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Structural property relationships for Canadian dimension lumber Liliefna, Leonard Dantje 1994

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STRUCTURAL PROPERTY RELATIONSHIPSFOR CANADIAN DIMENSION LUMBERByLEONARD DANTJE LILIEFNAB.F. Pattimura University, 1987A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE STUDIES(Department of Forestry)We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAMay 1994© Leonard Dantje Liliefha, 1994In presenting this thesis in partial fulfillment of the requirements for an advanced degree at theUniversity of British Columbia, I agree that the Library shall make it freely available forreference and study. I further agree that permission for extensive copying of this thesis forscholarly purposes may be granted by the head of my department or by his or herrepresentative. It is understood that copying or publication of this thesis for financial gainshall not be allowed without my written permission.Department of_____________________The University ofBritish ColumbiaVancouver, CanadaDate Cl /74ABSTRACTLumber property data from Canadian In-Grade Program for visually-graded dimensionlumber are used to model lumber property relationships. The lumber properties studied aremodulus of elasticity (MOE), modulus of rupture (MOR), ultimate tensile stress parallel to thegrain (UTS), and ultimate compression stress parallel to the grain (UCS) for Douglas-fir,Hem-Fir and Spruce-Pine-Fir species groups. Structural property relationships based onthree different approaches using Canadian dimension lumber have been modeled.The nonlinear models were adopted for the general stiffness-strength propertyrelationships. The fitted regression models for the general stiffhess-strength propertyrelationships then were used to model the strength property relationships.Band-width method was used to derive the relationships between modulus of elasticityand lower exclusion limits of strength values. The fitted models then were used to model thestrength property relationships. The resulted models represent the relationship between twostrength properties at the lower exclusion limit for lumber selected on the basis of modulus ofelasticity.The strength property relationships derived using equal-rank method agree with thatderived using general stiffness-strength property relationships. Therefore, for lumber selectedon the basis of modulus of elasticity, the models derived using equal-rank method yield anaverage or mean trend for the estimated properties.The results of the analysis show that there exist good relationships between lumberstrength properties. The strength property ratios for Canadian dimension lumber showsignificant species dependency particularly at the higher strength level. Property relationshipstrends are consistent across the species and methods of analyses.11The property ratio models are intended to provide property estimates of characteristicvalues for untested properties. The property ratios for Canadian dimension lumber aresignificantly higher than that proposed by the American Society for Testing and Materials(ASTM) standard D 1990.111TABLE OF CONTENTSgcABSTRACTTABLE OF CONTENTS ivLIST OF TABLES viiLIST OF FIGURES viiiLIST OF ABBREVIATIONS xiACKNOWLEDGMENTS xiii1. INTRODUCTION 11.1. Definition 11.2. Stress-Grade Lumber 21.2.1. Visually Graded Lumber 31.2.2 Mechanically Graded Lumber 51.3. Objectives 62. CURRENT MECHANICAL PROPERTY RELATIONSHIPS FOR DIMENSIONLUMBER 102.1. Relationships between MOE and Strength Properties 112.1.1. Relationships between MOE and MOR 122.1.2. Relationships between MOE and UTS 132.1.3. Relationships between MOE and UCS 142.2. Strength Property Relationships 153. THE CANADIAN N-GRADE DATA BASE FOR DIMENSION LUMBER 19iv3.1. Data Source .193.2. Sampling and Test Methods 204. PROPERTY ADJUSTMENTS FOR PROPERTY RELATIONSHIP STUDIES 214.1. Moisture Content Adjustment 214.2. Size Adjustment 235. DATA ANALYSIS 255.1. General Relationships between MOE and Strength Properties 265.2. Strength Property Relationships 315.2.1. Equal-RankMethod 325.2.2. Band-Width Method 386. RELATIONSHIPS BETWEEN MOE AND STRENGTH PROPERTIES 486.1. Relationships between MOE and MOR 486.2. Relationships between MOE and UTS 506.3. Relationships between MOE and UCS 517. RELATIONSHIPS BETWEEN STRENGTH PROPERTIES 537.1. Strength Property Ratios Based on MOR 537.1.1. Property Ratio Models from Equal-Rank Analysis 547.1.2. Property Ratio Models from General Relationships 567.1.3. Property Ratio Models from Band-Width Analysis 577.2. Property Ratios Based on UTS 607.2.1. Property Ratio Models from Equal-Rank Analysis 627.2.2 Property Ratio Models from General Relationships 637.2.3. Property Ratio Models from Band-Width Analysis 64V8. APPLICATION OF THE PROPERTY RELATIONSHIPS .668.1. MOE-Strength Property Relationships 668.2. Strength Property Relationships 679. CONCLUSIONS 71REFERENCE 74APPENDIX A: (TABLE 1 TO TABLE 17) 80APPENDIX B: (FIGURE 1 TO FIGURE 39) 114viLIST OF TABLESgc1. Relationships between MOE and MOR for the US In-grade data 812. Relationships between MOE and UTS for the US In-grade data 813. Width, test span and gauge length of bending, tension and compressionspecimens 814. General relationships between MOE and MOR 825. General relationships between MOE and UTS 836. General relationships between MOE and UCS 847. Strength property relationships based on MOR (Equal-Rank analysis) 858. Strength property relationships based on UTS (Equal-Rank analysis) 879. Goodness-of-fit analysis (1,000 MPa Band-Width) 8910. Relationships between MOE and MOR (Band-Width analysis) 9311. Relationships between MOE and UTS (Band-Width analysis) 9712. Relationships between MOE and UCS (Band-Width analysis) 10113. Analysis of variance table for the test on uniform regression line 10514. Strength property relationships (General relationships) 10615. Strength property relationships based on MOR (Band-Width analysis) 10716. Strength property relationships based on UTS (Band-Width analysis) 11017. Summary on the property relationships for the property estimates 113viiLIST OF FIGURES1. Histograms of MOE and strength propeies 1152. Graphical presentation of size adjustment on strength properties 1183. Relationships between MOE and MOR 1214. Relationships between MOE and UTS 1225. Relationships between MOE and UCS 1236a. Two strength properties at equivalent rank of percentile level 1246b. Relationship between two strength properties at percentiles of equivalentrank 1247. UTS and UTS/MOR as a function of MOR 1258. UCS and UCS/MOR as a function of MOR 1269. MOR and MORJUTS as a function of UTS 12710. UCS and UCS/UTS as a function of UTS 12811 a Illustration of the band-width effects 1291 lb Moving band-width (window) method 12912. Johnson Sb fit on MOE data (D-flr, 1,000 IVIPa Band-Width) 13013. Three parameter Weibull fit on MOR data (D-flr, 1,000 MPa Band-Width) 13314. Relationship between MOE and MOR (Band-Width = 1,000 MPa) 13415. Predicted MOR for each band-width and percentile level 13516. Grade increment factors for MOE-MOR (5-th percentile) 13617. Relationship between MOE and UTS (Band-Width = 1,000 MPa) 137viii18. Predicted UTS for each band-width and percentile level 13819. Grade increment factors for MOE-UT S (5-th percentile) 13920. Relationship between MOE and UCS (Band-Width = 1,000 MPa) 14021. Predicted UCS for each band-width and percentile level 14122. Grade increment factors for MOE-UCS (5-th percentile) 14223. Comparison of the 5 % exclusion lines between linear and nonlinear model 14324. Relationships of MOE to the 5-th percentile and mean trend of MOR 14325. Relationships of MOE to the 5-th percentile and mean trend of UTS 14426. Relationships of MOE to the 5-th percentile and mean trend of UCS 14427. Strength property ratios based on MOR 14528. UTS/MOR and UCS/MOR as a function of MOR (Band-Width = 1,000MPa) 14629. UTS/MOR and UCS/MOR as a function of MOE (Band-Width = 1,000MPa) 14930. MOE of MOR vs. MOE of UTS at 50-th percentile level 15231. Strength property ratios based on UTS 15332. MOR/UTS and UCS/UTS as a function of UTS (Band-Width = 1,000IVIPa) 15433. MOR/UTS and UCS/UTS as a function of MOE (Band-Width = 1,000MPa) 15734. Relationships between MOE and the predicted strength 16035. Comparison between 5-th percentile and mean trend for property ratiosbased on MOR 161ix36. Percentile-level effects on predicted strength ratios based on MOR 16237. Comparison between 5-th percentile and mean trend for property ratiosbased on UTS 1.6338. Percentile-level effects on predicted strength (based on UTS) 16439. Error in the prediction of MOR as the percentage of the given MOR 165xLIST OF ABBREVIATIONSASTM American Society for Testing and MaterialsB.C. British Columbiacdf cumulative distribution functionCOFI Council of Forest Industries of British ColumbiaCSA Canadian Standard AssociationCWC Canadian Wood CouncilD-fir Douglas-fir-Larch species groupEQRA Equal-rank assumptionf-E Allowable MOR and MOE for MSR lumberFPL Forest Products LaboratoryFPRS Forest Products Research SocietyGEN.REL General relationshipH-Fir Hem-Fir species groupK-S Kolmogorov-Smirnovksi kip per square inchLR Likelihood RatioM Moisture contentMOE Modulus of ElasticityMOR Modulus of RuptureMPa Mega PascalMSR Machine Stress RatedMSRD Maximum Strength Reducing DefectxNLGA National Lumber Grades AuthorityN/mm2 Newton per millimeter squareNo.2 Number 2 gradepsi pound per square inchSAS SAS System software for data analysisS-P-F Spruce-Pine-Fir species groupSPS Special Product StandardSS Select StructuralSSE Sum of Square ErrorUDL Uniformly Distributed LoadUCS Ultimate Compressive StressU.K. United KingdomUTS Ultimate Tensile StressUS United StatesWWPA Western Wood Products AssociationxiiACKNOWLEDGMENTSI would like to express my sincere gratitude to my supervisor, Dr. J. D. Barrett for hisinvaluable advice and patient guidance throughout the data analysis and preparation of thisthesis.I would like to express my appreciation to the members of my supervising committee,Dr. P. Steiner and Dr. S. Avramidis for their advice and support.The financial support from Eastern Indonesia University Development Project(ER.JDP) Simon Fraser University is gratefully acknowledge.My greatest thanks go to Mr. Wilson Lau for the development of the computerprogram without which this thesis would not have been possible. My greatest thanks also goto Dr. V. M. LeMay for her valuable help in statistical analysis. And last but not least, thecontributions of the fellow students and others not mentioned are gratefully acknowledged.XIII1. INTRODUCTION1.1. DefinitionKnowledge of material properties is essential in the design of structures. The designermust know the mechanical properties of a member under one or more applied load(s). Everystructural material, including wood, has its own certain physical and mechanical properties.Unlike man-made materials such as concrete and steel, wood is a natural fiber compositewhose properties are influenced by nature. Moreover, wood exhibits different strength valuesfor different grain directions. Therefore, wood can be described as an unisotropic material.Moreover, in the same direction e.g., parallel to the grain, wood also exhibits significantdifference between tension and compression properties.Based on the type of loading, there are three basic strength properties of wood whichare crucial to timber design engineers, i.e., bending strength, tensile strength parallel to thegrain and compression strength parallel to the grain. These are the strength propertiesobtained at maximum loads. For design and standard purposes, and in this thesis, the termsmodulus of rupture (MOR), ultimate tensile stress (UTS), and ultimate compression stress(UCS) are assigned to these three basic strength properties, respectively. Besides those threebasic strength properties, the elastic property or stifihess is also needed in the designapplication. The measure of the elastic property is called modulus of elasticity (MOE) and isdetermined from a static bending test. This property is used primarily for determining thedeflection of beams. The existing linkage between any two of these four mechanicalproperties is defined as property relationship. As outlined in the American Society for1Testing and Materials (ASTM) standard ASTM D 1990, the property relationships areintended to produce conservative estimates of characteristic values for untested properties(ASTM 1991). In this standard, the characteristic value is defined as the population mean,median or tolerance limit value estimated from the test data after it has been adjusted tostandardized conditions of temperature, moisture content, and characteristic size.Wood, for structural use, is available in a number of size categories such as lumber andtimber. Lumber is a general term which includes dimension lumber, timber, decking, boardsand finished lumber used as siding and flooring (CWC 1988a). Lumber properties, herein, aredetermined from dimension lumber. Dimension lumber is defined as surfaced softwoodlumber of thickness from 38 to 102 mm and is intended for use as framing members such asjoists, planks, rafters, studs and small posts or beams (CWC 1988a). Dimension lumber fromthe manufacturer is specified by species, grade and size.1.2. Stress-Grade LumberPieces of lumber of similar mechanical properties are placed in classes called stressgrades (FPL 1990). There are two type of stress-grading methods, i.e., visual grading andmechanical grading. Thus, there are two types of lumber, i.e., visually graded lumber andmechanically graded lumber. The purpose of grading is to provide material suited for theintended uses such as housing construction, etc. In Canada, the Standard Grading Rules forCanadian Lumber are published by Canadian Lumber Grades Authority (NLGA) and the rules2for stress grades are intended to provide a reliable measure for determining the strength valueof lumber (CWC 1991).1.2.1. Visually Graded LumberVisual grading is based on the premise that mechanical properties of lumber differ frommechanical properties of wood due to visual growth characteristics such as density, slope ofgrain, presence of knots etc. that affect the properties (FPL 1990). There are two methodsfor deriving mechanical properties for visually graded lumber, i.e., small clear specimensprocedure outlined in the ASTM standard ASTM D 143 “Standard Method of Testing SmallClear Specimens of Timber” and structural size (In-grade) procedure outlined in the ASTMstandards ASTM D 198 “Standard Method of Static Tests of Timbers in Structural Sizes” andASTM D 4761 “Standard Test Methods for Mechanical Properties of Lumber and Wood-Base Structural Material” (ASTM 1991).The property values that the design engineer uses in his design calculation are knownas allowable properties. The allowable engineering design properties must be either inferredor measured nondestructively (FPL 1990). Generally, the allowable properties depend uponthe particular sorting criteria and on additional factors that are independent of the sortingcriteria (FPL 1990).For small clear procedure, sorting criteria are handled with strength ratios for strengthproperties and with quality factors for modulus of elasticity as outlined in the ASTM standardASTM D 245 “Standard Practice for Establishing Structural Grades and Related Allowable3Properties for Visually Graded Lumber” (ASTM 1991). To account for variability in clearwood properties, the near minimum values, 5 % exclusion limit, and the mean value are usedfor strength properties and modulus of elasticity, respectively, as outlined in the ASTMstandard ASTM D 2555 ‘Standard Test Methods for Establishing Clear Wood StrengthValues” (ASTM 1991). In Canada, small clear test data for all commercially importantspecies are available in Forest Technical Report 21 (Jessome 1977). The similar test data forcommercially important North American softwood and hardwood species are summarized inthe ASTM D 2555 (ASTM 1991).The small clear procedure is less preferable because the design values do notnecessarily represent the true strength characteristics of structural lumber as used in service.Hence, testing full-size member is believed to provide a better representation of the strengthbehaviour of structural lumber in structures. This full-size testing, called In-grade testing, hasbeen conducted in Canada and US where visually-graded structural lumber, collected over awide geographic range within the respective countries, have been tested to destruction (FPRS1989). This research became known as the In-Grade Testing Program and was initiated toverifS’ the existing allowable design properties for softwood lumber and to provide a basis formore accurately estimating the mechanical properties of lumber for use in reliability-basedengineering design codes and standards (FPRS 1989). The standard practice for establishingallowable properties from visually-graded dimension lumber that resulted from Canadian andUnited States (US) In-Grade Program is outlined in the ASTM standard ASTM D 1990“Standard Practice for Establishing Allowable Properties for Visually-Graded DimensionLumber from In-Grade Tests of Full-Size Specimens” (ASTM 1991). In this standard, thecharacteristic values for strength properties are taken as the nonparametric 5-th percentile4point estimates of the test data. For MOE, the characteristic values for each grade are themean, and the lower 5 % tolerance limit (or other measure of dispersion).In Canada, the result of the In-grade testing on dimension lumber conducted by theCanadian Wood Council (CWC) has been incorporated in the 1989-edition of the CanadianStandard Association (CSA) CAN/CSA-086. 1-M89 “Engineering Design in Wood (LimitState Design)” in which the specified strength values for MOR, UTS, UCS, shear strength andMOE were derived based on the reliability-based design principles (CWC 1990).1.2.2 Mechanically Graded LumberMechanically grading, called machine stress rating (MSR), uses a machine to sortlumber based on flatwise bending MOE of the piece. The piece is then given the grade markincluding f-E classification indicating allowable stress values for MOR and MOE (CWC1 988b). MSR lumber is also required to meet certain visual requirements on defects such asedge-knot size, checks, etc. (NLGA 1987).Generally, for MSR lumber, the allowable stress for tension and compressionproperties are developed from the relationships with allowable bending stress rather than beingestimated directly by the nondestructive parameter, MOE (FPL 1990). Grades and theirmechanical property requirements for MSR lumber produced in Canada are described inNational Lumber Grades Authority (NLGA) Special Product Standard 2 (SPS 2), “ MachineStress-Rated Lumber” (NLGA 1987). Evaluations were conducted on MSR lumber in orderto evaluate the tension and bending property requirements for selected grades of mechanically5graded lumber (Barrett and Lau 1992). These evaluations confirmed that the propertyspecifications for MSR lumber given in SPS 2 are attained for tension and bending for Spruce-Pine-Fir. Allowable properties for 38 mm thick machine stress-rated (MSR) lumber arepublished in the CWC Datafile WP-5 Machine Stress-Rated Lumber (CWC 1988b). Inthis publication, the ratio of allowable tensile stress parallel to grain to bending stress follows asliding scale ranging from 0.39 for 900f-1.2E grade to 0.8 for the 2400f-2.OE grade, whereasthe ratio of allowable compression stress to bending stress is a constant factor 0.8 for eachgrade.1.3. ObjectivesAs mentioned above, the relationship between MOR and MOE has been used topredict bending strength (MOR) for MSR lumber. Hence, the prediction of UTS and UCSfrom bending MOE is of significant interest in the recent investigations of structural lumber.This is one of the reasons for the need of the stiffness-strength property relationships.Moreover, since the assignments of UTS and UCS are based on the relationships with MOR,studies on the relationships of UTS and UCS to MOR are crucial for the evaluation of theproperty assignments.For the evaluation of the property assignments, it is clear that MOE can be obtained ina nondestructive fashion, therefore the deterministic approach to the material propertyevaluation is possible only for MOE. Since the determination of MOR, UTS and UCS onfull-size lumber requires a destructive testing, one can only measure one strength property on6a single member. As a result, there are three pairs of stifThess-strength property relationshipsthat are of interest and which must be collected from tests, there are MOE-MOR, MOE-UT Sand MOE-UCS.As for MSR lumber, the property relationships for visually graded lumber are oftenused to estimate properties for which test data are unavailable. The In-grade data offer anopportunity to establish conservative property estimates for untested properties when only oneproperty is tested. Thus it allows for the development of models (property relationships) toestimate untested properties so that the amount of testing and cost to establish property valuesfor untested species, grades, or sizes could be greatly reduced (ASTM D 1 990-ASTM 1991).Moreover, the need for the property relationships is also prompted by the use of propertyratios for standardized property classification systems (stress class system) in internationalstandards (Green and Kretschmann 1990). In general, a better knowledge of lumber propertyrelationships could contribute to the development of more standardized grading systems andimproved property assignment in wood design standards (Barrett and Griffin 1989, Green andKsetschmann 1991)Strength property relationships, derived using the equal-rank method to analyze theUS In-grade and Canadian Spruce-Pine-Fir data, have been reported by Green andKretschmann (1991). The strength property relationships, derived using the same method oncombined data of Canadian and US Douglas-fir, Hem-Fir, Southern Pine and Spruce-Pine-Fir(North-American In-grade data) for the estimates of untested properties are described in theASTMD 1990 (ASTM 1991).According to the Council of Forest Industries of British Columbia (COFI), Canada isthe largest exporter of softwood lumber in the world with 50 % of the world market supply in71991 (COFI 1993). COFI also reported that British Columbia (B.C.) alone accounts for 34% of world exports of softwood lumber with the largest export customer being the UnitedStates which received 44.4 % of all wood products shipments from B.C. in 1992. Nearly 30% of the total value of $7.2 billion of the shipments of solid wood products in 1992 from B.C.were used within Canada (COFI 1993). Because Canada uses its own lumber and is a majorexporter, it is important to develop property relationships for Canadian species.Toward the international standards, the results from North American In-grade data(Barrett and Griffin 1989, Green and Kretschmann 1989, Green and Kretschmann 1990,1991) were compared with property ratios assumed in Eurocode 5 (Fewell and Glos 1989).The relationships adopted in Eurocode 5 were adopted from the work of Curry and Fewell(1977). Curiy and Fewell (1977) used an MOE based band-width approach to deriveproperty relationships. Therefore, it is important to examine the equal rank method used inNorth America and band-width method used by Curry and Fewell (1977) so that the resultscan be compared. In other words, it is important to evaluate how the results are affected bydifferent methods of analysis.Using the data base from CWC Full-Size Lumber Properties Program (Canadian In-Grade Program) which contains the three pairs of the stiffness-strength property values forgiven grades, sizes, and species, the objectives of this thesis are:1. To develop the relationships between lumber properties for the estimates ofcharacteristic values for untested properties,2. To evaluate the characteristics of property relationships developed by usingdifferent methods,83. To provide the information on the mechanical property relationships based onCanadian dimension lumber.It should be noted that even though the data used in this study are visually gradedmaterials, the results are relevant to MSR lumber. This is because basically the materials arethe same; only the grading methods are different.The reader is referred to Barrett and Griffin (1989) for the information on strengthproperty relationships for each grade, size and species, test configuration effects relative tothose assumed in Eurocode 5.92. CURRENT MECHANICAL PROPERTY RELATIONSHIPS FORDIMENSION LUMBERFor better design and standards, many small scale investigations as well as larger scaleinvestigations have been conducted to search for reliable property relationships for structurallumber. The review