{"Affiliation":[{"label":"Affiliation","value":"Forestry, Faculty of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."}],"AggregatedSourceRepository":[{"label":"AggregatedSourceRepository","value":"DSpace","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","classmap":"ore:Aggregation","property":"edm:dataProvider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","explain":"A Europeana Data Model Property; The name or identifier of the organization who contributes data indirectly to an aggregation service (e.g. Europeana)"}],"Campus":[{"label":"Campus","value":"UBCV","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","classmap":"oc:ThesisDescription","property":"oc:degreeCampus"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the name of the campus from which the graduate completed their degree."}],"Creator":[{"label":"Creator","value":"Xiong, Pingbo","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/creator","classmap":"dpla:SourceResource","property":"dcterms:creator"},"iri":"http:\/\/purl.org\/dc\/terms\/creator","explain":"A Dublin Core Terms Property; An entity primarily responsible for making the resource.; Examples of a Contributor include a person, an organization, or a service."}],"DateAvailable":[{"label":"DateAvailable","value":"2010-11-05T21:01:51Z","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"edm:WebResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"DateIssued":[{"label":"DateIssued","value":"1991","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"oc:SourceResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"Degree":[{"label":"Degree","value":"Master of Science - MSc","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","classmap":"vivo:ThesisDegree","property":"vivo:relatedDegree"},"iri":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","explain":"VIVO-ISF Ontology V1.6 Property; The thesis degree; Extended Property specified by UBC, as per https:\/\/wiki.duraspace.org\/display\/VIVO\/Ontology+Editor%27s+Guide"}],"DegreeGrantor":[{"label":"DegreeGrantor","value":"University of British Columbia","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","classmap":"oc:ThesisDescription","property":"oc:degreeGrantor"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the institution where thesis was granted."}],"Description":[{"label":"Description","value":"A localized E-simulation model and a strength simulation model, which are based on the theories of stationary random process and the bivariate standard normal distribution, have been developed.\r\nA group of 2x6 2100f-1.8E SPF MSR lumber have been tested to obtain the within-board compressive strengths. The test E-profiles and compressive strength data was used to provide the statistical information for E-simulation model and \u03c3-simulation model.\r\nThe comparison between the within-board compressive strength test data and the simulation results shows that the E-simulation approach and the \u03c3-simulation approach can model the localized stiffness and strength behaviours satisfactorily.\r\nUsing E-simulation model and \u03c3-simulation model three grades of MSR lumber have been generated with localized MOE, tension strength and compression strength profiles on each board. With these generated MSR lumbers, different sizes and layups of glulam beams have been built and the effect of beam sizes and layups on the strength of glulam beams has been simulated.\r\nThe results obtained from glulam beam simulation showed that the beam sizes and layups did have significant effect on the beam strength properties. With one or two layers of higher grade lamination on the outer layer of the beam and lower grade laminations in the rest of inner layers of the beam, the glulam beam bending strength could be improved significantly.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/description","classmap":"dpla:SourceResource","property":"dcterms:description"},"iri":"http:\/\/purl.org\/dc\/terms\/description","explain":"A Dublin Core Terms Property; An account of the resource.; Description may include but is not limited to: an abstract, a table of contents, a graphical representation, or a free-text account of the resource."}],"DigitalResourceOriginalRecord":[{"label":"DigitalResourceOriginalRecord","value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/29824?expand=metadata","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","classmap":"ore:Aggregation","property":"edm:aggregatedCHO"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","explain":"A Europeana Data Model Property; The identifier of the source object, e.g. the Mona Lisa itself. This could be a full linked open date URI or an internal identifier"}],"FullText":[{"label":"FullText","value":"MODELLING STRENGTH AND STIFFNESS OF GLUED-LAMINATED TIMBER USING MACHINE STRESS RATED LUMBER By PINGBO XIONG M. Eng., Anhui A g r i c u l t u r a l U n i v e r s i t y , 1985 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of F o r e s t r y We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September 1991 \u00a9 Pingbo Xiong, 1991 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada D a t e 0 O-T- ^ . I ^ S \\ DE-6 (2\/88) ABSTRACT A l o c a l i z e d E - s i m u l a t i o n model and a st r e n g t h s i m u l a t i o n model, which are based on the t h e o r i e s of s t a t i o n a r y random process and the b i v a r i a t e standard normal d i s t r i b u t i o n , have been developed. A group of 2 x 6 2100f-1.8E SPF MSR lumber have been t e s t e d to o b t a i n the within-board compressive strengths. The t e s t E - p r o f i l e s and compressive str e n g t h data was used to provide the s t a t i s t i c a l i n f o r m a t i o n f o r E- s i m u l a t i o n model and cr-simulation model. The comparison between the within-board compressive s t r e n g t h t e s t data and the s i m u l a t i o n r e s u l t s shows th a t the E - s i m u l a t i o n approach and the cr-simulation approach can model the l o c a l i z e d s t i f f n e s s and stre n g t h behaviours s a t i s f a c t o r i l y . Using E - s i m u l a t i o n model and cr-simulation model three grades of MSR lumber have been generated with l o c a l i z e d MOE, t e n s i o n s t r e n g t h and compression s t r e n g t h p r o f i l e s on each board. With these generated MSR lumbers, d i f f e r e n t s i z e s and layups of glulam beams have been b u i l t and the e f f e c t of beam s i z e s and layups on the st r e n g t h of glulam beams has been simulated. The r e s u l t s obtained from glulam beam s i m u l a t i o n showed t h a t the beam s i z e s and layups d i d have s i g n i f i c a n t e f f e c t on the beam st r e n g t h p r o p e r t i e s . With one or two l a y e r s of higher grade lam i n a t i o n on the outer l a y e r of the beam and lower grade laminations i n the r e s t of inner l a y e r s of the beam, the glulam beam bending s t r e n g t h c o u l d be improved s i g n i f i c a n t l y . i i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES v i LIST OF FIGURES v i i i LIST OF NOTATIONS xv ACKNOWLEDGEMENTS x v i i i 1. INTRODUCTION AND OBJECTIVES 1 1.1 Objectives 3 1.2 Previous Study 4 2. COFI MSR LUMBER TEST PROGRAM 7 2.1 I n t r o d u c t i o n 7 2.2 M a t e r i a l s 7 2.3 Test Procedures 8 2.4 R e s u l t s and A n a l y s i s 8 3. VITHIN-BOARD COMPRESSION EXPERIMENT 10 3.1 I n t r o d u c t i o n 10 3.2 M a t e r i a l s 10 3.3 Experiment Design and Procedures 11 3.4 R e s u l t s 12 4. E - RANDOM PROCESS SIMULATION MODEL 14 4.1 I n t r o d u c t i o n 14 4.2 T h e o r e t i c a l Basis 14 i v 4.2.1 Random Processes 14 4.2.2 F o u r i e r Transform and Power S p e c t r a l Density.. 15 4.2.3 Mode l l i n g the L o c a l i z e d E-functions 19 4.3 E - f u n c t i o n S i m u l a t i o n Example 24 4.3.1 MSR E - p r o f i l e Treatment 24 4.3.2 S p e c t r a l A n a l y s i s of E - p r o f i l e 25 4.3.3 Reconstruction of E-functions 26 5. BIVARIATE STANDARD NORMAL SIMULATION MODEL 29 5.1 I n t r o d u c t i o n 29 5.2 B i v a r i a t e Normal D i s t r i b u t i o n 30 5.3 Model Development 32 5.4 cr-Profile S i m u l a t i o n Example 35 6. GENERATION OF E- AND (7-PROFILES FOR THREE GRADES 38 6.1 I n t r o d u c t i o n 38 6.2 Generation of E - p r o f i l e s f o r Three Grades 38 6.3 Generation of (wt)) 113 17. Ensemble average of t e s t E - p r o f i l e 114 18. Ensemble standard d e v i a t i o n of t e s t E - p r o f i l e 115 19. The cumulative d i s t r i b u t i o n f u n c t i o n (CDF) of t e s t and ix f i t t e d board mean of MOE 116 20. The cumulative d i s t r i b u t i o n f u n c t i o n (CDF) of t e s t and f i t t e d board standard d e v i a t i o n of MOE 117 21. Ensemble average of the amplitude spectrum 118 22. Generated E - p r o f i l e of one board 119 23. Ensemble average of t e s t and generated E - p r o f i l e s 120 24. Ensemble standard d e v i a t i o n of t e s t and generated E - p r o f i l e s 121 25. Test and simulated ensemble average of amplitude s p e c t r a . 