. 1‘ .13....
«a?

 

 

 

 

 

 

:-
fixtvof.‘
.I! 2

3:. .
:99...

 

ix"
\
b

I

, . Fly!
912.-....

 

\ifinfi. (flat r . .r .81... ,wvwwtuf

3i ‘ . . ,
I

        

.Ewfihtweg‘qflbmfi. V, . . , H 2;... .
IPI | 1 x

 

1.45.515

This is to certify that the

dissertation entitled

Application of a Linear Viscoelastic Plate Theory
to Hygroscopic Warping of Laminates .\ .~.‘

presented by

 

Hong Xu

has been accepted towards fulfillment
of the requirements for

Ph. D. Forestry
degree in

 

 

am

'Major professor

Date Augusgl 993

MSU is an Affirmative Action/Equal Opportunity Institution 0-12771

 

M CHIGAN STATE UNIVERSITY LIBRARIES

lilililiiliWillWNWWilli\lliiil

3 1293 01025 357

 

LIBRARY
Michigan State
Unlverslty

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

APPLIC

APPLICATION OF A LINEAR VISCOELASTIC PLATE THEORY
TO HYGROSCOPIC WARPING OF LAMINATES

By

Hong Xu

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

DOCTOR OF PHILOSOPHY

Department of Forestry

1993

APPLI

A lineal
lamination T}
hiEiOSCOpic ma
equivalence Wa
equation and it
examined when
Yellow-Mm (9
”P were Com
high humidity c
radial direction
Characterizgd b1
mOdel is to m
VEco‘ilastic defi
was low to be
and the meme“
Component. nit
0Oefficienm of '

rfpreSEmed. Th

ABSTRACT

APPLICATION OF A LINEAR VISCOELASTIC PLATE THEORY
TO HYGROSCOPIC WARPING OF LAMINATES

By

Hong Xu

A linear viscoelastic plate theory (LVP) was formulated based on Classical
Lamination Theory (CLT) and the linear viscoelastic constitutive equation for
hygroscopic materials such as wood and wood-based materials. its solvable numerical
equivalence was arrived at by linear numerical integration of its integral governing
equation and its computation was automated on a desktop computer. Its validity was
examined when it was applied to the hygroscopic warping of a two-ply cross laminated
yellow-poplar (Liriodendron tulipifera) laminate and the theoretical predictions by the
LVP were compared with the measured warps suffered by the laminate subjected to a
high humidity environment. The viscoelastic creep properties of yellow-poplar in the
radial direction, needed as input in the application of the LVP, were tested and
characterized by a four-element Burger body. The non-Newtonian dashpot used in the
model is to account for the non-Newtonian behavior of the flow component of the
viscoelastic deformation. The general behavior of yellow-poplar in its radial direction
was found to be clearly nonlinearly viscoelastic, especially regarding the flow component
[and the recoverable component, while it is linearly elastic for the instantaneous elastic
component. The separate effects of moisture content, stress, and time on the four
coefficients of the four-element Burger body were investigated and mathematically

represented. The numerical form of the isothermal LVP theory was used to approximate

warping - a r
variations of it‘.
viscoelastic co
improved pred'
process satisfat
oi the warping
the mechanos<
and relaxation
into the relaxat
tfiort within [it

humidity mndl

warping - a nonlinear nonisothermal viscoelastic process - by accounting for the
variations of moisture content, stresses, and the four moisture content-stress dependent
viscoelastic coefficients in step-wise increments. The LVP theory resulted in much
improved predictions over its elastic counterpart. But, it failed to describe the warping
process satisfactorily in that it underestimated the drastic relaxation in the early stages
of the warping of the yellow-poplar laminate. Much of the error is very possibly due to
the mechano—sorptive effect which is known to cause far greater and more rapid creep
and relaxation and which is also prominent in warping and which was not incorporated
into the relaxation moduli input in the application. It is also demonstrated that any extra
effort within the elasticity realm in dealing with the warping problem in high and cyclic

humidity conditions will probably result in limited improvement.

Dedicated to my beloved Lan, Hope,
and my dear parents

The at
mentor. Dr.
encouragemer
each member

Specia.
for their undm

m this journe)

ACKNOWLEDGMENTS

The author wishes to express his sincere appreciation to his Ph.D. advisor and
mentor, Dr. Otto Suchsland, for his continued guidance, support, patience and
encouragement throughout the course of this study. Appreciation is also expressed to
each member of the committee for their counsels and recommendations.

Special thanks are extended to my dearest wife, Lan, and lovely daughter, Hope,
for their understanding, encouragement and warm companionship every step of the way

in this journey.

'—

 

TABLE OF C 05
LIST OF TABLE
LlST 0F HOUR
LIST 0F 8th B

CHAPTER
LINTRr

1.
l.

H.DEVE
FOR]

I“) {Q Is)

TABLE OF CONTENTS
LIST OF TABLES

LIST OF FIGURES
LIST OF SYMBOLS
CHAPTER

I. INTRODUCTION

TABLE OF CONTENTS

1.1 Background
1.2 Objectives of the Study

page
vi

ix

xiii

p—d

II. DEVELOPMENT OF A LINEAR VISCOELASTIC PLATE THEORY
FOR HYGROSCOPIC COMPOSITE LAMINATES

2.1 Introduction
2.2 Validity of the Classical Lamination Theory (CLT)
2.3 Introduction to the CLT '

2.3.1
2.3.2

2.3.3
2.3.4
2.3.5

Displacement Field and Strain Field
Elastic Lamina Constitutive Equations -
Stress-Strain Relations

Strain and Stress Variations in a Laminate
Resultant Laminate Forces and Moments
Hygroscopic Strains and Stress Analysis

2.4 Development of the Linear Viscoelastic Plate Theory (LVP)

2.4.1

2.4.2
2.4.3

2.4.4

Linear Viscoelastic Stress-Strain Relations
for Plane Stress

Linear Viscoelastic Governing Equation
Numericalization of Linear Viscoelastic
Governing Equation and Successive
Computation of Strains and Stresses
Linearity and Isothermal Requirements

vi

18
18

23
27
27
35
40

41
44

51
61

H1 CH

1V. APl

V- Sun

Compuu

2.4.5 Hygroscopic Strain Rates and Relaxation Moduli
llI. CHARACTERIZATION OF VISCOELASTIC BEHAVIOR

3.1 Introduction
3.2 Literature Reviews
3.2.1 The Nonlinearity - Relation Between
Viscoelastic Properties and Stress Levels
3.2.2 Relation Between Viscoelastic Properties and
Moisture Content, and Mechano—Sorptive Effect
3.2.3 Relation Between Viscoelastic Properties and
Temperature, and Thermal-Mechanical Coupling
3.2.4 Functional Forms and Linear Viscoelastic Models
3.2.4.1 Empirical Models
3.2.4.2 Linear Mechanical Models
3.2.4.3 Nonlinear Models
3.3 Creep Experimentation
3.3.1 Experimental Design
3.3.2 Experimental Results and Observations
3.4 Modelling Creep Behavior By Mechanical Model

IV. APPLICATION OF THE LVP THEORY TO HYGROSCOPIC
WARPING

4.1 The Process of Hygroscopic Warping of Wood
and Wood-Based Material Panels
4.2 Warping Experimentation
4.2.1 Panel Design and Experimentation
4.2.2 Warp, Moisture Content Gradient,
and Expansion Coefficients
4.2.3 Test Results
4.3 Application of the LVP Theory
4.3.1 Creep Compliances and Relaxation Moduli
4.3.2 Hygroscopic Strain Rates
4.3.3 Computer Programming of the Numerical
Form of the LVP Theory
4.3.4 Preparation of Inputs
4.4 Theoretical Predictions and Analysis

V. SUMMARY AND SUGGESTIONS ON FUTURE INVESTIGATIONS

APPENDIX

Computer Program of the LVP in Microsoft QuickBASIC

vii

62

63

63

65
68

72
74
75
76
83

107

123

123
131
132

136
142
150
150
163

165
166
166

171

176

REFERENCE

REFERENCES 2 15

viii

Table 1.

Table 2a.
Table 2b.
Table 3c_

Table 2d.

Table 3_

Table 4,
Table 5.
Table 6.
Table 7_
Table 3.

Treble 9.
Table to.

Table 11.

T.
Ye

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

1.

2a.

2b.

2c.

2d.

9.

Table 10.

Table 11.

LIST OF TABLES

Thermal and hygroscopic expansion coefficients for some wood
species [Forest Products Laboratory, 1987].

Maximum stresses and deflection in.3-ply laminate [Pagano, 1972].
Maximum stresses and deflection in 5-ply laminate [Pagano, 1972].
Maximum stresses and deflection in 7-ply laminate [Pagano, 1972].

Maximum stresses and deflection in 9-ply laminate [Pagano, 1972].

Relative humidity over saturated salt solutions.

Normalized creep compliance of yellow-poplar in the radial direction

at nine creep test conditions.

Instantaneous MOEs of yellow-poplar in the radial direction at nine

creep test conditions.

Composition of tension creep strain of yellow-poplar in the radial
direction.

Coefficients of normalized creep compliance of yellow—poplar in
the radial direction at nine creep test conditions.

Empirical functions of coefficients of normalized creep compliance
of yellow-poplar in the radial direction.

Measured moisture content gradients in the yellow-poplar laminate.

Static tension MOEs of yellow-poplar.

Theoretical predictions vs. measured vertical deflections of the
yellow-poplar laminate and beam.

ix

page

90

96

100

107

120

121

146

152

169

Figure

Figure

Figure

Figure

Figure

Figure

Figure
F1Euro

Figure

Figure 10.

Fig... 11.
Figure 12.

Figure 13.

Ix)

8.

9.

Figure

Figure

Figure

Figure

Figure

Figure

Figure
Figure

Figure

Figure 10.

Figure 11.
Figure 12.

Figure 13.

Figure 14.

8.

9.

LIST OF FIGURES

Flexural stress distribution for [+30°, ~30°] angle-ply laminate
at aspect ratio L/h = 10 [Lo, Christensen and Wu, 1977].

Flexural stress distribution for [+30°, -30°] angle-ply laminate
at aspect ratio L/h = 4 [L0, Christensen and Wu, 1977].

In-plane displacement for [+30°, -30°] angle-ply laminate
at aspect ratio L/h = 10 [L0, Christensen and Wu, 1977].

ln-plane displacement for [+30°, -30°] angle-ply laminate
at aspect ratio L/h = 4 [Lo, Christensen and Wu, 1977].

Flexural stress distribution for [0°, 90°, 0°] cross-ply laminate
at aspect ratio L/h = 10 [Lo, Christensen and Wu, 1977].

Flexural stress distribution for [0°, 90°, 0°] cross-ply laminate
at aspect ratio L/h = 4 [L0, Christensen and Wu, 1977].

Normal stress distribution in a laminate [Pagano, 1972].
Shear stress distributions in a laminate [Pagano, 1972].
In-plane displacement in a laminate [Pagano, 1972].

Convergence of exact elasticity solution to respective CLT
solution (Data of Pagano [1972]).

Geometry of deformation in the x-z plane [Jones, 1976].

Relation of material principle coordinates 1-2 with x-y coordinates.

Hypothetical variations of strain and stress across laminate
thickness [Jones, 1976].

Geometry of a n-layer laminate [Jones, 1976].

page

10

10

.11

ll

12

12

13

14

15

16

20

26

29

29

Figure 15.
Figure 16.
Figure 17.

Figure 18.

Figure 19.

Figure 20.

Figure 31.

Figure 22.
Iagure 23.
Figure 34a.

Figure 24b.

Figure 253

Figure 26.

Figure 27,

Figure 28.

FlEUre 29.

Flgtue 30

ln-
Ml
llll
Cc
Cl

Fc

Ct

Ct
en

Re
at

F].

Figure 15.
Figure 16.
Figure 17.
Figure 18.
Figure 19.
Figure 20.

Figure 21.

Figure 22.
Figure 23.
Figure 24a.

Figure 24b.

Figure 25a.
Figure 25b.

Figure 26.

Figure 27.

Figure 28.

Figure 29.

Figure 30.

In-plane forces on a flat laminate [Jones, 1976].

Moments on a flat laminate [Jones, 1976].

Illustration of linear finite difference approximation.
Coordinate positions of the yellow-poplar laminate and beam.
Classical linear viscoelastic models.

Four-element Burger body.

A typical creep curve [Bodig and Jane, 1982]:

(a) Load-time function
(b) Deformation-time function

Radial tension creep specimen of yellow-poplar.
Loading frame and extensometer assembly.
Complete test setup with flexible humidity chamber open.

Complete test setup with charged flexible humidity chamber
enclosing tension creep specimen.

Closeup of open flexible humidity chamber.
Closeup of closed charged flexible humidity chamber.

Normalized creep compliance 0f yellow-poplar in the radial
direction at nine creep test conditions.

Instantaneous MOEs of yellow-poplar in the radial direction.

Recovery creep strain of yellow-poplar in the radial direction
at nine creep test conditions.

Flow creep strain of yellow-poplar in the radial direction.

Components of normalized creep compliance of the four-element
Burger body.

xi

31

31

52

77

79

8O

86

87

88

89

91

101

103

105

108

110

Figure 31.
Figure 32.
Figure 33.

Figure 34.
Figure 35.
Figure 36.
1Figure 37.
Figure 38.
1Time 3921.
1Figure 39b.
Figure 40‘

Figure 41.

Figure 42.

Figure 44.
Figure 45.

Flgme 46

FL...)

In In

III

Figure 31. Obtaining retardation time by fitting single Kelvin element to
recovery creep strain. 117

Figure 32. Nonlinear regression on normalized creep compliance of yellow-poplar
in the radial direction at creep test condition 21. 118

Figure 33. Empirical function vs. experimental data of normalized creep

compliance of yellow-poplar in the radial direction. 122
Figure 34. Step-wise moisture content increase. 130
Figure 35. Manufacturing of edge grain yellow-poplar panel. 133
Figure 36. Specimen arrangement on the yellow-poplar laminate. 135
Figure 37. Static tension specimen of yellow-poplar. 137
Figure 38. Linear expansion specimen of yellow-poplar. 138
Figure 39a. Deflection measuring apparatus. 140
Figure 39b. Measuring apparatus on the warped yellow-poplar laminate. 141
Figure 40. Yellow-poplar laminate and beam in humidity chamber. 143

Figure 41. Theoretical predictions vs. measured vertical deflections of
the yellow-poplar laminate and beam. 145

Figure 42. Moisture content gradient development in the yellow-poplar laminate. 147

Figure 43a. Hygroscopic expansion coefficient of yellow-poplar in the
longitudinal direction. 148

Figure 43b. Hygroscopic expansion coefficient of yellow-poplar in the
radial direction. 149

Figure 44. Sorption isotherm of yellow-poplar based on linear expansion data. 150
Figure 45. Static tension MOEs of yellow-poplar. 153

Figure 46. Superposition scheme for flow compliance. 161

xii

Dar-1’)

E11! E22, Eb! En:

LIST OF SYMBOLS

coefficient in parabolic mechanical creep model

coefficient in the normalized creep compliance of a nonlinear
four-element Burger body

extensional stiffness
integrand related to extensional stiffness
constant in a logarithm mechanical creep model

coefficient in the normalized creep compliance of a nonlinear
four-element Burger body

coupling stiffness
integrand related to coupling stiffness

coefficient in the normalized creep compliance of a nonlinear
four-element Burger body

coefficient in the normalized creep compliance of a nonlinear
four-element Burger body

bending stiffness
integrand related to bending stiffness

respectively, Young’s modulus along the grain and across the
grain, Kelvin spring coefficient, and Maxwell spring coefficient

plane shear modulus

thickness of laminate
dummy indices on moduli, compliances, etc.

xiii

J(t), 1,04). 1
W

in). m)
M), 1M!)

i

U?)

J(t), 150-1), Jwfl-T)
1”“)

w), J,(t)
1M0. 1M!)
k

La)

m

MC, MCG
M, M, M”

MW». MW), M'a)
MW»

n

N

N, N, N”

N”(t), N”(t). N'a)

N”"(t)
1», pm
Q.» E
S

t

creep compliances

normalized creep compliance

respectively, recoverable and flow compliance
respectively, normalized recoverable and flow compliance
summation index on series

spectrum of retardation times

coefficient in parabolic mechanical model

respectively, moisture content and moisture content at which
mechanical property reaches a minimum

moments

respectively, mechanical, hygroscopic, and thermal moments
hygroscopic moment in reference to moisture content
number of layers in a laminate

number of Kelvin elements in a Kelvin chain

in-plane forces

respectively, mechanical, hygroscopic, and thermal in—plane
forces

hygroscopic in-plane force in reference to moisture content

respectively, creep load and mechanical load at MC moisture
content

respectively, reduced stiffness and transformed reduced stiffness
variable in Laplace domain

time

displacement in x-direction

xiv

“a

r

In

'9

I, j, 7,

m), ram)
Zr

"7-10

”'0
x, .V: 2

mt), Yaw-7)

£,(t)
EM!)

ea, e“

midplane displacement in x-direction
displacement in y-direction

midplane displacement in y-direction
midplane displacement in z-direction
cartesian coordinates

relaxation moduli

coordinate distance from reference surface (top surface of
laminate) to bottom of lamina

hygroscopic expansion coefficient

angular rotation of laminate cross section
mid-plane plane shear strain referred to x-y axes
transverse shear strains

instantaneous elastic strain

total strain

mid-plane normal strains referred to x-y axes
transverse normal strain

respectively, total strain, mechanical strain, and hygroscopic
strain

delayed elastic (recoverable) creep strain

normalized delayed elastic (recoverable) creep strain
hygroscopic strain in reference to moisture content
strains

XV

(,0)
till

lulu

en“)

e,(t)
’lw 11.4
0

‘x’ ‘7’ ‘17
v.0)
0"

an, 0",
all

0,, an
Ar

Ar,-

1

737

1,, 1,1

relative creep

viscous creep strain

respectively, Kelvin and Maxwell dashpot coefficient
angular orientation of a lamina

plate curvatures

Poisson’s ratio

time-wise constant creep stress
plane normal stresses

transverse normal stress

stresses

unit time interval

ith time interval

integration variable, retardation time
plane shear stress

transverse shear stresses

xvi

1.1 Backgrou
War-pi
initial state.

ptaCticat Case
1985],

Warn
results in sub
‘he years ha.
one Of the m.

and TCSeaJ-Ch

CHAPTER I

INTRODUCTION

1.1 Background

Warping may be defined as the deviation of the geometry of a panel from an
initial state. This initial state is almost always desired to be a state of flatness in
practical cases. The warped condition would thus be considered a defect [Suchsland,
1985].

Warping, probably one of the most pervasive problem in the wood industry,
results in substantial losses to both the manufacturers and users. Even though efforts over
the years have been made to solve it, two aspects of this phenomenon make it remain as
one of the most puzzling and frustrating problems to both the experienced manufacturing
and research professionals.

First, warping is a condition that can rarely be corrected or repaired, once it
occurs. Often the entire panel or the entire product of which the panel is a part must be
replaced with no real guarantee that the replacement will perform better than the original
[Suchsland, 1985].

Secondly, warping usually occurs subsequent to the manufacture of panels or

products of which panels are a part as indicated in the definition. This subsequent nature

indicates that ei

 

environments. u
to manifeSt the n-
in the panel its
process of the p
since the proces
are not all undc
Processing cond

Warping
Changes in panel
in wood and W0

[henna] exI’iittsio

 

(Table I), it Can
warping prmeSS'

Table 1_ Therm;
PrOduc:

\
Specie.
walnut
Red Oak
Red Pine

%

2

indicates that either the panel itself may have experienced some changes under changed
environments, or the changing surroundings may have caused certain inherent properties
to manifest themselves in the form of warping, or both. In any case, this is an instability
in the panel itself, and therefore the control must be directed to the manufacturing
process of the panel to avoid it. However, warping potentials are difficult to recognize
since the processing variables involved are complex and their contributions to warping
are not all understood. It is just difficult to relate the warping of a returned panel to
processing conditions at the time of manufacturing [Suchsland, 1985].

Warping is a manifestation of dimensional instability caused by linear dimensional
changes in panel elements. The environmental changes that cause dimensional changes
in wood and wood-based materials are temperature and humidity changes. Since the
thermal expansion coefficient is much smaller than the hygroscopic expansion coefficient
(Table 1), it can thus be recognized that the humidity change is the major factor in the
warping process.

Table 1. Thermal and hygroscopic expansion coefficients of some wood species [Forest
Products Laboratory, 1987].

 

 

Thermal Coefficient (fF‘) Hygroscopic coefficient (lMC(%))

 

 

 

 

Species Longitudinal Radial - Longitudinal Radial Tangential
Tangential
Walnut 0.17x10" 0.85X10" 0.3x10" 0.18)<10‘2 0.26X10’2
5 5 4
Red Oak 0.25x10 2.50x10 0.6x10 (“5qu 0.37x10'2
Red Pine for dry wood for most for most 0.13 X 10'2 0.24 x10‘2
species species
Douglas-fir 0.16x10‘2 0.25 x 10'2
m =

 

 

Consider
an eventual sol:
elastic approach
success. One ex
Norris [1942]. 2
results by far be!

However
“Md and wood-
mcse conditions
Wiping Process
approach overest

 

reSllllfi 31 10w am

A great
cha‘merllation
materials. But, h
be not been esra

‘1 is belie
“lemming of
approach that we

w .

3

Considerable research work has been directed towards a better understanding and
an eventual solution of warping. On the mathematical and theoretical side are many
elastic approaches which have been developed and have achieved moderate to good
success. One example is the popular beam approach used by Ismar and Paulitsch [1973],
Norris [1942], and Suchsland [1985]. The associated simplicity and its fairly reliable
results by far have made this approach the most useful analytical tool.

However, it is also recognized that the elastic approach is limited by the fact that
wood and wood-based materials behave elastically only under certain conditions. Once
these conditions are violated, the elastic approach would not be able to explain the
warping process with sufficient accuracy. Suchsland [1985] showed that the elastic beam
approach overestimated the warping of three-ply cross laminated red oak and three-ply
cross laminated loblolly pine beams at high humidities, while it provided agreeable
results at low and moderate humidities.

A great deal so far has been achieved regarding the understanding and
characterization of the significant viscoelastic behaviors of wood and wood-based
materials. But, how these viscoelastic properties are involved in the warping problems
has not been established.

It is believed that there are many avenues that could be taken to improve our
understanding of the warping mechanism. One of them is to develop a viscoelastic
approach that would account for the viscoelastic properties of constituent materials in

warping panels.

1.3 Objective

The fr
derive its sol
significant h).
WON-based :

The s
alilting it ti
high humidit
the measure(

l n do
for is viscoc

properties u

1.2 Objectives of the Study

The first goal of this research is to formulate a linear viscoelastic plate theory and
derive its solvable numerical form for panels of materials that are known to develop
significant hygroscopic deformations or strains. Examples of such materials are wood and
wood-based materials.

The second objective is to examine the applicability and validity of the theory by
applying it to a two-ply cross laminated yellow-poplar laminate and beam, subjected to
high humidity. Theoretical predictions of warping by the theory will be compared with
the measured values.

In developing the necessary inputs for the application, yellow-poplar will be tested
for its viscoelastic properties in radial tension creep, and proper characterization of these

properties will be sought.

DEVELI

 

2.1 lntroducuon

Laminate
titted composite
(”10mph or is
While homogenc

The app;
of Mo FFOCCdur
constitutive equ
‘FCOElastic co“
1967]. Therefo,

trons stress,“

Stich a ViSCOelau
validity Of its e

CHAPTER II

DEVELOPMENT OF A LINEAR VISCOELASTIC PLATE THEORY
FOR HYGROSCOPIC COMPOSITE LAMINATES

2.1 Introduction

Laminated wood and wood composite panels can be regarded as multi-layer struc-
tured composite plates comprised of any number of physical or imaginary layers of
orthotropic or isotropic character. Plane homogeneity is usually observed within all layers
while homogeneity or heterogeneity exists between layers.

The approach taken here in the development of a viscoelastic plate theory consists
of two procedures. First, an appropriate elastic plate theory is chosen. Then, the elastic
constitutive equations of the plate theory, namely Hooke’s Law, are substituted by linear
viscoelastic constitutive equations, namely Boltzmann Superposition Principle [Schapery,
1967]. Therefore, the key that distinguishes the two theories lies in the constitutive equa-
tions (stress-strain relations) while all other elements remain the same. The success of
such a viscoelastic theory inevitably and inherently depends on, among other things, the

validity of its elastic foundation.

2.2 Validity of the Classical Lamination Theory (CLT)

Exact solutions based on three dimensional elasticity are attainable for composite

plates [Pagano
plate structure.
of this approat‘:

Many pl
theories [Rehfie
the number of l;
and of good apt

C lassicai
meow in which
before deformat

displacement fie

 

 

dimensional sea]
are ”0‘ aCCountei
underesrimates ‘
meosnes‘ thesi
modulus miles ‘
typical isotroPic

To imprc
highfll‘der thew
examples have b

Sun [1973], Log

order Variations

6

plates [Pagano, 1969 and 1970]. However, certain restrictions have to be imposed upon
plate structure, boundary conditions, and loading patterns, thereby limiting the extension
of this approach to only a number of certain ideal cases.

Many plate theories have been developed as alternatives. While three-dimensional
theories [Rehfield and Valesetty, 1983; Pagano and Soni, 1983] tend to be intractable as
the number of layers becomes moderately large, two dimensional ones are less rigorous
and of good approximation accuracy.

Classical lamination Theory (CLT) is the most basic and simple two-dimensional
theory in which it is [Jones, 1975; Liu, 1987] assumed that normals to the mid-plane
before deformation remain straight and normal to the plane after deformation in its
displacement field. In this regard it is equivalent to beam theory, but on a two
dimensional scale. The implication is that the transverse normal and shear components
are not accounted for. The resulting errors are overestimates of natural frequencies, and
underestimates of transverse deflections of composite plates. In case of advanced
composites, these errors become substantial because of the large elastic modulus to shear
modulus ratios of these materials (e.g., of the order of 25 to 40, instead of 2.6 for
typical isotropic materials) [Reddy, 1984].

To improve upon CLT, shear deformation theories and further improvement -
high-order theories - which include the transverse components have been proposed. Some
examples have been offered by Mindlin [1951], Whitney & Pagano [1970], Whimey &
Sun [1973], Lo & Christensen & Wu [1977], and Reddy [1984], in which linear or high-

order variations of mid-plane displacements through thickness are assumed in their

respecnve disp'
in Figures 1 ti
Chrisnensen. ar‘
also n0ted that
siderably with 1

This last
by Pagano [196
showed [1972]
Specific compos

and highmder

 

[apatite CLT
Table 2 - Clean)
CLT and eXaCt 1
“peer ratio of
slower, yet fair]

The effei
inTable 2 dlSpl;

i .
Mime "1 num

7

respective displacement fields. They are generally much more accurate, as demonstrated
in Figures 1 to 6, where a very precise agreement of the high-order theory by Lo,
Christensen, and Wu [1977] with exact solutions is observed as opposed to CLT. It is
also noted that the poor agreement between CLT and exact solutions improved con-
siderably with the increase in aspect ratio.

This last observation is not incidental. It became very evident in the investigations
by Pagano [1969, 1970, 1971, and 1972] on CLT’s relations with exact solutions. He
showed [1972] that exact solutions converged to the respective CLT solutions for a
specific composite plate as its aspect ratio became very large (Shear deformation theories
and high-order theories, converging to exact solutions, subsequently converge to
respective CLT solutions under the same condition.). His results - Figures 7 to 10 and
Table 2 - clearly illustrate the rapid disappearance of the considerable deviation between
CLT and exact solutions in normal stress, shear stress, and in-plane displacement at an
aspect ratio of only 10. Convergence in the case of vertical deflection is relatively
slower, yet fairly good agreement is already achieved at an aspect ratio of 20.

The effect of number of layers was also examined by Pagano [1972]. His results
in Table 2 display a faster convergence for a composite plate of given thickness with the
increase in number of layers.

The described convergence property therefore defines the range of validity of
CLT with one parameter - aspect ratio. In practical applications, warping of wood and
wood composite plates, when identified as a problem, is normally associated with big

panels, whose large aspect ratios (larger than 20) thereby place themselves well into the

. Table 2a. Maximum stresses and deflection in 3-ply laminate [Pagano, 1972].

 

 

 

 

 

 

Aspect Ratio «1,, 0, Ta 1,, 1‘, w
S (a/2,a/2, $1/2) (a/2,a/2, $1/4) (0,a/2,0) (a/2,0,0) (0,0, $ 1/2) (a/2,a/2,0)
Elasticity Solution

2 1.388 0.835 0.153 0.295 -0.0863 11.767
-0.912 -0.795 0.264 0.298 0.0673

4 0.720 0.663 0.219 0.292 0.0467 4.491
-0.684 -0.666 0.222 0.0458

10 $0.559 0.401 0.301 0.196 -0.0275 1.709
-0.403 0.0276

20 $0.543 0.308 0.328 0.156 3F 0.0230 1.189

-0.309
50 $0.539 $0.276 0.337 0.141 3F 0.0216 1.031
100 $0.539 $0.271 0.339 0.139 430.0214 1.008
CLT Solution
$0.539 $0.269 0.339 0.138 3F0.0213 1.000

 

Table 2b. Maximum stresses and deflection in 5-ply laminate [Pagano, 1972].

 

 

 

 

 

 

 

 

 

Aspect Ratio 0, a, 1,, 1,, 1‘, w
S (a/2,a/2, $ 1/2) (a/2,a/2, $ 1/4) (0,a/2,0) (a/2,0,0) (0,0, $ 1/2) (a/2,a/2,0)
Elasticity Solution

2 1.332 1.001 0.227 0.186 0.0836 12.278
«0.903 -0.848 0.229 0.286 0.0634

4 0.685 0.633 0.238 0.229 00394 4.291
-0.651 -0.626 0.238 0.233 0.0384

10 $0.545 0.430 0.258 0.223 -0.0246 1.570
-0.432 0.223 0.0247

20 $0.539 $0.380 0.268 0.212 $00222 1.145

50 $0.539 $0.363 0.271 0.206 430.0214 1.023

100 $0.539 $0.360 0.272 0.205 «750.0213 1.006

CLT Solution
$0.539 $0.359 0.272 0.205 450.0213 1.000

Table 2c.

Amen Ratio
3

 

ha

 

Table 2c. Maximum stresses and deflection in 7-ply laminate [Pagano, 1972].

 

 

 

 

 

 

 

 

 

= = '-
Aspect Ratio 2. '5', 7n 7,, 7‘, W
S (a/2,a/2, $1/2) (a/2,a/2, $1/4) (0,a/2,0) (a/2,0,0) (0,0,$1/2) (a12,a/2,0)
Elasticity Solution
2 1.284 1.039 0.178 0.238 -0.0775 12.342
0880 08.8 0.229 0.239 0.0579
4 0.679 0.623 0.219 0.236 -0.0356 4.153
-0.645 -0.610 0.223 0.0347
10 $0.548 0.457 0.255 0.219 -0.0237 1.529
-0.458 0.255 0.0238
20 $0.539 0.419 0.267 0.210 $00219 1.133
-0.420
50 $0.539 $0.407 0.271 0.206 330.0214 1.021
100 $0.539 $0.405 0.272 0.205 430.0213 1.005
CLT Solution
$0.539 $0.404 0.272 0.205 $00213 1.000

 

Table 2d. Maximum stresses and deflection in 9-ply laminate [Pagano, 1972].

 

 

 

 

 

 

 

Aspect Ratio 0‘ a, 1,, 1,, 1‘, w
S (a/2,a/2, $1/2) (a/2,a/2, $ 1/4) (0,a/2,0) (a/2,0,0) (0,0, $ 1/2) (a/2,a/2,0)
Elasticity Solution

2 1.260 1.051 0.204 0.194 -0.0722 12.288
-0.866 -0.824 0.224 0.211 0.0534

4 0.684 0.628 0.223 0.223 -0.0337 4.079
-0.649 -0.612 0.223 0.225 0.0328

10 $0.551 0.477 0.247 0.226 -0.0233 1.512
0.226 0.0235

20 21:0.541 $0.444 0.255 0.221 10.0218 1.129

50 $0.539 $0.433 0.258 0.219 430.0214 1.021

100 $0.539 $0.431 0.259 0.219 430.0213 1.005

CLT Solution
:l:0.539 $0.431 0.259 0.219 430.0213 1.000

 

 

Figure 1. He
L/h

Figure 2- Flex
L/h

10

 

 

Ila/h
os _
04 _
0.3 _
oz _
0.1 .. '
l/
4 L L I 4 1 I l 1 1 ¥
«0 .oa -05 -04 -oz 0 oz 04 as as to 3x
p-Ot
L-oz
.413
b-O‘

exun’ ELASHC EIIUHON
L .05 -----..-- WEI m - MD FLAT! DEW

_-_W Mm MT! MW

 

 

(L/M' 10.0

 

Figure 1. Flexural stress distribution for [+30°, -30°] angle-ply laminate at aspect ratio
L/h = 10 [L0, Christensen, and Wu, 1977].

[Li/h

   
 

encr cusncm mum
.. ..... rue-ca moan twee PLATE mom
. -05 _ c. m me PLATE recon

 

(L/M'4

Figure 2. Flexural stress distribution for [+30°, -30°] angle-ply laminate at aspect ratio
L/h = 4 [L0, Christensen, and Wu, 1977].

P
Figure 3. [:1

 

 

.pa
HMM'ESI

——__ .

11

 

n
_ as
r. 04
t- 03
. 0.2
_ or
1 L 1 L 1 1 1 1 L 1 t
~24 -t.a on 16 as F
.OJ -1
'°2“ ._oucr ewmcm scum
..-.--nu¢n aux Lama run: new
'0'“ \ —W mum nun new
... . \
\ (LIN-10.0
.05.

 

Figure 3. In-plane displacement for [+30°, -30°] angle-ply laminate at aspect ratio L/h
= 10 [L0, Christensen, and Wu, 1977].

 

   

¥
7

I l
1.6 14 2.9 ;

EXACT ELASTOTY mum
..---.... W m mm PLATE W

 

_._CLm me PLATE m

(L/hi' 4.0

Figure 4. ln—plane displacement for [+30°, -30°] angle-ply laminate at aspect ratio L/h
= 4 [L0, Christensen, and Wu, 1977].

12

Elm

 

 

 

p1
”t
04
03
01 .
a. .l
1 L 1 0 2L 1 1 J >
no we ‘40 20 to 40 co 0 031.,
1"”
no:
___oucr W m
p.03 an..-“ m mm FLAT! m
___a.m m MT! um
F-oo
(LN-loo
F'QB

 

 

Figure 5. Flexural stress distribution for [0”, 90°, 0°] cross-ply laminate at aspect ratio
L/h = 10 [L0, Christensen, and Wu, 1977].

Elh

 

(L/hi'QO

 

Figure 6. Flexural stress distribution for [0“, 90°, 0°] cross-ply laminate at aspect
ratio L/h = 4 [Lo, Christensen, and Wu, 1977].

Figllre

 

13

 

LEGEND

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 7. Normal stress distribution in a laminate [Pagano, 1972].

Fig]

 

14

 

 

 

 

 

 

 

 

 

 

Figure 8. Shear stress distribution in a laminate [Pagano, 1972].

15

 

 

 

LEGEND

384

 

   

 

CCCCCC s..°
CPT

 

 

 

 

 

 

 

 

 

 

item/2.71
-.ora -.010 -.005 o -\.oos .oro
-.l'l ,
' : lNTERFACES

  

 

 

Figure 9. In-plane displacement in a laminate [Pagano, 1972].

Ht\. :CCZTCT. .55» ii.
235.215. >-....c.5v...<.< 2 .3
C _ - . . r. .- , L l. w:

 

 

03mm Seaman.

 

 

 

 

 

 

 

0.01
W .
Q
m.
w vol
m n
S
1.
3 war
9
S
S 1
)
I
O .
2 NT
d
m... -
mat

 

cm on ow om
F _ _ _ _ L _
$>.m\e.m\£ e ab ................... a ........... a,
a\_.m\s.m\£ @ are i .................................... a; ,
:10 .x.
w
8.m\e.m\£ e e u- .......... 9.0....
a...
W
W.

eosaom So
8:28.. 38:35 ore

 

 

rmfl

 

rm:

ARR: 28mm; .8 sue 5:28 .50 3:83.“: 3 .5238 33.8.0 .85 Co 8:032:80 .2 Bani

11011091190 ttun

validity range

lt is se:
that the elastic
of advanced or

in accuracy is

 

ratios. Further
I

since the anal}
development p
Finite 1
accuracy and a
FEM 10 the w
and Effort invc
that a Sound v
author laClts 3

Speed. Further
l3 P0550“.s r;
a Viscoelastic
dependent bet
"at“ itself is

of such a Vise

appTOaCh‘ In .