of the existing studies and reports on the mechanical propertyrelationships will emphasize the existing stifihess-strength property relationships and strengthproperty relationships for dimension lumber.As mentioned before, the relationship between MOE and MOR has been used as thebasis for MSR grading. Thus, the relationships between MOE and the other strengthproperties are also important since MOE can be measured directly. Because the variability instrength as a function of MOE is high, the information from studies with large data bases isvery important.The development of the strength property relationships was prompted by the need forestimating characteristic values of untested properties. The development of the strengthproperty relationships were also driven by the need for a better design property assignmentsfor MSR lumber. The information from the test results on full-size lumber from othersources, therefore, are needed in the evaluation and verification of models for the propertyrelationships derived from studies of Canadian visually graded lumber.102.1. Relationships between MOE and Strength PropertiesSince a good relationship exists between MOE and MOR, fl.irther studies have shownthat compression and tension parallel to grain also are related to MOE (Hoyle 1968). Testresults on stiffness-strength property relationships dating back to the years before 1966 weresummarized and presented by Hoyle (1968). He proposed the linear empirical models,MOR = 0.005 13 MOE - 2265, UCS = 0.00285 MOE + 480, and UTS = 0.00346 MOE -1850 measured in pound per square inch (psi) for bending, compression parallel to the grainand tension parallel to the grain, respectively, as an average for all North American species.Relationships between MOE and strength properties, generally, are modelled using asimple linear regression equation MOR=+ 1i (MOE), where 13° is the intercept and f3 isthe slope of the regression line (Hoyle 1968, Curry and Tory 1976, FPL 1977). For thesubsequent evaluation, the value of the coefficient of determination r2 will be emphasized.The coefficient of determination (r2), which is the square of coefficient of correlation, is themeasure of the goodness-of-fit of the model. This is a ratio that describes the relative amountof variation of the dependent variable that has been explained by the regression line.In relating MOE to a particular strength property, however, no unique value ofstrength exists. Because MOE can be measured in a nondestructive fashion, MOE can beused to study the relationship between two strength properties. Thus, it is important toevaluate the recent stiffiiess-strength property relationships for the comparative study withthat of In-grade results from Canadian species.112.1.1. Relationships between MOE and MORFor structural lumber, MOE is measured from a static bending test, therefore, it iscalled flexural modulus of elasticity. This MOE is found to be a good indicator of flexuralstrength or MOR. This relationship is the foundation of MSR grading (Hoyle 1961, Kramer1964, Sunley and Hudson 1964).The coefficient of determination (r2) for the relationships between MOE and MORranges from 0.32 to 0.76 for Douglas-fir, Hemlock, Spruce and Pine as calculated from thereport by Hoyle (1968). Curry and Tory (1976) reported the relation MOR = 3.576 x iOMOE - 1.66 (N/mm2) with r2 = 0.67 for European redwood and whitewood. Linearregression results derived from In-grade tests of US species (Green and Kretschmann 1991)are shown in Table 1.A perfect straight-line fit will have an r2 value of one, and as the r2 value decreasesfrom one, the proportion of the total variation in MOR which is explained by the regressionwith MOE decreases. The higher value ofr2, indicates that MOE is a good indicator variablefor MOR. As an example, for Hem-Fir in Table 1, 52 % of the total variation of the MORvalues is accounted for or explained by the linear relationship with MOE.In MSR grading, the lower exclusion limit (usually 5 %) of bending strength is usedrather than the mean value in order to account for variability along the linear regression line(Kramer 1964, Hoyle 1968, FPL 1977). According to Bodig and Jayne (1982), for somecases the variance is not constant along the linear regression line, therefore, the lower 5 %exclusion limit is not represented by straight line parallel to the regression line. It will beshown in the subsequent analysis that the band-width method introduced by Curry and Tory12(1976) indirectly takes into account the variability along the regression line in determining thelower exclusion limit. Another method to overcome this problem is to treat the standarderror of the estimate as a function of MOE (Woste et al. 1979).Because the relationship between strength property and MOE needs not always to belinear, other alternative procedures and models have been used to relate strength and elasticproperties of lumber. O’Halloran et al. (1972) reported that the nonlinear model,MOR= 13i (MOE)P2, gives a better fit on the scatter plot of MOE versus MOR particularlyfor the data at the extreme ends of MOE range and this model seems more realistic for thelower bound of MOR results for Lodgepole pine dimension lumber. Curry and Tory (1976)also reported that the minimum or lower exclusion limit for the relationship between MOE andMOR is fitted best by this nonlinear model for European species redwood and whitewood andCanadian hemlock.2.1.2. Relationships between MOE and UTSThe simple linear regression model is also adopted for the relationship between MOEand UTS. The coefficient of determination for the relationships between MOE and UTS ascalculated from the report by Hoyle (1968) was 0.55, 0.56 and 0.66 for Douglas-fir, white firand hemlock, respectively.By testing full-size structural lumber of two species namely Swedish redwood andwhitewood of three sizes 38 x 100, 150, and 200 mm, Curry and Fewell (1977) proposed themodel UTS = 0.00242 MOE - 1.51 (N/mm2)with r2 = 0.59.13The linear regression models for the relationships between MOE and UTS for the USIn-grade data reported by Green and Kretschmann (1991) are presented in Table 2. For eachspecies, the r2 is smaller than that observed for the relationship between MOE and MOR.Curry and Fewell (1977) also proposed the nonlinear model, UTS= J3 (MOE)132 forthe relationship between MOE and the lower exclusion limit of UTS.2.1.3. Relationships between MOE and UCSLike the relationship between MOE and UTS, the general relationship between MOEand UCS is represented typically by simple linear regression equation. The coefficient ofdetermination for this relationship, as calculated from the report by Hoyle (1968), was 0.61,0.71 and 0.45 for Douglas-fir, Grand fir and Southern pine, respectively. Curry and Fewell(1977) reported the model UCS = 0.00148 MOE + 10.41 (N/mm2) for Polish redwood(combined size 38 x 100 and 50 x 100 mm) with the r2 = 0.58.Based on tests of 2-inch Southern pine dimension lumber, Doyle and Markwardt(1966) proposed the model UCS = 0.0001767 MOEc + 1881 (1,000 psi) for all grades andsizes with coefficient of determination r2 = 0.45. MOEC is the modulus of elasticity incompression parallel to the grain entered in million psi units. They also found that thecompression modulus of elasticity is closely comparable to the flexural modulus of elasticityboth flatwise and edgewise.14The relationship between MOE and the lower exclusion limit of UCS has the samenonlinear form as the relationship between MOE and UTS proposed by Curry and Fewell(1977).2.2. Strength Property RelationshipsThe development of strength property relationships to estimate untested propertieswas prompted by the need for multiple property assignments (ASTM D 1990-ASTM 1991).In practice, however, bending and tension or bending and compression may occursimultaneously at a cross section. Therefore, different members require different types ofassigned stresses. As a consequence, the multiple allowable properties have to be assigned tothe product since the end use of the product is unknown in the outset.Measurement of the strength properties of lumber generally involve a destructive test.Normally, only one failure mode can be evaluated from a single piece of lumber. Becauseonly a single failure mode can be obtained, it is not possible to measure MOR, UTS and UCSat the same member cross-section. In other words, because one can not break a piece oflumber twice, the relationships between two strength properties, particularly for MOR, UTSand UCS, can only be described in probabilistic terms.Strength property assignments for visually graded lumber based on the full-size testingprocedure, practically, does not require property relationships for the tested properties. Theassignments of bending strength, tensile strength and compression strength parallel to the15grain are based on the 5 % lower exclusion limit (nonparametric fifth percentile estimate) asoutlined in the ASTM D 1990 (ASTM 1991).For MSR lumber, different rules apply. In machine grading, one can select lumberwith specified minimum MOR or minimum MOE. The assignments of the other propertiessuch as UTS and UCS were established as fixed proportions of bending stress. Historically,WWPA (1965) assigned both UTS and UCS as 80 % of allowable MOR. The work byLittleford (1967) showed that this allowable tensile stress was over estimated. The historicaloverview on the evolution of tensile stress assignment dating back to the years until 1979 wasreported by Galligan et al. (1979). According to their report, in 1969, a sliding tensionproperty scale factor, ranging from 0.39 to 0.8 of bending strength, was used to calculate theallowable tensile stress parallel to the grain for MSR lumber. Until now, similar scale factorsare maintained for the UTS/MOR ratios, while the UCS/MOR ratios are maintained atapproximately 80 % of the allowable bending stress for all grades as proposed by NLGA(1987) and CWC (1988b).Due to the need for the improvement of the grading and standards, several studieshave been carried out to model the property relationships for lumber (Curry and Fewell 1977,Johnson and Galligan 1983, Green et al. 1984, Bartlett and Lwin 1984, Evans et al. 1984,Green and Kretshmann 1991). There are several methods introduced by these investigatorsfor determining the property relationships.Curry and Fewell (1977) made use of the relationships between MOE and strengthproperties to establish the strength property relationships. Using European redwood andwhite wood, they showed that the ratio of the near minimum value (1 % and 5 % lower16exclusion levels) of UTS to MOR is approximately 0.60 in the design property range.Whereas the ratio of UCS to MOR is represented by:Rc,B = 4.93 M0R054 (1)where R, is the ratio of UCS to MOR and MOR is entered in N/mm2 (MPa). From thisequation, it is clear that the UCS/MOR property ratio decreases as bending strength increases.The property relationships, established on the basis of the equality of the percentilerank, have been proposed by Barrett and Griffin (1989) and Green and Kretschmann (1989)for Canadian In-grade data and US In-grade data, respectively. Using North-American In-grade data, Green and Kretschmann (1991) further developed the models for the strengthproperty relationships. Their results were incorporated in the ASTM standard D 1990(ASTM 1991) for North-American In-grade data.Green and Kretschmann (1991) proposed a constant factor 0.56 for the ratio of UTSto MOR for MOR values below 48.3 ]VIPa (7 ksi), for all grades, species and sizes, whereas aconservative factor of 0.45 is adopted in the ASTM standard ASTM D 1990 (ASTM 1991)for data adjusted to nominal 2 by 8 size and 15 % average moisture content. For the ratio ofUCS to MOR, Green and Kretschmann (1991) reported (for a 2 by 8 size, all species andgrades) a constant factor 0.596 for MOR > 49.6 IVIPa (7.2 ksi) and below this limit theUCS/MOR ratio is given by:Rc/B = 1.745—0.320 MOR+0.0223 MOR2 (2)where MOR is entered in ksi. The same model, except for the intercept is 1.55 and 0.22 forthe quadratic term, is adopted in the ASTM standard ASTM D 1990 (ASTM 1991) for allvalues of MOR adjusted to 2 by 8 and 15 % moisture content.Green and McDonald (1993) reported that the ratio model:17Re/B = 0.338+(3)for MOR 2.835 ksi (19.55 MPa) and the constant 1.06 if MOR is less than this limit for dataadjusted to 15 % moisture content was adopted for MSR softwood lumber by the AmericanLumber Standards Committee, Board of Review.Green and Kretschmann (1991) reported that (for a 2 by 8 size, all species and grades)for the property relationship set on the basis of tensile strength then the ratio of UCS to UTSis a constant factor 0.83 7 for UTS > 38.6 MPa (5.6 ksi) and below this limit is given by:Rc/T = 2.724—0.678 UTS+0.0608 UTS2 (4)where R is the ratio of UCS to UTS and UTS is entered in ksi. However, this ratio is saidto vary somewhat with species, lumber size and grade. A conservative model:Rc,T = 2.4—0.7 UTS+0.065 UTS2 (5)where UTS is entered in ksi, is adopted in the ASTM standard D 1990 based on UTS valuesadjusted to 2 by 8 size and 15 % moisture content (ASTM 1991). The ratio of MOR to UTSset on the basis of UTS is taken to be constant factor 1.2 in this standard for the sameadjustment conditions.The ASTM standard D 1990 (ASTM 1991) recommends that when both UTS andMOR data are available, the most conservative should be used for calculating UCS.183. THE CANADIAN IN-GRADE DATA BASE FOR DIMENSIONLUMBEROne of the objectives of the Canadian In-Grade Program is to determine themechanical properties of dimension lumber (CWC 1988c). With the introduction of the LimitState Design version of National Standard of Canadian CAN/CSA-086. 1 in 1984, the CSACommittee responsible for the Code Engineering Design in Wood adopted the philosophy thatdesign properties for structural wood products should be based on full-size structural tests(CWC 1990).3.1. Data SourceAs mentioned in the CSA Commentary to the 1989-edition of CAN/CSA-086. 1-M89by J. D. Barrett (CWC 1990), the Canadian Wood Council, through its Lumber PropertiesSteering Committee, conducted a lumber properties research program for bending, tensionand compression parallel to grain strength properties for 38 mm (nominal 2 inch) dimensionlumber of all commercially important species groups. In this program, short term bendingstiffness properties and bending, tension and compression parallel to grain strengths wereevaluated in accordance with the ASTM standard ASTM D 4761.The test data provided by Canadian Wood Council for this project will be used toestablish lumber property relationships for Canadian dimension lumber.193.2. Sampling and Test MethodsThe detailed description of the sampling, testing, moisture-content adjustmentprocedures are given in the report from the Council of Forest Industries of British Columbiaby Fouquet and Barrett (1989). The summary on the mechanical property data is presentedin the report by Canadian Wood Council (CWC 1988c). The following discussion provides abrief summary of key elements of the testing procedure for the mechanical properties (CWC1 988c).The experiments were conducted at the Vancouver laboratory of Forintek CanadaCorp. The specimens were conditioned to approximately 15 % moisture content prior tobending, tension parallel to the grain, and compression parallel to the grain tests. Moisturecontent of each specimen was measured at the time of testing.Prior to destructive testing the flatwise MOE profile was measured using a CookBolinders mechanical grading system. Edgewise bending MOE of each sample was measuredusing a third point loading system. The bending test was conducted on a 17 to 1 ratio of testspan (L) to member width (W) with the tension edge and the maximum strength reducingdefect (MSRD) randomly assigned in the test span. MOR, UTS, and UCS values weredetermined using the maximum load and the actual dimensions of the specimen. The gaugelengths of the test specimens are summarized in Table 3 (Barrett and Griffin 1989).The MOR, UTS and UCS data are available from three species combinations, i.e.,Douglas-fir-Larch, Hem-Fir and Spruce-Pine-Fir (hereafter, abbreviated to D-flr, H-Fir andS-P-F, respectively), two grades, (select structural (SS) and number 2 (No.2)) and 3 sizes(2x4, 2x8 and 2x10).204. PROPERTY ADJUSTMENTS FOR PROPERTY RELATIONSHIPSTUDIESPhysical properties of wood such as moisture content, density or specific gravityinfluence the mechanical properties of wood. Generally, moisture content of wood iscontrollable in the experiment. It is well known that moisture content influences bothstrength and stifihess of wood especially for bending and compression (Madsen 1992). Fordimension lumber, the testing procedure requires the test samples to be conditioned to thetarget moisture content prior to testing. However, moisture content at the time of test willvaiy in a narrow range.Testing the full size structural lumber has shown that the strength decreases with anincrease in member size. The early works on size effects by Bohannan (1966), Barrett(1974), and Kunesh and Johnson (1974) have shown that size has a very significant effect onstrength properties of lumber. Because of anisotropic nature of wood, Madsen and Buchanan(1986) suggested that different size parameters should be used to quantif,r for different sizeeffects in member width and length.4.1. Moisture Content AdjustmentBecause strength and stiffness of wood are influenced by moisture content, theindividual test results were adjusted to a common moisture content in order to reduce the biasdue to moisture content variations. Following the ASTM standard D 1990-90 (ASTM 1990)21MOE of each piece of lumber was adjusted to 15 % moisture content level using the In-gradeformula:11.8566 — 0.023 722 M2 1MOE2 = MOE1 1. 1.8566 — 0.023722M1 (6)where:MOE1 MOE (psi) at moisture content level 1,M1 = Moisture content level 1 (decimal),M2 Moisture content level 2 (decimal).This MOE value then was adjusted for uniformly distributed load (UDL) 21:1 span-to-depthratio according to ASTM D 2915-90 (ASTM 1990).MOR and UCS data (as tested) were adjusted to 15 % moisture content level using theLinear Surface Model (Barrett and Lau 1991 a, 199 lb) as follows:{i —D1(15 —M1)} —.f{(15—M1)D_i}2 —4P1D(15—M)= 2D(15—M (7)where:P1 = property value (ksi) at moisture content M1,P2 = property value (ksi) at moisture content 15 %,= -0.95689 for MOR, and -2.36662 for UCS,D2 = 0.2033 ksi’ for MOR, and -0.2 15548 ksi’ for UCS.M1 = actual moisture content.Moisture content adjustments were not made for the tension data in this study.224.2. Size AdjustmentTable 3 shows that the test gauge length for bending, tension and compressionspecimens vary by specimen widths. Bending specimens were tested at constant span-to-depth ratio 17:1. Since the test gauge lengths for tension and compression specimens aredifferent from bending specimens, the UTS and UCS test values were adjusted to the testlength of the bending specimens (i.e., the length is 17 x member width). Then, to account forwidth effects all test results were adjusted to a common width of 7.25 inches and a length of17 x 7.25 inches. The size adjustment procedure is similar to the standard procedure in theASTM D 1990 (ASTM 1991). The property adjustment formula is as follows:IL L(WRP2 = P1 I I— I (8)l17W) i.7.25Jwhere:P1 = Property measured at length L1 and width W1,P2 = Property adjusted to nominal 7.25 inch at length = 17 Wi,SL = Size effect factor for length,SR = Size effect factor for a member with a constant span to depth ratio 17:1.The factors, 5L and SR are 0.17 and 0,4 respectively, for bending and tension members and 0.1and 0.21 respectively, for compression members (Barrett, Lam and Lau 1992).After adjusting the 2x4, 2x8 and 2x10 data to nominal size 38 x 184 mm (1.5 x 7.25inch), preliminary analysis (not presented here), showed that there were no differences in themean values among the adjusted size groups. Therefore, for the subsequent analysis, the datafor the adjusted sizes were pooled. The same preliminary analysis showed that there were23significant differences in the mean values between the two grades, SS and No.2.Nevertheless, in order to have the results represent both low and high strength materials,select structural and No. 2 grades were pooled for the analysis on property relationships. Inaddition, the unit measurement of the data were converted from Imperial to Metric (SI) unitsafter all the necessary adjustments have been performed.The histograms of the pooled MOE and strength property data (adjusted size andmoisture content) are presented in Fig. 1. In this figure, the histograms for MOE and thestrength properties are presented on a common scale in order to allow direct comparisons ofthe shape of the distribution for each case. The height of the histogram is equal to the relativefrequency divided by base length called density scale (Devore 1991). Therefore, the ordinateis equal to the probability density function since the area under all of the histograms is equal tounity.Graphical presentations of 2x4, 2x8 and 2x 10 (15 % moisture content) data before andafter size adjustments (Eq. (8)) are given in Fig. 2. The graph shows that the size adjustmentquite successfully eliminated the effects of size variations especially for the lower and mediumstrength levels.245. DATA ANALYSISThe models for strength and stiffness-strength property relationships were derivedusing regression analysis and other modelling techniques, Regression analysis is defined as astatistical tool for evaluating the relationship of one or more independent variables to a singlecontinuous dependent variable (Kleinbaum et a!. 1988). Whereas, modelling refers to thedevelopment of mathematical expressions that describe in some sense the behaviour of arandom variable of interest (Rawlings 1988). Regression analysis is applied for severalreasons such as finding the quantitative formula or equation to describe the dependent variableas a function of the independent variable(s) or determining the best mathematical model fordescribing the relationship between a dependent variable and one or more independentvariables (Kleinbaum et a!. 1988). In the subsequent analysis, of the relationships betweenMOE and the strength property, MOE is considered as the independent variable whereas thestrength property, i.e., MOR, UTS, or UCS, is considered as the dependent variable.Likewise, when assessing the strength property relationship between two properties, one willbe considered as the independent and the other as the dependent variable depending on theintended application of the model.There are two main steps in the subsequent analysis. First, determining the empiricalmodels for the relationships and second, finding the best model by means of the regressionanalysis. There are many models that could be chosen to represent the relations betweenproperties. Some of the models can be developed based on assumptions about the underlyingwood property distributions. In other cases the choice of the model is based on experience orprevious work.255.1. General Relationships between MOE and Strength PropertiesThe general relationship is defined as the relationship between MOE and thecorresponding strength property obtained by fitting the regression model to the full data set.After the necessary adjustments have been carried out and the combination of the two grades,the regression analyses were performed for the general relationships between each strengthproperty (i.e., MOR, UTS, and UCS) and MOE.Two regression models, such as a linear and a nonlinear, were used to characterize therelationships between each strength property and MOE. As mentioned before, the simplelinear regression model has been used extensively to model the relationships between MOEand the strength properties. Green and Kretschmann (1991) also used this model for theirstudies of the US In-grade data. With the assumption of additive error, the linear model is ofthe form (Neter and Wasserman 1990, Rawlings 1988):= 13o + X1 +6, (9)where:= Strength property in the ith trial,Xl = MOE in the ith trial (assumed to be a set of known constants),i3 and I3 = Parameters,= Independent random error (assumed normally distributed with mean = 0and variance a2),I 1,2,...,n.The following are the nonlinear models that were evaluated:= 13o +f1x2 -i-61 (10)26= (11)= iX2133xi +c (12)where:Y1 = Strength property in the ith trial,X = MOE in the ith trial,130-133 = Parameters,= Random error (assumed independently and identically distributed withmean = 0 and variance = a2) (Judge et al. 1985),I = 1,2,...,n.The regression analyses were carried out using SAS package Release 6.03 (SAS1988). The starting points for the parameters in the nonlinear model of Eq. (10) were takenfrom the results of Eq. (9). Whereas, for Eqs. (11) and (12), the starting points wereestimated from the result of fitting a linear regression to the logarithmic transformation of themodels. All of the nonlinear models were fitted using the Gauss-Newton iterative method(SAS 1988). Tables 4, 5 and 6 show the results of these two regression analyses for allspecies.The following steps were taken in order to determine the best model for eachrelationship in general:1. Test if f3 = 0 in Eq. (9); if yes, then the model without intercept should be used;use Eq. (9) otherwise,2. Test if 132 = 1 in Eq. (10); if yes, then Eq. (9) is adequate; use Eq.