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Horizontal density distribution of particleboard: origin and implications Xu, Wei 1994

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HORIZONTAL DENSITY DISTRIBUTION OF PARTICLEBOARD:ORIGIN A1]) IMPLICATIONSbyWET XUB.Sc., The Central South Forestry University, 1983M.Sc., The Chinese Academy of Forestry, 1986A THESIS SUBMIITED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIES(Department of Forestry)We accept this thesis as conformingTHE UNIVERSITY OF BRITISH COLUMBIADecember, 1993© Wei Xu, 1993to the required standardIn presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)________________Department of_________________The University of British ColumbiaVancouver, CanadaDate3/774DE-6 (2/88)iiABSTRACTParticleboard products have been manufactured for over a half century.During this time, it has been recognized that a vast number of material andprocessing variables influence board properties. Little is known about theinternal structure of particleboard, and a fundamental principle or theoryinterrelating structure, processing and properties of particleboard has yet tobe developed. Such basic knowledge of particleboard structure is not onlynecessary to fully understand present particleboard technology, but alsoimportant for the future upgrading and development of wood composites.This study was designed to develop some of this knowledge base.The two major objectives of this study were: (1) to investigate theinfluence of raw material characteristics on horizontal density distribution(HDD), and (2) to determine the effect of particleboard nonuniformity asdefined by HDD, on some key board properties. Twenty six particleboardpanels made with precisely cut particles were used to study the firstobjective, while thirty boards involving different particle sizes anddistributions, and different wood species and combinations weremanufactured to study the second objective.In addressing the first objective, the Equation S = a(1/A)b was found tobe appropriate for relating standard deviation of density (S) and specimensize (A), where a and b are constants. At relatively larger specimen sizes,particleboard made with larger particles exhibited greater density variation,while particleboard made with smaller particles showed larger variations atsmaller specimen sizes. Two aspects of voids, namely number and size, wereidentified as factors contributing the relation between particle size and HDD.iiiIn addition, a layer concept was developed to relate particle thickness, wooddensity, board density and board thickness to HDD. This concept predicted adecrease of density variation as particle layers increased.In addressing the second objective, modulus of rupture (MOR), modulusof elasticity (MOE), internal bond (TB) and thickness swelling (TS) ofparticleboard were shown to be greatly controlled by nonuniformity of boardstructure. All these properties were improved as structure uniformityimproved. A nonuniformity effect concept, expressed as P = m(1/S) wasproposed in relating board properties (P) to standard deviation (S) ofhorizontal density, in which, m and n are constants. While TS was influencedmost by the high density portions, mechanical properties were dominated bythe low density areas.The concept of HDD was also used in this study to investigate therelationship of specimen size effect on TB and TS, for one commercialwaferboard. Both average values and standard deviations of TB and TSdecreased as specimen size increased. A criterion based on HDD concept wasproposed for the future establishment of testing standard in terms ofspecimen size selection.The relationship between HDD, raw material characteristics and boardproperties, demonstrated that HDD was a fundamental variable useful forcharacterizing particleboard structure and technology. The HDD concept hasthe potential of linking the effects of raw material characteristics andforming techniques to board properties in short-fiber wood composites.ivTABLE OF CONTENTSABSTRACT iiTABLE OF CONTENTS ivLIST OF TABLES viiiLIST OF FIGURES xABBREVIATIONS USED xviACKNOWLEDGEMENT xvii1. INTRODUCTION 12. LITERATURE REVIEW 52.1. Pressing strategy 52.2. Raw material characteristics 72.2.1. Particle geometry 72.2.2. Wood density 82.3. Resin content 102.4. Dimensional Stability 112.5. Model Development 122.6. Particleboard Standard 132.7. Particleboard Structure 152.8. Paper Structure 242.9 Summary 313. RESEARCH DIRECTION 324. METHODOLOGY 344.1. Materials 344.1.1. Roundwood 344.1.2. Wood particles 354.1.2.1. Specialty particles 354.1.2.2. Commercial and laboratory particles 35V4.1.3. Adhesive 384.1.4. Commercial wood products 384.1.4.1. Waferboard 394.1.4.2. Parallam 394.1.4.3. Medium density fiberboard (MDF) 394.1.4.4. Waferboard/OSB 404.1.5. Laboratory particleboard 404.2. Density Measurement 414.2.1. Density measurement methods 414.2.1.1. Gravimetric method 414.2.1.2. X-ray scanning method 444.2.1.3. y-ray method 464.2.2. Sampling of density specimens 464.2.2.1. Gravimetric method 464.2.2.1.1. Laboratory particleboard 464.2.2.1.2. Commercial waferboard 484.2.2.2. X-ray and y-ray methods 314.3. Determination of Board Properties 514.3.1. Modulus of rupture and modulus of elasticity 514.3.2. Internal bond 524.3.3. Thickness swelling 535. RESULTS AND DISCUSSION 555.1. Aspect of Horizontal Density Distribution 555.1.1. Phenomenon of horizontal density variation5.1.2. Relationship between standard deviationof density and specimen size 625.1.3. Estimation of S to A relationship 67vi5.2. Influence of Raw Material Characteristicson Horizontal Density Distribution 685.2.1. Particle size 685.2.1.1. Parameter b 775.2.2. Particle thickness, wood density,board thickness and board density 825.2.2.1 Layer concept 825.2.2.2. Verification of layer concept 835.2.2.2.1. Particle thickness 835.2.2.2.2. Wood density 865.2.2.2.3. Board thickness 895.2.2.2.4. Board density 915.2.3. Summary 935.3. Implication of Horizontal Density Distributionon the Selection of Specimen Size for SomeParticleboard Property Evaluations 955.3.1. Introduction 955.3.2. Internal bond 965.3.3. Thickness swelling 1125.4. Influence of Horizontal Density Distributionon Some Board Properties 1195.4.1. Introduction 1195.4.2. Application of adhesive 1195.4.3. Board formation 1225.4.4. Modulus of rupture and modulus of elasticity 1265.4.5. Internal bond 1335.4.6. Thickness swelling 133vii5.4.7. Summary 1376. SUMMARY AND CONCLUSIONS 1386.1. Future Developments 1407. LITERATURE CITED 142APPENDICES 149A. Description of Particles Used for the Studyof the Influence of Horizontal DensityDistribution on Board Properties 149B. Description of Obtainment of TB Specimensfor Commercial Waferboard 156C. Description of Obtainment of TS Specimensfor Commercial Waferboard 157D. Derivation of Equation (13) Var(Da)/Var(Db) = AilAa 158E. Derivation of Equation (19) Var(Db) = [(1+p)Var(D)]/2 159F. Derivation of Equation (21) Var(Db)/Var(Da) = (112)21) 1600. Significance Test of Parameter b in Equation (23) 161H. Derivation of Equation (24) Var(Db) = (la/lb)Var(Da) 164I. Derivation of Equation (38) a = x(S) 165J. Derivation of Equation (41) ç = iV(S)c 166viiiLIST OF TABLESTable 1. Roundwood information 34Table 2a. Particles used for studying particle size effecton horizontal density distribution 36Table 2b. Particles used for verifring layer concept 37Table 3. Particleboards used to verif5r layer concept 42Table 4. Particleboards used to study the influence of horizontaldensity distribution on board properties 43Table 5. Partitioning procedure for density determinationof commercial waferboard 50Table 6. Density determination of commercial waferboard 60Table 7. Density determination of laboratory particleboard 69Table 8. S to 1/A models for laboratory particleboard 70Table 9. Comparison of density variations in X and Y directions 82Table 10. Particleboards with different particle thicknesses 84Table 11. Particleboards with different wood species 87Table 12. Particleboards with different board thicknesses 89Table 13. Particleboards with different board densities 93Table 14. Internal bond results of commercial waferboardat different specimen sizes 97Table 15. Thickness swelling (%) of commercial waferboardat different specimen sizes and soaking times 113Table 16. Density measurement for laboratory particleboard 123Table 17. Modulus of rupture and modulus of elasticityof laboratory particleboard 129Table 18. Internal bond and thickness swellingixof laboratory particleboard .134Table G-1. Calculations for normal distribution test 162xLIST OF FIGURESFigure 1. A typical vertical density profile of a three layer particleboard(Data from Plath and Schnitzler, 1970) 6Figure 2. A schematic of a particle mat (Adapted from Suchsland, 1967) 16Figure 3. A schematic theoretical horizontal density distributionof particleboard. Values in brackets are particlethickness (Adapted from Suchsland, 1959) 18Figure 4. Particle distribution models in one layer (Kusian, 1968a)(a) Parallel deposition (b) Shifted deposition 19Figure 5. Relationship between mat density and particleaspect ratio as given by Equation (3)(Data from Kusian, 1968b) 21Figure 6. Relationship between probability of horizontaldensity distribution and particle dimension asgiven by Equation (4) (Adapted from Kusian, 1968a) 22Figure 7. Relationship between average particle overlapping lengthand particle dimension as given by Equation (5)(Adapted from Kusian, 1968a) 23Figure 8. A photograph of a 2.5 g/m2 sheet ofpaper (Kailmes and Corte, 1960) 25Figure 9. A random network of lines (Kallmes and Corte, 1960) 27Figure lOa. Distributuin characteristic ofvoid size of onepaper sheet with NIL = 69 (Adapted from Kailmesand Corte, 1960) 29Figure lOb. Variance of distribution of mass density ofone machine made paper as functions ofxispecimen size (Data from Corte,1970). 30Figure 11. Drill press set-up for density determination 45Figure 12. A schematic of the procedure for allocatingdrilling specimen 47Figure 13. Cutting pattern for preparing test specimensfor laboratory particleboard 54Figure 14. Horizontal density variation of one commercial waferboard 56Figure 15. Density distribution characteristic of one commercialwaferboard at specimen size of 29.16 cm2 57Figure 16. Density distribution characteristic of one commercialwaferboard at specimen size of 0.31 cm 58Figure 17. Standard deviation of density vs. specimensize of one commercial waferboard 61Figure 18. Scatter plot of standard deviation of density vs.1/’iKof one commercial waferboard 63Figure 19. Relationship between correlation coefficient of densityand Lag for one commercial waferboard/OSB 65Figure 20. Influence of particle length on standard deviationof density of particleboard at several specimensizes. Data sets with same particle width areconnected and labeled by width (cm) 71Figure 21. Influence of particle width on standard deviationof density of particleboard at several specimensizes. Data sets with same particle length areconnected and labeled by length (cm) 72Figure 22. Influence of particle size on standard deviationof density at several specimen sizes. The curvexiiis fitted by eye without regression analysis 73Figure 23. A schematic of particle and void distribution inone layer under hand-forming operation(a) Small particle (b) Large particle 75Figure 24. Influence of particle size on parameter b. The curveis fitted by eye without regression analysis 79Figure 25. Relationship between parameter b and density variationat specimen size of 0.141 cm2. The curve is fittedby eye without regression analysis 80Figure 26. Relationship between correlation coefficient ofdensity and Lag for MDF and Parallam 81Figure 27. Influence of particle thickness on density variation.The lines are model predictions, and points areexperimental measurements 85Figure 28. Influence of wood density on density variation.The lines are model predictions, and points areexperimental measurements 88Figure 29. Influence of board thickness on density variation.The lines are model predictions, and points areexperimental measurements 90Figure 30. Influence of board density on density variation.The lines are model predictions, and points areexperimental measurements 92Figure 31. 2-P and 3-P Weibull distribution fits to internalbond of one commercial waferboard at specimensize of 6.27 cm2 98Figure 32. 2-P and 3-P Weibull distribution fits to internalxiiibond of one commercial waferboard at specimensize of 225.79 cm2 99Figure 33. Influence of specimen size on 50th percentileand average strength of internal bond ofone commercial waferboard 101Figure 34. Relationship between average strength ofinternal bond and standard deviation ofdensity of one commercial waferboard 103Figure 35. Influence of specimen size on standard deviationof internal bond of one commercial waferboard 104Figure 36. Relationship between standard deviation ofinternal bond and standard deviation ofdensity of one commercial waferboard 106Figure 37. Relationship between density and internalbond of one commercial waferboard atspecimen size of 6.27 cm2 107Figure 38. Relationship between density and internalbond of one commercial waferboard atspecimen size of 225.79 cm2 108Figure 39. Influence of specimen size on coefficient ofvariation of internal bond of one commercialwaferboard. The curve is fitted by eyewithout regression analysis 109Figure 40. Influence of specimen size on 5th percentile,95th percentile and average of internalbond of one commercial waferboard 110Figure 41. Influence of specimen size on average thicknessxivswelling of one commercial waferboard atdifferent soaking times 114Figure 42. Influence of specimen size on standard deviationof thickness welling of one commercialwaferboard at different soaking times 115Figure 43. Thickness swelling vs. soaking time of one commercialwaferboard at specimen size of 4.26 cm2 117Figure 44. Influence of apparent resin content on internal bond 121Figure 45. Standard deviation of density vs. specimensize for laboratory particleboard 124Figure 46. Standard deviation of density vs. 1/Afor laboratory particleboard 125Figure 47. Reproducibility of board formation of board P1 127Figure 48. Relationship between modulus of rupture and standarddeviation of density of laboratory particleboard. Thespecimen size used to determine standard deviation ofdensity is different and indicated in each plot 130Figure 49. Relationship between modulus of elasticity and standarddeviation of density of laboratory particleboard. Thespecimen size used to determine standard deviation ofdensity is different and indicated in each plot 132Figure 50. Relationship between internal bond and standarddeviation of density of laboratory particleboard. Thespecimen size used to determine standard deviation ofdensity is different and indicated in each plot 135Figure 51. Relationship between thickness swelling and standarddeviation of density of laboratory particleboard. Thexvspecimen size used to determine standard deviation ofdensity is different and indicated in each plot 136Figure A-i. Distribution of dimensions of aspen commercial particle p1.n = number of observations, = average, s = standard deviation.. .. 150Figure A-2. Distribution of dimensions of aspen commercial particle p2.n = number of observations, 5 = average, s = standard deviation. . 15iFigure A-3. Distribution of dimensions of aspen commercial particle p3.n = number of observations, = average, s = standard deviation. . 152Figure A-4. Distribution of dimensions of aspen laboratory particle p5.n = number of observations, I = average, s = standard deviation. . 153Figure A-5. Distribution of dimensions of aspen laboratory particle p6.n = number of observations, = average, s = standard deviation. . . 154Figure A-6. Distribution of dimensions of birch laboratory particle p7.n = number of observations, 5 = average, s = standard deviation.. 155Figure D-1. Relationship between specimen sizes 158Figure E-1. Diagram showing Ab = 2Aa 159Figure G-1. Residual plot based on Equation (23) 161Figure H-i. Diagram showing the layer concept 164xviABBREVIATIONS USEDHDD - horizontal density distributionTB- internal bondMDD- mass density distributionM1)F - medium density fiberboardMOE - modulus of elasticityMOR - modulus of ruptureOSB- oriented strand boardRBA - relative bonded areaTS - thickness swellingxviiACKNOWLEDGEMENTI would like to thank Dr. Paul Steiner, Department of Wood Science,UBC, for his invaluable direction, supervision and patience throughout thisproject. I also wish to thank my supervisory committee, Drs. David Barrett,Simon Ellis, Valerie Lemay and Anoush Poursartip, UBC, for their guidanceduring this thesis study. Thanks also go to Dr. Ricardo Foschi, Department ofCivil Engineering, UBC, for his encouragement and suggestions.The assistance of Pansmill Woodenware Ltd., CAE Machinery Ltd. inproducing particles, and the permission of both Forintek Canada Corp.,forthe use of their particle screening device, and of Canadian Forest Productsfor the use of a y-ray density measurement device are all readilyacknowledged. The help and cooperation of technician staff, Department ofWood Science, UBC, especially Mr. Rob Johnson and Mr. Bob Myronuk areappreciated.I would also like to thank Weyerhaeuser Canada and Asa Johal for theirfinancial support of my Ph.D. studies.Finally, my greatest gratitude goes to my parents and my wife for theircontinuous support and patience during my educational studies.11. INTRODUCTIONThe concept for creating particleboard*, a sheet-like product of woodparticles bonded with an adhesive, has been known since the beginning ofthis century. The first plant for particleboard production was erected inGermany in 1941 (Kollmann et al., 1975). Since then, worldwide research anddevelopment efforts have resulted in the emergence of several majorparticleboard-type products and corresponding manufacturing techniques,together with dramatic improvements in board properties. The propertiesand performance of lamella based wood composites can be predicted from thelaminate theory (Agarwal and Broutman, 1980), by knowing the laminaeproperty and laminate lay-up. In contrast, a general theory on short-fibrewood composites is lacking, although some modelling efforts on OrientedStrand Board (OSB), a typical short-fibre wood composite in North America,have been attempted in the past decade (Higgins, 1989; Lau, 1982; Shalerand Blankenhorn, 1989). Consequently, particleboard technology has beenlimited to the evaluation of board properties as functions of intermediatecharacteristics, such as production parameters, rather than basic variables(Suchsland, 1959). Since any given process has a vast number of variableswhich often interact in a complex manner, the task of developing afundamental knowledge base about particleboard process or properties on thebasis of these intermediate characteristics becomes almost impossible(Kunesh, 1961). Therefore, a more basic analytical characteristic needs to be* Throughout this thesis, the term “particleboard” is used in the broad sense of the FAO(1957) and interchangeably with the term “short-fibre wood composite”. This covers all dryformed boards made with sawdusts, splinters, flakes or wafers. Commercial products aredesignated by their specific names.identified.It may be argued that for commodity type particleboard products, whichtypically have been developed through trial-and-error methods, a completeand thorough understanding of the effects of manufacturing variables is notnecessary. However, in the past, an improved understanding ofmanufacturing technology has helped the development of new products, newproduction techniques and new quality control methods. Such developmentsas OSB and steam-injection pressing technology and the use of verticaldensity profile measurement as a quality control method are evidences of thistrend. Improved understanding of the fundamental principles of woodcomposites is believed to be a key factor for the design of new generations ofproducts, and the upgrading of existing boards.Recent investigations of short-fibre wood composites have been directedat an improved understanding of particle behaviour during pressing,reducing board density and improving dimensional stability whilemaintaining or increasing strength properties, and basic mechanicalbehaviour of particleboard. These studies have