i a t i o n o f M o d e I I e n e r g y f o r t h e s t e e l - c o n c r e t e i n t e r f a c e . T h e n o n l i n e a r s o f t e n i n g

o f t h e i n t e r f a c e i n M o d e I I i s d e n e d i n E q u a t i o n ( 1 - b )

T h e d i e r e n t g r o u p s o f s i m u l a t i o n s a n d t h e i r p a r a m e t e r v a l u e s a r e s u m m a r i z e d i n T a b l e

1 .

N a m e C h a n g i n g P a r a m e t e r

a n d v a l u e s

A 1 R e f e r e n c e

A 2 0 . 2 b2A 3 1 . 8 b2B 1 0 . 5 GI

f

B 2 2 . 0 GI f

C 1 0 . 5 GII f

C 2 2 . 0 GII f

T a b l e 1 : T h e P a r a m e t e r v a r i a t i o n o f b2 - v a l u e o f t h e v e r t i c a l c r a c k a n d M o d e I + I I f r a c t u r e

e n e r g i e s o f s t e e l - c o n c r e t e i n t e r f a c e .

T h e r s t c a s e ( A ) c o n c e r n s t h e s i g n i c a n c e o f t h e t o u g h n e s s o f t h e v e r t i c a l c r a c k . T h e v e r -

t i c a l c r a c k i s c h a r a c t e r i s e d b y a b i l i n e a r c r a c k - o p e n i n g r e l a t i o n s h i p a c c o r d i n g t o e q u a t i o n

( 2 ) . T h e b2 - v a l u e m a y b e t a k e n a s a m e a s u r e o f t h e a m o u n t o f b e r u s e d i n t h e m a t e r i a l ,

w h e r e a h i g h b2 - v a l u e c o r r e s p o n d s t o h i g h b e r c o n t e n t . F i n a l l y , t h e s i g n i c a n c e o f d i e r -

e n t M o d e I a n d I I e n e r g i e s o f t h e s t e e l - c o n c r e t e i n t e r f a c e i s s t u d i e d t h r o u g h c a s e s B - C . I n

t h e r e f e r e n c e c a s e , A 1 , t h e f o l l o w i n g n u m e r i c a l v a l u e s o f t h e g e o m e t r y p a r a m e t e r s w e r e

u s e d , c f . F i g u r e 1 : L = 8 0 0 m m , hs = 8 m m , hc = 5 0 m m , b= 1 0 0 m m . T h e n u m e r i c a l v a l u e s

6



u s e d f o r t h e m a t e r i a l p a r a m e t e r s i n t h e b i l i n e a r σ−w r e l a t i o n o f t h e v e r t i c a l c r a c k w e r e :

f t = 3 M P a a1 = 1 0 m m

1, a2 = 0 . 1 m m

1, b2 = 0 . 5 . T h e n u m e r i c a l v a l u e s o f t h e s t e e l - c o n c r e t e

i n t e r f a c e w e r e , f t = 2 M P a , c= 3 M P a , t a n ( ϕ) = 0 . 5 , GI f = GII

f = 0 . 1 N / m m .

C h a n g i n g t h e t o u g h n e s s o f t h e v e r t i c a l c r a c k l e a d s t o a c h a n g e i n g l o b a l b e h a v i o u r . T h i s

i s p r e s e n t e d i n F i g u r e 5 ( a ) , s h o w i n g t h e m a x i m u m m o m e n t v e r s u s t h e o p e n i n g o f t h e

v e r t i c a l c r a c k . I t i s o b s e r v e d t h a t c h a n g i n g t h e b2 - v a l u e o f t h e v e r t i c a l c r a c k a e c t s t h e

g l o b a l d u c t i l i t y s i g n i c a n t l y .

T h e r e l a t i o n b e t w e e n t h e i n t e r f a c e c r a c k l e n g t h a n d C M O D i s s h o w n i n F i g u r e 5 ( b ) .

H e r e i t i s s h o w n t h a t t h e t o u g h n e s s o f t h e v e r t i c a l c r a c k h a s l i t t l e i n u e n c e o n t h e r e l a t i o n

b e t w e e n t h e i n t e r f a c e c r a c k l e n g t h a n d t h e o p e n i n g o f t h e v e r t i c a l c r a c k .

C h a n g i n g M o d e I a n d I I e n e r g y f o r t h e i n t e r f a c e h a s a l a r g e r e e c t o n t h e c r a c k l e n g t h -

C M O D r e l a t i o n t h a n t h e d u c t i l i t y o f t h e v e r t i c a l c r a c k . T h i s i s i l l u s t r a t e d i n F i g u r e 6 ( b ) .

F u r t h e r m o r e , a h i g h e r i n t e r f a c i a l M o d e I e n e r g y , h a s a n i n u e n c e o n t h e M o m e n t - C M O D

r e l a t i o n s h i p , c f . F i g u r e 6 ( a ) . C o m p a r i n g c a s e s B a n d C s h o w s t h a t - f o r t h e s e m a t e r i a l

p r o p e r t i e s - t h e e n e r g y c o n s u m e d a t t h e i n t e r f a c e p r i m a r i l y c o n s i s t o f M o d e I e n e r g y .

0 0.5 1 1.5 20

100

200

300

400

500

600

•

•

•


M o

m e n t [ k N m m ]

0 0.5 1 1.5 20

0.5

1

1.5

2


c r a c k l e

n g t h / h

c [ m m / m m ]

A2

A1

A3

Debondingstarts

A3

A2

A1

F i g u r e 5 : G r a p h i c a l r e p r e s e n t a t i o n o f t h r e e h y p o t h e t i c a l c a s e s : A 1 , - . - . A 2 , - - - A 3 . ( a )

T h e b e n d i n g m o m e n t M = P L/4v e r s u s t h e c r a c k o p e n i n g o f t h e v e r t i c a l c r a c k - C M O D ,

( b ) T h e i n t e r f a c e c r a c k l e n g t h n o r m a l i s e d w i t h r e s p e c t t o t h e c o n c r e t e h e i g h t hc v e r s u s

C M O D .

7



0 0.5 1 1.5 20

100

200

300

400

500

•••


M o m e n t [ k N

m m ]

0 0.5 1 1.5 20

0.5

1

1.5

2


c r a c k l e n g t h / h c [ m m / m m ]

Debondingstarts

B1

B2

A1,C1,C2

B1

A1,C1,C2

B2

F i g u r e 6 : G r a p h i c a l r e p r e s e n t a t i o n o f v e h y p o t h e t i c a l c a s e s c f . T a b l e 1 . ( a ) T h e b e n d i n g

m o m e n t M = P L/4 v e r s u s t h e c r a c k o p e n i n g o f t h e v e r t i c a l c r a c k - C M O D , ( b ) T h e

i n t e r f a c e c r a c k l e n g t h n o r m a l i s e d w i t h r e s p e c t t o t h e c o n c r e t e h e i g h t hc v e r s u s C M O D .

4 P a r a m e t r i c S t u d y I I - F a i l u r e S u r f a c e

I n o r d e r t o s t u d y t h e d e g r e e o f m i x e d m o d e p r o p a g a t i o n a s e c o n d p a r a m e t e r s t u d y i s

c a r r i e d o u t . S a m e r e f e r e n c e c a s e a s i n p r e v i o u s s e c t i o n A 1 i s c o n s i d e r e d . O n e g r o u p o f

s i m u l a t i o n s , d e n o t e d D , i s p e r f o r m e d . T h i s g r o u p i n v o l v e s a v a r i a t i o n o f t h e s l o p e o f

f a i l u r e s u r f a c e f 1 , c f . F i g u r e 2 . T h e p a r a m e t e r v a l u e s a r e s u m m a r i z e d i n T a b l e 2 .

N a m e C h a n g i n g P a r a m e t e r

a n d v a l u e

A 1 R e f e r e n c e

D 1 2 t a n ( ϕ)

D 2 3 t a n ( ϕ)

T a b l e 2 : P a r a m e t e r v a r i a t i o n o f s l o p e o f f a i l u r e s u r f a c e f 1 , t a n ( ϕ) a s d e n e d i n F i g u r e 2 .

F o r c o n s t a n t c o h e s i o n c a c h a n g e i n t h e s l o p e t a n ( ϕ) o f f 1 , w i l l c h a n g e t h e p o s i t i o n o f t h e

c o r n e r o f t h e f a i l u r e c r i t e r i o n . M i x e d s h e a r a n d t e n s i o n f a i l u r e t a k e s p l a c e i n t h e c o r n e r

w h e n f 1 a n d f 2 i n t e r s e c t s , a n d i s o f s p e c i a l i n t e r e s t , s i n c e t h e s t e e l - c o n c r e t e i n t e r f a c e c r a c k

p r o p a g a t e s i n a m i x e d m o d e .

T h e s t r e s s d i s t r i b u t i o n d u r i n g c r a c k p r o p a g a t i o n i s s h o w n i n F i g u r e 7 . R e s u l t s f o r t h e

c a s e s A 1 a n d D 1 f o r a v e r t i c a l c r a c k o p e n i n g o f C M O D = 0 . 5 m m a r e s h o w n . T h e i n t e r f a c e

c r a c k h a s p r o p a g a t e d a l e n g t h o f a r o u n d ( x - c o o r d i n a t e / hc ) ≈ 1 . T h e p e a k o f t e n s i l e

n o r m a l s t r e s s e s r e p r e s e n t s t h e i n t e r f a c i a l c r a c k t i p .

T h e c o r n e r v a l u e s f o r c a s e s A 1 a n d D 1 d i e r d u e t o a d i e r e n t s l o p e o f s u r f a c e f 1 . A s t h e

c r a c k p r o p a g a t e s a l o n g t h e i n t e r f a c e , i t w i l l t r y t o t r a n s f e r m a x i m u m n o r m a l - a n d s h e a r

s t r e s s a c r o s s t h e i n t e r f a c i a l c r a c k . D o m i n a t e d b y M o d e I f r a c t u r e , t h e s t r e s s s t a t e i n t h e

8



0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−80

−60

−40

−20

0

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80

100

crack tip propagation/hc

[mm/mm]

M i x e d m o d e a n g l e β [ D e

g r e e s ]

D2

D1

A1

( a ) ( b )

F i g u r e 8 : ( a ) M i x e d m o d e a n g l e β a s d e n e d i n E q u a t i o n ( 4 ) v e r s u s i n t e r f a c e c r a c k t i p

p o s i t i o n n o r m a l i s e d w i t h t h e c o n c r e t e h e i g h t

hcf o r t h r e e h y p o t h e t i c a l c a s e s : A 1 , - -

- D 1 , - . - . D 2 . T h e m i x e d m o d e a n g l e i s c a l c u l a t e d a t t h e p e a k o f n o r m a l s t r e s s , ( b )

S c h e m a t i c r e p r e s e n t a t i o n o f t h e t h r e e f a i l u r e s u r f a c e s c o n s i d e r e d .

5 C o n c l u s i o n

A m o d e l t o i n v e s t i g a t e t h e d e b o n d i n g p r o c e s s o f a c o n c r e t e o v e r l a y c a s t o n a s t e e l p l a t e h a s

b e e n d e v e l o p e d . B a s e d o n d i s c r e t e c r a c k t h e o r y , t h e m o d e l d e s c r i b e s t h e s i t u a t i o n w h e r e

a v e r t i c a l c r a c k p r o p a g a t e s t h r o u g h t h e c o n c r e t e o v e r l a y a n d c a u s e s d e b o n d i n g b e t w e e n

t h e s t e e l - c o n c r e t e i n t e r f a c e . T h e i n t e r f a c i a l m i x e d m o d e f r a c t u r e h a s b e e n m o d e l l e d u s i n g

a c o m p o s i t e i n t e r f a c e m o d e l t a k i n g i n t o a c c o u n t n o n l i n e a r s o f t e n i n g o f t h e i n t e r f a c e .

T h e d e b o n d i n g p r o c e s s i n t h e c a s e o f a n e g a t i v e b e n d i n g m o m e n t h a s b e e n i n v e s t i g a t e d

t h r o u g h t w o p a r a m e t e r s t u d i e s . F i r s t l y , t h e e e c t o f a d u c t i l e c o n c r e t e o v e r l a y h a s b e e n

i n v e s t i g a t e d a n d t h e r e s u l t s s h o w h o w i t i n u e n c e s t h e g l o b a l p e r f o r m a n c e . H o w e v e r ,

a d u c t i l e c o n c r e t e o v e r l a y h a s l i t t l e e e c t o n t h e r e l a t i o n b e t w e e n t h e i n t e r f a c i a l c r a c k

l e n g t h a n d t h e o p e n i n g o f t h e v e r t i c a l c r a c k C M O D . S e c o n d l y , M o d e I e n e r g y o f t h e

s t e e l - c o n c r e t e i n t e r f a c e h a s i n t h e c a s e c o n s i d e r e d , l a r g e r e e c t o n g l o b a l d u c t i l i t y t h a n

M o d e I I e n e r g y . F u r t h e r m o r e , i n t e r f a c i a l M o d e I e n e r g y h a s a s i g n i c a n t i n u e n c e o n t h e

M o m e n t - C M O D r e l a t i o n s h i p .

A s e c o n d p a r a m e t e r s t u d y h a s b e e n c a r r i e d o u t i n o r d e r t o s t u d y t h e s t r e s s d i s t r i b u t i o n

a l o n g t h e i n t e r f a c e . T h e s t r e s s d i s t r i b u t i o n d u r i n g c r a c k p r o p a g a t i o n h a s b e e n i n v e s t i g a t e d

f o r d i e r e n t s h a p e s o f t h e f a i l u r e c r i t e r i o n . I t c a n b e c o n c l u d e d t h a t t h e c r a c k i n i t i a t e s

i n a m i x e d m o d e , w h e r e t h e s t r e s s d i s t r i b u t i o n a l o n g t h e s t e e l - c o n c r e t e i s d e p e n d e n t o n

t h e i n t e r s e c t i o n b e t w e e n t h e s h e a r a n d n o r m a l s t r e s s f a i l u r e s u r f a c e s . D e p e n d e n t o n t h e

f a i l u r e c r i t e r i o n , t h e s t r e s s s t a t e a l o n g t h e i n t e r f a c i a l p r o c e s s z o n e t e n d s t o s t a y i n t h e

c o r n e r b e t w e e n s h e a r a n d t e n s i o n f a i l u r e . T h e m i x e d m o d e s t a t e c h a n g e s t h r o u g h c r a c k

p r o p a g a t i o n o f t h e i n t e r f a c e .

F r o m a p r a c t i c a l p o i n t o f v i e w t h e m o d e l c a r r i e d o u t a t t e m p t s t o i l l u s t r a t e t h e r o l e o f t h e

d i e r e n t p a r a m e t e r s , w h i c h m i g h t b e c o n s i d e r e d i n a d e s i g n s i t u a t i o n . I n a d d i t i o n t o t h e

1 0



s i g n i c a n c e o f h i g h f r a c t u r e e n e r g y o f t h e o v e r l a y i n t h e d e b o n d i n g s i t u a t i o n c o n s i d e r e d ,

t h i s s t u d y s h o w s t h e d i e r e n c e i n b e h a v i o u r f o r d i e r e n t M o d e I a n d I I e n e r g i e s o f t h e

s t e e l - c o n c r e t e i n t e r f a c e .

R e f e r e n c e s

d e r P l u i j m , R . V . ( 1 9 9 2 ) , ` M a t e r i a l p r o p e r t i e s a n d i t s c o m p o n e n t s u n d e r t e n s i o n a n d

s h e a r ' , P r o c e e d i n g o f 6 t h C a n . M a s o n r y S y m p . S a s k a t o n C a n a d a .



G r a n j u , J . ( 2 0 0 1 ) , ` D e b o n d i n g o f t h i n c e m e n t - b a s e d o v e r l a y s ' , J o u r n a l o f M a t e r i a l s i n

C i v i l E n g i n e e r i n g 1 3 ( 2 ) , 1 1 4 1 2 0 .

G r a n j u , J . L . ( 1 9 9 6 ) , ` T h i n b o n d e d o v e r l a y s : A b o u t t h e r o l e o f b e r r e i n f o r c e m e n t o n t h e

l i m i t a t i o n o f t h e i r d e b o n d i n g ' , A d v . C e m e n t B a s e d M a t 4 ( 1 ) , 2 1 2 7 .



R e s . 6 ( 6 ) , 7 7 3 7 8 2 .



L o u r e n ç o , P . B . , R o t s , J . G . & B l a a u w e n d r a a d , J . ( 1 9 9 8 ) , ` C o n t i n u u m m o d e l f o r m a s o n r y :

P a r a m e t e r e s t i m a t i o n a n d v a l i d a t i o n ' , J o u r n a l o f S t r u c t u r a l E n g i n e e r i n g 1 2 4 ( 7 ) , 6 4 2

6 5 2 .



W a l t e r , R . , S t a n g , H . , G i m s i n g , N . J . & O l e s e n , J . F . ( 2 0 0 3 ) , ` H i g h p e r f o r m a n c e c o m p o s i t e

b r i d g e d e c k s u s i n g s c s f r c ' , T h e F o u r t h I n t e r n a t i o n a l W o r k s h o p o n H i g h P e r f o r m a n c e

F i b e r R e i n f o r c e d C e m e n t C o m p o s i t e s , A n n A r b o r , M i c h i g a n p p . 4 9 5 5 0 4 .

1 1



Paper VI —

Method for Determination of Tensile Properties of ECC

Paper in the proceedings of : ConMat’05, Vancouver, Canada, 2005



M e t h o d f o r D e t e r m i n a t i o n o f T e n s i l e P r o p e r t i e s o f E n g i n e e r e d

C e m e n t i t i o u s C o m p o s i t e s ( E C C )

L e n n a r t Ø s t e r g a a r d , R a s m u s W a l t e r a n d J o h n F . O l e s e n


L y n g b y , D e n m a r k .

P a p e r i n : T h e p r o c e e d i n g s o f C o n M a t ' 0 5 , V a n c o u v e r , C a n a d a , 2 0 0 5 .

A b s t r a c t

T o d a y , E C C h a s b e e n d e v e l o p e d t o a s t a g e w h e r e t h e m a t e r i a l i s b e i n g u s e d i n e l d

t e s t i n g a n d l i m i t e d f u l l s c a l e c o n s t r u c t e d f a c i l i t i e s . H o w e v e r , t h e l a c k o f r e l i a b l e , p r a c -

t i c a l , s t a n d a r d i z e d t e s t m e t h o d s f o r d e t e r m i n a t i o n o f t h e s t r a i n h a r d e n i n g p r o p e r t i e s

o f t h e m a t e r i a l c o n s t i t u t e s a s i g n i c a n t o b s t a c l e f o r d a y - t o - d a y s t r u c t u r a l e n g i n e e r i n g

d e s i g n a n d m a t e r i a l q u a l i t y c o n t r o l d u r i n g c o n s t r u c t i o n e x e c u t i o n . I n p r a c t i c e a s i m p l e

a n d r o b u s t t e s t m e t h o d i s r e q u i r e d o n a c o n s t r u c t i o n s i t e . T h e u n i a x i a l t e n s i o n t e s t

( U T T ) i s r e g a r d e d t o o c o m p l i c a t e d t o s e r v e a s s u c h . I n s t e a d , t h e w i d e l y a d o p t e d f o u r

p o i n t b e n d i n g t e s t ( F P B T ) m a y b e u s e d f o r t h i s p u r p o s e , p r o v i d e d t h a t a n a p p r o p r i a t e

i n t e r p r e t a t i o n o f t h e t e s t d a t a i s a v a i l a b l e . T h e p r e s e n t p a p e r p r e s e n t s a c o n t r i b u t i o n

i n t h i s d i r e c t i o n . S e c t i o n a l a n a l y s i s o f E C C b e a m s i n t h e s t r a i n - h a r d e n i n g s t a g e i s u s e d

i n a p a r a m e t r i c a n a l y s i s a i m e d a t r e v e a l i n g t h e n e d e t a i l s o f c o r r e l a t i o n u n i q u e n e s s

b e t w e e n t h e t e n s i l e s t r e s s - s t r a i n ( σ - ε ) c u r v e p a r a m e t e r s o f a n E C C a n d i t s F P B T l o a d -

d e e c t i o n d i a g r a m . F u r t h e r , a n i n v e r s e a n a l y s i s d e s i g n a t e d f o r t h e d e t e r m i n a t i o n o f t h e

s t r a i n h a r d e n i n g p r o p e r t i e s i s d e s c r i b e d . F i n a l l y , e x p e r i m e n t a l r e s u l t s b a s e d o n F P B T

t e s t r e s u l t s a r e p r e s e n t e d .

K e y w o r d s : E C C , F P B T , i n v e r s e a n a l y s i s , h i n g e m o d e l , s t r a i n h a r d e n i n g , t e n s i l e p r o p -

e r t i e s

1 . I n t r o d u c t i o n

O v e r t h e p a s t d e c a d e , a n e w c l a s s o f c i v i l e n g i n e e r i n g m a t e r i a l s h a s b e e n d e v e l o p e d , o f t e n

r e f e r r e d t o a s H i g h P e r f o r m a n c e F i b e r R e i n f o r c e d C e m e n t i t i o u s C o m p o s i t e s ( H P F R C C ) .

A s u b c l a s s i s t h e E n g i n e e r e d C e m e n t i t i o u s C o m p o s i t e s ( E C C ) , w h i c h a r e e n g i n e e r e d

b y

d e t a i l e d d e s i g n o f t h e m i c r o s t r u c t u r e i n o r d e r t o o b t a i n e x t r a o r d i n a r y t o u g h n e s s a n d d u c -

t i l i t y . T h e f u n d a m e n t a l d i e r e n c e b e t w e e n E C C a n d a c o n v e n t i o n a l F i b e r R e i n f o r c e d

C o n c r e t e ( F R C ) i s i t s b e h a v i o r i n t e n s i o n . F o r F R C , f r a c t u r e l o c a l i z a t i o n o c c u r s i m m e d i -

a t e l y a f t e r t h e r s t c r a c k i s f o r m e d . I n t h e c a s e o f E C C , p r o p a g a t i o n o f t h e i n i t i a l c r a c k

w i l l b e p r e v e n t e d b y t h e b e r b r i d g i n g , c a u s i n g a s e c o n d m a t r i x c r a c k t o f o r m e l s e w h e r e .

T h i s r e s u l t s i n n o n - l o c a l i z e d m u l t i p l e c r a c k i n g , a l s o k n o w n a s m a c r o s c o p i c p s e u d o s t r a i n

h a r d e n i n g b e h a v i o r . E C C s h a v e p r e s e n t e d n e w a l t e r n a t i v e s i n t h e d e s i g n o f c i v i l s t r u c -

t u r e s . D u e t o t h e i r h i g h d u c t i l i t y t h e y a r e w e l l s u i t e d i n t h e d e s i g n o f s t r u c t u r e s e x p o s e d

1



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(a) (b)

F i g u r e 1 : A s s u m e d c o n s t i t u t i v e r e l a t i o n s h i p s f o r E C C : b i l i n e a r i n t e n s i o n a n d c o m p r e s s i o n p r i o r

t o c r a c k l o c a l i z a t i o n ( a ) , w h i l e a b i l i n e a r σ

- w

r e l a t i o n s h i p a f t e r c r a c k l o c a l i z a t i o n i s a s s u m e d ( b ) .

