10-Ship Structure - Introduction to Marine FRCs (2012)

Ship Structure − An Introduction to Marine FRCs SHIP STRUCTURE – AN INTRODUCTION TO MARINE FIBREREINFORCED COMPOSITES References & Extracts from: 10 MODULE Design Guide for Marine Applications of Composites [Final report for the Ship Structure Committee Commandant (USCG)] Eric Greene Associates, Annapolis 1995 Composite Material Concepts The marine industry has been saturated with the concept that it can build stronger and lighter vehicles through the use of composite materials This may be true but only if the designer fully understands how these materials behave Without this understanding, material systems cannot be optimised and indeed can lead to premature failures Metal construction involves an understanding of material specific properties and a knowledge of weld geometry and techniques Composite construction introduces a myriad of new material choices and process variables This gives the designer more latitude and avenues for optimisation, however, with this opportunity comes the greater potential for improper design Early fibreglass craft featured single-skin construction in laminates that contained a high percentage of resin Because these laminates were not as strong as those produced today and because constructors’ experience was limited, laminates tended to be very thick, made from numerous plies of fibreglass reinforcement These structures were very nearly isotropic (properties similar in all directions parallel to the surface) and were very forgiving In most cases craft were overbuilt from a strength perspective to minimise deflections With the emergence of sandwich laminates featuring thinner skins, the need to understand the structural response of laminates and failure mechanisms has increased Reinforcement and Matrix Behaviour The broadest definition of a composite material describes filamentary reinforcements supported in a matrix that starts as a liquid and ends up a solid via some curing process The reinforcement is designed to resist the primary loads that act on the laminate and the resin matrix serves to transmit loads between the plies, primarily via shear In compression loading, the resin can serve to stabilise the fibres for in-plane loads and to transmit loads via direct compression for out-ofplane loads Mechanical properties for dry reinforcements and resin systems differ greatly As an example, E-glass typically has tensile strength of 3450 MPa and an ultimate elongation of 4.8% A polyester resin typically has a tensile strength of 69 MPa and an ultimate elongation of 2% As Ship Structure – An Introduction to Marine FRPs laminates are stressed near their ultimate limits, resin matrix systems generally fail first The designer is thus required to ensure that a sufficient amount of reinforcement is in place to limit overall laminate stress Contrast this to a steel structure, which may have a tensile yield strength of 480 MPa, an ultimate elongation of 20% and a stiffness that is an order of magnitude greater than ‘conventional’ composite laminates Critical to laminate performance is the bond between fibres and resin, as this is the primary shear stress transfer mechanism Mechanical and chemical bonds transmit these loads Resin formulation, reinforcement sizing, processing techniques and laminate void content all influence the strength of this bond Directional Properties With the exception of chopped strand mat, reinforcements used in marine composite construction utilise bundles of fibres oriented in distinct directions Whether the fibres are aligned in a single direction or a combination thereof, the strength of the laminate will vary depending on the direction of the applied force When forces not align directly with reinforcement fibres, it is necessary for the resin system to then transmit a portion of the load Balanced laminates have a proportion of fibres in 0 and 90 directions Some newer reinforcement products include 45 fibres Triaxial knits have 45 fibres, plus either 0 or 90 fibres Quadraxial knits have fibres in all directions Design and Performance Comparison with Metallic Structures A marine designer with experience using steel or aluminium for hull structure will immediately notice that most composite materials have lower strength and stiffness values than the metal alloys used in marine applications Values for strength are typically reported as a function of cross-sectional area (MPa or GPa) Because composite materials are much lighter than metals, thicker plating (panels) can be used Because thicker panels are used for composite construction, panel stiffness can match or exceed that of metal hulls Indeed, frame spacing for composite vessels is often much greater For a given strength, composite panels may