recognized several commonconcepts referred to as “localized distribution of stress” (Humphrey, 1989),“voids”, “void volume”, “particle packing” (Steiner, 1989) and “local contact”(Hansel and Neumuller, 1988; Hansel and Niemz, 1989), which reflect thespatial nonuniformity of the structure of particulate composites. Spatialstructure is believed to be one of the most fundamental and importantcharacteristics affecting basic particleboard properties.The spatial structure of particleboard can be defined by the concept of athree-dimensional-density-distribution, which can be separated into a3vertical component and a horizontal component. The vertical component, i.e.,the vertical density profile, which indicates the density variation through thehorizontal layers in the thickness direction, has been extensively studied.The relationships between vertical density profile, pressing strategy andboard properties are well documented (Kelly, 1977). The concept of horizontaldensity distribution (HDD) or variation, proposed by Suchsland (1959), whichcharacterizes the horizontal micro-density nonuniformity in the plane of theboard, has never been determined directly (Suchsland and Xu, 1989). Therelation of HDD to raw material characteristics and board properties is notwell understood. It is this density distribution component which will be thesubject of this study.The hypotheses of this thesis were that raw material characteristicshave an overall effect on HDD of particleboard, and this distributioninfluences board properties. These hypotheses were addressed in threephases. The first phase undertook a quantitative determination of HDD inrelation to raw material characteristics.In the second phase, the effect of test specimen size on some boardproperties was studied to establish the importance of the HDD concept forfuture testing standard development, in terms of specimen size designationsfor two particleboard property evaluations.In the third phase, boards with different HDD characteristics weremade by choosing different particle size ranges, different wood species andcombinations of each. Physical and mechanical properties of these boardswere compared to determine the significance of HDD of particleboard onphysical and mechanical property variations.4Contrary to naturally formed products, such as solid lumber,particleboard is a manufactured material involving complicated syntheticprocesses. Developing this knowledge may provide a useful means torationalize choices of raw material characteristics and lay-up (formation)methods to improve the performance and efficiency of wood-based compositeproducts, and to develop a criterion for designating specimen sizes for someboard property determinations. The attainment of this knowledge may alsopresent a critical step towards establishing a three dmensional densitydistribution for short-fibre wood composites. The concept of a threedimensional density distribution would allow the structural characteristics ofparticleboard to be described in an improved fashion, and lead to the possibledevelopment of a general theory on short-fibre wood composites similar to thelaminate theory for continuous fibre composites.52. LITERATURE REVIEW2.1. Pressing strategyIn the development of particleboard technology, two subjects havereceived much attention. One of these was the effect of pressing strategy onboard properties. This included parameters such as press closing speed (afunction of pressure applied), temperature, moisture content and itsdistribution, and more recently steam-injection pressing. The second subjectconcerned the influence of raw material characteristics on board properties.Wood density, particle shape and particle geometry, including length, width,thickness, lengthJthickness ratio, length/width ratio and width/thicknessratio, were the main parameters.Studies emphasizing pressing strategy identified the vertical densityprofile as an important factor affecting many board properties. A typicalvertical density profile of particleboard is shown in Figure 1. This densityprofile is formed during the pressing process when wood particles in differentlayers of the mat experience different conditions of temperatures andmoisture contents, which affect their compressibility and stress relaxationbehaviour (Kelly, 1977). The contribution of these parameters to theformation of vertical density profile has been extensively studied and is wellunderstood. In general, increasing pressing temperature, particle moisturecontent and press closing speed or pressure leads to differences in densitybetween surface and core regions of the panel (Fahrni, 1956; Geimer, 1982;Geimer et a!., 1975; Heebink et a!., 1972; Strickler, 1959; Suchsland, 1962). Aclose relationship has been identified between vertical density profile andmost board properties. Specifically, modulus of rupture (MOR) and modulus6II1.00.90.80.70.60.50Thickness (mm)Figure 1. A typical vertical density profile of a three layer particleboard(Data from Plath and Sehnitzler, 1970).5 10 15 207of elasticity (MOE) are favourably affected by the increase of particleboardsurface density (Fahri, 1956; Geimer et al., 1975; Heebink et al., 1972; Plath,1971b; Plath and Schnitzler, 1974; Strickler, 1959); internal bond (TB) isimproved by increasing core density (Neusser, 1978; Plath and Schriitzler,1974); and layered torsion shear strength is positively related to individuallayer density (Shen and Carroll, 1969, 1970). As a result, the singularparameter of vertical density profile has been used extensively tocharacterize the effects of pressing strategy, which involves multi-parametereffects. For example, vertical density profile measurement has been used, to acertain extent, as a quality control method (Gibbon and Tundak, 1989;Lemaster, 1989). Thickness swelling (TS) is one property which does notseem to be strongly related to vertical density profile.2.2. Raw material characteristics2.2.1. Particle geometryThe effects of particle geometry on board properties have not been asconclusive as have pressing strategy parameters. The effects of particlelength, thickness and lengthJthickness ratio on MOR and MOE appear to bewell documented. However, most reports offered only general or conditionalstatements, such as MOR and MOE improve as particle length orlength/thickness ratio increases within certain ranges (Brumbaugh, 1960;Gatchell et al., 1966; Kimoto et al., 1964; Lehmann, 1974; Mottet, 1967; Post,1958, 1961; Rackwitz, 1963; Stewart and Lehman, 1973; Turner, 1954). Whilegeneral statements could be extracted from the literature that TB improves asthe particle configuration changes from a long, wide flake to planar shavingsor to slivers, some exceptions to this finding have been documented8(Brumbaugh, 1960; Gatchell et al., 1966; Heebink and Hann, 1959; Rackwitz,1963; Lehman, 1974; Stewart and Lehman, 1973; Talbott and Maloney,1957). Most reports were unanimous in stating that better thickness stabilityis obtained with boards produced from thin particles rather than from thickparticles. Much less agreement was found regarding the influence of particlelength and width on TS. Many studies utilized particles produced by avariety of production methods. These represented wide variations in particlegeometries which were often not documented. Consequently, it is uncertain ifpublished information concerning the particle geometry effect on boardproperties presents the true picture. Furthermore, no satisfactoryexplanation for describing how particle geometry may affect board propertiesis offered, although some qualitative analyses were made regarding the effectof particle length and width on MOR and TB (Kusian, 1968a; 1968b;Rackwitz, 1963;).Strictly speaking, it is difficult to separate and study one geometricparameter from another, since changes in one parameter will affect someother parameters. For example, changes in particle length will alterlength/thickness and length/width ratios. Therefore, a more basicparameter(s) or variable(s) is needed to characterize the effect of particlegeometry on board properties, similar to the manner in which vertical densityprofile is used to quantify the influence of pressing strategy on boardproperties.2.2.2. Wood densityWood composites have been manufactured from a variety of woodspecies. Usually, a particular wood composite is assembled from one or two9main wood species, or a combination of wood species. The typical examplewould be Waferboard/OSB production in North America, where aspen(Populus Spp.) is the main raw material of choice. The choice of a specificwood species or species combinations and the establishment of correspondingmanufacturing technology were usually accomplished through extensiveexperimental studies. Wood density was found to be the primary factorinfluencing board properties and for the selection of manufacturingparameters. At the same product density and manufacturing conditions,particleboard made from higher density wood species was always inferiorboth in mechanical and dimensional stability properties than particleboardmade from lower density wood species (Hse, 1975; Kehr, 1979; Stegmann andDurst, 1964; Stewart and Lehmann, 1973; Vital et al., 1974). Compactionratio, which is the ratio of board density to wood density, has been proposedas a concept to guide the selection of production parameters (Hse, 1975). It isgenerally accepted that a minimum range of compaction ratios of 1.3 - 1.5 arenecessary to achieve reasonable board properties. Consequently,substantially higher particleboard density results when high density woodspecies are used, provided that other board manufacturing conditions are thesame.It is recognized that the concept of compaction ratio was based onexperimental evidence, rather than on real composite structureconsideration. Limited studies in the past suggest that disbonding betweenparticle surfaces in tension and bending modes control the failure mechanismin particleboard (Laufenberg, 1984; Rackwitz, 1963; Suchsland, 1968). Thismay explain why the higher wood strength associated with high wood densitydoes not contribute to board properties as expected. This also suggests that10knowledge of particleboard structure is necessary to fully understand howwood density affects board properties, and subsequently the design andmanufacture of wood composites.2.3. Resin contentParticleboard adhesive level studies, traditionally expressed as theweight percentage of solid resin relative to the oven-dry weight of woodparticles, have been conducted mainly with liquid phenolic or ureaformaldehyde systems. Investigators were unanimous in their findings thatincreasing resin content increased all strength properties and improveddimensional stability of particleboard (Kelly, 1977). But the magnitude of thisimprovement was property dependent. Higgins (1990) reported that in aspenboards made with randomly oriented short strands, MOR increased accordingto a diminishing curvilinear relationship with increasing resin content. Onlysmall strength gains were achieved above 7-10 percent resin content. Thisgeneral trend has also been reported with other wood species at differentconditions (Adams, 1981; Kehr, 1967; Kimoto et al., 1964; Lehmann, 1970;Post, 1961; Price, 1974).Internal bond is the most sensitive property responding to resin contentchanges. The literature showed a continuous increase in TB as resin contentincreased in the ranges studied (Lehmann, 1970; Kehr, 1967). This suggeststhat the TB test is the most appropriate for studying bonding in short-fibrewood composites. Thickness swelling was also positively affected byincreasing resin content, but the same literature suggested that theimprovement was not as significant as that for TB. However, the effect of11resin content on TS was greatly influenced by test method, board density andother manufacturing conditions (Xu, 1989).Instead of expressing resin content as a weight percentage of wood, thecalculation of resin content based on particle surface area may be moreappropriate in studying adhesive level effects. The importance of particlethickness on adhesive requirements relative to particle surface area wasconsidered by Post (1961) and Gunn (1963). However, constant resin contentsbased on wood weight have been historically used as a basis for differentstudies. Thus, studies relating wood density and particle geometry, especiallyparticle thickness effects to particleboard properties are ambiguous, sinceactual quantities of resin on a particle surface area basis are inconsistent.While limited information has been reported on adhesive level effects forpowder adhesive, other adhesive characteristics, like flow properties andwood anatomy interactions were reviewed and studied by Ellis (1989).2.4. Dimensional stabilityThickness swelling in particleboard is usually taken as a primarymeasure of dimensional stability, although linear expansion properties arealso considered in some cases. Thickness swelling generally originates fromtwo sources. One is from the hygroscopic swelling of wood itself which is areversible process. The other is from the release of compressive stressincorporated into the particleboard during the pressing operation, which isan irreversible process and the major component of TS. Several importantstudies on the rheological behaviour of wood in compression perpendicular tothe grain were undertaken to better understand TS phenomenon (Kunesh,1961; Kollmann, 1962; Young, 1957; Wang, 1985, 1987a, 1987b). Some12possible strategies for improving dimensional stability and at the same timedecreasing the density of particleboard were proposed. These includedmaintaining a high moisture content during the pressing operation (Kunesh,1961), and increasing the compressibility of particle surfaces by chemicaltreatment (Wang, 1989). However, very limited experimental progress hasbeen reported on these aspects. A complicating factor is that the aggregate ofparticles in a mat respond to the applied force in a nonuniform manner, dueto the heterogeneous structure of the particle mat. Knowledge obtained fromstudying solid wood cannot be applied directly to the particleboard pressingoperation. Therefore, understanding particleboard structure may be aprerequisite to improve TS behaviour.2.5. Model developmentBecause of the continual reduction in quality of the available timberresources, short-fibre wood composites will become a more importantstructural material in the future. The successful modelling of their elasticand strength properties will be a requirement for acceptance in engineeringapplications. Some modelling efforts have been made for OSB (Higgins, 1989;Lau, 1982; Shaler and Blankenhorn, 1989). These studies have utilizedelasticity theory, the rule of mixtures (Agarwal and Broutman, 1980), theHalpin-Chai Equation (Halpin and Chai, 1967), a modified rule of mixturedeveloped for non-wood fibre-polymeric composites, and the Hankinsonformula (Bodig and Jayne, 1982), involving a simplified expression for off-axis strength estimation of wood. With all these theories, certainassumptions, such as a perfect alignment of particles, maximum bondingbetween particles and a continuous glueline were made. As some of these13assumptions are far from realistic for OSB, discrepancies were found betweenmodel predictions and experimental results.Particleboard made with randomly distributed particles is probablymore difficult to model than OSB, since its structure is even more difficult todefine. Just as knowledge of laminate lay-up is an essential element oflaminate theory, knowledge of particleboard structure is believed to be theprerequisite for the successful modelling of its material properties.2.6. Particleboard standardHistorically, wood-based panel products were manufactured according todescriptive product standards, such as PS 1-74 for commercial and industrialplywood (U.S. Department of Commerce, 1974), ANSI A208.1 for mat formedwood particleboard (American National Standards Institute, 1979) in theUnited States, CSA-0121-M78 for plywood (Canadian Standard Association,1978), and CAN3-0188.2-M78 for waferboard and OSB (Canadian StandardAssociation, 1978) in Canada. These standards are manufacturingprescriptions in which the raw material and method of assembly into largesheets are set down for minimum product requirements. They prescribed howthe products should be manufactured without being directly concerned abouttheir service applications (O’Halloran, 1980). As a consequence, change ormodification of manufacturing process, development of new products or use ofnew raw materials were hampered by these rigid standards. Acceptance bymajor building codes of products deviating from these product standards wasextremely difficult.By 1980, a new approach emerged for wood composites, with thedevelopment of performance standard, such as Performance Standards and14Policies for Structural-Use Panels by the American Plywood Association(1980) and Construction Sheathing by Canadian Standards Association(1988). In contrast to a product standard, a performance standard definesrequirements of specific end use application of the product. The objective wasto assure that for a particular end use the product would satisfSr therequirement of the application for which it was intended (O’Halloran, 1980).Since product manufacturing procedures were flexible (within a productcategory), products could be accepted by building codes as long as it satisfiedthe specific performance standard for the application. Therefore, performancestandards allow for innovations.It should be emphasized that the development of performance standardswas not the result of improvements in the fundamental understanding ofparticleboard technology. Beginning in the mid 1980’s, a reliability-basedlimit states design philosophy based on engineering computations for timberstructure emerged in Canada (Canadian Standard Association, 1984). A U.S.version is now being considered. Currently, short-fibre wood composites areused together with other products in timber structures. Although thismixture is accepted by building codes through performance standards, areliability level is not calculated either for wood composites or the wholesystem. This deficiency will be recognized more as reliability-based designbecomes more prevalent.It is believed that successful modelling of strength characteristics, andestablishment of design values are needed for reliability calculations of woodcomposites or structures involving wood composites. Since fundamentalknowledge of wood composites is limited, design properties for engineeringcalculations have not been established or recognized, even though some15efforts have been made to achieve this goal (McNatt, 1973; O’Halloran, 1988).Further knowledge of structure and other fundamental aspects of woodcomposites would help this effort.2.7. Particleboard structureThe concept of HDD in particleboard, was first proposed by Suchsland(1959) to analyze particleboard process. A schematic presentation of aparticle mat, which consists of wood particles and voids interspersed is shownin Figure 2. If very small vertical sections reaching from one surface of theboard to the other were isolated and their densities measured, one would findthat the densities vary to a smaller or larger extent about the average overallboard density (Suchsland and Xu, 1989). A model, based on a stack of veneerseach containing equal number of randomly distributed holes, was developedto characterize this density variation of particleboard (Suchsland, 1959).From this model structure, the distribution of the sums of the veneerthicknesses over any small area element followed a binomial distributiondefined as0(m) = (1)where0(m) = fraction of total area over which the number of solid veneer elementsis equal to m;n = total number of veneer layers;p = 1 - A = relative wood volume of each veneer layer;A = relative air volume of each veneer layer.16k\\\ F R 1 I\\\NRV \\\ \\1 k\’ F I\1 I\\\Ik\\i I\ \\\ k\I ‘I k\ \\N l\\\I k\\ k\ I k\\Ii\\ \“i \\\\ I \\1 \\\\ \I 1\\\N k\ \iJ\\\1 kIISi__h\i I\\I k\ i\\V1 1\\\\I h\\I \c\\H\\\ F1[\\I [\\\I l\ \\[ k-\I k\\ t\\I I\\\I k’\ \\\\I \\\J_____I\\\\1 t\\\1 T \\1 1\ \ \ \[ kN I\\\ k1I I\\1 I\I k\ f\\\I I\Ik\\I I\\J I\\ \\\\1\\\1 t\\\ I’\\ l\\jl\\\1 I\ I\1 ‘\\I I’\II\M k\\\I 1\ \S t\ I i\\i I\I \\ \1 t\\ET R\\ii\\\L____________________________j\ \‘1 l \ \\4 t\\’\I I\\I___\I_I\_\___\ I \\\\I\\ F\’\I j\ \1IR\iL\_.