shearstiﬀnessandcapacityofjoints betweenprecastwallelements1111597/fulltext01.pdf · in...

Shear Stiffness and Capacity of JointsBetween Precast Wall Elements

Semiha KayaDelvin Salim

June 2017TRITA-BKN. Master Thesis 516, 2017ISSN 1103-4297,ISRN KTH/BKN/EX–516–SE

c©Kaya, Salim 2017Royal Institute of Technology (KTH)Department of Civil and Architectural EngineeringDivision of Concrete StructuresStockholm, Sweden, 2017

Abstract

In this thesis an investigation of the shear stiffness and capacity of joints between pre-fabricated concrete elements regarding to different material properties is reported.Two different models of shear key joints, connected to prefabricated walls, were cre-ated in the non-linear finite element software, ATENA 3D, with the aim to estimatea realistic behaviour of the joints regarding to the external loads.

The literature study is summarized in the earlier parts of the thesis, to achieve awider knowledge about prefabricated wall elements and the non-linear behaviour ofthe connection between the walls, due to the external loads. Moreover, the shearcapacities of the joints were calculated according to Eurocode.

The work started by estimating external loads based on a real case by using a FE-model in ATENA 2D. The 2D model consisted of shear walls that illustrated a partof a ten-storey building exposed to in-plane horizontal loading. The loads werefurther applied on the shear key joint models created in ATENA 3D.

A parametric study was performed on the models to investigate the influence of thefriction coefficient, tensile strength and the cohesion in the interface between thejoint and the wall elements. The combination of the highest friction coefficient andthe highest tensile strength contributed to a better shear capacity of the joint.

The effect of the reinforcement in the shear key joint was also analysed. The resultsshowed that the shear capacity of the joint could be improved by placing transversalreinforcement bars in the shear key. The NLFEA of the unreinforced and reinforced3D shear key joint models gave shear capacities higher than the applied loads.

Hand calculations were performed according to EN 1992, for the shear resistancein interfaces between joints and wall elements. The hand calculations showed thatthe shear key joint without reinforcement did not resist the loads applied on themodel while the model with reinforcement resisted the applied loads with a godmarginal. The shear capacity obtained from NLFEA of the unreinforced modelwas higher than the shear capacity according to the Eurocode. For the reinforcedmodel the shear capacity obtained from the simulations was lower than the shearcapacity according to the Eurocode. The reason for the low shear capacity in thiscase was because the dowel actions were neglected in the NLFEA.

Keywords: Non-liner finite element analysis, prefabricated concrete elements, shearstiffness, capacity, shear key joint, reinforcement.

iii

Sammanfattning

Under detta arbete analyserades skjuvhållfastheten och kapaciteten av fogar mellanprefabricerade betongelement med hänsyn till olika material egenskaper. Två olikamodeller av fogar, kopplade till två prefabricerade väggelement, skapades i den icke-linjära finite element-programvaran ATENA 3D, med syfte på att uppskatta ettverklighetsbaserat beteende i fogen under belastning av yttre laster.

Litteraturstudien är sammafattad under tidigare delar av arbetet, för att ge bredarekunskaper inom prefabricerade väggelement och det icke-linjära beteendet av kop-plingen mellan väggarna under belastning av yttre laster. Vidare har skjuvhållfast-heten i fogen beräknats enligt Eurocode.

Arbetet började med att uppskatta yttre laster baserat på ett verkligt fall genomatt använda en FE-model i ATENA 2D. 2D modellen bestod av skjuvväggar somillustrerade en sektion av en tio-vånings byggnad utsatt för horisontella belastning iplanet. Lasterna var senare applicerade på en model av fogkärna skapad i ATENA3D.

En parameterstudie genomfördes för modellerna med hänsyn till effekterna av frik-tions koefficient, draghållfasthet och kohesion i gränsytan mellan fogen och väggele-menten. Kombinationen av den högsta friktions koefficienten och den högsta draghåll-fastheten bidrog till en bättre skjuvkapacitet i fogen.

Inverkan av armeringen i fogkärnan analyserades. Resultaten visade att skjuvka-paciteten i fogen kunde förbättras genom att placera transversella armerings byglari fogkärnan.Den icke-linjära finita element analyserna för den oarmerade och armer-ade 3D modellen av skjuvfogen gav högre skjuvkapacitet än de applicerade lasterna.

Handberäkningar var genomförda enligt EN 1992, för skjuvkapaciteten i gränsnit-tet mellan fogen och väggelementen. Handberäkningarna visade att fogkärnan utanarmerig klarade inte skjuvspänningarna som applicerades på modellen medan mod-ellen med armering klarade sig med en god marginal. Skjuvkapaciteten erhållenfrån de icke-linjära finita element analyserna för den oarmerade modellen var hö-gre än skjuvkapaciteten enligt Eurocode. Den erhållna skjuvkapaciteten från simu-leringarna för den armerade modellen var lägre än skjuvkapaciten enligt Eurocode.Orsken till den låga skjuvkapaciteten för det här fallet var på grund av att dymlingseffekterna var försummade i den icke-linjära finita element analysen.

Nyckelord: Icke linjära finita elementanalys, prefabricerde betongelement, skju-vhållfasthet, kapacitet, fogkärna, armering.

v

Preface

This thesis is the final work of the Degree of Master at the Department of Civil andArchitectural Engineering at the Royal Institute of Technology, KTH. The thesiswas conducted in cooperation with Tyréns AB. The work took place during thespring semester of 2017.

A special thanks to our supervisor Adjunct Professor Mikael Hallgren, for his in-structions and valuable advises during this thesis. We also thank Nils Ekroth, atTyréns AB, for his helpful guiding and discussions in the topic precast concreteelements.

We are also grateful to Anders Mattsson, for providing us with material regardingprecast, and Gunilla Baudin, for the opportunity to the visit at the construction siteof a precast project. Last but not least we thank our examiner Professor AndersAnsell.

Stockholm, June 2017

Semiha Kaya & Delvin Salim

vii

Symbols

αct Coefficient -αcc The angle of the crossing of the reinforcement in the

surface◦

β Reduction factor -β Return direction factor -γc Partial safety factor for concrete -µ Friction coefficient -εcrack Non-linear fracturing strain -εelastic Elastic strain -εtotal Total strain -εx Strain -ν Strength reduction factor for cracked concrete by shear -ξ Hydrostatic stress invariant MPaρ Deviatoric stress invariant MPaρ Reinforcement ratio -σn State of stress MPaσpij Plastic corrector MPaσtij Predictor state of stress MPa

a0 Crack propagation length mbi Interface width mc Cohesion factor -c1 Constant -c2 Constant -dmax Maximum aggregate size mfc Measured compressive strength of concrete MPafcd Design compressive strength of concrete MPafctd Design tensile strength of concrete MPafcm Mean compressive strength of concrete MPafcm0 Reference value for compressive strength of concrete MPafck Characteristic compressive strength of concrete MPafcu Compressive cube strength of concrete MPaft Concrete tensile strength MPafyd Design yield strength of reinforcement in tension MPa

ix

fyk Characteristic yield strength MPak Local stiffness N/mlp Length of fracture process zone mt Interface width mvEd Design value of the applied shear stress MPavRd,c Shear resistance MPavRd,max Maximum shear resistance MPaw Crack width mwc Critical crack width mz Lever arm of composite section m

Ai Joint area m2

As Reinforcement area m2

D Global displacement mE Modulus of elasticity of concrete MPaG Minimal Shear Modulus Nm/m2

Gf Fracture energy Nm/m2

Gf0 Base value of fracture energy Nm/m2

Gp Plastic potential of concrete MPaIa Internal forces NK Global stiffness N/mK0 Tangential stiffness MPaKnn Normal stiffness MPaKnn,min Minimal normal stiffness MPaKtt Shear stiffness MPaKtt,min Minimal shear stiffness MPaN Normal force NR Global force NRa Force residual NVEd Design value of the shear force MPaVEdi Shear stress in the interface MPaVRdi Design shear resistance MPa

x

Abbreviations

2D Two Dimensions3D Three DimensionsDOF Degrees of FreedomFE Finite ElementFEA Finite Element AnalysisFEM Finite Eelement MethodNLFEA Non-Linear Finite Element Analysis

xi

Contents

Abstract iii

Sammanfattning v

Preface vii

Symbols ix

Abbreviations xi

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Joints between Shear Walls . . . . . . . . . . . . . . . . . . . 2

1.3 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.5 Structure of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Shear Joints in Precast Concrete Walls 5

2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Precast Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Shear Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Vertical Joints between Shear Walls . . . . . . . . . . . . . . . . . . . 8

2.3.1 Continous Shear Key Joints . . . . . . . . . . . . . . . . . . . 11

2.3.2 The Structural Parameters affecting the Behavior of the ShearKey Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

xiii

2.4 Different Designs of Shear Key Joints . . . . . . . . . . . . . . . . . . 13

2.4.1 Unreinforced Shear Key Joints . . . . . . . . . . . . . . . . . . 14

2.4.2 Reinforced Shear Key Joints . . . . . . . . . . . . . . . . . . . 15

3 Theoretical Background 21

3.1 Loads on Precast Elements . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.1 Horizontal Loads . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.2 Vertical Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Shear Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1 Stability of Shear Walls . . . . . . . . . . . . . . . . . . . . . 23

3.2.2 Stiffness and Capacity of Shear Walls . . . . . . . . . . . . . . 25

4 Finite Element Analysis 27

4.1 Finite element method theory . . . . . . . . . . . . . . . . . . . . . . 27

4.1.1 Plane Stress Theory . . . . . . . . . . . . . . . . . . . . . . . 28

4.1.2 3D Continuum elements . . . . . . . . . . . . . . . . . . . . . 29

4.2 Non-linear Finite Element Analysis . . . . . . . . . . . . . . . . . . . 30

4.2.1 Iteration Procedure . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2.2 Nonlinearity in Concrete . . . . . . . . . . . . . . . . . . . . . 32

5 Finite Element Analysis in ATENA 2D/3D 37

5.1 ATENA Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2 Material Behaviour Definitions . . . . . . . . . . . . . . . . . . . . . . 37

5.2.1 Concrete material model . . . . . . . . . . . . . . . . . . . . . 38

5.2.2 Reinforcement Material Model . . . . . . . . . . . . . . . . . . 43

5.2.3 Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.3 Boundary Conditions and Loading . . . . . . . . . . . . . . . . . . . 45

5.4 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.5 Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.6 Solvers for Non-linear Analysis . . . . . . . . . . . . . . . . . . . . . . 47

xiv

6 Model used in the analysis 49

6.1 Models in ATENA 2D . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.1.1 Boundary Conditions and Loading of the Wall Model . . . . . 50

6.1.2 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6.2 Models in ATENA 3D . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6.2.1 Geometry of the walls . . . . . . . . . . . . . . . . . . . . . . 52

6.2.2 Assumptions and Limitations when Modelling . . . . . . . . . 56

6.2.3 Boundary Conditions and Loading . . . . . . . . . . . . . . . 56

6.2.4 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7 Results from the analyses 65

7.1 Shear and normal stresses from the 2D wall models . . . . . . . . . . 65

7.2 Simulations in 3D models . . . . . . . . . . . . . . . . . . . . . . . . 68

7.2.1 Parametric Study of the Unreinforced Models . . . . . . . . . 68

7.3 The results of the hand calculations . . . . . . . . . . . . . . . . . . . 80

8 Discussion and Conclusions 83

8.1 Shear and Normal stresses in the wall models . . . . . . . . . . . . . 83

8.2 The stiffness of the joint . . . . . . . . . . . . . . . . . . . . . . . . . 83

8.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

8.4 Needs for further research . . . . . . . . . . . . . . . . . . . . . . . . 86

Bibliography 87

Appendix A - Numerical Results 91

Appendix B - Hand Calculations 173

xv

Chapter 1

Introduction

New solutions are introduced in the construction business all the time. It is achallenge to find solutions, that are both cost and time effective. One solution thatfulfils these requirements is prefabricated construction. As the usage of these typeof constructions are getting all more usual, it is important to clarify how the jointsaffect the global shear stiffness and capacity of the whole structure. For a suchinvestigation the non-linear finite element method, NLFEM, can be used.

1.1 Background

This thesis is not directly based on a real case but it is a further research of a thesisnamed "Modelling Lateral Stability of Prefabricated Concrete Structures" writtenby Lindwall and Wester (2016). The thesis is about an investigation of how theassumptions of joints influence the total stability of the structure due to the loaddistribution. During the investigation the non-linear behaviour of the materials wasnot considered, therefore further research was recommended.

To estimate a more reliable total stability of a structure, the connections betweenthe prefabricated elements need to be analysed. An analysis of joints between pre-fabricated shear walls was chosen as topic in this thesis. The shear walls are the partof the structure that resist the seismic and wind loads and transfer the lateral forcesto the foundation. Therefore it is of great interest to analyse the load transmissionin the connections between shear walls.

Different joints were modelled between shear wall elements in non-linear FEM, usingthe ATENA 3D software. Two different models of joints were analysed, the firstmodel illustrated two prefabricated solid walls connected with cast in situ concrete.The second model was similar as the first model with reinforcement included in theconnection of the prefabricated walls. The models were analysed with different shearand normal forces. The initial forces applied on the models were obtained from awall model that was investigated in ATENA 2D.

1

CHAPTER 1. INTRODUCTION

1.2 Problem Description

1.2.1 Joints between Shear Walls

Precast concrete elements have a fixed size and several elements have to be jointedin order to build a complete structure. To achieve a structural interaction betweenelements it is important that the forces can be transferred from an element to anotherelement. This requires a good design and correct usage of joints (Elliott, 2013).

The properties, as stiffness and strength, of joints are different compared to thewall itself. The distribution of stress in walls depends on the stiffness of the joints(Olesen, 1975).

1.3 Aim

The purpose of this thesis is to estimate the importance of the connections betweenprecast wall elements regarding the stability and load distribution. The investigationconcerns the stiffness and capacity of the connection depending on design of thejoint. To achieve a result consistent to the reality the model should have sufficientreliability with respect to the actual behaviour of the material.

