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The Tsai-Wu strength theory for Douglas-fir laminated veneer Clouston, Peggi 1995

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The Tsai-Wu Strength Theory for Douglas-fir Laminated Veneer by Peggi Clouston B . A . S c , The University of British Columbia, 1989 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES The Department of Forestry We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December, 1995 © Peggi Clouston, 1995 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 DE-6 (2/88) Abstract This study investigates the determincrtion and use of a multi-axial failure criterion for Douglas-fir lcrrninated veneer. Unlike previous studies on failure theories, this study treats strength parameters as random variables as opposed to detenninistic variables. Also, size effect has been incorporated in the failure theory implementation. A comparison has been made between four established orthotopic failure theories based on off-axis tensile test data to determine the most appropriate theory of the four considered; Tscd-Hill, Norris, Tscri-Wu and Tan-Cheng theories. The Tsai-Wu tensor polynomial theory has been shown to best predict the mean values of the off-axis data considering both practicality and accuracy of the strength criteria. A non-linear mirurnization technique has been developed considering the strength parameters of the Tscri-Wu criterion to be random variables to approximate the mean and standard deviation of the interaction parameter, F12. The same statistical approach has been used to approximate a size effect adjustment factor to account for the difference in stressed volumes between the shear block specimens and the off-axis specimens. A sensitivity analysis has been conducted on the interaction parameter, F12. This ii study indicates that the data from the 15 degree off-axis tensile tests is more reliable than that of the other angles tested, 30, 45 and 60 degrees, in establishing the most accurate value for F12. Also, the first and second quadrants of the stress space are found to be the least sensitive to variations in F12. That is, small inaccuracies in the data obtained from tests producing these stress combinations could lead to significant errors in the effectiveness of the Tscd-Wu criterion in the third and fourth quadrants. The Tsai-Wu failure criterion has been coupled with finite element analyses in a simulation procedure to estimate the cumulative probability distribution for failure load of 30 and 45 degree off-axis 3 point bending specimens. A load configuration effect has been included in the prediction model to account for the brittle strengths, tension parallel and perpendicular to grain and shear having been developed in a uniform stress configuration. Two approaches, using Weibull weakest link theory, have been investigated to incorporate the load configuration effect. Both models provide reasonable accuracy in predicting the off-axis failure load when compared to experimental results. An alternative, less versatile approach to predicting failure load for the off-axis bending application has also been studied. This approach entails using the Tsai-Wu failure criterion to first predict off-axis tensile strengths and then, using Weibull formulation, adjusting these tensile strengths to predict off-axis bending strengths. This prediction model is also corroborated by the experimental results. iii Table of Contents Page Abstract ii Table of Contents iv List of Tables vii List of Figures viii Acknowledgement x Chapter 1 1. Introduction 1 Chapter 2 2. Background 8 2.1 Failure Theory Review 8 2.1.1 Maximum Distortional Energy Theory 8 2.1.2 Orthotropic Failure Theories 9 Hankinson Formula 9 Hill Theory 11 Tsai-Hill Theory 12 Norris Theory 13 Tsai-Wu Theory 14 Tan Theory 18 Tan-Cheng Theory 20 2.1.3 Comparison of Failure Theories 21 2.2 Interaction Term, F12of Tsai-Wu Theory 24 2.2.1 Significance of Interaction Term, F12 24 2.2.2 Evaluation of Interaction Term, F 1 2 25 2.3 Applications of Tensor Polynomial Strength Theory 29 iv Page Chapter 3 3. Determination of Most Appropriate Failure Theory for Laminated Veneer 31 3.1 Experimental 31 3.1.1 Material 31 3.1.2 Test Methods 34 Tension Tests . 34 Compression Tests 35 Shear Block Tests 36 3.1.3 Experimental Results 3 7 3.2 Analytical 46 3.2.1 Size Effects 46 3.2.2 Determination of Interaction Component, F12 53 Non-Linear Weighted Least Squares - Using SAS 53 Non-Linear Weighted Least Squares - Using F12FIT 56 Comparison of F12 Approximation Results 59 3.3 Comparison of Failure Theories 60 3.4 More on Tsai-Wu Theory 63 3.4.1 Size Effect for Shear Treated as a Random Variable 63 3.4.2 Sensitivity of Fj2 to Off-axis Experimental Data 66 3.4.3 F12 and the Strength Envelope of Laminated Veneer 73 Chapter 4 4. Verification of Tsai-Wu Failure Theory 75 4.1 Load Configuration Effect 75 4.2 Analytical Methods 76 4.2.1 Monte Carlo Simulation 76 4.2.2 Direct Approach - Coupling Finite Element Analysis with Tsai-WuTheory 77 Finite Element Analysis 77 Adjusting Brittle Strengths for Volume About Each Gauss Point (Method 1) 78 Adjusting Brittle Strengths for Volume Within One Finite Element (Method 2) 81 v Page 4.2.3 Indirect Approach - Load Configuration Effect Applied to Predicted Off-Axis Tensile Values (Method 3) 83 4.3 Experimental Off-Axis Bending Tests - Materials and Methods 89 4.3.1 Materials 89 4.3.2 Test Method 89 4.4 Results and Discussion 90 4.4.1 Off-Axis Bending Test Results 90 4.4.2 Comparison of Prediction Models and Experimental Results 91 Results of the Direct Approach 91 Results of the Indirect Approach 97 Chapter 5 5. Conclusion 100 5.1 Summary and Conclusions 100 5.2 Future Research 102 References 103 Appendix A 107 vi List of Tables Page Table 2.1 Comparison of Failure Theory Terms 21 Table 3.1 Statistical Summary of Experimental Results Table 3.2 Summary of F12 Approximations Table 3.3 Comparison of Strength Theories 38 59 60 Table 4.1 Statistical Summary for Off-Axis Bending Results 90 Table 4.2 Comparative Data Between Predicted and Experimental Results for Method 1 *(Shape Parameter from Maximum Likelihood -Corresponds to Figure 4.5) 93 Table 4.3 Comparative Data Between Predicted and Experimental Results for Method 1 *(Shape Parameter from COV Approximation -Corresponds to Figure 4.6) 93 Table 4.4 Comparative Data Between Predicted and Experimental Results for Method 2 *(Shape Parameter from Maximum Likelihood -Corresponds to Figure 4.7) 96 Table 4.5 Comparative Data Between Predicted and Experimental Results for Method 2 *(Shape Parameter from COV Approximation -Corresponds to Figure 4.8) 96 Table 4.6 Comparative Data Between Predicted and Experimental Results for Method 3 *(Shape Parameter from Maximum Likelihood Fit of Simulated Tension Strengths - Corresponds to Figure 4.9) 99 Table 4.7 Comparative Data Between Predicted and Experimental Results for Method 3 *(Shape Parameter from Maximum Likelihood Fit of Experimental Bending Strengths - Corresponds to Figure 4.10) 99 vii List of Figures Page Figure 1.1 Three Principal Axes of Laminated Veneer Lumber 4 Figure 1.2 Off Axis Tension TestforLVL 5 Figure 2.1 Rotation of Material Axes 16 Figure 2.2 Comparison of Failure Theory Envelopes 23 Figure 2.3 Theoretical Strength Envelopes of Paperboard with Different Values of Coefficient F 1 2 (from Suhling et. al., 1984) 25 Figure 2.4 Positive and Negative 45 Degree Off-Axis Shear 25 Figure 3.1 Specimen Cutting Layout 32 Figure 3.2 Perpendicular to Grain Compression Tests 33 Figure 3.3 Tension Specimen in Metriguard Testing Machine 35 Figure 3.4 Shear Block Dimensions and Configuration 36 Figure 3.5 Cumulative Distributions of Laminated Veneer Strength in the Longitudinal Direction 39 Figure 3.6 Cumulative Distributions of Laminated Veneer Strength in the Transverse Direction 40 Figure 3.7 Cumulative Distribution of Laminated Veneer Shear Block Strength 41 Figure 3.8 Cumulative Distribution of Laminated Veneer Off-Axis Strength 42 Figure 3.9 Tension Specimen Failure Modes 43 Figure 3.10 Sample Stress/Strain Diagram for Compression Parallel to Grain 44 Figure 3.11 Stress/Strain Diagram for Typical Shear Block Specimen 45 Figure 3.12 Finite Element Mesh for Shear Block Specimen 49 Figure 3.13 Shear Stress Distribution for Shear Block Specimen 51 Figure 3.14 Failure Theories vs Experimental Data 61 Figure 3.15 Tsai-Wu (Random Variable) Model vs Experimental Off-Axis Tension Results 57 Figure 3.16 Cumulative Distribution of F12 for Each Angle to Grain 67 Figure 3.17 Sensitivity of F12 for Each Angle to Grain 67 Figure 3.18 Influence of End Constraint in the Testing ofAnisotropic Bodies (from Pagano and Halpin, 1967) 68 viii Page Figure 3.19 Sensitivity of F12 with Variation in Tension Perpendicular to Grain Stress (for Each Angle to Grain) 70 Figure 3.20 Tsai-Wu (Random Variable) Model vs. Experimental Off-Axis Tension Results for 15 Degree Data Only 72 Figure 3.21 Tsai-Wu Failure Envelope with Varying Values for F12 74 Figure 4.1 Finite Element Mesh and General Setup for Off-Axis Bending Specimen Analysis 77 Figure 4.2 Regions Surrounding Gaussian Integration Points 79 Figure 4.3 Region of Uniform Stress for Size Effect Factor 81 Figure 4.4 Tsai-Wu (Random Variable) Model Prediction of Off-Axis Tension Strengths 87 Figure 4.5 Cumulative Distribution Function for Off-Axis Bending Failure Load Predicted vs Experimental (Method 1, k from Maximum Likelihood) 92 Figure 4.6 Cumulative Distribution Function for Off-Axis Bending Failure Load Predicted vs Experimental (Method 1, k from COVApproximation) 92 Figure 4.7 Cumulative Distribution Function for Off-Axis Bending Failure Load Predicted vs Experimental (Method 2, k from Maximum Likelihood) 95 Figure 4.8 Cumulative Distribution Function for Off-Axis Bending Failure Load Predicted vs Experimental (Method 2, kfrom COV Approximation) 95 Figure 4.9 Cumulative Distribution Function for Off-Axis Bending Failure Load Predicted vs Experimental (Method 3, k from Simulated Tension Strengths) 98 Figure 4.10 Cumulative Distribution Function for Off-Axis Bending Failure Load Predicted vs Experimental (Method 3, k from Experimental Bending Results) 98 i x Acknowledgement I would like to mcrnk Drs. J. D. Banrett and F. Lam for their guidance and support throughout the research. Also, gratitude is extended to Drs. R. Vaziri and H. Prion for reviewing and providing feedback on the thesis while serving on the final examining committee. Acknowledgement goes to both Weyerhaeuser Ltd. and Forintek Canada Corporation for providing financial support through their wood design and wood science fellowships. Finally, Ainsworth Lumber Canada Ltd. and the Department of Wood Science, U.B.C. are thanked for contributing materials and providing equipment for this research. 1. Introduction Lcrminated Veneer Lumber, LVL is a common structural wood composite manufactured by glue larrnnating rotary peeled wood veneers together with the grain of all plies oriented in the longitudinal direction. This process disperses the natural strength reducing flaws of wood, such as knots and slope of grcdn, throughout the composite resulting in a consistent material with highly uniform strength properties. Commercially made LVL is produced from a continuous veneer layup process enabling the production of relatively wide and/or long structural components when compared to that attcrmable by conventional lumber. With its high consistency and reduced restrictions for size, LVL is often utilized in high, complex stress applications, competing with materials traditionally considered for commercial construction: those being primarily steel and concrete. Most frequently, it is used in both residential and commercial building as mcdn carrying beams but it is also designed for use as secondary members, columns, truss chords, wood I-joist flanges and scaffold planking. Structural design of LVL, or any material, depends primarily on the ability to accurately quantify material strength and variability. The uniaxial and shear strength of LVL has been thoroughly investigated by numerous researchers since the early 1970s (J.C.Bohlen, 1974, 1975; R. Kunesh, 1978; J.A. Youngquist et. al., 1984; N. Hesterman & T. Gorman, 1992). In these types of experiments, specimens are subjected to uniaxial stress 1 (tension, compression or pure shear) along the material symmetry axes and the corresponding principal axis strength is the stress at which failure occurs. However, most practical applications, such as those described above, involve multi-axial stress states, in which parallel and perpendicular to the grain normal and shear stresses act simultaneously. In this case of multiaxial loading, member capacity can be predicted through use of a multiaxial strength criterion (or failure theory). There are numerous strength theories available in the literature which endeavour to predict material strength in the combined-stress state. Many of these theories have been compiled and compared in surveys by researchers such as Sandhu (1972), Rowlands (1985), and Nahas (1986). Failure theories are, in general, mamematical models which incorporate uniaxial strength data to provide a relatively simple method to predict the onset of material fracture in brittle materials or material yielding in ductile materials. Failure criteria may be determined theoretically by rational modelling of the material's physical characteristics or empirically by simply representing experimental observations. In all present cases, the theories are termed phenomenological 'treating the heterogeneous material as a continuum' and mcrking no attempt to explain the mechanisms which lead to material failure (Wu, 1974). 2 Ecrrly classical theories were based on luting a physical variable such as stress, strain, or strain energy assuming a homogeneous and isotropic1 material. For example, one of the first theories, the Maximum Normal Stress theory, mcrintains that material failure occurs when the mcrximum normal stress reaches a critical value independent of other stresses at that point. The critical value can be determined from a uniaxial tensile test. Three simple, independent equations result: = oc, o 2 = oc, o 3 =a c , where ai (i = 1,2,3) denote principal stresses and o c is the critical stress. These equations may be plotted with each normal stress component constituting an cods forrriing a triad in the stress space. This is known as a failure envelope (or surface). Combinations of stresses contained inside the failure envelope signify survival and on or beyond the envelope indicate material failure. The concept of a failure envelope is common for both isotropic and anisotropic failure criteria representing, in general, the boundary between survival and failure. For orthotropic materials, the plotted stresses usually correspond to those along the principal material axes. Isotropic materials possess infinitely many planes of material symmetry and thus assume the same mechanical properties in all material directions. In contrast, anisotropic materials exhibit no symmetry and have different properties in all six directions of the stress tensor. Finally, orthotropic materials have symmetry about three mutually perpendicular (orthogonal) planes and show different properties in these three directions. 