Response Analysis of Dynamically Loaded Composite Panelsorbit.dtu.dk/files/5436185/Riber.pdf · Response Analysis of Dynamically Loaded Composite ... of Dynamically Loaded Composite

Hans Jørgen RiberJune 1997

Response Analysis ofDynamically LoadedComposite Panels

Department ofNaval ArchitectureAnd Offshore Engineering

Technical University of Denmark

Response Analysis ofDynamically Loaded Composite

Panels

by

Hans Jørgen Riber

Department of Naval Architectureand Offshore Engineering

Technical University of Denmark

June 1997

Copyright © 1997 Hans Jørgen RiberDepartment of Naval Architectureand Offshore EngineeringTechnical University of DenmarkDK-2800 Lyngby, DenmarkISBN 87-89502-36-1

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The research of this thesis was carried out between October 1993 and March 1997 andsubmitted as partial fulfilment of the requirements for the Danish Ph.D. degree. Thework was carried out at the Department of Naval Architecture and Offshore Engineer-ing, the Technical University of Denmark, with Professor Preben Terndrup Pedersen andAss. Professor Jan Baatrup as supervisors.

The financial support from the Danish Technical Research Council (STVF) and the Nor-dic Fund for Technology and Industrial Development (NI) is gratefully acknowledged.

Special thanks to all my colleagues at the Department and especially my two supervi-sors, Preben and Jan, for giving me the opportunity to carry out this study.

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The background to this study is the need for handy design tools, which can, in a shorttime, calculate the most appropriate material composition and panel scantlings for FRPsandwich and single-skin vessels. Fibre-reinforced plastic (FRP) is a frequently usedmaterial for the building of high-speed light craft (HSLC). The scantlings of the hullpanels in these types of ships are often restricted by empirical and conservative designrules and it is of great interest to investigate whether a more rational calculation proce-dure will lead to better composite panels. With this in mind, analytical and numericalcalculation methods are developed, in order to permit the designer to use efficiently thecomposite materials in high-speed light craft.

Application of non-linear calculation methods to HSLC hull design seems meaningful,since the lateral load response of composite hull panels is characterised by remarkablegeometrical non-linearities, due to large panel sizes and high lateral impact loads(slamming), which is usually the dimensioning load.

In order to perform simple non-linear panel design without extensive computer applica-tion, two close-formed non-linear analytical solutions for laterally loaded compositeplates are developed by means of energy principles. The first method (6ROXWLRQ� �) isformulated as a complete solution. The second method (6ROXWLRQ��) is a simplification of6ROXWLRQ� �, dividing the governing equations into a linear part and a membrane part.This makes 6ROXWLRQ 2 suitable as a supplement to existing linear design rules in thisfield. The results calculated by use of both analytical methods are in good agreementwith experimental data and numerically calculated results.

A dynamic non-linear finite-difference-based program 3DQHO, dealing with orthotropicsandwich and single-skin panels, is developed. 3DQHO� calculates responses and failuremechanisms for composite plates subjected to various lateral time-dependent loads. Theresults of static as well as dynamic response are verified against the commercial finite-element-based software program Ansys. However, the present method is approximately50 times faster in CPU-time than Ansys.

A progressive damage model is developed and implemented in 3DQHO. This makes it pos-sible to improve the design by use of plots of the failure modes, loads and locations. Thefailure analysis uses the response from the non-linear analysis, leading to significantlyhigher ultimate failure loads than predicted by application of a linear response analysis.

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The ultimate failure loads predicted by 3DQHO are in good agreement with experimentsperformed on single-skin FRP panels subjected to high lateral loads.

Analyses of existing HSLC hull panels are presented in order to demonstrate 3DQHO. Adesign example is given to show the structural improvements which can be obtained byapplication of non-linear calculation methods.

Finally, the '19�High Speed Light Craft rule concerning FRP single-skin and sandwichpanels is discussed in the light of calculations with 3DQHO of hull bottom panels de-signed by application of the '19 rules. The single-skin rule, which is based on non-lin-ear theory, is found to be good. However, the maximum lateral deflection criterion ofZ�W equal to unity usually limits the design. The criterion seems unnecessary, since therule is based on a non-linear theory and, consequently, predicts accurately the panel re-sponses. A non-linear analytical method, 6ROXWLRQ��,�is suggested as a replacement to the'19 linear sandwich rule. In addition, the necessity of the maximum relative deflectioncriterion of Z�E equal to one percent should be further investigated. It seems reasonableto omit this criterion for particular ships since it often limits the design without apparentreason.

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Baggrunden for dette arbejde er behovet for et hurtigt og godt designværktøj, som påkort tid kan beregne optimale dimensioner og materialesammensætninger for skrogpane-ler. Fiberforstærkede matrialer (FRP) er ofte benyttet ved bygning af hurtiggående lettefartøjer (HSLC). Dimensioneringen af skrogpaneler i disse typer fartøjer er hoved-sageligt baseret på empiriske designregler. Derfor er nye og mere rationelle beregnings-metoder nødvendige for at kunne forbedre designet. Med dette som motivation er derudviklet analytiske og numeriske beregningsmetoder, som muliggør en bedre udnyttelseaf kompositmaterialer i hurtiggående lette fartøjer.

Geometriske ikke-lineære effekter (opbygning af membranspændinger) fra laterale ud-bøjninger, der skyldes store paneldimensioner i FRP skrog samt relativt høje lateraletryk (slamming), kræver ikke-lineære beregningsmetoder til korrekt responsberegning.

Simple analytiske ikke-lineære løsninger er udviklet til paneldesign uden brug aftidskrævende computerberegninger. Disse løsningsmetoder er fordelagtige i designfasen.Baseret på energimetoder præsenteres to forskellige løsninger. Den første, 6ROXWLRQ��, eren komplet løsning, hvor alle andenordensleddene indgår i pladeligningerne. Løsning to,6ROXWLRQ� �, er en simplificering af den første løsning. Her løses membrandelen(andenordensleddene) separat fra den linære del. Dette gør 6ROXWLRQ�� ideel som supple-ment til eksisterende linære beregningsmetoder. Resultater beregnet ved hjælp af beggemetoder er i god overensstemmelse med eksperimentelle data samt numeriskeberegninger. En undtagelse er dog spændingsberegninger for fast indspændte plader.

Dernæst er der udviklet en finite-difference baseret løsningsmetode til beregning af or-totropiske sandwich- og enkelt-skinds-paneler. Metoden er formuleret som et design-værktøj, 3DQHO, til paneler udsat for såvel statiske som dynamiske lastpåvirkninger. Re-sultater med 3DQHO er verificeret ved hjælp af det kommercielle finite-element baseredesoftware program Ansys. Der er god overensstemmelse mellem gensvarsresultater fra deto programmer, dog er beregningstiden med 3DQHO ca. 50 gange kortere end med Ansys.

En progressiv brudmodel er udviklet og indgår i 3DQHO, hvilket gør det muligt at forbed-re et design ud fra beregningsresultater. Denne del af programmet kan plotte geome-triske fordelinger af brudtyper og -laste. Den ikke-lineære responsberegning giver somresultat betydeligt større brudlaste end man finder ved linære beregninger. Det vises, at

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ultimative brudlaste beregnet med 3DQHO er i overensstemmelse med eksperimentieltbestemte brudlaste for en serie af enkelt-skinds-paneler.

Forskellige analyser og designeksempler udført med 3DQHO er vist for at demonstrereprogrammet, samt for at synliggøre mulighederne for at forbedre det strukturelle design.

Til slut diskuteres klassifikationsselskabet '19V�HSLC regler for FRP paneler ud fraberegninger af typiske skrogpaneler med henholdsvis reglerne og 3DQHO� Enkelt-skinds-reglen er begrænset af en relativ maximal udbøjning på een. Dette krav virker unød-vendigt, idet reglen er baseret på ikke-linær teori og derfor producerer nøjagtigeberegninger. For sandwichpaneler foreslås det at implementere ikke-lineære beregnings-udtryk i reglen samt at uddybe maximum udbøjningskravet og tillade, at reglen kanoverskrides i specificerede tilfælde.

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1.1 Motivation ......................................................................................................... 1

1.2 Organisation of the Thesis ................................................................................. 4

1.3 Bibliography...................................................................................................... 5

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2.1 Introduction ....................................................................................................... 7

2.2 Structural Design ............................................................................................... 8

2.2.1 Single-Skin Hull Design......................................................................... 9

2.2.2 Sandwich Hull Design.......................................................................... 13

2.3 Design Loads ................................................................................................... 17

2.3.1 Global Loads........................................................................................ 17

2.3.2 Local Loads ......................................................................................... 20

2.3.3 Slamming Loads .................................................................................. 20

2.4 Summary ......................................................................................................... 29

2.5 Bibliography.................................................................................................... 30

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3.1 Introduction..................................................................................................... 33

3.2 Theory............................................................................................................. 34

3.2.1 Assumptions and Configurations ......................................................... 34

3.2.2 Strain Displacement Relations ............................................................. 36

3.2.3 Equilibrium Equations ......................................................................... 39

3.3 Analytical Solutions ........................................................................................ 41

3.3.1 A Complete Analytical Solution, 6ROXWLRQ�� ......................................... 42

3.3.2 A Combined Analytical Solution, 6ROXWLRQ�� ........................................ 45

3.4 Results and Discussion .................................................................................... 56

3.5 Summary ......................................................................................................... 59

3.6 Bibliography ................................................................................................... 59

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4.1 Introduction..................................................................................................... 61

4.2 Integration Scheme in Time and Space ............................................................ 62

4.2.1 Central Finite Differences.................................................................... 63

4.2.2 Newmark’s Method ............................................................................. 64

4.2.3 Numerical Formulation of Equilibrium Equations ................................ 65

4.2.4 Boundary Conditions ........................................................................... 70

4.3 Solution Procedure .......................................................................................... 71

4.3.1 Iteration Loops and Time Steps............................................................ 71

4.3.2 Eigenfrequency and Added Mass ......................................................... 73

4.3.3 Formulation of Coefficient Matrix ....................................................... 75

4.4 Verification of the Method .............................................................................. 81

4.4.1 Static Response ................................................................................... 81

4.4.2 Dynamic Response .............................................................................. 88

4.5 Summary......................................................................................................... 91

4.6 Bibliography ................................................................................................... 91

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5.1 Introduction ..................................................................................................... 93

5.2 Failure Modes.................................................................................................. 94

5.2.1 Face Fracture ....................................................................................... 95

5.2.2 Local Buckling..................................................................................... 95

5.2.3 General Buckling ................................................................................. 96

5.3 Lamina Failure Analysis .................................................................................. 98

5.3.1 Principal Strains and Stresses............................................................. 100

5.3.2 Lamina Failure Modes and Criteria .................................................... 101

5.4 Laminate Failure Analysis ............................................................................. 108

5.4.1 Laminate Failure Model ..................................................................... 108

5.5 Core Failure Analysis .................................................................................... 109

5.5.1 Core Shear Failure ............................................................................. 110

5.5.2 Debonding of Core and Face .............................................................. 111

5.5.3 Shear Crimping .................................................................................. 112

5.5.4 Core Indentation ................................................................................ 112

5.6 Summary ....................................................................................................... 112

5.7 Bibliography.................................................................................................. 113

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6.1 Introduction ................................................................................................... 115

6.2 Plate Stiffness Reduction Model .................................................................... 116

6.3 Comparison of Damage Model and Experiments ............................................ 118

6.4 Failure Scenario Example .............................................................................. 121

6.5 Ultimate Strength, Linear and Non-Linear Analysis ....................................... 126

6.6 Summary ....................................................................................................... 128

6.7 Bibliography.................................................................................................. 129

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7.1 Introduction................................................................................................... 131

7.2 The Structure of 3DQHO................................................................................... 131

7.3 Analysis of Existing Design........................................................................... 134

7.3.1 Rescue Vessel LRB ........................................................................... 134

7.3.2 Mine Hunter SF300 ........................................................................... 139

7.3.3 Racing Yacht ILC40 .......................................................................... 141

7.4 Design Example ............................................................................................ 144

7.5 Summary ....................................................................................................... 145

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8.1 Introduction................................................................................................... 147

8.2 The Stiffened Single Skin Rule...................................................................... 147

8.3 The Sandwich Rule ....................................................................................... 151

8.4 Summary ....................................................................................................... 153

8.5 Bibliography ................................................................................................. 154

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Leonardo Da Vinci

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This thesis deals with the structural behaviour of laminated composite hull plates in high-speed light craft. Composites made of fibre-reinforced plastic (FRP) are often superior tosteel and aluminium as building material for high-speed light craft (HSLC) due to a lowweight/strength ratio. The high specific strength of glass fibres together with the superiorspecific stiffness offered by carbon and other high-modulus fibres has led to an increasinguse of these materials in fast marine vessels, such as ferries, special military ships andhigh-performance sailing and power boats. The knowledge of the material behaviour,strength and fatigue of FRP composites is still limited. Most designs are based on boatbuilding experience rather than structural analysis, which is often too expensive to per-form. The background for this study is the need of a handy design tool which, in a shorttime, is able to perform response and failure analysis of sandwich and single-skin FRPstructures.

Understanding of geometrical non-linear behaviour, due to large lateral deflections, is es-sential in order to produce correct and efficient composite designs. It has been known for anumber of years that the geometrical non-linearity of laterally loaded FRP plates is sig-nificant already at low load levels. This has been experimentally shown both for single-skin plates, Shenoi, Moy and Allen [7], and for sandwich plates, Bau, Kildegaard andSvendsen [1].

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At present, most of the dimensioning procedures for FRP hull plates rely on linear (smalldeflection) theory. In addition, the scantlings of composite plates in HSLC are often re-stricted by the empirical and somewhat arbitrary rule of a maximum lateral deflection re-lated to the panel span. This design criterion is imposed by many classification societiessuch as %9 [2] and '19 [3] and further discussed in Riber and Terndrup [6]. Since thiscriterion, in general, restricts the designs it is of interest to investigate alternative calcula-tion procedures.

Motivated by this, a dynamic finite difference model is developed. The model is based ongeometrical non-linear plate theory including the transverse shear deformation, which ispronounced for sandwich plates with relatively flexible core. It is formed into a Fortran-coded design tool 3DQHO dealing with orthotropic asymmetric composite single-skin andsandwich plates. Subjected to time-dependent lateral loads and with different boundaryconditions, the plates are analysed statically as well as dynamically with respect to lateraldeflections, strains and stresses. Furthermore, failure loads, locations and modes are cal-culated and visualised by use of a progressive damage model based on the appropriatefailure criteria.

The concept of the model is to provide a simple and a fast tool, which can be used in thedimensioning phase of hull panels. With a complete design based on calculations andanalyses with 3DQHO, more detailed information of the internal stress level can be obtained,if needed, by use of 3-D finite element (FEM) analyses of particular details in the struc-ture.

Various authors, among others Hildebrand and Visuri [5] and Falk [4], also using non-lin-ear approaches for FRP plate response analysis, suggest the use of larger panel fields inorder to eliminate errors introduced by incorrect boundary conditions. However, thosemethods are still based on time-consuming FEM calculations, and yet display the problemof defining the correct boundary conditions for the large panel field.

As an example the rescue boat (Fig. 1.1) is built of foam core sandwich with glass/epoxyskins. It is dimensioned for a vertical acceleration of 5 J, equivalent to a slamming load of125 .3D in order to withstand the rough weather conditions in the North Sea. However,the bottom part of the hull is conservatively dimensioned, since the structural design fol-lows the common classification rules, and moreover, the strict requirements for structuralsafety prescribed by the national authorities.

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Contrary to rescue boats, the requirements for structural safety in the design of high- per-formance sailing boats are low. These designs are governed by high performance ratherthan structural safety and endurance. This could be observed during the recent round-the-world solo regattas, where structural failures resulted in loss of boats and human lives.Too many designs among this type of boats are badly analysed with regard to structural re-sponse and safety.

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Navy vessels form a third group of FRP high-speed craft consisting of mine hunters, gunboats, patrol boats etc. These ships are normally well analysed with respect to the ultimatestrength. The hull structures are designed close to the structural limits, since the vessels,in general, need no classification approval. At present, the most modern ship in the Danish


Navy is the 54 P�multipurpose ship, 6)�� (Fig. 1.3), which is a glass/polyester foamcore sandwich design originating from the Swedish Navy. Among the most advanced shipsin this group are the Swedish high-speed craft 6P\JHQ and <6��. The first is a 30 PSES�test boat, whereas the latter is a 75 P multi-purpose ship with approximately the samedisplacement as the 6)��.

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The subject of this thesis is presented in 9 chapters composed as follows. Chapter 2 givesan overview and an introduction to FRP hull manufacturing and structural hull design,followed by the appropriate design loads with special focus on slamming pressures. InChapter 3 the general non-linear sandwich theory is presented and two analytical solutionsare derived in order to provide alternative simple design methods for FRP sandwichplates. Chapter 4 presents a numerical formulation of the theory given in Chapter 3. Theresult is programmed into a design tool 3DQHO, which is intended for preliminary design ofFRP hull panels. Chapter 5 discusses different failure criteria and failure modes, which areimplemented in a progressive damage model described in Chapter 6.

A brief introduction and a description of the design tool 3DQHO are given in Chapter 7 il-lustrated with examples of designs and analyses of FRP sandwich hull plates. Chapter 8discusses the '19 HSLC code in the light of results calculated with 3DQHO.

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[1] Bau-Madsen N.K., Svendsen K.H. and Kildegaard A. Large Deflections of SandwichPlates - an Experimental Investigation. &RPSRVLWH� 6WUXFWXUHV. Vol. 23, pp. 47-52,1993

[2] Bureau Veritas. Rules for the Construction and Classification of High Speed Craft.

17 bis, Place des Reflets, La Defense 2, 92400 Courbevoie, France, 1995. [3] DNV. Classification Rules for High Speed Light Craft, Det Norske Veritas Research

AS, Veritasveien 1, N-1322 Høvik, Norway, 1991. [4] Falk L. Membrane Stresses in Laterally Loaded Marine Sandwich Panels. 3URFHHG�

LQJV�RI� WKH��UG�,QWHUQDWLRQDO�&RQIHUHQFH�RQ�6DQGZLFK�&RQVWUXFWLRQV, Southampton,UK. Vol. 1 (4A), 1995.

[5] Hildebrand M. and Visuri M. The Non-linear Behaviour of Stiffened FRP-Sandwich

Structures for Marine Applications. 7HFKQLFDO� 5HSRUW� 977� 9$/%� ��, Espoo, Fin-land, 1996.

[6] Riber H.J. and Terndrup Pedersen P. Examination of Criteria for Panel Deflection in

DNV’s Rules for High Speed and Light Craft. Technical Report No. 96-2014, DetNorske Veritas Research AS, Veritasveien 1, N-1322 Høvik, Norway, May, 1996.

[7] Shenoi R.A., Allen H.G. and Moy S.S.J. Strength and Stiffness of FRP Plates. 3URF�

,QVW��&LYLO�(QJUV��6WUXFWXUHV�DQG�%XLOGLQJV, May 1996.


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Fibre-reinforced plastic (FRP) composites are among the most commonly used buildingmaterials for high speed light craft (HSLC) hulls. This is mainly because of the highstrength-to-weight ratio of the material, which is ideal for construction of ship hulls andmakes it a cost-efficient material. Further, the FRP is corrosion-resistant and has a lowmaintenance cost. Finally, the low magnetic characteristic of most FRPs makes themsuited for smaller naval ships assigned for special tasks, such as mine hunting.

In order to design a high-speed vessel the use of light materials in the structure is obvi-ously an advantage. However, for longer ships (/ZO�> 50 P) the hull flexibility must beconsidered as the relatively low hull beam stiffness of FRP ships compared to steel shipsmay introduce fatigue damages in the hull, Hansen et al. [8].

The common definition made by� 7KH� ,QWHUQDWLRQDO� 0DULWLPH� 2UJDQLVDWLRQ�� (,02), ofwhen a vessel is categorised as a high-speed craft is a minimum requirement for the serv-ice speed/displacement ratio, which states:

9 ≥ ⋅ ∇37 1 6. / (2. 1)

with the forward speed, 9, in knots and the displacement, ∇ , in tons. In general, the clas-sification societies use this definition for HSLC and apply special design rules for thesetypes of vessels. The leading classification societies providing rules for HSLC design areAmerican Bureau of Shipping ($%6), Bureau Veritas (%9), Det Norske Veritas ('19),Lloyds Register (/5) and Registro Italiano Navale (5,1D).

Most of the existing FRP high-speed craft are small (/ZO��P). This is mainly due to thelimitations of the technical aspects in the construction of large FRP hulls. However, this

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limit has been exceeded for several ships and, certainly, more FRP vessels beyond 50 P oflength will appear in the future. Currently, the Swedish Navy is building a multipurposewarship with a maximum speed of approximately 50 NQRWV� and a length of about 75 P,using foam core sandwich with skins of glass, carbon/aramid fibres in a vinylester resinfor the hull.

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A structural hull design is primarily based on the knowledge of the ultimate load condi-tions the particular ship will meet during its lifetime. From these design loads the prelimi-nary hull layout can begin and each structural member can be dimensioned in accordancewith the prescribed rules. When it is decided to use FRP composite as the building mate-rial, either sandwich or stiffened single skin can be selected for the hull structure. Often,the choice is determined from building traditions and design philosophy rather than simplytechnical considerations. E.g. the sandwich technology is widely used in the high-perform-ance craft built by the Swedish Navy. In contrast, the British Navy has a long tradition ofusing stiffened single skin for their HSLC marine vessels from the point of view thatshock loads may cause delamination of the skin from the core.

The primary structural design criteria, which should be taken into consideration in the de-sign phase of sandwich and single-skin hulls, are listed in the following:

◊ global hull bending, shear and torsion deformations◊ panel deflections◊ stresses in the skins or in the laminates◊ stresses in the core◊ skin wrinkling◊ global panel stability

Prior to the choice of hull type, the assets and the drawbacks involved in the manufactur-ing and design of either single skin or sandwich should be taken into consideration. Thetwo concepts are outlined in the following, in addition to the FRP design rules imposed bysome of the leading authorities, in order to provide the reader with an overview of the twodifferent building concepts and to give an idea of the limits of the design rules.

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The stiffened single-skin concept is technically the simplest way of building a FRP com-posite hull. Basically, it requires a female mould in which the fibre mats impregnated withresin are applied. Pre-fabricated stiffeners are then attached by use of additional fibre-re-

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inforced mats or by use of an adhesive (resin). Alternatively, the stiffeners are shaped di-rectly on the skin by use of a light type of foam as an inner mould for the stiffener (Fig.2.1).

The lay-up is usually done by hand, and more recently, with help by sophisticated vacuumtechniques and temperature-regulated moulds. This technique requires expensive toolssuch as special moulds, vacuum-bags and -pumps. However, improved and costly manu-facturing techniques are required in order to ensure sufficient quality of the hulls in mod-ern FRP high-performance vessels.

Single skin

Longitudinal stiffener

Resin, adhesiveor filler material

Foam core

Transverse stiffener

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The often complicated stiffener system is laborious to manufacture, especially as the stiff-ener attachment to the hull requires careful mechanical surface preparations. Delaminationof the stiffener from the hull is often observed in regions with high impact loads andwhere the stiffener has been badly assembled. The stress concentrations can be decreaseddrastically by rounding the corners of the stiffener reinforcement as illustrated in Fig. 2.2.High stress concentrations are introduced in the left stiffener attachment, since the radiusof curvature is very small, whereas the right stiffener evens out the stress level due to thelonger radius of curvature.


Stress concentrationsand delamination

Stiffener reinforcement Curvature 1/U

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The primary structural design criteria for single-skin plates in the bottom of the hull pro-vided by the classification societies, Bureau Veritas, %9 [3], and Det Norske Veritas, '19[5], are listed and commented on in the following (Eqs. 2.2-7). The numbers in bracketsrefer to the numbers in the respective rule set. Note that the units are in SI.

• Minimum thickness, Wmin, of the skin:

%9 W /ZOmin . .= ⋅ ⋅ +−15 10 0 97 103 (C.3.8.4.3.34) (2. 2)

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ZO

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min

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= ⋅+

⋅

−105 0 0 09

16 10

3

8

σ (A 202, Sec. 6) (2. 3)

where /ZO is the waterline length and σQX is the ultimate tensile stress.

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6

8

10

12

14

16

0 20 40 60 80 100

%9 Eq. 2.2

'19 Eq. 2.3

Length /ZO [P]

WPLQ [PP] Minimum hull thickness as function of hull length

σQX

03D= 160

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In the range of 10 - 100 metres of length the '19 requirement for the minimum skinthickness is approximately 12 % lower than the requirement for %9, (Fig. 2.3). The mini-mum thickness rule is intended for design against impact, however, it must be consideredin the design of laterally loaded panels.

• Maximum stress, σPD[, from a given load, T, on a square simply supported panel:

%9 σ σmax . .= ⋅

≤ ⋅0 313 0 22

2EW

T nu (C.3.8.4.3.35) (2. 4)

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+

+

≤ ⋅

2 42 6 47

30 23

0 302

ZWZW

EW

T��QX

2

(B 202-3, Sec. 6) (2. 5)

where Z� is the midpoint lateral deflection, W is the skin thickness, E is the plate breadthand T is the lateral pressure.

Eq. 2.5 is a combination of the '19�rules B 202-203, in order to make a better compari-son with the %9 rule (Eq. 2.4). The rules are presented for the special case of a plate withan aspect ratio equal to one and simply supported boundary conditions. However, both setsof rules provide correction factors depending on varying aspect ratios and boundary con-ditions. The '19 rules are based on non-linear theory and consequently, they are less con-servative than the %9 rules (Figs. 2.4-5). Furthermore, the maximum allowable stressvalue given by '19 is 35 % higher than suggested by %9.

0

5

10

15

20

25

30

35

0 0,5 1 1,5 2

%9 Eq. 2.4

'19 Eq. 2.5

Relative deflection Z�W

σPD[�[03D] Maximum stress as a function of relative deflection

T� ��.3DE�W� ��

Figure 2.4: 6WUHVV�UHVSRQVH�DFFRUGLQJ�WR�%XUHDX�9HULWDV�DQG�'HW�1RUVNH�9HULWDV�


The relative midpoint deflection is expressed for both the codes in Eqs. 2.6-7:

%9ZW

EW

T(

ZE

=

⋅ ≤−

42148 10

100. , (C.3.8.4.3.38) (2. 6)

'19ZW

ZW

EW

T(

ZW

+ ⋅

=

⋅

≤−

3 422 4 3 35 10 1. . , (B 202, Sec.6) (2. 7)

where ( is the elasticity-modulus of the plate. The formulae are given for a plate withclamped boundary conditions (only case provided in the %9 code).

0.0

0.5

1.0

1.5

2.0

0 5 10 15 20 25 30

%9, Eq. 2.6

'19, Eq. 2.7

Lateral load T [.3D]

Relative deflection as a function of lateral loadZ�W

(� ��*3DE�W� ��

Figure 2.5: 5HODWLYH ODWHUDO�GHIOHFWLRQ�DFFRUGLQJ�WR�%XUHDX�9HULWDV�DQG�'HW�1RUVNH�9HULWDV�

At the maximum deflection (Z�W� ��) the '19 rule, which is rewritten in the form of Eq.2.7, allows 42 % more lateral load than calculated by the %9 rule (Eq. 2.6) for a represen-tative GRP hull plate (Fig. 2.5). It is evident that the '19 code is more sophisticated thanthe %9 code concerning the design of FRP single-skin plates, since it takes into accountthe non-linearity from large deflections.

In the design of FRP stiffened single-skin plates, the above rules usually determine theminimum scantlings. The rules must be supplied by additional design formulae regardinglocal and global buckling, stress analyses at specific locations etc. in order to ensure acomplete structural analysis of the hull components.


��6DQGZLFK�+XOO�'HVLJQ

A sandwich consists of three main parts (Fig. 2.6): face (or skins), core and a bondingmaterial. The sandwich structure is defined by ASTM [2] as follows:

$�VWUXFWXUDO�VDQGZLFK�LV�D�VSHFLDO� IRUP�RI�D�ODPLQDWHG�FRPSRVLWH�FRPSULVLQJ�D�FRPELQD�WLRQ�RI�GLIIHUHQW�PDWHULDOV� WKDW�DUH�ERQGHG�WR�HDFK�RWKHU�VR�DV�WR�XWLOLVH� WKH�SURSHUWLHV�RIHDFK�VHSDUDWH�FRPSRQHQW�WR�WKH�VWUXFWXUDO�DGYDQWDJH�RI�WKH�ZKROH�DVVHPEO\�

%RQGLQJ�PDWHULDO )DFH�PDWHULDO

&RUH�PDWHULDO

Figure 2.6: ��6FKHPDWLF�GUDZLQJ�RI�D�VDQGZLFK�VWUXFWXUH�

The primary advantage of using the sandwich concept in a FRP hull instead of a stiffenedsingle-skin structure is the built-in flexural stiffness of the sandwich, which makes thestiffener system unnecessary. The bending and the in-plane stresses are mainly carried bythe faces, whereas the shear stresses are taken by the core. The building of an FRP sand-wich hull requires, however, more technical skills and advanced technology than buildinga single skin hull.

The most common production method of a sandwich hull is to make use of a female mouldand proceed as for the single skin hull. After the outer skin has been formed in the mouldthe core, usually PVC foams but also aluminium or resin-impregnated honeycomb, isbonded to the skin employing an adhesive, which is most often the resin used for theskins. Next, the core material is tapered before the inner skin is applied to the core.

Alternatively, the building process can be reversed, as done for the 6)��, in case oflarge hulls or when only a small series of hulls is built. The ship is built upside down byusing the transverse frames as a male mould on which the core is formed. Then, the outerskin is applied and the hull is turned around proceeding with the inner skin as for the sin-gle-skin hull production. In order to secure strong bonding between the skins and the corethe use of vacuum technique is an advantage.


WF� ��W

WF� ��W

Face thickness =�W��and��0DVVIDFH� ��0DVVFRUH

�W

�W

��W

( ), W W

0DVV 0DVV IDFH

13 3

1

1

122

2

32

= =

=

, W W W W ,

0DVV 0DVV 0DVV 0DVV

,0DVV

,0DVV

IDFH IDFH

2

23 3

1

2 1

2

2

1

1

23

22

1

12

14

37

22

25104

6 73

=

+ = =

= + =

=

.

.

, W W W W ,

0DVV 0DVV 0DVV 0DVV

,0DVV

,0DVV

IDFH IDFH

3

23 3

1

3 1

3

3

1

1

211

22

1

12

182

391

210

2512

758

=

+ = =

= + =

=

.

.

5HODWLYH�EHQGLQJ�VWLIIQHVV� ��



6WLIIQHVV�WR�ZHLJKW�UDWLR�RI�VDQGZLFK�YHUVXV�VLQJOH�VNLQ�SODWHV

Figure 2.7: %HQGLQJ�VWLIIQHVV�WR�ZHLJKW�UDWLRV�IRU�VDQGZLFK�DQG�VLQJOH�VNLQ�FURVV�VHFWLRQV�

The sandwich is a structurally efficient structure with regard to stiffness/weight ratio,which is illustrated in Figure 2.7. The example shows the moment of inertia, ,, the specificweight, :, and the stiffness/weight ratio, ,�:, for a representative GRP hull sandwich. Fora modern sandwich hull design the face/core thickness ratio is about 1/10, which gives arelative bending stiffness of almost 75 times the stiffness of the equivalent single skin. Itshould be noted that the comparison neglects the stiffener for the single skin. However,the example illustrates the structural efficiency of the sandwich concept.

The structural design criteria for sandwich plates in FRP hulls provided by Bureau Veri-tas, %9 [3], and Det Norske Veritas, '19 [5], are listed in the following (Eqs. 2.8-13).

• Minimum thickness, Wmin, of the faces:

%9 W /ZOmin . .= ⋅ ⋅ +−0 6 10 0 97 103 (C.3.8.4.4.42) (2. 8)


'19 W/

ZO

QX

min

. .

.

= ⋅+

⋅

−1015 0 09

16 10

3

8

σ (A 203, Sec. 5) (2. 9)

2

4

6

8

10

12

0 20 40 60 80 100

%9 Eq. 2.8

'19 Eq. 2.9

Length /ZO [P]

σQX

03D= 160

WPLQ [PP] Minimum face thickness as function of hull length

Figure 2.8: 0LQLPXP�IDFH�WKLFNQHVV�UHTXLUHG�E\�%XUHDX�9HULWDV�DQG�'HW�1RUVNH�9HULWDV.

The linear '19 rule for minimum face thickness penalises unnecessarily long ships. Thenon-linear formula given by %9 seems more reasonable.

• Maximum stresses, σmax, τPD[, and deflection, ZPD[, from a given pressure,�T:

%9 σ σmax . .= ⋅ ≤ ⋅0 052 0 222TE

: nu (C.3.8.4.4.43) (2. 10)

τ τmax . .= ≤ ⋅0 55 0 4TEG

nu (C.3.8.4.4.47)

'19 σ σmax . .= ⋅ ≤ ⋅0 050 0 302TE

: nu (B 201, Sec. 5) (2. 11)

τ τmax . .= ≤ ⋅0 34 0 35TEG

nu (B202, Sec. 5)

where : is the section modulus of the sandwich plate. For a sandwich with equal facethickness, we get :� GWIDFH, where G is the distance between the neutral axes of the twofaces. The rules are given for sandwich plates with aspect ratio equal to one and simply


supported boundary conditions. The '19 rules provide correction factors depending ondifferent aspect ratios and boundary conditions and, consequently, represent a more de-tailed set of rules, than the one of %9.

The face stress response is approximately the same for the two expressions (4 % higherpredicted by %9) but the maximum allowable stress given by '19 is 35 % higher than theone imposed by %9. The core shear stress predicted by %9 is almost 62 % higher than theone of '19. This is due to the simplification of the %9 rule, which covers all aspect ratiosin one single expression. For larger aspect ratio’s, the core shear stress (%9) is only 8 %higher than calculated by '19, hence the %9 is based on beam theory more than platetheory.

The relative midpoint deflection response is expressed for both of the codes below:

%9 ZTE

( G WTE* G

��ZE

I I F

= 2.47 + 75.6⋅ ≤4

2

2

100 (C.3.8.4.3.38) (2. 12)

'19 ZTE

( G WTE* G

��ZE

I I F

= 2.03 + 74⋅ ≤4

2

2

100(B 400, Sec. 5) (2. 13)

where (I is the E-modulus of the faces and *F is the shear-modulus of the core. The for-mulas are given for the case of a plate with clamped boundary conditions. In order to ex-press the two rules in the same form, the following approximations are made for eliminat-ing the moment of inertia, ,, in the %9 rule.

G W ,GWFRUH IDFH≈ ⋅ ≈ ≈11 0 25

2

2

. . ν

The %9 rule is the most cautious of the two and gives the deflection response for clampedboundary conditions and symmetric sandwich plates only. The '19 code also offers thepossibility of using different faces and simply supported boundary conditions.

The above discussion of the design rules for stiffened single-skin and sandwich plates,using the classification societies '19 and %9 as examples, shows that there is extensiveguidelines for making such structures. Yet, the sandwich rules need further investigationsince the rules in this field are based solely on linear theory. Furthermore, the maximumdeflection criterion, Z�E� �� 0.01, generally determines the scantlings of the plate, eventhough the stress levels are far below the allowable limits.