\_\\_I I\\\ILILd L\\\l I \\H\TI I \ \[\\\\ \ I ‘II\’L \\\I1\\l\\iI\\ 1\\!-Figure 2. A schematic of a particle mat (Adapted from Suchsland, 1967).17The resulting theoretical density distribution derived from this binomialdistribution would be like one of the curves shown in Figure 3. The possibleeffect of particle dimension on density distribution is also shown in Figure 3,which indicated a more uniform structure as particle thickness decreases,and particle shape changes from long slender to square ones. The significanceof relative area of compressed solid wood instead of the average board densityfor developing bending strength was also realized (Suchsland, 1959).Another model, consisting of narrow veneer strips arranged in mutuallyperpendicular layers so that veneer strip overlap area becomes the element ofa matrix representing the variation of the amount of wood material, wasdeveloped 30 years later (Suchsland and Xu, 1989). Direct measurements ofTB and TS on these matrix elements were used to study the effects ofnonuniformity of particleboard structure and other processing variables(Suchsland and Xu, 1989; 1991). Although these approaches were mainlyefforts to simulate the structure of particleboard using veneers or veneerstrips, it appears that at this time Suchsland was the principal researcherattempting to identify this relationship.In investigating the influence of particle size on the structural andstrength properties of particle materials, another researcher also realized thenonuniformity of particleboard structure (Kusian, 1968a; 1968b). Unlike the“brick” structure model in Figure 2, localized deposition of arbitrarilyoriented particles within circular areas was taken as a geometric model(Figure 4). As this model was based on plane projections, both particle lengthand width could be analyzed. First, the number of particles deposited in onelayer was calculated based on model (a) and a more compact model (b) in18Density (g/cm3)Figure 3. A schematic theoretical horizontal density distributionof particleboard. Values in brackets are particlethickness (Adapted from Suchsland, 1959)2.0C.?I0.0 0.4 0.8 1.2 1.619a bFigure 4. Particle distribution models in one layer (Kusian, 1968a).(a) Parallel deposition (b) Shifted deposition20Figure 4. Then, the overlapping and crossing of particles in between layerswere considered. Kusian (1968a) was able to relate void volume of particlemat, mat density, probability of horizontal density variation and averageparticle overlapping length to particle size.For model (a), Kusian (1968a) showed that the relationship between matdensity and particle size could be expressed as[m (m2Dmat2Dwii 2 I — I 2 (2)+lj \m +1in which, Dmat and D are the densities of the particle mat and woodparticles, m = 1/b is the aspect ratio between particle length 1 and width b.The probability (f) of horizontal density variation was given byf=k1m+1 (3)1where k = Wmat/(DwdFmat), W and Fmat are the weight and surface area ofparticle mat respectively and d is the particle thickness.For model (b), the average particle overlapping length L was expressedasL = 2b -- iJm2 +1 + m2 in4m +1+1+ in Jm + +It 3 12 m2+1_1 m .Jm2+1_m)These analyses predicted a particleboard structure with less horizontaldensity variation and longer particle overlapping length when particle lengthand width increased, and a particle mat with less void volume as aspect ratioincreased. Figures 5, 6 and 7 are graphical presentations of Equations (2), (3)and (4) respectively. Although only the mat density expression was compared21I0.60.50.40.30.20.10.07 14 21 28 350Aspect Ratio m = 1/bFigure 5. Relationship between mat density and particleaspect ratio as given by Equation (3)(Data from Kusian, 1968b).22C)C)CuC)Cu01.00.80.60.40.20.0Aspect Ratio m = 1/bFigure 6. Relationship between probability of horizontal density distributionand particle dimension as given by Equation (4)(Adapted from Kusian, 1968a).0 8 16 24 3223I2016128400Aspect Ratio m = lJbFigure 7. Relationship between average particle overlapping lengthand particle dimension as given by Equation (5)(Adapted from Kusian, 1968a).8 16 24 3224with experimental data (Figure 5), Kusian’s (1968a) mathematical analyseswas helpful in understanding the significance of particle size (length andwidth) in terms of particleboard inner structure.2.8. Paper structureFigure 8 is a reproduction of a photograph of a 2.5 g/m2 sheet of paper,roughly showing the arrangement of wood fibres in actual paper (Kailmesand Corte, 1960). Similar to particleboard products, the strength of papercomes from the interactive forces between fibres, although it is recognizedthat no adhesive is added during paper making. Two requirements arenecessary for such forces to operate. First of all, fibres must be brought intoclose contact. Secondly, sites for adhesion must be present on the surfaces ofthese fibres (Page, 1969). The pressing operation in particleboard is designedto achieve similar effects. An estimation of relative bonded area (RBA) or therelative contact area of fibres, achieved by measuring the light scatteringcoefficient of paper, was developed by Parsons (1942). This technique ispresently used in paper physics studies. The influence of fibre coarseness andthe extent of beating on RBA, and the contribution of RBA to the mechanicalproperties of paper have been studied extensively (Ingmanson and Thode,1959; Jones, 1972; Page, 1969; Parson, 1942; Rathff 1949; Seth, 1990).Generally, beating increases RBA, coarser fibres decrease RBA. While tensileand burst strength are positively related to RBA, tear strength seemsnegatively related to RBA. Page’s (1969) theory predicts a linear relationshipbetween tensile strength and RBA of paper.Another similarity between particleboard and paper products is thatboth products exhibit different sized voids and local variation of the areal25Figure 8. A photograph of a 2.5 gIm2 sheet ofpaper (Kailmes and Corte, 1960).26mass density in the direction of the plane. The term distribution of the massdensity (DMD) was coined to describe the latter for paper (Corte, 1969).These phenomenon arise from the fibrous network which makes up paperstructure. Figure 9 shows a random fibre network of the same number andaverage length of lines as in Figure 8, in which the coordinates of the linecentres were determined by random number pairs and the angle wasuniformly distributed (Kailmes and Corte, 1960). Some similarities betweenFigure 8 and Figure 9 are apparent. The application of geometric probabilityto study this random paper network structure was initiated in 1953 (LeCachenx, 1953), according to Corte (1982). For this random network, theprobability of finding r fibre centres in a square was given by the PoissonEquation (Kailmes and Corte, 1960; Corte and Kalimes, 1961)p(r) = e-mmr/r! (5)where, m is the average number of fibre centres in a square.The frequency of void having a size between v and v+dv was given by(Kailmes and Corte, 1960)N[Tf(v) = Pie dv (6a)2Lin which, N is the average number of fibres intersecting a scanning line withlength of L (cm).The variance of DM1) Var(d) was expressed as (Corte, 1969)Var(d) = lwDk/a2 (6b)in which, d is the variable areal mass density, 1 is the fibre length, w is theweight per unit length of the fibre, D is the average of d, a2 is the specimensize and k is a factor related to the size of fibre and specimen.I—328Figure lOa shows the results of the actual measurement of void sizedistribution for one paper sheet with NIL = 69 fibres/cm, and the theoreticalcalculations by Equation (6a). A good agreement between theoreticalconsideration and experimental measurement was apparent. This goodmatch indicates the randomness of fibre depositions during formation for thisvery low basis weight paper sheet (NIL = 69), which may not the case for realcommercial paper sheet. For real paper products, the permeability studyinvolving gas flow and mercury injection methods have to be used to estimatethe void size distributions (Corte, 1965).The development of the f3-ray absorption technique in the 1960’s madethe actual measurement of DMD possible. A comprehensive series ofmeasurements comparing 24 different machine made papers was publishedin 1970 (Corte, 1970). Significant differences of DMD were found amongthese papers. The measured variance of DMD for one paper sample is plottedin Figure lOb, together with the theoretical variance of DMD calculated byEquation (6b). It is noted that the actual variance of DMD was significantlylarger than the theoretical one based on the random network model. Thisprovided a strong support for the long time claim in paper physics study thata random structure would yield the most uniform paper, and it should be theaim of the paper maker if reducing the variance of DMD is the objective.Improved understanding of structure-property relationship in paperphysics study has recognized DMD as the most important and appropriateconcept to quantify paper structure. However, no systematic research on theeffects of raw material and processing variables on DMD have been reported,probably because of the relatively uniform distribution of fibre geometries inpaper making. Nevertheless, a few scattered examples examined by Corte29—‘ 40C.)C)I605030201000 10 20 30 40 50 60 70 80Void Size (10 cm2)Figure lOa. Distribution characteristic of void size of onepaper sheet with NIL = 69 (Adapted fromKailmes and Corte, 1960).3010.07.55.02.50.0Specimen Size (2)120I0 30 60 90Figure lOb. Variance of distribution of mass density ofone machine made paper as functions ofspecimen size (Data from Corte, 1970).31(1982) and later studies (Seth, 1990; Sosznski and Seth, 1985) did indicatethat many properties and the response of paper in different environments arerelated to DMD. According to Seth (1993), paper properties are alwaysadversely affected by increasing the variance of DMD.2.9. SummaryWhile structural nonuniformity in particleboard and paper materialswas probably recognized at approximately the same time, the concept ofquantifying structure or controlling its formation has yet to be established forparticleboard. In fact, a specimen size effect of the variance of HDD is notclearly defined yet for particleboard. As a result, our basic understanding ofother aspects of particleboard has been severely hampered, as indicated inthe literature review. It is believed that a detailed investigation on HDD maybetter understand the influence of raw material and forming method onparticleboard structure, and through it, on board properties.It should be mentioned that several recent studies have also recognizedthe spatial structure of particleboard (probably related to HDD) and itssignificance to board properties (Hansel and Neumuller, 1988; Hansel andNiemz, 1989; Humphrey, 1989; Steiner, 1989). The fact that nonuniformity inparticleboard structure has regained increasing attention from differentresearch groups indicates the need for a more systematic study of thisparameter. The main task of this thesis study was to examine the effect ofraw material characteristics on panel nonuniformity (HDD) and the effectthis nonuniformity has on selected particleboard properties.323. RESEARCH DIRECTIONUltimately, horizontal density variation of particleboard and otherphysical and mechanical properties associated with the plane of the boardcould be treated as random processes or random fields. This treatmentrequires the measurement of point board properties, because the densityvariation concerned in this study is caused by voids, and the differencesamong boards could probably be detected only when the specimen size fordetermining this density variation is very small. However, if such techniquescould be developed, within board variation of other point properties could bepredicted just by knowing the horizontal density process, and the crosscorrelation functions of these random processes. Furthermore, the propertiesdetermined at large specimen size, which is used to discriminate differentproducts, could also be obtained, provided that the relationship between pointand large specimen size properties were known. The random field theory hasbeen recently applied to model within board variations of MOE andcompressive strength properties of Glulam (Wang and Foschi, 1992; Xiong,1991).This thesis did not intend to quantify the within board point variationsof particleboard properties (although it is also very important). Rather, themain objective of this thesis was to use the concept of HDD to studyparticleboard technology, by comparing physical and mechanical propertydifferences among particleboards, which were themselves manufactured withdifferent board formations. This endeavor was probably similar, to a certainextent, with the comprehensive in-grade testing program started in the late197Os, with the objective to quantify the differences of various mechanicalproperties of different grades and sizes of lumber (Madsen, 1992). Therefore,33random field theory may not be necessary. Instead, an approach implied andlater used by Suchsland (1959, 1991) was adopted for this study. As Figure 3implies, given the same average board density, the fractional area thatpotential bonding may develop and the extent of micro nonuniformity ofboard structure could be characterized by the standard deviation of density.By studying this standard deviation parameter, determined at differentspecimen sizes, the contribution of raw material characteristics to boardstructural nonuniformity could be determined. The study of the relationshipbetween this parameter and board properties would identify the significanceof structural nonuniformity on physical and mechanical properties ofparticleboard. In the process of studying the structure of paper, imagerepresentation (Yang and Thorpe, 1977), spectrum presentation (Norman andWahren, 1973), and direct visual examination of beta radiographs anddensity histogram method (Corte, 1971) have been used to characterize thestructural nonuniformity. Corte (1982) concluded that “the characterizationof the nonuniformity of paper by the variance of its DMD permits a numericalformulation to explore the effect of material and process variables on theuniformity of paper and through it, on its performance”.344. METHODOLOGY4.1. Material4.1.1. RoundwoodTwo wood species, trembling aspen (Populus tremuloides Michx.) andwhite birch (Betula papyrifera Marsh.) were used to manufacture woodparticles for this study. Trembling aspen trees were cut from the Universityof British Columbia (UBC) Research Forest at Williams Lake, B.C. Whitebirch logs were obtained from the province of Alberta, while detailedinformation was not available. General species information is summarized inTable 1.Table 1. Roundwood information*Species Aspen White BirchLocation Williams Lake, B.C. AlbertaHarvest Time April, 1991 April, 1991Diameter (cm) 34-44 14-18n 100 100Density (g/cm3) 0.469 0.578s 0.029 0.041* n = number of observations; = average; s = standard deviation. The density wasdetermined based on approximately 1.5 cm x 1.5 cm x 10 cm wood specimens at an averagemoisture content of 6.8%. The actual volume was determined by the water displacementmethod (Heinrichs, 1954).354.1.2. Wood particles4.1.2.1. Specialty particlesPrecisely cut particles were used to study how raw materialcharacteristics affect HDD (measured by the standard deviation of density) ofparticleboard. Green trembling aspen and white birch logs were debarked byhand and cut into about 40 cm long discs. These discs were then rotary peeledinto veneers of predetermined thicknesses at Pansmill Woodenware Ltd. inVancouver, B.C.The green veneers were cut into particles of predetermined sizes using atable saw. These particles were then slowly dried in a laboratory oven attemperatures of 40-80 OC to maintain the flatness of particles. Final particledimensions at equilibrium moisture content of 6.8% were measured. Table 2alists the result of this measurement for particles used for the study of particlesize effect on HDD. Dimensions of particles used for the verification of layerconcept are provided in Table 2b.From these tables, the difference of actual thicknesses of particle groupsAl-Di, El and E2, Fl and F2, and Gl and G2 respectively could beconsidered negligible. Also for the purpose of analyses, the target length andwidth dimensions, which differed little from actual dimensions, were used forcalculations.4.1.2.2. Commercial and laboratory particlesCommercial and laboratory particles were used to make particleboardswith different HDD characteristics. Physical and mechanical properties ofthese boards were evaluated and related to the board HDD. The choice ofTable2a. Particlesusedforstudyingparticlesizeeffectonhorizontaldensitydistribution*ParticleCodeAlA2A3A4BiB2B3B4ClC2C3DlTarget4.06.08.010.04.06.08.010.06.08.010.08.0Lengthn303030303030303030303030(cm)4.005.958.059.954.005.958.0010.056.008.009.958.00s0.430.420.410.400.420.430.400.440.400.420.420.41Target2.02.02.02.04.04.04.04.06.06.06.08.0Widthn303030303030303030303030(cm)1.952.002.002.003.954.004.004.055.955.956.007.95s0.490.490.480.440.510.490.460.480.460.490.480.50n404040404040404040404040Thicknessi.640.636.637.636.640.635.641.637.639.640.637.640(mm)s.024.022.025.027.026.025.024.025.025.028.023.025*Alltheparticleswerepreparedfromaspenlogs; n-number ofobservation;I-average; s-standarddeviation.