3 Orthotopic failure theories are normally written in terms of material strengths and stresses in the principal material directions. Owing to the fact that orthotropic material strengths are directionally dependent, orthotropic failure theories are generally more complex than isotropic criteria. Where an isotropic material has three independent strengths, (tension, compression, and shear) an orthotropic material, assuming plane stress2, has five independent strengths in the principal material directions. Like wood, laminated veneer lumber is considered to be an orthotropic material with three natural axes at right angles to each other. Referring to Figure 1.1, the five independent strengths are: tension and compression parallel to the direction of the wood fiber Xt and tension and compression transverse to the direction of wood fiber, Y t and Y c, and shear in this same plane, S. axis 1 Qongitudincd) Figure 1.1 - Three Principal Axes ofLaminated Veneer Lumber 2 In assuming a plane state of stress, it is assumed that the thickness of the material is small enough to neglect any normal or shear stresses that may develop through it. 4 As LVL is both a wood product and a composite material, the failure theories considered for this study have either been proven effective in predicting wood failure or have been commonly used with composite materials. The composite failure theories are based on the material properties of the lomina which is under a state of plane stress. As such, the failure surfaces for this study are described in a two dimensional stress space. For composite materials, a biaxial stress state is often evaluated by a uniaxial off-axis test; that is, a test which measures either the compressive or the tensile strength of the material with the direction of applied stress at an angle to that of the material's ncrtural longitudinal axis. In so doing, the uniaxial stress produces two normal stresses and a shear stress component thereby acting in a complex stress state with respect to the material's principal axes. Referring to Figure 1.2, the applied stress a e produces the following stresses along the principal material directions: t a, = (1.1) Si Applied Stress, a t Figure 1.2 - Off-Axis Tensile Test for LVL 5 Evcduation of a matericd's fcrilure envelope is necessary to better understand the material's mechanical behaviour and leads to safer, more reliable structural design. In the case of composite materials, whose constituent materials can be manipulated, it may foster development of future composites. This thesis is intended as a preliminary investigation into assessing the failure envelope of LVL It should be noted, however, that unlike commercial LVL, the material tested in this study did not contain butt joints. As a result, the parallel and perpendicular to grain strength is understandably different from those in published studies of commercial LVL This study of simply 'lcrrninated veneer', is a requisite first step in assessing the mechanical behaviour of lcrminated veneer lumber. Future studies may incorporate these results, in concert with results for butt joint failure, to ultimately predict the strength characteristics of commercial LVL. 6 The overall objective of this study is to provide fundamental information for estabHshing an appropriate failure criterion for Douglas Fir Ixrminated Veneer Lumber. This was carried out in two stages : 1) Determination of an appropriate failure theory through comparison of off-axis tensile test results with several commonly used orthotropic failure theories. The theories evaluated were: Hcrnkinson3, Tscri-Hill, Norris, Tsai-Wu and Tan-Cheng theories. 2) Application and verification of the theory chosen in stage (1) to a separate condition of biaxial loading. Application of the failure theory involved computer simulations of an off-axis beam in 3-point bending. This application was verified through a series of experimental tests. 3 As discussed later, Hankinson's theory is not a rigorous failure theory for all loading combinations. It is included here for comparative as well as background information purposes. 7 2. Background 2.1 Failure Theory Review As previously alluded, early failure criteria, dcrting as far back as the 17th century, were written for homogeneous and isotropic materials. Failure criteria for anisotropic and orthotropic materials were explored subsequently (early 1900's) and were therefore often extensions of isotropic theories. Two classic composite material theories discussed in this paper are based on the well known Mcodmum Distortional Energy Theory. To gain a better understanding of these composite theories and to lend a historical as well as comparative perspective of the ensuing orthotropic failure criteria, this isotropic theory will be reviewed. 2.1.1 Maximum Distortional Energy Theory (circa 1920) The Huber-Henky-von Mises Maximum Distortion Energy theory predicts that an isotropic, ductile material will yield when a lirmting value for strain energy due to sheering distortion of an element is reached. The luting, or critical value may be found from a uniaxial tensile test and equated to the distortion strain energy for a combined stress situation. The resulting simplified equation in three dimensions is: 2 o c 2 - {a.-aj* ( o 2 - o 3 ) 2 * ^ - o , ) 2 (2.1) 8 where Oi (i = 1,2,3) are principal normal stresses and o c is the critical yield stress. This equation defines the values of the combined normal stresses at failure. The equation of the failure envelope under plane stress reduces to: a, o„ (2.2) 2.1.2 Orthotropic Failure Theories Hankinson Formula (1921) One of the earliest and perhaps most commonly used strength equations for wood is an empirically based formula developed by R.L. Hankinson (Hankinson, 1921). His original study set out to describe the compressive strength of spruce as it varies with angle to grain. Hankinson discovered that the formula could also apply to other wood species, as has been substantiated by several other studies since including those by Rowse(1923), Norris (1939) and Goodman/Bodig (1971). Hankinson's formula is : o XY 6 " Xsxf% * Y c o s n e *2-3) where (referencing Figure 1.1) X and Y are the parallel and perpendicular to grain compressive strengths, respectively, 0 is the angle between the direction of loading and 9 the loncritudincrl axis, o e is the compressive strength in the direction of loading, and 'n' denotes the trigonometric exponent which equals 2 in the original Hankinson's formula. As reported by Kollman and C6te (1968), a 1939 study by Kollman proposed a slight modification to this formula to render it applicable for tensile strength. It was shown that a modified Hankinson equation with the trigonometric exponent, n, between 1.5 and 2.0 could predict off-axis tensile strength with reasonable accuracy. While the Hankinson formula (both versions) has been widely accepted and used in wood design since its introduction in 1921, the formula is not a complete strength theory in that it only provides a prediction of failure for specimens tested at an angle to grain. Nevertheless, it is being recognized in this study because 1) it has had repeated success in describing the strength of wood as a function of grcrin angle, 2) it has been utilized in other failure criteria (as discussed in section 2.2.2), and 3) it provides an mteresting comparison for the off-axis tension data results. 10 Hill Theory (1948) R. Hill (1948) generalised the maximum distortion energy theory for use with ductile, specially orthotropic4 materials. Considering the three directions of material symmetry and assuming no difference in tension and compression strength, Hill reworked equation (2.2) for orthotropic material yielding under plane stress to produce the failure criterion: (2.4) where, X, Y and Z are the material strengths in the longitudinal, transverse and through-thickness directions respectively, S is the in-plane shear strength ,ax and a 2 are the applied normal stresses in the corresponding principal material directions and a6is the in-plane shear stress (reference Figure 1.1). Hill's theory is not commonly used for composite materials and is not being evaluated for lcariinated veneer in this study. It has been included in this discussion as a prelude to the Tsai-Hill criterion. 4 axes. For a specially orthotropic material, the applied stresses correspond to the principal material 11 Tsai-Hill Theory (1965) As previously noted, lcrminate strength is commonly characterised by lamina strength for composite theories. Azzi and Tscd adapted Hill's theory to composite materials by assuming the through-thickness strength of equation (2.4) to be equal to the perpendicular to grain strength, (ie: Y = Z ), which is appropriate for transversely isotropic laminae. The resulting strength criterion to predict lcmina failure is referred to as the Tsai-Hill (or Azzi-Tsai) theory. The failure criterion for in-plane stress is : / > 2 / \ _^2_ 2 / \ ,x t I x1 J ,y > ^ 1 (2.5) In contrast to Hill's theory, this criterion does account for differences in tension and compression strength by employing the corresponding strength to the prevailing stress components. By substituting the general transformation relationships of equation (1.1) into equation (2.5) one can calculate the Tsai-Hill off-axis strength, a e . cos4e s2 1 I i 2 o. _2 a sin4 0 — sin 6 cos2 0 + (2.6) 12 Norris Theory (1950) The mcrximum distortion energy theory was also used by C.B. Norris in 1950 to develop a strength criterion for orthotropic materials.5 His theory applied the isotropic failure criterion to a simplified geometrical model and defines failure through the following three expressions for a plane stress state: 2 ( \ 2 ( \ ° 2 ° 1 ° 2 l y I XY X i 1 or k 1 ^ 1 (2.7) Again, the parallel and perpendicular strengths may be either tension or compression depending on the sign of the corresponding applied load. By his own admission, Norris's approach was not 'rigorously correct', representing an orthotropic material by an isotropic material with regularly spaced voids. Nonetheless, his off-axis tensile strength equation, cos4 8 S2 X Y sin2 8 cos2 8 + sin4 8 (2.8) 5 plates. This theory succeeded his so called 'interaction formula' suggested in 1945 for use with plywood 13 produced good results when compared to data for 3 ply plywood and fiberglass laminate for which most of the glass fibers were aligned in one direction (Norris, 1962). Norris's theory has been recommended for design of glued-laminated beams in the Timber Construction Manual (1974). However, it was also criticized for being overly conservative for many beam and arch situations in practice (Kobetz and Krueger, 1976). Tsai-Wu Theory (1971) Another approach to developing an anisotropic strength criterion was proposed by Tscd and Wu in 19716. Their theory describes the combined-stress fcrilure surface as a quadratic polynomial, ¥iai + Fijaiai •> 1 , in index notation, where Fjcmd FyCrre strength tensors and i,j = 1,2, ... 6. According to this theory, the failure surface of an orthotropic material in a 2-dimensional stress state, (for example, the 1-2 plane in Figure 1.1) where o3= o4= o5=0, is calculated by the refined equation F i° . + F*° 8- F . X • F*°l + 2 F n ° i ° > * F*°\ - 1 (2.9) The coefficients Fj through F 6 6, with the exception of F 1 2, are described in terms of the strengths in the principal material directions. Considering a uniaxial tension load on a A more complex version of this criterion, using higher order terms, was suggested previously by Gol'denblat and Kopnov, 1966. 14 specimen in the 1 direction, the above equation at failure becomes (2.10a) and for compression is F i X c * F n x c • 1 (2.10b) letting the subscripts, t and c, represent tension and compression respectively. By solving the equcrtions 2.10a and 2.10b simultaneously and regarding the compression strength as negative, the expression for the strength parameters F, and F n are found to be F . . JL - -L X> X ° (2.11a) F - —-— 11 XtX t c Through similar mathematical manipulations, it can be shown that y , * (2.11b) F - _ L _ 1 22 F - J L 6 6 S 2 15 A significcmt feature of the tensor polynomial theory, unlike the other theories, is that the tensorial trcrnsfonnation laws can be applied to the strength tensors. In this way, it is invariant to the definition of the coordinate axes. In analyzing off-axis properties, one can either rotate the applied stresses to the principal material axes or rotate the material axes to the axes of the applied stresses thereby transforming the coefficients, F; to F/ and F^ to F^ ', as shown in Figure 2.1. Then, using the general transformation relationships, the transformed strength criteria, presented in matrix notation are: Figure 2.1 -Rotation of Material Axes • M r 2 f 6 0 0 0 0 0 K j F' M f 6 0 0 0 0 0 1 F' 0 -2T2 0 0 0 0 0 cos 20 F' M l 0 0 0 u> 2L76 Us u, sin 28 1 cos 20 F' R12 • 0 0 0 0 0 -u3 F' R16 0 0 0 0 2Ut -u, 2U7 -2U3 sin 20 F' R22 F' R26 0 0 0 -2U6 u3 u7 cos 40 sin 40 0 0 0 0 2U& -u2 -2U7 2U3 F' . 6 6 . 0 0 0 4f 5 0 0 -AU3 -4U7 (2.12) where; T, =(F,+ F2)/2 T2 = (F,-F2)/2 U, =(3Fu + 3F22 + 2F12 + F66)/8 U2 = (F„-F2,)/2 22' (Fn + F22-2F12-F66)/8 U4 = (Fn + F22+6F12-F66)/8 U5 = (Fu + F22-2F12+F66)/8 U6 = (F16 + F26)/4 U7 = (F16-F26)/4 (2.13) 16 To calculate the off-axis tensile strength, one can apply equation 2.9, in the transformed coordinate system, and solve for oe. A drawback in the application of the Tensor Polynomial theory is that there is no consensus among researchers for a method of determining the value of the combined stress parameter, F 1 2 which accounts for the interaction between normal stresses, a, and a 2. The only certainty is that in order for the failure surface to be a closed ellipsoid7, the coefficient F 1 2 must be bounded by the stability condition : F . I F M - F>* * 0 (2.15) The 'correct' determination of F 1 2 has been the topic of discussion for many researchers for many years. Section 2.2 of this thesis addresses this quandary. Plotting equation 2.9 with at and o2 as coordinates of a point will produce an ellipsoidal failure surface, providing F 1 2 is within the prescribed range. Else, the surface could become open-ended, meaning no matter how large the stresses became, failure would never ensue, which is physically impossible. 