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Doctoral Thesis

Model for interlaminar normal stresses in doubly curvedlaminates

Author(s): Roos, René

Publication Date: 2008

Permanent Link: https://doi.org/10.3929/ethz-a-005698254

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DISS. ETH NO. 17725

Model for Interlaminar

Normal Stresses in Doubly

Curved Laminates

A dissertation submitted to the

Swiss Federal Institute of Technology

Zurich

for the degree of

Dr. sc. ETH Zurich

presented by

Rene Roos

Dipl. Masch. Ing. ETHborn February 27, 1979citizen of Egolzwil (LU)

accepted on recommendation of

Prof. Dr. P. Ermanni, examinerProf. Dr. J. Botsis, co-examiner

Prof. Dr. K. Rohwer, co-examinerDr. G. Kress, co-examiner

2008

Abstract

This thesis investigates several aspects of interlaminar normal stressesin thick-walled curved laminates resulting from the need of accurateand numerical efficient calculation tools for composite structures.New mechanical models for the interlaminar normal stress calculationin thick-walled singly and doubly curved laminates are presented.The analytical models provide accurate results in consideration of thesevere simplifications. Moreover, the new model and solid finite ele-ment (FE) simulations are used to assess the maximum delaminationload which is provided by delamination strength tests.

A few concepts for modelling interlaminar shear and normal stresses inlaminated structures, caused by holding equilibrium to internal forces,edge and curvature effects, are explained. Especially the through-the-thickness distribution of the interlaminar normal stress is discussedand modelling rules for solid finite element models (FEM) are definedso that accurate stress results are achieved.

The new model for the interlaminar normal stress calculation insingly curved thick-walled laminates requires the in-plane strain in-formation which is calculated by using the classical laminate theory(CLT). Further, the model has to map the kinematical effects causedby the curved laminate shape where all shear effects are neglected.The interlaminar normal stress distributions, obtained from the newmodel, are compared with FEM results for unidirectional and cross-ply laminates.

Based on the existing model, the influence of the interlaminarshear is modelled by the first-order shear deformation theory and theimprovement of accuracy is validated.

The new model for the interlaminar normal stress calculation in thick-walled doubly curved laminates is derived from that for singly curvedlaminates. It is based on the through-the-thickness equilibrium con-dition for arbitrary doubly curved laminates and shares with the CLTthe assumption that the cross-sections remain plane. The agreementbetween the mechanical model and solid FEM results is investigatedfor different laminates and materials.

The model is developed for element-level post-processing and isimplemented in an in-house FE program with linear and quadraticshell element types. The required curvature radius evaluation is basedon the FE mesh. It is assumed that the reference surface of the shellFEs follows a parabolic function. A generic model assessment of thepost-processing method investigates the accuracy and robustness forboth element types, different laminate shapes, material properties,and angel-ply laminates.

In addition to the post-processing method, the interlaminar normalstrength of a unidirectional laminate is investigated within this the-sis. Failure is predicted by using Hashin’s failure criteria and there-fore a thick-walled singly curved specimen is designed which producessmoothly distributed interlaminar stresses. The mechanical tests pro-vided the maximum load of a unidirectional laminate and the failurelocation was observed by acoustic emission measurements. The dis-placement, measured at onset of delamination, is applied as load in asolid FEM which provides the interlaminar normal stresses.

The statistic spread of the interlaminar strength tests is discussedon the basis of (embedded) voids at the maximum interlaminar stresslocation. Moreover, the solid FE results are compared with perfor-mance data such as reaction force and strain. Finally, a sensitivity anderror analysis investigates the accuracy of the interlaminar strengthevaluation.

Zusammenfassung

Diese Dissertation befasst sich mit verschiedenen Aspekten der in-terlaminaren Normalspannung in dickwandigen und gekrummtenLaminaten, motiviert durch das Bedurfnis an genauen und numerischeffizienten Programmen fur die Berechnung von Kompositstrukturen.Es werden neue mechanische Modelle fur die Berechnung der inter-laminaren Normalspannung in dickwandigen einfach und zweifachgekrummten Laminaten vorgestellt. Die analytischen Modelle ergebengenaue Resultate unter Berucksichtigung der starken Vereinfachun-gen. Ausserdem werden Finite-Elemente-Simulationen und das neueModell verwendet um die maximale Delaminationsbeanspruchung zuuntersuchen, die durch Festigkeitstest bestimmt wurde.

Einige Konzepte fur die Modellierung von interlaminaren Schub- undNormalspannungen, verursacht durch das Erfullen des Gleichgewichtsder Schnittlasten oder durch Rand- und Krummungseffekte, werdenerlautert. Die Dickenverteilung der interlaminaren Normalspannungwird dabei ausfuhrlicher beschrieben und Modellierungsrichtlinien furdie Finite-Element-Modelle (FEM) werden aufgestellt, damit genaueSpannungsresultate erreicht werden.

Das neue Modell fur die Berechnung der interlaminaren Normal-spannung in einfach gekrummten dickwandigen Laminaten benotigtdie Laminatdehnungen (in-plane), die durch die klassische Lami-nattheorie (CLT) berechnet werden. Das Modell muss zudem die kine-matischen Effekte abbilden konnen, welche durch die gekrummte La-minatform verursacht werden, wobei alle Schubeinflusse vernachlassigtwerden. Die mit dem neuen Modell berechneten interlaminaren Nor-malspannungsverteilungen fur Kreuz- und unidirektionale Laminatewerden mit FEM Resultaten verglichen.

Basierend auf dem vorhandenen Modell wird der interlaminareSchubeinfluss mit der Schubtheorie erster Ordnung modelliert unddie Genauigkeitsverbesserung bestimmt.

Das Modell fur die Berechnung der interlaminaren Normalspannungin dickwandigen zweifach gekrummten Laminaten ist vom demjeni-gen fur einfach gekrummte Laminate abgeleitet. Es basiert auf der

Gleichgewichtsbedingung in Dickenrichtung fur beliebig zweifach ge-krummte Geometrien und geht wie die CLT von der Annahme aus,dass die Querschnitte eben bleiben. Die Ubereinstimmung zwischenden Ergebnissen des mechanischen Modells und der Berechnung mit3D finiten Elementen wird fur verschiedene Laminate und Materialienuntersucht.

Das neue Modell ist fur die elementbasierte Nachlaufrechnungentwickelt worden und ist in einem institutseigenen FE-Programmmit linearen und quadratischen Schalenelementen implementiert. Diedazu notwendige Berechnung der Krummungsradien basiert auf demFE-Netz. Es wird angenommen, dass die Referenzflache der finitenSchalenelementen einer parabolischen Funktion entspricht. Eine all-gemeine Beurteilung der Nachlaufrechnungsmethode untersucht dieGenauigkeit und Robustheit fur beide Elementtypen, verschiedeneLaminatformen, Materialeigenschaften und Winkellaminate.

Neben der Berechnung der interlaminaren Normalspannung wird imRahmen dieser Dissertation die interlaminare Normalfestigkeit einesunidirektionalen Laminates untersucht. Das Versagen wird mittelsHashins Versagenskriterien vorhergesagt und deshalb ist ein einfachgekrummter Probenkorper entwickelt worden, welcher kontinuierlichverteilte interlaminare Spannungen erzeugt. Die maximale Beanspru-chung wurde durch mechanische Versuche ermittelt und anhand vonakustischen Emissionsmessungen konnte der Versagensort festgestelltwerden.

Die statistische Streuung der interlaminaren Festigkeitsversuchewird mittels (eingebetteten) Fehlstellen am maximalen interlaminarenAnstrengungsort besprochen. Ausserdem werden die 3D FE-Resultatemit den Messergebnissen wie Kraft und Dehnung verglichen. Schlus-sendlich wird die Genauigkeit der interlaminaren Festigkeitsberech-nung anhand einer Sensitivitats- und Fehlerrechnung untersucht.

Acknowledgements

This thesis was accomplished at the Centre of Structure Technologiesof the Swiss Federal Institute of Technology in Zurich. The presentedwork was supported by the Swiss National Foundation (SNF-project200021-105142 / 1 and SNF-project 20020-113241 / 1).

I would like to thank:

• Professor Paolo Ermanni and Dr. Gerald Kress for giving methe opportunity of working in this interesting field of researchand their support during this thesis.

• Prof. John Botsis and Prof. Klaus Rohwer for being co-examiner of this thesis and their valuable suggestions.

• EVEN AG, particularly Oliver Konig and Nino Zehnder for thecollaboration and development of a composite post-processingtool.

• All the people at the mechanical systems engineering institute(EMPA), particularly Dr. Michel Barbezat for providing assis-tance with the testing program.

• Smartec AG, particularly Dr. Branko Glisic for introducing mein the field of optical fiber strain measurement.

• All the people at the Centre of Structure Technologies, particu-larly Mathias Giger, David Keller, Florian Hurlimann, BarbaraRohrnbauer, and Michael Sauter for being genial office mates.

• Martin Gamboni, Pascal Marti, Andri Bezzola, Marc Zoelly, andReto Eggimann for their semester theses which contributed tothe experimental part of this work.

• My parents and family for their lifelong support and encourage-ment.

List of Symbols

Greek

α Coefficient of thermal expansion, 33β Coefficient of moisture expansion, 33∆ Load change, 33η Element y-coordinate, 28γ Shear angle, 2γ Constant shear angle, 44κ Curvature, 18Φ Shape function, 64φ Angle, 30σ Direct stress, 18τ Shear stress, 18θ Lamina angle orientation, 12ε Direct strain, 18ϕ Second coordinate of the cylindrical

CS, 30ϑ Third coordinate of the modified

cylindrical CS, 51ξ Element x-coordinate, 28ζ Local thickness coordinate, 18

Latin

A In-plane stiffness matrix, 32a Parameter of the differential equation,

35

B Coupling stiffness matrix, 32b Parameter of the differential equation,

35C 3-D stiffness matrix, 19C Components of the 3-D stiffness ma-

trix, 33c Constant term, 130D Bending stiffness matrix, 18d Increment, 54E Young’s modulus, 18F Force, 37f Shear anisotropy factor, 44G Shear modulus, 44H Moisture / humidity, 33h Amplitude or extension, 36I Moment of inertia of an area, 18K Element stiffness matrix, 7k Shear correction factor, 3l Length, 36M Bending moment, 18N Number of layers in a laminate, 32~n Normalized shell normal vector, 28n Element node, 28P Particular part, 34Q Shear force, 18Q Reduced stiffness matrix, 18R Midplane curvature radius, 20r Radial coordinate R + z, 28rd Radius difference, 52s Anisotropy factor, 34T Temperature, 33t Laminate thickness, 6u In-plane discplacement in x-direction,

32

v In-plane discplacement in y-direction,32

w Through-the-thickness displacement,30

x Global x-coordinate, 18y Global y- and longitudinal coordinate

of the cylindrical CS, 30Z Rational curvature function, 28z Thickness coordinate, 18

Subscripts

ϕ Second coordinate of the cylindricalCS, 30

b Bottom, 44

, Derivative, 18

H Homogeneous solution, 34

ij Component index of the stiffness ma-trix, i,j=[1,6], 18

k Lamina number, 18

n Supporting point number, 54

P Particular solution, 34

r Radial coordinate of the cylindricalCS, 30

t Top, 44

x Global x-coordinate, 18

y Global y- and transverse coordinate ofthe cylindrical CS, 30

z Global z-coordinate, 18

Superscripts

~ Vector, 280 Midplane strain, 27* Modified term, 32F Sum of temperature and moisture ef-

fects, 33

List of Acronyms

2-D Twodimensional, 113-D Threedimensional, 3AE Acoustic Emission, 98BC Boundary Condition, 6CAD Computer Aided Design, 22CFRP Carbon Fiber Reinforced Plastics, 1CLT Classical Laminate Theory, 2CS Coordinate System

Cartesian: x, y, and zCylindrical: r, ϕ, and y, 30

CT Computer Tomography, 106CT-ratio Curvature-radius-to-thickness ratio,

63DCB Double Cantilever Beam, 12DKT Discrete Kirchhoff Theory, 8DoF Degree of Freedom, 7ECT End Crack Torsion, 12ELS End Load Split, 12ENF End Notch Flexure, 12ESF Extra Shape Function, 8ESL Equivalent Single Layer, 2ESPI Electronic Speckle Pattern Interfer-

ometry, 141FE Finite Element, 1FELyX Finite Element Library eXperiment,

63

xiv List of Acronyms

FEM Finite Element Model, 11FSDT First-order Shear Deformation The-

ory, 2HSDT Higher-order Shear Deformation The-

ory, 9ILPM Individual Layer Plate Model, 4INS Interlaminar Normal Stresses, 1IP Integration Point, 24ISS Interlaminar Shear Stresses, 3LSoE Linear System of Equation, 54LWM Layerwise Model, 4OF Optical Fiber, 107UD Unidirectional, 18

Contents

List of Symbols ix

List of Acronyms xiii

1 Introduction 1

1.1 State-of-the-art of layered shells and stress evaluation 2

1.2 Goals of the thesis . . . . . . . . . . . . . . . . . . . . 13

1.3 Thesis overview . . . . . . . . . . . . . . . . . . . . . . 14

2 Interlaminar stresses in shell structures 17

2.1 Interlaminar stresses caused by holding equilibrium tointernal forces . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Edge effect . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Solid brick FE simulation . . . . . . . . . . . . . . . . 22

3 Model for interlaminar normal stresses in singly

curved laminates 27

3.1 Assumptions and conventions . . . . . . . . . . . . . . 27

3.2 Analytical model . . . . . . . . . . . . . . . . . . . . . 31

3.3 Model assessment . . . . . . . . . . . . . . . . . . . . . 35

3.4 Enhanced model including shear . . . . . . . . . . . . 41

3.5 Comparison and conclusions . . . . . . . . . . . . . . . 45

4 Interlaminar normal stresses in doubly curved lami-

nates 51

4.1 Analytical model . . . . . . . . . . . . . . . . . . . . . 51

4.2 Model assessment . . . . . . . . . . . . . . . . . . . . . 55

xvi CONTENTS

5 Model implementation and evaluation 63

5.1 Shell finite elements . . . . . . . . . . . . . . . . . . . 635.2 Curvature radius evaluation . . . . . . . . . . . . . . . 675.3 General model assessment . . . . . . . . . . . . . . . . 705.4 Practical cases . . . . . . . . . . . . . . . . . . . . . . 84

6 Experiments 93

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 936.2 Manufacturing . . . . . . . . . . . . . . . . . . . . . . 946.3 Strength test results . . . . . . . . . . . . . . . . . . . 976.4 Investigation of the test results . . . . . . . . . . . . . 1056.5 Sensitivity and error analysis . . . . . . . . . . . . . . 1116.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 113

7 Conclusions and Outlook 115

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 1157.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 117

A Material properties 121

A.1 GY70/Epoxy . . . . . . . . . . . . . . . . . . . . . . . 121A.2 Elitrex EHKF 420-UD24k-40 T2 600mm . . . . . . . . 121A.3 IM7/8551-7 . . . . . . . . . . . . . . . . . . . . . . . . 123A.4 Aluminum alloy . . . . . . . . . . . . . . . . . . . . . . 123A.5 Tension tests: Elitrex EHKF . . . . . . . . . . . . . . 123

B Singly curved finite element 127

B.1 Stiffness matrix . . . . . . . . . . . . . . . . . . . . . . 127B.2 Through-the-thickness strain distribution . . . . . . . 127B.3 ISS distribution . . . . . . . . . . . . . . . . . . . . . . 131B.4 Solution of the differential equation . . . . . . . . . . . 132

C Doubly curved finite element 133

C.1 Stress equations . . . . . . . . . . . . . . . . . . . . . . 133C.2 Linear system of equations . . . . . . . . . . . . . . . . 134

D Experimental data 137

D.1 Specimen geometry . . . . . . . . . . . . . . . . . . . . 137D.2 CT images . . . . . . . . . . . . . . . . . . . . . . . . . 141D.3 Optical fiber measurement . . . . . . . . . . . . . . . . 142D.4 Strain sensors . . . . . . . . . . . . . . . . . . . . . . . 145

CONTENTS xvii

List of Tables 147

List of Figures 149

Bibliography 153

Own publications 167

Curriculum Vitae 169

Chapter 1

Introduction

This dissertation presents a new model for calculating interlaminarnormal stresses (INS)1 in curved laminates of moderate thickness andinvestigates its accuracy and predicting power for strength.

Composite materials like carbon fiber reinforced plastics (CFRP)are often used in fields like automotive, aerospace, and sport equip-ments. Traditionally, composites are most often used in lightweightstructures where the laminated shells tend to be thin with respect totheir in-plane extensions. In this case the interlaminar stresses actingupon these plates are small and often negligible. Increasingly often,composite materials are used in thick-walled curved structures whichare subjected to high mechanical and nonmechanical loadings wherethe interlaminar stresses play an important role concerning the failuremode of this structure. Air-intakes of formula race cars, which mustprotect the driver in roll-over situations, and strongly curved regionsof ship hulls provide examples for such thick-shell laminates designs.Owing to the fact that interlaminar strength is at least one order ofmagnitude smaller than that in fiber direction, the integrity of thickand curved laminates is often challenged by delaminations even whenthe interlaminar stresses are much smaller than those in the in-planedirections. Such delaminations reduce the bending stiffness and withit the buckling strength or vibration behavior [1].

1also known as transverse normal stresses σz

2 Introduction

This section presents the state-of-the-art of layered shell finite ele-ments (FE) and stress evaluation. Further, basic interlaminar stressmodels and mechanical tests regarding delamination are summarized.

1.1 State-of-the-art of layered shells and

stress evaluation

Layered shells models are used more and more in structural analysiswith new material systems such as CFRP. A survey of shell FEs isgiven by Yang et al. [2]; they cite 379 publications. The large amountof literature in this field indicates how many different problems andmechanical situations are addressed by shell analysis.

1.1.1 Laminate theories

Composite plates and shells are often modeled as an equivalent singlelayer (ESL) using the classical laminate theory (CLT). The CLTis described by Jones [3]. A better name for the CLT would be theclassical laminated plate theory because of the plane stress assump-tion for the stress-strain relations. A further assumption is that a lineoriginally straight and perpendicular to the middle surface of the lam-inate is assumed to remain straight and perpendicular to the middlesurface when the laminate is extended and bent. This implies that theplate or shell normal is not deformed and the transverse shear strainsγxz and γyz do not appear.

These assumptions of the CLT represent the well known Kirchhoffhypothesis for plates and the Kirchhoff-Love hypothesis for shells [4].It is justified if the thickness to in-plane extensions ratio is small. Ne-vertheless, composite plates are sensitive regarding thickness effectsbecause the transverse shear moduli are usually at least a magnitudesmaller than the in-plane elastic moduli. The natural frequencies canbe overrated by 25% for a plate with in-plane to thickness extensionsratio of 10 compared with a shear deformation theory. This factis shown by Bert and Chen [5] and Reddy and Kuppusamy [6].Furthermore, the CLT underestimates deflections and overestimatesbuckling loads.

Other ESL theories including shear effects are developed to overcomethis mismatch. They improve the response of the global deflections,


natural frequencies, and buckling loads compared with the CLT forthin and moderately thick composites. A theory overview is availablein [7]. Whitney and Pagano [8] developed a Mindlin-type first-order

shear deformation theory (FSDT) for multi-layered anisotropicplates. Similar FSDTs are developed for multi-layered shells and de-scribed in [9, 10, 11].

ESL theories can not model the warping of the cross-section ifthe in-plane displacements are approximated by linear functions.The assumption of constant shear angle γ overestimates the shearstiffness even for isotropic plates and requires shear correction factorsk44 and k55. A common way to calculate these factors is to comparethe shear energy calculated using the equilibrium equations with thestrain energy calculated using the stress-strain relations which waspresented by Whitney [12]. Kress [13] and Isaksson et al. [14] pre-sented a shear correction factor calculation method for symmetricaland unsymmetrical laminates under cylindrical bending, respectively.

Additionally, the assumption of a non-deformable normal resultsin incompatible interlaminar shear stresses (ISS)2 between adjacentlayers if the shear stresses are calculated by using the stress-strainrelations. Rolfes et al. [15] calculate the ISS by using the localequilibrium relations. This approach guarantees continuity of theinterlaminar stress distribution across layer interfaces and may, inmany situations, compete well with, or even outperform, higher-ordertheories as has been shown by Skytta [16]. The FSDT is still afamiliar model for multi-layered plate and shell simulations becausethe C0-continuity is sufficient for the shape functions. This allowsfor using linear approximation over the element area and locking canbe avoided using reduced integration. Further, the FSDT requireslow numerical cost. The timeliness of this research field is shown byAuricchio and Sacco and Kim and Cho. Auricchio [17] developed arefined FSDT where the out-of-plane stresses are considered as pri-mary variables of the problem. In particular, the shear stress profileis represented either by independent piecewise quadratic functions inthe thickness or by satisfying the threedimensional (3-D) equilibriumequations written in terms of midplane strains and curvatures. Thesolutions are in very good agreement with 3-D analytical solutions.Kim [18] assumed that the displacement and in-plane strain fields

2also known as transverse shear stresses τxz and τyz

4 Introduction

of FSDT can approximate, in an average sense, those of 3-D theory.The accuracy of the ISS increases with length-to-thickness ratio. TheISS in sandwich plates are not as accurate as those in a compositeplate. Both models do not consider INS.

Higher-order theories rely on shape assumptions that are higherthan first-order. The shape assumption can be made on either thedisplacement field or the stress field. The theories based on a specificdisplacement field assumption are usually simpler than the assumedstress field theories. The theories can be divided into the alreadymentioned ESL models and layerwise models (LWM). Reddy etal. [19, 20, 21] contributed third-order shear deformation theoriesfor ESL models where the midplane rotations of normals about thex- and y-axis provide first-order terms and second- and third-orderterms are included functions which are determined from the conditionthat the ISS vanish on the laminate top and bottom surfaces.

The LWMs (discrete layer plate or generalized laminate mod-

els) make displacement assumptions separately for each layer. Theyprovide much more realistic, or accurate, displacement fields but theyare also much more complicated and, from a numerical point of view,more costly because the number of unknowns increases with the num-ber of layers. Reddy [22] therefore suggests a FE mesh superpositiontechnique where an independent overlay mesh is superimposed ona global mesh to provide localized refinement for regions of interestregardless of the original global mesh topology.

Another class of theory is the individual layer plate model

(ILPM) [23, 24, 25] which in contrast to a LWM suggested byReddy [26] enforces the continuity of the interlaminar stresses a pri-ori. Masud and Panahandeh [27] present a shell FE theory that allowsfor the warping of the cross section. The kinematics are individuallyand independently represented for each layer. Transversal warping ofthe composite cross section is described by individual layer directorsthat simultaneously rotate and stretch. This allows for continuousinterlaminar stresses through the thickness of the laminate. The the-ory does not employ the zero normal stress shell hypothesis so thatINS are also calculated. Tanov and Tabiei [28] developed also a theoryfor the INS in a first- or higher-order shear deformation shell FE.The INS are represented by a continuous piecewise cubic function. A


model for the INS in FSDT shell FE is developed by Rolfes et al. [29].The INS rely on the ISS and are calculated in the second step basedon the derivatives of the ISS with respect to the in-plane coordinates.However, these theories are for flat plates and do no consider thecurved shell effects that are at the core of the present research interest.

Interlaminar stresses caused by thermal loads are discussed e.g. byRohwer et al. [30]. They investigated that higher-order approacheslead to much better results than the extended 2-D method in casesof temperature loads varying considerably in the direction of thereference surface.

1.1.2 Curvature effects in laminated structures

The curvature effect in a curved bar under pure bending is describedby Timoshenko [31]. Ascione and Fraternali [32] described a mecha-nical model for curved laminated beams. The laminated beam ismodeled as Timoshenko beams, perfectly bonded at the interfaces.The laminae3 of the composite can rotate separately what ensuresthe warping of the cross section. The penalty model gives a good ap-proximation of the interlaminar stresses (INS and ISS) and does notgenerate particular problems of convergence or stability.

Qatu [33] derived a complete and consistent set of equations forthe analysis of laminated composite curved beams and closed rings tomodel the effect of shear deformation, rotary inertia, curvature andthickness ratios, and material orthotropy on natural frequencies.

A non-linear model for laminated curved beams is presentedby Fraternali and Bilotti [34]. The theory accounts for moderatelylarge rotations, moderately large shear strains and a different elasticbehavior of the material in tension and in compression. The modelshows the potential of analyzing local effects, such as free edge stressconcentration and bimodular behavior of the material.

Shenoi and Wang [35] developed an elasticity-theory-based approachfor delamination and flexural strength of curved layered compositelaminates and sandwich beams. First, a model for characterizinglinear-static flexural behavior of a curved beam on an elastic foun-

3Lamina is a single layer of a laminate

6 Introduction

dation by classical beam theory is developed. The radial reactionforce and internal forces are then used as boundary conditions whenthe same beam is analyzed with elasticity equations. Finally, a so-lution for a thin or thick curved layered composite beam is achievedthat can map warped normals and calculate all interlaminar stresses.

A model for INS and ISS in cross-ply laminated composite andsandwich circular arches subjected to thermal and mechanical loadingis developed by Matsunaga [36]. A set of equilibrium equations of aone-dimensional higher-order theory for laminated composite circulararches is derived through the principal of virtual work. The globalhigher-order approximate theories can predict displacements andstresses in cross-ply laminated composite circular arches with a largenumber of layers within the small number of independent variablesaccurately. The curvature parameter (thickness to curvature radius)t/R ≪ 1 is assumed to be small. The model calculates the INS andISS by integrating the 3-D equations of equilibrium in the depthdirection, and satisfying the continuity conditions at the interfacebetween layers and stress boundary conditions (BCs) at the top andbottom surfaces of the arches.

1.1.3 Edge effects

Interlaminar stresses can also appear in angle-ply laminates near afree edge under tension, bending, or torsional load. The laminae havedifferent coupling behaviors in the same reference coordinate systemregarding Poisson’s ratio and direct strains and shear deformations.This fact introduces direct stresses in the transverse in-plane directionσy and τxy, which have to vanish at the free edge. Equilibrium causesinterlaminar stresses which vanish quickly with increasing distancefrom the free edge but can theoretically reach an infinite value. Thiseffect is known as the edge effect and has recently been dubbedPipes-Pagano problem [37, 38, 39]. In reality, material nonlinearitieslead to finite stresses at the free edge.