122 26. The r e g r e s s i o n p l o t of board mean of MOE vs. compressive str e n g t h (2100f-1.8E) 123 27. The r e g r e s s i o n p l o t of board standard d e v i a t i o n of MOE vs. compressive strength (2100f-1.8E) 124 28. Gra p h i c a l demonstration of the transformation between the r e a l space and standard normalized space 125 29. The r e g r e s s i o n p l o t of t e s t modulus of e l a s t i c i t y (E) vs. within-board compressive strength (O-Q) 126 30. The r e g r e s s i o n of t e s t and simulated modulus of e l a s t i c i t y (E) vs. within-board compressive str e n g t h (( ) = Phase angle; w = Angular frequency; oo = V a r i a b l e p e r t a i n i n g to i n f i n i t y ; ACKNOWLEDGEMENTS I would l i k e to express my s i n c e r e g r a t i t u d e to my s u p e r v i s o r , Dr. J.D. B a r r e t t f o r h i s i n v a l u a b l e advice and p a t i e n t guidance throughout the research work and i n pre p a r a t i o n of t h i s t h e s i s . I would l i k e to express my a p p r e c i a t i o n to Dr. R.O. Foschi f o r h i s advice and support. Thanks are a l s o due to Mr. Frank Lam and Mr. Yintang Wang f o r t h e i r h e l p f u l suggestions and a s s i s t a n c e during v a r i o u s stages of the work presented i n t h i s t h e s i s . The f i n a n c i a l support from the Department of Harvesting and Wood Science of the U n i v e r s i t y of B r i t i s h Columbia i s g r a t e f u l l y acknowledged. 1 1. INTRODUCTION Glued-laminated timber, commonly r e f e r r e d to as glulam, i s a s t r u c t u r a l timber product made of elements glued together from smaller pieces of wood, e i t h e r i n s t r a i g h t or curved form, w i t h the g r a i n of a l l the laminations e s s e n t i a l l y p a r a l l e l to the length of the member. In Canada, glulam i s manufactured i n accordance w i t h the requirements of Canadian Standards A s s o c i a t i o n (CSA) Standard 0122-M, Structural Glued-Laminated Timber. Complete S p e c i f i c a t i o n data f o r glulam i s given i n CVC d a t a f i l e VS-2 Glued-Laminated Timber Specifications. Design c r i t e r i a f o r glulam i n l i m i t s t a t e design format are contained i n N a t i o n a l Standard of Canada CAN3-086.1-M84, Engineering Design in Wood. In North America and Europe, glulam i s used i n a wide v a r i e t y of a p p l i c a t i o n s , ranging from headers or support beams i n r e s i d e n t i a l framing to major s t r u c t u r a l elements i n roof framing of domed stadiums. Glulam may be produced i n any s i z e and any shape d e s i r e d , ranging from l a r g e long-span s t r a i g h t beams to complex curved-arch c o n f i g u r a t i o n s . Current production l e v e l s by the glulam i n d u s t r y are approximately 13 MMFBM (1 MMFBM = 10 6 Foot Board Measure) per year i n Canada (Ainsworth, 1989). For glulam beams, the most common design a p p l i c a t i o n i s as a 2 bending member with the primary design loads a p p l i e d p e r p e n d i c u l a r to the wide face of the laminations. To more e f f e c t i v e l y u t i l i z e the a v a i l a b l e lumber resources and to enhance the competitive p o s i t i o n of glulam i n the market p l a c e , such bending members are produced using engineering layups or combinations, i n c o r p o r a t i n g a range of species and s t r u c t u r a l grades of lumber. In these engineered layups, the highest q u a l i t y m a t e r i a l i s p o s i t i o n e d i n the member where the s e r v i c e l o a d i n g w i l l create the highest s t r e s s . Conversely, lower grade laminations are p o s i t i o n e d i n areas or zones where the s t r e s s w i l l be lower. The Canadian glulam i n d u s t r y uses l a m i n a t i n g stock based on the v i s u a l c r i t e r i a and supplementary E - r a t i n g . S p r u c e - P i n e - F i r (SPF) machine s t r e s s - r a t e d lumber i s not permitted i n the CSA Standard 0122-M, Structural Glued-Laminated Timber. Since the SPF species group has a very l a r g e volume i n standing timber and i s more r e a d i l y a v a i l a b l e from domestic s u p p l i e r s , i t i s o b v i o u s l y a reasonable choice to consider SPF machine s t r e s s - r a t e d lumber as an a l t e r n a t e source and type of m a t e r i a l f o r the l a m i n a t i n g stock, e s p e c i a l l y s i n c e the e x i s t i n g v i s u a l l y graded lami n a t i n g stock must be E-rated p r i o r to use. The work done i n t h i s r e p o r t i s aimed a t a s s e s s i n g the use of MSR lumber as the l a m i n a t i n g stock i n the glulam beam production. In order to evaluate the glulam beam st r e n g t h and s t i f f n e s s behaviour, a fundamental understanding i s needed about the c o r r e l a t i o n and the v a r i a t i o n of modulus of e l a s t i c i t y (MOE) and s t r e n g t h values 3 w i t h i n and between the laminations used to f a b r i c a t e the glulam beams. 1.1 OBJECTIVES This study i s aimed a t a c h i e v i n g f i v e main o b j e c t i v e s , namely: 1. To analyze t e n s i l e s t r e n g t h , compressive s t r e n g t h and MOE data from e v a l u a t i o n s of three grades (1650f-1.5E, 2100f-1.8E and 2400f-2.0E) of 2 x 6 SPF MSR lumber provided by the C o u n c i l of Forest I n d u s t r i e s of B r i t i s h Columbia (COFI) and to o b t a i n the necessary d i s t r i b u t i o n parameters r e q u i r e d f o r developing an E - s i m u l a t i o n model and s t r e n g t h s i m u l a t i o n model. 2. To t e s t a group of 2 x 4 2100f-1.8E s p r u c e - p i n e - f i r (SPF) machine-stress-rated (MSR) lumber and develop a data base f o r studying p a i r e d MOE and compressive str e n g t h p a r a l l e l to g r a i n v a r i a t i o n s w i t h i n and between laminations to be used i n glulam beams. 3. To develop an E - s i m u l a t i o n model, from which MOE data p o i n t s along the board length can be generated. The generated MOE data can be used as the input data i n a t e n s i l e and compressive s t r e n g t h s i m u l a t i o n model. 4. To develop a strength s i m u l a t i o n model, from which compressive s t r e n g t h p a r a l l e l to g r a i n (crc) and t e n s i l e s t r e n g t h p a r a l l e l to g r a i n (cy) data p o i n t s can be generated corresponding to the c o r r e l a t e d MOE data p o i n t s generated by the E - s i m u l a t i o n model. 5. To simulate the e f f e c t of beam layups on the s t r e n g t h of 4 glulam beams f a b r i c a t e d using r e g u l a r grades of 2 x 6 SPF MSR lumber. The a n a l y s i s i s performed using i n a computer program c a l l e d GLULAM. 1.2 PREVIOUS RESEARCH The l o c a l i z e d values of modulus of e l a s t i c i t y E and st r e n g t h a along the length of lumber i s r e q u i r e d to a c c u r a t e l y determine the s t i f f n e s s , s t r e n g t h and the f a i l u r e modes of timber s t r u c t u r e s . In recent years, some researchers have turned t h e i r a t t e n t i o n to the v a r i a t i o n of l o c a l i z e d MOE along the length of the board ( B e c h t e l , 1985; F o s c h i , 1987). Based on an a p p l i c a t i o n of the F o u r i e r transform, a method which can produce s a t i s f a c t o r y approximations of the d i s t r i b u t i o n of the l o c a l modulus of e l a s t i c i t y E(x) along the length of a board was presented (1987, F o s c h i ) . I t was concluded t h a t the use of the minimum modulus of e l a s t i c i t y along the length of a board could improve the p r e d i c t i o n of stre n g t h through a b e t t e r s t i f f n e s s and str e n g t h c o r r e l a t i o n . Based on the theory of s t a t i o n a r y random processes, Vang et a l . (1990) developed a procedure which may be used to generate r e a l i s t i c E(z) f u n c t i o n s r e p r e s e n t i n g the v a r i a t i o n i n modulus of e l a s t i c i t y along the length of a specimen. Mo d e l l i n g c o r r e l a t e d lumber p r o p e r t i e s has been an area of a c t i v e research f o r many years. These models are u s e f u l i n Monte Ca r l o s i m u l a t i o n s t h a t p r e d i c t the r e l i a b i l i t y of wood s t r u c t u r e s . Several authors have discussed the importance of accounting f o r the 5 concomitance between lumber str e n g t h p r o p e r t i e s (Suddarth et a l . , 1978; G a l l i g a n et a l . , 1979; R o j i a n i and T a r b e l l , 1984). One approach to the problem was presented by Voeste et a l . (1979). They presented a b i v a r i a t e a n a l y s i s t h a t simulated the c o r r e l a t e d p r o p e r t i e s of modulus of e l a s t i c i t y (MOE) and t e n s i l e s t r e n g t h ( c T ) . T h e i r technique began w i t h f i t t i n g a set of MOE data w i t h an a p p r o p r i a t e p r o b a b i l i t y d i s t r i b u t i o n u s i n g the method of maximum l i k e l i h o o d e s t i m a t i o n . Then a weighted l e a s t squares r e g r e s s i o n was conducted to r e l a t e to MOE. Random values of MOE were then generated from the f i t t e d d i s t r i b u t i o n and s u b s t i t u t e