parametric modelling and fe analysis of architectural frp

Parametric modelling and FE analysisof architectural FRP modulesEstablishment of a computational design and analysisapproach for preliminary design

Masterrsquos thesis in Structural Engineering and Building Technology

HANNA ISABELLA NAumlRHI

Department of Industrial and Materials ScienceCHALMERS UNIVERSITY OF TECHNOLOGYGothenburg Sweden 2018

Parametric modelling and FE analysis ofarchitectural FRP modules

Establishment of a computational design and analysis approach forpreliminary design


Department of Industrial and Materials ScienceDivision of Material and Computational Mechanics

Chalmers University of TechnologyGothenburg Sweden 2018

Parametric modelling and FE analysis of architectural FRP modulesEstablishment of a computational design and analysis approach for preliminary designHANNA ISABELLA NAumlRHI

copy HANNA ISABELLA NAumlRHI 2018SupervisorDr-Ing Thiemo Fildhuth Knippers Helbig GmbH StuttgartExaminerDr Mats Ander Department of Industrial and Materials Science Chalmers Gothenburg

Department of Industrial and Materials ScienceDivision of Material and Computational MechanicsChalmers University of TechnologySE-412 96 GothenburgTelephone +46 31 772 1000

Cover Displacement plot for FRP module with beam elements modelled parametricallyin Rhinoceros 3D Grasshopper and SOFiSTiK The displacements have been importedinto Rhinoceros 3D and Grasshopper from SOFiSTiK using parametric tools developedfor this study

Typeset in LATEXGothenburg Sweden 2018

iv

Parametric modelling and FE analysis of architectural FRP modulesEstablishment of a computational design and analysis approach for preliminary designHANNA ISABELLA NAumlRHIDepartment of Industrial and Materials ScienceDivision of Material and Computational MechanicsChalmers University of Technology

Abstract

Despite low weight and high strength and stiffness Fibre Reinforced Polymers FRP areseldom used in building construction This could be due to complexity in dimensioningand manufacturing as well as lack of building design codes and standards for FRP TheUniversity of Stuttgart has developed an analysis and manufacturing method for FRPin architectural contexts This is the basis for a research project at Knippers HelbigGmbH where a new type of FRP module is developed for upcoming building projectsThe analysis method developed at The university of Stuttgart is however not applicablefor this FRP module and demands are also stricter since the modules will be used inbuilding design

In this masterrsquos thesis a new analysis method is developed For analysis SOFiSTiK wasrequired which imposed limitations thus simplifications were necessary The modulematerial is mainly uni-directional with mechanically coupled node zones Due to thethickness of the module cross sections it was assumed that coupling effects could beneglected and that the material behaviour will be transversely isotropic It was alsoassumed that beam elements could be used in analysis due to mainly uni-axial loadtransfer

To verify the assumptions case studies were carried out in MATLAB and SOFiSTiK withClassical Lamination Theory as a basis Two parametric FE models were created usingSOFiSTiK QUAD and beam elements The results were compared using parametric toolsdeveloped for this study There was a correspondence between the results obtained inthe MATLAB and SOFiSTiK analyses as well as between the QUAD and beam modelsexcept for in the case of torsion and shear where internal stresses were within a similarrange but displacements were not Here it was concluded that verification by mechanicaltests are required

The models were applied to variations of a full scale module with similar results Theperformance of the beam models were significantly better and it was concluded that theQUAD model could not be used in large scale models due to long computational timesDue to similarities in internal stresses and forces the beam model should be safe-sided andaccurate enough for preliminary design assuming that the models can be verified againstmechanical tests

Keywords FRP CFRP carbon fibre FEA parametric design structural designstructural analysis computational mechanics finite element method architecture

v

Parametrisk modellering och analys av arkitektoniska FRP modulerEtablering av modellerings- och analysmetod foumlr preliminaumlr dimensioneringHANNA ISABELLA NAumlRHIInstitutionen foumlr Industri- och materialvetenskapAvdelningen foumlr Material- och beraumlkningsmekanikChalmers University of Technology

Sammanfattning

Trots laringg vikt och houmlg haringllfasthet och styvhet aumlr fiberkompositer FRP saumlllan anvaumlndainom byggnadskonstruktion En anledning kan vara paring grund av komplexitet idimensionering och tillverkning samt att byggnadsnormer och standarder saknas foumlrFRP Institutionerna ITKE och ICD vid Stuttgarts Universitet har utvecklat en analys-och tillverkningsmetod foumlr FRP i arkitektoniska kontexter Detta ligger till grund foumlrett utvecklingsprojekt paring konstruktoumlrsfirman Knippers Helbig GmbH daumlr en ny typ avFRP-modul utvecklas foumlr kommande byggnadsprojekt Analysmetoden som utvecklatsvid Stuttgarts Universitet aumlr inte tillaumlmpbar foumlr den studerade FRP-modulen och kravenaumlr straumlngare eftersom modulerna ska anvaumlndas i byggnadsdesign

En ny analysmetod utvecklas i detta examensarbete Foumlr strukturanalys kraumlvdesSOFiSTiK vilket medfoumlrde begraumlnsningar foumlrenklingar var daumlrfoumlr noumldvaumlndigaModulmaterialet aumlr fraumlmst uni-directional (enkel-riktat) mekaniskt koppladenodzoner Paring grund av modultvaumlrsnittens tjocklek antogs dessa effekter foumlrsumbaraoch materialbeteendet transversellt isotropiskt Det antogs ocksaring att balkelement kananvaumlndas i strukturanalys eftersom lastoumlverfoumlringen aumlr fraumlmst enaxlig

Foumlr att verifiera antagandena utfoumlrdes studier i MATLAB och SOFiSTiK med KlassiskLaminatteori som teoretisk bas Tvaring parametriska modeller med SOFiSTiK QUAD-och balkelement utvecklades Resultaten jaumlmfoumlrdes med hjaumllp av parametriska verktygutvecklade foumlr den haumlr studien Resultaten fraringn MATLAB och SOFiSTiK oumlverensstaumlmdeoch aumlven QUAD- och balkmodellerna med undantag foumlr fallen med skjuvning och vridningdaumlr spaumlnningarna oumlverensstaumlmde men deformationer gjorde inte det Haumlr drogs slutsatsenatt kalibrering mot fysiska tester aumlr noumldvaumlndiga foumlr verifierering av resultaten

De parametriska modellerna tillaumlmpades paring variationer av en fullskalig modul Detobserverades att balkmodellen presterade avsevaumlrt baumlttre och att QUAD-modellendaumlrfoumlr inte aumlr tillaumlmpbar i storskaliga modeller paring grund av foumlr laringnga beraumlkningtiderBalkmodellen ansaringgs vara konservativ och tillraumlckligt korrekt foumlr preliminaumlrdimensionering eftersom snittkrafter och spaumlnningar staumlmde oumlverens foumlrutsatt attmetoden kan verifieras mot fysiska tester

Nyckelord FRP CFRP kolfiber FEA parametrisk design konstruktion strukturanalysberaumlkningsmekanik finitea elementmetoden arkitektur

vi

TABLEOFCONTENTS

Abstract v

Sammanfattning vi

Nomenclature ix

Terminology xii

Preface

1 Background 111 FRP in architectural and structural applications 212 New type of architectural FRP modules 6

2 Problem statement 921 Parametric design and FE analysis approach 1022 Limitations and excluded topics 1123 Hypotheses 13

3 Methods 1531 Methods 1632 Report outline 1833 Case studies 20

4 Preconditions 2341 FRP module definitions and parameters 2442 Parametric 3D modelling and structural analysis tools 29

5 Theoretical background 3351 Fibre reinforced plastic composites 3452 Mechanical properties of FRP laminae 3853 Constitutive equations for FRP laminates 4054 Mechanical coupling in FRP laminates 4555 Equivalent material parameters for Fully Orthotropic Laminates 48

6 Material parameters for structural analysis 5161 Analysis of FRP composites 5362 Mechanical response of FRP laminae 5563 Mechanical response of FRP laminates 5864 Equivalent material parameters 64

vii

7 Material parameters for FE analysis 6971 Parametric QUAD and beam models for verification of material parameters 7172 Adaptations of material parameters for FE analysis 7373 Verification of material parameters for QUAD elements 7774 Materials and cross sections for beam elements 88

8 Development of a parametric design and analysis approach 9781 Preliminary evaluation of detailed parametric models 9982 Comparison and evaluation of QUAD and beam models 102

9 Verification of the proposed parametric design and analysis approach 10791 Verification and calibration approaches 109

10 Application of the proposed parametric design and analysis approach 111101 Evaluation of FE models for full scale FRP module 113102 Comparison between QUAD and beam models 117

11 Discussion 121111 Establishment of design and analysis approach 123112 Evaluation of assumptions and results 125

12 Conclusion and recommendations 129121 Concluding remarks 130122 Suggestions for future research 131

References

List of Figures

List of Tables

Appendices

A Explanations and definitions

B Programs and parametric tools

viii

Nomenclature

Roman upper case letters

A Extensional stiffness matrixA Beam cross section areaAij Components of the A matrixAy Shear deformation area in the y-directionB Coupling stiffness matrixBij Components of the B matrixD Bending stiffness matrixDij Components of the D matrixEf Longitudinal modulus of fibre phaseEL Longitudinal modulus of a compositeEm Longitudinal modulus of matrix phaseET Transverse modulus of a compositeGLT Shear modulus of a compositeIy Area moment of inertiaIz Area moment of inertiaL Plate lengthM Moment vectorM Plate momentMT Beam torsional momentMy Beam bending momentMz Beam bending momentN Force vectorN Plate normal forceNx Beam normal forcePL Load carried by compositePfL Load carried by fibresPmL Load carried by fibresQ Lamina stiffness matrixQ Lamina stiffness matrix in reference coordinate systemQij Components of the Q matrixR Radius of plate curvatureS Lamina compliance matrixT Transformation matrixT1 Transformation matrixT2 Transformation matrix

ix

Vf Fibre volume fractionVm Matrix volume fractionWT Torsional constant for beam cross sections

Roman lower case letters

k Lamina number in laminaten Number of laminae in a laminatehk Bottom position of a lamina in a laminatet Laminate thicknesstk Lamina thicknessu Displacement in the x-directionu0 Mid-plane displacement in the x-directionv Displacement in the y-directionv0 Mid-plane displacement in the y-directionvc Composite volumevf Fibre volumevm Matrix volumew0 Mid-plane displacement in the z-direction

Greek upper case letters

∆ Difference operator

Greek lower case letters

αL Longitudinal thermal expansion coefficientαT Transverse thermal expansion coefficientϵ Strain vectorε Strainϵ0 Plate mid plane strainεc Composite strainεf Fibre strainεm Matrix strainη Parameter for estimation of composite transverse and shear stiffnessγ Shear strainκ Curvatureκ Curvatureν Poissonrsquos ratioϕx Shear angle in x direction

x

ϕy Shear angle in y directionψ Third Euler angleσ Stress vectorσ Normal stressσx Beam normal stressτ Shear stressτmax Maximum beam shear stressτT Torsional shear stress in beamτxy Maximum beam shear stress in the xy planeθ Orientation of laminaφx Slope of laminate mid-plane in x-directionφy Slope of laminate mid-plane in y-directionξ Parameter for estimation of composite transverse and shear stiffness

Miscellaneous symbols

part Partial derivative

xi

Terminology

Acronyms

CAD Computer Aided DesignCFRP Carbon Fibre Reinforced PolymerCLT Classical Laminate TheoryFEA Finite Element AnalysisFOL Fully Orthotropic LaminateFRP Fibre Reinforced PolymerGFRP Glass Fibre Reinforced PolymerLOC Laminate Orientation CodeOOP Object Oriented ProgrammingROM Rule of MixturesSAR Structural AreaSLN Structural LineSPT Structural Point

Terms

angle-ply laminate pg 46balanced laminate pg 46CADINP pg 31Classical Lamination Theory pg 41cross-ply laminate pg 46C pg 30fibre bundle pg 24fibre mesh pg 24Fully Orthotropic Laminate pg 48generally orthotropic lamina pg 38lamina pg 38laminate pg 40Laminate Orientation Code pg 40layup pg 40longitudinal direction pg 34matrix pg 34mechanical coupling pg 45mesh angle pg 26mesh density pg 26mesh segment pg 26

xii

node zone pg 26parametric modelling pg 29ply pg 38QUAD pg 31reinforcement pg 34specially orthotropic lamina pg 38structural area pg 31structural line pg 31structural point pg 31symmetric laminate pg 46transverse direction pg 34transversely isotropic pg 34uni-directional pg 34uni-directional laminate pg 46uni-directional zone pg 26volume fraction pg 35

xiii

Preface

This masterrsquos thesis is the conclusion of my studies at the Architecture and Engineeringprogram at Chalmers University of Technology in Gothenburg During my studies I havedeveloped an interest for computational and parametric design structural analysis andlightweight structures I have deepened my knowledge in these topics for the past twoyears in Stuttgart Germany where I have done internships related to architecture andstructural engineering and also conducted this masterrsquos thesis work

The masterrsquos thesis has been carried out at the engineering firm Knippers Helbig GmbHin Stuttgart under the supervision of Dr-Ing Thiemo Fildhuth The examiner andsupervisor at Chalmers in Gothenburg has been Dr Mats Ander at the Department ofIndustrial Materials Science I extend my gratitude to both Thiemo Fildhuth and MatsAnder who have offered invaluable guidance and feedback throughout this thesis work

I would like to also give my thanks to persons and organisations who have been helpfulduring the work on this thesis First I would like to thank the German-Swedish Chamberof Commerce for granting me a travel scholarship from Emil Possehlrsquos scholarship fundfor conducting my master thesis within the technical field in Germany And a thank youto my opponents Marina Mossberg and Samuel Eliasson

My colleagues and the management at Knippers Helbig in Stuttgart are also thanked Iwould like to thank my colleague Amin Li who has offered help and guidance on manyoccasions during this work and our office manager in Stuttgart Matthias Oppe for givingme the opportunity to conduct my thesis at the office And a big thank you to all mycolleagues who have been helpful and offered moral support or who have just been greatcompany during our lunch and coffee breaks

And finally to the two people who have always been a great support and have helped mekeep on track to reach my goal my mom Pirjo Naumlrhi and sister Saga Tinnevik -ett stort och oumldmjukt tack

Jag har vaumlrlden framfoumlr mina foumltterJag vill ut i denSlaumlpp mig fri

Jag har vaumlrlden framfoumlr mina foumltterVad haringller mig fastBara tankens kraft

Jag har vaumlrlden framfoumlr mina foumltterJag kaumlnner det slaumlppaTvekans grepp

Jag har vaumlrlden framfoumlr mina foumltterJag vaumlntar ett oumlgonblick tillSen aumlr den min

Hanna Isabella Naumlrhi Stuttgart September 2018

1BACKGROUND

1 BACKGROUND

11 FRP in architectural and structural applications

Despite their high strength low weight and high stiffness Fibre Reinforced PolymersFRP are not often used in building construction [1] even though FRP have been on thetopic in the building industries since the 1950rsquos and is widely used in for instance theautomotive and aerospace industries [2] see Figure 11 This could be because of thecomplexity in the dimensioning methods and manufacturing techniques which are notdirectly transferable to an architectural context as well as the lack of established designcodes and standards for the use of these types of materials in buildings [2]

Figure 11 Nearly half of the air frame of the Boeing 787 consists of carbonfibre reinforced plastics and other composites [3]From [4] Reproduced with permission CC-BY-SA 30 [5]

111 FRP in structural applications

In the building industry there is a lack of comprehensive and widespread knowledge ofthe mechanical behaviours of FRP which is necessary for a successful implementation[2] The anisotropic nature of FRP composites requires accurate modelling and analysistechniques through Finite Element Analysis FEA In the building industry FEAis common practice but the lack of knowledge and proper analysis strategies for FRPmaterials often results in over simplified models

Currently there exists no EN Eurocode which provide design rules for FRP materialin building design [6] Despite this several structures where FRP has been used asa structural material have been realised in Europe There is a growing interest forthese materials with several research projects related to modelling and testing of analysismodels and prototype testing Several international journals also exist which are dedicatedto the topic of FRP materials in building contexts Thus there is a current demand forstandardised guidelines as well as accurate modelling and analysis methods

Based on these developments and the practical experience gained there exists now enoughknowledge to establish a foundation for a new European guidelines A set of pre-normativeguidelines has been published [6] as a step in this process Prospect for New Guidance inthe Design of FRP After an evaluation process of these the CEN Technical Committee250 (CENT250) will determine if these can be converted into EN Eurocodes [7]

2

1 BACKGROUND

(a) (b)

Figure 12 Examples of existing structures in Europe where FRP compositeshave been used as structural materials(a) The pedestrian bridge at Kolding Denmark is made from

100 pultruded GFRP profiles [6]copyLuigi Ascione From [6] Reproduced with permission

(b) More than 80 of this Solar Charging station SUDI TMin Joueacute les Tours France is made from composites [6]copyLuigi Ascione From [6] Reproduced with permission

112 FRP in architectural applications

Some of the benefits of FRP composites such as low weight and the possibility to mouldthe material into a large variety of shapes can be a functional and aesthetic advantagefor architects and designers [1] Complex shapes can be achieved using FRP compositesand used as for instance facade panels were great materials savings can be done in thesupporting structure compared to if traditional materials are used see Figure 13

Figure 13 SFMOMA faccedilade of Snoslashhetta expansion photo copyHenrik Kamcourtesy SFMOMA Reproduced with permission From [8]More than 700 unique FRP panels make up the facade of theSan Francisco Museum of Modern Art SFMOMA [9] It wasestimated that 4536 tonnes of additional steel would have beenrequired in the supporting structure if the facade had beenconstructed using glass-reinforced concrete instead [1]

3

1 BACKGROUND

113 Architectural and structural collaboration

The computational and 3D modelling methods available today have made it possible todesign and analyse geometrically and structurally complex buildings The automotive andaerospace industries have successfully adopted these tools and methods and developeddedicated fabrication methods and simulation strategies [2] These methods however arerarely suitable for architectural or structural contexts as these require shaping mouldsand are based on serialised production of many similar parts which is rarely the case inarchitecture where each building is unique [10]

(a) (b)

(c) (d)

Figure 14 ICDITKE Research pavilions photos and development process(a) Photograph of research pavilion 2013-2014

From ITKE [11] Reproduced with permission(b) Analysis model for research pavilion 2013-2014

From ITKE [12] Reproduced with permission(c) Photograph of research pavilion 2014-2015

From ITKE [13] Reproduced with permission(d) Analysis model for research pavilion 2014-2015

From ITKE[14] Reproduced with permissionThrough a series of pavilion designs a new core-less free-winding robotic manufacturingmethod for glass and carbon FRP composites GFRPCFRP has been developedat the Institute of Building Structures and Structural Design (ITKE) and Institute forComputational Design and Construction (ICD) at the University of Stuttgart see Figure14 [10] Pavilions have historically driven technological development in architecture sincethey are temporary and donrsquot have to conform to the standard building codes

4

1 BACKGROUND

The manufacturing method which has been developed by the ITKE and ICD is a roboticwinding method see Figure 15 Thin FRP rovings are wound in a dense pattern withmultiple fibre directions forming a shell-like structure This eliminates the need of mouldsand allow for expressive free form shapes [15]

The load transfer in these modules is closely linked to the fibre directions but also thegeometry of the structure These multiple directions which give rise to a multi-axialstress state as well as the coupling between geometry and structural models require newmethods of analysis to be developed [2] Parametric tools which are becoming commonpractice in architecture made it possible to develop a context specific method of analysisfor these pavilions where the fibre orientations could be optimised through an iterativeanalysis process using parametric design and analysis tools

Figure 15 Thin FRP rovings are wound freely in the air by robotsFrom ITKE [16] Reproduced with permission

5

1 BACKGROUND

12 New type of architectural FRPmodules

The Stuttgart University research pavilions are the basis for a research and developmentproject at Knippers Helbig GmbH Stuttgart where the goal is to establish a designand analysis method which can be applied in upcoming projects for a new type of FRPmodule which is currently being developed see Figure 16

Figure 16 Principal geometry topology and layout of uni-directional fibrebundles of the proposed FRP module

This FRP module differs from the research pavilions in the way that the fibre mesh isperforated and the fibres are organised such that they form distinct fibre bundles withquite large cross sections which provide uni-axial load paths The structural behaviourof this module differs thus from the pavilion designs in which the fibres where distributedin multiple directions where they formed a shell like structure whereas in this module theload transfer is more beam-like and uni-axial The geometry and fibre bundles layout ofthis FRP module have been decided through an architectural design process rather thanby structural optimisation This poses a new challenge as the analysis method used forthe research pavilions no longer can be applied

The large fibre bundle cross sections form two distinctive zones in the module - zonesof uni-directional beam like sections with uni-axial load transfer and node zones wherethe fibre bundles overlap with multi-directional load transfer further there will be adiscontinuity zone due to a distinct change in geometry and stiffness see Figure 17

The manufacturing method of these modules will affect the geometry of the node whichin turn will affect the structural behaviour of the module The manufacturing method ofthese modules have not been determined in this stage however it is fairly certain therewill be layers of fibres stacked sequentially with alternating orientations as in Figure 18Two methods can be employed here the first where the fibre layers are continuous asin Figures 18a to 18c forming a distinct undulating node or as in Figures 18d to 18fwhere the layers are discontinuous and thus the node will remain flat

6

1 BACKGROUND

Figure 17 Discontinuity of the node zones(A) Uni-directional zone with uni-axial load transfer(B) Discontinuity region due to change of geometry and

stiffness(C) Node zone with multi-axial load transfer

(a) (b) (c)

(d) (e) (f)

Figure 18 Principal geometry and fibre layout of different types of nodes(a) Continuous FRP layers with different orientations at the

node zones of the FRP module(b) Undulating geometry at the node zone due to overlapping

FRP layers(c) Section view of undulating node(d) Discontinuous FRP layers with different orientations at the

node zones(e) Flat geometry at node due to discontinuous FRP layers(f) Section view of flat node

7

2PROBLEM STATEMENT

2 PROBLEM STATEMENT

21 Parametric design and FE analysis approach

The goal of the masterrsquos thesis work at Knippers Helbig GmbH is to use these modulesin upcoming projects and as such the requirements of these modules are much stricterthan for the research pavilions Several aspects must be considered such as applicabilityin practice manufacturing compliance and safety The lack of standards and norms alsoputs high demands on the accuracy of analysis and mechanical testing is required

The FRP modules are planned to be used in large scale projects where multiple of thesemodules are assembled to form a large structure This structure will be analysed usingconventional structural engineering software where the possibilities to analyse anisotropicmaterials are limited This means that the analysis method must be simplified withregard to material formulations number of elements and geometry while at the sametime efficiently provide accurate results which can be used for dimensioning in preliminarydesign

211 Aim and scope

The aim is to establish a simplified reliable and accurate parametric design and analysisapproach for preliminary design of the FRP modules described in Section 12 Theestablished design method should be applicable to varying geometries and mesh densitiesand must therefore be parametric The proposed design and analysis approach shouldfulfil a number of requirements which are listed below

Global and detailed analysis contextThe design approach should be applicable to a structural engineering context - meaningit should be possible to include and analyse the FRP modules in a large scale structuralmodel a global structural model Typically this kind of model would include a largenumber of structural elements and several load combinations The design approach shouldalso be applicable to small scale models for detailed design

Conservative approachThere are many uncertainties regarding this type of FRP module and the establish designand analysis method should therefore generate results on the safe side

Applicability of resultsIt should also be possible to extract the results namely deflections and displacements aswell as internal forces and stresses from such a global model in an easy manner and usethese for detailed design

Parametric design approachDue to the many uncertainties the method should be fully parametric so it is possibleto make geometrical and structural adjustments and apply the method to a variety ofgeometries and meshes

VerificationMechanical tests will be conducted on physical prototypes and the design approach shouldalso be established such that verification against test data can be done in a straight forwardmanner

10

2 PROBLEM STATEMENT

22 Limitations and excluded topics

A number of limitations are imposed on the study and are presented below Additionallysome topics have been excluded and are also presented below

Geometry and topology limitationsThe study will be limited to one type of geometry and topology but considerations willbe made so that the method can be applied to varying geometries and topologies

Computational limitationsThere are computational limitations with regard to processing power computational timeas well as softwares The analysis should therefore be simplified with regard to numberof FE elements to reduce computational times The main softwares which will be usedare SOFiSTiK MATLAB Rhinoceros 3D and Grasshopper

Structural analysis limitationsThe structural analysis is required to be performed in SOFiSTiK This is partly becauseit is the main software used for structural analysis at Knippers Helbig GmbH and alsodue to the fact that the established method should be applicable in a civil engineeringcontext using large scale global models which must comply with building standards anddesign codes for this purpose SOFiSTiK is well suited Additionally SOFiSTiK providebuilt in interfaces with multiple CAD softwares and has its own native scripting languagewhich is quite useful for parametric design and analysis

The structural analysis which is basis for establishment of simplified materialformulations will be limited to in-plane mechanical behaviour and the assumptionsof the Euler-Bernoulli beam theory and the Kirchhoff-Love plate theory will be usedIn SOFiSTiK the analysis will be conducted using Timoshenko beam theory andMindlin-Reissner plate theory since these are default settings for structural analysisThere will however be no consideration or adaptations of stiffness or material formulationswith regard to shear deformation plate thickness or out-of-plane stiffness

Material formulationsThe possibilities to analyse anisotropic and layered materials in SOFiSTiK exist but arecomplicated and limited Therefore it is preferred to use either isotropic or orthotropicas well as non-layered material for analysis in SOFiSTiK Thus the material formulationsused in structural analysis should be limited to orthotropic or transversely isotropicmaterials

Discontinuity zonesDiscontinuities especially at the FRP module boundary edges as well as the nodediscontinuity zones presented in Figure 17 will not be regarded In this study onlyone the flat node type will be regarded

Structural form finding and optimisation of topologymeshThere will be no structural optimisation or form finding of the geometry or topology norwill there be any optimisation with regard to dimensions Preliminary dimensions will beused as well as a predetermined geometry and topology

Residual stresses and imperfectionsResidual stresses and imperfections which may arise due to manufacturing will not beregarded in the study

11

2 PROBLEM STATEMENT

Interlaminar stressesEffects of interlaminar stresses which occur between the layers of FRP lamina will not beconsidered here

Fatigue and failureFatigue and failure mechanics will not be regarded in the study

Non-linearity and instabilityInstability and third order effects due to geometric discontinuity zones or imperfectionswill not be studied

12

2 PROBLEM STATEMENT

23 Hypotheses

Throughout the work a number of assumptions and hypotheses are made and are verifiedor discarded These are presented below

Transverse isotropyA simplified orthotropic material behaviour can be assumed for this type of FRP moduleand in zones where fibres overlap the material can be assumed to be transversely isotropic

Uni-axial load transferIt is assumed that the load transfer in the FRP modules is mainly uni-axial and thatanalysis using beam theory will be sufficient

Verification and calibrationIt is assumed that the material parameters and design and analysis method can becalibrated and verified against mechanical tests

Applicability of resultsThe results from a simplified global analysis can be used in a detailed design situation

13

3METHODS

3 METHODS

31 Methods

The principal methodology and work flow can be seen in Figure 31 The methodologycan be categorised into five parts which are briefly described below

Problem formulationThe problem is formulated and a number of requirements and limitations are statedThe geometry and boundary conditions for the analysis is defined and initial materialparameters are summarised

Theoretical material studiesA literature study in FRP mechanics and Classical Laminate Theory forms the basis offinidng suitable constitutive equations for FRP laminates The result of these studiesis the basis for a number of assumptions of simplifications that can be made for theconstitutive equations for specific types of FRP Laminates A number of studies areconducted to evaluate these assumptions and to derive material parameters which canbe used in a simplified analysis situation

The derived metarial parameters are uses in FE analysis to create equivalent materialswhich can be applied to shell and beam elements and the material parameters are adjustedand verified against the theory and assumptions

Development of parametric analysis approachA parametric design and analysis approach is developed for the specific topology studiedusing SOFiSTiK QUAD (shell) and beam elements

A number of case studies are conducted for different setups and strain and stress resultsof the different models are compared and evaluated The parametric FE model shouldalso be calibrated against mechanical tests

Evaluation of parametric analysis approachThe parametric FE analysis and design approach is applied to a full scale free form FRPmodule using both SOFiSTiK QUAD and beam elements The results are compared andevaluated

Recommendation of parametric analysis approachFinally a summary of the results is made and the results are evaluated The parametric FEand analysis approach is also evaluated and a recommendation is made and suggestionsfor future work is given

16

3 METHODS

Figure 31 Flowchart describing the method and report outline

17

3 METHODS

32 Report outline

The report outline can be seen in Figure 32 and can be used as a reading guide to thereport The report is structured chronologically and in six major parts which are describedbelow

IntroductionThe problem is introduced in Chapter 1 - Background and defined further in Chapter 2 -Problem statement

Methodology and outline of workThe methodology is introduced here in Chapter 3 - Methods where the methods chosenprincipal work flow and report outline are explained Following this the geometrytopology and initial material parameters for the FRP module as well as the tools usedare presented in Chapter 4 - Preconditions

Theoretical backgroundFollowing this a theoretical background to FRP mechanics and Classical Laminate Theorywhich is essential for understanding the following chapters is given in Chapter 5 -Theoretical background Here it is assumed that the reader is already familiar with strengthof materials structural mechanics as well as the Finite Element Method

Case studiesThe theoretical background is the basis for a number of assumptions In Chapter6 - Material parameters for structural analysis these assumptions are presented andevaluated using analytical analysis Following this equivalent material parameters whichcan be used in FE analysis are derived

In Chapter 7 - Material parameters for FE analysis the derived material parameters areadjusted and applied in FE analysis using shell and beam elements The results of theanalysis are evaluated against analytical results Material formulations and cross sectionsare established in SOFiSTiK for use in parametric analysis models

In Chapter 8 - Development of a parametric design and analysis approach a parametricdesign and analysis approach is developed for small FE models using shell and beamelements The models are evaluated and compared Following this in Chapter 9- Verification of the proposed parametric design and analysis approach methods forcalibration of the FE models are presented before finally applying the parametric analysisand design approach to a full scale free form FRP module in Chapter 10 - Application ofthe proposed parametric design and analysis approach

Evaluation and recommendationThe results of the case studies and the proposed parametric design and analysis approachare evaluated in Chapter 11 - Discussion and finally a recommendation is made andsuggestions for future work are given in Chapter 12 - Conclusion and recommendations

References explanations and appendicesThe final part of the report contains appendices describing the parametric studies ingreater detail summaries of input parameters and results as well as references and listof figures and tables

18

3 METHODS


19

3 METHODS

33 Case studies

A number of case studies were conducted to verify the assumptions made and to establisha parametric analysis and design approach The case studies are briefly described belowFurther descriptions and results are presented in Chapters 6 to 10

Material parameters for structural analysisIt was assumed that thick laminates with symmetric and balanced layups and alternatinglamina orientations will behave like fully orthotropic laminates To verify this assumptionthree case studies were conducted

Mechanical response of FRP laminaeMaterial studies are carried out on FRP laminae with varying orientations andfibre volume fractions to evaluate which laminae are most critical with regardto mechanical coupling

Mechanical response of FRP laminatesSymmetric and balanced laminates are studied to evaluate which orientationsare most critical with regard to mechanical coupling in these specific laminatetypes For the most critical orientations it is evaluated how mechanicalcoupling is affected when the number of laminae increases in order to evaluateif mechanical coupling can be neglected

Equivalent material parametersAssuming that the laminates are fully orthotropic efficient engineeringconstants were derived and used to construct equivalent transversely isotropiclaminates A study was conducted to compare and evaluate the mechanicalresponse of the symmetric balanced and equivalent laminates

Material parameters for FE analysisThe engineering constants which were derived were used to define nine transverselyisotropic materials for analysis in SOFiSTiK and these materials were compared againstthe analysis performed in MATLAB for the same materials

Adaptations of material parameters for FE analysisAdaptations of the engineering constants were necessary in order to be able touse the material in analyse in SOFiSTiK

Verification of material parameters for QUAD elementsA case study was conducted in SOFiSTiK using the adapted materialparameters for QUAD elements The same case study was conducted inMATLAB and the results of the studies were compared and evaluated

Adaptations of parameters for beam elementsA case study was conducted in SOFiSTiK comparing the mechanical responsein SOFiSTiK QUAD and beam elements with the adapted materials andadaptations were made to the beam cross sections The displacements andinternal forces and stresses were compared and evaluated for the QUAD modelsand the beam models with adapted cross sections

20

3 METHODS

Development of a parametric design and analysis approachThree parametric models were developed in Grasshopper for structural analysis of theFRP module The three models were compared and evaluated

Preliminary evaluation of detailed parametric modelsThe three models were developed and analysed in SOFiSTiK The QUADmodel was assumed to be the most accurate as this model was the only one totake into account the uni-directional as well as node zones The other modelswere compared using different parametric setups against the QUAD modeland disregarded if the results were not corresponding

Comparison and evaluation of QUAD and beammodelsA case study was conducted for the evaluated QUAD and beam models fordifferent geometrical setups and all materials and cross sections The resultswere evaluated and compared

Verification of the proposed parametric design and analysis approachThe assumptions made in previous chapters should be verified against mechanical testsand the parametric model should be adjusted and calibrated if possible Within this scopemechanical testing has not been possible This is however necessary in order to verify themodel

Verification and calibration approachesSuggestions for verification and calibration approaches are given and asummary of assumptions and aspects of the parametric design and analysisapproach which should be considered is given

Application of the proposed parametric design and analysis approachThe parametric models were applied to variations of a full scale FRP module wheremeshing and computation time as well as deflections and internal forces and stresseswere compared

Evaluation of FE models for full scale FRPmoduleThe meshing and computation times were compared and evaluated for threedifferent types of mesh densities

Comparison between QUAD and beammodelsCase studies were conducted for three mesh densities comparing the mechanicalresponse for compression and tension bending and torsion for the QUAD andbeam models The results were compared and evaluated

21

4PRECONDITIONS

4 PRECONDITIONS

41 FRPmodule definitions and parametersIn this chapter the geometry topology and initial parameters for the FRP module isexplained The parametric design and analysis approach which is developed in this studywill be applied to the module shown here but the proposed design and analysis methodcan be applied to any geometry with the the same geometrical and topological principalswhich are explained in this chapter

411 Geometry and topology

The FRP module can be divided into four geometrical elements the start edge (Figure41a) the end edge (Figure 41b) the center lines of the fibre bundles in the one direction(Figure 41c) and the fibre bundle curves in the other direction (Figure 41d) The finaltopology pattern will be referred to as the mesh or fibre mesh in the following textThe topology for this module has been defined as in Figure 41 and 42 and the definitionsand terminology is explained below

(a) Module start edge (b) Module end edge

(c) FRP curves in firstdirection

(d) FRP curves in seconddirection

Figure 41 The geometrical elements of the FRP module

Topology definitionSplit edge curvesThe edge curves or module boundaries are not planar and because of thisthe edges are first split at the kinks into two edge curve segments see Figure42a

