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This is to certify that the

dissertation entitled
THEORETICAL AND EXPERIMENTAL
INVESTIGATION ON HYGROSCOPIC WARPING

OF WOOD COMPOSITE PANELS
presented by

DANPING XU

has been accepted towards fulfillment
of the requirements for

Ph . D . degree in FORESTRY

Major professor

Otto Suchsland

January 11, 1994
Date

 

MS U is an Affirmative Action/Equal Opportunity Institution 0-12771

 

 

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LIBRARY
Mlchlgan State
Unlverslty

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

THEORETICAL AND EXPERIMENTAL
INVESTIGATION ON HYGROSCOPIC WARPIN G
OF WOOD COMPOSITE PANELS

By

Dan Ping Xu

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Forestry

1994

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ABSTRACT

THEORETICAL AND EXPERIMENTAL
INVESTIGATION ON HYGROS COPIC WARPING
OF WOOD COMPOSITE PANELS

By

Dan Ping Xu

This thesis deals with the hygroscopic warping of laminated wood and
wood composites. The theoretical analysis includes elastic, inelastic and
visco-elastic approaches. Experiments were conducted on narrow cross
laminated yellow-poplar beams and on two-ply beams constructed from
laminas of medium-density-fiberboard and particleboard. The elastic strains
and the swelling stresses that sustain warp under conditions of increasing
moisture content were determined experimentally, using a specially designed
restraining device.

In the case of the yellow-poplar laminates it was found that in the
transverse direction (tangential) only about 25 percent of the free
hygrosc0pic expansion is transformed into elastic strain under conditions of
complete restraint. In the longitudinal direction, yellow-poplar behaves
elastically. These results were used to modelling the inputs into the elastic
warping equation (inelastic approach) which greatly improved the accuracy
of the theoretical warping predictions.

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In the case of laminate composites, the elastic equation without
modification produced good agreements with measured warp.

The visco-elastic approach, although were time consuming produced
good agreement in both cases.

A detailed discussion of the elastic warping equation for laminated
beams is included. For a two-component multi-layer laminate warping is
found to be a function of the ratio of the elastic moduli of the two materials,
and to be proportional to the difference of the hygroscopic expansion of the
two materials.

This study will allow the formulation of relative simple guidelines for

the mathematical modeling of hygroscopic warping of wood based laminates.

iii

Dedicated to my beloved Ruogong, Shen, and

my dearest parents

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ACKNOWLEDGMENTS

The author wishes to express her deepest appreciation to Dr. Otto
Suchsland of the Department of Forestry, Michigan State University, for his
guidance and encouragement in the preparation of this thesis and for his
thought given throughout the course of this study. Thanks are also extended
to the other members of the author’s Ph. - D. Degree Committee: Professors
Alan Sliker of the Department of Forestry, Professor Frank Hatfield and
Robert Wen of the Department of Civil and Environmental Engineering for
their guidance.

She is particularly grateful to the financial support from the Michigan
State University Forestry Department’s Eastern Hardwood Utilization
Research Program, which is funded by the USDA / Cooperative State
Research Service.

Thanks are also due to Dr. Y. Feng, Dr. S. Hiziroglu and Dr. H. Xu for

their help in preparing the experimental set up and sample cutting.

TABLE OF CONTENTS

Page
LIST OF TABLES ix
LIST OF FIGURES xi
LIST or SYMBOLS . xvii
CHAPTER I INTRODUCTION 1
1.1 The Warping Problem . l
1.2 The Concept of Swelling Stress 4
1.3 Previous Work on Panel Warping
1.4 Need for More Sophisticated Approach 23
_ CHAPTER II THE MECHANICS OF WARPING OF

HYGROSCOPIC LAMINA’TES 30

2.1 Hygroscopic Behavior of Solid Wood and of Wood
Composites 30

2.2 Elastic Characteristics of Solid Wood and Wood
Composites 39

2.3 Prediction of Warping of Wood Laminates Based on
Elasticity 47
2.4 Nonelastic Behavior of Solid Wood and Wood Composite 55

2.4.1 General ' 55

vi

TABLE OF CONTENTS (continued)

2.5
2.6

2.4.2 Wood as Visco-elastic Material

2.4.3 Creep Behavior

2.4.4 Modelling of Creep

2.4.5 Visco-elastic Beam Theory

Development of Experiment on Restrained Swelling

Objectives and Scopes of the Study

CHAPTER HI EXPERIMENTAL METHODS

3.1
3.2

3.3

General

Specimen Preparation

3.2.1 Yellow-Poplar

3.2.2 Wood Composite Board

Test Methods

3.3.1 Relative Humidity Condition

3.3.2 Measurement of Hygrosc0pic Expansion
3.3.3 Measurement of Warping of Laminated Beams
3.3.4 Tension Test

3.3.5 Compression Test

3.3.6 Restrained Swelling Test

CHAPTER IV RESULTS AND ANALYSIS

4.1
4.2

4.3

General

Yellow-Poplar

4.2.1 Results of Experiment

4.2.2 Theoretical Estimation of Warping
MDF/PB

4.3.1 Results of Experiment

vii

55
56
58
61
73
81

82
82
84
84
86
94
94
95
96
98
100
102

123
123
124
124
126
139
139

TABLE OF CONTENTS (continued)
4.3.2 Theoretical Estimate of Warping

4.4 Application of Results to Plywood

4.5 Application of Results to Wood Composite
CHAPTER V SUMMARY AND CONCLUSION
LIST OF REFERENCES
APPENDICES

APPENDD{ A

APPENDIX B
APPENDIX C

viii

139
155
157

171

175

182

188
192

Table 1-1

Table 2-1

Table 2-2

Table 2-3
Table 2-4

Table 3-1
Table 4-1

Table 4-2

Table 4-3
Table 4-4

Table 4-5
Table 4-6

Table 4-7

LIST OF TABLES

Page
Swelling stresses (compression) in various species under
total axial restraint as functions of increasing average
moisture content [21] 29
Dimensional change of domestic woods [10] 37
Center deflection of two layer laminate (T2 = 1 in.

GI — a2 = 0.01 inJin. L = 48 in.) 54
Most often used empirical creep equations [3] 71
Flexural creep parameters of wood composite boards [1],

[30] 72
Test specimen outline ‘ 88
Modulus of elasticity in tension (ksi) of yellow-poplar in

longitudinal direction 129

Modulus of elasticity in compression (ksi) of yellow-poplar

in tangential direction 129
Properties of yellow-poplar 130
Measured warping of yellow-poplar laminated beams over

24-inch span with initial condition of 66 percent RH 130
Results from restraint test of yellow-poplar 131

Calculated warping of yellow-poplar laminated beams over
24inch span using elastic and inelastic approach 132
Visco-elastic estimation of yellow-poplar two-layer beams

over 24-inch span 133

ix

 

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LIST OF TABLES (continued)

Table 4-8
Table 4-9
Table 4-10
Table 411

Table 4-12

Table 4-13

Table 4—14

Table 4—15

Table 416

Table 4— 17

Table 4-18

Table 4-19

Table 4-20

Table 4-21

Comparison of estimated warping with measured warping
for yellow-poplar laminated beams over 24-inch span
Properties of MDF and PB

Results from restraint test of MDF

Results from restraint test of PB

Estimated warping of MDF/PB laminated beams over 30-
inch span using elastic approach

Estimated warping of MDF/PB laminated beams over 30-
inch span using inelastic approach by partial restraint
concept

Estimated warping of MDF/PB laminated beams over 30-
inch span using inelastic approach

Regression models of linear expansion (in/in.) of MDF and
PB

Regression models of swelling stress relaxation moduli
(psi) of PB and tension moduli of MDF

Comparison of estimated warping with measured warping
for MDF/PB laminated beams over 30-inch span

Random run inputs for yellow-poplar beams at 66 - 81%RH
interval

Random run inputs for yellow-poplar beams at 66 - 93%RH
interval

Random run inputs for yellow-poplar 3-ply balanced beams
using inelastic approach

Random run inputs for yellow-poplar 5-ply balanced beams

using inelastic approach

134
141
142
144

146

147

148

149

150

151

158

159

160

161

Figure 1-1
Figure 1-2

Figure 1-3

Figure 1-4
Figure 1-5

Figure 1-6

Figure 1-7

LIST OF FIGURES

Examples of difi'erent forms of warping of flat panels [4]
The concept of swelling stress. (A) free axial expansion; (B)
completely restrained expansion; (C) partially restrained
expansion

The free expansion of beam components (A) and resultant
expansion of laminated two-layer beam (B)

Warping of solid wood panel [54]

Stresses in axially restrained wood composite. Results of
one exposure cycle (36% to 93% to 36% RH). Graph
indicates axial stress and center deflection as functions of
average MC. Upon reversal of exposure condition (93% to
36% RH), center deflection does not fully recover [16]
Axial stresses and midpoint deflection of axially restrained
hardboard strips as function of moisture content (clamped
dry). Number 1 and 2 indicate sequence of exposure cycles.
Dashed line represents theoretical stresses [51]

Warping across face grain of unbalanced, 24 in. long three-
ply loblolly pine beams after exposure to two different
relative humidity intervals. The normal variation of the
results about the average values are due to assumed

variability of input variables of face layers [47]

Page
2

13

14

15

16

LIST OF FIGURES (continued)

Figure 1-8

Figure 1-9

Figure 1-10
Figure 1-11
.Figure 1-12
Figure 1-13

Figure 1-14

Figure 1-15

Figure 1-16

Figure 2—1
Figure 2-2

Figure 2-3.

Warping across face grain of balanced loblolly pine beams

after exposure to relative humidity intervals. Variability of
input variables of face layers causes warping in both
directions [47] 17
Effect of commotion on warping along face grain of 1/2 in.
loblolly pine plywood [47] 18
Effect of cross grain in t0p veneer on warping of 1/2 in., 3-

ply loblolly pine plywood [47] 19
Warping record of particleboard with I-IPL and backer
laminate. Exposure interval is 50% to 84% to 50% RH [42] 20
Warping record of particleboard with melamine laminate.
Exposure interval is 50% to 84% to 50% RH [42] 21
Theoretical predictions vs. measured vertical deflections of
yellow-poplar laminate and beam [57] 22
Time function of average axial swelling stress
(compression) for two different moisture content intervals

[21] 26
Radial and tangential swelling stress (compression) for two
different moisture content intervals (upper figure), and
stress gradients (lower figure) [21] 27
Free and restrained swelling deformations of beech as
function of two different initial moisture contents [21] 28
Hygroscopic isotherms for wood and particleboard [23] 34
Dimensional changes in solid wood pine and particleboard
(heavy line) versus moisture content [23] 35
Basic particle panel products: A particleboard, B
waferboard, and C oriented strand board [10] 36

xii

LIST OF FIGURES (continued)

Figure 2-4
Figure 2-5
Figure 2-6

Figure 2-7

Figure 2-8
Figure 2-9

Figure 2-10
Figure 2-11

Figure 2-12
Figure 2-13
Figure 2-14
Figure 2-15
Figure 2—16
Figure 3-1
Figure 3-2
Figure 3-3

The modulus (tension or compression) as function of grain
angle

Effects of wood species on bending strength in relation to
density of particleboard [23]

Effects of moisture content on various strength properties of
solid pine and of particleboard [23]

A laminated beam with n layers showing individual free
expansion, mi, and resultant expansion, am, (a), and
illustration of relationship between radius of curvature, R,
base length, L, and center deflection, W, of warped beam (b)
Illustration of dimensional variables of n -layer beam .
Center deflection of two layer beam ( T2 = 1 in. 0tl - a2 =
0.01 inJin. L = 48 in.)

Deformation response curve to constant load of elastic
materials

Deformation response curve to constant load of visco-
elastic materials

Effect of external factors on creep

Burger’s four-element visco-elastic model

Uniaxial beam element

Case 1: Complete restraint

Case H: Partial restraint

Designing of yellow-poplar board

Specimen arrangement on the yellow-poplar panel
Designing of unbalanced yellow-poplar laminated beams.
Upon increasing in relative humidity, beams warp

concavely upward

xiii

45

46

46

48
52

S3

68

68

69

70

62

79

80

91

92

LIST OF FIGURES (continued)

Figure 3-4
Figure 3-5
Figure 3-6
Figure 3-7
Figure 3-8

Figure 3-9

Figure 3-10
Figure 3-11

Figure 3-12

Figure 3- 13

Figure 3-14

Figure 3-15

Figure 4.1
Figure 4-2

Cutting diagram for l-inch thick MDF and PB boards
Conditioning chamber used for conditioning of specimens
Conditioning tank used for conditioning of specimens

A dial gage fixed on an aluminum frame used to measure
hygroscopic expansion of yellow-poplar

Optical comparator [50] used for hygrosc0pic expansion
determination

A dial gage fixed on an aluminum frame used to measure
warping of yellow-poplar laminated beam

Relative humidity conditions for MDF/PB laminated beams
Dimensions of tension specimens of yellow-poplar (a) and
MDF 0))

Device for measuring swelling stresses. Miniature load cell
measures swelling force, clip gage in center measures
length change of restrained specimens

Drawing of specimen placement: (a) restrained swelling; (b)
free hygroscopic expansion

Restraining test setup. All measurements are simultaneously
and continuously recorded by a data acquisition system
Record of restrained swelling test of MDF [43]. Specimens
reached equilibrium at M C2 after about 70 hours. Upon
removal of restraint the instantaneous elastic recovery a' of
the free expansion is determined. Swelling stress at that
point is 0"

Density profile of yellow-poplar board

Measured warping of yellow-poplar beams

xiv

93

108 -

110

112

114
98

116

117

119

120

122
135
136

LIST OF FIGURES (continued)

Figure 4-3

Figure 4-4

Figure 4-5
Figure 4-6
Figure 4—7

Figure 4—8

Figure 4-9

Figure 4-10

Figure 4.11

Figure 4-12

Figure 413

Figure 4—14

Nonlinear regression models of yellow-poplar for exposure
interval of 66 - 81%RH

Nonlinear regression models of yellow-poplar for exposure
interval of 66 - 93%RH

Nonlinear regression models of MDF expansion

Nonlinear regression models of PB expansion

Nonlinear regression models of PB swelling stress
relaxation modulus

Implication to the warping of plywood

Estimated warping of yellow-poplar two-ply (L-T) beam
from random run at the condition of 66 - 81%RH using
elastic and inelastic approach

Estimated warping of yellow-poplar three-ply beam (T-L-T)
from random run at the condition of 66 - 81%RH using
elastic and inelastic approach

Estimated warping of yellow-poplar three-ply beam (L-T-L)
from random run at the condition of 66 - 81%RH using
elastic and inelastic approach

Estimated warping of yellow-poplar two-ply (L-T) beam
from random run at the condition of 66 - 93%RH using
elastic and inelastic approach

Estimated warping of yellow-poplar three-ply beam (T-L-T)
from random run at the condition of 66 - 93%RH using
elastic and inelastic approach

Estimated warping of yellow-poplar three-ply beam (L-T-L)
from random run at the condition of 66 - 93%RH using

elastic and inelastic approach

XV

137

138
152
153

154

162

163

164

165

166

167

168

LIST OF FIGURES (continued)

Figure 4—15 Estimated warping (in.) of three-ply balanced yellow-poplar
beams with 48-inch span by introducing the variability on
all layers and face layers 169
Figure 4-16 Estimated warping (in.) of five-ply balanced yellow-poplar
beams with 48-inch span by introducing the variability on
all layers and face layers 170

xvi

a“ , bi 9

Cijkl 1

LIST OF SYMBOLS

m,D , y: constants

creep compliances

modulus of elasticity

the modulus at an angle 0 from the grain direction

the modulus in the grain direction

the modulus in the direction perpendicular to the grain

the modulus of elasticity of plywood in bending when the face
grain is parallel to the span

the modulus of elasticity of plywood in bending when the face
grain is perpendicular to the span

deformation modulus

instantaneous modulus of elasticity

delayed modulus of elasticity

resultant force

the moment of inertia of the 1'” layer about the neutral plane of
the plywood

the moment of inertia of the total cross section about its center line
curvature of a beam 1

length of a beam

xvii

LIST OF] SYMBOLS (continued)

Yuk]:

'1

X.y.z:

resultant moment

radius of curvature of a beam

= it. , is a dimensional parameter
um;

thickness of 1"" layer of a beam

total thickness

temperature change

center deflection or warping

uniaxial swelling stress relaxation modulus
relaxation moduli

coordinates

total free hygrosc0pic expansion value

resultant expansion of a laminated beam

hygroscopic or thermal expansion coefficient

the moisture content change brought about by a change in the
relative humidity of the air

time dependent strain

elastic portion of expansion or instantaneous recovery of expansion
delayed coefficient of viscosity

viscous coefficient of viscosity

stress

thermal stress

Poisson’s ratio

retardation time

xviii

CHAPTER I
INTRODUCTION

1.1 The Warping Problem

This thesis deals with a significant technological problem afflicting
panel materials made from solid wood or from small wooden elements
laminated to form sheet materials. These sheet materials, which will be
described in more detail later, may be overlaid with various thin decorative
layers like veneers or other rigid or flexible layers. In this form, the laminated
sheet materials are incorporated into furniture products such as tops, doors,
shelves, etc.

The aforementioned problem is related to the absorption or desorption
of water by these wood-based materials, which leads to dimensional changes
and in certain cases to geometric distortions known as warping, cupping, or
bowing. The geometric distortions can have different forms as indicated in
Figure 1-1 [4].

Warping of such panels or of furniture elements made from such
panels is both common and difficult to deal with. It is also a defect or
characteristic which is often not repairable, and may require the replacement

of the defective panel. It is difficult to deal with because it may only develop

 

 

 

.—---—--—----~----~-l ————————— .1 "

 

Figure 1-1 Examples of different forms of warping of flat
panels [4].

long after the product has left the manufacturing plant. Thus, the warping
potential of a panel can readily escape detection by standard quality control
systems. It is, therefore, difficult to adjust selection and manufacturing
precesses and procedures in order to reduce the occurrence of panel warping.

Warping is always caused by an imbalance of forces acting in the plane
of individual layers and resulting in bending moments. These forces are
generated whenever the free hygroscopic expansion or shrinkage of
individual layers is being restrained by adjacent layers.

Expansion or shrinkage difference between layers could be caused by a
- variety of factors, such as different hygroscopic equilibrium characteristics,
. differences in grain direction, species and other material characteristics. They
may also be caused by uneven moisture uptake or loss. The inherent
variability of wood material makes it almost impossible to consistently
produce perfectly balanced, and thus, warp-free panels.

It would be desirable to analyze practical panel constructions with
regard to their warping potentials and to assess the importance of various
contributing factors. Such analysis could be based on available information
on the physical and elastic properties and constants of the various materials
involved but, while relatively simple, would have the disadvantage of not
accounting for the distinct visco-elastic nature of wood materials, which
significantly moderates internal stresses, particularly at high moisture
contents.