��'HVLJQ�/RDGV��17

��'HVLJQ�/RDGV

The dimensioning loads for the hull of small high-speed craft are mainly impact loadsfrom vertical accelerations of the hull penetrating the water surface, i.e. slamming. Thestructural response from the global loads, such as hogging and sagging of the hull beams,are often minor compared to the response from slamming loads. The HSLC dimensioningrules from most classification societies neglect the global loads. If the ship is below acertain overall length. '19, for example, requires only analyses with local design loads ifthe ship is less than 50 P in length.

��*OREDO�/RDGV

For vessels exceeding the limit criteria of the small craft definition as specified by the in-dividual classification societies, the global hull strength must be taken into considerationas well as the local strength requirements. Thus, the following load situations (Figs. 2.9a-b) must be analysed with regard to global strength:

1. Crest landing moment2. Hollow landing moment3. Hogging moment4. Sagging moment5. Shear forces from longitudinal loading

For vessels with more than one hull, additional loads must be analysed:

1. Torsional moment2. Transverse bending moment3. Transverse shear force

Transverse stress resultants of twin hull

7UDQVYHUVH�EHQGLQJ��VKHDU�DQG�WRUVLRQ�

Figure 2.9a: ��&RQWLQXHG�RQ�WKH�QH[W�SDJH.


&UHVW�ODQGLQJ +ROORZ�ODQGLQJ

+RJJLQJ 6DJJLQJ

Slamming-induced global moments

Wave-induced global moments

Figure 2.9b: *OREDO�ORDGV�LQGXFHG�LQ�WKH�KXOO�EHDP�E\�ZDYHV�DQG�VODPPLQJ�

Rough estimates (from '19 [5]) of the above illustrated global bending moments andshear forces for mono-hulls are given here. The crest and hollow moments are derivedconsidering the hull as a simple beam (Fig. 2.10), where ∆ is the ship displacement, J� =9.81 P�V�, DG is the design acceleration and OV the extent of the longitudinal slamming area,(SI units).

/&* (longitudinal centre of gravity)

( )) J DGHVLJQ= +∆ 0

HZ HU

KROORZ FUHVW

OV)�� )��5HDFWLRQ 5HDFWLRQ 5HDFWLRQ

/&* for forwardand aft half of ship

Figure 2.10: %HDP�PRGHO�IRU�DSSUR[LPDWH�JOREDO�EHQGLQJ�PRPHQWV�LQ�VKLS�KXOO�


&UHVW�/DQGLQJ�0RPHQW

( )0 J D HO

FUHVW GHVLJQ Z

V= + ⋅ −

∆2 40 (2. 14)

+ROORZ�/DQGLQJ�0RPHQW

( )0 J DHH

HKROORZ GHVLJQ

Z

U

Z= + ⋅ −

⋅

∆2

10 (2. 15)

The hogging/sagging moments and the shear forces are derived by integration of the forcesfrom still-water analyses (buoyancy and body forces) in addition to the resultants from thewave contribution (hydrodynamic forces), Pedersen and Jensen [15]. Tentative designformulas are given below for ship length /ZO��P, '19 [5].

+RJJLQJ�0RPHQW��0KRJ (still water + wave)

0 / % &KRJ ZO ZO E= 24 3 (2. 16)

6DJJLQJ�0RPHQW��0VDJ�(still water + wave)

( )0 / % &9/VDJ ZO ZO E

ZO

= + ⋅ +

10 0 7 085 0 343 . . . (2. 17)

6KHDU�)RUFH��4E

40/E

WRWDO

ZO

= 4 103 (2. 18)

where /ZO, %ZO, &E and 9 are length, breadth, block coefficient and maximum speed, re-spectively, (SI units).

��/RFDO�/RDGV

The non-linear sandwich theory does not take into account local bending of the faces dueto vertical displacement of the core. According to the definition of a sandwich: ”WR�XWLOLVHWKH� SURSHUWLHV� RI� HDFK� VHSDUDWH� FRPSRQHQW� WR� WKH� VWUXFWXUDO� DGYDQWDJH� RI� WKH� ZKROH� DV�VHPEO\´ (Sec. 2.2), it should not be necessary to include analysis of local bending of thefaces, as a structure with a significant effect of local face bending is simply not a sand-wich. However, in real life local bending moments are sometimes introduced. As for fail-


ure prediction it is essential to analyse local bending effects in order to determine sometypes of delamination.

Thomsen [16] derives an analytical expression for approximate solutions of local bendingeffects in sandwich plates with orthotropic face layers subjected to localised loads. Thelocal loads can be concentrated external loads or line loads at the plate boundaries induc-ing large peeling stresses i.e a stress resultant in ] -direction, which may result inface/core delamination.

In his work the local bending analysis is based on the assumption that the relative deflec-tion of the loaded face against the core can be modelled by application of an elastic foun-dation model. This is achieved by introducing a two-parameter elastic foundation model,which takes into account the vertical and shear stress effects between the loaded face andthe core. The overall solution is completed by superposition of the linear sandwich theoryand the local solution.

Nevertheless, it is doubtful if the solution can be superimposed with the non-linear sand-wich theory presented in Chapter 3. For more detailed sandwich plate analysis concerningedge delamination, the method is recommended for small lateral deflections.

��6ODPPLQJ�/RDGV

A rather irregular load on high-speed craft is the slamming pressure, which is caused bythe impact of the bottom of the hull against the water surface resulting in a sudden changeof the relative acceleration of the boat. Slamming is an impulsive pressure during a veryshort period of time (milliseconds). For design of FRP hull panels the slamming pressureis generally the dimensioning load. A theoretical derivation of the slamming pressure isshown, followed by a simple approach to determine an equivalent static pressure as thedesign load.

It may be argued that the peak pressures have little importance for the panel responsesince they occur in a very short period of time. Thus, to compare slamming and strain re-sponse it is convenient to average the pressure over a period of time and a given area. Fi-nally, the strain response is dependent on the pressure variation in time and place (Eq.2.19).

ε η= ⋅I S W [ \( ) ( , , ) (2. 19)

where I ( )η is a response function.


Typical slamming measurements are shown in Fig. 2.11. The duration of the pressurepeaks is approximately 0.01 seconds, which requires a sample frequency of at least 100+].

-1

0

2

4

6

8

0 1 2 3 4

Pressure [.3D]

Time [VHF]

,PSDFW��Fig. 2.14

Slamming measurements on a 470 hull panel

+V ��P8Y ��P�VψFJ ��ψ� ��

Sample frequency 33 +]

Figure 2.11: 7\SLFDO�PHDVXUHG�WLPH�KLVWRU\�RI�VODPPLQJ�SUHVVXUH��5LEHU�>��@�

A simple way to model a hull slamming pressure is to consider the problem of a wedgepenetrating a liquid surface. Several two dimensional analyses of this type have been pub-lished, including those by Karman [9] and Wagner [17]. Among the more recent publica-tions are Szebehely [14], Chuang [4], Ochi and Bledsoe [11] and Payne [12], the latter isbased on the theory by Karman.

Hansen [7] compares the different slamming theories of the above-mentioned authors andconcludes that the simple theory by Karman produces adequate results. The followingderivation of the slamming pressure is based on the work by Karman.

A wedge-shaped body of mass 0 and of a dead-rise angle ψ strikes a horizontal surface ofwater with the velocity 8Y and generates a two-dimensional flow (Fig. 2.12). The wedge isconsidered to be rigid and to enter the liquid with a velocity normal to the liquid surface.Thus, neither hull flexibility nor forward speed is taken into account.

After the body has entered the water its velocity at time W is 9Y. The momentum, ,P, of thissystem becomes:

,P� 08Y� ��0�P�9Y��(2. 20)


neglecting the effect of gravity, buoyancy and skin friction, since they are considered neg-ligible in comparison with the unsteady hydrodynamic force according to Szebehely [14].

z

y

n

9Y

ψ

H

H’

piled up water

Figure 2.12: :HGJH�VKDSHG�ERG\�VWULNLQJ�WKH�ZDWHU�VXUIDFH�

The added mass, P, comes into existence when the body pushes away the fluid in front ofit, which creates a flow around the body. The added mass is found from the kinetic energyof the fluid put in motion by

( )P(

9 9JUDG G9RONLQ

Y Y9RO

= = ∫∫∫2

2 2

2ρφ (2. 21)

which is transformed by Green’s theorem to

P9 Q

G6Y

6

= ∫∫ρ

φ∂φ∂2 (2. 22)

where φ is the velocity potential, 6 is the boundary area between the water and the bodyand ρ is the density of the fluid. For a flat plate of semi-width H, where the upper part ofthe plate is not in contact with the water at the instant of impact, the added mass per unitlength becomes:

P H=1

22ρπ (2. 23)

as the potential φ for the flat plate is given by

φ( , )\ W 9 H \ HY

= − − <2 2 , y (2. 24)

The effect of piled-up water is neglected in Eq. 2.24. The true added mass is between

P H121

2= ρ and P H2

21

2= ′ρ (2. 25)


The ratio H�H = 1 is suggested by Karman [9] , whereas Wagner [17] uses H�H = π/2 forsmall dead-rise angles (tanψ ~ ψ). The phenomenon is profoundly discussed in Szebe-hely [14]. In the following derivation the piled-up water is neglected. Setting the velocity,9Y, as

9G]GW

G\GWY

= = tanψ (2. 26)

Eq. 2.20 becomes

0G\GW

\0

08Y

tan ( )ψρπ

12

2

+ = (2. 27)

yielding

G\GW

8Y

( ) cot1 1+ =µ ψ , µρπ

1

2

2=

\0

(2. 28)

which gives the relationship between velocity and depth as

G\GW

8Y=+cotψ

µ1 1

(2. 29)

Expressing the second derivative of \

G \GW

GGW

G\GW

2

212

2= ( ( ) ) (2. 30)

and combining this expression with Eq. 2.29, we get

G ]GW

G \GW

8 \0

Y

2

2

2

2

2 2

131

⋅ = =+

⋅ −cotcot

( )( )ψ

ψµ

ρπ (2. 31)

Finally, the expression from the force of impact, ), yields

) 0G ]GW

8 \Y= ⋅ − =

⋅+

( )cot

( )

2

2

2

131

ψ ρπµ

(2. 32)

The average pressure becomes

S)\

8Y= = ⋅

+2 2 1

2

13

ρ π ψµ

cot

( ) (2. 33)


and the maximum pressure is found at the moment of first contact for \ = 0:

S8

Y

max cot=ρ

π ψ2

2 (2. 34)

Eq. 2.33 averages the pressure over a given wet surface. In order to get the pressure varia-tion along the wet surface of the wedge immersed into the water, we combine the velocitypotential φ (Eq. 2.24) with Bernoulli’s equation for unsteady potential flow, neglecting theeffect of gravity, (Eq. 2.35).

S I W= − + +ρ∂φ∂

∂φ∂( ( ) ( ))

t y

1

22 and φ( , )\ W 9 H \ H

Y= − − <2 2 , y (2. 35)

Applying the substitutions and the differentiation (Eqs. 2.36-38) below to Eq. 2.35, an ex-pression for the pressure variation is obtained (Eq. 2.39).

∂φ∂

∂∂

∂∂

ψµψ µ

W9

WH \

9

WH \

9H

H \

H \

H

Y

Y

Y

= − − − −

= −−

+−

+ ⋅

( )

(tan tan ( )

)

2 2 2 2

2

2 2

12 2

1

1

2

2 1 4

1

(2. 36)

∂φ∂\ 9

\

H \Y

=−2 2

(2. 37)

where

98 H

0H

9 WY

Y Y=+

=1 21

1

2

µµ

ρπψ

, , tan

(2. 38)

Finally, the expression for the pressure variation becomes

S 8\H

\H H

\

8WHUP WHUP WHUP

Y

Y

=+

⋅

−

−+

− −−

=⋅ +

⋅ − −

1

2

2

1

1

1

2

11

1

21

11 2 3

2

12 2

2

1

1

2

2 2

2

2

12

ρψ µ

µµ

ψ

ρψ µ

tan ( ) ( )

tan

tan ( )( )

(2. 39)


The pressure variation along the wedge is illustrated for a planing 470 dinghy in Fig. 2.13for a moderate sea state.

0� �54�NJ�P 8Y� �3.1 P�V H� �0.21�P ψ� �tan17 = 0.31 ρ� �1015�NJ�P�.

The maximum pressure (Eq. 2.39) yields 8.7 .3D��Zhich has to be compared to 4.1 .3Dcalculated by Eq. 2.33, where the pressure is average over the plate width. This gives anestimated pressure approximately half the maximum pressure.

-6

-3

0

3

6

9

0 0.05 0.1 0.15 0.2 0.25

WHUP�

WHUP�

WHUP�

S

H� �0.21

3UHVVXUH [.3D]

Pressure variation along the bottom panel of a 470 sailing boat

( )1 2 11 1− +µ µLength from keel [P]

Figure 2.13: 3UHVVXUH�YDULDWLRQ�DORQJ�D��ERWWRP��5LEHU�>��@�

The first two terms come from the time derivation of φ , whereas the last term originatesfrom the gradient of φ, which is apparently of minor importance. It can be shown that thefirst term is greater than the third term, except very near to the edge. Therefore, not con-sidering the second term, which is finite, the pressure on the entire plate is positive. Thesecond term comes from the fact that the velocity of the plate 9Y is not constant in time.

If µ1 > 1 there is a negative pressure zone around the keel as the second term approaches

the value ( )− +2 11 1µ µ , since first term always contributes 1 and the last term nothing at

the keel. For small masses 0 a relatively small plate length H is required to make µ1 > 1(Eq. 2.38). Consequently, a higher probability of having negative pressures around thekeel can be expected for smaller masses than for large. The balance between the first andsecond term is physically explained as the balance between the deceleration of the bodyand the motion of the water mass around it.


Impact with Flat Bottomed Hull

If a flat bottom (ψ= 0) of the hull hits the surface, Eqs. 2.33 and 2.39 fail. The formulasyield infinite impact pressures, since the water has been assumed to be incompressible.Furthermore, neither the hull flexibility nor the damping effect from air cushions is takeninto account. By taking the compressibility of water into consideration, it is possible toobtain an approximate value for the maximum pressure occurring when a flat body strikesa horizontal water surface. The mass of fluid, GP, accelerated in the time, GW, is

GP 6FGW= ρ (2. 40)

where F is the speed of sound in the water (1440 P�V) and 6 the surface of fluid struck bythe body. If the dominating force acting on the fluid originates from the body, the equiva-lent force ) acting on the body is found from

)G,GW

9GPGW

PG9GW

9GPGW

P

Y

Y

Y= = + ≅ (2. 41)

Here�,P is the impulse from the mass of liquid surrounding the body and 9Y is the verticalvelocity, which is assumed to be constant at the impact phase where the slamming pres-sure happens. Eqs. 2.40-41 yield

) 6F8Y

= ρ (2. 42)

and the pressure averaged over the surface 6 becomes:

S)6

8 F8

F P VY

Y

= = ⋅ =ρ 2

2

21440, (2. 43)

Thus, the pressure turns out to be a factor 2⋅F�8Y� times the stagnation pressure, which isnot a reasonable result.

Design Method

A simple approach to providing an equivalent uniform static pressure for each structuralcomponent under localised water impact is proposed by Allen and Jones [1]. This methodis based on extensive full-scale trials conducted on a 65-ft and a 95-ft slender planing V-shaped hull and on large-scale structural models in the laboratory. The '19 rule con-cerning bottom hull slamming pressure for HSLC is partly based on the results from Allenand Jones [1] and given in the following for a mono-hull.


S$

7 DG FJ

F= ⋅ ⋅⋅

−−

13 1010

50

503

3

0 3

00 7.

.

.∆ ψψ g (2. 44)

in which $G is the load area of the element considered (for plates $G� � D×E), ψFJ� is thedead-rise angle at /&* (10 < ψFJ�< 30 [deg]), 7R is the draught at service speed and DFJ�isthe vertical design acceleration given as:

[ ]D9

/

I

/IFJ

ZO

J

ZO

J= ⋅ ∈0 76 58138 1 7. . , , (2. 45)

where IJ is an acceleration factor depending of the type of vessel and the service area, i.e.a safety factor depending on the probabilistic distribution of the sea-state in various areasfor a given type of vessel.

Accurate determination of the vertical design acceleration is difficult. In the design ofHSLC the acceleration levels for crew tolerance and structural design are most frequentlygiven as the average of the one-tenth highest acceleration, and the equivalent pressure isfound from this imposed or accepted acceleration level, without regard to any empiricallyor theoretically based design formulas. Table 2.1 from Koelbel [10] provides a generalguidance for selection of vertical accelerations for structural design.

[JR] Human affects Structural application

0.6 minor discomfort craft for passenger transport

1.0 maximum for military functionlong term (> 4hr)

1.5 maximum for military functionshort term (1-2 hr)

2.03.0

extreme discomfort patrol boats, crews, average owners, testcrews, anglers, long races

4.05.06.0

physical injurymedium length racesrace boat drivers, short racesmilitary crew under fire

Table 2.1: *HQHUDO�JXLGDQFH�IRU�VHOHFWLRQ�RI�YHUWLFDO�DFFHOHUDWLRQV�IRU�VWUXFWXUDO�GHVLJQ�$FFHOHUDWLRQ�OHYHOV�UHIHU�WR�WKH�DYHUDJH�RI� WKH��KLJKHVW�DW� WKH�FHQWUH�RI�JUDYLW\�RI� WKHFUDIW�

A serviceability design formula for a maximum allowable speed at a given significantwave height, +V, and the vertical design acceleration (Eq. 2.45) is given by '19 as


( )D+

%9

/

/ %FJ

V

ZO

FJ

ZO

ZO ZO=+

−

⋅

⋅

9 81

1650 0 08450

1852

10

22

3

.

.

.ψ

∆ (2. 46)

where %ZO is the waterline breadth at /ZO��.

Comparison of Formulas and Full Scale Tests

Results calculated by use of the above design formula (Eq. 2.44) and the theoretical de-rived expressions for the slamming pressures (Eqs. 2.33, 2.39, 2.43 and 2.44) are com-pared in Table 2.2 with experimental results from Riber [13], Fig. 2.14.

ψ = 17 0

pressure transducer\ = 0.2\

]

H = 0.21

,PSDFW��(Fig. 2.11)

Pressure [.3D]

Time [VHF]

Sample frequency IUHTXHQFH of ��+]

0

1

2

3

4

5

6

7

8

0,1 0,2 0,3 0,4 0,5

Full-scale slamming measurements on bottom panel of 470 sailing boat

+V = �� P8Y� ��P�VψFJ ��[GHJ]ψ� ��[GHJ]

Figure 2.14: 6ODPPLQJ�LPSDFW�RQ�D��KXOO�SDQHO��5LEHU�>��@�

The full-scale tests are carried out with a 470 sailing boat in protected water (IJ� �1.0). Apressure transducer is mounted in the bottom hull panel (Fig. 2.14) and the data are loggedwhile sailing. The constants in Eqs. 2.33, 2.40 and 2.47 are listed below.

0� �54�.J�P 8Y� �3.1�P�V H� �0.21�P ψ� �17 ψFJ� �10 ρ� �1015�NJ�P�

$G� 0.09�P� 7�� 0.10�P 9� �3.1�P�V IJ� �1.0 /ZO� �4.0 ∆ = 260�.J

The results with the calculations of the different slamming expressions and the full-scaletests are shown in Table 2.2.


0HWKRG 3UHVVXUH�>.3D@ &RPPHQWV(T�� 4.1 average(T�� 8.7 peak(T��GHVLJQ� 9.6 average7HVWV��)LJ�� 7.9 peak(T�� 4531 flat out 1

Table 2.2: 6ODPPLQJ�SUHVVXUH�RQ�ERWWRP�RI�KXOO�SDQHO�RI�D��VDLOLQJ�ERDW�

The highest pressure is obtained by the '19 design rule, which is used as a constant lat-eral load over the entire panel similar to the result obtained by Eq. 2.33, which is twotimes lower. The measured pressure (WHVW) and the pressure obtained by Eq. 2.40 both rep-resent peak values of the slamming. The above example indicates that the '19 rule pro-vides reasonable and safe design loads.

1 7KLV�SUHVVXUH�LV�KLJKHU�WKDQ�ZRXOG�DULVH�LQ�WKH�WHVWV�DV�WKH�HODVWLFLW\�RI�WKH�FRQVWUXFWLRQLV�JUHDWHU�WKDQ�WKDW�RI�WKH�ZDWHU�XQGHU�FRPSUHVVLRQ��7KH�DLU�SRFNHW�JLYHV�D�GDPSLQJ�HIIHFWZKLFK� UHGXFHV� WKH�SUHVVXUH��+RZHYHU�� LW� LV� HYLGHQW� WKDW�D� IODW� ERWWRPHG�ERDW�ZLOO� EH� OHVVIDYRXUDEOH�GXULQJ�LPSDFW�ZLWK�WKH�ZDWHU�

��6XPPDU\

An overview of FRP sandwich and stiffened single-skin hull manufacturing and structuraldesign is presented. In addition to this, the corresponding design rules provided by two ofthe leading classification societies (%9 and '19) are discussed. The '19 rules are moredetailed and less conservative (except for the minimum thickness) than the %9�rules. Fur-thermore, the '19 rules concerning stiffened single skin take into account the geometricalnon-linear behaviour for large deflections. However, the sandwich rules are still based onlinear theory for both the codes and need further investigation and development.

Global and local loads concerning FRP hull structural design are outlined with focus onslamming, as this is usually the dimensioning load for the design of hull panels in HSLC.Moreover, tentative rules for the design loads provided by '19 are presented. Resultsfrom full scale tests on a 470 sailing boat are compared to the '19 design formula and totheoretical derived expressions for the slamming pressures.


��%LEOLRJUDSK\

[1] Allen R.G. and Jones R.R. A Simplified Method for Determining Structural DesignLimit Pressures on High Performance marine Vehicles. $,$$�61$0(�$GYDQFHG�0D�ULQH�9HKLFOH�&RQIHUHQFH, 1978.

[2] ASTM. Annual book of ASTM standards. Technical report, American Society for

Testing and Materials, Philadelphia, Pennsylvania, USA, 1991. [3] Bureau Veritas. Rules for the Construction and Classification of High Speed Craft.

17 bis, Place des Reflets, La Defense 2, 92400 Courbevoie, France, 1995. [4] Chuang S. Experiments on Slamming of Wedge-shaped Bodies. -RXUQDO�RI�6KLSV�5H�

VHDUFK. Vol. 11 (4), pp. 190-198, 1967. [5] DNV. Classification Rules for High Speed Light Craft. Det Norske Veritas Research

AS, Veritasveien 1, N-1322 Høvik, Norway, 1991. [6] DNV. Response of Fast Craft Hull Structures to Slamming Loads. 3URFHHGLQJV�RI�WKH

6HFRQG� ,QWHUQDWLRQDO�&RQIHUHQFH� RQ�)DVW� 6HD� 7UDQVSRUWDWLRQ. Vol. 1, pp. 481-398,1991.

[7] Hansen A.M. Sammenligning af Slammingteorier. Department of Naval Architecture

and Offshore Engineering, DTU, Lyngby, Denmark, 1991, (in danish). [8] Hansen P.F., Juncher Jensen J. and Terndrup Pedersen P. Long Term Springing and

Whipping Stresses in High Speed Vessels. 3URFHHGLQJV� RI� WKH� 7KLUG� ,QWHUQDWLRQDO&RQIHUHQFH�RQ�)DVW�6HD�7UDQVSRUWDWLRQ. Vol. 1 (2,1C), pp. 473-485, 1995.

[9] Karman T. The Impact of Seaplane Floats during Landing. NACA TN 321, 1929. [10] Koelbel J.G. Comments on the Structural Design of High Speed Craft. 0DULQH�7HFK�

QRORJ\� Vol. 32 (2), pp. 77-100, April, 1995. [11] Ochi K.M. and Bledsoe M.D. Hydrodynamic Impact with Application to Ship Slam-

ming. )RXUWK�6\PSRVLXP�RI�1DYDO�+\GURG\QDPLFV. Washington DC, August. 1962. [12] Payne P.R. The Vertical Impact of a Wedge on a Fluid. 2FHDQ�(QJLQHHULQJ. Vol. 8

(4), pp. 421-436, 1981. [13] Riber H.J. Strength of a 470 Sailing Boat. MSc. thesis at the Department of Naval

Architecture and Offshore Engineering, Technical University of Denmark, 1993.


[14] Szebehely V.G. Hydrodynamics of Slamming of Ships. Navy Department Washing-ton DC, report 823, 1952.

[15] Terndrup Pedersen P. and Juncher Jensen J. Styrkeberegning af maritime konstruk-

tioner. Department of Naval Architecture and Offshore Engineering, Technical Uni-versity of Denmark, 1982, (in Danish).

[16] Thomsen O.T. Theoretical and Experimental Investigation of Local Bending Effects

in Sandwich Plates. &RPSRVLWH�6WUXFWXUHV. Vol. 30, pp. 85-101,1995. [17] Wagner V.H. Über Stoss und Gleitvorgänge an der Oberfläche von Flüssigkeiten.

ZAMM. Vol. 12, pp. 193-215,1939, (in German).


33

&KDSWHU��

1RQ�OLQHDU�6DQGZLFK�3ODWH�7KHRU\

��,QWURGXFWLRQ

This chapter focuses on analytical solution methods for the response of orthotropic sand-wich composite plates with large deflections due to high lateral loads, with special appli-cation to the design of composite panels in ship structures. A geometrical non-linear the-ory is outlined, on the basis of the classical sandwich plate theory expanded by the higher-order terms in the strain displacement relations, including shear deformation. By use ofthe principle of minimum potential energy, two different methods are derived for the sim-ply supported and the clamped cases. The solutions are presented as simple design formu-las. The results of the analytical calculations are discussed and compared to numericalnon-linear finite difference calculations and large-deflection experiments of equivalentplates. The presented methods (also described in Riber [13]) lead to good results for plateresponse and provide an alternative method for the design of sandwich plates subjected tohigh lateral loading.

Pronounced lateral deflections introduce in-plane displacements and membrane strains inthe faces, as well as shear deformation in the core. Thus, the classical Kirchhoff plate the-ory is not sufficient to describe this kind of response. Reissner [12] and Mindlin [8] intro-duced a theory governing finite deflections of sandwich plates with isotropic faces andcores. Based on Reissner’s theory, Alwan [2] solved the non-linear bending problem ofrectangular sandwich plates by means of double trigonometric series with simply sup-ported edges. Kan and Huang [7] derived a large-deflection solution of clamped sandwichplates by applying a perturbation technique. However, none of the above solutions areeasy to use in practice.

The non-linear theory for orthotropic single-skin and sandwich plates is outlined in Sec-tion 3.2, which concludes with the governing equations of the problem. In Section 3.3 ana-lytical solutions for the sandwich problem are derived and a new simple analytical design

34 &KDSWHU��1RQ�OLQHDU�6DQGZLFK�3ODWH�7KHRU\

rule is presented for predicting deflections, strains and stresses in sandwich panels withlarge deflections. The results are discussed and compared to experimental data obtained byBau, Kildegård and Svendsen [3] and equivalent numerical finite difference calculationsperformed by Riber [12] in Section 3.4, followed by a summary.

��7KHRU\

The present formulation is in accordance with the work of Whitney [15] and Zenkert [16],where the latter presents a simplification of the theory given in Allen [1] and Plantema[9]. The theory is based on the classical sandwich plate theory supplemented with thehigher-order terms in the strain displacement relations, which are usually neglected inplate analysis. The formulation is outlined for sandwich plates, but is also applicable tosingle-skin plates, where the two faces of the sandwich plate form the single-skin plateomitting the core. Hence, the term ‘plate’ refers to either the single-skin plate or thesandwich faces.

��$VVXPSWLRQV�DQG�&RQILJXUDWLRQV

A standard [�, \�, ]�co-ordinate system as shown in Fig. 3.1, is used to derive the equa-tions. The displacements in the [�, \�, and ]�directions are denoted X, Y, and Z, respec-tively. The origin of the co-ordinate system lies in the middle plane (for sandwich in thegeometrical symmetry plane of the core) with the positive ]-axis directed perpendicularlyto it and downwards. Consider a sandwich plate with its faces made of thin orthotropiclayers orientated with their material axes parallel to the plate sides and with the thicknessWI��and WI� and the core thickness WF. The following basic assumptions are made:

1. The plate is constructed of an arbitrary number of layers of orthotropic sheets of con-stant thickness bonded together.

2. The thickness of the core is constant. 3. The material is linearly elastic.

4. The out-of-plane transverse normal strain ε] is neglected. 5. Non-linear terms, i.e. the derivatives of the lateral deflection Z in the strain displace-

ment relations, are retained whereas the equivalent non-linear terms of the in-planedisplacement terms are omitted.

��7KHRU\��35

6. The deflection Z can be divided into two parts: Z�= ZEHQGLQJ + ZVKHDU (partial deflec-tion).

7. The position of the neutral axes for the [� and \�directions is the same, i.e. ][ = ]\ in

Eq. 3.1. 8. The Young’s modulus of the core is small compared with that of the face(s), i.e. (F <<

(I and the faces are thin compared to the core, i.e. WI << WF. This simplifies the stressdistributions in a structural sandwich to:�7KH�IDFHV�FDUU\�EHQGLQJ�PRPHQWV�DV� WHQVLOHDQG� FRPSUHVVLYH� VWUHVVHV�σ[[� DQG�σ\\�� DQG� WKH� FRUH� FDUULHV� WKH� WUDQVYHUVH� IRUFHV� DVVKHDU�VWUHVVHV�τ\]�DQG�τ[]�

9. The shear stresses are constant through the thickness of the core. 10. The core is considered isotropic.

lamina 2

lamina 1

lamina 1x

y

z

]�]�

]1

WI��

WI��

G =��WI�� WI��WF

[

\

]

WI�

WI�

WF��

WF�� E

D H

Figure 3.1: 6DQGZLFK�SODWH�DQG�IDFH�VLQJOH�VNLQ�FRQILJXUDWLRQ�

The distance between the geometrical symmetry plane and the neutral plane is denoted Hand the distances from the geometrical symmetry plane to the upper and lower face of plynumber N are denoted ]N�� and�]N (see Fig. 3.1). The number of plies in each face is denoted1L, where L � for sandwich plates and L � for single-skin plates.


��6WUDLQ�'LVSODFHPHQW�5HODWLRQV

The displacement field is assumed to be of the form

X [ \ ] X [ \ ] [ \

Y [ \ ] Y [ \ ] [ \

Z [ \ ] Z [ \

[ [

\ \

( , , ) ( , ) ( , )

( , , ) ( , ) ( , )

( , , ) ( , )

= += +=

ψψ (3. 1)

where X��Y��Z are the displacements in the [��\� and ]�directions, respectively, and ψ[��ψ\ are thecross-sectional rotations in the []� and \]-planes due to bending. Assuming that we may separatethe lateral displacement into contributions due to bending and shear and then superimpose themto give the total deflection, we have

Z Z ZE V

= + (3. 2)

The reason for introducing partial deflections is to uncouple the equilibrium equation de-rived later. This indeed speeds up the numerical finite difference solution, which is thebackbone of the non-linear design program 3DQHO (Chapter 7). The cross-sectional rota-tions may now be written as

ψ∂∂ ψ

∂∂

∂∂

∂∂ ψ

∂∂

∂∂ ψ[

E

\

E V

\

V

[

Z[

Z\

Z\

Z\

Z[

Z[

= − = − = + = +� � ��DQG�� , , (3. 3)

This means that the bending moments and the shearing forces will be independent of eachother, which is correct for panels with equal rigidities in both [- and \-directions or thesame neutral axis for both cross-sections. However, this also applies to orthotropic panelsand to most sandwich panels in general. Hence, bending causes the cross-section to rotate,whereas shearing is a sliding movement and does not add to any rotation. Using this sim-plification, we reduce the number of independent field variables from five to four:

X Y Z Y Z Z[ \ E V, , , , , , ,ψ ψ �� X �→ (3. 4)

The non-linear strain terms, which couple the in-plane and out-of-plane displacements, are usu-ally neglected in classical plate theory. However, for large deflection they cannot be omitted asthe coupling effect becomes significant. Eqs. 3.5a-b express the strains in terms of the displace-ment derivatives and the above partial deflections for Z as follows:

��7KHRU\��37

ε∂∂

∂∂

∂∂

∂∂

∂∂

ε∂∂

∂∂

∂∂

∂∂

∂∂

ε

[

E

\

E

]

X[

Z[

]Z

[X[

Y[

Y\

Z\

]Z

\X\

Y\

= +

−

≈

≈

= +

−

≈

≈

≈

12

2 2

2

2 2

12

2 2

2

2 2

0 0

0 0

0

, ,

, ,

(3.5.a)

and

2 2

2

2

2

ε∂∂

∂∂

∂∂

∂∂

∂∂ ∂

γ

ε∂∂

γ

ε∂∂

γ

[\

E

[\

[]

V

[]

\]

V

\]

X\

Y[

Z[Z\

]Z

[ \

Z

[Z\

= + + − =

= =

= =

(3. 5b)

If +RRNH¶V ODZ for an orthotropic material is applied and it is assumed that the stress com-ponent in the ]-direction vanishes everywhere, the constitutive relations for the Nth layerare given as

σστ

εεε

ττ

εε

[

\

[\

N N

[

\

[\

N

\]

[]

N N

\]

[]

N

4 4

4 4

4

4

4

=

=

11 12

12 22

66

44

55

0

0

0 0 2

0

0

2

2

(3. 6)

In the above expressions, the coefficients 4 NLM in the stiffness matrix are defined in Vinson

[14] for linear elastic materials. If the principal main material axes �� do not coincidewith the global plate axes [, \, the local stiffness matrix, defined in the material co-ordi-nate, is transformed into the global plate co-ordinate system by means of the transforma-tion matrix 7:

[ ] [ ] [ ] [ ] [ ]4 7 4 7 7N N N N N

N

= = −− −

−1

2 2

2 2

2 2

2

2,

cos sin cos sin

sin cos cos sin

cos sin cos sin cos sin

θ θ θ θθ θ θ θ

θ θ θ θ θ θ (3. 7)

where θ is the angle between the main fibre direction and the plate axis of ply number N.Combining Eqs. 3.5-6 and integrating over the thickness of the plate, we obtain the in-plane forces, the moments and the shear forces (see Fig. 3.2):


1

1

1

$ $

$ $

$

X[

Z[

Y\

Z\

X\

Y[

Z\Z[

% %

% %

%

[

\

[\

=

+

+

+ +

+

−11 12

12 22

66

12

2

12

2

11 12

12 22

66

2

0

0

0 0

0

0

0 0

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂ Z

[Z

\Z

[ \

E

E

E

∂∂∂∂∂ ∂

2

2

2

2

2

−

−

(3. 8)

4

4

N $

N $

Z

\Z

[

\

[

V

V

=

44 44

55 55

0

0

∂∂

∂∂

(3. 9)

0

0

0

' '

' '

'

Z

[Z

\Z

[ \

% %

% %

%

X[

Z[

Y\

Z\

X\

Y

[

\

[\

E

E

E

=

−

−

−

+

+

+

+

11 12

12 22

66

2

2

2

2

2

11 12

12 22

66

2

20

0

0 02

0

0

0 0

1

2

1

2

∂∂

∂∂∂∂ ∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂[

Z\Z[

+

(3. 10)

The matrices $LM,�'LM�(L��M�=1,2,6) and $LM (L��M = 4,5) represent the extensional, bending- andshear-stiffness, respectively. The coupling matrix %LM between in-plane forces and bendingdeformations vanishes in the case of plate symmetry. The relation between the transverseforces and the shear deflection (Eq. 3.9) becomes

4 N $Z[

4 N $Z\[

V

\

V= =5 55 4 44

∂∂

∂∂ and (3. 11)

where the NL�factors are dependent on the core material. For homogeneous isotropic plates,it can be shown that the value of N is 5/6 according to Reissner [11]. The stiffnesses, $LM,%LM and 'LM, are given below for a single-skin plate and for the faces and core (indices FRUH)of a sandwich plate, as follows:

( )$ 4 ] ] $ 4 L MLM LM

N

N NN

1

LM

FRUH

LM

FRUH= − = =−=

∑ ( ) , , , , ,11

1 2 4 5 6 t c (3. 12)

and

��7KHRU\��39

( )

( )

% 4 ] ] % L M

' 4 ] ] L M

LM LM

N

N N

N

1

LM

FRUH

LM LM

N

N

1

N N

= − = =

= − =

−=

=−

∑

∑

1

20 1 2 6

1

31 2 6

21

2

1

1

31

3

( )

( )

, , ,

, , ,

(3. 13)

The above expressions can be applied directly to a single-skin plate. As for the sandwichplate, assuming that the faces are thin and the shear is carried by the core (DVVXPSWLRQ��),we get the following expressions for the stiffness matrices:

$ $ $ �L�M � �

$ $ ��L�M �

LM

VDQ

LM

IDFH

LM

IDFH

LM

VDQ

LM

FRUH

= + =

= =

1 2 1 2 6

4 5

(3. 14)

and

( )%W

$ $ ��L�M � �

'W W

$W W

$ L�M � �

LM

VDQ F

LM

IDFH

LM

IDFH

LM

VDQ F I

LM

IDFH F I

LM

IDFH

= − =

=+

+

+

=

212 6

2 21 2 6

1 2

12

1 22

2

(3. 15)

where WI��refers to face 1 and WI�� refers to face 2. Here the coupling terms %LM do not arisedue to asymmetry in the faces since they are considered thin, but as a result of the differ-ent in-plane stiffness $LM of the two faces.