Subdivide start curveThe first edge of the curve segments is subdivided into segments of equallengths The intersection of two of these segments are shown in Figure 42b

Subdivide end curveThe end edge curves are also split and marked with points as the start curvesee Figure 42c

24

4 PRECONDITIONS

Geodesic curveA geodesic line is drawn between a point on the start curve and a point on theedge curve This is repeated for all points along the module boundaries seeFigure 42d

Apply rotationNext the points are moved in a rotating fashion in opposite directions on thetwo edges so that the curve is now extended on the surface as in Figure 42eThis is repeated for all the geodesic curves and points on the module edgesand surface

Reverse rotationFinally the rotation is reversed in order to create curves which are oriented inthe opposite direction as see Figure 42f

(a) Split edge curves (b) Subdivide start curve

(c) Subdivide end curve (d) Geodesic curve

(e) Apply rotation (f) Reverse rotationFigure 42 The topology of the FRP module is defined in these six steps

25

4 PRECONDITIONS

Topology terminology

Mesh segmentIn Figure 43a a mesh segment of the topology is shown

Uni-directional zoneIn Figure 43b a uni-directional element is shown These zones consists ofonly uni-directional fibre bundles

Node zoneIn Figure 43c a node zone is shown These zones consists of overlappingfibre bundles as previously explained in Chapter 1 Section 12

Mesh angleThe angle θ at the intersection of two uni-directional elements is called meshangle

Mesh densityThe number of points on the edge determines the mesh density in Figure44 three mesh densities are shown

(a) Mesh segment (b) Uni-directional zone

(c) Node zone (d) Mesh angleFigure 43 Terminology for the FRP module topology

(a) High (b) Medium (c) LowFigure 44 Different mesh densities of the FRP module

26

4 PRECONDITIONS

412 Coordinate system and directions

In Figure 45 the coordinate system which is used as reference coordinate system for theFRP module is shown the axes are denoted by x y and z In Figure 46 the referencecoordinate systems which are used as local reference coordinates for the uni-directionaland node zones respectively The axes for the uni-directional zones are denoted by L Tand T prime and the axes for the node zones are denoted by 1 2 and 3

(a) Perspective

(b) Side view (c) Top viewFigure 45 Reference coordinate system for the FRP module

Figure 46 Local reference coordinate systems for the uni-directionalelements (left) and the node zones (right) The orientation ofthe coordinate axes has been chosen based on the conventionsused Classical Laminate Theory see Chapter 5 - Theoreticalbackground

27

4 PRECONDITIONS

413 Cross sections

The cross sections which are used can be seen in Table 41 A rectangular cross section isused for the fibre bundle cross sections in analysis A description of fibre layups used inthis study is given in Appendix A11 The CFRP material parameters are explained inSection 414 below and further in Chapters 6 and 7 In FE analysis a solid steel crosssections is used as a boundary element for the FRP module see Chapter 10 The selfweights of the materials have been omitted in analysis

Table 41 Cross section parametersCross section Material Width [mm] Height [mm]Fibre bundle CFRP 20 10Module edge S355 45 45

414 Initial material parameters

The initial material parameters which are used in analysis can be seen in Tables 42 and 42and have been based on data sheet [17][18] and literature values [19 pg 2640][20][6] Theshear moduli in the tables below have been calculated from Equation 41 [21 pg 22] Theshear modulus for fibres is typically in a smaller range (27-50 GPa [6]) however this hassmall influence on the overall strength as the shear strength of the FRP composite is to alarge extent determined by the matrix see Chapter 5 Section 512 Characteristic valueswhich will be used in design should be determined through mechanical tests according toEN1990 [6] however the initial parameters below provide a sufficient initial guesses untilmaterial parameters can be determined through tests

G = E

2(1 + ν)[GPa] (41)

Table 42 Material properties of fibresMaterial property Variable Value UnitElastic longitudinal modulus EfL 238 GPaShear modulus Gf 91538 GPaPoissonrsquos ratio νf 03 minusThermal expansion coefficient αf minus04 10minus6degCFibre volume fraction Vf 052 minus

Table 43 Material properties of matrixMaterial property Variable Value UnitElastic longitudinal modulus EmL 31 GPaShear modulus Gm 1174 GPaPoissonrsquos ratio νm 032 minusThermal expansion coefficient αm 50 10minus6degCMatrix volume fraction Vm 1 minus Vf minus

28

4 PRECONDITIONS

42 Parametric 3Dmodelling and structural analysis toolsComputational tools make it possible to generate complex geometries and systems anddesigners and architects often use digital models to describe and visualise ideas Onedrawback of many of digital 3D modelling and structural analysis tools available is thelack of interactivity in design it can often be difficult to make changes to one aspect of amodel without making substantial changes to other parts [22] parametric modellingtools have made it possible to use parameters to create interactive models in which thedesigner can change a few parameters and the entire model will update Parametricdesign has been described by Wassin Jabi as lsquoA process based on algorithmic thinking thatenables the expression of parameters and rules that together define encode and clarifythe relationship between design intent and design responsersquo [23][22]

421 Parametric modelling with Rhinoceros 3D andGrasshopper

Rhinocerosreg sometimes abbreviated as Rhino is a 3D modelling tool and ComputerAided Design CAD tool developed by Robert McNeel amp Associates [24] As of version6 this software includes parametric capabilities with the built in plugin Grasshopperregwhich was previously a stand alone plugin for Rhino Grasshopper is a graphical algorithmeditor which is closely integrated with the Rhino interface and can be used withoutextensive knowledge about programming [25] It is also possible to extend the capabilitiesin both Rhino and Grasshopper by installing third party plugins or by programmingusing the built in or external programming editors

It is possible to directly export geometry for structural analysis using the interfacebetween SOFiSTiK and Rhinoceros 3D provided by SOFiSTiK [26] This interface isnot parametric and it can thus be tedious to set the structural properties for large orcomplex models Structural properties are stored in the 3D geometry objects as attributestherefore it can be practical to use the scripting and programming or external pluginsto automate the process of applying these attributes For this study a combination ofplugins from external sources grasshopper scripts and custom made plugins were usedto create parametric models and to establish a parametric design and analysis methodfor the studied FRP module type Some terms which are related to 3D modelling whichwill be used in the following text can be found in Appendix A2

4211 Parametric 3Dmodels

A large number of case studies were conducted and several parametric models were createdto simplify and automate the process a description of these models can be found inChapters 7 to 10

4212 Grasshopper tools for evaluation of results

In order to evaluate and compare the results in a straight forward and consistent mannera number of tools were developed in Grasshopper for this specific study These tools wereused to import displacements and internal stress and force results from SOFiSTiK intoRhino in order to calculate strains and curvatures and well as generate models showingdisplacements and stresses both as deformed meshes and with colour gradients Thesetools are described in Appendix B2

29

4 PRECONDITIONS

4213 Grasshopper plugins

The external plugins which were used in this study are listed and briefly described below

HumanThe plugin Human is a plugin which extends Grasshopperrsquos capability of creating andreferencing geometry and handling the Rhino document [27] In this study this pluginwas used to write object attributes (see Appendix A2) containing structural informationto geometrical objects in Rhino

WeaverbirdWeaverbird is a topological mesh editor plugin for Grasshopper [28] In this study thisplugin was used to rebuild free form surfaces NURBS surfaces into polygonised meshesTypically a polygonised mesh is less inaccurate than a NURBS surface but in thecases where precision has been of less importance this plugin has been used to reducecomputation times See Appendix A2 for a brief explanation of NURBs and polygonmeshes

TT toolboxTT Toolbox is a plugin developed by Thornton Tomasetti and features a range of toolswhich can be used in Grasshopper [29] In this study this plugin has been used to readdata from Excel spreadsheets into Grasshopper

LunchboxLunchbox is another plugin for Grasshopper which contains several tools for use in Rhinoand Grasshopper [30] This plugin has been used to automate the baking (see AppendixA2) or generation of geometry from Grasshopper to Rhino

ColorMeshThe plugin ColorMesh is a plugin which can be used to colour meshes to for instancevisualise data [31] This plugin has been used to visualise and compare structural analysisresults from SOFiSTiK which have been imported into Grasshopper

422 Object oriented programmingwith C

To automate the creation of structural attributes for geometrical objects a number ofcomponents were programmed using the Object Oriented Programming languageOOP C (pronounced C-sharp) which is developed by Microsoftreg[32] The pluginswere written using the built in C editor in Grasshopper and were compiled usingVisual Studio 2015 [33] using templates for compilation provided by McNeel [34] Someterms which are related to OOP which will be used in the following text can be found inAppendix A2

4221 Grasshopper components to extend SOFiSTiK interface

The interface provided between Rhinoceros and SOFiSTiK is not parametric thereforethree components were written using C in order to automate this process Thesecomponents create structural information which can be written as attributes togeometrical objects in Rhino These components are described in Appendix B2

30

4 PRECONDITIONS

423 Structural analysis in SOFiSTiK

SOFiSTiK is a FE analysis software for building and infrastructure projects [35] Oneof the benefits with SOFiSTiK is that it has a built-in interface with several BIM andCAD softwares as well as scripting possibilities with CADINP [36] the scripting languagenative to SOFiSTiK

The results from analysis in SOFiSTiK were evaluated in the program module Wingrafwhich is designed to display the analysis results visually [37] However this tool reads theresults directly from a database which is generated each time the analysis is run Dueto the parametric nature of this study the results were thus not saved when the inputparameters are changed This was solved by developing stand-alone tools for graphicalevaluation however Wingraf was used as a means to verify these tools

In SOFiSTiK it is possible to either define the FE model using CADINP code or tocreate geometrical elements in external CAD softwares and assign structural propertiesto these using SOFiSTiKrsquos built in interface with these softwares When assigningstructural properties it is necessary to understand the terminology for structural andgeometrical elements used in SOFiSTiK Below follows a short description of geometricaland structural elements in SOFiSTiK which are used in this study

SLN - Structural lineStructural lines SLN can either be straight lines or curves and are commonly usedcreate truss beam or cable elements

SAR - Structural areaStructural areas SAR are commonly used to create plate or shell elements but can alsobe used to define areas for load application

SPT - Structural pointStructural points SPT are often used for supports but can also be used to define springsor kinematic couplings

QUAD -Quadrilateral 2D elementsIn SOFiSTiK shell elements are modelled with QUAD elements These elements aredefined by four points and are typically analysed using the Mindlin-Reissner Plate Theory

424 Structural analysis usingMATLAB

MATLABregis a computer program and programming language [38] developed byMathWorks It was useful in his study due to the simplicity of performing matrixoperations and calculations In this study a number of already existing MATLABfunctions for analysis of FRP composites were used with permission [39] to establishconstitutive equations for different laminates and to perform simple structural analysissee Appendix B1

4241 MATLAB programs for structural analysis using Classical Lamination Theory

A number of MATLAB programs were written see Appendix B1 using the functionsmentioned above to evaluate assumptions made for different types of laminates

31

5THEORETICAL BACKGROUND

5 THEORETICAL BACKGROUND

51 Fibre reinforced plastic composites

A composite material is a material which consists of two or more distinct constituentmaterials or phases where the phases have significantly different properties [19 pg 1]In this text the focus will be on fibre composites or more specifically uni-directionalFRP composites

511 Uni-directional fibre composites

Uni-directional FRP composites consist of a discontinuous fibre phase dispersed in acontinuous polymer phase The fibre phase is referred to as reinforcement as it isusually the stronger phase and the polymer phase is usually called the matrix [19pg 2] Figure 51a shows a principal cross section of a uni-directional FRP compositewhere fibres vary in diameter and are randomly distributed throughout the matrix inanalysis it will however be assumed that the fibres have uniform properties and diametersand are evenly distributed throughout the matrix a in Figure 51b [19 pg 63]

(a) (b)

Figure 51 Principal cross sections of uni-directional fibre compositesAdapted from Figure 3-2 [19 pg 64](a) In reality the fibres have different diameters and are

randomly distributed throughout the matrix(b) When deriving elastic engineering constants for a

uni-directional FRP composite it is assumed that the fibreshave uniform properties and diameters and that they areevenly distributed throughout the matrix

In Figure 52 a principal representation of a uni-directional FRP composite is shown Thecomposite material can be described by three mutually perpendicular planes of symmetryand three main directions the longitudinal direction L(1) the transverse directionT (2) and the transverse direction perpendicular to the lamina plane T prime(3) [19pg 63] The two transverse directions T and T prime exhibit nearly identical mechanicalproperties due to the random distribution of fibres as in Figure 51 and the plane TT prime(23)can therefore be said to be isotropic The longitudinal and transverse directions havesignificantly different properties due to this the material can be said to be transverselyisotropic where the TT prime(23) plane is isotropic and the LT (12) and LT prime planes are equalin properties and orthotropic

34


Figure 52 Principal representation of a uni-directional fibre compositeAdapted from Figure 3-1 [19 pg 63](1) Longitudinal direction in composite plane denoted by L or

1 in the following text(2) Transverse direction in composite plane denoted by T or

2 in the following text(3) Transverse direction perpendicular to composite plane

denoted by Trsquo or 3 in the following text

512 Mechanical properties of FRP composites

The mechanical properties in the longitudinal and transverse directions are dependent onthe two constituent phases Each of these phases have significantly different mechanicalproperties which will affect the final structural behaviour of the composite materialHowever it is not only the mechanical properties which determine the final compositebut also the ratio of the two materials - the volume fractions of the fibre and matrixphases Vf and Vm The volume fractions are defined as ratios of the constituent materialvolumes vf and vm and the total composite volume vc see Equations 51 to 53 [19pg 64-67]

vc = vf + vm [m3] (51)

Vf = vf

vc

(52)

Vm = vm

vc

(53)

35


5121 Longitudinal stiffness

To determine the longitudinal stiffness EL of a uni-directional composite it is assumedthat the fibres are uniformly and continuously distributed throughout the compositeparallel to the longitudinal material axis with a uniform diameter as in Figure 52

Further it is assumed the material properties of the fibres are also uniform and thatthe bonding between the fibres and matrix is perfect which means that no slippage canoccur and the strains of the composite εc and the sum of the strains of the constituentmaterials εf and εm are equal see Equation 54 [19 pg 68]

εc = εf = εm (54)

For a longitudinally loaded composite the load carried in the composite PL must thusbe evenly distributed and the sum of the loads carried by the fibres and matrix PfL andPmL respectively must equal the total applied load as in equation 55

PL = PfL + PmL (55)

This means that the longitudinal stiffness of the laminate can be directly determined fromthe volume fractions and the stiffness of the fibre and matrix phases Ef and Em as inequation 56 This kind of relation is called the Rule of Mixtures ROM For derivationof this expression see Agarwal [19 pg 69]

EL = EfVf + EmVm [GPa] (56)

5122 Transverse stiffness

The transverse stiffness ET can be be derived in a similar manner as the longitudinalstiffness and is expressed as in Equation 57 Here it is assumed that the stress is constantover the cross section perpendicular to the direction of loading for a transversely loadedcomposite For derivation of this expression see Agarwal [19 pg 80]

1ET

= Vf

Ef

+ Vm

Em

(57)

Experiments have however shown that the transverse modulus is better determinedexperimentally as the actual distribution of fibres throughout the composite is not uniformas in Figure 52 but rather random as in Figure 51a [19 pg 83] A better estimationof the transverse modulus has been proposed by Halpin and Tsai and the equation fortransverse stiffness can be expressed using Equations 58 and 59 The term ξ is a factordepending on several parameters but can be assumed to be 2 for rectangular or circularcross sections [19 pg 85]

ET

Em

= 1 + ξηVf

1 minus ηVf

(58)

η = (EfEm) minus 1(EfEm) + ξ

(59)

36


5123 Shear stiffness

The in-plane shear stiffness GLT of a composite may be determined in the same wasas the transverse stiffness was determined in equation 57 For detailed derivation of thetransverse stiffness as given in Equation 510 see Agarwal [19 pg 91]

1GLT

= Vf

Gf

+ Vm

Gm

(510)

Again a better estimation for the shear stiffness has been proposed by Halpin and Tsaias in Equations 511 and 512 where it can be assumed that ξ = 1

GLT

Gm

= 1 + ξηVf

1 minus ηVf

(511)

η = (GfGm) minus 1(GfGm) + ξ

(512)

5124 Poissonrsquos ratio

The Major Poissonrsquos ratio νLT for the composite material can be determined from theRule of Mixtures for the Major Poissonrsquos ratio see Equation 513 The MinorPoissonrsquos ratio νLT can be expressed with the previously established values for thelongitudinal and transverse stiffness EL and ET see equation 514 For detailed derivationsee Agarwal [19 pg 95-96]

νLT = νfVf + νmVm (513)

νLT

EL

= νT L

ET

(514)

5125 Thermal expansion coefficients

Thermal expansion coefficients αL and αT for uni-directional FRP composites have beenderived by Schapery [19 pg 111] see equations 515 and 516

αL = 1EL

(αfEfVf + αmEmVm) [1degC] (515)

αT = (1 + νf )αfVf + (1 + νm)αmVm minus αLνLT [1degC] (516)

37


52 Mechanical properties of FRP laminae

One layer of a uni-directional FRP composite is called lamina or ply Several laminaeor plies are stacked on top each other to form a FRP laminate see Section 53 InSection 511 it was stated that all uni-directional FRP are transversely isotropic thismeans that four independent engineering constants are needed to relate stresses andstrains in a lamina [19 pg 160] It is practical to express this relation in terms ofa reference coordinate system as in Figure 53 When the longitudinal and transversematerial directions align with the axes of the reference coordinate system the lamina iscalled a specially orthotropic lamina when the axes do not align the lamina is calleda generally orthotropic lamina

(a) (b)

Figure 53 Specially and generally orthotropic laminae Adapted fromFigure 5-2 [19 pg 161](a) Specially orthotropic lamina(b) Generally orthotropic lamina

A common linear relationship which relates stresses σ and strains ϵ is the GeneralizedHookersquos Law [19 pg 175] and can be expressed as

σij = Eijklϵkl [GPa] (517)

where Eijkl is a fourth order tensor represented as a matrix which contains the elasticconstants of the material see Appendix A31 In the most general case this matrixcontains 21 independent components but in the case of transversely isotropic materialsthe number of components can be reduced to six where five are independent [19 pg177-180]

521 Specially orthotropic lamina under plane stress

For a specially orthotropic lamina loaded under plane stress stress strain relationship canbe expressed as in Equation 518 or on matrix form as in equation 519 [19 pg 182-186]

σ12 = Qϵ12 [GPa] (518)

38


σ1

σ2

τ 12

=

Q11 Q12 Q16

Q22 Q26

Sym Q66

ε1

ε2

γ12

[GPa] (519)

The Q matrix is called the lamina stiffness matrix for the composite The inverse ofthe stiffness matrix is called the compliance matrix S From the compliance matrix itis possible to derive equations in terms of the material engineering constants for thecomponents Qij of the Q matrix see Equations 520 to 524 and Appendix A31

Q11 = EL

1 minus νLTνT L

(520)

Q22 = ET

1 minus νLTνT L

(521)

Q12 = νLTET

1 minus νLTνT L

= νT LEL

1 minus νLTνT L

(522)

Q66 = GLT (523)

Q16 = Q26 = 0 (524)

522 Generally orthotropic lamina under plane stress

The stresses and strains for a generally orthotropic lamina oriented with the angle θ inthe reference coordinate system can be obtained by transformation as in Equations 525and 526[19 pg 189-192]

σL

σT

τLT

= σLT =[T1

]θσ12 (525)

εL

εT

γLT

= ϵLT =[T2

]θϵ12 (526)

The matrices T 1 and T 2 are transformation matrices with regard to θ see AppendixA32 [19 pg 190] The stresses for the lamina in the reference coordinate system cantherefore be expressed as in Equations 527 and 528 where Q is the transformed stiffnessmatrix for the lamina

σ12 =[T1

]minus1

θQ

[T2

]θϵ12 (527)

σ12 = Qθ ϵ12 (528)

39


53 Constitutive equations for FRP laminatesA FRP laminate consists of laminae or plies stacked on top of each other as in Figure54a A local coordinate system is used to describe each lamina in relation to the mid-planeof the laminate as in Figure 54b A laminate with thickness t is constructed from nnumber of plies where the laminate reference coordinate system is placed in the mid-planeof the laminate The bottom position related to the local z axis for a lamina k is givenas hk The thickness of an individual lamina is given as tk

(a) (b)

Figure 54 Description of a FRP laminate and terminology Adapted fromFigures 6-1 and 6-5 [19 pg 215219](a) A FRP laminate consists of FRP laminae stacked on top

of each oher(b) A laminate with thickness t is constructed from n number

of plies where the laminate reference coordinate systemis placed in the mid-plane of the laminate The bottomposition related to the local z axis for a lamina k is givenas hk The thickness of an individual lamina is given as tk

Since the laminae in a laminate may have unique orientations with regard to the referencecoordinate system it is necessary to establish a system which describes the individuallaminae and their orientations θ and position in a laminate a Laminate OrientationCode LOC [19 pg 225-226] which can also be referred to as layup The system isexplained below and an example is shown in Figure 55

Laminate Orientation Code LOC

OrientationThe orientation of a lamina is given as a positive or negative angle relative tothe reference coordinate system ranging from minus90deg to 90deg

SequenceThe laminae in the laminate are ordered from bottom to top where the startand end of the LOC is marked by brackets Each individual lamina is separatedfrom the others with a backslash if the orientations are different see Figure55

Number of laminaeIf two or more laminae in sequence have the same orientation the number oflaminae is given as a numerical subscript

40


Figure 55 A laminate with orientation code [090+45-45+45-45900]Adapted from Figure A3-1 [19 pg 543]

531 Classical lamination theory for FRP laminates

The constitutive equations for a laminate can be derived using Classical LaminationTheory CLT Here it is assumed that the bond between two laminae in a laminate isperfect thus the entire laminate will behave as a plate where the variation of the strainsvary linearly over the thickness of the laminate however due to the varying orientationsof each lamina the stresses will vary trough the thickness see Figure 56 (p 213 -216)

Figure 56 Strain and strain variation in a principal FRP laminate Adaptedfrom Figure 6-3 [19 pg 217](A) A principal laminae with three laminae(B) Strain variation of a principal laminate(C) Stress variation of a principal laminate

41


A laminated plate which undergoes a deformation is considered see Figure 57 It isassumed that the line AB is initially straight and perpendicular to the plate mid-planeand will remain straight and perpendicular to the mid-plane also in the deformed stateThis means that any shear deformations in the xz and yz planes are neglected

(a)

(b)

Figure 57 Deformation and displacements of a FRP laminate Adaptedfrom Figure 6-2 [19 pg 215](a) Deformation and displacements of a laminate in the xz

plane(a) Deformation and displacements of a laminate in the yz

plane

The displacement in the x-direction u of the point C (see Figure 57a) can be expressedin terms of the mid-plane displacement in the x-direction u0 and slope of the mid-planein the x-direction φx as in Equation 529 (p 214)

u = u0 minus z sinφx asymp u0 minus zφx = partw0

partx[m] (529)

Similarly the displacement of any point in the y-direction (see Figure 57b) v can beexpressed in terms of the mid-plane displacement in the y-direction v0 and slope of themid-plane in the y-direction φy as in Equation 530

v = v0 minus z sinφy asymp v0 minus zφy = partw0

party[m] (530)

42


It is also assumed that the stretching of the normal of the plate (AB) is insignificantand thus the displacements in the z-direction is assumed to be equal to the mid-planedisplacement in the z-direction w0 The strains of the laminate can therefore be expressedas in Equations 531 to 533

εx = partu

partx= partu0

partxminus z

partφx

partx(531)

εy = partv

party= partv0

partyminus z

partφy

party(532)

γxy = partu

party+ partv

partx= partu0

party+ partv0

partxminus z(partφx

party+ partφy

partx) (533)

The strain and displacement can now be expressed in terms of the mid-plane strains ϵ0and plate curvature κ in the laminate reference coordinate system as in Equations 534and 535

εx

εy

γxy

=

ε0

x

ε0y

γ0xy

+

κx

κy

κxy

z or

ε1

ε2

γ12

=

ε0

1

ε02

γ012

+

κ1

κ2

κ12

z (534)

ϵ = ϵ0 + κz (535)

The strains and stresses in each lamina can be independently expressed in relation to themid-plane strains and plate curvature of the laminate as in Equation 536

σ1

σ2

τ12

k

= Qk

ε0

1

ε02

γ012

+ Qk

κ1

κ2

κ12

z = Qkϵ0 + κz (536)

Due to the varying stress in each lamina it can be convenient to express the relationshipbetween stresses and strains in terms of resultant forces N and resultant moments M which can be summed by integration over the height of the laminate as in Equations 537and 538

N1

N2

N12

=int h2

minush2

σ1

σ2

τ12

dz =nsum

k=1

int hk

hkminus1

σ1

σ2

τ12

k

dz [Nm] (537)

M1

M2

M12

=int h2

minush2

σ1

σ2

τ12

zdz =nsum

k=1

int hk

hkminus1

σ1

σ2

τ12

k

zdz [Nmm] (538)

43


With Equation 536 these two equations can written as Equation 539 and 540

N =[sumn

k=1 Qk

int hkhkminus1 dz

]ϵ0 +

[sumnk=1 Qk

int hkhkminus1 zdz

]κ (539)

M =[sumn

k=1 Qk

int hkhkminus1 zdz

]ϵ0 +

[sumnk=1 Qk

int hkhkminus1 z

2dz]

κ (540)

By introducing

A =nsum

k=1Qk

int hk

hkminus1dz [Pamiddotm] (541)

B =nsum

k=1Qk

int hk

hkminus1zdz [Pamiddotm2] (542)

D =nsum

k=1Qk

int hk

hkminus1z2dz [Pamiddotm3] (543)

the constitutive equation for a laminate can thus now be expressed asN

M

=

A B

B D

ϵ0

κ

(544)

where A B and D are called the extensional stiffness matrix the coupling stiffnessmatrix and the bending stiffness matrix These matrices are 3 times 3 matrices and are allsymmetric see Equations 545 to 547

A =

A11 A12 A16

A22 A26

Sym A66

(545)

B =

B11 B12 B16

B22 B26

Sym B66

(546)

D =

D11 D12 D16

D22 D26

Sym D66

(547)

44


54 Mechanical coupling in FRP laminates

The components of the A B and D matrices are directly related to the orientationthickness and the layup sequence of the plies in the laminate The laminate can thereforebe designed in such a way that certain elements of these matrices can be made zero ornon-zero and thus exhibit or eliminate certain structural behaviours

Two special types of laminates which can be very useful are symmetric and balancedlaminates see Sections 541 and 542 because they eliminate mechanical couplingbetween in-plane extension and shear as well as out-of-plane coupling between extensionand bending see Figures 58 and 59

Laminates which have no coupling behaviours are called un-coupled laminates or FullyOrthotropic Laminates see Section 55 Even though symmetric and balanced laminatesare not entirely un-coupled these are often used in aerospace applications where it isassumed that the coupling effects may be safely disregarded in laminates with a largenumber of plies [40]

Figure 58 A laminate is said to be coupled with regard to in-planeextension and shear if an axially applied load gives rise to sheardeformation(A) An axially loaded laminate(B) Shear deformation in a laminate which is coupled with

regard to in-plane extension and shear

Figure 59 A laminate is said to be coupled with regard to in-plane extensionand out-of-plane bending if an axially applied load gives rise toplate bending or warping(A) An axially loaded laminate(B) Deformation in a laminate which is coupled with regard

to in-plane extension and out-of-plane bending

45


541 Symmetric laminates

A symmetric laminate is always symmetric about the mid-plane and for each laminawith an orientation θ and thickness tk at a certain distance z from the mid-plane theremust also exist one lamina with the same thickness tk with an orientation θ at a distanceminusz from the mid-plane [19] see Figure 510

Figure 510 A symmetric laminate with LOC [30-3015454515-3030]or [30-301545]s Adapted from Figure A3-1 [19 pg 543]

In these laminates the components Bij of the B matrix will always be zero and thematrix is therefore denoted as B0 [40] In some cases even a non-zero B matrix can beconsidered non-coupled if itrsquos small enough [41]

B0 =

B11 B12 B16

B22 B26

Sym B66

=

0 0 00 0 00 0 0

(548)

The B matrix is called the coupling stiffness matrix because it relates the extension andbending of a laminate mathematically see Equation 544 This means that it relates themid-plane strains and plate curvatures in the laminate Thus a load applied in the planeof the laminate can cause the laminate to bend or warp out of plane if the elements ofthe coupling matrix B are non-zero [19 pg 221] In a symmetric laminate there willtherefore be no out-of-plane extension and bending coupling

In these laminates the components Aij of the A matrix and components Dij of the Dmatrix may all be finite (non-zero) in which case these are denoted as Af and Df Themost general symmetric laminate is thus designated as AfB0Df [42]

46


542 Balanced laminates

A balanced laminate laminate can be constructed in three different ways [19 pg228-228] Angle-ply laminates are constructed such that there for every lamina witha thickness h and orientation θ exist an other lamina with the same thickness andorientation minusθ see Figure 511 In the following text the term balanced laminate willrefer to angle-ply laminates unless it is stated otherwise The other two balanced types areuni-directional laminates which consist only of laminae with the same orientation andcross-ply laminates which consist of laminae with the orientations plusmn0deg or plusmn90deg only

Figure 511 A balanced laminate with LOC [30-3015-154515-15-45]Adapted from Figure A3-1 [19 pg 543]

A laminate is said to be balanced if the A16 and A26 components of the A matrix arezero in which case it is denoted as As see Equation 549

As =

A11 A12 0

A22 0Sym A66

(549)

The A matrix describes the in-plane behaviour of a laminate If the terms A16 and A26are non-zero there will be in-plane shear coupling effects meaning that an axially loadedlaminate will also be subjected to shear deformation A balanced laminate will thereforenot have any in-plane extension and shear coupling

In these laminates the components Bij of the B matrix may all be finite (non-zero) inwhich case the matrix is denoted as Bf In the most general case the components Dij

of the D matrix will be non-zero but in some cases the D16 and D26 components maybe zero it will therefore be denoted as Ds The most general balanced laminate is thusdesignated as AsBfDf [42]

47


55 Equivalent material parameters for Fully OrthotropicLaminates

A Fully Orthotropic Laminate FOL will be fully un-coupled meaning it will notdisplay either in-plane extension and shear coupling nor out-of-plane extension andbending coupling A laminate of this type is designated as AsB0Ds [42] Theselaminates generally minimise distortion which may occur during manufacturing andincrease compression buckling strength in comparison with symmetric and balancedlaminates [40] This type of laminate will also have the same mechanical behaviour as auni-directional laminae and thus the mechanical properties of this type of laminate canbe described by elastic engineering constants which can be derived from the laminateconstitutive equation

551 Stress-strain relation in fully orthotropic laminates - Effectiveelastic engineering constants

Efficient elastic engineering constants for a FOL can be derived from the stress-strainrelation in a laminate the process for this is described below For the derivation it isassumed that the laminate is transversely isotropic meaning that both the strains andstress is linearly distributed over the laminate height This assumption is not true forFOL with laminae with orientations θ = 0deg however for the sake of the derivation ofefficient engineering constants this assumption is assumed to be valid as the mechanicalbehaviour of the laminate is transversely isotropic

By letting N1 = 0 and N2 = N12 = 0 the constitutive equation for this type of laminatecan be simplified as in Equation 550

N1 = A11ε1 + A12ε2 (550)

Since the stress is assumed to be uniform over the height this equation can be re-writtenas

σ1t = A11ε1 + A12ε2 (551)

where t is the laminate thickness By introducing Hookersquos Law into Equation 551 theexpression can now be formulated as in Equation 552

E1ε1t = A11ε1 + A12ε2 (552)

And now the elastic modulus in the main direction can be calculated as in Equation 553

E1 = A11

t+ A12ε2

ε1t(553)

48


The Major Poissonrsquos ration can be calculated as in Equation 554

ν12 = minusε2

ε1(554)

The transverse modulus and Minor Poissonrsquos ratio can be calculated in the same way byletting N2 = 0 and N1 = N12 = 0

E2 = A21ε1

ε2t+ A22

t(555)

ν21 = minusε1

ε2(556)

The shear modulus is calculated directly from the A matrix as in Equation 557

G12 = A66

t(557)

Finally the thermal expansion coefficient are calculated using the transformation matrixT2 and the αL and αT for the laminate see Equation 558

α1

α2

α12

= T2minus1

αL

αT

0

(558)

49

6MATERIAL PARAMETERS FOR

STRUCTURAL ANALYSIS

6 MATERIAL PARAMETERS FOR STRUCTURAL ANALYSIS

Figure 61 Flowchart describing the methodology used for the case studiespresented in Chapter 6 - Material parameters for structuralanalysis

52


61 Analysis of FRP composites

The possibilities to analyse anisotropic materials in SOFiSTiK are limited It is possibleto define fibrous or layered materials with orthotropic material properties for instanceply-wood [43] However in the case of uni-directional FRP composites it can be difficultto define suitable material formulations due to for instance a large number of laminae aPoissonrsquos ratio gt 05 or large differences in stiffness in different orientations This will beseen later in Chapter 7 Section 72 Due to these reasons it is convenient to use simplifiedmaterial definitions for analysis in SOFiSTiK in order to establish a stable and reliableparametric analysis approach

In Chapter 5 the constitutive equations for laminated FRP composites were derivedand the principal mechanical behaviours of balanced symmetric and fully orthotropiclaminates were explained It was seen that the components of the A B and D matricesare linked to the lamina orientations number of laminae as well as lamina thickness

The FRP module which is the topic of study as well as the node discontinuity zonesin Figure 17 are recalled from Chapter 1 It is clear that the theory presented in theprevious chapter mainly applies to the node zones at the intersections between the fibrebundles These areas are already potential instability and stress concentration zones dueto change of stiffness as well as geometric discontinuities In addition to this it is also nowknown that these zones may give rise to additional instability modes due to mechanicalcoupling effects

By design the fibre bundles of the FRP module cannot be constructed such that the nodezones will have fully orthotropic laminate layups however the layup will always be eithersymmetric or balanced with alternating lamina orientations plusmnθ see Appendix A1 If thenumber of laminae in these laminates is large enough it may be possible to neglect thecoupling effects see Section 54

Instabilities and stress concentrations due to geometric discontinuities have been excludedfrom the scope of this study however it is still necessary to determine if it is possible toalso exclude the coupling effects of the node zones in order to be able to simplify thematerial formulations in SOFiSTiK

The pre-normative design guidelines mentioned in Section 111 proposes methods inwhich characteristic material values for FRP composites can be determined It can bedone by using CLT conducting tests on a similar laminate or by laboratory tests whichare specified in the EN1990 [6] It is also recommended to apply a symmetric and balancedlaminate design which is not the case in the studied FRP module12

In this chapter a number of assumptions are presented and a number of case studies wereconducted in order to evaluate and verify these assumptions The principal method andwork flow for these case studies can be seen in Figure 61

1 In the pre-normative design guidelines the term balanced laminate (or bi-directional laminate) refersto only cross ply laminates (with plusmn0deg and plusmn90deg orientations) [6] This type of laminate will alwaysbe a FOL