This thesis attempts to take into consideration the visco-elastic
characteristics of wood and their effect on warping tendencies of simple

laminates made from veneer, wood particles, and wood fibers. The results of

such efforts, both experimental and theoretical, are then compared with
results based on elastic calculations, and with measured warping of laminated

beams exposed to changing relative hurnidities.
1.2 The Concept of Swelling Stress
The concept of swelling stress is illustrated in Figure 1-2 which shows

the free and restrained axial expansion of a single layer short wood column as

a result of a change in moisture content ( MC ) from MCl to MC2 ( AMC ).

 

 

 

 

 

 

 

 

 

 

 

 

 

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37i— , .............................................................. A . ..
w i:_ eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee Q
n l T
PI
P
I4
Pl P'l
A B C

Figure 1-2 The concept of swelling stress. (A) free axial expansion;
(B) completely restrained expansion; (C) partially
restrained expansion.

Figure 1-2 (A) shows the free axial expansion of the column, which

can be expressed as

AL = B AMC (1.1)

where B = the axial expansion coefficient (l/AMC )
AMC = the moisture content change brought about by a change in the
relative humidity of the air (%).
The column after expansion is stress free.
Figure 1-2 (B) shows the same column expanding against a rigid
clamp which prevents axial elongation. As a result of this restraint (in this
case complete restraint) an axial swelling stress develops, which can be

calculated as follows:

 

o=Ee
=59; (1.2)
.Eflgjvc
or,forL=l
o=EBAMC (1.3)

where E = the modulus of elasticity in compression of the column at the
end condition ( MC2 ),( psi).
Figure 1-2 (C) shows the same column under partial restraint, i.e. the

rigid clamp allows a certain amount of free expansion, b , to occur

(b<B AMC).

The stress is now

6' = E (B AMC-b) (1.4)
o'<o (1.5)

These calculations are based on the important assumption that
hygroscopic expansion converts to elastic strain.
This hygroscopic behavior of wood is analogous to the thermal

behavior of metals. The equivalent thermal stress is:

a, = E [3, AT . (1.6)

where [3,. = the thermal expansion coefficient ( 1/°F )
AT: the temperature change ( °F ).

The subject of thermal stress in metals has received considerable
attention [11, 15, 18]. In many cases, equations deve10ped for thermal stress
can be applied directly to hygroscopic swelling or shrinking of wood
materials [51].

Swelling stresses as depicted in Figure 1-2 can lead to buckling [51]
and possibly to the destruction of slender columns or thin plates of wood
composite materials by bending stress associated with buckling [16]. The
typical condition leading to warping is indicated in Figure 1-3. Here, a wood
material consisting of two layers having different expansion coefficients and
different moduli of elasticity and being attached to one another by a rigid

glue bond, is being subjected to a change in the relative humidity of the air.

Figure l-3(A) shows the two layers and their free expansion as it
would occur if the glue line did not exist. Although AMC is the same for
both layers, their total expansion is different on account of the difference
between their expansion coefficients ( [32 > Br ). The existence of the glue
line imposes the condition of partial restraint on both layers (Figure 1-3 (B)),
positive restraint in the case of layer 2 and negative restraint in the case of
layer 1. The laminate expands by BmAMC, which causes compressive stress

in layer 2 and tensile stress in layer 1, which could be expressed as follows:

e1

“1 = El W (1-7)
or, for unit length L = 1

o = E, e1 (1-8)

and

a = Q - (1.9)

If both layers have the same cross-section area, then the average stress

in tension and average stress in compression will be in equilibrium,

= 6 (1.10)

1 2

The two stresses form a bending moment which results in the warping

of the laminate.

Again, the assumption here is that hygroscopic expansion transforms

into elastic strain.

 

 

 

 

 

 

 

 

L

_lL___ .......................................... __
1 2 1 2
A B

Figure 1-3 The free expansion of beam components (A) and resultant
expansion of laminated two-layer beam (B).

1.3 Previous Work on Panel Warping

The elastic approach described above is essentially the approach taken ,
by most researchers who have tried to predict the warping of various types of
wood based laminates". These studies evaluate the end conditions of the
warp, i.e. the warp after all elements have reached their equilibrium moisture
contents. This kind of warp is always due to some structural imbalance and
will be sustained as long as the environmental end conditions are maintained.

Warp may also occur as a consequence of a transient imbalance such as
the development of an unbalanced moisture content gradient. If given enough
time, the moisture content gradient imbalance will disappear and the panel
will flatten out again. Such warp is described by Tong and Suchsland [54].
The theoretical warp of one quadrant of a Douglas fir board having an initial
moisture content of 12 percent and moisture content increases at the four
identical planes of

+ 8 percent at bottom surface
+ 6 percent at 1/3 thickness from bottom surface
+ 4 percent at 2/3 thickness from bottom surface
+ 2 percent at top surface
is shown in Figure 1—4. This study was based on a finite element model.

Iwashita and Stashevski [17] measured and calculated the warping of

panels consisting of various particleboard types overlaid on one side with

high pressure laminates (HPL). The distortions were measured on

 

* The equation used for predicting the warp of a multi-layered laminate will
be discussed in a later chapter.

 

 

apprl
mm

HPL.

3881111

reald

of [h]
Partic‘

 

 

10

approximately 90 cm x 90 cm square panels. Board substrates were 20
mm thick Some had been overlaid with veneer prior to the application of
HPL. Exposure conditions were:

12 percent equilibrium moisture content (65 percent relative humidity)
to 20 percent equilibrium moisture content (90 percent relative humidity)
to 12 percent equilibrium moisture content (65 percent relative humidity)
to 6 percent equilibrium moisture content (30 percent relative humidity)
to 12 percent equilibrium moisture content (65 percent relative humidity)

There was a fair agreement between computed values based on elastic
assumption (see Section 2.3) and measured values in some cases. In others,
real discrepancies existed.

Heebink and Haskell [13, 14] measured properties of HPL materials.
This information was later utilized in a follow-up study [32] on the warping
of three-ply laminates consisting of high pressure laminate (HPL) face,
particleboard, and HPL backer sheet. There was a fairly good agreement
between calculated and measured warping. ‘

The warping of a veneered cabinet door was analyzed by Suchsland in
1990 [44, 45]. By applying an elastic warping model to the exact construction
of the warped door, the effects of such variables as grain deviation in the
veneer layers, species differences, and veneer thickness differences were
investigated. The results showed warping of the same order of magnitude as
that exhibited by the real door. It was also found that warping caused by
imbalances in the panel construction could be amplified by large moisture
content changes in the core material. The study indicated that careful

monitoring of the moisture content of components at the time of assembly is

11

an essential prerequisite to the elimination of large moisture content
differences among layers and to the manufacture of stable panels.

A study of wood-based composite sheathing materials exposed to
fluctuating conditions was conducted by Hiziroglu [16]. Elastic analysis
indicated that even moderate moisture content changes in the materials could
lead to bending stresses exceeding the ultimate bending strength. But the
experimental investigation revealed that these materials are not elastic and
that relaxation at high relative humidities reduces the maximal bending
stresses to about 40 to 50 percent of the bending strength. A buckled OSB
beam does not return to its original straight configuration upon regaining its
initial moisture content, and at the initial moistr'rre content, the axial stress
does not disappear but turns from compression into a sizable tensile stress
(Figure 1-5).

Restrained hardboard columns upon moisture cycling showed similar
results [51] as shown in Figure 1-6.

Suchsland and McNatt conducted a comprehensive study of the
warping of laminated panels [47]. Emphasis in this study was on the
theoretical evaluations of the effect of the variability of panel component
properties, particularly of those of the face and back layers of multi-layered
laminates on the warping of structurally balanced and unbalanced panels.

Figure 1-7 indicates the range of predicted warping during two
different exposure changes as affected by a thickness imbalance and by
assumed ranges of face and back veneer variability as indicated by their
coefficients of variation. Figure 1-8 shows the predicted warping of the same

panel without thickness imbalance. These warping values seem to be

12

extremely high (up to 5 in. center deflection over 48 in. span). It must be
concluded that this warping response to a relatively small relative humidity
change would be greatly reduced in a real panel by considerable creep and
stress relaxation across the grain of face and back veneers.

Warping along the grain, on the other hand, again due to assumed
variation of lamina characteristics, is much smaller (Figure 1-9). Here the
elastic approach produces much more realistic results because the expansion
values along the grain are very small and the modulus of elasticity very high,
a combination resulting in more elastic behavior. This does not mean that the
practical or real panel shows no warping tendencies along the face grain. The
real panel does warp, because of its great sensitivity to even small grain
deviations from the true longitudinal orientation (Figure 1-10).

A study of the warping of overlaid particleboard was conducted by
Suchsland, Feng and Xu [42]. These laminates, consisting of different
particleboard substrates and of various types of plastic overlays, both in 2-ply
and 3-ply constructions, were exposed to one moisture cycle, and the warp
was measured and also calculated based on elastic assumptions. Residual
warping at the end of the cycle indicated viscoelastic behavior, at least on the
part of the particleboard substrates (Figure 1-11 and Figure 1-12).

A recent study by H. Xu [57] developed a visco-elastic plate theory,
taking into account the effects of changing moisture contents over time, and
applies this theory with good results to the prediction of the warp of a two-ply
* yellow-poplar laminate (Figure 1-13).

in
0.0031.

 

 

0.00140

 

a) Douglas-fir plate used as an example for finite

element approximation of panel warping.

13

sl

0.000698

 

92
.53
a.
H
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3
an
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o
O
Hut
0
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so
>.
.o
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so
Eu
0
DD
.3
3
’8

Maximum u, (0.00349in) occurs at bottom surface of

Displacement in Y-direction is indicated by shading.
lower comer.

b) Finite element mesh representing 6in by 6in quadrant
Figure 1.4

of Douglas-fir plate. Horizontal planes are planes of

equal moisture content change.

Warping of solid wood panel [54].

l4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

zoo , -
‘ l
3 I
’50 ».a E CENTER w /.r*' f !-
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AVERAGE MOISTURE court-m7, PERCENT
Figure 1-5 ‘ Stresses in axially restrained wood composite.

Results of one exposure cycle (36% to 93% to
36% RH). Graph indicates axial stress and
center deflection as functions of average MC.
Upon reversal of exposure condition (93% to
36% RH), center deflection does not fully
recover [16].

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

WDEFLECTION
30° \.\ 7%};‘4—300
\, V I ’d
-’ I
g ’/ ’4? I
I 4
l W:
8 200 f ‘\\ 400
r3 \
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3
(D
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S
’4 S 2 7
a /
w IOO
.—
zoo
o 4 a :2 us 20
HGSTURE CONTENT — V.
Figure 1-6 Axial stresses and midpoint deflection of axially

restrained hardboard strips as function of
moisture content (clamped dry). Number 1 and
2 indicate sequence of exposure cycles. Dashed
line represents theoretical stresses [51].

MIDPOINT DEFLECTION - IN.

16

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Figure 1-9

 

WARPING (II) — (IV/48W

Construction:

3-ply: 1/6, 1/6, 1/6
4—ply: 1/8 (1/8, 1/8) 1/8
5-ply: (1/10) x 5

9-ply: (1/45) x 9

Effect of construction on warping along face
grain of 1/2 in. loblolly pine plywood [47].

19

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

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as -.to -.5 o .5 10 t5
MRPWG-IN/«IN

 

Figure 1-10 Effect of cross grain in top veneer on warping of
1/2 in., 3-ply loblolly pine plywood [47].

0.30 P

20

HPL (PVA) - B - BCK (PVA)

35 CENTER DEFLECTION (INCH)

 

'9- Edge II
'8'— Edg. o

 

>
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RH: 50% ° 84% - 50%

Figure 1-11

TIME (DAYS)

Warping record of particleboard with HPL and

backer laminate. Exposure interval is 50% to
84% to 50% RH [42].

21

MEL-B

0 CENTER DEFLECTION (INCH)

 

 

 

 

 

 

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RH: 50% - 84$ - 50%

Figure 1.12

TIME (DAYS)

Warping record of particleboard with melamine
laminate. Exposure interval is 50% to 84% to
50% RH [42].

22

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23

1.4 Need for More Sophisticated Approach

As can be seen from the previous studies, estimating of warping based
on elastic assumptions has certain limitations.

Wood is known to be a visco-elastic material with a plastic response
component that will become significant at high stress levels and high
moisture contents. Therefore, stress relaxation is an important factor in the
mechanics of warping of laminated wood panels.

Keylwerth experimentally investigated the free and restrained swelling
of wood and reported his findings in a series of articles in 1962 [21, 22]. He
measured the development of compressive stresses in both radial and
tangential directions in small uniaxially restrained specimens of beech wood
as the moisture content increased. The swelling stresses quickly reached a
maximum value, but due to the increasing plastic deformation of the
restrained specimen, stress levels were considerably lower than would be
calculated under assumptions of elasticity (Figure 1-14 and 1-15). The free
expansion in the radial or tangential directions was then divided into elastic,
vised-elastic and plastic components as illustrated in Figure 1-16. This
separation was obtained by releasing the completely restrained specimens at
certain time intervals and measuring their immediate and time dependent
expansion, and comparing them to the total free expansion measured on
unrestrained control specimens. In Figure l-16, the upper portion shows
deformation in the tangential direction, which is also the direction of
restraint. The lower portion represents the radial direction. Restrained

swelling of various wood species of different specific gravities is summarized

 

wher

24

in Table 1-1. The exposure time for beech and pine was 24 hours, but it was
extended for bongossi to 48 hours because the diffusion in bongossi wood is
slower than in pine or beech.

It is interesting to note that the elastic component of the total restrained
expansion in each of the two directions is fairly constant in relative terms,
regardless of the species tested and of the moisture content range. There is
also a fair correlation between the maximum swelling stress and the static
modulus of elasticity measured at the initial condition.

It is clearly this elastic component of the restrained swelling which
determine the stress level and which sustain the potential warp. It should be
used to modify the elastic equation for panel warping.

This was attempted by Suchsland and McNatt [47] for the case of a
three layer plywood model in which the “deformation modulus” as

formulated by Keylwerth [21],

 

E. = 0' (1.11)

where E' = the deformation modulus
o' = the swelling stress measured when axially restrained
specimen has reached equilibrium with air condition
Salas: = the elastic portion of total swelling

was substituted for the static modulus of elasticity across the grain of the
veneer, and sum for the free expansion across the grain of the veneer.

It was not deemed necessary to make such modification in the

25

longitudinal direction of the veneer.

This modification of the input parameters produced more realistic
predictions than the elastic approach but still left considerable discrepancies.
This may be due to a rather unsophisticated method of determining swelling

stresses [48].

26

 

 

 

 

 

 

 

 

 

 

3'3 §
Swelling Stress (compression) 6

s
1

Moisture Content u
G

 

 

0 ’60 200 .700 m mm 500

Time

Figure 1-14 Time function of average axial swelling stress
(compression) for two different moisture
content intervals [21].

27

 

 

 

 

 

 

Swelling Stress (comp )
as
1°
f
1 so

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Gradient of
Swelling Stress (comp.)

 

Average moisture content

Figure 1-15 Radial and tangential swelling stress
(compression) for two different moisture
content intervals (upper figure), and stress
gradients (lower figure) [21].

28

 

*0”

Beech

f l

_—'l

_ I
o—oMC = 1.5%
Free Swelling-{._,Mcfi = 10.5%

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Q
N)
.5 ,
a I
3. .
10min ‘ ~I
a 5 i0 . o' 20
Average moisture content

Figure 1-16 Free and restrained swelling deformations of beech as
function of two different moisture content intervals [21].

29

 

 

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CHAPTER II
THE MECHANICS OF
WARPING OF HYGROSCOPIC LAMINATES

2.1 Hygroscopic Behavior of Solid Wood and of Wood Composites
W

Wood and all products derived from wood are hygroscopic. These
materials absorb and desorb water according to some functional relationship
between the relative humidity of the air and the materials moisture content
(Figure 2-1). This water resides in the cell wall as “bound” water. As the
relative humidity approaches 100 percent (approximately 30 percent
moisture content), water begins to appear in liquid form in the cell cavity.
The bound water causes cell walls to expand which results in gross
dimensional changes of the wood or wood products. Thus, changes in relative
humidity translate directly into dimensional changes. Increasing moisture
content to above 30 percent (fiber saturation point) does not cause further
swelling. The moisture content - relative humidity relationship of solid wood
is reversible, at least for most practical purposes. So is the relationship

between moisture content and dimensional change of wood (Figure 2-2).

30

31

The dimensional changes of solid wood in response to moisture
content changes are highly directional. They are greatest in the transverse
direction (radial and tangential) and smallest in the longitudinal direction as
shown in Table 2-1. The relationship between moisture content and
dimensional change is fairly linear so that it can be expressed as the product
of the dimensional change coefficient (change per 1 percent moisture content

change) and the moisture content change.

Writes

Conventional wood composite are sheet material composed of small
woody elements held together by an adhesives bond. The size and shape of
the element are the most important characteristics of the materials, ranging
from veneer sheets (plywood) to individual fibers (fiberboard). Most
composite sheet manufacturing processes have in common a randomization
of the grain direction in the plane of the sheet. This is accomplished by cross
lamination in the case of plywood and by random deposition of elements in
the case of particleboard and fiberboard. The distinction between fiberboard
and particleboard is in the size of the elements, fibers generally being much
smaller than particles, the former being derived from wood by a pulping
process, the latter by a mechanical cutting or milling operation.

Plywood is a glued wood panel made up of relatively thin layers of
veneer with the grain of adjacent layers at right angles. Each layer consists of
a single thin sheet, or ply. The usual constructions have an odd number of

layers. The outside plies are called faces or face and back plies, the inner

32

plies are called cores or centers.

Particleboard is a generic term for a panel manufactured from pieces of
wood smaller than veneer sheets but larger than wood fibers, combined with a
synthetic resin or other suitable binder and bonded together under heat and
pressure in a hot press by a process in which the interparticle bond is created
by the added binder. Densities range from 30 lb/ft3 to 60 lb/ft3 . Special
types of particle panel products are flakeboard, waferboard and oriented
strand board (OSB) as shown in Figure 2-3.

Medium-density fiberboard is a dry formed panel product
manufactured from 1i gnocellulosic fibers rather than particle combined with a
synthetic resin or other suitable binder. The panels are compressed to a
density of 31 lb/ft3 to 55 Ib/ft3 by the application of heat and pressure by
a process in which the interfiber bond is substantially created by the added
binder.

High-density fiberboard (about 55 to 70 lb/ft3 ) is called hardboard
[46].

Wood composition boards have some very desirable characteristics:
availability in large sheets, smooth surfaces, uniformity in properties from
sheet to sheet, and freedom from localized defects. However, some of the
products have been largely excluded from primary structural uses because
they have been unable to approach the longitudinal stiffness, dimensional
stability and long-term load-carrying ability of sawn lumber.