be significantly more flexible, however, which can lead to in-service deflections that are larger than those for metal hulls The above discussion pertains to panel behaviour when resisting hydrostatic and wave slamming loads If the structure of a large vessel is examined then consideration must be given to the overall hull girder bending stiffness Because structural material cannot be located further from the neutral axis (as is the case with thicker panels), the overall stiffness of large vessels is limited when quasi-isotropic laminates are used This has led to concern about main propulsion machinery alignment when considering construction of FRP vessels over 90 metres in length With smaller high performance craft such as racing sailboats, longitudinal stiffness is obtained through the use of longitudinal stiffeners, 0 reinforcements or high modulus materials, such as carbon fibre 4.1 Damage and Failure Modes Damage and failure modes for composites also differ significantly from those of metals Whereas a metal grillage will transition from elastic to plastic behaviour and collapse in its Ship Structure − An Introduction to Marine FRCs entirety, composite material will fail one ply at a time, causing a change in strength and stiffness, leading up to a catastrophic failure This would be preceded by warning cracks at ply failure points Crack propagation associated with metals typically does not occur with composites Interlaminar failures between successive plies is much more common This scenario has a much better chance of preserving watertight integrity Because composite laminates not exhibit the classic elastic to plastic stress-strain behaviour as metals do, safety factors based on ultimate strength are generally higher for composite construction, especially for compressive failure modes Properly designed composite structure see very low stress levels in service, which in turn should provide a good safety margin for extreme loading cases Many design and performance factors make direct comparison between composites and metals difficult However it is instructive to compare some physical properties of common marine industry materials Table 10.1 provides a summary of certain characteristics Tensile Strength Elastic Modulus Cost (1995) g/cm MPa GPa US$/lb ABS Grd A mild steel 7.85 400 206 0.29 ABS Grd AH high tensile steel 7.85 49 206 0.34 Aluminium 6061-T6 2.71 310 69 2.86 Aluminium 5086-H34 2.66 304 69 1.65 Orthophthalic Polyester 1.23 48 4.1 1.05 Isophthalic Polyester 1.21 71 3.9 1.19 Vinyl Ester 1.12 80 3.4 1.74 Epoxy (Gougen Proset) 1.20 60 3.7 3.90 Phenolic 1.15 35 3.7 1.10 E-glass (24 oz WR) 2.60 450 72 1.14 S-glass 2.49 589 87 5.00 Kevlar 49 1.44 623 124 20.00 Carbon-PAN 1.76 830 393 12.00 Material Density Metals Resins Fibres Cores End-grain balsa 0.11 9.1 2.55 3.70 0.08  0.1 1.4 0.06 5.20 Cross-linked PVC (Diab H-100) 0.10 3.1 0.12 5.95 Honeycomb (Nomex HRH-78) 0.10  0.41 13.25 Honeycomb (Nidaplast H8PP) 0.08   0.80 Linear PVC (Airex R62.80) Note: The values used in this table are for illustration only and should not be used for design purposes Table 10.1 Overview of marine structural construction materials Ship Structure – An Introduction to Marine FRPs Materials Materials form an integral part of the way composite structures perform Because the constructor is creating a structural material from diverse constituent compounds, material science concepts are essential to the understanding of how structural composites behave Reinforcements for marine composite structures are primarily E-glass due to its cost for strength and workability characteristics In contrast, the aerospace industry relies on carbon fibre as its backbone In general, carbon, aramid fibres and other specialty reinforcements are used in the marine industry where structures are highly engineered for optimum efficiency Architecture and fabric finishes are also critical elements to correct reinforcement selection Resin systems are probably the hardest material group for the designer and constructor to understand Although development of new formulations is ongoing, the marine industry has generally based its structures on polyester resins, with trends to vinyl ester and epoxy for structurally demanding projects and highly engineered products A particular resin system is affected by formulation, additives, catalysis and curing conditions Characteristics of a cured resin system as a structural matrix of a composite material system are therefore somewhat problematic Core materials form the basis for sandwich composite structures which clearly have advantages in marine construction A core is any material that can physically separate strong, laminated skins and transmit shear forces across the sandwich, and the dynamic behaviour of a