(10) otherwise,3. For f2 1 in Eq. (10), test if 10 = 0 in Eq. (10); if yes, then Eq. (11) is adequate;use Eq. (10) otherwise,274. Test if f33 = 1 in Eq. (12); if yes, then Eq. (11) is adequate; use Eq. (12) otherwise,5 If Eq. (10) and Eq. (12) are sufficient models, compare Eq. (10) with Eq. (12) interms of the lowest Sum of Square Error (S SE),The hypothesis test for the parameter is carried out using 2-tailed t-test (Gallant 1987):=[(j (13)where:O = Estimated parameter,o Known parameter,s () Standard error of the estimate.The null-hypothesis was rejected when Iti > t(1/2 n-p) where n is the sample size and p is thenumber of the parameter in the model. Since n is large, for a = 0.05, then t = 1.96.It can be seen from Tables 4, 5, and 6 (in the last column) that the ti value revealedthat the intercept in Eq. (9) was significantly different from 0 for each species and relationship.Therefore, the intercept in Eq. (9) is necessary at the level of significance a = 0.05.Parameter 132 in Eq. (10) was not significantly different from 1 for MOE-MORrelationships for Douglas-fir and Hem-Fir and MOE-UTS relationship for Douglas-fir,therefore Eq (9) is adequate to represent these relationships. However, this parameter wassignificantly different from 1 for MOE-MOR relationship for S-P-F, MOE-UTS for Hem-Firand S-P-F and MOE-UCS for all species. It indicates that the nonlinear rather than the linearmodel should be used.28For the case where parameter 132 in Eq. (10) was significantly different from 1, Tables4, 5 and 6 show that both MOE-MOR and MOE-UTS relationships are sufficientlyrepresented by Eq. (11) since the intercept (1) in Eq. (10) for each relationship was notsignificantly different from 0. This results were also justified by parameter 133 in Eq. (12)which in most cases was not significantly different from 1. Only for the MOE-UCSrelationship (Table 6) Eq. (10) is shown to best represent the data since parameter I3 wassignificantly different from 0 in all cases.As can be seen from Table 6 where the parameter test showed that Eq. (10) isgenerally adequate; Eq. (12) also gives good results for representing the data. Eq. (12), inmany cases, yielded a comparable SSE when the parameter f33 was significantly different from1. In this case, the proposed model should be chosen from these two equations based on theSSE and the simplicity of the model. For this reason, Eq. (10) is preferable than Eq. (12).In summary, the above analysis showed that generally MOE-MOR, MOE-UTS andMOE-UCS relationships were represented by, respectively, Eq. (9), Eq. (11) and Eq. (10).The complexity of Eq. (12) did not seem to be justified given the small improvement in themodel performance.In the subsequent analysis on modelling of strength properties, a single model for thestiffiess-strength relationships was adopted. Although, models like Eqs. (10) and (11) willperform nearly equal, Eq. (10) was adopted for the modelling purposes. The regressionmodels (Eq. (10)) for MOE-MOR, MOE-UTS, and MOE-UCS for each species are depictedinFigs. 3, 4and 5.It should be noted that, statistically, there are disadvantages in assuming a nonlinearmodel rather than the linear one for the MOE-strength relationships when the linear model is29adequate. If the model is linear and all the necessary assumptions concerning regressionprocedures are met, then the least-squares estimators of the parameters in Eq. (9) are optimalsince they are minimum variance unbiased estimators. However, when the model is nonlinear,there are no such best estimators of the parameters, i.e., none of these properties arepossessed by the least-squares estimators (Myers 1990). Nevertheless, if the error terms eare normally distributed, then the least squares estimator is the maximum likelihood estimatorand, under these conditions, the estimators posses asymptotic properties, i.e., the sample sizemust be large to approach the unbiasness and minimum variance (Myers 1990, Seber and Wild1989).Based on the underlying objective on the development of the property relationships oflumber, however, the purpose of using regression analysis is merely as a tool for obtaining theempirical model relating any two properties. Therefore, the above limitations on thenonlinear model are beyond the scope of this study.In this section four related regression models were evaluated for representing therelationship between strength properties and MOE. The analysis showed that a nonlinearmodel is best suited for representing the results considered in this study. A single nonlinearmodel was adopted for relating structural properties of lumber. This model will degenerateto the power-type models used in the United Kingdom (UK) (Curry and Fewell 1977) whenthe intercept is zero, and yields the common linear regression model if the power term on theindependent variable is not significant.305.2. Strength Property RelationshipsAs mentioned before, for a given piece of lumber only one type of failure mode isavailable from destructive measurement, thus the relationship between two strength propertiescan only be established indirectly. Several attempts have been made to relate one type ofstrength property (e.g., MOR) to the other (e.g., UTS or UCS). Johnson and Galligan(1983) introduced the method for estimating the concomitance or cofunction of lumberstrength properties. With their method, the choice can be made either with or withoutconsidering the relationships of MOE and knot size to a particular strength property. Similarwork, but without the information of knot size, has been presented by Bartlett and Lwin(1984). Other methods of estimating the correlation or degree of concomitance betweenlumber strength properties without using nondestructive information such as MOE and knotsize have been developed by Green et al. (1984) and Evans et al. (1984). However, all ofthese methods require proof loading of the materials, i.e., testing every board in the samplepopulation up to a pre-set load in one failure mode followed by testing the survivors in thesecond mode. In this case, the correlation between two failure modes or strength propertiesdepends on the choice of the proof load or cut-off point using in the proof loading.Relationships between structural properties have been estimated using MOE-basedregression method and the so-called equal rank method. The MOE-based regression methodwas used by Curry and Fewell (1977) and Green and Kretschmann (1991) to develop relationsbetween UTS and MOR. As has been discussed in the review for the strength propertyrelationships, the equal-rank method was used by Barrett and Griffin (1989) and Green andKretschmann (1989). It is the simpler method which allows the relationship between two31strength properties to be established on the basis of their empirical cumulative distributionfunctions.The applications of these methods rely on different underlying fundamentalassumptions about the relationship between property distributions which cannot be verifiedexperimentally.To date there are no published studies comparing the property relationship resultsderived using these two different approaches. In this study the In-grade data base will beanalyzed to establish property relationships using both the MOE-based regression method andthe equal-rank assumption method.5.2.1. Equal-Rank MethodThe equal-rank method is an extension of standard analysis procedures involvingcomparison of mean or percentile property results obtained from different strength propertyevaluations.Suppose, the strength data are obtained for a particular product and the cumulativedistribution functions are constructed for property A and B as shown in Fig. 6a. Then for anyselected cumulative probability level (F) the property levels A and B can be estimated fromthe data or computed from a fitted cumulative distribution function. The property A can beplotted as a function ofB for a range of cumulative probability levels as shown schematicallyin Fig. 6b.32Alternatively, property ratio A”1B can be derived from data or the cumulativedistribution function and presented as a function of PA or as appropriate.The concept of comparing property values at selected cumulative probability levels (orselected rank in the case of ranked data set) is called the equal-rank (equal probability)method.The results obtained from an equal-rank analysis will vary depending on the specificmethods adopted in the study. If the cumulative distributions for property A and B areknown, then the property relationships can be derived directly from the cumulative distributionfunctions. The procedure can be illustrated using the Weibull cumulative distributionfunction.Weibull distribution has been widely used to represent the distribution of lumberstrength data (Evans et al. 1989). The cumulative distribution function (cdi’) for a 2- and 3-parameter Weibull distribution can be written as follows (Barrett 1974, Bodig and Jayne1982), respectively:F x1Fw(x;m,k) 1 — (14)Fw(x;xo,m,k) 1 —[ V(x—O] (15)where:X = The strength of a given piece,V = Volume of the given piece,= A constant depending on the type of loading and the shape parameter,k = Shape parameter,m = Scale parameter,33= Location parameter.Eqs. (14) and (15) represent the probability of failure of a member having the strengthproperty X. If X is the bending strength (MOR) then, Eq. (15) can be rewritten withsubscripts b to indicate bending strength parameters as follows:Fw(M0R; xob,mb,kb)= 1 — K[_wv[M00bb](16)Similarly, for tensile strength and compression strength the Weibull cdfs are given by,respectively:Fw(uTs;xot,mt,kt) 1— L_v[UT0tj (17)Fw(ucs;xoc,mc,kc)=1_[_V[5x0j (18)where i = 1 for tension and compression strength parallel to the grain which have uniformstress distributions (Barrett 1974).Provided that the member has the same size and probability of failure (F) then, therelationship between UTS and MOR can be derived from Eqs. (16) and (17) as follows:1UTS_XOrt = (M0R_xobmt ) L\ mb )1 (19)UTS = XOt+(k/) .V/kt (M0R — XOb) kmbThus, the relationship between UTS and MOR can be derived if the scale (m), shape (k),location (x0) parameters, and the constant p of the bending and tension strength distribution34are known. Likewise, the relationship between UCS and MOR can be derived from Eqs. (16)and (18).Examining Eq. (19), it is apparent that the relationship between strength properties canbe written in a simplified form similar to that for regression analysis presented earlier. Thesimplified forms for the 2- and 3-parameter Weibull distributions can be expressed as follows:Y= 13 (x)2 . (20)‘‘= 130+ (x-132) (21)where:Y = UTSorUCS,X =MOR,- 133 = Parameters.Relationships between property values have been derived using the equal-rankassumption and Weibull cumulative distribution functions to represent the propertydistributions. The derived relationships are similar in form to the nonlinear regression model(Eqs. (11) and (12)). In fact, the equal-rank assumption applied to a 2-parameter Weibull cdfleads to exactly the same form of property model as Eq. (11).The 3-parameter Weibull cdf leads to a nonlinear model very similar in form to Eq.(12). Thus, the equal rank concept has provided a basis for deriving a property relationshipmodel which is consistent with test data if the individual data sets are adequately representedby Weibull cumulative distribution functions. Since Weibull models are widely used torepresent strength data, the models given in Eqs. (20) and (21) were chosen for subsequentproperty relationship studies.35The application of Eqs. (20) and (21) in property relationship analysis can beillustrated by considering UTS and MOR relations. Because the ranking was carried out toeach grade and size before pooling of the adjusted grades and sizes, the direct approach formodel parameter as per Eqs. (20) and (21) cannot be obtained. Therefore, the models of Eqs.(20) and (21) were fitted to UTS-MOR data using regression techniques (see Fig. 7).For each species and property, the data for select structural and No.2 grades, 2x4, 2x8and 2x 10 sizes were analyzed as follows:1. Adjust the strength data to 15 % moisture content using Eq. (7),2. Rank the data in ascending order,3. Estimate the non-parametric strength values for the corresponding percentilelevels 0.02, 0.05, 0.1,..., 0.95, 0.98,4. Adjust all data to a nominal size 38 x 184 mm (1.5 x 7.25 inch) and length L =3128 mm using Eq. (8),5. Fit the regression model of Eqs. (20), (21) and (12) to the combined SS andNo.2 grade data.Eq. (12) was also fitted to the data even though it does not have the basic assumptionabout the property distributions as for Eqs. (20) and (21). Nevertheless, this model ispreferred, as an alternative model to Eq. (21), if the argument of zero value for the dependentvariable for given zero value for the independent variable is to be maintained.All equations are fitted using the Gauss-Newton iterative procedure in SAS packagefor nonlinear regression. The starting points for Eq. (21) were found by firstly fitting a 3-parameter Weibull distribution to the strength data (full adjusted data set) to estimate36parameters 13o and 132, and secondly by fitting the regression to the reduced model afterknowing 13o and 132.The Sum of Square Error of Eq. (21) was tested against that of Eq. (20) for thesignificance of parameters f3 and 132 using Likelihood Ratio (LR) test (Judge et al. 1985)below:LR=n{lns(b*)_lns(b)} (22)where:s(b*)= Sum of Square Error of Eq. (20),s(b) = Sum of Square Error of Eq. (21),n = Sample size.The null hypothesis that parameters f3 and 132 (3-parameter Weibull) have significant effect inreducing the error variance is rejected if the statistic LR exceeds X2(a ,v) for a pre-specifiedlevel of significant c. For c = 0.05 and the number of the restricted parameters v = 2, then052) = 5.991. If 2LR < 5.991, then Eq. (20) is adequate to represent the relationship.However, the test shows that LR exceeds 5.991 except for the relationship between UTS andUCS for all species as shown in the last column (LR) of Table 7 and Table 8.The results of the fitted regression of Eqs. (12), (20) and (21) are presented in Table 7and Table 8 for all species and property relationships.375.2.2. Band-Width MethodDue to the need for establishing property relationships for MSR lumber, Curry andTory (1976) introduced a method based on analysis of strength and MOE data which involvescalculating strength properties for MOE subgroups or bands. In MSR grading, for a givengrade, the assigned bending stress is calculated for those pieces with an MOE in a pre-selectedMOE range. According to Curry and Tory (1976), this bending stress depends on the widthof the grade, i.e., the difference between the boundary value of MOE for one grade and thenext higher grade, and also on the location of the grade within the full range of MOE valuesfor the species. Their main objective was to evaluate the effects of these two factors on theassignment of bending stress. Fig. 11 a is a sketch that illustrates their method. In the studyreported herein, this method is called band-width method.The main objective here is to find the relationship between MOE and the lowerpercentile of the strength properties corresponding to the 1-St and 5-th percentile strengthlevels. Permissible stresses for machine stress-graded timber are constructed by taking the 5% exclusion limit for MOR as shown by the hypothetical 5 % exclusion line in Fig. 1 la. Thisline is chosen to insure that 95 % of all the strength values will be above this line. However,according to Curry and Tory (1976), this linear lower exclusion line can lead to zero ornegative values of a strength property associated with a non-zero value of MOE and often noaccount is taken of the effect of the range of MOE values included in a particular grade on thecorresponding 5-th percentile of strength values. Following their method, this problem issolved by sub-dividing the strength data into bands of MOE values in order to determine the38lower exclusion levels of the strength properties. This procedure is similar to the derivationof design properties for MSR lumber.It can be seen from Fig. ha that by setting the boundaries of MOE at points A and B,then the strength and MOE data that fall inside this boundary or band will have a certainfrequency distributions as illustrated by the bell-shape curves on their margins. The MOEvalues at points A and B are known at the outset once a range has been chosen as therepresentation of a grade increment. This boundary acts as a “window” in which thedistribution of MOE and the strength data have to be determined as shown in Fig. 1 lb. Thiswindow can be moved in a certain step-length or increment while a distribution model can befitted to the strength and MOE data in order to determine the point estimates.The method described above involves the following steps:1. Select the width of the band (window) for MOE and the step-length (increment),2. Start moving the window from zero MOE one step-length at a time,3. For each increment in point 2, rank the strength data in ascending order and fitthe appropriate distribution to estimate the values corresponding to the 1 -st, 5-thand 50-th percentile levels. Fit the appropriate distribution to the MOE data thatfall in the same window to estimate the value corresponding to the 50-thpercentile level.Following the method by Curry and Tory (1976), the band-widths chosen in step 1 are500, 690, 1,000, 1,500, 2,000 and 2,500 MPa. The 690 MPa band-width was added tocorrespond to the grade-increment i05 psi used in North America. The step-length is 100MPa irrespective of the band-width. Because the estimated values in the lower tail of thestrength distribution, particularly towards the extremes of MOE where the number of data39points are scarce, could be influenced by the initial location of the band or window, the initiallocation was set to zero and was advanced by increments (step-length) of 100 MPa along theMOE axis in order to minimize this effect. The number of data points in step 3 is arbitrarilychosen for which the minimum 30 data points is believed to be sufficient for the estimation.In order to determine the relationship between MOE and the minimum value of thestrength, one has to define what MOE value inside the window should be related to the 1 -st,5-th and 50-th percentile values of the strength data for a given band-width. There is nowarranty of an exact value of MOE for these percentile values of the strength. Curry andTory (1976) related minimum MOE for a given band-width to each corresponding 1-st and 5-th percentile strength values similar to what occurs in bending stress assignment in MSRlumber. Their results show higher strength values for wider MOE band-width.By comparing band or window A-B with A-C in Fig. 1 la in terms of theircorresponding 50-th percentile values (denoted by m1 and m2), it is clear that wider MOEband-width will have higher estimated strength value for the same minimum value of MOE.The regression line in this figure shows that, for a given MOE, the expected value of thestrength will lie on it provided that this is the best fitted line. Also the horizontal and verticallines corresponding to the mean values of the entire MOE and strength data, respectively, willintersect each other exactly on the regression line. In analogy, the 50-th percentile values ofthe strength distribution also will lie close to the regression line regardless of the band-widthsince any band-width must have the same mean trend values for the same data set. Thepreliminary analysis using the minimum values of MOE as the matching pair for the 50-thpercentile strength value for the band showed that there were no close results for the meantrends (50-th percentile) of strength as a function of MOE across the band-widths. The wider40band-width produced higher 50-th percentile strength values for the same minimum MOEvalue. Therefore, there was a shifting effect due to the width of the band. Thus, in order tomaintain the same predicted mean trend (50-th percentile) of the strength distribution acrossthe band-widths, the 50-th percentile of MOE was chosen as the corresponding point for the50-th percentile of the strength distribution as illustrated in Fig. 1 lb.The strength data that fall inside the window were fitted using a 3-parameter Weibulldistribution to estimate the strength values at the selected percentile levels, i.e., 1 -st, 5-th and50-th percentile. Whereas, the Johnson’s Sb distribution was fitted to the MOE data in thesame window in order to estimate the 50-th percentile value of MOE.The MOE data for the given window is fitted using Johnson’s Sb distribution withknown lower and upper bounds. These lower and upper bounds are known as the boundariesfor the band or window. The Johnson’s Sb distribution is given by (Johnson 1949):f(x;“‘= (x - e)(2± e — x)]xp{_o.5(z)2 } (23)for which,z+ni,J (cxc+)where:z = Standard normal distribution,x =MOE,e = MOE minimum for a given band-width,A = The width of the band or window,‘y and i = Parameters.41(For y = in’ (i = 1,2,..., n) then, the Maximum Likelihood estimatesfor y and i are given, respectively, by:yjwhere: y = 11If the relative measure of the skewness and kurtosis fall in the Sb region (see Shapiro andGross (1981)) then the distribution is Johnson’s 5b distribution. The relative measure of theskewness (‘Ib i) and kurtosis (b2) of MOE inside the window are calculated, respectively, asfollows:nnI/n jx1—X),and b2=111 I _4/ jxj—x)(25)For known lower and upper bounds, the MOE at any percentile level a can be calculated by:I (z_—lXm1EKP j+xXa= (i+EXPI Zcz Y11where:Xm = MOE maximum,‘1=n1/ I _27 jy1—y),and i=i=11in11/ / _\211711 yj—y)‘d i=1(24)11_l)(X _)2Ji=1xiwhere: = 1=111,Kn_l)(xi_)2i=1(26)42= MOE minimum.In order to verify if the Johnson’s Sb distribution fits the MOE data, the goodness-of-fit test was performed. Although there are six band-widths, the test on a single band-width isassumed adequate for this purpose. The result of the goodness-of-fit test on the MOEdistribution for a 1,000 MPa band-width is depicted in Table 9. This table shows that therelative measures of skewness and kurtosis, b1 and b2, fall in the 5b region (see Shapiro andGroos 1981). Moreover, the Kolmogorov-Smirnov (K-S) test (Neave and Worthington1988) shows that the hypothesis of the distribution originating from Johnson’s 5b distributioncan not be rejected for all cases at c = 0.05 significant level as can be seen on the last threecolumns in Table 9 where the K-S critical value is less than the tabulated values. Fig. 12shows the histogram and probability density function of Johnson’s 5b distribution for MOEdata from the Douglas-fir MOE-MOR relationship. It is clear from this figure that Johnson’s5b distribution represents the MOE data better than a Normal distribution.Due the large data base, the 3-parameter Weibull distribution is employed without anycomparison with other distributions through goodness-of-fit analysis. However, the 3-parameter Weibull has been widely used to represent the strength distribution of dimensionlumber (Pellicane and Bodig 1981, Taylor and Bender 1988, Heatwole et al. 1991). Curryand Tory (1976) reported that 3-parameter Weibull gave the best fit compared to normal,lognormal, and 2-parameter Weibull distributions for the strength data inside the boundary orband. Pellicane (1985) conducted the study on the goodness-of-fit of normal, lognormal, 3-parameter Weibull, and Johnson’s 5b distributions on dimension lumber data and concludedthat Johnson’s 5b distribution provided the best fit. However, he also reported that at thelower 5 % level no distribution seemed to be substantially superior to the others.43If the distribution of the sample is assumed from the 3-parameter Weibull distribution,then its cumulative distribution is given by (Bury 1975):Fw(x;xo,m,k) 1— (27)and the property value at any percentile level cx can be calculated by:x= xo+m{_ln(1_F)1’1C} (28)where:= Location parameter,k Shape parameter,m = Scale parameter.As for MOE data, the result of the goodness-of-fit test of Weibull parameter on thestrength data for a 1,000 MPa band-width are depicted in Table 9 for all species. TheKolmogorov-Smirnov test shows the hypothesis that the sample is from the 3-parameterWeibull distribution can not be rejected at the level of significant cx = 0.05 for all cases asshown in this table. The cumulative distribution function of the 3-parameter Weibulldistribution for the Douglas-fir MOR is depicted in Fig. 13.Finally, the 1-St , 5-th and 50-th percentile point estimates of the strength are thenplotted against the corresponding 50-th percentile MOE as shown in Figs. 14, 15 and 16 forthe MOE band-width of 1,000 MPa. Following Curry and Tory (1976), the minimum valuesor the lower tail of the strength distribution for the given band-width was fitted using theregression method to model the relationship between this minimum strength level and MOE.It should be noted that the data relating MOE to strength are correlated in cases whereoverlapping band-width occur. However, the overlapping bands provide more data which44smooth out the trend and provide a better regression result for which the bigger sample size isrequired. Therefore, the sampling effects should be ignored.The plot of the strength versus MOE shows nonlinear trend. The relationshipsbetween MOE and the lower tail of the strength distribution then were represented using Eqs.