2 It should be noted that a laminate which is simultaneously symmetric and balanced may not befully orthotropic due to the fact that the D matrix could be finite (Df ) An example of a laminatewhich is symmetric and balanced but not a FOL is [30-45-151545-30]s

53


611 Assumptions

The assumptions made in this chapter apply to thick laminates with balanced orsymmetric layups and alternating lamina orientations plusmnθ see Figure 62 and AppendixA1 These types of laminates have been chosen based on assumptions made regardingthe manufacturing of the FRP modules where layers of FRP will be either wound orstacked in such a manner that the orientations of the layers will be alternating In thecase of the symmetric laminate the number of laminae will always be uneven and in thecase of the balanced laminate the number of laminae will always be even

(a) (b)

Figure 62 Symmetric and balanced laminates with alternating laminaorientations(a) Symmetric laminate with alternating lamina orientations

and an uneven number of laminae LOC [plusmnθminusθ]s(b) Balanced laminate with alternating lamina orientations

and an even number of laminae LOC [plusmnθ]

Out-of-plane coupling effects can be neglectedIn balanced laminates the B matrix will be non-zero however the only non-zero termswill be B16 and B26 Due to the nearly symmetric layup only one lamina will contributeto these terms and as such it is assumed that the coupling effects will decrease for anincreasing number of laminae and will be so small that it can be neglected

Shear coupling effects can be neglectedIn symmetric laminates the A16 and A26 components of the A matrix are non-zerohowever the contributions to these terms will be cancelled out for all laminae exceptfor one in the laminates studied due to the nearly balanced layup Therefore only onelamina will contribute to in plane shear and extension coupling effects and for thicklaminates this is assumed to be so small that it can be neglected

612 MATLAB programs

Already existing MATLAB functions were used to create a number of MATLAB programswhich were written for the case studies in this chapter These programs are described inAppendix B1

54


62 Mechanical response of FRP laminae

The relation between fibre volume fraction stiffness and in-plane-shear coupling wasstudied to evaluate which fibre volume fractions and lamina orientations are most criticalwith regard to coupling The material parameters which were used to establish the laminaengineering constants can be found in Tables 42 and 43

621 Engineering constants for different laminae

Engineering constants were calculated using the MATLAB program EvalLaminam seeAppendix B1 The longitudinal modulus E1 was calculated using the Rule of Mixturesand the transverse modulus E2 and the shear modulus G were calculated using theHalpin-Tsai method see Chapter 5 The factor ξ for the transverse modulus was set to2 and for the shear modulus it was set to 1 The calculated engineering constants as afunction of the volume fraction Vf for the composite material can be seen in Figure 63

(a) Longitudinal transverse and shear moduli

(b) Major and Minor Poissonrsquos ratioFigure 63 Engineering constants for CFRP laminae as a function of the

volume fraction Vf

55


622 Mechanical coupling in different laminae

For each fibre volume fraction 005 le Vf le 095 and lamina orientation plusmn0deg le θ le plusmn45degthe Q matrix was calculated and the Q16 components were compared for the differentlaminae The largest value of the Q16 component was found for a fibre volume fraction of085 and lamina orientation plusmn30deg The Q16 for all other laminae were then compared tothis value see Equation 61 Figure 64 and Table 61

Q16rel =Q16plusmnθ

Q16plusmn30deg(61)

Figure 64 The calculated ratio Q16rel for plusmn30deg and plusmn45deg laminaorientations

Table 61 The calculated values for Q16rel which compares the Q16 term ofthe Q matrix for all lamina orientations and volume fractions tothe overall maximum Q16

θ

Vf 005 015 025 035 045 055 065 075 085 095

plusmn0deg 0 0 0 0 0 0 0 0 0 0plusmn5deg 2 5 9 13 16 19 23 25 27 23plusmn10deg 4 11 17 24 31 37 44 49 52 44plusmn15deg 5 15 25 34 44 53 61 68 73 61plusmn20deg 6 18 30 42 53 64 74 83 88 74plusmn25deg 7 20 33 46 59 71 82 92 97 82plusmn30deg 7 21 34 47 60 73 85 95 100 84plusmn35deg 7 20 33 46 59 71 82 92 96 80plusmn40deg 6 18 30 42 54 65 75 83 88 72plusmn45deg 5 16 26 36 46 56 64 72 75 60

56


For each lamina orientation and fibre volume fraction the Q16 component was alsocompared to the Q11 component of the same lamina It was seen here that for a laminaorientation of plusmn45deg the ratio between these two component was the highest for each fibrevolume fraction see see Figure 65 and Table 62 This was taken to mean that thecoupling tendency relative the lamina stiffness will always be most critical for laminaewith a plusmn45deg orientation

Qplusmnθrel =Q16plusmnθ

Q11plusmnθ

(62)

Figure 65 The calculated ratio Qplusmnθrel for plusmn30deg and plusmn45deg laminaorientations

Table 62 The calculated values for Qplusmnθrel which compares the Q16 and Q11terms of the Q matrix for each lamina orientation and fibre volumefraction

θ

Vf 005 015 025 035 045 055 065 075 085 095


57


63 Mechanical response of FRP laminates

Mechanical coupling in different types of laminates were studied to evaluate how thelamina orientations as well as the number of laminae affects the coupling tendencies of alaminate Here the fibre volume fraction Vf was set to 052 see Tables 42 and 43

631 Mechanical coupling in different types of laminates

The symmetric and balanced laminates presented in Tables 63 to 64 were studied usingthe MATLAB functions and the program MechanicalCouplingm see Appendix B1

Table 63 Thin laminates with lamina orientations plusmn0deg le θ le plusmn 45degThickness [mm] Count [-]

Laminate type Laminate Lamina Layers631 Symmetric thin 03 asymp 01 3631 Balanced thin 01 asymp 01 2

Table 64 Thick laminates with lamina orientations plusmn30deg and plusmn45degThickness [mm] Count [-]

Laminate type Laminate Lamina Layers631 Symmetric 10 10 33 ⪆ t ⪆ 01 3 le n le 101631 Symmetric 15 15 5 ⪆ t ⪆ 03 3 le n le 151631 Balanced 10 10 5 ⪆ t ⪆ 01 2 le n le 100631 Balanced 15 15 75 ⪆ t ⪆ 03 2 le n le 150

6311 Mechanical coupling in symmetric laminates

The A B and D matrices were calculated for the thin symmetric laminate in Table 63for lamina orientations plusmn0deg le θ le plusmn45deg In Figure 66 the components of the extensionalstiffness matrix A are plotted Here it can be seen that the largest contributions overallto the A16 and A26 components are for plusmn30deg orientations It can also be seen that thelargest contributions to these components relative the laminate stiffness are for plusmn45deg

The A B and D matrices were then calculated for the thick symmetric laminates inTable 64 with lamina orientations plusmn30deg and plusmn45deg for an increasing number of plies InFigure 67 it can be seen that the contribution to the A16 and A26 components of theextensional stiffness matrix for the 15 mm thick laminate decreased for an increasingnumber of plies and that these terms are small compared to the other components of theA matrix especially when the number of plies is larger than approximately 25 The sameresults were seen in the 10 mm laminate

In Figure 68 it can be seen that the A16 component appear to decrease exponentiallywith an increasing number of plies and that the reduction of these terms is approximately94 when the number of laminae has reached 51 and that the decrease rate appears toapproach a constant value The decrease rate was also calculated for the A26 componentand the same decrease rate was seen here For the 10 mm laminate the decrease ratefollowed the same pattern

58


Figure 66 Components of the extensional stiffness matrix A for a thinsymmetric laminates with varying ply orientations

Figure 67 Components of the extensional stiffness matrix A for asymmetric laminate with lamina orientations plusmn30deg for anincreasing number of plies

Figure 68 Decrease rate of A16 component of the extensional stiffnessmatrix A for a symmetric laminate with lamina orientationsplusmn30deg for an increasing number of plies

59


6312 Mechanical coupling in balanced laminates

The A B and D matrices were also calculated for a thin balanced laminate with twolaminae see Table 63 In Figure 69 the components of the coupling stiffness matrixfor this laminate is shown Here the coupling tendency can be observed in the B16 andB26 components of the B matrix The largest contribution overall to the B16 and B26components are for plusmn30deg orientations and it can be seen that the largest contribution tothese components relative the laminate stiffness is for plusmn45deg

The A B and D matrices were calculated for laminates with orientations plusmn30deg and plusmn45degand an increasing number of plies see Table 64 In Figures 610 and 611 it can be seenthat the contributions to the B16 and B26 components decreased for an increasing numberof plies for a 15 mm thick laminate In Figure 611 the decrease of the B16 componentis shown for the same laminate The reduction of this terms is approximately 96 fora number of 50 plies compared to 2 plies the same decrease rate was found for the B26component Again compared to the overall stiffness the contributions to the couplingterms B26 and B26 are very small for a number or laminae larger than approximately 25and the results of the 10 mm laminate followed the same pattern

Figure 69 Components of the coupling stiffness matrix B for a thinbalanced laminates with varying ply orientations

Figure 610 Components of the coupling stiffness matrix B for a balancedlaminate with lamina orientations plusmn30deg for an increasingnumber of plies

60


Figure 611 Decrease rate of B16 component of the coupling stiffness matrixB for a balanced laminate with lamina orientations plusmn30deg foran increasing number of plies

632 Mechanical response in different laminates

In order to evaluate weather the deformations which arise due to mechanical couplingare so small that these effects are negligible a study was conducted comparing the strainand curvature response in thin and thick (symmetric and balanced) laminates shown inTable 65 These studies were conducted using the MATLAB program EvalLaminatemsee Appendix B1

Table 65 Thin and thick symmetric and balanced laminate layups Theselaminates were studied for lamina orientations plusmn30deg and plusmn45deg

Thickness [mm] Count [-]Laminate type Laminate Lamina Layers632 Symmetric 1 1 asymp 01 11632 Symmetric 10 10 asymp 01 101632 Symmetric 15 15 asymp 03 51632 Symmetric 15 15 asymp 03 5632 Balanced 1 1 asymp 01 10632 Balanced 10 10 asymp 01 100632 Balanced 15 15 asymp 03 6632 Balanced 15 15 asymp 03 50

The laminates were subjected to axially applied loads uniformly distributed over thelaminate height and resulting strains were then calculated for each load case Equations63 and 64 to 66 The load magnitude was determined so that the results could be directlycompared for the thin and thick laminates The applied load was thus proportional to thelaminate height see Equation 63 where F = 1kN mminus1 and t is the laminate thicknessin metres

Nij = Ft [N] (63)

61


The subscript ij denotes the axis in which the load is applied in this case the longitudinal(1) and transverse directions (2) as well as the direction of shear (12) To differentiatethe results the loads and resulting strains and curvatures related to the thin laminateswill be denoted with a prime superscript (prime)

Load case 6321 N1 =

F

00

t (64)

Load case 6322 N2 =

0F

0

t (65)

Load case 6323 N3 =

00F

t (66)

6321 Mechanical response in symmetric laminates

In Table 66 the resulting strains for the 10 mm thick and 1 mm thin laminates subjectedto uniform loads N1 and N prime

1 are shown In both laminates the lamina thickness was setto approximately 01 mm Here it can be seen that the shear deformations which arisedue to coupling are small relative the extensional strains It can also be seen that thethin and thick laminate have nearly identical strain response in the longitudinal (1) andtransverse directions (2) Here it can also be seen that the reduction in shear deformationsis approximately 89 in the thick laminate compared to the thin laminate The resultsfor remaining load cases followed the same trend The study was also conducted forlaminates with 15 an 15 mm thickness with lamina thickness of approximately 03 mmwith similar results

6322 Mechanical response in balanced laminates

The curvature which occurs when a balanced laminate is loaded axially was evaluatedThe resulting curvatures for the 10 mm thick and 1 mm thin laminates subjected touniform loads N12 and N prime

12 are shown in Table 67 Here it can be seen that the out ofplane deformation in the thick laminates is less than 1 of the out-of-plane deformationfor thin laminates The results for remaining load cases followed the same trend Thestudy was also conducted for laminates with 15 an 15 mm thickness with lamina thicknessof approximately 03 mm with similar results

62


Table 66 Strains for a 10 mm thick symmetric laminate (ϵ1 ϵ2 γ12)compared to the strains (ϵprime

1 ϵprime2 γprime

12) for a 1 mm thick symmetriclaminate subjected to a uniform load N1 and N prime

1 respectivelyθ εprime

1 εprime2 γprime

12 ε1 ε2 γ12ε1εprime

1

ε2εprime

2

γ12γprime

12

deg [10minus9] [10minus9] [10minus9] [10minus9] [10minus9] [10minus9] [] [] []plusmn0deg 80 25 00 80 25 00 100 100 -plusmn5deg 81 31 17 81 31 02 100 100 11plusmn10deg 87 51 21 87 50 02 100 100 11plusmn15deg 99 86 21 98 85 02 100 100 11plusmn20deg 122 140 20 121 140 02 99 100 11plusmn25deg 165 220 19 164 220 02 100 100 11plusmn30deg 243 328 18 243 328 02 100 100 11plusmn35deg 374 458 17 373 458 02 100 100 11plusmn40deg 561 577 15 560 577 02 100 100 11plusmn45deg 765 628 11 765 629 01 100 100 11

Table 67 Curvatures for a 10 mm thick balanced laminate (κ1 κ2 κ12)compared to the curvatures (κprime

1 κprime2 κprime

12) for a 10 mm thicksymmetric laminate subjected to a uniform load N12 and N prime

12respectively

θ κprime1 κprime

2 κprime12 κ1 κ2 κ12

κ1κprime

1

κ2κprime

2

κ12κprime

12

deg [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [] [] []plusmn0deg 00 00 00 00 00 00 - - -plusmn5deg 109 47 00 01 00 00 099 099 -plusmn10deg 142 59 00 01 01 00 099 099 -plusmn15deg 141 53 00 01 01 00 098 098 -plusmn20deg 133 43 00 01 00 00 098 098 -plusmn25deg 128 31 00 01 00 00 098 098 -plusmn30deg 123 15 00 01 00 00 098 098 -plusmn35deg 116 08 00 01 00 00 098 098 -plusmn40deg 100 39 00 01 00 00 098 098 -plusmn45deg 72 72 00 01 01 00 098 098 -

63


64 Equivalent material parameters

In the possible layups which has been assumed for this FRP module see Table A1the smallest possible number of laminae will be 33 In the study presented in Section631 it was seen that the coupling effects are very small for a number of laminae largerthan approximately 25 In Section 632 it was seen that the shear deformation of thethick laminate was approximately 10 of the shear deformation of the thin laminate Itmay not seem like a significant decrease however considering that the shear deformationsare very small in comparison with the extensional strains it is here assumed that thesecoupling effects are insignificant It was therefore assumed that the laminates will be fullyorthotropic and efficient engineering constants can be derived following the procedure inSection 55

641 Engineering constants

Efficient engineering constants were calculated for the previously studied laminates usingthe MATLAB program EvalLaminatem see Appendix B1 The results were within thesame range for all laminates In Figures 612 and 613 as well as Table 68 the engineeringconstants which were derived for a 10 mm thick symmetric laminate are shown

Table 68 Derived engineering constants for a 10 mm thick symmetriclaminate with varying lamina orientations

θ E1 E2 G12 ν12 ν21 α12 α21

deg [GPa] [GPa] [GPa] [-] [-] [10minus6] [10minus6]plusmn0deg 12525 1241 359 031 003 02 3135plusmn5deg 12289 1233 447 038 004 044 3111plusmn10deg 11535 121 701 058 006 114 3041plusmn15deg 10178 1173 1089 087 01 229 2926plusmn20deg 8259 1126 1566 116 016 384 277plusmn25deg 6086 1076 2073 134 024 576 2578plusmn30deg 4122 1035 255 135 034 799 2356plusmn35deg 2679 1026 2939 123 047 1045 211plusmn40deg 1785 1093 3192 103 063 1307 1848plusmn45deg 1307 1307 328 082 082 1577 1577plusmn50deg 1093 1785 3192 063 103 1848 1307plusmn55deg 1026 2679 2939 047 123 211 1045plusmn60deg 1035 4122 255 034 135 2356 799plusmn65deg 1076 6086 2073 024 134 2578 576plusmn70deg 1126 8259 1566 016 116 277 384plusmn75deg 1173 10178 1089 01 087 2926 229plusmn80deg 121 11535 701 006 058 3041 114plusmn85deg 1233 12289 447 004 038 3111 044plusmn90deg 1241 12525 359 003 031 3135 02

64


Figure 612 Equivalent orthotropic elastic moduli derived for a 10 mm thicksymmetric laminate with varying lamina orientations

Figure 613 Equivalent orthotropic Poissonrsquos ratio derived for a 10 mm thicksymmetric laminate with varying lamina orientations

65


642 Verification of equivalent material parameters

A study was conducted to verify the calculated engineering constants using the MATLABprogram OrthotropicLaminatem see Appendix B1 For each lamina orientation in Table68 a fictional fully orthotropic laminate was constructed see Table 69

Table 69 Thin and thick fully orthotropic laminates studied with equivalentorthotropic material parameters derived for lamina orientationsplusmn0deg to plusmn45deg

Thickness [mm] Count [-]Laminate type Laminate Lamina Layers642 Orthotropic 1 1 asymp 01 11642 Orthotropic 10 10 asymp 01 101642 Orthotropic 15 15 asymp 03 51642 Orthotropic 15 15 asymp 03 5

For these fictional laminates the mechanical response was evaluated in the same manner asthe laminates studied in Section 632 In Figure 614 the strains for symmetric balancedand fully orthotropic laminates with a thickness of 10 mm are shown where each laminatewas subjected to a uniform loadN1 Here it can be seen that the strains are nearly identicalfor all laminates The same pattern could be seen for the remaining load cases andlaminate configurations The assumption that coupling effects can be neglected and thatthe laminates can be assumed to be transversely isotropic was thus considered sufficientlyaccurate

66


(a) Symmetric laminate

(b) Balanced laminate

(c) Equivalent orthotropic laminateFigure 614 Strains for different laminates with 10 millimeter thickness

subjected to a load N1

67

7MATERIAL PARAMETERS FOR FE

ANALYSIS

7 MATERIAL PARAMETERS FOR FE ANALYSIS

Figure 71 Flowchart describing the methodology used for the case studiespresented in Chapter 7 - Material parameters for FE analysis

70


71 Parametric QUAD and beam models for verification ofmaterial parameters

In Chapter 6 material parameters were derived for a CFRP material with varying fibreorientations These material parameters were analysed and verified using ClassicalLaminate Theory however the aim of this study is to establish a parametric approachfor FE analysis In order to verify that the material parameters could also be used inFE analysis for SOFiSTiK QUAD and beam elements a number of case studies wereconducted The outline of these case studies are illustrated in Figure 71

Two parametric models were developed a QUAD model and a beam model see Figures72 to 75 The geometry and FE topology were generated in Grasshopper using theseparametric model definitions and were then exported to SOFiSTiK as CADINP code Inboth model definitions weak springs were added as supports at the boundaries to preventrigid body motions in analysis

Figure 72 Definition of parametric model used for QUAD models Thegeometry and FE topology was generated in Grasshopper andRhinoceros 3D using this definition

Figure 73 Parametric FE QUAD model in SOFiSTiK Here the thicknessof the 2D quad elements is shown as well as springs (in greencolour) which are used as boundary supports

71


Figure 74 Definition of parametric model used for beam models Thegeometry and FE topology was generated in Grasshopper andRhinoceros 3D using this definition

Figure 75 Parametric FE beam model in SOFiSTiK Here the cross sectionsof the beam elements are shown as well as springs (in greencolour) which are used as boundary supports

72


72 Adaptations of material parameters for FE analysis

Nine orthotropic materials were defined in SOFiSTiK using the material parameters inTable 68 for the fibre orientations plusmn0deg to plusmn45deg The orthotropic FRP material wasmodelled as a timber material in SOFiSTiK because this material can be used to definetransversely isotropic materials In SOFiSTiK this material also allows a Poissonrsquos ratiolarger than 049 [43] Poissonrsquos ratio is typically theoretically limited to -1 to 05 howeverin the case of anisotropic materials such as CFRP it can be well above 05 [44] whichwas seen in Chapter 6 Before applying the materials in analysis it was however necessaryto make adaptations to the material parameters due to limitations in SOFiSTiK Therelation in Equation 71 must always be true otherwise SOFiSTiK will not be able touse the material in analysis see Figure 76

1 minus νxyνyx gt 0 =rArr

νxy gt 0 νyx lt1

νxy

νxy lt 0 νyx gt1

νxy

(71)

Figure 76 The error 447 will occur during analysis in SOFiSTiK if therelation in Equation 71 is not fulfilled

In SOFiSTiK the Minor Poissonrsquos ratio νyx is calculated from the elastic moduli of thematerial see Equation 72 [43] The relation in Equation 73 must also be true howeverin Figure 77a it can be seen that it is not the case for the material parameters in Table68 This can be solved by adapting the Poissonrsquos ratios as in Figure 77b (compare withFigure 613) and applying a material rotation ψ of 90deg see Figure 78 This rotationcorresponds to the third Euler angle [43] which describes a 90deg rotation about the localmaterial z-axis

νyx = ν12E1

E2(72)

ν12E1

E2lt

1ν12

(73)

73


000

500

1000

1500

2000

2500

3000

3500

plusmn0deg plusmn10deg plusmn20deg plusmn30deg plusmn40deg plusmn50deg plusmn60deg plusmn70deg plusmn80deg plusmn90deg

SofistikPoissonsreatios[]

Fibre angle [deg]

Sofistik Poissons ratio

E E

1

(a)

(b)

Figure 77 Diagrams showing the relation between the Poissonrsquos ratioscalculated for fully orthotropic laminates in Chapter 6 and theadaptations required for analysis in SOFiSTiK(a) The black solid line should always be below the black

dashed line it can be seen here that this is not the case forthe material parameters established in Chapter 6

(b) The Major and Minor Poissonrsquos ratios were adapted as in(b) and an Euler angle ψ of 90deg was added to the materialdefinition in SOFiSTiK

74


Figure 78 A rotation ψ of 90deg is applied in the cases where Poissonrsquos ratiohas been adjusted This rotation corresponds to the third Eulerangle [43] and describes a 90deg rotation about the local z-axis

Table 71 Nine materials were defined for FE analysis of QUAD elements inSOFiSTiK The name of the material in SOFiSTiK is defined bywhich fibre orientation the material corresponds to The elasticmoduli E and E90 correspond to E1 and E2 and G corresponds toG12 The Poissonrsquos ratios have been adapted so that ν correspondsto ν21 and ν90 corresponds to ν12 The thermal expansioncoefficient α has been taken as α1 The angle ψ corresponds tothe third Euler angle

E E90 G ν ν90 α ψ

Name [GPa] [GPa] [GPa] [-] [-] [10minus6] [deg]CFRP 100 12525 1241 359 003 031 02 90CFRP 105 12289 1233 447 004 038 044 90CFRP 110 11535 121 701 006 058 114 90CFRP 115 10178 1173 1089 01 087 229 90CFRP 120 8259 1126 1566 016 116 384 90CFRP 125 6086 1076 2073 024 134 576 90CFRP 130 4122 1035 255 034 135 799 90CFRP 135 2679 1026 2939 047 123 1045 90CFRP 140 1785 1093 3192 063 103 1307 90CFRP 145 1307 1307 328 082 082 1577 90

75


721 Mesh convergence study

A mesh convergence study was conducted to find a suitable mesh size before analysisThe parametric model definition for QUAD elements was used to model a plate withdimensions 100 times 100 times 10 mm The plate was fixed on one end and loaded with auniform load of on the other end see Figure 79 The mesh size was decreased in foursteps and displacements were compared for each mesh size see Figure 710 The increasein displacement for each mesh size was less than 0001 mm and the mesh was assumed tobe fine enough for a side length of 10 mm see Figure 710d

(a) Side view (xz plane) (b) Perspective viewFigure 79 Application of loads in mesh convergence study

(a) 50 mm mesh length (b) 25 mm mesh length

(c) 167 mm mesh length (d) 10 mm mesh lengthFigure 710 Displacement plots of mesh convergence study in SOFiSTiK

for a quadratic plate with dimensions 100 times 100 times 10 mm anddecreasing mesh side lengths

76


73 Verification ofmaterial parameters forQUADelements

A study was conducted in SOFiSTiK to verify the defined materials for QUAD elementsin SOFiSTiK For each material in Table 71 a plate with dimensions 100 times 100 times 10 mmwas evaluated in SOFiSTiK for six load cases see Figure 711 and Table 72 In each loadcase a displacement was applied to the boundary so that the resulting strain or curvatureof the in the load direction would be 1minus3

(a) Load case 731 (b) Load case 732

(c) Load case 733 (d) Load case 734

(e) Load case 735 (f) Load case 736Figure 711 Load cases used in QUAD model case studies

77


Table 72 Applied strains and curvatures in loadcases 731 to 736

Load case Strains [10minus3] Curvature [10minus3]Name εx εy εxy κx κx κxy

Load case 731 1 - - - - -Load case 732 - 1 - - - -Load case 733 - - 1 - - -Load case 734 - - - 1 - -Load case 735 - - - - 1 -Load case 736 - - - - - 1

731 Strains and curvatures for QUAD elements

For each load case the resulting strains and curvatures were compared to results from astudy conducted in MATLAB using the same materials load cases and laminate thicknessThe MATLAB study was conducted using the MATLAB program EquivalentLaminatemsee Appendix B1

7311 Calculation of strains and curvatures in Grasshopper

To simplify the comparison and evaluation of the strains and curvature results fromFE analysis in SOFiSTiK a parametric tool PlotStrainCurvature was developed forGrasshopper and Rhinoceros 3D see Appendix B2 The strains and curvatures werecalculated using the Equations 74 and 76 with the lengths L and radii R defined inFigures 712 and 713

(a) (b)

Figure 712

78


(a) (b)

Figure 713

ε = ∆LL

(74)

γ = ϕx + ϕy (75)

κ = 1R

(76)

7312 Evaluation of displacements in Grasshopper

The PlotStrainCurvature tool which was used to calculate strains and curvatures wasfurther developed and an additional tool PlotDisplacements was created to show thedisplaced FE mesh in Rhino With this tool it was also possible to visualise thedisplacements using colour gradients as was seen in Figure 710 This tool is describedfurther in Appendix B2

7313 Comparison of strains and curvature

In all six load cases and for all nine materials the calculated strains and curvatures inSOFiSTiK and MATLAB had a correspondence of nearly 100 The results for each loadcase and material can be found in Tables 73 to 78

The displacement results were imported into Grasshopper and plotted using thePlotDisplacements tool and it could also be seen here that the strains and curvatureswere uniform over the whole plate see Figures 714 to 716

79


(a) (b)

Figure 714 Displacement plots for axial strains for CFRP 130(a) Displacements for plate with a strain εx applied(b) Displacements for plate with a strain εy applied

(a) (b)

Figure 715 Displacement plots for plate curvatures for CFRP 130(a) Displacements for plate with a curvature κx applied(b) Displacements for plate with a cuvature κy applied

(a) (b)

Figure 716 Displacement plots for shear strain and curvature for CFRP130(a) Displacements for plate with a shear strain γxy applied(b) Displacements for plate with a shear curvature κxy applied

80


732 Stress distribution in QUAD elements

The stresses for QUAD elements were calculated in SOFiSTiK on the top bottom andcentre planes of the QUAD element In this study the normal stresses in the local x- andy-directions as well as shear stress in the xy-plane were evaluated

7321 Evaluation of stress results in Grasshopper

The PlotDisplacements tool were further developed into a new tool PlotStresses whichcould display stress results on the bottom top and centre planes of QUAD elements seeAppendix B2

7322 Comparison of stresses

The average stresses calculated in SOFiSTiK for each load case and material werecompared to the stresses calculated in MATLAB see Tables 73 to 78 Here it couldbe seen that the correspondence was nearly 100 for all materials and load cases Thestress results were imported into Grasshopper see Figures 721 to 723 and it couldbe seen here that the stress distribution is uniform over the whole plate except for inLoad case 736 However in this load case stress concentrations appeared due to tothe fact that the displacements were applied to the corner points and not uniformly alongthe boundary see Figure 711f The overall stress in this load case were however uniformsee Figure 717 and the average stress had a near 100 correspondence with the resultsfrom the MATLAB analysis

Figure 717 Stress plot showing small stress variations for plate subjectedto shear bending however it can be seen that these variationsare small and that the overall stress is uniform

81


(a) (b)

Figure 718 Stresses in bottom plane for the material CFRP 130(a) Load case 731(b) Load case 731

(a) (b)


(a) (b)


82


(a) (b)

Figure 721 Stresses in centre plane for the material CFRP 130(a) Load case 731(b) Load case 732

(a) (b)


(a) (b)


83


(a) (b)

Figure 724 Stresses in top plane for the material CFRP 130(a) Load case 731(b) Load case 732

(a) (b)


(a) (b)


84


Table 73 Resulting strains and stresses in MATLAB and SOFiSTiK forLoad case 731

Angle Strain ε2 [10minus3] Stress σ1 [MPa]θ MATLAB SOFiSTiK [] MATLAB SOFiSTiK []plusmn0deg 031 031 10000 125248 125248 10000plusmn5deg 038 038 10001 122892 122892 10000plusmn10deg 058 058 9999 115353 115352 10000plusmn15deg 087 087 9999 101787 101785 10000plusmn20deg 116 116 9999 82597 82591 10001plusmn25deg 134 134 9999 60866 60857 10001plusmn30deg 135 135 9999 41231 41221 10003plusmn35deg 123 123 9999 26796 26787 10003plusmn40deg 103 103 10000 17852 17847 10003plusmn45deg 082 082 10000 13074 13073 10000


Angle Strains ε1 [10minus3] Stress σ2 [MPa]θ MATLAB SOFiSTiK [] MATLAB SOFiSTiK []plusmn0deg 003 003 10000 12407 12407 10000plusmn5deg 004 004 10001 12329 12329 10000plusmn10deg 006 006 9998 12099 12099 10000plusmn15deg 010 010 9999 11732 11732 10000plusmn20deg 016 016 9999 11264 11263 10001plusmn25deg 024 024 9999 10761 10760 10001plusmn30deg 034 034 9999 10350 10348 10002plusmn35deg 047 047 9999 10267 10264 10003plusmn40deg 063 063 10000 10932 10930 10002plusmn45deg 082 082 10000 13074 13073 10000

85



Angle Strains γ12 [10minus3] Stress τ12 [MPa]θ MATLAB SOFiSTiK [] MATLAB SOFiSTiK []plusmn0deg 100 100 10000 3588 3590 9994plusmn5deg 100 100 10000 4469 4470 9997plusmn10deg 100 100 10000 7005 7010 9994plusmn15deg 100 100 10000 10892 10890 10002plusmn20deg 100 100 10000 15659 15660 10000plusmn25deg 100 100 10000 20733 20730 10001plusmn30deg 100 100 10000 25500 25500 10000plusmn35deg 100 100 10000 29387 29390 9999plusmn40deg 100 100 10000 31924 31920 10001plusmn45deg 100 100 10000 32805 32800 10001

Table 76 Resulting curvatures and stress in MATLAB and SOFiSTiK forLoad case 734

Angle Curvature κ2 [10minus3] Stress σ1 [MPa]θ MATLAB SOFiSTiK [] MATLAB SOFiSTiK []plusmn0deg 031 031 9994 62624 62624 10000plusmn5deg 038 038 9996 61446 61446 10000plusmn10deg 058 058 9995 57676 57676 10000plusmn15deg 087 087 9998 50894 50892 10000plusmn20deg 116 116 10002 41298 41296 10001plusmn25deg 134 134 10004 30433 30428 10002plusmn30deg 135 135 10004 20616 20610 10003plusmn35deg 123 123 10002 13398 13393 10004plusmn40deg 103 103 10000 8926 8923 10003plusmn45deg 082 082 9998 6537 6536 10001

86





Angle Curvature κ12 [10minus3] Stress τ12 [MPa]θ MATLAB SOFiSTiK [] MATLAB SOFiSTiK []plusmn0deg 100 100 10000 1794 1795 9994plusmn5deg 100 100 10000 2234 2235 9997plusmn10deg 100 100 10000 3503 3505 9994plusmn15deg 100 100 10000 5446 5445 10002plusmn20deg 100 100 10000 7830 7830 10000plusmn25deg 100 100 10000 10366 10365 10001plusmn30deg 100 100 10000 12750 12750 10000plusmn35deg 100 100 10000 14693 14695 9999plusmn40deg 100 100 10000 15962 15960 10001plusmn45deg 100 100 10000 16402 16400 10001

87


74 Materials and cross sections for beam elements

To verify if the material parameters also could be used for analysis using beam elementsa study was conducted comparing QUAD and beam elements in SOFiSTiK Four loadcases were defined see Table 710 and Figure 727 The study was conducted for twogeometries see Table 79

Table 79 Beam dimensions used in FE material case studies

Load case Length [m]Name Lx Ly Lz

Short beam 100 20 10Long beam 200 20 10

Table 710 Loads applied in Load case 741 to Load case 744

Load case Load [kNm]Name qx qy qz

Load case 741 5Load case 742 -5Load case 743 02Load case 744 02


(c) Load case 743 (d) Load case 744Figure 727 Loads applied in Load case 741 to Load case 744

88


741 Adaptations of parameters for beam elements

For each of the nine materials and load cases the maximum displacements were comparedfor the QUAD and beam models In Figures 728 to 730 the results of a study conductedfor a beam with cross section dimensions 20 times 10 mm and length of 100mm is shownHere it can be seen that displacements are similar for extension (Figure 728) and bendingabout the local z-axis (Figure 730) However for bending about the y-axis (Figure 729)the differences are very large and the the differences also increase for the materials basedon laminates with fibre orientations larger than plusmn0deg

Figure 728 Comparison of maximum displacements in the x direction forQUAD and beam elements with cross section dimensions 20 times10 mm and 100mm lengths for Load case 741

Figure 729 Comparison of maximum displacements in the y direction forQUAD and beam elements with cross section dimensions 20 times10 mm and 100mm lengths for Load case 744

89


Figure 730 Comparison of maximum displacements in the z direction forQUAD and beam elements with cross section dimensions 20 times10 mm and 100mm lengths for Load case 743

The large difference here is assumed to be due to the fact that the transverse beamdeflection are assumed to be uniform across the width of the beam and is a function of xin beam theory Due to the plate curvatures κy and κxy this deflection is not independentfrom y However for beams with a very high length to width ratio the effect of platecurvatures become so small that it can be neglected [19]

It was attempted to correct the difference in the transverse deflections by adapting thebeam cross sections in SOFiSTiK This was done by defining a shear deformation area Ay

[43] For each load case in the previous study the ratio between the transverse deflectionsfor beam and QUAD elements wb

y and wQy were calculated This was then applied as

a factor to the beam cross section area A see Equation 77 With these cross sectionadaptations the same study was conducted again and the results can be seen in Figure731 for load case Load case 744

Ay =wb

y

wQy

A (77)

Figure 731 Comparison of maximum displacements in y direction QUADand beam elements with adapted shear deformation areas forLoad case 744