composite structure is integrally related to the characteristics of the core material used 5.1 Reinforcements Fibreglass Glass fibres account for over 90% of the fibres used in reinforced plastics because they are inexpensive to produce and have relatively good strength to weight characteristics Additionally, glass fibres exhibit good chemical resistance and processability The excellent tensile strength of glass fibres, however, is somewhat susceptible to creep and has been shown to deteriorate when loads are applied for long periods of time Continuous glass fibres are formed by extruding molten glass to filament diameters of  25 micrometers (microns), i.e., 0.005  0.025 mm Individual filaments are coated with a sizing to reduce abrasion and then combined into a strand of either 102 or 204 filaments The sizing acts as a coupling agent during resin impregnation Eglass or ‘electrical glass’ was originally developed for the electrical industry because of it high resistivity S-glass was specifically developed for ‘structural’ applications with improved tensile strength The cost of S-glass is approximately  times that of E-glass which precludes a more widespread use of S-glass in the marine construction industry E-glass (lime aluminium borosilicate) is the most common reinforcement used in marine laminates because of its good strength properties and resistance to water degradation S-glass (silicon dioxide, aluminium and magnesium oxides) exhibits about 30% better tensile strength and in general, demonstrates better fatigue resistance Ship Structure − An Introduction to Marine FRCs Polymer Fibres The most common aramid fibre is Kevlar  developed by DuPont The outstanding features of aramids are low weight, high tensile strength and modulus, impact and fatigue resistance, and weaveability Compressive performance of aramids is not as good as glass as they display nonlinear ductile behaviour at low strain values Water absorption of un-impregnated Kevlar 49 is greater than other reinforcements, although ultra-high modulus Kevlar 149 absorbs almost 60% less than Kevlar 49 The unique characteristics of aramids can best be exploited if appropriate weave style and handling techniques are used Polyester and nylon thermoplastic fibres have recently been introduced to the marine industry as primary reinforcements and in a hybrid arrangement with fibreglass Hoechst-Celanese manufactures a product, Treveria, which is a heat-treated polyester fibre fabric designed as a gel coat barrier to reduce ‘print-through’, which occurs when the weave pattern of a reinforcement is visible at the laminate surface due to resin shrinkage during curing Although polyester fibres have fairly high strengths, their stiffness is considerably below that of glass Other attractive features include low density, reasonable cost, good impact and fatigue resistance, and potential for vibration damping and blister resistance .3 Carbon Fibres The terms ‘carbon’ and ‘graphite’ fibres are used interchangeably, although graphite technically refers to fibres that are greater than 99% carbon composition, versus 93  95% for PAN-base (polyacrylonitrile) fibres All continuous carbon fibres produced to date are made from organic precursors, which in addition to PAN, include rayon and pitches, with the latter generally used for low modulus fibres Carbon fibres offer the highest strength and stiffness of all common reinforcement fibres The fibres are not subject to stress rupture or stress corrosion, as with glass and aramids High temperature performance is particularly outstanding The major drawback to the PAN-base fibres is their relative cost, which is a function of high precursor costs and an energy-intensive manufacturing process Table 10.2 indicates some comparative fibre performance data Tensile Strength Elastic Modulus Ultimate Elongation kg/m MPa GPa % E-Glass 600 450 72 4.8 S-Glass 490 589 87 5.7 Aramid-Kevlar 49 440 623 124 2.9 Polyester-COMPET 360 030 10 22.0 720  800 400  800 228  393 0.38  2.0 Fibre Density  Carbon-PAN Table 10.2 Mechanical properties of reinforcement fibres Ship Structure – An Introduction to Marine FRPs 5.2 Reinforcement Construction Wovens Woven composite reinforcements generally fall into the category of cloth or woven roving The cloths are lighter in weight, typically 200  340 g/m2, and require about 40  50 plies to achieve a 25 mm thickness Their use in marine construction is limited to small parts and repairs Particular weave patterns include plain weave (the most highly interlaced), basket weave (paired warp and fill yarns) and satin weaves (exhibiting a minimum of interlacing) Satin weaves are produced in standard 4, 5 or 8harness configurations, which exhibit a corresponding increase in resistance to shear distortion