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Hans Jrgen RiberJune 1997

Response Analysis ofDynamically LoadedComposite Panels

Department ofNaval Architecture

And Offshore Engineering

Technical University of Denmark

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Response Analysis ofDynamically Loaded Composite

Panels

by

Hans Jrgen Riber

Department of Naval Architectureand Offshore EngineeringTechnical University of Denmark

June 1997

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Copyright 1997 Hans Jrgen RiberDepartment of Naval Architectureand Offshore EngineeringTechnical University of DenmarkDK-2800 Lyngby, Denmark

ISBN 87-89502-36-1

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i

The research of this thesis was carried out between October 1993 and March 1997 and

submitted as partial fulfilment of the requirements for the Danish Ph.D. degree. The

work 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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iii

The background to this study is the need for handy design tools, which can, in a short

time, calculate the most appropriate material composition and panel scantlings for FRP

sandwich and single-skin vessels. Fibre-reinforced plastic (FRP) is a frequently used

material 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 design

rules 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 numerical

calculation methods are developed, in order to permit the designer to use efficiently the

composite 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 remarkable

geometrical 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 composite

plates are developed by means of energy principles. The first method ( ) is

formulated as a complete solution. The second method ( ) is a simplification of

, dividing the governing equations into a linear part and a membrane part.

This makes 2 suitable as a supplement to existing linear design rules in this

field. The results calculated by use of both analytical methods are in good agreement

with experimental data and numerically calculated results.

A dynamic non-linear finite-difference-based program , dealing with orthotropicsandwich and single-skin panels, is developed. calculates responses and failure

mechanisms for composite plates subjected to various lateral time-dependent loads. The

results of static as well as dynamic response are verified against the commercial finite-

element-based software program Ansys. However, the present method is approximately

50 times faster in CPU-time than Ansys.

A progressive damage model is developed and implemented in . This makes it pos-

sible to improve the design by use of plots of the failure modes, loads and locations. The

failure analysis uses the response from the non-linear analysis, leading to significantly

higher ultimate failure loads than predicted by application of a linear response analysis.

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iv

The ultimate failure loads predicted by are in good agreement with experiments

performed on single-skin FRP panels subjected to high lateral loads.

Analyses of existing HSLC hull panels are presented in order to demonstrate . A

design example is given to show the structural improvements which can be obtained by

application of non-linear calculation methods.

Finally, the High Speed Light Craft rule concerning FRP single-skin and sandwich

panels is discussed in the light of calculations with of hull bottom panels de-

signed by application of the rules. The single-skin rule, which is based on non-lin-

ear theory, is found to be good. However, the maximum lateral deflection criterion of

equal to unity usually limits the design. The criterion seems unnecessary, since the

rule is based on a non-linear theory and, consequently, predicts accurately the panel re-sponses. A non-linear analytical method, ,is suggested as a replacement to the

linear sandwich rule. In addition, the necessity of the maximum relative deflection

criterion of equal to one percent should be further investigated. It seems reasonable

to omit this criterion for particular ships since it often limits the design without apparent

reason.

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v

Baggrunden for dette arbejde er behovet for et hurtigt og godt designvrktj, som pkort tid kan beregne optimale dimensioner og materialesammenstninger for skrogpane-

ler. Fiberforstrkede matrialer (FRP) er ofte benyttet ved bygning af hurtiggende lette

fartjer (HSLC). Dimensioneringen af skrogpaneler i disse typer fartjer er hoved-sageligt baseret p empiriske designregler. Derfor er nye og mere rationelle beregnings-

metoder ndvendige for at kunne forbedre designet. Med dette som motivation er derudviklet analytiske og numeriske beregningsmetoder, som muliggr en bedre udnyttelseaf kompositmaterialer i hurtiggende lette fartjer.

Geometriske ikke-linere effekter (opbygning af membranspndinger) fra laterale ud-bjninger, der skyldes store paneldimensioner i FRP skrog samt relativt hje laterale

tryk (slamming), krver ikke-linere beregningsmetoder til korrekt responsberegning.

Simple analytiske ikke-linere lsninger er udviklet til paneldesign uden brug af

tidskrvende computerberegninger. Disse lsningsmetoder er fordelagtige i designfasen.Baseret p energimetoder prsenteres to forskellige lsninger. Den frste, , eren komplet lsning, hvor alle andenordensleddene indgr i pladeligningerne. Lsning to,

, er en simplificering af den frste lsning. Her lses membrandelen(andenordensleddene) separat fra den linre del. Dette gr ideel som supple-ment til eksisterende linre beregningsmetoder. Resultater beregnet ved hjlp af begge

metoder er i god overensstemmelse med eksperimentelle data samt numeriskeberegninger. En undtagelse er dog spndingsberegninger for fast indspndte plader.

Dernst er der udviklet en finite-difference baseret lsningsmetode til beregning af or-

totropiske sandwich- og enkelt-skinds-paneler. Metoden er formuleret som et design-vrktj, , til paneler udsat for svel statiske som dynamiske lastpvirkninger. Re-

sultater med er verificeret ved hjlp af det kommercielle finite-element baseredesoftware program Ansys. Der er god overensstemmelse mellem gensvarsresultater fra deto programmer, dog er beregningstiden med ca. 50 gange kortere end med Ansys.