d i n t o the r e g r e s s i o n equation, adjusted by a randomly sampled r e s i d u a l , to generate a corresponding value of u^. This MOE -(Tj, p a i r was c o r r e l a t e d i n a manner s i m i l a r to the t e s t data. But a disadvantage w i t h t h i s approach i s th a t the dependent v a r i a b l e may need to be transformed to o b t a i n the d e s i r e d marginal p r o b a b i l i t y d i s t r i b u t i o n . The choice of transformations i s s u b j e c t i v e and o f f e r s only l i m i t e d f l e x i b i l i t y i n modelling marginal d i s t r i b u t i o n . T a y l o r and Bender (1988) presented an a l t e r n a t e method f o r s i m u l a t i n g c o r r e l a t e d lumber p r o p e r t i e s t h a t are not n e c e s s a r i l y normally d i s t r i b u t e d . This approach uses a tr a n s f o r m a t i o n of the m u l t i v a r i a t e normal d i s t r i b u t i o n to model the c o r r e l a t e d lumber p r o p e r t i e s , and had the advantages of e x a c t l y p r e s e r v i n g each marginal d i s t r i b u t i o n as w e l l as c l o s e l y approximating the c o r r e l a t i o n matrix of the v a r i a b l e s . Lam and Varoglu (1991a, 1991b) developed a model f o r the w i t h i n 6 member v a r i a t i o n of t e n s i l e s t r e n g t h p a r a l l e l to g r a i n i n nominal 38x89 mm No. 2 SPF lumber. They evaluated w i t h i n member t e n s i l e s t r e n g t h cumulative p r o b a b i l i t y d i s t r i b u t i o n s and the s p a t i a l c o r r e l a t i o n of the simulated data by window and semivariogram analyses. Foschi and B a r r e t t (1980) have developed a computer s i m u l a t i o n model of the stre n g t h and s t i f f n e s s of glued-laminated beams i n e i t h e r bending, compression or t e n s i o n . T h e i r approach was to use b a s i c data on the lami n a t i o n p r o p e r t i e s i n a f i n i t e element computer program to estimate v a r i a b i l i t y i n beam stre n g t h and s t i f f n e s s . The b a s i c data r e q u i r e d i n the model are MOE, t e n s i l e and compressive s t r e n g t h values and the d i s t r i b u t i o n of knots f o r va r i o u s l a m i n a t i n g grades i n a p a r t i c u l a r glulam beam layup. However, few experiments have been done to measure the v a r i a t i o n s of the l o c a l i z e d MOE and stre n g t h values along the board length f o r machine s t r e s s - r a t e d lumber. The model developed i n t h i s study incorporates some of the ideas by the previous r e s e a r c h e r s , and provides a general method to simulate the p a i r e d MOE and stre n g t h along the length of lumber. Using these generated MOE and stre n g t h values along the length of lumber, the str e n g t h and s t i f f n e s s p r o p e r t i e s of glulam beams w i l l be simulated to study the e f f e c t of beam layups on glulam beam strengths f a b r i c a t e d from MSR lumber. 7 2. COFI MSR LUMBER TEST PROGRAM 2.1 INTRODUCTION In the f a l l of 1987, the T e c h n i c a l S e r v i c e s Department of the C o u n c i l of F o r e s t I n d u s t r i e s of B r i t i s h Columbia (COFI) i n i t i a t e d a glulam beam research program to s p e c i f i c a l l y address an a l t e r n a t e source of raw m a t e r i a l supply f o r the glulam i n d u s t r y . The o b j e c t i v e of t h a t research program was to evaluate the s u i t a b i l i t y of machine s t r e s s - r a t e d (MSR) lumber f o r the manufacture of s t r u c t u r a l glulam products. This chapter o u t l i n e s some of the t e s t m a t e r i a l s , procedures and r e s u l t s obtained from COFI t e s t program (Ainsworth, 1989). 2.2 MATERIALS Three grades of 2x6 SPF MSR lumber, obtained from an MSR lumber producer l o c a t e d i n the i n t e r i o r of B r i t i s h Columbia, were s e l e c t e d by a Chief Grading Inspector. The grades s e l e c t e d were 1650f-1.5E, 2100f-1.8E and 2400f-2.0E. Two 6-foot t e s t specimens were s e l e c t e d from each of the 16-foot t e n s i o n and compression t e s t lumber groups. One specimen, c a l l e d Zone A, contained the minimum MOE zone lumber as determined by the Cook-B o l i n d e r s equipment. The other specimen, Zone B, was taken from the remaining p o r t i o n of the parent t e s t lumber (Fig u r e 1). 8 A d e s c r i p t i o n of the t e s t m a t e r i a l s and t e s t matrix from COFI i s provided i n Table 1. 2.3 TEST PROCEDURES The specimens from the t e n s i o n and compression t e s t groups were n o n - d e s t r u c t i v e l y t e s t e d i n the Cook-Bolinders machine to o b t a i n a f l a t w i s e bending E - p r o f i l e f o r each board. Jus t before t e s t i n g , an average moisture content of 16% was obtained from the randomly s e l e c t e d sample of ten p i e c e s . The t e n s i o n t e s t specimens from Table 1 and Figure 1 were t e s t e d i n a t e s t i n g machine with a gauge length of two f e e t (610 mm) between the g r i p s . The ramp load t e s t s were conducted at a r a t e of 4000 p s i per minute (27.8 MPa per minute). A l l specimens were t e s t e d to f a i l u r e . The compression t e s t specimens were t e s t e d i n a compression t e s t i n g machine with a gauge length of s i x f e e t (1830 mm). The ramp load t e s t s were conducted a t a displacement r a t e of 0.58 inches per minute (14.7 mm per minute). A l l specimens were t e s t e d to f a i l u r e . 2.4 RESULTS AND ANALYSIS A n a l y s i s of the t e n s i o n and compression specimens t e s t r e s u l t s c o n s i s t e d of a comparison between three MSR lumber grades (1650f-1.5E, 2100f-1.8E and 2400f-2.0E) using of cumulative d i s t r i b u t i o n f u n c t i o n s . 9 Summary s t a t i s t i c s f o r the tension and compression t e s t populations are provided i n Table 2 and Table 3. The cumulative d i s t r i b u t i o n functions f o r the three grades of tension and compression specimen t e s t r e s u l t s in Zone A and Zone B are provided from Figure 2 to 7. At f i r s t the COFI tension and compression t e s t data were planned to be used as the data base f o r the development of a strength simulation model. Later i t was recognized that i n s u f f i c i e n t strength data were a v a i l a b l e on each board in order to construct the simulation model f o r the within-board strength. Therefore, i t was decided that a more d e t a i l e d study of within-board compressive strength t e s t s should be completed in order to e s t a b l i s h a more complete l o c a l i z e d strength data base. The l o c a l i z e d strength experiment and the l o c a l i z e d strength and s t i f f n e s s simulation model developed from the d e t a i l e d compression studies w i l l be discussed i n the following chapters. 10 3. WITHIN-BOARD COMPRESSION EXPERIMENT 3.1 INTRODUCTION This chapter g i v e s a f u l l d e s c r i p t i o n of the procedures and r e s u l t s of the within-board compressive strength p a r a l l e l to g r a i n experiment. The purpose of t h i s experiment i s to develop a data base f o r p a i r e d within-board MOE and compressive str e n g t h p a r a l l e l to g r a i n ((Tgi) p r o p e r t i e s along the board length. The r e l a t i o n s h i p between MOE and compressive strength p a r a l l e l to g r a i n i^c) determined exp e r i m e n t a l l y w i l l provide a b a s i s f o r developing a s i m u l a t i o n method f o r generating c o r r e l a t e d property data f o r glulam beam stren g t h s t u d i e s . 3.2 MATERIALS The lumber f o r t h i s t e s t was s e l e c t e d from a MSR lumber producer l o c a t e d i n the i n t e r i o r of the province of B r i t i s h Columbia. Three grades of 2 x 4 inches (38mm x 89mm) SPF MSR lumber were s e l e c t e d . The s e l e c t e d grades were 1650f-1.5E, 2100f-1.8E, and 2400f-2.0E. Of these three grades, only grade 2100f-1.8E was t e s t e d f o r i t s within-board compressive s t r e n g t h p a r a l l e l to g r a i n . The other two grades were expected to be t e s t e d s e p a r a t e l y i n the f u t u r e . 