��(TXLOLEULXP�(TXDWLRQV

Referring to the sign convention in Fig. 3.2 below, we get the following equilibrium equa-tions including the body forces

]

[\

G\

G[

T�[�\�W�

4\

4[1\��0[

1[��0\

0\[��1\[

0[\��1[\

σ[

σ]

σ\

τ\]

τ\[

τ[]

τ[\

τ][τ]\

G[1[ 1[ +∂∂1[G[[

∂∂

∂∂

Z[

Z[G[+

2

2

Figure 3.2: 6LJQ�FRQYHQWLRQ�RI�WKH�SODWH.


∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

ρ∂∂

∂∂

∂∂

∂∂

∂∂

1[

1

\

1

\

1

[

4

[

4

\T

ZW

0[

0

\4

0

\

0

[4

[ \[

\ \[

[ \

[ [\

[

\ \[

\

+ =

+ =

+ = − +

+ + =

+ + =

∗ ∗

0

0

0

2

2

0

(3. 16)

where

, T T 1Z[

1Z\

1Z[ \

G] P[ \ [\

∗ ∗= + + + = + ′∫∂∂

∂∂

∂∂ ∂

ρ ρ2

2

2

2

2

2

Here, ′P is the added mass from the flow of the surrounding liquid. The above five equi-librium equations can be reduced to four by differentiating the last two equations and in-serting them in the third equation. In order to express the equilibrium equations in termsof the displacements, we combine Eq. 3.16 with Eqs. 3.8-10 and obtain four coupled non-linear differential equations in X, Y, ZV�and ZE, where Z� �ZE��ZV. The two in-plane equi-librium equations (Eqs 3.17-18):

( ) ( )$X[

$X\

$ $Y

[ \; %

Z[

% %Z

[ \11

2

2 66

2

2 12 66

2

11

3

3 12 66

3

22∂∂

∂∂

∂∂ ∂

∂∂

∂∂ ∂

+ + + = + + +∗ (3. 17)

where

( );Z[

$Z[

$Z\

$ $Z\

Z[ \

∗ = − +

− +

∂∂

∂∂

∂∂

∂∂

∂∂ ∂11

2

2 66

2

2 12 66

2

and

( ) ( )$Y[

$Y\

$ $X

[ \< %

Z\

% %Z

\ [66

2

2 22

2

2 12 66

2

22

3

3 12 66

3

22∂∂

∂∂

∂∂ ∂

∂∂

∂∂ ∂

+ + + = + + +∗ (3. 18)

where

( )<Z\

$Z\

$Z[

$ $Z[

Z[ \

∗ = − +

− +

∂∂

∂∂

∂∂

∂∂

∂∂ ∂22

2

2 66

2

2 12 66

2

��7KHRU\��41

The shear equation becomes

$Z

[$

Z

\Z

TV V

55

2

2 44

2

2

2

2

∂∂

∂∂

ρ∂∂

+ = −∗ ∗

�W (3. 19)

and finally the bending equation yields

( )

( )

'Z

['

Z

\' '

Z

[ \

ZW

T %X[

% %X

[ \Y

\ [%

Y\

E E E

11

4

4 22

4

4 12 66

4

2 2

2

2 11

3

3 12 66

3

2

3

2 22

3

3

2 2

2

∂∂

∂∂

∂∂ ∂

ρ∂∂

∂∂

∂∂ ∂

∂∂ ∂

∂∂

+ + + =

− + + + + ⋅ +

+∗ ∗

(3. 20)

The above set of equations must be combined with the appropriate boundary conditions ofthe specific problem. If we regard the right hand side of the equations as body forces andas lateral loads of magnitude� ; �� < �and T (�%LM�X�Y�),� the equations are identical to thegoverning equation for small deformations of an elastic plate. The numerical solution ofthese equations will be described in Chapter 4, whereas the analytical solutions based onenergy principles will be outlined in the following.

��$QDO\WLFDO�6ROXWLRQV

In this section two different analytical solutions of the non-linear differential equilibriumequations in Section 3.2 are presented. The methods provide closed-form approximate so-lutions for large deflections of orthotropic sandwich or single-skin plates. They are basedon the theorem for the minimum potential energy, which states:�7KH�WRWDO�SRWHQWLDO�HQHUJ\RI�D�V\VWHP�KDV�WKH�ORZHVW�VWDWLRQDU\�YDOXH�IRU�DOO�VPDOO�GLVSODFHPHQWV�ZKHQ�WKH�V\VWHP�LVLQ�HTXLOLEULXP. The energy introduced from a virtual displacement δZ due to an externalload�T corresponds to the equivalent strain energy in the plate. The total energy 8(X,Y,Z),which has a stationary value, is then minimised and the assumed deflection functions X, Yand Z are found by use of known boundary conditions together with the derivatives of thetotal energy 8 of the system, with respect to the unknown deflections X, Y and Z.

The author has not, so far, found simple non-linear analytical solutions for sandwichplates in the literature. Hence, the derivation of the equations to the final closed-form so-lutions will be described step by step for the reader in the following sections. Two differ-ent analytical solutions are described. A complete solution, 6ROXWLRQ� �, and a combinedsolution, 6ROXWLRQ��.

Many design rules concerning single-skin composite plates are already based on non-lineartheory. However, this is not the case in analytical design of sandwich structures, where the


existing design rules recommended by the classification societies are based on linear platetheory. The method presented in this paper provides an alternative and more accurate so-lution procedure for sandwich plates in the design phase. Moreover, the method takes intoaccount non-linear effects, without the need for costly and complex finite-element-basedcomputer models. These may, of course, be used in later structural verification and optimi-sation of the design or for problems with special boundary conditions.

��$�&RPSOHWH�$QDO\WLFDO�6ROXWLRQ��6ROXWLRQ��

The total energy 8 of the plate can be expressed as the sum of internal strain energy, 8�,and the potential energy, 8�, due to external loads T. Minimisation of the total energy 8� +8�, with respect to the parameters in the deflection functions, gives the following equa-tions:

( )∂∂

8 8

DL

1 20

+= (3. 21)

where DL present undetermined parameters in the deflection functions, which depend onthe given plate boundary conditions. The strain energy of an elastic plate in terms of an [��\��]- co-ordinate system is given by the relationship

( )8 G[G\G][ [ \ \ ] ] [] [] \] \] [\ [\9

� = + + + + +∫∫∫1

2σ ε σ ε σ ε τ γ τ γ σ γ (3. 22)

where the triple integration is performed over the volume of the plate. Taking into accountthe assumption of no strain in the ]�direction (DVVXPSWLRQ� �) together with the plystress/strain relations stated in Eq. 3.6, we obtain:

( )8 4 4 4 4 4 4 G[G\G][ \ [ \ [\ \] []9

1 = + + + + +∫∫∫1

2211

222

212 66

244

255

2ε ε ε ε γ γ γ (3. 23)

This relationship can be expressed in terms of the plate displacements X, Y, ZE and ZV�bysubstituting the strain-displacements relations of Eqs. 3.5 into the above equation. Integra-tion over the plate thickness yields the total strain energy of the plate (Appendix A, Eq.A.1).

In order to simplify the analytical expression, the in-plane bending terms %LM in the energyexpressions are omitted in the following. For general practical design purposes, it is rea-sonable to neglect these terms in the first place as most ship hull panels are close to beingsymmetric. The assumed deflection functions for X, Y, ZV� and ZE depend on the type of

��$QDO\WLFDO�6ROXWLRQV��43

boundary conditions. The complete solution for the plate response due to a lateral load Twill be derived for the simply supported and the clamped cases.

Simply Supported Plate

The simply supported edge is described by zero deflection Z and bending moments, 0[

and 0\. A third condition illustrated in Fig. 3.3 can either be zero “effective twisting mo-ment”:

40

\[

[\− =∂

∂ 0 and 40

[\

[\− =∂

∂0 (3. 24)

along the edges parallel to the [- and \�axes, allowing for shear, γ[] ≠ 0, γ\] ≠ 0, i.e. VRIWboundary conditions, or zero shear deformation γ[] , γ\], allowing for the existence of“effective twisting moments”, i.e. KDUG�boundary conditions�

γ[] 0[\

γ[] = 00[\ = 0

VRIW KDUG[

] \� ��E

Figure 3.3: 7UDQVYHUVH�VKHDU�ERXQGDU\�FRQGLWLRQ�IRU�VDQGZLFK�SODWHV�

For practical purposes, the hard boundary condition is more realistic since, in most cases,there will be an edge stiffener or some symmetry constraint preventing such shearing. Theplate edges are not allowed to move in the in-plane directions [� and \ which may, ofcourse, not be true in all practical cases. Thus, we need deflection functions which satisfythe following boundary conditions:

[� ��D Z� �Y� �0[� �γ\]� ��\� ��E Z� �X� �0\� �γ[]� �� (3.25a)

The following deflection functions satisfy these boundary conditions:


( )Z Z Z [ \

X X [ \D E

Y Y \ [

E V= +

= = =

=

sin sin

sin sin ,

sin sin

α β

α β απ

βπ

β α

2

2

, (3. 25b)

[

\

E

D

Figure 3.4: 6LJQ�FRQYHQWLRQ�IRU�WKH�SODWH�

Inserting the deflection functions in Eq. 3.25 into the energy expression Eq. A.1 and inte-grating over the plate, we obtain the total strain energy of the plate 8� and the potentialenergy 8� from the work of the external lateral load T expressed in the following andshown in detail in Appendix A.

( )8 X 8 X Y Z ZVV

L

L

VV

E V11

33

1= ==∑ , , , (3. 26)

( ) ( )8 TZG$ T Z Z [ \G$TDE

Z ZVV

$E V

$E V2 2

4= − = − + = +∫∫ ∫∫ sin sinα β

π (3. 27)

The in-plane displacements, X and Y, do not contribute to the potential energy of the exter-nal load as we only consider lateral load and no in-plane loads. Hence, minimisation of the

total energy, 8, with respect to Z Z X YE V, , , gives us adequate equations to determine these

coefficients. The final expressions yield

Z Z X Z Y Z

Z D Z D

V E E E

E E

= = =

+ + =

β β β7 8

2

9

2

3

2 3 0

, , (3. 28)

where the constants β7, β8, β9, D� and D� (Appendix A) are functions of the plate proper-ties, including length and breadth, D, E, and the stiffness matrices, $LM , %LM ,�'LM.

Clamped Plate

The procedure for the simply supported case is applied to the clamped case except for dif-ferent boundary conditions, which can be expressed as


[� ��D Z� �Y� ∂∂Z[� �γ\]� ��

\� ��E Z� �X� �∂∂Z\ �γ[]� �� (3.29a)

where the deflection functions satisfying these boundary conditions are

( )Z Z Z [ \

X X [ \D E

Y Y \ [

E V= +

= = =

=

sin sin

sin sin ,

sin sin

2 2

2

2

α β

α β απ

βπ

β α

, (3. 29b)

Using the same procedure as in the case of the simply supported plate, we obtain the totalstrain energy:

( )8 X 8 X Y Z ZFO FO

E V11

33

1= ==∑ , , , (3. 30)

The energy terms are outlined in Appendix A. The potential energy, 8�,�from the work ofthe external lateral load, T, is slightly smaller and becomes:

( ) ( )8 TZG$ T Z Z [ \G$TDE

Z ZFO

$E V

$V E2

2 2

4= − = − + = +∫∫ ∫∫ sin sinα β (3. 31)

Finally, we obtain the same relations as for the simply supported case expressed in Eq.3.26, with the constants β7, β8, β9, D� and D� given in Appendix A.

The strains and stresses can be derived from the displacement functions of Z, X and Y,which will be demonstrated along with the derivation of 6ROXWLRQ��.

��$�&RPELQHG�$QDO\WLFDO�6ROXWLRQ��6ROXWLRQ��

A complete non-linear analytical solution is demonstrated for the large deflection of sin-gle-skin and sandwich plates. Even though the final expression for the deflection functionsis simple, the coefficients in these expressions are quite complicated and not very practi-cal for simple analytical calculations. In order to simplify further the final expressions forthe non-linear plate response, an alternative method,�6ROXWLRQ��,�is presented here.

The idea is to use the linear solution for sandwich plates and combine it with the mem-brane solution, to give a good approximate result. By use of this method, the additional


membrane solution can be integrated into linear design rules given in standard textbooks,such as Hughes [6] and Zenkert [16], and in a simple way provide a non-linear plate solu-tion. The energy method provides a good means of obtaining an approximate solution forboth the membrane displacements and the bending/shear deflection of a plate. Large-de-flection solutions of the plate response are obtained by combining the two separate solu-tions.

To obtain an approximate large-deflection solution for a rectangular sandwich plate(simply supported or clamped with in-plane displacements fixed at the edges), a simplemethod consisting of a combination of the known theory of small deflections and themembrane theory solutions may be used. We assume that the load T can be resolved intotwo parts, T� and T�, so that T� is balanced by the bending and shearing stresses calculatedfrom the small-deflection theory and T� is balanced by the membrane stresses. Thus, weobtain:

T� �T��T�� 5��Z��5��Z3 (3. 32)

This third-order polynomial is solved for Z:

Z ) ( ) ) ( )

(5

5)

T5

= + + + − +

= =

3 23 3 23

1

2 23 2 ,

(3. 33)

Hence, T��and�T� are found from Eq. 3.32, where the corresponding stresses are calculatedby using T� for the small-bending/shear deflection and T� for the membrane deflection.The total strains and stresses are achieved by superposition of strains and stresses due tothe loads T��and� T�. The parameters 5��and 5� are found from the small-deflection platebending/shear theory and membrane theory, respectively. They are in the following ex-pressed as functions of the plate aspect ratio and the material properties.

The membrane solution is obtained by use of the strain energy expression and the princi-ple of virtual displacements with suitable expressions for the displacements X, Y and Z byapplication of the same procedure as for the previously demonstrated 6ROXWLRQ� �. Thestrain energy 8P of a membrane, which is due solely to stretching of its middle surface, isgiven by Eq. A.1 omitting the terms involving 'LM�and %LM.

( )( )

8 G[G\G]

$ $ $ $ G[G\

P [ [ \ \ [\ [\9

[ \ [ \ [\$

= + +

= + + +

∫∫∫∫∫

1

2

2112

222

12 662

σ ε σ ε σ γ

ε ε ε ε γ (3. 34)

The membrane parts of the strains, ε ε γ[

P

\

P, and xym , (Eq. 3.5 ) can be expressed as


ε∂∂

∂∂

ε∂∂

∂∂[

P

\

PX[

Z[

Y\

Z\

= +

= +

1

2

1

2

2 2

, (3. 35)

and

xymγ

∂∂

∂∂

∂∂

∂∂= + +

Y[

X\

Z[Z\

Substituting these strain expressions into Eq. 3.34, we obtain an energy expression 8P forthe membrane part, using the same procedure as in the previous section.

8 $X[

$Z[

$X[

Z[

$Y\

$Z\

$Y\

Z\

$X\

$Y[

$Z[Z\

$P

=

+

+

+

+

+

+

+

+

+

∫∫1

2

1

4

1

4

2

11

2

11

4

11

2

22

2

22

4

22

2

66

2

66

2

66

2

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

$X\Y[

$Z[Z\

X\

Y[

$X[Y\

$Z[

Z\

$X[

Z\

$Y\

Z[

66 66 12

12

2 2

12

2

12

2

2 2

1

2

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

+ +

+

+

+

+

��G[G\

(3. 36)

When the energy method is applied we must assume suitable expressions for the displace-ments X, Y and Z in order to satisfy the boundary conditions. A rectangular plate with itsedges fixed in the [�, \� and ]�directions behaves like a simply supported plate in all casesas the membrane has no bending stiffness. Thus, we obtain the same functions as the onesin Eq. 3.25. Inserting these functions into Eq. 3.36 and integrating over the plate area, weobtain

( )8 X 8 X Y ZP L

L

P= =

=∑

1

16

, , (3. 37)

where each of the 16 integrals in Eq. 3.37 is similar to the equivalent integrals for the sim-ply supported case in 6ROXWLRQ��, given in Appendix A. Application of the principle of vir-tual displacements leads to the following three equations:

∂∂

∂∂

∂∂

α β

8

X

8

Y

8

ZTZ [ \G[G\

P P

P

DE

= =

= ∫∫

0 0

00

�� DQG��

sin sin (3. 38)


After some reduction, the final expression for the parameters X, �Y� and Z in Eq. 3.37 be-comes

X Z Y Z ZT5

= = =α α11

2

12

2 3 2

2

, , (3. 39)

and 5� becomes

( )5DE2

2

1 11 2 12 34= + +

πβ α β α β (3. 40)

The αLM� and βL values are shown in Appendix A for the simply supported case, i.e. themembrane case.

With an isotropic core the solution can be expressed as a function of the magnitude of orthotropyof the faces and the plate aspect ratio. In order to solve the third-order polynomial (Eq. 3.32), it isnecessary to express the parameters 5��and 5� in a simple way. Thus, they are presented in dia-grams, which make a calculation by hand of the plate responses possible. The following assump-tions are made for the stiffness moduli in the faces.

$ . $ $ $ $ $

' . ' ' ' ' '

$ $ 6

11 22 12 22 66 22

11 22 12 22 66 22

44 55

1

21

2

= ⋅ = =−

= ⋅ = =−

= =

, ,

, ,

νν

νν

(3. 41)

Here, the factor . expresses the magnitude of orthotropy, i.e. stiffness ratio in the [� and\� directions. For sandwich plates, Eq. 3.41 has to be combined with Eqs. 3.14-15.

The variation of the membrane parameter 5� (Eq. 3.40) with the aspect ratio D�E and thestiffness ratio . is shown in Fig. 3.5. In practical terms, the parameter is independent ofPoisson’s ratio and we obtain a simple expression for 5�. An expression of the curve fitfor . = 1 is given in Eq. 3.42 below.

5$

E D ED E2

114

1165

0 50387=

++ ⋅ −

.

.. (3. 42)


0

10

20

30

40

50

60

70

0 1 2 3 4 5

.� = 1

.��= 2

.��= 3

.��= 5

.��= 10

5�Parameter 5� for the membrane part in 6ROXWLRQ��

. $ $

$ $

$ $

==

=−

11 22

12 22

66 22

1

2

νν

5$

E DE

DE2

114

1165

0 50387=

−+ −

.

..

Aspect ratio D�E

Figure 3.5: &RHIILFLHQW�5��PHPEUDQH�SDUW�RI�(T��)RU�GLIIHUHQW�DVSHFW�UDWLRV�DQG�PDJQL�WXGHV�RI�RUWKRWURS\�RI�WKH�IDFHV�

We have now obtained the non-linear part of the right side in Eq. 3.32, T� = 5� Z�, and

need the linear solution for the bending/shear deflection of a plate to obtain the linear termT� = 5�Z. This solution is described in various textbooks, e.g. Zenkert [16]. The procedureis in accordance with the one described in the above solution. Only the bending/shearingpart is retained in the general energy expression (Eq. A.1), and the deflection functions areassumed for both the simply supported and the clamped cases, as discussed in Section3.3.1. A simple expression for the coefficient 5� is presented as:

5DE

5DE

VV VV

FO FO

1 12

2

1 12

44

=

=

βπ

β

, simply supported

, clamped

(3. 43)

where the parameter β12 is expressed in Eq. A.2 and depends on the boundary conditions.

The coefficients, ZE and Z V , for the deflection functions become:

Z Z

Z Z

E

VV

V

VV

E

FO

V

FO

= −

= −

= −

= −

ββ

ββ

ββ

ββ

5

4

5

6

5

4

5

6

, ,

, ,

�� VLPSO\�VXSSRUWHG

�� FODPSHG

(3. 44)


By use of the assumption Eq. 3.41 5� can be expressed as:

5' 6

6 E ' EZ

T

5Z Z

T E

'

T E

6E V

E V

E V

111

411

21

1

14

11

12

=⋅

+= = + = +

α αα α

, (3. 45)

The parameter αV is dependent of the plate aspect ratio only (Fig. 3.6a), whereas αE is de-pendent of both stiffness ratio . and the aspect ratio D�E (Fig. 3.6b). Both of the parame-ters are invariant to Poisson’s ratio due to the assumption Eq. 3.43. The derivation of theparameters αV�and αE is shown in Eq. A.7.

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0 1 2 3 4 5

Simple supported

Clamped

Aspect ratio D�E

Parameter αV for simply and clamped b.c.αV

Figure 3.6a: 3DUDPHWHU�αV��IRU�DQ�LVRWURSLF�FRUH��VLPSO\�VXSSRUWHG�DQG�FODPSHG�E�F�

Curve fittings of Figs. 3.6a-b are shown in Eq. 3.46 as formulae for both the clamped andthe simply supported cases for .� ��.

αE

FO

D E=

−⋅ +

+

1

355 270 004.

αE

VV

D E=

−⋅ +

+

1

33 270 021. (3.46)

αV

FO

D E=

−⋅ +

+

1

10 3 2 00151

. ..

αV

VV

D E=

−⋅ +

+

1

85 160183

. ..


0

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0.050

0 1 2 3 4 5

.��= 1, ss

. = 2, ss

.��= 2, cl

.��= 3, ss

.��= 3, cl

Aspect ratio D�E

.� = 1, cl

.� �'��/�'��

'�� = '��ν)/�'�� = ν'��

Parameter αE for simple supported and clamped b.c.αE

Figure 3.6b: 3DUDPHWHU�αE� IRU� GLIIHUHQW� PDJQLWXGHV� RI� RUWKRWURS\� LQ� WKH� IDFHV�� VLPSO\VXSSRUWHG�DQG�FODPSHG�E�F�

The system, including Figs. 3.5-6 and Eq. 3.33 together with the given plate data, is suffi-cient to determine the non-linear plate response in a simple way by solving the third-orderpolynomial for Z. Thus, the strains and stresses can be found from Z, by means of thestrain-displacement relations (Eq. 3.5). The method provides fairly good results except forthe strains near the edges of clamped plates, because they are derivatives of the deflectionfunctions, which are themselves approximate. In order to complete the method and, in thisway, provide an alternative to the existing simple linear solution method suggested instandard textbooks (e.g. [6] and [16]), formulas are derived for the strains. The membranestrains, which are independent of the boundary conditions, can be found from Eqs. 3.25,3.35 and 3.39. Combining these three equations, we obtain

)

ε αα

α βα

α β

ε βα

β αβ

β α

γ αα

α β βα

β α

αβ α α β β

[

P

\

P

[\

P

ZE

[ \ [ \

ZE

\ [ \ [

ZE

[ \E

\ [

[ [ \ \

= +

= +

= +

+

2 11 2 2

2 12 2 2

2 12 11

2 22

2 22

2 2

cos sin cos sin

cos sin cos sin

cos sin cos sin

cos sin cos sin

(3. 47)


where the coefficients α�� and α�� are functions of the aspect ratio as shown in Fig. 3.7,and expressions for the curve fittings are presented in the following for different magni-tudes of orthotropy of the faces.

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0 1 2 3 4 5Aspect ratio D�E

α��α�� Parameters α��α�� for the membrane strains in 6ROXWLRQ��

.� ��, ν = 0.2

.� ��, ν = 0.3

.� ��

.� ��

.� ��, ν = 0.25

α ν11

0 25 1

2 63 0 060 09= =

−⋅ +

+

.

. ..D

E

α ν12

0 25 1

34 12 60 34= =

⋅ −−

.

..D

E

. $��$��$�� ν$��$�� $��ν��

Figure 3.7: 7KH�FRHIILFLHQWV�α��DQG�α��LQFOXGHG�LQ�WKH�VWUDLQ�H[SUHVVLRQV�(T��)RU�GLIIHU�HQW�PDJQLWXGHV�RI�RUWKRWURS\�RI�WKH�IDFHV�

The parameter α�� is, in practical terms, independent of Poisson’s ratio ν and the stiffnessratio .. The parameter α�� is dependent of Poisson’s ratio ν for stiffness ratios . close toone, whereas it becomes independent of ν for larger stiffness ratios. The parameters areexpressed in Eq. 3.48, where the equation for α�� is given for stiffness ratio equal to unity(isotropic faces) and for Poisson’s ratio equal to 0.25.

α

α

11

12

1

2 63 0 060 09

1

34 12 60 34

=−

⋅ ++

=⋅ −

−

. ..

..

DE

DE

(3. 48)

Similarly, we obtain the bending strains according to the boundary conditions (Fig. 3.3).


( )εα

α β[

EE

E E

] Z VV

] Z Z \ FO=

+ ⋅

− ⋅ −

2

2 22 2

,

, sin

( )εβ

β α\

E E

E E

] Z VV

] Z Z \ FO=

+ ⋅

− ⋅ −

2

2 22 2

,

, sin (3. 49)

γαβ α β

αβ α β α β[\

EE

E

] Z VV

] Z FO=

− ⋅

− ⋅

cos xcos y ,

cos xcos ysin xsin y , 4

The total strain is the sum of the bending and membrane strains. By application ofHooke’s law and the assumption Eq. 3.43, the stresses yield

( ) ( )σ ε ε ν σ ε νε

τν

ε τ γ τ γ

[ [ \ \ \ [

[\ [\ [] \]

4 .4

.4

.4 4

= + = +

=−

= =

11

11

11

55 44

1

2

, , xz yz

, (3. 50)

where

4(

4 4 *FRUH

11 2

44 55

1=

−= =

ν

The maximum core shear strains (at the middle of the edges) are underestimated by use ofthe derivatives of the deflection function of ZV. Thus, an alternative design formula formaximum shear strains is given in Eq. 3.51. The formula is based on simple linear sand-wich plate theory derived by means of energy methods, Allen [1], and produces very con-servative values of the peak shear strains, as non-linear effects are neglected. However, byreplacing the total load T with the linear part of the lateral load, T1 = T- T2 (Eq. 3.32), weobtain fairly accurate results. By so doing, we assume that the core is influenced only bythe linear load T1, which makes sense, as the non-linear part of the load, T2, is carried onlyby stretching of the faces.


τ κ τ κ

κ

κ

[] \]

T E

G

T E

GDE

DE

DE

DE

= =

= ⋅ −

−

= ⋅ +

−

11

12

1 10

2 10

0 3 47 29 0 04

115 17 0 04

,

. log .

. log . .

(3. 51)

The coefficients κL are plotted in Fig. 3.8.

κ − coefficients for shear strains

0.34

0.38

0.42

0.46

0.50

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0Aspect ratio a/b

κ

κ

κ

1 10

2 10

0 3 47 29 0 04

115 17 0 04

= ⋅ −

−

= ⋅ +

−

. log .

. log . .

DE

DE

DE

DE

Figure 3.8: &RHIILFLHQWV�κ��DQG�κ��LQFOXGHG�LQ�WKH�VKHDU�VWUDLQ�H[SUHVVLRQV��(T��)RU�VDQG�ZLFK�SDQHOV�ZLWK�LVRWURSLF�FRUHV�

The above solution method is easy to use and gives good results, which will be shown inSection 3.4, where the analytical solutions are compared to a numerical finite differencemethod. In order to understand the procedure of�6ROXWLRQ��, the method is summarised asfollows:

�� Find 5��and 5��using Figs. 3.5-6.�� Find the deflection Z�from Eq. 3.32.�� Find the strains and stresses using Figs. 3.7-8 and Eqs. 3.47-51.

The method is illustrated with an example where a symmetric simply supported or-thotropic sandwich plate is analysed for maximum deflections, strains and stresses. Mate-rial data and plate scantlings are listed below:

D� �1.4 P (�� 20 *3D *FRUH 85 03D νIDFH� �0.25E� �1.0 P WIDFH = 0.005 P WFRUH = 0.05 P T = 100 .3D . = 2

The in-plane, bending and shear stiffnesses (Eqs. 3.14-15) yield


$�� 213 03D '�� 161 .3D� 6� �4.25 03D

From Figs. 3.6a-b we obtain the parameters αE and αV (.� �2 and D�E = 1.4), which are in-serted in Eq. 3.45 yielding the parameter 5�, thus

αE = 0.013 αV = 0.110 5� = 9.38 ⋅106

The parameter 5� is found from Fig. 3.5, equal to 10 multiplied with $ E114 , as

5� = 2.13 ⋅109

The deflection amplitude, Z, and the load terms, T� and T�, are solved (Eqs. 3.32-3) usingthe values for 5� and 5�. Consequently,

T� �T��T�� 5��Z��5��Z3 ⇒ 105 = 9.38 ⋅106 Z + 2.13 ⋅109 Z3 ⇒

Z = 0.0438-0.0334 = 0.0104 P�⇒ T� = 97552 3D and T� = 2448 3D

The α�� and α�� parameters are obtained from Fig. 3.7 in order to get the membranestrains in the skins (Eq. 3.47), hence

α�� = -0.18 α�� = -0.32 ε[

P = 8.74⋅10-5 ε\

P = 2.17⋅10-4

The maximum bending strains in the upper and lower skins (] = ±0.03 P) are obtainedfrom Eq. 3.49 by use of the bending part of the deflection, ZE, found from Eq. 3.45, weget

ZE = 7.87 PP ε[

E = 1.18⋅10-3 ε \

E = 2.32⋅10-3

The total strains are the sum of the membrane and bending strains where the bendingstrains in the upper skin are negative. From Eq. 3.50 we find the stresses as

σ[

ORZHU = 33.8 03D σ \

ORZHU = 30.4 03D

σ[

XSSHU = -29.0�03D σ\

XSSHU = -25.3 03D

The maximum core shear stresses at the middle of the edges are obtained from Eq. 3.51,where the parameter κ1 and κ2 are found from Fig. 3.8, thus

κ1 = 0.41 κ2 = 0.36 τ[]

FRUH = 0.70 03D τ\]

FRUH = 0.80 03D


The above results are summarised in Table 3.1 where they are compared to results ob-tained by use of numerical calculations.

5HVXOWV [PP] Lower [03D] Upper [03D] Core [03D]Z ZE ZV σ[ σ\ σ[ σ\ τ[] τ\]

6ROXWLRQ�� 10.4 7.85 2.55 33.8 30.4 -29.0 -25.3 0.70 0.80

Numerical 10.0 7.75 2.25 31.1 28.6 -26.1 -23.0 0.66 0.76

Table 3.1: 5HVXOWV�RI�PD[LPXP�UHVSRQVHV�RI�VDQGZLFK�SODWH�REWDLQHG�XVLQJ�6ROXWLRQ��The method represents a simple alternative to non-linear plate response calculations. Fur-thermore, it is ideal for combination with existing linear design rules. This is so in thecase of the Det Norske Veritas, High Speed Light Craft Rule [5], which consists of onlythe linear part (the 5�-part) in the section on sandwich plates.

��5HVXOWV�DQG�'LVFXVVLRQ

Results of response calculations by application of the above methods are compared to nu-merically computed finite difference results from Riber [12] and experimental results fromBau et al. [3]. The following parameters are discussed: deflections, in-plane strains andshear strains (Z, ε[, γ[]) calculated in the middle of the plate and along the [-axis for \ =E/2. The results are shown in Figs. 3.9-14, where the terms QXPHULFDO, 6ROXWLRQ��, 6ROXWLRQ� and H[SHULPHQW refer to the central finite difference solution, the complete analyticalsolution, the combined analytical solution and experiments, respectively. The experimentin Bau [3] was performed with a clamped square sandwich panel only, and the analyticalsolution is obtained by use of the plate properties given (Bau [3]) as follows:

D� �E� �0.6 P, (IDFH� �73.4 *3D, νIDFH� �0.32, *FRUH 110 03D,νFRUH� �0.30, '� �5.901�.1P� �6� �1.440 01�P

The geometrical non-linear behaviour is most pronounced for the simply supported plateillustrated in Fig. 3.9, where the lateral deflection in the centre of the plate is plottedagainst the lateral load T. The analytical results are almost identical and quite accurate (~3%) compared to the numerical results and the experimental data.

��5HVXOWV�DQG�'LVFXVVLRQ��57

Midpoint deflection of the plate

0

2

4

6

8

10

12

0 20 40 60 80 100 120 140 160Lateral load T [.3D]

Z [PP]

clamped

simply supportedQXPHULFDO

6ROXWLRQ��6ROXWLRQ��H[SHULPHQW/LQHDU

Figure 3.9: 0LGSRLQW�GHIOHFWLRQ�Z�RI�D�FODPSHG�DQG�D�VLPSO\�VXSSRUWHG�VDQGZLFK�SODWH�If we consider the deflections along the [-axis (\ = E/2) in Fig. 3.10, the analytical solu-tions are not accurate for the clamped case, whereas, for the simply supported case, the de-flection curves agree well with the numerical results. This indicates that the strains (andstresses) derived from these analytical deflection functions are not reliable for the clampedcase.