Wood composite sheet materials such as particleboard and fiberboard
derive their important characteristics from solid wood. Their gross

mechanical and physical properties are similar to those of solid wood but are

33

modified because of the more or less randomized disposition in the plane of
the board of the individual particle or fiber elements.
Their hygroscopic behavior differs from that of solid wood in several
respects:
— The isotherm is lower and has a distinct hysteresis (Figure 2—1).
— The expansion and shrinkage in the plane of the sheet is the same in
all directions. It is relatively small compared with the transverse
dimensional change of solid wood (Figure 2-2 (a)).
-— Thickness swelling is large due to the effects of densification to
higher densities in manufacturing (Figure 2-2 (b)).
—— Both linear dimensional change and thickness swelling show
hysteresis which is particularly strong in thickness swelling.
Composition boards are often overlaid with either veneer or plastic
overlays such as melamine impregnated paper, high pressure laminates (resin
impregnated paper laminate), vinyl and veneer. Such overlaid panels are
common in the furniture, kitchen and store fixture industries. In many cases
the overlaid sheets are of an unbalanced construction potentially leading to

warping problems in service [29, 44, 45, 46].

34

 

30 —— Particleboard (Perkitny and Szymankieuicz 1962) I
'I. —--- Pinewoc'l - //
----- Port clebount (Weichett 1963) I,
/ I

 

 

 

N
m
r ”..-—.... —

 

 

N
D

 

 

 

 

 

-J
G

 

Equilibrium moisture content
at

 

(H

 

 

 

 

 

 

 

 

 

 

30 90 7. 100
Relative humidity

Figure 2-1 Hygrosc0pic isotherms for wood and particleboard
[23].

35

I

l
Pine .' Pine rooiol swelling
tongenliol swelli

/

.-‘
0

Average length swelling

Pine longitudinal swelling

 

0 3 6 9 12 15 18 21 '/. 24
Moisture content

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a). change in length
N.
'/.
all
12 / “
m} V. q
3. // ‘
g 5' 1 8
Pine tangential swelling
0 ,/
‘ . /p/'
o ./" ...—d
2 4.// ..¢-—"
.4 -....1 W
'/jlv'/.,../""" martial swelling
. . /"/ l I
n 3 ti SI 12 15 18 21 7. 24
Moisture content
(b). change in thickness

Figure 2-2 Dimensional changes in solid pine wood and
particleboard (heavy line) versus moisture content [23].

36

 

 

 

.nti l
9.3.! r.

fists L, .

 

waferboard, and C oriented strand board [10].

Figure 2-3 Basic particle panel products: A particleboard, B

 

 

 

37

 

 

 

Table 2-1 Dimensional change of domestic woods [10]
Minoan,” U‘ 30W M green :1
evendry content ovendry We corn-m
Radial Tltgenthl Volurneutc Redd Tm Volunetrtc
-------- Percent-unn- --------Percenr---.....
HARDWOODS

Alder. red 4.4 7.3 12.6 Hlakory, True:

Ash: Mockernut 7.7 1 1 .0 17.6
Black 5.0 7.6 15.2 Plgnut 7.2 1 1 .5 17.9
Blue 3.9 6.5 11.7 Shagbanr 7.0 10.5 16.7
Green 4.6 7.1 12.5 Shellbark 7.6 12.6 19.2
Oregon 4.1 6.1 132 Holly. American 4.9 9.9 16.9
Pumpkln 3.7 6.3 12.0 Honeylocuet 4.2 6.6 10.6
Whlt 4.9 7.6 13.3 Locust. black 4.6 7.2 10.2

Aspen: Medrone. Pacific 5.6 12.4 18.1

619166111 3.3 7.9 11.6 Maamllar
00111an 3.5 6.7 1 1.5 Cucumbertree 5.2 6.6 13.6

Basswood, Southern 5.4 6.6 12.3
American 6.6 9.3 15.6 Sweetba 4.7 6.3 12.9

Beech. Amerlcan 5.5 11 .9 17.2 Maple:

Birch: Blgleat 3.7 7.1 1 1 .6
Alaska paper 6.5 9.9 16.7 Black 4.6 9.3 . 14.0
Gray 5.2 — 14.7 _ Red 4.0 6.2 12.6
Paper 6.3 6.6 16.2 Sllver 3.0 7.2 12.0
River 4.7 9.2 1 3.5 Striped 3.2 6.6 12.3
Sweet 6.5 9.0 15.6 Sugar 4.6 9.9 14.7
Yellow 7.3 9.5 16.6 Oak. red:

Buckeye. yellow 3.6 6.1 12.5 Black 4.4 11 .1 15.1

Butternut 3.4 6.4 10.6 Laurel 4.0 9.9 19.0

Cherry, black 3.7 7,1 11,5 Northern red 4.0. 8.6 13.7

Chestnut. Pin 4.3 9.5 14.5

American 3.4 6.7 1 1.6 Scarlet 4.4 10.6 14.7

Cottonwood: Southern red 4.7 11.3 16.1
Balsam poplar 3.0 7.1 10.5 W819! 4.4 9.6 16.1
Black 3.6 6.6 12.4 Willow 5.0 9.6 16.9
Eastern 3.9 9.2 13.9 Oak. white:

Elm: Bur 4.4 6.8 12.7
American 4.2 9.5 14.6 Chestnut 5.3 10.6 16.4
Cedar 4.7 10.2 15.4 Live 6.6 9.5 14.7
Book 4.6 6.1 14.9 Overcup 5.3 12.7 16.0
Slippery 4.9 6.9 13.6 Poet 5.4 9.6 16.2
Wlnged 5.3 1 1 .6 17.7 Swamp

Hackberry 4.6 6.9 13.6 chestnut 5.2 10.6 16.4

Hickory. Pecan 4.9 6.9 13.6 White 5.6 10.5 16.3

38

Table 2-1 (continued)

 

 

 

 

 

 

Shrinkage irom green 10‘ Shrinkage hem green 10‘
cvendry m content ovendry moisture content
Radial Tangential Volunetnc Radial Tngenttal Volumetric
-------- Percent-------- --------Percent--------
HARDWOODS—continued
Persimmon, Tupelo:
common 7.9 11.2 19.1 Black 5.1 6.7 14.4
Sassatras 4.0 6.2 10.3 Water 4.2 7.6 12.5
Sweetgum 5.3 10.2 15.6 Walnut, black 5.5 7.6 12.6

Sycamore. Willow. black 3.3 6.7 13.9
American 5.0 6.4 14.1 Yellow-poplar 4.6 6.2 12.7

Tanoak 4.9 1 1.7 17.3

SOFTWOODS

Baldcypress 3.6 6.2 10.5 Hemlock (con):

Cedar: Western 4.2 7.6 12.4
Alaska- 2.6 6.0 9.2 Larch, western 4.5 9.1 14.0
Atlantic white. 2.9 5.4 6.6 Pine:

Eastern Eastern white 2.1 6.1 6.2
redcedar 3.1 4.7 7.6 Jack 3.7 6.6 10.3
Incense- 3.3 5.2 7.7 Loblolly 4.6 7.4 12.3
Northern Lodgepole 4.3 6.7 1 1.1
white- 2.2 4.9 7.2 Longleai 5.1 7.5 12.2
Port-Ortord- 4.6 6.9 10.1 Pitch 4.0 7.1 10.9
Western Pond 5.1 7.1 1 1.2
redcedar 2.4 5.0 6.6 Ponderosa 3.9 6.2 9.7

Douglas-fin? Red 3.6 7.2 1 1.3
Coast 4.6 7.6 12.4 Shortleal 4.6 7.7 12.3
Interior north 3.6 6.9 10.7 Slash 5.4 7.6 12.1
Interior west 4.6 7.5 11.6 Sugar 2.9 5.6 7.9

Fir: Virginia 4.2 7.2 1 1.9
Balsam 2.9 6.9 11.2 Western white 4.1 7.4 11.6

California red 4.5 7.9 11.4 Redwood:

Grand 3.4 7.5 11.0 Old-growth 2.6 4.4 6.6

Noble 4.3 6.3 12.4 Young-growth 2.2 4.9 7.0

Pacific silver 4.4 9.2 13.0 Spruce:

Subalpine 2.6 7.4 9.4 Black 4.1 6.6 1 1.3

White 3.3 7.0 9.6 Engelmann 3.6 7.1 1 1.0
Hemlock: Red 3.6 7.6 11.6

Eastern 3.0 6.6 9.7 Sitka 4.3 7.5 11.5

Mountain 4.4 7.1 1 1 .1 Tamarack 3.7 7.4 13.6

 

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CoastDouglas-firiederinedaeDouqu-flrgrmwingmmeStateaotOregonandWaeNngtonweetoimewmmnolmeCaacedeMountalns.
htenorWestincludeetheswummwulmmmwwmmmdmwmmmmmm.interim
mimmmrdmwwmmwmmmm.Monumenowming.

39

2.2 Elastic Characteristics of Solid Wood and Wood Composites

Wood

Wood is linearly elastic within certain load and load duration limits,
and follows Hooke’s law, that is, stress is linearly related to strain in direct
proportion.

As an orthotropic materials, wood has twelve constants. These are:

three moduli of elasticity: EL , ER , 12,
three shear moduli: Gm , Gm , G”

six Poisson’s ratios: 11”, “n , um , 11“., um, 11“.

The subscripts refer to the three principal directions in wood. L , T ,

R denote longitudinal, tangential and radial directions respectively. For the
Poisson’s ratios the first subscript refer to the direction of the applied stress
and the second subscript to the direction of the lateral strain. There is a set of
three equations relating some of the elastic constants, so that the number of

independent constants is reduced to nine. The equations are [10]:
“L7 _ “TL , “TR “RT , “fl = E51. (21)
ER °

TL‘E ’ ET ER ’ EL

The state of strain can be expressed as a function of the state of stress

by

 

40

 

 

 

 

 

l ll1'1. p'RL
“L7 1 ”RT
i :u ”TL 2;; ' ER 0 O 0 :LL
’7 _“_I-_R in: .1. "
1 ERR >= EL ET ER 0 0 0 i ORR >
2.2
in O 0 O __1_ O 0 1:nt ( )
7L2 Gm tLR
‘ 7LT 1 L 1LT J
O O O O '61:; 0
l
_ 0 () o 0 05;_

 

 

The elastic modulus in the longitudinal or the grain direction, EL , is
much higher and those in the transverse directions are lower. In any
directions relative to the grain direction, the modulus can be approximately

obtained from Hankinson-type formula as [10]

E _ ELEL .
°’E(mwr+5(mwv '(Z”
L .L

 

in which 15'9 = the modulus at an angle 6 from the grain direction
EL = the modulus in the grain direction
E J_ = the modulus in the direction perpendicular to the grain
n = an empirically determined constant, ranging from 1.5 to 2

for most wood species [10].

41

The modulus in any direction expressed as a fraction of the one in the
grain direction plotted against the angle 6 is shown in Figure 2-4.

The modulus of elasticity of solid wood is also affected by changes in
moisture content below the fiber saturation point. It increases with decreasing

moisture content and can be expressed as follows [10]

_(MC-12)

E
E = 5,43%) (2.4)

where E = the modulus of elasticity
MC = moisture content in percent
MCP =the moisture content at which the mechanical property
changes due to drying are first observed, usually around 20
to 30 percent depending on the wood species.
15'12 = the modulus of elasticity at 12 percent moisture content
15‘ = the modulus of elasticity for all moisture contents greater

than MC, (green condition).
Wires

Wood composites, due to the rearranging or randomization of wood
elements, leave elastic properties less directional or more uniform than those
of solid wood.

The elastic properties of plywood are orthotropic. As compared with

 

42

solid wood, the chief advantage of plywood is an equalization of elastic
properties in the plane of the panel. The strength along the grain of the face
veneers is of course remarkably reduced. Board such as particleboard or
fiberboard are generally considered to be transversely isotr0pic, which means
that there are only insignificant differences between machine direction and
cross-machine direction, but the elastic-mechanical pr0perties in the plane of
the board are generally different from those in the direction of the thickness.
The modulus of elasticity of plywood is equal to the weighted average
of the moduli of elasticity of all plies [10]. In edgewise tension or

compression,

5,, or Ex = ; 2 5,7,. (2.5)
1

where T = the thickness of plywood

T = the thickness of the veneer in the i‘“

ply
E = the modulus of elasticity of the veneer in the 1“” ply

EW and Ex is the modulus of elasticity of plywood parallel and
perpendicular to the face-grain respectively.

Modulus of elasticity at any angles to the face-grain direction is given

approximately by [10]
E}— = El—(c056)‘+E—1(sin6)4+—Wx(sin6)2(cos6)2 (2.6)
6 Ex

where E9 = the modulus of elasticity at an angle 6 to the face grain

43

G“: the shear modulus under edge-wise shear.
The relation of Ee/Ex is also plotted in Figure 2-4 for two different
veneer ratios.
The modulus of elasticity in flexure of plywood is equal to the average
of the moduli of elasticity parallel to the span of the various plies weighted

according to their moment of inertia about the neutral plane. That is

l n
Efw or Ef, = 7 2 6,1,.
l

where E1.” the modulus of elasticity of plywood in bending when the
face grain is parallel to the span
Efw = the modulus of elasticity of plywood in bending when the
face grain is perpendicular to the span
E. = the modulus of elasticity of the i"' layer in the span
direction

the moment of inertia of the 1"” layer about the neutral

N
II

plane of the plywood
I = the moment of inertia of the total cross section about its

center line

The important elastic strength properties of particleboard and
fiberboard include static bending, tensile strength parallel to the surface and
perpendicular to the surface (internal bond), compression strength and
bending strength parallel and perpendicular to the surface, etc. These
properties depend largely on the species of the material, the density, resin

44

content, resin type and the moisture content of the board.
Boards with higher density have higher mechanical strength. Figure 2-

5 shows the general trend of such relationship. Although the most important
species variable governing board properties is the density of the wood raw
material itself, the lower-density woods are widely used in the manufacturing
today. Their strength properties appear to be superior to the hi gher-density
species. This is because lower-density woods provide larger volume of wood
chips compared to the higher-density woods to make the same density of the
board. Therefore, a better bond among the chips occurs as the board is
compressed to its final thickness. In order to achieve a certain mechanical
strength of the board, the use of a higher-density wood willlead to a board

being much heavier.
The moisture content of the board has less effects on the strength of the

particleboard than on solid wood as indicated in Figure 2—6.

EO/EL

Figure 2-4 The modulus (tension or compression) as function

45

 

1.0

 

— Solid Wood
0.9- —— Plywood 1:1
—- Plywood 2:1
0.8-
0.7—\
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0.4-— \ \ /
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0 34 \§ // //
. \\\\.\' ////
0.2- “”
0.1-—
0.0

 

 

11111111
0 1020304050 60708090

Angle to the Grain (6°)

of grain angle.

 

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47

2.3 Prediction of Warping of Wood Laminates Based on Elasticity

Figure 2-7 shows a laminated beam with 71 layers. The differences
between the resultant expansion and free expansion (01m - or) indicate the
compressive or tensile strains which are proportional to the corresponding

forces that develop in the individual layers. These forces can give rise to
bending moments about the neutral axis of the beam. Only when these
bending moments are balanced, does the beam remain flat. The following
derivation of the warping equation is based on an article by CH. Norris [32].

If the radius of curvature is assumed as R , the total stress a in any

point can be determined as

- or) E- 115 (2.7)

a = (are: R

where or s = the resultant expansion of the beam.

re

The force resultant on a cross section of the beam can be found by

integrating the stress over the thickness as follows
F = 16d) = 1 (arm - or) Edy - 1%Edy (2.8)

and the bending moment resultant on a cross section of the beam is

determined by

M = joydy = “am-a) Eydy-I%Edy (2.9)

 

 

 

48

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

y
.4
L
I: “2 =
: L 2 L a“
1121..
K<<<<<u a
L l '
l
(a) “n
R
\ - i W
1: ,. 1
(b)

Figure 2-7 A laminated beam with 71 layers showing
individual free expansion, cg, and resultant
expansion, 01,“, (a), and illustration of relationship
between radius of curvature, R, base length, L, and
center deflection, W, of warped beam (b).

49

If it is assumed that there is no external force and moment on the

beam, Equations (2.8) and (2.9) can be integrated over each layer by using

summation
n n 1 n
aresZEiTi — 201,127,. — figs, (s,2 - 5,11) = o (2.10)
l l l
and
or

 

gszg,(s,.2-sg_,)-;2.,E,isg_ss-,1—;,—,zs.<s:-s:-.> = o (2.11)
r 1 1

The radius of curvature of the beam R can then be determined from

these two equations as

n n
225,(S?-S?_1) 251(52'512-1)
r r

 

 

n I!
3215,. (5,2 - sf_ 1) 225,7,
1

R = 1 (2.12)

 

n '8
291-51 (S? ' SI?- 1) 201,113,?“
r _ r

 

2E1(Sr2" 512-1) 2517:
r 1

where S, = ET, (in) , is the cumulated thickness (see Figure 2-8)

1
or, = the expansion value of the 1"” layer (in / in) over the range of

50

moisture change anticipated
E, = the modulus of elasticity of the 1"” layer (psi) at the end
condition

T, = the thickness of the 1"” layer (in) .

The center deflection or the warping W (Figure 2-7) is given by

n n
2011151“? '512- 1) 201,15],-
1 _ 1

 

 

2

II n

2 251' (S? " 512-1) 251T:
_l;_ r r

8

l"

(2.13)

 

S
11
I

2251' (S? ' 51-1) 251 (5:2 - 512- 1)
1 _ 1

II
1 1

00
k

 

 

where L = the length of the beam.

The radius of curvature of the beam, R , or the center deflection, W,
may have positive or negative values. We will define warp being positive if
the curve is concave when viewed from above as shown in Figure 2-7(b).

For a two-layer beam, it can be shown (see Appendix A) that W is a

function of the moduli ratio Til/1:?2 , the thickness ratio T,/T2 , and the
absolute value of thickness T2 , and is proportional to the difference of the

expansion values of the two layers:

51

2
(011-012)L

T, (2.14)

w = f(E,/E,, 71/1,)

 

For a given El/E, ratio, maximum warping Wm, occurs when

15,/E, = (T,/T,) 2 (2.15)

Equations (2.14) and (2.15) are true only for two-layer laminated

beams.

Figure 2-9 is the plot of center deflection or warping. W vs. various

El/E, , and T,/ T, ratios. The thickness of the second layer T, was set to be
1 inch, or1 -or, was set to be 0.01 in/r‘n (1 percent) and the span is 48
inches. The numerical results are listed in Table 2—2.

For any two layer laminate with arbitrary thickness of T, = T,’ ,
expansion difference of 01', -or', , and length of L = L' , the theoretical
corresponding warping value

(01'1 - a',) L'2

W‘ = W x 100
482 x T,’ (2'16)

 

where W is the value from Figure 2-9 or Table 2-2.
Equation (2.14) indicates that a larger difference (or1 — 01,) leads to
larger warping, but that larger T, leads to smaller warping providing that

Tl / T, and other parameters remain the same.

