Experimental and numerical determination of stiffness and strength og GRP/PVC sandwich structures

SIDSEL M. BECH

EXPERIMENTAL AND NUMERICAL

DETERMINATION OF STIFFNESS

AND STRENGTH OF GRP/PVC

SANDWICH STRUCTURES

Scheepshydrome"'ica

Archief

Mekeiweg 2, 2628 CD Deift

-Te1:-O-1S--7-868-7-3i-Fax: 78183 6

DOKTOR INGENIØRAVHANDLING 1994:127 INSTITUTF FOR MARTh KONSTRUKSJONER TRONDHEIM

UNIVERSITETET I TRONDHEIM

NTh-TRYKK

1994

www

Mekeiweg 2,

2B co Deift

T& 013-Th8873 Fait O15..781O

Experimental and Numerical Determination of

Stiffness and Strength of

GPIPVC Sandwich Structures

by

Sidsel M. Bech

Trondheim, January 1995

DEPARTMENT OF MARINE STRUCTURES

THE NORWEGIAN INSTITUTE OF TECHNOLOGY

Sandwich GRP skin/PVC core panels areattractive structural elements in high speed passenger vessels

and car femes due to their high strength to weight ratio, especially under lateral loads. With an

increasing ize and speed of such vessels, more knowledge about their performance is required. Especially lateral loading such as wheél Ïoads in car ferries may be critical.. Sandwich panels subjected

to coñcentrated loads from patch (wheel) loads adjacent to and above the panel support (bulkhead)

experiènce shear and compressive stresses which are particularly high in this area.

The main purpose of this work was twofold. The first objective was to develop and calibrate an experimental method for internal strain meastirement by means of strain gauges in a foam core

material. This thesis describes the test method and problems which arise during the process. The second objective was to examine the through thickness strain in a deck/bulkhead Joint (T-joint) configuration This problem is approached by means of both experimental and numerical methods.

The use of strain gauges makes it possiblé to measure the strain inside the foam, no other experimental method isavailable foE this purpose. The nature of the foam (porous maxerial,.low stiffness etc.) makes

it necessary with a good understanding of the subject to obtain satisfactory reralts. A detailed

explanation of each step in the procedure which was developed is therefore given. The numerical mOdels must include the features of the foam which are essential to the particular problems. The

behaviour of the foam is thus essential In both the eiq,erimental and numerical analysis and is therefore

considered as an important topic in this thesis.

The experimental technique for the T-joints mentioned above was developed in steps. As itis mainly compressive and shear stresses whiçh arise in the foam of the T-joints of interest, specimens of simple geometry exposed to loading which produce approximately pure compressive and pure shear strains

in the matenal have been tested The results which were achieved have been compared with FEM

analyses and analytical calculations to verify the use of stráth gauges in PVC core material. The

behaviour of the foam matenal is also discussed and the charactenstics of material properties of foam core material are determined.

ii Experimental and Numerical Determination of Stiffiiess and Strength of GRP/PVC Sandwich

Structures

The sandwich T-joints which were investigated represented common building practice of a

deck-bulkhead construction. Laboratory tests on joints with two different comer supports subjected to patch

(wheel) loads acting adjacent to and above the web have been accomplished. The core and laminate strains of the T-joints were recorded and the test results were compared with numerical solutions. A

strength analysis of the core material was performed and the influence of the different joint geometries and the different load situations on the stress and strain levels was investigated. Simplified method for stress and strain determination of the core is proposed.

The numerical results were obtained by the finite element method using the programs PATRAN and

ABAQUS. The model discretization was based on three dimensional quadrilateral and wedge solid

elements with quadratic displacement polynomial. A linear elastic material model was used for all the materials. The material properties of the total laminates were determined by means of tension tests and spreadsheet program (MIC-MAC) and specified by an orthotropic material card in the finite element model.

ACKNOWLEDGEMENT

This work has been carried out at the Department of Marine Strutures at the Norwegian Institute of Technology under the supervision of Professor Torgeir Moan His guidance and valuable comments

on my research are most gratefully acknowledged I wish to thank him for inspiring and encouraging me throughout my study.

I wish to thank Dr. Jon Taby and Professor Karl-Axel Olsson for their paiticipation as members of the

thesis committee, for valuable discussions dining the eaíly stage of my research and also for their

encouragement during the exciting end of my study.

I am particular grateful to Mr. Kjell Peflersen, Mr. Roar Schjethe, Mr. Eirik Fleischer and Mr. Terje Lines at the Marine Structures Laboratory Department of Manne Technology NTH for their invaluable

assistance and specific comments during the test work. Their encouragement and support are also

highly appreciated.

I would like to thank the companies Ulstein International A.S. and Eikefjord Marine A.S. represented

by Mr. Harald Nordal and Mr. Arnulf Aa for kindly providing the testspecimens.

I am also indebted to Dr. Brian Hayman, DnV, Professor ClaesGøran Gustafson, NTH and Mr. BjØrn

Høyning, FiReCo for helpful comments and ideas in coimetion with this work.

The Marine Technology Cenire has been an inspiring environment for these studies. My thanks go to

my colleagues at NTH and SINFEF who have been invariably helpful and encouraging Gratitude is especially expressed to Dr Elizabeth Passano för her invaluable support throughout my

study Mr

Hâvard Holm's assistance in connection with computer related problems and Mr. Egil Pettersen's

contribution with the draw ngs of Figures 1.1, 7.3 and 8.5 are also specially appreciated.

This study was made possible by grant from The Royal Nòrwegian Cöuncil for Scientificand Industrial Research, now part of the Research COuncil of Norway. Thànks are also extended to the Faculty of

iv Experimental and Numerical Determination of Stiffness and Strength of GRP/PVC Sandwich

Structures

I am also grateful to the Norwegian Supercomputing Committee (TRU) who supported the computer

time on CRAY Y-MP.

Lastly, I would like to thank my whole neglected family for their support. Special thanks go to my parents for their continuous encouragement, to my beloved Saeed who give me go-ahead spirit and strength and to my younger sister June who always understands and inspires me. Their patience and

impatience during this busy but also exciting and challenging period is very much appreciated.

Trondheim, January 17th, 1995

S.cJ '1Aa

rßDCM

TtuLE OF

CONTENTS

2. Sandwich 2.1 2.1 Generäl 2.1 2.2 Material 2.3 2.3 Foam 2.4 2.3.1 General 2.4

23.2

Defomiatiöñ Mechanisms of Foams 2.4 2.3.3 Mechanical Properties of Foam in Compression 2.6

2.4 Laminate 2.15

2.4.1 General 2.16

2.4.2 Elastic Constants 2.16

2.4.3 Restrictions on Elastic Constants 2.20

2.4.4 Laminate Behaviour 2.22

Abstract

Acknowledgements Lu Table of Contents V Nomenclature

Introduction

1.1 1.1 General 1.1

1.2 Objective and Scope ofWork 1.5

1.2 Previòus Work 1.4

vi Erperimenthi and Numerical Determination of Stiffness and Strength of GRP/PVC Sandwich