(A)37Table 2b. Particles used for verifying layer concept*Particle Code El E2 Fl F2 Gi G2Species A A A A B BTarget 10.0 6.0 10.0 6.0 10.0 6.0Length n 30 30 30 30 30 30(cm) 9.90 6.00 9.95 6.00 10.05 6.00s 0.43 0.43 0.41 0.40 0.41 0.42Target 2.0 6.0 2.0 6.0 2.0 6.0Width n 30 30 30 30 30 30(cm) 1.96 5.95 2.00 6.00 1.95 5.95s 0.47 0.45 0.49 0.50 0.44 0.47n 40 40 40 40 40 40Thickness 1.061 1.060 1.942 1.943 .696 .699(mm) s .030 .031 .038 .039 .028 .024* A - Aspen; B - Birch; n - number of observation; i - average;s - standard deviation.38commercial and laboratory particles for this purpose was based on theknowledge gained from HDD evaluations with specialty particles in the firstpart of this study.Three types of commercial wafers/strands from waferboard/OSB millswere obtained through the courtesy of Alberta Research Council inEdmonton, Alberta. Figures A-i, A-2, and A-3 in Appendix A show thedistribution of particle length, width and thickness for commercial particlesp1, p2 and p3. It can be seen from these figures that large variations exist intheir dimensions.Laboratory particles were cut on a laboratory-type-waferizer located atCAE Machinery Ltd. in Vancouver, B.C. The two species and severaldifferent waferizer settings were chosen to produce a variety of particles.Figures A-4, A-5 and A-6 in Appendix A show the distribution of laboratoryparticles. Variations in dimensions were expected, but they were less than forcommercial particles, especially as thickness is concerned.4.1.3. AdhesivePhenol-formaldehyde powder resin W3154 was used throughout thisstudy. It was provided by Borden Co. Ltd of Canada. This resin has been usedboth for face and core applications in commercial waferboardlOSBmanufacture.4.1.4. Commercial wood productsWaferboard was used for studies involving general HDD of particleboardand specimen size effect. Parallam, M1F and waferboard/OSB were used totest the X-ray density scanning machine. These products provided a wide39range of wood composite materials, in terms of the size of constituting woodelements. They were used in the sense that no attempt was made to sayanything about their whole productions.4.1.4.1. WaferboardCommercial waferboard panels were provided by Weldwood of CanadaLtd. Average board density was 0.67 g/cm3, and board thickness was 11 mm.The wood element sizes used to manufacture this waferboard were believedsimilar to that shown in Figure A-2, as both the waferboard and the particleswere from the same mill.4.1.4.2. ParallamParallam samples were obtained from MacMillan Bloedel Ltd., B.C. It ismade by aligning approximately 2.5 m long Douglas-fir strands in onedirection, and forming a board which is continuously pressed in a microwaveheating environment. The nominal cross section of the strands was 16 mm x3 mm, and the product used was measured to be 0.68 g/cm3 in density and 9.5mm in thickness.4.1.4.3. Medium density fibreboard (MDF)MDF was obtained from a local lumber supplier. Although the precisedimension of fibres was unknown, it is believed that the wood elements are inthe form of fibre bundles. Fibre bundles are generally 1-10 mm in length, and0.03-0.3 mm in diameter. MDF used in this study was determined to be 0.80g/cm3 in density and 9.7 mm in thickness.404.1.4.4. WaferboardlOSBOne waferboardlOSB product was also obtained from a local lumbersupplier. The particle strands are believed to be 10-75 mm long, 5-50 mmwide and 0.25-0.75 mm thick. It was measured to be 0.68 g/cm3 in densityand 11 mm in thickness.4.1.5. Laboratory ParticleboardBasic procedures for manufacturing particleboard in the laboratory wereas follows. First, a predetermined amount of wood particles and adhesivewere mixed in a laboratory blender for 10 minutes. A resin content of 2 %,based on oven-dry weight of particles, was used to produce boards forstudying the influence of raw material characteristics on HDD.The adhesive blended particles were then randomly hand-formed in a 71cm x 71 cm wooden box into a particleboard mat. During the forming process,mat height at the four corners and the middle of each side of the forming boxwas measured several times. This method was used to achieve a uniformlythick mat. The mat was then compressed and consolidated into a panel underheat and pressure in the hot-press. In this study, a pressing temperature of180 OC and a pressing time of 10 minutes were adopted. This relatively longpressing time was used to insure complete adhesive cure. A press closingtime of approximately 1 minute was also used.Two groups of boards were manufactured. The first group was designedto study the influence of raw material characteristics on HDD. In this group,one board was made with each type of specialty particles (Table 2a) to studythe effect of particle size. These boards had a target density of 0.67 g/cm3 and41thickness of 11 mm. In addition, boards were made to study other parametersof raw material characteristics, and these boards are summarized in Table 3.The second group of boards were manufactured mainly using laboratoryand commercial particles to determine the influence of HDD on selectedparticleboard properties. Table 4 describes the composition of these panels.After manufacture, all particleboards were stored for equilibration in anenvironmental room, with temperature controlled at 20±2 OC, and relativehumidity at 50±2 % for at least 4 weeks before testing.4.2. Density Measurement4.2.1. Density measurement methods4.2.1.1. Gravimetric methodA gravimetric method was the principal technique used to study HDD.Depending on the size of the specimen, either a sawing technique or a drillingtechnique was used. For specimen sizes above 25 cm2, a sawing techniquewas used with specimen cross-sectional area measured by a digital caliperwith an accuracy of ±0.01 mm and weight determined on an electronic digitalbalance of ±0.00 1 g accuracy. The thickness of the specimen was taken to beconstant and equal to the target thickness of the board. The density was thencalculated as Density = Weight / (Area x Thickness).When specimen size was less than 25 cm2, a drilling technique wasused. The cross-sectional area was taken to be the size of drill bit, and theweight was taken to be the weight loss of the specimen from before to after42Table 3. Particleboards used to verify layer concept*Board Particles Density Thickness(glcm3) (mm)El El 0.67 11E2 E2 0.67 11Fl Fl 0.67 11F2 F2 0.67 11Gi Gi 0.67 11G2 G2 0.67 11Ki 112E1-i-112F1 0.67 11K2 1J2E2-i-l/2F2 0.67 11Ri A4 0.86 11R2 Cl 0.89 11Si A4 0.67 6S2 Cl 0.67 20Wi A4 0.67 6W2 Cl 0.67 20* One board was made for each condition; birch furnishwas used for boards Gi and G2, other boardsused aspen furnish.43Table 4. Particleboards used to study the influence of horizontaldensity distribution on board properties*Board Particles SpeciesP1 p1 AP2 p2 AP3 p3 AP4 1/4p1÷114p3+1/4D1÷1/4p6 AP5 p5 AP6 p6 AP7 p7 BP8 213p2÷l/3p7 A+BP9 112p2÷112p7 A+BPlO 113p2+2/3p7 A+B* A- Aspen; B- Birch. Three boards were made for eachboard condition, with a target board density of0.75 g/cm3 and a board thickness of 11 mm.44drilling. The density was then calculated as Density = Weight / (Area xThickness). Figure 11 shows the set-up for the drill press.4.2.1.2. X-ray scanning methodAn X-ray technique, developed by VisionSmart in Edmonton, Alberta tomeasure density profile for lumber products, was also used to determinepanel density in a non-destructive manner. Although this instrumentationwas still under development, it provided a density profile at a resolution of 1mm x 3.4 mm size. Density at larger specimen sizes could be calculated basedon these density readings according tonDA= d/n (7)i=1where, DA is the density of specimen size A, d is the individual density atspecimen size of 1 mm x 3.4 mm, n is the number of adjacent 1 mm x 3.4 mmspecimens that are included in size A.Limited access to this technology during this thesis study enabled us toperform an analysis on the relationship between densities at certain distanceapart within a panel. The calculation of coefficient of correlation was asfollows,p = Cov(X,Y) I fS(X)S(Y)} (8)where, p is the coefficient of correlation between variables X and Y, Cov(X,Y)stands for the covariance between variables X and Y, while S(X) and S(Y) arestandard deviations of X and Y.45I.;cjFigure 11. Drill press set-up for density determination.464.2.1.3. y-ray methodThe y-ray measurement of density was performed on the Woodmatproduction line of Canadian Forest Products Ltd., New Westminster, B.C.This non-destructive technique measures the density of a circular specimenof 35 mm in diameter, and was used to characterize density variations at thissize for studying the influence of HDD on board properties.4.2.2. Sampling of density specimens4.2.2.1. Gravimetric method4.2.2.1.1. Laboratory particleboardAfter being trimmed, the laboratory-made particleboards wereapproximately 700 mm x 700 mm. As the particle mat spread outwardsapproximately 10 mm during pressing, final pressed particleboards werelarger in size than that of the forming box. Therefore, about 15 mm wastrimmed off from around the edges. Panels manufactured to study theinfluence of raw material characteristics on HDD each were cut into 36specimens of approximately 110 mm x 110 mm. Those panels made to studythe effect of HDD on selected board properties each were sectioned into 25specimens of approximately 110 nun x 110 mm, with the remaining portionsof these panels used for bending specimens (Figure 13, board 3). Densityvariation at this specimen size was determined.For assessing density variations at smaller specimen sizes, holes weredrilled into the 110 mm x 110 mm sections. A simple random samplingprocedure was used for selecting these holes. First, the maximum number ofdrilled holes arranged in squares as shown in Figure 12, for a specificRandom number = 11471 2 3 45 6 7 89 10 1213 14 15 16Figure 12. A schematic of the procedure for allocating drilling specimen.48diameter, that the section could accommodate was determined (some otherarrangements of the circles may end up with different maximum numbers ofcircles). Then, each potential hole was given a different but consecutivenumerical value. Figure 12 shows this procedure schematically for a hole sizeof 5.07 cm2. A random number then was chosen. The hole selected by thisrandom number was drilled and its density was measured as a randomobservation (Figure 12).After the density measurement for one hole size was completed fromeach 110 mm x 110 mm section, the next smaller sized measurement waschosen by a similar procedures as exhibited in Figure 12. One measurementwas taken from each drilled section for larger hole sizes (> 5 cm2), while twoor more holes were drilled when the specimen size (hole) was smaller.When this thesis was proposed, a systematic sampling method ofdensity specimens was also considered. This method would allow thecalculation of autocorrelation function of density process, which could be usedto simulate density process. Since the main hypothesis of this thesis was thatparticleboard technology could be better understood by studying themagnitude of HDD, a simple random sampling method was adopted. Notheory is available to estimate the statistics based on the systematicsampling method, and the variance calculated by using the formuladeveloped for the simple random sampling method is considered biased for asystematic sampling method (Cohran, 1977)4.2.2.1.2. Commercial waferboardSixteen commercial waferboard panels of size of 1220 mm x 606 mmwere obtained. The manufacturer indicated that these panels came from two49adjacent press loads and were cut into this size on the production line. Aswaferboard is formed and pressed in a continuous process, these panels couldbe viewed as coming from one big panel. Since it was not the objective fromthis limited sample to present the whole picture of density and other propertyvariations for waferboard production, these panels were treated as apopulation themselves in this study, just like the treament for laboratoryparticleboards.The standard deviation of density at this specimen size of 1220 mm x606 mm was determined by using 11 of these panels. Afterwards, five of thesepanels were randomly selected and partitioned in half to give 10approximately 536 mm x 536 mm sections, and the standard deviation ofdensity was determined at this specimen size. This partitioning process, asshown in Table 5, continued until a specimen size of approximately 117 mm x116 nim was reached. At this stage, twenty-nine of these specimens wererandomly selected (the rest were used later for TB test) and partitioned into58 smaller specimens measuring 116 mm x 54 mm, and density variation wasobtained at this specimen size. These specimens were also subsequentlypartitioned into 54 mm x 54 mm sections, and the standard deviation ofdensity again was estimated (These 54 mm x 54 mm specimens were reusedlater for TB test).Fifty waferboard sections of 117 mm x 116 mm were randomly selectedfrom the fifth partitioning stage and used as the basis for densitydeterminations at specimen sizes less than 25 cm2. The drilling techniquewas used and the sampling methodology used for the laboratory particleboardwas applied.50Table 5. Partitioning procedure for density determinationof commercial waferboardPartitioning Approximate Number ofStage Specimen Size Observation0 l22Ommx6O6mm 51 536mmx536mm 102 535mmx251mm 203 252mmx251mm 404 252mmxll7mm 785 ll7mmxll6mm 15351In one case, a sawing technique was used to partition a 570 mm x 570mm section of waferboard to specimen sizes of approximately 20 mm x 20 mmfor density profile presentation (Figure 14).4.2.2.2. X-ray and y-ray methodsWith the X-ray scanning method, the whole board was scanned and thedensity readings at each point were provided in a spreadsheet format. Allthese density readings were used for correlation analysis in this study.Consequently, no sampling was involved.With the y-ray method, only one measurement (38 mm in diameter) inthe middle of each approximately 110 mm x 110 mm sized specimen wasmade. This simple choice of measurement was due to the difficulty of movingaround the specimens on the production line.4.3. Determination of Board Properties4.3.1. Modulus of rupture and modulus of elasticityModulus of rupture and modulus of elasticity were determinedaccording to CAN3-0437.1-M85 (Canadian Standards Association, 1985).Specifically, the standard requires that the span/thickness ratio of thespecimen equals 24 and time to failure should be about 5 minutes.The calculations of MOR and MOE, determined on the same specimen,were as follows:MOR = 3LPmax/(2bt2) (9)MOE =[L3/(4bt)]p/y (10)where52Pmax = ultimate failure load (N);L = span between centres of supports (mm);b = width of the test specimen (mm);t = average thickness of the test specimen(mm);p = increment in load on the straight line portion of the load/deflection curves(N);y = increment in deflection at midspan corresponding to a p increment onload (mm).Three replicates were produced at each board condition to study theinfluence of H]JD on board properties. Two bending specimens were cut fromthe edge of each replicate to give a total of six bending specimens (Figure 13).4.3.2. Internal bondInternal bond was determined following CAN3-0437.1-M85 (CanadianStandard Association, 1985), except that a range of specimen sizes were used.A hot melt adhesive was used to bond the particleboard specimens toaluminum blocks. This test required that the aluminum block not be smallerin size than the particleboard specimen, and test results are valid only whenfailure does not occur in the glueline. Internal bond (MPa) was calculatedaccording toIBPmax/A (11)where, (N) represents the maximum failure load and A (mm2)the area ofspecimen.534.3.3. Thickness swellingThe method of submerging squared specimens horizontally to a 25 mmdepth in 20 OC water as specified in CAN3-0437.1-M85 (Canadian StandardAssociation, 1985) was followed. When specimen size was larger than 50 mmx 50 mm, specimen thickness was calculated as the average of thicknessesmeasured to an accuracy of ±0.05 mm at midway points along each side, 25mm from the edge of the specimen. With specimen size less than 50 mm x 50mm, thickness was measured only at the centre point of the specimen.Thickness swelling was then calculated to the nearest 1% in accordance withthe following formula:TS(t) = [T(t) -T(0)]100/T(0) (12)whereTS(t) = TS after t hours soak (%);T(t) = average thickness after t hours soak (mm);T(0) = average thickness before soaking (mm).The cutting pattern used to obtain TB and TS specimens for laboratoryparticleboard is shown in Figure 13. For commercial waferboard, samples forTB and TS tests were randomly chosen from the set of boards left from thedensity determinations. This sampling procedures are shown in Appendix Band Appendix C respectively.54board 1lBTSMOR + MOEboard2TSlBMOR + MOEboard 3HDDMOR + MOEFigure 13. Cutting pattern for preparing test specimensfor laboratory particleboard.555. RESULTS AND DISCUSSIONS5.1. Aspect of Horizontal Density Distribution5.1.1. Phenomenon of horizontal density variationThe density map for a commercial waferboard sample of 57 cm x 57 cmsize is presented in Figure 14. Individual densities were measured at aspecimen size of approximately 20 mm x 20 mm. Gaps of 3 mm, equivalent toa saw blade thickness, existed between adjacent specimens. This figuredemonstrates the phenomenon of HDD in particleboard showing the generalnonuniformity in structure.Figures 15 and 16 show two density distribution histograms for thecommercial waferboard determined at specimen sizes of 29.16 cm2 and 0.31cm2 respectively. A normal distribution curve was fitted to these histograms,together with a calculated Pearson CM-squared statistic (x2), a parameterindicating the goodness-of-fit. As statistics x2 were less than thecorresponding critical values ofX2(k.r.1 at significance level of 0.05, where kand r are respectively the number of groups used to arrange data and thenumber of statistics estimated for the hypothesized distribution, k-r-1 is thedegree of freedom, it was concluded that the waferboard densityapproximated a normal distribution. This was expected since individualdensity measured at any specimen size could always be taken as the averageof several smaller sized densities. The central limit theorem supports thisnormal approximation (Fisher, 1950).Figure 14 suggests that the density variation could be viewed as a.random field. For a random field, the variation of measurements decreases as56Ip0.130.6Figure 14. Horizontal density variation of one commercial waferboard.570C0.200.150.100.05Density (g/cm3)Figure 15. Density distribution characteristic of one commercialwaferboard at specimen size of 29.16 cm2.0.48 0.56 0.64 0.72 0.8058000.250.200.150.100.05Density (g/cm3)Figure 16. Density distribution characteristic of one commercialwaferboard at specimen size of 0.31 cm2.0.30 0.65 1.0059the measuring area (window) increases (Vanmarcke, 1980). The results ofdensity measurements for the same commercial waferboard at differentspecimen sizes are given in Table 6. The relationship between the standarddeviation of density and specimen size is shown in Figure 17, which agreeswith the random field theory with the result that standard devaition ofdensity decreases as specimen size increases.It was believed that three factors were associated with this densityvariation: variation in particle densities, nonuniformity of the formingprocess and the existence of voids (Suchsland and Xu, 1989). It can beobserved from Figure 17 that the sensitivity of density variation withspecimen size decreased dramatically as specimen size exceeded 25-50 cm2.This was referred to as the less sensitive range in this thesis. The significantsmaller variation in this specimen size range is believed to be the reflection ofuniformity of the forming process. The influence of voids and the variation inwood density is likely minimized in these relatively large specimen sizes.When specimen size was less than 25 cm2, all of the three factors wereprobably interactive. The effect due to the variation of particle densities isbelieved to decrease as board thickness increases, because the number ofparticle layers increases and the variation of average density of these layersdecreases. Furthermore, it is believed that the contribution of variation ofparticle densities to horizontal density variation is less profound compared tothat of voids. If the forming process is quite uniform, the density variationdetermined at relatively small specimen sizes could be considered to becaused mainly by voids. It is this density variation caused by voids that wasthe main focus of this thesis.60Table 6. Density determination of commercial waferboardSpecimen size Density (g/cm3)(cm2) Sample Size Mean Standard Deviation0.31 69 0.676 0.0930.71 65 0.699 0.0871.27 65 0.683 0.0745.07 55 0.680 0.06429.16 116 0.656 0.04562.64 58 0.659 0.039135.72 153 0.677 0.039294.84 78 0.677 0.036632.52 40 0.679 0.0321342.85 20 0.679 0.0282872.96 10 0.672 0.0227393.20 11 0.698 0.021610.10 I I I -I I0.08I0.060.00 I I I0 100 200 300 400 7400Specimen Size (cm2)Figure 17. Standard deviation of density vs. specimensize of one commercial waferboard.625.1.2. Relationship between standard deviation of density andspecimen sizeLet Aa and Ab represent respective specimen sizes (areas) in density setsA and B, Da and Db the density variables, and Var(Da) and Var(Db) thevariances of densities associated with density sets A and B. If the densities ofindividual points are independent, then the following relationship (AppendixD) exists according to basic statistics (Fisher, 1950),Var(Da)/Var(Db) = A1IAa (13)Taking the square root on both sides of Equation (13), we haveS(Da)/S(Db) = (14)where S stands for the standard deviation.Rearranging Equation (14) and letting the equality equal a constant c,we also haveS(Da)’JA = S(Db)”dX = c (15)which could be generalized asS = c(1/’IA) (16)When A = 1, c = S, the standard deviation of density at a unit specimensize. Equation (16) indicates a linear relationship between S and ii’1A Thescatter plot of S to 1/qA for the commercial waferboard (Figure 18), however,deviated clearly from a straight line. One possible explanation for this nonlinear trend is that the density of samples of various sizes are correlated. Forthe purpose of curve fitting, a curvilinear relationship of S to ii.IA was chosenas follows:S = c(1/J)bi (17)or630.10 I I..0.08 -.0.06 -0.04 -0.020.00 I I I0.0 0.4 0.8 1.2 1.6 2.0vxFigure 18. Scatter plot of standard deviation of density vs.lb/A of one commercial waferboard.64S = c(1/A)b (18)where, b1 = 2b, and both b and b1, and c are parameters to be determined byregression.The form of Equation (18) and the possible meaning and range forparameter b can be explained by considering statistical concepts. If aparticleboard production is under statistical control (quality control), thedensity of particleboard at every point should vary within a certain limitaround its mean (average board density). Therefore, this density variationcould be considered as a stationary process (Bendat and Piersol, 1980). For astationary process, the measurement within the process must be positivelycorrelated (Bendat and Piersol, 1980). In other words, the coefficient ofcorrelation (p) is larger than or equal