52

The resultant expansion arm of a 'multi-layered beam can be

calculated by

(1+ar.)

 

ll
2“
are: = -III——
iii-T

(2.17)

 

if the curvature of the beam is negligible. Otherwise, the resultant expansion

or should be determined by solving Equation (2.10) using known R .

res

 

Figure 2-8 . Illustration of dimensional variables of n -layer
beam

53

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55

2.4 Nonelastic Behavior of Solid Wood and Wood Composites

2.4.1 General

As has been pointed out in the first chapter of this study, the
assumption of elasticity in calculating the warping of wood based laminates

does not always lead to satisfactory results.
The non-elastic behavior shall therefore be briefly reviewed. This will

be followed by the analysis of a visco-elastic beam under stress. Finally, the
experimental approach used in the study shall be developed and the objective

shall be outlined.
2.4.2 Wood as Visco-elastic Material

Wood and wood-based materials are known as viscoelastic materials in
that they exhibit elastic as well as viscous strain properties. In wood, this
behavior is a consequence of its amorphous and crystalline components [33,
21, 1]. Wood is known to be a complex heterogenous material composed
mainly of cellulose, hemicelluloses, and lignin. Cellulose has both highly
arranged crystalline and rather disorganized amorphous regions in its
structure while lignin is mainly amorphous and hemicelluloses. Crystalline
materials usually are highly elastic and show little time-dependent strain
response to load over time. Amorphous materials normally have low

elasticity and a high time-dependent strain response to constant load as

indicated by Miller and Benicak [28].

 

 

56

Creep is defined as the time-dependent deformation exhibited by a
material under constant load. In contrast, if a displacement is applied
instantaneously and held constant over time, the stress will subside with time.
This phenomenon is called relaxation, the change in resistance to forces in a
material subjected to a constant deformation. The simplest type of time-
dependent behavior to observe is creep. A relaxation test usually requires
more specialized equipment [3]. Frequently the measurement of force
depends on the displacement of the sensing element. Consequently, force
measurements while displacement is presumably held constant can be subject
to error: although the change in displacement may be small, it may be

incorrect to assume that the sample is in a state of constant strain.

2.4.3 Creep Behavior

Figure 2-11 illustrates the characteristic deformation-time relationship

for visco-elastic materials during sustained loading and relaxation [1, 46, 3].
Unlike elastic material as shown in Figure 2-10, the deformation of visco.
elastic materials does not recover completely upon unloading. “In outlining
the curve, starting at the origin, the application of load at time to produces
an instantaneous elastic deformation DA, on maintaining the load to :1 , the
deformation increases at a decreasing rate, this increment is defined as creep.
With removal of the load at time tl , an instantaneous recovery BC occurs
which is approximately equal in magnitude to the initial elastic deformation
0A . As time elapses, there is a partial recovery CD of the creep deformation

at a decreasing rate until time t2 , where it becomes constant. Thus, the

mount of creep that occurred during loading is composed of a recoverable

57

component which is time-dependent and displays delayed elastic deformation

and an irrecoverable component which is permanent and is due to plastic or

viscous flow” [1].

Apart from the viscoelastic cell wall components, it was pointed out
[25, 24, 31] that factors affecting the amount of creep are mainly the
magnitude and duration of loading, moisture content and temperature as
illustrated in Figure 2-12.

The study of the time effect indicates that the deformation increases
very quickly following initial loading and slows down to a more or less
constant rate thereafter. Beyond some point in time, known as the “point of
inflection”, deformation increases at an increasing rate until rupture (failure)
occurs. '

Creep deflection increases, and time to failure decreases with

increasing level of stress. Beyond a given stress level, creep deflection
increases at a rapid rate and failure is imminent. The range for which creep of
solid wood can be assumed to be linear, i.e., proportional to stress, varies
from approximately 30 percent [31] to up to 50 and even 75 percent [38] of
ultimate strength.

The increase in creep deflection caused in wood by increasing in
moisture content or temperature is well documented, as indicated in the
literature reviewed by Schniewind [38]. Increases in these properties have the
effect of plasticizers that lower the viscosity of the matrix and allow
increased deformation to occur at a given stress level [41]. The effect of a

change in moisture content is much higher than that resulting from a change

in temperature.

58

Bodig and Jayne reported that in composites adhesive bonds between
fiber and between wood particles tend to weaken and may even rupture when
exposed to high relative humidity [3]. The rate of flexural creep of

particleboard was found to accelerate above approximately 75 percent

relative humidity according to Haygreen et a1 [12].
Cyclic relative humidity conditions have a greater effect on creep

properties of particleboard than steady state conditions, as reported by

McNatt and Hunt, Laufenberg [27, 25].

2.4.4 Modelling of Creep

Numerous models have been utilized in the literature to describe the
creep behavior of wood and wood-derived composite board. Essentially,
these fall into two categories: empirical curve fitting and mechanical

analogies in the form of linear viscoelastic models.
The power law model is most widely used in empirical curve fitting of

the creep response of wood materials. The equation takes the form:

so) = eo+ar’" (2.18)

where: e (t) = time dependent strain
so = elastic strain
1‘ = time

a, m = constants

59

Constants a and m can be determined by transforming the function to a

linear relationship by rearranging and taking common logarithms:

log (a - so) = loga + mlogt (2.19)

The power function is derived by taking the antilog of the linear
equation.

The power model is simple and describes the creep response of wood
and wood-derived materials well. Other possible empirical curve fitting
options are illustrated in Table 2-3 [3].

The viscoelastic behavior, as shown in Figure 2-11 could also be
simulated by mechanical analogies, i.e., a combination of springs and

dashpots. Springs are used to simulate the elastic component, where stress is
directly proportional to strain; and dashpots — pistons moving in cylinders
filled with a viscous fluid — are used to represent the plastic or viscous
components, where stress is directly proportional to the rate of strain.

The simplest linear model which has been used successfully for wood
and wood derived materials is the four-element or Burger model as shown in
Figure 2-13 [46, 1]. It is comprised of the Maxwell body and Kelvin body
connected in series. The Maxwell body consists of a spring and a dashpot

joined in series and accounts for elastic and viscous behavior, respectively.
The Kelvin body represents delayed elastic behavior (visco-elastic) and
consists of a parallel arrangement of spring and a dashpot. Together the two

models with the four elements account for the principal features of the time-

6O

dependent behavior of wood and wood composites.

In Figure 2-13, element A would respond instantly in accordance with
its modulus of elasticity E, , elements 8 and C cannot respond instantly
because of the “sluggishness” of the dashpots. They both will deflect,

however, with time (creep). Upon load removal, element A will recover
instantaneously and completely, element B will slowly but completely
contract to its original condition because of the restoring force of the spring.
Element C will remain extended (permanent plastic deformation).

The theoretical formula of the Burger model expresses the total strain

developed under stress with respect to time as the sum of the elastic,

viscoelastic, and viscous components [9]:

so) = atil- +% (1 -e“") + 11'1") (2.20)

e

where: e (t) = time dependent strain

 

o . . .
Z‘" = instantaneous elastic strain
8
a ’3/1 0 o ndg . o .
(1 — e ) = delayed elastic strain, 1 = —— 1s retardauon time
Ede E‘
a .
— = Viscous flow
'1»
E, = instantaneous modulus of elasticity
Ed, = delayed modulus of elasticity

11d, = delayed coefficient of viscosity

viscous coefficient of viscosity

.3
c
I

61

t = time
i+i(1_e-t/r) +i, in Equation (20) is called creep
E: Ede nv

compliance. For uniaxial loading, it can be used to define relaxation modulus
by inverting the creep compliance.
An example of these parameters for commercial sheathing plywood

and laboratory waferboard at three environmental conditions, and hardboard

at two moisture content levels are shown in Table 2-4.
2.4.5 Visco-elastic Beam Theory

The constitutive equations for classic linear viscoelastic anisotropic
material in a three dimensional state of stress can be written in a concise form
using tensor notation [37, 7, 57]

‘ ac“
can) = [Cyan-1);? dr i,j = 1,2,3 (2.21)
o.

The inverse of the above equation may be written as
‘ as
_ u . . __
60.0) _ [Yuan—fla- d1 1,, .. 1,2,3 (2.22)
0.
where Cm, , You , 8r] , and of]. are defined to be the creep compliances,

relaxation moduli, strains, and stresses, respectively. The stresses and strains

n Equations (2.21) and (2.22) are functions of coordinate position as well as

me.

62

 

 

 

 

 

I l
L - —l- ‘
Figure 2-14 Uniaxial beam element.

For an uniaxial beam element (Figure 2-14), Equations (2.21) and
(2.22) can be considerably simplified. Since the transverse shear strains are
negligible, therefore the transverse deformation in the axial direction is a

linear function of the z coordinate, and can be written as [19]

u = uo-zj—Z (2.23)

If the displacement w in the z direction is assumed to follow

quadratic curve, The curvature of the beam is

K = :12! (2.24)

(11:2

The signs of w and K follow the classic laminated theory.
The total axial strain which causes swelling stress due to hygroscopic

expansion ( 2" )-is in the form [35, ll, 57]

du
e = E34312!) = (ea-eh) +zK (2.25)

63

where (3° -e") = the transverse strain.

The total axial time-wise stress is then determined by

o’"(t) = j Y(t-t) (3%" (1:) '44?" (1:) +2%K(1) ) dr (2.26)
o-

t
_ d o d d
.. {YO—r) (fie (r)-I-zH--1-:-K(r)-31-:e"(1))dr

where Y(t - r) = the relaxation modulus in the axial direction

z = the distance between the neutral axis and the point considered
20(1) = the resultant strain at time 1:

K (1:) = R (r) is the radius of curvature of the beam at time 1

l
R (r) ’
5"(1) = the hygroscope free expansion due to change in moisture

content at time 'c .

Similar to the elastic theory, the total axial force over all layers is

h
-2
t n ht
= Y (r-r) (LI-e" (1:) +211? (1) -ie"(r) )dzdt (2-27)
LEM}; " d1: d1: d1 " .

t I l
= IQ“(t—t)e'°(1:)d1:+IQ12(t—I)K'(t)dt-J'Nh(t-t)dt
O O 0

64

and the moment at the end of the beam is

h

2
M"'(t) = jo'"(:)zdz
h

2

t t I
= [Q21 (r-r) e'” (1:)dT-l-JQ22 (t- r) K‘ (1:) d1: - IM” (t - 1:)d'c (2.28)
o o o

oo _ d o
where s (r) - 356 (r)

K' ('c) = filflr)

and Qll , Q12 , Q21 , and Q22 are functions of dimension and of relaxation
moduli Y(r) . N" and M’' are force and moment determined by dimension,

relaxation moduli Y (r) and the hygroscope free expansion 2" (1:)

Q11(t-T) = 2 Y(r-r)k(S,c—S,,_1) (2.29)
k=l

Qua-1) = 2 %Y(t-r)k(S:-S§_1) = Q21(t-1:) (2.30)
k=l

Qua—1:) = Z %Y(t-1:)k(S2-Si_l) (2.31)
k=l

N”(:-1.-) = 2Y(:-r)k(s,-s,_,)e'h(r), (2.32)

i=1

Y(:-r),(Si-S§-1)e"'(r), (2.33)

NI.—

M"(t—1:) = Z
k=l

65

The first derivative of 9"(1) can be approximated by

Aa(ri_l)

——_—(Tr _ T: - 1) (2.34)

2'” (IN) =

where AG (1,- 1) = the hygroscope free expansion value between time

“ti-1 and 1:-

Equation (2.27) and (2.28) can be combined in matrix form

1

NM“) = j‘ 911("‘) Q12("‘) 8'°(1:) ‘1‘“! N"(r-1:) d1: (235)
M'”(t) o Q21 (“1) 922(“19 [C(17) 0 M"(r—t) .

If there is no external force and moment, N'" (t) = M'" (t) = 0 , Equation
(2.35) becomes

l NM“) d'c =1[Q“(M) Q12(""][€'°<T>]d¢ (2 36)
o M"(t—r) 0 {2210-1) Qua—t) K‘(r) °

Using numerical analysis method,

(‘i’ti-i)

"' [Mu-1H)

i=1 MHz—rid)

 

m (2.37)
= 911(‘TTi-1) 912("‘.--1) yam-1) (ti-TH)
i= 1 Q21("tr-r) Q22(t-ti—l) [COR--1)

66

Taking 15-134 = constant, and '6", = t , to = 0 , the above equa-

tion canbewritten as
-1
8'°(T,,,-1) = Qll(t-tm-l) 9123-31-1)
K'(Tm_1) Q21(t‘1m_1) Q22(t-Tm_1)
(2.38)

i Nh(t'-ti_1) _mil Qll(t-Ti-1)Q12(t-ti-l) E'0(1i_l) fr“
i=1M"(r—'c‘._1) i=1 Q21(t—ti-1)Q22(t_ri-l) ”(Tr-1) '
Obviously, once the relaxation modulus and the hygroscopic

expansion are defined, all the parameters can be determined. The deformation

v0
rate [:2] at final time t = 1:", can therefore be obtained according to

Equation (2.38).
' The normal strain and the curvature [2 33)] can subsequently be
found by ...

e"(t,,,) = e'°(t,,,_,) (I -1: )+ 8°(r,,_,)
Km.) K'(r,,,_,) "‘ '"" K(r,,,_,)

= 2[3(t5-1)](1m i+l-Tm-i)+[€K(TO)] (2.39)

t l K'(ti-1) (To)

 

The deflection at any time t can be derived by setting the initial
values being zero: W(0.0) = 0 and W' (0.0) = 0 , and calculated by

wow) = %K(t)x2 (2.40)

Mllb

The center deflection can be obtained by substitutingx

67

"(mm = glam} (2.41)

The final center deflection is the summation of center deflections
determined for each time interval.

A program written in FORTRAN was generated for the calculation of
Wm” (Appendix B).

 

68

 

 

 

 

 

 

t:
.o
3
E
“-1
O
Q
§
[—
.— Time
t0 t1
Load Applied Load Removed

Figure 2-10 Deformation response curve to constant load
of elastic materials.

 

 

 

 

 

+ Elastic Deformation
:: B
.3 .
E Creep
8 A
a C
3
g D
0 PlaStiC Def.
’
to t1 t2 Tune
Load Applied Load Removed

Figure 2-11 Deformation response curve to constant load
of visco-elastic materials.

 

69

 

 

 

 

7 l

 

 

 

 

‘3' a.
3’2 ti
0 O
.. -
temperature rel. humidity

7 cyclic relative humidity
§‘
3 constant relative humidity

 

 

time

Figure 2-12 Effect of external factors on creep.

 

70

W

Ea A. elastic element

 

B. delayed elastic element
Tide Ede

 

11v C. plastic element

Figure 2-13 Burger’s four-element visco-elastic model.

'1

 

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73

2.5 Development of Experiment on Restrained Swelling

The experiment is based on the supposition that the warping of a
laminated panel can be predicted better by replacing two of the three inputs
( E, , T, , a, ) for each layer with parameters that allow for the viscoelastic
characteristics of wood. Instead of using a, which is the total free expansion
of the lamina for a given moisture content interval, we will use only the
“elastic portion” of it, i.e., that part which is actually transformed into elastic
strain. And instead of using E, which is the statically determined modulus

of elasticity as a function of moisture content, we will use a “deformation

modulus” defined as the maximum measured swelling stress divided by the

elastic portion of the free expansion

0' (1.11)

 

The thickness, T, , of course remains unaffected. Both a". and

must be determined by experiment.
As has been pointed out before, solid wood behaves sufficiently

elastically in the longitudinal direction, so that the above modification need

only be applied to layers stressed in the transverse direction (radial or

eielast

tangential).
This approach is illustrated in Figure 2-15 for the special case of a two-

ply laminate in which one lamina has a large hygroscopic expansion and a

low modulus of elasticity while the other has no or very little hygroscopic

"‘1

 

74

expansion and a very high modulus of elasticity. This would be typical of a
two-ply solid wood cross laminate.

The figure shows the free expansion in the horizontal axis and the
elastic portion of this expansion which will sustain the swelling stress under
complete restraint on the vertical axis.

One could view the condition of complete restraint of one lamina by
the other as the moisture content increases from one level to another as being

equivalent to the condition where the expanding lamina was allowed to move
freely and was then compressed back to its original length.
In a visco-elastic material, only a portion of this compressive strain e2
would be converted to elastic strain. In Figure 2-15 this portion is shown as
82‘1““ . It is that portion of the completely restrained hygroscopic expansion
which sustains a swelling stress at the end condition (hygroscopic
equilibrium). The graph assumes that this portion 22m“ increases linearly
from zero.
This elastic portion of the total compressive strain can be readily
measured for any end condition involving moisture content increase. The
condition of a shrinking laminate involving tensile strain is very similar, but
the experimental determination of the elastic strain portion would be much
more difficult. The experiments are therefore limited to moisture content
increases.
The elastic 92.1“: is thus substituted for the free expansion in the
warping formula. And instead of the modulus of elasticity E , measured at
the end condition by static test, the more realistic ratio of the swelling stress

at the end condition divided by e is used (deformation modulus). The

Zelasr

 

 

75

swelling stress is determined in the same experiment used for the

determination of 52¢ 1w.
Lamina I in Figure 2-15 is assumed to be elastic and requires no

substitution.

Upon increase in moisture content from the initial condition, the length
change of lamina I and therefore of the whole laminate is practically zero.
The free expansion a2 is being completely restrained ( a2 = e2 ). Rather

e2 sz for the calculation of the compression stress at stress
is the instantaneously

than using
balance, we shall use chume'z . where 82.10,,

recovered axial compression deformation after completely restrain 0:2 to

moisture content equilibrium

 

6|
5'2 = (2.42)

eZelast

where o' = the measured axial compression stress at complete restraint

of a2 at moisture content equilibrium.

Inputs for layerIwouldbe El and all since E1 is very large and at

very low that the elastic assumptions are justified.

The measurement of 2M“, and 13'2 requires a test apparatus capable

of continuous measurement of specimen weight, development of axial

swelling stress, and upon releasing of the restraint at moisture content

equilibrium, of the instantaneous recovery of the restrained expansion,

Additionally, the free expansion would also be measured. This

82:14:: °

apparatus will be described in detail below.

We will now consider the more general case of a two-ply laminate in

 

76

which both layers expand and have moderate moduli of elasticity. This is the
case presented in the introductory chapter (see Figure 1-3).This more general
case is illustrated in Figure 2-16. Both layers will be considered to be visco-
elastic.

The two layers have unequal free expansion (a2 > 0‘1 ) and unequal
moduli ( IE1 at 152 ). The elongation of the laminate after moisture content
increase is based on stress balance ( 0'l = cs2 ). This final elongation ( ctr“)
lies somewhere between all and a2 , which means that layer I will be
extended (tensile stress) and layer 11 be compressed (compression stress).