H o w e v e r , a l s o t h e l o c a l i z a t i o n a n d s o f t e n i n g o f E C C i s i m p o r t a n t w h e n a F P B T i s c o n -

s i d e r e d s i n c e i t m a y i n u e n c e t h e p r e - p e a k r e s p o n s e s i g n i c a n t l y . I n f a c t , u p t o 5 0 % o f

t h e p r e - p e a k r e s p o n s e m a y b e a s s o c i a t e d w i t h l o c a l i z a t i o n . T h i s i s d u e t o t h e d u c t i l e

b e h a v i o r o f t h e E C C w h i c h i n u e n c e s t h e σ - w r e l a t i o n s h i p , m a k i n g i t v e r y t o u g h . T h i s

i s i l l u s t r a t e d i n K a b e l e & H o r i i ( 1 9 9 7 ) , w h e r e t h e s o f t e n i n g w a s d e s c r i b e d u s i n g a l i n e a r

c u r v e f r o m p e a k s t r e s s t o z e r o s t r e s s w i t h w2 = Lf /2 , w h e r e w2 i s t h e c r a c k o p e n i n g f o r

w h i c h t h e s t r e s s t r a n s f e r i s e q u a l t o z e r o , a n d Lf i s t h e b e r l e n g t h . T h e t y p i c a l l e n g t h

o f t h e b e r s i n E C C i s 8 - 1 2 m m .

T h u s , t h e p r e s e n t m o d e l i n c l u d e s t h e i n u e n c e o f t h e σ - w r e l a t i o n s h i p , h o w e v e r , u s i n g

t h e b i l i n e a r a p p r o x i m a t i o n s h o w n i n F i g u r e 1 . C o n s i d e r a t i o n o f t h i s p a r t o f t h e m a t e r i a l

b e h a v i o r w i l l m a k e i t p o s s i b l e t o d e t e r m i n e t h e i n u e n c e o f t h e l o c a l i z a t i o n o n t h e F P B T

r e s p o n s e .

3 . T h e H i n g e M o d e l

T h e m o d e l l i n g w i l l f o l l o w t h e n o n l i n e a r h i n g e c o n c e p t d e s c r i b e d i n O l e s e n ( 2 0 0 1 ) . T h e

h i n g e i s i l l u s t r a t e d i n F i g u r e 2 a . T h e p r o p a g a t i n g c r a c k i s m o d e l l e d w i t h i n t h e e l e m e n t

b o u n d a r i e s a s a l a y e r o f i n d e p e n d e n t s p r i n g e l e m e n t s . T h e s e s p r i n g e l e m e n t s a r e f o r m e d

b y i n c r e m e n t a l h o r i z o n t a l s t r i p s , a n d a r e a t t a c h e d a t e a c h e n d t o a r i g i d b o u n d a r y . E a c h

b o u n d a r y m a y r o t a t e a n d t r a n s l a t e s u c h t h a t i t m a y b e j o i n e d w i t h a n u n c r a c k e d b e a m

m o d e l l e d a c c o r d i n g t o t h e c l a s s i c a l b e a m t h e o r y . F i g u r e 2 b s h o w s t h e s t r e s s d i s t r i b u t i o n i n

t h e h i n g e w h e n b o t h c r u s h i n g a n d l o c a l i z a t i o n w i t h s t r e s s f r e e t a i l h a s d e v e l o p e d . B a s e d

o n t h i s d i s t r i b u t i o n , t h e e n t i r e h i n g e b e h a v i o r f r o m u n s e t o f l o a d i n g t o f a i l u r e m a y b e

d e s c r i b e d . I t p r o v e s c o n v e n i e n t t o i n t r o d u c e t h e m e a n v a l u e s o f t h e c u r v a t u r e a n d t h e

d i s t r i b u t i o n o f l o n g i t u d i n a l s t r a i n s , κ∗ a n d ε∗ , r e s p e c t i v e l y :

3



s

h

2

M

N

M

N y

d

½ h

½ h

localized

crack

y0

( a )

y0

h

y-2

y-1

y1

y2 y3

y4d

( b )

F i g u r e 2 : G e o m e t r y , l o a d i n g a n d d e f o r m a t i o n o f t h e h i n g e e l e m e n t ( a ) a n d s t r e s s d i s t r i b u t i o n i n

t h e h i n g e e l e m e n t w h e n c r u s h i n g a n d l o c a l i z a t i o n o f t h e c r a c k h a s d e v e l o p e d ( b )

κ∗ = 2ϕ

s; ε∗ = 2(y − y0)κ∗ ( 1 )

T h e d e f o r m a t i o n o f a n i n c r e m e n t a l s t r i p i s g i v e n b y u(y) = sε∗(y) w h e r e s i s t h e w i d t h o f

t h e h i n g e . T h u s , g i v e n t h e m a t e r i a l p a r a m e t e r s i n F i g u r e 1 , t h e d e p t h s y−2 , y−1 , y1 a n d

y2 m a y b e d e t e r m i n e d :

y−2 = y0 − εc2s

2ϕ , y−1 = y0 − εc1s

2ϕ , y1 = y0 + εt1s

2ϕ , y2 = y0 + εt2s

2ϕ( 2 )

T h e t o t a l d e f o r m a t i o n , u(y) , i n t h e c r a c k e d s t a g e s m a y b e d e t e r m i n e d a s t h e s u m o f t h e

e l a s t i c d e f o r m a t i o n , t h e o p e n i n g o f t h e c r a c k a n d t h e e l a s t i c u n l o a d i n g d u e t o t h e s t r e s s

d r o p c a u s e d b y t h e l o c a l i z a t i o n :

u(y) = εt2s + w(y)−f t2 − σw(y)

E uls = εs + w(y) +

σw(y)

E uls ( 3 )

H e r e , t h e c r a c k o p e n i n g i n t h e c r a c k e d i n c r e m e n t a l s t r i p i n t h e d e p t h y , w(y), h a s b e e n

i n t r o d u c e d . T h e a s s o c i a t e d c r a c k b r i d g i n g s t r e s s , σw(w(y)) i s g i v e n b y σw(w(y)) = f t2(bi−

aiw(y)),

i = 1, 2w h e r e

bi = (1, b2), s e e F i g u r e 1 , a n d w h e r e

a1a n d

a2a r e t h e r s t a n d

s e c o n d s l o p e s o f t h e n o r m a l i z e d σ - w r e l a t i o n s h i p , r e s p e c t i v e l y . T h e u n l o a d i n g i s a s s u m e d

e l a s t i c a n d t h e s t i n e s s i s d e n o t e d E ul w h i l e ε i s t h e u n r e c o v e r e d s t r a i n a f t e r c o m p l e t e

u n l o a d i n g :

ε = εt2 −f t2E ul

( 4 )

C o m b i n i n g E q u a t i o n s 1 a n d 3 r e s u l t s i n a n e x p r e s s i o n f o r t h e c r a c k s t r e s s a t d e p t h y :

4



σw(w(y)) = (2(y − y0)ϕ− εs− w(y))E ul

s( 5 )

U t i l i z i n g t h e b i l i n e a r σ - w r e l a t i o n s h i p a s s u m p t i o n , a n e x p r e s s i o n f o r t h e c r a c k o p e n i n g ,

w(y), a n d t h e s t r e s s , σ(y) , m a y b e d e r i v e d :

w(y) =2(y − y0)ϕ− εs− ζ i

1− β i, σ(y) =

ζ i − 2(y − y0)ϕβ i + εsβ i1− β i

E uls

( 6 )

w h e r e t h e d i m e n s i o n l e s s p a r a m e t e r s β i a n d ζ i a r e d e n e d b y :

β i =f t2ais

E ul, ζ i =

f t2bis

E ul, i ∈ [1, 2] ( 7 )

T h e d e p t h s y3 a n d y4 m a y b e d e t e r m i n e d f r o m E q u a t i o n 6 b y s o l v i n g w1 = w(y3) a n d

w2 = w(y4):

yi+2 = y0 +1

2ϕ(ζ i + εs + (1 − β i)wi) , i ∈ [1, 2] ( 8 )

T h e c o m p l e t e s t r e s s d i s t r i b u t i o n m a y n o w b e e s t a b l i s h e d f o r a n y g i v e n r o t a t i o n o f t h e

h i n g e , ϕ, a n d t h u s , t h e n o r m a l f o r c e , N a n d t h e m o m e n t , M m a y b e d e r i v e d . T h i s i s

d o n e b y p i e c e w i s e n u m e r i c a l i n t e g r a t i o n i n t h e r a n g e y ∈ [0, h] u s i n g t h e p h a s e c h a n g e

p a r a m e t e r s yi, i ∈ [−2,−1,.., 4] . T h e c a l c u l a t i o n s p r o c e e d b y i n c r e m e n t i n g t h e h i n g e

r o t a t i o n , ϕ, a n d s u b s e q u e n t l y d e t e r m i n i n g r s t t h e d e p t h o f t h e n e u t r a l a x i s a n d t h e n t h e

h i n g e m o m e n t M .

4 . I m p l e m e n t a t i o n o f t h e h i n g e m o d e l i n t h e F P B T

F i g u r e 3 a s h o w s t h e F P B T . T h e c l e a r s p a n o f t h e b e a m i s Lt , w h i l e t h e d i s t a n c e f r o m

t h e e n d s t o t h e l o a d i n g p o i n t s i s Lc . T h e c o n s t a n t m o m e n t s p a n i s d e n o t e d b y Lm . T h e

d e e c t i o n a t m i d p o i n t o f t h e b e a m i s m e a s u r e d w i t h r e f e r e n c e t o t h e l o a d i n g p o i n t s . S i n c e

t h e m o m e n t i s c o n s t a n t i n t h e m i d d l e p a r t o f t h e b e a m , t h i s p a r t m a y b e m o d e l l e d w i t h

i n c r e m e n t a l h i n g e e l e m e n t s , a l l u n d e r g o i n g t h e s a m e r o t a t i o n . P r i o r t o l o c a l i z a t i o n o f t h e

c r a c k i n t h e h i n g e , t h e h i n g e r o t a t i o n a n d t h e m i d p o i n t d e e c t i o n w i l l b e g o v e r n e d b y t h e

i n i t i a l e l a s t i c p a r t a n d t h e s t r a i n h a r d e n i n g p a r t o f t h e σ−ε

c u r v e . T h e s e c o n t r i b u t i o n s a r e

b o t h n o n - l o c a l a n d m a y b e t e r m e d t h e n o n - l o c a l i z e d r o t a t i o n , ϕnl , a n d t h e n o n - l o c a l i z e d

d e e c t i o n ,

unl. T h i s i s t h e s i t u a t i o n i n t h e u p p e r p a r t o f F i g u r e 3 a . H o w e v e r , a f t e r t h e

p e a k m o m e n t , M p , i s r e a c h e d , a s t r u c t u r a l l o c a l i z a t i o n w i l l t a k e p l a c e , i . e . t h e c r a c k s w i l l

c o n t i n u e t o o p e n a t o n e o r t w o l o c a t i o n s , w h i l e t h e c r a c k s i n t h e r e m a i n i n g p a r t s o f t h e

b e a m w i l l c l o s e . O n e p o s s i b l e m e c h a n i s m i n t h i s s i t u a t i o n i s s h o w n i n t h e l o w e r p a r t o f

F i g u r e 3 a , w h i l e o t h e r s m a y b e c o n s t r u c t e d b y s u p e r p o s i t i o n o f s i m i l a r m e c h a n i s m s . A f t e r

t h e s t r u c t u r a l l o c a l i z a t i o n , t h e t o t a l r o t a t i o n o f t h e h i n g e a n d t h e m i d p o i n t d e e c t i o n w i l l

i n c l u d e a l o c a l i z e d p a r t , ϕl a n d ul . M o d e l l i n g o f t h e u n l o a d i n g r e q u i r e s t h a t a c e r t a i n

u n l o a d i n g b r a n c h i s d e n e d . F i g u r e 3 b s h o w s t h e a s s u m e d u n l o a d i n g b r a n c h f o r t h e

c l o s i n g h i n g e s w i t h a c o m m o n f o c u s p o i n t w i t h t h e c o o r d i n a t e s (ϕfp, M fp) . T h e o p e n i n g

5



h i n g e s w i l l d e t e r m i n e t h e m o m e n t c a p a c i t y o f t h e b e a m s i n c e h e r e t h e m a t e r i a l c o n t i n u e s t o

d e g r a d e . T h e m i d p o i n t d e e c t i o n , unl , a n d t h e l o a d , P , p r i o r t o t h e s t r u c t u r a l l o c a l i z a t i o n

a r e r e a d i l y o b t a i n e d a s :

unl = 18

κL2m = 1

4ϕnl

sL2m , P = 4M

Lt − Lm

( 9 )

A f t e r t h e s t r u c t u r a l l o c a l i z a t i o n h a s o c c u r r e d , t h e t o t a l h i n g e r o t a t i o n i s g i v e n b y : ϕtot =ϕnl + ϕl . F o r a g i v e n t o t a l h i n g e r o t a t i o n a n d c a l c u l a t e d m o m e n t , M (ϕtot), t h e s e c o n t r i -

b u t i o n s m a y b e d e t e r m i n e d a s :

ϕnl = ϕ p − (ϕ p − ϕfp)

M p −M

M p −M fp

, ϕl = ϕtot − ϕnl ( 1 0 )

w h e r e M p i s t h e p e a k m o m e n t a n d ϕ p t h e c o r r e s p o i n d i n g r o t a t i o n . T h e l o c a l i z e d d e f o r m a -

t i o n , ul , i s d e t e r m i n e d f r o m t h e a c t u a l l o c a l i z a t i o n m e c h a n i s m . F o r t h e s i t u a t i o n s h o w n

i n t h e l o w e r p a r t o f F i g u r e 3 a , t h e r e s u l t i s :

ul(ϕl) =1

2(La − Lc) ϕl f o r La ≤ Lb ( 1 1 )

T h e t o t a l d e e c t i o n i n t h e c a s e w h e r e s t r u c t u r a l l o c a l i z a t i o n h a s t a k e n p l a c e i s r e a d i l y

o b t a i n e d a s t h e s u m o f E q u a t i o n s 9 a n d 1 1 . N o t e , t h a t o t h e r l o c a l i z a t i o n m e c h a n i s m s

m a y b e m o d e l l e d b y s u p e r p o s i n g t h e m e c h a n i s m i n F i g u r e 3 u s i n g d i e r e n t l o c a l i z a t i o n

p o i n t s .

unl

Lc Lc Lm

nl

ul

l

La Lb

Lt

( a )

M

closing

hinge

opening hinge

onset of material

localization

fp

M fp

peak

moment M p

p

M

nl

l

tot

( b )

F i g u r e 3 : D e f o r m a t i o n s o f t h e F P B T d u r i n g l o a d i n g . I n i t i a l l y , n o s t r u c t u r a l l o c a l i z a t i o n i s

p r e s e n t , w h i l e l a t e r , a s t r u c t u r a l l o c a l i z a t i o n o c c u r s ( a ) . S c h e m a t i c m o m e n t - r o t a t i o n c u r v e f o r

t h e n o n - l i n e a r h i n g e s h o w i n g t h e i n c o r p o r a t i o n o f a s i m p l e e l a s t i c u n l o a d i n g c r i t e r i o n ( b ) .

6



0 50 100 150 200 2500

0.5

1

1.5

2

2.5

3

3.5

4

Normalized deflection u / uel [-]

N o r m a l i z e d l o a d P / P

e l

[ - ]

h = 10 mm50100

structural

localization

material

localization

onset of strain

hardening

F i g u r e 4 : F E M r e s u l t s ( t h i n l i n e s ) c o m p a r e d w i t h t h e h i n g e m o d e l r e s u l t s ( t h i c k l i n e s ) ( a ) f o r

a F P B T f o r t h r e e d i e r e n t b e a m h e i g h t s w i t h s/h= 0 . 9 . T h e w h i t e c i r c l e s m a r k t h e t r a n s i t i o n

p o i n t s o f t h e h i n g e m o d e l .

5 . F E M c a l i b r a t i o n o f t h e F P B T m o d e l

T h e h i n g e m o d e l i n t r o d u c e s t h e h i n g e w i d t h p a r a m e t e r , s , w h i c h n e e d s t o b e d e t e r m i n e d

t h r o u g h a n F E M c a l i b r a t i o n . I n v e s t i g a t i o n s o n n o r m a l c o n c r e t e a n d F R C a p p l y i n g t h e

h i n g e m o d e l o n t h e T h r e e P o i n t B e n d i n g T e s t ( T P B T ) i n d i c a t e t h a t s m a y b e s e l e c t e d

r o u g h l y a s h/2 f o r a l l m a t e r i a l c h o i c e s a n d b e a m g e o m e t r i e s , s e e e . g . Ø s t e r g a a r d ( 2 0 0 3 ) .

H o w e v e r , f o r t h e F P B T i t a p p e a r s t h a t t h e F E M r e s u l t s a r e a p p r o x i m a t e d m o s t p r e -

c i s e l y u s i n g s/h= 0 . 9 . F i g u r e 4 s h o w s a c o m p a r i s o n b e t w e e n t h e h i n g e m o d e l a n d t h e

F E M r e s u l t s f o r d i e r e n t b e a m h e i g h t s . T h e r e m a i n i n g p a r a m e t e r s f o r t h e b e a m s a r e :

Lt = 5 0 0 m m , Lm = 2 5 0 m m a n d t h i c k n e s s t= 1 0 0 m m . T h e m a t e r i a l p a r a m e t e r s h a v e b e e n

s e l e c t e d a s : f t1 = 4 M P a , f t2 = 6 M P a , E t1 = E c1 = 2 2 0 0 0 M P a , E t2 = 1 0 0 M P a a n d a1 = 0 . 0 8 3 3

m m

−1, w h e r e E c1 , E t1 a n d E t2 d e n o t e s t h e s l o p e s o f t h e σ - ε r e l a t i o n s h i p s . C o m p r e s s i v e

s o f t e n i n g a n d c r u s h i n g h a s b e e n d i s r e g a r d e d i n t h i s c o m p a r i s o n b y s e l e c t i n g l a r g e v a l u e s

o f f c1 a n d f c2 . T h e σ - w r e l a t i o n s h i p h a s b e e n a p p r o x i m a t e d w i t h a l i n e a r c u r v e b y s e -

l e c t i n g s m a l l v a l u e s o f a2 a n d b2 w h i l e t h e c a l c u l a t i o n o f a1 i s b a s e d o n t h e a s s u m p t i o n

t h a t w2 = Lf /2, Lf = 12 m m . T h e s t r a i n c a p a c i t y o f t h e m a t e r i a l i s a p p r o x i m a t e l y 2 %

(

εt2 ≈ 0.02) . T h e s t r u c t u r a l l o c a l i z a t i o n o f t h e F E M m o d e l o c c u r s s y m m e t r i c a l l y a r o u n d

t h e m i d s e c t i o n a t b o t h l o a d i n g p o i n t s ( F i g u r e 3 a w i t h La = Lc s u p e r p o s e d w i t h t h e s a m e

s i t u a t i o n m i r r o r e d ) . T h i s i s d u e t o a s m a l l s t r e s s c o n c e n t r a t i o n a t t h o s e p o i n t s c a u s e d b y

t h e c o n c e n t r a t e d l o a d i n g . T h e c h o i c e o f s i n F i g u r e 4 i s m a d e b a s e d o n a n a t t e m p t t o

t t h e p o i n t o f s t r u c t u r a l l o c a l i z a t i o n f o r a l l b e a m h e i g h t s . T h e c u r v e s i n F i g u r e 4 a r e

n o r m a l i z e d w i t h r e g a r d r e g a r d t o t h e e l a s t i c d e e c t i o n , uel , a n d l o a d , P el , w h i c h m a y b e

c a l c u l a t e d u s i n g E q u a t i o n s 2 a n d 9 .

7



T h e F E M a n d h i n g e m o d e l c u r v e s s h o w a h i g h d e g r e e o f c o r r e l a t i o n . P r i o r t o t h e m a t e r i a l

l o c a l i z a t i o n , w h e r e o n l y t h e t e n s i l e s t r a i n h a r d e n i n g a c c o u n t s f o r t h e n o n - l i n e a r i t y o f t h e

l o a d - d e e c t i o n r e s p o n s e , t h e r e s u l t s a r e e s s e n t i a l l y i d e n t i c a l . H o w e v e r , a f t e r t h e m a t e r i a l

l o c a l i z a t i o n i s i n i t i a t e d , t h e h i n g e m o d e l r e s u l t s b e c o m e d e p e n d e n t o n s , b u t w i t h t h e

o p t i m a l c h o i c e o f s, t h e d i e r e n c e s a r e s t i l l l i m i t e d .

6 . I n v e r s e a n a l y s i s

M e t h o d s f o r t h e e x t r a c t i o n o f t h e s t r a i n h a r d e n i n g p r o p e r t i e s o f E C C i s t h e s u b j e c t o f

o n l y o n e p a p e r i n t h e l i t e r a t u r e ( K a n a k u b o e t a l . 2 0 0 3 ) . H o w e v e r , t h i s p a p e r d o e s n o t

i n c l u d e t h e e e c t o f l o c a l i z a t i o n o f t h e c r a c k . U s i n g t h e h i n g e m o d e l d e s c r i b e d i n t h e

p r e v i o u s s e c t i o n s , l o c a l i z a t i o n i s i n c l u d e d , a n d a p r e c i s e r e s u l t i s o b t a i n e d b y s i m p l y u s i n g

t h e s q u a r e d s u m o f d i e r e n c e s b e t w e e n t h e e x p e r i m e n t a l r e s u l t a n d t h e m o d e l r e s u l t a s

o b j e c t f u n c t i o n :

min(f t1,f t2,E t1,E t2)

N i=1

P i − P i

P i

2

( 1 2 )

i n w h i c h P i i s t h e l o a d o b t a i n e d b y t h e h i n g e m o d e l f o r a c e r t a i n d e e c t i o n , ui , w h i l e

P i d e n o t e s t h e e x p e r i m e n t a l l y o b t a i n e d l o a d f o r t h e s a m e d e e c t i o n . T h e d a t a p o i n t s o n

t h e g r a p h m u s t b e e q u i d i s t a n t l y s p a c e d o n t h e c u r v e i n o r d e r t o g i v e e a c h p a r t t h e s a m e

w e i g h t i n t h e c a l c u l a t i o n s . N o t e t h a t t h e c o m p r e s s i v e p r o p e r t i e s a r e n o t i n c l u d e d i n t h e

o p t i m i z a t i o n s i n c e t h e y m a y b e d e t e r m i n e d i n i t i a l l y b y a s i m p l e c o m p r e s s i o n t e s t . A l s o

k n o w l e d g e o f t h e σ - w r e l a t i o n s h i p i s u n i m p o r t a n t - i n f a c t i t t u r n s o u t , i n c o m p a r i s o n w i t h

F E M r e s u l t s t h a t t h e s t r a i n h a r d e n i n g p r o p e r t i e s m a y b e d e t e r m i n e d w i t h o u t k n o w i n g

t h i s r e l a t i o n s h i p .