Analytical and experimental results of delamination in compositescaused by edge effects are discussed by Kim and Soni [40] andKress [41]. Salamon [42] mentioned that the interlaminar stressesaffect the boundary region of the order of one laminate thickness t(see Figure 2.3). Closed-form approaches of the interlaminar stresses


at the free edge and the corner of a laminate under mechanicalor thermal load are described by Becker et al. [43, 44, 45]. Theagreement between the analytical results and solid FE simulationsis good. Tahani and Nosier [46] developed a layerwise theory for ananalytical investigation of the edge effect for composite laminatesunder uniform axial extension or thermal load. The results dependon the number of numerical layers for each physical lamina. Ananalytical solution is presented by Nosier and Bahrami [47]. Thesolution is based an the FSDT and Reddy’s layerwise theory. Theyshowed that the convergence depends on the number of numericallayers for each lamina which depends on fiber direction.

Unlike the ISS, the sign of the INS affect the laminate strength prop-erties. The influence of interlaminar stresses on laminate strength isconsidered by Pipes [48] and Kress [41]. Delamination was identi-fied as a primary source of strength degradation for certain laminatesand therefore showed a significant variation in fatigue strength. Thisvariation could be assigned to positive INS at the free edge. It isalso mentioned that the fatigue strength of angle-ply laminates canbe significantly reduced by increases in lamina thickness. The staticstrength is also affected by the interlaminar stresses. The resultsshowed that angle-ply laminates which correspond to fiber orienta-tion of maximum shear coupling characteristics may exhibit dimin-ished static strength. The effect of the stacking sequence on the inter-laminar stress distribution is shown by Raju and Crews [49]. Thestress distributions in the cross-sections of a test specimen of CFRPare discussed by Rohwer [50].

1.1.4 Plate and shell finite elements

Plate and shell FEs are quite popular. The simplicity of their formu-lation, the adequate number of degree of freedoms (DoFs), and thequite accurate results allow their use in many different applications.The FE formulation for plates and shells are described by Cook etal. [51] and Zienkiewicz and Taylor [52]. Flat shell FEs are even usedfor modeling curved shells [53]. The first published shell FEs have fiveDoFs at each node, three in-plane deformations and two rotationalDoFs. The lack of an in-plane rotational DoF causes a singularelement stiffness matrix K if the element and global CS are parallel.Remedies have been found by Zienkiewicz and Taylor [52] and have

8 Introduction

been improved by MacNeal et al. [54]. The original approach wasfound to give erroneous results in the case of, for example the linearanalysis of a hook problem, termed the Raasch Challenge [55].

Another approach is to derive a FE with an in-plane rotational DoF.Allman [56] developed a membrane triangular element that has threeDoFs, two translations in-plane and one rotation, at each node.Ertas et al. [57] developed a three-node triangular element, termedAT/DKT, by combining an element similar to the Allman membranetriangular element with the discrete Kirchhoff theory (DKT). This el-ement was tested by Kapania and Mohan [58]. The results were foundto be in good agreement with those obtained by using other FEs. Theability of the AT/DKT element to model the in-plane rotational DoFmakes this element quite suitable to study large displacement analysisof laminated shells. Mohan and Kapania [59] extended the AT/DKTelement to study large-rotation static response, non-linear dynamicresponse, and thermal postbuckling analysis. The results obtainedfrom the AT/DKT element were found to be in excellent agreementwith those available in literature and/or those given by commercialFE software ABAQUS4. Including the in-plane rotational stiffness is,thus, important for large displacement analysis. The approach, FEwith an in-plane rotational DoF is also found in 4-node elements andis given by commercial FE code software. The theory is described byMacNeal and Harder [60].

Bilinear plate / shell FEs also have weaknesses. First, a full integra-tion method leads to the well known shear locking effect. This canbe avoided by using a reduced integration scheme which is discussedby Cook [51] and Prathap [61]. Another lack is that a pure bendingload case yields in-plane shear stresses. Such elements are called in-compatible elements5 and do not pass the so called patch test. Thisproblem is solved by introducing extra shape functions (ESF), whichare used to describe a pure bending load case. The patch test and theincompatible modes are described by Zienkiewicz and Taylor [52].

Quadratical plate and shell FEs are more robust regarding lockingand incompatible mode effects. Zienkiewicz [62] presented that thebest results are achieved for a 8-node shell FE using a 2x2 reduced

4www.abaqus.de/software/abaqus.html5also known as incompatible mode of an element


integration scheme. The reduced integration scheme shows a betterconvergence and accuracy than the full one.

Laminated composite plate elements

Laminated composite plate elements do by definition not include localcurvature effects but the literature results give interesting insightson modelling aspects that can be helpful for constructing models forshells. Especially the study of ISS in composites [16] is interestingfor the presented research. The conclusion is that even FSDT mayin certain situation give better ISS approximations than higher-ordertheories, when the shear stresses are not based on the kinematics inconjunction with material law but on the local equilibrium equationconditions [29]. Of course, when the stress-strain is overly simplifiedby the FSDT, the shear stiffness values need to be corrected by ashear correction factor.

Higher-order shear deformation theories (HSDT) are also imple-mented in plate elements. Gaudenzi et al. [63] evaluated a single-and multi-layer theory for the analysis of laminated plates. They al-low the normals to stretch but the multi-layer plate FE increases thenumber of global DoFs with the number of interfaces considered inthe plate model. This may make the use of such elements impractical.

Engblom and Ochoa [64] presented a FE formulation for a quadri-lateral plate element. The equilibrium equations are used to calculatethe INS and ISS. The results compare favorably with elasticity andnumerical solutions.

A consideration on higher-order FEs for multilayered plates is pre-sented by Ottavio et al. [65]. Different unified formulations like CLT,ESL, and ILPM theories are compared with the exact 3-D solutions(Pagano [66]) for different thickness ratios whereas the displacements,in-plane stresses and the ISS are of interest. In a second step, theconvergence of quadratical and bilinear elements is depicted. Ottavioverified that already ESL or first-order theories can give appropriateresults concerning transverse displacement and stress evaluation. INSare not considered in this publication.

10 Introduction

Laminated composite shell elements

A piecewise hierarchical p-version axisymmetrical model for lami-nated composite shell elements is presented by Liu and Surana [67]and Surana and Sorem [68]. The displacements in the thickness direc-tion are approximated layerwise by polynomials so that continuity ofinterlaminar stresses is realized, and good agreement with the exactsolution is demonstrated. The presented model is analyzed with theexact solution of a square plate and an infinitely long pressurizedcylindrical tube. All interlaminar stresses of the flat plate are closeto the exact solution for p ≥ 3. A higher polynomial order p = 8 isnecessary to get accurate results of the INS in the tube.

Wung [69] presented a continuum-based shell element with transversedeformation. The element is based on FSDT and fourth-order trans-verse deformation. It satisfies nonzero surface boundary conditionsand ISS continuity conditions and also includes INS and strain. Thepresented model gives accurate INS compared with the analytical so-lution.

Kulikov and Plotnikova [70] developed an effective shell FE. Theeffects of ISS and INS of laminated anisotropic material response areincluded. The fundamental unknowns consist of six displacements andeleven strains of the face surface of the shell, and 11 stress resultants.The quadrilateral 4-node element has bilinear shape functions for thein-plane displacements and linear approximations for the displacementin the thickness direction. The element passes the patch test and doesnot contain any spurious zero energy modes. The post-process results(stresses) are not compared with other solutions.

A nine-node shell element [71] with 6 DoFs at each node for theanalysis of a coupled electro-mechanical system differs from a degen-erate shell element because it does not assume that thickness doesnot change under load and may be properly used to model relativelythick structures. However, the assumption is made that all strainsvary linearly through the thickness.

A three-layer shell element [72] is proposed for analyzing sand-wich shells with composite sheets. The continuity of the interlaminarstresses are achieved by a postprocessing method where 3-D elasticityequations are used but it is simply assumed that the interlaminarstresses follow quadratic functions of the thickness coordinate.


A higher-order zic-zac theory of smart composite shells is presented byOh and Cho [73]. Smooth parabolic distribution through the thicknessis assumed in the out-of-plane displacement in order to consider trans-verse normal deformation and stress. The layer-dependent degrees offreedom of displacement fields are expressed in terms of reference pri-mary degrees of freedom by applying interface continuity conditionsas well as bounding surface free conditions of ISS. Thus the proposedtheory has only seven primary displacement unknowns and they donot depend upon the number of layers. For the accurate prediction ofinterlaminar stresses, the integration of 3-D local stress equilibriumequations is required in this theory. For an accurate prediction of theINS, higher-order polynomial up to the fifth order may be required inthe displacement field.

A model for radial stresses in singly curved prestressed shells likesilos is presented by Acharya and Menonb [74]. Two models arepresented whereas one is also applicable for doubly curved shells.The differential equation for the through-the-thickness displacementis derived from the classical Lame solution. Only isotropic shells areconsidered.

An equivalent single-layer model involving seven nodal DoFs is deve-loped by El-Abbasi and Meguid [75]. The layered model contains norestrictions on the number of layers, the thickness and the stackingsequence. The shell model accounts explicitly for the thickness changein the shell, as well as the normal stress and strain states through itsthickness. The results of the interlaminar stresses are not mentioned.

An improved finite element model (FEM) for the linear analysis ofanisotropic and laminated doubly curved, moderately thick compositeshells/shell-panels is presented by Hossain et al. [76]. The FSDT isused for the shell FE formulation and the ISS are obtained from theequilibrium equations. The results are compared with the availableanalytical and numerical solutions and the agreement between themis found to be excellent in both the cases of shallow and deep shells.

A post process method for interlaminar stress evaluation in plateand cylindrical shells is developed by Rohwer and Rolfes [77]. Thenecessary order of differentiation of the twodimensional (2-D) resultsis reduced by one. The method allows determining the ISS andINS just by means of the first and second derivatives of the shapefunctions, respectively. The method is compared with 3-D results forplates and shells under mechanical and thermal loads. The results

12 Introduction

show excellent agreement and indicate a wide range of applicability.

More details about shell FE formulations, ESF, locking effect, shearcorrection factors and reduced integration are presented in section 5.

1.1.5 Mechanical tests and delamination failuremodes

Traditionally, laminates are used as thin shells or plates exploiting thesuperior in-plane properties and not challenging the weak strengthin the unreinforced thickness direction. Nevertheless, delaminationscan occur as a result of impact load or edge effects and the topic hasreceived considerable research attention. The delamination behaviorof composite structures is discussed by Garg [78]. He points out thatdelamination affects strength, stiffness, and compressive behavior andthe prediction of onset of delamination requires extensive computa-tions.

A survey of standard test methods for delamination resistance of com-posite materials is presented by Davies et al. [79]. The first standardfracture test is the mode I (tensile opening) test. The crack propaga-tion is measured in a double cantilever beam (DCB) under externaltension force. A summary of mode I tests is given by Davies et al. [80].Pereira and Morais [81] and Saponara et al. [82] presented numericaland experimental studies of mode I interlaminar fracture of DCB andconfirm the persistent interest of delamination growth and fracture.

Further tests are the mode II (shear) and mode III (out-of-planeshear) test. Laksimi et al. [83] presented a detailed description ofmode II delamination in a ±θ-laminate. The tests with the end notchflexure (ENF) and the end load split (ELS) specimens showed thatthe angle orientation θ has a significant influence on the value of theenergy release rate.

An end crack torsion (ECT) specimen can be used to measurethe mode III interlaminar fracture toughness of composite lam-inates. Suemasu [84] presented a theoretically and numericallystudy for estimating the energy release rate. In addition, it is possibleto define specimens to obtain a I/II, I/III, and II/III mixed mode test.

1.2 Goals of the thesis 13

Delamination of tapered composites or delamination at terminatingplies are also of interest [85, 86, 87]. High ISS arise under in-planeloading at the end of the terminated plies or at dropped plies to pro-duce a tapper. As a result of the discontinuity, analysis gives a singu-larity.

The delamination of flat or curved composite structures undercompression load are discussed by Kultu and Chang [1] and Shortet al. [88, 89]. They describe the effect of delamination regardingbuckling and compression behavior.

Kruger [90] presented detailed delamination analysis in compositeskin/stringer specimens for various combinations of in-plane and out-of-plane loading conditions which can be used as parametric designstudies. He also published a survey of the crack closure technique,approach, and application [91]. Engineering problems of crack anddelaminations propagating between different materials are mentionedand different engineering problems are discussed to address damagetolerance of composite materials.

All these modes and their specimens predict critical loads fordelamination onset and growth or buckling caused by delamination.As a result of the fabricated delaminated specimens, discontinuityleads to high interlaminar stresses. Due to discontinuity, strain-energy-release concepts are necessary to predict delamination onsetand growth what is beyond our research.

1.2 Goals of the thesis

The goal of this dissertation is to develop new analytical models forcalculating INS in curved thick-walled laminates under mechanicalloading and to investigate their accuracy. These models are linkedwith well known shell FEs and are used for strength prediction ofcurved laminates under tension load.

The drawback of interlaminar stress evaluation in thick-walled curvedlaminated shells was pointed out by an air inlet of a Formula 1 carwhich delaminated under the critical load case. The 3-D stress fieldwas modeled with an accurate and fine solid FEM which exposed thehigh ISS. Accurate ISS models for curved laminated shell FEs are

14 Introduction

available in commercial FE softwares, but INS evaluation in doublycurved thick-walled laminates remains a lacking feature.

The main goal of this dissertation is segmented in three sub-goalswhich are derivation, implementation, and validation of the analyticalmodel. These goals are achieved by defining four workpackages:

1. The idea is to progress from the basic research to the refinementof models. In a first step, a model for singly curved thick-walledlaminates (tube) inspired from the solution of the compositetube problem [92, 93] is developed. The model includes curva-ture effects so that laminates with large curvature are correctlymodelled. It is necessary to find out the accuracy and limitsof the model and to extend the model to include shear effects.Finally, the model is compared with high-accuracy solid FEM.

2. After concluding the work on a model for singly curved thick-walled laminates, a model for doubly curved thick-walled lami-nates can be developed. This model is based on the same me-chanical approach that underlies the singly curved laminates butexpect more complicated mathematical equations and solutionproblems.

3. After the basic work, the model is linked with popular shell FEswhich can be used to predict delamination onset due to INS. Thepre-processing part is to ensure that all curvature effects and theshear compliance influence are well presented in the stiffnessmatrix so that the global response is as precise as possible. Thepost-processing part calculates the 3-D stress state from theprimary solution.

4. Finally, tests with singly curved thick-walled composite speci-mens will verify the mechanical model, FE solutions, and high-light critical interlaminar stresses and suitable failure criteria.Different monitoring systems help to get accurate and importantdata which can be compared with FE simulations.

1.3 Thesis overview

The thesis follows the four workpackages. As motivation a generaloverview about interlaminar stresses in composite structures is given

1.3 Thesis overview 15

in Chapter 2. The need of accurate models and basic aspects are de-scribed in detail. The first workpackage is presented in Chapter 3.The derivation of the INS formulation for singly curved laminatesis depicted and compared with solid FE simulations. The analyti-cal model without shear is compared with the enhanced model andthe results are discussed. Chapter 4 contains the formulation of thedoubly curved INS formulation and the model assessment. The thirdworkpackage is presented in Chapter 5. Different types of layeredshell elements are investigated and linked with the INS formulationfor doubly curved laminates. The INS evaluations in 3-, 4-, and 8-node shell FEs are compared with each other regarding mesh qualityand accuracy . In addition, a model assessment and the practicalityof the INS model are discussed. The mechanical experiments withlaminated curved specimens are presented in Chapter 6. The mecha-nical behavior, delamination onset, and accuracy of the FE simulationare investigated and critical interlaminar loads are discussed. Finally,the thesis is concluded in Chapter 7 and open issues and remaininginteresting tasks are discussed.

Chapter 2

Interlaminar stresses inshell structures

This chapter explains a few of the concepts for modelling interlaminarstresses which have been mentioned in the literature survey.

2.1 Interlaminar stresses caused by hold-

ing equilibrium to internal forces

The following illustrations are based on the assumption that the ISSand INS have to vanish at top and bottom of the laminate. Thisimplies that no shear forces, pressure, and point load act directly onthese surfaces. In contrast to the in-plane stresses, which are notcontinuous through the thickness, interlaminar stresses are continuousbut not continuously differentiable through the thickness.

A first example is a simple straight (laminated composite) beam whichis clamped at one end and loaded by a shear force Q at the other(see Figure 2.1). Hence, a constant shear force acts on the compositebeam in the length direction x.

The ISS are in balance with the shear force and have to vanish attop and bottom. The layerwise quadratic shear stress distribution isbased on an assumed layerwise linear normal stress distribution. Theequilibrium in axial direction then requires layerwise quadratic ISS.

18 Interlaminar stresses in shell structures

Q Q

M

x

x

y

z

Figure 2.1: Laminated composite beam and qualitative load distribu-tion.

This assumption is valid in regions where no clamping effects occur.The ISS derivation for an isotropic beam is described below [94].

Equilibrium of the moment M,x = Q = E · I · κ,x

Bending strain distribution ε(x, z),x = z · κ,x(x) = z QE·I

Local constitutive equation σ,x = E · ε,x

Local equilibrium τ,z = −σ,x

Integration τ(z) = −∫ z

−t2

σ,x dζ

= −QI

∫ z

−t2

ζ dζ

Stress distribution τ(z) = − Q2·I (z2 − ( t

2 )2)

The ISS evaluation for a symmetrical orthotropic plate under cylindri-cal bending is described by Kress [13]. In this case, the ISS distribu-tion is a result of the sum which derives its origin from the layerwiseintegration.

τxz(z) = −1

2· Qx

(Q11)k

D11(z2 − z2

k−1) + (τxz)k−1 (2.1)

Q11 and D11 are the first components of the reduced stiffness andbending matrix, respectively. Figure 2.2 illustrates the ISS distri-bution in three different composite plates with a symmetrical stack-ing sequence. The shear force Qx = 1 [N ] and laminate thickness

2.1 Interlaminar stresses caused by holding equilibrium to internalforces 19

t = 1 [mm] are constant. The laminates consist of a ultra-high modu-lus orthotropic material GY70/Epoxy (material properties are listedin Appendix A.1). The quadratic ISS distribution in the unidirectional(UD) laminate has a maximum of 1.5N/mm2. The ISS distributionin the [0,90]s laminate is similar to a trapezoid and the maximumis 1.29N/mm2. The absolute maximum ISS appears in the [90,0]slaminate and is 2.82 N/mm2. The explanation can be found in Equa-

tion 2.1. The term Q11

D11, which is proportional to the slope of the

ISS distribution, is much smaller in the 90-laminae compared withthe 0-laminae. Hence the ISS are near by zero in the 90 face sheets.Moreover, the shear stress has to be in equilibrium with the shear forceand the inner laminae balance this. The [0, 90]s laminate, which canbe compared with a sandwich plate, gives the best results (minimumISS).

−0.5 0 0.50

0.5

1

1.5

2

2.5

3

Thickness coordinate z [mm]

ISS

[N

/mm

2]

UD

[0,90]s

[90,0]s

Figure 2.2: ISS distribution in an orthotropic plate under cylindricalbending.

This simple model for calculating ISS in a symmetrical and bal-anced plate can be enhanced for a unsymmetrical plate where thestiffness matrix B causes a coupling between the in-plane forcesand bending moments. In addition, a unbalanced plate, wherethe stiffness terms C45 = C54 6= 0, effect coupling between the shearstresses τxz and τyz. The 3-D stiffness matrix is given in Equation 2.2.


σx

σy

σz

τxz

τyz

τxy

=

C11 C12 C13 0 0 C16

C21 C22 C23 0 0 C26

C31 C32 C33 0 0 C36

0 0 0 C44 C45 00 0 0 C54 C55 0

C61 C62 C63 0 0 C66

εx

εy

εz

γxz

γyz

γxy

(2.2)

where Cij are the components of the 3-D stiffness matrix expressedin reference coordinates. Interlaminar stresses also appear in curved

laminates which have under external load a non-linear strain dis-tribution through the thickness. The strain εx of a curved laminateunder cylindrical bending is smaller at the outer surface, if comparedwith the strain at the inner surface (see Figure 3.1). This effect causesa tension or compression in the transverse direction which causesINS. The INS have to vanish at top and bottom of the laminate andare proportional to the laminate thickness t and inverse proportionalto the curvature radius R. Of course, the INS depend also on thestacking sequence and material properties. This curvature effectand the corresponding INS are the central point of this thesis. Theanalytical models, INS distributions and critical loads are presentedin Chapter 3 and 4.

INS in a composite structure are also initiated by an external pressuredistribution or load introducing elements like in- or onserts1. In thesecases, the INS do not vanish at either the top or the bottom surface orpossibly both. The concerning through-the-thickness INS distributiondepends on stacking sequence, laminate thickness and geometry. Thisissue is not part of the thesis. Optimized onserts and the load dis-tribution are presented by Kress et al. [95, 96] and Keller et al. [97, 98].

2.2 Edge effect

A brief overview of the edge effect is given in the Subsection 1.1.3.The presented analytical models for INS in Chapter 3 and 4 do not

1Joining load introduction element which is simply bonded to the surface ofthe substrate.

2.2 Edge effect 21

include edge effects. They are compared with solid FEMs of singlyand doubly curved laminated beams where such effects cause inter-laminar stresses. The accuracy of the analytical models decrease inthe regions which are affected by edge effects. Regarding later studyof accuracy, a simple numerical illustration of the interlaminar stressescaused by the edge effect is given in this section. A [(±30)2, 90]s com-posite plate is used as specimen and loaded with a tension force. Asolid (20-node brick) FE simulation with a local mesh refinement atthe free edge exposes the interlaminar stress distribution. The mesh,stacking sequence and stress distribution are shown in Figure 2.3. Themesh size decreases near the free edge for a better reproduction of thesingularity.

The interlaminar stresses increase near the free edge and maycause delamination. The stresses affect the boundary region of theorder of one laminate thickness t. The INS are the critical stress termin this example, not only because of the maximum absolute valuebut also the positive sign. Already a small tension load can imposecritical stress states due to the low transverse strength properties ofcomposite materials.

-30

-30

-30

-30

+30

+30

+30

+30

9090

Width y/b 0 0.25 0.5 0.75 1−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Width y/b [−−]

INS [N/mm2]

INS

ISS xz

ISS yz

t

Figure 2.3: Interlaminar stresses in a [(±30)2, 90]s laminate. Quali-tative distribution between -30 and 90 laminae over the wide-sectionwidth.

The two examples illustrated in Figures 2.2 and 2.3 present possiblesources of interlaminar stresses in composite structures. It is also


mentioned that accurate and, from a numerical point of view, costlyFEMs are necessary for a detailed evaluation.

2.3 Solid brick FE simulation

This short overview of interlaminar stresses in composite structurespoints out the importance of accurate models for stress and failureprediction. Accurate ISS models in shell FEs are available in com-mercial software like ANSYS2 or ABAQUS, but the lack of accurateINS models in shell FEs complicates the design of such structures. Aselective review and survey of current developments of interlaminarstress evaluation in laminated composites is given by Kant and Swami-nathan [99]. Both analytical and numerical methods are considered.

One approach is to model such a laminate using a multilayerelement based on an assumed hybrid-stress FE model which is pre-sented by Chen and Huang [100]. The stress field in each layer isassumed from an in-plain strain field. The results (INS and ISS)agree excellently with other numerical and theoretical solutions. Another FE with accurate results regarding interlaminar stresses andedge effects is developed by Kim and Hong [101].

Layered solid and solsh (solid-shell) FEs are implemented in commer-cial software which can evaluate the 3-D stress tensor and an accuratedimensioning is possible. One disadvantage is that a volume computeraided design (CAD) model must be available for the structure repre-sentation what causes other problems:

• The CAD model must be adjusted to changing laminates be-cause the laminate thickness t is often unknown in the predesign.

• This increases human and computer resources in e.g. optimiza-tion cycle.

• A solid brick mesh is often not feasible for volumes with freeformsurfaces.

• A huge number of FEs through the thickness is necessary for aproper INS evaluation.

2www.ansys.com

2.3 Solid brick FE simulation 23

Such a FEM calculation is often not feasible due to the huge numberof DoFs and poor mesh quality.

As an example, the INS in a 10mm thick singly curved UD and [0, 90]slaminate are evaluated. A 8-node solid brick FE (solid185 in ANSYS10.0) is compared with the results of a 20-node solid brick (solid186)FE. The specimen and the FE mesh is illustrated in Figure 2.4. Thespecimen is clamped on the left side A and a longitude displacementis specified on the other side B.

A

B

Figure 2.4: Singly curved specimen and 8-node brick FE mesh with15 elements through the thickness.

Figure 2.5 illustrates the convergence of maximum INS of both solidFEs for the UD and cross-ply laminate. Both FEs converge to amaximum whereas the absolute values are higher in the 20-node FEM.The maximum differs by 5.5% in the UD and by 11.0% in the cross-ply laminate, respectively. An adequate fine FE mesh guaranteesan accurate mapping of curvature of the geometry, even using linearisoparametric elements. Therefore, the linear element length is one-third of the quadratic one. Figure 2.5 also shows that a few quadraticFEs through the thickness overestimate the maximum INS and linearFEs underestimate the INS because the stresses are extrapolated fromthe Gauss points to the corner nodes.

The through-the-thickness INS distribution in the UD laminateis shown in Figure 2.6. The first observation is that 10 FEs throughthe thickness give good results regarding the INS. Secondly, thestress results of the 20-node FEMs are more accurate at top and


1 2 3 4 5 10 15 20 2520

30

40

50

60

70

80

90

100

Elements through the thickness

INS

[N

/mm

2]

8−node Brick

20−node Brick

1 2 3 4 5 6 752

54

56

58

60

62

64

66

68

Elements per layer

INS

[N

/mm

2]

8−node Brick

20−node Brick

Figure 2.5: Convergence of maximum INS in a singly curved UD (left)and [0, 90]s (right) laminate.

bottom where the INS should vanish. The same results are noticedin the cross ply laminate and are plotted in Figure 2.7. Both themaximum of the INS and the stresses at the top and the bottomsurface change only by a small amount with further mesh refinements.

The conclusions of this simple example are:

• Quadratic 8-node plane or 20-node brick elements are used inthis work for solid FE simulations and as reference solution.Quadratic elements converge faster and satisfy the BC to ahigher extend.

• Convergence and BC tests have to be performed before a solidFE simulation can be used as reference solution.