Deflection along the x-axis, y = b/2

0,00

2,00

4,00

6,00

8,00

10,00

12,00

0 1/10 1/5 3/10 2/5 1/2x/a

w [mm]

numerical

Solution 1

Solution 2

experiment

simply supported

clamped

Figure 3.10: 'HIOHFWLRQ��Z��DORQJ� WKH�[�D[LV��\� �E��FODPSHG�DQG�VLPSO\�VXSSRUWHG��/DWHUDOORDG�T� ��.3D�


Strains along the [-axis, \�= E��

-1,50

-1,00

-0,50

0,00

0,50

1,00

1,50

2,00

2,50

0 1/10 1/5 3/10 2/5 1/2[�D

ε [PP�P]

inner face

outer face

Simply supported

QXPHULFDO

6ROXWLRQ��

6ROXWLRQ��

Figure 3.11: 6WUDLQV��ε[��DORQJ�WKH�[�D[LV��\� �E��VLPSO\�VXSSRUWHG��/DWHUDO�ORDG�T� ��.3D�

The analytical solutions for the strains along the centre line for the simply supported sand-wich plate in Figs. 3.11-12 show less good agreement with the numerical results. How-ever, in the centre of the plate on the outer face, where the maximum tension strains arefound (simply supported case), the values lie within 5 % compared to the numerical re-sults.

Midpoint strains in the faces

-1,50

-1,00

-0,50

0,00

0,50

1,00

1,50

2,00

2,50

0 40 80 120 160Lateral load T [.3D]

ε [PP�P]

Simply supported

Outer face

Inner face

QXPHULFDO

6ROXWLRQ��

6ROXWLRQ��

6ROXWLRQ��

DQG

6ROXWLRQ�

Figure 3.12: 0LGSRLQW�VWUDLQV��ε[��LQQHU�DQG�RXWHU�IDFHV�RI�VLPSO\�VXSSRUWHG�SODWH��/DWHUDO�ORDGT� ��.3D�

The highly non-linear behaviour of midpoint strains is seen in Fig. 3.12. The analytical so-lutions show good agreement with the numerical solutions, however, slightly conservative

��5HVXOWV�DQG�'LVFXVVLRQ��59

results for both the inner and outer faces. For the clamped case, the maximum strains arefound at the edge, and the analytical results are about 35 % lower than the experimen-tal/numerical results. In the centre of the plate, the analytical methods predict strains ap-proximately 20 % higher.

Fig. 3.13 shows the maximum core shear strain at the midpoints of the edges (\� ��E�DQG[� �D��) for the simply supported case. 6ROXWLRQ�� provides good results for the maximumcore shear strain. Taking into account that core shear failure is one of the most commonfailure modes in composite ship hull panels, the author suggests the same method may beused for the clamped case.

Shear strains at the middle of the edge

0

5

10

15

20

25

0 20 40 60 80 100 120 140 160Lateral load T�[.3D]

γxz

numerical

solution 1

solution 2

linear/solution 2

linear

[PP�P]

Figure 3.13: 6KHDU�VWUDLQV��γ[]��DW�PLGGOH�RI�HGJH��[� ��\� �E��VLPSO\�VXSSRUWHG�

��6XPPDU\

Two non-linear analytical methods, assigned 6ROXWLRQ�� and 6ROXWLRQ��, are derived for re-sponse calculations of laterally loaded sandwich plates. The simplest method - 6ROXWLRQ��-provides results, which are as good as the results from the more complex solution method- 6ROXWLRQ� �. 6ROXWLRQ� � is presented as a simple design procedure for sandwich plates.Both the methods give good predictions for the maximum deflection of the plate. For thesimply supported case, the strains are reasonably accurate, whereas for the clamped case,the strains are overestimated in the plate centre and underestimated near the plate edges.The maximum shear strains in the core are underestimated and an ordinary linear solutioncombined with 6ROXWLRQ�� is suggested in this case. The linear part of 6ROXWLRQ�� may bereplaced by other solutions based on more complex deflection functions (Cheng et al. [4])


than Eq. 3.29 resulting in accurate curvatures (strains) close to the edges for the clampedcase.

��%LEOLRJUDSK\

[1] Allen H.G. Analysis and Design of Structural Sandwich Panels. Pergamon Press, Ox-ford, UK, 1969.

[2] Alwan A.M. Bending of Sandwich Plates with Large Deflections. -�� (QJQJ�0HFK�

'LY��3URF��$6&(. Vol. 93 (EM3), pp. 83-93, 1967. [3] Bau-Madsen N.K., Svendsen K.H. and Kildegaard A. Large Deflections of Sandwich

Plates - an Experimental Investigation. &RPSRVLWH� 6WUXFWXUHV. Vol. 23, pp. 47-52,1993.

[4] Cheng Z., Wang X. and Huang M. Large Deflection of Rectangular Hoff Sandwich

Plates. ,QW��-��6ROLGV�6WUXFWXUHV. Vol. 30, pp. 2335-2346, 1993. [5] DNV. Classification Rules for High Speed Light Craft, Det Norske Veritas Research

AS, Veritasveien 1, N-1322 Høvik, Norway, 1991. [6] Hughes O. Ship Structural Design. John Wiley & Sons, New York, 1983. [7] Kan H.P. and Huang J.C. Large Deflections of Rectangular Sandwich Plates. $,$$

-O. Vol. 5(9), pp. 1706-1708, 1967. [8] Mindlin R.D. Influence of Rotatory Inertia and Shear on Flexural Motions of Iso-

tropic, Elastic Plates.�-RXUQDO�RI�$SSOLHG�0HFKDQLFV. Vol. 18, pp. 336-343, 1951. [9] Plantema F.J. Sandwich Construction. John Wiley & Sons, New York, 1966. [10] Reissner E. The Effect of Transverse Shear Deformation on the Bending of Elastic

Plates. -RXUQDO�RI�$SSOLHG�0HFKDQLFV. Vol. 12, pp. 69-77, 1945. [11] Reissner E. Finite Deflections of Sandwich Plates. -�� RI�$HURQDXW�� 6FL. Vol. 15� (7),

pp. 435-440, 1948. [12] Riber H.J. Rational Design of Composite Panels. 3URFHHGLQJV� RI� WKH� �UG� ,QWHUQD�

WLRQDO� &RQIHUHQFH� RQ� 6DQGZLFK� &RQVWUXFWLRQV, Department of Civil and Environ-mental Engineering, University of Southampton, UK. Vol. 1 (session 6B), 1995.

[13] Riber H.J. Non-linear Analytical Solutions for Laterally Loaded Sandwich Plates.

6XEPLWWHG�WR�&RPSRVLWH�6WUXFWXUHV, March, 1997.


[14] Vinson J.R. and Sierakowski R.L. The Behavior of Structures Composed of Compos-

ite Materials. Martinus Nijhoff Publishers, P.O. Box 163, 3300 AD Dordrecht, TheNetherlands, 1986.

[15] Whitney J.M. Structural Analysis of Laminated Anisotropic Plates. Technomic Pub-

lishing Company, Inc., 851 New Holland Avenue Box 3535, Lancaster, Pennsylva-nia, USA, 1987.

[16] Zenkert D. An Introduction to Sandwich Construction. Chameleon Press LTD, Lon-

don, 1995.

61

&KDSWHU��

1XPHULFDO�)RUPXODWLRQ�RI�'\QDPLF7KHRU\

��,QWURGXFWLRQ

As the time factor is of great importance in the preliminary design of ship structures, platescantlings are often calculated by use of simple analytical design rules. In some casesmore sophisticated numerical methods are applied, in order to obtain better design andmore detailed information about the response of the structure. These methods are mostfrequently computer programs based on linear finite element theory, requiring extensivetime-consuming geometrical modelling before the calculations can take place. If, in addi-tion, geometrical non-linear calculations are required, the computer time needed to solvethe equations increases drastically.

In order to perform such non-linear dynamic calculations, without using extensive com-puter time, a finite-difference-based solution method dealing with geometrical non-linearorthotropic sandwich and single-skin plates is developed. The solution method is imple-mented into a FORTRAN program 3DQHO (Chapter 7), aimed at preliminary design of shiphull panels.

The present formulation is based on Gorji [4]. Here, the non-linear terms of the lateral dis-placements are considered as an additional set of lateral loads, acting on the plate. The so-lution procedure is an iterative finite difference method applied to an equivalent plate, un-dergoing small deflections. The differential equations are uncoupled as the lateral deflec-tion, Z, is separated into a bending and a shear contribution, which indeed speeds up thenumerical solution. Hence, bending causes the cross-section to rotate, whereas shearing isa sliding movement and does not add to any rotations. Using this definition, we reduce thenumber of independent field variables from five to four.

62 &KDSWHU��1XPHULFDO�)RUPXODWLRQ�RI�7KHRU\

��,QWHJUDWLRQ�6FKHPH�LQ�7LPH�DQG�6SDFH

The dynamic analysis of the plate, consisting of inertia and damping forces as functions oftime (W), is solved by use of Newmark’s method [5], whereas the central finite differencemethod solves the governing equilibrium equations with regard to space ([, \, ]). Thecentral finite difference method, which is described in various text books, among othersTimoshenko [7], is used to express the derivatives of the four displacements X��Y��ZE�andZV, where the configuration is outlined in Figure 4.1. The choice of the central finite dif-ferences, alternatively to forward or backward finite differences, is due to the higher accu-racy, as the truncation error is of the order ∆2 compared to ∆, for forward or backward fi-nite differences. The grid distance ∆ is chosen to be equidistant in the [� and \�-directionsin order to simplify the linearised differential equations.

31 2

∆ [��M

∆

\��L

4 5 6

7 8 9

r

u

d

[� ��DM� ��Q

\� ��EL� ��P

Z�L�M�� Z��

Figure 4.1: &HQWUDO�ILQLWH�GLIIHUHQFH�QHW�FRQILJXUDWLRQ�

Newmark’s constant-average integration scheme, Newmark [5], is chosen partly due to itssimplicity. Nevertheless, the most important effect is that the accuracy is obtained withoutrequirement for the time increment δW, which is said to be an unconditionally stablescheme. The configuration is shown in Fig. 4.2.

��,QWHJUDWLRQ�6FKHPH�LQ�7LPH�DQG�6SDFH��63

( )1

2

2

21

2

2

∂∂

∂∂ δ

Z

W

Z

W WW W+

+

+

W W��

∂∂Z

WW

2

2

( )∂

∂ δZ

W WW +

+1

2

2

Figure 4.2: 1HZPDUN¶V�FRQVWDQW�DYHUDJH�DFFHOHUDWLRQ�VFKHPH�

��&HQWUDO�)LQLWH�'LIIHUHQFHV

By use of the central finite differences for the derivatives of X�� Y�� ZE�� ZV and Z(demonstrated for Z only), the following expressions for the derivatives are obtained:

( ) ( )

( ) ( )

( )

∂∂

∂∂

∂∂

∂∂

∂∂ ∂

Z[

Z ZZ\

Z Z

Z[

Z Z ZZ\

Z Z Z

Z[ \

Z Z Z Z

= − = −

= + − = + −

= + − −

6 4 8 2

2

2 6 4 52

2

2 8 2 52

2

9 1 7 32

2 2

2 2

4

/ /

/ /

/

∆ ∆

∆ ∆

∆

, ,

, , (4. 1)

and

( ) ( )

( )

( )

, ,

,

∂∂

∂∂

∂∂ ∂∂

∂ ∂

3

3 6 43

3

3 8 23

3

2 9 7 3 1 6 43

3

2 9 7 3 1 8 23

2 2 2 2 2 2

2 2 2

2 2 2

Z[

Z Z Z ZZ\

Z Z Z Z

Z[ \

Z Z Z Z Z Z

Z\ [

Z Z Z Z Z Z

U O G X= − − + = − − +

= − + − − +

= + − − − +

/ /

/

/

∆ ∆

∆

∆

(4. 2)

and


( )

( )

( )

∂∂∂∂∂

∂ ∂

4

4

4

4

4

2

x ,

y ,

y

ZZ Z Z Z Z

ZZ Z Z Z Z

Z[

Z Z Z Z Z Z Z Z Z

O U

X G

= − + − +

= − + − +

= + + + − − − − +

4 6 4

4 6 4

2 2 2 2 4

4 5 64

2 5 84

2 9 7 3 1 8 6 2 4 54

/

/

/

∆

∆

∆

(4. 3)

The notation of the indices in the above expressions refers to Fig. 4.1. The expressions areimplemented in the governing equilibrium equations (Eqs. 3.17-20), resulting in a set oflinearised non-linear equations, which are solved for each of the grid points in combina-tion with the known boundary conditions.

��1HZPDUN¶V�0HWKRG

The central finite difference method is combined with Newmark’s integration scheme, inorder to calculate the dynamic behaviour of the plate. The deflection, velocity and accel-eration at a time step W�δW�(denoted by indices�W��) are expressed by the values at the pre-vious time step W. The present formulation is in accordance with Bathe and Wilson [3],where the following assumptions are used:

Z ZZW

WZW

ZW

W

ZW

ZW

ZW

ZW

W

W W

W W W

W W W W

++

+ +

= + + +

= + +

12

2

2 1

22

1 2

2

2 1

2

1

4

1

2

∂∂

δ∂∂

∂∂

δ

∂∂

∂∂

∂∂

∂∂

δ (4. 4)

which is the so-called constant-average-acceleration method. In order to reduce the ex-pression the comma denotes differentiation with respect to time in the following. As weneed the acceleration and the velocity at time W�� the above expressions are manipulatedto

( )

( )

ZZW

Z Z Z Z

ZZW

Z Z Z

WW

W

W

W W

W

W

WW

W

W

W

W

W W

W

W

, , ,

, ,

++

+

++

+

= = − − −

= = − −

12 1

22 1

11

1

2∂

∂η η

∂∂

η ηδ

, =2

t

(4. 5)

These expressions (Eq. 4.5) are inserted in the equilibrium equations (Eqs. 3.17-20), re-ducing the unknown variables deflection� (ZW��), velocity (Z�W

W��) and acceleration (Z�WWW��)

at time step W�� to the unknown ZW��, Z�WW and Z�WW

W.


��1XPHULFDO�)RUPXODWLRQ�RI�(TXLOLEULXP�(TXDWLRQV

Inserting the finite differences and Newmark’s scheme in the governing equations (Eqs.3.17-20), we get four equilibrium equations. Here, the unknown values are�X��Y� ZE and�ZV

at time step W�δW (left side of the equations) expressed by known values at time step W(right side). The four equilibrium equations are expressed in condensed form below, wherematrices and vectors are denoted by [] and {}, respectively. By use of the central finitedifferences we obtain the bending/shearing equilibrium equations:

[ ]{ } [ ]{ } [ ]{ } { } [ ] ( ){ }∇ + + = + ∇+ + ++

∗ +4 1 1 11

3 1' Z & Z 0 Z T % X Y

E

W E

E W

W

WW

W

W

E W

αβ αβ αβ αβ, , , (4. 6)

and

[ ]{ } [ ]{ } [ ]{ } { }∇ − − = −+ + ++

∗2 1 1 11$ Z & Z 0 Z TV

V

W V

V W

W

WW

W

Wαβ αβ αβ, , (4. 7)

where

[ ] [ ] [ ] [ ]& ' & $E

E

V

V

V

αβ αβ αβ αβκ κ= ∇ = ∇4 2 and (4. 8)

Where the operators ∇ L are defined in Eqs. 4.10-16. The damping is related to the bend-ing- and shear stiffness with κE and κV, respectively. They are included in the equilibriumequations for bending and shear, although they are difficult to determine without doing in-situ tests with the particular plates. The inertia and the damping contributions from the in-plane displacements are neglected in the in-plane equilibrium equations, as they are re-garded as small in comparison with the out-of- plane contributions. Thus, the followingequations are obtained:

[ ] ( ){ } { } [ ]{ }[ ] ( ){ } { } [ ]{ }∇ = + ∇

∇ = + ∇

+ ∗ + +

+ ∗ + +

2 1 1 3 1

2 1 1 3 1

$ X Y ; % Z

$ Y X < % Z

W W W

W W W

αβ αβ

βα βα

,

, (4. 9)

The condensed notation of the equation keeps an acceptable length of the expression. Nev-ertheless, in order to understand each matrix term (indicated by []), they are written in fi-nite difference notation for an arbitrary node in the following. We start with the stiffnessoperator in the bending equilibrium equation (Eq. 4.6) at time W��:


( )

( ) ( )( )(( )( ) ( )

( ) ( ) )

∇ = + + +

= + + + − + +

− + + + +

+ + + + +

411

4

4 12 66

4

2 2 22

4

4

1 3 7 9 22 2 8

11 4 6 11

22 11 22 5

2 2

2 4

4

6 6 8

' Z 'Z

[' '

Z

[ \'

Z

\

' Z Z Z Z ' ' Z Z

' ' Z Z ' Z Z

' Z Z ' ' ' Z

E

E E E

E E E E E E

E E

O

E

U

E

X

E

G

E E

αβ∂∂

∂∂ ∂

∂∂

4∆

(4. 10)

where

' ' '= +12 662

The mass matrix operator yields

0 Z G]Z

WWWαβ ρ∂∂, = ⋅∫2

52 (4. 11)

whereas the bending/stretching operator yields

( )

( ) ( )((

))

∇ = + +

+

= − + − + − + −

+ ⋅ − + − + −

+ + − − + −

311

3

3

3

2

3

2 22

3

3

11 4 6 22 2 8

9 7 3 1 4 6

9 7 3 1 2 8

2 2 2 2

2 2

2 2

% X Y %X[

%X

[ \Y

\ [%

Y\

% X X X X % Y Y Y Y

X X X X X X

Y Y Y Y Y Y

E

U O G X

αβ∂∂

∂∂ ∂

∂∂ ∂

∂∂

,

B

2 3∆

(4. 12)

where

% % %= +12 662

The equivalent lateral load including the in-plane membrane strains becomes:

( ) ( )(( ))

T T 1 Z

T 1Z[

1Z\

1Z\ [

T 1 Z Z Z 1 Z Z Z

1 Z Z Z Z

[ \ [\

[ \

[\

∗ = + ∇

= + + +

= + + − + + −

+ − −

2

2

2

2

2

2

5 6 4 5 8 2 5

9 1 7 32

2

2 2

αβ

∂∂

∂∂

∂∂ ∂

�� ∆

(4. 13)


Written in finite difference form the shear equation (Eq. 4.7) yields

( ) ( )( )∇ = +

= + − + + −

255

2

2 44

2

2

55 6 4 5 44 8 2 522 2

$ Z $Z

[$

Z

\

$ Z Z Z $ Z Z Z

V

V

V V

V V V V V V

αβ∂∂

∂∂

∆ (4. 14)

Considering the two in-plane equations (Eq. 4.9), we obtain the following expressions forthe in-plane stiffness operators applied to the displacements, X and Y:

( )

( ) ( )(( ))

∇ = + + +

= + − + + −

+ + − −

211

2

2 66

2

2

2

66

11 6 4 5 22 8 2 5

9 1 7 3

2 2

$ X Y $X[

$Y\

$Y

[ \$

$ X X X $ X X X

$ Y Y Y Y

αβ∂∂

∂∂

∂∂ ∂, )��$ �$

��

��

�∆

(4. 15)

and

( )

( ) ( )(( ))

∇ = + + +

= + − + + −

+ + − −

222

2

2 66

2

2

2

66

66 6 4 5 22 8 2 5

9 1 7 3

2 2

$ Y X $Y\

$X[

$X

[ \$

$ Y Y Y $ Y Y Y

$ X X X X

βα∂∂

∂∂

∂∂ ∂, ) , A = (A

14 12

2∆

(4. 16)

whereas the two right sides in Eq. 4.9 become:

( )

( )( ) ( )( )(( )( ))

;Z[

$Z[

$Z\

Z\$

Z\ [

$ $

$ Z Z Z Z Z $ Z Z Z Z Z

$ Z Z Z Z Z Z

∗ = +

+ = +

= − + − + − + −

+ − − − +

∂∂

∂∂

∂∂

∂∂

∂∂ ∂11

2

2 66

2

2

2

14 12 66

11 6 4 6 4 5 66 6 4 8 2 5

8 2 9 7 3 1

2 2

��$

�� ∆

(4. 17)

and

( )

( )( ) ( )( )(( )( ))

<Z\

$Z[

$Z\

Z[$

Z\ [

$ $

$ Z Z Z Z Z $ Z Z Z Z Z

$ Z Z Z Z Z Z

∗ = +

+ = +

= − + − + − + −

+ − − − +

∂∂

∂∂

∂∂

∂∂

∂∂ ∂66

2

2 22

2

2

2

14 12 66

66 8 2 6 4 5 22 8 2 8 2 5

6 4 9 7 3 1

2 2

��$

�� ∆

(4. 18)


Finally, the stretching/bending operators in Eq. 4.9 become:

( )

( )(( ))

∇ = +

= ⋅ − + −

+ − + − + −

311

3

3

3

2

11 4 6

9 7 3 1 4 63

2 2

2 2 2

% Z %Z[

%Z

\ [

% Z Z X X

% Z Z Z Z Z Z

U O

αβ∂∂

∂∂ ∂

∆

(4. 19)

and

( )

( )(( ))

∇ = +

= ⋅ − + −

+ + − − + −

322

3

3

3

2

22 2 8

9 7 3 1 2 83

2 2

2 2 2

% Z %Z\

%Z

[ \

% Z Z X X

% Z Z Z Z Z Z

G X

βα∂∂

∂∂ ∂

∆

(4. 20)

By applying Newmark’s scheme (Eq. 4.5) to the equilibrium equations (Eqs. 4.6-9), weobtain the following time dependent set of non-linear differential equations after somesimplification:

( )[ ] [ ] [ ][ ]{ }[ ] [ ][ ]{ }

{ } { } { }{ }[ ]{ } { } { }{ }[ ] [ ] ( ){ }

κ η η

η

κ η

η η

αβ αβ αβ

αβ αβ

αβ

αβ αβ

E E

W

V

W

W E E

W

E W

W

W

W

W

WW

W E W

' 1 0 Z

P 1 Z

T Z Z '

Z Z Z 0 % X Y

+ ∇ − ∇ +

+ − ∇

= + + ∇

+ + + + ∇

+

+

+∗

+

1

2

4 2 2 1

2 2 1

14

2 3 1

,

, , ,

(4. 21)

and

( )[ ] [ ] [ ][ ]{ }[ ] [ ][ ]{ }

{ } { } { }{ }[ ]{ } { } { }{ }[ ]

1

2

2 2 2 1

2 2 1

12

2

− ∇ + ∇ −

+ − + ∇

= − − + ∇

− + +

+

+

+∗

κ η η

η

κ η

η η

αβ αβ αβ

αβ αβ

αβ

αβ

V

V

V

W

E

W

W V V

W

V W

W V

W

W

W

WW

W

$ 1 0 Z

P 1 Z

T Z Z $

Z Z Z 0

,

, ,

(4. 22)

Thus, the above set of equations, formulated by use of the finite difference method andNewmark’s method, expresses the lateral deflections, ZE and ZV, at the time W�δW. The in-plane displacements, X, Y, are coupled to the above equations through the right-side terms{ T } and [%E]. The bending/shearing equations (Eqs. 4.6-7) are de-coupled, in order to


speed up the numerical solution. In this way, we obtain two separate equations for each ofthe unknowns Z

E

W+1 and ZV

W+1 . The bending equation yields:

( )[ ] [ ] [ ][ ] { }{ } [ ] [ ][ ] { } [ ] { }

[ ] { } [ ] ( ){ }

κ η η

η κ

αβ αβ αβ

αβ αβ αβ

αβ αβ

E E

W

W V

W

E E

W

E

W W

' 1 0 Z

T 0 1 Z ' YHO

0 DFF % X Y

+ ∇ − ∇ + ⋅

= − − ∇ ⋅ + ∇ ⋅

+ ⋅ + ∇ ⋅

+

+∗ +

+

1 4 2 2 1

12 2 1 4

3 1,

(4. 23)

where

{ } { } { } { } { } { } { }YHO Z Z DFF Z Z ZE

W

E

W

E W

W

E

W W

W

W

WW

W= + = + +η η η, , , and 2 2

The shearing equation yields:

( )[ ] [ ] [ ][ ] { }{ } [ ] [ ][ ] { }

[ ] { } [ ] { }

1 2 2 2 1

12 2 1

2

− ∇ + ∇ − ⋅

= − + − ∇ ⋅

− ∇ ⋅ − ⋅

+

+∗ +

κ η η

η

κ

αβ αβ αβ

αβ αβ

αβ αβ

V

V

V

W

W E

W

V

V

V

W

V

W

$ 1 0 Z

T 0 1 Z

$ YHO 0 DFF

(4. 24)

where

{ } { } { } { } { } { } { }YHO Z Z DFF Z Z ZV

W

V

W

V W

W

V

W W

W

W

WW

W= + = + +η η η, , , and 2 2

The coupled in-plane equations (Eq. 4.9) are expressed in a combined matrix-vector equa-tion below:

∇∇

=

+∇∇

+

+

∗ +

∗ +

+

+

2

2

1

1

1

1

3

3

1

1

$

$

X

Y

;

<

%

%

Z

Z

W

W

W

W

W

W

αβ

βα

αβ

βα (4. 25)

The matrix-vector system of equations (Eqs. 4.23-25) forms the problem of the geometri-cal non-linear dynamic response of laterally loaded non-symmetric orthotropic compositeplates. By combining Eqs. 4.23-25 with the finite difference expressions in Eqs. 4.10-20we obtain a set of linear equations for each grid point. The system of equations is solvedin load and time steps, whereas the coefficient matrices must be updated, since it is still anon-linear problem.


��%RXQGDU\�&RQGLWLRQV

The solution of the system of equations (Eqs. 4.23-25) requires precise knowledge of thedisplacements at the boundaries. As for the fourth-order differentiation in the bendingequation (Eq. 4.10 or 4.23), we need to know the lateral bending deflection outside theboundaries. The boundary conditions will be explained for the simply supported and theclamped cases in the following on the assumption of symmetry.

]��Z

[��M

ZE�L�� ZE�L��

ZE�L��ZE�L�� ZE�L��

ZE�L��

Simply supported Clamped

X�Y�ZV�L�� X�Y�ZV�L��ERWK�FDVHV�

ZE

¶�L�� :E

¶¶�L��

Figure 4.3: )LQLWH�GLIIHUHQFH�DSSOLFDWLRQ�RI�ERXQGDU\�FRQGLWLRQV�

Fig. 4.3 shows the finite difference notation for both simply supported and clamped cases.Using the plate edges as symmetry lines, we have anti-symmetry for the simply supportedcase and symmetry for the clamped case with regard to the lateral deflection, Z. Symmetryis found for the in-plane deflections, X and Y, in both cases. In order to solve the equilib-rium equations (Eqs. 4.23-25) by use of the finite difference method, we assume knowndeflection fields, X, Y, ZE and ZV, at the plate edges, and the deflection field, ZE, outsidethe plate edges. For the two analysed boundary conditions (Fig. 4.3), the following deflec-tion fields are implemented:

Simply Supported

• Lateral displacement, Z

Z(1, M) = Z(P, M) = 0 , ZE(0, M) = -ZE(2, M) , ZV(0, M) = -ZV(2, M) , M� ��QZ(L,1) = Z(L, Q) = 0 , ZE(L,0) = -ZE(L,2) , ZV(L,0) = -ZV(L,2) , L� ��P

• In-plane displacements, X and Y, straight immovable edges

X(1, M) = X(P, M) = 0 , M� ��QX(L,1) = X(L, Q) = 0 , L� ��PY(1, M) = Y(P, M) = 0 , M� ��QY(L,1) = Y(L, Q) = 0 , L� ��P


• In-plane displacements, initial in-plane displacement,�X and Y

X(1, M) = I(M)⋅XR , X(P, M) =0 , M� ��QX(L,1) = I(L)⋅XR , X(L, Q) = 0 , L� ��PY(1, M) = J(M)⋅YR , Y(P, M) = 0 , M� ��QY(L,1) = J(L)⋅YR , Y(L, Q) = 0 , L� ��P

where I and J are shape functions depending on the prescribed in-plane initial displace-ments.

Clamped

• Lateral displacement, Z

Z(1, M) = Z(P, M) = 0 , ZE(0, M) = ZE(2, M) , ZV(0, M) = -ZV(2, M) , M� ��QZ(L,1) = Z(L�, Q) = 0 , ZE(L,0) = ZE(L,2) , ZV(L,0) = -ZV(L,2) , L� ��P

The in-plane boundary conditions are the same as for the simply supported case. Hence,in-plane initial forces, 1�[, 1�\ and 1�[\, are applied as equivalent in-plane displacements,X�, Y�.

��6ROXWLRQ�3URFHGXUH

By combining the equilibrium equations (Eqs. 4.23-25) with the finite difference expres-sions (Eqs. 4.10-20), we get three systems of equations, concerning ZE, ZV, and X, Y, re-spectively. They are solved by use of a sparse matrix solver, as each of the systems areformulated into global sparse matrices. In geometrical linear problems the coefficient ma-trices are constant, whereas in non-linear problems the matrices must be recalculated ateach integration step, which of course becomes more time-consuming. In the following,the solution procedure is outlined, focusing on the formulation of the matrices, iterationloops and load- and time-steps.

��,WHUDWLRQ�/RRSV�DQG�7LPH�6WHSV

Eqs. 4.23-25 are solved separately, reducing the computer time. In order to do so, iterationloops are applied within each time step. Fig. 4.4 illustrates the procedure. The time-stepintegration is straightforward by updating the accelerations and velocities at each time in-crement, whereas the “static” solution within each time step requires numeral iterations,until equilibrium is obtained for the whole set of equations. The system is tuned by the


relative allowable displacement error, δo, and the time increment, δW, where the latter de-termines the size of the load increment, δT(W).

Time step W�� δWW� �W�� δW

Time step W

Solving Eqs.4.23-25

by use ofMatrix solver

Initial values

ZE , �Eq. 4.23

0DWUL[�6ROYHU

X�Y�,� Eq. 4.25

ZV , Eq. 4.24

( ) ( )( )

Z Z X Y Z Z X Y

Z Z X Y

E V QHZ E V ROG

E V ROG

R

, , , , , ,

, , ,

−< δ

1R�

<HV�

8SGDWHVDFFHOHUDWLRQVYHORFLWLHV�DQG

ORDG

Figure 4.4: 6ROXWLRQ�SURFHGXUH�RI�(TV��VKRZLQJ�WLPH�VWHSV�DQG�LWHUDWLRQ�ORRSV�

In general, large load increments require many iterations, whereas small load incrementsrequire few iterations. Too large load increments will make the system of equations unsta-ble and the displacement increment, δ, will increase at each iteration.

Solving the bending equations (Eq. 4.23) for the first load increment gives us the trial val-ues of the deflection, Z, in each grid. Hence, we find the non-linear right-hand terms inEq. 4.25 (;*+ ∇3% ,�<*+ ∇3% )�and the in-plane equation is solved for X and Y. From the in-plane displacements we obtain the in-plane forces, which are included in the right-handside of Eq. 4.24, solving the lateral displacements, ZV. The right-hand side of Eq. 4.23 isnow updated and a new iteration loop takes place, until we reach an acceptable value ofthe relative displacement error δ (see Fig. 4.4).

��(LJHQIUHTXHQF\�DQG�$GGHG�0DVV

In order to avoid time increments which correspond to the eigenfrequency of the system asimple linear analytical solution for the lowest eigenfrequencies of the plate is used to in-dicate the critical value of δW. Prediction of the vibrations in hull plates is of great impor-tance. Bad design in this sense may result in uncomfortable sailing and in the worst case

��6ROXWLRQ�0HWKRG��73

intolerable vibrations leading to in fatigue damage and damage to the navigation equip-ment, loose fittings, delamination of panel faces, etc.

In order to simplify the expression, the non-linear terms and the bending-stretching termsdue to asymmetry are neglected in the derivation. Thus, a linear analytical solution to thelowest frequencies of a simply supported plate is presented. It gives us a conservative es-timate of the lowest eigenfrequency as the non-linear influence (the in-plane forces due tolateral deflection) increases the plate stiffness, which again increases the lowest eigenfre-quency. Thus, the non-linear terms at the right side in Eqs. 3.19-20 from the in-plane dis-placements, X and Y, are neglected and the governing equations of the problem are reducedto a bending and a shear equation only, expressed as:

$Z[

$Z\

ZW

TV V

55

2

2 44

2

2

2

2

∂∂

∂∂

ρ∂∂

+ = −∗ ∗

(4. 26)

and

( )'Z

['

Z

\' '

Z

[ \

ZW

TW5

Z

[

Z

\

E E E

E E

11

4

4 22

4

4 12 66

4

2 2

2

2

2

2

2

2

2

2

2 2∂∂

∂∂

∂∂ ∂

ρ∂∂

∂∂

∂∂

∂∂

+ + +

= − + + +

∗ ∗

(4. 27)

where the rotary inertia 5 and the mass ρ* are defined as:

5 ] G] G] P= = + ′∫ ∫∗ρ ρ ρ* 2 (4. 28)

The effect of the rotary inertia is small compared to the vertical inertia and for most sand-wich panels in marine structures it is about 50 -100 times less than the shear deformation(Zenkert [8]). By omitting the effect of rotary inertia in the following and by inserting Eq.4.26 in Eq. 4.27 and rearranging, we obtain:

∇ − −

⋅ −

∇∇

=42

2

4

21 0' Z TZW

' Z

$ ZV

V

V

αβαβ

αβρ

∂∂

* * (4. 29)

Free vibration is characterised by zero external force T and Eq. 4.29 yields

∇ + −∇∇

=42

2

4

21 0' ZZW

' Z

$ ZV

V

V

αβαβ

αβρ

∂∂

* (4. 30)

On the assumption of free harmonic excitations the deflection, Z([�\�W), can be written as


( ) ( ) ( ) ( ) ( )Z [ \ W W [ \ $ W % W Z ZP [D

Q \E

E V, , , cos sin sin sin= ⋅ = + ⋅ +Ψ Φ ω ωπ π

(4. 31)

This function satisfies the boundary conditions: zero moments and deflections at theedges. By inserting Eq. 4.31 in Eq. 4.30, we have:

ωρ

αβ

αβ

αβ

=∇

+∇∇

4

4

21

'

'

$ V

*

(4. 32)

where

( )∇ =

+ +

+

∇ =

+

411

4

12 66

2 2

22

4

244

2

55

2

2' 'PD

' 'PQDE

'QD

$ $PD

$QE

αβ

αβ

π π π

π π (4. 33)

Equation 4.32 gives the frequencies of a simply supported sandwich plate for the differentvibration modes, P, Q. If the shear stiffness� $ V

αβ becomes large, the stiffness ratio

' $ V

αβ αβ → 0 , and thus Eq. 4.32 corresponds to that of ordinary plate theory.

The added mass ′P arises from the water flow due to the motion of the plate and dependsof the vibration mode. From the theory of potential flow a simple expression (Eq. 4.34)for the added mass can be deduced (see Terndrup and Jensen [6]) as shown below for aplate with water on one side.

′ =

+

PPD

QE

ZDWHUρ

π2 2

(4. 34)

For water on both sides the expression (Eq. 4.34) must be multiplied by two.

��)RUPXODWLRQ�RI�&RHIILFLHQW�0DWUL[

The finite difference coefficient matrices in Eqs. 4.23-5 are outlined in this section. Theyare generated by use of do-loops implemented in the FORTRAN code program 3DQHO pre-sented in Chapter 7. They are formulated for both the simply supported and clampedboundary conditions, which in fact only has an influence on the bending equations (Eq.4.23) as discussed in Section 4.1.4.


The system of equations with the coefficient matrices, outlined in the following, is solvedin two steps, using existing matrix solvers from the commercial IMSL Math/Library [2],which is a collection of FORTRAN-coded subroutines for mathematical applications. Thefirst step computes the /8 factorisation of the coefficient matrix, whereas the second stepsolves the sparse system of linear equations given the /8 factorisation of the coefficientmatrix. By doing so the same /8 factorisation can be used in several steps, without up-dating the coefficient matrix in case of geometrical linear behaviour.