En progressiv brudmodel er udviklet og indgr i , hvilket gr 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-linere responsberegning giver somresultat betydeligt strre brudlaste end man finder ved linre beregninger. Det vises, at

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ultimative brudlaste beregnet med er i overensstemmelse med eksperimentielt

bestemte brudlaste for en serie af enkelt-skinds-paneler.

Forskellige analyser og designeksempler udfrt med er vist for at demonstrere

programmet, samt for at synliggre mulighederne for at forbedre det strukturelle design.

Til slut diskuteres klassifikationsselskabet HSLC regler for FRP paneler ud fra

beregninger af typiske skrogpaneler med henholdsvis reglerne og Enkelt-skinds-reglen er begrnset af en relativ maximal udbjning p een. Dette krav virker und-vendigt, idet reglen er baseret p ikke-linr teori og derfor producerer njagtige

beregninger. For sandwichpaneler foresls det at implementere ikke-linere beregnings-udtryk i reglen samt at uddybe maximum udbjningskravet og tillade, at reglen kan

overskrides i specificerede tilflde.

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

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

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

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..................................................................................................... 333.2 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, ......................................... 42

3.3.2 A Combined Analytical Solution, ........................................ 45

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

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

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

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

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

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

4.2.2 Newmarks 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 .....................................................................................................935.2 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

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................................................................................................... 1317.2 The Structure of................................................................................... 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

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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1

Leonardo Da Vinci

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 to

steel and aluminium as building material for high-speed light craft (HSLC) due to a low

weight/strength ratio. The high specific strength of glass fibres together with the superior

specific stiffness offered by carbon and other high-modulus fibres has led to an increasing

use of these materials in fast marine vessels, such as ferries, special military ships and

high-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 boat

building 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 short

time, is able to perform response and failure analysis of sandwich and single-skin FRP

structures.

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 a

number 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 and

Svendsen [1].

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At present, most of the dimensioning procedures for FRP hull plates rely on linear (small

deflection) 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 societies

such as [2] and [3] and further discussed in Riber and Terndrup [6]. Since this

criterion, 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 on

geometrical non-linear plate theory including the transverse shear deformation, which is

pronounced for sandwich plates with relatively flexible core. It is formed into a Fortran-

coded design tool dealing with orthotropic asymmetric composite single-skin and

sandwich plates. Subjected to time-dependent lateral loads and with different boundaryconditions, the plates are analysed statically as well as dynamically with respect to lateral

deflections, strains and stresses. Furthermore, failure loads, locations and modes are cal-

culated and visualised by use of a progressive damage model based on the appropriate

failure criteria.

The concept of the model is to provide a simple and a fast tool, which can be used in the

dimensioning phase of hull panels. With a complete design based on calculations and

analyses with, 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 in

order to eliminate errors introduced by incorrect boundary conditions. However, those

methods are still based on time-consuming FEM calculations, and yet display the problem

of 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/epoxy

skins. It is dimensioned for a vertical acceleration of 5 , equivalent to a slamming load of

125 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 structural

safety prescribed by the national authorities.

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Figure 1.1:

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 rather

than 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.

Figure 1.2:

Navy vessels form a third group of FRP high-speed craft consisting of mine hunters, gun

boats, patrol boats etc. These ships are normally well analysed with respect to the ultimate

strength. 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

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Navy is the 54 multipurpose ship, (Fig. 1.3), which is a glass/polyester foam

core sandwich design originating from the Swedish Navy. Among the most advanced ships

in this group are the Swedish high-speed craft and . The first is a 30 SES test boat, whereas the latter is a 75 multi-purpose ship with approximately the same

displacement as the .

Figure 1.3:

The subject of this thesis is presented in 9 chapters composed as follows. Chapter 2 gives

an overview and an introduction to FRP hull manufacturing and structural hull design,

followed by the appropriate design loads with special focus on slamming pressures. In

Chapter 3 the general non-linear sandwich theory is presented and two analytical solutions

are derived in order to provide alternative simple design methods for FRP sandwich

plates. Chapter 4 presents a numerical formulation of the theory given in Chapter 3. The

result is programmed into a design tool , which is intended for preliminary design ofFRP hull panels. Chapter 5 discusses different failure criteria and failure modes, which are

implemented in a progressive damage model described in Chapter 6.

A brief introduction and a description of the design tool are given in Chapter 7 il-

lustrated with examples of designs and analyses of FRP sandwich hull plates. Chapter 8

discusses theHSLC code in the light of results calculated with.

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[1] Bau-Madsen N.K., Svendsen K.H. and Kildegaard A. Large Deflections of SandwichPlates - an Experimental Investigation. . 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 ResearchAS, Veritasveien 1, N-1322 Hvik, Norway, 1991.

[4] Falk L. Membrane Stresses in Laterally Loaded Marine Sandwich Panels.

, Southampton,

UK. Vol. 1 (4A), 1995.

[5] Hildebrand M. and Visuri M. The Non-linear Behaviour of Stiffened FRP-SandwichStructures for Marine Applications. , Espoo, Fin-

land, 1996.

[6] Riber H.J. and Terndrup Pedersen P. Examination of Criteria for Panel Deflection inDNVs Rules for High Speed and Light Craft. Technical Report No. 96-2014, Det

Norske Veritas Research AS, Veritasveien 1, N-1322 Hvik, Norway, May, 1996.

[7] Shenoi R.A., Allen H.G. and Moy S.S.J. Strength and Stiffness of FRP Plates. , May 1996.