Structures

SttiiEl Analysis

3.1

3.1 General 3.1

3.2 Fmite Element Analysis 3.3

3.2.1 Modelling of Sand'vrich Structures 3.3

3.2.2 Computer Methods 3_5

3.2.3 Specific Formulations 3.6

3.2.4 Accuracy 3.13

Experimental Methods and Determinafion of Material Properties 4.1

4.1 Consideratións about Foam Measurements 4.1 4.2 Shear Strain Determined by Strain Gauges 4.4

4.3 Experimental Methods . 4.5

4.3.1 Foam in Compression 4.5

4.3.2 Foam in Shear 4.9

4.3.3 Sandwich Beam in Bending 4.12

4.3.4 Laminate; in Plane Properties 4.16 4.3.5 Laminate; out of Plane Properties 4.19

S Compressive Specimens 5.1

5.1 Introduction 5.1

52

Matenal Propertes . 5.2

5.2.1 Material of Compressive Test Specimens 5.2

5.2.2 Mechanisms of Foams in Compression 5.2

5.3 Test Method

52

5.3.1 Specimen Dimension 5.2 5.3.2 Adhesives 5.2 5.3.3 Strain Gauges 5.3 5.3.4 InstrUmentation 5.4 5.3.5 Testing Procedure 5.5

5.4 Behaviour, of the Foam . 5.6

5.5 Test.Results 5.7

5.5.1 Sources of Errors . 5.8

53.2

Compressive Properties 5.8

53.3

Measured Strain 5.8

5.5.4 Modulus of Elasticity 3.11

5.6 Conclusions and Recommendations for Further Work 5.12

6. Shear Specimens 6.1

6.1 Introduction 6.1

6.2 Material Properties 6.2

6.2.1 Material of Shear Test Specimens 6.2

6.3 Test Method IT 6.2

6.3.1 Standards 6.2

6.3.2 Specimens 6.2

6.3.3 Adhesives 6.2

6.3.4 Strain Gauges 6.4

6.3.5 Möunt ng of Strain Gauges 6.4

6.3.6 Instrumentation 6.5

6.3.7 Testing 6.6

6.4 Structural Analysis 6.7

6.5 Comparison of Experimental and Numerical Results 6.8

6.5.1 Sources of Errors 6.8

6.5.2 Shear Properties 6.9

6.5.3 Strain in the Foam Specimens 6.9 6.5.4 Stress in the Steel Plates 6.15 6.6 Conclusions and Reconimendation for Further Work 6.15

7. Flexural Specimens 7.1 7.1 Introduction 7.1 7.2 Test Method 7.1 7.2.1 Standards 7.1 7.2.2 Specimen 7.2 7.2.3 Adhesives 7.2

7.2.4 Stran Gauge Choice 7.2

7.2.5 Mounting of Strain Gauges 7.2

7.2.6 Instrumentation 7.4

7.2.7 Testing 7.6

7.3 StructuraI Analysis 7.6

7.4 Comparison of Experimental and Ñumèiiál Results 7.7

7.4.1 Sources of Errors 7.7

7.4.2 Analytical Results 7.8

7.4.3 Displacements 7.8

7.4.4 Strain in the Foam Core 7.9

7.4.5 Strain in the Laminates 7.11

7.5 Comparison of Shear and Bending Specimens 7.12

7.6 Conclusions and Recommendations for Fer Work 7.13

8. T-joints 8.1

8.1 Introduction 8.1

8.2 Test Method

86

8.2.1 Test Series 8.6

8.2.2 Preliminary Work 8.7

8.2.3 Stràin Gauge Choice 8.7

viii Experimental and Numerical Dererminafion of Stiffness and Strength of GRP/PVC Sandwich Structures

8.2.5 Mounting of Strain Gauges

88

8.2.6 Adhesive 8.10

8.2.7 Test Set Up 8.12

8.2;8 Testing Procedure 8.13

8.3 Determination of MatOrial PropertiCs 8.13

8.3.1 Laminate 8.13 8.3.2 Foam 8.19 8.3.3 Rubber 8.20 8.3.4 Steel 8.20 8.4 Structural Analysis 8.20 8.4.1 FEM Considerations 8.21 8.4.2 FEM Analyses 8.25

8.4.3 Accuracy of the FEM models 8.29

8.5 Comparison of Test Results and FEM Prediction for Foam 8x33 8.5.1 Sources of Eirors in the Test Results 8.34

8.5.2 Eccentriç Loading on T-joint with Triangular Foam Inset Support 8.35 8.5.3 Eccentriu Loading on T-joint with Filler Fillet Support 8.42

8.54

Centric Loading on T-joint with Triangular Foam Insert Support 8.46 8.5.5 Centric Loading on T-joint with Fifler Fillet Support 8.51

8.56

Concluding Remarks on Foam Strain Comparison 8.57

8.6 Comparison of Test Results and FEM Prediction of Laminates 8.61 8.6.1 Eccentric Loading on t-joint with Triangular Foam Insert Support 8.61

8.6.2 Eccentric Loading on T-joint with Filler Fillet Support 8.62 8.6.3 Centric Loading on T-joint with Triangular Foam Insert Support 8.63 8.6.4 Centric Loading on T-joint with Filler Fillet Support 8.63

8.7 Design Considerations 8.64

8..7. I Introduction 8.64

8.7.2 Strength Analysis of the Foam Core of the T-joints 8.65

8.7.3 Estiìnation of Stresses and Strains in the Foam 8.70 8.8 Conclusions and Recommendations for Further Work 8.83

9.

Condusions and Recommendations for Further Work

9.1

9.1 Behaviour of Simple Specimens 9.1

9.1.1 Compressive Loading 9.1

9.1.2 Shear Loading 9.3

9.1.3 Flexwal Loading 9.4

9.1.4 Comparison of Results for the Shear and Flexural Loading 9.5

9.2 Behaviour of T-joints 9.6

9.3 Final Remarks on the Experimental Technique 9.10

A. Compressive Speç mens

£1

A. I Tables of Test Results A.l

A.2 Load- Strain-Displacement Curves in Compression A.4 A.3 Additional Load- Strain-Displacement Curves in Compression A.lO

A.4 Curves of Modulus of Elasticity in Compression versus Displacement A.13

A.5 Material Data A.14

A.6 Producers/DistributOrs A.14

A.7 Standard Deviatiön A.14

B. Shear Specimens B.1

B.! Tables of Test and FEM Results B.l

B.2 Laboratory Results B.7

B.3 Finite Element Results B.lO

B.4 Material Properties B. 14

C. Flexuràl Specimens C.1

C.l

Tables of Test and FEM Results C.!

C.2 Laboratory Results C.5

C.3 Finite Element Results C.8

C.4 Material Properties C.l i

D. T-joints D.1

D. i Description of Specimen Production D.1

D.2 Material Properties of the T-joints D.5 D.3 Results of the FEM Accuracy Models D. 17

D.4

Tabulated Laboratory and FEM Results of the T-joints D.22

D.5 FEM Results of the T-joints

p.27

D.6 Laboratory Results of the F-joints D.45

D.7 Design Considerations D.59

x Experimental and. Numerical Determination of Stiffness and Strength of GRP/PVC Sandwich Srrucrures

NOMENCLATURE

Symbols Roman

a Compliance matrix term

a* Normalized compliance matrix term

A Area

A Effecthe area

A0 - Initial area [A] In-plane matrix

[A*] Nórmalized m-plane matrix

b - Compliance matrix term

b* _{Nrmalized compliance matrix term} [B] Coupliñg matrix

[B I - Normalized coupling matrix B - The "strain-displacement" matrix

c - Core thickness

C Constant

[C] Stiffness matix

C Stiffness matrix d Compliance matrix term

d1,d2,d3 Diagonal

d* Normalized compliance matrix tenu

D FleAural rigidity [D] Flexural matrix

[D I - Normalized flexural matrix

E Young's modulus

E Modulus of core

Ecomp Compressive modulus

xii Experimental and Numerical De:er,nination of Stiffness and Strength of GRP/PVC Sandwich Structures

E Flexural stiffness

Ef, Young's modulus of fibre in the longitudinal direction Efy Young's modulûs of fibre in the ttansverse direction E Contribution to Young's modulus from gas pressure in a cell Em Young's modulus of matrix