Background theory was obtained with help of a literature study about concretebehaviour, precast wall elements and non-linear finite element analysis, NLFEA, inthe usage of concrete. Some of the theories regarding precast elements were receivedby interviews and a site visit. After the literature study a series of NLFE simulationswere done in FE software ATENA 3D.

1.4 Limitations

The chosen models of the joints were based on the Swedish praxis. The modelsconsisted of only one shear key and parts of wall elements. The symmetry of themodels have been utilized and only half of the models were used in the simulations.The reasons for the simplifications were to avoid overloading of the software andthe computer during the simulations. Another reason for the simplifications was thelimitation of the time.

In material definition, for the both wall elements and the joint, the same strengthclass for concrete has been used.

The connection was replaced with a non-linear interface as a simplification in the2D simulations, when the shear- and normal stresses were estimated.

The analysed models were only details compared to the whole structure, thereforeit was assumed that the body force did not have any important influence on the

2

1.5. STRUCTURE OF THESIS

results and it could be neglected. In the 2D simulations only the horizontal loadwas of interest because it is the force that gives rise to shear forces and momentsin the wall. The magnitude of the horizontal load was obtained from a real case,presented in the thesis by Lindwall and Wester (2016).

Only one type and size of shear key was tested, the chosen models of the joints werebased on the Swedish praxis. The models consisted of only one shear key and partsof wall elements. The symmetry of the models have been utilized and only half ofthe models were used in the simulations.

In the simulations of the reinforced joints, only "reinforced shear key joints withtransverse U-bars" was investigated. The contribution of the dowel actions wereneglected in the simulations. More demarcations and limitations will be presentedfurther in the text.

1.5 Structure of Thesis

This thesis contains 6 chapters and 2 appendices, each part represents differentphases of the work. A short description for each chapter is given below.

Chapter 2 contains information about precast concrete structures with shear wallsand the connection between them as main focus.

Chapter 3 describes different loads that act on a structure. It also presents thebackground about stiffness and capacity in the shear walls.

Chapter 4 contains information due to the simplifications and assumptions madewhen modelling in ATENA 3D.

Chapter 5 presents input data for the material used in the model.

Chapter 6 presents the results obtained from the simulations in the wall model, inATENA 2D, and the models of joints, in ATENA 3D.

Chapter 7 discusses the results and how the these are influenced of assumptionsand limitations. It also presents the conclusions and recommended subjects forfurther research.

APPENDIX A presents the tests of the two models of the joints.

APPENDIX B contains the hand calculation of the shear resistance at the inter-face between the joint and the wall.

3

Chapter 2

Shear Joints in Precast ConcreteWalls

2.1 General

In the last decades the increasing population has forced the construction industryto implement faster and cheaper solutions as precast constructions. The advantagesof precast constructions are that it is timesaving, cost effective and it gives a betterpossibility to control the quality (Divan and Madhkhan, 2011).

There are some disadvantages with precast elements due to the transportation, in-stallation and storage of them. The transportation requires extra circumspectionsbecause of the risk for the precast elements to be damaged. Cranes and other equip-ments are necessary for the installation of the elements. The storage of the precastelements becomes another aspect to consider due to the size of the elements (Sitevisit, 2017). Besides the above-mentioned disadvantages a high attention must begiven to the connection between the elements. The connections between the ele-ments are weakest part of a structure, due to the interaction between the elements(Saviotti et al., 2012).

In prefabricated concrete frames, vertical joints and a series of horizontal jointsare used to attach the different parts of the structure. The vertical joints provideconnection between the walls laterally while the horizontal joints connect the panelsto each other (Divan and Madhkhan, 2011).

The behaviour of the precast structures differs, from the cast in situ structures,because of the connections between the elements. To achieve a good structuralstability it is necessary to analyse the behaviour of the joints between the precastelements (Dhankot and Sanghvi, 2015). The challenge is to design a connection thatfulfils the same requirements as the cast in situ solutions regarding to the structuralperformance (Sørensen et al., 2017).

A connection has several functions in a structure; it provides stability, preventsmovements and transfers loads. The design of the connection has an essential affect

5

CHAPTER 2. SHEAR JOINTS IN PRECAST CONCRETE WALLS

on its function .The capacity and the stiffness of a joint between the elements affectthe structural stability of the whole structure (Dhankot and Sanghvi, 2015).

2.2 Precast Walls

Precast concrete walls are manufactured in different categories, as solid walls, thin-shell walls and sandwich walls (PCI, 2007). The precast walls can be erected andpanelized in both vertical and horizontal position. The wall panels are designed asload bearing and nonloadbearing. The loadbearing walls have to carry lateral loads,as well as roof and floor loads (PCI, 2017).

(a) (b)

Figure 2.1: (a) Prefabricated solid wall (Indicon Solutions, 2015) (b) Prefabricatedthin-shell wall (PCI, 2017).

Figure 2.2: Prefabricated sandwich wall (Strängbetong, 2017).

6

2.2. PRECAST WALLS

2.2.1 Shear Walls

The shear walls are designed to resist the seismic and wind loads. The walls supportoverhead spanning components and the lateral loads. The shear walls work as verti-cal cantilever beams. The lateral forces that act parallel to the walls are transferredto the foundation from the superstructure. The walls are usually positioned to resistthe lateral loads along each principal axis of the structure (PCI, 2007). The typicaldimensions of a simple element is defined in table 2.1.

Figure 2.3: Simple wall element (Betongelementföreningen, 1998).

Table 2.1: The dimensions of a simple element (Betongelementföreningen, 1998).

Height, b [m] Thickness, c [m] Length, a [m]≤ 2.4(4.20) 0.08− 0.12(0.05− 0.25) ≤ 6(8.60)

The dimensions shown in the table table 2.1 are for the most manufactured typesof simple elements. The dimensions defined in the parenthesis are for elements thatare manufactured in only some factory (Betongelementföreningen, 1998).

The precast elements are designed to cooperate with each other, regardless to thedifferent deformation behaviour of the elements. The relationship between shearand flexural stiffness of these elements should be organised before estimating thedistribution of the horizontal loads (Bachmann and Steinle, 2011).

A homogeneous structure is stiffer and has a higher load carrying capacity than aseries of shear walls connected. The stiffer the joints are, the more homogeneousbehaviour the series of shear walls will have. The horizontal load acting on a shearwall causes shear force and internal moment, which in turn, induce shear and normalstresses in the horizontal and the vertical intersections of the shear wall. The verticalnon-uniform loading is another reason for the appearances of the shear stresses inthe joints. The shear joints are required to transfer the loads between componentsthat interact (BCA, 2001).

In figure 2.4 (a), the horizontal load transfers to vertical and horizontal shear joint,assumed that the components are interacting. The distribution of the vertical load

7


differs depending on the slab span, as shown in the figure 2.4 (b), this causes shearforces in adjacent connections (BCA, 2001).

(a) (b)

Figure 2.4: Shear and normal stresses acting on shear walls (BCA, 2001).

In the shear walls the shear forces are highest in the lower storeys, where openingsare common. The openings decrease the stiffness of the shear walls (Bachmann andSteinle, 2011). The precast shear wall is usually assumed to be homogeneous untilthe limiting shear stress is reached in the joint, the horizontal load can be increaseduntil this point (BCA, 2001).

2.3 Vertical Joints between Shear Walls

The wall elements are precast and delivered completely finished from the factory andmounted with help of cranes. The sides of the shear walls are designed to connectwith each other, the joints are completed when walls are placed side to side. Thejoint between two wall elements is casted in situ by filling the opening between thewall elements adjacent to each other. The concrete is filled either by pouring it fromthe top or spraying it horizontally from the bottom up with a hose (Site visit, 2017).

The connection between wall elements can be designed in different methods.A common method is castellated joints, with or without reinforcement (BCA, 2001).

It is assumed that the rigidity of a shear wall is proportional to the magnitude ofthe force taken by the wall. The behaviour of the vertical joints has an influenceon the distribution of the internal forces and the rigidity of the shear wall. Undershear loading, if the deformability of the joints and the precast elements do not

8

2.3. VERTICAL JOINTS BETWEEN SHEAR WALLS

differ, the joints are defined as rigid. Then the total wall is assumed to behave as ahomogeneous cantilever (Cholewicki, 1971).

If the deformability of the joints and the precast elements differ significantly the totalwall is assumed to have a multi-cantilever behaviour. As long as the load bearingcapacity is higher than the shear forces in the joints the wall strips are collaboratingwith each other. The wall strips start working independently if the load bearingcapacity is exceeded. The total system becomes more deformable compared to ahomogeneous cantilever (Cholewicki, 1971).

The behaviour of the total wall structure depends mostly on the shape and on thebasic structural parameters of the joint. A few basic structural parameters of thejoint are location and ratio of the reinforcement, strength and homogeneity of theinfill concrete, dimensions and shape of the shear keys and resistance to shrinkagecracks (Cholewicki, 1971).

There are several types of joints with different shapes, dependent on the profile ofthe wall edges and geometrical dimensions, as shown in figure 2.5.

(a) (b)

Figure 2.5: (a) Different types of joints, (a)plane joint (b)grooved joint (c)shearkey joint (Cholewicki, 1975) (b)((a)local connection in a key joint(b)continious key joint)(Cholewicki, 1975).

The shear key joints are the most practical alternative because they are capable totransfer the shear forces in a mechanical way even if there is no bond between the castin situ concrete and the prefabricated concrete (Cholewicki, 1971). The behaviourof the shear key is dependent on the mechanical interlock and the development of adiagonal compressive strut across the shear key, as shown in figure 2.6 (Elliot, 2017).

9


Figure 2.6: Diagonal compressive strut across the shear key. Recreated from (Elliot,2017).

The shear forces between wall elements are transferred by different mechanical ac-tions, it can be through interface joint friction, bond, interlocking by shear key,welding or dowel action of transverse steel bars or rods (BCA, 2001).

Shear transfer can be performed by interface joint friction if the normal compressiveforce exist permanently. The normal compressive force develops from prestressing,permanent gravity loads or by reinforcement bars settled across the joints (BCA,2001).

The shear transmission by bond between precast wall element and the cast in situjoint requires a low shear stress.

The shear forces between the wall elements are transferred by shear keys in thejoints. The shear key is the part of the joint with castellated surface, it behaves asmechanical locks to avoid significant slip at the interface.

When the joins are designed with steel bars or rods the dowel action contributes totransmission of shear forces between elements (BCA, 2001).

There are two types of shear key joints, local- and continuous shear key joints.The local shear key joints have keys of greater dimensions. The continuous shearkey joints have keys of smaller dimensions distributed along the total heigh of thestructure(Cholewicki, 1971).

The failure modes can appear differently in the joints, depended on the profile ofthe keys, compressive strength of the cast in situ concrete, bonding strength andadhesion of the contact surfaces.

In the figure 2.7 different failure modes are illustrated, given from left to right:diagonal tension cracks, shearing and crushing off of corner of shear key, verticalshearing cracks in the shear key, dislocation or slippage at the contact surface (BCA,2001).

10

2.3. VERTICAL JOINTS BETWEEN SHEAR WALLS

Figure 2.7: Different failure modes of shear key (BCA, 2001).

2.3.1 Continous Shear Key Joints

The continuous shear key joints have a working scheme as a band, which is exposedto the shear and normal forces. The forces are caused by factors as seismic forces andlateral wind loading, non-uniformly vertical loads on the wall, non-uniformly settle-ments of the structure, and rheological and thermal deformations of the structure(Cholewicki, 1971).

It is important to consider the wind pressure when designing the joints in multi-storey buildings due to the shear forces. The shear force influence the joints in twodifferent ways which are important in the estimation of deformability, load bearingcapacity and working conditions of the total wall. One way is when the bond ensuresthe structural homogeneity of the shear wall instead of the joint up to the failure ofthe joint, as long as the ultimate state of the shear wall is established by the bearingcapacity of the joint. Another, if there is no bond at or it is damaged by small shearforces (Cholewicki, 1971).

In figure 2.8 the behaviour of the joints are illustrated. Two phases are shown, onebefore and one after splitting (Cholewicki, 1971).

Figure 2.8: Working phases of a key joint (Cholewicki, 1975).

11


1. Phase I: This phase shows the behaviour of the joint when moment gives rise tosplitting along one edge of the wall.

2. Phase II: This phase shows the behaviour of the joint after splitting in two stages,before and after the appearance of the diagonal cracks.

The diagonal cracks correspond to the direction of longitudinal cracks or principalstresses caused by longitudinal shear stresses, in both phases. The splitting of thekey joint does not mean that it is a failure. The shear forces can be handled bytransversal reinforcement in the joint (Cholewicki, 1971).

2.3.2 The Structural Parameters affecting the Behavior ofthe Shear Key Joints

Shape of Shear Key Joints

The parameters that describe the form of the shear key joints, as illustrated in figure2.9, are key depth (b), gap width in the joint, inclination angel (α) of contact planesof the shear key, the ratio between the key height (hd) in the joint and the key height(hdp) in the wall (Cholewicki,1971).

Figure 2.9: Cracking pattern of a key joint. (a)longitidunal section. (b)Cross section.(Cholewicki,1971).

Reinforcement in the Shear Key Joints

The reinforcement in the joint affect the behaviour of the joint differently in thephases mentioned before (Cholewicki, 1971).

1. Phase I: In this phase, the reinforcement stresses are small independently fromthe joint shape. The increase of the splitting forces and the limitations of theincremental micro cracks along the contact surfaces in the joint, are the factors thatinfluence the behaviour of the joint by the reinforcement.

2. Phase II: The contribution of the reinforcement is dependent on the shape ofthe joint. The cross section area of the reinforcement, the anchorage and the ar-rangement are the affecting factors on the behaviour of the joint. The horizontal

12

2.4. DIFFERENT DESIGNS OF SHEAR KEY JOINTS

components of the shear forces are transferred by the transverse reinforcement inthe shear key joints after splitting. The contribution of the reinforcement in thetransfer of the shear forces is not easy to determine and it can be decreased whenthe anchorages crush or if the forces increase because of local failure of concretesurrounding the reinforcement.