3.3 EXPERIMENT DESIGN AND PROCEDURES The experiment was designed to t e s t the within-board compressive s t r e n g t h p a r a l l e l to g r a i n ( For a sample E - f u n c t i o n (En(x)) over the f i n i t e length 0< x < L, the 2-sided power s p e c t r a l d e n s i t y f u n c t i o n of the sample i s de f i n e d as (Bendat and P i e r s o l , 1 9 8 6 ) : S E(n, W ) 0 = J F n > , 0 F \u201e ( W , 0 (4.7) where Fn*(w, L) i s the complex conjugate of F n(w, L). The two-sided power s p e c t r a l d e n s i t y f u n c t i o n of the un d e r l y i n g s t a t i o n a r y E-process i s given by (Bendat and P i e r s o l , 1 9 8 6 ) : S E(w)= lim E[S E(n, u, 0 ] (4.8) L\u2014>oo where E[S E(n, u, \/.)] i s the expected value of the power s p e c t r a l d e n s i t y over the ensemble of E- f u n c t i o n s . S u b s t i t u t i n g Eq. (4.7) i n t o Eq. (4.8) y i e l d s : SE(w) = Um I E [ | F > , O l 2 ] (4.9) L\u2014*oo where | Fn(w, Z.) | i s the F o u r i e r amplitude spectrum. Since E n(z) i s r e a l , Sg(w) i s a r e a l , non-negative and even 18 f u n c t i o n : S E ( - W ) = SE(u>) ( 4.10 ) The one-sided power s p e c t r a l d e n s i t y f u n c t i o n GE(w) can be defi n e d as: G E( W) = 2 SE(u\/) = \\ lira E[ | F > , L) | 2 ] ( 4.11 ) where 0 < u < o o . A t r u e s t a t i o n a r y random process contains an i n f i n i t e number of sample f u n c t i o n s with i n f i n i t e l e n g t h , whereas r e a l E - f u n c t i o n s are few i n number and have f i n i t e length. Consequently, an estimate of the power s p e c t r a l d e n s i t y f u n c t i o n of the E-process GE(w) can be obtained by f i r s t computing the power s p e c t r a l d e n s i t y f u n c t i o n Gn(w, L) of each E - f u n c t i o n , and then averaging the ensemble of s p e c t r a l d e n s i t y components a t each frequency. The averaging i s intended to approximate the expected value i n Eq. (4.11), which can be replac e d by: G > , 0 = I | F > . O I 2 (4-12) G E ( \u00ab ) = i ! > > . o (4.i3) n = 1 where N i s the number of sample E-functions (sample s i z e ) . 19 4.2.3 MODELLING THE LOCALIZED E-FUNCTIONS A s t o c h a s t i c model f o r the s i m u l a t i o n of the E values along the length from MSR lumber data was set up. The model was developed from s p e c t r a l a n a l y s i s and random v i b r a t i o n theory, where the E - f u n c t i o n was t r e a t e d as s t a t i o n a r y , non-ergodic random process as discussed above. Using s t i f f n e s s t e s t data on lumber (MSR lumber d a t a ) , the power s p e c t r a l d e n s i t y of the u n d e r l y i n g E-process can be estimated. An ensemble of E-functions can then be r e c o n s t r u c t e d by combining the power s p e c t r a l d e n s i t y f u n c t i o n (or amplitude spectrum) of the process with the randomly s e l e c t e d phases. The r e c o n s t r u c t e d data have s i m i l a r s t a t i s t i c a l c h a r a c t e r i s t i c s and frequency content to the experimental data. The generated E-process can be modeled by the f o l l o w i n g cosine s e r i e s (Vang et a l . , 1990): N\/2 ( 4.14 ) i = 1 where N i s the number of the d i s c r e t e frequencies i n c l u d e d i n S E(w);the amplitude A(u>j) and frequency uii are d e t e r m i n i s t i c , but the phase angle ei(wt) i s assumed to be a random v a r i a b l e . The p r o b a b i l i t y d e n s i t y f u n c t i o n f o r 4>^i) i s taken to be d i s t r i b u t e d u n i f o r m l y between 0 and 2ir as shown i n Figure 16. That i s , otherwise 20 ( 4.15 ) The ensemble mean of the Eq. (4.14) a t a s p e c i f i c l o c a t i o n x i s N\/2 E l E ^ x ) ] = E [ \u00a3 A( WJcos(w , . x + # * , \u2022 ) ) ] i = 1 N\/2 f 27T N $NA(W ,-) cos(W,x + #w,.)) K*(w,0) \u00abW\",0) .\u2022 = i l o J = 0 ( 4.16 ) The ensemble mean square of the Eq. (4.14) i s N\/2 E[E3(x)] = E[( \u00a3A( W l.)cos( W l.x+^ W,.))) ] t = I N\/2 ( 2? ) = E S A V i ) cos 2(W,x + 0K>) i\u00bb(0(w,.)) < ^ K \u00bb \u00bb = H 2 i = 1 ( 4.17 ) Since the E-process i s a s t a t i o n a r y process, the a u t o c o r r e l a t i o n f u n c t i o n f o r the random process E(x) can be de f i n e d as the average value of the product E(x)E(x + r) as the f o l l o w i n g R B ( r ) = E[E(i)E(x + r ) ] 21 ( 4.18 ) where RE(T) i s the a u t o c o r r e l a t i o n f u n c t i o n f o r E(z) From Eq. (4.17), (4.18) and l e t r - 0 N\/2 .2\/ x E[E5-(x)] = \u00a3 = R \u00a3(0) ( 4.19 ) i = i A l s o , the F o u r i e r transform of RE(T), and i t s i n v e r s e , are given by (Bendat and P i e r s o l , 1 9 8 6 ) : ' oo ( 4.20 ) M O = i ' SE(w) rfw ( 4.21 ) where SE(u>) i s c a l l e d the power s p e c t r a l d e n s i t y f u n c t i o n of the E processes and i s a f u n c t i o n of angular frequency of u. I f r = 0, then M \u00b0 ) = SE(w) dw ( 4.22 ) Equating Eq. (4.19) and Eq. (4.22), and knowing t h a t SE(w) i s a 22 r e a l , non-negative and even f u n c t i o n : j2 A (\"\u2022\u2022) t = i oo dw = 2 N\/2 SE(u) du \u00ab 2 ^ S E ( W l ) 4 w (4.23) i = 1 from which the amplitude A(w f) can be c a l c u l a t e d to be: A(w,.) = 2^ SE(w,-) Au ( 4.24 ) For a random process with zero mean (E = 0 ) , the mean square value equals the standard d e v i a t i o n a ^ (x) of the process: 2 N\/2 .2\/ \\ E[E*.(x)] = E[(E(X) - E) ] = 4 ( x ) = \u00a3 i = l ( 4.25 ) ^ (*) i - 1 A\"( W.) ( 4.26 ) The E-process generated by Eq. (4.14) i s a s t a t i o n a r y , ergodic process wit h zero mean. To make the E-process to be non-ergodic with non-zero mean, Eq. (4.14) can be modified as f o l l o w s : N\/2 E j O ) = E i + \u00a3 A(w,-)cos(w,.x + #w,.)) \u00ab' = 1 ( 4.27 ) 23 where i s a random value chosen from the d i s t r i b u t i o n of beam mean MOE values. I f , f o r example, the mean data can be approximated by a 3-parameter Weibu l l d i s t r i b u t i o n , E^- i s given by Ej = 2 + 1 ( 4.29 ) where N i s an even number; or A( W (.) A( W t.) = | 0 MUN -\u00bb' +1) i < i > N + 1 2 N + 1 2 N + 1 2 ( 4.30 ) 24 where N i s an odd number. Therefore the amplitude spectrum A(w,) i s symmetrical to the Nyquist frequency. This property i s used to reduce the number of harmonics i n Eq. (4.27) without considerable e f f e c t on the accuracy of the r e s u l t s . 4.3 E-FUNCTION SIMULATION EXAMPLE Tests of l o c a l i z e d E - functions can be obtained by bending a board over consecutive short spans, using a concentrated load a p p l i e d a t the middle of each span. This i s done by grading machines based on the so c a l l e d \" d e f l e c t i o n method\" where the d e f l e c t i o n i s constant and the c e n t r e - p o i n t load i s measured. From the d e f l e c t i o n measurements and simple equations from the beam theory, an estimate of the bending E values along the length of the board can be obtained. The l o c a l i z e d MOE values along the length of the boards of grade 2100f-1.8E, which were obtained from the Cook-Bolinders machine i n F o r i n t e k Canada Corp. (Vancouver), were analyzed and used as the input data i n the f o l l o w i n g E - f u n c t i o n s i m u l a t i o n program. In t o t a l , 54 boards were n o n - d e s t r u c t i v e l y t e s t e d i n the Cook-Bolinders machine. 4.3.1 MSR E-PROFILE TREATMENT In order to implement the E - f u n c t i o n s i m u l a t i o n program, the machine s t r e s s - r a t e d MOE data p r o f i l e s should be t r e a t e d f i r s t to determine the process s t a t i s t i c s . 25 A f t e r averaging the MOE values a t the same l o c a t i o n s along the length of the boards, an ensemble average of MOE along the length of the board was obtained, as shown i n Figure 17. Fi g u r e 18 i s the ensemble average of the standard d e v i a t i o n of MOE along the length. Eq. (4.14) generates an E - p r o f i l e f o r a zero mean process. In order to perform the F o u r i e r transform of each board r e c o r d and c a l c u l a t e i t s power s p e c t r a l d e n s i t y f u n c t i o n and the r e l a t e d amplitude spectrum, the zero mean MOE records are cons t r u c t e d by s u b t r a c t i n g from each one the corresponding mean value E. The cumulative d i s t r i b u t i o n f u n c t i o n ( c d f ) of board mean MOE values and the within-board standard d e v i a t i o n of MOE i s shown i n Figure s 19 and 20 r e s p e c t i v e l y . The r e s u l t s i n Figures 19 and 20 have been f i t t e d w i t h normal, lognormal, 2-parameter Weibull and 3-parameter Weibu l l d i s t r i b u t i o n s . The 3-parameter Weibull d i s t r i b u t i o n was v i s u a l l y judged to provide the best f i t f o r the board mean of MOE values and the board standard d e v i a t i o n of MOE. The parameters f o r the d i s t r i b u t i o n s are summarized i n Table 5. 4.3.2 SPECTRAL ANALYSIS OF E-PROFILE By performing Fast F o u r i e r Transform (FFT), the amplitude spectrum of the MOE rec o r d i s obtained. Then the ensemble average of the amplitude spectrum by averaging the amplitudes a t each frequency, shown i n Figure 21, i s constructed. 26 4.3.3 RECONSTRUCTION OF E-FUNCTIONS The p r i n c i p l e of generation of E-functions i s th a t the rec o n s t r u c t e d E - f u n c t i o n should have s i m i l a r s t a t i s t i c a l c h a r a c t e r i s t i c s and frequency content as the experimental E values. The t y p i c a l E - f u n c t i o n can be rec o n s t r u c t e d as i n the form of Eq. (4.27), from which the f i r s t p a r t E^ i s the random mean of the sample E - f u n c t i o n obtained from a f i t t e d cumulative d i s t r i b u t i o n f u n c t i o n ( F i g u r e 19) through the t e s t mean of the E - f u n c t i o n s ; the N\/2 second p a r t , ^ A ( W - ) C O S ( W 1 J ; + ^ (w,-)) , i s the v a r i a t i o n of the E - f u n c t i o n \u00bb = 1 about the mean value along the length. From Eq. (4.26), the standard d e v i a t i o n crE(x) i n each generated board would be a constant value s i n c e the summation of the ensemble average of the amplitude spectrum i s a constant. Therefore we choose to normalize the ensemble average of amplitude spectrum by d i v i d i n g the standard d e v i a t i o n of the E-process o-g(x) . L a t e r , when r e c o n s t r u c t i n g E - f u n c t i o n s , the amplitude spectrum of each generated board i s c o r r e c t e d to achieve a random standard d e v i a t i o n cre(x). This random standard d e v i a t i o n ( ^ ) 2 } S i m p l i f y i n g the above equation by = \u00aby 1 1 - P2 ( 5.4 ) V* = A*y + P ^ ( * - Hx) ( 5.5 ) where a; i s a random v a r i a b l e from x \u2014 \\ix + RN \u2022 ax (0 < RN < 1 ) and RN i s a random normal number with mean 0 and standard d e v i a t i o n of 1. 