to zero.Figure 19 shows the relationship between the correlation coefficient ofdensities measured on approximately 3.0 mm x 3.4 mm specimens and theLag distance, performed on a commercial waferboardlOSB panel of 18 cm x18 cm (X-ray data). Although it was unknown how well this X-ray instrumentwas calibrated, a clear positive correlation existed among densities.Let us consider two specimen sizes, Aa and Ab again, but specially Ab =2Aa. Since densities were positively correlated, we have (Appendix E)Var(Db) = [2Var(Da) + 2pVar(D)]/4= [(1+p)Var(Da)]/2 (19)or,Var(Db)/Var(Da) = (l-t-p)/2 (20)By using Equation (18), we also have (Appendix F)Var(Db)/Var(Da) = (V2)2b (21)65SID00C.)000D1.00.80.60.40.20.0DD ID SD.c.VQ....csDC UCUICI0 20 40 60 80Lag (mm)Figure 19. Relationship between correlation coefficient of densityand Lag for one commercial waferboardJOSB.66Comparing Equations (20) and (21), the following relationship wasobtained,b = [ln{(l+p)/2}/ln(l/2)]/2 (22)Therefore, parameter b was related to p, coefficient of correlation ofadjacent specimens. Numerically, when p = 0, b = 1/2 according to Equation(22), or b1 = 1. This corresponds to Equation (16), for which no correlationexists between density measurements. When p = 1, b = 0, variance is not afunction of specimen size. This corresponds to the situation where boards areperfectly homogeneous, i.e., density is constant and therefore densities atadjacent points are completely correlated. Naturally, density is constant andspecimen size independent. Thus, parameter b possessed a physical meaningthat indicates the level of correlation among density points. As b increases to1/2, strength of correlation decreases.It is interesting to note that parameter b could not exceed 0.5. If b> 0.5,p < 0 according to Equation (22), which violates the property of stationaryprocess. As nonstationary process can usually be converted to stationaryprocess for analyses (Bendat and Piersol, 1980), the case of b > 0.5 can besafely excluded from the real world. Furthermore, b could not be negative. Ifb < 0, p > 1 according to Equation (22), which does not agree with thedefinition of correlation coefficient. In fact, if b < 0, Equation (18) predicts anincrease of standard deviation as specimen size increases, which is difficult toimagine.It should be mentioned that exactly the same expression as that ofEquation (17) was used to relate variance of crop yield per unit area to plotsize (Smith, 1938), and this relationship has been widely accepted and67applied to predict agricultural crop yields (Kuehi and Kittock, 1969; Nelson,1981). Recent comparison between geostatistics and Smith’s (1938)expression indicated the appropriateness of the latter, and a range between 0and 1 was also recognized for parameter b1 (Zhang et al, 1990). It was alsoworth mentioning that this model transformed the variable in such a fashionthat the relationship between standard deviation of density and specimensize could be better visualized, especially when the specimen size wasrelatively small. This was evident from the comparison of Figure 17 withFigure 18.5.1.3. Estimation of S to 1/A relationshipTaking natural logarithm of Equation (18) on both sides, we havelnS=lnc+bln(IJA) (23)Through the least square technique, b and c are obtained. For thecommercial waferboard, the values 0.1503 and 0.0795 were found. Statisticalanalyses (Appendix G) indicates that parameter b was significantly differentfrom 0.5 and 0, which were the boundaries for this parameter. Taking theseestimates back to Equation (18), it gaveS = 0.0795(IJA)0.1503 (24)The coefficient of determination (R2) between the model prediction andexperimental data was calculated asn nR2 = 1 - (s-) / (s1-) = 0.98 (25)i=1 i=1in which, is the mean, is the model estimate and Si 15 the actualobservation.685.2. Influence of Raw Material Characteristics on Horizontal DensityDistribution5.2.1. Particle sizeDensity variation results for laboratory boards made with specialtyparticles are given in Table 7. The fitted equations relating S and specimensize for different boards are listed in Table 8. It is noted in Table 8 that R2increased as b (the slope in the logarithmically transformed expression asEquation (23)) increased. This is consistent with the properties of regressionanalysis (Fisher, 1950). Figures 20 and 21 show the relationships betweenstandard deviation of density (S) with particle length and width, respectively,measured at several different specimen sizes. Since these trends as specimensize changes were similar for particle length and width, their product, whichis the particle size (area) in unit of cm2, was calculated and plotted against S.This is presented in Figure 22. It should be emphasized here that thespecimen size used to determine the density nonuniformity is different ineach small plot in this figure, but the same set of boards was usedthroughout. For example, the same particleboard (Al) made with particles ofsize of 8 cm2 (the first data point from left in each plot) was used for themeasurement of standard deviation of density at all six specimen sizes.It can be seen from Figure 22 that particleboard made with largerparticles is less uniform than particleboard made with smaller particles whenmeasured at specimen size larger than 1.267 cm2, as indicated by increased Sas particle size increased. While particleboard made with smaller particles isobserved to be less uniform at smaller specimen sizes, as demonstrated byincreased S as particle size decreased. It should also be mentioned here thatTable7.Densitydeterminationof laboratoryparticleboard*SpecimenSizeDensity(g/cm3)cm2ParticleboardAlA2A3A4B1B2B3B4ClC2C3Dln363636363636353636363636115.1021.653.677.674.673.671.688.686.674.670.658.670.663s.020.025.030.034.036.040.045.055.042.040.049.050n35363636363635363636363611.4011.681.707.708.699.701.696.700.700.701.704.700.666s.043.064.075.076.073.080.086.087.078.074.090.095n3636363536363536363636365.0671.666.689.671.692.647.668.650.682.651.662.658.675s.070.078.086.096.094.105.097.102.085.085.098.108n7070717272717069697072691.2671.645.679.679.663.696.693.686.666.681.665.657.680s.090.100.110.108.102.113.103.112.098.095.100.103n1051241081051081081021191081041081030.3171.680.703.688.658.685.671.700.700.685.669.669.689s.126.123.120.118.126.119.119.118.107.101.103.107ii1211321331201261311101251221201201200.1411.676.654.680.664.672.698.687.654.677.681.666.681s.184.179.169.167.166.160.152.147.149.134.129.130*Codeofpartideboardisthesameasforparticlesusedfortheboardmanufacture;n=numberof observations;i=average; s=standarddeviation.70Table 8. S to VA models for laboratory particleboard*Particleboard Modelc b R2Al 0.097 0.319 0.98A2 0.105 0.268 0.95A3 0.113 0.215 0.93A4 0.111 0.202 0.89Bi 0.110 0.208 0.94B2 0.114 0.182 0.90B3 0.111 0.160 0.92B4 0.113 0.132 0.93Cl 0.103 0.166 0.92C2 0.969 0.157 0.91C3 0.103 0.120 0.82Dl 0.107 0.117 0.76* R2 = coefficient of determination.Figure 20. Influence of particle length on standard deviation of density ofparticleboard at several specimen sizes. Data sets with sameparticle width are connected and labeled by width (cm).Size = 115.102 2I871ISize 11.401 cm2 ‘86424 6 8 10Length (cm)12I IISize= 1.267cm0.06 0.10•I 4..-.60.04 - 0.08oc, ecE E.> CC.)2. .0.02 0.06..-U2 CI)0.00 I I 0.042 4 6 8 10 12 2Length (cm)0.12 I 012., jSize=5.067cm2j+.—.‘8 CG 4 0.110.10 60c—’ 20.Q.G0.08. L 0.09.e -t.CI) CI)0.06 I I I 0.082 4 6 8 10 12Length (cm)0.13 I I I 0.20C.0.12 0.18-0C,—’ C—’ec. oc,:E0.1iCC.).@) NN.t010 0.14.t -t[Size=0.317cm’0.126422 4 6 8 10 12Length (cm)4Isize= 0.141 cm2I0.09 -2 4 6 8 10 12Length (cm)2 4 6 8 10 12Length (cm)Figure 21. Influence of particle width on standard deviation of density ofparticleboard at several specimen sizes. Data sets with sameparticle length are connected and labeled by length (cm).7210S.64Size = 115.102 2{64lSize= 11.401cm2I2 4 6 8 10Width (cm)0.06O.04ec.0A0.0000.12.0.100.06I0.060.130.12no.110.10I0.092 4 6 8 10Width (cm)[Size = 5.067 2I10880.10 -o 0.08ec0.06I0.0400.12I0.11E- c0.10.0.09Ci)0.080200.180.160J4I0.12640.4Size= 1.267cmI I I2 4 6 8Width (cm)2 4 6 8Width (cm)10 0 104 0 0 fize=0.317cm2I6810[Size = 0.14 1 cm2I468100 2 4 6 8 10Width (cm)I I I0 2 4 8 8 10Width (cm)73Figure 22. Influence of particle size on standard deviation of densityof particleboard at several specimen sizes. The curveis fitted by eye without regression analysis.00.04 C 0a0 000C0.10 -0.080.060ISize = 115.102 cmj0 20 40 60 80Particle Size (cm2)tfze = 11.40 1 cm220 40 60Particle Size (m2)800.06C)oc,0.oo0.12o 0‘—S10.080.060.13c0.110.10CI)0.09 -00CCaC0jjze = 5.067 cm10 20 40 60 80Particle Size (cm2)0.‘-SocF0.040.12O.11•0d0C) S.-’0.090.06020CC) 0.1800CIQ0.16.0.14C-SU)0.12I—00C0/KN\\.00ISize = 1.267 cm21I I00020 40 60Particle Size (cm2)080ISize = 0.317 2)0ISize = 0.141 cm2lCCC.020 40 60 80 0 20 40 80 60Particle Size (cm2) Particle Size (cm2)74the scale of S is different in each small plot of this figure (because densitynonuniformity is specimen size dependent). One possible explanation, asdicussed in the following, may provide insight for this trend of S to specimensize. During the hand-forming process, particles were deposited randomlythroughout the horizontal plane in a more or less layer by layer fashion.Voids can exist between any adjacent particles in any layer. As particle sizeincreased, the number of these voids was observed to decrease in a unit areawithin one layer (the number of particles also decreased), while the size ofvoids increased. Figure 23 schematically shows these two aspects of voids inone layer as affected by particle size under hand-forming operations. Thissimilar phenomenon was also observed in packing wooden rods, fibre glassand sphere shaped particles (Milewski, 1987). These two tendencies of voidsare then reflected in Figure 22.When specimen size was relatively large in comparison to the largestvoid size, an individual density specimen may contain several voids (orseveral particles) when the particleboard was made with relatively smallparticles, due to the relative small size of these voids (Figure 23). Underthese conditions, voids tended to be distributed more evenly among thedensity specimens. Less density variations resulted for particleboards madewith smaller particles than for particleboards made with larger particles.This was observed when specimen size was larger than 1.267 cm2, as in (a),(b) and (c) of Figure 22.When the specimen size for the determination of S was small incomparison to the smallest size of voids, void size may not be that important,compared to the number of voids, in determining the extent of densityvariations among different boards. As there were less numbers of voids75(a)ParticleVoid(b)ParticleVoidFigure 23. A schematic of particle and void distributionin one layer under hand-forming operation.(a) Small particle (b) Large particle76within particleboards made with larger particles, and since a densityspecimen consists of several particle layers, large particles may tend to coverthese voids more effectively in adjacent layers during the formation processleading to less density variation. For example, it is more unlikely for a smallhole to penetrate from one surface of particleboard to the other forparticleboard made with larger particles. This happened when specimen sizewas smaller than 1.267 cm2 and is shown in (e) and (f) of Figure 22. Kusian’s(1968) model analyses also predicted a decrease of probability of HDD asparticle size increased, which seems in accordance with the experimentalobservations of this study, although Kusian failed to bring specimen size intohis discussion.It is understood that this explanation was only based on visualobservation, a more rigorous experiment involving permeability test or modelanalysis (as in paper physics studies) might provide further insight.Furthermore, these void characteristics, and the dependence of therelationship between S and particle dimension on specimen size was based ona simple hand-forming method. It is expected that different forming methodsor even the same method but conducted by different persons will generatedifferent results. Consequently, specific values observed in this study are notto be taken as absolute. However, it is believed that the two aspects of voids(number of voids and size of voids) occur with present industrial formingmethods or other forming methods, and that the trend observed in this studywould be applicable. If a forming technique could be devised in such a waythat the void size is independent of particle size, density variation woulddecrease further for boards made with larger particles at relatively smallspecimen sizes. This suggests that forming method is as important as choice77of particle size in decreasing horizontal density variation, and effort shouldbe directed at this in the future. As these two aspects of voids exist andinteract in this study, density variations have to be measured at a range ofspecimen sizes, in order to understand the mechanism and influence ofparticle size on density variation and subsequent board properties.Besides particle size effect, particle shapes are also believed influencingHDD of particleboard. For example, Suchsland’s (1959) experiment suggestedthat square particles might produce more uniform structures thanrectangular ones. Two particles (with the same particle size) which couldprovide this comparison in this experiment were A3 (rectangular: 2 cm x 8cm) and Bi (square: 4 cm x 4 cm). However, Suchsland’s indication could notbe substantiated from this limited data (Table 7). A more comprehensiveexperiment is needed to investigate this aspect, not only comparing squareshaped with rectangular shaped particles, but also with other shapedparticles, like circular or triangular ones.5.2.1.1. Parameter bThe parameter b was used in Equation (18) to model the dependence ofdensity variation to specimen size. It was also recognized as an indicator ofthe strength of correlations among density measurements. Furthermore, theinfluence of parameter b on density variation could probably be observedmore clearly at relatively small specimen sizes, becaise the inverse ofspecimen size (1/A) and therefore the differences of standard deviations ofdensity due to different b values increase as specimen size decreases. Thus, itwas more appropriate to discuss the relationship between parameter b and78structural nonuniformity by confining the discussion to smaller specimensizes, say 0.14 1 or 0.3 17 cm2.The influence of particle size on b is shown in Figure 24. It is seen that bdecreased as particle size increased, indicating that correlation improves.This may also relate to void characteristics. When specimen size was small,the number of voids within a specimen decreased as particle size increased.When density measurements were closer to each other, stronger correlationresulted. Figure 25 shows the correspondence between b and standarddeviation of density at specimen size of 0.141 cm2.Figures 26 and 19 show the relationship between coefficient ofcorrelation (p) of density and separation distance (Lag), for Parallam, MDFand waferboard/OSB respectively. These three products exhibited different pto Lag relationships. A more striking feature of this diagram was thedirectional difference of these products, which testifies to the capability ofthis analysis technique. The difference between the machine direction (Ydirection) and cross machine direction (X direction) came from the intentionalalignment of strands for Parallam and waferboardlOSB products, and thetendency of self-alignment of fibre bundles for MDF in the machine directionduring formation. As alignment occurs in the machine direction, the numberof voids decreases and correlation strengthens in this direction.However, as these three products were produced with different woodspecies, and board thickness and density were different, a direct comparisonof density variation in relation to coefficient of correlation among theseproducts was not possible. But such a comparison was possible for eachproduct in two directions. This comparison is provided in Table 9, which was790.5 I I I0.4 -.D O.3\‘pE0.2-IJDD0.1 -0.0 I I0 25 50 75 100Particle Size (cm2)Figure 24. Influence of particle size on parameter b. The curve isfitted by eye without regression analysis.800.20 I— EJ018 /0.160.14/ D/I /Ci) 0.12 -0.10 I0.0 0.1 0.2 0.3 0.4Parameter bFigure 25. Relationship between parameter b and standard deviationof density at specimen size of 0.141 cm2. The curveis fitted by eye without regression analysis.81CG)CQC4-0C).C)C1.00.80.60.40.20.00 20 40 60 80 100Lag (mm)Figure 26. Relationship between correlation coefficient ofdensity and Lag for MDF and Parallam.82calculated by using the X-ray density readings. Results support therelationship that variance of HDD negatively relates to correlations.Table 9. Comparison of density variations in X and Y directionsStandard Deviation of Density (g/cm3)Product MDF WaferboardlOSB ParallamX 0.063 0.151 0.104DirectionY 0.033 0.135 0.063For this thesis, which used randomly hand-formed particleboard madein the laboratory to establish the concept of HDD, the direction dependence ofproperties should not be significant.5.2.2. Particle thickness, wood density, board thickness and boarddensity5.2.2.1. Layer conceptParticleboards could be viewed as a non-continuous layered structure.An idealized lay-up could be like that shown in Figure 2. An actual boardstructure will be less regular and more complex, but it should still have thefeatures of a discontinuous lay-up. In this structure, particles are depositedrandomly in each layer due to the nature of the forming operation. As adensity specimen is usually composed of several of these layers, the varianceof horizontal density variation would be expected to decrease as the numberof particle layers increases.83This concept can be represented statistically. Let la be the number oflayers of particles in particleboard A,‘b the number of layers withinparticleboard B, and the ratio of particle layers of board B and A is 11/la.Further, suppose other board conditions are the same, and that Db stands forthe density variable of a specimen of a certain size for particleboard B, Da thedensity variable of the same specimen size for board A. Then (Appendix H),Var(Db) = (la/lb)Var(Da) (26)or,S(Db) ‘117LS(Da) (27)Both Equations (26) and (27) show clearly that variance of HDDdecreases as number of particle layers increases.5.2.2.2. Verification of Layer ConceptThis layer concept was used to study the influence of particle thickness,wood species, board thickness and board density on HDD in this section.5.2.2.2.1. Particle thicknessLet t1 and t2 be the thicknesses of particles used for boards A and B,then the ratio of average particle layers within boards B and A is the ratio ofparticle thicknesses t2/t1, if other board characteristics are the same. Then,according to Equation (27), the following was obtained,S(Db) = JS(Da) (28)Experimental boards prepared for the verification of Equation (2)8 areindicated in Table 10. For boards where combinations of particles were used,an arithmetic average of thickness based on weight proportions was used.84The S to 1/A models developed earlier for boards A4 and Cl were used toestimate density variations of particleboards with different particlethicknesses.Table 10. Particleboards with different particle thicknessesParticleboard Particles usedA4 A4El ElFl FlKl 50% particle El + 50% particle FlCl ClE2 E2F2 F2K2 50% particle E2 + 50% particle F2Figure 27 shows the comparison between model predictions andexperimental measurements. For particleboards made of particle length andwidth of 60 mm (Cl, E2, F2) and boards El and Ki, the specified modelagreed well with experimental results. But for board Fl made of particlethickness of 1.942 mm (length of 100 mm and width of 20 mm), a consistentlower measured density variation than model predictions was seeminglyevident. This may be due to the greater efforts which were dedicated informing this particle mat, as it is usually more difficult to obtain a uniformmat structure with slender and thicker particles (particle Fl). Therefore, less850.360.30-0.240.180.120.06-Tujoard’— —— ———— ————*K.....aC/_‘boat1(a)I1.2 2.4 3.6VA (cm-2)4.8 6.0 7.20.00 —0.00.300.240.180.120.060.000.0 7.2VA (cm-2)Figure 27. Influence of particle thickness on density variation.The lines are model predictions, and points