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Fibre-reinforced plastic (FRP) composites are among the most commonly used building

materials for high speed light craft (HSLC) hulls. This is mainly because of the high

strength-to-weight ratio of the material, which is ideal for construction of ship hulls and

makes it a cost-efficient material. Further, the FRP is corrosion-resistant and has a low

maintenance cost. Finally, the low magnetic characteristic of most FRPs makes them

suited 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 (

> 50 ) the hull flexibility must be

considered as the relatively low hull beam stiffness of FRP ships compared to steel ships

may introduce fatigue damages in the hull, Hansen et al. [8].

The common definition made by (), of

when a vessel is categorised as a high-speed craft is a minimum requirement for the serv-

ice speed/displacement ratio, which states:

37 1 6. / (2. 1)

with the forward speed, , in knots and the displacement, , in tons. In general, the clas-sification societies use this definition for HSLC and apply special design rules for these

types of vessels. The leading classification societies providing rules for HSLC design are

American Bureau of Shipping (), Bureau Veritas (), Det Norske Veritas (),

Lloyds Register () and Registro Italiano Navale ().

Most of the existing FRP high-speed craft are small (

). This is mainly due to the

limitations 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 of

length will appear in the future. Currently, the Swedish Navy is building a multipurpose

warship with a maximum speed of approximately 50 and a length of about 75 ,using foam core sandwich with skins of glass, carbon/aramid fibres in a vinylester resin

for the hull.

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 simply

technical 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 of

using stiffened single skin for their HSLC marine vessels from the point of view that

shock 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. The

two 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 two

different building concepts and to give an idea of the limits of the design rules.

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 with

resin 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 vacuum

techniques and temperature-regulated moulds. This technique requires expensive tools

such 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

Figure 2.1:

The often complicated stiffener system is laborious to manufacture, especially as the stiff-

ener attachment to the hull requires careful mechanical surface preparations. Delamination

of the stiffener from the hull is often observed in regions with high impact loads and

where the stiffener has been badly assembled. The stress concentrations can be decreased

drastically 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 radius

of curvature is very small, whereas the right stiffener evens out the stress level due to the

longer radius of curvature.

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Stress concentrations

and delamination

Stiffener reinforcement Curvature 1/

Figure 2.2:

The primary structural design criteria for single-skin plates in the bottom of the hull pro-

vided by the classification societies, Bureau Veritas, [3], and Det Norske Veritas,[5], are listed and commented on in the following (Eqs. 2.2-7). The numbers in brackets

refer to the numbers in the respective rule set. Note that the units are in SI.

Minimum thickness, min , of the skin:

min . .= +15 10 0 97 103 (C.3.8.4.3.34) (2. 2)

min

. .

.

= +

105 0 0 09

16 10

3

8

(A 202, Sec. 6) (2. 3)

where

is the waterline length and

is the ultimate tensile stress.

4

6

8

10

12

14

16

0 20 40 60 80 100

Eq. 2.2

Eq. 2.3

Length

[]

[] Minimum hull thickness as function of hull length

= 160

Figure 2.3: .

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In the range of 10 - 100 metres of length the requirement for the minimum skin

thickness is approximately 12 % lower than the requirement for , (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,

, from a given load, , on a square simply supported panel:

max . .=

0 313 0 22

2

nu (C.3.8.4.3.35) (2. 4)

max

. .

.=

+

+

2 42 6 47

30 230 302

2

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

where is the midpoint lateral deflection, is the skin thickness, is the plate breadth

and is the lateral pressure.

Eq. 2.5 is a combination of the rules B 202-203, in order to make a better compari-

son with the rule (Eq. 2.4). The rules are presented for the special case of a plate with

an aspect ratio equal to one and simply supported boundary conditions. However, both sets

of rules provide correction factors depending on varying aspect ratios and boundary con-

ditions. Therules are based on non-linear theory and consequently, they are less con-

servative than the rules (Figs. 2.4-5). Furthermore, the maximum allowable stress

value given byis 35 % higher than suggested by.

0

5

10

15

20

25

30

35

0 0,5 1 1,5 2

Eq. 2.4

Eq. 2.5

Relative deflection

[] Maximum stress as a function of relative deflection

Figure 2.4:

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

=

4

21 48 10100

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

+

=

3 4

22 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 with

clamped boundary conditions (only case provided in the code).

0.0

0.5

1.0

1.5

2.0

0 5 10 15 20 25 30

, Eq. 2.6

, Eq. 2.7

Lateral load []

Relative deflection as a function of lateral load

Figure 2.5:

At the maximum deflection ( ) the rule, which is rewritten in the form of Eq.

2.7, allows 42 % more lateral load than calculated by the rule (Eq. 2.6) for a represen-

tative GRP hull plate (Fig. 2.5). It is evident that the code is more sophisticated than

the 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 the

minimum scantlings. The rules must be supplied by additional design formulae regarding

local and global buckling, stress analyses at specific locations etc. in order to ensure a

complete structural analysis of the hull components.

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A sandwich consists of three main parts (Fig. 2.6): face (or skins), core and a bonding

material. The sandwich structure is defined by ASTM [2] as follows:

Figure 2.6:

The primary advantage of using the sandwich concept in a FRP hull instead of a stiffened

single-skin structure is the built-in flexural stiffness of the sandwich, which makes the

stiffener system unnecessary. The bending and the in-plane stresses are mainly carried by

the 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 building

a single skin hull.