In analogy to the previous example the stress balance for the two

layers will be calculated as
61 = ylelast 5'1 (2.43)
and
02 = 8.2elast E'z (2'44)
at stress balance
01 = 02 (2.45)
or
e'm‘m El = E'Zelast 5'2 (2-46)

Here however the values 2' and e' are not the measured

lelast 2elast

77

recovered compression strain after complete axial restraint but only a certain
portion of that depending on the extent of the final remain restraint which is
equal to e2 in the case of layer II and r1 in the case of layer 1. These
portions are indicated within the shaded triangles.

It should be noted that layer 1 is not being restrained in a positive sense
(compression) but in a negative sense (tension). This would require that

e be determined by a tension test. However, such tension tests are

lelast

much more difficult to conduct and, therefore, the important assumption was
made that elelasr as determined by compression test is a valid approximation
of the corresponding value found by tension test.

The point of stress balance is found in Figure 2-16 as the point of

lntersectlon of the two hues representing .e'mamxlfi.”l and 2'2““sz .

Determination of the point of intersection Win yield 3' and e' the

lelast Zelast ’

desired inputs for the warping equation. E'l and B2 are determined as
described above.

The expression of r:1 , e2 , e' and 2' can be written in

lelast Zelast

terms of known values em“, , a a1 , a2 , E1 and E"2 asfollows

Zelast ’

__ 8Mom]? 2 (a2 ’ a1)
- 155’1 5'2 (2.47)
.,(. )

lelasta—l + shins-ta;

 

elelastE'l ((12 - (11)

e = .
2 E1 E 2
a1 elelastEI- + Salas-ta;

 

(2.48)

78

elelastE'ZelastE 2 ((12 - a1)

 

 

e' =
lelast '
elders-IE 10‘2 + gZtelastE‘Za'l
e' elelastehlastE 1 (a2 ‘ a1)
2elast - '
elelastE 1‘12 + e2elastE'2al

(2.49)

(2.50)

79

    

 

 

 

 

 

 

 

 

 

 

ASE I
T"
0'2
eZelast
JV COMPRESSION
from restraint test <—
e2
El ” Ez ‘
T1 8 T2 “1 - 0
T1 I I
T2 I II
a;
Input into warping equation: 51 . at
a.
E'Z = , 82:10::
8
Zeta."

Figure 2-15 Case 1: Complete restraint.

CASE II

    

80

 

 

 

 

 

 

 

 

 

 

 

 

 

 

am
stress balance
0'
a A
02\ /01
e'ZelastE'Z e 1‘“er 1
. a E O
COMPRESSION i
I 82cm." 5
from restraint test E "
: 8 lelast l
TENSIONE/fl eleltlsr
f—p g ) frun restraint
g a, E I test
T1 = T2 E1 ¢ E2 a] : C] 5 ‘2
Ti I I .—-3
T2 I II E“-———-ii
_. (12 m
Input lnto warping equation: 13' 1, 6 Was,
E'2. 8 Zelast

Figure 2-16

Case 11: Partial restraint.

81

2.6 Objectives and Scope of the Study

A qualitative understanding of the mechanics of panel warping is
needed if we are to improve the performance of wood and wood composite
products such as laminated wood, plywood or laminated wood composite
.panels.

The main objective of this study is to demonstrate the theoretical
analysis of the warping problem, elastically and non-elastically; to
investigate various factors involved in the hygroscopic warping; to evaluate
and compare the prediction power of different theories or approaches; and to
see the possibility of improving the existing theory by using alternative
inputs.

Theoretical analysis can be classified as elastic theory and visco-
elastic theory. Beyond the theoretical analysis, it is desirable to conduct some
simple but practical warping experiments using various materials with
different hygroscopic characteristics and conditions. This will allow
verification of analytical prediction.

The scope of the experiment is limited to narrow laminated beams,
since the results from beams can be used to simulate the maximum
deformation of a corresponding plate.

The secondary objective of this study is to experimentally investigate
the measurement of true swelling stresses on wood and wood composite
materials upon hygroscopic expansion. A specially designed device will be
used for this study. These experiments will also provide the inputs for the

theoretical treatments.

CHAPTER III
EXPERIMENTAL METHODS

3.1 General

The experiments were designed to provide the following

o hygroscopic and mechanical characteristics of the laminating

materials to be used as input into warping equations

- warping measurements on laminated beams of various

combination and exposed to a variety of relative humidity
intervals.

It was essential that the specimens used for the determination of
hygroscopic and mechanical characteristics were truly representative of the
characteristics of the laminas used for the manufacture of the laminated
beams. This Was especially critical in the case of the solid wood beams
because solid wood often shows considerable variation of its properties
between specimens coming from the same source.

The warping of the beams was observed over a number of different
relative humidity intervals (always increasing humidity). This required that

hygroscopic and mechanical tests were conducted at various levels of relative

82

83

humidity equilibrium and over various relative humidity intervals.
For the solid wood laminated beams, the warping was measured over
these intervals of relative humidity exposure:
66 - 81%
66 - 93%
and for the composite beams these intervals were
30 - 81%
50 - 81%
30 - 93%
50 - 93%

The testing program essentially breaks down into these elements:

0 Determination of modulus of elasticity in tension and
compression at the above relative humidity levels and
determination of hygroscopic expansion over the above relative
humidity intervals

0 Determination of the elastic portion of the free expansion and
determination of the actual swelling stress of restrained
specimens. This work required special apparatus 'which will be
described later.

Three types of wood materials were used throughout the study, namely,
solid wood, yellow-poplar (Liriodendron tulipifera), medium-density-
fiberboard (MDF), and particleboard (PB).

Tangentially sawn yellow-poplar lumber 4- to 6- inch wide, l-inch
thick and 6- ft long was purchased from a local sawmill. Its average density

was 0.483 g/ cm3 when moisture content is about 9 percent.

84

4-fr by 4-fr panels of MDF and PB of l-inch thickness were cut from
commercially manufactured panels provided by Georgia Pacific Corporation.
Both types of product had 8 percent average moisture content and the density
was 0.85 g/ cm3 for MDF and 0.82 g/ cm3 for the PB.

3.2 Specimen Preparation
3.2.1 Yellow-Poplar

In order to test the warping equation to the plywood structure, a
yellow-poplar board with only tangential and longitudinal direction in the
plane of the board was manufactured in the laboratory. The specimen
preparation and the experimental design were established based on the
preliminary study and experiments and is described in the following:

Clear and straight grained lumber were jointed and planned to 0.75-
inch thickness. The boards were then cut into l-inch wide and 5- ft long strips
on a table saw. '

After conditioning these strips at the initial condition of 66 percent

RH at 70°F, 26 strips which had perfect flat grain and were straight were
carefully selected and edge-glued to a panel as illustrated in Figure 3-1 using
epoxy resin (Duro—m Master MendTM TM61). The dimension of the panel at
this point was ZS-inch by 60-inch with a thickness of 0.75- inch.

The panel was then trimmed down to a size of 24- inch by 48- inch and
planned to a thickness of 0.25-inch. All the test specimens were cut from this

sheet (see Figure 3-2) and are described as follows:

85

Hygroscopic expansion specimens (24- inch by l-inch by 0.25-inch):
E-L (longitudinal)
E-T (tangential)
Beam lamina (24- inch by l-inch by 0.25-inch):
L1, L2, L3, L4 (longitudinal) '
T1, T2, T3, T4 (tangential)

These laminas were used to construct three types of
unbalanced cross-laminated beams as shown in Figure 3-3
and as described in Section 3.3.3.

Tension test specimens (IO-inch by 0.25- inch by 0.25-inch):
Ll-Tl, Ll-T2 ...... 1.4-T1, 1.4—T2 (longitudinal)

It will be noted that these specimens correspond directly

and are part of strips L1 to L4 used for the laminated beams
Compression test specimens (4-inch by l-inch by 0.75- inch)
Compl, Comp2, Comp3 (tangential)

These three strips dimensioned 24- inch by l-inch by 0.25-
inch were laminated to form a l-inch wide and 0.75-inch
thick beam, which is later cut into 4-in ch long specimens (see
Section 3.3.5).

Restraint test specimens (2- inch by 1- inch by 0.25- inch)
R1, R2, El, E2 (tangential)

These four strips dimensioned 24- inch by l-inch by 0.25-
inch were used to make restraint specimens (2- inch by l-inch
by 0.25- inch). Half of them were used for restrained swelling

test specimens, and the other half were used for hygroscopic

86

expansion test specimens as matched samples as described in

Section 3.3.6.

It will be noted that a special effort was made to characterize the strips
used for beam construction in terms of their important properties as closely as
possible by appropriate specimen selection from the edge glued panel.

For complete listing of test specimens see Table 3.1
3.2.2 Wood Composite Board

A similar experiment was conducted with wood composite board.
Again the warping was evaluated on two ply laminated beams. Since wood
composition boards are orthotropic in the plane of the board, the hygroscopic
and mechanical imbalance was created by laminating two dissimilar board
materials, one being an MDF board and the other a particleboard.

Most wood composite boards have a pronounced density variation
over their cross-section (thickness). This density variation is generally
balanced with higher density in the faces and lower density in the center.

In order to eliminate possible effects of this density variation on beam
balance, the beam laminas and all other specimens were cut from the full
thickness of the board (1 inch). The width of each specimens (0.25-inch)
became the thickness of the lamina (see upper right corner of Figure 3-4).

The cutting scheme of specimens from MDF and PB is presented in
Figure 3-4. A board was first cut into four equal size pieces to provide four

replications with randomization. On each piece, specimens were cut along

87

the machine direction of the board according to the cutting schedule shown in

Figure 3-4 and described as follows

Hygroscopic expansion specimens (12- inch by l-inch by 0.25-inch):
El-l, E1-2, E2-1, E2-2,..., E4-1, E4—2

Laminated beam lamina (30-inch by l-inch by 0.25-inch):
B1, B2, B3, B4
These laminas cut from MDF and PB were laminated to
construct four two-ply unbalanced laminated beams.

Tension test specimens (IO-inch by l-inch by 0.25- inch):
MTl-l, MTl-2,..., MT4—1, MT4-2
Only specimens cut from MDF were tested.

Compression test specimens (4-inch by l-inch by l-inch):
Cl-l, C1-2, C1-3, C1-4,..., C4—l, C4-2, C4-3, C4—4
Only specimens cut from PB were tested.

Restraint test specimens (2- inch by l-inch by 0.25- inch):
As in the case of yellow-poplar, some of them were used as
restrained swelling test specimens, and the rest were used as
matched samples for hygroscopic expansion test and weight

gain control samples as described in Section 3.3.6.

For complete listing of test specimens see Table 3.1.

 

88

 

 

 

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3.3 Test Methods
3.3.1 Relative Humidity Condition

To provide the various relative humidity conditions a conditioning
chamber was used to obtain 30’ and 50 percent relative humidity, while
different types of saturated salt solutions were employed for 66, 81, and 93
percent relative humidity: Sodium Nitrite, NaNO2 , for 66%, Ammonium
Sulfate, (NH4) 2SO4 , for 81%, and Mono Ammonium Phosphate,

NH4H2P04 , for 93%.

The conditioning chamber shown in Figure 3-5, which has a
volumetric capacity of 20- ft3 , was manufactured by Parameter Generation
Company. It is equipped with solid-state controls for precise adjustments and
provides constant temperatures ranging from 4°C to 88°C and constant
relative humidities ranging from 10 percent to 98 percent.

A series of tanks 12 x 30 x 12.5-inch in size charged with different
saturated salt solutions were used to condition the larger specimens.
Saturated salt solutions were maintained in two plastic u'ays placed in each
tank. Specimens were supported on a plexiglass frame. Each tank was
equipped with a fan to supply air circulation, as exhibited in Figure 3-6.

In addition to the tanks, several laboratory desiccators were also used
for conditioning the smaller specimens, and two stainless steel desiccators

were used for restraint tests.

CC

 

 

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95

3.3.2 Measurement of Hygroscopic Expansion

The hygroscopic free expansion is calculated by using the following

equation

 

(in/in)

where L, and L2 are the initial and the final length of the specimen,
respectively.

Determination of hygroscopic expansion was carried out using two
different measurement methods, namely a dial gage and an optical
comparator.

Yellow-poplar specimens were tested by a dial gage fixed on an
aluminum frame as illustrated in Figure 3-7. The dial gage has an accuracy of
0.001 inch being equivalent to 0.004% of the 24-inch long specimen. Both
specimens (longitudinal and tangential) were measured after exposure to
sequential RH levels of 66 percent, 81 percent and 93 percent.

Measurements were taken at each relative humidity condition after the
specimens had reached their equilibrium moisture contents. These conditions
were the same as those at which the laminated beams were conditioned. The
results were used as inputs to the warping equation.

Hygroscopic expansion of MDF and PB were determined on an optical
comparator as described by Suchsland [50] and shown in Figure 3-8.
Specimens were cut to 12- inch length, l-inch width and 0.25-inch thickness.
Gage length was 10.000 inches.

 

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the

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96

After an initial storage at 30 percent RH, specimens were sequentially
conditioned at 50 percent, 81 percent and 93 percent relative humidity. These
conditions were different from those used for the conditioning of yellow-
poplar specimens. However, they were the same as the conditions used for

the laminated beams of MDF and PB as explained in the following section.

3.3.3 Measurement of Warping of the Laminated Beams

In order to compare the analytical predictions with the experimental
results, several laminated beams with material 6r structural unbalance were

fabricated from yellow-poplar, MDF and PB strips.
W

Three types of yellow-poplar laminated beams were made introducing
structural and material unbalance (see Figure 3-3). These were:

L - T beam: a two-ply beam with longitudinal and tangential strips cut
from the yellow-poplar sheet. Each strip was 24- inch long and l-inch wide
and approximately 1/4~ inch thick

T - L - T beam: a three-ply beam with an additional l/l6-inch thick
tangential lamina added to the above L-T beam. '

L - T - L beam: a three-ply beam with an additional 1/16-inch thick
longitudinal lamina attached to the above L-T beam.

The thinner laminas in beams T-L-T and L-T-L were originally 0.25-

35

 

97

inch thick. Their thickness was reduced by planning after the beams had been

assembled.

MDF and PB

Four identical two-ply wood composite beams with 30-inch length and
1- inch width were laminated using MDF and PB. Each layer was cut from the
board as indicated in Figure 3-4. The laminas were l-inch wide and 0.25-inch
thick.

1.. lEl"'El

For all the yellow-poplar beams and MDF/PB beams, the strips or
lamina were first conditioned at the initial condition, then they were glued
together to the various combinations as explained previously by an epoxy
glue. The purpose of using the epoxy glue was to avoid the addition of water.
After glue lines were set, the beams were kept in the initial condition until
equilibrium was reached again. The initial warping (center deflection) of the
beams over the span was monitored on a device equipped with a dial gage
having an accuracy of 0.001 inch as shown in Figure 3-9. The beams were
then brought to the next higher RH atmosphere. The warping and the weight
change were measured thereafter until the measurements remained constant
and the beam was considered to have equalized. The effective measured
warping Wm is the difference between the final and the initial center

deflection.

98

All three yellow-poplar beams were exposed at the sequential relative
humidity levels of 66 percent, 81 percent and 93 percent.

For MDF and PB laminated beams, two pairs were initially
conditioned at 30 percent and 50 percent relative humidity levels
respectively. After they reached equilibrium moisture content, one of each
pair was exposed to 81 percent while the other was conditioned at 93 percent

relative humidity (see Figure 3-10).

30%RH 50%RH 81%RH 93%RH

 

0 O

O
O a '
l ' '
I I
I O

O
l ' D
O 0 O

 

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0 g ' :
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0 a '
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Figure 3-10 Relative humidity conditions for MDF/PB laminated beams.

3.3.4 Tension Test

Static tension tests were performed for those laminas which would be .
subjected to tensile stress during warping. In the case of yellow-poplar those

laminas are the longitudinal laminas, and in the case of the composites they

 

Fi

 

SC

in

il

99

are the MDF laminas, because MDF has a lower hygroscopic expansion than

PB, putting it in tension during moisture uptake.

Yellow-Poplg

Tension specimens of yellow-poplar were cut from the end of the strips
that were used as laminas for the unbalanced laminated beam as shown in
Figure 3-2. Two specimens were used from each lamina with a cross-
sectional area of 0.25- inch x 0.25- inch . The obtained modulus of elasticity
( MOE) represents the MOE of the lamina, and would be appropriate as input
for the tension layer in the elastic warping equation.

The dimensions and the construction of the tension specimens are
illustrated in Figure 3-11 (a). The ends of the specimen were reinforced with
two angled pieces of wood. Holes drilled through the center of these
assembles allowed load transfer by means of steel shafts. The axis of the hole
was aligned perfectly perpendicular to the axis of the specimen to prevent the
development of a bending moment during loading.

The tensile force was applied by dead load. An 1-inch su'ain gage
extensometer was mounted at the center part of the specimen. Readings were
recorded by using a 3800 Strain Indicator manufactured by Measurements
Group, Instruments Division with 1/63000 inch of precision over the 1-
inch span. After conditioning to equilibrium at 66 percent relative humidity,
a specimen was loaded by a 10 lb weight, initially. Increments of 10- lb
loads were added thereafter until the total load would develop a stress of l/3

of the pr0portional limit of the material. Therefore, the loading was always

100

within the elastic range of the material. The specimens were retested using
the same procedure after exposure to equilibrium at 81 percent and 93
percent relative humidity levels, respectively.

Only the MOE values in tension at the end conditions, 81% RH and
93% RH , were subsequently used as the inputs in the elastic warping
equation.

All specimens were tested in an environment of 66 percent relative
humidity and 70°F. The specimens conditioned at the higher relative
humidity levels may have lost some moisture during the testing. But these

losses were considered to be insignificant and were disregarded.

MDE

MDF tensile specimens were tested in a similar way. They were
randomly selected from the board. Six samples were tested. The full cross-
sectional area of the board was utilized as shown in Figure 3-11 (b). The
relative humidity conditions to which the specimens were exposed were 81

percent and 93 percent.
3.3.5 Compression Test
Compression tests were carried out for the materials which would be

under compression in an expanding unbalanced beam. That is, yellow-poplar

in the tangential direction and particleboard.

101

Yellow-Poplar

Three l-inch wide and 24- inch long strips in the tangential direction
were cut from the yellow-poplar sheet as explained previously. In order to
avoid buckling during the compression tests, three strips were glued together
to form a thick laminated strip by using epoxy glue. This thick laminated
strip was then cut into five 4-inch long specimens. In order to mount a strain
gage extensometer to the specimen, one of the vertical edges was chamfered
to provide an approximately 1/5 inch wide bearing area.

An Instron universal testing machine was employed to supply the
loading for the compression test. The deformation within an 1 inch gage
length was monitored by a strain gage extensometer connected to a strain
indicator.

The specimens were first tested with a continuously increasing load
within the elastic limit of the material at an initial relative humidity level of
66 percent, and then were retested after conditioning at 81 percent and 93
percent relative humidity levels, respectively. Loading speed was 0.015 in /
min.

