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

2.1.2.1 Hankinson Formula 9

2.1.2.2 Hill Theory 11

2.1.2.3 Tsai-Hill Theory 12

2.1.2.4 Norris Theory 13

2.1.2.5 Tsai-Wu Theory 14

2.1.2.6 Tan Theory 18

2.1.2.7 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

3.1.2.1 Tension Tests . 34

3.1.2.2 Compression Tests 35

3.1.2.3 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

3.2.2.1 Non-Linear Weighted Least Squares - Using SAS 53

3.2.2.2 Non-Linear Weighted Least Squares - Using F12FIT 56

3.2.2.3 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

4.2.2.1 Finite Element Analysis 77

4.2.2.2 Adjusting Brittle Strengths for Volume About Each Gauss Point

(Method 1) 78

4.2.2.3 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

4.4.2.1 Results of the Direct Approach 91

4.4.2.2 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

2.1.2.1 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

2.1.2.2 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

2.1.2.3 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

2.1.2.4 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).

2.1.2.5 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

sin 28 1

cos 20 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

F' R22 F' R26 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

2.1.2.6 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

2.1.2.6 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^ A 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

3.1.2.1 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

3.1.2.2 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.

3.1.2.3 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 C

O1

Stand. C

Mean O

0

CO

ft Location

Scale

Shape |

Scale |

Shape |

-=-, • 3

)ev (M

Pa)

(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/cm

3)

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 •

Com

pres

sion

Stre

ss

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.

3.2.2.1 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 e2 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

3.2.2.2 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

3.2.2.3 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 2.1.2.6, 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

Off-Axis Angle to

Grain Hankinson (n=1.7)>

Tsai-Hill Norris Tsai-Wu Tan- (MPa) Hankinson (n=1.7)>

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(st.dev.) = 0.00036, Size Factor (mean) = 1.68, and Size

Factor (st.dev.) = 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 3.1.2.1, 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

4.2.2.1 Finite Element Analysis

In this approach, the stresses induced by the applied loads were calculated using

finite element analysis. Similar to section 3.2.1.1, 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

4.2.2.2 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

4.2.2.3 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 L d /"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 4.4.2.2. 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 Statistics 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 2-P

Weibull 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%.

4.4.2.1 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 4.2.2.2). 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

1

30 50 2.20 2.11 4.3 0.81

1

30

95 2.39 2.44 2.1 0.25 1

45 5 1.00 0.96 4.2 0.16

1

45 50 1.12 1.29 15.2 2.89

1

45

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

1 30

50 2.32 2.11 10.0 4.41 1 30

95 2.51 2.44 2.9 0.49 1

45 5 1.08 0.96 12.5 1.44

1

45 50 1.20 1.29 7.5 0.81

1

45

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

2

30 50 2.13 2.11 0.9 0.04

2

30

95 2.35 2.44 3.8 0.81 2

45 5 0.95 0.96 1.1 0.01

2

45 50 1.08 1.29 19.4 4.41

2

45

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

2

30 50 2.23 2.11 5.7 1.44

2

30

95 2.44 2.44 0.0 0.00 2

45 5 1.02 0.96 6.3 0.36

2

45 50 1.15 1.29 12.2 1.96

2

45

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.

4.4.2.2 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

3

30 50 2.01 2.11 5.0 1.00 3

30

95 2.32 2.44 5.2 1.44 3

45 5 0.99 0.96 3.1 0.09

3

45 50 1.14 1.29 13.2 2.25

3

45

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

3

30 50 2.12 2.11 0.5 0.01

3

30

95 2.45 2.44 0.4 0.01 3

45 5 1.07 0.96 11.5 1.21

3

45 50 1.23 1.29 4.9 0.36

3

45

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

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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

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