T a b l e 1 : R e s u l t s f r o m i n v e r s e a n a l y s i s w i t h d i e r e n t b e a m h e i g h t s , h

[ m m ] , t o g e t h e r w i t h a n

a n a l y s i s o n t h e s e n s i t i v i t y o f t h e s e l e c t i o n o f a1 [ m m

−1] a n d s [ m m ] . T h e s t r a i n h a r d e n i n g

m a t e r i a l p a r a m e t e r s w e r e s e l e c t e d t o f t1 = 4 MPa, f t2 = 6 MPa , E t1 = 2 2 GPa a n d E t2 =

1 0 0 MPa . T h e n u m b e r s i n t h e t a b l e r e p r e s e n t t h e e r r o r o n t h e d e t e r m i n a t i o n o f t h e m a t e r i a l

p a r a m e t e r s i n p e r c e n t

h 1 0 4 0 5 0 6 0 1 0 0 5 0 5 0 5 0

s 1 1 1 1 1 1 4 5 9 0

a1 0 . 0 8 3 3 0 . 0 8 3 3 0 . 0 8 3 3 0 . 0 8 3 3 0 . 0 8 3 3 2 . 6 6 5 6 0 . 0 8 3 3 0 . 0 8 3 3

E t1 1 . 2 % + 1 . 1 % + 2 . 3 % + 4 . 4 % + 2 9 . 7 % + 2 . 3 % + 2 . 3 % + 2 . 3 %

E t2 + 9 . 3 % 0 . 8 % + 1 . 0 % + 2 . 1 % + 8 . 0 % 0 . 6 % 1 . 4 % 4 . 0 %

f t1 1 . 1 % 0 . 1 % 0 . 5 % 0 . 9 % 4 . 2 % 0 . 4 % 0 . 3 % 0 . 1 %

f t2 1 . 7 % 0 . 2 % 0 . 8 % 1 . 1 % 1 . 9 % + 1 . 4 % + 2 . 4 % + 5 . 4 %

T h i s i s s h o w n i n T a b l e 1 w h e r e v a r i o u s F E M r e s u l t s h a v e b e e n s u b j e c t t o t h e m e t h o d

o f i n v e r s e a n a l y s i s . T h e c o r r e l a t i o n b e t w e e n t h e r e s u l t s i s r e m a r k a b l y g o o d . I f m o d e r a t e

8



h/Lm v a l u e s a r e s e l e c t e d t h e i n v e r s e a n a l y s i s i s v e r y p r e c i s e - o n l y e x t r e m e l y t h i n b e a m s ,

h = 1 0 m m , o r v e r y t h i c k b e a m s , h = 1 0 0 m m g i v e l e s s n e r e s u l t s . T h i s i s p a r t i c u l a r s e e n

f o r t h e t h i c k b e a m w h e r e E t1 i s d e t e r m i n e d w i t h a n e r r o r o f 3 0 % . N o t e t h a t t h e s e l e c t i o n

o f s i s u n i m p o r t a n t - s m a l l a s w e l l a s h i g h v a l u e s o f s g i v e g o o d r e s u l t s , w h i c h s h o w s t h a t

a c l o s e t o f t h e p o i n t o f s t r u c t u r a l l o c a l i z a t i o n i s n o t i m p o r t a n t . I t i s a l s o i n t e r e s t i n g t o

n o t i c e t h a t t h e s t r a i n h a r d e n i n g p r o p e r t i e s c a n b e d e t e r m i n e d r e g a r d l e s s o f t h e f a c t t h a t

a p a r t o f t h e t o t a l d e e c t i o n o f t h e b e a m i s c a u s e d b y c r a c k l o c a l i z a t i o n . H o w e v e r , n o t e

t h a t t h i s i s o n l y t r u e i f t h e m a t e r i a l l o c a l i z a t i o n p o i n t i s p r e d i c t e d a c c u r a t e l y . F i n a l l y , a s

d e m o n s t r a t e d i n T a b l e 1 , e v e n i f a1 i s s e l e c t e d a s 3 2 t i m e s l a r g e r t h a n t h e v a l u e u s e d i n

t h e F E M m o d e l , a c c u r a t e r e s u l t s a r e o b t a i n e d .

7 . E x p e r i m e n t a l r e s u l t s

T h e m e t h o d f o r e x t r a c t i o n o f t h e σ - ε r e l a t i o n s h i p o f E C C h a s b e e n t e s t e d o n a n E C C m i x

w i t h t h e m i x p r o p o r t i o n s g i v e n i n T a b l e 2 . I n t o t a l 9 b e a m s w e r e c a s t w i t h v a r y i n g b e a m

h e i g h t s , h = 4 0 , 5 0 a n d 6 0 m m . T h e r e m a i n i n g b e a m d i m e n s i o n s w e r e Lm = 2 5 0 m m , t =

1 0 0 m m a n d Lt = 5 0 0 m m . T h e c a s t i n g d i r e c t i o n w a s p e r p e n d i c u l a r t o t h e b e a m h e i g h t

i n o r d e r t o a v o i d a n y m a t e r i a l v a r i a t i o n s t h r o u g h o u t t h e b e a m h e i g h t . T h e b e a m s w e r e

c a s t i n w o o d e n w a t e r - t i g h t m o l d s a n d s t o r e d i n t h e c o n c r e t e l a b f o r t h e r s t 2 4 h o u r s , c o v -

e r e d w i t h p l a s t i c . T h e n , t h e y w e r e w a t e r c u r e d u n t i l t h e t e s t i n g b e g a n a t a n a g e o f 7 d a y s .

T a b l e 2 : M i x d e s i g n . T h e c e m e n t t y p e w a s C E M I 5 2 . 5 . F A i s y a s h , w h i l e S P i s s u p e r p l a s t i c i z e r

M i x C e m e n t S a n d Q u a r t s F A ( m i l l e d ) F A W a t e r S P F i b e r s

kg/m35 8 1 2 3 2 2 3 3 2 0 9 4 8 9 3 1 5 8 2 6

T h e F P B T s e t u p w a s d e s i g n e d s u c h t h a t b o t h s u p p o r t s a n d l o a d i n g p o i n t s c o u l d r o t a t e

i n d e p e n d e n t l y a r o u n d t w o a x e s : a h o r i z o n t a l a x i s p a r a l l e l t o t h e b e a m a n d a h o r i z o n t a l

a x i s p e r p e n d i c u l a r t o t h e b e a m . F u r t h e r m o r e , e a c h l o a d i n g p o i n t c o u l d m o v e v e r t i c a l l y

i n d e p e n d e n t l y o f e a c h o t h e r . T h i s s y s t e m e n s u r e s t h a t n o r e d u n d a n t f o r c e s w e r e a c t i n g

o n t h e b e a m .

T h e r e s u l t s f r o m t h e e x p e r i m e n t s a r e s h o w n i n F i g u r e 5 a . A s e x p e c t e d , t h e r e i s s o m e

s c a t t e r , w h i c h u s u a l l y i s t h e c a s e f o r E C C . F i g u r e 5 b s h o w s a c o m p a r i s o n b e t w e e n o n e e x -

p e r i m e n t a l c u r v e a n d t h e c o r r e s p o n d i n g o u t p u t f r o m t h e i n v e r s e a n a l y s i s . T h e c o r r e l a t i o n

i s c o n v i n c i n g a n d i t d e m o n s t r a t e s t h e a b i l i t y o f t h e m o d e l t o c a p t u r e r e a l E C C b e h a v i o r .

T h e a v e r a g e o u t p u t f r o m t h e d i e r e n t b e a m h e i g h t s a r e s h o w n i n T a b l e 3 . I t i s e v i d e n t

t h a t a l l b e a m s h e i g h t g i v e s t h e s a m e r e s u l t , w h i c h i s r a t h e r p r o m i s i n g f o r t h e m e t h o d s i n c e

t h i s e x c l u d e s t h e p o s s i b i l i t y o f s t r u c t u r a l i n u e n c e o n t h e m a t e r i a l p a r a m e t e r s d e t e r m i n e d .

9



0 1 2 3 4 5 6 7 80

2000

4000

6000

8000

10000

12000

0 1 2 3 4 5 6 7 80

2

4

6

8

10

12

Deflection [mm]

L o a d [ k N ]

h = 60 mm

h = 50 mm

h = 40 mm

( a )

0 1 2 3 4 5 60

1000

2000

3000

4000

5000

6000

0 1 2 3 4 5 60

1

2

3

4

5

6

Deflection [mm]

L o a d [ k N ]

( b )

F i g u r e 5 : E x p e r i m e n t a l r e s u l t s ( a ) , a n d a c o m p a r i s o n b e t w e e n o n e e x p e r i m e n t a l c u r v e a n d t h e

o u t c o m e o f t h e i n v e r s e a n a l y s i s ( b ) , ( h

= 4 0 m m , s/h

= 0 . 9 , u n l o a d i n g a c c o r d i n g t o F i g u r e 3 a

w i t h La = Lc + 2 0 m m )

T a b l e 3 : E x p e r i m e n t a l r e s u l t s o b t a i n e d t h r o u g h t h e i n v e r s e a n a l y s i s . T h e r e s u l t f o r e a c h b e a m

h e i g h t r e p r e s e n t t h e a v e r a g e o f t h r e e e x p e r i m e n t s . T h e l a t t e r c o l u m n g i v e s t h e a v e r a g e a n d

s t a n d a r d d e v i a t i o n f o r a l l n i n e b e a m s .

h [ m m ] 4 0 5 0 6 0 A v e r a g e S t d . d e v .

E t1 [ G P a ] 3 4 3 3 3 4 3 3 7

E t2[ G P a ] 0 . 3 0 0 . 2 0 0 . 2 1 0 . 2 4 0 . 0 6

f t1 [ M P a ] 2 . 5 2 . 5 2 . 6 2 . 6 0 . 2

f t2 [ M P a ] 4 . 3 4 . 0 4 . 1 4 . 1 0 . 3

εt1 [ µε] 8 4 7 7 8 0 8 0 1 8

εt2 [ % ] 0 . 6 0 . 8 0 . 7 0 . 7 0 . 1

8 . C o n c l u s i o n s

T h i s p a p e r p r e s e n t s a n e w m e t h o d f o r i n t e r p r e t a t i o n o f t h e l o a d - d e e c t i o n r e s p o n s e o f t h e

E C C F P B T s e t u p , b y i n c o r p o r a t i n g c r u s h i n g , t h e b i l i n e a r c o m p r e s s i v e a n d t e n s i l e σ - εr e l a t i o n s h i p s a n d t h e b i l i n e a r

σ-

wr e l a t i o n s h i p o f t h e E C C m a t e r i a l . T h e m e t h o d p r o v e s

v e r y a c c u r a t e w h e n c o m p a r e d w i t h F E M r e s u l t s .

I t h a s b e e n d e m o n s t r a t e d t h a t l o c a l i z a t i o n o f t h e c r a c k o c c u r s p r i o r t o t h e p e a k l o a d , a n d

t h u s , o n l y a p a r t o f t h e t o t a l d e e c t i o n o f a n E C C b e a m m a y b e a t t r i b u t e d t o t h e s t r a i n

h a r d e n i n g r e s p o n s e .

H o w e v e r , a m e t h o d p r o p o s e d f o r i n v e r s e a n a l y s i s p r o v e s a b l e t o d i s t i n g u i s h b e t w e e n t h e

s t r a i n h a r d e n i n g a n d t h e c r a c k l o c a l i z a t i o n p a r t s o f t h e d e e c t i o n , t h u s e n a b l i n g e x t r a c -

1 0



t i o n o f t h e s t r a i n h a r d e n i n g p r o p e r t i e s . C o m p a r e d w i t h F E M , t h i s m e t h o d g i v e s v e r y

p r e c i s e r e s u l t s . T h e m e t h o d a l s o p e r f o r m s w e l l i n c o m p a r i s o n w i t h e x p e r i m e n t s . H o w -

e v e r , c o m p l e m e n t a r y U T T e x p e r i m e n t s a r e n e e d e d i n o r d e r t o n a l l y v a l i d a t e t h e m o d e l .

9 . A c k n o w l e d g e m e n t s

T h e K n u d H ø j g a a r d F o u n d a t i o n i s g r a t e f u l l y a c k n o w l e d g e d f o r s u p p o r t i n g t h e w o r k , w h i l e

K u r a r a y E u r o p e i s c r e d i t e d f o r t h e g e n e r o u s d e l i v e r y o f t h e P V A b e r s . P r o f . V . C . L i

a n d A s s o c . P r o f . H . S t a n g a r e a l s o h i g h l y r e c o g n i z e d f o r i n s p i r i n g t h i s w o r k .

R e f e r e n c e s

K a b e l e , P . & H o r i i , H . ( 1 9 9 7 ) , ` A n a l y t i c a l m o d e l f o r f r a c t u r e b e h a v i o u r o f p s e u d o s t r a i n -

h a r d e n i n g c e m e n t i t i o u s c o m p o s i t e s ' , C o n c r e t e L i b . o f J S C E 2 9 , 1 0 5 1 2 0 .

K a n a k u b o , T . , S h i m i z u , T . , K a t a g i r i , K . , K a n d r a , M . , F u k u y a m a , T . & R o k u g o , K .

( 2 0 0 3 ) , E v a l u a t i o n o f t e n s i l e p r o p e r t i e s f o r D F R C C - r e s u l t s o f r o u n d r o b i n t e s t b y

J C I t e c h n i c a l c o m m i t e e , i n ` P r o c . J C I S y m p . o n D F R C C ( i n J a p a n e s e ) ' , p p . 1 0 1 1 1 1 .

L i , V . ( 2 0 0 3 ) , ` O n e n g i n e e r i n g c e m e n t i t i o u s c o m p o s i t e s ( E C C ) ' , J o u r n a l o f A d v a n c e s

C o n c r e t e T e c h n o l o g y 1 ( 3 ) , 2 1 5 2 3 0 .

L i , V . , W u , C . , W a n g , S . , O g a w a , A . & S a i t o , T . ( 2 0 0 2 ) , ` I n t e r f a c e t a i l o r i n g f o r s t r a i n -

h a r d e n i n g P V A - E C C ' , A C I M a t e r i a l s J o u r n a l 9 9 ( 5 ) , 3 7 3 4 0 0 .

M a a l e j , M . & L i , V . C . ( 1 9 9 4 ) , ` F l e x u r a l / t e n s i l e s t r e n g t h r a t i o n i n e n g i n e e r e d c e m e n t i t i o u s

c o m p o s i t e s ' , A C S E J o u r n a l o f M a t e r i a l s i n C i v i l E n g i n e e r i n g 6 ( 4 ) , 5 1 3 5 2 8 .






S t a n g , H . & L i , V . C . ( 2 0 0 4 ) , C l a s s i c a t i o n o f b e r r e i n f o r c e d c e m e n t i t i o u s m a t e r i a l s f o r

s t r u c t u r a l a p p l i c a t i o n s , i n ` B E F I B ' , R I L E M , p p . 1 9 7 2 1 8 .

1 1



Paper VII —

Analysis of Steel Bridge Deck Stiffened with Cement-Based Overlay

Paper submitted to ASCE - Journal of Bridge Engineering



A n a l y s i s o f S t e e l B r i d g e D e c k S t i e n e d w i t h C e m e n t - B a s e d

O v e r l a y

R a s m u s W a l t e r * , J o h n F . O l e s e n * , H e n r i k S t a n g * , & T i n a V e j r u m * *

* D e p a r t m e n t o f C i v i l E n g i n e e r i n g T e c h n i c a l U n i v e r s i t y o f D e n m a r k D K - 2 8 0 0 K g s .


* * C O W I A / S C o n s u l t a n t E n g i n e e r s , P a r a l l e l v e j 2 , D K - 2 8 0 0 K g s . L y n g b y , D e n m a r k

P a p e r s u b m i t t e d t o A S C E - J o u r n a l o f B r i d g e E n g i n e e r i n g .

A b s t r a c t

O v e r t h e p a s t y e a r s , w i t h i n c r e a s i n g t r a c v o l u m e s a n d h i g h e r w h e e l l o a d s , f a t i g u e

d a m a g e i n s t e e l p a r t s o f t y p i c a l o r t h o t r o p i c s t e e l b r i d g e d e c k s , h a s b e e n e x p e r i e n c e d o n

h e a v i l y t r a c k e d r o u t e s . A d e m a n d e x i s t s t o n d a d u r a b l e s y s t e m t o i n c r e a s e t h e f a t i g u e

s a f e t y o f o r t h o t r o p i c s t e e l b r i d g e d e c k s . A s o l u t i o n m i g h t b e t o e n h a n c e t h e s t i n e s s o f

t h e t r a d i t i o n a l o r t h o t r o p i c b r i d g e d e c k b y u s i n g a c e m e n t - b a s e d o v e r l a y . I n t h i s p a p e r ,

a n o r t h o t r o p i c s t e e l b r i d g e d e c k s t i e n e d w i t h a c e m e n t - b a s e d o v e r l a y i s a n a l y z e d . T h e

a n a l y s i s i s b a s e d o n n o n l i n e a r f r a c t u r e m e c h a n i c s , a n d u t i l i z e t h e n i t e e l e m e n t m e t h o d .

T h e s t i n e s s o f t h e s t e e l d e c k r e i n f o r c e d w i t h a n o v e r l a y d e p e n d s h i g h l y o n t h e c o m p o s i t e

a c t i o n . T h e c o m p o s i t e a c t i o n i s c l o s e l y r e l a t e d t o c r a c k i n g o f t h e o v e r l a y a n d i n t e r f a c i a l

c r a c k i n g b e t w e e n t h e o v e r l a y a n d u n d e r l y i n g s t e e l p l a t e ( d e b o n d i n g ) . A s a n e x a m p l e , a

r e a l s i z e s t r u c t u r e , t h e F a r ø B r i d g e s l o c a t e d i n D e n m a r k , a r e a n a l y z e d . T h e s t e e l b o x

g i r d e r s o f t h e F a r ø B r i d g e s s p a n s 8 0 m e t e r s , a n d h a v e a d e p t h o f 3 . 5 m , a n d a w i d t h o f

1 9 . 5 m . T h e f o c u s o f t h e p r e s e n t s t u d y i s t h e t o p p a r t o f t h e s t e e l b o x g i r d e r s , w h i c h

i s c o n s t r u c t e d a s a n o r t h o t r o p i c d e c k p l a t e . N u m e r o u s f a c t o r s c a n i n u e n c e t h e c r a c k -

i n g b e h a v i o r o f t h e c e m e n t - b a s e d o v e r l a y s y s t e m . B o t h m e c h a n i c a l a n d e n v i r o n m e n t a l

l o a d i n g h a v e t o b e c o n s i d e r e d a n d e e c t s s u c h a s s h r i n k a g e , t e m p e r a t u r e g r a d i e n t s a n d

t r a c l o a d i n g a r e t a k e n i n t o a c c o u n t . T h e p e r f o r m a n c e o f f o u r o v e r l a y m a t e r i a l s a r e

i n v e s t i g a t e d i n t e r m s o f c r a c k w i d t h s . F u r t h e r m o r e , t h e a n a l y s i s s h o w s t h a t d e b o n d i n g

i s i n i t i a t e d f o r a c e r t a i n c r a c k w i d t h i n t h e o v e r l a y . T h e l o a d l e v e l w h e r e c r a c k i n g a n d

d e b o n d i n g i s i n i t i a t e d d e p e n d s o n t h e s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p o f t h e m a t e r i a l .

K e y w o r d s O r t h o t r o p i c b r i d g e d e c k , c e m e n t - b a s e d o v e r l a y , F E M , b r e r e i n f o r c e d c o n -

c r e t e , n o n l i n e a r f r a c t u r e m e c h a n i c s .


O r t h o t r o p i c s t e e l b r i d g e d e c k s a r e w i d e l y u s e d f o r l a r g e a n d m e d i u m s p a n b r i d g e s . C o m -

p a r e d t o c o m p o s i t e g i r d e r s t r u c t u r e s i n l o n g - s p a n s t r u c t u r e s , t h e a d v a n t a g e o f o r t h o t r o p i c

s t e e l d e c k s i s t h e c o n s i d e r a b l y r e d u c e d d e a d l o a d . T h e s u c c e s s o f o r t h o t r o p i c b r i d g e d e c k s

1



m a y b e d u e t o i t s h i g h s t r e n g t h t o w e i g h t r a t i o ( D o w l i n g 1 9 6 8 ) . H o w e v e r , c o m p a r e d t o a

r i g i d c o n c r e t e b r i d g e d e c k , a s t e e l d e c k p l a t e i s s u b j e c t e d t o c o n s i d e r a b l e l o c a l d e f o r m a t i o n s

c a u s e d b y t h e w h e e l l o a d s o n t h e d e c k . O v e r t h e y e a r s , i t h a s b e e n a c k n o w l e d g e d t h a t t h e

r e c o r d o f d u r a b i l i t y o n h e a v i l y t r a c k e d r o u t e s h a s n o t b e e n s a t i s f a c t o r y . F a t i g u e d a m -

a g e , e s p e c i a l l y i n w e l d s a n d s t e e l p l a t e s h a v e b e e n o b s e r v e d , ( K o l s t e i n & W a r d e n i e r 1 9 9 7 ) ,

( K o l s t e i n & W a r d e n i e r 1 9 9 8 ) . R e p a i r s , p a r t i c u l a r l y t o h i g h w a y s u r f a c i n g , h a v e b e e n n e c -

e s s a r y o n m a n y i m p o r t a n t b r i d g e s w i t h i n t h e l a s t 2 0 y e a r s o r l e s s , s e e e . g . S m i t h & B r i g h t

( 2 0 0 3 ) .

W i t h t h e a i m t o a n a l y z e t h e f a t i g u e p r o b l e m i n d e t a i l s , t h e m e c h a n i c a l b e h a v i o r o f o r -

t h o t r o p i c s t e e l b r i d g e d e c k s h a s b e e n t h e s u b j e c t o f n u m e r o u s r e s e a r c h p r o j e c t s o v e r t h e

r e c e n t y e a r s . T h e s u r f a c i n g , a n d i t s c o n t r i b u t i o n t o s t r e s s r e d u c t i o n h a s , b e e n i n v e s t i -

g a t e d b y s e v e r a l a u t h o r s . T h e c h a r a c t e r i s t i c p r o p e r t i e s o f t w o b a s i c s u r f a c i n g c a t e g o r i e s ,

b i t u m e n o u s a n d p o l y m e r i c m a t e r i a l s , h a v e b e e n a n a l y z e d b y W o l c h u k ( 2 0 0 2 ) . T h e s t u d y

s h o w s , t h a t f o r a h i g h e l a s t i c m o d u l i o f t h e s u r f a c i n g m a t e r i a l , t h e s t r e s s r a n g e i n t h e c r i t -

i c a l f a t i g u e p a r t s o f t h e s t e e l d e c k i s r e d u c e d c o n s i d e r a b l y . H o w e v e r , t r a d i t i o n a l s u r f a c i n g

m a t e r i a l s a r e g e n e r a l l y v i s c o e l a s t i c o r p l a s t i c a n d b e h a v e e l a s t i c a l l y a t l o w t e m p e r a t u r e s

o n l y . E x p e r i m e n t s c a r r i e d o u t b y J o n g e t a l . ( 2 0 0 4 ) p e r f o r m e d o n o r t h o t r o p i c b r i d g e d e c k s

w i t h d i e r e n t s u r f a c i n g m a t e r i a l s a t d i e r e n t t e m p e r a t u r e s , s h o w t h a t t h e s t r e s s r a n g e

c o m p a r e d t o a d e c k w i t h o u t s u r f a c i n g i s r e d u c e d b y a f a c t o r o f 1 t o 6 .

A p r o m i s i n g s o l u t i o n m i g h t b e t o u s e a c e m e n t - b a s e d o v e r l a y t o r e d u c e t h e s t r e s s r a n g e i n

t h e f a t i g u e s e n s i t i v e s t e e l p a r t s o f a n o r t h o t r o p i c b r i d g e d e c k . T h i s a p p r o a c h h a s a l r e a d y

b e e n i n v e s t i g a t e d b y s e v e r a l a u t h o r s , s e e e . g . B a t t i s t a & P f e i l ( 2 0 0 0 ) o r B r a a m e t a l .