17 Tan Theory (1990) Another tensor polynomial based failure criterion was proposed by Tan (1990). Tan suggested a 'fundamental' strength function to predict off-axis tensile or compressive strength in the form of a Fourier sine series: where Xgis the off-axis strength at an angle 0 to the 1 axis, and An, n = 0,1,2,... are the coefficients determined from fitting equation 2.16 to the data. The accuracy of this strength function depends on the number of terms to be used in the sine series and therefore depends on the data available for fitting purposes to determine An. The off-axis tensile strength, for example, for any angle can be found by knowing Xt, Yt and one off-axis strength U t, say at 6 = 30 degrees. For this condition the coefficients A„ are - i A o - £ A n s in 2 ne (2.16) 11-1,2,3... x, y, (2.17) I A + _L _ i 3 x * Y ' u Tan further suggested that a general strength theory could be established by utilising the results of the fundamental strength function in conjunction with the Tscd-Wu 18 tensor polynomial criterion. The strength parameters Fj and F^  (i=j) would be written in terms of Xg andYe (shear is not required). The interaction parameters Fy (i* j) would be described by the Fourier sine series approximation: F -*7 C0 - £ C sin 8n(|e|-45) n.1,2,3... JF~T~ V ™ Ti (2.18) where the strength parameters have subscripts in terms of x and y to signify a more general coordinate system. The interaction parameter can be found by ecrucrting formula 2.18 with the tensor polynomial criterion and using biaxial strength data (ox, a y), as follows: C o - £ C,8ln sn(|e|-45) n-1,2,3... In this criterion, the interaction terms would have a different value for each quadrant of the stress space. Since the criterion is essentially a curve fitting technique, the failure surface tends to fit experimental data very well. This criterion is not considered in this study due to its strongly empirical nature and also due to its distinct resemblance to the next criterion discussed. F o 7 7 F o 77 7 2a a JF~T I 7 V 1 1 Ti (2.19) 19 Tan-Cheng Theory (1993) This recently proposed theory is very similar in nature to Tan's theory above. In this theory, the off-axis strength is represented by a Fourier cosine series V A A I ( 2 - 2 0 ) 2^ n c o s n 0 n-0,2,4... The resulting off-cods tensile strength criterion using the 30 degree off-axis data (as used in this study) is ' _ L . _ L X c o s 20 + -6 KXt Y, U c o s 40 (2.21) Tan and Cheng's general strength theory also differs from Tan's theory in that the off-axis strengths, generated from the Fourier cosine series, are directly implemented in a quadratic interaction formula where the interaction parameter, F ,^ is treated as a function of 0. The resulting 2 dimensional failure criterion is: a j / a ^ 7 £ cr n cosn( |0 | -45) M>,2,4,.. I _ \ - 1 (2.22) Similar to Tan's theory, in order to deteiTnine a complete strength theory, the coefficients that define the interaction terms would have to be determined separately for each stress quadrant. 20 2.1.3 Comparison of Failure Theories As previously mentioned, all of these theories are phenomenological. That is, they all treat the material as macroscopically homogeneous and predict only the onset of failure. Mode of failure is not a consideration by any of these criteria. The tensor polynomial criteria can be viewed as a more general form of the quadratic interaction formulae. Both Tsai-Hill and Norris theories can be described in index notation as simply Fy a{ ai ^ 1 whereas the tensor polynomial theories also include a linear term FjO; . This linear term enables the strength equation to take into consideration differences between tension and compression strengths. Table 2.1 was recreated from Rowlands (1985) to compare the individual terms of each theory. Theory FJ F „ F 2 2 F F I F I F „ Tsai-Hill - - I x2 I y2 I s2 l " xi Norris - - I X2 I y2 I s2 1 " XY Tsai-Wu l l x,'xc _i i_ y y I xtxc I y y 1 indep. Tan-Cheng - - 1 4 I y2 - fcn( 1 ) Table 2.1 - Comparison of Strength and Interaction Parameters from each Failure Theory 21 The most obvious difference of all of these theories is the definition of the interaction term, F 1 2 . The Tsai-Hill, Norris and Tan-Cheng theories assume this term to be dependent on the principal strengths whereas the Tsai-Wu theory interprets it as an independent quantity to be established through experiments. (The Tan-Cheng theory also requires experimental determination, however, it remains a function of the principal strengths.) As a general visual comparison of the failure envelopes, a plot for a hypothetical material was created. (See Figure 2.2) The shear stress, which would normally form the third axis, was assumed constant for the convenience of a two-dimensional plot. Although the plot uses the same principal strengths for the four theories, some assumptions are made for the interaction terms in both Tsai-Wu and Tan and Cheng's theory which influence the envelope's shape. For the Tsai-Wu theory, the interaction parameter, F1 2 = 0, and for the Tan-Cheng theory, the interaction parameter is arbitrarily chosen to be -0.0003, 0.010, 0.019, and 0.037 for the 1st, 2nd, 3rd, and 4th quadrant, respectively. The following section will discuss the significance of the interaction coefficient in the Tsai-Wu theory. It is noted that only the first quadrant of the stress space will be examined in this study. 22 zz 2.2 Interaction Term, F 1 2 of Tsai-Wu Theory As previously mentioned, a standard method of determining the interaction term, F 1 2 has not been established and the topic has been in debate for some years among researchers. For this reason, this section reviews literature pertaining to the interaction component of the Tscri-Wu theory. 2.2.1 Significance of Interaction Term, F]2 The interaction coefficient, F 1 2 characterizes the interaction of the normal stresses, a, ando2. Because this term involves both normal stresses, its evaluation must occur under a biaxial loading situation unlike the other strength tensors whose values are determined from uniaxial loading. The Tscri-Wu failure envelope takes the form of an ellipse in the ax ,a2plane as shown in Figure 2.2. The strength tensors F,, F2, F n and F^ establish the axes intercepts whereas the component F 1 2 determines the rotation of the ellipsoid with respect to the principal material axes. For illustration, see Figure 2.3 from Suhling et al., 1984 which shows the influence of vcnying F 1 2 in equation 2.9 for paperboard. It has been argued that the value of F 1 2 "determines the effectiveness of tensorial-type failure criteria." (Suhling et al., 1984) 24 (A) F u = (B) f u = <C)Fn = (D) f u = (E) f u = (F) f u = Figure 2.3 - Theoretical Strength Envelopes of Paperboard with Different Values of Coefficient F12 (From Suhling et cd., 1984) 2.2.2 Evaluation of Interaction Term, F12 Tscd and Wu (1971) reported that a separate combined stress test was necessary for determining F 1 2 and they provided 6 viable test options in their original paper: a simple biaxial tension or compression test with the applied load equal in each direction; a 45 degree off-axis tension or compression test as outlined in this paper; or a 45 degree off-axis positive or negative shear test as illustrated in Figure 2.4. +Ve shear -Ve shear Figure 2.4 - Positive and Negative 45 Degree Off Axis Shear 25 Strict care must be taken in determining F 1 2 due to its high sensitivity to experimental variation. Tsai and Wu showed that for a graphite epoxy composite, slight inaccuracies in measurements of strength for most of the above tests would result in large inaccuracies in the calculated value of F12. This problem is accentuated by the fact that in order for F12to be physically admissible, it must satisfy the prescribed stability bounds (Eqn. 2.15). Other researchers carried out similar studies on different materials to deterrnine F 1 2 experimentally. Pipes and Cole, 1973 performed a limited number of off-axis tensile tests on boron epoxy composites to deterrnine F 1 2 . They tested 2 coupons at 60 degrees, 3 at 45 degrees, 4 at 30 degrees, and 3 at 15 degrees. Using the mean of each set they found that only the value obtained for the 15 degree off-axis tests satisfied the stability criterion. Consequently, they concluded that the off-axis tensile test was not an adequate method of deterrmning F 1 2 for boron epoxy composites. Siihling, et al. (1984) conducted a comprehensive study to establish F 1 2 for paperboard employing all of the six test methods suggested by Tsai and Wu (1971). They found the optimum value of F 1 2 by fitting a linear regression through experimental results for all four quadrants and compared this value with that obtcdned from the individual tests. The tension biaxial tests produced the closest value to the optimum value for a zero shear situation. They found that F 1 2 = 0 was a satisfactory solution for four shear levels considered; o6 = 0, 6.9, 10.3, and 15.9 MPa. They also submitted that due to the highly 26 sensitive and unstable nature of F l 2 when calculated using off-axis tests that this test was not a suitable method to determine the interaction parameter. Anticipating the problems that researchers might encounter when determining F12,, Wu published a paper in 1973 demonstrating a method to find an optimal biaxial stress ratio for ccdculcrting F 1 2 experimentally. This method was not strongly espoused because it involved a series of experimental iterations further compHcating the tensor polynomial theory. Difficulties in experimental detenxiination of F 1 2 prompted several researchers to seek a theoretical solution to the problem. Narctyanaswami and Adelman (1977) performed a numerical analysis to prove that, from a practiced standpoint, the arbitrary assignment F 1 2 = 0 was acceptable for filamentary composites. For the six tests described above, they computed the percentage error when setting F 1 2 = 0 for 10 different composite materials. In all cases the error was found to be less than 10 percent and they therefore concluded this to be an acceptable error for practical engineering applications. Cowin (1979) derived Hankinsoris formula using only the linear term of the tensor polynomial function, FjO;. By considering the quadratic terms as well, he derived a formula for F 1 2 : 2S 2 27 that produced a similar, yet more accurate, equation to Hankinson's criterion to represent bone strength with respect to angle to grain. Both van der Put (1982) and Liu (1984) derived an expression for F 1 2 that reduces the tensor polynomial theory into the Hankinson formula (when the trigonometric exponent is equal to 2): , f i i , 1^ (2.24) F - I 12 2 . XY X Y, S2 \ t c c t van der Put approached the problem by trcrnsforrriing the applied stresses to the principal material coordinate system while Liu transformed the strength tensors. As previously mentioned, Hankinson's formula has had remarkable and repeated success in predicting wood failure at an angle to grain. The foregoing survey depicts several negative findings towards performing off-axis tests for determining F12. However, the off-axis test is the easiest and most inexpensive test available. In fact, the practical limitations of biaxial or shear tests on materials such as wood or lominated veneer probably outweigh the potential inaccuracies one may-encounter when using an off-axis test for this purpose. It was therefore decided to use the off-cods test in this project both for the purpose of comparing the accuracy of the different orthotropic theories and for determination of the interaction term, F12. The sensitivity and accuracy of this coefficient for this study on lcrminated veneer will be addressed in section 3.4. 28 2.3 Applications of Tensor Polynomial Strength Theory The primary function of a strength theory is to provide a simple technique to estimate the load ccrnying capacity of a structure under specific loading conditions. This section reviews some previous applications of the Tsai-Wu criterion for this purpose. Leichti and Tang (1989) used the tensor polynomial theory together with finite element analysis to predict the ultimate load capacity, as well as failure mode and location, of wood-composite I-beams. The parameters Fj and F^  were determined for each component of the I-beam8 (ie. web and flanges) using small specimen tests. The strength parameters were assumed to be constant throughout the I-beam. Failure was detected by evaluation of the strength theory for failure at each Gaussian integration point of each component. Locations of points which indicated failure at relatively low loads were interpreted as "weak components" in the beam. The failure modes of these areas were reported as being: 1) crashing failure (adjacent to the supports) in the tension flange, 2) combined crushing/compression failure near the load point in the compression flange, 3) predominantly due to high o2 compression stresses near the butt joint tip in I-beams with butt joints and 4) due to high shear stresses in continuously webbed I joists. It was noted that full scale composite I-beam tests verified the failure location and mode predictions. However, accuracy of failure load prediction was less discernible. No precise method was presented to identify ultimate failure load. 8 The interaction coefficient was estimated by equation 2.23. 29 Triche and Hunt (1993) also used the Tscri-Wu theory in conjunction with finite element analysis in an attempt to model a wood composite - more specifically, a parallel aligned wood strand composite. In this case, the Tscri-Wu theory was one of five theories considered to predict the tensile strength of a small laminate. Although neither the accuracy nor the details of implementation of the theory was specifically discussed in the paper, the results summary table seems to indicate only fair accuracy with experimental results. Material strengths that exhibit brittle failure, such as tension or shear, are susceptible to a size effect, as discussed in section 3.2.1 of this thesis. It is unclear in the foregoing papers how, or if, size effects were considered for these two studies. 30 3. Determination of Most Appropriate Failure Theory for Laminated Veneer To evaluate the foregoing failure theories and establish the most appropriate theory for lcnriinated veneer, several experiments were required. Tests were performed to procure strengths in the principal material directions which are quantities required by all failure theories. Also, a series of off-axis tensile tests for load angles of 15, 30, 45 and 60 degrees with respect to the longitudinal axis were performed. These data were compared visually and numerically to each theory to establish which criterion provided the best fit. The following section describes all of these tests. 