17

validity range of CLT, in terms of accuracy.

It is seen in Table 4-1 in the Wood Handbook [Forest Products Laboratory, 1987]
that the elastic modulus to shear modulus ratios for wood can be large just as in the case
of advanced composites. Improved plate theories seem to be in order. However, the gain
in accuracy is very marginal due to the high accuracy of CLT approach at large aspect
ratios. Further, CLT has a critical advantage over high—order ones due to its simplicity
since the analytical complexity of an elastic foundation will inevitably complicate the
development process of a viscoelastic plate theory.

Finite Element Method (FEM) of composite plates is another approach providing
accuracy and automation capability. Tong and Suchsland [1992] made an attempt to apply
FEM to the warping of wood and wood composite panels. Based on the extensive time
and effort involved in the development of their elastic model, it is reasonable to speculate
that a sound viscoelastic FEM model may require a level of manpower and time that the
author lacks at this time. In addition, the numerical automation of their elastic model is
already beyond the capability of today’s desktop computers in terms of memory and
speed. Further, the elastic FEM model requires 9 independent elastic constants as inputs
(3 Poisson’s ratios, 3 moduli of elasticity, and 3 shear moduli), and it becomes clear that
a viscoelastic counterpart would need as inputs the quantitative descriptions of the time
dependent behaviors of these coefficients, which is not yet available and whose determi-
nation itself is an extensive and complicated task. Therefore, the practical applicability
of such a viscoelastic FEM model may be limited. In contrast a CLT-based viscoelastic

approach, in addition to possessing the validity of its elastic foundation at large aspect

ratio, is quite
memory can t
of inputs reqc

CLT. 1
field and thus
behavior of cl
Viscoelastic sit
readily availab
aPlllicability -.

ill SUmp

 

 

number of inp

devek’Pment of

7
“3 Immdllctio

All intrc

CLT-based W5C
similar to [he ‘

filtUl‘e [Cfere ”Ci

3 -
‘3'1 DlsPlacer

DisplaCe

or.
iterna] and,

l8

ratio, is quite easy to develop. A personal computer of adequate CPU and moderate
memory can carry out the computation automation at a fair speed. The smaller number
of inputs required renders it much more powerful.

CLT, by disregarding transverse shear and normal components in its displacement
field and thus making it inferior to improved plate theories in terms of defining the
behavior of composite plates, offers a blessing property in that the CLT and its
viscoelastic sibling are applicable where transverse normal and shear properties are not
readily available. Less stringent input requirement by a model characterizes its good
applicability - a trait always desired when good accuracy is already at hand.

In summary, the high accuracy at large aspect ratios, ease of analysis, and fewer
number of inputs required, justify CLT as being a good elastic foundation for the

development of a sound viscoelastic counterpart.

2.3 Introduction to the CLT

An introduction to the elastic CLT will be given prior to the development of the
CLT-based viscoelastic plate theory. The representations and derivations, which are quite
similar to the work by Liu [1987] and Jones [1975], are intended for convenience in

future referencing.

2.3.1 Displacement Field and Strain Field
Displacement field describes how a plate would deform under any combination

of external and/or internal forces and moments. Since it establishes a base for a plate

theory. it acci
theory. The c
displacement
theory.

The dis
“mothesis. lt

plane of a lami

 

iSdeformed. at
and after lamir
"01 zero (1“;
1ammate thickr
“0W1 after 1‘
across the thick
Change 115 leng

condlllm] 7 §
‘1‘

across the thicj

l9

theory, it accordingly fully governs the development, structure, and accuracy of such a
theory. The conditions under which such a description measures well against the actual
displacement behavior of plates determine the validity range for the resulting plate
theory.

The displacement field assumption for CLT is the familiar and classical Kirchhoff
Hypothesis. It assumes that: (1) a line originally straight and perpendicular to the mid-
plane of a laminate remains straight and perpendicular to the mid-plane after the laminate
is deformed, and (2) such a normal line has a constant length across the thickness before
and after laminate deformation, as shown in Figure 11. If transverse shear strains were
not zero (7,1;t0, '10:" 0), than transverse shear deformation would be present across the
laminate thickness, and a straight normal to the mid-plane would not remain straight and
normal after laminate deformation, violating assumption (1). If normal strain exists
across the thickness (euafi 0), causing plate thickness change, than a straight normal would
change its length across laminate thickness, violating assumption (2). Conversely, the
condition Yn=7yz=€u=0 (no transverse shear deformation and no normal deformation
across the thickness), must be satisfied in conformation to the Kirchhoff Hypothesis.

The bonds in the laminate can be presumed to be infinitesimally thin if they are
thin and few, and may therefore be excluded from consideration. However, they must
satisfy two additional requirements in compliance with the Kirchhoff Hypothesis. A
straight normal, remaining so under deformation, implies continuous displacements
across the laminate thickness, and thus requires that there be no relative interlamina slip

at the bonds. Absence of deformation across the thickness requires that the bonds be

 

FigUri

20

 

 

 

 

 

 

 

 

 

 

YN/
ii he

Figure 11. Geometry of deformation in the x-z plane [Jones, 1976].

non-sheardet

 

l
The l:

representation
direction fror

. , I
remains straig

ll‘ - “0-3;"

line ABCD al

the slope of th

 

21

non-shear-deformable, as well.

The laminate cross section in the xz plane shown in Figure 11 is a graphic
representation of the Kirchhoff Hypothesis. The displacement of point B in the x-
direction from the undeformed to the deformed mid-plane is u,. Since line ABCD

remains straight under deformation, the x-direction displacement for point C is,

ac - [lo—zdsinp 2..31
Line ABCD also remains normal to the mid-plane under deformation, indicating tanB is
the slope of the mid-plane in the x-direction, that is,

zanp - 3‘1" 2.3.2
ax

Small displacement assumption suggests that at very small 6, the following relations hold

0
sin ~ m - — 2.303
p p

Therefore, the x—direction displacement, u, at any 2 across the laminate thickness is

u - u, - zsinp ~ u, - z?” 2-3-4

By similar reasoning, the y-direction displacement, v, is

6w
v n v0 - z__0. 2.3.5

Therefore. th

6

u-%-zf

6*
V‘Vo-Z—
C

we ' W0(x, 1')

Since i
earlier, the tor;

“all Strainers

 

a“?!

61

(iv
83.5

all
Y-\+£
It 6y 6

22

Therefore, the displacement field is

0
““0“?
EIO
v-Vo-Zjay—

W0 ' wa(x9 y)

2.3.6

Since 7n=7yz=€u=0 by the virtue of the Kirchhoff Hypothesis as discussed

earlier, the total of 6 strains are reduced to 3 plane strains en, 6”, and 1”. By linear elas-

ticity strain-displacement relation

an
8'—
” ax
8v
EW-Ta-y-
, 31.2.
3? a), ax ’

three plane strains could subsequently be obtained as

 

. -3"_o_ziw_o
.. -5332
7? By By:
y -% 92—226sz
"’ 6y 61: exay

In matrix notation, the strain field is

2.3.7

2.3.8

fi—fi

9"
[03’ 3'09} 3],"

 

 

 

1:3»

or in short

lei ' ifo} 4

Where the firs
Plane as denc
rotations by l
are fully dete
the momma

”enslauons (

23

 

 

 

 

 

 

 

 

 

 

 

 

tauo ‘ :8sz ‘
ren‘ ax &2 reon‘ rxx ‘
6v
lent-+4 rut-62w" l-ie°”i+ztx,t 2.3.9
‘9’ ay’
.r... $.33» azw, .,, ’9’»
t 6y axl - axayj
or in short
{a} - {so} + ztx} 2.3.10

where the first term on the right hand side is the collection of plane strains of the mid-
plane as denoted by the zero superscript, the second term is the collection of curvature
rotations by the mid-plane as denoted by x multiplied by coordinate position z. Strains
are fully determined by the mid-plane motions (its stretch, shear, and its rotation) and
the coordinate location 2 as they are the sum of the mid-plane strains and linear

translations of the mid-plane curvature rotations at respective z coordinate positions.

2.3.2 Elastic Lamina Constitutive Equations - Stress-Strain Relations

With the strain field determined, the corresponding stress field could be obtained
by the elastic constitutive relations (stress-strain relations). A laminate under the
Kirchhoff Hypothesis results in 1n=1n=au=0 (no transverse shear stresses and normal
stress in the thickness direction) due to 7n=7yz=€u=0 (no transverse shear strains and
no transverse normal strains as implied by the Kirchhoff Hypothesis). The remaining

three nonzero stresses, an, a”, and r”, are confined within a plane geometry, char-

atterizing a p

 

an orthouopiw

l
”11 01.
022 ' 01:
’12 0

 

 

 

0r abbreviated

l0} ' [Oils

“here [Q]! [ht

CORStams. Eu

 

 

[01- .1,

 

24

acterizing a plane stress state. As such, the corresponding stress-strain relations are, for

an orthotropic lamina or layer in its material principle coordinates 1-2,

 

 

 

 

 

 

r a r 1 ,
”u on 012 0 [811

i022} " 012 022 0 i 822} 2.3.11
trlzl . 0 0 066. 3'12,

or abbreviated

{a} - [01m 2.3.12

where [Q], the reduced stiffnesses, are defined in terms of four elastic engineering

COIIStantS, E“, E2), 612’ P12.

 

 

 

 

Eu ”21511 0
1 ‘ V12 V21 1' V12 V21
[Q] - v1.5.2 522 0 2-3-13
1 ' v12 V21 1‘ V12 V21
0 o Gui

The four zero components resulted from stress-strain relations being described in material
principle coordinates 1-2 which in the case of wood is respectively the grain direction
and cross grain direction.

For most laminates such as those of cross lamination and angled lamination
schemes, the material principle coordinates of constituent laminas do not always coincide
with prescribed laminate coordinates. Some constituent laminas have their principle

coordinates deviate from the prescribed laminate coordinates by arbitrary angles. Since

the stress-sue
defining the s

For a

30f

 

 

F—-__-’
a” :10
a
n-
I
l O!

 

 

 

25

the stress-strain relations of all laminas must be defined in a unified coordinate system,
defining the stress-strain relations in arbitrary x-y coordinates is necessary.

For a lamina k, such stress-strain relations in arbitrary coordinates x-y are

r . ._ _ _. W
an 011 012 016 [3

 

 

 

 

 

 

{0”} ' 612 622 626 {8”} 2..314
11.011: _616 6215 666“, .yxrlk

or in short

{a}. - [5]., {8}, 2.3.15

The [a] are transformed reduced stiffnesses for arbitrary x-y coordinates as opposed to
the reduced stiffnesses [Q] for 1-2 material principle coordinates. The two are related by

the following transformation equations as found in Liu [1987] and Sims [1972],

a}? - Qucos‘fi + 2(Q,,+2Q,,)sin20 mm + Qnsin‘a

F2; - Qusin‘fi + 2(Q,,+2Q,,)sin20 mm + 0,, cos‘O

'6; - (011+Qn-4Q‘gsin’0cos’0 + Q,,(sin‘a+cos‘a) 2.3.16
5; - (011+sz-ZQn-ZQ“)sin20 mm + Q“(sin‘0+ cos‘O)

QT, - (0,, —Q,2-ZQ“)sin0 cos’O + (on-ounces) sin’BcosO

‘0'; - (Q1, -Qu-ZQ“)sin’0 case + (on-02,40“) sine cos’e

where 0 is the angle by which x-y coordinates rotate clock-wise to 1-2 material principle

coordinates, as illustrated in Figure 12.

FiEUre

26

 

 

7

 

 

Figure 12. Relation of material principle coordinates 1-2 with x-y coordinates.

2.3.3 Strain
The c
variation of 5

corresponding

 

 

 

coordinates) i

 

1.2 with x.,.

27

2.3.3 Strain and Stress Variations in a Laminate

The derived strain field verifies the Kirchhoff Hypothesis by implying a linear
variation of strains through thickness as characterized by Eq’s. 2.3.7 and 2.3.9. The
corresponding stress field based on Eq. 2.3.14 (linear stress-strain relations in x-y

coordinates) is

 

 

 

 

 

 

 

 

 

 

r i '— — — ‘ r r 0 w r 1 1

a,“ on 012 016 8 n 1",

40”} - Q12 Q22 Q26 Hewiq-zixy ii 2.3.17
.701); _Qrs 026 066,,z . (You. xx”! .

or in short

{0}. - [5].: {(30} +z{x}} 2.3.18

which in turn follows a linear variation across the k lamina. However, due to the
difference of transformed reduced stiffnesses among constituent laminas, the stress field
does not necessarily vary linearly across interlamina boundaries in a laminate, even
though the strain field behaves so. Such linear variation by strain field and stress field

are characteristic of CLT, as shown in Figure 13.

2.3.4 Resultant Laminate Forces and Moments
The stresses in a laminate must be balanced with resultant forces and moments
acting on the laminate by integration of stresses across laminate thickness. Resultant force

1-2 with x-y coordinatesN, and moment M, are related to stress a, by integrating a, over

thickness 11 o

 

in which the

section. l’BSpet

 

 

laminate as sh
in tems 0f Str
m
v ’42
‘y i - f
N ‘M
17‘ 1
0T
W2
{N} . f {c
‘NZ
Similar] re‘
M3]
N2
M) i .. f
A! ‘N‘
3)} <

 

0f

thickness In of a laminate as follows,

~ In

Nx- fond):
4512
m

Mx- fanzdx
-m

28

2.3.19

in which the units for N, and M, are force and moment per unit length of the cross

section, respectively. By applying 2.3.19 to all three stresses, 0,, 0,, and 1”, in a n-layer

laminate as shown in Figure 14 the entire collection of resultant forces can be defined

in terms of stresses:

f ‘

t dz

 

 

 

 

 

 

N: m (an n at an
(N, i - [40”idz-Zfia”
N 442 r k-I zl-l f
t ‘71 t 5’1 t ”1
01'
H2 n 2*
{N} - f {aldz - 2f tend:
“N2 k-I 44

Similarly, resultant moments are defined as

 

 

 

 

 

Of

 

r 1 r s r i

M3 m an n t: on

1M, > - fiantzdz-gfianizdz
-hl2 ' 1“

thyl .1391 17m:

2.3.20

2.3.21

2.3.22

29

 

 

' L
2 i
_ __..'

H ’ L
' ‘ A
VARIATION CHARACTERISTIC VARIATION
OF STRAIN MODUII OF STRESS

 

 

 

 

 

 

 

 

 

 

 

 

 

lAMINATE

Figure 13. Hypothetical variations of strain and stress across laminate thickness [Jones,
1976].

 

 

”I,
N
N
#1
N
N A
VAV‘T .—
N

 

 

>«.111... me!
i T:— ltj.‘ .7

lATlR m

 

 

 

 

 

 

 

 

 

 

 

Figure 14. Geometry of a n-layer laminate [Jones, 1976].

where z, and
in Figure 14,
after the integ

depiCted in F

 

 

 

 

 

30

hi2 .. I;
M - dz- dz 2.3.33
1 } £21012 1-. 11101.2

where z, and z“ determine the z position of lamina k in the x-y-z coordinates as defined
in Figure 14. The resultant forces and moments become independent of the z coordinate
after the integration, but are functions of x and y. Their positions and orientations are
depicted in Figures 15 and 16.

Substituting stress-strain relations of Eq. 2.3.15 into resultant—stress relations of

Eq’s. 2.3.21 and 2.3.23 generates resultant-strain relations

k-I

{N} - I: I {61.12:}. dz
2‘" 2.3.24

1M} - 2': f 161. {a}. zdz
’14

id

The reduced stiffness matrix is considered constant within each constituent lamina, and

therefore can be taken out of the integration to generate

k-I §.' k-I xi-)

{N} - Et‘é],f{{e°}+z{xi} dz - £161.{f{e°} dz+f1xt zdz}
‘“ 2.3.25

{M} - 231614 {{e°}+z{x}}zdz - £164 f {2°}zdz+ f {xiz’dz}

b1 111-: 1H 1M

As we recall, {e’} and {x} are mid-plane strains and curvatures and thus are independent

31

 

Figure 15. ln-plane forces on a flat laminate [Jones, 1976].

 

 

 

 

Figure 16. Moments on a flat laminate [Jones, 1976].

of; Theyt

the above 1

{Ni-12

I I .

‘12
k.

32

of z. They can be positioned outside of the summation and integration over z to simplify

the above relations to

{N} -

 

9:161. f 421120}

‘1-1

 

 

 

+[ZI5]. fzdz 1x}
k-I it-)
2.3.26
1M} - 27161, fz dzJM}
k-I ‘1—1
+ 2161,. fz’dz 1x}
k-l 1b!
Further, since
I dz ' (71 '71-: )
in
f 2 dz - gag-4,42) 2.3.27

1&4

It

1
fzzdz " gag-2,43)
'14

we obtain

 

 

Ol‘

33
i' n
{N} " Eli-6]; (Zk ‘qu )]{£0}
_ k-I

+ [ [[alk%(zk2-zk—12)

_ k-I

{K}

 

 

{M} ' [Z[6]k—:'(Zt2—Zk-12)J{€0}

k-l

+ [gl—Q'lkéai-zbflj {K}

01'

{N} - [AH-9°} + [Bllkl
{M} - [B]{8°} + {91116}

which can be jointly expressed as

{fl-l If}

01'

AIR

BID

 

 

2.3.28

2.3.29

2.3.30

 

 

 

 

 

 

 

‘Nx ‘ A11 A12 A16 311 312 Bus r8023
Ny A12 A22 A26 312 322 A26 90,,
ny A16 A26 A66 316 326 Ba; so”
i i - i i
M: Bu 312 316 Du Drz DIG X,
My BIZ 822 326 Dr: D22 D26 K,
”M BIG 326 366 Dre 026 Dari xxxy .
where

A1; ' Z (07;): (71 ‘zk-r)

k-l

'1 Z (0:3)]: (21:2 "-7-1: 12)

N

D. -§Z (antes-2H3)
k-I

2.3.31

2.3.32

The derived equations (Eq’s. 3.3.30 - 2.3.32) establish how the resultant forces

and moments acting on a laminate are related to the strains and curvatures at the mid-

plane surfaoe of the laminate deformed under such forces and moments. Usually called

the governing equations of CLT, they fully dictate the mechanical behavior of a laminate

under the Kirchhoff Hypothesis.

The relating matrix is called the general stiffness matrix because of its reflection

on the stiffnesses of a laminate. Its components as seen in Eq. 3.3.31 are summations

over all laminas of the product of each lamina’s transformed reduced stiffnesses and the

laminas z p
lamina are ti
its orientatio
determined b.
their geomet:

The [I
distinct impn

the extension;

 

 

 

only, and are
With C“l'V'éltur
Suggesrs that.
would Recess;
twist UDder jL
ted to bendin}

Plane, that is

2.35 Hng
Prevj

from mechat
. is mere n

prornin‘ént

35

lamina’s z position characteristics. Due to that the transformed reduced stiffnesses of a
lamina are functions of the lamina’s four engineering constants,E,,, En, G1,, it", plus
its orientation 0 with regard to the laminate coordinates. The stiffness matrix is fully
determined by the engineering properties (E,,, E”, G”, V") of all constituent laminas and
their geometric positions (coordinate orientation 0 and z coordinate position).

The [A], [B], and [D] in the combined stiffness matrix shown in Eq. 3.3.30 have
distinct implications. Relating only plane forces {N} to plane strains {5"}, [A] are called
the extensional stiffnesses. [D] relates bending moments {M} to bending curvatures {1:}
only, and are thus called the bending stiffnesses. [B] by bridging both plane forces {N}
with curvatures {x}, and plane strains {J} with bending moments {M}, however,
suggests that extensional plane forces {N} applied to a laminate with a nonzero [B] term
would necessarily generate a curvature {at} term, that is, the laminate would bend and/or
twist under just extensional plane forces. In addition, such a laminate can not be subjec-
ted to bending moments {M} without at the same time experiencing extension at the mid-
plane, that is, plane strains {60} would not be zero. [B] are therefore named the coupling

stiffnesses.

2.3.5 Hygroscopic Strains and Stress Analysis

Previous analysis of CLT considers mechanical strains, that are strains resulting
from mechanical forces and moments, while strains from other sources are disregarded.
This mere mechanical analysis does not suffice in cases where non-mechanical strains are

prominent. As is the case with laminates made of hygroscopic materials such as wood

andwood-b;
changes in .
analysis in c

Toral

expressed as.

 

36

and wood-based materials, which could deform due to hygroscopic strains from humidity
changes in their environments. CLT must account for the hygroscopic strains in its
analysis in order for it to be applicable to hygroscopic composites.

Total strains, including mechanical strains and hygroscopic strains, would be

expressed as the sum of both

{5C} _ {8M} + {8H} 2.3.33

where superscript C indicates total strains, superscript M mechanical strains and
superscript H hygroscopic strains. Subtraction of humidity strains from the total strains

gives the mechanical strains

{8"} - {.0} — {8"} 2.3.34

For laminates under the Kirchhoff Hypothesis, total strains are expressed as the sum of
the plane strains and the linear translations of curvatures at the mid—plane as defined in

Eq. 2.3.10. Hence, the mechanical strains are

{so -1e°1 + 21x} - {s"} 2-3-35

The corresponding mechanical stresses for lamina k, by Eq. 2.3.15, would be

{a}. - ['0']. {{e")+z{x}-{e”},} 2.3.36

where hygroscopic strains {12”} are assumed constant within each lamina. The associated

mechanical resultant forces and moments as denoted by superscript M would be, by Eq.

37
2.3.25

is

{N“}-ztoi.f{{e e"}+z{x}—{e”},,}dz

Zn
' W;[01k{f{ 8"} dz+f{x} zdz-f{e°}k dz}
1:1 1H 11.:

2.3.37
l-Zlol.f{{°}+ztx1-ts"},}zaz

k-I 1: l

- 2151.{ f {2°}zdz+ f {xiz’dz— f {8"}kzdz}
k-I it-) “-1

1H

which, in analogy to the derivation of Eq. 2.3.28 from Eq. 2.3.25, could be reduced to

{NM} " Euro—ll: (Zr: _zk-I )]{30}

k-I

Z[Q]kl 32kt "Zr: 12

k-I

- {it51.{f}.(z.-z,_,)}

 

){K}

k-I

2.3.38

k-I

{Mu} " ZIQJk—é (712 ‘11- 12)]{30}
+5 [Oh-(251,213)] {K}

k-I .

- { g; 161. {strafing-442)}

01

 

 

0i

‘s
s
“
~

“thh is the g
and M".
“‘9 Str.

V
gr-OSCOPIC en

38

01'

{N"} - {Alisa} + {310‘} - {NH}

2.3.39
{M”} - [B]{e°l + [D]{x} - {M”}
where {N”} and {M”}, defined as
W} - { £161. {er}.<z.-z,_,)}
"" 2.3.40

1M") -{5161. 1w}.§<z.2-z._,2)}

k-I

are called humidity forces and moments. They are therefore not real mechanical forces

and moments. Eq. 2.3.40 may be regrouped to

{N"} + {M”} - {Allen} + {310‘}

2.3.41
{14“} + {N”} - {3118”} + [Dim

or

N"+NH A | B so

______ _ -_ __ 2.3.42

which is the governing equation that includes hygroscopic effects with the terms of N”
and M”.
The strains and curvatures of a laminate under given mechanical actions and

hygroscopic effects can be obtained by inversion of the governing equation

In cases ssh

laminate are

 

 

 

39

2.3.43

In cases where the mechanical action is absent, hygroscopic effects imposed upon a

laminate are related to the strains and curvatures by

 

 

‘NH .4 I B 30
(---1_ -- -- __ 2.3.44
.MH B | D x
and vice versa by
so A | B " N”
2.3.45

It can be shown that [B] and {M”} are zero for laminates that are symmetric to
the mid-plane of the laminate in both geometry and material properties (both elastic
properties and hygroscopic properties). As such, curvatures must be zero by Eq. 2.3.45,
that is such laminates would not bend and/or twist under hygroscopic effects. However,
[3] and {Mn} are not necessarily zero with nonsymmetric laminates where curvatures
({x}) result, suggesting bending and/or twisting by the laminate under hygroscopic
effects.

Wood and wood-based materials respond to humidity changes with proportionate

changes in their moisture contents. It is, therefore, equivalent to express hygroscopic

effecrs or s:
”C become
Ther

procedure.

 

--—-—_.

and SUPCFSCI

2.4 DEVelop

(ConSiltutive

[Ime‘wiSe

mlmiOn, [10‘

thEory atte mI

relati
0n
1 e

Elam)“ in C
l

40

effects or strains with reference to either humidity or moisture content. Superscript " and
”0 become interchangeable in all expressions.

Thermal effects can also be included into the CLT analysis following the same
procedure. It can be shown that the resulting governing equation would then be

N”+N”+NT‘ A I B so
__________ t- __ __ 2.3.46

 

MM+MH+MK B l D x

where

{NT} " {i[6]k{81}k(zk-Zk_1)}

"' 2.3.47

k-I

{MT} " {flak {81}k%(zk2'Zk-12)}

and superscript 7 refers to thermal effects with other terms defined as before.

2.4 Development of the Linear Viscoelastic Plate Theory (LVP)

Viscoelastic behavior differs from elastic behavior in terms of stress-strain relation
(constitutive equation). The elastic relation, being time independent, is described by a
time-wise constant algebraic tensor equation, namely Hooke’s Law. The viscoelastic
relation, however, is not such a simple case due to the time dependence. Since any
theory attempting to represent material behavior must account for appropriate stress-strain
relation, the viscoelastic stress-strain relation must be substituted in place of the elastic

relation in CLT in the evolution of the viscoelastic CLT.

MJUM;
We
Wham:
vama
at the very
smshah
mmMLu
mmsma,
history as W:
ll has been
BOilZman fin
means 0f re;
anple ofs
The at
Weeding,

lion [ChriSten
I

:9. . [‘64le
o-
I

%.[§JL
0.

where walw) ‘

 

41

2.4.1 Linear Viscoelastic Stress-Strain Relations for Plane Stress

The definite one-to-one correspondence between stresses and associated strains,
while existing for elastic behavior, however, is not observed in viscoelastic behavior. The
strain variation with time under stress implies that strains depend not only on the stress
at the very current moment, but also on the time under stress. The past history under
stress is also contributing to the material response (strains), which suggests that the
material, aside from responding to the current stresses, remembers the past history of
stress states. This memory effect, observed of stresses in relation to strains and strain
history as well, is characteristic. of the stress-strain relation in viscoelastic behavior, and
it has been quantitatively represented by the Boltzman principle since 1874 when
Boltzman first proposed it. This principle is still considered one of the most important
means of representing material behavior. It is also known in the literatures as the
Principle of Superposition [Gross, 1953] and as the Hereditary Integral [Flugge, 1967].

The above principle, if expressed for a linear viscoelastic anisotropic material in
a three-dimensional stress state, gives the following stress-strain relations in tensor nota-

tion [Christensen, 1982; Schapery, 1967; Sims, 1972]]

8
60k,
80- " llfith-f) 37d?
Lflhl=Lz3 241

where Jfiu(t'7) and YMt—r) are defined to be the creep compliances and relaxation moduli

respectively. The 0’ in the lower integration limit suggests inclusion of a step discontinu-

ity of Stres

change the

cg. - Jim")

all - You“)

However. th

The'

Presumes the

’0' are Consi

 

 

 

8110)
£220) -

7,20) 1:

0110)
0220)

r12(1)
k

18(1)}k . r
1

{00m ‘ '
.f.

C

.1
C

42

ity of stresses or strains at t = 0, which upon separation from the integration would

change the relations to:

s- -Jfl(t)afl(0)+ [gyro—d:
i, j, k, 1 =1, 2, 3 2.4.2

0.. - Yu(t)eu(0)+ fYMKt—r)——dr

However, the notation in Eq. 2.4.1 is more compact and will be used here.
The viscoelastic plate theory which is to be developed on the frame of CLT
presumes the CLT plane stress state in which only the three plane stresses, 0;, 0,, and

r”, are considered. The resulting reduced stress-strain relations for lamina k are

 

 

 

 

 

 

 

 

 

 

 

r811“). , ’Juo‘f) 1120-7) 0 ‘ rank)"
(8220» - f 1,2(1- 1') 122(:- r) 0 (022(1') t at
1129).; 0- . 0 0 JAN“ 3,201, 2.4.3
iaua)‘ . 'Yua-r) Y12(t-r) o ' sum"
(022(0) - f Y,,(t- r) Y,,(t- r) 0 8,,(r) 1 dr
L1.120% k 0' . 0 0 Y4" 19.; 7120?” k
or
term. - f [Jo-o]. {arrn’tdr
°' 2.4.4

{00)}, - f [ya-1)], {e(r)}’,,dr
o-

where the i

“h

 

principle ct
Stress and 5
710-7) in p

obtained sin

 

8.10"

12,0) .
t.,(t) k

 

”3(1)

 

‘ an“) I -
k

4,0)

 

01’

43

where the prime (’) represents differentiation 6/61.

While the above viscoelastic stress-strain relations are with respect to material

principle coordinates 1-2, their expression in arbitrary coordinates x-y is desired. By

stress and strain transformation defined in Eq. 2.3.16 (with 150-7). 74M) or Yga-r),

Via-r) in place of Q6. 2,), the viscoelastic stress-strain relations in x-y coordinates are

obtained similarly to the elastic CLT analysis:

r 83(t) i
4 an“)

.717“) 1

V

 

 

r 0,0) ‘
4 an“)

. Tami

f

 

 

01'

I
9‘»

t

"mt-r) Jigs-r) Err-6'

.JZG- 1') 72—60— r) :60- r) J

PIC—IU- r) 230- 1') WI- 1').

 

71—20- 1') EG- r) 72;“- r)

 

I:

Y;(t- r) 320- r) E}:- r)

 

_f‘(t- r) 320- r) Elt- r)j k

{30)}; " f [79’7”]; 10“”,de

0"

{am}. - f [Kt-o]. {sun/.4:

0'

rank)

a”( 1:) i

 

 

.712“).

A

e,,( r)

 

53(1)‘

dr

1dr

 

.7xy( 91

2.4.5

2.4.6

where [Tm-1)], and [TI-(14)]; indicate that they are transformed relaxation moduli and

creep compliances in the arbitrary x-y coordinates.

The sharp contrast between Eq. 2.4.6 of linear viscoelastic CLT and Eq. 2.3.15

of elastic CLT is due to the time involvement and resulting complexity. The algebraic

44

time-wise constant tensor [Q] in elastic CLT states the definite match between current
stresses and strains, while the functional integration in time space in linear viscoelastic
CLT analysis accounts for not only the effect of current stresses or strains, but also the

characteristic memory effect (effect of strain history or stress history).

2.4.2 Linear Viscoelastic Governing Equation
When a hygroscopic effect is present, the mechanical strain rates, by differentia-

tion of Eq. 2.3.35, are

{fray - {20(61’ + z{x(t)}’ - {8"C(r)}/ 2.4.7

where the hygroscopic strain rates are expressed in reference to moisture content
(superscript M‘). If the hygroscopic strain rates are constant within each constituent
lamina, but different across the laminas in a laminate, the mechanical strain rates for a

particular lamina k are

{”1011 - {8"(r)}’ + zmoi’ - {2"C(r)}’. 2.4.8

The corresponding mechanical stresses in this particular lamina k which are related to

strains by linear viscoelastic stress-strains relations defined in Eq. 2.4.6, are

1am}. - f [Ytt- id]. {4'16}: dr
V 2.4.9

I

' f [7""914 {{‘"(')}’ + z {My — {Wren} d.

0-

CLT 211

results

OI

 

0i

45

The balance between the resultant forces-moments and the stresses in the elastic

CLT analysis defined in Eq’s. 2.3.19 - 2.3.21 must be maintained at any instant t, which

results in the following forces-stresses relation

 

 

 

 

 

 

'N, (o‘ m 'anm‘ ,t ’ano)‘
4N, (t)i - fia”(t)tdz - X f ta”(t)idz
4'] k-I .)

_N,,(t), 2 3,14), ‘* 3,141,
01'
M2 in 1:
{Nan - f {0(t)}dz - E f {0(t)},.dz
‘NZ k-l ‘k-l
and moments—stresses relation
“M, (0‘ 'ana)‘ 'ana)‘

"2 n 7"
(My(t)> - fia”(t)>zdz - [[10”(t)lzdz
4012 Zn

k-I
rug“). 1 I", (t) 1 - L t” (t).

 

 

 

 

 

 

 

OI'

"2 n 1*
{M(t)} - f {a(t)}zdz - Zf{a(t)},zdz
442

k-I H4

2.4.10

2.4.11

2.4.12

2.4.13

By viscoelastic stress-strain relations of Eq. 2.4.9, mechanical resultant forces

{N"(t)} for a n-layer laminate are expressed in terms of strains as

46

{}-N"(t) -E f {00)}.42

k-I
2.4.14

- 25110] [Ya-0]{{4:"(r)}’+z{k(r)}’-{6""C(1')}’,,}d¢'}dz

k-l

Switching t—integration with z-integration and moving them outside of summation results

in
{M44} - f { 2: f 1244-41,.{4 {castaway-4M4:>}’.}dz}dr
0' k) 1H

2

Elf Ytr-ril. {20461’ 42144- +

1t-

; u

k-I

the

2.4.15

9“...
D

4PM

JP n [Y(t-r)]k {dr)}’ zdz}dr —
k-I

Q"...

[N Y(t- a], {swam dz} d4-

h-Iztr

{60(7)}’, {x(r)}’ are strain and curvature rates at the mid-plane, respectively. Constant
with respect to z, they are moved outside of z—integration and summation. [7 (1'7”: and
hygroscopic strain rates {é‘c(1)}’k are only constant within the k lamina with respect to
z, and hence are only moved outside of z-integration. These manipulations, which are

similar to those in elastic analysis, reduce the above to

47

k-I

{N"(t)}- [[2126-4111] dz]{.-°(.)}'d. +
0

 

f[ [Y4t-r)],,f zdz may dr -
0‘ k- 1 lb,

[I n [URI-10],, f dz]{e"c(t)}’k}dr
0 k I 1H

which, due to Eq. 2.3.27, are further simplified to be
I
{-N"(t)} f [2
0-4-1

ii):

UT" 7)]1: (Zr: ‘ zk-I )) {30(1)}I dt +

k-I

[YG- 7)]; é“: " 2‘46) {#7)}, d7 -

{2: tar-411.42.-z.-,>te~64rn;}4.

k-I

4:3"...

Of

(114%)} - f [A(t— t)]{e°(r)}’dr +
a-

I

f 1844-6114424: - {It/“(0}

0-

where

2.4.16

2.4.17

2.4.18

48

[44:— 1)] - 2: {74:- r)], (z,-z,,_,)

k-I

[344- 4)] - E [20— 1)], go} - z,_,’)
k-I
2.4. 19

{Mom} - f {N"C(t- r.e"64r)’>} dr
,.

{Nuco- 7,8uc(f)l)} " E [?(t' 7)]‘5 (Z; ' zk-1){ ”0(7)”

k-I

{N"'c(t)} denotes hygroscopic forces in reference to moisture content change. Similarly,
mechanical resultant moments {M“c(t)} are expressed in terms of strains as

I

{We} - f [13(t-r)]{4:°(r)}’alt +

0'

2.4.20
f [Do-o] {xtrn’ dr - {M“C(t)}
0-
where
[BC-1)] ' Z [704)]; é<zi 'zkz-I)
k-l
[D(t-t)] - k): [Y'G- 19]; gag-1:4)
-1
2.4.21

(41%») - f {H‘Ca- r,e"c(r)’)}dr
a-

W‘fir- r.e"c(r)’)} - Z [70- o]. étzi - 41M 4mm};
k-I

49
{Mew} are the hygroscopic moments.

Eq’s. 2.4. 18 and 2.4.20 are jointly expressed as

N”(t) , A(t- r) | B(t- r)

 

80( I") / N MC“)
d r -

 

M“(t) 0- 30-7) 1 DU‘T) x(1:)

and abbreviated to

I

{NM“(t)} + {NM”C(t)} - f [ABD(t-1:)] {eox(t) }’ dr

where

N “(0 . Elmo—4246’)
{NM ”(0} ' ---- ' f ---------- d r
M “C(t) 0 MucO— r, e( r)’)

- f {Wma- r,e(r)’)}dr
a-
l.4(z- r) | B(t- r)

[ABD(t- r)] - ------------
. 86- r) l 00- r)

r 80(1) I
{3461 -4 -----

 

 

2.4.22

2.4.23

2.4.24

50

Eq’s. 2.4.22 - 2.4.24 express mechanical and hygroscopic forces-moments in terms of
the strain and curvature at the mid-plane, and are called the governing equations in linear
hygroscopic viscoelastic analysis. They are convolution-type integrals and are similar to
that developed by Sims [1972].