areexperimental measurements.1.2 2.4 3.6 4.8 6.086forming variation may result and contributed to the overall lower densityvariations.Suchsland’s (1959) analysis indicated that board uniformity improved asparticle thickness decreased (Figure 3), which agrees with this layer concept.But a specimen size effect was not recognized in that study. It should bementioned here that the difference of board structure (defined by standarddeviation of density) corresponding to the use of different particle thicknesseswas more observable in the small specimen size range than in the largespecimen size range (Figure 27). This suggests that relatively small specimensize should be used in order to really understand how board structure isaffected by raw material characteristics, and how structure relates to boardproperties.It is known that all particleboard properties are negatively influencedby an increase of particle thickness. The increase of board nonuniformity asparticle thickness increases may provide the structural explanation for thisobservation. Consequently, alternative methods to reduce nonuniformity mayneed to be considered when using thicker particles.5.2.2.2.2. Wood densitySuppose d2 and d1 are the densities of wood particles used for boards Band A. If other board conditions are the same, the ratio of layers of particlesof boards B and A is the ratio of wood densities. Then, the following based onEquation (27) was derived,S(Db) = ‘Jd2/diS(Da) (29)87Actual board conditions used to verify this concept are listed in Table11. As the thickness of birch particles was different from aspen particles, anadjustment factor based on Equation (28) was applied to Equation (29) toestimate density variations. A good agreement was obtained between theexperimental observations and the model (Figure 28).Table 11. Particleboards with different wood speciesParticleboard Particles usedA4 A4Gi GiCl ClG2 G2In reality, this experiment involved a combination of wood species andparticle thickness effects. The good agreement between experimental dataand model predictions demonstrates that the layer concept applies toindividual raw material characteristic, as well as to their combinations.The increase of structural nonuniformity may explain whyparticleboards made with high density wood species are inferior in propertiescompared to particleboards made with low density wood species. This mayalso explain why combining low and high density wood species is arecommended practice in the particleboard industry.88(a) 0.200.150.100.050.000.0(b) 0.18I 0.120.060.00hA (cm-2)7.27.2Figure 28. Influence of wood density on density variation.The lines are model predictions, and pointsare experimental measurements.1.2 2.4 3.6 4.8 6.00.0 1.2 2.4 3.6 4.8 6.0hA (cm-2)895.2.2.2.3. Board thicknessSuppose tb and ta are the thicknesses of boards B and A respectively.Then, the following based on Equation (27) was obtained,S(Db) =7b5Wa) (30)Table 12 presents board conditions while Figure 29 shows theverification. A good agreement was evident between this hypothesized modeland the experimental data.Table 12. Particleboards with different board thicknessesParticleboard Particles used Board thickness(nun)A4 A4 11Si A4 6Wi A4 20ci ci iiS2 Ci 6W2 Ci 20In furniture grade particleboard, it is noticed that thinner particles areused to produce thinner particleboard for furniture applications. Theinfluence of board thickness on board nonuniformity may also provide anexplanation for this practice.90(a) 0.240.16000.080.000.0(b) 0.21I-4 0.14CC0.070.00hA (cm-2)Figure 29. Influence of board thickness on density variation.The lines are model predictions, and points areexperimental measurements.1.2 2.4 3.6 4.8 6.0 7.2VA (cm-2)0.0 1.2 2.4 3.6 4.8 6.0 7.2915.2.2.2.4. Board densityLet db and da be the densities of boards B and A. For the purpose ofanalysis, board B could be viewed as being made from A in two steps. First,increase the thickness of board A from t to (dilda)t at the density of da, anddesignate this board C. Secondly, compress board C from thickness (dj,/da)t tot while maintaining the same board weight.For the first step, the following equation was valid based on Equation(30),Var(D) = (da/db)Var(Da) (31)For the second step,Var(Db) = Var(w/(at)) = Var(a(db/da)tDd(at))= (dijda)2Var(Dc (32)in which, w and a are respectively the weight variable and area of densityspecimen.Combining Equations (31) and (32), the following was derived,Var(Db) = (dijda)Var(Da) (33)or,S(Db) = ‘IdiJdaS(Da) (34)Experimental conditions are given in Table 13, while verification isshown in Figure 30. A good agreement was also obtained between thetheoretical model and experimental data.92(a)(b)0.00.210.140.070.000.200.150.100.050.00II1.2 2.4 3.6 4.8 6.01/A (cm-2)7.21.2 2.4 3.6 4.8 6.0 7.20.0VA (cm-2)Figure 30. Influence of board density on density variation.The lines are model predictions, and points areexperimental measurements.93Table 13. Particleboards with different board densitiesParticleboard Particles used Board density(glcm3)A4 A4 0.67Ri A4 0.86Cl Cl 0.67R2 Cl 0.895.2.3. SummaryIn this section, both a commercial waferboard and laboratory panelsmanufactured with specific particle geometries were studied to establish theinfluence of raw material characteristics on HDD. By examining differentsized specimens within a panel, it was found that the relationship betweenstandard deviation of density and particle size was specimen size dependent,with standard deviation of density increasing as specimen size decreased.Since particles from the same wood species were of similar density, and thelaboratory boards were formed in a consistent manner, this variation wasattributed to void quantity and size. In general, the number of voids per unitvolume of particleboard, decreased as particle size increased, while the size ofvoids increased. These two aspects of voids determine, in part, the effect ofparticle size on HDD.By further development of model equations and data analysis, it wasdemonstrated that a layer concept was suitable for relating particlethickness, wood density, board thickness and board density to HDD.94In the next two sections, this structural characteristic of HDD will beexamined in more detail. In particular, the influence of specimen size and itsinherent variation in density on the evaluation of particleboard properties,and the effect on some board properties brought about by altering HDD willbe investigated.955.8. Implication of Horizontal Density Distribution to the Selection ofSpecimen Size For Some Particleboard Property Evaluations5.3.1. IntroductionThe determination of appropriate specimen size for studying materialproperties has been a concern for researchers. The greatest change inspecimen size designation probably occurred in the late 1970’s, when an in-grade testing philosophy, based on actual structural lumber sizes, replacedthe traditional small clear wood specimen method in North America (Madsen,1992). This transition was the result of the improved understanding ofstructural behaviour of lumber. For particleboard, different countries usedifferent specimen sizes for property determinations. For example, specimensizes of 25 mm x 25 mm and 50 mm x 50 mm respectively are used for TS andTB evaluations in Europe (Heimeshoff, 1991), while sizes of 150 mm x 150mm and 50 mm x 50 mm are used in North America (American Society forTesting and Materials, 1960; Canadian Standard Association, 1985). Tn bothstandards, specimen sizes for TB and TS evaluations are different, eventhough these two properties are both measuring performance of particleboardin the same direction perpendicular to board surface. This designation ofspecimen size makes the preparation of specimens inconvenient.Present testing methods in North America to evaluate particleboardproperties are still more or less based on the small specimens of ASTMstandard D 1037 (American Society for Testing and Materials, 1960), whichwas originally intended for fibre-based panels. The deficiencies in thisstandard was reported in an investigation which examined some newproducts and new test methods based on different specimen sizes (McNatt,961984). Up to now, a criterion for designating such specimen sizes is yet to bedeveloped. In this section, the concept of HDD was used to study how someparticleboard properties are affected by specimen size changes, with theobjective of examining the potential of HDD as one possible criterion forspecimen size selections for TB and TS evaluations.5.3.2. Internal bondThe strength of a material follows a Weibull distribution (Weibull,1939), if the failure of the material is governed by the weakest link. Manymaterials exhibit this phenomenon. Since 1960, this strength theory has beenapplied to characterize strength properties of wood. Limited tests conductedon wood composites showed that MOR values from larger test specimen sizeswere lower than those for smaller test specimen sizes (McNatt, 1984; Post,1983; Szabo, 1980), indicating that the Weibull strength model might alsoapply to wood composites.Internal bond values for the commercial waferboard panel tested atseveral different specimen sizes are given in Table 14. The two parameter (2-P) and three parameter (3-P) Weibull distribution fits to the TB data atspecimen sizes of 6.27 cm2 and 225.79 cm2 are shown in Figures 31 and 32respectively. Visual examination indicates that both models match well withthe experimental data. Generally, the 3-P model fits data better, but the 2-Pmodel is simpler to use. In this study, the 2-P Weibull was used.One result of the Weibull distribution is size effect. That is, strengthdecreases as volume increases. This has been observed for wood in tensionparallel to grain (Madsen and Buchanan, 1986), in bending (Bohannan,1966), in shear (Foschi and Barrett, 1976) and in tension perpendicular to97Table 14. Internal bond results of commercial waferboardat different specimen sizes*Size Internal Bond (MPa)cm2 n s v2.45 120 0.411 0.112 0.2736.27 100 0.407 0.098 0.24117.85 100 0.364 0.078 0.21423.25 85 0.365 0.076 0.20829.28 75 0.332 0.062 0.18756.25 75 0.321 0.048 0.150101.23 70 0.310 0.044 0.142225.79 62 0.307 0.041 0.134398.78 29 0.31 0.033 0.106* n - number of observations; - average; s - standard deviation;v - coefficient of variation.981.00.8c 0.60C)0.40.20.00.0 0.8Stress (MPa)Figure 31. 2-P and 3-P Weibull distribution fits to internalbond of one commercial waferboard at specimensize of 6.27 cm2.0.2 0.4 0.6990.600C)..C.)1.00.8 2-P WeibullY = 1 - exp((XlO.33)8.41)3-P WeibullY = 1 - exp(.[(X-O.07Y0.2516s1)0.40.15 0.300.20.00.00Stress (MPa)Figure 32. 2-P and 3-P Weibull distribution fits to internalbond of one commercia’ waferboard at specimensize of 225.79 cm2.0.45100grain (Barrett, 1974). The expression of this size effect according to Weibulls(1939) 2-P distribution wasa=Ikm/V1k (35)where, a and V are the strength and volume of material, k and m are theshape and scale parameters of 2-P Weibull distribution model which can bedetermined experimentally, and‘k = re-dz was used by Weibull (1939) inthe statistical manipulations.As the thickness of specimens used for the determination of 113 wasconstant and equal to the thickness of the commercial particleboard,Equation (35) was changed toa = Ikmt/A’ (36)for 113 of particleboard, where A is the specimen size (area) and t is thethickness.Equation (36) could also be rewritten into a general form such asa = a(1/A)d (37)here, d = 1/k, a = IkmJtl]k. Both a and d could be determined by regressionanalyses of experimental data.Figure 33 shows the 50th percentile of TB strength at several specimensizes, together with a fit of Equation (37) to the experimental data. The 50thpercentile of TB decreased with size in the fashion that Equation (37)prescribed for the range being studied. However, the mean, rather than the50th percentile of TB is used exclusively in the particleboard industry for thepurpose of product evaluation and quality control. It was assumed thatEquation (37) would also be a good approximation of the relationship betweenaverage TB and specimen size, which is also shown in Figure 33.1010.45 I I• 50th percentile040‘ R2=0.89Y = 0.44 14(IJX)° 649Mean valueR2=0.89\x Y = O.4366(IJX)0.06C0.35 - •••.x— .x. .......x0.30 -0.25 I I0 100 200 300 400Specimen Size (cm2)Figure 33. Influence of specimen size on 50th percentile and averagestrength of internal bond of one commercial waferboard.102The similarities between Equations (37) and (18), and Figures 33 and 17were obvious. By combining Equation (37) with Equation (18), a generalexpression relating average strength and horizontal density variation wasobtained (Appendix I),a = (38)in which, ct and f are constants to be determined experimentally. Theexperimental results, together with a fit of Equation (38) are shown in Figure34.Another result of the Weibull distribution for defining strength is thatthe variation of strength is specimen size dependent, i.e., variation decreasesas specimen size increases. For the 2-P Weibull distribution, Weibull (1939)showed that the standard deviation of strength ç wasç = m(IIk2)l/Vilc (39)where, 1k/2 =re2dz.For lB property of the commercial waferboard studied, a general formbased on Equation (39) can also be derived asç = e(1/A) (40)here, f = Ilk and e = m(Iw2IkJ/tIJkare to be determined experimentally. Therelationship between standard deviation of TB and specimen size is shown inFigure 35, together with the fit of Equation (40).A general relationship between the standard deviation of 113 andstandard deviation of density similar to Equation (38), was obtained as(Appendix J)(41)1030.04 0.05 0.06 0.07 0.08Figure 34. Relationship between average internal bond and standarddeviation of density of one commercial waferboard.I I I I.0.450.400.350.300.250.. R2=0.89Y = 1.321(X)°0.03I I I LStandard Deviation of Density (glcm3)1040.120.100.080.060.040.02Specimen Size (2)Figure 35. Influence of specimen size on standard deviation ofinternal bond of one commercial waferboard.4000 100 200 300105here, N’ and are also constants to be determined experimentally. Figure 36shows the experiment results and the fit of Equation (41). Thus, variation inmechanical property (TB) decreased as variation of physical property (density)decreased.However, some discrepancies were evident as Equations (37) and (40),and the Weibull fits for the two specimen sizes (6.27 and 225.79 cm2) did notyield the same value for the shape parameter k, which violated a property ofWeibull theory. The large difference of k was due to the dependence of TB ondensity itself. It is widely accepted that TB increases as density increases.This relationship between TB and density for the commercial waferboard attwo specimen sizes, 6.27 cm2 and 225.79 cm2 respectively, are shown inFigures 37 and 38. As density variation decreased with increasing specimensize, this density dependence characteristic of TB would decrease thevariation of lB as specimen size increased. However, the average strength ofTB was not affected in this manner. Therefore, larger shape parametersresulted for Equation (37), and for the 2-P Weibull fit at specimen size of225.79 cm2.Two consequences of this discrepancy are shown in Figures 39 and 40.Figure 39 shows that coefficient of variation of TB decreases as specimen sizeincreases, rather than a constant value as one property of the 2-P Weibulldistribution. In Figure 40, the relationship between average strength, theestimated 5th percentile (estimated by both 2-P and 3-P models), 95thpercentile (estimated by 2-P model) of TB and specimen size is presented. Ttcould be expected from this figure that below certain percentiles, strengthwould increase with specimen size within the specimen size range studied.This seems to contradict the Weibull strength theory, but it is the direct1060.120.100.080.060.040.020.03 0.08Standard Deviation of Density (gfcm3)Figure 36. Relationship between standard deviation of internalbond and standard deviation of density ofone commercial waferboard.0.04 0.05 0.06 0.070.7 0.8 0.9Density (g/cm3)Figure 37. Relationship between density and internal bond of onecommercial waferboard at specimen size of 6.27 cm2.107I Ia •.a aa • a.II II IIaa0.70.60.50.4C‘ :;0.10.0I •I. ISS:‘IaaI.a• a. •a. ••S.a• I.SaaI aIIR2 = 0.30Y = -0.20 + 0.89 X0.5I I0.6C—Ce108I II II• ..III..1• •I •III R = 0.580.50.40.3 -0.20.1 -0.0 —0.5 0.9Density (g/cm3)Figure 38. Relationship between density and internal bond of onecommercial waferboard at specimen size of 225.79 cm2.Y = -0.28 ÷ 0.86 X0.6 0.7 0.81090.3 I I.g 0.2.oE0Q 0.1 -0.0 I0 100 200 300 400Specimen Size (cm2)Figure 39. Influence of specimen size on coefficient of variationof internal bond of one commercial waferboard. Thecurve is fitted by eye without regression analysis.C-.ê-•1110pA 3-P Weibull estimates0 • 2-P Weibull estimates00o.Mean value.0.60.50.40.30.20 200 400Specimen Size (2)Figure 40. Influence of specimen size on 5th percentile, 95th percentileand average of internal bond of one commercial waferboard.A A 5th percentile100 300111result of the density dependence of TB. It was observed in our experiments,that a noticeable number of specimens simply fell apart due to their lowdensities, when specimen size was smaller than 6.27 cm2. This was notobserved at larger specimen sizes. This problem of falling apart also existedin the study conducted by Suchsland and Xu (1991), when a small sizedmatrix element was used as the testing specimen. These observations suggestthat specimen size should be large enough to avoid direct testing of defects(low density points), if the influence of defects is to be studied. Tn fact, thisconcept is similar to that applied in the study of lumber where test specimensare chosen to be significantly larger than major strength reducing defects,such as knots and localized slope of grain.Equations (38) and (41) described positive relationships betweenaverage strength, standard deviation of TB with standard deviation ofdensity. According to these relationships, selection of specimen size in theless sensitive range of the S to A curve (Figure 17) would give averagestrength and standard deviation of TB which change less with specimen size.The selection of specimen size in this range will also avoid direct testing ofdefects, or low density points, as the variation of density above this specimensize is believed mainly to be controlled by nonuniformity in mat forming.Until now, the mean and standard deviation of particleboard properties areused almost exclusively for the purpose of product evaluation, comparisonand quality control. All these observations demonstrate that the S to A curveshould be used for the selection of appropriate specimen size for thesepurposes. The relationships observed in this study should be used as acriterion for the future selection of specimen sizes in relation to testingstandards.1125.3.3. Thickness swellingThe average TS of commercial waferboard, together with the standarddeviation of TS, at several different specimen sizes and soaking times arepresented in Table 15 and shown in Figures 41 and 42. Several importantobservations were made and summarized as follows.First, under the standard cold water soaking conditions, average TSdecreased as specimen size increased for all the soaking times except at theone week soaking condition. A plateau was reached once specimen sizeexceeded 25-50 cm2. Thickness swelling was a continuous process involvingthe release of compressive deformation incorporated into particleboardduring the pressing operation. Water up-take is a prerequisite for thisrelease. Obviously, smaller specimens more readily absorb water, therefore,rapidly swell in thickness. But this did not account for TS results whenspecimen size was larger than 25-50 cm2. A model concept developed bySuchsland (1973) provided insight on this phenomenon. According toSuchsland, high and low density areas in a TS specimen respond differentlyto swelling by moisture, and their opposing nature leads to differentbehaviours of particleboard at different testing conditions. Since densityvariations in the commercial waferboard at specimen sizes larger than 25-50cm2 were believed to be governed by forming uniformity, these sizedspecimens were considered not to be appreciably different as far as theirinterior structural characteristics were concerned. Therefore, a similar TSshould result according to this model concept. With specimen size less than25-50 cm2, density variations were considered to be a reflection of thestructural nonuniformity of particleboard (Figure 17), and according to113Table 15. Thickness swelling (%) of commercial waferboard at differentspecimen sizes and soaking times*Specimen size Thickness Swelling (%)(cm2)Hours 2 6 12 24 48 168n 48 48 48 48 48 484.26 13 30 38 43 45 46s 4.9 5.9 6.2 6.8 7.3 7.7n 48 48 48 48 48 488.39 6 21 33 38 41 43s 2.4 5.8 6.0 6.8 7.3 7.5n 44 44 44 44 44 4412.85 4 14 26 33 38 41s 2.0 5.3 5.8 6.7 6.8 7.0n 46 46 46 46 46 4617.49 3 9 19 30 38 43s 1.5 3.3 5.3 6.2 5.7 6.5n 32 32 32 32 32 3225.12 3 5 9 18 33 44s 0.8 1.1 2.2 4.7 6.2 6.2n 18 18 18 18 18 1856.25 3 5 8 13 25 46s 0.9 1.0 2.1 2.4 3.4 4.3n 18 18 18 18 18 18100.38 3 6 9 14 24 42s 0.6 0.8 1.0 1.5 2.2 4.5n 23 23 23 23 23 23224.50 3 8 9 17 28 41s 0.6 1.0 1.6 2.0 2.7 3.7* n = number of observations; i = average; s = standard deviation.11450 I I I I168 hrs.40 -‘ “30Cl)ri)20 -C)\10— —6h — — —2hrs.I I I00 50 100 150 200 250Specimen Size (cm2)Figure 41. Influence of specimen size on average thickness swelling of onecommercial waferboard at different soaking times..115\ A\\\\\• • Z68h.s.