PERFORMANCE OF LAMINATED VENEER WOOD PLATESIN DECKING SYSTEMSbyFrank Chung-Fat LamB.A.Sc. (Civil Engineering) University of British Columbia, 1982M.A.Sc. (Civil Engineering) University of British Columbia, 1985A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDepartment of ForestryWe accept this thesis as conformingto the required StandardThe University of British ColumbiaAugust, 1992© Frank Chung-Fat Lam, 1992In presenting this thesis in partial fulfilment of the requirements for an advanced degree at theUniversity of British Columbia, I agree that the Library shall make it freely available for reference andstudy. I further agree that permission for extensive copying of this thesis for scholarly purposes may begranted by the head of my department or by his or her representatives. It is understood that copying orpublication of this thesis thesis for financial gain shall not be allowed within my written permission.Faculty of Graduate StudiesDepartment of ForestryThe University of British ColumbiaVancouver, B.C.CanadaDate:^214j_q.st 5, 199211ABSTRACTA new laminated veneer wood plate has been developed which is a specialty product intendedfor highly engineered end-use including flat bed truck and dry freight van trailer decking systems. Theinvestigation described in this thesis represents the first known attempt to 1) develop a theoreticalframework for evaluating the performance of laminate veneer panels in prototype dry freight van trailerdecking systems, 2) develop a testing facility to generate an experimental database on these productthrough full scale testing, and 3) develop and model the fatigue behavior of this type of productthrough small specimen tests and damage accumulation laws.A structural analysis model has been developed to predict the structural behavior of aprototype decking system. A comprehensive database on the mechanical properties of 3.2 and 2.5 mm(gl and i6 inch) thick Douglas-fir veneers has been generated through experimental studies as input tothe model. The database includes information on bending, tension, and compression strengthproperties, shear moduli of rigidity, ultrasonic transmission time, and connection stiffness. Analyses ofvariance has indicated that the mean strength properties of 3.2 and 2.5 mm thick veneers aresignificantly different for the parallel to grain direction but not significantly different for theperpendicular to grain direction at the 95% probability level. Statistical information and distributionsparameters have been established for the various veneer strength properties so that simulations can beperformed to model the strength properties of the veneers.A trailer decking load simulator test facility has been developed so that full scale testing ofprototype dry freight van trailer decking systems can be performed. The experimental program hasbeen divided into two phases: 1) static test program and 2) cyclic test program. Four prototype deckingsystems have been considered. The static test program has generated information on the structuralbehavior of the panels in the prototype system. Experimental results agree well with predictions fromthe computer model. The cyclic test program has generated information on the performance of theprototype system under fatigue condition.iiiInformation on the fatigue behavior of the panels in a system has been established from testingof small specimens in bending mode with the appropriate stress history. The relationship between thefatigue life and failure mode of the small specimen tests and the full size panels in system has beenestablished. Damage accumulation laws have been developed from the small specimen tests resultswhich provide a basis for evaluation of the fatigue behavior of the material in decking systems.ivTABLE OF CONTENTSABSTRACT^ iiTABLE OF CONTENTS^ ivLIST OF TABLES viiiLIST OF FIGURES^ xiACKNOWLEDGEMENTS xv1. INTRODUCTION^ 12. BACKGROUND 72.1 Structural Analysis Models for Floor Systems^ 72.2 Material Properties^ 112.2.1 Elastic Properties of Laminated Veneer Wood Plates^ 112.2.2 Fatigue Data^ 122.3 Full Scale Testing of Dry Freight Van Trailer Decking Assembly^ 142.4 Modeling Fatigue Behavior in Wood Panel Products^ 153. STRUCTURAL ANALYSIS MODEL^ 183.1 Introduction^ 183.2 Strain Energy in the Cover^ 193.3 Strain Energy in the Supporting I-beams^ 223.4 Strain Energy in the Connectors^ 243.5 Potential Energy in the Applied Load 253.6 Finite Strip Formulation^ 263.6.1 Cover^ 273.6.2 Supporting I-beams^ 363.6.3 Connectors^ 363.6.4 Load Potential 37TABLE OF CONTENTS3.7 Minimization of Energy in System^ 393.7.1 Cover^ 403.7.2 Supporting I-beams^ 433.7.3 Connectors^ 433.7.4 Load Potential 443.8 Numerical Solution of Global System of Equations^ 453.9 Shear Deflection of the Supporting I-beams^ 473.10 Bending Stresses and Rolling Shear Stresses in Cover^ 483.11 Computing Environment and Efficiency^ 573.12 Sample Problems^ 583.12.1 Program Input 583.12.2 Program Output^ 614. VENEER MECHANICAL PROPERTIES TESTING PROGRAM^ 814.1 Ultrasonic Veneer Testing Program^ 814.2 Mechanical Properties Test Program 824.2.1 Materials and Methods^ 824.2.1.1 Bending Tests 844.2.1.2 Tension Tests^ 884.2.1.3 Compression Tests 904.2.1.4 Shear Modulus of Rigidity Tests^ 944.2.1.5 Connector Load Slip Tests 954.2.2 Results^ 994.2.2.1 Veneer Strength Properties Statistics^ 994.2.2.2 Effects of Veneer Thickness and Number of Plies^ 100viTABLE OF CONTENTS4.2.2.3 Correlations of Veneer Strength Properties^ 1274.2.2.4 Effectiveness of Ultrasonic Testing on Veneer Grading^ 1334.2.2.5 Connector Stiffness^ 1365. FULL SCALE TESTING OF DRY FREIGHT VAN TRAILER DECK ASSEMBLY^1415.1 Testing Facility^ 1415.2 Materials and Methods 1495.2.1 Static Test Program^ 1535.2.2 Cyclic Test Program 1555.2.3 Short Term Small Specimen Bending Tests^ 1565.3 Experimental Results^ 1565.3.1 Static Test Program 1565.3.1.1 Verification of Deck Analysis Model^ 1585.3.2 Cyclic Test Program^ 1726. CYCLIC TESTING OF SMALL SPECIMENS^ 1876.1 Materials and Methods^ 1876.1.1 Short Term Bending Tests^ 1886.1.2 Cyclic Bending Tests 1886.1.2.1 Applied Stress Levels^ 1906.2 Experimental Results^ 1926.2.1 Short Term Bending Tests^ 1926.2.2 Cyclic Bending Tests 1997. MODELING OF SMALL SPECIMEN FATIGUE PROPERTIES^ 2087.1 Ramp Load Case^ 2097.2 Piecewise Linear Representation of Stress History^ 210viiTABLE OF CONTENTS7.3 Representation of Stress History by a Series of Stress Pulses^ 2147.4 Model Calibration^ 2177.4.1 Calibration Results 2198. CONCLUSIONS^ 2318.1 Summary and Conclusions^ 2318.2 Future Research^ 2339. REFERENCES^ 235vu'LIST OF TABLESTable 1.^Descriptions of the various case studies in the example problem.^ 62Table 2.^Summary deflection results of the supporting I-beams in the eight case studies.^63Table 3.^Summary bending stress results of the supporting I-beams in the eight case studies.^64Table 4.^Maximum deflections and stresses in the cover of the eight case studies.^65Table 5.^Locations of maximum stresses in the cover of the eight case studies. 67Table 6.^Test span for the bending specimens.^ 85Table 7.^Specimen depths for the compression tests. 91Table 8.^Specimen sizes for the shear modulus of rigidity tests.^ 94Table 9.^Classification of veneer test specimens.^ 99Table 10.^Statistical data on the veneer elastic moduli. 101Table 11.^Statistical data on the veneer strengths.^ 102Table 12.^Analysis of variance results on veneer bending strength properties.^ 110Table 13.^Analysis of variance results on veneer tension strength properties. 111Table 14.^Analysis of variance results on veneer compression strength properties.^112Table 15.^Analysis of variance results on veneer modulus of rigidity.^ 113Table 16.^Duncan's multiple range test results for the various groups. 114Table 17.^Statistical data and distribution parameters of the veneer bending strengthproperties.^ 123Table 18.^Statistical data and distribution parameters of the veneer tension strengthproperties.^ 124Table 19.^Statistical data and distribution parameters of the veneer compression strengthproperties.^ 125Table 20.^Statistical data and distribution parameters of the veneer shear modulus of rigidity. 126ixLIST OF TABLESTable 21.^The dependent and independent variables considered in the various regressionmodels of veneer strength properties.^ 128Table 22.^Results of various regression models of 3.2 mm veneer strength properties forthe parallel to grain direction.^ 129Table 23.^Results of various regression models of 2.5 mm veneer strength properties forthe parallel to grain direction.^ 130Table 24.^Results of various regression models of 3.2 mm veneer strength properties forthe perpendicular to grain direction.^ 131Table 25.^Results of various regression models of 2.5 mm veneer strength properties forthe perpendicular to grain direction.^ 132Table 26.^Statistical data on the veneer strength properties of the various subgroups.^134Table 27.^Statistical data on the veneer elastic properties of the various subgroups.^135Table 28.^Duncan's multiple range test results for the various groups.^ 137Table 29.^Summary of connection load slip tests.^ 139Table 30.^Analysis of variance results and Duncan multiple range test results on connectorstiffness.^ 139Table 31.^Trailer decking load simulator calibration results.^ 146Table 32.^Peak midspan deflections under 40 kN front axle loading.^ 161Table 33.^Front axle load levels versus peak midspan prototype deformation.^162Table 34.^Veneer elastic moduli for DAP analyses.^ 164Table 35.^Input panel stiffness values for DAP analyses. 164Table 36.^Statistics of panel stiffness values for DAP analyses.^ 170Table 37.^Cyclic test results of full scale prototypes.^ 173LIST OF TABLESTable 38.^Regression parameters for the FR and Nf relationships of the four prototypeassemblies.^ 174Table 39.^Results of regression approach to analysis of covariance of the FR and Nfrelationships.^ 177Table 40.^Results of regression approach to analysis of covariance of the SR and Nfrelationships.^ 180Table 41.^Short term static bending test results of various prototypes.^ 185Table 42.^Short term small specimen static bending test results of various prototypes.^194Table 43.^Analysis of variance results on parallel to grain bending strengths.^ 197Table 44.^Small specimen cyclic bending test results.^ 201Table 45.^The parameters describing piecewise linear segments of a nondimensional stresscycle.^ 211Table 46.^The mean and standard deviation of the model parameters for prototypes 1 and 2. 222Table 47.^Model predicted small specimen fatigue performance.^ 225Table 48.^Model predicted full scale panel fatigue performance. 227xiLIST OF FIGURESFigure 1^Veneer lay up of transDeckTM .^ 4Figure 2^A prototype dry freight van trailer decking system.^ 10Figure 3^Finite strip representation of a section of decking assembly.^ 20Figure 4^Degrees of freedom for a supporting I-beam.^ 23Figure 5^Deformation profile of cover: case 7.^ 71Figure 6^Bending stress profile o in exterior ply of cover: case 7.^ 72Figure 7^Bending stress profile cry, in exterior ply of cover: case 7. 73Figure 8^Rolling shear stress profile ryz in cross ply of cover: case 7.^ 74Figure 9^Rolling shear stress profile 7-,,z in cross ply of cover: case 7. 75Figure 10^Deformation profile of cover: case 6.^ 76Figure 11^Bending stress profile cs, in exterior ply of cover: case 6.^ 77Figure 12^Bending stress profile ay, in exterior ply of cover: case 6. 78Figure 13^Rolling shear stress profile r ,, in cross ply of cover: case 6.^ 79Figure 14^Rolling shear stress profile Txz in cross ply of cover: case 6. 80Figure 15^Sonic transmission time cumulative probability distributions for 2.5 and 3.2 mmthick veneer.^ 83Figure 16^The veneer bending test set up.^ 86Figure 17^The veneer parallel to grain tension test set up.^ 89Figure 18^The veneer compression test set up.^ 92Figure 19^The veneer shear modulus of rigidity test set up.^ 96Figure 20^The connector load slip test set up.^ 97Figure 21^The cumulative probability distributions of veneer bending modulus of elasticity.^103Figure 22^The cumulative probability distributions of veneer bending strength.^104Figure 23^The cumulative probability distributions of veneer tension modulus of elasticity.^105xiiLIST OF FIGURESFigure 24^The cumulative probability distributions of veneer tension strength.^106Figure 25^The cumulative probability distributions of veneer compression modulus ofelasticity.^ 107Figure 26^The cumulative probability distributions of veneer compression strength.^108Figure 27^The cumulative probability distributions of veneer shear modulus of rigidity.^109Figure 28^The cumulative probability distributions and the 3-parameter Weibull distributionsof the veneer bending modulus of elasticity of each group.^ 116Figure 29^The cumulative probability distributions and the 3-parameter Weibull distributionsof the veneer bending strength of each group.^ 117Figure 30^The cumulative probability distributions and the 3-parameter Weibull distributionsof the veneer tension modulus of elasticity of each group.^ 118Figure 31^The cumulative probability distributions and the 3-parameter Weibull distributionsof the veneer tension strength of each group.^ 119Figure 32^The cumulative probability distributions and the 3-parameter Weibull distributionsof the veneer compression modulus of elasticity of each group.^ 120Figure 33^The cumulative probability distributions and the 3-parameter Weibull distributionsof the veneer compression strength of each group.^ 121Figure 34^The cumulative probability distributions and the 3-parameter Weibull distributionsof the veneer shear modulus of rigidity of each group.^ 122Figure 35^Connector load deformation curves.^ 138Figure 36^The trailer decking load simulator. 142Figure 37^Relationship between the front to rear axles load ratios and wheel cart load cellreadings.^ 147Figure 38^Relationship between the front axle loadings and wheel cart load cell readings.^148LIST OF FIGURESFigure 39^Prototype dry freight van trailer decking system used in the full scale test program. 150Figure 40^Prototype dry freight van trailer decking system during a static test.^154Figure 41^Midspan deformation profiles of the four prototype deck assemblies with the frontaxle located directly on Bay 9.^ 157Figure 42^Midspan deformation profiles of the four prototype deck assemblies with the frontaxle centered between Bays 2 and 3.^ 159Figure 43^Midspan deformation profiles of the four prototype deck assemblies with the frontaxle located directly on Bay 13.^ 160Figure 44^Comparisons of model predictions and measured midspan deformation profiles ofthe four prototype deck assemblies with the front axle located directly on Bay 9.^166Figure 45^^Comparisons of model predictions and measured midspan deformation profiles ofthe four prototype deck assemblies with the front axle centered between Bays 2and 3.^ 167Figure 46^Comparisons of model predictions and measured midspan deformation profiles ofthe four prototype deck assemblies with the front axle located directly on Bay 13.^168Figure 47^Upper and lower bounds of measured midspan deformation profiles of prototype 1with the front axle located directly on Bay 9.^ 171Figure 48^Performance of transDeckTM under fatigue loading. 175Figure 49^Normalized fatigue performance of transDeck TM with regular veneer.^183Figure 50^Normalized fatigue performance of transDeck TM with special veneer. 184Figure 51^Parallel to grain bending stress profiles for prototypes 1 and 2.^ 191Figure 52^Nondimensional profiles versus time for prototypes 1 and 2. 193Figure 53^Displacement controlled parallel to grain bending strength cumulative probabilitydistributions.^ 195xivLIST OF FIGURESFigure 54^Displacement controlled perpendicular to grain bending strength cumulativeprobability distributions.^ 196Figure 55^Load controlled parallel to grain bending strength cumulative probabilitydistributions.^ 198Figure 56^Small Specimen during a cyclic bending test.^ 200Figure 57^Typical rolling shear failure mode of a specimen under cyclic loading.^202Figure 58^Fatigue data of small specimens in bending.^ 203Figure 59^Comparisons of fatigue performance of small specimens and full size panels.^205Figure 60^Representation of stress cycle with stress pulses.^ 215Figure 61^Cumulative probability distribution of Nf for Prototype 1.^ 220Figure 62^Cumulative probability distribution of N f for Prototype 2. 221Figure 63^Comparisons of model predicted and actual cumulative probability distribution of1\11 for Prototype 1.^ 223Figure 64^Comparisons of model predicted and actual cumulative probability distribution ofNf for Prototype 2.^ 224Figure 65^Comparisons of model predicted and actual fatigue performance of small specimens. 228Figure 66^Comparisons of model predicted and actual fatigue performance of full size panels. 229XVACKNOWLEDGEMENTI would like to thank Drs. J.D. Barrett and R.O. Foschi for their guidance during the research.Dr. S. Avramidis is thanked for serving on the supervisory committee.Mr. R. J-M. Fouquet, the inventor of transDeck TM , is thanked for sharing his knowledge onthe product and his active role in setting up important linkages between this research project and theindustry.The following technical staff members of the Department of Harvesting and Wood Science areacknowledged for their invaluable contributions into the experimental program: Mr. G. Lee, Mr. J.Bernaldez, Mr. D. Trickett, Mr. R. Johnson, and Mr. B. Myronuk.Finally, Ainsworth Lumber Company Ltd., Forestry Canada, Natural Science and EngineeringResearch Council and the Department of Harvesting and Wood Science are thanked for their supportwith finances, equipment, and materials.11. INTRODUCTIONLaminated veneer wood products such as parallel laminated veneer lumber and plywood arecommon construction materials. Applications of parallel laminated veneer lumber include beams,flanges, headers, columns, and truss chord members. Plywood, on the other hand, is used mainly assheathing for floors, roofs and walls, web members for wooden I beams or box beams, stress-skinpanels, form work, and connection gussets. Laminated wood plates such as plywood play a veryimportant role in typical wooden floor and roof systems. These structural systems behave as stiffenedplates where the applied loads are carried by composite action. Interaction between the supportingframing members and the panel cover makes possible the sharing of loads amongst the supportingmembers. This is especially important when the stiffnesses of supporting members are nonuniform andwhen the ratio of stiffness between cover and framing members is not large.In some design codes, the design stress computed for a single beam in a system can bemultiplied by a system modification factor to account for system behavior. For example, Foschi et al.(1989) obtained an average system modification factor of 1.34 for bending members in joists supportedfloor and roof systems. Similarly Lam and Varoglu (1988) obtained a system modification factor of1.10 for tension members in parallel chord trusses supporting roof systems. Therefore in these structuralsystems where the parallel supporting members were spaced not more than 600 mm apart, significantincrease in load carrying capacity in the system over the load carrying capacity of individual memberscan be expected. It is evident that engineered application of laminated wood plates in new markets,where the demand for quality and performance is high, should be explored to take advantage of thecomposite action and load sharing behavior of stiffened plate systems.Decking systems for commercial trucks and trailers are required to perform in a demandingenvironment. In these systems high concentrated loads are repeatedly applied onto the decking by thewheels of lift trucks. Also the decking systems are exposed to weathering. The expected service life of2these systems is typically three years. Traditionally planks made from Apitong or Kerning species arewidely used in commercial truck and trailer decking. Apitong is a commercial species designation for agroup of over fifteen hardwood species grown in Malaysia with superior structural properties, wear anddecay resistance; therefore, it is widely accepted as the preferred species group for commercial truck andtrailer decking. Planks from certain North American hardwood and softwood species are also acceptedin truck and trailer decking applications where weight optimization is of primary concern and thelimited life expectancy of the softwood decking can be tolerated by the end user.In a limited volume, traditional softwood plywood panels are also being used as flooringmaterials for trucks and trailers. Factors limiting wider acceptance of the plywood structural panelsinclude: 1) inferior structural capabilities in truck and trailer decking application, 2) poor wearresistance, and 3) the inherent sensitivity of wooden panels to changes in moisture content. Increasingconcern about the continued availability of Apitong hardwood resource in suitable quality, theirregularity of its supply, its relatively high cost and sensitivity to edge and surface damage due toimpact, coupled with a need to reduce the dead weights for increased performance in vehicles have ledto an opportunity for softwood laminated veneer wood plates for decking material provided itsstructural performance, wear resistance performance, and moisture resistance characteristics canconform with the end-use requirements.Ainsworth Lumber Company Ltd. has recently developed a new innovative concept andtechnology for manufacturing a specialized value-added laminated veneer wood plate calledtransDeckTM . It is intended for highly engineered end-uses such as the market for truck and trailerdecking systems. A patent application was granted by the U.S. Patent Office on the manufacturingconcept of this product (Fouquet, 1991). The background research for the development of transDeck TMwas documented in two confidential research notes from Council of Forest Industries of BritishColumbia for Ainsworth Lumber Company Ltd. (Parasin and Nagy, 1989 and 1990).TransDeckTM is made from either 2.5 mm ( 1io inch) or 3.2 mm (8 inch) or combinations of 2.53mm and 3.2 mm thick veneers. The current study focuses on transDeckTM panels made from either 2.5mm or 3.2 mm thick veneers. Such panels consists of 11 plies. A typical veneer lay up scheme of thesetransDeckTM panels is shown in Figure 1. The lay up is a 3-1-3-1-3 scheme where the exterior andcenter of the panel consist of 3-ply veneers with the grain running in the longitudinal direction of thepanel. The two interior cross plies, each consisting of single ply veneer orientated 90° to the longitudinaldirection, are sandwiched by the 3-ply veneers. This particular lay up was judged to be mostappropriate for decking of truck and trailer systems based on results of the pilot research projects(Parasin and Nagy, 1989 and 1990).The durability and moisture resistance requirements of the prototype product for truck andtrailer decking systems were resolved by introducing proprietary surface treatments and sealants. Thesurfaces of prototype transDeckTM panels were covered by a proprietary paper product which wasnormally used in concrete forming panels. Since the propriety paper product performed very well inconcrete forms, it was judged that it would perform equally well in decking systems for trucks andtrailers. All edges were sealed by a moisture sealant which demonstrated only a 3% pick up of moisture(by weight) in a 24 hour soak test while untreated specimen showed over 20% water pick up (byweight). It was judged that transDeck TM would be able to meet or exceed the end-use requirements interms of durability and moisture resistance for truck and trailer decking systems.Structural performance is also a crucial requirement in truck and trailer decking applications.The trucking industry has developed performance standards based on loadings applied by lift trucksoperating on a prototype dry freight van trailer decking assembly. A cyclic proof load test is specifiedas structural performance test by the Truck Trailer Manufactures Association (TTMA) which is theindustry association of the U.S. trailer manufacturers (Truck Trailer Manufactures Association, 1989).In previous evaluations, both parallel-laminated veneer and plywood type constructions failed to satisfythese performance requirements. In parallel-laminated veneer products the perpendicular to grainstrengths were too low whereas in the traditional alternate cross-band plywood type constructions the4Figure 1 Veneer lay up of transDeckTM.5parallel to grain capacities were inadequate at the thicknesses required for the truck deck application.Previous in-house research by Ainsworth Lumber Company Ltd. demonstrated that thestructural requirements of TTMA were satisfied for certain specific lay-ups of Douglas-fir veneer. Atheoretical framework for evaluating the performance of the product and experimental data required forevaluation of alternative constructions and species combination are lacking. This thesis addresses theseissues to provide a basis for eventual optimization of the design and in-service performance oftransDeckTM .Improving the knowledge of the static structural behavior of laminated wood plates requiresthe development and verification of an appropriate structural analysis model to correctly predict thestructural behavior of a prototype decking system. Implementation of such a model requires as inputdata the appropriate elastic material properties of the panels, the elastic properties of the supportingsteel frame, and the connector stiffness between the panels and the steel frame.Model verification can be achieved by comparing model predictions to results from eitherexperiments or another verified model. In this case since no other verified model exists, full scale testingof prototype dry freight van trailer decking system is required. The cyclic testing program generatesimportant experimental data on the behavior of the panels in system and provides information to guidethe design and construction of the panels. Since full scale testing is very expensive and time consuming,it is practical to perform only a limited number of tests. This is especially true when the fatiguebehavior of the panels under cyclic load is of interest. In these cases, testing at a number of stress levelsis usually needed to develop a reliable database relating the load level and number of cycles to panelfailure. Therefore, an alternate approach is needed to develop the data on fatigue behavior of the panelsin a system. Such data can be established from testing of small specimens in the bending mode withthe appropriate span to depth ratio and stress history. The relationship between the fatigue life andfailure mode of the small specimen tests and the panels in system can be established to provide a basisfor evaluation of the fatigue behavior of the alternate panels in the deck system.6The overall objective of the study is to develop a framework to evaluate the performance oflaminated wood plates in decking systems. The overall objective can be divided into the followingsubobjectives:1) to develop a computer analysis program to evaluate the static behavior of the prototype dry freightvan trailer decking systems (Decking Analysis Program);2) to develop material properties data required as input for DAP for panels comprising of 2.5 mm and3.2 mm thick Douglas fir veneer;3) to experimentally study the structural behavior of prototype decking systems, with laminatedveneer panels made from 2.5 mm and 3.2 mm thick Douglas fir veneers, under static loading andto verify DAP using the experimental results;4) to experimentally study the structural behavior of prototype decking systems under cyclic loading;5) to develop a database on the structural behavior of small specimens under cyclic bending loads.6) to use a damage-accumulation model to predict the relationships between load level and time tofailure for cyclically loaded panels and beams specimens.72. BACKGROUND2.1 Structural Analysis Models for FloorsFinite element and finite strip methods are well known numerical methods commonly used instructural analysis problems. Both numerical methods can be used in the analysis of floor systems. Inthe finite element method, the displacement field is approximated by polynomial functions and thegoverning equations are replaced by a set of algebraic equations which are obtained by discretizing thecontinuum into a finite number of elements. The finite element method is very versatile as complicatedgeometry, boundary conditions, and loadings can be readily considered. The solution accuracy typicallydepends on the number of elements and the type of element used in the analysis.The finite strip method is suitable for analysis of structures with regular boundaries where thestructure can be divided in strips. Here the displacement field is approximated by continuouslydifferentiable analytical functions in one direction and polynomial functions in the other direction. Itshould be noted that the kinematic boundary conditions at the end of the strip must be satisfied by thecontinuously differentiable analytical functions a priori. The solution accuracy typically depends on thechoice of the continuously differentiable analytical functions.The major advantage of the finite strip method over the finite element method is that lessinput data, less core memory, and reduced execution times are required. The major disadvantage of thefinite strip method compared to the finite element method is that it is less versatile.General purpose finite element analysis programs are available to analyze orthogonallystiffened plate structures. These models typically ignore the slip between the cover and the supportingbeams; therefore, only special cases of wooden floor systems can be analyzed.Thompson, Goodman, and Vanderbilt (1975) developed a computer program, FEAFLO, basedon the finite element method to analyze the structural behavior of wooden floor systems. FEAFLO8considered the floor as a system of crossing beams which consisted of a series of T-beams and sheathingstrips running perpendicular to the T-beams. The web of each T-beam represented a joist in the floorsystem. The flange of a T-beam represented part of the cover which contained either one or two layersof sheathing. The crossing sheathing strips allowed the adjacent T-beams to interact which contributedto the load sharing behavior of the system. The T-beams also took into consideration the slip betweenthe cover and the joists by including the load slip behavior of the nails during the formulation of thestiffness matrix. Finite elements were used to model the T-beams and sheathing strips with theirdeformation matched at the points of intersection. The model was verified against test results througha complicated calibration procedure. The calibration steps included the estimation of the effectivewidth of the cover for the T-beam and the introduction of zones of low stiffness in the cover tocompensate for the restricted degrees of freedom in the model. Extraction of cover stresses from thismodel was considered not appropriate; therefore, FEAFLO was not adopted in the current study.Foschi (1982) introduced finite strip representation of wooden floor systems by developing acomputer model, Floor Analysis Program (FAP), which considered the strain energy of individualcomponents in a wooden floor system: the cover, the joists, and the nail connectors. The deformation ofthe floor system was assumed to be represented by Fourier series in the direction parallel to the joistsand by finite element approximation in the direction perpendicular to the joist. Gaps in the cover wereconsidered by removing the contribution of strain energy in the cover occupied by gaps. Foschi (1982)noted that the off-diagonal stiffness submatrices were coupled when gaps and/or discrete nailingpatterns were considered. This effectively increased execution time and reduced the major advantage ofapplying the finite strip method over the finite element method to structural analysis of floor systems.Despite of the reduced efficiency Foschi (1982) successfully implemented FAP in a microcomputerenvironment with small core memory. Finally results show that the model was still very efficient andprogram predictions agreed well with experiment results.Chen et al. (1990) developed the B-spline compound strip method to analyze stiffened plates9under transverse loading. In B-spline compound strip method, finite element method was firstemployed to partially discretize a problem to an ordinary differential equation. The Ritz-Galerkinmethod was then used to solve the equation with unequally spaced cubic B-spline functions. The use ofunequally spaced cubic B-spline functions allowed an accurate description of the response in regions ofhigh stress concentration which might result from wheel loads in the case of lift truck loading on dryfreight van trailer decking systems. Also the boundary conditions at the end of each spline can beprescribed to represent the fixity in the structure. The model, however, has not dealt with semi-rigidconnectors between the plate and the stiffeners nor gaps in the cover plate which are important featuresin dry freight van trailer decking systems.From these models, FAP analysis concepts, based on the finite strip method, appear to bemost suited for structural analysis of prototype dry freight van trailer decking system. However, asshown in Figure 2, the prototype decking system contains a gap along the center line C-C and theedges of the cover along lines A-A and B-B are not fully supported. If FAP was directly used to modelthe prototype decking system, the vertical deformation of the cover along the edges A-A and B-B(Figure 2) would be restricted by the use of Fourier Sine series to represent the vertical deformation ofthe cover in the direction parallel to the joist. Also the treatment of gaps in FAP would ensure that thevertical deformation of the cover across a gap to be continuous. However, due to the relatively largewheel loads which are concentrated in a small area, the different stiffness characteristics of the coverpanel on each side of the mid-span gap (one panel on each side of the gap), and the possibility ofunsymmetrical loading, it is expected that the gaps along the line C-C (Figure 2) will open under loadand the estimated stresses in the panel may not be accurate. Therefore, modification of FAP is neededto properly model the behavior of the prototype dry freight van trailer decking system. Specifically, thedevelopment of DAP would require modifications in the formulation of the strain energy terms in FAPby including the additional degrees of freedom to represent the boundary conditions and theunsupported gaps along the mid-span of the prototype dry freight van trailer decking system and the1 0B BCA Asx,y, v1 1 1 1 1 I i I 1wBay 1^2^3^4^5^6^7^8^9 10 11 12 13 14 15 16 17Figure 2 A prototype dry freight van trailer decking system.11unequal plate stiffnesses across the center line gap.2.2 Material