32 Then \/ ( H O = 0-N2T 1 exp (5.6) = %(0, 1) (5.7) y = a z + n (5.8) where cr* i s a constant; z* i s a random number ( 0 < z* < 1 ). Therefore, the c o n d i t i o n a l d i s t r i b u t i o n of Y, has the s i m p l i f i e d c o n d i t i o n a l p.d.f. \/ ( y \\ x), where 5.3 MODEL DEVELOPMENT A tra n s f o r m a t i o n of the b i v a r i a t e standard normal d i s t r i b u t i o n i s chosen to model the two c o r r e l a t e d lumber stre n g t h and s t i f f n e s s p r o p e r t i e s . The approach has the advantage of e x a c t l y p r e s e r v i n g each marginal d i s t r i b u t i o n as w e l l as the c o r r e l a t i o n between MOE and str e n g t h p r o p e r t i e s obtained from t e s t data. This s e c t i o n w i l l d e scribe the procedures which are used to generate a c o r r e l a t e d random p a i r (X, Y) or given X, to generate the random value Y, where X could be chosen as a MOE and Y the compressive (5.9) 33 s t r e n g t h a t a p a r t i c u l a r l o c a t i o n w i t h i n a board. The t h e o r e t i c a l c o n s i d e r a t i o n has already been discussed i n the above s e c t i o n . The study of within-board MOE and compressive s t r e n g t h property v a r i a t i o n showed t h a t both means and standard d e v i a t i o n s of MOE and compressive st r e n g t h vary from board to board. In the b i v a r i a t e standard normal d i s t r i b u t i o n s i m u l a t i o n model, we assume t h a t : a) . The c o r r e l a t i o n c o e f f i c i e n t p-~ f o r X mean and Y mean i s the \/ r xy same as p f o r X and Y, as shown i n Figure 26. b) . The c o r r e l a t i o n c o e f f i c i e n t pxy f o r X standard d e v i a t i o n and Y standard d e v i a t i o n i s zero, as shown i n Figure 27. The procedures f o r modelling the c o r r e l a t e d v a r i a b l e s are as f o l l o w s : 1. Obtain E - p r o f i l e s from the Cook-Bolinders grading machine and the s t r e n g t h p r o f i l e s from the within-board compressive s t r e n g t h t e s t s . 2. Estimate the parameters f o r the best f i t t i n g cumulative d i s t r i b u t i o n f u n c t i o n s of means and standard d e v i a t i o n s f o r both v a r i a b l e s X and Y. The mean and standard d e v i a t i o n cumulative d i s t r i b u t i o n f u n c t i o n s F(r) and F(j^) could be f i t t e d by normal, lognormal, 2-parameter Weibull and 3-parameter V e i b u l l d i s t r i b u t i o n s i n the model developed, as i n Figure 19, Figure 20, and Figure 12, Fig u r e 13. 34 3. Estimate the c o r r e l a t i o n c o e f f i c i e n t pxy i n the r e a l space by: E (** -1) (y* - y) i = 1 ( E ( y , - y ) 2 \u00a3 (y , -y) 2 ) ^ - ; ~ < 5- 1 0) 2 \u00ab = 1 \u00ab' = 1 where xi and are the observations of X and Y; x and j\/ are the means of the X and Y; n i s the number of the observations of J and Y. 4. Transform X from r e a l space i n t o standard normalized space X:-X w i t h i t s mean equals 0 and standard d e v i a t i o n equals 1, i.e. \u2014a \u2022 5. Transform Y from r e a l space i n t o standard normalized space Y.-Y with i t s mean equals 0 and standard d e v i a t i o n equals 1. i.e. Zy \"Y 6. Estimate the c o r r e l a t i o n c o e f f i c i e n t pN f o r Zx and Zy i n the standard normalized space. 7. Generate random board mean X and Y from the best f i t t i n g cumulative d i s t r i b u t i o n f u n c t i o n s of mean f o r X and Y assuming t h a t they have the c o r r e l a t i o n c o e f f i c i e n t pxy (as shown i n Fi g u r e 26). 8. Generate random board standard d e v i a t i o n cr- and <7* from the yi. Y best f i t t i n g cumulative d i s t r i b u t i o n f u n c t i o n s of standard d e v i a t i o n f o r cr- and cr- assuming t h a t t h e i r c o r r e l a t i o n c o e f f i c i e n t i s zero (as JC Y shown i n Figure 27). 9. Generate random Z-% and the c o r r e l a t e d random Zy by B i v a r i a t e Standard Normal D i s t r i b u t i o n S i m u l a t i o n Program (BNSIM). Or 35 10. Use Zx as input, generating the c o r r e l a t e d random ZY by B i v a r i a t e Standard Normal D i s t r i b u t i o n S i m u l a t i o n Program (BNSIM). 11. Transform Z% from standard normalized space back to r e a l space with i t s mean equals X and standard d e v i a t i o n equals a - , i . e . JL Xi = X + where = average compressive s t r e n g t h ; d ' The r a t i o of the transformed moment of i n e r t i a to the apparent moment of i n e r t i a can be determined: L E \/ ( 2 y t e ) 3 - tc3(E\/-Ec) I - E \/ ( 2 V t c ) 3 { 8 ' 4 } This r a t i o w i l l be c a l l e d the transformed s e c t i o n f a c t o r and denoted by T (Moody, 1974). Then, from Eq. (8.1) E can be c a l c u l a t e d from the f o l l o w i n g equation: EfV E \/ ( 2 t \/ + t c ) 3 - t c 3 ( E f - E c ) E = Jf- = \u2014 i \u2014 Z L vc_v f cJ ( 8.5 ) I 2 t \/ + t c ) 3 As Table 14 shows, the average simulated MOE values f o r the 24 beam layup groups were a l l w i t h i n 2% of t h e i r transformed MOE values (w i t h the average value of 0.4% h i g h e r ) , which v e r i f i e d the GLULAM program works w e l l to simulate the s t i f f n e s s p r o p e r t i e s of the glulam beams. Figure 43 compares the average simulated MOE values w i t h average transformed MOE values c a l c u l a t e d by using a transformed s e c t i o n a n a l y s i s f o r the twenty-four (24) beam layup groups. The average 53 simulated MOE values f o r the twenty-four (24) groups ranged from 1 . 4 7 8 x l 0 6 p s i to 2.054 x 10 6 p s i , while the transformed MOE values ranged, from 1 . 5 x l 0 6 p s i to 2.0 x l O 6 p s i (nominal v a l u e s ) . A r e g r e s s i o n a n a l y s i s suggested a l i n e of best f i t as: Y = 0.2276 + 0.8707X ( 8.6 ) where Y i s the simulated MOE and X i s the transformed MOE, both i n terms of m i l l i o n l b \/ i n 2 ( p s i ) . The c o e f f i c i e n t of determination (-R2) was 0.997. 8.3 BENDING STRENGTH P r e d i c t a b i l i t y of bending str e n g t h can be measured by comparing the p r e d i c t e d bending str e n g t h with the simulated bending s t r e n g t h (Table 15). The procedures of c a l c u l a t i n g the p r e d i c t e d bending s t r e n g t h w i l l be described i n the f o l l o w i n g s . For a two-zone beam (Figure 42), the transformed s e c t i o n f a c t o r , T, can be expressed as i n Eq. (8.4): E f ( 2 t \/ + t c ) 3 - t c 3 ( E f - E c ) T = -Z\u00b1\u2014f\u2014^- % V \/ c ( 8.7 ) E \/ ( 2 t \/ + t c ) 3 where: E^ and E c = moduli of e l a s t i c i t y f o r the face zone and core zone shown i n Figure 42; 54 t ^ and t c = depths shown i n Figure 42. The 5th p e r c e n t i l e MOR values f{ f o r grades of 1650f-1.5E, 2100f-1.8E and 2400f-2.0E are 3465, 4410 and 5040 p s i r e s p e c t i v e l y (obtained from \"SPS 2 NLGA S p e c i a l Products Standard f o r Machine S t r e s s Rated Lumber\"). In order to avoi d inner l a m i n a t i o n over s t r e s s e s i n the depths t c (ASTM, 1990): * * ( HThr-) ( i j ) 4 < 8 - 8 > I f f2 i s l e s s than the q u a n t i t y c a l c u l a t e d f o r the r i g h t s i d e of the equation (8.9), \/x i s l i m i t e d to a value t h a t w i l l s a t i s f y an e q u a l i t y . For use of with p r o p e r t i e s of the simulated p h y s i c a l s e c t i o n , f-y can be m u l t i p l i e d by T to y i e l d a value of a l l o w a b l e combination bending s t r e s s , or the p r e d i c t e d bending s t r e n g t h , \/ : f= AT ( 8.9 ) In order to account f o r the e f f e c t of depth a s i z e e f f e c t (12\/ef) 1\/ 9 was m u l t i p l i e d to the al l o w a b l e combination bending s t r e s s \/ where d i s the beam depth (ASTM, 1990). I t i s shown t h a t (see Table 15) most of the simulated bending s t r e n g t h from the 24 beam layup groups were w i t h i n 30% higher than the p r e d i c t e d bending s t r e n g t h except f o r the pure grade combinations by grade 1650f-1.5E. The simulated bending strengths i n those combinations a t t a i n e d only 85, 76 and 71 percent of the p r e d i c t e d 55 values a t the depths of 9, 12 and 18 i n c h , r e s p e c t i v e l y . Table 16 shows the hypothesis t e s t s of e q u a l i t y of means f o r bending s t r e n g t h of twenty-four (24) beam layup groups. I t i n d i c a t e d t h a t beam lay-up has a s i g n i f i c a n t e f f e c t on bending s t r e n g t h s . Within the beam layups of 9 inch depth, i t showed t h a t the d i f f e r e n c e s between beam9-A and beam9-B, and beam9-F and beam9-G are not s i g n i f i c a n t a t a = 0.05 l e v e l . This was expected because the outer l a y e r laminations (grades 2100f-1.8E and 2400f-2.0E) c o n t r i b u t e most to the beam bending s t r e n g t h , whereas the inner l a y e r laminations (grade 1650f-1.5E) do not have much i n f l u e n c e on the beam bending s t r e n g t h . S i m i l a r phenomena can a l s o be found i n the beam layups of 12 inch depth and 18 inch depth. The beam layup r e p r e s e n t a t i o n s i n Table 16 are the same as i n Tables 11, 12, 13 and Figures 39, 40, and 41. 