( 2 0 0 3 ) . I n p r a c t i c e , t h e r s t p i l o t t e s t a p p l y i n g a c e m e n t - b a s e d o v e r l a y t o a n o r t h o t r o p i c

s t e e l b r i d g e d e c k h a s a l r e a d y b e e n c a r r i e d o u t . A r e p l a c e m e n t o f a 6 0 m

2a r e a o f t h e

b a s c u l e p a r t o n t h e V a n B r e i n e n o o r d B r i d g e i n R o t t e r d a m , T h e N e t h e r l a n d s h a s b e e n

c a r r i e d o u t , c f . B u i t e l a a r ( 2 0 0 2 ) o r J o n g & K o l s t e i n ( 2 0 0 4 ) . H e r e i t i s c o n c l u d e d t h a t

s t r e s s e s i n f a t i g u e s e n s i t i v e d e t a i l s , c o m p a r e d t o a t r a d i t i o n a l o r t h o t r o p i c s t e e l b r i d g e

d e c k , a r e r e d u c e d f r o m 1 2 8 M P a t o 2 8 M P a . T h e e c o n o m i c a l i n v e s t m e n t f o r p l a c i n g t h e

o v e r l a y i s e q u a l t o t h e c o s t o f a t r a d i t i o n a l b i t u m i n o u s w e a r i n g c o u r s e o f m e l t e d a s p h a l t .

T h e p r e s e n t a n a l y s i s i n v e s t i g a t e s t h e p e r f o r m a n c e o f a n o r t h o t r o p i c s t e e l b r i d g e d e c k w i t h

a c e m e n t - b a s e d o v e r l a y a p p l i e d w i t h t h e a i m o f r e d u c i n g t h e s t r e s s r a n g e . A s t h e c o m p o s i t e

a c t i o n b e t w e e n t h e o v e r l a y a n d u n d e r l y i n g s t e e l d e c k i s c l o s e l y r e l a t e d t o c r a c k i n g , o v e r l a y

c r a c k i n g a n d d e b o n d i n g i s t h e m a i n f o c u s o f t h i s s t u d y . B a s e d o n n o n l i n e a r f r a c t u r e

m e c h a n i c s , m o d e l i n g o f c r a c k s i s c a r r i e d o u t u s i n g t h e s o - c a l l e d F i c t i t i o u s C r a c k M o d e l

( F C M ) r s t p r e s e n t e d b y H i l l e r b o r g e t a l . ( 1 9 7 6 ) . T h e a d v a n t a g e o f t h e F C M i s i t s

s i m p l i c i t y a n d g o o d c o r r e l a t i o n w i t h e x p e r i m e n t s .

E n v i r o n m e n t a l a n d m e c h a n i c a l l o a d i n g s u c h a s t r a c , e a r l y a g e s h r i n k a g e a n d t e m p e r a -

t u r e g r a d i e n t s , a r e a l l s i g n i c a n t f a c t o r s w h i c h c a n i n i t i a t e c r a c k i n g . H i g h c o n c e n t r a t e d

w h e e l l o a d s c a u s e c o n s i d e r a b l e l o c a l d e f o r m a t i o n s o f t h e d e c k . S i n c e t h e o v e r l a y i s b o n d e d

t o t h e d e c k , l o c a l d e f o r m a t i o n c a u s e d b e w h e e l l o a d s p r o d u c e s l a r g e t e n s i l e s t r e s s e s i n t h e

o v e r l a y a n d i n c r e a s e s t h e r i s k o f c r a c k i n g . T o o v e r c o m e t h e p r o b l e m o f h i g h t e n s i l e s t r e s s e s

i n t h e o v e r l a y , t h e a p p l i c a t i o n o f h i g h p e r f o r m a n c e c o n c r e t e m i g h t b e e e c t i v e . H o w e v e r ,

w h e n a p p l y i n g m o d e r n h i g h p e r f o r m a n c e c o n c r e t e s , c r a c k i n g s e n s i t i v i t y i s d r a m a t i c a l l y

2



i n c r e a s e d i n e a r l y a g e s a s w e l l a s i n t h e h a r d e n e d s t a g e . S i n c e t h e s e c o n c r e t e s h a v e l o w

w a t e r t o c e m e n t r a t i o s a n d o f t e n i n c l u d e s i l i c a f u m e , a u t o g e n o u s d e f o r m a t i o n s a r e i n -

c r e a s e d ( J e n s e n & H a n s e n 2 0 0 1 ) . T h u s , a n a n a l y s i s o f t h e s t r e s s h i s t o r y o f t h e o v e r l a y

i n e a r l y a g e s m i g h t b e i m p o r t a n t . A n o t h e r i m p o r t a n t a s p e c t , c o n c e r n s t h e i n u e n c e s o f

t e m p e r a t u r e g r a d i e n t s b e t w e e n t h e o v e r l a y a n d s t e e l p l a t e . S i n c e t h e o v e r l a y i s r e s t r a i n e d

f r o m m o v i n g d u e t o i t s b o n d t o t h e s t e e l d e c k , c o o l i n g o f t h e b r i d g e d e c k p r o d u c e s h i g h

t e n s i l e s t r e s s e s i n t h e c e m e n t - b a s e d o v e r l a y . T h e a b o v e c o n s i d e r a t i o n s a l l p o i n t t o w a r d s

a p p l y i n g d u c t i l e b e r r e i n f o r c e d c o n c r e t e a s o v e r l a y m a t e r i a l .

B a s e d o n n o n l i n e a r f r a c t u r e m e c h a n i c s , t h i s p a p e r a i m s t o a n a l y z e a n o r t h o t r o p i c b r i d g e

d e c k s t i e n e d w i t h a c e m e n t - b a s e d o v e r l a y . L o a d e e c t s s u c h a s m e c h a n i c a l a n d e n v i -

r o n m e n t a l l o a d i n g s a r e c o n s i d e r e d . T h e a i m o f t h e s t u d y i s t o d e s c r i b e a m e t h o d , w h i c h

i s c a p a b l e o f a n a l y z i n g t h e c e m e n t - b a s e d o v e r l a y s y s t e m b y m e a n s o f c r a c k i n g b e h a v i o r .

C r a c k w i d t h s a n d p a t t e r n s a r e i m p o r t a n t f a c t o r s i n t h e e s t i m a t i o n o f t h e d u r a b i l i t y a n d

h e n c e t h e s e r v i c e l i f e o f a g i v e n s t r u c t u r a l s y s t e m .

2 T h e F a r ø B r i d g e s

T h e p r e s e n t a n a l y s i s i s c a r r i e d o u t o n t h e F a r ø B r i d g e s , w h i c h i s a p a r t o f t h e S j æ l l a n d -

F a l s t e r c o n n e c t i o n l o c a t e d i n D e n m a r k . T h e F a r ø B r i d g e s w e r e b u i l t b e t w e e n 1 9 8 0 a n d

1 9 8 5 a n d c a r r i e s t h e " S o u t h M o t o r w a y " l e a d i n g f r o m C o p e n h a g e n t o t h e E u r o p e a n c o n -

t i n e n t o v e r w a t e r w a y s b e t w e e n t h e t w o i s l a n d s S j æ l l a n d a n d F a l s t e r , w i t h a s m a l l i s l a n d ,

F a r ø , i n t h e m i d d l e . T h e t w o b r i d g e s a r e c a l l e d t h e S j æ l l a n d - F a r ø B r i d g e a n d t h e F a r ø -

F a l s t e r B r i d g e , c f . F i g u r e 1 f o r a p i c t u r e o f t h e t w o b r i d g e s . F o r d e t a i l e d i n f o r m a t i o n , s e e

e . g . T h e D a n i s h M i n i s t r y o f T r a n s p o r t a t i o n ( 1 9 8 7 ) .

F i g u r e 1 : T h e F a r ø B r i d g e s .

F a t i g u e p r o b l e m s h a v e n o t y e t b e e n e x p e r i e n c e d i n t h e o r t h o t r o p i c s t e e l d e c k o f t h e F a r ø

3



B r i d g e s . T h e t r a c i n t e n s i t y o n t h e b r i d g e s i s s h o w n i n F i g u r e 2 f o r t h e p e r i o d f r o m

1 9 9 0 t o 2 0 0 4 . T h e t r a c i n t e n s i t y o n t h e F a r ø B r i d g e s c a n b e c o n s i d e r e d l o w c o m p a r e d

t o o t h e r b r i d g e s w h i c h h a v e e x p e r i e n c e d f a t i g u e d a m a g e i n o r t h o t r o p i c b r i d g e d e c k , c f .

( J o n g e t a l . 2 0 0 4 ) .

1990 1995 2000 200512.000

14.000

16.000

18.000

20.000

V e c h i c l e s p r . d a y

Year

F i g u r e 2 : T r a c i n t e n s i t y o n t h e F a r ø B r i d g e s , s o u r c e : ( D a n i s h M i n i s t r y o f T r a n s p o r t a -

t i o n , w w w . v d . d k ) .

T h e t w o F a r ø B r i d g e s a r e a l m o s t t h e s a m e t o t a l l e n g t h , a p p r o x i m a t e l y 1 6 0 0 a n d 1 7 0 0 m ,

r e s p e c t i v e l y , a n d a r e o n l y s e p a r a t e d b y t h e s m a l l i s l a n d F a r ø . T h e s t a t i c a l m a i n s y s t e m

o f t h e S j æ l l a n d - F a l s t e r B r i d g e c o n s i s t o f 1 8 s p a n s e a c h 8 0 m e t e r s l o n g a p a r t f r o m t h e t w o

e n d s p a n s w h i c h a r e 7 8 m e t e r s l o n g , c f . F i g u r e 3 . T h e s o u t h e r n b r i d g e , t h e F a r ø - F a l s t e r

B r i d g e , h a s a c a b l e s t a y e d p a r t w i t h a s p a n o f 2 9 0 m . F i g u r e 4 , s h o w s t h e s t a t i c a l m a i n

s y s t e m o f t h e F a r ø - F a l s t e r B r i d g e .

F i g u r e 3 : S t a t i c a l m a i n s y s t e m , t h e S j æ l l a n d - F a r ø B r i d g e .

T h e a p p r o a c h s p a n s o f t h e S j æ l l a n d - F a r ø B r i d g e a n d t h e F a r ø - F a l s t e r B r i d g e a l l h a v e a

s p a n o f 8 0 m . T h e a p p r o a c h s p a n s a r e t h e f o c u s o f t h e p r e s e n t s t u d y , a n d t h u s t h e c a b l e

s t a y e d p a r t i s n o t c o n s i d e r e d . T h e s t e e l b o x g i r d e r d e s i g n s f o r t h e t w o b r i d g e s a r e s i m i l a r .

T h e s t e e l b o x g i r d e r i s d e s i g n e d a s a c l o s e d b o x w i t h a t r a p e z o i d a l c r o s s - s e c t i o n a n d a n

4



F i g u r e 4 : S t a t i c a l m a i n s y s t e m , t h e F a r ø - F a l s t e r B r i d g e .

a l l - w e l d e d o r t h o t r o p i c r o a d w a y d e c k , w i t h a s t r u c t u r a l d e p t h o f 3 . 5 m . T r a c i s p l a c e d

i n t w o l a n e s i n e a c h d i r e c t i o n a n d t h e s t e e l g i r d e r h a s a t o t a l w i d t h o f 1 9 . 6 m , c f . F i g u r e

5 f o r a c r o s s - s e c t i o n a l v i e w .

F i g u r e 5 : C r o s s - s e c t i o n a l v i e w o f t h e s t e e l g i r d e r o f t h e F a r ø B r i d g e s .

T h e b r i d g e d e c k i s d e s i g n e d a s a n o r t h o t r o p i c 1 2 m m d e c k p l a t e p r o v i d e d w i t h l o n g i t u d i n a l

t r a p e z o i d a l r i b s o f 6 m m p l a t e t h i c k n e s s . T h e r i b s a r e 3 0 0 m m h i g h a n d p l a c e d a t 6 2 0 m m

c e n t r e s . T h e o r t h o t r o p i c d e c k i s s u p p o r t e d o n t r a n s v e r s e b u l k h e a d s a t e v e r y 4 m e t e r s .

F i g u r e 6 d i s p l a y s t h e s t r u c t u r a l l a y - o u t o f t h e b o x g i r d e r .

2 . 1 C e m e n t - B a s e d O v e r l a y S y s t e m

A w a y t o i n c r e a s e t h e l o c a l s t i n e s s o f t h e t o p o r t h o t r o p i c b r i d g e d e c k i n t h e s t e e l g i r d e r

m i g h t b e t o c o m b i n e a t h i n c e m e n t - b a s e d o v e r l a y w i t h t h e s t e e l p l a t e . I t i s s u g g e s t e d

i n t h i s s y s t e m t o a c h i e v e c o m p o s i t e a c t i o n t h r o u g h a d h e s i o n a n d l e a v i n g o u t m e c h a n i c a l

s h e a r c o n n e c t o r s . T h e m o t i v a t i o n f o r l e a v i n g o u t s h e a r c o n n e c t o r s i s m a i n l y b a s e d o n

t w o r e a s o n s : ( i ) u s i n g s h e a r c o n n e c t o r s c r e a t e s u n d e s i r a b l e l o c a l p e a k s t r e s s e s , a n d ( i i ) a

s y s t e m w i t h s m a l l s h e a r c o n n e c t o r s i n l a r g e n u m b e r s w i l l b e c o s t l y w i t h r e g a r d t o l a b o r .

F u r t h e r m o r e , s h e a r c o n n e c t o r s a r e o n l y a c t i v a t e d f o r l a r g e d i e r e n t i a l d e f o r m a t i o n s o

t h e t w o m a t e r i a l s .

A n u m b e r o f s m a l l s c a l e t e s t s h a v e b e e n p e r f o r m e d t o i n v e s t i g a t e t h e f r a c t u r e m e c h a n i c a l

5



F i g u r e 6 : B o x g i r d e r , s t r u c t u r a l p r i n c i p l e . B u l k h e a d s e v e r y 4 m .

b e h a v i o r o f a c e m e n t - b a s e d o v e r l a y b o n d e d t o a s t e e l p l a t e , c f . W a l t e r e t a l . ( 2 0 0 5 ) .

S m a l l s c a l e t e s t s g i v e a n i n i t i a l i d e a o f b o n d p r o p e r t i e s . T h e s t e e l p l a t e i s s a n d b l a s t e d

p r i o r t o c a s t i n g o f t h e o v e r l a y t o e n s u r e a g o o d b o n d . F u r t h e r m o r e , s a n d b l a s t i n g l e a v e s

a c l e a n m i c r o - r o u g h s u r f a c e a n d t h e r e b y t h e r i s k o f b o n d d e f e c t s i s m i n i m i z e d . B a s e d

o n e x p e r i e n c e f r o m s m a l l s c a l e t e s t s , i t i s b e l i e v e d t h a t t h e u s e o f a n o v e r l a y w i t h s e l f -

c o m p a c t i n g p r o p e r t i e s m i g h t c r e a t e a g o o d a n d s o u n d b o n d . V i b r a t i o n o f t h e o v e r l a y

m i g h t c a u s e w a t e r t o s e p a r a t e f r o m t h e m i x , w h i c h m i g h t r e s u l t i n a w e a k i n t e r f a c e .

S i m i l a r c o n c l u s i o n s h a v e b e e n d r a w n b y S c h i e s s l & Z i l c h ( 2 0 0 1 ) . S o f a r , l a b o r a t o r y t e s t s

o n b o n d b e t w e e n s t e e l p l a t e s a n d s e l f - c o m p a c t i n g c o n c r e t e h a v e s h o w n s u r p r i s i n g l y g o o d

r e s u l t s . T h e e x p e r i m e n t a l s t e e l - c o n c r e t e i n t e r f a c e s t u d i e s c o n c l u d e t h a t i t i s p o s s i b l e t o

a c h i e v e a s t e e l - c o n c r e t e b o n d , w h i c h p o s s e s s e s f r a c t u r e e n e r g y c l o s e t o t h a t o f c o n c r e t e .

I n t h e p r o p o s e d o v e r l a y s y s t e m , a t y p i c a l d e c k c o n s i s t s o f a 4 0 - 6 0 m m t h i c k c e m e n t - b a s e d

o v e r l a y b o n d e d t o t h e s t e e l p l a t e b y s e l f a d h e s i o n . A n e x a m p l e o f t h e p r o p o s e d o v e r l a y

s y s t e m , w i t h a c e m e n t - b a s e d o v e r l a y h a v i n g a t h i c k n e s s o f 5 0 m m i s s h o w n i n F i g u r e 7

a l o n g w i t h t h e t r a d i t i o n a l s t e e l d e c k .

O b v i o u s l y , a d i s a d v a n t a g e o f t h e o v e r l a y s y s t e m w o u l d b e i n c r e a s e d d e a d l o a d . A n i n -

c r e a s e i n t h e d e a d l o a d m i g h t , a t s o m e p o i n t , r e q u i r e s t r o n g e r m e m b e r s e l s e w h e r e i n t h e

s t r u c t u r a l s y s t e m . I n o r d e r t o k e e p t h e i n c r e a s e i n d e a d l o a d a t a m i n i m u m a n o p t i -

m i z a t i o n o f t h e w e a r i n g c o u r s e w o u l d b e n e c e s s a r y . I n t r a d i t i o n a l o r t h o t r o p i c s t e e l b r i d g e

d e c k s , t h e s u r f a c i n g c o n s i s t s o f a w e a r i n g c o u r s e ( 3 0 m m ) a n d a n u n d e r l y i n g i n t e r m e d i a t e

l a y e r ( 2 5 m m ) , c f . T h e D a n i s h M i n i s t r y o f T r a n s p o r t a t i o n ( 1 9 8 7 ) . T h e s t e e l p l a t e i s o f

c r u c i a l i m p o r t a n c e t o s e c u r e t h e s t r u c t u r a l s y s t e m a n d i s p r o t e c t e d b y t h e i n t e r m e d i a t e

l a y e r i n t h e o r i g i n a l s y s t e m . S i n c e a 5 0 m m t h i c k c e m e n t - b a s e d o v e r l a y m a y g i v e e n o u g h

p r o t e c t i o n t o t h e u n d e r l y i n g s t e e l p l a t e , i t c o u l d w e l l b e p o s s i b l e t o r e p l a c e t h e i n t e r m e d i -

a t e l a y e r b y a c e m e n t - b a s e d l a y e r . L e a v i n g o u t t h e i n t e r m e d i a t e l a y e r a n d u s i n g a s y s t e m

w i t h t h e c e m e n t - b a s e d o v e r l a y a n d a n a s p h a l t i c l a y e r o n t o p , m a y b e s u c i e n t t o k e e p

t h e d e a d l o a d a t a m i n i m u m . T h e t w o s u r f a c i n g s y s t e m s a r e s h o w n i n F i g u r e 8 .

6



F i g u r e 7 : ( a ) T y p i c a l s t e e l s y s t e m , w i t h a c e n t e r s p a n o f 3 0 0 m m b e t w e e n t h e s u p p o r t i n g

r i b s a n d a 1 2 m m s t e e l p l a t e . ( b ) C e m e n t - b a s e d o v e r l a y s y s t e m w i t h a 5 0 m m t h i c k

c e m e n t - b a s e d o v e r l a y .

F i g u r e 8 : ( a ) D e t a i l s h o w i n g t y p i c a l s u r f a c i n g f o r a n o r t h o t r o p i c s t e e l b r i d g e d e c k ( b )

P r o p o s e d s y s t e m u s i n g a s t e e l p l a t e a n d a l a y e r o f c e m e n t - b a s e d m a t e r i a l a c t i n g a s a

c o m p o s i t e a n d a 3 0 m m w e a r i n g c o u r s e o n t o p .

T h e w e a r i n g c o u r s e a n d i n t e r m e d i a t e l a y e r i n t h e t y p i c a l s t e e l d e c k s y s t e m h a v e a t o t a l

w e i g h t o f a p p r o x i m a t e l y 1 . 2 1 k N / m

2. I n t h e p r o p o s e d s y s t e m , t h e i n t e r m e d i a t e l a y e r

i s l e f t o u t a n d r e p l a c e d b y t h e c e m e n t - b a s e d o v e r l a y . T h e t o t a l s u m o f w e i g h t s i n t h e

p r o p o s e d s y s t e m i s 1 . 7 6 k N / m

2. T h u s , t h e o v e r l a y s y s t e m i n c r e a s e s t h e d e a d l o a d o f t h e

d e c k b y a p p r o x i m a t e l y 0 . 5 5 k N / m

2. C o m p a r e d t o t h e a v e r a g e d e a d l o a d o f t h e b o x g i r d e r

o n t h e F a r ø B r i d g e s w h i c h i s 8 . 6 k N / m

2, t h e c e m e n t - b a s e d o v e r l a y s y s t e m i n c r e a s e s t h e

d e a d l o a d b y 6 . 5 % .

3 M o d e l i n g

T h e p e r f o r m a n c e o f t h e o v e r l a y s y s t e m i s s t u d i e d b a s e d o n n o n l i n e a r f r a c t u r e m e c h a n i c s .

N u m e r i c a l c a l c u l a t i o n s a r e p e r f o r m e d u s i n g n i t e e l e m e n t s . T h e o r t h o t r o p i c b r i d g e d e c k

o f t h e 8 0 m e t e r l o n g a p p r o a c h s p a n s o f t h e F a r ø - F a l s t e r B r i d g e a n d t h e s i m i l a r s p a n s o f t h e

S j æ l l a n d - F a r ø B r i d g e a r e a n a l y z e d . T h e c a b l e s t a y e d p a r t i s n o t a n a l y z e d i n t h e p r e s e n t

s t u d y . T h e a n a l y s i s o f t h e o v e r l a y s y s t e m i s c a r r i e d o u t b y c o n s i d e r i n g a l o c a l m o d e l o f

7



3 . 9 m e t e r s t i m e s 8 m e t e r s . E x t e r i o r e e c t s f r o m t r a c a n d d e a d l o a d a r e c o n s i d e r e d i n

t h e m o d e l i n g .

3 . 1 L o a d i n g a n d B o u n d a r y C o n d i t i o n s

3 . 1 . 1 E x t e r i o r e e c t s

T h r e e t y p e s o f m e c h a n i c a l l o a d i n g a r e c o n s i d e r e d , d e a d l o a d , d i s t r i b u t e d t r a c l o a d a n d

c o n c e n t r a t e d t a n d e m l o a d i n g . W h e n c o n s i d e r i n g a l o c a l m o d e l , e x t e r i o r e e c t s f r o m

l o a d i n g a r e t a k e n i n t o a c c o u n t . E m p h a s i s i s p u t o n t h e s i t u a t i o n w h e r e t h e c e m e n t -

b a s e d o v e r l a y i s e x p o s e d t o m a x i m u m t e n s i o n , w h i c h i s c o n s i d e r e d t o b e t h e m o s t c r i t i c a l

s i t u a t i o n w i t h r e g a r d s t o t h e o v e r l a y . T h e o v e r l a y i s e x p o s e d t o m a x i m u m t e n s i o n w h e r e

t h e n e g a t i v e b e n d i n g m o m e n t w i l l a t t a i n i t s m a x i m u m . T h e e x t e r i o r f o r c e s o n t h e l o c a l

m o d e l c a n b e f o u n d u s i n g s i m p l i e d t w o - d i m e n s i o n a l b e a m m o d e l s a s s h o w n i n F i g u r e 9 .