• Stresses or strains are preferably evaluated at the integration(Gauss) points (IP) [102].

2.3 Solid brick FE simulation 25

5 0 −50

10

20

30

40

50

60

70

80

z−Coordinate [mm]

INS

[N

/mm

2]

1 Element

3 Elements

5 Elements

10 Elements

25 Elements

5 0 −5−20

0

20

40

60

80

100

z−Coordinate [mm]

INS

[N

/mm

2]

1 Element

3 Elements

5 Elements

10 Elements

25 Elements

Figure 2.6: INS distribution of linear (left) and quadratic (right) solidbrick FEs in a UD laminate. Mesh size: 1 to 25 elements through thethickness.

5 0 −50

10

20

30

40

50

60

z−Coordinate [mm]

INS

[N

/mm

2]

1 Element

3 Elements

5 Elements

7 Elements

5 0 −5−10

0

10

20

30

40

50

60

70

z−Coordinate [mm]

INS

[N

/mm

2]

1 Element

3 Elements

5 Elements

7 Elements

Figure 2.7: INS distribution of linear (left) and quadratic (right) solidbrick FEs in a [0, 90]s laminate. Mesh size: 1 to 7 elements per layer.

Chapter 3

Model for interlaminarnormal stresses in singlycurved laminates

The derivation of the INS formulation for singly curved laminates(tubes) is presented in this chapter. A detailed description of thecurvature effect and analytical model is given in Section 3.1 and 3.2,respectively. In Section 3.3 the accuracy of this model is investigatedby solid FE simulations. In a second step, the model is enhancedby a shear term (Section 3.4) and finally, Section 3.5 concludes thechapter.1

3.1 Assumptions and conventions

3.1.1 Strain distribution and curvature radius

A flat plate model based on the Kirchhoff assumptions (see Sec-tion 1.1.1) has a linear strain distribution

ε(z) = ε0 + z · κ (3.1)

1This chapter is based on the papers Model for interlaminar normal stress in

singly curved laminates [103] and Enhanced model for interlaminar normal stress

in singly curved laminates [104].

28 Model for interlaminar normal stresses in singly curved laminates

which is determined by the plate deformations ε0 and κ. The curva-ture radius R of a singly curved plate, illustrated in Figure 3.1, leads toa non-linear through-the-thickness distribution of strain εx [31] whichcan be described by the rational function Z

Z(z) =R

R + z=

R

r(3.2)

where z = [− t2 , t

2 ]. The through-the-thickness distribution of strain εx

in an incremental curved segment is shown in Figure 3.1. It is obviousthat the neutral axis in a positively curved plate, loaded by a bendingmoment, is displaced to the inner surface by some distance e. Inaddition, the maximum strain appears at the inner radius r = R−t/2.

+

-

t

zR

e

M M

dφ

εx

Figure 3.1: Constant curved finite segment and strain distribution.

The sign convention of the curvature radius R is illustrated in Fi-gure 3.3. It depends on the shell FE normal ~n direction which isdefined by the first three nodes. The reference element CS and thecoordinates of the nodes for a 4-node element are illustrated in Fi-gure 3.2. The specification of the nodes for a 3-, 6-, and 8-nodeelement is described by Cook [51].

The first (n1) and second (n2) nodes define the local ξ direction

~ξ = norm(~n2 − ~n1) (3.3)

3.1 Assumptions and conventions 29

and the ξη-plane is defined by the nodes n1, n2, and n3. The normal-ized shell normal ~n is obtained from the cross product of the planedirection vectors

~n = norm(~ξ × (~n3 − ~n1)) . (3.4)

ξ

η

n1 = (−1,−1) n2 = (1,−1)

n3 = (1, 1)n4 = (−1, 1)

Figure 3.2: CS of the reference element and the coordinates of thenodes (4-node element).

The curvature radius R is positive if the vectors ~R and ~n have thesame directions

R > 0 , if ~R · ~n > 0R < 0 , otherwise .

(3.5)

n1n1

n2n2

n3n3

n4n4 ~n

~n

R

R

R > 0 R < 0

Figure 3.3: Curvature radius convention: Positive curvature R leftand negative curvature R right.


The strain distribution Z and curvature radius R are defined in Equa-tion 3.2 and 3.5, respectively and the strain distribution of a singlycurved segment becomes

ε = Z · ε0 + Z · z · κ . (3.6)

3.1.2 Kinematical relations of a finite segment

The kinematical relations of a singly curved thick-walled plate arederived from the model of a thick-walled composite tube [92, 93].Angular sections of the tube are bounded by imaginary walls andany deformation depends on the distribution of displacement w(r)depending only on the spatial coordinate r in the radial direction.

The curved plate is divided into angular segments dφ with as-sumed constant curvature radius R. The imaginary walls still remainstraight, but can map displacements and rotations caused by line (nor-mal) forces and moments (see Figure 3.4). The radial displacementstill depends only on the spatial coordinate r; thus the following as-sumptions are defined:

• The strains are generally expressed in the cylindrical coordinatesystem (CS) by the variables r, ϕ, and y.

• Cylindrical bending of the curved shell about the y axis im-plies that the displacement v must be constant with respect tothe radial direction r and linear with respect to the generatordirection y.

• The shear strains γϕy and γyr do not appear in a UD or cross-plylaminate under cylindrical bending.

uu rr NϕNϕ

MyMyφ φ

Figure 3.4: Kinematics (left) and loads (right) of the curved laminate.

3.2 Analytical model 31

3.2 Analytical model

3.2.1 Modified ABD-Matrix

Considering the non-linear strain distribution (Equation 3.6), themodified strain distribution for a curved composite plate is

εx

εy

γxy

=

Z · ε0

x

ε0

y

Z · γ0

xy

+ z

Z · κx

κy

Z · κxy

=

Z 0 00 1 00 0 Z

ε0

x

ε0

y

γ0

xy

+ z

κx

κy

κxy

(3.7)

where the common notation of the CLT theory is used and the relationbetween the cylindrical and cartesian CS is: x = ϕ, y = y and z = r.In the limiting case of a straight plate (R → ∞) the function Z(z)becomes unity, Z = 1, and Equation 3.7 reduces to the kinematics ofthe CLT. The CLT assumes the individual layers of a laminate to bein a state of plane stress and writes the material law with the reducedstiffness matrix

σ = Q · ε (3.8)

where

Q =

Q11 Q12 Q16

Q21 Q22 Q26

Q61 Q62 Q66

. (3.9)

The integration of the stresses through the thickness combined withthe material law and the plate kinematics leads to the well knownABD-Matrix of the CLT.

NM

=N

∑

k=1

∫ zk

zk−1

σ

1z

dz

=

N∑

k=1

Qk

∫ zk

zk−1

[

1 zz z2

]

dz

ε0

κ

(3.10)


[A, B, D] =N

∑

k=1

Qk

[

(zk − zk−1) ,1

2

(

z2k − z2

k−1

)

,1

3

(

z3k − z3

k−1

)

]

(3.11)

The non-linear strain distribution (Equation 3.7) is combined withEquation 3.8 and the ABD-Matrix for a curved laminated plate be-comes

[

A∗ B∗

B∗ D∗

]

=

N∑

k=1

Qk

∫ zk

zk−1

[

1 zz z2

]

Z(z)dz . (3.12)

In a curved symmetrical plate a bending moment leads to an expan-sion of the centerline. This coupling between the bending momentM and the direct strain ε0 is represented by the B-matrix whichbecomes zero in a symmetrical flat laminate. The influence of thecurvature radius R on through-the-thickness distribution of strain εx,in-plane stiffness, coupling stiffness, bending stiffness, and through-the-thickness ISS distribution is shown in Appendix B.1 - B.3.

3.2.2 Derivation

The direct strains in the cylindrical CS depend on the displacementsby

εr = w,r

εϕ = wr +

u,ϕ

r

εy = v,y .

(3.13)

Following the imaginary-wall modelling idea, the displacement u canbe expressed by the laminate deformations ε0

ϕ and κϕ.

du(r) = R · dϕ(ε0ϕ + z · κϕ) = R · dϕ(ε0

ϕ + (r − R)κϕ) (3.14)

With this and the other kinematical assumptions explained in Sec-tion 3.1.2 the direct strains become

3.2 Analytical model 33

εr = w,r

εϕ = wr + R

r (ε0ϕ + (r − R)κϕ)

εy = v,y = const .

(3.15)

Since only direct strains are considered in this model, the materiallaw reduces to

σϕ

σy

σr

=

C11 C12 C13

C21 C22 C23

C31 C32 C33

εϕ − εF

ϕ

εy − εF

y

εr − εF

r

(3.16)

where Cij are components of the 3-D stiffness matrix expressed inreference coordinates which are parallel to the principal curvaturedirection (Section 5.2). The εF indicates free layer strains due tospatially constant changes of temperature T and moisture content H .

εF = α · ∆T + β · ∆H (3.17)

The thermal expansion and moisture swelling coefficients α and β areorthotropic and an example of α for a UD ply is given on Page 103.The radial equilibrium condition in the cylindrical CS, which is usedfor the INS calculation, is

1

r(rσr), r +

1

rτrϕ,ϕ + τry,y − 1

rσϕ = 0 . (3.18)

Upon neglecting the terms including shear-stress gradients the equi-librium condition (Equation 3.18) reduces to

(rσr),r − σϕ = 0 . (3.19)

The stresses in Equation 3.19 are expressed through deformations byusing the kinematical equations 3.15 and the material law 3.16.


σϕ = C11(wr + R

r (ε0

ϕ + (r − R)κϕ) − εF

ϕ)+

+ C12(ε0

y − εF

y ) + C13(w,r − εF

r )

σr = C31(wr + R

r (ε0

ϕ + (r − R)κϕ) − εF

ϕ)+

+ C32(ε0

y − εF

y ) + C33(w,r − εF

r )

(rσr),r = C31(w,r + R · κϕ − εF

ϕ)+

+ C32(ε0

y − εF

y ) + C33(w,r + r · w,rr − εF

r )

(3.20)

The result is a second-order differential equation of the through-the-thickness displacement w where the term C13w,r and C31w,r are can-celled out due to the symmetry of the stiffness matrix C: Cij = Cji.

C33 · r · w,rr + C33 · w,r − C11w

r+ P = 0 (3.21)

where P is the particular part.

P = − 1r C11 · R(ε0

ϕ − R · κϕ)+

− C11(R · κϕ − εF

ϕ) − C12(ε0

y − εF

y ) + C13 · εF

r +

+ C31(R · κϕ − εF

ϕ) + C32(ε0

y − εF

y ) − C33 · εF

r

(3.22)

Equation 3.21 is divided by r · C33 and the final differential equationfor the through-the-thickness discplacement w has the form

w,rr +w,r

r− s2 w

r2+ P ∗ = 0 (3.23)

whereas s2 = C11

C33and P ∗ = P

rC33. The general solution of Equa-

tion 3.23 is

w(r) = wH(r) + wP (r) = a · rs + b · r−s + wP (r) . (3.24)

The method of variation of the constants [105] finds the solution ofthe particular part P ∗

P ∗ = − 1r2 [ C11 · R(ε0

ϕ − R · κϕ)]+

+ 1r [ (−C11 + C13)R · κϕ + (C23 − C12)ε

0

y+

+(Ci1 − Ci3)εF

i ]

(3.25)

3.3 Model assessment 35

whereas Cij =Cij

C33. The solutions of P1 = 1

r2 and P2 = 1r are

wP =

wP1 = 1s2 wP2 = −r

1−s2 , if s 6= 1

wP1 = 1 wP2 = r2 [1 − ln(r)] , otherwise

(3.26)

and the particular part wP becomes

wP (r) = − wP1[ C11 · R(ε0

ϕ − R · κϕ)]+

+ wP2[ (−C11 + C13)R · κϕ + (C23 − C12)ε

0

y+

+(Ci1 − Ci3)εF

i ] .(3.27)

The free parameters a and b of the homogeneous solution wH areused to satisfy the BCs or interface continuity conditions in multilayerlaminates which are

wk+1(zk) = wk(zk)

σk+1(zk) = σk(zk)(3.28)

where σ stands for the INS (radial direct stress). The BCs are definedby the INS at the top and bottom surfaces which have to vanish ifno pressure P or point load acts on these surfaces. The boundaryand intersection conditions lead to 2N free parameters which haveto be adjusted. The through-the-thickness distribution of the INSis obtained by combining the solution of the through-the-thicknessdisplacement w with the radial stress Equation 3.20.

3.3 Model assessment

The new model is compared with 2-D FE simulations of a singlycurved thick-walled specimen which has two regions with relativelyhigh INS.

3.3.1 Specimen design and simulation

The geometry and the kinematical behavior of the specimens have toachieve the following requirements

• Singly curved.


• No discontinuities of the geometry.

• No shear stresses in the regions with INS.

The centerline of the specimen shape, which is the result of a struc-tural analysis, follows the function f

f(x) =

0 , −100 ≤ x < −50h2

[

cos(

2π·xl

)

+ 1]

, −50 ≤ x ≤ 500 , 50 < x ≤ 100

(3.29)

where h = 30mm is the amplitude and the length l is 100mm. Further,the curvature radius R of the centerline is

R(x) =

∞ , −100 ≤ x < −50“

1+f′

(x)2”

1.5

f ′′ (x), −50 ≤ x ≤ 50

∞ , 50 < x ≤ 100

. (3.30)

50mm

L=100mm

t

50mmIII

II

I

h

x

Figure 3.5: Singly curved specimen geometry.

The 10mm thick singly curved specimen is illustrated in Figure 3.5.The straight ends are used to model the BCs and to avoid clampingeffects in the curved part. The specimen is clamped at the left sideand a line load Fx = 500N/mm is applied at the other where therotations are suppressed. The external load causes internal line forcesFx, Qϕr and a line moment My. The corresponding distributionsalong the reference axis x are plotted in Figure 3.6.

These internal loads, which are obtained from a closed-formmodel for cylindrical plate bending including the curvature effect, are


used to evaluate the laminate deformations. The load distributionof the close-form model is compared with a numerical plane-strainmodel with an 8-node serendipity-type FE to asses the accuracyof the closed-form representation of the internal force and momentdistributions. The resultant forces obtained from the plane FEMare achieved by summing up the nodal forces, through the thicknessalong the interfaces of each element column, and transforming theminto the directions parallel and transverse to the centerline. The closeagreement between both models is illustrated in Figure 3.6. Thetransverse shear force Qx has the maximum in region II where thenormal force Fϕr is minimal and vanishes in region I and III wherethe maximal curvature radius R appears. The bending moment My

is maximal in region III.

−100 −80 −60 −40 −20 00

50

100

150

200

250

300

350

400

450

500

Length x [mm]

For

ce [N

/mm

]

N Beam ModelN Plane FEMQ Beam ModelQ Plane FEM

−100 −80 −60 −40 −20 0−5000

−2500

0

2500

5000

7500

10000

Length x [mm]

Mom

ent [

N/m

m2]

M Beam FEMM Plane FEM

Figure 3.6: Internal line forces Fx and Qϕr (left) and line moment My

(right) distribution in the singly curved specimen.

The displacements u and w of the centerline are plotted in Figure 3.7.The results between the beam and the plane model differ by up to15%. Using an artificially high value for the shear modulus G13 in theFEM removes the discrepancy because the closed-form model doesnot include shear deformation.

3.3.2 Interlaminar normal stress results

The new model is compared with the results obtained from a 2-Dsolid FEM. A quadratic 8-node plane element with 2 DoFs at eachnode, which is well suited to model curved boundaries, is used. This


−100 −80 −60 −40 −20 0−2

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

Length x [mm]

Dis

plac

emen

t [m

m]

U Beam ModelW Beam ModelU Plane FEMW Plane FEMU Plane FEM Shear RigidW Plane FEM Shear Rigid

Figure 3.7: Displacements of the singly curved specimen.

FE also considers orthotropic material properties. The curved partof the specimen’s geometry is meshed with 100 and 40 elements inlength and thickness direction, respectively. The straight parts aremeshed with 40×40 elements. The difference of the maximum INSbetween a FEM with 25 and 40 elements through the thickness is0.74h. In addition, the INS vanish at the top and the bottom sur-faces and therefore the convergence and BC tests can be consideredsuccessfully. The 10mm thick laminate is made of a UD reinforcedprepreg2 material with a cured thickness of about 0.19mm. Theproperties needed in this analysis are listed in Table 3.1 and all othersare given in Appendix A.2.

Table 3.1: Material properties Elitrex EHKF.

E1 = 110000 E2,3 = 10000 [MPa]G23 = 3846 G13,12 = 5000 [MPa]ν23 = 0.49 ν13,12 = 0.27 [–]

The global INS distribution of the plane FE simulation in the UDlaminate, loaded by 500 N/mm, is plotted in Figure 3.8. The INS

2preimpregnated


have two local maxima in region I and III where the ISS are close tozero.

-6.1 4.1 14.4 24.7 35.0 45.3 55.6 65.9 76.2 88.0

Figure 3.8: INS [N/mm2] distribution in the UD composite specimen.

The orientation of the thickness coordinate z in region I and III isdescribed in Figure 3.9. The inner surface is at z = t/2 and z = −t/2in region I and III, respectively.

III

I

Z

z = −t/2

z = t/2

z = −t/2

z = t/2R

Figure 3.9: z-coordinate in region I and III.

Unidirectional laminate [026]s

Figure 3.10 illustrates the INS distribution of the analytical modeland the FEM in region I and III. The maximum INS obtained fromthe FEM in region III (x = 0) is 87.86N/mm2 and is leaning towardsthe center of curvature. The absolute value of the analytical model is13.8% lower compared with the FE solution. The INS distribution ofthe analytical model looks similar to the FE solution.

The present model overestimates the INS in region I (x = 50) by30.4%. One reason for this difference is that the analytical modelobtains an abrupt change from the flat region to the curved beam


situation because the curvature radius jumps from infinite to its mi-nimum. In the 2-D FEM the transition appears much smoother andmore realistic. The maximum INS obtained from the FE simulationis also leaning towards the center of curvature and is 22.61N/mm2.In both locations, the analytical model predicts the maximum INScloser to the centerline.

−5 −2.5 0 2.5 50

10

20

30

40

50

60

70

80

90

z [mm]

INS

[N

/mm

2]

Present x=0

Present x=50

FEM x=0

FEM x=50

Figure 3.10: UD laminate: Comparison of the through-the-thicknessINS distribution between the analytical model and FEM.

Cross ply laminate [013, 9013]s

The through-the-thickness INS distribution in regions I and III isshown in Figure 3.11. The maximum INS is at the intersection ofthe 0- and 90-laminate which is closer to the center of curvature.Again, the INS distributions shows common features. The maximumINS in region III obtained from the analytical model is 68.05N/mm2

which is 25% lower compared with the plane FE solution. The max-imum INS in region I of the present model and the FEM differ onlyby 0.5% although the distributions differ more.

Cross ply laminate [9013, 013]s

The INS distribution obtained from the new model, plotted in Fi-gure 3.12, has a maximum of 140.2N/mm2 in region III and differsonly by 0.7% compared with the plane FEM. The situation in region

3.4 Enhanced model including shear 41

−5 −2.5 0 2.5 50

10

20

30

40

50

60

70

80

90

z [mm]

INS

[N

/mm

2]

Present x=0

Present x=50

FEM x=0

FEM x=50

Figure 3.11: [0, 90]s laminate: Comparison of the through-the-thickness INS distribution between the analytical model and FEM.

I appears similar to the UD laminate. The analytical model over-estimates the INS by 43.1% compared with the FE result. In bothregions, the maxima are close to the centerline.

The present model produces results whose agreement with those ofthe numerical results are quite good in view of the severe simplifyingassumptions made: The shear stresses are neglected and the cross sec-tion remains plane. The accuracy depends on the stacking sequenceand the load. The results of the new model agree better in region IIIwhere a smooth change of the curvature leads to the high INS com-pared with region I where an abrupt change of the curvature leads to astep in the INS distribution along the length direction x (Figure 3.16).

3.4 Enhanced model including shear

3.4.1 Kinematical relations of a finite segmentwith shear deformation

The model for INS in singly curved laminates presented in Sec-tion 3.2.2 is enhanced by a shear term. The kinematical model for afinite constant curved element is shown in Section 3.1.2. This modelis enhanced by a shear deformation which causes that plane cross sec-


−5 −2.5 0 2.5 50

20

40

60

80

100

120

140

z [mm]

INS

[N

/mm

2]

Present x=0

Present x=50

FEM x=0

FEM x=50

Figure 3.12: [90, 0]s laminate: Comparison of the through-the-thickness INS distribution between the analytical model and FEM.

tions perpendicular to the plate midplane remain plane but rotatewith respect to the deflection line by a shear angle γ. Interlaminarshear causes cross section warping which is not mapped in the kine-matics of the FSDT to which the present model belongs. Skytta [16]showed that FSDT compete well with, or even outperform, HSDT.

In the case of a plate the shear angle γ is constant through thethickness. In curved shells this would imply that the distance be-tween the top and bottom surfaces must change along the direction ϕas the sketch on the left of Figure 3.13 illustrates. It appears mechan-ically correct to assume that the thickness of a plate element is notkinematically coupled with shear deformation and this requires thatthe shear angle be variable through the thickness as indicated by thesketch on the right of the same figure. Specifically, the shear angle γis proportional to 1

r .

3.4.2 Derivation

Cylindrical bending of singly curved cross-ply laminates does notevoke the shear stresses τyr and τϕy so that the shear terms γyr andγϕy can be neglected. The kinematical relations expressed in cylin-drical coordinates become

3.4 Enhanced model including shear 43

rr

t

ϕϕ

AA

A’A’

BB

B’

B’

∆t

γ

γ(r)

γ(r)

Figure 3.13: Shear deformation in a curved beam. Constant shearangle γ (left) and constant thickness t (right).

εr = w,r

εϕ = wr +

u,ϕ

r

εy = v,y = const

γrϕ =w,ϕ

r + u,r − ur .

(3.31)

Equation 3.14 describes the displacement u, expressed with the lam-inate deformations, which is combined with the kinematical rela-tions 3.31. The direct strains are listed in Equation 3.15 and theshear strain becomes

γrϕ =w,ϕ

r + ϕ · R · κϕ − ϕRr

(

ε0

ϕ + (r − R)κϕ

)

=w,ϕ

r − ϕ·R·ε0ϕ

r +ϕ·R2

·κϕ

r

= 1r

[

w,ϕ − ϕ · R(

ε0

ϕ − R · κϕ

)]

(3.32)

whereas γrϕ = 2 · εrϕ. The shear stiffness C55 is now considered inthe constitutive equation

σϕ

σy

σr

τrϕ

=

C11 C12 C13 0C21 C22 C23 0C31 C32 C33 00 0 0 C55

εϕ − εF

ϕ

εy − εF

y

εr − εF

r

γrϕ

. (3.33)

The stress terms of the equilibrium condition

(rσr),r + (τrϕ),ϕ − σϕ = 0 (3.34)


are replaced by combining the Equations 3.15, 3.32, and 3.33 andthe differential equation for the through-the-thickness displacementw becomes

w,rr +w,r

r− s2 w

r2+ f2 w,ϕϕ

r2+ P ∗ = 0 (3.35)

where s2 is still an anisotropy factor C11

C33and f2 is the ratio between

the out-of-plane stiffness and shear stiffness C55

C33.

The goal is to capture the influence of the ISS on the INS distribu-tion although the cross sections of the singly curved laminate are stillmodeled to remain plane. Therefore, the shear term w,ϕϕ is replacedby using Rohwer’s approach [15]. The ISS distribution is well modeledwith this approach by integrating local equilibrium through the thick-ness. The corresponding warping of the cross section can be replacedby an average shear angle γ

γ = 1rt−rb

∫ rt

rb

γ(r)dr

= 1rt−rb

∫ rt

rb

τ

G55dr .

(3.36)

The integration of Equation 3.32 leads to the constant shear angleγ which has to be equal to the shear angle obtained from Rohwer’sapproach.

γ =

∫ rt

rb

γrϕdr

=

∫ rt

rb

1

r

[

w,ϕ − ϕ · R(

ε0

ϕ − R · κϕ

)]

dr

= ln(rt)−ln(rb)rt−rb

[

w,ϕ − ϕ · R(

ε0

ϕ − R · κϕ

)]

= 1rt−rb

∫ rt

rb

τ

G55dr

(3.37)

w,ϕ =1

ln(rt) − ln(rb)

∫ rt

rb

τ

G55dr + ϕ · R

(

ε0

ϕ − R · κϕ

)

(3.38)

3.5 Comparison and conclusions 45

This equation is used to determine the function w,ϕϕ appearing in thedifferential Equation 3.35 for the equilibrium in radial direction. Thederivative of w,ϕ is

w,ϕϕ =1

ln(rt) − ln(rb)

∫ rt

rb

τ,ϕ

G55dr + R

(

ε0

ϕ − R · κϕ

)

. (3.39)

This expression must be added to the other terms of the particularpart P ∗ (Equation 3.35) and the latter becomes

P ∗ = − 1r2

[

(C11 + C55)(

R · ε0

ϕ − R2 · κϕ

)

− f2 · w,ϕϕ

]

+

+ 1r

[

−(C11 − C13)R · κϕ + (C23 − C12)ε0

y+

+ (Ci1 − Ci3)εF

i

]

(3.40)where P ∗ = P

rC33. The homogeneous solution wH is equal to the

homogeneous solution of the analytical model without shear, given inEquation 3.24, and the particular solution wP is found again with themethod of variation of the constants, whereas the terms wP1 and wP2

are defined in Equation 3.26.

wP = −wP1

[

(C11 + C55)(

R · ε0

ϕ − R2 · κϕ

)

− f2 · w,ϕϕ

]

+

+wP2

[

−(C11 − C13)R · κϕ + (C23 − C12)ε0

y+

+ (Ci1 − Ci3)εF

i

]

(3.41)Two other solutions of the differential Equation 3.35 are found wherethe shear term w,ϕϕ is not replaced by a constant shear angle term.These solutions have four free parameters which have to be determinedusing BCs. They are given in Appendix B.4.

3.5 Comparison and conclusions

The enhanced model is compared with the model without shear termto illustrate the influence of the FSDT approach regarding the INSdistribution. The results disclose that the influence in the present


example is small for every layup in region I and III because thederivative of the shear force Q

′

x = −q(x) is small compared with theline load Fx and the bending moment M .