Coefficient Matrix in the Bending Equation

The structure of the coefficient matrix 'EHQGLQJ (Eq. 4.35) in the bending equation Eq. 4.23is illustrated in Eq. 4.36. It is a matrix with (P��)(Q��)×(P��)(Q��) elements consisting offour different types of sub-matrices: 'E�, 'E�, 'E�, 'E� forming a sparse matrix. The ma-trix is updated after each iteration loop (see Sec. 4.2.1) as the terms (Eq. 4.35) includingin-plane forces and vertical inertia depend on the last values of the unknown lateral bend-ing deflection ZE in each grid point. These terms can be identified in the matrix elementslisted after the sub-matrices shown in the following.

[ ] ( )[ ] [ ] [ ]' ' 1 0EHQGLQJ E= + ∇ − ∇ +κ η ηαβ αβ αβ1 4 2 2 (4. 35)

The sub-matrices are diagonal matrices and contain (Q��)×(Q��) elements. They are out-lined in Eq. 4.37-40 followed by the information of each element. Hence, the coefficientmatrix yields

[ ]

[ ] [ ] [ ][ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]

[ ] [ ]

'

' ' '

' ' ' '

' ' ' ' '

' ' ' ' '

' ' ' ' '

' ' ' '

' ' '

EHQGLQJ

E E

X

E

E

O

E E

X

E

E E

O

E E

X

E

E E

O

E E

X

E

E E

O

E E

X

E

E E

O

E E

X

E E

O

E

=

1 3 4

3 2 3 4

4 3 2 3 4

4 3 2 3 4

4 3 2 3 4

4 3 2 3

4 3

0 0 0 0 0

0 0 0 0

0 0 0

0 0 0

0

0 0

0 0 0

0 0 0 0

0 0 0 0 0

.

.

.

.

. . . . . . . .

. . . . . . .

.

.

. [ ]1

(4. 36)

and the (Q��)×(Q��)sub-matrices become:


[ ]'

G G G

G G G G

G G G G G

G G G G G

G G G G

G G G

E

K

K

K K

K K

K

K

1

0 4

4 5 4

4 5 4

4 5 4

4 5 4

4 0

0 0 0

0 0

0

0

0 0

0 0 0

=

.

.

.

. . . . . . .

.

.

.

(4. 37)

where

[ ] [ ]' ' GE E Y2 1 0= = , where (4. 38)

The sub-matrices at each side of the diagonal sub matrix yield:

[ ]'

G G

G G G

G G G

G G

E

X

3

2 6

1 2 6

1 2 6

1 6

0 0

0

0

0 0

=

.

.

. . . . .

.

.

(4. 39)

Here, the equivalent matrix below the diagonal in main matrix (Eq. 4.36) becomes

[ ] [ ]' ' G GE

O

E

X

3 3 1 6= ↔ , where (4. 40)

and the last sub-matrix yields

[ ]'

G

G

G

E

Y

Y

Y

4

0 0

0 0

0 0

=

.

.

. . . .

.

(4. 41)

The elements in the above sub-matrices are listed below:

( ) ( )G G G Q L M Q L MK [ \0 5 2= + + , ,

G G GY5 3= +

( ) ( ) ( )( )G ' ' ' 'E3 11 22 12 66

4 21 6 8 2= + + + + +κ η ρ η∆ *

( )( ) ( ) ( )G ' ' ' Q L M Q L ME [ [4 11 12 66

44 1 2= − + + + − −κ η ∆ , ,


( )( )G ' ' Q L ME [\1 12 6642 1 2= + + +κ η ∆ ( , )

( )( )G ' ' Q L ME [\6 12 6642 1 2= + + −κ η ∆ ( , )

( )( ) ( )G ' ' ' Q L ME \2 22 12 6644 1 2= − + + + −κ η ∆ ,

( )( )

G '

G '

K E

Y E

= +

= +

1

1

114

114

κ η

κ η

∆

∆ clamped boundary conditions.

( )( )

G '

G '

K E

Y E

= − +

= − +

1

1

114

114

κ η

κ η

∆

∆ simply supported boundary conditions .

Here, the terms Q[(L,M) and Q\(L,M) are elements in the matrix term ∇21αβ (Eq. 4.35), repre-

senting the in-plane forces at grid point (L,M) due to the in-plane displacements, X, Y, givenin the membrane equation Eq. 4.25.

Coefficient Matrix in the Shear Equation

In a similar manner we obtain the coefficient matrix $VKHDU�in the shear equation Eq. 4.24.The updating of the coefficient matrix with regard to the in-plane forces is time-consum-ing and, consequently, these terms are left on the right hand side of the shear equation.The in-plane forces have little influence on the shear coefficient matrix in comparisonwith the bending coefficient matrix, which allows us to move the in-plane terms to theright side of the equation without violating the solution.

In general, the system of equation converges, as the solution is an iterative process. Theupdating of the shear coefficient matrix is only necessary in the dynamic case, where thevertical inertia changes at each iteration loop within the time steps. The (P��)(Q��)×(P��)(Q��) shear coefficient matrix can be expressed in the form below:

[ ] ( )[ ] [ ][ ]$ $ 0VKHDU V

V= − ∇ −1 2 2κ η ηαβ αβ (4. 42)

Written in matrix form $VKHDU yields


[ ]

[ ] [ ][ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ][ ] [ ]

$

$ $

$ $ $

$ $ $

$ $ $

$ $

VKHDU

V V

V V V

V V V

V V V

V V

=

1 2

2 1 2

2 1 2

2 1 2

2 1

0 0 0

0 0

0 0

0 0

0 0 0

.

.

.

. . . . . .

.

.

(4. 43)

The sub-matrices contain (Q��)×(Q��) elements and are outlined in the following:

[ ]$

D D

D D D

D D D

D D

V

V V

V V V

V V V

V V

1

2 1

1 2 1

1 2 1

1 2

0 0

0

0

0 0

=

.

.

. . . . .

.

.

(4. 44)

and

[ ]$

D

D

D

V2

3

3

3

0 0

0 0

0 0

=

.

.

. . . .

.

(4. 45)

The elements in the above sub-matrices are listed below:

( )D $V1 55

21= −κ η ∆

( )( )D $ $V2 44 55

2 22 1= − − + −κ η ρ η∆ *

( )D $V3 44

21= −κ η ∆

Coefficient Matrix in the Membrane Equation

The membrane coefficient matrix $PHPEUDQH� in the membrane equation (Eq. 4.25) concern-ing the in-plane displacements X and Y is constant, as the vertical inertia is neglected andthe right hand side is assumed to be known from the previous iteration step. The coeffi-cient matrix is expressed below in Eq. 4.46.

[ ]$$

$PHPEUDQH=

∇∇

2

2αβ

βα (4. 46)

Hence, we obtain the following simple 2(P��)(Q��)×�(P��)(Q��) matrix:


[ ]

[ ] [ ][ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ][ ] [ ]

$

$ $

$ $ $

$ $ $

$ $ $

$ $

PHPEUDQH

P P

P P P

P P P

P P P

P P

=

1 3

2 1 3

2 1 3

2 1 3

2 1

0 0 0

0 0

0 0

0 0

0 0 0

.

.

.

. . . . . .

.

.

(4. 47)

The sub-matrices $P�, $P� and $P� are listed in the following:

[ ]$

D D

E D

D D D

D E D

D D D

D E D

D D

D E

P

P P

P P

P P P

P P P

P P P

P P P

P P

P P

1

1 2

1 4

2 1 2

4 1 4

2 1 2

4 1 4

2 1

4 1

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

=

.

.

.

.

.

. . . . . . . . .

.

.

.

(4. 48)

and

[ ]$

D D

D D

D D D

D D D

D D

D D D

D D

D D

P

P P

P P

P P P

P P P

P P

P P P

P P

P P

2

4 6

5 6

6 4 6

6 5 6

6 4

6 5 6

6 4

6 5

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

=

−−

−−

−

.

.

.

.

. .

. . . . . . . . .

.

.

.

(4. 49)

and


[ ]$

D D

D D

D D D

D D D

D D

D D D

D D

D D

P

P P

P P

P P P

P P P

P P

P P P

P P

P P

3

4 6

5 6

6 4 6

6 5 6

6 4

6 5 6

6 4

6 5

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

=

−−

−

−−

−

.

.

.

.

. .

. . . . . . . . .

.

.

.

(4. 50)

In which the elements DPL�in the above sub-matrices are as follows:

( )D $ $P1 11 66

22= − + ∆D $P2 11

2= ∆

( )D $ $P3 66 22

22= − + ∆D $P4 66

2= ∆D $P5 22

2= ∆

( )D $ $P6 12 66

24= + ∆

��9HULILFDWLRQ�RI�WKH�0HWKRG

The central finite difference method (FD) outlined in this chapter is verified by comparingthe results (denoted Riber in the figures) with results of calculations from the commer-cially available finite element based (FEM) program ANSYS [1] (denoted Ansys in thefigures). The finite element calculations with ANSYS use the “Shell91” element, which isan eight-node, higher-order, layered shell element made for sandwich structures. The ele-ment takes into account geometrical non-linear behaviour from large deflections as de-scribed in Chapter 3.

The responses from both static and dynamic loads are compared. For the static cases twodifferent examples of boundary conditions (simply supported -VV and clamped -FO) are ana-lysed verifying the deflections and strains (~ stresses). Both a sandwich plate and a single-skin plate are examined, as the shear effect is pronounced in the first type of plate,whereas membrane effects are dominant in the latter.

��9HULILFDWLRQ�RI�0HWKRG��81

In the dynamic case, midpoint deflections, Z, are compared for simply supported plates ofboth sandwich and single skin. The dynamic solution procedure in Ansys is also based onNewmark’s method, where each time step determines the size of the load, T. Hence thesolution within each time step is a quasi-static solution including the inertia and dampingterms, which are omitted in the static analysis.

Two types of symmetric plates are used in the comparison. A single-skin plate of 1×1 P�

and a sandwich plate of 2×2 P� with isotropic material properties as listed below:

6LQJOH�VNLQ�SODWH

D� �E� �1.0 P, (IDFH� �20.0 *3D, νIDFH� �0.30, WIDFH�= 0.01 P

6DQGZLFK�SODWH

D� �E� �2.0 P, (IDFH� �20.4 *3D, νIDFH� �0.30, WIDFH��= 0.003 P(FRUH� �143 03D, *FRUH �55.0 03D, νFRUH� �0.30, WFRUH�= 0.06 P

��6WDWLF�5HVSRQVH

Deflection and strain responses for two types of composite plates with a constant lateralload T are compared in the following by application of the methods Ansys ~ FEM andRiber ~ FD.

Clamped Single-Skin Plate

Midpoint strains and deflections, �[�\�� D��E��

-0,5

0,0

0,5

1,0

1,5

2,0

0 10 20 30 40 50 60 70 80 90 100


ε[ [PP�P]

Z��[FP]

Deflection

Strain top

Strain bottom

AnsysRiber

AnsysRiber

Ansys

Riber

Figure 4.5: 0LGSRLQW�GHIOHFWLRQ��Z��DQG�LQ�SODQH�VWUDLQ��ε[��FODPSHG�


Deflection along the [-axis, \� �E��

-25

-20

-15

-10

-5

00 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

[�� [P]

Z�[PP]

Ansys

Riber

T� ��.3D

Figure 4.6: 'HIOHFWLRQ��Z��DORQJ�WKH�[�D[LV��FODPSHG�

Strains along the [-axis, \� �E��

-6

-4

-2

0

2

4

6

8

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0[ [P]

εx [PP�P]

Bottom

TopRiber

Ansys

RiberAnsys

T� ��.3D

Figure 4.7: ,Q�SODQH�VWUDLQ��ε[��WRS�DQG�ERWWRP�RI�SODWH�DORQJ�WKH�[�D[LV��FODPSHG�

All of the Figures 4.5-7 show good agreement of the two solutions. Small variations arefound in Fig. 4.7, where the strains at the top and in the bottom of the plate have differentslopes towards the edges. The maximum strain in the bottom of the plate situated at ¼ D isapproximately 3 % higher calculated by Ansys in comparison with Riber. The rest of theresponse lies within this margin for this particular case and the FD method seems accu-rate.


Simply Supported Single-Skin Plate

Midpoint deflections and strains, ([�\�� D��E��

-0,5

0,0

0,5

1,0

1,5

2,0

0 10 20 30 40 50 60 70


εx [PP�P]Z��[FP]

Deflection

Strain bottom

Strain top

RiberAnsys

RiberAnsys

RiberAnsys

Figure 4. 8: 0LGSRLQW�GHIOHFWLRQ��Z��DQG�LQ�SODQH�VWUDLQ��ε[��VLPSO\�VXSSRUWHG�


0,0

0,5

1,0

1,5

2,0

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

[��[P]

ε[� [PP�P]

BottomRiber

Ansys

TopRiber

Ansys

T� ��.3D

Figure 4.9: ,Q�SODQH�VWUDLQV��ε[��WRS�DQG�ERWWRP�RI�WKH�SODWH�DORQJ�WKH�[�D[LV��VLPSO\�VXSSRUWHG�


Deflection along the [-axis, \� �E��

-25

-20

-15

-10

-5

00 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

[�� [P]

Z�[PP]

Ansys

Riber

T� ��.3D

Figure 4.10: 'HIOHFWLRQV��Z��DORQJ�WKH�[�D[LV��VLPSO\�VXSSRUWHG�

The results calculated by the two methods (Figs. 4.8-10) in the simply supported caseshow an agreement similar to that of the clamped case. The maximum strain in the bottomof the plate located at [,�\ = D/2, E/8, calculated by Ansys, is approximately 4.5 % higherthan when calculated by Riber. However, the slope calculated by Ansys in this maximumelicits a discontinuity with no physical meaning. In calculations by Ansys there are bothmembrane and bending strains at the edges, whereas in calculations by Riber there areonly membrane strains. The FEM shell element calculates the strains at Gauss nodes,which do not necessarily lie at the edges. This may cause the differences in the results bythe two methods. The results by the FD method seem more accurate in this particular case.

Clamped Sandwich Plate

Figures 4.11-13 show the response of the sandwich plate with clamped boundary condi-tions. Shear strains in the core along the [�axis are included in the sandwich analysis (Fig.4.12). The two methods are in good agreement, except for the transverse core shear strainsat the edges and the in-plane strains in the lower face, also at the edges.


Midpoint strains and deflections, �[�\�� D��E��

-4

-2

0

2

4

6

8

10

12

14

0 100 200 300 400 500 600 700 800 900 1000


ε[��[PP�P]Z [FP]

Deflection Z

Strains, bottom

Strains, top w/Riber

w/Ansys

Riber

Ansys

RiberAnsys

Figure 4.11: 0LGSRLQW�GHIOHFWLRQ��Z��DQG�LQ�SODQH�VWUDLQ��ε[��FODPSHG�

Shear strains and deflections along the�[-axis, \� �E��

-200

-150

-100

-50

0

50

100

150

0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0

[��[P]

RiberAnsys

Deflection

Shear strains Riber

Ansys

T� ��.3D

τ[] [PP�P]Z [FP]

Figure 4.12: 6KHDU�VWUDLQ��τ[]��DQG�GHIOHFWLRQ��Z��DORQJ�WKH�[�D[LV��FODPSHG�



-20

-10

0

10

20

30

40

0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0

[ [P]

Bottom

Top

RiberAnsys

RiberAnsys

T� ��.3D

ε[��[PP�P]

Figure 4.13: ,Q�SODQH�VWUDLQ��ε[��WRS�DQG�ERWWRP�RI�SODWH�DORQJ�WKH�[�D[LV��FODPSHG�

The shear strains as calculated by Riber are approximately 6 % lower than the ones ob-tained by Ansys, whereas the in-plane strains in the bottom of the plate at the edges are 16% higher when calculated by Riber. As discussed for the single-skin plate the strains cal-culated by the FEM shell element may not be exactly at the edges, but at an interpolatednode within the element. Except for this inconsistency at the edges, the methods are ingood agreement for this particular case.

Simply Supported Sandwich Plate

Midpoint strain and deflections, �[�\�� D��E��

-2

0

2

4

6

8

10

12

14

0 100 200 300 400 500 600 700 800

Lateral load T�[.3D]

Deflection

RiberAnsys

Strain bottom

Strain top

RiberAnsys

RiberAnsys

ε[��[PP�P]Z [FP]

Figure 4.14: 0LGSRLQW�GHIOHFWLRQV��Z��DQG�LQ�SODQH�VWUDLQV��ε[��VLPSO\�VXSSRUWHG�


Shear strains along the [-axis, \� �E��

-120

-80

-40

0

40

0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0

[ [P]

Shear strains RiberAnsys

AnsysRiber

Deflection

T� ��.3D

τ[] [PP�P]Z [PP]

Figure 4.15: 6KHDU�VWUDLQV��τ[]��DQG�GHIOHFWLRQV��Z��DORQJ�WKH�[�D[LV��VLPSO\�VXSSRUWHG�

Strains along the x-axis, y = b/2

0

2

4

6

8

10

12

14

16

18

0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0

x [m]

bottom

top

Riber

Riber

Ansys

Ansys

T� ��.3D

ε[��[PP�P]

Figure 4.16: ,Q�SODQH�VWUDLQV��ε[��WRS�DQG�ERWWRP�RI�SODWH�DORQJ�WKH�[�D[LV��VLPSO\�VXSSRUWHG�

The last case in the static analysis shows a simply supported sandwich plate (Figs. 4.14-16). Also here, the two methods are in good agreement, but still with some inconsistencynear the edges. The maximum strain occurs in the bottom face at approximately 1/8 of thelength from the edges and not in the centre of the plate, which is an important observation.


��'\QDPLF�5HVSRQVH

In order to verify the dynamic response, two simply supported sandwich and single skinplates are analysed according to Riber with respect to midpoint lateral deflection, Z, andthe results compared with results obtained by Ansys with similar plates. The plate proper-ties are identical to the ones for the static case (Section 4.4.1). Two different types ofloads are applied to the plates and the results are illustrated in Figs. 4.17-20 in the fol-lowing.

Simply Supported Sandwich Plate

Midpoint deflection Z, sandwich plate

-4

0

4

8

12

16

0 0,005 0,010 0,015 0,020 0,025 0,030 0,035 0,040 0,045 0,050

Time [VHF]

Riber

Ansys

load

Z [FP]T [105 3D]

Figure 4.17: '\QDPLF�UHVSRQVH��Z��RI�VLPSO\�VXSSRUWHG�VDQGZLFK�SODWH.


Midpoint deflection, Z, sandwich plate

-15

-10

-5

0

5

10

15

0 0,005 0,01 0,015 0,02 0,025 0,03 0,035 0,04 0,045 0,05

Time [VHF]

Riber

Ansys

load

Z [PP�P]T [105 3D]

Figure 4.18: '\QDPLF�UHVSRQVH��Z��RI�VLPSO\�VXSSRUWHG�VDQGZLFK�SODWH�

Simply Supported Single-Skin Plate

Midpoint deflection, Z, single skin

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

2,5

0 0,005 0,010 0,015 0,020 0,025 0,030

Time [VHF]

load

Riber

Ansys

Z [FP]T [105 3D]

Figure 4.19: '\QDPLF�UHVSRQVH��Z��RI�VLPSO\�VXSSRUWHG�VLQJOH�VNLQ�SODWH�


Midpoint deflection, Z, single skin

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

2,5

0 0,005 0,01 0,015 0,02 0,025 0,03

Time [VHF]

load

RiberAnsys

Z [FP]T [105 3D]

Figure 4.20: '\QDPLF�UHVSRQVH��Z��RI�VLPSO\�VXSSRUWHG�VLQJOH�VNLQ�SODWH�

The results obtained with the two methods are in good agreement for all the above loadand boundary cases. Nevertheless, for the sandwich plate the results from Ansys show aslightly higher vibration frequency.

��6XPPDU\

A numerical formulation of the non-linear plate theory for orthotropic sandwich and singleskin plates is deduced. The formulation is based on the central finite difference methodand Newmark’s constant acceleration method regarding space and time solutions, respec-tively.

The matrix system of the governing equations is presented in detail in order to understandthe numerical solution procedure. The method is programmed in FORTRAN code and im-plemented into a design software program 3DQHO (Chapter 7). The results of static as wellas dynamic response are verified against the commercial FEM-based software programAnsys.

The method yields results in perfect agreement with results obtained by Ansys, and fur-thermore, it is approximately 50 times faster in CPU-time than Ansys. This factor is foundby performing dynamic calculations on a sandwich plate, where the number of nodes isminimised for both methods, until the results lie within 3 % of the converged results, withrespect to number of nodes.


��%LEOLRJUDSK\

[1] ANSYS. FEM-Software Program. ANSYS, Inc.,Houston, PA 15342-1300, Texas,USA, 1994.

[2] IMSL Math/Library. FORTRAN Subroutines for Mathematical Applications. IMSL,

Inc., Houston, Texas, USA, 1991. [3] Bathe K.J. and Wilson E.L. Numerical Methods in Finite Element Analysis. Pren-

tice-Hall, Inc., Englewood Cliffs, New Jersey, 1976. [4] Gorji M. On Large Deflection of Symmetric Composite Plates under Static Loading.

3URF��,QVW��0HFK��(QJUV� Vol. 200 (C1), June 1986. [5] Newmark N.M. A Method of Computation for Structural Dynamics. -RXUQDO� RI� WKH

(QJLQHHU�0HFKDQLFV�GLYLVLRQ. Vol. 85 (EM 3), July 1959. [6] Terndrup Pedersen P. and Juncher Jensen J. Styrkeberegning af maritime konstruk-

tioner. Department of Naval Architecture and Offshore Engineering, Technical Uni-versity of Denmark, 1982

[7] S.P. Timoshenko and S. Woinowsky-Krieger. Theory of Plates and Shells. McGraw-

Hill, Inc. Singapore, 1959. [8] D. Zenkert. An Introduction to Sandwich Construction. Chameleon Press LTD, Lon-

don, 1995.


93

&KDSWHU��

)DLOXUH�RI�&RPSRVLWH�3ODWHV

��,QWURGXFWLRQ

The previous chapters focused on the non-linear response of sandwich and single skin FRPplates. The work presented in this chapter now addresses the broader objective: applica-tion of the acquired knowledge from the response analysis methodology to the design ofstructural safe systems. This task is illustrated in the design loop below (Figure 5.1).

Failure criteria&

Failure modes

'HVLJQ�GULYHU6WUXFWXUDO

FRQILJXUDWLRQ/RDGV

5HVSRQVH 3HUIRUPDQFH

5H�GHVLJQ

1

Figure 5.1: 'HVLJQ�ORRS�IRU�FRPSRVLWH�SODWHV�

As the structural quality of a composite sandwich or single skin may differ quite a lotcompared to similar constructions made of homogeneous materials, such as metals, the de-sign criteria for composite structures are often very conservative. Thus, because of insuf-ficient understanding of failure mechanisms in composites and because of the uncertainty

94 �&KDSWHU��)DLOXUH�RI�&RPSRVLWH�3ODWHV

in material properties within the composite, it is a difficult task to make safe and structur-ally optimal design.

The following sections discuss various failure criteria for fibre-reinforced plastics andcore materials, and fracture mechanism for sandwich and single-skin plates with their re-sulting failure modes. The result of this discussion is formulated as a progressive damagemodel (Chapter 6) and implemented in the design program 3DQHO (Chapter 7).

��)DLOXUH�0RGHV

Sandwich and single-skin plates can fail in several ways, and the failure modes are oftenquite different to those of isotropic metal plates such as steel and aluminium. Where steeland aluminium have a reserve when reaching the upper limit of the elastic behaviour(yielding), this stage of the loading is most critical for composite plates, as they are brittleand show little tendency to yield. Delamination between two plies in the faces anddebonding between the core and the face are both examples of complicated failure mecha-nisms, which are difficult to predict.

(a) (b) (c ) (d) (e) (f) (g) (h)

Figure 5.2: )DLOXUH�PRGHV�IRU�FRPSRVLWH�VDQGZLFK�SODWHV��=HQNHUW�>��@��7KH�IDLOXUH�PRGHV�DUHGHQRWHG��D��IDFH�\LHOGLQJ�IUDFWXUH��E��FRUH�VKHDU�IDLOXUH��F��IDFH�ZULQNOLQJ��G��IDFH�FRUH�VHSDUD�WLRQ��L�H��SHHO��H��JHQHUDO�EXFNOLQJ��I��VKHDU�FULPSLQJ��J��IDFH�GLPSOLQJ�DQG�K��ORFDO�LQGHQWDWLRQ�

��)DLOXUH�0RGHV��95

��)DFH�)UDFWXUH

Fracture of the laminate or the faces in either tension or compression can be predicted byuse of failure criteria based upon stress/strength ratios in fibres and matrix (Section 5.3).In addition to this we also deal with failure in the laminate initiated from more global fail-ure phenomena such as: Face wrinkling, global buckling, shear crimping, face dimpling.These failure modes must also be included in a failure damages model, in order to find thecorrect mode of the final failure of the structure.

Face dimpling or intercellular buckling, however, is only a problem in sandwich structureswith honeycomb or corrugated cores and is not included in the present damage model. It isdescribed in Zenkert [17] for both honeycomb and corrugated cores. Shear crimping is de-scribed later under core failure as it is failure due to a weak core.

��/RFDO�%XFNOLQJ

In the sandwich theory (Chapter 3) we assume infinite core stiffness in the out-of-plane di-rection i.e. the out-of-plane transverse strain, εz, is negligible. This assumption is based onseveral investigations (e.g. Allen [1] and Reissner [10]) and seems reasonable. However,for sandwich plates with thin faces and a weak core, we may have face wrinkling or localbuckling, illustrated in Fig. 5.3 of symmetrical or asymmetrical form.

&RXSOHG�V\PPHWULFDO

&RXSOHG�DV\PPHWULFDO

8QFRXSOHG

Figure 5.3: )DFH�ZULQNOLQJ�PRGHV�RI�VDQGZLFK�SODWH�VXEMHFWHG�WR�DQ�LQ�SODQH�ORDG�DQG�D�ODWHUDOEHQGLQJ�PRPHQW�

For bending of laterally loaded sandwich plates, wrinkling in the compressive face mayoccur (uncoupled wrinkling). For compressive in-plane loads combined with bending thecoupled wrinkling modes are prone to happen.

The critical face stress depends on the elastic stiffness of the core and the face. A detailedinvestigation of the face wrinkling phenomena is found in Allen [1], Hoff[7] and Zenkert[17] for both the uniaxial and the biaxial state of stress. A simple general formula for face


wrinkling based on experimental work and proposed by Hoff [7] gives good results. Theformula states

σ σ1 1 1 13

2 2 2 230 5 05IDFH IDFH FRUH FRUH IDFH IDFH FRUH FRUH( ( * ( ( *= ⋅ = ⋅. . and (5. 1)

For a face made of different plies and by use of the 4-stiffness notation used in Chapter 3,we get an average value for the critical face stress as

( ) ( ) ( ) ( )σ ν νDYHUDJH M

IDFH L MM

IDFH L

IDFH L

IDFH L MM

FRUH

FRUH

FRUH FRUH$

W4 4 4 L M−

−−

−−= ⋅ − ⋅ − =0 5 1 1 1 22 2

44 553. , , , (5. 2)

The wrinkling mode is determined by the thickness ratio Wc/Wf and the stiffness of the coreand face. The asymmetrical wrinkling mode is more likely to happen for small thicknessratios, where the deformations of the two faces are affected by each other by the displace-ment in the core, whereas for larger core thicknesses the displacements in the core aredamped out and will not affect the opposite face.

The wrinkling failure may happen as an indentation in the core, if the compressivestrength of the core is lower than the tensile strength of the core and the tensile strength ofthe bonding between face and core. Alternatively, it may happen as a tensile fracture in thecore if the compressive strength of the core is higher than the tensile strength of the coreor the bonding between face and core.

��*HQHUDO�%XFNOLQJ

If the core moduli (F and *F are high enough to suppress local buckling, the sandwichplate stability can be designed by classical theory of shell buckling. For single-skin platesthe global buckling is more prone to happen due to the lower plate stiffness. Global buck-ling occurs if the in-plane compressive load 3 > 3H, where 3H is the Euler buckling loadgiven for the specific structure.

Using the Euler load by assuming simply supported boundary conditions for a plate, weget a conservative estimate for the maximum in-plane load, solving the equilibrium equa-tion (Eq. 4.29) of the plate for an in-plane load 3. The sandwich plate equation yields

∇ − −

⋅ −

∇∇

=42

2

4

21 0' Z TZW

' Z

$ ZV

V

V

αβαβ

αβρ

∂∂

* * (5. 3)

where

��)DLOXUH�0RGHV��97

T T 1 Z

T 1Z[

1Z\

1Z\ [[ \ [\

∗ = + ∇

= + + +

2

2

2

2

2

2

2

αβ

∂∂

∂∂

∂∂ ∂

(5. 4)

The plate analysis in Chapter 4 concerns laterally loaded composite plates, whereas thiscase concerns a plate with an in-plane load force�and no lateral load, T. Thus, the in-planeinternal force, 1αβ, is now considered as the external force, whereas the lateral load, T, iszero. For an in-plane load 3[ = -E 1[, omitting the inertia forces, the above equations be-come

∇ + −∇∇

=4

4

2

2

21 0' Z3E

' Z

$ ZZZ

[ V

V

V

αβαβ

αβ

∂∂

(5. 5)

Inserting a solution of the form below

Z ZP [D

Q \E

= sin sinπ π

(5. 6)

which satisfies the simply supported boundary conditions, we get the limit of stability

( )

( )3E

DP

'PD

' 'PQDE

'QE

'PD

' 'PQDE

'QE

$PD

$QE

[ =

+ +

+

+

+ +

+

+

2 11

4

12 66

2

22

4

2

11

4

12 66

2

22

4

44

2

55

2

2

12

π

(5. 7)

In order to find the critical buckling load, we need the minimum value of 3[. Since theabove function has its minimum for Q = 1, we obtain

( )

( )3E

DE

' PED

' 'ED

'P

E'

PED

' 'PED

'

$PED

$

[ =

+ +

+

+

+ +

+

+

2 112

4

12 66

2

22 2

2 11

4

12 66

2

22

44

2

55

21

2

π

(5. 8)


The parameter P is found by solving the above equation for the minimum value of 3[. Bysolving with regard to the parameter P, we get the critical value for P = D�E. The criticalin-plane average face stress σ DYHUDJH

IDFH becomes:

( ) ( )( )

( )σ

π

DYHUDJH

IDFH [

IDFH IDFH IDFH IDFH

3

E W W W W

' ' ' '

E ' ' ' '

$ $

=+

=+

⋅+ + +

+

+ + ++

1 2 1 2

11 12 66 22

211 12 66 22

44 55

1 2

2 (5. 9)

In the case of isotropic material properties in a sandwich plate, we obtain the flexuralstiffness, ', of the faces and a shear stiffness, 6, of the core, hence

σπ

πDYHUDJH

IDFH

IDFH IDFHW W'

E'6

=+

⋅+

2

2

2

1 2 22 (5. 10)

For a single-skin plate with no shear deformation the critical buckling stress yields

σπ

DYHUDJH

ODQLQDWH

ODQLQDWHW'E

= ⋅2 2

2 (5. 11)

��/DPLQD�)DLOXUH�$QDO\VLV

We now focus on the failure mechanisms in composite laminae, which are the buildingblocks in the composite laminate forming a single-skin or a sandwich face. The mostcommon failure criteria are evaluated, in order to find the most appropriate criteria forlamina failure analysis.

Over the last three decades, the efforts have been continuous to develope failure criteriafor anisotropic composites. When FRP composites were introduced in structural design,the failure analysis of these structures was based on already known failure criteria devel-oped for isotropic metals, such as the von Mises and the Tresca criteria. Since then anumber of criteria have been presented in various textbooks such as Jones [8] and Vinson& Sierakowski [14].

Most experimental determinations of the strength of a lamina are based on uniaxial stressstates. However, the general practical problem involves at least a biaxial or even a triaxialstate of stress. Thus, a logical method for using uniaxial strength information in the analy-sis of multiaxial loading problems is required. A main problem of the design is the selec-tion of an appropriate failure criterion. For monolithic materials such as metals, it is suffi-

��/DPLQD�)DLOXUH�$QDO\VLV��99

cient to use one observable metric such as the ultimate tensile and compressive or shearstress to describe failure. For composites, however, the engineer must select a reasonablecriterion based on a number of observable stress metrics. Thus, one of the most difficultsubjects in the design of composite structures involves finding a suitable failure criterionfor the system.

All existing lamina failure criteria are phenomenological, where the macro-mechanicalfailure mechanisms are not described. The lamina strength is determined by evaluating theset of equations provided in each criterion, consisting of one or more combinations of theprincipal (material direction) stress or strain components, σ σ τ1 2 12, , or ε ε γ1 2 12, , .

The purpose of the lamina failure criterion is to determine the strength and the mode offailure of a unidirectional composite or lamina in a state of plane stress (σ3 = σ23 = σ32 =0). All lamina failure criteria require basic principal strength properties. Since most lami-nae can be considered as either transversely isotropic or orthotropic, we generally needonly five independent strength parameters to define the material system. These are definedin a composite system as follows (values in stress):

∗ ;: tensile strength of a unidirectional ply in the fibre direction (denoted �).There is no necking before failure. It is a sudden explosion in the case ofglass/epoxy composites, where matrix is removed from the fibre after fibre failure.For graphite/epoxy composites, the ply is often torn into parallel strips before theultimate failure.

∗ ;¶: compressive strength of a unidirectional ply. The failure is a shear-type fail-ure (45-degree) or a stability failure by a kink band formation. Compressivestrength is affected by both fibre and matrix properties and the interfacial strength.

∗ <: tensile strength of a unidirectional ply in the transverse direction (perpen-dicularly to fibres, denoted �). The failure is a cleavage type failure along the fi-bres, transverse to the applied uniaxial load. In terms of fracture, the failure iscaused by a crack opening mode.

∗ <¶: compressive strength of a unidirectional ply in the transverse direction(perpendicularly to fibres, denoted �). The failure is a shear type failure, along a45-degree plane to the applied uniaxial load.

∗ 6: shear strength of a unidirectional ply. The failure is a transverse crackingsimilar to the transverse tensile failure. In terms of crack propagation, it is deter-mined by a shear mode parallel to the axis of the fibres.

For strain based-analysis we have the equivalent ultimate strains, ;ε��;¶ε��<ε��<¶ε� and �6ε.


