THE EFFECT OF FATiGUE STRESS ON
SHEAR IN WOOD

Thain fa: the 00am 96 Ph. D.
MECHiGAN STATE UNWERSITY

£93m Dennis Suflivan
1958

 

tweets

This is to certify that the

thesis entitled

THE EFFECT a“ FATIGUES'I'RBSS WSHEAR IN WOOD

presented by
John Demis Sullivan

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in Forest Products

V1

[Major professor

 

O—l69

 

 

 

LIBRARY

Michigan State
University

 

THE EFFECT OF FATIGUE STRESS
ON SHEAR IN WOOD

BY

John Dennis Sullivan

AN ABSTRACT

Submitted to the School of Graduate Studies of Michigan
State University of.Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Forest Products

1958

ABSTRACT

The research reported here was conducted
primarily to study the effect of fatigue loads on
wood in shear parallel to the grain on a tangential
plane. To the best of the author's knowledge, this
research is the first of its kind. Samples of white
fir, western hemlock, and Douglas-fir were tested
under static loads and under fatigue loads in shear
parallel to the grain. Moisture content was held
constant and the specific gravity of the wood directly
adjacent to the plane of failure was measured. Data
were recorded and analysis was accomplished with the
use of acceptable statistical systems. The results

of the research are summarized in the following list:

1. Wood can be made to fail by fatigue loading

in shear parallel to the grain on a tangen-

tial plane.

2. There is a linear relationship between
shear stress and Specific gravity at the
line of failure for samples tested in static

shear parallel to the grain on a tangential

plane.

There is a significant regression of stress
level and specific gravity at the line of
failure on the number of cycles to failure
in shear parallel to the grain on a tangen-
tial plane.

A. linear relationship exists between shear
stress enui specific gravity at the line of
failure .hl wood tested by fatigue loading
in shear parallel to the grain on a tangen-
tial plane.

The early springwood offers the least re-
sistance to loading in shear parallel to the
grain on a tangential planeo

The stress level at failure for fatigue shear
samples is less than for samples of the same
species, with comparable specific gravity

values, tested under static loads.

THE EFFECT OF FATIGUE STRESS
ON SHEAR IN WOOD

By

John Dennis Sullivan

A THESIS

Submitted to the School of Graduate Studies of Michigan
State University of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Forest Products

1958

ii

ACKNOWLEDGEMENTS

The writer desires to express his appreciation
to Dr. A. J. Panshin for his guidance and criticism
in the preparation of this report. The guidance
offered by Dr. R. Swenson, Dr. W. Baten,

Dr. 0. Suchsland, Dr. P. Koch, and Dr. J. Creighton
is also deeply appreciated.

The author wishes to acknowledgment his
indebtedness to Dr. C. 0. Harris for the use of
testing equipment and to Mr. B. Radcliffe for the

photographic work.

TABLE 9}: CONTENTS

ACKNOWLEDGEMENTS . . . . . . .
LIST OF TABLES . . . . or. . .
LIST OF ILLUSTRATIONS . . . .
INTRODUCTION . . . . . . . . .

Historical Notes . . . .
Statement of the Problem

EXPERIMENTAL PROCEDURE . . . .
Selection and Preparation
Static Testing . . . . .
Fatigue Testing . . . .

Determination of Moisture
Determination of Specific

ANALYSIS AND RESULTS . . . . .

Specific Gravity

Moisture Content : : . :
Static Failure . . . . .
Fatigue Failure . . . . .
DISCUSSION OF RESULTS . . . .
Specific Gravity . . . .
Moisture Content . . . .
Static Failure . . . . .
Fatigue Failure . . . . .

CONCLUSIONS . . . . . . . . .

Theories of Fatigue Failure

Secondary Stresses .
HYStereSiS e o o e 0
Molecular Slip . . .

Application of Failure Theor e

of Samples
Content : :
Gravity . .

O O O O O O
O O O O O O
O O O O O O
O O O O O O

O O O O O
O O O O O O
O O O O O O
O O O O O O

i s . . .

iii

Page

ii

iv

Table of Contents.- Continued.

 

Page
SUMMARY . . . . .

O O O O O O O O O O O O O O 69

APPENDICES . . . . .

O O O O O O O C O O O O O 71

Appendix 1. List of Apparatus . . . . . 71
Appendis 11. Data . . . . . . . . . . . 73

LITERATURE CITED . . .

O O O O O O O O O O O O 79

gs: 9_1=_ TABLES
Table Page
1. Summary of Specific Gravity Analysis . . 27
2. Summary of Moisture Content Analysis . . 28

3. Summary of Regression Analysis for
Static Samples . . . . . . . . . . . . . 29

4. Summary of Multiple Regression Analysis
for Fatigue Samples . . . . . . . . . . . 40

APPENDIX II

a. Static Data Recorded for all White Fir
smpleSQQooeoooeeooooooe73

b. Static Data Recorded for the Western
Hemlock Samples . . . . . . . . . . . . . 74

c. Static Data Recorded for the Samples
of Douglas-fir . . . . . . . . . . . . . 75

d. Fatigue Data for White Fir . . . . . . . 76
e. Fatigue Data for Western Hemlock . . . . 77

f. Fatigue Data for Douglas-fir . . . . . . 78

LIST g_1_=_ ILLUSTRATIONS

 

Figure
1. Test Sample used for Static and
Fatigue Testing (photograph) . . . .
2. Sonntag Fatigue Machine and Load
Multiplier (photograph) . . . . . . .
3. Calibration Curves for the Fatigue
Machine . . . . . . . . . . . . . . .
4. Fatigue Loading Cycle (photograph) .
5. Stress Relaxation in Fatigue Samples
6. Calibration Curve for Velumeter . . .
7. Mercury Displacement Volumeter
(photograph) . . . . . . . . . . . .
8. Regression Curve of Shear Stress on
Specific Gravity for White Fir . . .
9. Regression Curve of Shear Stress on
Specific Gravity for Western Hemlock
10. Regression Curve of Shear Stress on
Specific Gravity for Douglas-fir . .
11. Regression Curves of Shear Stress on
Specific Gravity for all Three
Species Tested . . . . . . . . . . .
12. Migration of Static Failure Line
(photograph) . . . . . . . . . . . .
13. Straight Line Shear Failure
(photograph) . . . . . . . . . . . .
14. Failure Away from Plane of Maximum

Shear Stress (photograph) . . . . . .

vi

Page

15

16
18
19
23

25

31

32

33

34

36

37

38

List of Illustrations.-Continued.

 

Figure Page
15. S-N Curve for White Fir . . . . . . . . . 42
16. S-N Curve for Western Hemlock . . . . . . 43
17. S-N Curve for Douglas-fir . . . . . . . . 44
18. S-N Curves for all Species Tested . . . . 45

THE EFFECT OF FATIGUE STRESS ON SHEAR IN WOOD

INTRODUCTION

 

HISTORICAL NOTES

The fatigue strength or the endurance limit of a
material has been defined as the maximum stress that
can be applied repeatedly for a specified number of
stress cycles without producing rupture of the
material (2h). Actual testing to induce a fatigue
type failure was conducted in Germany in 1829 on
mine-hoist chains (15). The first-recorded, fatigue
research was followed initially by workers in France,
Germany, and England to the end of the nineteenth
century when the research in fatigue spread all over
the world. From the early interest in metals, fatigue
analysis was extended to plastics, glass, rubber,
flax, and wood.

The normal method of fatigue testing involves
the application of a cyclic load on the material in
question. Depending upon the type of analysis being
conducted, the stress may vary from zero to maximum
in one direction or it can involve complete reversal
of stresses. The end result of fatigue research is

usually expressed graphically to show the relationship

1

of stress level to number cn‘ cycles to failure.
Throughout the remainder of this paper this curve
will be called the SuN curve. Fatigue research in
metals revealed a scattering of data along the S-N
curveo This variation induced the use of statistical
analysis to establish values of fatigue strength at
certain levels of probability (31)(ll). The stress
levels used in such testing were based on a percent
of the static load at failure for the same material.
This design system produced admirable results for
isotropic materials. For anisotropic materials,
however, the same design has led to extreme variation
of data about the S-N curve. The reason for the vari-
ation lies in the numerous factors that influence all
the strength properties of wood regardless of the
loading method.

Research into the fatigue properties of wood
has been somewhat limited to compression and bending
for plywood and for solid wood specimens (20)(19)(2l)
(8)(18)(22)o An intensive search of the literature
revealed only one investigator who worked directly
on fatigue failure in shear parallel to the grain (23).
This research was conducted on glued-shear specimens.