Properties2.2.1 Elastic Properties of Laminated Veneer Wood PlatesDAP would require information on the elastic properties of transDeckTM as input. Necessarydata include the bending and axial moduli of elasticity in the direction perpendicular and parallel tograin. The modulus of rigidity in the plane of the panel, Poisson's ratios, and load-slip characteristic ofthe connector between the panel and the support members must also be known.Smith (1974) reported the strength properties of unsanded grades of Douglas-fir plywood.Parasin (1981) evaluated the mechanical properties of 3-ply 9.5 mm GI inch) and 5-ply 15.5 mminch) western hemlock and amabilis fir sheathing grade plywood. The strength properties of 3-ply 9.5mm (8 and 5-ply 15.5 mm (8-5 inch) western white spruce sheathing grade plywood from B.C. and8Alberta were also reported by Parasin (1983a, b). Although the data on the mechanical properties ofplywood are substantial, they are based on testing of plywood with the regular veneer lay upsencountered in plywood constructions. Comprehensive database on the mechanical properties oftransDeckTM was lacking.If information on the veneer properties were available, a more general approach can be taken inwhich the elastic properties of a laminated veneer panel can be built up analytically using data onindividual veneer properties. Such a model would provide a framework where the performance of thelaminated veneer panel can be optimized based on veneer placement during lay up.Limited experimental work on veneers strength properties has been reported. Kingston (1947)investigated the effect of grain direction on the tension strength and stiffness of Hoop pine veneer.McGowan (1974) tested Douglas fir veneers with and without knots to study the effect of prescribeddefects on the tensile properties of Douglas fir plywood strips. Booth and Hettiarachchi (1990)conducted an experimental program to evaluate the strength and stiffness of 1.55 mm (-A inch) thick12beech veneers in tension and compression. It is clear that limited information on veneer mechanicalproperties existed. More importantly information on veneer mechanical properties specific to the veneersource available to Ainsworth Lumber Company Ltd. to manufacture transDeck TM was lacking.2.2.2 Fatigue DataLimited information on the fatigue behavior of laminated veneer wood products was available.Kommers (1943) reported test data on solid Sitka spruce (593 specimens), solid Douglas-fir (424specimens), 5-ply yellow birch plywood (102 specimens), and 5-ply yellow poplar plywood (51specimens). Thermosetting phenol-formaldehyde Tego film glue were used as binding agent for theplywood. The specimen dimensions were 229 x 32 x 8 mm (9 x 14 x sinch) for the plywood and 229 x32 x 5 mm (9 x 14 x 4 inch) for the solid wood. They were conditioned and tested as a cantileveredplank over a 152 mm (6 inch) span at 75 °F and 65% relative humidity. Completely reversed bendingstress cycles (i.e., with zero mean stress per cycle) were applied to the plywood specimens. The solidwood specimens were stressed either with completely reversed bending stress cycles or repeated stressbending stress cycles (i.e., mean stress per cycle equals 0.5 the maximum stress). Maximum stress tonumber of cycles to failure curves (S-N curves) were developed. Kommers (1943) reported fatiguestrengths of 27% and 36% of the mean static modulus of rupture for 50 million cycles of reversed stress,for plywood and solid wood, respectively. The research was unable to detect an endurance limit even atthe fairly high load cycle of 50 million cycles.Kommers (1944) performed fatigue tests on solid Sitka spruce (593 specimens) and Douglas-fir(424 specimens). The specimen dimensions were 229 x 32 x 5 mm (9 x 1- 14- x sinch). The testconditions were similar to the first study by Kommers (1943) except completely reversed bending stresscycles were superimposed on a constant stress level. The results indicate that the S-N curves of solidwood were influenced by the shape of the applied stress cycle.Since S-N curves were influenced by the shape of the applied stress cycles, testing should be13performed using applied stress cycles which are similar in shape to that experience by the material inservice. Development of such data may not be practical for lumber or composite products in generalsince in-service load cases vary widely. However, this may be possible for a specialty product such astransDeckTM where the loading histories in dry freight van trailer decking systems are expected to bemore predictable compared to the loadings on typical roof or floor systems in buildings.McNatt (1970) studied the effect of rate, duration and repeated loading on the strengthproperties of 6 mm (1 inch) thick tempered hardboard. S-N curves for interlaminar shear (25specimens) and tension parallel to surface (27 specimens) for the material were reported. The fatiguestrength for 10 million stress cycles was found to be about 40% to 45% of the mean static strength intension and shear.McNatt and Werren (1976) reported the fatigue properties of three particleboards in tension(60 specimens) and interlaminar shear (61 specimens). The specimens were: 1) urea-bonded 16 mm (8inch) thick southern pine particle board; 2) phenolic-bonded 16 mm (& inch) thick Douglas-fir particleboard; 3) urea-bonded 13 mm (2 inch) thick Douglas-fir particle board. The specimens were conditionedto equilibrium at 73 T and 50% relative humidity. The resulting S-N curves for the three particlesboards indicated that the fatigue strengths for 10 million stress cycles were approximately 45% and40% of the mean static tensile and interlaminar shear strengths, respectively.A special case of the fatigue phenomenon in wood is known as static fatigue or load durationeffect on the strength properties of wood products. This phenomenon refers to the possibility of failureof a member carrying a constant load sustained over time. Palka and Rovner (1990) studied loadduration behavior of commercially available 16 mm CI inch) thick waferboard. McNatt and Laufenberg(1991) studied load duration behavior of commercially available 16 mm (8 inch) thick plywood andoriented strand board. In both studies, 300 x 1000 mm specimens were under third point loading withsimple support condition over a 900 mm span. Information on time to failure and creep were obtainedfor a range of stress levels. In the study by Palka and Rovner (1990), a range of environmental14conditions were also considered.2.3 Full Scale Testing of Dry Freight Van Trailer Decking AssemblyIn an in-house research and development study, Ainsworth Lumber company Ltd. tested oneprototype dry freight van trailer decking assembly under cyclic loading (Figure 2). This is a typical testprototype assembly which is recommended by TTMA for test decking material for dry freight vantrailers. The dimension of the assembly was 4.9 m x 2.4 m (8 x 16 feet). The transDeck TM panels wereconnected to the supporting I-beams using 8 mm (A inch) diameter bolts at a spacing of 102 mm (4inches) on center. The supporting members were 102 mm (4 inches) deep I beams with a 57 mm (21minches) wide flange. The I beams had a mass distribution of 4.78 kg ( 3.2 lb ). These beams wereftass spaced at 305 mm (12 inches) on center. The ends of each I beam were connected to two 158 mm (6.0inches) deep supporting channels using four steel bolts with a diameter of 9 mm inch) at eachconnection. The supporting channel in turn directly rested on the floor. At the supports, the I beamwere restricted from vertical displacement but allowed to rotate; therefore, simple support conditionscan be assumed for the I beams. The transDeck TM panels were made with 2.5 mm Douglas fir veneer.Prior to the manufacturing of the transDeck TM, the veneers were visually selected such that highquality material was used. A lift truck carrying 76 kN (17,000 lb) with a front wheel to back wheelweight distribution of 9:1 was repeatedly driven over the system. The foot print of a front wheel was229 x 229 mm (9 x 9 inches). Results from the test indicate rolling shear type failure in the cross plyafter 5000 load cycles which met the minimum TTMA requirement of 3000 load cycles without failure.Trailmobile Inc., a major truck trailer manufacturer in the United States, tested a similarprototype dry freight van trailer decking assembly under cyclic loading. The transDeck TM panels wereconnected to the supporting I-beams using 8 mm (A inch)-18 torx drive, flat head, type G, phosphateand oil coated self tapping screws at a spacing of 102 mm (4 inches) on center. The supportingmembers were 2.59 m (102 inches) long. A wheel cart carrying 77 kN (17,222 lb) with a front wheel to15back wheel weight distribution of 9:1 was repeatedly pushed and pulled over the system by a lift truck.The foot print of a front wheel was 89 x 229 mm (3.5 x 9 inches). Results from the test indicate failurein the panels after only 14 load cycles.Clearly, the limited research indicates great discrepancies in results. More testing is needed togain a better understanding of the performance of the laminate veneer wood plates in trailer deckingassemblies under repeated loading. For this type of research to proceed a trailer decking load simulatorfacility is needed.2.4 Modeling Fatigue Behavior in Wood Panel ProductsSince fatigue properties in wood panel products has not been a popular topic of research, therehas been little attention paid to developing models to describe the general fatigue properties in panelproducts. Damage accumulation models have been used in the past to describe fatigue behavior inother materials. The most widely used model for fatigue in metal is based on the Palmgren-Minerapproach. The model defines the cumulative damage, a, after the structural element experiences n icycles at a particular stress level as:= E N i [1]where Ni is the total number of cycles to failure. a = 0 denotes undamaged state and a = 1 denotesfailure. The advantage of Palmgren-Miner approach is its ease of application; however, itsdisadvantages include a) not accounting for the order in which stresses are applied and b) ignoring thepossible presence of a stress endurance limit.In the recent past, the special case of static fatigue has received much attention and severalresearchers have proposed alternative damage accumulation laws to model the duration of load (creep-rupture) phenomenon in dimension lumber (Foschi and Barrett, 1978; Foschi and Yao, 1986a and b;16and Gerhards and Link 1986; ).The Foschi and Barrett model takes the following form:da _ f r(t)^lbdt — a 1 '-s-- — a.0 1 + A a(t)^ [2]where a and A are independent lognormally distributed random variables, b and co are constants, r(t)is the applied stress, and Ts is the short term strength.This model was calibrated with duration of load data for dimension lumber in bending (Foschiand Barrett, 1978). However, Foschi and Yao (1986a and b) pointed out a situation when a has beenaccumulating to a significant level and 7(0 is infinitesimally greater than Ts ao , the damage will stillgrow exponentially regardless of the level of applied stress. The model will therefore predict failure ofsubstantially damaged specimen over time although little stress is applied after the initial damage.Foschi and Yao (1986a and b) extended the original model (Foschi and Barrett, 1978) bymaking the damage dependent term a function of the stress level. The Foschi and Yao model takes thefollowing form:dt^nda — a { T(t) — cors )b + c { 7(0 — crors 1 a(t) [3]where a, b, c, n and co are random model parameters which vary between members. 7(0 is the appliedstress and Ts is the short term strength. This model was satisfactorily calibrated using load durationtest results of dimension lumber in bending, tension, and compression (Karacabeyli, 1987) and loadduration test results of waferboard in bending (Palka and Rovner, 1990). The calibration procedure waspresented in details by Foschi, Folz, and Yao, (1987).The Gerhards and Link model is expressed as:da^a— + b Tsdt ewhere a, and b are model parameters. r(t) is the applied stress and Ts is the short term strength. Modelcalibration with load duration test results of dimension lumber was given by Gerhards and Link (1986)and Foschi et al., (1989). This model was also calibrated with load duration results of commerciallyproduced plywood and oriented strandboard (McNatt and Laufenberg, 1991) and waferboard (Palka,Rovner and Deacon, 1991) where satisfactory agreements were obtained. Since the rate of damage inthis model is independent of the previously sustained damage, the linearity in Gerhards and Linkmodel does not permit a good fit to the nonlinear upward trend in probability of failure versus thelogarithm of time to failure results for lumber (Foschi, Folz and, Yao, 1989).The major criticism of using damage accumulation laws to describe duration of behavior inlumber is that the model parameters lack physical meaning. Another approach to model the loadduration phenomenon in wood, based on fracture mechanics theory for viscoelastic bodies, wasproposed by Nielsen (1978, 1985). The model represents wood by a single crack with viscoelasticcharacteristics and considers the propagation of the crack under load through time. Duration of loadresults of dimension lumber were used to calibrate the model (Nielsen, 1985). Nielsen (1990) extendedthe model for fatigue behavior of wood products by considering crack closure. The model was calibratedusing fatigue data of wood products (Nielsen, 1990). However, the use of Nielsen's model is rathercomplicated because the rate of crack growth is controlled by a nonlinear first order equation; itssolution in general requires numerical integration procedures which is not suited for simulation studies.Furthermore, Nielsen's model also requires parameters which describe the viscoelastic behavior of thewood. However, these parameters cannot be obtained easily from experiments a priori; therefore, theyare typically obtained indirectly through model calibration. So, although Nielsen's model has morephysical meaning attached to its parameters in comparison with other damage accumulation models,the parameters for both types of model are obtained through model calibration.17[4]183. STRUCTURAL ANALYSIS MODEL3.1 IntroductionThe deformation and stress in an elastic body under load can be obtained from energymethods. Examples of energy methods include the principle of virtual work, the principle of minimumpotential energy and the Ritz method. Energy methods are based on the fact that the governingequation of a deformed elastic body is derivable by minimizing the energy associated with thedeformation and the loading through calculus of variation. Application of energy methods areparticularly effective in cases where irregular shapes, nonuniform loads, variable cross sections, andanisotropic materials are encountered. In this study analysis based on discrete models are used.This chapter describes the development of Deck Analysis Program (DAP) to evaluate thestructural performance of the prototype dry freight van trailer deck system shown in Figure 2. Here,the basic theory for the governing equations of the deformation of a deck assembly is formulated usingthe principle of minimum potential energy. First, the strain energy terms in the basic components ofthe deck assembly, which include the cover plate, the supporting I-beams, and the connectors betweenthe cover and the supporting I-beams, are developed. The load potential terms associated with thewheel loads from lift trucks are then derived. Application of the principle of minimum potential energyresults in a system of equations relating the applied load and the unknown displacement field. Thesystem of equations is then solved to obtain the deformations of the assembly. Based on the estimateddeformations in the assembly, the bending and rolling shear stress profiles in the cover can be derived.The critical locations where stresses are maximum can therefore be identified.Sample problems considering various case studies of a seventeen bay prototype dry freight vantrailer deck assembly are modeled and solved using DAP to illustrate the required program inputs andthe program outputs. The sensitivity of the DAP results to parameters such as connector stiffnesses andlocations of wheel loads are studied to provide guidance to the planning of the experimental phase of19the study.The development of DAP is based on a semi-analytical approach using the finite strip methodwhere the unknown displacement field is expressed as combinations of Fourier and polynomial series.As mentioned in the literature survey, Foschi (1982) used finite strip method to develop FAP to studywooden floor systems. The major differences in the formulation between DAP and FAP are: 1) theassumed displacement field in DAP includes additional degrees of freedom to model the unsupportededges and the mid-span gap in the prototype dry freight van trailer deck assembly; and 2) unequalelastic properties for the two sides of the cover in the deck system which corresponds to the use of twoindividual panels (one on each side of the mid-span gap) in the deck assembly are considered. Theseadditional issues are addressed so that dry freight van trailer decking systems sheathed withtransDeckTM panels can be properly modeled to predicted the structural responses such as deformationsand stresses in the system.3.2 Strain Energy in the CoverConsider the finite strip shown in Figure 3, it represents one bay in the prototype trailerdecking assembly shown in Figure 2. The strip contains a gap at midspan; i.e., x=. Also the edges of2the strip at x=0 and x=L which are not supported by the I beam are free to move in the verticaldirection. The x and y axes shown in Figure 3 are the principal material axes for the cover. In theapplication of transDeckTM in dry freight van trailer decking systems, the face grain of the cover istypically oriented parallel to the y-direction and the grain direction of the cross-ply is typically orientedperpendicular to the y-direction (i.e., in the x-direction).Assuming small deflections and orthotropic plate theory for a medium thick plate wherebending and membrane forces are considered, the strain energy of the cover under load is expressed as:20Figure 3 Finite strip representation of a section of the decking assembly.L2y2 _ay , 2^)^Dv2 (13.711c)(DTT) + 2 (ay + ax ) dx dyDx2 (all 2^ V^DG2 au aV)22^)+^`Ox) + D 2 ay [5]where u(x,y), v(x,y), w(x,y) represent the displacement field of the mid-plane of the cover in the x, yand z directions, respectively; s represents the width of a strip of the cover in the y-direction; and Lrepresents the length of the cover in the x-direction. The plate stiffnesses, K xi , Kyi, Kvi, KG1, Dx1,Dvi , and D Gi (i= 1 or 2) are given by:Exi d3^E ;Kxi12(1 - v v y ICY1 Kxi^Kvi^vxy Kx1 ; KGi = • .13 .xy yx xi GI 12'd^E ••D^• a—.D •—vDxi = 1 _ vxi v Exi,^— xy Dxi,^— G i d^y yxwhere the subscript [i] denotes plate stiffnesses and elastic properties of the i th section of the cover inwhich sections 1 and 2 represent the regions (0 < x < 5) and (--L2 —< x < L), respectively.Here, the moduli of elasticity and rigidity of the cover are equivalent values for homogeneousmaterial. Exi is the modulus of elasticity in the x-direction; Eyi is the modulus of elasticity in the y-direction; G i is the modulus of rigidity in the x-y plane of the i th section of the cover. Also v and vxy yxare Poisson's ratios where the first subscript denotes the direction of transverse strain and the secondsubscript denotes the direction of applied stress; and d is the thickness of the plate.[6]223.3 Strain Energy in the Supporting I-beamsWhen the finite strip shown in Figure 3 is under load, the strain energy in the supporting I-beam resulting from axial deformation, lateral bending (in the y-direction), vertical bending (in the z-direction), and torsional deformation can be expressed in terms of the displacements U(x), V(x), W(x),and 0(x). Here the displacements U(x), V(x), W(x) represent deformations of the supporting I-beam atits geometric center, as shown in Figure 4 (point A), in the x, y and z directions, respectively. Finally0(x) represents the rotation of the geometric center of the supporting I-beam about the x axis at pointA.The strain energy in the I-beam is given as:L= f E A (dU ) 2UI j^2 dx^E2Iz ( d2x-y)2 E2Iy (dd2)Acy)2 G J^) 2 dx0where E = the Young's modulus; G = the shear modulus; A = the cross sectional area; k = themoment of inertia about the z axis; Iy = the moment of inertia about the y axis; and J = the torsionalmoment of inertia of the I-beam, respectively. Given the specifications and physical dimensions of thesupporting steel I-beams, the elastic moduli and the geometrical parameters can be easily evaluated.In the prototype dry freight van trailer deck system, steel I-beams with E^200,000 MPa(29x106 psi) and G 77000 MPa (11x10 6 psi) are typically used. The moments of inertia Iz , IX Jand the cross sectional area A can be expressed in terms of the thickness of the flange (t f), the thicknessof the web (tw ), and the depth (h 1) and the width (b I) of the I-beam as:2 tf 13 /3^tw3 (hi — 2 tf)123^3= bi hi — (h I — 2 tf) (bi — t w )IY^12[8 a][8 b][7]23Figure 4 Degrees of freedom for a supporting I-beam.24(11 - tf) tf3 + 2 131 tw33^ [8 c]A = 2 131 tf^— 2 tf) tw^[8 d]3.4 Strain Energy in the ConnectorsThe construction of the prototype dry freight van trailer deck system involves initialpositioning of the cover panels over the supporting steel I-beams. Then the panels and the flange of theI-beam are pre-drilled with holes, 7 mm inch) in diameter, at the appropriate locations. Finally self-tapping screws, 8 mm (; inch) in diameter, are used to secure the panels to the I-beams to form acompleted assembly.The cover plate is therefore assumed to be semi-rigidly connected to the supporting I-beamthrough uniformly spaced connectors with linear load-slip relationship. The strain energy in theconnectors resulting from the relative movement between the cover and the supporting I-joist can beexpressed as:NAUN =^ (Kx (Au) i2 Kyc (,(Xv)i2 K9i=1[9 ]where (Au) i = the slip in the x-direction of the ith connector; (Av) i = the slip in the y-direction of thei th connector; Oi = the relative rotation between the cover and the supporting I-beam at the location ofthe ith connector; NA = the number of connectors per finite strip; Kx = the connector stiffness in thex-direction; K = the connector stiffness in the y-direction; and K = the rotational stiffness of theKy KBconnector.Since the screw connectors are unlikely to withdraw, it is reasonable to assume that there is noconnector withdrawal; therefore, (Au) ; and (Av)i can be expressed in terms of the displacements of thecover and the supporting I-beam as:25(Au)i = u(xi, 0) _ dw ( , \ (u(xi) + Sxj ))2 dx `—i/^ [10a](Av) i = v(xi, 0) — 4 z.i , 0 ) — (17 (xi) + 21 ^[lob]of = txi , 0 ) — 0(xi)^ pc,where xi = the location of the ith connector in the x-direction and d = the thickness of the cover.The discrete connector pattern modeled in Equation 9 can be converted to a equivalentcontinuous connector by considering the slips (Au) i , (6,v) i , and (fii as continuous functions such that UNcan be expressed as:UN = 21e j0whereL(' K.c (Au)2^Kyc (Av )2^Koc 02 dx^[11]Au = 0 ) — 2ddwx —^+ 2111 [12a]v(x, 0)^w-(x, 0) — + ht51) [12b]0-y— 2 a— Lv-(x,cb 0) — [12c]aye^= connector spacing. [12d]3.5 Potential Energy in the Applied LoadThe load potential can be expressed as:+i L^ 26QL- f I p(x, y) w(x, y) dx dy^ [13]_ s 02where p(x,y) corresponds to the loading onto a bay of the assembly due to a pair of wheels on a lifttruck which can either be the pair of front wheels or the pair of rear wheels. Note that the pair ofwheels is assumed to be aligned in the x-direction of the cover.Assuming the loading of each wheel is uniformly distributed over its foot print, the loadingp(x,y) is completely described by: 1) the magnitude of load on each wheel; 2) the dimensions of thewheel foot print; and 3) the location of the wheels on the deck which can be obtained from informationon the center to center distance between the two wheels and the location of one of the wheels on thebay.3.6 Finite Strip FormulationUsing a Fourier series expansion of the unknown functions in the x-direction and a onedimensional finite element approximation along the y-direction, the displacement field of the cover canbe represented by w(x,y), u(x,y) and v(x,y), which are expressed as:Nw(x,y) = E F in(y) sinciTrx) + Bn(y)(-1, ) + An(y)(1 _ __L )n=1Nw(x,y) = E F in (y ) sin(n—FT- ) + Dn (y)(2-e — 1) + Cn(y)(2 — 2_,),c)..n=1Nu(x,y)^n(y) cos(n-ir—:)n=1Nv(x,y) = E F3n(y) sin(T)n=1Lfor 0 < x < —2^[14a]for -2- < X < L^[14b][14 c][14d]where the functions Fi n(Y), F iji (y), An(y), Bn(y), Cn(y), Dn(y), F2n(y), and F3n(y) are one27dimensional finite element approximations in the form of fifth degree polynomials in the y-directions; N= the number of terms used in the Fourier series expansion. The chosen displacement fields of thecover in the x and y directions, u(x,y) and v(x,y), are described by continuous functions. Thedeformation of the cover in the y-direction at x=0 and x=L and the deformation of the cover in the x-direction at x4' are restricted; i.e., v(0, y)=0, v(L, y)=0 and u( 12—', y)=0. Also the polynomial functionsare chosen such that the edges of the cover (x=0 and x=L) are free to displace vertically except overthe supporting I-beam which properly models the boundary conditions of the prototype decking systemwhere the cover is indeed unsupported at the edges except over the I-beams.The deformation in the supporting I-beam U(x), V(x), W(x), and 0(x) can be expressed as:U(x) =^un cos(T)^ [15 a]n=1NV(X) = E Vn sin(n—P)^ [15 b]n=1vvn sin(ni-Lx )^ [15c]n=1n^Lsin(rir-Lc)^[15d]n= 1Here the coefficients Un, Vn, Wn, and On are displacement amplitudes of the supporting I-beam whichis assumed to be simply supported and restricted from lateral deformation and rotation at x=0 andx=L.3.6.1 CoverW(x) =0(x) =Let us consider individual terms in the expression of the strain energy of the cover shown inEquation 5 as:28(lc^uc ( 1) + uc (2) + uc (3) + uc ( 4) + uc ( 5) + uc (6) + ucco + uc ( 8 )where+.1 L2= f f Kxi (88x2W 2{ 2J 2^) dx + Lf I ___x22^x2(52w8 ) 2 dx dy_ s o L2^ 2+.1 j 2 ay21 IIUc(2)^ Kyi 102w , 2^1‘.^A2L „,1^) dx + ^---E9 (`'v-)2 dx} dy2 ayes 0 L_(3)^=uc +if ". L1(1/ a 2W a2W dx Kv2 a2w a2wax2 )( 5y2 )^+ f^2^( Ox2 )( ay2 ) dx} dys^0 L2+I 12'^L(./c (4)^= f { f 2 KG' Sx2 Vas; ) 2 dx +( K^2 dx dy12^G 2 ( aax2; )s^0 L2 2+i^1.-'" L(Jc (5 ) =_f {s^of Dxl^all 2 dx +LDx2 all 2(-8i) dx^dy---- (N) f2 2+i 2^Lf -÷ (g) } dyD2 A 2 dxs^02s LUc (7) = { Dui (R)(g) dx + Dv2 (k)(p dx} dys^02 2s L uc (8 ) [17h]uc (6) = f(g) dx +D n 22 22[16][17 a][17 b][17c][17d][17 e][17f][17g]29Let us define the functions R. and R. 2 as:L Si11(m -^sin(m n)^ sin(m - n)^sin(m n)^, ^ and R. =^,^(in + n) (in + n)Rnmi– 271-^(m - n) 2 Lir^(m -Also let the superscripts [] and ["] be defined as the first and second derivative with respect to y,respectively. Substituting Equations 14a to 14d into Equations 17a to 17h and performing theintegration with respect to y yields individual terms in the strain energy of the cover, expressed interms of the functions, F; n(y), (y), An(y), Bn(y), Cn(y), Dn(y), F2n(y), and F3n(y), as:N^4 4^2^2\Uc (i) =^E nJ 8 L3 (I(xl F ln^Kx2 F1 )[18]s n=1N N+ En=1 in=n m2 2 4n 2 L4 Rran1 (Kxi Fln Fim^Kx2 Fln Fim) dy[19 a]uc(2) = E L^F ln^2^N N R^yl^2 ± Ky2 F ln^E E 2 1 (1<y1 Fln11 Flm Ky2 F ln F imn=1 n=1 m=1—7 n mn=1 m=1(Kyi (A: Ain + B. Bm + B. Am ) Ky2 (Cn" Cm + D. Dm + D. Cm" ))• 11^11^r■^ 11nor Ky1 Fln (B. ( sin 2 — cos n-=-7 -r) + Am (1 — 2 • ni2^Tif- sin -T)K 2 F^C (cos '1' + 2 • nr)Y 1n^m^—2^n-72-r Sill -T — Dm (cos n71 + A- sin Irf)) dy^[19b]Uc (3) = – (Kv1 F ln F ln^Kv2 F ln F lnn--Tr2 sin 2))1 F1n (Bm nir^2 ( 2 sin^— cos n- r) + Am (1 –n7r^2Uc (4) =22^2(2) 2 (KG1 F1n + KG2 F1n ' )1 (An Am + BM BM — BM Am — An BM)+1 N(jc(5) = f E (n7)2(Dxi+Dx2) F 2 +8L^2ns n=1—7.N^2^(Dxi – Dx2)E^nm (1,r) Rrnni 2^) F2n F2. dy^[19e]n=1,m=1n r m30N^N^2^ • If+ E E ( lf) Rim1 (Kvi Fln Flin — Kv2 F1n F lm2 • n7r(CM (cos V + - .. sin -i-) — Dm (cos n7r + —2 sin n-i-r dy))+ Kv2 F1n^ nir^2N N+ E E (117) (11171L L R^(KG]. Fln Fim`"' nin \2 – KG2 F1n F lmn=1 ,m=1n r m+ KG2 (Cn Cm + Dm DM — Dm Cm — Cn Dni')+ 2 sin n÷r (KG1 F ln (Bin — Am) — KG2 Fin ^— CM)) dy^[19d]-ki2N „^ N NUc (6) = I. E --8, (Dy1+Dy2) F3i; + E ERnini (Dyl —2 Dy2) Fan F3m dy^ [19.4s n=i n=1 ,m=1—7^ n r inn=1,m1n r m[1 9 c]Uc(7) _ !INI(D 111 +4 D 112\1171"^) F 2n FJ s n=1—7Nn=1 1m=1n T m) R i (D /A — D v2) F2n F3m dy^[19g]t/c(8) = I E+i Nn=1 Ni+DG2) (F2n'2 + n7r 2 -F3n, (- 17) -^L^2n312 + ( 2n7r)N N^ 7E E Rnm2 (DG1 2 D G2) (j 2n F2m nm (r)2 F3n F3mn=1 ,m=1n m+ L F2r F3m^F3n Fan) ClY^[19h]The unknown polynomials and coefficients can be expressed in term of the unknowndisplacements and their derivatives for points along lines 0-0, 1-1 and 2-2, of the finite strip shown inFigure 3. The vectors of unknowns, 6. (n=1, 2, ..., N), associated with the n th Fourier term of a finitestrip are given by:(5337 f„• m • 1 Q W^W •• ,^W I W P^WQ^II^II s^III^III '^IV^'S1.""1n, "v1n^1n, 1n 1n, In 1n, W ln^w1n, w1n^W1n, W1n S, -11 1n , 11. 1 S,• ,^ I 'In S, llon , von ,^Un, Vn , On S,^ S,^ Wv1n ,^w S, Wn ,^W2n, W2n^W2n, W 2n 2n S ,_II^II '^III^III ' s w IV^IV ' S u^(n=1, 2, ..., N)^[20]"2n , W2n ^I W 2n^, 2n , w2n^2n, 112n S v2n, v^S}2nwhere the superscript [T] denotes the transpose of a vector, the superscript ['] denotes the derivativewith respect to y, the superscript [*] denotes section 1 of the cover where 0 < x < --I-' ' the superscript [12denotes section 2 of the cover where -2-1' < x < L, and the subscripts [o], [1], and [2] represent the pointsalong the lines, 0-0, 1-1, and 2-2, respectively. The superscripts [I], [II], and [Iv] denote the cornerpositions on the cover of a finite strip as shown in Figure 3 such that superscripts [I] and [ii] denote thepositions along the lines 1-1 and 2-2 at x=0 and X=7 -2-1' of section 1 of the cover, respectively; andsuperscripts [iii] and [Iv] denote the positions along the lines 1-1 and 2-2 at x= 2 x=L of section 22of the cover, respectively. Note that each vector of unknowns, b. (n=1, 2, ..., N), has 39 degrees offreedom.32The functions F;n(y),^(y), An(y), Bn(y), Cn(y), Dn(y), F 2n(y), and F3n(y) can be written interms of a nondimensional variable — 2y the shape function vectors, and the unknown displacementvectors b. (n=1, 2, ..., N) as:F1n(6 = MO( ) T Sn;^Fln(e) = m,(OT Sn;^An(0 = ma(OTBn(0 = Mb()T S.;^Cn(0 = me(OT S.;^Dn(0 = Md()TF2.(0 = m3air 8.;^F3n(0 = M5()T 15n^ [21]where the non-zero components of the shape function vectors, NI. (0, Afi, (0, ma(), mb(0, mc(0,MA), M3(4"), and M5 () are given as:A i(^=^= m^=^m^m 5 c3 c4^c5 [22 a]"b(7)"OM^0(3)^a(5) c(9)^"d(11) 4 2^42 e3 4 _F e5M^—M^—M^—M=M^ =MA [22 b]c0(2)^0(4)^—^b(8)^to^,/(12) 8_^2e + e5 [22 c]MO(19)^M0(19)= — 2M ^= 1 — 2e 2^e40(20)^0(20) [22d]M0(24)^M0(26) = Ma(28)^Mb(30)^Mc(32)^Md(34)2+c25^3^1^4^3^5^ e e — 4 ee3 ^e4 _f_ e5[22 e]MO^ = MO^ = M^=^M(25)^ (27)^a(29)^.(31)^e(33)^“(35) [22f]8A ir^ _ c^c2^1 c3^1 c4—"3 (13)^—5 (15)^4 's^'^"^'^4 s^2[22g]_F e2^e3^e4 —M3 [22h]M 5(14) =^(16) =^8M3 (17) = M5 (18) = 1 — 2e 2 [222]M3 (37) = M5 (39) =^8— e2^e433^3^2^1 3^1 4M =^=3 (36)^5 (38)^4 4^2The first and second derivatives of these functions with respect to e can be also expressed as:[22.2][22k]dF in(e)ded 2 F;n(e)dedBn (e) ded2Cn (e)deldF2n(e)de = M1()T-M2(4-)T 5n;= mbi coT (5n;= mc2( )Td2F in(e)deldAn(e)ded2Bn(e)d2dDn(e)dedF3n(e)de = 14;(0T 5n;= mai(e)T 5n;= mb2(0T 5.;= mdi coT= M6( )T (5ndFL(e) = m-;(4-)r 5.;ded2An(e) = ma2(0T c;de2dC d n(0 = mci (e)T 5n ;d2Dn(^=Md2 (e)T C;de - TA4(e)T 5n; [23]where the components of vectors, Ml (a) , M2 (0, Ml (0, M2(0 , mai(0, m.2(0, mbi(0, mb2M ,mc1(0 , me2(0, mai(0, md2(0, m4(0, and m6(4) are given as:m.^= dMo(k)^2.d2M0 (k).^(140(k)^.^o(k).i(k) — de^ , M2(k) = de2 , Ml(k) = de ,^M2(k) = d A/f2(k)^d2dMa(k)^d2Ma(k)^dMb(k)^d2Mboo.Mal (k) — de ;^Ma2(k)= de2^; Mbl (k) — de^1"-▪ b2(k)^de2^dMc(k)^d2Mc(k)^dMd(k)^d2Md(c).^Mcl(k) — de ;^Ivic2(k) de2 ; Mdl(k) = de ,^Md2(k) de2^dM3(k).^dM5(k).M4(k) = 4' otukK) = ^,^, for k=1, 2, ..., 39^[24] The various components in the strain energy of the cover can be expressed in terms of the34unknown displacement vectors, Sn (n=1, 2, ..., N), and the shape functions, M. (a), 1V1 1; (a), Ma(s),Mb(4") , Mc(E) , Md.(4") , 41(C) , M2(0, AC(f), M2(0, Mai(C), Ma2(C), Mbi(0, Mb2(C), Ma(0, Mc2(),Md1(0, and Md2 (4") as:1 N^S^ V` (n704 siiT (Kxi^T Kx2 40 140 T)16 L 1if-1=1+ N N 4 n2L R^T^M 111 . — K^4. T)nrn n x1 0 01 x2 0 0^m dE+1 NU (2) = (k)(2)3^E 4iSnT (Ky1 M2 2 T +2 Ky2 M2 42- T)n=1-1[25 a]n=1 Fi=1n mN+ En=1 jr1=1n m+ n=1 m=1 T {1 (-k- ^Ma2T + Mb2 Mb2T6 -'-'3r1 "-`a2 2 '^Mb2 Ma2T)6nT (Kyl M2 m2 ^-'- '372 M2mAA'^T — v Kir Ai2 T)(1b1 lUr T lutmc2 m-c2 mc12 mc12 + 's"-"c12 mc22 v m m ( 2117 —y1^(Mb2T( "'-` 2 — cos Pfr) + ma2T(1 —2^•^n71"sinKy2 2 (1%c2^ 2T (C0S 2 r-A-- sin —nir) — Md2T(cos ror 4- sin r+r)) om^[25 b]^AA --^T)- _LmO 'n •"35(y) 2 Rnmi 52' (Kul mo M2 T Kv2 MO M2 T)N NMa2T(1 —^sin 1 ))2 • n7r — cos^"'-a2T^sRA-^(^in^c^2T K M mb2 k(y)^vi^nn=1 m= 1n71"^2 • n7 — Md2 (cos n7r +1"(cos sin —)^R-27-, sin 1 -r)) (5. 4^[25 c]Kv2 MO (Mc2T^(V)(11±1:) R1m2 45nT (KG1 M1 M1 T KG2 M1^I5m2+ rN Nn=1 m=1551. (KG]. (Mal [mai— Mbl]T + Mbl [Mbl —U,0 )▪ KG2 (Mcl [Mcl Mdl]TMdl [Mdl MO] T )+ 2 sin rf (KG1 M1 [mbi— Mail T KG2 M1 [Ma - Ma] l)) 6m de+1I 1E1(n70 2 (Dx18+1,Dx2) 6,7 m3 m3T 6n— 2 1 n=1-2(Dx12 ) kg'Dx2)^M3 M3T bm den m (f) = s-^(Dy1+Dy2)2 5T m6 m6T on + N N (Dyi 2^ Dye) 5nT m-6 mfiT 5m. de [25.4n=1^ n=1 ,m=1-1 n mN N+n=1 m=1n m[25 d][25 e]36+1Uc (7) = — E^Dvi+Dv2\^,„ M6T snnor ^4 ) n M3 —6T -n=1-1N+n=11 pi=n *in(-y-) R.,„1 (D, 1 — Dii2) biLT M3 M6T 5 de [25g]+ 1rIc ( 8 )^f G +D G2) bnT ((&)2 M4 M4T (Y)2 M5 M5T ( LM M4 M5T)N N(DG1 DG2 T 2 2^ninir2 m m+ E E Rijn,^) 6n ((s) M4 M4T + 2 5 5T2 2^(25m17) M4 M5T (ff) M5 M4T) Sin^ [25h]3.6.2 Supporting 1-beamsSimilarly substituting Equations 15a to 15d into Equation 7 and performing the integration,the strain energy in the joist can be rewritten in terms of the coefficients U., V., W. and O n S whichare elements in the unknown displacement vectors Sn (n=1, 2, ..., N) as:(117)4^(n1.7 )2 un2^Wn2^-vn2) G (SL )2 on S)2^[26]3.6.3 ConnectorsFurthermore substituting Equations 14a to 14d and Equations 15a to 15d into Equation 11 andperforming the appropriate differentiation and integration, the strain energy in the connectors areexpressed in terms of the coefficients U., u.., W., V., v on, won' S and O. S which are also elements inthe unknown displacement vectors b. (n=1, 2, ..., N) as:n=1 ,m=1n mNUIn=1UN _ L— 8eN n=1\2+ Koc (won' s — on s) 1^ [27]3.6.4 Load PotentialTwo general load cases are considered in the treatment of the load potential: 1) two wheelswith equal loading (either the front or back wheels in a lift truck) located on section 1 of the finite stripwhich represent the region (0 < x < 4); and 2) two wheels with equaled loading (either the front orback wheels in a lift truck) with one wheel located on section 1 and the second wheel located on section2 of the finite strip where section 2 representing the region ( 12= < x < L).In the first load case the foot print and the associated loading from the two wheels are definedas:p(x,y)^= Po ( i )^if (x i. < x < x2 ) or (x3 < x < x4); and (y 1 < y < y2 )^[28 a]= 0^ otherwise^ [28 b]where po(i) represents the load per unit area from a wheel in which (0 < x 1 < l'); (0 < x2 < -IT'); (0 <L \^3 /^L \x3 < -E); anu 0 < x4 < y).Substituting Equations 14a, 28a, and 28b into Equation 13, the load potential for case 1 cantherefore be expressed in terms of the functions F 1n , An, and Bn as:Y2C2L(1)^=^Po(i) f N^n7:2> Flr,^1-( — —, 0 (cos illrx1 117x4 117x3cos^L^+ cosC cos L^)y i n=1xi 2±x42_ 3(32)^ dyThe load potential for case 1 can also be expressed in terms of the unknown displacement vectors b.