��3ULQFLSDO�6WUDLQV�DQG�6WUHVVHV

In the process of failure prediction, we need some measurable parameters from the loadresponse analysis of the composite structure. The principal strains in each constituent ofthe composite seem useful for this purpose, as they can be compared to ultimate strengthparameters for the various materials in the composite structure. From the lamina theory inChapter 3 we obtain the maximum global strains in each of the plies in the laminate (orfaces) and the strains in the core. These strains are transferred to the principal strains (inthe�� co-ordinate system) according to the material axes by means of the transformationmatrix 7 (Eq. 3.7). The relation between local and global stresses and strains are givenbelow as

[ ] [ ]

εεγγγ

εεγγγ

σσσσσ

σστττ

1

2

23

31

12

1

2

23

31

12

local global

k

local global

k

= and =

N

N

[

\

\]

[]

[\

N

N

[

\

\]

[]

[\

7 7 (5. 12)

where

[ ]7 N

N

= −− −

cos sin cos sin

sin cos cos sin

cos sin cos sin cos sin

2 2

2 2

2 2

2

2

θ θ θ θθ θ θ θ

θ θ θ θ θ θ (5. 13)

The principal stresses in each lamina and in the core are obtained from the stress-strainrelationship (Eq. 3.6) by use of the stiffness matrix 4N for each lamina, N, and the stiffnessmatrix, 4F, for the core. Assuming plane stress in the faces, that is σ σ σ3 4 5 0= = = , we

obtain the maximum stress by means of the local stiffness tensor, 4LM

N , for each lamina as

σ ε ε σ ε ε σ γ1 11 1 12 2 2 12 1 22 2 6 66 12N N N N N N N N N N N N N4 4 4 4 4= = =+ + , , (5. 14)

For the core with only transverse shear stresses and σ 3 0= , we get

σ γ σ γ4 44 23 5 55 31F F F F F F4 4= = , (5. 15)

The ultimate strength of various core materials, resins, fibres and FRP laminae is listed intextbooks and manuals, among others Smith [11] and Divinycell [4]. The strength proper-ties of a lamina depend mainly on the volume fraction of the fibres and the resin. In gen-eral, the limit of fracture for a lamina is different depending on the material being in com-


pression or in tension. The damage, which causes the principal stresses or strains to ex-ceed the maximum allowable failure criteria, eventually leads to failure of the plate.

��/DPLQD�)DLOXUH�0RGHV�DQG�&ULWHULD

Lamina failure can be divided into the following modes:

��)LEUH� EUHDNDJH� Principal stress, σ1, or principal strain, ε1, in fibre directiondominates lamina failure.

��7UDQVYHUVH�PDWUL[�FUDFNLQJ� Principal stress, σ2, or principal strain, ε2, in trans-verse fibre direction dominates lamina failure.

��6KHDU�PDWUL[� FUDFNLQJ� Principal shear stress, τ12, or principal shear strain, γ12,dominates lamina failure.

For analysis of the existing lamina failure criteria, they are categorised into four groups.Some of the most used criteria are listed within each group.

��/LPLW�FULWHULD (stress- or strain-dominated)

These criteria predict failure load and mode by comparing maximum principal stresses orstrains separately with the principal strength properties. There is no interaction betweenthe stresses (or strains), and the failure mode is defined by the criterion.

• Maximum stress

σ 1 1;

= , σ1 1

′=

;fibre failure (tension, compression)

σ 2 1<

= , σ 2 1

′=

<transverse matrix cracking (5. 16)

τ 21 16

= , shear matrix


• Maximum strain

ε

ε

1 1;

= , ε

ε

1 1′

=;

fibre failure (tension, compression)

ε

ε

2 1<

= , ε

ε

2 1′

=<

transverse matrix cracking (5. 17)

γ

ε

12 16

= , shear matrix

��,QWHUDFWLYH�FULWHULD

These criteria state that failure occurs under a combined set of stresses by application of asingle quadratic or higher-order polynomial, consisting of all stresses and the ultimatestrength. The mode of failure is determined by comparing the principal stress/strength ra-tios.

• Hill-Tsai

σ σ σ σ τ1

22

21 2 12

2

1; < ; ; 6

+

−

+

= (5. 18)

• Tsai-Wu

) ) ) ) ) )1 1 2 2 11 12

22 22

12 1 2 66 1222 1σ σ σ σ σ σ τ+ + + + + = (5. 19)

where

); ;

)< <

);;

)<<

)<< ;;

)6

1 2 11

22 12 66

1 1 1 1 1

1 1

2

1

= +′

= +′

=−

′

=−

′=

′ ′=

, , ,

, ,

��6HSDUDWH�PRGH�FULWHULD

These criteria separate matrix and fibre criteria, which can take the form of a limit crite-rion or an interactive criterion. The failure mode is defined by the actual failure criterion.Eqs. 5.20-21 express the Hashin-Rotem criterion and a modification of this extended by anextra parameter, µ, for the matrix failure proposed by Quinn & Sun [9]. They both exem-plify the concept of separating modes of failure while some stress interaction is maintain-ed.


• Hashin-Rotem

σ 1 1;

= , σ1 1

′=

;, fibre failure (tension, compression)

σ τ2

212

2

1< 6

+

= , matrix failure (tension) (5. 20)

σ τ2

212

2

1′

+

=

< 6, matrix failure (compression)

• Hashin-Rotem modified

σ 1 1;

= , σ1 1

′=

;, fibre failure (tension, compression)

σ τ2

212

2

1< 6

+

= , matrix failure (tension) (5. 21)

σ τ

µσ2

212

2

2

1′

+

−

=

< 6, matrix failure (compression)

where

0 4 0 5 0 0 0. .< < ≤ = ≥µ σ µ σ for and for 2 2

��([SHULPHQWDO

Various test procedures are described in textbooks for standards concerning testing,among others $670��',1��and ,62. However, a standardisation of material specifica-tions, processing and testing procedure is needed for, as discussed in Burke [3], sur-veying 1300 industries engaged in structural design and analysis of composite struc-tures.

The above failure criteria are graphically presented in Fig. 5.4 as a series of failure enve-lopes for combined stresses, in order to show the characteristics of the lamina failure cri-teria.


Failure envelopes for loads in principal axis

-4

-3

-2

-1

1

2

-1 0 1 2 σ1/X

σ2 /<

Max stress

Max strain

Hill-Tsai

Tsai-Wu

Hashin-Rotem (modified)

WHQVLRQ�WHQVLRQFRPSUHVVLRQ�WHQVLRQ

WHQVLRQ�FRPSUHVVLRQFRPSUHVVLRQ��FRPS�

Figure 5.4: �)DLOXUH�HQYHORSHV�RI�DQ�HSR[\�JODVV�ODPLQD�XQGHU�WKH�ELD[LDO�VWUHVV�σ��σ��

Figure 5.4 shows the six failure criteria for a biaxial state of stress. The envelopes arecomposed of failure stresses normalised by the tensile strength, ; and <, by use of a typi-cal unidirectional E-glass/epoxy lamina with the following material properties:

411 = 39.2 *3D� 422 = 8.39 *3D� 412 = 2.18 *3D466 = 4.14 *3D ; = 1062 03D ; ‘= -610 03D< = 31 03D <�‘=-118 03D 6 = 72 03D

In order to validate the failure criteria, comparison of the criteria with experimental testdata must be made. Quinn & Sun [9] recently performed a comprehensive evaluation andcomparison of different failure criteria with experimental tests by Swanson & Qian [12].

The test data by Swanson and Qian indicates that the shear strength becomes greater whenσ2 is compressive. Thus, a compressive fibre/matrix interfacial normal stress, which isproportional to σ2, will result in a greater fibre/matrix interfacial shear strength. To reflectthis behaviour Quinn and Sun came up with the modified Hashin-Rotem criterion in Eq.5.20. The failure envelope in the σ2-τ12 stress field with µ = 0.45 is in good agreementwith the experimental data of a T800/3900-2 carbon/epoxy (Fig. 5.5).


Hashin-Rotemµ = 0.45

Tsai-Wu

ExperimentalSwanson [11]

Hashin-Rotem,Hill-Tsai

Max stress,Max strain

-50

50

20 40 60 80 100

σ2 ultimate [03D]

τ12 ultimate [03D]

Figure 5.5: &RPSDULVRQ�RI�ODPLQD�IDLOXUH�FULWHULD�ZLWK�H[SHULPHQWDO�GDWD�XQGHU�σ��τ��ORDGLQJ�4XLQQ��6XQ�>�@�

The polynomial failure criterion (Eq. 5.19), commonly used in several classification so-cieties, may lead to misunderstanding in some cases. This is illustrated with two examplesin Figure 5.6. The first example shows that the biaxial tensile strength is increased, reduc-ing compression strength (micro -buckling or similar effects). The second example showshow to optimise the material used for submarine hulls, thus decreasing transverse-tensionstrength in the material, which apparently increases the biaxial compressive strength. Bothexamples show that, in some particular cases, the failure model lacks physical meaning.

?

σ

σ

x

y

σ

σ

x

y

?

� �

Reducing compressive strength

Increased tensile strength Reducing transverse tension

Increased compressive strength

Figure 5.6: /DFN�RI�SK\VLFDO�PHDQLQJ�LQ�WKH�SRO\QRPLDO�IDLOXUH�FULWHULD"


However, the polynomial criterion is mathematically more convenient than the other inter-active criterion (Eq. 5.18), as the failure envelope is described by a single equation.Hence, the polynomial criteria may not be useful for anisotropic materials with differentmaterial properties in tension and compression such as a fibre-matrix composite. For anisotropic homogeneous material, such as resins, polynomial failure criteria are appropriateand usually give good failure prediction as seen for the von Mises criterion applied tometals.

In order to validate further the criteria, each constituent of the composite should be stud-ied. Thus, a discussion of the failure behaviour of the fibres and the matrix is given in thefollowing section.

Fibre Failure

Assume that the stresses in the fibres are σ σ τ1 2 12I I I, and . Using a quadratic failure crite-

rion (Eq. 5.18), we get:

σ σ σ σ τ1

2

2

2

1 2 12

2

1I

I

I

I

I

I

I

I

I

I; < ; ; 6

+

−

+

= (5. 22)

where the strength properties, ;I , <I and 6I refer to the fibres. For unidirectional fibrecomposites with the fibre volume fraction &I , the fibre stresses are approximately relatedto the composite stresses as

σσ

σ σ τ τ11

2 2 12 12I

I

I I

&= = =, , (5. 23)

The longitudinal composite strength, ;, is related to the longitudinal fibre strength, ;I , as

;;&I

I

= (5. 24)

By use of Eqs. 5.23-24 we rewrite Eq. 5.22 in the form below:

σ σ σ σ τ1

2

2

2

1 2 12

2

1; < ; ; 6I I I

+

−

+

= (5. 25)

The values of σ2 and τ12 are limited by < and 6, respectively. Since < <I >> ≥ σ 2 ,

; <I >> > σ 2 and 6 6I >> > τ12 , the expression in Eq. 5.25 can be approximated to


σ σ12

11 1; ;

= = or (5. 26)

This explains the fibre failure criteria used in the limit criteria and the separate mode cri-teria.

Matrix Failure

Matrix failure is recognised as matrix cracking along the fibre direction. If cracking oc-curs in the matrix, all the three stress components, σ σ τ1 2

P P, and 12m , are included in the

quadratic failure criterion. Matrix cracking occurs along the fibre/matrix and failure isgoverned by the interfacial stresses, σ τ2

P and 12m . Thus, the matrix failure criterion can be

expressed as

σ τ σ τ2

2

12

2

2

2

12

2

1 1P P

< 6 < 6

+

=

+

= or (5. 27)

The above considerations of failure mechanisms in composite systems, i.e. fibre and ma-trix, indicate that the failure criteria for the lamina should be separated into a criterion forthe fibres and the matrix, respectively.

From the previous discussion it seems rational to assume that there are failure modes forboth fibre and matrix. As the stresses in the fibre and the matrix are different, their fail-ures are determined by different strains or stresses. Hence, a lamina failure criterion mustdistinguish between the stress states in the fibre and the matrix, which is the case for theseparate mode criteria consisting of the separate failure characteristics for fibre and matrixas illustrated in Figure 5.7.

Matrix failure criterion

Fibre failure criterion

Failure envelopefor "lamina"

σ

σ1

2

Figure 5. 7: )DLOXUH�HQYHORSH�IRU�D�ODPLQDWH�GHWHUPLQHG�E\�WKH�IDLOXUH�HQYHORSHV�RI�WKH�FRQVWLWX�HQWV�


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We now turn our attention from single-lamina analysis to failure analysis of multiplelaminae or a composite laminate. The objective of this analysis is to determine thestrength of the laminate by analysing the strength behaviour of each lamina in the laminateon the assumption that a plane state of stress exists for each lamina independent of its ori-entation and position within the laminate. The stresses of each lamina must be trans-formed to the material axes (Sec. 5.3.1) and compared to lamina failure criteria. In reality,the failure mechanisms in the laminate are more complicated than in a unidirectionallamina in plane stress. In addition to the failure mechanisms described in the previoussection, we also deal with delamination, which can be described as debonding of laminaeor as matrix failure between two laminae. Further stress concentrations may occur at thefree edges, producing new failure modes along the free edges.

In order to account for these complex failure mechanisms, the 2-D laminate analysis is notsufficient and a 3-D analysis is normally required. However, for single-skin and sandwichplates with simply supported or clamped boundaries there are no free edges, consequently,it is assumed that laminate failure does not initiate from the edges. Concerning the dela-mination we do not have sufficient stress information from the 2-D analysis. Nevertheless,by use of simple expressions for the peel stress, σ3, we are able to predict some types ofdelamination by applying an interactive failure criterion and assuming that the matrix is anisotropic material.

��/DPLQDWH�)DLOXUH�0RGHO

The damage effects to be included in the laminate strength analysis are the inter-laminafailures such as fibre breakage and transverse matrix cracking due to in-plane stress. Theloss of laminate stiffness during progressive lamina failure must be included in a laminatefailure model. The currently used methods of analysing laminate failure are summarisedbelow.

• 3O\�E\�3O\�'LVFRXQW�0HWKRG

The ply-by-ply discount method calculates the stresses and strains according to normallaminate theory, i.e. according to the methods outlined in Chapters 3-4. When a ply failsaccording to the applied lamina failure criterion, the ply still remains within the laminateas a volume, but it now carries load according to the failure criteria. A stiffness reductionmodel is used to reduce the stiffness of the laminate, due to that individual ply failure. Thelaminate is analysed again applying the reduced laminate stiffness properties. The next ply

��/DPLQDWH�)DLOXUH�$QDO\VLV��109

failure is found by use of the actual lamina failure criterion and the laminate stiffness isreduced accordingly. This cycle continues until ultimate laminate failure is reached.

The stiffness reduction model may reduce the whole lamina from carrying any load (411 =422 = 4�� 466 = 0), or it may use that the failure mechanism of the lamina is known. Forexample, if the matrix failure in a particular ply is known to occur, the transverse proper-ties of that ply can be omitted (422 = 466 = 0). Alternatively, if the fibre failure occurswithin a given ply, the ply can be treated for subsequent analysis as having zero stiffness(411 = 0).

Instead of a complete reduction of the lamina stiffness moduli, it might be more realisticto apply the stiffness reduction locally, since the stresses in a failure location are redistrib-uted through nearby intact fibre/resin material. Hence, the stiffness reduction takes placein the actual failure location only and the lamina stiffness is reduced accordingly by recal-culating the lamina stiffness moduli.

• 'LUHFW�/DPLQDWH�0HWKRG

In this method the laminate is analysed as a whole by use of effective laminate strengthvalues. Thus, lamina failure criteria become laminate failure criteria. This method requiresthe appropriate strength values for each laminate to be analysed. A variation of thismethod is used in fibre-dominated laminates, where the stiffness reduction due to progres-sive matrix failure is insignificant. Laminate failure coincides with the fibre failure of theload-carrying ply. In this analysis, a lamina failure criterion for the fibres is chosen andthe failure load is determined from the fibre failure in the load-carrying ply.

The Hart-Smith strain-based maximum shear stress criterion, [5] and [6], is a direct lami-nate method. He developed a graphical method based on the Tresca yield criterion to pre-dict failure in fibre-dominated composite laminates with a 45o-truncation at the shear-fail-ure lines. However, Quinn & Sun [9] state that the Tresca yield criterion for isotropic ma-terials cannot be directly adopted for anisotropic materials without further modificationsand point out that the failure strain envelope proposed by Hart-Smith can be obtained fromextension of Coulomb-Mohr’s criterion for brittle fracture to orthotropic materials.

��&RUH�)DLOXUH�$QDO\VLV

Usually the core material in a sandwich consists of foams or structural honeycomb madeof light metals. For sandwich in marine structures, the core material is usually made ofrigid close-celled PVC foams. They are considered almost linear elastic-ideal plastic incompression and brittle in tension according to Fig. 5.8 taken from Branner [2].


For composite plates in fast-moving hulls, core shear failure due to weak cores close tothe bulkheads or face delamination from the core is among the commonest failure modes.The latter caused by fracture in the core close to the face bonding material. In order toavoid these types of failure, heavier core materials, which are exposed to high lateral loads(slamming), are used in the lower part of the hull.

compression

tension

ρ = 100 kg/m3

GGW

Vε

= × − −4 5 10 4 1.

Strain, ε

Stress, σ [MPa]

1.0

0.2

0.2 0.4 0.6

Figure 5.8: (ODVWLF�EHKDYLRXU�RI��D�SRO\XUHWKDQH�IRDP�LQ�XQLD[LDO�ORDGLQJ��7KH�GHIRUPDWLRQ�LVVLPLODU�WR�39&�IRDPV��%UDQQHU�>�@�

��&RUH�6KHDU�)DLOXUH

The core material is mainly subjected to shear and carries basically the entire transverseforce. The interactive criteria are commonly used for failure prediction in isotropic linear-elastic metals and seem reasonable to use for failure prediction in PVC foams. The Tsai-Wu and the Hill-Tsai criteria are among the most commonly used criteria in analyses ofcores in composite sandwich structures. Applying the Hill-Tsai criterion to the core, weget

σ σ σ σ τ τ τ1

2

2

2

1 2 12

2

23

2

13

2

1F

F

F

F

F

F

F

F

F

F

F

F

F

F; < ; ; 6 6 6

+

−

+

+

+

= (5. 28)

��&RUH�)DLOXUH�$QDO\VLV��111

where the material strength is for the core only (subscript F). For the case of sandwichplate analysis, where the shear is taken solely by the core and the in-plane stresses by thefaces, the expression reduces to

τ τ23

2

13

2

1F

F

F

F6 6

+

= (5. 29)

��'HERQGLQJ�RI�&RUH�DQG�)DFH

Debonding is failure between the face and the core. It arises from peel stress, particularlyat fixed or loaded boundaries where the distribution of the transverse shear changes sud-denly. The resin or adhesive, which bonds the two materials together, fails and debondingappears. For many sandwich panels made of foam core and FRP skins, failure occurs inthe foam core rather than in the skins or the adhesive layer. The ultimate tensile strain ofthe adhesives, which is most often the matrix used in the skins, is 2-5 times larger than theultimate tensile strain of the laminate or the core. Thus, the debonding problems in struc-tural sandwich are in general due to low core strength properties (low density of the core)or poor manufacturing.

Local bending moments in the faces are sometimes introduced from local loads. Concern-ing failure prediction, it is essential to be able to analyse local bending effects in order todetermine debonding. A detailed investigation of local bending effects in sandwich plateswith orthotropic face layers subjected to localised loads is presented in Thomsen [13] andin Yoshii [15] -[16]. The analysis deals with the peeling stresses initiated by external localloads or line loads at the plate, which may result in face/core debonding.

Yoshii provides a semi-empirical formula for the peel stress σSHHO , which states

σ σ ξ3 2= = ⋅SHHO 4 / (5. 30)

where 4 is the maximum shear force per unit width. For a sandwich with flexural stiffness' and shear stiffness 6, the parameter ξ is defined as

ξ = 6 '2 (5. 31)

For some adhesives the ultimate bonding strength of the adhesive ;DGKHVLYH (often theresin) is given by the manufacturer, however, accurate values of ultimate bonding strengthare still difficult to obtain in the literature. Using the von Mises criterion for the adhesive,we obtain the failure limit expression for debonding as


( ) ( ) ( ) ( )σ σ σ σ σ σ τ τ τ1 2

2

1 3

2

3 2

2

122

232

132 26 2− + − + − + + + = ;

DGK (5. 32)

��6KHDU�&ULPSLQJ

Shear crimping occurs when the stress in the face exceeds the critical shear buckling load.The critical buckling stress for the plate (Eq. 5.10) can be divided into a shear bucklingstress and a flexural buckling stress as shown below.

1

2

1 12

2σ π σ σEXFN

I I

EXFN

EHQGLQJ

EXFN

VKHDU

W E

'

W

6= + = + (5. 33)

Hence, for a sandwich with isotropic core the critical shear buckling stress in the faces be-comes

σ σDYHUDJH

IDFH L

EXFN

VKHDU

IDFH IDFH

6W W

− = =+2

1 2

(5. 34)

��&RUH�,QGHQWDWLRQ

For sandwich plates with thin faces and a weak core we may get core indentation fromconcentrated loads. This can be avoided by applying the load over a sufficient large area$LQG, determined by the compressive stress of the core and the flexural stiffness of theface. Denoting the compressive strength of the core: ;

F

’ , we get the following conserva-tive criterion for the indentation:

3$

;

]

LQG

F

’ = 1 (5. 35)

��6XPPDU\

Failure modes and failure criteria for composite sandwich and single-skin plates have beendiscussed. General rules predicting the limit stress of the different failure modes are out-lined. The most commonly used failure criteria for fibre-reinforced composites are com-pared with regard to lamina and laminate level. The overall trend of failure prediction bythe different failure criteria discussed in this section is that the separate mode criteria,


which take into account the different failure modes of fibre and matrix, seem to outper-form the limit criteria and the interactive criteria. This is also confirmed by micro-me-chanical considerations, which indicate that fibre and failure are governed by differentfailure criteria.

��%LEOLRJUDSK\

[1] Allen H.G. Analysis and Design of Structural Sandwich Panels. Pergamon Press, Ox-ford, UK, 1969.

[2] Branner K. Capacity and Lifetime of Foam Core Sandwich Structures. Department of

Ocean Engineering, Technical University of Denmark, DK-2800 Lyngby, Denmark,1995.

[3] Burk R.C. Standard Failure Criteria Needed for Advanced Composites. $VWURQDXWLFV

DQG�$HURQDXWLFV. Vol. 21, pp. 58-62, 1983. [4] Divinicell. H-grade, High Performance Core Material for Sandwich Constructions.

Box 21 S-312 22, Laholm, Sweden, 1995. [5] Hart-Smith L.J. Predicting the Strength of Fibrous Composites by an Orthotropic

Generalisation of the Maximum-Shear-Stress (Tresca Criterion). 'RXJODV� $LUFUDIW&RPSDQ\��0F'RQQHOO�'RXJODV�&RUSRUDWLRQ, 1991.

[6] Hart-Smith L.J. A Scientific Approach to Composite Laminate Strength Prediction.

&RPSRVLWH�0DWHULDOV��7HVWLQJ�DQG�'HVLJQ��$670�673��, Glenn C. Grimes, Ed.,American Society for Testing and Materials, Philadelphia. Vol. 10, pp. 142 -169,1992.

[7] Hoff N.J. and Mautner S.E. Buckling of Sandwich Type Panels. -RXUQDO�RI�WKH�$HUR�

QDXWLFDO�6FLHQFH. Vol 12 (3), pp. 285-297, 1945. [8] Jones R.M. Mechanics of Composite Materials. Hemisphere Publishing Corp., USA,

1975. [9] Quinn B.J. and Sun C.T. A Critical Evaluation of Failure Analysis Methods for

Composite Laminates. 3URFHHGLQJV� RI� WKH� ��WK� 'R'�1$6$�)$$� &RQIHUHQFH� RQ� )L�EURXV�&RPSRVLWHV� LQ� 6WUXFWXUDO� 'HVLJQ, Hilton Head Island, South Carolina, USA,1993.


[10] Reissner E. Finite Deflections of Sandwich Plates. -RXUQDO�RI��$HURQDXWLFDO�6FLHQFH.Vol. 15 (7), pp. 435-440, 1948.

[11] Smith C.S. Design of Marine Structures in Composite Materials. Elsevier Science

Publishers Ltd. Crown House, Linton Road, Barking, Essex IG11 8JU, UK, 1990. [12] Swanson S.R. and Qian Y. Multi-axial Characterisation of T800/3900-2 Car-

bon/Epoxy Composites. &RPSRVLWHV�6FLHQFH�DQG�7HFKQRORJ\. Vol. 43, pp. 197-203,1992.

[13] Thomsen O.T. Theoretical and Experimental Investigation of Local Bending Effects

in Sandwich Plates. &RPSRVLWH�6WUXFWXUHV. Vol. 30, pp. 85-101, 1995. [14] Vinson J.R. and Sierasowski R.L. The Behavior of Structures Composed of Compos-

ite Materials. Martinus Nijhoff Publishers, Dordrecht, 1986. [15] Yoshii A. Optimum Design of Advanced Sandwich Composite Using Foam Core.

$GYDQFHG�&RPSRVLWH�0DWHULDOV. Vol. 2 (4), pp. 289-305, 1992. [16] Yoshii A. and Chikugo R. Peel Stress at Face-to-Core Interface in Sandwich. 3UR�

FHHGLQJV�RI�&RQIHUHQFH�RQ�0DWHULDOV�DQG�0HFKDQLFV��-60(�&RQIHUHQFH�1R��.Japan, Nov. 1991.

[17] Zenkert D. An Introduction to Sandwich Construction. Chameleon Press LTD, Lon-

don, 1995.

115

&KDSWHU��

Progressive Damage Analysis

��,QWURGXFWLRQ

From the discussion in the previous chapter of failure and failure modes a progressivedamage model (Fig. 6.1) is proposed and implemented in the program 3DQHO (Chapter 7).This damage model is demonstrated and tested against experimental data for orthotropicFRP single-skin plates, Shenoi et al. [5]. A failure analysis example is also given for asandwich plate until complete failure. Based on the failure history from the calculationsan improvement of the design is made. Furthermore, linear and non-linear response andfailure analyses are compared.

The non-linear plate analysis begins with the plate properties, the initial boundary con-ditions and the first step of the load history. The load is applied progressively accordingto the time-load history. Damage occurs in a lamina, core or the interface of thecore/face of the plate, when predicted by the appropriate failure criteria applied to theparticular constituent of the plate. Damage below the limit of complete failure of theplate will imply a change in the internal stress distribution of the laminate/core accord-ing to the stiffness reduction.

The loading continues until complete failure is obtained and it is followed by detectionof the failure mode, location and load. The failure location and type in each lamina andthe core are registered and plotted for better understanding. In the case of structural im-provement the failure types in each constituent of the plate are used to tailor the mate-rial (fibres) in each lamina in a more suitable way.

116� &KDSWHU��3URJUHVVLYH�'DPDJH�$QDO\VLV

Plate scantlingsmaterial propertiesmaterial strength

Analysis

Strains in each lamina

Strains and stresses in matrix and fibres in each lamina

Reduction instiffness moduli of

fibre, matrix and core

Load

Failure modesFailure criteria'DPDJH�"

No!

Yes!

Failure

Failure mode and location

Add load

Yes!

No!

Failure ofplate

3URJUHVVLYH�'DPDJH�0RGHO

Figure 6.1: 3URJUHVVLYH�GDPDJH�PRGHO�IRU�GHVLJQ�RI�VDQGZLFK�DQG�VLQJOH�VNLQ�SODWHV�

��3ODWH�6WLIIQHVV�5HGXFWLRQ�0RGHO

Based on the discussion of failure in laminates, core and the interface between face andcore (Chapter 5), the following stiffness reduction model is proposed for the entire com-posite plate (faces/core/adhesive).

• /DPLQDWH

a) fibre failure (tension or compression) in the ply N located in ([1�\1):

σ σ1 1

1 1N N

; ;≥

′≥, (6. 1)

��3ODWH�6WLIIQHVV�5HGXFWLRQ�0RGHO��117

resulting in a stiffness reduction of the ply N in the point ([1�\1):

( )4 [ \N

11 1 1 0, = (6. 2)

b) matrix failure in the ply N located in ([1�\1):

σ τ

σ τµσ

µσ

2

2

12

2

2

2

12

2

2

2

1

1

02

N N

N N

N

N

< 6

< 6

6

+

≥

+

−

≥

∈

tension

compression

,

(6. 3)

resulting a stiffness reduction of the ply N in the point ([1�\1):

( ) ( )4 [ \ 4 [ \N N

22 1 1 66 1 1 0, ,= = (6. 4)

• &RUH

c) core shear failure in the []� or \]�plane located in ([1�\1):

τ τ13

2

23

2

1F

F

F

F6 6

+

≥ (6. 5)

resulting in the following stiffness reduction of the core:

( )( )

4 [ \

4 [ \

F F

F F

44 1 1 13

55 1 1 13

0

0

,

,

= >

= ≥

,

,

23c

23c

τ τ

τ τ (6. 6)

• 'HERQGLQJ�IDFH�FRUH

d) Interfacial failure between core and face:

( ) ( ) ( )( )

σ σ σ σ σ σ

τ τ τDGK DGK DGK DGK DGK DGK

DGK DGK DGK DGK;

1 2

2

1 3

2

3 2

2

122

232

132 26 2

− + − + −

+ + + = (6. 7)


The flexural plate stiffness, 'LM

VDQ , (Eq. 3.15), is gradually reduced to the

minimum plate bending stiffness, 'LM

UHG . At this stage there is no sandwich

effect, i.e. no contact between face and core (the notation refers to Fig. 3.1and Eqs. 3.14-15):

'W

$W

$ L�M � �

'W W

G'

LM

UHG I

LM

IDFH I

LM

IDFH

LM

UHG IDFH IDFH

LM

VDQ

=

+

=

≈+

12

1 22

2

1 2

12 121 2 6

1

6

(6. 8)

Alternatively, Zenkert [7] presented a method, where the panel is divided into an intactpart and a debonded part. The additional plate deformation is determined by estimatingthe released energy from the debonded part.

Delamination or damage in the interface between two plies is a complex failure mecha-nism and difficult to predict. Comprehensive studies are presented in Point et al. [4] andHitchings et al. [3], where FEM models are proposed. For very detailed failure analysisthese types of FEM models are appropriate. However, in order to follow the philosophyof the present design tool, that is simple modelling and fast analysis of the structure,the present damage model is not capable of predicting this type of failures. A reasonabledelamination criterion could be when matrix between two plies fails over a certain area.The matrix failure can be predicted by a simple interactive failure criterion such as thevon Mises criterion.

In order to predict final failure and failure modes of the composite panel, the abovestiffness reduction model must be supplemented by analysis of the possible failuremodes. The progressive stiffness reduction model and the failure mode detection formthe progressive damage model.

��&RPSDULVRQ�RI�'DPDJH�0RGHO�DQG�([SHULPHQWV

Little experimental work has been done with regard to large deflection of compositeplates. The most accurate large-deflection tests have been made by Bau, Kildegaard, andSvendsen [2], who carried out experiments with clamped sandwich plates. These resultsare used to validate the non-linear numerical response analysis presented in Chapter 3.However, the experiments do not include the ultimate failure of the sandwich plate. Re-cently, Shenoi, Allen & Moy [5] performed large-deflection tests with FRP single-skinplates including ultimate failure of the plates. Results from these experiments are com-pared to those of the progressive damage model. Midpoint deflections of FRP single-

��&RPSDULVRQ�RI�'DPDJH�0RGHO�DQG�([SHULPHQWV��119

skin plates with different aspect ratios are shown for experiments [5] and for the nu-merically obtained results by 3DQHO (Fig. 6.2). Numerous tests are completed in Ref. [5]and the results used in this comparison are from the plates denoted 12, 13 and 14. Theyhave the aspect ratios 1, 1.5 and 2, respectively.

0

10

20

30

40

50

60

0 100 200 300 400 500 600 700

a.r. = 2

a.r. = 1.5

a.r. = 1

a.r. = 2

a.r. = 1.5

a.r. = 1

7HVWV

3DQHO

Lateral load T� [.3D]

Midpoint deflection�Z [PP]

Ultimate failure of single skin FRP plates

21.5

1 &ROODSVH

Figure 6.2: &RPSDULVRQ�RI�WKH�SURJUHVVLYH�GDPDJH�PRGHO�ZLWK�H[SHULPHQWDO�GDWD�IURP6KHQRL�HW�DO��>�@��6LQJOH�VNLQ�)53�SODWHV�ZLWK�GLIIHUHQW�DVSHFW�UDWLRV�

The material properties for the laminae, the ply orientation and the plate data for the ex-amined clamped single-skin plate are listed below:

• Ply orientation: [0/+45/90/-45/0]

• Ply properties, E-glass/polyester unidirectional:

4 *3D11 28 0= . 4 *3D22 592= . 4 *3D66 2 90= . 4 *3D12 156= . ; 03D= 260 ′ = −; 03D260

< 03D= 40 ′ = −< 03D120 6 03D= 75

• Plate scantlings:

Length D = 0.60 P��DVSHFW�UDWLRV: 1:1, 1:1.5, 1:2.Thickness Wface = 3.12 PP


Although a heavy steel plate was used to fix the edges in order to prevent in-planemovements in the experiments, this was not fully avoided and horizontal movements ofthe plate edges were observed and measured. This sliding movement loosens the mem-brane stresses, which results in a larger lateral deflection of the plate. In order to com-pensate for this sliding in the numerical response analysis, the measured in-plane edgemovements are applied successively during the load period. The maximum edge move-ments measure 1 PP at the midpoint of the edges for plate 12 in Shenoi et al. [5]. Thisvalue is applied over the entire plate edge in the numerical calculations (3DQHO) for thesquared plate. For the larger aspect ratios there is no such measurements and the edgemovements in the short direction are assumed to be the same, whereas the edge move-ments in the long direction are scaled inversely proportionally to the aspect ratio.

For the ultimate failure load of the three plates there is reasonable agreement betweenthe experimental data and the numerical calculations. Ultimate failure, predicted by3DQHO, occurs when the plate has no more load-carrying reserve left. This happens at aminimum value of the extensional, bending- and shear stiffnesses (Eqs. 3.12-12) of 15% of the initial stiffness values where the numerical calculations break down.

For aspect ratios 1, 1.5 and 2 the difference of the numerical calculations from the ex-periments is +3 %, -10% and -12, respectively. Thus, for the aspect ratio close to unity,we find the best agreement and more conservative failure prediction for larger aspect ra-tios in this specific example.

The actual failure mode for the plates in the tests with panel 12-14 in [5] was every-where observed as matrix cracking. This is also predicted by the damage model, wherethe matrix failure in each ply occurred in approximately 30 % of the entire matrix in theplate.

The numerical prediction of the load-deflection curves, which are influenced by the pro-gressive decrease in plate stiffness until the ultimate failure of the plates, lies within 5% of the experiments for the aspect ratio equal to one. For larger aspect ratios there isless good agreement, especially in the beginning of the load-deflection curves for all as-pect ratios, where the tests show a higher level of membrane stiffness. As the numericalmodel is verified against FEM codes and experiments (Chapters 3-4), this behaviour ofthe test plates is not fully understood by the author.

Although the validation of the damage model is based on few experiments and only forsingle-skin plates, the results obtained indicate that the damage model gives reasonableresults.

��)DLOXUH�6FHQDULR�([DPSOH��121

��)DLOXUH�6FHQDULR�([DPSOH

In order to illustrate the failure scenario and how to use the information for furtherstructural improvements of the plate, an example of failure analysis is given with aclamped symmetric sandwich plate (Fig. 6.3). The damage model is combined with anon-linear plate response analysis for prediction of the progressive damage in each con-stituent of the plate. The failure types in each ply, the core and the interface between thefaces and the core are visualised and examined to clarify the failure history of the entiresandwich plate. On the basis of the results from the failure scenario, the plate is struc-turally improved by changing the ply orientation.