The failure occurred primarily in the wood adjacent

to the glue line; however; the number of samples so
tested was considered inadequate by that investigator.
Previous studies of fatigue in wood have had
widely varying results. The S-N curves were approxi-
mately the same shape in these studies, inn; they
varied considerably iJI their height (Hi the stress
scale for identical species. All the workers appeared
to have their results limited by an insufficient
number of samples and by a large amount of variation
from the S~N curve. Another factor common to these
investigations was the complete absence of statistical
significance h] the results. This factor led the
author of this paper to the assumption that perhaps
these previous researchers had not taken some
fundamental sources of error under consideration.
These sources of error could be occasioned by such
variables as specific gravity, moisture content,

temperature, residual stresses, and damping capacity.

STATEMENT OF THE PROBLEM
The primary object of this research was to study
the effect of fatigue loading on wood in shear
parallel to the grain. This would obviously be
possible only if a testing method could be developed

that would induce failure in wood subjected to fatigue

loading. The secondary object of the problem was to
obtain an S~N curve with a minimum amount of variance.
It was the authoris opinion that a large source of
error could be eliminated if specific gravity could
either be accurately measured or held as a constant.
Since it is virtually impossible to hold Specific
gravity of wood constant from one sample to another,
the only remaining alternative was to measure specific
gravity as accurately as possible. Since failure was
to be related to specific gravity, it seemed natural
to assume that the specific gravity at the exact point
of failure would be a better measure than would be the
average specific gravity of an entire sample.

This assumption immediately posed two questions:
In which strength property of wood can the line of
failure be exactly located, and how is the Specific
gravity at the point of failure to be measured accu-
rately? Shear strength was selected because it is
simple to locate the failure line in shear and because
no previous fatigue research had been conducted on
solid wood specimens in shear fatigue.

The problem was then conducted with the solution

of the following objectives in mind:

1. To induce failure by fatigue in shear parallel

to the grain for solid samples of wood.

To accurately measure specific gravity at
the line of failure for Specimens tested.
To Show the effect of specific gravity on
the fatigue strength of wood in shear
parallel to the grain on a tangential
plane.

To show the effect of specific gravity
on the static strength of wood hi shear
parallel to the grain on a tangential
plane.

To obtain S-N curves with a minimum amount
of experimental variation for several wood
species.

To compare the shear strength of wood
under static loads to the shear strength
of wood subjected to fatigue loads.

To obtain statistical significance wherever

possible for the objectives listed above.

EXPERIMENTAL PROCEDURE

The procedures used in this experiment were
devised to minimize or eliminate experimental error.
Standard testing methods were applied wherever such
methods were applicable to the apparatus available.
A detailed list of apparatus is shown in Appendix I
of this report. Preliminary testing showed that some
modification of certain pieces of apparatus was
necessary or that different equipment would have to
be used. The appropriate modifications and the de-
sign of new equipment are described in detail in
later portions of this experimental procedure. The
experimental procedure, as described in this paper
and as used in the research, includes preparation
of samples, static testing, fatigue testing, deter-
mination of moisture content, and determination of

specific gravity.

SELECTION AND PREPARATION OF SAMPLES
The wood species selected for experimentation
were white fir (52123 grandis [Dougl.] Lindl.), western
hemlock (18333 heterophylla [Raf. ] Sarg. ), and

Douglas-fir (Pseudotsugg menziesii [Mirb.] Franco).

 

Throughout the remainder of this report the common

names, white fir, western hemlock, and Douglas-fir,
will be applied to the samples of the respective wood
species. The identity of all stock from which the
samples were obtained was confirmed by microscopic
inspection. Coniferous tree species were used because
of their relatively low resistance to failure in shear
parallel to the grain direction. White fir, western
hemlock, and Douglas-fir were selected because they
were readily available in the dimensions desired and
because these species are widely used in the various
forest products industries. Douglas-fir wood, which
is characterized in! the presence of small, sparsely
distributed resin canals, was tested as a portion of
this project because it was assumed that the small
size and number of resin canals would not significantly
affect the results.

The stock was carefully scrutinized to detect any
abnormalities that could influence the strength of the
samples. Any evidence of brashness, compression wood,
grain deviation, seasoning checks, or decay resulted
in the elimination of such material for use as sample
stock. The stock eventually used for samples was
defect-free, straight-grained wood.

The white fir was used to determine the plane
of minimum resistance to shear parallel to the grain.

Stock with wide growth increments CAppendix I, No. l)

 

was used in order to place the plane of maximum
shear in different Positions in the growth increments
of different samples. Western hemlock and Douglas-fir
stock were selected from material cut from large, slow
grown trees with very narrow growth increments (Appendix
1, No. 2 and No. 3). Such material insured the close
proximity of the plane of least resistance to the
plane of maximum shear stress parallel to the grain.
It was assumed that all the samples of western hemlock
and Douglas-fir would then fall at the weakest point in
the wood.

Preliminary testing showed thatithe shape of the
shear specimen could have a pronounced effect on the
magnitude of stress at failure. The ideal shear
specimen is characterized by shear only, without bending
moments on the plane of maximum shear. Unfortunately,
bending moments cannot be eliminated; they can only
be minimized (7)(32). The design of the shear block
specimen used for all samples tested in this research
is shown on the following page in Figure 1. The most
obvious advantage of the H-shapedsample is that the
stresses that induce failure by cleavage are minimal
at the plane of greatest shearing stress.

Test samples for static testing as well as for
fatigue testing were machined to form and dimension

on a band-saw (Appendix I, No. LI). All sammes were

Figure 1. Photograph of test sample used for

static and fatigue testing. The

sample in the picture is about two-
thirds of the actual size.

 

 

 

10

cut to the same shape and the same dimensions within
the limits of accuracy of the machine. The interior
faces of the testing-head groove in the samples were
machined to be as nearly parallel as possible. In
addition, the testing-head surface (A in Figure l),
the lower surface cu‘ the shear section (B in Figure
l), and the plane of the supports (C in Figure l)
were made mutually parallel within the machining
accuracy of the band-saw. All samples were fashioned
to make the plane of maximum shear tangent to the
growth increments in order to avoid the possibility
of failure across more than one growth increment.
.Matching fatigue samples were cut from material ad-
jacent in a tangential direction to the static samples.
The testing-head groove for each matched pair of
samples was cut into the same growth increment in

an attempt to have identical material along the
planes of maximum shear. The samples were marked

to insure their identity.

After machining, all test samples were conditioned
to a moisture content of about twelve percent. The
purpose of the conditioning period was to minimize
error induced by changes in moisture content. The
shear area of each sample was measured and recorded.
After conditioning, the samples were placed in

polyethylene bags to reduce the rate of change of

11

moisture in the wood. The samples were stored

until the mechanical tests were conducted.

STATIC TESTING

All static testing was performed on a Baldwin-

Emery, SR-h Testing Machine (Appendix I, No. S).

The samples were tested at amachine speed of

0.02h inches per minute (i). The testing head was

the same that was used for fatigue testing. A small,
flexible rubber strip was inserted between the testing
head and the specimen to obtain uniform distribution
of the load. Samples were supported on a universal
bearing to minimize the effect of horizontal components
of the load.

As static loading progressed on a sample, it was
closely inspected to determine the position of initial
failure. Samples were excluded from further considera-
tion when the initial failure was other than in shear
parallel to the grain at the plane of maximum shear.
A.common type of failure was in cleavage occurring at
the vertical center of the load-bearing section.
All samples that failed initially due to cleavage
exhibited secondary failure in shear parallel to the
grain. These samples were not included in the analysis
because the load-bearing section was not horizontally
continuous after cleavage. In addition, the magnitude

of the bending moment across the plane of maximum

l2

shear was different iuxmi one such sample to another.
Approximately thirty percent of the original number
of static samples were rejected because of failure
in cleavage.

The samples in which the initial failure occurred
in shear parallel to the grain at the plane of maximum
shear were considered for further analysis. The line
of failure was inSpected for the presense of any ab-
normalities that may have influenced the resistance of
the sample to the load. Samples were eliminated if the
failure plane was characterized by grain deviation, pin
knots, decay, or accumulations of resin. The latter
consideration applied only to the Douglas-fir samples.
This inspection eliminated about five percent of the
original samples in Douglas-fir and western hemlock
and about fifteen percent in white fir. The data
recorded for each acceptable samples were the stress
'at failure and the position of the line of failure with
respect to the summerwood and springwood (Appendix 11,

Tables a, b, and c).