The average of the MOE s thus obtained would represent the properties
of those strips in the tangential direction used in the beams, and were

subsequently used as inputs for those layers in the elastic warping equation.

102

Particle ard

Sixteen specimens of particleboard 1 inch by 1 inch in cross section
and 4 inches long were randomly selected from the board. The loading
direction was parallel to the surface of the board. Specimens were tested on
the Instron universal machine after conditioning at relative humidity levels of
30 percent, 50 percent, 81 percent and 93 percent with four replications at
each condition. Loading speed was 0.02 in /min

3.3.6 Restrained Swelling Test

This is the test that determined the elastic portion of the restrained

swelling and the actual swelling stress of the specimens expanding under
complete axial restraint. The tests were conducted on short columns which
after initial conditions at a low reference relative humidity were mounted in a
rigid clamp which prevented any axial expansion during exposure to higher
levels of relative humidity. When the specimens reached equilibrium at the
higher relative humidity, the clamp was opened and the instantaneous
recovery measured and expressed as a fraction. of the free expansion of
Control specimens expanding without restraint under the same test
Conditions. Incorporated in the clamping device was a load cell which
measured the developing swelling stress throughout the test.

The restraining device (see Figure 3-12) was specially designed and
built for this study. It consists of two stainless steel ball bearing guide blocks

faStened to a rigid aluminum beam and riding on two precision ground

103

stainless steel shafts. These shafts were fastened to a lower and an upper
horizontal steel plate. The position of the aluminum beam and guide blocks
could be finely adjusted by means of a spindle threaded through the center of
the upper horizontal plate, and connected to the aluminum beam. A spherical
depression was machined into the center of the upper aluminum beam. A
steel ball was placed between the beam and the spindle. A miniature load cell
was attached to the bottom face of the aluminum beam, its center coinciding
exactly with the center of the adjustment spindle (see Figure 3-12)

To reduce the possibility of eccentric loading which is almost
unavoidable with one small column, the test was conducted on two identical
columns, each two inches long, one inch wide, and 0.25 inch thick and

spaced two inches apart (Figure 3-13 (a)). The two columns were cut to
exactly the same length and were then glued between two aluminum blocks
which were notched to assure the exact spacing of the specimens.

An attached clip gage extensometer measured the distance between the
aluminum blocks, i.e., the length of the two columns. The entire assembly
was placed in a stainless steel desiccator in which the relative humidity could
be adjusted by means of saturated salt solutions maintained in shallow trays.
me air within the desiccator was continuously circulated by means of a small
electronic fan. Figure 3-14 shows two such desiccators, each with a restraint
deVice as described above. Measurements from load cell and clip gage were
recorded by electronic data acquisition equipment shown in left of Figure 3-

14-
The above mentioned control specimens were also mounted between

“"0 aluminum blocks. Their hygroscopic expansion was not restrained in any

104

way and was measured continuously by a clip gage extensometer (Figure 3-
13 (b)). An additional control specimen was suspended from a miniature
cantilever load cell, which continuously measured weight gain during the

test.

The test proceeded as follows:

Five identical short columns were conditioned at the low reference
relative humidity (66 percent for yellow-poplar, 30 and 50 percent for MDF
and PB). Two were then mounted between the aluminum blocks and placed
in the restrained device. A minimal lead was applied by twisting the spindle
to assure seating of the load cell and to remove all slack. The clip gage was

attached to pins projecting from the two aluminum blocks. Two more short
columns were mounted between another set of aluminum blocks. A clip gage
was attached and the assembly placed in the desiccator. A fifth column was
hung on the cantilever load cell within the desiccator. Salt solution were put
into place, the desiccator was closed, the fan started.
A sample record of a test is shown in Figure 3-15. Such test were
performed on tangential yellow-poplar columns for these test intervals:
66 - 81% relative humidity
66 - 93% relative humidity
They were performed on MDF and PB for these test intervals:
30 - 81% relative humidity
50 - 81 % relative humidity
30 - 93% relative humidity
50 - 93% relative humidity

105

In these cases, the columns were 0.25 inch thick cross-sections of the
1-inch thick board-Their length was 2.0 inch .

Five test replications were used for each of two condition of yellow-
poplar specimens, and six to eight replications were used for each of the four

conditions of MDF and PB, respectively.

106

Figure 3-5 Conditioning chamber used for conditioning of specimens.

107

 

 

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109

 

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110

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hygroscopic expansion of yellow-poplar.

111

 

 

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112

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determination.

113

 

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warping of yellow-poplar laminated beam.

 

 

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and MDF (b).

117

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measures swelling force, clip gage in center measures length

change of restrained specimens.

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free hygrosc0pic expansion.

120

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and continuously recorded by a data acquisition system.

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CHAPTER IV
RESULTS AND ANALYSIS

4.1 General

In this chapter, the results of the tests of the mechanical and physical
properties of the various materials used in the study are presented and
discussed. This includes the measured warp of the unbalanced laminated
beams exposed to various relative humidity conditions. These warp
measurements are then compared with three analytical predictions of warp

from:
- elastic beam equation
. - inelastic beam equation (modified elastic beam equation)
. visco-elastic beam equation.

Finally, the practical application of the results to plywood and

composite panel warping is discussed.

123

124

4.2. Yellow-Poplar
4.2.1 Results of Experiment

As illustrated in the previous chapter, a yellow-poplar sheet was
fabricated in our laboratory. The two principal axis in the plane of the sheet
are in the tangential and longitudinal directions. The sheet was made by edge
gluing selected wood strips.

The density of the individual strips that were used for the yellow-
poplar board was determined individually before edge gluing. The over-all
density profile is plotted in Figure 4-1, where X on the. horizontal axis
denotes the distance from one edge of the panel, while L is the width of the
panel. Each point represents the density of that particular strip. X/L = 0
corresponds to edge (9 - G) in Figure 3-2, and X/L = 1 corresponds to
edge @ - @. Twenty-six strips were used to form this panel. As can be
seen, the density across the panel is quite uniform.

The experimental results of tension, compression, and hygroscopic
expansion test in both longitudinal and tangential directions as well as the
results of the restrained swelling tests are listed in Tables 4—1 through 4—5.

The MOE values show the typical effect of moisture content. The
longitudinal values are considerably below the average MOE value in
bending as listed in the Wood Handbook (MOE bending: 1580.0 ksi at 12%
Moisture content). Tangential values are not listed. The measured
hygroscopic expansion values, if converted to a change from green to oven

dry (0 to 30 percent moisture content) are 0.1 percent longitudinally and 10.5

125

percent tangentially.

The longitudinal expansion value is not listed in the Wood Handbook.
The listed tangential value is 8.2 percent. While these variations from the
listed values seem large, the actual measured values are very consistent and
have a relatively small standard deviations.

The results of the restrained swelling test in the tangential direction are
summarized in Table 4—5 for the two relative humidity intervals 66 to 81
percent and 66 to 93 percent.

The elastic strain, a is that portion of the restrained expansion

alas: ’
which recovers instantly upon removing the restraint. It is this portion that
sustains the swelling stress and thus contributes to the warping. The swelling
stress 0' is the measured axial compressive stress at the end of the exposure
period, at which time the specimen has reached moisture content equilibrium
(weight constancy). The clamp is opened at this time and gala." is recorded.

It is interesting to note. that the “deformation modulus” E' = 579.1..."
is larger than the statically determined MOE at both 81 and 93 percent
relative humidity. This is so because what is being measured is the equivalent
of the stiffness of the elastic component of the Burger body alone.

It appears that the specimens become stifl'er at the higher moisture
content.

The elastic strain is about 25 percent of the total free expansion (or
total compressive strain). This is true for both relative humidity conditions
and also for many other specimens as shown by Keylwerth [21].

The 24 inches long laminated beams were conditioned in fish tanks

charged with saturated salt solutions. Two exposure intervals were used: 66

126

to 81 and 66 to 93 percent relative humidity. Their center deflection was
measured every day. The warping increased rapidly at first, then leveled off.
After seven days the beams had reached equilibrium and the warping
stabilized (Figure 4-2).

All these beams showed positive warp (concave) when viewed from
the top (Table 4—4). It may be noted that compared with beam T—L which is
severely unbalanced, the addition of a thin third layer in the tangential

direction (T-L-T) does not balance the construction, substantially reduces the

warp.

4.2.2 Theoretical Estimation of Warping

Three approaches were used for the theoretical estimation of the
warping of the laminated beams:

—— elastic approach

Standard beam equation (Equation 2-13) with

elastic inputs MOE and the free hygroscopic expansion
on for both directions.

— inelastic approach

Modified beam equation with MOE replaced by

“deformation modulus” E' = o'/s and with free

alas!

expansion on replaced by e

alas: °

These modification apply only to the tangential
direction.

127

The longitudinal direction is treated as in the elastic

approach.
— visco-elastic approach

Swelling stress relaxation modulus and free
expansion were used as inputs for the tangential
direction.

Relaxation modulus is defined as the ratio of
swelling stress to the free expansion value: Y = o'/a ,
as function of exposure time.

Static tension MOE and free expansion were used

for the longitudinal direction.

The visco-elastic approach required the establishment of time
functions for free expansion and for the relaxation modulus.

The experimentally obtained characteristics were subjected to a non
linear regression analysis which yielded continuous time functions of free
expansion and relaxation modulus.

The regression models are illustrated in Figures 4-3 and 4-4.

The final warping is the cumulative result of the warping calculated for
small time intervals. One hour time elements provided sufficiently accurate
results. Reducing the time element did not result in significant improvements.

Table 4-6 shows a summary of the inputs for the elastic and inelastic
approaches. The inputs for the visco-elastic model are shown in Table 4-7.
This approach was applied only to the two-layer laminates.

A summary of the results allowing a comparison of the theoretical

128

calculation with measured warping is given in Table 4-8.

It is apparent that the elastic approach leads to a substantial
overestimate of the warp of the laminated wood beams of all constructions.
The inelastic approach yields results that are very close to the measured
warp. The visco-elastic approach results in a slight overestimate.

It is known that moisture content gradients develop in solid wood
when exposure conditions are changed dramatically, i.e., portions of the
wood near the surface reach equilibrium moisture content much sooner than
the interior portions. Such moisture content gradients can have significant
effects on swelling stress and on warp development. No allowance was made
in this study for such effects. However, the cross-section of the lamina as well
as of the short columns used in the restrained compression test were
relatively small so disregarding the moisture content gradients may be

justified.

129

 

 

 

 

 

Table 4-1 Modulus of elasticity in tension (ksi) of yellow-poplar in the
longitudinal direction

Specimen ID 66%RH 81 %RH 93%RH

L1 1366.969 1288.556 1196.713

L2 1236.557 1159.427 1080.540

L3 1236.564 1152.939 1055.610

L4 1213.704 1 124.407 1039.997

AVG 1263.449 1181.332 1093.215
811) 69.850 73.083 70.990 ‘
C.V. 5.53% 6.19% 6.49%

Table 4-2 Modulus of elasticity in compression (ksi) of yellow-poplar in the
tangential direction

Specimen ID 66%RH 81%RH 93%RH

Cl 46.296 37.919 29.058

C2 45.029 37.055 27.045

C3 44.720 36.953 25.993

C4 49.404 41.445 31.226

C5 50.576 42.171 36.833

AVG 47.205 39.109 30.031

STD 2.643 2.506 4.297
C.V. 5.60% 6.41% 14.31%

 

130

 

 

 

 

 

Table 4-3 Pr0perties of yellow-poplar
66%RH 81%RH 93%RH
Moisture Content (%) 9.71 13.97 18.39
M OE tension longtudinal (ksi) 1263.45 1181.33 1093.22
MOE comp tangential (ksi) 47.21 39.11 30.03
66-81%RH 66-93%RH
Free Hygroscopic Expansion
Longitudinal (in/in) 0.00013 0.00025
. Free HygroscOpic Expansion
Tangential (in/in) 0.01492 0.02967
Free Hygroscopic Exp. Coefficient
Longitudinal (%/%) 0.00305 0.00288
Free Hygroscopic Exp. Coefficient
Tangential (%/%) 0.350 0.342
Table 4-4 Measured warping of yellow—poplar laminated beams over 24—inch

span with initial condition of 66 percent RH

 

 

NO. Beam ID Thickness 81 %RH 93%RH
l L - '1' 0.257 +0.296 +0.644
0.257
0.0643
2 T - L - '1' 0.257 +0.279 +0548
0.257
0.257
3 L - T - L 0.257 +0.041 +0.081

0.0643

 

131

 

 

 

 

 

Table 4-5 Results from restraint test of yellow-poplar
66 - 81%RH
Specimen Elastic Swelling Deformation Free
ID Strain em“ Stress o" Modulus E' Expansion a Eelar/ a
(in/In) (psi) (psi) (in/in)
1 0.003937 155.56 39512.1 0.016019 24.58%
2 0.003199 131.78 41199.2 0.015349 20.84%
3 0.003649 157.68 43214.6 0.015362 23.75%
4 0.003218 137.90 42857.1 0.013227 24.33%
5 0.003262 151.21 46355.7 0.014461 22.56%
AVG 0.003453 146.82 42627.7 0.014884 23.21%
STD 0.000328 11.40 2551.6 0.001079 0.01538
C.V. 9.49% 7.76% 5.99% 7.25% 6.63%
66 - 93%RH
Specimen Elastic Swelling Deformation Free
ID Strain em“ Stress o' Modulus E' Expansion 0t eem/ on
(in/in) (psi) (98') (in/in)
1 0.006933 319.09 46026.5 0.027674 25.05%
2 0.007224 316.39 43794.6 0.033381 21.64%
3 0.008480 381.51 44990.7
4 0.008381 375.55 44807.1 0.030392 27.58%
5 0.007576 314.13 41463.7 0.027230 27.82%
AVG 0.007719 341.33 44216.5 0.029669 24.76%
STD 0.000689 34.07 1730.6 0.002842 0.02808
C.V. 8.93% 9.98% 3.91% 9.58% 11.34%

 

132

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interval of 66 - 93%RH.

139

4.3 MDF/PB

4.3.1 Results of Experiment

The test results from MDF and PB tests are presented in Table 4-9 to
Table 4-11.

Results from the restrained swelling tests at various conditioning levels
are tabulated in Table 4.10 and Table 4.11 for MDF and PB respectively.

The elastic portion of the total swelling stress in the case of MDF is
near 40 percent for the exposure intervals 30 to 81 percent and 50 to 81
percent rdative humidity and between 20 and 25 for exposure intervals 30 to
93 and 50 to 93. The corresponding values for particleboard are 20 and 25
percent.

This may be related to the fact that the pure compression of MDF is

much lower than that of PB.

4.3.2 Theoretical Estimate of Warping

As in the case of the yellow-poplar laminate, three approaches were
used for the theoretical determination of the warp of the composite beams.
—— elastic approach
Standard beam equation (Equation 2-13) with MOE and
free hygroscopic expansion at inputs for both materials

(Table 4-12).

140

— inelastic approach
Modified beam equation for the condition of partial
restraint as discussed in chapter 11, Equation 2-47 to 2—50,
(Table 4-13)
— inelastic approach
Modified beam equation with MOE replaced by

“deformation modulus” E' = a'/e and with free

alas:
expansion 0: replaced by ea In” for both materials (Table 4—
l4).
— viscoelastic approach
This was based on the swelling stress relaxation moduli
and the free expansion of both materials. The regression
models are given in Table 4-15, 4—16 and Figure 4—5 to 4-7.
The results are listed in Table 4-17 together with the other
estimates for comparison.
This table shows that the elastic approach and the visco-elastic
approach give results that are in very good agreement with the actual
measured warping. The two inelastic approaches severely underestimating

the warp. It thus appears that the composite boards behave much like elastic

materials.

Table 4-9 Properties of MDF and PB

141

 

 

 

 

30%RH 50%RH 81%RH 93%RH
Moisture Content of MDF(%) 6.80 7.19 11.81 15.80
Moisture Content of PB (%) 7.80 8.16 13.66 17.21
MOE tension of MDF (ksi) - - 430.052 231.144
MOE comp of PB (ksi) 217.296 198.157 123.175 94.131
30-81%RH 50-81%RH 30-93%RH 50-93%RH
Free Hygroscopic Expansion
of MDF (in/in) 0.002495 0.001810 0.003553 0.002583
Free Hygroscopic Expansion
of PB (in/in) 0.005675 0.004338 0.009083 0.007100
Free Hygroscopic Expansion
Coefi. of MDF (%l%) 0.0498 0.0392 0.0395 0.0300
Free Hygroscopic Expansion
Coefl‘. of PB (%/%) 0.0968 0.0789 0.1045 0.0785
Warp of MDF/PB Beams +0.81 +0.70 +1.66 +1.33

 

142

 

 

 

 

 

Table 4-10 Results from restraint test of MDF
30 - 81%RH
Specimen Elastic Swelling Deformation Free
ID Strain eel“, Stress o" Modulus 5' Expansion (1 eelu/ 0t
(in/In) (psi) (psi) (in/in)
1 0.000866 219.75 2538534 0.002188 39.57%
2 0.001016 256.33 2524069 0.002363 42.97%
3 0.001020 227.42 2230464 0.002620 38.92%
4 0.001021 233.51 2287169 0.002296 44.47%
5 0.001024 269.03 2628100 0.002404 42.58%
6 0.000957 298.14 3113822 0.002795 34.25%
7 0.000939 302.83 3226475 0.002687 34.92%
8 0.001026 250.81 2443684 0.002606 39.38%
AVG 0.000983 257.23 2624040 0.002495 39.63%
STD ' 0.000058 31.15 36273.7 0.000212 0.037
C.V. 5.92% 12.11% 13.82% 8.50% 9.29%
50 - 81%RH
Specimen Elastic Swelling Deformation Free
ID Strain em” Stress 0' Modulus 5' Expansion a team} a
(Win) (981) (psi) (In/In)
1 0.000551 204.35 370578.0 0.002183 25.26%
2 0.000711 225.61 317399.] 0.002263 31.41%
3 0.000789 238.00 301697.0 0.001546 51.03%
4 0.000775 265.07 3422325 0.001817 42.63%
5 0.000598 176.74 2957333 0.001673 35.71%
6 0.000673 240.41 3574645 0.001657 40.58%
7 0.000708 229.14 3238288 0.001530 46.25%
AVG 0.000686 225.62 3298416 0.001810 38.98%
STD 0.000087 28.23 28054.9 0.000299 0.089
C.V. 12.71% 12.51% 8.51% 16.50% 22.71%

 

 

143

 

 

 

 

 