Another type of reinforcement in the joint is longitudinal reinforcement. The effectof this type of reinforcement is dependent on the cooperation in the transmittingof shear forces, resistance to horizontal cracks because of the shrinkage of concrete,strengthening of transverse reinforcement by designing in shape of loops and dis-tributed along the joint (Cholewicki, 1971).

Load Bearing Capacity of Shear Key Joints

Previous studies on the load bearing capacity of shear key joints performed byCholewicki (1971), who analysed the effect of the structural parameters in joints,concluded that the load bearing capacity of the joints should be estimated by phaseII, after splitting. A common failure in the joints is slipping of contact surfaces dueto unreinforced or under-reinforced joints.

Failure in joints may also occur due to shearing of key ends shaped by concrete filling,projecting from precast walls or diagonal trajectorial cracks (Cholewicki, 1971).

2.4 Different Designs of Shear Key Joints

There are both unreinforced and reinforced joints, the requirements of the loadbearing capacity of a joint vary depending on if the walls that will be connected areload bearing or not. The choice of shape and material parameters, as stiffness andamount of reinforcement in the joint, depend on the size and usage of the precastwall element.

According to the Swedish praxis, the most common strength class of the concretefor the joint is C30/37 with the maximum aggregate size of 8 mm (Strängbetong,2016).

13


2.4.1 Unreinforced Shear Key Joints

In smaller housing projects, there is usually no need of reinforcement in the joint.The joint consists only of cast in situ concrete (Site visit, 2017). The dimensions ofa standard joint according to Swedish Praxis is illustrated in figure 2.10. This typewas used in the NLFEA, in this thesis.

(a) (b)

Figure 2.10: (a) Elevation on vertical shear joint (Strängbetong, 2016) (b) Sectionof vertical shear key joint (Strängbetong, 2016).

14


2.4.2 Reinforced Shear Key Joints

Reinforcement in the joints is used to strengthen the connection between precast wallelements. Different types of reinforced joints are used depending on the requirementsof the load bearing capacity in the structures.

The transmission of the shear forces can be performed by dowel actions in thereinforced joints, as illustrated in figure 2.11. At the interface of the joint, thedowel is supported by contact stresses in the concrete and loaded by shear, whichcauses significant bending deformation in the dowel. In the ultimate state, localcrushing occur at the contact area and a plastic hinge shapes at the dowel. Eccentricshear loading decreases the capacity by dowel actions. Splitting reinforcement isneeded around the dowels when they are located close to the corner or edge of thecomponent. If the dowels are anchored by end anchorages or by bond it is possibleto obtain a combination of dowel action and shear friction (BCA, 2001).

Figure 2.11: Transmission of shear forces by dowel actions (BCA, 2001).

Reinforced Shear Key Joints according to Swedish Praxis

In figure 2.12 a common type of reinforced joint is illustrated, consisting two paralleltransverse U-bars protruding from the precast wall elements into the shear key. Twostraight reinforcement bars are placed in each U-bar to prevent sliding. A stirrupis enclosing the straight bars to keep them together and to prevent buckling of thestraight reinforcement bars due to shear loading. This type of reinforced shear keywas used in this thesis (Ekroth, 2017).

15


U-barStraight bar

Stirrup

Figure 2.12: Section of reinforced shear key joint (Ekroth, 2017).

In figure 2.13 another common type of reinforced joint is illustrated, containing threetransverse reinforcement bars connected by steel plates placed outside of the shearkey of the joint (Strängbetong, 2016).

Figure 2.13: Reinforced joints anchoraged with steel plates (Strängbetong, 2016).

Transversally Reinforced Shear Key Joints with Overlapping U-bars

Since the 1960s, transversally reinforced shear key joints with overlapping U-barshave been used, as illustrated in figure 2.14, with the purpose of establishing struc-tural continuity between precast wall elements. The ductility and strength of these

16


cast in situ connections are less than in the precast wall elements which leads todifficulties to obtain full structural continuity. This design requires additional workduring the assembling of precast wall elements because of the risk for rebar-clashingof the overlapping U-bars.

The problem can be avoided by bending up the U-bars protruding from the precastwalls before the installation of the wall elements. The bars can straightened afterthat the wall elements are placed.

This procedure limits the cross-section area of the bars, which in turn, limits thestrength of the joint. This connection is not suitable for tall buildings where ahigher load bearing capacity is required, due to an increase of horizontal loads. Abar diameter of 6-8 mm is commonly used for this type of connection (Sørensen etal., 2017).

Figure 2.14: (a) Conventional design of shear key joint. (b) Assembling process ofprecast wall elements. (Sørensen et al., 2017).

17


Figure 2.15: Connection of precast walls in a four-storey building (Herfelt et al.,2016).

Figure 2.16: Elevation and section through reinforced shear key joint (Herfelt et al.,2016).

Herfelt et al.(2016) analysed a design of shear key joint which is transversally rein-forced with overlapping U-bars, as shown in figures 2.15 and 2.16. In the previousinvestigations made on these type of joints the discontinuity of the overlapping U-bars was not considered which affected the ultimate capacity of the joints.

The analyses were implemented by several different test specimens, based on thelower bond theorem of plastic theory, to estimate the impact of the discontinuousreinforcement with help of finite element analysis. According to this theorem thematerial has a rigid-plastic behaviour and there is no occurrence of deformationsbefore yielding.

The investigation by Herfelt et al.(2016) showed that the U-bar shape and the dis-tance between adjacent overlapping U-bars have an important role for the ultimateshear capacity. The shear capacity estimated from the model was better than whatEurocode 2 design equations showed (Herfelt et al., 2016).

18


Transversally Reinforced Shear Key Joints with Rotated Parallel U-bars

Besides achieving an integrated structural system when connecting the precast el-ements there are other important factors, as time and cost efficiency, to consider.The challenge of designing connections is to achieve ductility and strength that cor-respond to cast in situ structures and have solutions that are easy to construct atthe same time (Sørensen et al., 2017).

Sørensen et al. (2017) have investigated a new model of reinforced key joints whichis construction-friendly and with a structural performance comparable to the cast insitu solutions. This design consists of transversal U-bars that have a plane parallelto the plane of the wall elements, with a double T-headed transverse bar placedinside each U-bar, as shown in figure 2.17. With this orientation of U-bars, thereis no risk of rebar-clashing. It is not necessary to post-straighten and pre-bend theU-bars, which gives possibility to use U-bars with diameter larger than 8 mm indesign (Sørensen et al., 2017).

Shear loading gives rise to diagonal compression struts between the shear key inter-faces. This compression can be handled by tension between the overlapping U-bars,which is transferred by the T-headed bar. The double T-headed bar is preferredinstead of straight reinforcement, to achieve increased anchorage of the short lacerreinforcement (Sørensen et al., 2017).

Sørensen et al. conducted an experimental program in order to analyse the structuralperformance of this design. The test results showed that this design has moreductile load-displacement response and higher load bearing capacity, compared tothe conventional design (Sørensen et al., 2017).

Figure 2.17: (a) The new design of shear key joint, with rotated U-bars (b) Assem-bling process of precast wall elements (Sørensen et al., 2017).

19

Chapter 3

Theoretical Background

3.1 Loads on Precast Elements

For the structural stability of a construction permanent and variable actions areconsidered. Permanent actions are actions as weight from the permanent mountedequipments and self-weight of the construction. Actions from snow, wind, accidentsand imposed loads on floors are counted as variable actions (EN 1991, 2005).

3.1.1 Horizontal Loads

Shear walls are vertical components that resist the horizontal actions exposed to thestructure. The wind actions are the horizontal actions that have the most impacton the overturning effects of the structure. The seismic actions are considered assignificant horizontal actions in many countries with high-seismic regions (Lorentsenet al., 2000).

According to Eurocode the influence of the actions from wind on the structuredepends on the dynamic properties, size and shape of the structure (EN 1991, 2005).

The wind actions vary with time and act directly as pressures on the exterior sur-faces, of enclosed structures. The pressure act indirectly on the interior surfacesbecause of the leaks of the exterior surfaces. In the open structures the wind actionscan also act directly on the interior surfaces. The wind actions act normal to sur-faces or to particular facing components of the structures. Tangential friction forceshave to be considered if the wind sweeps through bigger areas. The wind actionsare defined as variable fixed actions, if nothing else is specified (EN 1991, 2005).

The velocity of the wind pressure varies depending on the total height of the struc-ture and the distance from the ground to the part of the structure that is exposedto wind action (Isaksson et al., 2005).

The difference between the pressures on opposed surfaces, with signs considered, isdefined as net pressure. The wind actions give positive pressure, when acting di-

21

CHAPTER 3. THEORETICAL BACKGROUND

rected towards the surface, and negative suction, when acting away from the surface,as illustrated in figure 3.1 (EN 1991, 2005).

Figure 3.1: Positive and negative pressure of a structure. (EN 1991, 2005).

The impact of wind friction on a surface can be neglected if the total area of allsurfaces parallel with the wind is equivalent to or less than 4 times the total area ofall exterior surfaces normal to the wind (EN 1991, 2005).

3.1.2 Vertical Loads

The vertical actions on a structure consist of snow loads, imposed loads and deadloads (EN 1991, 2005). The vertical actions are summarized for each storey andtransferred by the bearing components of the structure (Lorentsen et al., 2000).

3.2 Shear Walls

In the frameworks the walls are used to stabilise the structure horizontally (Isaks-son et al., 2010). If the horizontal load, from the eccentricity and wind, is evenlydistributed the shear walls should be positioned in a way to minimize the rotationof the structure. The positions of the shear walls should be on at least three axesand in at least two non-parallel directions (Bachmann and Steinle, 2011).

The wind pressures received by the wall are transferred to floor and roof diaphragms.The loads are in turn transferred to the parts in the lateral-force-resisting systemwhich consist of shear walls that are parallel to the wind actions. The loads are thentransferred to the foundation through the shear walls (Ghosh and Fanella, 2003).

The factors that influence the distributed lateral loads on each shear wall separatelyare divided in four, that are described below:

- Supporting footings and soil- Stiffness of the roof and floor diaphragms- Relative shear and flexural stiffness of the connections and shear walls- Eccentricity of the lateral loads to the midspan of rigidity of the walls.

The first mentioned factor can generally be neglected in the shear force distribution(PCI, 2007).

22

3.2. SHEAR WALLS

The weakest parts of a wall are joints. To reduce the movements in joints it is notpreferred to reduce the element size, a better solution is to have a lower amounts ofjoints. The fewer joints the more economic it will be (Bachmann and Steinle, 2011).

3.2.1 Stability of Shear Walls

According to EN 1992 (2004), the shear stress at the interface between concrete castat different times must fulfil following condition:

VEdi ≤ VRdi (3.1)

The design value, VEdi is the shear stress in the interface which is defined as:

VEdi = βVEd/(zbi) (3.2)

where:

β, is the ratio of the longitudinal force in the new concrete area and the totallongitudinal force either in the compression or tension zone, both calculated for thesection considered.

VEd, is the transverse shear force.

z, is the lever arm of composite section.

bi, is the interface with.

The design shear resistance is defined as VRdi for the interface. The shear resistanceis given by:

VRdi = cfctd + µσn + ρfyd(µ sinα + cosα) ≤ 0.5vfcd (3.3)

where:

c, is the cohesion factor and µ is the friction coefficient. Both parameters are de-pendent on the roughness of the interface.

fctd, is the design tensile strength of the concrete. It is designed as fctd = αctfctk,0.05/γC .Where the γC , is the partial safety factor for concrete and the αct, is a coefficienttaking account for long term effects on the tensile strength and of unfavourableeffects, resulting from the way the load is applied.

σn, is stress per unit area caused by the minimum external normal force across theinterface that can act simultaneously with the shear force, positive for compression,such that σn < 0.6fcd and negative for tension. When σn is tensile cfctd should betaken as 0.

ρ, is the ratio between reinforcement area crossing the interface and the area of thejoint, defined as As

Ai.

23


α, is the angle of the crossing reinforcement in the surface, as shown in figure 3.3.It is limited to 45◦ ≤ α ≤ 90◦.

v, is the strength reduction factor.

Figure 3.2: Different interfaces. (EN 1992, 2004).

The surfaces are categorized as smooth, very smooth, indented and rough. Smoothsurfaces are extruded, slipformed or free from treatment after the vibration. Forthis type of surface c = 0, 20 and µ = 0, 6 are used.

In a very smooth surface, where the concrete is cast against plastic, steel or specialwooden moulds, c = 0, 025 to c = 0, 10 and µ = 0, 5 are used.

A indented surface is referred to a surface similar to the figure 3.3, where c = 0, 50and µ = 0, 9 are used.

A rough surface consist of at least 3 mm roughness around 40 mm space, the be-haviour is achieved by exposing of aggregate, raking or other method. For this typeof surface c = 0, 40 and µ = 0, 7 are used (EN 1992, 2004).

Figure 3.3: Design of shear key joint (EN 1992, 2004).

24

3.2. SHEAR WALLS

3.2.2 Stiffness and Capacity of Shear Walls

Olesen (1975) has investigated the behaviour of vertical shear key joints in precastwall elements. The strength and stiffness properties of the joints were estimatedby several test specimens. The test results showed that there is no linear relationbetween the displacement and the load in the joint due to the stiffness. The relationwas non-linear even before cracking.

The results from the test specimens were summarized in a stress-displacement curvefor vertical shear key joints, ash shown in figure 3.4. According to the results thestiffness was obtained as 2/3 of the maximum load (Olesen, 1975).

Figure 3.4: Stress-displacement relationship of vertical joints(Olesen, 1975).

The stiffness obtained from the test specimens showed a large variation of the stiff-ness between 2 − 30 GN/m3. The stiffness was calculated with following formula(Olesen, 1975):

k = v/δ (3.4)

where:

v is the shear force carried by jointδ is the displacement due to the force.

Bjuger (1976) has investigated the deformability of vertical shear joints in precaststructures with help of experiments. According to the investigation the Young´smodulus, of the interface area of the key and the concrete, could be used to obtainthe elastic stiffness of a shear key. For the elastic stiffness following formula wasused (Bjuger, 1976):

k = 1/λ = Eb · Fk/50[kg/cm] (3.5)

25


where:

λ is the deformability characteristic of a joint, which is a variable parameter.Eb is the smaller Young´s modulus of the in situ concrete or the precast wall[kg/cm2].Fk is the area of the interface of the key joint [cm2] (Bjuger, 1976).