90


Here it can be seen that this factor has a large influence on the transverse deflection forthe material defined by the lamina orientation plusmn0deg for remaining materials this factorhad little or no influence The displacements in the x and z directions were unaffected bythis change

The study was also conducted for a longer beam element with cross section dimensions20 times 10 mm and length of 200mm Here there was a much better correspondence for theplusmn0deg material The transverse deflections in the y were corresponding very well for allmaterials however the deflections in the z direction started to diverge for the two modelsaround plusmn20deg see Figure 732 to 734 No further attempts to find a better correspondencewere made and all materials with the exception of the plusmn0deg material were omitted foranalysis with beam elements in the following studies

The displacements were imported into Grasshopper and compared visually in order toevaluate the correspondence of the displacements over the whole width and length of thebeam see Figures 735 to 738 Here it was observed that the displacements were verysimilar overall



91



(a) Beam (b) QUADFigure 735 Comparison of displacements for Load case 741




92


742 Resulting forces and stresses in QUAD and BEAMelements

Stresses for beams can be calculated using the program module AQB in SOFiSTiK [45]and it is also possible to calculate stresses at certain points in the cross sections [43]In this study it was however chosen to calculate the stresses from sectional force resultssince it simplified the import of the results into Grasshopper with the developed toolPlotStresses and made it possible to directly compare stress results for beam and QUADmodels

7421 Calculation of stresses

Normal stresses due to bending about the local y and z axes in the beams were calculatedaccording to Equation 78 [46]

σx = Nx

A+ My

Iy

z + Mz

Iz

y (78)

The maximum shear stress in the centre point of the beam was according to Equation 79[46] Here the shear force V is taken as he largest of Vz and Vy This was compared tothe calculated absolute slab stress for QUAD elements in SOFiSTiK

τmax = 3V2A

(79)

Additionally the shear stress in the xy-plane was calculated for beams according toequation 710 for direct comparison with the shear stress in QUAD elements Shearstress in the xy-plane due to torsion was calculated according to Equations 711 and712 where β = 0246 and d and b are the beam cross section dimensions [46] Thecalculated stresses were verified against comparison with stresses calculated using AQBin SOFiSTiKrsquos Wingraf

τxy = 3Vy

2A(710)

τT = MT

WT

(711)

WT = βdb2 where b le d (712)

7422 Comparison betweenQUAD and beamsmodels

In Table 711 the minimum and maximum stresses for each load case and the materialCFRP 100 are shown Here it can be seen that there are differences in the minimumand maximum stresses between the two models due to local stress concentrations in theQUAD model but the average stress in all cases except for shear were corresponding well

The stresses were imported into Grasshopper and Rhino using the tool PlotStresses forvisual comparison of the results see Figures 739 to 744 Here it can be seen that thestress variation is very similar especially in the centre of the beam for all cases

93


(a) Beam (b) QUADFigure 739 Comparison of normal stress σx for Load case 741


(a) Beam (b) QUADFigure 741 Comparison of normal stress σx in top plane for

Load case 743

(a) Beam (b) QUADFigure 742 Comparison of normal stress σx in bottom plane for

Load case 743

(a) Beam (b) QUADFigure 743 Comparison of normal stress σx in centre plane for

Load case 744

(a) Beam (b) QUADFigure 744 Comparison of shear stress τxy in centre plane for

Load case 744

94


Table 711 Normal and shear stresses σx and τxy in QUAD and beammodels for Load case 741 to Load case 744

σx [MPa] τxy [MPa]Load case Position Min Max Min Max

Center 25862 102006 000 000Load case 741 - QUAD Top 25862 102006 000 000

Bottom 25862 102006 000 000Center 50000 50000 000 000

Load case 741 - Beam Top 50000 50000 000 000Bottom 50000 50000 000 000Center -102006 -25862 000 000

Load case 742 - QUAD Top -102006 -25862 000 000Bottom -102006 -25862 000 000Center -50000 -50000 000 000

Load case 742 - - Beam Top -50000 -50000 000 000Bottom -50000 -50000 000 000Center 000 000 -7180 7180

Load case 743 - QUAD Top -078 75077 000 000Bottom -75077 078 000 000Center 000 000 -7500 7500

Load case 743 - Beam Top 000 75000 000 000Bottom -75000 000 000 000Center -42003 42003 -6114 6114

Load case 744 - QUAD Top 000 000 000 000Bottom 000 000 000 000Center -37500 37500 -7500 7500

Load case 744 - BEAM Top 000 000 000 000Bottom 000 000 000 000

95

8DEVELOPMENTOFA PARAMETRICDESIGNANDANALYSIS APPROACH

8 DEVELOPMENTOFA PARAMETRICDESIGNANDANALYSIS APPROACH

Figure 81 Flowchart describing the methodology used for the case studiespresented in Chapter 8 - Development of a parametric design andanalysis approach

98


81 Preliminary evaluation of detailed parametric models

Three types of parametric models were developed for the topology of the studied FRPtype a QUAD model a beam model and a uniform QUAD model and case studies werecarried out as in Figure 81 The results from these case studied are presented in thischapter

811 Defining structural objects in Grasshopper

The structural information were defined as object attributes and three Grasshoppercomponents were written using C in order to generate these attributes for SPT SLNand SAR objects see Appendix B2 for a description of these tools

812 Parametric FEmodels in Grasshopper

The geometry and structural information for the FE models were defined in Grasshopperand a parametric model was developed see Figure 82 The models were defined for atopology mesh segment with the assumption that the principles applied in this smallscale model could also be applied to the actual FRP modules since the topology wouldbe the same The mesh segment was modelled assuming the mesh segment is planarIn the actual FRP module this is not the case this will however be further discussed inChapter 10 Comparison between the three models was done by comparing deflections inthe points indicated in Figure 82 as well as comparing forces and stresses mainly in theuni-directional zones

(a) Symmetric (b) AssymetricFigure 82 Principal illustration of the parametric model developed for a

mesh segment

8121 QUADmodel for topologymesh segment

The QUAD model was modeled as in Figure 83 Both the uni-directional and nodezones were considered by applying the CFRP 100 material in the unidirectional zonesand materials based on the angle θ in the node zones see Table 71

99


8122 Beammodel for topologymesh segment

The beam model was modelled as in Figure 84 The node zones were not accounted forand the beams were modelled as uniform beams with the material CFRP 100 see Table71 and cross section dimensions shown in Table 41 with a modified shear area as inEquation 77

8123 UniformQUADmodel for topologymesh segment

The uniform QUAD model was modelled using a uniform QUAD mesh over the wholemesh segment see Figure 85 The reason for this was to evaluate weather the QUADelements or materials could be adapted in a way such that this model would replicate thebehaviour of the QUAD model If possible this would mean that the parametric modelcould perhaps be simplified with regard to geometric complexity number of elements andmodel parameters

(a) Rhino surfacemodel

(b) SOFiSTiK model

Figure 83 Parametric QUAD models

(a) Rhino line model (b) SOFiSTiK modelFigure 84 Parametric beam models

(a) Rhino surface model (b) SOFiSTiK modelFigure 85 Parametric uniform QUAD models

100


813 Comparison and preliminary evaluation of parametric FEmodels

A preliminary evaluation of the different modelling approaches was conducted Themodels were analysed in SOFiSTiK and compared with regard to tensioncompressionbending and torsion see Section 82 The comparisons were made by adjusting theparametric model and importing the results into Grasshopper for visual evaluation ofdeflections and stresses The results were also compared in the SOFiSTiK environmentIn this section the initial findings are only discussed the results from detailed study whichfollowed this study is presented in Section 82 The QUAD model was assumed to be themost accurate model since this is the only model which takes into account the materialsfor the node zones This model was therefore used to compare and evaluate the two othermodels

8131 Comparison between detailed QUAD and beammodels

A similar correspondence between beam and QUAD models could be seen here as wasobserved in Chapter 7 The deflections were mostly corresponding with some differencesin the case of bending however the correspondence was assumed to be accurate enoughThe internal stresses and forces were also within range in the unidirectional zones withlocal differences in the node zones this is however expected due to the differences instructural models and material formulations in these zones

8132 Comparison between detailed QUAD and uniformQUADmodels

It was attempted to find a stiffness of the uniform QUAD which would simulate thestiffness of the QUAD model This was done by adapting the thickness of the QUADelements as well as adapting the stiffness of the elements in the local x and y directions[47]In symmetric cases see Figure 82a the thickness of the element and the stiffness factorscould be defined parametrically in Grasshopper as functions of the mesh area and densityWith the adapted stiffness a correspondence could be seen between the deflections ofthe QUAD and uniform QUAD models however in the case of stresses of forces it wasnot possible to directly relate the results of the uniform QUAD model to the detailedQUAD model In the asymmetric case see Figure 82b a way to adjust the stiffnessparametrically could not be found This modelling approach was thus abandoned

101


82 Comparison and evaluation of QUAD and beammodels

A parametric study was conducted to compare the QUAD and beam models Three loadcases were defined see Figures 86 to 88 and Table 82 The study was conducted forseveral parametric setups in this Chapter the results are presented for the parametricsetup shown in Table 81 The results for other parametric setups followed the sametrends

Table 81 Parameters used in comparison between QUAD and beam models

Model Length [m]Name L1 L2 θ

Symmetric 75 75 45deg

(a) Symmetric (b) AssymetricFigure 86 Load case 821


(a) Symmetric (b) AsymmetricFigure 88 Load case 823

102


Table 82 Load cases used for comparison between QUAD and beam modelsfor a mesh topology segment

Load case Force [kN] Moment [kNm]Name Px My

Load case 821 94Load case 822 01Load case 823 01

Table 83 Comparison of maximum and minimum displacements [mm] forLoad case 821 to Load case 823

Displacements [mm] Comparison []QUAD BEAM

Load case Min Max Min Max Min MaxLoad case 821 0180 0364 0204 0296 113 81Load case 822 0057 3952 0056 2895 98 73Load case 823 0222 0128 58

(a) Beam (b) QUADFigure 89 Comparison between displacements for Load case 821


103


821 Comparison of displacements

A comparison between the minimum and maximum displacements for the parametricsetup in Table 81 and the load case in Table 82 was done The results from thiscomparison can be seen in Table 83 At first glance the differences appear large howeverwhen visualised graphically in Grasshopper the displacements correspond well between thetwo models see Figures 89 and 810 It can also be considered more important to comparethe displacements on the points shown in Figure 82 especially in the uni-directional zonesand along the centre line of the beams It can also be seen in the QUAD models thatthe curvatures κx and κxy has some influence However overall the displacements wereassumed to be corresponding sufficiently well between the two models


The minimum and maximum stresses were compared between the two models and arepresented in Table 84 Here it can be seen again that there seems to be significantdifferences between the two models however when examining the overall stress visuallyin Grasshopper it can be seen that the differences are mainly due to local stressconcentrations in the node zones see Figures 811 to 813 Overall there was a goodcorrespondence between the stresses with slightly larger differences in shear stresses seeFigure 812

Table 84 Normal and shear stresses σx and τxy in QUAD and beammodels


Center -14215 15115 -7568 7568Load case 821 - QUAD Top 000 000 000 000

Bottom 000 000 000 000Center -17083 20417 -2500 2500

Load case 821 - Beam Top 000 000 000 000Bottom 000 000 000 000Center 000 000 000 000

Load case 822 - QUAD Top -7255 24508 -8338 8338Bottom -24508 7255 -8338 8338Center 000 000 000 000

Load case 822 - Beam Top 9999 10001 -6773 6773Bottom -10001 -9999 -6773 6773Center 000 000 000 000


Load case 823 - Beam Top -22008 22008 -333 333Bottom -22008 22008 -333 333

104



Load case 821


Load case 821


Load case 823

105

9VERIFICATIONOF THE PROPOSEDPARAMETRICDESIGNANDANALYSIS

APPROACH

9 VERIFICATIONOF PARAMETRICDESIGNDESIGNANDANALYSIS APPROACH

Figure 91 Flowchart describing the methodology used for the case studiespresented in Chapter 9 - Verification of the proposed parametricdesign and analysis approach

108


91 Verification and calibration approaches

Up until this point many assumptions have been made and two models have beendeveloped on the basis on these assumptions The two models have a good correspondencehowever the accuracy of the assumptions have not yet been determined and withoutmechanical tests it cannot be said whether the simplifications made with regard to thenodes and discontinuity zones are accurate

Further the material formulations considerations for material safety factors must be madeThe EN Eurocodes are used as standards for design of buildings where characteristicvalues and partial coefficients as well as methods for dimensioning can be found Howeverfor FRP materials such a European Standard does not yet exist A pre-normativedocument has been published by the European Commissionrsquos Joint Research Centrewhich summarises the current requirements which must be considered for the use of FRPmaterials in building construction Here it is stated that characteristic values should bedetermined by laboratory tests in accordance with EN 1990 [6]

Verification and calibration of material should thus be carried out through mechanicaltests a principal methodology for this can be seen in Figure 91 Mechanical tests andFE analysis should be carried out using the same configurations and the FE modelshould be adapted such that its results correspond sufficiently well with the results fromthe physical tests Unfortunately it has not been possible to conduct mechanical testswithin the time frame of this study however this may be an opportunity for future work

109

10APPLICATIONOF THE PROPOSED

PARAMETRICDESIGNANDANALYSISAPPROACH

10 APPLICATIONOF PARAMETRICDESIGNANDANALYSIS APPROACH

Figure 101 Flowchart describing the methodology used for the case studiespresented in Chapter 10 - Development of a parametric designand analysis approach

112


101 Evaluation of FEmodels for full scale FRPmodule

The parametric analysis method which was developed and presented in Chapter 8 wasapplied to a full scale FRP module and evaluated as shown in Figure 101 Even thoughthe method has not been verified see Chapter 9 it is still motivated to apply the methodto evaluate if the method is applicable to large scale models and if the assumptions madeup until now also hold for the full scale modules

1011 Parametric FEmodel in Grasshopper

A parametric model was developed in Grasshopper and Rhino for the FRP module forQUAD and beam elements see Figures 102 to 104

The material definitions used in the QUAD model have been determined by the FRPmodule geometry Here the uni-directional zones have been assigned the CFRP 100material see Table 71 and the materials in the node zones have been determined basedon the topology mesh angle

In the beam models the same material and cross section have been assigned to all beamsin the models where the shear area factor (see Section 741) has been determined by theaverage length of all beams

Figure 102 Surface model in Rhino which has been generated inGrasshopper for analysis of QUAD elements in SOFiSTiKAttributes with structural information has been written to thegeometric objects and the colours indicate which material hasbeen applied for the different zones Here red indicates that itis a uni-directional zone The materials in the node zones havebeen determined based on the mesh angle

113


Figure 103 Line model in Rhino which has been generated in Grasshopperfor analysis of beam elements in SOFiSTiK Here all beamshave been assigned the same materials and cross section

Figure 104 SOFiSTiK QUAD model for the FRP module

Figure 105 SOFiSTiK beam model for the FRP module

114


1012 Evaluation of non-planarity of mesh segment

In Chapter 8 the mesh segment was assumed to be planar however this is not the casein the actual FRP module A small study was conducted to evaluate the curvature ofthe FRP module see Figure 106 The mesh was drawn using curved lines as well asstraight lines and the two models were overlayed for comparison It can be seen here thatthe assumption that the lines defining the mesh are straight is mostly accurate In somezones there is a large difference however this large difference only occurs in modules withvery low mesh densities A mesh defined by straight lines was therefore used in analysisto simplify the model

(a)

(b)

Figure 106 The assumption that the mesh is planar was evaluated bycomparing the actual mesh with curved lines to a simplifiedmesh defined by straight lines It can be seen here that the twomeshes are similar except in zones with low density and highcurvature as in the zone marked in the image

115


1013 Evaluation of number of element numbers andmeshdependency

In Chapter 7 a mesh convergence study was conducted for simple QUAD model and anestablish mesh size was used in the case studies presented in Chapters 7 and 8 In the caseof the FRP modules there is a larger complexity especially at the element boundaries Asmall convergence study was therefore conducted for the parametric FRP module models

For the QUAD model of the FRP module the mesh was generated using SOFiSTiKrsquosautomatic meshing where it is possible to adjust the coarseness of the mesh The meshwas refined and the model analysed step-wise and here it was seen that the full FRPmodules required a finer mesh along the module boundary in order to reach convergenceThe mesh size in remaining areas did however not affect convergence and the mesh sizedetermined in Chapter 7 was sufficient here This indicates that the QUAD model is verymesh dependent in the case of the FRP modules and must be adjusted according to thegeometry

The same was done for the beam models where the models were analysed for an increasingnumber of subdivisions of the beam elements and here no significant difference inthe results could be observed This indicates that beams are independent of elementsubdivision however it should be noted here that the model was analysed using the samematerials and beam cross sections for all elements

1014 Evaluation of computational times

The meshing and computation times for each model were compared and evaluated theresults can be seen in Table 101 Here it can be observed that the QUAD modelsrequire significantly longer meshing and computation times compared to the beam modelsespecially in the case where the mesh density is higher

Table 101 Comparison of number of elements for QUAD and beam modelwith varying mesh densities

Element number [-] Meshing time [s] Calculation time [s]Density QUAD Beam QUAD Beam QUAD BeamLow 9046 864 9 3 25 17Medium 17365 3456 44 6 56 25High 34243 7776 170 9 147 42

116


102 Comparison betweenQUAD and beammodels

The FRP module was analysed using all three mesh densities for three different loadcases see Figures 107 to 109 The module was fixed on one end by coupling the moduleboundary to a fixed node see the yellow marking in Figures 107 to 109 The load wasthen applied to a node which was coupled to the other boundary edge The resultingdisplacements and internal stresses of the QUAD and beam models were compared Thedisplacements were compared by importing the SOFiSTiK results into Grasshopper Thestresses however were only compared in SOFiSTiK due to the fact that the Grasshoppertools which were used to import the stresses were not optimised for such a large numberof elements No attempt was made to further develop this tool at this point

Figure 107 Load case 1021 The FRP module was loaded with a normalforce of 12 kN

Figure 108 Load case 1022 The FRP module was loaded with a forceof 12 kN

Figure 109 Load case 1023 The FRP module was loaded with amoment of 12 kNm

117



The stresses were compared in SOFiSTiKrsquos interface for graphical representation ofresults Wingraf where it could be that there was a difference in stresses in the twomodels mainly at the module boundaries The QUAD model showed very large stressconcentrations at the module boundaries compared to the beam models It was ofinterest to evaluate and compare the stress in the elements not directly connected to theboundaries also In these non-boundary zones the stresses had a better correspondencebetween the QUAD and beam models It could be seen that there were stress peaks inthe QUAD models in the transition between the uni-directional and node zones whichis expected In the uni-directional areas the stresses has a good correspondence as wasalso seen in Chapters 7 and 8 In Table 102 the von Mises stress for each load case areshown for the QUAD and beam models both for the cases where the module boundarieswas considered and the cases where it was not

Table 102 Maximum von Mises Stress in FRP module for each load caseboth with and without consideration to module boundary

With boundary [MPa] Without boundary [MPa]Load case Quad Beam Quad BeamLoad case 1021 601 221 277 18Load case 1022 177 111 112 111Load case 1023 855 391 28 21


The maximum displacements for each load case can be seen in Table 103 InLoad case 1021 and Load case 1022 a similar correspondence can be seen as waspreviously observed in Chapter 8 however in Load case 1021 there is a much largerdifference The results were imported into Grasshopper and can be seen in Figures1010 to 1012 Here it can be seen again that the overall displacements are similar fortensioncompression and bending however in the case of torsion there is a large overalldifference as well as large local differences Similar results could be seen for the two othermesh densities

Table 103 Comparison of maximum and minimum displacements forLoad case 1021 to Load case 1023

Displacements [mm] Comparison []QUAD BEAM Comparison

Load case Min Max Min Max Min MaxLoad case 1021 0429 0376 88Load case 1022 1334 1166 87Load case 1023 0837 0252 30

118


(a) Beam (b) QUADFigure 1010 Comparison between displacements for load case

Load case 1021


Load case 1022


Load case 1023

119

11DISCUSSION

11 DISCUSSION

Figure 111 Flowchart describing the method for evaluation and comparisonof FE models

122

11 DISCUSSION

111 Establishment of design and analysis approachThis study was conducted in order to establish a new design and analysis approach forFRP modules in architectural and civil engineering contexts The underlying basis forthis study has been the research conducted at the University of Stuttgart In this chapterthe basis scope and limitations methodology assumptions and results are discussed inrelation to the previous research and the structural engineering context

1111 Importance of interdisciplinary cooperation

The building industry is far behind other industries when it comes to use of FRPas a structural material The material has many properties which are interesting inarchitectural and structural engineering contexts The ability to engineer the FRPmaterial to exhibit certain structural behaviours can for instance enable the architectto be more expressive in design and achieve shapes and forms which may be difficult toachieve using other materials This fact has also driven the structural engineer to fullyutilise her knowledge in mechanics and structural design in this study as well as attemptto adopt new strategies and tools

The research conducted at the University of Stuttgart is a very good example of wheninterdisciplinary collaboration has led to interesting and important architectural andstructural design developments The method developed at the University of Stuttgarthas however not been applicable in this study which indicates that there is still a needfor research and development in this field

Even though the aim has been derived from a combined architectural and structuralbackground the main content is technical and has adopted a strategy which may beconsidered a traditional civil engineering method - a structural analysis approach hasbeen developed based on architectural intent with little or no freedom in expression inform of the structure and there has been no interdisciplinary collaboration

Equally important as it is to develop new strategies for analysis is it to have a goodcollaboration between architects and engineers so that the full potential of the materialcan be utilised both aesthetically and structurally

1112 The need for simplification in analysis

Building standards and design codes are essential in the civil engineering context andas such it is logical that the tools used for building design are well adapted to takethese regulations into account Unfortunately this imposed many limitations in thepresented work Materials which are not defined in these standards and codes are also notincluded in SOFiSTiK Therefore it is difficult to implement FRP materials in analysis inSOFiSTiK Analysis could very well be carried out in other FE analysis software howeverthis may not be recommended as there are several more aspects to consider here otherthan material definitions For instance safety factors and load combinations defined inthe building codes are very easy to implement in softwares such as SOFiSTiK which aredeveloped specifically for the building industry

In this study it was required to use SOFiSTiK Therefore many aspects of the mechanicalbehaviour of FRP materials had to be omitted in analysis such as mechanical coupling or

123

11 DISCUSSION

inter-laminar stress distributions Within this context it may be more suited to performdetailed analysis using other analysis methods based on results from a simplified butsafe-sided analysis performed in SOFiSTiK Thus it may very well be motivated toexclude discontinuities and singularities in analysis for preliminary design if the analysisis safe-sided

1113 Evaluation of theory and references

Several literature references have been used throughout the work which have been thebasis for assumptions but also used to evaluate assumptions made in this study

The major theoretical basis here has been Classical Laminate Theory In the literaturereferences related to FRP mechanics it is clear that CLT is common practice in analysisof FRP materials It can be said that it is motivated to use CLT for analysis of FRPcomposites In most theory only plate theory is discussed as CLT is derived on the basisof plate theory little reference or previous research was found for the application of beamtheory and FRP materials

In structural analysis in SOFiSTiK the manuals provided by SOFiSTiK has been usedto find information about the theories used in analysis Some guess work had to be donehere especially in the case of analysis of orthotropic materials To evaluate the analysis inSOFiSTiK theoretical studies were conducted in MATLAB for evaluation and comparisonof the SOFiSTiK analysis

1114 Evaluation of methodology

The work has been carried out in a logical and step wise manner as was indicated inChapter 3 following the logic input - method - output In each step some input has beengiven or some assumption has been made this has been followed by structured studies inwhich the input or assumption has been evaluated From this some output data has beengenerated which in turn has served as input in the next step In this study where therehave been many uncertainties assumptions and a large numbers of structural models toevaluate this working method has been very well motivated

The work has been defined by investigation followed by evaluation and verification Ineach step the results have been evaluated and in the cases where results could not beverified assumptions have been disregarded or models and methods omitted from thework

124

11 DISCUSSION

112 Evaluation of assumptions and results

The assumptions made here have been made on basis on several literature referencesderivations as well as findings in this study The assumptions have been founded onmechanical principles with clearly stated simplifications or limitations and have beenevaluated through theoretical studies and analytical and numerical analysis In Chapter21 a number of hypothesis were also made these are discussed and evaluated here inrelation to the results of this study

1121 Transverse isotropy

It has been assumed that the material can be simplified in analysis such that a transverselyisotropic material formulation could be used This was evaluated through literaturestudies where it was found that the mechanical behaviour of a FRP laminate can beengineered by controlling the components of the A B and D matrices in the laminateconstitutive equation This can be done by adjusting thickness lamina orientations andlayup sequences of the laminate Certain laminates will be designed such that these arefully orthotropic laminates which means they will be transversely isotropic

The laminates in this study are not designed in such a way that it can be said thatthey are transversely isotropic by design however based on the literature studies it washypothesised that even these laminates it can be said to be transversely isotropic due tothe thickness of the laminates This assumption was also motivated by literature andverified through theoretical case studies

A limitation to this assumption is that it neglects the influence of the plate thicknesson the overall stiffness Depending on geometry and theory used in analysis this platethickness will have an influence It was seen in the case studies that it may not be entirelymotivated to neglect these effects and this should be further evaluated in future research

1122 Uni-axial load transfer

It was assumed that the load transfer in the modules is mainly uni-axial and thatbeam elements can be used in analysis It was seen in the small models that this wasmainly correct however when applying the method to the actual FRP module significantdifferences could be seen in the case of torsion and shear This indicates that there may bemulti-axial load transfer in the node zones and significant out-of-plane effects which maynot be properly simulated using beam elements The main limitation of beam elementsis the inability to account for plate curvatures In the case of FRP materials these effectsshould perhaps not be discounted due to material specific properties for certain layupswith for instance large variations in the Major and Minor Poissonrsquos ratios which may verywell be above 05

1123 Calibration and verification

It was assumed that the FE method and models could be verified and calibrated againstmechanical tests It has not been possible to verify this assumption since mechanicaltests could not be performed within the time frame of the study In the case studies

125

11 DISCUSSION

it was observed that mechanical tests for verification and calibration are necessary inorder to evaluate the simplifications made The influence of the node zones in the QUADmodels indicate that the simplifications made with regard to the discontinuity and nodezones may not be fully motivated Mechanical test should therefore be performed and theFE models should be adapted accordingly However in Chapter 9 potential adaptationsof the FE models were pointed out with indications on how the parametric modelsmaterials and cross sections can be adapted in a verification and calibration situations Itshould therefore be said that the parametric design and analysis method developed hereis adaptable and that it is likely that it can be verified and calibrated against mechanicaltests

1124 Applicability of results

It was assumed that results from a simplified global analysis of this FRP module can beused in a detailed design situation In all evaluated models the correspondence betweentheory and analysis as well as the correspondence between QUAD and beams modelscan be said to be sufficient with the exception of shear and torsional deformations andstresses where significant deviations could be seen between the QUAD and beam models

Overall and especially in the uni-directional zones the correspondence was good wherethe deflections and stress results in the beam analyses were generally larger This indicatedthat both methods can be used for analysis of the FRP module in this study providedthat mechanical tests can be performed with necessary adaptations to the parametricmodels

1125 Evaluation of tools

Three tools which were used to create structural objects in Grasshopper for analysis inSOFiSTiK were developed in this study The development of these tools was motivatedby the fact that the SOFiSTiK interface with Rhinoceros is not parametric an there wasthus a need for a tool to automate the creation of the structural models in Rhino Thetools were written using C and implemented in Grasshopper and could be used tocreate the structural elements in an easy and direct manner

Other tools which could be used to evaluate results from structural analysis in SOFiSTiKwere also developed and these were verified by comparing the imported results inGrasshopper with results which could be seen in SOFiSTiK These tools proved veryvaluable in comparison of the many models which were analysed since it made it simpleto make direct and qualitative comparison of structural analysis results in a 3D modellingenvironment This made it possible to for instance measure deformations directly in the3D software The interface in SOFiSTiK which can be used to evaluate results visuallydoes not have the same adaptability or interactivity Thus in a parametric study wheredirect comparison of many models were necessary these tools proved to be essential

1126 Evaluation of parametric FEmodels

The parametric models were developed in Grasshopper and Rhinoceros with the additionaluse of C and third party plugins The models have been verified against the MATLAB

126

11 DISCUSSION

functions which were used in this study The author of this thesis was part of thedevelopment of these functions as part of a student group project at Chalmers Universityof Technology where these functions where verified in theoretical studies These functionswere therefore considered to be accurate for verification of this study as well and thus itcan be said that the parametric models are also verified

The parametric models have been applied to a large variety of parametric setups andanalyses this indicated that these fulfil the purpose of this study in which an adaptableparametric model is required which can be applied to multiple geometries

1127 Evaluation of parametric design and analysis approach

The parametric design and analysis approach was developed on the basis on manylimitations simplifications and assumptions which have been discussed in this ChapterThe results of the structural analysis has mainly corresponded with the assumptionsmade this however only indicates if the assumptions made and analysis method areequally accurate or equally inaccurate Verification against mechanical tests still remainsHowever some results of the analysis give indications of the accuracy or inaccuracy of theparametric analysis approach

The method has been developed in a civil engineering context thus it has required severalsimplifications in order for successful implementation in the context of this study In thepublications related to the research at the University of Stuttgart which were studied hereit is stated that the lack of knowledge and suitable analysis tools for FRP composites inthe building industry often lead to over-simplified structural analysis models Here anin-depth study was made to gain sufficient knowledge however it may be said that themethod here has been over-simplified with regard to geometric and structural complexity

Some of the results of this study indicates inaccuracies in the assumptions made and themodelling and analysis approach For instance the large differences of deflections in theQUAD and beam models when comparing FRP modules subjected to torsion indicatethat the curvature of the mesh topology should perhaps be taken into account and thatQUAD elements may be more suitable for analysis due to this aspect

Stress concentrations due to geometric discontinuities and change of stiffness can bebetter simulated using QUAD elements however detailed analysis of inter-laminar stressis nevertheless still required With this in mind it may then be said that results acquiredfrom a simplified analysis using beam elements may be taken as preliminary values fordetailed design using analysis tools much more suited for analysis of laminated FRPcomposites This can be said especially due to the fact that SOFiSTiK is not suitedfor this type of analysis and also due to the fact that QUAD elements are inefficient inanalysis due to mesh size dependency as well as long meshing and computational times

The aim of this study was to develop a simplified reliable and sufficiently accurate ansafe-sided approach for design and analysis of the FRP module studied and as such itmay be said that the method established here fulfils this purpose however this remainsto be verified through mechanical tests

127

12CONCLUSIONAND

RECOMMENDATIONS

12 CONCLUSIONANDRECOMMENDATIONS

121 Concluding remarks

In this study a straight forward to use simple and reliable parametric design and analysismethod has been developed for a specific type of FRP module Based on the findings andresults a number of conclusions can be made for this approach with regard to the aimand scope of this study It should however be noted that these conclusions are valid forthe studied type of FRP module and only if the design and analysis can be verified andcalibrated against mechanical tests

Global and detailed analysis contextThe established design and analysis approach has proven to be applicable in a civilengineering context using SOFiSTiK as an analysis tool

The QUAD and beam models had very different performance and it can be said thatthe QUAD models are not recommended for analysis of large scale models due to longmeshing and computation times

Safe sided approachIt cannot be said whether the results are safe-sided until the method has been verified andcalibrated against mechanical tests The beam models however typically showed largerdisplacements and stresses and depending on analysis context considerations should memade for this fact

Applicability of resultsThe FE model was compared against theoretical case studies and an overallcorrespondence could be seen for both the QUAD and beam models It could thereforebe said that the results can be used in a detailed design situation

Parametric design approachThe parametric tools and models proved stable and reliable It was also seen that it couldbe applied to a variety of configurations with satisfactory results

CalibrationVerification and calibration could not be performed however it is likely that theparametric design and analysis method can be verified and calibrated due to reliabilityand adaptability in application as well as results indicating the both the QUAD andbeam material and cross section parameters can be adjusted in analysis to achievecorrespondence with desired mechanical behaviours

1211 Recommendation of parametric design and analysis approach

Assuming that mechanical tests and calibration can be performed it can be said thatit may be more suitable to use beam theory rather than plate theory in analysis of thestudied FRP module for preliminary design This is motivated by the fact that the beammodels proved to be more efficient than the QUAD models with regard to computationaltimes as well as due to the similarities between the two modelling approaches

130


122 Suggestions for future research

In this study some potential limitations and aspects which may be reconsidered has beenpointed out These are listed below and may offer opportunities for continued researchwithin the field of computational mechanics and parametric design for architecture andcivil engineering contexts

Mechanical tests for verification and calibrationsMechanical tests should be conducted for the studied FRP module to verify the accuracyof the parametric models The parametric analysis and design method should be adjustedto correspond to the mechanical tests and characteristic values for design should beestablished

Reconsideration and adaptations of discontinuity and node zonesWhen the parametric model was applied to the actual FRP module geometry somedifferences could be observed in the QUAD and beam models mainly in the case of torsionThis may indicate that there is multi-axial load transfer and thus the discontinuity andnode zones may have to be re-evaluated to account for this

Extendmethod to other geometries and topologiesThe method has been applied to one type of topology and geometry however the methodmay for instance also be adapted for triangulated mesh topologies where the load transferis mainly uni-axial

Engineeredmaterials for structural and architectural optimisationOne topic studied here was the in- and out-of-plane mechanical coupling which is amaterial property in FRP which can be design Thus the material can be engineeredto deform or carry load in a certain way This can be interesting in architectural andstructural applications where form-finding or expressive shapes may be of interest

Fatigue and failure and instability phenomenaNo consideration for fatigue or failure nor instability was taken into account in thisstudy however it may be of interest to study this in the case of this FRP module Due todistinctive unidirectional zones buckling tendencies may be especially important to study

Structural analysis and dedicatedmanufacturingIn this study a specific FRP module was studied and the structural analysis was dictatedby the assumed laminate layup in the node zones In many applications where FRPcomposites are used dedicated structural analyses and manufacturing methods havebeen developed for specific purposed This may be of interest in architectural andcivil engineering contexts where standardisation is of high importance Developmentof dedicated design and manufacturing methods may enable more use of FRP compositesin these contexts

131

BIBLIOGRAPHY

[1] M Legault ldquoArchitectural composites Rising to new challengesrdquo 2018[Online] Available https www compositesworld com articles architectural-composites-rising-to-new-challenges Accessed on May 052018

[2] F Waimer R La Magna S Reichert T Schwinn A Menges and J KnippersldquoIntegrated design methods for the simulation of fibre-based structuresrdquo 09 2013Proceedings of the Conference Design Modelling Symposium 2013 [Online]Available https www researchgate net publication 292745702 _Integrated _ design _ methods _ for _ the _ simulation _ of _ fibre-based _structures Accessed on May 5 2018

[3] J Hale ldquoBoeing 787 from the ground uprdquo Aero pp 17-24 2006 [Online]Available httpwwwboeingcomcommercialaeromagazinearticlesqtr_4_06indexhtml Accessed on May 05 2018