The most common production method of a sandwich hull is to make use of a female mould

and proceed as for the single skin hull. After the outer skin has been formed in the mould

the core, usually PVC foams but also aluminium or resin-impregnated honeycomb, is

bonded to the skin employing an adhesive, which is most often the resin used for the

skins. 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 , in case of

large hulls or when only a small series of hulls is built. The ship is built upside down by

using the transverse frames as a male mould on which the core is formed. Then, the outer

skin 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.

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14

Face thickness = and

( )

1

3 3

1

1

122

2

3

2

= =

=

2

2

3 3

1

2 1

2

2

1

1

23

22

1

12

14

37

22

25104

6 73

=

+ = =

= + =

=

.

.

3

2

3 3

1

3 1

3

3

1

1

211

22

1

12

182

391

210

2512

758

=

+ = =

= + =

=

.

.

Figure 2.7:

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 specific

weight, , and the stiffness/weight ratio,, for a representative GRP hull sandwich. For

a modern sandwich hull design the face/core thickness ratio is about 1/10, which gives a

relative bending stiffness of almost 75 times the stiffness of the equivalent single skin. It

should 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,[3], and Det Norske Veritas, [5], are listed in the following (Eqs. 2.8-13).

Minimum thickness, min , of the faces:

min . .= +

0 6 10 0 97 103 (C.3.8.4.4.42) (2. 8)

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15

min

. .

.

= +

1015 0 09

16 10

3

8

(A 203, Sec. 5) (2. 9)

2

4

6

8

10

12

0 20 40 60 80 100

Eq. 2.8

Eq. 2.9

Length

[]

= 160

[] Minimum face thickness as function of hull length

Figure 2.8: .

The linear rule for minimum face thickness penalises unnecessarily long ships. The

non-linear formula given byseems more reasonable.

Maximum stresses, max,

, and deflection,

, from a given pressure,:

max . .= 0 052 0 222

nu (C.3.8.4.4.43) (2. 10)

max

. .= 055 0 4

nu

(C.3.8.4.4.47)

max . .= 0 050 0 302

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

max . .= 0 34 0 35

nu (B202, Sec. 5)

where is the section modulus of the sandwich plate. For a sandwich with equal face

thickness, we get

, where is the distance between the neutral axes of the two

faces. The rules are given for sandwich plates with aspect ratio equal to one and simply

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16

supported boundary conditions. The rules provide correction factors depending on

different aspect ratios and boundary conditions and, consequently, represent a more de-

tailed set of rules, than the one of.

The face stress response is approximately the same for the two expressions (4 % higher

predicted by) but the maximum allowable stress given by is 35 % higher than the

one imposed by . The core shear stress predicted by is almost 62 % higher than the

one of. This is due to the simplification of therule, which covers all aspect ratios

in one single expression. For larger aspect ratios, the core shear stress () is only 8 %higher than calculated by , hence the is based on beam theory more than platetheory.

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

= 2.47 + 75.6 4

2

2

100 (C.3.8.4.3.38) (2. 12)

= 2.03 + 74 4

2

2

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

where

is the E-modulus of the faces and

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 therule.

11 0 252

2

. .

The rule is the most cautious of the two and gives the deflection response for clampedboundary conditions and symmetric sandwich plates only. The 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 and 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, 0.01, generally determines the scantlings of the plate, eventhough the stress levels are far below the allowable limits.

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17

The dimensioning loads for the hull of small high-speed craft are mainly impact loads

from vertical accelerations of the hull penetrating the water surface, i.e. slamming. The

structural 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 dimensioning

rules from most classification societies neglect the global loads. If the ship is below a

certain overall length. , for example, requires only analyses with local design loads if

the ship is less than 50 in length.

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 consideration

as 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 moment

2.Hollow landing moment

3.Hogging moment

4.

Sagging moment5.Shear forces from longitudinal loading

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

1.Torsional moment

2.Transverse bending moment

3.Transverse shear force

Transverse stress resultants of twin hull

Figure 2.9a: .

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18

Slamming-induced global moments

Wave-induced global moments

Figure 2.9b:

Rough estimates (from [5]) of the above illustrated global bending moments andshear forces for mono-hulls are given here. The crest and hollow moments are derived

considering the hull as a simple beam (Fig. 2.10), where is the ship displacement,

=

9.81 ,

is the design acceleration and

the extent of the longitudinal slamming area,

(SI units).

(longitudinal centre of gravity)

( )

= + 0

for forward

and aft half of ship

Figure 2.10:

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19

( )

= +

2 40

(2. 14)

( )

= +

2

10 (2. 15)

The hogging/sagging moments and the shear forces are derived by integration of the forces

from still-water analyses (buoyancy and body forces) in addition to the resultants from thewave contribution (hydrodynamic forces), Pedersen and Jensen [15]. Tentative design

formulas are given below for ship length

,[5].

(still water + wave)

= 24 3 (2. 16)

(still water + wave)

( )

= + +

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

= 4 103 (2. 18)

where

,

,

and are length, breadth, block coefficient and maximum speed, re-

spectively, (SI units).

The non-linear sandwich theory does not take into account local bending of the faces due

to vertical displacement of the core. According to the definition of a sandwich:

(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-

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20

ure prediction it is essential to analyse local bending effects in order to determine some

types of delamination.