26

Chapter 4

Finite Element Analysis

4.1 Finite element method theory

The finite element analysis (FEA) is a numerical method used to obtain approximatesolutions of field problems. FEA describes a field problem by an integral expressionor a differential equations. The finite element method is applicable to magneticfields, fluid mechanics, heat transfer, stress analysis and many other field problems(Cook et al. 2002).

In FEA, a field quantity is approximately interpolated piecewise. In modelling, thestructure is simplified stepwise and idealised into matematical model before it isdivided into small pieces, elements. The points that connect the elements are callednodes. The nodes can be subjects to possible displacement, translation and rotation,which are degrees of freedom, DOF of the structure (Malm, 2016). The elementsare arranged in a particular way defined as mesh (Cook et al. 2002).

Figure 4.1: Different steps in a FE analysis of a structure (Cook et al. 2002).

The input that have to be defined in structural engineering analysis are geometry,material properties and loads and boundary conditions. The material properties canbe defined as linear and/or non-linear.

27

CHAPTER 4. FINITE ELEMENT ANALYSIS

The general procedure of calculation with FEM is (Andersson, 2016):

1. Formulating the local stiffness matrix [k] for each element.

2. Connecting the elements to each other by transforming and assembling aglobal stiffness matrix [K].

3. Determining the global load vector {R}.

4. Reducing the system by applying boundary conditions.

5. Solving the equation [K] {D}={R} in order to determine the nodal values.

6. Calculating stresses or strain.

Different element types can be chosen for the field quantity. The element types aredepending on geometry and behaviour of the structure. Common element types are(Pacoste, 2016):

- Bar elements- Beam elements- 2D continuum elements (plane stress, plain strains)- 3D continuum elements- Axisymmetric elements- Plate shell & elements

Determining the global stiffness matrix for an engineering component or structurebecomes more complicated with increased amount of elements and nodes. Thematrix can be derived by hand calculations for simple systems but for more com-plex engineering problems it is preferred to use FEM-programs to simulations andsolutions. The FEM-programs turn the geometric models into mathematical mod-els with approximated behaviour, appropriate boundary conditions and differentialequations.

4.1.1 Plane Stress Theory

The plane stress theory is used in the case where stress vector in the z-direction canbe assumed as zero. This case is common for thin plates with smaller dimension inone direction compared to the other two, where the load acts perpendicular to thesmall direction. The stress-strain relationship is derived as (Cook et al., 2002):σxσyτxy

=

DXXXX DXXY Y 0DXXY Y DY Y Y Y 0

0 0 DXYXY

εxεyγxy

⇒ {σ} = [D] · {ε}

where:

[D] is the constitutive matrix{σ} is the stress vector{ε} is the strain vector.

28

4.1. FINITE ELEMENT METHOD THEORY

The stiffness matrix for one element is determined as (Andersson, 2016):

[k] =

∫[B]T [D][B]dv (4.1)

where:

[B] is the strain-displacement matrix, that is derived of the shape function.

4.1.2 3D Continuum elements

3D rectangular solid elements, brick elements, are extended rectangular plane ele-ments. The displacement in the x-direction for a eight-node solid is defined as:

u = a1 + a2x+ a3y + a4z + a5xy + a6yz + a7zx+ a8xyz (4.2)

The displacements in the y-direction, v, and z-direction, w, are defined in similarexpression. The displacement field for the element, {u} = [N]{d}, which also isexpressed as:

uvw

=

K1 0 0 K2 0 0 K3 0 0 ...0 K1 0 0 K2 0 0 K3 0 ...0 0 K1 0 0 K2 0 0 K3 ...

u1v1w1

u2...w8

The individual shape function is derived as:

Ni =(a± x)(b± y)(c± z)

8abc(4.3)

The shape functions of the element consist of product of three linear functionstherefore it can be called "trilinear". The eight-node solid has corner nodes and has24 degrees of freedom. As it is shown in figure 4.2 the twenty-node solid has bothcorner nodes and midside nodes. The twenty-node solid has 60 degrees of freedom(Cook et al. 2002).

29


Figure 4.2: (a) Eight-node solid element(b) twenty-node solid element (Cook et al. 2002).

4.2 Non-linear Finite Element Analysis

FE-models with linear behaviour give satisfactory approximations for many fieldproblems. It is common that the behaviour of a structure is non-linear. Non-linear behaviour aims on a variety of a phenomena that are complicated to define,where several phenomena can interact with each other. Some examples of non-linear behaviour in structural mechanics are cracking, material creep or yielding,local buckling and opening or closing of gaps (Cook et al., 2002).

Non-linearities in structural mechanics are:

- Material non-linearity, the material properties are functions of stress or strainwhere the relation of the stress and strain is non-linear. The crack growing inconcrete causes non-linearity in the material. The concrete experience cracking orsome sort of strain hardening when the elastic limit of the material is exceeded whichdecreases the stiffness of the concrete.

- Contact non-linearity, changes of the contact force can imply changes of thecontact area between adjacent parts, which give rise to sliding contact with frictionalforces or opening or closing of gaps.

- Geometric non-linearity, the longitudinal forces and bending moment exposedto the structure give rise to deformation which in turn influence the internal forces.The influencing eccentricities and bending moments are second order effects thatshould be considered when using the equilibrium equation regarding to the size ofthe deformations in the structure.

In non-linear problems the stiffness and perhaps the loads can be a function ofdeformation or displacement, as defined in the equation below (Cook et al., 2002):

[K]{D} = {R} (4.4)

30

4.2. NON-LINEAR FINITE ELEMENT ANALYSIS

where:

[K], is the coefficient matrix.{D}, is the deformation.{R}, is the load vector.

An iterative procedure is required to solve the equation for {D}. For FE-modelswith linear behaviour it is assumed that the deformation increases proportionalto the load. The linearity gives the possibility to apply the loads FE-model in asing step, which is called increment. This is not appropriate for non-linear cases,because of the independent variation of the size of the load and the deformation.The application of the load should be divided into small increments and for eachincrement the iterative procedure needs to be repeated to find {D} (Cook et al.,2002).

In general, all finite element softwares have steps as pre-processing, analysis andpost-processing. In the stage of pre-processing the material properties, geometry,loads, contact conditions, boundary conditions and mesh are defined. After definingall input data the defined model is submitted to analyse solver. The equation systemis solved for several time steps defined in the solver.

First the displacements are calculated for each node, based on this the reactionforces are calculated for the nodes with boundary conditions. The internal data, asstrains and stresses, of elements can be calculated based on the nodal data. Thesolved equilibrium equation, gives displacements, external forces, strains and stresses(Malm, 2016).

4.2.1 Iteration Procedure

In a non-linear analysis the equations can not be solved directly. The reason is thatthe deformation and the size of the load are not proportional, instead an iterativeprocess is used to obtain the solution. The desired load is reached gradually bydividing the load into increments (Malm, 2016).

The determination of the non-linear response of a structure subjected to the incre-ment of the load, ∆P , starts with the tangential stiffness K0. With help of thestiffness a displacement correction, ca, is obtained by an extrapolation. The dis-placement correction updates the displacement, from u0 to ua. The internal forces,Ia, are calculated by using the displacement, ua. The difference between the totalapplied load, P , and the internal forces, Ia, gives the force residual for the itera-tion, Ra. The aim of the iteration is to update the displacement until the tolerancecriterion, that has been defined beforehand, is reached by the force residual. Thestructure is in equilibrium when the criterion is fulfilled, the next load incrementwill then be proceeded in the iterative procedure (Malm, 2016).

31


Figure 4.3: Iteration of an increment in a NLFEA (Malm, 2016).

4.2.2 Nonlinearity in Concrete

It is necessary to define the non-linear material behaviour of concrete in more com-plicated designs to investigate the phenomena as the impact of cracks or the failuremode. A safety factor of a specific failure mode can be obtained with this approach.

In the FEM, the non-linear material behaviour of concrete can be defined in differ-ent theories as fracture mechanics, plasticity theory and/or damage theory (Malm,2016).

Fracture Mechanics

The non-linear behaviour of crack openings in concrete is usually described in frac-ture mechanics. There are three different failure modes that can occur in concreteas illustrated in figure 4.4. The failure modes are following (Malm, 2016):

- Mode I, failure mode in tensile.- Mode II, failure mode in shear.- Mode III, failure mode in tear.

Figure 4.4: Different failure modes that can occur in concrete (Malm, 2016).

The only failure mode that can occur in its pure form is mode I. Even mode II

32


initiates as mode I (Malm, 2016).

In figure 4.5 below, the stress distribution is illustrated near the crack tip for failuremode in tensile. A propagated macro element, with the length a0, is shown in thefigure. The macro and micro crack width are defined as wc and w, respectively. Thezone of the fracture process is presented as lp, where the macro cracks are developedfrom the micro cracks. The zone with macro and micro cracks together is definedas the crack length. The stress, σ(w), at the transition is equal to zero between themacro and micro crack, the macro crack is free from stress. As shown in the figure,the stress at the crack tip is equal to tensile strength, ft, and it increases in thefracture process zone (Malm, 2016).

Figure 4.5: Stress distribution for failure mode in tensile crack (Malm, 2016).

The fracture energy, Gf , is a material property that influences the the crack openingdisplacement. It is the amount of energy required for a crack, free from stress, toproceed. All energy that can not be restored regarding to elastic unloading is thefracture energy equal to the area under the stress displacement curve, as shown infigure 4.6 (Malm, 2016).

Figure 4.6: The process of crack propagation in concrete at uniaxial tensile loading(Malm, 2016).

Before the peak tensile strength point is reached the micro crack will not propagatefurther if the load is maintained. The propagation will become unstable when thepeak tensile strength point is passed. This means that the crack will propagateat loads lower than the maximum load. The part of the curve that is descending,following the peak tensile strength point, is presented as tensile softening. Thetensile softening describes the law of the crack opening, in propagation of the microcrack to the merge into a macro crack free from stress (Malm, 2016).

33


The fracture energy for normal weight concrete depends on the age of concrete,the maximum aggregate size and the water/cement ratio. The fracture energy isalso influenced by aggregate content and aggregate type for high strength concrete.The aggregate content and type have a more effect on the fracture energy than theaggregate size which is induced by the transition to the trans-aggregate fracturefrom interfacial fracture (MC10, 2012).

According to ModelCode 2010, MC10, the fracture energy for normal weight concreteis estimated as (MC10, 2012):

GF = 73 · f 0.18cm (4.5)

where:

fcm, is mean value of the compressive strength.

According to ModelCode 1990, MC90, the aggregate size is included in the estima-tion of the fracture energy, GF . It is calculated with the equation below (MC90,1993):

GF = GF0 · (fcmfcm0

)0,7 (4.6)

where:

fcm0 , is the reference value for the concrete compressive strength and it is equal to10MPa.GF0, is the base value of fracture energy and it varies depending on the maximumsize, dmax, of aggregate.

Table 4.1: Base values of fracture energy for different aggregate sizes.

dmax (mm) GF0 (Nmm/mm2)8 2.50·10−2

16 3.00·10−2

32 3.80·10−2

Smeared Crack

The cracking of concrete can be modelled in two fundamental approaches, discretecrack and smeared crack. The discrete crack approach is used for the elements whichare defined with linear elastic material behaviour. The crack openings are definedby an interface between the elements.

Both the elastic behaviour of the uncracked concrete and the behaviour of the crackare described for an element in the use of smeared crack approach. In this methodthere is no physical cracks, the actual crack is distributed, smeared, over the entireelement and the cracks appeared in the integrations points. The total strain in an

34


element contains a non-linear part, from the crack opening, and an elastic part, fromthe uncracked concrete. The total strain is estimated as (Malm, 2016):

εtotal = εelastic + εcrack (4.7)

where:

εelastic, is the elastic strain from the uncracked concrete.εcrack, is the non-linear fracturing strain from the crack opening.

In smeared crack method, there are two models defined, fixed crack model androtated crack model. In the both mentioned models the cracks start in the directionof the maximum principal, when the maximum principal stress is equal to the tensilestrength (Malm, 2016).

Figure 4.7: Fixed and rotated crack model respectively (Malm, 2016).

The crack, in the fixed model, is remained in the same direction regardless to thechanges in the strain and stress due to subsequent loading. The crack direction isdifferent from the direction of the principal stress, therefore shear stresses may occurat the crack surface (Malm, 2016).

The crack direction, in the rotated model, will rotate due to the subsequent loading,in a direction that coincides with the direction of the stess/strain direction. In therotated crack model the crack surface is free from shear stresses (Malm, 2016).

35


36

Chapter 5

Finite Element Analysis in ATENA2D/3D

5.1 ATENA Software

ATENA is a software for finite element analyses of both reinforced and unreinforcedconcrete structures. With this software it is possible to simulate the real behavior ofconcrete structures containing reinforcement yield, crushing and concrete cracking.The reasons behind choosing this software are that the reinforcement design in thecritical sections of the structure can be confirmed and internal force redistributionwith respect to cracking can be considered. By the possibility to consider these, theoverestimation of the amount of the reinforcement can be avoided (Červenka, 2017).

A realistic visualization of cracking pattern and cracks can be obtained in all differentstages of NLFEA. Both interior and surface cracks can be visualized and thereis an opportunity of filtering the visible cracks to get more realistic visualization(Červenka, 2017).

A series of NLFEA were performed, in ATENA 3D Engineering v.5 and ATENAStudio v.5 softwares, to estimate the effects of different parameters on the shearstiffness and capacity in joints between precast wall elements. The pre-processing ofthe analysis was performed in ATENA 3D Engineering and the analysis execution,post-processing and result extraction were performed in ATENA Studio.

A FEA was also performed in ATENA 2D in order to estimate the external loadingthat the joint is exposed to.