[4] Wikimedia contributor Spaceaero2 ldquoAll nippon airways boeing 787-8 (ja801a) landat okayama airport 2011rdquo [Electronic image] Available httpscommonswikimediaorgwikiFileAll_Nippon_Airways_Boeing_787-8_Dreamliner_JA801A_OKJjpg Accessed on May 05 2018

[5] Creative Commons Attribution-ShareAlike 30 Unported (CC BY-SA 30)[Online] Available httpscreativecommonsorglicensesby-sa30deeden Accessed on June 8 2018

[6] L Ascione J-F Caron P Godonou K van Ijselmuijden J KnippersT Mottram M Oppe M Gantriis Sorensen J Taby and L Tromp ldquoProspectfor new guidance in the design of frprdquo 2016 Publications Office of the EuropeanUnion Luxembourg EUR 27666 EN-2016 [Online] Available http publicationsjrceceuropaeurepositoryhandleJRC99714 Accessedon May 16 2018

[7] E Joint Research Centre (JRC) ldquoCentc250 produces the en eurocodesrdquo 2018[Online] Available httpeurocodesjrceceuropaeushowpagephpid=23Accessed on May 05 2018

[8] San Francisco Museum of Modern Art SFMOMA ldquoArchitecturerdquo 2018[Electronic image] Available https www sfmoma org about our-expansion-2016architecture Accessed on June 08 2018

[9] San Francisco Museum of Modern Art SFMOMA ldquoSan francisco museum of

133

BIBLIOGRAPHY

modern art architectural fact sheetrdquo 2018 [Electronic document] Availablehttpswwwsfmomaorgaboutour-expansion-2016architecture Accessedon June 08 2018

[10] D Prado M J M Solly A Menges and J Knippers ldquoElytra filamentpavilion Robotic filament winding for structural composite building systemsin fabricate rethinking design and constructionrdquo 2017 Proceedingsof the Fabricate Conference 2017 pp 224-233 [Online] Availablehttpswwwuclacukbartlettarchitectureevents2017aprfabricate-2017-international-peer-reviewed-conference-and-publicationAccessed on May 5 2018

[11] ICDITKE ldquoResearch pavillion process photo 2014rdquo [Electronic image] Availablehttp www itke uni-stuttgart de archives portfolio-type icd-itke-research-pavilion-2013-14 Accessed on May 05 2018

[12] ICDITKE ldquoResearch pavilion process photo 2014rdquo [Electronic image] Availablehttp www itke uni-stuttgart de archives portfolio-type icd-itke-research-pavilion-2013-14 Accessed on May 05 2018

[13] ICDITKE ldquoResearch pavilion process photo 2015rdquo [Electronic image] Availablehttp www itke uni-stuttgart de archives portfolio-type icditke-research-pavilion-2014-15 Accessed on May 05 2018


[15] S Reichert T Schwinn R L Magna F Waimer J Knippers and A MengesldquoFibrous structures An integrative approach to design computation simulationand fabrication for lightweight glass and carbon fibre composite structures inarchitecture based on biomimetic design principlesrdquo Computer-Aided Designvol 52 pp 27 ndash 39 2014 [Online] Available httpwwwsciencedirectcomsciencearticlepiiS0010448514000293 Accessed on May 05 2018

[16] ICDITKE ldquoDevelopment process research pavilion 2014rdquo [Electronic image]Available httpswwwitkeuni-stuttgartdearchivesportfolio-typeicd-itke-research-pavilion-2013-14 Accessed on June 08 2018

[17] SSIGRAFILregContinuous Carbon Fiber Tow Meitingen Germany SGLTechnologies GmbH 2018 [Online] Available httpswwwsglgroupcomcmsinternationalinfokorbDownloadcenterindexhtml__locale=en Accessedon May 16 2018

[18] SIGRAPREGregPre-impregnated Materials made from Carbon Glass and AramidFibers Meitingen Germany SGL Technologies GmbH 2018 [Online] AvailablehttpswwwsglgroupcomcmsinternationalinfokorbDownloadcenterindexhtml__locale=en Accessed on May 16 2018

[19] B D Agarwal L J Broutman and K Chanddrashekhara Analysis andPerformance of Fibre Composites Hoboken New Jersey USA John Wiley amp SonsInc 2006 Third Edition

134

BIBLIOGRAPHY

[20] I Krucinska and T Stypka ldquoDirect measurement of the axial poissonrsquos ratio ofsingle carbon fibresrdquo Composites Science and Technology vol 41 no 1 pp 1 ndash 121991 [Online] Available httpwwwsciencedirectcomsciencearticlepii026635389190049U Accessed on May 16 2018

[21] H Lundh Grundlaumlggande haringllfasthetslaumlra Stockholm Sweden KTHHaringllfasthetslaumlra 2013

[22] W Jabi Parametric Design for Architecture London England Laurence Kingpubl 2013

[23] J Frazer ldquoParametric computation History and futurerdquo Architectural Designvol 86 no 2 pp 18ndash23 2016 [Online] Available httpsonlinelibrarywileycomdoipdf101002ad2019 Accessed on May 31 2018

[24] Rhinoceros 3D ldquoRhino 5 version 5 sr12 64-bitrdquo 2017 [Software] Availablehttpswwwrhino3dcomAccessedonMay052018

[25] Grasshopper ldquoGrasshopper version 090076rdquo 2017 [Software] Available httpwwwgrasshopper3dcomAccessedonMay052018

[26] SOFiSTiK AG ldquoSofistik rhinoceros interfacerdquo [Online] Available httpswwwsofistikeusolutionsstructural-fearhinoceros-interface Accessed onMay 31 2018

[27] A Heumann ldquoHuman + tree frogrdquo 2018 [Software] Available httpwwwfood4rhinocomapphuman Accessed on May 05 2018

[28] G Piacentino ldquoWeaverbird - topological mesh editorrdquo [Software] Available httpwwwgiuliopiacentinocomweaverbird Accessed on May 05 2018

[29] J Schumacher ldquoTT toolbox 19rdquo 2017 [Software] Available httpwwwfood4rhinocomapptt-toolbox Accessed on May 05 2018

[30] N Miller ldquoLunchbox 201781rdquo 2017 [Software] Available http www food4rhinocomapplunchbox Accessed on May 05 2018

[31] H Naumlrhi ldquoColor a meshrdquo 2018 [Software] Available httpwwwfood4rhinocomappcolor-mesh Accessed on May 05 2018

[32] Microsoft ldquoC programming guiderdquo 2017 [Electronic] Available httpsdocsmicrosoftcomen-usdotnetcsharptour-of-csharpindex Accessed onMay 31 2018

[33] Microsoft ldquoVisual studio 2015rdquo 2015 [Software] Available https www visualstudiocom Accessed on May 31 2018

[34] McNeel ldquoGrasshopper assembly for v5rdquo 2017 [Electronic resource] Availablehttps marketplace visualstudio com items itemName = McNeel GrasshopperAssemblyforv5 Accessed on May 31 2018

[35] SOFiSTiK AG ldquoSofistik 2016 version 1613 -33_64 servicepack 2016 - 13rdquo 2017[Software] Available httpswwwsofistikcom Accessed on May 05 2018

[36] SOFiSTiK AG ldquoCadinp - text based inputrdquo 2017 [Online] Available httpswwwsofistikeuno_cachesolutionsstructural-feacadinp

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BIBLIOGRAPHY

[37] SOFiSTiK AG ldquoWingraf version 1813-33rdquo 2018 [Online] Available httpswwwsofistikdedocumentation2018entutorialslistoftutorialsgeneral-workflowswingrafhtml Accessed on June 06 2018

[38] Mathworks ldquoMatlab r2014ardquo 2017 [Software] Available https www mathworkscomAccessedonMay052018

[39] H Naumlrhi and M Stadler ldquoComputer assignment 1 - stress analysis ofthermomechanically loaded laminaterdquo 2016 Project in TME240 CompositeMechanics Unpublished

[40] C York and P Weaver ldquoBalanced and symmetric laminates new perspectives onan old design rule in 51st aiaaasmeasceahsasc structures structural dynamicsand materials conferencerdquo 2010 [Online] Available httpsarcaiaaorgdoiabs10251462010-2775 Accessed on May 16 2018

[41] E Valot and P Vannucci ldquoSome exact solutions for fully orthotropic laminatesrdquoComposite Structures vol 69 no 2 pp 157 ndash 166 2005 [Online] Available httpwwwsciencedirectcomsciencearticlepiiS0263822304002144 Accessedon May 16 2018

[42] C York ldquoStacking sequences for extensionally isotropic fully isotropic andquasi-homogeneous orthotropic laminatesrdquo Apr 2008 Proceedings of theConference 49th AIAAASMEASCEAHSASC Structures Structural Dynamicsand Materials Conference Schaumburg IL USA [Online] Available http eprintsglaacuk4526 Accessed on May 15 2018

[43] AQUA Materials and Cross Sections AQUA Manual 201614 SOFiSTiK AGOberschleissheim Germany 2016

[44] T C T Ting and T Chen ldquoPoissonrsquos ratio for anisotropic elastic materials canhave no boundsrdquo The Quarterly Journal of Mechanics and Applied Mathematicsvol 58 no 1 pp 73ndash82 2005 Accessed on May 16 2018

[45] AQB Design of Cross Sections AQUA Manual 201614 SOFiSTiK AGOberschleissheim Germany 2016

[46] K-J Schneider and K Berner Bautabellen fuumlr Ingenieure mitberechnungshinweisen und Beispielen Koumlln Germany Bundesanzeiger VerlagGmbH 2016 22nd Edition

[47] ASE General Static Analysis of Finite Element Structures ASE Manual 201612SOFiSTiK AG Oberschleissheim Germany 2016

136

List of Figures

11 Nearly half of the air frame of the Boeing 787 consists of carbon fibrereinforced plastics and other composites [3] From [4] Reproduced withpermission CC-BY-SA 30 [5] 2

12 Examples of existing structures in Europe where FRP composites havebeen used as structural materials [6] 3

14 ICDITKE Research pavilions photos and development process 416 Principal geometry topology and layout of uni-directional fibre bundles

of the proposed FRP module 617 Discontinuity of the node zones 718 Principal geometry and fibre layout of different types of nodes 7

31 Flowchart describing the method and report outline 1732 Flowchart describing the method and report outline 19

42 Definition of topology for FRP module 2543 Terminology for the FRP module topology 2644 FRP module mesh density 2646 Local reference coordinate systems for the uni-directional elements (left)

and the node zones (right) The orientation of the coordinate axes hasbeen chosen based on the conventions used Classical Laminate Theorysee Chapter 5 - Theoretical background 27

51 Principal cross sections of uni-directional fibre composites 3452 Principal representation of a uni-directional fibre composite 3553 Specially and generally orthotropic laminae 3854 Description of a FRP laminate and terminology 4055 A laminate with orientation code [090+45-45+45-45900]

Adapted from Figure A3-1 [19 pg 543] 4156 Strain and strain variation in a principal FRP laminate 4157 Deformation of a laminate 4258 FRP laminate which is coupled with regard to in-plane extension and

shear 4559 FRP laminate which is coupled with regard to in-plane extension and

out-of-plane bending 45510 A symmetric laminate 46511 A balanced laminate 47

61 Flowchart describing the methodology used for the case studies presentedin Chapter 6 - Material parameters for structural analysis 52

62 Symmetric and balanced laminates studied in MATLAB 5463 Engineering constants for FRP laminae as a function of the volume

fraction Vf 5564 The calculated ratio Q16rel for plusmn30deg and plusmn45deg lamina orientations 5665 The calculated ratio Qplusmnθrel for plusmn30deg and plusmn45deg lamina orientations 5766 Components of the extensional stiffness matrix A for a thin symmetric

laminates with varying ply orientations 59

137

LISTOF FIGURES

67 Components of the extensional stiffness matrix A for a symmetriclaminate with lamina orientations plusmn30deg for an increasing number of plies 59

69 Components of the coupling stiffness matrix B for a thin balancedlaminates with varying ply orientations 60

610 Components of the coupling stiffness matrix B for a balanced laminatewith lamina orientations plusmn30deg for an increasing number of plies 60

612 Equivalent orthotropic elastic moduli derived for a 10 mm thicksymmetric laminate with varying lamina orientations 65

613 Equivalent orthotropic Poissonrsquos ratio derived for a 10 mm thicksymmetric laminate with varying lamina orientations 65

614 Strains for different laminates with 10 millimeter thickness subjected to aload N1 67

71 Flowchart describing the methodology used for the case studies presentedin Chapter 7 - Material parameters for FE analysis 70

72 Definition of parametric model used for QUAD models 7173 Parametric FE QUAD model in SOFiSTiK 7174 Definition of parametric model used for beam models 7275 Parametric FE beam model in SOFiSTiK 7276 The error 447 will occur during analysis in SOFiSTiK if the relation in

Equation 71 is not fulfilled 7377 Diagrams showing the relation between the Poissonrsquos ratios calculated for

fully orthotropic laminates in Chapter 6 and the adaptations required foranalysis in SOFiSTiK 74

710 Mesh convergence study in SOFiSTiK for a rectangular plate 76711 Load cases used in QUAD case studies 77714 Displacement plots for axial strains for CFRP 130 80715 Displacement plots for plate curvatures for CFRP 130 80716 Displacement plots for shear strain and curvature for CFRP 130 80717 Stress plot showing small stress variations for plate subjected to shear

bending 81718 Stresses in bottom plane of QUAD element for the material CFRP 130 82719 Stresses in bottom plane of QUAD element for the material CFRP 130 82720 Stresses in bottom plane of QUAD element for the material CFRP 130 82721 Stresses in centre plane of QUAD element for the material CFRP 130 83722 Stresses in centre plane of QUAD element for the material CFRP 130 83723 Stresses in centre plane of QUAD element for the material CFRP 130 83724 Stresses in top plane of QUAD element for the material CFRP 130 84725 Stresses in top plane of QUAD element for the material CFRP 130 84726 Stresses in top plane of QUAD element for the material CFRP 130 84727 Loads applied in Load case 741 to Load case 744 88728 Comparison of maximum displacements in the x direction for QUAD and

beam elements with cross section dimensions 20 times 10 mm and 100mmlengths for Load case 741 89

729 Comparison of maximum displacements in the y direction for QUAD andbeam elements with cross section dimensions 20 times 10 mm and 100mmlengths for Load case 744 89

138

LISTOF FIGURES

730 Comparison of maximum displacements in the z direction for QUAD andbeam elements with cross section dimensions 20 times 10 mm and 100mmlengths for Load case 743 90

731 Comparison of maximum displacements in y direction QUAD and beamelements with adapted shear deformation areas for Load case 744 90

732 Comparison of maximum displacements in the x direction for QUAD andbeam elements with cross section dimensions 20 times 10 mm and 200mmlengths for Load case 741 91



735 Comparison of displacements for Load case 741 92736 Comparison of displacements for Load case 742 92737 Comparison of displacements for Load case 743 92738 Comparison of displacements for Load case 744 92

81 Flowchart describing the methodology used for the case studies presentedin Chapter 8 - Development of a parametric design and analysis approach 98

82 Principal illustration of the parametric model developed for a meshsegment 99

86 Load case 821 10287 Load case 822 10288 Load case 823 102

91 Flowchart describing the methodology used for the case studies presentedin Chapter 9 - Verification of the proposed parametric design andanalysis approach 108


102 Surface model in Rhino which has been generated in Grasshopper foranalysis of QUAD elements in SOFiSTiK 113

103 Line model in Rhino which has been generated in Grasshopper foranalysis of beam elements in SOFiSTiK 114

106 The assumption that the mesh is planar was evaluated by comparing theactual mesh with curved lines to a simplified mesh defined by straight lines 115

111 Flowchart describing the method for evaluation and comparison of FEmodels 122

A1 Grasshopper components which are connected by wires A3

B2 An overview of the Grasshopper routines used for evaluation of resultsfrom the case studies conducted in this study B3

139

LISTOF FIGURES

B1 Parametric Grasshopper definition The three types of models whichwere created for this study can be seen here as well as the part whereresults from SOFiSTiK was imported Note that the wires whichconnect the components have been hidden to give a clear overview ofthe Grasshopper definition B4

B3 MakeSPT Grasshopper component B5B4 MakeSLN Grasshopper component B6B5 MakeSAR Grasshopper component B7

140

List of Tables

41 Cross section parameters 2842 Material properties of fibres 2843 Material properties of matrix 28

61 The calculated values for Q16rel which compares the Q16 term of the Qmatrix for all lamina orientations and volume fractions to the overallmaximum Q16 56

62 The calculated values for Qplusmnθrel which compares the Q16 and Q11 termsof the Q matrix for each lamina orientation and fibre volume fraction 57

63 Thin laminates with lamina orientations plusmn0deg le θ le plusmn 45deg 5864 Thick laminates with lamina orientations plusmn30deg and plusmn45deg 5865 Thin and thick symmetric and balanced laminate layups These

laminates were studied for lamina orientations plusmn30deg and plusmn45deg 6166 Strains for a 10 mm thick symmetric laminate (ϵ1 ϵ2 γ12) compared to

the strains (ϵprime1 ϵprime

2 γprime12) for a 1 mm thick symmetric laminate subjected to

a uniform load N1 and N prime1 respectively 63

67 Curvatures for a 10 mm thick balanced laminate (κ1 κ2 κ12) comparedto the curvatures (κprime

1 κprime2 κprime

12) for a 10 mm thick symmetric laminatesubjected to a uniform load N12 and N prime

12 respectively 6368 Derived engineering constants for a 10 mm thick symmetric laminate

with varying lamina orientations 6469 Thin and thick fully orthotropic laminates studied with equivalent

orthotropic material parameters derived for lamina orientations plusmn0degto plusmn45deg 66

71 SOFiSTiK material parameters for QUAD elements 7572 Applied strains and curvatures in loadcases 731 to 736 7873 Resulting strains and stresses in MATLAB and SOFiSTiK for

Load case 731 8574 Resulting strains and stresses in MATLAB and SOFiSTiK for


Load case 733 8676 Resulting curvatures and stress in MATLAB and SOFiSTiK for



Load case 736 8779 Beam dimensions used in FE material case studies 88710 Loads applied in Load case 741 to Load case 744 88711 Normal and shear stresses σx and τxy in QUAD and beam models for

Load case 741 to Load case 744 95

81 Parameters used in comparison between QUAD and beam models 10282 Load cases used for comparison between QUAD and beam models for a

mesh topology segment 103

141

LISTOF TABLES

83 Comparison of maximum and minimum displacements [mm] forLoad case 821 to Load case 823 103

84 Normal and shear stresses σx and τxy in QUAD and beam models 104

101 Comparison of number of elements for QUAD and beam model withvarying mesh densities 116

102 Maximum von Mises Stress in FRP module for each load case both withand without consideration to module boundary 118

103 Comparison of maximum and minimum displacements forLoad case 1021 to Load case 1023 118

A1 Laminate variations of FRP module A1

142

AEXPLANATIONS ANDDEFINITIONS

APPENDIX A - CONTENTS

A1 Additional descriptions of FRP module A1A11 Laminate layup in node zones A1

A2 Explanations of 3D modelling and programming terms A2A21 3D modelling terminology A2A22 Grasshopper terminology A3A23 Programming terminology A4

A3 Additional theory A5A31 Stiffness and compliance matrices for FRP composites A5A32 Transformation matrices A7

Appendix A - References A8

Ai

A EXPLANATIONS ANDDEFINITIONS

A1 Additional descriptions of FRPmodule

In this section additional descriptions of the FRP module is presented

A11 Laminate layup in node zones

The laminate layup of the node zones in the FRP module is assumed to be eithersymmetric or balanced (angle-ply) with a total thickness of 10 or 15 mm Thethickness of the individual laminae are assumed to be approximately 01 or 03 mmThus a total of eight variations of laminates can be constructed see Table A1 Thelayups are either balanced or symmetric with alternating lamina orientations meaningthe layup sequence pattern will be [θminusθθminusθ] For symmetric laminates this layupsequence can be denoted as [plusmnθminusθ]s and for balanced the layup sequence can bedenoted as [plusmnθ]

Table A1 Laminate variations of FRP module

Thickness [mm] Count [-]Laminate type Laminate Lamina LayersSymmetric 10 asymp 01 101Symmetric 10 asymp 03 33Symmetric 15 asymp 01 151Symmetric 15 asymp 03 51Balanced 10 asymp 01 100Balanced 10 asymp 03 32Balanced 15 asymp 01 150Balanced 15 asymp 03 50

A1


A2 Explanations of 3Dmodelling and programming terms

Terms related to 3D modelling and programming which have been used throughoutthe report are described in this appendix Some aspects which could be important toconsider in relation to parametric modelling and analysis are also mentioned here

A21 3Dmodelling terminology

In this section some terms which are commonly used in 3D modelling especially in theRhino and Grasshopper environments will be briefly explained

NURBs

Non-Uniform Rational B-Spline NURBs are mathematical representations of 2Dor 3D geometry which are defined by a degree control points and knots [48] Thesegeometries are very accurate and can be used to describe complex and free formgeometries using only little information Benefits of NURBs geometries are that theyare easy to define easy to change and that they have high accuracy Downsides ofNURBs geometries could be that the format is not very compact meaning it uses alot of storage space when storing data [49] This can become an issue when a meshgeometry has been converted to a NURBs geometry or for instance when the NURBsgeometry is defined by a large number of control points

Polygonmesh

Polygon meshes is a collection of 2D and 3D surfaces [49] These surfaces are calledmesh facets and in Rhino these have either three or four edges The corner points ofthese facets are called vertices To represent a smooth and complex geometry manyfacets and vertices would therefore be required

One benefit of converting a NURBs surface into a mesh is that a mesh format is morecompact than a NURBs surface meaning it can store much more data in an efficientmanner However since NURBs geometries are mathematically defined these are muchbetter suited for representing smooth or complex geometries since a mesh geometrywould require a large amount of polygon faces to represent a smooth complex surface

In cases where it is important to have smooth surfaces which are easily editable NURBsgeometries are much better suited however in cases where geometric tolerances andgeometric complexity is of less importance meshes may be much more suited withregard to computational performance

Geometric tolerances

In Rhino geometric tolerances can be set to define how much accuracy is required orhow much error is allowed [50] Even though Rhino is NURBs modelling software it isstill not 100 correct and thus tolerances must be considered

In this study geometric tolerances have been an important aspect since it is necessaryto ensure that structural lines points and areas are properly intersecting Since thestructural analysis is conducted using FE analysis which is a numerical approximationit is very important to ensure that the geometric tolerance set in Rhino is set to thesame geometric tolerance used when defining the topology mesh

A2


In a parametric model directions and orientations are often described by relating onegeometry to an other and sometimes new geometry is defined by Boolean intersectionsHere it can be preferred to use meshes since these are more memory efficient And inthe case of Boolean intersections it is also important to set the tolerances correctlysince these operations are always approximated in Rhino [50]

A22 Grasshopper terminology

Grasshopper is a graphical programming interface for Rhino [51] Below is a briefdescription of terms which are commonly used in this contexts

ComponentThe Grasshopper components can be described like building blocks [52] Eachcomponent typically takes some input performs an action and generates an outputThe inputs and outputs of a component can be connected to other components by usingwires see Figure A1

CanvasThe components are places on the Grasshopper Canvas which is the graphicalvisualisation of the Grasshopper program

AGrasshopper definitionA network of components is called a Grasshopper definition [53] This is basicallythe solution or task which is carried out by the specific network of componentsSometimes this is also called Grasshopper routine solution or script

BakingThe geometry which has been defined in Grasshopper can be viewed in Rhino howeverthis is only a preview [53] To edit the geometry in Rhino the geometry must be bakedThe baking generates a copy of the Grasshopper geometry inside Rhino where it can beedited or exported to other software Once the geometry has been baked it is fix and itcannot be adjusted parametrically anymore in Grasshopper

Figure A1 Grasshopper components which are connected by wires

A3


A23 Programming terminology

A traditional computer program can be viewed as a series of operations which arecarried out in sequence in an input-processing-output order [54] In Object OrientedProgramming OOP this is no longer the case Here the order of operations can beallowed to vary and the actions of the user of the program may be unpredictable andthe program must therefore be written to take this into account Some terms related toOOP which have been used in this study are mentioned and briefly described here

Objects and classesSometimes the terms object and class are used to describe the same thing [55] A classis typically a description of a type and an object is an instance of a type To make itclear how this is used in programming one can use a blueprint analogy where the classis the blueprint and the object is what the blueprint describes

In this report the word object has been used to refer to geometrical objects in Rhinosuch as curves surfaces or points These are in fact instances of objects which contain ageometry and attribute part [56]

Attributes and propertiesIn this study the term attribute has been used to refer to properties which are assignedto an object In this study this was done by defining Object User Data which containsinformation that is linked to an object [57]

A4


A3 Additional theory

Some additional theory related to Classical Lamination Theory is presented in thissection

A31 Stiffness and compliancematrices for FRP composites

Hookersquos generalised law in three dimensions can be written on the extended matrixform as in Equation A1 where Eijkl is a fourth order tensor with 81 constantsdescribing the stiffness of the material however due to symmetry this can be reducedto 21 independent components

σ11

σ22

σ33

τ23

τ31

τ12

τ32

τ13

τ21

=

E1111 E1122 E1133 E1123 E1131 E1112 E1132 E1113 E1121

E2211 E2222 E2233 E2223 E2231 E2212 E2232 E2213 E2221

E3311 E3322 E3333 E3323 E3331 E3312 E3332 E3313 E3321

E2311 E2322 E2333 E2323 E2331 E2312 E2332 E2313 E2321

E3111 E3122 E3133 E3123 E3131 E3112 E3132 E3113 E3121

E1211 E1222 E1233 E1223 E1231 E1212 E1232 E1213 E1221

E3211 E3222 E3233 E3223 E3231 E3212 E3232 E3213 E3221

E1311 E1322 E1333 E1323 E1331 E1312 E1332 E1313 E1321

E2111 E2122 E2133 E2123 E2131 E2112 E2132 E2113 E2121

ϵ11

ϵ22

ϵ33

ϵ23

ϵ31

ϵ12

ϵ32

ϵ13

ϵ21

(A1)

In the case of orthotropic materials this can further be reduced to nine independentconstants see Equation A2

E1111 E1122 E1133 0 0 0E1122 E2222 E2233 0 0 0E1133 E2233 E3333 0 0 0

0 0 0 E2323 0 00 0 0 0 E1313 00 0 0 0 0 E1212

(A2)

By introducing new notation the stress-strain relation for an orthotropic material cannow be written as in Equation A3

σ1

σ2

σ3

τ23

τ13

τ12

=

C11 C12 C12 0 0 0C12 C11 C12 0 0 0C12 C12

C11minusC122 0 0 0

0 0 0 C11minusC122 0 0

0 0 0 0 C11minusC122 0

0 0 0 0 0 C11minusC122

ε1

ε2

ε3

γ23

γ12

γ12

(A3)

A5


For specially orthotropic materials the out-of-plane stress components σ1 τ23 and τ13are assumed to be zero The stress-strain relation can thus be expressed in terms of thelongitudinal and transverse material axes as in Equations A4 to A5 The Q matrix isthe stiffness matrix of the specially orthotropic material The inverse of the matrix QS is called the compliance matrix

σL

σT

τLT

=

Q11 Q12 0Q21 Q22 00 0 Q66

εL

εT

γLT

(A4)

εL

εT

γLT

=

S11 S12 0S21 S22 00 0 S66

σL

σT

τLT

(A5)

Now the components of S can be expressed in terms of the engineering constants ET EL GLT νLT and νT L see Equations A6 to A12

εL = S11σL + S12σT (A6)

εT = S12σT + S22σT (A7)

γLT = S66τLT (A8)

S11 = 1EL

(A9)

S22 = 1ET

(A10)

S12 = minusνLT

EL

= minusνT L

ET

(A11)

S66 = 1GLT

(A12)

Since there exists a one-to-one relation between the Q and S matrices the componentsof the stiffness matrix can be expressed directly in terms of the engineering constantsfor specially orthotropic laminae

A6


A32 Transformationmatrices

The stiffness matrix of a generally orthotropic laminae can be expressed in relation to areference coordinate system by applying a transformation using the matrices T1 and T2which are shown in Equations A13 and A14

T1 =

cos2(θ) sin2(θ) 2 sin(θ) cos(θ)cos2(θ) cos2(θ) minus2 sin(θ) cos(θ)

minus sin(θ) cos(θ) sin(θ) cos(θ) cos2(θ) minus sin2(θ)

(A13)

T2 =

cos2(θ) sin2(θ) sin(θ) cos(θ)cos2(θ) cos2(θ) minus sin(θ) cos(θ)

minus2 sin(θ) cos(θ) 2 sin(θ) cos(θ) cos2(θ) minus sin2(θ)

(A14)

A7

APPENDIX A - REFERENCES

[48] Robert McNeel amp Associatesreg ldquoWhat are nurbsrdquo 2018 [Online] Availablehttpswwwrhino3dcomnurbs Accessed on June 10 2018

[49] Robert McNeel amp Associatesreg ldquoThe meshtonurb commandrdquo 2018 [Online]Available httpswikimcneelcomrhinomeshtonurb Accessed on June10 2018

[50] Robert McNeel amp Associatesreg ldquoUnderstanding tolerancesrdquo 2018 [Online]Available httpswikimcneelcomrhinofaqtolerances Accessed on June11 2018


[52] TOI-Pedia ldquoGrasshopperrdquo 2018 [Online] Available httpwikibktudelftnltoi-pediaGrasshopper Accessed on June 12 2018

[53] TOI-Pedia ldquoGetting started with grasshopperrdquo 2018 [Online] Available httpwikibktudelftnltoi-pediaGetting_Started_with_GrasshopperAccessed on June 11 2018

[54] R A Mata-Toledo and P Gupta ldquoObject-oriented programmingrdquo 2014[Online] Available https www accessscience com content object-oriented-programming757337 Accessed on June 11 2018

[55] Microsoft ldquoObject-oriented programming (C)rdquo 2017 [Electronic] Availablehttpsdocsmicrosoftcomen-usdotnetcsharpprogramming-guideconceptsobject-oriented-programming Accessed on May 31 2018

[56] D Rutten ldquo7 geometryrdquo 2018 [Online] Available httpdeveloperrhino3dcomguidesrhinoscriptprimer-1017-geometry72-objects-in-rhinoAccessed on June 11 2018

[57] S Baer ldquoObject user datardquo 2018 [Online] Available httpdeveloperrhino3dcomguidesrhinocommonplugin-user-dataobject-user-dataAccessed on June 11 2018

A9

BPROGRAMSANDPARAMETRIC TOOLS

APPENDIX B - CONTENTS

B1 MATLAB functions and programs B1B11 Description of MATLAB functions B1B12 Description of MATLAB programs B2

B2 Grasshopper definitions B3B21 PlotStrainCurvature B3B22 PlotDisplacements B3B23 PlotStresses B3

B3 Grasshopper components B5B31 MakeSPT B5B32 MakeSLN B6B33 MakeSAR B7

Bi

B PROGRAMSANDPARAMETRIC TOOLS

B1 MATLAB functions and programs

The MATLAB functions and programs which were used in development of theparametric design and analysis approach are presented and briefly described here

B11 Description ofMATLAB functions

These MATLAB functions which are listed below were use to calculate engineeringconstants and constitutive equations for FRP composites as well as in analysis of FRPlaminates to evaluate strains and curvature as well as forces and stresses

ABDMatrixmThis function calculates the A B and D matrices for a laminate based on the layupand Q matrix

calcAlphamThis function calculates the thermal expansion coefficients of the laminate

calcEps0mThis function calculated the mid-plane strains and curvatures in a laminate subjectedto loading

calcMaxmThis function finds the maximum stress in a laminate and output in which lamina thisstress occurs

calcQmThis function calculates the Q from the composite engineering constants for speciallyorthotropic laminae

calcStressStrainmThis function calculates through the thickness strains and stresses for a laminate giventhe Q layup mid-plane strains and curvature and if a temperature is proved it willalso take into account thermal effects

compositePlotmPlots the layup of the laminate

compPropertiesmCalculates ET EL GLT νLT νT L αL and αT for a laminate given the mechanicalproperties of the constitutive materials and their volume fractions

plotStressStrainmPlots the strains and stresses though the thickness for a laminate

QbarmCalculates the transformed stiffness matrix for a generally orthotropic lamina given itsorientations relative a reference coordinate system

T1mCalculates the T1 matrix for a lamina given its orientations relative a referencecoordinate system

B1



thicknessmCalculates the coordinates for bottom and top positions for each lamina in a laminate

B12 Description ofMATLAB programs

A number of programs were written in MATLAB to evaluate the assumptions madeand to verify the SOFiSTiK materials which had been defined

EvalLaminamThis program was used to calculate engineering constants and stiffness matrces for FRPlaminae with varying fibre volume fractions and lamina orientations

MechanicalCouplingmThis program was used to evaluate mechanical coupling in thin and thick laminateswith varying lamina orientations This was done by calculating the A B and Dlaminates for laminates with varying lamina orientations and number or laminates

EvalLaminatemThe mechanical responses was evaluated for thin and thick laminates with thisprogram This was done by applying a load to the laminate and then calculating thestrain and curvature response

In this program engineering constants were also calculated for the laminates assumingthat these are fully orthotropic by neglecting any non-zero A16 and A26 components inthe A matrix The engineering constants were then exported as lists in text files

OrthotropicLaminatemThis program was used to evaluate the mechanical response in Fully OrthotropicLaminates The engineering constants which were used to calculate the Q matrix ofa fictional lamina were read from text files From this a laminate was constructed fromuni-directional layers of the fictional laminae

The strain and curvature response in these laminates was evaluated by applying a loadand calculating the resulting strains and curvatures

EquivalentLaminatemThis program was written to evaluate orthotropic laminates A strain or curvature wasapplied to the laminate and resulting strains and curvatures were calculated usingstatic condensation The stresses caused by the applied strains were also calculated

The results from this program was used to verify the materials defined in SOFiSTiK

B2


B2 Grasshopper definitions

A number of parametric tools were developed in this study to simplify comparisonof results and the creation of the parametric models These tools were created asGrasshopper routines In Figure B1 an overview of the Grasshopper routine createdfor this study The Results sections see Figure B2 of this Grasshopper routine containsthree tools which were developed for evaluation of results from the case studies Thesethree tools are briefly described in this section

Figure B2 An overview of the Grasshopper routines used for evaluation ofresults from the case studies conducted in this study

B21 PlotStrainCurvature

This tool was created to calculate and display strains and curvatures of a FE meshThe results can be visualised as a deformed mesh and the strains and curvatures arealso calculated and given as numerical values

B22 PlotDisplacements

This tool was created to display displacements of a FE mesh The displaced mesh canbe visualised both as a deformed mesh as well as a mesh with a colour gradient Thecolour of the mesh is determined by the displacement of each node

B23 PlotStresses

This tool was created to display stress results on a FE mesh The stresses can bevisualised as a colour gradient on either the non-deformed or deformed mesh