(n=1, 2, ..., N) asS^N2 I-0M En=1 ( —^(cos n':2nlrx1^nirx4^n7rx3^.^Tcos ^ ,cos ^cosL^ L ) M7()(x2 2 xi 2 _Fx42 x32) mb7(e)T2+x42^x32)x2 2 _where2S+ (X2 — Xi+X4 — X32y2^ gMal( ) T (5ny2[30]y22gM7(e) = f M0() dg; = Ma(f) ck; Mb7(e) = Mb(e) 4 [31]2^ 2gy1^ Sy1 S y1In the second load case the foot print and the associated loading from the two wheels aredefined as:P(x5Y)^= Po (2 )^if (x1 < x < x2 ) or (x3 < x < x4 ); and (y 1 < y < y2 )^[32 a]- = 0^ otherwise^ [32 b]where po(2) represents the load per unit area from a wheel in which (0 < x1 < 2); (0 < x 2 < 1 ); (5' <x3 < L); and (;-'- < x4 < L).38[29]+ B.Substituting Equations 14a to 14b and Equations 32a to 32d into Equation 13, load potentialfor case 2 can also be expressed in terms of the functions F 1., F 1., A., B., C., and D. as :39Y2^f N^—^E Fin^ nl,r) (cos niTx^n71x 1L 2 — cos L ) Bn X 22 XL 1 2) + An (x2 x1 X2 2 x12C2L(2)^Po(2)yi n=1n7rx^n7fx3^— x3 2▪ F1n^n---fik) (cos L 4 — COS ^) D ^ x4+x3)2^2+ 2 Cn (x4 — X3 2L—3C3 dv [33]Finally the load potential for case 2 can be expressed in terms of the unknown displacementvectors S. (n=1, 2, ..., N) as:^S L^n'71x2^117x1^T^ 3, M7(E _ h_rr ) {(cos L — cosCOS L ) 7(0 ± (cos 117x4^COS 1171-LxL(2) = —2 Po(2)^ )^011n=12x1 2+ (X2 ^x2 — x2)Mb7(e)T + (x2 x1^^1 Ma7(-s)T^2 ^2^, 2—x 2(X4 — X3 — ^ _^ x4+X3) Mci.7()T + 2 (x4 x3^42L Mc7(0T 5nwhere2^ 2^ 2g y2 Sy2 Vr2=^NIP (fl de;^Mc7(e) =^Mc(e) de;^Md7(e) =^Md(e) de2 2 28- gy1 gy1 y13.7 Minimization of Energy in the SystemThe total energy in a finite strip is given by:lI = Uc + UI ± UN — S2L^ [36]which is a minimum when the first variation of H with respect to the unknown displacement vectors SkN[34][35]40(k=1, 2, ..., N) is set to zero as:= 611 (lc]^6k[ U1]^6k[ UN] — 6k[C2d = 0^for (k=1, 2, ..., N)^[37]3.7.1 CoverConsider now the first variation of Uc with respect to bk (k= 1, 2, ..., N) to determine thecontribution from Uc into the system of equation. The matrix bk [ Uc] can be expressed as:8k[ (lc] = E 6k[uc (i)]where1ok [ (01)]^S8 L3 7,)(k_\.1MO mo T Kx2 40 40 T) 0 6k-1N^1(-1-(1) 2n=1 t^(Kxi Mo. Mo- T Kx2 140 40. T) d^ 6nn k^-16k [ uc (2)](K M M. T K 6ckyl 2 2 y2 M2 2 k+1N A+ E L Rknlf(Ky1 NI; 1V 2T — Ky 2 M2. M2 T) (k bnn=1n k^-1[38][39 a]+1n=1 -1 Ky1 [Ma2 Ma2T Mb2 Mb2T]^[Mb2 Ma2T Ma2 Mb2T])rm m^m+ m mr^+^AA 1) _L' "-`-c2 md2T-I)^' • • •+ K y2 3 L a c2 d2 MST] ' 3 I d2 c241+ for^T( 2 sinin^- cos —ior^yl 2^b2 kior Ma2T(1 - sin k4kir^2 j▪ Ky2 Atf2 (Mc2T (cos 2 + —2 sin —k7r^ir1 - Md2T (cos kkir^2 + err sin k+r))wa_ 4^AA-^2^• n71"' Wif(^( sin - 14-r sin n-fr )) M2 T- cos 2) Ma2 ( 1 ♦K^(cos n+r^sin 11:7r) - Md2 (cosn7r njr sin 2))42 T^isn+ y2^c2 [39 b]6k[ Uc (3) ]k 2 71.22SL^Kl[MO M2 T M2 MO T] -I-. Kv2 [MO M2 T + M2 MO- T]_1N +1n=1n^kf-1(Kv1 [MO M2 Tt4ior K^(M T P- sin k-11.vl^0^b2^kir^2n=1 -1M21] - Kv2 [mo T m2 T]) g- cos —1) Ma2T ( 1 - -- 22 sin k-Lr))kir ▪ K^Ovf T^kir^2 • kir \ (^cos kit +v2 0 c2 (cos^+ 0-r sin^- Md2 ( kT^err sin k-fr))▪ (74kn2)Kv12 n7rMb2^sin - COS n7r) M2^a2 (1 —n_27 sin^))▪ Kv2 ^(cos a 2 + 2 sin nrr) - Md2 T (cos^+ - sin ")) 1Vio T ck n [39c]2 6k [ uc (4) ]^2k272SL (KG1 M1 M1T KG2 M; M;^15k+1(k—) Rkn2 f (KG1 mi .^K G 2 M1.4nn=1n k^-1- 072 5 - 8L (Dxi+Dx2) M3 M3T dC+16k[ uc (5)]42N^+1(+ E ^ (KG1 (Mal [Mal — Mbl]T Mbl [Mbl Mal] TE-1▪ KG2 (Mc1 {Mcl MdliT Mdl [Mdl Mcl]T)+ sin k2 ^M1 [Mb]. — Mai] T KG2 M1 [ma — ma] l)+ sin 11-fr (KG1 [Mbl Mal] M; T KG2 [ma —^T)) 4 C [39d]( 5 k [^(6)]-1N^ +1+r` On) RL., (Dx1 — Dx2 ) J M3 m3 T dc do}z--, UcL) --.1.n=1n 0 k^ -1+1^ +1= L{( D j_D2S^yl ' y2) f M6 L kni yl^y2^T g^4 R (i) _ Dn=1-1 n k^ -1[39e]M6 M6T orc (54 [391]+ 1sk [ uc (7)]^— ir k4^Dv2)f [ M3 M6T + M6 M3T]^dk-1+1n=1 Rkn, (Dv1 — Dv2)1 [k M3 M6T n M6 M3T] CIC 6n}n k^ -1[39g]6k[ rfc (8) ]+1_ SL n +DG2) .1. 2 (M 2 TM4 M^flor)2^T) 4k— 16 ({ -- G 1^s ^4 1- VT/ M5 M5 + (--Lr) r^, S L' LM4 T M5 "14T1 uC °Ic-1N^ +1+^r Rkn2 (DV1 DV2) 2 ((212 4 M4T knR-2^m)24^nr7ik -1^L2^5 5+ (n+k)a-) [M4 M5T + M5 M4T ] cIC do^ [39h]433.7.2 Supporting I-BeamsThe contribution of the supporting I-beams to the system of equations is obtained by takingthe first variation of U/ with respect to the appropriate terms in the unknown displacement vectors b k(k= 1, 2, ..., N) which yields the following non-zero components of the matrices b k[ (k= 1, 2, ..., N) as:au, = E L (lor)4 Wkawk^`L= E L (hi)4 vaVk^2 U, IZ k8(4 — E A (kr)2 Ukauk — 2 L0U1^ _ G J (kr)2 saon s — 2 L S)3.7.3 ConnectorsSimilarly taking first variation of UN with respect to the appropriate terms in the unknowndisplacement vectors bk (k= 1, 2, ..., N), yields the contributions of the connectors to the system ofequations as the following non-zero components of the matrices 5k [ UN] (k= 1, 2, ..., N):au,„ _ Kx, L^((d-Fhi) kir- -Ok ^)— 2 e u0k Uk^2 L^ Wk)K L0 UN^xc^ (d-hhi) kr)auk (—^2 e u0k Uk 2 L ) Wk(d+111) kr (0 UN^K xc ^((d+hi) kr) Wku0k^2 L )awk^4 ea UN Kyc L^,2 e vOk k —2 S^— S SavOk[44 a][44 b][44 c][44 d]44K L^=^Y^c^—Ok — V k AV^wok' S^h 0a UNiaVk^2 e 2 S k s [44 e]auN^Kyc hi^K Laok s —^4 e S^ v^VOk^k 2 s wOkd^' S^hi e s^°c 2 wok S — Ok S2 S k 2 e S^ [44f]au^K L d^ h^KO LVk 2s wok S 2 Is Ok S w S—e sowo'k SN^4Yce S (vOk 2 e s2^Ok^k^ [44g]3.7.4 Load PotentialThe first variation of ClL(i) a= 1 or 2) with respect to dk (k= 1, 2, ..., N) for the two cases ofloading can be determined to obtain the contribution of SA, into the system of equation. The vectorsOk [C2L(j) ] (j= 1 or 2) can be expressed as:6k[C2L(i)] =^LPo ( i ) ( — k,;1_) (cos 1c7:2^kir{^ cos Lx1 + Cos kirx^kirx34L cos ^) m7(e)TL/X2 2 — X1 2±X4 2 — X3 2 \^T^ ) Mb7(02^2^2^2X2 — xi -Fx4 — X3+ (x2 —^— X3^ 1<g)11andS^L^l7rx2^krx1\ '^T^k7rx^kirx3Sk[ C2L(2)] — —2 Po(2) )—^{(cos L^cos L M7(e) - + (cos L^cos L ) M7(0 j22^ 2^2+ (x2 xi T^x2 — xL^ Mb7(e) x1^1^) M.7(0TX4 L— x3 2 X4 2 — x3 2))^ x4+X3) Md7(e)' + 2 (x4 x3 ^2L^Mc7(0T[ 4 5][46]45For a single bay system, Equation 37 can be expressed in matrix form as:bicE ucl + bkE^+ 6k [ UN] = bk [Qd^for (k=1, 2, ..., N)^[47]or alternately as:56( i , ) 96(1 , 2 )^. . .^96 (1 ,k)^. . .^96(1,N) bl 31196(2,1) 56(2,2)^. . .^96(2,k)^. . .^56(2,N) 62 912[48]96(k,1) 96(k,2)^. . .^56(k,k)^. . .^96(k,N) 6k Wtk96(N,1) 96(N,2)^. . .^56(N,k)^. . .^96(N,N) 6Nin which the load sub-vectors 91k (k= 1, 2,^N) correspond to the first variation of the load potential6k [C2L(i)] (j= 1 or 2) for the two load cases; the stiffness sub-matrices 96(i,j) (i,j = 1, 2, ..., N) correspond tocontributions from the first variation of the strain energies for the various Fourier terms. Note that96(iii) is a square 39 x 39 matrix and 96(i,j) = 560,0. Finally the integration needed in Equations 39a to39h and Equations 31 and 35 can be performed by a six point Gaussian quadrature procedure.3.8 Numerical Solution of Global System of EquationsFor a deck systems with multiple bays, the total energy in the system is given by:NJT"Total = E (uc +^"4- UN — C2L)ii=1[49]in which NJT represents the number of bays and the index [i] corresponds to a particular bay in thedeck system. Defining Ak ( k=1, 2, ..., N) as the global unknown displacement vectors, —ffTotal is46minimized by taking the first variation of 11 Totai with respect to the unknown global displacementvectors Ak (k=1, 2, ..., N) and setting it to zero as:NJT6k[ 11Totall =^(6k[ uc] + okE^+ ok[ UN] — 6k[QL^= 0i=ifor (k=1, 2 , ..., N)^[50]Therefore, the system of equation of individual bays can be coupled together by standardmethods to form a global system of equations of the assembly where the degrees of freedom along lines1-1 and 2-2 will be shared between adjacent member. The following thirty-two degrees of freedom perFourier term are shared between adjoining finite strips in a deck systems:I^Q^II^II^III^III^WS Q^IV^Q(5 Tnshared = { Win/ W.1n S, Wn/ W 1n. S7 W in/l ^Win/ W ln^W1n/ W ln 1n/ Wln 7j7^• , c^I ' c^II^II '111n, llin S, v1n, v in S, W2n/ W2n "/ W2n/ W2n. S, W2n/ W2n '37 W2n/ W2n S ,III w en' S^Iv S , u2n, u^S^2n ' , v^S}2n , v2n '^ [51Wzn W^W^W (n=1, 2, ..., N)^]2n 2n/ 2nThe global system of equations can be obtained when the appropriate terms are substitutedinto Equation 50. Defining K(i,i) = 1, 2, ..., N) as the global stiffness sub-matrices and R i (i 1, 2, ...,N) as the global load sub-vectors, the global system of equations can be expressed in matrix form as:K(1,1) K(1,2)^. . .^K(1,k)^. . .^K(1,N) Al R1K(2,1) K(2,2)^. . .^K(2,k)^. . .^K(2,N) A2 R2[52]K(k,1) K(k,2)^. . .^K(k,k)^. . .^K(k,N) RkK(N,1) K(N,2)^. . .^K(N,k)^. . .^K(N,N) AN RN47As a result of sharing of the 32 degrees of freedom between adjoining bays in the deck system,R., are (23 x NJT 16) x 1 vectors and KW) are (23 x NJT 16) x (23 x NJT 16) square matrices. Notethat K(i,i) are symmetrical with a bandwidth of 39 but K(i,j) (j i) are unsymmetrical. The globalstiffness matrix is symmetrical; i.e., K(i,j) =Foschi (1982) demonstrated that such a system of equation can be efficiently solved by theJacobi iterative procedure as follows:Ak = K(k,k) 1 {Rk — E^A2 1}nnA (k=1, 2, ..., N) [53]where the superscript i indicates the ith iteration and the initial vectors are taken as:0^ 1Ak = K(k,k) RkThe iteration is terminated when the following convergence criterion is met:[54]1i Aki - Ak" <II Hki 1 Ii(k=1, 2, ..., N) [55]where^represents the norm of the vector and e is taken as a small number (say 0.005).Given a particular loading pattern p(x,y), the solution of global system of equation yields Ak(k= 1, 2, ..., N); therefore, the deformed shape of the structure can be estimated which in turn will beused to predict the critical stresses in the cover.3.9 Shear Deflection of the Supporting I-beamsWhen the span to depth ratio of the supporting I-beams is small, the influence of shear on the48deflection of the I-beams may become important. It can be shown that the shear deflection W shear ofthe supporting I-beams satisfies the following condition approximately:dWshear — as { EI d3^KxW^2 e 01(x ' 0)^U^2^dxhI + d dWVdx^G A^Y dx3where as is a shape factor which depends on the geometry of the cross section and the distribution ofthe shear stresses. For an I-beam it can be shown that a can be approximated by:A (b^(hi — 2tf) 2i tf^tw)as — I t^2tf (h^) + ^8Y wSubstituting Equations 14c, 15a, and 15c into Equation 56 and performing the integrationyields the shear deflection in each supporting I beam as:v__N, a {^ i^Kx LWshear^G A^Iy (1,92 Kx h 4 e d Wn^11-71-n /I- 2 e (uml — U^sin Ln=13.10 Bending Stresses and Rolling Shear Stresses in CoverTwo common failure modes of the cover in truck deck applications are bending failure androlling shear failure. Rolling shear failures result from shear forces which roll wood fibers within a layerof veneer over each other. Consider again only one bay in a deck system, the solution of the globalsystem of equations yields Ak (k= 1, 2, ..., N) from which Sk (k= 1, 2, ..., N) can be obtained. Assumingthe face grain is oriented perpendicular to the direction of the supporting I-beams, the bending stressesof the exterior ply in the cover (at z=) in the parallel to face grain direction can be estimated as:22Ey (ay^au d (a w^a 21crs,(X, y) = 1 v^ay + VxY ax^2 ,93,2 + v " axe)xy yx [59][56][57][58]49where E denotes the modulus of elasticity of the exterior ply in the cover in the y-direction and ddenotes the thickness of the cover. Note that E is assumed to be equaled for both sections 1 and 2 ofthe cover. Similarly, the bending stresses of the exterior ply in the cover (at z=) in the perpendicular2to face grain direction can be estimated as:cri(x, y) = 1^Ex )^(au^av _ d2W^azw^ax^Pyx ay^ax2^liyx 8372) [60]xyV yxwhere Ex denotes the modulus of elasticity of the exterior ply in the cover in the x-direction. Note thatEx is also assumed to be equal for both sections 1 and 2 of the cover.Consider the bending stresses in y-direction in one bay of the deck system, substituting theshape functions and the unknown displacement vectors into Equation 59 and performing theappropriate differentiation for the location y= — -2-S , Equation 59 can be expressed in terms of elementsin the unknown displacement vectors 15. (n=1, 2, ..., N) for the location y= —; as:E^NCrY(X 2— —S) = 1 — v vIn - vxy L(-111)uln — d {— 46 win — 12 w 1 S — 16 won' Svxy yx n=1^ 2S22+32 Wn + 14 wen — 2 wen' S} + vxy (L) ^sin (n—f--, rx )_ 2x— —d {( —46 W ln — 12 Wln, S + 14 wen — 2 wen '2nT__,}(— 46 w1n — 12 win , S + 14 w 2n — 2 w2ini s) (2x)^for (0 < x < -1-21) [61 a]and50E^Ncr- (X, —)=^ 46^— 12 w 1. S —16 won' SY^2^1 — vxyvyx E vin'^(117) Uln 2S 2n=12+ 32 W. + 14 w2; —2 w2.'' S)^vxy^w1;1 sin (T)2xIII ,— — {(— 46 w 1n^ln— 12 w S 14 2n 2 w2n(^)S 2 --2S 2• (— 46 wInV — 12 wi.V S + 14 w2iv. — 2 w2i: 7 S)(f),c —1) for (4' < x < L) [61b]Similarly substituting the shape functions and the unknown displacement vectors intoEquation 59 and performing the appropriate differentiation for the location y=0, Equation 59 can beexpressed in terms of elements in the unknown displacement vectors b. (n=1, 2, ..., N) for the locationy=0 as:Eycr- -(x 0) —^Y^1 — v vxy yx{( 3^1^, s + 3^1 ^\ 1^(n7r\— -2" vin — 4 vin^, v2n — 4 v2n s) s- — vxy or ) nonS — 16 W. + 8 w 2.—2s2 {8 win +^ w2n. S} +— d {(Q^_L^8 I _ I' S) (1 2x)vy in W ln^-F W2n W2n2S22vxy^wn} sin (n1 jrx )II+( 8 win + win S + 8 w2n. W2Ini S) (r),)^for (0 < x < —2 ) [62 a]andEy^ NCf si (X, 0) = 1 —^ E ( — vin^yin' S^v2n^V2n S)^(V) uon + •vx3rvYx n=151{8 w + wln S — 16 Wn + 82- s2^In w2n^+ d ,-xy2W } sinL^n(n7Lrx)— d (8 ^+ ^S 8 win w ^s) (2 2 LX)win -1- 11' 1 n 2n^2n2S 2}+(8 wff, ± wff." , s ± 8 wI2Vn _ w2InV ' s) (2LX -1) for (5' < x < L) [62b]Here consider the bending stresses in x-direction in one bay of the deck system, substituting theshape functions and the unknown displacement vectors into Equation 60 and performing theappropriate differentiation for the location y= — 2, Equation 60 can be expressed in terms of elementsin the unknown displacement vectors Sn (n=1, 2, ..., N) for the location y= — as:S\ 1 - VEx N-2") '="^Exy Yx n=1 V x V ln ' - (111-1,) Uln - 2S2 Yx{^ d V { - 46 win —12 wln• ' S - 16 won' S2^+32 Wn + 14 inT ri —2^+ -(21 (ni-7,) win sin (n÷: -( )—2S2 vYx (— 46 w lI — 12 w I S + 14 w2I — 2 w2nI S) (1 L— 2x)^n ^n_ 46 w _ 12 w^s 14^_ 2 w2InI ' s a)^for (0 < x <) [63a]and(— 46 win' — 12 w S — 16 w Sxy yx n= icric(x ,^= 1 — vEx v^vyx yin — Er) nin 2:-1 2^2S Yx^In^Ond 11.7 win si2^}^nrxl+ 3 2 Wn + 14 win — 2 w2.n*^(17) sin i'TT' +52—^III^vv^(2 2 X)d vYx {(— 46 w in — 12 w in S + 14 VV2n2S2 2n L )46 wwin— 12 w ilily S + 14 w2Ivn^2^s)^_ 1) for (-IT' <^x < L) [631)]Similarly substituting the shape functions and the unknown displacement vectors intoEquation 60 and performing the appropriate differentiation for the location y=0, Equation 60 can beexpressed in terms of elements in the unknown displacement vectors 45 n (n=1, 2, ..., N) for the locationy=0 as:Excri (X, 0) — 1 — v v xy yx n= 1{vyx ( A Vin — j4. v in ' S + v2n —1 v 2n S) — (y) non- -d v {8 w • +2S2 yx^Win w in ' S —16 Wn + 8 wen — w2n s}2W } sin (11-7—rx )2 L^n^LI w2I '^2x1d^ I ' S + 8 w2n' W ln2S 2 Yx^I^n S)II^'^m II^IIwin + win S 8 '"2n W2ns) ()^for (0<x<2) [64a]andExo-5,(x, 0) — 1 — v vxy y n=13,^q1^,^3 v — 1 v ' s) 1 — (1-1 ' 9 uoPyx ( — v ln — T yin S + -2- 2n 4 2n S^L^n{— v {8 w" + wS — 16 Wn + 8 w2•n — w2.n.^+2S2 yx^In^In2sin ( L )(n7r) AAT } • nirx 2 L^vv rid v,ATT^s 8 vv i2ini^s) (2 L_ 2x)2- S2 YxIV ^S INT Q) (2x8 win + win 8 W2n W2n^—1) for^< x < L) [64b]2^2^z (a W a^- 8y2^xy ax22^2Z (a W + V ax2 yx a0yW2auOxavay[67 a][67h][67c]53From equilibrium, the rolling shear stresses at any ply in the cover in the direction parallel toface grain (ryz) and in the direction perpendicular to face grain (r xz ) can be expressed as:acry aTx„Tyz = — f (79-3-7^axj) dz^ [65]andTxz = — .1. 6—°Crxx 19.977) dzwhereEy^ (aycry(x, y) = 1 -v^ayxy yxcrx(x, y) = 1Ex ^(auvxyvyx Ox2Txy = Gc P-317 + ^- 2 z gc vavy) ax+ V xy+ 1/ yx[66]Note that Ex and Gc denotes the modulus of elasticity in the x-direction and the modulus of rigidity ofthe exterior ply in the cover, respectively. Here E x and G c are again assumed to be equaled for bothsections 1 and 2 of the cover.Consider the transDeck TM panel with the 3-1-3-1-3 lay up scheme shown in Figure 1, it is ofinterest to estimate the rolling shear stresses at the interface between the top three exterior parallelplies and the cross band. Assuming identical thickness (t) for each ply and identical elastic propertiesfor the three exterior parallel plies, the rolling shear stresses at the interface in the direction paralleland perpendicular to face grain can be expressed as:andT =xzh^ 54^22^3^3^ aV_L , all w Tyz = PlYcylly) 8372 ^u aXaY Z (58373 v" ax20y)5h222^2+ (911 j_OV^2 z ^w dzfaxay ax2^ax2ayh2a2V y}(^Ex^0211^ a3 IV ,^- vxyvyxl {aX2 + v ^Y x aXa^(53:5 ^Y3c aXay 2i5h222^2^Q3a V 2 z u w dz+ Gc^+ aOyll^x y^Oxay2Performing the integration yields the following:and1-XZ[68]T =Yz (^l ^EY^A2u.kaa2x2v) (1 — vxyvyx) vxy+ Gc) Ucay4 3 tEy^\ 1,93 \^Er^ n3{U. — V V ) w3) (U_ — v^Vxy 2Gc) u 2w )} 12 t 2xy yx y^xy yx aX ayEx \,a20 (^„2 ^E 1 — v v^ ) 2) Gc u) + x^ )1/ +G)( a Vxy yx ‘uX^c,372^— 1/x^yx^&coy)} 3 ty yx3 Ex {(1 — Vyx ) (a^—vxy vyx vyx + 2Gc w^2) 12 t 2xy^X ax0ySubstituting the shape functions and S. into Equations 70 and performing the appropriatedifferentiation for the location y=0, the rolling shear stresses at the interface between the cross andparallel ply for the parallel to face grain direction can be expressed in terms of elements in 15. as :[69][70][71]552Tyz(X, 0)^—^{((, _ 1 3r v^12 ) Vin^- 16 von + 8 v2n —S^v2n.' S) Gc^VOnn=1^xy yx S -( E^)^- V3, V ) VxY Gc) (S0 ^2 Uln^Uln S^U - 1 (xy yx 2n 4 ll2n S 3 t(G Ey v ) ( -j- ) ( - 60 win — 6 w in. ' S —48 won' S + 60 w2'n — 6 w 2n' S)xy yx S3E^\ 2+ v y v ) Vxy + 2 G)()() won S 12 t 2 sin (n÷-r, x )xy yx(G __ 1 31 vy) ( ) {( - 60 w iin —6 w in/ S + 60 w 2In —6 w2ni S) (1 —2'Ix)2x(— 60 w in —6 w in S + 60 w2n — 6 w2n S)()} 12 t 2) for (0 < x < 2) [72a]andTyz(x, 0) = —^{(u E v S2) Yin + /fin ' S - 16 von + 8 v2n — V2n S — G c —(nL 12 vonxy yx Sn=1((^YE ^) V + G ) ( IsLr ) ( - q^1^' S1 — v v / xY^c SL^2xy yx Uln - Li Uln -I- 3 U2n - 1-4 u2n S) 3 t((1 — / 3ryvy)( "§ )(— 60 win —6 w i .n. ' S —48 won' S + 60 w2n — 6 w 2.n' S)((1 - VxyVyx) VxY + 2 Gc)( e )(n-T-71 2 won ) 12 t 2} sin (T)( G __ 1 31 vyx) ( S3) {(_ 60 wins _6 wins ' S + 60 wI2III_ 6 w2IIIII s) (2 —N+ (— 60 win^ IV S)'^2x^)1 12 t for (2)^L < x < 0) [72b]—6 w in^2nS + 60 w^— 1 —— 6 w2n 256Substituting the shape functions and 5n. into Equations 71 and performing the appropriatedifferentiation for the location y=0, the rolling shear stresses at the interface between the cross andparallel ply for the perpendicular to face grain direction can be expressed in terms of elements in C as:^N^ 2x^ (111r) non Gc s-2 ) uln ui'rxz (x, 0)^— E^\1—v v /O.,/^1S — 16 110. + 8 112n —^n=1^xy yx 2n s)Exvxyvyx) ^± GC)^\Tin —14 V ln ' S^V2n^V2n s)) 3 t(^— ^vxy yx— 16 W. + 8ExWn^_ vxy yxvyx ± 2 Gc)^(8 ^win +^S—^S)) 12 t 2l cos (nirx)E^ 2 1^)—— 12 t 2 ((( j_ v x v )v ^2 Gc) () eVln^W2n W1n W2nxy yx 521.+ ( II , II ,win S — w2. S)— Wi Iii , S + w2n' s) for (0 < x < 2) [73a]andTxz(x, 0) = — En=1E^\ 2— v;:yvyx) (L)un Gc^) (8 u in +^S — 16 uo. + 8 u2„ 112n S)Ex \1 v2i, s)) 3 t+ ((i _ v^) vyx Gc (SL)^ yin — 1 yin' S + V2n( 3xy yx(^‘3Ex97)^(^Ex^) vyx + 2 G^nir^) IV + W 1 .1.1 S— V V )^-^— V__xy yx xyv yx c LS2— 16 W. + 8 w'r, —^12 t 2 cos (-11F-7) +57— 12 t2E. \1 — v v^v(((^xy yx) v x ± 2 Gc) (-1 ) (8S 2L(wipin ,^IV^III^III )1- W2n — w ln — w2n'^IV— w2n S — W1n^W2llnI s)^for^< x < L) [73b]Finally considering each bay in a deck system, 15. (n= 1, 2, ..., N) can be obtained from theglobal displacement vectors A. (n= 1, 2, ..., N). Therefore, the bending stresses of the exterior ply in thecover in the parallel to face grain direction can be obtained directly from Equations 61a, 61 b, 62a and62b. Similarly, the bending stresses of the exterior ply in the cover in the perpendicular to face graindirection can be obtained directly from Equations 63a, 63 b, 64a and 64b. Furthermore, the rolling shearstresses at the interface between the top three exterior parallel plies and the cross band in the directionparallel and perpendicular to face grain can also be obtained from Equations 72a, 72 b, 73a and 73b.Given a particular loading pattern p(x,y), the bending and rolling shear stress profile of thecover can be obtained. Furthermore, the location and magnitude of critical stresses in the cover cantherefore be predicted.3.11 Computing Environment and EfficiencyThe DAP was coded in Fortran-77 Language and implemented in a 80386 personal computerenvironment with a 33 MHz processor speed and a minimum of 2 Mbytes of random access memory(RAM). A special Fortran Language compiler, Lahey F77L EM/32, was used so that all the availableRAM in the minicomputer could be accessed by the program; standard Fortran compilers allowed only640 kbytes of RAM to be accessed by the program which was insufficient for DAP. The program canconsider up to seventeen supporting I-beams and 10 terms in the Fourier series.Considering solution of a case with 17 supporting beams and 8 terms in the Fourier series,typically 15 to 20 minutes of computing time was required. The execution time depends on the choiceof convergence requirement e and the coupling of the off-diagonal stiffness submatrices due to the58modeling of the midspan gap. During program execution typically 80 to 90 Jacobi iterations wererequired to satisfy the convergence requirements of e= 0.005. Therefore, the execution time required foreach iteration is reasonably short which is an indication of the efficiency of the finite strip formulation.3.12 Sample ProblemsThe deck system shown in Figure 2 was considered in the example problems. It containedseventeen bays with dimensions of 2.44 x 4.88 m (8 x 16 feet) in plan. The deck was sheathed by 11-ply 35 mm (4 inch) thick transDeck TM panels. The panels were supported by 2.44 m (8 feet) longsteel I-beams spaced at 305 mm (12 inches) on center. Eight cases which included various combinationsof: 1) two different wheel locations; 2) either four or six terms used in the Fourier Series; and 3) twodifferent connector stiffnesses were studied.3.12.1 Program InputThe dimensions of the transDeck TM panel were 1.22 x 2.44 m (4 x 8 feet) in plan. The facegrain and the long axis of the panels were oriented parallel to the long axis of the deck. Each end of thedeck was sheathed by two half sheets of panels, 1.22 x 1.22 m (4 x 4 feet) in dimension. The middleportion of the deck was sheathed by two full size panels, 1.22 x 2.44 m (4 x 8 feet) in dimension.Therefore, each deck contained 4 half sheets and 2 full sheet of panels in total. All panel edgesperpendicular to the long axis of the deck were fully supported by I-beams. However, the panel edgesparallel to the long axis of the deck were unsupported except over the I-beams.The elastic properties of a transDeck TM panel can be estimated from the elastic properties ofindividual veneer. Assuming the thickness of each ply equals to t, the stiffnesses of the panel can beobtained from:j=1N2^Ex t1 - v vj=1^xy yxE iv tya yx1 - v vxy yxDX =D =[74 a][74 b][74 c][74 d][74 e][74f][74g][74h]where N1 and N2 are the number of plies with the grain parallel to the x and y direction, respectively;IX and Iyl are the moment of inertia per unit of width, with respect to the center line of the panel, ofthe j th ply with the grain parallel to the x and y direction, respectively;^andd Ei are the axiala^Yamoduli of elasticity of the j th ply 0 = 1 to N 1 or N2 ) with the grain parallel to the x and y direction,respectively; and Eyfi are the flexure moduli of elasticity of the j th ply = 1 to N 1 or N 2 ) with thegrain parallel to the x and y direction, respectively; G i is the modulus of rigidity of the j th ply 0 = 1 toN 1 or N2 ) of the panel; and the vxy and vyx are the Poisson's ratio.In these example problems, the individual veneers in a panel were assumed to have identicalelastic properties. Here the axial moduli of elasticity of the veneer in the x and y directions were60assumed to equal 488.2 MPa (70811 psi) and 9828 MPa (1.425x10 6 psi), respectively. The flexuremoduli of elasticity of the veneer in the x and y directions were assumed to equal 430.9 MPa (62500psi) and 11566 MPa (1.678x10 6 psi), respectively. The moduli of rigidity of the veneer was assumed toequal 729.0 MPa (105738 psi). Furthermore, the Poisson's ratio v. y and vy. were taken as 0.02 and 0.4respectively. Finally, the thickness of each veneer was taken as 3.2 mm (0.125 inch).The stiffnesses of a panel in the cover were obtained from Equations 74a to 741/ as:K. =^4.48 kN • mKY = 38.48 kN • mK, =^1.25 kN • mKG = 2.59 kN -mD. = 76.97 1Y4Dy = 286.22 -1\---f-li\liDv = 25.44 MmNDG = 25.46 MmN(39618.1 lb • in);(340384.8 lb •in);(11042.6 lb • in);(22906.4 lb • in);(439554.3 R-3);in(1634468.0 !"));in(145305.9 P2);in(145389.5 N.inThe calculation of the stresses in the outer ply of the cover requires information on the Ey , E.and G c which were taken as 9828.0 MPa (1.425x10 6 psi), 488.2 MPa (70811 psi), and 729.0 MPa(105738 psi), respectively. The moduli of elasticity and rigidity of the supporting I-beams in the decksystem were taken as 200,000 MPa (29x10 6 psi) and 77000 MPa (11x10 6 psi), respectively. Also thesupporting I-beams were assumed to have a yield strength of 550 MPa (80000 psi). The thickness of theflange (tf) and the web (tw ) equaled 3.2 mm (0.125 inch). The depth (h1) and the width (b 1) of the I-beam were 102 mm (4 inches) and 57 mm (2.25 inches), respectively. The connectors between the coverand the supporting I-beams were considered uniformly spaced at 102 mm (4 inches) on center. The twocases of assumed connector stiffnesses were: 1) K K Vc and K equaled 1.75 -1V (10000 i-1ibi.); 2) K. ,cx ,^'^Oc cK3 and Koc equaled 3.50 ,1 -N (20000 i-I 1131) •1The loading on the deck was assumed to result from the wheels of a lift truck carrying a load61of 81.5 kN (18333 lb). The two front wheels were assumed to carry 90% of the total load, 73.4 kN(16500 lb). The two rear wheels therefore carried 8.2 kN (1833 lb).The front and rear axles were spaced at a distance of 1.22 m (4 feet) apart. Both axles wereoriented parallel to the direction of the supporting I-beams in the deck. The two different wheellocations considered were:1) the front axle was assumed to be centrally positioned between bays 8 and 9 of the deck and the rearaxle was therefore centrally positioned between bays 12 and 13 of the deck;2) the front axle was assumed to be positioned on bay 9 of the deck and the rear axle was thereforepositioned on bay 13 of the deck.The foot print of each front wheel was assumed to be 203 x 89 mm (8 x 3.5 inches) whichcovered an area of 18064 mm 2 (28 inches2 ). The rear wheels were smaller than the front wheels; eachrear wheel had a foot print of 89 x 80 mm (3.5 x 3.15 inches) which covered an area of 7113 mm 2 (11inches 2 ). The two front wheels were spaced at a distance of 965 mm (38 inches) apart and centered inthe x-direction in the deck. Similarly, the rear wheels were centered in the x-direction of the deck andspaced at a distance of 749 mm (29.5 inches) apart.3.12.2 Program outputShown in Table 1 are the eight case studies which included the various combinations of: 1) twodifferent wheel locations; 2) either four or six terms used in the Fourier Series; and 3) two differentconnector stiffnesses. Summary results from DAP on the maximum vertical deflection of the supportingI-beams in each bay for the eight cases are shown in Table 2. Similar summary results on themaximum bending stress in the supporting I-beams in each bay for the various cases are shown inTable 3.The results show that a maximum supporting I-beam deflection, W(x), of 20 mm (0.799 inch)occurred in case 4 at midspan of I-beam number 9. A maximum bending stress of 363 MPa (52631 psi)62in the supporting I-beams occurred in midspan of I-beam number 9 in case 6. In each finite strip, thedeflections of the cover in the vertical direction, w(x,y), were monitored along the lines 0-0 and 1-1 asshown in Figure 3, with a grid size of 61 mm (2.4 inches) in the x-direction. Table 4 shows themaximum cover deflections for the eight cases. In cases 2, 4, 6, and 8, when the front axle was directlyon top of I-beam Number 9, the maximum cover deflection was located at the midspan (x = 1.22 m) ofBay 9. When the front axle was centrally located between I-beams 8 and 9 for cases 1, 3, 5, and 7, themaximum cover deflection was located at the midspan (x = 1.22 m) and centered between Bays 8 and9. Within the eight cases, a maximum deflection of 20 mm (0.783 inch) was found in case 4.Table 1. Descriptions of the various case studies in the example problem.Case No. of termsin FourierSeriesKxc, Kyc, and Koc(MmN)Front Axle Location1 4 3.50 centered between bays 8 and 92 4 3.50 on bay 93 4 1.75 centered between bays 8 and 94 4 1.75 on bay 95 6 3.50 centered between bays 8 and 96 6 3.50 on bay 97 6 1.75 centered between bays 8 and 98 6 1.75 on bay 911b = 0.000175 MmNin63Table 2. Summary deflection results of the supporting I-beams in the eight case studies.Maximum Deflection in the Supporting I-beams (mm)Case No. 1 2 3 4 5 6 7 8Beam No. 1 -0.1 -0.0 -0.2 -0.0 -0.1 -0.0 -0.2 -0.0Beam No. 2 -0.4 -0.3 -0.5 -0.4 -0.4 -0.3 -0.5 -0.4Beam No. 3 -0.6 -0.6 -0.7 -0.7 -0.6 -0.6 -0.7 -0.7Beam No. 4 -0.4 -0.6 -0.5 -0.7 -0.4 -0.6 -0.5 -0.7Beam No. 5 1.0 0.1 1.1 0.1 1.0 0.1 1.1 0.1Beam No. 6 4.7 2.5 5.0 2.7 4.7 2.5 5.0 2.7Beam No. 7 11.0 7.5 11.8 8.1 11.1 7.5 11.8 8.1Beam No. 8 17.8 14.7 18.8 15.6 17.7 14.7 18.7 15.6Beam No. 9 18.0 19.2 19.0 20.3 17.9 19.0 18.9 20.1Beam No. 10 11.7 15.1 12.5 16.0 11.7 15.1 12.5 16.0Beam No. 11 6.1 8.5 6.6 9.2 6.1 8.5 6.6 9.2Beam No. 12 3.3 4.4 3.5 4.7 3.3 4.4 3.5 4.7Beam No. 13 1.8 2.5 1.9 2.7 1.8 2.5 1.9 2.6Beam No. 14 0.7 1.2 0.7 1.3 0.7 1.2 0.7 1.3Beam No. 15 0.1 0.4 0.1 0.4 0.1 0.4 0.1 0.4Beam No. 16 -0.1 -0.0 -0.1 -0.1 -0.1 -0.0 -0.1 -0.1Beam No. 17 -0.2 -0.3 -0.2 -0.3 -0.2 -0.3 -0.2 -0.31 inch = 25.4 mm64Table 3. Summary bending stress results of the supporting I-beams in the eight case studies.Maximum Bending Stress in the Supporting I-beams (MPa)Case No. 1 2 3 4 5 6 7 8Beam No. 1 -2.41 -0.29 -3.18 -0.63 -2.42 -0.29 -3.19 -0.63Beam No. 2 -8.33 -5.71 -9.48 -6.62 -8.32 -5.70 -9.47 -6.62Beam No. 3 -12.72 -10.97 -13.78 -12.17 -12.68 -10.95 -13.75 -12.15Beam No. 4 -9.23 -12.55 -9.16 -13.23 -9.23 -12.53 -9.18 -13.21Beam No. 5 16.14 -0.40 18.65 0.72 16.32 -0.35 18.83 0.76Beam No. 6 84.04 43.50 89.50 47.52 82.95 43.32 88.39 47.33Beam No. 7 203.68 138.49 210.31 144.95 208.97 139.14 215.78 145.64Beam No. 8 308.83 265.74 313.19 271.56 319.47 262.63 323.75 268.54Beam No. 9 311.93 324.95 316.60 328.59 322.22 362.86 326.77 366.48Beam No. 10 215.40 272.58 222.83 278.98 220.58 269.45 228.20 275.94Beam No. 11 109.88 156.56 116.32 163.97 109.24 157.25 115.67 164.71Beam No. 12 59.06 78.17 62.27 83.11 56.12 77.71 59.33 82.64Beam No. 13 33.61 46.43 34.36 48.19 31.49 43.54 32.20 45.14Beam No. 14 12.70 22.03 12.52 22.20 13.18 21.78 13.00 21.94Beam No. 15 2.03 6.37 1.55 5.96 1.95 6.44 1.49 6.03Beam No. 16 -2.27 -1.27 -2.81 -1.90 -2.26 -1.29 -2.79 -1.93Beam No. 17 -4.04 -5.48 -4.50 -6.15 -4.05 -5.51 -4.52 -6.181 ksi = 6.89 MPa65Table 4. Maximum deflections and stresses in the cover of the eight case studies.Case Maximum^Maximum^Maximum^Maximum^Maximumw(x,y)(mm)crs,(x,y)(MPa)cri(x,y)(MPa)7-3,z(x,y)(MPa)Txz(X,Y)(MPa)1 18.7 32.46 0.63 0.84 0.042 18.9 21.17 0.32 0.32 0.043 19.7 32.80 0.74 0.84 0.034 19.9 21.60 0.45 0.33 0.035 18.4 36.20 0.85 0.76 0.056 18.7 20.57 0.36 0.31 0.047 19.4 36.57 0.96 0.77 0.048 19.8 20.99 0.48 0.32 0.039 19.4 43.70 1.21 0.81 0.0510 19.6 44.96 1.37 0.88 0.0666The bending stresses in the exterior ply of the cover crs,(x,y) and u5c(x,y) were monitored at the samelocations as the cover deflections. Table 4 also shows the maximum cover bending stresses for the eightcases. Case 7 yielded the maximum cover bending stresses, a, and ak, of 36.57 MPa (5304 psi) and1.21 MPa (175 psi), respectively. Table 5 shows the maximum cover bending stresses locations for theeight cases. In cases 2, 4, 6, and 8, when the front axle was directly on top of I-beam Number 9, themaximum cr,(x,y) were located in Bay 9 between the foot prints of the front wheels. In cases 1, 3, 5,and 7, when the front axle was centered between Bays 8 and 9, the maximum as, values were locatedbetween Bays 8 and 9 under the foot prints of the front wheels. The maximum uk values were typicallylocated under the foot prints of the front wheels except in Cases 2 and 4 where the maximum stresseswere located between the foot prints of the front wheels.The rolling shear stresses in the cross ply of the cover in the x and y direction, Txz and Tyz ,were monitored over the supporting I-beams, with a grid size of 61 mm (2.4 inches) in the x-direction.Table 4 shows the maximum cover rolling shear stresses in the x- and y-direction for the eight cases.Within the eight cases, a maximum T yz of 0.77 MPa (111.9 psi) was found in case 7 and a maximumTxz of 0.05 MPa (7.2 psi) was found in case 5. Table 5 shows the locations of the maximum cover shearstresses for the eight cases. The maximum values were typically located between the foot prints ofthe front wheels except in Cases 5 and 7 where the maximum stresses were located under the footprints of the front wheels. In all cases the maximum Txz values were located near the edge of the panelsat x=0 and 2.44 m.The results from the case studies clearly show that the cover bending stress and the rollingshear stress in the x-direction were insignificant compared to the other stresses. Therefore, Cric and Txzwill not be further considered in detail when evaluating the impact of connector stiffness, wheellocation and number of Fourier terms used in the analysis.The influence of connector stiffness, which varies from 3.50 to 1.75 —MN on the deflection andmthe stresses of the supporting I-beams and the cover seems to be small. In general, less than three67Table 5. Locations of maximum stresses in the cover of the eight case studies.Case^y-location^ x-locationBending Stresses^ a-5,(x,y)^crk(x,Y)1 centered between Bays 8 and 9^0.79 and 1.65 m^0.79 and 1.65 m^2 Bay 9^ 1.22 m^1.16 and 1.28 m3 centered between Bays 8 and 9^0.79 and 1.65 m^0.79 and 1.65 m4 Bay 9^ 1.16 and 1.28 m^1.16 and 1.28 m5 centered between Bays 8 and 9^0.73 and 1.71 m^0.73 and 1.71 m6 Bay 9^ 