5.2 Material Behaviour Definitions

In Atena 3D there are three methods that can be chosen when defining the materialsand their properties. The methods are select from the catalogue, load from a fileand direct definition. In this thesis the first mentioned method was used but some

37

CHAPTER 5. FINITE ELEMENT ANALYSIS IN ATENA 2D/3D

properties for the selected material was changed (Červenka and Červenka, 2017).

5.2.1 Concrete material model

The fracture plastic constitutive material model,CC3DNonLinCementitious2, wasused to consider the non-linear behaviour of the concrete. It gives a material be-haviour according to the uniaxal stress-strain law, as illustrated in figure 5.1 (Čer-venka, et al., 2016).

Figure 5.1: Uniaxail stress-strain law for concrete (Červenka, et al., 2016).

Biaxial Stress Failure Criterion

The bi-axial stress failure criterion is utilized in the description of the bi-axial be-haviour (Kupfer et al., 1969). The material yields when the boundary of the bi-axialenvelope is reached by the state of the stress. The strength of the material increaseswith the bi-axial compressive stresses because of the confinement effect,as illustratedin figure 5.2. The effect can increase the strength up to 16% compared to the uniaxialcompressive strength (Malm, 2016).

When the concrete is subjected to compressive stress in one direction and tensilestress in the other direction the both compressive and tensile strengths are decreased(Malm, 2016).

38

5.2. MATERIAL BEHAVIOUR DEFINITIONS

Figure 5.2: Failure envelope at biaxial stress states for concrete (Červenka, et al.,2016).

Triaxial Stress Failure Criterion

In triaxial state of stress two different models, Rankine failure criterion and Menétrey-William failure criterion, are employed for the different behaviours of concrete. TheRankine failure criterion is an exponential softening model used to describe crack-ing, as fixed or rotated crack model. The Menétrey-William failure criterion is ahardening/softening plasticity model used to described the crushing of concrete.

In the plasticity model the predictor-corrector function, as illustrated in figure 5.3,is used to calculate the state of stress which is defined as (Červenka, et al., 2016):

Figure 5.3: Algorithm of plastic predictor-corrector (Červenka, et al., 2016).

39


σij = σtij − σpij (5.1)

where:

σtij, is the predictor-corrector.σpij, is the plastic corrector.

The plastic corrector is derived from the return mapping algorithm. The returnmapping algorithm returns the predictor stress state to the failure envelope, inreturn direction which is defined as (Červenka, et al., 2016):

Gp(σij) = β · ξ + ρ (5.2)

where: β, is the factor that determines the return direction.ξ, is the hydrostatic stress constant.ρ, is the deviatoric stress constant.

The factor β depends on the behaviour of the material during crushing. When thefactor is less than zero the material is compacted during the crushing. If the factorβ is equal to zero the material volume is preserved and for β higher than zero thematerial is dilated (Červenka, et al., 2016).

Concrete Cracking Models

The fracture plastic constitutive material model, CC3DNonLinCementitious2, isbased on crack band model and crack orthotropic smeared crack formulation. Inthe orthotropic smeared crack there are fixed crack model and rotated crack model,which are described in Chapter 4.2.2.

The crack opening function was derived by Hordijk (1991) experimentally describedas:

σ

ft=

{1 + (c1 ·

w

wc)3

}exp

(− c2 ·

w

wc

)− w

wc(1 + c31) exp(−c2) (5.3)

where:

w, is the crack opening.wc, is the critical crack opening free from stress.σ, is the normal stress in the crack opening.c1, is a constant with the value 3.c2, is a constant with the value 6.93.ft, is the tensile strength of the concrete.

The critical crack opening, wc, is estimated as:

40


wc = 5.14Gf

ft(5.4)

where:

Gf , is the fracture energy required for a crack, free from stress.

The fracture plastic constitutive material model, CC3DNonLinCementitious2, cal-culates the crack opening w from the sum of the current increment of fracturingstrain and the total fracturing strain, multiplied with the characteristic length. Thecharacteristic length, crack band width, which presents the element length where thecrack is distributed, was introduced by Bažant and Oh. The crack band width andthe curve derived by crack opening function, Eq. 5.3, which is illustrated in figure5.4 (Červenka, et al., 2016).

Figure 5.4: The crack band width and the exponential tensile softening (Červenka,et al., 2016).

Shear Retention Factor

The shear retention factor, crack shear stiffness factor, is the ratio between thereduced shear modulus, G, and initial concrete shear modulus, Gc, which is definesas:

rg =G

Gc

(5.5)

An increase of strain normal to the crack reduces the shear modulus which in turndecreases the shear stiffness as illustrated in figure 5.5.

41


Figure 5.5: Shear retention factor (Červenka, et al., 2016).

The shear retention factor is suitable only when the fixed crack coefficient is higherthan zero. The rotated crack gives a crack surface free from shear stresses (Červenka,et al., 2016).

Compressive Strength Redaction After Cracking

Compressive strength redaction is defined as a function of the maximum fractur-ing strain which is maximum tensile damage at the given point. The compressivestrength reduction after cracking with a parallel direction to the cracks is definedas:

fc′ef = rcf

′c (5.6)

where:rc = c+ (1− c)e−(128εu)2 (5.7)

For cases with zero normal strain, εν , the compressive strength reduction is zero.The strength is reaching the maximum value, fc

′ef = cf ′c, for large strains, as shownin figure 5.6. The constant c is the maximum strength reduction during a largetransverse strain (Červenka, et al., 2016).

Figure 5.6: Compressive strength reduction of cracked concrete (Červenka, et al.,2016).

42


Aggregate Interlock

The load between different sides of a crack is transferred by the crack through thecontact between fragments of the aggregate. This mechanism is called aggregateinterlock.

Inclined cracks occur due to the tensile stresses and propagate as a result of thecombination of tension and shear. The aggregate interlocking should be consideredbecause it can have a significant influence on these cracks (Malm, 2009).

The shear strength of a cracked concrete is dependent on the compressive strength,the maximum aggregate size and the maximum crack width (Červenka, et al., 2016).

5.2.2 Reinforcement Material Model

There are two options of modelling the reinforcement in ATENA 3D. It can be mod-elled as smeared or discrete. The smeared reinforcement is describes as compositematerial and it has no impact on the geometrical model (Červenka and Červenka,2017). Layers are used when defining the reinforcement, each layer contains barswith same direction. The discrete reinforcement is reinforcing bars and it is modelledwith 1D truss elements (Červenka, et al., 2016).

The discrete reinforcement approach was used to model the reinforcement in thejoint. The definition of the material properties was performed with Reinforcementmaterial model with the behaviour of bi-linear elastic-perfectly plastic stress-strainlaw, as illustrated in figure 5.7.

Figure 5.7: The law of bi-linear stress-strain for reinforcement (Červenka, et al.,2016).

The bond between the reinforcement and concrete can be defined either by theoption Perfect Bond or by creating a Bond for Reinforcement. In the analysis a

43


bond for reinforcement was created to achieve a more reality based behaviour of thereinforcement in concrete.

Reinforcement Bond Models

The bond-slip relationship is the main property of the reinforcement bond model.The relationship describes the cohesion τb, bond strength, depending on the currentslip value between the reinforcement and the concrete around. There are three differ-ent models for bond-slip in ATENA: the user defined law, slip law by Bigaj and themodel based on ModelCode 1990. The last two models are based on the compres-sive strength of concrete, reinforcement type and diameter. The quality of castingconcrete and confinement conditions are other important parameters ((Červenka, etal., 2016).

5.2.3 Interface

To simulate the contact between materials the Interface material model can be usedin ATENA 3D. For instance the contact between a joint and precast concrete elementcan be modelled with interface material model. The Mohr-Coulomb criterion withtension cut off is the basis of the interface material model(Červenka, et al., 2016).

A fundamental three-dimensional case is described by the tractions on interfaceplanes, relative sliding and opening displacements (Červenka, et al., 2016).τ1τ2σ

=

Ktt 0 00 Ktt 00 0 Knn

4v14v24u

The initial elastic normal and the shear stiffness are defined as Knn and Ktt, respec-tively. In the case of an interface with a contact thickness of zero the stiffnessesapproach high penalty numbers. Extremely high values should be avoid in order toprevent numerical instabilities. The stiffness values can be defined as:

Knn =E

t(5.8)

Ktt =G

t(5.9)

where:

E, is the minimal elastic modulus.G, is the minimal shear modulus.t, is the width of the interface.

The stiffness Knn and Ktt are used for mathematical aims after the element failureto retain the global system of the equations positive. The stiffness of the interfaceshould be equal to zero after the failure of the interface. This means that the global

44

5.3. BOUNDARY CONDITIONS AND LOADING

stiffness becomes undefined. The minimal stiffnesses Knn,min and Ktt,min are around0.001 or 0.01 times the normal stiffness, Knn, and shear stiffness ,Ktt, respectively(Červenka, et al., 2016).

Figure 5.8: Behaviour of interface model in shear and tension, respectively (Čer-venka, et al., 2016).

5.3 Boundary Conditions and Loading

In Atena 3D the boundary conditions and loads are defined as load cases. The loadcases are applied on macroelements, surfaces, lines or joints. In this thesis it wasimportant to define the boundary conditions correctly to allow the movement in thejoint to capture the shear stiffness.

The boundary conditions and loads applied on the models are presented in chapter6.2.3.

5.4 Mesh

The mesh is very important in the analysis of finite element methods. It influencesthe quality of the memory requirements, analysis results and the speed.

The geometrical model consist of three-dimensional solid regions defines asmacroele-ments. It is possible to mesh each macroelement independently. A macroelementmesh can be generated in three different ways, brick element, tetrahedral and mixedmeshes. The brick element is only suitable for macroelements with six boundarysurfaces. For other macroelements the mixed and tetrahedral meshes can be used.

In the usage of the mixed meshes the program creates a uniform brick mesh insideof the model. The other parts of the model close to the boundary will be meshedwith tetrahedral and pyramid elements. The mixed mesh is suitable only for the

45


sufficiently small mesh sizes. If the program can not create brick mesh uniformly inthe interior of big macroelements, tetrahedral meshes will be used instead (Červenka,et al., 2016).

The mesh can be generated as linear or quadratic. The nodes are placed in eachcorner of elements for the linear elements, therefore this type of generation is in loworder. For the quadratic generation there are extra nodes placed on each elementedge and it is in higher DOF. Nodes can exist inside the element and in the centerof element sides for some quadratic elements (Červenka, et al., 2016).

In this thesis the analysed model was meshed in brick elements in the beginning andlater some simulations were performed in tetrahedral elements, shown in figure 5.9.Both linear and quadratic elements were implemented in the mesh generations fordifferent tests.

(a) (b)

Figure 5.9: Meshed model created in ATENA Engineerig, (a) Brick mesh (b) Tetra-hedral mesh.

5.5 Elements

The model in the thesis was meshed with 8-noded CCIsoBrick elements, as illus-trated in figure 5.10 (a). Some tests were meshed in CCIsoTetra which is suitablefor 4 to 10-noded elements, as shown in figure 5.10 (b).

46

5.6. SOLVERS FOR NON-LINEAR ANALYSIS

(a) (b)

Figure 5.10: (a) The geometry of a CCIsoBrick element (b) The geometry of aCCIsoTetra element (Červenka, et al., 2016).

5.6 Solvers for Non-linear Analysis

For the non-linear analysis in ATENA 3D the iterative solvers such as Newton-Raphson or Arc-Length, illustrated in figure 5.11, can be used. The algorithms for thecalculation of the stiffness matrix are different in these two methods. The methodsare effective solvers for calculation for geometrical and material non-linearity.

Figure 5.11: Iterative solver for Newton-Raphson method and Arc-Length method,respectively (Červenka and Červenka,2017)

In the Newton-Raphson method the load increment is unchanged and an iteration forthe displacement is done until the equilibrium is reached, within the given tolerance.This method is suitable for the cases when the values of the load must be preciselymet. It is used in cases with loading types as pre-stressing, body forces, shrinkageand temperature. The method is stable for load controlled analyses (Červenka andČervenka,2017).

In the Arc-Length method the solution path is constant. It iterates the increaseof both the forces and displacements. The Arc-Length method is a more generalmethod compared to Newton-Raphson. The method is not suitable for problems

47


that requires the precisely intensities as shrinkage, pre-stressing, temperature andbody forces (Červenka and Červenka, 2017).

In this thesis, Newton-Raphson method was performed in all model analyses. Forsome of the tests the Newton-Raphson and Arc-Length methods were combined.Some of the analyses performed with Newton-Raphson method were load controlled.

48

Chapter 6

Model used in the analysis

6.1 Models in ATENA 2D

To estimate the distribution of shear and normal stresses in the joint between shearwalls a simplified 2D ATENA model was used. The model consisted shear walls thatillustrated a part of a ten-story building, as shown in figure 6.1.

Figure 6.1: 2D wall model.

The elements was assumed to be solid and the reinforcement was neglected in themodel. To consider the effects of the reinforcement the concrete was defined as linearelastic.

Each shear wall model consisted of ten pieces of 3m high and 2.5m wide cantilever

49

CHAPTER 6. MODEL USED IN THE ANALYSIS

walls with a thickness of 0.25m.

The wall model and the distribution of the horizontal load were taken from themaster thesis by Lindwall and Wester. The model was based on a real project andit was investigated to estimate the stiffness and the influence of the joints betweenshear walls (Lindwall and Wester, 2016).

6.1.1 Boundary Conditions and Loading of the Wall Model

The structure was supported in the ground, as described in table 6.1. The horizontalpoint forces applied on the each floor were with a magnitude of 35kN (Lindwall andWester, 2016), as shown in figure 6.1. The supports of the foundation of the wallmodel were defined as fixed in both x- and y-direction.

6.1.2 Material

Concrete

The material used in the wall elements was Plane Stress Elastic Isotropic. The prop-erties were defined specific for concrete class C30/37 (EN 1992, 2004), as presentedin table 6.1.

Table 6.1: Chosen strength values for concrete.