The difference between the elastic stress-strain relation and governing equation
(Eq’s. 2.3. 14 - 2.3.15) and the linear viscoelastic counterparts (Eq’s. 2.4.5 - 2.4.6) lies,
as expected, in the time domain - reflecting the distinction between elastic behavior and
viscoelastic behavior. The parallelness becomes more conspicuous in the Laplace domain
where the linear viscoelastic stress-strain and governing equation are transformed to be,

respectively

{56)} - s 1%)] {36)}

2.4.24.A
{3 4s» - s17(s)1{3ts)}
and
117%) + N—"é(s) Z(s) I its) :56)
____________ _ s ______ ______ ______ 2.4.24.3
F43) 4W6) ‘54s) I 134s) Ets)

With time involvement hidden, they resemble their respective elastic counterparts (Eq’s.
2.3.14 - 2.3.15 and Eq’s. 2.4.5 - 2.4.6) in the algebraic form. Such resemblance, which
is always observed between corresponding elastic and linear viscoelastic equations, is the
basis of the poplar and useful Correspondence Principle.

Theoretically, history of strain and curvature rates at mid-plane {£x(r)}’ can be

5 1
obtained by Eq. 2.4.23 given mechanical actions and hygroscopic effect, namely {N”(t)},

{Wm}, {N”c(t)}, and {Mom}. Yet, an exact analytical solution is probably only
possible in certain simple and ideal situations, presenting little practical usage for the

analytical governing equation. A numerical alternative has to be attempted.

2.4.3 Numericalization of Linear Viscoelastic Governing Equation and Successive Com-
putation of Strains and Stresses
A convolution-type integral like the following can be subdivided into a series of

integrals,

fflt- r)g( r)’dt
0-

- Meta) + f flt- r)g( r)’dr
0

2.4.25
- 14:)ng + [ flt-r)s(t)’dr + f + f + ------ + f + jar-rimri’dr
fo-O r, r, 13.4 r__,

- 116240) + 2'; fan.- r)s(r)’dr

where integration limits 10=0 through r,=t are m+l t-points along the time domain and

(70'0) < TI< 12< 1'3< ....< rm-2< tin-1 < (1"-t) 2.4.26

As shown in Figure 17, if the time interval is very small, that is

471-(ri- rH) - R..Ar < 1 2.4.27

52

f(t-T)g(7’)'
l

f(t-T.)g(r,)’

ftt—mgmn' /

 

 

 

 

 

 

 

Figure 17. Illustration of linear finite difference approximation.

53

the area under the curve could be approximated by the shaded area of a trapezoid,

fflt- r)g(r)’dr - -:—( r. - r.._,)[/(t- r._,)g( r._,)’ +f(t- r,)g( r,)’]
’1-1 2.4.28

.. g. R, A tMI- r,,,)g( r,_,)’ +flt- 4984411

As such, integral of Eq. 2.4.25 becomes

fflt- r)g(r)’dr
0' 2.4.29

- 1108(0) + $4 72 R.Lflf..~f.--,)8(f,--,Y +11%; f.)8(f.-)’]
i-I

The integral has been degenerated into a summation of algebraic expressions. Note that
all individual time intervals A1, are measured against and expressed in terms of a unit
time interval Ar.

By Eq. 2.4.29, the hygroscopic forces-moments in governing equation Eq. 2.4.23

can be degenerated to

{NM‘Cm} - [ {W‘Ca— r,e”c(r)’)} 1. dr
0.

- {W'c(r,.-0.e"c(0))} 2‘4'30

+
k

2-472"R11{WMC(7.‘Tt-1"Mc(71-I)I)} + {Wuc(rfl_r"£uc(t‘)/)})
1-1

and the right side of the governing equation Eq. 2.4.23 to

54

I- f.

f [ABD(t- 1:)]{ 40x4 1)}, dr

_ [(430%)] {201(0)} + 2.3.31
%4 five, ( [43044,- r,_,)] {4°14 r._,)} + [430m- r1)]1‘4‘0’4‘10)
1-1
The governing equation could then be written in algebraic form as
{ mum} + {Wuc( r~_0’£uc(0))} +
$- ArERi({WuC rm-r1_,,z"c(t,,,)l)} + {IV—MuC Tn‘rt’euc(TI)/)}) '
4-1
2.4.32
[ABD(¢',_)] { 3010(0)} +
g A 121%,“ ”DU..." 3.4)] { “fin-1)}, + {.4300}.- f)] {501411)}7)
4-1
where
(Ta‘o) < 1'1< 1'2< 73< ""< rfl-2< Tm—1<(tn-t) 2.4.33

Next, successive computation is employed to obtain the history of strain-curvature
and their rates at the (+1 discrete time points, 1,, through 7,.

At m=1, which is t=r,=r,, there is

Ni

:43.

[in

(’1‘:

Sino

SECOI

55

{NM"(:,)} + {W”C(r,-0.e"040))} +

% ArRI({NMMC rl— rwsMC(ro)’)}+ { NMMCU'I- rpe eM-C(rl)/)})
2.4.34
[ABD(f,)]{8 {801(0)}7‘
£4 rR,([ABD(r,-ro)] { compy + [43045-19] { e°x(r1)}’)
As 1, approaches 70:0, it is obtained that
{NM“40)} + {W”C(0.e”‘(0))} +
(in: l ArRI({WuC(rI-ra,ch(ro)’)} + {W”C(r,-r,,euc(r,)’)}) -
ITI‘VJTO

[43040)] { e040» +

lint éArR([ABD(t,- to)]{ shaggy + [.4er,- 4,)” e (11»)

(71" rO)-0

Since no sudden hygroscopic strain input at t=0 is possible ({eM"(0)},,z = {0}). the
second term on the left hand side is zero. Assuming no mechanical action ({NMM(0)}

= {0}) and no initial strains at t=0 ({e°x(0)} = {0}). we then have

4 11:34—21?-ArR,({N—M“C(r,—ra,e"c(ro)/)}+ {NM”C(r,-r,,e“‘(r,y)}) -
t' f 2.4.36

1"," ‘1'47R1([ABD(71' to)“ 8W} + [ABDUI- 1'1)” 3W)

(rl- to)~0 2

and subsequently arrive at

W

lhes

{ can

when
curva
I: 7”:

“pm

56

{W‘Qoflqm} - [ABD(0)] { campy 2.4.37

The strain and curvature rates at t=10=0 are then determined to be

{6014(0)}’ - [ABD(0)]"{WMC(0,zMC(0)’)} 2.4.38

The strains and curvatures at t=1, can subsequently be approximated by

{30411)} " {80481-1)}'R.-47+{e°x(r,-,)} at i-1 2.4.39

where small interval condition cf Eq. 2.4.27 is to be maintained, and initial strains and
curvatures at t=10=0 ({e°x(ra)}) are assumed given. With strain and curvature rates at
t=10=0 ({e"x(1a)}’) known, the rates at t=1, ({e°x(1,)}’) can then be determined by
expanding Eq. 2.4.32 to be

{3:01er}, - [ABD(t,—r,)]"

(i({NMM(ri)} " {Wmfipe‘mfimi ' [430(941‘0'4‘0)» *

A 1' 2.4.40
R,{WMC(1,-ro,eMc(ro)’)} + a, {W'C4r,-r,,e"c(r,)’)} -
R, [mom-10)] {comp}’)
The strains and curvatures at t=1, are then
{2014(9)} ~ {eoqr,_,)}’R, Ar+{ 801441,,» at i-2 2.4.41

Eq. 2.4.32 could be further expanded to obtain the rates at t=1,, subsequently. Similarly,

rates at t=1,, 1,, and till 1,, could be successively determined, as well as the strains and

C1111

atd

51181;

beco

57

curvatures in analogy to Eq. 2.4.39. The general form for this successive computation

at discrete time point, t=1,,,, is derived to be

{ 801(1)}, - [ABD(rm- r~)]“’

 

(%({NMN(1E)} + {Wilcaweuchw} _ [4300)“ 30,470)» +

2.4.42
EU?” +R1)[ {Wm(.i.Ri A t,£uc( 714”} ’ AB“: R} A 0] { 80“ 71-1)}l]
i-I 1-3 I"

+ Rm {muck 1,"- 1.,eMc( fly} )

where
Ra-O
"14.23;” 2.4.43
in n -1
211,41 - X RjAr- ‘2 RjAt
i-i 1-1 1-1

The second term within the enclosing parenthesis on the right hand side of Eq.
2.4.42 which results from step-wise elevation of hygroscopic strains at t=1,=0
({eMc(1a)},,) must be zero since {eMc(1,)},, = {0} (It is not possible to elevate hygroscopic
strains instantaneously as discussed in the derivation of Eq. 2.4.36). As such, Eq. 2.4.42

becomes

lllllf

Kill

the li
tun,a
meter
degen
points

58

{soggy - [ABD(1m—1m)]“'

(fillmfiwl - [4300.911 2°44») +
2.4.44

ABDdER, A r)

j-i

 

{ ”a“ 71-1”)

+ Rm { WUC(1.M_ tm’EMC( 75),} )

 

2::‘(12H + R,)( {firs—{“42}, A r,cMC( r,_,)/)} —
i- j-I

{e’x(1a)}, the sudden step-wise elevation of strains and curvatures at t=10=0,
must also be zero ({e‘x(1,,)}={0}) if there is no provocation by initial mechanical
actions. In addition, if mechanical actions are never introduced ({NMM(1.)}={0}). Eq.

2.4.44 is reduced to

{8041.)}, - [213043; my!

i-l

[2‘01“ +R})[ {WIRE-5&4 t,e"c(1,_,)’)} - [ABDé’ij 413] { ”471-183 2.4.45

+ R” {WUC tuc- Tn’ EMC( Thy} )

The associated stresses could be computed from the strain and curvature rates by
the linear viscoelastic stress-strain relations defined in Eq. 2.4.6. However, the strain and
curvature rates are known only at discrete time points rather than continuously, and
therefore the integral linear viscoelastic stress-strain relations of Eq. 2.4.6 needs to be
degenerated to express stresses in terms of strain and curvature rates at those discrete
points. By the same numericalization technique as specified in Eq’s. 2.4.25 - 2.4.29, the

stress for lamina k is degenerated to

whici

Strait

59

t- 1.

tom}.- f [744-4)],teu4:)}'.44

0'

.. [?(;)]k{gfl(a)}k + 2.4.46

j-i +1 1""

g]: g R, A r[ [24 Z“ R, A 1)L{ #419}; - [74212, A 1)] 1{ £M(r,_,)}’k]

which, by further expansion and regrouping and by assuming zero sudden step input of

strains and curvatures at t=10=0 ({eM(0)},,={0}). becomes

1442"}. ~

2.4.47
£211 [gm-1"R91715f44213 #45,»; + R'[7(0)]k{e"(r_)}’k
where
{ yup}; - { 80(r,)}’ +z{x(r,)}’-{ #6419}; 2.4.48
as in Eq. 2.4.8.

For a laminate as depicted in Figure 18, its vertical deflection at an instant I could

be determined by

W609”) " - £0,101!” + 402’ + x,,(t)xy) 2.4.49

if the laminate’s geometrical center (x/2, y/2, z, t) is fixed to the coordinate reference

point, and the mid-plane of the flat laminate coincides with the z=0 x-y plane. This is

 

 

 

 

 

 

 

 

 

 

 

 

Figure 18. Coordinate positions of the yellow-poplar laminate and beam.

1831

11611

3.4.

the t
We
prob
00m;
Such
Over
The

60nd.

1501114

61

readily verified by use of the curvature-vertical deflection relation defined in the strain
field of Eq. 2.3.9.

In summary, Eq’s. 2.4.45, 2.4.41, 2.4.47, and 2.4.49 form the set of formulas
of the developed linear viscoelastic plate theory for solving numerically for the history
of strain and curvature rates, strains, stresses, and vertical deformations of a laminate
described in Figure 18 subjected to hygroscopic effects. Upon the added presence of
mechanical actions and sudden elevation of mechanical strains and curvatures at t=0,

Eq’s. 2.4.44 and 2.4.46 replace Eq’s. 2.4.45 and 2.4.47 respectively in the formula set.

2.4.4 Linearity and Isothermal Requirements

The linear isothermal viscoelastic constitutive equations of Eq. 2.4.3 or 2.4.4,
the basis upon which the LVP theory is developed, determined that the analytical
governing equation of Eq. 2.4.23 is only applicable to linear isothermal viscoelastic
problems which are defined by the stress independence of the relaxation moduli or creep
compliances (linear), and the steady and uniform temperature distribution (isothermal).
Such application restrictions on the integral governing equation (Eq. 2.4.23), once carried
over to its numerical equivalence (Eq’s. 2.4.43 - 2.4.45), however are much lessened.
The first step (Eq. 2.4.25) in the numericalization process which establishes the
conditional numerical-integral equivalence, clearly determines that the linearity and
isothermal requirements only need to be imposed within each time segments 1,. - 71-1: R,
A1, while nonlinearity and nonisothermal conditions would be allowed across time

segments in the numerical equivalence (Eq’s. 2.4.43 - 2.4.45). Further, since time

teas
1501
3911
the

ther

mois
Whit

inan

relax
h1g1
CXpa

P101).

62

segments are sufficiently small due to the small interval assumption (Eq. 2.4.27), it is
reasonable to approximate nonlinearity and nonisothermal conditions with linear and
isothermal behavior within each segment. Therefore, the linearity and isothermal
application restrictions are no longer imposed on the numerical equivalence outlined in
the formula set, and the LVP theory is therefore applicable to nonlinear and noniso-
thermal viscoelastic problems such as hygroscopic warping.

If parallelness can be drawn regarding the effects of temperature and that of
moisture content, the aforementioned isothermal requirement extends to moisture content,
which would require a uniform and steady moisture content distribution for hygroscopic

materials.

2.4.5 Hygroscopic Strain Rates and Relaxation Moduli

It is evident in the set of formulas that the hygroscopic strain rates and linear
relaxation moduli denoted as {eMc(t)} ’, and [Y(t)],, are two of the essential inputs. While
hygroscopic strain rates {eM°(t)}’,, are the results of changing moisture contents and
expansion coefficients, linear relaxation moduli [Y(t)],, are determined by the viscoelastic
properties of constituent laminas which are the subject of the next chapter. As the
theoretical development of the CLT-based linear viscoelastic plate theory is the focus of
this chapter, the discussion and formulation of all required inputs will be furnished in the
last chapter when the numerical form of the LVP theory is applied to an actual physical

laminate.

3.1

ma
the

des

0UP

reqi

lime

c011:

and

Dav

”1014

$3110

CHAPTER 111

CHARACTERIZATION OF VISCOELASTIC BEHAVIOR

3.1 Introduction

The proper characterization of the viscoelastic behavior of the constituent
materials in a laminate in the form of relaxation moduli is the key to the application of
the theory to practical problems. A perfect and complete description, though ideally
desired, may not be obtained since it is, in many cases, not only painstakingly difficult
to achieve, but also unnecessary as the associated efforts and complexity may far
outweigh any benefits and the limited gains may prove to be overkill under given
requirements. Therefore, an adequate description which must be contemplated in light
of balancing factors should give a sufficiently accurate characterization, but at the same
time should be as simple an expression as possible, and achievable with moderate effort.

It is known that wood and wood-based materials behave viscoelastically as a
consequence of their amorphous and crystalline components, cellulose, hemicellulose,
and lignin, and the structures revolving among them [Alexopoulos, 1989; Pentoney and
Davidson, 1962]. Characterization of the viscoelastic behavior based on a complete
molecular theory, though theoretically feasible, is rather difficult to develop however,
due to the complexity and lack of understanding of the wood components and the

structures of both wood and wood-based materials. As a result, the most dominant

63

all)“
is th

that

“004
501111

14131

and

line;

Othe
all p

014

64

approach has been from the phenomenological point of view, that is the macro behavior
is the subject of investigation while transcending the micro-mechanism responsible for
the viscoelastic behaviors. It is not surprising that the same approach is adopted here.

It is widely proven in the enormous literature and very evident in applications that
wood and wood-based materials generally behave nonlinearly viscoelastically. The
sometimes so-called linear regions are no more than regions where nonlinearity is
relatively insignificant as a result of minimum influence from low and/or moderate levels
of stress, moisture content and temperature, and can be approximated to be linear. This
assumption is usually necessary for achieving a successful analysis, yet at the sacrifice
of accuracy that may be large when nonlinearity becomes relatively pronounced.

An attempt is made in the following to quantitatively characterize the nonlinear
and nonisothermal viscoelastic behavior of wood and wood-based materials, but with
linear concepts and linear mechanical viscoelastic models for simplicity and ease of

analysis.

3.2 Literature Reviews

New ideas and additional work are always conceived and build upon the work by
others of today and yesterday. This thesis is no exception. A review and discussion of
all pertinent information is the foundation for the development of a quantitative depiction

of the nonlinear viscoelastic behavior.

3.3.

C0ll'.

Whli

she
and

tatit

‘11

ll 01
I01

Wit

65

3.2.1 The Nonlinearvity - Relation Between Viscoelastic Pr0perties and Stress Levels
Viscoelastic behavior is defined as being linear if stress independence of creep
compliance J(t) is observed in a creep test (or relaxation modulus in a relaxation test),

which is defined as

1(4) - 3—(5- 3.2.1

g

0’

where J(t) and e(t) are creep compliance and strain history, respectively and or}, the
applied time-wise constant creep stress in the test. The relative creep is defined as the

ratio of creep strain over the initial instantaneous strain,
dfl-dw
8(0)

- 119‘” ”(OW 3.2.2
J(0)a'

_ J(t) - 1(0)
no

8 R0) '

It obviously observes the same stress independence, and is therefore a criteria often used
to check for linearity. The stress independence is the equivalent to strain being linear
with stress, as indicated in Eq. 3.2.1.

Reported limits of stress levels within which wood may be assumed to be linearly
viscoelastic vary widely due to differences in testing environments, designs, species, and
loading with respect to grain direction. Nielsen [1972] reported 30 percent of the ultimate

static strength to be the limit of linearity, yet according to a review by Schniewind [1968]

11131

pert

Clea

iaCl

vise

no

intt:
app
litn

10]

tel:

311

CO

81;

66

many authors in their experiments arrived at values from 38 percent to as high as 75
percent. It was shown that once above those limits wood and wood-based materials
clearly exhibited nonlinear viscoelastic behavior. It is a well documented and established
fact that a region where wood and wood-based materials can be treated as being linearly
viscoelastic definitely exists. Due to the simplicity of linear viscoelastic analysis, it is
of great practical significance if wood and wood-based materials in use would usually fall
into such linear regions and therefore can be treated so. Since actual load levels in
application are rarely greater than 50 percent of the static strength which is about the
limit of linearity found by many authors [Schniewind, 1968], linear treatment may seem
to be a rather promising approach as suggested by Schniewind [1968].

In further examining the test conditions under which those limits of linearity were
arrived at, one may raise some doubts as to the validity of the linear treatment. The
relative humidity and temperature in those tests were low and moderate, and maintained
at constant levels. In contrast, they vary continuously in actual applications and could rise
to very high levels. Bach and Pentoney [1968] found that the deformation-stress relation
of maple subjected to longitudinal tension was definitely nonlinear at high humidity and
temperature values.

Bach and Pentoney [1968] succeeded in expressing total compliance as well as its
components as quadratic functions of stress, moisture content, and temperature in
longitudinal tension tests on maple in which the separate effects of the three variables
were included. They showed that wood responds to longitudinal tension stress as an

elastic body at low temperatures and moisture contents (total compliance being constant).

 

 

(1111

1110

the

c0n

”11111

int

and

Greg

67

At very low stress, wood approximates the response of a linear viscoelastic body (total
compliance stress independent) even if the temperatures and moisture contents are
relatively high, and changing. However, stress dependence of the compliance (character
of nonlinear viscoelasticity) at moderate and higher stress levels when varying moisture
content and temperature is evident in the compliance expressions by Bach and Pentoney
[1968] who pointed out that wood must be treated as a nonlinear viscoelastic material to
be realistic.

Compliance results by Bach and Pentoney [1968] were obtained for the grain
direction which is the most crystalline and elastic since the orientation of microfibrils are
mostly in the grain direction. Nonlinearity is expected to be much more prominent across
the grain of wood.

Though functions of moisture content, temperature, and stress levels, the
compliance results by Bach and Pentoney [1968] do not represent the combined actions
of the involved variables on actual compliances since Bach and Pentoney did not really
vary moisture content and temperature under stress during their tests, but kept them
constant in each creep test and varied their levels between tests. The actual effects of
changing moisture content and temperature under stress are much more severe and
complex. Numerous references have indicated that creep rates and the total creep were
much larger when moisture content and temperature were changed under load rather than
prior to loading and kept constant under load. The interaction between mechanical stress
and moisture content-temperature changes obviously contributes a great deal more to

creep than if they acted alone separately. Consequently, a creep compliance which would

01
1111
161

181

Wh

and

3,1

SOr

I0 1

68

otherwise have been stress independent (linear viscoelasticity) will be stress dependent
(nonlinear viscoelastic behavior) in addition to being dependent on moisture content. To
reflect the coupling effects of stress and moisture content changes, the interaction is often
referred to as the mechano—sorptive effect. The prevalent view regarding the mechanism
[Gibson, 1965] suggests structural changes at the micro level. Since structural changes
occur under temperature changes [Davidson, 1962] as well, temperature changes under
load are speculated to exert similar thermal-mechanical coupling effects.

In short, the region in which wood and wood-based materials may be assumed to
be linearly viscoelastic, though it exits, unfortunately has not as much practical
significance as expected, largely due to the pervasive humidity and temperature changes
(mechano-sorptive and thermal-mechanical coupling effects) in application environments
which violate the conditions under which the linear assumption is valid. In reality wood

and wood-based materials inevitably exhibit nonlinearity in their viscoelastic behavior.

3.2.2 Relation Between Viscoelastic Properties and Moisture Contents, and Mechano-
Sorptive Effects

Moisture acts as a plasticizer, and increases in moisture content will usually lead
to increases in creep compliance and decreases in relaxation modulus [Schniewind,
1968]. Bach and Pentoney [1968] found in longitudinal tension on maple that total
compliance, as well as its components, were proportional to quadratic terms of moisture
content. Increase in creep at increasing moisture content was also observed in bending

[Nemeth, 1964; Ota and Tsubota, 1966], in tension and compression parallel and

1511"
in to
tep01
netet
tesrs.

direc

011111
1111411
Stres:

beha

loade
Whicfi
beam
tteeF
While

16st,

creep
and 1

11951

69
perpendicular to the grain of beech [Schniewind, 1968] and red oak [Youngs, 1957], and

in torsion of Hinoki [Norimoto, Hiyano and Yamada, 1965]. Sharp stress relaxation was
reported with increases in moisture content [Ugolev, 1964; Kunesh, 1961]. It is to be
noted that moisture content changes were not introduced during creep and relaxation
tests, but prior to the tests. For this reason, this phenomenon is sometimes called the
direct effect of moisture content as in a review by Schniewind [1968].

The moisture content changes in wood and/or wood-based materials as a result
of the frequent and sometimes drastic humidity changes in their application environment
unfortunately coincide with the materials being in one or another state of loading or
stress. As briefly mentioned earlier, the resulting coupling effect on the viscoelastic
behavior is much greater than if moisture content changes had acted alone.

Armstrong and Kingston [1960] noted that small beams made of three species
loaded green and kept so during creep test showed about the same relative creep as those
which were loaded and kept dry, but the relative creep doubled when initially green
beams were allowed to dry under load during the creep test. They also indicated that
creep increased markedly in wood subjected to increasing moisture content during test
while under load as compared to the case where moisture content was raised before the
test. Similar trends were observed in relaxation test of beams [Armstrong and Kingston,
1960]. Cyclic changes under load amplify the above effect and lead to much larger total
creep as observed by Armstrong and Christensen [1961] and Hearmon and Paton [1964],
and to greater stress relaxation rate and residual deformation, as reported by Lawniczak

[1958], Takemura and Ikeda [1963], and Takemura [1966 & 1967]. For example,

70

extensive cycling of moisture content at a given load caused a beam to deflect over 20
times more than keeping the beam conditioned at the extreme moisture content at the
same load throughout the test [Hearmon and Paton, 1964].

Some manifestations of this mechano-sorptive effect are actually very common
phenomena which have been used to advantage in practice. During either kiln or air
drying, timber tends to distort because of spiral grain, growth stresses, reaction wood,
and even its own gravity weight. However, good stacking and application of loads to the
stack while drying takes place, can prevent most of such distortions [Koch, 1971]. In
contrast, stacking and application of load even for an extended period of time after the
timber is dried eliminates little of such drying distortions. In restrained swelling, the final
pressure required to keep a wood body within its original dimensions as the body absorbs
or desorbs moisture in the restraining process is much lower than that required at the end
of the restraining process if the body’s moisture content is raised or lowered preceding
the restraint [Perkitny and Kingston, 1972]. Therefore, the resistance force exerted to
a structure by a restrained wood member due to its hygroscopic expansion or shrinkage
is greatly minimized.

Hearmon and Paton [1964] noticed that the known moisture content-MOE relation
was not changed by cycling moisture content under load, suggesting that the elastic
component is only affected by moisture content, regardless of whether and how moisture
content is changed. Grossman [1976] indicated that upon removal of a load maintained
during moisture content changes as much immediate and delayed recovery of shape

occurred as would occur had the moisture content changes preceded the application of

71

load, indicating that both the elastic and the recoverable components of the deformation
do not suffer from the mechano-sorptive effect. Only the flow portion of deformation are
susceptible to the mechano-sorptive coupling.

One of the explanations regarding the mechanism is that the coupling effect results
from interaction between the applied stress and stresses arising from the swelling or
shrinkage differential caused by moisture content gradient. Armstrong and Christensen
[1961] in testing both 1mm thick beams so as to eliminate the influence of moisture
content gradient and 2cm thick beams for comparison observed similar effects on beams
of both sizes, which suggested that stresses arising from nonuniform expansion or
shrinkage were not likely involved. The most attractive hypothesis which is capable of
explaining most aspects of this effect is that during the breaking and remaking of
hydrogen bonds that must occur during moisture content changes in either direction,
stress bias will favor slippage, resulting in new positions and shape [Schniewind, 1968;
Gibson, 1965; Grossman, 1976].

There are many other aspects and characters to this mechano—sorptive effect as
thoroughly discussed by Grossman [1976]. It is a phenomenon of considerable interest
and practical significance, but nevertheless remains very perplexing with a great lack of
understanding regarding its description and interpretation. A number of postulates were
proposed over the years, yet fail to explain all the features of this effect cohesively.
Finding a characterization that is compatible with all the observations remains a big
challenge to wood scientists, and may take years of extensive research effort cooperative-

ly put forward by all interested researchers.

72

Significant as this effect is, its lack of understanding also presents an impasse at
the moment. For this reason, it was not included in the creep experiments and
characterization of nonlinear viscoelastic behavior in this study. However, the direct
effect of moisture content and stress level changes will be accounted for as they, even
in the absence of mechano-sorptive effects, still exert great effects on the behaviors of
wood and wood-based materials (hygroscopic materials).

An interesting characteristic displayed by wood and wood-based materials stands
alone among materials in that the permanent deformation resulting from direct .and
mechano-sorptive effects of varying moisture contents under load are not really totally
permanent, but at least a large portion can be recovered when the unloaded material is
taken through another moisture content cycle. This is revealed by permanent defamation
due to mechano-sorptive effect by Armstrong and Christensen [1961] and Christensen
[1962]. It is demonstrated that the strain energy is not dissipated in the loading process
and that some component within the structure ”remembers” its original length and shape
[Grossman, 1976]. This perplexing phenomenon, while a very significant and live
component in the viscoelastic behavior of wood and wood-based materials because of the
constant irregular humidity variation (cycling) in actual applications, can not be

characterized yet due to the little information and understanding of it.

3.2.3 Relation Between Viscoelastic Properties and Temperature, and Thermal-
Mechanical Coupling

As in the case of moisture content, we can distinguish between direct effect of

73

temperatures and interaction effect of temperature changes while under stress [Schniew-
ind, 1968].

In general, increase in temperature leads to increase in creep compliance and
decrease in relaxation modulus. In hard maple subjected to tensile creep parallel to grain
in the temperature range from 30 70° C, the total creep compliance as well as its
components were reported to be proportional to quadratic terms of temperature [Bach and
Pentoney, 1968]. In bending creep test on green specimens, the deflection increased
exponentially with increasing temperature in the range of 5 to 70° C [Kitahara and
Okabe, 1959]. Similar relation is observed in compression creep on oven-dry Hinoki in
the temperature range from 100 to 180° C [Arima, 1967]. In torsional creep of Hinoki,
the temperature effect seemed somewhat inconsistent, but in general caused an increase
in creep compliance with increasing temperature [Norimoto, Hiyano and Yamada, 1965].
The results of Norimota and Yamada [1965] showed a decrease in relaxation modulus
as temperature rose from 15 to 625° C, though the relaxation rate was affected little
within this range. Other authors also found increased relaxation at elevated temperatures
[Youngs, 1957; Ugolev and Pimenowa, 1963]. The results in many other references
show the same general trend, but are rather diverse and inconclusive in terms of a
quantitative description of the temperature effect.

More profound than the direct effect of temperature is the interaction arising from
temperature changes occurring under load during test (thermal-mechanical coupling). It
is noted in creep bending [Kitahara and Yukawa, 1964] that when the temperature was

raised from 20 to 50° C during a creep test, the creep response was greater than that

74

occurring under constant temperature at the highest level throughout the test, though
cooling during the test resulted in a sharp reduction of creep.

Temperature changes when coupled with moisture content changes while under
load result in even more complex viscoelastic behavior [Schniewind, 1968]. This,
unfortunately, occurs commonly in the applications of wood and wood-based materials.
Very limited information regarding this phenomenon is available and much lack of
understanding remains.

To isolate the direct effect of moisture content changes on the viscoelastic
behavior, both the direct and interactive effects by temperatures variations are not
included in this work. The results of the direct effect of increasing moisture content may
be extended to the direct effect of increasing temperature to some degree of approxima-
tion according to Bach and Pentoney [1968], who noted an equivalence between one
percent moisture content increase and 6° C temperature increase regarding their direct

effects on the viscoelastic properties of hard maple tensioned parallel to the grain.

3.2.4 Functional Forms and Linear Viscoelastic Models

One of the basic aims of the study of viscoelastic behavior is the determination
of mathematical representation of material response in relation to material parameters and
time. Numerous models have been proposed in the literature which simulate the real
materials with varying degree of success. In essence, they fall into two categories:

empirical curve fitting, and mechanical models (springs and dashpots).

75
3.2.4.1 Empirical Models
Clouser [1959] in his bending creep work found that over relatively long time
spans, the power law (parabolic) model was rather useful. Other researcher have had
good success in using this model under mild and high humidity conditions. The model
takes the form
3.2.3

2(1) - £0 + at”

where

e(t) - total strain
e0 - instantaneous elastic strain 3 2 4
t - elapsed time

a,m - constants

A variant is

8(t) - at" 3.2.5

which was also employed by some others. These two models prove to be the most

popular representations.
King, Jr. used another form for very short term experiments in tension parallel

to grain [King, Jr., 1961]:

e(t) - ‘0 + blog(t+ 1) 3.2.6

The sum of Eq’s. 3.2.5 and 3.2.6 was also used to fit experimental results [Yamada,

76
Takemura and Kadita, 1961]. Bodig and Jane [1982] compiled a list of commonly used

empirical equations in their publication on wood and wood mechanics.

3.2.4.2 Linear Mechanical Models

The empirical representations attempt to achieve the best mathematical fit to
existing experimental data. They are simple and successful, but, only valid within the test
conditions under which they are obtained. Different models may have to be employed
once test conditions are changed to produce good fit. It is obvious that this test condition
dependence does not allow comparison across tests in different environments, thus
reducing the empirical models’ application significance. And moreover the empirical
method does not help in providing insights into the underlying processes controlling the
viscoelastic behavior.

Rheological models or mechanical models, however, do not have those
limitations. They are comprised of springs and dashpots interconnected in various
fashions and model linear viscoelastic behavior if the springs and dashpots are linear
elements. To be specific, linear springs (Hookean springs) which feature linear propor-
tionality between stress and strain are used to simulate elastic components. Linear
dashpots (Newtonian dashpots) possessing linear proportionality between stress and strain
rates represent the flow components.

The most significant classical linear models are the Kelvin element, the Maxwell
element, and the various combinations of the two, shown in Figure 19. A Kelvin element

consists of a linear spring and a linear dashpot in parallel, while a Maxwell element

77

Ems

 

 

 

 

 

 

1

W.

 

 

 

 

 

IV»— Maxwell Element

Kelvin Element

Maxwell Chain

Kelvin Chain

 

Figure 19. Classical linear viscoelastic models.

78

consists of a linear spring and a linear dashpot in series. A Kelvin chain is any number
of Kelvin elements connected in series, whereas a Maxwell chain is any number of
Maxwell elements linked in parallel.

The most convenient and simple linear model which has been used successfully
to describe wood and wood-based materials is the four-element Burger body found in the
book by Bodig and Jane [1982]. It is a Kelvin and a Maxwell element chained in series,
as seen in Figure 20. By and large, it is accepted as capable of sufficiently and
qualitatively explaining some of the fundamental aspects of the viscoelastic behavior of
wood and wood-based materials. The Kelvin element accounts for the recoverable
component of the behavior, while the spring and dashpot in the Maxwell element
represent instantaneous elastic and flow behavior, respectively.

A good illustration of the creep deformation process by this Burger body is given
in Figure 21. As seen, when a load is suddenly applied at to, only the spring responds
by an instantaneous deformation equal to the product of the spring constant (E) and
applied load (1’) while the others remain in their original positions. With passage of time
under the action of the load (P), both the Kelvin element and the Maxwell dashpot begin
to extend gradually. Upon the removal of the applied load (P) at (t,), the Maxwell
spring instantaneously returns to its original position. The Kelvin element will return to
its undeformed position too (elastic), however over a period of time, showing delayed
character. The lone Maxwell dashpot will remain in its extended and irrecoverable
position. For a more detailed discussion refer to Bodig and Jane [1982].

The mathematical form of the model, expressed in terms of creep compliance, is

79

 

'lmd

 

 

’1“ Eu

 

 

 

Figure 20. Four-element Burger body.

Applied Load

Deformation

Figure 21.

4

80

 

 

 

1_

—.

O

:3
:I
3
(D

flow

4

   
 

 

(b)

A typical creep curve [Bodig and Jane, 1982].

(a) load-time function.
(b) deformation-time function.

Me
(to

the

i521

"162

into

81

J(t)'-€(-t:)-- I +(1—e-dt)+ t

0 En: Eb "m4 3.2.7

 

 

- J(0) + J,(t) + 1p)

where J(t), J,(t), and J,(t) are the total compliance, the recoverable compliance, and the
flow compliance, respectively. e(t) and o" are the total strain and time-wise constant
creep stress. E”, E” "to and 1),, are respectively the Kelvin and Maxwell spring
constants, the Kelvin and Maxwell dashpot coefficients. The first term represents the
Maxwell spring (elastic compliance), the second term represents the Kelvin element
(recoverable compliance), and the last the Maxwell dashpot (flow compliance). Clearly,
the total compliance is the sum of the contributions of the three elements.

The retardation time for the Kelvin element, 1, defined as

r - 22 3.2.3

is determined by the relative ratio of the Kelvin spring constant over the Kelvin dashpot
coefficient. With t=1 time elapsed, a Kelvin element has extended about 1/3 (almost one
e“) of or recovered about 2/3 of the total possible extension depending on whether a load
is applied at zero extension or released at full extension. It is therefore a convenient
measure of the retardation of the element’s response.

The Burger body may be extended by inserting any number of Kelvin elements
into the serial linking. Each element may have a different retardation time (defined in

Eq. 3.2.8) which characterizes the time response of the element. Such models with

rel

82

discrete retardation times have been applied by several authors [Ugolev, 1963; Rose,
1965; Ethington and Youngs, 1965]. Higher degree of approximation is expected of the
extended Burger body in describing the recoverable component.