FATIGUE TESTING
All fatigue testing was performed on a Sonntag
fatigue testing machine. (Appendix 1, No. 6). The
machine consisted of a large stationary platen and a
smaller reciprocating platen. The oscillating force

was produced through the reciprocating platen by means

13

of an unbalanced rotating mass operating at a constant
speed of 1800 cycles per minute. The machine was
capable of operating either with or without a static
preload. To satisfy the requirements of this research
the machine was to be operated on the tension side
with the vibratory load fluctuating from zero to the
maximum load desired. It was therefore necessary to
place a static preload one-half the magnitude of the
total load on the sample. Preliminary testing showed
that the maximum load capacity of the machine was not
sufficiently high to satisfy the testing requirements
of this research. Therefore a load multiplier was
designed that was capable of delivering the loads
desired.

Fortunately, a torsion load multiplier for the
Sonntag machine was available for study as a prototype
(2). The torsion fixture was a fixed-pivot lever arm
designed to produce an alternating torque from an
alternating force. It was obvious that the use of
a freely-pivoting lever arm would produce a large
alternating force from a relatively small alternating
force. This was the principle that was used in the
design of the load multiplier for this problem. One
pivot was rigidly attached to the reciprocating platen
and the second pivot was securely fastened to the

stationary Platen. The exact position of the test

11+

specimen on the stationary platen between the pivots
determined the load multiplication factor of the
fixture. A photograph of the Sonntag machine with
the attached fixture is shown in Figure 2 on the
following page.

Additional preliminary testing revealed that
the load multiplier required calibration of load for
complete satisfaction. Strain gages (Appendix 1, No.
7) were cemented to opposite sides of the rear pivot
post. The purpose for the opposite placement of gages'
was to compensate for bending of the post. Predeter-
mined tensile, static loads of incresing magnitude were
applied and the strain was measured with a strain
amplifier (Appendix 1, No. 8). The calibration curve
of the strain system is shown in Figure 3, page 16.
The strain was then measured on the fatigue machine
for the same loads and was compared to the static load
calibration. The curves in Figure 3 show that a portion
of the load applied through the machine was lost in the
load-multiplier. This was probably due to frictional
losses at the pivots and at the testing-head guides.
The load on a sample to be tested and the load
on the reciprocating platen could then be easily
calculated by use of moments. The reciprocating
Platen load was divided equally into two components,

a static preload and a dynamic oscillating load.

15

Figure 2.

 

Photograph of Sonntag Fatigue Machine
used hi this research. The loadm

multiplier‘ is nwunted on the platens
of the machine. The oscilloscope was
used to calibrate the loading system.

 

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The loading cycle is shown in Figure 14., page 18.
The dynamic load varies sinusoidally from zero to
maximum stress and the static load is equal in magnitude
to one-half the total load.

Some difficulty was encountered in maintaining
the static component at a constant level. The static
preload diminished as a function of time when the
samples were loaded. This phenomenon was the result
of creep in a vertical direction and of local crushing
under the testing head in the teSt specimens. It was
necessary to stop the machine and readjust the preload
at frequent intervals during testing. Figure 5 illus-
trates clearly that as the number of cycles increased
it was not necessary to readjust the preload as
often.

The initial fatigue testing was based on the
use of stress levels that were a predetermined per-
centage of the stress at static failure of a matched
sample. .A short analysis showed that if any relation-
ship existed, it was drastically influenced by the
specific gravity on the line of failure. Since it
was physically impossible to measure the specific
gravity at the line of failure before failure occurred,
it was impossible to predict failure at any predeter-
mined stress level. Therefore, it was decided to do the

fatigue testing initially at very low stress levels

18

Figure A.

Fatigue loading cycle at a machine load
of thirty pounds static preload and
thirty pounds dynamic load. The median
line represents the static component

and the dynamic component oscillates
from maximum load to zero.

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and to increase the stress level after a certain
number of cycles if failure did not occur. The stress
level was elevated at 100,000 cycles. The data
recorded for each sample were the maximum shear
stress at failure, number of cycles to failure, and
the position of the line of failure with respect to
springwood and summerwood (Appendix 11, Tables d, e,
and f). Any samples that exhibited evidence of grain
deviation, pin knots, decay, accumulations of resin,
or failure in clevage were eliminated from further
consideration. This final inspection eliminated
about fifteen percent of the original number of white
fir samples and approximately five percent in Douglas-

fir and western hemlock.

DETERMINATION OF NDISTURE CONTENT
All samples were weighed on a triple-beam

balance (Appendix 1, No. 9) immediately after failure.
The balance pan and pivots were cleaned and the
instrument adjusted for zero before any measurements
were made. The zero balance position was checked
periodically during the course of the data collection.
The weight immediately after failure was recorded for
each sample. The samples were then placed in a
drying-oven (Appendix 1, No. 10) and allowed to dry
until no further loss of weight occurred. The drying

temperature varied from 100 to 102 degrees Centigrade.

21

The constant weight that was eventually attained was
recorded for each sample. The moisture content at
the time of failure was computed on the basis of weight
loss expressed as a percentage of the oven—dry weight
for each samfie. This moisture content was then re-

corded for each sample (Appendix II).

DETERMINATION OF SPECIFIC GRAVITY

The design of this research problem called for a
measurement of specific gravity of the wood directly
adjacent to the line of failure. Therefore, it was
necessary that very thin chips or shavings be obtained
from the failure line. Ideally the thickness of the
sections should have been the diameter of a woody
plant cell. This would have been very difficult to
determine because the cell is characterized by decreaSing
diameter from large springwood cells to very small
summerwood cells. The average tangential diameter
of a tracheid of white fir and Douglas-fir is from
35-145 microns while western hemlock tracheids, with
an average tangential diameter of 30-140 microns, are
somewhat smaller (1;). Micro-sectioning was not
possible since the ‘lines of failure were usually
anything but straight and flat. There remained the
one method that was actually used; sections sliced
with an exceedingly sharp knife. The sections varied

from sixty to one-hundred microns in thickness and

22

usually contained two or three tangential rows of cells.
The thickness measurements were made with a micrometer.

Preliminary tests showed that ttm: usual methods
for obtaining specific gravity were not suitable for
very thin sections. The problenx was 'to determine
the volume of the sections quickly and accurately to
avoid error due to volume change resulting from intake
or exodus of water. It was eventually decided to use
the method of liquid mercury displacement. .A mercury'
displacement volumeter (Appendix 1, No. 11) was de-
signed with the aid of a prototype (Appendix 1, No. 12)
loaned to the author. The volumeter was calibrated by
measuring the displacement of precision ground ball
bearings of varying diameters. The calibration curve
for the volumeter is shown in Figure 6 on the following
page.

Before measurements were made for the specific
gravity determination, the wood samples were again
conditioned in a constant temperature-humidity room
to a moisture content of approximately twelve percent.
Thin sections were cut from each side of the line of
failure and weighed on a grammatic balance (Appendix 1,
No. 13). Material from one side of the failure was
measured for weight and volume separately from material

from the opposite side of the same line of failure.

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The sections were placed in the volumeter, the
mercury was moved up to a reading-line on the sight
tube, and the micrometer reading was recorded to the
nearest ten-thousandths of an inch (Figure 7, page
25). The chips were extracted from the micrometer
and the mercury was displaced up to the reading line,
and the zero micrometer displacement was recorded.
The difference between the two micrometer readings
was converted directly into cubic centimeters. The
specific gravity was then computed for each side of
each line of failure in every sample. The average
specific gravity at the line of failure was computed
arithmetically and recorded (Appendix II).

The specific gravity as referred to in this paper
is based on weight and volume of the wood at about
twelve percent moisture content. All samples were
measured under the same conditions and the resulting
specific gravity data were intended to show only the
possible effect of this variable within one species.
The average specific gravity figures in this paper
should not be compared to Species averages from any
source unless such measurements were made in exactly

the same manner as in this research.

25

Figure 7. Mercury dISplacement volumeter.

_ A indicates the reading-line on the
sight tube, B is the micrometer and
C is the mercury reservoir.

 

 

 

 

 

 

 

26

ANALYSI 8 AND RESULTS

All portions of the analysis for this study
were completed with the use of acceptable statis-
tical systems. Computations were worked out and
individually checked on an automatic calculator
(Appendix I, No. ILL). Each species was analysed
separately for variation in specific gravity at line
of failure and moisture content at time of failure.
In addition the relationship between specific gravity
and shear stress at failure was explored for the
statically loaded specimen. The data obtained from
fatigue testing were used to determine the influence
of the specific gravity and the shear stress at failure

on the number of cycles to failure.