(10) and (11). A preliminary analysis showed that Eq. (10) resulted in significant SSEreduction compared to that given by Eq. (11) for every case. Therefore only the results forEq. (10) are presented. The fitted model of Eq. (10) (1-st and 5-th percentile) for a 1,000MPa band-width for each species and property relationship is depicted in Figs. 3, 4 and 5 sothat it can be compared with the fill data set. The results of the regression analyses for allband-widths are presented in Tables 10, 11, and 12 for all species, property relationships andpercentile levels.It has been shown that the 50-th percentile point of MOE is the corresponding pointfor the 50-th percentile of the strength for each window position. For consistency, the 1-stand 5-th percentile points of the strength were related to the 50-th percentile of MOE. Sincethe 50-th percentile values of MOE rather than the minimum value for the band is used, theprocedure is different from the one introduced by Curry and Tory (1976). Nevertheless, theminimum value of MOE for the grade can be used to calculate the assigned strength valuesonce the MOE-strength relationship has been obtained. The purpose of using the 50-thpercentile MOE value for the band is to eliminate the shifting effects which would beintroduced by changing the MOE reference point. The band-width selected may, however,affect the lower exclusion limits (1-st and 5-th percentile) of strength values since a widerMOE band will include a wider range of strength values.45Since each band-width has been given the same weight, such as using the 50-thpercentile value of MOE for the band, the relationship between MOE and the 50-th percentilestrength should be the same for all band-widths. This can be seen from Figs. 17, 18, and 19where the predicted lines for the 50-th percentile generally overlap each other. The fittedmodels for the general relationships between strength and MOE (abbreviated GEN.REL) aredepicted in these figures as well. It can be seen that, for each species, this generalrelationship agrees very well with the fitted models for the 50-th percentile strength for allband-widths. Because the fitted model for the 50-th percentile strength from each bandwidth gives similar results, it shows that the expected mean trend (50-th percentile) for thestrength value is less affected by the band-width by changing the MOE reference points. Theexpected values for the lower 1-St and 5-th percentiles strength levels are affected by bandwidth. This is because wider band-width will give wider strength distribution for the samevalue of the 50-th percentile MOE. The effect of band-width, called grade increment factor,is depicted in Figs. 20, 21 and 22. This is a factor by which the minimum strength value,determined on the basis of a 1,000 MPa band-width, should be multiplied if the actual bandwidth is different from this value.The band-width method can be used to derive the relationship between strength andMOE. The strength property relationships can be derived at a selected MOE level. Theresults depend on the model chosen for the MOE-strength relationships. The relationshipbetween UTS and MOR, for instance, can be derived analytically if the MOE-strength modelsrelating MOE to UTS and MOR, respectively, both have the forms that allow MOE to beeliminated in relating MOR to UTS. Models with this property have been selected. Other46models, for example Eq. (12), are not suitable because they do not allow MOE to beeliminated to derive a strength property relationship.Another drawback from band-width method is that the narrow band-width would notcover the extremes (the lower and upper ranges) of the entire MOE data for the species withsufficient data points. In other words, the number of data points in the extremes are not largeenough to guarantee a good estimation and as a consequence, the extremes will be excluded inthe moving band-width process. The larger the width of the band the more likely to cover theextremes of MOE data for the species.476. RELATIONSHIPS BETWEEN MOE AND STRENGTHPROPERTIESThe regression analysis shows that the general relationship between MOE and MOR isdefined adequately by the simple linear model of Eq. (9). However, the relationships betweenMOE and UTS or between MOE and UCS are nonlinear and are best defined by, the nonlinearmodel of Eqs. (11) and (10), respectively.For modelling the strength property relationship based on the MOE-strengthrelationships, the proper MOE-strength models should be chosen. By using the linear modelfor the MOE-strength relationships to formulate UTS/IVIOR ratio as a function of MOR,Green and Kretschmann (1991) found that UTS/MOR ratios did not agree with the resultsfrom the equal-rank analysis at the lower MOR levels. Thus, the following discussions will befocused on the establishment of the MOE-strength relationships in order to develop thestrength property relationships.6.1. Relationships between MOE and MORThe general relationship between MOE and MOR is defined, commonly, by the simplelinear model of Eq. (9). The coefficients of determination, r2, obtained for the In-grade dataare 0.58, 0.47 and 0.60 for Douglas-fir, Hem-Fir and S-P-F, respectively. These values fall inthe range reported by Hoyle (1968) and they are a slightly larger than the values reported byGreen and Kretschmann (1991) for the US In-grade data set. However, the nonlinear model48Eq. (10) gives the lowest Sum of Square Error. In order to be consistent for the subsequentanalysis on the strength property relationships, the model of Eq. (10) was chosen to representthe general relationship between MOE and MOR. The resulting regression model is showngraphically in Fig. 3 for each species. All the graphs are presented on the same scale so thatthe comparisons among the species can easily be made.For the band-width analysis, Eq. (10) is found to be a better model for therelationships between MOE and MOR at the lower 1 % and 5 % exclusion levels. The modelparameters for all band-widths and species are presented in Table 10. The predicted lines, fora 1,000 MPa band-width, are depicted in Fig. 3 for every species. It can be seen from thegraph for each species that the predicted lines both for the 1-St and 5-th percentile generallyfollow the lower trend (margin) of the MOR scatter.At the lower MOE levels, it seems that the 5-th percentile relation from band-widthanalysis is not conservative compared to the 5 % exclusion line from simple linear model.This can be seen from the result for Douglas-fir in Fig. 23. In this figure, the nonlinear 5-thpercentile trend from the band-width analysis gives higher MOR values at the lower MOElevels. However, the 5-th percentile trend from band-width analysis gives lower MOR valuesat higher MOE levels compared to that of linear 5 % exclusion limit. The 5-th percentiletrend from band-width analysis reflects the pattern of the changes of variance for each level ofMOE. This figure shows that at MOE below about 7,000 MPa, 5 % exclusion line fromlinear model has negative values. This is one of the reasons why Curry and Tory (1976)introduced the band-width method. They found the power model of Eq. (11) to be a bettermodel for relating minimum values of MOR to MOE. Unlike their model, the model of Eq.49(10) can result in negative values of MOR, however Fig. 24 shows that the 5-th percentile linefrom the nonlinear model still gives positive values for MOR even for low MOE values.The general relationships or mean trends for MOE-MOR vary little by species (Fig.24). This trend is similar to that found in the US In-grade data (Green and Kretschmann1991) which was intended to test if the single regression line can represent all species. Forthis, the proposed Eq. (10) was fitted to the pooled data for all species (Ratkowsky 1983).The F-test in Table 13 shows that the hypothesis of single regression line for all species wasrejected at the confidence level a = 0.05. Fig. 24 shows that there are minor differences inthe 5-th percentile level of MOR regression from band-width analysis.61. Relationships between MOE and UTSThe regression analysis shows that the general relationship between MOE and UTS forHem-Fir and S-P-F is in a nonlinear form. Statistically, Eq. (11) is adequate for representingthe general relationship between MOE and UTS. However, Eq. (10) is chosen to representthis relationship in order to be consistent with the subsequent analysis on the strength propertyrelationships. Furthermore, it gives the lowest SSE. The general relationships betweenMOE and UTS are shown graphically in Fig. 4 for each species.The general relationship between MOE and UTS is species dependent as shown byFig. 25. For the same MOE, S-P-F has the highest value of UTS especially at higher MOElevels. The test for a single regression line for all species in Table 13 showed no evidence for50rejecting different regression lines for different species. In other words, the regression lineswere not similar.As for the MOE-MOR relationship, the band-width analysis also shows that Eq. (10) isa better model for MOE-UTS relationship at the lower 1 % and 5 % exclusion levels. Themodel parameters are presented in Table 11 for all band-widths and species. The models, fora 1,000 MPa band-width, are depicted in Fig. 4. It can be seen from the figure for eachspecies that the predicted lines for the lower 1 % and 5 % exclusion levels successfully followthe lower margin of the UTS scatter.The MOE-UTS relationships for the 5 % lower exclusion level also reveal differencesby species particularly at high MOE levels as shown by Fig. 25.6.3. Relationships between MOE and UCSThe regression analysis shows that MOE-UCS has a strong nonlinear relationship.The statistical test on the parameters and SSE showed that Eq. (10) is a better model forrelating MOE and UCS. This model is shown graphically in Fig. 5 for each species.The band-width analysis also showed that Eq. (10) is the best model for relating MOEand UCS at the lower 1 % and 5 % exclusion levels. The results of the regression analysisare presented in Table 12 for all band-widths and species. Again, the equations, for a 1,000MPa band-width, are depicted in Fig. 5 for each species. The figures reveal that the fittedlines follow the lower boundary of the UCS scatter for each species.51The mean trend in MOE-UCS relationships are similar for Douglas-fir and Hem-Fir asshown in Fig. 26. S-P-F shows lower UCS value for the same given MOE compared toDouglas-fir and Hem-Fir. Again, the test for a single regression line for all species in Table13 showed that the hypothesis of single regression line for all species was rejected atconfidence level c = 0.05.For the lower 5 % exclusion level, all species show little variation in the higher MOElevels.527. RELATIONSHIPS BETWEEN STRENGTH PROPERTIESThe relationship between two strength properties can be described in the form of theratio of one property to the other. Since MSR predicts bending strength, UTS and UCS areassigned as a proportion or ratio to MOR. Based on the failure mode of lumber in bending, itwill be shown that it is reasonable to assign MOR as a proportion of UTS.Using the results from equal-rank method, MOE-strength based general relationshipsand band-width method, the following analysis will be focused on the development andevaluation of the property ratio relationships based on the Canadian In-grade data. The resultfrom equal-rank method will be discussed first followed by MOE-strength general relationshipand band-width method so that the evaluation and comparison among these methods can bemade.7.1. Strength Property Ratios Based on MORThere is no fundamental reason, based on the mechanical behaviour of lumber, toexpect that the ultimate tensile strength and compression strength parallel to the grain dependon MOR. Since the invention of mechanical grading (MSR), it was found that there is a highcorrelation between fiexural strength (MOR) and the fiexural stiffness (MOE). However,experimental studies have shown that strength properties generally increase as MOE increases.Therefore, for design and grading purposes, tensile strength and compression strength parallelto the grain are often expressed as a function of MOR. The assignments of UTS and UCS53are expressed as the ratio or percentage of MOR and MOR can be predicted from flexuralstiffness (MOE).7.1.1. Property Ratio Models from Equal-Rank AnalysisThe equal-rank method allows the strength property relationships to be formulated byassuming a 3-parameter Weibull distribution for each property which yields an expression inthe form of Eq. (21). The property ratio based on MOR is constructed by dividing the modelin Table 7 by MOR. For example, the ratio of UTS to MOR as a function of MOR forDouglas-fir is as follows:6.6678 (M0R —6.02 15)1.3274RT/B = + 0.1373 (29)MOR MORFig. 7 contains the plots of this equation and the similar results for Hem-Fir and S-P-F withthe scatter plots of data points.The clear picture of the property ratio as the function of MOR across the species canbe seen in Fig. 27. It is clear from this figure that UTS/MOR ratio is species dependentespecially for higher bending strength. Below 48.3 MPa, the ratio of UTS to MOR, for eachspecies, agrees well with the average value of 0.56 for all grades, species and sizes asproposed by Green and Kretschmann (1991), and is slightly lower than the 0.60 as proposedby Curry and Fewell (1977). It is clear that the constant factor 0.45 proposed in ASTM D1990 (ASTM 1991) is conservative for these species.54The ratio of UCS to MOR is also depicted in Fig. 27 so that the comparison betweenUTS and UCS as a function of MOR can be made. Like the UTS/MOR ratio, the UCS/MORratio is also constructed by dividing the model results in Table 7 by MOR. The property ratiofor Douglas-fir is given by:15.5812 (MOR _6.0215)0.926Rc/B = + 0.63 17 (30)MOR MORThe plots of this equation and similar results for Hem-Fir and S-P-F are depicted in Fig. 8.Fig. 27 also contains the models for UCS/MOR ratios reported by Curry and Fewell(1977), Green and Kretschmann (1991) and proposed by ASTM D 1990 (ASTM 1991). TheUCS/MOR ratio for S-P-F is very close to that reported by Curry and Fewell (1977) andGreen and Kretschmann (1991). It is clear from Fig. 27 that the proposed model fromASTM D 1990 (ASTM 1991) is more conservative than that of S-P-F whose UTS/MOR ratiois the lowest among that of the three species groups.The differences in the property relationships are inevitable since there are somedifferences in the property adjustments, size, test span and the method of testing used by thevarious authors (see Curry and Fewell 1977, Green and Kretschmann 1991 and ASTM D1990-ASTM 1991). It should be noted that the proposed property ratios in ASTM D 1990(ASTM 1991) were accumulated from the North American (US and Canada) In-grade database, whereas, Green and Kretschmann (1991) used only the US In-grade data and CanadianS-P-F data to derive the property ratio models.The strength property relationships based on MOR show that the relationship betweenUTS and MOR is not as good as that found between UCS and MOR. This finding was alsoreported by Green and Kretschmann (1991). For each species, UTS-MOR relationshipshows higher variance than that shown by UCS-MOR relationship as can be seen in Table 7.55The mean trend for the ratio of UCS to MOR for clear wood is included in Fig. 27.This trend is formulated by fitting Eq. (10) to the mean values of UCS and MOR data forspecies included in Douglas-fir, Hem-Fir and S-P-F groups (Jessome 1977). The UCS andMOR data were not adjusted for size effects, therefore the absolute values of the propertyratio is not directly comparable with the structural lumber results. The clear wood samplesare defect-free wood, therefore, one would expect the stronger commercial lumber to have theproperty ratio trends close to that of clear wood. Fig. 27 shows that, for UCS/MOR ratios,all species have the tendency to reach a lower limit as predicted by clear wood property data.7.1.2. Property Ratio Models from General RelationshipsStrength-MOE equations can be used to formulate the relationship between twostrength properties. For instance, if Eq. (10) is employed, the general relationship betweenUTS and MOR can be formulated as follows:(UTS-13 2tUTS= f3 +I3it (MOE)2t, thus MOE= OtJMOR=ob+lb(MOE)2b , thus MOE=[M00b2bsolving for UTS gives:(UTS—(MOR— ObI3it ) I1b JUTS=ot )(tI2b)}MoR_Ob)2t256or more generally,UTS = (31)In other words, for a piece of lumber with a certain MOE, the mean trend relationshipbetween bending strength and tensile strength is expected to follow a trend given by Eq. (31).It is clear that the parameters for this equation are obtained directly from MOE-UTS andMOE-MOR equations and are presented in Table 14 for each species. The property ratio isderived by dividing this equation by MOR. The plots of these property ratios (abbreviatedGEN.REL.) are illustrated in Fig. 28 for every species in order to compare them with theresults from the band-width analysis.7.1.3. Property Ratio Models from Band-Width AnalysisFollowing the same procedure as for the property ratios from general relationship (Eq.(31)) the results from band-width analysis can be used to formulate the relation between UTSand MOR. The model parameters for all band-widths and percentile levels are presented inTable 15 for all species. Fig. 28 contains the scatter plots for the 5-th and 50-th percentilestrength levels for a 1,000 MPa band-width. The scatter plot of UTS/MOR as a function ofMOR, for example, is constructed by plotting UTS/MOR ratio against MOR for the samepercentile level inside the same band-width.The predicted models derived from the MOE-strength general relationship and fromthe result of equal-rank analysis (abbreviated GEN,REL and EQRA, respectively) are depictedin Fig. 28. It is clear from this figure that the property ratios derived using these two57different methods are similar and they agree very well with the scatter of the median (50-thpercentile) from band-width analysis. At low MOR levels the relationships based on equal-rank analysis yield property ratios close to the 5-th percentile data derived from the bandwidth analysis. Therefore, it shows that the equal-rank method gives results which areconsistent with the predictions of the band-width analysis at low MOR levels.There are slight differences between GEN.REL and EQRA in Fig. 28, especially at thelower and higher levels of MOR data. These deviations at the extremes can be explained bylooking at how each model was constructed. The property ratio resulted from MOE-strengthgeneral relationship is formulated by solving two regression equations i.e., two MOE-strengthequations. In this case, the fitted regression line depends on the range of MOE, that is, thedistance between the minimum and maximum. The number of data points are small in theextremes. On the other hand, the equal-rank procedure is independent of MOE, namely, itdepends on the range of strength data, such as, the distance between the minimum andmaximum. Then, by examining the scatter plot of MOE versus strength (Figs. 3,4 and 5), it isclear that the lower strength values are not always in the lower MOE range and, likewise, thehigher strength values are not always in the higher MOE range. MOE range is the horizontaldistance whereas the strength range is the vertical distance. Therefore, if the increase of thestrength values does not follow the increase of MOE, then there will be differences in theextremes. That is because, if the MOE-strength relationship has a perfect relationship, whichmeans that the plot of strength versus MOE is just a straight line, the two methods willprovide the same results.In Fig. 28, the property ratio resulted from MOE-strength based general relationshipsgives close results to the 50-th percentile (median) data from band-width analysis. There are58slight differences in the extremes most likely due to cut-off points in the extremes for bandwidth analysis and the distribution of strength values along the regression line for the MOE-strength general relationships.Design stresses for lumber are obtained from estimates of minimum values for eachgrade, traditionally at the 5 % lower exclusion level. Visually, the difference between thislower exclusion level and the mean trend on the property ratios is small if presented on thestrength basis especially for UCS/MOR ratios as illustrated in Fig. 28 for each species.In Fig. 28, for band-width method, it is obvious that UTS/IVIOR and UCS/MOR ratiodata points at 5 % lower exclusion limit (5-th percentile) do not go up to the higher MORlevels as that at 50-th percentile. Hence, the fitted lines for the property ratios, in this case at5-th percentile, were limited to the maximum values permitted by the data. For example, inthe case of Douglas-fir, the maximum data values of MOR at 5-th percentile is about 50 MPa.The difference between the 5 % lower exclusion level and the mean trend (50-thpercentile) for the property ratio is significant if presented on the basis of MOE as showngraphically in Fig. 29 for each species. Each data point for the scatter plot of UTS/MOR as afunction of MOE, for example, is constructed by plotting the ratio of UTS to MOR againstthe 50-th percentile MOE data point inside the same band or window. This MOE data pointcan be chosen from either the MOE-MOR or the MOE-UTS relationship. Either one of theseMOE data points can be used only if the value from the MOE-MOR is not different from theMOE-UTS for the same band or window. Fig. 30 was generated to check if there is anysignificant difference between these two MOE values for the same band or window. Eachgraph in this figure shows that there is a perfect correlation between these two 50-thpercentile values of MOE.59In Fig. 29, each predicted line for the 5 % lower exclusion level is just the ratiobetween the predicted UTS and the predicted MOR both as the function of MOE as shown bythe following equation:UTS ot+it(MOE)2tMOR I3ObI31b (MOE)2bwhere the subscripts t and b represent UTS and MOR, respectively. This equation can becalculated when MOE is given.The same predicted line (abbreviated GEN.REL) but from the general relationshipbetween strength and MOE is also plotted in Fig. 29. This line agrees very well with thescatter of the 50-th percentile data points. This agreement indicates that the 50-th percentilestrength values fall around the regression line in the band-width analysis discussed in Fig. 11 a.In Fig. 29, the UCS/MOR ratios as a function of MOE shows a wider differencebetween the 5-th and 50-th percentiles than that shown by the UTS/MOR ratios especially forDouglas-fir and Hem-Fir.7.2. Property Ratios Based on UTSThere are at least two common failure modes found in a bending member. Themember compression zone may exhibit compression failure whereas the tension zone alwaysexhibits tension failure parallel to the grain at ultimate bending capacity. In the directionparallel to the grain, defect-free wood, tested in compression will exhibit a linear stress-strainrelationship up to the proportional limit after which yielding will take place. Whereas, in60tension, the stress-strain relationships is almost linear up to failure. The test results show thatclear wood, in the direction parallel to the grain, is much stronger in tension than incompression (Maholtra and Bazan 1980, Anderson 1981). According to Schniewind (1962),tensile strength is approximately two to three times as great as compression strength.Lumber most often contain some defects such as knots, depending on