E Young's modulus out of the laminate plane

E° In plane stiffness

E Young's modulus of the solid of the foam

Tensile secant modulus of elasticity at x% strain

Young's modulus of a unidirectional ply in longitudinal direction Young's modulus of a unidirectional ply m the transverse direction

Ei* _{Young's modulus of fóam normal to the rise direction} - Young's modulus of foam in the rise direction E Young s modulus of the foam

f

Body force vector

F Load

F1 Load at the yield point

Fm Load at the yield point

F Load coiresponding to x% strain

F

Concentrated force vector

G Shear modulus

G* - _{- Shear thodulus of foam}

of

Shear modulus of fibre

Gm Shear modtilus of matrix

G0 Shear modulus out of the laminate plane Gp Shear modulus of unidirectional ply

h Total thickness Height of cell

h0 Portion of the thickness I SecOnd moment of area

J - Jacobian

J

JacObian operator matrix

[k] Curvature vector

k Element stiffness matrix

K Bulk modulus

Length

1,m,n - Direction cosines Span length

L Length

L0 Gauge length of extensometer

L Span length

M Arithmetic iiieañ

M Moment

M., Moment of plastic collapse

n - Numbers of observations

n Number of nodes

[NI - In-plane load vector

N Interpolation polynomial manu

p Gas pressure after compression Po Initial gas pressure

P Load

[Q) Stiffness matrix in local coordinates

[Q'] Stiffness matrix in global coordmates R Shape-an sotropy rátio

R Magnification ratio of extensometer

R

NOdal force vector Body force vector

- Surface force vector Initial strain force vector

ic,°

InitiaJ stress force vector

s Length

S Stiffness of cell edges

S Core shear stress

S - Surface

[SI Compliance matrix S Compliance matrix

SD Standard deviation

t Thickness

te Thickness of cell edge tf Thickness of cell face

tf Laminate thickness

T Length of end pieces of a tensile test specimen Virtual displacement

u,v,w Displacement components Volume fraction of fibre Volume fraction of matrix -Nodal displacement vector

V

- Virtual displacement veçtor

V Volume

Compressed volume

Vf - Volume of solid in a cell face Vg Compressed volume of gas Vg° Initial volume of gas V0 Initial volume

xiv Experimental and Numerical Determination of Stzffiwss and Strength of GRP/PVC Sandwich Strüctures. w Width We Effective width w2 Midspan deflectiòn W Work

Wb Work from edge bending in the foam Wf Work fröm face stretching in the fòam Wh - Work from plastic hinges in the foam

x,y,i Cartesian coordinates

X Value of a single observation

X1,X2,X3 - Coordinate system X1 through X6 - Constants z z-dimensiôn z _{Norma1izd z-dimension} Greek Angle a, J3 Constants

a',

'

-.

Constants

y

Shearstrain

Shear strain in the x-y plane Small change operator matrix

6 - Deflectión, displacement Deflection due to bending Displacement at the yield póint Deflection due. to shear Totál deflection

e Strain

Ei Strain at the yield point

Initial strain

ç

Strain in the x-direction Strain in the y-direction

0 45, 90 Strain in the 00, 450, 90° directions Virtual strain

[e] - Strain vector

[f1'] _{Flexural strain}

[efl] In plane strain

[e'] Strain vector in global coordinates

Neutral coordinate system

- Stress partitioning parameter Stress part tioning parameter

O Rotation

vi

Flexural Póisson's ratio

Vf Poisson's ratio of fibre

Vm Poisson's ratio of matrix

v0,

Poisson's ratio out of the laminate plane y0 In pläne Poissin's ratio

v, Poisson's ratio of unidirectional ply Pòisson's ratio of foam

Density of the solid of which the foam is made p*

Density of the foam

G Stress

GB Stress in the facing which are likely to cause wrinkling Gel* Elastic collapse stress

G1 Compressive strength Gm - Stress at maximum force

- Facing proportional limit stress Plastic collapse strength ax Stress in x-direction ay - Stress in y-direction

Yield strength of cell wall material G0 Initial stress matrx

[a]

-

Stress vector [ai] - Flexural stress

[a°] - In plane stress

[G'] Stress vectór in global coordinates 't

-

Shear stress

Core shear strength

- Shear stress in the core that causes shear failure

tm Maximum shear stress - Shear strèss in the x-y plane - Fraction of solid in the cell edges

tl) - Surface traction Abbreviations

CN Cyanoaciylate

CSM Chopped strand mat FEM Finite element method FRP Fibre reinforced plastic GRP Glass fibre reinforced plastic LVDT Linear displacement transducer NDT Non-destructive testing PVC Polyvinyl chloride

SES Surface effect ship

xvi ExperimensEzi wzd Numerical Determination of Stzffñess and Strength of GRP/PVC Sandwich Structures UD Unidirectional WR Woven roving 2-D Two dimensional 3-D - Three dimensional

1.

INtRODUCTlON

1.1

General

In the society of today where time is an important factor and people travel more frequently and longer distances the demand for fast and safe conveyance has mcreased High speed vehicles on

land air or

sea require the use of lightweight materials, The focus herewillbe on fast going vessels which are characterized by size speed power behaviour in a seaway and relative cost The high speed vessel

commonly called suiface effect ship (SES) has shown good characteristics and are used widely as passenger passenger-car and military vessels The SES has a catamaran type of hull At high speed

air cushion supports the majority of the weight, this reduces the resistance so much that the sum of lift

and propulsión power is significantly less than the propulsipn power of the equivalent conventional craft. At off-cushion a SES behaves ve!)' much like a convCntional catamaran. When on-cushion these

high speed crafts are sensitive to weight, and w order to keep acceptable power levels and payloads strict attention to weight has to be paid throughout design. Hull matenáis úsed in design are steel,

aluminum alloy and glass-reinforced plastic (ORP). In thé case of GRP both single skin and sand ch axe used. See Lavis et al. (1991) and Butler (1985). A sketch of a SES passenger-carferry is shown

in Figure 1.1.

Cirrus, Norway has for designed a passenger-car ferry, CIRR 200P. This SES has a length

of 60 meter a full load-displacement of close to 500 tons and a capability of carrying a payload of 364

passengers and 56 car units with up to 5 ton per axis. The hull material is cored GRP.

Clearly, there is a great demandfor low cost materials with high strength co weight.ratio for both hulls and interiors of marine vessels Sandwich matenals are competitive m this relation Compared to steel and aluminium, sandwich material is rather new in manne applications Advantages of FRP/sandwich structures are (Brødrene AA):

- high strength to weight ratio

- flexibthty in design (different shapes are easy tomake) - high thermal insulation

1.2 Experimental and Numerical Derermiñation of Stiffness and Strength of GRP/PVC Sandwich Structures

Figure 1.1

illustration of SES passenger-car ferry.

- maximum use of interior volume and "clean" internal sides

- no corrosion, or waterabsorption rotting - superior noise suppreSSiòn

- attractive finish

- two shells and excessive reserve buoyancy in the core reduce risk of siiikage in the case of

grouñding

In addition GRP sandwich is easy to repair and has good ability of stiffness to weight optimizing

A sandwich construction also has sorne disadvantages, like (Bau-Mailsen et al. 1990): problems with production control

- problems in the çpntact point of the load (like crushing of cote material), usually

enforcements have to be used

- bonding problems between the different layers

- poor heät resistance

- heavy smoke development at elevated temperatures

- lack of non-destructive methods (NDT) for sandwich material control - poor recycling possibIlities

However this field is under continuous development and the disadvantages are being rtduced. For example research with new methóds of detecting core fáilure by NDT shows promising tesults

(Gullberg et al. 1990).