The results of the INS distribution of the analytical models differonly by 0.14% and 0.06% in the UD laminate in region I and III,respectively. The accuracy of the INS distribution in the [0, 90]slaminate obtained from the enhanced analytical model increases by0.8% in region III compared with the model without shear term anddecreases by 0.5% in region I. The accordant INS distributions ofthe analytical models and the FEM are plotted in Figure 3.14. Theresults in the [90, 0]s laminate of both analytical models diverge onlyby 0.19% in region I and by 0.08% in region III. Generally, the benefitis small regarding the low improvement and the additional numericalcost.

−5 −2.5 0 2.5 50

10

20

30

40

50

60

70

80

90

z [mm]

INS

[N

/mm

2]

No shear x=0

No shear x=50

FEM x=0

FEM x=50

Shear x=0

Shear x=50

Figure 3.14: [0, 90]s laminate: Comparison of the analytical modelwith and without shear.

The previous results reveal that the shear term does not have anysignificant influence regarding the accuracy of INS evaluation basedon the mechanical model. The shear influence is investigated for anelliptical beam which is illustrated in Figure 3.15. It is clampledat one end and a normal force F = 1000N and a bending momentMz = 20000Nmm are applied at the other end. The bending moment


is adjusted so that only a shear force Q = 1000N (internal force) actsat point A. The semi-major axes of the midplane are 100 and 20mmand the position of point A is x = 100 and y = 0mm.

FMz

x

y

A

t=10mm

Figure 3.15: Geometry of the 10mm thick ellipse and loads.

The INS distribution is obtained from a 8-node plane FEM with 40elements in thickness direction where the material properties are givenin Appendix A.2. The FE results confirm that the INS is close tozero (−0.01N/mm2) and is not affected by the shear force. Onlythe derivatives of the shear terms appear in the radial equilibriumcondition 3.18 which become zero in the present evaluation at pointA.

Moreover, the derivative of the shear force is small in the singlycurved geometry at region III because the shear distribution appearssmoothly and the INS is therefore affected moderately.

3.5.1 Conclusions

The new analytical models are based on an ESL theory, whereonly the laminate deformations of the centerline are known, andare inspired by the exact solution of the thick-walled compositetube problem. A differential equation of the through-the-thicknessdisplacement w is derived from the radial equilibrium equation.The approach is based on cylindrical bending and that the crosssection remains straight and is linked with a beam FEM where thecurvature effect is included. The agreement between the results ofthe analytical models and those of the FE simulation depends on the


stacking sequence of the laminate. The distributions of the INS sharethe same features and appear generally very similar. The values atmaximum stress location differ by between 0.7% and 25% whereas theanalytical model underestimates the maximum stress. The enhancedmodel with the FSDT approach improves the accuracy only slightly.

The curvature radius is coupled with the beam FEM which leads toan abrupt change of the curvature in region I from infinite to a smallcurvature-radius-to-thickness ratio R

t = 1.69. Figure 3.16 shows theeffect of the INS distribution at the centerline in the UD laminatealong the length direction x. The maximum INS jumps from zero tothe first maximum in region I unlike the INS distribution in regionIII. The curvature radius distribution along the centerline is given inEquation 3.30. The INS, obtained from the analytical model, becomezero in the straight parts due to infinite curvature radius R. At x =−50mm the curvature radius jumps from infinite to −16.9mm andthe INS increases abruptly. The changing of the curvature radius inthe curved part is continuously differentiable and therefore the changeof the INS along the centerline is much smoother.

The plane FEM simulates the transition at x = −50mm muchsmoother because the thickness deformation is represented by thenodal deformations which is continuous.

−100 −75 −50 −25 0

0

10

20

30

40

50

60

70

80

x [mm]

INS

[N/m

m2]

Beam modelPlane model

Figure 3.16: INS distribution of the centerline along length directionx.


This model gives good results considering its simplicity and efficiency.The run-time ratio between the plane model and the mechanical modelis about 130. These facts support that this approach is used for an INSmodel in doubly curved laminates which is presented in Chapter 4.

Chapter 4

Interlaminar normalstresses in doublycurved laminates

This chapter contains the derivation of the INS evaluation for dou-bly curved laminates. A detailed description of the analytical modelis given in Section 4.1 and a model assessment is presented in Sec-tion 4.2.1

4.1 Analytical model

4.1.1 Derivation

An arbitrary doubly curved shell panel cannot be described with aspherical CS and therefore a modified cylindrical CS approach isused [107]. The curved shell geometry, illustrated in Figure 4.1, isdescribed by the coordinates (r, ϕ, ϑ) and it is subdivided into angu-lar segments with the apex angles dϕ and dϑ and constant curvatureradii of the centerline R1 and R2. The sign convention for both cur-vature radii is described in Section 3.1. Finally, the radial equilibriumequation becomes

1This chapter is based on the paper A post-processing method for interlaminar

normal stresses in doubly curved laminates [106].

52 Interlaminar normal stresses in doubly curved laminates

t

R2R1

rd

dϕ

dϑ

z

R

Figure 4.1: Doubly curved FE geometry.

σr,r +1

rσϕr,ϕ +

1

r + rdσϑr,ϑ +

σr − σϕ

r+

σr − σϑ

r + rd= 0 (4.1)

where r = R1 + z, rd = R2 − R1 and z = [−t/2, t/2]. Each segmentis embedded in between four cross-sections which are still assumedto remain straight and perpendicular to the midplane. Based on thegood results of the singly curved analytical model and the small effectof the shear terms in UD and cross-ply laminates (Section 3.5), theshear terms are neglected. Equation 4.1 is reduced to

σr,r +σr − σϕ

r+

σr − σϑ

r + rd= 0 (4.2)

where only direct stresses appear and the material law reduces to

σϕ

σϑ

σr

=

C11 C12 C13

C21 C22 C23

C31 C32 C33

εϕ − εF

ϕ

εϑ − εF

ϑ

εr − εF

r

. (4.3)

The goal is to expressed the direct strains in Equation 4.3 through thedisplacements u, v, and w. The kinematical relations in the modifiedCS are

4.1 analytical model 53

εϕ = 1r (w + u,ϕ)

εϑ = 1r+rd

(w + v,ϑ)

εr = w,r .

(4.4)

The in-plane deformations u and v are expressed by the laminate de-formations ε0 and κ analog Equation 3.14. The through-the-thicknesscoordinate z is replaced with the radial coordinate r and the curvatureradii of the midplane.

du = R1 · dϕ(

ε0

ϕ + (r − R1)κϕ

)

dv = R2 · dϑ (ε0

ϑ + (r − R2 + rd)κϑ)(4.5)

The direct strains become

εϕ = 1r

(

w + R1

(

ε0

ϕ + (r − R1)κϕ

))

εϑ = 1r+rd

(w + R2 (ε0

ϑ + (r − R2 + rd)κϑ))

εr = w,r .

(4.6)

The combination of the material law (Equation 4.3) with the kine-matical relations 4.4 leads to the direct stresses expressed by the de-formations.

σϕ = C11

r

(

w + R1

(

ε0

ϕ + (r − R1)κϕ

))

+

+ C12

r+rd(w + R2 (ε0

ϑ + (r − R2 + rd)κϑ))+

+C13 · w,r − C1i · εF

i

σϑ = C21

r

(

w + R1

(

ε0

ϕ + (r − R1)κϕ

))

+

+ C22

r+rd(w + R2 (ε0

ϑ + (r − R2 + rd)κϑ))+

+C23 · w,r − C2i · εF

i

σr = C31

r

(

w + R1

(

ε0

ϕ + (r − R1)κϕ

))

+

+ C32

r+rd(w + R2 (ε0

ϑ + (r − R2 + rd)κϑ))+

+C33 · w,r − C3i · εF

i

(4.7)

Equation 4.7 is combined with the radial equilibrium Equation 4.2 andthe differential equation of the through-the-thickness displacement is


0 = w,rr + w,r

(

1r + 1

r+rd

)

+

+w(

C13+C23−2C12

r(r+rd) − C11

r2 − C22

(r+rd)2

)

+ P ∗

(4.8)

where Cij =Cij

C33and P ∗ is

P ∗ = u,ϕ

(

C13−C12

r(r+rd) − C11

r2

)

+ u,ϕrC13

r +

+v,ϑ

(

C23−C12

r(r+rd) − C22

(r+rd)2

)

+ v,ϑrC23

r+rd+

+εF

i

(

C1i−C3i

r + C2i−C3i

r+rd

)

. (4.9)

The expressions σr,r, σr − σϕ, σr − σϑ, and the derivatives of thedisplacements are given in the Appendix C.1.

4.1.2 Numerical solution

The solution of the differential Equation 4.8 is found with the finite-differences method [108]. The differential equation represents a linearsecond order boundary value problem.

w,rr + p(r) · w,r + q(r) · w(r) = g(r) (4.10)

where the derivatives are replaced with

w,r = wn+1−wn−1

2d

w,rr = wn+1−2wn+wk−1

d2

(4.11)

whereas wn are the displacements at the supporting points throughthe thickness and d is the distance between two consecutive supportingpoints. Their placement scheme for a single-layer laminate is shown onthe left of Figure 4.2. The BCs lead to a non-singular linear systemof equations (LSoE) and are represented by the INS which have tovanish at the top and bottom surfaces of the laminate.

The through-the-thickness INS distribution is obtained by com-bining Equations 4.3 and 4.4.

σr =C13

r(u,ϕ + w) +

C23

r + rd(v,ϑ + w) + C33 · w,r (4.12)


This equation can be transformed to

w,r(r) + α · w(r) = β (4.13)

and is integrated in the LSoE whereas α and β modify the first andthe last row of the left and right side of the LSoE, respectively.

w1 w1

wIP+1

wn−1

wn

wn+1

w7

w8

w15

w16

layer 1

layer 2

layer 3

σ1 = 0

σ3 = 0

σ1 = σ2w1 = w2

σ2 = σ3w2 = w3

Figure 4.2: Integration scheme in a single layer (left) and multilayerlaminate (right) where the indices 1, 2, and 3 count the layers and σmeans σr.

Every additional layer leads to two more interface-continuity condi-tions (Equation 3.28) which have to be fullfilled. The first deriva-tive of the through-the-thickness displacements w is found in the INSEquation 4.12. This causes that additional supporting points, whichare placed outside the layer, are necessary to evaluate the INS atthe layer intersections. An integration scheme for a three layer lami-nate is plotted on the right of Figure 4.2 where the supporting pointsn = [7, 8, 15, 16] guarantee the through-the-thickness continuity of theINS.

The detailed LSoE and the components are described in the Ap-pendix C.2.

4.2 Model assessment

The doubly curved post-process model is compared with fully 3-D so-lutions obtained from a 20-node brick FEM (solid186 ANSYS 10.0)


for different geometries, lay-ups, and materials. The laminate defor-mations of the CLT, described in Section 3.2.1, are obtained from a8-node shell FEM (shell91 ANSYS 10.0).

4.2.1 Evenly curved plate

The doubly curved specimen is based on the singly curved one de-scribed in Section 3.3. The centerline is described by a cosine functionwith an amplitude h = 30mm and total length 2l = 200mm whichis rotated around the point of symmetry. The cross-section of thespecimen is plotted in Figure 4.3.

A tension load is applied by a 2mm radial displacement atthe outer edge where all other DoFs are fixed. The through-the-thickness INS distribution is compared at the center of the specimen(x = y = 0).

uu

t

hx

y

z

l

Figure 4.3: Cross-section of the doubly curved shell.

An accuracy and convergence check of the INS distribution obtainedfrom the solid FEM with 20 elements through the thickness pointsout that a mesh refinement (Section 2.3 and 5.3.1) leads to changes ofthe INS distribution which are less than 1%. The meshes of the solidand shell FEM, which have the same size, are plotted in Figure 4.4.Symmetry planes are used to reduce the model size and computationalcost.

Isotropic material

Aluminum alloy is used as isotropic material with the material prop-erties given in Table 4.1. Figure 4.5 shows the through-the-thickness


Figure 4.4: Solid (left) and shell (right) FE mesh.

INS distribution in the center of the 10mm thick specimen obtainedfrom the solid FEM and the new model. The distributions do notexactly share the same features. The distribution obtained from thesolid FEM has the lower slope at the inner surface z = −5mm andthe maximum is closer to the outer surface which is 5.3% highercompared with the analytical model. The maximum INS in thecenter is compared for different thicknesses t between 5 and 20mm.The results are shown in Figure 4.6. The accuracy of the analyticalmodel is quite good. The difference between both methods is 5.5% inthe 5mm thick specimen and decreases to 3.3% for the 20mm thickone whereas the new model underestimates the INS.

Table 4.1: Aluminum alloy material properties.

E = 70000 [MPa]G = 26923.1 [MPa]ν = 0.3 [–]

The only exceptional case is the specimen with thickness 7.5mm. Thethrough-the-thickness INS distribution obtained from the solid FEMhas a change of sign what is not represented by the new model andis illustrated in Figure 4.7. Hence, the deviation between the maxi-mum INS increases in this case where the absolute values are smallcompared with the other maxima. The total normal forces (integral


−5 −2.5 0 2.5 5

0

5

10

15

20

25

z [mm]

INS

[N/m

m2]

PresentFEM

Figure 4.5: Through-the-thickness INS distribution of the 10mm thickisotropic specimen.

of the INS through the thickness) of the analytical and FEM are -3.4and -3.2 N , respectively.

Anisotropic material

The same comparisons are done for a UD and [0, 90]s cross-plylaminate with a fiber reinforced material where the fiber referencedirection (0-direction) is parallel to the global x-direction. Thesymmetry of the structures reduces from rotation-symmetric to amodel with two symmetry axes which are parallel to the global x- andy-directions due to the fiber orientation. The materials properties aregiven in Table 4.2. Again, the results of both models are comparedat the center (x = y = 0).

The global INS distribution of an 10mm thick UD laminate is illustra-ted in Figure 4.8 where the layer with the maximum INS is shown.High INS appear at the outer edge where the fiber orientation is pa-rallel to the applied displacement and in the symmetry plane x = 0.An INS distribution symmetrical to the x- and y-planes is expectedwhat is quite well represented by the analytical model.


5 7.5 10 12.5 15 17.5 20

0

50

100

150

200

Thickness t [mm]

Max

. IN

S [N

/mm

2]

PresentFEM

Figure 4.6: Maximum INS in the center of the isotropic specimen.

Table 4.2: UD material properties.

E1 = 140000 E2,3 = 10000 [MPa]G23 = 3350 G13,12 = 6000 [MPa]ν23 = 0.49 ν13,12 = 0.3 [–]

The left graph in Figure 4.9 shows the maximum INS obtainedfrom the solid FEM and the analytical model for different laminatethicknesses. The results agree well for all thicknesses whereas theerror of the new model increases with increasing laminate thickness t.It is 6.4% in in the 5mm thick model and has a maximum of 15.6% inthe 15mm thick specimen. In these evaluations, the analytical modelunderestimates the INS.

The graph on the right of Figure 4.9 illustrates the assessment ofthe present model for an [0, 90]s laminate. The laminate thicknessis either 5, 10, 15, or 20mm. Once again, the analytical model un-derestimates the maximum INS in the center. The error is 14.5% inthe 5mm thick laminate and decreases to 7.2% in the 20mm thick one.


−4 −2 0 2 4−5

−4

−3

−2

−1

0

1

2

3

Thickness z [mm]

INS

[N/m

m2]

PresentFEM

Figure 4.7: Through-the-thickness INS distribution of the 7.5mmthick specimen.

INS [N/mm2]

40

35

30

25

20

15

10

5

0

X

Y

Z

Figure 4.8: Global INS distribution in a UD 10mm thick laminate atthe thickness-direction level where the maximum INS appears.

The agreement of the analytical model with accurate solid FEM isquite good either for isotropic or orthotropic materials and UD orcross-ply laminates. In the isotropic structure, the thickness of thespecimen does not have any significant influence on the accuracy.The influence increases in the orthotropic specimen where the errorincreases in the UD structure and decreases in the cross-ply laminate.The present results differ compared with those presented in [104].The difference is caused by an improved curvature radius evaluation.Previously it was assumed that the FEs map a sphere what leads


to a different curvature radius evaluation which is based on the FEmesh. The new approach of a general parabolic surface maps thereality better and therefore leads to better results. More details arepresented in Section 5.2.

5 7.5 10 12.5 15 17.5 200

20

40

60

80

100

120

140

Laminate thickness t [mm]

INS

[N/m

m2]

Presentsolid FEM

5 10 15 20

0

20

40

60

80

100

120

Laminate thickness t [mm]

INS

[N/m

m2]

Presentsolid FEM

Figure 4.9: Maximum INS in the UD laminate (left) and the cross-plylaminate (right).

A more general model assessment and limits of the doubly curvedmodel are presented in Chapter 5.

Chapter 5

Model implementationand evaluation

The post-processing method for calculating INS in doubly curvedlaminates, described in Chapter 4, is implemented with commercialavailable linear and quadratic shell FEs described in Section 5.1. Themodel agreement is evaluated for different shell FE types, materialproperties, curvature-radius-to-thickness ratios R

t (CT-ratio), stack-ing sequences, and mesh qualities.

The INS are evaluated only at the element center (ξ = η = 0) andare constant in the element in-plane directions. Therefore, the lami-nate deformations (ε and κ) and the curvature radii are also computedat the center.

5.1 Shell finite elements

An in-house FE tool has been developed at the Centre of StructureTechnologies in the past years. The basic concept of this FE tool,called Finite Element Library eXperiment (FELyX), is described byKonig [109], Wintermantel [110], and Giger [111]. FELyX supportslinear and quadratic shell FEs whose results (deformations, strains,and stresses) agree with the results obtained from ANSYS 10.0 andwhich are named shell181 and shell91 in the ANSYS user manual.FELyX first evaluates the deformation solution and then it uses thepost-processing results to compute the INS.

64 Model implementation and evaluation

5.1.1 Linear shell FEs

Linear quadrilateral shell FEs are mostly used in industrial sim-ulations of shell structures. First of all, the number of DoFs issmaller than in quadratic FEMs and secondly it is easier to mesha free-form surface because the curvature-to-thickness ratio has notto be considered. The new model is implemented for 3- and 4-nodeshell elements although linear 3-node elements may give inaccuratedisplacement, strain, and stress results.

The present 3-node element uses linear displacement shape functionsand the constant strain and stress state will often model structuralstiffness far too high. This effect can be reduced by a global meshrefinement. Nevertheless, whenever possible linear 3-node shell FEsshould not be used for structural FE analysis.

The 3-node element implemented in FELyX is described by Er-tas [57, 112], Cook [51], and Jeyachandrabose [113]. It has threedisplacement and three rotational DoFs at each node. The shapefunctions are described in area coordinates [51] and are

Φ1 = L1 = 1 − ξ − ηΦ2 = L2 = ξΦ3 = L3 = η .

(5.1)

The 4-node shell FE, which is implemented in FELyX, is described byCook [51]. It has 3 displacement and 2 rotational DoFs at each node.To avoid singular stiffness matrices, if the element CS is parallel tothe global CS, a virtual in-plane rotational stiffness term is added tothe element stiffness matrix K [114]. The element has bi-linear shapefunctions

Φ1 = 0.25(1 − ξ)(1 − η)Φ2 = 0.25(1 + ξ)(1 − η)Φ3 = 0.25(1 + ξ)(1 + η)Φ4 = 0.25(1 − ξ)(1 + η)

(5.2)

and therefore linear in-plane strain and stress distributions. The in-plane stiffness matrix K is evaluated by a 2 × 2 integration scheme.This element description causes in-plane shear stresses if it is loaded byan in-plane bending moment Mz (parallel to the thickness direction)what does not correspond to the real stress state. This can be avoided

5.1 Shell finite elements 65

by adding two extra shape functions which are used to describe in-plane bending. Taylor [115] and Piltner [116] presented a quadrilateralelement with two enhanced strain modes

Φ5 = (1 − J2

J0η)(1 − ξ2) + J1

J0ξ(1 − η2)

Φ6 = (1 − J1

J0ξ)(1 − η2) + J2

J0η(1 − ξ2)

(5.3)

where J0, J1, and J2 are terms of the in-plane Jacobian matrix de-scribing the coordinate transformation from the element CS to theglobal CS. The out-of-plane shear stiffness matrix K is evaluated byan 1-point integration scheme to avoid shear locking. The constantthrough-the-thickness shear-angle distribution overestimates the shearstiffness. Therefore, the out-of-plane shear stiffnesses C44, C55, andC45,54 (Equation 2.2) are multiplied by the shear correction factorsk44 and k55.

C∗

44 = k44 · C44

C∗

55 = k55 · C55

C∗

45,54 =√

k44 ·√

k55 · C45,54

(5.4)

The shear correction factors become 5/6 in an isotropic or UDlaminate analog to beams with rectangular cross-sections [13]. Anevaluation method of the shear correction factors is presented byIsaksson et al. [14] and implemented in FELyX.

As an example, the deformation solutions of a 10×10×1mm aluminumalloy plate meshed with 3- and 4-node shell FEs are compared. Themesh size is 1mm. The plate is clamped at one side and either anin-plane shear force Qy, out-of-plane shear force Qz, or a bendingmoment My about the y-axis is applied as Figure 5.1 illustrates. Themaximum deformations, obtained from each element type, are listedin Table 5.1.

a

b xyz

Qy

Qz

My

Figure 5.1: Loads and 10x10x1mm aluminum alloy plate.


Table 5.1: Maximum deformations (mm) obtained from a 3- and 4-node shell FEM with a reduced (RI) and full integration (FI) scheme.

Load 3-node 4-node RI 4-node FIQy = 10N 1.16E-3 1.098E-3 1.058E-3Qz = 1N 5.466E-3 5.455E-3 5.452E-3My = 1Nmm -8.337E-3 -8.3372E-3 -8.3372E-3

The deformation solutions, caused by in-plane shear, differ by 7.5%where the 3-node element underestimates the deflection. The bestsolution is obtained from the 4-node element with the full integrationscheme. The difference between this solution and the 3-node elementis 4.0%.

The deflection of the plate loaded by a out-of-plane shear force Qz

is 5.452E-3mm. The solutions obtained from the FEMs differ onlyby 0.2%. All three FEMs compute equal deformations if the plate isloaded by a bending moment My.

FE analysis is more sensitive regarding to the strains and stresses(post-processing results) because the secondary solution convergesslower than the primary solution. The best deformations, strains,and stresses are obtained from the 4-node element with full integrationwhere the post-processing results are not constant over the element.

5.1.2 Quadratic shell FEs

Quadratic elements have two basic disadvantages: the numerical effortincreases due to the DoFs and the meshing of a free-form surfaceis more complex because the thickness-to-curvature ratio has to beconsidered. Recently improvements in computing power, memory,and meshing algorithms make these elements more useful. The gainof higher shape-function approaches are better displacement, strain,and stress results. In addition, curved surfaces are mapped better(see Section 5.2 and 5.3.1) because the shape functions are also usedto describe the element geometry (isoparametric elements).

The element stiffness matrices K of the 6-node and 8-node shellFEs are evaluated with 3-point and 2 × 2 integration schemes, res-pectively. The element descriptions are presented by Cook [51]. The

5.2 Curvature radius evaluation 67

quadratic shape functions of the 6-node and 8-node element are givenin Equation 5.5 and 5.6, respectively.

Φ1 = L1(2L1 − 1)Φ2 = L2(2L2 − 1)Φ3 = L3(2L3 − 1)Φ4 = 4L1 · L2

Φ5 = 4L2 · L3

Φ6 = 4L3 · L1

(5.5)

where L1 = 1 − ξ − η, L2 = ξ, and L3 = η.

Φ1 = 0.25(1 − ξ)(1 − η)(−ξ − η − 1)Φ2 = 0.25(1 + ξ)(1 − η)(ξ − η − 1)Φ3 = 0.25(1 + ξ)(1 + η)(ξ + η − 1)Φ4 = 0.25(1 − ξ)(1 + η)(−ξ + η − 1)Φ5 = 0.5(1 − ξ2)(1 − η)Φ6 = 0.5(1 + ξ)(1 − η2)Φ7 = 0.5(1 − ξ2)(1 + η)Φ8 = 0.5(1 − ξ)(1 − η2)

(5.6)

The out-of-plane stiffness C44, C55, and C45,54 are multiplied by shearcorrection factors. Both quadratic shell FEs have 3 displacement and2 rotational DoFs at each node. Again, a virtual in-plane bendingstiffness term is added to the element stiffness matrix K to avoidsingularity.

The quadratic triangular element is more accurate regarding thein-plane shear deformation and therefore more suitable for accurateFE analysis than the linear one.

5.2 Curvature radius evaluation

All data, which are necessary for the new method, are stored in aFEM except for the curvature radii. These are therefore evaluateddirectly or indirectly (adjacent elements) depending on the FE shapefunction approach.

The curvature radius R is defined as

R =(1 + g

′2)1.5

g′′(5.7)


where g describes a surface and ()′

denotes its derivative with re-spect to the local 1- and 2-direction of the reference CS. It is assumedthat the reference surface of a shell FE follows a parabolic functiong(ξ, η) [117]

z = g(ξ, η) = a · ξ + b · η + 2c · ξ · η + d · ξ2 + e · η2 (5.8)

where ξ and η are the first and second direction of the reference ele-ment CS. This function can be expressed in matrix form

z = g(ξ, η) = [ξ, η]

[

ab

]

+ [ξ, η]

[

d cc e

] [

ξη

]

. (5.9)

The constant terms in the polynomial vanish because the origin ofthe reference CS is at the element center (ξ = η = 0) and lies onthe approximated surface. The curvature radii R1 and R2 in theprincipal strain directions are obtained from a matrix rotation. Thetransformation is described with a rotation matrix T

T =

[

cos(α) sin(α)−sin(α) cos(α)

]

(5.10)

where α is the angle between the local ξ direction and the principalstrain direction where the in-plane shear strain is zero. The Equa-tion 5.9 becomes

z = g(ξ, η) = [ξ, η] T

[

ab

]

+ [ξ, η] T T

[

d cc e

]

T

[

ξη

]

= [ξ, η]

[

a∗

b∗

]

+ [ξ, η]

[

d∗ c∗

c∗ e∗

] [

ξη

] . (5.11)

Hence, the curvature radii R1 and R2 at the element center ξ = η = 0are

R1 =(1+a∗2)1.5

d∗

R2 =(1+b∗2)

1.5

e∗

(5.12)

and R1 or R2 become infinite if d∗ or e∗ are zero, respectively.