1.16 and 1.28 m^0.79 and 1.65 m7 centered between Bays 8 and 9^0.73 and 1.71 m^0.73 and 1.71 m8 Bay 9^ 0.98 and 1.46 m^0.79 and 1.65 mShear Stresses^Tyz(x,Y)^Txz(x,Y)1 Bay 8 1.22 m 0.43 and 2.01 mBay 9^ 1.22 m^0.43 and 2.01 m2 Bay 8 1.16 and 1.28 m^0.00 and 2.44 mBay 10^ 1.16 and 1.28 m^0.00 and 2.44 m3 Bay 8 1.22 m^0.49 and 1.95 mBay 9^ 1.22 m 0.49 and 1.95 m4 Bay 8 1.16 and 1.28 m^0.00 and 2.44 mBay 10^ 1.16 and 1.28 m^0.00 and 2.44 m5 Bay 8 0.73 and 1.71 m^0.43 and 2.01 mBay 9^ 0.73 and 1.71 m^0.43 and 2.01 m6 Bay 8 1.04 and 1.40 m^0.00 and 2.44 mBay 10^ 1.04 and 1.40 m^0.00 and 2.44 m7 Bay 8 0.73 and 1.71 m^0.49 and 1.95 mBay 9^ 0.73 and 1.71 m^0.49 and 1.95 m8 Bay 8 1.04 and 1.40 m^0.06 and 2.38 mBay 10^ 1.04 and 1.40 m^0.06 and 2.38 m68percent difference in cover stresses, as; and ryz , and and less than six percent difference in coverdeflection can be observed between the cases with the two connector stiffnesses.However the wheel location and the number of Fourier terms used in the analysis significantlyinfluence the estimated stresses of the supporting I-beams and the cover. Rolling shear stresses andbending stresses in the cover in the y-direction, Tyz and us„ were much higher when the wheels werecentrally positioned between the supporting I-beams compared to the cases when the wheels weredirectly over the supporting I-beams. Also bending stresses of the supporting I-beams, andd v i,, werehigher by as much as 10% when the six rather than four terms in the Fourier series were used. Theinfluence of the number of terms used in the Fourier series on the bending deflections of the supportingI-beams and the deflection of the cover was insignificant because less than two percent difference inresponse was observed between the cases with the four and six terms in the Fourier series.Finally two more case studies were conducted to investigate the convergence of the deflectionsand stresses. Here similar to case 7, the connector stiffness was considered to be 1.75 IV and the frontwheels were considered to be centrally placed between Bays 8 and 9. However, in cases 9 and 10, eightand ten terms were used in the Fourier series, respectively. In case 9, the results indicate a maximumW(x) and supporting I-beam bending stress of 18.9 mm (0.744 inch) and 332.5 MPa (48227 psi),respectively. In case 10, the results indicate a maximum W(x) and supporting I-beam bending stress of18.9 mm (0.744 inch), respectively. The maximum w(x,y), o -y,(x,y), crTe (x,y), ryz (x,y), and rxz (x,y)estimates for these cases are shown in Table 4.Comparing the results from cases 9 and 10, the absolute percentage difference in estimatingmaximum W(x), supporting I-beam bending stress, w(x,y), o-y,(x,y), cry,(x,y), ryz (x,y), and rxz (x,y)were 0.05%, 0.92%, 0.72%, 2.82%, 11.76%, 7.85%, and 14.46%, respectively. Large differences occur inthe estimation of maximum cr,7 (x,y) and rxz (x,y) of the cover. The magnitude of these stresses weresmall compared to the stresses in the other directions such as the maximum o- y (x,y) and 7-3,,(x,y) of thecover. It is possible that a few more terms were needed for convergence of these stresses because of the69concentrated applied loads and the strong directional elastic properties of the cover. This is especiallytrue when the loads were applied in between two bays rather than directly on top of the supporting I-beams. Since the deflections and the other major stresses in the system converged rapidly, eight termsin the Fourier series were deemed sufficient to achieve the desired accuracy for estimation of stressesand deflections.Finally Figures 5 to 9 show the w(x,y) profile, the o y (x,y) profile, the cr(x,y) profile, theryz (x,y) profile and the^profile the of the cover for case 7. Figures 10 to 14 show the w(x,y) profile,the cry (x,y) profile, the cr(x,y) profile, the ryz (x,y) profile and the profile the of the cover for case8. From Figures 5 and 10, it can be seen that the shapes of both cover deformation profiles wereconsistent with the peak deflections occurring directly under the front wheels. From Figures 6 to 9 and11 to 14, it is seen that the stress profiles were complicated and dependent on the wheel location. Whenthe front wheels were located in between the supporting I-beam, large stress concentrations occurredwhich could initiate damage in the cover. It can also be noted that the stresses at the gap (x= 1 ) arenon-zero. With a large number of Fourier terms, it is expected that the normal and shear stresses onthe free edges at the gap will converge to zero. In terms of bending and rolling shear stresses in the y-direction, the critical locations were clearly directly under the front wheels and in midspan of the deckalong the direction of the front axle, respectively. Considering a lift truck moving across the decksystem, the stresses for any wheel position (i.e., the stress history) in these critical points can beestimated from DAP.DAP is a general purpose deck analysis program which considers the deck as a plate stiffenedby supporting beams. The versatility of DAP can be illustrated by the following examples:1) Both the supporting beams and the cover plate can be made up of any material as long as thestrength properties remain linear elastic within the load range of interest. Materials for cover mayinclude composite wood products, fiber glass products, metal products, or composite wood and fiberglass products provided that their strength properties are known.702) With statistical modeling of the cover strength properties, simulation and reliability studies can beperformed to evaluate the performance of the decking.3) Finally different shapes and dimensions of supporting beams and different types of connector canalso be considered to improve the structural performance of the deck.Although these studies are beyond the scope of the current program, it is clear that DAP can easily beused to consider other problems.71Figure 5 Deformation profile of cover: case 7.72Figure 6 Bending stress profile us, in exterior ply of cover: case 7.73Figure 7 Bending stress profile ci in exterior ply of cover: case 7.74Figure 8 Rolling shear stress profile r yz in cross ply of cover: case 7.75Figure 9 Rolling shear stress profile Txz in cross ply of cover: case 7.76Figure 10 Deformation profile of cover: case 6.77Figure 11 Bending stress profile a-, in exterior ply of cover: case 6.78Figure 12 Bending stress profile (75, in exterior ply of cover: case 6.79Figure 13 Rolling shear stress profile ryz in cross ply of cover: case 6.80Figure 14 Rolling shear stress profile Tx z in cross ply of cover: case 6.814. VENEER MECHANICAL PROPERTIES TESTING PROGRAMThe elastic properties of the cover are required by DAP as input. Here a database on themechanical properties on 2.5 mm ( 1.1--0 inch) and 3.2 mm ($ inch) thick Douglas-fir veneer wasdeveloped. The veneers were sampled from all the sources available to Ainsworth Lumber CompanyLtd. for transDeckTM manufacturing. Using this information, the elastic properties of laminated veneerpanels can be estimated from Equations 74a to 74h.Since it is difficult to test the mechanical properties of single ply veneer, 3-ply and 4-ply panelswith the face grain in each veneer oriented in the longitudinal direction of the panels were considered toextrapolate single ply veneer mechanical properties. However, this approach required that themechanical properties of the veneers within each 3-ply or 4-ply panel not be significantly different.Therefore, a non-destructive testing program based on ultrasonic techniques was initiated to sort theveneers into groups with similar mechanical properties prior to making the 3-ply and 4-ply panels.The elastic properties of the 3-ply and 4-ply veneers in the directions parallel and perpendicularto grain were then obtained from bending, tension, and compression tests. Also a plate twisting testwas performed to obtain the shear modulus of rigidity tests of the 3-ply and 4-ply veneers.4.1 Ultrasonic Veneer Testing ProgramMetriguard Inc. developed a continuous ultrasonic veneer tester, Metriguard Model 2600 veneergrader, to sort veneer for structural applications (Logan, 1987). This technology is based on the wellknown principle that sonic transmission velocity and the density of the transmission medium arestrongly correlated. Since the density of veneer is also correlated to the mechanical properties of theveneer, it is reasonable to use the machine to grade the veneers for structural use. Currently, more than10 machines are in service in North America.82Sonic propagation time in the longitudinal direction of each veneer sheet was monitored aseach sheet of veneer was transversely passed through the veneer tester. The ratio between the numberof signals received and transmitted by the transducers was also continuously monitored which indicateddefects such as splits in the veneer.Each sheet of veneer sampled was tested with the Metriguard veneer grader. For the twoveneer thicknesses, the cumulative probability distributions of sonic transmission time are shown inFigure 15. Based on the cumulative probability distributions of sonic transmission time of the veneers,the 2.5 mm ($ inch) and 3.2 mm (I inch) thick veneers were sorted into the following three groups:1) group A contained veneers from lower 25 th percentile of the sonic transmission time distribution;2) group B contained veneers in the middle 25 th to 75 th percentiles of the sonic transmission timedistribution;3) group C contained veneers from upper 75 th percentile of the sonic transmission time distribution.Group C was expected to contain the worst material while group A was expected to contain the bestquality veneer.4.2 Mechanical Properties Test Program4.2.1 Materials and MethodsIn the veneer mechanical properties test program, a total of eighty 3-ply and eighty 4-ply, 1.2x 2.4 m (4 x 8 feet), specimens were made. For each veneer thickness, in the 3-ply case, 15 sheets ofveneer were randomly selected from group A to be made into 5 panels; 30 sheets of veneer wererandomly selected from group B to be made into 10 panels; and 15 sheets of veneer were randomlyselected from group C to be made into 5 panels. Similarly in the 4-ply cases, 20, 40 and 20 sheets ofveneer were randomly selected from groups A, B and C, respectively. The veneers from groups A, B831.00.8• ■••.S3co 0.620a)= 0.4E0.20.0350 400 450 500 550 600 650 700Sonic Transmission Time (microsecond)Figure 15 Sonic transmission time cumulative probability distributions for 2.5 and 3.2 rnm thick veneer.84and C were made into 5, 10 and 5 four-ply panels, respectively. The specimens were then conditionedat a temperature of 20 ± 3°C and relative humidity of 65 th 5% for more than four weeks untilequilibrium moisture content was reached.There are no standard test methods to determine the mechanical properties of veneers.Therefore, the 3-ply and 4-ply veneers were tested in the directions parallel and perpendicular to grainunder bending, tension, and compression by following as closely as possible ASTM D3043A, ASTMD3500B, and ASTM D3501B, respectively (ASTM 1990). The shear modulus of rigidity of the 3-plyand 4-ply veneers were also tested as closely as possible to ASTM D3044 (ASTM 1990).4.2.1.1 Bending Tests:Two specimens were obtained from each panel for the bending tests. The first specimen was cutwith its long axis perpendicular to the grain and the second specimen was cut with its long axis parallelto the grain. The lengths of the specimens were chosen such that the minimum span to depth ratios of48:1 and 24:1 were maintained when the veneers were oriented parallel to span and perpendicular tospan, respectively. The test spans of various plies/thickness combinations of the specimens excludingthe 25 mm (1 inch) of overhang from each end are given in Table 6. The specimen width was 51 mm(2 inches).The specimens were simply supported at two ends with roller bearing plates and loaded undera center point load. A MTS model 810 hydraulic control close loop universal testing machine with acapacity of 222.4 kN (50000 lb) applied the load in a deflection control mode. An uniform rate of crossinch)head motion of 1.37 mm (0.054 w used which resulted in specimen failure between 7 to 10mm. mm.minutes of loading. Figure 16 shows a specimen being tested in bending.85Table 6. Test span for the bending specimens.No. of^Veneer Thickness^ Orientation of VeneerPlies^(mm)^Parallel to span^Perpendicular to spanTest Span (mm)3 3.2 457.2 228.64 3.2 609.6 304.83 2.5 368.3 190.54 2.5 495.3 254.086Figure 16 The veneer bending test set up.87The specimen thicknesses at mid-span and at two points near each edge were measured,averaged, and recorded. The specimen width at mid-span was also recorded. A computer based dataacquisition system and software were used to acquire the load versus deformation (cross head motion)data. A load cell with a 25 kN (5620 lb) capacity was used to monitor the loads.The bending modulus of elasticity in the perpendicular and parallel to grain directions (Ex f andEyd were based on estimating the slope of the load deformation curve between two preset points on thelinear portion of the load-deformation curve. Here, the two preset points were taken as the loads at fiveand twenty percent of the peak load, respectively. Linear regression of the data between the presetpoints yielded the slope of load-deformation curve. The bending moduli of elasticity were estimated as:E^(Px) L 3xf — Ax 48 IfPY) L3EYf - - (AY f 48 IPwhere I = the moment of inertia of the entire cross section, (' = the slope of the linear portion ofAThe moduli of rupture in the perpendicular and parallel to grain directions, S and S yb , forxb^Yip'each specimen were obtained from the peak loads as:PxL DSxb — 8 1^[76 a]P,8 LDf IsYb = J^[76 b][75 a][75 b]xf^P,the load deflection curve in the perpendicular to grain bending tests, A—'- I = the slope of the linearY fportion of the load deflection curve in the parallel to grain bending tests, and L = the span.where D = depth of the specimen, Pxf the peak load in the perpendicular to grain bending tests, and88PYf = the peak load in the parallel to grain bending tests.4.2.1.2 Tension Tests:A specimen, 254 mm (10 inches) wide and 1219 mm (48 inches) long, was cut from each panelfor the parallel to the grain tension tests. For the perpendicular to grain tension tests, a specimen, 50mm (2 inches) wide and 406 mm (16 inches) long, was cut from each panel.The parallel to grain tension tests used a 444.8 kN (100000 lb) capacity Metriguard hydraulic\tension testing machine (model 412) with an uniform loading rate of 1.524 mm (0.06 inch)in. min. )typically failed between 7 to 10 minutes of loading. The applied load was monitored by a 222.4 kN(50000 lb) capacity load cell. The two ends of a specimen were gripped by self aligned urethane frictionpads which applied uniformly distributed loads along and across the cross section. The distancebetween the grips was 610 mm (24 inches). Figure 17 shows a specimen during the tension test.In the perpendicular to grain tension tests, an Instron mechanical testing machine with acapacity of 48.9 kN (11000 lb) was used to apply the load in a deflection control mode. A uniform rateof cross head motion of 0.5 mm (0.020 inch ) was used which resulted in specimen failure between 3 tomin. min.5 minutes of loading. The applied load was monitored by a 222.4 kN (1000 lb) capacity load cell. Thetwo ends of a specimen were gripped by self aligned and tightened wedge type jaws. The distancebetween the grips was 305 mm (12 inches). The perpendicular to grain specimens were not neckedbecause pilot tests indicated that failure zones were not in the gripping area.Two linear variable differential transducers (LVDT) were mounted on the two opposite sides ofeach parallel and perpendicular to grain tension specimens. Gauge lengths of 152 and 178 mm (6 and 7inches) were used in the perpendicular and parallel to grain tension tests, respectively. The two sets ofLVDT readings were averaged to obtain the overall deflection. The specimen thickness and width atmid-span were measured and recorded. A computer based data acquisition system and software wereused to acquire the load versus overall deformation information.89Figure 17 The veneer parallel to grain tension test set up.90The tension moduli of elasticity in the perpendicular and parallel to grain directions (Ext andEye) were based on the estimating the slope of the load deformation curve between two preset points onthe linear portion of the load-deformation curve. Here, the two preset points were taken as the loads atfive and fifty percent of the peak load, respectively. Linear regression of the data between the presetpoint yielded the slope of the load-deformation curve. The tension moduli of elasticity in theperpendicular and parallel to grain directions were estimated as:Ext = (eP ) kt.= P(L) LE y t^AyY t[77a][77 b]where A = cross sectional area, 7--Px^Py J = slope of the linear portion of the load deflection(Llx t^Y tcurve in the perpendicular and parallel to grain compression tests, respectively, and L = gauge length.The tensile strengths in the perpendicular and parallel to grain directions S xt and SYt of eachspecimen were obtained from the peak loads as:PxSxt = At^ [78 a]PYSYt^At= ^ [79 b]where Pxt = the peak load in the perpendicular to grain tension tests, and P3,t = the peak load in theparallel to grain tension tests.4.2.1.3 Compression Tests:The compression test specimens were 191 mm (7.5 inches) wide and 381 mm (15 inches) long.No. of^Veneer Thickness^Total No. of Specimen ThicknessPlies (mm)^Plies^(mm)3 3.2 15 46.74 3.2 16 50.83 2.5 18 45.74 2.5 16 40.691To eliminate buckling, a length to depth ratio of 10 was used; therefore, specimens cut from the samepanels were glued face to back together using a polyvinyl acetate resin to form the final specimens. Thespecimens were cut slightly over sized and then trimmed to final size so that all adjacent edges were atright angles. The nominal specimen depths of the various ply/thickness combinations are given inTable 7.Table 7. Specimen depths for the compression tests.In the perpendicular to grain compression tests, a MTS model 810 hydraulic universal testingmachine with a capacity of 222.4 kN (50000 lb) was used to apply the load in a deflection control^ (0.035 inch )^mode. A uniform rate of cross head motion of 0.89 mm was used. Specimen typicallymin.^min.failed between 7 to 10 minutes of loading. The applied load was monitored by a 222.4 kN (50000 lb)capacity load cell. Figure 18 shows a specimen being tested in compression.Since the applied loads were considerable larger in the parallel to grain compression tests, aBaldwin hydraulic universal testing machine with a capacity of 1779.2 kN (400000 lb) was used. Anapproximate rate of cross head motion of 0.07 mm (0.003min.^inch) was used which resulted in failuremin.within 10 minute of loading. The applied load was monitored by a 1779.2 kN (400000 lb) capacity loadcell.92Figure 18 The veneer compression test set up.93Two LVDT units were mounted on the two opposite faces of each parallel and perpendicular tograin compression specimen over a gauge length of 127 mm (5 inches). The two sets of LVDT readingswere averaged to obtain the overall deflection. The specimen thickness and width at mid-span weremeasured and recorded. A computer based data acquisition system and software were used to acquirethe load versus overall deformation information.The compression moduli of elasticity in the perpendicular and parallel to grain directions (E Xcand EYc ) were based on the estimating the slope of the load deformation curve between two presetpoints on the linear portion of the load-deformation curve. Here, the two preset points were taken asthe loads at five and twenty percent of the peak load, respectively. Linear regression of the databetween the preset point yielded the slope of the load-deformation curve. The compression moduli ofelasticity in the perpendicular and parallel to grain directions were estimated as:Px) LExc = ( 217,PY) LE =Yc —y Acwhere A = cross sectional area,^and -- = the slope of the linear portion of the load1-1 xdeflection curve in the perpendicular and parallel to grain compression tests, respectively, and L = thegauge length.The compression strengths in the perpendicular and parallel to grain directions S and S yc ofeach specimen were obtained from the peak loads as:[80 a][80 b]No. ofPliesVeneer Thickness(mm)Specimen Size(mm)Loading Rate(mm)`min.'3 3.2 381.0 x 381.0 4.574 3.2 508.0 x 508.0 6.103 2.5 304.8 x 304.8 3.664 2.5 406.4 x 406.4 4.8894where P and P = the peak load in the perpendicular and parallel to grain compression tests,xc^3'crespectively.4.2.1.4 Shear Modulus of Rigidity Tests:A single square specimen was taken from each panel for the shear modulus of rigidity tests.The specimen width and length were chosen as 40 times the nominal thickness. The specimen size forthe various ply/thickness combinations are given in Table 8.Table 8. Specimen sizes for the shear modulus of rigidity tests.The MTS model 810 hydraulic control close loop universal testing machine with a capacity of222.4 kN (50000 lb) was used to apply the load in a deflection control mode. The specimen wassupported on the opposite corners of one of its diagonals and loaded with a uniform rate of loadingfrom the opposite corners of the other diagonal. Table 8 also shows the rates of cross head motion inthe tests for the various ply/thickness combination.A 25 kN (5620 lb) capacity load cell was used to monitor the loads. The deflection on quarterpoints on each diagonal relative to the center of the specimen was measured using an LVDT mounted95on a special yoke apparatus. Under this arrangement, the measured deflection was twice that of thedeflection relative to the center of the specimen. The thicknesses at six locations in the specimen weremeasured, averaged, and recorded. A computer based data acquisition system and software were usedto acquire the load versus deformation information. Figure 19 shows the experimental set up and aspecimen during the shear modulus of rigidity test.The specimen was loaded to a maximum load of 89 N (20 lb). To eliminate the effects of slightinitial curvature, the test was repeated with the specimen rotated 90° about an axis through the centerof the plate and perpendicular to the plane of the plies. The results from the two tests were averaged toobtain an overall shear modulus of rigidity for the specimen.The slope of the load deformation curve was estimated from linear regression of the load-deformation data. The shear modulus of rigidity was estimated according to ASTM D3044 (ASTM1990) as:G = 3 u^2 (P)2 h3 Ag(where AP I = slope of the load deflection curve, u = distance the panel center to the reference point4.2.1.5 Connector Load Slip Tests:The load-slip relationship of the connector between the cover and the supporting member weretested using 11-ply transDeck TM specimens which were built from random sampling the veneers fromthe entire sonic propagation velocity distribution. A 305 mm (12 inch) long section of a typical dryfreight van trailer deck supporting I-beam was sandwiched between two 152 x 254 mm (6 x 10 inch)transDeckTM specimens as shown in Figure 20. The moduli of elasticity and the yield strength of the I-beams were taken as 200,000 MPa (29x10 6 psi) and 550 MPa (80000 psi), respectively. The thickness of[83]gfor deflection measurement, and h = mean thickness of specimen.96Figure 19 The veneer shear modulus of rigidity test set up.97Figure 20 The connector load slip test set up.98the flange (tf) and the web (t w) equaled 3.2 mm (0.125 inch). The depth (h1) and the width (b1) of theI-beam were 102 mm (4 inches) and 57 mm (2.25 inches), respectively. Each transDeck TM panel wasconnected through one of the flanges of the I-beam using two 8 mm (A inch)-18 torx drive, flat head,type G, phosphate and oil coated self tapping screws. The two connectors were staggered with verticaland horizontal spacings of 102 mm (4 inch) and 38 mm (1.5 inch), respectively. Pilot holes with adiameter of 7 mm inch) were pre-drilled through the panels and the I-beams prior to applying theself tapping screws. A ratchet was used used to tighten the connector until its head was flush with thepanel surface. The technique to assemble the test specimen is consistent with the procedures used in theconstruction of commercial dry freight van trailer decking with the exception that air driven torquewrench which are not regulated for any torque value are usually used in the industry.With each veneer thickness, five specimens with the face grain of the panel oriented parallel tothe direction of the load and five specimens with face grain of the panel oriented perpendicular to thedirection of the load were tested in compression to estimate the load slip characteristic of theconnectors. A compression load of up to 14 kN (3147 lb) was applied using a MTS model 810 hydrauliccontrol close loop universal testing machine with a capacity of 222.4 kN (50000 lb). The machineoperated in a deflection control mode with an uniform rate of cross head motion of 0.46 mm (0.018mm.inchAssuming each connector carries one quarter of the load in the assembly, the stiffness of eachconnector was estimated from the slope of the load deformation curve as:K ( _ 1 Pxx, 4 Ax1 (PYK =Yc 4 A,)con(Pwhere (-371) x ) and A^= the slope of the linear portion of the load deflection curve in thex con^Y)concon[84a][84 b]99perpendicular and parallel to face grain connection tests, respectively.The linear portion of the load-deformation curve was taken between two preset points of 2 and7 kN (450 and 1574 lb), respectively. Linear regression of the data between the preset point yielded theslope of the load-deformation curve.4.2.2 Results4.2.2.1 Veneer Strength Properties Statistics:From each modulus of elasticity test program, the specimens were classified by: the twodirections of testing; the four combinations of number of plies/veneer thickness; and the three groups ofultrasonic test results. Shown in Table 9 is the classification system for the veneer test specimens.Table 9. Classification of veneer test specimens.Group No. ofPliesVeneerThickness(mm)Direction UltrasoundSubgroup310A 3 2.5 Parallel Al B1 Cl410A 4 2.5 Parallel A2 B2 C2308A 3 3.2 Parallel A3 B3 C3408A 4 3.2 Parallel A4 B4 C4310E 3 2.5 Perpendicular A5 B5 C5410E 4 2.5 Perpendicular A6 B6 C6308E 3 3.2 Perpendicular A7 B7 C7408E 4 3.2 Perpendicular A8 B8 C8100Statistical information on the bending, compression, tension and shear elastic moduli of eachgroup is shown in Table 10. Similar statistical information on the bending, compression, and tensionstrength of each group is shown in Table 11. Figures 21 and 22 show the cumulative probabilitydistributions of veneer bending modulus of elasticity and strength, respectively. The cumulativeprobability distributions of tension modulus of elasticity and strength of the veneers are also shown inFigures 23 to 24, respectively. Figures 25 and 26 show the cumulative probability distributions ofveneer compression modulus of elasticity and strength, respectively. Finally, Figure 27 shows thecumulative probability distribution of modulus of rigidity of the veneers.4.2.2.2 Effects of veneer thickness and number of plies:For each strength property, regression approach to analysis of variance was performed on the 8data groups and the 24 data subgroups. In each case, either the elastic modulus or the strength wastreated as dependent variables. Indicators (0 or 1) were used as independent variables in the regressionanalysis to represent the various groups and subgroups.Results of the analysis of variance for the veneer bending, tension, compression and shearmodulus of elasticity strength properties are shown in Tables 12 to 15, respectively. The results showthat there was a significant relationship between the dependent variable (either modulus of elasticity orstrength) and the independent variables at 95% probability level. The results also show that bothgroup effect and subgroup effect were significant at 95% probability level in all cases except for themodulus of rigidity results where the subgroup effect was found to be not significantly different.With each strength property, the mean values between the various groups were compared usingDuncan's multiple range test. Table 16 shows the results of the comparisons. First, consider the shearmodulus of rigidity, the results indicate that the mean values of the 310 and 410 groups were notsignificantly different from each other and the mean values of the 410, 308, and 408 groups were notsignificantly different at the 95% probability level.101Table 10. Statistical data on the veneer elastic moduli.Group^310A^410A 308A 408A 310E 410E 308E 408EBending Modulus of ElasticityMean (MPa)^12825^13091 11106 12026 424.3 435.3 408.6 455.4STDV (MPa)^2337^2265 1712 2703 135.9 99.0 81.7 161.9Tension Modulus of ElasticityMean (MPa)^13677^13485 12378 11941 420.7 434.4 332.1 303.9STDV (MPa)^1729^2494 1896 2291 94.1 104.9 60.8 105.8Compression Modulus of ElasticityMean (MPa)^11864^11380 9536 10120 491.2 550.6 411.2 499.8STDV (MPa)^3272^2770 3161 2049 79.0 82.3 44.9 106.2Shear Modulus of RigidityMean (MPa)^815.8^780.2 733.7 724.3STDV (MPa)^119.2^82.0 87.2 98.4Count^20^20 20 20 20 20 20 20Note: STDV = Standard deviation102Table 11. Statistical data on the veneer strengths.Group 310A 410A 308A 408A 310E 410E 308E 408EBending StrengthMean (MPa) 89.4 85.8 75.8 80.3 3.52 2.95 2.59 3.04STDV (MPa) 16.6 18.7 9.9 15.5 0.80 0.72 0.75 0.72Tension StrengthMean (MPa) 42.3 41.6 36.8 35.8 0.72 0.91 0.67 0.44STDV (MPa) 16.6 18.7 9.9 15.5 0.80 0.72 0.75 0.72Compression StrengthMean (MPa) 50.2 50.2 43.8 46.4 7.59 7.66 7.67 6.94STDV (MPa) 5.2 5.4 7.3 6.1 1.20 1.02 1.05 0.53Count 20 20 20 20 20 20 20 20Parallel to Grain Direction- 3-Ply 2.5 mm Veneer—h- 4-Ply 2.5 mm Veneer- 4-Ply 3.2 mm Veneer-e- 3-Ply 3.2 mm Veneer0.80.20.0Perpendicular to Grain Direction- 3-Ply 2.5 mm Veneer-4— 4-Ply 2.5 mm Veneer- 4-Ply 3.2 mm Veneer-8- 3-Ply 3.2 mm Veneer1031.00.8.o.0▪ 0.6a.a3 0.4E0.20.05^10^15^20^25Bending Modulus of Elasticity (x 1000 MPa)1 . 00^200^400^600^800^1000Bending Modulus of Elasticity (MPa)Figure 21 The cumulative probability distributions of veneer bending modulus of elasticity.1.00.80.60.40.20.01.00.8.o2a.E.