the assignedgrade. Unlike defect-free or clear wood, low strength beams will exhibit tension failurebefore the beam reaches its proportional limit stress in compression zone leaving the linearstress-strain distribution (Ramos 1961). Higher grade beam, mostly defect-free, will reach itsproportional limit stress in compression zone. Further loading will cause yielding or bucklingin this zone and the neutral axis will shift toward the tension zone resulting in higher stress inthat zone. By increasing the load, the beam will fail in tension at its maximum load capacity(Maholtra and Bazan 1980).It is clear now that the ultimate bending strength (MOR) is governed by the strength ofthe tension zone of the beam. In other words, one would expect the tension failure even forhigher grade material. This behaviour of bending member provides the foundation forexpressing bending strength as a function of tensile strength.Unlike bending strength and tensile strength, compression strength and tensile strengthare fundamentally different properties. However, since bending strength is related to tensilestrength, it seems appropriate to express compression strength as a function of tensilestrength.617.2.1. Property Ratio Models from Equal-Rank AnalysisThe model parameters for the property relationships based on UTS resulting from theequal-rank analysis are presented in Table 8. Following the approach for deriving theproperty ratios based on MOR, the property ratios based on UTS are formulated by dividingthe models in Table 8 by UTS. For instance, the ratio of MOR to UTS for Douglas-fir is asfollows:6.0215 (uTs _6.6180)0.7299RB,T + 4.7865 (33)UTS UTSThis equation and the similar results for Hem-Fir and S-P-F are depicted in Fig. 9. Thescatter plots follow the pattern shown in the report by Green and Kretschmann (1991).The ratio of UCS to UTS based on UTS, for example, for Douglas-fir is as follows:14.5265 (uTs — 5.7852)0.703RrIT = + 2.5256 (34)UTS UTSThis equation and similar results for Hem-Fir and S-P-F are illustrated in Fig. 10.The property ratios based on UTS across the species are illustrated in Fig. 31. Theproposed model by Green and Kretschmann (1991) and ASTM D 1990 (ASTM 1991) forUCS/UTS ratio are depicted as well. The UCS/UTS ratio from S-P-F is very close to thatreported by Green and Kretschmann (1991). It is clear from this figure that the UCS/UTSratio suggested by ASTM D 1990 (ASTM 1991) is lower than that given by each species.The constant factor 1.2 for MOR/UTS ratio suggested by ASTM D 1990 (ASTM 1991) ismore conservative than that given by each species especially in the middle range of UTSvalues.62Fig. 31 shows that the MOR/UTS ratios are species dependent, as well as theUCS/UTS ratios. UCS/UTS ratios show similar pattern across the species. For the samegiven UTS value, the UCS/UTS ratio from one species to the other almost can be factored bya constant.It is expected to have a good relationship between MOR and UTS as a function ofUTS, however, the relationship between MOR and UTS is not as good as that found betweenUCS and UTS. This finding was also reported by Green and Kretschmann (1991) for the U.SIn-grade data. For every species, MOR-UTS relationship shows higher variance than thatshown by UCS-UTS relationship as can be seen in Table 8.7.2.2 Property Ratio Models from General RelationshipsStrength property relationships based on UTS resulted from solving their generalrelationships with MOE can be constructed in the same way as that shown for the propertyrelationships based on MOR. It can be shown as per Eq. 31, that MOR as a function of UTS,for example, is as follows:MOR= (35)The model parameters for this type of equation are presented in Table 14 for all species.Again, this means that given a piece of lumber with certain MOE value, the mean trendrelationship between bending and tensile strength, the former as a function of the latter, isgiven by this equation. The property ratio is taken by dividing the model in Table 14 by UTS.63This property ratio (abbreviated GEN.REL.) for each species is depicted in Fig. 32 for thecomparison with that of band-width analysis.7.2.3. Property Ratio Models from Band-Width AnalysisThe results from band-width analysis for the property relationships based on UTS arepresented in Table 16 for all band-widths, percentile levels and species. The scatter plots forthe 5-th and 50-th percentile values for a 1,000 IVIPa band-width are illustrated in Fig. 32 foreach species. The scatter plots are constructed the same way as that shown for the propertyrelationships based on MOR.The predicted models derived from MOE-strength general relationship and from equal-rank analysis (denoted by GEN.REL and EQRA, respectively) are depicted in Fig. 32 as well.Since they both agree with the scatter of the 50-th percentile data points, again it is proventhat the equal-rank procedure gives about the mean trend for the property relationships.Similar to the property ratios based on MOR in Fig. 28, Fig. 32 shows that at low UTS levels,the relationships based on equal-rank analysis yield property ratios close to the 5-th percentiledata derived from band-width analysis.Fig. 32 shows that UCS/UTS ratios give very consistent results compared to that ofMOR/UTS ratios because the UCS/UTS ratios have less scatter than that of MOR/UTSratios.As for the property ratios based on MOR, in Fig. 32, the fitted lines for the 5-thpercentile strength level were limited to the maximum values permitted by the data.64The property ratios as a function of MOE are depicted in Fig. 33. It can be seen thatthe effect of taking the lower exclusion level (5-th percentile) is very significant; i.e., there is alarge difference in property values between 5-th and 50-th percentile levels. The scatter plotsof MOR/UTS as a function of MOE, for example, is constructed by plotting the MOR/UTSratio against the 50-th percentile value of MOE (from MOE-UTS relationship) inside the sameband or window. The predicted line for the 5-th percentile level is the ratio between thepredicted MOR and the predicted UTS both as the function of MOE as shown for Eq. (32),but in reverse order.The property ratios presented on the basis of MOE also indicate that the median (50-thpercentile) values of the strength fall around the regression line postulated in the band-widthmethod (see Fig. 1 la) because the plot of the property ratios from the general relationshipbetween strength and MOE agrees with the scatter of the 50-th percentile data points.658. APPLICATION OF THE PROPERTY RELATIONSHIPS8.1. MOE-Strength Property RelationshipsModulus of elasticity can be measured directly, therefore having known the MOE, onecan predict the desired strength values using the empirical models developed in this study.Since the design value for structural application of lumber is traditionally based on the lower 5% exclusion limit, the analysis has been emphasized on finding this limit by means of the bandwidth method.It has been shown that the equation for the 5 % lower exclusion level from band-widthmethod is a better model for estimating the 5 % lower exclusion limit than the traditional 5 %lower exclusion line from the linear model which is adequate but conservative (see Fig. 23).The nonlinear model is more realistic for the lower exclusion level. However, the model forthe 5 % exclusion level from band-width analysis depends on the boundary or the width of theband used in the analysis. Nevertheless, the results show that the width of the band has alittle effect on the estimation of this 5 % lower exclusion value. The band-width effects(called grade increment factors) determined on the basis of a 1,000 MPa band-width are lessthan 8 % for all band-widths.Because the width of the band is unknown at the outset, a conservative approachwould be to use 1,000 MPa as the standard. It is wide enough to cover the number of datapoints of the strength data, even on the extremes of MOE range, for estimating the lower 5-thpercentile point estimates. It can be seen from Figs. 14, 15 and 16 by comparing to the full66data range in Figs. 3, 4 and 5 that the 1,000 MPa band-width successfully covers the extremesof MOE values for the species.The relationships between MOE and the predicted MOR, UTS and UCS for eachspecies group can be seen from Fig. 34. For the same given MOE, generally each speciesshows that MOR is the highest one follows by UCS and lastly UTS, however, the differencesare not in the same degree. Therefore, the property ratios vary with MOE as shown by thefigures for the property ratios as a function of MOE (see Fig. 35).It has been mentioned before that there is a strong relationship between flexuralstrength (MOR) and flexural stifThess (MOE) which is used in MSR grading system. As ananalogy, tensile strength has to be related to its tensile modulus of elasticity and likewisecompressive strength has to be related to its modulus of elasticity parallel to the grain.However, Doyle and Markwardt (1966) reported that MOE in compression is about equal tothe flexural MOE. GOtz et al. (1989) stated that those three different elastic properties arepractically equal below the proportional limit. If the flexural MOE alone is believed to be areliable indicator of strength properties, then by measuring MOE one can calculate any desiredstrength property for the property assignments using the proposed models in this study.8.2. Strength Property RelationshipsIf MOR is available, like in MSR practice where MOR is predicted from measuredMOE, the models from equal-rank analysis can be used to calculate the assignments of tensilestrength and compressive strength. In this case, these two strength properties are based on67the assumption that each of them has the same probability of failure (equal rank) with MORbut are independent of the nondestructive parameter MOE. Similar results will be obtainedby using the models derived from the general relationships between MOE and strengthproperties.Because the models from the equal-rank method and MOE-strength based generalrelationship provide the mean trend or average values for the predicted strength values, themodel for the 5 % lower exclusion level from the result of band-width method should be usedfor lumber selected on the basis of MOE. The comparisons between 5 % lower exclusionlevel (5-th percentile) and the mean trend for the property ratios based on MOR are depictedin Fig. 35. From this figure, it can be seen that Douglas-fir and Hem-Fir show higher valuesof property ratios for 5 % lower exclusion level for higher MOR levels and vice versa forlower MOR levels. However, S-P-F shows lower values of UTS/MOR ratios for 5 % lowerexclusion level for all levels of MOR and about equal for UCS/MOR ratios for MOR 20MPa. The maximum MOR values in Fig. 35 are set to that of maximum 5-th percentilevalues derived in the band-width analysis (see Fig. 14).The difference, called percentile-level effect, between using the mean trend and 5 %lower exclusion level, as the percentage of the values of the 5 % lower exclusion level isshown in Fig. 36 for every species. Depending on the species and strength levels, thedifference can be up to 30 % higher or lower if one would use the model from the MOEstrength based general relationship or the equal-rank analysis result rather than the 5-thpercentile model from band-width analysis.As proposed in ASTM D 1990 (ASTM 1991), UTS can be used for the estimates ofuntested properties. Moreover, according to Green and Kretschmann (1991), there has been68an interest in determining the UTS for MSR lumber in quality control programs. If theproperty relationships are based on UTS, then the models for the property relationships basedon UTS proposed in this study can be implemented.The comparisons between 5 % lower exclusion level (5-th percentile) and the meantrend for the property ratios based on UTS are depicted in Fig. 37. The MOR/UTS ratios inthis figure basically are the inverse of UTS/MOR ratios in Fig. 35. In Fig. 37, the UCS/UTSratios show a trend of decreasing function with the increase in UTS similar to UCS/MORratios in Fig. 35. For the 5 % lower exclusion level in Fig. 37, Douglas-fir and S-P-F showhigher ratios for higher levels of UTS, whereas Hem-Fir shows higher ratios at the lowerlevels of UTS and slightly lower ratios for higher levels of UTS. The maximum UTS valuesin Fig. 37 are set to that of maximum 5-th percentile strength values derived in the band-widthanalysis (see Fig. 17).The percentile-level effects are shown in Fig. 38. Depending on the species andstrength values, the difference can be up to 25 % higher or lower if one would use the modelfrom the general relationship or the equal-rank analysis result rather than the 5-th percentilemodel from band-width analysis.The empirical models for the stiffness-strength property relationships and for thestrength property relationships are summarized in Table 17 for the estimation of untestedproperties.Table 17 shows that there are three equations expressing strength as a function ofMOE, i.e., Strength =f(MOE), two equations expressing strength as a function of MOR, i.e.,UTS =f(MOR) and UCS =f(MOR), and two equations expressing strength as a function ofUTS, i.e., MOR =f(UTS) and UCS =f(UTS).69Because MOR can be expressed as a function of UTS and vice versa, it should benoted that their relationships resulted from equal-rank analysis were from different fittedregressions. For Eq. 21, however, one can simply works out to find the inverse function ofthe given equation, therefore one fitted regression is adequate. However, Fig. 39 shows thatthe effects (called error) of using the result of two different fitted regressions are less than 5 %(as the percentage of MOR) for MOR 20 MPa if one would use the inverse function ratherthan the fitted regression. For the models resulting from the MOE-strength generalrelationships and the band-width analysis, the error is zero since MOR as a function of UTS isexactly the inverse function for UTS as the function of MOR.709. CONCLUSIONSThe relationships between modulus of elasticity and the strength properties have beenmodeled. The traditional linear relationship has been found to still be a good model for themean trend or general relationship between MOE and MOR. It also shows that the generalrelationships between MOE and MOR vary little by species especially for Hem-Fir and S-P-Fbut there is no statistical justification for using a single regression equation for all species.However, the relationship between MOE and the minimum values of MOR (5 % lowerexclusion level) determined using band-width method shows slightly different values fordifferent species. The differences depend on the variability of MOR for the species.The general relationships between MOE and UTS, as well as, UCS are found to bewell represented by the nonlinear models. The relationship between MOE and UTS showssignificant species effects especially for higher MOE values. S-P-F shows the highest tensilestrength values follows by Hem-Fir among the three species groups. Unlike the relationshipsbetween MOE and UTS, the relationships between MOE and UCS, especially for the lower 5% exclusion level, show significant species effect at the lower MOE levels.The modified band-width method, such as, by changing the reference point of MOE tothe 50-th percentile values determined using Johnson’s Sb distribution for the band, in thisstudy shows that the effects of the width of the band on the estimation of the 5 % lowerexclusion limit is insignificant. For all band-widths, from 500 MPa to 2,500 MPa, thedifferences are less than 8 % determined on the basis of a 1,000 MPa band-width.Strength property relationships formulated on the basis of the relationship betweenmodulus of elasticity and strength properties have been modeled. The relationships have also71been modeled based on the equality of the probability of failure (equal rank) for the strengthproperties. There exists good relationships between lumber strength properties.The models, both resulting from the equal-rank analysis and general MOE-strengthbased relationships, show similar results. Therefore, it can be concluded that the equal-rankmethod yields the mean trend or average values for the strength property relationships. Theband-width method justifies this conclusion by showing that the models from equal-rankanalysis give very close results to the strength value at the median or 50-th percentile level.The difference (called percentile-level effects) between using the model from equal-rank method and the model for the 5 % lower exclusion limit from band-width method can beup to 30 % as the percentage of the values given by the model for 5 % lower exclusion limitfrom band-width method. This means that the model from equal-rank analysis will give about30 % lower or higher values if one would use this model rather than the model for 5 % lowerexclusion limit from band-width analysis. The effects of percentile level on the propertyratios are very significant if presented on the basis of modulus of elasticity especially for theUCS/MOR and UCS/UTS ratios.The mean trend of the strength property ratios for the Canadian In-grade data showstrong species dependency. The mean trends of UCS/MOR and UCS/UTS ratios forDouglas-fir and Hem-Fir are higher than those reported by Green and Kretschmann (1991) aswell as those suggested by ASTM D 1990 (ASTM 1991). In this case, only S-P-F showcloser results with that reported by Green and Kretschmann (1991) for these two propertyratios.72UTS/MOR ratios for MOR below 48.3 IVIPa for all species agree very well with theones reported by Curry and Fewell (1977), and Green and Kretschmann (1991) and areslightly higher than the ones suggested by ASTM D 1990 (ASTM 1991).MOR/(JTS ratio for every species shows much higher value than that suggested byASTMD 1990 (ASTM 1991).As per ASTM D 1990 (ASTM 1991), the models developed in this study can be usedfor the estimates of characteristic values for untested properties of Canadian dimensionlumber.73REFERENCEAmerican Society for Testing and Material (ASTM). 1990. Designation D 1990-90 (P. 321),Designation D 2915-90 (p. 430). Annual Book of ASTM Standards, Wood, Vol.04.09. American Society for Testing and Material, Philadelphia. PA._____1991. Designation D 143-83 (p. 37), Designation D 198-84 (p. 82), Designation D245-88 (p. 101), Designation D 1990-91 (p. 321), Designation D 2555-88 (p. 393),Designation D 4761-88 (p. 533). Annual Book of ASTM Standards, Wood, Vol.04.09. American Society for Testing and Material, Philadelphia. PA.Anderson, J. A. 1981. Stress-strain relationship for defect-free timber beams. WoodScience, Vol. 14. No. 1 Pp. 23-3 1.Barrett, J. D. 1974. Effect of size on tension perpendicular to grain strength of Douglas-fir.Wood and Fiber, 6(2): 126-143.Barrett, J. D. and W. Lau. 1991a. Compression strength adjustments or moisture content inDouglas-fir structural lumber. Wood and Fiber Science, 23(4): 543-557.and. 199 lb. Bending strength adjustments for moisture content for structurallumber. Wood science and Technology, 25:433-447.Barrett, J. D., F. Lam and W. Lau. 1992. Size effects in visually graded softwood structurallumber. International Council for Building Research Studies and Documentation.Working Commission Wi 8-Timber Structures. Meeting twenty-five. Sweden.Barrett, J. D. and H. Griffin. 1989. Size effect and property relationship for Canadian 2-inchdimension lumber. Proceedings CIB-W18A/22-6-1. Berlin.Barrett, J. D. and W. Lau. 1992. Bending and tension properties of Spruce-Pine-Fir 2x4machine stress-rated lumber. Prepared for Canadian Wood Council. Ontario.Bartlett, N. R. and T. Lwin. 1984. Estimating a relationship between different destructivetests on timber. Applied Statistics, 33, No. 1, pp. 65-72.Bodig, J. and B. A. Jayne. 1982. Mechanics of wood and wood composites. Van NostrandReinhold Co. NY.Bohannan, B. 1966. Effect of size on bending strength of wood members. USDA ForestService. Research Paper FPL 56. Madison WI.Bury, K. V. 1986. Statistical models in applied science. Robert E. Krieger Publishing Co.Malabar, Florida.74Canadian Wood Council (CWC). 1988a. Lumber specification. Canadian WoodConstruction. CWC Datafile WS- 1._____1988b. Machine stress rated lumber. Canadian Wood Construction. CWC DatafileWP-5.•1988c. Mechanical properties of visually graded lumber. A summary. CanadianWood Council. Ottawa, Canada.1990. Wood design manual; the complete reference for wood design in Canada.Canadian Wood Council. Ottawa, Ontario, Canada.1991. Introduction to design in wood; the complete text for instruction in wooddesign. Canadian Wood Council. Ottawa, Ontario, CanadaCouncil of Forest Industries of British Columbia (COFI). 1993. British Columbia ForestIndustry Fact Book. Vancouver, B.C. Canada.Curry, W. T. and A. R. Fewell. 1977. The relations between the ultimate tension andultimate compression strength of timber and its modulus of elasticity. BuildingResearch Establishment. Current Paper CP 22/77. Princes Risborough LaboratoryAylesbury Buck. U.K.Curry, W. T. and J. R. Tory. 1976. The relation between the modulus of rupture (ultimatebending stress) and modulus of elasticity of timber. Building Research Establishment,Current Paper CP 3 0/76. Princes Risborough Laboratory Aylesbury Buck. U.K.Devore, J. L. 1991. Probability and statistics for engineering and the sciences. Thirdedition. Brooks/Cole Publishing Company, Pacific Grove, California.Doyle, D. V. and L. J. Markward. 1966. Properties of Southern pine in relation to strengthgrading of dimension lumber. Res. Pap. FPL 64. US Department of Agriculture,Forest Service, Forest Product Laboratory, Madison, WI.Evans, J. W., R. A. Johnson, and D. W. Green. 1984. Estimating the correlation betweenvariables under destructive testing, or how to break the same board twice.Technometrics, Vol. 26, No. 3, (August).1989. Two and Three-parameter Weibull goodness-of-fit tests. United StatesDepartment of Agriculture, Forest Service, Forest Products Laboratory. Researchpaper, FPL-RP-493.Fewell, A. R. and P. Glos. 1989. The determination of characteristic values of mechanicalproperties and density for timber. Fifth draft of CEN standard xxxl. DocumentCEN/TC 124/WG 2 N82E. British Standards Institution.75Forest Product Laboratory (FPL). 1990. Wood Engineering Handbook. Second edition.Prentice Hall, Englewood Cliffs, N.J._____1977. Machine stress rating: practical concerns for lumber producers. USDAForest Service General Technical Report FPL 7. Forest Service, Madison, Wis.Fouquet, R. and J. D. Barrett. 1989. Physical and mechanical properties of Canadian visuallystress-graded lumber established by in-grade testing. Council of Forest Industries ofBritish Columbia. Vancouver.Forest Product Research Society (FPRS) 1989. In-grade testing of structural lumber.Government, University, Industry. Proceeding 47363. Forest Product ResearchSociety. USGallant, A. R. 1987. Nonlinear statistics model. John Wiley & Sons Inc.Galligan, W. L., C. C. Gerhards and R. L. Ethington. 1979. Evolution of tensile designstresses for lumber. Gen. Tech. Rep. FPL 28. US Department of Agriculture, ForestService, Forest Product Laboratory, Madison, WI.Glos, P. and A. R. Fewell. 1989. The determination of characteristic strength values forstress grades of structural timber. Part I. International Council for Building ResearchStudies and Documentation Working Commission W18A-Timber Structures. Meetingtwenty-one, Vancouver Island, Canada.GOtz, K-H, D. Hoor, K. MOhler and J. Natterer. 1989. Timber design and constructionsource book. McGraw-Hill Publishing Company.Green, D. W. and D. E. Kretschmann. 1990. Stress class systems. An idea whose time hascome? Res. Pap. FPL 500. U.S Department of Agriculture, Forest Service, ForestProduct Laboratory, Madison, WI.and_____.1991. Lumber property relationships for engineering design standards.Wood andFiber Science, Vol. 23(3) :436-456.Green, D. W. and K. A. McDonald. 1993. Investigation of the mechanical properties of redoak 2 by 4’s. Wood and Fiber Science Vol. 25(1):35-45.Green, D. W., J. W. Evans, and R. A. Johnson. 1984. Investigation of the procedure forestimating concomitance of lumber strength properties. Wood and Fiber Science,Vol. 16(3):427-440.Hahn, G. J and S. S. Shapiro 1967. Statistical models in engineering. John Wiley & SonsInc. NY.76Heatwole, E L, F. E. Woeste, and D. W. Green. 1991. Allowable bending strengthenhancement of 2 by 4 lumber by tension and compression proofloading. Wood andFiber Science, Vol. 23(1): 1-14.Hoyle, R. J. 1961. A nondestructive test for stiffness of structural lumber. Forest ProductJournal (June) pp. 25 1-254._____1968. Background to machine stress grading. Forest Product Journal, Vol.1 8(4):87-97.Jessome, A. P. 1977. Strength and related properties of woods grown in Canada. EasternForest Products Laboratory, Forestry Technical Report 21. Ottawa.Johnson, N. L. 1949. System of frequency curves generated by methods of translation.Biometrika, 36, 149-176.Johnson, R. A. and W. L. Galligan. 