5.3 General model assessment 69

Five conditions have to be defined for evaluating the parametersin Equation 5.8. In an isoparametric FE, the shape functions areused to describe the geometry of the element and the displacementfield. Therefore, a quadratic element follows a parabolic surface. Toreduce numerical cost, only five nodes are used for the parameterevaluation. In the present results, the first three corner nodes and thefirst and second mid-nodes define the parabolic surface. Figure 5.2illustrates the corner and mid-nodes of a 6- and 8-node element.In both elements the mentioned corner nodes are n1,2,3 where themid-nodes are n4,5 and n5,6 in the 6- and 8-node element, respectively.

n1n1

n2n2

n3n3n4

n4

n5

n5

n6

n6

n7

n8

ξξηη

Figure 5.2: Corner and mid-nodes of a 8- and 6-node element.

The shape functions of 4-node shell FEs map bi-linear surfaces. Hence,the surface is approximated using adjacent elements. Adjacent ele-ments share an edge with the reference element where an edge isdefined by two consecutive corner nodes. The parameters of the sur-face equation are defined by the element centers of all adjacent ele-ments. The inaccuracy of this surface approximation is illustrated inFigure 5.3 and it is obvious that the curvature radii are overestimatedin this example.

The peak of the parabolic surface is not represented by the elementsbecause none of the nodes, which are placed on the surface, lies nearthe zenith ([x, y, z] = [0, 0, 0]). The equation of the above illustratedsurface is g(x, y) = x2 + y2. In this case, the surface approximation,based on 4-node elements, leads to the surface equation g∗(x, y) =23

(

x2 + y2)

and the curvature radii at the element center become 1.5.This is 50% higher if compared with the curvature radii of the originalsurface. The overestimated curvature radii would lead to smaller INSin this region.

This problem can be reduced by a local mesh refinement. Never-theless, linear shell FEs often lead to inaccurate INS results.


−1 −0.5 0 0.5 1

−0.50

0.510

0.5

1

1.5

2

xy

z

Reference element

Adjacent element

Figure 5.3: Approximation of a parabolic surface based on 4-nodeelements.

5.3 General model assessment

5.3.1 Element size and type

The accurate mapping of the curvature radii is a basic issue foraccurate results of the INS obtained from the new model. It dependson the element size and type and is discussed for a singly and doublycurved structure where element size refers to the length of the elementedges.

The singly curved structure, illustrated in Figure 3.5, is loaded intension. The midplane is meshed with either 3-, 4- or 8-node shellFEs with a size of 3 or 6mm. Figure 5.4 illustrates the global INSdistribution for each configuration. The best result is obtained fromthe 8-node 3mm mesh which represents the reference solution.

The maximum INS, obtained from the 4-node 3mm mesh, differsonly by -0.8% and the global INS distributions are equal. In contrast,the 3-node mesh overestimates the INS in the curved regions. A3-node mesh causes, on a singly-curved surface, a zic-zac pattern inthe original straight y-direction. This zic-zac pattern of the meshleads to inaccurate curvature radii and therefore to inaccurate INS asillustrated in Figure 5.5. In this case, the curvature radii are smallercompared with those of the original surface and the INS increase.


X

YZ

INS [N/mm2]

70

60

50

40

30

20

10

0

8-node

6mm

70.7

Element type:

Element size:

Max. INS [N/mm2]

4-node

6mm

68.9

8-node

3mm

71.9

4-node

3mm

71.3

3-node

3mm

108.5

Figure 5.4: Influence of the element type and size regarding the globalINS distribution (nodal results). Layer where the maximum INS ap-pears.

XY

Z

A

A

Z

Y

A - A

Figure 5.5: A 3-node mesh on a singly curved shell causes a zic-zacpattern in the original straight y-direction.

The influence of the element size is shown in the two left plots of Fi-gure 5.4. The results of the 8-node 3 and 6mm mesh differ only by−1.7% at the maximum stress location and the global stress distribu-tions agree well. The accuracy of the INS results, based on linear shellFEM, is more sensitive to the element size. The INS at the maximumstress location differs by -4.2% and -3.4% compared with the 8-node3mm and 4-node 3mm FE results, respectively.

The linear quadrilateral shell FEs provide good results in a singlycurved structure whereas a quadratic element type is more robust


regarding the element size. The accuracy of both element typesincreases with decreasing element size.

The INS distributions based on a 4- and 8-node mesh in the doublycurved structure, illustrated in Figure 4.3, are compared. The surfacecan not be meshed with rectangular elements due to the complex geo-metry. The mesh is irregular and a few 3-node elements are includedin the linear FE mesh. Nevertheless, the element size is adjustedhaving a regular mesh where only small element distorsions appear.

X

Y

Z

INS [N/mm2]

45

40

35

30

25

20

15

10

5

8-node

4-node

4-node

Figure 5.6: Influence of the element type and size regarding the globalINS distribution in a doubly curved geometry (nodal solution).

A radial displacement is applied at the outer edge where all otherDoFs are suppressed. The laminate has a UD stacking sequence andthe fiber direction is parallel to the global x-direction. The two sym-metry axes of the INS distribution, plotted on the left of Figure 5.6,are well represented in the 8-node shell FE solution. The maximumINS at the specimen center (x = y = 0) is 46.9 N/mm2.

Contrary to the results of the 4-node shell FE in the singly curvedstructure, those in the doubly curved structure differ much moreif compared with the quadratic shell FEM. The symmetry of thestress distribution is hardly fullfilled and the maximum INS at thecenter is underestimated by 67.2%. A local mesh refinement at thecenter is applied to have a better mapping of the geometry. The im-provement is small compared with the additional pre-processing and


computational costs. The maximum INS at the center increases to20.8 N/mm2 but still differs by -55.7% compared with the quadraticFEM. The inaccurate mapping of the curvature radii, illustrated inFigure 5.3, has an increasing influence in doubly curved shells and theconsequential INS. A global mesh refinement leads to better resultsbut can not compensate this disadvantage.

Quadrilateral linear shell FEs can give good results in singly curvedstructures and the accuracy can be improved by a local mesh refine-ment in the curved regions (see Figure 5.4). The results in a doublycurved structure are inaccurate and not recommended for an INSevaluation. Shell FEs with quadratic shape functions are able to mapparabolic surfaces what explains the accurate results in singly anddoubly curved structures. In addition, they are robust regarding theelement size. 3-node shell FEs give inaccurate results in singly as wellas doubly curved shells.

5.3.2 Singly curved geometry

The results of the doubly curved analytical model, described in Sec-tion 4.1.1, are compared with 3-D solid FE simulations for differentmaterials, geometries, and stacking sequences. The 8-node shell FE(shell91 ANSYS 10.0) is used due to the accurate results presented inthe previous section. The volume models are meshed with a 20-nodesolid FE (solid186 ANSYS 10.0).

The first evaluation aims to investigate the geometry influence regard-ing the accuracy of the analytical model. The global maximum INS inthe singly curved structure, illustrated in Figure 5.7, appears at thecenter x = 0mm where the geometry can be described by a CT-ratioRt . The midplane of the curved part follows a cosine-function

f(x) =h

2

[

cos

(

2π · xl

)

+ 1

]

(5.13)

and the curvature radius at the center x = 0 becomes

R =2 · l2

4 · h · π2. (5.14)

Different CT-ratios can be represented by varying the amplitude hand thickness t of the specimen, illustrated in Figure 5.7, where


t = [1, 16.9]mm and h = [1, 60]mm. The geometries on the left ofFigure 5.8 have an amplitude h = 1mm and the thickness t is 1mmand 16.9mm, respectively. Increasing the amplitude h to 60mm gener-ates the geometries on the right side. Therefore, all CT-ratios between507 (flat plate) and 0.5 (cylinder) can be modelled.

50mm 50mm

l=100mm

h

t

x

Figure 5.7: Singly curved specimen with the parameters amplitude hand thickness t.

116.9

507 8.45

Thic

knes

st

[mm

]

Curvature radius [mm]

R/t=507

R/t=0.5R/t=30

R/t=8.45

Figure 5.8: Feasible geometries for different parameter values.

Aluminum alloy

The solid FEM is meshed with 20 elements through the thickness andthe INS distribution in the cross-section, where the highest INS ap-pear, is shown in Figure 5.9. The maximum INS in this cross-sectionobtained from the solid FE analysis and the new post-processingmethod, described in Section 4.1, are compared for all different


CT-ratios. Aluminum alloy is used for the first evaluation and thematerial properties are given in the Appendix A.4. A displacement of1.5mm is applied on the right side and the left side is clamped whatinduces positive INS.

Figure 5.9: FE mesh and INS distribution in the cross-section of in-terest (element solution).

The maximum INS, obtained from the solid FEM and post-processingmethod, are illustrated in Figure 5.10 where the x-axis represents thecurvature radius R and the y-axis the thickness t. These contour plotsshow the maximum INS at the previous mentioned cross-section foreach feasible combination of t and h (CT-ratio).

The contour plots share the same features. The absolute maximumINS (CT-ratio=0.5) is similar and the contour lines have the sameappearance. The maximum INS increases with decreasing CT-ratio.

The percent difference between the solid and the analytical modelis plotted in Figure 5.11. The new model gives accurate results forall CT-ratios between 1 and 50. The difference is between 5 and 10%where the analytical model underestimates the INS. The differenceincreases in geometries with a high curvature radius R > 50. Thepost-processing method overestimates the maximum INS in geome-tries with a CT-ratio between 0.5 and 0.75 where the results are stillgood. It has to be mentioned that the laminate deformations ε andκ, which are obtained from the shell model, are more accurate forshell structures with smaller CT-ratios because this shell FE (shell91ANSYS 10.0) is not recommended for thick curved structures.


Curvature radius R [mm]

Th

ickn

ess

t [

mm

]

507 50.7 25.3 16.9 12.7 10.1 8.451

5.0

8.95

12.9

16.9

Curvature radius R [mm]T

hic

kne

ss t

[m

m]

507 50.7 25.3 16.9 12.7 10.1 8.451

5.0

8.95

12.9

16.9

50

100

150

200

250

Figure 5.10: Maximum INS [N/mm2] in a isotropic structure for dif-ferent CT-ratios. Solid FE results (left) and new model (right).

UD laminate

The same evaluations are repeated for a UD laminate. The Young’smoduli in fiber and transverse direction are E1 = 165′000MPa andE2,3 = 8′400MPa, respectively. The material properties are alsogiven in the Appendix A.3. The maximum INS for all CT-ratios ofboth models are illustrated in Figure 5.12. Contrary to the isotropicstructure, the contour plots do not share the same features. The solidFEM predicts the maximum INS in the structure with a CT-ratio of1.3. The new model has the maximum where the CT-ratio is 2.25 andthe maxima differ by 21.7%. The INS, obtained from the new model,decreases for a certain thickness t with decreasing curvature radius Rwhere this effect is smaller in the solid FEM.

The difference between both models is shown in Figure 5.13. Theaccuracy of the new model decreases with decreasing CT-ratio andit differs by −10% if the CT-ratio is between 3 and 50. The −30%contour line represents a CT-ratio of 1.3± 0.06. Again, the results ofthe specimens with a CT-ratio between 0.5 and 1 have to be treatedcarefully due to the inaccurate results of the shell FEM for thick-walled high curved structures. In addition, the agreement decreasesif the curvature radius is higher than 100.


−−

−− −

−−

25

−2

0−

20

−1

5−

15

−10

−1

0

−1

0

−5

−5

−5

−5

0

5


Th

ickn

ess

t [

mm

]

507 50.7 25.3 16.9 12.7 10.1 8.451

5.0

8.95

12.9

16.9

−50

−40

−30

−20

−10

0

Figure 5.11: Percent difference of the post-processing method inisotropic structures.

Curvature radius [mm]

Th

ickn

ess

t [

mm

]

507 50.7 25.3 16.9 12.7 10.1 8.451

5.0

8.95

12.9

16.9


Th

ickn

ess

t [

mm

]

507 50.7 25.3 16.9 12.7 10.1 8.451

5.0

8.95

12.9

16.9

10

20

30

40

50

60

70

80

90

100

110

Figure 5.12: Maximum INS [N/mm2] in a UD laminate for differentCT-ratios. Solid FE results (left) and new model (right).

Anisotropy

The model assessment in the singly curved structure shows that thematerial properties have a severe influence on the accuracy of the post-


−80

−70

−60

−50

−50

−4

0

−40

−40

−3

0−30

−30

−2

0

−20

−20−1

0−

10

−10

−10

−10


Th

ickn

ess

t [

mm

]

507 50.7 25.3 16.9 12.7 10.1 8.451

5.0

8.95

12.9

16.9

−80

−70

−60

−50

−40

−30

−20

−10

0

Figure 5.13: Percent difference of the post-processing method in UDlaminates.

processing method. Therefore, the influence of the material propertiesis investigated. The specimen, illustrated in Figure 5.7, has a CT-ratio of 1.69 (h = 30mm and t = 10mm) where the agreement ofthe new model is -19.5% (E1/E3 = 19.6). This specimen represents athick-walled curved laminate which suffers delamination as first failureloaded in tension (Chapter 6) and is used for the next evaluation.

An anisotropy factor is defined as reference and it makes senseto define it as E1

E3because a plane strain (εy = 0) situation can

be assumed at the maximum INS location. The properties of atransversal-isotropic material depend on the resin and the fiber prop-erties, and the fiber volume content. The fiber properties are given inTable 5.2 and the resin has a Poisson’s ratio of ν = 0.35. The laminaproperties are evaluated using Chamis micromechanics equations,presented in [118], where the resin Young’s modulus is between 4’000and 45′000MPa and the fiber volume content is between 0 and 90%.

The left contour plot of Figure 5.14 illustrates the distribution of theanisotropy factor for all feasible variations where the x-axis repre-sents the fiber volume content and the y-axis the resin Young’s mod-ulus. The lamina has isotropic material properties if the fiber volume


Table 5.2: Fiber material properties/

E1 = 276000 E2 = 19000 [MPa]G12 = 27000 G23 = 7000 [MPa]ν12 = 0.2 [–]

content is zero and the anisotropy increases for increasing fiber vol-ume content and decreases with increasing resin stiffness (E). Theanisotropy factor is between 1 and 16.5 and the accordant agreementof the post-processing method is shown on the right side of the samefigure.

The correlation between the material anisotropy and the accuracyof the model, compared with the solid FEM, is obvious. The bestresults are obtained for isotropic material properties and the agree-ment decreases with increasing anisotropy. The difference is −4.2%and −19.5% for E1

E3 = 1 and E1

E3 = 16.5, respectively. The correla-tion between the anisotropy and the accuracy, in this singly curvedstructure loaded in tension, is close to Difference = −4% − E1

E3 .

2

22

4

44

6

66

8

88

10

10

10

12

12

12

1416

Fiber volume content

Re

sin

[M

Pa

]

0 0.2 0.4 0.6 0.80.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

4

−18−16

−1

4

−14

−12

−1

2

−12

−1

0−

10

−10

−8

−8

−8

−6

−6

−6

−4

−4

−4

−4

−4

Fiber volume content

Re

sin

[M

Pa

]

0 0.2 0.4 0.6 0.80.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

4

−18

−16

−14

−12

−10

−8

−6

−4

2

4

6

8

10

12

14

16

Figure 5.14: Anisotropy distribution and percent difference of the newmodel.


Angle-ply laminates

So far, only UD and cross-ply laminates are investigated because theshear stresses are not considered in the mechanical model which donot appear in these laminates under cylindrical bending. In fact,laminates have often angle-plies to satisfy complex loads, strength,safety margins etc. The agreement of the present model for angle-plylaminates is evaluated for a 10mm thick laminate with a stackingsequence [0, α, β, 0] where each lamina is 2.5mm thick and α and βare between -90 and 90. The geometry, shown in Figure 5.7, andthe load are equal to the previous example. The lamina properties aregiven in Appendix A.3. Again, the maximum INS in the cross-section,plotted in Figure 5.9, as calculated by the analytical and solid FEmodels are compared with each other.

The model agreement is shown in Figure 5.15. The mechanical modelassumes cylindrical bending and considers only UD and cross-ply lam-inates. The accuracy decreases if the stacking sequence does not rep-resent neither a UD or cross-ply laminate. The best results are ob-tained in the [0, 0, 90, 0] or [0, 0,−90, 0] laminate where the differenceis -16%. The accuracy decreases by 5% if the second and third layerare interchanged. The difference is between -20% and -17% where αand β are close to zero.

The model accuracy decreases in the laminates where α is between±[45, 75] and β between ±[30, 90]. The model does not consider sheareffects which increase in angle-ply laminates with the mentionedstacking sequences. Especially the in-plane shear γxy has the sameorder of magnitude as the direct strains and it becomes smaller incross-ply laminates where also the model accuracy increases.

The previous model assessment shows that the accuracy correlateswith the stacking sequence. Angle-ply laminates causes in-plane shearwhich is not considered in the constitutive law (Equation 4.3) wherethe interlaminar shear terms are neglected in the radial equilibriumcondition (Equation 4.1). Both, the intra- and interlaminar shearterms affect the accuracy of the mechanical model.

The laminate deformations εx,y and κx,y do not describe sheardeformations. Assuming that the in-plane strains εx,y and γxy aretransformed in the direction where only shear appears, the consideredlaminate deformations can not be used to describe the stress state.


α [°]

β [

°]

−90 −60 −30 0 30 60 90−90

−60

−30

0

30

60

90

−25

−24

−23

−22

−21

−20

−19

−18

−17

−16

Figure 5.15: Percent difference of the new model for angle-ply lami-nates [0, α, β, 0].

Therefore, the difference between the post-processing method andsolid FE results increases in angle-ply laminates.

This mismatch can be corrected by rotating the reference CS pa-rallel to the principal strain direction. Unfortunately, the accuracydecreases in the present singly curved structure because also the cur-vature radii have to be transformed in the reference CS. The princi-pal strain and curvature directions are not equal in the singly curvedstructure made up of angle-ply laminates and the model accuracy de-creases. They coincide only in UD or cross-ply laminates. Therefore,the reference CS of the INS evaluation can be rotated either in theprincipal strain or curvature radius direction. In the singly curvedstructures, the reference CS is set to the principal curvature direc-tions.

5.3.3 Doubly curved geometry

The agreement of the analytical model in an evenly doubly curvedgeometry, discussed in Section 4.2, is good for different laminates andthicknesses where the curvature radii at the center are R1 = R2 =67.5mm.

The next model assessment investigates the correlation betweenboth curvature radii and the model accuracy. Only UD and cross-plylaminates are considered where the principal strain direction is parallel


to the curvature radius direction. The doubly curved geometry has anelliptical plan view with the two semi-major axes a and b illustratedin Figure 5.16. The length of these axes is either 50, 100, or 200mmand their z-coordinate is defined as

f1(x) = h2

[

cos(

2π·x2a

)

+ 1]

f2(y) = h2

[

cos(

2π·y2b

)

+ 1]

(5.15)

where the amplitude h is 30mm. A 10mm thick UD laminate is spe-cified with the lamina properties given in Table 4.2 where the fiberdirection is parallel to the global x-direction. A radial displacement(normal to the boundary) of 2% is applied at the outer edge where allother DoFs are suppressed. The fiber orientation f and the displace-ment u are also illustrated in Figure 5.16.

ab

u

u fx

y

Figure 5.16: Doubly curved geometry and plan view.

Again, the maximum INS at the center (x = y = 0), where the shearstrains are at least one order of magnitude smaller than the directones, is compared with the results obtained from a solid FEM and theagreement is listed in Table 5.3.

The accuracy of the analytical model is good for the first threegeometries where the curvature radius R1, which is parallel to thefiber orientation, is small. The difference increases moderately withincreasing curvature radius R2.

The results differ to a greater extend in the geometries where thecurvature radius in fiber direction is higher than that in matrix di-rection. The model underestimates the maximum INS by 49.5% andby 30.0% in the geometry R1,2 = [67.5, 16.9] and R1,2 = [270.2, 67.5],respectively. The agreement in the geometry R1,2 = [270.2, 16.9] is14.0% where the model overestimates the maximum INS.

The error of the analytical model is between -11.3% and 6.6% ifthe curvature radius in fiber direction is equal or smaller than that


Table 5.3: Doubly curved model assessment.

a b R1 R2 Solid Present Diff.[mm] [mm] [mm] [mm] [N/mm2] [N/mm2] %

50 50 16.9 16.9 89.5 84.9 -5.150 100 16.9 67.5 119.0 125.5 5.550 200 16.9 270.2 134.4 143.4 6.6

100 50 67.5 16.9 20.0 10.1 -49.5100 100 67.5 67.5 45.9 40.7 -11.3100 200 67.5 270.2 54.4 53.0 -2.6200 50 270.2 16.9 -10.9 -11.4 14.0200 100 270.2 67.5 7.0 4.9 -30.0200 200 270.2 270.2 14.6 13.3 -8.9

in matrix direction.

A correlation between curvature radii R1,2 and the model accuracycan be observed. The best results are obtained in the geometrieswhere R1 is equal or smaller than R2. Contrary to the previouslypresented results, the analytical model overestimates the maximumINS in certain geometries.

5.3.4 Conclusions

The analytical model for INS in doubly curved thick-walled laminates,presented in Chapter 4, is implemented for linear and quadratic shellFEs. Independent of the element size and type, the post-processingmethod points out the maximum INS regions. Good results areobtained in singly curved structures from quadratic or linear quadri-lateral shell FEs. Quadratic elements are more robust on the elementsize regarding the accuracy of the INS. In doubly curved structures,accurate results are only obtained from quadratic shell FEMs witha mesh refinement at the region of interest. Linear shell FEs leadto inaccurate curvature radii, especially in doubly curved structures,what correlates directly with the INS results.


The accuracy of the model is also evaluated for different materials,geometries, and stacking sequences. In the singly curved structure,the accuracy of the model shows a close correlation between materialproperties and CT-ratio. The model accuracy is good in singly curvedstructures with a moderate CT-ratio (1.7 - 50).

The correlation between the CT-ratio and accuracy increases inanisotropic structures. The agreement decreases with decreasing CT-ratio. Further, the accuracy correlates with the material anisotropywhich is defined as E1/E3 and is close to Difference = −4%−E1/E3.

The agreement of the new model decreases in angle-ply laminates.Angle-ply laminates cause intra- and interlaminar shear stresses whichare not considered. The best results are obtained if the principalstrain and curvature directions are equal.

In doubly curved structures made up of a UD laminate, good resultsare obtained in regions where the curvature radius in fiber directionis equal or smaller than that in transverse direction. In these ge-ometries, the new model agrees well with solid FE results and differsbetween by -11.3% and by 6.6%.

The model assessments point out that the error is lower than 20% if

• E1/E3 ≤ 19.6

• 1.7 ≤ CT-ratio ≤ 50.0

• Rfiber direction ≤ Rtransverse

• Principal strain and curvature directions are coincidental

5.4 Practical cases

The gain and risk of this post-processing method are shown with twoother structures. The effect of shear stresses, edge effects, stackingsequence, and mesh quality are discussed at regions with high INS.

5.4.1 Doubly curved L-Probe

The probe, illustrated on the top of Figure 5.17, can be partitionedinto four regions:

5.4 Practical cases 85

• Flat region where the loads are applied.

• Singly curved region with the curvature parallel to the fiberdirection.

• Doubly curved region where the global maximum INS appear.

• Intersection region which is curved in the transverse direction.

Five specimens were manufactured and tested where delamination oc-curred prior to in-plane failure. The delamination grew, under tensionload, from the doubly curved region I. The thickness of the [±45, 015]slaminate is 6.2mm and the global x-direction is parallel to the fiberdirection. In the present evaluation, the specimen is clamped at theleft side and a longitudinal displacement of 3.6mm is applied at theother side where all other DoFs are suppressed.

The reaction forces obtained from the shell and solid FEM differby 1.0% what ensures accurate laminate deformations obtained fromthe shell FEM. The through-the-thickness mesh size is seven 20-nodeelements for each weave layer and 24 for the UD laminate. Thisensures fullfilling the convergence and BC of the interlaminar stresses.The UD and weave material properties are given in Table 4.2 and 5.4,respectively.

Table 5.4: ±45 weave material properties.

E1,2 = 17664 E3 = 17664 [MPa]G12 = 35090 G13,23 = 4347.7 [MPa]ν12 = 0.766 ν13,23 = 0.077 [–]

The global INS distribution in the layer with the highest INS, ob-tained from the mechanical model, is illustrated on the bottom ofFigure 5.17. The local and global maxima of the INS are in region IIand I, respectively and are compared with solid FE results.The results are listed in Table 5.5. The points A, B, C, and D arein the midplane of the laminate (z = 0). The INS obtained from thesolid FEM at point A and B are 80.7 and 72.0N/mm2, respectivelywhere A is at the end face and B at the center (y = 0). At thesame locations (column Loc), the present model differs by -17.2% atpoint A and by -10.6% at B. The higher deviation at the end face


L

h

RI

II

I

II

t

b

X

Y

Z

INS [N/mm2]

70635649423528211470

Detail - A

A

C

B

D

Figure 5.17: L-probe and global INS distribution.

Table 5.5: Maximum INS in the doubly curved L probe.

Location Solid PresentLoc Diff. Max Diff.

[N/mm2] [N/mm2] % [N/mm2] %A 80.7 66.8 -17.2 69.3 -14.1B 72.0 64.4 -10.6 69.6 -3.3C 41.5 37.1 -10.6 40.2 -3.1D 47.7 39.5 -17.2 41.7 -12.6

can be caused by ISS, which have a maximum of 32.2N/mm2, andedge effects which decrease to zero in center direction. The error inregion II is higher at the end face (point C) and increases to the center(point D). The error agrees with the previous mentioned assessmentsfor singly curved structures.

The accuracy of the present model increases if the absolute max-ima (column max ) are compared. The agreement increases between3.1% and 7.5%. The maximum INS of the new model is always closeto the transition where the curvature radii jump from infinite to arelatively small value because the curvature radii are coupled with


the shell FE. The INS depend directly on the curvature radii and thedetails (Detail-A) are shown in Figure 5.17. The solid FEM maps thistransition much smoother. The same effect is already noticed in thesingly curved model (Section 3.5.1).

Again, the results are more accurate than those published in [106].The new curvature radii evaluation has a positive effect in both thesingly and doubly curved regions.