Since each Kelvin element’s contribution to creep compliance is

1,0) - £1112 3.2.9
Eh

the creep compliance of a Burger body with N Kelvin elements, in analogy to Eq. 3.2.7,

is

N -dr
J0) - 3.“. - _I_+E.§1_1__')_+_‘_
0' E,” i Eb! ",4 3.2.10

- 1(0) + 1,0) + 1,0)

The extreme is reached when the number of Kelvin elements approaches infinity, at
which time retardation times are continuously distributed. The summation in Eq. 3.2.10

becomes an inteng and the total compliance is

J(t) - fl - 31.4 fur)(1-e'7)dznr+; 3211

0 "M

- 1(0) + 1,0) + 1,0)

where L(r) is the spectrum of retardation times.
It is possible to approximate both the retardation spectrum and its counterpart the
relaxation spectrum from experimental data if they do exist. Both Alfrey [1948] and

Leaderman [195 8] developed a first approximation method. The result is quite good , but

YI

pr

83

only when the distribution is constant or changing slowly. Schwarzl and Staverman
[1953] have extended this method to yield a better approximation. Pentoney and
Davidson [1962] found in their study that the spectrum was quite flat, while Noritomo
did find spikes in the spectra. The experimental work in obtaining the spectra is no easy
task. As the moisture content and the temperature change, the spectra will both shift and
change as a result of their sensitivity to moisture content [Norimoto, Hiyano and
Yamada, 1965] and temperature [Davidson, 1962]. Obtaining spectra functions in terms
of moisture content and temperature should be more challenging.

It should be emphasizedthat nonlinearity in a recoverable component would void
all the effort on spectrum analysis which is based on linear mechanical models. The
recoverable component may possibly be characterized as being linearly viscoelastic only
at low stress levels and at temperature below 50° C [Davidson, 1962], and at low and
moderate moisture content [Bach, 1968]. As significant as the contribution of the infinite
Kelvin chain is in the extended Burger body, the extensive effort involved results in no
more than a better depiction of the recoverable component, while a much larger part of

the viscoelastic deformation is due to the flow element.

3.2.4.3 Nonlinear Models

It has been clearly indicated that at least the flow component in wood is non-
Newtonian (nonlinear) [Davidson, 1962; Ethington and Youngs, 1965; Bach, 1965;
Youngs and Hilbrand, 1963]. Among the attempts to model nonlinearity, Kiihne [1961]

proposed a hinged frame containing linear springs and dashpots where nonlinearity arises

84

out of frame geometry. Kingston and Clarke [1961] had moderate success with reaction
rate theory and a model following the hyperbolic sine law. Ylinen [1965] devised a
model consisting of a Maxwell element paralleled with a nonlinear spring to be applied
to compressive loading parallel to the grain and to the long term load effects on columns
[Ylinen, 1966]. The most complete treatment of nonlinearity is an empirical one by Bach
[1965] in which total creep compliance, as well as its three components (instantaneous
compliance, recoverable compliance, and flow compliance) were expressed as a function
of linear and quadratic terms of logarithmic time, stress, moisture content, temperature,

and their products.

3.3 Creep Experimentation

As stated in the literature review section, only direct effects of moisture content
and stress levels will be tested.

Yellow-poplar (Liriodendron tulipifera) is the species chosen for the creep
experiments. However, only viscoelastic properties in the radial direction will be tested
and characterized. This will satisfy the input requirements for the application of the LVP

theory, as detailed in the last chapter.

3.3.1 Experimental Design
Tension creep tests were carried out in the radial direction at three stress levels,
approximately 25 % , 50% , and 90% of the static strength, and at three moisture contents,

11.5%, 15.7%, and 21.5%, respectively. Temperature was maintained constant. With

85

two replications, there were a total of 3 X3 x3 = 18 creep tests.

Edge grained boards were cut from large flat sawn yellow-poplar lumber free of
juvenile wood. After being jointed and planed to appropriate thickness, they were edge
glued with epoxy to form larger edge grained panels which after conditioning at 70° F
and 65 % R.H. were jointed and planed again to achieve flatness and a final thickness of
1/4 inch.

Long and slender radial tension specimens of 1/4 inch by 1/4 inch cross section
and about 12 inch in length were cross cut from the prepared flat edge grained panel.
The specimen ends were inserted into tension grips, leaving a usable specimen length of
about 9 inches. The sample dimension and configuration are illustrated in Figure 22.

Holes were drilled through the tension grips at the ends of the tension specimen
to fit loading pins. One of them anchored the creep sample to a frame, while the other
transferred the static creep load (consisting of dead weights) to the creep specimen.
Figure 23 shows the loading and extensometer assembly.

The three stress levels were achieved by combinations of different weights. The
three moisture content levels were achieved by a light and simple apparatus affixed to
the loading assembly. As shown in Figure 24a and Figure 24b, the apparatus consists of
a rectangular plexi glass plate and two layers of thick plastic bags. The plate is supported
by the top pin by means of four wires attached to the plate’s four corners, and the hole
in the plate center allows the weight hanger rod to pass through so that the plate does not
interfere with the loading mechanism. The two-ply plastic bag with a hole in its closed

end for the hanger rod to go through encloses the plexi glass plate and the entire

86

 

 

 

 

 

 

 

 

L/l/
\/\\ \\

 

 

 

 

\J/

Figure 22. Radial tension creep specimen of yellow-poplar.

87

Figure 23. Loading frame and extensometer assembly.

 

88

Figure 24a. Complete test setup with flexible humidity chamber open.

88A

“T.*°~o-o»o-—o
“ o 0'9 3'0
3......“ .-

 

89

Figure 24b. Complete test setup with charged flexible humidity chamber enclosing
tension creep specimen.

cum-o.

3

0
0..

9

0-
l

a~o~

 

9O

specimen. Tying up the open end of the plastic bag tight above the top pin creates an
insulated humidity chamber in which the tension specimen could be placed and loaded.
The flexible chamber is very small and is sealed at the top and at the bottom. By placing
two dishes of saturated salt solutions on the plexi glass plate inside the flexible chamber,
appropriate relative humidities could be created and maintained. (Figure 25a and Figure
25b are closeups of the flexible humidity chamber.) The three nominal relative humidities
for achieving the three moisture content levels are 66% RH, 86% RH, and 93% RH

which were created over saturated salt solutions listed in Table 3.

Table 3. Relative humidity over saturated salt solutions.

 

Relative Humidity (96)

 

Chemical @ 20° C

Sodium Nitrite 66

Potassium Chloride 86
_Ammonium Phosphate 93

 

The specimen was placed into the loading mechanism and the flexible humidity
chamber charged with appropriate salt solution with no load applied. As shown in Figure
24a, an calibrated lnstron extensometer of 1 inch gauge length was affixed to the center
portion of the specimen. Any extension within the gauge length was detected and
recorded by the connected strain indicator.

To reduce possible slippage between the contact edges of the lnstron extensometer
and the specimen surface, the specimen was first sanded and then coated with a very

narrow thin smooth layer of super glue where the extensometer edges contacted the

91

Figure 25a. Closeup of open flexible humidity chamber.

91A

 

92

Figure 25b. Closeup of closed charged flexible humidity chamber.

 

93

specimen. To further eliminate any possible slack in the assembly, a 20 psi load was
applied to the whole setup for 2 seconds, which increased reproducability.

While the stress-free specimen was being conditioned in the flexible humidity
chamber to the equilibrium moisture content, the strain indicator indicated the
hygroscopic expansion of the specimens. The extensometer-indicator system was
calibrated to a sensitivity of 1/50000 extension per inch per scale unit indicator reading,
that equals to an expansion strain of ((1/50000)/1)100 = 1/500% = 0.002%. (The
extensometer-indicator system can be calibrated to higher sensitivity, but readings
become unstable at sensitivities finer than 0.001% expansion strain.) When the indicator
reading did not change more than 2 units (a 0.004% extension strain) over 12 hours,
equilibrium moisture content was assumed to have been reached. At that point, the
indicator was reset to zero and predetermined load was applied to the creep specimen by
lowering a laboratory jack. The instantaneous elastic response was recorded immediate-
ly. Subsequent creep extension readings were recorded at selected time intervals.

Creep was allowed to continue in the charged flexible humidity chamber for about
600 hours, after which time the load was released by raising the laboratory jack. The
instantaneous recovery was immediately recorded. The delayed recovery was recorded
intermittently till the indicator reading became stable. The whole assembly was then
taken down and preparations were made for the next creep test.

The modulus of elasticity (MOE) of a sample can be tested nondestructively with
the extensometer-indicator mechanism. By subjecting the specimen to increments of load

and recording corresponding strains on the indicator, a linear stress-strain relation was

94
obtained. The tangent of the linear line is the MOE. However, this test procedure by this

must be completed within a couple minutes to avoid the onset of time dependent
behavior.

All specimens, after having been conditioned and equalized at room conditions
of 70° F and 65% RH, had their MOE so tested. Those with very similar MOE values
were selected for the 18 creep tests to assure uniformity among test samples. The
statistical basis for this selection was the normal distribution of the MOEs. Samples
whose MOEs were around the average of the sample population were retained with
others eliminated.

A small piece of control sample of yellow-poplar was placed into the charged
flexible humidity chamber at the time a test sample was fitted into the loading set up. It
was weighed both at the end of the test and after subsequent overnight oven-drying at
103° C, after which the equilibrium moisture content were computed.

The test setup is very simple and worked well, yet it is also very delicate and
sensitive. Even the slightest disturbances to any part of it, like, repositioning of the
extensometer wire, caused at least small changes of the indicator reading, and therefore
had to be avoided. It is far from fool-proof, but care and practice reduce failures and

mistakes.

3.3.2 Experimental Results and Observations
The recorded indicator readings were converted to extension strains based on the

calibrated extensometer-indicator sensitivity. Dividing the extension strains by the applied

95

constant creep stress yields creep compliance

1(2) - fii 3.3.1

1

0’

The initial creep compliance J,=J(0) is the elastic instantaneous strain divided by creep
stress, which is the elastic compliance. Creep compliance can be normalized by dividing

it by the corresponding initial creep compliance, that is

.. fl _ fl
Mt) 1(0) Jo 3.3.2

JN(t), the normalized creep compliance which is a compliance ratio, is related to relative
creep 63(1) by

1”“) - 6R0) +1 3.3.3

according to Eq’s. 3.3.2 and 3.2.2.

For comparability, data of all tension creep tests compiled in Table 4 were
expressed in terms of normalized creep compliance. They are the average of two
replications of tests for each of the nine test conditions outlined in Table 5. Nine sets of
normalized creep compliances are presented in Figure 26.

It is shown in Figure 26 that the three normalized creep compliances for 11.5%
moisture content almost coincide with each other. The stress level has no effect on the
creep compliances (the criteria for linear viscoelasticity). At the two higher moisture

content levels, however, the normalized creep compliances increase substantially with

96

Table 4. Normalized creep compliance of yellow-poplar in the radial direction at nine
creep test conditions.

 

Test Condition Number
1 1 12 13

 

 

 

 

 

 

 

 

 

 

 

 

 

 

t (hour) J(t)/J(0) t (hour) J(t)/J(0) t (hour) J(t)/J(0)
0 1 0 1 0 1
0.42 1.043 0.5 1.1 0.17 1.062
1.25 1.1 l 1.17 0.33 1.085
2.92 1.143 2 1.157 0.5 1.101
9.25 1.193 9.67 1.22 1 1.127

19.42 1.25 41.51 1.318 1.33 1.144

32.92 1.293 118.81 1.425 1.67 1.149

42.92 1.314 184.22 1.473 2 1.158

67.5 1.357 ‘ 253.1 1.51 3.33 1.175

104.17 1.4 335.14 1.54 5.08 1.189

152.08 1.443 440.21 1.57 6.75 1.203

209.1 1.478 541.31 1.586 9.67 1.22
207.17 1.521 600 1.598 19.83 1.26
423.58 1.55 600 0.618 31.5 1.294
544 1.578 600.54 0.592 41.5 1.319
600 1.585 602 0.58 66.08 1.364
600 0.6 619.5 0.552 118.8 1.427
600.17 0.586 654 0.536 184.08 1.474
600.5 0.578 745.19 0.528 253.83 1.511
602 0.564 335.08 1.54
619 0.535 440.93 1.571
643.6 0.521 541.5 1.588
725 0.514 600 1.599
600 0.619
600.19 0.607
611.54 0.593
602.5 0.579
618.7 0.554
654.2 0.537
765 0.528
Test Condition Number
21 22 23

t (hour) I(t)/J(0) t (hour) J(t)/J(0) t (hour) J(t)/J(0)
0 1 0 1 0 1
0.5 1.08 0.08 1.1 0.07 1.15

5.5 1.292 0.17 1.128 0.15 1.205

97

Table 4 (cont’d)

 

 

 

 

 

 

25 1.604 0.25 1.15 0.23 1.26
49.67 1.792 0.42 1.182 0.4 1.291
76 1.979 0.67 1.215 1.07 1.474
101 2.104 2 1.332 1.73 1.574
148 2.25 2.67 1.364 2.73 1.685
189.5 2.354 3.67 1.407 3.73 1.771
241.67 2.458 4.67 1.439 4.73 1.84
292.5 2.5 5.67 1.472 5.73 1.904
351.67 2.604 6.67 1.504 6.82 1.949
404 2.625 7.75 1.536 9.07 2.018
471.25 2.667 10 1.579 10.73 2.073
509.42 2.708 11.67 1.611 11.73 2.109
559.13 2.729 12.67 1.632 22.4 2.42
600 2.75 23.33 1.804 48.57 2.767
600 1.771 46.5 2.041 69.4 3
600.17 1.75 71.33 2.233 135.42 3.465
600.5 1.708 94.33 2.393 207.9 3.77
602 1.688 154.67 2.67 265.37 3.965
619 1.625 203.33 2.8 312.5 4.048
643.67 1.604 257 2.93 353.83 4.131
667 1.583 304 3.004 447.5 4.27
766.48 1.563 353.5 3.079 516.27 4.325
418.67 3.154 569.47 4.375
474.27 3.209 620 4.408
521.07 3.241 620 3.424
610 3.284 620.5 3.41
610 2.265 620.8 3.408
610.53 2.19 621.75 3.362
610.78 2.146 627.85 3.296
611.78 2.093 638.75 3.258
617.78 2.041 675.1 3.195
628.78 1.976 709.2 3.17
653.45 1.921 755.1 3.16
676.78 1.889 830.8 3.15
708.28 1.869
798 1.846
Test Condition Number
31 32 33
t (hour) J(t)/J(0) t (hourL J(t)/J(0) t (hour) J(t)/J(0)
0 1 0 1 0 1

0.08 1.059 0.07 1.121 0.05 1.151

98

Table 4 (cont’d)

 

0.33 1.078 0.23 1.204 0.13 1.263
0.67 1.137 0.57 1.287 0.3 1.368
1 1.157 0.73 1.316 0.63 1.528
1.33 1.196 1.23 1.371 0.97 1.644
2.12 1.235 3.23 1.566 1.3 1.735
2.75 1.274 5.23 1.678 1.72 1.844
3.5 1.314 6.23 1.704 2.08 1.918
4.5 1.373 10.23 1.891 2.72 2.05
5 1.412 22.9 2.242 3.47 2.177
6.67 1.471 46.23 2.613 4.47 2.352
9.17 1.627 74.23 2.946 4.97 2.422
10 1.647 84.23 3.021 6.63 2.623
12.97 1.784 103.73 3.197 9.13 2.918
22.5 2.039 118.73 3.271 9.97 2.97
24.75 2.078 142.73 3.392 10.13 2.988
28 2.137 166.73 3.502 12.88 3.191
30.75 2.157 176.4 3.55 22.47 3.613
34 2.216 180.18 3.567 24.72 3.669
47.5 2.412 189.68 3.596 25.97 3.75
54 2.431 214.68 3.66 30.75 3.815
82.75 2.627 237.93 3.725 33.97 3.887
97.5 2.745 252.68 3.8 47.47 4.093
119.83 2.784 274.35 3.826 53.97 4.131
143.67 2.882 298.18 3.892 82.72 4.333
166.25 2.941 357.42 4.03 97.47 4.444
190.75 3.019 442.02 4.179 119.8 4.5
237.27 3.137 509.2 4.272 143.63 4.585
287.5 3.255 562.02 4.327 237.17 4.844
347.9 3.333 610 4.392 297.05 4.968
407.5 3.451 610 3.392 333.33 5.055
467.9 3.529 610.05 3.335 385.1 5.131
528.5 3.568 610.18 3.289 437.5 5.2
600 3.647 610.93 3.216 516.77 5.313
600 2.686 611.93 3.16 620 5.387
600.07 2.314 615.43 3.076 620 4.433
600.22 2.294 633.1 2.975 620.02 4.431
600.97 2.255 659.43 2.891 620.05 4.4
601.47 2.235 730.18 2.742 620.3 4.354
603.97 2.196 871.93 2.65 620.88 4.313
649.47 2.118 991.93 2.613 624.22 4.229

692.93 2.078 1052.67 2.604 631.3 4.179

99

Table 4 (cont’d)

795.4 2.039 641.72 4.132
861.87 2.02 650.22 4.1

911.37 2 672.22 4.055

696.22 4.011

742.72 3.988

809.22 3.946

937.22 3.917

__ 1179.75 3.901

 

 

stress levels, evidencing the nonlinear viscoelastic behavior of yellow-poplar in the
radialdirection at moderate and high moisture content levels.

The two MOEs were obtained from each creep test based on the instantaneous
elastic strain from load application and from load removal, respectively. The former is
called the initial MOE and the latter the recovery MOE. It is seen in Table 5 that there
are slight increases in the range of 0 to 5 percent in the recovery MOE over the
corresponding initial MOE. The elastic component showed time dependent creep at
constant moisture content. Such time dependence is larger at high stress and moisture
content levels. To substantiate the above finding, further investigation on larger numbers
of observations is necessary. Nevertheless, the observed time dependence is very
marginal and can probably be disregarded. Actually, both Herman and Paton [1964],
and Grossman [1976] indicated that the elastic component quantified by MOE only,
increases or decreases directly according to moisture content (temperature not considered
here) under moisture content changes and/or moisture content cycling during stress,
suggesting no such time dependent variation is experienced by MOE under either the

direct effect of moisture content or mechano-sorptive coupling effect.

Table 5. Instantaneous MOEs of yellow-poplar in the radial direction at nine creep test

conditions.

 

Instantaneous MOE

 

 

 

 

 

 

 

(10’ psi)
Test Condition Moisture Creep Stress Initial MOE Recovery MOE MOE Change
Number Content (psi) @ 600 hrs (96)
(91»)

11 11.5 100 145.96 146.92 0.66

12 11.5 310 148.72 149.96 0.83

13 11.5 650 145.54 148.59 $10
Average: 146.74 148.49
Predicted': 145.25 148.12
Difference (%)2: -1.02 —0.25

21 15.7 133 135.89 138.57 1.97

22 15.7 314 136.93 140.57 2.65

23 15.7 785 135.56 142_.22 4.91
Average: 136.13 140.45
Predicted: 136.94 140.66
Difference (96): 0.59 0.15

31 21.5 130 124.80 129.89 4.08

32 21.5 321 124.58 130.65 4.88

33 21.5 600 124.02 131.00 5.62__

Average: 124.47 130.51
Predicted: 122.72 130.06
Difference (96): -1.4 -0.35

 

1: MOE computed according to Eq. 3.3.5 on the basis of the MOEs at two other moisture content levels.
2: Difference between the computed MOE and the measured MOE.

101

 

 

 

 

 

 

 

 

 

 

made:
8.8 8.2 can 2% cm; 0mm 0
2.8.; To 100
“W mlm 4141}
Nw $118
8 n..-
«1 Q m. U” 1 H 0 f®.H
mm Twirl
2 I .
Tiff; u u
I \ tom
111111..
R \1 if.
4i.\§\hL r

 

 

 

 

 

 

 

 

 

 

10.6

gage—.8 3.8 80.8 9:: a 8.82% 3:68 05 E EnoaéB—g .6 3.5358 30.6 322.82 .cm 2:»:—

1/1‘)
991:) pazneuuoN

0

aauendmog (<1

102

As expected, both MOEs decrease with increase in moisture content, as shown

in Figure 27. The experimental data were checked for their agreement with the known

moisture content-MOE relationship given in the Wood Handbook [Forest Products

Laboratory, 1987]. A mechanical property at moisture content MC is found to be

 

MC-IZ)

P12 12-uc
PMC'P12(T( a

G

where
Pm - mechanical property at moisture content of MC percent
Pu - mechanical property at moisture content of 12 percent

PG - mechanical property at green condition

3.3.4

MCG is the moisture content at which the mechanical property ceases to decrease with

moisture content. A variant form of the same relation is

P MCx - 11c,
uc,) 110, are,

P -P

 

11c: MC, (

P are,

where

PM“ - mechanical property at moisture content of Mex percent
Pm, — mechanical property at moisture content of MC, percent

I",m - mechanical property at moisture content of MC, percent

3.3.5

This version allows for the calculation of the mechanical property at any moisture content

(Mex) given the property at moisture content MC, and MC, The MOE at the third

103

on

.5302? .32: 05 5 338-323» we mmOE 33:5:38. Hm oeswE

mm

c _

33 2:23:00 magmflcz

o

I-
h—
)—

N n

H

_ _

ofi

 

use; ,
@306..er

883 TU
vcuocqoem Elm

 

//

@934
. E

 

 

 

 

 

 

 

 

 

tom

rm:

1G3

 

10m:

(rsd 1:01) How snoauequeqsul

104

moisture content calculated based on the MOEs at two other moisture content levels was
found to be within 2% of the actual MOE value. Table 5 and Figure 27 illustrate such
close agreement.

The recoverable component is the part of creep deformation that can recover over
time following the instantaneous recovery upon the load release. For comparability, it

was expressed as normalized recoverable strain, defined as

e,(t) e,(t)
8(0) 90

3.3.6

 

 

emit) -

where

e,N(t) = normalized recoverable strain in percent

e,(t) = recoverable strain

e, = instantaneous elastic strain at the time of load application

The normalized recoverable strain is equivalent to the normalized recoverable compliance

as demonstrated in

e(t)--———--————-——-1,(t) 3.3.7

In Figure 28 which shows recovery strain at nine test conditions, it may be noted
that increases in both stress level and moisture content result in an increase of the
normalized recoverable compliance (or normalized recoverable strain) and in longer
recovery time, which is more profound at higher stress levels and moisture contents. The

observed moisture content dependence, and the stress dependence of the normalized

105

owe

»

omv

mcso:
own OWN

lr

 

 

 

Ma

fl
am
Nm

Iii.

illnummmmmm

P

 

4a
451

ll

 

 

 

 

 

 

 

 

 

 

gauze—.8 33 30.8 05: a 5585c 3!: 05 5 338-32....» he £83 988 @383— .w~ 2:»:—

fl
LO
N

700

7
LO
1"

 

100M

/A19Aooaa

011921 1119418

(1%)

9118913 19111111

106

recoverable compliance - indicating nonlinear behavior of the recoverable component -
agree with the findings of Bach [1968] that recoverable compliance is a nonlinear
function of stress levels and moisture content.

The effect of mechano-sorptive coupling on the recoverable component was not
investigated in this work. The indication by Grossman [1976] is that the coupling should
impose no further influence on the recoverable component.

Measurement at the end of the creep test (about 600 hours) does not reveal
whether the recoverable component is time dependent during the creep test maintained
at constant moisture content. Investigation of this time dependence would require the
release of the creep load at different times during the creep tests on a number of
matching specimens.

The recoverable component is, therefore, assumed to be time independent in this
study. Consequently, any such time dependence is being factored into the flow
component. Characterization is made much simpler this way as there is only one time
dependent component - the flow component - to deal with. The rationale rests in the
relative insignificant contribution by the recoverable component as outlined in Table 6
which shows that the recoverable component accounts for only about 20 percent of the
total creep, while 80% is due to the flow component.

The flow component results in permanent deformation that does not recover after
the release of load. It is apparent in Figure 29 that the normalized flow strain increases
with an increase in both stress and moisture content, as found by Bach and Pentoney

[1968] that the flow component is clearly nonlinearly viscoelastic.

107
3.4 Modelling Creep Behavior by Mechanical Model

As opposed to empirical models, the mechanical ones provide a unified approach

to the analysis of viscoelastic behavior in that the behavior of a material can be described

Table 6. Composition of tension creep strain of yellow-poplar in the radial direction.

 

 

 

 

 

Normalized Creep Strain (%)
Test Total Recoverable Permanent Recoverable Permanent
Condition
Number
11 0.5857 0.0747 0.511 12.75 87.25
12 0.598 0.084 0.514 14.05 85.95
13 0.5989 0.0709 0.528 11.84 88.16
21 1.75 0.188 1.563 10.71 89.29
22 2.28 0.211 2.069 9.25 90.75
23 3.41 0.258 3.15 7.57 92.43
31 2.65 0.65 2 24.53 75.47
32 3.39 0.61 2.78 17.99 82.01
33 4.39 0.77 3.62 17.54 82.46

 

by a finite number of parameters that can be compared to those of other materials as well
as of the same material subject to different environmental conditions [Bodig and Jane,
1982]. Therefore, a unified mechanical model must be used for the nine test conditions
in this study.

The model proposed is based on the ever popular 4-element Burger body defined
in Eq. 3.2.7. Since the initial creep compliance is the elastic compliance — the reciprocal

of modulus of elasticity -

108

£2826 3.5.05 5 Eucaéo=oa be 59.8 398 3c."— dm Bani

2mm: mmcim Q0015

 

 

 

 

 

82 com com cow com o
r . _ . g . _ _ _ . _
o: No: I. no
u: was: cum
0: .805 E .1 6 8
12:
\\1 loom

 

t\\\\\\ room

 

 

 

 

 

 

 

 

 

roov

13 lgillul/Mold

(2) 011921 1119118

:31qu

109

1(0) " Jo " E— 1 3.3.8

the normalized total creep compliance is

1(1) 1(1)
1 1 - —— - —
”0 1(0) 10

'1
_1_E .LEQE ._‘__5 3.3.9

- 1 + J,(t)Em + J,(t)Em

- 1 + 1,N(1) + 1,N(1)

where

1”“) = normalized total creep compliance

J,N(t) = normalized recoverable compliance

J,,,(t) = normalized flow compliance

with others defined in the same manner as specified in Eq. 3.2.7. The total creep
compliance can then be easily obtained by dividing a given normalized creep compliance
by the initial MOE or the reciprocal of the initial creep compliance.

The normalized total creep compliance and its components described in Eq. 3.3.9
are shown in Figure 30 to demonstrate relations among them. The instantaneous
component is normalized to unity. The recoverable component rises and levels off
relatively quickly, and recovers completely after load release at t, evidencing the

behavior of the Kelvin element. The flow component increases linearly with time and

110

.33 59.5 258204. 2.. we coca—9:8 goo... gangsta: he 3:32.800 6m Bani

@EME

 

 

3.5565. 53me 3
.cmcanoo m_nc..c>ocmm -93.me

 

 

 

a...

\\

-\ Awaram :c3xm2v .cccanoo mzoccmaccamaz

w ufifl
HQ
760$

l
O

 

 

39.13 pazneumoN

1/1)

0

aouendmog ((1

111

remains permanent after load release, indicating Newtonian behavior (linear dashpot).
(The Newtonian dashpot, by definition, maintains linear proportionality of its strain rate

with stress

111/(1) 3.3.10
dt

 

0" - 0,... 11(1)’ - 0...,

Integration of the above results in

1,0) - i 1 0‘ 3.3.11
0..., -

The creep compliance is then

 

1,0) - 84f) - —’—1 3.3.12

0 17...,

and its normalization over the initial creep compliance J, is

I
J t E
Jf(t) - ll - l"!— - —""-1 3.3.13
N Jo _1_ "In
E”

This shows that the normalized creep compliance of the Newtonian dashpot is represented
by a straight line.). In the superimposition of the three elements, the instantaneous part
establishes a starting base at unity, the recoverable component raises it to another level,
and the flow segment rides atop. The only transient phase is before the recoverable

component reaches its equilibrium level. This phase whose length depends on the Kelvin

112

element’s retardation time is however generally short in comparison to the total creep
span. Therefore, the slope behavior of normalized total compliance is dominated by that
of the Newtonian flow component. The tangent of the slope for the normalized total

creep compliance could be approximated by that of its flow component as

111,11) _ d“ + 11,0) t 15(1)) __ 1115(1) _ 53 3.3.14

dt dt dt ".14

 

 

 

In absolute creep compliance, the slope tangent of a Newtonian dashpot is then just the

reciprocal of the dashpot’s coefficient

111(1) 4 WW1) 401075...) 1 3.3.15

dt dt dt 17”

However, it is shown in Figure 26 that the actual normalized total compliances never
reach straight slopes. In another words, the flow component is not linear and can not be
represented by a Newtonian dashpot. This is neither new nor surprising as many others
have arrived at the same finding [Davidson, 1962; Ethington and Youngs, 1965; Bach,
1965; Youngs and Hilbrand, 1963]. Realistic characterization must employ a non-
Newtonian dashpot.

Before addressing the mathematical form of an appropriate non-Newtonian flow
component, we first explore possible relations between Newtonian and non-Newtonian
dashpots. If one agrees that a curve (nonlinear) can be divided into an infinitely large
number of infinitely short straight lines and that each line segment is the slope or

derivative of the nonlinear curve within the respective segment, one could in the same

113

way divide a non-Newtonian compliance curve into an infinite number of infinitely short
straight compliance lines. Since each individual straight compliance segment corresponds
to a Newtonian dashpot whose coefficient is the reciprocal of the tangent of the straight
segment within the respective infinitesimal time interval, a non-Newtonian dashpot can
be viewed as a Newtonian one with its coefficient varying in accordance with the tangent
or the differentiation of the non-Newtonian compliance curve.

By trial, the parabolic function (at') which has been a successful empirical model
for wood and wood-based materials defined in Eq. 3.2.5 is found to be an appropriate
non-Newtonian dashpot that best fits the experimental data of the normalized creep
compliance in this study. The reciprocal of the coefficient of its Newtonian equivalence

is therefore

—-!— - amt""l 3.3.16
"and

which is the differentiation of the parabolic function. The coefficient

1
’7 ' 3.3.17
"d amt""

 

is no longer a constant as it is in a Newtonian dashpot, but time dependent.
The recoverable component which makes only a small contribution to the total
creep (Table 6) is approximated with a single Kelvin element which contributes a

compliance of

114

5h

-—r

(I-e "“)
Eb

 

The proposed four element linear Burger body model is then modified to

 

3.3.18

 

 

in terms of the normalized compliance. It differs from the proposed linear Burger body
(Eq. 3.3.9) in the flow term. Equation 3.3.18 will be used to characterize the
experimental normalized creep compliance data in Figure 26 in order to determine the
coefficients, namely Em, Eu, 1)“, and 11.4-

3,, is easily determined from the instantaneous elastic strain at the start of the
creep test - load application.

The Kelvin element with which E,” and 1),“, are associated is governed by the

differential equation

d e,(t)
d1

 

Ebe,(t) - nu (- 0') 3.3.19

115

as a result of its parallel arrangement of spring and dashpot. At the initial condition of

zero extension, the solution of the differential equation results in the normalized

 

compliance
8,0) £5:
Jr (1) - 1'9 .. L - 35(1). - ENLISLD 3.3.20
" Jo 1(0) 8(0) E.
a!

At the initial condition of full extension, the normalized compliance would be

 

3292 -3.
Jr (1) .. 11E) - J; .. 3&2 .. Ewe a“ 3.3.21
” Jo 6(0) 8(0) Eb
a.

It also follows from Eq. 3.3.20 that when t goes to infinity, which is when the Kelvin

model is fully extended,

J (on) - £2 3.3.22
1'" Eb
01'
1
LG») - — 3.3.23
Eb

Therefore, the Kelvin spring constant, E”, and the Kelvin dashpot coefficient, flu,

can be obtained from experimental recovery strain data. First, Eb was found by dividing

116

the creep stress by the total recoverable strain according to Eq. 3.3.23. Next, Eq.
3.3.21 was forced on the recovery portion of the normalized creep compliance curve
(Figure 31) to arrive at 31./flu (the reciprocal of the retardation time 1/1) to obtain flu-
It is apparent in Figure 31 that one Kelvin element does not suffice for good approxima-
tion of the behavior, but can account for the most important two aspects of the
recoverable component -the total recovery and the total recovery time.

The a, and m coefficients in the flow term were obtained by nonlinear regression
fitting of Eq. 3.3. 18 to the nine sets of normalized total creep compliances in Figure 26.
An example is given in Figure '32 for the normalized creep compliance at test condition
21.

To express the normalized creep compliance as function of moisture content,

stress, and time, Eq. 3.3. 18 may be rewritten as

shame) 1

MW”) 3.3.24
’ + E.(MC>a(Mc.a)1'-<"¢~>

JN(MC, 11,1) - 1 + EN(MC)(1 ‘ ‘
Milan)

 

where
MC - moisture content
0 - stress

Alternatively, it could be alternatively written as

MMCN) - 1 + AlMC.a‘)(1 - e") + C(MC,a)t"“‘c") 3.3.25

where

117

.50.... 30.0 E0500. 2 2.0.5.0 5202 0.9.: 9.2.... .3 0...: 5.3350. 9.2.350 gm 05w...

 

 

 

 

 

 

 

MADOI
com om: ON. om 0v 0
F _ _ _ . _ _ _ .

v0.5m00w14 lo

ESLEEM
in.
1.;
ifim

Hm I GOEECOU 50,—. -
QEQoizxmawomu»

 

 

 

 

 

 

 

 

1mm

[WNW/419400311
011921 1119118

13

(

911891

._~ Egg—Eco .8. .n 5:00.... 3...... o... E 58932.0» .6 3:39:00 moo... v3.2.5.9. :o .5383... 32.2.52 .Nm 2:3".

 

 

 

 

 

m.:oI
com. om... ow. 2.? 3m o
_ _ _ _ _ F _

flog

N

O

I .J

m

[m N W

1 T p

I )0

.m 556.80 69.. Pi

5 w W%

58%... u .me (d

I Q

.23.... u Em m

.oumm.mwmm.. H mm m.

AmvmthSmi+2x*mu.ouvn.xmu.rmo.m. n > , w

8

 

 

 

 

 

 

 

 

ION.

119

 

E M
Eb(MC,a)
E MC,
B(MC,a) - _E'_(___i)
nhKMCm)
3.3.26
1 _ C(MC,a)D(MC,o)t(D(”c"')'D
0(MC,a,t) E...

C(MC,0) - Em(MC) a(MC,a)

D(MC, a) - m(MC,a)

Given A, B, C, D, and E”, Eh, flu, and 1)., can be readily computed. The A, B, C, and
D values for the 9 conditions were obtained and are listed in Table 7. Though multi-
variable regression analysis is the best choice for the procurement of the empirical A, B,
C, and D functions of stress and moisture content, the author opted for two steps of
single variable empirical curve fitting. Because there is no coupling between stress and
moisture content changes, their effects on A, B, C, and D may be dealt with indepen-
dently. The A, B, C, and D functions of stress at each of the three moisture content
levels were obtained in the first empirical fitting. The variations of the coefficients in the
acquired A, B, C, and D functions of stress with the three levels of moisture content
were then captured in the second empirical fitting, which yields A, B, C, and D functions
of both stress and moisture content as presented in Table 8.

The normalized creep compliances based on the empirical functions of A, B, C,
and D at the three stress levels and three moisture contents were drawn against the
experimental normalized creep compliances in Figure 33 to show the fairly good

agreement between them.

120

Table 7. Coefficients of normalized creep compliance of yellow-poplar in the radial
direction at nine creep test conditions.

 

 

 

Coefficient
Test Moisture Creep Stress A B C D
Condition Content (psi)
Number (%)
11 11.5 100 0.08566 0.035 0.10927 0.2366
12 11.5 310 0.08566 0.035 0.11526 0.2367
13 11.5 650 0.08927 0.035 0.11526 0.2318
21 15.7 133 0.2102 0.030 0.19538 0.3280
22 15.7 314 0.2503 0.030 0.38578 0.2582
23 15.7 785 0.3038 0.030 0.51460 0.2866
31 21.5 130 0.4302 0.015 0.28780 0.3267
32 21.5 321 0.7420 0.011 0.54078 0.2472

33 21.5 600 0.9210 0.010 0.70318 0.2622

 

In summary, the creep behavior of yellow-poplar in its radial tension as
influenced by stress levels and moisture contents has been mathematically characterized
by a four element Burger body containing a non-Newtonian Maxwell dashpot. The
coefficients of the Burger body are successfully expressed as functions of their dependent
variables, namely, stress, moisture content, and time. Though there are many available
choices for the function forms, the chosen one presumably is the simplest and did
produce good conformity with the actual experimental data. It is to be kept in mind that
the current mathematical characterization only includes the direct effects of stress and
moisture content changes during moisture content gain. Its validity is not certain beyond

the tested stress and moisture content ranges.