E� ��P

;

<

=

D� ��P

6DQGZLFK�SODWHFODPSHG

Load T Unidirectional E-glass/epoxy

DivinycellH-100

Epoxyadhesive

WFRUH� ��PP

WIDFH�� PP��SOLHV��PP

WIDFH�� PP��SOLHV��PP

12

3

4

12

3

4

Figure 6.3: �7RSRORJ\�DQG�SODWH�VFDQWOLQJV�IRU�ODWHUDOO\�ORDGHG�FODPSHG�VDQGZLFK�SODWH�

The material properties and ultimate strength for the plies and the core are listed below.

• Ply orientation [0/90/0/90]s, E-glass/epoxy unidirectional:

411 = 39.2 *3D� 422 = 8.39 *3D 412 = 2.18 *3D466 = 4.14 *3D ; = 1062 03D ; ‘= -610 03D< = 31 03D <�‘=-118 03D 6 = 72 03D

• Core properties: H-100 divinycell:

6c� 1.4�03D 411 = 121 03D 444 = 455 = 40 03D

• Adhesive properties, epoxy: ;DGK� ��03D


This sandwich plate is analysed with 3DQHO up to the maximum lateral load, To = 208.3D, at complete failure of the plate. The damage in each of the plies and in the core isschematically shown in Figs. 6.4-6. Each of the plies suffers from fibre and matrix fail-ure, the latter failure type dominates. Core failure is located near the edges. Along theedges wrinkling is seen in the lower face due to high compressive stresses.

8SSHU�IDFH��ILEUH�DQG�PDWUL[�IDLOXUH

No failure

Matrix failure

Fibre failure

Ply 1, [0]

No failure

Matrix failure

Fibre failure

Ply 2, [90]

No failure

Matrix failure

Fibre failure

Ply 3, [0]

No failure

Matrix failure

Fibre failure

Ply 4, [90]

Figure 6.4: )DLOXUH�W\SHV�LQ�HDFK�SO\�LQ�WKH�XSSHU�IDFH�RI�WKH�ODWHUDOO\�ORDGHG�FODPSHGVDQGZLFK� SODWH� �)LJXUH� �� ZLWK� IDFH� SOLHV� DQG� FRUH� RULHQWDWHG� DV� >��@�FRUH�>��@� 0D[LPXP�ORDG�FDUU\LQJ�FDSDFLW\�RI�WKH�SODWH�LV�TR� ��.3D�


&RUH��\]��DQG�[]�VKHDU�IDLOXUH /RZHU�IDFH��ZULQNOLQJ

\]- shear failure

[]- shear failure

No failure

Core, isotropic

wrinkling y-dir.

wrinkling x-dir.

No failure

Lower face

Figure 6.5: &RUH� VKHDU� IDLOXUH� DQG� ZULQNOLQJ� LQ� ORZHU� IDFH� RI� WKH� ODWHUDOO\� ORDGHGFODPSHG� VDQGZLFK� SODWH� �)LJXUH� �� ZLWK� IDFH� SOLHV� DQG� FRUH� RULHQWDWHG� DV>��@�FRUH�>��@� 0D[LPXP�ORDG�FDUU\LQJ�FDSDFLW\�RI� WKH�SODWH�LV�TR� ��.3D�

/RZHU�IDFH��ILEUH�DQG�PDWUL[�IDLOXUH

No failure

Matrix failure

Fibre failure

Ply 1, [90]

No failure

Matrix failure

Fibre failure

Ply 2, [0]

No failure

Matrix failure

Fibre failure

Ply 3, [90]

No failure

Matrix failure

Fibre failure

Ply 4, [0]

Figure 6.6: )DLOXUH�W\SHV�LQ�HDFK�SO\�LQ�WKH�ORZHU�IDFH�RI�WKH�ODWHUDOO\�ORDGHG�FODPSHGVDQGZLFK� SODWH�� )LJXUH� �� ZLWK� IDFH� SOLHV� DQG� FRUH� RULHQWDWHG� DV� >��@�FRUH�>��@� 0D[LPXP�ORDG�FDUU\LQJ�FDSDFLW\�RI�WKH�SODWH�LV�TR� ��.3D�


Thus, to improve the load capacity of the plate, the matrix failure in the plies should bereduced and the strength of the fibres should be used more efficiently. By rotating themiddle plies (ply number 2 and 3) in each face the matrix cracking in the plies is re-strained. This increases the maximum load capacity to� To = 288 .3D, i.e. an improve-ment of 26 % of the ultimate load capacity. The matrix failure is limited by the fibres inthe direction ±45 degrees. This alteration gives a better structure, where the material isbetter balanced with respect to failure capacity.

Figs. 6.7-8 show the failure scenario of each ply. The new ply orientation results inmore fibre breakage and less matrix cracking in a comparison of Figs. 6.4-6 with Figs.6.7-8. The shear core failure and the wrinkling near the edges are almost the same forface configuration and consequently not shown.

8SSHU�IDFH��ILEUH�DQG�PDWUL[�IDLOXUH

Ply 1, [0]

No failure

Matrix failure

Fibre failure

No failure

Matrix failure

Fibre failure

Ply 2, [-45]

Matrix failure

Fibre failure

No failure

Ply 3, [+45] Ply 4, [90]

No failure

Matrix failure

Fibre failure

Figure 6.7: )DLOXUH�W\SHV�LQ�HDFK�SO\�LQ�WKH�XSSHU�IDFH�RI�WKH�ODWHUDOO\�ORDGHG�FODPSHGVDQGZLFK� SODWH� �)LJXUH� �� PRGLILHG� E\� URWDWLQJ� WKH� PLGGOH� SOLHV�±�� GHJUHHV�� )DFHSOLHV� DQG� FRUH� RULHQWDWHG� DV� >��@�FRUH�>��@� 0D[LPXP� ORDG�FDU�U\LQJ�FDSDFLW\�RI�WKH�SODWH�LV�TR� ��.3D�


/RZHU�IDFH��ILEUH�DQG�PDWUL[�IDLOXUH

No failure

Matrix failure

Fibre failure

Ply 1, [90]

No failure

Matrix failure

Fibre failure

Ply 2, [+45]

No failure

Matrix failure

Fibre failure

Ply 3, [-45]

No failure

Matrix failure

Fibre failure

Ply 4, [0]

Figure 6.8: )DLOXUH�W\SHV�LQ�HDFK�SO\�LQ�WKH�ORZHU�IDFH�RI�WKH�ODWHUDOO\�ORDGHG�FODPSHGVDQGZLFK� SODWH� �)LJXUH� ��PRGLILHG� E\� URWDWLQJ� WKH�PLGGOH� SOLHV�±�� GHJUHHV� 0D[L�PXP�ORDG�FDUU\LQJ�FDSDFLW\�RI�WKH�SODWH�LV�TR� ��.3D�

The load deflection curve (Fig. 6.9) shows that the [0,-45,+45,90] ply orientation, re-sults in a better tailored plate with respect to the ultimate strength than by use of[0,90,0,90] orientation resulting in a 35 % higher total load carrying capacity of the [0,-45,+45,90] plate at complete failure.


0

20

40

60

80

100

120

140

160

0 50 100 150 200 250 300

Lateral load T�[.3D]

Midpoint deflection Z [PP]

Ultimate load-carrying capacity of sandwich plate

[0/-45/+45/90]

[0/90/0/90])DFH�FRQILJXUDWLRQV

&ROODSVH

Figure 6.9: /RDG�GHIOHFWLRQ�RI�D�FODPSHG�V\PPHWULF� VDQGZLFK�SODWH�ZLWK�GLIIHUHQW�SO\RULHQWDWLRQ�LQ�WKH�IDFHV��>��@�DQG�>��@�

The failure scenario together with the subsequent change of the structure based on thefailure types in each ply, as illustrated above, is a simple way to improve the design.The study can be supplemented by more theoretical optimisation routines concerning thenumber of plies, the order of the lay-up, the fibre direction, the core weight etc. Morerefined optimisation tools have been investigated by numerous authors, among othersAbrate [1] and Yoshi [6].

��8OWLPDWH�6WUHQJWK��/LQHDU�DQG�1RQ�/LQHDU�$QDO\VLV

As failure prediction requires both a strength response analysis and a failure model itseems reasonable to study the difference in the failure prediction using linear and non-linear plate response analysis. The following example uses the sandwich plate fromSection 6.4. Figure 6.10 shows the midpoint deflections and the ultimate failure loadcalculated by the non-linear and the linear solution methods.

��8OWLPDWH�6WUHQJWK��/LQHDU�DQG�1RQ�OLQHDU�$QDO\VLV��127

0

20

40

60

80

100

120

140

160

0 50 100 150 200 250 300

Z�non-linZ�linσ linσ non-lin


Z��[PP]σ 107 [3D]

Ultimate failure load of sandwich plate, linear >< non-linear

&ROODSVH

Figure 6.10: 8OWLPDWH�IDLOXUH�ORDG�� ORDG�GHIOHFWLRQ�DQG� ORDG�VWUHVV�FXUYHV� IRU� WKH�FHQ�WUH�RI�WKH�VDQGZLFK�SODWH��)LJ��>��@��FDOFXODWHG�E\�XVH�RI�OLQHDU�DQG�QRQ�OLQHDU�PHWKRGV�FRPELQHG�ZLWK�WKH�SURJUHVVLYH�GDPDJH�PRGHO�

For this example the linear analysis predicts failure at a much lower load compared tothe non-linear analysis. The non-linear ultimate load is approximately 2.5 times higherthan the linear calculated failure load. The damage distribution (Fig. 6.11) shows amuch higher percentage of damage in the fibres for the linear case (almost 50 % morefibre breakage in the lower face than for the non-linear case). The linear midpoint stresslevel in the lower face increases rapidly at a load level above ~100 .3D, whereas thenon-linear stress level reaches this value much later, at approximately twice the loadlevel of the linear case. The progressive failure depends on the stress level in the plate.For the non-linear analysis, this is apparently lower compared to the linear analysis for agiven load, which is due to the reduced lateral deflection by application of the non-lin-ear response analysis.

Wrinkling is introduced in the middle of the upper face due to compressive stresses ac-cording to linear analysis. In the non-linear case the compressive stress level is de-creased by the membrane stresses resulting in no wrinkling.


0

10

20

30

40

50

1 2 1 2 1 2

ILEUHV PDWUL[ ZULQNOLQJ[��\ [��\

% damage of constituent Damage distribution for ultimate load

FRUH

Linear, T� ��.3D

Non-linear, T� ��.3D

1 Lower face2 Upper face

Figure 6.11: 'DPDJH� WR�HDFK�FRQVWLWXHQW� LQ� WKH� VDQGZLFK�SODWH�� OLQHDU�!��QRQ�OLQHDUFDOFXODWLRQ�PHWKRGV�

The example indicates that the failure prediction is highly dependent on whether a linearor non-linear analysis is used. If FRP plates are designed with ultimate failure as crite-rion, the non-linear response analysis is highly recommended.

��6XPPDU\

A progressive damage model is presented (and implemented as part of the responseanalysis program 3DQHO) based on the discussion in Chapter 5. The model takes into ac-count the different failure modes of the fibres and the matrix. Local damages are intro-duced as local stiffness reductions according to the actual failure type.

The failure model is used in connection with the non-linear response analysis and com-pared to experimental data from FRP single-skin plates, Ref. [5]. The model gives goodresults with the specific examples. A failure scenario of a sandwich plate is illustratedand the results of different failure types are used to improve the actual design by rotat-ing the fibre direction in some of the laminae.

Finally, prediction of failure and ultimate strength is discussed in the light of linear andnon-linear response analysis. The linear analysis gives much lower ultimate failureloads, in the example 2.5 times lower than in the non-linear analysis. Hence, accuratefailure prediction of composite plates requires non-linear response analyses.


��%LEOLRJUDSK\

[1] Abrate S. Optimal Design of Laminated Plates and Shells. &RPSRVLWH� 6WUXFWXUHV.Vol. 29, pp. 269-286, 1994.

[2] Bau-Madsen N.K., Svendsen K.H. and Kildegaard A. Large Deflections of Sand-

wich Plates - an Experimental Investigation. &RPSRVLWH� 6WUXFWXUHV. Vol. 23, pp.47-52, 1993.

[3] Hitchings D., Robinson P. and Javidrad F. A Finite Element Model for Delamina-

tion Propagation in Composites. &RPSXWHUV�DQG�6WUXFWXUHV. Vol. 60 (6), pp. 1093-1104, 1996.

[4] Point N. and Sacco E. A Delamination for Laminated Composites. ,QW�� -RXUQDO�RI

6ROLG�6WUXFWXUHV. Vol. 33 (4), pp. 483-509, 1996. [5] Shenoi R.A., Allen H.G. and Moy S.S.J. Strength and Stiffness of FRP Plates.

3URF��,QVW��&LYLO�(QJUV��6WUXFWXUDO��%XLOGLQJV. May 1996. [6] Yoshii A. Optimum Design of Advanced Sandwich Composite Using Foam Core.

$GYDQFHG�&RPSRVLWH�0DWHULDOV. Vol. 2 (4), pp. 289-305, 1992. [7] Zenkert D. Strength of Sandwich Beams with Interface Debondings. &RPSRVLWH

6WUXFWXUHV. Vol. 17, pp. 331-351, 1991.


131

&KDSWHU��

Analysis and Design with Panel

��,QWURGXFWLRQ

In order to make efficient and correct design and analyses of FRP HSLC hull panels, asuitable tool is needed. A number of commercial programs are capable of doing non-lin-ear response analysis. However, the pre-processing of modelling the structure and thefollowing calculations are often a time-consuming process in these programs. These twophases are drastically reduced in 3DQHO, which is a design tool for response and failureanalysis of composite sandwich and single-skin panels in HSLC hulls. It is based on thegeometrical non-linear sandwich theory (Chapter 3). The numerical formulation of thetheory is based on a combination of the central finite difference method and the New-mark method, (Chapter 4). The tool is provided with a progressive damage model(Chapter 6), which enables the user to predict failure types and failure loads.

The program is demonstrated by analyses of hull panels from different types of HSLC,comparing linear and non-linear methods for static and dynamic load cases. Further-more, a design example is given, where the panel weight is improved by making use ofthe advantage of doing non-linear response calculations.

��7KH�6WUXFWXUH�RI�3DQHO

3DQHO consists of a main program and 21 individual routines in addition to in- and out-data files. The structure of 3DQHO�and a short description of each subroutine are given inTable 7.1 and Fig. 7.1. The entire program is developed by the author except for subrou-tines nos. 20-21, which are borrowed from a collection of Fortran codes for mathemati-cal applications.

132� &KDSWHU��$QDO\VLV�DQG�'HVLJQ�ZLWK�3DQHO

Name Description No.

0DLQ The source program, calling all subroutines. 1

$QDO\� Analytical solution (Chapter 4, Solution 1). 2

$QDO\� Analytical solution (Chapter 4, Solution 2). 3

(LJHQ Calculates eigenfrequencies (Chapter 5). 4

)DLO� Predicts failure load, locations and modes. 5

,QSXW Reads the data from the input file generated for the problem. 6

/RDG Generates time-load history according to load type. 7

0DWUL[ Computes plate stiffness, initial and during failure. 8

0HPEUD Calculates resultant moments, in-plane and shear forces. 9

2XWSXW[ Provides relevant output data, Output 1-3. 10

5LJKW Right side of in-plane equations in Solve2, (Chapter 4). 11

6ROYH� Solves bending equations (Chapter 5). 12

6ROYH� Solves in-plane equations (Chapter 5). 13

6ROYH� Solves shear equations (Chapter 5.) 14

6WUHVV Calculates strains and stresses from known displacements. 15

7ULDO� In-plane initial displacements, u, v, at boundaries. 16

7ULDO� Progressive in-plane displacements, u, v, at boundaries. 17

8YZUDQ Generates displacements at boundaries. 18

:HLJKW Calculates weight and added mass of plate. 19

'/�/;* Computes the /8 factorisation of the coefficient matrix. 20

'/)6;* Solves the system of equations given the /8 factorisation. 21

,Q�GDWD File with the plate scantlings, material properties, loads etc. 22

RXW�GDWD File with the calculated response and failure data. 23

,QFOX Defines and includes variables for all subroutines. 24

Table 7.1: /LVW�RI�WKH�GLIIHUHQW�VXEURXWLQHV�DQG�GDWD�ILOHV�LQFOXGHG�LQ�3DQHO�

��7KH�6WUXFWXUH�RI�3DQHO��133

Main,Q�GDWD

2XW�GDWD

,QSXW

/RDG

0DWUL[

:HLJKW(LJHQ

7ULDO�2XWSXW�

6WUHVV

2XWSXW�

,QLWLDO�FDOFXODWLRQV��3DUW��

6ROYHU��3DUW�6ROYH�

6ROYH�

6ROYH�

,WHUDWLRQ

3RVWSURFHVVLQJ��3DUW��6WUHVV

0HPEUD

)DLO�

$QDO\�

$QDO\�

8YZUDQ

7ULDO�

2XWSXW�

0DWUL[

3$1(/

3URJUDP�3DQHO

Figure 7.1: )ORZ�GLDJUDP�RI�3DQHO�


��$QDO\VLV�RI�([LVWLQJ�'HVLJQ

For response and failure analysis of FRP panels with 3DQHO, the user must provide theprogram with the appropriate data such as material properties of each ply and the core,fibre directions, panel scantlings, boundary conditions, load conditions, number of grid-points etc. An example of such an in-data file is given in Appendix B.

The input of data is probably the most time-consuming part of making response analysiswith 3DQHO. The actual calculations take seconds or few minutes depending on the typeof analysis, i.e. linear/non-linear and static/dynamic. The output data can be selected ac-cording to the desired information level, which can be in the range from maximum lat-eral deflections to the deflections, all the stress components and the failure informationin each grid-point.

��5HVFXH�9HVVHO�/5%��

An analysis of a bottom panel of the rescue boat /5%�� (Fig. 7.2) is demonstrated. Thedesign load is found from the '19 rule Eq. 2.47 with an acceleration of 5 JR given bythe authorities (classified by %9). The plate scantlings and material properties are givenin Table 7.2. The boatyard Mathis in Denmark has built this rescue vessel and they areabout to finish the last order in a series of six.

Figure 7.2: 5HVFXH�YHVVHO�/5%��HSR[\�JODVV�VDQGZLFK�KXOO��0D[�VSHHG��NQRWV�

��$QDO\VLV�RI�([LVWLQJ�'HVLJQ��135

5HVFXH�%RDW�/5%��Faces: E-glass/aramid/epoxyInner 3.0 PP [aramid(0/90)/-45/45/0/90] (towards the core)Outer 3.9 PP [90/0/45/-45/aramid(0/90)/csm] (from the core)E-glass, (woven rowing)411 = 39.2 *3D� 422 = 8.39 *3D 412 = 2.18 *3D466 = 4.14 *3D ; = 450 03D ; ‘= -310 03D< = 31 03D <�‘= -118 03D 6 = 72 03DAramid, 90/0 (unidirectional)411 = 76.6 *3D� 422 = 5.55 *3D 412 = 1.89 *3D466 = 2.30 *3D ; = 1.00 *3D ; ‘= -235 03D< = 12 03D <�‘= -53 03D 6 = 34 03DCore 40 PPDivinycell H 250.6c� 4.5�03D 444 = 455 = 108 03DAdhesive (epoxy) ;DGK� ��03DDimensions D�� 1.4�P, E� �0.8�PDesign load TR� �65�.3D ('19 Eq. 2.47)Vertical acc. DG� 5 JR , given

Table 7.2: +XOO�SDQHO�GDWD�IRU�WKH�UHVFXH�ERDW�/5%��

The calculated responses and the applied loads are illustrated as functions of the time inFig. 7.3-5, where linear and non-linear calculations are plotted for the dynamic and thestatic cases, respectively. The linear and the non-linear static cases represent the exist-ing design rules and the non-linear analytical method 6ROXWLRQ� � (Chapter 3), respec-tively. In addition, results from non-linear dynamic calculations including the addedmass are plotted, which represent the most correct responses.

The added mass (Eq. 4.34) contributes significantly to the total mass of the plate and re-duces the frequency of the system and, thus, increases the dynamic response amplifica-tion factor.

In the static calculations, where the body forces are neglected in the equilibrium equa-tions, the load is applied successively over the entire plate. The slamming load is dis-tributed over the entire plate with an impact duration of 0.05 seconds and a peak valueof 0.01 seconds, which is the period of a typical slamming impact.

Hence, Figures 7.3-5 show the responses calculated by use of the following three meth-ods: the conventional linear static solution (GHVLJQ� UXOH), the non-linear static solutionsuggested as improvement of the existing rule (6ROXWLRQ� �) and the non-linear dynamicsolution including the added mass (3DQHO). Ultimate failure loads calculated by use ofnon-linear and linear methods are illustrated in Fig. 7.6, where the non-linear methodshows considerably higher ultimate loads.


-0,4

-0,2

0

0,2

0,4

0,6

0,8

1,0

0 0,02 0,04 0,06 0,08 0,10

Design rule

6ROXWLRQ��

3DQHO

Static load

Dynamic load

Z�E [%]T [��3D]

Relative midpoint deflection and load history

Time [seconds]

Figure 7.3: 0LGSRLQW�GHIOHFWLRQ�RI�D�KXOO�SDQHO�LQ�WKH�/5%��UHVFXH�ERDW�FDOFXODWHG�E\XVH�RI�GLIIHUHQW�PHWKRGV�

-100

-50

0

50

100

150

200

0 0,02 0,04 0,06 0,08 0,10

design rule

Solution 2

Panel

Time [seconds]

Maximum stresses in the inner face and load historyσPD[ [03D]T [.3D]

Dynamic load

Static load

Figure 7.4: '\QDPLF� DQG� VWDWLF� UHVSRQVH� DQDO\VLV� RI� KXOO� SDQHO� LQ� WKH� /5%�� UHVFXHERDW��0LGSRLQW�VWUHVVHV�LQ�WKH�LQQHU�VNLQ�DW�WKH�VXUIDFH�


-0,4

-0,2

0,0

0,2

0,4

0,6

0,8

1,0

0 0,02 0,04 0,06 0,08 0,10

design rule

Solution 2

Panel

τPD[ [03D]T [��3D]

Maximum core shear stress and load history

Time [seconds]

Dynamic load

Static load

Figure 7.5: '\QDPLF� DQG� VWDWLF� UHVSRQVH� DQDO\VLV� RI� KXOO� SDQHO� LQ� WKH� /5%�� UHVFXHERDW��&RUH�VKHDU�VWUHVVHV�DW�WKH�PLGGOH�RI�HGJH�

The following observations are made from Fig. 7.3 concerning the relative midpoint de-flection: The deflection response behaves linearly within the design load TR�equal to 65.3D. For this design load, the relative deflection Z�E equals 0.52, whereas the maximumresponse is 0.99 by use of dynamic calculations applying a typical slamming load his-tory. Hence, there is a pronounced deviation of the response calculated statically anddynamically. This dynamic amplification factor, 1.92 in the particular case, depends onthe slamming impact period. The relative maximum deflection is below the� '19 1 %design rule for all the calculation methods.

The maximum stress behaves almost linearly for the applied maximum design load (Fig.7.4). The dynamic response is 2.1 times the static response for the maximum midpointstress σPD[ at the inner face of the panel, but it is still within the value imposed by the'19 design rule of 0.3�;DUDPLG equal to 300 03D. The maximum core shear stress τPD[

equal to 0.51 03D at the middle of the longest edge is below the '19 design rule of0.356F equal to 1.35 03D (Fig. 7.5). No wrinkling or debonding is observed from the re-sults of the calculations.

The results are summarised in Table 7.3 below where the maximum responses are com-pared to the '19� design criteria concerning laterally loaded panels. The most signifi-cant result is the dynamic amplification factor of approximately 2. The non-linear re-sults are almost equal to the linear results due to the relatively small deflection of thepanel. The ratio between the linear method (design rule) and the non-linear dynamic


method (3DQHO) is given in the last column of Table 7.3 indicating the error made by alinear static calculation for the given design.

Method----------Criteria

1. Linearstatic

('HVLJQ�UXOH)

2. Non-linearstatic

(6ROXWLRQ��)

3. Non-lineardynamic(3DQHO)

3

1

Z�E��[%] 0.52 0.51 0.99 1.9

σPD[�0.30; 0.30 0.31 0.64 2.1

τPD[��6F 0.32 0.32 0.59 1.9

Table 7.3: 3DQHO� UHVSRQVHV� FRPSDUHG� WR�'19� FULWHULD� IRU� WKH� /5%�� KXOO� SDQHO� VXE�MHFWHG�WR�D�GHVLJQ�ORDG�T�� .3D�

The /5%�� hull panel is conservatively dimensioned according to the above analysis.An ultimate failure analysis with 3DQHO is given in Figure 7.6 for a static load calcula-tion using linear and non-linear methods. The ultimate load calculated by the linearmethod is 7.5 times larger than the design load and 15 times larger than the design loadusing non-linear calculations. Thus, a factor of 15 is incorporated in the panel designregarding ultimate failure.

0

1

2

3

4

5

0 100 200 300 400 500 600 700 800 900 1000

Linear

Non-linear

Z�E %LRB rescue boat, ultimate failure of hull panel


480 .3D 960 .3D

&ROODSVH

&ROODSVH

Figure 7.6: 8OWLPDWH�IDLOXUH�ORDG�RI� WKH�/5%��KXOO�SDQHO�DFFRUGLQJ�WR�³3DQHO´�XVLQJOLQHDU�DQG�QRQ�OLQHDU�VWDWLF�PHWKRGV�


The failure analysis depends highly on the ultimate strength assigned to each ply (Table7.2) and test data of the ply properties are surely an advantage. The dominating failuremodes, which causes final collapse, are fibre and matrix breakage.

��0LQH�+XQWHU�6)��

The mine hunter SF300 class is currently one of the most advanced types of vessel in theDanish Navy. It is built upside down by applying the foam core onto a mould of tempo-rary transverse bulkheads and lists. After manual taping of the core the outer skin isbuilt upon the core. When curing has been brought about the hull is turned upwardsagain and the transverse bulkheads are removed and lay-up of the inner skin can takeplace.

Figure 7.7: 'DQLVK�PLQH�KXQWHU�6)��3RO\HVWHU�JODVV�VDQGZLFK�KXOO��0D[� VSHHG��NQRWV�

A vertical design acceleration of 3 J� gives a static design load T� equal to 145 .3D('19 Eq. 2.47). Data for the analysed hull bottom panel is given in Table 7.3. The re-sults are summarised in Table 7.4 and an ultimate failure analysis is given in Fig. 7.8.


0LQH�+XQWHU�6)��Faces: E-glass/polyesterInner 8.0 PP [csm/90/0/-45/45] (towards the core)Outer 9.4 PP [45/-45/0/90/csm] (from the core)E-glass, (woven rowing)411 = 39.2 *3D� 422 = 8.39 *3D 412 = 2.18 *3D466 = 4.14 *3D ; = 450 03D ; ‘= -310 03D< = 31 03D <�‘= -118 03D 6 = 72 03DCore 60 PPDivinycell H 200.6c� 3.3�03D 444 = 455 = 85 03DAdhesive (epoxy) ;DGK� ��03DDimensions D�� 2.0�P, E� �1.6�PDesign load TR� �145�.3D ('19 Eq. 2.47)Vertical acc. DG� 3 JR , given

Table 7.3: +XOO�SDQHO�GDWD�IRU�WKH��PLQH�KXQWHU�6)��

The hull panel is assumed to be simply supported and the stresses from global bendingof the hull are neglected in order to simplify the example. Hull bending from still waterand wave loads may introduce significant stresses, which are usually included in analy-ses of long FRP hulls as well as shock waves from underwater explosions, which may bethe dimensioning design load for this type of vessel. However, they are omitted in thisexample.


1. Linearstatic

('HVLJQ�UXOH)

2. Non-linearstatic

(6ROXWLRQ��)


3

1

Z�E��[%] 1.20 1.14 1.92 1.6

σPD[�0.30; 0.36 0.40 0.76 2.1

τPD[��6F 1.20 1.13 1.70 1.4

Table 7.4: 3DQHO� UHVSRQVHV� FRPSDUHG� WR�'19� FULWHULD� IRU� WKH� 6)�� KXOO� SDQHO� VXE�MHFWHG�WR�D�GHVLJQ�ORDG�T�� .3D�

As for the /5%�� rescue boat the geometrical non-linear behaviour is small for thegiven design load. Nevertheless, the non-linear static response is 11 % higher than cal-culated by the conventional linear theory concerning the maximum stress, whereas thedynamic stress response is 2.1 times higher. The ultimate failure load (Fig. 7.8) is 3.4and 6.5 times higher than the design load for linear and non-linear static calculations,respectively. Thus, the panel is designed much closer to the ultimate failure limit thanthe /5%�� panel.


0

1

2

3

4

5

0 100 200 300 400 500 600 700 800 900 1000

Non-linear

Linear

490 .3D 950 .3D

&ROODSVH

&ROODSVHZ�E % SF300 mine hunter, ultimate failure of hull panel

Lateral load T [KPa]

Figure 7.8: 8OWLPDWH� IDLOXUH� ORDG�RI� WKH�6)��KXOO�SDQHO�DFFRUGLQJ� WR�³3DQHO´�XVLQJOLQHDU�DQG�QRQ�OLQHDU�VWDWLF�PHWKRGV�

��5DFLQJ�<DFKW�,/&��

The third analysis of existing HSLC hull panel design with 3DQHO uses the high perform-ance racing yacht “Okyalos” as an example. It is designed according to the new ,/&��rules, which have taken over from the old IOR rules. The building of the boat is themost advanced technique of the three vessels discussed.

Hull and deck are built in one structure around a mould of transverse bulkheads andlists. The mould is shaped by an automatic milling machine operating from a mobileunit attached to rails in the ceiling. The foam is applied to the mould and shaped by themilling machine proceeding as for the 6)��. Then, the core is taped into the correctthickness and the lay-up and the curing of the epoxy-impregnated aramid/glass mats aredone in vacuum at a high temperature. After curing the mould is folded together and re-moved from the inside hull and deck structure through the entrance to the cabin. Thehull finish is made smooth by use of adequate milling and polish tools.


Figure 7.9: <DFKW�,/&��(SR[\�JODVV�DUDPLG�VDQGZLFK�KXOO��0D[�VSHHG��NQRWV�

5DFLQJ�<DFKW�,/&��Faces: E-glass/aramid/epoxyInner 1.1 PP [csm/aramid(0/-45/45/0)] (towards the core)Outer 1.1 PP [aramid(90/0/45/-45)/csm] (from core and out)E-glass, (woven rowing)411 = 39.2 *3D� 422 = 8.39 *3D 412 = 2.18 *3D466 = 4.14 *3D ; = 450 03D ; ‘= -310 03D< = 31 03D <�‘= -118 03D 6 = 72 03DAramid, 90/0 (unidirectional)411 = 76.6 *3D� 422 = 5.55 *3D 412 = 1.89 *3D466 = 2.30 *3D ; = 1.40 *3D ; ‘= -235 03D< = 12 03D <�‘= -53 03D 6 = 34 03DCore 30 PPDivinycell H 100.6c� 1.4�03D 444 = 455 = 40 03DAdhesive (epoxy) ;DGK� ��03DDimensions D�� 1.8�P, E� �1.2�PDesign load TR� �25�.3D ('19 Eq. 2.47)Vertical acc. DG� 3 JR , given

Table 7.5: +XOO�SDQHO�GDWD�IRU�WKH�UDFLQJ�\DFKW�,/&��


The analysis is summarised in Table 7.6. The difference between linear static and non-linear dynamic calculations is less pronounced than in the previous analyses, since thedynamic amplification is retained by the high membrane stresses in the faces. The stresslevel calculated by use of the non-linear method is 1.9 times higher than the linearlycalculated stress level, which is due to the high membrane stresses.


1. Linearstatic

('HVLJQ�UXOH)

2. Non-linearstatic

(6ROXWLRQ��)


3

1

Z�E��[%] 1.58 1.0 1.58 1.0

σPD[�0.30; 0.51 0.54 0.97 1.9

τPD[��6F 0.78 0.67 0.98 1.24

Table 7.6: 3DQHO�UHVSRQVH�FRPSDUHG�WR�'19�FULWHULD�IRU�D�,/&��KXOO�SDQHO�VXEMHFWHGWR�D�GHVLJQ�ORDG�T�� .3D�

The ultimate failure load (linear method) of the panel, illustrated in Fig. 7.10, is twiceas high as the design load, while the non-linear ultimate failure load is almost 11 timeshigher. This significant increase in ultimate load capacity is caused mainly by the highstrength aramid fibres in the faces providing an extra strength reserve. The dominantnon-linear failure modes are matrix and fibre breakage, whereas the linear analysis alsopredicts a significant core shear failure along the edges of the panel.

0

1

2

3

4

5

6

7

0 50 100 150 200 250 300

Non-linear

Linear

Z�E�� ILC40 racing yacht, ultimate failure of hull panel


&ROODSVH

&ROODSVH

48 .3D 270 .3D

Figure 7.10: 8OWLPDWH�IDLOXUH�ORDG�RI��WKH�,/&��KXOO�SDQHO�DFFRUGLQJ�WR�³3DQHO´�XVLQJOLQHDU�DQG�QRQ�OLQHDU�VWDWLF�PHWKRGV�


��'HVLJQ�([DPSOH

An HSLC hull panel is designed and improved with regard to weight/strength with3DQHO in order to show the possible design advantages by use of non-linear calculations.The example (Table 7.3) shows the relative deflection calculated by linear and non-lin-ear calculations (row 1-4) for a given lateral pressure according to '19� (Chapter 2).The limit design criterion for each analysis is given in the last column. These criteria aredeflections, tensile stresses, shear stresses and wrinkling/compressive stresses. The spe-cific panel weight is the design parameter for comparison (grey column). The weight isreduced by decreasing the face and core thickness and by changing the fibre material.The material properties for the faces are the same as given in Table 7.2.

0HWKRG 0DWHULDOface/core

6FDQWOLQJVface/core/face

3UHVVXUH*105[Pa]

6SHFLILFPDVV

Z�E�

/LPLW

� /LQHDU glass/H200 5/90/5 1 1 1 Z�E

� 1RQ�OLQHDU glass/H200 4/90/4 1 0.92 1 Z�E

� 1RQ�OLQHDU glass/H200 4/60/4 1 0.74 2 σWHQ

� 1RQ�OLQHDU aramid/H130 1.25/50/1.25 1 0.28 3.9 τ\]

Table 7.7: 'HVLJQ�RI�D�VDQGZLFK�SDQHO�LQ�WKH�ERWWRP�SDUW�RI�DQ�+6/&�KXOO�

Within the 1 % rule (maximum deflection/shortest panel-span < 0.01) the non-linearanalysis does not give remarkable improvements compared to the linear analysis, hencea simple linear approach is adequate (row 1-2).

If the 1% rule is omitted (row 3-4), a pronounced non-linear behaviour is allowed for,which may be used as a design advantage. This is shown in row 3, where the core thick-ness is minimised in order to reduce the specific weight to 0.74. The 1% relative deflec-tion is exceeded and the tensile stress becomes the limit criterion. By exchanging theglass fibres for high-strength aramid fibres in the faces and by decreasing the face andcore thickness and the core density, the specific weight is reduced to 0.28. The limitcriterion is then the core shear stress.

The above example shows the potential of using a non-linear approach in the design ofFRP sandwich panels. This, of course, implies that the 1 % rule can be neglected in par-ticular cases.