>00.60.40.2Parallel to Grain Direction3-Ply 2.5 mm Veneer4-Ply 2.5 mm Veneer* 4-Ply 3.2 mm Veneer-9- 3-Ply 3.2 mm VeneerPerpendicular to Grain Direction---- 3-Ply 2.6 mm Veneer4-Ply 2.6 mm Veneer* 4-Ply 3.2 mm Veneer-8- 3-Ply 3.2 mm Veneer0.010440^60^80^100^120^140Bending Strength (MPa)2^3^4^5^6Bending Strength (MPa)Figure 22 The cumulative probability distributions of veneer bending strength.Parallel to Grain Direction3-Ply 2.5 mm Veneer-f-- 4-Ply 2.5 mm Veneer* 3-Ply 3.2 mm Veneer- 9- 4-Ply 3.2 mm VeneerPerpendicular to Grain Direction3-Ply 2.5 mm Veneer±- 4-Ply 2.5 mm Veneer3-Ply 3.2 mm Veneer-9-- 4-Ply 3.2 mm Veneer1051.00.8a.0 0.62a.co• 0.4E0.20.05^10^15^20^25Tension Modulus of Elasticity ix 1000 MPa)100^200^300^400^500^600^700^800Tension Modulus of Elasticity (MPa)1.00.8too 0.6a.ws 0.4E0.20.0Figure 23 The cumulative probability distributions of veneer tension modulus of elasticity.Parallel to Grain Direction3-Ply 2.5 mm Veneer4-Ply 2.5 mm Veneer--Ws- 3-Ply 3.2 mm Veneer-9- 4-Ply 3.2 mm VeneerPerpendicular to Grain Direction--- 3-Ply 2.5 mm Veneer—H 4-Ply 2.5 mm Veneer)1( 3-Ply 3.2 mm Veneer-8- 4-Ply 3.2 mm Veneer1.00.8.o0.6E0.210670 800.010^20^30^40^50^60Tension Strength (MPa)1.00.8:0o 0.6o.3 0.4EC.)0.20.00^0.5^1^1.5^2Tension Strength (MPa)Figure 24 The cumulative probability distributions of veneer tension strength.Parallel to Grain Direction3-Ply 2.5 mm Veneer—H 4-Ply 2.5 mm Veneer3-Ply 3.2 mm Veneer-9- 4-Ply 3.2 mm Veneer1.00.00.8:aal.0L' 0.60.2Perpendicular to Grain Direction3-Ply 2.5 mm Veneer- 4-Ply 2.5 mm Veneer* 3-Ply 3.2 mm Veneer-9- 4-Ply 3.2 mm Veneer1.00.01074^8^12^16^20Compression Modulus of Elasticity (x 1000 MPa)0.8.00 0.6a.S 0.4E00.2300^400^500^600^700^800Compression Modulus of Elasticity (MPa)Figure 25 The cumulative probability distributions of veneer compression modulus of elasticity.1.00.80.60.40.20.0Parallel to Grain Direction3-Ply 2.6 mm Veneer-4— 4-Ply 2.6 mm Veneer- 3-Ply 3.2 mm Veneer- B- 4-Ply 3.2 mm VeneerPerpendicular to Grain Direction--- 3-Ply 2.5 mm Veneer—I— 4-Ply 2.5 mm Veneer-1 3-Ply 3.2 mm Veneer-9- 4-Ply 3.2 mm Veneer10830^40^50^60^70Compression Strength (MPa)1.00.8.02 0.6r.g 0.4E0.20.04^5^6^7^8^9^10^11^12Compression Strength (MPa)Figure 26 The cumulative probability distributions of veneer compression strength.500 600 700^800 900 1000^1100 12001.00.8--- 3-Ply 2.6 mm Veneer--H 4-Ply 2.6 mm Veneer* 3-Ply 3.2 mm Veneer--13- 4-Ply 3.2 mm Veneer0.20.0109Shear Modulus of Rigidity (MPaiFigure 27 The cumulative probability distributions of veneer shear rnodulus of rigidity.110Table 12. Analysis of variance results on veneer bending strength properties.Source^Degree of^Sum of^Mean^F Value^Probability > FFreedom^Squares^SquareDependent Variable: Bending Modulus of ElasticityModel 23 5877784086 255555830 210.84 0.0001Group 7 5645685375 806526482 665.41 0.0001Subgroup 16 232098711 14506169 11.97 0.0001Error 136 164842145 1212075Total 159 6042626231Dependent Variable: Bending StrengthModel 23 263143.5431 11411.0236 128.96 0.0001Group 7 256879.1005 36697.0144 413.63 0.0001Subgroup 16 6264.4426 391.5277 4.41 0.0001Error 136 12065.7523 88.7188Total 159 275209.2954111Table 13. Analysis of variance results on veneer tension strength properties.Source^Degree of^Sum of^Mean^F Value^Probability > FFreedom^Squares^SquareDependent Variable: Tension Modulus of ElasticityModel 23 6523595987 283634608 349.33 0.0001Group 7 6290282873 898611839 1106.74 0.0001Subgroup 16 233313114 14582070 17.96 0.0001Error 136 110424541 811945Total 159 6634020528Dependent Variable: Tension StrengthModel 23 62054.16735 2698.00728 77.16 0.0001Group 7 59779.55558 8539.93651 244.23 0.0001Subgroup 16 2274.61177 142.16324 4.07 0.0001Error 136 4755.44448 34.96650Total 159 66809.61183112Table 14. Analysis of variance results on veneer compression strength properties.Source^Degree of^Sum of^Mean^F Value^Probability > FFreedom^Squares^SquareDependent Variable: Compression Modulus of ElasticityModel^23^4513900934^196256562^72.66^0.0001Group^7^4261854575^608836368^225.42^0.0001Subgroup^16^252046359^15752897^5.83^0.0001Error^136^367321926^2700897Total^159^4881222860Dependent Variable: Compression StrengthModel^23^66934.28599^2910.18635^367.84^0.0001Group^7^65127.91607^9303.98801^1175.99^0.0001Subgroup^16^1806.36993^112.89812^14.27^0.0001Error^136^1075.97628^7.91159Total^159^68010.26228113Table 15. Analysis of variance results on veneer modulus of rigidity.Source^Degree of^Sum of^Mean^F Value^Probability > FFreedom^Squares^SquareDependent Variable: Modulus of rigidityModel 11 226945.2864 20631.3897 2.31 0.0180Group 3 108806.5482 36268.8494 4.06 0.0103Subgroup 8 118138.7382 14767.3423 1.65 0.1265Error 68 607837.0589 8938.7807Total 79 834782.3752114Table 16. Duncan's multiple range test results for the various groups.Strength Properties GroupsBending Modulus of Elasticity 410A 310A 408A 308A 410E 310E 408E 308EBending Strength 310A 410A 408A 308A 310E 308E 410E 408ETension Modulus of Elasticity 310E 308E 408E310A 410A 308A 408A 410ETension Strength 310A 410A 308A 408A 410E 310E 408E 308ECompression Modulus of Elasticity 310A 410A 408A 308A 410E 408E 310E 308ECompression Strength 310A 410A 308A 408A 408E 410E 310E 308EShear Modulus of Rigidity 310 410 408 308Note: The mean values of the underlined groups were not significantly different at 95% probabilitylevel. The groups were arranged in descending order with respect to the mean value of each group.115When considering each strength property in the perpendicular to grain direction, the results indicatethat the mean values of the various groups were not significantly different at the 95% probability level.For each strength property in the parallel to grain direction, the results also indicate that the meanvalues of the 310 and 410 groups were not significantly different at the 95% probability level. Finally,the mean values of the 308 and 408 groups were not significantly different at the 95% probability levelfor all strength properties except for the bending modulus of elasticity and compression strength in theparallel to grain direction.Based on these results, for the shear modulus of rigidity data, the 310 and 410 groups wereconsidered as a single group and the 308 and 408 groups were considered as another single group. Withthe other strength properties, the 4 groups in the perpendicular direction (410E, 310E, 408E, and 308E)were considered as a single group. Each strength property in the perpendicular to grain direction wastherefore represented by a probability distribution where the effects of number of plies and veneerthickness were assumed insignificant. Similarly, the 310A and 410A groups in the parallel to graindirection were considered as a single group. Finally for practical reason, the 308A and 408A groups inthe parallel to grain direction were also considered as another single group. Therefore, each strengthproperty in the parallel to grain direction was represented by two probability distributions (one foreach veneer thickness) where the effects of number of plies were assumed insignificant.Ignoring the effects of the number of plies, the normal, 2-parameter Weibull, and 3-parameterWeibull probability distributions were fitted to individual data groups following the maximumlikelihood estimation approach (Lawless, 1982). Both the normal and the 3-parameter Weibulldistributions were visually judged to provide the good fit to the data. Figures 28 to 34 shows thecumulative probability distributions, the normal distributions and the 3-parameter Weibulldistributions of each group for the various strength properties. The statistical information and thedistribution parameters of the elastic moduli and strengths of the various groups are summarized inTables 17 and 20.Parallel to Grain Direction2.5 mm Veneers-I- 3.2 mm VeneersNormal Distribution3-Parameter Weibull1161.00.8:a-a 0 ' 62 a)2 0.4'E00.20.05^10^15^20^25Bending Modulus of Elasticity (x 1000 MPa)1.00.8:a.0• 0.62aea▪ 0.4E00.20.0Perpendicular to Grain Direction• Combined Groups- - Normal Distribution— 3-Parameter Weibull 0^200^400^600^800^1000Bending Modulus of Elasticity (MPa)Figure 28 The cumulative probability distributions and the 3-parameter Weibull distributionsof the veneer bending modulus of elasticity of each group.Parallel to Grain Direction' 2.5 mm Veneers3.2 mm Veneers--- Normal Distribution— 3-Parameter Weibull-+1.00.8.o• 0.600.70'▪ 0.4E00.21170.040^60^80^100^120 140^160Bending Strength (MPa)1 .00.8.o▪0.60a.0.4E00.20.0Perpendicular to Grain Direction" Combined Groups-_ ^Distribution- 3-Parameter Weibull 2^3^4^5^6Bending Strength (MPa)Figure 29 The cumulative probability distributions and the 3-parameter Weibull distributionsof the veneer bending strength of each group.Parallel to Grain Direction• 2.5 mm Veneers-1- 3.2 mm Veneers- Normal Distribution— 3-Parameter Weibull1181.00.81E.0 0.60I; 0.4E0.20.05^10^15^20^25Tension Modulus of Elasticity (x 1000 MPa)1.00.8.0.o 0.60is 0.4E0.20.0Perpendicular to Grain Direction' Combined Groups-- Normal Distribution— 3-Parameter Weibull 100^200^300^400^500^600^700^800Tension Modulus of Elasticity (MPa)Figure 30 The cumulative probability distributions and the 3-parameter Weibull distributionsof the veneer tension modulus of elasticity of each group.Parallel to Grain Direction2.5 mm Veneers3.2 mm VeneersNormal Distribution3-Parameter WeibullPerpendicular to Grain Direction• Combined GroupsNormal Distribution— 3-Parameter Weibull1191.00.80.60o.to' 0.4E00.20.010^20^30^40^50^60^70^80Tension Strength (MPa)1.00.80.60a.aco 0.4E0.20.000^0.5^1.0^1.5^20Tension Strength (MPa)Figure 31 The cumulative probability distributions and the 3-parameter Weibull distributionsof the veneer tension strength of each group.Parallel to Grain Direction' 2.5 mm Veneers3.2 mm Veneers- Normal Distribution— 3-Parameter WeibullPerpendicular to Grain Direction• Combined GroupsNormal Distribution— 3-Parameter Weibull1201.00.8>.es.o 0.60a.r..al 0.4E0.20.04^6^8^10^12^14^16^18^20^22^24Compression Modulus of Elasticity (x 1000 MPa)1.0a0.80.60o.0.4E00.20.0300^400^500^600^700^800Compression Modulus of Elasticity (MPa)Figure 32 The cumulative probability distributions and the 3-parameter Weibull distributionsof the veneer compression modulus of elasticity of each group.0.20.030^35^40^45^50^55^600.81.065 70Parallel to Grain Direction' 2.5 mm Veneers+ 3.2 mm Veneers— Normal Distribution— 3-Parameter Weibull121Compression Strength (MPa)1.00.8>._aco 0.600.a.711 • 0.4E0.20.0Perpendicular to Grain Direction• Combined Groups-- Normal Distribution3-Parameter Weibull 4^5^6^7^8^9^10^11^12Compression Strength (MPa)Figure 33 The cumulative probability distributions and the 3-parameter Weibull distributionsof the veneer compression strength of each group.1200500^600^700^800^900^1000^11001.00.80.60.42.6 mm Veneers3.2 mm Veneers- Normal Distribution— 3-Parameter Weibull0.20.0122Shear Modulus of Rigidity (MPa)Figure 34 The cumulative probability distributions and the 3-parameter Weibull distributionsof the veneer shear modulus of rigidity of each group.123Table 17. Statistical data and distribution parameters of the veneer bending strength properties.DirectionThickness (mm)Modulus of ElasticityParallel^Perpendicular2.5^3.2^CombinedStrengthParallel^Perpendicular2.5^3.2^CombinedMean (MPa) 12957.9 11563.3 430.90 87.58 78.07 3.025Median (MPa) 13130.8 11600.8 439.29 87.24 80.19 3.059STDV (MPa) 2275.8 2280.7 122.51 17.53 13.05 0.805Count 40 40 80 40 40 802-Parameter WeibullShape 5.604 5.295 3.767 5.461 6.793 4.095Scale (MPa) 13905.7 12501.4 475.68 94.61 83.52 3.3293-Parameter WeibullShape 3.092 2.645 3.033 3.686 3.971 2.429Scale (MPa) 7464.7 6307.5 385.79 64.53 50.22 2.050Location (MPa) 6246.1 5950.8 85.42 29.24 32.60 1.207124Table 18. Statistical data and distribution parameters of the veneer tension strength properties.DirectionThickness (mm)Modulus of ElasticityParallel^Perpendicular2.5^3.2^CombinedStrengthParallel^Perpendicular2.5^3.2^CombinedMean (MPa) 13580.8 12159.4 372.76 41.98 36.28 0.688Median (MPa) 13340.0 11792.7 376.17 43.47 36.31 0.638STDV (MPa) 2120.4 2087.8 107.33 9.47 9.53 0.295Count 40 40 80 40 40 802-Parameter WeibullShape 6.604 6.429 3.877 5.065 4.072 2.493Scale (MPa) 14487.3 13034.7 412.15 45.64 39.89 0.7773-Parameter WeibullShape 2.684 4.346 3.126 2.512 3.117 1.607Scale (MPa) 5920.6 6570.6 337.75 39.51 25.09 0.513Location (MPa) 8307.1 6280.9 70.76 5.97 13.98 0.228125Table 19. Statistical data and distribution parameters of the veneer compression strength properties.DirectionThickness (mm)Modulus of ElasticityParallel^Perpendicular2.5^3.2^CombinedStrengthParallel^Perpendicular2.5^3.2^CombinedMean (MPa) 11622.0 9827.6 488.22 50.17 45.09 7.34Median (MPa) 11548.7 9484.7 472.34 49.75 44.76 7.27STDV (MPa) 3002.5 2646.0 94.09 5.24 6.81 1.26Count 40 40 80 40 40 802-Parameter WeibullShape 4.278 3.866 5.385 8.981 6.969 6.870Scale (MPa) 12769.1 10830.3 527.52 52.57 48.06 7.7553-Parameter WeibullShape 2.232 1.556 2.188 2.916 2.034 6.870Scale (MPa) 7047.1 4526.6 219.21 16.21 14.74 7.755Location (MPa) 5378.6 5745.6 294.05 35.65 32.03 0.000126Table 20. Statistical data and distribution parameters of the veneer shear modulus of rigidity.Modulus of RigidityThickness (mm) 2.5 3.2Mean (MPa) 798.02 729.00Median (MPa) 804.78 709.32STDV (MPa) 102.58 91.86Count 40 402-Parameter WeibullShape 7.983 7.424Scale (MPa) 843.10 770.993-Parameter WeibullShape 2.486 1.281Scale (MPa) 265.64 129.83Location (MPa) 562.12 608.221274.2.2.3 Correlations of veneer strength properties:Regression analyses were performed to examine the correlation amongst the various strengthproperties of the 2.5 and 3.2 mm veneer groups in the parallel and perpendicular to grain directions.Table 21 shows the dependent and independent variables in the various regression models considered.Tables 22 and 23 show the regression results for the 3.2 and 2.5 mm veneer groups in the parallel tograin direction. Tables 24 and 25 show the regression results for the 3.2 and 2.5 mm veneer groups inthe perpendicular to grain direction.When considering the 3.2 mm and 2.5 mm thick veneer in the parallel to grain direction,results indicate significant relationships existed at the 95% probability level for bending, tension andcompression modulus of elasticity versus the respective strengths. The coefficient of determination, r 2 ,for these relationships ranged from 0.26 to 0.69. For the 3.2 mm thick veneer in the parallel to graindirection, results also indicate significant relationships existed at the 95% probability level forcompression versus tension modulus of elasticity (r 2 =0.36) and compression versus tension strength(r2=0.44). With the 2.5 mm thick veneer in the parallel to grain direction, results also indicatesignificant relationships existed at the 95% probability level for bending versus compression modulus ofelasticity (r2=0.37), bending versus tension modulus of elasticity (r 2 =0.53), tension versus compressionmodulus of elasticity (r 2 =0.13), bending versus compression strength (r 2 =0.55), bending versus tensionstrength (r 2=0.31), and compression versus tension strength (r 2=0.36).For the 3.2 mm thick veneer in the perpendicular to grain direction, significant relationshipsexisted at the 95% probability level for bending modulus of elasticity versus strength, and compressionmodulus of elasticity versus strength with r 2 ranging from 0.17 to 0.21. With the 2.5 mm thick veneerin the perpendicular to grain direction, significant relationships existed at the 95% probability level forbending modulus of elasticity versus strength, tension modulus of elasticity versus strength, bendingversus tension modulus of elasticity, compression versus tension modulus of elasticity, and compressionmodulus of elasticity versus shear modulus of rigidity with r 2 ranging from 0.12 to 0.33.128Table 21. The dependent and independent variables considered in the various regression models ofveneer strength properties.Model DependentVariableIndependentVariable1 Bending Strength Bending Modulus of Elasticity2 Compression Strength Compression Modulus of Elasticity3 Tension Strength Tension Modulus of Elasticity4 Bending Modulus of Elasticity Compression Modulus of Elasticity5 Bending Modulus of Elasticity Tension Modulus of Elasticity6 Bending Modulus of Elasticity Shear Modulus of Rigidity7 Compression Modulus of Elasticity Tension Modulus of Elasticity8 Compression Modulus of Elasticity Shear Modulus of Rigidity9 Tension Modulus of Elasticity Shear Modulus of Rigidity10 Bending Strength Compression Strength11 Bending Strength Tension Strength12 Compression Strength Tension Strength129Table 22. Results of various regression models of 3.2 mm veneer strength properties for the parallel tograin direction.Model NumberofObservationsF-value Probability> Fr2 Intercept(MPa)Slope1 40 58.966 0.0001 0.6081 26.496 0.0044602 40 46.748 0.0001 0.5516 26.311 0.0019113 40 34.562 0.0001 0.4763 -2.0244 0.0031514 40 0.581 0.4507 0.0151 10524 0.1057535 40 2.747 0.1057 0.674 8114.7 0.2836166 40 1.015 0.3201 0.0260 14482 -4.0045657 40 21.866 0.0001 0.3653 514.33 0.7659348 40 0.931 0.3406 0.0239 13075 -4.4547889 40 3.286 0.0778 0.0796 16834 -6.41214910 40 3.954 0.0540 0.0942 51.552 0.58813211 40 0.533 0.4700 0.0138 72.235 0.16091412 40 29.283 0.0001 0.4352 27.992 0.471331130Table 23. Results of various regression models of 2.5 mm veneer strength properties for the parallel tograin direction.Model NumberofObservationsF-value Probability> Fr2 Intercept(MPa)Slope1 40 85.806 0.0001 0.6931 4.4843 0.0064132 40 18.333 0.0001 0.3254 38.605 0.0009953 40 13.670 0.0007 0.2646 10.778 0.0022974 40 21.945 0.0001 0.3661 7628.0 0.4586045 40 42.744 0.0001 0.5294 2352.6 0.7809046 40 0.075 0.7853 0.0020 12171 0.9866217 40 5.473 0.0247 0.1259 4798.9 0.5024038 40 0.564 0.4574 0.0146 8797.8 3.5390269 40 0.191 0.6644 0.0050 12414 1.46254910 40 46.618 0.0001 0.5509 -37.050 2.48206211 40 17.385 0.0002 0.3139 44.045 1.03705912 40 21.634 0.0001 0.3628 36.187 0.333130131Table 24. Results of various regression models of 3.2 mm veneer strength properties for theperpendicular to grain direction.Model NumberofObservationsF-value Probability> Fr2 Intercept(MPa)Slope1 40 7.976 0.0075 0.1735 1.7531 0.0024612 40 10.313 0.0027 0.2135 5.2580 0.0044963 40 1.097 0.3015 0.0281 0.4313 0.0003964 40 2.096 0.1559 0.0523 286.408 0.3196085 40 0.289 0.5940 0.0075 390.791 0.1295956 40 0.138 0.7127 0.0036 370.600 0.0842207 40 2.156 0.1503 0.0537 376.911 0.2472428 40 1.791 0.1888 0.0450 300.410 0.2127779 40 0.242 0.6253 0.0063 372.514 -0.07483510 40 0.198 0.6589 0.0052 2.3698 0.06110311 40 1.220 0.2764 0.0311 3.1827 -0.65798212 40 0.271 0.6060 0.0071 7.5118 -0.369602Model Number^F-value Probability^r2^Intercept^Slopeof > F (MPa)Observations1 40 5.185 0.0285 0.1201 2.2141 0.0023702 40 1.594 0.2145 0.0402 5.4689 0.0036603 40 18.354 0.0001 0.3257 0.0364 0.0018294 40 1.278 0.2654 0.0325 300.124 0.2489595 40 4.981 0.0316 0.1159 256.280 0.4058626 40 1.286 0.2640 0.0327 264.419 0.2072547 40 10.882 0.0021 0.2226 346.676 0.4075318 40 10.125 0.0029 0.2104 217.089 0.3807339 40 2.750 0.1055 0.0675 228.346 0.24964210 40 0.081 0.7774 0.0021 3.4090 -0.02388011 40 1.039 0.3146 0.0266 3.5725 -0.41492512 40 0.213 0.6468 0.0056 7.0749 0.367330132Table 25. Results of various regression models of 2.5 mm veneer strength properties for theperpendicular to grain direction.133Although these relationships were found to be statistically significant, in most cases their lowr2 values indicate the relationships were weak. This is especially true for the relationships betweenveneer elastic properties with the exception of the relationship between bending and tension moduli ofelasticity for 2.5 mm thick veneer in the parallel to grain direction (r 2=0.53). For practical purposes,the elasticity properties of the veneers were considered as independent from each other. Therefore,based on the distribution parameters shown in Tables 17 to 20, each veneer elastic property can beindependently generated in simulation which can be substituted into Equations 74a to 74h to evaluatethe elastic properties of the transDeckTM panels .The relationships between the various veneer strengths and elastic properties seem to bestronger than the relationships between the various veneer elastic properties as indicated by the higherr2 values. Therefore, when simulations of veneer strength data are required, the correlation betweenveneer strength and elastic properties should be considered.4.2.2.4 Effectiveness of Ultrasonic Testing on Veneer Grading:Tables 26 and 27 show the statistical data on the strength and elastic properties of the varioussubgroups, respectively. With each parallel to grain strength property, the ratio between the meanstrength value in subgroup A and the mean strength value of the combined groups A to C ranged from1.07 to 1.14 for the 310 and 410 groups and 1.05 to 1.26 for the 308 and 408 groups. Similarly for eachelastic property in the parallel to grain direction, the ratio between the mean elastic value of subgroupA and the mean elastic value of the combined groups A to C ranged from 1.11 to 1.21 for the 310 and410 groups and 1.13 to 1.40 for the 308 and 408 groups. In the perpendicular to grain direction, theratio between the mean strength or elastic property in subgroup A and the mean strength or elasticproperty of the combined groups A to C ranged from 0.67 to 1.11. Finally when considering the shearmodulus of elasticity, the ratio between the mean value in subgroup A and the strength value of thecombined groups A to C ranged from 0.93 to 1.05 for the 310, 410, 408, and 308 groups.134Table 26. Statistical data on the veneer strength properties of the various subgroups.Group 310 410 408 308Subgroup Count Mean^STDV(MPa) (MPa)Mean(MPa)STDV(MPa)Mean(MPa)STDV(MPa)Mean(MPa)STDV(MPa)SYbC 5 77.1 11.2 65.4 18.3 70.0 16.3 75.0 9.7B 10 91.0 13.7 90.9 12.7 76.9 11.7 74.4 10.5A 5 98.5 21.4 96.0 15.9 97.5 5.6 79.5 10.3SYcC 5 46.5 0.9 44.0 4.9 38.9 3.3 37.7 1.9B 10 50.1 3.8 51.5 3.6 46.1 2.0 41.7 3.8A 5 53.9 7.9 53.7 4.2 54.6 2.4 54.1 5.4SYtC 5 37.8 4.9 31.5 9.8 29.0 10.3 31.8 3.5B 10 41.7 11.6 44.6 8.1 34.6 7.9 34.4 6.5A 5 48.2 2.9 45.6 7.4 45.0 6.7 46.4 12.5SxbC 5 3.11 0.47 3.17 0.35 2.70 0.88 2.67 0.87B 10 3.69 0.90 2.81 0.83 2.62 0.83 3.16 0.65A 5 3.57 0.84 3.01 0.80 2.42 0.52 3.19 0.75Sxc C 5 6.53 0.91 6.95 1.06 7.20 0.67 7.01 0.24B 10 7.74 0.97 7.82 1.06 7.53 1.24 6.77 0.57A 5 8.37 1.32 8.04 0.67 8.42 0.48 7.20 0.60Sxt C 5 0.93 0.32 1.10 0.39 0.82 0.08 0.46 0.05B 10 0.68 0.25 0.89 0.28 0.71 0.21 0.41 0.09A 5 0.60 0.26 0.76 0.29 0.45 0.14 0.49 0.17135Table 27. Statistical data on the veneer elastic properties of the various subgroups.Group 310 410 408 308Subgroup Count Mean(MPa)STDV(MPa)Mean(MPa)STDV(MPa)Mean(MPa)STDV(MPa)Mean(MPa)STDV(MPa)Eb' C 5 10269 547 10543 2109 8876 1401 10392 1388B 10 13228 809 13354 1547 11794 1005 10747 1501A 5 14574 3432 15112 1071 15619 1344 12538 1844E3' , C 5 8750 1239 9921 2236 7963 750 7607 1748B 10 12238 3263 10987 2835 10430 2006 8588 1765A 5 14231 2426 13623 2008 11656 1144 13359 3499EYtC 5 12262 1585 10567 1040 9285 791 10461 1665B 10 13593 1330 13506 1084 11645 715 12349 1051A 5 15260 1432 16360 2200 15187 980 14352 1536Exb C 5 300.1 139.9 512.0 63.2430.2 162.3 447.4 106.4B 10 470.7 116.4 389.5 94.9 474.8 196.3 375.3 52.5A 5 455.7 108.8 450.1 97.1 441.8 97.3 436.1 92.3Exc C 5 468.4 71.2 559.9 74.9 533.7 73.8 401.0 67.9 B 10 519.3 79.3 573.4 88.4 514.0 133.5 429.1 27.9A 5 457.7 80.3 495.9 61.8 437.5 35.2 385.7 38.5Ext C 5 442.4 140.5 491.8 57.3 241.5 40.4 309.4 28.2B 10 423.6 78.4 420.0 103.1 350.6 114.7 330.3 77.8A 5 393.2 83.1 405.7 139.3 272.6 102.0 358.1 40.7G C 5 833.1 135.6 795.9 111.4 842.6 119.8 749.4 106.4B 10 833.0 124.3 773.7 90.8 673.7 49.9 728.4 87.5A 5 764.2 99.6 777.7 23.9 707.3 41.6 728.5 84.5136For each strength property, the mean values between the 24 subgroups were compared using Duncan'smultiple range test. Table 28 shows the results of the comparisons amongst the three subgroups withineach of the eight groups for each strength property. In the shear modulus of rigidity data, the resultsindicate that the mean values of the three subgroups within each group were not significantly differentat the 95% probability level except in the 408 group where subgroup C4 was significantly different.With the other strength properties in the parallel to grain direction, the mean values in subgroup Cwere typically lower than those in subgroups A and B. Also subgroup A had the highest mean valueexcept in 1 out of 24 cases. The ultrasonic procedure seems to be most effective in sorting the veneerfor the parallel to grain tension modulus of elasticity as the results indicate that the mean values in allthree subgroups within each group were significantly different at the 95% probability level. In the otherstrength properties, the distinction amongst the three subgroups were not as clear but subgroup C wastypically identified as significantly different from the subgroups A and B at the 95% probability levelexcept in 8 out of 24 cases. In the perpendicular to grain direction, the strength properties in thesubgroups seem to be confounded. Therefore, it seems that the ultrasonic test procedure was able tosort the veneers into strength groups for the parallel to grain direction but could not deliver the samelevel of performance for strength properties in the perpendicular to grain direction.4.2.2.5 Connector Stiffness:Figure 35 shows the load deformation curves of four typical test specimens. Within the rangeof loading, the load slip curve seems relatively linear for the cases where the face grain was parallel tothe direction of loading. When the face grain was perpendicular to the direction of loading, both theconnector stiffness and the linearity of the load slip curves were clearly reduced. The connection loadslip test results are summarized in Table 29. Analysis of variance and Duncan multiple range test wereperformed on the four test groups. Table 30 shows the results which indicate that the four groups werestatistically different at the 95% probability level. Also the mean stiffness values of the 10A and 08A137Table 28. Duncan's multiple range test results for the various groups.Strength Properties^ SubgroupsBending Modulus of Elasticity^Al B1 Cl; A2 B2 C2; A3 B3 C3; A4 B4 C4;B5 A5 C5; B6 A6 C6; C7 A7 B7; B8 A8 C8;Bending Strength^ Al B1 Cl; A2 B2 C2; A3 B3 C3; A4 B4 C4;B5 A5 C5; C6 A6 B6; A7 B7 C7; B8 A8 C8;Tension Modulus of Elasticity^Al B1 Cl; A2 B2 C2; A3 B3 C3; B4 A4 C4;C5 B5 A5; C6 B6 A6; AT B7 C7; B8 A8 C8;Tension Strength^ Al B1 Cl; A2 B2 C2; A3 B3 C3; A4 B4 C4;C5 B5 A5; C6 B6 A6; A7 C7 B7; B8 C8 A8;Compression Modulus of Elasticity^Al B1 Cl; A2 B2 C2; A3 B3 C3; A4 B4 C4;B5 C5 A5; C6 B6 A6; B7 C7 A7; C8 B8 A8;Compression Strength^ Al B1 Cl; A2 B2 C2; A3 B3 C3; A4 B4 C4;A5 B5 C5; A6 B6 C6; A7 C7 B7; A8 B8 C8;Shear Modulus of Rigidity^Cl B1 Al; C2 A2 B2; C3 A3 B3; C4 A4 B4;Note: Mean values of the underlined groups were not significantly different at 95% probability level.The groups were arranged in descending order with respect to the mean value of each group.1 61 21:3 80-J4000138* • •'!diAlII41AImVrk;'.4rtil:4-^', Ng 16INiN■+ :no_R-.^gg G4- ° ft"+ ° !Li r"r!brmt+^,+ ,,,.. •72.5 mm Veneer (Par.) Ilk79+m-VIPkopIiW + 3.2 mm Veneer (Par.)II * 2.5 mm Veneer (Per.)PI ^ 3.2 mm Veneer (Per.)Ir..1k: 1 I0.4^0.8^1.2^1.6Deformation (mm)Figure 35 Connectors load deformation curves.Group Sample Veneer^Specimen^Face Grain^Mean stiffnessSize Thickness^Thickness^Orientation^per connector(mm) (mm) ( kN )Nanmi10A 5 2.5 28 parallel 4.15708A 5 3.2 35 parallel 4.10910E 5 2.5 28 Perpendicular 3.12408E 5 3.2 35 Perpendicular 2.691STDV stiffnessper connector(rin(4)0.6140.4700.6670.421139Table 29. Summary of connection load slip tests.Table 30. Analysis of variance results and Duncan multiple range test results on connector stiffness.Source^Degree of^Sum of^Mean^F Value^Probability > FFreedom^Squares^SquareDependent Variable: Connector StiffnessModelErrorTotal316197.975185394.8762009112.851386302.658395130.304762568.72 0.0012Duncan multiple range test results:^Group 08A 10A 08E 10E140groups were not statistically different at the 95% probability level. Similarly the mean stiffness valuesof the 10E and 08E groups were not statistically different at the 95% probability level. However themean stiffness values between the parallel and perpendicular to face grain groups were statisticallydifferent.1415. FULL SCALE TESTING OF DRY FREIGHT VAN TRAILER DECK ASSEMBLYThis chapter describes the development of a Trailer Decking Load Simulator (TDLS) to testfull scale deck assemblies. The structural behavior of full scale prototypes was evaluated using theTDLS in static and fatigue modes and the test results were used to verify DAP.5.1 Testing FacilityShown in Figure 36 is the TDLS designed and constructed to perform full scale testing of theprototype dry freight van trailer decking systems. The facility can test a 2.44 x 4.88 m (8 x 16 feet) dryfreight van trailer deck assembly under either static or cyclic loading of up to 177.9 kN (40000 lb)which can simulate the range of loading expected from lift trucks on the dry freight van trailer decking.The TDLS consisted of four main components: a structural frame, a wheel cart apparatus, a verticalloading apparatus to apply simulated lift truck loading, and a lateral drive apparatus to cyclically pullthe wheel cart along the long axis of the deck assembly. All forces were internally taken by the steelmembers in the structural frame which was designed with the criterion that the top beams deflect notmore than 1 mm (0.045 inch) when a maximum load of 177.9 kN (40000 lb) was applied at midspan.The requirement was intended to allow proper tracking of the wheel cart apparatus when reactingagainst the top beams.The loading systems were pneumatically driven by three compressors arranged in parallel: 1) aPowerrex 10 HP compressor Model CT 1031211, 2) a Ingersoll-Rand 10 HP compressor Model 71T2,and 3) a Devilbiss 5 HP compressor Model TAP5050. The system can deliver a maximum flow rate of(7.23 ft3_ee )^0.205 m3for a pressure range of 861.8 to 1378.9 kPa (125 to 200 psi). A system of airmin.^min.valves and regulators was installed to control the pressure and flow rate for the various components inthe loading system.The vertical loading apparatus consisted of four Firestone two ply bellows air stoke actuators142Figure 36 The trailer decking load simulator.143Model IT19G-5. The capacity of each actuator was 57.8 kN (13000 lb) at 827.3 kPa (120 psi). Theactuators required a flow rate of 0.17 —m3 (6 feet3) at 689 kPa (100 psi). The operating height of themin.^min.actuators was 178 to 254 mm (7 to 10 inches); therefore, a stroke range of 76.2 mm (3 inches) wasachieved. Within this stroke range, the actuator delivered a constant load of less than 1% difference inload at 689 kPa (100 psi). Therefore, in the cyclic tests of the decking assembly a constant load wasapplied to the deck while it underwent a range of deformation. A Interface Precision-Universal load cell(Model 1220) was installed between the vertical loading apparatus and the wheel cart to monitor theapplied loads on the decking. The capacity of the load cell was 222 kN (50000 lb) and nonlinearity wasexpected when measuring loads at ± 0.05% of full range. When pulling the wheel cart apparatuscyclically across the deck, extraneous lateral forces of up to 17.5 kN (3927 lb) might be applied to theload cell. The load cell can resist up to 88.8 kN (20000 lb) of extraneous lateral forces without damageto the electrical or mechanical components. These extraneous lateral forces can cause a maximum errorof 0.1% in measuring the vertical loads.The lateral drive apparatus consisted of a Greenco air cable cylinder Model CD50 108A-FTP.The cylinder had a diameter of 127 mm (5 inches) and a maximum stroke of 2.74 m (9 feet). Itrequired a flow rate 0.71 m3 (25 feet3 ) at a maximum air pressure of 1379 kPa (200 psi). The lateralmin. 5 min.driving system was designed to apply a maximum 5,000 cycles within 12 hours. A cycle is defined asthe returned travel of the wheel cart front axle along the length of a 2.44 m (8 feet) long decking panel.Magnetic sensors were installed to detect the location of the wheel cart and to trigger the valvingsystem for reversal the air flow to change the direction of wheel cart. A cable extending from each endof the cylinder was connected to each end the wheel cart. Each cable was pre-tensioned through aspring system with a maximum load 4.448 kN (1000 lb) to prevent fatigue of the cable system.A wheel cart was designed and built such that the ratio between the front and rear wheelloadings was closed to 9:1. The wheel cart was connected to the loading apparatus through two 38 mm(1.5 inch) diameters pins. The pins were positioned such that the bottom part of the wheel cart144apparatus rotated freely about these moment free connections thus equal loading on both the frontwheels was expected and equal loading on both the rear wheels was also expected after the wheel cartapparatus were properly aligned. The wheels were made by Industrial Tires Limited. The outsidediameter, the width, and the inside diameter of the front wheel were 533, 229, and 381 mm (21, 9, and15 inches), respectively. The front wheels were made from polyurethane with a load capacity of 53.4 kN(12000 lb) per wheel. The foot print of each front wheel was 203 x 89 mm (8 x 3.5 inches) whichcovered an area of 18064 mm 2 (28 inches2 ). The outside diameter, the width, and the inside diameterof each rear wheel were 407, 127, and 267 mm (16, 5, 10.5 inches), respectively. It was made from SNrubber with a load capacity of 13744 kN (3090 lb) per wheel. Each rear wheel had a foot print of 89 x80 mm (3.5 x 3.15 inches) which covered an area of 7113 mm 2 (11 inches 2 ).The front and rear axles were spaced at a distance of 1.22 m (4 feet) apart. Both axles wereoriented parallel to the direction of the supporting I-beams in the deck. The two front wheels werespaced at a distance of 965 mm (38 inches) apart and centered in the x-direction in the deck. Similarly,the rear wheels were centered in the x-direction of the deck and spaced at a distance of 749 mm (29.5inches) apart.The wheel cart also contained a system of guide wheels to ensure proper alignment andtracking of the wheel cart. A guide bushing apparatus was also installed to prevent shearing of the airactuators when pulling the cart laterally while allowing the proper transfer of vertical loads betweenthe air actuators and the wheels.Prior to testing, the loading mechanism was calibrated by supporting the wheel cart on threeload cells while applying a load of up to 88.9 kN (20,000 lb) on the system. This calibration procedurewas designed to establish: 1) the relationship of the load cell readings within the wheel cart againstwheel loads; 2) the relationship between load cell readings within the wheel cart and pressure gaugereadings; 3) the distribution of loads between the front and rear axles; and 4) the dead weight of thesystem. A special mounting device was built to attach two 45 kN (10,000 lb) capacity load cells145directly under the front axle. The rear wheels were supported temporary by a steel channel which inturn rested on a 222.4 kN (50,000 lb) capacity load cell. The dead weight of the system was measuredfirst. The air actuators were then activated to increase the total load in the system from its deadweight to 88.9 kN (20,000 lb). The loading mechanism was fine tuned by adjusting the location of twoload transfer brackets so that a front to back wheel load distribution of approximately 9:1 was achievedduring loading of 80 kN (18,000 lb).Results of the calibration are shown in Table 31. The results show the wheel cart assembly hada dead weight of 14.63 kN (3289 lb) with a front to rear axles dead load distribution of approximately7 to 3. Since the applied loads were positioned closed to the front axle, the front to rear axles loaddistribution would shift as the load from the air actuators was increased. Shown in Figure 37 is therelationship between the front to rear axles load ratio and the wheel cart load cell readings. Resultsfrom regression analysis using a combined hyperbola and second order polynomial model is described asfollows:RA = 0.2088 — 0.0032 TD + 0.000023 TD2 + 0.3815TD [85]where RA represents the front to rear axles load ratio and TD represents the wheel cart load cellreadings in kN. The coefficient of determination (r 2 ) of this relationship is 0.999.Figure 38 shows the relationship between the front axle loadings and wheel cart load cellreadings. This relationship can be described by a simple linear regression model as:FR = 3.374 + 1.11688 TD^ [86]where FR represents the front axle loadings in kN. The coefficient of determination (r 2 ) of thisrelationship is 0.9999.146Table 31. Trailer decking load simulator calibration results.Pressure Wheel Cart Rear Front Total Front toGauge Load Cell Axle Axle Load Rear AxlesReadings Readings Loading Loading (kN) Load Ratio(kPa) (kN) (kN) (kN) (%)0 6.883 3.610 11.020 14.631 24.6834.5 13.549 4.402 18.243 22.646 19.4469.0 19.288 5.288 24.961 30.249 17.48103.4 23.824 5.800 30.048 35.848 16.18137.9 31.647 6.359 38.719 45.078 14.11172.4 38.219 6.918 46.043 52.962 13.06206.9 45.255 7.431 54.398 61.829 12.02234.5 50.994 7.803 60.255 68.059 11.47248.3 53.864 8.083 63.475 71.557 11.30275.9 59.928 8.595 70.013 78.609 10.93289.7 62.659 8.782 73.407 82.189 10.68303.4 65.343 9.014 76.350 85.364 10.56317.2 68.768 9.341 80.207 89.548 10.4335%O 30%OCC 25%co00 20%cl)5%0%1470^10^20^30^40^50^60^70Wheel Cart Load Cell Readings (kN)Figure 37 Relationship between the front to rear axles load ratios andwheel cart load cell readings.100148800al 60X402U-2000^10^20^30^40^50^60^70Wheel Cart Load Cell Readings (kN)Figure 38 Relationship between the front axle loadings and wheel cart load cell readings.149Clearly, RA and FR were successfully represented by the models shown in Equations 85 and 86,respectively. Therefore, from the wheel cart load cell readings, RA and FR can be evaluated toaccurately estimate the loadings on both the front and rear wheels.5.2 Materials and MethodsIn the full scale test program, prototype deck assemblies were constructed on the structuralsteel frame and loaded by the TDLS apparatus. Each prototype deck assembly consisted of a seventeenbay steel deck frame, two full sheets of 1.22 x 2.44 m (4 x 8 feet) transDeck TM panels and four halfsheets of 1.22 x 1.22 m (4 x 4 feet) transDeckTM panels. The transDeck TM panels were mounted on thesteel frame, 2.44 x 4.98 m (96 x 196 inches) in plan, with a pattern shown in Figure 39. The face grainof the panels was parallel to the long axis of the steel frame.The steel frame was made with two C shape standard steel channels (C150x12) and 17 highstrength light weight steel I-beams. The C channels were 152 mm (6 inches) deep and 4.98 m (196inches) long. The weight and length of the steel I-beams were 46.7 .-1 (3.2 1. 