B3


Figure

B1

Parametric

Grasshopper

definitionT

hethree

typesofm

odelsw

hichwere

createdfor

thisstudy

canbe

seenhereas

wellasthe

partw

hereresults

fromSO

FiSTiK

wasim

portedN

otethat

thew

iresw

hichconnect

thecom

ponentshave

beenhidden

togive

aclear

overviewofthe

Grasshopper

definition

B4


B3 Grasshopper components

A number of Grasshopper components were developed in this study to automateand simplify the creation of FE models for SOFiSTiK These components are brieflydescribed below

B31 MakeSPT

Figure B3 MakeSPT Grasshopper component

This component creates structural information for point objects in GrasshopperRhinofor export to SOFiSTiK The inputs and outputs of the component can be seen inFigure B5

B5


B32 MakeSLN

Figure B4 MakeSLN Grasshopper component

This component creates structural information for line and curve objects inGrasshopperRhino for export to SOFiSTiK The inputs and outputs of the componentcan be seen in Figure B5

B6


B33 MakeSAR

Figure B5 MakeSAR Grasshopper component

This component creates structural information for surface objects inGrasshopperRhino for export to SOFiSTiK The inputs and outputs of the componentcan be seen in Figure B5

B7

Abstract

Sammanfattning

Nomenclature

Terminology

Preface

Background

FRP in architectural and structural applications

New type of architectural FRP modules

Problem statement

Parametric design and FE analysis approach

Limitations and excluded topics

Hypotheses

Methods

Methods

Report outline

Case studies

Preconditions

FRP module definitions and parameters

Parametric 3D modelling and structural analysis tools

Theoretical background

Fibre reinforced plastic composites

Mechanical properties of FRP laminae

Constitutive equations for FRP laminates

Mechanical coupling in FRP laminates

Equivalent material parameters for Fully Orthotropic Laminates

Material parameters for structural analysis

Analysis of FRP composites

Mechanical response of FRP laminae

Mechanical response of FRP laminates

Equivalent material parameters

Material parameters for FE analysis

Parametric QUAD and beam models for verification of material parameters

Adaptations of material parameters for FE analysis

Verification of material parameters for QUAD elements

Materials and cross sections for beam elements

Development of a parametric design and analysis approach

Preliminary evaluation of detailed parametric models

Comparison and evaluation of QUAD and beam models

Verification of the proposed parametric design and analysis approach

Verification and calibration approaches

Application of the proposed parametric design and analysis approach

Evaluation of FE models for full scale FRP module

Comparison between QUAD and beam models

Discussion

Establishment of design and analysis approach

Evaluation of assumptions and results

Conclusion and recommendations

Concluding remarks

Suggestions for future research

References

List of Figures

List of Tables

Appendices

Explanations and definitions


Additional descriptions of FRP module

Explanations of 3D modelling and programming terms

Additional theory

Appendix A - References

Programs and parametric tools


MATLAB functions and programs

Grasshopper definitions

Grasshopper components

Parametric modelling and FE analysis ofarchitectural FRP modules

Establishment of a computational design and analysis approach forpreliminary design


Department of Industrial and Materials ScienceDivision of Material and Computational Mechanics

Chalmers University of TechnologyGothenburg Sweden 2018






iv


Abstract






v


Sammanfattning






vi

TABLEOFCONTENTS

Abstract v

Sammanfattning vi

Nomenclature ix

Terminology xii

Preface







vii







References

List of Figures

List of Tables

Appendices



viii

Nomenclature



ix








x




xi

Terminology

Acronyms


Terms


xii


xiii

Preface











1BACKGROUND

1 BACKGROUND








2

1 BACKGROUND

(a) (b)







3

1 BACKGROUND



(a) (b)

(c) (d)






4

1 BACKGROUND




5

1 BACKGROUND







6

1 BACKGROUND



(a) (b) (c)

(d) (e) (f)





7

2PROBLEM STATEMENT

2 PROBLEM STATEMENT




211 Aim and scope







10

2 PROBLEM STATEMENT











11

2 PROBLEM STATEMENT




12

2 PROBLEM STATEMENT

23 Hypotheses






13

3METHODS

3 METHODS

31 Methods









16

3 METHODS


17

3 METHODS

32 Report outline










18

3 METHODS


19

3 METHODS

33 Case studies










20

3 METHODS









21

4PRECONDITIONS

4 PRECONDITIONS











24

4 PRECONDITIONS







25

4 PRECONDITIONS










26

4 PRECONDITIONS



(a) Perspective



27

4 PRECONDITIONS

413 Cross sections





G = E

2(1 + ν)[GPa] (41)



28

4 PRECONDITIONS









29

4 PRECONDITIONS












30

4 PRECONDITIONS













31







(a) (b)





34








vc = vf + vm [m3] (51)

Vf = vf

vc

(52)

Vm = vm

vc

(53)

35





εc = εf = εm (54)


PL = PfL + PmL (55)





1ET

= Vf

Ef

+ Vm

Em

(57)


ET

Em

= 1 + ξηVf

1 minus ηVf

(58)


(59)

36




1GLT

= Vf

Gf

+ Vm

Gm

(510)


GLT

Gm

= 1 + ξηVf

1 minus ηVf

(511)


(512)




νLT

EL

= νT L

ET

(514)



αL = 1EL



37




(a) (b)







σ12 = Qϵ12 [GPa] (518)

38


σ1

σ2

τ 12

=

Q11 Q12 Q16

Q22 Q26

Sym Q66

ε1

ε2

γ12

[GPa] (519)


Q11 = EL

1 minus νLTνT L

(520)

Q22 = ET

1 minus νLTνT L

(521)

Q12 = νLTET

1 minus νLTνT L

= νT LEL

1 minus νLTνT L

(522)

Q66 = GLT (523)

Q16 = Q26 = 0 (524)



σL

σT

τLT

= σLT =[T1

]θσ12 (525)

εL

εT

γLT

= ϵLT =[T2

]θϵ12 (526)


σ12 =[T1

]minus1

θQ

[T2

]θϵ12 (527)

σ12 = Qθ ϵ12 (528)

39



(a) (b)









40






41



(a)

(b)



plane



partx[m] (529)



party[m] (530)

42



εx = partu

partx= partu0

partxminus z

partφx

partx(531)

εy = partv

party= partv0

partyminus z

partφy

party(532)

γxy = partu

party+ partv

partx= partu0

party+ partv0


party+ partφy

partx) (533)


εx

εy

γxy

=

ε0

x

ε0y

γ0xy

+

κx

κy

κxy

z or

ε1

ε2

γ12

=

ε0

1

ε02

γ012

+

κ1

κ2

κ12

z (534)

ϵ = ϵ0 + κz (535)


σ1

σ2

τ12

k

= Qk

ε0

1

ε02

γ012

+ Qk

κ1

κ2

κ12

z = Qkϵ0 + κz (536)


N1

N2

N12

=int h2

minush2

σ1

σ2

τ12

dz =nsum

k=1

int hk

hkminus1

σ1

σ2

τ12

k

dz [Nm] (537)

M1

M2

M12

=int h2

minush2

σ1

σ2

τ12

zdz =nsum

k=1

int hk

hkminus1

σ1

σ2

τ12

k

zdz [Nmm] (538)

43



N =[sumn

k=1 Qk

int hkhkminus1 dz

]ϵ0 +

[sumnk=1 Qk

int hkhkminus1 zdz

]κ (539)

M =[sumn

k=1 Qk

int hkhkminus1 zdz

]ϵ0 +

[sumnk=1 Qk

int hkhkminus1 z

2dz]

κ (540)

By introducing

A =nsum

k=1Qk

int hk


B =nsum

k=1Qk

int hk


D =nsum

k=1Qk

int hk



M

=

A B

B D

ϵ0

κ

(544)


A =

A11 A12 A16

A22 A26

Sym A66

(545)

B =

B11 B12 B16

B22 B26

Sym B66

(546)

D =

D11 D12 D16

D22 D26

Sym D66

(547)

44










45






B0 =

B11 B12 B16

B22 B26

Sym B66

=

0 0 00 0 00 0 0

(548)



46






As =

A11 A12 0

A22 0Sym A66

(549)




47







N1 = A11ε1 + A12ε2 (550)


σ1t = A11ε1 + A12ε2 (551)


E1ε1t = A11ε1 + A12ε2 (552)


E1 = A11

t+ A12ε2

ε1t(553)

48



ν12 = minusε2

ε1(554)


E2 = A21ε1

ε2t+ A22

t(555)

ν21 = minusε1

ε2(556)


G12 = A66

t(557)


α1

α2

α12

= T2minus1

αL

αT

0

(558)

49


STRUCTURAL ANALYSIS



52












53


611 Assumptions


(a) (b)






612 MATLAB programs


54








volume fraction Vf

55




Q16rel =Q16plusmnθ

Q16plusmn30deg(61)



θ

Vf 005 015 025 035 045 055 065 075 085 095


56




Q11plusmnθ

(62)



θ

Vf 005 015 025 035 045 055 065 075 085 095


57














58





59







60








Nij = Ft [N] (63)

61



Load case 6321 N1 =

F

00

t (64)

Load case 6322 N2 =

0F

0

t (65)

Load case 6323 N3 =

00F

t (66)







62



1 ϵprime2 γprime



1 εprime2 γprime


1

ε2εprime

2

γ12γprime

12



1 κprime2 κprime


12respectively

θ κprime1 κprime


κ1κprime

1

κ2κprime

2

κ12κprime

12

deg [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6


63







θ E1 E2 G12 ν12 ν21 α12 α21


64




65







66






67


ANALYSIS



70







71




72





νxy gt 0 νyx lt1

νxy

νxy lt 0 νyx gt1

νxy

(71)



νyx = ν12E1

E2(72)

ν12E1

E2lt

1ν12

(73)

73


000

500

1000

1500

2000

2500

3000

3500



Fibre angle [deg]


E E

1

(a)

(b)




74






75








76







77









(a) (b)

Figure 712

78


(a) (b)

Figure 713

ε = ∆LL

(74)

γ = ϕx + ϕy (75)

κ = 1R

(76)






79


(a) (b)


(a) (b)


(a) (b)


80









81


(a) (b)


(a) (b)


(a) (b)


82


(a) (b)


(a) (b)


(a) (b)


83


(a) (b)


(a) (b)


(a) (b)


84






85






86






87












88






89








Ay =wb

y

wQy

A (77)


90







91







92






σx = Nx

A+ My

Iy

z + Mz

Iz

y (78)


τmax = 3V2A

(79)


τxy = 3Vy

2A(710)

τT = MT

WT

(711)





93





Load case 743


Load case 743


Load case 744


Load case 744

94





Bottom 25862 102006 000 000Center 50000 50000 000 000








95




98









mesh segment



99







(b) SOFiSTiK model




100








101










102










103









Bottom 000 000 000 000Center -17083 20417 -2500 2500






104



Load case 821


Load case 821


Load case 823

105


APPROACH



108






109





112









113





114




(a)

(b)


115










116







117











118



Load case 1021


Load case 1022


Load case 1023

119

11DISCUSSION

11 DISCUSSION


122

11 DISCUSSION










123

11 DISCUSSION









124

11 DISCUSSION











125

11 DISCUSSION










126

11 DISCUSSION









127

12CONCLUSIONAND

RECOMMENDATIONS












130










131

BIBLIOGRAPHY










133

BIBLIOGRAPHY












134

BIBLIOGRAPHY


















135

BIBLIOGRAPHY












136

List of Figures
















137

LISTOF FIGURES















138

LISTOF FIGURES


















139

LISTOF FIGURES



140

List of Tables










1 κprime2 κprime















141

LISTOF TABLES







142







Ai








A1






NURBs


Polygonmesh







A2










A3







A4






σ11

σ22

σ33

τ23

τ31

τ12

τ32

τ13

τ21

=

E1111 E1122 E1133 E1123 E1131 E1112 E1132 E1113 E1121

E2211 E2222 E2233 E2223 E2231 E2212 E2232 E2213 E2221

E3311 E3322 E3333 E3323 E3331 E3312 E3332 E3313 E3321

E2311 E2322 E2333 E2323 E2331 E2312 E2332 E2313 E2321

E3111 E3122 E3133 E3123 E3131 E3112 E3132 E3113 E3121

E1211 E1222 E1233 E1223 E1231 E1212 E1232 E1213 E1221

E3211 E3222 E3233 E3223 E3231 E3212 E3232 E3213 E3221

E1311 E1322 E1333 E1323 E1331 E1312 E1332 E1313 E1321

E2111 E2122 E2133 E2123 E2131 E2112 E2132 E2113 E2121

ϵ11

ϵ22

ϵ33

ϵ23

ϵ31

ϵ12

ϵ32

ϵ13

ϵ21

(A1)


E1111 E1122 E1133 0 0 0E1122 E2222 E2233 0 0 0E1133 E2233 E3333 0 0 0

0 0 0 E2323 0 00 0 0 0 E1313 00 0 0 0 0 E1212

(A2)


σ1

σ2

σ3

τ23

τ13

τ12

=

C11 C12 C12 0 0 0C12 C11 C12 0 0 0C12 C12

C11minusC122 0 0 0

0 0 0 C11minusC122 0 0

0 0 0 0 C11minusC122 0

0 0 0 0 0 C11minusC122

ε1

ε2

ε3

γ23

γ12

γ12

(A3)

A5



σL

σT

τLT

=

Q11 Q12 0Q21 Q22 00 0 Q66

εL

εT

γLT

(A4)

εL

εT

γLT

=

S11 S12 0S21 S22 00 0 S66

σL

σT

τLT

(A5)


εL = S11σL + S12σT (A6)

εT = S12σT + S22σT (A7)

γLT = S66τLT (A8)

S11 = 1EL

(A9)

S22 = 1ET

(A10)

S12 = minusνLT

EL

= minusνT L

ET

(A11)

S66 = 1GLT

(A12)


A6




T1 =



(A13)

T2 =



(A14)

A7












A9






Bi

















B1














B2









B23 PlotStresses


B3


Figure

B1

Parametric

Grasshopper

definitionT

hethree

typesofm

odelsw

hichwere

createdfor

thisstudy

canbe

seenhereas

wellasthe

partw

hereresults

fromSO

FiSTiK

wasim

portedN

otethat

thew

iresw

hichconnect

thecom

ponentshave

beenhidden

togive

aclear

overviewofthe

Grasshopper

definition

B4




B31 MakeSPT



B5


B32 MakeSLN



B6


B33 MakeSAR



B7

Abstract

Sammanfattning

Nomenclature

Terminology

Preface

Background



Problem statement



Hypotheses

Methods

Methods

Report outline

Case studies

Preconditions



























Discussion




Concluding remarks


References

List of Figures

List of Tables

Appendices





Additional theory












iv


Abstract






v


Sammanfattning






vi

TABLEOFCONTENTS

Abstract v

Sammanfattning vi

Nomenclature ix

Terminology xii

Preface







vii







References

List of Figures

List of Tables

Appendices



viii

Nomenclature



ix








x




xi

Terminology

Acronyms


Terms


xii


xiii

Preface











1BACKGROUND

1 BACKGROUND








2

1 BACKGROUND

(a) (b)







3

1 BACKGROUND



(a) (b)

(c) (d)






4

1 BACKGROUND




5

1 BACKGROUND







6

1 BACKGROUND



(a) (b) (c)

(d) (e) (f)





7

2PROBLEM STATEMENT

2 PROBLEM STATEMENT




211 Aim and scope







10

2 PROBLEM STATEMENT











11

2 PROBLEM STATEMENT




12

2 PROBLEM STATEMENT

23 Hypotheses






13

3METHODS

3 METHODS

31 Methods









16

3 METHODS


17

3 METHODS

32 Report outline










18

3 METHODS


19

3 METHODS

33 Case studies










20

3 METHODS









21

4PRECONDITIONS

4 PRECONDITIONS











24

4 PRECONDITIONS







25

4 PRECONDITIONS










26

4 PRECONDITIONS



(a) Perspective



27

4 PRECONDITIONS

413 Cross sections





G = E

2(1 + ν)[GPa] (41)



28

4 PRECONDITIONS









29

4 PRECONDITIONS












30

4 PRECONDITIONS













31







(a) (b)





34








vc = vf + vm [m3] (51)

Vf = vf

vc

(52)

Vm = vm

vc

(53)

35





εc = εf = εm (54)


PL = PfL + PmL (55)





1ET

= Vf

Ef

+ Vm

Em

(57)


ET

Em

= 1 + ξηVf

1 minus ηVf

(58)


(59)

36




1GLT

= Vf

Gf

+ Vm

Gm

(510)


GLT

Gm

= 1 + ξηVf

1 minus ηVf

(511)


(512)




νLT

EL

= νT L

ET

(514)



αL = 1EL



37




(a) (b)







σ12 = Qϵ12 [GPa] (518)

38


σ1

σ2

τ 12

=

Q11 Q12 Q16

Q22 Q26

Sym Q66

ε1

ε2

γ12

[GPa] (519)


Q11 = EL

1 minus νLTνT L

(520)

Q22 = ET

1 minus νLTνT L

(521)

Q12 = νLTET

1 minus νLTνT L

= νT LEL

1 minus νLTνT L

(522)

Q66 = GLT (523)

Q16 = Q26 = 0 (524)



σL

σT

τLT

= σLT =[T1

]θσ12 (525)

εL

εT

γLT

= ϵLT =[T2

]θϵ12 (526)


σ12 =[T1

]minus1

θQ

[T2

]θϵ12 (527)

σ12 = Qθ ϵ12 (528)

39



(a) (b)









40






41



(a)

(b)



plane



partx[m] (529)



party[m] (530)

42



εx = partu

partx= partu0

partxminus z

partφx

partx(531)

εy = partv

party= partv0

partyminus z

partφy

party(532)

γxy = partu

party+ partv

partx= partu0

party+ partv0


party+ partφy

partx) (533)


εx

εy

γxy

=

ε0

x

ε0y

γ0xy

+

κx

κy

κxy

z or

ε1

ε2

γ12

=

ε0

1

ε02

γ012

+

κ1

κ2

κ12

z (534)

ϵ = ϵ0 + κz (535)


σ1

σ2

τ12

k

= Qk

ε0

1

ε02

γ012

+ Qk

κ1

κ2

κ12

z = Qkϵ0 + κz (536)


N1

N2

N12

=int h2

minush2

σ1

σ2

τ12

dz =nsum

k=1

int hk

hkminus1

σ1

σ2

τ12

k

dz [Nm] (537)

M1

M2

M12

=int h2

minush2

σ1

σ2

τ12

zdz =nsum

k=1

int hk

hkminus1

σ1

σ2

τ12

k

zdz [Nmm] (538)

43



N =[sumn

k=1 Qk

int hkhkminus1 dz

]ϵ0 +

[sumnk=1 Qk

int hkhkminus1 zdz

]κ (539)

M =[sumn

k=1 Qk

int hkhkminus1 zdz

]ϵ0 +

[sumnk=1 Qk

int hkhkminus1 z

2dz]

κ (540)

By introducing

A =nsum

k=1Qk

int hk


B =nsum

k=1Qk

int hk


D =nsum

k=1Qk

int hk



M

=

A B

B D

ϵ0

κ

(544)


A =

A11 A12 A16

A22 A26

Sym A66

(545)

B =

B11 B12 B16

B22 B26

Sym B66

(546)

D =

D11 D12 D16

D22 D26

Sym D66

(547)

44










45






B0 =

B11 B12 B16

B22 B26

Sym B66

=

0 0 00 0 00 0 0

(548)



46






As =

A11 A12 0

A22 0Sym A66

(549)




47







N1 = A11ε1 + A12ε2 (550)


σ1t = A11ε1 + A12ε2 (551)


E1ε1t = A11ε1 + A12ε2 (552)


E1 = A11

t+ A12ε2

ε1t(553)

48



ν12 = minusε2

ε1(554)


E2 = A21ε1

ε2t+ A22

t(555)

ν21 = minusε1

ε2(556)


G12 = A66

t(557)


α1

α2

α12

= T2minus1

αL

αT

0

(558)

49


STRUCTURAL ANALYSIS



52












53


611 Assumptions


(a) (b)






612 MATLAB programs


54








volume fraction Vf

55




Q16rel =Q16plusmnθ

Q16plusmn30deg(61)



θ

Vf 005 015 025 035 045 055 065 075 085 095


56




Q11plusmnθ

(62)



θ

Vf 005 015 025 035 045 055 065 075 085 095


57














58





59







60








Nij = Ft [N] (63)

61



Load case 6321 N1 =

F

00

t (64)

Load case 6322 N2 =

0F

0

t (65)

Load case 6323 N3 =

00F

t (66)







62



1 ϵprime2 γprime



1 εprime2 γprime


1

ε2εprime

2

γ12γprime

12



1 κprime2 κprime


12respectively

θ κprime1 κprime


κ1κprime

1

κ2κprime

2

κ12κprime

12

deg [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6


63







θ E1 E2 G12 ν12 ν21 α12 α21


64




65







66






67


ANALYSIS



70







71




72





νxy gt 0 νyx lt1

νxy

νxy lt 0 νyx gt1

νxy

(71)



νyx = ν12E1

E2(72)

ν12E1

E2lt

1ν12

(73)

73


000

500

1000

1500

2000

2500

3000

3500



Fibre angle [deg]


E E

1

(a)

(b)




74






75








76







77









(a) (b)

Figure 712

78


(a) (b)

Figure 713

ε = ∆LL

(74)

γ = ϕx + ϕy (75)

κ = 1R

(76)






79


(a) (b)


(a) (b)


(a) (b)


80









81


(a) (b)


(a) (b)


(a) (b)


82


(a) (b)


(a) (b)


(a) (b)


83


(a) (b)


(a) (b)


(a) (b)


84






85






86






87












88






89








Ay =wb

y

wQy

A (77)


90







91







92






σx = Nx

A+ My

Iy

z + Mz

Iz

y (78)


τmax = 3V2A

(79)


τxy = 3Vy

2A(710)

τT = MT

WT

(711)





93





Load case 743


Load case 743


Load case 744


Load case 744

94





Bottom 25862 102006 000 000Center 50000 50000 000 000








95




98









mesh segment



99







(b) SOFiSTiK model




100








101










102










103









Bottom 000 000 000 000Center -17083 20417 -2500 2500






104



Load case 821


Load case 821


Load case 823

105


APPROACH



108






109





112









113





114




(a)

(b)


115










116







117











118



Load case 1021


Load case 1022


Load case 1023

119

11DISCUSSION

11 DISCUSSION


122

11 DISCUSSION










123

11 DISCUSSION









124

11 DISCUSSION











125

11 DISCUSSION










126

11 DISCUSSION









127

12CONCLUSIONAND

RECOMMENDATIONS












130










131

BIBLIOGRAPHY










133

BIBLIOGRAPHY












134

BIBLIOGRAPHY


















135

BIBLIOGRAPHY












136

List of Figures
















137

LISTOF FIGURES















138

LISTOF FIGURES


















139

LISTOF FIGURES



140

List of Tables










1 κprime2 κprime















141

LISTOF TABLES







142







Ai








A1






NURBs


Polygonmesh







A2










A3







A4






σ11

σ22

σ33

τ23

τ31

τ12

τ32

τ13

τ21

=

E1111 E1122 E1133 E1123 E1131 E1112 E1132 E1113 E1121

E2211 E2222 E2233 E2223 E2231 E2212 E2232 E2213 E2221

E3311 E3322 E3333 E3323 E3331 E3312 E3332 E3313 E3321

E2311 E2322 E2333 E2323 E2331 E2312 E2332 E2313 E2321

E3111 E3122 E3133 E3123 E3131 E3112 E3132 E3113 E3121

E1211 E1222 E1233 E1223 E1231 E1212 E1232 E1213 E1221

E3211 E3222 E3233 E3223 E3231 E3212 E3232 E3213 E3221

E1311 E1322 E1333 E1323 E1331 E1312 E1332 E1313 E1321

E2111 E2122 E2133 E2123 E2131 E2112 E2132 E2113 E2121

ϵ11

ϵ22

ϵ33

ϵ23

ϵ31

ϵ12

ϵ32

ϵ13

ϵ21

(A1)


E1111 E1122 E1133 0 0 0E1122 E2222 E2233 0 0 0E1133 E2233 E3333 0 0 0

0 0 0 E2323 0 00 0 0 0 E1313 00 0 0 0 0 E1212

(A2)


σ1

σ2

σ3

τ23

τ13

τ12

=

C11 C12 C12 0 0 0C12 C11 C12 0 0 0C12 C12

C11minusC122 0 0 0

0 0 0 C11minusC122 0 0

0 0 0 0 C11minusC122 0

0 0 0 0 0 C11minusC122

ε1

ε2

ε3

γ23

γ12

γ12

(A3)

A5



σL

σT

τLT

=

Q11 Q12 0Q21 Q22 00 0 Q66

εL

εT

γLT

(A4)

εL

εT

γLT

=

S11 S12 0S21 S22 00 0 S66

σL

σT

τLT

(A5)


εL = S11σL + S12σT (A6)

εT = S12σT + S22σT (A7)

γLT = S66τLT (A8)

S11 = 1EL

(A9)

S22 = 1ET

(A10)

S12 = minusνLT

EL

= minusνT L

ET

(A11)

S66 = 1GLT

(A12)


A6




T1 =



(A13)

T2 =



(A14)

A7












A9






Bi

















B1














B2









B23 PlotStresses


B3


Figure

B1

Parametric

Grasshopper

definitionT

hethree

typesofm

odelsw

hichwere

createdfor

thisstudy

canbe

seenhereas

wellasthe

partw

hereresults

fromSO

FiSTiK

wasim

portedN

otethat

thew

iresw

hichconnect

thecom

ponentshave

beenhidden

togive

aclear

overviewofthe

Grasshopper

definition

B4




B31 MakeSPT



B5


B32 MakeSLN



B6


B33 MakeSAR



B7

Abstract

Sammanfattning

Nomenclature

Terminology

Preface

Background



Problem statement



Hypotheses

Methods

Methods

Report outline

Case studies

Preconditions



























Discussion




Concluding remarks


References

List of Figures

List of Tables

Appendices





Additional theory








Abstract






v


Sammanfattning






vi

TABLEOFCONTENTS

Abstract v

Sammanfattning vi

Nomenclature ix

Terminology xii

Preface







vii







References

List of Figures

List of Tables

Appendices



viii

Nomenclature



ix








x




xi

Terminology

Acronyms


Terms


xii


xiii

Preface











1BACKGROUND

1 BACKGROUND








2

1 BACKGROUND

(a) (b)







3

1 BACKGROUND



(a) (b)

(c) (d)






4

1 BACKGROUND




5

1 BACKGROUND







6

1 BACKGROUND



(a) (b) (c)

(d) (e) (f)





7

2PROBLEM STATEMENT

2 PROBLEM STATEMENT




211 Aim and scope







10

2 PROBLEM STATEMENT











11

2 PROBLEM STATEMENT




12

2 PROBLEM STATEMENT

23 Hypotheses






13

3METHODS

3 METHODS

31 Methods









16

3 METHODS


17

3 METHODS

32 Report outline










18

3 METHODS


19

3 METHODS

33 Case studies










20

3 METHODS









21

4PRECONDITIONS

4 PRECONDITIONS











24

4 PRECONDITIONS







25

4 PRECONDITIONS










26

4 PRECONDITIONS



(a) Perspective



27

4 PRECONDITIONS

413 Cross sections





G = E

2(1 + ν)[GPa] (41)



28

4 PRECONDITIONS









29

4 PRECONDITIONS












30

4 PRECONDITIONS













31







(a) (b)





34








vc = vf + vm [m3] (51)

Vf = vf

vc

(52)

Vm = vm

vc

(53)

35





εc = εf = εm (54)


PL = PfL + PmL (55)





1ET

= Vf

Ef

+ Vm

Em

(57)


ET

Em

= 1 + ξηVf

1 minus ηVf

(58)


(59)

36




1GLT

= Vf

Gf

+ Vm

Gm

(510)


GLT

Gm

= 1 + ξηVf

1 minus ηVf

(511)


(512)




νLT

EL

= νT L

ET

(514)



αL = 1EL



37




(a) (b)







σ12 = Qϵ12 [GPa] (518)

38


σ1

σ2

τ 12

=

Q11 Q12 Q16

Q22 Q26

Sym Q66

ε1

ε2

γ12

[GPa] (519)


Q11 = EL

1 minus νLTνT L

(520)

Q22 = ET

1 minus νLTνT L

(521)

Q12 = νLTET

1 minus νLTνT L

= νT LEL

1 minus νLTνT L

(522)

Q66 = GLT (523)

Q16 = Q26 = 0 (524)



σL

σT

τLT

= σLT =[T1

]θσ12 (525)

εL

εT

γLT

= ϵLT =[T2

]θϵ12 (526)


σ12 =[T1

]minus1

θQ

[T2

]θϵ12 (527)

σ12 = Qθ ϵ12 (528)

39



(a) (b)









40






41



(a)

(b)



plane



partx[m] (529)



party[m] (530)

42



εx = partu

partx= partu0

partxminus z

partφx

partx(531)

εy = partv

party= partv0

partyminus z

partφy

party(532)

γxy = partu

party+ partv

partx= partu0

party+ partv0


party+ partφy

partx) (533)


εx

εy

γxy

=

ε0

x

ε0y

γ0xy

+

κx

κy

κxy

z or

ε1

ε2

γ12

=

ε0

1

ε02

γ012

+

κ1

κ2

κ12

z (534)

ϵ = ϵ0 + κz (535)


σ1

σ2

τ12

k

= Qk

ε0

1

ε02

γ012

+ Qk

κ1

κ2

κ12

z = Qkϵ0 + κz (536)


N1

N2

N12

=int h2

minush2

σ1

σ2

τ12

dz =nsum

k=1

int hk

hkminus1

σ1

σ2

τ12

k

dz [Nm] (537)

M1

M2

M12

=int h2

minush2

σ1

σ2

τ12

zdz =nsum

k=1

int hk

hkminus1

σ1

σ2

τ12

k

zdz [Nmm] (538)

43



N =[sumn

k=1 Qk

int hkhkminus1 dz

]ϵ0 +

[sumnk=1 Qk

int hkhkminus1 zdz

]κ (539)

M =[sumn

k=1 Qk

int hkhkminus1 zdz

]ϵ0 +

[sumnk=1 Qk

int hkhkminus1 z

2dz]

κ (540)

By introducing

A =nsum

k=1Qk

int hk


B =nsum

k=1Qk

int hk


D =nsum

k=1Qk

int hk



M

=

A B

B D

ϵ0

κ

(544)


A =

A11 A12 A16

A22 A26

Sym A66

(545)

B =

B11 B12 B16

B22 B26

Sym B66

(546)

D =

D11 D12 D16

D22 D26

Sym D66

(547)

44










45






B0 =

B11 B12 B16

B22 B26

Sym B66

=

0 0 00 0 00 0 0

(548)



46






As =

A11 A12 0

A22 0Sym A66

(549)




47







N1 = A11ε1 + A12ε2 (550)


σ1t = A11ε1 + A12ε2 (551)


E1ε1t = A11ε1 + A12ε2 (552)


E1 = A11

t+ A12ε2

ε1t(553)

48



ν12 = minusε2

ε1(554)


E2 = A21ε1

ε2t+ A22

t(555)

ν21 = minusε1

ε2(556)


G12 = A66

t(557)


α1

α2

α12

= T2minus1

αL

αT

0

(558)

49


STRUCTURAL ANALYSIS



52












53


611 Assumptions


(a) (b)






612 MATLAB programs


54








volume fraction Vf

55




Q16rel =Q16plusmnθ

Q16plusmn30deg(61)



θ

Vf 005 015 025 035 045 055 065 075 085 095


56




Q11plusmnθ

(62)



θ

Vf 005 015 025 035 045 055 065 075 085 095


57














58





59







60








Nij = Ft [N] (63)

61



Load case 6321 N1 =

F

00

t (64)

Load case 6322 N2 =

0F

0

t (65)

Load case 6323 N3 =

00F

t (66)







62



1 ϵprime2 γprime



1 εprime2 γprime


1

ε2εprime

2

γ12γprime

12



1 κprime2 κprime


12respectively

θ κprime1 κprime


κ1κprime

1

κ2κprime

2

κ12κprime

12

deg [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6


63







θ E1 E2 G12 ν12 ν21 α12 α21


64




65







66






67


ANALYSIS



70







71




72





νxy gt 0 νyx lt1

νxy

νxy lt 0 νyx gt1

νxy

(71)



νyx = ν12E1

E2(72)

ν12E1

E2lt

1ν12

(73)

73


000

500

1000

1500

2000

2500

3000

3500



Fibre angle [deg]


E E

1

(a)

(b)




74






75








76







77









(a) (b)

Figure 712

78


(a) (b)

Figure 713

ε = ∆LL

(74)

γ = ϕx + ϕy (75)

κ = 1R

(76)






79


(a) (b)


(a) (b)


(a) (b)


80









81


(a) (b)


(a) (b)


(a) (b)


82


(a) (b)


(a) (b)


(a) (b)


83


(a) (b)


(a) (b)


(a) (b)


84






85






86






87












88






89








Ay =wb

y

wQy

A (77)


90







91







92






σx = Nx

A+ My

Iy

z + Mz

Iz

y (78)


τmax = 3V2A

(79)


τxy = 3Vy

2A(710)

τT = MT

WT

(711)





93





Load case 743


Load case 743


Load case 744


Load case 744

94





Bottom 25862 102006 000 000Center 50000 50000 000 000








95




98









mesh segment



99







(b) SOFiSTiK model




100








101










102










103









Bottom 000 000 000 000Center -17083 20417 -2500 2500






104



Load case 821


Load case 821


Load case 823

105


APPROACH



108






109





112









113





114




(a)

(b)


115










116







117











118



Load case 1021


Load case 1022


Load case 1023

119

11DISCUSSION

11 DISCUSSION


122

11 DISCUSSION










123

11 DISCUSSION









124

11 DISCUSSION











125

11 DISCUSSION










126

11 DISCUSSION









127

12CONCLUSIONAND

RECOMMENDATIONS












130










131

BIBLIOGRAPHY










133

BIBLIOGRAPHY












134

BIBLIOGRAPHY


















135

BIBLIOGRAPHY












136

List of Figures
















137

LISTOF FIGURES















138

LISTOF FIGURES


















139

LISTOF FIGURES



140

List of Tables










1 κprime2 κprime















141

LISTOF TABLES







142







Ai








A1






NURBs


Polygonmesh







A2










A3







A4






σ11

σ22

σ33

τ23

τ31

τ12

τ32

τ13

τ21

=

E1111 E1122 E1133 E1123 E1131 E1112 E1132 E1113 E1121

E2211 E2222 E2233 E2223 E2231 E2212 E2232 E2213 E2221

E3311 E3322 E3333 E3323 E3331 E3312 E3332 E3313 E3321

E2311 E2322 E2333 E2323 E2331 E2312 E2332 E2313 E2321

E3111 E3122 E3133 E3123 E3131 E3112 E3132 E3113 E3121

E1211 E1222 E1233 E1223 E1231 E1212 E1232 E1213 E1221

E3211 E3222 E3233 E3223 E3231 E3212 E3232 E3213 E3221

E1311 E1322 E1333 E1323 E1331 E1312 E1332 E1313 E1321

E2111 E2122 E2133 E2123 E2131 E2112 E2132 E2113 E2121

ϵ11

ϵ22

ϵ33

ϵ23

ϵ31

ϵ12

ϵ32

ϵ13

ϵ21

(A1)