8.3.1 GRADE EFFECT Figures 44 to 46 are the cumulative d i s t r i b u t i o n f u n c t i o n s ( c d f ) of bending str e n g t h of three grades (1650f-1.5E, 2100f-1.8E and 2400f-2.0E) under three depths (9, 12, and 18 in c h e s ) . Figures 47 to 48 show the v a r i a t i o n i n 5th p e r c e n t i l e and 50th p e r c e n t i l e values of bending strength of three grades (1650f-1.5E, 2100f-1.8E and 2400f-2.0E) with depths. C l e a r l y i n d i c a t e d from Table 16 and Figur e s 44 to 48, the 56 bending strengths of the pure grade combinations, made up of 1650f-1.5E, 2100f-1.8E, and 2400f-2.0E lumber r e s p e c t i v e l y , are d i f f e r e n t s i g n i f i c a n t l y a t the 0.05 l e v e l of s i g n i f i c a n c e . 8.3.2 DEPTH EFFECT Figures 49 to 51 are the cumulative d i s t r i b u t i o n f u n c t i o n s ( c d f ) of bending s t r e n g t h of three depths (9, 12, and 18 inches) f o r the three grades (1650f-1.5E, 2100f-1.8E and 2400f-2.0E). Figures 52 to 53 show the v a r i a t i o n of the 5th p e r c e n t i l e and 50th p e r c e n t i l e values of bending strength of three depths (9\", 12\", and 18\") as a f u n c t i o n of grade. From Table 16 and Figures 49 to 53, the bending strengths of pure grade combinations, made up of 1650f-1.5E, 2100f-1.8E, and 2400f-2.0E r e s p e c t i v e l y , are s i g n i f i c a n t l y d i f f e r e n t ( a t a = 0.05 l e v e l ) a t the three beam depths (9, 12 and 18 inc h e s ) . 8.3.3 EFFECT OF MIXED GRADE LAYUPS Twenty-four d i f f e r e n t beam combinations, which can be d i v i d e d i n t o two groups, were assigned. One group was the combinations among grades 1650f-1.5E and 2100f-1.8E, i n which grade 2100f-1.8E i s used f o r the outer laminations and grade 1650f-1.5E f o r the inner laminations. The other group was the combinations among grades 1650f-1.5E and 2400f-2.0E, i n which grade 2400f-2.0E i s used f o r the outer laminations and grade 1650f-1.5E f o r the inner laminations. The e f f e c t 57 of mixed grade layups on the bending strength f o r the two groups w i l l be discussed s e p a r a t e l y i n the f o l l o w i n g s e c t i o n s . 8.3.3.1 1650f-1.5E AND 2100f-1.8E COMBINATIONS Figures 54 to 56 are the cumulative d i s t r i b u t i o n f u n c t i o n s ( c d f ) of bending str e n g t h w i t h the combinations of grades 1650f-1.5E and 2100f-1.8E under three depths (9, 12, and 18 in c h e s ) . Figures 57 to 58 are the 5th p e r c e n t i l e and 50th p e r c e n t i l e values of bending str e n g t h as a f u n c t i o n of the outer l a y e r percent of 2100fl-1.8E. G e n e r a l l y speaking, the 5th and 50th p e r c e n t i l e values of bending str e n g t h increased with i n c r e a s i n g outer l a y e r percent of grade 2100f-1.8E (see Figure 57 and Figure 58). From Table 16, Figure 39 to Figure 41, i t shows th a t the f o l l o w i n g group t e s t s of bending strength mean values are not s i g n i f i c a n t l y d i f f e r e n t a t the 0.05 l e v e l of s i g n i f i c a n c e : BEAM9-F vs. BEAM9-G (depth - 9 inches) and BEAM18-H vs. BEAM18-I (depth = 18 inches).That means as the outer l a y e r percent of grade 2100f-1.8E increased, the bending strength would not be improved too much a t those two beam depth groups (see Figures 54 to 56). 8.3.3.2 1650f-1.5E AND 2400f-2.0E COMBINATIONS Figures 59 to 61 are the cumulative d i s t r i b u t i o n f u n c t i o n s ( c d f ) 58 of bending s t r e n g t h w i t h the combinations of grades 1650f-1.5E and 2400f-2.0E under three depths (9, 12, and 18 i n c h e s ) . F i g u r e s 62 to 63 are the 5th p e r c e n t i l e and 50th p e r c e n t i l e values of bending str e n g t h as a f u n c t i o n of the outer l a y e r percent of 2400fl-2.0E. S i m i l a r as i n the above s e c t i o n , the 5th and 50th p e r c e n t i l e values of bending str e n g t h increased w i t h i n c r e a s i n g outer l a y e r percent of grade 2400f-2.0E (see Figures 62 and 63). From Table 16, Figures 49 to 51 and Figures 59 to 61, i t shows t h a t the f o l l o w i n g t e s t of bending str e n g t h mean values are not s i g n i f i c a n t l y d i f f e r e n t a t the 0.05 l e v e l of s i g n i f i c a n c e : BEAM9-A vs. BEAM9-B (depth = 9 i n c h e s ) , BEAM12-A vs. BEAM12-B, BEAM12-B vs. BEAM12-C (depth = 12 i n c h e s ) , BEAM18-A vs. BEAM18-B, BEAM18-B vs. BEAM18-C, and BEAM18-A vs. BEAM18-C (depth = 18 in c h e s ) . That was because t h a t the outer l a y e r laminations (grades 2400f-2.0E) c o n t r i b u t e most to the beam bending s t r e n g t h , whereas the inner l a y e r laminations (grade 1650f-1.5E) do not have much i n f l u e n c e on the beam bending s t r e n g t h . As discussed i n the l a s t two s e c t i o n s , we found t h a t as long as the inner laminations (grade 1650f-1.5E) was w i t h i n 50% of the t o t a l depth the beam bending str e n g t h would not be i n f l u e n c e d adversely. I t i s i n t e r e s t i n g to see t h a t , w i t h one l a y e r of grade 2100f-1.8E or 2400f-2.0E on the outer l a y e r of the beam and the r e s t inner 59 l a y e r of grade 1650f-1.5E, the beam bending s t r e n g t h can be increased 197. or 59% (from 5287 x 10 6 p s i to 6293 x l O 6 p s i or 8406 x 10 6 p s i ) r e s p e c t i v e l y when the beam depth i s 9 inch. The beam bending s t r e n g t h can be increased 35% or 61% (from 4384 x 10 6 p s i to 5902 x 10 6 p s i or 7 0 6 4 x l O 6 p s i ) r e s p e c t i v e l y when the beam depth i s 12 in c h . With two l a y e r of grade 2100f-1.8E or 2400f-2.0E on the outer l a y e r of the beam and the r e s t inner l a y e r of grade 1650f-1.5E, the beam bending s t r e n g t h can be increased 46% or 76% (from 3771 x 10 6 p s i to 5492 x l O 6 p s i or 6655 x l O 6 p s i ) r e s p e c t i v e l y when the beam depth i s 18 in c h . The above r e s u l t s i n d i c a t e d t h a t w i t h one or two l a y e r s of higher grade lam i n a t i o n on the outer l a y e r of the beam and lower grade laminations i n the r e s t of inner l a y e r s of the beam, the glulam beam bending s t r e n g t h could be improved s i g n i f i c a n t l y . 8.3.4. COMPARISON OF BENDING STRENGTH FOR GLULAM From c u r r e n t glulam standard (CAN3-086-M84), lodgepole pine-spruce and D. f i r - l a r c h glulam beams with v i s u a l l y graded l a m i n a t i o n , grade 20f-EX and 24f-EX, have the assigned maximum bending s t r e s s f 6 = 2000 psi and f 6 = 2400 psi. The corresponding 5th p e r c e n t i l e values are 2.1x2000 = 4200 psi and 2.1x2400 = 5040 psi, r e s p e c t i v e l y (see Figures 57 and 62, h o r i z o n t a l l i n e ) . In F igure 57, i t i s shown th a t the combinations BEAM9-E, BEAM12-G and BEAM18-H (see Figures 39, 40 and 41), which are made up of 33%, 50% and 66% grade 2100f-1.8E with corresponding depths of 9, 12 and 18 60 inches r e s p e c t i v e l y , w i l l produce higher bending s t r e n g t h values than grade 20f-EX f b . I f the depth i s 9 inches, the combination BEAM9-F, which i s made up of 66% grade 2100f-1.8E, can even produce higher bending s t r e n g t h values than grade 24f-EX f f c. S i m i l a r l y , In Figure 62, i t i s shown t h a t the combinations BEAM9-C, BEAM12-C and BEAM18-D (see Figures 39, 40 and 41), which are made up of 33%, 50% and 50% grade 2400f-2.0E with corresponding depths of 9, 12 and 18 inches r e s p e c t i v e l y , w i l l produce higher bending s t r e n g t h values than grade 24f-EX ff,. The s i m u l a t i o n r e s u l t s show th a t the Spruce - P i n e - F i r MSR lumber can be used to produce layups which achieve the f 6 = 2000 psi and f 6 = 2400 psi design s t r e s s l e v e l . 8.4 TENSILE STRENGTH Table 17 shows the hypothesis t e s t s of e q u a l i t y of means f o r t e n s i l e s t r e n g t h p a r a l l e l to g r a i n of twenty-four (24) beam layup groups. I t i n d i c a t e d t h a t beam layup has a s i g n i f i c a n t e f f e c t on mean t e n s i l e strengths . 8.4.1 GRADE EFFECT Figures 64 to 66 are the cumulative d i s t r i b u t i o n f u n c t i o n s ( c d f ) of t e n s i l e s t r e n g t h of three grades (1650f-1.5E, 2100f-1.8E and 2400f-2.0E) under three depths (9, 12, and 18 in c h e s ) . 