Woven roving reinforcements consist of flattened bundles of continuous strands in a plain weave pattern with slightly more material in the warp direction This is the most common type of reinforcement used for large marine structures because it is available in fairly heavy weights A weight of 810 g/m2 is the most common, which allows for a rapid build-up of thickness Impact resistance is enhanced because the fibres are continuously woven Figure 8.1 Examples of weave patterns used for composite reinforcements .2 Knits Knitted reinforcement fabrics were first introduced in 1975 to provide greater strength and stiffness per unit thickness as compared to woven rovings A knitted reinforcement is constructed using a combination of unidirectional reinforcements that are stitched together with a non-structural synthetic, such as a polyester A layer of mat may also be incorporated into the construction The process provides the advantage of having the reinforcing fibre lying flat, versus the crimped orientation of the woven roving fibre Additionally, reinforcements can be orientated along any combination of axes Superior glass to resin ratios are also achieved which makes overall laminate costs competitive with traditional materials Ship Structure − An Introduction to Marine FRCs Figure 8.2 Examples of woven and knitted rovings used for composite reinforcements .3 Omnidirectional Reinforcements Omnidirectional reinforcements can be applied during hand lay-up as prefabricated mat or via the spray-up process as chopped strand mat Chopped strand mat consists of randomly oriented glass fibre strands that are held together with a soluble resinous binder Continuous strand mat is similar to the chopped strand form, except that the fibre is continuous and laid down in a swirl pattern A chopper gun takes roving and chops it up as it is sprayed with resin to create a structure similar to chopped strand mat Both hand lay-up and spray-up produce plies with equal properties along the x and y axes and good interlaminar shear strength This is a very economical way to build up thickness, especially with complex moulds In-plane mechanical properties are low because fibres are randomly orientated and plies are resin-rich .4 Unidirectional Reinforcements Pure unidirectional construction implies no structural reinforcement in the fill direction Ultra high strength/modulus material, such as carbon fibre, is sometimes used in this form due to its high cost and specificity of application Material widths are generally limited due to the difficulty of handling Some unidirectionals are held together with thermoplastic web binders that are compatible with thermoset resin systems making for easier handling Typical applications for unidirectionals include stem and centreline stiffening as well as the tops of stiffeners Entire hulls are fabricated from unidirectional reinforcements when an ultra high performance laminate is desired and load paths are well defined 5.2 Resins Polyester Polyester resins are the simplest, most economical resin systems that are easiest to use and display good chemical resistance Polyester resins have long been considered the least toxic thermoset to personnel, although recent scrutiny of styrene emissions in the work place has led to Ship Structure – An Introduction to Marine FRPs the development of alternate formulations The basic polyester resins used in the marine industry are orthophthalic and isophthalic The ortho resins were the original group of polyesters developed and are still in widespread use They have somewhat limited thermal stability, chemical resistance and processability characteristics The iso resins generally have better mechanical properties and chemical resistance Their increased resistance to water permeation has prompted many constructors to use this resin as a gel coat or barrier coat in marine applications The rigidity of polyester resins can be lessened by increasing the ratio of the saturated to unsaturated acids contained Flexible resins may be advantageous for increased impact resistance, however, this comes at the expense of stiffness Non-structural laminate plies, such as gel coats and barrier veils are sometimes formulated with more flexible resins to resist local cracking On the end of the spectrum are the low-profile resins that are designed to minimise reinforcement print-through Curing of polyester without the addition of heat is accomplished by adding accelerator along with the catalyst Gel times can be carefully controlled by modifying formulations to match ambient temperature conditions and laminate thickness Other resin additives can modify the viscosity of the resin if vertical or overhead surfaces are being laminated .2 Vinyl Ester The handling and performance