F i g u r e 9 : ( a ) G l o b a l m o d e l t o c a l c u l a t e e x t e r i o r e e c t s o n t h e l o c a l m o d e l f r o m t r a c a n d

d e a d l o a d . A b e n d i n g m o m e n t M global a n d s h e a r f o r c e V global a r e f o u n d . A t o t a l n u m b e r

o f 6 s p a n s a r e t a k e n i n t o a c c o u n t a n d 4 t r a c l a n e s a r e u s e d . ( b ) M o d e l t o c a l c u l a t e

e x t e r i o r f o r c e s , M bulkhead a n d V bulkhead , c a u s e d b y b u l k h e a d s p l a c e d a t e v e r y 4 m e t e r s . A

t o t a l n u m b e r o f 6 s p a n s o f 4 m e t e r s a n d o n e t r a c l a n e i s t a k e n i n t o a c c o u n t .

8



T h e e x t e r i o r f o r c e s f r o m t r a c a n d d e a d l o a d o n t h e b r i d g e c a n b e d i v i d e d i n t o t w o g r o u p s

a s s h o w n i n F i g u r e 9 ( a ) a n d ( b ) . A g l o b a l b e n d i n g m o m e n t M global a n d s h e a r f o r c e V global

a r e f o u n d b y p l a c i n g u n i f o r m d i s t r i b u t e d t r a c a n d d e a d l o a d o n t h e b r i d g e , c f . F i g u r e

9 ( a ) . A s t a t i c a l m a i n s y s t e m o f 6 s p a n s , e a c h h a v i n g a s p a n o f 8 0 m e t e r s i s u s e d t o n d

t h e g l o b a l f o r c e s c a u s e d b y t r a c a n d d e a d l o a d a l o n g t h e b r i d g e . A b e n d i n g m o m e n t

M bulkhead a n d s h e a r f o r c e V bulkhead c a u s e d b y b e n d i n g l o c a l l y i n t h e o r t h o t r o p i c d e c k d u e

t o t h e b u l k h e a d s p l a c e d e v e r y 4 m e t e r s , a r e a l s o t a k e n i n t o a c c o u n t , c f . F i g u r e 9 ( b ) . T h e

e x t e r i o r f o r c e s a t t h e b u l k h e a d a r e f o u n d b y c o n s i d e r i n g a s t a t i c a l s y s t e m o f 6 s p a n s o f 4

m e t e r s e a c h . F o r c e s f o u n d f r o m t h e t w o d i m e n s i o n a l b e a m m o d e l s a r e t h e n l a t e r a p p l i e d

t o t h e l o c a l t h r e e d i m e n s i o n a l m o d e l .

3 . 1 . 2 L o a d

T r a c l o a d , d e n o t e d U n i f o r m D i s t r i b u t e d L o a d ( U D L ) , a n d t a n d e m l o a d a r e c h o s e n i n

a c c o r d a n c e t o E N V 1 9 9 1 - 3 ( 1 9 9 1 ) . I n t h e g l o b a l b r i d g e m o d e l , c f . F i g u r e 9 ( a ) , a t o t a l

n u m b e r o f f o u r l a n e s a r e c o n s i d e r e d , w h e r e e a c h l a n e h a s a w i d t h o f 3 m . T h r e e l a n e s

h a v e a n u n i f o r m d i s t r i b u t e d t r a c l o a d o f 1 . 7 k N / m

2, a n d o n e l a n e h a s a t r a c l o a d o f 6

k N / m

2. T h e a v e r a g e d e a d l o a d o f t h e s t e e l g i r d e r , i n c l u d i n g b r i d g e e q u i p m e n t , h a s b e e n

c a l c u l a t e d t o 8 . 6 k N / m

2, c f . ( T h e D a n i s h M i n i s t r y o f T r a n s p o r t a t i o n 1 9 8 7 ) . T h e t a n d e m

s y s t e m a p p l i e d c o n s i s t s o f t h r e e d o u b l e - a x l e t a n d e m s w i t h t h e f o l l o w i n g l o a d v a l u e s : 1 0 0 ,

2 0 0 a n d 3 0 0 k N .

E x t e r i o r e e c t s c a u s e d b y t h e b u l k h e a d s a t e v e r y 4 m e t e r s , c f . F i g u r e 9 ( b ) , a r e f o u n d b y

c o n s i d e r i n g 6 s p a n s o f 4 m e t e r s e a c h . B o t h t h e u n i f o r m l y d i s t r i b u t e d l o a d a n d a t a n d e m

s y s t e m a r e t a k e n i n t o a c c o u n t . I n t h i s c a s e t h e w i d t h o f 3 m e t e r s ( t h e w i d t h o f o n e s i n g l e

l a n e ) i s u s e d . F r o m e a c h o f t h e t w o c a s e s i n F i g u r e 9 ( a ) - ( b ) a b e n d i n g m o m e n t a n d a

s h e a r f o r c e i s f o u n d .

3 . 1 . 3 L o c a l M o d e l

T h e l o c a l t h r e e - d i m e n s i o n a l m o d e l i s a p a r t o f t h e o r t h o t r o p i c d e c k , a s w e l l a s p a r t o f t h e

b u l k h e a d . T h e l o c a l o r t h o t r o p i c d e c k p a r t c o n s i d e r e d h a s a h e i g h t o f 1 . 5 m , a w i d t h o f

3 . 9 m e t e r s a n d a l e n g t h o f 8 m , c f . F i g u r e 1 0 ( a ) . D e a d l o a d , u n i f o r m d i s t r i b u t e d l o a d

a n d t h e d o u b l e - a x l e t a n d e m s y s t e m a r e a p p l i e d a s l o a d s . P l a c e m e n t o f t h e d o u b l e - a x l e

t a n d e m i s s h o w n i n F i g u r e 1 0 ( a ) . A s i l l u s t r a t e d , t h e t a n d e m p r e s s u r e i s a p p l i e d o v e r a n

a r e a o f 0 . 4 x 0 . 4 m

2. D u e t o s y m m e t r y , t h e a c t u a l m o d e l i n g i s r e d u c e d t o a q u a r t e r o f t h e

s e c t i o n c o n s i d e r e d , c f . F i g u r e 1 0 ( b ) . T h e e x t e r i o r e e c t s f o u n d p r e v i o u s l y a r e a p p l i e d t o

t h e m o d e l a s b o u n d a r y c o n d i t i o n s .

T h e b o u n d a r y c o n d i t i o n s o f t h e m o d e l a r e s h o w n i n F i g u r e 1 1 . A s o b s e r v e d i n t h e g u r e ,

t h e U D L a n d o n e q u a r t e r o f t h e d o u b l e - a x l e t a n d e m a r e a p p l i e d a s l o a d s t o t h e m o d e l .

O n e e n d o f t h e o r t h o t r o p i c d e c k , o p p o s i t e t h e b u l k h e a d , i s m o d e l e d a s a r i g i d p l a n e .

E x t e r i o r e e c t s , f o u n d f r o m F i g u r e 9 ( a ) - ( b ) , a r e a p p l i e d t o t h e r i g i d p l a n e .

9



( a ) ( b )

F i g u r e 1 0 : F i g u r e s h o w i n g p a r t o f b r i d g e d e c k t h a t i s m o d e l e d u s i n g F E . ( a ) D e t a i l e d

d r a w i n g s h o w i n g m e a s u r e m e n t s a n d s h o w i n g p l a c i n g o f w h e e l l o a d s . ( b ) D u e t o s y m m e t r y

o n l y o n e q u a r t e r i s m o d e l e d ( N o t e a l l m e a s u r e m e n t s a r e i n m i l l i m e t e r s ) .

F i g u r e 1 1 : B o u n d a r y c o n d i t i o n s o f b r i d g e d e c k m o d e l . T h e e x t e r i o r b e n d i n g m o m e n t s

a n d s h e a r f o r c e s a r e a p p l i e d t o a r i g i d p l a n e .

1 0



3 . 2 F i n i t e E l e m e n t M o d e l i n g

A t h r e e d i m e n s i o n a l F i n i t e E l e m e n t ( F E ) m o d e l u s i n g s o l i d e l e m e n t s h a s b e e n e s t a b -

l i s h e d . A c o m m e r c i a l n i t e e l e m e n t s o f t w a r e p a c k a g e , ( D I A N A 2 0 0 3 ) , h a s b e e n a p p l i e d .

M o d e l i n g o f t h e s t e e l a n d c e m e n t - b a s e d o v e r l a y h a s b e e n u t i l i z e d u s i n g s t a n d a r d 2 0 - n o d e

s o l i d e l e m e n t s . T h e c o n n e c t i o n b e t w e e n t h e o v e r l a y a n d u n d e r l y i n g s t e e l d e c k h a s b e e n

m o d e l e d u s i n g a n 8 - n o d e i n t e r f a c e e l e m e n t a v a i l a b l e i n D I A N A ( 2 0 0 3 ) , t o g e t h e r w i t h a

u s e r - s u p p l i e d c o n s t i t u t i v e m o d e l a s d e s c r i b e d i n S e c t i o n 3 . 5 . T h e a p p l i e d e l e m e n t m e s h

i s s h o w n i n F i g u r e 1 2 .

F i g u r e 1 2 : A p p l i e d n i t e e l e m e n t m e s h o f t h e d e c k p a r t . T o m o d e l s t e e l a n d o v e r l a y

a s o l i d 2 0 - n o d e e l e m e n t i s u s e d . T h e c o n n e c t i o n b e t w e e n t h e o v e r l a y a n d s t e e l p l a t e i s

m o d e l e d u s i n g a 8 - n o d e i n t e r f a c e e l e m e n t .

T h e d e n s i t y o f t h e m e s h h a s b e e n v e r i e d u s i n g a c o n v e r g e n c e t e s t c o m p a r i n g t h e d e f o r -

m a t i o n m o d e a n d s t r e s s e s . T h e m e s h s h o w n i n F i g u r e 1 2 , h a s s h o w n t o b e a d e q u a t e . I n

t o t a l t h e F E m o d e l o f t h e b r i d g e d e c k c o n s i s t s o f 4 1 0 8 e l e m e n t s .

3 . 3 S t e e l

A c c o r d i n g t o T h e D a n i s h M i n i s t r y o f T r a n s p o r t a t i o n ( 1 9 8 7 ) , t h e m a i n s t r u c t u r a l e l e m e n t s

h a v e b e e n d e s i g n e d i n a h i g h y i e l d s t e e l q u a l i t y , g r a d e S t . 5 2 - 3 N , w i t h a y i e l d s t r e s s o f

3 6 0 M P a a n d a u l t i m a t e t e n s i l e s t r e n g t h o f 5 5 0 M P a .

E n g i n e e r i n g c o n s t a n t s o f t h e s t e e l u s e d i n t h e m o d e l i n g w o r k a r e a n e l a s t i c m o d u l u s o f

2 1 0 G P a a n d a p o i s s o n r a t i o o f 0 . 3 . I t i s a s s u m e d , t h a t t h e s t e e l s t r e s s e s i n t h e p r e s e n t

a n a l y s i s a r e k e p t b e l o w i t s y i e l d v a l u e . F u r t h e r m o r e , b u c k l i n g o f s t e e l p a r t s i s n o t t a k e n

i n t o c o n s i d e r a t i o n .

1 1



3 . 4 O v e r l a y M a t e r i a l

T h e w e l l - k n o w n a d v a n t a g e o f b r e r e i n f o r c e d c o n c r e t e i s i t s a b i l i t y t o c a r r y l o a d a f t e r

t h e r s t c r a c k i s f o r m e d . T h e b r e s w i l l t y p i c a l l y s t a y u n b r o k e n a f t e r t h e r s t c r a c k

i s f o r m e d a n d t h e b r e s , t h a t c r o s s a c r a c k , w i l l r e s i s t f u r t h e r o p e n i n g . D e p e n d i n g o n

t h e c r a c k b r i d g i n g e e c t , b r e r e i n f o r c e d c o m p o s i t e s c a n s h o w d i e r e n t f a i l u r e m o d e s ( L i

& L e u n g 1 9 9 2 ) . I f t h e a v e r a g e b r e b r i d g i n g e e c t i s i n c r e a s i n g d u r i n g c r a c k i n i t i a t i o n

a n d p r o p a g a t i o n , t h e n m u l t i p l e c r a c k s c a n f o r m . T h i s b e h a v i o r i s a l s o k n o w n a s s t r a i n

h a r d e n i n g . O n t h e o t h e r h a n d , i f t h e b r e s c a n n o t c a r r y m o r e l o a d a f t e r t h e f o r m a t i o n o f

t h e r s t c r a c k , f u r t h e r d e f o r m a t i o n i s g o v e r n e d b y o p e n i n g o f a s i n g l e c r a c k . T h i s b e h a v i o r

i s c a l l e d t e n s i o n s o f t e n i n g . I n t h e f o l l o w i n g , b o t h o v e r l a y s o f t e n s i o n s o f t e n i n g a n d s t r a i n

h a r d e n i n g m a t e r i a l s a r e a n a l y z e d u s i n g t h e F E m o d e l .

3 . 4 . 1 C o n s t i t u t i v e M o d e l i n g o f T e n s i o n S o f t e n i n g M a t e r i a l s

T h e t e n s i o n s o f t e n i n g b e h a v i o r c a n b e m o d e l e d , c o n c e p t u a l l y , b y m o d e l i n g a l o c a l i z e d

c r a c k . A s i m p l e m e c h a n i c a l d e s c r i p t i o n o f t h e l o c a l i z e d c r a c k i n c o n c r e t e w a s s u g g e s t e d

b y H i l l e r b o r g e t a l . ( 1 9 7 6 ) , k n o w n a s t h e F i c t i t i o u s C r a c k M o d e l ( F C M ) . A c o m p r e h e n s i v e

o v e r v i e w o f t h e u s e o f n o n l i n e a r f r a c t u r e m e c h a n i c s f o r c o n c r e t e s t r u c t u r e s u s i n g m o d e l i n g

c o n c e p t s f o r t e n s i o n s o f t e n i n g m a t e r i a l s i s g i v e n i n K a r i h a l o o ( 1 9 9 5 ) .

T h e r e q u i r e d c o n s t i t u t i v e i n p u t f o r m o d e l i n g a t e n s i o n s o f t e n i n g m a t e r i a l i s t h e s o - c a l l e d

s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p . T h i s r e l a t i o n s h i p , e x p r e s s e s t h e a m o u n t o f c r a c k o p e n -

i n g w a s a f u n c t i o n o f t h e s t r e s s a c r o s s t h e c r a c k σ . I t t u r n s o u t t h a t a b i l i n e a r s h a p e , a s

s h o w n i n F i g u r e 1 3 , c o r r e l a t e s w e l l t o e x p e r i m e n t a l d a t a .

F i g u r e 1 3 : C o n s t i t u t i v e m o d e l i n g o f a t e n s i o n s o f t e n i n g m a t e r i a l . T h e m a t e r i a l i s d e -

s c r i b e d i n t w o p h a s e s : a l i n e a r e l a s t i c r a n g e , a n d a b i l i n e a r s t r e s s - c r a c k o p e n i n g r e l a t i o n -

s h i p .

T h e s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p i n F i g u r e 1 3 i s s h o w n n o r m a l i z e d w i t h r e s p e c t t o

t h e u n i a x i a l t e n s i l e s t r e n g t h f t . T h e b i l i n e a r s h a p e i s d e s c r i b e d b y t w o l i n e s e g m e n t s ,

h a v i n g t h e n e g a t i v e s l o p e s a1 a n d a2 , r e s p e c t i v e l y . T h e i n t e r s e c t i o n b e t w e e n t h e s e c o n d

l i n e s e g m e n t a n d t h e y - a x i s i s d e n o t e d b2 , w h i c h i s a d i m e n s i o n l e s s p a r a m e t e r .

1 2



A p p l i c a t i o n o f t h e F C M i n a F E f o r m u l a t i o n i s r a t h e r s t r a i g h t f o r w a r d . M o d e l i n g o f

t h e c r a c k c a n b e c a r r i e d o u t e i t h e r b y t h e u s e o f s t a n d a r d i n t e r f a c e e l e m e n t s o r i n a

c o n t i n u o u s f o r m u l a t i o n . T h e c o n t i n u o u s f o r m u l a t i o n , a l s o k n o w n a s t h e s m e a r e d c r a c k

a p p r o a c h , w a s f o r m u l a t e d f o r c o n c r e t e b y B a º a n t & O h ( 1 9 8 3 ) . I n c o n t r a s t t o t h e m o d e l i n g

o f d i s c r e t e c r a c k s b y u s i n g i n t e r f a c e e l e m e n t s , t h e s m e a r e d c r a c k a p p r o a c h s m e a r s o u t

l o c a l i z e d d e f o r m a t i o n o v e r a c h a r a c t e r i s t i c l e n g t h . T h e F E f o r m u l a t i o n i s d e s c r i b e d i n

t e r m s o f s t r a i n s i n s t e a d o f r e l a t i v e d i s p l a c e m e n t s .

E x p e r i m e n t a l s t u d i e s i n v e s t i g a t i n g t h e b e h a v i o r o f t e n s i o n s o f t e n i n g m a t e r i a l s i n c o m p o s -

i t e a c t i o n w i t h s t e e l h a v e b e e n c a r r i e d o u t i n W a l t e r e t a l . ( 2 0 0 4 ) . T h e s t u d i e s i n c l u d e d

t w o t e n s i o n s o f t e n i n g m a t e r i a l s , t h e w h i c h c o n s t i t u t i v e p a r a m e t e r s o f w h i c h w e r e o b t a i n e d

u s i n g a u n i a x i a l t e s t s e t - u p s i m i l a r t o R I L E M ( 2 0 0 1 ) . T h e u n i a x i a l t e s t s w e r e u s e d t o

o b t a i n t h e s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p i n t h e b i l i n e a r f o r m s h o w n i n F i g u r e 1 3 . T h e

t e n s i o n s o f t e n i n g m a t e r i a l s t e s t e d w e r e F i b r e R e i n f o r c e d C o n c r e t e ( F R C ) a n d F i b r e R e -

i n f o r c e d D e n s i t ( F R D ) . T h e a v e r a g e c o n s t i t u t i v e p a r a m e t e r s o b t a i n e d i n t h e u n i a x i a l

t e s t s f o r t h e t w o t e n s i o n s o f t e n i n g m a t e r i a l s a r e l i s t e d i n T a b l e 1 . A d d i t i o n a l l y , t y p i c a l

s o f t e n i n g p a r a m e t e r s f o r p l a i n n o r m a l s t r e n g t h c o n c r e t e a r e u s e d f o r c o m p a r i s o n .

M a t e r i a l f t a1 a2 b2 Gf

[ M P a ] [ m m

−1] [ m m

−1] [ - ] [ N / m m ]

F R C 2 . 3 2 7 0 . 5 0 0 . 3 5 0 . 3 0

F R D 4 . 8 6 4 0 . 5 7 0 . 4 6 0 . 9 0

C o n c r e t e 2 . 0 5 0 1 . 0 0 . 1 0 . 0 3

T a b l e 1 : C o n s t i t u t i v e p a r a m e t e r s o f t h r e e t e n s i o n s o f t e n i n g m a t e r i a l s a n a l y z e d a s o v e r l a y

m a t e r i a l . P a r a m e t e r s a r e b a s e d o n v a l u e s f r o m W a l t e r e t a l . ( 2 0 0 4 ) .

I n a d d i t i o n t o t h e b i l i n e a r s t r e s s - c r a c k o p e n i n g p a r a m e t e r s , t h e f r a c t u r e e n e r g y Gf i s

s h o w n i n T a b l e 1 . A s o b s e r v e d , t h e b r e r e i n f o r c e d m a t e r i a l s , F R C a n d F R D , a r e c h a r a c -

t e r i z e d b y a m u c h h i g h e r f r a c t u r e e n e r g y t h a n p l a i n c o n c r e t e d u e t o t h e i r b r e c o n t e n t s .

T h e F R C m a t e r i a l i n t h e e x p e r i m e n t s c o n t a i n e d 1 % , 3 0 m m l o n g , h o o k e d e n d s t e e l b e r s

w i t h a d i a m e t e r o f 0 . 5 m m , a n d t h e F R D m a t e r i a l c o n t a i n e d 2 % s t r a i g h t s t e e l b e r s w i t h

a l e n g t h o f 6 m m a n d a d i a m e t e r o f 0 . 1 6 m m .

3 . 4 . 2 C o n s t i t u t i v e M o d e l i n g o f S t r a i n H a r d e n i n g M a t e r i a l s

T h e c o n s t i t u t i v e b e h a v i o r o f a s t r a i n - h a r d e n i n g m a t e r i a l c a n b e d i v i d e d i n t o t h r e e p h a s e s :

( i ) a l i n e a r e l a s t i c p h a s e , ( i i ) a m u l t i p l e c r a c k i n g p h a s e , a n d ( i i i ) a l o c a l i z a t i o n p h a s e . T h e

b e h a v i o r i s i l l u s t r a t e d i n F i g u r e 1 4 .

T h e s e c o n d p h a s e , t h e m u l t i p l e c r a c k i n g p h a s e , c a n b e d e s c r i b e d i n a s t r e s s v s . s t r a i n

r e l a t i o n s h i p , w i t h a n u l t i m a t e t e n s i l e s t r e n g t h f u a t t h e s t r a i n u . T h e t h i r d a n d n a l

p a r t i s d e s c r i b e d i n a s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p , w i t h a n u l t i m a t e c r a c k o p e n i n g

d e n o t e d wu .

A s t r a i n h a r d e n i n g m a t e r i a l d e n o t e d E n g i n e e r e d C e m e n t i t i o u s C o m p o s i t e ( E C C ) h a s a l s o

b e e n a n a l y z e d e x p e r i m e n t a l l y i n W a l t e r e t a l . ( 2 0 0 4 ) . T h e f r a c t u r e b e h a v i o r o f E C C w e r e

1 3



F i g u r e 1 4 : C o n s t i t u t i v e m o d e l i n g o f a s t r a i n h a r d e n i n g m a t e r i a l . T h e m a t e r i a l i s d e s c r i b e d

i n t h r e e p h a s e s : a l i n e a r e l a s t i c p h a s e , a s t r a i n h a r d e n i n g p h a s e , a n d a l o c a l i z a t i o n p h a s e .

c h a r a c t e r i z e d u s i n g a u n i a x i a l t e s t s e t - u p a c c o r d i n g t o L i e t a l . ( 2 0 0 2 ) . T h e n o t a t i o n

f o r t h e c o n s t i t u t i v e p a r a m e t e r s i s i n a c c o r d a n c e w i t h F i g u r e 1 4 . T h e a v e r a g e p a r a m e t e r s

o b t a i n e d i n t h e t e s t s a r e s h o w n i n T a b l e 2 .

M a t e r i a l f t f u u wu

[ M P a ] [ M P a ] [ % ] [ m m ]

E C C 4 . 2 5 . 3 1 . 9 4 *

T a b l e 2 : C o n s t i t u t i v e p a r a m e t e r s o f a s t r a i n h a r d e n i n g m a t e r i a l E C C ( * N o t e t h i s v a l u e

i s a n a s s u m p t i o n ) . P a r a m e t e r s a r e b a s e d o n v a l u e s f r o m W a l t e r e t a l . ( 2 0 0 4 ) .