Thomsen [16] derives an analytical expression for approximate solutions of local bending

effects in sandwich plates with orthotropic face layers subjected to localised loads. The

local 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 in

face/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 theory

and 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 concerning

edge delamination, the method is recommended for small lateral deflections.

A rather irregular load on high-speed craft is the slamming pressure, which is caused by

the impact of the bottom of the hull against the water surface resulting in a sudden change

of the relative acceleration of the boat. Slamming is an impulsive pressure during a very

short period of time (milliseconds). For design of FRP hull panels the slamming pressure

is generally the dimensioning load. A theoretical derivation of the slamming pressure is

shown, followed by a simple approach to determine an equivalent static pressure as the

design load.

It may be argued that the peak pressures have little importance for the panel response

since 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).

= ( ) ( , , ) (2. 19)

where ( ) is a response function.

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21

Typical slamming measurements are shown in Fig. 2.11. The duration of the pressure

peaks 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 []

Time []

Fig. 2.14

Slamming measurements on a 470 hull panel

Sample frequency 33

Figure 2.11:

A simple way to model a hull slamming pressure is to consider the problem of a wedge

penetrating 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 is

based on the theory by Karman.

Hansen [7] compares the different slamming theories of the above-mentioned authors and

concludes that the simple theory by Karman produces adequate results. The following

derivation of the slamming pressure is based on the work by Karman.

A wedge-shaped body of mass and of a dead-rise angle strikes a horizontal surface ofwater with the velocity

and generates a two-dimensional flow (Fig. 2.12). The wedge is

considered 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 is

. The momentum,

, of this

system becomes:

(2. 20)

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22

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

piled up water

Figure 2.12:

The added mass, , comes into existence when the body pushes away the fluid in front of

it, which creates a flow around the body. The added mass is found from the kinetic energy

of the fluid put in motion by

( )

= = 2

2 2

2 (2. 21)

which is transformed by Greens theorem to

=

2 (2. 22)

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

length becomes:

=1

2

2 (2. 23)

as the potential for the flat plate is given by

( , )

=

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23

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

, as

= = tan (2. 26)

Eq. 2.20 becomes

tan ( )

12

2

+ = (2. 27)

yielding

( ) cot1 1+ = ,

1

2

2=

(2. 28)

which gives the relationship between velocity and depth as

=+cot

1 1 (2. 29)

Expressing the second derivative of

2

2

12

2= ( ( ) ) (2. 30)

and combining this expression with Eq. 2.29, we get

2

2

2

2

2 2

1

31 = =

+ cot

cot

( )( )

(2. 31)

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

= =

+

( )cot

( )

2

2

2

1

31

(2. 32)

The average pressure becomes

= =

+2 2 1

2

1

3

cot

( ) (2. 33)

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24

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

max cot=

2

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 velocity

potential (Eq. 2.24) with Bernoullis equation for unsteady potential flow, neglecting theeffect of gravity, (Eq. 2.35).

= + +

( ( ) ( ))t y

1

2

2

and ( , )

= 1 there is a negative pressure zone around the keel as the second term approaches

the value ( ) +2 11 1 , since first term always contributes 1 and the last term nothing atthe keel. For small masses a relatively small plate length is required to make 1> 1(Eq. 2.38). Consequently, a higher probability of having negative pressures around the

keel can be expected for smaller masses than for large. The balance between the first and

second term is physically explained as the balance between the deceleration of the body

and the motion of the water mass around it.

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26

Impact with Flat Bottomed Hull

If a flat bottom (= 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 taken

into account. By taking the compressibility of water into consideration, it is possible to

obtain an approximate value for the maximum pressure occurring when a flat body strikes

a horizontal water surface. The mass of fluid, , accelerated in the time, , is

= (2. 40)

where is the speed of sound in the water (1440 ) and the surface of fluid struck by

the body. If the dominating force acting on the fluid originates from the body, the equiva-

lent forceacting on the body is found from

= = + (2. 41)

Here

is the impulse from the mass of liquid surrounding the body and

is the vertical

velocity, which is assumed to be constant at the impact phase where the slamming pres-

sure happens. Eqs. 2.40-41 yield

= (2. 42)

and the pressure averaged over the surface becomes:

= = = 2

2

21440, (2. 43)

Thus, the pressure turns out to be a factor 2

times the stagnation pressure, which is

not a reasonable result.

Design Method

A simple approach to providing an equivalent uniform static pressure for each structural

component under localised water impact is proposed by Allen and Jones [1]. This method

is 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 rule con-

cerning bottom hull slamming pressure for HSLC is partly based on the results from Allen

and Jones [1] and given in the following for a mono-hull.

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27

=

13 10

10

50

50

3

3

0 3

0

0 7.

.

.

g (2. 44)

in which

is the load area of the element considered (for plates

),

is the

dead-rise angle at (10 <

< 30 [deg]),

is the draught at service speed and

is

the vertical design acceleration given as:

[ ]

= 0 76 58138 1 7. . , , (2. 45)

where

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 of

HSLC the acceleration levels for crew tolerance and structural design are most frequently

given as the average of the one-tenth highest acceleration, and the equivalent pressure is

found from this imposed or accepted acceleration level, without regard to any empirically

or theoretically based design formulas. Table 2.1 from Koelbel [10] provides a general

guidance for selection of vertical accelerations for structural design.