1-) and 2432 mm (95.75inches), respectively. The specification moduli of elasticity and yield strength of the I-beams were200,000 MPa (29x10 6 psi) and 550 MPa (80000 psi), respectively. The thickness of the flange (t f) andthe web (C) equaled 3.2 mm (0.125 inch). The depth (II I) and the width (130 of the I-beam were 108mm (44 inches) and 57 mm (2.25 inches), respectively. A steel connection plate, 3.2 mm (0.125 inch)thick, was welded to each end of a I-beam. The steel I-beams were spaced a distance of 305 mm (12inches) on center and each end of a I-beam was bolted through the connection plate to the web of thechannel with four symmetrically placed 9 mm (8 inch) diameter bolts. The horizontal and vertical boltspacings were 32 x 57 mm (1.25 and 2.25 inches), respectively. The I-beams were mounted to the Cchannels such that the top flange of each I-beam was leveled with the top flanges of the two channels.Each C channel rested on a 305 mm (12 inches) deep steel I-beam member in the TDLS structural steelframe. The C channels were not clamped to the structural steel frame and were free to rotate about150Figure 39 Prototype dry freight van trailer decking system used in the full scale test program.151their respective axes along the long direction when the prototype deck system was under load.TransDeckTM panels were connected through the flanges of the I-beams using 8 mm ( IA inch)-18 torx drive, flat head, type G, phosphate and oil coated self tapping screws. Two staggered rows ofconnectors were used in each I-beam. In each row, a total of twelve connectors were uniformly spacedat 203 mm (8 inch). The end distances of the connectors in each row were 51 and 152 mm (2 and 6inches) from the two ends of a I-beam, respectively. The two rows of staggered connectors werecentrally spaced a distance of 38 mm (1.5 inch) about the web. The stagger between the two rows ofconnectors was 102 mm (4 inch). Since it was not feasible to replace the steel I-beams after each test,repeated installation of self tapping screws through the I-beams was difficult without damaging the I-beam and altering the stiffness characteristics of the connection. A bolt and nut connection system wastherefore used in the experimental program rather than relying solely on the self tapping screws.9Slightly oversized pilot holes with a diameter of 8 mm (-5 inch) rather than 7 mm ( 3-2 mm) were pre-16drilled through the panels and the I-beams. The self tapping screws were inserted into the panels andthrough the flanges of the steel I-beams. Using the self tapping screw as a bolt, a nut was then appliedonto each connector from the bottom of the deck. An air driven torque wrench without torqueregulation was used to fasten the connector until its head was flush with the panel surface.A special connection system was recommended by Ainsworth Lumber Company Ltd. forjoining transDeckTM panels along the perpendicular to face grain direction. In the prototype assembly,two such joints existed along Bays 5 and 13 which were the junctions between the full and half sizepanels. A connection steel band, 5 mm (A inch) thick, 76 mm (3 inches) wide and 2.4 m (8 feet) long,was installed at the each junction. A 5 x 38 x 1220 mm ( T6-3 x 1.5 x 48 inches) section was machinedusing a router from the top surface along one edge of each transDeckTM panel where the full and halfsize panels were butted against each other. The steel band was fitted into this slot so that the topsurfaces of the steel band and the original panel were flush. In each junction, two rows of connectors, 8mm (6 inch)-18 torx drive self tapping screws, were uniformly spaced at 102 mm (4 inches) and152applied through the steel connection band, the panels and the I-beam. The end distance of theconnectors was 51 mm (2 inches) from the each end of a I-beam. The two rows of connectors werecentrally spaced a distance of 32 mm (1.25 inch) about the web. Standard pre-drilling and fasteningprocedures were followed during application of the self tapping screws and nut system.The following 4 prototype dry freight van trailer decking assemblies built from 11-ply Douglas-fir veneer transDeckTM panels were considered in the full scale testing program:1) built from regular 11-ply transDeck TM panels with 3.2 mm thick veneer;2) built from regular 11-ply transDeck TM panels with 2.5 mm thick veneer;3) built from special 11-ply transDeckTM panels with 3.2 mm thick veneer;4) built from special 11-ply transDeck TM panels with 2.5 mm thick veneer.All the veneers for the panels in the full scale testing program were graded using theMetriguard Model 2600 veneer grader. Similar to the veneer testing program, the veneers were dividedinto three groups based on their propagation time: group A (best quality material); group B (mediumquality material); and group C (worst quality material). The regular 11-ply transDeck TM panels weremade from veneers randomly and proportionally sampled from the three groups. In the special 11-plytransDeckTM panels, three exterior plies on each face were randomly selected from group A (the bestquality veneer) while the interior veneers were randomly sampled from groups B and C. Sevenreplicates were considered in each of the two prototype decking assembly with regular transDeck TMpanels. Only four replicates were available in each of the two prototype decking assembly with specialtransDeckTM panels.Seventeen Duncan Model 606 linear motion position sensors were mounted under the deckingsystem to monitor the vertical deformations in the system. Nine of the seventeen transducers wereplaced under the midspan of the supporting I-beams in Bays 5 to 13 of the prototype decking system.The other eight transducers monitored the vertical deformation at midspan of the transDeck TM panels.Each of these eight transducers were centrally mounted between the adjacent supporting I-beams and153at midspan in Bays 5 to 13. The transducers had a maximum travel of 152 mm (6 inches) andnonlinearity was expected when measuring deformations at ± 0.12% of full range. An eighteen channel(17 transducers and 1 load cell) data acquisition system was set up to amplify, condition and recordthe signals. Each channel recorded at a frequency of 300 readings per second.5.2.1 Static test programIn each static test, the loading mechanism simulating the action of a lift truck with a frontaxle loading of up to 40 kN (9,000 lb) was used. The wheel cart was first lifted off the panels to obtainreadings on the displacement transducers corresponding to zero load. The wheel cart was positioned atthe following three locations in turn in the prototype seventeen bay decking system:1) the front axle directly on top of Bay 9;2) the front axle centered between Bays 6-7;3) the front axle directly on top of the connection steel band in Bay 13.At each of these loading positions, both axles in the wheel cart were centered across the widthof the prototype deck system. When the wheel cart was positioned at the first loading position and theair bag actuators were pressurized with the loading in the front axle increased from its dead weight to40 kN (9,000 lb). During the loading sequence, the load deformation characteristics of the prototypedecking system were monitored and recorded using the eighteen channel data acquisition system. Thewheel cart was then repositioned at the second loading position with the loading and data acquisitionsequences repeated. This procedure was repeated until all three loading positions were considered. Fromthe wheel cart load cell readings, the loading in each axle was estimated using the load distributioninformation collected during loading mechanism calibrations (Equations 85 and 86). The deformationprofiles of the prototype decking system under various static load levels for a particular wheel cartlocation were therefore obtained. Figure 40 shows the prototype deck assembly during a typical staticloading test.154Figure 40 Prototype dry freight van trailer decking system during a static test.1555.2.2 Cyclic test programSince the number of available specimens in each prototype was limited, it was decided tocharacterize the fatigue behavior of the prototype test assemblies by obtaining information on a rangeof number of cycles to failure by testing the replicates at various load levels. Typically, the desiredrange of number of cycles to failure was between 10 to 3000 cycles. Testing at too low a load level canlead to an unpractically large number of cycles to failure. Contrary, testing at too high a load levelmay result in catastrophic failure under static load. With a new product such as transDeck TM , limitedinformation existed on the fatigue behavior; therefore, the choice of the load level for the first specimenwas based on 60 to 70% of the estimated static load capacity of the prototype assembly. The loadlevels for subsequent tests were chosen based on the relationship between load level and number ofcycles to failure information from previous tests.After each set of static tests, the front axle of the wheel cart apparatus was positioned directlyon Bay 9 the air bag actuators were pressurized while the load cell within the wheel assembly wasmonitored until the desired load level was reached. The applied load was maintained by closing thevalves for the air bag actuators. Then the wheel cart apparatus was cycled along the long axis of theprototype decking assembly with near constant front and rear axle loadings. A half cycle of wheel carttravel is defined by the front wheels moving from the two half size panels over the connection steelband onto the adjacent full size panels and traveling 2.44 m (96 inches) along the long axis of theprototype assembly to the other end of the full size panels and moving off over the connection steelband onto the adjacent half size panels. The front wheels traversed a total distance of 2.54 m (100inches) during each half cycle. Typically, the period of wheel cart travel was 18 seconds per cycle. Thewheel cart was cycled until transDeck TM panels failure was encountered. A counter device was installedto monitor the total number of cycles of motion. The load and deformation of the prototype deckingsystem during each cycle of movement were monitored by the eighteen channels data acquisitionsystem. The maximum value in each channel monitored during a cycle was stored. The decking was156considered failed when punch through type failure was visually detected from the top surface. At thisstage, the prototype assembly would have accumulated sufficient damage that system collapse wouldbe imminent.5.2.3 Short Term Small Specimen Bending TestsOne 102 x 406 mm (4 x 16 inches) specimen, with face grain oriented parallel to its long axis,was cut from each full size test panel evaluated in the cyclic test program. The specimen was takenfrom a bay where failure occurred at a location near the side of the prototype decking assembly wherethe estimated stresses experienced by the specimen during cyclic testing were minimal. The short-termbending strengths of these specimens were then tested under a center point load with simply supportconditions at two ends obtained using roller bearing plates. A test span of 305 mm (12 inches) waschosen such that the span to depth ratio resembled the prototype decking assembly. A MTS model 810hydraulic control close loop universal testing machine with a capacity of 222.4 kN (50000 lb) was usedto apply the load in a deflection control mode. An uniform rate of cross head motion of 1.37 mm min.(0.054 -inch) was used which resulted in specimen failure between 3 to 5 minutes of loading.min.The specimen dimensions at mid-span were measured. A computer based data acquisitionsystem was used to acquire the load versus deformation (cross head motion) data. A load cell with a 25kN (5620 lb) capacity was used to monitor the loads. The time to failure information was alsorecorded. The bending strengths, based on the measured peak loads, were estimated according toEquations 76a and 76b, respectively.5.3 Experimental Results5.3.1 Static Test ProgramShown in Figure 41 are the experimentally measured midspan deformation profiles of the fourprototype deck assemblies with the front axle located directly on Bay 9. The experimentally measured12^13115^6^7^8^9^10Bay Number13121120* 40 kN 0 30 kN X 20 kN 0 10 kNE 150010CIaa•13505^6^7^8^9^10Bay Number1111111* 40 kN 0 30 kN^X 20 kN^0 10 kN//),/ 1IIii200EE 150Uaa 10coa035 6* 40 kN 0 30 kN X 20 kN^0 10 kN20EE 150U7 101;a0a50157Prototype 1^Prototype 220E 1 50t,O)100aaaV5 60Prototype 3111111 11* 40 PIN^0 30 kN^X 20 kN^4 10 kN*CX0t15^6^7^8^9^10^11^12^13Bay NumberPrototype 45^6^7^8^9^10^11^12^13Bay NumberFigure 41 Midspan deformation profiles of the four prototype deck assemblies with thefront axle located directly on Bay 9.158midspan deformation profiles of the four prototype deck assemblies with the front axle centrally locatedbetween Bays 2 and 3 are shown in Figure 42. Finally shown in Figure 43 are the experimentallymeasured midspan deformation profiles of the four prototype deck assemblies with the front axlelocated directly on top of the steel band in Bay 13. For each loading position, the average and standarddeviation of the peak midspan deflection in each prototype assembly under a front axle loading of 40kN (8992 lb) are also shown in Table 32. The consistency of the deformation profiles indicates that thevariability in the elastic properties within each prototype assembly was small. An increase in peakmidspan deformation was noted the when wheel cart apparatus was located at loading position 3 (frontaxle directly on top of the steel band). This increase was due to the discontinuity at Bays 5 and 13where the half and full panels were connected to the steel I beams through the steel band. Apparentlythat the steel band could not completely compensate for the discontinuity at the joint by restrictingpanel rotation about its neutral axis over the supporting I beam; therefore, reduced assembly stiffnessat the special connection was expected. The relationship between front axle load levels and peakmidspan deformation is shown in Table 33 for the four prototype assemblies. The coefficient ofdetermination values for the various cases range from 0.921 to 0.999 which indicate that linear load-deformation relationships under static front axle loading of up to 40 kN (8992 lb).5.3.1.1 Verification of Deck Analysis Model:The four prototype test assemblies were modeled using the DAP. The program predictedresponses were compared to the experimental results for model verification. The modeling of the fourprototype test assemblies required as input the elastic properties of the veneers in the transDeck TMpanels. The axial and flexural moduli of elasticity in the x and y directions (E , E , E , E ) and the(EX EX ^yf yamoduli of rigidity (G) of the veneers were obtained from the results of the veneer mechanical testprogram.In prototypes 1 and 2, the individual veneers within a panel were assumed to have identical11 - 11111* 40 ka^0 30 ka^X 20 ka^<> 10 kNginIIIIII 16^7^8^9^10^11^12^13Bay Number0520EisCOUaCaaa5[1111114( 40 ka^0 30 014^X 20 kl4^0 10 k N1111 In 01111 10 V6^7^8^9^10^11^12^13Bay Number20 i^I^r^i^i^r^i^ 20EE 150Ua7:-0 10OCcoa605l 40 kN^0 30 kN^X 2000^0 10 1114E 15C0UN 1 0a36^7^a^9^10^11^12^13Bay Number159Prototype 1^Prototype 2.............III1111111111111111116^7^8^9^10^11^12^13Bay NumberPrototype 3^Prototype 420E 15CO5aai 100a.05Figure 42 Midspan deformation profiles of the four prototype deck assemblies with thefront axle centered between Bays 2 and 3.20EE 1 5OU12 13115^6^7^8^9^10Bay NumberI^I^I^I^I^I^I* 40 kN^0 30 kN^X 20 kN^0 10 kN0 e• 81141 .0 iLill I i"It^ !001 6i*x. 0X)( $15^6^7^8^9^10^11^12^13Bay Number20020EE 150O120 100..1350EE 150Oa10a5* 40 kN^0 30 kN^X 20 kN^0 10 kN16005Prototype 11111111* 40 kN^0 30 kN^X TO kN^0 10 kN1^ii 0 Ie46^7^8^9^10^11^12^13Bay NumberPrototype 2I^I^I^I^I^I^1 if 40 kN^0 30 kN^x 20 kN^0 10 kNØot1I Im X•g6^7^8^9^10^11^12^13Bay Number20E15Oa:011 10IS Q125 5Prototype 3^Prototype 4Figure 43 Midspan deformation profiles of the four prototype deck assemblies with thefront axle located directly on Bay 13.161Table 32. Peak midspan deflections under 40 kN front axle loading.^Prototype Loading^Peak Midspan DeflectionPosition^Experimental Results^Model Predictions PredictionAverage(mm)STDV(mm)N (mm) Error(%)1 1 10.393 0.233 7 10.147 -2.372 10.175 0.073 7 10.037 -1.363 11.691 0.744 7 10.267 -12.182 1 11.938 0.103 7 11.871 -0.562 11.418 0.364 7 11.672 -2.223 12.283 0.345 7 11.931 -2.863 1 9.914 0.052 4 9.770 -1.482 9.879 0.031 4 9.689 -1.923 10.931 0.107 4 9.918 -9.264 1 11.713 0.213 4 11.559 -1.332 11.070 0.075 4 11.385 -2.763 12.690 0.468 4 11.635 -9.07162Table 33. Front axle load levels versus peak midspan prototype deformation.Prototype LoadingPositionLoad (kN) = a Deformation (mm) + ba^b^r2( la)^(kN)1 1 3.98 -1.21 0.9962 4.08 -1.37 0.9993 3.35 1.03 0.9732 1 3.48 -1.36 0.9992 3.63 -1.26 0.9933 3.27 0.33 0.9873 1 3.73 -0.11 0.9942 4.38 -3.09 0.9963 4.34 -2.90 0.9994 1 3.56 -1.55 0.9982 3.84 -2.19 0.9983 3.11 1.22 0.921163elastic properties. Here, E and E were obtained from the veneer bending test results. E and E xaYf^ XYawere obtained from the veneer compression test results rather than the veneer tensile test results. Thisis because accurate measurement of veneer tensile elastic properties in the perpendicular to graindirection was difficult. For the parallel to grain direction, the mean elastic moduli of the 3-plies and 4-plies specimens were averaged for each veneer thickness. For the perpendicular to grain direction, themean elastic moduli of the 3-plies and 4-plies specimens with various veneer thickness were averaged.The modulus of rigidity properties (G) were obtained from the veneer shear modulus of rigidity testresults where the mean rigidity moduli of the 3-plies and 4-plies specimens were averaged for eachveneer thickness.In prototypes 3 and 4, the three exterior plies on the face of each panel were made with veneerfrom subgroup A (best quality veneer) whereas the interior plies were made from subgroups B and C;therefore, the elastic properties of the exterior and interior plies were different. Using the mean elasticproperties results of the various subgroups shown in Table 27, Ex f , Ex , EYf, , EYa , and G for the interiorand exterior plies of panels in prototypes 3 and 4 were obtained. Table 34 shows the input veneerelastic properties for the four prototype assemblies. The Poisson's ratio v xy and vyx of the veneers weretaken as 0.02 and 0.4 respectively. The veneer thickness for prototypes 1 and 3 were taken as 3.2 mm(0.125 inch) and for prototypes 2 and 4 were taken as 2.5 mm (0.1 inch). The stiffnesses of a panel inthe cover were obtained from Equations 74a to 74h for the various prototypes and the results are shownin Table 35.A seventeen bay deck system was modeled. The moduli of elasticity and rigidity of thesupporting I-beams in the deck system were taken as 200,000 MPa (29x10 6 psi) and 77000 MPa(11x106 psi), respectively. Also the supporting I-beams were assumed to have a yield strength of 550MPa (80000 psi). The thickness of the flange (t f) and the web (tw ) equaled 3.2 mm (0.125 inch). Thedepth (III) and the width (b I) of the I-beam were 102 mm (4 inches) and 57 mm (2.25 inches),respectively. The connectors between the cover and the supporting I-beams were considered uniformlyPrototype^E^Ev^E^Ex^GJ f j aX aExterior Interior^Exterior Interior(MPa)^(MPa)^(MPa)^(MPa)^(MPa)^(MPa)^(MPa)1 11566 11566 9828 9828 430.9 488.2 7292 12958 12958 11622 11622 430.9 488.2 7983 14079 10681 12508 8932 430.9 488.2 7294 14843 12246 13927 10653 430.9 488.2 798164Table 34. Veneer elastic moduli for DAP analyses.Table 35. Input panel stiffness values for DAP analyses.Prototype Kx Ky Kv KG DX Dy DI, DG(kN • m) (kN • m) (kN • m) (kN • m) ( 1i14) ( 1‘#) (MA(IN)1 4.48 38.46 1.25 2.59 76.97 286.22 25.45 25.462 2.48 22.05 0.71 1.43 70.77 270.32 24.03 22.043 4.24 46.54 1.15 2.59 71.24 329.07 23.15 25.464 2.38 25.16 0.68 1.43 65.80 298.29 22.05 22.04spaced at 102 mm (4 inches) on center. The connector stiffnesses K y , Kxc^c, and KB were assumed tocequal 4.133 -MmN (23600 +.3il ), 2.908 IV (16600 113.1! ), and 3.502 MmN (20000 lb), respectively. The Kyc andKxc values were obtained by taking the average of the mean stiffness values of groups 10A and 08Aand groups 10E and 08E from Table 29, respectively.Four levels of front axle wheel loads were considered: 40 kN (8992 lb), 30 kN (6775 lb), 20 kN(4496 lb), and 10 kN (2248 lb). In the prototype test assemblies the front and rear axle load165distribution depended on the load level and Equation 85 was used to estimate the load distribution andthe loading on the rear wheel for the various load levels. The front and rear axles, oriented parallel tothe direction of the supporting 1-beams, were spaced at a distance of 1.22 m (4 feet) apart. Three wheellocations corresponding to the static tests were also considered. The foot print of each front wheel wasassumed to be 203 x 89 mm (8 x 3.5 inches) which covered an area of 18064 mm 2 (28 inches 2 ). Eachrear wheel has a foot print of 89 x 80 mm (3.5 x 3.15 inches) which covered an area of 7113 mm2 (11inches 2 ). The two front wheels were spaced at a distance of 965 mm (38 inches) apart and centered inthe x-direction in the deck. Similarly, the rear wheels were centered in the x-direction of the deck andspaced at a distance of 749 mm (29.5 inches) apart.Shown in Figure 44 are the comparisons of the model predicted response and theexperimentally measured midspan deformation profiles of the four prototype deck assemblies with thefront axle located directly on Bay 9. Similar comparisons between model predicted and experimentallymeasured midspan deformation profiles of the four prototype deck assemblies with the front axlecentrally located between Bays 2 and 3 are shown in Figure 45. Finally shown in Figures 46 are thecomparisons of model predictions and experimentally measured midspan deformation profiles of thefour prototype deck assemblies with the front axle located directly on top of the steel band in Bay 13.Good agreement between model predictions and experimentally measured response wasobtained. DAP could predict the system response for a region within 3 Bays of the front axlereasonably well. When considering the system response at a distance from the front wheel load, thelevel of agreement between model predictions and measured responses reduces. However, at theselocations, small deformations were found; therefore, the lack of model agreement was consideredrelatively insignificant.Also shown in Table 32 are the model predicted peak midspan deflection under a load of 40 kN(8992 lb) for each loading position. At 40 kN (8992 lb) front axle loading, a maximum prediction errorof 2.76 % was found for loading positions 1 and 2. A maximum prediction error of 12.18 % was foundE 150(0 100ao.co016620E 15001; 10ao.a2 50Prototype 1 — Model^* 40 kN^0 SO ONX 20 kN^0 10 kNaBay NumberPrototype 2— Model^* 40 kN^0^30 kNX^20 kN^5^10 kNEN11161r&P11111109^10^11Bay Number10^11^720E 15001- 10aOD220 2050Prototype 3I I I— Model^*X^20 kN^040 kN10 kN0^30 kN10 0 011(7^a^9Bay NumberPrototype 4I^I— Model^* 40 kN^0^30 kNX^20 kN^0^10 kN411111khe.Pill.5114AowN1011Mh1V. 1111a^10^11Bay Number10^11^7Figure 44 Comparisons of model predictions and measured midspan deformation profilesof the four prototype deck assemblies with the front axlelocated directly on Bay 9.20E 15C00a4.-01 10OCasaa2 5— Model^*X^20 kN^040 kN^0113 kN30 kNilig -AL.Noir5^6^7Bay NumberI^I— Model^* 40 kN^0^30 kNX^20 kN^0^10 kNpi„„,„„Houg,Airdrit7^8^9Bay NumberBE 15C0010CaaaX 52008^9^52015C0"€-D 10CaaaVX 50167EE 15C00a;-.0w 10CaaaV5200105Prototype 1I^I^I— Model^* 40 kN^0^30 kNX^20 kN^0^10 kN..gallignilhhi..9....-^-.Mt,ift:^'^1:m„, ...^.5^6^7^8^9Bay NumberPrototype 2— Model^* 40 kNX^20 kN^0^10 kN0^30 kN1110111.1.jir."1"0 111/1 417^8^9Bay NumberPrototype 3^Prototype 4Figure 45 Comparisons of model predictions and measured midspan deformation profilesof the four prototype deck assemblies with the front axlefront axle centered between Bays 2 and 3.131220— Model^4( 40 kN^0 30 kNX 20 kN^0 10 kNEEi5OU15 10CaaM 510^11Bay Number09200EE 15C010CaaM 5168Prototype 1^Prototype 2—XModel^*20 kN^040 kN10 kN0^30 kN.•9^10^11^12^13Bay NumberE 15U15 10CaaV5200909Prototype 3— Model^* 40 kN^0^30 kNX^20 kN^0^10 kNIlllirdniimmwm„,woo^mu_._..... ....amia.10^11^12^13Bay NumberPrototype 4—XModel^*20 kN^040 kN10 kN0 30 kN10^11^12^13Bay Number20EE 15C0Ua0) 10aCaVM 5Figure 46 Comparisons of model predictions and measured midspan deformation profilesof the four prototype deck assemblies with the front axlelocated directly on Bay 13.169for loading position 3. This error can be attributed to the inability of DAP to consider thediscontinuity and the special connection steel bands in Bays 5 and 9. Based on the low prediction errorsin estimating peak deflections and reasonable agreement between predicted and measured deformationprofiles, DAP was considered successfully verified.Using the input parameters for the four prototypes, DAP was used to estimate the maximumbending stress of the exterior ply in the panel (o --37 ) considering -1.- cycle of wheel cart travel with a frontaxle load level of 73.39 kN (16500 lb) and a front to rear axles load ratio of 9:1. The maximum o-Y wasfound to occur at under the foot print of the front wheels when the front axle was centrally locatedbetween Bays 5 and 6. The maximum cry was found to be 43.22 MPa, 67.15 MPa, 52.55 MPa, and76.05 MPa for prototypes 1 to 4, respectively.The variability in the measured deformation profiles can be examined by considering thevariability in the veneer elastic properties shown in Table 17, 19 and 20. For each prototype, the elasticmoduli (Eye Eya, Exf, Exa , and G) for 11000 sheets of veneer were randomly simulated assumingnormal distributions and perfect correlation in the rank of the various elastic moduli within each sheetof veneer. Panel stiffness values for 1000 replicates of each prototype were computed according toEquations 74a to 74h with their statistics shown in Table 36. The mean panel stiffness values from thesimulation study agreed with the input panel stiffness values for DAP shown in Table 35.Two sets of upper and lower bound panel stiffness values were obtained as: the mean panelstiffness ± 2 x the standard deviation of panel stiffness. Each of the two sets of panel stiffness valueswas used as input into DAP to approximate the upper and lower bounds of the deformation profiles.Figure 47 shows the deformation bounds for prototype 1 with the front axle located directly on Bay 9.The measured and DAP predicted deformation profiles were within the estimated deformation bounds.Similar results were obtained for other prototypes and other loading conditions170Table 36. Statistics of panel stiffness values for DAP analyses.Prototype^K.^Ky(kN •m) (kN • m)Kv(kN • m)KG(kN •m)D.(1111)Dy(MN)Dv(Mf-)DG(1\4)Mean Values1 4.48 38.34 1.25 2.58 77.31 285.62 25.59 25.232 2.48 22.00 0.72 1.43 71.07 269.77 24.16 21.843 4.25 46.42 1.16 2.58 71.52 328.29 23.27 25.234 2.39 25.09 0.68 1.43 66.08 297.59 22.16 21.84Standard Deviations1 0.45 3.20 0.17 0.13 12.05 25.54 4.82 3.282 0.23 1.63 0.09 0.07 10.93 23.19 4.37 2.933 0.39 3.26 0.14 0.13 10.58 24.56 4.23 3.284 0.23 1.80 0.08 0.07 10.09 22.92 4.04 2.9310 118^9Bay Number720......E 1 5E•_2 50Model^Bounds^K 40 kN^ 30 kN^X 20 kN 0 10 kN171Figure 47 Upper and lower bounds of measured midspan deformation profilesof prototype 1 with the front axle located directly on Bay 9.1725.3.2 Cyclic Test ProgramTable 37 shows the applied front axle load level (F R) and the number of cycles to failure (N f)information for the four prototypes. Since it was difficult to maintain a completely constant load levelduring testing, the mean and standard deviation of the applied load were calculated and shown inTable 37. The results indicate that the applied loads were near constant during each test. A maximumstandard deviation of 1.065 kN can be found in specimen 4 of prototype 2 which was caused bytechnical problems in the cable cylinders.The failure locations in the panels are also shown in Table 37. It can be noted that the failureoccurred only under the foot print of the front wheels. Over 50 % of the time, failure occurs in betweeneither Bays 5 and 6 or Bays 12 and 13 in the prototype systems. These locations were adjacent to thetwo steel connection bands in Bays 5 and 13 where discontinuity in the panel existed. Therefore, thesefrequent failure location should be considered as critical zones in the prototype systems.The final failure mode was punch through type failure visible from upper surface of the panel.The stiffness of the prototype system was significantly reduced after punch through type failuresoccurred. Therefore, at this stage the tests were terminated before the final collapse of the system wherepermanent damage to the steel beams might result. However, prior to punch through type failure beingvisibly detected, a series of secondary failure modes such as tension parallel and perpendicular to grainfailures in the exterior plies and rolling shear type failure in the interior plies were detected. Althoughthe stiffness of the prototype system was reduced when secondary failures occurred, the overall loadcarrying capacity of the system remained available.The midspan deformation information monitored by the seventeen transducers during cyclictesting showed a progressive increase of midspan deformation as the deck undergoes creep andaccumulates damage under the cyclic load. The ratio between the initial and final midspan deformationrecorded by each transducer (Rd) was calculated. The maximum Rd in each specimen is shown inTable 37 which ranged from 1.233 to 1.066. This information was originally intended to signal the173Table 37. Cyclic test results of full scale prototypes.Prototype Specimen FR^Nf^Maximum^Failure Location Number Mean(kN)STDV(kN)Rd Under FrontWheel #1 andBetween BaysUnder FrontWheel #2 andBetween Bays1 1 72.248 0.904 10 1.127 5 & 62 70.249 0.596 10 1.104 12 & 133 55.160 0.480 265 1.123 8 & 9, 10 & 114 55.473 0.122 313 1.208 7 & 8, 11 & 125 44.346 0.182 1517 1.233 9 & 10, 11 & 136 42.137 0.102 1434 1.145 7 & 87 39.918 0.098 5436 1.160 5 & 62 1 66.793 0.734 3 1.100 6 & 82 60.404 0.290 10 1.066 7 & 8, 12 & 133 55.499 0.161 47 1.098 12 & 134 49.317 1.065 92 1.169 12 & 13 5 & 65 37.828 0.092 1226 1.146 12 & 136 35.497 0.162 2536 1.119 7 & 107 35.493 0.083 2198 1.119 11 & 123 1 60.234 0.115 20 1.094 8 & 92 55.267 0.064 86 1.082 5& 6 6& 73 48.707 0.126 223 1.112 9 & 104 44.419 0.139 214 1.128 9 & 104 1 55.566 0.537 7 1.095 5& 62 55.270 0.118 40 1.089 12 & 133 48.830 0.131 96 1.095 5 & 6, 12 & 134 44.296 0.076 180 1.107 6 & 7, 8 & 9, 11 & 12^Prototype Number of^b1i^boi^r2 Standard Error FR^FR^X1s 3000^Specimens of Estimate^(kN) (kN)X21 7 -12.136 83.665 0.980 2.0237 87.32 41.47 1 12 7 -10.851 71.861 0.994 1.0669 75.13 34.13 -1 13 4 -13.317 78.504 0.872 3.0620 82.51 32.20 1 -14 4 -7.695 63.850 0.751 3.1349 66.17 37.09 -1 -1174initiation of failure and provide a guide to the termination fatigue test. The relatively wide range inmaximum Rd indicated the insensitivity of this parameter to the occurrence of final failure and made itunsuitable to be considered as a signal of the onset of final failure.The relationships between FR and Nf for the four prototype assemblies can be given by:FR =^bli login (N1)^(i = 1,..,4)^[87]where 130. and b 1. denote the slope and intercept of the relationships for the ith prototype, respectively.Figure 48 shows a plot of the relationships between FR and Nf (in log 10 scale) for the four prototypeassemblies. Table 38 shows the b 1. , 130. , and r2 values for the various relationships. The r 2 values forthe relationships between FR and Nf (in log10 scale) ranged from 0.75 to 0.99 which clearly indicatedEquation 87 successfully represented the fatigue performance of transDeck TM. It was noted thatprototypes 3 and 4 had lower r 2 values compared to prototypes 1 and 2 which was attributed to thenarrower range of load levels considered in the cyclic testing program for prototypes 3 and 4.Table 38. Regression parameters for the FR and Nf relationships of the four prototype assemblies.Prototype 210080604020Y 80aOax 800UiNNN^1004020a0Oax 600u_401002 8020 ^175100Prototype 180aZ0 0-Jax 80 X0 0LL402010^100^1000^10000^ 10^100^1000^10000^Number of Cycle to Failure Number of Cycle to Failure10000Prototype 310^100^1000Number of Cycle to FailurePrototype 410^100^t000^10000Number of Cycle to FailureFigure 48 Performance of transDeck TM under fatigue loading.176The static strength of a prototype assembly can be defined as the maximum front axle loadlevel which the panel can carry for -2-1 cycle of wheel cart travel. Shown in Table 38 are the estimatedmean static capacity (FR ) and the estimated mean load capacity at 3000 cycles of wheel cart travels(FR^) for each prototype using the relationships in Equation 87.3000The relationships between FR and Nf (in log10 scale) for the four prototype assemblies can betested using analysis of covariance to check whether their slopes can be considered statistically equal.Here, regression approach to analysis of covariance was used in which the full and reduced regressionmodels for the relationships between FR and Nf are respectively given by:FR = b0 ± bi X i + b2 X2 + b3 X i X 2 + b4 Z + b5 X i Z + b6 X2 Z^ [88]andFR = 130 + b1 X 1 + b 2 X2 + b3 X1 X 2 + b4 Z^ [89]where Z is the covariate which equals (log 10 (Nf) — log10 (NO) and the indicator variables X 1 and X2for the various treatments were assigned the (-1, 0 ,1) scheme as shown in Table 38.The equality of slopes between treatments were checked using a partial F test that there wasno interaction between the covariate and treatment. The partial F between the full and reduced modelsis given by:(SSR —^= 1.67SST, ) DF,''Partial F = ^f2 SS '' fEf[90]where SS Rr equals the sum of squares of regression of the reduced model; SSR f , SSEf and DFf equals,the sum of squares of regression, sum of squares of error and degree of freedom of error of the fullmodel, respectively. The test statistics Fc = F2, 15; 1-a = 3.68 for a=0.05. Since partial F < F c , it wasconcluded that there was interaction between treatments and covariate; therefore, the slopes of the177relationship between FR and 1\11 (in log 10 scale) between various treatment groups were consideredstatistically equal at the 95% probability level. The results are summarized in Table 39.Table 39. Results of regression approach to analysis of covariance of the FR and N 1 relationships.Model inEquationb0 b1 b2 b3 b4 b5 b6 DFf SSR SSE88 51.14 3.232 2.354 1.352 -10.798 -0.870 -0.716 15 2281.127 74.43489 50.81 3.359 2.509 1.044 -11.267 17 2264.518 91.04391 50.81 3.644 2.510 -11.272 18 2242.309 113.2593 50.85 2.315 1.939 -10.680 18 2040.755 314.8194 51.50 3.262 1.047 -10.621 18 2143.837 211.7297 51.51 1.674 -0.060 1.091 18 109.7751 2245.8Interaction between the veneer thickness effect and the effect of ultrasonic graded specialveneers was checked using partial F test. Here the regression model given in Equation 89 was treated asthe full model and the following regression model was treated as the reduced model:FR = bo bi^b2 X2 + 134 Z^ [91]The results are summarized in Table 39. The partial F between the full and reduced models is givenby:(SSR —^) DFfPartial F = f SS^= 4.15Ef[92]The test statistics Fc= F1, 17; 1-a = 4.45 for a=0.05. Since partial F < F c , it was concluded that there178was no interaction between treatments.The veneer thickness effect and the effect of ultrasonics graded special veneers were alsochecked using partial F tests. Again the regression model given in Equation 89 was treated as the fullmodel and the following regression models were treated as the reduced models for veneer thicknesseffect and the effect of ultrasonic graded special veneers, respectively:FR = 