1983. Estimating the concomitance of lumber strengthproperties. Wood andFiber Science, Vol. 15(3): 23 5-244.Judge, G. G., W. E. Griffits, R. C. Hill, H. Lutkepohl, and T. C. Lee. 1985. The theory andpractice of econometrics. Second edition. John Wiley & Sons. NY.Kleinbaum, 0. G, L. L. Kupper, and K. E. Muller. 1988. Applied regression analysis andother multivariable methods. University of North Carolina at Chapel Hill. PWSKent publishing Comp. Boston.Kramer, P. R. 1964. Correlation of bending strength and stiffness of Southern pine. ForestProduct Journal, (October 1964:495- 496).Kunesh, R. H. and J. W. Johnson. 1974. Effect of size on tensile strength of clear Douglas-Fir and Hem-Fir dimension lumber. Forest Product Journal, Vol. 24, No. 8.Littleford, T. W. 1967. Tensile strength and modulus of elasticity of machine graded 2x6Douglas-fir. Information report VP-X-12. Forest Product Laboratory. Vancouver,British Columbia, Canada.Madsen, B. 1992. Structural behaviour of timber. Timber Engineering Ltd. BritishColumbia, Canada.Maholtra, S. K. and I. M. M. Ba.zan. 1980. Ultimate bending strength theory for timberbeams. Wood Science, Vol. 13. No. 1 pp. 50-63.Myers, R. H. 1990. Classical and modern regression with application. Second edition.PWS-KENT Publishing Company. Boston.Neave, H. R. and P. L Worthington. 1988. Distribution-Free test. Unwin Hyman Ltd.London.77National Lumber Grades Authority (NLGA). 1987. NLGA special product standard formachine stress rated lumber SPS 2. National Lumber Grades Authority. Vancouver,B.C.Neter, J., W. Wasserman, and M. H. Kutner. 1990. Applied linear statistical models;Regression analysis of variance, and experimental designs. Third edition. Irwin,Boston, USO’Halloran, M. R., J. Bodig, and D. C. Bowden. 1972. Nondestructive parameters ofLodgepole pine dimension lumber in bending. Forest Product Journal, 22(2):44-5 1.Pellicane, P. J. 1985. Goodness-of-fit analysis for lumber data. Wood Science andTechnology, Vol. 19:117-129.Pellicane, P. J. and J. Bodig. 1981. Sampling error in the bending strength distribution ofdimension lumber. Wood Science and Technology, Vol. 15: 211-225.Ramos, A. N. Jr. 1961. Stress-strain distribution in Douglas-fir beams within the plasticrange. Forest Products Laboratory, Forest Service, U.S Department of Agriculture.Report No. 2231.Ratkowsky, D. A. 1983. Nonlinear regression modeling. A unified practical approach.Statistics, textbooks and monographs; Vol. 48. Marcel Decker Inc. NY and Basel.Rawlings, J. 0. 1988. Applied Regression Analysis: A research tool. Wadsworth &Brooks. Pacific Gove, California.SAS. 1988. SAS/STAT User guide. Release 6.03 edition. SAS Institute Inc. Cary, NC.USA.Schniewind, A. P. 1962. Research on the mechanical properties of wood as it relates tostructural utilization Prospects and New Directions. Reprinted from the Journal ofThe Institute of Wood Science, No. 9 (April).Seber, G. A. F. and C. J. Wild. 1989. Nonlinear regression. John Wiley & Sons. NY.Shapiro, S. S. and A. J. Gross. 1981. Statistical modeling techniques. Statistics, textbooksand Monographs, v. 38. Marcel Decker Inc. NY & Basel.Sunley, J. G. and W. M. Hudson. 1964. Machine grading of lumber in Britain. ForestProduct Journal (April 1964, 155-158).Taylor, S. E. and D. A. Bender. 1988. Simulating correlated lumber properties using amodified multivariate normal approach. Transaction of the American Society ofAgricultural Engineers, Vol. 31(1): 182-186.78Western Wood Product Association (WWPA). 1965. Allowable stress for machine stressrated lumber. W.W.P.A. Standard Grading Rules.Woeste, F. E., S. K. Suddarth, and W. L. Galligan. 1979. Simulation of correlated lumberproperty data-A regression approach. Wood Science, Vol. 12(2):73-79.79APPENDIX A: (Table 1 to Table 17)80Table 1. Relationships between MOR and MOE for the US In-grade data (Greenand Kretschmann 1992)*Species groups Intercept ] Slope r (r2)Southernpine 0.012 4.249 0.72 (0.521)Douglas-fir-Larch -0.394 4.341 0.73 (0.538)Hem-fir -0.175 4.299 0.72 (0.52)* MOR is in ksi, MOE is in million psi, data adjusted to 2 by 8 size 15% moisture content.Table 2. Relationships between UTS and MOE for the US In-grade data (Greenand Kretschmann 1992)*Species group Intercept D Slope 0 r (r2)Southern pine -1.258 3.420 0.67 (0.442)Douglas-fir-Larch -0.515 2.878 0.64 (0.405)Hem-fir -0.867 3.363 0.65 (0.421)* MOR is in ksi, MOE is in million psi, data adjusted to 2 by 8 size 15% moisture content.Table 3. Width, test span and gauge length for bending, tension and compressionspecimens (Barrett and Griffin 1989).Width (mm) Test snan Gauge lengthBending (mm) Tension (mm) Compression (mm)89 1510 2640 2440184 3130 3680 3660235 3990 3680 427081Table 4. General relationships between MOE and MORSpecies Sample Model * SSE. Parameter Estimate Asymptotic Asymptotic 95% r2Size Variance Std. Error Confidence Interval** Lower I Upper** —D-FIR 2229 1 333788.89 BO -13.2271 1.0925 -15.3684 -11.0858 0.58 iiTi149.88 Bi 4.9333 0.0887 4.7594 5.10712 333687.11 BO -18.9570 7.9839 -34.6140 -3.3000 2.37149.90 BI 6.7651 2.7241 1.4230 12.1072B2 0.9111 0.1119 0.6917 1.1304 0.803 335383.60 BI 1.9385 0.1203 1.7025 2.1744150.60 B2 1.2705 0.0238 1.2238 1.31724 332818.02 BI 0.6539 0.1800 0.3009 1.0069149.50 B2 1.9694 0.1739 1.6285 2.3104B3 0.9484 0.0124 0.9242 0.9727 4.16H-FIR 2295 1 289313.94 BO -9.0746 1.2159 -11.4578 -6.6915 0A7 7.46126.17 Bi 4.7233 0.1045 4.5186 4.92812 289306.92 BO -11.2593 10.0043 -30.8782 8.3595 1.13126.23 Bi 5.4160 3.2559 -0.9690 11.8011B2 0.9604 0.1728 0.6215 1.2993 0.233 289565.56 Bi 2.4078 0.1667 2.0809 2.7347126.28 B2 1.1993 0.0276 1.1452 1.25334 289131.70 Bi 1.2738 0.4492 0.3928 2.1547126.15 B2 1.6338 0.2376 1.1679 2.0997B3 0.9644 0.0190 0.9271 1.0016 1.87S-P-F 3192 1 186720.03 BO -8.0930 0.6778 -9.4216 -6.7645 060 11.9458.53 BI 4.5865 0.0668 4.4556 4.71742 186174.67 BO 2.2346 2.7475 -3.1525 7.6217 0.8158.38 BI 1.7435 0.5534 0.6584 2.8286B2 1.3057 0.1033 1.1032 1.5082 2.963 186210.51 Bi 2.2216 0.0988 2.0279 2.415258.37 B2 1.2279 0.0187 1.1913 1.26464 186201.92 BI 2.2847 0.4470 1.5082 3.261358.39 B2 1.1746 0.1386 0.9028 1.4463B3 1.0051 0.0132 0.9792 1.0311— 0.39* Model 1 = Eq. (9)Model2 Eq.(1O)Model3 Eq.(11)Model 4 Eq. (12)** Standard error and confidence interval for model I82Table 5. General relationships between MOE and UTSSpecies Sample Model* SSE. Parameteil Estimate Asymptotic Asymptotic 95% r2Size Variance I Std. Error Confidence IntervalI ** Lower**I Upper**D-FIR 2232 1 165237.78 BO -7.0968 0.7628 -8.5920 -5.6016 0A7 9.3074.10 BI 2.7366 0.0617 2.6157 2.85752 165170.06 50 -3.4928 3.5865 -10.5261 3.5405 0.9774.10 Bi 1.7929 0.8178 0.1890 3.3967B2 1.1218 0.1329 0.8611 1.3824 0.923 165264.51 BI 1.1052 0.0828 0.9429 1.267674.11 B2 1.2631 0.0287 1.2069 1.31934 165031.94 Bi 0.6530 0.2008 0.2592 1.046774.04 B2 1.5954 0.1905 1.2218 1.9690B3 0.9759 0.0135 0.9494 1.0024 1.79H-FIR 2245 1 162757.68 BO -12.5818 0.8693 -14.2856 -10.8779 049 14.4772.56 51 3.3752 0.0732 3.2318 3.51872 161370.87 BO 2.1844 2.3207 -2.3666 6.7354 0.9471.98 Bi 0.4624 0.2119 0.0468 0.8780B2 1.6090 0.1462 1.3223 1.8957 4.173 161431.53 Bi 0.6859 0.0567 0.5748 0.797171.97 B2 1.4853 0.0322 1.4221 1.54854 161431.46 BI 0.6785 0.2376 0.2126 1.144572.00 B2 1.4923 0.2238 1.0534 1.9312B3 0.9995 0.0172 0.9658 1.0331 0.03S-P-F 2694 1 138818.92 BO -10.2807 0.7084 -11.6692 -8.8921 046 145151.57 BI 3.2747 0.0686 3.1402 3.40922 137800.40 BO 1.6290 1.9654 -2.2249 5.4830 0.8351.21 BI 0.5228 0.2293 0.0732 0.9723B2 1.5923 0.1477 1.3027 1.8819 4.013 137836.25 BI 0.7301 0.0567 0.6189 0.841351.20 B2 1.4817 0.0322 1.4185 1.54494 137819.37 Bi 0.5974 0.2070 0.1914 1.003451.21 B2 1.6278 0.2492 1.1392 2.1164B3 0.9867 0.0225 0.9426 1.0308— 0.59* Model 1 = Eq. (9)Model 2 Eq. (10)Model 3 Eq.(1l)Model 4 = Eq. (12)** Standard error and confidence interval for model 183Table 6. General relationships between MOE and UCSSpecies Sample Model* SSE. Parameterj Estimate Asymptotic Asymptotic 95% r2 [IiSize Variance I Std. Error Confidence IntervalI ** Lower** I Upper**D-FIR 2237 1 61664.17 BO 7.1133 0.4584 6.2148 8.0117 063 15.5227.59 BI 2.3376 0.0380 2.2632 2.41202 61279.88 BO 14.8683 1.5448 11.8389 17.8977 9.6327.43 BI 0.6300 0.2143 0.2098 1.0501B2 1.3908 0.1045 1.1858 1.5958 3.743 62172.20 81 4.8072 0.1594 4.4946 5.119827.82 B2 0.8026 0.0131 0.7770 0.82824 61269.17 BI 9.8312 1.2377 7.4039 12.258427.43 B2 0.3253 0.0823 0.1639 0.4867B3 1.0388 0.0067 1.0256 1.0520 5.79H-FIR 2289 1 65535.61 BO 3.3500 0.5583 2.2557 4.4443 0.58 6.0028.66 BI 2.6642 0.0479 2.5703 2.75812 65053.05 BO 13.4110 1.7873 9.9060 16.9159 7.5028.46 BI 0.5112 02091 0.1011 0.9213B2 1.5065 0.1304 1.2508 1.7622 3.893 65742.84 BI 3.6891 0.1530 3.3892 3.989128.75 B2 0.9098 0.0166 0.8772 0.94234 65155.16 Bi 8.2866 1.4399 5.4630 11.110328.50 B2 0.3538 0.1176 0.1232 0.5845B3 1.0479 0.0103 1.0277 1.0681 4.65S-P-F 2602 1 37603.70 BO 3.5721 0.3771 2.8329 4.3113 061 9A714.46 BI 2.3522 0.0371 2.2794 2.42502 37322.98 BO 11.0819 1.2064 8.7163 13.4475 9.1714.36 Bi 0.5112 0.1726 0.1727 0.8496B2 1.4895 0.1126 1.2687 1.7103 4.353 37778.29 BI 3.6305 0.1207 3.3938 3.867314.53 B2 0.8737 0.0141 0.8461 0.90144 37338.29 BI 7.3907 0.9559 5.5161 9.265214.37 B2 0.3298 0.0970 0.1395 0.5201B3 1.0546 0.0099 1.0351 1.0740 — 5.52* Model 1 = Eq. (9)Model 2= Eq.(10)Model 3 Eq.(11)Model 4 Eq.(12)** Standard error and confidence interval for model 184Table 7. Strength property relationships based on MOR (Equal-Rank analysis)Relation Species Sample Model* SSE Parameter Estimate Asymptotic Asymptotic 95 % LRSize Variance Std. Error Confidence IntervalLower I UpperUTS-MOR D-FIR 126 1 799.72 BO 6.6678 9.6509 -12.4373 25.7729 TYT66.56 BI 0.1373 0.2152 -0.2887 0.5633B2 6.0215 30.5591 -54.4740 66.5170B3 1.3274 0.2914 0.7505 1.90432 918.58 BI 0.3481 0.0362 0.2764 0.41987.41 B2 1.1226 0.0253 1.0724 1.17273 831.64 Bi 1.0067 0.2982 0.4165 1.59706.76 B2 0.7609 0.0991 0.5647 0.9572B3 1.0067 0.0018 1.0031 1.0102H-FIR 126 1 845.48 BO 8.4913 4.5178 -0.4521 17.4347 23296.93 Bi 0.0548 0.0989 -0.1409 0.2505B2 8.8262 18.4406 -27.6791 45.3316B3 1.5938 0.3506 0.8998 2.28792 1017.14 Bi 0.1641 0.0207 0.1230 0.20515.20 B2 1.3296 0.0312 1.2678 1.39153 904.92 Bi 0.9063 0.3776 0.1590 1.65377.36 B2 0.7426 0.1419 0.4617 1.0234B3 1.0113 0.0028 1.0059 1.0168S-P-F 126 1 351.94 BO 6.2877 4.4008 -2.4242 14.9997 27.112.89 Bi 0.0409 0.0877 -0.1326 0.2145B2 4.5935 18.2172 -31.4696 40.6565B3 1.6981 0.4250 0.8568 2.53952 436.43 BI 0.1437 0.0144 0.1153 0.17223.52 B2 1.3912 0.0260 1.3397 1.44283 364.38 BI 0.8143 0.2766 0.2668 1.36182.96 B2 0.7506 0.1243 0.5046 0.9966B3 1.0153 0.0030 1.0094 1.0212 —85Table 7. ContinuedRelation Species Sample Model* SSE Parameter Estimate symptotic Asymptotic 95 % LRSize Variance Std. Error Confidence IntervalLower I UpperUCS-MOR D-FIR 126 1 654.07 BO 15.5812 69.5356 -122.0725 153.2349 iT95.36 Bi 0.6317 1.1308 -1.6070 2.8703B2 6.0215 130.1654 -251.6560 263.6990B3 0.9260 0.3203 0.2920 1.55992 764.12 Bi 3.9738 0.2433 3.4922 4.45546.16 B2 0.5717 0.0154 0.5411 0.60223 636.90 BI 8.3015 1.2976 5.7330 10.87005.17 B2 0.3051 0.0550 0.1963 0.4138B3 1.0057 0.0011 1.0034 1.0079H-FIR 126 1 428.16 BO 16.6192 21.6001 -26.1406 59.3791 33443.51 Bi 0.2146 0.4384 -0.6533 1.0826B2 6.0996 63.0865 -118.7875 130.9867B3 1.1910 0.3802 0.4383 1.94382 558.31 Bi 2.6262 0.1685 2.2927 2.95974.50 B2 0.6746 0.0163 0.6423 0.70693 423.89 Bi 7.7605 1.3654 5.0577 10.46323.45 B2 0.2825 0.0624 0.1589 0.4060B3 1.0086 0.0014 1.0059 1.0113S-P-F 126 1 89.01 BO 14.0910 8.1171 -1.9778 30.1598 90.770.73 81 0.1953 0.2323 -0.2646 0.6552B2 7.0520 25.7198 -43.8632 57.9672B3 1.2201 0.2339 0.7571 1.68322 182.94 Bi 2.2987 0.1092 2.0825 2.51491.48 B2 0.6831 0.0127 0.6579 0.70833 81.10 BI 8.2709 0.8575 6.5735 9.96830.66 B2 0.1856 0.0395 0.1074 0.2639B3 1.0133 0.0011 1.0112 1.0154 —* Model 1 = Eq. (21)Model 2 = Eq. (20)Model 3 = Eq. (12)86Table 8. Strength property relationships based on UTS (Equal-Rank analysis)Relation Species Sample Model* SSE Parameter Estimate Asymptotic Asymptotic 95 % LRSize Variance Std. Error Confidence IntervalLower UpperMOR-UTS D-FIR 126 1 1745.89 BO 6.0215 5.1755 -4.2241 16.2671 27.1214.31 BI 4.7865 1.3074 2.1983 7.3746B2 6.6180 0.9814 4.6752 8.5609B3 0.7299 0.0597 0.6117 0.84812 2165.20 BI 2.8489 0.1794 2.4938 3.203917.46 B2 0.8584 0.0178 0.8231 0.89373 1578.75 BI 0.9419 1.3284 0.0742 1.181512.84 B2 1.3284 0.0742 1.1815 1.4754B3 0.9851 0.0023 0.9806 0.9896H-FIR 126 1 1168.50 BO 6.0996 8.8200 -11.3605 23.5597 44.659.58 BI 7.5577 3.1746 1.2732 13.8421B2 8.0101 1.3776 5.2830 10.7372B3 0.5786 0.0827 0.4148 0.74242 1665.43 Bi 4.0596 0.2296 3.6052 4.514013.43 B2 0.7385 0.0161 0.7066 0.77033 1259.74 BI 1.5656 0.2623 1.0463 2.084910.24 B2 1.1460 0.0692 1.0090 1.2830B3 0.9869 0.0021 0.9826 0.9911S-P-F 126 1 436.89 BO 7.0520 6.6974 -6.2063 20.3103 39.803.58 BI 5.1927 1.9231 1.3858 8.9996B2 5.9813 1.5774 2.8586 9.1041B3 0.6394 0.0757 0.4895 0.78922 599.16 BI 4.0447 0.1651 3.7181 4.37144.83 B2 0.7168 0.0122 0.6926 0.74103 384.86 BI 1.8607 0.1931 1.4785 2.24293.13 B2 1.0724 0.0458 0.9818 1.1630B3 0.9865 0.0017 0.9832 0.9898 —87Table 8. ContinuedRelation Species Sample Model* SSE Parameter Estimate Asymptotic Asymptotic 95 % LRSize Variance Std. Error Confidence IntervalLower Upper —UCS-UTS D-FIR 126 1 311.32 BO 14.5265 6.9739 0.7208 28.3322 2242.55 B1 2.5256 1.3053 -0.0584 5.1096B2 5.7852 3.7598 -1.6577 13.2281B3 0.7030 0.1025 0.5002 0.90582 316.91 Bi 6.1348 0.1853 5.7679 6.50162.56 B2 0.5394 0.0088 0.5219 0.55703 312.44 BI 6.8239 0.5868 5.6623 7.98552.54 B2 0.4924 0.0367 0.4198 0.5650B3 1.0016 0.0012 0.9992 1.0041H-FIR 126 1 167.44 BO 16.7095 2.5017 11.7570 21.6619 5.081.37 BI 2.2696 0.5838 1.1139 3.4252B2 8.2496 1.2126 5.8491 10.6500B3 0.7076 0.0542 0.6002 0.81492 174.33 BI 5.8757 0.1385 5.6020 6.14981.41 B2 0.5397 0.0068 0.5261 0.55323 173.72 Bi 6.1461 0.4453 5.2649 7.02741.41 B2 0.5200 0.0307 0.4594 0.5807B3 1.0007 0.0010 0.9987 1.0027S-P-F 126 1 144.10 BO 14.0910 0.7706 12.5656 15.6164 5.181.18 BI 1.6714 0.2846 1.1079 2.2348B2 6.5370 0.1597 6.2210 6.8530B3 0.7436 0.0408 0.6628 0.82442 150.15 Bi 5.5494 0.1515 5.2496 5.84921.21 B2 0.5123 0.0084 0.4958 0.52883 144.61 Bi 6.4628 0.4836 5.5055 7.42021.18 B2 0.4405 0.0339 0.3734 0.5077B3 1.0029 0.0013 1.0003 1.0055 —*MjeI I = Eq.(21)Model I = Eq. (20)Model 1 = Eq. (12)88Table9.Goodness-of-fitanalysis(1,000 MPaBand-Width)SpeciesIPropertyMOEMOESampleMOERel.skew.Rel.kurt.ISbparameterWeibull parameterK-Stestcriticalvaluesmm.max.size50-thpctl.B_iB2GammaEtaShapeIScaleILocationSbI3-pWTableD-FIRMOE-MOR67396.65140.22272.0888-0.38210.61131.274610.447010.66370.07950.08980.2130781107.64290.18881.8490-0.31360.53351.635016.443910.00000.04970.06340.1297891738.55060.03291.7625-0.10080.49612.128122.17258.45330.06410.05950.10349102639.51840.00261.7512-0.04260.57962.187425.166311.35650.03990.04000.0839101132310.49890.00831.81670.00230.53062.666231.46529.43990.06510.02790.0757111228811.47470.01371.80550.05590.55233.090838.29439.01700.04430.05630.0801121326012.48100.00261.75960.04390.57794.300453.97300.00000.05730.05530.0843131422213.45560.04821.89860.10350.58073.650443.408016.25130.05020.04340.0913141518914.46960.00621.86560.06830.56135.021863.25770.00000.07980.04300.0989151613915.49130.00421.77320.02020.58254.456865.01644.23920.04540.04720.115416178216.54670.00011.8435-0.09350.49903.873258.002014.63760.10350.06810.150217185817.44140.12751.56330.13370.56805.487179.66680.00000.13820.09340.1786________18194318.48220.00611.77130.03330.46904.719882.84780.00000.12430.07040.2074MOE-UTS67346.59160.05871.5833-0.17310.46681.49077.70495.53620.06130.07110.227078977.65540.13132.0339-0.39280.61112.366511.40524.03460.07880.10650.1381891598.64130.08071.8274-0.30410.52321.917613.47574.23980.07400.11120.10799102689.53480.03301.8497-0.07620.54681.913613.02097.22830.05710.05890.0831101132210.51060.00051.7495-0.02570.60831.942714.39697.66990.02650.03960.0758111232611.53980.00181.8228-0.08610.53931.887818.93798.30510.05650.05570.0753121327112.47690.01651.76640.05530.59732.206920.89788.79020.03270.05130.0826131421213.45960.02011.88890.08380.51692.212423.01199.33420.06280.05310.0934141518514.51590.00041.8437-0.03860.60773.139234.38032.40730.07660.06720.1000151611715.49250.00332.19070.01830.61292.429227.195611.77310.09320.04730.125716179316.52960.00001.5896-0.06510.54892.021121.617616.54010.06040.04470.141017184717.40720.01091.83260.18300.48702.367126.997216.77740.13620.08580.198418193318.38040.08981.66220.34880.71491.928425.348122.02170.09350.07330.2310Table9.ContinuedCSpeciesIPropertyMOEMOESampleMOERel.skew.Rel.kurt.SbparameterWeibullparameterK-Stestcriticalvaluesmm.max.size50-thpctl.B1B2GammaIEtaShapeIScaleILoc.Sb3-pWTableD-FIRMOE-UCS67496.70180.15191.8616-0.36750.42944.173512.338511.75430.10730.07650.1943781207.62330.11871.9124-0.31630.62792.536312.482713.95770.03860.06620.1242892148.50700.00031.8910-0.01820.64745.382120.62788.18300.04170.05710.09309102789.52280.00441.8884-0.05480.59904.430820.158311.16060.04390.05100.0816101134210.49550.00031.78760.00920.51227.770132.75140.35300.04530.06100.0735111231411.48870.00761.81550.02760.60786.488331.38474.67460.04350.05320.0767121323312.51500.00531.8122-0.03610.60074.722724.313113.71910.05080.07670.0891131419813.52300.00801.7531-0.05700.61803.186220.962219.34600.03410.07710.0967141514414.50280.00031.7491-0.00660.58944.929633.004810.76090.04080.07460.1133151613415.50570.00461.7518-0.01200.52834.556331.388614.53760.08200.08130.117516178016.48500.00011.72900.03200.53343.201120.494027.64790.07380.06990.152117186317.41330.02561.98400.19650.56082.587316.195432.89400.14230.08770.1713H-FIRMOE-MOR781007.60450.10542.0455-0.25170.59322.254222.58857.02180.07870.06750.1360891868.58750.09171.9926-0.21140.59782.664827.65786.83770.04370.03550.09979103259.56870.04391.9251-0.16810.60782.812729.83258.93310.03620.02530.0754101138910.50180.00021.8655-0.00420.57663.114834.06839.57950.04400.04600.0690111238711.53080.00831.8491-0.06870.55634.229846.58123.13230.05050.04930.0691121331712.49460.01211.77900.01140.53113.656644.298410.27800.04510.04140.0764131423213.44920.02351.86280.12590.61785.668159.28310.00000.04150.07340.0893141516814.39790.02351.77380.23140.55855.183564.54630.00000.06120.09190.104915169815.40290.03171.81890.23730.60336.489269.26750.00000.06520.06150.137416174916.53140.01161.9155-0.06750.53705.076972.36660.00000.10500.10520.1943MOE-UTS781057.53460.02082.0049-0.08230.59332.544211.77974.22990.08730.09100.1327891938.56290.01151.7457-0.14590.57712.067213.28535.17800.04900.09350.09799102739.54380.06681.8570-0.10000.57002.320514.96756.51170.06760.06330.0823101132810.52790.00731.7680-0.06100.54692.266117.39385.88560.03890.05870.0751111238411.51600.00381.6989-0.03400.53122.485924.23454.20840.03080.07410.0694121332412.49020.00141.87080.02270.58152.590827.38845.53050.04880.03730.0756131424413.47390.00771.85390.05600.53492.657028.37627.52080.05130.04460.0871141515214.41550.05141.94540.20800.60972.271523.976713.23100.06020.03250.1103151611215.39550.00841.76040.25250.59494.917945.70750.00000.07660.06130.128516175316.41150.00081.70910.18480.51673.896034.130615.00520.12030.11730.1868Table9.ContinuedSpeciesPropertyMOEMOESampleMOERel.skew. Re!.kurt.Sbparameter-Weibull parameterK-Stestcritical valuesmm.max.size50-thpct!.B_iB2GammaEtaShapeIScaleILoc.SbI3-pWITableH-FIRMOE-UCS781007.60400.06252.0663-0.25610.60662.584012.349013.26710.08570.07920.1360891968.54450.00381.7549-0.09420.52753.317814.346313.06380.06370.02970.09719103269.53090.00001.7660-0.07380.59723.300414.831015.14000.03740.04330.0753101138910.53820.01651.8647-0.08680.56723.822718.506914.07780.04330.03210.0690111236011.48620.00431.78570.03030.54663.677720.471215.52420.04600.04010.0717121330412.43040.09392.02690.17250.61513.317721.121317.38620.06480.04850.0780131426013.53140.00051.8451-0.07980.63543.362919.975921.38960.05320.04910.0843141516114.44180.00111.73970.13360.57154.028026.598617.92060.05640.05920.107215168315.40320.03662.16770.22280.56813.932128.348419.06990.10420.10830.149316174516.41660.04701.79010.24950.74082.968022.216827.39570.06550.08850.2027S-P-FMOE-MOR56575.64000.24981.9295-0.35820.62251.879910.95419.49320.08640.06410.1801671536.60830.08901.7549-0.23320.52981.738712.426111.86760.05680.04750.1099783327.56480.04421.9219-0.15170.58242.603719.43578.74730.05030.03550.0746894898.52420.00321.7839-0.05520.56983.719727.76715.69480.03450.02480.06159105969.47500.00151.82190.05650.56354.251331.63346.33990.05340.02710.0557101156410.47320.00601.87100.06160.57424.960138.53064.44900.04610.02110.0573111248411.43570.01361.79210.13800.53384.795836.521811.08160.04050.03490.0618121329112.43570.03011.80980.13740.53094.992941.049711.42610.04480.05350.0797131414113.40200.13761.81130.24060.60573.947335.780221.18680.05250.07090.114514154914.36750.17001.98190.28550.52566.775162.67770.00000.08220.10050.1943MOE-UTS56375.64960.06551.8869-0.41010.66421.00005.23295.72090.08550.10160.2180671116.61240.18142.1980-0.26790.58551.27326.54036.65740.08090.08150.1291782327.55750.04672.0274-0.13600.58882.211111.18854.07160.06250.08750.0893893848.56100.03031.8435-0.13390.54602.327713.51035.07450.05010.05790.06949105099.52250.00211.7871-0.05100.56702.485817.33134.83430.03240.05430.0603101154410.48780.00341.71660.02590.52872.648821.28694.98100.03640.05820.0583111240311.47510.00591.85620.05670.56752.248619.410610.40890.05080.02980.0677121325112.50540.00071.7172-0.01220.56512.732725.08567.89970.03680.02830.0858131413513.39180.11691.94610.25510.58003.556731.68978.03510.05080.04600.117114155314.45680.08071.88940.10180.58783.598437.85023.00830.09300.05100.1868Table9.ContinuedSpeciesPropertyMOEMOESampleMOERel.skew.Rd.kurt.SbparameterWeibull_parameterK-Stestcriticalvaluesmm.max.size50-thpctl.B1B2GammaIEtaShapeScaleLoc.SbI3-pWITableS-P-FMOE-UCS56395.72090.20591.9196-0.51420.54171.63014.939513.53060.09060.06920.2100671176.60030.04641.9107-0.25040.61575.720818.36352.41340.05640.05350.1257782587.52120.01691.8067-0.04700.55504.730815.67626.94150.06650.05930.0847894278.56490.03871.8529-0.14480.55463.575413.107311.84320.03980.07080.06589104959.49760.00131.88490.00560.58005.670620.77916.28370.04810.05770.0611101148010.46250.00891.82830.08860.58993.737314.122115.26080.03790.05200.0621111238011.45140.02861.86890.10900.55883.471313.500718.16410.04050.04900.0698121322212.48020.00831.77420.04520.56963.341017.069917.61970.04100.06040.0913131410613.42570.04521.82090.16490.55112.983915.860421.60100.04670.10530.1321_____________14154414.34410.39802.05140.37500.58116.708331.77168.69720.10730.07250.2050Table 10. Relationships between MOE and MOR (Band-Width analysis)Species Band- Sample PCTL Parameter Estimate Asymptotic Asymptotic 95%Width Size Std. Error Confidence IntervalLower UpperD-FIR 500 106 1-st 80 2.571055 3.644842 -4.657675 9.799785Bi 0.292945 0.31 7370 -0.336486 0.922377B2 1.652017 0.347744 0.962345 2.3416895-th BO -8.322920 5.308494 -18.851132 2.205293BI 1.943278 1.187438 -0.411740 4.298296B2 1.164390 0.182746 0.801955 1.52682550-th BO -37.485075 8.355181 -54.055711 -20.914440BI 13.409229 3.894428 5.685499 21.132958B2 0.736805 0.076054 0.585968 0.887641690 117 1-st 80 1.469774 3.324246 -5.115580 8.055128Bi 0.420124 0.361308 -0.295630 1.135878B2 1.526763 0.269673 0.992540 2.0609865-th BO -7.438555 4.344389 -16.044818 1.167709BI 1.782691 0.914939 -0.029809 3.595191B2 1.186753 0.152916 0.883826 1.48968050-th BO -28.467992 5.545797 -39.454254 -17.481731Bi 9.427161 2.192234 5.084331 13.769991B2 0.832443 0.062975 0.707690 0.9571961000 128 1-st 80 2.729538 2.605000 -2.426122 7.88519781 0.298591 0.227350 -0.151367 0.74854982 1.626803 0.239856 1.152093 2.1015135-th 80 -6.181606 3.390445 -12.891772 0.528560Bi 1 .578368 0.667756 0.256785 2.899950B2 1.217588 0.126157 0.967906 1.46727050-th 80 -30.079164 3.826081 -37.651514 -22.506814B1 10.227783 1.567528 7.125426 13.33014082 0.808537 0.040983 0.727427 0.8896481500 133 1-st BO 5.155194 1.424008 2.337941 7.97244881 0.104857 0.060583 -0.015000 0.22471482 1.965553 0.186808 1.595973 2.3351335-th 80 -2.374785 2.049975 -6.430450 1.68088081 0.854583 0.279221 0.302172 1.406994B2 1.407217 0.100284 1.208815 1.60562050-th 80 -28.712142 3.008615 -34.664377 -22.75990781 9.559147 1.192668 7.199577 11.91871782 0.828437 0.033605 0.761 954 0.8949202000 139 1-st 80 6.266269 0.960674 4.366461 8.16607781 0.050140 0.023823 0.003028 0.09725182 2.213721 0.155768 1.905677 2.5217655-th 80 -0.059217 1.407603 -2.842861 2.724427Bi 0.521047 0.140149 0.243892 0.79820382 1.565577 0.084184 1.399096 1.73205850-th BO -26.494196 2.437466 -31.314475 -21.673917BI 8.572840 0.919723 6.754015 10.391665_____ _____ _____B2 0.859810 0.029254 0.801958 0.91766193Table 10. ContinuedSpecies Band- Sample PCTI Parameter Estimate Asymptotic Asymptotic 95%Width Size Std. Error Confidence IntervalLower UpperD-FIR 2500 145 1-st BO 8.785936 0.970455 6.867513 10.704359Bi 0.004979 0.003694 -0.002324 0.012282B2 3.009873 0.248480 2.518671 3.5010755-th BO 3.250637 1.389436 0.503959 5.997314B1 0.200167 0.069915 0.061956 0.33837782 1.882847 0.111933 1.661574 2.10412050-th BO -27.718570 2.323216 -32.311172 -23.12596981 8.866346 0.880559 7.125631 10.607061B2 0.852897 0.026924 0.799672 0.906122H-FIR 500 97 1-st BO 5.988787 1.571110- 2.869295 9.108279BI 0.042854 0.032474 -0.021623 0.107332B2 2.360331 0.258383 1.847304 2.8733595-th BO 2.745504 1.968412 -1.162846 6.653854Bi 0.299710 0.135087 0.031489 0.567930B2 1.787788 0.148065 1.493800 2.08177650-th BO -17.178204 5.038259 -27.181839 -7.174570B1 7.003413 1.833555 3.362828 10.64399982 0.896915 0.073578 0.750822 1.043007690 101 1-st 80 1.861866 2.312249 -2.726743 6.45047681 0.275442 0.185206 -0.092097 0.642980B2 1.712137 0.219228 1.277084 2.1471905-th BO -2.006605 2.748602 -7.461149 3.447939Bi 0.860304 0.380687 0.104839 1.615768B2 1.433515 0.139715 1.156254 1.71077650-th BO -16.114421 4.426557 -24.898831 -7.330011B1 6.790301 1.599185 3.616753 9.963849B2 0.902537 0.066329 0.770909 1.0341651000 102 1-st BO 0.376822 2.229768 -4.047548 4.801193B1 0.419569 0.239108 -0.054876 0.894015B2 1.570087 0.183472 1.206037 1.9341365-th BO -3.830542 2.662910 -9.114365 1.453282BI 1.123835 0.436860 0.257004 1.990666B2 1.348737 0.121486 1.107680 1.58979450-th BO -14.912589 3.497610 -21.852646 -7.97253181 6.286618 1.220784 3.864303 8.708932B2 0.926075 0.055224 0.816499 1.0356521500 108 1-st BO -2.495990 2.863008 -8.172842 3.180862BI 0.898215 0.505165 -0.103441 1.899872B2 1.312329 0.174849 0.965633 1.6590245-th BO -8.111302 3.377821 -14.808940 -1.413663BI 2.055066 0.795233 0.478255 3.631 877B2 1.154753 0.116832 0.923096 1.38641150-th BO -16.293184 2.965933 -22.174120 -10.412248BI 6.706326 1.068714 4.587250 8.825402B2 0.908949 0.045107 0.819509 0.99838894Table 10. ContinuedSpecies Band- Sample PCTI4 Parameter Estimate Asymptotic Asymptotic 95%Width Size Std. Error Confidence IntervalLower I UpperH-FIR 2000 116 1-st BO -4.264076 3.143135 -10.491239 1.963086BI 1.336127 0.711390 -0.073275 2.745529B2 1.176650 0.161361 0.856963 1.4963385-th BO -12.317054 3.978593 -20.199421 -4.434686B1 3.255466 1.204902 0.868321 5.642612B2 1.011739 0.107944 0.797882 1.22559750-th BO -22.746187 3.058526 -28.805722 -16.68665281 9.021110 1.257058 6.530634 11.511586B2 0.828661 0.038318 0.752746 0.9045762500 120 1-st BO -4.287888 2.713924 -9.662712 1.086937BI 1.280502 0.597156 0.097856 2.463148B2 1.189456 0.141520 0.909181 1.4697305-th BO -12.544163 3.638584 -19.750241 -5.338084BI 3.213171 1.087544 1.059331 5.367012B2 1.016835 0.098718 0.821329 1.21234250-th BO -24.009176 3.200023 -30.346702 -17.671650BI 9.340913 1.325164 6.716475 11.965351B2 0.821805 0.038855 0.744854 0.898756S-P-F 500 86 1-st5-th50-thBOBIB2BOBIB2BO81B27.0445580.0646562.2827633.4226490.4183241 .699077-6.7443493.8925151.0580351.5294160.051 7250.2875931.6224650.1679980.1382531.4799420.4655630.0370454. 