A s a l l r e a d y m e n t i o n e d t h e E C C m a t e r i a l e x h i b i t s s t r a i n h a r d e n i n g p r o p e r t i e s w h e n s u b -

j e c t e d t o t e n s i o n . T h e m a t e r i a l h a s b e e n e n g i n e e r e d t h r o u g h a m i c r o m e c h a n i c a l d e s i g n

a p p r o a c h , s e e e . g . L i ( 2 0 0 2 ) . T h e m a t e r i a l t e s t e d c o n t a i n e d 2 % b y v o l u m e p o l y - v i n y l -

a l c o h o l ( P V A ) b e r s , w i t h a b r e l e n g t h o f 8 m m .

3 . 5 O v e r l a y - S t e e l I n t e r f a c e

T h e o v e r l a y - s t e e l i n t e r f a c e b e h a v i o r i s b a s e d o n l o c a l i z e d c r a c k i n g a n d m o d e l e d u s i n g

s t a n d a r d t h r e e - d i m e n s i o n a l i n t e r f a c e e l e m e n t s . A n a p p l i e d c o n s t i t u t i v e m i x e d m o d e m o d e l

h a s b e e n i m p l e m e n t e d i n D I A N A ( 2 0 0 3 ) , u s i n g u s e r - s u p p l i e d s u b r o u t i n e s . T h e m o d e l i s

b a s e d , c o n c e p t u a l l y , o n t h e c t i t i o u s c r a c k m o d e l b y H i l l e r b o r g e t a l . ( 1 9 7 6 ) . H o w e v e r ,

t h e m o d e l b y H i l l e r b o r g o n l y i n c l u d e s n o r m a l c r a c k i n g ( M o d e I c r a c k i n g ) a n d d o e s n o t

t a k e i n t o a c c o u n t t h e i n u e n c e o f s h e a r ( M o d e I I c r a c k i n g ) . T h e c u r r e n t m o d e l i s a n

e x p a n s i o n o f t h e F C M i n t h r e e d i m e n s i o n s , b a s e d o n a m o d e l b y W e r n e r s s o n ( 1 9 9 4 ) a n d

l a t e r i n t h e t h r e e - d i m e n s i o n a l c a s e b y S e r r a n o ( 2 0 0 0 ) .

T h e i n t e r f a c e i s a s s u m e d t o h a v e a t h i c k n e s s o f z e r o , s o t h e s t r e s s e s t h a t s h o u l d b e t a k e n

i n t o a c c o u n t a r e t h e n o r m a l s t r e s s a n d t w o i n - p l a n e s h e a r s t r e s s e s . P r i o r t o c r a c k i n g , l i n e a r

e l a s t i c b e h a v i o r i s a s s u m e d , w h e r e a s a f t e r c r a c k i n g , t h e m o d e l d e s c r i b e s t h e c r a c k i n g

b e h a v i o r a s d e c r e a s i n g s t r e s s f o r i n c r e a s i n g n o r m a l a n d t a n g e n t i a l c r a c k o p e n i n g . T h e

1 4



t h r e e s t r e s s c o m p o n e n t s a r e , i n t h e p u r e M o d e I o r I I c a s e , d e s c r i b e d a s s t r e s s v s . c r a c k

o p e n i n g i n e i t h e r n o r m a l o r t a n g e n t i a l d i r e c t i o n . T h e s e t h r e e c u r v e s a r e c o u p l e d i n t h e

t h r e e d i m e n s i o n a l s p a c e , s o t h a t t h e c o r r e s p o n d i n g s t i n e s s a n d s t r e s s v a l u e s c a n b e

o b t a i n e d f o r a n y s t a t e o f m i x e d m o d e c r a c k i n g .

T h e n o r m a l s t r e s s c o m p o n e n t ( σ ) a n d t h e t w o s h e a r s t r e s s c o m p o n e n t s ( τ x a n d τ y ) a r e

s h o w n i n F i g u r e 1 5 a l o n g w i t h t h e i r c r a c k o p e n i n g d i s p l a c e m e n t c o m p o n e n t s ( δ n , δ tx , a n d

δ ty ) .

F i g u r e 1 5 : T h r e e d i m e n s i o n a l i n t e r f a c e c o n g u r a t i o n . T h e i n t e r f a c e i s d e s c r i b e d b y o n e

n o r m a l s t r e s s a n d t w o s h e a r s t r e s s c o m p o n e n t s .

T h e r e l a t i o n b e t w e e n t h e g r a d i e n t s o f t h e s i x c o m p o n e n t s i s d e s c r i b e d t h r o u g h t h e f o l l o w -

i n g f o r m u l a : στ x

τ y

=

D11 D12 D13

D21 D22 D23

D31 D32 D33

δ n˙δ tx

˙δ ty

( 1 )

w h e r e t h e Dij c o m p o n e n t s d e s c r i b e t h e r e l a t i o n b e t w e e n t h e g r a d i e n t s o f s t r e s s a n d c r a c k

d e f o r m a t i o n . I f t h e c u r r e n t s t a t e i s e l a s t i c , t h e r e s p o n s e i s l i n e a r a n d u n c o u p l e d , h e n c e ,

t h e o - d i a g o n a l t e r m s o f Dij a r e e q u a l t o z e r o . T h e d i a g o n a l e l e m e n t s a r e , i n t h e e l a s t i c

s t a g e , a s s i g n e d l a r g e v a l u e s t o m o d e l i n i t i a l c o n t i n u o u s g e o m e t r y .

T h e i n t e r f a c e b e h a v i o r a f t e r p e a k l o a d i s o b t a i n e d t h r o u g h t h e c o n s t r u c t i o n o f a t h r e e -

d i m e n s i o n a l s t r e s s - c r a c k d e f o r m a t i o n r e l a t i o n . T h e c o n s t r u c t i o n i s b a s e d o n a n a s s u m p -

t i o n r e g a r d i n g t h e c o u p l i n g o f s t r e s s a n d c r a c k d e f o r m a t i o n s i n t h r e e d i m e n s i o n s . T h e

c o e c i e n t s i n E q u a t i o n 1 c a n t h e n b e d e r i v e d b y d i e r e n t i a t i o n . F u r t h e r m o r e , a g e n -

e r a l a s s u m p t i o n i n t h e p r e s e n t s t u d y i s i s o t r o p y i n t h e s h e a r p l a n e , s o t h a t D22 = D33 ,

D32 = D23 , D21 = D31 , a n d D12 = D13 .

A m i x e d m o d e s t a t e c a n b e d e n e d u s i n g t w o m i x e d m o d e a n g l e s φxy a n d φtn :

φxy = arctanδ tx

δ ty( 2 )

φtn = arctanδ tδ n

( 3 )

w h e r e

1 5



δ t =

δ 2tx + δ 2ty ( 4 )

T h e c o u p l i n g i s b a s e d o n t h e t w o b i l i n e a r c u r v e s i n p u r e M o d e I a n d I I , c f . F i g u r e 1 6 .

F i g u r e 1 6 : B i l i n e a r s t r e s s - c r a c k c u r v e s i n p u r e M o d e I a n d I I . U n t i l p e a k s t r e s s t h e c u r v e s

a r e u n c o u p l e d , a n d t h e s l o p e s a r e a s s i g n e d l a r g e v a l u e s t o m o d e l c o n t i n u o u s g e o m e t r y .

T h e k i n k p o i n t s o f t h e p u r e M o d e I a n d I I c u r v e s a r e c o u p l e d w i t h t h e k i n k p o i n t s f o r t h e

c u r r e n t m i x e d m o d e s t a t e , b y a s s u m i n g t h e f o l l o w i n g c r i t e r i a :

δ knδ max k

n

p

+ δ ktx

δ maxktx

n

+ δ kty

δ max kty

m

= 1.0( 5 )

w h e r e δ max kn , δ max k

tx a n d δ max kty d e n o t e a k i n k p o i n t ( d e n e d b y i n d e x k ) o n e i t h e r t h e

p u r e M o d e I o r I I c u r v e . T h e t e r m s δ kn , δ ktx a n d δ kty d e n o t e a k i n k p o i n t o n t h e c u r r e n t c r a c k

d e f o r m a t i o n c u r v e f o r a g i v e n m i x e d m o d e s t a t e . T h e p o w e r s p, n a n d m a r e m a t e r i a l

c o n s t a n t s , a n d a r e i n t h e p r e s e n t s t u d y a s s u m e d t o b e :

p = n = m = 2 ( 6 )

F i n a l l y , t h e k i n k p o i n t r e l a t i o n s h i p b e t w e e n c r a c k d e f o r m a t i o n s a n d s t r e s s e s , i n t h e p u r e

M o d e I o r I I c a s e a n d t h e m i x e d m o d e c a s e , a r e a s s u m e d t o b e :

σk = δ knσmax k

δ max kn

( 7 )

τ ktx = δ ktx

τ max ktx

δ max ktx

( 8 )

τ kty = δ kty

τ max kty

δ max kty

( 9 )

1 6



T h e c o u p l i n g c a n t h e n b e f o u n d u s i n g e q u a t i o n s ( 5 ) a n d ( 7 ) - ( 9 ) . E v e r y k i n k p o i n t o n t h e

p u r e M o d e I a n d I I c u r v e s a r e l i n k e d i n t h e t h r e e - d i m e n s i o n a l s p a c e , s o t h a t t h e c u r r e n t

s t r e s s s t a t e c a n b e f o u n d v i a l i n e a r i n t e r p o l a t i o n . T h e r e q u i r e d i n p u t i s t h e t w o b i l i n e a r

c u r v e s a l o n g w i t h t h e p o w e r s p, n a n d m. T h e r e q u i r e d b i l i n e a r c u r v e s a r e t h e p u r e M o d e

I c u r v e ( σ - δ n ) , a n d t h e b i l i n e a r p u r e M o d e I I c u r v e , w h i c h a r e t h e s a m e i n t h e x a n d yd i r e c t i o n , s o t h a t t h e p u r e τ x - δ tx a n d τ y - δ ty c u r v e s a r e g i v e n t h e s a m e i n p u t d u e t o i s o t r o p y

i n t h e p l a n e . A f t e r t h e t h r e e d i m e n s i o n a l s t r e s s v e r s u s c r a c k d e f o r m a t i o n r e l a t i o n s h i p s

h a v e b e e n c o n s t r u c t e d a s d e s c r i b e d a b o v e , t h e Dij t e r m s i n E q u a t i o n ( 1 ) c a n b e f o u n d b y

d i e r e n t i a t i o n .

T h e i n p u t o f t h e p u r e M o d e I a n d I I c u r v e s a r e i n t h e c u r r e n t s t u d y b a s e d o n e x p e r i m e n t a l

w o r k c a r r i e d o u t o n c o n c r e t e - s t e e l i n t e r f a c e s . I t h a s b e e n s h o w n i n a p r e v i o u s s t u d y ,

( W a l t e r e t a l . 2 0 0 5 ) , t h a t i n t e r f a c i a l f r a c t u r e p r o p e r t i e s a r e c l o s e t o t h o s e m e a s u r e d f o r t h e

c o n c r e t e i t s e l f . T h e s o - c a l l e d w a l l e e c t h a s t o b e t a k e n i n t o a c c o u n t , s i n c e b i g a g g r e g a t e s

a r e n o t p r e s e n t c l o s e t o t h e s t e e l p l a t e w h e r e t h e i n t e r f a c i a l f r a c t u r e o c c u r s . E x p e r i m e n t a l

s t u d i e s h a v e b e e n c a r r i e d o u t t o c h a r a c t e r i z e a c o n c r e t e - s t e e l i n t e r f a c e e x p o s e d t o m i x e d

m o d e l o a d i n g , s e e e . g . W a l t e r & O l e s e n ( 2 0 0 5 ) . T h e n o t a t i o n f o r t h e p u r e M o d e I a n d I I

c u r v e s i s g i v e n i n F i g u r e 1 6 .

T h e c o n s t i t u t i v e p r o p e r t i e s a p p l i e d a r e t h e s a m e f o r a l l t h e m a t e r i a l s a n a l y z e d i n t h e

p r e s e n t s t u d y , a n d a r e l i s t e d i n T a b l e 3 . T h i s i s a n a s s u m p t i o n a n d l i m i t a t i o n o f t h e

p r e s e n t s t u d y , s i n c e i t w o u l d b e m o r e c o r r e c t t o m e a s u r e t h e f r a c t u r e p r o p e r t i e s o f e a c h

i n d i v i d u a l m a t e r i a l , F R C , F R D a n d E C C . T h e c o n s t i t u t i v e p a r a m e t e r s a p p l i e d i n t h e

s t u d y a r e b a s e d o n e x p e r i m e n t a l w o r k o n a s t e e l - c o n c r e t e i n t e r f a c e , c l o s e t o w h a t c o u l d

b e m e a s u r e d f o r c e m e n t - b a s e d m a t e r i a l s s i n c e b r e s a r e n o t l i k e l y t o i n u e n c e t h e o v e r l a y -

s t e e l i n t e r f a c e c r a c k i n g . N u m e r i c a l s t u d i e s , a n a l y z i n g t h e r o l e o f f r a c t u r e t o u g h n e s s o f t h e

i n t e r f a c e a n d h o w d e b o n d i n g i n u e n c e s t h e g l o b a l r e s p o n s e , h a s a l r e a d y b e e n c a r r i e d

o u t , s e e e . g . W a l t e r e t a l . ( 2 0 0 3 ) . T h e c o n c l u s i o n i s t h a t t h e f r a c t u r e e n e r g y o f t h e s t e e l -

c o n c r e t e i n t e r f a c e p l a y s a m i n o r r o l e a n d d e b o n d i n g i s i n i t i a t e d f o r a c r a c k m o u t h o p e n i n g

o f a p p r o x i m a t e l y 0 . 0 3 t o 0 . 1 m m .

M o d e I σmax1[ M P a ] σmax2

[ M P a ] δ max2n [ m m ] δ max3

n [ m m ] Gf [ n / m m ]

n - d i r e c t i o n 3 . 0 0 . 4 0 . 0 2 0 . 5 0 . 1 2

M o d e I I τ max1x [ M P a ] τ max2

x [ M P a ] δ max2tx [ m m ] δ max3

tx [ m m ] Gf [ n / m m ]

t x - d i r e c t i o n 3 . 5 0 . 5 0 . 0 2 0 . 7 7 0 . 2 3

M o d e I I τ max1y [ M P a ] τ max2

y [ M P a ] δ max2ty [ m m ] δ max3

ty [ m m ] Gf [ n / m m ]

t y - d i r e c t i o n 3 . 5 0 . 5 0 . 0 2 0 . 7 7 0 . 2 3

T a b l e 3 : P u r e M o d e I a n d I I p a r a m e t e r s ( F i g u r e 1 6 ) b a s e d o n v a l u e s f r o m W a l t e r & O l e s e n

( 2 0 0 5 ) , u s e d a s c o n s t i t u t i v e i n t e r f a c e p a r a m e t e r s . N o t e t h a t δ max1n a n d δ max1

ty = δ max1tx a r e

a s s i g n e d a s m a l l v a l u e f o r n u m e r i c a l r e a s o n s .

1 7



4 R e s u l t s a n d D i s c u s s i o n

T h e r s t s t u d y i s b a s e d o n l i n e a r e l a s t i c i t y a n d n o n l i n e a r e e c t s s u c h a s o v e r l a y c r a c k i n g

a n d d e b o n d i n g a r e n o t c o n s i d e r e d i n t h i s p a r t . T h e l i n e a r e l a s t i c s t u d y i s c a r r i e d o u t t o

g i v e a s i m p l e e s t i m a t e o n h o w m u c h t h e o v e r l a y c o n t r i b u t e s t o s t r e s s r e d u c t i o n i n t h e

f a t i g u e s e n s i t i v e s t e e l p a r t s , w h e n e x p o s e d t o t r a c l o a d . L a t e r i n t h e p r e s e n t s e c t i o n a n

e s t i m a t i o n o f c r a c k w i d t h a n d d e b o n d i n g b a s e d o n n o n l i n e a r f r a c t u r e m e c h a n i c s i s g i v e n .

T h e e s t i m a t i o n i s g i v e n i n t e r m s o f c r a c k w i d t h a n d d e b o n d i n g . T h e n o n l i n e a r s t u d y t a k e s

i n t o a c c o u n t m e c h a n i c a l l o a d i n g a l o n g w i t h t e m p e r a t u r e g r a d i e n t s b e t w e e n t h e o v e r l a y

a n d s t e e l s t r u c t u r e . F i n a l l y , e a r l y - a g e s h r i n k a g e i s m o d e l e d a n d d i s c u s s e d . F a t i g u e a n d

s e r v i c e l i f e i s n o t p a r t o f t h e p r e s e n t s t u d y , h o w e v e r , t h e a s p e c t s o f c y c l i c l o a d i n g a r e

d i s c u s s e d l a t e r w i t h r e g a r d s t o f u r t h e r d e v e l o p m e n t s .

4 . 1 L i n e a r E l a s t i c S t u d i e s

O n e o f t h e p r o b l e m s c a u s i n g f a t i g u e i n t r a d i t i o n a l o r t h o t r o p i c s t e e l b r i d g e d e c k s , i s l a c k o f

t r a n s v e r s e b e n d i n g s t i n e s s . L a c k o f t r a n s v e r s e b e n d i n g s t i n e s s a n d t h e c o n t r i b u t i o n t o

b e n d i n g s t i n e s s f r o m t h e c e m e n t - b a s e d o v e r l a y c a n w e l l b e i l l u s t r a t e d i n a l i n e a r e l a s t i c

s t u d y . C o n s i d e r a d o u b l e - a x l e t a n d e m w i t h a n a x l e l o a d o f 2 6 0 k N p l a c e d a s s h o w n

p r e v i o u s l y i n F i g u r e 1 0 , t o g e t h e r w i t h g l o b a l t r a c a s o u t l i n e d i n F i g u r e 9 ( a ) - ( b ) . T h e

g e o m e t r y o f t h e F a r ø B r i d g e s i s u s e d i n t h e a n a l y s i s a n d t h e p a r t m o d e l e d i s s h o w n i n

F i g u r e 1 0 . P l a c i n g a d o u b l e - a x l e t a n d e m o n t h e s t r u c t u r e c r e a t e s a t r a n s v e r s e b e n d i n g

m o m e n t , a n d i n i t i a t e s l a r g e s t r e s s e s i n t h e w e l d e d j o i n t b e t w e e n t h e s t e e l p l a t e a n d r i b .

T h i s i s i l l u s t r a t e d i n F i g u r e 1 7 , s h o w i n g a d e f o r m a t i o n p l o t o f t h e n i t e e l e m e n t m o d e l .

(a) (b)

F i g u r e 1 7 : D e f o r m a t i o n p l o t s f r o m l i n e a r e l a s t i c F E c a l c u l a t i o n s . ( a ) W h e e l l o a d c a u s i n g

t r a n s v e r s e b e n d i n g . ( b ) T h r e e d i m e n s i o n a l v i e w o f d e f o r m a t i o n c a u s e d b y w h e e l l o a d .

T h e p o s i t i o n o f t h e w h e e l l o a d i s s h o w n o n t h e d e f o r m a t i o n p l o t , c f . F i g u r e 1 7 . A s

o b s e r v e d , t h e w h e e l l o a d c a u s e s c o n s i d e r a b l e t r a n s v e r s e b e n d i n g . P a r a m e t r i c s t u d i e s a s -

s u m i n g l i n e a r e l a s t i c i t y , u s i n g n i t e e l e m e n t c a l c u l a t i o n s , h a v e b e e n c a r r i e d o u t . T h r o u g h

1 8



a p a r a m e t r i c s t u d y , t h e i n u e n c e s o f : ( i ) d i e r e n t s t e e l p l a t e t h i c k n e s s e s , a n d ( i i ) d i e r e n t

o v e r l a y t h i c k n e s s e s o n a d e c k w i t h a s t e e l p l a t e t h i c k n e s s o f 1 2 m m , a r e i n v e s t i g a t e d . I n

t h i s s t u d y , f u l l c o m p o s i t e a c t i o n i s a s s u m e d , a n d n o d e b o n d i n g o r c r a c k i n g o f t h e o v e r l a y

i s m o d e l e d . T h e m a x i m u m v o n M i s e s s t r e s s i n t h e i n t e r s e c t i o n b e t w e e n t h e s t e e l p l a t e

a n d t h e r i b i s i n f o c u s . T h e r e s u l t s f r o m t h e l i n e a r e l a s t i c s t u d y a r e p r e s e n t e d i n F i g u r e

1 8 .

40 60 80 1000

100

200

300

Overlay thickness [mm]

0 10 20

0

100

200

300

Steel plate thickness [mm]

v o n M i s e s s t r e s s [ M P a ]

Steel plate thicknessvariation (no overlay)

Overlay thicknessvariation, steel plate

thickness 12 mm

F i g u r e 1 8 : R e s u l t o f l i n e a r e l a s t i c s t u d i e s f o r t r a c a n d a n a x l e l o a d o f 2 6 0 k N . T h e l e f t

m o s t l i n e r e p r e s e n t s t h e v o n M i s e s s t r e s s o f d i e r e n t s t e e l p l a t e t h i c k n e s s e s w i t h o u t a n

o v e r l a y . T h e o t h e r l i n e r e p r e s e n t s r e s u l t s f r o m a s t e e l p l a t e o f 1 2 m m a n d d i e r e n t o v e r l a y

t h i c k n e s s e s .

T h e g r a p h s h o w s t h e m a x i m u m v o n M i s e s s t r e s s i n t h e i n t e r s e c t i o n b e t w e e n t h e s t e e l p l a t e

a n d r i b f o r d i e r e n t o r t h o t r o p i c d e c k g e o m e t r i e s . T h e r s t p a r a m e t r i c s t u d y i n v e s t i g a t e s

a p u r e o r t h o t r o p i c s t e e l d e c k w i t h o u t a n y o v e r l a y . I n t h i s s t u d y , t h e g e o m e t r y o f t h e F a r ø

B r i d g e s i s a p p l i e d , a n d t h e o n l y m o d i c a t i o n t o t h e o r i g i n a l d e s i g n i s t h e t h i c k n e s s o f t h e

t o p s t e e l p l a t e w h i c h i s t h e v a r i a b l e o f t h e r s t p a r a m e t r i c s t u d y . A s s e e n i n F i g u r e 1 8 , t h e

v o n M i s e s s t r e s s e s h a v e b e e n c a l c u l a t e d f o r s t e e l p l a t e t h i c k n e s s e s o f 8 , 1 2 , 1 4 , 1 6 , 1 8 , a n d

2 0 m m . I n t h e s e c o n d p a r a m e t r i c s t u d y , v o n M i s e s s t r e s s e s h a v e b e e n c a l c u l a t e d i n t h e

s i m i l a r i n t e r s e c t i o n , w h e r e t h e o n l y m o d i c a t i o n t o t h e o r i g i n a l d e s i g n i s t h e a p p l i c a t i o n

o f a n o v e r l a y .

T h e r e s u l t s f r o m F i g u r e 1 8 s h o w a r e d u c t i o n i n t h e v o n M i s e s s t r e s s f o r i n c r e a s i n g s t e e l

p l a t e t h i c k n e s s . T h e v o n M i s e s s t r e s s i s a r o u n d 3 0 0 M P a w h e n h a v i n g a s t e e l p l a t e

t h i c k n e s s o f 8 m m a n d a r o u n d 2 5 M P a w h e n h a v i n g a s t e e l p l a t e t h i c k n e s s o f 2 0 m m .