Property ValuesModulus of elasticity, E [GPa] 33Poisson´s ratio, ν [-] 0.2Specific material weight, ρ [KN/m3] 23Thermal expansion coefficient, α [1/K] 1.2· 10−5

The modulus of elasticity, E, was calculated with the following formula (EN 1992,2004):

E = 22 · (fcm10

)0,3 (6.1)

where:

fcm is the mean compressive strength, which is 38 [MPa] (EN 1992, 2004).

50

6.1. MODELS IN ATENA 2D

Interface

The vertical connection between the wall elements was defined as interface in the2D model. The material properties of the interface are described in table 6.2.

Table 6.2: Chosen stiffness based on Ec and element size.

Property ValuesNormal stiffness, Knn [MN/m3] 1.10 · 106

Tangential stiffness, Ktt [MN/m3] 1.10 · 106

Minimal normal stiffness, Knn,min [MN/m2] 1.10 · 104

Minimal tangential stiffness, Ktt,min [MN/m2] 1.10 · 104

The stiffness values were calculated according to Modelcode (MC90, 1993), theformulas are presented below:

Knn =Econcrete

elementsize · 10(6.2)

Ktt =Econcrete

elementsize · 10(6.3)

Knn,min =Knn

1000or Knn,min =

Knn

100(6.4)

Ktt,min =Ktt

1000or Ktt,min =

Ktt

1000(6.5)

The minimal normal and the minimal tangential stiffnesses are calculated with themultiple 100. The element size is set to 0.3m.

The influence of the parameters friction and cohesion, on the shear and normal stresshave been analysed. Different tests have been performed with different parametersas described in table 6.3.

Table 6.3: The parameters used in the test.

Test CohesionC [MPa]

Frictioncoefficientµ [−]

1 4.35 ·10−1 0.32 5.80 ·10−1 0.43 7.25 ·10−1 0.5

The cohesion is greater than or equal to µ · fct, where µ is the friction coefficientand fct is the tensile strength of the interface.

51


According to ATENA, the friction coefficient is almost always higher than 0.1. Therecommended values are 0.3-0.5 for cases where good measurement is missing (Pryland Červenka, 2017).

The tensile strength of the interface can be set to 1/4 or 1/2 times the tensilestrength of the weakest material in the connection (Pryl and Červenka, 2017). Inthe test presented in table 6.4 the half of the tensile strength of the material wasused, which was calculated to 1.45 MPa.


Test CohesionC [MPa]

Frictioncoefficientµ [−]

4 2.18 ·10−1 0.35 0 0.3

In the test 5 presented in table 6.5, 1/4 of the tensile strength was used, which wascalculated to 0.725 MPa. The last test was performed without tensile strength andcohesion.

6.2 Models in ATENA 3D

Two different shear keys were modelled, one reinforced and one unreinforced model,in order to estimate the capacity of the joint and the contribution of the reinforce-ment to the bearing capacity. Ten different tests were performed for the unreinforcedshear key model which are named as test A - test J. For the reinforced shear keymodel, four different tests were performed which are named test RA− test RD.

The models were created in ATENA Engineering 3D but because of the size of themodels the calculations were performed in ATENA Studio. ATENA Studio is moretime-effective compared to ATENA Engineering.

6.2.1 Geometry of the walls

Unreinforced shear key joint

To receive a reality based model the cross-section dimensions were taken from aprefab manufacturers catalogue.

The geometry of the model was divided in several parts also called macro elements.When the model consist of several macro elements it is important to consider thecontact surfaces. In ATENA one surface of a macro element can only be connectedto one surface of another macro element. In this study shape and size of the macroelements were adapt to the shear key geometry. The geometry of the model ispresented in table 6.5 and figures 6.2 and 6.3.

52


Table 6.5: The dimensions of the shear key joint without reinforcement.

Label Size [mm]h 170h1 55h2 5h3 50b 152b1 12b2 24b3 64c 250c1 70c2 60c3 5

Figure 6.2: The unreinforced shear wall model. Figure 6.3: The shear key from above.

Figure 6.4: The unreinforced model, presented in ATENA 3D

An additional model with a frame on the left side of the model, as illustrated infigure 6.5, was designed for the deformation controlled analyses.

53


Figure 6.5: The unreinforced model with frame, presented in ATENA 3D.

Reinforced shear key joint

The cross-section dimensions for the shear key and the reinforcement were takenfrom a real project. The reinforced joint was designed to connect prefabricatedinterior walls in the project.

The shape of the joint in this model is similar to the unreinforced joint mentionedbefore. The space between the wall elements was extended due to the size of thereinforcement. The geometry of the model is presented in table 6.6 and figures 6.6and 6.7.

Table 6.6: The dimensions of the shear key joint with reinforcement.

Label Size [mm]h 170h1 55h2 5h3 50b 288b1 12b2 160b3 64c 350c1 170c2 160c3 5

54


Figure 6.6: The shear wall mode without reinforcement. Figure 6.7: The shear key from above.

An additional model with a frame on the left side of the model, as illustrated infigure 6.8, was designed for the deformation controlled analyses.

Figure 6.8: The reinforced model with frame, presented in ATENA 3D.

The diameters of the reinforcement are defined in table 6.7. The straight bars arecontinuous in along the whole vertical joint and the U-bar are continuous in to thewalls. As mentioned before the model is a section of a connection and therefore thereinforcement bars are also cut, as shown in figure 6.9. To retain the continuity ofthe bars the ends are prevented from slip in both beginning and end of the barswhen modelling.

Table 6.7: The diameters of the reinforcement bars.

Reinforcement type Diameter [mm]U-bar loops 12Longitudinal bars 12Bended U-bar loop 8

55


Figure 6.9: The reinforced model, presented in ATENA 3D

6.2.2 Assumptions and Limitations when Modelling

Several limitations were made due to the size of the precast walls and the verticaljoint, in order to decrease the size of the output files and the process time.

The 2D model was limited, the joint was replaced with a linear elastic interfacebetween the walls.

The 3D model was limited. Instead of modelling two whole shear walls with a verticaljoint, only sections of the connected walls were considered. The sections representedparts of wall elements connected with one shear key. Additional limitations weremade where the symmetry of the model was utilized to reduce the size of the model.

In this thesis the wind load was the only load considered, while the other externaland internal loads were neglected.

6.2.3 Boundary Conditions and Loading

The both models in 3D were defined with same boundary conditions and loading.The boundary condition and loading of the models were chosen to achieve the desiredresults in the shear key.

The boundary conditions applied to the model are described in table 6.8 and figure6.10. The vertical locking was applied to prevent the vertical movement of the rightwall element. The aim of the locking was to analyse the load transmission from theleft wall element to the rest of the model through the joint. To prevent the horizontalmovement of the model the right wall was fixed in one surface in order to obtainthe load distribution in the joint due to the external loads. The boundary conditionsymmetry was imposed along the longitudinal surface of the model because of the

56


utilization of the symmetry.


Boundary condition DirectionX Y Z

Vertical locking Free Free FixedHorizontal locking Free Fixed FreeSymmetry Fixed Free Free

(a) (b)

(c)

Figure 6.10: Boundary conditions applied to the model, (a) Vertical locking (b) Hor-izontal locking (c) Symmetry

57


The loads applied on the model was obtained from the FEA on the 2D wall model.The estimated vertical and horizontal pressures were of magnitude 1.04MN/m2 and2.63 · 10−2MN/m2, respectively. In the beginning, some of the tests were analysedwith a lower horizontal pressure of magnitude 9.90·10−3MN/m2, as shown in figures6.11 and 6.12. The tests A − H and RA − RB were performed with the lowerhorizontal pressure while the tests I − J and RC − RD were performed with thehorizontal pressure obtained from the 2D wall model. The increase of the horizontalpressure was proportional to the load increments in tests A-I and for tests RA−RC .

Figure 6.11: The vertical and horizontal loading obtained from the 2D wall model,presented in ATENA 3D

Figure 6.12: The horizontal loading with lower magnitude, presented in ATENA 3D

The models in the tests J and RD were with frames around the left precast wall andhad an additional boundary condition combined with loading called deformation

58


control as illustrated in figure 6.13. The deformation control were of magnitude2.4 · 10−4m and it was applied up to the failure of the model. The total horizontalpressure was applied before the deformation control.

Figure 6.13: The model with deformation control, presented in ATENA 3D.

59


6.2.4 Material

Concrete

The material used in the wall elements was CC3DNonLinCementitious2. The con-crete properties were based on the strength class C30/37, the mean values were used(EN 1992, 2004), as presented in table 6.9.

Table 6.9: Chosen strength values for concrete of the model.

Property ValuesModulus of elasticity, E [GPa] 33Poisson´s ratio, ν [-] 0.2Tensile strength, ft [MPa] 2.9Compressive strength, fc [MPa] 38Specific fracture energy, GF [MN/m] 6.4 ·10−5

Aggregate size in the joint, [m] 0.008Aggregate size in the wall, [m] 0.016Specific material weight, ρ [KN/m3] 23Thermal expansion coefficient, α [1/K] 1.2· 10−5

Crack shear stiffness factor, SF [−] 20Compressive strength reduction, rc, lim [−] 0.8Fixed crack model coefficient, [−] 1.0Multiplier for plastic flow direction,β [−] 0.0

The specific fracture energy, GF , was calculated according to Eq. 4.4. The aggregatesizes of the joint and the wall were based on the Swedish praxis (Ekroth, 2017).

For the deformation controlled tests, the frame was defined as 3D Elastic Isotropic.The elastic modulus was set to 33GPa and the poisson’s ratio was set to 0.2.

Interface

The connection between the wall elements and the joint was defined as interface inthe 3D model. The material properties used in the model are defined in table 6.10.The stiffnesses were calculated according Eq. 6.2-6.5. The element size is based onthe aggregate size, which was set to 0.016.

Table 6.10: Chosen stiffness based on Ec and element size.

Property ValuesNormal stiffness, Knn [MN/m3] 1.65·107

Tangential stiffness, Ktt [MN/m3] 1.65·107

Minimal normal stiffness, Knn,min [MN/m3] 1.65·105

Minimal tangential stiffness, Ktt,min [MN/m3] 1.65·105

A parameter study was performed for the tensile strength, cohesion and frictioncoefficient, as described in table 6.11. The tests A-C and D were generated in

60


quadratic elements. The tensile strength was set to 1/2, for tests A-C, and 1/4, fortest D, of the material tensile strength which are 1.45MPa and 7.25 · 10−1MPa,respectively. The last test described in table was simulated without tensile strength.The model was meshed with brick elements. The horizontal pressure for the testsA-H were of magnitude 9.90 · 10−3MN/m2.


Test Friction coefficientµ [−]

CohesionC [MPa]

A 0.3 4.35 · 10−1

B 0.5 7.25 · 10−1

C 0.7 1.02D 0.3 2.18 · 10−1

E 0 0

Different tests were performed for models generated in linear elements and meshedin tetrahedral elements. For the tests F-H, the friction coefficient, cohesion andtensile strength are 0.3, 4.35 · 10−1 MPa and 1/2 of the tensile strength in concrete.


Test Generation type Mesh typeF Linear BrickG Linear BrickH Linear Tetrahedral

The tests I and J were generated in linear elements and meshed in brick elements.The horizontal pressure for the tests were of magnitude 2.63 · 10−2MN/m2. Thetest J is deformation controlled of magnitude 2.4 · 10−4m. The tensile strength inthe tests was set to 1/2 of the tensile strength in concrete which is 1.45MPa. Thefriction coefficient and the cohesion were set to 0.5 and 7.25 ·10−1MPa, respectively.

Reinforcement

The material type Bilinear was defined for the reinforcement. The properties of thereinforcement bars were set to default values for steel recommended by ATENA, asshown in table 6.13.

Table 6.13: Properties of the reinforcement.

Property ReinforcementModulus of elasticity, E [GPa] 200Yield strength, σ [MPa] 550Specific weight, ρ [kN/m3] 78.5

61


Bond for Reinforcement

The contact between the reinforcement bars and the concrete was defined as ma-terial type Bond for Reinforcement. In the definition of the material the generatorCEB-FIB Model Code 1990 was chosen. The properties were defined as shown intable 6.14. When creating the bond for reinforcement it was required to define thespecified characteristic cube compressive strength, fcu, which was calculated accord-ing to MC90 (1993):

fcu =fcm0.85

(6.6)

Table 6.14: Properties of the reinforcement.

PropertyReinforcement type Hot rolled barsConcrete confinement ConfinedBond quality Good

The tests for the reinforced models were generated in both quadratic and linear meshelements, as show in table 6.15. For all the reinforced tests, the tensile strength wasset to 1/2 of the material tensile strength, which is of magnitude 1.45MPa. Thefriction coefficient and cohesion were set to 0.3 and 4.35 · 10−1MPa in tests RA andRB, respectively.

62



Test Generation type Mesh typeRA Quadratic BrickRB Linear BrickRC Linear BrickRD Linear Brick

For tests RC and RD the friction coefficient and cohesion were set to 0.5 and7.25 · 10−1MPa, respectively. The horizontal pressure applied on the models was ofmagnitude 2.63 · 10−2MPa. The test RD was deformation controlled of magnitude2.4 · 10−4m.

63

Chapter 7

Results from the analyses

7.1 Shear and normal stresses from the 2D wallmodels

In the 2D wall models a parameter study was performed in order to analyse theeffect of the friction, cohesion and tensile strength on the shear and normal stresses.

Different shear and normal stresses were obtained in the nodes along the verticalconnection between the wall elements. The maximum and minimum shear stressesand the related normal stresses were of interest. The stresses obtained from thetests 1-5 are presented in tables 7.1 and 7.2.

Table 7.1: The maximum shear stresses and the related normal stresses for the dif-ferent frictions.

Test Shear stress τ [MPa] Normal stress σ [MPa]1-4 3.92·10−1 2.63·10−2

5 0.0 2.66·10−2 **The maximal normal stress.

Table 7.2: The minimum shear stresses and the related normal stresses for the dif-ferent frictions.

Test Shear stress τ [MPa] Normal stress σ [MPa]1-4 1.07·10−2 7.34·10−2

5 0.0 1.78·10−2 **The minimal normal stress.