Table 4-10 (continued)
30 - 93%RH
Specimen Elastic Swelling Deformation Free
1D Strain scum Stress 0' Modulus 5' Expansion a eelm/ a
(in/In) (as!) (980 (in/in)
1 0.000743 257.12 345976.1 0.003597 20.66%
2 0.000705 243.11 3450000 0.003331 21.16%
3 0.000743 229.02 3081673 0.003731 19.92%
4 0.000774 241.26 311606.1 0.003908 19.81%
5 0.000669 230.52 3444912 0.003435 19.48%
6 0.000773 317.88 411341.0 0.003316 23.30%
AVG 0.000735 253.15 3444303 0.003553 20.72%
STD 0.000041 33.29 37073.4 . 0.000236 0.014
C.V. 5.57% 13.15% 10.76% 6.65% . 6.78%
50 - 93%RH
Specimen Elastic Swelling Deformation Free
ID Strain eel“, Stress o" Modulus 5' Expansion (1 eem/ a.
(In/In) (Psi) (psi) (In/In)
1 0.000679 191.15 2816851 0.002242 30.27%
2 0.000697 161.80 2322462 0.002874 24.24%
3 0.000602 274.47 4560250 0.002618 22.99%
4 0.000618 179.36 2900243 0.002723 22.71%
5 0.000679 204.83 3018404 0.002317 29.29%
6 0.000642 183.78 2860422 0.002723 23.59%
AVG 0.000653 199.23 3079772 0.002583 25.51%
STD 0.000038 39.48 76401.2 0.000250 0.034
C.V. 5.79% 19.81% 24.81% 9.68% 13.16%

 

144

 

 

 

 

 

Table 4-11 Results from restraint test of PB
30 - 81%RH
Specimen Elastic Swelling Deformation Free
ID Strain sum Stress 0" Modulus 5' Expansion a 8d“/ a
(in/in) (psi) (Psi) (in/in)
1 0.001383 353.40 2555482 0.005990 23.09%
2 0.001491 319.60 2144151 0.005489 27.15%
3 0.001351 371.87 2752413 0.005853 23.08%
4 0.001495 291.87 1952130 0.006369 23.48%
5 0.001483 295.28 1991104 0.006081 24.39%
6 0.001538 320.08 2081692 0.005140 29.91%
7 0.001427 350.51 2456429 0.005626 25.36%
8 0.001463 347.93 2377720 0.004852 30.16%
AVG 0.001454 331.32 2288890 0.005675 25.83%
STD 0.000062 29.02 28979.0 0.000504 0.029
C.V. 4.30% 8.76% 12.66% 8.88% 11.34%
50 - 81%RH
Specimen Elastic Swelling Deformation Free
ID Strain scum Stm 0" Modulus 1:" Expansion 0 eelm/ a
(in/In) (Psi) (Psi) (In/In)
1 0.000864 213.62 2473559 0.004447 19.42%
2 0.001190 217.07 1824337 0.004189 28.40%
3 0.001272 253.81 1995007 0.003880 32.79%
4 0.001020 258.98 2539394 0.004189 24.34%
5 0.001190 253.43 2130114 0.004790 24.84%
6 0.001190 247.37 207907.6 0.004533 26.25%
7 0.001197 248.36 2075406
AVG 0.001132 241.80 2159558 0.004338 26.01%
STD 0.000141 18.51 25713.5 0.000319 0.045
C.V. 12.42% 7.65% 11.91% 7.35% 17.13%

 

145

 

 

 

 

 

Table 4-11 (continued)
30 - 93%RH
Specimen Elastic Swelling Deformation Free
ID Strain sum Stress 0' Modulus E' Expansion at can”! a
(In/In) (96') (psi) (in/in)
1 0.001929 441.27 2287701 0.008766 22.00%
2 0.001914 466.38 2436424 0.008747 21.88%
3 0.001839 350.58 190616.9 0.009900 18.58%
4 0.001907 415.45 2178695 0.008491 22.46%
5 0.001748 358.81 2053089 0.009223 18.95%
6 0.001742 416.15 2388655 0.009370 18.59%
AVG 0.001846 408.11 220846.6 0.009083 20.41%
STD 0.000085 45.51 203535 0.000516 0.019
C.V. 4.58% 11.15% 9.22% 5.69% 9.22%
50 - 93%RH
Specimen Elastic Swelling Deformation Free
ID Strain 3cm: Stress 0' Modulus E' Expansion (1 eem/ a
(in/in) (psi) (98') (in/in)
1 0.001395 236.75 1697403 0.007509 18.57%
2 0.001518 259.57 1709655 0.006126 24.78%
3 0.001317 228.18 1732125 0006340 20.78%
4 0.001523 253.69 166551.4 0.007147 21.31%
5 0.001321 259.91 1968080 0.007707 17.14%
6 0.001693 273.05 1613035 0.007772 21.78%
AVG 0.001461 251.86 1730968 0.007100 20.73%
STD 0.000145 16.53 12328.6 0.000709 0.027
C.V. 9.95% 6.56% 7.12% 9.99% 12.84%

 

' ‘1

146

Table 4-12 Estimated warping of MDF/PB laminated beams over 30-inch span
using elastic approach ~

 

Condition Layer Thickness Modulus Expansion EsLWarp Meas.Warp

 

 

 

 

Level ID (in) (psi) (in/in) (in) (in)
MDF 0.25 430052 0.002459

30 - 81%RH 0.9659 0.81
PB 0.25 123175 0.005675
MDF 0.25 430052 0.001810

50 - 81%RH 0.7679 0.70
PB , 0.25 123175 0.004338
MDF 0.25 231144 0.003553

30 - 93%RH 1.7709 1.66
PB 0.25 94131 0.009083
MDF 0.25 231144 0.002583

50 - 93%RH 1.4465 1.33
PB 0.25 94131 0.007100

 

1) Using MOEtension. a for MDF; MOEcomp., a for PB

147

Table 4-13 Estimated warping of MDF/PB laminated beams over 30-inch span
using inelastic approach by partial restraint concept”

 

LayerID Parameter 30-81%RH 30-81%RH 30-81%RH 30-81%RH

 

81cm (in/in) 0.000983 0.000686 0.000735 0.000653

E (psi) 430052 430052 231144 231144
MDF 01(in/in) 0.002495 0.001810 0.003553 0.002583

el 0.0008173 0.0006492 0.0026788 0.00171112

8 ’ Mm 0.000322152 0.000246124 0.0005538 0.0004324

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suntan/1n) 0.001454 0.001132 0.001846 0.001461

E' (psi) 228889 215956 220846 173097
PB 01(in/in) 0.005675 0.004338 0.009083 0.007100
02 000236273 0.0018788 0.0028512 000280588
8 ' 2,1,3; 0000605281 000049013 0000579624 000057745
Estimated Warping (in) 0.1648 0.1442 0.4474 0.3335
Measured Warping (in) 0.81 0.70 1.66 1.33

 

1) Described in Section 2-5. 8 ’1“, e’ M, eland e; are calculated by Equation 2-47 to 2-50; substitute negative
8' 1m and positive 8 ' 2111.: as the expansion values in warping equation 2-13.

148

Table 4-14 Estimated warping of MDF/PB laminated beams over 30-inch span
using inelastic approach

 

Condition Layer

Thickness Deformation Restrained Est.Warp Meas.Warp

 

 

 

 

Modulus Swelling

Level ID (in) (psi) (in/in) (in) (in)
MDF 0.25 262404 0.000983

30 - 81%RH 0.1588 0.81
PB 0.25 228889 0.001454

MDF 0.25 329848 0.000686 .

50 - 81%RH 0.1488 0.70
PB 0.25 215956 0.001132
MDF 0.25 344430 0.000735

30 - 93%RH 0.3703 1.66
PB 0.25 220847 0.001846
MDF 0.25 307977 0.000653

50 - 93%RH 0.2670 1.33
PB 0.25 173097 0.001461

 

1) Using elastic strain and deformation modulus for both MDF and PB

 

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155

4.4 Application of Results to Plywood

It is clear from the results that the warping of cross-laminated solid
wood cannot be predicted with a reasonable degree of accuracy by applying
elastic beam equation. It can, however, be approximated remarkably well by
either the modified elastic beam equation or by the visco-elastic approach. Of
the two, the modified elastic approach is by far the easier one to use. Its
success derives from the fact that a solid wood lamina can in the longitudinal
direction be treated as an elastic material (high MOE and very small a)
This results in almost complete expansion restraint in the transverse direction
of adjacent layers (low MOE, large a ). The model illustrated in Figure 2-15
is therefore applicable. Its application, however, requires the experimental
determination of the deformation modulus and of the elastic strain
component. There is evidence, however, that the elastic strain component
may be a constant fraction of the free expansion for many species (see
Keylwerth [21]). Also the deformation modulus is not drastically different
from the statically determined MOE so that fairy reasonable results could
possibly be obtained using inputs readily available for many species from the
Wood Handbook [10].

It should be noted here, however, that these discussion and predictions
are limited to the condition of increasing moisture contents and to one species
(yellow-poplar).

Figure 4—8 illustrates the relationship between the warp measured on
and calculated for narrow laminated beams and the warp of a square plate of

the same construction.

156

In the following, the elastic and inelastic approximating of cross
laminated solid wood is applied to the beams tested and to conventional
plywood of various construction with the additional feature of including in
the inputs the natural variation of these properties (see [47]).

The generated computer program creates random variables for each
inputs controlled by arbitrary coefficients of variation. 200 runs were
employed for each of the beam and plywood construction. The results are
shown in the form of normal distribution curves of the warping, indicating
the effect on warping of the natural variability of the input variables (Figure
4—9 to 4-14). .

The computer model for the two yellow-poplar plywood sheets shown
are average warp of zero. This is because the plywood is structurally
balanced. The warp on each side of the average is due to natural variation in
the input variable. (Figure 4—15 and 4-16).

The plywood sheets are constructed of 0.125-inch thick veneer
sheets. Inputs and their assumed variations are shown in Table 4-20 and 4—21.

Also shown in Figure 4-15 and 4-16 is the effect of assuming property
variability only in the faces and constant average properties in the core layers
veneers and assumed property variability in all layers.

It is apparent that any variation of properties in the core layers has a

negligible effect on the computed warping.

157

4.5 Application of Results to Wood Composites

The results clearly indicate that both MDF and PB behave like elastic .
materials. The reasons for this unexpected behavior are not clear.

It seems justified, therefore, to apply the elastic beam equation to
materials consisting of or containing PB or MDF as substrates.

This is supported by experimental and theoretical results of warping
study involving particleboard substrates and veneer or plastic overlays ([42]
and [44]).

158

 

 

 

 

 

Thble 4-18 Random run inputs for yellow-paplar beams at 66 - 81%RH interval
Elastic Approach
Thickness Modulus Expansion
(in) (psi) (in/in)
ID AVG C.V. AVG C.V. AVG C.V
L - T
L: 0.257 5% 1181332 6.19% 0.00013 8%
T: 0.257 5% 39109 6.41 % 0.01492 8%
T - L - T
T: 0.0643 5% 39109 6.41 % 0.01492 8%
L: 0.257 5% 1181332 6.19% 0.(X)013 8%
T: 0.257 5% 39109 6.41% 0.01492 8%
L - T - L
L: 0.257 5% 1181332 6.19% 0.(X)013 8%
T: 0.257 5% 39109 6.41 % 0.01492 8%
L: 0.0643 5% 1181332 6.19% 0.00013 8%
Inelastic Approach
Thickness Modulus Expansion
(in) (psi) (in/in)
ID AVG C.V. AVG C.V. AVG C.V
L - T
L: 0.257 5% 1181332 6.19% 0.00013 8%
T: 0.257 5% 42628 5.99% 0.003453 9.49%
T - L - T
T: 0.0643 5% 42628 5.99% 0.003453 9.49%
L: 0.257 5% 1181332 6.19% 0.00013 8%
T: 0.257 5% 42628 5.99% 0.003453 9.49%
L - T - L
L: 0.257 5% 1181332 6.19% 0.00013 8%
T: 0.257 5% 42628 5.99% 0.003453 9.49%
L: 0.0643 5% 1181332 6.19% 0.00013 8%

 

“I

159

 

 

 

 

 

Table 4.19 Random run inputs for yellOw-poplar beams at 66 - 93%RH interval
Elastic Approach
Thickness Modulus Expansion
(in) (Psi) (in/in)
ID . AVG C.V. AVG C.V. AVG C.V
L - T
L: 0.257 5% 1093215 6.49% 0.00025 8%
T: 0.257 5% 30031 14.31% 0.02967 8%
T-L-T
'1‘: 0.0643 5% 30031 14.31% 0.02967 8%
L: 0.257 5% 1093215 6.49% 0.(I)025 8%
T: 0.257 5% 30031 14.31% 0.02967 8%
L-T-L
L: 0.257 5% 1093215 6.49% 0.00025 8%
'1‘. 0.257 5% 30031 14.31% 0.02967 8%
L: 0.0643 5% 1093215 6.49% 0.00025 8%
Inelastic Approach
Thickness Modulus Expansion
(pSi) (in/in)
ID AVG C.V. AVG C.V. AVG C.V
L - T
L: 0.257 5% 1093215 6.49% 0.00025 8%
T: 0.257 5% 44217 3.91% 0.007719 8.93%
T-L-T
T: 0.0643 5% 44217 3.91% 0.007719 8.93%
L: 0.257 5% 1093215 6.49% 0.00025 8%
'1‘: 0.257 5% 44217 3.91% 0.007719 8.93%
L-T-L
L: 0.257 5% 1093215 6.49% 0.00025 8%
'I‘. 0.257 5% 44217 3.91% 0.007719 8.93%
L: 0.0643 5% 1093215 6.49% 0.00025 8%

 

160

Table 4-20 Random run inputs for yellow-poplar 3-ply balanced beams using
inelastic approach

 

 

 

 

 

66 - 81 % RH
Thickness Modulus Expansion
(in) (Psi) (in/in)
1]) AVG C.V. AVG C.V. AVG C.V
All layers have variability:
L: 0.125 5% 1181332 6.19% 0.00013 8%
'1‘. 0.125 5% 42628 5.99% 0003453 9.49%
L: 0.125 5% 1181332 6.19% 0.00013 8%
Only face layers have variability:
L: 0.125 5% 1181332 6.19% 0.00013 8%
'I‘. 0.125 0% 42628 0.00% 0.003453 0%
L: 0.125 5% 1181332 6.19% ' 0.00013 8%
66 - 93 % RH
Thickness Modulus Expansion
(in) (psi) (in/in)
ID AVG C.V. AVG C.V. AVG C.V
All layers have variability:
L: 0.125 5% 1093215 6.49% 0.00025 8%
'I‘. 0.125 5% 44217 3.91% 0.007719 8.93%
L: 0.125 5% 1093215 6.49% 0.00025 8%
Only face layers have variability:
L: 0.125 5% 1093215 6.49% 0.00025 8%
'1‘. 0.125 0% 44217 0.00% 0.007719 0%

L: 0.125 5% 1093215 6.49% 0.00025 8%

 

161

 

 

 

 

 

Table 4-21 Random run inputs for yellow-poplar 5-ply balanced beams using
inelastic approach
66 - 81%RH
Thickness Modulus Expansion
(in) (psi) (in/in)

ID AVG C.V. AVG C.V. AVG C.V

All layers have variability:
L: 0.125 5% 1181332 6.19% 0.00013 8%
'1‘. 0.125 5% 42628 5.99% 0.003453 9.49%
L: 0.125 5% 1181332 6.19% 0.00013 8%
T: 0.125 5% 42628 5.99% 0.003453 9.49%
L: 0.125 5% 1181332 6.19% 0.00013 8%

Only face layers have variability: ,
L: 0.125 5% 1181332 6.19% 0.00013 . 8%
T: 0.125 0% 42628 0. 00% 0.003453 0%
L: 0.125 0% 1181332 0. 00% 0.00013 0%
“I: 0.125 0% 42628 0. 00% 0.00345 3 0%
L: 0.125 5% 1181332 6.19% 0.00013 - 8%

66 - 93%RH
Thickness Modulus Expansion
(in) (Psi) (in/in)

ID AVG C.V. AVG C.V. AVG C.V

All layers have variability:
L: 0.125 5% 1093215 6.49% 0.00025 8%
1‘: 0.125 5% 44217 3.91% 0.007719 8.93%
L: 0.125 5% 1093215 6.49% 0.00025 8%
'I‘. 0.125 5% 44217 3.91% 0.007719 8.93%
L: 0.125 5% 1093215 6.49% 0.00025 8%

Only face layers have variability:
L. 0.125 5% 1093215 6.49% 0.00025 8%
’I‘. 0.125 0% 44217 0.00% 0.007719 0%
L: 0.125 0% 1093215 0.00% 0.00025 0%
1‘. 0.125 0% 44217 0.00% 0.007719 0%
L: 0.125 5% 1093215 6.49% 0.00025 8%

 

162

 

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Warping of T-L-T beam

Figure 4-8 Implication to the warping of plywood

163

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—— Var in All 66 - 81%RH
—- Var in Faces

 

 

 

 

r 1 - l
-O 2 0.2
66 - 93%RH
r I l I
--O.2 -O.1 0.0 0.1 0.2

Warping (//) of S—ply Balanced Beam

Figure 4-15 Estimated warping (in.) of three-ply balanced yellow-poplar
beams with 48-inch span by introducing the variability on all

layers and face layers.

170

 

 

 

 

 

—— Var in All 66 - 81%RH
—- Var in Faces .-
l
r r 1 l
—0 2 0.2
66 - 93%RH
/\
/
r r 1 1
-0.2 -0.1 0.0 0.1 0.2

Warping (//) of 5—ply Balanced Beam

Figure 4-16 Estimated warping (in.) of five-ply balanced yellow-paplar
beams with 48-inch span by introducing the variability on all

layers and face layers.

“l-

CHAPTER V
SUMMARY AND CONCLUSION

All hygroscopic warping of panel materials made from solid wood or
wood composites is due to the imbalance of ensuing swelling stresses over
the cross section of the panel. This stress imbalance in turn is the result of an
unsymmetrical distribution over the cross section of the combination of
hygroscopic expansion characteristics and of the stiffness (modulus of
elasticity). In the case of plywood, this distribution is discrete and can readily
be investigated by tests described in this study. In the case of wood
composites, the distribution of these characteristics is more gradual and less
easy determined.

Given the above characteristics: hygroscopic expansion, modulus of
elasticity and their spatial distribution over the cross section, the Warp upon
moisture content change can be calculated. The accuracy of the predicted
warp is dependent to a large extent on how well the materials in question
follows the assumption made by the theoretical analysis or on how well the
theoretical treatment can be modified to accommodate the specific
characteristics of the materials involved.

This study used essentially two approaches for the prediction of the

warping of composite beams, namely the application of the standard equation

171

172

for the deformation of an elastic laminated beam including some
modification to accommodate the less than elastic behavior of solid wood
across the grain and a visco elastic approach which accounts for the time
dependence of the mechanical characteristics of the materials involved.

The elastic equation is discussed in some detail in Appendix A. It is
shown there that the warping of a two component beam, which could have
more than two layers is proportional to the difference between the expansion
value of the two components ( a2 —a1 ) and is a nonlinear function of the
moduli ratio ( til/1'52 ).

The equation assumes that any restraint imposed by one layer on the
hygroscopic expansion of another layer is transfdrmed into elastic strain.