\864200 250Specimen Size (cm2)Figure 42. Influence of specimen size on standard deviation of thicknessswelling of one commercial waferboard atdifferent soaking times.-_±It__.N.“N 6hrs._•50 100 150 200116Suchsland’s model concept, individual TS specimens swelled more freely asspecimen size decreased.Secondly, as found with horizontal density variation, the standarddeviation of TS decreased as specimen size increased, and this decrease wasnot obvious when specimen size was larger than 25-50 cm2, which was in theless sensitive range of the density variation versus specimen size plot (Figure17). This can be readily explained by the application of Suchsland’s (1973)concept. When specimen size was larger than 25-50 cm2, density variationswere small and less sensitive to specimen size changes, therefore, small andfairly constant variations of TS resulted. When specimen size was less than25-50 cm2, density variation increased as specimen size decreased, andbecause individual specimens swelled more freely at this specimen sizerange, large variation of TS resulted.Thirdly, one week of soaking probably released all the TS inparticleboard as determined by this cold water soaking method, since theaverage TS values for all the specimen sizes were fairly consistent. This canbe clearly seen from the plot of TS versus soaking time for specimen size of4.26 cm2, shown in Figure 43. Since TS did not change significantly from 48hours to one week of soaking, and since TS of all the specimen sizes wereconsistent at one week of soaking condition, it was concluded that furthersoaking after one week would not increase the TS significantly for thespecimen size studied.Similar to the TB test, the average and standard deviation of TS areused almost exclusively for the purpose of quality control and productcomparison. The selection of specimen size from the less sensitive range of117I50403020100Soaking Time (hrs.)Figure 43. Thickness swelling vs. soaking time of one commercialwaferboard at specimen size of 4.26 cm2.1800 30 60 90 120 150118the horizontal density variation versus specimen size plot (Figure 17) willyield stable and consistent results with less variation, according to theobservations made above. These qualitative findings could also be used as onecriterion for the future development of TS testing method for particleboard,with regard to specimen size selection. For example, the change of specimensize from 25 mm x 25 mm to 50 mm x 50 mm for furniture gradeparticleboard in Europe as a result of CEN-standardization resulted in thereduction of variation of this property (Heimeshoff, 1991). Finally, one weeksoaking should be applied to determine the TS property, especially for thepurpose of product evaluation and development, as it was the most reliableand true index of dimensional stability.1195.4. Influence of Horizontal Density Distribution on Some BoardProperties5.4.1. IntroductionIn section 4.2, a detailed investigation on particleboard structureidentified some principles regarding the influence of raw materialcharacteristics on HDD. In section 4.3, HDD was found to be a potentialconcept or criterion in selecting specimen size for some board propertyevaluations. In this section, particleboards with different HDDcharacteristics were made using particles more representative of commercialsized distributions. The intention was to establish the relationship betweenHDD and some common board properties. If this objective was met, theconcept of HDD could be used to better understand present and futureparticleboard technology.5.4.2. Application of adhesiveAs discussed earlier, particleboard properties were improved byincreasing resin content, with TB being the most sensitive property. Thedeficiency of using a fixed percentage of resin content based on wood weight,as practiced traditionally in particleboard studies, is well recognized. Sincechanges in HDD were achieved by using different particle geometries, woodspecies or combination of both in this study, this resin deficiency needs to beremoved in order to accurately study the influence of horizontal densityvariation on particleboard properties.With the application of powder adhesives, it was believed that onlycertain quantities of resin were being picked up by wood particles. Two to120three percent based on wood weight was believed to be normal for currentcommercial wafers/strands (Steiner, 1992). Therefore, provided a surplus ofresin is applied in the blending operation, maximum and uniform resin pickup could be consistently achieved throughout all the different particles basedon particle surface area. Thus, any possible variation of particleboardproperties due to resin application is minimized.To test this proposition, an experiment was carried out to blend severaldifferent resin contents with commercial particles p1, using TB as theproperty for evaluation. The same procedures of making laboratoryparticleboard were followed, and a board density of 0.72 g/cm3 and athickness of 11 mm were targeted. The TB mean, together with the calculated95% confidence interval based on an assumption of normal distribution arepresented for several resin content levels in Figure 44. It was concluded thatTB was not benefited by increasing apparent resin content above 6%, anindication that a maximum resin pick-up was achieved.It should be mentioned that 6% was not necessarily the actual amountof resin picked by the particles, nor should it be taken as the level to apply toany experimental conditions. This was the resin level at which an obviousamount of extra resin was observed in this experiment to be left in theblender, after the rest was picked up by the particles and the blender surface.The presence of this extra resin is an indication that maximum pick-up ofresin has been reached. This indicator was used to manufacture subsequentparticleboards for studying the influence of HDD on particleboard properties.Obviously, the actual resin content will depend on moisture content, surfacearea and surface conditions of particles, size and surface conditions of theblender and other factors.1210.600.45p.1-d0 . -0.150.00 -0 12Apparent Resin Content (%)I I I I1-3 9 15Figure 44. Influence of apparent resin content on internal bond.1225.4.3. Board formationParticleboards used to study the influence of HDD on board propertieswere manufactured in the laboratory. The composition of these boards ispresented in Table 4, and the respective distributions of particle sizes areshown in Appendix A. Table 16 presents the density variations for theseboards measured at several different specimen sizes. As these particleboardswere made using different distributions of particle length, width andthickness, and sometimes a combination of different species, a thoroughinterpretation of the structural differences of these boards in terms ofindividual raw material characteristic was difficult. However, theexamination of two extreme board cases (P5 and P7) demonstrates that theobservations made in section 4.2 were valid. Board P5 was made by using thelargest particle size (an average of 119 mm x 88 mm) with smallest particlethickness (an average of 0.52 mm). Accordingly, a much lower standarddeviation of density was observed at small specimen sizes. Board P7 wasmanufactured with the thickest birch particles (an average of 0.93 mm),therefore, much higher density variations were observed.The relationship between the standard deviation of density with bothspecimen size and the inverse of specimen size are shown respectively inFigures 45 and 46 for these particleboards. In order to present the data moreclearly, the boards were divided into two portions for presentations in thefigures. Most importantly, this data showed that these boards differedconsiderably in standard deviation of density at relatively small specimensizes. This provided the material base for studying the influence of HDD onboard properties.123Table 16. Density measurement for laboratory particleboard*Specimen Size Density (glcm3)(cm2)Board P1 P2 P3 P4 P5 P6 P7 P8 P9 P10n 12 12 12 12 12 12 12 12 12 12278.48 .749 .758 .748 .748 .754 .756 .758 .754 .740 .752s .026 .018 .031 .032 .038 .037 .038 .025 .027 .032n 25 25 25 25 25 25 25 25 25 25134.560 .747 .744 .752 .753 .758 .759 .755 .746 .747 .751s .034 .026 .039 .046 .050 .040 .041 .032 .035 .045n 25 25 25 25 25 25 25 25 25 2555.080 1 .731 .751 .754 .737 .760 .743 .767 .740 .742 .752s .048 .031 .060 .052 .051 .065 .076 .042 .060 .046n 25 25 25 25 25 25 25 25 25 2511.401 .746 .745 .757 .758 .764 .755 .736 .752 .761 .748s .057 .045 .071 .055 .086 .082 .091 .078 .088 .066n 25 25 25 25 25 25 25 25 25 255.067 .752 .749 .757 .730 .744 .764 .724 .752 .743 .743s .069 .06 .075 .081 .088 .09 .089 .084 .09 .104n 50 50 50 50 50 50 50 50 50 501.267 1 .748 .752 .757 .764. 760 .769 .776 .740 .738 .744s .096 .100 .100 .092 .105 .132 .124 .110 .122 .132n 75 75 75 75 75 75 75 75 75 750.317 1 .768 .752 .745 .773 .744 .777 .766 .767 .766 .764s .129 .116 .125 .113 .110 .148 .163 .131 .140 .150n 100 100 100 100 100 100 100 100 100 1000.126 1 .765 .765 .761 .722 .776 .724 .731 .728 .732 .765s .142 .138 .135 .124 .117 .168 .181 .144 .159 .171* . — .n = number of observations; x = average; s = standard deviation. Thedensity at specimen size of 5.067 cm2 was measured by y-ray method.124I.0G)aSI00.200.150.100.050.000.200.150.100.050.0070 140 210Specimen Size (cm2)2800 70 140 210 280Specimen Size (cm2)Figure 45. Standard deviation of density vs. specimensize for laboratory particleboard.125I0.200.15j 0.100.053Ci)0 41/A (cm-2)6 80.0w0.200.150.100.050.000 2 4 6 8hA (cm-2)Figure 46. Standard deviation of density vs. 1/Afor laboratory particleboard126From Figure 45, it is striking to notice that the standard deviation ofdensity is less size sensitive when specimen size is larger than approximately50 cm2 for all particleboards, which also coincides with the less sensitiverange for the commercial waferboard (Figure 17). This observation suggeststhat the less sensitive range may not be influenced much by wood species andparticle sizes, within the particle size range being used and the formingmenthods being utilized. By using the observations in section 4.3, it wasdecided to select a specimen size of 100 mm x 100 mm, rather than 50 mm x50 mm, as specified in published standard (Canadian Standard Association,1985), for lB and TS tests. A specimen width of 100 mm, rather than 75 mm,was also chosen for bending tests.As three replicates were made for each board condition, and only oneboard was used to determine HDD characteristics, board reproducibilityneeds to be examined. Figure 47 shows this examination for board P1. It wasassumed that the reproducibility was acceptable for this study.5.4.4. Modulus of rupture and modulus of elasticityThe essence of Weibull theory of strength is that the worst defect orweakest link controls the strength of a material. By applying this theory todifferent sizes of the same material, a size effect is recognized. Strengthdecreases as specimen size increases, simply because more defects areexpected in a bigger volume. Therefore, volume could be viewed as an indexof defect in the size effect formula of Equation (35). Now, let us considersimilar materials having different strength properties measured at the samespecimen size. If the defects inside a material could be quantified, the1270.16 I0.12 -0.08-x X boardiC) D“ board2CI) 0.04 -0.00 I I0 2 4 6 81/A (cm-2)Figure 47. Reproducibility of board formation ofboard P1.128underlying principle in size effect formula should be applicable to develop aconcept to explain the strength differences among materials.Specifically, for particleboards with different HDD characteristics, ifthis density variability is the major strength reducing defect, then densityvariability could be taken as the quantifying defect, and used to relate tostrength properties of particleboard.A concept of nonuniformity effect, similar to that of size effect formulawas proposed here to account for the difference of strength properties bystandard deviation of density, that was:P = m(1/S) (42)where, P and S are the property and standard deviation of density ofparticleboard respectively, and m and n are constants.Modulus of rupture and modulus of elasticity results for laboratoryboards are presented in Table 17. The relationship between MOR andstandard deviation of density is shown in Figure 48, together with a fit ofEquation (42). It should be recognized that the specimen size used todetermine the standard deviation of density is different in each small plot,but MOR of particleboards were determined at a fixed specimen size and thesame MOR values were used throughout the figure to examine the influenceof density nonuniformity. For example, P5 has a value of 46.8 MPa (thehighest data point in each plot) for MOR, but the standard deviation ofdensity changes when specimen size used to determine this variationchanges. As Figures 45 and 46 suggested, particleboard structure differenceswere detected more clearly and consistently when specimen size was smaller,129Table 17. Modulus of rupture and modulus of elasticityof laboratory particleboardMOR (MPa)Board P1 P2 P3 P4 P5 P6 P7 P8 P9 P1026.9 25.5 32.1 38.5 42.1 26.5 17.8 19.1 27.4 18.925.1 24.2 35.1 32.1 48.5 14.5 15.4 24.6 30.2 16.931.6 26.2 34.2 28.2 43.7 12.3 12.0 17.2 19.1 18.025.5 27.8 36.3 34.7 44.9 27.5 12.4 25.0 29.6 27.829.4 29.1 41.9 40.5 49.3 17.5 14.4 20.8 25.2 23.833.1 25.3 37.6 37.9 52.3 20.5 20.5 20.8 23.2 16.3Average 28.6 26.3 36.2 35.3 46.8 19.8 15.4 21.2 25.8 20.3MOE (MPa)Board P1 P2 P3 P4 P5 P6 P7 P8 P9 P104000 3100 4300 4600 6000 3900 1900 3100 3100 16003800 2900 5000 4000 6100 3000 1600 3800 3100 20004700 3300 5100 4000 5700 3400 1800 3300 3900 20003600 3700 5100 4200 5700 3600 1600 3800 3700 24004300 3500 5300 5100 6300 3400 1300 3600 3800 25004600 3800 5400 5000 6600 3600 2200 3500 3000 1700Average 4200 3400 5000 4500 6100 3500 1700 3500 3400 2000Figure 48. Relationship between modulus of rupture and standarddeviation of density of laboratory particleboard. Thespecimen size used to determine standard deviationof density is different and indicated in each plot.130JSize= 1.267cm2j0[ize=5.067 cm2aMOR = 5.124(1,)o.663R2 = 0.09a0060400604020—0000—0.04 0.06 0.08 0.10 0.120MOR = 0.537(IiS)’7R2 = 0.49Standard Deviation of Density(g/cm3)6040I06040I20000.08 0.10 0.12Standard Deviation of Density(g/cm3)0.14[Size = 0.3 17 cmj0100Size= 0.126cm2MOR = 0.273(L)224°R2 = 0.770—0.1000.12 0.14 0.16MOR = 0.414CL)2’6°R2 = 0.840.18Standard Deviation of Density(g/cm3)0—0.10 0.12 0.14 0.16 0.18 020Standard Deviation of Density(glcm3)131therefore, only the four smallest specimen sizes (used to measure densitynonuniformity) were used in the presentations in Figure 48.As expected, MOR decreased as standard deviation of density or densitynonuniformity increased. It is believed that the low density portionscontrolled the bending failure of particleboard, as poor or no bondingoccurred in these portions. As density nonuniformity increased, these lowerdensity portions increased, and a lower MOR resulted. Moreover, the fit ofEquation (42) improved as specimen size used to detect the densitynonuniformity decreased (R2 increased as specimen size decreased). Thissuggests that a certain size resolution is needed, if a procedure is going to bedeveloped to detect structural nonuniformity.Particleboards used to study the relationship between HDD and MOR inthis thesis were manufactured with different particle length, width, thicknessdistributions, different wood species (aspen and birch) and combinationsthereof. A direct analyses of MOR in terms of these raw materialcharacteristics, which was the traditional approach, is not feasible. Therationalization of MOR in terms of standard deviation of density (R2 are 0.77and 0.84 when specimen size are 0.317 cm2 and 0.126 cm2 respectively)demonstrates that density nonuniformity or HDD is a true structural featureof particleboard. The close relationship between HDD and raw materialcharacteristics indicates that a singular parameter of HDD is sufficient tostudy the effects of the latter on MOR, just as vertical density profile is usedto characterize pressing strategy parameters.A similar analysis for MOE is shown in Figure 49. Although MOE is nota strength property, Equation (42) was still used to relate MOE to standardFigure 49. Relationship between modulus of elasticity and standarddeviation of density of laboratory particleboard. Thespecimen size used to determine standard deviationof density is different and indicated in each plot.[Size = 5.067 2l013200I Size = 1.267 cm2l00000 -MOE = 339.65(1IS)°-937R2=0.1380006000-00020000-0.04800060O011000•200000 -00.06 0.08 0.10Standard Deviation of Density(g/cm3)0.12MOE • 448()1.987R2 = 0.4380006000400020000—0.08800060004000—200000.10 0.12Standard Deviation of Density(g/cm3)0140I Size = 0.3 17 cm200Size = 0.126 cm2]0 0aaMOE = 18.75(t)2.582R2 = 0.70000 I 0—0.10 0.12 0.14 0.16 0.18 0.10Standard Deviation of Density(g/cm3)0MOE = 29.22(1j)2 497R2= 0.780.12 0.14 0.16 0.18Standard Deviation of Density(glcm3)0.20133deviation of density. Again, a significant relationship (R2 = 0.78 at a specimensize of 0.126 cm2) existed between MOE and density nonuniformity, and thefit of Equation (42) improved (R2 increased) as specimen size used todetermine density variation decreased. The close relationship between HDDand this elastic property further demonstrates that HDD is a parameter tocharacterize particleboard structure, and is capable of describing the effectsof raw material characteristics, which involve multi-parameters.5.4.5. Internal bondInternal bond values are provided in Table 18. Figure 50 shows theinfluence of standard deviation of density on TB, together with a fit ofEquation (42). A significant relationship (R2 0.77 at a specimen size of0.126 cm2) between micro-density nonuniformity and TB was obtained. Apartfrom the explanation given in 4.4.3, there may be another reason that 113decreased as nonuniformity increased. That is, at the same board density,relative bonded area (section 2.1.8) decreased as density variation increased.It is believed that TB is positively related to RBA.The same particleboards used to study the effect of HDD on MOR andMOE were used in the study of TB. The reason why raw materialcharacteristics affect IB, which was not understood well in the past, wasreadily explained by HDD, a structural phenomenon of particleboard.5.4.6. Thickness swellingThickness swelling results of these particleboards based on one weekcold water soaking are also presented in Table 18. The influence of horizontaldensity variation on TS is shown in Figure 51. Although TS is not a strength134Table 18. Internal bond and thickness swellingof laboratory particleboard*BoardInternal Bond (MPa)P1 P2 P3 P4 P5 P6 P7 P8 P9 P10n 25 25 25 25 25 25 25 25 25 25.430 .400 .410 .501 .631 .261 .289 .440 .441 .329s .044 .042 .044 .046 .042 .047 .048 .051 .038 .043Thickness Swelling (%)Board P1 P2 P3 P4 P5 P6 P7 P8 P9 P10n 25 25 25 25 25 25 25 25 25 255 44 42 42 31 25 55 63 52 58 54s 5.1 3.2 4.4 4.8 4.1 5.0 5.3 4.6 4.5 5.1*n = number of observations; = average; s = standard deviation.135Figure 50. Relationship between interna:1 bond and standarddeviation of density of laboratory particleboard. Thespecimen size used to determine standard deviationof density is different and indicated in each plot.