��6XPPDU\��145

��6XPPDU\

The structure of the design tool 3DQHO is outlined and the use of the program is demon-strated by different examples of analyses and design. The first three examples show re-sponse and failure analyses of hull panels in different HSLCs, whereas the fourth exam-ple demonstrates a new design of an HSLC hull panel.

In the examples of the existing HSLC panel designs, static and dynamic calculations arecompared for linear and non-linear analyses. The results for the two first panels (/5%��and 6)��) show that the dynamic response of the panels is noticeably increased for atypical slamming load. Especially for the /5%��, since this panel has an eigenfrequencyvery close to that of the slamming load. The non-linear behaviour for these two panels isinsignificant for the given design loads.

The light- and high-strength panel of the ,/&�� racing yacht displays a more pro-nounced non-linear behaviour, since it has thin high strength faces and, consequently,the faces behave more like membranes than the other sandwich panels. The dynamicamplification is not obvious as it counteracts the non-linear behaviour.

The last design example demonstrates the advantage of using non-linear calculations,which results in a significant reduction of the panel weight to 28 % of the originalweight for the same strength.


147

&KDSWHU��

Discussion of the DNV Code

��,QWURGXFWLRQ

Although the FRP HSLC design rules, imposed by the classification societies, have beenprogressively improved over the past two to three decades, they are still in the phase ofdevelopment compared to the equivalent design rules concerning metals, which arebased on a much longer history of practice and experience. Many of the FRP rules arelimited by safety considerations from empirical experience rather than theoreticalknowledge of the structure, which often leads to unnecessarily conservative designs.

The '19 HSLC code [1], concerning structural design of composite panels, is one ofthe more advanced sets of rules among the classification societies, (Chapter 2). Typicalsingle-skin and sandwich hull panels, designed by use of the '19�code, are discussed inthe light of non-linear results calculated with 3DQHO. The rule of a maximum relative lat-eral deflection of 1 % is usually the limiting criterion, '19 [2], and consequently, theuse of non-linear design formulas may be of advantage.

��7KH�6WLIIHQHG�6LQJOH�6NLQ�5XOH

The '19 design rules Eqs. 8.1-2, for single-skin panels concerning maximum lateral de-flections and stresses are based on geometrical non-linear theory. The expression for therelative deflection (Eq. 8.1a) is given as

( ) [ ]W ET

( & &PP=

+

178

12

2

0 25

δ δ

.

(8.1a)

148 &KDSWHU��'LVFXVVLRQ�RI�WKH�'19�&RGH

Expressed by the relative lateral deflection, δ =�Z�W, we get

ZW

&&

ZW &

EW

T(

ZW

+ ⋅

=

≤

31

2 2

410041

. , (8.1b)

The rule concerning maximum stress response yields

( ) [ ]σ δ δmax/=

+

WE

( & & & & 03D1000

2

1 3 4 22 3 (8.2a)

or rewritten in terms of Z�W, we obtain

σ σmax

/

.=+

+

≤ ⋅

& &ZW

& &

&ZW

&

EW

T��QX

4 22 3

1 3

2

2

1

0 302

(8.2b)

where

T design pressure [.3D]E the length of the shortest side of the panel [P]δ the deflection to thickness ratio, Z�W( Young’s modulus [03D]&��&� factors depending on the panel aspect ratio and its boundary conditions&��&� factor depending on the panel aspect ratio

The rules are compared to calculations with 3DQHO (Figs 8.1-2) using an FRP single-skinpanel (3ODWH��) with the following properties:

3ODWH��:

Material CSM E-glass/polyester(-module ( = 20 *3DPoisson ratio ν = 0.3Thickness W = 10 PPDimensions D = E = 1.00�PMax loading T = 20 .3D, simply supported boundary conditionsMax loading T = 40 .3D, clamped boundary conditions

��7KH�6WLIIHQHG�6LQJOH�6NLQ�5XOH��149

0,0

0,2

0,4

0,6

0,8

1,0

1,2

0 5 10 15 20 25

'19, &� = 72, cl

'19, ss

3DQHO, ss

3DQHO, cl

'19, &� = 63, cl

Linear, ss

Linear, cl


Z�W '19 rule (Eq. 8.1b) >< 3DQHO, single-skin

Figure 8.1: 0LGSRLQW� GHIOHFWLRQ� RI� 3ODWH� �� DFFRUGLQJ� WR� '19� >�@� DQG� 3DQHO�� VLPSO\VXSSRUWHG�DQG�FODPSHG�ERXQGDU\�FRQGLWLRQV�

The results from calculations with 3DQHO and the '19 rule Eq. 8.1b (Fig. 8.1) are ingood agreement for both types of boundary conditions. In this particular case, they showconsiderably geometrical non-linear behaviour of the FRP single-skin plate subjected tohigh lateral loads for both boundary conditions.

For the clamped boundary conditions, the rule predicts slightly non-conservative resultscompared to the results calculated with 3DQHO. It seems that &� is overestimated with 15%, i.e. 72 instead of 63 according to Fig. 8.1. The factor is plotted as a function of theplate aspect ratios in the '19 code (Ch. 4, Sec. 6, B202, Fig. 1) and a validation of thiscurve is suggested. The non-linear method allows 50 % more load than predicted by alinear calculation for a relative deflection of 1 %.

For the plate with simply supported boundary conditions, the rule predicts very accurateresults. In the given example the lateral load is increased by 2.4 times the linear, calcu-lated load for a relative deflection of 1 %.

Thus, design advantages are obtained by use of non-linear solution methods provided bythe '19 code. Similar HSLC codes, Bureau Veritas, Lloyds and ABS, use linear expres-sions for the deflection response, and in general, the maximum deflection criterion ofZ�W < 1 or Z�E < 0.01 becomes the limiting criterion.

Fig. 8.2 compares the maximum stresses in the centre of the plate. The rule (Eq. 8.2b)shows good agreement with the non-linear calculations with 3DQHO in the simply sup-

150 &KDSWHU��'LVFXVVLRQ�RI�WKH�'19�&RGH

ported case. For the clamped plate the stress level is overestimated by the rule, and itseems that the &� factor is wrong (0.31 instead of 0.13 for aspect ratio equal to unity).The factor is plotted as function of plate aspect ratio in the '19 code (Ch. 4, Sec. 6,B202, Fig. 2) and a validation (correction) of this curve can be obtained by use of theanalytical non-linear method described in Chapter 3.

0

10

20

30

40

50

0 5 10 15 20 25

Linear, ss

Linear, cl

3DQHO, ss

3DQHO, cl

'19, ss

'19, &��= 0.31, cl

'19, &� = 0.13, cl


σPD['19 rule (Eq. 8.2b) >< 3DQHO, single-skin

Z�W� ��QRQ�OLQHDUVV��FO

Figure 8.2: 0LGSRLQW�VWUHVVHV�RI�3ODWH��DFFRUGLQJ�WR�'19�>�@�DQG�3DQHO��VLPSO\�VXS�SRUWHG�DQG�FODPSHG�ERXQGDU\�FRQGLWLRQV�

At the maximum load level, where Z�W� ��according to the non-linear calculations, therule predicts a maximum stress level which is 35 % and 15 % lower than predicted bylinear calculations for simply supported and clamped boundary conditions, respectively.

It seems appropriate to ask for an explanation of the reason for the maximum deflectioncriterion and to allow for exceptions in particular cases. Since the code has implementedthe non-linear response formulas for stresses and deflections, it offers already an accu-rate method for response prediction for larger Z�W ratios.

��7KH�6DQGZLFK�5XOH��151

��7KH�6DQGZLFK�5XOH

The '19 design rule concerning FRP sandwich plates is based on linear theory. Figs.8.3-4 compare the rule with non-linear calculations (3DQHO) using a typical sandwichplate from a high-speed hull (3ODWH��). The '19 formulas are given below for maximumstress as:

[ ]σQ 1

TE:

& & 03D=160 2

1 (8.3)

where

T design pressure [.3D]E length of shortest side of the panel [P]: section modulus of the panel per unit width [PP2]&1 &��ν&� for stresses parallel to the longest edge, and&1 &��ν&� for stresses parallel to the shortest edge&� factor depending on the boundary conditions

and the maximum deflection as

( ) [ ]ZTE'

& & & PP= +106 4

26 8 7ρ (8.4)

where

D2 flexural rigidity of the sandwich panel per unit width [1PP]ρ nominal ratio between bending and shear stiffnessC6 factor depending on the panel aspect ratioC7 factor depending on the panel aspect ratioC8 factor depending on the panel aspect ratio and its boundary conditions

Calculations are performed by use of a sandwich plate (3ODWH��) with the properties:

3ODWH��

Skin as for 3ODWH��Material core Divinycell H-160*-module core * = 66 03DCore thickness G = 60 PPSkin thickness W = 3 PPDimensions D� �E�= 2.00 PMax loading T = 300 .3D, simply supported boundary conditionsMax loading T = 400 .3D, clamped boundary conditions

152 &KDSWHU��'LVFXVVLRQ�RI�WKH�'19�+6/&�&RGH

0,0

0,4

0,8

1,2

1,6

0 20 40 60 80 100

3DQHO, cl

3DQHO, ss

'19, cl

'19, ss

Lateral load T��[.3D]

Z�E�[%] '19 rule (Eq. 8.4) >< �3DQHO� Sandwich

Figure 8.3: 0LGSRLQW� GHIOHFWLRQ� RI� 3ODWH� �� DFFRUGLQJ� WR� '19� >�@� DQG� 3DQHO�� VLPSO\VXSSRUWHG�DQG�FODPSHG�ERXQGDU\�FRQGLWLRQV�

-120

-80

-40

0

40

80

120

0 20 40 60 80 100

Panel, lower face, ss

Panel, upper face, ss

Panel, lower face, cl

Panel, upper face, cl

DNV, cl

DNV, ss

σPD[��[03D] '19 rule (Eq. 8.3) >< 3DQHO, sandwich


Figure 8.4: 0LGSRLQW�VWUHVVHV�LQ�XSSHU�DQG�ORZHU�VNLQV��VLPSO\�VXSSRUWHG��DQG VWUHVVHVDW�WKH�PLGGOH�RI�WKH�HGJH��FODPSHG��RI�3ODWH��DFFRUGLQJ�WR�'19�>�@�DQG�3DQHO�

��7KH�6DQGZLFK�5XOH��153

The plate deflection (Fig. 8.3) behaves linearly within the relative maximum allowedlateral deflection Z�E equal to 1 % and the rule (Eq. 8.4) seems reasonable.

The stress level in the centre of the plate responds non-linearly according to the Fig. 8.4.For the simply supported panel, the maximum stress (in the middle of the lower face)calculated according to Eq. 8.3 is 9 % lower than calculated by 3DQHO. For the clampedpanel, the maximum stresses (at the middle of the edge in the upper face) are almostidentical for both methods.

As the stress values in upper and lower skins differ significantly, anti-symmetric platedesigns will be of great advantage. This will imply a distinction between upper andlower skins in the rule. This may be done by use of the analytical non-linear method de-scribed in Chapter 3. However, within the 1 % rule the present linear rule gives reason-able results concerning maximum stresses in the faces.

If the 1 % rule is exceeded, which seems reasonable to permit in particular cases, non-linear calculation methods for maximum deflection and stresses in the faces are essen-tial.

��6XPPDU\

The single-skin FRP deflection rule (Eq. 8.1b) gives good results for small and large de-flections. Eq. 8.2 concerning stresses is not correct for the clamped boundary conditioncase using the current &� and &� values, which need to be validated. This may be doneby application of the method outlined in Chapter 3. It is not clear why the maximum lat-eral deflection rule is necessary for the single-skin plate, since the rules concerningstresses and deflections are based on non-linear theory and predict the responses well.

The '19� HSLC rule concerning sandwich needs to be reconsidered, if the relativemaximum deflection of 1 % has to be exceeded. Within small deflections there is a sig-nificant difference in the stress levels of the upper and lower faces. It is suggested thatthe expressions for the maximum deflections and stresses (Eqs. 8.3-4) should be supple-mented with a non-linear part. The analytical method, 6ROXWLRQ��, (Chapter 3) would besuitable for this purpose, as it combines linear and non-linear expressions. Thus, theoriginal rule can be kept and the non-linear part may act as a supplement to the rule usedfor special occasions to be specified by the rule.

154 &KDSWHU��'LVFXVVLRQ�RI�WKH�'19�+6/&�&RGH

��%LEOLRJUDSK\

[1] DNV. High Speed and Light Craft. Classification Rules for High Speed LightCraft. Det Norske Veritas Research AS, Veritasveien 1, N-1322 Høvik, Norway,1991.

[2] DNV. Examination of Criteria for Panel Deflection in DNV’s Rules for High

Speed and Light Craft. Technical Report No. 96-2014. Det Norske Veritas Re-search AS, Veritasveien 1, N-1322 Høvik, Norway, May, 1996.

155

Conclusion

The objective of this study is to develop a suitable tool for the design of composite hullpanels in high-speed light craft (HSLC). These panels are made of fibre-reinforce plas-tics (FRP), which are manufactured into either stiffened skin panels or sandwich panels.The latter are manufactured by use of light core materials with faces of FRP skins.

In order to achieve this objective the following key subjects are investigated and cap-tured: Behaviour of orthotropic materials, failure and fracture mechanisms, loading andresponse characteristics. Furthermore, existing rules and criteria are examined in orderto improve the present rules and to develop new and more rational methods for the de-sign and analysis of FRP panels.

The results of the investigation are formulated in the design program 3DQHO and in twosimple analytical solution methods, 6ROXWLRQ�� and 6ROXWLRQ��. It is common to the abovemethods that they are based on geometrical non-linear orthotropic sandwich plate the-ory. This is due to the FRP panels in HSLC being characterised by large panel sizes andhigh lateral loads from the vertical decelerations of the vessel hitting the water surface.

Although the theory behind the analytical solutions has been known for years, 6ROXWLRQ��represents a new way to calculate non-linear responses for sandwich plates in a verysimple way. In addition, the solution is suitable for use with existing linear design rules,as it is divided into a linear part and a non-linear part. Both of the analytical solutionsshow good agreement with numerical calculations and experiments. The solutions omitthe dynamic terms in the equilibrium equations, which is important when slammingloads are dealt with. However, this is captured in the numerically formulated solutionmethod, 3DQHO.

3DQHO is based on the finite difference method and Newmark’s method. The advantagesof 3DQHO compared to existing commercial non-linear dynamic finite-element programs,such as Ansys, are a significantly lower CPU-time and a short pre-processing phase.However, for a more detailed level of response information. Of course Ansys is a moregeneral program than 3DQHO, able to treat a variety of panel shapes and boundaryconditions.A progressive damage model is developed and implemented in 3DQHO by application ofexisting failure criteria for FRP laminates and foam cores. Non-linear damage analyses

156� ��&RQFOXVLRQ

with� 3DQHO show notably higher ultimate loads than obtained by use of linear damageanalysis. This circumstance is the most significant result in the non-linear analysis ap-proach.

Dynamic response calculations, with existing HSLC hull panels subjected to typicalslamming loads, show that there is a significant increase in responses compared to theresponses obtained by use of static calculations, where the panels are subjected to con-stant uniform loads.

The design advantages by application of non-linear methods are demonstrated by a con-ventionally designed sandwich panel. This simplified example illustrates the possi-bilities of improvements of the present design methods.

Further research on the subject is suggested below:

• Introduction of curved panels.• Large stiffened panel fields.• Panels with combined loading.• Probabilistic calibration of design codes.

7KRVH�ZKR�IDOO�LQ�ORYH�ZLWK�VFLHQFH�ZLWKRXW�SUDFWLVHDUH�OLNH�D�VDLORU�ZKR�VWHHUV�D�VKLS�ZLWK�RQO\�KHOP�DQGFRPSDVV��DQG�ZKR�QHYHU�FDQ�EH�FHUWDLQ�ZKLWKHU�KH�LV�JRLQJ�

'��)LQH

157

$SSHQGL[�$

This Appendix shows the derivation of the non-linear analytical solutions performed inChapter 3. The reader who needs an analytical tool ready for implementation in designrules as a simple computer program may find it relevant.

$��,QWHJUDOV�DQG�3DUDPHWHUV

In the following, detailed derivations of the terms in the strain energy expression arepresented for both the simply supported and clamped cases. The general expression forthe strain energy in the plate is shown below, where the higher-order terms in the strainfunctions are retained.

8 $X[

$Z[

$X[

Z[

$Y\

$Z\

$Y\

Z\

$X\

$Y[

$Z[

Z\

$�

=

+

+

+

+

+

+

+

+

+

∫∫1

2

1

4

1

4

2

11

2

11

4

11

2

22

2

22

4

22

2

66

2

66

2

66

2

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

$X\

Y[

$Z[

Z\

X\

Y[

$X[

Y\

$Z[

Z\

$X[

Z\

$Y\

Z[

%Z

[Z[

%Z

[X[

%E E

66

66 12 12

2 2

12

2

12

2

11

2

2

2

11

2

2 22

2

2 21

2

2

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂

+ +

+ +

+

+

−

− −

Z

\Z\

%Z

\Y\

%X\

Y[

Z[ \

%Z[ \

Z[

Z\

%X[

Z

\

%Y\

Z

[%

Z[

Z

\'

Z

['

Z

E

E E E E

E E E E

∂∂∂

∂∂

∂∂

∂∂

∂∂

∂∂ ∂

∂∂ ∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂

2

2

22

2

2 66

2

66

2

12

2

2

12

2

2 12

2 2

2 11

2

2

2

22

2

2 4 4 2

2

− − +

− −

− −

+

+

∂

∂∂ ∂

∂∂

∂∂

∂∂

∂∂

\

'Z

[ \'

Z

[

Z

\$

Z

\$

Z

[E E E V V

2

2

66

2 2

12

2

2

2

2 44

2

55

2

4 2

+

+ +

+

�G$

(A. 1)

158� $SSHQGL[�$

$��6LPSO\�6XSSRUWHG�&DVH

By inserting the deflection functions (Eq. 3.25b) in the energy expression Eq. A.1 andintegrating over the plate area D×� E, we obtain thirty-two terms XL, forming the totalstrain energy 8� = ∑XL, as shown below (subscripts VV and FO refer to the simply sup-ported and clamped cases, respectively):

X $ X [ \G[G\ $ DEX X XVV

$

VV

1 112 2 2 2

112 2 21

24 2

1

21= = = ⋅∫∫ α α β αcos sin

X $ Z [ \G[G\ $ DEZ X ZVV

$

VV

2 114 4 4 4

114 4 41

2

1

4

9

5122= = = ⋅∫∫ α α β αcos sin

X $ XZ [ [ \G[G\ $ DXZ X XZVV

$

VV

3 112 2 2 3

11

32 21

22 2

1

33= = = ⋅∫∫ α α α β

αβ

cos cos sin

X $ Y \ [G[G\ $ DEY X YVV

$

VV

4 222 2 2 2

222 2 21

24 2

1

24= = = ⋅∫∫ β β α βcos sin


$

VV

5 224 4 4 4

224 4 41

2

1

4

9

5125= = = ⋅∫∫ β α β βcos sin

X $ YZ \ [ [G[G\ $ EXZ X YZVV

$

VV

6 222 2 2 3

22 3

2 21

22 2

1

36= = = ⋅∫∫ β β α α

βα

cos cos sin

X $ X [ \G[G\ $ DEX X XVV

$

VV

7 662 2 2 2

662 2 21

22

1

87= = = ⋅∫∫ β α β βsin cos

X $ Y \ [G[G\ $ DEY X YVV

$

VV

8 662 2 2 2

662 2 21

22

1

88= = = ⋅∫∫ α β α αsin cos

X $ Z \ \ [ [G[G\ $ DEZ X ZVV

$

VV

9 662 2 4 2 2 2 2

662 2 4 41

2

1

1289= = = ⋅∫∫ α β β β α α α βsin cos sin cos

X $ XY \ \ [ [G[G\ $ XY X XYVV

$

VV

10 66 66

1

22 2 2

16

910= = = ⋅∫∫ αβ β β α αsin cos sin cos

X $ XZ [ [ [ \ \G[G\ $ DXZ X XZVV

$

VV

11 66

2 266

2 21

22 2

1

611= = = ⋅∫∫ αβ α α α β β αβsin cos sin cos sin

X $ YZ \ \ \ [ [G[G\ $ EYZ X YZVV

$

VV

12 66

2 266

2 21

22 2

1

612= = = ⋅∫∫ αβ β β β α α αβsin cos sin cos sin

X $ XY \ \ [ [G[G\ $ XY X XYVV

$

VV

13 12 12

1

28 2 2

16

913= = = ⋅∫∫ αβ β β α αcos sin cos sin

X $ Z \ \ [ [G[G\ $ DEZ X ZVV

$

VV

14 122 2 4 2 2 2 2

122 2 4 41

2

1

2

1

25614= = = ⋅∫∫ α β β β α α α βsin cos sin cos

$��,QWHJUDOV�DQG�3DUDPHWHUV��159

X $ XZ [ [ \ \G[G\ $ DXZ X XZVV

$

VV

15 122 2 2 2

12

2 21

22 2

1

615= = − = ⋅∫∫ αβ α α β β αβcos sin cos sin

X $ YZ \ \ [ [G[G\ $ EYZ X YZVV

$

VV

16 122 2 2 2

12

2 21

22 2

1

616= = − = ⋅∫∫ α β β β α α αβcos sin cos sin

If symmetry is assumed in order to reduce the final algebraic expression for the deflec-tion functions, the stretching/bending-stiffness terms %LM in the energy expressions X�� →X��, become zero. It is no problem to take these terms into account. However, when thisis done the final analytical expressions become too large for practical use, except incomputer programs based on these types of analytical solutions. Thus

X ' Z [ \G[G\ ' DEZ X ZVV

E$

E

VV

E28 114 2 2 2

114 2 21

2

1

828= = = ⋅∫∫ α α β αsin sin


E$

E

VV

E29 224 2 2 2

224 2 21

2

1

829= = = ⋅∫∫ β α β βsin sin


E$

E

VV

E30 662 2 2 2 2

662 2 2 21

24

1

230= = = ⋅∫∫ α β α β α βcos cos


E$

E

VV

E31 122 2 2 2 2

122 2 2 21

22

1

431= = = ⋅∫∫ α β α β α βsin sin


V$

V

VV

V32 442 2 2 2

442 2 21

2

1

832= = = ⋅∫∫ β α β βsin cos

X $ Z \ [G[G\ $ DEZ X ZVV

V$

V

VV

V33 552 2 2 2

552 2 21

2

1

833= = = ⋅∫∫ β β α αsin cos

( ) ( )( ) ( )

( )

β β

β β

βπ

β ββ β

β β

1 2

3 4

5 2 6 124 6

4 6

4 2 5 9 14 2 3 11 15

2 6 12 16 2 28 29 30 31

42 32 33

= + + + = + +

= + + = + + +

= − = + =+

X X X X X X X

X X X X X X X

TDEX X

VV VV

VV VV

VV

,

,

, ,

(A. 2)

αβ

α α

αβ

α α

12

43

210 13 2 1 2 7

210 13 2 4 2 8

= = + = +

= = + = +

, ,

, ,

2 3

5 6

X X X X

X X X X (A. 3)

160� $SSHQGL[�$

D D2 4 10 3 5 10 7 4 6

8 11 11 7 11 72

9 12 12 7 12 72

10 1 1 7 1 72

1 73

2 8 3 9 2 7 8 3 7 9

2 2

3 3

= = =

= + + = + +

= + + + + + + +

β β β β β β ββ α α β α β β α α β α ββ β β β β β β β β β β β β β β β β β

, ,

, (A. 4)

αα α α αα α α α α

α α α αα α α α11

6 1 4 2

5 2 6 312

4 3 5 1

5 2 6 3

=−−

=−−

, (A. 5)

5DE DE

' 6' 6

' ' . ' ' ' ' 6 $ $

VV VV

V E

1 12

24 6

4 6

211

11

11 22 12 22 66 22 44 55

4 4

1

2

= =+

=+

= = =−

= =

βπ β β

β βπ

α α

νν

, , ,

(A. 6)

( )

( )

β α βυ

α β ν α β

β α β

απ α β

υα β ν α β

απ α β

4 114 4 2 2 2 2

62 2

2 4 4 2 2 2 2

2 2 2

21

8

14

1

2

12

1

21

816

14

1

2

12

1

16

= ⋅ + +−

+

= ⋅ +

=+ +

−+

=+

' DE. . .

6 DE

. . .

E

V

(A. 7)

$��&ODPSHG�&DVH

Using the same procedure as for the simply supported plate, by combination of Eq. A.1and Eq. 3.29, we obtain the following energy terms which form the strain energy of theclamped plate.

X $ X [ \G[G\ $ DEX X XFO

$

FO

1 112 2 2 2

112 2 21

24 2

1

21= = = ⋅∫∫ α α β αcos sin

X $ Z [ [ \G[G\ $ DEZ X ZFO

$

FO

2 114 4 4 4 8

114 4 41

2

1

416

105

81922= = = ⋅∫∫ α α α β αcos sin sin

$��,QWHJUDOV�DQG�3DUDPHWHUV��161

X $ XZ [ [ [ \G[G\ XZ X XZFO

$

FO

3 113 2 2 2 5 2 21

28 2 0 3= = ⋅ = ⋅∫∫ α α α α βcos sin cos sin

X $ Y \ [G[G\ $ DEY X YFO

$

FO

4 222 2 2 2

222 2 21

24 2

1

24= = = ⋅∫∫ β β α βcos sin

X $ Z [ \ \G[G\ $ DEZ X ZFO

$

FO

5 224 4 8 4 4

224 4 41

2

1

416

105

81925= = = ⋅∫∫ β α β β βcos cos sin

X $ YZ \ \ \ [G[G\ XZ X XZFO

$

FO

6 222 2 2 2 5 2 21

28 2 0 6= = ⋅ = ⋅∫∫ β β β β αcos sin cos sin

X $ X [ \G[G\ $ DEX X XFO

$

FO

7 662 2 2 2

662 2 21

22

1

87= = = ⋅∫∫ β α β βsin cos

X $ Y \ [G[G\ $ DEY X YFO

$

FO

8 662 2 2 2

662 2 21

22

1

88= = = ⋅∫∫ α β α αsin cos

X $ Z \ \ [ [G[G\ $ DEZ X ZFO

$

FO

9 662 2 4 6 2 6 2

2

662 2 4 41

216 8

5

1289= =

= ⋅∫∫ α β β β α α α βsin cos sin cos

X $ XY \ \ [ [G[G\ $ XY X XYFO

$

FO

10 66 66

1

22 2 2

16

910= = = ⋅∫∫ αβ β β α αsin cos sin cos

X $ XZ [ [ [ \ \G[G\ $ DXZ X XZFO

$

FO

11 66

2 3 2 366

2 21

24 2

2

1511= = = ⋅∫∫ αβ α α α β β αβsin sin cos cos sin

X $ YZ \ \ \ [ [G[G\ $ EYZ X YZFO

$

FO

12 66

2 3 2 366

2 21

24 2

2

1512= = = ⋅∫∫ αβ β β β α α αβsin cos sin cos sin

X $ XY \ \ [ [G[G\ $ XY X XYFO

$

FO

13 12 12

1

28 2 2

16

913= = = ⋅∫∫ αβ β β α αcos sin cos sin

X $ Z \ \ [ [G[G\ $ DEZ X ZFO

$

FO

14 122 2 4 6 2 6 2

2

122 2 4 41

28 4

5

12814= =

= ⋅∫∫ α β β β α α α βsin cos sin cos

X $ XZ [ [ \ \G[G\ $ DXZ X XZFO

$

FO

15 122 2 4 2 3

12

2 21

28 2

4

1515= = − = ⋅∫∫ αβ α α β β αβcos sin cos sin

X $ YZ \ \ [ [G[G\ $ EYZ X YZFO

$

FO

16 122 2 4 2 3

12

2 21

28 2

4

1516= = − = ⋅∫∫ α β β β α α αβcos sin cos sin

X ' Z [ [G[G\ ' DEZ X ZFO

E$

E

FO

E28 114 2 4 2

114 2 21

24 2

3

828= = = ⋅∫∫ α β α αsin cos

X ' Z [ \G[G\ ' DEZ X ZFO

E$

E

FO

E29 224 2 4 2

224 2 21

24 2

3

829= = = ⋅∫∫ β α β βsin sin

162� $SSHQGL[�$

X ' Z [ \ [ \G[G\ ' DEZ X ZFO

E$

E

FO

E30 662 2 2 2 2 2 2

662 2 2 21

264

1

230= = = ⋅∫∫ α β α β α β α βcos cos sin sin

X ' Z [ \ [ \G[G\ ' DEZ X ZFO

E$

E

FO

E31 122 2 2 2 2

122 2 2 21

28 2 2

1

431= = = ⋅∫∫ α β α β α β α βsin sin cos cos

X $ Z [ \ \G[G\ $ DEZ X ZFO

V$

V

FO

V32 442 2 2 2 4

442 2 21

24

3

3232= = = ⋅∫∫ β α β β βsin cos sin

X $ Z \ [ [G[G\ $ DEZ X ZFO

V$

V

FO

V33 552 2 2 2 4

552 2 21

24

3

3233= = = ⋅∫∫ β β α α αsin cos sin

163

$SSHQGL[�%

For response and failure analysis with the program 3DQHO the user has to fill in a datafile with the appropriate information. An example of such an in-data file concerninganalysis of a sandwich panel in the hull of the rescue vessel LRB12 (chapter 7) is givenbelow. The actual input are written in regular style and the comments in bold.

��5HVFXH�%RDW�/5%��)DFHV� E-glass/aramid/epoxyInner 3.0 PP [aramid(0/90)/-45/45/0/90] (towards the core)Outer 3.9 PP [90/0/45/-45/aramid(0/90)/csm] (from the core)(�JODVV��ZRYHQ�URZLQJ�411 = 39.2 *3D� 422 = 8.39 *3D 412 = 2.18 *3D466 = 4.14 *3D ; = 450 03D ; ‘= 310 03D< = 31 03D <�‘=118 03D 6 = 72 03D$UDPLG��XQLGLUHFWLRQDO�411 = 76.6 *3D� 422 = 5.55 *3D 412 = 1.89 *3D466 = 2.30 *3D ; = 1.00 *3D ; ‘= 235 03D< = 12 03D <�‘=53 03D 6 = 34 03D&RUH��PP�Divinycell H 250.6c� 4.5�03D 444 = 455 = 108 03D$GKHVLYH��HSR[\� ;DGK� ��03DDimensions D�� 1.4�P, E� �0.8�PDesign load TR� �65�.3D ('19 Eq. 2.47)Vertical acc. DG� 5 JR , given

Table B.1: +XOO�SDQHO�GDWD�IRU�WKH�UHVFXH�ERDW�/5%��

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Ph.d. ThesesDepartment of Naval Architecture and O�shore Engineering, DTU

1961 Str�m-Tejsen, J.: \Damage Stability Calculations on the ComputerDASK".

1963 Silovic, V.: \A Five Hole Spherical Pilot Tube for three DimensionalWake Measurements".

1964 Chomchuenchit, V.: \Determination of the Weight Distribution ofShip Models".

1965 Chislett, M.S.: \A Planar Motion Mechanism".

1965 Nicordhanon, P.: \A Phase Changer in the HyA Planar Motion Mecha-nism and Calculation of Phase Angle".

1966 Jensen, B.: \Anvendelse af statistiske metoder til kontrol af forskelligeeksisterende tiln�rmelsesformler og udarbejdelse af nye til bestemmelseaf skibes tonnage og stabilitet".

1968 Aage, C.: \Eksperimentel og beregningsm�ssig bestemmelse af vind-kr�fter p�a skibe".

1972 Prytz, K.: \Datamatorienterede studier af planende b�ades fremdriv-ningsforhold".

1977 Hee, J.M.: \Store sideportes ind ydelse p�a langskibs styrke".

1977 Madsen, N.F.: \Vibrations in Ships".

1978 Andersen, P.: \B�lgeinducerede bev�gelser og belastninger for skib p�al�gt vand".

1978 R�omeling, J.U.: \Buling af afstivede pladepaneler".

1978 S�rensen, H.H.: \Sammenkobling af rotations-symmetriske og generelletre-dimensionale konstruktioner i elementmetode-beregninger".

1980 Fabian, O.: \Elastic-Plastic Collapse of Long Tubes under CombinedBending and Pressure Load".

1980 Petersen, M.J.: \Ship Collisions".

1981 Gong, J.: \A Rational Approach to Automatic Design of Ship Sec-tions".

1982 Nielsen, K.: \B�lgeenergimaskiner".

1984 Rish�j Nielsen, N.J.: \Structural Optimization of Ship Structures".

1984 Liebst, J.: \Torsion of Container Ships".

1985 Gjers�e-Fog, N.: \Mathematical De�nition of Ship Hull Surfaces usingB-splines".

1985 Jensen, P.S.: \Station�re skibsb�lger".

1986 Nedergaard, H.: \Collapse of O�shore Platforms".

1986 Junqui, Y.: \3-D Analysis of Pipelines during Laying".

1987 Holt-Madsen, A.: \A Quadratic Theory for the Fatigue Life Estima-tion of O�shore Structures".

1989 Vogt Andersen, S.: \Numerical Treatment of the Design-AnalysisProblem of Ship Propellers using Vortex Latttice Methods".

1989 Rasmussen, J.: \Structural Design of Sandwich Structures".

1990 Baatrup, J.: \Structural Analysis of Marine Structures".

1990 Wedel-Heinen, J.: \Vibration Analysis of Imperfect Elements in Ma-rine Structures".

1991 Almlund, J.: \Life Cycle Model for O�shore Installations for Use inProspect Evaluation".

1991 Back-Pedersen, A.: \Analysis of Slender Marine Structures".

1992 Bendiksen, E.: \Hull Girder Collapse".

1992 Buus Petersen, J.: \Non-Linear Strip Theories for Ship Response inWaves".

1992 Schalck, S.: \Ship Design Using B-spline Patches".

1993 Kierkegaard, H.: \Ship Collisions with Icebergs".

1994 Pedersen, B.: \A Free-Surface Analysis of a Two-Dimensional MovingSurface-Piercing Body".

1994 Friis Hansen, P.: \Reliability Analysis of a Midship Section".

1994 Michelsen, J.: \A Free-Form Geometric Modelling Approach with ShipDesign Applications".

1995 Melchior Hansen, A.: \Reliability Methods for the Longitu-dinalStrength of Ships".

1995 Branner, K.: \Capacity and Lifetime of Foam Core Sandwich Struc-tures".

1995 Schack, C.: \Skrogudvikling af hurtigg�aende f�rger med henblik p�as�dygtighed og lav modstand".

1997 Cerup Simonsen, B.: \Mechanics of Ship Grounding".

1997 Riber, H.J.: \Response Analysis of Dynamically Loaded CompositePanels".

Department of Naval Architecture

And Offshore Engineering

Technical University of DenmarkBuilding 101EDk-2800 LyngbyDenmarkPhone +45 4525 1360Telefax +45 4588 4325

Email

Internet

[email protected]://www.ish.dtu.dk

ISBN 87-89502-36-1

Documents

Response Analysis of Dynamically Loaded Composite Panelsorbit.dtu.dk/files/5436185/Riber.pdf · Response Analysis of Dynamically Loaded Composite ... of Dynamically Loaded Composite