A SES structure is subjected to different types of loading like slamnüng, sea pressure and weight of equipment and dry cargo. In addition bending moments and shear forces arise due to the passage of the ship through the waves, see Hayman et al. (1989) Typiçal components of sandwich vessels are

a)

Panel / beam

L11i1

b) T-joints

Figure 1.2 Typical components of sandwich vessels. shown in Figure 1.2.

A sandwich material is cOmposed of different materials whiçh form a structural membeì where two thin skins are bonded to a core material. For maiiiia applications the skins are often lamina built up in layers of glass fibre mats embedded m plastics whereas the core is often a foam, that is a solid

material which hàs got ceiltilar structure under a foaming process.

Thê materials has different

mecharncal properties but the composition gives a very good utilization of each material and thereby a lightweight and strong sandwich construction However the material properties of the core are the single most dominating factor which influences the overall behaviour of a sandwich structure Desirable

core properties and their major influence

óñ the pùthince of a sandwich panel are according to

Olssön et. al. (1992);

- Low density; lightwéight structure

- High shear strength; high transverse load carrying capability

- High shear modulus; little out-of'plàne défOrmation and high m-plane load carrying capability

(prevents panel bùkliiig)

- High maximum shear strain; high tolerance to stress concentrations and high damage

tolerance

- High modulus of elasticity;high in-plane load carrying capability (prevents local buckling

of skin)

- High tensile and compressive strength perpendicular to the faces high in plane load carrying capability (prevents local buckling of skin due to failúre iñ core)

- High fracture toughness; hiwi tolérance to inherent flaws and high damage tolerance.

The performance of sandwich panéls is for lateral loading (bending behaviour).. In marine vessels structural panel members like hüul, bulkhead and deck will meet in joints. The joints are

4-1.4 Eîperiinenftl and Numerical Determination of Stiffness an4 Strength of GRP/PVC Sandwich

Stnctures

SW panel

Figure 13

Typ'cal sandwich T-joint.

however CTitiCal parts of the strncture due to highstress level and the existence of stress concentrations. The loadcarrying ability of the joints depend on its configuration. In this thesis the strength of T-joints is of particular concern. An example of a common joint configuration is shown in Figure 1.3.

1.2

Objective and Scope of Work

This work has two main objectives. The first one was to develop an experimental technique for internal strain measurement by means of strain gauges in a foam core material. The method was intended for

marine application so the material of the test specimens was representative for this purpose. The experimental technique was developed and tested out on simple standard specimens which experienced either pure compressive- or shear- strain.

The second objective was to use the experimental method to investigate a critical part in a sandwich

marine structure An investigation of the local strain in the area of panel support (T-joint) caused by

wheel loads from a car on board a car ferry has as far as the author knows not yet been performed.

The problem was approached by means of experimental and numerical methods. Specimens with two

different joint configurations were examined for two critica] loadcases where one predothinantly

supported shear load and one compressive load respectively.

The T-joints were subjected to load from a car wheel only; any other possible loads like moment and

shearfrom the passage of the ferry through the sea must be added.

The T-joints which were investigated, experienced mainly compressive

and shear strains. The

experimental method was therefore not tested out for the tensile case.

The use of strain gauges on the foam material is limited to the elastic region asfailure mechanisms

in the foam will lead to fracture in the auges if the mat rial is further strained. Linearelastic. models

were used in the numerical analyses.

The experimental technique was developed for a specific material. As many parameters (foam structure

and density, presence of strain gauges and adhesive joints, temperature and humidity etc.) might influence the results the procedure is only valid for the materials and conditions at which t is tested

out In this thesis the foam materials Divinycell H100 and H130 were ued. However with the detailed

description of the procedüre which is given in this thesis the experimental method can easily be extended and tested out for other types of foam materials.

When the load actS centric to the web the stress an

strain in the foam core of the flange do

approximately not change if the width of the T-jomt is changed. If however the lOad acts eccentric to the wèb the vertical and shearstress depends on the width of the test specimen. In this case the results

are not general but limited to specimens with thiS particular width.

1.3

PreviOus work

The use of sandwich material in marine industry is new, compared to that of more conventional

materials, like wood, steel and aluminum. Some years ago not much systematic work had been done

m this field, but with the increasing popularity of the matenal over

the last years the amount of

research has increased. But still many unsolved problems and hence challenges remain in this field.

In this section relevant recent literature is reviewed. Even though manyof these references describe

work which -is beyond the scope of this thesis, they contribute to a general background which is

beneficial during the wò±k with sandwich structures.

Using strain gauges in the foam is a rather new method to determine the response of the. material. Rusnak et al. (1989) has developed a method to measure the strain in a sandwich bean subjected to

three point bending by mounting strain gauges on the foam suiface The report described the specific problems encountered and procedural soiutions deñvdto ¿Íéui with them. Hayman et al. (1991) studied

the response of bottom stnicfl of fast craft under slasuuliing impact load where measurements by means of strain gauges were carried out on the foam core of GRP sandwich hull of costal rescue craft while in service and on models-representing variôus hull constructions droppitig into the water. Strain

gauges were also used by Moyer et al. (1992) to experiìnentally determine the strains in the core of

a GRP sandwich panel subjected to shock loading due to an underwater explosion. In these experiments internal strains in the foam were determined by attaching strain gauges inside a foam plug which were inserted into the core of the sandwich structure.

Sandwich bearíis and plates subjected to bending is a basic load carrying element in a sandwich

structure. Various investigations have been carried oUt on such specimens. The advantage of using a beam.is that its simple geometry and behaviour, compared to for example panels and joints simplifies the matter and at the same time important information which more easily can be verified are achieved In addition the results can give experience and background to more complex problems.

1.6 Experimental and Numerical Determination of Stzffi2ess and Strength of GRP/PVC Sandwich Structures

may easily change the propertós of the specimens and thereby affect the readings measured in the

strain gauges. Rusnak et al. (1989) and Buene et al. (1990) have examined the effect of adhesive joints

in foam in sandwich beams. The first reference investigated longitudinal adhesive joints in the foam and expeiienced a change in the response of the beam test specimens. The second showed that for a

beam in four point bending subjected to static load the bean properties may change due to the

longitudinal or transverse adhesive joints. The tests were camed out on specimens with both Divinycell H100 and H200 with and without joints of some common polyester filler adhesive. Further work has

been done on this subject by Buene et al. (1991) where beams in four-point bending have been

subjected to simulated slamming loads. Shear properties of the beams were determined experimentally and in the case of beams having adhesive joints in the core, the experimental work was supplemented

by evaluation of stresses in the core and adhesive tising

finite element analysis. A substantial

amplification of shear stress occurred in the adhesive joints because of the mismatch in modulus

between adhesive and core material.

It might be difficult to find the correct material properties for foam materials. McGeorge et al. (1991 a) says that the difference from one source of information toanother may be as large as 20% for one core material and propose a procedure for core property determination based on FRP-sandwich beams.

McGeorge (1991 b) have also done another interesting work on beams in bending. This reference

describes analytic methods for deflection and stress calculations of sandwich beams subjected to four point bending and developed a spreadsheet program for parameter studies. Analytical solutions of both beam and panel structures are thoroughly described in several textbooks like Allen (1969), Stain et al.

(1974) and Tet (1989).

Olsson et al. (1989) descnbe test. methods for mechani al property determination. Feichtinger (1989) also describe static mechanical test methods and include comparison of the response of some structural

core materials which are used in sandwich constraction. The evaluation of accurate shear properties

of the core material is particularly important in the determination of overall sandwich beam and plate behaviour as these constructiOns experience significant shear deformation.