3.1 Experimental 3.1.1 Material Nineteen individual boards of lcrminated veneer were utilized. Each board consisted of 11 laminated sheets of 3.2 mm x 1220 mm x 2440 mm Douglas Fir veneer oriented with the face grain in the longitudinal direction. The sheets were lominated with a phenol-formaldehyde resin and pressed at 146°C and 1.38 MPa pressure for approximately 12 minutes. For hcrndling, both faces of the board were overlaid with % mm thick, resin impregnated paper. It is suggested that due to the brittle nature of this relatively thin overlay, its contribution to the board's strength is insignificant. However, 31 it would be prudent to perform a series of tests without overlay to assess its true contribution prior to extrapolating the data in this project for any laminated veneer design. Specimens were cut from each board as shown in Figure 3.1. The tensile specimens were cipproximately 63 mm wide by 35 mm thick with a test length of 610 mm as permitted by ASTM D198-84. A few tension specimens experienced failure in the gripping area. Replacement specimens were taken from the opposite side of the board, as shown, and prepared in the same manner as their original counterparts. . ( d i r e c t i o n o f g r a i n ) , Figure 3.1 - Specimen Cutting Layout Compression strengths in the principal directions were necessary for the tensor polynomial theory. The width of both the parallel and perpendicular to grain specimens was fixed at 63 mm to be consistent with the tensile specimens whereas the lengths were 32 dictated by other factors. The parallel to crrain specimen length of 160 mm met the requirements of ASTM D198-84 for short columns with no lateral support. That is, the length to radius of gyration, 'r1 ratio, where r - ^j^> I = moment of inertia and A = cross-sectional area, was less than 17. The perpendicular to grain compression specimens were initially tested with a length of 150 mm as per ASTM D143-84 and loaded at mid-length by a 57 mm bearing plate. However, upon reflection, it was decided that in order to establish an unbiased perpendicular to grain strength, _ _ L U H i (ie. without influence from the ends) the specimen length 4 J-was reduced to the 57 mm length of the bearing plate. + l i J m a Both strengths are recorded in the results section of this Figure 3.2 - Perpendicular to Grain Compression Tests chapter. The shear block specimens were prepared in accordance with ASTM D143-83, except that the specimen thickness was 35 mm as opposed to 51 mm because of the set board thickness. (See Figure 3.4) All specimens were conditioned in a controlled humidity chomber at a relative humidity of 50 ± 2% and temperature at 20± 1 °C for a minimum period of 6 weeks. 33 3.1.2 Test Methods Tensile Tests All tensile tests were performed in a Metriguard hydrcrulic testing machine (capacity 444.8 kN, accuracy ± .02 kN) with self cdigning, non-rotating grips. (See photograph 3.3) The use of non-rotating clamped ends will create non-uniform stresses in the off-axis specimen from a shear coupling effect as shown by Pagano & Halpin (1968). However, the specimen length to width ratio of approximately 9.7 should miriimize the non-uniformity of the stress field. (R. Rizzo 1969). (Further discussion of these potential influences is found in section 3.4.2). The loading rate was set to produce failure in no less than 5 minutes and no greater than 10 minutes in compliance with ASTM. Ultimate load and description of failure were recorded. As well, cross sectional dimensions near the failure location were measured using calipers with an accuracy of ± 0.005 mm to calculate ultimate stress. Moisture content (oven dry method per ASTM D2016-83) and specific gravity (method A - ASTM D2395-83) were both measured from a small cube cut near the location of failure of each specimen. 34 Figure 3.3 - Tension Specimen in Metriguard Testing Machine Compression Tests Compression specimens, both parallel and perpendicular to grain, were performed on a Syntech 30/D testing machine with a 145 KN load cell linked to a computerized data acquisition system. Specimen dimensions were taken using calipers prior to loading. For the parallel to grain specimens, one spherical bearing block was used to prevent eccentric loading on the specimen. The loading rate was set to induce failure in a time frame of 8 to 12 minutes as permitted by ASTM. For the perpendicular to grain specimens, both top and bottom spherical bearing blocks were used with a 57 35 mm becrring plate. These specimens were loaded at a rate of 0.305 mm/min. up to a maximum of 2.5 mm as per ASTM D143-83. Maximum load and description of failure were recorded for each specimen. The shear specimens were tested in a standard ASTM shear block tester in which the intended failure plane was parallel to grain and perpendicular to the glueline as shown in Figure 3.4. The corresponding shear plane perpendicular to grain and perpendicular to the glueline is the stronger of the two shear planes. Since the actual shear stress applied is symmetric (ie. a 1 2 = a21), failure will occur on the weaker of the two planes first. Hence, the shear strength of the material is the ultimate shear stress associated with the weaker plane. Ultimate shear and exact shear plane dimensions were recorded in order to calculate ultimate shear stress. Shear Block Tests 1 Figure 3.4 - Shear Block Dimensions and Configuration (Dimensions in mm) 36 3.1.3 Experimental Results A comprehensive sunimcrry of all measured strength data is given in Table 3.1. Descriptive statistics are provided for each set of data. Two and three parameter Weibull distributions were fitted to the strength data using a Maximum Likelihood technique and their respective parameters are given as are the normal and lognormal distribution parameters. Figures 3.5 to 3.8 illustrate the various distributions fitted to the data. The graphs are grouped together to visually compare strengths in the two principal material directions and also to compare off-axis strengths. It is noted that where possible, the scales were kept the same, to compare material variability in the different failure modes. Only compression perpendicular to grain with the latter specimen length of 57 mm was plotted. It was observed that, in general, the normal and lognormal distributions describe the principal strength data relatively well. For the off-cods data, normal or lognormal distributions were most appropriate and the three parameter Weibull was favoured over the two parameter case. The moisture content was approximately 8% with a standard deviation of 0.26% and the average specific gravity was 0.53 g/cm3 with a standard deviation of 0.02 g/cm3. 37 9i ip 8. C<5 i a . 3 s 6" N H cn cn TJ CD (D f) O 0 (D 5T (T CQ CQ & 5-II II Cn Weibull 3-P Weibull 2-P CO1 Stand. C Mean O 0 CO ft Location Scale Shape | Scale | Shape | -=-, • 3 )ev (MPa) (MPa) istics 35.55 22.27 2.10 59.48 5.79 18.28 10.11 55.31 00 0 0 16.69 2.42 1.42 19.55 15.40 7.35 1.39 18.92 cn 0 Tens 5.64 0.94 2.31 6.60 18.27 6.18 0.40 6.47 CO 0 0 ion by gra: 3.16 0.64 2.55 3.85 14.30 6.42 0.24 3.74 CO £>> Cn 0 in angle 2.43 0.27 1.15 2.76 16.12 6.72 0.18 2.68 CD CD O 0 (MPa) 0.00 2.33 15.89 2.33 15.89 9.78 0.22 2.25 CD O 0 48.98 9.28 3.27 58.56 20.74 5.11 2.93 57.29 CO (MPa) Comp. 13.75 3.21 2.26 17.22 12.78 8.32 1.38 16.59 00 Comp. Perp. 1 (MPa) 9.33 3.03 2.09 12.63 8.99 11.48 1.38 12.02 CO ScVc? t o 2.37 9.13 9.10 11.53 10.61 10.62 1.17 11.02 CO Shear Strength (MPa) 3.29 0.26 7.91 0 CO Moisture Content (%) 3.77 0.02 0.53 0 CO Specific Gravity (g/cm3) 38 44 50 56 62 68 74 80 Tension parallel to grain strength, Xt (MPa) Compression parallel to grain strength, Xc (MPa) Figure 3.5 Cumulative Distibutions of Laminated Veneer Strength in the Longitudinal Direction 39 Data ~ Normal Log Normal - Weibull 2 Par - Weibull 3 Par 0 1 2 3 4 5 6 Tension perpendicular to grain strength, Yt (MPa) Compression perpendicular to grain strength, Yc (MPa) Figure 3.6 Cumulative Distibutions of Laminated Veneer Strength in the Transverse Direction 40 Shear Block Strength,S (MPa) Figure 3.7 Cumulative Distibution of Laminated Veneer Shear Block Strength 41 The stcmdcrrd deviation of the on-axis specimens, (ie. 0° specimen) is noticeably higher than those of the off-axis specimens. It is speculated that this may be attributed to the different failure modes. The latter group consistently displayed a failure plane coincident with the grain angle, failing primarily by combined tension perpendicular to grain and shear (See Figure 3.9). The failure mode of the 0° specimens, entirely tension parallel to grain failure, was more splintered. This may indicate that parallel to grain tension failure, ie. failure within individual wood fibres, is naturally more variable than combined tension perpendicular to c/rcrin and shear, ie. failure between wood fibers. Smooth Failure Plane I I 111 Splintered Failure Off-Axis Failure Mode On-Axis Failure Mode (B = 15,30,45,60&90) (9 = 0) Figure 3.9 - Tension Specimen Failure Modes The most prevalent failure mode of the compression parallel to grain specimens involved buckling of the outside one or two lcrminations. Occasionally, the failure mode was shear across the lcariinations which, apparently, is typical for LVL in the absence of 43 lcrpjoints (Hesterman and Gorman, 1992). Stress/strain diagrams typically became nonlinear as failure approached (See Figure 3.10). 60 -, 1 2 7 50 • 2 (3 S AO -9 2 S. 30 • Compression Stress 3 o 8 < ) 1 1 1 t 1 1 1 1 1 0XX>2 0X304 OJ006 OJ008 OJ01 0J012 OJOU 0J016 0J018 Strain (mm/mm) Figure 3.10 - Sample Stress/Strain Diagram for Compression Parallel to Grain Compression strength perpendicular to grain with a specimen length of 57 mm was less than that with the longer specimen length. This obvious difference is attributed to the influence of the continuous wood fibers which are only partially loaded in the longer specimen (See Figure 3.2). This loading regime produces a shear support at the ends of the plate. A visual inspection of these specimens showed that these fibers were sheared. The fibers in the short specimen, however, experienced only crushing and were thus significantly weaker in compression perpendicular to grain strength. 44 The shear block specimens failed approximately along the intended shear plane, across lcrminations, in a brittle manner (See stress/strain Figure 3.11). Alternatively, a short beam test could be conducted G.C. Bohlen, 1975) to evaluate shear in this orientation. However, the compression perpendicular to grcrin stresses at the becrring would obscure measurement of the critical shear stresses. The shear block test was used to achieve, as closely as possible, a pure shear state. Unfortunately, the use of small specimens for measuring shear stress raises an issue of a size effect that must be considered. The following section addresses this concern. OM 0.011 0.014 OMt OMB OM 0J022 0.024 0.020 0.028 Strain (mm/mm) Figure 3.11 - Stress/Strain Diagram for Typical Shear Block Specimen 45 3.2 Analytical In order to apply the strength theories with some degree of certainty, all strengths used in the equations should be representative of a specific material condition. This is especially true for wood as it is known that wood strength varies with many factors; for example, moisture content, temperature, load time, or specimen size. In the foregoing experiments, all factors were kept crpproximcrtely constant with the exception of specimen size. Before employing these failure theories, the difference in specimen size must be taken into account. 3.2.1 Size Effects It has long been recognized that large brittle members tend to display lower strengths than smaller ones of the same material when subjected to the same environmental and loading conditions. Numerous papers have been written on this subject dealing with many different materials. (Barrett et al., 1975; Lam & Varaglu, 1990, Sharp and Suddarth, 1991, Zweben, 1994) This is known as a size effect and has been rationalized using various size effect theories such as the Weibull wecdcest-link theory. 46 Weibull (1939) proposed that a variation in strength properties existed due to the statistical probability of a strength controlling defect occurring in a given volume. He used the weakest link concept to show how strength of a 'perfectly brittle' material could be described by a specific cumulative distribution function. Weibull's theory enabled the prediction of the probability of failure of a homogeneous isotropic material at a given volume according to the following: F (x) = probability of failure x = material strength tm i n = minimum material strength (location parameter) m = scale parameter k = shape parameter (tmin, m, and k are material constants and VQ is a reference volume) Commonly, xmin is accepted as being equal to zero, simplifying the problem, and the resulting formulation is called a two parameter Weibull Distribution. Weibull's theory was used to explain the strength difference observed in bodies subjected to the same stress distribution but differing in volume which, as formerly noted, is a size effect. According to Weibull theory, it is argued that the larger member has a higher probability of contcrining a larger flaw (or weaker zone) than does the smaller member and thus has a lower strength. In general, the strength of a volume of material at a given probability of failure can be predicted given the strength and shape parameter of a (3.1) where: 47 common material when both are subjected to the same stress distribution. The solution is found by equating the probability of failure of one volume, V, (corresponding to z,), to that of another volume, V2 (corresponding to T 2 ) , which yields: l -(3.2) for the two parameter Weibull theory. Equation 3.2 simplifies to (3.3) Based on equation 3.3, the shear strengths corresponding to the volume of the larger off-axis tension specimens were estimated from the experimental data of the smaller shear block specimens. Although the compression specimens also differed in size, it was assumed that the size effect for this ductile failure mode would not be significant. Application of Weibull Weakest-Link Theory Let: T I = T A S T M " shear strength of shear block specimen T 2 = = shear strength corresponding to the volume of the rectangular off-axis specimen 48 then from equation 3.3 : (3.4) Consider the left hand side (LHS) of equation 3.4 first. To evaluate this integral, finite element analyses was performed