SPECIFIC GRAVITY
Analysis was conducted on each species to determine
the variability in specific gravity that was observed
between the statically loaded samples and the fatigue
samples, The null hypothesis was assumed and simple
group Comparison with the t test for significance was

aPplied. The statistical results are summarized in
Table 1.

27

Table 1. Summary cfi‘ the statistical analysis to test
the mean difference of specific gravity between static
and fatigue samples in each species.

 

 

 

Western
Species White Fir Hemlock Douglas-fir
Testing
Method Static Fatigue Static Fatigue Static Fatigue
Mean 0.306 0.316 0.357 0.393 0.h28 0.h37

Variance 0.0891 0.0676 0.0h27 0.03h0 0.0372 0.0366

Pooled
Variance 0.00h12h 0.002017 0.0019h3
Standard

Error 0.0203 0.01ul9 0.0139h
t 0.680 2.297% 1.000

 

*statistical significance at the 95 percent level.

 

The t_values shown in Table I were interpreted to yield

the results listed below:

1. The null hypothesis was accepted for white fir and
Douglas-fir. It may be assumed that all the spe-
cific gravity data were obtained from a common
population and that the average specific gravity
at the line of failure for the fatigue samples
was not different from that of the static samples.

2. The null hypothesis was rejected for western
hemlock. It may be assumed that the specific
gravity at the line of failure for the fatigue
samples was significantly greater than for that

of the Static samples.

MOISTURE CONTENT

28

An analysis was made in each species to study the

variation in moisture content between the statically

loaded samples and the fatigue samples.

The null

hypothesis was assumed and simple group comparison

with the t test for significance was applied. The

results are summarized below in.Table 2.

Table 2.

Summary of the statistical analysis to test

the mean difference of moisture content between static
and fatigue samples in each species.

 

 

 

Western

Species White Fir Hemlock Douglas-fir
Testing

.Method. {Static Fatigue Static Fatigue Static Fatigue
Mean 10.88 10.23 13.h3 11.77 12.3h 11.13
Variance 1.821 7.300 2.989 22.207 0.827 1.071
Pooled
Variance 0.2h00 0.6631 0.0h996
Standard -

Error 0.15119 0.2575 0.0707
.E 11.18% 6.1.117“- 17.086”

 

*fistatistical significance at the 99 percent level.

 

The summary shown

in Table 2 indicates that in

all three species the moisture content at the time of

29
static failure was significantly greater than the

moisture content at the time of fatigue failure.

STATIC FAILURE

vThe method of linear regression was employed to
determine the effect of specific gravity on the shear
stress at failure parallel to the grain in each of
the species tested. This method was applicable since
the variance was homogeneous, specific gravity was
measured without error, and the shear stress was random.
The results of the regression analysis are shown in
Table 3.
Table 3. Summary of regression analysis to demonstrate

the effect of specific gravity on the magnitude of
shear stress at failure.

 

 

 

Species Regression

Coefficient t-value
White fir 2606.0 9.112**
Western hemlock 3282.3 5.659**
Douglas-fir 3581.5 6.707**

 

 

Regression Equation

 

White fir Y -336.6 + (2606.0) x
'Western hemlock Y = - 26.0 + (3282.3) x

Douglas-fir Y = -460.0 + (3581.5) x

 

30
Table 3.-Continued.

 

** statistical significance at the 99 percent level.

Y' estimated shear stress for any value of specific
gravity.

X any value of specific gravity.

 

 

The _1_:_ values shown in Table 3 denote a statis-
tically significant9 linear, positive regression of
shear stress on the average specific gravity at the
line of failure. Figures 8, 9, and 10 show the regres-
sion and the 95 percent fiducial limits of estimated
shear stresses for white fir, western hemlock, and
Douglas-fir respectively. The regression line is
based on the premise that the sum of squares of dev-
iations from linear regression are minimum (27).

The confidence interval of the stresses was determined
by direct calculation over a range of specific gravity
figures (29). In figure 11 the regression lines of

all three species are shown on one graph. The greatest
shear stress at the average specific gravity was
developed in western hemlock, followed by Douglas-fir
and white fir in that order. This successive relation-
ship corresponds closely to published values of shear
Parallel to the grain.

The shear line of failure for western hemlock
and Douglas-fir occurred at the point of maximum shear

stress in all samples included in the analysis. In

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35
addition, the line of failure was parallel to the grain
direction in all samples. The line of failure character-
istically occurred either in.the springwood or on the
line between the summerwood cfi'an increment and the
springwood of an adjacent increment. No failure took
place entirely in the summerwood of the samples. The
white fir samples were more variable than the other
species with respect to the location of the line of
failure. For the majority of the samples, shear
failure began at the point of maximum shear and then
migrated toward the early springwood. A photograph
showing this phenomenon can be seen in Figure 12.
page 36. The migration of the failure line always
occurred at an angle to the grain and never progressed
into a summerwood portion of a sample. The resulting
oblique failure line was found only in samples that
had summerwood at the point of maximum shear directly
under the edge of the testinguhead. If the early
springwood was at the point of maximum shear stress,
the failure line was characteristically straight and
parallel to the grain direction. This more normal
type of failure is shown in Figure 13, page 37.
Several samples in all three species exhibited another
type of failure line when summerwood was at the point
of maximum shear. The failure occurred between the

summerwood of one increment and the springwood of the

 

36

Figure 12. Migration of the static failure line.
The arrow indicates the failure line
slanted towards low-density springwood.

 

 

 

 

37

Figure 13. Straight-line shear failure in static
specimen. The failure in this specimen
occurred at the plane of maximum shear.

 

 

 

 

 

38

Figure 14. Failure occurring in a statically
loaded specimen away from the plane
of maximum shear stress.

 

 

 

 

39

following increment at a plane removed from the point
of maximum stress. This resulted in material being
sheared from the side of the testing-head groove as

shown in Figure 14, page 38.

FATIGUE FAILURE

The method of multiple regression was used to
determine the effect of specific gravity and stress
level on the number of loading cycles to failure.
.A graph of the data on linear paper evidenced an
exponential relationship between the three variables.
Semi-logarithmic plotting and analysis did not yield
significant results. Log-log plotting finally showed
a relationship that could be subjected to analysis by
the method of multiple regression. A search of the
literature revealed that multiple logrithmic regres-
sion can be appropriately used as a method of statis-
tical analysis (26). In order to facilitate the use
of this analysis, the data for specific gravity were
converted from three-place decimals to three-digit,
'whole numbers. The purpose for the translocation of the
decimal point was to avoid the use of negative loga-
rithms in the subsequent analysis; All data were con-
verted to logarithms and analysed with the standard
multiple regression procedure. The results are

summarized in Table 4.

Table 4.

40

Summary of the results of the regression

of number of cycles to failure on shear stress and
specific gravity for white fir, western hemlock,
and Douglas-fir.

 

 

White Western

 

 

 

Fir Hemlock Douglas-fir
Mean
specific 316' 393' 437'
gravity
Mean
shear 315 psi 486 psi 480 psi
stress
Mean no.
of cycles 1569 888 1175
byl°2 0.597 8.105 10.249
t for
‘5y1‘2 0.066 3.742** 3,034..
by2~l -7.424 -8.481 -10.625
t for
‘Byzoi 1.901 13.252** 7.601**
R. 0.821** 0.942** 0.838**
Species Equations of Multiple Regression

 

White Fir

Y
Western Hemlock Y = (50828) X1
Douglas-fir Y

R

i

-
= (1.814)(10)20 x10'357 x2 7'35:
2
10.249 x -10.625
2

(31160) X1

the mean specific gravity figures are actually

0.316, 0.393, and 0.437.

**

statistical significance greater than the 99

percent level.

41
Table 4.-Continued.

 

by1°2 is the regression of number of cycles on
specific gravity independent of shear stress.

byZ-l is the regression of number of cycles on
shear stress independent of specific gravity.

R is the multiple correlation coefficient.
Y is the estimated number of cycles to failure.
X1 is specific gravity expressed as a three-digit

whole number.

X2 is shear stress at failure.

 

 

The multiple regression of number of cycles on
specific gravity and stress was significant for all
three species. .A closer inspection of the partial
regression coefficients reveal that in all three species
the number of cycles to failure varies directly with
exponential functions of specific gravity and varies
inversely with exponential functions of shear stress.
The equations of multiple regression define a curved,
three-dimensional plane. In order to show the S-N
curve graphically, it was necessary to hold one variable
constant. Because it was the S-N curve that was
desired, the relationships shown in Figures 15, l6,
l7, and 18 were obtained for the average specific
gravity in each species. The resulting two-dimensional
curves are line intercepts on their respective three-
dimensional, curved planes. The broken lines in

Figures 15, 16, and 17 represent one standard deviation

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46
of estimated number of cycles. The magnitude of the
deviation was obtained by the direct count of the
number of logarithmic units separating the regression
line from the plotted data points. The square root of
the quotient of the sum of the deviations squared,
divided by the degrees of freedom, yielded the
logarithmic magnitude of the standard deviation.
Inspection of Figure 18 reveals that the S-N curves
are approximately the same shape for all three species.
In addition the relationship between the stress levels
for the three species at less than 100 cycles is about
the same as under static loading.