The overall rigidity of the hull and deck assembly depends on the out-of-plane carrying members such

as girders and bulkhead, which also allow the hull to retain in shape after loading. The loadcariying

capability of the members depend on the efficiency of the joints. Important factors for stress

distribution, load carrying capacities and failure mechanisms are the size and type of the joint. An

investigation of a T-joint, which actually is a part of for example bulkhead to hull or bulkhead to deck joints can contribute with useful information.

Kildegaard (1993) has. carried out static pull-out test on T-joints to experimentally investigate the

influence of the different joint configurations on the static strength and the failure me hanism of the jòint, and experienced that the strength of T-joints with triangular foam nsert was independent of the

size of the insert whereas the strength of joints with quarter-circular insert increased linearly with increasing radius of the insert. The strength of these T-joints were higher than the strength of those

laminate of the flange. The nfluence'ofíhejòht geometry oñiheàbility to transfer outof-plane loads for hull-bulkhead joints in sthall boats has been investigated by Shenoi et al. (1990). In addition a study of the failure mechanisms for T-joints were done. Also this investigation too showed that the

strength and failure mechanism may depend on the configuration and size of the joint. Dodkins et al. (1992) has numerically and experimentally studied T-joint single sIm configurations subjected to pull-off force of the web at 45° to the flange, and found that increasing the thinkness of overlamination has a detrimental effect on joint performance and that increasing fillet radius enables the joint to withstand higher loads. A slight reduction in the pull out strength of sandwich Tjoints with increasing thickness of the attachment lap was also experienced by Fagerheim et ai. (1989). An investigation of the joints

subjected to shear load was also performed, with the conclusion that the size of the attachment lap

seemed to be an insignificant parameter in this case. Theotokoglou et al. (1991) has supplemented this work by FEM analysis of stresses in the joint with particular attention to the stress picture in the area of the attachment lap, glue and laminate.

Sandwich structures are very sensitive to failure by application of loads of strongly localized nature,

like point loads, line loads or distributed loads of very high intensity. Such loads may for example

come from cars or bear fittings attached by bolt arrays. Because the core materialsusually are lighter,

more flexible and weaker than the facings, loads may result in core crushing or delamination at the interface of core and facings.

Allison (1990) has investigated the stress induced by such loadings of the faces and the effect of

varying the facing thickness by testing a photoelastic model sandwich beam and concluded that stiff

facings reduce the pressure which would otherwise be developed in the core material and thus provide effective protection against localised n Ormai loads. Also Thomsen (1991, 1992) considered localized loading. Based on the assumption that the core behaves like an elastic foundation, an analytical method

to estimate local, bending of the loaded facing of a FRP-sandwich beam subjected to concentrated

loading was developed.

Care should be taken when a panel is modelled as a beam. Beam equations do not account for the membrane effect of the FRP skins. This can be an important factor for sandwich panels constructed

with resilient core materials (such as low-density foams) and thin skins. The shear stress levels in the

core material of sandwich beams are muçh higher than those foúnd in the core of sandwich panels. Thus the flexural response of a sandwich beam is more dependent on core material properties than the flexural response of a sandwich panai, see Reichard (1988). Difference in data from beam and panel subjected to pressure load are illustrated by Reichard (1989). The deflections from beam models were

significant higher than what occurred for a panel. The magnitude of the discrepancy varies with the

fibre orientations in the laminate, the properties of the core material and dimensions of the beam/plate, therefore beam models data can not fôr example be used to select the optimum laminate for the panel.

Foams are widely used as core material in SW construction. These materials are constructed by thin

cell walls, and thedeforrnation and failure mechanisms and thereby the mechanical properties depends on properties of the base material and the geometrical configuration of the core Gibson et al. (1988) discuss these mechanisms for different types of fOams.

¡.8

Experimental and Numerical Determination of Stiffness and Strength of GRP/PVC Sandwich

Structures

Allen (1989) ask an important question: 'How efficient are today's core materials, and can any improvements be expected in the future?. The conclusion is that deficiencies exists both on the

structural and theoretical side and that improvements are looked for.

1.4

Organization and Presentation of Work

This thesis is organized in three main parts. Chapters 2 through 4 provide background for the

experimental, numerical and analytical work of Chapters 5 through 7. In Chapter 8 T-joints are investigated theoretically and experimentally based on the experience reported in the preceding

chapters.

Chapter 2 reviews some basic concepts of sandwich structures and materials. This includes theoretical

models concerning both behaviour and property detennination. The finite element method (FEM),

which is a powerful tool in the structural analysis of a sañdwich construction, is presented in Chapter In addition special aspects which have to be taken into consideration in the modelling of the crrent constructions are discussed. Experimental test methods of standard specimens are described in Chapter

In addition analytical- and numerical (spreadiheet program)- methods for result evaluation and

material property detennination of these specimens are outlined. Furthermore considerations aboutfoam measurements are discussed.

Test of simple standard compressive-, shear- and bending- specimens were performed to develop an

experimental technique for internai strain measurements by means of strain gauges. These test specimens were exposed to loading which produced either approximate pure compressive strain or approximately pure shear strain in the core material. In Chapters 5 through 7 these experiments are

described and the test results are compäid with analytical and numerical calculations. The behaviour

of the foam material is also discussed and the characteristics of material properties of foam core

mâterial with and without skins were determined.

The experimental method was applied to a deck-bulkhead section (T-joint) of a car feny subjected to

wheel loads. The T-joints were also analyzed numerically by means of the finite element method

(FEM). Chapter 8 contains both the experimental and numerical investigation together with a

comparison of the results of the two methods. Furthermore a section of material property determination for the FEM analysis is included and a strength 'analysis of the foam core of the T-joints is performed. At the end simplified methods to estimate the compressive- and shear- stresses and strains in the foam

of the T-joints are proposed.

Finally, the main conclusions of the present work and suggestions for further studies are presented in Chapter 9.

SANDWICH

In this sect on a brief overview of sandwich constrection and the materials is presented. The sandwich

principle is shown by reference to a beam Theoretical and mathematical models which explain the material behaviour and property determination are desçribed. This give useful knowledge when

analytical calculations and experimental tests are going to be performed.

2.1

General

A sandwich is astructtiínl component which consists of two stiff skiiis separated by a low density core.

All thre layers are firnily bonded together The face material is much stiffer than the core material. The skins take up the normal suesses in the stmcture and under bending oneskin will be in tension while the other in compression Another important feature of theskin is to provide durable surfaces

The foam takes up the shearing forces and contribute to the bending stiffness by separating the

axialstiff skins In addltlofl it also stabilizes the skins and thereby work against buckling of the faces

The bonding is needed to combine the facings and core material and prevent them from moving

relative to each other, otherwise the sandwich effectwilldisappear and the facings will behave like two thin separate plates. This will redüce the stiffness and strengt1 significantly.

As the shear modulus of the core generálly is low, the defle tion due to shear can notbe neglected.

The total deflection of a sandwich beani may be found by superimposing the bending and shear

deflection Figure 2 1 shows the deflection due to the shear in the core only 6 and due to the bending of the faces only 6b Thetotal deflection 8isthe sum of the two 6t=6+8b

& is zeromthe parts of

the beam where there is no shear load.

The stràin distributión always varies continuously in a sandwich construction. But due to different stiffnesses of the lamina and foam this is not the case for the stresses The normal stressand strain

distribution across the depth of a sandwich beam subjected to bending load is shown in Figure 22 Here the average stiffness of the laminate is cònsideiad. The plies in the laminates usually have

different stresses due to different elàsticity modúli. The shear distribution for the same problem is also shown in the figure.