[ ] Human affects Structural application

0.6 minor discomfort craft for passenger transport

1.0 maximum for mili tary function

long term (> 4hr)

1.5 maximum for mili tary function

short term (1-2 hr)

2.0

3.0

extreme discomfort patrol boats, crews, average owners, test

crews, anglers, long races

4.0

5.0

6.0

physical injury

medium length races

race boat drivers, short races

military crew under fire

Table 2.1:

A serviceability design formula for a maximum allowable speed at a given significant

wave height,

, and the vertical design acceleration (Eq. 2.45) is given by as

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28

( )

=

+

9 81

1650 0 084

501852

10

22

3

.

.

.

(2. 46)

where

is the waterline breadth at

.

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.

= 17 0

pressure transducer= 0.2

= 0.21

(Fig. 2.11)

Pressure []

Time []

Sample frequencyof

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

=

[]

[]

Figure 2.14:

The full-scale tests are carried out with a 470 sailing boat in protected water (

1.0). A

pressure transducer is mounted in the bottom hull panel (Fig. 2.14) and the data are logged

while sailing. The constants in Eqs. 2.33, 2.40 and 2.47 are listed below.

54

3.1 0.21 17

10 1015

0.09

0.10 3.1

1.0

4.0 = 260

The results with the calculations of the different slamming expressions and the full-scale

tests are shown in Table 2.2.

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29

4.1 average

8.7 peak

9.6 average

7.9 peak

4531 flat out1

Table 2.2:

The highest pressure is obtained by the 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 two

times lower. The measured pressure () and the pressure obtained by Eq. 2.40 both rep-resent peak values of the slamming. The above example indicates that the rule pro-

vides reasonable and safe design loads.

1

An overview of FRP sandwich and stiffened single-skin hull manufacturing and structural

design is presented. In addition to this, the corresponding design rules provided by two of

the leading classification societies ( and ) are discussed. The rules are moredetailed and less conservative (except for the minimum thickness) than the rules. Fur-

thermore, therules concerning stiffened single skin take into account the geometrical

non-linear behaviour for large deflections. However, the sandwich rules are still based on

linear theory for both the codes and need further investigation and development.

Global and local loads concerning FRP hull structural design are outlined with focus on

slamming, as this is usually the dimensioning load for the design of hull panels in HSLC.

Moreover, tentative rules for the design loads provided by are presented. Results

from full scale tests on a 470 sailing boat are compared to the design formula and to

theoretical derived expressions for the slamming pressures.

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30

[1] Allen R.G. and Jones R.R. A Simplified Method for Determining Structural Design

Limit Pressures on High Performance marine Vehicles.

, 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.

. 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 Hvik, Norway, 1991.

[6] DNV. Response of Fast Craft Hull Structures to Slamming Loads. . Vol. 1, pp. 481-398,1991.

[7]

Hansen A.M. Sammenligning af Slammingteorier. Department of Naval Architectureand Offshore Engineering, DTU, Lyngby, Denmark, 1991, (in danish).

[8] Hansen P.F., Juncher Jensen J. and Terndrup Pedersen P. Long Term Springing andWhipping Stresses in High Speed Vessels. . 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.

Vol. 32 (2), pp. 77-100, April, 1995.

[11] Ochi K.M. and Bledsoe M.D. Hydrodynamic Impact with Application to Ship Slam-ming. . Washington DC, August. 1962.

[12] Payne P.R. The Vertical Impact of a Wedge on a Fluid. . Vol. 8(4), pp. 421-436, 1981.

[13] Riber H.J. Strength of a 470 Sailing Boat. MSc. thesis at the Department of NavalArchitecture and Offshore Engineering, Technical University of Denmark, 1993.

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31

[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. . Vol. 30, pp. 85-101,1995.

[17] Wagner V.H. ber Stoss und Gleitvorgnge an der Oberflche von Flssigkeiten.ZAMM. Vol. 12, pp. 193-215,1939, (in German).

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33

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 of

the 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 numerical

non-linear finite difference calculations and large-deflection experiments of equivalent

plates. The presented methods (also described in Riber [13]) lead to good results for plate

response and provide an alternative method for the design of sandwich plates subjected to

high lateral loading.

Pronounced lateral deflections introduce in-plane displacements and membrane strains in

the 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 and

cores. Based on Reissners 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

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34

rule is presented for predicting deflections, strains and stresses in sandwich panels with

large deflections. The results are discussed and compared to experimental data obtained by

Bau, Kildegrd and Svendsen [3] and equivalent numerical finite difference calculationsperformed by Riber [12] in Section 3.4, followed by a summary.

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 the

higher-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 the

sandwich faces.

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 , , and , respec-tively. The origin of the co-ordinate system lies in the middle plane (for sandwich in the

geometrical 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 thickness

and

and the core thickness

. 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

is neglected.

5. Non-linear terms, i.e. the derivatives of the lateral deflection in the strain displace-ment relations, are retained whereas the equivalent non-linear terms of the in-planedisplacement terms are omitted.

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35

6. The deflection can be divided into two parts: =

+

(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 Youngs modulus of the core is small compared with that of the face(s), i.e.