130 b2 X2 + b3 X1 X2 b4 Z^[93]andFR = bo^Xi+ b3 X i X2 + b4 Z^ [94]The results are summarized in Table 39. The partial F values between the full and reduced models forEquation 93 and 94 are respectively given by:(SSR — SSR ) DFfPartial F =^f^r SS^= 41.78E(SSR SS— SSR)DFfPartial F = ^ = 22.53EfThe test statistics F c = F1, 17; 1-a= 4.45 for a=0.05. Since partial F > 1', in both cases, it wasconcluded that both treatment effects were significant.Finally the justification of using analysis of covariance was considered by checking the effect ofcovariate using a partial F test. Again the regression model given in Equation 89 was treated as the fullmodel and the following regression model was treated as the reduced model for covariate effect:[95][96]FR = bo^X1+ b2 X2 + b3 X 1 X 2^[97]179The results are summarized in Table 39. The partial F value between the full and reduced models forEquation 97 is given by:(SSRf SS— SSR)DFfPartial F = ^ = 402.35Ef[98]The test statistics F c = F1; 17; i_a = 4.45 for a=0.05. Since partial F > F c , it was concluded thatcovariate effect was significant; therefore, analysis of covariance was justified.The relationship between FR and Nf (in log 10 scale) for the each prototype assembly wasnormalized with respect to its estimated static capacity F s to yield information on applied stress ratio(SR) to number of cycles to failure. The relationships between SR and Nf (in log 10 scale) for the fourprototype assemblies were tested using analysis of covariance to check whether their slopes can beconsidered statistically equal. Again regression approach to analysis of covariance was used in whichthe full and reduced regression models for the relationships between SR and Nf are respectively givenby:SR = 130 + b i X i ± b2 X2 + b3 X1 X2 ± b4 Z ± b5 X1 Z + b6 X2 ZandSR = 130 + b 1 X 1 + b 2 X2 + b3 X 1 X 2 + b4 ZThe partial F between the full and reduced models is given by:(SSRf — SSR) DFPartial F = ^ = 0.432 SSEf[99][100][101]The test statistics I', = F2; 15; i _ ct = 3.68 for a=0.05. Since partial F < F c , it can be concluded thatthere was interaction between treatments and covariate; therefore, the slopes of the relationship180between SR and Nf (in log10 scale) between various treatment groups were statistically equal at the95% probability level. The results are summarized in Table 40.Table 40. Results of regression approach to analysis of covariance of the S R and Nf relationships.Model inEquation130 b1 b2 b3 b4 b5 b6 DFf SSR SSE99 0.697 -0.012 -0.040 0.018 -0.146 0.007 0.005 15 0.45709 0.01552100 0.698 -0.012 -0.040 0.019 -0.143 17 0.45620 0.01641102 0.698 -0.007 -0.041 -0.140 18 0.44906 0.02354103 0.698 -0.040 0.016 -0.143 18 0.45337 0.01923104 0.687 -0.012 0.021 -0.153 18 0.42566 0.04695105 0.707 -0.011 -0.072 -0.002 18 0.11015 0.36245Interaction between the veneer thickness effect and the effect of ultrasonic graded specialveneers was checked by considering the regression model given in Equation 100 as the full model andthe following regression model as the reduced model:SR = bo + b i X i + b2 X2 + 134 Z [102]The results are summarized in Table 40. The partial F between the full and reduced models was 7.39.The test statistics 1', = F 1, 17; i _ ce = 4.45 for a=0.05. Since partial F > F c , it can be concluded thatthere was interaction between treatments.The veneer thickness effect and the effect of ultrasonic graded special veneers were also checkedby considering the regression model given in Equation 100 as the full model and the following181regression models as the reduced models for veneer thickness effect and the effect of ultrasonic gradedspecial veneers, respectively:SR = 1)0 ± b2 X2 + b3 X1 X2 + b4 Z^[103]andSR = bo + bi Xi+ b3 Xi X2 ± b4 Z^ [104]The results are summarized in Table 40. The partial F values between the full and reduced models forEquation 103 and 104 were 2.93 and 31.6, respectively. The test statistics F c = F1, 17; i_ a = 4.45 fora=0.05. Since partial F < F c for veneer thickness effect, it was considered not significant. Howeversince partial F > F c for ultrasonic effect, it was considered significant.Finally the justification of using analysis of covariance was considered by treating theregression model given in Equation 100 as the full model and the following regression model as thereduced model for covariate effect:SR = bo + b1 X 1 + b2 X2 + b3 X 1 X 2^ [105]The results are summarized in Table 40. The partial F value between the full and reduced models forEquation 105 is 358. The test statistics F c = F1, 17; i _ a = 4.45 for a=0.05. Since partial F > F c ,covariate effect was significant and analysis of covariance was justified.The overall normalized relationship between SR and Nf (in log io scale) for the variousprototypes is given by:SR = 1.00009 — 0.14278 log io (N1) — 0.01178 X 1 — 0.03991 X2 + 0.018971 X 1 X2^[106]182where the r2 value is 0.965, the standard error of estimate is 0.031, and the degree of freedom is 17.Since the effect of veneer thickness was not significant and the slopes of the normalizedrelationships of the various prototypes were not significantly different, 1) the data from prototypes 1and 2 were combined to obtain the normalized relationships of the regular veneer and 2) the data fromprototypes 3 and 4 were combined to obtain the normalized relationships of the special veneer.The relationship between SR and N1 using combined data from prototypes 1 and 2 is given by:SR = 0.956 — 0.14102 log io (Nf)^ [107]where the r 2 value is 0.986, the standard error of estimate is 0.019, and the degree of freedom is 12.The relationship between SR and Nf using combined data from prototypes 3 and 4 is given by:SR = 1.017 — 0.13029 log io (Nf)^ [108]where the r 2 value is 0.635, the standard error of estimate is 0.058, and the degree of freedom is 6.Figures 49 and 50 show the relationships between SR and Nf (in log io scale) for the combinedprototypes 1 and 2 data and combined prototypes 3 and 4 data, respectively. Based on Equations 107and 108, the fatigue performance of transDeck TM can therefore be related directly to the staticstrength. The estimated applied stress ratios corresponding to 3000 cycles to failure were 0.466 and0.564 for the regular and special veneer panels, respectively.The short term small specimen parallel to grain bending test results using specimens cut fromeach full size test panel from the cyclic test program is presented in Table 41. The relatively lowstandard deviations clearly indicated the uniformity in bending strength of transDeckTM. Thedifference of mean bending strengths amongst the four prototypes were consistent with the veneermechanical test program results.1.01830.90ctS 0.8CCcoa)4— 0.7U) •N-aD.0.60.5° Prototypes 1 and 20.41^10^100^1000^10000Number of Cycles to FailureFigure 49 Normalized fatigue performance of transDeck TM with regular veneer.0.9O•_0.8CCcoco22'''^7'' 0cn -Va)•_1841.00.5° Prototypes 3 and 40.41^10^100^1000^10000Number of Cycles to FailureFigure 50 Normalized fatigue performance of transDeckTM with special veneer.185Table 41. Short term static bending test results of various prototypes.Prototype Number ofi^SpecimensBending Strength (MPa)Mean^STDVFs.(kN)FR(kN)Error(%)1 14 54.115 4.345 91.89 87.32 5.232 13 65.437 6.384 71.52 75.13 -4.803 8 53.362 4.733 74.53 82.51 -9.674 8 73.374 6.235 70.81 66.17 7.01For each transDeckTM prototype, the mean small specimen bending strength (S r ) wascompared to the DAP predicted maximum bending stress o--y for a front axle loading of 73.39 kN(section 5.3.1) to estimate the mean static capacity of the prototypes (F s.) as:gvFS ,,‘ = ^.13 73.39 kNCI -3'[109]The predicted mean static capacity of the each prototype, F s., were further compared to thecorresponding experimentally projected mean static capacity F R . The absolute predicted errors rangedSfrom 4.8% to 9.67% which was a further successful verification of DAP. The results of this analysis areshown in Table 41. Note that the small specimens used in the bending tests were obtained from the fullsize cyclic test specimens at the bays where failures occurred. It was assumed that size effect did notplay a role when the small specimen bending test results were used to estimate the capacity of full scaleassembly because the small specimens were sampled from the weakest bay within each panel.For the development of a new prototype, pilot specimens can be made and tested to establishthe small specimen short term bending strength. Since it is not possible to know the weakest locationwithin each panel a priori, size effect models can be invoked to estimate the bending strength of the186full size panels from the small specimen test data. DAP can then be used to predicted the mean staticcapacity of the new prototype. From Equations 107 and 108, the applied stress ratios corresponding to3000 cycles to failure were approximately 0.47 and 0.56 for the regular and special veneer panels,respectively. This factor can be applied to the estimated mean short term static capacity to rate thenew product at 3000 cycles of loading. This simple scheme provides a practical method for gaining apreliminary assessment of the static and fatigue performance of new prototype transDeckTMconstructions.1876. CYCLIC TESTING OF SMALL SPECIMENSThe full scale test program provided information on the performance of the prototype deckingsystems under static and cyclic loading. However, it is a very expensive and time consuming exercise;therefore, only a limited number of tests can practically be performed. A companion test program ofsmall specimens is presented to develop the panel fatigue data. Here, static and cyclic bending testshave been performed with the appropriate span to depth ratio and stress history to determine therelationship between the fatigue life and failure mode of the small specimen and the full scale testsprogram results.6.1 Materials and MethodsIn this test program, a total of twenty 11-ply, 1.2 x 2.4 m (4 x 8 feet), transDeck TM panelswith either 2.5 mm (—I inch) or 3.2 mm (k inch) thick veneer were used. All the veneers for the panels10were graded using the Metriguard Model 2600 veneer grader. Similar to the veneer testing program, theveneers were divided into three groups based on their propagation time: group A (best qualitymaterial); group B (medium quality material); and group C (worst quality material). The 11-plytransDeckTM panels were made from veneers randomly and proportionally sampled from the threegroups; i.e., the specimens with 3.2 mm ( g and 2.5 mm (-110 inch) thick veneer corresponded to8prototypes 1 and 2, respectively.From each panel, eight 102 x 356 mm (4 x 14 inches) transDeck TM specimens were obtained.Seven of the eight specimens were oriented with face grain parallel to the long axis of the specimen.The other specimen was oriented with face grain perpendicular to long axis of the specimen. Thespecimens were then conditioned at a temperature of 20 ± 3°C and relative humidity of 65 ± 5% formore than four weeks where equilibrium was reached.1886.1.1 Short Term Bending TestsFrom each panel, two specimens with face grain oriented parallel to specimen long axis andone specimen with face grain oriented perpendicular to specimen long axis were tested on the flat instatic bending to establish the short term bending strengths. The specimens were simply supported attwo ends with roller bearing plates and loaded under a center point load. A test span of 305 mm (12inches) was used such that the span to depth ratio resembled dry freight van trailer deck applications.A MTS model 810 hydraulic control close loop universal testing machine with a 222.4 kN (50000 lb)capacity was used. From each panel, one specimen with face grain oriented parallel to specimen longaxis and one specimen with face grain oriented perpendicular to specimen long axis were tested underh)^deflection control mode. An uniform rate of cross head motion of 1.37 mm (0.054 inc ^usedmin.^mm.which resulted in specimen failure between 3 to 5 minutes of loading. The other specimen with facegrain oriented parallel to long axis was tested in a load control mode. An uniform rate of loading ofkip44.44 iN (9.99 --) was used which resulted in specimen failure between 0.5 and 1.0 second of loading.The specimen thicknesses at mid-span and at two points near each edge were measured,averaged, and recorded. The specimen width at midspan was also recorded. A computer based dataacquisition system and software were used to acquire the load versus deformation (cross head motion)data. A load cell with a 25 kN (5620 lb) capacity was used to monitor the loads. The time to failureinformation was also recorded. The bending strengths based on the measured peak loads in theperpendicular and parallel to grain directions were estimated from Equations 76a and 76b, respectively.6.1.2 Cyclic Bending TestsFrom each prototype, ten of twenty panels were randomly chosen for the small specimen cyclictests. The five specimens from each panel, with face grain oriented parallel to the specimen long axis,were tested flat wise in cyclic bending to establish their fatigue characteristics at five different stresslevels. The five stress levels were chosen such that the peak stress within each cycle was approximately189100%, 90%, 85%, 80%, and 75% of the mean short term parallel to grain bending strengths of the twoprototypes which were obtained from the load controlled small specimen short term bending testresults. This range of load levels was chosen so that the number of cycles to failure, in the lowest stresslevel (75%), was not prohibitively high so that testing program could be completed within a reasonabletime frame.Since fatigue data were influenced by the magnitude and frequency of loading; i.e., the shapeof the applied stress cycles, the cyclic tests were performed using stress cycles which were expected to besimilar in shape to that experienced by the material in service. As DAP results indicated, the panel atthe critical location of the decking system (under the front wheel foot print centered between Bays 5and 6) experienced stress reversal as the wheels traverse over the prototype deck assembly; therefore, aspecial fatigue bending test jig was designed and built to allow application of cyclic loading capable ofinducing stress reversal in the specimen under simple support condition and center point loading.The bending test jig consisted of a bearing system at each simple support and a loading head.At each simple support the specimen was free to move along its long axis and to rotate about thehorizontal axis normal to its narrow face. Vertical movement of the specimen at the support wasrestricted to provide proper support when the loads were reversed. This was achieved by sandwichingthe specimen between two bearing plates and bolting the plates together. The reactions weretransmitted from the wood to the bearing plates through the bolt into the steel fame of the testingmachine. A steel spacer block, slightly thicker than the specimen, was inserted between the platesduring set up so that tightening the bolts did not damage the specimen. The loading head consisted oftwo channels in between which the specimen was sandwiched. The contact faces of the channels wererounded with a radius of curvature of 53 mm (2.06 inches). The channels were also bolted togetherwith the proper spacer block to prevent specimen damage during set up and to ensure reversibleloading. A MTS model 810 hydraulic control close loop universal testing machine with a capacity of222.4 kN (50000 lb) was used to apply the load in a load control mode. The machine was programmed190to repeat a prescribed load cycle until specimen failure. The load history of each specimen was recordedusing a sampling rate of 5 Hz. Knowing the period of each load cycle and the sampling rate, thenumber of cycles to failure was estimated.6.1.2.1 Applied Stress Levels:The computer program DAP was used to generate the parallel to face grain bending stresshistories in the both prototypes 1 and 2 at the critical location of the decking system during one cycleof wheel cart travel. The front and rear axle loadings were 73.39 kN (16500 lb) and 8.15 kN (1833 lb),respectively. First the front and rear wheels were positioned on Bay 5 and Bay 1, respectively. For eachtype of panel, the DAP yielded the maximum parallel to face grain bending stress at the criticallocation (under the front wheel foot print at x=0.731 m and centered between Bays 5 and 6). Then thefront and rear wheels were repositioned 152.4 mm (6 inches) along the long axis of the decking systemfrom original position (front axle centered between Bay 6 and 7). DAP was executed to yield themaximum parallel to face grain bending stress at the critical location for the wheel cart secondlocation. This procedure was repeated until the front and rear wheels were positioned on Bay 9 andBay 5, respectively. The parallel to grain bending stress of the panel at the critical location of thedecking system during the returning half cycle of the wheel cart travel was obtained directly from theDAP results for the first half cycle of the wheel cart travel. Since the period of wheel cart motionduring the full scale cyclic test was 18 seconds, the critical parallel to grain bending stress histories ofthe two type of panels under a front and rear axle loadings of 73.39 kN and 8.15 kN were estimated.Figure 51 shows one cycle of the parallel to grain bending stress history for each of the twoprototypes at the critical location under a front and rear axle loadings of 73.39 kN and 8.15 kN,+vT1918 0it S0-M6 00)u)'il-)C.6Cuc 40'Elca)coaal 20L00TI)05 0itiCL— Prototype 1 —I- Prototype 2-20 0 2^4^6^8^10^12^14^16^18Time (Second)Figure 51 Parallel to grain bending stress profiles for prototypes 1 and 2.192respectively. Each stress history was normalized with respect to its peak stress and yielded anondimensional profile versus time. This was achieved by dividing each stress value by the peak stresswithin the cycle. Figure 52 shows the nondimensional profiles versus time for the two prototypes.Clearly two nondimensional profiles were very similar in shape; therefore, the average of the twoprofiles was judged to represent the prototypes well.Given a target stress level, for example 85% of the mean short term parallel to grain bendingstrength of one of the prototypes, the average of the two nondimensional profiles was used to estimatethe bending stress profile for the target stress level. From beam bending theory, the load profile for asimple supported beam under center point loading for the target stress level was also estimated. Thisload profile was programmed into the MTS controller to drive the fatigue test specimens. Differenttarget stress levels were achieved by scaling the load level controls on the MTS controller. Since thiswas a load controlled experiment, the loading head would move in a particular direction until theprescribed load level was reached. Therefore, deflection and load limits was set up to keep the loadinghead from contacting the test bed after specimen failure.6.2 Experimental Results6.2.1 Short Term Bending TestsSummary statistics from both the displacement and load controlled small specimen parallel tograin bending tests are shown in Table 42. Similar summary statistics from the displacement controlledsmall specimen perpendicular to grain bending tests are also shown in Table 42. The differences inmean parallel to grain bending strengths between the load controlled and displacement controlled testswere 3.67% and 4.05% for prototypes 1 and 2, respectively. The small difference indicates that withinthis range of loading rates the estimation of mean static bending strength was not significantlyinfluenced by loading mode.1.21.0a)= 0.8O0WI 0.6C00 0.20.02^4^6^8^10^12^14^16^18193Time (Second)Figure 52 Nondimensional profiles versus time for prototypes 1 and 2.^Prototype Number of^Displacement Controlled^Load Controlled^Specimens^Bending Strength (MPa) Bending Strength (MPa)Mean STDV Mean STDVParallel to grain direction1^20^51.413 4.557 53.299 5.9702^20 66.466 7.605 69.157 8.537Perpendicular to grain direction1^20^14.633 2.9362^20 20.176 3.082194Table 42. Short term small specimen static bending test results of various prototypes.Three failure modes were observed from the bending test specimens: 1) rolling shear failure atthe cross ply (RS), 2) tension failure of the outer ply (B), and 3) both rolling shear failure at the crossply and tension failure at the outer ply (RS-}-B). In the parallel to grain direction displacementcontrolled tests, the number of specimens with failure modes RS, B, and RS-FB were 7, 7, and 6,respectively for prototype 1 and 13, 4, and 3, respectively for prototype 2. In the perpendicular to graindirection displacement controlled tests, only failure mode B was observed. In the parallel to graindirection load controlled tests, it was difficult to differentiate the various failure modes becausetypically the failed specimens were severely damaged.Figure 53 shows the displacement controlled parallel to grain bending strength cumulativeprobability distributions for Prototypes 1 and 2. The perpendicular to grain bending strengthcumulative probability distributions for Prototypes 1 and 2 are shown in Figure 54. The data clearlyshow that: 1) the bending strengths of both Prototypes were weak in the perpendicular to graindirection, 2) both Prototypes had relatively low variability, and 3) Prototype 2 had significantly.••..+• +•+±+-I-. ++. +. +.. +- +• +• ++. +. +. +,.,.•.±--I-++++. -1-. ++• + ° Prototype 1. +• + + Prototype 2• + I1.00.00.8Ez0.219540^50^60^70^80Modulus of Rupture (MPa)Figure 53 Displacement controlled parallel to grain bending strengthcumulative probability distributions.1.00.8al 0 6-CI •20U)>0.4E00.2..+•1+....+++1.. 4-1+4•.1+++....+++.+Prototype 1Prototype 2I10^20^30^400.00196Modulus of Rupture (MPa)Figure 54 Displacement controlled perpendicular to grain bending strengthcumulative probability distributions.Source^Degree of^Sum of^Mean^F Value^Probability > FFreedom^Squares^SquareDependent Variable: Parallel to Grain Bending Modulus of RuptureModel 3 4844.265657 1614.755219 34.62 0.0001Prototype 1 4738.175019 4738.175019 101.60 0.0001Mode 1 102.897074 102.897074 2.21 0.1416Prototype*Mode 1 3.193563 3.193562 0.07 0.7943Error 75 3497.683176 46.635776Total 78 8341.9848833197higher bending strengths when compared to Prototype 1. The information was consistent with resultsobtained in the veneer mechanical test program.Figure 55 shows the load controlled parallel to grain bending strength cumulative probabilitydistributions for Prototypes 1 and 2. Clearly the cumulative distributions from the load controlled testscompared closely with the distributions from the displacement controlled tests. Table 43 shows resultsof analyses of variance performed on the parallel to grain bending strength data. Results indicate thatthe difference between load and displacement controlled test data was not statistically significant at the95% probability level. However, the difference between the bending strengths of Prototypes 1 and 2 wasstatistically significant at the 95% probability level. Finally, interaction effect between the mode oftesting and the type of prototype was not statistically significant at the 95% probability level.Table 43. Analysis of variance results on parallel to grain bending strengths.1981.00.8ISE;tts-0 0.600a)co 0.4E0.20.0.,°^Prototype 1+^Prototype 2— 2-Par. WeibullLognormalI^I40^50^60^70^80^90^100Modulus of Rupture (MPa)Figure 55 Load controlled parallel to grain bending strength cumulative probability distributions.199Also shown in Figure 55 are the lognormal and two parameter Weibull distributions fitted tothe parallel to grain bending strength data following the maximum likelihood estimation approach(Lawless, 1982). Visual inspection show both distributions fit the data well. The means of thelognormal parallel to grain bending strength distribution for Prototypes 1 and 2 were 52.325 MPa and65.677 MPa, respectively. The standard deviations of the natural logarithm of parallel to grain bendingstrengths for Prototypes 1 and 2 were 0.0875 and 0.1062, respectively. The shape parameters of thetwo parameter Weibull distribution were 11.050 and 9.411 for Prototypes 1 and 2, respectively. Thescale parameters of the two parameter Weibull distribution were 55.853 MPa and 72.825 MPa forPrototypes 1 and 2, respectively.6.2.2 Cyclic Bending TestsFigure 56 shows a small specimen under load during a cyclic bending test. Results from thecyclic test program are summarized in Table 44. Within each group, the variability in the number ofcycles to failure (Nf) was found to be large. The large variability was consistent with fatigueperformance of plywood (Kommers, 1943 and 1944). The peak loads in Groups A to E corresponded tobending stress levels of approximately 80%, 90%, 85%, 75% and 100%, respectively, of the meanbending strengths obtained from the load controlled bending tests. Figure 57 shows a typical rollingshear failure of the cross ply which was the predominant failure mode at the low peak load levels. Atthe high peak load levels, more tension failures at the outer ply were found.The peak loads were normalized with the mean load controlled bending strengths to computethe bending stress level (S R) for each group. Figure 58 shows a plot of the SR versus Nf (in log10 scale)relationships for the two prototypes. Here, the range of stress ratios was limited to accommodate areasonable testing time. Direct regression analysis with such a narrow data range might not beappropriate. Instead the mean N f (in log10 scale) and the mean stress level (Sr) in each group were usedto establish the relationship between stress level and mean fatigue life for each prototype.200Figure 56 Small Specimen during a cyclic bending test.201Table 44. Small specimen cyclic bending test results.Prototype Number of Group^Peak Load Mean Number of Cycles Number of Specimensi^Specimens^Mean^STDV Stress^to Failure^with Failure Mode(kN)^^(kN) Level Mean STDV RS B RS-I-BSr1 10 A 11.253 0.022^0.779 292.9 289.6 8 0 210 B 12.559 0.022^0.868 55.1 60.4 10 0 010 C 11.936 0.006^0.826 78.2 58.0 7 1 210 D 10.707 0.017^0.741 710.7 426.2 10 0 010 E 13.936 0.094^0.965 24.3 33.1 4 3 32 10 A 9.083 0.047^0.757 470.8 332.3 8 0 210 B 10.291 0.044^0.858 85.9 129.1 7 0 310 C 9.577 0.025^0.798 355.5 355.4 10 0 010 D 8.574 0.027^0.715 1775.8 1652.0 8 2 010 E 11.194 0.075^0.933 72.6 86.7 5 0 5202Figure 57 Typical rolling shear failure mode of a specimen under cyclic loading.1.00.904c-ti 0.8CC0.6.• •.\I..,.,,&,.•AA,..,..1Al, AIAL_Ac.n,,ct.'\*A-L\°^Small Spec. Proto.A^Small Spec. Proto._*^Mean Res. Proto. 10^Mean Res. Proto. 2^ Reg. Res. with Means1^1^1111111^1^1^[WM10 100^1000^10000Number of Cycles to Failure0.51203Figure 58 Fatigue data of small specimens in bending.204The relationship between S R, and Nf for the ith prototype is given by:SRi = Ii — Si logio (N1)^[110]where I i and 12 equal 1.128 and 1.161 respectively, S 1 and S 2 equal -0.14106 and -0.14207 respectively.The r2 values for Prototypes 1 and 2 were 0.912 and 0.900, respectively. The standard error of estimatevalues for Prototypes 1 and 2 were 0.030 and 0.031, respectively. Regression approach to analysis ofcovariance was also performed and partial F tests show:1) the slopes S 1 and S2 were not significantly different at 95% probability level;2) the effect of prototypes was not significantly different at 95% probability level;3) the effect of covariate was significant at 95% probability level which means the use of analysis ofcovariance was justified.The results suggest that combining the fatigue performance of the small specimen of the twoprototypes into a single relationship was justified.Figure 59 compares the normalized small specimen fatigue data to the normalized fatigueperformance of full size TransDeckTM panels. Clearly the small specimens sustained a significantlyhigher stress level for a given N 1 in comparison with the full size panels. This phenomenon could beexplained by a size effect analysis. The full size test results were obtained by cycling a load over a testassembly which contained two sides with eight 304.8 mm (12 inches) wide bays. The span in the smallspecimen cyclic tests was only 304.8 mm (12 inches) long; therefore, the stressed volume in the full sizepanels of the test assembly was approximately 16 times that of the small specimen.Using the Weibull's weakest link theory, the bending strength ratio between the smallspecimens and the full size specimens can be estimated as:1Ssmall = 16 kSfull size205*IIN r;<1XEXC.a11,^ ^Pred. Full Scale Full Scale Results—^*^Small Spec. Prot. 1^^^Small Spec. Prot. 2X^Size Adjusted DataI^1^1^1^11111^I^I^I^11111110 100^1000^10000Number of Cycles to FailureFigure 59 Comparisons of fatigue performance of small specimens and full size panels.1 .00.90.80cr2 0.70.60.50.41206where k is the shape parameter, and S small and Sfull size are the bending strengths of the small and fullsize specimens, respectively. From Figure 55 the small specimens bending strength data were fittedsatisfactorily by a 2 parameter Weibull distribution with shape parameters of 11.050 and 9.411 forPrototypes 1 and 2, respectively. Therefore, using Equation 111, the parallel to grain bending strengthsof the small specimens were adjusted to those of the full size panels. New stress ratios relating to thefull size panels could therefore be established by normalizing the peak loads in the fatigue data with themean size adjusted bending strengths. The relationship between the size adjusted S R, and Nf for the ithprototype is given by:SRi = h - Si log io (Nf)^ [112]where I I and 1 2 equal 0.878 and 0.865 respectively, S i and S 2 equal -0.10976 and -0.10581 respectively.The r2 values for Prototypes 1 and 2 were 0.912 and 0.900, respectively. The standard error of estimatevalues for Prototypes 1 and 2 were 0.023 and 0.023, respectively.Analysis of covariance was performed on the two prototypes of size adjusted SR and Nf (inlog io scale) data. The partial F test results show that the slopes of the relationship between SR and Nf(in log io scale) of the two prototypes were statistically equal at the 95% probability level. Also theeffect of prototype was not significant but the effect of covariate was significant at the 95% probabilitylevel. Therefore, the size adjusted relationships between SR and Nf (in logio scale) for both prototypeswere combined. The relationship between SR and Nf projected for full size panels is given by:SR = 0.874 — 0.10908 log 10 (Nf)^ [113]where the r 2 value was 0.917, the standard error of estimate was 0.020, and the degree of freedom waseight.207Finally the size adjusted SR and Nf (in log10 scale) data from the small specimen fatigue testswere combined with the SR and Nf (in log10 scale) data of prototypes 1 and 2 from the full sizespecimen fatigue tests in regression approach to analysis of covariance. The two treatment groups were1) veneer thickness effect and 2) size adjustment procedure effect. The partial F test results show thatthe slopes of the relationship between SR and Nf (in log10 scale) between the size adjusted and the fullsize treatment groups was statistically equal at the 95% probability level. Also both the interactioneffect between treatment groups and the effects of either treatment groups were not significant at 95%probability level. Finally the effect of covariate was significant at 95% probability level.Using the full size relationship from Equation 107, the applied stress ratios corresponding to I2cycle and 3000 cycles to failure were approximately 1.00 and 0.47, respectively. Using the size adjustedrelationship from Equation 113, the applied stress ratios corresponding to 1 cycle and 3000 cycles to2failure were approximately 0.91 and 0.50, respectively. The reasonable agreement indicates the sizeadjustment procedure explains the differences in the fatigue information on full size panels and smallspecimens.2087. MODELING OF SMALL SPECIMEN FATIGUE PROPERTIESThe fatigue behavior of the small specimens in bending has been shown to be related to thefatigue behavior of the full size specimens through size effect adjustment procedure. It is desirable toidentify and calibrate a damage accumulation model so that the relationship between load levels andnumber of cycles to failure can be predicted for small specimen in bending.Based on the literature survey, the damage accumulation model developed by Foschi and Yao(1986a and b) was considered most appropriate for this investigation. The damage accumulation modeltakes the form:dadt = a NO — cors}b + c NO — crars}na(t)^[114]where a, b, c, n, and c o are random model parameters which are constants for a given member butvary from member to member. a is the damage where a=0 means no damage and a > 1 means failure.7(0 is the stress history experienced by the specimen. r s is the short term strength of the specimensobtained from load controlled tests. g o is the threshold stress ratio where damage accumulates only ifr(t) > uors .Let f1 = {7(0 — cars}b and f2 = {T(t) — crorsr, Equation 114 can be rewritten as follows:dadt e(- f c f2 dt)= {a fi + c f2 a(t)} e(- f c f2 dt)Aidt a e(- f c f2 dt)} = a fl e(- f c f2 dt)t[115][116]Integrating Equation 116 yields:Ta e(- f c f2 dt) i = f _ (- f c f2 dt) dt0I = j a fl e0Given a stress history r(t), Equation 117 can be evaluated by performing the integration over theintervals when r(t) — curs > 0 to estimate the damage in the specimen if the model parameters areknown.7.1 Ramp Load CaseConsider now a ramp load test where the load increases at a constant rate K s . Assuming thestress also increases at the same rate, the stress history can be expressed as r(t)= K st. This is the typeof loading used to determine the load controlled parallel to grain bending strength of the smallspecimens. Since damage only occurs when r(t) — c ons > 0, let us define a time t o at which r(t o) =crors . Substituting the stress history from the ramp load tests into Equation 117 and integrating fromt=to to t=T yields:{Ks(n-cF1) [Kst (7ors](n+1)IT^T I — c [K^cr 7t —^](11+1)}a etI = i a [Kst — aors]b e j(s(n+1) s 0 s dt^[118]o^toLet us define the time to failure of a specimen during a short term ramp test as T s. At t=Ts , a=1 andKsTs = rs also at t=to , a=0 and Ksto = co ; therefore, Equation 118 becomes:e{ICs(n_Fc 1 ) [75 croi- n+1)}^Ts— c=^a [Ks t — crorsib e{K s (n+1 ) [Kst — crol- j(n+1)}ftodt^[119]Foschi and Yao (1986a and b) showed that Ks is typically large compared to the model parameter c;209[117]therefore, Equation 119 yields an approximate relation between the model parameters a, b, co and theramp rate Ks as:Ks (b+1)a a-. ^(r8 — cor5)(b+1)This relationship suggests that the model parameter a cannot be chosen independently from the othermodel parameters b, co , the ramp rate Ks and the short term strength Ts .7.2 Piecewise Linear Representation of Stress HistoryNow consider one cycle of the nondimensional stress history experienced by the TransDeckTMsmall specimens during the fatigue tests shown in Figure 52. This stress history can be defined ascomposed of piecewise linear segments with equal intervals of 0.5625 second. The stress history in theith interval is given by:T(t) =[^Si(t — tsi — tj)] Tmax^ tsi < t < tei and (i=1,.., k)^[121]where Tmax the peak applied stress within a cycle, ti is the starting time of the j th cycle, and t si andtei are the starting and ending time of the i th segment, respectively. Table 45 shows the parameters /iand Si for the ith segment during the first cycle.Let us now evaluate the accumulated damage of a specimen after the first piecewise linearsegment in the first stress cycle where tj =0 (j=1). Define tot as the time when 17- (t01 )I just exceedscrors and note that S1 >0, the damage can be expressed as in terms of the model parameters as:^f tel^tela e 3 = f fa Pi + Si (t — tsi)] Tmax — Cord' e 3 dt^ (i=1)^ [122]^tot^t210[120j211Table 45. The parameters describing piecewise linear segments of a nondimensional stress cycle.SegmentiInterceptIiSlopeSiSegmentiInterceptIiSlopeSi1 0.260639 1.314419 17 0.034787 -0.004512 1.766239 -1.3622 18 0.03042 -0.004053 0.599725 -0.3253 19 0.0129 -0.002324 0.329665 -0.16526 20 0.005047 -0.001595 0.104534 -0.0652 21 -0.06938 0.0050256 -0.02553 -0.01896 22 -0.15386 0.0121777 -0.16293 0.021748 23 -1.19411 0.0962378 -0.38812 0.078941 24 1.979518 -0.149069 -0.70369 0.149067 25 1.032817 -0.0789410 0.538161 -0.09623 26 0.228536 -0.0217411 0.065321 -0.01217 27 -0.36684 0.01896112 0.021069 -0.00502 28 -1.06921 0.06520813 -0.02359 0.001591 29 -2.64513 0.16526614 -0.02896 0.002325 30 -5.25571 0.32530215 -0.04259 0.004056 31 -22.7534 1.36220316 -0.04644 0.004512 32 23.92018 -1.31441212wheref = c^^([/.