E1111 E1122 E1133 0 0 0E1122 E2222 E2233 0 0 0E1133 E2233 E3333 0 0 0

0 0 0 E2323 0 00 0 0 0 E1313 00 0 0 0 0 E1212

(A2)


σ1

σ2

σ3

τ23

τ13

τ12

=

C11 C12 C12 0 0 0C12 C11 C12 0 0 0C12 C12

C11minusC122 0 0 0

0 0 0 C11minusC122 0 0

0 0 0 0 C11minusC122 0

0 0 0 0 0 C11minusC122

ε1

ε2

ε3

γ23

γ12

γ12

(A3)

A5



σL

σT

τLT

=

Q11 Q12 0Q21 Q22 00 0 Q66

εL

εT

γLT

(A4)

εL

εT

γLT

=

S11 S12 0S21 S22 00 0 S66

σL

σT

τLT

(A5)


εL = S11σL + S12σT (A6)

εT = S12σT + S22σT (A7)

γLT = S66τLT (A8)

S11 = 1EL

(A9)

S22 = 1ET

(A10)

S12 = minusνLT

EL

= minusνT L

ET

(A11)

S66 = 1GLT

(A12)


A6




T1 =



(A13)

T2 =



(A14)

A7












A9






Bi

















B1














B2









B23 PlotStresses


B3


Figure

B1

Parametric

Grasshopper

definitionT

hethree

typesofm

odelsw

hichwere

createdfor

thisstudy

canbe

seenhereas

wellasthe

partw

hereresults

fromSO

FiSTiK

wasim

portedN

otethat

thew

iresw

hichconnect

thecom

ponentshave

beenhidden

togive

aclear

overviewofthe

Grasshopper

definition

B4




B31 MakeSPT



B5


B32 MakeSLN



B6


B33 MakeSAR



B7

Abstract

Sammanfattning

Nomenclature

Terminology

Preface

Background



Problem statement



Hypotheses

Methods

Methods

Report outline

Case studies

Preconditions



























Discussion




Concluding remarks


References

List of Figures

List of Tables

Appendices





Additional theory








Sammanfattning






vi

TABLEOFCONTENTS

Abstract v

Sammanfattning vi

Nomenclature ix

Terminology xii

Preface







vii







References

List of Figures

List of Tables

Appendices



viii

Nomenclature



ix








x




xi

Terminology

Acronyms


Terms


xii


xiii

Preface











1BACKGROUND

1 BACKGROUND








2

1 BACKGROUND

(a) (b)







3

1 BACKGROUND



(a) (b)

(c) (d)






4

1 BACKGROUND




5

1 BACKGROUND







6

1 BACKGROUND



(a) (b) (c)

(d) (e) (f)





7

2PROBLEM STATEMENT

2 PROBLEM STATEMENT




211 Aim and scope







10

2 PROBLEM STATEMENT











11

2 PROBLEM STATEMENT




12

2 PROBLEM STATEMENT

23 Hypotheses






13

3METHODS

3 METHODS

31 Methods









16

3 METHODS


17

3 METHODS

32 Report outline










18

3 METHODS


19

3 METHODS

33 Case studies










20

3 METHODS









21

4PRECONDITIONS

4 PRECONDITIONS











24

4 PRECONDITIONS







25

4 PRECONDITIONS










26

4 PRECONDITIONS



(a) Perspective



27

4 PRECONDITIONS

413 Cross sections





G = E

2(1 + ν)[GPa] (41)



28

4 PRECONDITIONS









29

4 PRECONDITIONS












30

4 PRECONDITIONS













31







(a) (b)





34








vc = vf + vm [m3] (51)

Vf = vf

vc

(52)

Vm = vm

vc

(53)

35





εc = εf = εm (54)


PL = PfL + PmL (55)





1ET

= Vf

Ef

+ Vm

Em

(57)


ET

Em

= 1 + ξηVf

1 minus ηVf

(58)


(59)

36




1GLT

= Vf

Gf

+ Vm

Gm

(510)


GLT

Gm

= 1 + ξηVf

1 minus ηVf

(511)


(512)




νLT

EL

= νT L

ET

(514)



αL = 1EL



37




(a) (b)







σ12 = Qϵ12 [GPa] (518)

38


σ1

σ2

τ 12

=

Q11 Q12 Q16

Q22 Q26

Sym Q66

ε1

ε2

γ12

[GPa] (519)


Q11 = EL

1 minus νLTνT L

(520)

Q22 = ET

1 minus νLTνT L

(521)

Q12 = νLTET

1 minus νLTνT L

= νT LEL

1 minus νLTνT L

(522)

Q66 = GLT (523)

Q16 = Q26 = 0 (524)



σL

σT

τLT

= σLT =[T1

]θσ12 (525)

εL

εT

γLT

= ϵLT =[T2

]θϵ12 (526)


σ12 =[T1

]minus1

θQ

[T2

]θϵ12 (527)

σ12 = Qθ ϵ12 (528)

39



(a) (b)









40






41



(a)

(b)



plane



partx[m] (529)



party[m] (530)

42



εx = partu

partx= partu0

partxminus z

partφx

partx(531)

εy = partv

party= partv0

partyminus z

partφy

party(532)

γxy = partu

party+ partv

partx= partu0

party+ partv0


party+ partφy

partx) (533)


εx

εy

γxy

=

ε0

x

ε0y

γ0xy

+

κx

κy

κxy

z or

ε1

ε2

γ12

=

ε0

1

ε02

γ012

+

κ1

κ2

κ12

z (534)

ϵ = ϵ0 + κz (535)


σ1

σ2

τ12

k

= Qk

ε0

1

ε02

γ012

+ Qk

κ1

κ2

κ12

z = Qkϵ0 + κz (536)


N1

N2

N12

=int h2

minush2

σ1

σ2

τ12

dz =nsum

k=1

int hk

hkminus1

σ1

σ2

τ12

k

dz [Nm] (537)

M1

M2

M12

=int h2

minush2

σ1

σ2

τ12

zdz =nsum

k=1

int hk

hkminus1

σ1

σ2

τ12

k

zdz [Nmm] (538)

43



N =[sumn

k=1 Qk

int hkhkminus1 dz

]ϵ0 +

[sumnk=1 Qk

int hkhkminus1 zdz

]κ (539)

M =[sumn

k=1 Qk

int hkhkminus1 zdz

]ϵ0 +

[sumnk=1 Qk

int hkhkminus1 z

2dz]

κ (540)

By introducing

A =nsum

k=1Qk

int hk


B =nsum

k=1Qk

int hk


D =nsum

k=1Qk

int hk



M

=

A B

B D

ϵ0

κ

(544)


A =

A11 A12 A16

A22 A26

Sym A66

(545)

B =

B11 B12 B16

B22 B26

Sym B66

(546)

D =

D11 D12 D16

D22 D26

Sym D66

(547)

44










45






B0 =

B11 B12 B16

B22 B26

Sym B66

=

0 0 00 0 00 0 0

(548)



46






As =

A11 A12 0

A22 0Sym A66

(549)




47







N1 = A11ε1 + A12ε2 (550)


σ1t = A11ε1 + A12ε2 (551)


E1ε1t = A11ε1 + A12ε2 (552)


E1 = A11

t+ A12ε2

ε1t(553)

48



ν12 = minusε2

ε1(554)


E2 = A21ε1

ε2t+ A22

t(555)

ν21 = minusε1

ε2(556)


G12 = A66

t(557)


α1

α2

α12

= T2minus1

αL

αT

0

(558)

49


STRUCTURAL ANALYSIS



52












53


611 Assumptions


(a) (b)






612 MATLAB programs


54








volume fraction Vf

55




Q16rel =Q16plusmnθ

Q16plusmn30deg(61)



θ

Vf 005 015 025 035 045 055 065 075 085 095


56




Q11plusmnθ

(62)



θ

Vf 005 015 025 035 045 055 065 075 085 095


57














58





59







60








Nij = Ft [N] (63)

61



Load case 6321 N1 =

F

00

t (64)

Load case 6322 N2 =

0F

0

t (65)

Load case 6323 N3 =

00F

t (66)







62



1 ϵprime2 γprime



1 εprime2 γprime


1

ε2εprime

2

γ12γprime

12



1 κprime2 κprime


12respectively

θ κprime1 κprime


κ1κprime

1

κ2κprime

2

κ12κprime

12

deg [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6


63







θ E1 E2 G12 ν12 ν21 α12 α21


64




65







66






67


ANALYSIS



70







71




72





νxy gt 0 νyx lt1

νxy

νxy lt 0 νyx gt1

νxy

(71)



νyx = ν12E1

E2(72)

ν12E1

E2lt

1ν12

(73)

73


000

500

1000

1500

2000

2500

3000

3500



Fibre angle [deg]


E E

1

(a)

(b)




74






75








76







77









(a) (b)

Figure 712

78


(a) (b)

Figure 713

ε = ∆LL

(74)

γ = ϕx + ϕy (75)

κ = 1R

(76)






79


(a) (b)


(a) (b)


(a) (b)


80









81


(a) (b)


(a) (b)


(a) (b)


82


(a) (b)


(a) (b)


(a) (b)


83


(a) (b)


(a) (b)


(a) (b)


84






85






86






87












88






89








Ay =wb

y

wQy

A (77)


90







91







92






σx = Nx

A+ My

Iy

z + Mz

Iz

y (78)


τmax = 3V2A

(79)


τxy = 3Vy

2A(710)

τT = MT

WT

(711)





93





Load case 743


Load case 743


Load case 744


Load case 744

94





Bottom 25862 102006 000 000Center 50000 50000 000 000








95




98









mesh segment



99







(b) SOFiSTiK model




100








101










102










103









Bottom 000 000 000 000Center -17083 20417 -2500 2500






104



Load case 821


Load case 821


Load case 823

105


APPROACH



108






109





112









113





114




(a)

(b)


115










116







117











118



Load case 1021


Load case 1022


Load case 1023

119

11DISCUSSION

11 DISCUSSION


122

11 DISCUSSION










123

11 DISCUSSION









124

11 DISCUSSION











125

11 DISCUSSION










126

11 DISCUSSION









127

12CONCLUSIONAND

RECOMMENDATIONS












130










131

BIBLIOGRAPHY










133

BIBLIOGRAPHY












134

BIBLIOGRAPHY


















135

BIBLIOGRAPHY












136

List of Figures
















137

LISTOF FIGURES















138

LISTOF FIGURES


















139

LISTOF FIGURES



140

List of Tables










1 κprime2 κprime















141

LISTOF TABLES







142







Ai








A1






NURBs


Polygonmesh







A2










A3







A4






σ11

σ22

σ33

τ23

τ31

τ12

τ32

τ13

τ21

=

E1111 E1122 E1133 E1123 E1131 E1112 E1132 E1113 E1121

E2211 E2222 E2233 E2223 E2231 E2212 E2232 E2213 E2221

E3311 E3322 E3333 E3323 E3331 E3312 E3332 E3313 E3321

E2311 E2322 E2333 E2323 E2331 E2312 E2332 E2313 E2321

E3111 E3122 E3133 E3123 E3131 E3112 E3132 E3113 E3121

E1211 E1222 E1233 E1223 E1231 E1212 E1232 E1213 E1221

E3211 E3222 E3233 E3223 E3231 E3212 E3232 E3213 E3221

E1311 E1322 E1333 E1323 E1331 E1312 E1332 E1313 E1321

E2111 E2122 E2133 E2123 E2131 E2112 E2132 E2113 E2121

ϵ11

ϵ22

ϵ33

ϵ23

ϵ31

ϵ12

ϵ32

ϵ13

ϵ21

(A1)


E1111 E1122 E1133 0 0 0E1122 E2222 E2233 0 0 0E1133 E2233 E3333 0 0 0

0 0 0 E2323 0 00 0 0 0 E1313 00 0 0 0 0 E1212

(A2)


σ1

σ2

σ3

τ23

τ13

τ12

=

C11 C12 C12 0 0 0C12 C11 C12 0 0 0C12 C12

C11minusC122 0 0 0

0 0 0 C11minusC122 0 0

0 0 0 0 C11minusC122 0

0 0 0 0 0 C11minusC122

ε1

ε2

ε3

γ23

γ12

γ12

(A3)

A5



σL

σT

τLT

=

Q11 Q12 0Q21 Q22 00 0 Q66

εL

εT

γLT

(A4)

εL

εT

γLT

=

S11 S12 0S21 S22 00 0 S66

σL

σT

τLT

(A5)


εL = S11σL + S12σT (A6)

εT = S12σT + S22σT (A7)

γLT = S66τLT (A8)

S11 = 1EL

(A9)

S22 = 1ET

(A10)

S12 = minusνLT

EL

= minusνT L

ET

(A11)

S66 = 1GLT

(A12)


A6




T1 =



(A13)

T2 =



(A14)

A7












A9






Bi

















B1














B2









B23 PlotStresses


B3


Figure

B1

Parametric

Grasshopper

definitionT

hethree

typesofm

odelsw

hichwere

createdfor

thisstudy

canbe

seenhereas

wellasthe

partw

hereresults

fromSO

FiSTiK

wasim

portedN

otethat

thew

iresw

hichconnect

thecom

ponentshave

beenhidden

togive

aclear

overviewofthe

Grasshopper

definition

B4




B31 MakeSPT



B5


B32 MakeSLN



B6


B33 MakeSAR



B7

Abstract

Sammanfattning

Nomenclature

Terminology

Preface

Background



Problem statement



Hypotheses

Methods

Methods

Report outline

Case studies

Preconditions



























Discussion




Concluding remarks


References

List of Figures

List of Tables

Appendices





Additional theory







TABLEOFCONTENTS

Abstract v

Sammanfattning vi

Nomenclature ix

Terminology xii

Preface







vii







References

List of Figures

List of Tables

Appendices



viii

Nomenclature



ix








x




xi

Terminology

Acronyms


Terms


xii


xiii

Preface











1BACKGROUND

1 BACKGROUND








2

1 BACKGROUND

(a) (b)







3

1 BACKGROUND



(a) (b)

(c) (d)






4

1 BACKGROUND




5

1 BACKGROUND







6

1 BACKGROUND



(a) (b) (c)

(d) (e) (f)





7

2PROBLEM STATEMENT

2 PROBLEM STATEMENT




211 Aim and scope







10

2 PROBLEM STATEMENT











11

2 PROBLEM STATEMENT




12

2 PROBLEM STATEMENT

23 Hypotheses






13

3METHODS

3 METHODS

31 Methods









16

3 METHODS


17

3 METHODS

32 Report outline










18

3 METHODS


19

3 METHODS

33 Case studies










20

3 METHODS









21

4PRECONDITIONS

4 PRECONDITIONS











24

4 PRECONDITIONS







25

4 PRECONDITIONS










26

4 PRECONDITIONS



(a) Perspective



27

4 PRECONDITIONS

413 Cross sections





G = E

2(1 + ν)[GPa] (41)



28

4 PRECONDITIONS









29

4 PRECONDITIONS












30

4 PRECONDITIONS













31







(a) (b)





34








vc = vf + vm [m3] (51)

Vf = vf

vc

(52)

Vm = vm

vc

(53)

35





εc = εf = εm (54)


PL = PfL + PmL (55)





1ET

= Vf

Ef

+ Vm

Em

(57)


ET

Em

= 1 + ξηVf

1 minus ηVf

(58)


(59)

36




1GLT

= Vf

Gf

+ Vm

Gm

(510)


GLT

Gm

= 1 + ξηVf

1 minus ηVf

(511)


(512)




νLT

EL

= νT L

ET

(514)



αL = 1EL



37




(a) (b)







σ12 = Qϵ12 [GPa] (518)

38


σ1

σ2

τ 12

=

Q11 Q12 Q16

Q22 Q26

Sym Q66

ε1

ε2

γ12

[GPa] (519)


Q11 = EL

1 minus νLTνT L

(520)

Q22 = ET

1 minus νLTνT L

(521)

Q12 = νLTET

1 minus νLTνT L

= νT LEL

1 minus νLTνT L

(522)

Q66 = GLT (523)

Q16 = Q26 = 0 (524)



σL

σT

τLT

= σLT =[T1

]θσ12 (525)

εL

εT

γLT

= ϵLT =[T2

]θϵ12 (526)


σ12 =[T1

]minus1

θQ

[T2

]θϵ12 (527)

σ12 = Qθ ϵ12 (528)

39



(a) (b)









40






41



(a)

(b)



plane



partx[m] (529)



party[m] (530)

42



εx = partu

partx= partu0

partxminus z

partφx

partx(531)

εy = partv

party= partv0

partyminus z

partφy

party(532)

γxy = partu

party+ partv

partx= partu0

party+ partv0


party+ partφy

partx) (533)


εx

εy

γxy

=

ε0

x

ε0y

γ0xy

+

κx

κy

κxy

z or

ε1

ε2

γ12

=

ε0

1

ε02

γ012

+

κ1

κ2

κ12

z (534)

ϵ = ϵ0 + κz (535)


σ1

σ2

τ12

k

= Qk

ε0

1

ε02

γ012

+ Qk

κ1

κ2

κ12

z = Qkϵ0 + κz (536)


N1

N2

N12

=int h2

minush2

σ1

σ2

τ12

dz =nsum

k=1

int hk

hkminus1

σ1

σ2

τ12

k

dz [Nm] (537)

M1

M2

M12

=int h2

minush2

σ1

σ2

τ12

zdz =nsum

k=1

int hk

hkminus1

σ1

σ2

τ12

k

zdz [Nmm] (538)

43



N =[sumn

k=1 Qk

int hkhkminus1 dz

]ϵ0 +

[sumnk=1 Qk

int hkhkminus1 zdz

]κ (539)

M =[sumn

k=1 Qk

int hkhkminus1 zdz

]ϵ0 +

[sumnk=1 Qk

int hkhkminus1 z

2dz]

κ (540)

By introducing

A =nsum

k=1Qk

int hk


B =nsum

k=1Qk

int hk


D =nsum

k=1Qk

int hk



M

=

A B

B D

ϵ0

κ

(544)


A =

A11 A12 A16

A22 A26

Sym A66

(545)

B =

B11 B12 B16

B22 B26

Sym B66

(546)

D =

D11 D12 D16

D22 D26

Sym D66

(547)

44










45






B0 =

B11 B12 B16

B22 B26

Sym B66

=

0 0 00 0 00 0 0

(548)



46






As =

A11 A12 0

A22 0Sym A66

(549)




47







N1 = A11ε1 + A12ε2 (550)


σ1t = A11ε1 + A12ε2 (551)


E1ε1t = A11ε1 + A12ε2 (552)


E1 = A11

t+ A12ε2

ε1t(553)

48



ν12 = minusε2

ε1(554)


E2 = A21ε1

ε2t+ A22

t(555)

ν21 = minusε1

ε2(556)


G12 = A66

t(557)


α1

α2

α12

= T2minus1

αL

αT

0

(558)

49


STRUCTURAL ANALYSIS



52












53


611 Assumptions


(a) (b)






612 MATLAB programs


54








volume fraction Vf

55




Q16rel =Q16plusmnθ

Q16plusmn30deg(61)



θ

Vf 005 015 025 035 045 055 065 075 085 095


56




Q11plusmnθ

(62)



θ

Vf 005 015 025 035 045 055 065 075 085 095


57














58





59







60








Nij = Ft [N] (63)

61



Load case 6321 N1 =

F

00

t (64)

Load case 6322 N2 =

0F

0

t (65)

Load case 6323 N3 =

00F

t (66)







62



1 ϵprime2 γprime



1 εprime2 γprime


1

ε2εprime

2

γ12γprime

12



1 κprime2 κprime


12respectively

θ κprime1 κprime


κ1κprime

1

κ2κprime

2

κ12κprime

12

deg [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6


63







θ E1 E2 G12 ν12 ν21 α12 α21


64




65







66






67


ANALYSIS



70







71




72





νxy gt 0 νyx lt1

νxy

νxy lt 0 νyx gt1

νxy

(71)



νyx = ν12E1

E2(72)

ν12E1

E2lt

1ν12

(73)

73


000

500

1000

1500

2000

2500

3000

3500



Fibre angle [deg]


E E

1

(a)

(b)




74






75








76







77









(a) (b)

Figure 712

78


(a) (b)

Figure 713

ε = ∆LL

(74)

γ = ϕx + ϕy (75)

κ = 1R

(76)






79


(a) (b)


(a) (b)


(a) (b)


80









81


(a) (b)


(a) (b)


(a) (b)


82


(a) (b)


(a) (b)


(a) (b)


83


(a) (b)


(a) (b)


(a) (b)


84






85






86






87












88






89








Ay =wb

y

wQy

A (77)


90







91







92






σx = Nx

A+ My

Iy

z + Mz

Iz

y (78)


τmax = 3V2A

(79)


τxy = 3Vy

2A(710)

τT = MT

WT

(711)





93





Load case 743


Load case 743


Load case 744


Load case 744

94





Bottom 25862 102006 000 000Center 50000 50000 000 000








95




98









mesh segment



99







(b) SOFiSTiK model




100








101










102










103









Bottom 000 000 000 000Center -17083 20417 -2500 2500






104



Load case 821


Load case 821


Load case 823

105


APPROACH



108






109





112









113





114




(a)

(b)


115










116







117











118



Load case 1021


Load case 1022


Load case 1023

119

11DISCUSSION

11 DISCUSSION


122

11 DISCUSSION










123

11 DISCUSSION









124

11 DISCUSSION











125

11 DISCUSSION










126

11 DISCUSSION









127

12CONCLUSIONAND

RECOMMENDATIONS












130










131

BIBLIOGRAPHY










133

BIBLIOGRAPHY












134

BIBLIOGRAPHY


















135

BIBLIOGRAPHY












136

List of Figures
















137

LISTOF FIGURES















138

LISTOF FIGURES


















139

LISTOF FIGURES



140

List of Tables










1 κprime2 κprime















141

LISTOF TABLES







142







Ai








A1






NURBs


Polygonmesh







A2










A3







A4






σ11

σ22

σ33

τ23

τ31

τ12

τ32

τ13

τ21

=

E1111 E1122 E1133 E1123 E1131 E1112 E1132 E1113 E1121

E2211 E2222 E2233 E2223 E2231 E2212 E2232 E2213 E2221

E3311 E3322 E3333 E3323 E3331 E3312 E3332 E3313 E3321

E2311 E2322 E2333 E2323 E2331 E2312 E2332 E2313 E2321

E3111 E3122 E3133 E3123 E3131 E3112 E3132 E3113 E3121

E1211 E1222 E1233 E1223 E1231 E1212 E1232 E1213 E1221

E3211 E3222 E3233 E3223 E3231 E3212 E3232 E3213 E3221

E1311 E1322 E1333 E1323 E1331 E1312 E1332 E1313 E1321

E2111 E2122 E2133 E2123 E2131 E2112 E2132 E2113 E2121

ϵ11

ϵ22

ϵ33

ϵ23

ϵ31

ϵ12

ϵ32

ϵ13

ϵ21

(A1)


E1111 E1122 E1133 0 0 0E1122 E2222 E2233 0 0 0E1133 E2233 E3333 0 0 0

0 0 0 E2323 0 00 0 0 0 E1313 00 0 0 0 0 E1212

(A2)


σ1

σ2

σ3

τ23

τ13

τ12

=

C11 C12 C12 0 0 0C12 C11 C12 0 0 0C12 C12

C11minusC122 0 0 0

0 0 0 C11minusC122 0 0

0 0 0 0 C11minusC122 0

0 0 0 0 0 C11minusC122

ε1

ε2

ε3

γ23

γ12

γ12

(A3)

A5



σL

σT

τLT

=

Q11 Q12 0Q21 Q22 00 0 Q66

εL

εT

γLT

(A4)

εL

εT

γLT

=

S11 S12 0S21 S22 00 0 S66

σL

σT

τLT

(A5)


εL = S11σL + S12σT (A6)

εT = S12σT + S22σT (A7)

γLT = S66τLT (A8)

S11 = 1EL

(A9)

S22 = 1ET

(A10)

S12 = minusνLT

EL

= minusνT L

ET

(A11)

S66 = 1GLT

(A12)


A6




T1 =



(A13)

T2 =



(A14)

A7












A9






Bi

















B1














B2









B23 PlotStresses


B3


Figure

B1

Parametric

Grasshopper

definitionT

hethree

typesofm

odelsw

hichwere

createdfor

thisstudy

canbe

seenhereas

wellasthe

partw

hereresults

fromSO

FiSTiK

wasim

portedN

otethat

thew

iresw

hichconnect

thecom

ponentshave

beenhidden

togive

aclear

overviewofthe

Grasshopper

definition

B4




B31 MakeSPT



B5


B32 MakeSLN



B6


B33 MakeSAR



B7

Abstract

Sammanfattning

Nomenclature

Terminology

Preface

Background



Problem statement



Hypotheses

Methods

Methods

Report outline

Case studies

Preconditions



























Discussion




Concluding remarks


References

List of Figures

List of Tables

Appendices





Additional theory













References

List of Figures

List of Tables

Appendices



viii

Nomenclature



ix








x




xi

Terminology

Acronyms


Terms


xii


xiii

Preface











1BACKGROUND

1 BACKGROUND








2

1 BACKGROUND

(a) (b)







3

1 BACKGROUND



(a) (b)

(c) (d)






4

1 BACKGROUND




5

1 BACKGROUND







6

1 BACKGROUND



(a) (b) (c)

(d) (e) (f)





7

2PROBLEM STATEMENT

2 PROBLEM STATEMENT




211 Aim and scope







10

2 PROBLEM STATEMENT











11

2 PROBLEM STATEMENT




12

2 PROBLEM STATEMENT

23 Hypotheses






13

3METHODS

3 METHODS

31 Methods









16

3 METHODS


17

3 METHODS

32 Report outline










18

3 METHODS


19

3 METHODS

33 Case studies










20

3 METHODS









21

4PRECONDITIONS

4 PRECONDITIONS











24

4 PRECONDITIONS







25

4 PRECONDITIONS










26

4 PRECONDITIONS



(a) Perspective



27

4 PRECONDITIONS

413 Cross sections





G = E

2(1 + ν)[GPa] (41)



28

4 PRECONDITIONS









29

4 PRECONDITIONS












30

4 PRECONDITIONS













31







(a) (b)





34








vc = vf + vm [m3] (51)

Vf = vf

vc

(52)

Vm = vm

vc

(53)

35





εc = εf = εm (54)


PL = PfL + PmL (55)





1ET

= Vf

Ef

+ Vm

Em

(57)


ET

Em

= 1 + ξηVf

1 minus ηVf

(58)


(59)

36




1GLT

= Vf

Gf

+ Vm

Gm

(510)


GLT

Gm

= 1 + ξηVf

1 minus ηVf

(511)


(512)




νLT

EL

= νT L

ET

(514)



αL = 1EL



37




(a) (b)







σ12 = Qϵ12 [GPa] (518)

38


σ1

σ2

τ 12

=

Q11 Q12 Q16

Q22 Q26

Sym Q66

ε1

ε2

γ12

[GPa] (519)


Q11 = EL

1 minus νLTνT L

(520)

Q22 = ET

1 minus νLTνT L

(521)

Q12 = νLTET

1 minus νLTνT L

= νT LEL

1 minus νLTνT L

(522)

Q66 = GLT (523)

Q16 = Q26 = 0 (524)



σL

σT

τLT

= σLT =[T1

]θσ12 (525)

εL

εT

γLT

= ϵLT =[T2

]θϵ12 (526)


σ12 =[T1

]minus1

θQ

[T2

]θϵ12 (527)

σ12 = Qθ ϵ12 (528)

39



(a) (b)









40






41



(a)

(b)



plane



partx[m] (529)



party[m] (530)

42



εx = partu

partx= partu0

partxminus z

partφx

partx(531)

εy = partv

party= partv0

partyminus z

partφy

party(532)

γxy = partu

party+ partv

partx= partu0

party+ partv0


party+ partφy

partx) (533)


εx

εy

γxy

=

ε0

x

ε0y

γ0xy

+

κx

κy

κxy

z or

ε1

ε2

γ12

=

ε0

1

ε02

γ012

+

κ1

κ2

κ12

z (534)

ϵ = ϵ0 + κz (535)


σ1

σ2

τ12

k

= Qk

ε0

1

ε02

γ012

+ Qk

κ1

κ2

κ12

z = Qkϵ0 + κz (536)


N1

N2

N12

=int h2

minush2

σ1

σ2

τ12

dz =nsum

k=1

int hk

hkminus1

σ1

σ2

τ12

k

dz [Nm] (537)

M1

M2

M12

=int h2

minush2

σ1

σ2

τ12

zdz =nsum

k=1

int hk

hkminus1

σ1

σ2

τ12

k

zdz [Nmm] (538)

43



N =[sumn

k=1 Qk

int hkhkminus1 dz

]ϵ0 +

[sumnk=1 Qk

int hkhkminus1 zdz

]κ (539)

M =[sumn

k=1 Qk

int hkhkminus1 zdz

]ϵ0 +

[sumnk=1 Qk

int hkhkminus1 z

2dz]

κ (540)

By introducing

A =nsum

k=1Qk

int hk


B =nsum

k=1Qk

int hk


D =nsum

k=1Qk

int hk



M

=

A B

B D

ϵ0

κ

(544)


A =

A11 A12 A16

A22 A26

Sym A66

(545)

B =

B11 B12 B16

B22 B26

Sym B66

(546)

D =

D11 D12 D16

D22 D26

Sym D66

(547)

44










45






B0 =

B11 B12 B16

B22 B26

Sym B66

=

0 0 00 0 00 0 0

(548)



46






As =

A11 A12 0

A22 0Sym A66

(549)




47







N1 = A11ε1 + A12ε2 (550)


σ1t = A11ε1 + A12ε2 (551)


E1ε1t = A11ε1 + A12ε2 (552)


E1 = A11

t+ A12ε2

ε1t(553)

48



ν12 = minusε2

ε1(554)


E2 = A21ε1

ε2t+ A22

t(555)

ν21 = minusε1

ε2(556)


G12 = A66

t(557)


α1

α2

α12

= T2minus1

αL

αT

0

(558)

49


STRUCTURAL ANALYSIS



52












53


611 Assumptions


(a) (b)






612 MATLAB programs


54








volume fraction Vf

55




Q16rel =Q16plusmnθ

Q16plusmn30deg(61)



θ

Vf 005 015 025 035 045 055 065 075 085 095


56




Q11plusmnθ

(62)



θ

Vf 005 015 025 035 045 055 065 075 085 095


57














58





59







60








Nij = Ft [N] (63)

61



Load case 6321 N1 =

F

00

t (64)

Load case 6322 N2 =

0F

0

t (65)

Load case 6323 N3 =

00F

t (66)







62



1 ϵprime2 γprime



1 εprime2 γprime


1

ε2εprime

2

γ12γprime

12



1 κprime2 κprime


12respectively

θ κprime1 κprime


κ1κprime

1

κ2κprime

2

κ12κprime

12

deg [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6


63







θ E1 E2 G12 ν12 ν21 α12 α21


64




65







66






67


ANALYSIS



70







71




72





νxy gt 0 νyx lt1

νxy

νxy lt 0 νyx gt1

νxy

(71)



νyx = ν12E1

E2(72)

ν12E1

E2lt

1ν12

(73)

73


000

500

1000

1500

2000

2500

3000

3500



Fibre angle [deg]


E E

1

(a)

(b)




74






75








76







77









(a) (b)

Figure 712

78


(a) (b)

Figure 713

ε = ∆LL

(74)

γ = ϕx + ϕy (75)

κ = 1R

(76)






79


(a) (b)


(a) (b)


(a) (b)


80









81


(a) (b)


(a) (b)


(a) (b)


82


(a) (b)


(a) (b)


(a) (b)


83


(a) (b)


(a) (b)


(a) (b)


84






85






86






87












88






89








Ay =wb

y

wQy

A (77)


90







91







92






σx = Nx

A+ My

Iy

z + Mz

Iz

y (78)


τmax = 3V2A

(79)


τxy = 3Vy

2A(710)

τT = MT

WT

(711)





93





Load case 743


Load case 743


Load case 744


Load case 744

94





Bottom 25862 102006 000 000Center 50000 50000 000 000








95




98









mesh segment



99







(b) SOFiSTiK model




100








101










102










103









Bottom 000 000 000 000Center -17083 20417 -2500 2500






104



Load case 821


Load case 821


Load case 823

105


APPROACH



108






109





112









113





114




(a)

(b)


115










116







117











118



Load case 1021


Load case 1022


Load case 1023

119

11DISCUSSION

11 DISCUSSION


122

11 DISCUSSION










123

11 DISCUSSION









124

11 DISCUSSION











125

11 DISCUSSION










126

11 DISCUSSION









127

12CONCLUSIONAND

RECOMMENDATIONS












130










131

BIBLIOGRAPHY










133

BIBLIOGRAPHY












134

BIBLIOGRAPHY


















135

BIBLIOGRAPHY












136

List of Figures
















137

LISTOF FIGURES















138

LISTOF FIGURES


















139

LISTOF FIGURES



140

List of Tables










1 κprime2 κprime















141

LISTOF TABLES







142







Ai








A1






NURBs


Polygonmesh







A2










A3







A4






σ11

σ22

σ33

τ23

τ31

τ12

τ32

τ13

τ21

=

E1111 E1122 E1133 E1123 E1131 E1112 E1132 E1113 E1121

E2211 E2222 E2233 E2223 E2231 E2212 E2232 E2213 E2221

E3311 E3322 E3333 E3323 E3331 E3312 E3332 E3313 E3321

E2311 E2322 E2333 E2323 E2331 E2312 E2332 E2313 E2321

E3111 E3122 E3133 E3123 E3131 E3112 E3132 E3113 E3121

E1211 E1222 E1233 E1223 E1231 E1212 E1232 E1213 E1221

E3211 E3222 E3233 E3223 E3231 E3212 E3232 E3213 E3221

E1311 E1322 E1333 E1323 E1331 E1312 E1332 E1313 E1321

E2111 E2122 E2133 E2123 E2131 E2112 E2132 E2113 E2121

ϵ11

ϵ22

ϵ33

ϵ23

ϵ31

ϵ12

ϵ32

ϵ13

ϵ21

(A1)


E1111 E1122 E1133 0 0 0E1122 E2222 E2233 0 0 0E1133 E2233 E3333 0 0 0

0 0 0 E2323 0 00 0 0 0 E1313 00 0 0 0 0 E1212

(A2)


σ1

σ2

σ3

τ23

τ13

τ12

=

C11 C12 C12 0 0 0C12 C11 C12 0 0 0C12 C12

C11minusC122 0 0 0

0 0 0 C11minusC122 0 0

0 0 0 0 C11minusC122 0

0 0 0 0 0 C11minusC122

ε1

ε2

ε3

γ23

γ12

γ12

(A3)