The failure line in fatigue samples occurred at
the plane of maximum shear. The failure line was
straight and parallel to the grain in all cases.
Unlike the static samples9 no migration of the line of
failure toward low density material occurred in the
fatigue samples.

It was extremely difficult to compare the shear
stresses at failure obtained from static testing to
those obtained from fatigue testing. The specific
gravity dependence and the varying number of cycles
to failure makes a direct comparison meaningless.

The results indicated, however, that for samples in
the same species with comparable specific gravity

figures, the fatigue stress at failure was always

47
lower than the static stress at failure. The ratio

of fatigue strength to static strength in shear parallel
to the grain varied from 0.90 to 0.24 in Douglas-fir,
0.88 to 0.17 in western hemlock and from 0.95 to 0.36

in white fir. The above variations are averages and

have no statistical significance between species.

48

DISCUSSION QE RESULTS

 

The purpose of this section of the thesis is to
review the sources of error with respect to the results
obtained by the analysis. The results of this research
are not in themselves the total answer to the questions
probed during the course of the study. Careful
examination must be made of the sources of error and
of the underlying factors that occasion the variables
that were measured as a portion of this research.

To insure clarity, the origin of error in specific
gravity and moisture content determination will be
discussed in separate portions of this section. The
relationship between specific gravity and shear stress
at failure for the statically loaded specimens will
also be covered in the discussion. Lastly, the
factors that affected the regression of number of
cycles to failure on the specific gravity and on the

shear stress in the fatigue specimens will be reviewed.

SPECIFIC GRAVITY
To satisfy the purposes of this research, the
sections of wood used to determine specific gravity
were cut from both sides of the failure line. These
sections were very thin. As a result, the volume of

such sections was correspondingly small and the ratio

49
of surface area to volume decreased rapidly with an
increase of thickness of the section. It was antici»
pated that the percent possible error in specific
gravity of a thin section would be many times greater
than the percent possible error for a relatively
thick section. In addition, it was obvious that any
accurate control of thickness for the specific gravity
sections was impossible. The percent possible error
in specific gravity was estimated with the partial
differential method and was found to vary from almost
100 percent for very thin sections to approximately
one percent for the thickest sections. An effort
was made to cut the specimens at a thickness that
would yield an error of about fifteen percent. This
figure allowed for sections thin enough to obtain
material directly adjacent to the line of failure
and of sufficient thickness to measure with an
acceptable amount of error.

The specific gravity at the line of failure was
greater in the fatigue specimens than in the static
specimens for all three species tested in this research
(Table 1, page 27). This difference was caused by the
tendency for the failure line to migrate toward
low density springwood in the statically loaded
samples. The average for a group of such samples

would necessarily be lower than for the fatigue

50
samples in which failure occurred in the early spring-
wood only if such material was at the plane of
maximum shear stress. Because the western hemlock
stock had a large proportion of summerwood, the
specific gravity for fatigue samples was significantly
greater than for the static samples. Douglas~fir
and white fir stock had a greater proportion of
springwood and, as a result, a nonnsignificant
difference in specific gravity appeared between the
static and fatigue samples.

The physical sources of error in the determination

of specific gravity were as listed below:

1. Thickness of sections. The thickness of
each section was variable due to small
deviations in the plane of the failure.

It was not possible to cut a section so that
the plane of the cut was equidistant from
the plane of the failure at all points.

The resulting error could be either positive
or negative.

2. Roughness of failure plane. In many samples
the plane of failure was characterized by
alternating grooves and ridges. The size
of the grooves limited the mercury contact
since, at standard temperature and pressure,

mercury will not pass through an opening

51
smaller than twowthousandths inches. The

resulting error in specific gravity was
negative in all cases.

Fuzziness of failure_plane. The failure

plane in western hemlock was characterized

by a fuzzy appearance caused by the partial
separation of some cells from the mass

of material. These cells inhibited full
contact by the mercury and induced a negative
error in specific gravity figures.

Change of moisture content. The time required
for the measurement of the weight and volume
of each section was approximately three
minutes. Since this work was conducted under
controlled temperature and humidity conditions,
it was exceedingly unlikely that any large
error was introduced by changes in weight

or volume.

Trapped air in mercury reservoir. Trapped

air could be avoided by using the same mercury
for only ten or twelve specific gravity
measurements. Minute particles of wood

caused air bubbles to appear in the volumeter
sight-tube if more than a dozen determinations
were made without changing or cleaning the

mercury. In this research, the mercury was

52
changed when air bubbles began to appear.
The presence of air bubbles induced either
a positive or negative error in the specific
gravity.

6. Change of temperature. The mercury volumeter
was extremely sensitive to changes in
temperature due to the relatively high
coefficient of thermal expansion of mercury
(13).

The sources of error listed above were controlled
as closely as possible during the experimental work
and it may be said with assurance that the physical
errors were relatively unimportant in the summation
of their effect on the specific gravity of a specimen.
The mercury displacement volumeter provided the writer
with a useful, accurate instrument for the determination

of volume for small, hygroscopic pieces of material.

MOISTURE CONTENT
The results of the moisture content determinations
showed that in each species the average moisture
content of the static samples was greater than that
of the fatigue samples. The difference was caused
by the time required for testing. Each static sample
was tested over a time interval of one to two minutes

in contrast to the fatigue samples that required from

53
several minutes to ten hours. The moisture conditions
at the time of fatigue testing were such that water
was lost from the wood. As the results in Table 2,
page 28, point out, this loss of moisture was signifi~
cant for fatigue samples. The moisture loss was un-
doubtedly accelerated by higher temperatures in the
wood induced by friction of the testing=head against
the wood.

Despite the significance of the moisture content
difference between static and fatigue samples, the mag-
nitude of the difference was quite low. In western
hemlock, where the difference was the greatest, the
average moisture content of the fatigue samples was only
1.7 percent higher than that of the static samples.

The physical sources of error in the moisture
content determinations were as listed below:

1. Change in humidity. Static and fatigue
testing were conducted day and night during
the early summer months when the relative
humidity differential can be unusually high.
Samples that were tested and weighed at
night exhibited a higher moisture content
than those measured at mid-day. The resulting
change in moisture content was either pos»
itive or negative.

2. Balance inaccuracies. The samples were weighed

on a triple beam balance before and after

54
oven-drying. Any inaccuracy intrinsic to the
balance would produce an error in the moisture
content. The balance was zeroed periodically
to minimize error. The change in moisture
content was either positive or negative.

3. Moisture movement during weighing. A small
amount of error was introduced by the addition
of moisture during the time required to
obtain the ovenndry weight of the specimens.
The error was the same for each sample and
induced a negative effect on the moisture
content.

The maximum, possible, physical error was cal~
culated by the partial differential method at the exu
tremities of the moisture content range. The percentage
possible error due to physical measurement varied from
seven percent to ten percent. Inspection of Table 2,
page 28, shows that the moisture content variance for
each test method in each species is relatively low.
This indicates that the error due to physical measure-
ment was insignificant and it can be assumed that all
the moisture content figures for either test method

in each species, were drawn from a common population.

STATIC FAILURE
The results summarized in Table 3, page 29,

indicate that the stress level at shear failure under

55

a static load varies directly with the specific grav»

ity of the wood at the line of failure. The variance

of this relationship was influenced by error in speci-

fic gravity (see page 50) and by errors made in the

determination of stress level at failure for the

statically loaded shear specimens. The sources of

error for loading are listed below:

1.

Stress concentration. The possibility existed
that the specimen did not support the load
evenly at all points of the contact area
between the testing-head and the sample.

This could occasion failure at a stress

level considerably lower than was anticipated.
The error due to stress concentration was
minimized by the use of a rubber mat under

the testing-head to distribute the load

over the shear section of the sample.

Loading. The testing machine was recalibrated
previous to this research to give an error

of about one percent at the load range for

the static testing to be done as a portion

of this problem. The error induced by loading
could have been either positive or negative.
Shear area. The shear area could not be
measured exactly until after failure had

occurred and could not be measured exactly

56
when the plane of failure was rough. The
result was higher stress values than were
actually present.