2.2 Ecperimental and Numerical Determination of Stiffness and Strength of GRPIPVC Santhvich Structures

¿F

¿F

TF

Shear deflection

Bending defledioü

Strain

Sandwich

Shear sfr

tf

tf

_/

Figure 2.2

Stress and strain distribution in a SW beam.

The basic principle of a sandwich construction is shown above. The material is however used in many different constructions therefore the laminate and foam must also be able to withstand other types of loadings.

In sandwich constructions plates might meet in a joint. Load carrying joints like bulkhead to hull and

bulkhead to deck jomts may be critical ..reas m a vessel There is a discontinuity m the sandwich

material and a geometry change in the corners where stresses are transferred and stress concentrations arise. The design of the joint is of importance for the distribution and level of stresses. Different joint configurations are shown in Figure 2.3.

Emodule

Perpendicular to the

Undeformed middle plane

Perpendicular to the

deformed middle plane Figure 2.2 Shear afld bending deflection of a beam in four point bending.

Stress

tf

Figure 23

SW bulkhead to hull/bulkhead to deck joint configuratiöns (Sbenòi et al.

1990).

2.2

Material

The use of FRP/sandwich in marine structures requires knowledge about fibre, matrix, foam.and ifiler with reference to mechanical properties, durability in the current environment, cost, fabricability and possibly therthal, electrical and chemical properties.

A wide range of materials is available for marine use. Rèsm includes polyesters, epoxies, phenolics änd

venylester. Filer reinforcement coüld be E or S glass, high strength and high modUlus carbon or aran,.ids such as Kevlas. Core choice could be foams, balsa or honeycombs. In the case of foams,

expanded polyvinylcloride (PVC), polyester, epoxy and polymethacrylicimid are some which can be

mentioned. A short description of some of these aspects for the material of interest to this thesis are

described below. For further information see the literature (POI 1988, Smith 1990).

E-glass, which got the name due to its high electrical resistance, is commonly used as reinforcement in marine structures. The main characteristics are low cost, high strength and rather low stiffness. The diameter of the fibres ranges from 4-20 .Irn.

The most widely used matrix materiál for laminates in marine structure application is polyester. It is

a thermosetting resin, that is a resin which harden permanently during the curing process. The main advanages of polyester are moderate cost, good fabricability forhand lay-up and spray-up and good

2.4 Experimental and Numerical Determination of Stiffness and Strength of GRP/PVC Structures

As core material in marine sandwich stnictures polyvenylcioride (PVC) are widely used. It is available in a range of densities. PVC foams shows good resistance to water penetration, good thermal, electiical

and acoustic insòlatiow Their main deficiencies are reduction of strength and stiffness at elevated temperature and chemical break down with emission of RCI vapour at temperatures over 200° C.

Structuïal fillet för FRP sandwich material need high adhesion to cured FR1' brniinatfs and core

materials, good performance in marine environment and good mechanical properties. Applications of

fillet materials are structural interfaces, contour joints in FRP components and to build up damaged

23

Foam

A foam is a cellular material. Load deformation curves show how the material behave, while

mathematical models explain what happen during loading and which parameters the material properties depend on. Theoretical knowledge about the foam behaviour is therefore advantageous when analytical calculations or experimental tests are going to be performed.

23.1

General

Almost any material (metals, ceramics, glasses) can be foamed, but polymers are the most common.

The solids are foamed by different techniques to get its cellulàr structure where the cells can be open

or closed. Polymers aie usually produced through an expansion process, where gas bubbles are

introduced into a liquid or plastic phase. These bubbles are then increased to the right sae before this

cell structure are stabilized through a physical or chemical method (FGI 1988). Figure 2.4 shows a

schematic production of the foam Divinycell to both soft and hard qualities.

Foams are usually anisotropic due to the production process where the foaming agent rise. The cells are usually elongated in this rise direction., The symmetry of the cells depend on the production

process. Anisotropy in the modùli, strength and toughness of foams is a common observation.Young's

modulus is greatest in the direction of greatest elongation of the cella.

The available properties, which are dramatically extended by foaming, depends of geometric structure

of the foam and the intrinsic propert s of the material of which the cell walls are made. Salient

structural features of a foam are its relative density (or porosity), the degree to whether its cells are

open or closed, and the mean cell diameter and Is shape isotropy

ratios. 'The crucial cell-wall

properties are the solid density, the Youngs modulas, the yield strength, the fracture strength and the creep parameters. Factors such as strain-rate, temperature, anisotropy and multiple loading all influence the properties too.

2.3.2

Deformation Mechanisms of Foams.

The following two subsections gives a descnption of a big topic it is therefore impossible to cover everything. These pages focus on the aspects whiçh is of most interest to this thesis. For additional information see Gibson et al. (1988) which is the source of this and the next section.

Mng

250 kp/cm2

Pressing in heat

fiar

Sawing

Ready for use in constructions Press-gluing core and

skins to a sandwith

Post hardening

Figure 2.4 Scbematic production of Divinycefi foam. 200°C

-p-Expansion

Soft Divinycell

product

Figure 2.5 show elastomeric, elastic-plastic and elastic-brittle behaviour of foams in compression. The stress-strain curves show linear elastic behaviour at low stress. In this region the cell walls bend. The

next stage on the curves is the long collapse plateau where the cells experience elastic buckling in

elastomeric foams, formation of plastic hinges in a foam which yield and brittle crushing in a brittle foam. When the cells have almost completely collapsed opposing cell walls touch, and further strain compresses the solid itself giving the final densification region where the stress rise steeply.

In the case of closed-cell foam, effects of cell face stretching and

enclosed fluid pressure are

superimposed. In man-made foams the fluid is usually a gas and sometimes liquids.

Youngs modulus of the foam, E* is the initial slope of the stress strain curve. When the density of the foam, p is increased relative to the density of the solid of which it is made, p the Youngs modulus

increase, the plateau stress raises and the strain at whichdensification starts reduces.

Examples of materials which experience ela.stomeric, elastic-plastic and elastic-brittle behaviour in compression are rubber, metal and rigid polymer foams, and ceramic respectively.

Using the deformation and fracture diagrams, it is possible to identify the dominant mechanisms of deformation and failure and to select a constitutive equation (see Section 2.3.3) which describe the

2.6 Experimental and Nwnerical Determination of Stiffizess and Strength of GRP/PVC Sdndwich Srrùctzues a) ELASflC-BRITItE FOAM NPRESS4 DENS&F1CAI1O

PtATEnJ (8Rfl1.E cna.

LINEAR Et.STIOTYIBENDN) -i STRAIN. e b) Elastic-plastic foam. o

0

StRAIN, c) Elastic-brittle foam.

Figure 2.5 Schematic compressive stress-strain curves for foams (Gibson et al. 1988).

Foams in tension also experience elastomeric, elastic-plastic and elastic-brittle behaviour. Some foáms

(for example rigid polymer foams) are plastic in compression but brittle in tension. For deformation

mechanisms of foams in tension see Gibson et al. (1988).

2.3.3

Mechaflical Properties of Foam in Compression.

The behaviour of foam can be explained by theoretical models. See Figure 2.5 to relate thefollowing discussion to the stress strain curves for different foam behaviour in compression. Depending on the

shape of the stress-strain curves a foam can be elastomeric, dlasticplastic or brittle. For a thorough

description of the models see GibsOn et al. (1988) which is the source of this section. This reference also include the tensile case.

linear elasticity.

Linear elasticity is limited to small strains. The mechanism depends on whether the cells are open or closed. For both cell types the deformation is controlled by cell-wall bending, whereas in the case of closed cells, the effect of enclosed gas pressure and membrane stretching are also of importance.