121

Table 8. Empirical functions of coefficients of normalized creep compliance of yellow-
poplar in the radial direction.

 

JN(t) - 110—) - 1+A(1-e"’)+CtD
0

1

-A30
A 1e + A3

 

A...

B - -0.0014e°-“2”C + 0.0424

c - c,(1 - e'c’a)

D - 0.154756e‘°'°°°“5"c + 0.25

A, - -2.85 + 0.467931MC - 0.00799Mc:2

A, - -0.00085912 + 0.00004962MC + 00000165th2
A, - 247.08e'027lm

C1 _ l.01(1 _e-0.151(uc-105))

C, - 0.00019(1 - e°-35WC-2°-4)) - 0.0031

 

 

 

MC - Moisture Content

£2.02... 3...... 0... E 3.08-325. .0 005......8 30.0 30:08.9. .0 0.0.. 3.8.5.098 .9. 3.8.5. .8.....Em .mn 0...»...

0.303

com. com own. owv
_ . . r _

b b

 

 

'fll“ 0

 

 

1
\

T
C?
89.13 DSZIIQUIJON

0

;
aouendmog C(l

 

 

r/r)

11111111 on
11111111 mm
11111111 .m

4
a.
4
mm I
m.
U
o

 

mm
.m
m"..m..:
.00....QEM. 1..

 

 

 

 

 

 

 

 

CHAPTER IV

APPLICATION OF THE LVP THEORYTO HYGROSCOPIC WARPING

In this chapter, the hygroscopic warping is viewed as being a highly viscoelastic,
as well as a dynamic process. With the viscoelastic properties characterized in Chapter
3 as one of the necessary inputs, the LVP theory developed in Chapter 2 is to be applied
to a cross laminated yellow-poplar wood plate and beam. Theoretical predictions of the
warping development of the plate and beam are to be compared with the measured values

along the experimentally determined moisture content path.

4.1 The Process of Hygroscopic Warping of Wood and Wood-based Material Panels
Hygroscopic warping of a panel occurs when the hygroscopic expansions or
shrinkages resulting from moisture content changes at any pair of planes that are
symmetrical with respect to the mid-plane of the panel differ and generate bending
moments. Such differentiation can come from either the structural unsymmetry such as
difference in species - difference in hygroscopic expansion or shrinkage coefficient - or
unsymmetry in moisture content gradient across the thickness of the panel. It is
theoretically possible for a panel of structural unsymmetry to become flat at some point
if subjected to opposing and equal moisture content gradient unsymmetry. In reality, this

temporary stability however can not be maintained. A panel of perfect structural

123

124

symmetry can still warp when the moisture content gradient is unbalanced, but the
problem is not inherent in the structure. Therefore, such panel is considered hygro-
scopically stable. Perfect structural symmetry is ideal, but difficult to achieve in
manufacturing practice. The best effort can only result in the reduction of any unsymme-
try to a minimum, according to potential application requirements. However, it often has
to be compromised due to cost, availability of species, etc..

The viscoelastic nature of warping in wood and wood-based panels is soundly
evidenced in the following manifestation. When the moisture content of an unsymmetrical
panel is elevated to moderate and high levels, the resulting warp is much less than the
elastic estimate. Reversing the moisture content to its original level which would have
reversed the warp backward to its initial value were the panel elastic, however, leaves
a significant part of the original warp unrecovered or permanent. This is nevertheless
expected as the panel behavior must reflect the viscoelastic properties of constituent wood
and wood-based materials.

In the following thought process which attempts to analyze a viscoelastic warping
process, a wood panel cross laminated of two very thin plies of the same thickness and
species is selected. The layers are so thin that the moisture content is assumed to change
uniformly across the panel thickness so that there is no differential expansion or
shrinkage within each of the layer of the panel. The plate is presumed to be at some
initial moisture content where the panel is flat, and therefore stress free (due to its
structural instability). The hygroscopic expansion or shrinkage differential between the

two cross laminated layers in each of the two coordinate directions is the cause of

125

potential mutual restraint between them. This restraint is triggered by any deviation of
the moisture content from its initial value and will result in a saddle shaped warp of the
paneL

Suppose the panel’s moisture content is going to be increased along a path to
MC+AMC over a time period of At. As soon as the moisture content strays away from
its initial point, both layers start to expand. The much larger hygroscopic expansion
across the grain relative to that along the grain results in mutual restraint, leading to
tension along the grain and compression across the grain in the panel. The resulting
bending moment would then cause the panel to warp away from its original flat position.
Immediately at the onset of tension and compression stresses, the viscoelastic property
of the constituent would emerge in the process (viscoelastic properties are dormant in the
stress free state). It asserts itself in that the tension layer would tend to creep-stretch, and
the compression layer tend to compression-creep under their respective stresses. This
creep, here called deformation creep, would lessen the mutual restraint between the two
layers, resulting in lower stresses (stresses relaxed), and consequently smaller bending
moment and warp than would be the case if the constituents were completely elastic.
Here, the onset of stresses and deformation creep (plus stress relaxation) are related in
a sequential manner. These two are actually simultaneous, intertwined, and inseparable
in time space. No sooner than with the onset of the tension and compression stresses
will deformation creep begin and stresses start to relax, resulting in a reduced potential
for bending moment and warp.

As moisture content further increases, the expansion differential becomes larger,

126

causing greater mutual restraint and restraining stresses. In the meantime, it is known
that the hygroscopic constituent layers will creep and relax more at higher moisture
contents and under higher stresses. Therefore, the deformation creep and stress relaxation
in the two layers are expected to occur to a greater extent and at a faster rate with each
increment of moisture content and stress increase, thereby lessening the relative restraint
and slowing down the otherwise much larger and faster elastic warp development. The
concurrent development of the two phases (restraint and restraining stresses versus
deformation creep and stress relaxation) represents a balancing mechanism which
minimizes the restraint and resulting warp. Due to larger and faster creep at higher stress
of wood and wood-based materials, the more severe and the faster the condition of
restraint develops, the more significant and the faster the deformation creep and stress
relaxation occur. Conversely, the less severe the restraint is, the less significant the
deformation creep will be and the slower the stress relaxation will take place. It should
be emphasized that these events take place along a continuous moisture content path as
it is impossible to elevate moisture content instantaneously.

As moisture content stabilizes, no further hygroscopic expansion is introduced.
Yet the mutual restraint and restraining stresses, and the warp resulting from the previous
moisture content path are still present. Therefore the deformation creep and stress
relaxation will continue. As the restraint is further lessened, less deformation creep and
stress relaxation occur due to less creep by wood under lower stresses. This downward
trend would continue until a balance between the lessening of the restraint and the

development of the deformation creep has been reached. It needs to be emphasized that

127

the laminate is not necessarily free of stress or restraint free at this point. It is merely
in a temporary stable state. The permanent portion of the deformation creep (the
Maxwell dashpot) would be embedded in such a stable warp by manifesting itself as the
irrecoverable portion of the warp if the panel is subjected to the reverse of the moisture
content path back to its initial moisture content level.

In real panels there exist moisture content gradients since moisture content can
not be raised uniformly across a thickness, just like temperature. It is by the gradient that
the moisture gets diffused and uniform moisture content is reached over a period of time.
The presence of a moisture content gradient poses a more complicated scenario in that
aside from the mutual restraint as caused by the directional expansion differential
between the two layers, there is mutual restraint everywhere within each layer, caused
by moisture content differentials. Every imaginary layer as thin as necessary is restrained
by its adjacent ones, and therefore two phases of the warping process must be taking
place concurrently in the involved layers.

Another source of mutual restraint for this two-layer panel could come from the
warp itself. It is known that when the panel warps, a stress distribution develops in such
a way that the stresses in the outer layers are the largest, and that they gradually diminish
towards the mid-plane. The stress differential between adjacent imaginary layers imposes
constraint between them.

In short, the hygroscopically induced mutual restraint due to either imbalance in
structure and/or moisture content gradient and the simultaneously resulting deformation

creep and stress relaxation due to the viscoelastic nature of the constituent wood and

128

wood-based materials are the two inherently cohesive phases in the warping process.
Though conceptually conceived and separated here for the analysis of the warping
process, they actually are inseparable, interdependent, counter balancing and canceling
each other. Consideration of just one of them in a fixed time frame in any analysis is not
feasible.

The complexity of this warping process, is further increased by the fact that
moisture content and stresses not only vary across the thickness of each physical layer
in a panel due to moisture content gradients and stress distributions, but also that such
gradients and distributions do not remain constant, but vary with time. These two varia-
tions have to be captured somehow in any realistic approach.

The LVP theory is not capable of capturing them in a continuous manner since
it must treat a panel as composed of a finite number of layers, however thin they may
be, and consider the process within a finite number of discrete time intervals. The
moisture content within a layer must be uniform as required by the CLT structure and
constant within a time interval as required by the isothermal requirement of the LVP
theory (We can extend the isothermal concept to moisture content if moisture content and
temperature are considered to have equivalent effects. Warping where moisture content
gradient changes with time actually falls in the realm of thermal viscoelasticity.
However, the isothermal LVP theory is a lot simpler, and the isothermal restriction is
only to be enforced within each of, but not across the discretized time intervals in the
numerical form of the LVP theory achieved by linear finite difference approximation.

Therefore, the isothermal LVP theory in its numerical form can be and is used in this

129

study to approximate the thermal viscoelastic problem.). If we treat moisture content
increase as a step-wise process, that is moisture content within a layer is uniform and
remains constant during time interval At, before advancing to a new level at the next time
interval At2 as shown in Figure 34, the LVP theory is able to numerically capture
moisture content variations in the thickness direction along a time scale. This treatment
is feasible provided that the layers are specified to be very thin and time intervals are
sufficiently small so that the moisture content gradient within a layer is small enough to
be viewed as uniform, and its variation with time in a single time interval so little as to
be considered constant.

Secondly, the viscoelastic properties of wood and wood-based material such as
yellow-poplar for each imaginary thin layer in the panel are stress and moisture content
dependent, and therefore would change accordingly with the changes in moisture content
and stresses in the warping process. Such changes should be accounted for, if not
precisely, then at least numerically to a good approximation. The moisture content
dependent changes in viscoelastic coefficients can be numerically captured since moisture
content. variation across the thickness along the time scale can be numerically captured
as just discussed and the viscoelastic coefficients-moisture content relations have already
been characterized mathematically for yellow-poplar in Chapter 3. To account for the
stress dependent changes in viscoelastic coefficients, a similar numerical approach is
applied where the stress distribution within each thin layer would be assumed to be
uniform and to change in the same step wise manner as the moisture content. Stress

distributions exist inside each layer, however thin the layer may be. But, they are

130

0000.0... ...0...00 0.0.0.0...

2E.

v

.<

00.3-..3m .3. 0.5.".

.< .4. .<

 

 

U ‘1.

E

 

 

\
\
\
\
\
‘
\
\

 

 

 

in. .033. ..... t

 

(04) quaquog aansroW

131

averaged within a layer for the purpose of determining the stress effect on the
viscoelastic coefficients of the layer during a time interval according to the viscoelastic
coefficients-stress relationships defined in Chapter 3. (The CLT structure of this LVP
theory determined that the viscoelastic coefficients must be uniform within a layer.
Therefore the stress dependence of the coefficients must be computed based on one stress
within a layer.). This account of stress dependence would be a close approximation if
the layer was thin enough.

It is seen that dividing a panel into thin imaginary plane layers is the basis of the
LVP theory. The thinner and the larger the number of layers, the closer the theoretical
predictions should be to actual results, if the LVP theory proves itself valid. An exact
account of the actual behavior would necessarily resort to analytical solutions which
consider the continuous nature of the warping process, but this is not achievable. A
discrete method has to be the alternative, just as in so many other science and
engineering problems where numerical methods such as the finite element method are
widely used. Actually in a sense, the thin imaginary layer strategy resembles the finite
element method in a sense as the panel is divided into a finite number of parallel adjacent

layers.

4.2 Warping Experimentation
A yellow-poplar panel was constructed in the laboratory of 65 % RH and subjected
to 91% relative humidity. Its subsequent warp development was measured and compared

with the theoretical predictions by the LVP theory to examine validity of this theory.

132

4.2.1 Panel Design and Manufacturing

A two-ply cross laminate of yellow-poplar layers of identical thickness was
selected for it is of the simplest unsymmetrical structure, and requires the least effort in
lamination and manufacturing. The most important reason however is that the structural
unsymmetry would not allow any stresses in it without manifesting the stresses in the
form of warping of the panel. Every effort was made to minimize any possible residual
stresses arising from the laminating process.

As shown in Figure 35, edge grained strips of yellow-poplar of 3/4 by 1/2 by 50
in were cut from flat sawn yellow-poplar lumber and edge-glued with conventional white
glue to form two large edge grain panels with dimension 50 by 23 by 1/2. The panels
were then conditioned at a room condition of 70° F and 65 % RH for three weeks before
they were planed to a final thickness of 0.25 inches. These panels were monitored to
remain flat at further conditioning at the prior room conditions, indicating dimensional
stability. Three 1 in wide strips were cross cut from each of the two panels for use as
radial linear expansion samples. One of the panels was then cut in half and the two
halves were turned around for 90 degrees before being cross laminated to the other panel
with Epoxy AW-106/HV-953 made by Ciba-Geigy. To avoid any potential moisture
content change, the glue spreading and laminating operation were conducted in the same
room condition. The Epoxy which is 100% chemical reactive in its curing process does
not contain water or other solvents thus eliminating any hygroscopic expansions during
the lamination and curing. The AW-106/HV-953 Epoxy was recommended by the

manufacturer based on our requirements that the glue line must be able to cure under

133

 

 

 

 

1/2"
/ Flat sawn lumber a L

3M"

 

 

 

 

 

 

 

 

 

 

 

 

 

 

<1. ~,- ' “‘\~ :3.
/ x / ‘/’- "‘\
/ 4m
Ulmle11[1111i<1fl[11111111.
Edgegrainpanel

Figure 35 . Manufacturing of edge grain yellow-poplar panel.

134

room temperatures as hot press curing would introduce thermal stresses, must be
relatively rigid with as little creep and slippage as possible so that glue line could be
considered as being of zero thickness and in compliance with the Kirchhoff deformation
field, and must be highly moisture or humidity resistant as it is to be subjected to high
humidity for extended periods of time, and must have moderate viscosity for ease of
handling during the lamination process. The laminate assembly was then transferred to
a plywood press to cure under pressure. As suggested by the manufacturer, a relatively
low press pressure (15 psi) was used to minimize the development of residual stresses,
while maintaining sufficiently good contact. The laminate was allowed to remain under
such pressure to cure for over 18 hours before it was taken out and placed in the prior
room condition. As the press is situated in a room of lower humidity, some moisture
along the four edges of the laminate may have escaped during the 18 hour pressing time,
and some stresses may have been introduced along the edge areas. For that reason, the
laminate was trimmed to a final size of 42 by 19 by 0.5 inches. The samples prepared
from this piece are a l by 19 in beam, and a 19 by 29 in plate for the warping test, and
eight 4 by 4 in square blocks for the determination of moisture content gradients, as
shown in Figure 36.

The laminate should have remained flat when returned to the prior room condition
since its constituent panels had been equalized under such conditions. However, the
laminate when placed in the same room conditions warped as though the constituent
panels were conditioned at a higher relative humidity condition, though it came out of

the press flat. It was later confirmed that on the day of the lamination the valve system

135

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Beam 1” 1

29"

Laminate
42”
MC ..

Gradient ‘

‘l'

4"
2 19! 9

Figure 36. Specimen arrangement on the yellow-poplar laminate.

136

controlling the conditioning room malfunctioned for a short time without the author’s
knowledge. As a result, the actual room humidity did go up somewhat above 65% RH.
When the laminate was placed in a conditioning chamber maintained at 70% RH, it
straightened out eventually and remained flat. Therefore, 70° F and 70% RH are
considered the initial condition at which the laminate was both stress free and free of
warp.

Longitudinal tension samples for testing static MOEs of the laminating material
and longitudinal linear expansion samples were prepared from some of the randomly
selected strips cut from the flat sawn lumber. Radial tension samples for radial MOEs
and radial linear expansion samples were cross cut from the edge glued radial panels
shown in Figure 35 prior to laminating. A total of 11 longitudinal and 10 radial tension
samples were made. Their shape and dimensions are shown in Figure 37. A total of 10
longitudinal and 4 radial linear expansion samples were prepared. Figure 38 shows the

their dimensions and shape.

4.2.2 Warp, Moisture Content Gradient, and Expansion Coefficients

To avoid horizontal moisture content gradients due to faster moisture penetration
through the four edge surfaces, the edge surfaces of the prepared samples - the laminate
of 42 by 20 by 0.5 in, the beam of l by 19 in, and 10 4 by 4 in control squares - were
sealed with wax before transfer into the humidity chamber. After the laminate and the
beam had straightened out at 70% RH in the humidity chamber, the relative humidity in

the chamber was raised to 91% RH within only 5 minutes. As a consequence, moisture

137

 

 

 

 

NI.
1
l
L l

.4.

 

 

 

 

 

 

 

 

 

 

_W-01—

1/4" 1/4

 

 

 

 

 

 

 

hindr—c—i
O _—p.-.qu——1

 

 

 

 

 

 

 

 

Figure 37. Static tension specimen of yellow-poplar.

138

3.000-323» .0 008.00% 00.8.2.5 .00.... .mm 0...»...

 

 

 

 

 

 

 

=3.

 

 

 

 

139
content gradients developed and increased. Laminate and beam started to warp.

In this study warp was recorded as the vertical deflection of a laminate in
reference to the original flat position of the panel. In Figure 18 which shows the
positioning of the laminate in x-y-z coordinate system, the geometrical center of the
laminate is fixed to the coordinate origin, and the mid-plane of the original flat laminate
coincides with the z=0 x-y plane, as implied in Eq. 2.4.49. The vertical deflection of
the laminate at coordinate position (x, y, 0) therefore is in reference to the z=0 x-y
plane. Positive vertical deflection indicates a downward deflection, while a negative one
identifies a upward deflection. Due to cross lamination of two identical yellow-poplar
plies, the vertical deflection is symmetrical to x-z and y-z planes, that is the vertical
deflections at (0, y, 0) and (0, -y, 0), or at (x, 0, 0) and (-x, 0, 0) are identical.

At certain time intervals, vertical deflections relative to the geometrical center
were measured at (14.1, 0, 0) and (0, 9.15, 0), respectively on the laminate, and at (0,
9.15, 0) on the 19in beam shown in Figure 18. The measuring device is an aluminum
beam with contact points at each end and with a conventional dial gauge affixed to its
center as shown in Figure 39a. By placing the two end pins at (114.1, 0, 0) or at (0,
$9.15, 0) where the vertical deflections are identical, the dial gauge which touches the
geometrical center of the laminate or the beam thereby indicates the vertical deflections
at the two end pin locations, as shown in Figure 39b. The device was first leveled by
adjusting the heights of the two end pins.

The focus of this study is on only the hygroscopic warp, and therefore any other

sources affecting the warp including the effect of the laminate own gravity force on

140

Figure 39a. Deflection measuring apparatus.

 

141

Figure 39b. Measuring apparatus on the warped yellow-poplar laminate.

141A

y.

. .11.
. n1 «ivy/A

...»

.211 ..

. ‘
1N vVnOlmfi

 

142

vertical deflection had to be avoided. For this reason, the laminate and the beam were
placed on their edges both inside the humidity chamber (Figure 40) and when being
measured.

The development of the moisture content gradient in the thickness direction is the
necessary input for the application of the LVP theory. The measurement of moisture
content gradient in the laminate was conducted on 4 by 4 in square blocks cut from the
original laminate. The blocks were place in the humidity chamber simultaneously with
the laminate. Each block was taken out at certain time interval after the chamber
humidity was raised to 91% RH and tested by the method developed by Feng and
Suchsland [1993]. The depth increment used is 0.1 inches and only three layers of
sampling were taken as the moisture content gradient in this laminate can be considered
symmetrical to the mid-plane of the laminate. The new method using Forstner drill bit
was demonstrated to be superior in accuracy to the conventional layer sawing technique
[Feng and Suchsland, 1993].

The linear expansion samples were first conditioned at 65% RH and then
subjected to a humidity cycle of from 65% to 86% to 93% RH. Measurements were
taken when the specimens had equalized at 86% and 93 % RH, respectively on an optical

comparator developed by Suchsland [1970].

4.2.3 Test Results
The developments of vertical deflections at locations (114.1, 0, 0) and (0,

$9.15, 0) on the laminate and (0, 19.15, 0) on the beam as indicated in Figure 18, are

143

Figure 40. Yellow-poplar laminate and beam in humidity chamber.

143A

 

144

presented in Figure 41. The vertical deflections for these particular locations may be
interpreted as the center deflections over a span of 14.1 X2 =28.2 in one direction and
9.15 X2=18.3 in the other direction for the laminate, and a span of 9.15 x2=18.3 for
the beam. Time zero is when the humidity in the chamber was at 91 % RH. The jump
from 70% to 91% RH took only 5 minutes during which time no noticeable warp
occurred in either the laminate or the beam.

The vertical deflections are seen in Figure 41 to rise quickly, slow down about
50 hours into the exposure, and eventually level off at 200 hours. The beam shows larger
vertical deflection than the laminate over the same span, which is probably due to the
absence of two dimensional restraint on the beam.

In Table 9 and Figure 42 which both indicate moisture content gradient progress,
it is seen that the moisture content gradient is uniform at the beginning and end of the
exposure test. As expected, the moisture contents in the outer layers rise first and rather
quickly, while they lag behind for the inner layers. Describing the empirical gradients
with quadratic equation resulted in a good fit (see regression coefficients in Table 9).

The expansion coefficients for both longitudinal and radial directions were
obtained by the method introduced by Xu and Suchsland [1992] as a function of moisture
content in the considered moisture content range which in this case is from 11.5% to
21.5%. As shown in Figure 43, expansion coefficients in both directions experience a
reduction with increase in moisture content. Figure 44 is the sorption isotherm obtained
based on measurements performed on expansion specimens.

The tension test for MOE was performed on the prepared longitudinal and radial

145

00.880 .050 0. 0000 0. Nam

 

 

 

 

 

 

 

 

 

 

 

   

 

 

1
co
1

1
1—4
|

 

(U1) 11011081180 190111811

00000.0 000 0. 000... 0. fix. .0>0 00.0000... 00.82.00 "0.00.60.
00.... 0. Wm. .0>0 00.0000... 00.82.00 .E00m
0.00.. 000 200.00. 8.000-328» 0... .0 00280.80 .00...0> 00.00000 .0> 80.8.00... 0.0.8.000... .2. 0.00.".
0.503
coo. com com 000 com o
_ _ _ . . _ . _ . _ L
mmll 00.50002 1
06.8.00... 2.00.0 .....
0030.000n. 2.00.0000; in
W .3005... 500m. 0300.... 0
m. :1 .
S ...............................................................................
“IT I I I I
n , ,
J
a
O .1 11111111111111111 /H./
O O 111111111111111111111 X...
W /,8 .o 0...... 0 0.00.0.3 /
m \8 0.0.. .8 0 0.00.0.3 1
1. 1.0 0.00 .0. 0 E000 11-11111111.111-111.00.
) fl H i. ...........................................................................................
mm 0
.93.. 000...:
/.:0...ou 0.3.0.02 .0>0. 00...
mm 1

146

Table 9. Measured moisture content gradients in the yellow-poplar laminate.

 

Distance From Surface (in)

 

 

 

 

 

 

 

 

0.05 0.15 0.25 0.35 0.45
Exposure Time Moisture Content (96)
(X) (Y)
0.00 hours 11.72 11.68 11.66 11.68 11.72
1.33 hours 13.63 12.27 11.67 12.27 13.63
7.00 hours 14.81 12.93 12.05 12.93 14.81
12.50 hours 15.71 13.36 12.38 13.36 15.71
27.50 hours 16.30 13.97 13.06 13.97 16.30
47.50 hours 16.60 14.64 13.33 14.64 16.60
97.00 hours 17.37 15.59 14.68 15.59 17.37
6.00 days 17.64 16.29 15.44 16.29 17.64
12.00 days 19.09 19.09 19.09 19.09 19.09
Regression Curve Fitting of Moisture Content Gradients
Quadratic Function: Y = 3(0) + B(1)X + B(2))C2
Coefficient
13(0) 8(1) 3(2) 1'2

0.00 hours 11.737 -0.503 0.893 0.999
1.33 hours 14.713 -23.714 47.428 0.998
7.00 hours 16.312 -33.000 66.000 0.994
12.50 hours 17.545 -40.500 81.000 0.997
27.50 hours 18.097 -39.785 79.572 0.999
47.50 hours 18.338 -37.343 74.685 0.973
97.00 hours 18.824 -31.823 63.686 0.992
6.00 days 18.810 -25.357 50.714 0.978
12.00 days 19.090 0.000 0.000 1.000

 

147

0.00.0.0. 0.000-328. 0... 0. .000...0.0>00 .00.00.w 80.000 0.0.0.0.). .3 0.00.”.

.00 0.000 08000.0...

 

 

0.0 0.0 m.o Nd ..o o.
T _ _ . _ . . . _
0.00: mm. 0
0.00: b 0
0.00.. m.m. D
0.00.. mfim m.

0.00.. my... I

0.00: um 0

0%00 w 0
\Im No.1 0.00: o//

 

 

 

\\Im &O®

 

 

0000 0.1\

 

 

o

 

[ON

(04) 1110in;) aanstoW

148

.00.80..0 80.000000. 0... 0. 00.00-32...» .0 00.0....000 00.000000 0.000020»: .09 0.00.".

mm

mm

AN. .00.:00 0.0.0.0.).

om

. .

b.
.

 

 

 

30080.08

00.000..me

oo.x020\q

1.0.0.0“;

 

 

 

 

 

0-0:

 

010.;
3500000.

x*©©mm.©*ml
rx*©w.mmm+m

 

00000....

.>
.0000|.u>

 

iood

Io...

imod

imod

 

 

 

fivod

éxg

I

x3WP/(“1/1)Utp
111813111803 uotsue

001
(z/z)

149

$0. 8.0.000 0.0.0.02

mm o

N n... 0

..

 

300.0...000

COBCMQXWV

III/ll
co . x020\~.

XIII.

 

 

\

 

 

.\.\

010 H LX301

 

00:00...
500...]...0...

 

00000100000003”...

$6630.49;

 

.0000:.n>

 

 

 

.00.80..0 .0.00. 0... 0. 0.000-328» .0 .00.0....000 00.000000 0.000020»: .09 0.00....

00.0

led

IN.

rm.

 

:0.N

éxa

xowp/(°'I/'I)UIP = ,
100101;;003 uotsue

001
/%)

(is

.83 5359.0 .32.: co comma Egonéo=o> .3 83:33 coceom .3. 23E

c5 bags: 332%

mm Wm mm; flu ow
_ . -.

 

\

 

150

 

 

 

 

 

 

 

 

 

2

TON

1mm

 

(04) 11191qu ajmfsyow

151

tension samples using the previously described extensometer-strain indicator loading
setup. The results at three moisture content levels are listed in Table 10 and plotted in
Figure 45. The projected indicates the MOE-moisture content relation according to the

Wood Handbook [Forest Products Laboratory, 1987].

4.3 Application of the LVP Theory
4.3.1 Creep Compliances and Relaxation Moduli

One of the primary inputs for the LVP theory are the relaxation moduli described
in a two dimensional plane stress state - namely [Y].

When Onsager’s principal is applicable, one can draw upon results directly from
elasticity theory for each type of geometric symmetry of interest [Schapery, 1967].
Therefore, for an orthotropic material, there are 9 independent relaxation moduli. Upon

reducing the stress state down to plane stress, that number is reduced to 4, that is

 

 

 

 

 

 

’a,,(:)‘ Yum Yam 0 ‘ rel}

+0220» - Yum Yam 0 «cg,» 4.3.1
L012“! . 0 0 Y“(t)‘ [8:2

01'

{00)} - [Y(t)]{e‘} 4.3.2

Superscript "‘ indicates time-wise constant strains in a two dimensional relaxation test.

The counterpart of the relaxation moduli m, the creep compliances [J] in a two

Table 10. Static tension MOEs of yellow-poplar.

 

Specimen #

Moisture Content (96)

 

 

1R

5
8R
12
14R
15
17
18
20
23

Average:
STD Coeff.l (%):

Projectedzz
Difference’ (96):

 

 

 

R 1
R2
R3
R4R
R5
R6
R7
R8R
R9

3.10

Average:
STD Coeff. (%):

Projected:
Difference (96):

1: Deviation coefficient.

 

 

11.63 16.85 23.08
Longitudinal Tension MOE (103 psi)
2081.04 1712.90 1587.63
1529.95 1324.50 1125.91
2075.81 1794.06 1488.51
1917.83 1689.10 1424.80
1330.26 1202.30 1079.40
1780.15 1544.26 1287.23
1748.54 1536.81 1308.21
1616.92 1452.02 1288.86
1467.14 1237.54 1109.24 '
1586.09 1436.22 1260.47
1709.05 1502.36 1212.30
1712.98 1493.82 1297.60
13.42 12.20 11.65
1680.91 1509.27 1268.66
-1.87 1.03 -2.23
Radial Tension MOE (10’ psi)
155.96 126.54 111.60
127.48 115.21 104.19
127.10 113.40 105.40
111.22 105.28 98.49
150.21 132.28 114.67
153.03 124.13 110.5
126.97 114.90 106.81
156.23 138.14 121.33
169.02 149.63 128.88
188.09 169.55 146.77
146.53 128.93 114.86
15.02 14.28 11.75
142.03 131.14 110.66
1.71 -3.66

-3. 97

 

 

2: MOE computed according to Eq. 3.3.5 on the basis of the M0133 at two other moisture content levels.
3: Difference between the computed MOE and the measured MOE.

153

5.8..-323» we mmOZ .8123 055m .9 053m

35 “53:00 33202

 

 

 

 

 

 

om. mm ON mm o_ m
.1 . _ _ e _ _ L _ . _
oat 63mg. . fioc _
Umaooqem
1 .633... -
nd 35039.n—
Q 1 1 .
m. 1 a
m
3 2L1 row;
\I/
03 -
M. 9:1 lab;

 

 

 

 

 

 

 

 

 

81 . loom

(18d 901) How [QUIPUNSUOT

154

dimensional creep test is expressed as

F ‘ ‘

1811(0‘ 1,110) 1’20) 0 1 011
12,20» - 1120) 122(1) 0 1 02:2. 4.3.3
£8120) _ 0 0 J“(t)‘ £01.21

 

 

 

 

 

 

where superscript * refers to time-wise constant creep stresses. If Poisson’s ratio is

introduced as in elasticity theory, then [J] can be expressed as

1 J11“) ' V21“) 122“) 0 1
" v12(t)Ju(t) 1220) 0 4.14
0 0 16,01

 

 

If the two viscoelastic quantities are linear, they may be related through Laplace

transformation

[Ytsn - 993: 4.3.5
S2
Where s is the variable in the Laplace domain. Thus the relaxation moduli may be
obtained if all components of the creep compliances are known, namely 1,,(1), the creep
compliance in the grain direction, 1,,(t), the creep compliance across the grain, 1,,(0,
a compliance of the nature of Poisson’s ratio, and 1““), the plane shear creep compli-
ance. However, only the creep compliance across the grain (radial) of yellow-poplar
J,,(t) was tested and known. This is sufficient for the purpose of this study as shown in

the following.

155

Wood is viscoelastic radially as well as longitudinally as has been shown in the
literature. It is also known that wood is relatively very rigid in the grain direction
compared to the radial direction with a MOE ratio of 10:1 for yellow-poplar (Wood
Handbook) [Forest Products Laboratory, 1987]. The mutual restraint due to cross
lamination must result in much larger viscoelastic activity in the radial direction. It is
therefore reasonable to disregard the relatively small viscoelastic activity in the
longitudinal direction and assume complete elasticity in the grain direction in a cross
laminated structure. J,,(t) is thus known to be 1,,(0) — longitudinal elastic compliance,
easily obtainable by a static test. If a simple radial tension creep test is performed on a

radial sample which creates a stress state of

0
. 4.3.6

the strain in the longitudinal direction by Eq’s. 4.3.3 and 4.3.4 would be

sum - -v21J22(t)a' 4.3.7

Due to creep associated with longitudinal strain, e,,(t) should not change with time.

Therefore,

811“) - 811(0) - ‘V21(t).,22(t) - -V21(0).,22(0) 4.18

and

156

122(0)
V21(t) '36)— V21(0) 4.3.9
122(0) is the initial radial creep compliance which happens to be the radial elastic
compliance. 1121(0), the initial creep Poisson’s ratio at t=0 happens to be the elastic

Poisson’s ratio. Since both of them plus the radial compliance 13,“) are all known, v,,(t)

is easily found by Eq. 4.3.9. By compliance symmetry

-v,,(t)12,(t) - -v12(t)J“(t) , 4.3.10

Poisson’s ratio 7,,(t) can be easily found to be

122“)
111(0)

v12(t) - v21(t) 4.3.11
J(t),, defined in 4.3.3 and 4.3.4 can subsequently be obtained.

In a cross lamination structure, the plane shear stress and strain are absent. The
shear compliance term J“(t) therefore is irrelevant and could take any nonzero value.
This may be an important for the wood industry where cross lamination is the most
widely practiced lamination scheme for wood and wood-based composite panels.

Obtaining the relaxation moduli [Y] from the creep compliances [J] by Laplace
transformation is only valid if both quantities are linear and isothermal. But, they are not,
since warping is a nonlinear nonisothermal viscoelastic phenomenon. However, we have
become familiar with the technique of discretization of the time domain in the numerical

form of the LVP theory, the viscoelastic behavior and thus these two quantities may be

157

considered as linear and isothermal within a small enough time interval, where stresses,
moisture content, and Poisson’s ratio all vary so little that it is both practical and realistic
for them to be assumed constant.

The expressions for creep compliance terms depend on the chosen mechanical
model which in this study is the four element Burger body with an non-Newtonian
Maxwell dashpot, as discussed in Chapter 3. Such a nonlinear model may be linearized
if considered in a sufficiently small time interval where the non-Newtonian (nonlinear)
dashpot would have such a minor variation in its coefficient that it can be regarded as
a Newtonian (linear) dashpot with constant coefficient.

For a linear four element model, the creep compliance in terms of the coefficients

of its four elements is

E.“

-—:
1.1:) - ELU—e "“~>+—1—+ ‘

bu ms, ”and“

ii - 11.22.66 43'”

 

If Poisson’s ratio is assumed constant, the corresponding relaxation moduli by Laplace

transformation is found to be

 

 

 

Emuxua) - v21 ENIIR" 0 l
(1 ‘ V12 V21) (1 ‘ "12 v21)
[“0] ' -VIZEmaR22(t) EWRuO) 0 4.3.13
(1 ’ V12 V21) (1 ‘ V12 V21)
0 o EmaRJt)‘

 

 

where

158

"P 3 ‘Puz‘
R110) - Aue "' +Bae

 

 

””1 '10“: 4.3.14

 

c. - 5252
"u“ "m,
ii - 11,22,66

It is to be remembered that this conversion is made under the condition that the Poisson’s
ratio is constant and is therefore only valid during a small time interval where Poisson’s
ratio can be regarded as constant. It is seen that both creep compliances and relaxation
moduli are functions of the coefficients of the four elements of the proposed mechanical
model. In other words, the study of creep compliances and relaxation moduli is reduced
to that of the coefficients of the elements comprising the model.

Due to the continuous variations of moisture contents and stresses during
hygroscopic warping, the four coefficients in the relaxation moduli would vary

accordingly. The numerical form of the LVP theory due to its discrete nature allows only

159

discretized descriptions of those variations. Within each thin layer, moisture content and
stresses are assumed to be the averages of their distributions in the layer, and remain so
during the current time interval before they jump to new levels at the beginning of the
next time interval. The moisture content dependence of the coefficients would subse-
quently follow such step-wise pace with the moisture content.

The stress dependence of the coefficients is more difficult to deal with. In order
to account for the stress dependence of the coefficients, stresses must be known. Stresses,
however, can not be obtained for the current interval before the determination of the
coefficient values. This indetermination results from the nonlinearity of the viscoelastic
behavior where the coefficients (coefficients determine compliance) depend on the stress
levels. The approach taken here to break the impasse is to use the stresses already found
at the previous time interval as the basis for computation of the stress dependence of the
coefficients for the current interval, and then obtain strain rates, strains, and stresses.
It should be a good approximation since the time interval is very small and thus the step-
wise stress increase from the previous time interval to the current one is not significant.