A5



σL

σT

τLT

=

Q11 Q12 0Q21 Q22 00 0 Q66

εL

εT

γLT

(A4)

εL

εT

γLT

=

S11 S12 0S21 S22 00 0 S66

σL

σT

τLT

(A5)


εL = S11σL + S12σT (A6)

εT = S12σT + S22σT (A7)

γLT = S66τLT (A8)

S11 = 1EL

(A9)

S22 = 1ET

(A10)

S12 = minusνLT

EL

= minusνT L

ET

(A11)

S66 = 1GLT

(A12)


A6




T1 =



(A13)

T2 =



(A14)

A7












A9






Bi

















B1














B2









B23 PlotStresses


B3


Figure

B1

Parametric

Grasshopper

definitionT

hethree

typesofm

odelsw

hichwere

createdfor

thisstudy

canbe

seenhereas

wellasthe

partw

hereresults

fromSO

FiSTiK

wasim

portedN

otethat

thew

iresw

hichconnect

thecom

ponentshave

beenhidden

togive

aclear

overviewofthe

Grasshopper

definition

B4




B31 MakeSPT



B5


B32 MakeSLN



B6


B33 MakeSAR



B7

Abstract

Sammanfattning

Nomenclature

Terminology

Preface

Background



Problem statement



Hypotheses

Methods

Methods

Report outline

Case studies

Preconditions



























Discussion




Concluding remarks


References

List of Figures

List of Tables

Appendices





Additional theory







Nomenclature



ix








x




xi

Terminology

Acronyms


Terms


xii


xiii

Preface











1BACKGROUND

1 BACKGROUND








2

1 BACKGROUND

(a) (b)







3

1 BACKGROUND



(a) (b)

(c) (d)






4

1 BACKGROUND




5

1 BACKGROUND







6

1 BACKGROUND



(a) (b) (c)

(d) (e) (f)





7

2PROBLEM STATEMENT

2 PROBLEM STATEMENT




211 Aim and scope







10

2 PROBLEM STATEMENT











11

2 PROBLEM STATEMENT




12

2 PROBLEM STATEMENT

23 Hypotheses






13

3METHODS

3 METHODS

31 Methods









16

3 METHODS


17

3 METHODS

32 Report outline










18

3 METHODS


19

3 METHODS

33 Case studies










20

3 METHODS









21

4PRECONDITIONS

4 PRECONDITIONS











24

4 PRECONDITIONS







25

4 PRECONDITIONS










26

4 PRECONDITIONS



(a) Perspective



27

4 PRECONDITIONS

413 Cross sections





G = E

2(1 + ν)[GPa] (41)



28

4 PRECONDITIONS









29

4 PRECONDITIONS












30

4 PRECONDITIONS













31







(a) (b)





34








vc = vf + vm [m3] (51)

Vf = vf

vc

(52)

Vm = vm

vc

(53)

35





εc = εf = εm (54)


PL = PfL + PmL (55)





1ET

= Vf

Ef

+ Vm

Em

(57)


ET

Em

= 1 + ξηVf

1 minus ηVf

(58)


(59)

36




1GLT

= Vf

Gf

+ Vm

Gm

(510)


GLT

Gm

= 1 + ξηVf

1 minus ηVf

(511)


(512)




νLT

EL

= νT L

ET

(514)



αL = 1EL



37




(a) (b)







σ12 = Qϵ12 [GPa] (518)

38


σ1

σ2

τ 12

=

Q11 Q12 Q16

Q22 Q26

Sym Q66

ε1

ε2

γ12

[GPa] (519)


Q11 = EL

1 minus νLTνT L

(520)

Q22 = ET

1 minus νLTνT L

(521)

Q12 = νLTET

1 minus νLTνT L

= νT LEL

1 minus νLTνT L

(522)

Q66 = GLT (523)

Q16 = Q26 = 0 (524)



σL

σT

τLT

= σLT =[T1

]θσ12 (525)

εL

εT

γLT

= ϵLT =[T2

]θϵ12 (526)


σ12 =[T1

]minus1

θQ

[T2

]θϵ12 (527)

σ12 = Qθ ϵ12 (528)

39



(a) (b)









40






41



(a)

(b)



plane



partx[m] (529)



party[m] (530)

42



εx = partu

partx= partu0

partxminus z

partφx

partx(531)

εy = partv

party= partv0

partyminus z

partφy

party(532)

γxy = partu

party+ partv

partx= partu0

party+ partv0


party+ partφy

partx) (533)


εx

εy

γxy

=

ε0

x

ε0y

γ0xy

+

κx

κy

κxy

z or

ε1

ε2

γ12

=

ε0

1

ε02

γ012

+

κ1

κ2

κ12

z (534)

ϵ = ϵ0 + κz (535)


σ1

σ2

τ12

k

= Qk

ε0

1

ε02

γ012

+ Qk

κ1

κ2

κ12

z = Qkϵ0 + κz (536)


N1

N2

N12

=int h2

minush2

σ1

σ2

τ12

dz =nsum

k=1

int hk

hkminus1

σ1

σ2

τ12

k

dz [Nm] (537)

M1

M2

M12

=int h2

minush2

σ1

σ2

τ12

zdz =nsum

k=1

int hk

hkminus1

σ1

σ2

τ12

k

zdz [Nmm] (538)

43



N =[sumn

k=1 Qk

int hkhkminus1 dz

]ϵ0 +

[sumnk=1 Qk

int hkhkminus1 zdz

]κ (539)

M =[sumn

k=1 Qk

int hkhkminus1 zdz

]ϵ0 +

[sumnk=1 Qk

int hkhkminus1 z

2dz]

κ (540)

By introducing

A =nsum

k=1Qk

int hk


B =nsum

k=1Qk

int hk


D =nsum

k=1Qk

int hk



M

=

A B

B D

ϵ0

κ

(544)


A =

A11 A12 A16

A22 A26

Sym A66

(545)

B =

B11 B12 B16

B22 B26

Sym B66

(546)

D =

D11 D12 D16

D22 D26

Sym D66

(547)

44










45






B0 =

B11 B12 B16

B22 B26

Sym B66

=

0 0 00 0 00 0 0

(548)



46






As =

A11 A12 0

A22 0Sym A66

(549)




47







N1 = A11ε1 + A12ε2 (550)


σ1t = A11ε1 + A12ε2 (551)


E1ε1t = A11ε1 + A12ε2 (552)


E1 = A11

t+ A12ε2

ε1t(553)

48



ν12 = minusε2

ε1(554)


E2 = A21ε1

ε2t+ A22

t(555)

ν21 = minusε1

ε2(556)


G12 = A66

t(557)


α1

α2

α12

= T2minus1

αL

αT

0

(558)

49


STRUCTURAL ANALYSIS



52












53


611 Assumptions


(a) (b)






612 MATLAB programs


54








volume fraction Vf

55




Q16rel =Q16plusmnθ

Q16plusmn30deg(61)



θ

Vf 005 015 025 035 045 055 065 075 085 095


56




Q11plusmnθ

(62)



θ

Vf 005 015 025 035 045 055 065 075 085 095


57














58





59







60








Nij = Ft [N] (63)

61



Load case 6321 N1 =

F

00

t (64)

Load case 6322 N2 =

0F

0

t (65)

Load case 6323 N3 =

00F

t (66)







62



1 ϵprime2 γprime



1 εprime2 γprime


1

ε2εprime

2

γ12γprime

12



1 κprime2 κprime


12respectively

θ κprime1 κprime


κ1κprime

1

κ2κprime

2

κ12κprime

12

deg [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6


63







θ E1 E2 G12 ν12 ν21 α12 α21


64




65







66






67


ANALYSIS



70







71




72





νxy gt 0 νyx lt1

νxy

νxy lt 0 νyx gt1

νxy

(71)



νyx = ν12E1

E2(72)

ν12E1

E2lt

1ν12

(73)

73


000

500

1000

1500

2000

2500

3000

3500



Fibre angle [deg]


E E

1

(a)

(b)




74






75








76







77









(a) (b)

Figure 712

78


(a) (b)

Figure 713

ε = ∆LL

(74)

γ = ϕx + ϕy (75)

κ = 1R

(76)






79


(a) (b)


(a) (b)


(a) (b)


80









81


(a) (b)


(a) (b)


(a) (b)


82


(a) (b)


(a) (b)


(a) (b)


83


(a) (b)


(a) (b)


(a) (b)


84






85






86






87












88






89








Ay =wb

y

wQy

A (77)


90







91







92






σx = Nx

A+ My

Iy

z + Mz

Iz

y (78)


τmax = 3V2A

(79)


τxy = 3Vy

2A(710)

τT = MT

WT

(711)





93





Load case 743


Load case 743


Load case 744


Load case 744

94





Bottom 25862 102006 000 000Center 50000 50000 000 000








95




98









mesh segment



99







(b) SOFiSTiK model




100








101










102










103









Bottom 000 000 000 000Center -17083 20417 -2500 2500






104



Load case 821


Load case 821


Load case 823

105


APPROACH



108






109





112









113





114




(a)

(b)


115










116







117











118



Load case 1021


Load case 1022


Load case 1023

119

11DISCUSSION

11 DISCUSSION


122

11 DISCUSSION










123

11 DISCUSSION









124

11 DISCUSSION











125

11 DISCUSSION










126

11 DISCUSSION









127

12CONCLUSIONAND

RECOMMENDATIONS












130










131

BIBLIOGRAPHY










133

BIBLIOGRAPHY












134

BIBLIOGRAPHY


















135

BIBLIOGRAPHY












136

List of Figures
















137

LISTOF FIGURES















138

LISTOF FIGURES


















139

LISTOF FIGURES



140

List of Tables










1 κprime2 κprime















141

LISTOF TABLES







142







Ai








A1






NURBs


Polygonmesh







A2










A3







A4






σ11

σ22

σ33

τ23

τ31

τ12

τ32

τ13

τ21

=

E1111 E1122 E1133 E1123 E1131 E1112 E1132 E1113 E1121

E2211 E2222 E2233 E2223 E2231 E2212 E2232 E2213 E2221

E3311 E3322 E3333 E3323 E3331 E3312 E3332 E3313 E3321

E2311 E2322 E2333 E2323 E2331 E2312 E2332 E2313 E2321

E3111 E3122 E3133 E3123 E3131 E3112 E3132 E3113 E3121

E1211 E1222 E1233 E1223 E1231 E1212 E1232 E1213 E1221

E3211 E3222 E3233 E3223 E3231 E3212 E3232 E3213 E3221

E1311 E1322 E1333 E1323 E1331 E1312 E1332 E1313 E1321

E2111 E2122 E2133 E2123 E2131 E2112 E2132 E2113 E2121

ϵ11

ϵ22

ϵ33

ϵ23

ϵ31

ϵ12

ϵ32

ϵ13

ϵ21

(A1)


E1111 E1122 E1133 0 0 0E1122 E2222 E2233 0 0 0E1133 E2233 E3333 0 0 0

0 0 0 E2323 0 00 0 0 0 E1313 00 0 0 0 0 E1212

(A2)


σ1

σ2

σ3

τ23

τ13

τ12

=

C11 C12 C12 0 0 0C12 C11 C12 0 0 0C12 C12

C11minusC122 0 0 0

0 0 0 C11minusC122 0 0

0 0 0 0 C11minusC122 0

0 0 0 0 0 C11minusC122

ε1

ε2

ε3

γ23

γ12

γ12

(A3)

A5



σL

σT

τLT

=

Q11 Q12 0Q21 Q22 00 0 Q66

εL

εT

γLT

(A4)

εL

εT

γLT

=

S11 S12 0S21 S22 00 0 S66

σL

σT

τLT

(A5)


εL = S11σL + S12σT (A6)

εT = S12σT + S22σT (A7)

γLT = S66τLT (A8)

S11 = 1EL

(A9)

S22 = 1ET

(A10)

S12 = minusνLT

EL

= minusνT L

ET

(A11)

S66 = 1GLT

(A12)


A6




T1 =



(A13)

T2 =



(A14)

A7












A9






Bi

















B1














B2









B23 PlotStresses


B3


Figure

B1

Parametric

Grasshopper

definitionT

hethree

typesofm

odelsw

hichwere

createdfor

thisstudy

canbe

seenhereas

wellasthe

partw

hereresults

fromSO

FiSTiK

wasim

portedN

otethat

thew

iresw

hichconnect

thecom

ponentshave

beenhidden

togive

aclear

overviewofthe

Grasshopper

definition

B4




B31 MakeSPT



B5


B32 MakeSLN



B6


B33 MakeSAR



B7

Abstract

Sammanfattning

Nomenclature

Terminology

Preface

Background



Problem statement



Hypotheses

Methods

Methods

Report outline

Case studies

Preconditions



























Discussion




Concluding remarks


References

List of Figures

List of Tables

Appendices





Additional theory














x




xi

Terminology

Acronyms


Terms


xii


xiii

Preface











1BACKGROUND

1 BACKGROUND








2

1 BACKGROUND

(a) (b)







3

1 BACKGROUND



(a) (b)

(c) (d)






4

1 BACKGROUND




5

1 BACKGROUND







6

1 BACKGROUND



(a) (b) (c)

(d) (e) (f)





7

2PROBLEM STATEMENT

2 PROBLEM STATEMENT




211 Aim and scope







10

2 PROBLEM STATEMENT











11

2 PROBLEM STATEMENT




12

2 PROBLEM STATEMENT

23 Hypotheses






13

3METHODS

3 METHODS

31 Methods









16

3 METHODS


17

3 METHODS

32 Report outline










18

3 METHODS


19

3 METHODS

33 Case studies










20

3 METHODS









21

4PRECONDITIONS

4 PRECONDITIONS











24

4 PRECONDITIONS







25

4 PRECONDITIONS










26

4 PRECONDITIONS



(a) Perspective



27

4 PRECONDITIONS

413 Cross sections





G = E

2(1 + ν)[GPa] (41)



28

4 PRECONDITIONS









29

4 PRECONDITIONS












30

4 PRECONDITIONS













31







(a) (b)





34








vc = vf + vm [m3] (51)

Vf = vf

vc

(52)

Vm = vm

vc

(53)

35





εc = εf = εm (54)


PL = PfL + PmL (55)





1ET

= Vf

Ef

+ Vm

Em

(57)


ET

Em

= 1 + ξηVf

1 minus ηVf

(58)


(59)

36




1GLT

= Vf

Gf

+ Vm

Gm

(510)


GLT

Gm

= 1 + ξηVf

1 minus ηVf

(511)


(512)




νLT

EL

= νT L

ET

(514)



αL = 1EL



37




(a) (b)







σ12 = Qϵ12 [GPa] (518)

38


σ1

σ2

τ 12

=

Q11 Q12 Q16

Q22 Q26

Sym Q66

ε1

ε2

γ12

[GPa] (519)


Q11 = EL

1 minus νLTνT L

(520)

Q22 = ET

1 minus νLTνT L

(521)

Q12 = νLTET

1 minus νLTνT L

= νT LEL

1 minus νLTνT L

(522)

Q66 = GLT (523)

Q16 = Q26 = 0 (524)



σL

σT

τLT

= σLT =[T1

]θσ12 (525)

εL

εT

γLT

= ϵLT =[T2

]θϵ12 (526)


σ12 =[T1

]minus1

θQ

[T2

]θϵ12 (527)

σ12 = Qθ ϵ12 (528)

39



(a) (b)









40






41



(a)

(b)



plane



partx[m] (529)



party[m] (530)

42



εx = partu

partx= partu0

partxminus z

partφx

partx(531)

εy = partv

party= partv0

partyminus z

partφy

party(532)

γxy = partu

party+ partv

partx= partu0

party+ partv0


party+ partφy

partx) (533)


εx

εy

γxy

=

ε0

x

ε0y

γ0xy

+

κx

κy

κxy

z or

ε1

ε2

γ12

=

ε0

1

ε02

γ012

+

κ1

κ2

κ12

z (534)

ϵ = ϵ0 + κz (535)


σ1

σ2

τ12

k

= Qk

ε0

1

ε02

γ012

+ Qk

κ1

κ2

κ12

z = Qkϵ0 + κz (536)


N1

N2

N12

=int h2

minush2

σ1

σ2

τ12

dz =nsum

k=1

int hk

hkminus1

σ1

σ2

τ12

k

dz [Nm] (537)

M1

M2

M12

=int h2

minush2

σ1

σ2

τ12

zdz =nsum

k=1

int hk

hkminus1

σ1

σ2

τ12

k

zdz [Nmm] (538)

43



N =[sumn

k=1 Qk

int hkhkminus1 dz

]ϵ0 +

[sumnk=1 Qk

int hkhkminus1 zdz

]κ (539)

M =[sumn

k=1 Qk

int hkhkminus1 zdz

]ϵ0 +

[sumnk=1 Qk

int hkhkminus1 z

2dz]

κ (540)

By introducing

A =nsum

k=1Qk

int hk


B =nsum

k=1Qk

int hk


D =nsum

k=1Qk

int hk



M

=

A B

B D

ϵ0

κ

(544)


A =

A11 A12 A16

A22 A26

Sym A66

(545)

B =

B11 B12 B16

B22 B26

Sym B66

(546)

D =

D11 D12 D16

D22 D26

Sym D66

(547)

44










45






B0 =

B11 B12 B16

B22 B26

Sym B66

=

0 0 00 0 00 0 0

(548)



46






As =

A11 A12 0

A22 0Sym A66

(549)




47







N1 = A11ε1 + A12ε2 (550)


σ1t = A11ε1 + A12ε2 (551)


E1ε1t = A11ε1 + A12ε2 (552)


E1 = A11

t+ A12ε2

ε1t(553)

48



ν12 = minusε2

ε1(554)


E2 = A21ε1

ε2t+ A22

t(555)

ν21 = minusε1

ε2(556)


G12 = A66

t(557)


α1

α2

α12

= T2minus1

αL

αT

0

(558)

49


STRUCTURAL ANALYSIS



52












53


611 Assumptions


(a) (b)






612 MATLAB programs


54








volume fraction Vf

55




Q16rel =Q16plusmnθ

Q16plusmn30deg(61)



θ

Vf 005 015 025 035 045 055 065 075 085 095


56




Q11plusmnθ

(62)



θ

Vf 005 015 025 035 045 055 065 075 085 095


57














58





59







60








Nij = Ft [N] (63)

61



Load case 6321 N1 =

F

00

t (64)

Load case 6322 N2 =

0F

0

t (65)

Load case 6323 N3 =

00F

t (66)







62



1 ϵprime2 γprime



1 εprime2 γprime


1

ε2εprime

2

γ12γprime

12



1 κprime2 κprime


12respectively

θ κprime1 κprime


κ1κprime

1

κ2κprime

2

κ12κprime

12

deg [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6


63







θ E1 E2 G12 ν12 ν21 α12 α21


64




65







66






67


ANALYSIS



70







71




72





νxy gt 0 νyx lt1

νxy

νxy lt 0 νyx gt1

νxy

(71)



νyx = ν12E1

E2(72)

ν12E1

E2lt

1ν12

(73)

73


000

500

1000

1500

2000

2500

3000

3500



Fibre angle [deg]


E E

1

(a)

(b)




74






75








76







77









(a) (b)

Figure 712

78


(a) (b)

Figure 713

ε = ∆LL

(74)

γ = ϕx + ϕy (75)

κ = 1R

(76)






79


(a) (b)


(a) (b)


(a) (b)


80









81


(a) (b)


(a) (b)


(a) (b)


82


(a) (b)


(a) (b)


(a) (b)


83


(a) (b)


(a) (b)


(a) (b)


84






85






86






87












88






89








Ay =wb

y

wQy

A (77)


90







91







92






σx = Nx

A+ My

Iy

z + Mz

Iz

y (78)


τmax = 3V2A

(79)


τxy = 3Vy

2A(710)

τT = MT

WT

(711)





93





Load case 743


Load case 743


Load case 744


Load case 744

94





Bottom 25862 102006 000 000Center 50000 50000 000 000








95




98









mesh segment



99







(b) SOFiSTiK model




100








101










102










103









Bottom 000 000 000 000Center -17083 20417 -2500 2500






104



Load case 821


Load case 821


Load case 823

105


APPROACH



108






109





112









113





114




(a)

(b)


115










116







117











118



Load case 1021


Load case 1022


Load case 1023

119

11DISCUSSION

11 DISCUSSION


122

11 DISCUSSION










123

11 DISCUSSION









124

11 DISCUSSION











125

11 DISCUSSION










126

11 DISCUSSION









127

12CONCLUSIONAND

RECOMMENDATIONS












130










131

BIBLIOGRAPHY










133

BIBLIOGRAPHY












134

BIBLIOGRAPHY


















135

BIBLIOGRAPHY












136

List of Figures
















137

LISTOF FIGURES















138

LISTOF FIGURES


















139

LISTOF FIGURES



140

List of Tables










1 κprime2 κprime















141

LISTOF TABLES







142







Ai








A1






NURBs


Polygonmesh







A2










A3







A4






σ11

σ22

σ33

τ23

τ31

τ12

τ32

τ13

τ21

=

E1111 E1122 E1133 E1123 E1131 E1112 E1132 E1113 E1121

E2211 E2222 E2233 E2223 E2231 E2212 E2232 E2213 E2221

E3311 E3322 E3333 E3323 E3331 E3312 E3332 E3313 E3321

E2311 E2322 E2333 E2323 E2331 E2312 E2332 E2313 E2321

E3111 E3122 E3133 E3123 E3131 E3112 E3132 E3113 E3121

E1211 E1222 E1233 E1223 E1231 E1212 E1232 E1213 E1221

E3211 E3222 E3233 E3223 E3231 E3212 E3232 E3213 E3221

E1311 E1322 E1333 E1323 E1331 E1312 E1332 E1313 E1321

E2111 E2122 E2133 E2123 E2131 E2112 E2132 E2113 E2121

ϵ11

ϵ22

ϵ33

ϵ23

ϵ31

ϵ12

ϵ32

ϵ13

ϵ21

(A1)


E1111 E1122 E1133 0 0 0E1122 E2222 E2233 0 0 0E1133 E2233 E3333 0 0 0

0 0 0 E2323 0 00 0 0 0 E1313 00 0 0 0 0 E1212

(A2)


σ1

σ2

σ3

τ23

τ13

τ12

=

C11 C12 C12 0 0 0C12 C11 C12 0 0 0C12 C12

C11minusC122 0 0 0

0 0 0 C11minusC122 0 0

0 0 0 0 C11minusC122 0

0 0 0 0 0 C11minusC122

ε1

ε2

ε3

γ23

γ12

γ12

(A3)

A5



σL

σT

τLT

=

Q11 Q12 0Q21 Q22 00 0 Q66

εL

εT

γLT

(A4)

εL

εT

γLT

=

S11 S12 0S21 S22 00 0 S66

σL

σT

τLT

(A5)


εL = S11σL + S12σT (A6)

εT = S12σT + S22σT (A7)

γLT = S66τLT (A8)

S11 = 1EL

(A9)

S22 = 1ET

(A10)

S12 = minusνLT

EL

= minusνT L

ET

(A11)

S66 = 1GLT

(A12)


A6




T1 =



(A13)

T2 =



(A14)

A7












A9






Bi

















B1














B2









B23 PlotStresses


B3


Figure

B1

Parametric

Grasshopper

definitionT

hethree

typesofm

odelsw

hichwere

createdfor

thisstudy

canbe

seenhereas

wellasthe

partw

hereresults

fromSO

FiSTiK

wasim

portedN

otethat

thew

iresw

hichconnect

thecom

ponentshave

beenhidden

togive

aclear

overviewofthe

Grasshopper

definition

B4




B31 MakeSPT



B5


B32 MakeSLN



B6


B33 MakeSAR



B7

Abstract

Sammanfattning

Nomenclature

Terminology

Preface

Background



Problem statement



Hypotheses

Methods

Methods

Report outline

Case studies

Preconditions



























Discussion




Concluding remarks


References

List of Figures

List of Tables

Appendices





Additional theory










xi

Terminology

Acronyms


Terms


xii


xiii

Preface











1BACKGROUND

1 BACKGROUND








2

1 BACKGROUND

(a) (b)







3

1 BACKGROUND



(a) (b)

(c) (d)






4

1 BACKGROUND




5

1 BACKGROUND







6

1 BACKGROUND



(a) (b) (c)

(d) (e) (f)





7

2PROBLEM STATEMENT

2 PROBLEM STATEMENT




211 Aim and scope







10

2 PROBLEM STATEMENT











11

2 PROBLEM STATEMENT




12

2 PROBLEM STATEMENT

23 Hypotheses






13

3METHODS

3 METHODS

31 Methods









16

3 METHODS


17

3 METHODS

32 Report outline










18

3 METHODS


19

3 METHODS

33 Case studies










20

3 METHODS









21

4PRECONDITIONS

4 PRECONDITIONS











24

4 PRECONDITIONS







25

4 PRECONDITIONS










26

4 PRECONDITIONS



(a) Perspective



27

4 PRECONDITIONS

413 Cross sections





G = E

2(1 + ν)[GPa] (41)



28

4 PRECONDITIONS









29

4 PRECONDITIONS












30

4 PRECONDITIONS













31







(a) (b)





34








vc = vf + vm [m3] (51)

Vf = vf

vc

(52)

Vm = vm

vc

(53)

35





εc = εf = εm (54)


PL = PfL + PmL (55)





1ET

= Vf

Ef

+ Vm

Em

(57)


ET

Em

= 1 + ξηVf

1 minus ηVf

(58)


(59)

36




1GLT

= Vf

Gf

+ Vm

Gm

(510)


GLT

Gm

= 1 + ξηVf

1 minus ηVf

(511)


(512)




νLT

EL

= νT L

ET

(514)



αL = 1EL



37




(a) (b)







σ12 = Qϵ12 [GPa] (518)

38


σ1

σ2

τ 12

=

Q11 Q12 Q16

Q22 Q26

Sym Q66

ε1

ε2

γ12

[GPa] (519)


Q11 = EL

1 minus νLTνT L

(520)

Q22 = ET

1 minus νLTνT L

(521)

Q12 = νLTET

1 minus νLTνT L

= νT LEL

1 minus νLTνT L

(522)

Q66 = GLT (523)

Q16 = Q26 = 0 (524)



σL

σT

τLT

= σLT =[T1

]θσ12 (525)

εL

εT

γLT

= ϵLT =[T2

]θϵ12 (526)


σ12 =[T1

]minus1

θQ

[T2

]θϵ12 (527)

σ12 = Qθ ϵ12 (528)

39



(a) (b)









40






41



(a)

(b)



plane



partx[m] (529)



party[m] (530)

42



εx = partu

partx= partu0

partxminus z

partφx

partx(531)

εy = partv

party= partv0

partyminus z

partφy

party(532)

γxy = partu

party+ partv

partx= partu0

party+ partv0


party+ partφy

partx) (533)


εx

εy

γxy

=

ε0

x

ε0y

γ0xy

+

κx

κy

κxy

z or

ε1

ε2

γ12

=

ε0

1

ε02

γ012

+

κ1

κ2

κ12

z (534)

ϵ = ϵ0 + κz (535)


σ1

σ2

τ12

k

= Qk

ε0

1

ε02

γ012

+ Qk

κ1

κ2

κ12

z = Qkϵ0 + κz (536)


N1

N2

N12

=int h2

minush2

σ1

σ2

τ12

dz =nsum

k=1

int hk

hkminus1

σ1

σ2

τ12

k

dz [Nm] (537)

M1

M2

M12

=int h2

minush2

σ1

σ2

τ12

zdz =nsum

k=1

int hk

hkminus1

σ1

σ2

τ12

k

zdz [Nmm] (538)

43



N =[sumn

k=1 Qk

int hkhkminus1 dz

]ϵ0 +

[sumnk=1 Qk

int hkhkminus1 zdz

]κ (539)

M =[sumn

k=1 Qk

int hkhkminus1 zdz

]ϵ0 +

[sumnk=1 Qk

int hkhkminus1 z

2dz]

κ (540)

By introducing

A =nsum

k=1Qk

int hk


B =nsum

k=1Qk

int hk


D =nsum

k=1Qk

int hk



M

=

A B

B D

ϵ0

κ

(544)


A =

A11 A12 A16

A22 A26

Sym A66

(545)

B =

B11 B12 B16

B22 B26

Sym B66

(546)

D =

D11 D12 D16

D22 D26

Sym D66

(547)

44










45






B0 =

B11 B12 B16

B22 B26

Sym B66

=

0 0 00 0 00 0 0

(548)



46






As =

A11 A12 0

A22 0Sym A66

(549)




47







N1 = A11ε1 + A12ε2 (550)


σ1t = A11ε1 + A12ε2 (551)


E1ε1t = A11ε1 + A12ε2 (552)


E1 = A11

t+ A12ε2

ε1t(553)

48



ν12 = minusε2

ε1(554)


E2 = A21ε1

ε2t+ A22

t(555)

ν21 = minusε1

ε2(556)


G12 = A66

t(557)


α1

α2

α12

= T2minus1

αL

αT

0

(558)

49


STRUCTURAL ANALYSIS



52












53


611 Assumptions


(a) (b)






612 MATLAB programs


54








volume fraction Vf

55




Q16rel =Q16plusmnθ

Q16plusmn30deg(61)



θ

Vf 005 015 025 035 045 055 065 075 085 095


56




Q11plusmnθ

(62)



θ

Vf 005 015 025 035 045 055 065 075 085 095


57














58





59







60








Nij = Ft [N] (63)

61



Load case 6321 N1 =

F

00

t (64)

Load case 6322 N2 =

0F

0

t (65)

Load case 6323 N3 =

00F

t (66)







62



1 ϵprime2 γprime



1 εprime2 γprime


1

ε2εprime

2

γ12γprime

12



1 κprime2 κprime


12respectively

θ κprime1 κprime


κ1κprime

1

κ2κprime

2

κ12κprime

12

deg [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6

rad ] [10minus6


63







θ E1 E2 G12 ν12 ν21 α12 α21


64




65







66






67


ANALYSIS



70







71




72





νxy gt 0 νyx lt1

νxy

νxy lt 0 νyx gt1

νxy

(71)



νyx = ν12E1

E2(72)

ν12E1

E2lt

1ν12

(73)

73


000

500

1000

1500

2000

2500

3000

3500



Fibre angle [deg]


E E

1

(a)

(b)




74






75








76







77









(a) (b)

Figure 712

78


(a) (b)

Figure 713

ε = ∆LL

(74)

γ = ϕx + ϕy (75)

κ = 1R

(76)






79


(a) (b)


(a) (b)


(a) (b)


80









81


(a) (b)


(a) (b)


(a) (b)


82


(a) (b)


(a) (b)


(a) (b)


83


(a) (b)


(a) (b)


(a) (b)


84






85






86






87












88






89








Ay =wb

y

wQy

A (77)


90







91







92






σx = Nx

A+ My

Iy

z + Mz

Iz

y (78)


τmax = 3V2A

(79)


τxy = 3Vy

2A(710)

τT = MT

WT

(711)





93





Load case 743


Load case 743


Load case 744


Load case 744

94





Bottom 25862 102006 000 000Center 50000 50000 000 000








95




98









mesh segment



99







(b) SOFiSTiK model




100








101










102










103









Bottom 000 000 000 000Center -17083 20417 -2500 2500






104



Load case 821


Load case 821


Load case 823

105


APPROACH



108






109





112









113





114




(a)

(b)


115










116







117











118



Load case 1021


Load case 1022


Load case 1023

119

11DISCUSSION

11 DISCUSSION


122

11 DISCUSSION










123

11 DISCUSSION









124

11 DISCUSSION











125

11 DISCUSSION










126

11 DISCUSSION









127

12CONCLUSIONAND

RECOMMENDATIONS












130










131

BIBLIOGRAPHY










133

BIBLIOGRAPHY












134

BIBLIOGRAPHY


















135

BIBLIOGRAPHY












136

List of Figures
















137

LISTOF FIGURES















138

LISTOF FIGURES


















139

LISTOF FIGURES



140

List of Tables










1 κprime2 κprime















141

LISTOF TABLES







142







Ai








A1






NURBs


Polygonmesh







A2










A3







A4






σ11

σ22

σ33

τ23

τ31

τ12

τ32

τ13

τ21

=

E1111 E1122 E1133 E1123 E1131 E1112 E1132 E1113 E1121

E2211 E2222 E2233 E2223 E2231 E2212 E2232 E2213 E2221

E3311 E3322 E3333 E3323 E3331 E3312 E3332 E3313 E3321

E2311 E2322 E2333 E2323 E2331 E2312 E2332 E2313 E2321

E3111 E3122 E3133 E3123 E3131 E3112 E3132 E3113 E3121

E1211 E1222 E1233 E1223 E1231 E1212 E1232 E1213 E1221

E3211 E3222 E3233 E3223 E3231 E3212 E3232 E3213 E3221

E1311 E1322 E1333 E1323 E1331 E1312 E1332 E1313 E1321

E2111 E2122 E2133 E2123 E2131 E2112 E2132 E2113 E2121

ϵ11

ϵ22

ϵ33

ϵ23

ϵ31

ϵ12

ϵ32

ϵ13

ϵ21

(A1)


E1111 E1122 E1133 0 0 0E1122 E2222 E2233 0 0 0E1133 E2233 E3333 0 0 0

0 0 0 E2323 0 00 0 0 0 E1313 00 0 0 0 0 E1212

(A2)


σ1

σ2

σ3

τ23

τ13

τ12

=

C11 C12 C12 0 0 0C12 C11 C12 0 0 0C12 C12

C11minusC122 0 0 0

0 0 0 C11minusC122 0 0

0 0 0 0 C11minusC122 0

0 0 0 0 0 C11minusC122

ε1

ε2

ε3

γ23

γ12

γ12

(A3)

A5



σL

σT

τLT

=

Q11 Q12 0Q21 Q22 00 0 Q66

εL

εT

γLT

(A4)

εL

εT

γLT

=

S11 S12 0S21 S22 00 0 S66

σL

σT

τLT

(A5)


εL = S11σL + S12σT (A6)

εT = S12σT + S22σT (A7)

γLT = S66τLT (A8)

S11 = 1EL

(A9)

S22 = 1ET

(A10)

S12 = minusνLT

EL

= minusνT L

ET

(A11)

S66 = 1GLT

(A12)


A6




T1 =



(A13)

T2 =



(A14)

A7












A9






Bi

















B1














B2









B23 PlotStresses


B3


Figure

B1

Parametric

Grasshopper

definitionT

hethree

typesofm

odelsw

hichwere

createdfor

thisstudy

canbe

seenhereas

wellasthe

partw

hereresults

fromSO

FiSTiK

wasim

portedN

otethat

thew

iresw

hichconnect

thecom

ponentshave

beenhidden

togive

aclear

overviewofthe

Grasshopper

definition

B4




B31 MakeSPT



B5


B32 MakeSLN



B6


B33 MakeSAR



B7

Abstract

Sammanfattning

Nomenclature

Terminology

Preface

Background



Problem statement



Hypotheses

Methods

Methods

Report outline

Case studies

Preconditions



























Discussion




Concluding remarks


References

List of Figures

List of Tables

Appendices





Additional theory







parametric modelling and fe analysis of architectural frp

Documents