characteristics of vinyl ester resins are similar to polyesters Some advantages of the vinyl esters, which may justify their higher cost, include superior corrosion resistance, hydrolytic stability and excellent physical properties such as impact and fatigue resistance It has been shown that an interlayer with a vinyl ester resin matrix can provide an excellent permeation barrier to resist blistering in marine laminates .3 Epoxy Epoxy resins display the best performance characteristics of all the resins used in the marine industry and additionally, exhibit the least shrinkage upon cure of all the thermosets Aerospace applications use epoxy almost exclusively, except when high temperature performance is critical The high cost of epoxies and handling difficulties have, however, limited their use for large marine structures Phenolic These thermosetting resins have typically been cured at high temperatures (140  180C) and usually under high pressures Developments in the late 1970’s led to a new range of phenolic resins that were designed to cure at lower temperatures and pressures through the use of catalysts The processing of the resins has been advanced such that all the processes normally used for composite production are commercially available Two categories of phenolic resin are novolacs and resoles Novolacs are thermoplastic materials often referred to as 2-stage resins since they need to be heated with additional formaldehyde in order to cross-link to their final infusible form Resoles are thermosetting resins often referred to as 1-stage resins Resole resins for laminating are usually dissolved in alcohol prior to distribution These resins have a sufficiently high formaldehyde content for them to cross-link on further heating Curing can also be brought about through the addition of strong acids which Ship Structure − An Introduction to Marine FRCs will, however, produce an extremely exothermic reaction Phenolic resins perform much better than polyesters and epoxies in fires, showing reduced flame spread characteristics and increased time to ignition Because phenolic resin is very promising for applications where fire is a threat, resin manufacturers have devoted effort to improve processability and strength characteristics .5 Thermoplastics Thermoplastics generally come in a form of moulding compound that softens at high temperatures Polyethylene, polystyrene, polypropylene and nylon are examples of thermoplastics Their use in the marine context has generally been limited to small craft and recreational items, although their use has been investigated for the large scale production of structural components Some attractive features include no exotherm upon cure which plagued filament winding or extremely thick sections with thermosets, and enhanced damage tolerance Processability and strengths compatible with reinforcement material are key areas under development Table 10.3 provides a comparative summary of certain mechanical properties of thermosetting resins Tensile Strength Elastic Modulus Ultimate Elongation MPa GPa % Orthophthalic Polyester 48.3 4.1 0.91 Isophthalic Polyester 77.1 3.9 2.0 Resin 76  83 3.4 56 Epoxy (Gougen Proset) 54.9 3.7 7.7 Phenolic 48.3 3.8 1.75 Vinyl Ester Table 10.3 Comparative data for thermoset resin systems 5.3 Core Materials Balsa End-grain balsa has a closed-cell structure consisting of elongated prismatic cells with a length (grain direction) approximately 16 times the diameter In densities between 0.1 and 0.25 g/cm 3, the material exhibits excellent stiffness and bond strength Stiffness and strength characteristics are much like aerospace honeycomb cores Although the static strength of balsa panels will generally be higher than the PVC foams, impact energy absorption is lower Local impact resistance is very good because stress is efficiently transmitted between sandwich skins Endgrain balsa is available in sheet form for flat panel construction or in block arrangements that conforms to complex curves .2 Thermoset Foams Foamed plastics such as cellular cellulose acetate (CCA) polystyrene and polyurethane are very light (55 g/cm3) and resist water, fungi and decay These materials have very low mechanical properties and polystyrene will be attacked by polyester resin These foams will not conform to Ship Structure – An Introduction to Marine FRPs complex curves unless they are blown in place Use is generally limited to buoyancy rather than structural applications Polyurethane is often foamed in place when used as a buoyancy material .3 Syntactic Foams Syntactic foams are made by mixing hollow microspheres of glass, epoxy and phenolic into fluid resin with additives and curing agents to form a mouldable, curable, lightweight fluid mass Sprayable syntactic core is available