A better method however is by iteration of computation to reach convergence.
Specifically, stress dependence of the coefficients for the current time interval is first
calculated based on the stress values from the previous time interval, followed by the
computation of the stress values using the obtained coefficient values. The computed
stress value for this time interval is the first approximation. In the next iteration, stress
dependence of the coefficients for the current interval are again computed, but this time

on the basis of the first approximation of the stress values. The resulting new coefficient

160

values when used to compute the stress should result in new stress values called the
second approximation which are different from those of the first approximation. This
iteration computation could carry on until the difference of the stress values of the newest
iteration from those of the previous iteration is relatively insignificant indicating that
convergence has been achieved. Such technique should yield better results. However,
it is not adopted here because the intensive interaction computation was expected to take
too much computing time, slowing down the computing speed on a desktop PC. It should
pose no such problem when the numerical LVP is implemented on mainframes. Such
implementation of the computer program is not difficult.

Of the four coefficients (Em, Eb, fl...» flu) which determine the relaxation moduli

[W0]. 7...: defined as

t1 - WC,”
a(MC,o)m(MC,o)

thMCm) - 4.3.15

 

needs special attention since it is also dependent on time in addition to moisture content
and stresses, as indicated in Chapter 3. Therefore, time into the warping process must
somehow be factored into the value of this coefficient. As we only know of the variation
of the coefficient with time under given constant stress and moisture content, certain
treatment must be adopted to approximate its variation under changing moisture content
and stress during the warping process.

If it is assumed that the stress a and the moisture content MC are to increase step-
wise along the discretized time scale shown in Figure 46, the flow component of the

creep compliance due to Ad, and MC, in the first time interval is by Eq. 3.3.24

161

 

 

 

 

 

 

At At At

1 2 3

MC MCZ MCB

 

 

 

 

Figure 46. Superposition scheme for flow compliance.

162

111110,, Aa,,t)|ml - a(MC,, Aa,)t”(uc"M’) 4.3.16

Therefore, the reciprocal of the coefficient I/"u which is assumed constant in the small
time interval At, takes the following value at the midpoint At,/2, which is the differentia-

tion of 4.3.16 at At,/2

I At “MCI,AUI)'I

1 -a(MC,,A a,)m(MC,,A a,)(—2—’)

"ma

 

'4‘). 4.3.17

1
wimp“)

In the next interval where moisture content is elevated to MC, and stress incremented by
another amount of Ad, on top of A0,, the part of the flow compliance due to the stress
Aar, at the second interval (At) is not directly available since moisture content and stress

have changed, but could be approximated by

 

J Mc,xm, +MC2xAt2

’40,: 4.3.18
At,+At2 1 )lm’

where the creep is supposedly to occur under the weighted average moisture content of
the two intervals. Another part of the flow compliance is due to the A0, at the beginning

of the second interval which contributes an amount of

110110,, 402,1)”, 4.3.19

Superimposing the two parts results in the flow compliance for the second interval. The
reciprocal of the flow coefficient for the second interval is then the differentiation of the

sum compliance at (At/2)

163

1 1 I

- +
nfilac. MCIxAt1+MC2xAt2 A IAt 4,2)lm, MCZA 2 4,2) I»,
, 0 , +—— , 0 ,
"J 4‘1”"; ’ 2 "J —2 4.3.20

 

 

 

At the next or third time interval, there will be three parts of the coefficient summed up
due to the presence of three stress jumps, A0,, A0,, and A0,. By this superimposing
scheme, the coefficient 11,, for every discretized time interval along the time scale of the
warping process may be approximately and numerically obtained.

The resulting relaxation moduli [Y(t)] as input for the application of the LVP
theory therefore accounts for the changing of its four coefficients with moisture content,

stress level, and time in the warping process.

4.3.2 Hygroscopic Strain Rates

The hygroscopic strain {eMc(t)} may be expressed as

 

 

 

 

 

 

rei'fa)‘ m) ‘a,,(:)‘ t ’a,,(:)l

leffa)» - f 4022(t)rdMC - f4 d22(t)tMC(t)’d1 4.3.21

Legca)‘ "0(0) , “120), 0 L612“),

01'

{8"C(t)} - f { 0(t)}MC(t)’dt 4.3.22
0

where {emm} and {am} are hygroscopic strains and hygroscopic expansion coefficients,

respectively. MC(t)’ is the rate of moisture content variation. It follows that the strain

164

 

 

 

 

rates are

r8150)" ra,,(t)‘

«ego» - 4a,,(t) lMca)’ 4.3.23

#200), 312“).

due to
x /

00/ ' [[1110 dx] - fix) 4.3.24
0

For a small time increment At, hygroscopic strain rates could be approximated by

 

 

 

 

 

r i / r ‘
amt) a,,(t)

legged» - 4 and» 431‘: 4.3.25
.4360) .“12(‘)l

since

MC(t)’ ~ 5%? 4.3.26

a,,(t) and a,,(t) are the expansion coefficients in the grain and across the grain
direction, respectively, while a,,(t) would be zero. If the x-y coordinates are rotated
clock-wise by an angel 0 relative to the 1-2 coordinates as indicated in Figure 12, then

the strain rates in x-y coordinates become

 

 

[8:60)‘ {030$
l exec» ~ i an“)
L age“! . an“).

 

 

165

} AMC 4.3.27

 

At

The {a(t}),,, (hygroscopic expansion coefficients in x-y coordinates), are related to

{a(t)},,, (hygroscopic expansion coefficient in 1-2 material principle coordinates) through

coordinate transformation [’1]

 

 

Fa ‘
“u n

a
a2; ' [Tu W}
0 .52

h 2 J
where

r 00829

[T] - s' 20

 

4.3.28

sin’fl 2cosasin0
c0320 —2cos&sin0 43'”
_—sin0ms'0 sinacosfi cosZO-sinza

4.3.3 Computer Programming of the Numerical Form of the LVP Theory

The numerical form of the LVP was implemented in the Microsoft QuickBASIC

language. The computation is automatically carried out on a 486DX50 IBM compatible

desktop computer, given the necessary inputs. The detailed programming codes are

presented in Appendix A. It is to be pointed out that the programming focus was on the

main computing body. The preprocessing which handles the accepting of inputs and

166

postprocessing which handles the output function are merely functional and sufficient for
the author, but by no means fancy and user friendly. However, these two parts can be
refined to a user friendly and efficient level without much difficulty if necessary in the

future.

4.3.4 Preparation of Inputs

Aside from the relaxation moduli and the hygroscopic expansion strain rates
already formulated in previous sections, the remaining necessary inputs are the thickness
and grain orientations of constituent layers. The yellow-poplar laminate in this study
consisted of two cross laminated lamina of equal thickness, each assumed to be

comprised of an equal number of imaginary thin layers.

4.4 Theoretical Predictions and Analysis

In Figure 41, theoretical predictions by the LVP on the vertical deflections are
plotted against the measured values for the yellow-poplar laminate and beam (refer to
Figure 18 for the coordinate positions and locations). Moisture content developments
with respect to time are presented as a reference on the progress of the warping process.
Also included for comparison are the elastic predictions which are achieved by replacing
the viscoelastic relaxation moduli with the elastic moduli for inputs in the LVP, equiva-
lent to using a mechanical model of only a spring.

The laminate and beam are seen to deflect very quickly in the early stages of the

warping process where the moisture content gain is rather steep. Their vertical

167

deflections, however, level off at about 200 hours into warping process - 80 hours before
moisture content equilibrium.

The overestimate by the elastic prediction is expected because the viscoelastic
nature of the warping process is not accounted for in the elastic moduli. The leveling
off of the elastic predictions occurs exactly where anticipated as the hygroscopic strains
ceases to increase at moisture content equilibrium. The elastic prediction gave somewhat
better estimates than the elastic beam theory [Suchsland, 1985]], possibly due to the
incorporation of the effect of moisture content on elastic moduli during warping. The
latter, however, is much simpler in that it only requires inputs of the total moisture
content increase for the whole warping process, the average expansion coefficients, and
the elastic moduli at the end condition. The small improvement by the former whose
complexity results from accounting for the moisture content effect on the elastic moduli
during the warping process only proves that the elastic beam theory yields a sufficiently
precise and practical estimate. It further demonstrates that extra effort within the realm
of elasticity is of very limited reward, and such effort should bedirected towards the
study of viscoelasticity. It is worth mentioning that the elastic beam theory has provided
warping estimates that are sufficiently accurate at low and moderate moisture contents
[Suchsland, 1985].

The viscoelastic predictions show quick deflection increase due to rapid moisture
content increase, and reach a peak at about 130 hours of exposure when they start to
relax to final values that are within 5% of the measured at 800 hours. The LVP theory

reflects the early domination by the large hygroscopic expansion resulting from rapid

168

moisture content increase. Even though deformation creep and stress relaxation must be
occurring simultaneously and picking up due to the rise in stresses, they are still
overshadowed. However, as the moisture content increase slows down, deformation
creep and stress relaxation increase their share. When the predicted vertical deflection
peaks at 130 hours,deformation creep and stress relaxation start to dominate. As stresses
are relaxed, less deformation creep and stress relaxation take place, which is seen as the
decrease in the relaxation rate. When the moisture content is completely equalized at 300
hours, no further hygroscopic strains are introduced. The LVP responds with a inflection
point in the theoretical predictions. From that point on, only deformation creep and
stress relaxation take place. They slow themselves down due to the stress reduction till
the leveling off point at 800 hours.

Though the LVP theory is clearly able to simulate the warping process and
provide much improved predictions over its elastic counterpart, it deviates from the
actual warping development, especially in the early stages. In the theoretical predictions,
peaks are observed and it is well after moisture content equalization that predictions relax
to stable levels, while actual vertical deflections have no such peaks and reach a stable
levels at about 200 hours, well before the point of moisture content equalization (about
300 hours). It is evident that there exists in the warping process a mechanism
responsible for the much earlier and faster relaxation than described in the LVP theory.
This disparity could possibly be due to either the limitations of the LVP theory itself or
the inaccurate description of the constituent’s (yellow-poplar) viscoelastic response. The

former can not be verified before the latter is investigated and corrected. Regarding the

169

latter, the effects of moisture content and stresses on wood and wood-based materials are
not as simple as characterized in this study, where only the separate effects of stress and
moisture content are considered. When moisture content is changed under stress, creep
and relaxation are much larger and faster than if moisture content is changed prior to
stressing as has been discussed in detail in the review section. Moisture content and
stresses change simultaneously in a warping process, and assuming separate effects of
moisture content and stress is obviously not realistic, very possibly resulting in the large
variance between the theoretical predictions and the measured vertical deflections.

As the LVP theory requires the discretization of the panel into a number of
layers, theoretical predictions are computed based on both a 10 layer and a 6 layer
discretization. Slight differences resulted, as listed in Table 11. Theoretically, more

layer elements generate better results, but require more computing time.

Table 11. Theoretical predictions vs. measured vertical deflections of the yellow-poplar
laminate and beam.

 

 

 

Vertical Deflection

 

 

 

Measured Viscoelastic Prediction Elastic Prediction Elastic Beam
Theory
(In) (in) (in) (In)
6-layer 10-layer 6clayer 10-layer

Laminate

@ (31:14.1, 0, 0) -0.623 -0.625 -0.642 -2.699 -2.698 -3.050

@ (0, $9.15, 0) 0.246 0.263 0.270 1.137 1.136 1.285

Beam

@ (0, $9.15, 0) 0.291 0.263 0.270 1.137 1.136 1.285

m

170

Since the agreement of theoretical predictions and the measured vertical
deflections was checked for only discrete locations ($14.1, 0) and (0, $9.15) of the
laminate, it is not known yet how well the overall warped shape of the laminate is
simulated by the LVP theory. One way to examine this is to check the vertical
deflections at as many locations on the laminate as necessary. However, this can be
achieved by a much simpler test.

The Kirchhoff Hypothesis (the displacement field) - the foundation of both the
CLT and the LVP theory - assumes a quadratic surface for laminates as defined in Eq.
2.4.9. For the particular 2-ply cross laminated yellow-poplar laminate, the LVP theory
predicts that curvatures x, and x, are of equal magnitude but of opposite sign, while the
twist curvature rotation at” is zero. The cross lamination of the two identical layers
determines that the same degree of curvature but in opposite direction should develop in
the x and y directions, respectively, and that no twist should occur in the laminate.
Therefore, by Eq. 2.4.9, no vertical deflections should occur along the two 45° diagonal
lines, if the warped shape of the laminate agrees with the quadratic surface. Placing a
straight edge along the diagonal lines on the laminate showed little vertical deflections,
thereby confirming that the actual deformation of the laminate does approximately follow

a quadratic surface as the LVP theory assumes.

CHAPTER V

Summary and Suggestions on Future Investigations

This work is probably the first attempt ever to formulate a viscoelastic plate
theory in solvable numerical form integrated with viscoelastic properties of constituent
materials to theoretically approach the hygroscopic warping phenomenon - a nonlinear
nonisothermal viscoelastic process.

The LVP theory was developed on the basis of CLT and linear viscoelasticity
theory. The viscoelastic properties of yellow-poplar were characterized using a four
element Burger body model with a non-Newtonian Maxwell dashpot to account for the
non-Newtonian behavior of the flow component of the viscoelastic deformation. The
numerical form of the LVP theory was achieved by applying linear finite approximation
to the integral viscoelastic governing equation of the LVP theory (discretization of time
domain). Numerical computations were implemented in QuickBASIC and conducted on
a desktop PC computer. The treatment taken in applying this linear isothermal theory
to nonlinear nonisothermal viscoelastic warping problem is that the viscoelastic
coefficients determining the viscoelastic properties of the constituent materials were
allowed to vary accordingly with moisture content, stresses, and time during warping.
The validity of this theory and its superiority over elastic prediction in describing the

viscoelastic warping process was demonstrated at least on a preliminary level in its

171

172

application to a two-ply cross laminated yellow-poplar laminate.

In characterizing the viscoelastic properties of yellow-poplar in the radial direction
as inputs for the application, it is verified what many authors have found, namely that
wood exhibits nonlinear viscoelasticity of its flow and recoverable components, while the
instantaneous elastic component remains linearly elastic. The elastic component varies
with moisture content according to the known moisture content effect on mechanical
properties, while the other two components are shown to be dependent on both moisture
content and stresses.

The development and computer implementation are somewhat involved. But once
carried to completion as in this study, the application is reduced to running the program
with required inputs on a desktop computer and retrieve the result files from the
designated disk. The theory and its numerical form are material independent and could
be applicable to a variety of panels of different materials and structures suffering from
hygroscopic deformation, as long as the information regarding the structures and the
viscoelastic properties of the composing materials are given.

The LVP theory loses its applicability when the actual displacement field gets
more complicated than that of the Kirchhoff Hypothesis. It has been proven that beams
and plates of small aspect ratios do not deform according to the Kirchhoff assumption
because of the relative significance of transverse shear and normal components.
Fortunately, most industrial wood and wood-based composite panels that are susceptible
to hygroscopic warping have large aspect ratios which ensures convergence to the

displacement field specified by the Kirchhoff Hypothesis, as supported by the simple

173

straight edge test.

The applicability of the described LVP theory very much depends on the
availability and accuracy of the required inputs. Of the greatest importance is the
formation of the two dimensional stress state relaxation moduli that characterize the
viscoelastic properties of the involved materials. How accurately and completely the
relaxation moduli account for the viscoelastic properties should greatly influence the
quality of the theoretical predictions of the LVP theory. In this study only the separate
effects of moisture content and stress were characterized into the relaxation moduli,
which is why the theoretical predictions, though much better than the elastic approach,
still deviate from the measured results. This shortcoming can be attributed in part to the
exclusion of the mechano-sorptive behavior (coupling between stress and moisture content
change). The mechano—sorptive effects are known to cause as much as many times more
creep and faster relaxation than would occur when moisture content is changed prior to
stressing. Such effect is very prevalent in a warping process where the moisture content
changes during stress development, and therefore must be factored into the viscoelastic
relaxation moduli for better results.

The moisture content gradient development and the hygroscopic expansion
coefficients which are necessary for calculating the cause of hygroscopic deformation -
hygroscopic strains - are also expected to influence theoretical predictions. As both the
LVP theory and warping are process dependent, the in-process variation path of moisture
content development and coefficients will definitely affect the final outcome.

How moisture content gradient development, expansion coefficients, material

174

viscoelastic properties, and laminate structures affect the hygroscopic warping have up
to now been either qualitatively analyzed or experimentally determined. This could now
be evaluated and analyzed relatively quickly on a computer by entering different variables
into the LVP theory. The described method could serve as an efficient research tool in
aiding further study and providing understanding of the hygroscopic warping process.

Several assumptions are essential in this study. Transverse normal and shear
components were not considered as a result of the CLT structure of the LVP theory (the
Kirchhoff Hypothesis). Their definite presence has been proven to exert minimum
influences on the overall behavior of laminates of large aspect ratios (Chapter 2).
Mechano-sorptive coupling of wood and wood-based materials (varying moisture content
under creep load) is known to be heavily involved in the viscoelastic warping process,
yet has not been well understood and mathematically defined. It was therefore not tested
and characterized into the relaxation moduli input for the application of the LVP. Plane
shear component was assumed to be zero in the yellow-poplar laminate due to the cross
lamination structure. It was also assumed that there is no viscoelastic activity in the grain
direction because of the relatively highly crystalline and elastic nature in the grain
direction.

As to further improvements of the applicability of the LVP theory to the warping
problem, the first to be suggested is a characterization approach that can integrate
mechano-sorptive effects into the relaxation moduli. The mechano-sorptive effect in the
warping process has been shown to be too significant to be ignored.

The applicability of the LVP theory was examined on a preliminary level in this

175

study. Further verification on a larger scale is necessary. In the meantime, it can offer
some new insights into the warping process.

The programming of its numerical computation needs to be refined, especially the
preprocessing and postprocessing portions that handle processing of inputs and outputs.
Increased capability and versatility of the preprocessing portion would allow inputs in a
variety of forms. Improved postprocessing would provide better illustration and

interpretation of outputs.

 

l‘.

APPENDIX

Computer Program of LVP in Mirosoft QuickBASIC

Computer Program of LVP in Microsoft QuickBASIC

oecuae sun mm m. n. emsm
oecune sue eamocm (low
oecme sun uxmmm out 8)
oecune sue minim (m S)
oecune sue name»: (m not
oecune sun mums (All. All»
DECLARE sus puma (All. no. net» ~
necune sue masses 0
oecune sue mscnu mu. m
necune sue mm: M. om)
oecme sus nurses at on, TT. cars new)
uecune sun HOTMF no
oecune sue wrecm mum, mutt mom
oecuse sun mo out v0. con»
oecune sun umsrune at on. u m
oecune sue nonetcoe at XFLABOO. PAM»
oecune sue expanse: (K. on. ll
oecune sue new (It on on Ch
oecune sun coeeem m. P1. K1. om. ll
necuxe sue mum mu rm CEFlOl
oecune sun mm m. m an em
oecune sun euuccoee ma mt cent»
oecune sun animate «on men um»
necune sue smeom (SXYO. men. 3120)
oecune sun mum m. Pl. Kl. om)
necune sue mom M. P!. m. nun
uecuute sun scream at m A. a. c. n. e. Fl
necune WON mm: mm. m
oecune euucnou ammo! m)
necune auction LEFTWOROO amt
oecuae Function uncum ms. to
oecuse mm menw as»
oecune FUNCTION memes rrvu
oecune auction RELMDU mm. as. M. K8. m. It
comes v. on
com semen mo. meme. flexes. srnesso. mun. mum. mum. name. means
cm semen mam. 0mm. mm. mists, tsmess. czt cv. sot. set. 9!. 91.. not. mes. sea:
on semen mum" to at tsmmn 1'0 ll
com lSTllAlml. memo
cm smen scm. nseu
KEYlll on
on mm some Item
KEYlZl on
on xevm ensue otseuv
CALL unscmntr. m
cxu unscnu
IF men: - 'Y' men
men on
use
men are

 

176

177

END IF

CTN - O

OH MRI 1) 608110 DISPLAY
OLO - THEVALUITNEH

OPEN OIRIPO o ”LE8 FORWASII
00
LE WT '1. "TO
LOOP UHTI. LEFTVOROHRITH - m.
00
LIE NUT #1.!th
LOOP UNTIL LEFTWOROOIITH - 7 OR EOFIII
1F EOFIII - -1 THEN CALL ERRLOCTTK'2 I WOULE'l
L“ “PDT #1. “TO
K - VALIDITS)
M - MTNUHRIOLT, TTl 'RETUN TOTAL 3 OF ms
ON SHARED 1110 TO K1
ON SHARED AHOLEII TO K1. '11 TO K1
OII SHARED M0011 TO 201. TIMI TO 201
ON SHAREDUCM + 1. ITO K1
0“ SHARED AVW e 1. 1 TO K1
0" SHARED EXPCOHI TO K. I TO 2. I TO 41
OH SHARED EXWALIO TO 3. ITO K. 1T0 3. 1 T011
ON SHARED “E1211TO K. I TO 31
ON SHARED KENI TO K. ITO 31
ON SHARED 01211 TO K1
ONSHAREO RIOTOI + ILMTOI +11
ON SHARED PARIAII TO K. I TO 3. 1 TO 01
OH SHARED PAM" TO K. 1T0 3.1TO 31
ON SHARED PAMI TO K. 1 TO 3. 1 TO 101
DNSHAREDW 01,1T0K.1TO31
DH SHARED “AVE“ 9 I. I TO K. 1 TO 31
ON SHARED K81". I TO K. ITO 31
ON SHARED MN. I TO K. I TO 31
OH SHARED RKVM. 1 TO K. I TO 31
OH SHARED FLABOII TO K. I TO 31
OH SHARED VTRII TO K. ITO 3.1TO 31
DOA SIARED YEJII TO K. 1 TO 3.1TOI1
DISHAREOYKJII TOK. 1TO3. ITO 11
OH SHARED YHJII TO K. ITO 3. ITO 11
DNSHARED STRESW. 1 TOK. I T031
FOR J - I TO K
FOR I - I TO 3
STRESSO. J. '1 - O
KXTI
NEXT .1
DH SHARED AROOII TO 8. I TO 01
ON SHARED HUI TO 3. ITO 11
ON SHARED GRMII TO 0. I TO 11
ON SHARED Elm. I TO 01
ON SHARED EKfl + I. 1 TO 01
FOR 11 - 1 TO 3
EKIO.1|1 - O
EXT 11
OH SHARED PSKII TO 01. PSYII TO 01
ON SHARED POXI I TO 01. PDYII TO 01
ON SHARED FHXII TO 61. PHYII TO 01
OH SHARED PPXII TO 01. "Y1 I TO 01
DH SHARED M11 TO 31. M11 TO 01
FOR 11 - I TO 6

178

PSXIIII - O: PSYIIII - 0
POXIIII - O: PDYIII) - O
PHXIII) - O: M1111 - O
PPXIII) - 0:1?le - O
PMXIlll - O: NYIIII - 0
NEXT 11
CHIS - 0
CT”? - 0
CALL WTSECIK. DLT. T'T. CH8. mm
CALL ”“1110“. TT1 'COWTE 00 A“ M
CLOSE [I

CALL PLOTCWHKl

CALL FILEIJITIK. DLT. CX. 0“

CALL MOISTUREIK. DLT. M. CUIS- 11
CALL EXPAHSNK. OLT. H1

PORN-0T0“
FORJ- ITOK
FORII- ITO3
YEJIJJLII-O
YKJIJ.11.11-O
YHJIJJLIl-O
HEXTII
HEXTJ
FORI- ITOO
CRAlDllll-O
HEXTII
lEI-OTHEH
CALL COEFFNIIDJ. K.DLT.'A1
FORI- ITOO
FORII- ITOC
ARODIlM-O
HEXTIII
Mll-O
HEXTII
CALL STFHHSII0.0.K. 0LT1
ELSEFH>OTHEH
FORI-ITOH+1
CALL COEFFIIILIKOLT.H1
FORR- ITOB
FORII- ITOO
AOOOIlli-O
KXTII
”111-O
HEXTI
CALLSTFIHSIILIKDLU
EXTI
E1011:
CALLMOMIMMKDLTI
WU

LOCATE 24. 411- Ward M tint '1
FRUIT 'Total m tun: ':
FRI“ TKSTRBNTOAEVALUITIEOI - mm

00
LOCATE 24. 40 ~ LENI'SM m [N]: '1
INPUT '8". mm [H]: ', CHOICES
IF CHOICES - 'Y' THEN

 

179

SHELL 1'C:'1
SHELL 1' WW
SIELL 1' COIVPTIOFIOIITPUT'I
SHELL M ' e PREFIX: 0 mu
SHELL I’COIVPTIOHOLITPUTI' e PREFIX! e “~81
SHELL I'OEL ' e PREFIX! + "."1
SHELL I'COPY O:' o PREFIX! + "."1
SHELL I'OEL Oz' 4» PREF1X8 o "."1
SHELL I'OEL O:' + PREFIX. o ".PLT'1
SHELL I'COI')
SHELL I'COIVPTIPF'I
EXIT 00

ELSEIF CHOICE! - 11' OR CHOICES - " THEH
EXIT OO

EHO F

BEEP

LOCATE 20. I

PRIT STRISOIN. ' '1

LOOP

KEYO:
9H1
RETUM

DISPLAY:
If CTN - 0 THE!
LOCATE 25. 2
PRIT 'Tu “and: ';
Ell) F
ROW - ML.
COL - POSIOI
m PRIT 13 TO 25
LOCATE 25. 18
PRIIT TKSTRCOMALIRTKH - OLDI.’
m PRNT 13 TO 24
LOCATE ROW. COL
CTN - CTN o I
RETURN

SUBCOEFFII 11. P. K. 0LT.”
lEl-P+ ITHEHEXITNC
DHSSIZIITO31

IEI- OWP-OTHEH

LOCATEI7.1

PRITSTRHO!I73."1

LOCATE17.22

PRIT'AMWWI'

FORTH -OTOI¢I

FORJ - ITOK
FORII -1TO3
WJJII - ME121J. 111' MEIZIJ. IIHIEW. 1111‘ (IIZ-WJmml- 1211

180

MAVGITN. J.1|1- ME121J.111' M120. mmesuu. 1111‘ 1112 - AW J111MJ1- 1211
KXTII
NEXTJ
NEXTTI
EXIT SUB
ELSE
IE TYPE! - '8' THEN
EXIT SUB 'COWLETE ELASTICITY
ELSE VISCOELASTIC"?
IFI - 1 THEN 'ONLY THE IST CYCLE "TH“ THE HER LOOP.
FORJ - ITO X
MCI - INCIPJI e '00- 1,J1112
N62 - MCI ' MCI
FOR 11 - ITO 3
IE FLASIIJ. 111 - 'V' THEN VISCOELASTIC LAYER
SI - ABSISTRESSIP- I.J. 1111
AAA - PARNAIJ. II. 11 0 PARHAIJ.1|.21'W1¢ PAMJJVNCZ
AAO - PARNAIJ. ll. 41* PARIAIJ. I. 51' m1+ PM. I. 01 ° m2
AAC - PARMAIJ. ll. 71' EXPIPARNAIJ. 1101' N011
KSIP, J. 111 - ”SUCH. J. 111' (AAA ' EXPI~AAI ' 811 + AAC1
ORR - PAMIJJI. 11' EXPIPAMIJJLZVUIH PAWJJI
RKVIP. J. 111 - 0001K”. J. I)

lETYPEI-‘KS‘ORI<>PTHEN
EXITSIIO
ELSE
fI-OTHEN
EORJ-ITOK
WI-NCIJJJI
m-NCI'NCI
FORl-ITO3
1E FLABHJJI-‘V'THEN ‘VlSCOELASTELAYER
CCA-PAMUJLH'II-EXP1PARNCU.121'm1-Pm.lm
CCO-PARNCIJJIJI'II-EXP1PARWIJ.K.51'MI-PAW.IOII11-Pm.l71
CC - CCA '11 - EXPICCO ' AOSISTRESSP -1. J. 111111
OO-PARMCIJJIJI'EXPIPARUCIJJJVADGSTRESSIP-LJJMOPMJJOI
lEl-ITHEN
MILJJII-O 'ZEROWOATISTHRCYCLEDEFOREWM
ENDIE
M11101-CC'DD'IP'OLTI‘IDD-IHMIJJJJI
E105
KXTI
KXTJ
ELEFI-ITHEN
FORJ-ITOK
EORJJ-ITOI
NCI-AVGICHJJJI
WI-MIJJ.J1+NCIJJ~I.J1112
NC2-AICI'IEI
FORII-ITO3
IEFLAGOIJJII- 'V'THEN VISCOELASTIC LAYER
lEJJ-ITHEN
SD - ABSISTRESSIO. J. 1111
SIGN-1
ELSE
SD-STRESSIJJ-1.J,|11-STRESSIJJ-2.J.Il1
SIGN-1

 

181

IESTRESSIJJ-I.J.II<-OAMSTRESSJJ-2.J.II1 <-OTHEN
IESO < OTHEN
SfiN-I
ELSEIESO > OTHEN
SIGN-“I
ENDIE
ELSEIESTRESSJJ-IJ.”<-OAMSTRE881JJ-2.J.l1>-OTHEN
SIGN-J
ELSEIFSTREsaJJ-IJ.”>-OAIISTRESfiJJ-2.J.KI<-OTHEN
SIGN-I
ELSEFSTRESSIJJ-IJJO>-OAMSTRESflJJ-2.J.I1>-OTHEN
ISO < OTHEN 'STRESSREDDCTIM

CCA-PAWJLI1'11-EXP1PAMIJ.I.21'H1~PMU.I.31111
CCO-PAMIJJJPII-EXPIPAMU.I51'w1-PAW.KM-Pm.l71
CC - CCA' 11 .EXPICCO’ “$180111
DO-PARICIJ.11.01'EXHPANCIJ.II1'AIK8TREW-I.J.I1H+“mull”
'OO - PARNCUJII'EXPIPAWJJI'MIHPMUJ101
lEl-ITHEN
MILJJIl-O 'ZEROWOATISTKRCYCLEOEFOREWTM.
SIM-RM”

NUT-CC'OO'M°OLT1‘100~IHMV01JJ.J.II

EMF

FMILJ.l1-OAMS$N'MT<OTHEN
“(1.101-MT

EMF

EMILJJIHISSN'UWK-OTHH

ELSE
WJJl-RUVILJJHISBN'MTI

EU!

'PRITLJJLRHVILJJII

'FMILJ.I1<OTHENSTOP

EMIE
KXTI
KXTJJ
KXTJ
EMIE
EMF
ENDIE
ENDIE

EXIT 81.10
ENO SUB
SUB ERRLOCTN ILOC81

CLS
COLOR 7. I

 

182

LOCATE 15. 15

PRIT 'ERROR J"; LOCO; '1'
LOCATE IO. 15

PRIIT ’PROSRAM TEMATEDI'
STOP

ENO 81.10
8110 EXPANSIJ 1K. DLT. M1
OIA EXPCII TO 3,1TOILEXPP11TO 3. ITO 11

LOCATE 17, I

PRRIT STRHStUS. ' '1

LOCATE 17.23

HINT 'Cflnlfl'lll um QM?

FORlI-OTOM
FORJ- ITOK
MCI -WI.J1
MC2-MCI'W1
m-MQ'WI

EXPCII.11- EXPCOFIJ.I.11’W3 # EXPCOFIJ. 1.21'
EXPCIZ. Il-EXPCOEU.2.11'U3+EXPCOFIJ.2.21'

EXPCI3.11- 0
CALL OTRAFOMEXPCO. ANSLEIJI. EXW
FORIII - ITO3
EXPVALII. J. I11- EXPPII. 11
KXTN
KXTJ
NEXTII

END SUI
SUB EILEMIT 1K. DLT. CX. CYI

LOCATE 17. I
FRUIT mm. ' '1
LOCATE 17. 2O

PRNT'M" ' flal'

OPEN OMOUT! o STRAU! EOROUTPUTASII
CLOSE II
OPENOMOUTO + STRA'O FORAPPEM ASII
PRNT lI.TA3111;STRAM8
PRITII, 'IMT - IOMIIM'
PRIT II. TAOI 11: 'HOUR‘:
.PRIT l1. TABISI: CHR812331 3' 'I";
PRNT I1,TA31211:CHR1123I1 e 'V":
PRUT l1. TAOI331; CHR812301 * 'IV";
PRIT II. "01451; 'Kx";
PRO“ II. TA01571: 'Kv":
FRUIT II. TAOIOOI; 'ny"
CLOSE #1

FOR 11 - 1 TD 0
SELECT CASE 11
CASE 1
SYMO - CHR312301 + '1"
CASE 2

meEXPCOEIJ. 1.31'U1 eEXPCOFIJ. 1.41
U2+EXPCOEIJ.2.31'UI OEXPCOEIJJ.”

183

SYN: - 01111812381 e 'y"
CASE 3
SYMI - CHR812301 4 'IV"
CASE 4
SIM: - 'Kx"
CASE 5
SYMO - ‘Ky"
CASE 8
SYM: - 'ny"
ENO SELECT

OPEN OIROUTO + LSTRANPOM FOR OUTPUT AS #1
CLOSE #1
OPEN OIROUT! 0 LSTRANPSIRI FOR APPM AS #1
PRNT II.TA8111; LSTRAM’OIII
PRUT l1. 'UMT - 10.M1h1'
PRRIT II. TAOI 11; 'HOUR';
PRIIT II. TAOIO1: SYMO
CLOSE #1
NEXT 11

 

OPEN OIROUT8 + STRAUS FOR OUTPUT AS '1
CLOSE #1
OPEN DIROUTt 4 STRAHO FOR m AS '1
ma #1. STRAIO
PRIIT #1.'UN1T - 10.W11'
PRNT II. TARIII; 'HOUR':
PRNT II. TADIOI; CHR812301 + '1':
PRNT '1. TADI211: CHR812301 * '1':
PRNT II. TAII331: CHR312301 * 'IV':
FRUIT II. TAOI‘SI: 'Kl';
FRUIT ll. TAOIS71; 'KV';
PRIIT (I. TARIOOI: 'KIV'

PRNT II. TAMI); USMC 'Mll'; 0:
FOR 11 - I TO 5
PRIIT '1. TM 4 II - 11 ' 121; US“ 'IW"": 0;
KXT 11
PRMT #1. “01001; USMC 'rmr““; 0
CLOSE #1

FOR ll - I TO 8
SELECT CASE A
CASE I
SIM: - CHROI2301+ '1'
CASE 2
SYMS - “113123310 ‘v'
CASE 3
SW: - CHlltmIl + '1"
CASE 4
SYMO - 11'
CASE 5
SYMO - 'Kv'
CASE 8
SYN: - 'ny'
EHO SELECT

OPEN OIROUTI + LSTRAMSIIII FOR OUTPUT AS ll
CLOSE #1
OPEN OIROUTQ + LSTRARIIIIII FOR APPEND AS '1

 

 

184

PRNTII,TA0111: LSTRAl!1|11
PRRJT #1, 'UNIT - 10.1X111'
PRIIT #1,TAOII1;'HOUR';
PRNT #1. TABLE); SYM!