H o w e v e r , c o m p a r i n g t h e s t r e s s l e v e l s f o r t h a t o f a p u r e s t e e l d e c k t o t h e o v e r l a y c a s e , i t

i s o b s e r v e d t h a t e v e n a p p l y i n g a t h i n o v e r l a y r e s u l t s i n a s i g n i c a n t r e d u c t i o n o f t h e v o n

1 9



F i g u r e 1 9 : C r a c k p a t t e r n s o f t h e o v e r l a y ( c o n c r e t e ) f o r t r a c a n d a n a x l e l o a d o f : ( a ) 5 0

k N , ( b ) 1 8 0 k N , ( c ) 2 6 0 k N , a n d ( d ) 2 9 0 k N .

0 0.02 0.04 0.06 0.08 0.10

100

200

300

400

500

600

700

•

•

•⊕

⊕

Crack width [mm] or Strain (ECC) in percent

A x l e

l o a d

[ k N ]



ConcreteFRCFRD

ECC

F i g u r e 2 0 : N u m e r i c a l r e s u l t s o f t h r e e o v e r l a y m a t e r i a l s f o r x e d t r a c l o a d a n d v a r i a b l e

a x l e l o a d i n g . T h e r e s u l t s a r e p r e s e n t e d i n a l o a d v s . c r a c k w i d t h d i a g r a m f o r c o n c r e t e ,

F R C a n d F R D . I n t h e c a s e o f E C C m a t e r i a l t h e r e s u l t s a r e p r e s e n t e d a s l o a d v s . s t r a i n

i n p e r c e n t .

2 1



a l w a y s l o c a t e d i n c r a c k a r e a n o . 1 a s d e n e d i n F i g u r e 1 9 ( a ) . I n a d d i t i o n t o t h e v a l u e

o f t h e c r a c k w i d t h , i n i t i a t i o n o f c r a c k i n g i n c r a c k a r e a n o . 2 ( a s d e n e d i n F i g u r e 1 9 ( d ) )

i s m a r k e d . I n t h e l o a d r a n g e c o n s i d e r e d , c r a c k i n g o n l y i n i t i a t e s i n c o n c r e t e a n d F R C i n

c r a c k a r e a n o . 2 . I n t h e c a s e o f t e n s i o n s o f t e n i n g m a t e r i a l s , i n c r e a s i n g c r a c k w i d t h w i l l

a t s o m e p o i n t l e a d t o d e b o n d i n g . D e b o n d i n g i s i n i t i a t e d f o r a c e r t a i n c r a c k w i d t h w h i c h

i s a l s o m a r k e d i n F i g u r e 2 0 . I t i s n o t e d t h a t d e b o n d i n g i s i n i t i a t e d a t a p p r o x i m a t e l y t h e

s a m e c r a c k w i d t h f o r t h e m a t e r i a l s : C o n c r e t e , F R C a n d F R D .

4 . 3 T e m p e r a t u r e G r a d i e n t s

A t e m p e r a t u r e g r a d i e n t b e t w e e n t h e o v e r l a y a n d u n d e r l y i n g s t e e l p l a t e m a y g i v e r i s e

t o t e n s i l e s t r e s s e s i n t h e o v e r l a y . S i n c e t h e o v e r l a y i s r e s t r a i n e d f r o m m o v i n g d u e t o

i t s b o n d t o t h e s t e e l p l a t e , t e m p e r a t u r e g r a d i e n t s m i g h t p l a y a s i g n i c a n t r o l e o n t h e

c r a c k i n g b e h a v i o r o f t h e o v e r l a y s y s t e m . T e m p e r a t u r e g r a d i e n t s h a v e t o b e t a k e n i n t o

a c c o u n t i n o r d e r t o e s t i m a t e t h e m a x i m u m c r a c k w i d t h o f t h e o v e r l a y . O f s p e c i a l c o n c e r n

i s c o o l i n g o f t h e b r i d g e d e c k ( b o t t o m w a r m e r t h a n t o p ) . S i n c e t h e o v e r l a y i s r e s t r a i n e d

f r o m m o v i n g , c o o l i n g o f t h e b r i d g e d e c k r e s u l t s i n a s i t u a t i o n w i t h t e n s i l e s t r e s s e s i n

t h e o v e r l a y . F o r t h i s s t u d y , t e m p e r a t u r e g r a d i e n t s a n d t h e r m a l e x p a n s i o n c o e c i e n t s

h a v e b e e n c h o s e n a c c o r d i n g t o E N V 1 9 9 1 - 2 - 5 : 1 9 9 7 ( 1 9 9 7 ) . T h u s , a t h e r m a l e x p a n s i o n

c o e c i e n t o f 1 0 ·1 0

−6/0 C h a s b e e n a p p l i e d t o t h e o v e r l a y . F o r a c o m p o s i t e s t r u c t u r e w i t h

a s l a b h e i g h t o f 0 . 2 m , t h e t e m p e r a t u r e d i e r e n c e i n a c o o l i n g s i t u a t i o n d e p e n d s o n t h e

s u r f a c e t h i c k n e s s . I t r a n g e s f r o m 5 . 9

0C ( f o r u n s u r f a c e d ) t o 1 . 6

0C ( 2 0 0 m m s u r f a c e ) . I n

t h e p r e s e n t a n a l y s i s a p a r a m e t r i c s t u d y i s c a r r i e d o u t l o w e r i n g t h e t e m p e r a t u r e o f t h e

o v e r l a y t o s t u d y c r a c k i n g b e h a v i o r . T h e a p p l i e d t e m p e r a t u r e d i e r e n c e a n d d i s t r i b u t i o n

a p p l i e d a r e s h o w n i n F i g u r e 2 1 .

F i g u r e 2 1 : T e m p e r a t u r e d i s t r i b u t i o n i n b r i d g e d e c k .

T h e t e m p e r a t u r e i n t h e s t e e l p a r t ,

T s, i s a l w a y s s e t t o z e r o d e g r e e s . T h e t e m p e r a t u r e

d i e r e n c e i s e x p r e s s e d b y :

∆T = T s − T o ( 1 0 )

w h e r e

T s ≡ 00C ( 1 1 )

2 2



T h e c r a c k i n g b e h a v i o r o f t h e o v e r l a y i s s t u d i e d f o r a t e m p e r a t u r e d i e r e n c e ∆T o f : 0

0C , 2 . 5

0C , 5

0C , 7 . 5

0C , a n d 1 0

0C . T h e a n a l y s i s i s c a r r i e d o u t b y , r s t , s e t t i n g t h e

t e m p e r a t u r e i n t h e s t e e l p a r t , T s , t o z e r o d e g r e e s a n d c h a n g i n g t h e t e m p e r a t u r e i n t h e

o v e r l a y . A f t e r a p p l y i n g t h e t e m p e r a t u r e , x e d t r a c l o a d s a r e a p p l i e d , a n d n a l l y t h e

t a n d e m l o a d i s i n c r e a s e d s t e p w i s e .

0 0.02 0.04 0.06 0.080

100

200

300

400

500

•

•

•

•

•

⊕

⊕

⊕

⊕

⊕

Crack width [mm]

A x l e

l o a d [ k N ]



00C

2.50C

50C

7.50C

100C

F i g u r e 2 2 : A x l e l o a d v s . c r a c k w i d t h f o r a t e m p e r a t u r e s g r a d i e n t o f ∆T = 0

0C , 2 . 5

0C , 5

0C , 7 . 5

0C , a n d 1 0

0C b e t w e e n t h e F R C o v e r l a y ( l o w e s t t e m p e r a t u r e ) a n d t h e s t e e l d e c k

( h i g h e s t t e m p e r a t u r e ) .

T h e r e s u l t o f t h e a n a l y s i s f o r a F R C o v e r l a y i s p l o t t e d i n a n a x l e l o a d v s . c r a c k w i d t h

d i a g r a m f o r d i e r e n t t e m p e r a t u r e d i e r e n c e s i n F i g u r e 2 2 . A s t h e t e m p e r a t u r e d i e r e n c e

i n c r e a s e s , t h e v a l u e o f a x l e l o a d t h a t i n i t i a t e s c r a c k i n g i s l o w e r e d . F o r t e m p e r a t u r e d i e r -

e n c e s l a r g e r t h a n 5

0C , c r a c k i n g i s i n i t i a t e d b y t h e t e m p e r a t u r e l o a d i n g i t s e l f . M o r e o v e r ,

t h e l o a d l e v e l s w h e r e d e b o n d i n g a n d c r a c k i n g i n t h e d i r e c t i o n o f t h e b r i d g e a x i s i n i t i a t e s

a r e a l s o l o w e r e d . T h e a n a l y s i s s h o w s t h a t t e m p e r a t u r e d i e r e n c e s b e t w e e n t h e o v e r l a y

a n d s t e e l p l a t e i n u e n c e s t h e c r a c k i n g b e h a v i o r c o n s i d e r a b l y a n d s h o u l d b e t a k e n i n t o

a c c o u n t i n a d e s i g n s i t u a t i o n . I t s h o u l d a l s o b e n o t e d , t h a t t h e w a v y s h a p e d c u r v e s i n

F i g u r e 2 2 a r e d u e t o n u m e r i c a l u n c e r t a i n t i e s a n d c o u l d b e i m p r o v e d b y d e c r e a s i n g t h e

e l e m e n t s i z e .

4 . 4 E a r l y A g e S h r i n k a g e

I m m e d i a t e l y a f t e r c a s t i n g o f t h e c e m e n t - b a s e d o v e r l a y , t h e c e m e n t s t a r t s t o h y d r a t e . A s

a c o n s e q u e n c e o f h y d r a t i o n , t h e m a t e r i a l e x p e r i e n c e s a m a c r o s c o p i c v o l u m e r e d u c t i o n

2 3



k n o w n a s a u t o g e n o u s s h r i n k a g e . A u t o g e n o u s s h r i n k a g e c a n b e q u i t e s i g n i c a n t i n h i g h

p e r f o r m a n c e c o n c r e t e s d u e t o t h e i r l o w w a t e r t o b i n d e r r a t i o ( w / b r a t i o ) . A u t o g e n o u s

s h r i n k a g e p l a y s a n i m p o r t a n t r o l e f o r w a t e r - c e m e n t r a t i o s b e l o w 0 . 4 0 - 0 . 4 5 ( J e n s e n &

H a n s e n 2 0 0 1 ) . T h e m a g n i t u d e o f a u t o g e n o u s s h r i n k a g e h a s b e e n m e a s u r e d , o n c e m e n t

p a s t e , o f m o r e t h a n 1 0 0 0 m i c r o s t r a i n s a f t e r a f e w w e e k s o f h a r d e n i n g , ( J e n s e n & H a n s e n

2 0 0 1 ) . S i n c e t h e o v e r l a y i s r e s t r a i n e d f r o m m o v i n g , d u e t o i t s b o n d w i t h t h e s t e e l p l a t e ,

s h r i n k a g e m a y c a u s e c r a c k i n g a n d e a r l y a g e p e r f o r m a n c e h a s t o b e t a k e n i n t o a c c o u n t .

P r o p e r c u r i n g c o n d i t i o n s s h o u l d b e e s t a b l i s h e d , s i n c e p o o r c u r i n g c o n d i t i o n s m a y c a u s e

a d d i t i o n a l s h r i n k a g e , k n o w n a s d r y i n g s h r i n k a g e . I n t h e f o l l o w i n g , a u t o g e n o u s s h r i n k a g e

i s t h e m a i n f o c u s , a s t h i s i s h a r d t o a v o i d , w h e n u s i n g h i g h p e r f o r m a n c e c o n c r e t e s .

I n t h i s s e c t i o n , a s h o r t i n t r o d u c t i o n t o t h e a p p l i e d m o d e l i n g f r a m e w o r k w i l l b e g i v e n ,

w i t h t h e a i m t o a n a l y z e t h e s h r i n k a g e b e h a v i o r a n d r i s k o f c r a c k i n g a t e a r l y a g e s . G i v e n

a s h r i n k a g e h i s t o r y , i t i s t h e a i m o f t h e s t u d y t o g i v e a n e s t i m a t e o f t h e r i s k o f c r a c k i n g

a n d t h e s t r e s s l e v e l i n t h e o v e r l a y a t a g i v e n t i m e . C r e e p o f t h e o v e r l a y i s a s i g n i c a n t

f a c t o r a t e a r l y a g e a n d h a s t o b e t a k e n i n t o a c c o u n t .

4 . 4 . 1 S h r i n k a g e F u n c t i o n

A n e x p r e s s i o n f o r t h e a u t o g e n o u s s h r i n k a g e o v e r t i m e , b a s e d o n e x p e r i m e n t a l d a t a , h a s

b e e n p r o p o s e d b y t h e J a p a n e s e C o n c r e t e I n s t i t u t e ( T a z a w a e t a l . 2 0 0 0 ) . S h r i n k a g e a s a

f u n c t i o n o f t i m e c a n b e f o u n d b y t h e f o l l o w i n g e x p r e s s i o n :

shrinkage(t) = [γinf (w/b)β (t)]10−6( 1 2 )

w h e r e γ i s a c o e c i e n t t o d e s c r i b e t h e e e c t o f c e m e n t t y p e ( γ = 1 f o r o r d i n a r y P o r t -

l a n d C e m e n t ) ,

inf (w/b)i s t h e u l t i m a t e a u t o g e n o u s s h r i n k a g e . T h e u l t i m a t e a u t o g e -

n o u s s h r i n k a g e i s a f u n c t i o n o f t h e w a t e r - b i n d e r r a t i o w/b , a n d c a n b e f o u n d f r o m

inf (w/b) = 3070exp(−7.2(w/b)) . T h e d e v e l o p m e n t o f s h r i n k a g e o v e r t i m e i s d e s c r i b e d

b y β (t) i n t h e f o l l o w i n g f o r m :

β (t) = [1 − exp(−a(t− t0)b)] ( 1 3 )

w h e r e t0 i s t h e i n i t i a l s e t t i n g t i m e . T h e t w o r e m a i n i n g v a r i a b l e s a a n d b a r e c o n s t a n t s .

4 . 4 . 2 M o d e l i n g o f C r e e p

A s d i s c u s s e d e a r l i e r , a u t o g e n o u s s h r i n k a g e i n t h e o v e r l a y c a u s e s t e n s i l e s t r e s s e s i n t h e

o v e r l a y . M o r e o v e r , w h e n a p p l y i n g a c o n s t a n t s t r e s s o v e r t i m e i n a c e m e n t - b a s e d m a t e r i a l ,

r e l a x a t i o n o f t h e m a t e r i a l n e e d s t o b e t a k e n i n t o a c c o u n t . A s t h e c e m e n t - b a s e d m a t e r i a l i s

e x p o s e d t o a c o n s t a n t s t r e s s o v e r t i m e , t h e s t r e s s e s w i l l d e c l i n e w i t h t i m e . T h i s b e h a v i o r

c a n b e d e s c r i b e d b y v i s c o e l a s t i c i t y a n d v a r i o u s m o d e l s c a n b e a p p l i e d t o d e s c r i b e t h e

v i s c o e l a s t i c b e h a v i o r o f c e m e n t - b a s e d m a t e r i a l s . I n t h i s s t u d y , a K e l v i n c h a i n i s a p p l i e d ,

f o r m o d e l i n g o f s h r i n k a g e a n d c r e e p . B a º a n t ( 1 9 8 8 ) g i v e s a f u l l d e t a i l e d r e v i e w o f c r e e p

m o d e l i n g i n c e m e n t - b a s e d m a t e r i a l s . F o r c l a r i t y o f t h e m o d e l i n g a n d t o m a k e t h e p a p e r

s e l f - c o n t a i n e d , a s h o r t i n t r o d u c t i o n i s g i v e n t o t h e a p p l i e d m o d e l . T h e K e l v i n c h a i n m o d e l

2 4



i s b a s e d o n a u n i q u e c o m p l i a n c e f u n c t i o n J (t, t) , w h i c h p r e d i c t s t h e t i m e d e p e n d e n t

d e f o r m a t i o n s a s f u n c t i o n o f t i m e o f l o a d a p p l i c a t i o n t a n d t h e d u r a t i o n o f l o a d (t − t).

T h e c o m p l i a n c e f u n c t i o n i s e x p r e s s e d a s :

J (t, t) = 1E 0(t)

+ 1E 1(t)

H 1(t, t) + 1E 2(t)

H 2(t, t) + ... + 1E n(t)

H n(t, t) ( 1 4 )

w h e r e t i s t i m e , E 0(t, t) i s t h e i n s t a n t a n e o u s e l a s t i c d e f o r m a t i o n , f o r t h e l o a d a p p l i e d a t

a n a g e o f t . T h e f u n c t i o n H n(t, t) i s g i v e n b y :

H n(t, t) = 1 − exp

t − t

τ n

( 1 5 )

w h e r e τ n i s t h e s o - c a l l e d r e t a r d a t i o n t i m e o f c h a i n u n i t n. T h e l i n k b e t w e e n t h e r e t a r d a t i o n

t i m e a n d t h e v i s c o s i t y o f t h e d a s h p o t ηn , i s g i v e n b y :

τ n = ηnE n

( 1 6 )

T h e d e v e l o p m e n t o f t h e m o d u l u s o f e l a s t i c i t y o v e r t i m e , a s i n c l u d e d i n E q u a t i o n ( 1 4 ) , c a n

b e e x p r e s s e d b y t h e f o l l o w i n g f u n c t i o n :

E (t) = A1[1 − e−(t/τ 1)n1 ] + A2[1 − e−(t/τ 2)n2 ]( 1 7 )

w h e r e ni i s a d i m e n s i o n l e s s f a c t o r r e l a t e d t o g e o m e t r y o f t h e p r o b l e m .

E x p e r i m e n t a l c r e e p p r o p e r t i e s o b t a i n e d b y Ø s t e r g a a r d ( 2 0 0 3 ) a r e a p p l i e d i n t h i s s t u d y .

H e c o n d u c t e d e x p e r i m e n t s a t t = 1 , 3 a n d 5 d a y s . T h e t t o e x p e r i m e n t a l v a l u e s h a v e

b e e n o b t a i n e d b y u s i n g a 4 c h a i n u n i t , c f . F i g u r e 2 3 .

F i g u r e 2 3 : K e l v i n c h a i n u s e d i n t h e a n a l y s i s , c o m p o s e d 4 K e l v i n u n i t s i n s e r i a l c o n n e c t i o n .

T h e K e l v i n C h a i n c o n s i s t s o f 4 u n i t s , a l l o f t h e m w i t h a s p r i n g , a n d t h r e e u n i t s h a v e

a d a s h p o t . T h e s p r i n g v a l u e s , a c c o r d i n g t o E q u a t i o n ( 1 7 ) , a r e l i s t e d i n T a b l e 4 . T h e

d a s h p o t p a r a m e t e r s t o t h e t h r e e u n i t s a r e g i v e n i n T a b l e 5 .

4 . 4 . 3 F E M o d e l i n g o f S h r i n k a g e a n d C r e e p

V i s c o e l a s t i c i t y o f c e m e n t - b a s e d m a t e r i a l s c a n b e m o d e l e d u s i n g n i t e e l e m e n t s . I n t h e

p r e s e n t s t u d y m o d e l i n g i s c a r r i e d o u t u s i n g a K e l v i n c h a i n , w h i c h i s i m p l e m e n t e d i n

2 5



E 0 E 1 E 2 E 3A1 [ G P a ] 4 3 . 4 1 8 0 2 8 0 3 6 0

τ 1 [ h o u r s ] 2 3 . 9 5 5 0 5 0 0

n1 [ - ] 2 . 1 9 1 1 1

A2 [ G P a ] 2 0 . 0 - - -

τ 2 [ h o u r s ] 1 0 0 - - -

n2 [ - ] 4 - - -

T a b l e 4 : F i t t i n g p a r a m e t e r s o f t h e s p r i n g s i n t h e a p p l i e d K e l v i n C h a i n

H 1 H 2 H 3τ n [ h o u r s ] 5 0 5 5 0 0

T a b l e 5 : F i t t i n g p a r a m e t e r s o f t h e t h r e e d a s h p o t s i n t h e a p p l i e d K e l v i n C h a i n

D I A N A ( 2 0 0 3 ) . T o i l l u s t r a t e a n o v e r l a y c a s t o n a s t e e l p l a t e a s m a l l e x a m p l e i s a n a l y z e d .

A n i n n i t y l o n g s p e c i m e n i s c o n s i d e r e d , w i t h a t h i c k n e s s o f 5 0 m m , b o n d e d t o a s t e e l p l a t e ,

e x p o s e d t o s h r i n k a g e o v e r t i m e . T h e i n v e s t i g a t i o n a i m s a t c a l c u l a t i n g t h e s t r e s s e s a s a

f u n c t i o n o f t i m e . S i n c e t h e m o d u l u s o f e l a s t i c i t y a n d c r e e p p r o p e r t i e s d e v e l o p o v e r t i m e ,

t h e m a t e r i a l i s g i v e n t h e c r e e p p r o p e r t i e s s h o w n p r e v i o u s l y . T h e i n n i t y l o n g s p e c i m e n

i s a n a l y z e d t h r o u g h a p a r a m e t r i c s t u d y . T h e s p e c i m e n i s g i v e n f o u r d i e r e n t s h r i n k a g e

h i s t o r i e s u s i n g E q u a t i o n ( 1 2 ) . A s a n e x a m p l e , a w a t e r b i n d e r r a t i o n i s c h o s e n t o 0 . 4 5 ,

a n o r d i n a r y P o r t l a n d C e m e n t ( γ = 1) , a n d a c o n s t a n t b v a l u e ( b = 1 ) . T h e c o n s t a n t a i s

v a r i e d a c c o r d i n g t o :

a = 0.3, 0.5, 0.7, 0.9.

0 2 4 6 8 100

20

40

60

80

100

t [Days]

S h r i n k a g e [ M i c r o s t r a i n ]

0 5 10 15 200

0.5

1

1.5

2

t [Days]

S t r e s s [ M P a ]

a=0.3, 0.5, 0.7, 0.9

a=0.3

a=0.5

a=0.7

a=0.9w/b=0.5

b=1.0γ =1

F i g u r e 2 4 : ( a ) S h r i n k a g e o v e r t i m e u s i n g E q u a t i o n ( 1 2 ) . ( b ) S t r e s s e s i n o v e r l a y f o r d i e r e n t

s h r i n k a g e h i s t o r i e s .

2 6



T h e r e s u l t s o f t h e a n a l y s i s , v a r y i n g t h e s h r i n k a g e h i s t o r y , a r e s h o w n i n F i g u r e 2 4 . F i g u r e

2 4 ( a ) , s h o w s t h e f o u r d i e r e n t s h r i n k a g e h i s t o r i e s , c o m p u t e d v a r y i n g t h e c o n s t a n t a i n

f o r m u l a ( 1 2 ) . A s o b s e r v e d , i n a l l t h e c a s e s , t h e c u r v e s a l l e n d a t t h e s a m e u l t i m a t e

s h r i n k a g e v a l u e . H o w e v e r , a l o w v a l u e o f t h e c o n s t a n t a, p r o d u c e s a m o r e s l o w a p p r o a c h

t o t h e u l t i m a t e s h r i n k a g e v a l u e . T h e c o r r e s p o n d i n g r e s p o n s e s o f t h e s t r e s s l e v e l i n t h e

s p e c i m e n a r e s h o w n i n F i g u r e 2 4 ( b ) . A s s e e n , t h e f o u r c a s e s p r o d u c e q u i t e d i e r e n t

r e s p o n s e s . I n t h e c a s e o f a l o w v a l u e o f t h e c o n s t a n t a , t h e m a x i m u m s t r e s s l e v e l b e c o m e s

q u i t e l a r g e c o m p a r e d t o t h e o t h e r c a s e s . M o r e o v e r , t h e r a t e o f s t r e s s d r o p i s s l o w e r o v e r

t i m e , w h i c h i s c a u s e d b y t h e f a c t t h a t r e l a x a t i o n o f a c e m e n t - b a s e d m a t e r i a l i s l a r g e s t i n

t h e r s t f e w h o u r s a f t e r c a s t i n g . S i n c e t h e m a g n i t u d e o f r e l a x a t i o n i s l a r g e a t t h i s s t a g e ,

i t i s i d e a l t o h a v e a u t o g e n o u s s h r i n k a g e d e v e l o p i n g h e r e .