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36

The displacement field is assumed to be of the form

( , , ) ( , ) ( , )

( , , ) ( , ) ( , )

( , , ) ( , )

= += +

=

(3. 1)

where are the displacements in the anddirections, respectively, and

are the

cross-sectional rotations in theand-planes due to bending. Assuming that we may separate

the lateral displacement into contributions due to bending and shear and then superimpose them

to give the total deflection, we have

= + (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 the

backbone of the non-linear design program (Chapter 7). The cross-sectional rota-

tions may now be written as

= = = + = + , , (3. 3)

This means that the bending moments and the shearing forces will be independent of each

other, which is correct for panels with equal rigidities in both - and -directions or the

same neutral axis for both cross-sections. However, this also applies to orthotropic panels

and 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:

, , , , , , , (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 as

the coupling effect becomes significant. Eqs. 3.5a-b express the strains in terms of the displace-

ment derivatives and the above partial deflections for as follows:

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37

= +

= +

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

= + + =

= =

= =

(3. 5b)

Iffor an orthotropic material is applied and it is assumed that the stress com-

ponent in the -direction vanishes everywhere, the constitutive relations for the th layer

are given as

=

=

11 12

12 22

66

44

55

0

0

0 0 2

0

0

2

2

(3. 6)

In the above expressions, the coefficients

in the stiffness matrix are defined in Vinson

[14] for linear elastic materials. If the principal main material axes do not coincide

with 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 :

[ ] [ ][ ] [ ] [ ]

= =

1

2 2

2 2

2 2

2

2,

cos sin cos sin

sin cos cos sin

cos sin cos sin cos sin

(3. 7)

where is the angle between the main fibre direction and the plate axis of ply number .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):

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=

+

+

+ +

+

11 12

12 22

66

12

2

12

2

11 12

12 22

66

2

0

0

0 0

0

0

0 0

2

2

2

2

2

(3. 8)

=

44 44

55 55

0

0

(3. 9)

=

+

+

+

+

11 12

12 22

66

2

2

2

2

2

11 12

12 22

66

2

20

0

0 0

2

0

0

0 0

1

2

1

2

+

(3. 10)

The matrices , ( =1,2,6) and ( = 4,5) represent the extensional, bending- andshear-stiffness, respectively. The coupling matrix

between in-plane forces and bending

deformations vanishes in the case of plate symmetry. The relation between the transverse

forces and the shear deflection (Eq. 3.9) becomes

= =5 55 4 44 and (3. 11)

where the

factors are dependent on the core material. For homogeneous isotropic plates,

it can be shown that the value of is 5/6 according to Reissner [11]. The stiffnesses,

,

and

, are given below for a single-skin plate and for the faces and core (indices )of a sandwich plate, as follows:

( )

= = ==

( ) , , , , ,11

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

and

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( )

( )

= = =

= =

=

=

1

20 1 2 6

1

31 2 6

2

1

2

1

1

3

1

3

( )

( )

, , ,

, , ,

(3. 13)

The above expressions can be applied directly to a single-skin plate. As for the sandwich

plate, assuming that the faces are thin and the shear is carried by the core ( ),

we get the following expressions for the stiffness matrices:

= + =

= =

1 2 12 6

4 5

(3. 14)

and

( )

= =

= +

+

+

=

212 6

2 212 6

1 2

1

2

1 2

2

2

(3. 15)

where

refers to face 1 and

refers to face 2. Here the coupling terms

do not arise

due to asymmetry in the faces since they are considered thin, but as a result of the differ-

ent in-plane stiffness of the two faces.

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

tions including the body forces

+

+

2

2

Figure 3.2: .

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+ =

+ =

+ = +

+ + =

+ + =

0

0

0

2

2

0

(3. 16)

where

,

= + + + = +

2

2

2

2

2

2

Here, 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 terms

of the displacements, we combine Eq. 3.16 with Eqs. 3.8-10 and obtain four coupled non-

linear differential equations in , ,

and

, where

. The two in-plane equi-

librium equations (Eqs 3.17-18):

( ) ( )

11

2

2 66

2

2 12 66

2

11

3

3 12 66

3

22

+ + + = + + + (3. 17)

where

( )

= +

+

11

2

2 66

2

2 12 66

2

and

( ) ( )

66

2

2 22

2

2 12 66

2

22

3

3 12 66

3

22

+ + + = + + + (3. 18)

where

( )

= +

+

22

2

2 66

2

2 12 66

2

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The shear equation becomes

55

2

2 44

2

2

2

2

+ =

(3. 19)

and finally the bending equation yields

( )

( )

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

+ + + =

+ + + + +

+

(3. 20)

The above set of equations must be combined with the appropriate boundary conditions of

the specific problem. If we regard the right hand side of the equations as body forces and

as lateral loads of magnitude and

(

), the equations are identical to the

governing equation for small deformations of an elastic plate. The numerical solution of

these equations will be described in Chapter 4, whereas the analytical solutions based on

energy principles will be outlined in the following.

In this section two different analytical solutions of the non-linear differential equilibrium

equations 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 based

on the theorem for the minimum potential energy, which states:

. The energy introduced from a virtual displacement due to an externalloadcorresponds to the equivalent strain energy in the plate. The total energy (,,),

which has a stationary value, is then minimised and the assumed deflection functions ,

and are found by use of known boundary conditions together with the derivatives of thetotal energy of the system, with respect to the unknown deflections , and .

The author has not, so far, found simple non-linear analytical solutions for sandwich

plates 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, , and a combined

solution, .

Many design rules concerning single-skin composite plates are already based on non-linear

theory. However, this is not the case in analytical design of sandwich structures, where the

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42

existing design rules recommended by the classification societies are based on linear plate

theory. 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-based

computer models. These may, of course, be used in later structural verification and optimi-

sation of the design or for problems with special boundary conditions.