FRUIT II. TACII 1: USMC 'fl"."': 0:
PRIIT lI.TAO(91:USN6 warm.- 0
CLOSE 11
NEXT 11

OPEN OIROUT! + STRESS! FOR OUTPUT AS #1
CLOSE [1
OPEN OIROUT! + STRESS! FOR APPEM AS 31
PRIJT l1. STRESS!
PRNT 1‘1.'UNIT - IIM'II'I'
PRIJT #1, TABI11; 'HOUR';
PRIIT l1. TAOIOI.‘ 'LAYER':
PRNT ll. TAOIIStCHR812201 4' 'u';
PRNT II. TABI201: CHR!12201 * 'w':
PRIIT II. TAOIAOI: CHR!12311 * 'lv'

PRMTII, TAOIII. USMC 'MIII'; 0:

FOR 11 - I TO K
PRNT II. TAOIOI; USU um I;
FOR I - I TO 2

PRITII. TANIO + II ~11'121: US“ 'IIM'""; 0:

NEXTII
PRNT II. TAMI: US” 'IW“"; 0

NEXT 11

CLOSE '1

FOR J - 1 TO K .
OPEN DIROUT! o LSTRES! * LTRNNSTRNJII + '.PM' FOR OUTPUT AS II
CLOSE #1
OPEN OIROUT! + LSTRES! 0 LTRUNSTRW” * 'Pfl' FOR APPEM AS II
PRNT l1. LSTRES! + LTRI!ISTR!IJ11 + {PIN'
PROJT #1. 'UNIT - (IM'IIT
PRIJT II. 'LAYER' + LTRH!ISTR!1JI1: TAIIII: 'TOP sumce'; mm; 'ROTTW SURFACE'
PRNT II, 'HOUR';TA0101: CHR!12201 + 'u'.‘ "01201: CHR!12201 * 'W'.‘ TAII311; CHR!12311 * 'lv'.‘
PRNT II. TAOIMI: CHR!12201 + 'u';TA31551; CHR!12201 * 'w': ”.1071: 31111312311 * 'lv'
PRIIT I1. USMC mm»; P ' 0LT;
FOR I - I TO 3
PRMT II. TAILS #10 -11'111;USU 'uur““; 0:
NEXT I
FOR 11 - 1 TO 3
PRIIT II. TAOIII +111. 3 ~11'111: US" 'IW'“': 0:
NEXT 11
PRIIT #1.
CLOSE '1
NEXT J

FOR 11 - I TO 6
SELECT CASE 11
CASE 1
SM! - 'TI'
OIR! - 'u'
CASE 2
SYMO! - 'TZ'

185

DIR! - 'yy'
CASE 3

SM! - 'T3'

DIR! - 'xy'
CASE 4

M! - 'OI'

DIR! - 'n'
CASE 5

SM! - '82'

OIR! - 'yy'
CASE 0

SYMO! - '83'

DIR! - 'ry'

END SELECT

 

 

OPEN DIROUT! + LSTRES! 4 SM! 6 '.PM' FOR OUTPUT AS #1
CLOSE #1
OPEN DIROUT! + LSTRES! + M! o '.PRN' FOR APPEU ASII
PRIIT PI, LSTRES! + SYMD! + '.PRN'
PRIIT #1. 'UNIT - (IM'IIII'
PRIJT #1, 'HOUR'; TADIOI: CHR!12231 + DUI!
PRIT l1. U803 'our; P ' DLT.‘
PRHT II,TA0101;USNS 1'1”!” ""; 0
CLOSE #1
NEXT 11

FOR 11 - I TO 2
OPEN DIROUT! + OW! 4» LTRIHSTRHIII * 'm FOR OUTPUT ASII
CLOSE ll
OPEN DIROUT! o OW! + LTRNHSTRHIII + 'PU' FOR APPED AS '1
PRIIT 11. OM! + LTRI!ISTR!II11 e '.PRN'
PRNT II. TAOI 11; 'HOUR';
PRNT ll. TARIOI; 'OISPLACEAENT MHI'
1F 11 - 1 THEN
X - CX: Y - CY
ELSE
X - ~CX
END IF
PRIJT 31. TADISI; '1':
PRIIT I 1. USMC '"I': X;
FRUIT II. '. ';
PRNT 31.11805 'ur; Y:
PRIIT I1. '1'

PRHT II, TAOI 11: O:
PRIT PI. TADISI; US“ 'lllll‘"": O
CLOSE II
IEXT ll

END SUB

SUD FUNCCOEF 1X11. Y0. CEFIII
X12 - X1111 X121

X23 - X121! X131

M - 1X111~ X121111X121- X1311
Y12 - Y11.111Y12.11

Y23 - Y12.111Y13.11

CEFIZ. Il - LOGIY1211Y23 ' M111L061X1211X23 ' M11

186

CEF13.11'L061Y1211X12 ' CEF12.111111X111- X1211
CEFII.11- Y11,1111X111' CEF12.11° EXPICEF13.11° X11111

END SUB
SUB NITSCRN
SHARED OLD

SCREEN 1
SCREEN 9
CLS

COLOR 7, I

LOCATE 5. 4

PRIIT 'm M! ';

LOCATE 7, 4

PRIT 'FI -Eal F2-I'n'

LOCATE 9. 4

PRlI'T Tun-Pam Emu-Conflu'

IF TKR! - 'Y' THEN
LOCATE 25. 2
PRIIT Tu “and: ';
END IF

VIEWPRIT 13 TO 24
V1EW1200. 0183.161. 1.2
NONI-40. 4101-1520. 1101
LIE 10. 0115”. 01. 7

LIIEIO. 4mm. INI. 7

FORII-ITOIO
LIEN'SOJHI'SOJIJ
NEXTI

FORII-ITO4
LME10.25'IIH5.25'111.7
“10,-25'111-15p25'1117

NEXTII

E10 SUD

WION ”AUTO 111

'EDUATION TO AUTOKATICALLY 0mm THE T“ mm YET TO BE DMD.
“TO ACTIVATE OR MACE AUTOMATIC DETERMIATION. THE MAWAL DETENTION
'SUDROUTK CALL 0! "TSCM SURROUTK HIST 0E Tum OFF.

EU euucnou

SUD mm (DLT. "1

DH TKI TO 151. T011 TO 151

II - O: SUM - O

RPL! - SED!

OED-Rel

T11111 - VALILEFTVIORDHRPL!”
TDIIII - VAULEFTWOROHRPL!”

 

 

187

SUM - SUN + TD11|1
LOOP UNTIL TD1||1 - 0
TDIIII - UT - SUM1
R101 - 0
BED - 0
FORI - I T0 11

NU“ - 110111 I T111111 DLT

FOR 111 - I TO NU“

111111 + 0E61 - T1111

NEXT 111

BED - DEG + NI.“
NEXTI
R1DEO + 11 - I

 

RSUM101- 0
RSUW +11- 0
FORI - ITO 0E0
RSUHII - Ram-11+ R111
NEXTI

 

END SUD
FUNCTION m 1DLT. "1

SIM - 0

RES - 0

RIM - SE04

00
TI - VAULEFNDRDOMO”
T0 - VALOEFTWORDOIRI'LW
SUN - SIN + TD
IF T0 - 0 THEN EXIT 00
DEC - REG 4 TDITIIDLT

LOOP

TD - TT - SIN

BEG - 0E6 + TDIITI'OLTI

”INN-0E0
WWW
wamscmmu.m

5N COLOR 7. I
CLS

ROW-4

OO
LOCATE ROW. 40 - LENI'bI! 4m Ii. [N ' .I'IINI: '1
WT'mauIiuIlI ' .PRNI: '. W4
If IIFNUIAS < > " TIIEN
llFILEt - 'II‘ + ms + '.PRN‘
EXIT 00
E140 IF
BEEP
LOCATE ROW. 1
PIIIIT STRICWO. ' '1
LOOP

188

LOCATE ROW . 1. 40 . [Whit W [C:1VF“DMTU: '1
RIPUT 'm 111mm IC:\VPTIDFIIIPUT\1: '. DIRIP!
IF DIRIIP! - " THEN
DIRRIP! - 'CzIVPTIDFINPUTI'
LOCATE ROW 6 1. 40
PRNT DIRIIP!
END IF

LOCATE ROW + 2. 40 - LEN1'OImIn My 10:]: '1
“PUT '0m New 10:1: '. DIROUT!
IF DIROUT! - " THEN
DIROUT! - 'O:'
LOCATE ROW + 2. 40
PRIIT DIROUT!
END IF

LOCATEROWO 3.40-LEIH'PufiqumII-m'1
W'fiufilfvmm: '. PREFIX!

STRAI! - PREFIX! + ‘N' + W! 4* '.PIII'

STRANP! - PREFIX! + 'P' + ms + '.PRN'

FOR 11 - I TO 0
LSTRADIHIII - PREFIX! + 'N' + m: 4» LTRN!ISTR!1R11 1» '.PM'
LSTRANPHII - PREFIX! + 'P‘ + W! + LTRIHSTRNIII * 'mr

NEXT II

STRESS! - PREFIX! o 'S' o M! o '.PII'

NOISTU! - PREFIX! 4» 'N' 0 “I"! + KW

LST RES! - PREFIX! + 'S' + IFNUM!

LAIOIST! - PREFIX! ¢ 'N' + IFNUN!

0M1! - PREFIX! + '0' + M!

PLTCID! - PREFIX! + W! + '.PLT‘

LOCATE ROW * 5. 40 - LEIN'DELTA II "I: '1
“PUT '0“ [1 m1: ', OLT
IF 0LT - 0 THEN
0LT - 1
LOCATE RON + 5. 40
PRO“ 0LT
END IF

00
LOCATE ROW + 0. 40 - LEN'EI“ flu [0M ”81: '1
WT 'Ellifi. u‘u [coo M81: ', TT
IF TT - 0 THEN
IT - 01X!
LOCATE ROW + O. 40
PRIIT TT
EXIT 00
ELSEIF 1T > 0LT THEN
EXIT 00
END IF
DEEP
LOCATE ROW 0 8. 1
FRUIT ST RICH”. ' '1
LOOP

LOCATE ROW 9 7. 4O - LENI'I’m m m: '1
INPUT 'Tmm m: '. SEO!
IF SEO! - "' THEN

SEO! - '1 102204205100101502510050'

__

189

LOCATE ROW 4‘ 7. 40
FRUIT SEO!
END IF

OD
LOCATE ROW + l. 40 - LENl'Pu-I in IZO'I in'inI: '1
NPLIT 'Pnd in IZO'I in'inl: ', TEA?!
IF TEIAPI - " TIIEN
TEMP! - '20'1'
LOCATE ROW + 8. 4O
PRIIT TEMP!
END IF
N - DISTRITEMPL "'1
IF N < > 0 THEN
CX - VALILEFTIFTEIAP!. N- III! 2
CY - VALIRICIITIITEIl’8. LEIATEAN’fl-NIIIZ
EXIT 00
END IF
LOOP

00
LOCATE ROW 0 9. 40 - [m m. m 125$I: '1
IIPUT 1mm Ilium “125“: '. NCL
IF NCL - 0 THEN '
NCL - 25
LOCATE 110' + 9. 40
PRIIT IACL
EXIT 00
ELSEIF ”CL < - 30 THEN
EXIT 00
em I:
BEEP
LOCATE ROW + 0. 1
PRIIT STRICWO, ' '1
LOOP

OO
LOCATE 110' + 10.40 - LEM'Nu'l-sm'n mo IOMIM: '1
[OUT W m m [0.M1h1: '. SPL
IF SPL - 0 THEN
SPL - .OWI
LOCATE RON + 10.40
PRIT SPL
EXIT 00
ELSEIF SPL < - .IIIIS THEN
EXIT 00
END IF
BEEP
LOCATE RON 4» IO, 1
PRIIT STRIGHTO. ' '1
LOOP

DO
LOCATE ROW ! 11. 40 - Will “10W”: '1
“PUT 1mm min [ONT]: '. 91L
IF a - 0 THEN
M - .M7
LOCATE ROW + II. 40
PRIT $1
EXIT 00

 

 

 

 

190

ELSEIF 91L < .01 THEN
‘ EXIT 00

END IF

DEEP

LOCATE ROW 0 11. I

PRIIT STRNGIITO. ' '1
LOOP

00
LOCATE ROW + 12. 4O - W m 0112]: '1
I'UT m m 1.11211'.SDL
IF SOL - 0 THEN
SOL - 2
LOCATE ROW 0 12. 40
PRWT SOL
EXIT 00
ELSE! SOL < 5 THEN
EXTT DO
END IF
DEEP
LOCATE ROW + 12. 1
PRllT STRWCIITO. ' '1
LOOP ‘

 

00
LOCATE ROW o 1140-me11
WPIIT'Nu'n-Imbit'.8&
IFSSL > OWSSL < IMTHBIEXITOO
BEEP
LOCATEROW+13.1
PRWTS‘I'RKHNRO
LOOP

OO
LOCATE ROW o14.40-LEH'EI~Im-IS.N.XS.NXNI:'1
WTMWISIKSOIM'. TYPE!
IFTYPE! - 'H'ORTYPE! - 'XS‘ORTYPE! - 'XI'ORTYPE! - 'S'THEN
EXIT 00
ENO IF
BEEP
LOCATE ROW 0 14. 1
PRWT STRIIGIOO. ' 'I
LOOP

00
LOCATE ROW + 15. 40 - m. [N]: '1
MT '1'. IN]: '. TKR!
IF T‘R! - " THEN
TKR! - ‘N'
LOCATE ROW + 15. 40
PROIT TNER!
EU F
1FTKR!- 'Y'ORTHER! - 'N'THEN
EXIT 00
E111 IF
DEEP
LOCATE ROW + 15. 1
PRIIT STRIGIUO. ' '1
LOOP

 

 

191

00
LOCATE ROW ¢ 16. 40 - LENI'EGI m [N]: '1
WPUT'EiImINI:', CAIT!
IF CMT! - " TIIEN
COAT! - 'N'
LOCATE ROW + 18. 40
PRWT CMT!
END IF
IFCAIT! - 'Y‘OR CIT! - 1' THEN
EXIT 00
END IF
BEEP
LOCATE ROW + 16, I
PRWT STRW6!179. ' '1
LOOP

DO
LOCATE ROW + 17, 40 - LEH'PI'II on m: '1
RIPUT 'Pr'ut on m: ', SCRN!
IF SCRN! - " THEN
SCRN! - 'Y'
LOCATE ROW + 17, 40
PRWT SCRN!
EM 5
IFSCRN! - 'Y'ORSCIH! - 'N'THEN
EXIT 00
EU I:
DEEP
LOCATE ROW 0 17. I
PRNT mun. ' '1
LOOP

DO
LOCATE ROW +10. 40 - WM 0' m: '1
NIPUT 'Diahy on m: '. OSPL!
1F DSPL! - " THEN
OSPL! - 'Y'
LOCATE ROW 6 10. 40
PRI'T DSPL!
E10 AF
IF DSPL! - 'Y' 0R OSPL! - 'N' THEN
EXIT 00
E” If
BEEP
LOCATE ROW # 10. 1
PRIIT STRWCHTO. ' '1
LOOP

DO
LOCATE ROW + 10. 40 ~ mm m: '1
RIPUT 'CcncI 1Y1: '. YIN!
IF YIN! - 'Y' 0R YIN! - '7' 0R YIN! - "THEN
IF CUT! - 'Y' THEN
SHELL IDIIOUTN
SHELL I'PE2 ' + PREFIX! 0 WFNUM! 0 'DAT'I
END IF
EXIT 00
ELSEIF YIN! - ‘N' OR YIN! - '0' THEN
GOTO SW

192

ELSE
DEEP
LOCATE ROW + 19. 30
PRWT ' '.
END IF
LOOP

END SUB
SUB NPUTSEC 1K. DLT. TT. WIS. CTNPI
ON THXII TO X1

LOCATE 17. I

PRWT STRIIC!I79. ' '1

LOCATE I7. 23

PRWT '11“ and m huts!“

LIE WPUT #1. WT!
LIIE NPUT 41. IT!
FOR 11 - I TO K
ANGLEIIII - VALILEFTWDRDH'T !11
NEXT 11

LWE "UT“. IT!
LIIE MT '1. IT!
FOR I - 1T0 X
THKIII - VALILEFTWOROOIIITOII
NEXTII
TTHK - 0
FOR 11 - I TO X
TTHK - ITHX ¢ THX1|II
NEXTII
H101 - ~TTHX12
FOR 11 - 1T0 X
H1111- H1I.I1+THXIIII
NEXT 11

LNEIPUTII.IT!

LKIIPUTII,WT!

FORR -1TOX
MRI-VAULEFTWDRDNWT!”

NEXTII

LIIEHUTIIJIT!
LIEHUTIIJIT!
FORJ -1TDX
“UNTIL”!
LWI! - LEFTWORD!1IT!1
IFVALILW1!1- J THEN
LW2! -LEFIWORD!1WT!1
1F LW2! - 'SAK' THEN
FOR 11 - 1T0 7
IFII <- OTHEN
“121.1.” - K1211 1.111
ELSEIFII >-4AINJH <-0THEN
HEWJ-OI-KW-IJI-OI
ELSEIFII - 7THEN
VIZIJI - VIZIJ- 11
ENDIF

193

NEXT II
ELSE
INT!-LW2!+"+IIT!
FOR II - ITO 7
IF II < - 3 THEN
NE 121.1. 111 - VALILEFTWDRDHIIT!”
ELSEIFII <-BANOII >-4THEN
NECNIJ. II - 31 - VALILEFTWDROHNT 111
ELSEIF II - 7 THEN
VIZIJI - VALILEFTWDRDHIIT!»
E10 IF
NEXT II
Em IF
ELSE
FOR JJ - J TO K
FOR II - ITO 7
IF I < - 3 THEN
NE12IJJ. III - “121.1 ~ I. 111
ELSEIFII >- 4AM! <- BTHEN
NECNIJJJI-(II - NEW-IJ-OI
ELSEIF II - 7THEN
VIZIJJI - VIZIJ - 11
END IF'
NEXT II
NEXTJJ
EXIT FOR
END IF
NEXTJ

 

LIE BOUT '1. WT!
LIIE WPUTII. NT!
FDRJ - 1 TO K
LIE WPUT II. 011!
LWI! - LEFTWDROHIITH
IFVALILWIII - J THEN
LW2! - LEFTWDRDHIIT!)
IF LW2! - 'SAK' THEN
FOR II - ITO 2
FOR III - ITO 4
EXPCDFIJ. II. III - EXPCOFIJ - 1.1LIII
NEXTIII
IIXTI
ELSE
WT!-LW2!#"+IIT!
FOR I - ITO 2
FOR III - ITO 4
EXPCOFIJ. IL '1 - VALILEFTWDRDHWTHII 1m

ELSE
FOR JJ - .I TO K
FOR 11 - 1T0 2
FOR 111 - ITO 4
EXPCDFIJJ. 11. 1'1 - EXPCOF1J - 1.11. 1111
NEXT III
NEXTII
NEXTJJ
EXIT FDR
END IF

194

NEXT J

LIE WPUT 31. WT!
CALL NOOELCOFIX. FLASH). PARUCIII
LNE NPUT #1. WT!
CALL MODELCOHX. FLAC!II. PAMAIII
LIIE DIPUT II. IT!
CALL MODELCOFIX. FLASHI. PAMDIH

LIIE NPUT II. HIT!
LINE IPUT II. WT!
LIIE WPUT '1. IT!
FOR 111 - I TO 0
EXIO. 1111 - VALILEFTWDRDHNT 111
NEXT III

LIE MT '1. WT!
LNE WPUT [1. NT!
LIIE WPUTII. NT!
NS - 0
00
M MT '1, WT!
COPY! - NT!
IF LEFTWDRO!ICOPY!I - 'EIII' THEN EXIT 00
CH8 - CHIS o 1
MCHCHSI - WT!
LOOP

DO
LOCATE 17. 1
PRIIT STRIGHTO. ' '1
LOCATE 17. 40 ~ LENI'PI'II'QIIIINI: '1
NPUT 'PflIL input: 1le ', W!
IF I! - " THEN
W! - 11'
LOCATE 17. 40
PRWT II!
Ell) 1F
IFW! - 'Y'ORN! - 'N'THEN
IF II! - 'N' THEN
INN - I
ELSE
NW - 2
END IF
LOCATE 17. I
PRWT STRIIC!179, ' '1
EXIT 00
END IF
LOOP
FORI - 1 T0 N1."
IF WM - 2 THEN
OPEN 'LPTI:' FOR OUTPUT AS '2
ELSEIF NIH - 1 THEN
OPEN OHIOUT! 4» PREFIX! + IIFNUN! ¢ 1" FOR OUTPUT ASH
END IF
LOCATE I7, 1
PRIIT '2. TADISI; 'MIT FILE'; TADIIOI. NILE!
PRIIT #2. TADISI. 'IIPUT DIR'; TADIIOI; DfillP!
PRIIT #2. TABISI. 'OUTPUT DIR';TA01101; DIROUT!

 

 

 

195

PRIIT '2. TAOISI; 'FlEPREFIX';TA01101: PREFIX!
PRIIT t2.TA0151:'TYPE';TA01101; TYPE!
PRNT l2. STRICHN. ' '1
PRI'T I2. TARISI: 'X'; TADI 101: X
PRII'T I2. TA0151; 'DLT';TA01101: DLT:TA01301: 'MI'
PRIIT l2. TAOISI; 'ENDIIG'; TAOI 101; IT; TAMI; 'MI'
PRWT l2. TABISI: 'ICRENENT SEOUENCE'
PRWT I2. TAOIIOI; SEO!
PRW'T l2. TADISI; 'PAKL fiZE'; TADI 101; CX ' 2: 'Uv': CY ' 2: TAIISOI; 'I'I'iI'
PRIT I2. TAOISI; 'AKLE (“'1'
FOR II - I TO K

PRIIT I2. TAOIIO 9 0 ' III -111:USI!B 'mu'; AISLEIRI:
NEXT 11
PRIT '2.

PRWTIZTAIGI: 'THXIiII':
FORII -1TOX
PRITA'LTAOIIB o 0 ' 111-111$“ 'IJW:THXIIII:
NEXTII
mun.

PRIIT '2. TA0151; 'HI'II':

FORII-OTOX
mnrmuumxumvmnm

KXTR

PRWTIZ.

PRITIZTAIEE'WIINI':

FORI- ITOK
PRITI2.TA011|¢0'II~I11:USM'MM';WI:

KXTI

mun.

PRWTIZ

PRWT I2. TAOISI; WE 11in Ifl'
PRWT I2. TAOIIOI; 'LAVER'; TA01101: 'E1 1'; TA01301; 'EZZ'; TA01421: 'EI2': "0641; 'V12'
PRWT 4‘2. TA0151: '12N':
FOR J - 1 TO K

PRWT I2. TADIIOI; .1:

FOR 11 - 1T0 3

PRIT l2. TADIIO + 12 ' II -111: usus 'IW“"; ‘121-1. 111:

KXT I

PRIT '2. mm USIC 'IJW; V1201
KXTJ

PRIT I2. TA0151; 'SH';
FORJ - 1T0 K
PRNT l2. TA01101; J;
FOR I - ITO 3
PRWT ’2. TAIIIB o 12 ' 1| - 111: US“ 'IW“": KW. l1:
IKXT I
PRWT '2. TAIISAI; US“ 'umr; VIZIJI
IEXTJ
PRWT '2.

PRWT 32. TADISI: 'EXPANSION COEFFICIENT PARAKTERS'
PRWT l2. TAOI 101: 'AAI'; TA01301: 'AAZ':TA01421; 'AA3';TA01541: 'AA4'
PRWT 4‘2. TAOISI: 'EXP';
FOR J - I TO K
PRWT I2. TADI 101; J:

 

196

FOR 11 - I TO 2
IF I: - 1mm : mu :2 nun» 'II';
IF I: - 2 mm : mu :2. man; 'I_ ';
ran III - 1 TO 4

mu :2. mm + 12 ° «I- III: usus 'unm'; EXPCOFIJ. 11. "O:
um nu
mu 22.
um 11
um J
mu :2.

CALL PRNTPAMX. 'C'. PAMOI
CALL PRN'TPARWX. 'A'. PARNAOI
CALL PRNTPARHX. '0'. PAWI

PRWT l2. TADISI: 'HTAL TOTAL STRAIS'
PRWT I2. TAOIIOI: 'El';TA01101: 'Ey': “01201: 111': "01341: 'XI'; TAII421: 'Xy'; "01501: 'Kly'
FOR I - 1 TO 0
PRIITIZ. TAOIIO + 111- 11' 01; EX10. l1:
IEXTII
PRIITIZ

PRWT P2. TADISI; W'z TA01201: 'HOUR';TAIm1: 'LAYER':

FORI -1TOX
mnrm1 +1I-11'71:I:

KXTII

PRWTH.

FORI- nouns
mnrmmucsm

IEXTR

murmmr;

FORI- ITOCT'
mnrmmmsm

IEXTI

CLOSE
IEXTI

FIINCTION LEFT CHAR! 1W". N1
00 WHNE LEFT!IWT!. 11 - ' '

WT! - RICHT!1IIT!. MN- 11
LOOP ’

TEN! - LEFT!1WT!. N1
LT! - TEIH!

EU FUNCTIM

Fm LEFTWORO! INT !1

'00 “IE LEFT!1WT!. 11 - ' '

' WT! - RICHT!IWT!. LENIT !1 - 11
'LOOP

NIT! - LTRH!1IIT!1

IF INSTRIIITI. ' '1 > 1 THEN

 

 

197

TEMP! - LEFT!INT!. ”THIN". ' '1- 11
NT! - RICHT!IRIT!. LENWT !1 - LENITEIAPIII
LEFTWORO! - TEMP!
00 WHILE LEFT!1|IT!. 11- ' '
NT! - RIGHT!1NT!. W” - 11
LOOP
ELSEIF WT! < > " THEN
LEFTWORO! - NT!
NIT! - "’
ELSE
LEFTWORO! - IT!
END IF

END FUNCTION
SUB WWI I“. P. X. NU

ON ABOOINII TO 0.1T0 01. EKPWTII TO 0. ITO 11

DNA YWTII TO 3. ITO 31. EPII TO 3.11011.XP11T0 3.1T011
DH EXPPII TO 3.1T011

OHYEII T03. 1 TO ILYXII T03. 1 T0 ILYHII T03. ITO 11
OH TEWII TO 31.8161211T0 31. WARPII T0 21

IF P - 0 THEN

LOCATE 17, I

PRWT STRIGHN. ' '1
END IF

CALL NATRMSIAOOOO. AOOOWVOI
CALL MAMLTIAOOOIYO. 6M0. EXPWTIII

FOR II - ITOO
EXPPIP. 111 - EXPW'I’IIL 11
NEXT 11

OPEN DIROUT! 0 STRM FOR APPEM ASII 'cm
PRIIT I1. TADIII: US“ 'IIIIII'; am ' 0LT:
FOR II - I TO 0 -
PRITII. TAOIO +1I-11'121; US“ 'IJIII'“".‘ EKPI'TII. 11' 1M:
NEXT I
PRIT II.
CLOSE I1
FOR 11 - I TO 0 'EACH LAYER
OPEN DIROUT! o LSTRANPHII FOR APPEIII AS I1
PRIT II. TAOIII: USN 'IIIIII'; am ' 0LT:
PRWT II. TAOIOI: USIO 'IIIII'“": EXPWTIII. 11' 1M
CLOSE II
NEXT II

IF P - A! THEN
NCT - 0LT 'STRAW RATES
ICTSLN - 111% + 11 'STRAII
OLTT - IRSUHP1 + 11' OLT ‘WARPACE
ELSE
DICT - RIP 6 11' DLT 'STRAI RATES
RICTSUN - RSIMP # 11 'STRAI
OLTT - RSIMP o 11 ' 0LT ‘WARPACE
END IF

FORII-1106

 

l 98
mp . 1,111 - EKIP, m + exp-mu. 11° nor
um 1:

OPEN DIROUT! 0 ST RAN! FOR APPENO AS I1 ‘COHIED
PRNT I1. TADI 11; USMC 'IIIIII'.‘ NCTSIM ' 0LT;
FOR 11 - I TO 8
PROIT I1. TA019 0111-11' 121; USND 'IIIII'""; EXIP o I. 111' IIXXI:
NEXT 11
PRIIT I1.
CLOSE II
FOR II . 1 T0 8 'EACH LAYER
OPEN DIROUT! + LSTRAWHIII FOR APPEM AS I1
PRWT I1. TAOIII: USMC 'IIII.II'; IICTSUN ' DLT;
PRIIT I1. TA0191;USRIO 'IIIII‘""; EXIP + I. 111' IM
CLOSE I1
NEXT 11

OPEN DIROUT! + STRESS! FOR APPENO AS I1 'AOOREOATEO
PRIIT II.TA0111;USOIC 'IIIIII'.‘ RSUHPI ' 0LT;
FOR J - I TO K
PRIIT II.TA0191: USlIO 'III'; J:
OIAC - MCIP + 1.JI~IACIP.J1111R1P + 11 ' DLT1
FOR II - 1T0 3
FOR 111 - I TO 3
mm. 1111 - YTRIJ. 11.1111
IEXT 1H
NEXTII
IFP > OTHEN 'EPII. XPIIAIDEXPPOARESTRAIRATES.
FOR 11 - 1 TO 3
EPIII. 11 - EXPI'TIII. 11
XPIR. 11- EKPIITIII + 3.11
EXPP1111- EXPVALIP. J. 11. 11' M
NEXT 11
ELSEE P - 0 THEN 'THE ABOVE DUANTITIES ARE STRAIS AT P-O
FOR 11 - I TO 3
EPIR. 11 - EX10.111
XP111.11-EX10.II¢ 31
EXPPIII. 11 - 0
NEXT 11
END IF
CALL NATRNULTIYIIT 11. EPII. YE111
CALL MATRNULTIYNT 11. XPII. YK111
CALL MATRMULTIYIITII. EXPPII. YH111
IF P - 0 THEN
RPLC - I 'STRESSES AT P-O
ELSE
RPLC -.5'R1P1'DLT
END IF
FOR 11 - 1 TO 3
YEJIJ.11.11- YEJIJ.II.11¢ RPLC' YE111. 11
YXJIJ. II. 11 - YXJIJ. 11. 11 * RPLC ' YXIII.11
YHJIJJLII-YHJIJJLII" RPLC'YHIlII
NEXT 11
FOR 11 - I TO 3 'AT THE CENTER OF EACH LAYER
TEWIIII- IYEJIJ.11.II+ YXJ1J.II. 11' .5'1H1J1+ HIJ-I11-YHJ1J.I.111' IIXXI
PRIIT I1. TAOIIB + 111- 11' 121: USRIC 'I.IIII““";TEW1111:
NEXT 11
CALL STRAFORIAITEMPII. ANGLEIJI. 81612111
FOR 11 . I TO 3
ST RESSIP. J. 111 - 816121111

 

199

NEXT 11
PRIIT II.
NEXT J
CLOSE I1

FOR J - I TO K 'AT THE SURFACE OF EACH LAYER
OPEN DIROUT! 2 LSTRES! + LTRH!ISTR!1J11 + '.PRN' FOR APPEM ASII
PRIIT II. USIIC 'IIIIII';RSUHP1' 0LT:
HH - HIJ - 11 * IHIJI - III-I ~ 11112 "' FOR STRESS AT THE “FONT OF EACH LAYER
FOR 11 - ITO 3
TIAP - 1YEJIJ.II. 11* YXJIJ.II.11' HH-YHJIJ.11.111' 1M
PRWT II.TA010 * III -11' IIIzUSNS 'IIIII““'; TWI 1M:
NEXT 11
FOR 11 - ITO 3
TNP - IYEJIJ.II.11* YKJIJ. IL II'HIJI-YHJIJJL 111' IM
PRWT I1. TADIII + II + 3 - 11 ' 111; US“ 'IIIII““'; 1UP! 1M:
NEXTII
PRIITII.
CLOSE I1
NEXTJ
FOR I - I TO 0 'TOP AM COTTON SURFACES
SELECT CASE I
CASE 1
SYNC! - 'TI'
CASE2
SM! - 'T2'
CASE 3
SM! - 'T3'
CASE 4
SYU! - '01'
CASE 5
SM! - '02'
CASES
SYU! - '03'
E111 SELECT
OPEN DHIOUT! 0 LSTRES! 0 SM! + '.PRN' FOR APPEU ASII
PRWT II. USNG 'IIIIII': RSIMPI ' DLT.‘
IFR < 4THEN 'TOPLAYER
M - 1YEJ11.II.11+ YXJII. l 11' H101~YHJ11. [111' IM
PRWT I1,TA0101; USMC 'IIII “'1'”! 1m
ELSEFI > 3 THEN 'OOTTOM LAYER
M- 1YEJ1X.II-3. 110 YXJIKI-3.II'HIK1-YHJIX.I-3.111'Im
PRNT I1, TAOIOI; U805 'IIIII““': MI I“
m f
CLOSE I1
NEXTII

FOR I - ITO 2
OPEN DIROUT! . OW! 4» LTRH!ISTR!1|I11 + '.PRN' FOR APPEM AS I1
PRWT I1. TAOI 11: USWC 'IIIII'.‘ OLTT:
1F H - 1 THEN
X - 14.1: Y - 0
ELSE I - 2
X-&Y-lfi
an IF
WARPIIII-I-.SI'1EXIP¢1.4I'X'X oEXIP¢1.51'Y'Y+EXIP 01.81'X'Y1
PRII'T II. ”0191:0918 'IIIII‘"";WARP1HI:
PRWT I1.
CLOSE II
NEXT 11

 

200

IFSCRN! - 'Y'ORISCRN! - 'N'ANOP - NITHEN
PRIIT ' SP'; TAOICI; USHS 'IIIII';RSUHP1 ' 0LT;
FOR 11 - I TO 0
PRIT TAOIIS +111-11' IIIUSUG 'IIII‘“"; EXPRITIII. 11:
NEXT 11
PRNT

PRWT ' 91'; TADIOI: USUG 'IIIII'.‘ NCTSUI ' 0LT;
FOR 11 - I TO 0
PRIIT TADIIS + 111- 11' III. USIC 'IIII‘““': EXIP + 1.111:
NEXT 11
PRIIT

PRIIT'SS'.‘ TAOIOII USIG 'IIIII';RSIHPI ' OLT:

FORII-1T03
TIP-IYEJIIJI.11+YXJII.II.11'1'1101-YHJII.R.111'1M
PRWT TADIIS +111 - II ' III; USUG 'IIII’““'; I";

NEXTII

FORII- ITO3
TWGNEJIKIIHYKJIKIIJI'HIXLYHJIKIIII'1M
PRIITTADIIS +111 + 3- II ' 111; US“ 'IIII‘“"; T":

IEXTII

PRIIT

 

PRWT'OS':TADIOI;USM'III.II';OLTT:
FOR 11 - ITO 2
PRWT TAOIIS + III-11' IIkUSUC 'IIII'"";WARP1111:
NEXTII
PRWT
PRNT STRIB!17B. ' '1
EIIIIF

IF DSPL! - 'Y' THEN
FOR 11 - ITO 3
CALL SCR'LOTII. 14. m Rm ml ‘ IO. NCIP. H1 ' IO. PIXIII. PIYIIII
NEXT 11

FOR 11 . 1 TO 4 STEP 3
IF II - 1 THEN
RPLSPL - SPI.
ELSEIF I - 4THEN
RPLSPL - SPL ' 0
EM 1F
CALL SCR'LOTII. 15. MI. swam RPLSPL EXPIITII. II. PPXIIII. PP'Y11111
NEXT!

FOR 11 - ITO 4 STEP 3
IE 11 - 1 THEN
RPLSI. - 9L
ELSEIF II - 4THEN
RPLGIL - 94L ' 10
END IF
CALL SCRNPLOTII. 7. am RSUHPI. RPLSH. EXIP + 1. H1. PNXIIII. PNYIIIII
NEXT 11

FORII- ITOZ
IFII < 4THEN

201

T0 - 0
ELSEIF II > 3 THEN
T0 - X
END IF
T'MP - 1YEJ11.II. 11¢ YXJ11.II. 11' "(101-W11. 11.111' 1“”
CALL SCRNPLOTII, 2. RSUMIMI. RSUMIPI. 331.. M. PSXIIII. PSYIIII
NEXT 11

FOR 11 - 1 TD 1
CALL SCRNPLOTII. I4. RSUINMI. RSIMPI. SOL WARPIIII. PDXIIII. PDYIIIII
NEXT 11
END IF

END SUD

SOB IAATRWVS IAII. A1111
'Mxmmwmmm
'Imm' [Alhmmlfllbm

’IimlAl-LN'LN minIBI-LN'ZLN
'finmmmulnmbhdlfluhm

LN - UO0UNDIA. 11
ON 011 TO LN. I TO 2 ' LN1

FOR IN - ITO LN
FOR JN - 1 TO LN
011N. JN +LN1- O
01IN.JN1- AON.JN1
IIXTJN
DON. IN 0 LN1- I
KXTIN

'mmmmumumu
'silIaIIOIIIINUN-mm. mmdlAIll
'Mhuhm.

FOR KXN - I TO LN
IF XXN - LN THEN SOTO 42424
MN - XXN

Faith-mill“
FORIN - XXN + ITDLN

IF A081011N. KXNII > ABSIIM. XXNIITHENIAN - IN
NEXTIN

IF MN - KXN THEN SOTO 42424
FORJN - KXNTOZ' LN
0 - 01XXN. JN1
DIXXN. JN1 - CNN. JN1
OIMN. JN1 - 0
NEXT JN

' ii“ "I X
42424: FOR JN - XXN o I TO 2 ' LN
OIXXN. JN1 - 01XKN. JN1 I OIXXN. KXNI
NEXT JN

1F XXN - 1THEN COTO 42434
FORIN- ITOXXN-I
FORJN-XXN+ ITOZ'LN

202

011N.JN1 - 011N. JN1- 011N. XXNI ' 01XKN.JN1
NEXT JN
NEXT IN

IF XKN - LN THEN OOTO 42430
42434: FOR IN - XXN ¢ 1 TO LN
FORJN - KXN + ITOZ'LN
011N.JN1 - 011N. JN1. 011N. XXNI ' OIXXN. JN1
NEXT JN
KXT IN
42430: NEXT XXN

'mmmmmiams]
FORIN - I TOLN
FORJN - ITO LN
AX1N.JN1- 011N. JN + LN1
NEXTJN
NEXTIN

END SUD
SUO MATRWLT IAIII. 0111. A0101

'A0-A'0 A is LN'MN 0 is MN'NN A0 is LN'NN
'CW THE PRODUCT MATRIX
'MATRIX A011 DOES NOT KEO TO 0E NTIALIZED.