61 Figures 67 to 68 show the v a r i a t i o n of the 5th p e r c e n t i l e and 50th p e r c e n t i l e values of t e n s i l e s t r e n g t h of three grades (1650f-1.5E, 2100f-1.8E and 2400f-2.0E) as a f u n c t i o n of depths. I t i n d i c a t e d from Table 17 and Figures 64 to 68 th a t a l l of the t e n s i l e strengths of the pure grade combinations, made up of 1650f-1.5E, 2100f-1.8E, and 2400f-2.0E lumber r e s p e c t i v e l y , are d i f f e r e n t s i g n i f i c a n t l y a t the 0.05 l e v e l of s i g n i f i c a n c e . 8.4.2 DEPTH EFFECT Figure s 69 to 71 are the cumulative d i s t r i b u t i o n f u n c t i o n s ( c d f ) of t e n s i l e s t r e n g t h of three depths (9, 12, and 18 inches) under three grades (1650f-1.5E, 2100f-1.8E and 2400f-2.0E). Figures 72 to 73 are the 5th p e r c e n t i l e and 50th p e r c e n t i l e values of t e n s i l e s t r e n g t h of three depths (9, 12, and 18 inches) as a f u n c t i o n of grades. From Table 17 and Figures 69 to 73, i t shows t h a t the t e n s i l e s t r e n g t h mean values of the pure grade combinations, made up of 1650f-1.5E, 2100f-1.8E, and 2400f-2.0E r e s p e c t i v e l y , are s i g n i f i c a n t l y d i f f e r e n t ( a t or = 0.05 l e v e l ) with three beam depths (9, 12 and 18 in c h e s ) . Figure 69 and f i g u r e 72 show t h a t the 5th p e r c e n t i l e values of grade 1650f-1.5E are very c l o s e among the three depths. 62 8.4.3 EFFECT OF MIXED GRADE LAYUPS The mixed grade beam layups, shown i n Figure 49 to Figure 51, have two kinds of grade combinations among the three grades. That i s grade 1650f-1.5E combined with grade 2100f-1.8E or with grade 2400f-2.0E. I t i s discussed i n the f o l l o w i n g s e c t i o n . F i g u r e s 74 to 76 are the cumulative d i s t r i b u t i o n f u n c t i o n s ( c d f ) of t e n s i l e s t r e n g t h w i t h the combinations of grades 1650f-1.5E and 2100f-1.8E under three depths (9, 12, and 18 i n c h e s ) . Figures 77 to 79 are the cumulative d i s t r i b u t i o n f u n c t i o n s ( c d f ) of t e n s i l e s t r e n g t h with the combinations of grades 1650f-1.5E and 2400f-2.0E under three depths (9, 12, and 18 i n c h e s ) . Figures 80 to 81 are the 5th p e r c e n t i l e and 50th p e r c e n t i l e values of t e n s i l e s t r e n g t h as a f u n c t i o n of the outer l a y e r percent of grade 2100fl-1.8E. F i g u r e s 82 to 83 are the 5th p e r c e n t i l e and 50th p e r c e n t i l e values of t e n s i l e s t r e n g t h as a f u n c t i o n of the outer l a y e r percent of grade 2400fl-2.0E. Although the 5th and 50th p e r c e n t i l e values of t e n s i l e s t r e n g t h increased with i n c r e a s i n g outer l a y e r percent of grade 2100f-1.8E and 2400f-2.0E (see Figures 74 to 83), the amount of increase suddenly went up when the outer l a y e r percent of grade 2100f-1.8E and 2400f-2.0E tend towards 100 percent. This i s because t h a t t e n s i o n p a r a l l e l to g r a i n high s t r e s s e s are d i s t r i b u t e d across the t o t a l c r o s s - s e c t i o n . 63 The t e n s i l e s t r e n g t h i s dependent on the stre n g t h p r o p e r t i e s of a l l l a m i n a t i o n s , and to a c e r t a i n degree the weakest laminations c o n t r o l the s t r e n g t h . Therefore, any lower grade laminations i n the centre of the beam could reduce the beam t e n s i l e strength d r a m a t i c a l l y . 8.5 COMPRESSIVE STRENGTH Table 18 shows the hypothesis t e s t s of e q u a l i t y of means f o r compressive s t r e n g t h p a r a l l e l to g r a i n of twenty-four (24) beam layup groups. I t i n d i c a t e d t h a t the beam layup has a s i g n i f i c a n t e f f e c t on the mean compressive strengths. 8.5.1 GRADE EFFECT Figure s 84 to 86 are the cumulative d i s t r i b u t i o n f u n c t i o n s ( c d f ) of compressive str e n g t h of three grades (1650f-1.5E, 2100f-1.8E and 2400f-2.0E) under three depths (9, 12, and 18 in c h e s ) . F i g u r e s 87 to 88 are the 5th p e r c e n t i l e and 50th p e r c e n t i l e values of compressive strength of three grades (1650f-1.5E, 2100f-1.8E and 2400f-2.0E) as a f u n c t i o n of depths. I t i n d i c a t e d from Table 18 and Figures 84 to 88 t h a t a l l of the compressive strengths of the pure grade combinations, made up of 1650f-1.5E, 2100f-1.8E, and 2400f-2.0E r e s p e c t i v e l y , are d i f f e r e n t s i g n i f i c a n t l y a t the 0.05 l e v e l of s i g n i f i c a n c e . 64 8.5.2 DEPTH EFFECT Figure s 89 to 91 are the cumulative d i s t r i b u t i o n f u n c t i o n s ( c d f ) of compressive str e n g t h of three depths (9, 12, and 18 inches) under three grades (1650f-1.5E, 2100f-1.8E and 2400f-2.0E). Figures 92 to 93 are the 5th p e r c e n t i l e and 50th p e r c e n t i l e values of compressive str e n g t h of three depths (9, 12, and 18 inches) as a f u n c t i o n of grades. From Table 18 and Figures 89 to 93, i t shows t h a t the compressive str e n g t h mean values of pure grade combinations, made by 1650f-1.5E, 2100f-1.8E, and 2400f-2.0E r e s p e c t i v e l y , are s i g n i f i c a n t l y d i f f e r e n t ( a t a = 0.05 l e v e l ) with three beam depths (9, 12 and 18 inc h e s ) . 8.5.3 EFFECT OF MIXED GRADE LAYUPS The mixed grade beam layups, shown i n Figure 49 to Figure 51, have two kinds of grade combinations among the three grades. That i s grade 1650f-1.5E combined with grade 2100f-1.8E or with grade 2400f-2.0E. I t i s discussed i n the f o l l o w i n g s e c t i o n . Figures 94 to 96 are the cumulative d i s t r i b u t i o n f u n c t i o n s ( c d f ) of compressive str e n g t h with the combinations of grades 1650f-1.5E and 2100f-1.8E under three depths (9, 12, and 18 in c h e s ) . F i g u r e s 97 to 99 are the cumulative d i s t r i b u t i o n f u n c t i o n s ( c d f ) of compressive str e n g t h with the combinations of grades 1650f-1.5E and 65 2400f-2.0E under three depths (9, 12, and 18 i n c h e s ) . F i g u r e s 100 to 101 are the 5th p e r c e n t i l e and 50th p e r c e n t i l e values of compressive s t r e n g t h as a f u n c t i o n of the outer l a y e r percent of 2100fl-1.8E. Figures 102 to 103 are the 5th p e r c e n t i l e and 50th p e r c e n t i l e values of compressive s t r e n g t h as a f u n c t i o n of the outer l a y e r percent of 2400fl-2.0E. The 5th and 50th p e r c e n t i l e values of compressive s t r e n g t h increased w i t h i n c r e a s i n g outer l a y e r percent of grade 2100f-1.8E and 2400f-2.0E (see Figures 71 to 80). The amount of increase was not suddenly changed as i n the case of t e n s i l e s t r e n g t h r a t h e r a very smooth increase occurred when the outer l a y e r percent of grade 2100f-2.8E and 2400f-2.0E tend towards 100 percent. 66 9. SIZE EFFECTS ANALYSIS 9.1 INTRODUCTION In t h i s chapter, the s i z e e f f e c t s i n bending, t e n s i l e and compressive strengths have been considered. S i z e e f f e c t i n lumber have been observed by Madsen from bending t e s t performed a t d i f f e r e n t lengths, f o r constant depths and the same load i n g p a t t e r n . Madsen concluded t h a t the observed s i z e dependence could be e x p l a i n e d by the changes i n length, and proposed an a d j u s t i n g f a c t o r based on the a p p l i c a t i o n of Weibull weakest l i n k theory. Thus, lengths and i 2 - The parameter k was found to be approximately 4.2 by c a l i b r a t i o n of Eq. (9.1) to Madsen's experimental data. In recent years, the use of volume-effect f a c t o r i s g e t t i n g more and more popular. This volume-effect f a c t o r accounts f o r a l l three parameters of volume: width, depth and length. For glulam beams of equal width subjected to the same loa d i n g c o n f i g u r a t i o n , the strengths and L\u2014 +-CO > \"co co a> k_ Q. E o O o oo 3 0 40 60 Test data Normal fit Lognormal fit 2P Weibull fit 3P Weibull fit _ i 6 0 70 Compressive strength (MPa) Figure 12 CDF of test and fitted board mean of compressive strength 0 2 4 6 8 10 Standard deviation (MPa) Figure 13 CDF of test and fitted board S.D. of compressive strength M x ) t Length of the board Figure 14 Ensemble of MOE along the length of the board E\u201e(x) 0.0 mi Ai 1 \/V n WOT \u2022 E (x) 0.0 J. k. Length of the board Figure 15 Ensemble of zero mean MOE along the length zn 113 1 2 3 4 Length (1000 mm) Figure 17 Ensemble average of test E-profile 2 0 0 0 Modulus of Elasticity (1000 MPa) Figure 19 CDF of test and fitted board mean of MOE I\u2014\u00bb ON Figure 20 CDF of test and fitted board standard deviation of MOE r\u2014\u00bb r\u2014' -0 6 0 0 5 0 0 Figure 21 Ensemble average of the amplitude spectrum 00 Figure 22 Generated E-profile of one board 16 Test data Generated data 0 1 2 3 4 5 Length (1000 mm) Figure 23 Ensemble average of test and generated E-profiles 500 0 1 2 3 4 5 Length (1000 mm) Figure 24 Ensemble standard deviation of test and generated E-profiles to 7 0 0 6 0 0 5 0 0 4 0 0 3 0 0 2 0 0 1 0 0 Test data Simulated data 10 2 5 3 0 15 2 0 F r e q u e n c y (co) F i g u r e 2 5 T e s t a n d s i m u l a t e d e n s e m b l e a v e r a g e o f a m p l i t u d e s p e c t r a 3 5 65 10 0 500 1000 1500 2000 Board standard deviation of MOE (MPa) Figure 