and a numerical integration carried out. Finite Element Analyses Finite element analyses were conducted using a linear elastic, two-chrnensional finite element program written by Foschi (1974). The finite elements were cruadratic, isoparametric and arranged in a mesh as shown in Figure 3.12. Numerical integration was carried out using a Gauss quadrature order 3 rule producing 9 stresses and strains in each element. average failure load / mm (node points, typ.) Boundary Conditions 1 ^AuAuiuAuA 4 Aiuououo ' Figure 3.12 - Finite Element Mesh for Shear Block Specimen 49 Assumed boundary conditions can influence the stresses in the specimen; therefore, four analyses with differing boundary conditions were performed and compared. A legend of these boundary conditions is provided in Figure 3.12. The shear stress distribution in the specimen for boundary condition 2 is illustrated in Figure 3.13. It can be seen from this 3-dimensional plot that high stress concentrations exist around the two elements shaded in Figure 3.12. The stress contributions from these elements have been ignored for the integration of shear block stresses in equation 3.4. Shear Strength Corresponding to Volume of Off-Axis Specimen Given the integration of shear block shear stresses, which is performed by the finite element program, one can solve for the shear strength corresponding to the volume of the off-axis specimen using equation 3.4. Assuming the shear stress is constant for the off-axis tension specimen, a sample calculation procedure follows: Given: let I 50 51 and rect rect rect Thus, T I I rect \ ' rect) ' I' is computed by the finite element program, k is given in Table 3.1 and is simply the volume of the rectangular off-axis specimen. The differing boundary conditions yield relatively consistent values for off-cods strength averaging 6.7 MPa. This shear strength will be used as a basis for comparing the failure theories. Limitations to this procedure It is recognised that the foregoing analysis provides an approximation for the off-cods shear strength which is dependent on several assumptions made: removal of two elements with high stress concentrations from integration of stresses, linear elastic material behaviour, and the choice of boundary conditions. Nonetheless, the resulting value should provide a fair estimation of shear strength for the purpose of comparing failure theories. A statistically based method to evaluate the size effect will be implemented subsequently in this thesis with the chosen failure theory for which the off-axis shear strength of 6.7 MPa will be used as a comparison. 52 3.2.2 Determination of Interaction Component, F12 for Tsai-Wu Theory All of the vcrriables to be used in the equcrtions of the failure theories can be calculated with relative ease with the exception of the interaction parameter F 1 2 for the Tsai-Wu theory. In the ensuing section, two methods are used to determine this value. Both of these methods are statistically based accounting for the variability of the material. The material strengths are treated as random variables as opposed to detenriinistic values.9 The detenriinistic solutions of F12, using the methodologies discussed in section 2.2.2, will be presented and compared to the statistical solutions at the end of this section. Non-Linear Weighted Least Squares Approach Using SAS In this approach, a nonlinear equation consisting of independent variables; Xj, X,., Y t, Yc, S (adjusted for size effect), and 0, was fitted to the off-axis strength data, a e (the dependent variable). All variables were obtained from the experiments outlined in section 3.1. The formulation was set up as follows: By substituting equcrtions 1.1 (p.5) into 2.9 (p. 14) we get F^gco^e • F2oesdn2e • F u o e 2 c o s 4 e • F^OgWe • 2F l s o 9 2 cos 2 8s in 2 e • F 6 6 o e 2 d n 2 ecos 2 e - 1 (3.5) 9 With deterministic values, one definitive value represents an entire distribution. 53 Recrrrcrnging, o 8 2 ( F n c o s 4 6 • F^sln'e • 2FBcos2esdni!e • F 6 6cos 26sin 2e) • a9{F1cos iQ • F2sin2e) - 1 - 0 ^ Q) Designate, Xx - F^os'Q * F2sin2e Xt - Fncos*e • F^sin^ • FMcos29 sin26 X3 - 2 cos2 9 sin2 9 and using the standard quadratic formula, (3.7) °s - — ' ' (3.8) 2(XS • F 1 8 X.) The left hand side (LHS) of the equation is the vector contcdning four off-axis strengths for each of the nineteen boards (with some exceptions for discarded specimens). The right hand side (RHS) of the equation consists entirely of complex functions of the principal material strengths for the corresponding board and the random variable F12. This formulation was programmed into the mainframe SAS (Statistical Analysis System) computer program to deterrnine a non-linear least squares approximation of F12. The program solves for F12, such that the function $ - £ (a<r* - o^)2 is minimized where the superscript act refers to actual off-cods strength (LHS of eqn. 3.8) and the superscript pred refers to the predicted off-axis strength (RHS of eqn. 3.8). The SAS 54 source code and data input file are provided in Appendix A. It was noted that the dependent variable, a e , was heteroskedastic10 with respect to the angle to grcrin. To correct this problem, both sides of equation 3.8 were weighted using the inverse of the standard deviation for the individual groupings of angles. For example, all cases of 15 degree off-axis values were multiplied by 1/1.39 (MPa"1) and 30 degrees by 1/0.40 (MPa"1) etc. This is a standard procedure to remedy heteroskedasticity to enable inferences to be made about the estimated coefficients as outlined in Neter et. al. (1990). The result of this analysis is an average value of F 1 2 = + 0.0032 with an asymptotic11 standard deviation of 0.0015. The asymptotic 95% confidence interval12 is +0.00024 and +0.0062. A deterministic approxrmation of the upper and lower bounds using the mean strength values and equation 2.15 is ± 0.0034. Thus, the calculated mean of + 0.0032 is bounded and physically possible. It is noted, however, that the standard deviation of F 1 2 is relatively high (coefficient of variation = 47%). 10 Heteroskedasticity is the condition when the variance of the error terms, (o e a o t - o e p r e d) is not constant for all cases. Generally, a pattern is evident in which the variance increases or decreases over the independent variable(s). n The interaction parameter, F 1 2 is assumed to have an asymptotically normal distribution and inferences from these tests are only accurate when the sample size is large. It is assumed, in this case, with a sample size of 68, that this is sufficiently large to produce relatively accurate results. 12 The 95% confidence interval is the interval in which, with a 95% probability, the actual value of F 1 2 is contained. 55 Non-Linear Least Squares Approach using 'F12FIT' Procedure An estimation of the distribution parameters of F12was obtained by considering the principal strengths to be lognormally distributed random variables instead of specific data points as in the previous procedure. The program DOLFTT, for fitting four parameters to the Foschi-Yao damage accumulation model (R.O. Foschi, 1990), was adapted to perform the non-linear least square 'F12FIT procedure. This procedure requires the following information: the mean and standard deviation of all principal strengths, sorted off-axis experimental data with corresponding probability of failure and random seeds for random number generation. This information is used to carry out the following iterative analysis. 56 A matrix of random numbers for all strengths is generated (see illustration below) using one pair of random seeds. For each replication (which is comparable to one board), the off-axis failure stress, o e is calculated for one angle to grain, using equation 3.8. Random Variables No. of Replications S -> 1 2 3 NREPL The failure stress, o e is then ranked (ie. sorted in ascending order and given an appropriate probability of failure) and the corresponding principal strengths sorted. For each off-axis data point, the function $ 1 a p r a d is calculated, where the superscripts are as previously described. The 'predicted' value is at the same probability of failure as the 'actual' experimental off-axis values. This procedure is then repeated, using the same randomly generated data above, for the next angle to grain and so on. The function, 0 and its gradient (ie. the 1 s t derivative of the function with respect to oepred) are summed over all points for all angles. The final value for the function and the gradient are minimized using a least squares minimization process. (In this case, the 57 subroutine DFMin was utilised.) In this process, the mean and standard deviation for F 1 2 is slightly modified and the entire procedure above is repeated until the difference between the final function values of subsequent iterations is within a prescribed tolerance. In summary, the procedure produces a mean and standard deviation of F 1 2 assuming a normal ch^ tribution, that minimizes the function 0 . As this procedure is based on a minimization technique, there is potential for the solution to be found at a local minimum rather than a global minimum; therefore, it is sensitive to the initial input values. For this reason, several initial values were checked and the solution yielding the smallest function O was deemed to be the final solution. With 2500 replications, this analysis produced a mean value of F 1 2 = 0.0013 and a standard deviation of 0.00030 (averaged over 10 sets of random seeds). Again, this value is within the deterministic bounds, ± 0.0034. 58 Comparison of Fn Approximation Results For interest, the value of F 1 2 was calculated using various methods discussed in this thesis and summarised in Table 3.2. It is obvious that the methods do not lead to the same conclusion. The value according to Liu's formulation is beyond the stability bounds and is therefore unacceptable. Cowin's equation produces a negative value very near zero which is consistent with Narayanaswami and Adelmen's assertion that it could be arbitrarily set to zero. The statistical results based on the off-axis data are similar, as would be expected. It is speculated that the value obtained using the F12FTT procedure is the more reliable as it is fitted to many probability levels as opposed to just the mean value as was done using SAS. Method Mean St. Dev. Narayanaswami and Adelman 0 — Cowin (Eqn. 2.23) -0.000042 — Liu (Eqn. 2.24) -0.0072 — SAS +0.0032 0.0015 F12FTT +0.0013 0.00030 Table 3.2 - Summary ofF12 Approximations 59 3.3 Comparison of Failure Theories Both Table 3.3 and Figure 3.14 illustrate the differences between the strength theories. The best fitting curve, as determined by the smallest weighted sum of squared errors, was obtained by the curve fitting formula of Tan and Cheng (total SSE = 78.9). This is predictable since the method is a curve fitting technique which fitted the mean of the 30 degree off-axis data exactly. Although the criterion produced the best results, it is not practical in this study because, as already stated in section, in order to have a complete strength theory, tests for all four quadrants are required. It was included in this project as an interesting comparison, and, owing to its accuracy, perhaps could be considered for future LVL strength studies. Off-Axis Angle to Grain Predicted Mean (MPa) [Weighted Sum of Squared Error, SSE] Exp. Mean Hankinson (n=1.7)> Tsai-Hill Norris Tsai-Wu Tan- (MPa) F12=0.00322 F12=0.00133 Cheng 15° 16.19 [81.5] 19.86 . [23.8] 22.06 [103.0] 16.35 [74.1] 17.04 [46.9] 17.68 [29.7] 18.92 30° 6.62 [17.9] 7.75 [189.1] 8.11 [300.8] 6.90 [35.3] 7.08 [54.9] 6.47 [15.7] 6.47 45° 3.89 [24.3] 4.26 [102.9] 4.34 [129.9] 4.04 [45.6] 4.09 [56.01 3.69 [17.6] 3.74 60° 2.82 [25.3] 2.94 [48.8] 2.96 [53.8] 2.88 [35.5] 2.90 [38.5] 2.72 [15.9] 2.68 Total SSE 149.1 364.6 587.5 190.5 196.3 78.9 1 'n' denotes the trigonometric exponent of the Hankinson Equation 2 from SAS analysis 3 from F12FTT analysis Table 3.3 - Comparison of Strength Theories 60 Figure 3.14 - Failure Theories vs Experimental Data Hcmkinson formula with n= 1.7 also predicted the experimental mean with good accuracy (SSE = 149.1). This is in keeping with findings of other studies on wood. However, this result is also included only for reference purposes. As previously mentioned, it is an off-axis strength criterion, not a complete strength theory. The Tscd-Wu theory also produced good results for this comparison, considering both practicality and accuracy. Since the weighted sum of squared error is rriinimized in the SAS procedure, it is understandable that the curve with this F 1 2 value produced the smallest SSE (190.5). It is only slightly less than that with the F12 value from the more intensive F12FTT procedure (196.3). The Norris and Tsai-Hill equations are not adequate, particularly in the intermediate range of 0 (ie. 30° and 45°). In this range the theoretical failure mode is a result of an interaction between longitudinal tension, shear and transverse tension. These two theories assume that the normal stress interaction is dependent on the principal material strengths, whereas the Tsai-Wu theory treats this interaction as an independent material strength. The latter appears to be more appropriate and may explain the significant difference between the curves in this region. At 0 £ 60°, where transverse tensile fracture is dominant, all of the theories become almost mdistinguishable. 62 3.4 More on Tsai-Wu Theory Given the results of section 3.3, more study was performed on the Tscd-Wu theory in an attempt to improve the accuracy of the final failure criterion. This section provides some variations on the F12FTT method to determine F 1 2, explores the sensitivity of F 1 2 with respect to inaccuracies in experimental data for this study, and details the importance of the accuracy of the F 1 2 value specifically for lcrminated veneer. 3.4.1 Size Effect for Shear Treated as a Random Variable Size effect adjustments for shear strength were made as outlined in section 3.2.1 to provide a common basis upon which to compare all failure theories. This method involved several variables that were difficult to confirm, treating the adjustment factor as a deterministic value so that the average shear strength was 6.7 MPa. The adjustment factor was calculated as the quotient of the average shear block strength and the shear strength corresponding to the volume of the off-axis specimen, 11.02 MPa-r- 6.7 MPa = 1.64. It is proposed that this adjustment factor could also be a random variable. A new method was developed to deterrnine a nonlinear least square fit of the Tsai-Wu criterion considering both F 1 2 and the size adjustment factor as normally distributed random numbers. This new procedure worked on the same premise as the F12FIT procedure except the mean and standard deviation of the two variables were estimated simultaneously. With 2500 replications and several different initial values tried, O was minimized when F,2 (mean) = 0.00063, F12( = 0.00036, Size Factor (mean) = 1.68, and Size Factor ( = 0.17. 