The percent possible error for shear stress under
static loading was computed by the method of partial
differentials. The error in stress varied from six
percent for the greatest load to nine percent for the
smallest load. The scatter of the data shown in
Figures 8, 9, and 10 was influenced more by error in
specific gravity than by error in stress. Since the
regression of shear stress at failure on specific
gravity was significant for all three species, the

combined experimental error may be considered unimportant.

FATIGUE FAILURE

The multiple regression of number of cycles to
failure on specific gravity and stress level was
influenced by physical error in the measurement of
all three factors. The error contributed by specific
gravity is discussed on pages 48 to 52 of this paper.
The physical error involved in determining the number
of cycles to failure was extremely small and may be
ignored. The factors listed below are those that
influenced the magnitude of the measurement error for
shear stress at failure:

1. Stress concentration. The stress concentration

factor for fatigue loading was the same as

57
for static loading (see page 55).
Loading. The magnitude of the load at
failure constituted the most significant
source of error in the estimation of the
fatigue strength in shear. The error was
occasioned by the stress relaxation that
occurred in the specimens and by the
subsequent decrease in the static preload on
the testing machine. The error was more
pronounced at a relatively low number of
cycles to failure because the reduction of
the static preload was most pronounced at
the early stage of loading. This source of
error would have been virtually eliminated
if the testing machine had an automatic
preload adjustment. Unfortunately, such
equipment was not available for this research.
The decrease of static preload resulted in
a negative error in shear stress and a
positive error in the number of cycles to
failure.
Bending moment. The presence of a bending
moment across the shear section produced error
in the shear stress and, consequently, in the
number of cycles to failure. Because the bending

moment at the plane of maximum shear was

58
small, the error was small when the failure
occurred at the plane of maximum shear.

4. Horizontal load components. The testing«
head guides were designed to make the loado
multiplier transfer only the vertical com~
ponent of the machine load. However it was
unreasonable to assume that the horizontal
loading component was zero. The horizontal
components of load resulted in positive error
in shear stress and negative error in the
number of cycles to failure. This factor
somewhat compensated for the error induced
by the decrease in the static preload.
The percent possible error for fatigue shear
stress was determined for the highest and lowest
values of load by the method of partial differentiation.
The percent error varied from eight percent for the
highest load to seventeen percent for the smallest
load. Because change of stress is reflected directly
in the number of cycles to failure, a small change
in stress at the end of the S~N curve would yield a
relatively unimportant change in the number of cycles
to failure. The same change of stress at the beginning
of the S—N curve could lead to an extremely significant

change in the number of cycles to failure.

59

CONCLUSIONS

THEORIES OF FATIGUE FAILURE

The exact mechanism of fatigue failure in wood

is an unknown quantity. This is not surprising since

there exist many schools of thought concerning fatigue

failure for materials that have received a great deal

more attention than has wood. There is no doubt that

the mechanism of fatigue in wood is similar in some
respects to that for metals, plastics, and glass.
This portion of the paper is devoted to a review of

the more important theories of failure and a discussion

of their utility for wood. The theories mentioned in

this section are secondary stresses, hysteresis, and
molecular slip.

Secondary Stresses

This theory states that a difference exists between
the actual strengths of solid materials and the strengths

they ought to possess by reason of molecular cohesive

forces (6). All materials contain sub-microscopic

defects that reduce the actual strength. These defects

occur on both the interior and exterior of the material
and furnish a position for failure to begin before the

maximum stress can be developed. In addition, the

60
presence of internal stresses can reduce the magnitude

of stress at failure. When applied stresses are

imposed on pre-existing residual stresses. the resultant

stresses are impossible to determine before failure

actually occurs (12).

The theory of secondary stresses appears at least
partially significant to explain the mechanism of

fatigue failure in wood. There are innumerable factors

that can introduce microscopic defects and internal
stresses to wood, both in the growing tree and in the

preparation of wood for any particular use. In the

growing tree, defects and internal stresses are

occasioned by such factors as the weight of the tree,

wind loads, snow loads, and frost action. In addition,

all the many environmental conditions that influence

the growth rate of a tree could also affect the amount

and distribution of microscopic defects and residual

stresses in the wood. These defects and stresses can

be magnified and increased numerically by the logging
and machining methods that are necessary to convert

the tree into a wood product. The inevitable drying

of the wood complicates the picture further by the

addition of still more defects and residual stresses.
If all the microscopic defects and residual

stresses could be somehow eliminated, the natural

morphology of wood itself contains factors that

61

account for increased stress. Some of the more

important characteristics of wood that may be stress-

raisers are listed below:

1.

Specific gravity. Difference in specific
gravity in adjacent portions of wood are a
function of cell wall thickness. The
relatively high specific gravity of
summerwood in softwoods results from thick
cell walls in contrast to the low specific
gravity of wood resulting from thin-walled
springwood cells. When the cells are supporting
a load parallel to their long axis, there is
a greater cell wall area sustaining the load
in summerwood with a resulting lower stress.
At the same load, the thin-walled springwood
cells, with a smaller cell wall area, must
withstand a greater stress. As a result, it
may be theorized that the weakest portion of
the wood is in the early springwood where
the specific gravity is lowest. The results
of the research reported here appear to
substantiate this theory.

Type of cells. The type and the amount of
cells that are supporting or resisting a
load on wood can have a pronounced influence

on the stress at failure in shear parallel to

62
the grain. Some cell types are character-
istically thin or thick walled in certain
species. The number of cells of a particular
kind will produce a specific gravity depen-
dence as described above. Parenchyma cells,
for example, are normally thin-walled in
some species and, if present in a sufficient
number, could introduce a source of stress
concentration.
Slope of fibrils. The secondary cell wall
of woody plant tissue exhibits fibril ori-
entation at an angle to the long axis of its
cellular components (5). The fibril angles
in the three layers of the secondary wall are
generally totally different from one another,
resulting in a complex stress formation inside
the cell wall. Stress concentration could be
expected to be maximum where the deviation
of fibril alignment was greatest between
adjacent layers of the secondary cell wall.
In addition, the fibril alignment is variable
from one cell to another. This could be a
significant factor between summerwood and
springwood where the difference in fibril

alignment is maximum in adjacent cells.

63

4. Pits. Woody plant cells are characterized
by the presence of pits between adjacent
cells. These recesses in the secondary
cell wall distort the normal fibril
alignment in the cell walls and are effective
stress raisers. The number, size, and distri-
bution of the pits vary from one species to
another and, as a result, may have a greater
or lesser influence on the magnitude of the
stress.

5. Sub-microscopic structure. The pattern of
the cellulose in plant cell walls is not
continuous. The long chain molecules of
cellulose within the fibrils have alternate
regions of parallel and non-parallel
arrangement. The parallel regions are called
crystalline and the disorganized regions
are known as the amorphous areas (5).

The discontinuity formed by the amorphous
regions may serve as a stress raiser in shear

parallel to the grain.

Hysteresis
Wood is not a perfectly elastic material. The

slightest load on a wood member will induce some

64
permanent deformation too small to measure. In
addition, if stress-strain diagrams are made for the
loading and unloading of a sample, the points will
not coincide exactly but will form a loop, called the
hysteresis loop (16)(28). The area inside the loop
is directly proportional to the amount of energy lost
in the cycle, and the strain after one complete cycle
is the permanent deformation. The energy loss is
probably due to friction and is given off as heat.

The hysteresis loops occur even within the elastic
regions of wood. The hysteresis effect is very slight
in wood for a single loading and unloading cycle at
very low stress. However, if the stress is greater
and if the stresses are repeated with sufficient
rapidity, the strain can accumulate until failure
eventually occurs. For very low, alternating stresses,
‘ the hysteresis loop may be a straight line. As the
fatigue loading progresses, the hysteresis loop

broadens more and more to eventual failure.

Molecular glip

Repeated stresses in metals result in the
formation of "slip bands" in the crystals (9). These
slip bands increase with the number of stress appli-
cations until failure occurs when the bands cover the

entire surface of the crystal. An analogous feature called

65
the slip plane, has been found in the cellulose of
wood that has been subjected to mechanical stress
(l4)(l7)(25)(30). There is some doubt regarding the
origin of the slip planes in wood, but the possibility
certainly exists that these planes represent minute
failure in the crystalline region of the fibril.
Continued fatigue loading would cause the formation
of new slip planes and the enlargement of already

existing slip planes to failure.