The same results were obtained for the tests 1-4. In test 5, where the tensile strengthand the cohesion were set to zero, only normal stresses were proceeded. The shearstresses are dependent on the tensile strength and therefore no shear stresses wereobtained in test 5. The stress distribution and the deformation of the model fortest 1 and 5 are shown in figures 7.1 and 7.2. It should be observed that the stress

65

CHAPTER 7. RESULTS FROM THE ANALYSES

distribution in the figures have the same signs even if the stresses seems to changedirection in the figures.

(a) (b)

Figure 7.1: Test 1, presented in ATENA 2D, (a) Deformed shape with normal stressdistribution (b) Deformed shape with shear stress distribution.

The translation of the stresses obtained from the 2D model corresponding the loadingapplied on the 3D models are presented in appendix B.

66

7.1. SHEAR AND NORMAL STRESSES FROM THE 2D WALL MODELS

(a) (b)

Figure 7.2: Test 5, presented in ATENA 2D, (a) Undeformed shape (b) Deformedshape with normal stress distribution.

67


7.2 Simulations in 3D models

In the 3D models a parameter study was performed, in tests A-E, in order to analysethe effect of the friction, cohesion and tensile strength on the stiffness of the contactsurface between the joint and the wall. The model of the unreinforced joint wasused for the study of the parameters, with the brick mesh and quadratic generation.

In the test F, the same input parameters were used as in test A, but it was generatedin linear elements to analyse the difference in results due to the generation type. Anadditional test, test G, with the same input parameters as in test F, was analysedwith only shear stresses applied.

To analyse the influence of the mesh type, test H was performed with the samematerial parameters as in test F and it was meshed with tetrahedral.

The horizontal pressure was increasing proportional to the load increment for thetests A-H. An additional test, I, was performed in the same way but with a higherhorizontal pressure that was obtained from the 2D wall model. The same horizontalpressure was used for test J with a frame around the left precast wall. For the test Jthe total horizontal pressure was applied before the vertical deformation control ofthe precast wall, on the left side of the joint. The deformation control was appliedup to the failure of the model. The frame and the deformation control were used toprevent the rotation of the left precast wall.

The model of reinforced shear key joint, RA, was generated with quadratic meshelement. The other reinforced models, RB, RC and RD were generated with linearmesh element. The horizontal pressure was increasing proportional to the loadincrement for the tests RA and RB. The test RC was performed in the same waybut with a higher horizontal pressure that was obtained from the 2D wall model. Inthe test RD, the same horizontal pressure was used as in test RC . The test RD wasmodelled with a frame around the left precast wall and it was deformation controlledafter that the total horizontal pressure was applied.

All the tests were calculated with Newton-Raphson method. Some of the tests wererecalculated with Newton-Raphson and Arc-length solution methods combined.

7.2.1 Parametric Study of the Unreinforced Models

Unreinforced Model with Quadratic Elements

The results for tests A-E, are presented in table 7.3. In phase 1 the joint started tolose contact with the wall. In phase 2 the first cracks were developed and in phase3 the software failed to find convergence and the calculations were interrupted. Thevalues presented in the table show the loading that was applied on the model inrespective phase. The first and third phases are shown in figure 7.3 for the test A.The results for the other tests are presented in appendix A.

68

7.2. SIMULATIONS IN 3D MODELS

Table 7.3: The results obtained from the tests A-E, with the brick element genera-tion.

Test Loading1

for phases2

[MN ]3

A,µ = 0.3, C = 4.35 · 10−1 1.67·10−3 3.50·10−3 6.33·10−3

B, µ = 0.5, C = 7.25 2.33·10−3 4.33·10−3 6.91·10−3

C, µ = 0.7, C = 1.02 · 10−1 2.92·10−3 4.08·10−3 6.50 ·10−3

D, µ = 0.3, C = 2.18 · 10−1 1.25 ·10−3 5.41·10−3 7.91·10−3

E, µ = 0.0, C = 0.0 8.33·10−5 2.25·10−3 5.41·10−3

According to the hand calculations the maximum load increment applied on themodels was of magnitude 8.33 ·10−3MN , as presented in appendix B. As it is shownin the table above the maximum load increment was not reached in the unreinforcedtests A-E.

(a) (b)

Figure 7.3: Test A, presented in ATENA Studio, (a) Phase 1 (b) Phase 3.

A comparison between the tests A,B and C with different friction coefficients areassembled in a load increment-deformation curve, as shown in figure 7.4. The higherthe friction coefficient was, the higher load was received by the joint before it loosedthe contact with the wall.

69


Figure 7.4: Deformation and external load for tests A-C, with different frictions.

In the tests A, D and E different tensile strengths were used. A load increment-deformation curve of the tests is illustrated figure 7.5. The joint with a lower tensilestrength in the interface, started to loose the contact with the wall under a lowerloading compared to the other model. The test without tensile strength showed thatthe joint loosed the contact in the first load increment.

Figure 7.5: Deformation and external load for tests A, D and E, with different tensilestrength values.

70


Unreinforced Model with Linear Elements

For the tests F-J the models were generated in linear elements. The results for thetests are shown in tables 7.4 and 7.5. The other results are presented in appendix A.In both tests G and H the same material parameters were used. The tests F-G andI-J were generated in brick elements while the test H was generated in tetrahedralelements. In the test G, only shear stress was applied on the model.

Table 7.4: The results obtained from the tests F, G and H.

Test Loading1

for phases2

[MN ]3

F, µ = 0.3, C = 4.36 · 10−1 1.67·10−3 3.33 ·10−3 4.17 ·10−3

G, µ = 0.3, C = 4.36 · 10−1 1.33 ·10−3 3.25 ·10−3 6.75 ·10−3

H, µ = 0.3, C = 4.36 · 10−1 1.67 ·10−3 3.33 ·10−3 1.27 ·10−2

A higher horizontal pressure was applied on the models for tests I and J. In test Ja deformation control was used.

Table 7.5: The results obtained from the tests I and J.

Test Loading1

for phases2

[MN ]3

I, µ = 0.5, C = 7.25 · 10−1 2.50·10−3 4.17 ·10−3 1.03 ·10−2

J, µ = 0.5, C = 7.25 · 10−1 3.52 ·10−3 5.24 ·10−3 7.60 ·10−3

A comparison of quadratic and linear mesh type is presented in figure 7.6. Thesoftware converged shorter with quadratic element because of the high degrees offreedom but the simulation time was longer. The analyses with this type of elementsare in higher accuracy and the analysis interrupts due to high convergence failures.The analyses with linear elements continues despite of the high convergence failures.

71


Figure 7.6: Deformation and external load for tests A and F, with quadratic andlinear elements, respectively.

In the test G, only shear stresses were applied on the model and the effects of thenormal stresses were analysed by comparing it with test F, as shown in figure 7.7.

Figure 7.7: Deformation and external load for tests F and G, with different loading.

72


The mesh types brick and tetrahedral were compared and as shown in figure 7.8,almost the same results were obtained. The model with tetrahedral mesh convergedlonger.

Figure 7.8: Deformation and external load for tests H and F, with brick and tetra-hedral mesh, respectively.

A comparison between tests I and J is presented in figure 7.9 to analyse the effectof the deformation control and the frame, which were used to prevent the rotationof the left precast wall. Another difference between these tests was the horizontalpressure. For the test I the horizontal pressure increased propotional to the loadincrement while for the test J the total horizontal pressure was applied before thedeformation control. The test I converged longer than test J . The curve for thetest J reached a peak point before it changed the direction. The test I resisted themaximum load increment in phase 3 while the test J resisted the maximum loadincrement at the peak point.

73


Figure 7.9: Deformation and external load for tests I and J.

Reinforced Model with Quadratic and Linear Elements

The reinforced models were generated in quadratic elements, for the test RA, andin linear elements, for the tests RB − RD. The results are shown in tables 7.6 and7.7. The rest of the results are presented in appendix A.

Table 7.6: The results obtained from the tests RA and RB, with the brick elementgeneration.

Test Loading1

for phases2

[MN ]3

RA, quadratic elements 1.52 ·10−3 5.30 ·10−3 1.11 ·10−2

RB, linear elements 1.17 ·10−3 9.33 ·10−3 1.40 ·10−2

A higher horizontal pressure was applied on the models for tests RC and RD. Intest RD a deformation control was used.

Table 7.7: The results obtained from the tests RC and RD, with the brick elementgeneration.

Test Loading1

for phases2

[MN ]3

RC , quadratic elements 1.75 ·10−3 1.06 ·10−2 1.66 ·10−2

RD, linear elements 2.37 ·10−3 1.68 ·10−2 2.76 ·10−2

74


(a) (b)

Figure 7.10: Test RA, presented in ATENA Studio, (a) Phase 1 (b) Phase 2, alsoshown in appendix A.

The quadratic and linear mesh were compared in figure 7.10, the test RA and thetest RB were meshed in quadratic and linear elements, respectively. The result werealmost similar in the tests.

The effect of the reinforcement was analysed by comparing the test A and RA,which were meshed in quadratic elements. An additional comparison was performedbetween test F andRB, which were meshed in linear elements. The both comparisonsare shown in figure 7.12 and 7.13. In the both cases the joint lost contact with thewall element under less loading in the reinforced models. The simulations convergedlonger for the reinforced models.

A comparison between tests RC and RD is presented in figure 7.14 to analyse theeffect of the deformation control, frame and horizontal pressure. The test RC con-verged longer and as it is shown in the figure it reached a quite constant behaviour.The curve for the test RD reached a peak point before it changed the direction. Theboth tests resisted the maximum load increment.

The stress distributions of the reinforcement for the last steps of the simulations inthe tests RA −RD are presented in figures 7.15- 7.18 and also in appendix A. As itcan be seen in the figures, the reinforcement did not yield in phase 3.

75


Figure 7.11: Deformation and external load for tests RA and RB.

Figure 7.12: Deformation and external load for tests A and RA, shear key jointwithout and with reinforcement.

76


Figure 7.13: Deformation and external load for tests F and RB, shear key joint with-out and with reinforcement

Figure 7.14: Deformation and external load for tests RC and RD.

77


Figure 7.15: The stresses in the reinforcement in the test RA, obtained in phase 3.

Figure 7.16: The stresses in the reinforcement in the test RB, obtained in phase 3.

78


Figure 7.17: The stresses in the reinforcement in the test RC , obtained in phase 3.

Figure 7.18: The stresses in the reinforcement in the test RD, obtained in phase 3.

79


7.3 The results of the hand calculations

Hand calculations for the shear resistance at the interface were performed accordingto EN 1992, the results obtained are presented in table 7.8, for the unreinforcedshear key joint, and in table, for the reinforced shear key joint. The calculations arepresented in appendix B.

Table 7.8: The results obtained from the hand calculations for the unreinforced andreinforced shear key joints.

Shear resistance [MN ]νRdi,EC2,UNREIN 5.99 ·10−3

νRdi,EC2,REIN 3.24 ·10−2

80

7.3. THE RESULTS OF THE HAND CALCULATIONS

Figure 7.19: Deformation and external load for tests I and RC , obtained in phase 3compared to the hand calculations according to EN 1992.

Figure 7.20: Deformation and external load for tests J and RD, obtained in phase 3compared to the hand calculations according to EN 1992.

81


As shown in figure 7.19 the load increment-deformation curve for test I exceededthe curve based on the hand calculation. The NLFEA gave a better shear capacityfor the unreinforced model than the Eurocode in this case. The curve, for shearresistance at the interface with reinforcement according to the hand calculation wasmuch higher than the curve obtained for test, RC . The capacity obtained from thehand calculations was not reached by the NLFEA of the reinforced model.

The peak points were obtained for the tests with deformation control, as illustratedin figure 7.20. The magnitude of the peak points were 1.1 · 10−2MN for the un-reinforced test and 2.91 · 10−2MN for the reinforced test. According to the handcalculation the shear capacity is higher than the the results obtained from the sim-ulations for the reinforced model. The curve for the test J exceeded the curve forthe shear resistance according to the hand calculation, for unreinforced model. Theboth unreinforced and reinforced, deformation controlled tests, resisted the maxi-mum load increment that were of magnitude 8.33 · 10−3MN and 1.17 · 10−2MN ,respectively.

82

Chapter 8

Discussion and Conclusions

8.1 Shear and Normal stresses in the wall models

The 2D wall model was used to estimate reality based shear and normal stressesexposed to the joint. In the reality the vertical joint is locked by the floor slab ateach floor which was neglected in the 2D analyses. Due to all simplifications, theresults are over conservative. The aim was to apply reasonable loadings in the 3Dmodels. The simulations of the 3D models were performed with loading incrementswith the opportunity to analyse the behaviour of the models with both higher andless loading.

8.2 The stiffness of the joint

The development of the shear friction mechanism starts with the occurrence of thefirst cracks and sliding between interfaces. The friction coefficient influences theshear capacity of the joint, the joint loosed the contact with the wall under a higherloading in the test where the highest friction coefficient was used. A high friction inthe interface prevents the sliding between the wall element and the shear key joint.The friction coefficient is dependent on the normal force across the joint thereforethe normal force has a positive impact on the shear capacity of the joint.

According to the results obtained in the test the tensile strength in the interfaceis significant for the bearing capacity of the joint. It strengthens the cooperationbetween the wall elements and the joint.

In the ATENA software the cohesion of the interface is dependent of the frictioncoefficient and the tensile strength of the interface. An increase of friction coefficientand the tensile strength give automatically a higher cohesion which contributes toa higher capacity in the interface.

The quadratic mesh element is in a higher DOF and it should give a higher accuracywhich was the reason of why this element type was chosen in the simulations in

83

CHAPTER 8. DISCUSSION AND CONCLUSIONS

the beginning. The disadvantage of the quadratic mesh was that it was not timeeffective and it gave an over rigid behaviour in this case, therefore the simulationswere performed in linear mesh element after a number of simulations. By using thelinear elements the time of the simulations was decreased to half. The tests withlinear elements converged longer. This can depend on that the Newton-Raphsonmethod tries to find the equilibrium by iterating each node. In quadratic elementsit is higher risk to have cracks close to the nodes, due to the higher amount of nodes,which can impact the convergence negatively and give rise to additional cracks whichis not the case in the reality.