63 The mean value of F 1 2 (0.00063) is slightly less than that predicted with the Fl 2FTT procedure (0.0013). However, the mean size factor of 1.68 is slightly higher than the previously assumed deterministic size factor of 1.64. It was also noted that the coefficient of variation of the size factor is approximately 10%. This is not unreasonable considering there is some variation in the specimen sizes for both the shear block and the off-axis specimens. The off-axis strengths were randomly generated according to the Tscd-Wu criterion (ecfuation 3.8) assuming lognormally distributed values for the principal strengths and normal values for F 1 2 and the size factor. Four hundred replications were executed with one pair of random seeds, which is analogous to 400 tested specimens. The results were ranked and plotted as cumulative probability Distributions in Figure 3.15 for visual interpretation. Although the simulated failure loads for each angle to grain reasonably predicted the off-axis failure loads, the 15 degree data were underestimated while the 30, 45 and 60 degree data were overestimated. A more in-depth study of the off-axis data and its influence on the parameter F12was deemed necessary to help explain this result. 64 3.4.2 Sensitivity of F12 to Off-Axis Experimental Data Solving ecruation 3.5 for F 1 2 yields the foUowing equation: 2o„ 1 sin2 8 cos2 8 sin2e cos^ e ) tan28 • F^tan2 8 (3.9) Using this equation, values for F 1 2 were obtained for the 4 angles to grcrin for each of the 19 boards tested (with some exceptions for flawed specimens ). The strength parameters, F, F 2, F n , F^ and F 6 6 are the same for each board since they are functions of each board's principal strengths; however, the angle to grain, 0 and the corresponding experimentally determined off-axis strength, a e differ for each F 1 2 calculation. The size adjustment factor for shear was assumed to be a deterministic value of 1.68. The results are displayed as cumulative probability distributions for each angle to grcrin in Figure 3.16. The probability distribution for each angle to grain is quite distinct. It was found that the mean values and the standard deviations are significantly different for each group at a 0.05 level of significance. The Fl 2 value associated with the 15 degree data is fairly consistent, (standard deviation = 0.0037) with a negative mean value of -0.0035. In contrast, the values from the larger angles are less consistent with positive mean values. This curiosity can be explained partially by a high sensitivity of the interaction parameter to variations in experimental data. In Tsai and Wu's original 1971 paper, the effect of variations in data on F 1 2 obtained from different combined stress tests on graphite-epoxy was demonstrated with a stress vs. F 1 2 plot. A similar plot has been generated for the differing angles to grain of this study. See Figure 3.17. 66 Figure 3.16 Cumulative Distribution Function ofF12 for Each Angle to Grain Angle F12 (mean) (stdev.) 15 -0.0035 0.0037 30 0.0079 0.0067 45 0.0172 0.0124 60 0.0369 0.0488 s I -S 05 CDF P 4 • A A -0.12 -0.09 -0.06 -0.03 0 0.03 ,-2\ 0.06 Fl2 (MPa2) ® 15 degrees * 30 degrees A 45 degrees a 60 degrees —I— 0.09 0.12 Figure 3.17 Sensitivity ofF12 for Each Angle to Grain 15 degrees -0.008 Fl2 (MPa2) 0.006 0.008 67 Curves for 30, 45 and 60 degrees are nearly horizontal. This means that if a small inaccuracy is made in measuring the off-axis strength from one of these angles (from human or systematic error), the value for F 1 2 would vary extensively and would be completely obscured in the stability region. This is likely the reason for the increased variation for the three angles, shown in Figure 3.16. The 15 degree curve is slightly more inclined and therefore has more tolerance for inaccuracies in the experimental strength results. It is also noted that the mean F 1 2 values, shown in Figure 3.16, become successively larger as the angle to grain increases. As mentioned in section, the off-axis tests were conducted with the use of non-rotating clamped grips. Pagano and Halpin (1968) showed that these end constraints could induce shearing forces and bending couples at the ends of the specimens per Figure 3.18. Further to this, however, Rizzo, 1969 showed how these non-uniform influences could be minimized by providing an adequate length to width ratio. He found that for long specimens with l/w " 10, "a high degree of test accuracy (could) be obtained". The specimens in this study had a l/w ~ 9.7. This should be adequate to provide sufficiently accurate results; however, based on the sensitivity of F12, a small grip effect may cause the difference in F 1 2 mean values. Figure 3.18 Influence of End Constraint in the Testing of Anisotropic 5o<i/e5^ om Pagano and Halpin, 1968) 68 If the off-axis specimens were subjected to additional forces at the specimen ends, then the specimens could be subjected to higher tension perpendicular to grain stresses. (The parallel to grain and shear stresses could also be affected, but the specimen response to these stresses would not be as significant.) Figure 3.19 was created to visually interpret the effect of potentially higher tension perpendicular to grain forces on the final solution for F 1 2 for each angle to grain. The curves of Figure 3.19 were calculated with equation 2.9, assuming the average values for all principal strengths, constant values for o , anda6 (computed from equation 1.1 for average off-axis stress) and vcrrying the value for o 2 for each angle to grcrin. The tension perpendicular to grain stress at failure, assiiming a uniform stress distribution across the specimen, for all specimens varied between approximately 1.1 and 2.5 MPa. The significance of this plot is as follows: if there were a small influence from the non-rotating grips, a higher than expected value for perpendicular to grcrin stress could be present in the specimen. Calculating F 1 2 based on a mistakenly lower value of o 2 would result in very large values for F 1 2 for 45 and 60 degree specimens and moderately high values for 30 degree specimens. There is comparatively little influence on the 15 degree specimens. This could explain the successively larger mean values for F12for increasing angle, obtained from the individual off-axis tests shown in Figure 3.16. It appears then, that the best of the four tests performed, in terms of establishing the most accurate value for F 1 2 , is the 15 degree off-axis test. 69 Based on these findings, another analysis to deterrnine F 1 2 was carried out using only the off-axis test results for 15 degree data. The solution was a mean value for F 1 2 very close to zero, +0.00003 with a standard deviation of 0.000015. The size factor changed from the previous mean value of 1.68 to 1.38 (standard deviation from 0.17 to 0.063) which translates to a mean shear strength of 8.0 MPa from the previous 6.7 MPa. These values will be utilized in the application of the Tsai-Wu theory in the following chapter. The cumulative probability distributions for the simulated off-axis failure stresses is reproduced using the above parameters for F 1 2 and size factor in Figure 3.20. A visual comparison of the simulated and experimental results shows the random variable Tsai-Wu model fits the experimental 15 degree data very well. The 30, 45 and 60 degree data are still reasonably accurate however they are consistently overestimated. 71 3.4.3 Influence of Fn on the Strength Envelope of Laminated Veneer Prior to implementing the Tscd-Wu strength criterion, it is prudent to understand the influence of the interaction parameter, F 1 2 on the failure envelope in all four quadrants of the stress space because the F,2 parameter determines how well the strength criterion fits the multiaxial strength data in all four quadrants. A plot of the failure envelope with vcrrying F 1 2 values, assuming zero shear for convenience, was created in Figure 3.21. This plot is similar to Figure 2.3, except the average values for principal strengths obtained from this study were utilised. It is evident from Figure 3.21 that the curves in quadrant I are very similar. This means that an inaccuracy in F 1 2 would not affect the results significantly in this quadrant; however, the consequential results in the third and fourth quadrants are magnified. Combined stress tests from either quadrant HI or IV would be more suitable to determine the value for F 1 2 for laminated veneer. There are, however, practical limitations to these tests as mentioned previously in section 2.2.3. 73 4. Verification of Tsai-Wu Failure Theory Using the strength parameters established in Chapter 3, the Tsai-Wu theory is applied to predict the cumulative probabiTity distribution for the failure load of off-axis three point bending specimens. The loading application is illustrated in Figure 4.1, pg. 77. Since the 'test' application is in bending, the strengths for the brittle failure modes, ^ , Yt and S are susceptible to a load configuration effect. This effect must be addressed accordingly in the failure theory application. Three approaches are considered which use Monte Carlo simulations to account for the random nature of the strength variables in the strength theory. The models are then verified by comparison with experimental results. 4.1 Load Configuration Effect Similar to the aforementioned size effect, load configuration effect can be quantified using Weibull weakest link theory. Weibull theory predicts that when comparing brittle members of the same size but subjected to different stress distributions, the observed strength decreases as the percentage of volume that is highly stressed, increases. This is rationalized as follows: the member with more volume highly stressed has a higher probability of containing a larger flaw (or weaker zone) than does the member with a lower portion of material highly stressed. In this study, the strengths, Xj, Yv and S are acquired through tests on specimens of equal size, or are adjusted for size 75 effect, in the case of shear. The specimens for the off-cads bending tests also conform to this specimen size. However, the stress distributions for the bending test differ from that of the tensile test. For the uniaxial tensile test, tensile stress is uniform across the test length whereas the nominal tensile stress Distribution for a bending specimen increases linearly to the highest value at center span on the tension edge. (The distribution for shear stress in the two cases also differ). It is necessary then, that the brittle strengths of Tscd-Wu criterion, X,, Yt, and S, include some load configuration effect when applied to a different stress configuration, such as bending. The three models which apply the theory differ in their approach to incorporating this load configuration effect. Each model entails Monte Carlo simulations. 4.2 Analytical Methods 4.2.1 Monte Carlo Simulation Monte Carlo simulation procedures involve substantial repetition of a simulation process using random variables generated from assumed probability distributions. For each set of randomly generated values, a simulated solution to a problem is obtained. The simulated results can therefore represent a sample of experimental observations. The prediction models of this thesis utilize simulation procedures to estimate the probability of failure of an off-cods bending specimen at different loads through application of equation 2.9. The random variables are the principal material strengths, 76 X(, Yt, X,., Yc, S, F 1 2 and the size factor for the shear block strength, represented by the probability distribution parameters established earlier. The other variables in equation 2.9 (ie. the applied stresses in the principal material directions, a,, o 2, and o6 ) are assumed to be deterministic and evaluated either from finite element models or classical mechanics. 4.2.2 Direct Approach - Coupling Finite Element Analysis with Tsai- Wu Theory Finite Element Analysis In this approach, the stresses induced by the applied loads were calculated using finite element analysis. Similar to section, the formulation again used quadratic, isoparametric elements with a 3 x 3 Gauss quadrature. The off-axis bending specimen was discretized as shown in Figure 4.1. There are, in total, 261 nodal points and 72 elements. The stresses at the Gaussian integration points were transformed to the stresses in the principal material directions. 