APPLICATION OF FAILURE THEORIES

This study has shown that wood appears to have
a definite fatigue limit in shear parallel to the grain.
The exact fatigue limit for a particular sample is
difficult to determine in advance because of the strong
influence of specific gravity at the plane of maximum
shear. The fatigue strength of wood in shear parallel
to the grain is different for different species and
must be determined experimentally for each species.

The exact mechanism of fatigue failure appears
to be different from the mechanism of static failure.
Previous research has shown that static failure of
wood in compression occurs along a shear plane between
the outer and central layers of the secondary cell wall
and that the failure spirals along the length of the

cell (3). This indicates that the maximum stress

66
position rotates as failure progresses. If this holds
true for shear failure under static load, it partially
explains the migration of the static, shear failure
observed in this experiment. The maximum stress could
be transferred through the isotropic compound middle
lamella from a strong cell of high specific gravity
to a weaker cell of lower specific gravity. This may
have taken place since in all cases the migration was
toward a region of low specific gravity. It is
evident that investigation into the anatomy of shear
failure under a static load is essential to the
eventual solution of this portion of the problem.

If it is assumed that the theory, postulated
above, for the mechanism of static, shear failure
is correct, then it is evident that the mechanism
of fatigue shear failure is different. The evidence to
support this contention lies in the fact that there
was no migration of the shear line of failure in the
fatigue specimens. Since the line of failure was
straight there was little lateral transformation of
shear stresses. This suggests several possibilities.
First, the possibility exists that the stress occurred
primarily in the isotropic region of the compound
middle lamella. If this actually happened, then the
mechanism of fatigue failure was probably a combination

of secondary stresses and hysteresis. The hysteresis

67
loops would be progressively broadened by the energy
loss attributable to failure by secondary stresses in
the middle lamella. Another possibility for failure
in fatigue is that it may occur within the cell wall.
The mechanism of failure is then a combination of
secondary stresses, hysteresis and molecular slip.

In this study, as in many others in wood
research, emperical equations were obtained through
laboratory testing and statistical analysis. In orig-
inal research it is important to first demonstrate the
presence of a phenomenon; in this study the phenomenon
was shear failure parallel to the grain in wood
subjected to fatigue loading. This occurrence was
shown to be inclusive by the method of testing and
by the manner of failure. If the concent of fatigue in
shear for wood is accepted, the next portion of the
research should properly be an effort to determine
the influence of various agencies on fatigue in shear.
In this study, the influence of stress level
and specific gravity were studied in an effort to
estimate their contributiOn to variation in fatigue.
When all the sources of variation have been explored,
it should be possible to begin further research to
learn the exact origin and mechanism of fatigue failure
in shear for wood. Such research is not possible

until all the sources of error have been estimated,

68
to enable the researcher to either elimate or
account for them. Such errors may be occasioned by
variables that have not been evaluated in this study.
These include moisture content, temperature, residual
stresses, presence of extractives, degree of lignifica-
tion, damping capacity, and elastic and plastic defor-
mation. All these variables, however, can be called
superficial. They are merely outward manifestations
of some internal change in the wood itself. The
internal changes constitute the real failure due to
fatigue loading. It is the author's intention to

continue this research to its logical conclusion.

69

SUMMARY
Wood can be made to fail by fatigue loading in
shear parallel to the grain on a tangential plane.
There is a linear relationship between shear
stress and specific gravity at the line of
failure for samples tested in static shear parallel
to the grain on a tangential plane.
There is a significant regression of stress
level and specific gravity at the line of failure
on the number of cycles to failure in shear
parallel to the grain on a tangential plane.
A linear relationship exists between shear stress
and specific gravity at the line of failure in
wood tested by fatigue loading in shear
parallel to the grain on a tangential plane.
The early springwood offers the least resistance
to loading in shear parallel to the grain.
The mechanism of fatigue failure in shear is
different from that of static shear failure.
The stress level at failure for fatigue shear
samples is less than for samples tested under a
static load for samples of the same species with

comparable specific gravity values.

8.

Additional research into the mechanism of

fatigue failure in wood is necessary.

70

10.

71

W1

LIST OF.APPARATUS

White fir stock from which samples were
prepared contained about four and one-half
growth increments per inch.

western hemlock from which test samples were
‘prepared contained about twenty growth
increments per inch.

Douglas-fir stock from which test samples were
prepared contained about twenty-one growth
increments per inch.

Band-saw. 36 inch band-saw, Model B-l8388,
made by the Yates-American Machine Company,
Beloit, Wisconsin.

Static Testing Machine. Baldwin-Emery, SR-4
Testing Machine, Model F6T, manufactured by
the Baldwin-Lima-Hamilton Corporation,
Philadelphia.

Fatigue Testing Machine. Sonntag Universal
Fatigue Testing Machine, Model SF-OléU,
designed by the Sonntag Scientific Corporation,
Greenwich, Connecticut.

Strain Gages. SR-4 Strain gages, Type AD-7
with a resistance of 120 ohms and gage factor
of 1.95. Gages manufactured by the Baldwin-
Lima-Hamilton Corporation, Philadelphia.

Strain Amplifier. Baldwin SR-4, Type M,
Portable Strain Indicator manufactured by the
Baldwin-Lima-Hamilton Corporation, Philadelphia.

Triple Beam Balance. Cenco Agate Bearing,
Triple Beam Balance, made by the Central
Scientific Company, Chicago.

Drying Oven. Fisher Senior Forced Draft

, Isotemp Oven, made by the Fisher Scientific

Company, New York.

72

List of Apparatus.-Continued.

11.

12.

13.

14.

Volumeter. Mercury displacement volumeter
designed by the author from a prototype and
made by the Metal Machining Company, Lansing.

velumeter. Mercury displacement volumeter
designed by Dr. Everett Ellis. Associate
Professor, University of Michigan and made
at the University of Idaho machine shop,
Moscow, Idaho.

Grammatic Balance. Gram-matic Balance made
by E. Mettler, Zurich, Switzerland, and
distributed in the United States by the
Fisher Scientific Company. New York.

Calculator. Friden Automatic Calculator
Model STW, manufactured by Friden Inc. San
Leandro, California.

APPENDIX 1;

DATA

Table a. Data recorded for white fir samples
tested in static shear parallel to the grain.

 

 

 

Moisture Specific Shear
Content Gravity Stress
(7») (psi)
9.46 0.338 492
10.64 0.452 800
9.43 0.317 456
9.66 0.348 575
9.74 0.268 427
10.79 0.439 879
10.69 0.365 678
9.76 0.250 260
10.63 0.304 309
10.75 0.284 564
10.70 0.250 208
9.30 0.232 232
10.73 0.206 204
9.47 0.328 656
10.67 0.306 425
9.40 0.346 435
10.66 0.250 295
10.76 0.228 315
10.78 0.242 385
10.67 0.374 643

 

 

74

Table b. Data recorded for western hemlock samples
tested in static shear parallel to the grain.

 

 

 

Moisture Specific Shear
Content Gravity Stress
(%) (psi)
13.87 0.275 910
14.05 0.344 1130
14.06 0.368 1129
13.55 0.296 896
13.63 0.280 817
13.80 0.360 1234
13.77 0.420 1386
13.76 0.446 1498
13.59 0.323 1114
13.56 0.326 1057
13.43 0.338 1249
13.13 0.382 1212
13.09 0.412 1540
12.95 0.301 922
12.95 0.375 1407
12.84 0.398 1251
13.13 0.352 991
13.03 0.374 1084
12.89 0.372 1276
13.44 0.405 833

 

 

Table c. Data recorded for Douglas-fir samples
tested in static shear parallel to the grain.

 

 

 

Moisture (Specific Shear
Content Gravity Stress
(7») (psi)
12.27 0.441 1071
12.44 0.408 825
12.62 0.380 883
12.46 0.404 941
12.59 0.360 756
12.55 0.414 1022
12.27 0.454 1162
12.24 0.389 1263
12.55 0.470 1270
12.56 0.496 1307
12.55 0.391 1011
12.31 0.416 1023
12.01 0.449 1155
12.10 0.505 1397
12.26 0.464 1268
12.40 0.336 781
12.38 0.426 979
12.16 0.473 1323
12.04 0.448 1093
11.94 0.446 1050

 

 

76

Table d. Data recorded for white fir samples
tested in fatigue.