Our experiments show that this is not the case for solid wood in the
cross grain direction.

These tests yielded results that allowed appropriate modification of the
inputs to the above equation (inelastic approach), which greatly improved the
accuracy of the predicted warp. The important modification is the reduction
of the free hygroscopic expansion to about 25% of its measured value, which
represents the elastic portion of the total free expansion, i.e., that portion that
converts into elastic strain. The other less important modification is the
calculation of a “deformation modulus”, which, however, does not differ too
much from the measured modulus of elasticity across the grain.

These results present the very attractive possibility of calculating the
warp of plywood from listed values 5,, and E i , and 1/4 of the free
expansion. These values would, of course, have to be adjusted for a specific

moisture content interval. But even if it was necessary to measure the free

 

173

expansion, it would be an easy test to perform. In the grain direction
(longitudinal), the elastic assumption are valid and no modification are
necessary.

The visco elastic approach in the case of solid wood laminates yielded
very good results, but the generation of the necessary inputs is complicated
and time consuming.

In the case of laminated composition boards, MDF and PB, the results
were unexpected. It turned out that the beam equation fitted the warping
problem very well without modification. Modification based on experiments
resulted in considerable error of the estimates. This means that these
materials act like elastic materials or more like solid wood in the grain
direction (longitudinal).

This is supported by research on overlaid particleboard [42].

The following speculative explanation is offered. MDF and PB have
far smaller expansion value than solid wood across the grain. Also the
difference in their expansion values are very small so that the mutual restraint
is relatively minor and the resulting stresses relatively small.

Further investigation of these phenomena would be desirable.

In any event, the warping of such laminate can well be calculated using
standard inputs into the elastic beam equation.

In conclusion, the experiments appear to support the following
statements:

—— The uniaxial restraining of swelling stresses of wood across
the grain is a viscoelastic process. Only a fraction of the total

swelling restraint can be converted into elastic strain.

174

— The elastic portion of the free expansion across the grain of
solid wood is about 25%.
— Along the grain, solid wood can be assumed to behave
elastically under conditions prevailing in a laminate subjected to
moisture content increase.
—— Laminates consisting of composite boards behave
“elastically” and their reaction to moisture content increase can be
described by standard beam equations.

These results and conclusions will allow the formulation of relatively

simple guidelines for the mathematical modeling of wood based laminates.

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Suchsland, O. 1965. Swelling stresses and swelling deformation in
hardboard. Michgan Agriculture Experiment Station. Michgan State
University, E. Lansing. Quarterly Bulletin 47(4): 591-605.

181

LIST OF REFERENCES (continued)

52.

53.

54.

55.

56.

57.

Timoshenko, S. P. and J. N. Goodier. 1951. Theory of Elasticity 2nd. ed.
McGraw-I-Iill, Inc. New York, NY.

Timoshenko, S. P. 1925. Analysis of bi-metal thermostats. J. of Optic.
Soc. Amer.: 235-255.

Tong, Y. and O. Suchsland. 1993. Application of finite element analysis
to panel warping. Holz gig Roh- uno Workstoff (51): 55-57.

Tsai, S. W. and T. H. Hahn. 1980. W.
Techromic Pub.

Whitney, J. M. 1969. Analysis of heterogeneous anisotropic plates,
WV01 36 PP 251 266

Xu, H. 1993. Application of a linear visco-elastic plate theory on
hygroscopic warping of laminates. Ph. D. dissertation, .Department of
Forestry, Michigan State University, E. Lansing, MI.

APPENDIX A

APPENDIX A
ON WARPIN G EQUATION FOR TWO—FLY BEAM

For a two-ply laminated beam, it can be derived that the center
deflection W is proportional to the difierence of the expansion value

between the two layers. In fact, by letting

A1

51(51- 5%)

A2 = 52(Sg-Sf)
(A-l)

51
ll

EITI

B2

5272

Equation (2-13) leads to the center deflection of the beam over length

.aIA1+azA2-a181+a232
2 _ L2 A1+A2 Bl-HB2
' T

 

E
u
lb

 

2 2
2251(S?-S?-1) 25.-($412.1)
1 _ 1

on
a

(A-Z)

 

 

2 2
3215,.(32- s?_ l) 2212,13.
1 1

182

 

183

The numerator in the above equation

A1+A2 31w, " (A1+A2) (1914-32)

 

 

 

= (518261-53)Tz-ElEzwi-Sifll) (a 412) (A-3)

2 2 1
mag—saggy,
1 1

 

Therefore, the difference of the expansion value between the two

layers is proportional to the center deflection, that is

2
W £85: (onl - a2) (A‘4)

where C is a constant for given dimension and moduli of the materials

 

 

 

 

 

determinedby
2 2 2 2
C_ K 2 Elm.finalize-SOT: (A-S)

22E,(S?-S?_1) 25.-(5:2—512-1) 2 2

1

2 ' 1 2 251(51'2‘Siz-1)ZEiTi
K3212,.(sf-s§_1) 2251R- )1 1

1 1

It can be shown that W is also a function of 51/ E2 ,

184

reference thickness, say T2.

By letting T2 = T and T1 = cT2 = cT

one get

s0 =0

3:
ll

sg-sf

— 11+ 1'2: (c+l)T

c2T2

(2c+1)T2

c3T3

(3c2+3c+1)7"

Tl/T2 and a

(A-6)

185

The center deflection may be expressed as

0115ch +0125, (2c+ 1) 7d alElcT+ 0:21:27

 

 

 

 

 

 

 

 

 

w _ L2 _ L2 Elczrz+Ez(2c+1)1Q - 516T+EzT
- B—R - 72(Elc31'3+52(3c2+3c+1)1°) 1'21c21¢+152(2c+1)r2
.3(E1c2T2+Ez(2c+l)T2) 2051674527) -
alflc2+a2(2c+1) alfic+a2
= E: Bc2+(2c+l) BC+1 (A-7)
8T2(Bc3+ (3c2+3c+ 1)) _'Bc2+ (2c+ 1)
3(Bc2+(2c+1)) 2(1364'1)
01'
D D -

w = 7‘ = 7‘; (A 8)

2

(al-a2)L
w = f(E1/E2,T1/T2) T2 (A-9)

where _D is a function of B( = IE‘I/E2 ) and c ( = Tl/Tz ). and is a
constant for given materials.

The relationship between W and 151/E2 , a -a canbe extended

1 2
to any two-component multi-layer laminated beams.
The effect of 195/152 and Tl/T2 on W is not linear. Certain

combination of these two would leads to maximum warping. In order to find

186

out 51/52 ratio at a point when warping reaches to the maximum for

certain Tl/T2 , rewrite Equation (A-7) as

W _ 6L2 (Bc+1)(ach2+a2(2c+l))-(Bc2+ (2c+1)) (ach+a2)

8T4(Bc+1)(BcI3-1-7(3r:2-1-3c+1))-3(Bc2'l-(2fv"l'1))(562"‘(2‘v"*'1))

 

= 3_,_2(13c+1) (a18c2+a2(2c+1))- (Bc2+2c+l) (alflc+a2)

4T

 

4(Bc+l) (Bc3+3c2+3c+l) -3(Bc2+2c-1-1)2

3L2(a2-al)c(1+c) 13
= 4r 2

 

4(Bc-l-l) (Bc3+3c -l-3c-1-l)--3(Bc2+2c+l)2

To get a B value when warping reaches the extreme, consider T , c
be constants. Take the first partial derivative of W respect to B , and set to

zero, solve for B ,

g-g, = O (A'IO)

Only the numerator needs to be zero. Therefore, we have

4(Bc+1) (Bc3+3c2+3c+1) -3(B€2+2c+1)2

-B[4c(nc3+3c2+3c+1)+4c3(Bc+1)-6c2(Bc2+2c+1)] = o

187

After manipulating, we get

4(Bc3+3c2+3c+ 1) -4Bc3 (Bc+ 1) + (Bc2+2c+ 1) (35c2—6c25-3) :0

This can be simplified as

l-Bzc‘ = 0 (A-ll)

or take the positive root, when B = -13, warping reaches the maximum. In
C

other words,

 

= (:2 )2 (A-12)

When Tl/T2 = l , the thickness of both layers are the same, the
maximum warping occurs if El/E2 reaches 1.

Equation (A-12) implies that one should avoid to use a til/IF:2 value
which leads to maximum warping if desired. For instance, if the structure
requires Tl/T2 = l , the maximum warping occurs when a value of

3
131/152 = 9 . Either El/Ez>9 or El/E2<9 would reduce the

warp.

APPENDIX B

 

00000000

200
10

210
220
230

240

APPENDIX B
COMPUTER PROGRAM OF
VISCO-ELASTIC BEAM THEORY

PROGRAM NONLWARP

*************************************************

* AK, BK, CK, DK, AFK, BTK ARE CONSTANTS FROM

* FREE HYGROSCOPE EXPANSION REGRESSION: *
* EPH = A + B * T ** AF

* Y = C + D * EXP(BT * T)

* K USED FOR DIFFERENT LAYERS

*

*

'k

*

*************************************************

IMPLICIT REAL*8 (A-H,O-Z)

DIMENSION AK(2),BK(2),CK(2),DK(2),AFK(2),BTK(2),
H(2),S(3)

DIMENSION ST(2),F(2),E(2).SE(2),SN(2,2),SNI(2,2)
DIMENSION EP(100),RK(IOO).EPHO(2)

OPEN (UNIT = 1, FILE = ’input',STATUS = ’OLD’)°
OPEN (UNIT = 2, FILE = ’output',STATUS = 'NEW')
++++++++INITIALIZE++++++++++

WRITE (2,230)

DO 10 I = 1,2

READ (1,200) AK(I),BK(I),CK(I),DK(I),AFK(I),
BTK(I),H(I)

WRITE (2,240) AK(I),BK(I),CK(I),DK(I),AFK(I),
BTK(I),H(I)

FORMAT(' ’,7F15.12)

CONTINUE

READ (1,210) AL,DT

WRITE (2,220) AL,DT

FORMAT(’ ’,2F10.5)

FORMAT(//’ ’,'SPAN LENGTH: ’,F7.1,//’ ’,

'DT = ',F7.3)

FORMAT(’ ’,SX,’AK(I)',7X,'BK(I)',7X,’CK(I)’,7X.
’DK(I)',5X,'AFK(I)’,5X,’BTK(I)’,5X,’H(I)'/)
FORMAT(/’ ',2F12.9,2F12.1,2F10.6,F10.2)

188

20

189

8(1) = 0.

8(2) = H(l)

5(3) = H(1) + H(2)
EP(1) = 0.

RK(1) = 0.

RKI = 0.

TT = 168.

DO 5 K = 1,2
EPHO(K) = 0.
++++++++CALCULATE+++++++++++
NT = INT(TT/DT)
DO 80 M = 1, NT
DO 20 K = 1, 2
E(K) = 0.

F(K) = 0.
CONTINUE

DO 70 N = 1, M
AYT =

E U
K

a a

I n n
c>c>c>c>o

T = N * DT

Tl (N-l) * DT
T2 = T - T1
DO 50 K = l, 2
A

B

C

D

AK(K)

BK(K)

CK(K)

DK(K)

= AFK(K)

T = BTK(K)

IF (N.EQ.1) THEN
EPHI = EPHO(K)

mu:
m

ELSE

EPHI = A + B * Tl ** AF

END IF

EPH = A + B * T ** AF

DEP = (EPH - EPHI)/DT

Y = C + D * EXP(BT * T2)

AY = Y * (S(K+1) - S(K))

BY = Y * (S(K+1)**2 - S(K)**2)/2
DY = Y * (S(K+1)**3 - S(K)**3)/3
HN = Y * (S(K+l) - S(K)) * DEP
HM = Y * (S(K+l)**2 - S(K)**2)/2 * DEP

AYT = AYT + AY

50

100

70

6O

80

250

10

190

BYT = BYT + BY
DYT = DYT + DY
HNT = HNT + HN
HMT = HMT + HM
CONTINUE

IF (N.EQ.M) GO TO 100

E(1) = E(1) + AYT * EP(N) + BYT * RK(N)
E(2) = E(2) + BYT * EP(N) + DYT * RK(N)
F(l) = F(1) + HNT

F(2) = F(2) + HMT

CONTINUE

DO 60 I = 1, 2
ST(I) = F(I) - E(I)

SN(111) = AYT
SN(1,2) = BYT
SN(2,1) = BYT
SN(2,2) = DYT

CALL INVERSE (SN,SNI)

CALL MULTIPL (SNI,ST,SE)

EP(M) = SE(1)

RK(M) = SE(2)

RKI = RKI + RK(M) * DT

EPI = EPI + EP(M) * DT
CONTINUE

W = RKI * AL**2 / 8.

WRITE (2,250) W,EPI

FORMAT (//' ’,’WARPING OVER 30 INCH: ’,F7.4,//
’ ’,’NORMAL EXPANSION: ’,F9.7)
END

SUBROUTINE INVERSE(SN,SNI)
DIMENSION SN(2,2),SNI(2,2)
DO 10 I = l, 2

DO 10 J = 1, 2

SNI(I,J) = 0.
CONTINUE
DETM = SN(l,l)*SN(2,2)-SN(1,2)*SN(2,1)

SNI(1,1) = SN(2,2)/DETM '
SNI(2,2) = SN(1,1)/DETM

SNI(1,2) = - SN(1,2)/DETM

SNI(2,1) = - SN(2,1)/DETM

RETURN

END

10

20

191

SUBROUTINE MULTIPL(AM,BM,CM)
DIMENSION AM(2,2),BM(2).CM(2)
DO 10 I 1, 2

CM(I)
DO 20
DO 20
CM(I)
CONTINUE
RETURN
END

ll 0"

1, 2
= l, 2
CM(I) + AM(I,J) * BM(J)

llC-JHII

 

APPENDIX C

000000

0000

10

+ + + +

APPENDIX C
COMPUTER PROGRAM OF
WARPRAND

PROGRAM WARPRAND
**************************************************
THE PROGRAM CALCULATES THE WARPING OF A.MULTILAYER
BEAM

USING ELASTIC THEORY WITH DEFINED MEAN AND COEFF. OF
VAR.
**************************************************
PARAMETER (LTEMP = 10, NRAND = 1000)

DIMENSION MOE(LTEMP),THICK(LTEMP),EXPAN(LTEMP)I
CVMOE(LTEMP),CVTHICK(LTEMP),CVEXPAN(LTEMP),
STDMOE(LTEMP),STDTHICK(LTEMP),STDEXPAN(LTEMP),
MOER(NRAND,LTEMP).THICKR(NRAND,LTEMP)a
EXPANR(NRAND,LTEMP), + WARP(NRAND

REAL*8 MOE, MDER
OPEN(UNIT = 1, FILE
OPEN(UNIT = 2, FILE

'input', STATUS = ’OLD')
'output', STATUS = 'OLD’)

INPUT NO. OF LAYERS, LENGTH OF THE BEAM (IN), NO. OF
RANDOM RUN

READ (1, *) NLAYER, ALENGTH, NR
WRITE (2, 1) NLAYER, ALENGTH, NR

DO 10 I = 1, NLAYER
MOE(I) = 0.
THICK(I)
EXPAN(I)
CVMOE (I)
CVTHICK(I)
CVEXPAN(I)
STDMOE(I) =
STDTHICK(I)
STDEXPAN(I)
CONTINUE

II"
"11000
IIIO --
..o
c>c> -

192

20

30

40

193

INPUT MOE(PSI), THICK(IN), EXPAN(IN/IN), CV(%) FOR
EACH LAYER

DO 20 I = 1, NLAYER

READ (1, *) MOE(I), CVMDE(I), THICK(I), CVTHICK(I),
EXPAN(I), CVEXPAN(I)

STDMOE(I) = CVMDE(I) * MOE(I) / 100.

STDTHICK(I) = CVTHICK(I) * THICK(I) / 100.
STDEXPAN(I) = CVEXPAN(I) * EXPAN(I) / 100.

WRITE (2, S) MOE(I), THICK(I), EXPAN(I),

CVMOE(I), CVTHICK(I), CVEXPAN(I).

STDMOE(I), STDTHICK(I), STDEXPAN(I)

CONTINUE

DO 40 N = 1, NR

S = 0
SS =
SUMI
SUMJ
SUMK
SUML
SUMM
DO 30 I = 1, NLAYER

R1 = RAND(ARGUMENT) - 0.5

MOER(N,I) = R1 * STDMOE(I) / 0.258 + MOE(I)

R2 = RAND(ARGUMENT) - 0.5

THICKR(N,I) = R2 * STDTHICK(I) / 0.258 + THICK(I)
R3 = RAND(ARGUMENT) - 0.5

EXPANR(N,I) = R3 * STDEXPAN(I) / 0.258 + EXPAN(I)
SS = S

S = S + THICKR(N, I)

D = MOER(N,I) * (S**3 - SS**3)

E = MOER(N,I) * (S**2 - SS**2)

F = MOER(N,I) * THICKR(N,I)

G

H

S

II "C-

II
00000

= MOER(N,I) * EXPANR(N,I) * (S**2 - SS**2)
= MOER(N,I) * EXPANR(N,I) * THICKR(N,I)

UMI = SUMI + D
SUMJ = SUMJ + E
SUMK=SUMK+F
SUML = SUML + G
SUMM = SUMM + H
CONTINUE
RADIUS = ((2*SUMI)/(3*SUMJ)-SUMJ/(2*SUMK))/
((SUML/SUMJ) - (SUMM/SUMK))
WARP(N) = ALENGTH ** 2 / (8 * RADIUS)
CONTINUE
WRITE (2,6)

DOSON=11NR

194

WRITE (2,*) (MOER(N,I), I = 1,NLAYER)
WRITE (2,7)

DO 60 N = 1, NR

WRITE (2,*) (THICKR(N,I), I = 1,NLAYER)
WRITE (2,8)

DO '70 N = 1, NR

WRITE (2,*) (EXPANR(N,I), I = 1,NLAYER)

WRITE (2.9)

D080N=1,NR

WRITE (2,*) WARP(N)

FORMAT (' ','NO. OF LAYER: ’,IS/’LENGTH OF BEAMz’,
FS.0/’NO. OF RANDOM RUNS: ',15/)

FORMAT (' ’,15X,’MOE(I)',SX,’THICKNESS(I)',SX,
'EXPANSION(I)’,/2X,’MEAN: ',F13.2,F13.3,3X,F13.6
/2X,’C.V.(%):’,2F13.3,F16.3/2X,'STD: '. 2F13.3.
3x, F13.6/)

FORMAT (' ’,/12X,'MOE (1)',12X,'MOE (2)',12x,
'MDE (3)'/)

FORMAT (' ’,/9X,'THICK(1)',9X,'THICK(2)’,
9X,'THICK(3)'/)

FORMAT (' ’,/9X,'EXPAN(1)’,9X,'EXPAN(2)',9X,
'EXPAN(3)’/)

FORMAT (' './5x,'WARP'/)

END