63.4 i 1 \ y y y r f y y y y 12.7 ±19 19 25.4 r / ^ o o ^ i \\\ >Symmetrical 6 ® 3 8 ' 1 2@25.4 7 ^ 7 !2.7 610 Figure 4.1 - Finite Element Mesh and General Setup for Off-Axis Bending Specimen Analysis (All units in mm) 77 Adjusting Brittle Strengths for Volume About Each Gauss Point (Method 1) Monte Carlo simulations with size adjustment factors for the strengths X,, Yt, and S were performed. Using Weibull weakest-link principles , the three brittle strengths were adjusted from representing the strength of the original rectangular test volume to representing that of a small volume in the beam. A sample calculation to determine a size adjustment factor for follows: Given equation 3.3: tj = o r e c t = parallel to grain tensile strength of the rectangular test specimen x 2 = ° B E A M = parallel to grain tensile strength of small beam volume then It is assumed that the stress in both sides of the equation is uniform. Therefore, Let: V, red a k BEAM BEAM 78 Figure 4.2 shows a typical finite element and the associated 9 Gaussian integration points. The region surrounding each Gaussian integration point was assumed to be uniformly stressed. The boundaries of a region were defined by the midpoint between adjacent Gaussian integration points. For example, consider region 1: V B E A M © = 7.8 x 3.9 x 35 mm3 1064.7 mm3 35 3£ © x © x © x © x ® X © x ® x @ x © X 3.9 4.9 3.9 -+-7.8 9.8 7.8 Figure 4.2 Regions Surrounding Gaussian Integration Points The size factor for tensile strength parallel to grain, X,, is : V r e c t= 1.36 x 106mm3 k = 5.79 for Xj (Table 3.1) \ > 1.36x106 V 1064.7 3.44 Assuming the brittle strength values to be lognormally distributed, they were initially generated using the mean and standard deviation strength values based on the original rectangular volume. They were then adjusted by the appropriate size factor to obtain the random brittle strengths of the corresponding Gaussian integration point 79 region of interest. The other strength variables, X,., Yc, and F 1 2 were assumed random between beams but constant within each member. Therefore, they were generated once for each beam. The Tscd-Wu criterion was evaluated for each Gaussian integration point, given the set of principal strengths and the corresponding set of applied stresses.13 If any one of these points indicated failure according to the Tscri-Wu criterion, it was assumed that the entire beam incured failure. This analysis was replicated 2500 times. The probability of any beam failing under a specific load P was taken as the ratio of the number of beams that failed in the above simulations and the total number of replications (ie. 2500). The stresses were determined for one load P = 1.1 kN, using finite element analysis. For other loads, the original stresses were multiplied by a load ratio factor, assuming the analysis to be linear elastic. In this way, the probability of failure was computed for many loads enabling the generation of a cumulative probability distribution for failure load, P. 13 It was noted that failure occurred predominantly at the bottom midspan Gaussian integration points. To obtain a more accurate prediction, the applied stresses at these points were adjusted to reflect that of the extreme bottom edge of the beam. 80 Adjusting Brittle Strengths for Volume Within One Finite Element (Method 2) It was noted in the analysis of the previous model that failure consistently occurred at midspan on the tension edge of the beam. Referring to Figure 4.3, the shaded region within each finite element along the bottom of the beam, in the failure zone, was deemed to be uniformly stressed. As a result, each region was assumed to have one strength. Using this premise, an alternative size adjustment factor was investigated. The stresses at the three Gaussian integration points contained in the shaded region, differed at most by approximately 6%, considering both 30 and 45 degree results. The width of the region is the width of one element and the height is from the bottom of the beam to the midpoint between the Gaussian integration points. H-- T n T 4= i I I i —L J >7 '-— V —y 1S5 Failure Zone 35 ?3 12.7 3.9 4.9 3.9 25.4 Figure 4.3 - Region of Uniform Stress for Size Effect Factor 81 Now, V B E A M = 25.4 x 3.9 x 35 mm3 = 3467.1 mm3 Thus, the new size factor for tensile strength parallel to grcrin, , for this entire lower region is: BEAM net V \ 'BEAM/ 1.36xl06 \ > V 3467.1 ) 2.80 The same calculation was carried out for the strengths Yt and S, using the corresponding shape parameters for these values. These size factors were incorporated into the theory in the same manner as those for method 1. However, the Tsai-Wu criterion was evaluated for each region containing 3 Gaussian integration points instead of each region surrounding each Gaussian integration point. The brittle strengths were assumed to be constant within a region and the stresses within a region in the failure zone were averaged. Again, beam failure was assumed upon detection of first failure and the probability of failure was determined by the number of failures in 2500 replications. 82 4.2.3 Indirect Approach -Load Configuration Effect Applied to Predicted Off-Axis Tensile Values (Method 3) A less direct and perhaps less versatile approach to obtaining the probability of failure of the off-axis specimen was investigated to provide a comparison to the former methods. Here the Tscri-Wu failure criterion was used to simulate random strengths of off-axis tension specimens first. Then, using Weibull formulation, the off-axis tensile strengths were adjusted for a load configuration effect to yield off-axis bending strengths at the corresponding angle to grain. The failure loads in bending were derived through elementary beam theory and then ranked to obtain the cumulative probability distribution. The methodology for adjusting the off-cods tensile strengths to the off-cods bending strengths follows. Equation 3.2 states that: Let: x i = o b = tension stress in off-cods bending T 2 = ot = tension stress in off-cods tension 83 Considering the bending specimen first: 4* Mr Let: max(b) mcodmum tension stress in bending = bending moment = Px M m a x = mcrximum bending moment = PL then, „, _ max mar (b) ~ Z PL/' 6 4 ( b e * 2 , thus; p - A b d 2 6L °mai(b) Now, M v 2 Jbd3 84 Substituting for P; o b ( r.y) = - m a z i b ) 6L2bd 3 which simplifies to b Id Let P^ , = probability of failure in bending; 1 f /"mat4 \* b L 2 0 2 ° - ^ .ft 0 "2 "2 0 2 1 0 2 L -2b / ° m a r 4 \ ( ry ) 1 V Q \ L d n J / ( l . l ) 2 0 -2b / 0mar4\ 4 V „ U d ' n / (t.l)2 finally; -2bLd  /"max y M m 1 P/b= 1 - e 85 Considering the off-cods tension specimen ; V0 \ m J X —I Equating the two probabilities; therefore; / \ max(t) i ( \ max(b) r 1 k 1 1 1 > V o m > 2 (i: * 1 )2 maz(f) i(i>)/ maz(t) lib) Vt2(i:* l)2 Using this relationship, the predicted 30 and 45 degree off-cods tensile strengths, obtained from simulations with the Tsai-Wu theory, were adjusted to predict the off-cods bending strengths. The cumulative probability distributions for the simulated off-cods tensile strengths is shown in Figure 4.4. A sample calculation to adjust the 50th percentile 30 degree off-cods tensile strength follows. Given, Vb = 35mm x 68mm x 559mm (center to center bearing) V, = 35 mm x 68mm x 610mm (grip to grip length) 86 from Figure 4.4: amcDt(t) =7.47 MPa then; k (shape parameter for 2-P Weibull obtained from simulated results for 30 degree) = 11.85 x(t) [ V, 2(ir* 1)2J 7.47 559 V 610x2x(11.85* l) 2 12.28 MPa 11.8! Using elementary beam theory; 4Jbd2 6L 4(35)(63)2 6(559) 2034.4 N (12.28) - 2.03 KJV This calculation was repeated for all probabilities, yielding a cumulative probability distribution for the failure load in bending. 88 4.3 Experimental Off-Axis Bending Tests - Materials and Methods The accuracy of the previous prediction models was checked by small, off-axis, three point bending tests. The following section reviews the details of these tests. 4.3.1 Materials The specimens for these tests were taken from the same original boards from which the principal and the off-axis tensile strengths were obtained. The fabrication specifications and board treatment are therefore as outlined in section 3.1.1. The cutting layout for these specimens is shown in Figure 3.1. The specimens were dimensioned to adhere as closely as possible to that of the analytical models of section 4, which, in turn, conformed to the size of the tension specimens. Thus, the specimens were approximately 63mm wide by 35mm thick and 61 Omm long (Figure 4.1). 4.3.2 Test Method The bending tests were performed on a Syntech 30/D testing machine with a 8.9 kN load cell capacity. The bearing block used to impart load was of the form and size designated by ASTM D143-83. Also, the support apparatus conformed to this standard. The speed of testing was at a constant rate of 1.3 mm/min which induced failure between 5 and 10 minutes. The cross sectional dimensions of the specimens were measured with 89 calipers at the two ends of the specimen and averaged to ensure rninimal deviation from that assumed for the prediction models. 4.4 Results and Discussion 4.4.1 Off-Axis Bending Test Results Descriptive statistics for both the 30 and 45 degree data are summarised in Table 4.1. Also the 2 parameter Weibull coefficients are provided as they are used for later analysis in section The failure mode for all specimens was tension perpendicular to grain with the failure plane coincident with the grain angle. Failure occurred within 51mm of midspan with one exception in the 45 degree data set which initiated failure at approximately 100 mm from midspan. This behaviour is consistent with that shown in the prediction models. Statistics Off-Axis Bending Failure Load 30 Degrees 45 Degrees Count 17 18 Mean (kN) 2.09 1.27 Stand. Dev. (kN) 0.24 0.16 Coefficient of Variation (%) 11.5 12.6 2-P Weibull Shape 10.05 9.01 Scale 2.19 1.34 Table 4.1 - Statistical Summary for Off-Axis Bending Tests 90 4.4.2 Comparison of Prediction Models and Experimental Results Figures 4.5 - 4.10 compare the cumulative distributions of the failure load from the various prediction models with that of the experimental results. Also, Tables 4.2 - 4.7 summarize the comparative data which excrmines the accuracy of each method at three failure probability levels; 5%, 50% and 95%. Results of the Direct Approach Method 1 The results of method 1 are illustrated in both Figures 4.5 and 4.6. Tables 4.2 and 4.3 quantify the accuracy of each graph respectively. The two graphs differ solely in the values used for the Weibull shape parameters (used when ccdculating size adjustment factor as in section For Figure 4.5, the shape parameters for the variables Xj, Yt, and S, shown in Table 3.1, were determined using a maximum likelihood approach. Whereas, for Figure 4.6 the shape parameters are derived by the respective variables' coefficient of variation (k = COV "1 085). This approximation was shown to be acceptable by R.H. Leicester (1973). Comparing Figure 4.5 with 4.6, it is apparent that the prediction model is dependent on the shape parameters chosen. This is especially important for the shape parameters associated with the perpendicular to grain tensile strength, kYt . The maximum likelihood and COV approach yield kYt = 15.89 and kYt = 12.46, respectively. 91 Method 1 Cumulative Distribution Function for Off Axis Bending Failure Load Predicted vs Experimental Figure 4.5 - k from Maximum Likelihood Failure Load, P (KN) Figure 4.6 - k from COV Approximation Failure Load, P (KN) 92 Method Angle to Grain Prob. of Failure % Predicted (KN) Experimental(KN) (From 2 Parameter Weibull Fit) % Difference SSE (x Iff2) 1 30 5 1.99 1.63 22.0 12.96 50 2.20 2.11 4.3 0.81 95 2.39 2.44 2.1 0.25 45 5 1.00 0.96 4.2 0.16 50 1.12 1.29 15.2 2.89 95 1.21 1.51 24.8 9.00 Total SSE 26.07 Table 4.2 - Comparative Data Between Predicted and Experimental Results for Method 1, *(Shape Parameter from Maximum Likelihood - Corresponds to Figure 4.5) Method Angle to Grain Prob. of Failure % Predicted (KN) Experimental(KN) (From 2 Parameter Weibull Fit) % Difference SSE (xl(T2) 1 30 5 2.10 1.63 28.8 22.09 50 2.32 2.11 10.0 4.41 95 2.51 2.44 2.9 0.49 45 5 1.08 0.96 12.5 1.44 50 1.20 1.29 7.5 0.81 95 1.30 1.51 16.2 4.41 Total SSE 33.65 Table 4.3 - Comparative Data Between Predicted and Experimental Results for Method 1, * (Shape Parameter from COV Approximation - Corresponds to Figure 4.6) 93 The smaller 'k' of Figure 4.6 produces higher tension perpendicular to grcrin strengths and thus produces higher failure loads for the same probability of failure, shifting both curves to the right. Figure 4.5 provides a better overall fit comparing the sum of squared errors for the three probability levels for both 30 and 45 degree graphs in Figure 4.5 (SSE = 0.26) to that of Figure 4.6 (SSE = 0.34). Both the 30 and 45 degree prediction curves under-estimate the variability of the experimental results. Using strength parameters from a more intense testing scheme may provide a better fit to the extreme probability levels. Considering the 50th percentile results only, Figure 4.5 (SSE = 0.037) is again more accurate than Figure 4.6 (SSE =0.052). Method 2 The results of method 2 are shown in both Figures 4.7 and 4.8. Tables 4.4 and 4.5 summarize the analysis of each graph respectively. Again, the two graphs differed in the set of shape parameters used, as in method 1. Figure 4.8, using the 'k' from the COV approximation, provides a more accurate fit than Figure 4.7, comparing the total SSE of the two graphs. The same is true when comparing just the 50th percentile. 94 Method 2 Cumulative Distribution Function for Off Axis Bending Failure Load Predicted vs Experimental Figure 4.7 - k from Maximum Likelihood Failure Load, P (KN) Figure 4.8 - k from COV Approximation 0.5 1.5 Failure Load, P (KN) 2.5 95 Method Angle to Grain Prob. of Failure % Predicted (KN) ExperimentalfKN) (From 2 Parameter Weibull Fit) % Difference SSE (xl(T2) 2 30 5 1.90 1.63 16.6 7.29 50 2.13 2.11 0.9 0.04 95 2.35 2.44 3.8 0.81 45 5 0.95 0.96 1.1 0.01 50 1.08 1.29 19.4 4.41 95 1.20 1.51 25.8 9.61 Total SSE 22.17 Table 4.4 - Comparative Data Between Predicted and Experimental Results for Method 2, *(Shape Parameter from Maximum Likelihood - Corresponds to Figure 4.7) Method Angle to Grain Prob. of Failure % Predicted (KN) Experimental(KN) (From 2 Parameter Weibull Fit) % Difference SSE (x 1(T2) 2 30 5 1.98 1.63 21.5 12.25 50 2.23 2.11 5.7 1.44 95 2.44 2.44 0.0 0.00 45 5 1.02 0.96 6.3 0.36 50 1.15 1.29 12.2 1.96 95 1.28 1.51 18.0 5.29 Total SSE 21.30 Table 4.5 - Comparative Data Between Predicted and Experimental Results for Method 2, *(Shape Parameter from COV Approximation - Corresponds to Figure 4.8) 96 As in method 1, method 2 is not accurate in predicting the extreme probabilities. However, method 2 does provide a slightly better depiction of the experimental variability than does method 1. In comparing the two methods (ie. the four graphs) of the direct approach, it was found that method 2, in which the stresses were averaged over a section of an element, was more accurate. Results of the Indirect Approach Method 3 The results of the mdirect approach, using shape parameters obtained by fitting a 2 parameter Weibull distribution to the simulated tensile strengths are summarized in Figure 4.9 and Table 4.6. The results are quite reasonable with a low SSE of 0.11. The relatively small sample sizes for determining principal strengths may influence the variability of the predicted tensile strengths and hence influence the shape parameter. Therefore, to evaluate the sensitivity of method 3 to shape parameter used, another shape parameter, obtained by fitting the experimental bending tests, was used. The results of this analysis are summarized in Figure 4.10 and Table 4.7. Figure 4.10 illustrates very good agreement between method 3 and the experimental results. This finding is reiterated in Table 4.7 where the total SSE = 0.08 is the best among all methods tried. 