[= 5.067 2]0lB = O.1513(1IS)°R2=O.05ISize= 1.267cm2100 00.60.6C0.4C)020.0 —0.040.8 -0.600.4—C)02 -0B0lB = 0.0214(IJS)1.S288R2=0.470.06 0.08 0.10 0.12Standard Deviation of Density(glcm3)0.800.4C)02 -0.0 —0.080.8 —00.4C)1-4020.0 —0.1000.10 0.12Standard Deviation of Density(g/cm3)[Size = 0.317 2]0.14I II Size = 0.126 cm20.0 0-0 0lB = 0.0138(IJS)1.6610R2= 0.700000.0 —0.10 0.1200.14 0.16 0.18Standard Deviation of Density(g/cm3)lB =R2=0.770.12 0.14 0.16 0.18Standard Deviation of Density(g/cm3)020Figure 51. Relationship between thickness swelling and standarddeviation of density of laboratory particleboard. Thespecimen size used to determine standard deviationof density is different and indicated in each plot.136lSize= L267cx180I Size = 5.067 21ae60b 0 0..C4Ou,.C)0.20TS = 146(1iS)°47R2 = 0.05806040 -200Ba0—0.04 0.06 0.08 0.10Standard Deviation of Density(g/cm3)0.120—0.08T5 1244(1)’6’R2 = 0.480.10—C)Ci)U)8060C200.12Standard Deviation of Density(glcm3)FSize=0.317Cn1!j000.1480rCi) 40U)U)C)2000ISize=0.i26cm2I0000TS = 2496(1/s)1.98R2= 0.790—0.10 0.12 0.14TS = 1713(j.)1.89I 0—0.16 0.18 0.10Standard Deviation of Density(g/cin3)R2= 0.860.12 0.14 0.16 0.18 020Standard Deviation of Density(g/cm3)137property, Equation (42) was still adopted to relate TS to standard deviation ofdensity. A good fit was obvious between this physical property and structuralproperty (R2 are 0.86 and 0.79 when specimen sizes were 0.126 cm2 and 0.317cm2 respectively), which also demonstrates that HDD is an importantstructural phenomenon. This can be readily rationalized by usingSuchsland’s (1973) model concept. Suchsland reasoned that at highermoisture content, higher density portions dominated TS. As densitynonuniformity increased, higher density portions increased. Therefore,higher TS resulted according to Suchsland’s concept, and Equation (42), ifused for TS property, agrees with this concept.5.4.7. SummaryWhile TS was controlled by the higher density portions of a panel,strength properties (MOR and TB) and MOE were believed to be controlled bythe lower density portions. In either case, standard deviation of density was agood index to distinguish among particleboards. This was probably whyEquation (42) provided a good fit between micro-density variation anddifferent properties. Although this equation may not necessarily best fit thedata points in terms of goodness-of-fit, the simplicity of this equation helps tobetter understand the significance of the horizontal density phenomenon.After all, Equation (42) was proposed on the basis of the concept of Weibull(1939) theory of strength.The associations between HDD, raw material characteristics and someboard properties, demonstrates that HDD is a singular structural parametercapable of describing the effects of raw material characteristics and possiblyforming technology.1386. SUMMARY AND CONCLUSIONSThe concept of HDD was first proposed by Suchsland in 1959 to analyzeparticleboard technology. A gravimetric method involving a drillingtechnique was shown to be capable of detecting the micro-density variation inparticleboard, as well as distinguishing particleboards in this regard. It wasfound that this variation decreased as specimen size, used in determiningthis density nonuniformity, increased. If standard deviation of density wasused as the index of nonuniformity, the expression S = a(IJA)b was shown tobe appropriate to relate density variation (S) to specimen size (A). Parameterb related to correlations of density points, and a range between 0 and 0.5 wasdetermined for this parameter. It was found that b decreased as particle sizeincreased.Raw material characteristics influenced HDD. Generally, with largerspecimen sizes, particleboards made with larger particles exhibited greaterdensity variations. With smaller specimen sizes, particleboards made withsmaller particles showed larger density variations. The size and number ofvoids were identified as responsible for these results for mats provided underhand-forming operations. The recognition of these two aspects of voidssuggests that forming method plays a significant role in determining themagnitude of HDD. Any method which reduces the size of voids in betweenparticles would improve board uniformity.A layer concept was developed and used to study the effects of particlethickness, board thickness, board density and wood species on HDD ofparticleboard. This concept was based on the observation that an increase inparticle layers reduces density variations.139Particleboard properties were shown to be greatly influenced byhorizontal density nonuniformity. While TS was considered to be controlledby high density portions of the boards, MOR, MOE and TB were believed to becontrolled by low density portions. A concept of nonuniformity effectexpressed as P = m(1/S), similar to the size effect formula, was proposed andwas shown to be feasible in relating conventional particleboard properties (P)to standard deviation of horizontal density (S).The concept of HDD was also shown to be a potential criterion for thefuture development of particleboard testing standards in relation to specimensize selections. Both the average values and standard deviations of TB and TSdecreased as specimen size increased. The selection of specimen size for TBand TS tests from the less sensitive range in the density variation vs.specimen size curve, was capable of producing consistent and stable testresults.In the thickness swelling tests, one week of cold water soaking probablyreleased most of the thickness expansion. One week soaking is recommendedfor evaluation and comparison of dimensional stability of particleboard.The relationships between HDD, raw material characteristics and boardproperties, demonstrate that HDD is a fundamental variable for studyingparticleboard structure and technology. HDD can be used to describe rawmaterial characteristics effects, just as vertical density profile is used forstudying pressing strategy parameters. This study showed that:- raw material characteristics affect HDD,- conventional board properties are controlled by HDD,140- the concept of HDD should be a criterion for the selection of testspecimen size,6.1. Future DevelopmentsThe gravimetric method using the drilling technique was capable ofdetecting micro-density nonuniformity, but it is destructive and timeconsuming. A non-destructive method with good resolution, probably basedon VisionSmart X-ray technology, is required for future development of theconcepts presented in this thesis. The availability of that technique will makeit possible for both physical structure and board properties to be determinedon the same board.The finding that the number and size of voids influence the horizontalmicro-density nonuniformity, suggests that a future priority should be givento research and development of an improved forming technology, in whichsize of voids could be minimized and board uniformity improved.Ultimately, particleboard could be viewed as a three dimensionalnonuniform structure. With the establishment of the concept of horizontaldensity distribution, a concept of a three dimensional density distributioncould be developed by combining the knowledge of vertical density profile.This concept could lead to the development of a general theory on short-fibrewood composites, similar to the laminate theory for continuous fibrecomposites.This thesis did not examine how well particle bonding was achieved.Particleboard is a highly complex material, as physical, structural andchemical mechanisms all contribute to its ultimate properties. Particleboard141structure is not uniform, nor is the distribution of adhesive bonds. Theunderstanding of how structure interacts with adhesive bonding is not onlyimportant to fully comprehend the influence of HDD, but to the developmentof future particleboard design values for engineering calculations andreliability analyses.1427. LITERATURE CITEDAgarwal, B. D.; C. J. Broutman. 1980. Analyses and performance of fibrecomposites. 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For.Prod. J. 41(11/12):55-60.148Szabo, T. 1980. Flexual properties of waferboard. Technical Report 505ER,Forintek Canada Corp.Talbott, J. W.; T. M. Maloney. 1957. Effect of several production variables onthe modulus of rupture and internal bond strength of boards made ofgreen Douglas-fir planer shavings. For. Prod. J. 7(10):395-398.Turner, H. D. 1954. Effect of particle size and shape on strength anddimensional stability of resin bonded wood-particle panels. For. Prod.J. 4(5):210-222.U.S. Department of Commerce, National Bureau of Standards. 1974.Voluntary product standard for construction and industrial plywood,PS 1-74.Vanmarcke, E. H. 1984. Random fields: analysis and synthesis. The MITpress, Cambridge, Massachusetts.Wang, P. 1985. Determination of rheological parameters of wood incompression perpendicular to grain. Scientia Silvae Sinicae. 21(4):404-413.Wang, P. 1987a. The rheological behaviour of poplar wood in compressionperpendicular to grain. I. Viscoelasticity. Scientia Silvae Sinicae.23(2): 182-190.Wang, P. 1987b. The rheological behaviour of poplar wood in compressionperpendicular to grain. II. Plasticity. Scientia Silvae Sinicae. 23(3):357-363.Wang, P. 1989. The rheological behaviour of poplar chips in compressionperpendicular to grain. Scientia Silvae Sinicae. 25(6):522-528.Wang Y. T.; R. 0. Foschi. 1992. Random field stiffness properties andreliability of laminated wood beams. Structural Safety. 11:191-202.Weibull, W. 1939. A statistical theory of the strength of materials. SwedishRoyal Inst. Eng. Res. Proc., Stockholm,Xiong, P. 1991. Modelling strength and stiffness of Glulam. M.Sc. thesis. TheUniversity of British Columbia.Xu, W. 1989. Thickness swelling of particleboard. Unpublished directedstudy. The University of British Columbia.Young, C.; J. Thorpe. 1977. Density distribution vs. wet strain in papersheets. Tappi. 60(12):141-145.Youngs, R. L. 1957. The perpendicular-to-grain mechanical properties of RedOak as related to temperature, moisture content, and time. U.S. For.Service Report FPL No. 2079.Zhang, R.; A. W. Warrick; D. E. Myers. 1990. Variance as a function ofsample support size. Mathematical Geology. 22(1):107-121.149APPENDIX ADescription of Particles Used for the Study of the Influence ofHorizontal Density Distribution on Board Properties150n = 100= 71.6mm0.5 -s=15.7mm0.4-0.3-0__•10.2-0.1-______________20 50 80 110 140Particle Length (mm)n = 1001=17.6mm0.3 s = 10.3 mm0.21I Ig 0.1-1___0 20 40 60 80Particle Width (mm)n = 1001=0.69mm8=0.23mm00.2.1a o.i-0 I0.0 0.4 0.8 1.2 — 1.6Particle Thickness (mm)Figure A-i. Distribution of dimensions of aspen commercial particle p1.n = number of observations, 1= average, s = standard deviation.151n=71= 46.2 mms = 6.5 mm0.4•. 0.31a 0.2-I0.1120 50 ,80Particle Length (mm)0.4. n=71o 1= 19.5 mm0.30.1.0 30 60Particle Width (mm)n=71.2 0.3 Y= 0.56 mm0.2 1 s=0.2lmme 0.10.0 0.8 1.6Particle Thickness (mm)Figure A-2. Distribution of dimensions of aspen commercial particle p2.= number of observations, I = average, s standard deviation.152= 1001=82.8mms=23.8 mmg 0.4C0s.. 0.1—•1I I I20 50 80 110 140Particle Length (mm)n = 1000.4 1=13.2mm8=7.9 mmParticle Width (mm)n = 1001=0.66mms = 0.19 mm0.00.0 OA 0 1.2 — 1.6Particle Thickness (mm)Figure A-3. Distribution of dimensions of aspen commercial particle p3.n = number of observations, = average, s = standard deviation.153Particle Width (mm)0.40.30.200.2 0.6n =75i= 0.52 mms=0.lOmm1.0Particle Thickness (mm)Figur€ A-4. Distribution of dimensions of aspen laboratory particle p5.n = number of observations, = average, s = standard deviation.0..00s-I0.40.30.20.1Ti =75x= 118.8 mms = 6.1mm90 120 150Particle Length (mm)0.4to.20.15-4n = 75= 87.6 mms8.0 mm60 74 88 100 120154n=1360.7- 62.1 mm0.6 -g s... s=9.9mm04-0.3-p 0.2-0.1•I —F- -i20 50 80 110 140.Particle Length (mm)n = 13333.7 mms=15.2mm-i---r-—i—-_—i60 80Particle Width (mm)0.5- n=1200.4 =0.96mm_s=mm0.00 0.32 0.64 0.96 1.28 1.60Particle Thickness (mm)Figure A-5. Distribution of dimensions of aspen laboratory particle v6.n = number of observations, = average, s = standard deviation.155I0.40.30.20.1Particle Width (mm)Particle Thickness (mm)Figure A-6. Distribution of dimensions of birch laboratory particle p7.n = number of observations, = average, s = standard deviation.0.50.4000n= 110=58.8mm8=9.0mm0.130 50 70 90 110Particle Length (mm)n= 110Y3L4mm8 13.8 mm0.2________rm-rrrH II’’ I0 I0 20 40 60 80n = 1101=0.93mms=0.llmm1.3 1.6156APPENDIX BDescription of Obtainment of TB Specimens for Commercial Waferboard1. Nine commercial waferboard panels of size of 1220 mm x 606 mmwere cut into 18 smaller panels of size of 608 mm x 606 mm. Three of thesesmaller panels were randomly selected to make approximately 200 mm x 200mm sized TB specimens;2. lB specimens at a target size of 150 mm x 150 mm were preparedfrom 4 panels of 608 mm x 606 mm, which were also randomly selected;3. Approximately 100 mm x 100 mm sized lB specimens were preparedfrom 117 mm x 116 mm sized density specimens;4. One panel of 608 mm x 606 mm was randomly chosen and used tomake specimens at a target size of 75 mm x 75 mm;5. Density measurement specimens of approximately 54 mm x 54 mmwere also used for TB test at this size;6. lB specimens at a target size of 48 mm x 48 mm were cut from onerandomly selected panel of size of 608 mm x 606 mm;7. One panel of size of 608 mm x 606 mm was randomly chosen forpreparing specimens of approximately 42 mm x 42 mm for 113 evaluations;8. Approximately 25 mm x 25 mm TB specimens were prepared fromforty unused density specimens of size of 54 mm x 54 mm;9. One panel of size of 608 mm x 606 mm was randomly chosen for TBspecimens at a target size of 15 mm x 15 mm.157APPENDIX CDescription of Obtainment of TS Specimens for commercial Waferboard1. Two panels of size of 608 mm x 606 mm were randomly chosen fromrest of these sized panels to make TS specimens at a target size of 150 mm x150 mm;2. One panel of 608 mm x 606 mm was used for TS test at a specimensize of approximately 100 mm x 100 mm;3. TS specimens at a target size of 75 mm x 75 mm were prepared fromone randomly selected panel of 608 mm x 606 mm;4. One panel of 608 mm x 606 mm was used to make TS specimens ofsizes of approximately 50 mm x 50 mm and 42 mm x 42 mm respectively;5. TS specimens at target sizes of 36 mm x 36 mm, 29 mm x 29 mm and21 mm x 21 mm respectively were prepared from one panel of 608 mm x 606mm.158(D-1)(D-2)Aa AaAa AaAPPENDIX DDerivation of Equation (13) Var(Da)Nar(Db) = A1JAaLet Aa and Ab be the sizes of density sets A and B, Da and Db the densityvariables, and Var(Da) and Var(Db) the variances of density associated withdensity sets A and B. Further, suppose Ab be multiples of Aa, i.e., Ab = m.Aa(see the following diagram).AbAb = 4Aa (m=4)Figure D-1. Relationship between specimen sizes.According to basic statistics, we have,mdb = urn daii=1mVar(Db) = urn2 Var(Dai)i=1where, db and d are the individual observations of variables Db and Da.Assume Var(Dai) = ... = Var(D) = Var(Da), Equation (D-2) becameVar(Db) = jIm Var(Da) (D-3)orVar(Da)/Var(Db) = rn (D-4)As m = Ab/Aa, Equation (D-4) was generalized asVar(Da)/Var(Db) = Ab/Aa (D-5)159APPENDIX EDerivation of Equation (19) Var(Db) = [(1+p)Var(DaYI/2Given the same notations as in Appendix B, and let Ab = 2Aa (see thefollowing diagram).Aa AaAb2AaFigure E-1. Diagram showing Ab = 2AaWe have,db = (dai + da2)/2Var(Db) = [Var(Dai) + Var(Da2)+ 2COV(Dai, Da2)]14where, COV(Dai, Da2) is the covariance between the adjacent specimens.As Cov(D81, Da2) = p’JVar(Dai)Var(Da2),where p is the correlationcoefficient between adjacent densities Dai and Da2, Equation (C-2) thenbecameVar(Db) = [Var(Dai) + Var(Da2)+ 2p’JVar(D81)Var(D)J (C-3)Assume Var(Dai) = Var(D) = Var(Da), Equation (C-3) was changed toVar(Db) = [2Var(Da) + 2pVar(D)1/4= [(1+p)Var(Da)]/2 (C-4)Ab(C-i)(C-2)160APPENDIX FDerivation of Equation (21) Var(Db) = Var(Da) = (1/2)2Given the same notations as in Appendix D, and according to Equation(18), we haveVar(Db) =c2(1/Ab)b (F-i)Var(Da) = C2(i/Aa)b (F-2)Dividing Equation (F-i) by Equation (F-2), the following relationshipwas obtained,Var(Db)/Var(Da) = (Aa/Ab)21 (F-3)As Ab = 2Aa, Equation (F-3) was changed toVar(Db)/Var(Da) = (i/2)2b (F-4)161APPENDIX GSignificance Test of Parameter b in Equation (23)Null hypotheses H0: b = 0.5, b = 0.Alternative hypotheses H1: b 0.5, b 0.In order to test these hypotheses based on Equation (23), severalassumptions about this model need to be satisfied. Two of these were thatvariances of dependent variable ln(S) are homogeneous at different values ofindependent variable ln(1/A), and in(S) was taken from a normal distribution.These two assumptions were first checked.1. Homogeneous variance.The following diagram is the residual plot based on Equation (23). Theresidual at each value of independent variable ln(11A1) was the differencebetween the experimental data ln(S) and the model estimate in(S1). As theseresiduals could be assumed reasonably uniform based on this plot, ahomogeneous variance was assumed for the model based on this relativelysmall sample (Conover, 1980).0.2 I I010 00 0 00.00 00 0 00-0.1-02—10 —7 —4 —1 2ln(1/A)Figure G-1. Residual plot based on Equation (23).1622. Normal distribution.The lilliefors test for normality was used (Conover, 1980). First, the rawdata was normalized as z, = (x1- )/s, x1 is the individual data (in(S1)), is thearithmetic mean, and s the standard deviation of raw data. Second, theempirical cumulative probability function F(z) based on the normalized data(z1), and the cumulative probability function Ø(z) of standard normaldistribution were calculated. The difference T1 between F(z1) and Ø(z) wascalculated and the maximum difference was taken as the test statistic. Thefollowing table shows this procedure.Table G-1. Calculations for normal distribution testln(S) z F(z) T1-3.863 -1.472 0.083 0.071 0.013-3.817 -1.376 0.167 0.085 0.082-3.576 -0.879 0.250 0.189 0.061-3.442 -0.604 0.333 0.272 0.061-3.324 -0.362 0.417 0.359 0.057-3.244 -0.197 0.500 0.422 0.079-3.244 -0.197 0.583 0.422 0.162-3.101 0.097 0.667 0.748 -0.081-2.749 0.823 0.750 0.794 -0.044-2.604 1.121 0.833 0.869 -0.036-2.442 1.455 0.917 0.927 -0.010-2.375 1.592 1.000 0.945 0.050163The maximum difference was 0.162. As it was less than the criticalvalue of 0.242 (obtained from Conover, 1980) at significance level of 0.05, itwas concluded that ln(S) follows a normal distribution.Now, back to the original hypotheses tests. The regression analyses ofEquation (23) yielded a value of 0.1503 for b, and 0.0049 for standarddeviation of b. For the first null hypothesis H0: b = 0.5, the test statistic t =(0.1503- 0.5)/0.0049 = -71.37, was less than the critical value ta(fl2) = -1.812 ata = 0.05 (n is the number of observations used for the model, n-2 is the degreeof freedom). The first null hypothesis was rejected, i.e., b 0.5.For the second null hypothesis H0: b = 0, the test statistic t =0.1503/0.0049 = 30.67, was larger than the critical value of 1.8 12 at a = 0.05.The second null hypothesis was also rejected, i.e., b 0.164APPENDIX HDerivation of Equation (26) Var(Db) = (laIlb)Var(Da)Given the same notations as in section 4.2.2.1., and further let 1b bemultiples Of la, i.e., 1b = mla (see the following diagram).lamiaboard A(H-i)(H-2)(H-3)board BFigure H-i. Diagram showing the layer concept.The expected value of Db is just an expectation of the average of severalDa’S,mE(Db) = E(1/m D)i=iandmVar(Db) = tim2 Var(D8)i=ihere, E stands for the expectation.Suppose Var(Dai) = ... = Var(Dam) = Var(Da), Equation (H-2) becameVar(Db) = Var(Dj/mAs m = ‘IJ’a’ Equation (H-3) was generalized asVar(Db) = (la/lb)Var(Da) (H-4)165APPENDIX IDerivation of Equation (38) a = a(S)PFrom Equation (18),S = c(1/A) (I-i)we have1/A = (S/c) (1-2)Substituting Equation (1-2) into Equation (37), the followingrelationship was obtained,a = a(SIc) = a(S) (1-3)where, = dlb, x = ac”.166APPENDIX JDerivation of Equation (41) ç=From Equation (18)S = c(1IA) (J-1)we have1/A = (S/c)11b (J-2)Substituting Equation (J-2) into Equation (40), the following equationwas obtained,ç = e(S/c) = NJ(S) (J-3)where,= N’ = ec”.

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