27 Regression plot of board S.D. MOE vs. compressive strength Z X | in standard Xi in real distribution normal distribution ( X , , Y , ) 1 \"\"' 1 - r \" i \u2022 -Y, in real distribution Z Y l in standard normal distribution Figure 28 Graphical demonstration of the transformation between the real space and standard normalized space 65 CO Q_ c > to CD Q. E o O 60 55 50 45 40 35 30 \u2022 \u2022 \u2022 Fitted regression line C = 8.0343 + 0.0027 x E P = 0.654 25 10 12 14 16 18 Modulus of elasticity (10 MPa) 20 Figure 29 Regression plot of test MOE vs. within-board compressive strength ON CO D) C 2 C O > 00 00 CD CL E o O 65 60 55 5 0 45 40 3 5 3 0 25 \u2022 10 Jo\"*1 \u2022 qb \u2022 \u2022 \u2022 A Ig A (DD \u2022 \u2022 AP QA \u2022 HA A A A \u2022 \u2022 \u2022 Test data (p = 0.654 ) A Simulated data (p = 0.649 ) 12 14 16 18 2 0 Modulus of elasticity (103 MPa) Figure 30 Regression plot of test and simulated MOE vs. compressive strength to 1 0.8 \"8 0.6 O I 0.4 \u2022 E o 0.2 R c at 50 percentile - 1.15 1.2 Ratio R c Figure 31 Cumulative distribution function of ratio R c 1.4 S 3 CXI \/ R T at 50 percentile - 1.20 1.4 Ratio R T Figure 32 Cumulative distribution function R T 1.9 Tensile strength (MPa) Figure 33 CDF of test and generated minimum tension data (1650f-1.5E) h-\u00bb O 0 20 40 60 Tensile strength (MPa) Figure 34 CDF of test and generated minimum tension data (2100f-1.8E) Tensile strength (MPa) Figure 35 CDF of test and generated minimum tension data (2400f-2.0) Compressive strength (MPa) Figure 36 CDF of test and generated minimum compression data (1650f-1.5E) Compressive strength (MPa) Figure 37 CDF of test and generated minimum compression data (21 OOf-1.8E) 4^ J ' '\/ Compressive strength (MPa) Figure 38 CDF of test and generated minimum compression data (2400f-2.0E) Figure 39 Beam layups for three grade combinations - 9 in. beam Figure 40 Beam layups for three grade combinations - 12 in. beam Figure 41 Beam layups for three grade combinations - 18 in. beam Figure 42 Beam with two stiffness zones 2.2 1.4 1.6 1.8 2 2.2 Trnasformed MOE (10* psi) Figure 43 Comparison of simulated MOE with transformed MOE Bending strength (1000 psi) Figure 44 CDF of bending strength for three grades (depth = 9 inch) 1 3 5 7 9 11 13 15 17 19 Bending strength (1000 psi) Figure 45 CDF of bending strength for three grades (depth = 12 inch) Bending strength (1000 psi) Figure 46 CDF of bending strength for three grades (depth = 18 inch) 10 9 8 7 6 5 4 3 2 1 sa 2400f-2.0E + 2100M.8E o 1650f-1.5E J L J I I L 11 13 Depth (inch) 15 17 Figure 47 The 5th percentile value of bending strength vs. depth 4^ m 2400f-2.0E + 2100f-1.8E o 1650M.5E J I I I L _ I I I I L 9 11 13 15 17 Depth (inch) Figure 48 The 50th percentile value of bending strength vs. depth Bending strength (1000 psi) Figure 49 CDF of bending strength for 1650f-1.5E beams as a function of beam depth ON Bending strength (1000 psi) Figure 50 CDF of bending strength for 2100f-1.8E beams as a function of beam depth 1 3 5 7 9 11 13 15 17 19 Bending strength (1000 psi) Figure 51 CDF of bending strength for 2400f-2.0E beams as a function of beam depth CC ii DEPTH=9\" + DEPTH=12\u00b0 o DEPTH 18\" 2400f-2.0E 2100f-1.8E Grade 1650M.5E Figure 52 The 5th percentile value of bending strength vs. grade 10 I _ J _ : I I I 2400f-2.0E 2100M.8E 1650M.5E Grade Figure 53 The 50th percentile value of bending strength vs. grade Figure 54 CDF of bending strength with the combination of 1650f-1.5E and 2100f-1.8E (depth = 9 inch) Bending strength (1000 psi) Figure 56 CDF of bending strength with the combinations of 1650M.5E and 2100f-1.8E (depth = 18 inch) Figure 57 The 5th percentile value of bending strength vs. the outer layer percent of 2100f-1.8E U l ii DEPTH=9\" + DEPTH=12\" o DEPTH=18\" J i i i i i i i i i i 0 20 40 60 80 100 Outer layer percent of 2100f-1.8E (%) Figure 58 The 50th percentile value of bending strength vs. the outer layer percent of 2100f-1.8E Bending strength (1000 psi) Figure 59 CDF of bending strength with the combination of 1650M.5E and 2400f-2.0E (depth = 9 inch) U l ON Bending strength (1000 psi) Figure 60 CDF of bending strength with the combination of 1650f-1.5E and 2400f-2.0E (depth = 12 inch) Bending strength (1000 psi) Figure 61 CDF of bending strength with the combination of 1650f-1.5E and 2400f-2.0E (depth = 18 inch) Figure 62 The 5th percentile value of bending strength vs. the outer layer percent of 2400f-2.0E vo 01 DEPTH=9\" + DEPTH=12\" o DEPTH=18\" J I I I I I I I I L _ 0 20 40 60 80 100 Outer layer percent of 2400f-2.0E (%) Figure 63 The 50th percentile value of bending strength vs. the outer layer percent of 2400f-2.0E Tensile strength (1000 psi) Figure 64 CDF of tensile strength for three grades (depth = 9 inch) Tensile strength (1000 psi) Figure 65 CDF of tensile strength for three grades (depth = 12 inch) 2400f-2.0E CT18-A) 2100f-1.8E (T18-I) 1650M.5E rri8-F) 3-P Weibull fitting 1 3 5 7 9 Tensile strength (1000 psi) Figure 66 CDF of tensile strength for three grades (depth = 18 inch) SB 2400f-2.0E + 2100M.8E o 1650M.5E Depth (inch) Figure 67 The 5th percentile value of tensile strength vs. depth 2400f-2.0E + 2100f-1.8E o 1650f-1.5E 11 13 15 Depth (inch) 17 Figure 68 The 50th percentile value of tensile strength vs. depth U l Tensile strength (1000 psi) Figure 69 CDF of tensile strength under three depth (1650f-1.5E) Tensile strength (1000 psi) Figure 70 CDF of tensile strength under three depth (2100f-1.8E) Figure 72 The 5th percentile value of tensile strength vs. grade Figure 73 The 50th percentile value of tensile strength vs. grade o Tensile strength (1000 psi) Figure 77 CDF of tensile strength with the combination of 1650f-1.5E and 2400f-2.0E (depth = 9 inch) ^ Tensile strength (1000 psi) Figure 78 CDF of tensile strength with the combination of 1650M.5E and 2400f-2.0E (depth = 12 inch) Tensile strength (1000 psi) Figure 79 CDF of tensile strength with the combination of 1650M.5E and 240W-2.0E (depth = 18 inch) Figure 80 The 5th percentile value of tensile strength vs. the outer layer percent of 2100f-1.8E Figure 81 The 50th percentile value of tensile strength vs. the outer layer percent of 2100f-1.8E i\u2014> oo 20 40 60 80 Outer layer percent of 2400f-2.0E (%) 100 Figure 82 The 5th percentile value of tensile strength vs. the outer layer percent of 2400f-2.0E \u2022 DEPTH=9\" + DEPTH=12\" o DEPTH=18\" Figure 83 The 50th percentile value of tensile strength vs. the outer layer percent of 2400f-2.0E Compressive strength (1000 psi) Figure 85 CDF of compressive strength for three grades (depth = 12 inch) Figure 87 The 5th percentile value of compressive strength vs. depth oo + 2100f-1.8E o 1650f-1.5E \u2014 \u201e J i i \u00ab 1 7 Depth (inch) Figure 88 The 50th percentile value of compressive strength vs. depth Compressive strength (1000 psi) Figure 89 CDF of compressive strength under three depth (1650f-1.5E) 00 ON Compressive strength (1000 psi) Figure 90 CDF of compressive strength under three depth (2100f-1.8E) Compressive strength (1000 psi) Figure 91 CDF of compressive strength under three depth (2400f-2.0E) 0 0 oo 7 6 5 4 3 2 h \u2022 DEPTH=9\" + DEPTH = 12\" o DEPTH 18\" 2400f-2.0E 2100M.8E Grade 1650f-1.5E Figure 92 The 5th percentile value of compressive strength vs. grade 10 8 7 6 5 4 3 2 h 1 \u2022 DEPTH=9\" + DEPTH= 12\" o DEPTH 18\" 24001-2.0E 2100M.8E Grade 1650f-1.5E Figure 93 The 50th percentile value of compressive strength vs. grade VO o Compressive strength (1000 psi) Figure 94 CDF of compressive strength with the combination of 1650f-1.5E and 2100M.8E (depth = 9 inch) Compressive strength (1000 psi) Figure 95 CDF of compressive strength with the combination of 1650M.5E and 2100M.8E (depth = 12 inch) Compressive strength (1000 psi) Figure 96 CDF of compressive strength with the combination of 1650M.5E and 2100M.8E (depth = 18 inch) vo Compressive strength (1000 psi) Figure 97 CDF of compressive strength with the combination of 1650f-1.5E and 2400f-2.0E (depth = 9 inch) vo 4*. Compressive strength (\\0Q0 psO Figure 98 CDF of compressive strength with the combination of 1650M.5E and 2400f-2.0E (depth = 12 inch) Compressive strength (1000 psi) Figure 99 CDF of compressive strength with the combination of 1650f-1.5E and 2400f-2.0E (depth = 18 inch) ON Figure 100 The 5th percentile value of compressive strength vs. the outer layer percent of 2100f-1.8E Figure 101 The 50th percentile value of compressive strength vs. the outer layer percent of 2100f-1.8E B DEPTH=9' + DEPTH=12\" o DEPTH 20 40 60 80 Outer layer percent of 2400f-2.0E (%) Figure 102 The 5th percentile value of compressive strength vs. the outer layer percent of 2400f-2.0E Figure 103 The 50th percentile value of compressive strength vs. the outer layer percent of 2400f-2.0E ","attrs":{"lang":"en","ns":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","classmap":"oc:AnnotationContainer"},"iri":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","explain":"Simple Knowledge Organisation System; Notes are used to provide information relating to SKOS concepts. 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