 

 

 

Moisture Specific Shear Cycles to
Content Gravity Stress Failure
(%) (psi)
10.66 0.306 340 1,000
10.56 0.324 448 100
11.11 0.313 294 1,000
10.74 0.285 347 200
10.66 0.338 170 1,650,000
10.93 0.310 429 200
10.67 0.279 242 4,000
11.22 0.297 190 1,250,000
11.32 0.363 225 1,000
11.25 0.282 219 190,000
11.09 0.384 656 50
11.11 0.281 500 300
10.83 0.542 400 10
11.00 0.318 400 40
11.04 0.322 300 400
10.69 0.306 225 4,000
10.71 0.279 200 49,000
10.67 0.294 300 2,000
11.29 0.313 400 5,000
10.08 0.268 350 300

 

 

77

Table e. Data recorded for western hemlock samples
tested in fatigue.

 

 

 

Moisture Specific Shear Cycles to

Content Gravity Stress Failure
(%) (psi)
12.60 0.347 544 100
12.17 0.380 657 175
12.67 0.390 876 5
12.55 0.420 706 500
12.82 0.340 716 10
13.06 0.436 384 10,000
12.81 0.366 604 30
12.84 0.370 439 1,000
11.78 0.366 414 1,000
11.85 0.368 268 200,000
11.66 0.422 334 98,000
11.22 0.410 450 5,000
11.68 0.448 450 2,000
11.74 0.342 450 1,500
9.84 0.358 450 1,000
9.53 0.380 450 3,000
10.74 0.326 634 30
9.65 0.486 734 100
11.69 0.394 782 50
12.33 0.444 450 16.000
11.77 0.384 453 100
11.82 0.350 . 564 10
12.24 0.354 763 100
12.19 0.379 572 50
12.11 0.512 500 3.000
10.63 0.410 217 750,000
11.69 0.409 466 100
12.47 0.412 784 10
11.33 0.380 191 3,000,000
11.05 0.491 268 3,000,000

 

 

78

Table f. Data recorded for Douglas-fir samples
tested in fatigue.

 

 

 

Moisture Specific Shear Cycles to
Content Gravity Stress Failure
(%) (psi)
11.50 0.532 536 2,000
10.86 0.424 660 50
11.32 0.493 442 9,000
10.55 0.471 847 20
10.91 0.450 454 100
11.35 0.444 511 1,000
11.28 0.386 581 20
11.00 0.432 505 2,500
11.02 0.394 700 10
11.55 0.418 408 5,000
11.19 0.390 404 1,000
11.21 0.442 358 52,000
10.83 0.456 520 200
11.20 0.449 367 1,250,000
11.14 0.402 824 10
11.09 0.420 514 400
11.28 0.491 392 1,200,000
11.06 0.384 464 300
11.22 0.530 602 2,500
10.98 0.439 253 1,500,000
10.87 0.466 428 1,500
10.67 0.440 495 60
9.68 0.396 706 40
10.82 0.430 565 50
10.95 0.504 378 22,000
11.11 0.382 307 2,200,000
11.32 0.313 349 500
10.92 0.520 505 1,000
10.87 0.418 381 21,000
11.13 0.485 498 500

 

 

79

LITERATURE CITED

American Society for Testing Materials. "Testing
Small Clear Specimens of Timber," Book of
Standards, Pt. 4, Serial Designation D I43-52,
Am. Soc. Test. Mat., Philadelphia, 1955.

 

Baldwin-Lima-Hamilton Corporation. Instruction
Manual for Universal Testing Machin . An
Instructioananual Distributed By tfie Sonntag
Scientific Corporation, Greenwich, Conn., 1951.

 

Bienfat, L. J., "Relation of the Manner of Failure
to the Structure of Wood under Compression Parallel
to the Grain," Journal of A ricultural Research,
Washington, D. C., p. 3:_l9§6.

Brown, H. P., Panshin, A. J. and Forsaith, C. C.,
Textbook gf'Wood Technology, Vol. 1, McGraw-Hill
Book Co., New Tork, .

Brown, H. P., Panshin, A. J. and Forsaith, C. C.,
Textbook 2f Wood Technology, Vol. 2, McGraw-Hill
Book Co., New York, .

Cazaud, R., Fati ue of Metals, Philosophical
Library Inc., New TOTE, I953.

Coker, E. G. and Coleman, G. P., "Photoelastic
Investigations of Shear Tests of Timber,"

Selected Engineering Pagers, Institution of Civil
Engineers, 0. , Lon on, 1935.

Dietz, A. G. H. and Grinsfelder, H., "Behavior of
Plywood under Repeated Stresses," Transactions,
American Society of Mechanical Engineers, April,
1943.

 

Ewing, J. A. and Humfrey, J. C. W., "The Fracture
of Metals under Repeated Alternations of Stress,"

Philosoghical Transactions, Royal Society,
0 o , 3.

 

80

Literature Cited.-Continued.

10.

11.

12.

13.

14.

15.

16.

17.‘

18.

19.

20.

 

Forest Products Laboratory. Wood Handbook,
Agricultural Handbook No. 72, United States
Government Printing Office, Washington, D. C.,
1955.

 

Freudenthal, A. M., "Planning and Interpretation
of Fatigue Tests," Statistical Aspects of

Fati ue-A Symposium, American Society of_Testing
Materials, Stp. No. 121, 1951.

 

 

Gilchrist, J., "On Wohlers Laws," Engineering,
Vol. 90, 1900.

 

Hodgman, C. D., Weast, R. C. and Selby, S. M.,
Handbook of Chemistry and Ph sics, Ed. 39,
CEemicaI Rfibber Publishing Co., Cleveland,
1958.

 

Hohnel, von P., "Ueber den Einfluss des Rinden-
druckes auf die Beschaffenheit der Bastfasern
der Dicotylen," Jahrbucher fur Wissenschaftliche
Botanik, No. 15, Berlin, p.-3I1, 1884.

 

 

HOPPG, 0., "Albersts Versuche und Erfindungen,"
Stahl und Eisen, Vol. 16, Berlin, 1896.

 

Houwink, R., Plasticity Elasticity and Structure
‘gf Matter, Harren Press, Washington, D. C., I953.

 

Kisser, J. and Steininger, A., "Makroskopische
und Mikroskopische Strukturanderungen bei der
Beigebeanspruchung von Holz," Holz gig RohéUnd
Werkstoff, No. 10, p. 415, 1952.

 

 

Kollmann, P., Technologie des Holzes und der
Holzwerkstoffe, V01. 1, Springer-Verlag,
Berlin, 1951.

 

Kommers, W. J., "Effect of 5000 Cycles of Repeated

Bending Stresses on Five-ply Sitka Spruce Plywood,"
United States Forest Products Laboratory, F. P. L.

mimeo. 1305, 1943.

, "Effect of Ten Repitions of
Stress on the Bending and Compressive Strengths
of Sitka Spruce and Douglas-fir," United States
Forest Products Laboratory, F. P. L. mimeo.
1320, 1943.

 

81

Literature Cited.-Continued.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

 

,Jf, , "The Fatigue Behavior of Wood and
Plywood Subjected to Repeated and Reversed
Bending Stresses," United States Forest Products
Laboratory, F. P. L. mimeo. 1327, 1943.

 

Leggett, J. L.9 "Investigation of Fatigue
Strength of Railroad Timber Bridge Stringers,"
Bulletin, American Railway Engineering Associ-
ation, No. 510, 1953.

Lewis, W. C., "Fatigue of Wood and Glued Joints
used in Laminated Construction," Proceedin 8,
Forest Products Research Society, p. 22!, I951.

Marin, J., Engineering Materials, Prentice Hall
Inc., Englewoo s,7NEwTJersey, 1952.

Searle, G. 0., "The Microscopic Structure of‘
Tendered Fibers," gournal, Textile Institute,
No. 15, p. 371, 19 .

Severo, N. C. and Olds, E. G., "A Comparison
of Tests on the Mean of a Logarithmico-normal
Distribution with known Variance," Annals of
Mathematical Statistics, v61. 27, N‘_36. T 1936.

 

Snedecor, G. W., Statistical Methods, Ed. 4,
The Iowa State College Press, Ames, Iowa, 1946.

 

Timoshenko, S., Strength 2f Materials, Lancaster
Press Inc., Lancas er, Penn., 1930.

Walker, H. M. and Lev, J., Statistical Inference,
Henry Holt and Company, New York, 1953.

Wardrop, A. B. and Dadswell, H. E., Council
of Scientific and Industrial Research, Bulletin
221, 1947.

Weibull, W., "A Statistical Representation of
Fatigue Failures in Solids," Transactions, Royal
Institute of Technology, No. 22, 1949.

 

Yavorsky, J. M., Cunningham, J. H. and Hundley,
N. G.,"Survey of Factors Affecting Strength
Tests of Glued Joints," Iournal, Forest Products
Research Society, Vol. 5, o. , 1955.

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