^Si lt — t^— Cfors)(n+1)}3^Si(n+1)7-ina.tsi)] TmaxSince a(toi ) = 0, the damage at the end of the first segment is:to//lI a [1; Si (t — tsi)] Tmax — cors] b ef3 dto(tei)=tO1 er An+1)TmaxC^([1i + Si(tei tsi)] Tmax Cr O TSPI+1)}(i=1) [123]Consider now the second segment in the first cycle and define t 02 as the time when 17(t02 )1 justexceeds crc,rs and note that S2 <0, the damage accumulated during this segment can be expressed as interms for the model parameters as:f toe^toe/ae 3 I = a [J^Si (t — tsi)] Tmax — crors] b ef3 dt^ (1=2)^ [124]ts2 "s2where 17(t s2 )1> o-ors and t02 te2 .Since t s2=te1 , the total damage accumulated at the end of the second segment of the first cycle is:to/2I a^Si(t — ts1)] Tmax CrOrslb ef3 dt^a(tei ) ef4a(te2)= ts2fsi(n+l^ ([1i + Si( t02 tsi)] Tmax CrO TS) (11+11e^)rmax(i=2) [125]213wheref4 = Si(n+1 c^([/. + Si ( tei — %O]l-max c0rs)(n+1)}) 71/1ax^1Repeating the above procedure for all the segments within the first cycle, the damage accumulatedduring the first cycle a l can be evaluated.Let the damage after I th cycle be expressed in form of a recurrence relationship as :aI =^Ko(I) + K1(I)^[126]Since each stress cycle is identical in shape, K oW=Ko (J)=Ko and K 1 (I)=K 1 (J)=K 1 where I J.Knowing ao=0, the two unknown coefficients K o and K1 can be evaluated by considering theaccumulated damage after the first two stress cycles; i.e, K1 = a1 and Ko =^— 1). With K o andKiK1 known, the recurrence relationship of Equation 126 can be evaluated to find the accumulateddamage after the I+N cycle as:ceI+N = a1-1 KoN K1 (K0N-1 K0N-2+ + K0 + 1)when 1=1, a0=0; therefore,a11-N = K 1 (KON-1 + KoN-2+^KO + 1 )which can be rewritten as:1 — KoN\= K1 K^ )0[127][128][129]214Let failure occur at N+1 cycle, where a 1+N=1. Therefore, the number of cycles to failure N f can beestimated from Equation 129 as:=^log (Ki Ko —^K1^)log(Ko)^+ 1 [130]Treating the stress history as composed of piecewise linear segments requires numericalintegration for each segment when 17(01— a ors > 0. If the model parameters of a specimen are known,the number of cycles to failure can be estimated. However, if the model parameters are unknown, it isnecessary to calibrate the damage model to fatigue data using nonlinear function minimizationprocedures. This may be difficult when the stress history is assumed to be composed of piecewise linearsegments because Equations 123 or 125 are very sensitive to the choice of model parameters. Duringthe search for solution with a nonlinear function minimization procedure the parameters may fluctuatesubstantially which may lead to unstable solutions unless initial choice of model parameters are close tothe solution. Therefore it is inappropriate to consider the stress history as composed of piecewise linearsegments when model parameters are required to be calibrated to fatigue data. Alternately the stresshistory can be considered to be composed of a series of stress pulses.7.3 Representation of Stress History by a Series of Stress PulsesConsider several piecewise linear segments of a stress cycle shown in Figure 60. A piecewiselinear segment can be subdivided into m sections each with a constant stress pulse of magnitude Ti(1=1," m) and duration At. The magnitude of the stress pulse Ti (1=1,..,M) is taken as the averagestress within the i th section and the duration At is taken as 0.5625 seconds.The accumulated damage sustained by a specimen during the i th load pulse, from t = t i_ 1 to tiwhen irii > aors, can be expressed as in terms of the model parameters as:31^2Time (Second)1.21.0O0.80CL(71 0.60Z 0.20.0-0.20tfififiTTfi215Figure 60 Representation of stress cycle with stress pulses.216t ia ef6 t tli = f a[7-1 — o-ors]b e16 t dtwhere1.6 = — c [71 — 6ors]nEvaluation of Equation 131 yields:ai „i_i e{ — c [71 — (Tors] llAt} + c[Ti _ crors](b — n) (e { — c {71 — °-ors]n At}Equation 132 can be rewritten as:ai = ai_ 1 k0 (i) -Fk i (i)^ if hi > crorswhere-2=ko (i) = el — c [Ti — cors]nAt) and k i (i) = [Ti — trors] (b —n) (ko (i) — 1) when Ir i l > Cr 0 7 sandko (i) = 1 and k 1 (1) = 0 when Jri < o -orsFor one stress cycle with k piecewise linear segments, the accumulated damage can be evaluated as:a1 =00a2 = al k0 (2) +k 1 (2)[134]am*k = Cxm*k_ 1 k0 (m*k) + k 1 (m*k)^m*k^m*k-1^m*k^= a1117 ^+ > k1(p) 1 1 ko(j) + k 1 (m*k)j=2^p=2^j=p+1t • _.1 i^t i_ i[131][132][133]217Let the subscript I denotes the stress cycle, the damage after I th cycle is given by:^aI = aI-1 KO + K 1^ [135]wherem*kKo= II ko(j)j=2andm*k-1^m*kK 1= E k1(p)^ k (j) k 1 (m*k)p=2^j=p-1-1From Equation 130, the number of cycles to failure of a specimen can be estimated if the modelparameters are known.7.4 Model CalibrationThe damage model was calibrated against the test data following the procedures presented byFoschi, Folz, and Yao, (1987). As shown in Figure 55, the lognormal distribution fits the short termstrength data rs well; therefore, T s was assumed lognormally distributed. Each of the four independentmodel parameters (b, c, n, a n) were also assumed to be lognormally distributed. A nonlinear functionminimization procedure using the quasi-Newton method was employed to estimate the mean andstandard deviation of each independent lognormal distribution of the model parameters. Therefore, anvector X with eight unknowns corresponding to the mean and the standard deviation of the fourindependent parameters, XT={b, c , n , co, STDb , STDc, STDn, STD0.0 }, was estimated by theminimization procedure. The procedure is summarized as follows:1) Initial estimates of the mean and standard deviation of each independent lognormal distribution ofthe model parameters were provided;2) Based on the initial distribution parameters, a random sample of 1000 number of cycles to failure218(N?) was generated using Equations 130 and 135;3) The cumulative probability distribution of the randomly generated sample of number of cycles tofailure was obtained and compared to the experimental data by computing a objective function 4,as:= 2 -i=1Nf si.) 2Nf ,[136]where L is the number of probability levels considered, N f s is the simulated number of cycles tofailure at the i th probability level, and N1 a is the actual number of cycles to failure at the sameprobability level obtained from the experimental results;4) The gradient of the objective function with respect to unknown vector X, V 4,={^°II ITOX1 ' .. •'0X8J 'was required by the minimization procedure and was numerically computed. First each unknowndistribution^ d theobjective function was reevaluated by repeating step 2 and 3 using the same sequence of randomnumbers as 4,+. Then each unknown distribution parameter X. (i=1,..,8) was perturbed in turn bya negative increment -AX and the objective function is reevaluated by repeating step 2 and 3using the same sequence of random numbers as^In this study, the increment AXi was chosen as0.001X.. The partial derivative of 4, with respect to the unknown distribution parameters X.(i=1,..,8) is given as:00 —OXi = 20X (i=1,..,8)^ [137]5) Equation 136 was minimized following the quasi-Newton method in which the initial choices ofdistribution parameters were modified automatically through an iteration procedure. An optimalsolution was considered obtained when changes in X i of magnitude eAXi did not reduce theobjective function value for i= 1,..,8 where the convergence criterion of e=0.001 was used.219The relatively small convergence target, step size and the reasonably large sample size werechosen to avoid wide fluctuations in the estimated model parameters when a new set of random seedswere considered.7.4.1 Calibration ResultsFigures 61 and 62 show the cumulative distributions of Nf (in log io scale) of the smallspecimens under bending at five stress levels for prototypes 1 and 2, respectively. It can be observedthat the distributions were highly skewed. Although some weak specimens failed during uploading,failure was still considered to have occurred at the first cycle. Note also that each load level had alimited sample size of 10 and the variability in the number of cycles to failure was large; therefore,model calibration with these two families of five cumulative distributions proved to be rather difficult.When all five load levels were considered during model calibration, regardless of the initialmodel parameters, it seems that the calibrated results fitted the cumulative distributions with thehigher mean bending stress level well but failed to provide good fit to the cumulative distributions withthe lower mean bending stress level. The objective function in Equation 136 evaluated the square of therelative difference between the actual and the simulated N f, which weighted the cumulativedistributions with the lower N1 in favor of the cumulative distributions with higher Nf. If the objectivefunction in Equation 136 was formulated to evaluate the square of the absolute difference between theactual and the simulated Nf, the cumulative distributions with the higher N f would be weighted infavor over the the cumulative distributions with lower N f.Rather than modifying the objective function, it was decided to calibrate the model only to thecumulative distribution with the lowest mean bending stress level in each prototype (Sr=0.741 andSr=0.715 for prototypes 1 and 2, respectively). For each prototype, the calibrated model was then usedto generate simulated Nf for the five stress levels. The model was verified by comparing modelpredictions and experimental data at the various stress levels.1.0220XAD0.8X0XLI0.010.210^100^1000^10000 100000Number of Cycles to Failureo Sr • 0.741• Sr • 0.779* Sr • 0.826^ Sr • 0.869X Sr • 0.965I^Il11111^1^11111LI AFigure 61 Cumulative probability distribution of Nf for Prototype 1.0.8>14••••cts0.61"a.a)1.15 0.4E0.010.21.0D<X1*0 XAX .0 X A Sr - 0.715A^Sr • 0.757X A *^Sr • 0.798^^Sr - 0.858>,G1( X^Sr • 0.9331^I I 11111^111 ^1 1110^100^1000^10000 100000Number of Cycles to Failure221Figure 62 Cumulative probability distribution of N f for Prototype 2.222Table 46 shows the mean and standard deviation of the model parameters for prototypes 1 and2. Also shown in Table 46 are the minimum (I) from each prototype obtained from the minimizationprocedure.Figures 63 and 64 show the cumulative distributions of the simulated and actual N f (in log ioscale) of the small specimens under bending at five stress levels for prototypes 1 and 2, respectively.The matching between model predicted and actual number of cycles to failure was reasonableconsidering the limited sample size and the large variability in the number of cycles to failure.Table 46. The mean and standard deviation of the model parameters for prototypes 1 and 2.Parameter Mean STDVPrototype 1: Minimum (1, = 2.02488b 10.9536 2.58713 x 10-1c 1.25549 x 10-6 1.81542 x 10 -8n 5.57889 x 10 -1 2.27 x 10-2°-0 9.47904 x 10-2 6.29348 x 10-2Prototype 2: Minimum 0 = 1.92129b 11.3832 4.30419 x 10 -1c 6.445890 x 10 -7 1.84566 x 10 -8n 4.34451 x 10 -1 2.01685 x 10 -2(70 8.47986 x 10-2 2.61574 x 10-2It can be observed that the both the simulated and the actual cumulative probabilitydistributions were highly skewed. The fatigue performance of small specimen shown in Figure 59 wasobtained from the mean number of cycles to failure test data at various mean stress levels. Since each°^Proto. 1 Data• •Damage Model•10^100^1000^10000 100000Number of Cycles to Failure1.00.80.20.0120_-CI 0 6a)•_c"5 0.4E223Figure 63 Comparisons of model predicted and actual cumulative probabilitydistribution of Nf for Prototype 1.1.00.8a)E0.210^100^1000^10000^100000Number of Cycles to Failure0.0224Figure 64 Comparisons of model predicted and actual cumulative probabilitydistribution of Nf for Prototype 2.225load level contained only 10 data points, the mean and standard deviation of the number of cycles tofailure reported in Table 43 were the average and standard deviation of the test data from theprobability levels of approximately 0.07 to 0.93 assuming the rank of the ith data point from a samplesize of m was given by mi +0.4.— " The mean and standard deviation of the model predicted number ofcycles to failure were also obtained from the probability levels of 0.07 to 0.93 to avoid the influencefrom the skewness in the cumulative distributions when making comparisons between model predictionsand actual fatigue performance. Table 47 shows the model predicted fatigue performance of the twoprototypes.Table 47. Model predicted small specimen fatigue performance.PrototypeiStress LevelSrNfMean STDV120.7790.8680.8260.7410.9650.7570.8580.7980.7150.933414.28102.98195.06787.4428.84609.00120.30305.221299.2627.43473.94114.54219.53915.9530.95787.49151.53391.091690.4433.00The relationship between the mean stress level Sr i and the model predicted Nf for the ithprototype is given by:Sri = fis — sis login (N1)where Its and 12S equal 1.187 and 1.126 respectively, S i ' and S ts equal -0.15574 and -0.13187respectively. The r 2 values of the model predicted fatigue performance for Prototypes 1 and 2 were0.997 and 0.998, respectively. The standard error of estimate values of the model predicted fatigueperformance for Prototypes 1 and 2 were 0.005 and 0.004, respectively.Figure 65 compares the model predicted and actual fatigue performance of small specimenunder cyclic loading. Clearly the model predicted and actual fatigue performance of small specimenunder cyclic loading agreed reasonably well.The size adjustment procedure given in Equation 111 was used to convert the short termstrength (Ts) of small specimens in the damage model to that of the full size panels. In each prototype,five stress levels (0.9, 0.8, 0.7, 0.6 and 0.5) were chosen. For each stress level, the number of cycles tofailure of 1000 replicates were randomly simulated. The model predicted mean number of cycles tofailure were obtained from the probability levels of 0.07 to 0.93 to avoid the influence from theskewness in the cumulative distributions when making comparisons between model predictions andactual fatigue performance. Table 48 shows the model predicted fatigue performance of the twoprototypes.The relationship between the stress ratios SR d and 1\11 from the size adjusted damage model forithe ith prototype is given by:SR di = Ii — S i log io (Nf) [139]where I i , 1 2 , S i and S 2 equal 0.942, 0.908, -0.12794 and -0.13055, respectively. The r 2 values forPrototypes 1 and 2 were 0.983 and 0.975, respectively. The standard error of estimate values forPrototypes 1 and 2 were 0.024 and 0.029, respectively.226[138]Prototype^Stress Leveli^dSR iMean Nf1^0.9^3.3393^0.8 10.8580.7^52.6170.6 367.970.5^3974.32^0.9^2.01680.8 5.41560.7^24.9200.6 175.390.5^1924.8227Table 48. Model predicted full scale panel fatigue performance.Finally, the size adjusted relationships between SRd and Nf (in log io scale) from the damagemodel for both prototypes were combined to yield the fatigue performance of full size panels as:SRd =0.921— 0.12684 logio (Nf)— 0 . [140]where the r 2 value was 0.961, the standard error of estimate was 0.031, and the degree of freedom waseight.Figure 66 compares the damage model predicted fatigue performance of full size panels withthe actual fatigue performance of full size prototype 1 and 2 panels. Again the model predictions andthe full size test results under cyclic loading agreed reasonably well.0.510.90.61.0o Prot. 1 Test DataO Prot. 2 Test Data* Prot. 1 Damage Model• Prot. 2 Damage Model^ Reg. Results - DataReg. Results - ModelI^111111II^I^I^I ^1 100^1000Number of Cycles to Failure228O-1-' 0.8ccci)C/)(7) 0.710000Figure 65 Comparisons of model predicted and actual fatigue performance of small specimens.> <x^x^ '\>t+C+> ^^ Pred. Full ScaleFull Scale Results* Small Spec. Prot. 1III Small Spec. Prot. 2X Size Adjusted DataI^1^1^111111^I^I^1^)11111CI2291.00.90.80•_16CCcio) 0.70.60.50.41 10 100^1000^10000Number of Cycles to FailureFigure 66 Comparisons of model predicted and actual fatigue performance of full size panels.230Using the full size relationship from Equation 107, the applied stress ratios corresponding to 12cycle and 3000 cycles to failure were approximately 1.00 and 0.47, respectively. Using the size adjustedrelationship from Equation 113, the applied stress ratios corresponding to 1 cycle and 3000 cycles to2failure were approximately 0.91 and 0.50, respectively. Using the damage model and size adjustedrelationship from Equation 140, the applied stress ratios corresponding to 1 cycle and 3000 cycles to2failure were approximately 0.96 and 0.48, respectively. The reasonable agreement indicates the damagemodel and size adjustment procedure used to obtain fatigue information on full size panels from smallspecimen fatigue results was valid. It should be noted that damage model and size adjustmentprocedure may be generally applied to other material; however, the calibrated damage modelparameters are restricted only to transDeckTM panels.2318. CONCLUSIONS8.1 Summary and ConclusionsThe formulation of a structural analysis model (DAP) based on finite strip method waspresented to predict the structural behavior of prototype dry freight van trailer decking systems. Theassumed displacement field in DAP took into consideration of the degrees of freedom to model theunsupported edges and the mid-span gap in the prototype dry freight van trailer deck assembly. DAPalso considered the unequal elastic properties for the two sides of the cover in the deck system whichcorresponded to the use of two individual panels (one on each side of the mid-span gap) in the deckassembly.DAP treats the decking as a cover plate stiffened by supporting beams. The cover plate can bemade up of any material as long as it behaves linearly within the load range of interest. Materialsincluding composite wood products (e.g., TransDeckTM, parallel stranded lumber, laminated veneerlumber), fiber glass products, metal products, or composite wood and fiber glass products can beconsidered if their elastic properties are known. In this study TransDeck TM panels were considered toprovide the necessary database for program verification. Given as inputs the elastic properties of coverand the applied wheel load, DAP predicted the deformations in the system, the bending stresses in thesupporting I-beams, the parallel and perpendicular to grain bending stresses in the extreme fiber of thepanel, and the rolling shear stresses at the interior fiber of the panel.A comprehensive database on the mechanical properties of 3.2 and 2.5 mm ( lg and T-1-0 inch)thick Douglas-fir veneers was developed. The database included information on bending, tension, andcompression strength properties, shear moduli of rigidity, ultrasonic transmission time, and connectionstiffness. Analyses of variance indicated that the mean strength properties of 3.2 and 2.5 mm thickveneers were significantly different for the parallel to grain direction but not significantly different forthe perpendicular to grain direction at the 95% probability level. Statistical information and232distributions parameters was established for the various veneer strength properties so that simulationscould be performed to model the strength properties of the veneers.A trailer decking load simulator test facility was developed to perform full scale testing ofprototype dry freight van trailer decking systems. A static and a cyclic test program were performed onfour prototype decking systems using TransDeckTM panels. In the static test program, load versusdeformation relationships of the various prototypes were obtained. Results agreed well with DAPpredictions. A maximum peak deformation prediction error of 12% was observed. In the cyclic testprogram, the fatigue performance of the four prototype decking systems were established. The staticstrengths of the prototypes were also projected from the data. A maximum error of 10% was foundwhen comparing static strengths predicted by DAP with those projected from test data. It wasconcluded that 1) DAP was successfully verified by the full scale test program and 2) the firstgeneration TransDeckTM did not meet the structural requirements for use in typical dry freight vantrailer decking systems where a fatigue life of 3000 cycles under a front axle loading of 73 kN (16500lb) was required.The fatigue performance of full scale panels was normalized with respect to the static capacityand expressed in terms of stress ratio. The results indicate the applied stress ratio corresponding to3000 cycles to failure was approximately 0.47 and 0.56 for the regular and special veneer panels. DAPcould be used to estimate the mean short term static capacity of a second generation TransDeckTM .The fatigue performance of the new product at 3000 cycles of loading could be estimated using anapproximate stress ratio of 0.5.A companion small specimen testing program was conducted to establish their fatigueperformance in bending mode with the appropriate stress history. Failure modes of the small specimensand the full size panels agreed well. The fatigue performance of small specimens were adjusted to fullsize panels through a size adjustment procedure using the Weibull weakest link theory. The sizeadjusted small specimen fatigue data and the full size fatigue data agreed well.233A damage model which took into consideration the stress history was calibrated to the smallspecimen fatigue results. This damage model parameters are specific to the TransDeckTM although themethod can be generalized to other decking material. Good agreement between damage modelpredictions and small specimen fatigue performance was obtained. Using Weibull weakest link theory,the damage model predicted small specimen fatigue performance was converted to full size panelfatigue performance. The size adjusted model predictions and the full size fatigue data agreed well.8.2 Future ResearchThere is an obvious need to develop a product which can meet the structural requirements foruse in typical dry freight van trailer decking systems where a fatigue life of 3000 cycles under a frontaxle loading of 73 kN (16500 lb) is required. Based on the theoretical framework developed in thisstudy, guidance can be provided to arrive at a product with increased capacity. For example, consider asecond generation laminated veneer wood product, it is possible that 15 to 16 ply panels made with 2.5mm thick veneers are needed. Fiber glass reinforcements in both the parallel and perpendicular to facegrain directions may also be required. If sufficient fiber glass reinforcement in the perpendicular tograin direction is available, the cross ply should be eliminated to avoid the predominant rolling shearfailure mode at the cross ply of the first generation product. Such a new product should be also testedusing both full scale and small specimens so that proper technical information is available for productmarketing. If the new product can be proven commercially viable, simulation studies and reliabilitystudies should also be performed to better define the product performance.The static and fatigue performance of currently acceptable decking system using hardwoodsshould be evaluated so that a baseline performance level can be established. Since the hardwood decksare not moisture protected, the effect of moisture on the static and fatigue performance should beexamined.DAP has been shown to accurately predict the response of dry freight van trailer prototype234decking system. Decking systems for flat bed trailers and containers have a different configurationswhere the supporting I-beams are in turn supported at approximately the third points. The simplesupport conditions assumed for dry freight van trailer decking systems are no long valid. Major work isrequired to formulate a new structural analysis program to address the decking systems for flat bedtrailers and containers. Such a program may involve the B-spline compound strip analysis.2359. REFERENCESAmerican Society for Testing and Materials (ASTM). 1990. Annual Book of ASTM Standards. Volume04.09 Wood. ASTM. Philadelphia, PA.Booth, L.G., and Hettiarachchi, M.T.P. 1990. Predicting the bending strength of structural plywood.Journal of the Institute of Wood Science. 12(1):48-58.Chen, C.J., Gutkowski, R.M., and Puckett, J.A. 1990. B-spline compound strip analysis of stiffenedplates under transverse loading. Computer and Structures. 34(2):337-347.Foschi, R.O. 1982. Structural analysis of wood floor systems. Journal of the Structural Division. ASCE.108(7):1557-1574.Foschi, R.O. and Barrett J.D. 1982. Load duration effect in western hemlock lumber. Journal ofStructural Division. ASCE. 108(7):1494-1510.Foschi, R.O. Folz, B. and Yao, F.Z. 1989. Reliability-based design of wood structures. StructuralResearch Series, Report No. 34. Department of Civil Engineering, University of British Columbia,Vancouver, Canada.Foschi, R.O. and Yao, Z.C. 1986a. Duration of load effect and reliability based design (single member).In Proceedings of IUFRO Wood Engineering Group meeting, Florence, Italy.Foschi, R.O. and Yao, Z.C. 1986b. Another load at three duration of load models. In Proceedings ofIUFRO Wood Engineering Group meeting, Florence, Italy.236Fouquet, R.J-M. 1991. Flooring panels for flat platform trailers. U.S. Patent Number 5041322.Gerhards, C.C. and Link C.L. 1986. Effect of loading rate on bending strength of Douglas-fir 2 by 4's.Forest Products Journal. 36(2):63-66.Karacabeyli, E. 1987. Duration of load - lumber. Report Prepared for the Canadian Wood CouncilLumber Properties Steering Committee. Forintek Canada Corp.Kingston, R.S.T. 1947. The variation of tensile strength and modulus of elasticity of Hoop Pine veneerwith the direction of the grain. Journal of Council for Scientific and Industrial Research. 20(3):338-345.Kommers, W.J. 1943. The fatigue behavior of wood and plywood subjected to repeated and reversedbending stresses. U.S. Forest Products Laboratory Report 1327. Madison, Wis.Kommers, W.J. 1944. The fatigue behavior of Douglas-fir and Sitka spruce subjected to reversedstresses superimposed on steady stresses. U.S. Forest Products Laboratory Report 1327-A. Madison,Wis.Lam, F. and Varoklu, E. 1988. Load sharing factors for truss systems. Discussion paper prepared forCSA-086 Sawn Lumber Sub-Committee. Forintek Canada Corp. Vancouver, B.C. Canada.Lawless, J.F. 1982. Statistical Models and Methods for Lifetime Data. John Wiley and Sons, N.Y. pp.522-527.237Logan, J.D. 1987. Continuous ultrasonic veneer testing: sorting veneer for structural applications.Paper presented at the Sawmill Clinic. Portland, Oregon.Nielsen, L.F. 1978. Crack failure of dead-, ramp-, and combined-loaded viscoelastic materials. InProceedings of First International Conference on Wood Fracture. Banff, Alberta, Canada. pp. 187-200.Nielsen, L.F. 1985. Wood as a cracked viscoelastic material, part I: theory and application; part II:sensitivity and justification of a theory. In Proceedings of International Workshop on Duration of Loadin Lumber and Wood Products, Richmond, B.C., Canada. pp. 67-89.Nielsen, L.F. 1990. Lifetime and fatigue of wood and other building materials subjected to static andrepeated loads. In Proceedings of IUFRO Timber Engineering Group Conference. Saint John, NewBrunswick, Canada.McGowan, W.M. 1974. Effect of prescribed defects on tensile properties of Douglas fir plywood strips.Forest Products Journal. 24(5):39-44.McNatt, J.D. 1970. Design stresses for hardboard-effect of rate, duration, and repeated loading. ForestProducts Journal. 20(1):53-60.McNatt, J.D. and Laufenberg, T.L. 1991. Creep and creep-rupture of plywood and orientedstrandboard. In Proceedings of the 1991 International Timber Engineering Conference. London, U.K.3:457-464.McNatt, J.D. and Werren, F. 1976. Fatigue properties of three particleboards in tension andinterlaminar shear. Forest Products Journal 26(5):45-48.238Miner, M.A. 1945. Cumulative damage in fatigue. Journal of Applied Mechanics. American Society ofMechanical Engineer. 12(3):532-543.Palka, L.C. and Rovner, B. 1990. Long-term strength of Canadian commercial waferboard 5/8-inchpanels in bending. Short and long term test data. Report prepared for Forestry Canada. DSS File No.055SS.4Y002-9-0285. Forintek Canada Corp. Vancouver, B.C.Palka, L.C., Rovner, B., and Deacon, W. 1991. Long-term strength of Canadian commercialwaferboard 5/8-inch panels in bending. Creep and creep-rupture data and interpretation. Reportprepared for Forestry Canada. DSS File No. 066SS.4Y002-9-0284. Forintek Canada Corp. Vancouver,B.C.Parasin, A.V. 1981. Strength properties of 9.5 mm - 3 ply and 15.5 mm 5 ply western hemlock andamabilis fir sheathing grade plywood. Council of Forest Industries of B.C. Report 121. Vancouver, B.C.Parasin, A.V. 1983a. Strength properties of 9.5 mm - 3 ply and 15.5 mm 5 ply western white sprucesheathing grade plywood - B.C. Council of Forest Industries of B.C. Report 124. Vancouver, B.C.Parasin, A.V. 1983b. Strength properties of 9.5 mm - 3 ply and 15.5 mm 5 ply western white sprucesheathing grade plywood - Alberta. Council of Forest Industries of B.C. Report 170. Vancouver, B.C.Parasin, A.V. and Nagy, N.J. 1989. Experimental evaluation of plywood made by Ainsworth LumberCompany Ltd. for truck trailer decking. Council of Forest Industries of B.C. Technical Note 89.2.Vancouver, B.C.239Parasin, A.V. and Nagy, N.J. 1990. Experimental evaluation of transdeck. Council of Forest Industriesof B.C. Technical Note 90.1. Vancouver, B.C.Smith, G.R. 1974. The Derivation of allowable unit stresses for unsanded grades of Douglas-fir plywoodfrom in-grade test data. Council of Forest Industries of B.C. Report 105. Vancouver, B.C.Thompson, E.G., Goodman, J.R., and Vanderbilt, M.D. 1975. Finite element analysis of layered woodsystems. Journal of the Structural Division. ASCE. 101(12):2659-2672.Truck Trailer manufacturers Association. 1989. TTMA recommended practice: rating of van trailer andcontainer floors for lift truck loading. RP No. 37-89. Alexandria, Virginia.
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Performance of laminated veneer wood plates in decking systems Lam, Frank C.F. 1992
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Title | Performance of laminated veneer wood plates in decking systems |
Creator |
Lam, Frank C.F. |
Date Issued | 1992 |
Description | A new laminated veneer wood plate has been developed which is a specialty product intended for highly engineered end-use including flat bed truck and dry freight van trailer decking systems. The investigation described in this thesis represents the first known attempt to 1) develop a theoretical framework for evaluating the performance of laminate veneer panels in prototype dry freight van trailer decking systems, 2) develop a testing facility to generate an experimental database on these product through full scale testing, and 3) develop and model the fatigue behavior of this type of product through small specimen tests and damage accumulation laws. A structural analysis model has been developed to predict the structural behavior of a prototype decking system. A comprehensive database on the mechanical properties of 3.2 and 2.5 mm (gl and i6 inch) thick Douglas-fir veneers has been generated through experimental studies as input to the model. The database includes information on bending, tension, and compression strength properties, shear moduli of rigidity, ultrasonic transmission time, and connection stiffness. Analyses of variance has indicated that the mean strength properties of 3.2 and 2.5 mm thick veneers are significantly different for the parallel to grain direction but not significantly different for the perpendicular to grain direction at the 95% probability level. Statistical information and distributions parameters have been established for the various veneer strength properties so that simulations can be performed to model the strength properties of the veneers. A trailer decking load simulator test facility has been developed so that full scale testing of prototype dry freight van trailer decking systems can be performed. The experimental program has been divided into two phases: 1) static test program and 2) cyclic test program. Four prototype decking systems have been considered. The static test program has generated information on the structural behavior of the panels in the prototype system. Experimental results agree well with predictions from the computer model. The cyclic test program has generated information on the performance of the prototype system under fatigue condition. Information on the fatigue behavior of the panels in a system has been established from testing of small specimens in bending mode with the appropriate stress history. The relationship between the fatigue life and failure mode of the small specimen tests and the full size panels in system has been established. Damage accumulation laws have been developed from the small specimen tests results which provide a basis for evaluation of the fatigue behavior of the material in decking systems. |
Extent | 14473298 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2008-09-15 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0075205 |
URI | http://hdl.handle.net/2429/2015 |
Degree |
Doctor of Philosophy - PhD |
Program |
Forestry |
Affiliation |
Forestry, Faculty of |
Degree Grantor | University of British Columbia |
GraduationDate | 1993-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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