that can be applied in thicknesses up to 9.5 mm at densities between 0.48 and 0.69 g/cm3 Syntactic cores can be used instead of laminate ‘bulkers’ to build up laminate thickness to increase flexural strength .4 Cross-linked PVC Foams Polyvinyl chloride (PVC) foam cores have cell diameters ranging from 0.07 to 0.69 mm The material is thermoplastic, enabling the material to conform to compound curvature of a hull PVC foams have almost exclusively replaced urethane foams as a structural core material, except in configurations where the foam is blown in place A number of manufacturers market crosslinked PVC products to the marine industry in sheet form with densities ranging from 0.03 to 0.19 g/cm3 and as with the balsa products solid sheets or block configurations are available .5 Linear PVC Foams Linear PVC foam core produced for the marine industry has unique mechanical properties that are a result of a non-connected molecular structure which allows significant displacements before failure In comparison with the cross-linked (non-linear) PVC’s, linear foam will exhibit less favourable static properties and better impact absorption capability Individual cell diameters range from 0.5 to 2.0 mm .6 Linear Structural Foam ATC Chemical of the USA developed a linear polymer foam Core-Cell with the aim of achieving the impact strength of linear PVC foam and approach the static stiffness of crosslinked foams Core-Cell exhibits toughness and rigidity with high shear elongation and good impact strength Densities range from 55 to 220 g/cm3 and thicknesses from 6.35 to 38 mm .7 Honeycomb Various types of honeycomb cores are used extensively in the aerospace industry Constituent materials include aluminium, phenolic impregnated fibreglass, polypropylene and aramid fibre phenolic treated paper Densities range from 0.016 to 0.1 g/cm3 and cell sizes from to 9.5 mm Physical properties vary in a near-linear fashion with density Although the fabrication of extremely lightweight panels is possible with honeycomb cores, applications in a marine environment are limited due to the difficulty of bonding to complex face geometry and the potential for significant water absorption .8 PMI Foam PMI foam was developed specifically for the aerospace industry The material requires minimum laminating pressures to develop a good bond to face skins (peel strength) The most 10 Ship Structure − An Introduction to Marine FRCs attractive feature of this material is its ability to withstand curing temperatures in excess of 175C This feature is essential when considering prepreg construction with autoclave cure Tensile Strength Compressive Strength Shear Strength Shear Modulus g/cm MPa MPa MPa MPa 112  145 9.12  12.3 8.19  11.9 2.17  2.81 120  151 Termanto C70.75 75 2.21 1.41 1.11 11 Klegecell II 75 1.21 1.10  11 Divinycell H-80 80 1.79 1.17 1.00 30 Termanto C70.90 91 2.21 1.78 1.16 13 Divinycell H-100 96 2.48 1.79 1.59 45 Linear PVC Foam Airex 80  96 1.38 0.86 1.17 29 Linear Structural Foam Core-Cell Core Material Density Balsa End-grain balsa Cross-linked PVC Foam 55  210 0.81  2.27 0.40  1.45 0.56  1.75 12  35 PMI Foam Rohacell 71 75 2.74 1.47 1.28 30 Rohacell 100 111 3.40 2.94 2.35 49 96  7.76 1.38 41 Phenolic Resin Honeycomb Table 10.4 Comparative data for sandwich core materials Mechanical Properties of Fibre-reinforced Composites Difficulties in applying numerical methods of analysis such as finite element analysis to composite structure mean that they not yield a precise solution Empiricism based upon experiments has been introduced for the prediction of properties of composites and are currently embodied in the draft ISO International Standard for Small Craft – Hull Construction & Scantlings A high proportion of the empiric formulae contained therein involve the use of volume and weight fractions for the fibre reinforcement of the composite material This module of reference notes is intended to introduce the student to these concepts and their calculation in order to enable the student to apply the ISO Standard in the design of composite small craft Rule of Mixtures 2.1 Case – Force Applied Parallel to Fibres For the following derivation of the rule of mixtures the following subscript nomenclature is used: F for fibre R for resin (matrix) C for the composite laminate 11 Ship Structure – An Introduction to Marine FRPs If it is assumed that a good bond exists between resin and fibres, the strains in the fibres and resin will be equal, i.e.:  C = F = R (10.1) F F RESIN FIBRE FIBRE Figure 10.1 The force (F) is shared between fibres and resin, i.e.: FC = FF  FR Since:  = F A or (10.2) F = A Then equation 10.2 may be written:  C AC =  F AF   R AR Since: E =   or (10.3)  = E Then equation 10.3 may now be written: EC C AC = EF F AF  ER  