In Figure 2.6a open cell foam is modelled as a cubic array of members. The relative density of the foam p/p and the second moment of area of a member, I aie related to the dimensions of the cells:

(2.1)

ELASrICPLAST1C FOAM

cOMPRESSI1

b

w DO(SF1CATI PLATEAJ (PLA&TC IELnNG} LINEAR O.A5T(QT(8ENOTNG)

b

w

a) Undeforined open-cell foam.

If

-r

F

c) Undeformed closed-cell foam. d) Stretching of the faces of a

cloed'.oeH foam.

F

F

_{rigid corners}

F

buckled

edges

e) Elastic buckling in the cell

walls of an open-cell foam.

b) Cell edge bending during linear

elastic deformation Of an open-cell foam.

f) The formatton ot pias¡Chinges an opèncell foam.

F broken cell

edge thiCklieSs te

I111111L4

face tlñ kneSs I

1!Il

ip LI

F

F

_j

g) Plastic stretching of the cell h) Cell wall fracttfre during crushing

faces of a closed-cell foam. of a brittle opencell foam.

2.8 Experimental and Numerkal Determznation of Snffness and StrEngth of GRP/PVC Sandivich

Structures

and Poisson's ratio y is

and

¡cc (2.2)

where is the density of the foam, p the density of the solid of which it is made, t the thickness of

the square cross-section and 1 the length of a member.

Under a uniaxial stress G, the oeil edges transmit a force F and the edges bend like bedms. This is

shown in Figure 2 6b The linear deflection of the structure as a hole is proportional to the deflections

of the beams, 8c'cFl3/E5i. E5 is the Young's modulas for tbe material of the beam. F is the load and

is related to the remote stress FocGl2 and the strain to the displacement eoc5jl. Young's modUlus for the foam follows:

E=G=

(2.3)

c

This gives:

(2.4)

E p5

Where C1 includes all of the geometri constants of proportionality.

This is the modülus due to asmall load F which produce Small strains. At higher load P an additional moment on the bent edge will arise, even before the Euler load is reached and where the edge buckles. This bedUi-column interaction lowers the modulus, E and increases it in tension. Therefore the "linear elastic" region is not truly linear but concave downwards.

The shearmodulus can be achieved from similar considerations. Thç oeil members respond by bending

when the shedr stress 'r is applied to the foam. The shear stress'rand shear strain y are proportional to 2 and M.

_-C2(_)

G*_

P*

E5 p5

(2.5)

For an isotropic material with linear behaviour the following relation exist

G= E (2.6)

CI

C3 2C2

(2.7)

where C2 and C3 are geometnc constants of proportionality and G* is the shear modulus of the foam.

The Poisson's ratio of the foam v is the negative ratio of the lateral to the axial strain. These strains are both proportional to a bending deflection per oeil length. Their ratio is therefore constant and

Poisons ratio independent Of density and solely a function of oeil geometry.

Below the closed cell foamis considered. For some fòams the cell walls are so thin that the influence

of this can be neglected. For other foams the effect of cell walls give an essential contribution. In

Figure 2.6c a fraction of the solid iscontained in the cell edges and the remaining fraction (l-4) in

the faces.

The Young's modulus can be determined by the saine considerations as above The relation between the thickness of the cell edge te, 4, and (p*Ips) can be shown to be approximately:

.=O.934,It2(_.)1/2

I p5

With IOcte4 thiS gives

E5 p5

This means that the effect of the cell edge bending contributes to the Young's modulus with a factor

of (O.864,)2 compared to the open cells.

In a closed cell foam effect from compression of the céll fluid must be taken into account. When a foam which contain a gas is compressed axially by a strain e, the volume decrease from V0 to VC,

where

V

(2.10)

vo

The gas occupies the cell space excluding the volume occupied by the solid cell edge and faces, so its volume decreases from vg0 to Vg where

Vg l_e(l_2v*)_P*/Ps

(2.11)

1-p/p5

The contribution to the modulus of elasticity is calculated from Boyle's law:

(2.8)

2.10 Experimental änd Numerical DeterminatiOn of Stiffness and Strength of GRP/PVC Sandwich

Structures

pVp0V'

(2.12)

where Po1 the initial gas pressure and p the pressure after a strain n is applied. The pressure which mustbe overcome by the applied Stress is:

pp-p

This gives

p0E(1-2V)

-i -n(l _2v*)_P/Ps

Taking the limir at small E the contribution to the modulus become

E*=

p0(1-2v)

g _{(1_p*/p5)}

The contribution of cell fluid is usually small when Po is atmospheric pressure.This effect can however not be neglected if Po is much larger than atmospheric pressure or the cell fluid is not air but a liquid.

The läst contribution comes from membrane stresses in the cell faces as the bending of the cell edges

causes the cell faces to stretch. See Figure 2.6d. A force F causes the cell edge to deflect a distance 6. The work done against the restoring force caused by cell edge bending and face stretching is

w%

F8 (2.16)

Work from edge bending is:

Wb4

(2.17)

where the stiffness of the cell edgesSocE5JJI3. The work from face stretching is:

W1OCZ/z E5E2V1 (2.l8

where thé strain caused by stretching a cell face ccM and the volume of solid in a cell face Vfcl2tf.

Thus

cxEI5

₆₂₂

34F&= SE()lt1

Using Icct4 andE*.c (F112)/(611)gives:

(2.19) (2.13)

(2.14)

(2.20)

E5 ¡4 ¡

where the relation between relative densities and dimensions of the cells are approximately:

.t'1.4(l 11)._.

I p5

1/2

_.=O.934

1/2(P)

I p5

This give an equation which describes the combined effect of cell bending and cell face stretching:

._=C142(L.)2+C1(l -4)!_

E5 p5 p5

To this equation the còntribgtióñ from gas within cells should be added when compression of gas is important a, f3, a, f3' and C1' are constants of propornality.

The shear modulus G* of a closed cell foam is, as for Young's modulus influenced by cell-face stretchm.and can be achieved by similar considerations.

._=C2d2(L.)+C2(l -4)).a_

(2.24)

E5 p5 p5

There. is no contribution of gas pressure as pure shear produces no volume change.

Here too, as for the case of open cells, the Posson's ratio is the ratio of two strains, and thüs depends

on the details of the cell shape but not on the relative density.

The discussion above was limited to defórmations at small strains.

Non-linear elasticity.

For élastômeric foams the deformation at high strains is still recoverable an4 is thus elastic but it is

nonlinear.. The stress strain curve in Figure 2.5a shows the eladtic collapin stress ;i* and the extensive plateau of the post-collapse behav òur. The mechanism of nonlinear elasticity and densificalion depends

on whether the foani has opeñ or closed cells. Figure 2.6e illustrate this mechanism. As the foam

ôonsidered in this thesis do not experience this behaviour, it is not considered any further. (2.23) (2.21)

2.12 Erperimenral and Numerical Determination of Suffiiess and Strength of GRP/PVC SàiithSich Structures

Plastic collapse.

When loaded beyond the hnear-elastic region plastic collapse occur for foams made from materials like

rigid polymer or metals which have a plastic yield point. After the plastic collapse stress is

reached the stress strain curve in Figure 2 5b shows an extensive plateau but the strain is no longer recoverable. In the case of open cell foam, the collapse occurs when the moment exerted by a force F on the cell walls exceeds the fully plastic moment creating hinges like those shown in Figure 2.6f; For a beam with cross Section area of t t, this moment is:

M=!a3/

(2.25)

where,5 is the yield strength of the cell material. For beams in theplane normal to the load direction

the maximum bending moment is proportional to FI and the stress on the foam is proportional toF/I2.