In addition to the post-processing method, 3-D failure criteria areimplemented in FELyX. Different failure modes have to be regardedin composites, namely fiber, matrix, out-of-plane failure, wrinkling,and core shear. The last two are not considered in this discussionbecause they address sandwich structures.

Strength evaluation of composite materials can be complex andtime consuming. The layer design leads to a high data volume. Thestrains and stresses are stored at bottom, mid, and top of everylayer and the stacking sequence may often includes, say, 20 layers ormore. A summarization of all these data in one or two files (plots)helps getting the necessary information and saves time. Thereforethe present stress and failure evaluation is user-friendly implementedfor composite structure analysis and can be used as composite post-processing tool. The inverse reserve factors of the failure criteria ofinterest are calculated in a first step for every integration point andevery layer. In a second step, the maximum inverse reserve factor ofall layers of the element and the failure mode are stored.

Figure 5.18 illustrates the maximum inverse reserve factor of Hashin’sfailure criterion [119] for each element for all layers and the absolutemaximum is 1.77 where all six stress components are considered. Themaximum failure index is close to the laminate midplane (0 − 0 in-terface). The strength properties of the lamina are listed in Table 5.6.The specimen is loaded in tension and failure is predicted in the dou-bly curved region. The maximum inverse reserve factor reduces to0.80 if only the in-plane stresses are attended.

The maximum stress criterion is suited to evaluate the criticalstress component. It is a linear failure criterion and therefore theinverse reserve factor is smaller than Hashin’s if it is higher than one.The critical stress components of part A are shown in Figure 5.19.The INS is the critical stress component in the doubly curved region.


A failure in the intersection regions would be addressed by the fiberstresses and by the ISS in the shoulder region.

0.00 0.00 0.251 0.25 0.501 0.50 0.752 0.75 1.00 1.00 1.25 1.25 1.50 1.50 1.75 1.75

Part A

Figure 5.18: Maximum inverse reserve factor of Hashin’s failure cri-teria for all layers.

0.00 0.20 0.40 0.61 0.81 1.01 1.21 1.41

s1c

s1c s1c

s1c s1c

s1t s1t

s1t s1t s1ts3t

s3t

s3t

s3t

s3t

s3t

s3t

s3t

s3t

s3t

s3t

s3t

s3t

s3t

s3t

s3t

s3t

s3t

s3t

s3t

s3t

s3t

s3t

s3t

s3t

s3t

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s13

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s13

s13

s13

Figure 5.19: Critical stress component of part A in each element forall layers: s3t = positive INS; s1c = fiber compression; s1t = fibertension; s13 = ISS.


Table 5.6: Lamina strength properties [N/mm2].

XT = 1450 YT = ZT = 55XC = 1400 YC = ZC = 170SXY = 90 SXZ = SY Z = 90

5.4.2 Rudder of a sail boat

The post-processing method is used to evaluate the INS distributionin a rudder made up of thick-walled curved laminates. It is only aqualitative discussion to point out the risk and gain of this method.The rudder is about 3.5m long, 80mm high, and 200mm deep. Itis clamped at the root and loaded by a pressure force which is notconsidered in the BCs of the INS distribution. The surface is meshedwith a linear 4-node shell FE (shell181). Figure 5.20 illustrates theINS distribution in the fourth layer which is discussed at the pointsA, B, and C.

The mesh is regular at the trailing edge (point A) where the surfaceis only curved in the global y-direction. The fiber direction of theUD laminate is parallel to the global y-direction. Hence, the refer-ence CS of the present model is parallel to the principal curvaturedirection where only small shear terms appear. The maximum INSin the 14mm thick laminate is 11.5N/mm2. The stacking sequenceand the orientation of the fibers and curvature radii should ensureaccurate results. However, the linear FE mesh does not fullfill all re-quirements. The effect of the element size in a singly curved structureis shown in Figure 5.4. Therefore, it can be assumed that the INS areunderestimated in this region because the element size is too high foran accurate mapping of the curvature radius. Regarding a strengthanalysis, further investigations should be done at this point and bet-ter results could be obtained from a mesh refinement or an alternativesolid FEM.

The maximum INS is smaller at point B even the laminate thick-ness is about 18mm. The laminate has only fibers parallel to theglobal x- and y-direction. The principal curvature direction is pa-rallel to the global x-direction as well to the reference CS of thepost-processing method. The mesh quality regarding the curvature


is better at this point because the curvature appears much smootherthan at point A. The maximum INS, which is 4.5N/mm2, is not crit-ical and it can be assumed that the lay-up, fiber orientation, and themoderate curvature radius is well modelled by the new method.

One drawback of this post-processing method, combined withlinear shell FEMs, is pointed out at point C. The inaccurate INSare caused by the curvature radius evaluation based on the distortedmesh. The mechanical model is not recommended to evaluate the INSin doubly curved structures meshed with linear shell FEs (Figure 5.6).At point C, the curvature radii are overestimated and probably alsothe INS. In addition, the uncertainty increases at point C becausethe principal strain direction is not parallel to the principal curvaturedirection.

XY

Z

A

B

C

Detail - A

Detail - C

Detail - B

Figure 5.20: Global INS distribution in a thick-walled curved lami-nate.

The analytical model points out the maximum INS regions wheredelamination could occur. But it can not be used as automatedstress analysis tool because mesh distortion, laminate, and curvatureorientation has to be considered. Critical regions have to be studied


regarding curvature, element size and type, stacking sequence, orien-tation, and edge effect. The influence of element type and size canbe reduced using quadratic shell FEs. Nevertheless, this numericallyefficient tool can be used as a first dimensioning tool pointing out thecritical INS regions.

The run time ration of the solid and shell FEMs, including the presentpost-processing method, is about 100. The number indicates thatmuch efficiency could be gained over solid FEMs if combining thenew method with shell FEs. It can be used as first stress or failureanalysis tool when INS plays a crucial part.

Chapter 6

Experiments

6.1 Introduction

A few experiments on singly curved specimens which suffereddelamination due to INS and ISS are presented in the paper [104].Based on these tests and results, improved specimens were manu-factured and other monitoring systems were used to investigate theagreement of the simulations and the test results.

Several delamination strength tests are discussed in Section 1.1.5.They were used to investigate critical loads, delamination onset andgrowth. Delaminations, whether they result from faulty manufactu-ring or stressing, create discontinuity and induce additional stressingat the delamination crack front. It is important to note that thedelamination occurring in the present specimens are caused by diffe-rent mechanisms. The specimens produce interlaminar stresses whichare smoothly distributed as long as the specimens remain undamaged.It is thus possible to predict critical loads for delamination onset byusing stress failure criteria which is addressed by this work. The edgesof a delamination, once it has been formed, raise stress concentrationsso that a progressing damage analysis [120] will have to invoke frac-ture mechanics concepts such as a strain-energy-release concept [121]what is beyond of the scope of this work as well. Following theseideas, the specimens have to meet the following requirements:

• Cause delamination by INS or ISS.

94 Experiments

• No break points or delamination layers which could harm oraffect the mechanical behavior.

• Only UD and cross-ply laminates are considered.

The manufacturing process is described in Section 6.2. The test pro-cedure, results, and failure mode identification is presented in Sec-tion 6.3. Section 6.4 discusses the correlation between delaminationonset and voids and the accuracy of the strength analysis is inves-tigated. Further, the dependency of the through-the-thickness INSdistribution on thickness, curvature radius, and material properties isexamined in Section 6.5.

6.2 Manufacturing

6.2.1 Specimen specification

High INS occur in thick-walled curved laminates due to external loadssuch as normal force and bending moment. A preliminary studysearched for specimen geometries and stacking sequences so that ini-tial failure should be delamination rather than in-plane failure. Theresult is a singly curved specimen, plotted in Figure 6.1, whose mid-plane follows the function

f(x) =

0 , −130 ≤ x < −50h2

[

cos(

2π·xl

)

+ 1]

, −50 ≤ x ≤ 500 , 50 < x ≤ 130

(6.1)

III

II

I

80mm 80mm

l

h

t

x

Figure 6.1: Geometry of the singly curved specimen.

6.2 Manufacturing 95

where h = 30mm is the amplitude and l = 100mm the length ofthe curved part. The laminate thickness t is 10mm where UD and[0, 90]s laminates are considered. The specimen can be loaded intension and compression. Tension causes high positive INS in regionIII and compression causes negative INS so that the ISS, whose signdoes not matter, can contribute to delamination failure probability.Critical INS and ISS are predicted in region III and II, respectively.The depth of the specimen is 40mm to reduce edge effects at themaximum stress locations.

The first test results revealed that the UD laminated specimenssuffered delamination as first failure but fabrication errors causedhigh statistic spread. The geometry was therefore adjusted aimingto increase the manufacturing quality and decrease the strengthmeasurement scatter. The adjusted parameters are obtained froman iterative process using FE simulations. Finally, the thickness t,amplitude h, and length l are reduced to 8mm, 20mm, and 80mm,respectively. The basic improvement is obtained from the reducedthickness t because the probability of air locks and folds is decreased.

6.2.2 Lamination and curing process

The specimens were made from a UD prepreg material describedin Appendix A.2. A good specimen quality was achieved usingan aluminum mold as positive form of the inner surface which isillustrated in Figure 6.2. Prepreg sheets were laid into the aluminummold to produce a singly curved plate from which the specimens werecut. The mold was prepared with a release agent to ease separationafter the curing process. Otherwise, the plate could not be demoldedproperly. The prepreg sheets were cut out by a cutting machine whatincreased the quality and accuracy of the manufacturing process.

The 8 and 10mm thick plates were made up of 42 and 52 prepreg-sheets, respectively. They were laid into the mold one by one andafter every four sheets the form, covered in a breather, was put intoa vacuum bag for 10-15 minutes. The breather provided the exhaustof the air. This process reduced the locked-air content and increasedthe fiber volume content.

96 Experiments

Figure 6.2: Positive aluminum alloy mold and vacuum bag.

The plates were cured in an autoclave at the Centre for StructureTechnologies, ETH Zurich. They were covered in a release fabric,bleeder fabric, and breather as plotted on the right of Figure 6.2.Further, the part was enveloped by a fleece and put into a vacuumbag. The curing conditions are given in Table 6.1.

Table 6.1: Curing conditions.

Vacuum bag -0.9 barPressure 5 barTemperature 130 CTime 3 h

The viscosity of the resin decreased during the heating. In this phase,where the resin was in a uncured state, it was pressed out of theplate and was exhausted by the breather. Later, the curing reactionof the resin begun and the viscosity increased until the duroplast wascompletely cured. After this reaction, the autoclave was shut downto room conditions and the plate could be separated from the mold.

The straight parts of the specimens were meant to be placed in theclamps of a testing machine. They had to be absolutely plane avoid-ing any relative movements between the clamps and the specimen.In the next manufacturing phase, woven fabrics were therefore laidonto the top surface of the cured plate. A metallic plate pressedthese layers against the top surface so that a good planeness and

6.3 Strength test results 97

bonding between the laminated plate and woven-fabrics was achieved.

In order to have specimens with similar material properties and ma-nufacturing conditions, the unmachined laminated plates had a depthof 200mm so that several specimens could be cut out of the same part.The fiber volume fraction was determined by the so called volumetricmethod. The volume fraction Vf depends on the volume ν, the massm, and the densities of the fiber ρf and matrix ρn materials where Vf

is defined as

Vf =νf

ν=

m − νρm

ν (ρf − ρm). (6.2)

The measured fiber volume fraction Vf was 59% what is close tothe value given by the prepreg supplier. Although the fiber volumefraction was close to the given one, additional material tests wereperformed to measure the stiffness in fiber direction which are de-scribed in Section 6.3.1.

The manufactured specimens disclosed two discrepancies. The totallaminate thickness t was not equal for each probe and also not constantalong the longitudinal x-direction. It tends to be smaller in regionIII and larger in region I (Figure 6.1). Measured values of it areconsidered in the FE analysis to achieve accurate INS results at themaximum stress location. The details of the specimen geometries aregiven in Appendix D.1.

6.3 Strength test results

6.3.1 Identification of material properties

The material stiffness and strength values in fiber direction, providedby the supplier, were double-checked with measurements of straighttest specimens which were cut from unidirectionally reinforced plates.These plates were made up of six UD layers and the length anddepth were 200mm and 260mm, respectively. The plates were vac-uum bagged and an extra constant outside pressure of five bar wasapplied in the autoclave. The vacuum pressure in the autoclavereached a value of about 920mbar below surrounding pressure. Afterthree hours, the autoclave shut down and the plates were cut into

98 Experiments

20 × 200mm tension specimens. The specimens had a thicknessbetween 1.15 and 1.20mm.

The tension tests were performed at the Centre of Structure Technolo-gies using a 100kN electro-mechanical testing machine. The resultsare listed in Appendix A.5 and provided a higher value of Young’smodulus E1 in fiber direction if compared with that given by the sup-plier. Young’s modulus E1 is set to 144′134MPa for the FE analysisto achieve accurate results.

6.3.2 Delamination tests

The singly curved specimens were loaded in tension and compressionup to failure through controlled displacement of the cross-head at aconstant rate of 0.5−2.0mm per minute where the reaction force wasmeasured by a load cell. These tests were performed at the EMPA1

or at the Centre of Structure technologies.

Figure 6.3 shows an instrumented and damaged test specimen,still placed in the testing machine. The delaminations at severalinterfaces are clearly visible. The white points serve as markers forthe displacement measuring video device. The aluminum cylinderson the right and left side, clamped by an spring, are the acousticemission (AE) sensors.

Figure 6.3: Experiment facility and delaminated specimen.

1Federal materials science and technology research institution


Figure 6.4 illustrates the force-time diagram of a tension test whichclearly indicates the first and subsequent failure events by the suddenforce drops. Since the cross-head of the testing machine moves withconstant speed, time can also be taken as cross-head displacement.The tension force is divided by the cross-section area of the specimento achieve a better comparability between all tests. An AE measure-ment device was used to record and locate AE failure events alongthe longitudinal axis. The graph on the right of Figure 6.4 shows theagreement between the first force drop and the sudden increase of theAE failure events per unit time. It reveals an increase of AE activitybefore delamination occurs which is a sign of an early damage growth.In contrast to the tension tests, the specimen loaded in compressionfailed suddenly without preceding AE activity of increasing damage.

0 50 100 150 200 250 3005

10

15

20

25

30

35

40

45

50

Time [sec]

Fo

rce

[N

/mm

2]

0 50 100 150 200 250 3000

50

100

150

200

250

300

350

400

Time [sec]

Nu

mb

er

of

rea

din

gs

[−−

]

Delamination

Figure 6.4: Tension test: Force-time diagram and acoustic emissionevents.

AE failure events, which were recorded on both sides, could be locatedalong the longitudinal specimen axis. The acoustic velocity along thespecimen was provided by snapping of a lead on the top layers. Theuncertainty of the localization is ±1cm. Figure 6.5 shows all locatedfailure events along the longitudinal axis. Few failure events werelocated near the clamping before delamination occurred.

The failure, which induced the first observable load drop, causeda rising of the AE events which were mostly located in the middleof the specimen where delamination was predicted. In this region,only the INS challenged the strength. Hence, the force drop was

100 Experiments

clearly caused by delamination onset at region III (see Figure 6.1).Once failure occurred, the delamination propagated to region II andI because relatively high ISS appeared in region II which drove thecrack propagation.

0 50 100 150 200 250 300

−5

−3

−1

1

3

5

7

9

11

13

Time [sec]

x [cm

]

Specimen

Figure 6.5: Localisation of AE events along the longitudinal specimenaxis.

Delamination due to INS

All UD laminated specimens loaded in tension suffered delaminationin region III. The critical loads, displacements, and maximum INS inregion III are listed in Table 6.2. Although care was taken to achieveconstant properties the specimens showed varying ultimate load.

The INS are obtained from a solid FE analysis. The geometryis meshed with 20 quadratic brick elements through the thicknessand model symmetries are taken in order to reduce numerical cost.The displacement, measured at onset of failure, is applied as loadin the FEM. The thickness variation of each specimen is mapped toimprove the model accuracy and agreement of the simulated reactionforces Ftot with the measured ones. The geometrical parameters


of each specimen are given in Appendix D.1. The global and localINS distributions in a singly curved specimen loaded in tension areillustrated in Figure 3.8 and 5.9.

Table 6.2: Displacements, loads, percent differences of Ftot, and max-imum INS.

Test FE analysisSpec. Ftot Disp. Ftot Diff. max. INS

[N/mm2] [mm] [N/mm2] % [N/mm2]

T2 49.25 1.865 48.51 -1.5 86.63T6 42.74 1.709 44.02 3.0 79.38T7 35.10 1.435 37.70 7.4 66.65T1.1 59.66 1.314 67.54 13.2 87.16T1.2 70.73 1.312 70.36 -0.5 89.93T1.3 76.65 1.459 80.18 4.6 101.89T1.4 68.81 1.414 70.77 2.9 91.87T3.1 50.97 0.991 54.46 6.8 69.20T3.2 70.40 1.290 74.27 5.5 93.30T3.3 77.40 1.361 75.39 -2.6 95.62T3.4 63.09 1.153 63.87 1.2 81.01R1.1 43.85 0.954 51.68 17.9 66.98R1.4 42.00 0.750 43.16 2.8 55.10R1.5 53.93 0.828 47.65 -11.6 60.83R2.1 40.74 0.745 45.05 10.5 56.74R2.2 49.71 0.829 52.13 4.9 65.01

It is obvious that the deviation of the maximum load, displacement,and INS is high. The test results of specimen R1.1 are not consideredin the evaluation due to the high deviation between the mechanicaltest and the FEM. Although care was taken to reduce uncertaintiesand fabrication faults, the delamination strength varies between55.10 and 101.89N/mm2. The mean value is 78.69N/mm2 and thestandard deviation is 15.26N/mm2. The average percent differenceof the FE simulations, if compared with the tests, is 3.1% and thestandard deviation is 5.9%.

102 Experiments

The localization of the highest INS values, at region III, amplifiesthe scatter of the strength measurement results. This is because itmatters whether a material flaw occurs there or in a less stressedlocation. In addition to the weakening of the material a flaw canalso cause additional stresses. Once a sufficiently large delaminationhas formed, it propagates if the energy release rate exceeds a criticalvalue. Faults can result from air locks or folds and, depending ontheir location and size, affect the interlaminar stress more or less.An investigation of material properties of (carbon/epoxy) laminatesis discussed by Ghiorse [122] and Judd [123]. They showed that theinterlaminar strength decreases by 7% for each 1% increase of thevoid content.

The UD laminate loaded in tension had a higher INS strength, ifcompared with the literature values [124, 125]. This can be causedbecause the first delaminations in the maximum INS location may notpropagate and could not be observed in the force-time diagram. TheINS decreases fast with increasing distance from the maximum stresslocation. The energy release rate is therefore not high enough for thefirst delaminations to propagate.

Delamination due to ISS

In addition, delamination tests caused by ISS were performed to ap-prove the failure mode prediction based on the FE simulations. Thespecimens, loaded in compression, suffered delamination in regionII caused by ISS. The delamination onset and growth could not berecorded by the AE sensors because the specimens failed suddenlywithout preceding AE activity of increasing damage. Nevertheless,delamination occurred due to ISS.

Some of the specimens failed when the force-displacement diagramshowed a non-linear behavior what is considered in the FE analysis.The mean value of the maximum ISS is 83.31N/mm2 and the stan-dard deviation is 4.82N/mm2. The minimal and maximal ISS are74.53 and 88.57N/mm2, respectively. The lower standard deviationis ascribed to the higher laminate quality in region II and to the moreuniform ISS distribution. Hence, local fabrication errors affected theISS strength less than those in region III. The test and FE results arelisted in Table D.2 and D.3, respectively. The average difference of theFE reaction forces, if compared with the tests, is 14.0% and the stan-


dard deviation is 12.8%. This deviation is caused by the non-linearityand the compression fiber stiffness which differs from the tensile one.These results have to be handled with caution because of the highdeviation and only seven tests were performed.

Matrix failure

The analytical model and FE simulation predicted delamination,caused by INS in region III, as first failure mode in the [013, 9013]slaminate loaded in tension. Unfortunately, matrix failure occurredfirst under small external load due to thermal stresses. The failureonset occurred in the 90 layers between region II and III and prop-agated to the top layers. Thenceforward, it propagated between the90 and 0 intersections. Figure 6.6 illustrates the failure and thepropagation is shown in the framed plot.

Figure 6.6: Matrix failure in the [0, 90]s laminate.

The thermal stresses are caused by different coefficients of thermalexpansion in the principal material directions 1, 2, and 3. Figure 6.7illustrates the matrix stress distribution with and without thermalstresses. A difference between curing and room temperatures of 110Cis assumed for the calculations. The coefficients of thermal expansionare α1,2,3 = [−0.6 · 10−6, 0.3 · 10−4, 0.3 · 10−4]1/K. In this exam-ple, the matrix stresses increase to 51.9 N/mm2 what is close to thematrix strength. Due to the thermal stresses, no more cross-ply lami-nates were manufactured and tested. It was not considered worth the

104 Experiments

effort because the investigations could be carried out with unidirec-tional specimens where thermal stresses play a much lesser role. Themaximal loads and displacements when matrix failure occurred aregiven in Table D.4 and D.5, respectively.

-13.8 -6.5 0.8 8.1 15.4 22.7 30.0 37.3 44.6 51.9 -4.7 2.8 -1.0 0.9 2.8 4.7 6.6 8.4 10.3 12.2

Figure 6.7: Transverse matrix stresses in region II-III with (left) andwithout (right) temperature effect.

6.3.3 Strength analysis

In the present evaluations, failure is predicted using Hashin’s failurecriteria [119]. These criteria address different failure modes of the UDcomposite separately. Further, interaction coefficients were neglectedwhat facilitates the experimental determination of the strength para-meters. Hashin distinguishes between tensile and compression fiberand tensile and compression matrix failure modes. The present UDspecimens loaded in tension suffered tensile matrix failure which comesinto effect if σ2 + σ3 ≥ 0, where the stresses are expressed in the ma-terial CS. The tensile matrix failure criterion is defined as

1

σ+2T

(σ2 + σ3)2

+1

τ2T

(

τ223 − σ2 · σ3

)

+1

τ2A

(

τ212 + τ2

13

)

= 1 (6.3)

where σT and τT,A are UD failure stresses. The stresses in the trans-verse in-plane direction are small compared to the others in the singlycurved specimen loaded in one-axial tension because cylindrical bend-ing is assumed and only UD laminates are considered. Hence, thetensile matrix failure criterion is approximately

6.4 Investigation of the test results 105

(

σ3

σT

)2

+

(

τ23

τT

)2

+

(

τ13

τA

)2

= 1 . (6.4)

This simplified failure criterion gives good results in the present me-chanical tests listed in Table 6.2. The failure plane is perpendicular tothe laminate thickness-direction (z) what is confirmed by the failurepropagation and the delamination cracks plotted in Figure D.3.

6.4 Investigation of the test results

The spread of the INS strength is investigated in Section 6.4.1 anda detailed comparison between the mechanical tests and the FEMresults is presented in Section 6.4.2.

6.4.1 Specimen quality

The INS strength of the specimens, listed in Table 6.2, has a highspread. It is mentioned that the maximum INS appears in regionIII and manufacturing errors in this region cause this spread. Thecorrelation between INS strength and voids at the maximum INS lo-cation is investigated by embedding a break point in the singly curvedspecimen.

The length of the curved part was 80mm and the amplitude h was20mm. The UD stacking sequence was made up of 45 layers wherea teflon plate with a diameter of 4mm was integrated in the 23rdlayer as illustrated in Figure 6.8. Finally, the cured UD plate wascut into nine specimens, four with and five without teflon plate, toachieve equal curing and manufacturing conditions where the manu-facturing is described in Section 6.2. In addition, a silicon stamp wasmanufactured which was used as second positive form. This stampdistributed the applied pressure uniformly over the laminated plateand ensured a better quality (more uniform thickness). The geometryparameters of each specimen are given in Table D.6.

The specimens were loaded in tension and delamination occurred priorto in-plane failure. All specimens had a similar stiffness what confirmsthe comparison between the specimens with and without embeddedteflon plate. Further, the maximum load of each specimen type was

106 Experiments

Figure 6.8: Specimens with integrated teflon plates at the maximumINS location.

constant which indicates the increased uniformity of specimen qual-ity. The force-displacement diagram of all specimens of this series isillustrated in Figure 6.9.

The maximum INS strength is clearly affected by the small teflonplate. The maximum tension load was reduced to 25 − 29N/mm2

where the maximum load of the specimens without break point wasbetween 51 and 52N/mm2. The only exception is probe 7 whichfailed by 41N/mm2. The break point reduced the maximum INS by43% and clearly indicates that the strength is affected by errors atthe maximum stress location.

A computer tomography (CT) of this series was made at the EMPAbefore they were tested to investigate the correlation between manu-facturing errors and INS strength. The resolution of the CT dependson the cross-section area and in this case one pixel represented 40µm.Therefore, air locks and folds of about 120µm could be detected. Spec-imen 7 suffered delamination before the average maximum load wasachieved. The CT of this specimen, plotted in Figure 6.10, shows oneair lock in the cross-section where high INS appear. This air lockcould cause local stress peaks and delamination. The dimension ofthe air lock can be defined by density profile plots which are givenin Appendix D.2. The length of the air lock is 2mm and it has adiameter of 70µm.

The other specimens without break point had also air locks andfolds which were not at the maximum INS location. The INS strength


0 0.2 0.4 0.6 0.8 1

0

10

20

30

40

50

60

Displacement [mm]

Fo

rce

[N

/mm

2]

Probe 1

Probe 3

Probe 4

Probe 5

Probe 6

Probe 7

Probe 8

Probe 9

Figure 6.9: Force-displacement diagram of the specimens with breakpoint (−) and without (- -).

was therefore not affected in the same extent if compared with spec-imen 7. Further, CT images of a specimen with an embedded teflonplate are given in Appendix D.2.

Figure 6.10: CT images of specimen 7 which show an air lock inlongitudinal direction. Side view on the left and top view on theright.

108 Experiments

6.4.2 Strain gauge measurement

The accuracy of the stress results, obtained from the solid FEM,is investigated by comparing the strain results obtained from themechanical tests and FE analysis. The FE reaction force results, pre-sented in Table 6.2, are close to the mechanical test results exceptingspecimen R1.1 which is not considered in the present evaluation.Strain gauge sensors were applied to record the strains at the bottomand top surfaces. Further, optical fibers (OF) were embedded in thespecimens to measure the strains at the midplane. The results arepresented in the Appendix D.3.