002598-0.0382241.71 07490.1956180.0841 811.424096-9.6879062.9665250.98435310.0865190.1675372.8547766.6496800.7524671 .974058-3.8007934.81 85051.131 716690 91 1-st BO 7.275396 1.089795 5.109647 9.441145BI 0.049557 0.030344 -0.010745 0.109859B2 2.379522 0.220925 1.940479 2.8185655-th BO 4.090538 1.180774 1.743988 6.437089BI 0.332991 0.105473 0.123385 0.542597B2 1.779949 0.109858 1.561629 1.99826950-th BO -5.259637 0.953320 -7.1541 68 -3.3651 05Bi 3.41 31 80 0.281073 2.854603 3.971 75782 1.100349 0.025792 1.049094 1.1516051000 99 1-st5-th50-thBOB1B2BOBiB2BOBI827.6738590.0290762.5818755.1083970.2212651.928540-2.5082482.5767671.1929480.7445680.01 34960.1680330.801 7660.0537430.0849830.7469020.1892820.0234486.1958940.0022872.2483313.5168950.1145851.759849-3.9908442.201 0441.1464059.151 8230.0558652.9154196.6998990.3279462.097230-1.0256522.9524911.23949295Table 10. ContinuedSpecies Band- Sample PCT1 Parameter Estimate Asymptotic Asymptotic 95%Width Size Std. Error Confidence IntervalLower I UpperS-P-F 1500 107 1-st BO 7.434986BI 0.028694B2 2.5877105-th BO 5.413194B1 0.178988B2 2.00731950-th BO -1 .569846B1 2.242168B2 1.2426830.5576910.0098250.1234840.6009580.0342950.0671 970.6140230.1428430.0204956.3290560.0092112.3428354.2214650.1109791.874065-2.7874851.9589041.2020408.5409160.0481 772.8325856.6049230.2469972.140573-0.3522062.5254321.2833252000 112 1-st BO 7.335125BI 0.025825B2 2.6282225-th BO 5.435238Bi 0.161641B2 2.04563350-th BO -1 .679381BI 2.217766B2 1.2486180.4904920.0079170.1106750.5039570.0266270.0578970.4533940.1043280.01 51476.3629790.01 01 342.4088674.4364050.1088671.930882-2.5779992.0109891.21 85968.3072710.041 5172.8475776.4340710.2144152.160383-0.7807642.4245441.2786402500 117 1-st BO 7.619745BI 0.016478B2 2.7981435-th BO 5.588123BI 0.139342B2 2.10016650-th BO -1 .683456Bi 2.171703B2 1.2576360.4514790.0050560.1113010.4483500.0211780.0535790.4548110.1028740.01 52756.7253630.0064622.5776554.6999380.0973881.994026-2.5844401.9679091.2273768.5141280.0264943.01 86326.4763080.1812962.206307-0.7824722.3754971.28789696Table 11. Relationships between MOE and UTS (Band-Width analysis)Species Band- Sample PCTL. Parameter Estimate Asymptotic Asymptotic 95%Width Size Std. Error Confidence IntervalLower I UpperD-FIR 500 101 I-st BO 4.870822 0.885183 3.114194 6.627449BI 0.004279 0.005400 -0.006436 0.014995B2 2.876210 0.433478 2.015982 3.7364375-th BO 3.091072 1.027859 1.051307 5.130837BI 0.080350 0.048643 -0.016180 0.176881B2 1.943165 0.199134 1.547987 2.33834350-th BO -12.817155 6.154476 -25.030582 -0.603727Bi 4.682104 2.414845 -0.110105 9.474314B2 0.839767 0.140721 0.560508 1.119025690 109 1-st BO 4.585838 0.740336 3.118041 6.053636BI 0.005732 0.005625 -0.005420 0.016885B2 2.768894 0.334240 2.106226 3.4315625-th 80 3.435984 0.747549 1.953885 4.918083BI 0.060579 0.028746 0.003586 0.117572B2 2.036539 0.156472 1.726316 2.34676350-th BO -7.419808 3.652154 -14.660607 -0.17901081 2.847667 1.133232 0.600909 5.094425B2 0.977321 0.113385 0.752523 1.2021191000 123 1-st BO 4.891729 0.510018 3.881922 5.901536BI 0.002662 0.001912 -0.001125 0.006448B2 3.034269 0.242870 2.553400 3.51 51 395-th BO 4.159771 0.434311 3.299859 5.019682Bi 0.032058 0.009944 0.012369 0.051746B2 2.250622 0.102248 2.0481 76 2.45306750-th BO -3.1661 94 1.862004 -6.852861 0.520473Bi 1.656225 0.426827 0.811131 2.50131982 1.136985 0.075747 0.987009 1.2869611500 137 1-st BO 4.258485 0.486880 3.295512 5.221457BI 0.008516 0.004937 -0.001248 0.018280B2 2.614979 0.193496 2.232275 2.9976835-th BO 4.160421 0.282753 3.601180 4.719661BI 0.031882 0.006655 0.018721 0.045044B2 2.246680 0.068759 2.110685 2.38267550-th BO 1.026110 1.014719 -0.980845 3.033065B1 0.786889 0.153050 0.484179 1.089599B2 1.368061 0.059702 1.249980 1.4861422000 147 1-st BO 4.324355 0.415126 3.503819 5.144891BI 0.006558 0.003251 0.000132 0.012984B2 2.698128 0.164192 2.373587 3.0226685-th BO 4.572160 0.242603 4.092632 5.051687Bi 0.018948 0.003630 0.011773 0.026123B2 2.422681 0.062943 2.298269 2.54709450-th BO 2.355579 0.777990 0.81 7810 3.893348Bi 0.557161 0.091357 0.376586 0.737736_____ _____ _____B2 1.481541 0.050672 1.381384 1.58169897Table 11. ContinuedSpecies Band- Sample PCTL. Parameter Estimate Asymptotic Asymptotic 95%Width Size Std. Error Confidence IntervalLower I UpperD-FIR 2500 156 1-st BO 4.715891 0.352542 4.019406 5.412375BI 0.002497 0.001190 0.000146 0.004849B2 3.026342 0.157842 2.714508 3.3381755-th 80 4.976790 0.222741 4.536741 5.41684081 0.010362 0.001972 0.006466 0.014257B2 2.628880 0.062488 2.505427 2.75233350-th BO 2.569025 0.696529 1.192956 3.945094BI 0.505602 0.074655 0.358112 0.653092B2 1.516380 0.045539 1.426413 1.606348H-FIR 500 91 1-st BO 6.704616 0.314761 6.079092 7.330139BI 0.000015 0.000016 -0.000017 0.000047B2 4.989596 0.397311 4.200020 5.7791725-th BO 8.058023 0.265309 7.530775 8.585270BI 0.000038 0.000024 -0.000010 0.00008682 4.755550 0.228398 4.301655 5.20944550-th B0 8.658697 1.022116 6.627445 10.689948BI 0.042020 0.01 9239 0.003787 0.080253B2 2.428779 0.158193 2.114403 2.743154690 101 1-st BO 6.812487 0.219617 6.376662 7.248312BI 0.000002 0.000001 -0.000001 0.000005B2 5.737293 0.257871 5.225554 6.2490335-th BO 7.770990 0.202654 7.368827 8.173153Bi 0.000058 0.000022 0.000014 0.00010282 4.602322 0.134833 4.334749 4.86989550-th BO 7.446784 0.887646 5.685270 9.208298BI 0.073958 0.023785 0.026756 0.121159B2 2.231843 0.108545 2.016437 2.4472491000 106 1-st BO 6.416685 0.207420 6.005315 6.828056BI 0.000006 0.000004 -0.000001 0.000013B2 5.325898 0.208005 4.91 3367 5.7384295-th BO 7.416872 0.178397 7.063063 7.770682BI 0.000103 0.000031 0.000042 0.000164B2 4.395493 0.105215 4.186823 4.60416350-th BO 6.864771 0.769678 5.338286 8.391256Bi 0.090226 0.023377 0.043863 0.13658882 2.166317 0.086816 1.994137 2.3384971500 112 1-st BO 5.946716 0.195834 5.558576 6.334856Bi 0.000027 0.000013 0.000002 0.00005282 4.770560 0.164973 4.443586 5.0975345-th 80 6.963850 0.177353 6.612341 7.31536081 0.000231 0.000060 0.000112 0.00035082 4.103058 0.091313 3.922076 4.28403950-th BO 6.195801 0.603487 4.999700 7.391901Bi 0.112075 0.021117 0.070221 0.153929B2 2.095021 0.062697 1.970756 2.21928598Table 11. ContinuedSpecies Band- Sample PCTh. Parameter Estimate Asymptotic Asymptotic 95%Width Size Std. Error Confidence IntervalLower UpperH-FIR 2000 118 1-st BO 5.889656 0.158387 5.575921 6.203392BI 0.000025 0.000010 0.000006 0.000045B2 4.791856 0.138127 4.518252 5.0654605-th BO 6.909466 0.145293 6.621665 7.197266Bi 0.000212 0.000046 0.000120 0.000303B2 4.132754 0.076307 3.981603 4.28390650-th BO 5.984979 0.451950 5.089746 6.880211BI 0.116007 0.016198 0.083921 0.148093B2 2.086596 0.046457 1.994573 2.1786182500 124 1-st BO 5.763927 0.136301 5.494081 6.033773BI 0.000023 0.000007 0.000008 0.000037B2 4.827503 0.113866 4.602073 5.0529325-th BO 6.643247 0.145050 6.356080 6.930415BI 0.000289 0.000058 0.0001 75 0.000404B2 4.022266 0.069714 3.884247 4.16028650-th BO 5.476390 0.397496 4.689436 6.263344BI 0.133550 0.015352 0.103157 0.163943B2 2.041717 0.037989 1.966507 2.116926S-P-F 500 85 1-st BO 3.070890 1.808084 -0.525980 6.667759BI 0.058393 0.093789 -0.128184 0.244971B2 2.035668 0.563417 0.914847 3.1564895-th BO 3.283926 1.178172 0.940157 5.627695Bi 0.050529 0.042302 -0.033624 0.134681B2 2.221839 0.297146 1.630718 2.81296050-th BO 1.920224 1.258342 -0.583028 4.423476BI 0.345769 0.112724 0.121523 0.570015B2 1.751353 0.111717 1.529111 1.973595690 90 1-st BO 3.655284 1.013582 1.640670 5.669898Bi 0.028510 0.033167 -0.037413 0.094434B2 2.282979 0.415683 1.456761 3.1091975-th BO 3.361493 0.779641 1.811866 4.911120BI 0.045477 0.026921 -0.008031 0.098986B2 2.256109 0.211223 1.836279 2.67594050-th BO 1.971697 0.937985 0.107342 3.836052B1 0.354180 0.087871 0.179526 0.528834B2 1.741275 0.085298 1.571735 1.9108141000 98 1-st BO 4.206708 0.515611 3.183085 5.230331BI 0.008869 0.006784 -0.004600 0.022337B2 2.708079 0.275984 2.160178 3.2559805-th BO 3.362588 0.51 8033 2.3341 57 4.391 019Bi 0.043980 0.017158 0.009917 0.078043B2 2.262104 0.138217 1.987707 2.53650050-tb BO -0.145353 1.149766 -2.427939 2.137233BI 0.667303 0.160879 0.347916 0.98669082 1.515688 0.080378 1.356116 1.67526099Table 11. ContinuedSpecies Band- Sample PCTL. Parameter Estimate Asymptotic Asymptotic 95%Width Size Std. Error Confidence IntervalLower UpperS-P-F 1500 106 1-st BO 4.155866 0.333148 3.495141 4.816590B1 0.005471 0.002940 -0.000359 0.011301B2 2.895961 0.194627 2.509962 3.2819615-th BO 3.441 043 0.330689 2.7851 95 4.09689281 0.034250 0.009085 0.01 6232 0.052268B2 2.356515 0.094330 2.169432 2.54359950-th BO 0.006193 1.009469 -1.995863 2.008249BI 0.672739 0.143737 0.387669 0.957809B2 1.510425 0.071228 1.369160 1.6516902000 111 1-st 80 4.128789 0.277944 3.577851 4.679726BI 0.003945 0.001862 0.000253 0.007636B2 3.023383 0.171 550 2.683338 3.3634285-th BO 3.468405 0.267243 2.938681 3.998130BI 0.028816 0.006411 0.016108 0.041523B2 2.422322 0.079345 2.265046 2.57959850-th BO 0.027042 0.901674 -1.760245 1.81432881 0.655024 0.125726 0.405813 0.904236B2 1.521664 0.064073 1.394659 1.6486692500 115 1-st BO 4.056976 0.256635 3.548482 4.565469BI 0.003454 0.001550 0.000383 0.006525B2 3.075922 0.163594 2.751779 3.4000665-th 80 3.4701 56 0.271 934 2.931 349 4.00896381 0.024516 0.005772 0.013079 0.035954B2 2.484889 0.084348 2.317764 2.65201450-th BO 0.527670 0.782025 -1.021822 2.077163BI 0.558417 0.098344 0.363560 0.753275B2 1.580004 0.059294 1.462518 1.697489100Table 12. Relationships between MOE and UCS (Band-Width analysis)Species Band- Sample PCTL. Parameter Estimate Asymtotic Asymtotic 95%Width Size Std. Error Confidence IntervalLower I UpperD-FIR 500 103 1-st BO 14.319708 0.829260 12.674470 15.964945BI 0.004141 0.004211 -0.004213 0.012496B2 2.982119 0.352566 2.282634 3.6816035-th BO 14.280493 1.043247 12.21 0708 16.350278BI 0.067876 0.041551 -0.014561 0.150312B2 2.045655 0.204246 1.640434 2.45087650-th BO 10.314038 1.330582 7.674184 12.953891BI 1.548303 0.330031 0.893527 2.203079B2 1.116504 0.063478 0.990564 1.242444690 115 1-st BO 15.501505 0.435396 14.638817 16.364192BI 0.000627 0.000426 -0.000217 0.001471B2 3.633793 0.234888 3.168390 4.0991 975-th BO 15.609150 0.567982 14.483758 16.734542BI 0.026629 0.010855 0.005121 0.048136B2 2.358114 0.136498 2.087658 2.62856950-th BO 7.309948 1.534782 4.268952 10.350945BI 2.457620 0.478160 1 .51 0200 3.405039B2 0.977673 0.055473 0.867759 1.0875861000 121 1-st BO 15.843633 0.281890 15.285409 16.401857BI 0.000272 0.000136 0.000002 0.000542B2 3.919065 0.173051 3.576375 4.2617555-th 80 15.846048 0.382029 15.089520 16.602576BI 0.020385 0.006017 0.008470 0.03230082 2.447311 0.099117 2.251031 2.64359050-th 80 7.466283 1.079322 5.328916 9.603650Bi 2.395523 0.333675 1.734750 3.056296B2 0.985765 0.039858 0.906834 1.0646961500 129 1-st BO 15.981458 0.214433 15.557097 16.405819Bi 0.0001 77 0.000072 0.000035 0.00031982 4.062718 0.140080 3.785501 4.3399355-th 80 15.909763 0.289500 15.336845 16.48268081 0.017006 0.003984 0.009122 0.02489082 2.509377 0.078748 2.353536 2.66521850-th BO 8.271287 0.719965 6.846486 9.696088BI 2.093486 0.207995 1.681867 2.50510482 1.027746 0.028774 0.970802 1.0846902000 141 1-st 80 15.506693 0.214354 15.082847 15.930539Bi 0.000510 0.0001 56 0.000201 0.000819B2 3.685973 0.103886 3.480558 3.8913885-th BO 15.939766 0.231188 15.482632 16.396899Bi 0.014869 0.002620 0.009689 0.02004982 2.554945 0.058448 2.439374 2.67051550-th BO 11.371619 0.607346 10.170701 12.572537BI 1.203962 0.124719 0.957353 1.45057182 1.198673 0.030782 1.137806 1.259539101Table 12. ContinuedSpecies Band- Sample PCTL. Parameter Estimate Asymtotic Asymtotic 95%Width Size Std. Error Confidence IntervalLower I UpperD-FIR 2500 147 1-st BO 15.540000 0.183461 15.177372 15.902627BI 0.000412 0.000113 0.000188 0.000635B2 3.755150 0.093038 3.571252 3.9390495-th BO 15.957414 0.196076 15.569852 16.344976Bi 0.01 3450 0.002068 0.009362 0.01 7538B2 2.587700 0.051 053 2.486789 2.68861050-th BO 11.304761 0.528300 10.260527 12.348995Bi 1.184385 0.107100 0.972692 1.396077B2 1.206616 0.026915 1.153416 1.259817H-FIR 500 87 1-st BO 1.739825 6.859341 -11.900783 15.380434BI 1.943710 2.261295 -2.553142 6.440563B2 0.962590 0.336008 0.294397 1.6307825-th BO 0.748564 4.097333 -7.399467 8.896594BI 2.440054 1.397499 -0.339039 5.219147B2 0.942342 0.164395 0.615424 1.26925950-th BO 9.139132 1.467774 6.220290 12.057974BI 1.154010 0.290275 0.576764 1.731256B2 1.253008 0.077799 1.098295 1.407722690 96 1-st BO 5.870451 3.084131 -0.254054 11.994957Bi 0.793218 0.618814 -0.435629 2.022066B2 1.231997 0.238130 0.759116 1.7048775-th BO 3.822761 2.222452 -0.590610 8.236133Bi 1.455692 0.570398 0.322989 2.588394B2 1.096308 0.116259 0.865440 1.32717650-th BO 9.680853 1.304746 7.089873 12.271833B1 1.064108 0.240658 0.586208 1.542009B2 1.277393 0.069603 1.139174 1.4156121000 104 1-st BO 9.269859 1.282473 6.725763 11.813955Bi 0.246164 0.126218 -0.004221 0.496549B2 1.603547 0.164845 1.276537 1.9305575-tb BO 5.137106 1.249322 2.658772 7.615440BI 1.103193 0.275752 0.556172 1.650214B2 1.180484 0.075448 1.030815 1.33015450-th BO 9.219007 0.966494 7.301731 11.136284Bi 1.187807 0.190908 0.809096 1.566519B2 1.240888 0.049079 1.143528 1.3382471500 112 1-st BO 11.563760 0.699232 10.177896 12.949624Bi 0.066061 0.027756 0.011049 0.121073B2 2.042838 0.139751 1.765853 2.3198225-th 80 8.668061 0.798599 7.085252 10.250870BI 0.421313 0.098518 0.226052 0.616575B2 1.486325 0.073965 1.339727 1.63292350-th BO 10.275347 0.569551 9.146507 11.40418681 0.947618 0.097679 0.754020 1.141216_____B2 1.314820 0.031814 1.251766 1.377875102Table 12. ContinuedSpecies Band- Sample PCTL. Parameter Estimate Asymtotic Asymtotic 95%Width Size Std. Error Confidence IntervalLower I UpperH-FIR 2000 119 1-st BC 12.093802 0.589258 10.926694 13.260911Bi 0.037177 0.01 5161 0.007149 0.067206B2 2.244106 0.136998 1.972762 2.5154515-tb BO 9.752055 0.694314 8.376870 11.127241Bi 0.261090 0.062249 0.137797 0.384383B2 1.647178 0.076734 1.495196 1.79916150-th BO 10.399676 0.450416 9.507564 11.291788BI 0.886779 0.073734 0.740739 1.03281982 1.339489 0.025747 1.288493 1.3904852500 124 1-st BO 11.910396 0.538171 10.844935 12.975856Bi 0.036814 0.013622 0.009846 0.063782B2 2.250211 0.124278 2.004168 2.4962535-th BO 10.061189 0.596300 8.880646 11.241732Bi 0.213822 0.046466 0.121830 0.305815B2 1.715485 0.070372 1.576164 1.85480650-th BO 10.548875 0.432877 9.691875 11.40587551 0.834697 0.067847 0.700376 0.96901982 1.361444 0.025244 1.311467 1.411421S-P-F 500 88 1-st 80 6.645319 1.866935 2.933337 10.35730281 0.236926 0.193866 -0.148534 0.62238682 1.685655 0.279864 1.129208 2.2421025-th BC 6.777005 1.535020 3.724961 9.829050BI 0.455337 0.233079 -0.008089 0.91 876282 1.477281 0.170795 1.137692 1.81687050-Ui BO 9.423833 0.730945 7.970512 10.877154Bi 0.759385 0.136117 0.488746 1.03002482 1.361950 0.058798 1.245042 1.478857690 88 1-st BO 6.327167 1.860141 2.628693 10.02564281 0.244505 0.198684 -0.150534 0.63954482 1.677959 0.278666 1.123893 2.2320255-th BO 6.691020 1.426772 3.854204 9.52783681 0.453349 0.21 8619 0.01 8673 0.88802482 1.478863 0.161481 1.157794 1.79993150-th BO 10.062145 0.531569 9.005238 11.119052BI 0.658968 0.092451 0.4751 50 0.842786B2 1.407212 0.046500 1.314756 1.4996671000 95 1-st BC 4.246146 2.381444 -0.483632 8.975924BI 0.647732 0.476919 -0.299475 1.59494082 1.322331 0.239865 0.845937 1.7987255-th BO 6.386674 1.133229 4.135971 8.63737781 0.538696 0.194697 0.152008 0.92538482 1.409824 0.119374 1.172736 1.64691250-th 80 11.555058 0.396663 10.767248 12.342868Bi 0.413296 0.051300 0.311410 0.51518282 1.567950 0.041863 1.484806 1.651094103Table 12. ContinuedSpecies Band- Sample PCTL. Parameter Estimate Asymtotic Asymtotic 95%Width Size Std. Error Confidence IntervalLower I UpperS-P-F 1500 103 1-st BO -1.800864 3.493775 -8.732456 5.130729Bi 2.324027 1.348474 -0.351322 4.999377B2 0.922596 0.171717 0.581913 1.2632805-th BO 4.637319 0.954450 2.743706 6.530931Bi 0.831609 0.208983 0.416990 1.246228B2 1.268792 0.080922 1.108244 1.42933950-th BO 11.908701 0.291163 11.331038 12.486363BI 0.354444 0.033746 0.287493 0.421 395B2 1.624480 0.032223 1.560549 1.6884102000 109 1-st BO -0.780178 2.657068 -6.048108 4.487752Bi 1.788484 0.905244 -0.006262 3.583230B2 1.005102 0.153841 0.700095 1.3101085-th BO 3.841471 0.832707 2.190538 5.492405Bi 0.934129 0.194218 0.549070 1.319189B2 1.235090 0.066634 1.102981 1.36719950-th BC 12.004631 0.246442 11.516032 12.493230Bi 0.330463 0.027288 0.276361 0.384564B2 1.652523 0.028064 1.596882 1.7081642500 117 1-st BO -3.257293 3.429507 -10.051169 3.536583BI 2.700002 1.409464 -0.092155 5.492159B2 0.878604 0.151763 0.577960 1.1792475-th BO 2.852842 0.944509 0.981 762 4.723921Bi 1.149292 0.243521 0.666874 1.631709B2 1.167869 0.066669 1.035796 1.29994150-th BO 11.421461 0.207555 11.010294 11.832628BI 0.390287 0.024871 0.341 018 0.439556B2 1.598578 0.021411 1.556163 1.640994104Table 13. Analysis of variance table for the test on uniform regression lineNumber of Degress 0: Sum of Mean F RatioDescription of test parameter freedom Square SquareError (S SE) Error(MSE)A Individual 13o, 13i, 132MOR 9 7707 809168.7 104.99UTS 9 7162 464341.33 64.83UCS 9 7119 163655.91 22.99B Common13p, 13i, 132MOR 3 7713 814177.4UTS 3 7168 480368.05UCS 3 7125 180365.87Test of common 13o, 13i 132 Change in(B-A) SSEMOR 6 5008.7 834.78 795*UTS 6 16026.72 2671.12 41.2*UCS 6 16709.96 2784.99 121.14**significant at c = 0.05105Table 14. Strength property relationships (General relationships)Relationships Species ParameterBO I Bi I B2 I B3UTS-MOR D-FIR -3.4930 0.1702 -18.9570 1.2316H-FIR 2.1844 0.0272 -11.2590 1.6760S-P-F 1.6290 0.2655 2.2350 1.2193UCS-MOR D-FIR 14.8683 0.0340 -18.9570 1.5269H-FIR 13.4110 0.0361 -11.2590 1.5693S-P-F 11.0820 0.2711 2.2350 1.1412MOR-UTS D-FIR -18.9570 4.2110 -3.4930 0.8119H-FIR -11.2590 8.5855 2.1844 0.5966S-P-F 2.2350 2.9669 1.6290 0.8202UCS-UTS D-FIR 14.8683 0.3055 -3.4930 1.2398H-FIR 13.4110 1.0534 2.1844 0.9363S-P-F 11.0820 0.9377 1.6290 0.9359106Table 15. Strength property relatiosnhips based on MOR (Band-Width analysis)UTS-MOR D-FIRRelations Species Band- PCTL. ParameterWidth BO I Bi I B2 I B350069010001500200025001-st5-th50-th1-St5-th50-th1-st5-th50-th1-st5-th50-th1-St5-th50-th1-St5-th50-ti,4.8708223.091 072-12.81 71 554.5858383.435984-7.4198084.8917294.159771-3.1661 944.2584854.1604211.0261104.3243554.5721602.3555794.71 58914.9767902.5690250.0362850.0265140.2429360.0276280.0224630.2044220.0253640.01 37900.0629710.171 0940.0409740.01 89180.2517840.0519620.01 37440.5164000.09791404422.571 055-8.322920-37.4850751.469774-7.438555-28.4679922.729538-6.181606-30.0791645.1551 94-2.374785-28.71 21426.266269-0.059217-26.4941968.7859363.250637-27.7185701.7410291.6688261.1397411.81 35721.7160601.1740391.8651 731.8484261.4062241.3304041.5965401.651 3761.2188201.5474691.7231 031.0054721.3962261.777917H-FIR 50069010001500200025001-St5-th50-ti,1-St5-th50-th1-st5-th50-th1-st5-ti,50-th1-St5-th50-ti,1-St5-ti,50-th6.7046168.0580238.6586976.8124877.7709907.4467846.4166857.4168726.8647715.9467166.9638506.1958015.8896566.9094665.9849795.7639276.6432475.4763900.0114700.0009360.0002160.0001460.0000940.0006480.0001140.0000700.0012240.0000400.0000180.0013950.0000080.0000020.0004560.0000080.0000030.0005195.9887872.745504-17.1782041.861 866-2.006605-16.1144210.376822-3.830542-14.912589-2.495990-8.111302-16.2931 84-4.264076-12.31 7054-22.746187-4.287888-12.544163-24.0091762.1139392.6600192.7079263.3509553.21 05162.4728543.3921 043.2589702.3392453.6351873.5531902.3048834.0724554.0848012.51 80334.0585823.9556712.484430107Table 15. ContoinuedRelations Species Band- PCTL. ParameterWidth BO Bi I B2 I B3UTS-MOR S-P-F 500 1-st 3.070890 0.671439 7.044558 0.8917565-tb 3.283926 0.157934 3.422649 1.30767450-th 1.920224 0.036457 -6.744349 1.655289690 1-st 3.655284 0.509281 7.275396 0.9594275-tb 3.361493 0.183281 4.090538 1.26751350-th 1.971697 0.050759 -5.259637 1.5824751000 1-st 4.206708 0.362596 7.673859 1.0488815-tb 3.362588 0.258016 5.108397 1.17296250-tb -0.145353 0.200464 -2.508248 1.2705391500 1-st 4.155866 0.291058 7.434986 1.1191215-tb 3.441043 0.258115 5.413194 1.17396250-th 0.006193 0.252131 -1.569846 1.2154552000 1-st 4.128789 0.264669 7.335125 1.1503535-th 3.468405 0.249353 5.435238 1.18414350-th 0.027042 0.248140 -1 .679381 1.2186782500 1-st 4.056976 0.315063 7.619745 1.0992735-th 3.470156 0.252445 5.588123 1.18318750-tb 0.527670 0.210779 -1 .683456 1.256328UCS-MOR D-FIR 500 1-st 14.319708 0.037991 2.571055 1.8051385-th 14.280493 0.021125 -8.322920 1.75684750-th 10.314038 0.030301 -37.485075 1.515333690 1-st 15.501505 0.004937 1.469774 2.3800645-th 15.609150 0.008442 -7.438555 1.98703050-th 7.309948 0.176255 -28.467992 1.1744621000 1-st 15.843633 0.005004 2.729538 2.4090595-th 15.846048 0.008146 -6.181606 2.00996650-th 7.466283 0.140695 -30.079164 1.2191951500 1-st 15.981458 0.018726 5.155194 2.0669595-th 15.909763 0.022506 -2.374785 1.78321950-th 8.271287 0.127226 -28.712142 1.2405842000 1-st 15.506693 0.074452 6.266269 1.6650585-th 15.939766 0.043085 -0.059217 1.63195150-tb 11.371619 0.060219 -26.494196 1.3941142500 1-st 15.540000 0.307327 8.785936 1.2476115-tb 15.957414 0.122705 3.250637 1.37435550-tb 11.304761 0.054037 -27.718570 1.414726108Table 15. ContinuedRelations Species Band- PCTL. ParameterWidth BO I Bl I B2 I B3UCS-MOR H-HR 500 1-st 1.739825 7.023133 5.988787 0.4078205-th 0.748564 4.605004 2.745504 0.52709950-th 9.139132 0.076084 -17.178204 1.397021690 1-st 5.870451 2.005990 1.861866 0.7195675-th 3.822761 1.633224 -2.006605 0.76476950-th 9.680853 0.070727 -16.114421 1.4153361000 1-st 9.269859 0.597668 0.376822 1.0213115-th 5.137106 0.996034 -3.830542 0.87525250-th 9.219007 0.101138 -14.912589 1.3399421500 1-st 11.563760 0.078076 -2.495990 1.5566515-tli 8.668061 0.166708 -8.111302 1.28713650-th 10.275347 0.060409 -16.293184 1.4465292000 1-st 12.093802 0.021392 -4.264076 1.9071995-th 9.752055 0.038214 -12.317054 1.62806650-th 10.399676 0.025333 -22.746187 1.6164502500 1-st 11.910396 0.023061 -4.287888 1.8917995-th 10.061189 0.029841 -12.544163 1.687083_______ _______50-th 10.548875 0.020603 -24.009176 1.656651S-P-F 500 1-st 6.645319 1.790151 7.044558 0.7384285-th 6.777005 0.971432 3.422649 0.86946150-th 9.423833 0.132036 -6.744349 1.287245690 1-st 6.327167 2.034498 7.275396 0.7051665-th 6.691020 1.130361 4.090538 0.83084650-th 10.062145 0.137094 -5.259637 1.2788771000 1-st 4.246146 3.965616 7.673859 0.5121595-th 6.386674 1.622681 5.108397 0.73103250-th 11.555058 0.119115 -2.508248 1.3143491500 1-st -1.800864 8.243011 7.434986 0.3565305-th 4.637319 2.467156 5.413194 0.63208350-th 11.908701 0.123351 -1 .569846 1.3072362000 1-st -0.780178 7.240354 7.335125 0.3824265-th 3.841471 2.807108 5.435238 0.60376950-th 12.004631 0.115162 -1.679381 1.3234822500 1-st -3.257293 9.800470 7.619745 0.3139955-th 2.852842 3.438682 5.588123 0.556084_50-th 11.421461 0.145639 -1.683456 1.271098109Table 16. Strength property relationships based on UTS (Band-Width analysis)Relations Species Band- PCTL. ParameterWidth BO I B1 I B2 I B3MOR-UTS D-FIR 500 1-st 2.571055 6.718249 4.870822 0.5743735-tb -8.322920 8.804249 3.091072 0.59922450-th -37.485075 3.460701 -12.817155 0.877392690 1-st 1.469774 7.234936 4.585838 0.5513985-th -7.438555 9.133810 3.435984 0.58273050-th -28.467992 3.866029 -7.419808 0.8517601000 1-st 2.729538 7.170769 4.891729 0.5361435-th -6.181606 10.150768 4.159771 0.54100150-th -30.079164 7.144313 -3.166194 0.7111241500 1-st 5.155194 3.769988 4.258485 0.7516525-tb -2.374785 7.397123 4.160421 0.62635450-th -28.712142 11.052215 1.026110 0.6055562000 1-st 6.266269 3.100528 4.324355 0.8204665-tb -0.059217 6.759955 4.572160 0.64621750-th -26.494196 12.037721 2.355579 0.5803482500 1-st 8.785936 1.929532 4.715891 0.9945585-th 3.250637 5.281693 4.976790 0.716217______ _______50-th -27.718570 13.011857 2.569025 0.562456H-FIR 500 1-st 5.988787 8.278111 6.704616 0.4730515-th 2.745504 13.759848 8.058023 0.37593750-th -17.178204 22.576131 8.658697 0.369286690 1-st 1.861866 13.955459 6.812487 0.2984225-th -2.006605 17.965825 7.770990 0.31147650-tb -16.114421 19.465324 7.446784 0.4043911000 1-st 0.376822 14.535501 6.416685 0.2948025-th -3.830542 18.804660 7.416872 0.30684550-th -14.912589 17.579298 6.864771 0.4274881500 1-st -2.495990 16.197808 5.946716 0.2750895-th -8.111302 21.700072 6.963850 0.28143750-th -16.293184 17.332725 6.195801 0.4338612000 1-st -4.264076 18.017616 5.889656 0.2455525-tb -12.317054 25.828325 6.909466 0.24481050-th -22.7461 87 21 .222006 5.984979 0.3971 352500 1-st -4.287888 17.857618 5.763927 0.2463915-th -12.544163 25.209186 6.643247 0.25280250-th -24.009176 21.005026 5.476390 0.402507110Table 16. ContinuedRelations Species Band- PCTL. ParameterWidth BO I Bi I B2 I B3MOR-UTS S-P-F 500 1-st 7.044558 1.563119 3.070890 1.1213825-th 3.422649 4.101453 3.283926 0.76471650-th -6.744349 7.393681 1.920224 0.6041 24690 1-st 7.275396 2.020387 3.655284 1.0422885-th 4.090538 3.813819 3.361493 0.78894650-th -5.259637 6.576796 1.971697 0.631 9221000 1-st 7.673859 2.630537 4.206708 0.9533975-th 5.108397 3.173924 3.362588 0.85254250-th -2.508248 3.542785 -0.145353 0.7870671500 1-st 7.434986 3.012769 4.155866 0.8935585-th 5.413194 3.169768 3.441043 0.85181750-th -1.569846 3.106741 0.006193 0.8227372000 1-st 7.335125 3.175732 4.128789 0.8692985-tb 5.435238 3.231 359 3.468405 0.84449350-tb -1 .679381 3.138252 0.027042 0.8205612500 1-st 7.619745 2.859588 4.056976 0.9096925-th 5.588123 3.200915 3.470156 0.84517550-th -1.683456 3.453136 0.527670 0.795970UCS-UTS D-FIR 500 1-st 14.319708 1.183016 4.870822 1.0368225-th 14.280493 0.964902 3.091 072 1.05274450-tb 10.314038 0.198828 -12.817155 1.329541690 1-st 15.501505 0.548281 4.585838 1.3123635-th 15.6091 50 0.684377 3.435984 1.15790250-th 7.309948 0.862704 -7.41 9808 1.0003601000 1-st 15.843633 0.575961 4.891729 1.2916015-th 15.846048 0.858909 4.159771 1.08739350-th 7.466283 1.546764 -3.1661 94 0.8669991500 1-st 15.981458 0.290889 4.258485 1.5536335-th 15.909763 0.798036 4.160421 1.11692750-th 8.271287 2.506480 1.026110 0.7512432000 1-st 15.506693 0.489943 4.324355 1.3661235-th 15.939766 0.974428 4.572160 1.05459450-th 11.371619 1.932558 2.355579 0.8090722500 1-st 15.540000 0.697804 4.715891 1.2408225-th 15.957414 1.208398 4.976790 0.98433550-th 11.304761 2.037880 2.569025 0.795721111Table 16. ContinuedRelations Species] Band- PCTLJF ParameterIWidth BO I Bi I B2 I B3UCS-UTS H-FIR 500 1-st 1.739825 16.629585 6.704616 0.1929195-th 0.748564 18.339703 8.058023 0.1981 5650-th 9.139132 5.920668 8.658697 0.515901690 1-st 5.870451 13.367453 6.812487 0.2147355-th 3.822761 14.873319 7.770990 0.23820850-th 9.680853 4.724127 7.446784 0.5723491000 1-st 9.269859 9.197338 6.416685 0.3010855-th 5.137106 12.989039 7.416872 0.26856750-Ui 9.219007 4.711330 6.864771 0.5728101500 1-st 11.563760 5.959610 5.946716 0.4282185-th 8.668061 8.7531 35 6.963850 0.36224850-th 10.275347 3.742451 6.195801 0.6275932000 1-st 12.093802 5.310379 5.889656 0.4683175-th 9.752055 7.606896 6.909466 0.39856750-th 10.399676 3.534836 5.984979 0.641 9492500 1-st 11.910396 5.383614 5.763927 0.4661235-th 10.061189 6.908103 6.643247 0.42649750-th 10.548875 3.195661 5.476390 0.666813UCS-UTS S-P-F 500 1-st 6.645319 2.489654 3.070890 0.8280605-th 6.777005 3.31 3889 3.283926 0.66489150-th 9.423833 1.734314 1.920224 0.777656690 1-st 6.327167 3.340716 3.655284 0.7349875-Hi 6.691020 3.437461 3.361493 0.65549350-th 10.062145 1.524612 1.971697 0.8081501000 1-st 4.246146 6.507892 4.206708 0.4882915-th 6.386674 3.774994 3.362588 0.62323650-th 11.555058 0.628052 -0.145353 1.0344811500 1-st -1.800864 12.213822 4.155866 0.3185805-th 4.637319 5.115492 3.441043 0.53841950-th 11.908701 0.542876 0.006193 1.0755112000 1-st -0.780178 11.263624 4.128789 0.3324435-th 3.841471 5.699152 3.468405 0.50987950-th 12.004631 0.523199 0.027042 1.0859982500 1-st -3.257293 13.630885 4.056976 0.2856395-th 2.852842 6.566996 3.470156 0.46998850-th 11.421461 0.703720 0.527670 1.011756112Table 17. Summary of the property relationships for the property estimatesGiven property Predicted propertyMOE MOR UTS UCSMOR- UTS UCSUTS MOR- UCS113APPENDIX B: (Figure 1 to Figure 39)1140.2400.2CLL 016‘90.120.08.0‘90.04a-0.24C00.2C0.1600.120.08.0‘90.04a-0.07C0.060C0.050.04C00.030.02‘9.0e 0.01a0 07 -0.060.050.04C‘90.030.02‘9.00.01a-0 L3.5 5.5 7.5 9.5 11.5 13.5 15.5 17.5 19.5 21.5 23.5MOE (x 1000 MPa)6 14 22 30 38 46 54 62 70 78 86 94 102 110MOR (MPa)0Figure 1. Histograms of MOE and strength propertiesD-FIR1153.5 5.5 7.5 9.5 11.5 13.5 15.5 17.5 19.5 21.5 23.5MOE (x 1000 MPa)6 14 22 30 38 46 54 62 70 78 86 94 102 110UTS (MPa)C00.2CLL 0.1600.120.08.0‘9- 0.04F0 I .,....... I I3.5 5.5 7.5 9.5 11.5 13.5 15.5 17.5 19.5 21.5 23.5MOE (x 1000 MPa)0.07C0.060CLE 0.05>.0.04C‘90030.02‘9.0e 0.01a.11111.11 I6 14 22 30 38 46 54 62 70 78 86 94 102 110UCS (MPa)Figure 1. Continued_t_F:3.5 5.5 7.5 9.5 11.5 13.5 15.5 17.5 19.5 21.5 23.5MOE (xl 000 MPa)0.070.060.050.040.030.02to2 0.0100J0.07C0.060C0050.04Cto0030.022 0.010H-FIR0.24C002C0.1600.120.08to0.04a00.07C0.06UC0.050.04C0)0.030.02to2 0.01a-lIIl6 14 22 30 38 46 54 62 70 78 86 94 102 110MOR (MPa)0.24C00.2CU.. 0.16.‘00.12:‘ 0.08.to• 004035 5.5 7.5 9.5 11.5 13.5 15.5 17.5 19.5 21.5 23.5MOE (x 1000 MPa)[6 14 22 30 38 46 54 62 70 78 86 94 102 110UTS (MPa)0.24C00.2CLL. 0.16 —>,00.120.08.to• 0.0400 I r.,3.5 5.5 75 9.5 11.5 13.5 15.5 17.5 19.5 21.5 23.5MOE (x 1000 MPa)1IIIIIIL liii6 14 22 30 38 46 54 62 70 78 86 94 102 110UCS (MPa)116Figure 1. Continued3.5 5.5 7.5 9.5 11.5 13.5 15.5 17.5 19.5 21.5 23.5MOE (x 1000 MPa)0.24C0tO 0.2CLL. 0.162’0012a0.08.000.040S-P-F0.07 -C2 0.06C,CL 0052’0.04CC,o 0.032’0.020.09 0.0100 24C00.2CLL 0.1600.1200.08.000.0400.07C2 0.06CL 0.050.04C0o 0.032’0.020.09 0.010 Jr p.6 14 22 30 38 46 54 62 70 78 86 94 102 110MOR (MPa)0- —3.5 5.5 7.5 9.5 11.5 13.5 15.5 17.5 19.5 21.5 23.5MOE (x 1000 MPa)14 22 30 38 46 54 62 70 78 86 94 102 110UTS (MPa)0.24C00.2C0LL. 0.1600.120>‘0.08.00.040.070.060.050.04CC0 0.03>10,02C,.09 0.01003.5 5.5 7.5 9.5 11.5 13.5 15.5 17.5 19.5 21.5 23.5MOE (x 1000 MPa)-J6 14 22 30 38 46 54 62 70 78 86 94 102 110UCS (MPa)117Figure 2. Graphical presentation of size adjustment on strength propertiesUnadjusted D-FIR Adjusted100909070000C050ci040C3020100100909070060CWooC)0 4003020109 9 101520257035404979556095797590959995100995919MOR (MPa)0 5 10 19 20 25 30 35 40 45 50 55 00 65 70 75 90 05 90 05990105110MOR (MPa)901070060CWooC)WooC-302010CC0C)Ca-0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75UTS (MPa)0 5 10 15 20 25 30 35 40 45 50 55 60 69 70 75UTS (MPa)10090907007= 00C0 soci0 40C-3020100C0C)0C-0 5 10 15 20 25 30 35 40 47 50 55 60 65UCS (MPa)0 5 10 15 20 25 30 35 40 45 50 75 60 65UCS (MPa)118Figure 2. Continued100906070000C050C,0400302010Unadjusted H-FIR Adjusted0C0C,0a-05101520253055404550550065707500059093100105MOR (MPa)0C0(.30a-100900070060C059C,403020100 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75UTS (MPa)0 5 10 23 20 29 50 35 40 45 50 55 00 65 70 75UTS (MPa)0C0(30a-0 5 10 15 16 25 30 35 40 45 50 55 60 65UCS (MPa)0 5 19 15 20 25 36 35 40 45 50 55 90 65UCS (MPa)119100900070060OsoC.,0400302010UnadjustedFigure 2. ContinuedS-P-F Adjusted00C.,0a-100908070066CC.,0403020to0 5 10 15 20 25 30 33 46 43 50 55 00 65 70 73 00 09 90MOR (MPa)0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 15 80 85 90MOR (MPa)0C0C.,00 5 10 15 20 23 30 35 40 45 56 55 66UTS (MPa)0 3 10 15 20 25 36 35 40 45 56 55 60UTS (MPa)0C00a-0C0C.)0a-to 15 10 15 30 35 40 45 56UCS (MPa)4SS: 2 x 4 Select Structural4N2: 2x4No.210 15 20 25 30 35 40 45 50UCS (MPa)120Figure 3. Relationships between MOE and MOR100908070O 605004030201001009080706001234567504030201008 910111213141516171819202122232425MOE (xl000MPa)100 0 00 MOR o@ g9II 0 0-AR1-STPCIL.0 0o0 400070REG060______50LINE_ ____ _ _0805-Th00Q403020100 I I I I I I I I I I I I I I I I I I0 1 2 3 4 5 6 7 8 910111213141516171819202122232425MOE (xl000 MPa)0 MOR1-ST PCL5-111 PCrL- — —- REG. LINEH-FIR00.V000o0I00000— 0V000 0cPo00 MOR1-ST PUlL5-’HIPCILREG. LINE00S-P-F0 00 jOg0O00 :—7900 00/V000 1 2 3 4 5 6 7 8 9 10 1112 13 14 1516 17 18 19 20 21 22 23 24 25MOE (xl000MPa)121Figure 4. Relationships between MOE and UTS100908070o 6050U)1-40D3020100012345 6 7 8 910111213141516171819202122232425MOE (xl000 MPa)100900 1 2 3 4 5 6 7 8 910111213141516171819202122232425MOE (xl000MPa)o U]SD-FIR1-ST PCI!..0 05-ThPCrL0 00 0- - - RF- L0 00 0000 00 - - -0 0 000U)I—D10090807060504030201000 U-IS1-STPCIL5-TN PCIL- — —- REQ. LINE000OcH-FIR00,9 p0,08070U)I-.DS-P-F00 P-—6050403020100I-STPCIL5-TN PUll.REQ. LINE0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25MOE (x 1000 MPa)122Figure 5. Relationships between MOE and UCS100908070cL6O50340D30201000123456 7 8 9 10111213141516171819202122232425MOE (x 1000 MPa)10090807060504030201009 10 1112 13 14 15 16 17 18 1920 21 2223 2425MOE (xl000MPa)0 1 2 3 4 5 6 7 8 910111213141516171819202122232425MOE (xl000 MPa)o uc D-F1R1-ST PCfl.5-111 PCTL.@---REG.L1NE 0 o‘ç °°o6&00o00 uc1-ST PCL5-flIPCILREQ. LINED1009080706050403020100H-FIR0- -_c2-’?01234567800C.)D0 ucs1-ST PCQ.5-THPCrLREQ. LINEoo-QS-P-F123Figure 6a. Two strength properties at equivalent rank of percentile level403530<25200 1510500 10 20 30 40 50 60 70.2 0.90.80.700.6.0.50.4.? 0.3•10.2E 0.1C00 10 20 30 40 50 60Strength (MPa).000co0’00000000000Property BFigure 6b. Relationship between two strength propertiesat percentiles of equivalent rank124Figure 7. UTS and UTS/MOR as a function of MORD-FIR0000070 0.860 0.70.6 0 UTS(S.S)500.5 UTS(No2)D_40 0___PRD. UTS04Ci) UTS/MOR (S.S)300.3 UTs/MOR(No.2)I-200.2 PRD.UTS/MOR10 0.10 I I I I I I I I 00 10 20 30 40 50 60 70 80 90 100MOR (MPa)60 0.80.7500.6 0 UTS(S.S)40‘V 0.5 UTS(NO.2)0 030 0.4PRD.UTSI 0 UTh/MOR (S.S)0.3 I UThIMOR(No2)200.2 ------------ PRD.UTS/MOR100.10 00 10 20 30 40 50 60 70 80 90MOR (MPa)S-P-F055 0.950 0.8450.7° UTS (SS)400.6..- 35 0 UTS(No.2)0 5 PRD.UTS30.25 0.4 UTS/MOR(S.S)I-200.3 UTS/MOR(NO2)150.2PRD.UTS/MOR105 0.10 I 00 8020 40MOR (MPa)60125Figure 8. UCS and UCS/MOR as a function of MORS-P-F 0,7060504030D201001 .61 .41.2 0 UCS (S.S)1 cr UCS (NO.2)PRD.UCS0.8UCS/MOR (S.S)0.6 D • UCS/MOR(N0.2)0.4 PRO. UCS/MOR0.200 10 20 30 40 50 60 70 80 90 100MOR (MPa)H-FIR 00°060 1.450 1.20 UCS(S.S)400_ 0.8 0_____30PRD.UCS0.6 UCS/MOR(S.S)20 D UCS/MOR(N0.2)0.4PRO. UCS/MOR10 0.20 I I I I I I I I 00 10 20 30 60 70 80 9040 50MOR (MPa)454035302520D105001.41 .2cr.0C)0 UCS(SS)U CS (NO 2)PRO. UCSUCS/MOR (S.S)UCS/MOR (No.2)PRO. UCS/MOR0.80.60.40.2020 40MOR (MPa)60 80126Figure 9. MOR and MOR/UTS as a function of UTSf,GO 000H-FIRI I I I I100 2.22901.880° MOR(S.S)1.6701.4 MOR(NO.2)PRO. MOR601.2MORmTS(s.S)00 400.8 • MORJUTS(NO.2)30 0.6 PRD.MOR/UTS20 0.410 0.20 I I I I I I 00 10 20 30 40 50 60 70UTS (MPa)D-FIR90807060a 504030201000 10 20 30 40 50 60UTS (MPa)2.221.81.61.4 wI-1.200.80.60.4MOR(SS)M OR (NO .2)PRD. MORMOR/UTS (S.S)M OR/U TS (NO .2)PROMOR/UTS0.2070 2.2260 1.8MOR (S.S)50 1.6 01.4 MOR(NO.2)40 1.2 D PRO.MORMOR/UTS (S.S)30 1 00.8 • MOR/UTS(NO.2)20 0.6 PRO.MOR/UTS0.4100.20 I I I I I I I I I 00 5 10 15 20 25 30 35 40 45 50 55UTS (MPa)S-P-F127Figure 10. UCS and UCS/IJTS as a function of UTS00 00D-FIR70 360 2.5UCS (S.S)502 UCS (No.2)O..40___1 5PRD.UCSUCS/UTS (S.S)330 Q= 1 • UCS/UTS(NO.2)20PRO. UCS/UTS10 0.50 00 10 20 30 40 50 60 70UTS (MPa)6050- 40030U,C.)= 201002.221.81.60 UCS (S.S)1.4 u, 0 UCS (No.2)1.2 PRD.UCS1 UCS/UTS (S.S)0.8 = • UCS/UTS(NO.2)0.6 PRD.UCS/UTS0.40.200 10 20 30UTS (MPa)40 50 60S-P-F45 2.42.240235 1.8 0 UCS(S.S)30 1.625 1.4UCS (NO 2)= PRD.UCS12—20 1• 3 UCS/UTS(S.S)= UCS/UTS(NO.2)= 15 0.80.6 PRD.UCS/UTS10__ _ ____________0.45 0.20 I I I I I I 00 5 10 15 20 25 30 35 40 45 50 55UTS (MPa)128Figure 11 a. Illustration of the band-width effectsFigure 1 lb. Moving band-width (window) method129I—CDzwI—Cl)5% exclusion lineAII-(9zwI—(I)MOEpctl.BProbabilitydensityfunction0000U‘0-0-m a 0 o‘a_________Probabilitydensityfunction0J0\00—ProbabiiltydensityfunctionProbabilitydensityfunction______f:C0 m 0 0 0 SProbabilitydensityfunctionProbabiiltydensityfunctionC 000m 0 0 -oF0 m x 0 0 0 SProbabilitydensityfunctionoo00————0t.3QQQ—3•000t3Probabilitydensityfunction0t.aO00—ra0Probabilitydensityfunction—Io6/0 m S 0 0 0 •0 St:f*iS%i£ LZLA IL1I[.::j11H_.:Probabilitydensityfunction0000———Probabilitydensityfunction0000——(2 0 In.S.:0___________m_ot zProbabilitydensityfunctionU—o0Im_f...SO0_I /I...00•0//Probabilitydensityfunctioncccc——-—ct3.O00—‘3.C00i’.)IF’00U0_________m xg.o ocoC//00 00 00 00 00 00(Th 0 (D13380706050-400 302010080706050400 3020100Figure 14. Relationship between MOE and MOR (Band-Width 1,000 MPa)H-FIR50-th pctl.6 7 8 9 10 11 12 13 14 15 16 17MOE (xl,000 MPa)6 7 8 9 10 11 12 13 14MOE (xl,000 MPa)15 16 17 18 19706050201005 6 7 8 9 10 11 12 13 14 15MOE (x 1,000 MPa)134Figure 15. Relationship between MOE and UTS (Band-Width = 1,000 MPa)5045403530!2520D15105055504540030LI)25201510506 7 8 9 10 11 12 13 14 15 16 17 18MOE (xl,000 MPa)D-F1R50-th pctl.6 7 8 9 10 11 12 13 14 15 16 17 18 19MOE (xl,000 MPa)H-FIR135Figure 16. Relationship between MOE and UCS (Band-Width = 1,000 MPa)50454035Q-30-25D1510506 7 8 9 10 11 12 13 14 15 16 17MOE (xl,000 MPa)454035.301:1 2520D 1510505 6 7 8 9 10 11MOE (x 1,000 MPa)12 13 14 1550454035..30-25U)o 20D1510506 7 8 9 10 11 12 13 14 15 16 17 18 19MOE (xl,000 MPa)H-FiR 50-thpctl1-st poti.S-P-F50-th P136Figure 17. Predicted MOR for each band-width and percentile level807005004O30•2001006 7 8 9 10 11 12 13 14 15 16 17MOE (xl,000 MPa)8070050040302001006 7 8 9 10 11 12 13 14 15 16 17 18 19MOE (xl,000 MPa)H-FIRGEF4 RELrpcfl.70160I50Q40D 301:200.. 1005 6 7 8 9 10 11MOE (x 1,000 MPa)12 13 14 151375045Q40-3530D 25201510505550‘4540(1)30252015& 10504540o_ 35• 3025D20a)15100506 7 8 9 10 11 12 13MOE (xl ,000 MPa)14 15 16 17 18Figure 18. Predicted UTS for each band-width and percentile level50-tli pcit.GEN. REL.D-FIRI I I I I I I6 7 8 9 10 11 12 13 14 15 16 17 18 19MOE (xl,000 MPa)pctI.H-FiR1-st pctl.I I I I I I I I I IS-P-F1-st pctl.5 6 7 8 9 10 11 12MOE (xlOOO MPa)13 14 15 16138Figure 19. Predicted UCS for each band-width and percentile levelD-F1R45 GEN. R.40EL30D 2520: 151050 I I I I I I I I6 7 8 9 10 11 12 13 14 15 16 17 18 19MOE (xl,000 MPa)50H-FiR454030thI.D 25205 15S 1050 I I I I6 7 8 9 10 11 12 13 14 15 16 17MOE (xl,000 MPa)45S-P-F40 50-tb pctl35‘3025D20a)15S io050 I I I I I5 6 7 8 9 10 11 12 13 14 15MOE (xl,000 MPa)1390E0=U0S-P-F—-Figure 20. Grade increment factors for MOE-MOR (5-th percentile)1.11.081.061.041.020.980.960.940.920.9H-FIR//1 /, ///6 7 8 9 10 II 12 13 14MOE (x 1000 MPa)15 16 17UEU=U-o01.11.081.061.041.020.980.960.940.920.9B-W=500B-W=690B-W=1000--•-•-B-W=15OOB-W=2000-———B-W=25005 6 7 8 9 10 11 12 13 14 15MOE (x 1,000 MPa)140Figure 21. Grade increment factors for MOE-UT S (5-th percentile)1.1108 D-FIR0 1.06B-W=500— 1.04B-W=6901.02______B-W’lOO(1-.-.B-W=15O(098-- ----BW=200(ci) 096 - ——-———B-W=250C0.940.920.9 I I I I I I I I I I I I6 7 8 9 10 11 12 13 14 15 16 17 18 19 20MOE (x 1,000 MPa)ci)Eci)-1.11.081.061.041.02o 980.960.940.920.9H-FIR6 7 8 9 10 11 12 13 14 15 16 17MOE (x 1,000 MPa)181.11.08 S-P-F1.06B-W=5001.04B-W=690102- BW=10001BW=l5OC0.98:_..B-W=2OOCW 0.96-———B-W=25000.94 —(90.920.9 I I I I I I I I5 6 7 8 9 10 11 12 13 14 15 16MOE (x 1,000 MPa)141Figure 22. Grade increment factors for MOE-UCS (5-th percentile)1.11.081.061.041.02E 10.98C) 0.96-D0.940.920.95 6 7 8 9 10 11 12 13 14MOE (x 1,000 MPa)15D-FIR1.11.081.06B-W=5001.04______B-W=6901.02B-W=lOO0.98-..-..-..- B-W=200C) 0.96 - — —— B-W=2506 7 8 9 10 11 12 13 14 15 16 17 18 19MOE (x 1,000 MPa)H-FIR-1.11.081.06B-W=5001.04--—.. B-W=6901.02B-W=100(0.98-..-..-..- B-W=200(C) 0.96 - — —— B-W=250(0.94___ ___________0.920.96 7 8 9 10 11 12 13 14 15 16 17MOE (x 1,000 MPa)S-P-FV14210080.060040UI20UI00-200 5 10 15 20 25MOE (x 1,000 MPa)Fig. 23. Comparison of the 5 % exclusion lines between linear and non linear model800605040U30020UO 100MEAN5-TM PCTL SLR1-ST PCTL5-THPCTL5 6 7 8 9 10 11 12 13 14 15 16 17 18 19MOE (x 1,000 MPa)Figure 24. Relationships of MOE to the 5-th percentile and mean trend of MOR143U)I—DLiiLii005 6 7 8 9 10 11 12 13 14 15 16 17 18 19MOE (x 1,000 MPa)Figure 25. Relationships of MOE to the 5-th percentile and mean trend of UTS.IU)0DwIii005 6 7 8 9 10 11 12 13 14 15 16 17 18 19MOE (x 1,000 MPa)D-FIRH-FIR- --.- S-P-F5045403530252015105- 5-TH PCTL.D-FIRH-FIRS-P-F5550454035302520151055-TH PCTL.Figure 26. Relationships of MOE to the 5-th percentile and mean trend of UCS.1440 (I) U D 0 U, I D C&FCurryandFewell(1977)G&KGreenandKretschmann(1991)ASTM:ASTMD1990(ASTM1991)Figure27.Strengthpropertyratios basedonMORcM0102030405060708090100110MOR(MPa)Figure28.UTS/MORandUCS/MORasafunctionof MOR(Band-Width=1,000MPa)1.81.7 1.61.5 1.4 1.3 1.2 1.110.90.80.70.60.50.40 C’) C-) D 0 Cl) I DA5-THPCTL50-THPCTLPRD. UTS/MOR(5-THPCTL)PRD. UTS/MOR(GEN.REL)—--—-.PRD. UTS/MOR(EQRA))(5-THPCTL050-THPCTLPRD.UCS/MOR(5-THPCTL)PR]).UCS/MOR(GEN. REL.)—--.PRD.UCS/MOR(EQRA)510152025303540455055606570758085MOR(MPa)Figure28.Continued1.4 1.3 1.201.1(1)1C., D .0.9ci) 0.7 0.6 0.50.4510152025305.THPCTL50-THPCTLPRD.UFS/MOR(5.ThPCTL)PRD.UTS/MOR(GEN.REL)PRD.UTS/MOR(EQRA))5-THPCTL50.THPCTLPRD.UCS/MOR(5.THPCTL)PRD.UCS/MOR(GEN.REL.)PRD.UCS/MOR(EQRA)354045505560657075MOR(MPa)Figure28.Continued0 Cl) 0 D 0 C’) I D001.41.3 1.21.110.90.80.70.60.50.45-THPCTL50-THPCTLPRD. UTS/MOR(5-THPCTL)PRD. UTS/MOR(GEN. REL.)PRD. UTS/MOR(EQ))4(5-THPCTL050-THPCTLPRD.UCS/MOR(5-THPCTL)PRD.UCS/MOR(GEN. REL.)PRD. UCS/MOR(EQRA)510152025303540455055606570MOR(MPa)Figure29.UTS/MORandUCS/MORasafunctionofMOE(Band-Width =1,000MPa)67891011MOE141516171819MPa)A5-THPCTL50-THPCTLPRD.UTS/MOR(5-THPCTL)-————-PRD.UTS/MOR(GEN.REL)(5-THPCTL50-THPCTLPRD.UCS/MOR(5-THPCTL)PRD.UCS/MOR(GEN.REL.)1.81.61.4Cl) C)1.2D 01Cl)I D0.80.60.41213(x1,000Figure29.Continued0 C’, 0 D 0 C,) I DC1.41.3 1.2 1.11.0.90.80.70.60.5 0.4A5-THPCTL5O-THPCTLPRO.UTS/MOR(5-THPCTL)PRO.UTS/MOR(GEN.REL))(5-THPCTL50-THPCTLPRO.UCS/MOR(5-THPCTL)PRO.UCS/MOR(GEN. REL)67891011121314151617MOE(x1,000MPa)Figure29.Continued1.4 1.3 1.2 1.110.7 0.60.5 0.4S-P-F 15UCS/MOR0 Cl) C.) D0.9Cl) I DUTS/MORA5-THPCTL50-THPCTLPRD.UTS/MOR(5-THPCTL)P1W.UTS/MOR(GEN.REL))4(5-THPCTL050-THPCTLPRD.UCS/MOR(5-THPCTL)PRD.UCS/MOR(GEN. REL)IIIIIIIIIIIIIII567891011MOE(x1,000MPa)III121314Figure 30. MOE of MOR vs. MOE of UTS at 50-th percentile level1915141312111098766 7 8 9 10 11 12 13 14 15 16 17 18 1950-thpctl. MOE0fMOR17161514° 1310CI4766 7 8 9 10 11 12 13 14 15 16 1750-th pcdL MOE of MOR151413121110Lfl655 6 7 8 9 10 11 12 13 14 1550-th pclL MOE of MOR152Figure31.StrengthpropertyratiosbasedonUTSU, H D 0 0.41020UTS(MPa)6070G&K:Green&Kretschmaim(1991)ASTM:ASTMD1990(ASTM1991)2.6•2.4c,2.2H2D b%1.801.2 1 0.80.6U’D-FIRH-FIRS-P-FG&KASTM/1.20.2 0•+UCS/UTS030II4050Figure32.MOR/UTSandUCS/UTSasafunctionofUTS(Band-Width1,000MPa)3.A5-THPCTLCl)50-THPCTLDPRD.MOR/UTS(5-THPCTL)2.2PRD. MORIUTS(GEN.REL.)D—-—-.PRD.MORIUTS(EQRA)Cl))l(5-THPCTLI D..50-THPCTL0PRD. UCS/UTS(5-THPCTL)1.6PRD.UCS/{flS(GEN.REL.)14—-—-.PRD. UCS/UTS(EQRA)1.2III5101520502.82.6AD-Fm‘AAI-.,1MOR/UTS1UCS/UTSIIII25 UTS30(MPa)354045Figure32.Continued51015202530UTS(MPa)MOUTS404550A5-THPCTL50-THPCTLPRD.MORJUTS(5-THPCTL)PRD.MORIUTS(GEN.REL)PRD.MORIUTS(EQRA))K5-THPCTL50-THPCTLPRD.UCS/UTS(5-THPCTL)PRD.UCS/UTS(GEN.REL)PRD.UCS/UTS(EQRA)AAH-FIRcji U’Cl) I D Cl) C-) D Cl) I D 02.42.2 2 1.8 1.6 1.4 1.21-IIIIIII.1,JCS/UTSIIII35Figure32.ContinuedCl)I D Cl)C-)D Cl)I D 0cJ2.62.42.2 2 1.8 1.6 1.4 1.210.8A5-THPCTL5O-THPCTLPRD. MORJUTS(5-THPCTL)PRD.MORJUTS(GEN.REL.)PRD.MORIUTS(EQRA))5-THPCTLD’50-THPCTLPRD.UCS/UTS(5-THPCTL)PRD.UCS/UTS(GEN. REL)PRD.UCS/UTS(EQRA)510152025UTS(MPa)303540Cl)I D Cl) C-) D Cl) I D 01678910111213141516171819MOE(x1,000MPa)Figure33.MOR/UTSandUCS/UTSasafunctionofMOE(Band-Width=1,000MPa)32.82.62.42.2 2 1.8 1.6 1.4 1.25-THPCTL50-THPCTLPRD. MORIUTS(5-THPCTL)PRD. MORIUTS(GEN.REL.))<5-THPCTL050-THPCTLPRD. UCS/UTS(5-THPCTL)PRD.UCS/UTS(GEN.REL.)Figure33.ContinuedCl)I D Cl) C-) D Cl) I D 067891011121314MOE(x1,000MPa)151617002.42.2 2 1.8 1.6 1.4 1.21A5.THPCTL50-THPCTLPRO.MORIUTS(5.THPCTL)PRD.MORJUTS(GEN.REL))5.THPCTL50.THPCTLPRD.UCS/UTS(5.THPCTL)PRD.UCS/UTS(GEN. REL)Figure33.ContinuedCl)I D Cl)C) D Cl) I D 02.62.42.2 2 1.8 1.6 1.4UI1.210.8A5-THPCTL•50-THPCTLPRD.MORJUTS(5-THPCTL)PRD.MORIUTS(GEN.REL.))K5-THPCTL050-THPCTLPRD.UCS/UTS(5-THPCTL)PRD.UCS/UTS(GEN.REL)567891011MOE(x1,000MPa)12131415Figure 34. Relationships between MOE and the predicted strength(General Relationship and 5-th percentile)160Figure 35. Comparison between 5-th percentile and mean trend forproperty ratios based on MOR1.61.41 2CS1ORS-P-F::010 15 20 25 30 35 40 45 50 55 60 65MOR (MPa)0.80.2UTS/MOR (5-HI PCTL)- --. UTS/MOR (GEN. REL)UTS/MOR (EQRA)UCS/MOR (5-Eli PCTL)- - --- UCS/MOR (GEN. REL)UCS/MOR (EQRA)1.61.41.2C,)C-)00.6C,)I—0.40.2UTS/MOR (5-UI PCTL)-. - --. UTS/MOR (GEN. REL).UTS/MOR (EQRA)UCS/MOR (5-TH PCTL)-- --. UCS/MOR (GEN. REL)UCS/MOR (EQRA)10 15 20 25 30 35 40 45 50 55 60 65MOR (MPa)1.6H-FIR1.4 \ucs/M0R1.2UTS/MOR (5-TH PCTL)1- --- UTSfMOR (GEN.REL)UTSIMOR (EQRA)0.8UTS/MOR ---- UCS/MOR (5-TH PCTL)0.6- - .-. UCS/MOR (GEN.REL)UCS/MOR(EQRA)D 0.40.20 I I I I I I I I10 15 20 25 30 35 40 45 50 55 60 65MOR (MPa)0Cl)C)0I—D161Figure 36. Percentile-level effects on predicted strength ratios based on MOR2520151-5-104)-15-20-25DFTRI I I I I I I I IUTS. (GEN. REL).TJTS. (RQRA)-. - --TiCS. (GEN. REL)TiCS. Q)5 10 15 20 25 30 35 40MOR (MPa)45 50 55 60 65H-FIR201510IJFS. (GEN. REL.)I ------ - (EQRA)>0 I4 — . - - - -. UCS. (GEN. REL.)-5_... UCS. (EQRA)-10-15—20 I I I10 15 20 25 30 35 40 45 50MOR (MPa)3025201511:-5UTS. (GEN. REL.)UTS. (EQRA)UCS. (GEN.REL.)H----- TiCS. Q)10 15 20 25 30MOR (MPa)35 40 45 50162Figure 37. Comparison between 5-th percentile and mean trend forproperty ratios based on UTS2.62.42.22IU) 16C-)1.41.2Q 0.80.60.40.20MORIUTS (5-TM PCTL)- - -.• MORJUTS (GEN.REL)MOR!UTS (EQRA)UCS/UTS (5-TH PCTL)- - --. UCS/UTS (GEN.REL)UCS/UTS (EQRA)10 15 20 25 30 35UTS (MPa)H-FIR2.62.42.221.8 MOR/UTS (5-TM PCTL)1.6- --. MOR/UTS (GEN.REL)1.4 MOR/UTS (EQRA)1.2 UCS/UTS (5-TM PCTL)1- - -.. UCS/UTS (GEN.REL)0.8 UCSJUTS (EQRA)0.60.40.20- I I I I I5 10 15 20 25 30 35UTS (MPa)UCS/UTSMOR!UTSS-P-FI I IUCS/UTS2.62.42.221.8 MORJUTS (5-TH PCTL)1.6- - - - --. MOR/UTS (GEN.REL)1.4 MOR/UTS (EQRA)1.2 UCS/UTS (5-TM PCTL)1-. - .UCS/UTS (GEN.REL)0.8 UCS/UTS (EQRA)0.60.40.205 10 15 20 25 30 - 35UTS (MPa)163Figure 38. Percentile-level effects on predicted strength (based on UTS)15— 105-5-10-15-20-25D-FIRMOR. (GEN. REL).MOR. (EQRA)- -- UCS. (GEN. REL.)UCS. (EQRA)5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30UTS (MPa)S-P-F1015___________0 MOR. (GEN. REL).wMOR. (EQRA)>-.- ... UCS. (GEN. REL.)-10-..-..-... UCS. (EQRA)QJ -15-20 I I I I I I5 7.5 10 12.5 15 17.5 20 22.5 25UTS (MPa)164Figure 39. Error in the prediction of MOR as the percentage of the given MOR165161412__100’•—‘ 8LU20-2-45 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100MOR (MPa)\.. ‘.‘I I I I I I I

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