LN - UBOINRIIA. 11
MN - mm 21
NN - UOOLWOIO. 21

ON C11 TO IN. I TO NNI

FORI-ITOLN
FORJ-ITONN
FORK-ITOMN -
CILJI-CILJHAILXI'DIKJI
A01LJI-CILJ1

END SUO
SUO MATRPRNI 1X11. 31
L - UOOUNDIX. 11

IF 8 - 0 THEN
FORI - I TOL
PRITTADIII; 08040 'IIII““'; X111
NEXT 1
PRIIT
ELSE
CLOSE
OPEN 'LPTIz' FOR OUTPUT AS I1
FORI - I TOL
PRIIT II, TA0111: USMC 'I.III“'"; X111
NEXT I

203

P110“ II.
PRNT II.
CLOSE #1
END IF

mm
mo sue
sun mm (x11 81

L - UBOUNDIX. 11
A4 - UODW 21

IFS-OTNEN
FOOI-ITDL
FORJ- ITDM
PRITTADII + IJ-II' IDLUSM'IM‘“"; X11. J1:
NEXTJ

CLOSE
OPEN ’LPTIz‘ FOR OUTPUT AS 41
F001 - ITOL
F011 J - I TO I
HUT I1. TAIII + U - 11' 101: m 'IM‘“"; X11. J1.“
KXTJ
POM”.
NEXTI
PIINT #1.
PORT II.
CLOSE '1
END IF

END SUD
SUD MATRTIIAN 1X0. XT111

D - UODWD1X.11
IEUOOUNOIX. 21 < > OOIIUO0M1XT. 11 < > DDRUO0M1XT,21< > OTNENCALLENNLDCTH'I IMATRTIIAN'I

FORI - I TO D

FOII J - I'TO D

XT 11. J1 - XIJ. 11

IEXT J
um I
an SUD
SUD NDDELCDE 1K. XHACIO. PAMI
NUAI - UODUUIPAIN. 31

LNE INPUT #1, NT.
LNE HPUT [1.01“

FORJ - ITOK
LNENPUTIIJJTS

 

204
mu - LEFTWOROIMII
IF VALILVIIII - J THEN
um - LEFTWOROIIIITII
IF um - 'SAIIE'THEII
FORI . I TO 3
IFIIUII > OTHEII
XFLASIIJ. II - xrusm . 1. n
END IF
FOR I - ITO um
PAMJ. I. III - PAW-1,1,")
IIEXTII
NEXTI
ELSE
FOIII - ITO 3
m: - LEFTWOROIIIITII
Irma - 'E' THEN
IFIIOII > OTHEII ‘OIIYUIIIEAONPAIWETEIIC
XFLABIIJ. II - um
an IF
ELSEIF W38 - ‘II‘TIIEI
IFIIIIIA > OTNEII 'OIAYOIIIIEAOMPARAKTEIIC
XFLASIIJ. II - [W38
END IF
ELSE
IFIIIIA > STIIEII 'OIYONIIEAON PM“
XFLASIIJ. II - 'V'
an IF
mz-Lwasuulm
FOR I - ITO MIN
PAW. I. III - VAULEFTWOIIOIIITIII
IKXTI
EIII IF
IFI < OWEN
LIE MIT“. IT:
['28 - LEFTWOROSIIITII
EIII f
IEXTI
END IF
ELSE
FOII JJ - J TO K
FOIII - ITO 3
IFII” > ITIIEII
XFLABIIJJ. II - XFLASOU - I. II
EIII IF
FOR I - ITO“
PAMJJJIII- PAW-1.1m
IIXTII
IfXTI
IIEXTJJ
EXIT FOR
EIIO IF
«an

an SUD
SUB MOISTURE 1X. DLT. I. PTSI
OH MCCDFII TD X.1TD ("S 11. 1 TO 31

O" VII TD PTSI. W11 TD X. 51
ON RCOFII TO 1PTS- 11 ' 3.11

 

 

205

DNA TIAEII TO PTSI
004011 TO PTS. 1 T031

LOCATE I7, I

PHIIT 81110068179, ' 'I

LOCATE 17. 28

PRO" '9...“ m M?

FOR I - 1 TD PTS
TNEIII - VALILEFWDHDIMCOIIUI
FOR 11 - I TO 3
011. 111 - VAULEFTWOHDOMCHHII
NEXT 11
NEXTI

FORJ-ITOK
OEPTH - Assmu-n-mo» . mama-1m:
FORII- ITOPTS
mn-mmmzvosm .mavosrm'oem
um"
cm manna m neon»
FORI- ITOIPTS-II
mau- no:
uccom.u1- mono-"'3 + II. II
NEXTII
um:
NEXTJ

FDHJ -1TOK
FORI-OTDNOI
IEI-M+1THEN
DLTI -IHSl~I-11+11°OLT
ELSE
DLTI - M'DLT
EIIJIF
DLT2 - DLTI ' DLTI
IF DLTI < - MWSITHEN
C-I
DO
IFDLTI <- MIC + IITHEN
WILJI - HCCDFIJ. C. 11' DLT2 9 ”OF“. C. 21' DLTI o uccom. C. 31
EXITDD

ELSE

FOR J - ITO X
If TYPES - 'XU' OH TYPES - 'IA' THEN
FOR 11 - ITO I
AVGIAC - “0111..” 4' HCIII- I.J1112
DHSUIA - IRSUNIII - RSIMII - 111’ OLT
AVBIACMII. J1 - AVGMCWIIL 1. J1 ' RSUHII ~11' OLT 0 AVCHC ' DHSUI
AVGMCMH. J1 - AVCIACWIII. J11 (HSUWIII ' DLTI

In

 

206

um In
sum IF
OPEN 0mm: . moms . Lmuusmu» + 'mr FOII am as n
moss :1
OPEN omoun . moms + Lmuumm» ‘ 'mr FOII am As n
mu :1. LMOISTI . LTRIIIISTRIIJII + 'mr
mu :1, muswnt comm mr
mu :1, «our; TABIIOI; 'uvtn' . 3mm
.FOIIII- own .1
IE 11 - M + 1 THEN
Tm - nsumu . n + 1
ELSE
mi - nsuum
an IF
mu :1. usus 'mu'; ms ' on;
mu :1, mm»; usus 'mur; 00111. .1)
mm u
moss I:
um .1

END SUB
SUB PARAMTHI 1M1). Y1). CEHII

O - UOOUNOIM. 11
IF O < > UOOUNDIY. "THEN CALL EHRLOCTN'I I PARAMTIII'I
If D < > UOOMICEF, II THEN CALL EHHLOCm I PAHAMTIII'I

OHMCTII TOO. ITO D1
OIMCTIII TO D. I TO DI

FOHL- ITOO
FOHM- ITOO
MCTILMI-WU‘IWMI
NEXTM
NEXTL

CALL MATRIVSIMCTII. MCT101
CALL MATHMULTIMCT 111. VI). CEF111

END SUI
SUD PARAMTR2 M1. 111. S11. CEFOI

O - UOOLMIM. 11
IF D < > UOOUNOIY. 11 THEN CALL EHHLOCTM'I I PAHAMTH2'1

OH MCTII TO D. I TO DI
ON MCTIII TD 0. 1 TD DI
ONSTSIITOD. ITO D1
ONSTSHI TOO.1TODI
DH STSIT1I TO O, ITO D1
OH XVII TO O. I TO 01

FOOL - ITOO
FORM - ITOO
MCTIL. M1 - MIL) ‘ ID - M1
STSILMI- SILI‘ID-MI
NEXTM
NEXTL

 

 

CALL WRITECMDIZ. O. F101111

CALL WRITECMOIII, X + 0. F1081”
CALL \VRITECMOIII. X o 2. H0801
CALL WRITECMOIO. X + 5. F1080)

207

'I WARP CURVE
'3 NORMAL STRAII CURVES
'3 NORMAL ST RAN RATE
'3 CURVATURE STRAII RATE

CALL WRITECMDIO. X + 11. H0101 '3 CURVATURE ST RAIL

CALL WRITECM010. 2 ' X ¢ I4. F10I111

'3 TOP SURFACE STRESS CURVES
'3 OOTTOM SURFACE STRESS CURVES

SPF! - SUODIRO ¢ PREFIX! 0 mm + '.SPF'

PRNT II.

PRNT #1
PRIIT II

PRIIT #1.
PRNT #1.

P1101101.
PRIIT #1.
PRNT #1.
PRDIT 41.
PRIJT AI.
PRITII.

'M‘ 'COHENT
. 'RESET 1.7'

. SPF!

'W'

'E'

'C'
'SAVE'
SPFS
'PLDTIT.I,' 'CREATE °.PIC

81100011 + PREFIX1 0 IFNUMO + '.P1C'
SPF.

'EOP
'SAVE '.SPF

 

mu n. mar
mu :1. sm
mu :1. '1'
ems: n

EWSUO
SUO PMTPAIUIX. P1. PM

PRIT ’2. PI + ' FARMERS?

PRITIZ. 'LAVER OIR‘:

FOR 11 - I TO UOOLHDIPARM. 31
PRITRTAOIM + 111-11°5th

NEXTII

PRITIZ.

FOI'IJ - ITOK
PRIITnJ:
FORI-ITO3
010- II 'I2FI-3THENOM- 12
PRIITIZJAOIOLOII:
FOR! - ITO UOOIMOIPARM. 31 STEPZ
manna" + III- "'51; US“ 'IN‘““;PAMJ.|.111:
NEXTII
PRITIZ.
FORI - ZTO UBOUIIIIPARM. 31 STEPZ
PRITHJAOIIO + (1121’ 51:09.0 'lll““‘; PAMIMJII;
NEXTII
mun
NEXTI
NEXTJ

ENOSUO

SUO OTRAFORM 1011. ANGLE. OXYIII

FOIM - UO0UN010. 11
SD“ - UOOUNDID. 21

 

208

IF F0" < > UOOUNDIDXY. 11 THEN CALL ERRLOCTNI'I I TRANSFORM'I
IF 800A < > UROUNOIOXV, 2) THEN CALL ERRLOCTIH'2 I TRANSFORM'I

THETA - ANOLEI 100 ' 3141502054!
C-COSITHETAI

S-SIITHETAI

CS - C'S

C2 - C'C

$2 - S'S

 

lFSDIA-3THEN
C3-C2'C A
C4-C3‘C
S3-S2'S
S4-S3'S
C2S2-C2'S2
TC2$2-2°C2S2
FC282-4'C282
C3S-C3'S
CS3-C'S3

OXYII.II-$1.11'C4+1l2.21'84+U1.21'TC282+1X3.31°FC282

OXY12.21-1l1, 11'84011211'04’1X1.21’TC282+1X3.31°F0232
DXVII,21-1X1,II'C2S2+UZZI'C282+fl1.21'IC40841+1X3.31'-FC282
OXY13.31-1l1. II'C2S2+U221'C2820G1.21°-2°C282+G3.31'1C2-SZI'1C2-S21
0XYII.31-III.II'C38¢1l2.21'-C9*111.21'ICS3-C331+1I3.31'2'1C9-C381
OXVI2.31-UI.II'CSD0“ZIP-C380111.21'IC38-CS31+U3.31’2'1C38-C$1
DXY12.II-OXVII.21

0XYI3.II-OXYII,3I

OXY13. 21 - 0XY12. 31

ELSE? SDI - I THEN
OITII TD 3. I TO 31. TII TO 3. 1 T031

T11. "-02
T12.21-C2
TII.21-SZ
T12.11-S2
T13.11-CS
"121-CS
T11.31- Z’CS
T12. 31- '2' CS
T13. 31- C2-S2

CALL MATRIVSIT 11. T1111
CALL MATRMULT1'1'111. 111. 0XV111
0XY13.I1- 0XVI3.11' 2
ELSE
CALL ERRLOCTN1°3 I TMIIIIII'I
EM) IF

E10 $10
FUNCTION RELXMDDU ITVPEI. MS. RMV. XS. RXV. T1

'IF RMV - 0 THEN FREE OF MAXWELL OASHPOT
‘IF RKV - 0 THEN FREE OF KELVI ELEKNT

SELECT CASE TYPE!
CASE 'XM'

 

209

UX-XS'RXV
UM-MS'RMV
OI-UKOMS'RKVH.“
CI-UX'UM
PII -.5'101+101“2-4'CII‘.51
PI2-.5'101-101"2-4'C11‘.51
IFPII -P12AMPII -OTHEN
RELXMODU-I
ELSEIFPII -P12THEN
RELXMDDU - EXPIPII 'TI
ELSE
A-IPII-UXIIIPII-PIZI
0-1UX-PI2111PII-P121
RELXMODU-A'EXH-HI'U¢0'EXHP12'T1
ENOIF
CASE'M'
UM-MS'RMV
RELXMDDU - EXPI-UM'TI
CASE'XS'
A-XSIMSOXSI
0-MSIIMS+X81
C-IMS+XSI'RXV°T
RELXMDDU - A + 0'EXP1-C1
CASEELSE
CALL ERRLOCTH'I IRELXIDDU'I
EMSELECT

ENOFLICTION
$IOSCRNPLOTIIM1M.A.0.C.0.E.F1

CSX-SMIA'O
CSY- IMIC'O
SELECTCASEII
CASEI
STYLE-OHFFFF
CASE2
STYLE-OICCCC
CASE3
STYLE-81110000
ENDSELECT
LIEIEIHCSXISVLMJTVLE
E-CSX
F-CSV

an $10
$.10 SPLICUO 1X11. Y0. COR»

0 - UOOUNDIX. 11

IF 0 < > UO0UN01V. 110R 10131 ' 4 < > UODLNICOF. 110R UROMICOF, 21 < > 1 THEN
_ CALL ERRLOCTNI'I I SPLICUO'I

END IF

OIAXXII T010131'4.IT010131’41

0HXXIIIT010131°4. IT010131'41

OHWII TOIDI3I'4. I TO 11

FORI-ITOIDI31
XXII 04'II-II.I+4'11-111-X111'X111'X111
XXII04'11-11204'11-111-X111'X111

 

 

 

XMI+4°U~
XMI+4'"~
XX12+4°11-
xn2.4'u.
XX1294'11-
XX12+4'11-
XM3¢4°U-
Xfl394'fl-
XM3F4'0-
Xfl3+4°fl~
Xfl4o4'fl-
XM4+4'U-
Xfl4o4°fl-

113+4'fl-
114+4'U-
III+4°0-
112+4’0-
113+4‘fl-
II4+4'fl-
ILI+4'0-
11294'0-
113*4'0-
IL4+4°fl~
111.4'0-
ILZF4°0-
113+4'fl-

210

111-X111

111-I
111-XIIFII'X11¢II'XII+11
111-X11+II'XII+11
III-X11011

III-1
111-X11021°X11*21'XII+21
111-X11+21'X1|+21
111-XII*21

111-I
111-3'XII¢21'X114>21
111-2'XII+21

111-I

 

IFI<0-ITHEN
XXI4+4'1|-11.5+3'11-11)--3°X11*21°X11*21
XX14+4'11-11.0+3'11-111--2'X11*21
XX14+4'II-11.793'11-111--I

ENOIF

YYII +4'11-1111-V11 011-111

YYI2¢4°11-11.11-VI2*11-111

WI3+4'11-11.11-YI3*11-111

W14+4'II.11.11-0

NEXTI ‘

CALL MATRIVSIXXII. XXIOI
CALL MATRMULTIXXXI. W11. COF111

EM) $10
SUO SPLIOUD 1X11. Y1). COHII

0 - U001I01X. 11

IFO < > 000100". IIORID- 11' 3 < > UOOLNICDF.IIDRU001NICOF,21< > 1 THEN
CALL ERRLOCTN'I I mm

ENOIF

DIXXII T010-11'3.IT010-11’31

DIIXXIIITDID-11'3.IT010-1)'31

ONYVIITOID-II'IIJTOII '

FORI-ITDD-I

XXII #3’11-
XXII +3'II—
XX11+3'1|-
XX12+3'1I-
XX12‘3'11-
XX1293’II-
XXI3+3°11-
XX13+3'II-

111+3'fl-
I12‘3'fl-
113+3'fl-
111*3'0-
112+3'0-
11303'0-
ILI+3'fl-
11203'0-

IFI<O-ITIIEN

111- X111° X111

111- X111

111- I

111-XI“ 11'X1|+ I)
111- X11 +11

111- I

III-2'X114' 11

111- 1

XXI3*3“11-11.4+3'11-111--2°X1I¢11
XX13+3'11-II.5¢3'11'111--I
ENOF
W11 ‘3'11-11.11-YII +11-III
W12+3'1|-11.11-Y12*11-111
YVI3¢3°II-II,II-O 'ISTDIFFERENTIATIONATLASTPOITISZERO.
NEXTI

CALL MATRIVSIXXII. XXX11
CALL MATRMULTIXXIII. W0. COHII

211

END SUO
$10 STFIMSI 11. P. K. DLTI

ORA RRII TO 31

00A VII TO 3.1TO 31. VITII T0 3. ITO 31

ORA EXPITII TO 3.1T01LVEXPIITO 3.1T011

00A EPII TO 3. I T011. XPII TO 3.1T011

00A VEII TO 3. ITO II. VXII T0 3. 1 TO 11. YHII T0 3.1TOII
DI A00OEXPII TD 0. I TD 11. EXPII TO 0. 1 T011

 

IFP > OAN01<-PTHEN
FOR 11 - 1 TD 0
EXP111. II - EKPP11- 1.111 'PREVIDUS STRAI RATES
NEXT 11 , _
FORII - ITO3
EPIII.II-EKP111.11
XPIII.11- EXPIII 0 3.11
NEXTII
EWIF

IFP-DAMI-OTHEN
RPLT-O
RPLEXP-D
Rm-I

ELSE
RPLT-IRm-RSUHI-III'DLT
RPLEXP-I-I
RPLW-I

EWIF

FOR J - I TO 11
FOR II - I TO 3
IF RPlT < > OTIIEN
IF TYPE: < > '8‘ THEN VISCOELASTIC
IF FLAGIIJ. III - 'E' TIIEN
RRIIII - I
ELSEIF FLACIIJ. III - 'N' THEN
RRIIII - I 'SIIEAR RELAXATION MOOOLIIS YET TO OE OEFIEO.
ELSEIF FLAOIIJ. III - 'V' T1181
RRIRI - RELXMOOUITYI’EI. mm. J. R1. RMVIL J. II. XSIP. J. III. RRVIP. J. l1. RPLTI
Ell) 1F
ELSE 'COIDLETE ELASTIC
RRIIII - I
END IF
ELSEIF RPlT - 0 THEN 'AT T-O RELAXATION IAOOIJLUS OESEIERATES TO NOE.
RRIIII - I
END IF
NEXT 11
V21 - IRRIZI ' MSIACII. J. ZIIIIRRIII ' IASMCII. J. III ' V1201
VIZI - IHI -VI21JI'V211
III. II - IASMCII. J. II ' RRIIIIVIZI
111.21- YII.11°I-\I211
111.31 - 0
112.21- MIL J. 21 ' RRIZIIVIZI
VIZ. II - 112.21'I-V121JII
VIZ 31 - O
YI3. 11 - O
YI3. 21 - O
113. 31 - AISIACII. J. 31 ' RR131

 

 

212

CALL DTRAFORMIYII. ANGLEIJ). YITIII
FORII- ITO3
FORIII-ITO3
YTRIJ.II.IIII - VITIILIIII
NEXTIII
EXPITIII. II - EXPVALIRPLEXP. J. IL 11
NEXTII .
CALL MATMILTIVIT 11. EXPITII. YEXPIII
OMC - IMCIRPIMC.J1-MCIRPLMC- I.J1111R1RPL.MC1' DLTI
FORII- ITO3
YHIIII-VEXPINII'M
'EXTII
IFI <- PAMP > OWEN'EXCLUOEISTRWDFMI ATP-O
'EXCLUOE FIALITERATION IHICHISI-PoI
'THE FIALSUMWILLRETAXENIMAICDM1$IOROUTIE
CALL MATRMULTIVITII.EP11. YE111
CALL MATRHILTIVIT 11. XPII. YX111
OR - .5'1RII-II+ 01111'0LT
FORII- ITO3
YEJ1J.I.II-VEJ1J.I.II+DR'VEII.11
YXJIJ.IL11-VKJ1J.1|.II+OR'YX1N.II
VHJIJJJI-VHJIJJLIHOR'YHIIII
IEXTII
ENDIF -
HJI -H1JI:HJ2-HJI'HJI:HJ3-HJ2'IIJI
HI -H1J-11:H2-H1'HI:H3-H2'H1
FDRI- ITO3
FDRM- ITO3
AOODMMI-A0001lM1*YTRIJ.lI1'IHJI-NII
A0001lM*31-A0001lll+31*VTIJ.I.I1'5’1HJ2-H21
A00010+3.I1-A0001[Il+31
A000093.I+31-A0001!03.I¢31+VTRIJ.I.I1'III31'1HJ3-H31
ml
MII-MIIFMII'IHJI-HIII
“1103.11-M+3.II*YHII.II’.5'IHJ2vH21
I‘XTI
NEXTJ

IF P > DTHEN 'EXCLUOE IST RLI OF m1 AT P-O
IFI < - PTHEN 'EXCEPT FIAL ITERATIM
CALL MATRHILT1A00011. EXPO. A000EX1'111
RPLR 011111-110 111111101111
FOR 11 - ITO 0
GRAWIH. II - 0Rm11+ RPLR' m 11-A000EKP10. 111
NEXTI

ELSE '0UTFIALITERATIM
FORI- ITOO
SRANOLII-ORAUMIDMII
NEXTII
EMF
ELSE
FDRII- ITOO
WWII-“11.11
NEXTII
EWIF
ENO$10

SUO STRAFORM ISXVII. ANGLE. 812111

 

 

 

213

THETA - ANGLE] 100 ' 3141502054!
C-COSITHETAI

S-SI1THETAI

CS - C ‘8

C2 - C 'C

32 - S'S

$12111 - C2 ' SXYIII + $2 ' SXY121+ 2 ' CS ' SXV131
812121- S2 ' SXYIII + C2 ' SXV121-2 ' CS ' SXVI31
$12131 - .CS ' SXY111+ CS ' SXV121 * 102 - S21 ' SXVI31

END SUO
FUNCTION TIAEST 1108 1TVL1

1118- '0'
H28 - '0'
M18 . '0'
M28 - '0'
$18 - 'O'
828 - '0'

IF TVI. > - moo THEN
HI: - LEFTWOROIISTRIIITVL 1 m1»
TVL - TVI. MOO moo

END IF

IF TVI. > - woo THEN
H28 - LEFTWOROIISTRIITVI. 1 380011
TVL - TVL MOO 3000

END IF

IF TVL > - coo THEN
MII - LEFTWOROIISTRIII'VLIOIIIII
TVL - TVl MOO m

END IF

1F TVL > - BO THEN
M28 - LEFTWOROIISTRIITVLIOOII
TVL - TVl MOO GO

END IF

IF TVL > - IO THEN
SI: - LEFTWOROIISIRIITVLI 1011
WI . TVl MOO IO

END IF

S28 - LEFTWORO8ISTR8ITVL11

TIESTR08 - (H18 ¢ H28 9 '2' + M18 ¢ M28 + 'z' + 818 6 S281

END FLICTION

FUNCTION TWALU 1TS81

H18 - LEFT81TS8. IL TS8 - RISHT81TS8.LEN1TS81o11

H28 - LEFT 81T S8. 11: T88 - RBHT81TS8. LEN1TS8I - 21

M18 - LEFT81TS8. 11: T88 - RISHT81TS8. LEMTS81- 11

M28 - LEFT 811' S8. 11: T88 - RICHT8ITS8. LEHT S81 - 21

$18 - LEFT811'S8. 11: 828 - RIOHT81TS8.11

TIEVALU - VALIH181' IO ' 00 ' 00 0 VALIH281' 00 ' 00 0 VAL1MI81' IO ' 00 + VALIM281° 00 9 VALISI81' IO * VALISZ8I

END FUNCTION

 

214

$10 IRITECMD INUM. RAM. NAME 8111

IFRANX-DTHEN
X-I

ELSE
X-2

ENOIF

FOR 11 - I TO X
PRIT II. 'C'
NEXT 11
PRIT II. 'M'
FOR 11 - I TO NUM
PRIT II. 'RESET ' + STR81R1 + '.I4' lINCUIIVES
PRIT II. 'CzIVPTIOFIOUTPUTI' + PREFIX8 0 W8 + '1' + M8111“ 0 RI * '.SDF'
NEXT 11 '
PRIT II. 'I'
PRIT II. 'E'

 

 

REFERENCES

 

REFERENCES

Alfrey, Jr., T.. 1948. The Mghapig! Behavior of High Polymers. Interscience, New
York.

Alexopoulos, J.. 1989. Effect of Resin Content on Creep and Other Properties of
Waferboard. M.S. Thesis, Department of Forestry, University of Toronto, Toronto,
Canada.

Arima, T.. 1967. The Influence of High Temperature on Compressive Creep of Wood.
Journal of Japan Wood Research Society, 13(2):36—40.

Armstrong, L. D. and R. S. T. Kingston. 1960. Effect of Moisture Changes on Creep
in Wood. Nature, 185(4716):862-863.

Armstrong, L. D. and G. N. Christensen. 1961. Influence of Moisture Changes on
Deformation of Wood Under Stress. Nature, 19l(4791):869-870.

Bach, L. and R. E. Pentoney. 1968. Nonlinear Mechanical Behavior of Wood. Forest
Products Journal, 18(3):60-68.

Bodig. J. and B. A. Jane. 1982. W Van
Nostrand Reinjold Co., New York.

Christensen, G. N.. 1962. The Use of Small Specimens for Studying Effects of Moisture
Content Changes on the Deformation of Wood Under Load. Australian Journal of
Applied Science, 13(4):242.

Christensen, R. M.. 1982. W. Academic Press. New York.

Clouser, W. S.. 1959. Creep of Small Wood Beams Under Constant Bending Loads.
Forest Products Laboratory, Madison, Report No. 2150.

Davidson, W. R.. 1962. The Influence of Temperature on Creep of Wood. Forest
Products Journal, 12(8):377-381.

215

 

216

Ethington, R. L. and R. L. Youngs. 1965. Das rheologische Verhalten von Roteiche
(Quercus rubra L.) bei Beanspruchung quer zur Faserrichtung. Holz als Roh- und
Werkstoff, 23(5): 196-201.

Feng, Y. and O. Suchsland. 1993. Improved Technique for Measuring Moisture Content
Gradients in Wood. Forest Products Journal, 43(3):56-58.

Flugge, W.. 1967. W. Blaisdell Publishing Co., Waltham, Massachusetts.

Forest Products Laboratory. 1987. MW (Agriculture Handbook 72). Forest
Service, US. Agriculture Department, Washington D.C..

Gibson, E. J.. 1965. Role of Water and Effects of a Changing Moisture Content. Nature,
206(4980):213-215.

Gross, B..1953. M m i I re of ti fVi 1 ii .Herman.

Paris, France.

Grossman, P. U. A.. 1976. Requirements for a Model that Exhibits Mechano—Sorptive
Behavior. Wood Science and Technology, 10(3): 163-168.

Heramon, R. F. S. and M. M. Paton. 1964. Moisture Content Changes and Creep of
Wood. Forest Products Journal, l4(8):357-379.

Ismar, H. and M. Paulitsch. 1973. Effect of Climate on the Residual Stress and
Deformation of Multi-layered Particleboard. Holz als Roh- und Werkstoff, 31(22):469-
474.

Jones, R. M.. 1975. WWW. Scripta Book Co., Washington.

King, E. G. Jr.. 1961. Time Dependent Behavior of Wood in Tension Parallel to the
Grain. Forest Products Journal, 11(3):]56-165.

Kingston, R. S. T. and L. N. Clarke. 1961. Some Aspects of the Rheological Behavior
of Wood. 1, 11. Australian Journal of Applied Science, 12(2):211-226, 227-240.

Kitahara, K. and N. Okabe. 1959. The Influence of Temperature on Creep of Wood by
Bending Test. Journal of Japan Wood Research Society, 5(1):]2-18.

Kitahara, K. and K. Yukawa. 1964. The Influence of the Change of Temperature on
Creep in Bending. Journal of Japan Wood Research Society, 10(5):169-175.

Koch, P.. 1971. Process for Straightening and Drying Southern Pine 2 by 4’s in 24
Hours. Forest Products Journal, 21(5):]7.

 

217

Kiihne, H.. 1961. Beitrag zur Theorie des mechanischen Formanderungsverhaltens von
Holz. Holz als Roh- und Werkstoff, 19(3):81-82.

Kunesh, R. H., 1961. The Inelastic Behavior of Wood. A New Concept for Improved
Panel Forming Processes. Forest Products Journal, 11(9):395-406.

Lawniczak, M.. 1958. Investigations on Creep Deformation and Stress Relaxation in
Steamed Beech Wood. CSIRO, Australia, Translation No. 4395, from lnstytut
Technologii Drewna. Prace 5(1):61-81.

Leaderman, 11.. 1958. Viscoelastic Phenomena in Amorphous High Polymeric Systems.
Rheology: Theory and Applications. Edited by F. R. Eirich. Chapterl, vol. 2. Academic
Press, New York.

Liu, D.. 1987. Unpublished. Class Notes for Nonmetallic Composite Materials (MMM4-
44). MMM Department, Michigan State University.

Lo, K. H., R. M. Christensen, and E. M. Wu. 1977. A Higher-Order Theory of Plate
Deformation, Part2: Laminated Plates. Journal of Applied Mechanics, 44(4):669-676.

Mindlin, R. D.. 1951. Influence of Rotary Inertia and Shear on Flexural Motions of
Isotropic, Elastic Plates. Journal of Applied Mechanics, 18(1):31-38.

Nemeth, L. L. J.. 1964. The Influence of Moisture on Creep in Wood. M.S. Thesis,
University of California at Berkeley.

Nielsen, A.. 1972. Rheology of Building Materials. Document D6: 1972. National Swedish
Institute Bldg. Res.:213.

Norimoto, M. and T. Yamada. 1965. The Effect of Temperature on the Stress Relaxation
of Hinoki. Wood Research Bulletin, Wood Research Institute, Kyoto University, 35 :44-50.

Norimoto, M., H. Hiyano and T. Yamada. 1965. On the Torsional Creep of Hinoki
Wood. Bulletin of Wood Research Institute, Keyoto University, 34:37-44.

Norris, C. B.. 1942. W. I. F. Laucks, Inc., Seattle, Washington.

Ota, M. and Y. Tsubota. 1966. Several Investigations on the Static Viscoelasticity of
Wood Subjected to Bending Test. Journal of Japan Wood Research Society, 12(1):26-29.

Pagano, N. J .. 1970. Exact Solution for Rectangular Bidirectional Composites and
Sandwich Plates. Journal of Composite Materials, 4(January):20-34.

 

218

Pagano, N. J.. 1970. Influence of Shear Coupling in Cylindrical Bending of Anisotropic
Laminates. Journal of Composite Materials, 4(July):330-343.

Pagano, N. J .. 1969. Exact Solutions for Composite Laminates in Cylindrical Bending.
Journal of Composite Materials, 3(July):398—411.

Pagano, N. J. and A. S. D. Wang. 1971. Further Study of Composite Laminates Under
Cylindrical Bending. Journal of Composite Materials, 5(October):521-528.

Pagano, N. J. and S. R. Soni. 1983. Global-local Variational Model. International Journal
of Solids & Structure, 19(3):207-228.

Pagano, N. J .. 1972. Elastic Behavior of Multilayered Bidirectional Composites. AIAA
Journal, 10(7):931-933.

Perkitny, T. and R. s. T. Kingston. 1972. Wood Science and Technology, 6(3):215.

Pentoney, R. E. and R. W. Davidson. 1962. Rheology and the Study of Wood. Forest
Products Journal, 12(5):243-248.

Reddy, J. N .. 1984. A Simple Higher-Order Theory for Laminated Composite Plates.
Journal of Applied Mechanics, 51(4):745-752.

Rehfleld, L. W. and R. R. Valesetty. 1983. A Comprehensive Theory for Planar Bending
of Composite Laminates. Computer Structures, 16(1-4):441-447.

Rose, G.. 1965 . Das mechanische Verhalten des Kiefernholzes bei dynamischer Dauerbea-
nspruchung in Abhangigkeit von Belastungsart, Belastungsgréfle, Feuchtigkeit und
Temperatur. Holz als Roh- und Werkstoff, 23(7):271-284.

Schapery, R. A.. 1967. Stress Analysis of Viscoelastic Composite Materials. Journal of
Composite Materials, 1(3):228-267.

Schniewind, A.. 1968. Recent Progress in the Study of the Rheology of Wood. Wood
Science and Technology, 2(3): 188-206.

Schwarzl, F. and A. J. Staverman. 1953. Higher Approximation Methods for the Relax-
ation Spectrum from Static and Dynamic Measurements of Viscoelastic Materials. Applied
Science Research, A4(2):127.

Sims, D.. 1972. Viscoelastic Creep and Relaxation Behavior of Laminated Composite
Plates. Ph.D. Dissertation. Southern Methodist University.

 

 

219

Suchsland, 0.. 1970. Optical Determination of Linear Expansion and Shrinkage of Wood.
Forest Products Journal, 20(6):26-29.

Suchsland, O. and J. D. McNatt. 1985. On the Warping of Laminated Wood Panels.
Report on Cooperative Study Between the Forest Products Laboratory; the Forestry
Department and the Agricultural Experiment Station of Michigan State University.

Takemura, T. and Y. Ikeda. 1963. The Stress Relaxation in Relation to the Non-equilibri-
um State in the Wood-water System. Bulletin of Shimane Agricultural College, Matsue,
11A:105-133.

Takemura, T.. 1967. Plastic Properties of Wood in Relation to Non-equilibrium States
of Moisture Contents. Journal of Japan Wood Research Society, 13(3):77-8l.

Takemura, T.. 1966. Plastic Properties of Wood in Relation to Non-equilibrium States
of Moisture Contents. Memoirs of the College of Agriculture, Keyoto University, 88:31-
48.

Tong, Y. H. and O. Suchsland. 1992. Application of Finite Element Method to Panel
Warping. Holz als Roh- und Werkstoff, 51(1993) 55-57.

Ugolev, B. N.. 1964. Determination of Rheological Properties of Wood. CSIRO,
Australian Translation 7161.

Ugolev, B. N.. 1963. Determination of the Rheological Properties of Wood. Derev.
Prom. 12(2):17-19. (CSIRO, Australia, Translation 7161).

Ugolev, B. N. and W. I. Pimenowa. 1963. Investigation of the Effect of Temperature
and Moisture Content on the Rheological Behavior of Beech Wood. Derev. Prom.,
12(6):]0—12.

Whitney, J. M. and C. T. Sun. 1973. A Higher Order Theory for Extensional Motion
of Laminated Composites. Journal of Sound and Vibration, 30(1):85-97.

Whitney, J. M. and N. J. Pagano. 1970. Shear Deformation in Heterogeneous Anisotropic
Plates. Journal of Applied Mechanics, 37(4):1031-1036.

Yamada, T., T. Takemura and S. Kadita. 1961. On the Rheology of Wood 111. The
Creep and Relaxation of Buna. Journal of Japan Wood Research Society, 7(2):63-67.

Ylinen, A.. 1957. Zur Theorie der Dauerstandfestigkeit des Holzes. Holz als Roh- und
Werkstoff, 15(5):213-215.

 

220

Youngs, R. L.. 1957. The Perpendicular-to-Grain Mechanical Properties of Red Oak as
Related to Temperatures, Moisture Content, and Time. U. S. Forest Products Laboratory,
Madison, Wisconsin, Report 2079.

Youngs, R. L. and H. C. Hilbrand. 1963. Time Related Flexural Behavior of Small
Douglas-fir Beams Under Prolonged Loading. Forest Products Journal, 13(6):227-232.

Xu, H and Otto Suchsland. 1991. The Expansion Potential: A New Evaluator of the
Expansion Behavior of Wood Composites. Forest Products Journal, 41(6):39-42.