Accurate strain measurements can be achieved if a strain sensor isplaced at the maximum-strain location. Strain gauges were used for anew set of experiments because they provide accurate results and arecheaper than OFs. Even though the sensor area is larger than that ofan OF, accurate results are achieved because the strain is more or lessconstant in the fiber and transverse direction at the region of interest(region III Figure 6.1). The dimension of the strain gauge sensor is1.2 × 1.5mm.

Strain gauges were attached to the top and bottom surfaces of thespecimens named R (Table 6.2) and the specifications of the straingauges are given in Table D.7. Figure 6.11 shows a strain gauge onthe top surface at the cross-section where the maximum INS appears.The cables and all other electrical devices of the strain gauges wereinsulated from electrical radiation by strands.

The strain-time diagram for the top and bottom surfaces is shown inFigure 6.12. The absolute value of the strain at the inner (bottom)surface is higher than that at the outer surface. The vertical linemarks the delamination which caused another through-the-thicknessdistribution of the strain εx (top surface) where the strain gauge atthe bottom surface broke due to the delamination.

The strains obtained from the mechanical tests and solid-model FEsimulations are compared to investigate the accuracy of the INSstrength evaluation. The accuracy of the FE stress results correlateswith the accuracy of the FE strain results because the stress-strainrelation is defined by the constitutive law. The results are listed inTable 6.3. The percent difference of the evaluated strains, if compared


1cm

Figure 6.11: 1.2× 1.5mm strain gauge on the top surface at the max-imum strain location.

0 5 10 15 20−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

7

Time [sec]

Bottom

Top

Delamination

X103

Strain ε

[µm/m]

x

Figure 6.12: Strain gauge results at the bottom and top surfaces ofspecimen R2.2.

with the mechanical tests, is between -7.4% and 12.0% where specimenR1.1 is not considered because the thickness of this specimen was notconstant in the transverse direction what caused inaccurate results.The average percent difference of the strains at the bottom and topsurface is 7 and 2%, respectively where the FEM overestimates thestrains.

110 Experiments

The non-linear through-the-thickness distribution of the strain εx

is shown on the left of Figure D.4. The strain highly increases at theinner surface (z = −4.15) and small changes of the thickness wouldlead to other results. The strain at the outer surface (z = 4.15) isless sensitive regarding the thickness. The test results reveal thatthe solid-model FE simulations give accurate strain results if thethickness is well defined.

Table 6.3: Measured and evaluated strains εx at the top and bottomsurfaces.

Load Strain gauges Solid FEMSpec. Disp. Top Bottom Top Bottom

[mm] [µm/m] [µm/m] [µm/m] [µm/m]

R1.1 0.954 -3612 6007 -3744 6964R1.4 0.572 -2155 3882 -2328 4349R1.5 0.828 -3127 5849 -3293 6191R2.1 0.745 -3115 no data -3007 5737R2.2 0.829 -3586 6481 -3398 6604R2.4 0.573 -2200 4155 -2321 4443

6.4.3 Material properties

An additional uncertainty of the INS results is induced by the varyingmaterial properties. The maximum difference of the reaction forcesobtained from the mechanical tests and the solid FEMs is 13.2% (spec-imen R1.1 is not considered). In the following FE analysis, the fiberYoung’s modulus E1 is adjusted in the FEM that the applied dis-placement results in the same reaction force if compared with themechanical test. E1 is reduced by 13% and the percent difference ofthe reaction forces become −0.6% where the maximum INS decreasesby 9.6% if compared with the result listed in Table 6.2.

The average percent difference of the reaction forces obtained fromthe solid FEM, if compared with the mechanical tests, is 3.1% and thestandard deviation is 5.9%. The correlation between the maximumINS and the material properties is not investigated in detail. The INS

6.5 Sensitivity and error analysis 111

depends linearly on the Young’s modulus in isotropic structures andbecomes non-linear in anisotropic structures where the linear ratiooutweighs the non-linear part.

Further, it can be assumed that the longitudinal Young’s modulusE1 is overestimated by 3.1% because the average percent differenceof the reaction force is 3.1%. The INS strength is therefore overesti-mated and should be reduced regarding strength analysis to satisfysafety margins.

6.5 Sensitivity and error analysis

6.5.1 Sensitivity analysis

The previous investigations of the accuracy of the INS strength resultsreveal that the results are sensitive regarding thickness and materialproperties. A FEM analysis is always an approximation of a real struc-ture where loads, geometry, material etc. are simplified. Differentmonitoring devices can be used to compare the results of mechanicaltests and simulations except for stresses which can not be measureddirectly. The following evaluations aim to investigate the sensitivity ofthe INS results obtained from the analytical model regarding geome-try (thickness and curvature radius) and material properties (Young’smoduli).

It is assumed that the deformations and strains remain constantand one of the parameters thickness t, curvature radius R, or Young’smodulus E1,2,3 vary between −5% and 5%. The changing of themaximum INS against the variation of the parameters represents thesensitivity of the INS evaluation.

The singly curved specimen, illustrated in Figure 6.1, is used for theevaluation. The specimen is made from a UD laminate and the ma-terial properties are given in Table A.2 where the fiber modulus E1 isadjusted to 144′134MPa. The first evaluation investigates the influ-ence of the curvature radius R and thickness t. Figure 6.13 illustratesthe changing of the maximum INS for a ±5% variation of one of theseparameters which is the result of a structural analysis. The maximumdiffers by ±7.0% if the thickness t varies by ±5%. The maximum INSchanges by 2.0% if the curvature radius R differs by 5%.

112 Experiments

The results of the material properties are illustrated on the rightside of Figure 6.13. The maximum INS is sensitive regarding theYoung’s moduli in thickness direction E3 and that in curvaturedirection E1. The INS differs by 3.3% and 2.0% if E1 and E3 vary by5%, respectively. In this example, the Young’s modulus in transversedirection E2 has a low influence, if compared with E1 and E3, becausethe curvature radius in this direction is infinite. If all parametersdiffer by ±5% compared with the assumed real model, the maximumINS either decreases by 13.8% or increases by 14.8%.

−5 −4 −3 −2 −1 0 1 2 3 4 5−8

−6

−4

−2

0

2

4

6

8

Variation [%]

Ch

an

gin

g o

f th

e m

ax.

INS

[%

]

Curvature R

Thickness t1

−5 −4 −3 −2 −1 0 1 2 3 4 5−4

−3

−2

−1

0

1

2

3

4

Variation [%]

Ch

an

gin

g o

f th

e m

ax.

INS

[%

]

E

E

E

1

2

3

Figure 6.13: Sensitivity of the INS regarding thickness, curvature ra-dius, and Young’s moduli.

6.5.2 Error analysis of the INS evaluation

The results of Section 6.4.1 and 6.5.1 are used to investigate the totaluncertainty of the INS strength evaluation listed in Table 6.2. Theerrors, introduced from the testing machine and strain gauge device,are neglected. It is assumed that these errors are much smaller thanthose listed in the next paragraph. Further, the curvature radius Rof the geometry is well defined due to the molding (aluminum) andthe error is set to zero.

The INS tests result in an INS strength of 78.7N/mm2. The averagepercent difference between the measured and evaluated reaction forcesis 3.1% what is assumed as the average variation of the fiber Young’smodulus E1. No tests were performed to measure the Young’s modu-

6.6 Conclusion 113

lus E3 and the uncertainty is therefore assumed as 10%. The thicknesst was measured using a caliper or polished cross-section images andthe accuracy is ±0.2mm what corresponds to ±2.5% spread of thethickness. The average deviation of the FE strain results, if comparedwith the test results, is approximately 4%. Adding all uncertainties,the total spread of the INS strength evaluation is approximately be-tween -12 and −3%. The INS strength is therefore between 69.2 and76.3/mm2.

6.6 Conclusions

A singly curved laminated specimen type was designed and manu-factured which delaminated due to INS caused by a tension load.The specimens achieve the specification that no break points ordelamination layers are necessary for the delamination growth. AnAE events device was used to measure and localize the failureevents. Failure events were identified from the beginning where theywere mostly close to the clampings. The few failure events in themaximum stress region did not cause a uncontrolled delaminationgrowth till a certain maximum INS (energy rate) was achieved. Thedelamination growth due to positive INS, which caused the first ob-servable load drop in the force-time diagram, was localized by an AEdevice and the most failure events were located at the maximum INSlocation. The variation of the strength is caused by air locks and folds.

Embedded break points and CT images were used to investigate thevariation of the INS strength. The maximum load of the specimenswith embedded teflon plates reduces by 50%. CT images reveal thatalso small fabrication errors at the maximum INS location reducesignificantly the INS strength what caused the variation of the maxi-mum load.

Solid FEMs are used to evaluate the stress distributions and theaverage INS strength is 78.69N/mm2 where delamination onset,growth, or energy methods are not considered.

A strain gauge device is used to investigate the accuracy of the FEMresults compared with the mechanical tests. The strains on the topand bottom surfaces at the maximum stress location agree well with

114 Experiments

those obtained from the solid FEMs. In addition, the sensitivity of theINS evaluation concerning laminate thickness, curvature radius, andmaterial properties is investigated. The uncertainties of the materialproperties, laminate thickness, and strain results cause approximatelya failure deviation between -12 and −3%.

Chapter 7

Conclusions andOutlook

7.1 Conclusions

This thesis investigates the INS calculation in singly and doublycurved thick-walled laminates based on the CLT and investigatesdelamination strength tests on singly curved specimens which suffereddelamination due to positive INS.

7.1.1 Analytical model

New models for the INS calculation in singly and doubly curved thick-walled laminates have been derived. The singly curved model isbased on the exact solution of the thick-walled composite tube prob-lem [92, 93]. In a first step, the shear terms are neglected which donot appear in UD and cross-ply laminates under cylindrical bend-ing. Further, the mechanical model belongs to the ESL theory wherethe kinematics of the laminate is reduced to the midplane deforma-tions ε and κ. The analytical model is compared with 2-D solid FEMresults. A closed-form model for cylindrical plate bending, includ-ing the curvature effect, provides the laminate deformations ε and κwhere a good agreement of the internal forces between both models isachieved. Finally, INS results obtained from the 2-D solid FEM andthe mechanical model are compared which each other.

116 Conclusions and Outlook

The analytical model provides accurate results in consideration ofthe severe simplifications. The analytical model is enhanced by anout-of-plane shear term which is considered in the radial equilibriumcondition. The enhanced model shows moderate improvements inUD and cross-ply laminates.

A mechanical model for the calculation of INS in doubly curved thick-walled laminates is derived from the singly curved model withoutshear term. Again, only UD and cross-ply laminates under cylindricalbending are considered. The second-order differential equation of thethrough-the-thickness displacement w is solved with a finite differencemethod. Accurate INS results are already achieved using a five-pointintegration scheme through the thickness per layer.

The analytical model is developed for element-level post-processingand is implemented for linear and quadratic shell FEs in an in-housefinite element library (FELyX). The accuracy of the post-processingmodel, based on linear and quadratic shell FEs, is discussed for singlyand doubly curved structures. The results reveal that quadratic shellFEs provide accurate results independent of the geometry and elementdistortion where it is assumed that the laminate reference surfacefollows a parabolic function.

A generic model assessment discloses the agreement between thepost-processing method and accurate solid FEMs. The accuracy ofthe new model correlates with the material properties, CT-ratio, layerorientation, and the principal curvature and strain directions where aclose interrelation between the accuracy and the previous mentionedparameters is observed. The percent difference between the analyticaland solid FEM INS results is lower than 20% if

• E1/E3 ≤ 19.6

• 1.7 ≤ CT-ratio ≤ 50.0

• Rfiber direction ≤ Rtransverse

• Principal strain and curvature directions are coincidental

• UD or cross-ply laminate

7.2 Outlook 117

7.1.2 Mechanical tests

The present models are applied for INS strength testing and tointerpret the results of singly curved specimens which suffereddelamination due to positive INS prior to in-plane failure. Thespecimens were loaded in tension and an AE event device indicatedthat the delamination onset was in the region where only positiveINS challenged the strength.

The results of the mechanical tests are discussed with solid-modelFE simulations which provide the INS distribution. The mechanicaltests revealed that the INS strength is sensitive regarding air locksand folds which occur in hand-made laminates. Specimens withembedded break points and CT images are used to investigate thesensitivity of the INS strength regarding voids. They indicated thatalready small (2 × 0.07 × 0.07mm) voids can significantly affect thestrength.

Results of measurements with load cells and strain gauges are com-pared with FEM simulations with solid finite elements. The meanpercent difference of the reaction forces, obtained from the FEM andload cell, is 3.1% and the strain results differ by 7 and 2% at thebottom and top surface, respectively.

A sensitivity analysis assesses the accuracy of the INS strengthevaluation. The changes of the maximum INS with respect to thevariation of the laminate thickness t, curvature radius R, and materialproperties E1,2,3 discloses the sensitivity of the INS evaluation. Thesum of all uncertainties provides approximately an inaccuracy of theINS strength evaluation between −12% and −3%.

7.2 Outlook

7.2.1 Analytical model

The analytical model is based on the laminate deformations ε and κobtained from the CLT. It gives good approximations if consideringthe numerical efficiency and severe simplifications. The inaccuracyincreases if the principal strain and curvature directions are notcoincidental, and ISS or edge effects appear. The accuracy of the


analytical model can be increased by further investigations:

The analytical model can be completed by including the in-plane

shear γϕϑ and then the constitutive law becomes

σϕ

σϑ

σr

τϕϑ

=

C11 C12 C13 C16

C21 C22 C23 C26

C31 C32 C33 C36

C61 C62 C63 C66

εϕ

εϑ

εr

γϕϑ

(7.1)

where the kinematical relation for the shear strain in the reference CSis

γϕϑ =1

r + rdu,ϑ +

1

rv,ϕ . (7.2)

In addition to the constitutive law, the assumed expressions for thein-plane deformations u and v (Equation 4.5) have to be adjusted. Fi-nally, the analytical model should be more accurate in regions wherethe principal strain and curvature directions are not coincidental.

The out-of-plane shear term in the radial equilibrium condition(Equation 3.18) is considered in the enhanced singly curved modelusing a FSDT. The benefit is small regarding the accuracy improve-ment and the increased numerical effort. It can be assumed thatthe shear influence increases in angle-ply laminates. Consideringangle-ply laminates, a HSDT often provides more accurate resultsthan the FSDT. Some approaches for a zic-zac or HSDT are given byKim [18], Liu [67], Oh [73], Reddy [20], and Surana [68].

Recently developed layered shell elements are based on layerwise

theories where the displacement assumption holds separately foreach layer. The analytical model should provide more accurate INSresults if the layerwise strains are considered. Layerwise plate andshell models are described by Sciuva [23], Bhaskar [24], Cho [25], andReddy [26] where some of these models enforce the continuity of theinterlaminar stresses a priori. It should be considered that the numberof DoFs in ESL shells does not depend on the number of layers andremains constant where the numerical cost of layerwise shells oftenincreases because the number of DoFs increases with the number oflayers. An idea to overcome this drawback is presented by Reddy [22].

7.2 Outlook 119

The accuracy of the post-processing method depends on the cur-

vature radius evaluation which is based on the shape functionapproach of the shell FEs. Quadratic elements follows a parabolicsurface and good results are provided by the assumed parabolicsurface approach. The accuracy of the curvature radius evaluation,provided by linear shell FEs, depends on the geometry, mesh distor-tion, and element size (length of the element edges). The accuracy ofthe post-processing method for linear shell FEs can be increased byan improved curvature radius evaluation.

7.2.2 Mechanical tests

The mechanical tests provided useful results considering the hand-made specimens. However, the repeatability and accuracy of the testscan be improved by increasing the manufacturing quality. First,automated processes can replace the hand-made process. Further, thethickness and fiber-volume content can be better adjusted using twoaluminum molds. The reduction of uncertainties and voids decreasesthe scatter of the delamination strength results and a more accuratestrength analysis can be performed.

An AE event device disclosed the location of failure onset where onlypositive INS challenged the strength. In addition, CT images showeda correlation between delamination onset and voids. The presentINS strength exceeds the literature value because the maximumINS appeared in a limited region. It can be therefore assumed thatdelamination onset occurred which could not be observed in theforce-time diagram because the cracks did not propagate. A crackpropagates if the energy release rate exceeds the critical value forthe delamination growth. Energy-release-rate theories can be used toinvestigate the delamination onset and growth and to specify to INSstrength more accurate. Ideas and approaches are presented in theliterature [90, 91].

Different monitoring devices were used to record the strain εx

in fiber direction where the solid FEMs show a good agreementcompared with the test results. Fiber bragg sensors can also be usedto measure the through-the-thickness strain field of εx in laminatedplates [126] which can be compared with the results obtained from a


solid FEM. OFs were also used to investigate the delamination onsetand growth [127, 128, 129]. Further, ESPI or strain gauge pattern canbe used to measure the strain distribution of εz at the end face of thesingly curved specimen. These performance data can be combinedwith energy-release-rate methods. Moreover, the FE analysis resultscan be better investigated.

Appendix A

Material properties

A.1 GY70/Epoxy

Young’s Poisson’s Shearmodulus [MPa] ratio [–] modulus [MPA]E1 290000 ν12 0.41 G12 5000E2 5000 ν23 0.49 G23 2083E3 5000 ν13 0.41 G13 5000

Fiber volume 60%Source Dornier Systems GmbH

A.2 Elitrex EHKF 420-UD24k-40 T2

600mm


122 Material properties

Description

Preimpregnated Carbond-UD Modified hot-melt epoxy resinUD weight 210 ± 10 g

m2

Density 1.8 gcm3

Prepreg weight 350 ± 20 gm2

Volatile 135C30min < 1%

Proceedings Press, Autoclave, Vacuum-bagTackiness Light tacky till tacky

Curing conditions

Temperature 120C 130CCure time 45 min 30 minSpec. Pressure 0.4 ±0.1 MPa 0.4 ±0.1 MPa

Heat-up / Cool-down ∼ 10Kmin ∼ 10

Kmin

Remove material at < 100C < 100CCured prepreg Thickness 0.19 - 0.2 mm

Mechanical properties at 23C

Fiber volume 60%Units Normal Min. µ95 Test method

Tensile strength MPa 1750 1600 DIN EN 2561 ATensile modulus MPa 110000 100000 DIN EN 2561 AFlexural strength MPa 1550 1400 DIN EN 2562 BILSS (short beam) MPa 85 75 DIN EN 2563

Source Stesalit AG Business Unit KasselOtto-Hahn-Str. 5D-34123 KasselPhone: 0049 (0)561 / 998563 - 0

A.3 IM7/8551-7 123

A.3 IM7/8551-7


Strengthstensile [MPa] compressive [MPa] shear [MPa]XT 2560 XC 1590 S12 90YT 73 YC 185 S23 57ZT 63 ZC 185 S13 90

Fiber volume 60%Fiber type IM7Matrix 8551-7

A.4 Aluminum alloy

Young’s Poisson’s Shearmodulus [MPa] ratio [–] modulus [MPA]E 70000 ν 0.3 G 26923.1

A.5 Tension tests: Elitrex EHKF


Table A.1: Series 1.

Curing conditionsTime Temperature Pressure3 [h] 130 [C] 5 [bar]

Nr Cross-section Young’s modulus Strength[mm2] [MPa] [N/mm2]

1 23.00 141017 24552 23.00 154644 25593 23.00 151037 25954 23.00 140020 24055 23.00 138520 25506 24.00 147646 23767 24.00 139733 24218 23.88 140530 2411

Statistical valuesX 23.36 144143 2459S 0.50 6105 84

A.5 Tension tests: Elitrex EHKF 125




1 22.00 147202 17912 24.72 183959 20253 24.72 145619 19354 24.72 159160 20265 24.72 140159 23286 24.48 138303 24407 23.98 137434 24728 23.98 138669 23859 25.75 155520 253710 25.75 155494 2423

Statistical values

X 24.48 150152 2236S 1.06 14316 265





1 23.88 145319 22322 24.00 136792 22363 24.00 157806 25254 25.00 144475 24825 25.00 133292 22336 24.00 135336 22257 24.06 134683 22318 24.00 157533 23849 25.06 152952 242310 25.00 148071 2272

Statistical valuesX 24.40 144626 2324S 0.52 9410 118

Appendix B

Singly curved finiteelement

B.1 Stiffness matrix

The curvature effect in a singly curved plate increases the stiffnesscompared with a flat plate. The qualitative influence is shown in Fi-gure B.1 for a homogeneous plate with the thickness t = 1 and Young’smodulus E = 1. The CLT stiffnesses of this flat plate are A11 = 1,B11 = 0, and D11 = 1

12 . The stiffness increases with decreasingcurvature radius R and becomes infinite for the curvature-radius-to-thickness ratio R

t = 0.5 which resembles the limiting case of a solidcylinder. The difference between the modified terms A∗ and D∗ andthe CLT is less than 1% for a curvature-radius-to-thickness ratio of3. In addition, the curvature effect leads to a negative coupling effectbetween the line force Nx and line moment Mx.

B.2 Through-the-thickness strain distri-bution

The influence on the strain and stress distributions is greater thanthat on the plate stiffness. The linear strain assumption (Equa-tion 3.1) of the CLT assumes equal absolute strains at the top andbottom surfaces in a plate loaded by a bending moment. However,

128 Singly curved finite element

100

101

102

15

10

20

30

40

50

60

70

log(R/t)

Cha

nge

in %

101

0

1

2

3

4

log(R/t)

Details: 2 < R/t < 10

AD

100

101

102

−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

log(R/t)

Stif

fnes

s

B

Figure B.1: Curvature effect: Reinforcement of A and D matrix (left)and negative coupling effect (right).

the absolute strain in a singly curved plate is greater in the innersurface. This effect is proportional to the curvature radius R and isshown with the following example.

The modified ABD-Matrix for a singly curved plate (Equation 3.12) issimplified so that only the line loads Nx and Mx and the deformationsεx and κx appear. The curvatures κy and κxy do not appear undercylindrical bending. In addition, it is assumed that Ny = Nxy = 0.Finally, Equation 3.12 is reduced to

Nx

00

Mx

=

A∗

11 A∗

12 A∗

16 B∗

11

A∗

21 A∗

22 A∗

26 B∗

21

A∗

61 A∗

62 A∗

66 B∗

61

B∗

11 B∗

12 B∗

16 D∗

11

ε0

x

ε0

y

γ0

xy

κx

. (B.1)

The stiffness coupling terms A∗

12, A∗

16, B∗

12, and B∗

16 vanish if onlycross-ply laminates are considered and the Poisson ratios ν12 of thelayers are assumed to be zero. Then Equation B.1 becomes

Nx

Mx

=

[

A∗

11 B∗

11

B∗

11 D∗

11

]

ε0x

κx

. (B.2)

Under pure bending, the plate deformations ε0

x and κx are

ε0

x

κx

=1

A∗

11 · D∗

11 − (B∗

11)2

[

D∗

11 −B∗

11

−B∗

11 A∗

11

]

0Mx

. (B.3)

B.2 Through-the-thickness strain distribution 129

The direct strains at the top and bottom surfaces are shown inFigure B.2. The strain at the bottom surface becomes infinite for acurvature-radius-to-thickness ratio of R

t = 0.5. Contrary, the absolutestrain at the top surface decreases and becomes zero. The strain atthe bottom surface is 5% greater for a curvature radius to thickness ra-tio of 6 compared with the CLT and is still 1% greater for a ratio of 25.

100

101

102

−30

−25

−20

−15

−10

−5

0

5

10

log(R/t)

Direct strain [-]

Top

Bottom

Figure B.2: Curvature effect.

In contrast to the strain distribution caused by a bending load, thestrain distribution caused by a line load is still constant through thethickness in a homogeneous curved beam. The plate is now loadedwith a line force Nx and Equation B.2 becomes

ε0

x

κx

=1

A∗

11 · D∗

11 − (B∗

11)2

[

D∗

11 −B∗

11

−B∗

11 A∗

11

]

Nx

0

(B.4)

and the laminate deformations are

ε0

x = DA∗

11·D∗

11−(B∗

11)2 Nx

κx = −BA∗

11·D∗

11−(B∗

11)2 Nx .

(B.5)

These terms are combined with Equation 3.6


ε(z) =R

R + zNx

(

D∗

11

A∗

11 · D∗

11 − (B∗

11)2

+ z−B∗

11

A∗

11 · D∗

11 − (B∗

11)2

)

(B.6)

and is simplified whereas c = NxR

A∗

11·D∗

11−(B∗

11)2 .

ε(z) =c

R + z(D∗

11 − z · B∗

11) . (B.7)

The bending and coupling stiffness terms D∗

11 and B∗

11 are obtainedfrom the integration of the reduced stiffness term Q11 through thethickness.

B∗

11 = Q11

∫ t/2

−t/2

z · Z(z)dz

= RQ11

(

R log(

R−t/2R+t/2

)

+ t)

(B.8)

D∗

11 = Q11

∫ t/2

−t/2

z2 · Z(z)dz

= −R2Q11

(

R log(

R−t/2R+t/2

)

+ t)

(B.9)

These terms are combined with Equation B.7 and simplified.

ε(z) = cR+z

[

−R2Q11

(

R log(

R−t/2R+t/2

)

+ t)

+

−z · RQ11

(

R log(

R−t/2R+t/2

)

+ t)]

= Q11c

R+z

[

−R3 log(

R−t/2R+t/2

)

− R2t +

−z(

R2 log(

R−t/2R+t/2

)

− Rt)]

= Q11c

R+z

[

−(R + z)(

R2 log(

R−t/2R+t/2

)

+ Rt)]

= Q11 · R · c[

−R log(

R−t/2R+t/2

)

− t]

(B.10)

The final strain distribution caused by a normal force is independentof the through-the-thickness variable z and therefore constant.

B.3 ISS distribution 131

B.3 ISS distribution

The modified ABD matrix is used to compute the through-the-thickness ISS distribution in a singly curved plate using Rohwer’sapproach [15] and is compared with the original model for a flat plate.The stress distributions are evaluated in a UD, [0, 90]s, and [90, 0]slaminate with a curvature-radius-to-thickness ratio R

t of 1.69 whatcorresponds to the situations in region I and III of the specimen plot-ted in Figure 3.5. The maximum stress increases by 0.10% in the UDlaminate, and decreases by 0.12% and 0.64% in the [0, 90]s and [90, 0]slaminate compared with the model for a flat plate. The through-the-thickness ISS distribution in the [90, 0]s laminate is illustrated inFigure B.3. The modified approach is implemented in the analyticalmodels although the effect is smaller than 1% in symmetrical cross-plylaminates.