R AR (10.4) From equation 10.1, if all strains are equal, then equation 10.4 may be reduced to EC AC = E F AF  E R AR (10.5) 12 Ship Structure − An Introduction to Marine FRCs Dividing through by AC (i.e the cross-sectional area of the composite) AF A  ER R AC AC EC = EF (10.6) If: AF AC = volume fraction of fibres,  F then: EC = E F F  E R and since AR AC = (1  F ) (10.7) then: EC = EFF  ER (1  F ) (10.8) AR AC Equation 10.8 is the rule of mixtures for forces applied parallel to the fibres and fibres in one direction It should be noted, however, that the draft ISO International Standard for Small Craft – Hull Construction & Scantlings (2004) modify this form of the rule of mixtures by accounting for alignment of fibres as manufactured, i.e., allowing for misalignment, cross-plies and chopped strand mat The ISO form of the rule of mixtures where forces are applied parallel to the fibres is: E C =  EFF  ER (1  F ) where:  (10.9) = fibre alignment coefficient = 0.9 for unidirectional fibres = 0.5 for cross-plies (balanced 0°/90°) = 0.33 for chopped strand mat 2.2 Case – Force Applied Perpendicular to Fibres RESIN F FIBRE FIBRE F Figure 10.2 13 Ship Structure – An Introduction to Marine FRPs In this case, with forces applied across the fibres:  C =  F F   R R =  FF   R (1  F ) Since: E =    or (10.10) =  E Then equation 10.10 may be written: C EC Since:  = F EF F = A F  R ER (1   F ) (10.11) the stress is equal in fibres and resin, i.e.: = R C = F Thus equation 10.11 may be rewritten: or: 1 = F  (1   F ) EF ER EC (10.12)  (1   F ) = F  EF ER EC (10.13) EC =  F (1   F )  EF ER (10.14) This is the rule of mixtures for forces applied perpendicular to the fibres and for unidirectional fibres The ISO International Standard for Small Craft – Hull Construction & Scantlings (2004) use the form of the expression employing densities: ( EC = E R where: F   2)  2 F ( F  1) R R F     F ( F  1) R R  F = fibre density kg/m3  R = resin density kg/m3 (10.15) 14 Ship Structure − An Introduction to Marine FRCs Volume & Weight Fractions One of the primary factors that determine the properties of a composite material is the relative proportion of fibres and of resin matrix The relative proportion can be in terms of weight or volume, and either can be quoted as a fraction Weight fractions are easier to obtain during fabrication or by experimental method after fabrication Volume fractions on the other hand are more convenient for theoretical calculations Hence it is desirable to determine expressions for conversion between the two Increasing volume fraction of the fibre reinforcement can increase the fibre load fraction However, there is a practical limit beyond which fibres cannot be packed in a resin matrix (approximately 80%) although in theory cylindrical fibres can be packed to almost 90% The draft ISO International Standard for Small Craft – Hull Construction & Scantlings suggests that volume fraction for fibre reinforcement typically ranges from 20 to 52% and that the weight fraction for fibres typically ranges from 30 to 70% 3.1 Expressions for Volume & Weight Fractions, and Densities Where: v = volume (m3) w = weight (kg)  = density (kg/m3)  = volume fraction ‘phi’  = weight fraction ‘psi’ Volume: vC = vF  vR F = vF vC R = vR vC =  F (10.16) (since F  R = 1) Weight: wC = wF  wR F = wF wC R = wR wC =  F Density: w thus: (10.17) (10.18) (since  F   R = 1) (10.19) = v wC = wF  wR 15 Ship Structure – An Introduction to Marine FRPs vC  C = vF  F  vR  R C = (10.20) vF v F  R R vC vC = F  F  R  R = F  F  (1  F )  R (10.21) Thus equation 10.21 is an expression for the density of a laminate in terms of volume fraction and component densities w Since: v =  vC = v F  v R thus: wC C = = = C C C wF F  wR R wF  R F R  vR  F vC  F wF  R  wR  F F R = wF  R  wR  F wC  F  R =  F R  R F F R =  F  R  (1   F )  F F R = F R  F  R   F (1   F ) = F R  F  R   F   F F = F R  F   F ( R   F ) (10.22) (10.23) 16 Ship Structure − An Introduction to Marine FRCs Thus equation 10.23 is an expression for the density of a laminate in terms of weight fraction and component densities Similar techniques are used to produce the following two alternative expressions for fibre volume and weight fractions respectively: F =  F R  F   F ( R   F ) (10.24) F = F  R  R  F ( R   F ) (10.25) 17 ... referred to as 2-stage resins since they need to be heated with additional formaldehyde in order to cross-link to their final infusible form Resoles are thermosetting resins often referred to as 1-stage... laminating pressures to develop a good bond to face skins (peel strength) The most 10 Ship Structure − An Introduction to Marine FRCs ... grillage will transition from elastic to plastic behaviour and collapse in its Ship Structure − An Introduction to Marine FRCs