The total energy of the plate can be expressed as the sum of internal strain energy,

,

and the potential energy,

, due to external loads . Minimisation of the total energy

+

, with respect to the parameters in the deflection functions, gives the following equa-tions:

( )

1 20

+= (3. 21)

where

present undetermined parameters in the deflection functions, which depend on

the given plate boundary conditions. The strain energy of an elastic plate in terms of an

- co-ordinate system is given by the relationship

( )

= + + + + +1

2 (3. 22)

where the triple integration is performed over the volume of the plate. Taking into account

the assumption of no strain in the direction ( ) together with the ply

stress/strain relations stated in Eq. 3.6, we obtain:

( )

1= + + + + +1

2211

2

22

2

12 66

2

44

2

55

2 (3. 23)

This relationship can be expressed in terms of the plate displacements , ,

and

by

substituting 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

in the energy

expressions 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 being

symmetric. The assumed deflection functions for , ,

and

depend on the type of

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43

boundary conditions. The complete solution for the plate response due to a lateral load

will be derived for the simply supported and the clamped cases.

Simply Supported Plate

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

and

. A third condition illustrated in Fig. 3.3 can either be zero effective twisting mo-ment:

=

0 and

= 0 (3. 24)

along the edges parallel to the - and axes, allowing for shear,

0,

0, i.e. boundary conditions, or zero shear deformation

,

, allowing for the existence of

effective twisting moments, i.e. boundary conditions

= 0

= 0

Figure 3.3:

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, of

course, not be true in all practical cases. Thus, we need deflection functions which satisfythe following boundary conditions:

(3.25a)

The following deflection functions satisfy these boundary conditions:

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( )

= +

= = =

=

sin sin

sin sin ,

sin sin

2

2

, (3. 25b)

Figure 3.4:

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

and the potential

energy

from the work of the external lateral load expressed in the following and

shown in detail in Appendix A.

( )

11

33

1= == , , , (3. 26)

( ) ( )

2 2

4= = + = + sin sin (3. 27)

The in-plane displacements, and , 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, , with respect to

, , , gives us adequate equations to determine these

coefficients. The final expressions yield

= = =

+ + =

7 82

9

2

3

2 3 0

, ,

(3. 28)

where the constants 7, 8, 9,

and

(Appendix A) are functions of the plate proper-

ties, including length and breadth, , , and the stiffness matrices,

,

,

.

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

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(3.29a)

where the deflection functions satisfying these boundary conditions are

( )

= +

= = =

=

sin sin

sin sin ,

sin sin

2 2

2

2

, (3. 29b)

Using the same procedure as in the case of the simply supported plate, we obtain the total

strain energy:

( ) 11

33

1= == , , , (3. 30)

The energy terms are outlined in Appendix A. The potential energy,

,from the work of

the external lateral load, , is slightly smaller and becomes:

( ) ( )

2

2 2

4= = + = + sin sin (3. 31)

Finally, we obtain the same relations as for the simply supported case expressed in Eq.

3.26, with the constants 7, 8, 9,

and

given in Appendix A.

The strains and stresses can be derived from the displacement functions of , and ,

which will be demonstrated along with the derivation of .

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 functions

is 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 for

the non-linear plate response, an alternative method, ,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

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46

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 simple

method consisting of a combination of the known theory of small deflections and the

membrane theory solutions may be used. We assume that the load can be resolved into

two parts,

and

, so that

is balanced by the bending and shearing stresses calculated

from the small-deflection theory and

is balanced by the membrane stresses. Thus, weobtain:

3 (3. 32)

This third-order polynomial is solved for :

= + + + +

= =

3 23 3 23

1

2 23 2 ,

(3. 33)

Hence,

and

are found from Eq. 3.32, where the corresponding stresses are calculated

by using

for the small-bending/shear deflection and

for the membrane deflection.

The total strains and stresses are achieved by superposition of strains and stresses due to

the loads

and

. The parameters

and

are found from the small-deflection plate

bending/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 , and byapplication of the same procedure as for the previously demonstrated . The

strain energy

of a membrane, which is due solely to stretching of its middle surface, is

given by Eq. A.1 omitting the terms involving

and

.

( )

( )

= + +

= + + +

1

2

2112

22

2

12 66

2

(3. 34)

The membrane parts of the strains,

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

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= +

= +

1

2

1

2

2 2

, (3. 35)

and

xym

= + +

Substituting these strain expressions into Eq. 3.34, we obtain an energy expression

for

the membrane part, using the same procedure as in the previous section.

=

+

+

+

+

+

+

+

+

+

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

66 66 12

12

2 2

12

2

12

2

2 2

1

2

+ +

+

+

+

+

(3. 36)

When the energy method is applied we must assume suitable expressions for the displace-

ments , and in order to satisfy the boundary conditions. A rectangular plate with its

edges fixed in the,anddirections behaves like a simply supported plate in all cases

as the membrane has no bending stiffness. Thus, we obtain the same functions as the ones

in Eq. 3.25. Inserting these functions into Eq. 3.36 and integrating over the plate area, we

obtain

( )

= ==

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 , given in Appendix A. Application of the principle of vir-

tual displacements leads to the following three equations:

= =

=

0 0

00

sin sin

(3. 38)

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After some reduction, the fina