97 Method 3 Cumulative Distribution Function for Off Axis Bending Failure Load Predicted vs Experimental Figure 4.9 - k from Simulated Tension Strengths Failure Load, P (KN) Figure 4.10 - k from Experimental Bending Results Failure Load, P (KN) 98 Method Angle to Grain Proh. of Failure % Predicted (KN) Experimental(KN) (From 2 Parameter Weibull Fit) % Difference SSE (xW2) 3 30 5 1.78 1.63 9.2 2.25 50 2.01 2.11 5.0 1.00 95 2.32 2.44 5.2 1.44 45 5 0.99 0.96 3.1 0.09 50 1.14 1.29 13.2 2.25 95 1.32 1.51 14.4 3.61 Total SSE 10.64 Table 4.6 - Comparitive Data Between Predicted and Experimental Results for Method 3, *(Shape Parameter from Maximum Likelihood Fit of Simulated Tension Strengths -Corresponds to Figure 4.9) Method Angle to Grain Proh. of Failure % Predicted (KN) Experimental(KN) (From 2 Parameter Weibull Fit) % Difference SSE (xlO-2) 3 30 5 1.87 1.63 14.7 5.76 50 2.12 2.11 0.5 0.01 95 2.45 2.44 0.4 0.01 45 5 1.07 0.96 11.5 1.21 50 1.23 1.29 4.9 0.36 95 1.43 1.51 5.6 0.64 Total SSE 7.99 Table 4.7 - Comparitive Data Between Predicted and Experimental Results for Method 3, *(Shape Parameter from Maximum Likelihood Fit of Experimental Bending Strength-Corresponds to Figure 4.10) 99 5. Conclusion 5.1 Summary and Conclusions A statistically based method for determining the interaction parameter, F 1 2 of the Tsai-Wu theory is described which allows this parameter to be represented by a probability distribution. This approach is more rational for dealing with problems containing random variables. Previous studies evaluated the parameter based on deterministic values which often lead to ineffectual results. Also, with regards to the parameter F 1 2 , a sensitivity study revealed that the 15 degree off-axis tensile test is the most reliable of the four angles considered. It was found that small inaccuracies in the 15 degree off-axis data would have less impact on the interaction parameter than for the higher degree angles evaluated in this study. A comparison of four orthotropic failure theories; Tscd-Hill, Norris, Tsai-Wu and Tan-Cheng theories was conducted based on the off-axis tensile test results. Comparing weighted sum of squared errors for each theory, it was found that considering practicality and accuracy, the Tscd-Wu criterion was the best theory. This theory was then applied to another biaxial stress situation to verify its accuracy. The Tscd-Wu strength theory was applied in conjunction with finite element analysis to predict failure load cumulative probability distribution, of 30 and 45 degree 100 off-axis 3 point bending specimens. Since the brittle strengths of the failure theory, tension parallel and perpendicular to grain, and shear strength, were established in a uniform stress state, a load configuration effect had to be incorporated into the prediction model. Two methods to accomplish this were investigated using Weibull weakest link theory. In both cases, the brittle strengths were adjusted from describing the strength of the original rectangular test volume to describing the strength of a small volume of material in the beam. The prediction models were corroborated by experimental bending tests. An alternative, less adaptable method to apply the failure criterion, and incorporate load configuration effect was explored. Instead of directly applying the theory to the bending application with finite element analysis, the theory was used to predict the off-axis tensile strengths and these strengths were adjusted for load configuration effect to ultimately predict failure load in bending. The results of this approach were also in good agreement with the experimental results. In summary, this study reveals several heretofore unknown aspects of predicting strength of wood products. Most significantly is the fact that size effects are an integral component of strength prediction and must be addressed in failure theory implementation. Also, statistically based methods are provided for both determination and application of the Tsai-Wu strength theory. Unlike previous studies, this study treats strength parameters as random variables providing an overall more reliable, improved basis for predicting strength of wood products. 101 5.2 Future Research It was stated at the outset of this thesis that the analysis contained within dealt only with lconinated veneer as opposed to lominated veneer lumber, which is the commonly used structural material. The two materials are significantly different in that LVL contains butt joints. In order to model LVL these joints must be taken into consideration. This thesis may be used in conjunction with a future study that considers butt joints to ultimately predict multi-axial strength of LVL In reviewing the effect that the interaction parameter, F 1 2 of the Tsai-Wu theory has on the failure envelope, it was discovered that the first quadrant of the stress space, the quadrant represented in this study, is the least sensitive to variations in this parameter. This means that in determining F 1 2 from this quadrant, a small error could induce much larger errors in other quadrants. Although tests for these other quadrants are accepted as being very difficult to implement for lcrminated veneer, further consideration should be given to this area for this purpose. The influence of a size effect on the brittle strength parameters of the Tscd-Wu theory was considered in several phases of this thesis assuming the Weibull weakest link concept. However, it was assumed that the more ductile failure modes did not require any size adjustment. A future study could evaluate this assumption. 102 References ASTMStandard D-5456 (1993) "Standard Specification for Evaluation of Structural Composite Lumber Products" American Society for Testing and Materials, Philadelphia, Pa. Vol. 04.09 ASTMD-143-83 "Standard Methods of Testing Small Clear Specimens of Timber" ASTMD-198-84 "Standard Methods of Static Tests of Timbers in Structural Sizes" ASTM D-2016-83 "Standard Test Methods for Moisture Content of Wood" ASTM D-2395-83 "Standard Test Methods for Specific Gravity of Wood and Wood-Base Materials" American Institute of Timber Construction. Timber Construction Manual. (1974) John Wiley & Sons, New York Barrett, J.D., Foschi, R.O., and Fox, S. P. (1975). "Perpendicular-to-grain strength of Douglas-fir." Canadian J. Civ. Engrg., 2(1): 50-57 Bodig, J., and Jayne, B. A. (1993) Mechanics of Wood and Wood Composites, Krieger Publishing Co. Malabar, Florida Bohlen JC, (1974) "Tensile Strength of Douglas-Fir Laminated-Veneer Lumber," Forest Prod J. 24(1) pp. 54-58 (1975) "Shear Strength of Douglas-Fir Laminated-Veneer Lumber," Forest Prod J. 25(2) pp. 16-23 Cowin, S.C, (1979) " On the Strength Anisotropy of Bone and Wood," J. of Applied Mechanics, ASME Transactions, 46(4), 1979, pp. 832-837. Goodman, J. R., and Bodig, J., (1971) "Orthotropic Strength of Wood in Compression," Wood Science, 4(2), pp. 83-94. 103 Griffith, J.E. and Baldwin, W.M. (1962) "Failure Theories for Generally Orthotropic Materials," Developments in Theoretical and Applied Mechanics, Vol, 1 pp. 410-420 Hankinson, R. L. (1921) "Investigation of Crushing Strength of Spruce at Varying Angles of Grain," Air Service Information Circular No. 259, U.S. Air Service Hesterman, N.D. and Gorman, T.M. (1992) "Mechanical Properties of Laminated Veneer Lumber made from Interior Douglas-Fir and Lodgepole Pine," Forest Prod. J. 42(11/12) pp. 69-73 Hill, R. (1948) "A Theory of the Yielding and Plastic Flow of Anisotropic Metals," Proceedings of the Royal Society, Series A, Vol. 193, pp. 281-297 Kobetz, RW. and Krueger, G. P. (1976) "Ultimate Strength Design of Reinforced Timber-Biaxial Stress Failure Criteria," Wood Science, 8(4), pp. 252-261 Kollman, F.F.P. and Cote, W.A. Jr.,(1968) Principles of Wood Science and Technology, I Solid Wood, New York: Springer-Verlag Inc., pp. 326 Lam, F., and Varoglu, E. (1990) "Effect of Length on the Tensile Strength of Lumber." Forest Products J., 40(5): 37-42 Leicester, RH. (1973) " Effect of size on the Strength of Structures" CSIRO, Div. Build. Res., Tech. Pap. No. 71, Melbourne, Aust. Leichti, R.J. and Tang, R.C. (1989) "Predicting the Load Capacity of Wood Composite I - beams Using the Tensor Polynomial Theory," Wood Science Technology, 23, pp. 109-121 Liu, J.Y. (1984) "Evaluation of the Tensor Polynomial Strength Theory for Wood," J. Composite Material, Vol.18, pp. 216-225 Nahas, M.N. (1986) " Survey of Failure and Post-Failure Theories of Laminated Fiber-Reinforced Composites" J. Composites Technology and Research. Vol. 8, No.4, pp. 138-153 Narayanaswami, R. and Adelman, H. M., (1977) "Evaluation of the Tensor Polynomial and Hoffman Strength Theories for Composite Materials," J. Composite Materials,V ol. 11, pp. 366-377 104 Neter, J., Wasserman, W. and Kutner, M.H. (1990) (3rd Ed.) Applied Linear Statistical Models. Richard D. Irwin. INC., Burr Ridge, Illinois. Norris, C.B., (1962) "Strength of Orthotropic Materials Subjected to Combined Stress," U.S. Forest Products Lab. Rep. 1816, FPL, Madison, Wis. Pagano, NJ. and Halpin, JC. (1968) "Influence of End Constraint in the Testing of Anisotropic Bodies," J. Composite Materials 2(1) pp. 18-31 Perkins, R.W. (1967) "Fundamental Concepts Concerning the Mechanics of Wood Deformation: Strength and Plastic Behaviour," Forest Prod J. 17(4), pp. 57-68 Pipes, R.B. and Cole, B.W. (1973) "On the Off-Axis Strength Test for Anisotropic Materials," J. Composite Materials, Vol. 7, pp. 246-256 Rizzo, R.R. (1969) "More on the Influence of End Constraints on Off-Axis Tensile Tests," J. Composite Materials, Vol. 3, pp. 202-219 Rowlands, R.E. (1985) "Strength (Failure) Theories and Their Experimental Correlation," Handbook of Composites. Vol.3 - Failure Mechanics of Composites, pp.71-125 Sandhu, R.S. (1972) "A Survey of Failure Theories of Isotropic and Anisotropic Materials, "U.S. Air Force Technical Report No. AFFDL-TR-72-71, Wright Patterson AFB, OH Sharp, D.J., and Suddarth, S.K. (1991) "Volumetric Effects in Structural Composite Lumber." Proc, Inc. Timber Engrg. Conf., London, England, 3: pp. 427-433 Suhling, J.C., Rowlands, R.E., Johnson, M.W. and Gunderson, D.E. (1984) "Tensorial Strength Analysis of Paperboard," Experimental Mechanics, 25(1), pp. 75-84 Tan, S.C. (1990) "A New Approach of Three-Dimensional Strength Theory for Anisotropic Materials," International J. of Fracture, Vol. 45, pp35-50 Tan, S.C. and Cheng, S. (1993) "Failure Criteria for Fibrous Anisotropic Materials," J. of Materials in Civil Engineering, 5(2) pp. 198-211 Triche, M.H. and Hunt, M.O. (1993) "Modelling of Parallel-Aligned Wood Strand Composites", Forest Prod J. 43(11/12), pp. 33-44 105 Tsai, S. W., and Wu, E. M. (1971) "A General Theory of Strength for Anisotropic Materials," Journal of Composite Materials, Vol. 5, pp. 58-80. van der Put, T.AC.M. (1982) "A General Failure Criterion for Wood," IUFRO Timber Engineering Group Meeting, Paper 23 [Sweden], IUFRO, Vienna. Weibull, W. (1939). "A Statistical Theory of the Strength of Materials." Proc., Royal Swedish Inst., No. 151, Stockholm, Sweden Wu, E. M., (1974) "Phenomenological Anisotropic Failure Criterion," Mechanics of Composite Materials (ed. G.P. Sendeckyj) New York: Academic Press Youngquist, J.A., Laufenberg, T.L., and Bryant, B.S. (1984) "End Jointing of Laminated Veneer Lumber for Structural Use," Forest Prod. J. 33(11/12), pp. 25-32 Zweben, C. (1994) "Size Effect in Composite Materials and Structures: Basic Concepts and Design Considerations", NASA Conference Publication (Vol/Iss:3271) pp. 197-217 106 Appendix A 107 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * SAS Source Code -* Non-linear l e a s t square * determination of F12 ******************************************** 'ilename f i l e l '570a.dat'; o p t i o n ps=60 ls=70; Data data; i n f i l e f i l e l ; input angle s t r e s s Xt Yt S Xc Yc grpvar; wt=l/(grpvar**0.5); wtstress=wt*stress; Fl=l/Xt-1/Xc; F2=l/Yt-1/Yc; F l l = l/(Xt*Xc) ; F22 = l / (Yt*Yc) ; F66 = l / (S/l.64)**2; X1=F1*(cos(3.1416/180*angle))**2+F2*(sin(3.1416/180*angle))**2; a l = Fll*(cos(3.1416/180*angle)) **4; a2 = F22* (sin(3.1416/180*angle)) **4; a3 = F66* (sin(3.1416/180*angle))**2*(cos(3.1416/180*angle))**2; X2=al+a2+a3; X3 = 2*cos-(3.1416/180*angle) **2*sin(3 .1416/180*angle) **2; Proc NLIN; parms b0=0; model wtstress=wt*((-X1+(Xl**2 + 4*(X2+bO*X3) ) **(1/2) ) / (2*(X2+bO*X3))); output out=pout p=yhat r=resid; Proc p l o t data=pout; p l o t r e s i d * y h a t ; *********************************************** data grouped; set data; i f angle=0 then group=l; 15 then group= 2; 30 then group= 3; 45 then group= 4; 60 then group= 5; 90 then group= 6; by group; proc means data=sorted var n mean; var s t r e s s ; by group; 108 Data Input File for SAS Angle Stress 15 20.19 15 19.51 15 21.44 15 17.83 15 20.31 15 19.67 15 18.46 15 17.54 15 18.62 15 20.07 15 19.50 15 17.57 15 16.69 15 17.25 15 17.38 30 7.17 30 6.29 30 6.29 30 5.80 30 6.77 30 6.76 30 6.84 30 6.03 30 6.24 30 7.03 30 6.20 30 6.78 30 6.50 30 6.16 45 3.56 45 3.79 45 3.72 45 3.26 45 3.78 45 3.56 45 3.68 45 3.72 45 3.95 45 3.70 45 3.60 45 3.61 45 4.38 45 4.08 45 3.52 45 3.76 60 2.73 60 2.99 60 2.60 60 2.44 60 2.97 60 2.82 60 2.85 60 2.65 60 2.52 60 2.57 60 2.43 60 2.89 60 2.65 60 2.68 Xt Yt 39.21 2.48 50.98 2.26 79.71 2.40 55.90 1.71 56.95 2.41 44.96 2.06 67.68 2.20 56.34 2.42 63.86 2.32 56.39 2.06 39.80 2.42 45.27 2.23 53.39 2.36 62.47 2.36 61.87 2.29 39.21 2.48 50.98 2.26 79.71 2.40 55.90 1.71 44.96 2.06 67.68 2.20 56.34 2.42 63.86 2.32 56.39 2.06 39.80 2.42 45.27 2.23 53.39 2.36 62.47 2.36 51.11 2.41 39.21 2.48 50.98 2.26 79.71 2.40 55.90 1.71 56.95 2.41 44.96 2.06 67.68 2.20 56.34 2.42 63.86 2.32 56.39 2.06 39.80 2.42 45.27 2.23 53.39 2.36 62.47 2.36 51.11 2.41 61.87 2.29 39.21 2.48 50.98 2.26 55.90 1.71 56.95 2.41 44.96 2.06 67.68 2.20 56.34 2.42 63.86 2.32 56.39 2.06 39.80 2.42 45.27 2.23 53.39 2.36 51.11 2.41 61.87 2.29 S Xc 10.88 60.42 10.49 55.06 9.87 53.55 11.11 58.74 11.48 55.28 12.47 62.41 10.67 58.92 8.20 56.18 12.90 59.63 12.09 56.99 11.19 58.31 12.28 51.72 10.82 61.44 11.02 55.24 9.14 60.10 10.88 60.42 10.49 55.06 9.87 53.55 11.11 58.74 12.47 62.41 10.67 58.92 8.20 56.18 12.90 59.63 12.09 56.99 11.19 58.31 12.28 51.72 10.82 61.44 11.02 55.24 11.26 57.47 10.88 60.42 10.49 55.06 9.87 53.55 11.11 58.74 11.48 55.28 12.47 62.41 10.67 58.92 8.20 56.18 12.90 59.63 12.09 56.99 11.19 58.31 12.28 51.72 10.82 61.44 11.02 55.24 11.26 57.47 9.14 60.10 10.88 60.42 10.49 55.06 11.11 58.74 11.48 55.28 12.47 62.41 10.67 58.92 8.20 56.18 12.90 59.63 12.09 56.99 11.19 58.31 12.28 51.72 10.82 61.44 11.26 57.47 9.14 60.10 Yc Variance 15.27 1.97 9.71 1.97 13.42 1.97 13.06 1.97 10.95 1.97 12.04 1.97 13.19 1.97 10.84 1.97 13.17 1.97 12.02 1.97 12.92 1.97 10.97 1.97 11.46 1.97 10.67 1.97 10.44 1.97 15.27 0.16 9.71 0.16 13.42 0.16 13.06 0.16 12.04 0.16 13.19 0.16 10.84 0.16 13.17 0.16 12.02 0.16 12.92 0.16 10.97 0.16 11.46 0.16 10.67 0.16 12.94 0.16 15.27 0.06 9.71 0.06 13.42 0.06 13.06 0.06 10.95 0.06 12.04 0.06 13.19 0.06 10.84 0.06 13.17 0.06 12.02 0.06 12.92 0.06 10.97 0.06 11.46 0.06 10.67 0.06 12.94 0.06 10.44 0.06 15.27 0.03 9.71 0.03 13.06 0.03 10.95 0.03 12.04 0.03 13.19 0.03 10.84 0.03 13.17 0.03 12.02 0.03 12.92 0.03 10.97 0.03 11.46 0.03 12.94 0.03 10.44 0.03 "109 


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