The plastic collapse strength of the foam is

(j)

Using p*dpscc(tJl)2 the following expression for the plastic collapse strength arise:

ev *

p1_,-.P

3/2

*,yS _P

(2.2)

where C5 contains all the constants of pmportipnality.

This model has limitations. In the case of large relative densities, the cell walls are so short and squat

that they will yield axially before they bend and if the relative density is verylow the oeil walls will buckle elastically prior to plastic collapse.

In closed-cell foam,the plastic collapse is in addition to cell wall stretching alsó influenced by cell wnll bending anTd by the presence of a fluid within the cells. Figure 2.6g shows the plastic stretching of the cell faces of a closed-cell fóam. The force required to crumple the thiñ membrane in the compression direction is small but at right angles to this direction the membranes are stretched and the plastic work

Sequi to extend them contribute significantly to the yieldstrength of the foam.

The force F give a compressive plastic displacement 8. The work done against the restoring force caused by plastic hinges at corners and cell stretching is:

Wc'F6 (2.28)

Mö

Whc'c_.L

Usmg Equations 2.21 and2.22:

L=C5(4_)312C5 _(1 -)(.-)

yS PS P

This equation describe the combined effect of plastic bending of cell edges and plastic stretching of their faces. To it should be added the contribution from cells containing fluid. l'bis term is usually

small when the flúid is a gas, but if the fluid has a pressure much higher than this the contribution is important.

Brittle crushing strength.

Figure 2.6h shows cell wall fracture during crushing ofa brittle open cell foam. This model is not

considered any fu±ther here.

Cell anisotropy, linear elasticity.

Due to production process the foam are ahnöst always anisotropic. The degree of anisotropy in a cell

shape is measured by the shape-anisotropy ratio R, which is the rato of the largest celldimension to the smallest. For polymers this ratio is typical 1.3. The case of axisymmetric foam is discussed briefly

below. Figure 2.7 shows a model of an axisyminetric cell with the X3 axis in the rise direction.

The saine beam consideratiOn as done for an cubic cell is done andFcca3l2, &3F 131E51 and E30c&3Thc where h is the height of the cell. Young's modulus in the rise direction follows:

E*_G3__CE

(t)4hc

3i;-

-

57

f

(2.29)

The cell faces is stretched a distance proportional to & and gives the work

WcY,,58t/ (2.30)

F&=aM!t3c58zf1

(2.31)

with Focl2 and Maa5t3/4 this gives

(232)

(2.33)

i'

IA

Figure 2.7 An axisymmetric unit cell with R=1.5 (Gibson. et al. 1988)!

2.14 Experimental and Numerical Determination of Suffness and Strength of GRP/PVC Sandwich Structures

When h=l in Equation 2.3, Equation 2.34 results. In the X1 direction all fóur beams experience. the

sanie deflection thórefore the load Carried by the larger beams is less than that carried by the shorter

onós. The first is proportional to ESI/h3 while the second to E51/13. The total load is proportional to crhl and the strain c1=61/l.

The Young's modulus in the two directions normal to the rise direction is: É*

c1 2Jz, _{l h} 2 1 h h

With R=h/l the Young's modulus of anisotropy ratio is:

E3 2

E'

-

(1 +(l/R)3)

and depend strongly on anisoffopy. Shape anisotropy of 2. have a modulus anisotropy of nearly &

For closed cells the membrane stresses contribute co the anisotropy with an additional term in Equation 2.36 is:

(l-4)_2R

I +Çf/R)

The ratio of shear modulus is found in a similar way:

(2.37) (2.35)

2

r'

l+R

This is much less sensitive to anisotropy.

Poison's ratio is the negatLve ratio ofa lateral to an applied strain. It is.independeht of relative density and depends Only On the cell geometry.

Cell anisotropy, plastic collapse..

Plastic collapse for an atiisotropic matérial is done by the same considerations as for cubic cell material

explained above When loaded in the X3 direction edges of length 1 must collapse and MacFl and

FocG3l2. This gives:

* CM

I

-

P

" pl.'3

(238)

(2.39)

When loaded in X1 diréction two edgés Of length i and two of length h must collapse. The forces are

proportional to M.1,/l and M1/li and total force is a1lli. This gives:

CM _i

(a1)1

I

With R=h/l the plastic collapse ratio becomes:

(a;1)3 _2R

(a1)1 1+(.!.)

The cells are thus stronger in the rise direction, but the anisotropy in str ngth is not so large as in

stiffness. A shape anisotropy of 2 shOuld have a strength anisotropy around 2.6.

2.4

Lnrninte

A laminate is a composed material. Knowledge of the mathematical models is important to be able to understand its complex behaviour and to determine reliable material input data for a FEM program

This section give a mathematical description of laniinate behaviour by means of micro- and macro-mechanical models The calculation of engineering constants of a laminate is included A practical

method for material property determination is giveninSections 434 and 4 3 5

Some FEM programs have special orthotropic material card for a laminate in plane stress. In other

stress cases a full orthotropic matenal model might have to be used The elastic engineering constants

of an orthotropic matenal must usually be input the FEM program by means of the stiffness matrix

(2.40)

2.16 Experimental and Nwnerical Determination of Stiffness and Strength of GRP/PVC Sandwich

Structures

components where the material relations are within allowable limits of the mathématical models. The isotropic material model is most common, generally speaking, and is therefore included below.

The sections about material constants and its restrictions have general validity. For foam material, for example they are especially useful in the case when an isotropic material do not satisfy the necessary

restrictions and therefore have to be expressed in terms of an orthotropic model. 2.4.1

General

In a laminate, fibres of high stiffness and strength are embedded in a matrix materiàl of considerably lowerdensity, stiffness and strength. Embedding fibres in the matrix changes the fibres ability to carry

load. An unimpregnased fibre, also called strand, are only able to carry tensile load, while ised as

reinforcement in a bminte it contribute to most of the tensile, compressive, flexural and shear stiffness

and strength. The purpose of the matrix is support, protection, stress transfer etc. Layers of fibre-reinforced material axe built up with fibre direction of each layer typically oriented in different

directiOns to give different strength and stiffuesses in the various directions. In alaminate the strains vary lihearly through the thinkness while the stresses are piece-wise linear due to the different stiffness

of the plies. See Figure 2.8.

In manne relation fibre mats with different fibre amount andorientatiòns are build up layer by layer to form a laminate. Chopped strand mat (CSM) is made from strands chopped into short lengths and loosely bond togetherto form a mat with fibres in arbitrary directions. Parallel roving may be used to form unidirectional cloths (UD) or it may be woven or knitted to form woven roving (WR) or knitted roving.

In different laminates the number of plies, stacking sequence and plyorientation varies, lithe laminate

possess midplane symmetry it is symmetric, if this is not present it is unsymmetric. And further the

laminate is balanced when plies with positive angles are balanced by equaì plies with negative angles.

The mechanical properties of a laminate depends on the properties of the fibres and matrix materials, the fibre orientation, volume fraction of fibres and the adhesion between fibres and matrix.

Fibre reinfórced plastic (FRP) has wide application alone and as skins of sandwich constructions (Smith

1990).

2.4.2

Elastic Constants

In a FEM model the laminate material data is often given as orthotropic properties in terms of stiffness

matrix To keep track of the many elastic constants in this material model, it is important with a consistent notation. The contracted notation used in Tsai (1988) is followed

in this section. The

engineering shear strain is used. Tsai (1988) and Jones (1975) are the sources of this section.

Orthotropicandisotropic material symmetnes are common when dealing with compositematerials. M orthotropic material has properties that are different in three mutually perpendicular directions at a