I n r e l a t i o n t o t h e p r e s e n t o v e r l a y a p p l i c a t i o n , t h e d e v e l o p m e n t o f t h e t e n s i l e s t r e s s e s a n d

t h e r i s k o f c r a c k i n g h a v e t o b e t a k e n i n t o a c c o u n t . T o o l a r g e s h r i n k a g e v a l u e s a t e a r l y

a g e s m i g h t e x c e e d t h e t e n s i l e s t r e n g t h a n d t h e r e b y c a u s e c r a c k i n g . A n o t h e r a s p e c t i s t h e

s t r e s s l e v e l w h e n t h e b r i d g e i s o p e n e d f o r t r a c . A c o n s i d e r a b l e s t r e s s l e v e l m a y h a v e

t o b e c o n s i d e r e d a l o n g w i t h t r a c a n d t e m p e r a t u r e l o a d i n o r d e r t o g i v e a n e s t i m a t e o f

t h e o v e r l a y p e r f o r m a n c e . I n a g i v e n d e s i g n s i t u a t i o n , d i e r e n t s h r i n k a g e l e v e l s s h o u l d b e

a p p l i e d t o t h e o v e r l a y t o g e t h e r w i t h t r a c a n d t e m p e r a t u r e l o a d i n g .

5 F u r t h e r I n v e s t i g a t i o n s

C y c l i c l o a d i n g h a s n o t b e e n a n a l y z e d i n t h e p r e s e n t s t u d y , b u t s h o u l d b e a p a r t o f a f u l l

a s s e s s m e n t o f t h e c e m e n t - b a s e d o v e r l a y s y s t e m . T h e m a x i m u m c r a c k w i d t h w i l l i n c r e a s e

f o r a c e r t a i n n u m b e r o f c y c l e s , w h i c h h a s n o t b e e n t a k e n i n t o a c c o u n t . A t p r e s e n t , a

n u m b e r o f f r a c t u r e m e c h a n i c a l s t u d i e s o n t h e f a t i g u e b e h a v i o r o f c e m e n t - b a s e d m a t e r i a l s

h a v e b e e n c a r r i e d o u t , s e e e . g . M a t s u m o t o & L i ( 1 9 9 9 ) o r Z h a n g e t a l . ( 2 0 0 1 ) . I n o r d e r

t o e s t i m a t e t h e s e r v i c e l i f e o f t h e o v e r l a y s y s t e m , t h e i n u e n c e o f l o a d c y c l e s s h o u l d b e

i m p l e m e n t e d i n t h e c o n s t i t u t i v e m o d e l s a p p l i e d .


O n e o f t h e m a i n c a u s e s o f f a t i g u e i n o r t h o t r o p i c s t e e l b r i d g e d e c k s i s t h e i r l a c k o f s t i n e s s .

I n a l i n e a r e l a s t i c s t u d y u s i n g n i t e e l e m e n t s i t h a s b e e n s h o w n , t h a t a p p l y i n g a c e m e n t -

b a s e d o v e r l a y r e s u l t s i n a s i g n i c a n t a n d i n m a n y c a s e s s u c i e n t r e d u c t i o n o f s t r e s s e s i n

t h e f a t i g u e s e n s i t i v e s t e e l r e g i o n s .

T h e c e m e n t - b a s e d o v e r l a y s y s t e m f o r s t i e n i n g a t r a d i t i o n a l o r t h o t r o p i c s t e e l b r i d g e d e c k

h a s b e e n a n a l y z e d a p p l y i n g n o n l i n e a r f r a c t u r e m e c h a n i c s . T h e i n v e s t i g a t i o n c o n c e r n s t h e

o r t h o t r o p i c b r i d g e d e c k o f t h e s t e e l b o x g i r d e r s o f t h e a p p r o a c h s p a n s o f t h e F a r ø B r i d g e s ,

D e n m a r k . T h e f o c u s i s o n t h e s i t u a t i o n w i t h m a x i m u m n e g a t i v e b e n d i n g ( t e n s i o n i n t h e

o v e r l a y ) , s i n c e t h i s s i t u a t i o n i s b e l i e v e d t o b e c r i t i c a l t o t h e o v e r l a y . P e r f o r m a n c e o f t h e

o v e r l a y m i g h t b e c l o s e l y r e l a t e d t o c r a c k i n g a n d d e b o n d i n g . C r a c k i n g b e h a v i o r o f t h e

o v e r l a y s y s t e m h a s b e e n a n a l y z e d f o r f o u r d i e r e n t m a t e r i a l s . T h e m a t e r i a l s i n v e s t i g a t e d

2 7



a r e : p l a i n c o n c r e t e , F i b r e R e i n f o r c e d C o n c r e t e ( F R C ) , F i b r e R e i n f o r c e d D e n s i t ( F R D ) ,

a n d E n g i n e e r e d C e m e n t i t i o u s C o m p o s i t e ( E C C ) . C o n c r e t e , F R C a n d F R D c a n b e c l a s -

s i e d a s t e n s i o n s o f t e n i n g m a t e r i a l s , w h e r e a s E C C i s c h a r a c t e r i z e d a s a s t r a i n h a r d e n i n g

m a t e r i a l .

C r a c k i n g a n d e v e n t u a l l y d e b o n d i n g i s i n i t i a t e d d u e t o m e c h a n i c a l a n d e n v i r o n m e n t a l l o a d -

i n g . M e c h a n i c a l l o a d i n g ( t r a c ) a n d e n v i r o n m e n t a l l o a d i n g s u c h a s e a r l y a g e s h r i n k a g e

a n d t e m p e r a t u r e g r a d i e n t s h a v e b e e n a n a l y z e d . A d o u b l e a x l e t a n d e m s y s t e m p l a c e d

a c r o s s t h e b u l k h e a d i n i t i a t e s c r a c k i n g o f t h e o v e r l a y d u e t o l a r g e t e n s i l e s t r e s s e s . A t

s o m e p o i n t ( f o r a c e r t a i n c r a c k w i d t h i n t h e o v e r l a y ) d e b o n d i n g i s i n i t i a t e d a n d d e v e l o p s

p e r p e n d i c u l a r a n d a w a y f r o m t h e b u l k h e a d . W h e n c o m p a r i n g t h e p e r f o r m a n c e o f t h e

d i e r e n t t e n s i o n s o f t e n i n g m a t e r i a l s , d e b o n d i n g i s i n i t i a t e d a t a p p r o x i m a t e l y t h e s a m e

c r a c k m o u t h o p e n i n g i n t h e o v e r l a y . D e b o n d i n g i s n o t i n i t i a t e d f o r t h e s t r a i n h a r d e n i n g

m a t e r i a l a n a l y z e d , E C C . F u r t h e r l o a d i n g o f t h e t a n d e m s y s t e m i n i t i a t e s c r a c k i n g p e r p e n -

d i c u l a r t o t h e b r i d g e a x i s . I n t h e l o a d r a n g e c o n s i d e r e d , c r a c k i n g p e r p e n d i c u l a r t o t h e

b r i d g e a x i s i s o n l y i n i t i a t e d f o r t h e t e n s i o n s o f t e n i n g m a t e r i a l s . T h e a n a l y s i s s h o w s t h a t

a p p l y i n g a c e m e n t - b a s e d m a t e r i a l t h a t e x h i b i t s m u l t i p l e c r a c k i n g i s d e s i r a b l e d u e t o t h e

f a c t t h a t d e b o n d i n g c a n b e a v o i d e d i n t h e l o a d r a n g e c o n s i d e r e d .

T e m p e r a t u r e g r a d i e n t s m i g h t h a v e a s i g n i c a n t i n u e n c e o n t h e p e r f o r m a n c e o f t h e

c e m e n t - b a s e d o v e r l a y s y s t e m . C o o l i n g o f t h e o v e r l a y ( b o t t o m w a r m e r t h a n t o p ) , p r o -

d u c e s t e n s i l e s t r e s s e s i n t h e o v e r l a y s i n c e i t i s r e s t r a i n e d f r o m m o v i n g b e c a u s e o f i t s b o n d

t o t h e s t e e l d e c k . F o r d i e r e n t t e m p e r a t u r e g r a d i e n t s a n d t a n d e m l o a d t h e p e r f o r m a n c e

o f t h e o v e r l a y s y s t e m i s s h o w n i n t e r m s o f c r a c k w i d t h . A g a i n , d e b o n d i n g i s i n i t i a t e d

a c r o s s t h e b u l k h e a d f o r a c e r t a i n c r a c k m o u t h o p e n i n g .

F i n a l l y , i t i s d e m o n s t r a t e d h o w e a r l y a g e s h r i n k a g e c a n b e t a k e n i n t o a c c o u n t . A f t e r

c a s t i n g o f t h e o v e r l a y , h i g h t e n s i o n s t r e s s e s d e v e l o p i n t h e o v e r l a y d u e t o m a c r o s c o p i c

v o l u m e c h a n g e s . I n t h e w o r s t c a s e , c r a c k i n g m i g h t b e i n i t i a t e d d u e t o e a r l y a g e s h r i n k a g e ,

o r h i g h t e n s i l e s t r e s s e s m a y b e p r e s e n t w h e n t h e b r i d g e i s o p e n e d f o r t r a c .

R e f e r e n c e s

B a t t i s t a , R . & P f e i l , M . ( 2 0 0 0 ) , S t r a n g h e n i n g f a t i g u e c r a c k e d o r t h o t r o p i c d e c k s w i t h

c o m p o s i t e l a y e r s , i n ` A n n u a l T e c h n i c a l S e s s e i o n , a n d M e e t i n , S t r u c t u r a l S t a b i l i t y

R e s e a r c h C o u n c i l ' , p p . 3 7 6 3 8 9 .

B a º a n t , Z . P . ( 1 9 8 8 ) , M a t h e m a t i c a l M o d e l i n g o f C r e e p a n d S h r i n k a g e o f C o n c r e t e , U K :

J o h n W i l e y .

B a º a n t , Z . P . & O h , B . H . ( 1 9 8 3 ) , ` C r a c k b a n d t h e o r y f o r f r a c t u r e o f c o n c r e t e ' , M a t e r i a l s

a n d S t r u c t u r e s 1 6 , 1 5 5 1 5 7 .

B r a a m , C . R . , B u i t e l a a r , P . & K a p t a i j n , N . ( 2 0 0 3 ) , R e i n f o r c e d h i g h p e r f o r m a n c e c o n -

c r e t e o v e r l a y s y s t e m f o r s t e e l b r i d g e s , i n ` I n t h e 5 t h I n t e r n a t i o n a l C R O W - W o r k s h o p

o n F u n d a m e n t a l M o d e l l i n g o f t h e D e s i g n a n d P e r f o r m a n c e o f C o n c r e t e P a v e m e n t s ,

I s t a n b u l , T u r k e y ' .

2 8



B u i t e l a a r , P . ( 2 0 0 2 ) , U l t r a t h i n h e a v y r e i n f o r c e d h i g h p e r f o r m a n c e c o n c r e t e o v e r l a y s , i n

` 6 t h I n t e r n a t i o n a l S y m p o s i u m o n U t i l i z a t i o n o f H i g h S t r e n g t h / H i g h P e r f o r m a n c e

C o n c r e t e , L e i p z i g , G e r m a n y ' , p p . 1 5 7 7 1 5 9 0 .

B u i t e l a a r , P . , B r a a m , R . & K a p t i j n , N . ( 2 0 0 4 ) , R e i n f o r c e d h i g p e r f o r m a n c e c o n c r e t e

o v e r l a y s y s t e m f o r r e h a b i l i t a t i o n a n d s t r e n g t h e n i n g o f o r t h o t r o p i c s t e e l b r i d g e d e c k s ,

i n ` O r t h o t r o p i c B r i d g e C o n f e r e n c e , S a c r e m e n t o , U S A ' , A S C E , p p . 3 8 4 4 0 1 .



D o w l i n g , P . J . ( 1 9 6 8 ) , T h e B e h a v i o u r o f S t i e n e d P l a t e B r i d g e D e c k u n d e r W h e e l L o a d i n g ,

P h D t h e s i s , I m p e r i a l C o l l e g e L o n d o n .

E N V 1 9 9 1 - 2 - 5 : 1 9 9 7 ( 1 9 9 7 ) , E u r o c o d e 1 - A c t i o n s o n S t r u c t u r e s - P a r t 1 - 5 : G e n e r a l A c t i o n s

- T h e r m a l A c t i o n s , E u r o p e a n C o m m i t e e f o r S t a n d a r d i s a t i o n .

E N V 1 9 9 1 - 3 ( 1 9 9 1 ) , E u r o c o d e 1 - B a s i s o f D e s i g n a n d A c t i o n s o n S t r u c t u r e s , P a r t 3 :

T r a c L o a d s o n B r i d g e s , E u r o p e a n C o m m i t e e f o r S t a n d a r d i s a t i o n .



R e s . 6 ( 6 ) , 7 7 3 7 8 2 .

J e n s e n , O . M . & H a n s e n , P . F . ( 2 0 0 1 ) , ` A u t o g e n o u s s h r i n k a g e a n d R H - c h a n g e i n p e r s p e c -

t i v e ' , C e m e n t a n d C o n c r e t e R e s e a r c h 3 1 ( 1 2 ) , 5 6 7 5 7 5 .

J o n g , F . B . P . & K o l s t e i n , M . H . ( 2 0 0 4 ) , S t r e n g h e n i n g a b r i d g e d e c k w i t h h i g h p e r f o r m a n c e

c o n c r e t e , i n ` O r t h o t r o p i c B r i d g e C o n f e r e n c e , S a c r e m e n t o , U S A ' , A S C E , p p . 3 2 8 3 4 7 .

J o n g , F . B . P . , K o l s t e i n , M . H . & B i j l a a r d , F . S . K . ( 2 0 0 4 ) , S t r a i n m e a s u r e m e n t t e s t s a t

o r t h o t r o p i c s t e e l b r i d g e d e c k s w i t h a h e a v y v e h i c l e s i m u l a t o r , i n ` P r o o c e d d i n g s o f t h e

1 0 t h N o r d i c S t e e l C o n s t r u c t i o n C o n f e r e n c e , C o p e n h a g e n , D e n m a r k ' , p p . 4 0 1 4 1 2 .

K a r i h a l o o , B . L . ( 1 9 9 5 ) , F r a c t u r e M e c h a n i c s a n d S t r u c t u r a l C o n c r e t e , C o n c r e t e D e s i g n

a n d C o n s t r u c t i o n S e r i e s , L o n g m a n S c i e n t i c a n d T e c h n i c a l , H a r l o w , E s s e x , E n g l a n d .

K o l s t e i n , M . H . & W a r d e n i e r , J . ( 1 9 9 7 ) , ` S t r e s s r e d u c t i o n d u e t o s u r f a c i n g o n o r t h o t r o p i c

s t e e l d e c k s ' , I A B S E W o r k s h o p , I A B S E R e p o r t s , V o l . 7 6 , Z u r i c h , L a u s a n n e .

K o l s t e i n , M . H . & W a r d e n i e r , J . ( 1 9 9 8 ) , A n e w t y p e o f f a t i g u e f a i l u r e s i n s t e e l o r -

t h o t r o p i c b r i d g e d e c k s , i n ` P r o c e e d i n g s o f t h e f t h P a c i c S t r u c t u r a l C o n f e r e n c e ,

K o r e a ' , p p . 4 8 3 4 8 8 .

L i , V . ( 2 0 0 2 ) , ` A d v a n c e s i n E C C r e s e a r c h ' , A C I S p e c i a l P u b l i c a t i o n o n C o n c r e t e : M a t e r i a l

S c i e n c e t o A p p l i c a t i o n s S P 2 0 6 ( 2 3 ) , 4 6 3 4 7 2 .

2 9



L i , V . C . & L e u n g , C . K . Y . ( 1 9 9 2 ) , ` S t e a d y - s t a t e a n d m u l t i p l e c r a c k i n g o f s h o r t r a n d o m

b e r c o m p o s i t e s ' , J o u r n a l o f E n g i n e e r i n g M e c h a n i c s 1 1 8 ( 1 1 ) , 2 2 4 6 2 2 6 4 .

L i , V . , W u , C . , W a n g , S . , O g a w a , A . & S a i t o , T . ( 2 0 0 2 ) , ` I n t e r f a c e t a i l o r i n g f o r s t r a i n -

h a r d e n i n g P V A - E C C ' , A C I M a t e r i a l s J o u r n a l 9 9 ( 5 ) , 3 7 3 4 0 0 .

M a t s u m o t o , T . & L i , V . C . ( 1 9 9 9 ) , ` F a t i g u e l i f e a n a l y s i s o f b e r r e i n f o r c e d c o n c r e t e w i t h a

f r a c t u r e m e c h a n i c s b a s e d m o d e l ' , C e m e n t a n d C o n c r e t e C o m p o s i t e s 2 1 ( 4 ) , 2 4 9 2 6 1 .




R I L E M ( 2 0 0 1 ) , ` T e s t a n d d e s i g n m e t h o d s f o r s t e e l b e r r e i n f o r c e d c o n c r e t e . r e c o m m e n -

d a t i o n s f o r u n i a x i a l t e n s i o n t e s t ' , M a t e r i a l s a n d S t r u c t u r e s 3 4 ( 3 6 ) . P r e p a r e d b y

R I L E M - C o m m i t t e e - T D F - 1 6 2 , C h a i r l a d y L . V a n d e w a l l e .

S c h i e s s l , A . & Z i l c h , K . ( 2 0 0 1 ) , ` T h e e e c t o f t h e m o d i e d c o m p o s i t i o n o f S C C o n s h e a r

a n d b o n d b e h a v i o r ' , P r o c e e d i n g s o f t h e S e c o n d I n t e r n a t i o n a l S y m p o s i u m o n s e l f -

C o m p a c t i n g C o n c r e t e , T o k y o p p . 5 0 1 5 0 6 .

S e r r a n o , E . ( 2 0 0 0 ) , ` A d h e s i v e j o i n t s i n t i m b e r e n g i n e e r i n g - m o d e l l i n g a n d t e s t i n g o f f r a c -

t u r e p r o p e r t i e s ' , D o c t o r a l T h e s i s , L u n d U n i v e r s i t y , D i v i s i o n o f S t r u c t u r a l M e c h a n i c s

.

S m i t h , J . W . & B r i g h t , S . ( 2 0 0 3 ) , U p g r a d i n g o r t h o t r o p i c b r i d g e d e c k s w i t h b e r r e i n f o r c e d

c o m p o s i t e s , i n ` H i g h P e r f o r m a n c e M a t e r i a l s i n B r i d g e s ' , p p . 4 6 3 4 7 2 .

T a z a w a , E . , S a t o , R . , S a k a i , E . & M i y a z a w a , S . ( 2 0 0 0 ) , W o r k o f J C I c o m m i t e e o n a u t o -

g e n o u s s h r i n k a g e , i n V . B a r o g h e l - B o u n y & P . C . A i t c i n , e d s , ` S h r i n k a g e o f C o n c r e t e ,

I n t e r n a t i o n a l R I L E M W o r k s h o p ' , V o l . P R O 1 7 .

T h e D a n i s h M i n i s t r y o f T r a n s p o r t a t i o n ( 1 9 8 7 ) , ` T h e s u p e r s t r u c t u r e o f t h e f a r ø b r i d g e s

1 . d e s i g n o f t h e s t e e l s u p e r s t r u c t u r e ' , T h e D a n i s h M i n i s t r y o f T r a n s p o r t a t i o n , T h e

R o a d D i r e c t o r a t e , I S B N 8 7 - 7 4 9 1 - 2 3 8 - 0 .

W a l t e r , R . & O l e s e n , J . F . ( 2 0 0 5 ) , ` C o h e s i v e m i x e d m o d e f r a c t u r e m o d e l l i n g a n d e x p e r i -

m e n t s ' , P a p e r s u b m i t t e d t o : J o u r n a l o f E n g i n e e r i n g F r a c t u r e M e c h a n i c s .

W a l t e r , R . , O l e s e n , J . F . , L i , V . C . & S t a n g , H . ( 2 0 0 4 ) , ` C e m e n t - b a s e d o v e r l a y i n n e g a t i v e

b e n d i n g ' , P a p e r s u b m i t t e d f o r p u b l i c a t i o n ( i n c l u d e d i n P h D T h e s i s b y R . W a l t e r ) .


s t e e l - c o n c r e t e i n t e r f a c e ' , J o u r n a l o f E n g i n e e r i n g F r a c t u r e M e c h a n i c s 7 2

( 1 7 ) , 2 5 6 5

2 5 8 3 .

3 0





1 9 1 - 2 0 0 .



W o l c h u k , R . ( 2 0 0 2 ) , ` S t r u c t u r a l b e h a v i o u r o f s u r f a c i n g o n s t e e l o r t h o t r o p i c d e c k s a n d

c o n s i d e r a t i o n s f o r p r a c t i c a l d e s i g n ' , S t r u c t u r a l E n g i n e e r i n g I n t e r n a t i o n a l 2 , 1 2 4 1 2 9 .

Z h a n g , J . , S t a n g , H . & L i , V . C . ( 2 0 0 1 ) , ` C r a c k b r i d g i n g m o d e l f o r b r e r e i n f o r c e d c o n c r e t e

u n d e r f a t i g u e t e n s i o n ' , I n t e r n a t i o n a l J o u r n a l o f F a t i g u e 2 3 ( 8 ) , 6 5 5 6 7 0 .

N o t a t i o n

β (t) f u n c t i o n t o d e s c r i b e t h e d e v e l o p m e n t o f s h r i n k a g e

δ n c r a c k o p e n i n g i n n o r m a l p l a n e

δ tx t a n g e n t i a l c r a c k o p e n i n g i n x - d i r e c t i o n

δ ty t a n g e n t i a l c r a c k o p e n i n g i n y - d i r e c t i o n

δ t r e s u l t i n g t a n g e n t i a l c r a c k o p e n i n g i n t - p l a n e

e e l a s t i c s t r a i n l i m i t f o r s t r a i n h a r d e n i n g m a t e r i a l

inf (w/b) u l t i m a t e a u t o g e n o u s s h r i n k a g e

shrinkage s h r i n k a g e s t r a i n

u u l t i m a t e s t r a i n l i m i t f o r s t r a i n h a r d e n i n g m a t e r i a l

η v i s c o s i t y

γ c o e c i e n t i n s h r i n k a g e f o r m u l a t i o n

φtn m i x e d m o d e a n g l e i n t h e t - n p l a n e

φxy m i x e d m o d e a n g l e i n t h e x - y p l a n e

σ n o r m a l s t r e s s

τ r e t a r d a t i o n t i m e


τ x s h e a r s t r e s s i n x - d i r e c t i o n

τ y s h e a r s t r e s s i n y - d i r e c t i o n

A m a t e r i a l c o n s t a n t

a c o n s t a n t

a1, a2, b1, b2 p a r a m e t e r s i n t h e b i l i n e a r s t r e s s - c r a c k o p e n i n g r e l a t i o n s h i p

3 1

Cement-Based Overlay for Orthotropic Steel Bridge Decks

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