In the tests where brick mesh elements were used the results showed that upper leftedge of the shear key was totally crushed in the last iteration step of the simulations.This could have been related to the inappropriate mesh between the sharp edges ofthe shear key and the adjacent wall element. To check if that was an affecting factorthe tetrahedral mesh generation was used in some tests. The results showed that,the tests with the tetrahedral mesh converged longer.

The affects of the reinforcement in the joints were analysed by the comparisons forthe models with quadratic and linear meshes, which showed that the shear capacityof the reinforced model was higher the the unreinforced models. As it was illustratedin figures 7.12 and 7.13, the first separation was occurred before the stresses weretransferred to the reinforcement. After the first opening the reinforcement resistedthe loading and started to contribute to the shear capacity. Afterwards the behaviourof the curve changed which could have been because of the insufficient boundaryconditions.

The wall element on the left side of the joint was exposed to the external forcesand started to bend downwards against the joint. The loading applied on the wallelement induced a rotation with a long lever arm instead of shear stresses. Thelower edge of the wall element pressed the joint from the bottom and gave the jointresistant.

In the reality the rotation can not occur because the wall is fixed in the whole loweredge. In the model it was not possible to apply the same boundary conditions,locking the lower edge from movement would influence the stress distribution injoint. The behaviour of the model was improved by a deformation control a framearound the prefabricated wall element, on the left side of the joint. The deformationcontrol and the frame reduced the rotation of the wall element significantly but therewas still a small rotation of the joint, which can not occur in the reality due to thecontinuity of the joint.

In the first phase of the tests the joints started to loose the contact with the wall ele-ment under different loading and the micro cracks were developed. The convergencecriteria of the calculations, as shown in figure A.5, showed that there was a largeconvergence failure under these loadings where the separation of the wall elementsand joints began. It did not take long time for the software to find convergenceagain which showed that the joints still had the capacity to resist more loading thanapplied. The reason of the separation can depend on the geometry of the macro

84

8.2. THE STIFFNESS OF THE JOINT

elements in the model. The second large convergence failure as shown in figure A.5,occurred in phase 2 where visible cracks were developed.

In the model the indented macro elements of the joint and the adjacent macroelements in the wall were modelled with sharp edges. The sharp edges are difficultto mesh together and it causes complications in the simulations. In phase 3, thesoftware could not converge further and the calculations were aborted. This canbe interpreted in two ways for the models without deformation control. The firstalternative is that the maximal load capacity of the joint is reached and the jointcan not receive more loads, the failure occurs.

The other alternative is that the convergence failure depends on that the macroelement, the upper shear key edge of the joint starts to crush totally. The crack widthin the element reaches around 1 mm which makes it difficult for the unreinforcedconcrete to handle. In this case it becomes impossible to keep the joint edge onplace, it collapse completely. However, it might be possible for the joint to still havebearing capacity. For some of the tests the software could perform the calculationsfor a load increment higher than the loading applied on the model. This can beinterpreted as that those models have a higher shear capacity than the applied load.

In ATENA software the cases without deformation control for the load increment upto failure or further it is necessary to use Arc-Length method. In the all simulationsof the tests the Newton-Raphson method were used. The results of the tests wereassembled in external force-displacement curves. All the curves were continuedupward until the simulations were aborted, it was not possible to capture peakpoints for the maximum shear capacity of the joints. For some tests the Newton-Raphson and Arc-Length were combined to obtain the peak point. The Arc-Lengthcould not found equilibrium, instead it attempted to achieve equilibrium with allthe macro elements which resulted in cracks in the entire model which differs fromthe reality. The peak points, that gave the maximum shear capacity of the shearkey joint, were captured in the simulations with deformation control.

A comparison between tests with different normal loads was performed, whichshowed that the normal loads contribute the shear capacity positively.

According to the results of the hand calculations the shear key without reinforcementcould not resist the shear stresses from the external loads. The reinforced shear keyjoint had higher shear resistance in the interface than what was applied.

The shear capacity, peak point, for the unreinforced test with deformation controlwas higher than the shear capacity obtained in the hand calculations which also havebeen observed from investigation by Herfelt et al.(2016). This can be interpreted asthat the Eurocode underestimates the shear capacity of the joint. For the reinforcedtest with deformation control the shear capacity was lower than the shear capacityobtained in the hand calculation. The reason for this can be the effects of the dowelaction that were neglected in the NLFEA.

85

CHAPTER 8. DISCUSSION AND CONCLUSIONS

8.3 Conclusions

The conclusion of all tests is that the joints with deformation control gave themost reality based behaviour. To capture the shear capacity for the joint withreinforcement it is necessary to consider the dowel actions which was not possiblein ATENA software.

The shear capacity for the unreinforced shear key joint with deformation controlwas of magnitude 1.1 · 10−2MN which is higher than the obtained shear capacityaccording to the Eurocode, that was calculated to 5.99 · 10−3MN .

For the reinforced shear key joint the shear capacity was of magnitude 2.91·10−2MNwhich is lower than the obtained shear capacity according to the Eurocode, whichwas calculated to 3.24 · 10−2. In this case the dowel actions are neglected whichmeans that a higher shear capacity would be obtained in the simulations if thedowel actions were included.

8.4 Needs for further research

The aim of this thesis was to analyse the shear stiffness and capacity of the shearkey joints regarding to different material properties and assumptions. The loadsapplied on the model were taken from a real project and the analysed model wasa section of a shear key connected to parts of wall elements. One suggestion is toanalyse a bigger section of the connection of the joint and the walls.

To capture a shear capacity higher than the calculated according to Eurocode thedowel actions must be considered. The ATENA software does not consider theeffects automatically it should be investigated how to consider these effects in themodelling.

More investigations can be performed about different solutions of reinforced shearkeys, for example the rotated transversal U-bars can be analysed.

86

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89

Appendix A - Numerical Results

Figure A.1: Test A, Phase 1.

91



92



93


Figure A.4: Test A, Deformation and external load.

94


Figure A.5: Test A, Iteration convergence.

95


Figure A.6: Test A, Castellated parts of shear key joint in phase 3.

96


Figure A.7: Test A, Castellated parts of shear key joint in phase 3.

Figure A.8: Test A, Shear key joint in phase 3.

97


Figure A.9: Test A, Prefabricated wall on the right side of the joint in phase 3.

98


Figure A.10: Test A, Prefabricated wall on the left side of the joint in phase 3.

99


Figure A.11: Test A, The interface between the prefabricated wall elements and joint.

100


Figure A.12: Test A, Upper edge of the shear key joints in phase 3.

101


Figure A.13: Test B, Phase 1.

102



103



104


Figure A.16: Test B, Deformation and external load.

105


Figure A.17: Test B, Iteration convergence.

106


Figure A.18: Test C, Phase 1.

107



108



109


Figure A.21: Test C, Deformation and external load.

110


Figure A.22: Test C, Iteration convergence.

111


Figure A.23: Test D, Phase 1.

112



113



114


Figure A.26: Test D, Deformation and external load.

115


Figure A.27: Test D, Iteration convergence.

116


Figure A.28: Test E, Phase 1.

117



118



119


Figure A.31: Test E, Deformation and external load.

120


Figure A.32: Test E, Iteration convergence.

121


Figure A.33: Test F, Phase 1.

122



123



124


Figure A.36: Test F, Deformation and external load.

125


Figure A.37: Test F, Iteration convergence.

126


Figure A.38: Test G, Phase 1.

127



128



129


Figure A.41: Test G, Deformation and external force.

130


Figure A.42: Test G, Iteration convergence.

131


Figure A.43: Test H, Phase 1.

132



133



134


Figure A.46: Test H, Deformation and external load.

135


Figure A.47: Test H, Iteration convergence.

136


Figure A.48: Test I, Phase 1.

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Figure A.51: Test I, Deformation and external load.

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Figure A.52: Test I, Iteration convergence.

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Figure A.53: Test J, Phase 1.

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Figure A.55: Test J, Peak point.

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Figure A.57: Test J, Deformation and external load.

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Figure A.58: Test J, Iteration convergence.

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Figure A.59: Test RA, Phase 1.

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Figure A.62: Test RA, Deformation and external load.

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Figure A.63: Test RA, Iteration convergence.

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Figure A.64: Test RB, Phase 1.

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Figure A.67: Test RB, Deformation and external load.

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Figure A.68: Test RB, Iteration convergence.

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Figure A.69: Test RC , Phase 1.

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159



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Figure A.72: Test RC , Deformation and external load.

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Figure A.73: Test RC , Iteration convergence.

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Figure A.74: Test RD, Phase 1.

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Figure A.76: Test RD, Peak point.

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Figure A.78: Test RD, Deformation and external load.

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Figure A.79: Test RD, Iteration convergence.

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Figure A.80: The stresses in the reinforcement in the test RA, obtained in phase 3.

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Figure A.81: The stresses in the reinforcement in the test RB, obtained in phase 3.

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Figure A.82: The stresses in the reinforcement in the test RC , obtained in phase 3.

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Figure A.83: The stresses in the reinforcement in the test RD, obtained in phase 3.

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Appendix B - Hand Calculations

The Translation of Stresses from 2D to 3D

Figure B.1: Dimensions of the unreinforced model.

The stresses obtained in 2D are presented below.

Table B.1: Dimensions of the model.

Shear stress τ (MPa) Normal stress σ (MPa)3.92·10−1 2.63 ·10−2

The dimensions of the model:

h = 0.17mcunrein = 0.25mcrein = 0.35mb3 = 0.064m

The area of the vertical surface:

Av,unrein = h · cunrein = 0.17 · 0.25 = 4.25 · 10−2m2

Av,rein = h · crein = 0.17 · 0.35 = 5.95 · 10−2m2

The horizontal and vertical pressures applied on the 3D models:

σn = σ = 2.63MN/m2

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σv,unrein =τ ·Av,unrein

cunrein·b3 = 3.92·10−1·4.25·10−2

0.25·0.064 = 1.04MN/m2

σv,rein =τ ·Av,rein

crein·b3 = 3.92·10−1·5.95·10−2

0.35·0.064 = 1.04MN/m2

The Shear Resistance at the Interface According to Eurocode

The total capacity of the interface between the shear key is defined in the Eq.3.3,which i presented below.

νRdi = cfctd + µσn + ρfyd(µ sinα + cosα) ≤ 0.5vfcd

Input parameters:

In the software ATENA the mean values of the material were used in the simulationstherefore the mean values were used in the hand calculation.

The cohesion and friction coefficients for indented surface (EN 1992, 2004):

c = 0.5

µ = 0.9

The material parameters for C30/37:

fck = 30MPa

fcd = fck1.5

= 20MPa

fctm = 2.9MPa

fctd = fctm,ATENA = 1.45MPa which was calculated according to ATENA, (Pryland Červenka,2017) as:

fctm,ATENA = fctm2

The strength reduction factor:

v = 0.6(1− fck250

) = 0.6(1− 30250

) = 5.28 · 10−1

Geometry:

α = 90◦

bi,unreinforced = c1unreinforced = 0.07m

bi,reinforced = c1reinforced = 0.18m

c− c = 0.115m

rs = 0.006m

As = 2 · (rs2 · π) = 2 · (62 · π) = 2.26 · 10−4m2

Ai = bi,reinforced · c− c = 0.18 · 0.115 = 2.07 · 10−2m2

174


ρ = As

Ai= 2.26 · 10−4/2.07 · 10−2 = 1.09 · 10−2

The material parameters for reinforcement, B500:

fy = 550MPa

The stresses caused by the minimum external force across the interface:

σn = 2.63 · 10−2MN/m2

The design shear resistance at the interface without reinforcement:

Figure B.2: Dimensions of the unreinforced model, seen from above.

νRdi,unrein = cfctd + µσn = 0.5 · 1.45 + 0.9 · 2.63 · 10−2 = 7.49 · 10−1MN/m2

0.5vfcd = 0.5 · 5.28 · 10−1 · 20 = 5.28MN/m2

7.49 · 10−1MN/m2 ≤ 5.28MN/m2

The shear transferred from the external loads:

Vexernalload =τ ·Av,unrein

cunrein·bi,unreinforced= 3.92·10−1·4.25·10−2

0.25·0.07 = 9.52 · 10−1MN/m2

7.49 · 10−1MN/m2 < 9.52 · 10−1MN/m2

This shows that the unreinforced model does not have enough shear capacity toresist the applied loads.

The maximum load increment in the model:τ ·Av,unrein

2= 3.92·10−1·4.25·10−2

2= 8.33 · 10−3MN

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The design value of the shear stress in the interface with reinforcement:

(a)

c-c

(b)

Figure B.3: (a) Dimensions of the reinforced model, seen from above.(b) C-c distance between the transversal U-bars.

νRdi,rein = cfctd + µσn + ρfyd(µ sinα + cosα) = 0.5 · 1.45 + 0.9 · 3.63 · 10−2 + 1.09 ·10−2 · 550(0.9 · sin(90) + cos(90)) = 2.89MN/m2

0.5vfcd = 0.5 · 5.28 · 10−1 · 20 = 5.28MN/m2

2.89MN/m2 ≤ 5.28MN/m2

The shear transferred from the external loads:

Vexernalload =τ ·Av,rein

crein·bi,reinforced= 3.92·10−1·5.95·10−2

0.35·0.18 = 3.70 · 10−1MN/m2

2.89MN/m2 < 3.70 · 10−1MN/m2

This shows that the reinforced models have enough shear capacity to resist theapplied loads.

The maximum load increment in the model:τ ·Av,rein

2= 3.92·10−1·5.95·10−2

2= 1.17 · 10−2MN

The design shear resistance presented in the figures 7.19 and 7.20:

The design shear resistance at the interface according to Eurocode is converted toloads in order to compare it with the results from the simulations. The values aredivided in half due to the utilization of the symmetry in the models.

VRdi,EC2,UNREIN =νRdi,unrein·(b3·cunrein)

2= 7.49·10−1·(0.25·0.064)

2= 5.99 · 10−3MN

VRdi,EC2,REIN =νRdi,rein·(b3·crein)

2= 2.89·(0.35·0.064)

2= 3.24 · 10−2MN

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