This effect increases in a [0, 90] laminate in which the 0 layers areplaced toward the center of curvature and the 90 layers are placedin the outer region. The maximum ISS increases by 6.94% in thisconfiguration if compared with the flat plate model.

−5 −2.5 0 2.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

z [mm]

INS

[N/m

m2]

Curved plateFlat plate

Figure B.3: Through-the-thickness ISS distribution including the cur-vature effect in the [90, 0]s laminate.


B.4 Solution of the differential equation

The shear term w,ϕϕ in Equation B.11 is replaced by a FSDT approachin Section 3.4.

w,rr +w,r

r− s2 w

r2+ f2 w,ϕϕ

r2+ P ∗ = 0 (B.11)

Two different homogeneous solutions including the shear term aregiven in Equation B.12

wH = a · rs + b · r−s + a′ · sinh(f∗ϕ) + b

′ · cosh(f∗ϕ)

wH = [a + b · log(r)][

a′ · sinh(f∗ϕ) + b

′ · cosh(f∗ϕ)] (B.12)

where a, b, a′

, and b′

are free parameters and f∗ =√

C55

C11. The

parameters a and b are used to fullfill the INS BC at the top andthe bottom surfaces of the laminate and the interface continuityconditions of the through-the-thickness displacement w and the INS.Unfortunately, no approach is found to determine the other two freeparameters which is the reason why the shear term is replaced by aFSDT approach.

Appendix C

Doubly curved finiteelement

C.1 Stress equations

The derivative of σr,r is

σr,r = − C31

r2

(

w + R1

(

ε0

ϕ + (r − R1)κϕ

))

+

+ C31

r (w,r + R1 · κϕ)+

− C32

(r+rd)2 (w + R2 (ε0

ϑ + (r − R1)κϑ))+

+ C32

r+rd(w,r + R2 · κϑ) + C33 · w,rr

(C.1)

and the terms σr − σϕ and σr − σϑ are

σr − σϕ = C31−C11

r

(

w + R1

(

ε0

ϕ + (r − R1)κϕ

))

+

+ C32−C12

r+rd

(

w + R2

(

ε0

ϕ + (r − R2)κϕ

))

+

+(C33 − C13)w,r + (C1i − C3i)εF

i

(C.2)

σr − σϑ = C31−C21

r

(

w + R1

(

ε0

ϕ + (r − R1)κϕ

))

+

+ C32−C22

r+rd

(

w + R2

(

ε0

ϕ + (r − R2)κϕ

))

+

+(C33 − C23)w,r + (C2i − C3i)εF

i

(C.3)

134 Doubly curved finite element

The derivatives of the in-plane displacements of Equation 4.9 are

u,ϕ = R1

(

ε0

ϕ + z · κϕ

)

(C.4)

u,ϕr = z · R1 · κϕ (C.5)

v,ϑ = R2 (ε0

ϑ + z · κϑ) (C.6)

v,ϑr = z · R2 · κϑ (C.7)

where the terms r − R1 and r − R2 + rd are replaced with z.

C.2 Linear system of equations

The derivatives of Equation 4.10 are replaced by the differential co-efficient (Equation 4.11) and the differential equation of the through-the-thickness displacement w becomes

(

−1 +d

2

)

wn−1 + (2 − d2qn)wn +

(

−1 − h

2pn

)

wn+1 + h2gn = 0

(C.8)whereas pn = p(rn), qn = q(rn), and gn = g(rn). The functions p(r),q(r), and g(r) of the linear second-order boundary value problem,given in Equation 4.10, are

p(r) =1

r+

1

r + rd(C.9)

q(r) =1

C33

(

− C11

r2− C22

(r + rd)2+

C13 + C23 − 2C12

r(r + rd)

)

(C.10)

g(r) = −P ∗ (C.11)

whereas P ∗ is given in Equation 4.9. The BCs for the INS at the topand bottom surfaces are expressed by the difference approximation

wn+1 − wn−1

2d+ α · wn = β . (C.12)

This equation is combined with Equation C.8 and the supportingpoints wn−1 and wn+1 are eliminated in the bottom and top BCs,respectively. The approximation differential equations at the bottomand top surfaces become

C.2 Linear system of equations 135

0 = −2wn+1 + wn

[

2 − d2 · qn + α(

−1 + d2 · pn

)]

+ (C.13)

+h2 · gn + β(

1 − h2 · pn

)

0 = −2wn−1 + wn

[

2 − d2 · qn + α(

1 + d2 · pn

)]

+ (C.14)

+h2 · gn − β(

1 + h2 · pn

)

(C.15)

where α and β are

α =2d

C33

(

C13

rn+

C23

rn + rd

)

(C.16)

β =2d

C33

(

−C13u,ϕ

rn− C23

v,ϑ

rn + rd

)

. (C.17)

Appendix D

Experimental data

D.1 Specimen geometry

The following tables list the exact geometry parameters of each speci-men. Clamping means the distance between the clamps of the testingmachine which is used for the FE simulation. The other parametersare illustrated in Figure 6.1.

138 Experimental data

Table D.1: Specimen geometry: Delamination due to INS.

Spec. Amplitude Length Thickness Depth Clampingh [mm] L [mm] t [mm] [mm] [mm]

T2 30 100 10.10 40 185T6 30 100 10.20 40 185T7 30 100 10.00 40 185

T1.1 20 80 7.60 40 172T1.2 20 80 7.85 40 172T1.3 20 80 8.00 40 172T1.4 20 80 7.45 40 172T3.1 20 80 8.00 40 172T3.2 20 80 8.30 40 172T3.3 20 80 8.05 40 172T3.4 20 80 8.05 40 172R1.1 20 80 8.10 32.5 190R1.4 20 80 8.50 32.6 190R1.5 20 80 8.50 32.3 190R2.1 20 80 8.85 32.4 190R2.2 20 80 9.15 32.6 190R2.4 20 80 8.90 32.6 190

Table D.2: Specimen geometry: Delamination due to ISS.

Spec. Amplitude Length Thickness Depth Clamping[mm] [mm] [mm] [mm] [mm]

C1 30 100 10.05 40 185C2 30 100 10.10 40 185C3 30 100 10.15 40 185C4 30 100 10.05 40 185

C2.1 20 80 9.55 40 172C2.2 20 80 9.55 40 172C2.3 20 80 9.65 40 172

D.1 Specimen geometry 139

Table D.3: Results of the ISS tests. Difference between the reactionforces obtained from the tests and FE results and maximum ISS.

Spec. Force Disp. Diff. max. ISS[N ] [mm] % [N/mm2]

C1 28473 4.669 12.3 87.65C2 24439 3.475 1.7 74.53C3 26105 5.493 39.9 79.87C4 28815 4.670 11.0 88.57

C2.1 23596 1.117 19.5 83.14C2.2 25866 1.077 5.1 84.52C2.3 26294 1.100 8.2 84.92

Table D.4: Specimen geometry: Matrix failure.


CP1 30 100 10.10 40 185CP2 30 100 10.20 40 185CP3 30 100 10.00 40 185CP4 30 100 10.00 40 185

Table D.5: Maximum load in the [013, 9013]s laminate.

Spec. Force Disp.[N ] [mm]

CP1 2911 0.366CP2 6457 0.549CP3 7159 0.610CP4 4011 0.422


Table D.6: Specimen geometry: Series with teflon plate.


M1.1 20 80 8.70 19.50 175M1.3 20 80 8.90 19.50 175M1.4 20 80 8.75 19.60 175M1.5 20 80 8.80 16.45 175M1.6 20 80 8.85 19.70 175M1.7 20 80 8.60 19.55 175M1.8 20 80 8.50 19.35 175M1.9 20 80 8.50 19.50 175

D.2 CT images 141

D.2 CT images

0 100 200 300 400 50050

100

150

200

250

Pixels [−−]

Side view: "ber direction

0 50 100 150 2000

50

100

150

200

250

Pixels [−−]

Side view: thickness direction

0 100 200 300 400 5000

50

100

150

200

250

Pixels [−−]

Top view: transverse direction

0 100 200 300 400 5000

50

100

150

200

250

Pixels [−−]

Top view: "ber direction

Air lock

Air lock

Air lock Air lock

Figure D.1: Density profile plots of the air lock detected in the singlycurved specimen.

Figure D.2: CT images of a specimen with embedded break point(teflon plate). Side view on the left and top view on the right.


D.3 Optical fiber measurement

These tests aimed to record the strain εx inside the specimen where themaximum INS appears. The strain distribution at the end face can bemeasured using strain gauge pattern or an electronic speckle patterninterferometry (ESPI) method but were not performed because theINS decreases towards the end face.

OFs have a diameter of 0.145mm and should not affect the stiff-ness and strength of a specimen. Furthermore, fiber bragg gratingsensors allow to measure the strain at a certain point and can be po-sitioned accurately. OFs are immune to electromagnetic interferenceand have a high temperature resistance (∼ 200C). An overview ofOF bragg grating sensors in composite laminated plates is given byBotsis [130, 131].

First, it had to be confirmed that the OFs do not affect the specimenproperties (stiffness and strength) and if the OFs overcome themanufacturing process. A specimen series with and without OFs wasmanufactured. The UD laminate was made up of 45 layers wherethe OFs were embedded between the 23rd and 24th layer. The OFswere manufactured by Smartec SA1 and a polyimide coating wasused to achieve a high temperature resistance. The center wavelengthof the OFs was ±1532µm. The OF specification is given in Table D.8.

After the curing process and the cutting, it was confirmed that a beamof light still passes through the OFs. The losses did not exceed 1.71dBwhich is far below the allowed losses of 30dB for the MuST2 system.Smartec’s long experience confirms that a fiber without coating cannot survive embedding and polyimide coating provides excellent straintransfer between the monitored material and the optical fiber. Thusit is not necessary to remove the polyimide coating nearby the bragggrating what does not decrease the accuracy of the measurement.

The specimens with an embedded OF were loaded in tension tomeasure the stiffness and strength. The maximum cross-section loadwas between 52 and 60N/mm2 which was equal or even surpassedthe results of the specimens without break point illustrated in Fi-gure 6.9. Further, the specimens with and without OFs had the

1http://www.smartec.ch2Multiplexed strain and temperature monitoring system based on fiber bragg

gratings

D.3 Optical fiber measurement 143

same force-displacement relation. These results indicate that theoptical fibers did not alter the specimens structural properties andstrength significantly. Therefore, specimen with embedded OF weremanufactured where the bragg grating was placed at the maximumINS location.

As mentioned before, the thickness of the specimens varies. In addi-tion, it can not be assumed that all layers have the same thicknessand that the OF is exactly placed in the midplane of the specimen.Micrographs were made at the maximum INS location to define thelocation of the OF and the images are plotted in Figure D.3. Thelayer interfaces are still discernible and the OF is good surroundedby the fibers. In addition, these polished cross-section images revealthat the delaminations run horizontally.

0.1mm

1mm

1mm

Figure D.3: Micrograph of laminates with an embedded OF betweenthe 23th and 24th layer.

The specimens were loaded in tension and the strains in fiber directionat the maximum INS location, obtained from the OFs and FEMs, arecompared. The through-the-thickness position of the OF is definedbased on the polished cut images. Six specimens were tested and theminimum and maximum strain εx was −270 and −57µm/m, respec-


tively. The strains differ by factor 2 up to 16 if compared with theFEM results. Two principal reasons cause this high deviation. First,the absolute measured strain is very small. The resolution of OFbragg grating is 1µ/m what is in some cases only one order of magni-tude smaller than the measured number. Further, the strain in fiberdirection εx is sensitive regarding the through-the-thickness direction.The through-the-thickness distribution of εx is illustrated on the leftof Figure D.4. The maxima are at the top and bottom surfaces dueto the bending moment and the strain is close to zero at the midplane.

−4.15 −2.08 0 2.08 4.15

−2

−1

0

1

2

3

4

x 10−3

Thickness z [mm]

Str

ain

ε

[−−

]

Strain in !ber direction

x

−0.2 −0.1 0 0.1 0.2

−14

−12

−10

−8

−6

−4

−2

0

2

4

6x 10

−5

Distance [mm]

Str

ains

x [−

−]

Thickness directionFiber directionTransverse directionOFBoundary OF

Figure D.4: Through-the-thickness distribution of εx obtained fromsolid FE analysis (left) and the sensitivity in thickness, fiber, andtransverse direction (right).

The right figure shows the strain εx against the thickness, fiber, andtransverse direction. The strain changes moderately in the fiber di-rection and is constant in the transverse direction what indicates auniform strain field in these directions. It can be therefore assumedthat the error is not caused by a wrong positioning of the bragg gratingin these directions.

The boundary of the OF (Figure D.4) represents the diameter ofthe OF where the measured value (−80µm/m) is placed at the cen-ter. The correlation between the strain and the thickness direction,obtained from the FE analysis, is −40µm/m per 0.1mm at this lo-cation. Therefore, the difference of the strain between the top andthe bottom of the OF is 52µm/m what is more than the half of themeasured value. The strain distribution in the thickness direction dis-closes a non-uniform strain field where a peak shift measurement byoptical fibers implies a uniform strain field.

D.4 Strain sensors 145

However, the principal issue is that the exact through-the-thickness location of the OF at the maximum stress location has to bemeasured to achieve accurate results. Polished cross-section imagessuggest one possibility but are time consuming and the delaminationcracks also affect the thickness of the probe.

D.4 Strain sensors

Table D.7: Strain gauge specification.

Type 1.5/350LY11Resistance 350 Ω ±0.35%Gauge factor 1.93 ±1.5%Transverse sensitivity 0.1%Temp. coefficientof gauge factor 104 ±10 [10−6/C]Gauge pattern 1.5x1.2mmCompany HBM

Table D.8: Optical fiber specification.

System MuSTCoating PolyimideBragg-grating length 5mmDiameter without coating 0.125mmDiameter with coating 0.145mmCenter wavelength 1532µm± 0.35µmReflection 85% ± 5%Product code SF-S36P-0004Company Smartec

List of Tables

3.1 Material properties Elitrex EHKF. . . . . . . . . . . . 38

4.1 Aluminum alloy material properties. . . . . . . . . . . 574.2 UD material properties. . . . . . . . . . . . . . . . . . 59

5.1 Maximum deformations (mm) obtained from a 3- and4-node shell FEM with a reduced (RI) and full integra-tion (FI) scheme. . . . . . . . . . . . . . . . . . . . . . 66

5.2 Fiber material properties/ . . . . . . . . . . . . . . . . 795.3 Doubly curved model assessment. . . . . . . . . . . . . 835.4 ±45 weave material properties. . . . . . . . . . . . . . 855.5 Maximum INS in the doubly curved L probe. . . . . . 865.6 Lamina strength properties [N/mm2]. . . . . . . . . . 89

6.1 Curing conditions. . . . . . . . . . . . . . . . . . . . . 966.2 Displacements, loads, percent differences of Ftot, and

maximum INS. . . . . . . . . . . . . . . . . . . . . . . 1016.3 Measured and evaluated strains εx at the top and bot-

tom surfaces. . . . . . . . . . . . . . . . . . . . . . . . 110

A.1 Series 1. . . . . . . . . . . . . . . . . . . . . . . . . . . 124A.2 Series 2. . . . . . . . . . . . . . . . . . . . . . . . . . . 125A.3 Series 3. . . . . . . . . . . . . . . . . . . . . . . . . . . 126

D.1 Specimen geometry: Delamination due to INS. . . . . 138D.2 Specimen geometry: Delamination due to ISS. . . . . . 138D.3 Results of the ISS tests. Difference between the reac-

tion forces obtained from the tests and FE results andmaximum ISS. . . . . . . . . . . . . . . . . . . . . . . 139

148 LIST OF TABLES

D.4 Specimen geometry: Matrix failure. . . . . . . . . . . . 139D.5 Maximum load in the [013, 9013]s laminate. . . . . . . 139D.6 Specimen geometry: Series with teflon plate. . . . . . 140D.7 Strain gauge specification. . . . . . . . . . . . . . . . . 145D.8 Optical fiber specification. . . . . . . . . . . . . . . . . 145

List of Figures

2.1 Laminated composite beam and qualitative load distri-bution. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 ISS distribution in an orthotropic plate under cylindri-cal bending. . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Interlaminar stresses in a [(±30)2, 90]s laminate. Qual-itative distribution between -30 and 90 laminae overthe wide-section width. . . . . . . . . . . . . . . . . . . 21

2.4 Singly curved specimen and 8-node brick FE mesh with15 elements through the thickness. . . . . . . . . . . . 23

2.5 Convergence of maximum INS in a singly curved UD(left) and [0, 90]s (right) laminate. . . . . . . . . . . . 24

2.6 INS distribution of linear (left) and quadratic (right)solid brick FEs in a UD laminate. Mesh size: 1 to 25elements through the thickness. . . . . . . . . . . . . . 25

2.7 INS distribution of linear (left) and quadratic (right)solid brick FEs in a [0, 90]s laminate. Mesh size: 1 to7 elements per layer. . . . . . . . . . . . . . . . . . . . 25

3.1 Constant curved finite segment and strain distribution. 283.2 CS of the reference element and the coordinates of the

nodes (4-node element). . . . . . . . . . . . . . . . . . 293.3 Curvature radius convention: Positive curvature R left

and negative curvature R right. . . . . . . . . . . . . . 293.4 Kinematics (left) and loads (right) of the curved laminate. 303.5 Singly curved specimen geometry. . . . . . . . . . . . . 363.6 Internal line forces Fx and Qϕr (left) and line moment

My (right) distribution in the singly curved specimen. 373.7 Displacements of the singly curved specimen. . . . . . 38

150 LIST OF FIGURES

3.8 INS [N/mm2] distribution in the UD composite specimen. 39

3.9 z-coordinate in region I and III. . . . . . . . . . . . . . 39

3.10 UD laminate: Comparison of the through-the-thicknessINS distribution between the analytical model and FEM. 40

3.11 [0, 90]s laminate: Comparison of the through-the-thickness INS distribution between the analyticalmodel and FEM. . . . . . . . . . . . . . . . . . . . . . 41

3.12 [90, 0]s laminate: Comparison of the through-the-thickness INS distribution between the analyticalmodel and FEM. . . . . . . . . . . . . . . . . . . . . . 42

3.13 Shear deformation in a curved beam. Constant shearangle γ (left) and constant thickness t (right). . . . . . 43

3.14 [0, 90]s laminate: Comparison of the analytical modelwith and without shear. . . . . . . . . . . . . . . . . . 46

3.15 Geometry of the 10mm thick ellipse and loads. . . . . 47

3.16 INS distribution of the centerline along length directionx. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.1 Doubly curved FE geometry. . . . . . . . . . . . . . . 52

4.2 Integration scheme in a single layer (left) and multilayerlaminate (right) where the indices 1, 2, and 3 count thelayers and σ means σr. . . . . . . . . . . . . . . . . . . 55

4.3 Cross-section of the doubly curved shell. . . . . . . . . 56

4.4 Solid (left) and shell (right) FE mesh. . . . . . . . . . 57

4.5 Through-the-thickness INS distribution of the 10mmthick isotropic specimen. . . . . . . . . . . . . . . . . . 58

4.6 Maximum INS in the center of the isotropic specimen. 59

4.7 Through-the-thickness INS distribution of the 7.5mmthick specimen. . . . . . . . . . . . . . . . . . . . . . . 60

4.8 Global INS distribution in a UD 10mm thick laminateat the thickness-direction level where the maximumINS appears. . . . . . . . . . . . . . . . . . . . . . . . 60

4.9 Maximum INS in the UD laminate (left) and the cross-ply laminate (right). . . . . . . . . . . . . . . . . . . . 61

5.1 Loads and 10x10x1mm aluminum alloy plate. . . . . . 65

5.2 Corner and mid-nodes of a 8- and 6-node element. . . 69

5.3 Approximation of a parabolic surface based on 4-nodeelements. . . . . . . . . . . . . . . . . . . . . . . . . . 70

LIST OF FIGURES 151

5.4 Influence of the element type and size regarding theglobal INS distribution (nodal results). Layer wherethe maximum INS appears. . . . . . . . . . . . . . . . 71

5.5 A 3-node mesh on a singly curved shell causes a zic-zacpattern in the original straight y-direction. . . . . . . . 71

5.6 Influence of the element type and size regarding theglobal INS distribution in a doubly curved geometry(nodal solution). . . . . . . . . . . . . . . . . . . . . . 72

5.7 Singly curved specimen with the parameters amplitudeh and thickness t. . . . . . . . . . . . . . . . . . . . . . 74

5.8 Feasible geometries for different parameter values. . . 745.9 FE mesh and INS distribution in the cross-section of

interest (element solution). . . . . . . . . . . . . . . . 755.10 Maximum INS [N/mm2] in a isotropic structure for

different CT-ratios. Solid FE results (left) and newmodel (right). . . . . . . . . . . . . . . . . . . . . . . . 76

5.11 Percent difference of the post-processing method inisotropic structures. . . . . . . . . . . . . . . . . . . . 77

5.12 Maximum INS [N/mm2] in a UD laminate for differentCT-ratios. Solid FE results (left) and new model (right). 77

5.13 Percent difference of the post-processing method in UDlaminates. . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.14 Anisotropy distribution and percent difference of thenew model. . . . . . . . . . . . . . . . . . . . . . . . . 79

5.15 Percent difference of the new model for angle-ply lam-inates [0, α, β, 0]. . . . . . . . . . . . . . . . . . . . . . 81

5.16 Doubly curved geometry and plan view. . . . . . . . . 825.17 L-probe and global INS distribution. . . . . . . . . . . 865.18 Maximum inverse reserve factor of Hashin’s failure cri-

teria for all layers. . . . . . . . . . . . . . . . . . . . . 885.19 Critical stress component of part A in each element for

all layers: s3t = positive INS; s1c = fiber compression;s1t = fiber tension; s13 = ISS. . . . . . . . . . . . . . 88

5.20 Global INS distribution in a thick-walled curved laminate. 90

6.1 Geometry of the singly curved specimen. . . . . . . . . 946.2 Positive aluminum alloy mold and vacuum bag. . . . . 966.3 Experiment facility and delaminated specimen. . . . . 986.4 Tension test: Force-time diagram and acoustic emission

events. . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

152 LIST OF FIGURES

6.5 Localisation of AE events along the longitudinal spec-imen axis. . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.6 Matrix failure in the [0, 90]s laminate. . . . . . . . . . 1036.7 Transverse matrix stresses in region II-III with (left)

and without (right) temperature effect. . . . . . . . . . 1046.8 Specimens with integrated teflon plates at the maxi-

mum INS location. . . . . . . . . . . . . . . . . . . . . 1066.9 Force-displacement diagram of the specimens with

break point (−) and without (- -). . . . . . . . . . . . 1076.10 CT images of specimen 7 which show an air lock in

longitudinal direction. Side view on the left and topview on the right. . . . . . . . . . . . . . . . . . . . . . 107

6.11 1.2 × 1.5mm strain gauge on the top surface at themaximum strain location. . . . . . . . . . . . . . . . . 109

6.12 Strain gauge results at the bottom and top surfaces ofspecimen R2.2. . . . . . . . . . . . . . . . . . . . . . . 109

6.13 Sensitivity of the INS regarding thickness, curvatureradius, and Young’s moduli. . . . . . . . . . . . . . . . 112

B.1 Curvature effect: Reinforcement of A and D matrix(left) and negative coupling effect (right). . . . . . . . 128

B.2 Curvature effect. . . . . . . . . . . . . . . . . . . . . . 129B.3 Through-the-thickness ISS distribution including the

curvature effect in the [90, 0]s laminate. . . . . . . . . 131

D.1 Density profile plots of the air lock detected in thesingly curved specimen. . . . . . . . . . . . . . . . . . 141

D.2 CT images of a specimen with embedded break point(teflon plate). Side view on the left and top view onthe right. . . . . . . . . . . . . . . . . . . . . . . . . . 141

D.3 Micrograph of laminates with an embedded OF be-tween the 23th and 24th layer. . . . . . . . . . . . . . 143

D.4 Through-the-thickness distribution of εx obtained fromsolid FE analysis (left) and the sensitivity in thickness,fiber, and transverse direction (right). . . . . . . . . . 144

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Own publications

[1] G. Kress, R. Roos, M. Barbezat, C. Dransfeld, and P. Ermanni.Model for interlaminar normal stress in singly curved laminates,Composite Structures, 69:458-469, 2005.

[2] R. Roos, G. Kress, M. Barbezat, and P. Ermanni. Enhancedmodel for interlaminar normal stress in singly curved laminates,Composite Structures, 80:327-333, 2007.

[3] R. Roos, G. Kress, M. Barbezat, and P. Ermanni. A post-processing method for interlaminar normal stresses in doublycurved laminates, Composite Structures, 81:463-470, 2007.

[4] R. Roos. Interlaminar normal stresses in composite materialsusing shell element for strength analysis. Proceedings of Com-

posite & Polycon, American Composites Manufacturers Associ-

ation, Tampa, FL USA, October 2007.

[5] R. Roos. Model for interlaminar normal stresses in doublycurved thick-walled laminates. Proceedings of NAFEMS Semi-

nar, Bad Kissingen, Germany, October 2007.

[6] O. Konig, U. Mennel, R. Roos, and M. Wintermantel. Finite-element postprocessing of laminated composite structures. Pro-

ceedings of NAFEMS Seminar, Bad Kissingen, Germany, Oc-tober 2007.

Curriculum VitaeName: Rene Roos

Date of Birth: February 27, 1979Nationality: Switzerland

Address: Allruti 4CH-6343 Rotkreuz

EDUCATION

Ph.D., ETH Zurich Jun04 – Dec07

Institute: Centre of Structure Technologies

Study Mechanical Engineering, ETH Zurich Oct99 – Feb04

Subjects: - Aerospace and lightweight structures- Composite structures- Power generation and mobility

Diploma thesis: Interlaminar stresses and strength

WORKING EXPERIENCE

20% employment, Marz Ofenbau AG, Zurich Nov02 – Mar03

Construction and engineering

Internship, Marz Ofenbau AG, Zurich Jul02 – Oct02

ConstructionPreliminary design of a limestone charging process

Internship, Aebi MFH AG, Hochdorf Feb00 – Apr00

Product development

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